Handbook of Materials Modeling

HANDBOOK OF MATERIALS MODELING HANDBOOK OF MATERIALS MODELING Part B. Models Editor Sidney Yip, Massachusetts Institu...

Author: Sidney Yip

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HANDBOOK OF MATERIALS MODELING

HANDBOOK OF MATERIALS MODELING Part B. Models Editor Sidney Yip, Massachusetts Institute of Technology

A C.I.P. Catalogue record for this book is available from the Library of Congress.

ISBN-10 1-4020-3287-0 (HB) Springer Dordrecht, Berlin, Heidelberg, New York ISBN-10 1-4020-3286-2 (e-book) Springer Dordrecht, Berlin, Heidelberg, New York ISBN-13 978-1-4020-3287-5 (HB) Springer Dordrecht, Berlin, Heidelberg, New York ISBN-13 978-1-4020-3286-8 (e-book) Springer Dordrecht, Berlin, Heidelberg, New York

Published by Springer, P.O. Box 17, 3300 AA Dordrecht, The Netherlands.

Printed on acid-free paper

All Rights Reserved

© 2005 Springer No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Printed in The Netherlands

CONTENTS PART A – METHODS Preface

xii

List of Subject Editors

ix

List of Contributors

xi

Detailed Table of Contents

xxix

Introduction

1

Chapter 1.

Electronic Scale

7

Chapter 2.

Atomistic Scale

449

Chapter 3.

Mesoscale/Continuum Methods

1069

Chapter 4.

Mathematical Methods

1215

PART B – MODELS Preface

xii

List of Subject Editors

ix

List of Contributors

xi

Detailed Table of Contents

xxix

Chapter 5.

Rate Processes

1565

Chapter 6.

Crystal Defects

1849

Chapter 7.

Microstructure

2081

Chapter 8.

Fluids

2409

Chapter 9.

Polymers and Soft Matter

2553

Plenary Perspectives

2657

Index of Contributors

2943

Index of Keywords

2947 v

PREFACE This Handbook contains a set of articles introducing the modeling and simulation of materials from the standpoint of basic methods and studies. The intent is to provide a compendium that is foundational to an emerging field of computational research, a new discipline that may now be called Computational Materials. This area has become sufficiently diverse that any attempt to cover all the pertinent topics would be futile. Even with a limited scope, the present undertaking has required the dedicated efforts of 13 Subject Editors to set the scope of nine chapters, solicit authors, and collect the manuscripts. The contributors were asked to target students and non-specialists as the primary audience, to provide an accessible entry into the field, and to offer references for further reading. With no precedents to follow, the editors and authors were only guided by a common goal – to produce a volume that would set a standard toward defining the broad community and stimulating its growth. The idea of a reference work on materials modeling surfaced in conversations with Peter Binfield, then the Reference Works Editor at Kluwer Academic Publishers, in the spring of 1999. The rationale at the time already seemed quite clear – the field of computational materials research was taking off, powerful computer capabilities were becoming increasingly available, and many sectors of the scientific community were getting involved in the enterprise. It was felt that a volume that could articulate the broad foundations of computational materials and connect with the established fields of computational physics and computational chemistry through common fundamental scientific challenges would be timely. After five years, none of the conditions have changed; the need remains for a defining reference volume, interest in materials modeling and simulation is further intensifying, the community continues to grow. In this work materials modeling is treated in 9 chapters, loosely grouped into two parts. Part A, emphasizing foundations and methodology, consists of three chapters describing theory and simulation at the electronic, atomistic, and mesoscale levels, and a chapter on analysis-based methods. Part B is more concerned with models and basic applications. There are five chapters describing basic problems in materials modeling and simulation, rate-dependent phenomena, crystal defects, microstructure, fluids, polymers and soft matter. In vii

viii

Preface

addition this part contains a collection of commentaries on a range of issues in materials modeling, written in a free-style format by experienced individuals with definite views that could enlighten the future members of the community. See the opening Introduction for further comments on modeling and simulation and an overview of the Handbook contents. Any organizational undertaking of this magnitude cans only be a collective effort. Yet the fate of this volume would not be so certain without the critical contributions from a few individuals. My gratitude goes to Liesbeth Mol, Peter Binfield’s successor at Springer Science + Business Media, for continued faith and support, Ju Li and Xiaofeng Qian for managing the websites and manuscript files, and Tim Kaxrias for stepping in at a critical stage of the project. To all the authors who found time in your hectic schedules to write the contributions, I am deeply appreciative and trust you are not disappointed. To the Subject Editors I say the Handbook is a reality only because of your perseverance and sacrifices. It has been my good fortune to have colleagues who were generous with advice and assistance. I hope this work motivates them even more to continue sharing their knowledge and insights in the work ahead. Sidney Yip Department of Nuclear Science and Engineering, Department of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA

LIST OF SUBJECT EDITORS Martin Bazant, Massachusetts Institute of Technology (Chapter 4) Bruce Boghosian, Tufts University (Chapter 8) Richard Catlow, Royal Institution, UK (Chapter 6) Long-Qing Chen, Pennsylvania State University (Chapter 7) William Curtin, Brown University (Chapter 1, Chapter 2, Chapter 4) Tomas Diaz de la Rubia, Lawrence Livermore National Laboratory (Chapter 6) Nicolas Hadjiconstantinou, Massachusetts Institute of Technology (Chapter 8) Mark F. Horstemeyer, Mississippi State University (Chapter 3) Efthimios Kaxiras, Harvard University (Chapter 1, Chapter 2) L. Mahadevan, Harvard University (Chapter 9) Dimitrios Maroudas, University of Massachusetts (Chapter 4) Nicola Marzari, Massachusetts Institute of Technology (Chapter 1) Horia Metiu, University of California Santa Barbara (Chapter 5) Gregory C. Rutledge, Massachusetts Institute of Technology (Chapter 9) David J. Srolovitz, Princeton University (Chapter 7) Bernhardt L. Trout, Massachusetts Institute of Technology (Chapter 1) Dieter Wolf, Argonne National Laboratory (Chapter 6) Sidney Yip, Massachusetts Institute of Technology (Chapter 1, Chapter 2, Chapter 6, Plenary Perspectives)

ix

LIST OF CONTRIBUTORS Farid F. Abraham IBM Almaden Research Center, San Jose, California [email protected] P20

Robert Averback Accelerator Laboratory, P.O. Box 43 (Pietari Kalmin k. 2), 00014, University of Helsinki, Finland; Department of Materials Science and Engineering, University of Illinois at Urbana-Champaign, Illinois, USA [email protected] 6.2

Francis J. Alexander Los Alamos National Laboratory, Los Alamos, NM, USA [email protected] 8.7

D.J. Bammann Sandia National Laboratories, Livermore, CA, USA [email protected] 3.2

N.R. Aluru Beckman Institute for Advanced Science and Technology, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA [email protected] 8.3

K. Barmak Department of Materials Science and Engineering, Carnegie Mellon University, Pittsburgh, PA 15213, USA [email protected] 7.19

Filippo de Angelis Istituto CNR di Scienze e Tecnologie Molecolari ISTM, Dipartimento di Chimica, Universit´a di Perugia, Via Elce di Sotto $, I-06123, Perugia, Italy [email protected] 1.4

Stefano Baroni DEMOCRITOS-INFM, SISSA-ISAS, Trieste, Italy [email protected] 1.10

Emilio Artacho University of Cambridge, Cambridge, UK [email protected] 1.5

Rodney J. Bartlett Quantum Theory Project, Departments of Chemistry and Physics, University of Florida, Gainesville, FL 32611, USA [email protected] 1.3

Mark Asta Northwestern University, Evanston, IL, USA [email protected] 1.16

Corbett Battaile Sandia National Laboratories, Albuquerque, NM, USA [email protected] 7.17

xi

xii Martin Z. Bazant Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, USA [email protected] 4.1, 4.10 Noam Bernstein Naval Research Laboratory, Washington, DC, USA [email protected] 2.24 Kurt Binder Institut fuer Physik, Johannes Gutenberg Universitaet Mainz, Staudinger Weg 7, 55099 Mainz, Germany [email protected] P19 Peter E. Bl¨ohl Institute for Theoretical Physics, Clausthal University of Technology, Clausthal-Zellerfeld, Germany [email protected] 1.6 Bruce M. Boghosian Department of Mathematics, Tufts University, Bromfield-Pearson Hall, Medford, MA 02155, USA [email protected] 8.1 Jean Pierre Boon Center for Nonlinear Phenomena and Complex Systems, Universit´e Libre de Bruxelles, 1050-Bruxelles, Belgium [email protected] P21

List of contributors Russel Caflisch University of California at Los Angeles, Los Angeles, CA, USA [email protected] 7.15 Wei Cai Department of Mechanical Engineering, Stanford University, Stanford, CA 94305-4040, USA [email protected] 2.21 Roberto Car Department of Chemistry and Princeton Materials Institute, Princeton University, Princeton, NJ, USA [email protected] 1.4 Paolo Carloni International School for Advanced Studies (SISSA/ISAS) and INFM Democritos Center, Trieste, Italy [email protected] 1.13 Emily A. Carter Department of Mechanical and Aerospace Engineering and Program in Applied and Computational Mathematics, Princeton University, Princeton, NJ 08544, USA [email protected] 1.8

Iain D. Boyd University of Michigan, Ann Arbor, MI, USA [email protected] P22

C.R.A. Catlow Davy Faraday Laboratory, The Royal Institution, 21 Albemarle Street, London W1S 4BS, UK; Department of Chemistry, University College London, 20 Gordon Street, London WC1H 0AJ, UK [email protected] 2.7, 6.1

Vasily V. Bulatov Lawrence Livermore National Laboratory, University of California, Livermore, CA 94550, USA [email protected] P7

Gerbrand Ceder Department of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, MA, USA [email protected] 1.17, 1.18

List of contributors

xiii

Alan V. Chadwick Functional Materials Group, School of Physical Sciences, University of Kent, Canterbury, Kent CT2 7NR, UK [email protected] 6.5

Marvin L. Cohen University of California at Berkeley and Lawrence Berkeley National Laboratory, Berkeley, CA, USA [email protected] 1.2

Hue Sun Chan University of Toronto, Toronto, Ont., Canada [email protected] 5.16

John Corish Department of Chemistry, Trinity College, University of Dublin, Dublin 2, Ireland [email protected] 6.4

James R. Chelikowsky University of Minnesota, Minneapolis, MN, USA [email protected] 1.7 Long-Qing Chen Department of Materials Science and Engineering, Penn State University, University Park, PA 16802, USA [email protected] 7.1 I-Wei Chen Department of Materials Science and Engineering, University of Pennsylvania, Philadelphia, PA 19104-6282, USA [email protected] P27 Sow-Hsin Chen Department of Nuclear Engineering, MIT, Cambridge, MA 02139, USA [email protected] P28 Christophe Chipot Equipe de dynamique des assemblages membranaires, Unit´e mixte de recherche CNRS/UHP 7565, Institut nanc´een de chimie mol´eculaire, Universit´e Henri Poincar´e, BP 239, 54506 Vanduvre-l`es-Nancy cedex, France 2.26 Giovanni Ciccotti INFM and Dipartimento di Fisica, Universit`a “La Sapienza,” Piazzale Aldo Moro, 2, 00185 Roma, Italy [email protected] 2.17, 5.4

Peter V. Coveney Centre for Computational Science, Department of Chemistry, University College London, 20 Gordon Street, London WC1H 0AJ, UK [email protected] 8.5 Jean-Paul Crocombette CEA Saclay, DEN-SRMP, 91191 Gif/Yvette cedex, France [email protected] 2.28 Darren Crowdy Department of Mathematics, Imperial College, London, UK [email protected] 4.10 G´abor Cs´anyi Cavendish Laboratory, University of Cambridge, UK [email protected] P16 Nguyen Ngoc Cuong Massachusetts Institute of Technology, Cambridge, MA, USA [email protected] 4.15 Christoph Dellago Institute of Experimental Physics, University of Vienna, Vienna, Austria [email protected] 5.3

xiv J.D. Doll Department of Chemistry, Brown University, Providence, RI, USA Jimmie [email protected] 5.2 Patrick S. Doyle Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, MA, USA [email protected] 9.7

List of contributors Diana Farkas Department of Materials Science and Engineering, Virginia Tech, Blacksburg, VA 24061, USA [email protected] 2.23 Clemens J. F¨orst Institute for Theoretical Physics, Clausthal University of Technology, Clausthal-Zellerfeld, Germany [email protected] 1.6

Weinan E Department of Mathematics, Program in Applied and Computational Mathematics, Princeton University, Princeton, NJ 08544-1000, USA [email protected] 4.13

Glenn H. Fredrickson Department of Chemical Engineering & Materials, The University of California at Santa, Barbara Santa Barbara, CA, USA [email protected] 9.9

Jens Eggers School of Mathematics, University of Bristol, University Walk, Bristol BS8 1TW, UK [email protected] 4.9

Daan Frenkel FOM Institute for Atomic and Molecular Physics, Amsterdam, The Netherlands [email protected] 2.14

Pep Espanol ˜ Dept. Física Fundamental, Universidad Nacional de Educaci´on a Distancia, Aptdo. 60141, E-28080 Madrid, Spain [email protected] 8.6 J.W. Evans Ames Laboratory - USDOE, and Department of Mathematics, Iowa State University, Ames, Iowa, 50011, USA [email protected] 5.12 Denis J. Evans Research School of Chemistry, Australian National University, Canberra, ACT, Australia [email protected] P17 Michael L. Falk University of Michigan, Ann Arbor, MI, USA [email protected] 4.3

Julian D. Gale Nanochemistry Research Institute, Department of Applied Chemistry, Curtin University of Technology, Perth, 6845, Western Australia [email protected] 1.5, 2.3 Giulia Galli Lawrence Livermore National Laboratory, CA, USA [email protected] P8 Venkat Ganesan Department of Chemical Engineering, The University of Texas at Austin, Austin, TX, USA [email protected] 9.9 Alberto García Universidad del País Vasco, Bilbao, Spain [email protected] 1.5

List of contributors C. William Gear Princeton University, Princeton, NJ, USA [email protected] 4.11 Timothy C. Germann Applied Physics Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA [email protected] 2.11 Eitan Geva Department of Chemistry, University of Michigan, Ann Arbor, Michigan 48109-1055, USA [email protected] 5.9 Nasr M. Ghoniem Department of Mechanical and Aerospace Engineering, University of California, Los Angeles, CA 90095-1597, USA [email protected] 7.11, P11, P30 Paolo Giannozzi Scuola Normale Superiore and National Simulation Center, INFM-DEMOCRITOS, Pisa, Italy [email protected] 1.4, 1.10 E. Van der Giessen University of Groningen, Groningen, The Netherlands [email protected] 3.4 Daniel T. Gillespie Dan T Gillespie Consulting, 30504 Cordoba Place, Castaic, CA 91384, USA [email protected] 5.11 George Gilmer Lawrence Livermore National Laboratory, P.O. box 808, Livermore, CA 94550, USA [email protected] 2.10

xv William A. Goddard III Materials and Process Simulation Center, California Institute of Technology, Pasadena, CA 91125, USA [email protected] P9 Axel Groß Physik-Department T30, TU M¨unchen, 85747 Garching, Germany [email protected] 5.10 Peter Gumbsch Institut f¨ur Zuverl¨assigkeit von Bauteilen und Systemen izbs, Universit¨at Karlsruhe (TH), Kaiserstr. 12, 76131Karlsruhe, Germany and Fraunhofer Institut f¨ur Werkstoffmechanik IWM, W¨ohlerstr. 11, D-79194 Freiburg, Germany [email protected] P10 Fran¸cois Gygi Lawrence Livermore National Laboratory, CA, USA P8 Nicolas G. Hadjiconstantinou Department of Mechanical Engineering, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA [email protected] 8.1, 8.8 J.P. Hirth Ohio State and Washington State Universities, 114 E. Ramsey Canyon Rd., Hereford, AZ 85615, USA [email protected] P31 K.M. Ho Ames Laboratory-U.S. DOE and Department of Physics and Astronomy, Iowa State University, Ames, IA 50011, USA 1.15

xvi

List of contributors

Wesley P. Hoffman Air Force Research Laboratory, Edwards, CA, USA [email protected] P37

C.S. Jayanthi Department of Physics, University of Louisville, Louisville, KY 40292 [email protected] P39

Wm.G. Hoover Department of Applied Science, University of California at Davis/Livermore and Lawrence Livermore National Laboratory, Livermore, California, 94551-7808 [email protected] P34

Raymond Jeanloz University of California, Berkeley, CA, USA [email protected] P25

M.F. Horstemeyer Mississippi State University, Mississippi State, MS, USA [email protected] 3.1, 3.5 Thomas Y. Hou California Institute of Technology, Pasadena, CA, USA [email protected] 4.14 Hanchen Huang Department of Mechanical, Aerospace and Nuclear Engineering, Rensselaer Polytechnic Institute, 110 8th Street, Troy, NY 12180-3590, USA [email protected] 2.30 Gerhard Hummer National Institutes of Health, Bethesda, MD, USA [email protected] 4.11 M. Saiful Islam Chemistry Division, SBMS, University of Surrey, Guildford GU2 7XH, UK [email protected] 6.6 Seogjoo Jang Chemistry Department, Brookhaven National Laboratory, Upton, New York 11973-5000, USA [email protected] 5.9

Pablo Jensen Laboratoire de Physique de la Mati´ere Condens´ee et des Nanostructures, CNRS and Universit´e Claude Bernard Lyon-1, 69622 Villeurbanne C´edex, France [email protected] 5.13 Yongmei M. Jin Department of Ceramic and Materials Engineering, Rutgers University, 607 Taylor Road, Piscataway, NJ 08854, USA [email protected] 7.12 Xiaozhong Jin Beckman Institute for Advanced Science and Technology, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA [email protected] 8.3 J.D. Joannopoulos Francis Wright Davis Professor of Physics, Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA [email protected] P4 Javier Junquera Rutgers University, New Jersey, USA [email protected] 1.5 Jo˜ao F. Justo Escola Polit´ecnica, Universidade de S˜ao Paulo, S˜ao Paulo, Brazil [email protected] 2.4

List of contributors Hideo Kaburaki Japan Atomic Energy Research Institute, Tokai, Ibaraki, Japan [email protected] 2.18 Rajiv K. Kalia Collaboratory for Advanced Computing and Simulations, Department of Physics & Astronomy, University of Southern California, 3651 Watt Way, VHE 608, Los Angeles, CA 90089-0242, USA [email protected] 2.25 Raymond Kapral Chemical Physics Theory Group, Department of Chemistry, University of Toronto, Toronto, Ont. M5S 3H6, Canada [email protected] 2.17, 5.4 Alain Karma Northeastern University, Boston, MA, USA [email protected] 7.2 Johannes K¨astner Institute for Theoretical Physics, Clausthal University of Technology, Clausthal-Zellerfeld, Germany [email protected] 1.6 Markos A. Katsoulakis Department of Mathematics and Statistics, University of Massachusetts - Amherst, Amherst, MA 01002, USA [email protected] 4.12 Efthimios Kaxiras Department of Nuclear Science and Engineering and Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA [email protected] 2.1, 8.4

xvii Ronald J. Kerans Air Force Research Laboratory, Materials and Manufacturing Directorate, Wright-Patterson Air Force Base, Ohio, USA [email protected] P38 Ioannis G. Kevrekidis Princeton University, Princeton, NJ, USA [email protected] 4.11 Armen G. Khachaturyan Department of Ceramic and Materials Engineering, Rutgers University, 607 Taylor Road, Piscataway, NJ 08854, USA [email protected] 7.12 T.A. Khraishi University of New Mexico, Albuquerque, NM, USA [email protected] 3.3 Seong Gyoon Kim Kunsan National University, Kunsan 573-701, Korea [email protected] 7.3 Won Tae Kim Chongju University, Chongju 360-764, Korea [email protected] 7.3 Michael L. Klein Center for Molecular Modeling, Chemistry Department, University of Pennsylvania, 231 South 34th Street, Philadelphia, PA 19104-6323, USA [email protected] 2.26 Walter Kob Laboratoire des Verres, Universit´e Montpellier 2, 34095 Montpellier, France [email protected] P24

xviii David A. Kofke University at Buffalo, The State University of New York, Buffalo, New York, USA [email protected] 2.14 Maurice de Koning University of S˜ao Paulo, S˜ao Paulo, Brazil [email protected] 2.15 Anatoli Korkin Quantum Theory Project, Departments of Chemistry and Physics, University of Florida, Gainesville, FL 32611, USA 1.3 Kurt Kremer MPI for Polymer Research, D-55021 Mainz, Germany [email protected] P5

List of contributors C. Leahy Department of Physics, University of Louisville, Louisville, KY 40292, USA P39 R. LeSar Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA [email protected] 7.14 Ju Li Department of Materials Science and Engineering, Ohio State University, Columbus, OH, USA [email protected] 2.8, 2.19, 2.31 Xiantao Li Program in Applied and Computational Mathematics, Princeton University, Princeton, NJ 08544, USA [email protected] 4.13

Carl E. Krill III Materials Division, University of Ulm, Albert-Einstein-Allee 47, D-89081 Ulm, Germany [email protected] 7.6

Gang Li Beckman Institute for Advanced Science and Technology, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA [email protected] 8.3

Ladislas P. Kubin LEM, CNRS-ONERA, 29 Av. de la Division Leclerc, BP 72, 92322 Chatillon Cedex, France [email protected] P33

Vincent L. Lign`eres Department of Chemistry, Princeton University, Princeton, NJ 08544, USA 1.8

D.P. Landau Center for Simulational Physics, The University of Georgia, Athens, GA 30602, USA [email protected] P2 James S. Langer Department of Physics, University of California, Santa Barbara, CA 93106-9530, USA [email protected] 4.3, P14

Turab Lookman Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM, USA [email protected] 7.5 Steven G. Louie Department of Physics, University of California at Berkeley and Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA [email protected] 1.11

List of contributors

xix

John Lowengrub University of California, Irvine, California, USA [email protected] 7.8

Richard M. Martin University of Illinois at Urbana, Urbana, IL, USA [email protected] 1.5

Gang Lu Division of Engineering and Applied Science, Harvard University, Cambridge, Massachusetts, USA [email protected] 2.20

Georges Martin ´ Commissariat a` l’Energie Atomique, Cab. H.C., 33 rue de la F´ed´eration, 75752 Paris Cedex 15, France [email protected] 7.9

Alexander D. MacKerell, Jr. Department of Pharmaceutical Sciences, School of Pharmacy, University of Maryland, 20 Penn Street, Baltimore, MD, 21201, USA [email protected] 2.5

Nicola Marzari Department of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, MA, USA [email protected] 1.1, 1.4

Alessandra Magistrato International School for Advanced Studies (SISSA/ISAS) and INFM Democritos Center, Trieste, Italy 1.13

Wayne L. Mattice Department of Polymer Science, The University of Akron, Akron, OH 44325-3909 [email protected] 9.3

L. Mahadevan Division of Engineering and Applied Sciences, Department of Organismic and Evolutionary Biology, Department of Systems Biology, Harvard University Cambridge, MA 02138, USA [email protected] Dionisios Margetis Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA [email protected] 4.8

V.G. Mavrantzas Department of Chemical Engineering, University of Patras, Patras, GR 26500, Greece [email protected] 9.4 D.L. McDowell Georgia Institute of Technology, Atlanta, GA, USA [email protected] 3.6, 3.9

E.B. Marin Sandia National Laboratories, Livermore, CA, USA [email protected] 3.5

Michael J. Mehl Center for Computational Materials Science, Naval Research Laboratory, Washington, DC, USA [email protected] 1.14

Dimitrios Maroudas University of Massachusetts, Amherst, MA, USA [email protected] 4.1

Horia Metiu University of California, Santa Barbara, CA, USA [email protected] 5.1

xx R.E. Miller Carleton University, Ottawa, ON, Canada [email protected] 2.13 Frederick Milstein Mechanical Engineering and Materials Depts., University of California, Santa Barbara, CA, USA [email protected] 4.2 Y. Mishin George Mason University, Fairfax, VA, USA [email protected] 2.2 Francesco Montalenti INFM, L-NESS, and Dipartimento di Scienza dei Materiali, Universit`a degli Studi di Milano-Bicocca, Via Cozzi 53, I-20125 Milan, Italy [email protected] 2.11 Dane Morgan Massachusetts Institute of Technology, Cambridge MA, USA [email protected] 1.18 John A. Moriarty Lawrence Livermore National Laboratory, University of California, Livermore, CA 94551-0808 [email protected] P13 J.W. Morris, Jr. Department of Materials Science and Engineering, University of California, Berkeley, CA, USA [email protected] P18 Raymond D. Mountain Physical and Chemical Properties Division, Chemical Science and Technology Laboratory, National Institute of Standards and Technology, Gaithersburg, MD 20899-8380, USA [email protected] P23

List of contributors Marcus Muller ¨ Department of Physics, University of Wisconsin, Madison, WI 53706-1390, USA [email protected] 9.5 Aiichiro Nakano Collaboratory for Advanced Computing and Simulations, Department of Computer Science, University of Southern California, 3651 Watt Way, VHE 608, Los Angeles, CA 90089-0242, USA [email protected] 2.25 A. Needleman Brown University, Providence, RI, USA [email protected] 3.4 Abraham Nitzan Tel Aviv University, Tel Aviv, 69978, Israel [email protected] 5.7 Kai Nordlund Accelerator Laboratory, P.O. Box 43 (Pietari Kalmin k. 2), 00014, University of Helsinki, Finland; Department of Materials Science and Engineering, University of Illinois at Urbana-Champaign, Illinois, USA 6.2 G. Robert Odette Department of Mechanical Engineering and Department of Materials, University of California, Santa Barbara, CA, USA [email protected] 2.29 Shigenobu Ogata Osaka University, Osaka, Japan [email protected] 1.20

List of contributors Gregory B. Olson Department of Materials Science and Engineering, Northwestern University, Evanston, IL, USA [email protected] P3 Pablo Ordej´on Instituto de Materiales, CSIC, Barcelona, Spain [email protected] 1.5 Tadeusz Pakula Max Planck Institute for Polymer Research, Mainz, Germany and Department of Molecular Physics, Technical University, Lodz, Poland [email protected] P35 Vijay Pande Department of Chemistry and of Structural Biology, Stanford University, Stanford, CA 94305-5080, USA [email protected] 5.17 I.R. Pankratov Russian Research Centre, “Kurchatov Institute”, Moscow 123182, Russia [email protected] 7.10 D.A. Papaconstantopoulos Center for Computational Materials Science, Naval Research Laboratory, Washington, DC, USA [email protected] 1.14 J.E. Pask Lawrence Livermore National Laboratory, Livermore, CA, USA [email protected] 1.19 Anthony T. Patera Massachusetts Institute of Technology, Cambridge, MA, USA [email protected] 4.15

xxi Mike Payne Cavendish Laboratory, University of Cambridge, UK [email protected] P16 Leonid Pechenik University of California, Santa Barbara, CA, USA [email protected] 4.3 Joaquim Peir´o Department of Aeronautics, Imperial College, London, UK [email protected] 8.2 Simon R. Phillpot Department of Materials Science and Engineering, University of Florida, Gainesville, FL 32611, USA [email protected] 2.6, 6.11 G.P. Potirniche Mississippi State University, Mississippi State, MS, USA [email protected] 3.5 Thomas R. Powers Division of Engineering, Brown University, Providence, RI, USA thomas [email protected] 9.8 Dierk Raabe Max-Planck-Institut f¨ur Eisenforschung, Max-Planck-Str. 1, D-40237 D¨usseldorf, Germany [email protected] 7.7, P6 Ravi Radhakrishnan Department of Bioengineering, University of Pennsylvania, Philadelphia, PA, USA [email protected] 5.5

xxii Christian Ratsch University of California at Los Angeles, Los Angeles, CA, USA [email protected] 7.15 John R. Ray 1190 Old Seneca Road, Central, SC 29630, USA [email protected] 2.16 William P. Reinhardt University of Washington Seattle, Washington, USA [email protected] 2.15 Karsten Reuter Fritz-Haber-Institut der Max-Planck-Gesellschaft, Faradayweg 4-6, D-14195 Berlin, Germany [email protected] 1.9 J.M. Rickman Department of Materials Science and Engineering, Lehigh University, Bethlehem, PA 18015, USA [email protected] 7.14, 7.19

List of contributors Tomonori Sakai Centre for Computational Science, Queen Mary, University of London, Mile End Road, London E1 4NS, UK 8.5 Deniel S´anchez-Portal Donostia International Physics Center, Donostia, Spain [email protected] 1.5 Joachim Sauer Institut f¨ur Chemie, Humboldt-Universit¨at zu Berlin, Unter den Linden 6, D-10099 Berlin, Germany 1.12 Avadh Saxena Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM, USA [email protected] 7.5 Matthias Scheffler Fritz-Haber-Institut der Max-Planck-Gesellschaft, Faradayweg 4-6, D-14195 Berlin, Germany [email protected] 1.9

Angel Rubio Departamento Física de Materiales and Unidad de Física de Materiales Centro Mixto CSIC-UPV, Universidad del País Vasco and Donosita Internacional Physics Center (DIPC), Spain [email protected] 1.11

Klaus Schulten Theoretical and Computational Biophysics Group, Beckman Institute for Advanced Science and Technology, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA [email protected] 5.15

Robert E. Rudd Lawrence Livermore National Laboratory, University of California, L-045 Livermore, CA 94551, USA [email protected] 2.12

Steven D. Schwartz Departments of Biophysics and Biochemistry, Albert Einstein College of Medicine, New York, USA [email protected] 5.8

Gregory C. Rutledge Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA [email protected] 9.1

Robin L.B. Selinger Physics Department, Catholic University, Washington, DC 20064, USA [email protected] 2.23

List of contributors Marcelo Sepliarsky Instituto de Física Rosario, Facultad de Ciencias Exactas, Ingenieria y Agrimensura, Universidad Nacional de Rosario, 27 de Febreo 210 Bis, (2000) Rosario, Argentina [email protected] 2.6 Alessandro Sergi Chemical Physics Theory Group, Department of Chemistry, University of Toronto, Toronto, Ont. M5S 3H6, Canada [email protected] 2.17, 5.4 J.A. Sethian Department of Mathematics, University of California, Berkeley, CA, USA [email protected] 4.6 Michael J. Shelley Courant Institute of Mathematical Sciences, New York University, New York, NY, USA [email protected] 4.7 C. Shen The Ohio State University, Columbus, Ohio, USA [email protected] 7.4 Spencer Sherwin Department of Aeronautics, Imperial College, London, UK [email protected] 8.2 Marek Sierka Institut f¨ur Physikalische Chemie, Lehrstuhl f¨ur Theoretische Chemie, Universit¨at Karlsruhe, Kaiserstraße 12, D-76128 Karlsruhe, Germany [email protected] 1.12 Asimina Sierou University of Cambridge, Cambridge, UK [email protected] 9.6

xxiii Grant D. Smith Department of Materials Science and Engineering, Department of Chemical Engineering, University of Utah, Salt Lake City, Utah, USA [email protected] 9.2 Fr´ed´eric Soisson CEA Saclay, DMN-SRMP, 91191 Gif-sur-Yuette, France [email protected] 7.9 Jos´e M. Soler Universidad Aut´onoma de Madrid, Madrid, Spain [email protected] 1.5 Didier Sornette Institute of Geophysics and Planetary Physics and Department of Earth and Space Science, University of California, Los Angeles, California, USA and CNRS and Universit´e des Sciences, Nice, France [email protected] 4.4 David J. Srolovitz Princeton Materials Institute and Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, USA [email protected] 7.1, 7.13 Marcelo G. Stachiotti Instituto de Física Rosario, Facultad de Ciencias Exactas, Ingenieria y Agrimensura, Universidad Nacional de Rosario, 27 de Febreo 210 Bis, (2000) Rosario, Argentina [email protected] 2.6 Catherine Stampfl Fritz-Haber-Institut der Max-Planck-Gesellschaft, Faradayweg 4-6, D-14195 Berlin, Germany; School of Physics, The University of Sydney, Sydney 2006, Australia [email protected] 1.9

xxiv

List of contributors

H. Eugene Stanley Center for Polymer Studies and Department of Physics Boston, University, Boston, MA 02215, USA [email protected] P36

Meijie Tang Lawrence Livermore National Laboratory, P.O. Box 808, Livermore, CA 94550 [email protected] 2.22

P.A. Sterne Lawrence Livermore National Laboratory, Livermore, CA, USA [email protected] 1.19

Mounir Tarek Equipe de dynamique des assemblages membranaires, Unit´e mixte de recherche CNRS/UHP 7565, Institut nanc´eien de chimie mol´eculaire, Universit´e Henri Poincar´e, BP 239, 54506 Vanduvre-l`es-Nancy cedex, France 2.26

Howard A. Stone Division of Engineering and Applied Sciences, Harvard University, Cambridge, MA 01238, USA [email protected] 4.8 Marshall Stoneham Centre for Materials Research, and London Centre for Nanotechnology, Department of Physics and Astronomy, University College London, Gower Street, London WC1E 6BT, UK [email protected] P12 Sauro Succi Istituto Applicazioni Calcolo, National Research Council, viale del Policlinico, 137, 00161, Rome, Italy [email protected] 8.4 E.B. Tadmor Technion-Israel Institute of Technology, Haifa, Israel [email protected] 2.13 Emad Tajkhorshid Theoretical and Computational Biophysics Group, Beckman Institute for Advanced Science and Technology, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA [email protected] 5.15

DeCarlos E. Taylor Quantum Theory Project, Departments of Chemistry and Physics, University of Florida, Gainesville, FL 32611, USA [email protected] 1.3 Doros N. Theodorou School of Chemical Engineering, National Technical University of Athens, 9 Heroon Polytechniou Street, Zografou Campus, 157 80 Athens, Greece [email protected] P15 Carl V. Thompson Department of Materials Science and Engineering, M.I.T., Cambridge, MA 02139, USA [email protected] P26 Anna-Karin Tornberg Courant Institute of Mathematical Sciences, New York University, New York, NY, USA [email protected] 4.7 S. Torquato Department of Chemistry, PRISM, and Program in Applied & Computational Mathematics, Princeton University, Princeton, NJ 08544, USA [email protected] 4.5, 7.18

List of contributors Bernhardt L. Trout Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, MA, USA [email protected] 5.5 Mark E. Tuckerman Department of Chemistry, Courant Institute of Mathematical Science, New York University, New York, NY 10003, USA [email protected] 2.9 Blas P. Uberuaga Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA [email protected] 2.11, 5.6 Patrick T. Underhill Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, MA, USA 9.7 V.G. Vaks Russian Research Centre, “Kurchatov Institute”, Moscow 123182, Russia [email protected] 7.10 Priya Vashishta Collaboratory for Advanced Computing and Simulations, Department of Chemical Engineering and Materials Science, University of Southern California, 3651 Watt Way, VHE 608, Los Angeles, CA 90089-0242, USA [email protected] 2.25 A. Van der Ven Department of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, MA, USA 1.17 Karen Veroy Massachusetts Institute of Technology, Cambridge, MA, USA [email protected] 4.15

xxv Alessandro De Vita King’s College London, UK, Center for Nanostructured, Materials (CENMAT) and DEMOCRITOS National Simulation Center, Trieste, Italy alessandro.de [email protected] P16 V. Vitek Department of Materials Science and Engineering, University of Pennsylvania, Philadelphia, PA 19104, USA [email protected] P32 Dionisios G. Vlachos Department of Chemical Engineering, Center for Catalytic Science and Technology, University of Delaware, Newark, DE 19716, USA [email protected] 4.12 Arthur F. Voter Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA [email protected] 2.11, 5.6 Gregory A. Voth Department of Chemistry and Henry Eyring Center for Theoretical Chemistry, University of Utah, Salt Lake City, Utah 84112-0850, USA [email protected] 5.9 G.Z. Voyiadjis Louisiana State University, Baton Rouge, LA, USA [email protected] 3.8 Dimitri D. Vvedensky Imperial College, London, United Kingdom [email protected] 7.16 G¨oran Wahnstr¨om Chalmers University of Technology and G¨oteborg University Materials and Surface Theory, SE-412 96 G¨oteborg, Sweden [email protected] 5.14

xxvi

List of contributors

Duane C. Wallace Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA [email protected] P1

Brian D. Wirth Department of Nuclear Engineering, University of California, Barkeley, CA, USA [email protected] 2.29

Axel van de Walle Northwestern University, Evanston, IL, USA [email protected] 1.16

Dieter Wolf Materials Science Division, Argonne National Laboratory, Argonne, IL 60439, USA [email protected] 6.7, 6.9, 6.10, 6.11, 6.12, 6.13

Chris G. Van de Walle Materials Department, University of California, Santa Barbara, California, USA [email protected] 6.3

C.Z. Wang Ames Laboratory-U.S. DOE and Department of Physics and Astronomy, Iowa State University, Ames, IA 50011, USA [email protected] 1.15

Y. Wang The Ohio State University, Columbus, Ohio, USA [email protected] 7.4

Yu U. Wang Department of Materials Science and Engineering, Virginia Tech., Blacksburg, VA 24061, USA [email protected] 7.12

Hettithanthrige S. Wijesinghe Department of Mechanical Engineering, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA [email protected] 8.8

Chung H. Woo The Hong Kong Polytechnic University, Hong Kong SAR, China [email protected] 2.27 Christopher Woodward Northwestern University, Evanston, Illinois, USA [email protected] P29 S.Y. Wu Department of Physics, University of Louisville, Louisville, KY 40292, USA [email protected] P39 Yang Xiang Department of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong [email protected] 7.13 Sidney Yip Department of Physics, Harvard University, Cambridge, MA 02138, USA [email protected] 2.1, 2.10, 6.7, 6.8, 6.11 M. Yu Department of Physics, University of Louisville, Louisville, KY 40292, USA P39

List of contributors H.M. Zbib Washington State University, Pullman, WA, USA [email protected] 3.3 Fangqiang Zhu Theoretical and Computational Biophysics Group, Beckman Institute for Advanced Science and Technology, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA [email protected] 5.15

xxvii M. Zikry North Carolina State University, Raleigh, NC, USA [email protected] 3.7

DETAILED TABLE OF CONTENTS PART A – METHODS Chapter 1. Electronic Scale 1.1

Understand, Predict, and Design Nicola Marzari 1.2 Concepts for Modeling Electrons in Solids: A Perspective Marvin L. Cohen 1.3 Achieving Predictive Simulations with Quantum Mechanical Forces Via the Transfer Hamiltonian: Problems and Prospects Rodney J. Bartlett, DeCarlos E. Taylor, and Anatoli Korkin 1.4 First-Principles Molecular Dynamics Roberto Car, Filippo de Angelis, Paolo Giannozzi, and Nicola Marzari 1.5 Electronic Structure Calculations with Localized Orbitals: The Siesta Method Emilio Artacho, Julian D. Gale, Alberto García, Javier Junquera, Richard M. Martin, Pablo Ordej´on, Deniel S´anchez-Portal, and Jos´e M. Soler 1.6 Electronic Structure Methods: Augmented Waves, Pseudopotentials and the Projector Augmented Wave Method Peter E. Bl¨ochl, Johannes K¨astner, and Clemens J. F¨orst 1.7 Electronic Scale James R. Chelikowsky 1.8 An Introduction to Orbital-Free Density Functional Theory Vincent L. Lign`eres and Emily A. Carter 1.9 Ab Initio Atomistic Thermodynamics and Statistical Mechanics of Surface Properties and Functions Karsten Reuter, Catherine Stampfl, and Matthias Scheffler 1.10 Density-Functional Perturbation Theory Paolo Giannozzi and Stefano Baroni

xxix

9 13

27

59

77

93 121 137

149 195

xxx

Detailed table of contents

1.11 Quasiparticle and Optical Properties of Solids and Nanostructures: The GW-BSE Approach Steven G. Louie and Angel Rubio 1.12 Hybrid Quantum Mechanics/Molecular Mechanics Methods and their Application Marek Sierka and Joachim Sauer 1.13 Ab Initio Molecular Dynamics Simulations of Biologically Relevant Systems Alessandra Magistrato and Paolo Carloni 1.14 Tight-Binding Total Energy Methods for Magnetic Materials and Multi-Element Systems Michael J. Mehl and D.A. Papaconstantopoulos 1.15 Environment-Dependent Tight-Binding Potential Models C.Z. Wang and K.M. Ho 1.16 First-Principles Modeling of Phase Equilibria Axel van de Walle and Mark Asta 1.17 Diffusion and Configurational Disorder in Multicomponent Solids A. Van der Ven and G. Ceder 1.18 Data Mining in Materials Development Dane Morgan and Gerbrand Ceder 1.19 Finite Elements in Ab Initio Electronic-Structure Calculations J.E. Pask and P.A. Sterne 1.20 Ab Initio Study of Mechanical Deformation Shigenobu Ogata

215

241

259

275 307 349

367 395 423 439

Chapter 2. Atomistic Scale 2.1 2.2 2.3 2.4 2.5 2.6

2.7 2.8

Introduction: Atomistic Nature of Materials Efthimios Kaxiras and Sidney Yip Interatomic Potentials for Metals Y. Mishin Interatomic Potential Models for Ionic Materials Julian D. Gale Modeling Covalent Bond with Interatomic Potentials Jo˜ao F. Justo Interatomic Potentials: Molecules Alexander D. MacKerell, Jr. Interatomic Potentials: Ferroelectrics Marcelo Sepliarsky, Marcelo G. Stachiotti, and Simon R. Phillpot Energy Minimization Techniques in Materials Modeling C.R.A. Catlow Basic Molecular Dynamics Ju Li

451 459 479 499 509

527 547 565

Detailed table of contents 2.9 2.10 2.11

2.12

2.13 2.14 2.15 2.16

2.17 2.18

2.19 2.20

2.21 2.22

2.23 2.24 2.25

2.26

Generating Equilibrium Ensembles Via Molecular Dynamics Mark E. Tuckerman Basic Monte Carlo Models: Equilibrium and Kinetics George Gilmer and Sidney Yip Accelerated Molecular Dynamics Methods Blas P. Uberuaga, Francesco Montalenti, Timothy C. Germann, and Arthur F. Voter Concurrent Multiscale Simulation at Finite Temperature: Coarse-Grained Molecular Dynamics Robert E. Rudd The Theory and Implementation of the Quasicontinuum Method E.B. Tadmor and R.E. Miller Perspective: Free Energies and Phase Equilibria David A. Kofke and Daan Frenkel Free-Energy Calculation Using Nonequilibrium Simulations Maurice de Koning and William P. Reinhardt Ensembles and Computer Simulation Calculation of Response Functions John R. Ray Non-Equilibrium Molecular Dynamics Giovanni Ciccotti, Raymond Kapral, and Alessandro Sergi Thermal Transport Process by the Molecular Dynamics Method Hideo Kaburaki Atomistic Calculation of Mechanical Behavior Ju Li The Peierls–Nabarro Model of Dislocations: A Venerable Theory and its Current Development Gang Lu Modeling Dislocations Using a Periodic Cell Wei Cai A Lattice Based Screw-Edge Dislocation Dynamics Simulation of Body Center Cubic Single Crystals Meijie Tang Atomistics of Fracture Diana Farkas and Robin L.B. Selinger Atomistic Simulations of Fracture in Semiconductors Noam Bernstein Multimillion Atom Molecular-Dynamics Simulations of Nanostructured Materials and Processes on Parallel Computers Priya Vashishta, Rajiv K. Kalia, and Aiichiro Nakano Modeling Lipid Membranes Christophe Chipot, Michael L. Klein, and Mounir Tarek

xxxi 589 613

629

649 663 683 707

729 745

763 773

793 813

827 839 855

875 929

xxxii

Detailed table of contents

2.27 Modeling Irradiation Damage Accumulation in Crystals Chung H. Woo 2.28 Cascade Modeling Jean-Paul Crocombette 2.29 Radiation Effects in Fission and Fusion Reactors G. Robert Odette and Brian D. Wirth 2.30 Texture Evolution During Thin Film Deposition Hanchen Huang 2.31 Atomistic Visualization Ju Li

959 987 999 1039 1051

Chapter 3. Mesoscale/Continuum Methods 3.1 3.2

3.3 3.4 3.5 3.6 3.7 3.8 3.9

Mesoscale/Macroscale Computational Methods M.F. Horstemeyer Perspective on Continuum Modeling of Mesoscale/Macroscale Phenomena D.J. Bammann Dislocation Dynamics H.M. Zbib and T.A. Khraishi Discrete Dislocation Plasticity E. Van der Giessen and A. Needleman Crystal Plasticity M.F. Horstemeyer, G.P. Potirniche, and E.B. Marin Internal State Variable Theory D.L. McDowell Ductile Fracture M. Zikry Continuum Damage Mechanics G.Z. Voyiadjis Microstructure-Sensitive Computational Fatigue Analysis D.L. McDowell

1071

1077 1097 1115 1133 1151 1171 1183 1193

Chapter 4. Mathematical Methods 4.1 4.2

4.3

4.4

Overview of Chapter 4: Mathematical Methods Martin Z. Bazant and Dimitrios Maroudas Elastic Stability Criteria and Structural Bifurcations in Crystals Under Load Frederick Milstein Toward a Shear-Transformation-Zone Theory of Amorphous Plasticity Michael L. Falk, James S. Langer, and Leonid Pechenik Statistical Physics of Rupture in Heterogeneous Media Didier Sornette

1217

1223

1281 1313

Detailed table of contents 4.5 4.6

4.7 4.8 4.9 4.10 4.11 4.12

4.13 4.14

4.15

Theory of Random Heterogeneous Materials S. Torquato Modern Interface Methods for Semiconductor Process Simulation J.A. Sethian Computing Microstructural Dynamics for Complex Fluids Michael J. Shelley and Anna-Karin Tornberg Continuum Descriptions of Crystal Surface Evolution Howard A. Stone and Dionisios Margetis Breakup and Coalescence of Free Surface Flows Jens Eggers Conformal Mapping Methods for Interfacial Dynamics Martin Z. Bazant and Darren Crowdy Equation-Free Modeling for Complex Systems Ioannis G. Kevrekidis, C. William Gear, and Gerhard Hummer Mathematical Strategies for the Coarse-Graining of Microscopic Models Markos A. Katsoulakis and Dionisios G. Vlachos Multiscale Modeling of Crystalline Solids Weinan E and Xiantao Li Multiscale Computation of Fluid Flow in Heterogeneous Media Thomas Y. Hou Certified Real-Time Solution of Parametrized Partial Differential Equations Nguyen Ngoc Cuong, Karen Veroy, and Anthony T. Patera

xxxiii 1333

1359 1371 1389 1403 1417 1453

1477 1491

1507

1529

PART B – MODELS Chapter 5. Rate Processes 5.1 5.2 5.3 5.4 5.5

5.6

Introduction: Rate Processes Horia Metiu A Modern Perspective on Transition State Theory J.D. Doll Transition Path Sampling Christoph Dellago Simulating Reactions that Occur Once in a Blue Moon Giovanni Ciccotti, Raymond Kapral, and Alessandro Sergi Order Parameter Approach to Understanding and Quantifying the Physico-Chemical Behavior of Complex Systems Ravi Radhakrishnan and Bernhardt L. Trout Determining Reaction Mechanisms Blas P. Uberuaga and Arthur F. Voter

1567 1573 1585 1597

1613 1627

xxxiv 5.7 5.8

5.9 5.10

5.11 5.12

5.13 5.14 5.15

5.16 5.17

Detailed table of contents Stochastic Theory of Rate Processes Abraham Nitzan Approximate Quantum Mechanical Methods for Rate Computation in Complex Systems Steven D. Schwartz Quantum Rate Theory: A Path Integral Centroid Perspective Eitan Geva, Seogjoo Jang, and Gregory A. Voth Quantum Theory of Reactive Scattering and Adsorption at Surfaces Axel Groß Stochastic Chemical Kinetics Daniel T. Gillespie Kinetic Monte Carlo Simulation of Non-Equilibrium Lattice-Gas Models: Basic and Refined Algorithms Applied to Surface Adsorption Processes J.W. Evans Simple Models for Nanocrystal Growth Pablo Jensen Diffusion in Solids G¨oran Wahnstr¨om Kinetic Theory and Simulation of Single-Channel Water Transport Emad Tajkhorshid, Fangqiang Zhu, and Klaus Schulten Simplified Models of Protein Folding Hue Sun Chan Protein Folding: Detailed Models Vijay Pande

1635

1673 1691

1713 1735

1753 1769 1787

1797 1823 1837

Chapter 6. Crystal Defects 6.1 6.2 6.3 6.4 6.5 6.6 6.7

Point Defects C.R.A. Catlow Point Defects in Metals Kai Nordlund and Robert Averback Defects and Impurities in Semiconductors Chris G. Van de Walle Point Defects in Simple Ionic Solids John Corish Fast Ion Conductors Alan V. Chadwick Defects and Ion Migration in Complex Oxides M. Saiful Islam Introduction: Modeling Crystal Interfaces Sidney Yip and Dieter Wolf

1851 1855 1877 1889 1901 1915 1925

Detailed table of contents 6.8 6.9 6.10

6.11 6.12 6.13

Atomistic Methods for Structure–Property Correlations Sidney Yip Structure and Energy of Grain Boundaries Dieter Wolf High-Temperature Structure and Properties of Grain Boundaries Dieter Wolf Crystal Disordering in Melting and Amorphization Sidney Yip, Simon R. Phillpot, and Dieter Wolf Elastic Behavior of Interfaces Dieter Wolf Grain Boundaries in Nanocrystalline Materials Dieter Wolf

xxxv 1931 1953

1985 2009 2025 2055

Chapter 7. Microstructure 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8

7.9

7.10 7.11 7.12 7.13

Introduction: Microstructure David J. Srolovitz and Long-Qing Chen Phase-Field Modeling Alain Karma Phase-Field Modeling of Solidification Seong Gyoon Kim and Won Tae Kim Coherent Precipitation – Phase Field Method C. Shen and Y. Wang Ferroic Domain Structures using Ginzburg–Landau Methods Avadh Saxena and Turab Lookman Phase-Field Modeling of Grain Growth Carl E. Krill III Recrystallization Simulation by Use of Cellular Automata Dierk Raabe Modeling Coarsening Dynamics using Interface Tracking Methods John Lowengrub Kinetic Monte Carlo Method to Model Diffusion Controlled Phase Transformations in the Solid State Georges Martin and Fr´ed´eric Soisson Diffusional Transformations: Microscopic Kinetic Approach I.R. Pankratov and V.G. Vaks Modeling the Dynamics of Dislocation Ensembles Nasr M. Ghoniem Dislocation Dynamics – Phase Field Yu U. Wang, Yongmei M. Jin, and Armen G. Khachaturyan Level Set Dislocation Dynamics Method Yang Xiang and David J. Srolovitz

2083 2087 2105 2117 2143 2157 2173

2205

2223 2249 2269 2287 2307

xxxvi

Detailed table of contents

7.14 Coarse-Graining Methodologies for Dislocation Energetics and Dynamics J.M. Rickman and R. LeSar 7.15 Level Set Methods for Simulation of Thin Film Growth Russel Caflisch and Christian Ratsch 7.16 Stochastic Equations for Thin Film Morphology Dimitri D. Vvedensky 7.17 Monte Carlo Methods for Simulating Thin Film Deposition Corbett Battaile 7.18 Microstructure Optimization S. Torquato 7.19 Microstructural Characterization Associated with Solid–Solid Transformations J.M. Rickman and K. Barmak

2325 2337 2351 2363 2379

2397

Chapter 8. Fluids 8.1 8.2

8.3

8.4 8.5

8.6 8.7

8.8

Mesoscale Models of Fluid Dynamics Bruce M. Boghosian and Nicolas G. Hadjiconstantinou Finite Difference, Finite Element and Finite Volume Methods for Partial Differential Equations Joaquim Peir´o and Spencer Sherwin Meshless Methods for Numerical Solution of Partial Differential Equations Gang Li, Xiaozhong Jin, and N.R. Aluru Lattice Boltzmann Methods for Multiscale Fluid Problems Sauro Succi, Weinan E, and Efthimios Kaxiras Discrete Simulation Automata: Mesoscopic Fluid Models Endowed with Thermal Fluctuations Tomonori Sakai and Peter V. Coveney Dissipative Particle Dynamics Pep Espa˜nol The Direct Simulation Monte Carlo Method: Going Beyond Continuum Hydrodynamics Francis J. Alexander Hybrid Atomistic–Continuum Formulations for Multiscale Hydrodynamics Hettithanthrige S. Wijesinghe and Nicolas G. Hadjiconstantinou

2411

2415

2447 2475

2487 2503

2513

2523

Chapter 9. Polymers and Soft Matter 9.1 9.2

Polymers and Soft Matter L. Mahadevan and Gregory C. Rutledge Atomistic Potentials for Polymers and Organic Materials Grant D. Smith

2555 2561

Detailed table of contents 9.3 9.4 9.5 9.6 9.7

9.8 9.9

Rotational Isomeric State Methods Wayne L. Mattice Monte Carlo Simulation of Chain Molecules V.G. Mavrantzas The Bond Fluctuation Model and Other Lattice Models Marcus M¨uller Stokesian Dynamics Simulations for Particle Laden Flows Asimina Sierou Brownian Dynamics Simulations of Polymers and Soft Matter Patrick S. Doyle and Patrick T. Underhill Mechanics of Lipid Bilayer Membranes Thomas R. Powers Field-Theoretic Simulations Venkat Ganesan and Glenn H. Fredrickson

xxxvii 2575 2583 2599 2607

2619 2631 2645

Plenary Perspectives P1 P2 P3 P4 P5 P6

P7

Progress in Unifying Condensed Matter Theory Duane C. Wallace The Future of Simulations in Materials Science D.P. Landau Materials by Design Gregory B. Olson Modeling at the Speed of Light J.D. Joannopoulos Modeling Soft Matter Kurt Kremer Drowning in Data – A Viewpoint on Strategies for Doing Science with Simulations Dierk Raabe Dangers of “Common Knowledge” in Materials Simulations Vasily V. Bulatov

Quantum Simulations as a Tool for Predictive Nanoscience Giulia Galli and François Gygi P9 A Perspective of Materials Modeling William A. Goddard III P10 An Application Oriented View on Materials Modeling Peter Gumbsch P11 The Role of Theory and Modeling in the Development of Materials for Fusion Energy Nasr M. Ghoniem

2659 2663 2667 2671 2675

2687

2695

P8

2701 2707 2713

2719

xxxviii

Detailed table of contents

P12 Where are the Gaps? Marshall Stoneham P13 Bridging the Gap between Quantum Mechanics and Large-Scale Atomistic Simulation John A. Moriarty P14 Bridging the Gap between Atomistics and Structural Engineering J.S. Langer P15 Multiscale Modeling of Polymers Doros N. Theodorou P16 Hybrid Atomistic Modelling of Materials Processes Mike Payne, G´abor Cs´anyi, and Alessandro De Vita P17 The Fluctuation Theorem and its Implications for Materials Processing and Modeling Denis J. Evans P18 The Limits of Strength J.W. Morris, Jr. P19 Simulations of Interfaces between Coexisting Phases: What Do They Tell us? Kurt Binder P20 How Fast Can Cracks Move? Farid F. Abraham P21 Lattice Gas Automaton Methods Jean Pierre Boon P22 Multi-Scale Modeling of Hypersonic Gas Flow Iain D. Boyd P23 Commentary on Liquid Simulations and Industrial Applications Raymond D. Mountain P24 Computer Simulations of Supercooled Liquids and Glasses Walter Kob P25 Interplay between Materials Theory and High-Pressure Experiments Raymond Jeanloz P26 Perspectives on Experiments, Modeling and Simulations of Grain Growth Carl V. Thompson P27 Atomistic Simulation of Ferroelectric Domain Walls I-Wei Chen

2731

2737

2749 2757 2763

2773 2777

2787 2793

2805 2811

2819 2823

2829

2837 2843

Detailed table of contents

xxxix

P28 Measurements of Interfacial Curvatures and Characterization of Bicontinuous Morphologies Sow-Hsin Chen

2849

P29 Plasticity at the Atomic Scale: Parametric, Atomistic, and Electronic Structure Methods Christopher Woodward P30 A Perspective on Dislocation Dynamics Nasr M. Ghoniem P31 Dislocation-Pressure Interactions J.P. Hirth P32 Dislocation Cores and Unconventional Properties of Plastic Behavior V. Vitek P33 3-D Mesoscale Plasticity and its Connections to Other Scales Ladislas P. Kubin P34 Simulating Fluid and Solid Particles and Continua with SPH and SPAM Wm.G. Hoover P35 Modeling of Complex Polymers and Processes Tadeusz Pakula P36 Liquid and Glassy Water: Two Materials of Interdisciplinary Interest H. Eugene Stanley P37 Material Science of Carbon Wesley P. Hoffman P38 Concurrent Lifetime-Design of Emerging High Temperature Materials and Components Ronald J. Kerans P39 Towards a Coherent Treatment of the Self-Consistency and the Environment-Dependency in a Semi-Empirical Hamiltonian for Materials Simulation S.Y. Wu, C.S. Jayanthi, C. Leahy, and M. Yu

2865 2871 2879

2883 2897

2903 2907

2917 2923

2929

2935

5.1 INTRODUCTION: RATE PROCESSES Horia Metiu University of California, Santa Barbara, CA, USA

We can divide the time evolution of a system into two classes. In one, a part of the system changes its state from time to time; chemical reactions, polaron mobility, diffusion of adsorbates on a surface, and protein folding belong to this class. In the other, the change of state takes place continuously; electrical conductivity, the diffusion of molecules in gases, and the thermoelectric effect in doped semiconductors belong to this class. Chemical kinetics deals with phenomena of the first kind; the second kind is studied by transport theory. It is in the nature of a many-body system that its parts share energy with each other, creating a state of approximate equality. This leads to stagnation: each part tends to hover near the bottom of a bowl in the potential energy surface. Occasionally, the inherent thermal fluctuations put enough energy in a part of the many-body system to cause it to escape from its bowl and travel away from home. But the tendency to lose energy rapidly, once a part acquires more than its average share, will trap the traveler in another bowl. When this happens, the system has undergone a chemical reaction, or the polaron took a jump to another lattice site, or an impurity in a solid changed location. The rate of these events is described by well known, generic, phenomenological rate equations. The parameter characterizing the rate of a specific system is the rate constant k. In the past 30 years great progress has been made in our ability to calculate the rate constant by atomic simulations. The machinery for performing such calculations is described in the first articles in this chapter. Doll presents the modern view on the old and famous transition state theory, which is still one of the most useful and widely used procedures for calculating rate constants. The atomic motion in a many-body system takes place on a scale of femtoseconds, while the lifetime of a system in a potential energy bowl is much longer. This discrepancy led to the misconception that the dynamics of a chemical reaction is slow. 1567 S. Yip (ed.), Handbook of Materials Modeling, 1567–1571. c 2005 Springer. Printed in the Netherlands.

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The main insight of TST is that a system acquires enough energy to undergo a reaction only “once in a blue moon”. If enough energy is acquired, in the right coordinates, the dynamics of the reaction is very rapid. The rate of reaction is low not because its dynamics is slow, but because the system has enough energy very rarely. In modern parlance the reaction is a rare event. This causes problems for a brute-force simulation of a reaction. One can follow a group of atoms, in the many-body system, for a nanosecond, because of limitations in computing power, and not observe a reactive event. The second insight of TST is that the only parameter out of equilibrium, in a chemical kinetics experiment, is the concentration. Each molecule participating in the reaction is in equilibrium with its environment at all times. Therefore, one can calculate, from equilibrium statistical mechanics, the probability that the system reaches the transition state and the rate with which the system crosses the ridge separating the bowl in which the system is initially located from the one that is the final destination. This is all it takes to build a theory of the rate constant. The only approximation is the assumption that once the system crosses the ridge, it will turn back only in a long time, on the order of k−1 . This late event is part of the backward reaction and it does not affect the forward rate constant. Given the propensity of many-body systems to share energy among degrees of freedom, this is not a bad assumption: once it crosses the ridge the system has a high energy in the reaction coordinate and it is likely to lose it. There are, however, cases in which the shape of the potential energy around the ridge is peculiar or the reaction coordinate is weakly coupled to the other degrees of freedom. When this happens, recrossing is not negligible and TST makes errors. In my experience these errors are small and rarely affect the prefactor A, in the expression k =A exp[−E/RT], by more than 30%. Given the fact that we are unable to calculate the activation energy E accurately and that the latter appears at the exponent it seems unwise to try to obtain an accurate value for A when one makes substantial errors in E (a 0.2-eV error is not rare). This is why TST is still popular in spite of the fact that one could calculate the rate constant exactly, sometimes without a great deal of additional computational effort. The TST reduces the calculation of k to the calculation of partition functions, which can be performed by Monte Carlo simulations. There is no longer any need to perform extremely long Molecular Dynamics calculations in the hope of observing a transition of the system from one bowl to another. Because recrossing is neglected, the rate constant calculated by TST is always larger than the exact rate constant. This does not mean that the TST rate constant is always larger than the measured one. It is only larger than the rate constant calculated exactly on the potential energy surface used in the TST calculations. This inequality led to the development of variational transition state theory, developed and used extensively in Truhlar’s work. In this procedure one

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varies the position of the surface dividing the initial and the final bowls, until the transition theory rate constant has a minimum. The rate constant obtained in this way is more accurate (assuming that the potential energy is accurate) than the one calculated by placing the dividing surface on the ridge separating the two bowls. These issues are discussed and explained in Doll’s article. The next two articles, by Dellago and by Ciccotti, Kapral and Sergi, describe the methods used for exact calculations of the rate constant k. Here “exact” means the exact rate constant for a given potential energy surface. If the potential energy surface is erroneous, the exact rate constant has nothing to do with reality. However, it is important to have an exact theory, since our ability to generate reasonable (and sometimes accurate) potential energy surfaces is improving each year. The exact theory of the rate constant is based on the so-called correlation function theory, which first appeared in a paper by Yamamoto. Since this theory does not assume that recrossing does not take place, it must use molecular dynamics to determine which trajectories recross and which do not. It does this very cleverly, to avoid the “rare event” trap. It uses equilibrium statistical mechanics to place the system on the dividing surface, with the correct probability. Then it lets the system evolve to cross the dividing surface and follows its evolution to determine whether it will recross the dividing surface. If it does, that particular crossing event is discarded. If it does not, it is kept as a reactive event. Averages over many such reactive events, used in a specific equation provided by the theory, give the exact rate constant. The advantage of this method, over ordinary molecular dynamics, is that it must follow the trajectory only for the time when the reaction coordinate loses energy and the system becomes unable to recross the dividing surface. As many experiments and simulations show, this time is shorter than a picosecond, which is quite manageable in computations. Moreover, the procedure generates a large of number of reactive trajectories with the appropriate probability. Since reactive trajectories are very improbable, a brute-force molecular dynamics simulation, starting with the reactants, will generate roughly one reactive trajectory in 100 000 calculations, each requiring a very long trajectory. This is why brute-force calculations of the rate constant are not possible. The two articles mentioned above discuss two different ways of implementing the theory. The theory presented by Dellago is new and has not been extensively tested. The one presented by Ciccotti, Kapral, and Sergi is the workhorse used for all systems that can be described by classical mechanics. While in principle the method is simple, the implementation is full of pitfalls and “small” technical difficulties, and these are clarified in the articles. Application of the correlation function theory to the diffusion of impurities in solids is discussed by Wahnstrom.

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The statements made above, about the time scales reached by molecular dynamics, were true until a few years ago, when Voter proposed several methods that allow us to accelerate molecular dynamics to the point that we can follow the evolution of a complex system for microseconds. This has brought unexpected benefits. To use the transition state theory, or the correlation function theory of rare events, one must know what the events are; we need to know the initial and final state of the system. There are systems for which this is not easy to do. For example, Johnsson discovered, while studying the evolution of the shape of an “island” made by adsorbed atoms, that six atoms move in concert with relative ease. It is very unlikely that anyone would have proposed the existence of this “reaction” on the basis of chemical intuition. In general, in the complex systems encountered in biology and materials science, a group of molecules may move coherently and rapidly together in ways that are not intuitively expected. The accelerated dynamics method often finds such events, since it does not make assumptions about the final state of the system. The article of Blas, Uberuaga, and Voter discusses this aspect of kinetics. Since Kramers’ classic work, it has been realized that in many systems chemical reactions can be described by a stochastic method that involves the Brownian motion of the representative point of the system on the potential energy surface. Since then, the theory has been expanded and used to explain chemical kinetics in condensed phases. Its advantage is that it expresses chemical kinetics in complex media in terms of a few parameters, the strength of thermal fluctuations in the system and the “friction” causing the system to lose energy from the reaction coordinate. This reductionist approach appeals to many experimentalists who have used it to analyze chemical kinetics of molecules in liquids. Much work has also been done to connect the friction and the fluctuations to the detailed dynamics of the system. Nitzan’s article reviews the status of this field. All theories mentioned above assume that the motion of the system can be described by classical mechanics. This is not the case in reactions involving proton or electron transfer. The generalization of the correlation function theory of the rate constant to a fully quantum theory has been made by Miller, Schwartz, and Tromp, who extended considerably the early work of Yamamoto. Some of the first computational methods using this theory were proposed by Wahnstrom and Metiu. Since then, approximate methods, that allow calculations for systems with many degrees of freedom, have been invented. These are reviewed by Schwartz and Voth, who have both contributed substantially to this field. The review of quantum theory of rates is rounded off by an article by Gross, on reactive scattering and adsorption at surfaces. This discusses the dynamics of such reactions in more detail than usual in kinetics, since it examines the rate of reaction (dissociation or adsorption) when the molecule approaching the surface has a well-defined quantum

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state. One can obtain the rate constant from this information, by averaging the state-specific rates over a thermal distribution of initial states. Many people familiar with statistical mechanics have realized that chemical kinetics is, like any other phenomenon in a many-body system, subject to fluctuations that might be observable if one could detect the kinetic behavior of a small number of molecules. It was believed that light scattering may be able to study such fluctuations, since it can detect the evolution of concentration in the very small volume illuminated by light. It turned out that the volume was not small enough and, as far as I know, the fluctuations have not been detected by this method. Undaunted by this lack of experimental observations, Gillespie went ahead and developed the methodology needed for studying the stochastic evolution of the concentration in a system undergoing chemical reactions. This methodology assumed that the rate constants are known and examined the evolution of the concentrations in space and time. Later on, scanning tunneling microscopy studies of the evolution of atoms deposited on a surface and a variety of single molecule kinetic experiments provided examples of systems in which fluctuations in rate processes play a very important role. Gillespie’s article reviews the methods dealing with fluctuating chemical kinetics. Evans reviews the stochastic algorithms needed for studying the kinetics of adsorbates, with applications to crystal growth and catalysis. Jensen’s article studies specific kinetic models used in crystal growth. The chapter ends with three articles on kinetic phenomena of interest in biology. The rate of protein folding, studied with minimalist models that try to capture the essential features causing proteins to fold, is reviewed by Chan. Pande examines the use of detailed models in which the interatomic interactions are treated in detail. The two approaches are complementary and much can be learned by comparing their conclusions. Tajkhorshid, Zhu, and Schulten review the transport of water through the pores of cell membranes. A dominant feature of this transport is that water forms a quasi one-dimensional “wire”. For this reason, transport in biological channels is closely related to water transport through a carbon nanotube and the article reviews both. Kinetics, one of the oldest and most useful branches of chemical physics, is undergoing a quiet revolution and is penetrating in all areas of materials science and biochemistry. There is a very good reason for this: most systems we are interested in are metastable. To understand what they are, we need to use kinetics to simulate how they are made. Moreover, we need to use kinetics to understand how they function and how they are degraded by outside influences or by inner instabilities. Finally, a well-formulated kinetic model contains thermodynamics as the long-time limit.

5.2 A MODERN PERSPECTIVE ON TRANSITION STATE THEORY J.D. Doll Department of Chemistry, Brown University, Providence, RI, USA

Chemical rates, the temporal evolution of the populations of species of interest, are of fundamental importance in science. Understanding how such rates are determined by the microscopic forces involved is, in turn, a basic focus of the present discussion. Before delving into the details, it is valuable to consider the general nature of the problem we face when considering the calculation of chemical rates. In what follows we shall assume that we know: • • • •

the relevant physical laws (classical or quantum) governing the system, the molecular forces at work, the identity of the chemical species of interest, and the formal statistical-mechanical expressions for the desired rates.

Given all that, what is the “problem?” In principle, of course, there is none. “All” that we need do is to work out the “details” of our formal expressions and we have our desired rates. The kinetics of any conceivable physical, chemical, or biologic process are thus within our reach. We can predict fracture kinetics in complex materials, investigate the effects of arbitrary mutations on protein folding rates, and optimize the choice of catalyst for the decomposition/storage of hydrogen in metals, right? Sadly, “no.” Even assuming that all of the above information is at our disposal, at present it is not possible in practice to carry out the “details” at the level necessary to produce the desired rates for arbitrary systems of interest. Why not? The essential problem we face when discussing chemical rates is one of greatly differing time scales. If, for example, a species is of sufficient interest that it makes sense to monitor its population, it is, by default, generally relatively “stable.” That is, it is a species that tends to live a “long” time on the scale 1573 S. Yip (ed.), Handbook of Materials Modeling, 1573–1583. c 2005 Springer. Printed in the Netherlands.

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of something like a molecular vibration. On the other hand, if we are to understand the details of chemical events of interest, then we must be able to describe the dynamics of those events on a time scale that is “short” on the molecular level. If we do otherwise , we risk losing the ability to understand how those detailed molecular motions influence and/or determine the rates at issue. What happens then when we confront the problem of describing a rate process whose natural time scale is on the order of seconds? If we are not careful we end up drowning in the detail imposed by being forced to describe events on macroscopic time scales using microscopic dynamical methods. In short, we spend a great deal of time (and effort) watching things “not happen.” Is there a better way to proceed? Fortunately, “yes.” Using methods developed by a number of investigators [1–9], it is possible to formulate practical and reliable methods for estimating chemical rates for systems of realistic complexity. While there are often assumptions involved in the practical implementation of these approaches, it is increasingly feasible to quantify and often remove the effects of these assumptions albeit at the expense of additional work. It is our purpose to review and illustrate these methods. Our discussion will focus principally on classical level implementations. Quantum formulations of these methods are possible and are considered elsewhere in this monograph. While much effort has been devoted to the quantum problem, it remains a particularly active area of current research. In the present discussion, we purposely utilize a sometimes nonstandard language in order to unify the discussion of a number of historically separate topics and approaches. The starting point for any discussion of chemical rates is the identification of various species of interest whose population will be monitored as a function of time. While there are many possible ways in which to do this, it is convenient to consider an approach based on the Stillinger/Weber inherent structure ideas [10, 11]. In this formulation, configuration space is partitioned by assigned each position to a unique potential energy basin (“inherent structure”) based on a steepest descent quench procedure. The relevant mental image is that of watching a “ball” roll slowly “downhill” on the potential energy surface under the action of an over-damped dynamics. In many applications the Stillinger/Weber inherent structures are themselves of primary interest. Although the number of such structures grows rapidly (exponentially) with system size [12], this type of analysis and the associated graphical tools it has spawned [13], provide a valuable language for characterizing potential energy surfaces. Wales, in particular, has utilized variations of the technique to great advantage in their study of the minimization problem [14]. In our discussion, it is the evolution of the populations of the inherent structures rather than the structures themselves that are of primary concern. Inherent structures, by construction, are associated with local minima in the

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potential energy surface. They thus have an intrinsic equilibrium population that can, if desired, be estimated using established statistical–mechanical techniques. Since the dynamics in the vicinity of the inherent structures is locally stable, the inherent structure populations tend to be (relatively) slowly varying and thus provide us with a natural set of populations for kinetic study. If followed as a function of time under the action of the dynamics generated by potential energy surface to which the inherent structures belong, the populations of the inherent structures will, aside from fluctuations, tend to remain constant at their various equilibrium values. Fluctuations in these populations, on the other hand, will result in a net flow of material between the various inherent structures. Such flows are the mechanism by which such fluctuations, either induced or spontaneous, “relax.” Consequently, they contain sufficient information to establish the desired kinetic parameters. To make the discussion more explicit, we consider the simple situation of a particle moving on the bistable potential energy depicted in Fig. 1. Performing a Stillinger/Weber quench on this potential energy will obviously produce two inherent structures. Denoted A and B in the figure, these correspond to the regions to the left and right of the potential energy maximum, respectively. We now imagine that we follow the dynamics of a statistical ensemble of N particles moving on this potential energy surface. For the purposes of discussion, we assume that the physical dynamics involved includes a solvent or “bath” (here unspecified) that provides fluctuating forces that act on the system

V(x)

A

B x

Figure 1. A prototypical, bistable potential energy. The two inherent structures, A and B, are separated by an energy barrier.

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of interest. The bath dynamics acts both to energize the system (permitting it to acquire sufficient energy to sometimes cross the potential barrier) as well as to dissipate that energy once it has been acquired. It is important to note that these fluctuations and dissipations must, in some sense, be balanced if an equilibrium state is to be produced and sustained [7]. Were the dynamics in our example purely conservative and one-dimensional in nature, for example, the notion of rates would be ill-posed. We now assume in what follows that we can monitor the populations of the inherent structures as a function of time. Denoting these populations NA (t) and NB (t), we further assume, following Chandler [7], that the overall kinetics of the system can described by the phenomenological rate equations dNA (t) = −kA→B NA (t) + kB→A NB (t) dt (1) dNB (t) = +kA→B NA (t) − kB→A NB (t). dt If the total number of particles is conserved, then the two inherent structure populations are trivially related: the fluctuation in the population of one inherent structure is the negative of that for the other. Assuming a fixed number of particles, it is thus a relatively simple matter to show that dδ NA (t) = −(kA→B + kB→A )δ NA (t), (2) dt where δ NA (t) indicates the deviation of NA (t) from its equilibrium value. The decay of a fluctuation in the population of inherent structure A, relative to an initial value at time zero, is thus given by δ NA (t) = δ NA (0) e−keff t ,

(3)

where keff is given by the sum of the “forward” and “backward” rate constants keff = (kA→B + kB→A ).

(4)

As noted by Onsager [15], it is physically reasonable to assume that if they are small, fluctuations, whether induced or spontaneous, are damped in a similar manner. Accepting this hypothesis, we conclude from the above analysis that the decay of the equilibrium population autocorrelation function, denoted here by , is given in terms of keff by δ NA (0)δ NA (t) = e−keff t . δ NA (0)δ NA (0)

(5)

Equivalently, taking the time derivative of both sides of this expression, we see that keff is given explicitly as keff = −

δ NA (0)δ N˙ A (t) . δ NA (0)δ NA (t)

(6)

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Equations (5) and (6) are formally exact expressions that relate the sum of the basic rate constants of interest to various dynamical objects that can be computed. Since we also know the ratio of these two rate constants (it is given by the corresponding ratio of the equilibrium populations), the desired rate parameters can be obtained from either expression provided that we can obtain the relevant time correlation functions involved. Although formally equivalent, Eqs. (5) and (6) differ with respect to their implicit computational demands. Computing the rate parameters via Eq. (5), for example, entails monitoring the decay of the population autocorrelation function. To obtain reliable estimates of the rate parameters from Eq. (5), we have to follow the system dynamics over a time-scale that is an appreciable fraction of the reciprocal of keff . If the barriers separating the inherent structures involved are “large”, this time scale can become macroscopic. Simply stated, the disparate time-scale problem makes it difficult to study directly the dynamics of infrequent events using the approach suggested by Eq. (5). Equation (6), on the other hand, offers a more convenient route to the desired kinetic parameters. In particular, it indicates that we might be able to obtain these parameters from short as opposed to long-time dynamical information. If the phenomenological rate expressions are formally correct for all times, then the ratio of the two time correlation functions in Eq. (6) is time-independent. However, since it is generally likely that the phenomenological rate expressions accurately describe only the longer-time motion between inherent structures, we expect in practice that the ratio on the right hand side of Eq. (6) will approach a constant “plateau” value only at times long on the scale of detailed molecular motions. The critical point, however, is that this transient period will be of molecular not macroscopic duration. With Eq. (6), we thus have a route to the desired kinetic parameters that requires only molecular or short time-scale dynamical input. A valuable practical point concerning kinetic formulations based on Eq. (6) is that for many applications the final plateau value of the correlation function ratio involved is often relatively well approximated by its zero time value. Because the correlation functions required depend only on time differences, such zero-time quantities are purely equilibrium objects. Consequently, an existing and extensive set of equilibrium tools can be invoked to produce approximations to kinetic parameters. The approach to the calculation of chemical rates based on Eq. (6) has several desirable characteristics. Most importantly, it has a refinable nature and can be implemented in stages. At the simplest level, we can estimate chemical rate parameters using purely zero-time, or equilibrium methods. Such approximate methods alone may be adequate for many applications. We are, however, not restricted to accepting such approximations blindly. With additional effort we can “correct” such preliminary estimates by performing additional dynamical studies. Because such calculations involve “corrections” to

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equilibrium estimates of rate parameters, as opposed to the entire rate parameters themselves, the dynamical input required is only that necessary to remove the errors induced by the initial equilibrium assumptions. Because such errors tend to involve simplified assumptions concerning the nature of transition state dynamics, the input required to estimate the corrections is of a molecular, not macroscopic time scale. We now focus our discussion on some of practical issues involved in generating equilibrium estimates of the rates. We shall illustrate these using the simple two-state example described above. We begin by imagining that we have at our disposal the time history of a reaction coordinate of interest, x(t). As a function of time, x(t) moves back-and-forth between inherent structures A and B, which we assume to be separated by the position x = q. Using one of the basic properties of the delta function, δ(ax) =

1 δ(x), |a|

(7)

it is easy to show that N (τ , [x(t)]), defined by N (τ, [x(t)]) =

τ

dx(t) δ(x(t) − q), dt

dt

0

(8)

is a functional of the path whose value is equal to the (total) number of crossings of the x(t) = q surface in the interval (0,τ ). Every time x(t) crosses q, the delta function argument takes on a zero value. Because the delta function in Eq. (8) is in coordinate space while the integral is with respect to time, the Jacobian factor into Eq. (8) creates a functional whose value jumps by unity each time x(t) − q sweeps through a value of zero. If we form a statistical ensemble corresponding to various possible histories of the motion of our system and bath, we can compute the average number of crossings of the x(t) = q surface in the (0,τ ) interval, N(τ , [x(t)]), using the expression N (τ, [x(t)]) =

τ

dt x(t) ˙ δ(x(t) − q) .

(9)

0

Here represents the time derivative of x(t). Because are dealing with a “stationary” or equilibrium process, the time correlation function that appears on the right hand side of Eq. (9) can be function only of time differences. Consequently, the integrand on the right hand side of Eq. (9) is time-independent and can be brought outside the integral. The result thus becomes τ dt, N (τ, [x(t)]) = x˙ δ(x − q) 0

(10)

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where the (now unnecessary) time labels have been dropped. We thus see that the number of crossings of the x(t) = q surface in this system per unit time is given by N (τ, [x(t)]) = x˙ δ(x − q) . (11) τ Recalling that N measures the total number of crossings, the number of crossings per unit time in the direction from A to B (the number of “up zeroes” of x(t) − q in the language of Slater) is half the value in Eq. (11). Thus, the equilibrium estimate of the rate constant for the A to B transition, (i.e., the number of crossings per unit time from A to B per atom in inherent structure A) is given by

1 TST = kA→B 2

x˙ δ(x − q)

NA

.

(12)

Equation (12) gives an approximate expression to the rate constant that involves an equilibrium flux between the relevant inherent structures. Because the relevant flux is associated with the “transition” of one inherent structure into another, the approach to chemical rates suggested by Eq. (12) is typically termed “transition state” theory (TST). Along with its multi-dimensional generalizations, it represents a convenient and useful approximation to the desired chemical rate constants. Being an equilibrium approximation to the dynamical objects of interest, it permits the powerful machinery of Monte Carlo methods [16, 17] to be brought to bear on the computational problem. The significance of this is that the required averages can be computed to any desired accuracy for arbitrary potential energy models. One can proceed analytically by making secondary, simplifying assumptions concerning the potential. Such approximations are, however, controllable in that their quality can be tested. Furthermore, Eq. (12) provides a unified treatment of the problem that is independent of the nature of the statistical ensemble that is involved. Applications involving canonical, microcanonical and other ensembles are treated within a common framework. It is historically interesting in this regard to note that if the reaction coordinate of interest is expressed as a superposition of normal modes, Eq. (12) leads naturally to the unimolecular reaction expressions of Ref. [4]. There is a technical aspect concerning the calculation of the averages appearing in Eq. (12) that merits discussion. In particular, it is apparent from the nature of the average involved that, if they are to be computed accurately, the numerical methods involved must be capable of accurately describing the reactant’s concentration profile in the vicinity of the transition state. If we are dealing with with activated processes where the difference between transition state in inherent structure energies are “large”, then such concentrations can become quite small and difficult to treat by standard methods. This is

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simply the equilibrium, “sparse-sampling” analog of the disparate time-scale dynamical problem. Fortunately, there are a number well-defined techniques for coping with this technical issue. These include, to name a few, umbrella methods [18], Bennett/Voter techniques [19, 20], J-walking [21, 22], and parallel tempering approaches [23]. These and related methods make it possible to compute the the required, transition-state-constrained averages. The basic approach outlined above can be extended in a number of ways. One immediate extension involves problems in which there are multiple, rather than two states involved. Adams has considered such problems in the context of his studies on the effects of precursor states on thermal desorption [24]. A second extension involves using the fundamental kinetic parameters produced to study more complex events. Voter, in a series of developments, has formulated a computationally viable method for studying diffusion in solids based on such an approach [25]. In its most complete form (including dynamical corrections), this approach produces a computationally exact procedure for surface or bulk diffusion coefficients of a point defect at arbitrary temperatures in a periodic system [26]. In related developments, Voter [25] and Henkelmen and J´onsson [27] have discussed using “on-the-fly” determinations of TST kinetic parameters in kinetic Monte Carlo studies. Such methods make it possible to explore a variety of lattice dynamical problems without resorting to ad hoc assumptions concerning mechanisms of various elementary events. In a particularly promising development, they also appear to offer a valuable tool for the study of long-time dynamical events [28, 29]. An important practical issue in the calculation of TST approximations to rates is the identification of the transition state itself. In many problems, such as the simple two-state problem discussed previously, locating the transition state is trivial. In others, it is not. Techniques designed to locate explicit transition states in complex systems have been discussed in the literature. One popular technique, developed by Cerjan and Miller [30] and extended by others [31–33], is based on an “eigenvector following” method. In this approach, one basically moves “up-hill” from a selected inherent structure using local mode information to determine the transition state. Other approaches, including methods that do not require explicit second-order derivatives of the potential, have been discussed [34]. It is also important to mention a different class of methods suggested by Pratt [35]. Borrowing a page from path integral applications, this technique attempts to locate transition states by working with paths that build in proper initial and final inherent structure character from the outset. Expanding upon the spirit of the original Pratt suggestion, recent efforts have considered sampling barrier crossing paths directly [36]. We wish to close by pointing out what we feel may prove to be a potentially useful link between inherent structure decomposition methods and the problem of “probabilistic clustering” [37, 38]. An important problem in applied mathematics is the reconstruction of an unknown probability distribution given a

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known statistical sampling of that distribution. So stated, the probabilistic clustering problem is effectively the inverse of the Monte Carlo sampling problem. Rather than producing a statistical sampling of a given distribution, we seek instead to reconstruct the unknown distribution from a known statistical sampling. This clustering problem is of broad significance in information technology and has received considerable attention. Our point in emphasizing the link between probabilistic clustering and inherent structure methods is that our increased ability to sample arbitrary, sparse distributions would appear to offer an alternative to the Stillinger/Weber quench approach to the inherent structure decomposition problem. In particular, one could use clustering methods both to “identify” and to “measure” the concentrations of inherent structures present in a system.

Acknowledgments The author would like to thank the National Science Foundation for support through awards CHE-0095053 and CHE-0131114 and the Department of Energy through award DE-FG02-03ER46704. He also wishes to thank the Center for Advanced Scientific Computing and Visualization (TCASCV) at Brown University for valuable assistance with respect to some of the numerical simulations described in the present paper.

References [1] M. Polanyi and E. Wigner, “The interference of characteristic vibrations as the cause of energy fluctuations and chemical change,” Z. Phys. Chem., 139(Abt. A), 439, 1928. [2] H. Eyring, “Activated complex in chemical reaction,” J. Chem. Phys., 3, 107, 1935. [3] H.A. Kramers, “Brownian motion in a field of force and the diffusion model of chemical reactions,” Physica (The Hague), 7, 284, 1940. [4] N.B. Slater, Theory of Unimolecular Reactions, Cornell University Press, Ithaca, 1959. [5] P.J. Robinson and K.A. Holbrook, Unimolecular Reactions, Wiley-Interscience, 1972. [6] D.G. Truhlar and B.C. Garrett, “Variational transition state theory,” Ann. Rev. Phys. Chem., 35, 159, 1984. [7] D. Chandler, Introduction to Modern Statistical Mechanics, Oxford, New York, 1987. [8] P. H¨anggi, P. Talkner, and M. Borkovec, “Reaction-rate theory: fifty years after Kramers,” Rev. Mod. Phys., 62, 251, 1990. [9] M. Garcia-Viloca, J. Gao, M. Karplus, and D.G. Truhlar, “How enzymes work: analysis by modern rate theory and computer simulations,” Science, 303, 186, 2004.

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[10] F.H. Stillinger and T.A. Weber, “Dynamics of structural transitions in liquids,” Phys. Rev. A, 28, 2408, 1983. [11] F.H. Stillinger and T.A. Weber, “Packing structures and transitions in liquids and solids,” Science, 225, 983, 1984. [12] F.H. Stillinger, “Exponential multiplicity of inherent structures,” Phys. Rev. E, 59, 48, 1999. [13] O.M. Becker and M. Karplus, “The topology of multidimensional potential energy surfaces: theory and application to peptide structure and kinetics,” J. Chem. Phys., 106, 1495, 1997. [14] D.J. Wales and J.P.K. Doye, “Global optimization by basin-hopping and the lowest energy structures of Lennard–Jones clusters containing up to 110 atoms,” J. Phys. Chem. A, 101, 5111, 1997. [15] L. Onsager, “Reciprocal relations in irreversible processes, II,” Phys. Rev., 38, 2265, 1931. [16] M.H. Kalos and P.A. Whitlock, Monte Carlo Methods, Wiley-Interscience, New York, 1986. [17] M.P. Nightingale and C.J. Umrigar, Quantum Monte Carlo Methods in Physics and Chemistry, Kluwer, Dordrecht, 1998. [18] J.P. Valleau and G.M. Torrie, “A guide to Monte Carlo for statistical mechanics: 2. byways,” In: B.J. Berne (ed.), Statistical Mechanics: Equilibrium Techniques, Plenum, New York, 1969, 1977. [19] C.H. Bennett, “Exact defect calculations in model substances,” In: A.S. Nowick and J.J. Burton (eds.), Diffusion in Solids: Recent Developments, Academic Press, New York, pp. 73, 1975. [20] A.F. Voter, “A Monte Carlo method for determining free-energy differences and transition state theory rate constants,” J. Chem. Phys., 82,1890, 1985. [21] D.D. Frantz, D.L. Freeman, and J.D. Doll, “Reducing quasi-ergodic behavior in Monte Carlo simulations by J-walking: applications to atomic clusters,” J. Chem. Phys., 93, 2769, 1990. [22] J.P. Neirotti, F. Calvo, D.L. Freeman, and J.D. Doll, “Phase changes in 38 atom Lennard-Jones clusters: I: a parallel tempering study in the canonical ensemble,” J. Chem. Phys., 112, 10340, 2000. [23] C.J. Geyer and E.A. Thompson, “Anealing Markov chain Monte Carlo with applications to ancestral inference,” J. Am. Stat. Assoc., 90, 909, 1995. [24] J.E. Adams and J.D. Doll, “Dynamical aspects of precursor state kinetics,” Surf. Sci., 111, 492, 1981. [25] J.D. Doll and A.F. Voter, “Recent developments in the theory of surface diffusion,” Ann. Revi. Phys. Chem., 38, 413, 1987. [26] A.F. Voter, J.D. Doll, and J.M. Cohen, “Using multistate dynamical corrections to compute classically exact diffusion constants at arbitrary temperature,” J. Chem. Phys., 90, 2045, 1989. [27] G. Henkelman and H. J´onsson, “Long time scale kinetic Monte Carlo simulations without lattice approximation and predefined event table,” J. Chem. Phys., 115, 9657, 2001. [28] A.F. Voter, F. Montalenti, and T.C. Germann, “Extending the time scale in atomistic simulation of materials,” Ann. Rev. Mater. Res., 32, 321, 2002. [29] V.S. Pande, I. Baker, J. Chapman, S.P. Elmer, S. Khaliq, S.M. Larson, Y.M. Rhee, M.R. Shirts, C.D. Snow, E.J. Sorin, and B. Zagrovic, “Atomistic protein folding simulations on the submillisecond time scale using worldwide distributed computing,” Biopolymers, 68, 91, 2003.

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[30] C.J. Cerjan and W.H. Miller, “On finding transition states,” J. Chem. Phys., 75, 2800, 1981. [31] C.J. Tsai and K.D. Jordan, “Use of an eigenmode method to locate the stationary points on the potential energy surfaces of selected argon and water clusters,” J. Phys. Chem., 97, 11227, 1993. [32] J. Nichols, H. Taylor, P. Schmidt, and J. Simons, “Walking on potential energy surfaces,” J. Chem. Phys., 92, 340, 1990. [33] D.J. Wales, “Rearrangements of 55-atom Lennard–Jones and (C60) 55 clusters,” J. Chem. Phys., 101, 3750, 1994. [34] G. Henkelman and H. J´onsson, “A dimer method for finding saddle points on high dimensional potential surfaces using only first derivatives,” J. Chem. Phys., 111, 7010, 1999. [35] L.R. Pratt, “A statistical method for identifying transition states in high dimensional problems,” J. Chem. Phys., 85, 5045–5048, 1986. [36] P.G. Bolhuis, D. Chandler, C. Dellago, and P.L. Geissler, “Transition path sampling: throwing ropes over rough mountain passes, in the dark,” Ann. Rev. Phys. Chem., 53, 291, 2002. [37] B.G. Mirkin, Mathematical Classification and Clustering, Kluwer, Dordrecht, 1996. [38] D. Sabo, D.L Freeman, and J.D. Doll, “Stationary tempering and the complex quadrature problem,” J. Chem. Phys., 116, 3509, 2002.

5.3 TRANSITION PATH SAMPLING Christoph Dellago Institute of Experimental Physics, University of Vienna, Vienna, Austria

Often, the dynamics of complex condensed materials is characterized by the presence of a wide range of different time scales, complicating the study of such processes with computer simulations. Consider, for instance, dynamical processes occurring in liquid water. Here, the fastest molecular processes are intramolecular vibrations with periods in the 10–20 fs range. The translational and rotational motions of water molecules occur on a significantly longer time scale. Typically, the direction of translational motion of a molecule persist for about 500 fs, corresponding to 50 vibrational periods. Hydrogen bonds, responsible for many of the unique properties of liquid water, have an average lifetime of about 1 ps and the rotational motion of water molecules stays correlated for about 10 ps. Much longer time scales are typically involved if covalent bonds are broken and formed. For instance, the average lifetime of a water molecule in liquid water before it dissociates and forms hydroxide and hydronium ions is on the order of 10 h. This enormous range of time scales, spanning nearly 20 orders of magnitude, is a challenge for the computer simulator who wants to study such processes. In general, the dynamics of molecular systems can be explored on a computer with molecular dynamics simulation (MD), a method in which the underlying equations of motion are solved in small time steps. In such simulations the size of the time step must be shorter than the shortest characteristic time scale in the system. Thus, many molecular dynamics steps must be carried out to explore the dynamics of a molecular system for times that are long compared with the basic time scale of molecular vibrations. Depending on specific system properties and the available computer equipment, one can carry out from 10 000 to millions of such steps. In ab initio simulations where interatomic forces are determined by solving the electronic structure problem on the fly, total simulation times typically do not exceed dozens of picoseconds. Longer simulations of nanosecond, or, in some rare cases, microsecond length can be achieved if forces are determined from computationally less expensive empirical force fields often 1585 S. Yip (ed.), Handbook of Materials Modeling, 1585–1596. c 2005 Springer. Printed in the Netherlands.

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used to simulate biological systems. But many interesting and important processes still lay beyond the time scale accessible with MD simulations even on today’s fastest computers. Indeed, an ab initio molecular dynamics simulation of liquid water long enough to observe a few dissociations of water molecules would require a multiple of the age of the universe of computing time even on state-of-the-art parallel high performance computers. The computational effort needed to study many other interesting processes, ranging from protein folding to the nucleation of phase transitions and transport in and on solids, in straightforward molecular dynamics simulations with atomistic resolution may be less extreme, but still surpasses the capabilities of current computer technology. Fortunately, many processes occurring on long time scale are rare rather than slow. Consider, for instance, a chemical reaction during which the system has to overcome a large energy barrier on its way from reactants to products. Before the reaction occurs, the system typically spends a long time in the reactant state and only a rare fluctuation can drive the system over the barrier. If this fluctuation happens, however, the barrier is crossed rapidly. For example, it is now known from transition path sampling simulations that the dissociation of a water molecule in liquid water takes place in a few hundred femtoseconds once a rare solvent fluctuation drives the transition between the stable states, the intact water molecule and the separated ion pair. As mentioned earlier, the waiting time for this event, however, is of the order of 10 h. Other examples of rare but fast transitions between stable states include the nucleation of first order phase transitions, conformational transitions of biopolymers, and transport in and on solids. In such cases it is computationally advantageous to focus on those segments of the time evolution during which the rare event takes place rather than wasting large amounts of computing time following the dynamics of the system waiting for the rare event to happen. Several computational techniques to accomplish that have been put forward [1–4]. One approach consists in locating (or postulating) the bottleneck separating the stable states between which the rare transition occurs. Molecular dynamics trajectories initiated at this bottleneck, or transition state, can then be used to study the reaction mechanism in detail and to calculate reaction rate constants [5]. In small or highly ordered systems transition states can often be associated with saddle points on the potential energy surface. Such saddle points can be located with appropriate algorithms. Particularly in complex, disordered systems such as liquids, however, such an approach is frequently unfeasible. The number of saddle points on the potential energy surface may be very large and most saddle points may be irrelevant for the transition one wants to study. Entropic effects can further complicate the problem. In this case, a technique called transition path sampling provides an alternative approach [6]. Transition path sampling is a computational methodology based on a statistical mechanics of trajectories. It is designed to study rare transitions between

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known and well defined stable states. In contrast to other methods, transition path sampling does not require any a priori knowledge of the mechanism. Instead, it is sufficient to unambiguously define the stable states between which the transition occurs. The basic idea of transition path sampling consists in assigning a probability, or weight, to every pathway. This probability is a statistical description of all possible reactive trajectories, the transition path ensemble. Then, trajectories, are generated according to their probability in the transition path ensemble. Analysis of the harvested pathways yields detailed mechanistic information on the transition mechanism. Reaction rate constants can be determined within the framework of transition path sampling by calculating “free energies” between different ensembles of trajectories. In the following, we will give a brief overview of the basic concepts and algorithms of the transition path sampling technique. For a detailed description of the methodology and for practical issues regarding the implementation of transition path sampling simulations the reader is referred to two recent review articles [7, 8].

1.

The Transition Path Ensemble

Imagine a system with two long-lived stable states, call them A and B, between which rare transitions occur (see Fig. 1). The system spends much of its time fluctuating in the stable states A and B but rarely transitions between A and B occur. In the transition path sampling method one focuses on short

B

A

Figure 1. Several transition pathways connecting stable states A and B which are separated by a rough free energy barrier.

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trajectories x(T ) of length T (in time) represented by a time-ordered discrete sequence of states: x(T ) ≡ {x0 , xt , x2t , . . . , xT }.

(1)

Here, xt is the state of the system at time t. Each trajectory may be thought of as a chain of states obtained by taking snapshots at regular time intervals of length t as the system evolves according to the rules of the underlying dynamics. If the time evolution of the system follows Newton’s equations of motion, x ≡ {r, p} is a point in phase space and consists of the coordinates, r, and momenta, p, of all particles. For systems evolving according to a high friction Langevin equation or a Monte Carlo procedure the state x may include only coordinates and no momenta. The probability of a certain trajectory to be observed depends on the probability ρ(x0 ) of its initial point x0 and on the probability to observe the subsequent sequence of states starting from that initial point. For a Markovian process, that is for a process in which the probability of state xt to evolve into state xt +t over a time t depends only on xt and not on the history of the system prior to t, the probability P[x(T )] of a trajectory x(T ) can simply be written as a product of single step transition probabilities p(xt → xt +t ): P[x(T )] = ρ(x0 )

T /t −1

p(xit → x(i+1)t ).

(2)

i=0

For an equilibrium system in contact with a heat bath at temperature T the distribution of starting points is canonical, i.e., ρ(x0 ) ∝ exp{−H (x)/kB T }, where H (x) is the Hamiltonian of the system and kB is Boltzmann’s constant. Depending on the process under study other distributions of initial conditions may be appropriate. The path distribution of Eq. (2) describes the probability to observe a particular trajectory regardless of whether it connects the two stable states A and B. Since in the transition path approach the focus is on reactive trajectories, the path distribution P[x(T )] is restricted to the subset of pathway starting in A and ending in B: PAB [x(T )] ≡ h A (x0 )P[x(T )]h B (xT )/Z AB (T ).

(3)

The functions h A (x) and h B (x) are unity if their argument x lies in region A or B, respectively, and they vanish otherwise. Accordingly, only reactive trajectories starting in A and ending in B can have a weight different from zero in the path distribution PAB [x(T )]. The factor Z AB (T ) ≡

Dx(T ) h A (x0 )P[x(T )]h B (xT ),

(4)

which has the form of a partition function, normalizes the path distribution of Eq. (3). The notation Dx(T ), familiar from path integral theory, denotes a

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summation over all pathways. The function PAB [x(T )], which is a probability distribution function in the high dimensional space of all trajectories, describes the set of all reactive trajectories with their correct weight. This set of pathways is the transition path ensemble. In transition path sampling simulations care must be exercised in defining the stable states A and B. Both A and B need to be large enough to accommodate most equilibrium fluctuations, i.e., the system should spend the overwhelming fraction of time in either A or B. At the same time, A and B should not overlap with each other’s basin of attraction. Here, the basin of attraction of region A consist of all configurations that relax predominantly into that region. The basin of attraction of region B is defined analogously. If state A is incorrectly defined in such a way that it contains also points belonging to the basin of attraction of B, the transition path ensemble includes pathways only apparently connecting the two stable states. This situation is illustrated in Fig. 2. In many cases the stable states A and B can be defined through specific limits of a one-dimensional order parameter q(x). Although there is no general rule guiding the construction of such order parameters, this step in setting up a

q'

TS

A

B qA

qB

q

Figure 2. Regions A and B must be defined in a way to avoid overlap of A and B with each other’s basin of attraction. On this two dimensional free energy surface region A defined through q < q A includes points belonging to the basin of attraction of B (defined through q > q B ). Thus, the transition path ensemble PAB [x(T )] contains paths which start in A and end in B, but which never cross the transition state region marked by TS (dashed line). This problem can be avoided by using also the variable q in the definition of the stable states.

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transition path sampling simulation can be usually completed quite easily with a trial and error procedure. Note, however, that an appropriate order parameter is not necessarily a good reaction coordinate capable of describing the whole transition. In general, finding such a reaction coordinate is a difficult problem.

2.

Sampling the Transition Path Ensemble

In the transition path sampling method a biased random walk through path space is performed in such a way that pathways are visited according to their weight in the transition path ensemble PAB [x(T )]. This can be accomplished in an efficient way with Monte Carlo methods proceeding in analogy to a conventional Monte Carlo simulation of, say, a simple liquid at a given temperature T [9]. In that case a random walk through configuration space is constructed by carrying out a sequence of discrete steps. In each step, a new configuration is generated from an old one, for instance by displacing a single particle in a random direction by a random amount. Then, the new configuration, also called trial configuration, is accepted or rejected depending on how the probability of the new configuration compares to that of the old one. This is most easily done by applying the celebrated Metropolis rule [10], designed to enforce detailed balance between the move and its reverse. As a result, the trial move is always accepted if the energy of the new configuration is lower than that of the old one and accepted with a probability exp(−E/kB T ) if the trial move is energetically uphill (here, E is the energy difference between the new and the old configuration). Execution of a long sequence of such random moves followed by the acceptance or rejection step yields a random walk of the system through configuration space during which configurations are sampled with a frequency proportional to their weight in the canonical ensemble. Ensemble averages of structural and thermodynamics quantities can then straightforwardly computed by averaging over this sequence of configurations. In a transition path sampling simulation one proceeds analogously. But in contrast to a conventional Monte Carlo simulation, the random walk is carried out in the space of all trajectories and the result is a sequence of pathways instead of a sequence of configurations. In each step of this random walk a new pathway x (n) (T ), the trial path, is generated from an an old one, x (o) (T ). Then, the trial pathway is accepted or rejected according to how its weight PAB [x (n) (T )] in the transition path ensemble compares to the weight of the old one, PAB [x (o) (T )]. Correct sampling of the transition path ensemble is guaranteed by enforcing the detailed balance condition which requires the probability of a path move from x (o) (T ) to x (n) (T ) to be balanced exactly by the probability of the reverse path move. This detailed balance condition can be satisfied using the Metropolis criterion. Iterating this procedure of path generation followed by acceptance or rejection, one obtains a sequence of pathways in which

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each pathway is visited according to its weight in the transition path ensemble. It is important to note that while pathways are sampled with a Monte Carlo procedure, each single pathway is a genuinely dynamical pathway generated according to the rules of the underlying dynamics. To implement the procedure outlined above, one needs to specify how to generate a new pathway from an old one. This can be done efficiently with algorithms called shooting and shifting. For simplicity we will explain these algorithms for Newtonian dynamics (as used in most MD simulations) although they can be easily applied to other types of dynamics as well. So, imagine a Newtonian trajectory of length T as obtained from a molecular dynamics simulation of L = T /t steps starting in region A and ending in region B (see Fig. 3). From this existing transition pathway a new trajectory is generated by first randomly selecting a state of the existing trajectory. Then, the momenta belonging to the selected state are changed by a small random amount. Starting from this state with modified momenta the equations of motion are integrated forward to time T and backward to time 0. As a result, one obtains a complete new trajectory of length T which crosses (in configuration space) the old trajectory at one point. By keeping the momentum displacement small the new trajectory can be made to resemble the old one closely. As a consequence, the new pathway is likely to be reactive as well and to have a nonzero weight in the transition path ensemble. Any new trajectory with starting point in A and ending point in B can be accepted with high likelihood (in fact, for constant energy trajectories with a microcanonical distribution of initial conditions all new trajectories connecting A and B can be accepted). If the new trajectory does not begin in A or does not end in B it is rejected. For optimum efficiency, the magnitude of the momentum displacement should be selected such that the average acceptance probability is in the range from 40 to 60%. Shooting moves can be complemented with shifting moves, which consist in shifting the starting point of the path in time. This kind of move is computationally inexpensive since typically only a small part of the pathway needs to

A

B

Figure 3. In a shooting move one generates a new trajectory (dashed line) from an old one (solid line) by integrating the equations of motion forward and backward starting from a point with random momenta randomly selected along the old trajectory. The acceptance probability of the newly generated path can be controlled by varying the magnitude of the momentum displacement (thin arrow).

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be regrown. If the starting point of the path is shifted forward in time, a path segment of appropriate length has to be appended at the end of the path by integration of the equation of motion. If, on the other hand, the starting point is shifted backward in time, the trajectory must be completed by integrating the equations of motion backward in time starting from the initial point of the original pathway. Depending on the time by which the path is shifted, the new path can have large parts in common with the old path. Since ergodic sampling is not possible with shifting moves alone, path shifting always needs to be combined with path shooting. Although shifting moves cannot generate a truly new path, they can increase sampling efficiency especially for the calculation of reaction rate constants. To start the Monte Carlo path sampling procedure one needs a pathway that already connects A with B. This initial pathway is not required to be a high-weight dynamical trajectory, but can be an artificially constructed chain of states. Shooting and shifting will then rapidly relax this initial pathway towards regions of higher probability in path space. The generation of an initial trajectory is strongly system dependent and usually does not pose a serious problem.

3.

Analyzing Transition Pathways

Pathways harvested with the transition path sampling method are full dynamical trajectories in the space spanned by positions and momenta of all particles. In such high-dimensional many-particle systems it is usually difficult to identify the relevant degrees of freedom and to distinguish them from those which might be regarded as random noise. In the case of a chemical reaction occurring in a solvent, for instance, the specific role of solvent molecules during the reaction is often unclear. Although direct inspection of transition pathways with molecular visualization tools may yield some insight, detailed knowledge of the transition mechanism can only be gained through systematic analysis of the collected pathways. In the following, we will briefly review two approaches to carry out such an analysis: the transition state ensemble and the distribution of committors. In simple systems of a few degrees of freedom, for instance a small molecule undergoing an isomerization in the gas phase, one can study transition mechanisms by locating minima and saddle points on the potential energy surface of the system. While the potential energy minima are the stable states in which the system spends most of its time, the saddle point are configurations the system must cross on its way from one potential energy well to another. These so called transition states are the lowest points on ridges separating the stable states from each other. From the transition states the system can relax

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into either one of the two stable states depending on the initial direction of motion. In a high dimensional complex system local potential energy minima and saddle points do not carry the same significance as in simple systems. In a large, disordered system many local potential energy minima and saddle points may belong to one single stable state, and free energy barriers may not be related to a single saddle point. Nevertheless, the concept of a transition state is still meaningful if defined in a statistical way. In this definition, configurations are considered to be transition states if trajectories started from them with random initial momenta have equal probability to relax to either one of the stable states between which transitions occur. Naturally, along each transition pathway there is at least one (but sometimes several) configuration with this property. Performing such an analysis for many transition pathways yields the transition state ensemble, the set of all configurations on transition pathways which relax into A and B with equal probability. Inspection of this set of configurations is simpler than scrutiny of the set of harvested complete pathways. As a result of the analysis described above one may be led to guess which degrees of freedom are most important during the transition, or, in other words, which degrees of freedom contribute to the reaction coordinate. Such a guessed reaction coordinate, q(x), can be tested with the following procedure. The first step consists in calculating the free energy F(q), for instance by using umbrella sampling [9] or constrained molecular dynamics [11]. The free energy profile F(q) will possess minima at values of q typical for the stables states A and B and a barrier located at q = q ∗ separating these two minima. If q is a good reaction coordinate, trajectories started from configurations with q = q ∗ relax into A and B with equal probability. To verify the quality of the postulated reaction coordinate, a set of configurations with q = q ∗ is generated. Then, for each of these configurations one calculates p B , the probability to relax into state B, also called the committor. This can be done by initiating many short trajectories at the configuration and observing which state they relax to. As a result, one obtains a distribution P( p B ) of committors. For a good reaction coordinate, this distribution should peak at a value of p B ≈ 1/2. If this is not case, other degrees of freedom need to be taken into account for a correct description of the transition [7].

4.

Reaction Rate Constants

Since trajectories collected in the transition path sampling method are genuine dynamical trajectories, they can be used to study the kinetics of reactions. The phenomenologic description of the kinetics in terms of reaction rate constants is related to the underlying microscopic dynamics by time correlation functions of appropriate population functions that describe how the system

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relaxes after a perturbation [12]. In particular, for transitions from A to B the relevant correlation function is h A (x0 )h B (xt ) , (5) C(t) = h A where the angular brackets · · · denote equilibrium averages. The correlation function C(t) is the conditional probability to observe the system in region B at time t provided it was in region A at time 0. To understand the general features of this function, let us imagine that we prepare a large number of identical and independent systems in a way that at time t = 0 all of them are located in A. Then, we let all systems evolve freely and observe the fraction of systems in region B as a function of time. This fraction is the correlation function C(t). Initially, all systems are in A and, therefore, C(0) = 0. As time goes on, some systems cross the barrier due to random fluctuations and contribute to the population in region B. So C(t) grows and it keeps growing until equilibrium sets in, i.e., until the flow of systems from A to B is compensated by the flow of system moving from B back to A. For very long times, correlations are lost and the probability to find a system in B is just given by the equilibrium population h B . For first order kinetics C(t) approaches its asymptotic value exponentially, C(t) = h B [1 − exp(−t/τrxn )], where the reaction time τrxn can be written in terms of the forward and backward reaction rate constants, τrxn = (k AB + k B A )−1 . For times short compared to the reaction time τrxn (but longer than the time necessary to cross the barrier) the correlation function C(t) grows linearly, C(t) ≈ k AB t, and the slope of this curve is the forward rate constant k AB . Thus, to determine reaction rate constants one has to calculate the correlation function C(t). To determine the correlation form C(t) in the transition path sampling method we rewrite it in the suggestive form

C(t) =

Dx(t) h A (x0 )P[x(t)]h B (xt ) . Dx(t) h A (x0 )P[x(t)]

(6)

Here, both numerator and denominator are integrals over path distributions and can be viewed as partition functions belonging to two different path ensembles. The integral in the denominator has the form of a partition function of the ensemble of pathways starting in region A and ending somewhere. The integral in the numerator, on the other hand, is more restrictive and places a condition also on the final point of the path. This integral can be viewed as the partition function of the ensemble of pathways starting in region A and ending in region B. Thus, the path ensemble in the numerator is a subset of the path ensemble in the denominator. The ratio of partition functions can be related to the free energy difference F between the two ensembles of pathways, C(t) ≡ exp(−F).

(7)

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This free energy difference is the generalized reversible work necessary to confine the endpoints of pathways starting in A to region B. Exploiting this viewpoint, one can calculate the time correlation function C(t) and hence determine reaction rate constants by adapting conventional free energy estimation methods to work in trajectory space. So far, reaction rate constants have been calculated in the framework of transition path sampling with umbrella sampling, thermodynamic integration, and fast switching methods. In principle, the forward reaction rate constant k AB can be determined by carrying out a free energy calculation for different times t and taking a numerical derivative. In the time range where C(t) grows linearly, this derivative has a plateau which coincides with k AB . Proceeding in such a way one has to perform several computationally expensive free energy calculations. Fortunately, C(t) can be factorized in a way so that only one such calculation needs to be carried out for a particular time t . The value of C(t) at all other times in the range [0, T ] can then be determined from a single transition path sampling simulation of trajectories with length T . Thus, calculating reaction rate constants in the transition path sampling method is a two-step procedure. First, C(t ) is determined for a particular time t using a free energy estimation method in path space. In a second step, one additional transition path sampling simulation is carried out to determine C(t) at all other times. The reaction rate constant can finally be calculated by determining the time derivative of C(t).

5.

Outlook

Transition path sampling is a practical and very general methodology to collect and analyze rare pathways. In equilibrium, such rare but important trajectories may arise due to free energetic barriers impeding the motion of the system through configuration space. Transition path sampling, however, can be used equally well to study rare trajectories occurring in non-equilibrium processes such as solvent relaxation following excitation or rare pathways arising in new methodologies for the computation of free energy differences. Different types of dynamics ranging from Monte Carlo and Brownian dynamics to Newtonian and non-equilibrium dynamics can be treated on the same footing. To date, the transition path sampling method has been applied to many processes in physics, chemistry and materials science. Examples include chemical reactions in solution, conformational changes of biomolecules, isomerizations of small cluster, the dynamics of hydrogen bonds, ionic dissociation, transport in solids, proton transfer in aqueous systems, the dynamics of non-equilibrium systems, base pair binding in DNA, hydrophobic collapse, and cavitation between solvophobic surfaces. Furthermore, the transition path sampling has been combined with other approaches such as parallel tempering, master equation methods, and the Jarzynski method for the computation of free energy

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differences. Due to the generality of the transition path sampling method it is likely that in the future this approach will be used fruitfully to study new problems in a variety of complex systems.

References [1] D. Wales, Energy Landscapes, Applications to Clusters, Biomolecules and Glasses, Cambridge University Press, Cambridge, 2003. [2] R. Elber, A. Ghosh, and A. C´ardenas, Long time dynamics of complex systems, Acc. Chem. Res., 35, 396, 2002. [3] H. J´onsson, G. Mills, and K.W. Jacobsen, “Nudged elastic band method for finding minimum energy paths of transitions,” In: B.J. Berne, G. Ciccotti, and D.F. Coker, (eds.), Computer Simulation of Rare Events and Dynamics of Classical and Quantum Condensed-Phase Systems – Classical and Quantum Dynamics in Condensed Phase Simulations, World Scientific, Singapore, p. 385, 1998. [4] W.E.W. Ren and E. Vanden-Eijnden, String method for the study of rare events, Phys. Rev. B, 66, 052301, 2002. [5] D. Chandler, “Barrier crossings: classical theory of rare but important events,” In: B.J. Berne, G. Ciccotti, and D.F. Coker (eds.), Computer Simulation of Rare Events and Dynamics of Classical and Quantum Condensed-Phase Systems – Classical and Quantum Dynamics in Condensed Phase Simulations, World Scientific, Singapore, p. 3, 1998. [6] C. Dellago, P.G. Bolhuis, F.S. Csajka, and D. Chandler, “Transition path sampling and the calculation of rate constants,” J. Chem. Phys., 108, 1964, 1998. [7] C. Dellago, P.G. Bolhuis, and P.L. Geissler, “Transition path sampling,” Adv. Chem. Phys., 123, 1, 2002. [8] P.G. Bolhuis, D. Chandler, C. Dellago, and P.L. Geissler, “Transition path sampling: throwing ropes over mountain passes in the dark,” Ann. Rev. Phys. Chem., 53, 291, 2002. [9] D. Frenkel and B. Smit, Understanding Molecular Simulation: From Algorithms to Applications, 2nd edn. Academic, San Diego, 2002. [10] N. Metropolis, A.W. Metropolis, M.N. Rosenbluth, A.H. Teller, and E. Teller, “Equation of state calculations for fast computing machines,” J. Chem. Phys., 21, 1087, 1953. [11] G. Ciccotti, “Molecular dynamics simulations of nonequilibrium phenomena and rare dynamical events,” In: M. Meyer and V. Pontikis (eds.), Proceedings of the NATO ASI on Computer Simulation in Materials Science, Kluwer, Dordrecht, p. 119, 1991. [12] D. Chandler, Introduction to Modern Statistical Mechanics, Oxford University Press, Oxford, 1987.

5.4 SIMULATING REACTIONS THAT OCCUR ONCE IN A BLUE MOON Giovanni Ciccotti1 , Raymond Kapral2 , and Alessandro Sergi2 1

INFM and Dipartimento di Fisica, Universit`a “La Sapienza”, Piazzale Aldo Moro, 2, 00185 Roma, Italy 2 Chemical Physics Theory Group, Department of Chemistry, University of Toronto, Toronto, ON, Canada, M5S 3H6

The computation of the rates of condensed phase chemical reactions poses many challenges for theory. Not only do condensed phase systems possess a large number of degrees of freedom so that computations are lengthy, but typically chemical reactions are activated processes so that transitions between metastable states are rare events that occur on time scales long compared to those of most molecular motions. This time scale separation makes it almost impossible to determine reaction rates by straightforward simulation of the equations of motion. Furthermore, condensed phase reactions often involve collective degrees of freedom where the solvent participates in an important way in the reactive process. Consequently, the choice of a reaction coordinate to describe the reaction is often far from a trivial task. Various methods for determining reaction paths have been devised (see Refs. [1, 2] and references therein). These methods have the goal of determining how the system passes from one metastable state to another and thus finding the reaction path or reaction coordinate. In many situations one has some knowledge of how to construct a reaction coordinate (or set of reaction coordinates) for a particular physical problem. One example is the use of the many-body solvent polarization reaction coordinate to describe electron or proton transfer in solution. In almost all situations investigated to date the dynamics of condensed phase activated reaction rates can be described in terms of a small number of reaction coordinates (often involving collective degrees of freedom). In this chapter, we describe how to simulate the rates of activated chemical reactions that occur on slow time scales, assuming that some set of suitable reaction coordinates is known. In order to compute the rates of rare reactive 1597 S. Yip (ed.), Handbook of Materials Modeling, 1597–1611. c 2005 Springer. Printed in the Netherlands.

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events we need to be able to sample regions of configuration space that are rarely visited since the interconversion between reactants and products entails passage through such regions of low probability. We show that by applying holonomic constraints to the reaction coordinate in a molecular dynamics simulation we can force the system to visit unfavorable configuration space regions. Through such constraints we can generate an ensemble of configurations (the Blue Moon ensemble) that allows one to efficiently estimate the rate constant for activated chemical processes [3].

1.

Reactive Flux Correlation Function Formalism

We begin with a sketch of the reactive flux correlation function formalism in order to specify the quantities that must be computed to obtain the reaction rate constant. In order to simplify the notation, we consider a molecular system containing N atoms with Hamiltonian H = K (p) + V (r), where K (p) is the kinetic energy, V (r) is the potential energy and (p, r) denotes the 6N momenta and coordinates defining the phase space of the system. A chemical reaction A B is assumed to take place in the system. The reaction dynamics is described phenomenologically by the mass action rate law dn A (t) = −kf n A (t) + kr n B (t), dt

(1)

where n A (t) is the mean number density of species A. The task is to compute −1 the forward kf and reverse kr = kf K eq (K eq is the equilibrium constant) rate constants by molecular dynamics simulation. (The formalism is easily generalized to other reaction schemes.) To this end, we assume that the progress of the reaction can be characterized on a microscopic level by a scalar reaction coordinate ξ(r) which is a function of the positions of the particles in the system. A dividing surface at ξ ‡ serves to partition the configuration space of the system into two A and B domains that contain the metastable A and B species. The microscopic variable corresponding to the fraction of systems in the A domain is n A (r) = θ(ξ ‡ − ξ(r)), where θ is the Heaviside function. Similarly, the fraction of systems in the B domain is n B (r) = θ(ξ(r) − ξ ‡ ). The time rate of change of n A (r) is n˙ A (r) = −ξ˙ (r)δ(ξ(r) − ξ ‡ ).

(2)

The rate at which the A and B species interconvert can be determined from the well-known reactive flux formula for the rate constant [4–6]. Using this formalism the time-dependent forward rate coefficient can be expressed in terms of the equilibrium correlation function of the initial flux of A with the

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A species density at time t as 1 1 ˙ A (r)n A (r, t) = eq ξ˙ δ(ξ(r) − ξ ‡ ) θ(ξ(r(t)) − ξ ‡ ) . (3) eq n nA nA

kf (t) =

Here, the angular brackets denote an equilibrium canonical average, · · · = eq Q −1 dr dr exp{−β H } · · · , where Q is the partition function and n A is the equilibrium density of species A. The forward rate constant can be determined from the plateau value of this time-dependent forward rate coefficient [6]. We can separate the static and dynamic contributions to the rate coefficient by multiplying and dividing each term on the right-hand side of Eq. (3) by δ(ξα (r) − ξ ‡ ) to obtain kf (t) =

 

 ξ˙ δ(ξ(r) − ξ ‡ )θ(ξ ‡ − ξ(r(t)))  δ(ξ(r) − ξ ‡ ) 



δ(ξ(r) − ξ ‡ )

eq

nA

.

(4)

The equilibrium average δ(ξ(r) − ξ ‡ ) = P(ξ ‡ ) is the probability density of finding the value of the reaction coordinate ξ(r) = ξ ‡ . We may introduce the free energy W(ξ ) associated with the reaction coordinate by the definition W(ξ ) = − β −1 ln(P(ξ )/Pu ), where Pu is a uniform probability density of ξ . For an activated process the free energy will have the form shown schematically in Fig. 1. A high free energy barrier at ξ = ξ ‡ separates the metastable reactant and product states. The equilibrium density of species A is

n A = θ(ξ ‡ − ξ(r)) = eq

=

dξ θ(ξ ‡ − ξ ) δ(ξ(r) − ξ )

dξ P(ξ ).

(5)

ξ <ξ ‡

B

W(ξ)

A

ξ‡

ξ

Figure 1. Sketch of the free energy versus ξ showing the free energy maximum at ξ = ξ ‡ and specification of the A and B domains.

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Using these results the expression for the time-dependent rate coefficient may be written as  cd 

‡



 e−β W (ξ ) , kf (t) = ξ˙ θ(ξ ‡ − ξ(r(t)) ‡ ξ  ‡ dξ e−β W (ξ )  ξ <ξ

(6)

where · · · cd‡ defines an average conditional on ξ(r) = ξ ‡ ξ

· · · cd ξ =

· · · δ(ξ(r) − ξ ) . δ(ξ(r) − ξ )

(7)

Often it is useful to represent the results in terms of the time-dependent transmission coefficient κ(t) which is defined as κ(t) = kf (t)/kfTST , where the transition-state theory value of the rate constant is given by the limit t → 0+ of Eq. (3) as [5] kfTST =

1 ‡ ˙ ˙ ξ δ(ξ(r) − ξ )θ( ξ ) . eq nA

(8)

The transmission coefficient κ(t) measures the deviations from kfTST due to dynamical recrossing events. From Eq. (6) we see that the computation of the rate coefficient requires the calculation of conditional averages depending on specified, rarely visited, values of the reaction coordinate. The ensemble of such configurations which are visited “once in a blue moon” is termed the blue moon ensemble [3]. In Section 2, we describe how the conditional averages which play an essential role in the computation of the rate constant may be determined from constrained molecular dynamics trajectories which allow one to efficiently sample rarely visited regions of configuration space.

2.

The Blue Moon Ensemble

The presence of the delta function δ(ξ(r) − ξ ) in the conditional equilibrium averages fixes the reaction coordinate to have the specified value ξ . We shall now show that such conditional averages of observables depending only on configuration space variables can be computed by applying holonomic constraints to the equations of motion. For simplicity, we assume that no other bond constraints are present but this more general case is easily treated [3]. To this end, we consider a system described by the Cartesian coordinates (r, p) subject to a holonomic constraint σ (r) = ξ(r) − ξ = 0 on the reaction coordinate. When there are constraints one usually introduces a set of generalized coordinates q and conjugate momenta pq such that r = r(q). In general, it is not possible to invert this relation since there are more r coordinates than q

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coordinates. However, by adding the expression for the constraint there is an extra generalized coordinate and the one to one correspondence r = r(q, σ ) is recovered. The statistical mechanics of the system may be formulated in terms of the generalized coordinates q; however, it is useful to have an equivalent formulation in terms of the original Cartesian coordinates. The dynamics of the system is described in Cartesian coordinates by the Lagrangian L(r, r˙ ) = K (˙r) − V (r) =

N 1 i=1

2

m i r˙ 2i − V (r),

(9)

to which we add the constraint σ = 0. The set of 3N − 1 generalized coordinates q plus σ can be taken as a new set of equivalent coordinates denoted collectively by u. We have r = r (q, σ ) = r (u), or u = u (r). In the new variables the Lagrangian is given by ˙ = 12 u˙ T Mu˙ − V (u), L (u, u)

(10)

where uT is the transpose of vector u and M = JT mJ is the metric matrix with elements given by Mµν =

N

mi

i=1

∂ri ∂u µ

·

∂ri ∂u ν

.

(11)

Here, J is the Jacobian matrix of the transformation r ↔ u and m is a diagonal matrix of the masses. The Lagrangian of the constrained motion is easily obtained by putting σ = σ˙ = 0 ˙ σ˙ = 0). ˙ ≡ L (q, σ = 0, q, Lc (q, q)

(12)

To derive the statistical mechanical ensemble we need the Hamiltonian description of the dynamical system. The Hamiltonian corresponding to the description with coordinates u is given by H (u, pu ) = 12 pu T M−1 pu + V (u),

(13)

where pu =

∂L ∂u˙

˙ = Mu.

(14)

The inverse of the metric matrix M−1 can be written explicitly as

M−1

µν

=

N 1 ∂u µ i=1

m i ∂ri

·

∂u ν ∂ri

.

(15)

To obtain the constrained motion we have to compute the Hamiltonian at σ = 0 and pσ satisfying the constraints σ = σ˙ = 0. Since

σ˙ = M−1 pu

3N

= Epq + Z pσ ,

(16)

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where the subscript 3N denotes that element of the vector M−1 pu and E and Z are submatrices in the block form of M−1 M−1 =

ET

E , Z

(17)

˜ q , where the tilde the above condition corresponds to taking pσ = − Z˜ −1 Ep means that the matrices have to be evaluated at σ = 0. The explicit form of Z is needed below and is given by Z=

N 1 ∂σ i=1

m i ∂ri

·

∂σ ∂ri

.

(18)

The constrained Hamiltonian may now be written as

˜ q . Hc (q, pq ) ≡ H q, σ = 0, pq , pσ = − Z˜ −1 Ep

(19)

Note that Eq. (16) implies pσ + Z −1 Epq = Z −1 σ˙ . Letting ρc (q, pq ) be the probability density for the constrained dynamical system, we have

ρc q, pq dq dpq = ρ u, pu δ(σ )δ( pσ + Z −1 Epq )du dpu = ρ r, pr δ(σ )δ(Z −1 σ˙ )dr dpr ≡ ρξ r, pr dr dpr . (20) In the penultimate equality, we used the fact that the point contact transformation (u, pu ) ↔ (r, pr ) is a canonical, phase space volume conserving, transformation [7]. We may rewrite this probability density as

p

ρξ r, pr = ρξr (r) ρξ pr |r ,

(21)

where the configurational probability density, ρξr (r), is obtained by performing the momentum integration of the full probability density ρξ (r, pr ) and the conditional probability density of the momenta given the configuration, p ρξ (pr |r), is defined in Eq. (21). In the canonical ensemble, the configurational probability density is given by 1/2 −β V (r) 1/2 −β V (r) e δ(σ )dr = Q −1 e δ(ξ(r) − ξ )dr, ρξr (r) dr = Q −1 c |Z | c |Z | (22)

where Q c is the partition function of the constrained system. The factor |Z |1/2 arises from performing the momentum integration of Eq. (20). The conditional probability density of the momenta given the configuration is

ρξ pr |r dpr = |Z |−1/2 e−β K δ(Z −1 σ˙ )dpr . p

(23)

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The physical interpretation of Z which enters this expression for the probability density has been discussed by several authors [8, 9] and has its origin in the restriction imposed in momentum space by the constraint σ = 0 which, holding at all times, implies that the generalized velocity σ˙ must vanish. The configurational probability density in Eq. (22) should be compared with the joint configurational probability density to be at r and at ξ = ξ ρ r (r) δ(ξ(r) − ξ )dr = Q −1 e−β V (r) δ(ξ(r) − ξ )dr.

(24)

We may express the conditional average of any configurational property of our system in terms of the ξ -constrained ensemble introduced above. While the value ξ we wish to sample is rare in the original ensemble, only configurations with ξ = ξ are sampled in the ξ -constrained ensemble. This feature is illustrated schematically in Fig. 2. By comparison of Eqs. (22) and (24) we may write O(r)δ(ξ(r) − ξ ) |Z |−1/2 O(r)ξ = , δ(ξ(r) − ξ ) |Z |−1/2 ξ

(25)

where the observable O(r) is any function of the configuration space, · · · denotes a canonical ensemble average and · · · ξ denotes an average over the constrained ensemble with ξ = ξ . Equation (25) allows one to estimate the conditional average on the left-hand side in terms of averages in the constrained ensemble with evident statistical advantages. The constrained ensemble we have constructed can be generalized to give the biased (whose bias may be removed) configurational sample determined above and the correct distribution of momenta. This Blue Moon ensemble can be easily obtained by multiplying the ξ -constrained configurational probability

ξ(r)⫽ξ'

Figure 2. Schematic representation of the sampling procedure in the Blue Moon ensemble. The bold line depicts the constrained (ξ(r) = ξ ) dynamical evolution in phase space. The unconstrained natural evolution of the system is shown as a dashed line. The open circles represent common points in configuration space which are the initial conditions of the activated trajectory sampling. These points are not real crossings in phase space since the two trajectories differ in the momentum space. Interruptions of the dashed line denote lengthy segments in the natural trajectory and indicate that “crossings” are rare events. The dynamics represented by the solid line segments of the unconstrained trajectory in the vicinity of the crossing points provide the dynamical information needed to compute averages.

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density times the correct (Maxwellian) conditional (most often the momenta and coordinates are independent) probability of the momenta ρBM (r, pr ) = ρξr (r)ρ p (pr |r).

(26)

This ensemble provides the basis for a natural method that can be used to compute time correlation functions. Given two arbitrary dynamical variables O (r, pr ) and O (r, pr ) we may write O (r, pr )O (r(t), pr (t))δ(ξ(r) − ξ ) δ(ξ(r) − ξ ) −1/2 O (r, pr )O (r(t), pr (t))ξ |Z | . = |Z |−1/2 ξ

(27)

The result in Eq. (27) allows one to compute the contribution to the timedependent rate coefficient in the first set of brackets in Eq. (4) using the Blue Moon ensemble. To complete the calculation of the time-dependent rate coefficient, we must be able to calculate Pξ (ξ ) = δ(ξ − ξ ) = Pu exp(−βW(ξ )) in an efficient manner. The free energy W(ξ ) is the reversible work needed to bring the system from a given reference state to ξ = ξ . The associated thermodynamic force F(ξ ) = −

dW(ξ ) , dξ

(28)

can be expressed as the conditional average of a suitable observable and from the thermodynamic integration of F(ξ ) over ξ , we can obtain the potential of mean force W(ξ ) =

ξ

dW(ξ dξ dξ

)

ξ

=

dξ

(∂ H/∂ξ ) δ(ξ − ξ ) . δ(ξ − ξ )

(29)

The explicit form of the thermodynamic force can be obtained by performing the derivative in Eq. (28) to obtain (∂ H/∂ξ )δ(ξ −ξ ) (β −1 (∂/∂ξ ) ln |J |−(∂ V /∂ξ ))δ(ξ − ξ ) = δ(ξ −ξ ) δ(ξ − ξ ) ˆ ξ ˆ Fδ(ξ − ξ ) Z −1/2 F = , (30) ≡ δ(ξ − ξ ) Z −1/2 ξ

F(ξ ) = −

where |J | is the Jacobian of the transformation r → u resulting from the explicit integration over the momenta. The quantity Fˆ whose conditional average determines the mean force is the sum of two terms: the first term, β −1 (∂/∂ξ ) ln|J |, represents the apparent forces acting on the system due to the use of generalized (non-inertial) coordinates, while the second term,

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−(∂ V /∂ξ ), corresponds to the component along the generalized coordinate ξ of the force coming from the potential V . This result expresses the thermodynamic force as a conditional average which can be computed numerically by using the Blue Moon ensemble as indicated in the last line of Eq. (30). It is possible to obtain another, more convenient, expression for the mean force that obviates the need to compute the Jacobian and the ξ derivative of the potential, two quantities that are often difficult to determine [10]. The alternative form involves the Lagrange multiplier associated with the constraint on ξ , one of the variables in the (u, pu ) representation of phase space u˙ =

∂H ∂pu

,

p˙ u = −

∂H ∂u

− λδξ u .

(31)

In this approach, one keeps the momentum dependent observable (∂ H/∂ξ ) and computes the difference between the configurationally unbiased constrained average and the corresponding conditional average. The (negative of the) mean force can be written as dW(ξ ) (∂ H/∂ξ ) δ(ξ − ξ ) = dξ δ(ξ − ξ ) −1/2 Z − λ − p˙ξ + (1/2β) (∂ ln |Z |/∂ξ ) ξ = Z −1/2 ξ −1/2 Z (1/β)G − λ ξ = , Z −1/2 ξ

(32)

where G=

N 1 ∂ξ ∂2 ξ ∂ξ 1 · · . 2 Z i, j =1 m i m j ∂ri ∂ri ∂r j ∂r j

(33)

The computations leading to this result are straightforward but lengthy [10]. This formula provides a much more convenient route for the computation of the mean force and, hence, the potential of mean force which uses quantities that are automatically provided by SHAKE [11, 12] in the constrained molecular dynamics simulation. From these considerations, we see that all quantities needed to estimate the rate coefficient may be determined efficiently in the Blue Moon ensemble. It is a straightforward to include other constraints, such as bond constraints, in the formalism [3]. In particular, the expression for the correlation function takes the form O (r, pr )O (r(t), pr (t))δ(ξ(r) − ξ ) δ(ξ(r) − ξ ) −1/2 O (r, pr )O (r(t), pr (t))ξ ,M D , = D −1/2 ξ ,M

(34)

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where the subscript M refers to the other bond constraints and D = |Z|/|Z | with Z defined as in Eq. (18) but extended to include all constraints Z mn =

N 1 ∂σm i=1

m i ∂ri

·

∂σn ∂ri

,

(35)

while Z is defined by a similar equation with the restriction to only bond constraints. To illustrate how the formalism can be applied we consider the adiabatic transfer of a proton in an [AH A]− complex, with fixed internuclear separation between the A− anions, solvated by a polar liquid [13] A − H · · · A− A− · · · H − A.

(36)

Let rp be the quantum coordinate of the proton and R the remainder of the complex and solvent classical degrees of freedom. The total potential energy of the system is V =Vps (rp , R)+Vs (R), where Vps and Vs are the proton–solvent and solvent–solvent interactions, respectively. In the adiabatic approximation, ¨ the proton wave function satisfies the following Schrodinger equation

h¯ 2 2 − ∇ + Vps (rp , R) n (rp ; R) = n (R) n (rp ; R), 2m p rp

(37)

where m p is the mass of the proton, 2πh¯ is Planck’s constant, n (R) is the nth adiabatic eigenvalue and the corresponding wave function is n (rp ; R). The ¨ = −∇R (Vps + classical coordinates follow Newton’s equations of motion m i R

n (R)) on the nth adiabatic energy surface. In particular, the adiabatic dynamics on the ground-state surface can be calculated easily by solving the ¨ Schrodinger equation for each solvent configuration in order to obtain the ground-state energy 0 (R) and wave function 0 . A convenient reaction coordinate for this problem is the solvent polarization, ξ(R) = E(R) E(R) =

i

zi

1 1 − , |Ri − u| |Ri − u |

(38)

where z i is the charge on site i and u and u are two chosen reference positions. The Blue Moon expression for the time-dependent transmission coefficient is κf (t) =

˙ D −1/2 ( E)θ( E(t) − E ‡ )ξ,M . ˙ ˙ D −1/2( E)θ( ( E)) ξ,M

(39)

Here, D −1/2 is the Blue Moon unbiasing factor with D = (2m)−1 i (∇i E)2 , where the sum extends over all classical degrees of freedom assumed to have equal mass m. The time-dependent transmission coefficient κ(t) was calculated by constraining the system to E(t) = E ‡ to generate the Blue Moon ensemble and releasing the constraint to determine the time evolution of

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1

κ(t )

0.8

0.6

0.4

0

1

2

3

4

5

t (ps) Figure 3.

The time-dependent transmission coefficient for adiabatic proton transfer.

θ( E(t) − E ‡ ) that appears in the correlation function expression for κ(t). The results of this calculation are shown in Fig. 3 [13]. The simulations show that a plateau is reached on the order of 1 ps and that the forward rate constant kf is about 0.6 of its kfTST value as a result of recrossings of the E(t) = E ‡ surface. The transition-state rate constant can also be determined in the Blue Moon ensemble using the expression

kfTST = (2πβ)−1/2 D −1/2

−1

ξ,M

e−β W ( E

E< E ‡

‡)

d Ee−β W ( E)

.

(40)

The free energy at the barrier top E ‡ was estimated from quadratic approximations to the free energy function near the metastable states and the expectation value of D −1/2 in the above formula was evaluated in the Blue Moon ensemble. The resulting value for the transition-state rate constant is kfTST = 4 × 109 s−1 . The direct molecular dynamics simulation of the rate constant would be a difficult task without the use of a rare event sampling technique in view of the activated nature of this proton transfer reaction.

3.

Vectorial Reaction Coordinate

In some instances, the reaction path the system takes to go from reactants to products may not be simply related to a single scalar reaction coordinate which is chosen on physical grounds. An inappropriate choice of a reaction coordinate can lead to difficulties in the computation of the rate. The underlying structure of the reaction path may often be revealed by extending the description to vectorial reaction coordinates. For example, if the free energy surface as a function of two reaction coordinates ξ1 and ξ2 has the structure

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shown in Fig. 4, then a description based on projections of the free energy along ξ1 will lead to misleading results. The discovery of the appropriate set of reaction coordinates, or more generally the reaction path, may not be a simple task for some systems. To describe the free energy of such more complex situations, we suppose that the system can be characterized by a set of reaction coordinates which are a functions of the positions of the particles in the system, ξ(r) = {ξ1 , ξ2 , . . . , ξn }. The probability density of finding ξ(r) = ξ is

P(ξ ) = δ(ξ(r) − ξ ) =

n

δ(ξα (r) −

α=1

ξα )

.

(41)

The free energy W(ξ ) associated with the vectorial reaction coordinate is W(ξ ) = −β −1 ln(P(ξ )/Pu ),

(42)

where Pu is again a uniform probability density of ξ . This free energy W or reversible work needed to take the system from the vectorial reaction coordinate value ξa to ξb can be calculated by means of a n-dimensional line integral

W(ξb ) − W(ξa ) =

dξ ·

∂W

C (ξ a ,ξ b )

∂ξ

,

(43)

where C(ξ a , ξ b ) is the path taken from ξa to ξb . Using Eqs. (41) and (42) and the properties of the delta function one may show that −

∂W ∂ξ

=−

∂H ∂ξ

cd

= Fξ ,

(44)

ξ

ψ2

B

‡

ψ2

X

A ψ1‡

ψ1

Figure 4. Sketch of a free energy surface showing two metastable regions depicted as shaded domains in the figure. To obtain such a plot, the free energy is computed for specified values ‡ ‡ of two reaction coordinates, ξ1 (r) and ξ2 (r). The saddle point at (ξ1 , ξ2 ) is indicated by a × symbol.

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where Fξ is the mean force associated with ξ. Following the procedure outlined in Section 2 we may write the (negative of the) mean force in the form [14] ∂W ∂ξ

||−1/2 1/βG − λ

=

||−1/2

ξ‡

ξ‡

,

(45)

where αβ =

1 ∂ξα ∂ξβ k

m k ∂rk ∂rk

.

(46)

and we have defined the vector G with elements Gα =

i,n

1 −1 ∂ξµ ∂2 ξγ ∂ξν −1 , m i m n µγ ν µα ∂ri ∂ri ∂rn ∂rn γ ν

α = 1, . . . , n,

(47)

and λ is the vector of Lagrange multipliers appearing in the constrained equations of motion. The formalism for the most general treatment where there are both other constraints and vectorial reaction coordinate constraints has been given by [15]. The Blue Moon ensemble has been used to compute the free energy as a function of several reaction coordinates for ionization reactions of [NaCl2 ]− ion complexes in water clusters [16]. The multidimensional reaction coordinate formalism described above has also been applied to study the interaction between monomers in a superoxide dismutase protein, Photobacterium leiognathi [14], which we now briefly describe. This protein provides a good example of macromolecular recognition since the monomers are able to form the dimeric enzyme in water. Calculation of the binding force for different mutant proteins, obtained by substituting the amino acids at the monomer–monomer interface, and structural analysis could provide insight into the recognition process. In Fig. 5, we give a pictorial view of the protein as found in nature.

Zn Cu

Figure 5. Photobacterium leiognathi Cu,Zn SOD structure. The ribbon shows the fold of the two identical subunits constituting the dimer. Arrows represent the β-strands while thin wires represent the random-coil structure and the turns. The copper and zinc ions are shown as dark and light gray labelled spheres, respectively.

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W(kcal/mol)

0 ⫺10 ⫺20 ⫺30 ⫺40 ⫺50 16

17

18

19

20

21

22

23

24

ψ1(Å) Figure 6.

The graph shows the potential of mean force as a function of the separation.

The separation of the two monomers can be studied as a function of their relative displacement and orientation. This requires a six-dimensional reaction coordinate. A complete sampling of the phase space as a function of this sixdimensional coordinate is not feasible for the system under investigation (there are 2694 atoms in the proteins which were solvated by 9944 water molecules). However, the principal component of the binding force can be calculated by freezing the slow orientational modes of the two monomers and studying their separation at a fixed orientation. One can choose the initial orientation of the monomers to be that minimizing the energy of the system; hence, the mean force along the relative separation distance can be calculated. The result of such a calculation is shown in Fig. 6. While the separation path one obtains is not fully realistic, it can be used to perform a series of identical numerical experiments on different mutants of Photobacterium leiognathi to investigate the important structural and dynamical features in the recognition process.

4.

Outlook

The description of condensed phase activated rate processes is a challenging problem. The choice of a suitable reaction coordinate or set of reaction coordinates is a central feature of such descriptions. Often this choice is made on physical grounds but schemes for determining reaction paths are needed to provide results when physical considerations are inadequate. Once such reaction coordinates are known the methods described in this chapter provide algorithms for the computation of reaction rates. As briefly described in the text, the methods presented here have been used to investigate also adiabatic quantum rate processes. Non-adiabatic reaction rates may also be treated using the techniques developed here [17].

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References [1] C. Dellago, P.G. Bolhuis, and P.L. Geissler, “Transition path sampling,” Adv. Chem. Phys., 123, 1–78, 2002. [2] W.E and E. Vanden Eijnden, “Conformational dynamics and transition pathways in complex systems,” In: S. Attinger and P. Koumoutsakes (eds.), Lecture Notes in Computational Science and Engineering, Springer, Berlin, vol. 39, to be published, 2004. [3] E. Carter, G. Ciccotti, C. Hynes, and R. Kapral, “Constrained reaction coordinate dynamics for the simulation of rare events,” Chem. Phys. Lett., 156, 472–477, 1989. [4] T. Yamamoto, “Quantum statistical mechanical theory of the rate of exchange chemical reactions in the gas phase,” J. Chem. Phys., 33, 281–289, 1960. [5] D. Chandler, “Statistical-mechanics of isomerization dynamics in liquids and transition-state approximation,” J. Chem. Phys., 68, 2959–2970, 1978. [6] R. Kapral, S. Consta, and L. McWhirter, “Chemical rate laws and rate constants,” In: B. Berne, G. Ciccotti, and D. Coker (eds.), Classical and Quantum Dynamics in Condensed Phase Systems, World Scientific, Singapore, pp. 583–616, 1998. [7] H. Goldstein, Classical Mechanics, Addison-Wesley, Reading, MA, 1980. [8] M. Fixman, “Classical statistical-mechanics of constraints – theorem and application to polymers,” Proc. Natl. Acad. Sci. USA., 71, 3050–3053, 1974. [9] N.G. van Kampen and J.J. Lodder, “Constraints,” Am. J. Phys., 52, 419–424, 1984. [10] M. Sprik and G. Ciccotti, “Free energy from constrained molecular dynamics,” J. Chem. Phys., 109, 7737–7744, 1998. [11] J.P. Ryckaert, G. Ciccotti, and H.J.C. Berendsen, “Numerical-integration of Cartesian equations of motion of a system with constraints – molecular-dynamics of n-alkanes, J. Comput. Phys., 23, 327–341, 1977. [12] G. Ciccotti and J.P. Ryckaert, “Molecular dynamics simulation of rigid molecules,” Comput. Phys. Rep., 4, 345–392, 1986. [13] D. Laria, G. Ciccotti, M. Ferrario, and R. Kapral “Molecular dynamics study of adiabatic proton transfer reactions in solution,” J. Chem. Phys., 97, 378–388, 1992. [14] A. Sergi, G. Ciccotti, M. Falconi, A. Desideri, and M. Ferrario, “Effective binding force calculation in a dimeric protein by molecular dynamics simulation,” J. Chem. Phys., 116, 6329–6338, 2002. [15] I. Coluzza, M. Sprik, and G. Ciccotti, “Constrained reaction coordinate dynamics for systems with constraints,” Mol. Phys., 101, 2885–2894, 2003. [16] S. Consta and R. Kapral, “Ionization reactions of ion complexes in mesoscopic water clusters,” J. Chem. Phys., 111, 10183–10191, 1999. [17] A. Sergi and R. Kapral, “Quantum-classical dynamics of non-adiabatic chemical reactions,” J. Chem. Phys., 118, 8566–8575, 2003.

5.5 ORDER PARAMETER APPROACH TO UNDERSTANDING AND QUANTIFYING THE PHYSICO-CHEMICAL BEHAVIOR OF COMPLEX SYSTEMS Ravi Radhakrishnan1 and Bernhardt L. Trout2 1 Department of Bioengineering, University of Pennsylvania, Philadelphia, PA, USA 2

Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, MA, USA

Many physico-chemical processes such as nucleation events in phase transitions, chemical reactions, conformational changes of biomolecules, and protein folding are activated processes that involve rare transitions between stable or metastable states in the free energy surface. Understanding the underlying mechanism and computing the rates associated with such processes is a central to many applications. For instance, the familiar process of nucleation of ice from supercooled water is encountered in several scientific and technologically relevant processes. The formation of ice microcrystals in clouds via nucleation is a phenomenon that has a large impact in terms of governing global climatic changes. The key to the survival of Antarctic fish and certain species of beetles through harsh winters is their ability to inhibit nucleation of intracellular ice with the aid of antifreeze proteins. At the other end of the spectrum, certain protein assemblies called ice-nucleation agents are believed to be responsible for catalyzing ice nucleation, a phenomenon, which is exploited by certain bacteria to derive nutrients from their host plants. Controlling the formation and propagation of intracellular ice is finding importance in cryopreservation of natural and biosynthetic tissues. Similarly, one can cite many technologically relevant self-assembly and transport processes in the context of fabrication of advanced materials for specific applications in drug delivery, biosensing, and chemical catalysis. A unifying feature among various activated events is that they can be understood in terms of transitions between a series of stable (global minimum) 1613 S. Yip (ed.), Handbook of Materials Modeling, 1613–1626. c 2005 Springer. Printed in the Netherlands.

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or metastable (local minima) basins in the free energy landscape separated by free energy barriers or bottlenecks (also known as transition states). The metastable states represent high-probability regions and the transition states represent low-probability regions of phase space. For a system consisting of N atoms, the free energy landscape F = −kB T ln Q is 3N-dimensional and related the configurational partition function Q given by,

Q=

dr1 dr2 · · · dr N exp(−β H (r1 , r2 , . . ., r N )).

(1)

Here, kB is the Boltzmann constant, T is the temperature, r1 ,r2 , . . . , r N are coordinates of the N atoms in the system, β = 1/kB T , and H is the classical Hamiltonian giving the energy of the system for a given configuration. An activated process can be described by a set of pathways connecting the relevant metastable states in the free energy landscape (see Fig. 1). A Monte Carlo path or a molecular dynamics trajectory that captures an activated process is likely to be representative of a pathway for the transition in the sense that the trajectory will show many characteristics that are unique to the pathway. However, an ensemble of molecular dynamics trajectories (or Monte Carlo paths) connecting the metastable states of the free energy landscape, rather than a single trajectory or path best describes the mechanism of transition. Consequently, the intermediate states along the pathway can be

H (r1,r2)

1.5 1 0.5 0 ⫺0.5 ⫺1 ⫺1.5 ⫺2 80

A

60 r2 40

B 20 0

0

10

20 30

40 r1

50

60

70

Figure 1. Model Hamiltonian for a system with two degrees of freedom. The stable (A) and metastable (B) states are shown along with the transition state. Three different paths connecting states A and B are shown on the contour projection.

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characterized by unifying patterns (structural or energetic) that are common to the ensemble of molecular dynamics trajectories. In a statistical sense, identifying the dynamical variables to quantify the patterns and averaging over the different molecular configurations along the transition pathway can yield insight into the relationship between the evolution of the patterns and the free energy landscape. These dynamical variables, also referred to as order parameters, are quantities that can classify different metastable states according to their distinguishing characteristics (such as symmetries associated with different phases). In addition, the order parameters depend on the nature of intermolecular forces, solvent degrees of freedom, etc., and consequently, the chemical environment will impact their choice. In the simple example in Fig. 1, the intermediate states and the transition pathway are easily described in terms of the variables r1 and r2 , which serve as good order parameters. As we shall see throughout this article, the information about the free energy landscape obtained from such an approach is useful in quantifying the rate of the activated process while the evolution of the order parameters along the pathway is useful in understanding the underlying mechanism. Furthermore, equilibrium properties of the system, and how they depend on the control variables, can be inferred from the relationship between the order parameters and the free energy landscape. Conceptual understanding of the nature of the process, which arises from the relationship between the order parameters and the free energy landscape, generally goes by the name of “a phenomenological theory”. More specifically, the phenomenology arises from identifying how the order parameters (and hence the rate of the process) are influenced by the chemical environment, state variables, and control variables, so that the order parameters can themselves be ascribed physical meaning and interpretation.

1.

Relationship between the Order Parameters and the Free Energy Function

The use of order parameters to construct phenomenological theories (top-down approach) of phase transitions in condensed matter and solid-state systems was pioneered by Landau, Ginzburg, De Gennes, and others. In this approach, the order parameter is chosen based on physical grounds and intuition, and the free energy functional (as a function of the order parameter) is constructed based on symmetry arguments. The literature in this class of problems is extensive, with applications ranging from superconductivity and superfluidity, magnetic and liquid-gas transitions, and theory of liquid crystals. A comprehensive treatise on the study of phase transitions using

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phenomenological approaches, including universality associated with critical behavior and the re-normalization group method is provided in Refs. [1, 2]. Order parameters can also be used to construct the free energy functional by the coarse graining (bottom-up approach) of the microscopic Hamiltonian. For complex systems, for which the construction of an analytic free energy functional may be nontrivial, the free energy as a function of the order parameters can be obtained via density functional theory [3], or via molecular simulations. In this article, we discuss the latter approach. Starting from a set of n order parameters (φ1 , φ2 , . . . , φn ), the free energy density, [φ1 , φ2 , . . . , φn ] (also called the potential of mean force or Landau free energy), along the order parameters, are related to the microscopic Hamiltonian by [1]

exp(−β[φ1 , φ2 , . . . , φn ])=

dr1 dr2 · · · dr N exp(−β H (r1 , r2 , . . . , r N )) ×δ(φ1 − φ1 )δ(φ2 − φ2 ) · · · (φn − φn ).

Here, δ is the Dirac delta function, and [φ1 , φ2 , . . . , φn ] is to be interpreted as [φ1 = φ1 ,φ2 = φ2 , . . . , φn = φn ]. The free energy F is then given by

exp(−β F) =

dφ1 dφ2 · · · dφn exp(−β[φ1 , φ2 , . . . , φn ]).

(3)

The domain of integration in the above equation covers the range of order parameter values characterizing the particular state. In the example given in Fig. 1, [φ1 , φ2 ] ≡ [r1 , r2 ] = exp(−β H (r1 , r2 )), and the free energy of state A is obtained by integrating over r1 in the range (0–30), and over r2 in the range (30–60). The value of the free energy is insensitive to the exact values defining the domain of integration, as long as the region containing the minimum of the function is included. In the course of a microscopic simulation (such as molecular dynamics or Monte Carlo), the free energy density is calculable by collecting histograms of the distribution of the order parameters. If the sampling in the simulations is ergodic (i.e., encompasses the relevant phase space) and sufficiently long, these histograms are proportional to the joint probability distribution of the order parameters, P[φ1 , φ2 , . . . , φn ]. The free energy density is related to the P[φ1 , φ2 , . . . , φn ] by β[φ1 , φ2 , . . . , φn ] = −ln(P[φ1 , φ2 , . . . , φn ]) + Constant.

(4)

In order to circumvent the problem associated with ergodicity, the histograms are evaluated in separate windows of the order parameter ranges using the procedure of umbrella sampling [4]. The umbrella sampling can be understood as performing the simulations in an extended ensemble whose free

Order parameter approach of complex systems

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energy is related to the original thermodynamic ensemble (with a free energy F) by = F −

h i φi ,

(5)

i

where i h i φi is chosen a priori as a weighting function W {φ1 , φ2 , . . . , φn }. The umbrella sampling scheme amounts to simulations being performed using a modified Hamiltonian H = H + βW {φ1 , φ2 , . . . , φn }). The probability distribution P [φ1 , φ2 , . . . , φn ] in the modified ensemble defined by is the related to P[φ1 , φ2 , . . . , φn ] by −kB T ln(P [φ1 , φ2 , . . . , φn ]) = −kB T ln(P[φ1 , φ2 , . . . , φn ]) −W {φ1 , φ2 , . . . , φn }.

(6)

In the simplest case, choosing W {φ1 , φ2 , . . . , φn } = 0 for φ1,min < φ1 < φ1,max , and ∞ otherwise, enables the calculation of P[φ1 , φ2 , . . . , φn ] in the range φ1,min < φ1 < φ1,max . Performing such calculations over several windows covering the entire range of φ1 (of relevance) enables an accurate calculation of P[φ1 , φ2 , . . . , φn ]. In addition, enhanced sampling methods such as configurational bias sampling [5], parallel tempering [6], density of states Monte Carlo [7], and methods based on Tsallis statistics [8] can be used to improve the accuracy of the calculations.

2.

Types of Order Parameters

Order parameters have been extensively used in conjunction with molecular simulations in applications involving solid and liquid-crystalline (LC) phases, in which one of the phases is characterized by long-range order. In such cases, the order parameter can be chosen on the basis of the symmetry of the ordered phase. A few examples are given in Fig. 2. For phases with longrange order (i.e., φ(0)φ(r → ∞) = nonzero constant), the order parameter assumes a nonzero value. For disordered phases (i.e., φ(0)φ(r) ∼ exp(−r/λ), λ being the correlation length), the order parameter is zero for an infinite system. For phases with quasi-long-range order (i.e., φ(0)φ(r) ∼ r −η ), the order parameter in a finite system assumes a value intermediate between the disordered and ordered phases, with system size dependence characterized by the exponent η. The Mermin order parameter [9] is introduced to quantify order in a twodimensional crystal of circular disks, where the only close-packing possible is hexagonal, i.e., leading to a triangular lattice. More generally, for an “N-atic” order in two-dimensional systems, the pair correlation function g(r) ≡ g(r, θ) in a cylindrical coordinate system (r ≡ reiθ ) can be expressed in terms of a Fourier series in angular (θ) space

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R. Radhakrishnan and B.L. Trout

Order parameters describing bond-orientational order in condensed phases.

g(r, θ) =

g j (r) exp(i N j θ),

(7)

j

where the summation over j runs from 0 to ∞. The coefficients of expansion (i.e., the g j (r)s ) are suitable order parameters. For a hexatic (6-fold) symmetry, the dominant order parameter is given by the first term in the expansion

Order parameter approach of complex systems

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evaluated at the nearest neighbor distance, i.e., g1 (r = rnn ), which is the same as the Mermin order parameter 6 (r) in Fig. 2. The Steinhardt order parameters [10] are a generalization of the above definition to three-dimensional systems. The pair correlation function g(r) ≡ g(r, θ, φ) in spherical polar coordinates is expanded in terms of a Fourier series,

g(r) = go (r) 1 +

l

lm (r) lm (θ, φ) ,

(8)

m

where the summation over l runs from 0 to ∞, and that over m runs from −1 to +1. The coefficients lm are related to the Steinhardt order parameters (Fig. 2), which are useful in differentiating between various crystal types in three-dimensional systems. For water-like molecules (which have a propensity for tetrahedral coordination, owing to their hydrogen-bonding nature), the tetrahedral order parameter [11] in Fig. 2 measures the degree to which the nearest-neighbor molecules are tetrahedrally coordinated with respect to a given molecule. The tetrahedral order parameter is a three-body order parameter, which ensures local tetrahedral symmetry around each (water-like) molecule. The tetrahedral order parameter is sensitive to formation of structures of crystalline structures in water. Similarly, the nematic order parameter [1] in Fig. 2 quantifies the degree of nematic order (i.e., parallel ordering of anisotropic molecules along their longitudinal axis) in liquid-crystalline systems. In fact, the nematic order parameter is characterized by a 2-fold symmetry (the longitudinal axis is a headless director, i.e., with no up or down direction), and therefore is closely related to Y2m term of the Steinhardt order parameter. In the examples given in Fig. 2, the definitions of order parameters are based on bond-orientational order, i.e., orientation of nearest neighbor (or molecular) bonds. In each case, the order parameters quantify the degree of crystalline order in the system; therefore, the order parameters assume nonzero (distinct) values in the crystalline phase, which reduce (mostly to zero) in the disordered phase.

3. 3.1.

Applications of Order Parameters Quantification of Disorder

Torquato and coworkers [12], have used the three-dimensional Steinhardt order parameters along with a translational order parameter t (based on the radial distribution function g(r)) to quantify the degree of disorder in dense packed materials. For a system of hard spheres in three dimensions, the authors

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computed order parameter maps (Q 6 vs. t) for the liquid and crystal phases at equilibrium, and series of metastable states with jammed configurations (i.e., those configurations in which a given particle cannot be displaced when the rest of the particles in the system are fixed). The authors found that the translational order parameter was always positively correlated with the bond orientational order parameter (i.e., an increase in one led to an increase in the other) for the hard sphere system. Additionally, the bond orientational order parameter (being the more sensitive measure of the two) increased monotonically with increasing volume packing fraction for the jammed structures, for the entire range of packing fraction between the equilibrium liquid and crystal. The authors concluded that the concept of random close-packing is ill-defined based on the observation that an infinitesimal increase in the bond-orientational order parameter can lead to an infinitesimal increase in the packing fraction. The results also supported the view that glassy structures were not merely liquid-like structures with “frozen-in” disorder, because they were characterized by distinctly different values of Q 6 , intermediate between liquid and crystal phases.

3.2.

Anomalies of Liquid Water

Errington and Debenedetti [13] have advanced a formalism to understand the structure–property relationship in liquid water on the basis of order parameter maps. Based on the values of the translational order parameter t and the tetrahedral order parameter ξ for liquid water at equilibrium, the authors traced paths of the system in ξ − t space. Each path was obtained at constant temperature as the density was gradually increased in their computer simulations. Unlike the hard sphere case (where Q 6 and t are positively correlated and monotonically increase with increasing volume fraction), the authors found structurally anomalous regions (of state space, e.g., in the temperature–density plot) in water where “order” decreased with increasing density. The boundary of the structurally anomalous region was identified by ξ and t extrema on the ξ − t traces at different temperatures. The authors also found that the structural anomaly in water was correlated with a transport anomaly (when the diffusion coefficient increases with increasing density), and with a thermodynamic anomaly (when the coefficient of thermal expansion is negative); in particular, the anomalous regions occur as a cascade, i.e., the structurally anomalous region encompasses the region characterized by the transport anomaly, which encompasses the region showing the thermodynamic anomaly. In a following work, the authors also investigated the order parameter maps in a system of Lennard-Jones particles and found that the LJ system of particles displayed the same qualitative behavior as the system of hard spheres, which further

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supports the view that the anomalies in water are directly related to the chemical structure of the water molecule.

3.3.

Nucleation of Crystalline Forms from Liquid

In a pioneering study, Frenkel and coworkers [14] calculated the free energy barrier to crystal nucleation in a system of Lennard-Jones particles in three dimensions, by employing the Steinhardt order parameters and using the formalism in Section 2. The authors showed that the path to nucleation of the stable face centered cubic phase (when the liquid is supercooled below the freezing temperature) can be described in terms of increasing values of Q 6 , while simultaneously suppressing the increase of W4 . The path in order parameter space along which both Q 6 and W4 simultaneously increase, leads the system into a metastable body centered cubic phase. Radhakrishnan and Trout [15–17] extended the above approach to describe ice-nucleation under a variety of homogeneous and inhomogeneous environments including hexagonal ice in the bulk, cubic ice under an external electric field and in a confined system, and clathrate hydrates in a super-saturated aqueous solution containing the hydrophobic solute, CO2 . In order to calculate the free energy barrier to nucleation, the authors employed the two-body Steinhardt order parameters and the three-body tetrahedral order parameters for the one-component systems, and additionally used a translational order parameter based on g(r) for the two-component aqueous solution of CO2 . The authors found that as the successive density modes in the liquid (quantified by peaks in the direct correlation function) became more correlated – owing to a decrease in temperature, influence of an external potential, or increased inhomogeneity – the free energy barrier to nucleation decreased. Interpreting their results in light of density functional theory of freezing, the authors discovered an inverse correlation between the degree of coupling of the successive density modes in the liquid phase and the free energy barrier to nucleation. Rutledge and coworkers [18], using molecular dynamics simulations, have studied crystal nucleation in a polymer melt. The time-scale to observe nucleation of a crystal phase in a polymer melt is normally beyond the scope of molecular dynamics simulations. The authors, however, found that under an applied a uniaxial (extensional) stress, the barrier to nucleation reduces considerably, to the extent that crystal nucleation can be captured in the simulations. The crystalline domains in the simulations were identified based on the local value of the nematic order parameter. The authors concluded that the addition of a large deforming stress accelerates the crystallization process by driving the individual chains into a low energy torsional conformation and by aligning them in a single direction, which leads to a lowering of the nucleation barrier.

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Shetty and coworkers [19] have proposed a new formalism to construct new types of order parameters to quantify local order in inhomogeneous systems (such as a crystal–melt interface) based on pattern recognition and genetic algorithm.

3.4.

Solvation in Biomolecules

The contribution of hydration in molecular assembly and enzyme catalysis has long been recognized. Many biomolecules are characterized by surfaces containing extended nonpolar regions, and the aggregation and subsequent removal of such surfaces from water is believed to play a critical role in the biomolecular assembly in cells. Conventional views hold that the hydration shell of small hydrophobic solutes is clathrate-like, characterized by local cage-like hydrogen-bonding structures and a distinct loss in entropy. Using molecular dynamics simulations on the solvated polypeptide melittin, Cheng and Roskky [20] found that the hydration of extended nonpolar planar surfaces appears to involve structures that are orientationally inverted relative to clathrate-like hydration shells, with unsatisfied hydrogen bonds that are directed towards the hydrophobic surface. The authors employed bond-orientational order parameters to classify the local structuring of the solvent. Based on the correlation between the observed values of the order parameters and the average binding energy (i.e., the interaction of a molecule with all other molecules in the system) of proximal water molecules in each surface set, they concluded that the clathrate-like and inverted clathrate-like structures are distinguished by a substantial difference in the water–water interaction enthalpy, and that their relative contributions depended strongly on the surface topography of the melittin molecule. Clathrate-like structures dominate near convex surface patches, whereas the hydration shell near flat surfaces fluctuates between clathrate-like and less-ordered or inverted structures. The strong influence of surface topography on the structure and free energy of hydrophobic hydration is likely to be a generic feature, which may be important for many biomolecules.

3.5.

Freezing of Inhomogeneous Fluids in Porous Media

Molecular simulations for simple fluids confined in model systems of slit-shaped pores show a freezing behavior that is governed by the relative strength of the fluid–wall interaction to the fluid–fluid interaction (quantified by a parameter α), and the pore width H . The shift in the freezing temperature, Tf,pore − Tf,bulk, is found to be positive if the fluid–wall interaction is more strongly attractive than the fluid–fluid interaction and negative if the fluid–wall interaction is less attractive than the fluid–fluid interaction.

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Using the Mermin order parameters and the free energy formalism in Section 2, Radhakrishnan and coworkers [21] discovered the presence of several thermodynamically stable intermediate phases lying between the liquid phase and the solid phase in computer simulation studies of Lennard-Jones molecules in slit-pores. Their studies led to the conclusion that the contact layers, i.e., the layers closest to the pore walls, freeze at a higher temperature than the inner layers for strongly attractive pores (large values of α), and thus the intermediate phase has the structure termed “contact-crystalline”, i.e., the contact layers are crystalline while the inner layers are liquid-like. For moderate values of α, the contact layers are liquid-like while the inner layers are crystalline, and the intermediate phase exists as a “contact-liquid” phase. The authors also found that for repulsive and weakly attractive walls, the intermediate (contact-layer) phase is at best metastable, and thus, only the liquid and crystal phases are stable. Based on these observations, the authors constructed “global phase diagrams” which present a unifying picture of confined phase freezing behavior in terms of the parameter α and the pore width H . In a later study [22], the authors also found evidence for the existence of a hexatic phase as an intermediary between the fluid and crystalline ones. The hexatic phase is a manifestation of the fact that, in a continuous symmetry breaking transition such as the freezing transition, the translational symmetry and the rotational symmetry can break at two different temperatures. Thus, in the liquid to hexatic phase transition, the rotational symmetry is broken and in the hexatic to crystalline transition the translational symmetry is broken. Hexatic phases, which retain long-range orientational, but not positional order, are known to occur in infinite quasi-two-dimensional systems; the authors established their presence in the simulations using a system size scaling analysis. They also found that for pore sizes accommodating more than three adsorbed molecular layers, a “contact-hexatic” phase was stable phase (in the simulations), where the contact layers are hexatic, while the inner layers are liquid-like.

4.

Assumptions in the Order Parameter Approach and Outlook

As illustrated in this article, the order parameter approach can be gainfully employed in studies of complex systems. In problems where an obvious symmetry is involved, defining a suitable set of order parameters becomes an easy task. The various examples described here demonstrate that these order parameters can be used to associate a defining characteristic of the free energy landscape to the observed phenomenon. At the expense of increased physical insight and phenomenological understanding, several inherent assumptions go into the order parameter approach described here, which we discuss below.

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4.1.

R. Radhakrishnan and B.L. Trout

Physical Significance of the Order Parameters

Based on the relationship between the order parameters and the free energy landscape, it is tempting to associate the order parameters with physical variables. However, in most cases, the order parameters appear as coefficients of expansion of an extensive thermodynamic variable. Consequently, depending on the choice of expansion, the definitions of the order parameters vary and are certainly not unique. Therefore, it is not guaranteed a priori that the order parameters should represent physical variables. In the examples described here, the physicality of the order parameters can be understood on the basis of density functional theory. Since the free energy F is a unique functional of the spatially varying density ρ(r), and the order parameters in Fig. 2 are all based on expansions of g(r) (where ρg(r) = ρ(r)), the order parameters can be ascribed to physical variables, if the particular density mode they characterize can be identified. In general, for other choices of order parameters, this connection must be established. Another underlying assumption is that the phenomena of interest (happening in 3N-dimensional space) can be described in terms of a small number of order parameters. Although a rigorous proof in support of this assumption may not be possible, an argument based on a few general characteristics of physical systems may be put forward in its defense. If our objective is to correlate the equilibrium properties of the system using the order parameters, the order parameters can once again be chosen as the coefficients of expansion of g(r). On first glance, it appears that one needs to include an infinite set of order parameters to have a rigorous theory. However, most often, correlations in g(r) die out by molecular length scales (even in the ordered phase, at finite temperatures), and therefore only order parameters corresponding to density modes relevant to these length-scales need to be included, which reduces their number to a few. If the objective is to describe an activated process, only those dynamical variables corresponding to the slowest modes (which are identifiable by principal component analysis of a molecular dynamics trajectory) in the transition pathway need to be included, therefore greatly reducing the number of relevant order parameters.

4.2.

Rate Processes

Within order parameter framework, we need to invoke additional assumptions to compute the rates of activated processes. A straightforward and conceptually appealing description is using transition state theory, according to which, the rate k is given by k = A exp(−β F),

(9)

Order parameter approach of complex systems

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where F is the free energy barrier along the reaction pathway (i.e., the free energy of the transition state relative to the reactants), and A is a pre-factor related to the inherent frequency of barrier crossing. For an ideal gas, A = (βh)−1 , h being the Planck’s constant. More generally, A is given by the inverse of the time-scale over which the order parameter correlation function φ(0)φ(t) decays when calculated at the transition state. Both terms (i.e., F and A) can be important in calculating the rate. In closing, we note that methods for verifying the existence of a transition state in a multidimensional free energy landscape independent of the order parameter formalism exist, which can validate the findings of the order parameter approach [23]. Alternative approaches to treat activated processes, which are independent of the existence of order parameters (and hence related approximations) have also been developed [24]. Nevertheless, the order parameter approach continues to be widely employed because of its simplicity, computational efficiency, and phenomenological appeal.

References [1] P.M. Chaikin and T.C. Lubensky, Principles of Condensed Matter Physics, Cambridge University Press, Cambridge, 1995. [2] N. Goldenfeld, Lectures on Phase Transitions and the Renormalization Group, Addison-Wesley, New York, 1992. [3] D. Henderson (ed.), Fundamentals of Inhomogeneous Fluids, Marcel Dekker, New York, 1992. [4] D. Chandler, Introduction to Modern Statistical Mechanics, Oxford University Press, Oxford, 1987. [5] D. Frenkel and B. Smit, Understanding Molecular Simulations: From Algorithms to Applications, 2nd edn., Academic Press, San Diego, 2001. [6] R.H. Zhou, B.J. Berne, and R. Germain, “The free energy landscape for beta hairpin folding in explicit water,” Proc. Natl. Acad. Sci. USA, 98, 14931–14936, 2001. [7] F.G. Wang and D.P. Landau, “Efficient, multiple-range random walk algorithm to calculate the density of states,” Phys. Rev. Lett., 86, 2050–2053, 2001. [8] C. Tsallis, “Possible generalization of the Boltzmann–Gibbs Statistics,” J. Stat. Phys., 52, 479–487, 1988. [9] N.D. Mermin, “Crystalline order in 2 dimensions,” Phys. Rev., 176, 250, 1968. [10] P.J. Steinhardt, D.R. Nelson, and M. Ronchetti, “Bond-orientational order in liquids and glasses,” Phys. Rev. B, 28, 784–805, 1983. [11] P.L. Chau and A.J. Hardwick, “A new order parameter for tetrahedral configurations,” Mol. Phys., 93, 511–518, 1998. [12] S. Torquato, T.M. Truskett, and P.G. Debenedetti, “Is random close packing of spheres well defined?” Phys. Rev. Lett., 84, 2064–2067, 2000. [13] J.R. Errington and P.G. Debenedetti, “Relationship between structural order and the anomalies of liquid water,” Nature, 409, 318–321, 2001. [14] R.M. Lyndenbell, J.S. Van Duijneveldt, and D. Frenkel, “Free-energy changes on freezing and melting ductile metals,” Mol. Phys., 80, 801–814, 1993.

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[15] R. Radhakrishnan and B.L. Trout, “A new approach for studying nucleation phenomena using molecular simulations: application to CO2 hydrate clathrates,” J. Chem. Phys., 117(4), 1786, 2002. [16] R. Radhakrishnan and B.L. Trout, “Nucleation of crystalline phases of water in homogeneous and inhomogeneous environments,” Phys. Rev. Lett., 90, 2003, 2003. [17] R. Radhakrishnan and B.L. Trout, “Nucleation of hexagonal ice Ih in liquid water,” J. Am. Chem. Soc., 125, 7743, 2003. [18] M.S. Lavine, N. Waheed, and G.C. Rutledge, “Molecular dynamics simulation of orientation and crystallization of polyethylene during uniaxial extension,” Polymer, 44, 1771–1779, 2003. [19] R. Shetty, F. Escobedo, D. Choudhary, and P. Clancy, “Characterization of order in simple materials. A pattern recognition approach,” J. Chem. Phys., 117, 4000–4009, 2002. [20] Y.K. Cheng and P.J. Rossky, “Surface topography dependence of biomolecular hydrophobic hydration,” Nature, 392, 696–699, 1998. [21] R. Radhakrishnan, K.E. Gubbins, and M. Sliwinska-Bartkowiak, “Global phase diagrams for freezing in porous media,” J. Chem. Phys., 116, 1147–1155, 2002. [22] R. Radhakrishnan, K.E. Gubbins, and M. Sliwinska-Bartkowiak, “Existence of a hexatic phase in porous media,” Phys. Rev. Lett., 89, 076101, 2002. [23] P.G. Bolhuis, C. Dellago, and D. Chandler, “Reaction coordinates of biomolecular isomerization,” Proc. Natl. Acad. Sci. USA, 97, 5877–5882, 2000. [24] P.G. Bolhuis, D. Chandler, C. Dellago, and P. Geissler “Transition path sampling: throwing ropes over rough mountain passes, in the dark,” Ann. Rev. Phys. Chem., 53, 291–318, 2002.

5.6 DETERMINING REACTION MECHANISMS Blas P. Uberuaga and Arthur F. Voter Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA

The articles in this chapter review methods for determining reaction rates for systems in which transitions from state to state are infrequent. Building on seminal concepts such as transition state theory, advances in the last 25 years have given us efficient techniques for finding saddle points and computing accurate rate constants, both classically and quantum mechanically. One motivation for calculating rate constants is that they can be supplied as input to higher-level simulation models, such as the kinetic Monte Carlo (KMC) method, which are the subject of the next chapter. For a KMC simulation to provide dynamical predictions at a desired level of quality, the accuracy of the rate constants should be of equal quality, but, just as important, the list of rates needs to be complete; it should contain all of the relevant reaction mechanisms. This latter challenge is often more daunting than the former. In this article, we discuss specific examples illustrating the difficulty in identifying the important reaction mechanisms. Through these examples, we demonstrate the use of methods for searching for these mechanisms and the importance of doing so. Realistic systems often behave in ways we least expect and if we rely on our intuition about what reaction pathways exist, we are almost certain to miss important mechanisms. We focus on infrequent-event systems in which each state corresponds to a basin in the potential energy surface, the typical case for materials processes, although some of the concepts can be generalized to free-energy basins (i.e., entropically trapped states). As we consider making up a list of rates, these systems generally fall into one (or more) of four categories: Category 0: systems in which all the reaction mechanisms are known, and can be specified in advance. Category 1: systems with reaction mechanisms that are unexpected, lying outside our intuition. Category 2: systems with reaction mechanisms that only become important (or come into existence) when the system conditions are varied. 1627 S. Yip (ed.), Handbook of Materials Modeling, 1627–1634. c 2005 Springer. Printed in the Netherlands.

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Category 3: systems with reaction mechanisms that become important (or come into existence) after evolution of the system causes it to change its character. Systems from category 0 are ideal for kinetic Monte Carlo simulation. Knowing all the reaction mechanisms, we can, in principle, compute the rate constant for each one, forming a complete rate catalog [1]. If the rate constants provided to the KMC algorithm are exact, the resulting dynamics will also be exact – an extremely appealing situation. However, in essence, category 0 exists only for model systems. Systems for which all the reaction mechanisms are known in advance are ones we have made that way by construction. Pure model systems play an important role in understanding materials properties and testing methodology, but often we are interested in accurately modeling realistic systems or obtaining accurate results for dynamics in a particular interatomic potential. As physical systems are allowed to evolve freely, they inevitably show us unexpected reaction mechanisms. This puts them into category 1 at least, and varying the conditions may also reveal additional mechanisms (category 2). The problem is that, often, a system must evolve for times longer than those accessible via MD before this behavior becomes evident. Worse, if the system also belongs to category 3, which we typically do not know in advance, we will need to follow the dynamics for much longer times. Frustratingly, this time scale issue is the very reason we appealed to KMC in the first place, and now we are faced with it again just to find the relevant rate constants.

1.

Approaches to Exploring the System

Accelerated molecular dynamics (AMD) methods [2], described elsewhere in this handbook [3], offer a way around this dilemma. They give us a way to probe the system dynamics without the limitations of our imposed intuition. In the accelerated dynamics approach, the system is allowed to evolve according to the classical equations of motion, as in a direct molecular dynamics simulation. However, by design, during this evolution, state-to-state transitions are coaxed into occurring more rapidly, though in a way still faithful to the dynamics of the system. The result is that, with these methods, the system evolution can be followed over much longer time scales than are accessible to direct molecular dynamics simulation. Reaction mechanisms that are too slow to be observed with direct MD occur naturally during the evolution of an AMD simulation. An alternative to AMD for discovering reaction mechanisms and probing long-time dynamics is to collect a set of escape paths from a given potential

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basin. One way of doing this is to search for saddle points bounding the basin. For example, in the dimer method, described elsewhere in this chapter [4], efficient mode-following searches are initiated at random positions in the basin. Such methods offer a convenient way to scan for possible reaction mechanisms out of a given state and can be employed to probe long-time dynamics. By performing a large number of saddle searches, the majority of relevant escape paths can be found. If the escape rate is then computed for each of these paths, one can be chosen in an on-the-fly kinetic Monte Carlo (OFKMC) step. This takes the system to a new state, and the procedure is then repeated. One must keep in mind, however, that if any of the escape paths are missed in the saddle searches, then the OFKMC step to the next state may not be advancing the system in a dynamically correct way. If low-barrier escape paths are missed, the dynamical evolution will almost certainly be corrupted. The philosophy of this saddle-finding approach differs from that of the accelerated dynamics methods. In AMD, no attempt is made to determine all possible mechanisms for escape from a state (potential basin). Instead, the goal is to propagate the system from state to state as fast as possible while maintaining accuracy in the sense that the probability for choosing a given escape path from a state is proportional to the rate constant for that path. In exact evolution (e.g., via MD), the system chooses an appropriate escape path without knowing about the other possible escape paths, and the power of AMD comes from adopting this philosophy. As tools for determining reaction mechanisms to be supplied as input to KMC, the AMD approach and the saddle-finding approach are complimentary. To probe long times accurately, e.g., for category-3 systems, the higher fidelity of the AMD approach may be advantageous. To generate a list of reaction mechanisms out of a given type of state, the saddle-finding approach is typically more efficient. To illustrate the main point of this discussion, we now give examples of systems for each of the three categories listed above; in each case, an incomplete (intuition-based) KMC table would result in incorrect dynamics. The category-1 example is historical, concerning the diffusion behavior on the fcc(100) surface, perhaps the simplest of metal surfaces. For the surfacediffusion community, at least, this was where the issue of mechanism complexity and rate-catalog incompleteness first became apparent. The second example, the diffusion of interstitial hydrogen in an fcc lattice of fullerenes, illustrates a category-2 system in which parallel replica dynamics (an AMD method) was instrumental in uncovering an unexpected mechanism that, in turn, qualitatively altered the KMC predictions. In the last (category-3) example, taken from a study of radiation damage annealing in MgO, a surprising and unexpected mechanism turned up only after a temperature accelerated dynamics simulation was used to evolve interstitial clusters for seconds.

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Category 1: Surface Exchange on FCC(100)

The simplicity of the fcc(100) surface has long made it a prototype case for studies of metallic surface diffusion and growth. An isolated adatom rests in a four-fold hollow with four available 011 directions for hopping to an adjacent site. Because atoms in the surface layer make four nearest-neighbor bonds, i.e., the surface is tightly packed, it was originally assumed that these hopping events were the only diffusive mechanism available to the adatom, and kinetic Monte Carlo models [1] were based on this assumption. However, density functional theory calculations on A1 [5] showed that an adatom can exchange with a surface layer atom, leading to a 100 displacement to a second-nearest-neighbor position, as shown in Fig. 1. For a number of fcc metals (e.g., Al, Pt, Au), this mechanism is not only available, but substantially favored over hopping. The exchange event can be easily observed in a hightemperature molecular dynamics simulation using embedded-atom method potentials [6], but up to that time the community was so confident in its intuition about adatoms hopping on fcc(100) that no such simulation had been performed. In subsequent years, exchange processes were discovered to play a dominant role, providing low-energy paths for surface smoothing, diffusion over step edges and interface mixing (e.g, see Refs. [7, 8]), even for the metals where hopping is favored for an isolated adatom. Many more complicated processes have been discovered as well (e.g., [9] and Refs. [30–44] in Ref. [2]). Clearly, exchange events should be included in kinetic Monte Carlo simulations to obtain a proper description of the long-time behavior during growth or annealing.

Figure 1. Adatom diffusion mechanisms on Ag(100). Top row: adatom hop. Bottom row: adatom exchange mechanism. In each case, themiddle frame is the saddle point and the initial adatom is highlighted for clarity.

Determining reaction mechanisms

3.

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Category 2: H2 Diffusion in FCC C60

Understanding the behavior of H2 and other light gases in carbonaceous materials is of great interest because of the potential technological applications, such as fuel cells. Using parallel replica dynamics, the diffusive behavior of H2 in crystalline C60 was examined in an effort to determine the important diffusive events [10]. As expected, H2 moves through the interstitial sites of the C60 lattice by single molecular hops, jumping between octahedral and tetrahedral sites in the lattice. The parallel replica simulations revealed, however, that under higher H2 loading, more than one H2 molecule can occupy a single octahedral site, and this fact dramatically changes the diffusive behavior. The dependence of the diffusive behavior on loading makes this system an example of a category2 system. After the parallel replica simulations had explored the system long enough so that all of the relevant mechanisms were uncovered, the rates of these mechanisms were provide as input to a KMC simulation. If multiple occupancy of octahedral sites is not allowed, the self-diffusivity of H2 in C60 falls off quickly as the loading of H2 is increased, as seen in Fig. 2. However, once the fact that multiple H2 can share an interstitial site is included in the KMC simulation, the behavior reverses and the self-diffusivity increases substantially with loading. The system had to be allowed to fully explore the state-space available to it under high loading conditions.

100

D/D0

10

1

T=700K T=500K T=300K T=500K (restricted model) T=300K (restricted model)

0.1

0.01

0

0.2

0.4

0.6

0.8

1

Fractional loading

Figure 2. Self-diffusivity of H2 predicted by a lattice-gas KMC model with and without the existence of interstitial dimers and trimers (filled and open symbols, respectively). At each temperature, the self-diffusivity is shown normalized by the self-diffusivity of an isolated interstitial H2 , D0 . Error bars, shown for the 700 K case with dimers, are similar for the other cases. The overall self-diffusivity changes completely with the addition of reactions involving interstitial dimers (after Ref. [11]).

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Category 3: Radiation-damage Annealing in MgO

Oxide ceramics are important materials in nuclear applications, as, for example, fuels in light water reactors and host materials for storing nuclear waste. Thus, an understanding of the radiation damage behavior is critical for predicting the aging properties of these materials. In a study using pairwise Coulombic potentials for MgO, and temperature accelerated dynamics (TAD) augmented by lowest-barrier information from the dimer method (dimer-TAD [3]), the room-temperature annealing of cascade-generated defects was investigated [11]. Because the barriers in this system were typically high (relative to T = 300 K), dimer-TAD yielded substantial boost factors, allowing simulations on very long time scales. With this interatomic potential, interstitials and vacancies formed in lowenergy cascades are charged, and thus interact strongly. Interstitials, mobile on the ns time scale, annihilate with vacancies (which are immobile) or aggregate with other interstitials to form clusters In , containing n interstitials. Dimers (I2 ) are mobile on the time scale of seconds at T = 300 K, and two dimers encountering each other form an I4 cluster, which is both stable against dissociation and essentially immobile. This suggested that all larger clusters would be stable and immobile as well, so that only the mobility of smaller clusters (n < 4) needed to be accounted for in order to understand the long time behavior of damage in MgO. However, pursuing the cascade annealing evolution further showed this was not true, and several unusual characteristics probably would not have been discovered in a static study. As shown in Fig. 3, a diffusing I2 that encounters an I4 can create a highly mobile I6 entity. After the initial encounter, the I6 system passed through 32 states over a time of 2.9 s before the onset of mobility. Further, this mobile state travels one-dimensionally along a single 110 direction, and diffuses faster than a single interstitial. Each diffusive jump is a concerted event involving 12 atoms – the six interstitial atoms and six lattice atoms. Finally, this kinetically formed state is not the ground state, but a metastable state with a lifetime of years at T = 300 K. This is a clear example of category-3 behavior. Simulations following the coalescence of interstitials into dimers, which in turn formed tetramers and finally hexamers, on the time scale of seconds, were required before the unexpectedly mobile I6 cluster emerged. This behavior clearly impacts the cascade annealing dynamics, and an accurate KMC model will need to include the formation and diffusion kinetics of I6 and perhaps larger clusters as well.

5.

Conclusions

As these examples illustrate, care must be taken in constructing a rate table for higher-level simulations such as KMC. Often, human intuition is not

Determining reaction mechanisms (a)

1633 (b)

8 ps

(c)

4.1 s

1.2 s

(d)

4.1 s ⫹ 41 ns

Figure 3. TAD simulation of the formation of interstitial cluster I6 (three Mg cation interstitials and three O anion interstitials) from I2 and I4 at T = 300 K. Only defects in the lattice are shown. The scheme is: large spheres are interstitials and small spheres arevacancies, light spheres are O and dark spheres are Mg, where a vacancy is defined to be any lattice site with no atom within 0.83 Å and, conversely, an interstitial is any atom not within 0.83 Å of a lattice site. (a) An I2 and I4 begin about 1.2 nm apart. (b) By t = 1.2 s, the I2 approaches the immobile I4 . (c) By t = 4.1 s, the combined cluster anneals to form the metastable I6 , (d) which diffuses on the ns time scale with a barrier of 0.24 eV (after Ref. [11]).

adequate for determining what reaction mechanisms might be important. A much more reliable approach is to allow the system to explore the space available to it, considering both the varying initial conditions that might be relevant as well as the states it might enter (unexpectedly) at much longer times. Finally, we note that for some systems, the behavior will be so complex as to defy condensation into an accurate rate catalog. Highly concerted mechanisms can make even the specification of events difficult. Moreover, if the rates depend strongly on the environment beyond a short distance, the size of the catalog, which grows exponentially with the neighborhood size [1], becomes prohibitively large. This becomes a serious problem more quickly for alloy systems. In this situation, direct simulation using accelerated dynamics or OFKMC methods offers an alternative.

Acknowledgments This work was supported by the United States Department of Energy (DOE), Office of Science, Office of Basic Energy Sciences, Division of

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Materials Sciences; and through a cooperative research and development agreement (CRADA) with Motorola, Inc.

References [1] A.F. Voter, “Classically exact overlayer dynamics – diffusion of rhodium clusters on Rh(100),” Phys. Rev. B–Condens. Matter., 34, 6819–6829, 1986. [2] A.F. Voter, F. Montalenti, and T.C. Germann, “Extending the time scale in atomistic simulation of materials,” Annu. Rev. Mater. Res., 32, 321–346, 2002. [3] B.P. Uberuaga, F.Montalenti, T.C. Germann, and A.F. Voter, “Accelerated molecular dynamics methods,” Handbook of Materials, 2004. [4] H. J´onsson, Handbook of Materials, 2004. [5] G.L. Kellogg and P.J. Feibelman, “Surface self-diffusion on Pt(001) by an atomic exchange mechanism,” Phys. Rev. Lett., 64, 3143–3146, 1990. [6] M.S. Daw, S.M. Foiles, and M.I. Baskes, “The embedded-atom method: a review of theory and applications,” Mater. Sci. Rep., 9, 251–310, 1993. [7] T. Ala-Nissila, R. Ferrando, and S.C. Ying, “Collective and single particle diffusion on surfaces,” Adv. Phys., 51, 949–1078, 2002. [8] A.K. Schmid, J.C. Hamilton, N.C. Bartelt, and R.Q. Hwang, “Surface alloy formation by interdiffusion across a linear interface,” Phys. Rev. Lett., 77, 2977–2980, 1996. [9] H. J´onsson, “Theoretical studies of atomic-scale processes relevant to crystal growth,” Ann. Rev. Phys. Chem., 51, 623–653, 2000. [10] B.P. Uberuaga, A.E. Voter, K.K. Sieber, and D.S. Sholl, “Mechanisms and rates of interstitial H2 diffusion in crystalline C60 ,” Phys. Rev. Lett., 91, 105901, 2003. [11] B.P. Uberuaga, R. Smith, A.R. Cleave, F. Montalenti, G. Henkelman, R.W. Grimes, A.F. Voter, and K.E. Sickafus, “Structure and mobility of defects formed from collision cascades in MgO,” Phys. Rev. Lett., 92, 115505, 2004.

5.7 STOCHASTIC THEORY OF RATE PROCESSES Abraham Nitzan Tel Aviv University, Tel Aviv, 69978, Israel

1. 1.1.

Stochastic Modeling of Physical Processes Introduction

This chapter is written under the assumption that the reader has basic knowledge of probability theory such as needed in an elementary course in statistical mechanics, and at least an intuitive feeling for random numbers as those generated by tossing a coin or throwing a die. A random function is a function that assigns a random number to each value of its argument. Using this argument as an ordering parameter, each realization of this function is an ordered sequence of such random numbers. When the ordering parameter is time we have a time series of random variables, which is called a stochastic process. For example, the random function F(t) that assign to each time t the number of cars on a given highway segment is a random function of time, i.e., a stochastic process. Time is a continuous ordering parameter, however if observations of the random function z(t) are made at discrete time 0 < t1 < t2 , · · · , < tn < T , then the sequence {z(ti )} is a discrete sample of the continuous function z(t). Stochastic processes are ubiquitous in descriptions of observed phenomena. Here we focus on systems of classical particles. Given the initial conditions of a classical system of N particles (i.e., all initial 3N positions and 3N momenta) its time evolution is determined by the Newton equations of motion. For a quantum system, the corresponding N -particle wavefunction is determined by evolving the initial wavefunction according to the Schr¨odinger equation. In fact these initial conditions are generally not known but can be often characterized by a probability distribution (e.g., the Boltzmann distribution for an equilibrium system). The (completely deterministic) time evolution 1635 S. Yip (ed.), Handbook of Materials Modeling, 1635–1672. c 2005 Springer. Printed in the Netherlands.

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associated with any given initial state should be averaged over this distribution. This is the usual starting point of non-equilibrium statistical mechanics. In many cases, however, we seek simplified descriptions of physical processes by focusing an small subsystems or on a few observables that characterize the process of interest. These observables can be macroscopic, e.g., the energy, pressure, temperature, etc. of the system, or microscopic, e.g., the center of mass position, a particular bond length or the internal energy of a single molecule. In the reduced space of these “important” observables, the microscopic influence of the other ∼1023 degrees of freedom appears as random fluctuations that give these observables an apparently random character. For example, the energy of an individual molecule behaves as a random function of time (i.e., a stochastic process) even in a closed system whose total energy is strictly constant. Lets consider a particular example in which we are interested in the center of mass position ri of an isotopically substituted molecule i in an equilibrium homogeneous fluid containing a macroscopic number N of normal molecules. The trajectory ri (t) of this molecule shows an erratic behavior, changing direction (and velocity) after each collision. Indeed, this trajectory is just a projection a deterministic trajectory in the 6N dimensional phase space on the coordinate of interest, however solving this 6N -dimensional problem may be intractable and, moreover, may constitute a huge waste of effort because it yields the time dependence of 6N momenta and positions of all N particles while we are interested only in ri (t), the position of a single particle i. Instead we may look for a reduced description of ri (t) only. We may attempt to get it by a systematical reduction the 6N -coupled equations of motion. Alternatively we may construct a phenomenological model for the motion of this coordinate under the influence of all other motions. As we shall see, both ways lead to the characterization of ri (t) as a stochastic process. As another example consider the internal vibrational energy of a diatomic solute molecule, e.g., CO, in a simple atomic solvent (e.g., Ar). This energy can be monitored by spectroscopic methods, and we can follow processes such as thermal (or optical) excitation and relaxation, energy transfer and energy migration. The quantity of interest may be the time evolution of the average vibrational energy per molecule, where the average is taken over all molecules of this type in the system (or in the observation zone). At low concentration these molecules do not affect each other and all dynamical information can be obtained by observing (or theorizing on) the energy Ej (t) of a single molecule j . Average over many such molecules or over repeated independent observations on a single molecule is an ensemble average. Following vibrational excitation, it is often observed that the subsequent relaxation is exponential, E(t)=E(0) exp(−γt). A single trajectory Ej (t) (also observable by a method called single molecule spectroscopy) is however much more complicated. As before, to predict its exact course of evolution we need to know the initial

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positions and velocities of all the particles in the system, then to solve the Newton or the Schr¨odinger equation with these initial conditions. Again, the resulting trajectory in phase space is completely deterministic, however Ej (t) appears random. In particular, it will look different in repeated experiments because in setting up such experiments only the initial value of Ej is specified, while the other degrees of freedom are subjected only to a few conditions (such as temperature and density). In this reduced description Ej (t) may be viewed as a stochastic variable. The role of the theory is to set up its statistical properties and to investigate its consequences. Obviously, once the usefulness of descriptions of physical processes as stochastic processes in a small subspace of variables is realized, different models of this type can be examined, in which different subspaces are considered. Being interested in the time evolution of the vibrational energy of a single diatomic molecule, we may focus just on this variable, or on the coordinate (the internuclear distance) and momentum of the intramolecular nuclear motion, or on the atomic coordinates and velocities associated with the molecule and its nearest neighbors, etc. These increasingly detailed reduced descriptions lead to greater accuracy at the cost of bigger calculations. The choice of level of reduction is guided by the information designated as relevant based on available experiments, and by considerations of accuracy based on physical arguments. In particular, timescale and interaction-range considerations are central to the theory and practice of reduced descriptions. The relevance of stochastic descriptions brings out the issue of their theoretical and numerical evaluation. Instead of solving the equations of motion for ∼6×1023 degrees of freedom we now face the much less demanding, but still challenging need to construct and to solve stochastic equations of motion for the few relevant variables. The next section describes a particular example.

1.2.

An Example: The Random Walk Problem

An example of a stochastic process associated with a reduced molecular description is the random walk process, in which a particle starts from a given position and moves randomly. This is a model for the motion of a molecule that is assumed to change direction randomly after each collision, then moves a certain length l (of the order of the mean free path) before the next collision. For simplicity lets consider a 1-dimensional (1D) model: during a time interval t the particle is assumed to move to the right with probability pr = kr t and to the left with probability pl = kl t, so that the probability that its stays in its original position is 1 − pr − pl . kl and kr are rate coefficients, measuring the probabilities per unit time that the corresponding properties will occur. In a homogeneous system the rates to move to the right and the left are the

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same, kl = kr . Inequality may reflect the existing of some force that makes the probability to move in a particular direction larger than in the opposite one. Obviously pl and pr are linear in t only for t sufficiently small so that these numbers are substantially less then 1. Starting from t = 0, we require the probability P(n, t) = P(n, N t) that the particle has made a net number of n steps to the right (A negative n implies that the particle has actually moved to the left), i.e., if it starts at position n = 0 we ask what is the probability that it is found at position n (i.e., at distance nl from the origin) at time t, after making at total of N steps. An equation for P(n, t) can be found by considering the propagation from time t to t +t: P(n, t + t) = P(n, t) + kr t (P(n − 1, t) − P(n, t)) + kl t (P(n + 1, t) − P(n, t))

(1.1)

In Eq. (1.1) the terms that add to P(n, t) on the right are associated with different random steps. Thus, for example, kr t P(n − 1, t) is the increase in P(n, t) due to the possibility of jump, during a time interval t, from position n − 1 to position n, while −kr t P(n, t) is the decrease in P(n, t) resulting from transitions from n to n + 1 in the same period. Rearranging Eq. (1.1) and deviding by t we get, when t → 0, ∂ P(n, t) = kr (P(n − 1, t) − P(n, t)) + kl (P(n + 1, t) − P(n, t)) ∂t (1.2) Note that in (1.2) time is a continuous variable while position, expressed by n is discrete. We may also go into a continuous representation in position space by substituting n → nx = x, n − 1 → x − x, n + 1 → x + x to get ∂ P(x, t) = kr (P(x − x, t) − P(x, t)) + kl (P(x + x, t) − P(x, t)) ∂t (1.3) Here P(x, t) may be understood as the probability to find the particle in an interval of length x about x. Introducing the density f (x, t) so that P(x, t) = f (x, t)x and expanding the right hand side of (1.3) up to second order in x we obtain ∂ f (x, t) ∂ 2 f (x, t) ∂ f (x, t) = −v +D (1.4) ∂t ∂x ∂x2 where x (1.5) v = (kr − kl )x = ( pr − pl ) t and where 1 x 2 (kr + kl )x = ( pr + pl ) (1.6) D= 2 (2t)

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Note that even though in (1.4) we use a continuous representation of position and time, the nature of our physical problem implies that x and t are finite, of the order of the mean free path and the mean free time, respectively. To get a feeling for the nature of the solution of Eq. (1.4) consider first the case D = 0. The solutions of the equation ∂ f /∂t = −v∂ f /∂ x have the form f (x, t) = f (x − vt) i.e., any structure defined by f is moving forward (to the right) with speed (drift velocity)) v. This is what is expected under the influence of a constant force that makes kr and kl different. The first term of (1.4) is seen to reflect the effect of the systematic motion resulting from this force. Next consider Eq. (1.4) for the case v = 0, i.e., when kr = kl . In this case Eq. (1.4) becomes the diffusion equation ∂ 2 f (x, t) ∂ f (x, t) =D (1.7) ∂t ∂x2 The solution of this equation for the initial condition f (x, 0) = δ(x − x0 ) is

1 (x − x0 )2 exp − f (x, t|x0 , t = 0) = 4Dt (4π Dt)1/2

(1.8)

(Note that the left hand side is written as a conditional probability density: We have found the probability density about point x at time t given that the particle was at x 0 at time t = 0. Note also that the initial density f (x, 0) = δ(x − x0 ) reflects the initial condition that the particle was, with probability 1, in a neighborhood of length x about x0 taken in the limit x → 0). The diffusion process is the actual manifestation of the random walk that leads to a symmetric spread of the density about the initial position. Equation (1.7) implies that the solution of (1.4) under the initial condition f (x, 0) = δ(x − x0 ) is

1 (x − vt − x0 )2 exp − f (x, t|x0 , t = 0) = 4Dt (4π Dt)1/2

(1.9)

showing both the drift and the diffusion spread. Further insight into the nature of the drift-diffusion process that we are studying can be obtained by considering moments of the probability distribution. Equation (1.2) readily yields equations that describe the time evolution of these moments. Both sides of Eq. (1.4) yield zero when summed over all n from −∞ to ∞, while multiplying this equation by n and n 2 then performing the summation lead to d n = kr − kl (1.10) dt and dn 2 = 2n(kr − kl ) + kr + kl (1.11) dt

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Assuming n(t = 0) = n 2 (t = 0) = 0, i.e., that the particle starts its walk from the origin, n = 0, Eq. (1.10) results in nt = (kr − kl )t

(1.12)

while Eq. (1.11) leads to n 2 t = (kr − kl )2 t 2 + (kr + kl )t

(1.13)

From Eqs. (1.12) and (1.13) it follows that t = ( pr + pl )N (1.14) t for a walker that has executed a total of N steps during time t = N t. When we are interested in length scales that are large relative to x we may consider a continuous representation of these results. Putting x = nx, Eq. (1.12) becomes δn 2 t = n 2 t − n2t = (kr + kl )t = ( pr + pl )

xt = vt

(1.15)

while Eq. (1.14) gives δx 2 t = 2Dt

(1.16)

where v and D are given by Eqs. (1.5) and (1.6), respectively. Together Eqs. (1.5) and (1.6) express the essential features of biased random walk: a drift with speed v associated with the bias kr =/ kl , and a spread with a diffusion coefficient D. The linearity of the spread δx 2 with time is a characteristic feature of normal diffusion. Note that for a random walk in an isotropic 3-dimensional space the corresponding relationship is δr 2 = δx 2 + δy 2 + δz 2 = 6Dt

2.

(1.17)

Some Concepts from the General Theory of Stochastic Processes

In Section 1 we have defined a stochastic process as a time series of random variable(s). If observations are made at discrete time 0 < t1 < t2 , · · · , < tn < T , then the sequence {z(ti )} is a discrete sample of the continuous function z(t). In the random walk problem discussed in Section 1 z(t) was the position at time t of a particle that executes such a walk. We can measure and discuss z(t) directly, keeping in mind that we will obtain different realizations (stochastic trajectories) of this function from different experiments performed under identical conditions. Alternatively we can characterize the process using the probability distributions associated with it. We can consider many such distributions: P(z, t)dz is the probability that the

Stochastic theory of rate processes

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realization of the random variable z at time t is in the interval between z and z + dz. P2 (z 2 t2 ; z 1 t1 ) dz 1 dz 2 is the probability that z will have a value between z 1 and z 1 + dz 1 at t1 and between z 2 and dz 2 at t2 , etc. The time evolution of the process, if recorded in times t0 , t1 , t2 , . . . , tn is most generally represented by the joint probability distribution P(z n tn ; . . . ; z 0 t0 ). The concept of conditional probability is useful: P1 (z 1 t1 |z 0 t0 )dz 1 =

P2 (z 1 t1 ; z 0 t0 )dz 1 P1 (z 0 t0 )

(2.1)

is the probability that the variable z will have a value in the interval z 1 , . . . , z 1 + dz 1 at time t1 if it assumed the value z 0 at time t0 . Similarly P2 (z 4 t4 ; z 3 t3 |z 2 t2 ; z 1 t1 )dz 3 dz 4 =

P4 (z 4 t4 ; z 3 t3 ; z 2 t2 ; z 1 t1 ) dz 3 dz 4 P2 (z 2 t2 ; z 1 t1 )

(2.2)

is the conditional probability that z is in z 4 , . . . , z 4 + dz 4 at t4 and is in z 3 , . . . , z 3 + dz 3 at t3 , given that its values were z 2 at t2 and z 1 at t1 . In the absence of time-correlations, the values taken by z(t) at different times are independent. In this case P(z n tn ; z n−1 tn−1 ; . . . ; z 0 t0 )= nk=0 P(z k , tk ) and time correlation functions, e.g., C(t2 , t1 ) = z(t2 )z(t1 ), are given by products of simple averages C(t2 , t1 )=z(t2 )z(t1 ), where z(t1 )= dz z P1 (z, t1 ). This is often the case when the sampling times tk are placed far from each other – farther than the longest correlation time of the process. More generally, the time correlation function C(t1 , t2 ) can be obtained from the distribution P2 (z 2 t2 ; z 1 t1 ) by the obvious expressions

C(t2 , t1 ) = dz 1 dz 2 z 2 z 1 P2 (z 2 t2 ; z 1 t1 )

(2.3a)

C(t3 , t2 , t1 ) = dz 1 dz 2 dz 3 z 3 z 2 z 1 P2 (z 3 t3 ; z 2 t2 ; z 1 t1 )

(2.3b)

In practice, numerical values of time correlations functions are obtained by averaging over an ensemble of realizations. Let z (k) (t) be the kth realization of the random function z(t). Such realizations are obtained by observing z as a function of time in many experiments done under identical conditions. The correlation function C(t2 , t1 ) is then given by C(t2 , t1 ) = lim

N→∞

N 1

z (k) (t2 )z (k) (t1 ) N k=1

(2.4)

If the stochastic process is stationary, the time origin is of no importance. In this case P1 (z 1 , t1 ) = P1 (z 1 ) does not depend on time, while P2 (z 2 t2 ; z 1 t1 ) = P2 (z 2 , t2 − t1 ; z 1 , 0) depends only on the time difference t21 = t2 − t1 . Also in this case the correlation function C(t2 , t1 ) = C(t21 ) can be obtained by taking

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a time average over different origins along a single stochastic trajectory according to N 1

z(tk + t)z(tk ) C(t) = lim N→∞ N k=1

(2.5)

Here the average is over a sample of reference times that span a region of time that is much larger than the longest correlation time of the process. Further progress can be made by specifying particular kinds of processes of physical interest. In particular we focus on two such processes: (1) Markovian stochastic processes. The process z(t) is called Markovian if the knowledge of the value of z (say z 1 ) at a given time (say t1 ) fully determines the probability of observing z at any later time P(z 2 t2 |z 1 t1 ; z 0 t0 ) = P(z 2 t2 |z 1 t1 );

t2 > t1 > t0

(2.6)

Markov processes have no memory of earlier information. Newton equations describe deterministic Markovian processes by this definition, since knowledge of system state (all positions and momenta) at a given time is sufficient for determining it at any later time. The random walk problem discussed in Section 1.2 is an example of a stochastic Markov process. The Markovian property can be expressed by P(z 2 t2 ; z 1 t1 ; z 0 t0 ) = P(z 2 t2 |z 1 t1 )P(z 1 t1 ; z 0 t0 );

for t0 < t1 < t2

(2.7)

or P(z 2 t2 ; z 1 t1 |z 0 t0 ) = P(z 2 t2 |z 1 t1 )P(z 1 t1 |z 0 t0 );

for t0 < t1 < t2

(2.8)

because, by definition, the probability to go from (z 1 , t1 ) to (z 2 , t2 ) is independent of the probability to go from (z 0 , t0 ) to (z 1 , t1 ). The above relation holds for any intermediate point between (z 0 , t0 ) and (z 2 , z 2 ). As with any joint probability, integrating the left hand side of Eq. (2.8) over z 1 yields P(z 2 t2 |t0 z 0 ). Thus for a Markovian process P(z 2 , t2 |z 0 , t0 ) =

dz 1 P(z 2 , t2 |z 1 , t1 )P(z 1 , t1 |z 0 , t0 )

(2.9)

This is the Chapman Kolmogorov equation. What is the significance of the Markovian property of a physical process? Note that the Newton equations of motion as well as the time dependent Schrodinger equation are Markovian in the sense that the future evolution of a system described by these equations is fully determined by the present (“initial”) state of the system. Non-Markovian dynamics results from the same reduction procedure that we use in order to focus on the “relevant” subsystem, that, as argued above, leads us to consider stochastic time evolution. To see

Stochastic theory of rate processes

1643

this consider a “universe” described by two variables, x1 and x2 , that satisfy the Markovian equations of motion dx1 = F1 (x1 (t), x2 (t), t) dt

(2.10a)

dx2 = F2 (x1 (t), x2 (t), t) dt

(2.10b)

We have taken F2 to depend only on the instantaneous value of x1 for simplicity. If x1 is the “relevant” subsystem, a description of the dynamics in the subspace of this variable can be achieved if we integrate Eq. (2.10b) to get x2 (t) = x2 (t = 0) +

t

F2 x1 (t ), t

(2.11)

0

Inserting this into (2.10a) gives 

dx1 = F1 x1 (t), x2 (t = 0) + dt

t



F2 x1 (t ), t , t 

(2.12)

0

This equation describes the dynamics in the x1 subspace, and its non-Markovian nature is evident. Starting at time t, the future evolution of x1 is seen to depend not only on its value at time t, but also on its past history, since the right hand side depends on all values of x1 (t ) starting from t = 0.1 Why has the Markovian time evolution (2.10) of a system of two degrees of freedom become a non-Markovian descripting in the subspace of one of them? Equation (2.11) shows that this results from the fact that x2 (t) responds to the historical time evolution of x1 , and therefore depends on past values of x1 , not only on its value at time t. More generally, consider a system A+B made of a part (subsystem) A that is relevant to us as observers, and another part B, “the environment”, that is uninteresting to us except for its effect on the relevant subsystem A. A non-Markovian behavior of the reduced description of the physical subsystem A reflects the fact the at any time t subsystem A interacts with the rest of the total system, i.e., with B, whose state is affected by its past interaction with A. In effect, the present state of B carries the memory of past states of the relevant subsystem A. This observation is very important because it points to a way to consider this memory as a qualitative attribute of system B (the environment or the bath) that 1 Equation (2.12) shows also the origin of the stochastic nature of reduced descriptions. Focusing on x , 1 we have no knowledge of the initial state x2 (t = 0) of the “rest of the universe”. At most we may know the distribution (e.g., Boltzmann) of different initial states. Different values of x2 (t = 0) correspond to different realizations of the “relevant” trajectory x1 (t). When the number of “irrelevant” degrees of freedom becomes huge their initial state and therefore also this trajectory assume a stochastic character.

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A. Nitzan

determines the physical behavior of system A. In the example of Eq. (2.10), where system B comprises one degree of freedom x2 , its dynamics is solely determined by its interaction with system A, coordinate x1 and the memory can be as long as the duration of the observed motion. In practical applications however system A represents only a few degrees of freedom, while B is the rest of the universe. B is so large relative to A that its dynamics may be dominated by interactions between B particles. To be more precise, suppose first that the interaction between subsystems A and B suddenly disappeared at some time t . From this time on B evolves under its internal interactions, and our physical experience tells us that it will eventually, say after a characteristic relaxation time τB , reach thermal equilibrium. Therefore at time of order t + τB subsystem B will carry no memory of its interaction with A at times up to t . This implies that also in the real case, where A and B interact continuously, the state of B at time t does not depend on states of A at times earlier than t = t − τB . Consequently, dynamics in the A subspace at time t will depend on the history of A at earlier times going back only as far as this t . The relaxation time τB can be therefore identified with the memory time of the environment B. We can now state the condition for the reduced dynamics of subsystem A to be Markovian: This will be the case if the characteristic timescales of the evolution of A are slow relative to the characteristic relaxation time associated with the environment B. When this condition holds, measurable changes in the A subsystem occur slowly enough so that on this relevant timescale B appears to be always at thermal equilibrium, and independent of its historical interaction with A. Reiterating, denoting the characteristic time for the evolution of subsystem A by τA , the condition for the time evolution within the A subspace to be Markovian is τB τA .

(2.13)

(2) Gaussian stochastic Processes. The special status of the Gaussian (“normal”) distribution in reduced descriptions of physical processes stems from the central limit theorem of probability theory and the fact that in a coarsegrained description a “time evolution step” taken by the system is affected by many more or less independent random events. A succession of such evolution steps, whether Markovian or not, constitutes a Gaussian stochastic process. As a general definition, a stochastic process z(t) is Gaussian if the probability distribution of its observed values z 1 , z 2 , . . . , z n at any n time points t1 , t2 , . . . , tn (for any value of the integer n) is an n-dimensional Gaussian distribution. − 12

Pn (z 1 t1 , z 2 t2 , . . . , z n tn ) = ce

n n j =1 k=1

a j k (z j −m j )(z k −m k )

;

−∞ < z j < ∞ (2.14)

where the matrix (a j k ) = A is symmetric and positive definite (i.e., u† Au > 0 for any vector u) and where c is a normalization factor.

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1645

A Gaussian process can be Markovian. Consider, for example, the process defined by Eq. (2.8) together with

1 (z k − zl )2 P(z k , tk |zl , tl ) = √ exp − 22kl 2π kl

(2.15)

This process is Gaussian by definition since (2.14) is satisfied for any pair of times. It follows that

1 dz 1 P(z 2 , t2 |z 1 , t1 )P(z 1 , t1 |z 0 , t0 ) = 2π 221 + 210

(z 2 − z 0 )2 × exp 2 2 21 + 210

(2.16)

so the Markovian property (2.9) implies 220 = 221 + 210

(2.17)

If we further assume that kl is a function only of tk − tl , it follows that its form must be √ (t) = Dt (2.18) where D is some constant. Noting that 210 is the variance of the probability distribution in Eq. (2.15), we have found that in a Markovian process described by (2.15) this variance is proportional to the elapsed time. Comparing this result with (1.8) we see that we have just identified diffusion as a Gaussian Markovian stochastic process. Taken independently of the time ordering, the distribution (2.14) is a multivariable, n-dimensional, Gaussian distribution Pn (z 1 , z 2 , . . . , z n ). We state without proof two important properties of this distribution: z j = m j ;

δz j δz k = (A)−1

j,k

;

where

δz = z − z

(2.19)

These relationships imply that a Gaussian distribution is completely characterized by the first two moments of its variables. In particular, since in (2.14) z j and the matrix element z k are associated with the times t j and tk , respectively, −1 are seen to be the time correlation function δz(t j )δz(tk ) . (A) j,k

3.

Stochastic Equations of Motion

We have already observed that the full phase space description of a system of N particles (taking all 6N coordinates and velocities into account) requires the solution of the Newton (or Schr¨odinger) equations of motion in this phase

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A. Nitzan

space, and is deterministic in nature, while the time evolution of a small subsystem is stochastic in nature. Focusing on the latter, we would like to derive or construct appropriate equations of motion that will describe this stochastic motion. Two routes can be taken towards this goal: (1) Derive such equations from first principles. In this approach we start with the equations of motion for the entire system, and derive from it equations of motion for the subsystem under consideration. The stochastic nature of the latter stems from the fact that the state of the complementary system, “the rest of the world”, is not known precisely, and is given only in probabilistic terms. (2) Construct such equations using physical arguments, experimental observations and intuition. The resulting equations must be considered phenomenological. We have already encountered an example for such a phenomenological model in Eq. (1.2) for the random walk problem ∂ P(n, t) = kr (P(n − 1, t) − P(n, t)) + kl (P(n + 1, t) − P(n, t)) ∂t (3.1) Such an equation has been termed master equation. The Master equation is a generalized kinetic equation, giving the time-evolution of the probability distribution in terms of the transition rates between different “states” of the system. In this section we take the second route. The equations introduced and discussed below should be viewed as models for physical processes. The input to these models is the nature of the process and the choice of stochastic variables, available information such as the temperature or known average rates, some knowledge of the time and length scales involved (such as needed to choose between a Markovian or non-Markovian description) and basic principles such as any known symmetries and conservation rules, and when applicable, the requirement that at long time the system described should approach thermal equilibrium.

3.1.

Langevin Equations

3.1.1. Langevin equation for 1 particle in 1 dimension Sometimes we find it advantageous to focus our stochastic description not on the probability but on the random variable itself. This makes it possible to describe in a more direct way the source of randomness in the system and its effect on the time evolution of the interesting subsystem. In this case the basic stochastic input is not a set of transition probabilities, but the actual

Stochastic theory of rate processes

1647

effect of the “environment” on the “interesting subsystem”. Obviously this effect is random in nature, reflecting the fact that we do not have a complete microscopic description of the environment. As discussed above, we could attempt to derive these stochastic equations of motion from first principle, i.e., from the full Hamiltonian of the system + environment, Alternatively we can construct the equation of motion using intuitive arguments and as much available physical information as possible. Again, we are taking the second route. As a simple example consider the equation of motion of a particle moving in a 1-dimensional potential, x¨ = −

1 ∂ V (x) m ∂x

(3.2)

and consider the effect on this particle’s dynamics of putting it in contact with a “thermal environment”. Obviously the effect depends on the strength of interaction between the particle and it environment. A useful measure of the latter within a simple intuitive model is the friction force, proportional to the particle velocity, which acts to slow the particle: x¨ = −

1 ∂ V (x) − γ x˙ m ∂x

(3.3)

The effect of friction is to damp the particle energy. This can most easily be seen by multiplying Eq. (3.3) by m x, ˙ and using m x˙ x¨ + x˙ (∂ V (x)/∂ x) = (d/dt)[E K + E P ] = E˙ to get E˙ = −2γ E k . Here E, E K and E P are the total energy of the particle and its kinetic and potential components, respectively. Equation (3.3) thus describes a process of energy dissipation, and leads to zero energy (when measured from a local minimum on the potential surface) at infinite time. It therefore cannot in itself describe the time evolution in a thermal system. What is missing is the random “kicks” that the particle can occasionally obtain from the surrounding thermal particles. These kicks can be modeled by an additional random force in Eq. (3.3) x¨ = −

1 1 ∂ V (x) − γ x˙ + R(t) m ∂x m

(3.4)

The function R(t) describes the effects of random collisions between our particle (“system”) and the molecules of the thermal environment (“bath”). This force is obviously a stochastic process, and a full stochastic description of our system is obtained once we define its statistical nature. What can be said about the statistical character of the stochastic process R(t)? First, from symmetry arguments valid for stationary systems, R(t)=0, where the average can be either time or ensemble average. Secondly, since Eq. (3.3) seems to describe the relaxation of the system at temperature T = 0, R should be related to the finite temperature of the thermal environment. Next, at T = 0, the time evolution of x is Markovian (knowledge of x and x˙ fully

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determines the future of x), so the system-bath coupling introduced in (3.3) is of Markovian nature (i.e., what the bath does to the system at time t does not depend on history of the system or the bath; in particular, the bath has no memory of what the system did in the past). The additional, finite temperature term R(t) has to be consistent with the Markovian form of the damping term since both arise from the same source, the bath motion. Finally, in the absence of further knowledge and because R is envisioned as a combined effect of many environmental motions, it makes sense to assume that, for each time t, R(t) is a Gaussian variable, and that the stochastic process R(t) is a Gaussian process. We have already argued that the Markovian nature of the system evolution implies that the relaxation dynamics of the bath is much faster than that of the system. The bath loses its memory fast relative to the timescale of interest for the system dynamics. Still the timescale for the bath motion is not unimportant. If, for example the sign of R(t) would change infinitely fast it will make no effect on the system. Indeed, in order for a finite force R to move the particle it has to have a finite duration. It is convenient to introduce a timescale τ B , which characterizes the bath motion, and to consider an approximate picture in which R(t) is constant in the interval τ B , while R(t1 ) and R(t2 ) are independent Gaussian random variables if |t1 − t2 | ≥ (1/2)τ B . Accordingly, R(t1 )R(t1 + t) = C S(t)

(3.5)

where S(t) is 1 if |t| < (1/2)τ B , and is 0 otherwise. Since R(t) was assumed to be a Gaussian process, the first two moments specify completely its statistical nature. The assumption that the bath is fast relative to the timescales that characterize the system implies that τ B is much shorter than all timescales (inverse frequencies) derived from the potential V (x) and much smaller than the time γ −1 for the energy relaxation. In Eqs. (3.4) and (3.5), both γ and C are related to the strength of the system-bath coupling, and should therefore be somehow related to each other. In order to obtain this relation it is sufficient to consider Eq. (3.4) for the case where the potential V is a position independent constant. In this case the equation u˙ = −γ u +

1 m

R(t)

(3.6)

(here u= x˙ is the particle velocity) can be solved as a first order inhomogeneous differential equation, to yield

u(t) = u(t = 0)e

−γ t

1 + m

t 0

dt e−γ (t −t ) R(t )

(3.7)

Stochastic theory of rate processes

1649

For long times, as the system reaches equilibrium, only the second term on the right of (3.7) contributes. For the average u at thermal equilibrium this gives zero, while for u 2 we get 1 u = 2 m

t

2

dt 0

t

dt e−γ (t −t )−γ (t −t ) C S(t − t )

(3.8)

0

Since for non-vanishing integrand, |t − t"| ≤ τ B 1/γ , u 2 in Eq. (3.8) can be approximated by 1 u = 2 m

t

2

−2γ (t −t )

t

dt e 0

0

dt C S(t − t ) =

1 Cτ B 2m 2 γ

(3.9)

where to get the final result we took the limit t → ∞. Since in this limit the system should be in thermal equilibrium we have u 2 = k B T/m, whence C=

2mγ k B T τB

(3.10)

Using this result in Eq. (3.5) we find that the correlation function of the Gaussian random force R has the form R(t1 )R(t1 + t) = 2mγ k B T

S(t) τ B →0 −−−→ 2mγ k B T δ(t) τB

(3.11)

The last result indicates that, since for the system’s motion to be Markovian τ B has to be much shorter than the relevant system timescales, its actual magnitude is not important and the random force may be thought of as δ-correlated. The limiting process described above indicates that mathematical consistency requires that as τ B → 0 the second moment of the random force diverge, and the proper limiting form of the correlation function is a Dirac δ function in the time difference. Usually in analytical treatments of the Langevin equation this limiting form is convenient. In numerical solutions however, the random force is generated at time intervals t, determined by the integration routine. The random force then is generated as a Gaussian random variable with zero average and variance equal to 2mγ k B T/t. The result (3.11) shows that the requirement that the friction γ and the random force R(t) together act to bring the system to thermal equilibrium at long time, naturally leads to a relation between them, expressed by Eq. (3.11). This is a relation between the fluctuations and the dissipation in the system. It constitutes an example of the fluctuation-dissipation theorem of non-equilibrium statistical mechanics.

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3.1.2. The high friction limit The friction coefficient γ defines the timescale, γ −1 , of the thermal relaxation of the system described by (3.4). A simpler stochastic description can be obtained for a system in which this time is shorter than any other characteristic timescale of our system. This situation is often referred to as the overdamed limit. In this limit of large γ , the relaxation of the velocity is so fast that we can assume that v quickly reaches a steady state for any value of the random force R(t), i.e., v˙ = x¨ = 0. This statement is not obvious, and a supporting (though not rigorous) argument is provided below. If it is true then Eqs. (3.4) and (3.11) yield 1 dx = dt γm

−

dV + R(t) ; dx

R = 0;

R(0)R(t) = 2mγk B T δ(t) (3.12)

This is a Langevin type equation that describes strong coupling between the system and its environment. Obviously, the limit γ → 0 of deterministic motion cannot be identified here. Why can we, in this limit, neglect the acceleration term in (3.4)? Consider a particular realization of the random force in this equation and denote −(1/m)dV/dx + R(t) = F(t). Consider then Eq. (3.4) in the form v˙ = −γ v + F(t)

(3.13)

If F(t) is constant than after some transient (short for large γ ) the solution of (3.13) reaches the constant velocity state v = F/γ

(3.14)

The neglect of the v˙ term in (3.13) is equivalent to the assumption that Eq. (3.14) provides a good approximation for the solution of (3.13) even when F depends on time. To find the conditions under which this assumption holds consider the solution of (3.13) for a particular Fourier component of the time dependent force F(t) = Fω eiωt

(3.15)

Disregarding any initial transient amounts to looking for a solution of (3.13) of the form v(t) = v ω eiωt .

(3.16)

Inserting (3.15) and (3.16) into (3.13) we find vω =

Fω 1 Fω = iω + γ γ 1 + iω/γ

(3.17)

Stochastic theory of rate processes

1651

which implies F(t) (1 + O(ω/γ )) (3.18) v(t) = γ We found that Eq. (3.14) holds, with corrections of order ω/γ . It should be emphasized that this argument is not rigorous because the random part of F(t) is in principle fast, i.e., contain Fourier components with large ω. More rigorously, the transition from Eq. (3.4) to (3.12) should be regarded as coarsegraining in time that leads to a description in which the fast components of the random force are averaged to zero and the velocity distribution follows the instantaneous applied force.

3.2.

Master Equations

As discussed at the beginning of Section 3, a phenomenological stochastic evolution equation can be constructed using a model that describes the relevant states of the system and the transition rates between them. For example, in the one-dimensional random walk problem discussed in Section 1.2 we described the position of the walker using a discrete set of equally spaced points mx(m = −∞ . . . ∞) on the real axis. Denoting by P(m, t) the probability that the particle is at position m at time t and by kr and kl the probabilities per unit time (i.e., the rates) that the particle would move from a given site to the neighboring site on its right and left respectively, we obtained a kinetic equation for the time evolution of P(m, t): ∂ P(n, t) = kr (P(n − 1, t) − P(n, t)) + kl (P(n + 1, t) − P(n, t)) ∂t (3.19) This is an example of a master equation.2 More generally, transition rates between any two states can be given, and the master equation then takes the form

∂ P(m, t)

= kmn P(n, t) − knm P(m, t) (3.20) ∂t n n n= /m

n= /m

where kmn ≡ km←n is the rate to go from state n to state m. Note that we may write (3.20) in the form ∂ P(m, t)

∂P K mn P(n, t); i.e., = = KP (3.21) ∂t ∂t n 2 Many science texts refer to a 1928 paper by W. Pauli [W. Pauli, Festschrift zum 60. Geburtstage A.

Sommerfelds (Hirzel, Leipzig, 1928) p. 30] as the first derivation of this type of Kinetic equation. Pauli has used this approach to construct a model for the time evolution of a many-state quantum system, using expression derived from quantum perturbation theory for the transition rates.

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provided we define K mn = kmn for m =/ n;

K mm = −

knm

(3.22)

n

n= /m

Note that (3.22) implies that m K mn = 0 for all n. This is compatible with the fact that m P(m, t) = 1 is independent of time. The nearest neighbor random walk process is described by a special case of this master equation with K mn = kl δn,m+1 + kr δn,m−1 − (kl + kr ) δn,m

(3.23)

In what follows we consider several examples.

3.2.1. The random walk problem We first consider the 1-dimensional random walk problem described by Eq. (3.19). It can be easily seen that summing either side of this equation over all m from −∞ to ∞ yields zero, while multiplying this equation by m and m 2 then performing the summation yields ∂m = kr − kl (3.24) ∂t and ∂m 2 = 2m(kr − kl ) + kr + kl (3.25) ∂t Therefore (assuming m(t = 0) = m 2 (t = 0) = 0, i.e., that the particle starts its walk from the origin, m = 0) mt = (kr − kl )t = ( p − q)t/τ = ( p − q)N m 2 t = (kr − kl )2 t 2 + (kr + kl )t

(3.26)

and δm 2 t = m 2 t − m2t = (kr + kl )t = ( p + q) τt = ( p + q)N

(3.27)

for a walker that has executed a total of N steps during the time t = N τ , with probabilities p and q to jump to the right and to the left, respectively. We can do more by introducing the generating function, defined by F(s, t) =

∞

P(m, t)s m ;

(0 < |s| < 1)

(3.28)

m=−∞

Its usefulness stems from the fact that it can be used to generate all moments of the probability distribution according to:

∂ s ∂s

k

= m k

F(s, t) s=1

(3.29)

Stochastic theory of rate processes

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We can get an equation for the time evolution of F by multiplying the master ∞ m m and summing over all m. Using equation by s m = −∞ s P(m − 1, t) = s F(s) ∞ m and m = −∞ s P(m + 1, t) = F(s)/s leads to ∂ F(s, t) = k+ s F(s, t) + k− 1s F(s, t) − (k+ + k− )F(s, t) ∂t whose solution is F(s, t) = Ae[k+ s+(k− /s)−(k+ +k− )]t

(3.30)

(3.31)

If the particle starts from m = 0, i.e., P(m, t = 0) = δm,0 , Eq. (3.28) implies that F(s, t = 0) = 1. In this case the integration constant in Eq. (3.31) is A=1. It is easily verified that using (3.31) in Eq. (3.29) with k = 1, 2 leads to Eqs. (3.26). Using it with larger k’s leads to the higher moments of the time dependent distribution.

3.2.2. Chemical kinetics k

Consider the simple 1st order chemical reaction, A −→ B. The corresponding kinetic equation, dA = −kA ⇒ A(t) = A(t = 0)e−kt (3.32) dt describes the time evolution of the average number of molecules A in the system. Without averaging, the time evolution of this number is a random process, because the moment at which each specific A molecule transforms into B is undetermined. (The stochastic nature of radioactive decay, which is described by a similar 1st order kinetics, can be realized by listening to a Geiger counter). In addition, fluctuations from the average can also be observed if we monitor the reaction in a small enough volume, e.g., in a biological cell. We can derive a master equation for the probability P(n, t) that the number of A molecules is n at time t using considerations similar to those we used above: P(n, t + t) = P(n, t) + k(n + 1)P(n + 1, t)t − kn P(n, t)t ∂ P(n, t) = k(n + 1)P(n + 1, t) − kn P(n, t) (3.33) ⇒ ∂t Unlike the random walk problem, the rate at which the probability to be in a given state n changes depends on the state: The probability per unit time to go from n + 1 to n is k(n + 1), and the probability per unit time to go from n to n − 1 is kn. The process described by Eq. (3.33) is an example of a birthand-death process. In this particular example there is no source feeding A molecules into the system, so only death steps take place.

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The solution of Eq. (3.33) is easily achieved using the generating function method. The random variable n can take only non-negative integer values, and the generating function is therefore F(s, t) =

∞

s n P(n, t)

(3.34)

n=0

Multiplying (3.33) by s n and summing leads to ∂F ∂F ∂F ∂ F(s, t) =k − ks = k(1 − s) ∂t ∂s ∂s ∂s

(3.35)

Here we have used identities such as ∞

s n n P(n, t) = s

n=0

∂ F(s, t) ∂s

(3.36)

and ∞

s n (n + 1)P(n + 1, t) =

n=0

∂F ∂s

(3.37)

If P(n, t = 0) = δn,n0 then F(s, t = 0) = s n0 . For this initial condition the solution of Eq. (3.35) is

F(s, t) = 1 + (s − 1)e−kt

n0

(3.38)

This again gives all the moments using Eq. (3.29). For example it is easy to get nt = n o e−kt

(3.39)

δn 2 t = n o e−kt (1 − e−kt )

(3.40)

The first moment gives the familiar evolution of the average A population. The second moment shows that the variance of the fluctuations from this average is zero at t = 0 and t = ∞, and goes through a maximum at some intermediate time.

3.3.

The Fokker Planck Equation

In many practical situations the random process under observation is continuous in the sense that (a) the space of possible states x is continuous (or it can be transformed to a continuous-like representation by a coarse graining procedure), and (b) the change in the system state during a small time interval is small, i.e., if the system is found in a state x at time t then the

Stochastic theory of rate processes

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probability to find it in state y =/ x at time t + δt vanishes when δt → 0.3 When these, and some other conditions detailed below, are satisfied, we can derive a partial differential equation for the probability distribution. The result is the Fokker–Planck equation. As an example without mathematical justification consider the master equation for the random walk problem ∂ P(n, t) = kr P(n − 1, t) + kl P(n + 1, t) − (kr + kl )P(n, t) ∂t = −kr (P(n, t) − P(n − 1, t)) − kl (P(n, t) − P(n + 1, t)) (3.41) = −kr (1 − e−∂/∂n )P(n, t) − kl (1 − e∂/∂n )P(n, t) In the last step we have regarded n as a continuous variable and have used the Taylor expansion ea(∂/∂n) P(n) = 1 + a

1 ∂2 P ∂P + a 2 2 + · · · = P(n + a) ∂n 2 ∂n

(3.42)

In practical situations n is a very large number – it is the number of microscopic steps taken on the timescale of a macroscopic observations. This implies that ∂ k P/∂n k ∂ k + 1 P/∂n k + 1 .4 We therefore expand the exponential operators according to ∂

1 − e± ∂n = ∓

1 ∂2 ∂ − ∂n 2 ∂n 2

(3.43)

and neglect higher order terms, to get ∂ P(n, t) ∂ 2 P(n, t) ∂ P(n, t) = −A +B ∂t ∂n ∂n 2

(3.44)

where A = −(kr − kl ) and B = kr + kl . We can give this result a more physical form by transforming from the number-of-steps variable n to the position variable x = nx, using Pn (n) = Px (x) · x. Here x is the step length, and the subscripts n and x denote the probability in the space of position indices and the probability density on the axis x (we omit these subscripts above and below when the nature of the distribution is clear from the text). This leads to ∂ P(x, t) ∂ 2 P(x, t) ∂ P(x, t) = −v +D ∂t ∂x ∂x2

(3.45)

3 In fact we will require that this probability vanishes faster than δt when δt → 0. 4 For example if f (n) = n a then ∂ f /∂n = an a−1 which is of order f /n. The situation is less obvious in cases such as the Gausssian distribution f (n) ∼ exp((n − n)2 /2δn 2 ). Here the derivatives with respect to n adds a factor ∼ (n − n) /δn 2 that is much smaller than 1 as long as n − n n because δn 2 is of

order n.

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where v = x A and D = x 2 B. Note that we have just repeated, using a somewhat different form, the derivation of Eq. (1.4). The result (3.45) (or (1.4)) is a Fokker–Planck type equation. As already discussed below Eq. (1.4), these equations describe a drift diffusion process: For a symmetric walk, kr = kl , v = 0 and (3.45) becomes the diffusion equation with the diffusion coefficient D = x 2 (kr + kl ) = x 2 /τ. Here τ is the hopping time defined by τ = (kr + kl )−1 . When kr =/ kl the parameter v is non-zero and represents the drift velocity that may be induced by an external force that creates an asymmetry in the flow through the system. Additional insight can be obtained by rewriting Eq. (3.45) in the form: ∂ P(x, t) ∂ J (x, t) =− ∂t ∂x J (x, t) = v P(x, t) − D

(3.46a) ∂ P(x, t) ∂x

(3.46b)

Equation (3.46a) expresses the fact that the probability distribution P is a conserved quantity and therefore its time dependence can only stem from boundary fluxes. Indeed, Eq. (3.46a) implies that Pab (t) = ab P(x, t); a < b satisfies dPab (t)/dt =J (a, t) − J (b, t). This identifies J (x, t) as the probability flux at point x: J (a, t) is the flux entering the [a,b] interval (for positive J ) from a, J (b, t) – the flux leaving the interval (if positive) from b. In 1-dimension, J is of dimensionality t −1 , and when multiplied by the total number of walkers it gives the number of such walkers that pass the point x per unit time in a direction determined by the sign of J . Equation (3.46b) shows that J is a combination of the drift flux, v P, associated with the net local velocity v, and the diffusion flux, D∂ P/∂ x associated with the spatial inhomogeneity of the distribution. In a 3-dimensional system the analog of Eq. (3.46) is ∂ P(r, t) = −∇ · J(r, t) ∂t J(r, t) = v P(r, t) − D∇ P(r, t)

(3.47)

Now P(r, t) is of dimensionality l −3 and the vector J has the dimensionality l −2 t −1 . J expresses the flux of walkers (number per unit time and area) in the J direction.5 The derivation of the Fokker–Planck (FP) equation described above is far from rigorous since the conditions for neglecting higher order terms in the 5 It is important to emphasize that, again, the first of Eqs. (3.47) is just a conservation law. Integrating it over some volume s enclosed by some surface S and denoting P (t) = dP(r, t) we find, using the

divergence theorem of vector calculus, dP (t)/dt =− S dS · J(r, t) where dS is a vector whose magnitude is a surface element and its direction is a vector normal to this element in the direction outward of the volume .

Stochastic theory of rate processes

1657

expansion of exp(±∂/∂ x) where not established. A rigorous derivation of the Fokker–Planck equation for a Markov process can be obtained from the Chapman–Kolmogorov equation, Eq. (2.9), under fairly general continuity conditions that are satisfied in most physical situations. Since the latter describes a continuous stochastic process, a Fokker–Planck equation is indeed expected in the Markovian case. Here we outline this derivation for the simpler case of the high friction limit, where the Langevin equation is (c.f. Eq. (3.12)) 1 dx = dt γm

−

dV + R(t) dx

(3.48)

where R(t) again satisfies Eq. (3.11). In this case we will derive an equation for P(x, t), the probability density to find the position at x (the velocity distribution is assumed equilibrated on the timescale considered). It is convenient to redefine the time scale τ = t/(γm)

(3.49)

Denoting the random force on this timescale by ρ(τ ) = R(t), we have ρ(τ1 )ρ(τ2 ) = 2mγk B T δ(t1 − t2 ) = 2k B T δ(τ1 − τ2 ). The new Langevin equation becomes dV (x) dx =− + ρ(τ ) dt dx ρ = 0;

(3.50a)

ρ(0)ρ(τ ) = 2k B T δ(τ )

(3.50b)

The friction γ does not appear in these scaled equations, however any rate evaluated from this scheme will be inversely proportional to γ when described on the real (i.e., unscaled) time axis. The starting point of the derivation of an equation for the time evolution of the probability density P(x, t) is the statement of the fact that the integrated probability is conserved. As already discussed this implies that the time derivative of P should be given by the gradient of the flux xP, ˙ i.e., ∂ ∂ ∂ P(x, τ ) = − (x˙ P) = − ∂τ ∂x ∂x

−

∂V + ρ(τ ) P ∂x

Rewrite this in the form ∂ P(x, t) ˆ )P = (τ ∂t ∂ ∂V ˆ − ρ(τ ) (τ ) = ∂x ∂x

(3.51)

(3.52)

and integrate between τ and τ + τ to get P(x, τ + τ ) = P(x, τ ) +

τ +τ τ

dτ1 (τ1 )P(x, τ1 )

(3.53)

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The operator contains the random function ρ(τ ). Repeated iterations in the integral and averaging over all realizations of ρ lead to P(x, τ + τ ) − P(x, τ ) =

τ +τ

+

dτ1 (τ1 )

τ τ +τ

τ1

dτ2 (τ1 )(τ2 )+· · · P(x, τ )

dτ1 τ

τ

(3.54) Our aim now is to take these averages using the statistical properties of ρ and to carry out the required integrations keeping only terms of order τ . To this ˆ ˆ ) = Aˆ + Bρ(τ end we note that is of the form (τ ) where Aˆ and Bˆ are deterministic operators. Since ρ = 0 the first term in the square brackets is ˆ simply Aτ = ∂/∂ x(∂ V (x)/∂ x)τ , where the operator ∂/∂ x is understood to operate on everything on its right. The integrand in the second term inside the square brackets contains terms of the kind A A, ABρ=0 and B 2 ρ(τ1 )ρ(τ2 ). The deterministic AA terms with the double integral are of order τ 2 and may be neglected. The only contribution of order τ come from the BB terms (recall Bˆ = ∂/∂ x) using Eq. (3.50b) τ1

τ +τ

dτ1 τ

τ

∂2 dτ2 ρ(τ1 )ρ(τ2 ) 2 = ∂x

τ +τ

dτ1 k B T τ

∂2 ∂2 = k T τ B ∂x2 ∂x2 (3.55)

With a little effort we can convince ourselves that higher order terms in the 2 expansion (2.15) contribute only terms of order τ τ2 or higher. Consider for τ +τ τ1 dτ1 τ dτ2 τ dτ3 (τ1 )(τ2 )(τ3 ). example the third order term τ Evaluating the integrals with a deterministic AAA term will yield a result of order τ 3 that can be disregarded. The AAB and BBB terms appear in products with terms like ρand ρρρ which are zero. The only terms that may potentially contribute are of the type ABB. However they do not: such terms appear with a function like ρ(τ1 )ρ(τ2 ) that yields a δ-function that eliminates one of the three time integrals, however the remaining two yield a τ 2 term and do not contribute to order τ . Similar considerations show that all higher order terms in Eq. (3.54) may be disregarded. Equations (3.56) and (3.55) finally lead to ∂ P(x, τ ) = ∂τ

∂2 ∂ dV + k B T 2 P(x, τ ) ∂ x dx ∂x

(3.57)

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Transforming back to the original time variable t = γ mτ results in the Smoluchowski equation

∂ ∂ ∂V ∂ P(x, t) =D + β P(x, t); ∂t ∂x ∂x ∂x D=

β=

kB T mγ

1 kB T

(3.58) (3.59)

The Smoluchowski equation (3.58) is the overdamped limit of the original Fokker–Planck equation. When the potential V is constant it becomes the well-known diffusion equation. Equation (3.59) is a relationship between the diffusion constant D and the friction coefficient γ and is known as the Einstein relationship. Let’s consider some properties of Eq. (3.58). First note that it can be rewritten in the form ∂ ∂ P(x, t) = − J (x, t) ∂t ∂x

(3.60)

where the probability flux J is given by

∂V ∂ +β J = −D ∂x ∂x

P(x, t)

(3.61)

As discussed above (see Eq. (3.46) and the discussion below it) Eq. (3.60) has the form of a conservation rule, related to the fact that the overall probability is conserved.6 The 3-dimensional generalization of (3.58), ∂ P(r, t) = D∇ · (β∇ V + ∇)P(r, t), ∂t

(3.62)

can similarly be written as a divergence of a flux ∂ P(r, t) = −∇ · J ∂t

(3.63a)

J = −D (β∇ V + ∇) P(r, t)

(3.63b)

Again, Eq. (3.63a) is just a conservation low: it may be shown to be equivalent to the integral form dP =− dt

J(r) · ds

(3.64)

S

6 If N is the total number of particles NP(x) is the particles number density. The conservation of the inte

grated probability, i.e., dx P(x, t) = 1 is equivalent to the conservation of the total number of particles: In the process under discussion particles are neither destroyed nor created, only move in position space.

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where the probability P to be in the region of space surrounded by a closed surface S

d3 P(r)

P =

(3.65)

and where ds is a vector whose magnitude equals that of the surface element and its direction is normal to that element in the outward direction. (The flux through a surface that defined a closed subspace is defined to be positive when the flux is directed in the outward direction). Equation (3.64) expresses the fact that the change in the probability to be in the region is associated with the flux(s) that enter or leave through the boundary S of .The equivalence of the differential form (3.63a) and the integral form (3.64) of the conservation condition is a direct result of the divergence theorem of vector calculus

J(r) · ds =

S

(∇ · J)d3 r

(3.66)

S

that implies

d3r

∂ P(r, t) =− ∂t

d3r(∇ · J) ⇒

∂P = −∇ · J ∂t

(3.67)

Secondly, the flux is seen to be a sum of two terms, J = JD + JF , where JD = −D∂ P/∂ x (or, in 3D, J D = −D∇ P) is the diffusion flux, while JF = Dβ (−∂ V /∂ x) P (or , in 3D, J F = β D (−∇ V ) P) is the flux caused by the force F = −∂ V /∂ x (or F = −∇ V ). The latter corresponds to the term v P in (3.46b), where the drift velocity v is proportional to the force, i.e., JF = u FP. This identifies the mobility u as u = β D = (mγ )−1

(3.68)

Finally note that at equilibrium the flux should be zero. Equation (3.61) then leads to a Boltzmann distribution: ∂V ∂P = −β P ⇒ P(x) = const · e−β V (x) ∂x ∂x

(3.69)

The procedure to find the evolution equation for the probability distribution equivalent to the general 1D Langevin Eq. (3.4) is similar in spirit to that described for the derivation of the Smoluchowski equation and will not be repeated here. It leads to the Fokker–Planck equation

1 ∂V ∂ ∂ ∂ ∂P(x, v, t) = −v + +γ ∂t ∂x m ∂ x ∂v ∂v

kB T ∂ v+ m ∂v

P(x, v, t) (3.70)

Stochastic theory of rate processes

1661

To understand the physical content of this equation consider first the case where γ vanishes. In this case the Langevion Eq. (3.4) becomes the deterministic Newton equation (1/m) ∂ V /∂ x = −v˙ and Eq. (3.70) is seen to take the form ∂P ∂P ∂P = −x˙ − v˙ (3.71) ∂t ∂x ∂v This is, again, an expression for the conservation of probability since it implies dP ∂P dx ∂P dv ∂P = + + =0 (3.72) dt ∂t dt ∂ x dt dv Equation (3.72) is known as the Liouville equation (written here for a 1D single particle system), and is completely equivalent to the Newton equation of motion. Next consider the term proportional to γ in Eq. (3.70). This term is responsible for the system-bath coupling. We see that γ (v + (k B T /m) ∂/∂v) P(x, v, t) is an expression for a dissipative probability flux: it is a flux in the v direction that results from the coupling of our particle to the thermal environment. This coupling cannot change P if P is already an equilibrium distribution. Indeed kB T ∂ 2 v+ e−β1/2mv = 0 (3.73) m ∂v i.e., the dissipative flux vanishes at equilibrium just as the deterministic flux does. Finally note that the dissipative flux does not depend on the potential.

4.

Applications to Chemical Reactions in Condensed Phases

In this section we apply the mathematical apparatus developed above to study the dynamics of chemical reactions in condensed phase. Only two simple examples will be handled with some detail while other physical situations will only be briefly outlined. A chemical reaction may generally be viewed as taking place in two principal stages. In the first the reactants are brought together and in the second the assembled chemical system undergoes the structural/chemical change. In a condensed phase the first process involves diffusion, sometimes (e.g., when the species involved are charged) in a force field. The second stage often involves the crossing of a potential barrier. In unimolecular reactions the species that undergoes the chemical change is already assembled and only the latter process is relevant. Also, when the barrier is high the latter process is rate determining. On the other hand in bi-molecular reactions, if the barrier is low (of order k B T or less) the diffusion process that brings the reactants together may be rate determining. In this case the reaction is controlled by this diffusion process. In what follows we treat these two processes separately.

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A. Nitzan

Diffusion Controlled Reactions

To treat the case where the reaction rate is determined by the rate at which reactants approach each other we model the molecular motion by the Smoluchowski equation. To be specific we consider two species, A and B, where the A molecules are assumed to be static while the B molecules undergo diffusion characterized by a diffusion coefficient D.7 A chemical reaction in which B disappears occurs when B reaches a critical distance R ∗ from A. We will assume that A remains intact in this reaction. The macroscopic rate equation is d[B] = −k[B][A] (4.1) dt where [A] and [B] are molar concentrations. We want to relate the rate coefficient k to the diffusion coefficient D. It is convenient to define A = A[A] and B = A[B], where A is the Avogadro number and A and B are molecular number densities of the two species. In terms of these quantities Eq. (4.1) takes the form dB −k B A = (4.2) dt A Macroscopically the system is homogeneous. Microscopically however, as the reaction proceeds, the concentration of B near any A center becomes depleted and the rate becomes dominated by the diffusion process that brings fresh supply of B into the neighborhood of A. Focusing on one particular A molecule we consider the distribution of B molecules, B(r) = N B P(r) in its neighborhood. Here N B is the total number of B molecules and P(r) is the probability density for finding B molecules at position r given that an A molecule resides at the origin. P(r), and therefore B(r), satisfy the Smoluchowski equation ∂ B(r, t) = −∇ · J ∂t J = −D(β∇ V + ∇)B(r, t)

(4.3)

where V is the A − B interaction potential. At long time a steady state is established, in which B disappears as it reaches a distance R ∗ from A and a constant diffusion flux of B molecules is maintained towards A. If D is small (i.e., if the reaction is diffusion controlled) this steady state is established much before B is consumed, and we are interested in the corresponding steady state rate. For simplicity we assume that the A and B molecules are spherical, so that the interaction between them depends only on their relative distance rand 7 It can be shown that if the molecules A diffuse as well, the same formalism applies, with D replaced by

D A + DB .

Stochastic theory of rate processes

1663

the steady state distribution is spherically symmetric. This implies that only the radial part of J is non zero

J (r) = −D β

d d B(r) V (r) + dr dr

(4.4)

Furthermore, conservation of the B density in each spherical shell surrounding A implies that the integral of J(r) over any sphere centered about A is a constant independent of the sphere radius. Denoting this constant by − J0 gives J (r) = −

J0 4πr 2

(4.5)

Using this in Eq. (4.4) leads to

d d V (r) + B(r) J0 = 4π Dr β dr dr 2

(4.6)

It is convenient at this point to change variable, putting B(r) = b(r) exp(−βV (r)). Equation (2.6) then becomes J0 eβ V (r) db(r) = dr 4πD r 2

(4.7)

which may be integrated from R ∗ to ∞ to yield J0 ; b(∞) − b(R ) = 4πDλ ∗

with λ

−1

≡

∞

dr R∗

eβ V (r) r2

(4.8)

λ is a parameter of dimension length. Note that in the absence of an A − B interaction (i.e., V (r) = 0) λ = R ∗ . We will assume that V (r) → 0 as r → ∞. This implies that b(∞) = B(∞) = B, the bulk number density of the B species. Denoting B ∗ = B(R ∗ ) and V ∗ = V (R ∗ ), Eq. (4.8) finally gives

B∗ = B −

J0 ∗ e−β V 4πDλ

(4.9)

Consider now Eq. (4.2). The rate dB/dt at which B is consumed (per unit volume) is equal to the integrated B flux towards any A center, multiplied by the number of such centers per unit volume −kB

A = −4πr 2 J (r) A = −J0 A A

(4.10)

kB A

(4.11)

whence J0 =

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A. Nitzan

Using this in Eq. (4.9) leads to ∗

B = Be

−β V ∗

k 1− 4πDλA

(4.12)

If upon reactive contact, i.e., when r = R ∗ , reaction occurs instantaneously with unit probability, then B ∗ = 0. The steady state rate is then k = 4πADλ

(4.13)

The result (4.13) is the simplest form of rate expression for diffusion controlled reactions. Note again that λ = R ∗ if V (r) = 0. More generally, it is possible that B disappears at R ∗ with a rate that is proportional to B ∗ , i.e., dB A A = −k B = −k ∗ B ∗ , dt A A Using this in Eq. (4.12) leads to k=

i.e., k B = k ∗ B ∗

4π DλA 1 + (4π DλA/k ∗ e−β V ∗ )

(4.14)

(4.15) ∗

which yields the result (4.13) in the limit k ∗ e−β V → ∞. V ∗ is the interaction potential between the A and B species at the critical separation distance R ∗ (on a scale where V (∞) = 0), and can be positive or negative. A strongly positive ∗ V ∗ amounts to a potential barrier to reaction. In the limit k ∗ e−β V → 0 we ∗ get k = k ∗ e−β V . In this case however we need independent information about the rate k ∗ , of the process (usually barrier crossing) that takes place once the reactants have been assembled. This rate is, the subject of our discussion next.

4.2.

Barrier Crossing Processes

Once the reactants are assembled, the reaction may take place. This usually involves some configurational change within the assembled species. In many cases the initial and final configurations are relatively stable (if not than this configurational change will take place almost instantaneously and we would be in the diffusion controlled limit). This implies that on route between the reactant to the product configurations the system has to go through a maximum potential energy (in fact free energy). The reaction rate then becomes the rate of the barrier crossing process. If we model the process as the stochastic motion of a particle along the “reaction coordinate” with a potential characterized by minima for the reactant and product species and a barrier between them, we can treat the dynamics of this process using the Fokker–Planck equation, Eq. (3.70). Before we do this we discuss a particular situation in which knowledge of this dynamics is not needed.

Stochastic theory of rate processes

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4.2.1. Transition state theory (TST) Consider a system of particles moving in a box at thermal equilibrium, under their mutual interactions. In the absence of any external forces the system will be homogenous, characterized by the equilibrium particle density. From the Maxwell velocity distribution for the particles, we can easily calculate the equilibrium flux in any direction inside the box, say in the positive x direction, Jx = ρυx , where ρ is the density of particles and υx = 0∞ dυx υx exp(−βmυx /2). Obviously this quantity has no relation to the kinetic processes observed in the corresponding non-equilibrium system. For example, if we disturb the homogeneous distribution of particles, the rate of the resulting diffusion process is associated with the net particle flux (difference between fluxes in opposing directions) which is zero at equilibrium. There are, however, situations where the equilibrium current calculated as described above, through a carefully chosen surface, provides a good approximation for an observed non-equilibrium rate. In fact, for many chemical processes characterized by transitions through high energy barriers, this approximation is so successful that dynamical effects result in relatively small corrections to the so called transition state theory, which is based on the calculation of just that equilibrium flux. To understand the reason for this success we need to carefully examine the assumptions underlying this theory: (1) The rate can be calculated for a system in thermal equilibrium. This assumption is based on the observation that the timescale of processes characterized by high barriers is much longer than the timescale of achieving local thermal equilibrium in the reactants and product regions. The only quantity which remains in a non-equlibrium state on this long timescale is the relative concentrations of reactants and products. (2) The rate is associated with the equilibrium flux across the boundary separating reactants and products. The fact that this boundary is characterized by a high energy barrier is again essential here. Suppose that the barrier is impenetrable at first and we start with all particles in the “reactant state”, say to the left of the barrier in Fig. 1. On a very short timescale thermal equilibrium is achieved in this state. Assumption (1) assures us that this thermal equilibrium is maintained also after we lower the barrier to its actual (large) height. Assumption (2) suggests that if we count the number of barrier crossing events per unit time in the direction reactants → products using the (implied by assumption (1)) Maxwell distribution of velocities, we get a good representation of the rate. For this to be true, the event of barrier crossing has to be the deciding factor concerning the transformation of reactants to products. This is far less simple then it sounds: If the particle is infinitesimally to the left or to the right of the barrier its identity as reactant or product is

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A. Nitzan V

EB

x⫽0

xB

x

Figure 1. A schematic view of the potential surface for a unimolecular reaction. x = 0 is the bottom of the reactant well. X B is the position of the barrier.

not assured. It is only after subsequent relaxation leads it towards the bottoms of the corresponding wells that its identity is determined. Assumption (2) in fact states that all equilibrium trajectories crossing the barrier are reactive, i.e., indeed go from well defined reactants to well defined products. For this to be approximately true two conditions should be satisfied: (a) the barrier region should be small relative to the mean free path of the particles along the reaction coordinate, so that their transition from a well defined left to a well defined right is undisturbed and can be calculated from the thermal velocity, and (b) once the particles cross the barrier they relax quickly to the equilibrium reactant or product states. These conditions are inconsistent with each other: The fast relaxation required by the latter will make the mean free path small, in contrast to the requirement of the former. Indeed, this is the origin of the failure of assumption (2) in a barrierless process. Such a process proceeds by diffusion, which is defined over lengthscales large relative to the mean free path of the particles. Therefore, the transition region needed to define the “final” location of the particle to the left or to the right cannot be smaller then this mean free path. For a transition region located at the top of a high barrier we have a more favorable situation: Once the particle has crossed the barrier, it gains kinetic energy as it goes into the well region, and since the rate of energy loss due to friction is proportional to the kinetic energy, the particle may lose energy quickly and become identified as a product before it is reflected back to the reactant well. This fast energy loss from the reaction coordinate is strongly accelerated in real chemical systems where the reaction coordinate is usually strongly coupled, away from the barrier region, to other non-reactive molecular degrees of freedom.

Stochastic theory of rate processes

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We next derive the transition state rate of escape for a simple example. Consider an activated rate process represented by the escape of a particle from a 1D potential well (Fig. 1). The Hamiltonian of the particle is H=

p2 + V (x) 2m

(4.16)

where V (x) is characterized by a potential well with a minimum at x = 0 and a potential barrier peaked at x = x B > 0, separating reactants (x < x B ) from products (x > x B ) (Fig. 1). Under the above assumptions the rate coefficient for the escape of a particle out of the well is given by the forward flux at the transition state x = x B kTST = v f P (x B )

(4.17)

where P(x B )dx is the equilibrium probability that the particle is within dx of x B , P(x B ) = E B

−∞

exp (β E B ) dx exp (−βV (x))

;

E B = V (x B )

(4.18)

and where υ f is the average of the forward velocity ∞

1 dv ve−(1/2)βmv √ = 2 −(1/2)βmv 2πβm −∞ dve 2

v f = 0∞

(4.19)

Note that the fact that only half the particles move in the forward direction is taken into account in the normalization of Eq. (4.19). For a high barrier, most of the contribution to the integral in the denominator of (4.18) comes from regions of the coordinate x for which V (x) is well represented by an harmonic well, V (x) = (1/2)mω02 x 2 . Under this approximation this denominator then becomes

∞

−(1/2)βmω02 x 2

dxe −∞

=

2π βmω02

(4.20)

Inserting Eqs. (4.18)–(4.20) into (4.17) leads to kT ST =

ω0 −β E B e 2π

(4.21)

The transition state rate is of a typical Arrenius form: a product of a frequency factor that may be interpreted as the number of attempts, per unit time, that

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A. Nitzan

the particle makes to exit the well, and an activation term associated with the height of the barrier. It is important to note that it does not depend on the coupling between the molecule and its environment, only on parameters that determine the equilibrium distribution. A similar derivation can be done in a multidimensional case. For a molecular system with N nuclear degrees of freedom represented below by their coordinates x1 , x2 , . . . , the analog of the frequency ω0 are N frequencies that characterize the bottom of the well. These are the normal modes frequencies obtained from diagonalizing the force constant matrix ∂ 2 V /∂ xi ∂ x j evaluated at the position of the well bottom. The barrier in this case is a saddle point on this potential surface, and the normal modes frequencies evaluated at the saddle configuration include at least one imaginary frequencies. It is the normal mode coordinate associated with this frequency that defines the reaction coordinate at the barrier. mode frequencies evaluated Denoting the normal at the well bottom by ω0 j ; j = 1, . . . , N and the frequencies of the stable normal modes evaluated at the saddle point by ω B j ; j = 1, . . . , N − 1 the generalization of Eq. (4.21) reads

N 1 i=0 ω0i −β E B e kTST = N 2π i=1 ω Bi

(4.22)

Some observations: In view of the simplifying assumptions which form the basis for transition State Theory, its success in many practical situations may come as a surprise. Bear in mind however that transition state theory accounts quantitatively for the most important factor affecting the rate – the activation energy. Dynamical theories which account for deviations from TST often deal with effects which are orders of magnitude smaller than that associated with the activation barrier. Environmental effects on the dynamics of chemical reactions in solution are therefore often masked by solvent effect on the activation free energy. Transition State Theory is important in one additional respect: It is clear from the formulation above that the rate (4.21), or (4.22) constitutes an upper bound to the exact rate. The reason for this is that the correction factor discussed above, essentially the probability that an escaping equilibrium trajectory is indeed a reactive trajectory, is smaller than unity. This observation forms the basis to the so called variational Transition State Theory, which exploit the freedom of choosing the dividing surface between reactants and products: Since any dividing surface will yield an upper bound to the exact rate, the best choice is that which maximizes the TST rate. Corrections to TST arise from dynamical effects on the rate. They arise when the coupling to the thermal environment is either too large or too small. In the framework of the Fokker–Planck Eq. (3.70) these are the limits γ → ∞ and γ → 0, respectively, where the friction γ measures the strength of the

Stochastic theory of rate processes

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system coupling to its environment. In the first case the total outgoing flux out of the reactant region is not a good representative of the reactive flux because most of the trajectories cross the dividing surface many times – a general characteristics of a diffusive process. In the extreme strong coupling case the system cannot execute any large amplitude motion, and the actual rate vanishes even though the transition state rate is still given by the expressions derived above. In the opposite limit of a very small coupling between the system and its thermal environment it is the assumption that thermal equilibrium is maintained in the reactant region that breaks down.8 In the extreme limit of this situation the rate is controlled not by the time it take a thermal particle to traverse the barrier, but by the time it take the reactant particle to accumulate enough energy to reach the barrier. As just stated, both limits can be obtained from the general solution of Eq. (3.70). The rate obtained from this analysis is found to vanish like γ when γ → 0 and like γ −1 when γ → ∞. In the following subsection we show how this result is obtained in one of these limits, that of large friction.

4.2.2. Barrier crossing in the high friction limit The theory of unimolecular rates based on the Fokker–Planck Eq. (3.70) is known as the Kramers theory. The Fokker–Planck equation

∂P kB T ∂2 P ∂ ∂ P(x, v; t) 1 dV ∂ P = −v +γ (v P) + ∂t m dx ∂v ∂x ∂v m ∂v 2

(4.23)

with a potential V (x) characterized by a barrier (Fig. 1) is sometimes referred to in the present context as the Kramers equation. In the high friction limit this equation yields the Smoluchowski equation, Eq. (3.58), that we write in the form ∂ ∂ P(x, t) = − J (x, t) (4.24) ∂t ∂x dV ∂ +β P(x, t) (4.25) J = −D ∂x dx Denoting by A the amount of reactant in the well9 A≡

dx P(x, t)

(4.26)

well

8 If the product space is bound, another source of error is the breakdown of the assumption of fast equilibration in the product region. Unrelaxed trajectories may find their way back into the reactant subspace. 9 If N is the total number of particles, their number in the well region is NA. A is the fraction of particles in the well, or the probability per particle to be in the well.

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A. Nitzan

The rate of escape from this initial well is given by k = A−1 (−dA/dt). The reaction, i.e., the escape from the well depletes the number of particles in the well. However, the rate coefficient k is well defined as a time independent constant when it does not depend on this number. We can therefore evaluate the rate by considering an artificial situation in which A is maintained strictly constant (so the quasi-steady-state is replaced by a true one) by imposing a source at the bottom of the well and a sink outside it. For the mathematical derivation it may be more convenient to put the source at x = −∞ and the sink at x = +∞, where the geometry of Fig. 1 is considered. In fact, this source does not have to be described in detail: We simply impose the condition that the population (or probability) inside the well, far from the barrier region, is fixed, while outside the well we imposed the condition that it is zero. Under such conditions the system will approach, at long time, a steady state in which ∂ P/∂t = 0 but J =/ 0. The desired rate k is then given by J/A. To carry out this program we start from the Smoluchowski (3.58) written in the form ∂ ∂ P(x, t) = − J (x, t) ∂t ∂x

dV ∂ J = −D +β ∂x dx

(4.27)

P(x, t)

(4.28)

At steady state J is constant and

D β

d dV + dx dx

Pss (x) = − J

(4.29)

J can be viewed as one of the integration constants to be determined by the boundary conditions. Without extra effort the case D = D(x) can be considered. The equation for Pss (x) is then

β

d dV J + Pss (x) = − dx dx D(x)

(4.30)

Looking for a solution of the form Pss (x) = f (x)e−β V (x)

(4.31)

we find J β V (x) df =− e dx D(x) f (x) = − J

x ∞

β V (x ) e dx D(x )

(4.32) ∞

=J x

β V (x ) e dx D(x )

(4.33)

Stochastic theory of rate processes

1671

The choice of ∞ as an integration limit amounts to the choice of boundary condition f (x → ∞) = 0, which represents a sink at infinity. This leads to Pss (x) = J e−β V (x)

∞

dx

x

eβ V (x ) D(x )

(4.34)

Integrating both sides from x = −∞ to x = x B finally leads to 

J = k = xB dx P (x) ss −∞

x B

dxe−β V (x)

−∞

∞ x

−1

eβ V (x )  dx D(x )

(4.35)

This result can be further simplified by using the high barrier assumption, β(V (x B )−V (0)) 1, that was already recognized as a condition to get a meaningful unimolecular behavior with a time independent rate constant. In this case the largest contribution to the inner integral comes from the" neighborhood of the barrier, x =# x B , so exp [βV (x)] can be replaced by exp β(E B − (1/2) mω2B (x − x B )2 ) , while the main contribution to the outer integral comes from the

bottom of the well at x = 0, so the exp[−βV(x)] can be replaced by exp −1/2βmω2B x 2 . This then gives 

k=

∞

dxe−1/2βmω0 x

−∞

2 2

"

∞

β E B −12 mω2B (x−x B )2

dx −∞

e

D(x B )

# −1 

(4.36)

The integrals are now straightforward, and the result is (using D = (βmγ )−1 ) ωB ω0 ω B −β E B e (4.37) = kTST . k= 2πγ γ The resulting rate is expressed as a corrected TST rate. Recall that we have considered a situation where the damping γ is faster than any other characteristic rate in the system. Therefore the correction term is smaller than unity, as expected.

4.2.3. Other results The simplicity of the 1D Smoluchowski Eq. (3.58) has made it possible for us to obtain a practically exact rate expression for the large friction high barrier case. Other situations can be handled by solving the Fokker–Planck Eq. (3.70). We have already mentioned that the rate obtained in the low friction limit (γ → 0) is proportional to γ . Since the transition state theory result, Eq. (4.21), which does not depend on γ , is an upper bound to the rate, the overall picture regarding the dependence of the rate on the molecule-environment coupling as expressed by the friction has to look like Fig. 2, where the dotted line interpolates between the limiting cases.

1672

A. Nitzan k k⫽kTST⫽(ω/2π)exp[⫺βEB] k~γ

k~γ⫺1

γ Figure 2. A schematic view of the dependence of an the reaction rate k on the friction coefficient γ .

A full stochastic theory of chemical reactions has to consider further generalizations of this model. First, many dimensional models have to be considered. Secondly, the Markovian approximation, that was founded on the assumption that the bath dynamics is much faster than all other degrees of freedom in the system is not a good approximation for many molecular processes since periods of molecular frequencies are considerably shorter than characteristic timescales of a typical thermal environment. Finally, for some reactions, in particular those involving motions of hydrogen atoms at low temperature, tunneling may be important. The scope of this chapter does not allow the coverage of these fascinating issues. Rather, it provides the introduction and the necessary conceptual ingredients for further studies of these subjects.

5.8 APPROXIMATE QUANTUM MECHANICAL METHODS FOR RATE COMPUTATION IN COMPLEX SYSTEMS Steven D. Schwartz Departments of Biophysics and Biochemistry, Albert Einstein College of Medicine, New York, USA

1.

Introduction

The last 20 years have seen qualitative leaps in the complexity of chemical reactions that have been studied using theoretical methods. While methodologies for small molecule scattering are still of great importance and under active development [1], two important trends have allowed the theoretical study of the rates of reaction in complex molecules, condensed phase systems, and biological systems. First, there has been the explicit recognition that the type of state to state information obtained by rigorous scattering theory is not only not possible for complex systems, but more importantly, not meaningful. Thus, methodologies have been developed that compute averaged rate data directly from a Hamiltonian. Perhaps the most influential of these approaches has been the correlation function formalisms developed by Bill Miller et al. [2]. While these formal expressions for rate theories are certainly not the only correlation function descriptions of quantum rates [3, 4], these expressions of rates directly in terms of evolution operators, and in their coordinate space representations as Feynman Propagators, have lent themselves beautifully to complex systems because many of the approximation methods that have been devised are for Feynman propagator computation. This fact brings us to the second contributor to the blossoming of these approximate methods, the development of a wide variety of approximate mathematical methods to compute the time evolution of quantum systems. Thus the marriage of these mathematical developments has created the necessary powerful tools needed to probe systems of complexity unimagined just a few decades ago.

1673 S. Yip (ed.), Handbook of Materials Modeling, 1673–1689. c 2005 Springer. Printed in the Netherlands.

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S.D. Schwartz

The structure of this chapter will be as follows: in Section 2, we will briefly review correlation function theories of chemical reaction rates. In particular attention will be paid to specific features that allow approximation. Section 3 will begin the description of approximation methods for rate computation. The first approach studied will be that developed in our group – the quantum Kramers methodology coupled with approximate propagator computation via operator expansion and resummation. This is a fully quantum mechanical approach which gains efficiency both by approximation of the system by a simplified model, and by evaluation of the necessary quantum mechanical propagators using approximate methodologies. This approach has proven to yield results that are accurate and relatively simple to implement. The next set of approximation methods will be semiclassical computations of the propagator. Broadly classed, semiclassical mechanics is among the earliest of approximate quantum methodologies, stretching back to the JWKB approximation [5]. New realizations of semiclassical mechanics have seen a significant expansion of viability of the concept. We will include recent advances in the initial value representation (IVR) approach and wavepacket propagation including coherent state representations. Next, we will examine mixed quantum classical time evolution methodologies – so called surface hopping approaches. Finally we will briefly describe other methods for approximate quantum computation, such as transition state theory (TST). We then conclude with challenges faced by these approaches in the future.

2.

Correlation Function Theories of Rates

As was mentioned above, for a complex system, such as a large molecular reaction, a reaction in a condensed phase, or a reaction in a macromolecule such as an enzyme, one would like to compute quantum rate data at least approximately, but this will never be possible from summing and averaging state-to-state information. For this reason it is desirable to have a direct quantum rate methodology. Bill Miller recognized many years ago that in classical mechanics, TST provided such a formalism – albeit an approximate one [6]. Such considerations led to a fully quantum theory of reaction rates expressed as correlation functions. Three correlation functions were derived which correlated flux across a dividing surface with itself, a flux-side correlation function (by side we mean reactant or product), and a side–side correlation function; and in fact all three were shown to be related by simple derivative relations. The two forms that have seen the most practical application have been the flux autocorrelation function and the flux-side correlation function. The rate

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1675

of a reaction is directly expressed as a time integral of the flux autocorrelation function: k(T ) = Q r (T )−1

∞

dtC f f (t),

(1)

0

where

C f f (t) = tr e

−β H /2

Fe

−β H /2 i H t /1h

e

Fe

−i H t /1h

.

(2)

F is the quantum flux operator given by (i/1h ) H , h , and h is a quantum operator which is just the Heaviside function of the quantum position operator corresponding to a reaction coordinate or order parameter. Similarly the rate is defined via the flux side correlation function as a long time limit k(T ) = Q r (T )−1 lim C f s (t),

(3)

t →∞

with the flux side correlation function given by

C f s (t) = tr e

−β H /2

Fe

−β H /2 i H t /1h

e

he

−i H t /1h

.

(4)

Application of this formalism reduces to evaluation of matrix elements of the correlation functions that in turn reduces to matrix elements of the evolution operator, or the Feynman Propagator [7]. Thus, using these types of expressions for the rate, approximations are needed for the Feynman propagator. We now begin our brief survey of these approximation methods for both Feynman propagators, and more generally rates.

3.

The Quantum Kramers Methodology and Evolution Operator Expansions and Resummations

While there have been great strides made in the exact evaluation of Feynman propagators for multiple degrees of freedom, exact evaluation of propagators in real time is still largely limited to few degree of freedom problems. For this reason we have evaluated rates using an approximation to a full many body potential employing a system bath approach. It is known that for a purely classical system [8, 9], an accurate approximation of dynamics of a tagged degree of freedom in a condensed phase can be obtained through the use of a generalized Langevin equation. The generalized Langevin equation is given by Newtonian dynamics plus the effects of the environment in the form of a memory

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S.D. Schwartz

friction and a random force [10]. The Fluctuation–Dissipation theorem yields the relation between the friction and the random force. The quantum Kramers approach is dependent on an observation of Zwanzig [11], that given a specific set of mathematical requirements on an interaction potential for a condensed phase system, the Generalized Langevin equation can be formally related to a microscopic Hamiltonian in which the system is coupled to an infinite set of harmonic oscillators via simple bilinear coupling

P2 1 ck s P2 k + m k ωk2 qk − H = s + Vo + 2m s 2m k 2 m k ωk2 k

2

,

(5)

and a discrete spectral density gives the distribution of bath modes in the harmonic environment weighed by their coupling to the reaction coordinate J (ω) =

π ck2 [δ(ω − ωk ) − δ(ω + ωk )]. 2 k m k ωk

(6)

The direct relation between the physical system represented by the generalized Langevin equation and the harmonic Hamiltonian shown above is, via the friction on the reaction coordinate, with the spectral density obtained from the cosine transform of the time dependent friction on the reaction coordinate. Implicit in this assumption is that the friction that will be felt by the reaction coordinate is independent of the position along the reaction coordinate. Thus, the way this approach is implemented in a classical system [8], is a molecular dynamics (MD) calculation is performed with the reaction coordinate fixed at a specific location. For a reactive system, it is usually assumed that a region close to the transition state dominates the dynamics [12, 13], and so the reacting particle is clamped at the transition state. Because the rigorous transition state may not be at hand for a complex reaction, the location is often defined as the top of the potential barrier to reaction. The idea of the quantum Kramers approach we have employed is that, if a classical Langevin equation accurately describes the classical dynamics, and is equivalent to a microscopic Hamiltonian, then a quantum analogue is found by solving for the distribution of bath modes classically to yield a spectral density, but then solving the dynamics of the resulting Zwanzig Hamiltonian quantum mechanically. Thus, if one wants to evaluate a rate, then a Zwanzig Hamiltonian in which the distribution of bath modes is obtained from an accurate classical MD simulation on the exact system replaces the exact Hamiltonian in the evaluation of the flux autocorrelation function of Eq. 2. Even with this approximation, exact solution of the quantum mechanical propagator, for this many degree of freedom system, is not possible, and so we evaluate the propagator using an operator methodology we developed that allows an approximate adiabatic evolution operator to be easily corrected to an almost exact many

Approximate quantum mechanical methods

1677

body propagator. The details are given in the literature [14–19], but in short, we write a general Hamiltonian as

H = H a + H b + f (a, b),

(7)

where a and b are shorthand for any number of degrees of freedom. f (a, b) is a coupling, usually only a function of coordinates, but this is not required. The operator resummation idea rests on the fact that because these three terms are operators, the exact evolution operator may not be expressed as a product

e−i H t =/ e−i H a t e−i H b t + f (a,b),

(8)

but in fact equality may be achieved by application of an infinite order product of nth order commutators

e−i H t =/ e−i H a t e−i H b t + f (a,b)ec1 ec2 ec3 · · · .

(9)

This is usually referred to as the Zassenhaus expansion or the Baker Campbell Hausdorf theorem [20]. As an aside a symmetrized version of this expansion terminated at the C1 term results in the Feit and Fleck [21] approximate propagator. We have shown that an infinite order subset of these commutators, may be resummed exactly as an interaction propagator U (t)resum = U (t) Ha U (t) Hb + f (a,b)U −1 (t) Ha + f (a,b)U (t) Ha .

(10)

The first two terms are just the adiabatic approximation, and the second two terms the correction. For example, if we have a fast subsystem labeled by the “coordinate” a, and a slow subsystem labeled by b; then the approximate evolution operator to first order in commutators with respect to the slow suborder in system b f a b Hb , and infinite the commutators of the “fast” Hamiltonian with the coupling: f a b Ha is given by e−i(Ha +Hb + f (a,b))t /1h ≈ e−i Ha t /1h e−i(Hb + f (a,b))t /1h e+i(Ha + f (a,b))t /1h e−i Ha t /1h . (11) The advantage to this formulation is higher dimensional evolution operators are replaced by a product of lower dimensional evolution operators. This is always a far easier computation. In addition, because products of evolution operators replace the full evolution operator, a variety of mathematical properties are retained, such as unitarity, and thus time reversal symmetry. Combination of the quantum Kramers idea with the resummed evolution operators results in a largely analytic formulation for the flux autocorrelation function for a chemical reaction in a condensed phase. After a lengthy but not complex computation the quantum Kramers flux autocorrelation function has been shown to be [22, 23]. Cf =

C 0f

B1 Z bath −

∞ 0

dωκ 0f J (ω)B2 Z bath .

(12)

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S.D. Schwartz

Here C 0f is the gas phase (uncoupled) flux autocorrelation function, Z bath is the bath partition function, J (ω) is the bath spectral density (computed as described above from a classical MD computation), B1 and B2 are combinations of trigonometric functions of the frequency ω and the inverse barrier frequency, and finally κ 0f =

2 1 −iHs tc /1h s = 0|e |s = 0 . 4m 2s

(13)

As in other flux correlation function computations, tc is the complex time t − (i1h β)/2. Thus, given the quantum Kramers model for the reaction in the complex system, and the resummed operator expansion as a practical way to evaluate the necessary evolution operators needed for the flux autocorrelation function, the quantum rate in the complex system is reduced to a simple combination of gas phase correlation functions with simple algebraic functions. This methodology has now been applied to models of condensed phase rates [22, 23], proton transfer in benzoic acid crystals [24], proton transfer in polar solvent [25, 26], quantum control of reaction rates in solution [27, 28], a quantum theory of polar solvation [29], and chemical reaction rates in enzymes [30]. It can produce results of experimental accuracy, and has allowed the identification of crucial physical features not previously identified in complex systems such as enzymes. One example is found in the concept of promoting vibrations [31] we now briefly review. The Hamiltonian of Eq. (5) contains only antisymmetrically coupled environmental modes. We have described how these modes are mathematically equivalent to the well symmetrizing modes of Marcus theory [32, 33]. These modes cause well bottoms to fluctuate in depth, and so as in the case of electron transfer theory, environmental reorganization energy is required to bring well depths into equivalence, and results in the activation barrier. Symmetrically coupled modes, on the other hand move the position of well bottoms closer and further away. We first noticed such an addition to the theory was needed in our work on proton transfer in benzoic acid crystals where the physical motion is obvious: the motion of carboxylic oxygens (see Fig. 1). Motions of this type can be incorporated into the framework of the Hamiltonian of Eq. (5) with the addition of another oscillator to the Hamiltonian

P2 1 ck s P2 k + m k ωk2 qk − H = s + Vo + 2 2m s 2m 2 m k k ωk k

1 + M2 2

Cs 2 Q− M2

2

+

PQ2 2M

2

.

(14)

Approximate quantum mechanical methods

1679

H

O

O

O

O

H

C

C O

H

O

Figure 1. A benzoic acid dimer showing how the symmetric motion of the oxygen atoms will affect the potential for hydrogen transfer.

Again, employing operator expansion and resummation techniques we were able to compute rates from correlation functions for this physical system. Interestingly, various properties indicative of chemistry in regimes where quantum effects are large, such as tunneling reactions, are dramatically affected by the addition of a single symmetrically coupled mode. For example, in tunneling regimes, where kinetic isotope effects (KIEs) are expected to be large, such a symmetrically coupled vibration suppresses KIEs even in the presence of deep tunneling. This odd physical effect is due to differential corner cutting in hydrogen and deuterium reactions, and has been seen now in a variety of materials including enzymatically-catalyzed reactions. These computations have been among the first to explain this puzzling feature of enzymatic hydrogen transfer reactions: that is how can it be that certain indicators of tunneling such as secondary and tertiary KIEs strongly indicate tunneling of a reacting hydrogen, while the primary KIE on that particular hydrogen transfer is quite low, seemingly classical. A current topic of some controversy is, if in fact enzymes might have evolved to maximize tunneling in some cases as the primary mechanism of catalysis, rather than the more standard biochemical view of barrier lowering through transition state stabilization [34, 35].

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S.D. Schwartz

Semiclassical Propagators and Rate Theory

As was mentioned previously, semiclassical mechanics is one of the oldest methodologies for the computation of quantum dynamics. It has gone through a variety of incarnations, each more usable than the previous. The current revivification revolves around the IVR. The utility of this method as compared to previous methods is simple to see. All modern semiclassical treatments [36–39] end up deriving a quantum propagator from classical trajectories. Not surprising, given that the full coordinate space matrix element of the evolution operator, the Feynman propagator, is built from classical trajectories weighted by the phase factor of the action integral. The difficulty is found in the boundary conditions of these trajectories. The early approaches of semiclassical mechanics required a search in phase space for trajectories that connected initial and final states. This can easily be seen from the well-known Van Vleck propagator [40] ∂ x 2 1/2 iS(x ,x ;t )/1h e 2 1 . ∂p

K (x2 , x1 ; t) = (2πi1h ) F

(15)

1

Here, S is the classical action integral for a trajectory that goes from x1 to x2 in time t. The practical difficulty in implementation is that one must find a trajectory that evolves from an initial condition in phase space (x1 , p1 ) to the final point x2 . As is well known this is a non-linear search, difficult to implement in multiple dimensions. The IVR approach pioneered by Miller [41] and then discussed by Marcus [42] relies on exact integrations over the x2 coordinate (rather than semiclassical stationary phase integration). This in turn allows a change of variables from x2 to p1 , in other words – all initial conditions. The resulting wavefunction matrix element of the evolution operator is given by:

|∂ x˜t (x˜0 , p˜0 )/∂ p˜0 | 1/2 K n2 ,n1 (t) = dx˜0 d p˜0 (2πi1h ) F × eiSt (x˜0 , p˜0 )/1h ψ∗n2 (x(t)) ˜ ψn1 (x(0)). ˜

(16)

Thus an exact numerical integral is traded for a semiclassical stationary phase one, but the root search is avoided. This is of the utmost importance in complex, many dimensional systems, where the integrals needed are well suited to Monte Carlo methodologies, but the older “primitive” semiclassical approach would present a root search that was essentially impossible to accomplish. Other approaches to semiclassical time evolution that end up having a similar functional form are important to mention. Perhaps one of the most influential is the semiclassical propagator of Herman and Kluk [43]. Though it ends up in a similar form, the approach was derived from a very different approach.

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They investigated another semiclassical approximation – that of frozen Gaussians. This simple approach had been used to great effect by Heller et al. [44]. The idea here is to expand any initial wavefunction which must be propagated in a nonorthogonal, overcomplete set of coherent states g(x, t; xt , pt ) = N e(−γ (x−xt )

2 +i p

t (x−x t ))

.

(17)

Here N is a normalization factor, and xt and pt are the center and momentum of the wavepacket. The center and momentum may be thought of as a continuous quantum number space, and one may resolve the identity as

δ(x − x ) = (2π )−1 dx 1 d p1 g(x, t; x1 , p1 )g ∗ (x , t; x1 , p1 ).

(18)

The idea behind the frozen Gaussian approach is to assume that after an initial decomposition of a wavefunction into a swarm of Gaussians, then time evolution of the actual wavefunction is given simply be classical motion of the individual wavepackets via classical mechanics. Thus all evolution is found in the classical motions of the centers and momenta of the wavepackets. It is certainly well known that in a non-harmonic potential, a wavepacket will spread and deform, but the frozen Gaussian approach is successful because the collective motion of the wavepackets approximates the evolution of the initial wavefunction [44]. The Herman–Kluk propagator was derived as an attempt to improve on the frozen Gaussian approximation. That is, an initial wavefunction is expanded in a coherent state basis, but rather than use purely classical propagation of the centers, a semiclassically exact propagation with steepest descent integration is used to compute the evolution. The Herman–Kluk version of IVR results in the expression for the time evolved wavefunction x|e−iH t /1h |x = (2π 1h )

−1

dx0 d p0 g(x, t; xt , pt )g ∗ (x , t; x0 , p0 )eiSt (x0 , p0 )/1h C H K . (19)

Here st (x0 , p0 ) is the classical action for the unique trajectory that evolves from the initial point x0 , p0 , and C H K is the Herman–Kluk prefactor given by the monodromy matrix elements

CHK =

1 2

Mx x + M pp +

1h γ i Mq p + M pq i 1h γ

1/2

.

(20)

Here Mx x = M px

∂ xt ∂ x0

∂ pt = ∂ x0

Mx p =

∂ xt ∂ p0

∂ pt M pp = ∂ p0

.

(21)

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It is apparent from the structure of Eq. (16) that this is again an initial value semiclassical method – the propagator is built up from the coherent state basis weighted by the classical action along a trajectory beginning at a specific phase point in combination with the partial variations in position or momenta at intermediate time t with variation in initial phase point. In fact, Miller has shown a formal relation between the Herman–Kluk formulation, more standard IVR, and an exact coherent state representation [45]. It is also worth mentioning that Filinov filtering has been used to great effect to damp out unwanted oscillations in the integrand of the propagator expression [46]. As with the resummed operator expansion method, a semiclassical expression for the coordinate space matrix element of the propagator may be used in correlation function rate expressions. Currently, such methods have not been applied to chemical reaction in truly complex materials such as condensed phases or enzymes, but they have been applied to computations of rates in systems of hundreds of degrees of freedom [47], and such computations make application to real material systems seems possible in the foreseeable future. It should be pointed out that there have also been recent approaches that employ coherent state expansions to exactly rewrite the Schrodinger Eq. [48]. The motivating idea is then that either new approximations are presented given the new mathematical formulation, or the equivalent but mapped set of equations are more amenable to numerical solution than the original Schrodinger equation. There is of course a long history of such approaches – time dependent perturbation arises from rewriting the Schodinger equation in terms of time dependent basis set occupations. It is also worth mentioning the recent applications of Bohm’s quantum formulation [49]. Wyatt et al. have made great strides in this area [50]. Writing the time evolution of the wavefunction as the time evolution of a prefactor and a phase allows one to cast the quantum evolution in terms of fluid mechanics like equations, and so, one may hope to take advantage of the large body of approximations built up in this area of investigation.

5.

Mixed Quantum Classical Propagation Methods

The final general approach to rate computation in complex systems we describe are actually a broad class of approaches that are unified by a mixing of quantum and classical mechanics. This is in contradistinction to the previously described methods that handle all degrees of freedom equivalently, but approximately. The idea behind the mixed quantum classical methods is simply that there are either degrees of freedom or more generally components of a computation that are essentially classical in nature, while there are other degrees of freedom which are inherently quantum mechanical. This approach is quite appealing. No one would seriously question the lack of quantum

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character of say a large protein molecule, but it is equally well true that individual chemical events within that protein molecule could exhibit significant quantum character. In addition there are some formulations of rate theory that are inherently classical, TST being the most prominent, to which one would like to add quantum corrections to maintain simplicity, but improve accuracy. One of the more successful mixed quantum classical methods is the classical trajectory with quantum transitions method pioneered by Tully et al. [51]. This method derives from the original surface hopping idea again of Tully [52], designed to include electronic nonadiabaticity in dynamics computations. The idea here was that in the simulation of an electronically non-adiabatic system, rather than attempt to solve the Schrodinger equation for both the electronic and nuclear degrees of freedom, classical mechanics is used but equations of motion are propagated on two or more adiabatic Born–Oppenheimer potential surfaces. The inherent nonadiabaticity in the problem is included by allowing the trajectories to “hop” from one Born–Oppenheimer surface to another at regions of strong non-adiabatic coupling – avoided crossings. The probability of hops is given by the off diagonal elements of the electronic Hamiltonian. In some sense, it may seem inappropriate to include one type of quantum effect – nonadiabaticity in the electronic degrees of freedom, while ignoring all the others, tunneling, zero-point motions, interference of trajectories, but this is in fact exactly the goal. In many systems not involving for example, hydrogen transfer, one expects these other quantum effect to be minimal in magnitude, while the mixing of electronic surfaces can actually have a significant impact on the computed rate. It is well known that given the large disparity in nuclear and electronic mass, electronic non-adiabatic effect tend to be quite small and localized to specific regions of adiabatic potential surfaces, and so the localized surface hopping approach is well justified. For problems in which electronic nonadiabaticity does not play a role, there may still be degrees of freedom for which quantum mechanics is crucial, but other degrees of freedom may be treated classically. The difficulty is to find a way to mix these two inherently dissimilar descriptions, although it is well known that classical mechanics may be derived from quantum mechanics as an 1h → 0 limit, it is not clear how to take only certain degrees of freedom to this limit. In fact, there is no “basic” derivation to do this, but rather each prescription must be tested. Metiu et al. [53] proposed just such a method that proved successful in computing thermal rate constants. They also start from a correlation function definition for the rate constant. The full Hamiltonian operator is partitioned into a reaction coordinate plus coupling to all other degrees of freedom, with the rest of the degrees of freedom labeled as “spectator” coordinates. The assumption is made that the reaction coordinate Hamiltonian and the spectator Hamiltonian (soon to be taken to a classical limit) commute, and this evolution operator factors. Then the algorithm proceeds by assigning classical methodologies to all manipulations of the classical degrees of

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freedom. For example, traces over quantum operators become integrals over phase space. Classical degrees of freedom appearing in the quantum Hamiltonian as parameters, are treated as time dependent quantities evolving according to classical mechanics – a fairly common classical path assumption. There are a variety of levels of possible approximation – for example allowing the quantum systems to feedback self-consistently to the classical system, and the reader is referred to the original work. More recently, there have been advances in the surface hopping technology that allow the concept to be applied to nuclear quantum effects when there is no electronic nonadiabaticity. In many ways this is a more complex problem because the quantum effects are not limited to a specific region(s) of potential energy surfaces such as avoided crossings or seams, but rather are present through out the entire phase space. In fact, the way to handle this problem was to apply a more sophisticated surface-hopping algorithm again developed by Tully [51]. The central feature of this methodology is the “fewest switches” algorithm. This algorithm determines for a given trajectory at each step whether a switch should be made to another electronic potential energy surface. This algorithm is designed so that the statistical distribution of the different electronic states is maintained. The switches are sudden, in quantum language, all probability density is suddenly transferred from one electronic state to another, but because the actual implementation involves the propagation of swarms of trajectories, flux slowly evolves from one state to another. The method also is designed to conserve energy, and so an electronic transition requires alteration of the momentum of a specific trajectory. Once the possibility exists for quantum transitions at any point on the potential energy surface, it is a straightforward extension to allow quantum transitions for specific nuclear degrees of freedom. For example in the initial application to proton transfer in polar solvent [54], the proton is treated quantum mechanically, while all other degrees of freedom classically. Once the dynamics is run, then quantum expectation values for the position of the quantum degree of freedom may be computed as a function of time. A reaction if said to have occurred if the system is started in the reactant region, and then undergoes a transition to the product region. Coker et al. [55, 56] have been carried out extensive simulations, using the advanced surface hopping technologies for electronically non-adiabatic chemical reactions. In these reactions dozens of electronic potential energy surfaces are involved. In addition, Coker has shown formally how one may rigorously justify the Tully approach from “first principles.” [57, 58] Recently Kapral and Ciccotti have provided more formal first principles work deriving surface hopping methodologies [59]. Rossky and co-workers [60] have extensively investigated the importance of quantum decoherence in mixed quantum classical simulations. This effect needs to be “reintroduced” to quantum classical simulations, as it is certainly not present in classical mechanics. MD with

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quantum transitions has also found use in the simulation of chemical reactions in biological materials. Hammes–Schiffer and her group have recently published papers applying this method to liver alcohol dehydrogenase and dihydrofolate reductase [61, 62]. In addition to application of the Tully method, this group mixes quantum and classical mechanics in a variety of other ways as well. Computations of classical TST rates, the transmission coefficient, and KIEs augment the surface hopping computations. Quantum mechanics is included in the TST results via a partially quantum computed free energy profile from which the TST Boltzmann factor is derived. It is to be admitted that these reactions in biological materials are monumentally complex, and in this case the methods that have been devised to deal with them are a complex mixture of classical, dynamical, and statistical. Having described the previous approaches to quantum corrected TST, our final section of mixed quantum classical techniques would not be complete without a description of some of the more recent applications of TST to reactions in complex materials. TST is an inherently classical theory, but there are a variety of ways in which it can be augmented with quantum information to give fairly simple to use, but fairly accurate computations of rates. Standard gas phase TST gives the rate as k=

Q ‡ k B T −Vo /k B T e , Q reacts h

(22)

where Q ‡ is the partition function for the transition state, similarly Q reacts is that for the reactants, and the Boltzmann factor is of the gas phase barrier height. For a reaction in a complex material, a more reasonable approach is to employ the potential of mean force in the Boltzmann factor, thereby taking account of statistical fluctuations from the surrounding extended environment. In such cases, the “theory” is really more of an empirically tested algorithm, and one may write a condensed matter transition state rate expression as κk B T − G pmf /RT T e . (23) h Truhlar, Hillier and Nicoll [63] employed such a formula recently. The question of practical application then is what should the appropriate free energy of reaction be. There are a great variety of ways to decide on this. For example, one may compute this either variationally, or based on surface energetics. There are other sections of this book that describe exactly this problem, so we would not describe the alternatives in detail. The final required component is the factor labeled as κ. This factor contains everything else that is not included in TST. It might for example include barrier recrossings, or quantum tunneling through the barrier. For a complex, multidimensional system there is really no way to rigorously compute such a “tunneling correction”, but lack of derivational rigor has not prevented the application of what has become an k(T ) =

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effective method. There have been a great many tunneling schemes derived that provide rate constants of accuracy of an entirely reasonable nature. As another example, Gao and Truhlar et al. [64] have applied TST analysis to hydride transfer in an enzyme system with a potential energy surface derived via mixed quantum classical approaches with central active site atoms treated quantum mechanically, and the surrounding atoms modeled via a molecular mechanics potential.

6.

Conclusions and Future Directions

This brief section has provided an overview of some of the dominant approaches to the computation of rates in complex materials. The vast increase of complexity of systems studied has not simply been due to the advancement of mathematical technique or the increase in computer power. It has also been due to investigators increased willingness to study, using approximate techniques, problems for which there can never be an exact solution. Care must, of course, be continuously exercised to check results against appropriate experiment when possible. It is also important to realize that the use of theoretical prediction is at times more valid for distinguishing between different physical models rather than exact quantitative prediction. This is a paradigm shift from the early gas phase days of rate computation, where matching of experimental numbers was the dominant goal. This having been said, we assert that the main challenge to rate computation in particular and dynamics in general for complex material systems over the next 20 years, will be continued to development of basic methodologies such as the Flux Correlation function formalism, operator resummations for evolution operators, and the IVR semiclassical methodologies. These formal theories are not always clearly applicable to new more complex materials (as was the case for the Flux Correlation function formalism) but it is only through basic development that new more complex materials will be studied in the decades to come.

References [1] Y.M. Li and J.Z.H. Zhang, “Theoretical dynamical treatment of chemical reactions,” In: Modern Trends In Chemical Reaction Dynamics Part I: Experiment and Theory by Xueming Yang & Kopin Liu (eds.), 2003. [2] Wm.H. Miller, S.D. Schwartz, and J.W. Tromp, “Quantum mechanical rate constants for bimolecular reactions,” J. Chem. Phys., 79, 4889–4898, 1983. [3] T. Yamamoto, “Quantum statistical mechanical theory of the rate of exchange chemical reactions in the gas phase,” J. Chem. Phys., 33, 281, 1960. [4] D. Chandler, “Statistical mechanics of isomerization dynamics in liquids and the transition state approximation,” J. Chem. Phys., 2959–2970, 1978.

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[5] For an older but excellent review see: R.B. Bernstein, “Quantum effects in elastic molecular scattering,” Adv. Chem. Phys., 10, 75, 1966. [6] Wm.H. Miller, “Quantum mechanical transition state theory and a new semiclassical model for reaction rate constants,” J. Chem. Phys., 61, 1823–1834, 1974. [7] R.P. Feynman and A.R. Hibbs, Quantum Mechanics and Path Integrals, McGraw Hill, New York, 1965. [8] J.E. Straub, M. Borkovec, and B.J. Berne, “Molecular dynamics study of an isomerizing diatomic in a Lennard–Jones fluid,” J. Chem. Phys., 89, 4833, 1988. [9] B.J. Gertner, K.R. Wilson, and J.T. Hynes, “Nonequilibrium solvation effects on reaction rates for model SN2 reactions in water,” J. Chem. Phys., 90, 3537, 1988. [10] E. Cortes, B.J. West, and K. Lindenberg, “On the generalized langevin equation: classical and quantum mechanical,” J. Chem. Phys., 82, 2708–2717, 1985. [11] R. Zwanzig, “The nonlinear generalized langevin equation,” J. Stat. Phys., 9, 215, 1973. [12] Wm.H. Miller, “Quantum mechanical transition state theory and a new semiclassical model for reaction rate constants,” J. Chem. Phys., 61, 1823–1834, 1974. [13] P. Pechukas in “Dynamics of molecular collisions,” Part B Wm.H. Miller, (ed.), Plenum, New York, 1976. [14] S.D. Schwartz, “Accurate quantum mechanics from high order resummed operator expansions,” J. Chem. Phys., 100, 8795–8801, 1994. [15] S.D. Schwartz, “Vibrational energy transfer from resummed evolution operators, J. Chem. Phys., 101, 10436–10441, 1994. [16] D. Antoniou and S.D. Schwartz, “Vibrational energy transfer in linear hydrocarbon chains: new quantum results,” J. Chem. Phys., 103, 7277–7286, 1995. [17] S.D. Schwartz, “The interaction representation and non-adiabatic corrections to adiabatic evolution operators,” J. Chem. Phys., 104, 1394–1398, 1996. [18] D. Antoniou and S.D. Schwartz, “Nonadiabatic effects in a method that combines classical and quantum mechanics,” J. Chem. Phys., 104, 3526–3530, 1996. [19] S.D. Schwartz, “The interaction representation and non-adiabatic corrections to adiabatic evolution operators II: nonlinear quantum systems,” J. Chem. Phys., 104, 7985–7987, 1996. [20] W. Magnus, “On the exponential solution of differential equations for a linear operator,” Comm. Pure and Appl. Math. VII, 649, 1954. [21] M.D. Feit and J.A. Fleck Jr., “Solution of the schrodinger equation by a spectral method II: vibrational energy levels of triatomic molecules,” J. Chem. Phys., 78, 301, 1983. [22] S.D. Schwartz, “Quantum activated rates – an evolution operator approach,” J. Chem. Phys., 105, 6871–6879, 1996. [23] S.D. Schwartz, “Quantum reaction in a condensed phase – turnover behavior from new adiabatic factorizations and corrections,” J. Chem. Phys., 107, 2424–2429, 1997. [24] D. Antoniou and S.D. Schwartz, “Proton transfer in benzoic acid crystals: another look using quantum operator theory,” J. Chem. Phys., 109, 2287–2293, 1998. [25] D. Antoniou and S.D. Schwartz, “A Molecular dynamics quantum kramers study of proton transfer in solution,” J. Chem. Phys., 110, 465–472, 1999. [26] D. Antoniou and S.D. Schwartz, “Quantum Proton transfer with spatially dependent friction: phenol-amine in methyl chloride,” J. Chem. Phys., 110, 7359–7364, 1999. [27] P. Gross and S.D. Schwartz, “External field control of condensed phase reactions,” J. Chem. Phys., 109, 4843–4851, 1998.

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[28] R. Karmacharya, P. Gross, and S.D. Schwartz, “The Effect of coupled nonreactive modes on laser control of quantum wavepacket dynamics,” J. Chem. Phys., 111, 6864–6868, 1999. [29] R. Karmacharya, D. Antoniou, and S.D. Schwartz, “Nonequilibrium solvation and the quantum Kramers problem: proton transfer in aqueous glycine,” J. Phys. Chem. (Bill Miller festschrift), B105, 2563–2567, 2001. [30] D. Antoniou, S. Caratzoulas, C. Kalyanaraman, J.S. Mincer, and S.D. Schwartz, “Barrier passage and protein dynamics in enzymatically catalyzed reactions,” European Journal of Biochemistry, 269, 3103–3112, 2002. [31] D. Antoniou and S.D. Schwartz, “Internal enzyme motions as a source of catalytic activity: rate promoting vibrations and hydrogen tunneling,” J. Phys. Chem., B105, 5553–5558, 2001. [32] R.A. Marcus, “Chemical and electrochemical electron transfer theory,” Ann. Rev. Phys. Chem., 15, 155–181, 1964. [33] V. Babamov and R.A. Marcus, “Dynamics of Hydrogen Atom and Proton Transfer reactions: Symmetric Case,” J. Chem. Phys., 74, 1790, 1981. [34] V.L. Schramm, “Enzymatic transition state analysis and transition-state analogues,” methods in enzymology 308, 301–354, 1999. [35] R.L. Schowen, Transition States of Biochemical Processes, Plenum Press, New York, 1978. [36] Wm.H. Miller, “Classical Limit Quantum Mechanics and the Theory of Molecular Collisions,” Adv. Chem. Phys., 25, 69–177, 1974. [37] P. Pechukas, “Semiclassical scattering theory I,” Phys. Rev., 181, 166–173, 1969. [38] P. Pechukas, “Semiclassical scattering theory II atomic collisions,” Phys. Rev., 181, 174–181, 1969. [39] R.A. Marcus, “Theory of Semiclassical transition probabilities (S matrix) for inelastic and reactive collisions,” J. Chem. Phys., 54, 3965, 1971. [40] M.C. Gutzwiller, “Chaos in classical and quantum mechanics,” Springer New York, 1990. [41] Wm.H. Miller, “Classical S Matrix: Numerical application to inelastic collisions,” J. Chem. Phys., 53, 3578–3587, 1970. [42] R.A. Marcus, “Theory of Semiclassical transition probabilities (S matrix) for inelastic and reactive collisions,” J. Chem. Phys., 56, 3548, 1972. [43] M.F. Herman and E. Kluk, “A semiclassical justification for the use of non- spreading wavepackets in dynamics calculations,” Chem. Phys., 91, 27–34, 1984. [44] E.J. Heller, “Frozen Gaussians: a very simple semiclassical approximation,” J. Chem. Phys., 75, 2923–2931, 1981. [45] Wm.H. Miller, “On the Relation between the semiclassical initial value representation and an exact quantum expansion in time-dependent coherent States,” J. Phys. Chem. B, 106, 8132–8135, 2002. [46] V.I. Filinov, Nucl. Phys., B271, 717–725, 1986. [47] H. Wang, X. Sun, and Wm.H. Miller, “Semiclassical approximations for the calculation of thermal rate constants for chemical reactions in complex molecular systems,” J. Chem. Phys., 108, 9726–9736, 1998. [48] D.V. Shalashin and M.S. Child, “Nine-dimensional quantum molecular dynamics simulation of intramolecular vibrational energy redistribution in the CHD3 molecule with the help of coupled coherent states,” J. Chem. Phys., 119, 1961–1969, 2003. [49] D. Bohm, “A suggested interpretation of the quantum theory in terms of “hidden” variables I,” Phys. Rev., 85, 166, 1952.

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[50] E.R. Bittner and R.E. Wyatt, “Integrating the quantum Hamilton–Jacobi equations by wavefront expansion and phase space analysis,” J. Chem. Phys., 113, 8888–8897, 2000. [51] J.C. Tully, “Molecular dynamics with electronic transitions,” J. Chem. Phys., 93, 1061–1071 1990. [52] J.C. Tully, In: Wm.H. Miller (ed.), Dynamics of Molecular Collisions, Part B, Plenum, New York, pp. 217, 1976. [53] G. Wahnstrom and H. Metiu, “The calculation of the thermal rate coefficient by a method combining classical and quantum mechanics,” J. Chem. Phys., 88, 2478–2491, 1988. [54] S. Hammes-Schiffer and J.C. Tully, “Proton transfer in solution: molecular dynamics with quantum transitions,” J. Chem. Phys., 101, 4657–4667, 1994. [55] N. Yu, C.J. Margulis, and D.F. Coker, “Influence of solvation environment on excited state avoided crossings and photo-dissociation dynamics”, J. Phys. Chem. B, 105, 6728–2737, 2001. [56] C.J. Margulis and D.F. Coker, “Nonadiabatic molecular dynamics simulations of photofragmentation and geminate recombination dynamics in size-selected I2-(CO2)n cluster ions,” J. Chem. Phys., 110, 5677–5690, 1999. [57] D.F. Coker and L. Xiao, “Methods for molecular dynamics with non-adiabatic transitions,” J. Chem. Phys., 102, 496–510, 1995. [58] H.S. Mei and D.F. Coker, “Quantum molecular dynamics studies of H2 transport in water,” J. Chem. Phys., 104, 4755–4767, 1996. [59] S. Nielsen, R. Kapral, and G. Ciccotti, “Mixed quantum-classical surface hopping dynamics,” J. Chem. Phys., 112, 6543–6553, 2000. [60] B.J. Schwartz, E.R. Bittner, O.V. Prezdo, and P.J. Rossky, “Quantum decoherence and the isotope effect in condensed phase nonadiabatic molecular dynamics simulations,” J. Chem. Phys., 104, 5942–5955, 1996. [61] S.R. Billeter, S.P. Webb, P.K. Agarwal, T. Iordanov and S. Hammes-Schiffer, “Hydride transfer in liver alcohol dehydrogenase: quantum dynamics, kinetic isotope effects, and role of enzyme motion,” J.A.C.S., 123, 11262–11272, 2001. [62] P.K. Agarwal, S.R. Billeter, and S. Hammes Schiffer, “Nuclear quantum effects and enzyme dynamics in dihydrofolate reductase catalysis,” J. Phys. Chem. B, 106, 3238–3293, 2002. [63] R.M. Nicoll, I. Hillier, D.G. Truhlar, “Quantum mechanical dynamics of hydride transfer in polycyclic hydroxy keytones in the condensed phase,” J.A.C.S., 123, 1459–1463, 2001. [64] C. Alhambra, J.C. Corchado, M.L. Sanchez, J. Gao, and D.G. Truhlar, “Quantum dynamics of hydride transfer in enzyme catalysis,” J.A.C.S., 122, 8197–8203, 2000.

5.9 QUANTUM RATE THEORY: A PATH INTEGRAL CENTROID PERSPECTIVE Eitan Geva1, Seogjoo Jang2, and Gregory A. Voth3 1

Department of Chemistry, University of Michigan, Ann Arbor, Michigan 48109-1055, USA 2 Chemistry Department, Brookhaven National Laboratory, Upton, New York 11973-5000, USA 3 Department of Chemistry and Henry Eyring Center for Theoretical Chemistry, University of Utah, Salt Lake City, Utah 84112-0850, USA

1.

Introduction

The dynamics of many important processes that take place in condensed phase hosts can be described in terms of rate kinetics, with well-defined rate constants. The calculation of such rate constants from first principles has presented theoretical chemistry with an ongoing challenge. Nonequilibrium statistical mechanics provides a framework within which one can derive explicit expressions for those rate constants, via either linear response theory or Fermi’s golden rule. In both cases, one finds that the rate constants are given in terms of equilibrium correlation functions [1–6]. Those correlation functions can be evaluated with relative ease from classical molecular dynamics (MD) simulations, even for complex anharmonic many-body systems such as molecular liquids and biopolymers. However, classical mechanics is not valid in the case of many important processes, such as electron and proton transfer and vibrational relaxation. In those cases, one needs to compute the quantummechanical correlation functions. A numerically exact calculation of the latter lies far beyond the reach of currently available computer resources, due to the exponential scaling of the computational effort with the number of degrees of freedom [7, 8]. The challenge therefore lies in finding ways to compute quantum mechanical rate constants which are based on either bypassing the explicit simulation of the quantum dynamics (e.g., transition state theory (TST)), or by using reliable and computationally feasible approximate techniques for computing quantitatively accurate quantum mechanical correlation functions. 1691 S. Yip (ed.), Handbook of Materials Modeling, 1691–1712. c 2005 Springer. Printed in the Netherlands.

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Feynman’s path integral formulation of quantum mechanics [9–11] provides a powerful framework for understanding the statics and dynamics of quantum-mechanical condensed phase systems [12]. The propagator, which corresponds to the probability amplitude for going from one position to another over a time period t, is the central quantity in this formulation. The propagator is given by a sum, over all possible classical paths that satisfy those boundary conditions, of eiS/h¯ , where S is the corresponding classical action. The very same propagator can also be related to statistical-mechanical averages at thermal equilibrium, by replacing the real time, t, with an “imaginary time” −iβ h¯ (β = 1/k B T ). The propagator is known in closed form only for a small number of relatively simple systems. Unfortunately, a numerical calculation of the real-time propagator of a general many-body system is not possible in practice, due to the highly oscillatory nature of the integrand, eiS/h¯ , in the real-time path integral (the so called “sign problem”). It is possible to simplify the calculation by using semiclassical approximations in order to minimize the number of classical paths summed over, and thereby extend the applicability of this approach to somewhat larger systems [13]. However, the application of those techniques to condensed phase systems has proven to be extremely difficult, and requires rather drastic additional approximations. The situation is quite different in the case of the imaginary-time pathintegral propagator, where a numerically exact calculation is possible, even in the case of relatively complex and anharmonic many body systems. This is attributed to a fascinating equivalence between the imaginary-time propagator and the partition function of a classical system of cyclic chains of beads, which are connected by harmonic springs (each quantum degree of freedom give rise to one chain, and the effective number of beads in a chain increases as the temperature and mass are decreased). This equivalence between quantum degrees of freedom and classical chains implies that one can obtain the imaginarytime path-integral propagator from scalable classical MD and Monte Carlo (MC) simulations of the corresponding cyclic chains [14, 15]. The center of mass of those chains is known as their centroid. Feynman [9, 10] was the first to point out that the centroid can be thought of as a classical-like variable moving on a classical-like effective potential, and thereby serving as a basis for a classical-like formulation of quantum-mechanical equilibrium statistical mechanics. Several workers have later used the centroid perspective in order to construct variational approximations for the quantum partition function [16, 17]. The first application of the centroid concept to rate theory was in the context of quantum mechanical TST [18–29], and led to the development of path-integral QTST (PI-QTST) [21, 22, 24, 25, 28]. The structure of PI-QTST is similar to that of classical TST [30], except that the classical positions are replaced by the centroids of the corresponding chains. The path integral centroid approach was later extended to the calculation of time-dependent correlation functions, with the introduction of centroid

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molecular dynamics (CMD) by Cao and Voth [25, 31–37]. CMD is based on the hypothesis that the centroid follows classical-like dynamics, and that quantum effects can be incorporated by modifying the initial sampling and the force fields, as well as by representing dynamical observables by suitably defined “centroid symbols”. Important progress has been made over the last years in clarifying the assumptions underlying CMD [36–41]. CMD has also been shown to be useful and computationally feasible for realistic, complex, many-body systems (see, e.g., Refs. [42–51]. This chapter provides an overview of recent progress in the application of the path integral centroid approach to the calculation of quantum mechanical rates. Section 2 provides an overview of the formal theory of the path integral centroid and rate processes. Applications to the calculation of reaction rate constants, diffusion constants, and vibrational energy relaxation rate constants are described in Sections 4, 4.1, and 4.2, respectively. We close in Section 5, with conclusions, and some discussion of future prospects and open problems.

2. 2.1.

Formal Theory The Centroid Formulation of Quantum Statistical Mechanics

In its most recent formulation [36, 37], centroid dynamics has been shown to be based on the following phase–space operator (given here in 1D, for simplicity) ˆ c , pc ) = h¯ φ(x 2π

∞

∞

dξ −∞

ˆ

ˆ c )+iη( p− ˆ pc )−β H dηeiξ(x−x ,

(1)

−∞

where xc and pc are the centroid position and momentum, respectively, and β = 1/k B T is the inverse temperature. A central role is reserved for the trace of this operator, which corresponds to the centroid density ˆ c , pc )]. ρc (xc , pc ) = T r[φ(x

(2)

The centroid approach also associates a classical-like centroid symbol, ˆ which is Ac (xc , pc ), with each quantum dynamical observable, A(xˆ , p), defined by Ac (xc , pc ) =

ˆ c , pc ) A(xˆ , p)] ˆ T r[φ(x . ρc (xc , pc )

(3)

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The centroid density, ρc (xc , pc ), turns out to have a classical-like form, which is similar to that of the classical Boltzmann distribution ρc (xc , pc ) = e−β pc /2m ρc (xc ) ≡ e−β pc /2m e−β Vcm (xc ) . 2

2

(4)

Vcm (xc ) = − ln[ρc (xc )]/β in Eq. (4) is called the centroid potential. It is distinctly different from the classical potential and can be written in terms of a constrained imaginary-time path integral

e

−β Vcm (x c )

≡ ρc (xc ) =

2πβ h¯ 2 m



× δ xc − (β h¯ )−1

= lim

P→∞

1/2

Dx(λ) x(0)=x(β h¯ )

β h¯



dλx(λ) exp {−S[x(λ)]/h¯ }

0

2πβ h¯ 2 m

1/2

mP 2πβ h¯ 2

P/2

dx1 · · ·

dx P

P 1

× δ xc − xk exp {−S[x1 , . . . , x P ]/h¯ }, P k=1

(5)

with 1 1 S[x(λ)] = lim S[x1 , . . . , x P ] P→∞ h h¯ ¯ 1 = h¯ and

β h¯

dλ 0

1 2 m [x(λ)] ˙ + V [x(λ)] 2

P P

1

mP 1 2 S[x1 , . . . , x P ] = β (x − x ) + V (xk ) k k+1 h¯ P k=1 2β 2 h¯ 2 k=1

(6)

(7)

In Eq. (7), x P + 1 = x1 . It should be noted that ρc (xc ) is proportional to the probability density of finding a classical cyclic chain polymer consisting of P beads, which are connected by harmonic springs and subject to the potential V (x)/P, with their center of mass (the centroid) at x = xc . The centroid also corresponds to the zero-frequency normal mode of the Fourier representation of the imaginary time propagator. The constrained imaginary-time propagator in Eq. (5) can be computed using classical MD or MC simulations (PIMD and PIMC, respectively) for relatively complex many-body systems, [14, 15]. The above definitions form the basis for an exact classical-like formulation of quantum statistical mechanics, which is summarized in Table 1. The last line in Table 1 is of particular importance since it relates the classical-like

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Table 1. The centroid formulation of quantum statistical mechanics. Bˆ is an arbitrary linear combination of xˆ and pˆ Standard

ˆ

T r e−β H

Z

ˆ

T r e−β H Aˆ

ˆ A

β C Kubo B A (t)

Z

ˆ

−β H Bˆ A(t ˆ + iλ/h¯ ) dλ T r e

β

Z

0

Centroid dxc d pc ρc (xc , pc ) 2π h¯ dxc d pc ρc (xc , pc )A c (xc , pc ) 2π h¯ Z dxc d pc ρc (xc , pc )Bc A c [xc , pc ; t] 2π h¯ Z

two-time centroid correlation function with the exact Kubo-transformed quantum-mechanical correlation function. More specifically, in the case where Bˆ = xˆ or pˆ (or any linear combination of xˆ and p), ˆ the following identity holds 1 2π h¯

dxc 1 = β

β

d pc ρc (xc , pc )xc Ac [xc , pc ; t]

ˆ ˆ + i hλ) dλT r e−β H xˆ A(t . ¯

(8)

0

It should be noted that Kubo-transformed correlation functions can be related to the corresponding regular correlation functions via a well known identity [39]. However, the relationship in Eq. (8) is of little practical use since the exact time dependence of the centroid symbol Ac [xc , pc ; t] is given by

Ac (xc , pc ; t) = T r e

ˆ −i Hˆ t /h¯ φc (x c ,

pc ) i Hˆ t /h¯ ˆ e A , ρc (xc , pc )

(9)

which requires the same amount of numerical effort to compute as its standard quantum mechanical analogue. Several computationally feasible approximations of the centroid dynamics have been proposed [37]. In all of them, the centroid is assumed to move on an effective potential, obtained by averaging over the higher normal modes of the imaginary-time path. Hence, the centroid is assumed to be effectively decoupled from the higher normal modes [38]. The CMD method, which is by far the most popular of those methods, is based on the following approximation [37]. φˆc [xc (t), pc (t)] φˆc (xc , pc ) i Ht ˆ ˆ e /h¯ ≈ , (10) e−i H t /h¯ ρc (xc , pc ) ρc (xc (t), pc (t)) such that Ac [xc , pc ; t] ≈ Ac [xc (t), pc (t)].

(11)

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Here, xc (t) and pc (t) are propagated as classical-like position and momentum variables on the centroid potential, Vcm (xc ) (Cf. Eqs. (4) and (5)). We also note for later use that a centroid correlation function similar to that in Eq. (8), except for the fact that xc is replaced by xcn , where n is an integer, can be shown to be identical to the corresponding high order Kubo-transformed correlation function [39]. For example, 1 2π h¯

dxc 2 = 2 β

d pc e−β[ pc /2m+Vcm (xc )] xc2 Ac [xc , pc ; t] 2

β1

β

dβ1 0

ˆ

ˆ dβ2 T r e−β H x(−iβ ˆ ˆ . ¯ )x(−iβ ¯ ) A(t) 1 /h 2 /h

(12)

0

The CMD approximation for the correlation function in Eq. (12) can then be obtained by applying Eq. (11) to Ac [xc , pc ; t]. The calculation of correlation functions which do not have the same form as the ones considered above require one or more of the following additional approximations: [25, 32, 34, 39, 44, 52–54] (1) approximate analytic continuation; (2) a second order cumulant approximation; (3) approximate semiclassical representation of nonlinear operators; and (4) approximate classical representation of nonlinear operators. The results obtained via those methods should be used and interpreted with care. For example, the cumulant approximation will fail if the dynamics is not Gaussian, and will not lead to the correct classical limit. In a recent paper, Reichman et al. have derived a formal relationship between nonlinear centroid time correlation functions and highorder Kubo-transformed quantum correlation functions (Cf. Eq. (12)) [39]. However, in practice, a numerically exact transformation of these high-order Kubo-transformed correlation functions into standard ones, can become very difficult, and particularly so in the case of highly nonlinear and/or many-body operators.

3. 3.1.

Reaction Rate Constants Path Integral Quantum Transition State Theory (PI-QTST)

Activated barrier crossing events usually satisfy rare event statistics. In those processes, the exponentially small probability of activation, i.e., the probability of visiting a reactive configuration, is the dominant factor in determining the rate of the reaction. Advances in importance sampling methodology and MD simulation techniques, have made the calculation of classical reaction rate constants into a routine procedure, which is able to provide accurate

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results in condensed phase systems [55]. The situation is far less satisfactory when quantum effects such as tunneling and zero-point energy, are significant, which is particularly relevant in the case of proton and electron transfer reactions. Numerous theoretical attempts have been made to devise approximate, yet reliable, methods for calculating quantum-mechanical reaction rate constants. One of the most successful among these is the PI-QTST, which was proposed by Voth, Chandler, and Miller [21], and partially based on an idea due to Gillan [19, 20]. PI-QTST stands out in its unique ability to introduce an effective procedure for carrying out importance sampling in the context of quantum barrier crossing. It has also been found to be surprisingly reliable in applications to a wide range of condensed matter systems. The purpose of the present section is to introduce the essential characteristics of PI-QTST, and overview various ways of improving it that were introduced over the past decade. We will restrict ourselves, for the sake of simplicity, to the case of a Cartesian one-dimensional reaction coordinate. In the case where the coupling to the other degrees of freedom of the overall system (the bath) is weak, such that the motion along the reaction coordinate is almost ballistic near the reactive zone, one can apply the following PI-QTST rate expression kPI−QTST =

1 −β Vcm (xc∗ ) e , hβ Z r

(13)

where xc∗ corresponds to the position of the barrier top in the centroid potential, Vcm (xc ), as defined by Eq. (4), and Z r is the reactant partition function. Test calculations [21] show that this expression is virtually exact if the quantum tunneling is confined to the close vicinity of the barrier top. A simple mathematical criterion of this situation is that β < 2π/(h¯ ωb ), where ωb is the barrier frequency (i.e., the angular frequency of the inverted harmonic oscillator potential that can be fitted to the barrier top). In the low temperature deep tunneling regime of β > 2π/(h¯ ωb ), unless the potential barrier is very asymmetric, Eq. (13) still provides good qualitative estimates of the exponential factors, although the prefactors tend to be underestimated by about 2π/(β h¯ ωb ). In many respects, the role of PI-QTST in the quantum regime is similar to that of the TST in the classical regime. The evaluation of Eq. (13) does not require any quantum dynamics, and only requires a feasible imaginary time path integral simulation for the calculation of Vcm (xc ) as a function of the reaction coordinate centroid. Thus, PI-QTST represents a rather accurate and affordable approach to calculating quantum reaction rate constants in realistic condensed phase systems. It is expected to provide quantitative predictions, if the conditions underlying its validity are met, and valuable qualitative insight otherwise. In this respect, it is not surprising that attempts have been made to improve the PI-QTST in ways analogous to those employed for improving classical TST. For example, such improvements in the spirit of variational TST

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(VTST) [22, 56, 57] and the reactive flux formulation [58] have been incorporated into PI-QTST, and have been shown to lead to improvements. Also, corrections due to quantum friction, within the Kramers model, have also been incorporated into the PI-QTST [22, 59]. However, unlike the classical TST, the dynamical basis underlying the PI-QTST has not been clear from the beginning. More specifically, the assumptions underlying PI-QTST are somewhat ambiguous, which makes it difficult to systematically correct for its shortcomings in the deep tunneling regime and for the case of a very asymmetric barrier. Suggestions were made for quantum dynamical correction of Eq. (13) based on a rigorous quantum dynamical formulation [21, 60], which has not been tested even for model systems, or based on an empirical relation [61] that does not seem to have general validity. Notable progress has been made in two respects. The first is the formulation of a unified rate theory [62], which is based on a rate expression suggested by Affleck [63]. When implemented within the path integral centroid formalism, this theory provides an improved dynamical correction factor. The second is the modification of the PI-QTST for very asymmetric barriers [64], which also clarified the source of errors within the semiclassical approximation.

3.2.

A Centroid Linear-response Approach to Rate Constants

It has recently been argued that linear response theory provides an alternative route for calculating rate constants from CMD simulations, without resorting to additional approximations [40, 41]. In this approach, one starts with the system at a nonequilibrium state, which, in its most general form, is given by ˆ

ρ(0) ˆ =

e−β H ˆ + . Z

(14)

ˆ must obviously satisfy T r( ) ˆ = 0,

ˆ † = , ˆ The deviation from equilibrium, , ˆ to equiand keep ρ(0) ˆ positive. The relaxation of the quantity of interest, A, librium is then given by ˆ ˆ ˆ A(t)], δ A(t) = T r[ δ

(15)

ˆ ˆ eq . Assuming that the relaxation of δ A(t) is exponential, where δ Aˆ = Aˆ − A −kt ˆ ˆ i.e., that δ A(t)/δ A(0) = e , then leads to to the following expression for the rate constant k=−

˙ˆ ˆ A(t T r{

p )} . ˆ ˆ A} T r{ δ

(16)

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˙ˆ = i[ Hˆ , A]/ ˆ h¯ is the operator that represents the flux of A, and 0 < t p where A −1 k is a relatively short transient time preceding the onset of rate kinetics. The important point is that the actual value of the rate constant is, by definition, insensitive to the details of the initial state. This translates into flexˆ which one can take advantage of when it comes to ibility in the choice of , methods like CMD that are more directly applicable to specific types of corre ˆ ˆ = 0β dλe−(β−λ) Hˆ δ xe lation functions. More specifically, substituting

ˆ −λ H /Z , into Eq. (16) yields an expression for the rate constant in terms of a correlation function that can be evaluated directly from CMD simulations k=−

K ubo Cδx, (t) A˙ K ubo Cδx,A

=−

dxc d pc ρc (xc , pc )δxc A˙ c [xc pc ; t] . dxc d pc ρc (xc , pc )δxc δ Ac [xc pc ]

(17)

In cases where another type of perturbation to take is needed, it is possible advantage of the relationship between dxc d pc ρc (xc , pc )xcn Ac [xc pc ; t] and higher order Kubo-transformed correlation functions [39]. For example, substituting ˆ

−β H ˆ =e

Z

β1

β

dβ1 0

dβ2 δ x(−iβ ˆ ˆ ¯ )δ x(−iβ ¯) 1 /h 2 /h

(18)

0

into Eq. (16) would yield β

β1

˙ˆ dβ2 δ x(−iβ ˆ ˆ ¯ )δ x(−iβ ¯ ) A(t) 1 /h 2 /h eq β1 ˆ ˆ ˆ ¯ )δ x(−iβ ¯ )δ Aeq 1 /h 2 /h 0 dβ1 0 dβ2 δ x(−iβ 2 ˙ dxc d pc ρc (xc , pc )δxc Ac [xc pc ; t] =− . dxc d pc ρc (xc , pc )δxc2 δ Ac [xc pc ] dβ1

k = − β 0

0

(19)

To summarize, the linear response approach outlined above allows us to express any quantum mechanical rate constant in terms of a correlation function that can be obtained directly from centroid dynamics simulations. It should be noted that the above formulation is completely general, and can be incorporated into any approximate scheme for simulating the centroid dynamics. It should also be emphasized that the accuracy of the result will generally depend on the ability of the particular approximation of the centroid dynamics to capture the quantum aspects which are relevant to the specific relaxation process under consideration.

3.3.

Reaction Rate Constants from CMD Simulations

In this subsection, we consider the application of the linear response approach outlined in Section 3.2, to the calculation of reaction rate constants.

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In the case of reaction kinetics, the observed quantity is the instantaneous mole fraction of the product. Within the linear response framework, this dictates that Aˆ = h(ˆs ) in Eq. (15), where sˆ is the operator that represents the reaction coordinate and h(ˆs ) is the heaviside operator. The expectation value of h(ˆs ) corresponds to the mole fraction of the product, and its flux is given ˙ s ) = [ pδ(ˆ by h(ˆ ˆ s ) + δ(ˆs ) p]/2m. ˆ The standard reactive flux method [1, 18, 65, ˆ = 0β dλe−(β − λ) Hˆ 66] can be derived by choosing an initial state, such that

ˆ δh(ˆs )e−λ H /Z . It leads to equivalent expressions for the reaction rate constant in terms of either the flux-heaviside or flux-flux correlation functions, which cannot be obtained directly from centroid dynamics simulations. However, we may choose another initial state since the rate constant is independent ˆ = 0β dλe−(β − λ) Hˆ δ sˆ e−λ Hˆ /Z , leads to the of it. More specifically, setting

following expression for the reaction rate constant k=−

(t p ) CsKubo ˆ , Fˆ CδKubo (0) sˆ ,δ hˆ

.

(20)

The position-flux correlation function in Eq. (20) can then be evaluated directly from simulations of the centroid dynamics (cf. Eq. (17)). Quantum corrections are introduced into the resulting centroid approximation of the reaction rate constant in two distinctively different ways: • The centroid symbol of the flux, which is given explicitly in Ref. [40], requires that the initial value of the reaction coordinate centroid, sc , is sampled from a distribution of finite width. This should be contrasted with the classical analogue, where all trajectories start at the barrier top. This initial distribution of sc reflects quantum delocalization, and becomes wider as the temperature and friction are lowered. • The dynamics takes place on the centroid potential, rather than the classical potential. The barrier on the centroid potential is lower than its classical counterpart, and increasingly so as the temperature and friction are decreased, which is a reflection of tunneling and zero-point energy effects. The linear response centroid methodology outlined above for calculating the unimolecular reaction rate constant from CMD simulations has been applied to a symmetrical [40] and asymmetrical [67] double-well potential, both of which are bilinearly coupled to a bath of harmonic oscillators. It was found that CMD is able to quantitatively capture most of the quantum enhancement to the reaction rate, over a wide range of temperatures and frictions. It was also found that the reaction rate constants obtained from CMD coincide with these obtained from PI-QTST at high frictions. At intermediate frictions, the predictions of PI-QTST were found to be in a slightly better agreement with the exact result, which is likely to be accidental. However, CMD, being a

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dynamical method, could capture the turnover behavior at low frictions, which PI-QTST, being a TST method, could not. The delocalized nature of the initial distribution of the reaction coordinate centroid makes it increasingly more difficult to calculate the rate constant at very low temperatures. As the temperature decreases, the initial distribution acquires a bi-modal shape with sharp peaks on both sides of, and far away from, the barrier top. This distribution results in a situation where the large majority of the trajectories start far from the barrier top and therefore have a very small likelihood of crossing the barrier. At the same time, a small minority of the trajectories, which start in the close vicinity of the barrier top, are very likely to cross the barrier. As it turns out, the two types of trajectories make comparable contributions to the rate constant (the low likelihood of crossing the barrier is compensated for by the high probability of starting far from the barrier top). Efficient sampling of the trajectories that start in the close vicinity of the barrier top is possible via umbrella sampling. However, sampling of the trajectories that start far from the barrier top is made increasingly more demanding due to the inherent rare event statistics. In other words, more and more trajectories need to be sampled in order to obtain good statistics by having enough of them cross the barrier. Shi and Geva have shown that, at least for the case of bilinear coupling to a harmonic bath, one can overcome this problem by resorting to classicallike sampling, where all the trajectories start at the barrier top, as defined with respect to the centroid potential [68]. Although this is an approximation, it was shown that the error is given by a factor whose value is of the order of unity (except at extremely low temperatures and frictions). The expression for the reaction rate constant, within this approximation, is identical to the classical one, except for the fact that the classical potential is replaced by the centroid potential. This approximation was found to perform well when tested on the above mentioned benchmark problems [67, 68], and extends the applicability of the CMD-based method to very low temperatures that would have been difficult to access via the original method.

3.4.

Relationship Between PI-QTST and CMD

The formulation of path integral centroid dynamics [36] and identification of the mathematical procedure leading to CMD approximation, [37] raises a natural question about the possibility to understand the dynamical basis of the PI-QTST in a similar manner. The major theoretical difficulty in this effort has been that the standard expression for the reaction rate involves population–flux or flux–flux time correlation functions, all of which are nonlinear functions of coordinate. According to the formulation of real time centroid dynamics, [36] at least one physical observable should be linear in the position or momentum,

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for the calculation of time correlation function based on the centroid approach to be possible. An alternative formulation was developed by Jang and Voth, [28] where the reaction rate is defined as the steady state decay rate of the reactant population. Approximation of the reactant state in terms of an effective reactant centroid density leads to a rate expression given by the correlation between position centroid and time dependent flux operator. Application of the path integral centroid dynamics formalism, [36] and the application of the CMD approximation for the time evolution of the flux operator, then leads to the following, surprisingly simple, expression for the reaction rate constant: kCMD = κ(d)kPI−QTST ,

(21)

which involves a new quantum transmission factor defined as

κ(d) =

∞

dxc −∞

d|φˆc (xc , pc )|d ρc (xc , pc )

(22)

where the dividing surface is positioned at d. The occurrence of κ(d) in Eq. (21) and its dependence on the dividing surface is due to the approximate nature of CMD. In the high temperature limit, numerical tests show that κ(d) is independent of the value of d. In the low temperature limit, κ(d) depends on the choice of d. It was argued that the best choice is the value of d maximizing κ(d), which was based on the rationale that the CMD approximation tends to underestimate the true tunneling rate. Test calculations [28] for symmetric and cut-off asymmetric Eckart barriers showed that this choice indeed provides improved estimates in comparison to PI-QTST. As was shown in Section 3.2, the linear response CMD expression for the reaction rate constant can be approximated by the classical expression for the rate constant, as long as we replace the classical potential with the centroid potential. Taking advantage of the fact that the rate constant is independent of the initial condition, one may then revert from an expression which is given in terms of the position–flux correlation function, to one that is given in terms of the heavyside-flux correlation function. The resulting approximation coincides with that previously proposed by Schenter et al., as a way of introducing dynamical corrections into PI-QTST [58]. One can then go in the opposite direction, and establish a general relationship between the CMD-based expression for the rate constant, and PI-QTST. This result extends the relationship between CMD and PI-QTST developed by Jang and Voth in Ref. [28] to situations that involve bounded reactive potentials and coupling to a bath.

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3.5.

1703

Applications of PI-QTST

In the present subsection, we provide some examples of realistic systems where PI-QTST or its variations has been applied. Our intent is not to present an extensive list of references, but rather to emphasize the range of systems for which the application of PI-QTST is possible. Our choice of examples is thus rather subjective, but we believe these cases are representative of the range of various applications of PI-QTST to date.

3.5.1. Hydrogen diffusion on metallic surfaces One of the first applications of PI-QTST was to study hydrogen diffusion on metal surfaces [69]. Because of the general nature of the PI-QTST method, the problem could be studied for the first time in a highly realistic fashion, i.e., by simultaneously including the effects of anharmonic interactions, lattice distortions, and phonon fluctuations. It was found that these effects were large and cannot be neglected (e.g., through frozen surface approximations). Subsequently, Rick et al. [70] applied PI-QTST for the calculation of hydrogen and deuterium diffusion rates on the Pd(111) surface. They found significant quantum effects for the surface and subsurface transitions. Their calculations showed that the quantum effects for hydrogen increase the diffusion rate by a factor of two even at room temperature. Mattsson and Wahnstr¨om [71] also calculated the isotope effects for the quantum diffusion of hydrogen on the Ni(001) surface. They found the calculated results are in quantitative agreement with experimental results at room temperature and the crossover to the temperature-independent deep tunneling regime occurs at about T = 120 K.

3.5.2. Diffusion of helium atoms in zeolites Murphy et al. [72] have calculated the diffusion rate of helium atoms in zeolites. Between the two competing effects of tunneling and zero-point motion, they found that surprisingly the latter dominates below 100 K. The resulting quantum effect was found to lower the diffusion rate by about a factor of 9 at 50 K.

3.5.3. Recombination of atomic impurities in solid hydrogen Jang and Voth [73] applied PI-QTST for the calculation of the recombination rate of two lithium atoms in solid para-hydrogen at 4 K. Both the lithium atoms and the hydrogen molecules exhibit significant quantum behavior in this low temperature limit. The result of the calculation was also found to be

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consistent with experimental observations. A comparison with a calculation for classical lithium atoms showed that the quantum effects of the lithium atoms alone enhances the recombination rate by about a factor of 200 at 4 K. Similar calculations were also performed for boron atom recombination in solid hydrogen [74].

3.5.4. Proton transfer in water Schmitt and Voth, [43, 75, 76] have presented an extensive study of the hydrated proton in liquid water using their multi-state empirical valence bond model. In these studies, they performed calculatios of the PI-QTST quantum free energy barrier along the proton transfer reaction coordinate that represents the variation between the two distinct forms of the solvated proton, called the Eigen and Zundel cations. Comparison with classical calculations showed that the quantum effects lower the free energy barrier by about 0.4 kcal/mol, thereby predicting that the quantum effect enhances the proton transport rate by about a factor of two at 300 K. This enhancement was confirmed by direct CMD simulation [43, 76]. Earlier work by Lobaugh and Voth [77, 78] had already demonstrated the power and generality of the PI-QTST approach for elucidating the various complex features of proton transfer reactions in polar solvents.

3.5.5. Enzymatic reactions In principle, one of the most significant future applications of PI-QTST will be in the calculation of accurate isotope effects for enzymatic reactions, especially those involving proton or hydride transfers, as increasingly accurate potential energy functions or hybrid “QM/MM” methods become available for describing such reactions. As a recent example, Feierberg et al. [79] have applied an approximate version of PI-QTST to a rate limiting proton abstraction reaction in the enzyme glyoxalase I (see also references cited therein for other examples). They found significant isotope effects exist even at physiological temperatures. Namely, the H/D kinetic isotope effect was found to be about a factor of five. However, these authors also found a similar similar isotope effect for the uncatalyzed reaction, and thus they concluded that the quantum effect, while significant, does not have a net effect in the enzyme catalysis.

3.5.6. Intramolecular proton transfer Iftimie and Schofield [80] have applied PI-QTST to the tautomerization reaction of the enol form of malonaldehyde, and studied the effects of the

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1705

quantization of the proton and also other secondary nuclear degrees of freedom, i.e., those involving carbon and oxygen. They found that the quantization of the proton lowers the free energy barrier by 2.5 kcal/mol at 300 K. This study was similar to the earlier work published by Hinsen and Roux on proton transfer in acetylacetone [81].

3.5.7. Completely first-principles reaction rates In a “proof of concept” paper, Pavese et al. [49] showed how ab initio MD could be combined with CMD to calculate completely “first principles” rate constants and other dynamical properties (albeit at a considerable computational cost). In this approach, both the quantization of the electrons (in their adibatic ground state) and the quantization of the nuclear motions are included within a single computational algorithm. This method was subsequently used by Tuckerman and Marx [82] to study the simultaneous effects of skeleton atom quantization and proton tunneling in an intramolecular hydrogen transfer reaction.

4.

Related Topics

In this section, two related topics (self-diffusion rates and vibrational energy relaxation rates) will be discussed.

4.1.

Self-diffusion Rates

The accurate determination of quantum transport properties, specifically the self-diffusion coefficient, of many-body systems remains at the forefront of scientific effort. The velocity time auto-correlation function, which serves as one of the most relevant quantities which can be used to characterize the behavior of disordered liquids, provides one means of uncovering the self-diffusion coefficient through the well know Green–Kubo relation, 1 D= 3

∞

dtCv (t).

(23)

0

As such, a great deal of theoretical effort has been applied to developing methods which are capable of accurately determining the velocity time autocorrelation function in sytems which exhibit quantum properties. CMD time correlation functions, which are exact for quadratic potentials in the classical limit and provide well defined approximations to the exact Kubo-transformed

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time correlation functions in the anharmonic case, allow one to obtain accurate approximate quantum velocity autocorrelation functions through the frequency space identity relating the Kubo-transformed time correlation function to its quantum counterpart. The unique quantum characteristics of liquid hydrogen have led to intensive studies both experimentally and theoretically [46–48, 51, 83–95]. Liquid hydrogen exhibits important nuclear quantum effects and yet particle exchange is negligible. While much progress has been made, an accurate characterization of the dynamical behavior of such systems remains a challenging problem. The CMD method has proven to be an invaluable tool in accurately characterizing the dynamical behavior of liquid hydrogen systems. Indeed, recent CMD studies [95] have accurately determined the self-diffusion coefficients for several such systems as compared to the measured experimental values. For example, these CMD studies have determined the following self-diffusion coefficients: 0.35 Å2 /ps for liquid para-hydrogen at T = 14 K; 1.52 Å2 /ps for higher temperature, T = 25 K, liquid para-hydrogen; and 0.40 Å2 /ps for liquid ortho-deuterium at T = 20.7 K. The experimental values are 0.4, 1.6, and 0.36 Å2 /ps, respectively. The good agreement between the CMD results and those of experiment is a reflects the abiltity of CMD to account for the most prominent dynamical quantum effects where classical MD simulations are known to fail. It is also worth noting that the self-diffusion coefficients predicted by CMD are more accurate than those predicted by other theoretical methods [84, 85, 88]. The experimental determination of the actual velocity time autocorrelation functions is a difficult problem due to the difficulties in measuring the dynamic structure factor at low wave vectors. The CMD studies have provided reliable approximations for these functions as well and have shwon that hard, velocity reversing collisions are more prominent in colder, more dense, liquid para-hydrogen [42, 45, 95]. The relaxation time for the liquid para-hydrogen and ortho-deuterium systems was also established. The frequency representation of the velocity autocorrelation function further elucidated the distribution of the single phonon density of states in both liquid para-hydrogen and ortho-deuterium. Experimentally, neutron scattering techniques have become a prominent source of obtaining dynamical information for liquid hydrogen. Recently, both the coherent and incoherent dynamic structure factors of liquid hydrogen have been determined experimentally [51, 83, 92]. Celli et al. [83] compared their experimentally determined incoherent dynamic structure factor of liquid parahydrogen with that predicted by CMD and found excellent agreement for all values of wave vectors and frequencies considered. Similarly, good agreement was found between the experimentally determined coherent dynamic structure factors and those predicted by CMD over a range of wave vectors

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[48, 51]. Of particular interest was the ability of CMD to predict well-defined collective density fluctuations in liquid para-hydrogen which are consistent with quantum effects in low temperature regimes. Furthermore, CMD was able to reasonably predict the de Gennes narrowing phenomenon resulting from the reduction of collective excitations as the wave vector increases. CMD has been equally successful in determining the dynamic structure factors in liquid ortho-deuterium [95].

4.2.

Vibrational Energy Relaxation Rate Constants

The problem of vibrational energy relaxation (VER) in the condensed phase has received much attention over the last few decades [96–103]. The VER rate provides a sensitive probe of intramolecular dynamics and solute-solvent interactions, which are known to have a crucial impact on other important properties, such as chemical reactivity, solvation dynamics, and transport coefficients. The standard approach to VER is based on the Landau–Teller (LT) formula, which gives the VER rate constant in terms of the Fourier transform (FT), at the vibrational frequency, of a certain short-lived force–force correlation function (FFCF), which can be calculated from equilibrium MD simulations with a rigid solute. It should be noted that the derivation of the LT formula is based on several assumptions, namely: weak coupling between the solute and solvent, separation of time scales (such that the VER life-time is much longer than the correlation time of the FFCF), and the rotating wave approximation (RWA) [104]. The fact that the frequency of most molecular vibrations is high in the sense that h¯ ω/k B T 1, dictates that the quantum-mechanical FFCF, rather than the classical FFCF, should be used in the LT formula. The most popular approach for dealing with this difficulty is to first evaluate the FT of the classical FFCF, and then multiply the result by a frequency-dependent quantum correction factor (QCF) [96, 105–120]. Various approximate QCFs have been proposed in the literature. Unfortunately, estimates obtained from different QCFs can differ by orders of magnitude, and particularly so when high-frequency vibrations are involved. Thus, finding more rigorous ways for computing VER rate constants is clearly highly desirable. Previous attempts [44, 52–54] to apply CMD for the calculation of VER rate constant were complicated by the fact that the force in the FFCF involves a highly nonlinear function of the coordinates, and therefore cannot be directly obtained from CMD simulations without additional approximations for the nonlinear operators. Shi and Geva have recently proposed a linear–response-based approach to VER [104], which made it possible to calculate the VER rate constant from

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CMD simulations, without the introduction of further approximations [41]. As was shown in Refs. [41] and [104], a LT-type VER requires that the initial deviation from equilibrium be quadratic in the vibrational coordinate, q. To ˆ this end, one may make the following particular choice of

ˆ

−β H ˆ =e

Z

β

β1

dβ1 0

0

dβ2 δ q(−iβ ˆ ˆ ¯ )δ q(−iβ ¯) 1 /h 2 /h

ˆ −δ q(−iβ ˆ ¯ )δ q(−iβ ¯ )eq . 1 /h 2 /h

(24)

ˆ is obviously quadratic in q, and at the same time leads to an expression This

for the VER rate constant in terms of a correlation function that can be directly obtained from CMD simulations (cf. Eq. (19)). Geva and Shi have also derived a centroid LT formula for the VER rate constant, by making the same assumptions as in the derivation of the original LT formula. The resulting centroid LT formula turned out to be similar to the classical LT formula, except for the fact that the force is replaced by its centroid symbol, and the dynamics takes place on the centroid potential, rather than on the original classical potential (see Ref. [41] for more details). By virtue of this approach, the centroid VER rate constant has been calculated for several models: (1) a vibrational mode coupled to a harmonic bath, with the coupling exponential in the bath coordinates; (2) a diatomic molecule coupled to a short linear chain of Helium atoms; and (3) a “breathing sphere” diatomic molecule in a two-dimensional monoatomic Lennard–Jones liquid. It was confirmed that CMD is able to capture the main features of the quantum mechanical force–force correlation function rather well, in both time and frequency domains. At the same time, it was observed that CMD was unable to accurately predict the high-frequency tail of the quantum-mechanical power spectrum of this correlation function, which limits its usefulness for calculating VER rate constants of high-frequency molecular vibrations. Interestingly, a recent calculation of the VER rate constant which was based on the linearized-semiclassical initial-value-representation (LSC-IVR) method, revealed that the high-frequency tail of the FFCF power spectrum is dominated by a non-classical term which is very sensitive to quantum fluctuations of the force around its average value [121]. The centroid symbol of the force corresponds to the average force over the corresponding imaginary-time cyclic path, and as a result seems to miss this effect. Another fact in favor of this interpretation comes from a recently established relationship between LSCIVR and CMD [38], where it was shown that the centroid correlation function can be obtained from the LSC-IVR correlation function, by decoupling the centroid, which corresponds to the zero-frequency normal mode of the corresponding imaginary-time cyclic path, from the higher normal modes. These higher normal modes appear to be responsible for the very same quantum

Quantum rate theory: a path integral centroid perspective

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fluctuations that seem to play a key role in VER. This suggests that more advanced centroid dynamics approaches in the future should take these issues into consideration.

5.

Conclusions, Future Prospects and Open Problems

The path integral centroid approach combines the feasibility of imaginarytime path integral simulations with classical-like centroid dynamics, with the result that real-time quantum dynamical information can be obtained from imaginary time simulations. The development of these methods has taken a rather unusual step where its practical applications came first and the justifications behind these applications followed up later. These theories have clearly established the conditions under which the application of path integral centroid methods is reliable, but, on the other hand, they have also opened up theoretical issues to be resolved for these methods to be even more applicable. Our current level of understanding is that all of the existing centroid methods for actual calculations can be derived from the formulation of Ref. [36] and the application of the CMD approximation. Model calculations and test for realistic systems show that these practicable centroid methods provide reliable information on the incoherent and stationary quantum dynamics for a broad range of condensed phase systems. Considering the generality of this approach, which does not rely on the specific nature of the system, this clearly constitutes an important advance. However, as is the case with any newly developing methodology, important theoretical issues are pending, which need to be resolved in order to make the centroid methods even more accurate and general. We will therefore conclude this chapter by listing the following four prominent issues: (i) developing more accurate, yet feasible, schemes for simulating the time evolution of centroid variables; (ii) extending the methodology to include quantum statistics and electronically nonadiabatic dynamics (however, signifcant progress has already been made on both the former [122–128] and latter [129] fronts); (iii) extension of the existing centroid theories for general nonlinear time correlation functions; and (iv) generalization of the centroid theories for nonequilibrium scenarios.

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5.10 QUANTUM THEORY OF REACTIVE SCATTERING AND ADSORPTION AT SURFACES Axel Groß Physik-Department T30, TU M¨unchen, 85747 Garching, Germany

1.

Introduction

The interaction of atoms and molecules with surfaces is of great technological relevance [1]. Both advantageous and harmful processes can occur at surfaces. Catalytic reactions at surfaces represent a desired process while corrosion is an unwanted process. If light atoms and molecules such as hydrogen or helium are interacting with the surface, then quantum effects in the interaction dynamics between the incoming beam and the substrate have to be taken into account. First of all there are quantum effects in the energy transfer to the substrate vibrations, the phonons. While classically there will always be an energy loss of the incident particles to the substrate, quantum mechanically there is a certain probability for elastic scattering, i.e., without any energy transfer between the substrate and the scattered particles. This has also important consequences on the sticking probabilities of weakly bound species such as rare gases at low kinetic energies. Furthermore, in elastic scattering at a periodic surface, the wave vector parallel to the surface can only be changed by reciprocal lattice vectors because of the quasi-momentum conservation. If the de Broglie wavelength of the incident particles is of the order of the lattice spacing of the substrate, the angular distribution of the scattered particles exhibits a characteristic pattern of wellresolved reflection peaks. The resulting diffraction pattern depends only on the geometry of the surface. Therefore it has been used extensively as a tool to determine surface structures [2, 3]. I first address quantum effects in the sticking of weakly bound species, namely rare gas atoms, at surfaces. Depending on the mass of the rare gas atoms, the whole range between almost purely classical and almost purely 1713 S. Yip (ed.), Handbook of Materials Modeling, 1713–1733. c 2005 Springer. Printed in the Netherlands.

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quantum behavior can be observed [4, 5]. The lighter the atom, the higher the probability for elastic scattering and therefore the lower the trapping probability. We also briefly mention quantum effects in the adsorption dynamics which, in fact lead to a vanishing trapping probability in the limit of very low incident kinetic energies and surface temperatures [6–9]. As far as quantum effects in the dynamics of the scattered particles are concerned, I use the interaction of hydrogen with palladium surfaces as a model system. This system has been well-studied both experimentally and theoretically. Initially these studies were motivated among other reasons by the fact that bulk palladium can absorb huge amounts of hydrogen. This made it a possible candidate for a hydrogen storage device in the context of the fuel cell technology. Besides, palladium is also used as a catalyst material for hydrogenation and dehydrogenation reactions. The strong corrugation and anisotropy of the H2 /Pd(100) potential energy surface (PES) lead to significant off-specular and rotationally inelastic diffraction intensities [10]. These effects have been verified for related reactive systems [11, 12]. Furthermore, the diffraction intensities exhibit a pronounced oscillatory structure because of threshold effects and resonances in the scattering process. This structure is also visible in the quantum mechanically determined adsorption probability of H2 /Pd(100) [10, 13, 14]. This, however, has not been found in experiments yet [15]. Further quantum effects in activated systems are due to the localization and quantization of the wave function in the barrier region [16, 17] which causes a steplike structure in the reaction probabilities. This chapter is structured as follows. In Section 2, a general introduction into the phenomena occuring in the quantum scattering at surfaces is given. Then quantum effects in the trapping at surfaces and diffraction are addressed. The influence of quantum phenomena in the reaction dynamics at surfaces is discussed in Section 5. The chapter ends with some concluding remarks.

2.

Quantum Scattering at Surfaces

A schematic summary of possible collision processes in the scattering of atoms and molecules at surfaces is presented in Fig. 1. We consider a monoenergetic beam of atoms or molecules impinging on a periodic surface. In the following, we refer to both atoms and molecules by just calling them molecules. A monoenergetic incident beam is characterized by the wave vector K i = Pi /, where Pi is the initial momentum of the particles. When the incoming particles hit the surface, the substrate atoms will recoil. Therefore, classically there will always be a certain energy transfer from the molecules to the substrate. Quantum mechanically, however, there will be a certain probability for elastic scattering, i.e., with no energy transfer to the substrate. This probability is given by the so-called Debye–Waller factor.

Quantum theory of reactive scattering and adsorption at surfaces

1715

Specular Ioo

in

λ

om

c In

Diffraction Imn g be am

Inelastic

(Dissociative) adsorption

Selective adsorption

a

Figure 1.

Summary of the different collision processes in reactive scattering at surfaces.

Furthermore, if the de Broglie wavelength λ = 1/| K i | of the incident beam is of the order of the lattice spacing a, quantum effects in the momentum transfer parallel to the surface become important. In the case of elastic scattering, the component of the wave vector parallel to the surface can only be changed by a reciprocal lattice vector of the periodic surface. This means that the wave vector K f after the scattering is given by mn , K f = K i + G

(1)

mn is a reciprocal lattice vector of the periodic surface. The conservawhere G tion of the quasi-momentum parallel to the surface in elastic scattering leads to diffraction which means that there is only a discrete number of allowed scattering angles. The intensity of the elastic diffraction peak mn according to Eq. (1) is denoted by Imn . The scattering peak I00 with K f = K i is called the specular peak. From the diffraction pattern the periodicity and lattice constant of the surface can be derived. The coherent scattering of atoms or molecules from surfaces has been known as a tool for probing surface structures since 1930 [18]. In particular helium atom scattering (HAS) has been used intensively to study surface crystallography (see, e.g., [2] and references therein). The main source for the energy transfer between the impinging molecules and the substrate is the excitation and deexcitation of substrate phonons. Phonons also carry momentum. Then the conservation of quasi-momentum parallel to the surface reads mn + K f = K i + G

exch.phon.

± Q,

(2)

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is a two-dimensional phonon-momentum vector parallel to the where Q surface. The plus-signs in the sum correspond to the excitation or emission of a phonon while the minus signs represent the deexcitation or absorption of a phonon. The energy balance in phonon-inelastic scattering can be expressed as 2 K 2f 2 K i2 = + ±ω Q, j, 2M 2M exch.phon.

(3)

and where ω Q, j corresponds to the energy of the phonon with momentum Q mode index j . In fact, helium atom scattering has been used extensively in order to determine the surface phonon spectrum in one-phonon collisions via Eqs. (2) and (3) [2, 19]. The excitation of phonons usually leads to a reduced normal component of the kinetic energy of the back-scattered atoms or molecules. Thus the reflected beam is shifted in general to larger angles with respect to the surface normal compared to the angle of incidence. The resulting supraspecular scattering is indicated in Fig. 1 as the inelastic reflection event. In the case of the scattering of weakly interacting particles at smooth surfaces, often resonances in the intensity of the specular peak as a function of the angle of incidence are observed [20]. These so-called selective adsorption resonances which are also indicated in Fig. 1 occur when the scattered particle can make a transition into one of the discrete bound state of the adsorption potential [21]. This is only possible if temporarily the motion of the particle is entirely parallel to the surface. The interference of different possible paths along the surface causes the resonance effects. Energy and momentum conservation yields the selective adsorption condition mn )2 2 K i2 2 ( K i + G = − |El |, 2M 2M

(4)

where El is a bound level of the adsorption potential. From the scattering resonances, bound state energies can be obtained using Eq. (4) without any detailed knowledge about the scattering process. The coherent elastic scattering of molecules is more complex than atom scattering. Additional peaks may appear in the diffraction pattern. They are a consequence of the fact that in addition to parallel momentum transfer the internal degrees of freedom of the molecule, rotations and vibrations, can be excited during the collision process. The total energy balance in the molecular scattering is 2 K 2f 2 K i2 = + E rot + E vib + ±ω Q, j. 2M 2M exch.phon.

(5)

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1717

Usually the excitation of molecular vibrations in molecule-surface scattering is negligible, in contrast to the phonon excitation. This is due to the fact that the time-scale of the molecular vibrations is usually much shorter than the scattering time or the rotational period. This leads to high frequencies of the molecular vibrations whose energies are too high to become excited in a typical scattering experiment. Molecular rotations, on the other hand, can be excited rather efficiently in the scattering at highly corrugated and anisotropic surfaces. Because of the energy conservation, the rotational excitation in scattering reduces the kinetic energy perpendicular to the surface. This leads to additional peaks in the diffraction spectrum, the rotationally inelastic diffraction peaks. Experimentally, rotationally inelastic diffraction of hydrogen molecules has been first observed in the scattering at inert ionic solids such as MgO [22] or NaF [23]. At metal surfaces with a high barrier for dissociative adsorption, the molecules are scattered at the tails of the metal electron density which are usually rather smooth. In addition, the interaction of the molecules with these tails does not depend very strongly on the orientation of the molecules. Hence relatively weak diffraction and hardly any rotationally inelastic diffraction has been observed for, e.g., the scattering of H2 from Cu(001) [24, 25]. This is different for the case of HD scattering, where the displacement of the center of mass from the center of the charge distribution leads to a strong rotational anisotropy [26]. At reactive surfaces where non-activated adsorption is possible, the scattering occurs rather close to the surface where the potential energy surface is already strongly corrugated and anisotropic. For such systems, rotationally inelastic peaks in the diffraction pattern have been clearly resolved experimentally [11, 12] and predicted theoretically in six-dimensional quantum dynamical calculations [10] as is discussed below. At reactive surfaces, the particles can of course also adsorb. As it is indicated in Fig. 1, molecules can adsorb both molecularly which means intact or dissociatively. In the case of the atomic or molecular adsorption, the particles can only remain trapped at the surface if their initial kinetic is transfered to the surface and dissipated. For light projectiles, the quantum nature of the substrate phonons becomes important. This is the topic of Section 3.

3.

Quantum Effects in the Trapping at Surfaces

Let us consider an atom impinging on a surface. Even in the absence of any chemical binding, there will always be an attractive interaction between the surface and the atom due to the van der Waals forces [27]. Let us further assume that there is no energetic barrier for the access of the atomic adsorption well. A particle impinging on a surface can only become trapped

1718

A. Groß

in an attractive adsorption well if it transfers its entire initial kinetic energy to the surface because then it cannot escape back to the gas phase. Hence the sticking probability can be expressed as ∞

PE () d,

S(E) =

(6)

E

where PE () is the probability that an incoming particle with kinetic energy E transfers the energy to the surface. If the adsorption process is treated purely classical, no matter how small the adsorption well, no matter how small the mass ratio between the impinging atom and the substrate oscillator, for E → 0 and Ts → 0 the sticking probability will always reach unity if there is no barrier before the adsorption well. This is due to the fact that every impinging particle will transfer energy to the substrate at zero temperature. In the limit of zero initial kinetic energy any energy transfer will be sufficient to keep the particle in the adsorption well. Quantum mechanically, however, there is a non-zero probability for elastic scattering at the surface. Hence the sticking probabilities should become less than unity in the zero-energy limit, in particular for light atoms impinging on a surface. This has indeed been observed in the sticking of rare gas atoms at cold Ru(0001) surfaces [4, 5]. In order to reproduce elastic scattering, the quantum nature of the phonon system has to be taken into account. In the simplest approach, the substrate phonons can be modeled by an ensemble of independent quantum surface oscillators. Since the oscillators are assumed to be independent, the essential physics can be captured by just considering an atomic projectile interacting via linear coupling with a single surface oscillator. In the so-called trajectory approximation, the particle’s motion is treated classically. Assuming that the motion of the atom is hardly influenced by the excitation of the surface oscillator, the equations of motion are solved without taking the coupling to the oscillator into account. The classical trajectory then introduces a time-dependent force in the Hamiltonian of the oscillator. In this forced oscillator model [28], the energy transfer probability PE () can be evaluated. In fact, a compact expression can be derived for the energy distribution in the scattering of an atom at a system of phonon oscillators with a Debye spectrum at a temperature Ts [29, 30]. Assuming some analytical form for the potential, this expression depends on a small set of parameters such as the potential well depth, the potential range, the mass of the surface oscillator and the surface Debye temperature. This model was used in order to reproduce the measured sticking probabilities of rare gas atoms on a Ru(001) surface at a temperature of Ts = 6.5 K [5]. A comparison between the measured and calculated sticking probabilities for Ne, Ar, Kr, and Xe on Ru(001) is shown in Fig. 2. The lighter the atoms, the

Quantum theory of reactive scattering and adsorption at surfaces

1719

smaller the sticking probability. At small energies, the sticking probabilities do not reach unity due to the quantum nature of the substrate phonons except for the heaviest rare gas atom Xe. Indeed, attempts to reproduce the measured sticking probabilities with purely classical methods have failed, at least for Ne and Ar [4, 5]. A classical treatment of the solid is only appropriate if the energy transfer to the surface is large compared to the Debye energy of the solid [6]. At even lower kinetic energies than reached in the experiments [5] shown in Fig. 2, the quantum nature of the impinging particles cannot be neglected any longer. Hence the trajectory approximation cannot be applied any more. In fact, in the limit E → 0 the de Broglie wavelength of the incoming particle tends to infinity. In the case of a short-range attractive potential this means that the amplitude of the particle’s wave function vanishes in the attractive region [6, 7]. Thus there is no coupling and consequently no energy transfer between the particle and the substrate vibrations. Therefore the quantum mechanical sticking probability also vanishes for E → 0. However, in order to see this effect extremely small kinetic energies corresponding to a temperature below 0.1 K are required [6]. Nevertheless, this quantum phenomenon in the sticking at surfaces has been verified experimentally for the adsorption of atomic hydrogen on thick liquid 4 He films [31]. There is yet another effect that also leads to zero sticking at very low energies. Quantum mechanically particles can also be reflected at attractive

1.0

∗∗ ∗

Trapping probabilty

0.8

0.6

∗∗

∗ ∗∗Xe

∗ ∗

∗ Kr ∗∗∗ ∗ Experiment ∗ Ar ∗∗ Theory ∗ ∗ ∗ ∗∗ 0.2 ∗ ∗ ∗ ∗∗ ∗ ∗∗ Ne ∗ ∗ ∗∗∗ ∗ ∗ 0.0 ∗0.02∗ 0.00 0.04 0.06 0.4

0.08

Kinetic energy (eV)

Figure 2. Sticking probability of rare gas atoms on Ru(0001) at a surface temperature of Ts = 6.5 K. Stars (*): experiment; lines: theoretical results obtained with the forced oscillator model (after [5], not all measured data points are included).

1720

A. Groß

parts of the potential. If the potential falls off asymptotically faster than 1/Z 2 , then the reflection amplitude R exhibits the universal behavior [9, 32] |R| −→ 1 − bk, k→0

(7)

where k is the wave number corresponding to the asymptotic kinetic energy E =2 k 2 /2M. This means that in the low energy limit the reflection probability |R|2 goes to unity even if the particle does not reach a classical turning point. In fact, such a quantum reflection has been observed in the scattering of an ultracold beam of metastable neon atoms from silicon and glass surfaces [8]. In order to reproduce the measured reflectivities, an 1/Z 4 dependence of the potential has to be assumed [8, 9], where Z is the distance to the surface. This indicates that the atoms are scattered at the long-range tail of the so-called Casimir–van der Waals potential.

4.

Diffraction

In order to describe diffraction, the wave nature of the scattered molecules has to be taken into account by solving the appropriate Schr¨odinger equation. Either the time-dependent Schr¨odinger equation i

∂ t) = H ( R, t) ( R, ∂t

(8)

or the time-independent Schr¨odinger equation = E( R, t) H ( R)

(9)

may be considered to treat the scattering process. The time-dependent Schr¨odinger equation is typically solved on a numerical grid using the wavepacket formalism [33–36]. In the time-independent formulation, the wavefunction is usually expanded in some suitable set of eigenfunctions leading to so-called coupled-channel equations [27, 37]. One important prerequisite for the determination of scattering intensities is the knowledge of the interaction potential between the scattered particles and the surface. While about one decade ago most interaction potentials had to be guessed based on experimental information, it has now become possible to map out whole potential energy surfaces by ab initio total-energy calculations [37, 38] which is illustrated in Fig. 3. This development has been caused by the progress in computer power and by the development of efficient electronic structure codes (see, e.g., Refs. [39–42]). We use the scattering of H2 at a metal surface as an exemplary system to discuss the quantum effects in the scattering at surfaces. While a decade ago it was also not possible to perform full quantum dynamical simulations in all

Quantum theory of reactive scattering and adsorption at surfaces

1721

Figure 3. Contour plots of the PES along two two-dimensional cuts through the sixdimensional coordinate space of H2 /Pd (100) (from [49]). The contour spacing is 0.1 eV per H2 molecule. The considered coordinates are indicated in the inset. The lateral position of the H2 molecule and its orientation are indicated above the contour plots.

hydrogen degrees of freedom, this can now be routinely done [13, 43–47]. In particular, hydrogen/palladium represents a system that is well-suited for both experiments under ultra-high vacuum conditions as well as for a theoretical treatment in the framework of modern electronic structure methods. There is a wealth of microscopic information which is well established and doublechecked through the fruitful combination of state-of-the-art experiments with ab initio total-energy calculations and related simulations. The interaction potential of hydrogen interacting with palladium surfaces has been determined in detail by several total-energy calculations [48–51] based on density-functional theory. Parametrizations of the ab initio potential energy surfaces have been used for quantum and classical molecular dynamics simulations of the scattering and adsorption of H2 on Pd(100) [10, 13, 14], Pd(111) [52–54] and Pd(110) [55]. In this contribution, I will particularly focus on H2 /Pd(100). Two so-called elbow potentials of this system which were determined by DFT calculations [49, 56] are shown in Fig. 3. The H2 /Pd(100) PES which is a six-dimensional hyperplane according to the H2 degrees of freedom when the substrate atoms

1722

A. Groß

are kept fixed is usually analysed in terms of these elbow potentials. They correspond to two-dimensional cuts through the six-dimensional PES as a function of the molecular distance from the surface Z and the interatomic H–H distance r for fixed lateral center-of-mass coordinates and molecular orientation. Hydrogen molecules usually adsorb dissociatively on metal surfaces [57, 58]. As Fig. 3(a) indicates, H2 dissociates spontaneously at Pd(100), i.e., there are non-activated paths for dissociative adsorption. However, dissociative adsorption corresponds to a bond making-bond breaking process that depends sensitively on the local chemical environment. Consequently, the PES is strongly corrugated which means the interaction strongly varies as a function of the lateral coordinates of the molecule. This is illustrated in Fig. 3(b). If the molecule comes down over the on-top site, the shape of the elbow looks entirely different. Along this pathway, the adsorption is no longer non-activated. We will see that the strong corrugation leads to significant intensities in the off-specular peaks. The PES does not only depend on the lateral position of the H2 molecule, i.e., the PES is not only corrugated, but it is also strongly anisotropic. Only molecules with their axis parallel to the surface can dissociate, for molecules approaching the Pd surface in an upright orientation the PES is purely repulsive [49]. Because of the anisotropy of the PES, in addition to elastic diffraction peaks there will be large intensities in rotationally inelastic diffraction peaks which correspond to rotational transitions in the collision process. The six-dimensional ab initio PES of H2 /Pd(100) has been parametrized using some suitable analytical form [14]. Using this fit, the six-dimensional quantum dynamics of H2 interacting with a fixed substrate have been determined [10] by solving the time-independent Schr¨odinger equation in a coupledchannel scheme using the concept of the so-called local reflection matrix (LORE) [59, 60]. This is a numerically very efficient and stable scheme that is based on a fine step-wise representation of the PES. One typical calculated angular distribution of H2 molecules scattered at Pd(100) is shown in Fig. 4 [10]. The total initial kinetic energy is E i = 76 meV. ¯ direction which The incident parallel momentum equals 2G along the 011 ◦ corresponds to an incident angle of θi = 32 . The molecules are initially in the rotational ground state ji = 0. Figure 4(a) shows the so-called in-plane scattering distribution, i.e., the diffraction peaks in the plane spanned by the wave vector of the incident beam and the surface normal. The label (m, n) denotes the parallel momentum transfer G = (mG, nG). The specular peak is the most pronounced one, but the first order diffraction peak (10) is only a factor of four smaller. Note that in a typical helium atom scattering experiment the off-specular peaks are about two orders of magnitude smaller than the specular peak [2]. This is due to the fact that the chemically inert helium atoms are scattered at the smooth tails of the surface electron distribution.

Quantum theory of reactive scattering and adsorption at surfaces (a)

1723

0.3

Scattering intensity

(00)

0.2

0.1

(20) ∆j⫽2 (40) ∆j⫽2 (50)

0.0 ⫺90

⫺60

(30) ∆j⫽2 (40) (30)

⫺30

(10) ∆j⫽2

(00) (10) ∆j⫽2

(10) 0

30

60

90

Scattering angle θf(˚)

(b)

90

Final angle θy(˚)

60 30 0

⫺30 ⫺60 ⫺90 ⫺90

⫺60

⫺30

0

30

60

90

Final angle θx(˚)

Figure 4. Six-dimensional quantum results of the rotationally inelastic scattering of H2 on Pd(100) for a kinetic energy of 76 meV at an incidence angle of 32◦ along the [10] direction of the square surface lattice. Panel (a) shows the in-plane diffraction spectrum where all peaks have been labeled according to the transition. Both in-plane and out-of-plane diffraction peaks are plotted in panel (b), the open and filled circles correspond to the rotationally elastic and rotationally inelastic scattering, respectively. The radius of the circles is proportional to the logarithm of the scattering intensity (after [10]).

In addition, rotationally inelastic diffraction peaks corresponding to the rotational excitation j = 0 → 2 are plotted in Fig. 4(a). They have been summed up over all final azimuthal quantum numbers m j . Note that the excitation probability of the so-called cartwheel rotation with m = 0 is for all peaks approximately one order of magnitude larger than for the so-called helicopter rotation m = j , since the polar anisotropy of the PES is stronger than the azimuthal one. The intensity of the rotationally inelastic diffraction peaks in Fig. 4 is comparable to the rotationally elastic ones. Except for the specular peak they

1724

A. Groß

are even larger than the corresponding rotationally elastic diffraction peak with the same momentum transfer (m, n). Because of the particular conditions with the incident parallel momentum corresponding to the reciprocal lattice vector ¯ diffraction peaks fall = (2G, 0), the rotationally elastic and inelastic (20) G upon each other. The out-of-plane scattering intensities are not negligible, which is demonstrated in Fig. 4(b). The open circles represent the rotationally elastic, the filled circles the rotationally inelastic diffraction peaks. The radii of the circles are proportional to the logarithm of the scattering intensity. The sum of all out-ofplane scattering intensities is approximately equal to the sum of all in-plane scattering intensities. Interestingly, some diffraction peaks with a large parallel momentum transfer still show substantial intensities. This phenomenon is well known from helium atom scattering and has been discussed within the concept of so-called rainbow scattering [61]. The intensity of the scattering peaks for normal incidence are analysed in detail in Fig. 5. The intensities of four diffraction peaks are plotted as a function of the kinetic energy for rotationally elastic (Fig. 5(a)) and rotationally inelastic (Fig. 5(b)) scattering. In molecular beam experiments, the beams are not monoenergetic but have a certain velocity spread. In order to allow a better comparison with the experiment, an initial velocity spread of v/v = 0.05 typical for experiments [12] has been applied to the results of the quantum dynamical simulations. The theoretical results still exhibit a rather strong oscillatory structure which is a consequence of the quantum nature of H2 scattering. Let us first focus on the specular peak. An analysis of the energetic position of the oscillations reveals that they occur whenever new diffraction channels open up. This process is illustrated in Fig. 6. For a particular kinetic energy, there is a discrete number of diffraction peaks. The final wave vectors K f differ by multiples of the reciprocal lattice. At certain threshold energies of the unit vectors G E threshold the energy becomes large enough that additional diffraction channels open up. At exactly E = E threshold, the new channel corresponds to a wave that propagates paralllel to the surface. Thus the oscillations in the scattering intensities are a consequence of the fact that at the threshold energies the number of diffraction peaks changes discontinuously. In detail, the first pronounced dip in the intensity of the specular peak at E i = 12 meV coincides with the emergence of the (11) diffraction peak, the small dip at E i = 22 meV with the opening up of the (20) diffraction channel. The huge dip at approximately E i = 50 meV reflects the threshold for rotationally inelastic scattering. Interestingly, the rotational elastic (10) and (11) diffraction peaks show pronounced maxima at this energy. This indicates a strong coupling between parallel motion and rotational excitation. Figure 5(b) shows the intensities of the rotationally inelastic diffraction peaks. The specular peak is still the largest, however, some off-specular peaks

Quantum theory of reactive scattering and adsorption at surfaces

1725

Scattering intensity

(a) (00) (10) (20) (11)

10⫺1

10⫺2

j⫽0→0

10⫺3 0.00

0.05

0.10

0.15

0.20

0.25

Total incident kinetic energy Ei (eV)

Scattering intensity

(b)

10⫺1

10⫺2

j⫽0→2

10⫺3 0.00

0.05

0.10

0.15

0.20

0.25

Total incident kinetic energy Ei (eV)

Figure 5. Calculated scattering intensity versus kinetic energy for H2 molecules in the rotational ground state impinging under normal incidence on Pd(100) with an initial velocity spread of v/v = 0.05 (after [10]).

G||

Kf

E < Ethreshold

E⫽Ethreshold

Figure 6. Schematic illustration of the opening up of new scattering channels for normal incidence as a function of the energy.

1726

A. Groß

become larger than the (00) peak at higher energies. This is due to the fact that the rotationally anisotropic component of the potential is more corrugated than the isotropic component [49]. Besides, it is apparent that the oscillatory structure for rotationally inelastic scattering is somewhat smaller than for rotationally elastic scattering. Not all peaks in the scattering amplitudes can be unambiguously attributed to the emergence of new scattering channels. As already mentioned in Section 2, additional structures in the scattering intensities can also be caused by selective adsorption resonances: molecules become temporarily trapped into metastable molecular adsorption states at the surface due to the transfer of normal momentum to parallel and angular momentum which resonantly enhances the scattering intensities. Such resonances have been clearly resolved, e.g., in the physisorption of H2 on Cu [62]. For the strongly corrugated and anisotropic H2 /Pd(100) system it is difficult to identify the nature of possible scattering resonances from the quantum calculations. Classically, one observes dynamic trapping in the H2 /Pd interaction dynamics [52, 53, 63] which is the equivalent of selective adsorption resonances: impinging molecules do neither directly scatter not dissociate but transfer energy from the translation perpendicular to the surface into internal degrees of freedom and motion parallel to the surface. In this transient state, they can spend several picoseconds at the surface. Although most of the dynamically trapped molecules eventually dissociate, this process still influences the reflection probabilities. Oscillatory structures have been known for years in He and H2 scattering [20] and also in low-energy electron diffraction (LEED) [64]. For reactive systems such as H2 /Pd(100), the experimental observation of diffraction is not trivial. Because of the reactivity, an adsorbate layer builds up very rapidly during the experiment. These layers destroy the perfect periodicity of the surface and thus suppress diffraction effects. In order to keep the surface relatively clean, one has to use rather high surface temperatures so that adsorbates quickly desorb again. High surface temperatures, on the other hand, also smear out the diffraction pattern. Still experimentalists managed to clearly resolve rotationally inelastic peaks in the diffraction pattern of D2 /Ni(110) [12] and D2 /Rh(110) [11] in addition to rotationally elastic peaks.

5.

Quantum Effects in Reaction Probabilities

While elastic scattering and diffraction are purely quantum phenomena that cannot be understood and reproduced within classical physics, reaction probabilities can be calculated by both classical and quantum molecular dynamics calculations. In a multidimensional situation, classical reaction probabilities are obtained by averaging over molecular dynamics simulations for statistically distributed initial conditions. For example, to determine the probability

Quantum theory of reactive scattering and adsorption at surfaces

1727

for dissociative adsorption, trajectories with different initial lateral positions within the surface unit cell and different molecular orientations have to be run. A quantum wave function, on the other hand, is always delocalized to a certain degree. One could say that quantum reaction probabilities correspond to a coherent average over initial conditions while classically the average is done incoherently. This coherent averaging causes quantum effects for example in the dissociative adsorption probability that will be discussed in this section. We continue to focus on the system H2 /Pd(100). In the determination of the diffraction pattern, we had neglected the substrate motion. This approximation is indeed justified in the description of the interaction of hydrogen with densely packed metal surfaces. There is only a small energy transfer from the light hydrogen molecule to the heavy substrate atoms. Furthermore, usually no significant surface rearrangement occurs during the interaction time. Even in the description of the dissociative adsorption of H2 on metal surfaces, in contrast to molecular adsorption, the substrate motion can be safely neglected [36, 37, 58]. The crucial process in the dissociative adsorption dynamics is the bond-breaking process, i.e., the conversion of translational and internal energy of the hydrogen molecule into translational energy of the atomic fragments on the surface relative to each other. The fragments will of course eventually thermalize at the surface by transfering their excess energy to the substrate, but this only occurs after the dissociation step. Thus the dissociation dynamics can be described by a six-dimensional PES which takes only the molecular degrees of freedom into account. In this framework, the dissociation probability can be regarded as a quantum transmission probability from the entrance channel of the impinging molecule to the dissociation channel at the surface. Because of the conservation of the particle flux, the adsorption probability for some particular initial state i can be evaluated by Si = 1 −

|R j i |2 ,

(10)

j

where the R j i are the amplitudes of all final scattering states. The calculated dissociative adsorption probability of H2 /Pd(100) as a function of the kinetic energy is shown in Fig. 7 and compared to the results of molecular beam experiments [15, 65]. The inset shows more recent results using an improved ab initio potential energy surface [45]. Because of the unitarity relation Eq. (10), scattering and adsorption probabilities are closely linked to each other. Indeed, the adsorption probability also exhibits a pronounced oscillatory structure at exactly the same kinetic energies as the scattering intensities. This means that this structure is also due to threshold effects because of the emergence of new scattering channels. In addition, resonance phenomena contribute to the oscillatory structure. However, if one again assumes a velocity spread of the incident molecules typical

1728

A. Groß 1.0

Sticking probability

0.8

1.0 0.8

Exp.. Rendulic et al 6D QD, ji⫽0 (Eichler et al.)

0.6

Exp. Rettner and Auerbach

0.4

0.6

0.2 0.0 0.0

0.1

0.2

0.3

0.4

0.5

0.4

0.2 Exp. Rendulic et al. 6D QD,ji⫽0 (Gross et al.) 6D QD, beam simulation

0.0 0.0

Exp. Rettner and Auerbach

0.1

0.2

0.3

0.4

0.5

Kinetic energy Ei(eV)

Figure 7. Sticking probability of H2 /Pd(100) as a function of the initial kinetic energy. Circles: H2 molecular beam adsorption experiment under normal incidence (Rendulic et al. [65]); dash–dotted line: H2 effusive beam scattering experiment with an incident angle of θi = 15◦ (Rettner and Auerbach [15]); dashed and solid line: theory according to H2 initially in the ground state and with a thermal distribution appropriate for a molecular beam [13]. The inset shows the theoretical results using an improved ab initio potential energy surface [45].

for molecular beam experiments [65], the calculated sticking probability becomes rather smooth (solid line in Fig. 7). This means that the quantum effects in the dissociative adsorption probability are hardly visible. The predicted quantum oscillations have been searched for experimentally by an effusive beam experiment for an angle of incidence of 15◦ [15, 66], but no oscillations could be detected. As already pointed out, surface imperfections at a reactive substrate such as adatoms or steps reduce the coherence of the scattering process and thus smooth out the oscillatory structure [10, 67]. But more importantly, also the angle of incidence has a decisive influence on the symmetry and the scattering intensities [14]. The calculations were done for normal incidence while the experiment was done for non-normal incidence [15, 66]. At non-normal incidence, the number of symmetrically equivalent diffraction channels is reduced compared to normal incidence. This makes the effect of the opening up of new diffraction channels less dramatic and thus also smoothes the adsorption probabilities. Experiment [65] and theory agree well, as far as the qualitative trend of the adsorption probability as a function of the kinetic energy is concerned. First there is an initial decrease, and after a minimum the sticking probability rises again. The initial decrease of the sticking probability is typical for H2 adsorption at transition metal surfaces [15, 65, 68–70]. In these systems, the PES shows purely attractive paths towards dissociative adsorption, but the majority of reaction paths for different molecular orientations and impact points exhibits energetic barriers hindering the dissociation. However, at low

Quantum theory of reactive scattering and adsorption at surfaces

1729

kinetic energies most impinging molecules are steered towards the attractive dissociation channel leading to a high adsorption probability. This steering effect [13, 71, 72] is suppressed at higher kinetic energies causing the decrease in the adsorption probability. While diffraction is a consequence of the periodicity of the surface, there are also more local quantum effects occurring within the surface unit cell, in particular if the wave function has to propagate through a narrow transition state. The consequences of such a situation will be illustrated using simple lowdimensional model calculations [17]. In Fig. 8(a), an idealized two-dimensional potential energy surface for activated adsorption is plotted as a function of one lateral coordinate and a reaction path coordinate. This PES has features

(a) 1.0

Reaction path coordinate s (Å)

0.8 0.6

0.3

0.1

0.5 0.9

0.4 0.7

0.2 0.0

⫺0.1

⫺0.2

⫺0.3

⫺0.4

⫺0.5 ⫺0.7

⫺0.6 ⫺0.8

⫺0.9

⫺1.0 0.0

1.0

2.0

3.0

4.0

5.0

Surface coordinate X (Å)

(b) 0.06

Sticking probability

classical dynamics H2 quantum dynamics D2 quantum dynamics

0.04

0.02

0.00

0.1

0.2 0.3 Kinetic energy (eV)

0.4

Figure 8. Activated dissociation of molecular hydrogen at a two-dimensional corrugated surface. (a) potential energy surface, (b) sticking probability vs. kinetic energy for a hydrogen beam under normal incidence. Full line: classical sticking probability which is independent of the mass as a function of the kinetic energy [14]; dashed line: H2 quantum sticking probability; long-dashed line: D2 quantum sticking probability [17].

1730

A. Groß

appropriate for, e.g., the hydrogen dissociation at the (2×2) sulfur covered Pd(100) surface [48, 73]: The minimum barrier has a height of 0.09 eV, the adsorption energy is E ad = 1 eV, and the square surface unit cell has a lattice constant of a = 5.5 Å. The calculated dissociation probability at such a surface is plotted in Fig. 8(b). The classical sticking probability is compared to the quantum sticking probabilities of the hydrogen isotopes H2 and D2 . Please note that there is no isotope effect in the dissociation probability as a function of the kinetic energy for hydrogen moving classically on a PES as long as there are no energy transfer processes to, e.g., substrate phonons [14]. This is caused by the fact that at the same kinetic energy H2 and D2 follow exactly the same trajectories in space. The quantum results show a very regular oscillatory structure as a function of the kinetic energy. This is not due to any resonance phenomenom but rather to the existence of quantized states at the transition state [16, 74]. At the minimum barrier position the wave function has to pass through a narrow valley of the corrugated PES. This leads to a localization of the wave function and thereby to a quantization of the allowed states that can pass through this valley. In the harmonic approximation the energy levels correspond to harmonic oscillator eigenstates which are equidistant in energy. Their spacing ω is determined by the curvature of the PES in the coordinates perpendicular to the reaction path. For H2 passing through the transition state shown in Fig. 8(a), the curvature of the potential perpendicular to the reaction path corresponds to a frequency of ω = 104 meV. And indeed, the oscillations in the H2 sticking probability exhibit a period of about 100 meV. The level spacing of the quantized states depends on the mass of the traversbetween the quantized states at ing particles. For D2 the energetic separation √ the transition state is smaller by a factor of 1/ 2 compared to H2 . This is indeed reflected in Fig. 8(b) by the smaller period of the oscillations in the D2 quantum sticking probability. The existence of quantized states is closely related to the zero-point energies. Because of the Heisenberg uncertainty principle, there is a minimum energy required for any localized quantum state, namely the zero-point energy. For a harmonic oscillator, this zero-point energy is given by ω/2. It leads to an effectively higher minimum barrier for the quantum propagation through a transition state region. Consequently, the onset of sticking occurs at higher energies in the quantum calculation than in the classical calculations (see Fig. 8(b)). However, this onset is not shifted by ω/2, but by a smaller amount. This is caused by another quantum phenomenom, tunneling. Quantum mechanically particles can also traverse a barrier region for energies below the minimum barrier height. This promoting effect partially counterbalances the hindering effect of the zero-point energies.

Quantum theory of reactive scattering and adsorption at surfaces

1731

Figure 8(b) demonstrates that the quantum sticking probabilities oscillate around the classical result which means that tunneling and quantization effects almost cancel each other on the average. In addition, if more degrees of freedom are considered, there will be further quantization effects. The combined effect will be a smoothing of the oscillatory structure. Indeed, in six-dimensional quantum calculations of the dissociative adsorption of H2 on Cu(100) [43], hardly any steplike structure is visible in the adsorption probability. Therefore it is almost very hard to detect these quantum effects in molecular beam experiments because of limited energetic resolution of the beams and the unavoidable existence of surface imperfections.

6.

Conclusions

In this review, we have presented an overview over the quantum effects in the interaction dynamics of atoms and molecules with surfaces. They are of particular importance for light atoms and molecules such as helium or hydrogen. The quantum nature of the substrate phonons leads to the phenomenom of elastic scattering at surfaces. This leads to trapping probabilities that are less than one in the non-activated sticking of weakly bound species at surfaces. Another quantum effect, namely diffraction, is a consequence of the periodicity of surfaces together with elastic scattering. It occurs when the de Broglie wavelength of the incident beam is of the order of the lattice spacing of the substrate and can be used as a tool to determine surface structures. The opening up of new scattering channels leads to an oscillatory structure in the intensities of the diffraction peaks and in the dissociative adsorption probabilities of H2 at reactive surfaces. Furthermore, there are quantum effects due to the existence of quantized states at the transition states of the multidimensional potential energy surface. However, all these additional quantum effects are suppressed by substrate imperfections and surface temperature effects. Hence they can hardly be resolved in experiments.

References [1] C.B. Duke and E.W. Plummer (eds.), Frontiers in Surface and Interface Science, North-Holland, 2002. [2] E. Hulpke (ed.), Helium Atom Scattering from Surfaces, volume 27 of Springer Series in Surface Sciences, Springer, Berlin, 1992. [3] D. Farías and K.-H. Rieder, Rep. Prog. Phys., 61, 1575, 1998. [4] H. Schlichting, D. Menzel, T. Brunner, W. Brenig, and J.C. Tully, Phys. Rev. Lett., 60, 2515, 1988. [5] H. Schlichting, D. Menzel, T. Brunner, and W. Brenig, J. Chem. Phys., 97, 4453, 1992.

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A. Groß R. Sedlmeir and W. Brenig, Z. Phys. B, 36, 245, 1980. W. Brenig, Z. Phys. B, 36, 227, 1980. F. Shimizu, Phys. Rev. Lett., 86, 987, 2001. H. Friedrich, G. Jacoby, and C.G. Meiter, Phys. Rev. B, 65, 032902, 2002. A. Groß and M. Scheffler, Chem. Phys. Lett., 263, 567, 1996. D. Cvetko, A. Morgante, A. Santaniello, and F. Tommasini, J. Chem. Phys., 104, 7778, 1996. M.F. Bertino, F. Hofmann, and J.P. Toennies, J. Chem. Phys., 106, 4327, 1997. A. Groß, S. Wilke, and M. Scheffler, Phys. Rev. Lett., 75, 2718, 1995. A. Groß and M. Scheffler, Phys. Rev. B, 57, 2493, 1998. C.T. Rettner and D.J. Auerbach, Chem. Phys. Lett., 253, 236, 1996. A.D. Kinnersley, G.R. Darling, S. Holloway, and B. Hammer, Surf. Sci., 364, 219, 1996. A. Groß, J. Chem. Phys., 110, 8696, 1999. I. Estermann and O. Stern, Z. Phys., 61, 95, 1930. B. Gumhalter, Phys. Rep., 351, 1, 2001. R. Frisch and O. Stern, Z. Phys., 84, 430, 1933. M. Patting, D. Farías, and K.-H. Rieder, Surf. Sci., 429, L503, 1999. R.G. Rowe and G. Ehrlich, J. Chem. Phys., 4648, 1975. G. Brusdeylins and J.P. Toennies, Surf. Sci., 126, 647, 1983. J. Lapujoulade, Y. Lecruer, M. Lefort, Y. Lejay, and E. Maurel, Surf. Sci., 103, L85, 1981. M.F. Bertino and D. Farías, J. Phys.: Condens. Matter, 14, 6037, 2002. K.B. Whaley, C.-F. Yu, C.S. Hogg, J.C. Light, and S. Sibener, J. Chem. Phys., 83, 4235, 1985. A. Groß, Theoretical Surface Science – A Microscopic Perspective, Springer, Berlin, 2002. R.W. Fuller, S.M. Harris, and E.L. Slaggie, Am. J. Phys., 31, 431, 1963. W. Brenig, Z. Phys. B, 36, 81, 1979. J. B¨oheim and W. Brenig, Z. Phys. B, 41, 243, 1981. I.A. Yu, J.M. Doyle, J.C. Sandberg, C.L. Cesar, D. Kleppner, and J.T. Greytak, Phys. Rev. Lett., 71, 1589, 1993. C. Eltschka, H. Friedrich, and M.J. Moritz, Phys. Rev. Lett., 86, 2693, 2001. R. Newton, Scattering Theory of Waves and Particles, 2nd edn., Springer, New York, 1982. J.A. Fleck, J.R. Morris, and M.D. Feit, Appl. Phys., 10, 129, 1976. H. Tal-Ezer and R. Kosloff, J. Chem. Phys., 81, 3967, 1984. G.-J. Kroes, Prog. Surf. Sci., 60, 1, 1999. A. Groß, Surf. Sci. Rep., 32, 291, 1998. A. Groß, Surf. Sci., 500, 347, 2002. G. Kresse and J. Furthm¨uller, Phys. Rev. B, 54, 11169, 1996. G. Kresse and D. Joubert, Phys. Rev. B, 59, 1758, 1999. B. Hammer, L.B. Hansen, and J.K. Nørskov, Phys. Rev. B, 59, 7413, 1999. B. Kohler, S. Wilke, M. Scheffler, R. Kouba, and C. Ambrosch-Draxl, Comput. Phys. Commun., 94, 31, 1996. G.-J. Kroes, E.J. Baerends, and R.C. Mowrey, Phys. Rev. Lett., 78, 3583, 1997. A. Groß, C.-M. Wei, and M. Scheffler, Surf. Sci., 416, L1095, 1998. A. Eichler, J. Hafner, A. Groß, and M. Scheffler, Phys. Rev. B, 59, 13297, 1999. Y. Miura, H. Kasai, and W. Di˜no, J. Phys.: Condens. Matter, 14, L479, 2002. W. Brenig and M.F. Hilf, J. Phys. Condens. Mat., 13, R61, 2001.

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[48] S. Wilke, D. Hennig, R. L¨ober, M. Methfessel, and M. Scheffler, Surf. Sci., 307, 76, 1994. [49] S. Wilke and M. Scheffler, Phys. Rev. B, 53, 4926, 1996. [50] W. Dong and J. Hafner, Phys. Rev. B, 56, 15396, 1997. [51] V. Ledentu, W. Dong, and P. Sautet, Surf. Sci., 412, 518, 1998. [52] H.F. Busnengo, W. Dong, and A. Salin, Chem. Phys. Lett., 320, 328, 2000. [53] C. Crespos, H.F. Busnengo, W. Dong, and A. Salin, J. Chem. Phys., 114, 10954, 2001. [54] H.F. Busnengo, E. Pijper, M.F. Somers, G.J. Kroes, A. Salin, R.A. Olsen, D. Lemoine, and W. Dong, Chem. Phys. Lett., 356, 515, 2002. [55] H.F. Di Cesare, M.A. Busnengo, W. Dong, and A. Salin, J. Chem. Phys., 118, 11226, 2003. [56] S. Wilke and M. Scheffler, Surf. Sci., 329, L605, 1995. [57] K. Christmann, Surf. Sci. Rep., 9, 1, 1988. [58] G.R. Darling and S. Holloway, Rep. Prog. Phys., 58, 1595, 1995. [59] W. Brenig, T. Brunner, A. Groß, and R. Russ, Z. Phys. B, 93, 91, 1993. [60] W. Brenig and R. Russ, Surf. Sci., 315, 195, 1994. [61] U. Garibaldi, A.C. Levi, R. Spadacini, and G.E. Tommei, Surf. Sci., 48, 649, 1995. [62] S. Andersson, L. Wilzen, M. Persson, and J. Harris, Phys. Rev. B, 40, 8146, 1989. [63] A. Groß and M. Scheffler, J. Vac. Sci. Technol. A, 15, 1624, 1997. [64] E.G. McRae, Rev. Mod. Phys., 51, 541, 1979. [65] K.D. Rendulic, G. Anger, and A. Winkler, Surf. Sci., 208, 404, 1989. [66] C.T. Rettner and D.J. Auerbach, Phys. Rev. Lett., 77, 404, 1996. [67] A. Groß and M. Scheffler, Phys. Rev. Lett., 77, 405, 1996. [68] K.D. Rendulic and A. Winkler, Surf. Sci., 299/300, 261, 1994. [69] M. Beutl, M. Riedler, and K.D. Rendulic, Chem. Phys. Lett., 256, 33, 1996. [70] M. Gostein and G.O. Sitz, J. Chem. Phys., 106, 7378, 1997. [71] D.A. King, CRC Crit. Rev. Solid State Mater. Sci., 7, 167, 1978. [72] M. Kay, G.R. Darling, S. Holloway, J.A. White, and D.M. Bird, Chem. Phys. Lett., 245, 311, 1995. [73] C.M. Wei, A. Groß, and M. Scheffler, Phys. Rev. B, 57, 15572, 1998. [74] D.C. Chatfield, R.S. Friedman, and D.G. Truhlar, Faraday Discuss., 91, 289, 1991.

5.11 STOCHASTIC CHEMICAL KINETICS Daniel T. Gillespie Dan T Gillespie Consulting, 30504 Cordoba Place, Castaic, CA 91384

The time evolution of a well-stirred chemically reacting system is traditionally described by a set of coupled, first-order, ordinary differential equations. Obtained through heuristic, phenomenological reasoning, these equations characterize the evolution of the molecular populations as a continuous, deterministic process. But a little reflection reveals that the system actually possesses neither of those attributes: Molecular populations are whole numbers, and when they change they always do so by discrete, integer amounts. Furthermore, in excusing ourselves from the arduous task of tracking the positions and velocities of all the molecules in the system, which we hope to justify on the grounds that the system is “well-stirred”, we preclude a deterministic description of the system’s evolution; because, a knowledge of the system’s current molecular populations is not by itself sufficient to predict with certainty the future molecular populations. Just as rolled dice are essentially random or “stochastic” when we do not precisely track their positions and velocities and all the forces acting on them, so is the time evolution of a well-stirred chemically reacting system for all practical purposes stochastic. That said, discreteness and stochasticity are usually not noticeable in chemical systems of “test-tube” size or larger, and for most such systems the traditional continuous deterministic description seems to be adequate. But if the molecular populations of some reactant species are very small, as is often the case for instance in cellular systems in biology, discreteness and stochasticity can sometimes play an important role. Whenever that happens, the ordinary differential equations approach will not be able to accurately describe the true behavior of the system. Stochastic chemical kinetics attempts to describe the time evolution of a well-stirred chemically reacting system as an overtly discrete, stochastic process, evolving in real (continuous) time. And it tries to do this in a way that accurately reflects how chemical reactions physically occur at the molecular level. This article will outline the theoretical foundations of stochastic chemical kinetics, and then derive and interrelate its principle equations and 1735 S. Yip (ed.), Handbook of Materials Modeling, 1735–1752. c 2005 Springer. Printed in the Netherlands.

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computational methods. It will also show how it happens that the resulting discrete, stochastic description usually gives way to the traditional continuous, deterministic description in a special limiting approximation.

1.

Microphysical Foundations of Stochastic Chemical Kinetics

We consider a well-stirred system of molecules of N chemical species {S1, . . . , S N }, which interact through M chemical reaction channels {R1 , . . . , R M }. We assume the system to be confined to a constant volume , and to be in thermal (but not necessarily chemical) equilibrium at some constant absolute temperature T . We let X i (t) denote the number of molecules of species Si in the system at time t. Our goal is to estimate, as best we can, the state vector X(t) = (X 1 (t), . . . , X N (t)), given that the system was in state X(t0 ) = x0 at some initial time t0 < t.1 Each reaction channel R j is assumed to be “elemental” in the sense that it describes a distinct physical event which happens essentially instantaneously. This assumption restricts us to two general types of reaction: Unimolecular reactions of the form Si → product(s); and bimolecular reactions of the form Si + Si → product(s), where in the latter i may or may not be the same as i.2 A given reaction channel R j is characterized mathematically by two quantities. The first is its state-change vector ν j = (ν1 j , . . . , ν N j ), where νi j is defined to be the change in the Si molecular population caused by one R j reaction event; thus, if the system is in state x and an R j reaction occurs, the system immediately jumps to state x + ν j . The two-dimensional array νi j is commonly known as the stoichiometric matrix. Its elements are practically always confined to the values 0, ±1 and ±2. The other defining quantity for reaction channel R j is its propensity function a j . It is defined as follows:

a j (x) dt = the probability, given X(t) = x, that one R j reaction will occur somewhere inside in the next infinitesimal time interval [t, t + dt).

(1)

1 Boldface variables will always be understood here to be N-component vectors, with the components

corresponding to the N chemical species in the system.

2 A set of three elemental reactions of the form S + S → S and S + S → S can often be regarded as 1 2← 4 3 4 5 the single trimolecular reaction S1 + S2 + S3 → S5 if the first two reactions are much faster than the third.

But this is always an approximation.

Stochastic chemical kinetics

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This definition might be said to be the fundamental premise of stochastic chemical kinetics, because everything else follows from it. It is important to recognize that this probabilistic definition has a solid basis in physical theory, more solid in fact than the reasoning that is traditionally used to justify the deterministic differential equations mentioned earlier. Since the microphysical basis of Eq. (1) ultimately determines the forms of the propensity functions, it is appropriate to describe it briefly here. If R j is the unimolecular reaction S1 → product(s), the underlying physics, which might be quantum mechanical, generally dictates the existence of some constant which we shall call c j such that c j dt gives the probability that any particular S1 molecule will so react in the next infinitesimal time dt. If there is currently a finite number x1 of S1 molecules in the system, we can take dt to be so small that no more than one of them will undergo that reaction in the next dt. This allows us to invoke the addition law of probability theory for mutually exclusive events, and so calculate the probability for any S1 molecule in the system to undergo the R j reaction by simply summing the individual reaction probabilities. That sum gives x1 × c j dt, from which we may conclude that the propensity function in Eq. (1) is a j (x) = c j x1 . If R j is a bimolecular reaction of the form S1 + S2 → product(s), stochasticity manifests itself in two ways, both stemming from the fact that we do not know the exact position and velocity of any molecule in the system: First, we can predict only the probability that an S1 molecule and an S2 molecule will collide in the next dt. And second, we can predict only the probability that such a collision will actually produce an R j reaction. Consider a randomly chosen pair of S1 and S2 molecules. The assumption of thermal equilibrium implies that the S2 molecule √ will see the S1 molecule moving with an average relative speed v¯ 12 = 8kB T /π m 12, where kB is Boltzmann’s constant and m 12 = m 1 m 2 /(m 1 + m 2 ). Denote the effective collision cross section of the molecular pair by σ12 (which would equal π(r1 + r2 )2 if the molecules were hard spheres with radii r1 and r2 ). In the next infinitesimal time dt, the S1 molecule will sweep out relative to the S2 molecule an infinitesimally small “collision volume” of size (v¯12 dt)σ12 – so called because if the center of the S2 molecule happens to lie inside that volume then the two molecules will collide in the next dt. (We take dt to be so small that there is virtually no chance that the collision will be preempted by an earlier collision with some third molecule.) By our assumption that the system is “well-stirred” – a condition that can be secured either by an externally driven stirrer or by the inevitable self-stirring effects of the many non-reactive (bounce-off) molecular collisions that typically occur in such a system – the probability that the center of the S2 molecule will lie inside the collision volume is just the ratio of that volume to the total system volume: (v¯12 dt)σ12 / . This ratio is therefore the probability that the pair will collide in the next dt. Denoting by p j the probability that a colliding S1-S2 molecular pair will actually

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react according to R j , we conclude by the multiplication law of probability theory that

v¯12 σ12 p j (v¯12 dt)σ12 dt = c j dt (2) × pj = gives the probability that a randomly chosen S1-S2 molecular pair will und-ergo the R j reaction in the next dt. Now taking dt to be so small that no more than one of the x1 x2 S1-S2 pairs in the system will react in the next dt, we can invoke the addition law of probability for mutually exclusive events to compute the probability for some pair to so react as x1 x2 × c j dt. Thus we conclude that the propensity function in Eq. (1) is a j (x) = c j x1 x2 . If this bimolecular have reckoned the reaction had instead been S1 + S1 → product(s), we would number of distinct S1 molecular pairs to be x1 (x1 − 1) 2, and so obtained for the propensity function a j (x) = c j (1/2)x1 (x1 − 1), which properly vanishes if there is only one S1 molecule. The foregoing analysis shows two things: First, an elemental reaction channel R j can indeed be described by a function a j (x) in the manner prescribed by Eq. (1). And second, a j (x) can usually be written as the product of some constant c j , called the specific reaction probability rate constant, times the number of distinct combinations of R j reactant molecules that are available when the system’s state is x. Our subsequent work here will depend critically on the first point, but will tolerate considerable variance with respect to the second; hence, we shall be less concerned here with the forms of the propensity functions than with the fact that they exist and satisfy Eq. (1). But we should note in passing that the task of evaluating c j entirely from first principles is a very challenging one. An interesting result for bimolecular reactions emerges in the idealized case in which the colliding molecules will react if and only if the kinetic energy associated with the component of their relative velocity along their line of centers at contact exceeds some threshold value ε ∗ ; in that case, it can be proved from elementary kinetic theory that the conditional reaction probability p j in Eq. (2) is given by p j = −ε ∗/kB T , thus providing a physically transparent interpretation of the familiar Arrhenius factor.3 If it were also the case that the reaction can occur only if the point of collisional contact between the two molecules lies inside specific solid angles ω1 on molecule 1 and ω2 on molecule 2, then in the absence of any orienting forces p j would contain the additional probability factors (ω1 /4π ) and (ω2 /4π ). It turns out that c j for a unimolecular reaction is numerically equal to the reaction rate constant k j of conventional deterministic chemical kinetics, while c j for a bimolecular reaction is equal to k j / if the reactants are 3 See R. Present, Kinetic Theory of Gases (McGraw-Hill, New York, 1958), and D. Gillespie, Physica A 188, 404–425, 1992.

Stochastic chemical kinetics

1739

different species, or 2k j / if they are the same. Contemplating this result by itself, one might be tempted to conclude that the fundamental premise (1) and the mathematical forms of the propensity functions all follow from some simple heuristic “stochastic extrapolation” of the mass-action equations of deterministic chemical kinetics. But the foregoing analysis shows that that is not the case: The existence and forms of the propensity functions are rooted in the realities of molecular dynamics. The equations of stochastic chemical kinetics cannot be derived in a logically rigorous way from the equations of deterministic chemical kinetics; rather, as we shall see later, the derivation goes the other way. In what follows, we shall simply assume that the propensity functions a j (x), like the state-change vectors ν j , are all given.

2.

The Chemical Master Equation

Although the probabilistic nature of Eq. (1) precludes making an exact prediction of X(t) given that X(t0 ) = x0 for any t > t0 , we might reasonably hope to infer the probability

P(x, t | x0 , t0 ) = Prob {X(t) = x, given X(t0 ) = x0 } .

(3)

In technical terms, P(x, t | x0 , t0 ) is the “probability density function” of the time-dependent “random variable” X(t), and X(t) in turn is, by virtue of the dynamics prescribed by Eq. (1), a “jump Markov process”.4 It is not difficult to deduce a time-evolution equation for the function (3) by using the laws of probability theory to write P(x, t + dt | x0 , t0 ) as the sum of the probabilities of all the mutually exclusive ways in which the system could evolve from state x0 at time t0 to state x at time t + dt, via specified states at time t: 

P(x, t + dt | x0 , t0 ) = P(x, t | x0 , t0 ) × 1 −

M



a j (x)dt 

j =1

+

M

P(x − ν j , t | x0 , t0 ) × a j (x − ν j )dt .

j =1

4 A stochastic process – a random variable that depends on time – is said to be Markov if its future values depend on its past values only through its present value. (A Markov process is distinguished from a Markov chain by the fact that time is a real or continuous variable in a process and an integer variable in a chain.) A jump Markov process changes discontinuously at isolated instants in time, and remains constant between such jumps. There are also continuous Markov processes, which evolve in a way that is mathematically continuous but often not differentiable.

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Here, the first term on the right is the probability that the system is already in state x at time t and then no reaction of any kind occurs in [t, t + dt). And the generic second term is the probability that the system is one R j reaction removed from state x at time t and then one R j reaction occurs in [t, t + dt). That these M + 1 routes to the final state x are mutually exclusive and collectively exhaustive is ensured by taking dt to be so small that no more than one reaction of any kind can occur in [t, t + dt). Subtracting P(x, t | x0 , t0 ) from both sides of the above equation, dividing through by dt, and then taking the limit dt → 0, we obtain what is know as the chemical master equation (CME): M

∂ P(x, t | x0 , t0 ) = a j (x − ν j )P(x − ν j , t | x0 , t0 ) ∂t j =1

− a j (x)P(x, t | x0 , t0 ) .

(4)

In principle, the CME completely determines the function P(x, t | x0 , t0 ). But a closer inspection of Eq. (3) reveals that it is actually a set of coupled, ordinary differential equations in t; in fact, there is one equation for each possible value (0,1,2, . . .) of each of the M components of the variable x – roughly as many equations as there are combinations of molecules in the system! So it is perhaps not surprising that the CME can be solved analytically for only a very few very simple systems, and numerical solutions are usually prohibitively difficult in other cases. One might hope, less ambitiously, to learn something from the CME about the behavior of functional averages like f (X(t)) ≡ x f (x)P(x, t | x0 , t0 ), but this too turns out to be practically impossible if any of the reaction channels are bimolecular. For example, it can be proved from Eq. (4) that M dX(t) = ν j a j (X(t)) . dt j =1

(5)

Now, if all the reactions were monomolecular, the propensity functions would all be linear in the state variables, and we would have a j (X(t)) = a j (X(t)). Equation (5) would then become a closed ordinary differential equation for the first moment or mean X(t). But if any reaction is bimolecular, the right hand side of Eq. (5) will contain at least one quadratic moment of the form X i (t)X i (t), and Eq. (5) would then be merely the first of an infinite, openended set of equations for all the moments. In the hypothetical case in which there are no fluctuations, i.e., if X(t) were a deterministic or sure process, we would have f (X(t)) = f (X(t)) for all functions f . Equation (5) would then reduce to M dX(t) = ν j a j (X(t)). dt j =1

(6)

Stochastic chemical kinetics

1741

This is just the well known reaction rate equation (RRE) of traditional deterministic chemical kinetics – a set of coupled first-order ordinary differential equations for the components X i (t), which are now continuous (real) variables. The RRE is more commonly written in terms of the concentration vari able Z(t) = X(t)/ , but that simple scalar transformation is inconsequential for our purposes here. Although the foregoing analysis shows that the deterministic RRE (6) would be valid if all fluctuations were simply ignored, it does not tell us how or why the fluctuations might ever be “ignorable”. We shall later prove that the RRE can actually be derived from Eq. (1) through a series of physically transparent approximating assumptions.

3.

The Stochastic Simulation Algorithm

Since the CME (4) is practically never useful for calculating the probability density function of X(t), we need another approach. Let us look for a way to construct a numerical realization of X(t), i.e., a simulated trajectory of X(t) vs. t. Note that this is not the same as solving the CME numerically; however, much the same effect can be achieved by either histogramming or averaging the results of many realizations. For example, the nth n moment X (t ) , which would be given in terms of the solution to the CME 1 i as x xin P(x, t1 | x0 , t0 ), can also be estimated by generating L trajectories x(1) (t), . . ., x(L) (t) from state x0 at time t0 to time t1 , and then computing n L L −1 l=1 xi(l) (t1 ) . This estimate will have an associated uncertainty which decreases with the number of realizations L like L −1/2 . In practice, it is often found that as few as two or three simulated trajectories can convey as good a picture of the dynamical behavior of X(t) as would be afforded by an exact expression for P(x, t | x0 , t0 ). The key to generating simulated trajectories of X(t) is actually not the CME, but rather a new probability function, p(τ, j | x, t), which is defined as follows:

p(τ, j | x, t) dτ = the probability, given X(t) = x, that the next reaction in the system will occur in the infinitesimal time interval [t + τ, t + τ + dτ ), and will be an R j reaction. (7) Formally, this function is the joint probability density function of the two random variables “time to the next reaction” (τ ) and “index of the next reaction” ( j ), given that the system is currently in state x. If we can derive an analytical expression for this function, we could use Monte Carlo techniques

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D.T. Gillespie

to generate simultaneous samples of τ and j , and that would enable us to advance the system in time from one reaction event to the next. Happily, it turns out that we can do all this fairly easily, and without having to make any approximations. To derive an analytical expression for p(τ, j | x, t), we begin by introducing yet another probability function, P0 (τ | x, t), which is defined as the probability, given X(t) = x, that no reaction of any kind occurs in the time interval [t, t +τ ). By the definition (1) and the laws of probability theory, this function satisfies 

P0 (τ + dτ | x, t) = P0 (τ | x, t)× 1 −

M



(a j (x)dτ ),

j =1

since the right side gives the probability that no reaction occurs in [t, t + τ ) and then no reaction occurs in [t + τ, t + τ + dτ ) (as usual we take the infinitesimal time span dτ to be so small that it can contain no more than one reaction). A simple algebraic rearrangement of this equation and passage to the limit dτ → 0 results in the differential equation dP0 (τ | x, t) = −a0 (x) P0 (τ | x, t), dτ where we have defined

a0 (x) =

M

a j (x).

(8)

j =1

The solution to this differential equation for the initial condition P0 (τ = 0 | x, t) = 1 is P0 (τ | x, t) = exp (−a0 (x) τ ). Now we observe that the probability defined in Eq. (7) can be written p(τ, j | x, t) dτ = P0 (τ | x, t)×(a j (x)dτ ), since the right side gives the probability that no reactions occur in [t, t + τ ) and then one R j reaction occurs in [t + τ, t + τ + dτ ). When we insert the above formula for P0 (τ | x, t) into this last equation and cancel the dτ ’s, we obtain p(τ, j | x, t) = a j (x) exp (−a0 (x) τ ),

(9a)

Stochastic chemical kinetics

1743

or equivalently p(τ, j | x, t) = a0 (x) exp (−a0 (x) τ ) ×

a j (x) . a0 (x)

(9b)

Equation (9a) is the desired explicit formula for the joint probability density function of τ and j . The equivalent form (9b) shows that this joint density function can be factored as the product of a τ -density function and a j -density function; more precisely, it shows that τ is an exponential random variable with mean and standard deviation 1/a0 (x), while j is a statistically independent integer random variable with point probabilities a j (x)/a0 (x). There are several exact Monte Carlo procedures for generating samples of these random variables. Perhaps the most direct is the procedure that follows by applying to each of the two probability density functions in Eq. (9b) the so-called inversion generating method:5 Draw two random numbers r1 and r2 from the uniform distribution in the unit-interval, and take

1 1 ln , τ= a0 (x) r1 j = the smallest integer satisfying

(10a) j

a j (x) > r2 a0 (x).

(10b)

j =1

And so we arrive at the following exact procedure for constructing a numerical realization of the process X(t), a procedure called the stochastic simulation algorithm (SSA): 0. Initialize the time t = t0 and the system’s state x = x0 . 1. With the system in state x at time t, evaluate all the a j (x) and their sum a0 (x). 2. Generate values for τ and j using Eqs. (10) (or an equivalent procedure). 3. Effect the next reaction by replacing t ← t + τ and x ← x + ν j . 4. Record (x, t) as desired. Return to Step 1, or else end the simulation. The X(t) trajectory that is produced by the SSA might be thought of as a “stochastic version” of the trajectory that would be obtained by solving the RRE (6). (But note that the time step τ in the SSA is exact, and is not a finite approximation to some infinitesimal dt, as is the time step in most numerical solvers for the RRE.) If it is found that every SSA-generated trajectory is practically indistinguishable from the RRE trajectory, then we may conclude that microscale randomness is negligible for this system. But if the SSA trajectories are found to deviate significantly from the RRE trajectory, then we must 5 It can be proved that a sample x of the random variable X can be obtained from a sample r of the unitx interval uniform random variable by solving −∞ P(x )dx = r, where P is the density function of X. This is known as the “inversion” generating procedure.

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conclude that microscale randomness is not negligible, and the deterministic RRE does not provide an accurate description of the system’s true behavior. The SSA and the CME are logically equivalent to each other, since each is derived without approximation from premise (1). But even when the CME is completely intractable, the SSA is quite straightforward to implement. In fact, as a numerical procedure, the SSA is even simpler than the procedures that are typically used to numerically solve the RRE (6). The catch is that the SSA is often very slow. The source of this slowness can be traced to the factor 1/a0 (x) in Eq. (10a), which as mentioned earlier is the mean of the random variable τ : Since a0 (x) is at least linear and more commonly quadratic in the reactant populations, a0 (x) can be very large, and τ correspondingly very small, whenever any reactant species is present in large numbers, and that is nearly always the case in practice. One notable attempt to speed up the SSA is the Gibson-Bruck procedure, which advances the system in exact accord with the function p(τ, j | x, t) in Eq. (9) but using a different scheme than Eqs. (10).6 Although this procedure is more complicated to code than the procedure described above, it is significantly faster and more efficient for systems having many species and many reaction channels. But any procedure that simulates every reaction event, exactly and one at a time, will inevitably be too slow for many practical applications. This prompts us to consider the possibility of giving up some of the exactness of the SSA in return for greater simulation speed.

4.

Tau Leaping

One approximate accelerated simulation strategy is tau-leaping, which tries to advance the system by a pre-selected time interval τ that encompasses more than one reaction event. To properly accomplish that feat when the system is in state x at time t, we need to know how to generate sample values of the M random variables

K j (τ ; x, t) = the number of times reaction channel R j fires in [t, t + τ ), given that X(t) = x ( j = 1, . . . , M).

(11)

For then, we could simply insert those sample values into the update formula X(t + τ ) = x +

M

K j (τ ; x, t ) ν j

j =1

6 For details, see M. Gibson and J. Bruck, J. Phys. Chem., 104, 1876–1889, 2000.

(12)

Stochastic chemical kinetics

1745

to “leap” the system ahead by the chosen time τ . Unfortunately, that is easier said than done. In general, the M random variables (11) are statistically dependent, and it is not altogether clear even how to calculate their joint probability density function, much less generate random samples according to that function. Suppose, however, that τ is chosen small enough that the following Leap Condition is satisfied: The expected state change induced by the leap is sufficiently small that no propensity function changes its value by a significant amount. In that case, we should be able to approximate each K j (τ ; x, t) by a statistically independent Poisson random variable: K j (τ ; x, t ) ≈ P j (a j (x), τ ) ( j = 1, . . . , M).

(13)

This is because the generic Poisson random variable P(a, τ ) is by definition the number of events that will occur in time τ , given that a dt is the probability that an event will occur in any infinitesimal time dt, where a may be any positive constant (hence the need for the Leap Condition).7 Therefore, if we can find a value for τ that is small enough that the Leap Condition is satisfied, yet large enough that many reaction events occur in time τ , we may indeed have a faster, albeit approximate, simulation strategy. The practical question arises, how can we determine in advance the largest value of τ that is compatible with the Leap Condition? Although there is as yet no unequivocal answer to this question, the following recipe for choosing τ will approximately ensure that no propensity function is likely to change its value in the leap by more than εa0 (x), where ε is some pre-chosen accuracy control parameter satisfying 0 < ε 1: With

f (x) = j j

N ∂a j (x) i=1

and M

∂ xi

νi j

( j, j = 1, . . . , M) 

µ j (x) =

  f j j (x) a j (x)  

σ j2 (x) =

  f j2j (x) a j (x)  

j =1 M j =1

take8

τ = Min

j ∈[1,M]

(14a)

( j = 1, . . . , M),

(14b)

ε a (x) ε 2 a02 (x) 0 , . µ j (x) σ 2 (x) j

(15)

7 It can be shown that the probability that the random variable P(a, τ ) as so defined will equal any non negative integer n is e−aτ (aτ )n n!, and also that the mean and variance of P(a, τ ) are both equal to

aτ . 8 For a derivation, see D. Gillespie and L. Petzold, J. Chem. Phys., 119, 8229–8234, 2003.

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The explicit tau-leaping simulation procedure thus goes as follows: 1. In state x at time t, and with a value chosen for ε, evaluate τ from Eq. (15). R j in time 2. For j =1, . . . , M, generate the number of firings k j of reaction

τ as a sample of the Poisson random variable P a j (x), τ .9 3. Leap, by replacing t ← t + τ and x ← x + M j =1 k j ν j . Smaller values of ε will result in a better satisfaction of the Leap Condition, and hence a leap that is more accurate, but of course shorter. In the limit ε → 0, tau-leaping becomes mathematically equivalent to the SSA; however, tau-leaping will be very inefficient in that limit because all the k j ’s will usually be zero, giving a very small time step without any change of state. Therefore, it is advisable to abort the above procedure after Step 1 if τ is found to be less than a few multiples of 1/a0 (x), the mean time to the next reaction, and instead use the SSA to step directly to that next reaction. A variation on the foregoing tau-leaping procedure allows us to advance the system to the moment of the next firing of some particular reaction channel Rα , which perhaps initiates some critical sequence of events in the system. To do that, we start by computing a tentative τ from Eq. (15), and then computing aα (x) τ , the expected number of Rα firings in that time τ . If aα (x) τ < 1, we should not try to leap ahead to the next Rα reaction because that would violate the Leap Condition. But if aα (x) τ ≥ 1, then a leap with kα = 1 should be okay. In that case, wewould generate the actual time τ to the next Rα reaction as τ = aα−1 (x) ln (1 r),where r is a unit-interval uniform random number. Using that value for τ , we would then generate Poisson values for all the other k j =/ α as in Step 2, and finally effect the leap as in Step 3. If the system happens to be “dynamically stiff” – meaning that it has widely varying dynamical modes, the fastest of which are stable – the explicit tauleaping procedure will be computationally unstable for time steps that are larger than the fastest time scale, and that may severely restrict the size of τ . Stiffness is very common in chemical systems. Recently, an implicit tauleaping procedure has been proposed which shows promise of overcoming the instability problem for stiff systems.10 It should be noted that tau-leaping is not as foolproof as the SSA. If one takes leaps that are too large, bad things can happen; e.g., some species populations might be driven negative. The underlying philosophy of tau-leaping is to leap over “unimportant” reaction events but not the “important” ones, and 9 Numerical procedures for generating Poisson random numbers can be found, for instance, in W. Press, B. Flannery, S. Teukolsky, and W. Vetterling, Numerical Recipes: The Art of Scientific Computing, Cambridge University Press, New York, 1986. 10 For details, see M. Rathinam, L. Petzold, Y. Cao, and D. Gillespie, J. Chem. Phys., 119, 12784–12794, 2003.

Stochastic chemical kinetics

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in some circumstances special measures must be taken to ensure that outcome. Much more work in this area is needed.

5.

The Chemical Langevin Equation

In the previous section we noted that, when the system is in state x at time t, if we choose a time-step t that is small enough that none of the propensity function values changes significantly during t, then the system’s state at time t + t can be decently approximated by

. X(t + t) = x + P j a j (x), t ν j , M

(16)

j =1

where the P j ’s are statistically independent Poisson random variables. Suppose the system admits a t that satisfies not only that condition, but also the condition that the expected (or mean) number of firings of each reaction channel in time t is 1; i.e., a j (x) t 1 for all j = 1, . . . , M.

(17)

It will usually be possible to find such a t if the molecular populations of all reactant species are “sufficiently large”. Now, it is well know that the Poisson random variable P(a, τ ), which has mean and variance aτ , can be approximated when aτ 1 by the normal random variable with the same mean and variance.11 Therefore, denoting the normal random variable with mean m and variance σ 2 by by N (m, σ 2 ), condition (17) allows Eq. (16) to be further approximated as follows:

. N j a j (x)t, a j (x)t ν j , X(t + t) = x + M

(18a)

j =1

=x+

M

a j (x)t +

a j (x)tN j (0, 1) ν j ,

j =1

√ . X(t + t) = x + ν j a j (x)t + ν j a j (x)N j (0, 1) t , M

M

j =1

j =1

(18b)

where the second line invokes the fact that N (m, σ 2 ) = m + σ N (0, 1). 11 That e−aτ (aτ )n /n! ≈ (2π aτ )−1/2 exp(−(n − aτ )2 /2aτ ) when aτ 1 follows from Stirling’s approx-

imation and the small-x approximation for ln(1 + x).

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We have thus established the following result: If the system admits a macroscopically infinitesimal time increment dt, defined so that during dt (i) no propensity function changes its value significantly yet (ii) every reaction channel fires many more times that once, then we can approximate the t to t + dt system evolution by √ . ν j a j (X(t)) dt + ν j a j (X(t)) N j (t) dt, X(t + dt) = X(t) + M

M

j =1

j =1

(19) where the N j (t) are M statistically independent, temporally uncorrelated, normal random variables with means 0 and variances 1. Equation (19) is called the chemical Langevin equation (CLE). The dot over its equal sign reminds us that it is an approximation, valid only to the extent that dt is small enough to satisfy condition (i) and simultaneously large enough to satisfy condition (ii). It is usually possible to find such a dt if all the reactant populations are sufficiently large. But if that is not possible, Eq. (19) has no basis and should not be invoked. The approximate character of the CLE (19) is underscored in the fact that the state vector X(t) therein is no longer discrete (integer-valued), but instead is continuous (real-valued); in fact, the name “Langevin” is applied because Eq. (19) has the exact mathematical form of the like-named generic equation that governs the time-evolution of any continuous Markov process. For the sake of completeness, two pertinent but unobvious results from the formal theory of continuous Markov processes should be noted here:12 First, Eq. (19) can be written in the mathematically equivalent form M M dX(t) . = ν j a j (X(t)) + ν j a j (X(t)) j (t) . dt j =1 j =1

(20)

independent “Gaussian white noise” processes Here, j (t) are statistically satisfying j (t) j (t ) = δ j j δ(t − t ), where the first delta function is Kronecker’s and the second is Dirac’s. Equation (20) is called the “white noise form” of the CLE. Second, the time evolution of X(t) prescribed by Eq. (19)

12 For a proof of the equivalence of the mathematical forms (19), (20) and (21), see D. Gillespie, Am. J.

Phys., 64, 1246–1257, 1996.

Stochastic chemical kinetics

1749

induces a time evolution in the probability density function of X(t) according to the partial differential equation 





N M ∂  ∂ P(x, t|x0 , t0 ) . =− νi j a j (x) P(x, t | x0 , t0 ) ∂t ∂ x i i=1 j =1

+

+



N 1 ∂2

2

i=1

N i,i =1 (i
∂ xi2



M





νi2j a j (x) P(x, t | x0 , t0 )

j =1



M





∂  νi j νi j a j (x) P(x, t | x0 , t0 ). ∂ xi ∂ xi j =1 2

(21) This equation is called the chemical Fokker–Planck equation (CFPE). Essentially, we have approximated the jump Markov process governed by the master Eq. (4) by the continuous Markov process governed by the Fokker–Planck Eq. (21). All this somewhat complicated and possibly unfamiliar mathematics should not be allowed to obscure the genuine simplicity of the logical arguments underlying the foregoing derivation of the CLE (19): Condition (i) allowed us to infer, essentially from the fundamental premise (1), the Poisson approximation (16), and condition (ii) then allowed us to make the normal approximation (18), whence the CLE (19).13 Before examining some interesting theoretical implications of the CLE, we should note that it has a very practical numerical application: In either of its forms (18), the CLE enables us to approximately advance the system in time by a macroscopically infinitesimal time increment t. By virtue of condition (17), that would allow us to leap over very many individual reactions, thus producing a very substantial increase in simulation speed over the SSA. The Langevin update formula (18) is computationally more attractive than the explicit tau-leaping update formula (16) simply because normal random numbers are easier to generate than Poisson random numbers.14 But it should be clear that the Langevin update formula (18) is really just a limiting approximation of the explicit tau-leaping update formula (16): Whenever conditions (17) hold, tau-leaping inevitably reduces to Langevin leaping.

13 Note that this derivation of the CLE does not proceed in the ad-hoc manner of many Langevin equation derivations, in which the forms of the coefficients of j (t) in Eq. (20) are simply assumed with an eye to obtaining some pre-conceived outcome. 14 See the reference cited in footnote 9.

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D.T. Gillespie

The Reaction Rate Equation Limit

In practice, most chemical systems contain huge numbers of molecules, and are thus well on their way to the so-called thermodynamic limit, in which the species populations X i and the system volume all approach infinity in such a way that the species concentrations X i / remain constant. The large molecular populations of such systems usually means that their dynamical behavior is well described by the CLE (19). An inspection of the CLE (19) shows that it separates the state increment X(t + dt) − X(t) in a macroscopically infinitesimal time step dt into two components: a deterministic component proportional to dt, and a fluctu√ ating component proportional to dt. The deterministic component is evidently linear in the propensity functions, while the fluctuating component is proportional to the square root of the propensity functions. Now it happens that all propensity functions grow, in the thermodynamic limit, in direct proportion to the size of the system. For a unimolecular propensity function of the form c j xi this is obvious; for a bimolecular propensity function of the form c j xi xi this follows because c j is inversely proportional to the system volume [cf. Eq. (2)], which offsets one of the population variables. Therefore, as the thermodynamic limit is approached, the deterministic component of the state increment in the CLE (19) grows like the system size, whereas the fluctuating component grows like the square root of the system size. The fluctuating component thus scales, relative to the deterministic component, as the inverse square root of the system size. This establishes, in a logically deductive way, the conventional rule-of-thumb that relative fluctuations in a chemically reacting system typically scale as the inverse square root of the system size. This scaling behavior also implies that, in the full thermodynamic limit, the fluctuating term in the CLE (19) usually becomes vanishingly small compared to the deterministic term, and hence can be dropped. The CLE therefore becomes in the full thermodynamic limit, . ν j a j (X(t)) dt. X(t + dt) = X(t) + M

(22)

j =1

This is just the conventional RRE (6). But we have now derived it within the theoretical framework of stochastic chemical kinetics. Notice how our description of the system’s dynamical behavior has progressed: The CME (4) and the SSA (9) describe X(t) as a discrete stochastic process. The CLE (19) and the CFPE (21) describe X(t) as a continuous stochastic process. And the RRE (22) describes X(t) as a continuous deterministic process. At each level, the description is an approximation of the description at the previous level, valid only under certain specific conditions.

Stochastic chemical kinetics

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One instance in which the limiting form (22) can be misleading is when the sum on the right hand side is zero, which happens whenever the system evolves to a “stable state”. In such a circumstance, the fluctuating term in the CLE (19) will inevitably be larger than the deterministic term, and hence not entirely negligible. Another instance of inadequacy of the RRE concerns the long-time behavior of an open or driven system that has more than one stable state: Such a system will in fact perpetually visit all of those stable states, whereas the RRE contrarily implies that the system will go to the nearest (downhill) stable state and stay there forever. But in the many cases where the approximating assumptions leading to the RRE are warranted, the RRE provides a very efficient description of the system’s temporal behavior.

7.

The Chemical Kinetics Modeling Hierarchy

We conclude by summarizing the hierarchy of schemes that are available for modeling the time evolution of a chemically reacting system, proceeding from the slowest and most accurate to the fastest and most approximate. The most exact procedure for simulating the time evolution of a chemically reacting system is molecular dynamics (MD), wherein the position and velocity of every molecule in the system are tracked precisely. This results in a simulation of every molecular collision that occurs in the system, not only the reactive collisions but also the non-reactive (elastic) collisions. MD is thus able to show very accurately the evolution of not only the species populations, but also their spatial distributions. But of course, this essentially exact approach requires an enormous investment of computation time and resources. If the system is such that reactive collisions are usually separated in time by many non-reactive collisions, and the predominant effect of the latter is simply to “stir” the system, then we may back away from an MD simulation and use instead the SSA. The SSA simulates only the reactive collisions. Because it skips over all the non-reactive collisions, and also avoids computing spatial distributions (which are assumed to be uniform in the statistical sense), the SSA is computationally much faster than MD. Tau-leaping is based on the same assumptions as the SSA, but it proceeds approximately from those assumptions; more specifically, it uses a special Poisson approximation to advance the system by a pre-selected time τ during which more than one reaction event may occur. The size of τ is restricted by the condition that no propensity function may change its value during τ by a “significant” amount. Whenever that condition is satisfied and at least some of the reaction channels fire very many times in τ , tau-leaping will be faster than the SSA. A tau-leap in which all of the reaction channels fire many more times than once is approximately described by the CLE. In a Langevin-leap, the number

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of firings of each reaction channel is approximated by a normal random number instead of a Poisson random number. Since by hypothesis many reaction events are skipped over, and since also normal random numbers are easier to generate than Poisson random numbers, Langevin-leaping is faster than ordinary tau-leaping. Finally, if the system admits a description by the CLE and is for all practical purposes at the thermodynamic or “large system” limit, then the random term in the CLE will usually be negligibly small compared to the deterministic term. The CLE then reduces to the deterministic RRE. This RRE limit is usually justified for macroscopic systems, and when it is, it provides the most efficient way to simulate the evolution of the system.

Acknowledgments This work was supported by the Air Force Office of Scientific Research and the California Institute of Technology under DARPA Award No. F3060201-2-0558, and also by the Molecular Sciences Institute under Contract No. 244725 with the Sandia National Laboratories and the Department of Energy’s “Genomes to Life Program.”

5.12 KINETIC MONTE CARLO SIMULATION OF NON-EQUILIBRIUM LATTICE-GAS MODELS: BASIC AND REFINED ALGORITHMS APPLIED TO SURFACE ADSORPTION PROCESSES J.W. Evans Ames Laboratory – USDOE, and Department of Mathematics, Iowa State University, Ames, Iowa, 50011, USA

For many growth, transport, or reaction processes occurring on the surfaces or in the bulk of crystalline solids, atoms reside primarily at a discrete periodic array or lattice of sites, actually vibrating about such sites. These atoms make occasional “sudden” transitions between nearby sites due to diffusive hopping, or may populate or depopulate sites due to adsorption and desorption, possibly involving reaction. Most of these microscopic processes are thermally activated, the rates having an Arrhenius form reliably determined by transition state theory [1]. In general, these rates will depend on the local environment (i.e., the occupancy of nearby sites) thus introducing cooperativity into the process, and they may vary over many orders of magnitude. Such systems are naturally described by lattice-gas (LG) models wherein the sites of a periodic lattice are designated as either occupied (perhaps by various types of particles) or vacant. A specification of all possible transitions between different configurations of particles, together with the associated rates, completely prescribes the evolution of the LG model for the process of interest. Such models are called Interacting Particle Systems (IPS) in the mathematical statistics community [2]. They correspond to stochastic Markov processes for evolution between different possible configurations of the system, and their evolution is rigorously described by appropriate master equations [3]. Since it is typically not possible to precisely determine the behavior of the solutions of these equations with analytic methods, Kinetic Monte Carlo (KMC) simulation is the most common tool for analysis. This approach, described below, implements on computer the “typical” evolution of the LG model 1753 S. Yip (ed.), Handbook of Materials Modeling, 1753–1767. c 2005 Springer. Printed in the Netherlands.

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J.W. Evans

through a specific sequence configurations using a random number generator to select processes with the appropriate weights [4]. The great advantage of KMC is that usually it can treat these processes on the physically relevant time and length scales, contrasting conventional Molecular Dynamics. Another advantage of KMC is its versatility with respect to model modification, allowing systematic testing of the effect of various processes on behavior. The focus here is on simulation of non-equilibrium processes, hence the “kinetic” in KMC. This contrasts conventional Monte Carlo (MC) simulation for equilibrium (Gibbs) states of Hamiltonian systems. For the latter, the independence of the equilibrium state on the history or dynamics of the system provides considerable flexibility in optimizing the simulation procedure [4]. For example, to speed up the simulation, one can use artificial dynamics provided that is consistent with detailed-balance. Also, other tools are available for analysis of simulation data, e.g., histogram re-weighting, based on the features of the Gibbs distribution. These techniques are not available for non-equilibrium systems, where the simulation must incorporate the physical dynamics to correctly predict the possibly non-trivial competition between various kinetic pathways. This requirement was fully not appreciated in earlier studies of the approach to equilibrium where generic rules for rates were often used. The philosophy adopted here is that the atomistic LG model should first be clearly defined, as distinct from the simulation algorithm used to analyze the model. Thus, below we first give a general description of non-equilibrium LG models, together with the master equations which describe their evolution, and only then describe the two types of generic KMC simulation algorithms. It is most instructive to illustrate these basic algorithms and various refinements to them in the context of specific classes of examples. We choose models for evolution of homoepitaxial thin film systems and for catalytic surface reactions.

1.

Evolution of Stochastic Lattice-Gas Models: Master Equations

The basic master equation formulation described below applies to the case of finite systems, i.e., lattices with L d sites where d is the spatial dimension. Any simulation is of course also restricted to such finite systems. Usually, finite-size effects are minimized by choosing periodic boundary conditions. Below, we let n j denote the occupancy of site j , {n j } the configuration of the entire system, and P({n j }, t) the probability for the system to be in this configuration at time t. Implicitly, these probabilities involve ensemble averaging which, in the context of KMC simulation, may correspond to averaging over a

KMC simulation of non-equilibrium LG models

1755

large number of simulation trials. Then, evolution is described exactly by the master equations [3] d/dt P({n j }, t) =

{n j }

−

W ({n j } → {n j })P({n j }, t)

{n j }

W ({n j } → {n j })P({n j }, t),

(1)

where W ({n j }→{n j }) denotes the prescribed rate of transitions from configuration {n j } to {n j }. These two configurations will differ only in the occupancy of a single site for adsorption or desorption, but in the occupancy of a pair of sites for diffusion. On the right hand side of (1), the first (second) term reflects gain (loss) in the population of configuration {n j }. As an aside, we note that for a Markov process, specifying a rate for each microscopic process actually means there is an exponential waiting-time distribution between events associated with this process, with the mean waiting-time between consecutive events given by the inverse of the rate. The solutions of the master equations satisfy conservation of probability, positivity, etc. The eigenvalues of the evolution matrix associated with these linear equations have non-positive real parts to avoid blow-up of probabilities, but they can in general be complex-valued. The latter scenario corresponds to time-oscillatory solutions which can occur in open non-equilibrium systems. In cases where the energy of configuration {n j } is described by a Hamiltonian, H ({n j }), selecting rates to satisfy the detailed-balance condition (Landau and Binder, 2000) W ({n j } → {n j }) exp(−H ({n j })/kT ) =W ({n j } → {n j }) exp(−H ({n j })/kT ),

(2)

guarantees that the solution will evolve to the Gibbs equilibrium state Peq ({n j })∝ exp(−H {n j }/kT ). In this case, one can also show that the evolution matrix has only real (non-positive) eigenvalues, so solutions of (1) exhibit only decay in time, not oscillatory behavior [3]. In both analytic and simulation studies, the P({n j }, t) typically contain too much information to be manageable. It is thus common to focus on reduced quantities such as the probability that a single site k is occupied, P(n k , t) = P({n j }, t), and higher-order quantities such as spatial pair-correlations. nj = /k From (1), one can obtain a hierarchy of rate equations for these, which can be analyzed using approximate factorization relations to truncate the hierarchy at some low order, or by exact Taylor series expansions for short-time behavior [5]. Often, one has translational invariance due to periodic boundary conditions, so site quantities are independent of location, and pair-correlations depend only on separation of the pair of sites. Furthermore, behavior in the

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J.W. Evans

limit of infinite system size, L → ∞, is of primary interest. Then, in the context of KMC simulation, such reduced quantities can be obtained precisely from a single simulation for a sufficiently large system, rather than by averaging over several trials.

2.

Generic Kinetic Monte Carlo Simulation Algorithms

We first describe the two types of generic KMC algorithms applied to LG models. We then compare features of the two algorithms, and give an example of their application to a simple deposition process. Finally, we discuss some issues associated with the finite size of the simulation system. Below, we assume that these models incorporate a variety of distinct atomistic processes, which we label by α (e.g., α = adsorption, desorption, diffusion, reaction, etc.). Furthermore, we suppose that each process, α, occurs with only a finite number of microscopic rates, Wα (m), for m = 1, 2, . . . , depending on the local environment.

2.1.

Basic Algorithm

Here, we let Wα (max) denote the maximum of the Wα (m), for each α. We then set Wtot = α Wα (max), and define pα = Wα (max)/Wtot , so that α pα = 1. In the basic algorithm, one first randomly selects a site, then, selects a process, α, with probability, pα , reflecting the maximum rate for that process α. Finally, one implements this process (if allowed) with a probability, qα = Wα /Wα (max)≤1, where Wα is the actual rate for process α at site j. This means that Wα is one of the Wα (m), with m determined by the local environment of site j. It is also essential to connect the “simulation time,” i.e., the number of times a site is chosen, with the “physical time” in the stochastic LG model. On each occasion a site is chosen, we increment the physical time by δt, where Ld Wtot δt = 1. Thus, after one attempt per site, the physical time has increased by 1/Wtot .

2.2.

Bortz Algorithm

Here, we let Nα (m) denote the (finite) number of particles which can partake in process α with the mth rate, Wα (m). Then, the total rate for all particles in the system associated with this process α occurring at the mth rate is Rα (m) = Wα (m)Nα (m), and the total rate for all processes is Rtot = β n Rβ (n). The Bortz (or Bortz–Kalos–Lebowitz) simulation algorithm [6] maintains a list of these particles for each α and m. The simulation proceeds by selecting a sub-process (α, m) with probability pα (m) = Rα (m)/Rtot , then

KMC simulation of non-equilibrium LG models

1757

randomly selecting one of the Nα (m) particles capable of making this move from the corresponding list, and then implementing the process (after which lists have to be updated). Again, one must connect the “simulation time,” i.e., the number of times a process is chosen, with the “physical time” in the stochastic LG model. On each occasion when a process is implemented, one increments the physical time by δt = 1/Rtot .

2.3.

Comparison of Algorithms

In comparing standard and Bortz algorithms, it is appropriate to first note that often the rates Wα (m), described above, vary over many orders of magnitude. Furthermore, processes with high rates often have a low population of available particles, a feature which can apply not just under quasi-equilibrium conditions, but more generally. Thus, in the basic algorithm, after selecting a site, usually one selects a process α with a large Wα (max), but then typically fails to implement this process due to the small population of particles in this class. Thus, the basic algorithm is simple, but possibly inefficient due to the large fraction of failed attempts. In contrast, in the Bortz algorithm, one always implements the chosen process, so in this sense the algorithm has optimal efficiency. However, there is a substantial “book keeping” penalty in that one must maintain and continually update lists of length Nα (m) of particles for each sub-process (labeled by α and m). In practice, for complex models where processes have many rates, one may compromise between the two approaches accepting some fraction of failed attempts to avoid substantial additional complexity or cost in book-keeping.

2.4.

A Simple Example

To illustrate these features, consider irreversible island formation during submonolayer deposition [7]. Here, atoms deposit randomly at rate F per unit time at the adsorption sites on the surface represented by an L × L site square lattice (so d = 2) with coordination number z = 4, and with periodic boundary conditions. Adsorbed atoms (adatoms) then hop randomly to adjacent sites at rate h per unit time (in each of z = 4 directions) until meeting other diffusing atoms and irreversibly nucleating new (immobile) islands, or until irreversibly aggregating with existing islands. We assume some simple rule for incorporating into islands adatoms which land on top of islands, or which diffuse to island edges, where this rule does not involve additional processes with finite rates. Thus, the model is characterized by just two rates. Typically, F ∼ 10−2 /s, but h ∼ 105 −107 /s is many orders of magnitude higher, and this leads to a very low density of diffusing adatoms on the surface (∼10−5 −10−7 atoms per site).

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For this deposition model, we write α = “dep” (deposition), or “hop” (diffusive hopping), where each process is described by a single rate. In the basic algorithm, one has pdep = F/(F + zh) 1, and phop = zh/(F + zh) ≈ 1. Thus, after choosing a site, typically one attempts to hop, but fails due to the very low probability of that site being occupied by a diffusing adatom. Also, one increments time by δt = (F + zh)−1 L−2 . In the Bortz algorithm, one maintains a list of the Nhop diffusing adatoms and their positions. Then, one has Rdep = FL2 (as all sites are adsorption sites) and Rhop = zhNhop . Thus, at each Monte Carlo step, one chooses either to deposit with probability pdep = FL2 /(FL2 + zhNhop ), or to hop with probability phop = zhNhop /(FL2 + zhNhop ). For deposition, one randomly chooses a lattice site and deposits. For hopping, one randomly chooses one of the Nhop diffusing atoms from the list, and then implements the hop in a randomly chosen direction. After either event, the list of diffusing adatoms is updated. In particular, one must check for incorporation into an island, which leads to removal of the adatom from the list.

2.5.

Finite Size Effects

For large systems, the time increments δt described above are small. Thus, the above algorithms accurately represent the continuous-time dynamics of the stochastic lattice gas models. These algorithms also automatically produce an exponential waiting-time distribution between consecutive events for each particle. However, for small systems, the increments δt become significant on the time scale of the slowest process. To recover an accurate description of continuous kinetics and waiting-time distributions, in the basic algorithm, one could simply reduce all the pα by some factor ε 1, and correspondingly reduce all the δt by the same factor. Analogous refinements are possible in the Bortz algorithm. Instead, one can recover the exponential waiting-time distribution by setting δt = −ln(x)L−d /Wtot (basic algorithm), or δt = −ln(x)/Rtot (Bortz algorithm), where x is a random number chosen uniformly in [0,1]. For KMC simulation (in finite systems), there are fluctuations between different runs or trials in predictions of quantities at some specific time. Simplistically, fluctuations in some number, N (e.g., of adsorbed √ particles, of islands, etc.) should vary like the square root of the number, N. Such numbers typically scale linearly with the system size (i.e., the number of sites = Ld ), so the corresponding densities ρ = N/Ld are roughly size-independent. Thus, it follows that uncertainties in numbers (densities) should scale like Ld/2 (L−d/2 ). A more sophisticated analysis comes from applying general fluctuationcorrelation relations (the presentation in Landau and Binder, 2000, for equilibrium systems is readily generalized): (δN)2 = Ld Ctot, or equivalently that (δρ)2 = L−d Ctot , where Ctot represents the pair-correlations for the quantity of interest (e.g., adsorbed atoms, islands, etc.) summed over all separations.

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Finally, we discuss the effects of finite system size on mean behavior of quantities of interest. Usually, the choice of periodic boundary conditions is motivated by the desire to minimize such effects, and specifically to remove “edge effects”. In general, one expects finite size effects to be negligible when the linear system size, L, significantly exceeds the relevant spatial correlation length, L c . This condition is violated near “critical points” where L c → ∞.

3.

Simulation of Homoepitaxial Thin Film Growth and Relaxation

Homoepitaxial growth [8] involves random deposition of adatoms on a surface, and their subsequent diffusion. Adatom diffusion mediates nucleation of new islands, when suitable number of adatoms meet, in competition with growth of existing islands, when adatoms aggregate with island edges. In addition, the details of interlayer transport are critical in determining multilayer morphologies. Post-deposition relaxation often occurs on a much longer timescale than growth, and different processes may dominate, e.g., 2D evaporationcondensation at island edges.

3.1.

Tailored Models and Algorithms

Rather than developing generic models which might handle both growth and relaxation, often a more effective strategy is to develop “tailored” models. These focus on the essential atomistic processes (for the conditions of interest) which are described by a few key parameters. As an example, we describe a simple but effective model for metal(100) homoepitaxial growth with irreversible island formation [9]. As in the simple example used above, deposition occurs at rate F and subsequent hopping to adjacent sites at rate h. Diffusing adatoms irreversibly nucleate new islands upon meeting, and irreversibly aggregate with existing islands. Islands have compact near-square shapes in these systems due to efficient edge diffusion and kink rounding. Thus, once a diffusing atom reaches an island edge, it is immediately moved to a nearby doubly-coordinated kink site. This produces near-square individual islands, and describes reasonably growth coalescence shapes for impinging islands. Atoms landing on top of islands diffuse until nucleating new islands in higher layers, or until reaching island edges. In the latter case, adatoms can hop down to lower layers also with rate h if the step edge is kinked, but with reduced rate h < h, for a straight close-packed step edge. Finally, we incorporate “downward funneling” of atoms deposited right at step edges to adsorption sites in lower layers. See Fig. 1 for a schematic of these processes.

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F NUCLEATION

h'

RESTRICTED DOWNWARD TRANSPORT (h'
AGGREGATION

TERRACE DIFFUSION

h

Figure 1. Schematic of metal(100) homoepitaxy with irreversible island formation. The square grid represents the lattice of substrate adsorption sites. Adatoms reaching island edges are moved immediately to nearby kink sites [9].

Thus, the model has only three rates, h, h , and F. One would naturally apply a Bortz-type algorithm maintaining a list of all hopping atoms in all layers. Rather than maintain a separate list of atoms just above close-packed step edges which can hop down at a distinct reduced rate h , it is easier to include them in a single list of “hoppers”, but if hopping down is selected, then implement this process with probability pdown = h / h < 1. One can determine h by matching the observed submonolayer island density, and h by matching, e.g., the second layer population after deposition of 1 ml. Corresponding activation barriers come from assuming an Arrhenius form with a prefactor of ∼1013 /s. Then, matching F to experiment, the model has no free parameters. How does it do? For Ag/Ag(100) homoepitaxy at 300 K, a purported classic case of smooth quasi-layer-by-layer growth, it predicts initial smooth growth up to ∼30 ml, but then extremely rapid roughening up to ∼1500 ml. For lower temperatures, initial growth is rougher (as expected), but growth of thicker films is smoother than at 300 K (contrasting expectations). These predictions are supported by recent experiments, i.e., the tailored model works!

3.2.

Classically Exact Models and Algorithms (with Look-up Tables for Rates)

In contrast to tailored models, one could attempt to describe exactly adatom diffusion in all possible local environments during or after growth. Typically, the barrier for intra-layer diffusion will depend only on the occupancy of sites which are neighbors or next-neighbors to either the initial or final site of the hopping particle. For metal(100) surfaces represented by a square

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lattice of adsorption sites, there are 10 such sites. Then, one should specify rates or barriers for 210 =1024 possible local environments, ignoring symmetries [10, 11]. Thus, in the simulation algorithm, if hopping is chosen, one must assess the local environment of the selected adatom, and determine the relevant rate which will be stored in a large look-up table. It is not possible to precisely determine so many barriers, and in fact film morphology may not be sensitive to the precise values of many of these: too low means the process is essentially instantaneous, too high means inoperative on the relevant time scale. For efficiency in simulation with look-up tables, it is reasonable to not implement processes with barriers above a certain threshold value, and perhaps to divide up all diffusing particles into a few classes (fast, medium, slow diffusers) for Bortz-type treatment [11]. This approach was introduced by Voter [10] to treat post-deposition diffusion of 2D islands in metal(100) homoepitaxial systems, and then adapted to treat film growth [11]. Originally, the values of barriers for rates were determined from Lennard–Jones or semi-empirical many-body potentials. Effort has been made to decompose this large set of diffusion processes into a few basic classes (which can aid simulation, as indicated above), and to develop reliable approximate formulae for barriers in various environments. Recently, at least a subset of key rates have been extracted from higher-level DFT calculations. However, we caution that even DFT may not have the accuracy to allow quantitative prediction of film morphologies.

3.3.

Self-Teaching or On-the-Fly Algorithms

There are a vast number of possible local configurations and rates for diffusing adatoms, but how many of these are practically important? Usually, most of these processes are associated with diffusion and detachment of adatoms at step edges, and one has some idea as to which are the most dominant processes. Thus, one strategy is to start with a smaller look-up table containing these key rates. Then, run the simulation using these rates, and any time a new local environment is generated in which an atom attempts to hop, stop the simulation, calculate the rate, insert the configuration and barrier value into the table, and continue the simulation [12, 13]. This approach could even be utilized to search for possible many-atom concerted moves in addition to probing single-atom hops.

3.4.

Hybrid Algorithms

Many thin film deposition systems exhibit large characteristic lateral lengths (e.g., large island separations). Consequently, rather than atomistic

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simulation of deposition and diffusion-mediated aggregation at island edges, it makes sense to adopt a continuum PDE description of the adatom density [14, 15]. The local nucleation rate can be determined from this density, and the nucleation process implemented stochastically with this rate. This approach has a significant advantage for reversible island formation with a high density of diffusing adatoms, as it is computationally expensive to follow all these particles in KMC. However, a continuum description of island growth can be problematic. Growth shapes are very sensitive to noise in the aggregation process for inefficient shape relaxation (the Mullins–Sekerka or DLA instability), and reliable continuum formulations are lacking for compact growth shapes due to efficient shape relaxation. Thus, it is natural to combine a continuum description of deposition, diffusion, and aggregation with an atomistic description of island shape evolution [16]. To grow islands, one tracks the cumulative total aggregation flux, and adds an atom when this reaches unity at a location chosen with a probability reflecting the local aggregation flux. Treatment of detachment from island edges is similar. Edge diffusion is treated atomistically as in a standard simulation.

3.5.

Other Algorithms

For island formation during deposition, island growth rates can be characterized precisely in terms of the areas of “capture zones” (CZs) which surround islands [14]. Combining this CZ-based formulation of island growth, together with a reliable characterization of the spatial aspects of nucleation, e.g., as primarily along CZ boundaries, one could imagine implementing a purely Geometry-Based Simulation (GBS) algorithm for island formation. As in the above hybrid approach, here one retains a stochastic component to the prescription of island nucleation [17]. Finally, we discuss tailored simulation algorithms for post-deposition coarsening of submonolayer island distributions in metal(100) homoepitaxial systems, where coarsening is mediated by the diffusion and coalescence of islands. Given the diffusion rates versus island size, one could develop the following simulation algorithm [18]: adopt a simple characterization of islands as squares with various sizes; let these undergo random walks with the appropriate diffusion rates; after each collision, replace two islands by a single island so as to preserve size.

4.

Simulation of Catalytic Surface Reactions

In catalytic surface reaction systems, the reactants are continually introduced as a gas above the surface. They adsorb (sometimes reversibly), usually diffuse across the surface, and react with coadsorbed species, producing

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product(s) which desorb. The reactant and product species are continually removed from the system by pumping. Thus, one has an open system which might achieve a time-independent steady-state, but this is not a Gibbs state. In simple models, it may also be possible to develop absorbing (or poisoned) states where the surface is completely covered by some non-desorbing species [19]. In more complex models, one may develop oscillatory states, although fluctuations preclude perfect periodic behavior.

4.1.

Basic Algorithms

If adsorption, desorption, diffusion, and reaction rates are comparable, then the basic KMC algorithm is effective. Consider the canonical monomer (A)– dimer (B2 ) reaction [19–21], which mimics CO-oxidation (A=CO and B2 =O2 ): A adsorbs reversibly at single empty sites; B2 adsorbs dissociatively and irreversibly at nearby pairs of empty sites; A may diffuse on the surface; adjacent A and B react to produce the product AB (=CO2 ) which immediately desorbs. For limited (non-reactive) desorption of A, upon increasing the adsorption rate of A relative to B2 , one finds a discontinuous non-equilibrium phase transition from a reactive steady state with low A-coverage, θA− , to a nearly A-poisoned steady state with high θA+ . This discontinuous transition disappears at a nonequilibrium critical point upon increasing the A desorption rate. See Fig. 2 for a schematic of the monomer-dimer reaction model and its steady-state behavior. As an aside, in the absence of desorption of A, this model exhibits a completely A-poisoned absorbing state [19]. From the general properties of finitestate Markov processes, any finite system must eventually evolve to such a state [3], while infinite systems can avoid such states indefinitely by remaining in other non-trivial steady states. Thus, KMC simulation must eventually reach such absorbing states (there are no true non-trivial steady states). However, in practice, this can take an immense amount of time, and the system resides in a pseudo-steady state which accurately reflects the true steady state of the corresponding infinite system. B

A

pA

dA

B B

B

A

A

pB2

A

k AB B

A h A

DISCONTINUOUS TRANSITION ⫹

θA θA

θA⫺

Low hA Low dA pA

BISTABILITY

θA

High hA

Low dA pA

Figure 2. Schematic of the monomer (A)–dimer (B2 ) surface reaction model which mimics CO-oxidation. Also shown is the variation of the steady-state coverage of A with adsorption rate, pA . Note the emergence of bistability with increasing hop rate, h A , of A.

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The above example illustrates that non-equilibrium steady states can exhibit phase transitions analogous to classic equilibrium systems. One cannot apply thermodynamic concepts geared to Hamiltonian systems, but KMC simulation combined with finite-size-scaling ideas borrowed from equilibrium theory is an effective tool to analyze their behavior. This remains true for more a realistic reaction model which incorporate rapid diffusion of CO, and interactions between adsorbed CO and O, although refined algorithms are needed for efficient simulation [20].

4.2.

“Constant-Coverage” Simulation Algorithms

In the conventional “constant-adsorption-rate” simulations of the above monomer-dimer model, if adsorption is selected as the process to be implemented, one chooses between attempting deposition of A or of B2 with probabilities reflecting their adsorption rates. A distinct “constant-coverage” simulation approach was suggested by Ziff and Brosilow [22]. Here, the structure of the conventional simulation algorithm is retained, except that now if adsorption is selected, one attempts to adsorb A (B2 ) if the current coverage is below (above) some prescribed target “constant-coverage” value, θA∗ , say. Furthermore, during the simulation, one tracks the fraction of attempts to adsorb A (rather than B2 ). The long-time value of this fraction determines the A adsorption rate corresponding to the prescribed coverage θA∗ . Thus, it determines the adsorption rate exactly at the discontinuous transition if one chooses θA− < θA∗ < θA+ . In summary, in conventional simulations of steady state behavior, one prescribes the A adsorption rate, and extracts the A coverage. In constant-coverage simulations, one prescribes the A coverage and extracts the A adsorption rate. Other variations are possible. Are the constant-adsorption-rate and constant-coverage simulations entirely consistent? Clearly, for conventional simulations in a small finite system, there are significant fluctuations in the steady-state A coverage. Such fluctuations are “artificially” removed in the constant-coverage simulation approach, so one also should expect some differences in mean values of various quantities. However, in the limit of large system size where fluctuations in conventional simulations diminish, the two simulation approaches should converge.

4.3.

Hybrid Algorithms

In “real” CO-oxidation or related reactions, the surface diffusion or hop rate for CO is often many orders of magnitude above other rates. Also, since removal of CO from the surface is not diffusion-limited, but reaction-limited, there is a significant build-up of rapidly hopping CO molecules. This makes

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conventional simulation inefficient. However, rapid mobility and reactionlimited removal of CO also mean that the CO should be quasi-equilibrated within the complex geometry of the relatively immobile coadsorbed reactant O. This suggests a hybrid approach wherein the distribution of CO is described by some simple analytic equilibrium procedure, and the O distribution is described by conventional LG KMC simulations [20, 23]. Here, reaction of a specific O to form CO2 is determined from the equilibrium probability of finding an adjacent CO. Next, we discuss application of the hybrid approach to the monomer–dimer reaction with infinitely mobile adsorbed A, which does not interact with other adsorbed A or B (other than through reaction with B). Now A will be randomly distributed on sites not occupied by B. Thus, in our hybrid simulation procedure, we track the location of all adsorbed Bs with a LG simulation, but just the total number of adsorbed A. From this number, one can readily determine the (spatially uniform) probability that any non-B site is occupied by A, and thus determine reaction rates, etc. The most dramatic consequence of replacing finite mobility of A with infinite mobility is that the discontinuous transition described above is replaced by bistability, i.e., stable reactive and near-poisoned states coexist for a range of A adsorption rates [24]. Bistability is also obtained from a mean-field rate equation treatment of the chemical kinetics. This is not surprising since mean-field equations apply to a well-stirred system (i.e., rapid surface diffusion). In this mean-field treatment, the two stable steady states are smoothly joined by a coexisting unstable state, all of which are readily determined from a steady-state rate equation analysis. In our hybrid model, one expects that an unstable steady state may exist. However, it will have a non-trivial distribution of adsorbed O, and cannot be readily analyzed by conventional (constant-adsorption-rate) simulations for which the system will always evolve away from the unstable state. However, efficient analysis of the non-trivial unstable state behavior is possible by simply implementing a constant-coverage version of the hybrid simulation code [20, 24]. By varying the target θ A∗ , one maps out both stable and unstable steady states.

5.

Outlook

KMC simulation has proved a tremendously successful tool for analyzing and elucidating the evolution of non-equilibrium LG models for a broad variety of cooperative phenomena (not just in physical sciences). This approach will continue to be applied effectively to analyze more complex and realistic models in traditional areas of investigation, as well as in new areas of cooperative phenomena. Recent variations and hybrid algorithms show great promise not only in more efficiently connecting atomistic processes with

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resulting behavior on far larger length scales, but just as significantly in providing fundamental insight into the key physics.

Acknowledgments Prof. Evans is supported by the USDOE BES SciDAC Computational Chemistry program (simulation algorithms) and Division of Chemical Sciences (surface reactions), and by NSF Grant CHE-0414378 (thin films). This work was performed at Ames Laboratory, which is operated for the USDOE by Iowa State University under contract No. W-7405-Eng-82.

References [1] A.F. Voter, F. Montalenti, and T.C. Germann, “Extending the time scale in atomistic simulation of materials,” Ann. Rev. Mater. Sci., 32, 321, 2002. [2] T. Liggett, Interacting Particle Systems, Springer-Verlag, Berlin, 1985. [3] N.G. Van Kampen, Stochastic Processes in Physics and Chemistry, North-Holland, Amsterdam, 1981. [4] D.P. Landau and K. Binder, A Guide to Monte Carlo Simulations in Statistical Physics, Cambridge UP, Cambridge, 2000. [5] J.W. Evans, “Random and cooperative sequential adsorption,” Rev. Mod. Phys., 65, 1281, 1993. [6] A.B. Bortz, M.H. Kalos, and J.L. Lebowitz, “A new algorithm for Monte Carlo simulation of Ising spin systems,” J. Comp. Phys., 17, 10, 1975. [7] M.C. Bartelt and J.W. Evans, “Nucleation and growth of square islands during deposition: sizes, coalescence, separations, and correlations,” Surf. Sci., 298, 421, 1993. [8] Z. Zhang and M.G. Lagally, (eds.), Morphological Organization in Epitaxial Growth and Removal, World Scientific, Singapore, 1998. [9] K.J. Caspersen, A.R. Layson, C.R. Stoldt, V. Fournee, P.A. Thiel, and J.W. Evans, “Development and ordering of mounds in metal(100) homoepitaxy,” Phys. Rev. B, 65, 193407, 2002. [10] A.F. Voter, “Classically exact overlayer dynamics: diffusion of rhodium clusters on Rh(100),” Phys. Rev. B, 34, 6819, 1986. [11] H. Mehl, O. Biham, I. Furman, and M. Karimi, “Models for adatom diffusion on fcc(001) metal surfaces,” Phys. Rev. B, 60, 2106, 1999. [12] G. Henkelman and H. Jonsson, “Long time scale kinetic Monte Carlo simulations without lattice approximation and predefined event table,” J. Chem. Phys., 115, 9657, 2001. [13] O. Trushin, A. Kara, and T.S. Rahman, “A self-teaching KMC method,” to be published, 2005. [14] M.C. Bartelt, A.K. Schmid, J.W. Evans, and R.Q. Hwang, “Island size and environment dependence of adatom capture: Cu/Co islands on Ru(0001),” Phys. Rev. Lett., 81, 1901, 1998. [15] C. Ratsch, M.F. Gyure, R.E. Caflisch, F. Gibou, M. Petersen, M. Kang, J. Garcia, and D.D. Vvedensky, “Level set method for island dynamics in epitaxial growth,” Phys. Rev. B, 65, 195403, 2002.

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[16] G. Russo, L.M. Sander, and P. Smereka, “Quasicontinuum Monte Carlo: a method for surface growth simulations,” Phys. Rev. B, 69, 121406, 2004. [17] M. Li, M.C. Bartelt, and J.W. Evans, “Geometry-based simulation of submonolayer film growth,” Phys. Rev. B, 68, 121401, 2003. [18] T.R. Mattsson, G. Mills, and H. Metiu, “A new method for simulating late stages of coarsening in island growth: the role of island diffusion and evaporation,” J. Chem. Phys., 110, 12151, 1999. [19] R.M. Ziff, E. Gulari, and Y. Barshad, “Kinetic phase transitions in an irreversible surface reaction model,” Phys. Rev. Lett., 56, 2553, 1986. [20] J.W. Evans, M. Tammaro, and D.-J.Liu, “From atomistic lattice-gas models for surface reactions to hydrodynamic reaction-diffusion equations,” Chaos, 12, 131, 2002. [21] E. Loscar and E.Z. Albano, “Critical behavior of irreversible reaction systems,” Rep. Prog. Phys., 66, 1343, 2003. [22] R.M. Ziff and B.J. Brosilow, “Investigation of the first-order transition in the A-B2 reaction model using a constant-coverage kinetic ensemble,” Phys. Rev. A, 46, 4630, 1992. [23] M. Silverberg and A. Ben Shaul, “Adsorbate islanding in surface reactions: a combined Monte Carlo – lattice gas approach,” J. Chem. Phys., 87, 3178, 1989. [24] M. Tammaro, M. Sabella, and J.W. Evans, “Hybrid treatment of spatiotemporal behavior in surface reactions with coexisting immobile and highly mobile reactants,” J. Chem. Phys., 103, 10277, 1995.

5.13 SIMPLE MODELS FOR NANOCRYSTAL GROWTH Pablo Jensen Laboratoire de Physique de la Mati`ere Condens´ee et des Nanostructures, CNRS and Universit´e Claude Bernard Lyon-1, 69622 Villeurbanne C´edex, France

1.

Introduction

Growth of new materials with tailored properties is one of the most active research directions for physicists. As pointed out by Silvan Schweber in his brilliant analysis of the evolution of physics after World War II [1] “An important transformation has taken place in physics: As had previously happened in chemistry, an ever larger fraction of the efforts in the field were being devoted to the study of novelty rather than to the elucidation of fundamental laws and interactions [. . .] The successes of quantum mechanics at the atomic level immediately made it clear to the more perspicacious physicists that the laws behind the phenomena had been apprehended, that they could therefore control the behavior of simple macroscopic systems and, more importantly, that they could create new structures, new objects and new phenomena [. . .] Condensed matter physics has indeed become the study of systems that have never before existed. Phenomena such as superconductivity are genuine novelties in the universe.” Among these new materials, those obtained as thin films are of outstanding importance. Indeed, the possibility of growing thin films with desired properties is at the heart of the electronics technological revolution (for a nice introduction to the history of that revolution, see Ref. [2]). Thin film technology combines the three precious advantages of miniaturization, assembly line production (leading to low cost materials) and growth flexibility (depositing successively different materials to grow complex devices). Recently, the search for smaller and smaller devices lead to the new field of nanostructure growth, where one tries to obtain structures containing a few hundred atoms. As a consequence, an impressive quantity of deposition techniques have been developed to grow carefully controlled thin films and nanostructures from atomic deposition [3]. 1769 S. Yip (ed.), Handbook of Materials Modeling, 1769–1785. c 2005 Springer. Printed in the Netherlands.

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While most of these techniques are complex and keyed to specific applications, Molecular Beam Epitaxy (MBE) [4] has received much attention from physicists [5], mainly because of its (relative) simplicity. A younger technique, which seems promising to grow nanostructured materials with tailored properties is cluster deposition. Here, instead of atoms, one uses a beam of preformed large “molecules” containing typically 10–2000 atoms, the clusters. This technique has been shown to produce films with properties different from those obtained by the usual atomic beams. It is reviewed in Ref. [6] and will be considered no further in this short chapter. Due to the technological impetus, a tremendous amount of both experimental and theoretical work has been carried out in this field, and it is impossible to summarize every aspect of it here. I will therefore concentrate on simple models adapted to understand the first stages of growth (the submonolayer regime).

2.

Nanostructures: Why and How

As argued in the Introduction, the miniaturization logic naturally leads to trying to grow devices at the nanometer scale. This domain is very fashionable nowadays and the interested reader can find several information sources: for a simple and enjoyable introduction to the progressive miniaturization of electronics devices, see Ref. [7]. For more technical discussions, see for example Refs. [8, 9] and the journals entirely devoted to this field [10]. The reader is also referred to the enormous number of World Wide Web pages, especially those quoted in Ref. [13]. Besides the obvious advantages of miniaturization (for device speed and density on a chip), it has been argued [9] that the (magnetic, optical and mechanical) properties of nanostructured films can be intrinsically different from their macrocrystalline counterparts. For example, recent studies of the mechanical deformation properties of nanocrystalline copper [11] have shown that high strain can be reached before the appearance of plastic deformation, thanks to the high density of grain boundaries. Nanoparticles are also interesting as model catalysts [12]. The usual technology to grow thin films is deposition of atoms from a vapor onto a flat substrate. This technique was mainly used to grow relatively thick films (thickness larger than 100 nm typically). Recent developments with MBE allowed to control the growth at the atomic level, and for several materials it is possible to grow atomically flat surfaces over many micrometers. The same is true for interfaces in multilayer films, which are interesting for applications in electronics and magnetism. I refer the reader interested in the techniques and applications of atom deposition to several reviews [3]. I will focus here on a particular direction: the control of the submonolayer regime, i.e., before deposition of a single monolayer. There are two main

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interests: from the fundamental point of view, this regime allows a clearer determination of the atomic processes present during growth (the “elementary” processes to be described below). The models presented in this paper are useful in this regime and have allowed to understand and quantify many aspects of this regime of growth. One can also justify the study of the formation of the first layer since it is a template for the subsequent growth of the film [14, 15]. To grow a periodic array of nanometer islands of welldefined sizes, a promising direction seems to be the growth of strained islands by heteroepitaxy, stress being an ordering force which can lead to order [16]. However, growth in presence of elastic forces is beyond the capabilities of the present models which only take into account some of their effects (see below and Ref. [17]). Therefore I will not discuss this important subfield further. There are already many good reviews on atomic deposition with different emphasis: for a simple introduction, see Refs. [5], for more technical presentations, see Refs. [18–21]. One can find also a comprehensive compilation of measurements and analysis of atomic diffusion [22] or one more specific to metal surfaces [23] or to metal atoms deposited on amorphous substrates [24] or on oxides [25].

3. 3.1.

Models of Atom Deposition Introduction to Kinetic Monte Carlo Simulations

Given an experimental system, how can one predict the growth characteristics for a given set of parameters (substrate temperature, incoming flux of particles . . . )?

3.1.1. A bad idea: molecular dynamics simulations A first idea – the “brute-force” approach – would be to run a molecular dynamics simulation (see Ref. [26]). It should be clear however that such an approach is bound to fail since the calculation time is far too large. The problem is that there is an intrinsically large time scale in the growth problem: the mean time needed to fill a significant fraction of the substrate with the incident atoms. An estimate of this time is fixed by tML , the time needed to fill a monolayer: tML 1/F where F is the atom flux expressed in monolayers per second (ML/s). Typically, the experimental values of the flux are lower than 1 ML/s, leading to tML ≥ 1 s. Therefore, there is a time span of about 13 decades between the typical vibration time of an atom (approximately given

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by the Debye frequency 10−13 s, the lower time scale for the simulations) and tML , rendering hopeless any “brute-force” approach.

3.1.2. Choosing clever elementary processes To reduce this time span, a more clever approach is needed. The idea is to “coarsen” the description by using elementary processes such as those sketched in Fig. 1. This idea is similar to the usual renormalization technique, but here one hides the shortest times (as one hides the highest energies) in “effective” parameters (see [1, 27] for a simple discussion on this point). For a discussion of the most relevant elementary processes for atomic deposition, see below and [18]. The rates of the different processes could in principle be calculated using the empirical or ab initio potentials, or be taken as parameters in the analysis. However, given the high number of possible processes it is more convenient to choose only some of them in the analysis. The advantage of this approach is that using a limited number of elementary processes allows to understand in detail their respective role in determining the growth characteristics. Moreover, a model with too many parameters can reproduce almost any experiment and it is dubious that meaningful comparisons can be obtained. The drawback of the “elementary processes” approach is that before interpreting an experiment in the framework of one of these models one has to be sure that no other process than those chosen is present, for otherwise the interpretation could be meaningless. The case-in-point example for warning against a too rapid interpretation of experiments by elementary processes is

(e) (a) (d) (b) (c)

Figure 1. Main elementary processes considered in this paper for the growth of films by atom deposition. (a) adsorption of a atom by deposition; (b) and (d) diffusion of the isolated atoms on the substrate; (c) formation of an island of two monomers by juxtaposition of two monomers (nucleation) (d) growth of a supported island by incorporation of a diffusing atom (e) evaporation of an adsorbed atom.

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the growth in the Pt/Pt(111) system. The initial experimental observations by the Comsa group had been thoroughly interpreted with a variety of elementary processes, only to discover, after several years, that the experimental results were determined by an unexpected process, not included in any of the simulations: contamination by CO adsorbates. . . See the full story in Ref. [28]. A simple physical rationale for choosing only a limited set of parameters is the following (see Fig. 2). For any given system, there will be a “hierarchy” of time scales, and the relevant ones for a growth experiment are those much lower than tML 1/F. The others are too slow to act and can be neglected. The problem is that which processes are relevant or not depends on the precise system under study. For example, for typical metal on metal systems, the evaporation time is larger than the time needed to break a single bond. Thus, evaporation can be neglected in the analysis even at high temperatures where atoms can detach from islands. For metal atoms deposited on some insulating surfaces, the contrary might be true: since the bond between an adatom and the substrate may be weaker than the bond between two metal adatoms,

characteristic time

diffusion inside substrate

island diffusion detachment evaporation 1/F edge diffusion diffusion on substrate Figure 2. Time scales of some elementary processes considered in this paper for the growth of films by atomic deposition. The relevant processes are those whose timescale are smaller than the deposition time scale shown by the arrow in the left. In this case, models including only atom diffusion on the substrate and along the island (or step) edges are appropriate.

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evaporation from the substrate occurs even at low temperatures for which islands are still stable and there is no adatom detachment.

3.1.3. Combining the elementary processes: Kinetic Monte Carlo Now, given a set of elementary processes, there are two possibilities to predict the growth. The oldest is to write “rate-equations” which describe in a mean-field way the effect of these processes on the number of isolated atoms (called monomers) and islands of a given size. The first author to attempt such an approach for growth was Zinsmeister [29] in 1966, but the general approach is similar to the old rate-equations, first used by Smoluchovsky for particle aggregation [30]. Recently, Bales and Chrzan [35] have developed a more sophisticated self-consistent rate-equations approach which gives better results and allows to justify many of the approximations made in the past. However, these analytical approaches are mean-field in nature and cannot reproduce all the characteristics of the growth. Two known examples are the island morphology and the island size distribution (see [35] and also recent developments to improve the mean-field approach in [31]. There is an alternative approach to predict the growth: Kinetic Monte Carlo (KMC) simulations. Here one simply implements the processes chosen in a computer program with their respective rates and lets the computer simulate the growth. KMC simulations are an exact way to reproduce the growth, in the sense that they avoid any mean-field approximation. Given the calculation speed of present-day computers, systems containing up to 4000 × 4000 lattice sites can be simulated in a reasonable time (a few hours), which limits the finite size effects usually observed in this kind of simulation. Let me now discuss in some detail the way KMC simulations are implemented to reproduce the growth, once a set of processes has been defined, with their respective rates νpro taking arbitrary values or being derived from known potentials. There are two main points to discuss here: the physical correctness of the dynamics and the calculation speed. Concerning the first point, it should be noted that, originally [32], Monte Carlo simulations aimed at the description of the equations of state of a system. Then, the MC method performs a “time” averaging of a model with (often artificial) stochastic kinetics: time plays the role of a label characterizing the sequential order of states, and need not be related to the physical times. One should be cautious therefore on the precise Monte Carlo scheme used for the simulation when attempting at describing the kinetics of a system, as in KMC simulations. Note that the KMC approach is fundamentally different from the usual Monte Carlo algorithm, where one looks for the equilibrium properties of a system, using the energy differences of the different configurations. Instead, in KMC, one is interested in the kinetics, using the different energy barriers for the transitions between the different configurations.

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Let me address now the important problem of calculation speed. One could naively think of choosing a time interval t smaller than all the relevant times in the problem, and then repeat the following procedure: (1) choose one atom randomly (2) choose randomly one of the possible processes for this atom (3) calculate the probability ppro of this process happening during the time interval t ( ppro = νpro t) (4) throw a random number pr and compare it with ppro : if ppro < pr perform the process, if not go to the next step (5) increase the time by t and go to (1) This procedure leads to the correct kinetic evolution of the system but might be extremely slow if there is a large range of probabilities ppro for the different processes (and therefore some ppro 1). The reason is that a significant fraction of the loops leads to rejected moves, i.e., to no evolution at all of the system. Instead, Bortz et al. [33] have proposed a clever approach to eliminate all the rejected moves and thus reduce dramatically the computational times. The point is to choose not the atoms but the processes, according to their respective rate νpro and the number of possible ways of performing this process (called pro ). This procedure can be represented schematically as follows: (1) update the list of the possible ways of performing every possible process pro (2) randomly select one of the process, weighting the probability of selection by the process rate νpro and pro : ppro = (νpro pro ) processes pro νpro (3) randomly select a atom for performing this process (4) move the atom (5) increase the time by dt = (6) goto (1)

processes pro νpro

−1

This procedure implies a less intuitive increment of time, and one has to create (and update) a list of all the pro constantly, but the acceleration of the calculations is worth the effort. A serious limitation of KMC approaches is that one has to assume a finite number of local environments to obtain a finite number of parameters. This confines KMC approaches to regular lattices, thus preventing a rigorous consideration of elastic relaxation, stress effects . . . everything that affects not only the number of first or second nearest neighbors but also their precise position. Indeed, considering the precise position as in MD simulations introduces a continuous variable and leads to an infinite number of possible configurations or processes. Stress effects can be introduced approximately in KMC simulations [17] by allowing a variation of the bonding energy of an atom to an island as

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a function of the island size (the stress depending on the size), but it is unclear how meaningful these approaches are.

3.2.

Basic Ingredients of the Growth

What is likely to occur when atoms are deposited on a surface? I will analyze in detail the following elementary processes: deposition, diffusion and evaporation of the atoms and their interaction on the surface (Fig. 1). The influence of surface defects which could act as traps for the atoms is also addressed. The first ingredient of the growth, deposition, is quantified by the flux F, i.e., the number of atoms that are deposited on the surface per unit area and unit time. The flux is usually uniform in time, but in some experimental situations it can be pulsed, i.e., change from a constant value to 0 over a given period. Chopping the flux can affect the growth of the film significantly [36]. The second ingredient is the diffusion of the atoms which have reached the substrate. I assume that the diffusion is brownian, i.e., the atom undergoes a random walk on the substrate. To quantify the diffusion, one can use both the usual diffusion coefficient D or the diffusion time τ , i.e., the time needed by an atom to move by one diameter. These two quantities are connected by D ∼ d 2 /(4τ ) where d is the hop length. The diffusion is here supposed to occur on a perfect substrate. Real surfaces always present some defects such as steps [37], vacancies or adsorbed chemical impurities. The presence of these defects on the surface can significantly alter the diffusion of the atoms and therefore the growth of the film. A third process which could be present in growth is re-evaporation √ of the atoms from the substrate after a time τe . It is useful to define X S = Dτe the mean diffusion length on the substrate before desorption. The last simple process I will consider is the interaction between atoms. The simplest case is when (a) atoms ignore each other as long as they are not immediate neighbors (b) atoms attach irreversibly upon contact. Point (a), commonplace in all simulations until recently, has been challenged by precise calculations of the potential felt by an atom approaching another atom or an island [38]. It has been shown that, for some systems, past the short range, a repulsive ring is formed around the adatoms (Fig. 3). The magnitude of the repulsion can be comparable to the diffusion barrier. Therefore, not taking this repulsive effect into account can lead to island densities much larger than experimentally observed. It remains to be seen how general this repulsive ring is. Point (b) is not correct at high temperatures, because atom-atom bonds can be broken. This situation is discussed in Section 4.2. The usual game for theoreticians is to combine these elementary processes and predict the growth of the film. However, experimentalists are interested in

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⫺1.0 ⫺1.5

log10(NX), ML

⫺2.0 ⫺2.5 ⫺3.0 ⫺3.5

ε R →∞

⫺4.0

DFT-kMC ε R ⫽25 meV Nucleation Theory

⫺4.5 1.4

1.8 1/T, K

2.2

2.6⫻10⫺2

Figure 3. Arrhenius plot of the island density as a function of temperature from an impermeable repulsive ring (squares), a KMC model including the repulsion (circles), a simplified KMC model including a repulsive ring with 25 meV (diamonds), and nucleation theory (not including the repulsion efect (triangles). After Ref. [38].

the reverse strategy: from (a set of) experimental results, they wish to understand which elementary processes are actually present in their growth experiments and what are the magnitudes of each of them (this is what physicists call “understanding a phenomenon”). The problem, of course, is that with so many processes, many combinations will reproduce the same experiments. Then, some clever guesses are needed to first identify which processes are present. I gave several hints in a previous review [6] and will not address this question in detail here.

4.

Predicting Growth with Computer Simulations

“Classical” studies [19] have focused on the evolution of the concentration of islands on the surface as a function of time, and especially on the

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saturation island density, i.e., the maximum of the island density observed before reaching a continuous film. The reason is of course the double possibility to calculate it from rate-equations and to measure it experimentally by conventional microscopy. I will show other interesting quantities such as island size distributions which are measurable experimentally and have been recently calculated by computer simulations [40, 41]. Since I am only interested in the submonolayer regime, there is no need to take into account atoms falling on preexisting islands, except for the asymptotic case of strong evaporation discussed in Ref. [40]. Most metal on metal growth corresponds to this case, while metal on insulating surfaces grows by forming 3d islands (this is called the Wolmer–Weber growth mode, see for example Refs. [42]).

4.1.

Two Dimensional Growth: Irreversible Aggregation

I first study the formation of the islands in the limiting case of irreversible aggregation, for two growth hypothesis: negligible or important evaporation.

4.1.1. Complete condensation Let me start with the simplest case where only diffusion takes place on a perfect substrate (no evaporation). Figure 4a shows the evolution of the monomer (i.e., isolated atoms) and island densities as a function of deposition time. We see that the monomer density rapidly grows, leading to a rapid increase of island density by monomer-monomer encounter on the surface. This goes on until the islands occupy a significant fraction of the surface, roughly 0.1%. Then, islands capture efficiently the monomers, whose density decreases. As a consequence, it becomes less probable to create more islands, and their number increases more slowly. When the coverage reaches a value close to 15%, coalescence starts to decrease the number of islands. The maximum number of islands at saturation Nsat is thus reached for coverages around 15%. Concerning the dependence of Nsat as a function of the model parameters, it has been shown that the maximum number of islands per unit area formed on the surface scales as Nsat (F/D)1/319. Simulations [6, 35, 39] and theoretical analysis [34] have shown (Fig. 6) that the precise relation is Nsat = 0.53(Fτ )0.36 for the ramified islands produced by pure juxtaposition. This relation is very important since it allows, from an experimental measure of Nsat , to determine the value of τ (F is generally known), provided one knows that the simple hypothesis made are appropriate for the experiments. To show that this limiting case is not only of theoretical interest, let me show an experimental example. Thanks to a technological innovation, a scanning

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3 islands 4

5 monomers

6

5

4

3

2

log ( Ft (ML))

1

0

condensation 2 islands 4

6

3

2 1 log ( Ft (ML))

0

Figure 4. Evolution of the monomer and island densities as a function of the thickness (in monolayers), for islands formed by irreversible aggregation: (a) complete condensation, F = 10−8 , τe = 1010 , τ = 1 (leading to X S = 105 and CC = 22) (b) important evaporation, F = 10−8 , τe = 600 (τ = 1) (X S = 25 and CC = 22). CC represents the mean island separation at saturation for the given fluxes when there is no evaporation [40]. The length units correspond to the atomic diameter. In (b) the “condensation” curve represents the total number of particles actually present on the surface divided by the total number of particles sent on the surface (Ft). It would be 1 for the complete condensation case, neglecting the monomers that are deposited on top of the islands. The solid line represents the constant value expected for the monomer concentration (equal to Fτe ).

tunneling microscope operating a very low temperatures, a group in Lausanne University could observe, for the first time, the beginning of the growth of a film at the atomic scale [43]. Working at very low temperatures (50 K) is essential to “hide” many elementary processes (which cannot be thermally excited) and render the growth simple enough, so that the naive models of theoreticians can be relevant (for an introduction to the strategies used by physicists to understand nature, see [44]). Figure 5 shows that simple models as the ones presented in this paragraph are able, in these conditions, to reproduce in detail the experimental results.

4.1.2. Evaporation What happens when evaporation is also included? Figure 4b shows that now the monomer density becomes roughly a constant, since it is now mainly determined by the balancing of deposition and evaporation. As expected, the constant concentration equals Fτe (solid line). The number of islands increases linearly with time, since the island creation rate is given by the probability of atom-atom encounter, which is roughly proportional to the square atom concentration. We also notice that only a small fraction (1/100) of the monomers do effectively remain on the substrate, as shown by the low condensation

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(a)

(b)

Figure 5. Comparison of the morphologies of experimental (silver atoms deposited on platinium, a–c) and predicted with KMC models (d–f) submonolayer films of different thicknesses (see text). These figures show a small portion of the surface, 160 atomic diameters wide. To adjust the experimental results, we had to take the following rates for the elementary processes: a diffusion hop every 2 ms, thirty atoms being deposited every second on this square.

coefficient value at early times. This can be understood by noting that the islands grow by capturing only the monomers that are deposited within their “capture zone” (comprised between two circles of radius R and R + X S ). The other monomers evaporate before reaching the islands. As in the case of complete condensation, when the islands occupy a significant fraction of the surface, they capture rapidly the monomers. This has two effects: the monomer density starts to decrease, and the condensation coefficient starts to increase. Shortly after, the island density saturates and starts to decrease because of island-island coalescence. Figure 6 shows the evolution of the maximum island density in the presence of evaporation. A detailed analysis of the effect of monomer evaporation on the growth is given in Ref. [40], where is also discussed the regime of “direct impingement” which arises when X S ≤ 1: islands are formed by direct impingement of incident atoms as first neighbors of adatoms, and grow by direct impingement of adatoms on the island boundary. An experimental observation of the evaporation regime can be found in Ref. [45].

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log10(Nsat (per site))

2

3

no evap

4

evap, τe⫽100 τ mobile islands 5

14

10

6 log10(F)

2

Figure 6. Saturation island density as a function of the flux for different growth hypothesis indicated on the figure, always in the case of island growth by irreversible aggregation. “no evap” (circles) means complete condensation. Triangles show the densities obtained if τe = 100τ . In the preceding cases, islands are supposed to be immobile. This hypothesis is relaxed for the last set of data, “mobile islands” (squares) , where island mobility is supposed to decrease as the inverse island size [39] (there is no evaporation). The dashed line is an extrapolation of the data for the low normalized fluxes. Fits of the different curves in the low-flux region give: “no evap” (solid line): Nsat = 0.53(Fτ )0.36 ; “evap”(dotted line): Nsat = 0.26F 0.67 τ −1/3 τe (for the τ and τe exponents, see [40]) and “mobile islands” (dashed line): Nsat = 0.33(Fτ )0.42 .

4.2.

Reversible Aggregation

Previous results were obtained by assuming that atom-atom aggregation is irreversible. It is physically clear that at high temperatures atoms can detach from islands, and this has to be included in the models. The rate-equations approach [19] introduce a critical size i ∗ defined as follows: islands containing up to i ∗ atoms decay, while larger islands are stable. This means that only the concentration of sub-critical islands is in equilibrium with a gas of monomers. The concept of critical size was adopted for practical reasons (it simplifies the mathematical treatment) even if the macroscopic thermodynamical notions implicitly employed are difficult to justify for such small systems [18]. A more

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satisfactory approach was developed with the help of KMC simulations [41, 46]: instead of defining arbitrarily a critical size, one uses binding energies for atoms and studies which islands grow and decay. KMC simulations have shown that the morphology of the submonolayer films change dramatically from ramified to compact islands as the ratio of bond energy to substrate temperature is varied (Fig. 7) and that the critical size is ill defined, the control parameter being the ratio of the dimer dissociation rate to the rate of adatom capture by dimers [41, 46]: λ=

N2 /τ1 Dρ N2

(1)

where τ1 is the mean time for a dimer to dissociate, D is the diffusion constant for monomers and ρ, N2 represent the densities of adatoms and dimers respectively. The case λ ∼ 0 represents irreversible aggregation whereas large λ values mean that islands can dissociate easily. In the case of reversible atomic aggregation, the scope is to determine the aggregation parameter λ (defined in Eq. 1). This can be done in several ways [41]: (1) By studying the flux dependence of Nsat : the exponent depends on λ; (2) By measuring the island size distribution which also uniquely depends on λ; (3) By measuring the nucleation rate and studying its dependence on the incident flux.

(a)

(b)

(c)

Figure 7. Morphology of the films obtained with reversible aggregation for atomic deposition with different atom-atom bond energies. The temperature is fixed to 400 K, the activation energy for diffusion of isolated atoms to 0.45 eV, the flux to 1 ML/s and the thickness to 0.03 ML. The bond energies are: (a) 0.5 eV, (b) 0.2 eV and (c) 0.1 eV.

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Once λ has been found, it is in principle possible to extract the microscopic parameters, even if in practice uncertainties remain because of the limited amount of experimental data generally available and the high number of fit parameters (for examples of such fits see [15, 41, 46].

5.

Conclusion

Modeling crystal growth is a rapidly evolving field. This is due to rapid developments in the experimental side: near-field microscopy (for example, scanning tunneling microscopy), control of the growth conditions (low temperature, vacuum). Thanks to all these improvements, experiments can be carried out on “theoretical” surfaces, namely carefully controlled surfaces similar to those that theoreticians can study. On the theoretical side, better algorithms to combine the different growth ingredients have been developed, and we now have better methods to predict atom-atom interaction (mainly the ab initio approach). For a recent informal review, see [47]. Many challenges remain, however: predicting, from atomistic level simulations, the behavior of the system on a macroscopic scale, which is difficult mainly when several intermediate scales are relevant (for example if elastic interactions are important); predicting, from precise simulations carried out over static configurations or, at best, nanoseconds, the behavior of a system over seconds or hours. These are not challenges only for surface science but also for physics in general (modeling of brittle or ductile fracture, ageing phenomena. . .), which leaves some hope that other fields will help us solving our problems!

References [1] S.S. Schweber, Physics Today, pp. 34, November, 1993. [2] Michael Riordan and Lillian Hoddeson, Crystal Fire: The Invention of the Transistor and the Birth of the Information Age, Sloan Technology Series, W.W. Norton, 1998. [3] For an introduction to this enormous field, see for example: F. Rosenberger, Fundamentals of Crystal Growth, Springer, 1979; J.A. Venables, Surf. Sci., 299/300, 798, 1994. [4] M.A. Herman and H. Sitter, Molecular Beam Epitaxy, Springer-Verlag, Berlin, 1989. [5] M. Lagally, Physics Today, 46(11), 24 (1993); Z. Zhang and M.G. Lagally, Science, 276, 377, 1997; P. Jensen, La Recherche, 283, 42, 1996; A.-L. Barabasi and H.E. Stanley, Fractal Concepts in Surface Growth, Cambridge University Press, 1995. A. Pimpinelli and J. Villain, Physics of Crystal Growth, Cambridge University Press, 1998. [6] P. Jensen, Rev. Mod. Phys., 71, 1695, 1999. [7] R. Turton, The quantum dot, W.H. Freeman and Company Ltd., 1995. [8] H. Gleiter, Nanostructured Materials, 1, 1, 1992. [9] G.W. Nieman, J.R. Weertman, and R.W. Siegel, J. Mater. Res., 6, 1012, 1991.

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[10] Nanostructured Materials, Pergamon Press; Physica E, Elsevier Science; NanoLetters, American Chemical Society; Materials Today, www.materialstoday.com is a free magazine, which often deals with nanoscience. [11] J. Schiotz, T. Rasmussen, and K.W. Jacobsen et al., Phil. Mag. Lett., 74, 339, 1996. [12] C.R. Henry, Surf. Sci. Rep., 31, 235, 1998. [13] http://nano.gov; http://nanoweb.mit.edu; http://www.nano.org.uk [14] M.C. Bartelt and J.W. Evans, Phys. Rev. Lett., 75, 4250, 1995. [15] J.W. Evans and M.C. Bartelt, In: Surface Diffusion and Collective Processes, M.C. Tringides (ed.), Plenum, New York, 1997. [16] H. Brune et al., Phys. Rev. B, 52, R14380, 1995; H. Ibach, Surf. Sci. Rep., 29, 195, 1997. [17] M. Schroeder and D.E. Wolf, Surf. Sci., 375, 129, 1997; C. Ratsch et al., Phys. Rev. B, 55, 6750, 1997. [18] S.-L. Chang and P.A. Thiel, Critical Reviews in Surface Chemistry, 3, 239–296, 1994. [19] J.A. Venables, G.D.T. Spiller, and M. Hanb¨ucken, Rep. Prog. Phys., 47, 399, 1984. Note that some of the growth regimes predicted in this paper have been shown to be wrong (see [40]). [20] G.L. Kellogg, Surf. Sci. Rep., 21, 1, 1994. [21] H. Brune, Surf. Sci. Rep., 31, 125, 1998. [22] E.G. Seebauer and C.E. Allen, Prog. Surf. Sci., 49, 265, 1995. [23] R. Gomer, Rep. Prog. Phys., 53, 917, 1990. [24] A.A. Schmidt, H. Eggers, and K. Herwig et al., Surf. Sci., 349, 301, 1996. [25] C.T. Campbell, Surf. Sci. Rep., 27, 1, 1994. [26] D. Frenkel and B. Smit, Understanding Molecular Simulation, Academic Press, 1996; M.P. Allen and D.J. Tildesley, Computer Simulation of Liquids, Clarendon Press, Oxford, 1987. [27] P. Jensen, Phys. Today, July, 58, 1998. [28] P. Feibelman, Phys. Rev. B, 60, 4972, 1999; J. Wu et al., Phys. Rev. Lett., 89, 146103, 2002. [29] G. Zinsmeister, Vacuum, 16, 529, 1966; Thin Solid Films, 2, 497, 1968; Thin Solid Films, 4, 363, 1969; Thin Solid Films, 7, 51, 1971. [30] M. Smoluchovsky, Phys. Z., 17, 557 and 585, 1916. [31] J.W. Evans and M.C. Bartelt, Phys. Rev. B, 66, 235410, 2002. [32] N. Metropolis, et al., J. Chem. Phys., 21, 1087, 1953. [33] A.B. Bortz, M.H. Kalos, and J.L. Lebowitz, J. Comp. Phys., 17, 10, 1975. [34] J. Villain, A. Pimpinelli, and L.-H. Tang, et al., J. Phys. I, France 2, 2107, 1992; J. Villain, A. Pimpinelli, and D.E. Wolf, Comments Cond. Mat. Phys., 16, 1, 1992. [35] G.S. Bales and D.C. Chrzan, Phys. Rev. B, 50, 6057, 1994. [36] P. Jensen and B. Niemeyer, Surf. Sci. Lett., 384, 823, 1997. [37] Hyeong-Chai Jeong and Ellen D. Williams, Surf. Sci. Rep., 34, 171, 1999. [38] Kristen A. Fichthorn and Matthias Scheffler, Phys. Rev. Lett., 84, 5371, 2000. [39] P. Jensen, A.-L. Barab´asi, and H. Larralde, et al., Nature 368, 22, 1994; Phys. Rev. B, 50, 15316, 1994. [40] P. Jensen, H. Larralde, and A. Pimpinelli, Phys. Rev. B, 55, 2556, 1997, Note that in this paper, a mistake was made in the normalization of the island size distributions (Fig. 9), This mistake is corrected in [6]. [41] C. Ratsch, P. Smilauer, and A. Zangwill et al., Surf. Sci. Lett., 329, L599, 1995. [42] A. Zangwill, Physics at surfaces, Cambridge University Press, Cambridge, 1988. [43] H. R¨oder et al., Nature 366, 141, 1993.

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[44] P. Jensen, Entrer en mati`ere: les atomes expliquent-ils le monde? in French, Seuil, 2001. [45] R. Anton and P. Kreutzer, Phys. Rev. B, 61, 16077, 2000. [46] M.C. Bartelt, L.S. Perkins, and J.W. Evans, Surf. Sci. Lett., 344, L1193, 1995. [47] P.J. Feibelman, J. Vac. Sci. Tech., A21, S64, 2003.

5.14 DIFFUSION IN SOLIDS G¨oran Wahnstr¨om Chalmers University of Technology and G¨oteborg University Materials and Surface Theory, SE-412 96 G¨oteborg, Sweden

A knowledge of diffusion in solids is necessary in order to describe the kinetics of various solid state reactions such as phase transformations, creep, annealing, precipitation, oxidation, corrosion, etc., all fundamental processes in materials science. There are two main approaches to diffusion in solids [1–5]: (i) the atomistic approach, where the atomic nature of the diffusing entities is explicitly considered; and (ii) the continuum approach, where the diffusing entities are treated as a continuous medium and the atomic nature of the diffusion process is ignored. Many useful results and general relations can be obtained within the continuum approach, but a more complete picture is obtained if the atomic motions are considered. Macroscopic quantities, such as diffusion fluxes, can then be related to microscopic quantities, such as atomic jump frequencies. Knowledge of how atoms move in solids is also intimately connected with the study of defects in solids.

1.

The Diffusion Equation

In the continuum approach the diffusion coefficient D is introduced through the Fick’s law which expresses the flux of particles j(r, t) in terms of the gradient of the particle concentration n(r, t) at the same position r and time t j(r, t) = −D∇n(r, t)

(1)

To arrive at the standard diffusion equation Fick’s law is combined with the equation which describes the conservation of particles, ∂n(r, t) + ∇ · j(r, t) = 0 (2) ∂t which implies that ∂n(r, t) (3) = D∇ 2 n(r, t) ∂t 1787 S. Yip (ed.), Handbook of Materials Modeling, 1787–1796. c 2005 Springer. Printed in the Netherlands.

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We have here assumed that D itself is independent on concentration. The solution to this equation is obtained exploiting the Fourier transform method. It can be written on the form 1 2 n 0 (r )e−(r−r ) /4Dt dr (4) n(r, t) = 3/2 (4π Dt) where n 0 (r) = n(r, t = 0) is the initial particle concentration. If boundary conditions have to be specified at finite distances the Fourier series expansion method has to be used. Several different diffusion coefficients can be defined. The tracer or selfdiffusion coefficient Ds describes the diffusive behavior of a given, or tagged, particle. Experimentally, that can be measured using a small amount of radioactive isotopes. The density of the tagged particle is described by the probability p(r, t)dr to find the particle at time t in the volume element dr at r, and is given by ∂ p(r, t) (5) = Ds ∇ 2 p(r, t) ∂t which is identical to Eq. (3) except for that D is replaced by Ds . The probability to find the tagged particle at position r at time t, given that it was located at r = 0 at time t = 0, can be obtained from the general solution (4), i.e., 1 2 e−r /4Ds t (6) p(r, t) = (4π Ds t)3/2 This Gaussian function describes the diffusive spreading of the probability distribution. The width is equal to the mean squared displacement of the tagged particle motion, R2 (t) = 6Ds t, and can be used as a definition of the selfdiffusion coefficient. 1 (7) Ds = R2 (t) 6t where the notation · · · is used for the averaging procedure. Equation (5) is based on the assumption that the motion is diffusive. For short times, the particle motion deviates from purely diffusive behavior and Eq. (7) becomes invalid. Therefore, the definition of Ds should be supplemented with the condition that t > τ0 , where τ0 is a suitable microscopic time-scale. The various diffusion coefficients depend on the thermodynamic variables, i.e., temperature, pressure and composition. It is well known that diffusion coefficients in solids generally depend rather strongly on temperature, being very low at low temperatures but appreciable at high temperatures. Empirically, this dependence can often be described by the Arrhenius formula D = D0 e−Ea /kB T

(8)

where D0 is commonly referred to as the pre-exponential factor and E a the activation energy for diffusion.

Diffusion in solids

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The Continuum Approach

In the general case the situation can be quite complicated [1–3]. In a multi-component system one has to introduce material fluxes for each component i, ji (r, t) = −

Di j ∇n j (r, t)

(9)

j

The gradient in the concentration of one species may contribute to the flux of another species, described by the off-diagonal components of Di j . The diffusion coefficient is a function of composition as well as of temperature and pressure (or more generally stress). If a temperature or pressure gradient is present that may also introduce material fluxes and the Fick’s law of diffusion has to be generalized. Thermodynamic equilibrium demands not only that temperature and pressure be the same throughout a system but also that the chemical potential be everywhere the same. Therefore, the gradient of the chemical potential should enter in a more general description of diffusion. The theory of non-equilibrium thermodynamics is used to derive the general formalism for diffusion [3]. The theory put different phenomenological diffusion treatments together into a coherent structure. It is a linear theory and expresses the fluxes of the different species Ji in terms of suitable defined forces X j acting on these species, according to Ji =

Lij X j

(10)

j

where the phenomenological coefficients L i j are the basic kinetic parameters in the theory. In general, they will be functions of the usual thermodynamic variables, but they are independent on the forces X j . An important theorem, the Onsager reciprocity theorem, states that the matrix L is symmetric, i.e., L i j = L j i . This relation derives from the underlying atomic dynamics of the system and ultimately from the principle of detailed balance in statistical mechanics. In an isothermal, isobaric system the appropriate force is the gradient of the chemical potential X j = −∇µ j , and the corresponding transport coefficient L i j is related, but not equal, to the diffusion coefficient Di j . For instance, although by Onsager’s theorem L i j = L j i , it does not follow that Di j = D j i . In a non-isothermal system the equations must also include the heat flow Jq and a corresponding thermal force X q = −∇T /T . The corresponding set of coupled diffusion equations are derived by supplementing Eq. (10) with the particle conservation law. Numerical software packages for solution of multi-component diffusion equations have been developed [6]. An important application is the simulation of diffusion controlled transformations in alloys of practical importance. Necessary input is kinetic and thermodynamic data. These are derived by collecting and selecting

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0.70

WEIGHT-PERCENT C

0.65 0.60 0.55 0.50 0.45 0.40 0.35 0.30 L.S. Darken: AIME 180(1948)430-438

0.25 0

E-3

5

10

15

20 25 30 DISTANCE

35

40

45 50

Figure 1. Simulated carbon concentration profile in a weld between two steels with initially similar carbon but different silicon contents see (http://www.thermocalc.com/Products/ Dictra.html).

experimental data from the literature. The progress of various solid state phase transformations can then be simulated. In Fig. 1 a result from a diffusion simulation is shown produced by the software package DICTRA [6].

3.

The Atomic Mechanism of Diffusion

The continuum approach is phenomenological. It does not give information on the nature of the diffusive motion. In order to describe the diffusion phenomena properly a knowledge of the underlying atomic mechanisms is required. Atoms in a solid vibrate around their equilibrium positions. Occasionally these oscillations become large enough to allow an atom to change site. It is these jumps from one site to another which gives rise to diffusion in a solid. The atomic jumps in a solid are rare on a microscopic time scale. The self-diffusion coefficient is about 10−8 cm2 /s near the melting point in most closed packed metals. The lattice spacing is of the order 10−8 cm which implies, using Eq. (7), that the atoms change site about 107 times/s. This should be compared with the vibrational frequency which is 1013 –1014 per second. Thus even near the melting point the great majority of the time the atom is oscillating about its equilibrium position in the crystal. It changes site only on one oscillation in 104 or 105 .

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There are two common mechanisms by which atoms can diffuse through a crystalline solid, the vacancy and the interstitial mechanism. These are schematically illustrated in Fig. 2. For bulk diffusion in closed packed metals the vacancy mechanism is most important. Near the melting point the vacancy concentration is about 10−3 –10−4 site fraction in most metals. These vacancies allow the atoms to move, and this mechanism is operating in most cases with jumps to nearest neighbor sites or also to next nearest neighbor sites in bcc crystals. At high temperatures vacancy aggregates as divacancies may be present and influence the diffusivity. Curvature in the Arrhenius plot of self-diffusion is commonly interpreted as resulting from a monovacancy jump process at low temperatures with an increasing contribution from a divacancy jump process at higher temperatures [1]. That interpretation has recently been questioned based on computer simulations and it is argued that the curvature could be equally well interpreted by a single vacancy mechanism with a temperature-dependent activation energy [7]. At high temperatures interstitials may also be present but due to the high formation energy these defects are in most cases assumed to give no contribution at equilibrium. Substitutional atoms usually also diffuse by the vacancy mechanism. Other mechanisms as various exchange mechanisms have been suggested [1]. At the present there is no experimental support for any such mechanisms in crystallized metals and alloys. However, in disordered solids these more cooperative motions are more likely operating. In the interstitial mechanism the atoms move from interstitial site to interstitial site. Usually small interstitial atoms, like hydrogen or carbon atoms in metals, diffuse through the lattice by this mechanism. The surrounding solvent atoms are not greatly displaced from the normal lattice sites. If the interstitial atom is nearly equal in size to the lattice atoms diffusion is more likely to occur by the interstitialcy mechanism [1]. Here the interstitial atoms does not move directly to another interstitial site. Instead it moves into a normal lattice site and the atom that was originally at the lattice site is pushed into a neighboring interstitial site.

a

b

Figure 2. Mechanisms of diffusion in crystals: (a) the vacancy mechanism (b) the interstitial mechanism.

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The Random Walk Model

The aim of the random walk model is to describe the observed macroscopic diffusion in terms of the atomic jumps which are the elementary processes in diffusion. It has been noted that the atomic jumps in a solid are rare on a microscopic time scale. The actual duration of an atomic jump is, however, short and can be neglected compared with the mean residence time at a site. This justifies an assumption of randomness of the atomic jumps. On the other hand, the total number of jumps over the period of hours or days is immense, about 108 each second, and a statistical treatment becomes justified. In the random walk models these aspects of the diffusive motion are taken into account. Consider a random walk on a simple cubic lattice with lattice spacing a. We assume that all sites are equally available and that the diffusing entities perform a series of uncorrelated jumps, i.e., we assume that interaction between diffusing entities and correlation effects can be neglected. If the jump vector for the ith jump is denoted by si , the total displacement after N jumps can be written as RN =

N

si

(11)

i=1

From symmetry considerations the mean displacement will be zero, R N = 0, while the mean-squared displacement is proportional to the number of jumps R2N =

N N i=1 j =1

si · s j =

i

si · si +

si · s j = N a 2

(12)

i= /j

where the last equality follows from the fact that we have assumed the jumps to be uncorrelated, i=/ j si · s j = 0. In many situations this is not the case and the analysis becomes much more complicated [5]. We can also write this in terms of the jump rate k between two neighboring sites R2N = ka 2 t

(13)

The jump rate is related to the mean residence time τ at a site according to 1/τ = nk, where n is the number of nearest neighboring sites. Furthermore, it can be related to the self-diffusion coefficient by comparing with Eq. (7), i.e., Ds =

a2 k 6

(14)

This very simple random walk model can be extended in many different directions [8] and the more complicated models are most often solved using the numerical Monte Carlo (MC) simulation technique.

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The random walk modeling can also be generalized by writing down the equation for the rate of change of the probability distribution directly. We obtain the following rate equation, or Master equation ∂ p(ri , t) = k j →i p(r j , t) − ki→ j p(ri , t) ∂t j

(15)

with ki→ j equal to the transition rate from i to j . If nearest neighboring jumps are assumed with the same jump rate it simplifies to 1 ∂ p(r, t) = p(r + ak , t) − p(r, t) ∂t nτ k

(16)

where ak is the set of vectors which connects a site with its nearest neighboring sites. This equation gives a more detailed spatial information of the diffusive motion compared with ordinary diffusion Eq. (5). To recover the latter equation we may expand the probability distribution around r, and use the symmetry. The diffusion Eq. (5) is then obtained with Ds = (1/6) · (a 2 /τ n). Equation (16) is most easily solved in Fourier space. We obtain I s (q, t) ≡

and S s (q, ω) ≡

dreiq·r p(r, t) = e−(q)t (q) 1 dt −iωt s e I (q, t) = 2 2π π ω + 2 (q)

(17)

(18)

with (q) =

1 (1 − e−iq·ak ) nτ k

(19)

Quasi-elastic neutron scattering can be used to study diffusion [3]. In that case the incoherent scattering cross-section is directly related to S s (q, ω) and by determining the width of the quasi-elastic peak as function of the scattering wave-vector a very detailed description of the diffusive motion may be obtained. In practise only relatively fast diffusing atoms can be studied with neutrons. Interstitial solutions of hydrogen in metals and fast ion conductors are among those which have been extensively studied in this way. The same quantities can also be obtained using the numerical moleculardynamics (MD) simulation technique. In Fig. 3 results from a MD simulation for hydrogen diffusion in palladium are compared with quasi-elastic neutron scattering data [9]. The width of the quasi-elastic peak is shown as function of wave-vector. The temperature is 623 K and a classical description of the hydrogen motion should be quite reasonable. The simulation data agree with experiments provided energy dissipation to both the lattice vibrations and the electron excitations are taken into account.

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(a)

(b) 18 The dimensionless half-width ∆

The dimensionless half-width ∆

18 15 12 9 6 3 0

0

4

8

12

aq

15 12 9 6 3 0

0

4

8 aq

12

16

Figure 3. The half-width ( ≡ (q)a 2 /Ds ) of the quasi-elastic peak as function of wavevector, in units of a, along (a) the 100 direction and along (b) the 110 direction at T = 623 K. •: quasi-elastic neutron scattering data; : molecular-dynamics simulation data (with coupling to lattice vibrations); : molecular-dynamics simulation data (with coupling to lattice vibrations and electronic excitations). The dotted line shows the results from Eq. (19). Reprinted with permission from Ref. [9]. Copyright (1992) by the American Physical Society.

5.

The Atomic Jump Frequency

The random walk model relates the atomic jumps to the macroscopic diffusion phenomena. An understanding of parameters entering the expression for the atomic jump frequency and related quantities is therefore of great interest. Direct calculations of those parameters are important, in particular, if accurate calculations can be performed without fitting to experimental data, so called first-principles or ab initio calculations. In vacancy and interstitial diffusion the diffusion coefficient will depend on the concentration of defects and the atomic jump frequency k. In vacancy diffusion the relevant jump frequency is the one of an atom into an adjacent vacancy and in interstitial diffusion it is the jump rate between different interstitial sites. Using equilibrium statistical mechanics the defect concentration can be expressed in terms of formation entalpies and entropies. The atomic jump frequency k is most often approximated using the absolute rate theory, or transition state theory, according to k=

kB T Q # h Q

(20)

where Q and Q # are the statistical mechanical partition functions evaluated for the system at a stable site and at the transition site, respectively. The

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transition site is defined as the hypersurface separating two stable sites. Assuming harmonic lattice vibrations Vineyard [10] derived the following expression for the transition rate N

j =1 ν j −E/kB T e ∗ j =1 v j

k = N−1

(21)

where the activation energy E (cf Eq. (8)) is the energy difference between the system located at a stable site and at the transition site or saddle point configuration. ν j are the N normal mode frequencies of the entire system at the stable site and ν ∗j the N −1 normal mode frequencies of the system constrained in the transition site. The various parameters entering the expressions for the defect concentration and the jump frequencies can be evaluated from first principles. In particular, the density functional theory has been applied extensively. Dynamics and finite temperature effects have also been considered from first principles. In Fig. 4 we show the result from such a calculation [7]. It is found that for aluminum the mono-vacancy diffusion alone dominates over diffusion due to divacancies and interstitials for all temperatures up to the melting point. The calculated diffusion rate agrees with experimental data over 11 orders of magnitude.

1000 900 ↑ Tm

800

←T(K) 700 10

10⫺12

D(m2/s)

10 10

600

v

10⫺14

⫺13

10

⫺16

2v

10⫺18

⫺14

i 1.2

10⫺15 10⫺16 10⫺17 10⫺18 1

500

⫺12

1.4

1.6

Tracer NMR Present work

1.2

1.4

1.6

1.8

2

1000/T(K⫺1)

Figure 4. Temperature dependence for the self-diffusion coefficient in aluminum as function of the inverse temperature. Open and filled circles are experimental data and the lines are from molecular-dynamics simulations. The inset shows calculated diffusion coefficients for vacancies (v), divacancies (2v), and interstitials (i). The contribution from divacancies and interstitials is less than 1% of that from mono-vacancies at the melting temperature. Reprinted with permission from Ref. [7]. Copyright (2002) by the American Physical Society.

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Outlook

In the past diffusion studies have been dominated by various experimental techniques and the development of the theoretical description. Software has been developed for accurate simulation of diffusion in solids based on experimental input. More recently ab initio computations and computer simulations have gained in importance. The first-principles or ab initio methods can be used to get insight and to obtain data for various elementary properties in relation to diffusion. If the diffusivity is high the MD simulation technique can be used to study diffusion in a very direct way. It provides well-controlled “experiments” and allows a proper check of the validity of the various theoretical descriptions. The method requires a description of the inter-atomic interaction as input and if that is sufficiently reliable the method provides a fairly reliable substitute to actual experiments. The Monte Carlo simulation technique can also be used to study diffusion. In that case a model for the kinetic description has to be established. The method is particularly useful for the study of diffusion in complex systems, like concentrated alloys and disordered materials. To conclude; it is not unlikely that the present time of diffusion studies will be characterized as the computational period.

References [1] J.L. Bocquet, G. Brebec, and Y. Limoge, “Diffusion in metals and alloys,” In: R.W. Cahn and P. Haasen (eds.), Physical Metallurgy, 4th edn., Elsevier Science BV, Amsterdam, pp. 535–668, 1996. [2] C.P. Flynn, Point Defects and Diffusion, Clarendon Press, Oxford, 1972. [3] A.R. Allnatt and A.B. Lidiard, Atomic Transport in Solids, Cambridge University Press, 1993. [4] P. Shewmon, Diffusion in Solids, 2nd edn., The Minerals, Metals and Materials Society, Pennsylvania, 1989. [5] J.R. Manning, Diffusion Kinetics for Atoms in Crystals, D. van Nostrand, Princeton, 1968. [6] A. Borgenstam, A. Engstr¨om, L.H¨oglund et al., “DICTRA, a tool for simulation of diffusional transformations in alloys,” J. Phase Equilibria, 21, 269, 2000. [7] N. Sandberg, B. Magyari-K¨ope, and T.R. Mattsson, “Self-diffusion rates in Al from combined first-principles and model-potential calculations,” Phys. Rev. Lett., 89, 065901, 2002. [8] J.W. Haus and K.W. Kehr, “Diffusion in regular and disordered lattices,” Phys. Rep., 150, pp. 263–416, 1983. [9] Y. Li and G. Wahnstr¨om, “Nonadiabatic effects in hydrogen diffusion in metals,” Phys. Rev. Lett., 68, 3444, 1992. [10] G.H. Vineyard, “Frequency factors and isotope effects in solid state processes,” J. Phys. Chem. Solids, 3, 121, 1957.

5.15 KINETIC THEORY AND SIMULATION OF SINGLE-CHANNEL WATER TRANSPORT Emad Tajkhorshid, Fangqiang Zhu, and Klaus Schulten Theoretical and Computational Biophysics Group, Beckman Institute for Advanced Science and Technology, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA

Water translocation between various compartments of a system is a fundamental process in biology of all living cells and in a wide variety of technological problems. The process is of interest in different fields of physiology, physical chemistry, and physics, and many scientists have tried to describe the process through physical models. Owing to advances in computer simulation of molecular processes at an atomic level, water transport has been studied in a variety of molecular systems ranging from biological water channels to artificial nanotubes. While simulations have successfully described various kinetic aspects of water transport, offering a simple, unified model to describe trans-channel translocation of water turned out to be a nontrivial task. Owing to its small molecular size and its high concentration in the environment, water is able to achieve significant permeation rates through different membranes, including biological cell membranes which are primarily composed of lipid bilayers. As such, water is generally exchangeable between various compartments of living organisms. However, due to the hydrophobic nature of the core of lipid bilayers, high permeation rates can only be achieved through devising additional pores in the bilayer that increase the permeability of water. These pores, known as channels, are primarily formed by folding and aggregation of one or more polypeptide chains inside the membrane. Aquaporins (AQPs) are the most prominent family of biological channels that facilitate transport of water across membranes in a selective manner. Other porins and channels also allow water molecules to pass, but they are either nonselective channels or mainly used for transport of other substrates, i.e., water is co-transported with other substrates through these channels.

1797 S. Yip (ed.), Handbook of Materials Modeling, 1797–1822. c 2005 Springer. Printed in the Netherlands.

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Water permeation through biological channels, such as AQPs, has been the subject of theoretical and experimental studies for many years [1]. Molecular dynamics (MD) simulations provide an ideal tool for investigating water transport through channels [2–5], since the movement of every single water molecule can be closely monitored in the simulations. A large body of evidence, including the recently solved structures of these channels and extensive MD simulations, have indicated that the pore region of selective water channels confines water molecules to a single file configuration, in which a highly correlated motion of neighboring, hydrogen-bonded water molecules governs the rate of diffusion and permeation of water through the channel. A very similar behavior of water has been reported in artificial water channels formed by carbon nanotubes (CNTs). This chapter presents a detailed description of water motion and permeation through water channels, through a comprehensive survey of the theory associated with single-channel water transport, methodologies developed to simulate such events, and comparison of experimental and calculated observables. The main objective is to provide the reader with a clear description of experimentally measurable properties of water channels. Our description links these properties to the microscopic structure and dynamics of channels. We show how observables like channel permeabilities can be examined by computer simulations and we present a mathematical theory of single-channel water transport.

1.

Structurally Known Biological Water Channels

AQPs are a family of membrane water channels for which crystallographic structures are available. They are present in nearly all life forms. In human, AQPs have been found in multiple tissues, such as the kidneys, the eye, and the brain. They form homo-tetramers in cell membranes, each monomer forming a functionally independent water pore, which does not conduct protons, ions or other charged solutes (Fig. 1a). A fifth pore, formed in the center of the tetramer, has been proposed to conduct ions under certain circumstances [6]. However, passive transport of water across cell membranes remains to be the main established physiological function of AQPs. Atomic resolution structures of aquaporin-1 (AQP1) [7–9] and the E. coli glycerol channel (GlpF) [10] have been employed in MD simulations characterizing the structure–function relationship of these channels in particular, regarding their selectivity [2–4, 11–14].

Kinetic theory and simulation of single-channel water transport (a)

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(b)

Figure 1. (a) Top view of a tetrameric AQP surrounded by lipidmolecules of a membrane. Each monomer constitutes an independent water pore. (b) An array of CNTs as a simplified model for the study of single channel water transport.

2.

Nanotubes as Simple Models of Water Channels

Synthetic pore-forming molecules, such as nanotubes, have attracted a great deal of attention recently. Due to their chemical simplicity, these artificial channels have been the subject of numerous experimental and theoretical studies [15, 16]. Simulation studies have employed CNTs as models for complicated biological channels, as they can be investigated more readily by MD simulations [17, 18] due to their simplicity, stability, and small size (Fig. 1b). Biological water channels are much more complex than CNTs, with irregular surfaces and highly inhomogeneous charge distributions. For example, MD simulations have revealed that water molecules in AQPs adopt a bipolar orientation which is induced electrostatically and is linked to the need that proton conduction must be prevented in AQP channels [3]. CNTs are electrically neutral, and may not reproduce some important features of biological channels. However, one may modify CNTs through the introduction of charges [18] to mimic various aspects of biological water channels. Computational studies have suggested that CNTs can be designed as molecular channels to transport water. Single-walled CNTs (with a diameter of 8.1 Å) have been studied recently by MD simulations. Simulations revealed that the CNTs spontaneously fill with a single file of water molecules and that water diffuses through the tube concertedly at a fast rate.

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Experimental Measurement of Transmembrane Water Transport

The key characteristics accounting for transport through water channels are the osmotic permeability ( pf ) and the diffusion permeability ( pd ) [1], both measurable experimentally. pf is measured through application of osmotic pressure differences, while pd is measured through isotope labeling, e.g., use of heavy water. In this section, we explain how water transport is characterized experimentally, and what are the most important properties used to characterize the rate of transport of water through channels. We will introduce and define pf and pd of water channels, and, in particular, investigate the relationship between the two for single-file water channels. When the solutions on the two sides of a membrane have different concentrations of an impermeable solute, water flows from the low concentration side to the other side. In dilute solutions, the net water flux through a singlewater channel, jW (mol/s), is linearly proportional to the solute concentration difference CS (mol/cm3 ): jW = pf CS ,

(1)

where CS (mol/cm3 ) is the concentration difference of the impermeable solute between the two reservoirs connected by the channel, jW (mol/s) is the net molar water flux through the channel, and pf (cm3 /s) is defined as the osmotic permeability of the channel [1]. In contrast, no net water flux is expected in equilibrium, i.e., when no solute concentration difference is present. It is, however, still of interest to study water diffusion through the channels for CS = 0. For this purpose, experiments have been designed where a fraction of water molecules is labeled, e.g., by isotopic replacement or by monitoring nuclear spin states, so that they can be traced. Assuming that the interactions of these so-called tracers with the membrane and with other water molecules are identical to those of normal water molecules, tracers can be used to study diffusion of water molecules through channels at equilibrium conditions. When the reservoirs on the two sides of a membrane have different concentrations of tracers, a diffusional tracer flux will be established down the concentration gradient, although the average net water flux (consisting of both tracers and normal water molecules) remains zero. The tracer flux jtr (mol/s) through a single channel is linearly proportional to the tracer concentration difference Ctr (mol/cm3 ): jtr = pd Ctr ,

(2)

where pd (cm3 /s) is defined as the diffusion permeability of the channel [1] (Fig. 2).

Kinetic theory and simulation of single-channel water transport

jw solute

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water

jtr tracer Figure 2. Schematic presentation of experimental procedures to measurediffusion and osmotic permeability of channels. (Top) Addition of an impermeable solute to one side of the channel establishes a chemical potential difference of water that drives water transport to the solute-rich side. (Bottom) In the absence of a chemical potential difference of water across the channel, labeled water molecules (tracers) can be used to monitor random diffusion of water from one side to the other side of the channel.

Different experimental techniques are used for measurement of Pf and Pd [19]. It is important to note that, due to difficulties in measuring water transport through single channels, almost all of the experimental setups measure water permeation through a membrane, and the measured permeabilities (Pf and Pd with capital P) are those of the entire membrane. To obtain singlechannel permeabilities, pf or pd , one needs to know the density of the channel in the membrane, i.e., the number of channels per unit area. However, the ratio pf / pd can be measured without the knowledge of the channel density [20]. Pd is measured in the absence of a chemical potential difference of water (balanced osmotic/hydrostatic pressure on the two sides of the membrane). There is no net transport of water under these conditions. In order to monitor

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random translocation of water molecules from one side to the other side of the membrane, special water molecules (tracers) are needed. Isotopic water (such as 3 H2 O) or water molecules with different nuclear spin states can be used for this purpose. Immediately after introduction of tracers, tracer concentrations of the two reservoirs are monitored directly or indirectly over time [19], and Pd can be determined from the decay rate of the concentration difference. Pf is usually measured in the presence of an osmotic pressure difference, i.e., a difference in solute concentration. Typical Pf measurements are performed on cells or liposomes (small lipid vesicles with embedded water channels), by exploiting the stopped-flow technique. In this setup, the solute concentration of the extracellular solution is suddenly changed, resulting in volume changes of the cells (or vesicles) due to the net water flux. The volume change can be inferred by monitoring light scattering from the suspension [21], and the net water flux determined from the rate of volume change. Pf for a planar membrane can be determined by measuring the ionic concentration distribution near the surfaces of the membrane [22].

4.

Theory of Single-file Water Transport

The theory and derivations presented in this section closely follow Zhu et al. [5, 23]. We define a permeation event as a complete transport of a water molecule through the channel from one reservoir to the other. Let q0 be the average number of such permeation events in one direction per unit time; the number of permeation events in either direction should be identical, resulting in a total number of 2q0 . q0 is an intrinsic property of a water channel and is independent of tracer concentration. Let us assume that one reservoir has a tracer concentration of Ctr , and (for the sake of convenience) that the other reservoir has zero tracer concentration. The ratio of tracers to all water molecules in the first reservoir is Ctr /CW , where CW = 1/ VW is the concentration of water, and VW (18 cm3 /mol) is the molar volume of water, which is usually assumed to be constant. Since according to our assumption tracers move just like normal water molecules, the same proportion (i.e., Ctr /CW ) should characterize water molecules permeating the channel. Consequently, the tracer flux can be related to the total number of water molecules permeating the channel (q0 ) by jtr = (1/NA )q0 (Ctr /CW ), where NA is Avogadro’s number. Therefore, pd and q0 are related by a constant factor: pd =

VW q0 = v W q0 , NA

where v W = VW /NA is the average volume of a single water molecule.

(3)

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Within narrow channels water molecules form a single file, and their movement along the channel axis accordingly is highly correlated. Recently, a continuous-time random-walk (CTRW) model was proposed [24] to describe the transport of single-file water in channels. This model assumes that the channel is always occupied by N water molecules, and the whole water file moves in hops (translocations that shift all water molecules by the distance separating two neighboring water molecules) simultaneously and concertedly, with leftward and rightward hopping rates kl and kr , respectively. In equilibrium, kl and kr have the same value, denoted as k0 . Due to strong coupling between the water molecules, local effects (energetic barriers arising from interaction with certain parts of the channel wall, access resistance at channel entrances, etc.) contribute to the hopping rate of the whole water file. Consequently, all factors affecting the kinetics of water movement are effectively integrated into this single parameter (k0 ). In the following, we will show that both pd and pf can be predicted by this model, in terms of N and k0 . Since the complete permeation of a water molecule from one end of the channel to the other end includes at least N + 1 hops (shifts) of the single file, one expects the rate of permeation events at equilibrium to be smaller than the hopping rate. Indeed, the number of uni-directional permeation events per unit time, q0 , is given by q0 =

k0 . N +1

(4)

Equation (4) has been proven from kinetics [24] as well as using a state diagram [18], and its validity was verified by MD simulations of CNTs [17, 18]. Combining Eqs. (3) and (4), pd can be expressed as: pd =

v W k0 . N +1

(5)

pf is measured when a net water flux is induced by different solute concentrations in two reservoirs. In this case, the chemical potentials of water in the two reservoirs are different (the difference denoted as µ). Consequently, the hopping rates (kr and kl ) of the two directions are no longer the same. We note that the yield of a hop is the transfer of one water molecule from one reservoir to the other, resulting in a free energy change of µ in the system. In analogy to the forward and backward rates of a chemical reaction, the ratio of kr to kl can be expressed by [25]:

−µ , kr /kl = exp kB T

(6)

where kB is the Boltzmann constant and T is the temperature. We note now that kr and kl are both functions of µ/kB T . Since under physiological conditions, µ is much smaller than kB T (e.g., µ/kB T = 0.0036

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for a 200 mM solution of sucrose, according to Eq. (10)), we can expand kr and kl to first order:

kr = k0 1 + α

µ , kB T

kl = k0 1 + β

µ kB T

(7)

(for a symmetric channel also holds α = −β). The net water flux can be expressed by the difference between kr and kl : jW =

1 k0 (α − β) µ . (kr − kl ) = NA NA kB T

(8)

Substituting Eq. (7) into Eq. (6) and comparing the first order terms in µ/kB T leads to β − α = 1. The net water flux is then: jW = −

k0 µ . NA k B T

(9)

For dilute solutions, µ is linearly proportional to the solute concentration difference [1]: µ = −kB T VW CS .

(10)

From Eqs. (9) and (10), we obtain jW = k0 v W CS

(11)

and using Eq. (1), pf = v W k0 .

(12)

According to Eqs. (5) and (12), the ratio of pf to pd predicted by the CTRW model is pf / pd = N + 1.

(13)

The difference between pf and pd can be further elaborated as follows. For single-file water transport, a hop results in the net transfer of one water molecule from one side of the channel to the other side. pf is related to the rate of net water transfer under a chemical potential difference and, therefore, is determined by the hopping rate (see Eq. (12)). In contrast, pd is determined by the rate of permeation events (see Eq. (3)). A permeation event requires an individual water molecule to traverse all the way through the channel, and is not the same as a hop. Actually, the pf / pd ratio is exactly determined by the relative rates of hops and permeation events. Most models proposed for single-file water transport predict this ratio to be N or N + 1 [1]. As stated before, most experimental techniques take advantage of osmotic pressure to establish a chemical potential difference that is needed for the determination of pf . A hydrostatic pressure difference P between the two

Kinetic theory and simulation of single-channel water transport

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reservoirs can also give rise to a difference in the chemical potential of water [1] µ = v W P.

(14)

In fact, the osmotic pressure difference between two solutions is defined as the hydrostatic pressure difference that would generate the same µ. Therefore, the osmotic pressure difference between two dilute solutions is given by van’t Hoff’s law [1]: P = RT CS ,

(15)

where R = kB NA is the gas constant. It is also known experimentally that equal osmotic and hydrostatic pressure differences produce the same water flux through water channels [19]. The hopping rates and, hence, the water flux are functions of µ alone (Eqs. (7) and (9)), regardless of whether µ arises from osmotic or hydrostatic pressure differences. According to the CTRW model, when an osmotic or hydrostatic pressure difference exists, the water file performs a biased random walk, characterized by the hopping rates kr and kl . In this section, we will determine the statistical distribution of hops as a function of time. Within any infinitesimally small time dt, the probability of the water file to make a rightward hop is kr dt, independent of its history, i.e., when and how many rightward hops were made before. Such a process is referred to as a Poisson process, and the total number of rightward hops within time t, m r (t), obeys the well-known Poisson distribution, whose mean and variance are both kr t. Similarly, the number of leftward hops, m l (t), also obeys the Poisson distribution, with kl t being its mean and variance. The net number of hops, m(t), is defined as the difference of the numbers of rightward and leftward hops, i.e., m(t) = m r (t) − m l (t). Since the probabilities of making rightward and leftward hops are independent of each other, we obtain: m(t) = (kr − kl )t,

(16)

Var[m(t)] = (kr + kl )t,

(17)

where Var[m] = m 2 − m2 . Equations (16) and (17) show that both the mean and the variance of m(t) increase linearly with time. These expressions show that monitoring the average number of hops and its variance permits one to determine both kr and kl [5].

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E. Tajkhorshid et al.

Collective Diffusion Model of Single-channel Water Transport

Following its definition, pf is measured in experiments under nonequilibrium conditions, for systems with nonzero µ. In principle, the same conditions (a chemical potential difference across the channel) can be established in MD simulations of water transport. Two of the techniques for doing so are (1) introduction of solutes to one side of the membrane to generate an osmotic pressure difference [25], and (2) application of a hydrostatic pressure difference across the channel through mechanically manipulating individual water molecules in the bulk [4, 5]. Through adjustment of the salt concentration or of the pressure difference, one may reach different values of µ in the simulations. Due to the presently accessible (ns) time scale of MD simulations, however, one has to adopt a large µ to obtain sufficient statistics of water permeation. This leads to situations that are far from actual experimental conditions, and it is not clear whether the results represent the normal kinetics of the water channel under study. If one can establish a quantitative relationship between water conduction under equilibrium and nonequilibrium conditions, this problem can be circumvented. In this section we demonstrate that water permeation obeys a linear current – µ relationship over a very wide range of µ values and that equilibrium MD simulations (µ = 0) can be used to characterize the osmotic permeability of a channel. Water permeation usually involves multiple water molecules in a channel whose movements are coupled to each other. As a result, a complicated multidimensional representation seems to be necessary to model this process. In the following, we introduce a collective coordinate, n, which offers a much simplified description of water translocation in channels. The derivation follows closely Zhu et al. [23]. Consider a channel (of length L) aligned along the z-direction. The collective coordinate n is defined in its differential form as follows: let S(t) denote the set of water molecules in the channel at time t, and let us assume that the displacement of water molecule i in the z-direction during dt is dz i ; then we define dz i dn = . (18) L i∈S(t ) By demanding n = 0 at t = 0, n(t) can be uniquely determined by integrating dn. Note that S(t) changes with time, and that a water molecule i contributes to n only when it is in the channel, i.e., if i ∈ S(t) at time t. We further note that every water molecule crossing the channel from one reservoir to the other contributes to n a total increment of exactly +1 or −1. Therefore, n quantifies the net amount of water permeation, and the trajectory n(t) describes the time evolution of the permeation.

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An important scenario is the stationary state in which a steady water flux through the channel exists. In this case, n(t) on average grows linearly with t, and the water flux is given by jW =

1 1 n(t) , jn = NA NA t

(19)

where NA is Avogadro’s number, and jn = NA jW is the water flux in the unit of number of water molecules/s. At equilibrium, the net amount of water permeation through the channel vanishes on average, i.e., n(t) = 0. Spontaneous, random water transport, however, may occur due to thermal fluctuation. Such microscopic fluctuations may not be detectable in experiments, but can be readily observed in MD simulations through n(t). At equilibrium, n(t) can be described as a onedimensional unbiased random walk, with a diffusion coefficient Dn that obeys

n 2 (t) = 2Dn t.

(20)

Dn has dimension t −1 since n is dimensionless. Intuitively, Dn is related to the rate at which the net transport of one water molecule happens spontaneously. All factors affecting water kinetics contribute to Dn and are effectively integrated into this single parameter. In the presence of a chemical potential difference (µ) of water between the two reservoirs, n obeys a biased random walk. We note that the net transport of one water molecule from one reservoir to the other results in a change of ±µ in the free energy, and that the total free energy change is proportional to the net amount of water transported. The free energy can be expressed then as a linear function in n: U (n) = µn.

(21)

Consequently, the trajectory of n can be described as a one-dimensional diffusion in a linear potential. Therefore, on average n is drifting with a constant velocity [23]: n(t) = −

µ Dn t, kB T

(22)

which corresponds to a stationary water flow through the channel. According to Eq. (19), the water flux is given by jn = −

µ Dn . kB T

(23)

From Eqs. (1), (10), (19) and (23) one obtains then for the osmotic permeability of the channel pf = v W Dn .

(24)

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Equation (24) shows that one can determine pf using the Dn value obtained from equilibrium MD simulations (cf. Eq. (20)) [23]. The CTRW model proposed for single-file water channels assumes that the whole water file moves in discrete hops simultaneously and concertedly, with rightward and leftward hopping rates kr and kl , respectively. k0 , defined as the value of kr or kl at equilibrium, is the major kinetic parameter in the model. Since each hop changes the collective coordinate, n, by +1 or −1, it holds n(t) = m r (t) − m l (t), where m r (t) and m l (t) are the number of rightward and leftward hops during time t, respectively. Because m r (t) and m l (t) obey a Poisson distribution (see alsoabove) whose mean and variance are both k0 t at equilibrium [5], one obtains n 2 (t) = 2k0 t. Comparison with Eq. (20) yields Dn = k0 . Therefore, for the discrete water movement described by the CTRW model, Dn is identical to the hopping rate k0 , and the expression derived from the CTRW model, namely, pf = v W k0 [5], is actually equivalent to Eq. (24) in the collective diffusion model [23]. However, while the CTRW model is only valid for single-file channels, the collective diffusion model is applicable to any water channel since it makes no assumption regarding water configuration or water movement inside the channel. In the CTRW model, in order to determine the net water flux ( jn = kr − kl ) as a function of µ, the rate theory expression kr /kl = exp(−µ/kB T ) was exploited [5, 25], along with the linear response approach which assumes that µ is much smaller than kB T [5]. The model, however, is not able to predict how jn relates to µ when µ is comparable or larger than kB T . In contrast, the collective diffusion model (Eq. (23)) predicts a linear relationship between jn and µ even when µ exceeds kB T [23].

6.

Simulation of Water Transport and Calculation of pd and pf

Equilibrium MD simulations provide an ideal tool to study free water diffusion through channels, since all water molecules can be easily traced in the simulations, and q0 counted [3, 23]. pd can then be calculated according to Eq. (3) from the simulations. In order to determine pf in a fashion similar to experiments, one needs to produce different osmotic or hydrostatic pressures on the two sides of the membrane. Figure 3 illustrates a scheme to induce a hydrostatic pressure difference in MD simulations [4, 5]. In order to avoid inaccuracies at the boundaries, applying periodic boundary conditions has become a common practice in MD simulation of molecular systems, particularly those that involve a considerable amount of solvents like water. In a periodic system, the unit cell is replicated in three dimensions; therefore, water layers and membranes alternate along the z-direction, defined as the membrane normal. Figure 3 shows a water layer

Kinetic theory and simulation of single-channel water transport

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membrane region II (P2)

unit cell

f

region III

region I (P1)

membrane

Figure 3. Illustration of the method to produce a pressuredifference in MD simulations. The two membranes shown in the figure are “images” of each other under periodic boundary conditions. A constant force f is applied only to water molecules in region III.

sandwiched by adjacent membranes. We define three regions (I, II, III) in the water layer, as shown in the figure. Region III is isolated from the two sides of the membrane by regions I and II, respectively. A constant force f along the z-direction is exerted on all water molecules in region III, generating a pressure gradient in this region that, consequently, results in a pressure difference between regions I and II, i.e., on the two sides of the membrane [4] nf , (25) A where n is the number of water molecules in region III, and A the area of the membrane. Consequently, a net water flux jW through the membrane channels embedded in the membrane can be induced, and pf calculated from jW and P. We note that the membrane needs to be held in its position, e.g., by constraints, to prevent an overall translation of the whole system along the direction of the applied forces. Assuming that the thickness of region III is d, the number of water molecules in this region is n = Ad/v W . Substituting this into Eq. (25) and the result into Eq. (14), we obtain for the chemical potential difference of water between regions I and II: P = P1 − P2 =

µ = f d.

(26)

The external force field generates a mechanical potential difference of f d between regions I and II, which must be exactly balanced by the chemical

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potential difference µ under a stationary population distribution of water, therefore also giving Eq. (26). In an earlier approach [4], all water molecules in the bulk region, including those adjacent to the entrances of the channels, were subject to external forces, a setup which might artificially affect the number of water molecules permeating the channel. This shortcoming was overcome later [5] through application of external forces only to water molecules in region III (Fig. 3), which leaves regions I and II under uniform hydrostatic pressures, and, hence, represents experimental conditions more closely. In order to keep the membrane in place, one can either apply constant counter forces on the membrane to balance the effect of hydrostatic pressure gradients experienced by the membrane [4], or constrain the membrane in the z-direction to prevent an overall translation of the system. The latter is superior, because the number of water molecules (n) in region III, and, therefore, the total external force to water (n f ), experience slight fluctuations during the simulation, and application of a fixed counter force on the membrane may not always exactly balance n f . Moreover, for very long simulations, applying constraints can also eliminate drifting of the membrane along the z-direction that may happen due to thermal motion. Too strong constraints, however, may restrict the dynamics of channel lining groups, which might, particularly in proteins, influence the kinetics of water transport, and one must carefully choose the constraints as to minimize this undesired effect. An interesting method, which we refer to as the “two-chamber setup”, has also been used to study osmotically driven water flow in MD simulations [25], where the unit cell consists of two membranes and two water layers containing different concentrations of solutes. We chose our proposed method rather than the two-chamber setup for two reasons. First, in order to observe on the ns time scale a statistically significant water flux through channels, one has to induce in the two-chamber setup a large chemical potential difference (µ) of water. However, it is noteworthy that Eq. (10) is valid only for dilute solutions; when the solute concentration is high, µ is no longer linearly proportional to the concentration difference. In contrast, in our method, µ can be linearly controlled (see Eq. (26)). Second, the osmotic water flux in the two-chamber setup will decrease with time and eventually stop [25], while application of a hydrostatic pressure gradient maintains a stationary flux that permits sampling for as long as one can afford.

7.

Calculation of Water Permeability of Aquaporins

As discussed earlier, using the CTRW model [24], one can demonstrate that pf and pd of a single-file water channel are related, but differ in value. Equilibrium MD simulations yield the pd value, and applying hydrostatic pressure

Kinetic theory and simulation of single-channel water transport

1811

differences across the membrane allows one to determine pf of membrane channels from MD simulations. We will now present the application of the described method to the example of a real biological channel, namely AQP1. Pressure-induced water permeation will be used to determine the channel’s pf value, which, as we will see, is found to agree well with experimental measurements. The simulations presented in this section are taken from [5]. The AQP1 [9] tetramer was embedded in a POPE lipid bilayer and solvated by adding layers of water molecules on both sides of the membrane. The whole system (shown in Fig. 4) contains 81 065 atoms. The system was first equilibrated for 500 ps with the protein fixed, under constant temperature (310 K) and constant pressure (1 atm) conditions. Then the protein was released and another 450 ps equilibration performed. Starting from the last frame of the equilibration, four simulations were initiated. In these simulations (sim1, sim2, sim3 and sim4), a constant force ( f ) was applied to the oxygen atoms of water molecules in region III, defined as a 7.7 Å-thick layer (shown in Fig. 4) in our system, to induce a pressure difference across the membrane. In principle, the position and thickness of region III can be arbitrarily defined and should not affect the results, as long as the induced pressure difference is set to the same value (by choosing a proper f ); in practice, one would partition the bulk water in such a way that

III I

II III Figure 4. Side view of the unit cell including the AQP1 tetramer (tuberepresentation), and lipid and water molecules (line representation). Hydrogen atoms of lipids are not shown and the phosphorus atoms are drawn as vdW spheres. Water molecules in region III (see Fig. 3) are drawn in a slightly darker shade.

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each of the three regions (I, II, III) has a sufficiently large thickness (relative to the diameter of a water molecule). The constant forces used in the four simulations differ in their direction or magnitude, generating four pressure differences, as summarized in Table 1. The simulations were performed under constant temperature (310 K) and constant volume conditions. As mentioned earlier, the membrane needs to be constrained to prevent the overall translocation of the system under the external forces. This was done by applying harmonic constraints to the Cα atoms of the protein and the phosphorus atoms of the lipid molecules, with spring constants of 0.12 kcal/mol/Å2 and 0.8 kcal/mol/Å2 , respectively. These spring constants are chosen to fully balance the external forces when the whole membrane is displaced by about Å along z from its reference position under a pressure difference of 200 MPa (as in sim1 and sim4). The constraints are applied only in the z-direction, and all atoms are free to move in the x- and y-directions. Note that the constraints on the protein are fairly weak and act only on the backbone Cα atoms; therefore, significant flexibility of protein side chains, which may influence the kinetics of water permeation, was maintained during the simulations. All simulations were performed using the CHARMM27 force field [26], the TIP3P water model, and the MD program NAMD2 [27]. Full electrostatics was employed using the Particle Mesh Ewald (PME) method [28]. Simulations sim1, sim2, sim3 and sim4 were each run for 5 ns, with the first 1 ns discarded and the remaining 4 ns used for analysis. 1 ns of simulation took 22.4 h on 128 1-GHz Alpha processors. During the simulations, the water density distribution in regions I, II, and III exhibited different patterns, as shown in Fig. 5, where the dashed lines are the boundaries separating these regions. In region III, where the external forces are applied, a gradient of water density is observed; in regions I and II, the density of water is roughly constant, indicating that the hydrostatic pressure in these regions is uniform. The water density gradient in region III and, hence, the density difference between regions I and II, differ in the four Table 1. Summary of the four simulations reported in this studya sim1 sim2 sim3 sim4

f (pN)

P (MPa)

µ (kcal/mol)

−7.36 −3.68 3.68 7.36

−195 −97 97 195

−0.814 −0.407 0.407 0.814

a The thickness of region III is d = 7.68 Å, containing on average

2470 water molecules. f is the constant force applied on individual water molecules. The area of the membrane in the unit cell is A = 9.35 × 10−17 m2 . The induced pressure difference P and chemical potential difference µ of water are calculated according to Eqs. (25) and (26), respectively.

Kinetic theory and simulation of single-channel water transport 61

I

III

1813

II

60

Density (mol/l)

59 58 57 56 55 54 53

35

40

45

50

z(Å)

Figure 5. Water density distribution along the z-direction in region III (bracketed by the dashed lines) and part of regions I and II. Data points marked by circles, diamonds, stars, and squares represent sim1, sim2, sim3, and sim4, respectively. The density is measured by averaging the number of water molecules within a 1 Å-thick slab over the last 4 ns of each trajectory.

simulations. From the observed water density difference and the calculated pressure difference (see Table 1) in these simulations, the compressibility of water is estimated to be 4.9 × 10−5 atm, which is in satisfactory agreement with its experimental value of 4.5 × 10−5 atm [19]. Water molecules in the channels were usually found in the single-file configuration (as shown in Fig. 6a) and moved concertedly during the simulations (Fig. 6b). Occasionally, larger number of water molecules were accommodated in the channel, or the water file appeared broken in part of the channel. Nevertheless, the CTRW model can be used to provide a simplified quantitative description of water movement in AQP1 channels, as demonstrated in [5]. The net water fluxes, directly determined from the simulations, are shown in Table 2. These values are plotted vs. the applied pressure difference in Fig. 7. From their best-fit slope, and according to Eqs. (1) and (15), the osmotic permeability was determined to be pf = (7.1 ± 0.9) × 10−14 cm3 /s. Different experiments have reported pf values for AQP1 monomers in the range of 1–16 × 10−14 cm 3 /s, the variation being probably due to uncertainties in the number of channels per unit membrane area; typically referenced pf values range from 5.43 × 10−14 cm3 /s [29] to 11.7×10−14 cm3 /s [21]. In light of this, the pf value calculated from our simulations agrees satisfactorily with experiments. In equilibrium MD simulations of AQP1, a total of 16 permeation events (in four AQP1 monomers in either direction) were observed in 10 ns [11]. Therefore, the rate of uni-directional permeation events in a monomer is

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(a)

(b)

z (Å)

5

0

⫺5

⫺10 0

100

200 300 Time (ps)

400

500

Figure 6. (a) An AQP1 monomer with channel water and nearby bulk water. Water molecules in the constriction (single-file) region, the vestibules of the channel, and in the bulk are rendered in vdW, CPK and line representations, respectively. (b) Trajectories (from sim1) of seven water molecules in the constriction region during 500 ps. Table 2. Water flux observed in the four simulationsa M1 sim1 sim2 sim3 sim4

−13.5 −9.5 11.5 11.5

Water count/4 ns M2 M3 −14.5 −6 8.5 9

−15 −1 5 10.5

M4 −17.5 −12.5 8 7

Flux (# /ns) Mean SD −3.8 −1.8 2.1 2.4

0.4 1.2 0.7 0.5

a To obtain the net water transfer through a channel, a plane normal to its axis is defined, and

when a water molecule crosses the plane, a count of +1 or −1 is accumulated, depending on its crossing direction. Two such planes were defined in the central part of the channel, and the average of their net counts is listed as the water count of the channel. The mean and standard deviation (SD) of the flux were calculated from the water counts of the four AQP1 monomers (M1 to M4) during 4 ns.

q0 = 0.2 H2 O/ns. According to Eq. (3), this q0 value translates into a diffusion permeability of pd = 6.0 × 10−15 cm3 /s. Using this pd value and the calculated pf value of this study, one obtains a pf / pd ratio of 11.9, in good agreement with the experimentally measured ratio of 13.2 for AQP1 [20]. The ratio corresponds to the number of effective steps in which a water molecule needs to participate to cross AQP1. The number (∼12) of effective steps in a complete permeation event should be interpreted as follows. In the bulk, water conduction is essentially uncorrelated, i.e., the bulk phase does not contribute to the pf / pd ratio. In the

Kinetic theory and simulation of single-channel water transport

1815

4 3

Water flux (#/ns)

2 1 0 ⫺1 ⫺2 ⫺3 ⫺4 ⫺200

⫺100

0 ∆P (MPa)

100

200

Figure 7. Relation of water flux and the applied pressure gradient. Values of pressure differences and water fluxes are taken from Tables 1 and 2, respectively. A line with the best-fit slope for the four data points is also shown in the figure.

constriction region of the channel, however, on average N = 7 water molecules move essentially in single file, i.e., in a correlated and concerted fashion, such that N + 1 = 8 steps are needed to transport a water molecule through. Water molecules in the vestibules (also shown in Fig. 6a) at the termini of the channel are not forming a single file, but nevertheless move in a somewhat concerted fashion, accounting for the remainder of the pf / pd ratio. For AQP1, the average number of water molecules in the single-file region is about 7 corresponding to a pt/pd ratio of 8, but the experimentally measured ratio of pf / pd is 13.2 [20]. In order to understand the difference, we note that water molecules in an AQP1 channel may occasionally deviate from the single-file configuration due to conformational fluctuation of the protein. Furthermore, the behavior of water in the vestibule regions of AQP channels [3, 13, 30] suggests that the single-file model is too simple and that water transport effectively involves vestibular water at the channel entrances, such that the latter water cannot be counted as bulk water (see Fig. 6a).

8.

Nanotube Simulations and the Collective Diffusion Model

In order to illustrate the validity of the collective diffusion model we consider MD simulations performed on two channels [23], denoted as a and b, and shown in Fig. 8. The simulations and systems results presented in this section

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E. Tajkhorshid et al.

a

b

Figure 8. Side view of the unit cells in systems a andb, with dimensions of 18.0 Å × 18.0 Å × 41.4 Å and 46.0 Å × 46.0 Å × 42.1 Å, respectively. Half of the CNT channels and the membranes are removed in order to reveal water molecules in the channels. The dashed lines and the bars indicate the layers where constant forces were applied to the water molecules in nonequilibrium simulations (see text).

are taken from [23]. In each system, two layers of carbon atoms mimicking a membrane partition the bulk water and a CNT serves as a water channel. The CNT in system a is of (6,6) armchair type with a C–C diameter of ∼8 Å. Previous simulations [17, 18] showed that this CNT conducts water strictly in single-file manner. The CNT in system b is of (15,15) armchair type with a C–C diameter of ∼20 Å, and with disordered, bulk-like water molecules in it. Systems a and b contain 276 (∼5 in pore) and 1923 (∼90 in pore) water molecules, respectively. The length of the channel is L = 13.2 Å in both systems. All nanotube simulations were performed under periodic boundary conditions with constant volume. The temperature was kept constant (T = 300 K) by Langevin dynamics with a damping coefficient of 5/ps. The CNT and the membrane were fixed in all simulations. The TIP3P model was used for water molecules. We employed the MD program NAMD2 [27] for the simulations, with full electrostatics calculated by the PME method. The channels were fixed and kept rigid throughout the simulations. This ensured that the channel maintains its structure under the large pressure from bulk water molecules. Equilibrium MD simulations of 40 ns and 20 ns were performed on systems a and b, respectively, with coordinates recorded every picosecond. We took the sum of one-dimensional displacements of all water molecules in the channel, divided by L, as the displacement n in each picosecond (cf. Eq. (18)). If a water molecule enters or exits the channel within a picosecond, only the portion of its displacement within the channel contributes to the sum. The trajectories of n(t), as shown in Fig. 9, were obtained by summing up (integrating) the n values as explained above. The mean square deviation (MSD) of n(t)

Kinetic theory and simulation of single-channel water transport

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80

b

n

40

0

a

⫺40

⫺80

0

10

20 t (ns)

30

40

Figure 9. Trajectories of n for equilibrium MD simulations of systems a and b.

for each system is presented in Fig. 10. According to Eq. (20), the diffusion coefficient Dn is one-half of the slope of the MSD–t curve. From the best-fit slopes, the Dn values were determined to be (16.5 ± 2.1)/ns and (524 ± 40)/ns for systems a and b, respectively. In order to test the key aspect of the collective diffusion model, namely, Eq. (23), we need to perform nonequilibrium simulations in the presence of a chemical potential difference (µ) of water across the membrane. This was achieved by application of a hydrostatic pressure difference, which corresponds to a chemical potential difference µ = f d across the membrane. The defined layers in systems a and b are shown in Fig. 8, with thicknesses d = 7.4 and 8.1 Å, respectively. By choosing a proper f , one can select any desired value for µ. For each system, we performed six nonequilibrium simulations, with µ set to 0.2 kB T , 0.5 kB T , 1 kB T , 2 kB T , 5 kB T , and 10 kB T . The simulation times (1–40 ns) varied in different simulations, but were long enough to observe a net transport of at least 100 water molecules in each case. Figure 11 shows both the predicted water fluxes (solid lines) from Eq. (23) and the observed water fluxes (squares) in the simulations, from which one can discern excellent agreements between predictions and simulations. The results demonstrate the validity of the collective diffusion model. It is remarkable that the water flux induced by a µ as large as 10 kB T can still be predicted by the Dn value determined from equilibrium simulations. In light of this, one is not surprised that the calculated osmotic permeability ( pf ) of AQP1 obtained from

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MSD

15

10

5 a

0

0

20

40

60

80

100

t (ps)

Figure 10. Mean square deviations (MSDs) of n for systems a and b. For each system, the trajectory n(t) shown in Fig. 9 was evenly divided into M (400 for system a, 1000 for system b) short time-periods. n(t) in each period was treated as an independent sub-trajectory n 2i (t), and was shifted so that n i (t)|t =0 = 0. The average over n i (t) (i = 1, . . . , M) was then taken as MSD(t). A line with the best-fit slope was superimposed on each MSD curve.

20

200

a

100

j n (/ns)

10

0 0 600

0.5

1

0

4000

200

2000

0

0

6000

b

400

0

a

0.5

1

0

5

10

5

10

b

0

∆µ/k B T

Figure 11. The dependence of water flux ( jn ) on the chemical potential difference (µ) of water. Each data point (marked as a square) represents the jn value obtained from a nonequilibrium simulation, by dividing the total displacement of n in the simulation by the simulation time (cf. Eq. (19)). The solid lines show the jn –µ relations predicted from Eq. (23), with Dn = 16.5 / ns for system a and Dn = 524 / ns for system b, both values being determined from the equilibrium simulations.

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nonequilibrium simulations [5], which were reported in the previous section, agrees with the experimental data despite the fact that the µ values (∼1 kB T ) in the simulations were much larger than experimental values (e.g., a solute concentration difference of 200 mM, as is typical in actual measurements, corresponds to a µ of 0.0036 kB T ). In this section we have mainly focused on the collective movement of water inside the channel. The movement of individual water molecules also deserves attention. In particular, some water molecules may permeate all the way through the channel, an event described as a full permeation event. One can count the number of such permeation events in each direction in unit time, denoted as q0 , from equilibrium simulations. We observed q0 values of about 3 and 110/ns from our equilibrium simulations for systems a and b, respectively. While the Dn value, which quantifies the collective water movement, determines the osmotic permeability pf (see Eq. (24)), the q0 value determines another experimental quantity for water channels, namely, the diffusion permeability pd [5]. The ratio pf / pd is actually equal to Dn /q0 . We obtained Dn /q0 ratios of 5.5 and 4.8 for systems a and b, respectively. The pf / pd ratio for a single-file channel can be interpreted as the number of effective steps a water molecule needs to take to completely cross the channel, i.e., the number of water molecules inside the channel plus 1 [5]; interestingly, despite the much larger number of water molecules in the pore region of system b, the pf / pd values for the two channels turned out to be similar. It is of interest to determine this ratio for different types of water channels in future studies. The collective diffusion model establishes a quantitative relationship between the spontaneous water transport at equilibrium and the stationary water flux under nonequilibrium conditions. Using this model, pf can be determined readily from equilibrium MD simulations. Since the model does not make specific assumptions on water channels, it can be used to characterize water permeation in any channel.

9.

Outlook

Biological water channels, even though only recently discovered, have evolved rapidly to a level of rather complete characterization, both through observation and theory. A key reason for the successful investigations was the fact that the structure of the channel has been solved for key members of the AQP family. Other reasons for the success are the relatively simple function, water transport, the lack of significant motion needed for function, and the related very rigid structure of water channel proteins. Yet, there are still fascinating research problems connected with water channels. Most pressing is an understanding of the mechanism of proton exclusion that is vital for the biological function, since the channels must not

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dissipate cell membrane potentials. Much success has been achieved recently [3, 31–34]. Another interesting aspect of water channel research is to develop an understanding of the diversity of water channels in the whole kingdom of life. Humans have 11 different AQPs [34] in various tissues, some being pure water channels, others being water as well as glycerol channels. The differences in the human AQPs might be related to their function, e.g., possibly to their different ability to gate the channel, but more likely connected with the transport, storage, and deployment of the channels in cells, e.g., as controlled through the antidiuretic hormone. Likewise, existence of many different AQPs in other species, such as plants, yeast, and bacteria, and their involvement in membrane transport of materials ranging from O2 and CO2 gases to substrates like nitrate pose important questions in terms of their selectivity that need to be understood. A fascinating opportunity for the study of AQPs has been opened up recently through the solvation of the structures of both an aqualglyceroprin (GlpF) and a pure water channel (AqpZ) for a single organism, namely, E. coli [10, 36]. A comparison of the two structures provides a fundamental chance to understand the design of this important class of membrane channels in terms of selectivity, transport rates, and role in the survival of cells.

Acknowledgments We acknowledge grants from the National Institutes of Health NIH P41RR05969 and R01-GM067887 and from the National Science Foundation NSF CCR 02-10843. The authors also acknowledge computer time provided at the NSF centers by the grant NRAC MCA93S028. F.Z. acknowledges a graduate fellowship awarded by the UIUC Beckman Institute. Molecular images in this paper were generated with the molecular graphics program VMD [37].

References [1] A. Finkelstein, Water Movement Through Lipid Bilayers, Pores, and Plasma Membranes, John Wiley & Sons, New York, 1987. [2] F. Zhu, E. Tajkhorshid, and K. Schulten, “Molecular dynamics study of aquaporin-1 water channel in a lipid bilayer,” FEBS Lett., 504, 212–218, 2001. [3] E. Tajkhorshid, P. Nollert, M.Ø. Jensen, L.J.W. Miercke, J. O’Connell, R.M. Stroud, and K. Schulten, “Control of the selectivity of the aquaporin water channel family by global orientational tuning,” Science, 296, 525–530, 2002. [4] F. Zhu, E. Tajkhorshid, and K. Schulten, “Pressure-induced water transport in membrane channels studied by molecular dynamics,” Biophys. J., 83, 154–160, 2002.

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[5] F. Zhu, E. Tajkhorshid, and K. Schulten, “Theory and simulation of water permeation in aquaporin-1,” Biophys. J., 86, 50–57, 2004. [6] A.J. Yool and A.M. Weinstein, “New roles for old holes: ion channel function in aquaporin-1,” News Physio. Sci., 17, 68–72, 2002. [7] K. Murata, K. Mitsuoka, T. Hirai, T. Walz, P. Agre, J.B. Heymann, A. Engel, and Y. Fujiyoshi, “Structural determinants of water permeation through aquaporin-1,” Nature, 407, 599–605, 2000. [8] G. Ren, V.S. Reddy, A. Cheng, P. Melnyk, and A.K. Mitra, “Visualization of a waterselective pore by electron crystallography in vitreous ice,” Proc. Natl. Acad. Sci. U.S.A., 98, 1398–1403, 2001. [9] H. Sui, B.-G. Han, J.K. Lee, P. Walian, and B.K. Jap, “Structural basis of waterspecific transport through the AQP1 water channel,” Nature, 414, 872–878, 2001. [10] D. Fu, A. Libson, L.J.W. Miercke, C. Weitzman, P. Nollert, J. Krucinski, and R.M. Stroud, “Structure of a glycerol conducting channel and the basis for its selectivity,” Science, 290, 481–486, 2000. [11] B.L. de Groot and H. Grubm¨uller, “Water permeation across biological membranes: mechanism and dynamics of aquaporin-1 and GlpF,” Science, 294, 2353–2357, 2001. [12] M.Ø. Jensen, E. Tajkhorshid, and K. Schulten, “The mechanism of glycerol conduction in aquaglyceroporins,” Structure, 9, 1083–1093, 2001. [13] M.Ø. Jensen, S. Park, E. Tajkhorshid, and K. Schulten, “Energetics of glycerol conduction through aquaglyceroporin GlpF,” Proc. Natl. Acad. Sci. U.S.A., 99, 6731– 6736, 2002. [14] M.Ø. Jensen, E. Tajkhorshid, and K. Schulten, “Electrostatic tuning of permeation and selectivity in aquaporin water channels,” Biophys. J., 85, 2884–2899, 2003. [15] S. Iijima, “Helical microtubules of graphitic carbon,” Nature, 354, 56–58, 1991. [16] R. Saito, G. Dresselhaus, and M.S. Dresselhaus, Physcial Properties of Carbon Nanotubes, Imperial College Press, 1998. [17] G. Hummer, J.C. Rasaiah, and J.P. Noworyta, “Water conduction through the hydrophobic channel of a carbon nanotube,” Nature, 414, 188–190, 2001. [18] F. Zhu and K. Schulten, “Water and proton conduction through carbon nanotubes as models for biological channels,” Biophys. J., 85, 236–244, 2003. [19] N. Sperelakis, Cell Physiology Source Book, Academic Press, San Diego, 1998. [20] J.C. Mathai, S. Mori, B.L. Smith, G.M. Preston, N. Mohandas, M. Collins, P.C.M. van Zijl, M.L. Zeidel, and P. Agre, “Functional analysis of aquaporin-1 deficient red cells,” J. Biol. Chem., 271, 1309–1313, 1996. [21] M.L. Zeidel, S.V. Ambudkar, B.L. Smith, and P. Agre, “Reconstitution of functional water channels in liposomes containing purified red cell CHIP28 protein, Biochemistry, 31, 7436–7440, 1992. [22] P. Pohl, S.M. Saparov, M.J. Borgnia, and P. Agre, “Highly selective water channel activity measured by voltage clamp: analysis of planar lipid bilayers reconstituted with purified AqpZ,” Proc. Natl. Acad. Sci. U.S.A., 98, 9624–9629, 2001. [23] F. Zhu, E. Tajkhorshid, and K. Schulten, “Collective diffusion model for water permeation through microscopic channels,” Phys. Rev. Lett., 2004, submitted. [24] A. Berezhkovskii and G. Hummer, “Single-file transport of water molecules through a carbon nanotube,” Phys. Rev. Lett., 89, 064503, 2002. [25] A. Kalra, S. Garde, and G. Hummer, “Osmotic water transport through carbon nanotube membranes,” Proc. Natl. Acad. Sci.U.S.A., 100, 10175–10180, 2003. [26] A.D. MacKerell Jr., D. Bashford, M. Bellott et al., “All-hydrogen empirical potential for molecular modeling and dynamics studies of proteins using the CHARMM22 force field,” J. Phys. Chem. B, 102, 3586–3616, 1998.

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[27] L. Kal´e, R. Skeel, M. Bhandarkar, R. Brunner, A. Gursoy, N. Krawetz, J. Phillips, A. Shinozaki, K. Varadarajan, and K. Schulten, “NAMD2: greater scalability for parallel molecular dynamics,” J. Comp. Phys., 151, 283–312, 1999. [28] U. Essmann, L. Perera, M.L. Berkowitz, T. Darden, H. Lee, and L.G. Pedersen, “A smooth particle mesh Ewald method,” J. Chem. Phys., 103, 8577–8593, 1995. [29] T. Walz, B.L. Smith, M.L. Zeidel, A. Engel, and P. Agre, “Biologically active twodimensional crystals of aquaporin CHIP,” J. Biol. Chem., 269, 1583–1586, 1994. [30] D. Lu, P. Grayson, and K. Schulten, “Glycerol conductance and physical asymmetry of the Escherichia coli glycerol facilitator GlpF,” Biophys. J., 85, 2977–2987, 2003. [31] B.L. de Groot, T. Frigato, V. Helms, and H. Grubm¨uller, “The mechanism of proton exclusion in the aquaporin-1 water channe,” J. Mol. Biol., 333, 279–293, 2003. [32] A. Burykin and A. Warshel, “What really prevents proton transport through aquaporins,” Biophys. J., 85, 3696–3706, 2003. [33] N. Chakrabarti, E. Tajkhorshid, B. Roux, and R. Pom`es, “Molecular basis of proton blockage in aquaporins,” Structure, 12, 65–74, 2004. [34] B. Ilan, E. Tajkhorshid, K. Schulten, and G.A. Voth, “The mechanism of proton exclusion in aquaporin channels,” Proteins: Struct. Func. Bioinf., 55, 223–228, 2004. [35] J.B. Heymann and A. Engel, “Aquaporins: phylogeny, structure, and physiology of water channels,” News Physio. Sci., 14, 187–193, 1999. [36] D.F. Savage, P.F. Egea, Y. Robles-Colmenares, J.D. O’Connell III, and R.M. Stroud, “Architecture and selectivity in aquaporins: 2.5 Å x-ray structure of aquaporin Z,” PLoS Biol., 1, 334–340, 2003. [37] W. Humphrey, A. Dalke, and K. Schulten, “VMD – visual molecular dynamics,” J. Mol. Graphics, 14, 33–38, 1996.

5.16 SIMPLIFIED MODELS OF PROTEIN FOLDING Hue Sun Chan University of Toronto, Toronto, Ont., Canada

Protein folding is one of the most basic physico-chemical self-assembly processes in biology. Elucidation of its underlying physical principles requires modeling efforts at multiple levels of complexity [1]. As in all theoretical endeavors, the degree of simplification in modeling protein behavior depends on the questions to be addressed. The motivations for using simplified models to study protein folding are at once practical and intellectual. Realistically, a truly ab initio solution to the Schr¨odinger equation for a protein and its surrounding solvent molecules is currently out of the question. Although classical (Newtonian) descriptions based on geometrically high-resolution all-atom molecular dynamics have provided much useful insight, these models are computationally costly. Moreover, it is unclear whether common empirical potential functions used in such all-atom approaches are ultimately adequate. In this context, simplified models offer a complementary and efficient means for posing questions and testing hypotheses. Similar in spirit to the Ising model of ferromagnetism, simplified models of protein folding are designed to capture essential physics, and are geared towards the discovery of higher organizing principles [2] while omitting details deemed unimportant for the question at hand.

1.

Lattice Protein Models

Experimental protein folding data has long been analyzed in terms of “native”, “denatured”, and “intermediate” states, often without a clear delineation of the relationship between the empirically defined states and their underlying conformational ensembles. While enlightening, the physical pictures emerged from such interpretations are far from complete. Proteins are chain molecules. Obviously, a microscopic physical picture of their folding 1823 S. Yip (ed.), Handbook of Materials Modeling, 1823–1836. c 2005 Springer. Printed in the Netherlands.

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must involve at least a rudimentary account of chain connectivity, conformational freedom, and the excluded volume constraints that two amino acid residues cannot be at the same place at the same time [3]. Lattice protein models fulfill this minimal requirement by representing protein chains as selfavoiding walks on a lattice. The multitude of lattice walks correspond to the many conformations accessible to a protein molecule. The most commonly applied lattice protein models are based upon two-dimensional (2D) square and three-dimensional (3D) simple cubic lattices (Fig. 1). Bond angles in these models are restricted to either 0◦ and 180◦ . Lattices with higher coordination numbers that allow for a larger set of possible bond angles have also been used extensively to provide more realistic representations of protein geometry (see review by Chan et al. [1]). Lattice protein models are closely related to lattice approaches in many areas of polymer physics. Although homopolymer models (with identical monomer units along the chain) are valuable in addressing certain issues in protein folding, the main emphasis in lattice modeling of proteins is on heteropolymer models (wherein monomers along the chain can be different). This is because protein sequences are made up of different types of amino acids. Intraprotein interactions are heterogeneous. A number of different schemes have been used to model intraprotein interactions. These include allowing different numbers of possible monomer types (different numbers of letters in an alphabet), ranging from reduced two-letter models designed for simplicity and tractability (e.g., Fig. 1) to 20-letter models that aim to better capture the energetics in real proteins. Some interaction schemes are general and transferrable in that they depend solely on the model sequence, while others have explicit biases towards a particular target structure (see below). The field of simplified lattice modeling of proteins has expanded dramatically in the last decade. Many of the models and their applications have been reviewed (e.g., [4–8]). One distinct advantage of these self-contained polymer models with explicitchain representations [1] is that they provide a clear deductive relationship between the premises of a model and its predictions. Ideas and hypotheses can be efficiently verified or falsified by simplifed protein folding models because of this logical clarity and their computational tractability.

2.

Energetics, Chain Moves and Density of States

Besides simplified lattice models, continuum (off-lattice) models with simplified representations of the polypeptide chain have also been used to study protein folding [6, 9–11]. The rationale for both simplified on- and offlattice models is to enhance the capability to broadly sample conformational space by sacrificing geometrical accuracy of the model chains. A central quantity governing the energetics of a simplified protein model is its density of

Simplified models of protein folding

1825 DENATURED ENSEMBLE

lng(h) 0 1 2 3 4 5 6 7 8 9

h2 g(2) 855549

h1 g(1) 2059356

h0 g(0) 2565772

h4 g(4) 61831

h3 g(3) 252356

h

h5 g(5) 11629

NATIVE

h9 g(9) 1

h8 g(8) 9

h7 g(7) 162

h6 g(6) 1670

Figure 1. A 2D HP lattice model illustration of protein folding thermodynamics. Filled and open circles correspond, respectively, to H and P residues. The given model protein sequence can have a maximum of nine HH contacts, achievable only by a single native conformation (lower left). The double arrow indicates the thermodynamic equilibrium between the single native conformation and the denatured ensemble (right) that consists of a total of 5,808,334 conformations, schematically depicted using one representative conformation for each h value. The corresponding logarithmic density of states is shown at the upper left.

states g(E), defined to be the number of conformations as a function of the intraprotein interaction energy E. In general, intraprotein energies in simplified models should be viewed as an effective potential. In addition to the direct interactions between chemical groups along the protein chain, an effective

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potential also includes energetic contributions arising from protein–solvent and solvent–solvent interactions by implicitly averaging over solvent degrees of freedom. As a result of solvent averaging, effective potentials can depend on temperature [1]. For many applications, however, one may simplify the calculation by taking E as temperature independent as long as the above consideration is taken into account in relating model predictions to experiments. As for any physical system, a model protein’s thermodynamics is controlled by its partition function Q=

g(E)e−E/kB T ,

(1)

E

where kB is Boltzmann constant and T is absolute temperature. The summation here, which may be replaced by an integration for continuum models, is over all possible energies E. In protein folding, a quantity of central interest is the ratio between native (N) and denatured (D) populations:

−E/kB T [N] E∈{E N } g(E)e = , −E/kB T [D] E∈{E D } g(E)e

(2)

where E ∈ {E N } and E ∈ {E D } indicate that the respective summations are over conformations defined to be in the native and denatured states. In formulations with T -independent E’s, the native energies are lower than the denatured energies, such that chain population is concentrated in the native (folded) state at low T but shift to the denatured state at high T . This provides a model description for the protein folding/unfolding transition. In general, the denatured state is comprised of many more conformations than that of the native state. In some highly simplified lattice models (e.g., Fig. 1), the native state is defined to be a single conformation ({E N } has only a single energy). The average energy

Eg(E)e−E/kB T −E/ kB T E g(E)e

E = E

(3)

and (constant-volume) heat capacity 1 ∂E = CV (T ) = ∂T kB T 2

E 2 g(E)e−E/kB T − −E/kB T E g(E)e E

−E/kB T 2 E Eg(E)e −E/k T E

g(E)e

B

(4) of the model protein can be readily computed once the density of states is determined. The unfolding transition of a protein is often associated with a prominent peak of the heat capacity as a function of T . This peak of heat absorption may be viewed as a finite-system analog of latent heat, whereby energy is taken up to propel protein conformations from the lower-energy native state to the higher-energy denatured state. The heat capacity of a model

Simplified models of protein folding

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protein provides important information about the cooperativity of the protein folding/unfolding transition, i.e., to what degree it can be viewed as an “allor-none” process [12]. It should be noted that most calorimetric data provides the constant-pressure heat capacity CP rather than CV . Therefore, to facilitate comparison with experiments, an expression for CP (T ) may be obtained by replacing energy E in Eq. (4) with enthalpy H . In lattice models, the energy E in Eq. (4) is taken to be the potential energy alone (since kinetic energy is not defined). In continuum models with Newtonian or Langevin dynamics, the energy in Eq. (4) should be the total energy that includes both the potential and kinetic energies, although the effect of including kinetic energy on the heat capacity function is small in the transition peak region. If the effective energy E is temperature dependent, the differentiation in Eq. (4) would lead to extra terms in the heat capacity expression. This can result in different intrinsic heat capacities for the native and denatured states, a feature similar to that observed experimentally for real proteins [13]. For short-chain lattice models, g(E) may be enumerated exactly (see below). For longer chains, exact enumeration is not practical. In such cases, g(E) is estimated instead by conformational sampling using Monte Carlo techniques (for continuum as well as lattice models) or Newtonian/Langevin dynamics (for certain continuum models). A common Monte Carlo technique is known as the Metropolis algorithm, which consists of two basic steps: (i) starting with any conformation of the model protein, attempt to randomly change its conformation by applying a chain move from a set of elementary coordinate transformations. Move sets are chosen primarily for their efficiency in reaching every region of conformational space. (ii) Evaluate the attempted conformational transition. In the Metropolis algorithm, the probability of accepting an attempted transition from conformation a with energy E a to conformation b with energy E b (b =/ a) is given by Pab = min{1, exp[−(E b − E a )/kB T ]}.

(5)

If the attempted conformational transition is accepted, b is added to the conformational sample. If the attempted transition is not accepted, a is counted one more time in the conformational sample. After sufficient sampling, the resulting collection of conformations visited is expected to converge to a Boltzmann distribution, from which an estimate of the density of states g(E) can be readily extracted. It should be emphasized that the Metropolis procedure was originally designed for thermodynamic sampling, not for kinetic simulation. The sequence of events in a Monte Carlo run does not necessarily correspond to a physical kinetic process. For certain applications, a computationally efficient move set may bear little semblance to actual chain dynamics. Nonetheless, when the set of moves and their relative probabilities are judiciously chosen so one can intuitively assume that the sampling moves correspond to physically plausible elementary chain motions, the series of conformations

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sampled in a Metropolis Monte Carlo run may be taken as a model kinetic trajectory. Indeed, this working assumption has been heavily utilized in simplified lattice models of protein folding kinetics.

3.

Reduced Alphabets and Exact Lattice Enumerations

Exact enumeration of lattice conformations has been an investigative tool in polymer physics since the late 1940s. The methodology has contributed to the development of renormalization group analyses of excluded volume effects (see review by Chan and Dill [3]). Since the late 1980s, simple exact models of protein folding have inherited this rigorous polymer physics technique of exhaustively accounting for all possible conformations [5]. For certain simplified lattice protein models with few-letter alphabets such as the HP model described below, the method can be extended to exhaustively accounting for all possible sequences as well [14]. A widely applied simplified lattice protein model is the two-letter HP (hydrophobic–polar) model. Sequences in this model are made up of two residue types – hydrophobic (H) and polar (P). Chains are configured on 2D square or 3D simple cubic lattices. The HP model is designed to capture the interplay between chain conformational freedom and hydrophobic interactions. For this purpose, the 2D HP model is exceptionally versatile and instructive, its reduced number of spatial dimensions notwithstanding. Hydrophobic effect is a main driving force for protein folding because hydrophobic residues tend to cluster together to avoid water and hence drive a protein to adopt a globular folded form. In the HP model, this effect is minimally modeled by assigning a favorable energy (< 0) to each non-bonded nearest-neighbor HH (hydrophobic–hydrophobic) contact. All other contacts are neutral (have zero energy). It follows that the energy of a conformation equals E = h, where h is the number of HH contacts in the conformation. Hence, the density of states of a model sequence is given by g(h), the number of conformations as a function of h. The number of lowest-energy (maximum-h) conformations is often denoted by g, termed ground-state degeneracy. Only a small fraction of approximately 2.5% of 2D HP sequences have a unique ground-state conformation (g = 1), they are used as model proteins. Most HP sequences have g > 1. This echoes experimental observations that an overwhelming majority of random amino acid sequences do not fold like natural proteins. Despite its simplicity, the HP model exhibits many proteinlike properties, and has provided numerous insights into protein structure and stability [3, 5]. An example HP model protein sequence and its exactly enumerated g(h) are shown in Fig. 1.

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Together with the HP model, a wide diversity of simplified lattice protein models have been extensively investigated during the past decade. Because these models are highly simplified, caution has to be used to relate their predictions to experiments. Simple-minded interpretations of simplified model results can be misleading. Notably, the mere fact that a model alphabet has 20 letters does not by itself mean that its interactions resemble the physical interactions among the 20 real amino acid types. To address issues of model interpretation, a physical evaluation of the energetics embodied by some of the recent simplified lattice protein models was provided by Chan et al. [1].

4.

Sequence-Structure Mapping and Evolution

Although simplified lattice models with pairwise additive contact energies (these include the HP model) have been instrumental in making fundamental conceptual advances, their minimalist interaction schemes are not sufficient to provide quantitative rationalizations for several key generic thermodynamic and kinetic properties of protein folding ([12, 15] see below). Nevertheless, despite this limitation, the HP model continues to be valuable particularly to evolutionary studies because it offers an exactly enumerable yet physically motivated mapping between sequences and their ground-state conformations. Energetic contributions in a natural protein are “minimally frustrated” in that they tend to consistently favor the same native structure [4]. It follows that even though the HP potential is incomplete, the correspondence between a model sequence’s H/P pattern and its ground-state conformation(s) is expected to mimic that for real proteins. This interpretative framework is supported by the observation that H/P patterns among short 2D HP model protein sequences are similar to that observed among real proteins (reviewed in Ref. [14]). A useful evolutionary concept elucidated by HP and other simplified lattice models is that of the neutral net (Fig. 2). A neutral net is a set of sequences interconnected by single-point mutations and encoding for the same native structure. For the examples in Fig. 2, the top left conformation is identical to the native conformation in Fig. 1 with the same sequence, whereas the top right conformation (encoded by a different sequence) is structurally identical to the h = 6 example conformation in Fig. 1. Figure 2 shows how a sequence encoding for the top left conformation may evolve into a sequence encoding for the top right conformation by undergoing only two single-point mutations (dashed lines). Not surprisingly, one of the changes necessary for favoring the top right conformation is replacing the P residue in the core of the h = 6 conformation in Fig. 1 with an H residue. Other aspects of sequence-structure mapping can be subtle and less obvious. For instance, the above P → H substitution alone leads only to the g = 4 sequence in Fig. 2. This sequence has

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H.S. Chan NEUTRAL NETS

H H H H H H H H H H H H

P P P P P P P P P P P P

H H H H H H H H H H H H

P P P P P P P P P P P P

PH PH HH PH HH PH PH HH PH PH PH HH

H H H H H H H H H H H H

H H H H H H H H H H H H

PH PH HH HH PH PH HH PH PH HH PH HH

PP H PHH HP H HP H HP H HP H PPH PPH P HH PPH PPH PPH

P P P P P P P P P P P P

HP HP HP HP HP HP HP HP PP PP PP HP

H H H H H H H H H H H H

H H H H H H H H H H H H

H H H H H H H H H

H H H H H H H H H

H H H H H H H H H

P P P P HP HP P P P P P P P P P P

H H H H H H H H H

HH PH P H P H P H P H P H P P P P

P P P P P P P P P

H H H H H H H H H

P P P P P P P P P

P P P P P P P P P

P P P P P P P P P

P H HH P H P H P H HH P H P H P H

P H HH HH P H HH P H P H P H HH

H H H H H H H H H

CROSSOVER

HHHPPHPPPHPPHPHPHH

Figure 2. Modeling protein evolution. The two lattice conformations at the top are encoded by two neutral nets (shown below the conformations) consisting, respectively, of the 12 (left) and 9 (right) unique (g = 1) HP sequences listed under the nets. Net topologies are depicted by lines connecting pairs of sequences (represented by diamonds) that differ by a single-point H → P or P → H substitutive mutation. Sequences of the same neutral net are inter-connected by solid lines. The two neutral nets in this figure are linked via a g = 4 sequence (diamond between two dashed lines), which is connected by single-point substitutions (dashed lines) to one sequence in each of the two neutral nets. The latter sequences correspond to the two that are first on the two unique sequence lists, and are shown with their respective native conformations (top). In this example, a crossover between the pair of sequences that are last on the two sequence lists begets a recombined sequence that encodes for a different conformation, which is shown at the bottom of the figure with a dotted box highlighting the 10-residue sub-sequence that originates from the first 10 residues of the parent sequence on the right. The rest of the recombined sequence originates from the last eight residues of the parent sequence on the left.

Simplified models of protein folding

1831

four ground-state conformations, one of which is the top left conformation, but the top right conformation is not among them. A further single-point H → P mutation is needed in this case to create a sequence encoding for the top right conformation. HP and other model results indicate that sequences in individual neutral nets often conform to a “superfunnel” paradigm in that native stability tend to increase as a sequence’s mutational difference with a centrally located prototype sequence decreases. A hallmark of prototype sequences is their mutational stability. The prototype sequences of the two neutral nets in Fig. 2 correspond, respectively, to the first listed sequence on the left (shown with the top left conformation) and the seventh listed sequence on the right (which is one single-point mutation away from the sequence shown with the top right conformation). The exhaustive coverage of sequence-structure mappings in simplified lattice protein models provide a means to explore the effects of sequence-space topology on evolutionary population dynamics. Recombinatoric evolution has been modeled by simplified protein lattice model as well (Fig. 2, bottom). Here it is noteworthy that in the recombined sequence’s native structure (bottom center), the conformation adopted by a sub-sequence (enclosed in the dotted box in Fig. 2) is identical to that of the same sub-sequence in a different native structure (top right) encoded by the parent sequence on the right. More generally, HP model results suggest that a sequence’s power to encode for a unique structure resides partly in its sequentially local H/P patterns. This observation is reminiscent of, and provides a rationalization for the autonomous folding units in real proteins. Further details of simple exact models of protein evolution can be found in a recent review [14].

5.

Generic Protein Properties as Stringent Modeling Constraints

An approach for probing real protein energetics is to apply generic experimental protein folding properties as constraints on postulated simplified model interaction schemes. This investigative protocol is effective because even some apparently mundane features of protein folding turned out to be extremely stringent modeling constraints. Therefore, by rigorously evaluating model interactions against experiments, inferences can be made about the functional form of the interactions operating in real proteins (i.e., their “energy landscapes”). A prime example is the thermodynamic cooperativity associated with thermal denaturation of many small proteins, quantified by an van’t Hoff to calorimetric enthalpy ratio HvH /Hcal ≈ 1 deduced from experimental CP (T ). This condition implies that a protein’s density of state is sharply bimodal [10, 12]. It is noteworthy that several simplified lattice protein models deviate significantly from this quantitative criterion. Although not all

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real proteins exhibit thermodynamic cooperativity, this observation suggests strongly that the energetics of certain simplified models differ substantially from that of a large class of real proteins that do fold and unfold cooperatively. In general, thermodynamic cooperativity tends to be facilitated by enhancing interaction heterogeneity, which is often achievable by adopting a larger model alphabet [12]. The folding kinetic cooperativity of a growing number of small, singledomain proteins provide an even more stringent modeling constraint. The hallmarks of these proteins’ folding/unfolding kinetics are: (i) single-exponential kinetic relaxation, (ii) the logarithm of the folding and unfolding rates (ln kf and ln ku ) at constant T are essentially linear in chemical denaturant concentration, i.e., both arms of the “chevron plot” are linear, and (iii) the equilibrium [N]/[D] = kf /ku . Taken together, conditions (ii) and (iii) imply that for these proteins, ln kf and ln ku are linear in the free energy of unfolding (native stability) G u ≡ kB T ln([N]/[D]). Apparently, thermodynamic cooperativity is a prerequisite of kinetic cooperativity. The kinetic cooperativity conditions are highly discriminating modeling constraints. Figure 3 shows that these conditions are not satisfied by the common G¯o model. Even though this model has explicit energetic biases favoring only contact interactions present in the native structure (see below), kinetic trapping remains significant. A popular 20-letter lattice model as well as continuum G¯o models with pairwise additive contact interactions also fail to exhibit folding kinetic cooperativity [11, 12]. This is a likely cause of the fact that the diversity in the folding rates of these models does not exhibit trends similar to that observed experimentally for real, small, single-domain proteins [12, 15, 16]. In contrast, Fig. 3 shows that a chevron plot with essentially linear arms and single-exponential relaxation is obtainable from a model with a local–non-local coupling mechanism involving many-body interactions. Here “local” or “non-local” interaction refers, respectively, to a small or large separation along the protein chain between chemical groups involved in a given interaction [3]. The postulated coupling effect in Fig. 3 stipulates that contact interactions (mostly non-local) are strongly favorable only when short stretches of the chains around the pair of contacting residues adopt nativelike local conformations. The success of models embodying this mechanism (e.g., Fig. 3) in producing generic protein thermodynamic and kinetic behavior suggests that similar mechanisms are likely operative in real proteins [12, 16]. From these and related analyses, thermodynamic and kinetic cooperativities emerge as powerful modeling constraints on simplified protein folding model. They complement, and for the case of small, single-domain proteins, supersede several previous proposed modeling criteria such as the ratio between folding temperature and glass transition temperature. The earlier criteria tackled similar energetic issues insightfully but made less quantitative connections with experiments [12].

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ⴚ10

In(rate)

ⴚ12 ⴚ14 ⴚ16 ⴚ18

15

10

5

0

ⴚ5

ⴚ10

∆Gu/kBT

Figure 3. Modeling protein folding cooperativity. Chevron plots showing the dependence of ln kf (squares and circles) and ln ku (triangles and diamonds) on native stability in units of kB T are computed for two 3D lattice 27mer models using Metropolis Monte Carlo dynamics. The models have the same unique ground-state conformation (structure shown on the right) but different interaction schemes. The upper plot is obtained using the common pairwise additive G¯o potential, whereas the lower plot incorporates a many-body mechanism with local–non-local coupling. The G u /kB T scale is for the lower plot. A lack of discrepancy between the open and filled symbols indicates that kinetic relaxation is essentially single-exponential. Kinetic trapping causes the folding arm of the common G¯o-model chevron plot to deviate significantly from linearity. In contrast, the chevron plot of the model with local–non-local coupling exhibits features similar to that of real, small, single-domain proteins. A hypothetical rate dependence (fitted V-shape) consistent with the apparent two-state equilibrium thermodynamics of the latter model is shown to coincide substantially with the qusai-linear regime of the lower chevron plot (see Ref. [16] for further details).

6.

Continuum (Off-lattice) Models: Limitations of Native-centric Methods

As for simplified lattice models, simplified continuum chain models of protein folding have utilized general, transferrable potentials [6, 10] as well as “native-centric” (G¯o-like) potentials with explicit energetic biases towards a particular target native structure [8, 9, 11]. Models with transferrable potentials are reductionist in nature. Their starting point is the general microscopic interactions presumed by a model. In that sense they conform closer to a deductive general physical theory. However, it is not yet computationally feasible to address many thermodynamic and kinetic questions of interest using

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explicit-solvent all-atom molecular dynamics models. At the same time, simplified continuum models with general transferrable potentials often lead to much more kinetic trapping than that observed experimentally in real, small proteins, indicating that such model potentials might not have captured certain key ingredients in real protein energetics. In this context, native-centric continuum models emerge as a complementary approach. These models stipulate that a protein’s known native structure contains significant information about its actual energetics, even if their precise nature remains to be elucidated. Assuming that this energetics can be approximately captured by a G¯o-like potential that explicitly favors a set of native contacts, these models are then applied to explore aspects of protein folding that are not immediately obvious from the presumed G¯o-like potential itself. Native-centric continuum modeling has provided much useful insight into folding kinetics, especially the transition state barrier to protein folding [8, 9]. However, because a G¯o-like potential cannot be totally realistic, and there are inevitably many arbitrary features in the definition of a native-centric model, extra caution has to be used to assess the robustness of their predictions [11]. Figure 4 examines this question by comparing two alternate definitions of native contacts in the literature. It shows that the free energy profiles predicted by two native-centric Langevin dynamics models with the same Cα chain representation for the same protein are sensitive to the choice of native contact set. Whereas the upper profile has a single peak region between the N and D states, the lower profile exhibits a dip on top of the overall peak. This difference may be interpreted to imply that while a high-energy folding intermediate is present on the lower profile, it is absent on the upper profile. This discrepancy underscores that predictions from a particular formulation of G¯o-like potential can only be regarded as tentative.

7.

Outlook

Simplified models have proven to be a powerful analytical tool for conceptual development and semi-quantitative rationalization of protein folding. Their ability to capture essential physics is well appreciated. At the same time, their intrinsic limitations should be recognized. Simplified models are most effective when attention is directed towards both their failures and their successes in reproducing experimental protein properties. As in the case of cooperativity discussed above, rigorous evaluations of model predictions against experiments and persistent efforts to resolve their discrepancies are indispensable for advance. As the field progresses, it is expected that an increasingly positive feedback between simplified and detailed modeling would ensue. Mesoscopic organizing principles emerging from simplified models should be used to provide novel ideas for detailed modeling. Detailed

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N

Figure 4. Continuum G¯o-like model predictions are sensitive to the definition of native contacts. This figure compares the native contact sets NCS1 and NCS2 given in Ref. [11] for the 64-residue truncated form of chymotrypsin inhibitor 2. Results here are obtained using the “with-solvation” contact potential that incorporates rudimentary pairwise desolvation barriers. Thin lines connecting Cα positions along the protein’s backbone traces (thick line segments) show native contacts common to both NCS1 and NCS2 (middle drawing) as well as those that are in NCS2 but not in NCS1 (left drawing), and those in NCS1 but not in NCS2 (right drawing). The upper and lower curves spanning the denatured (D) and native (N) minima are free energy profiles for NCS2 and NCS1, respectively, as functions of fractional number of native contacts Q. (Figure courtesy of H¨useyin Kaya; see Ref. [11] for further details.)

modeling and experiment in turn are necessary for verifying or falsifying those very ideas, and for determining whether the proposed organizing principles are atomistically feasible. Thus, a solution to the protein folding problem may be incrementally approached by systematically bridging the gap between simplified and detailed models.

References [1] H.S. Chan, H. Kaya, and S. Shimizu, “Computational methods for protein folding: scaling a hierarchy of complexities,” In: T. Jiang, Y. Xu, and M.Q. Zhang (eds.), Current Topics in Computational Molecular Biology, The MIT Press, Cambridge, MA, pp. 403–447, 2002.

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[2] R.B. Laughlin and D. Pines, “The theory of everything,” Proc. Natl. Acad. Sci. USA, 97, 28–31, 2000. [3] H.S. Chan and K.A. Dill, “Polymer principles in protein structure and stability,” Annu. Rev. Biophys. Biophys. Chem., 20, 447–490, 1991. [4] J.D. Bryngelson, J.N. Onuchic, N.D. Socci, and P.G. Wolynes, “Funnels, pathways, and the energy landscape of protein folding – a synthesis,” Proteins Struct. Funct. Genet., 21, 167–195, 1995. [5] K.A. Dill, S. Bromberg, K. Yue, K.M. Fiebig, D.P. Yee, P.D. Thomas, and H.S. Chan, “Principles of protein folding – a perspective from simple exact models,” Protein Sci., 4, 561–602, 1995. [6] D. Thirumalai and S.A.Woodson, “Kinetics of folding of proteins and RNA,” Acc. Chem. Res., 29, 433–439, 1996. [7] J.N. Onuchic, H. Nymeyer, A.E. García, J. Chahine, and N.D. Socci, “The energy landscape theory of protein folding: insights into folding mechanisms and scenarios,” Adv. Protein Chem., 53, 87–152, 2000. [8] L. Mirny and E. Shakhnovich, “Protein folding theory: from lattice to all-atom models,” Annu. Rev. Biophys. Biomol. Struct., 30, 361–396, 2001. [9] C. Clementi, H. Nymeyer, and J.N. Onuchic, “Topological and energetic factors: what determines the structural details of the transition state ensemble and ‘en-route’ intermediates for protein folding? An investigation for small globular proteins,” J. Mol. Biol., 298, 937–953, 2000. [10] T. Head-Gordon and S. Brown, “Minimalist models for protein folding and design,” Curr. Opin. Struct. Biol., 13, 160–167, 2003. [11] H. Kaya and H.S. Chan, “Solvation effects and driving forces for protein thermodynamic and kinetic cooperativity: how adequate is native-centric topological modeling?,” J. Mol. Biol., 326, 911–931, 2003. [Corrigendum: 337, 1069–1070, 2004]. [12] H.S. Chan, S. Shimizu, and H. Kaya, “Cooperativity principles in protein folding,” Meth. Enzymol., 380, 350–379, 2004. [13] S. Shimizu and H.S. Chan, “Anti-cooperativity and cooperativity in hydrophobic interactions: three-body free energy landscapes and comparison with implicit-solvent potential functions for proteins,” Proteins: Struct. Funct. Genet., 48, 15–30, 2002. [14] H.S. Chan and E. Bornberg-Bauer, “Perspectives on protein evolution from simple exact models,” Appl. Bioinformat., 1, 121–144, 2002. [15] H.S. Chan, “Protein folding: matching speed with locality,” Nature, 392, 761–763, 1998. [16] H. Kaya and H.S. Chan, “Contact order dependent protein folding rates: kinetic consequences of a cooperative interplay between favorable nonlocal interactions and local conformational preferences,” Proteins Struct. Funct. Genet., 52, 524–533, 2003.

5.17 PROTEIN FOLDING: DETAILED MODELS Vijay Pande Department of Chemistry and of Structural Biology, Stanford University, Stanford, CA 94305-5080, USA

1.

Goals and Challenges of Atomistic Simulation

Proteins play a fundamental role in biology. With their ability to perform numerous biological roles, including acting as catalysts, antibodies, and molecular signals, proteins today realize many of the goals that modern nanotechnology aspires to. However, before proteins can carry out these remarkable molecular functions, they must perform another amazing feat – they must assemble themselves. This process of protein self-assembly into a particular shape, or “fold” is called protein folding. Due to the importance of the folded state in the biological activity of proteins, recent interest from misfolding related diseases [1], as well as a fascination of just how this process occurs [2–4], there has been much work performed in order to unravel the mechanism of protein folding [5]. There are two approaches one can take in molecular simulation. One direction is to perform coarse grained simulations, using simplified models. These models typically make either simplifying assumptions (such as Go models which use simplified Hamiltonians [6]) or coarse grained representations (such as an alpha-carbon only model for the protein [7]) or potentially both. While these methods are often first considered due to their computational efficiency, perhaps an even greater benefit of simplified models is their ability to potentially yield insight into general properties involved in protein folding. However, with any model or approach, there are limitations, and the cost for potential insight into general properties of folding is the limitation of restricted applicability to any particular protein system. Insight from folding simulations from simplified models is discussed in detail in Hue Sun Chan’s section. Alternatively, one can examine more detailed models. These models typically have full atomic detail, often for both the protein as well as the solvent. Detailed models have the obvious benefit of potentially greater fidelity to 1837 S. Yip (ed.), Handbook of Materials Modeling, 1837–1848. c 2005 Springer. Printed in the Netherlands.

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experiment. However, this comes at two great costs. First, the computational demands for purely performing the simulation become enormous. Second, the added degrees of freedom lead to an explosion of extra detail and simulationgenerated data; the act of gleaming insight from this sea of data is no simple task and is often underestimated, especially in light of the more straightforward (although still often overwhelming) task of simply performing the simulations.

1.1.

Why are Detailed Models Worth This Enormous Effort in Both Simulation and Analysis?

First, quantitative comparison between theory and experiment is critical for validating simulation as well as lending interpretation to experimental results. While it is generally held that experiments will not be able to yield the detail and precision available in simulations (and that simulations may likely be the only way one can fully understand the folding mechanism [8]), without quantitative validation of simulations, there is no way to know whether the simulation model or methodology are sufficiently accurate to yield a faithful reproduction of reality. Indeed, without a quantitative comparison to experiment, there is no way to decisively arbitrate the relative predictive merits of one model over another. Second, detailed models potentially have a greater predictive power. In principle, a detailed model should allow one to start purely from the protein sequence and by simulating the physical dynamics of protein folding, yield everything that one can measure experimentally, including folding and unfold rates, free energies, and the detailed geometry of the folded state. In practice, the ability for detail models to achieve these lofty goals rests both on the ability to carry out the computationally demanding kinetics simulations as well as the ability for current models (force fields) to yield sufficiently accurate representations of inter-atomic interactions.

1.2.

Why are the Challenges for Atomistic Simulation?

First, one must consider the source of the great computational demands of molecular simulation at atomic detail. To simulate dynamics, typically one simulates molecular dynamics by numerically integrating Newton’s equations for all the atoms in the system. By choosing to model with atomic degrees of freedom, one must simulate the dynamics at the timescales of atomic motion. Typically, this means that one must include fast timescales up to the femtosecond timescale. Indeed, if the timestep involved in numerical integration

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is pushed too high (without constraining degrees of freedom), the numerical integration becomes unstable. This leads to the simple problem that if one wants to reach the millisecond timescale by taking femtosecond steps, many (1012 ) steps must be taken. While modern molecular dynamics codes are extremely well optimized and perform typically millions of steps per CPU day, this clearly falls short of what is needed (see Fig. 1). However, even if one could reach the relevant timescales, the next question is whether our models would be sufficiently accurate. In particular, would we reach the folded state, would the folded state be stable (with free energy of stability comparable to experiment), and would we reach the folded state with a rate comparable to experiment. Indeed, if one could quantitatively predict protein folding rates, free energy of stability, and structure, one would be able to predict essentially everything that one can measure experimentally. While rates and free energies themselves can only indirectly detail the nature of how proteins fold, clearly the ability to quantitatively predict all experimental observables is a necessary perquisite for any successful theory or simulation of protein folding. However, a quantitative prediction of all experimental observables is not sufficient. If a simulation could only reproduce experiments, the simulation would not yield any new insight, which is the goal of simulations in the first place, of course. This leads to a third important challenge for simulation: gaining insight from simulations. Indeed, as one adds detail to simulations, the burden of analysis becomes greater and greater. Atomistic simulations can easily generate gigabytes of data to be processed and the vast number of degrees of from time-resolved protein and water coordinates can obscure any simple, direct analysis of the folding mechanism.

Figure 1. Relevant timescales for protein folding. While detailed simulations must start with femtosecond timesteps, the timescales one would like to reach are much longer, requiring billions (microseconds), to trillions (milliseconds) of iterations. Typical fast, modern CPUs can do approximately a million iterations in a day, posing a major challenge for detailed simulation.

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Models: Atomistic Models for Protein Folding Atomic Force Field

Atomistic models for protein folding typically utilize a classical force field which attempts to reproduce the physical interaction between the atoms in the protein and solvent. The energy of the system is defined as the sum of interatomic potentials, which consist of several terms: E = E LJ + E Coulomb + E bonded

(1)

The van der Waals interaction between atoms is modeled by a Lenard– Jones energy (ELJ)

E Coulomb = ij εij

σij rij

12

−

σij rij

6

(2)

where σij is related to the size of the atoms i and j and εij is related to the strength of interaction. While van der Waals attraction is relatively weak, the LJ potential also serves an important role in providing the hard core repulsion between atoms. The bonded interactions modeled in E bonded handle the specific stereochemistry of the molecule – in particular, the nature of the covalent bonds and steric constraints in the angles and dihedral angles of the molecule. While these interactions are clearly local, they play a very important role in determining the conformational space of the molecule. Finally, E Coulomb corresponds to the familiar Coulomb’s law: qj qi (3) E Coulomb = rij ij where qi is the charge on atom i and rij the distance between atoms i and j . It is perhaps most natural to handle the pairwise interactions explicitly as the equation above, but this of course leads to simulation codes whose performance scales like N 2 , where N is the number of atoms. Clearly, this is very computationally demanding and ideally the calculation can be made O(N ). For inherently short range interactions, it is natural to do this with cutoffs and long-range corrections, i.e., to set the potential to zero smoothly beyond some cutoff distance, e.g., 12A. However, this cutoff procedure has been shown to lead to qualitatively incorrect results for Coulomb interactions [9] and instead reaction field or Ewald-based methods have been suggested with significantly better results [10]. Clearly, there are many parameters in the above formulae. Indeed, these numbers grow further when one considers the fact that the chemical environment of atoms renders even the same type of chemical element (e.g., carbon) to act very differently in different environments. For example, carbon in a hydrocarbon chain will behave fundamentally differently than carbon in an aromatic

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ring. In order to handle this purely quantum mechanical effect in a classical model, one creates multiple atom types (corresponding to the different relevant environments) for each physical atomic element. In this example, one would define different carbon atom types. Thus, while there are only a handful of relevant physical atoms involved (primarily carbon, hydrogen, oxygen, and nitrogen), there can be tens of different atom types. While this is clearly the natural way to handle the role of chemical environment in a classical model, this leads to an explosion of parameters needed in the model, leading to a modeling challenge of the determination of these parameters. Several groups have risen to this challenge and have developed parameterizations for the force field functionals similar to the form above. Typically, these parameterizations are divided into terms for proteins (such as AMBER [11], CHARMM [12], and OPLS [13]) and for the solvent (e.g., TIP3P or SPC). These force fields are typically also associated with molecular dynamics codes themselves and thus one should not confuse the two.

2.2.

Implicit Solvation Models

With the parameterization above for the physical forces between atoms, one can simulate the relevant interactions: protein–protein, protein–solvent, and solvent–solvent. However, in typical simulations with solvent represented explicitly (i.e., directly simulating the solvent atom by atom), the number of solvent atoms is much larger than the number of protein atoms and thus most (e.g., 90%) of the computational time goes into simulating the solvent. Clearly, the solvent plays an important role since the hydrophobic and dielectric properties of water play a fundamental role in protein stability. However, an alternative to explicit simulation of water is to include these properties implicitly; i.e., by using a continuum model. Typically, these models account for hydrophobicity in terms of some free energy price for solvent exposed area on the protein. These surface area (SA) based methods vary somewhat in terms of how the SA is calculated. We stress that one should not a priori expect that a simpler (and perhaps less accurate) calculation of the SA yield worse results than a more geometrically accurate SA calculation. Indeed, since SA is itself an approximation, what is important for the fidelity of the model is not the geometric accuracy of the SA but rather whether the SA term faithfully reproduces the physical effect as judged by comparison to experiment. The dielectric contribution of water to the free energy is in some way a more difficult contribution to consider. The canonical method follows the Poisson–Boltzmann equation. Consider a protein in solvent. We can model the protein as a dielectric medium with a dielectric of εin and water as a medium with a dielectric of εout (thus making the dielectric a function of spatial

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position, ε(x)). Also, consider that the protein will likely have charges with a spatial density ρprotein(x) and that there will be counter ions in the solvent with a charge density ρcountert(x). In this case, we can describe the interaction between the charges and the dielectrics as: ∇[ε(x)∇φ] = −4πρ(x) = −4π [ρprotein(x) + ρcountert(x)] where the total charge density ρ(x) is comprised of both the protein and counter ion charges. If one assumes that the counter ion density is driven thermodynamically to its free energy minimum, we can make the mean field like approximation and state that ρcountert(x) =

i

n i qi exp

−qi φ(x) kT

(4)

where n i is the bulk number density of counter ion species i and qi is its charge. Thus, this method hands counter ions implicitly as well as the water. Including this term leads to the so-called nonlinear Poisson–Boltzmann equation. If the Boltzmann term is Taylor expanded for small cφ(x)/kT (i.e., high temperature, low counter ion concentration, or low potential strength), one gets the so-called linearized Poisson–Boltzmann equation. In general, the Poisson–Boltzmann equation is considered by many to be the “gold standard” for implicit solvation calculation. It can be used for both energy and force calculation [14] and thus can be used for molecular dynamics calculation. However, PB calculation is typically very computationally demanding and there has been much effort to develop more computationally tractable, empirical approximations to the PB equation. For example, Still and coworkers developed an empirical approximation to PB [15]. Based on a generalization of the Born equation for the potential of atoms, Still’s Generalized Born (GB) model (and its subsequent variants from Still’s group and other groups) have been shown to be both computationally tractable and quantitatively accurate for some problems, including the solvation free energy of small molecules [15] and protein folding kinetics [16].

2.3.

How Accurate are these Models?

Any question of accuracy must consider the desired quantity to calculate. While the spirit of this model has in some ways been empirically derived, it has been shown to agree reasonably well with PB calculation. More importantly, GB models have been able to accurately predict experiment, such as the solvation free energy of small molecules [15, 17]. In the end, experiment must of course be the final arbiter of any theoretical method. Moreover, while PB is on much more firm mathematical footing (i.e., one can derive it directly from the Poisson equation), one must consider that PB itself is empirical in

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nature in some respects. Clearly, the concept of a dielectric is a macroscopic quantity and it is an approximation of sorts to apply these macroscopic concepts to the microscopic world of small molecules and proteins (hundreds to thousands of atoms). However, PB’s great success as a predictive tool demonstrates the validity (or at the very least predictive power) of such methods and approximations.

3.

Sampling: Methods to Tackle the Long Timescales Involved in Folding

Simulating the mechanism of protein folding is a great computational challenge due to the long timescales involved. Below, we briefly summarize some methods which have been used to address this challenge. As in any computational method, each has its own limitations and it is natural to consider the regime of applicability of each method.

3.1.

Diffusion–Collision Models

Due to the intrinsic hierarchical nature of the structure of proteins, it is intriguing to consider whether folding kinetics may also be hierarchical [18, 19]. Indeed, a natural hypothesis for the mechanism of protein folding is that secondary structural elements (such as alpha helices and beta hairpins) form first and then diffuse and consequently collide to form tertiary contacts and thus finally form the folded state. A generalization of this idea is that one can drastically simplify the conformational space of proteins into a series of specific states, perhaps determined by the formation of secondary structure, but not necessarily so. In this model, protein elements form, diffuse, and then collide to form larger tertiary structure. Weaver and Karplus first proposed this model for folding and its ability to predict folding kinetics [20]. With this simplification, one can model the folding mechanism (and predict folding rates) by kinetically connecting these states by including the rates between these states in a diffusion equation. This allows one to predict the overall rate (to compare with experiment) as well as the ability to predict the rate of formation of any relevant accumulating intermediate states [21, 22].

3.2.

High Temperature Unfolding

While folding times are clearly long from a simulation point of view, unfolding (especially under high denaturation conditions) can be very fast – on the nanosecond timescale. Under extreme denaturing conditions (e.g., ∼400 K

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temperature), one would expect that the folded state would become only metastable, with a low barrier to unfolding. If one is considering an energy (not free energy) barrier, increasing the temperature speeds kinetics in a relatively simple manner, in that the thermal energy to cross the barrier is simply higher and increases the rate of barrier crossing. Protein unfolding at high temperatures is more complex since entropy plays a dominant role in protein folding and unfolding. Thus, as one changes the temperature, one changes the nature of the underlying free energy landscape and barrier. Daggett and Levitt [24] first took advantage of this scenario, and Daggett’s group has subsequently used this method to examine a variety of proteins and compare their results to experiment, especially with a comparison of phi values calculated high temperature folding vs. experimental measurements [8, 25]. It is interesting to ask what is the regime of applicability of high temperature unfolding? Clearly, if the temperature is too high, it can lead to qualitative changes of the underlying free energy landscape. For example, at very high temperatures, the free energy barrier will be completely lost, and unfolding will occur as a “downhill” process. However, at less severe temperatures, the free energy barrier will still be present, although it is not guaranteed that alternate pathways would not become the rate limiting step in unfolding and one may expect that the transition state may shift to be more native-like. Nevertheless, this method has been used to predict experimental phi values, demonstrating agreement to experiment [8, 25]. The interesting next step is to understand the reasons for this intriguing level of agreement.

3.3.

Low-Viscosity Simulation

Another common means to try to tackle long timescales is to use an implicit solvation model with low viscosity. In implicit solvation models, one typically uses the Langevin equation for dynamics and employs a damping term consistent with water-like viscosity. However, water is relatively quite viscous and one need not use full-water viscosity. Instead, many groups have proposed to use viscosities 1/100 to 1/1000 that of water – or even no viscosity at all. Clearly, lowering the viscosity will speed the kinetics, but potentially at the risk of altering the nature of the kinetics, not only the rate [26]. Indeed, consider the case of protein folding from a random-walk configuration. In a water-viscosity simulation, collapsed is slowed due to the viscosity and the protein anneals its conformation (presumably forming native-like structure) along the way to collapse, thus collapsing into a partially native (or potentially completely native) structure. In a low-viscosity simulation, collapse is no longer rate limiting, and thus one quickly reaches a collapsed, but random globule; folding then proceeds via conformational rearrangements of this collapsed globule, which is a very different mechanism than that of the

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water-like viscosity simulations. Moreover, this change in mechanism is likely responsible for the nonlinear relationship between folding time and Langevin damping constant γ found in recent simulation [26]: near water-like viscosity, there is a linear correlation between folding time and γ . However, at low γ values, this correlation becomes nonlinear, and thus one cannot reliably use an extrapolation in γ from low-viscosity simulation to predict water-like viscosity rates [26]. Nevertheless, for applications in which kinetic information is not needed, this means may potentially be a significant speed increase (and a competitive advantage of implicit solvation models in general over explicit solvation models).

3.4.

Sampling of Paths Using Relatively Short Trajectories

One fundamental property of all barrier crossing systems is that the ratio of the time to cross the barrier on an activated trajectory will be considerably faster than the mean folding time. Indeed, such a property is also directly a consequence of single exponential kinetics. In this case, the probability of a given molecule to fold at time t is given by p(t) = k exp(−kt), where k is the rate of folding. This probability can be integrated to get the more familiar fraction of molecules which fold by time t: f (t) = 1–exp(−kt). By either measure, we see that the probability of a molecule folding at short times (much shorter than 1/k) is very high. Of course, eventually, this single exponential approximation will break down, leaving the more direct question of what is the ratio of the barrier crossing time to 1/k. In simple systems, such as molecular isomerization, this time is likely very short, on the picosecond timescale [27]. For protein folding, these timescales are likely much longer, on the nanosecond timescale [28–30]. For systems in which the barrier crossing time is amenable to simulation (but perhaps simulating timescales at 1/k is not), path sampling methodologies can likely play a useful role [31–33]. The path sampling methods pioneered by Pratt’s and Chandler’s groups seek to sample a series of uncorrelated paths connecting reactant and product species (unfolded and folded states in the case of protein folding, for example). These paths can be generated by using an initial molecular dynamics trajectory and then “shooting” off new trajectories by starting a new trajectory from an existing part of the original trajectory (i.e., with the same coordinates), but with a variation in the momenta, such that the new trajectory takes a different path. Thus, shooting leads to a series of decorrelated trajectories. Path sampling-based methods have successfully used by Bolhuis’ group to simulate the folding of the C-terminal beta hairpin from protein G [34]. Bolhuis’ work agreed with the mechanism previously proposed, as well as a quantitative agreement of predicted rate with experiment.

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One caveat with these path sampling methods is the need to characterize the reactant and product states for the generation of folding trajectories. Indeed, this characterization itself is difficult and a change in this characterization would require rerunning the path sampling simulation. Also, one must typically choose a timescale for barrier crossing trajectories and this quantity is difficult to assess a priori, especially in the case of protein folding. To avoid these complications, one can take a far simpler approach and simply run many, uncoupled trajectories. This method is another application of the concept that the barrier crossing time, since many, relatively short trajectories can be naturally analyzed in the single-exponential folding regime to yield information of folding mechanisms and rates. This method has been applied by Pande’s group on several, small, fast folding proteins. By running tens of thousands of atomistic molecular dynamics simulations, each on the tens of nanosecond timescale, Pande’s group has been able to simulate the folding of small, fast folding proteins with a quantitative agreement with experiment [29, 35–37]. The primary benefits of this method are that it (1) can fold directly from sequence, since it does not require the knowledge of the native state to generate the simulations (although knowledge of the native state was used in the analysis of the data) and (2) can yield quantitative agreement with experimental rates, typically within the experimental uncertainty. There are a few important caveats of this method. First, one must make sure that each trajectory exceeds the typical barrier crossing time (typically on the nanosecond timescale) [38]; this may or may not be possible with a given system and it is difficult to know a priori. Second, while the analysis of multiple short trajectories need not assume single exponential kinetics, the usefulness of this method is strongest in cases where the second most rate limiting step is faster than the simulated trajectory timescale (i.e., faster than tens of nanoseconds). This will likely pose a challenge for larger, more complex proteins, which may appear single exponential experimentally, but have second most rate limiting steps with long timescale phases (e.g., hundreds of nanoseconds). Finally, one major caveat with this method is its great computational demands, typically requiring distributed computing methods [39]. However, with grid and distributed computing resources becoming popular, one might expect that this would become an overwhelming burden in the coming years, and will always be more computationally efficient than traditional parallel molecular dynamics.

References [1] C.M. Dobson, Trends Biochem. Sci., 24, 329–32, 1999. [2] V. Grantcharova, E.J. Alm, D. Baker, and A.L. Horwich, Curr. Opin. Struct. Biol., 11, 70–82, 2001.

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[3] C.M. Dobson, A. Sali, and M. Karplus, Angew. Chem. Int. Ed. Engl., 37, 868–893, 1998. [4] V.S. Pande, A. Grosberg, T. Tanaka, and D.S. Rokhsar, Curr. Opin. Struct. Biol., 8, 68–79, 1998. [5] M. Levitt, Nature Str. Biol., 8, 392–393, 2001. [6] H. Abe and N. Go, Biopolymers, 20, 1013, 1981. [7] V.S. Pande, and D.S. Rokhsar, Proc. Natl. Acad. Sci. U.S.A., 95, 1490–1494, 1998. [8] U. Mayor, N.R. Guydosh, C.M. Johnson, J.G. Grossmann, S. Sato, G.S. Jas, S.M. Freund, D.O. Alonso, V. Daggett, and A.R. Fersht, Nature, 421, 863–867, 2003. [9] P.J. Steinbach and B.R. Brooks, Journal of Computational Chemistry, 15, 667–683, 1994. [10] I.G. Tironi, R. Sperb, P.E. Smith, and W.F. Vangunsteren, J. Chem. Phys., 102, 5451– 5459, 1995. [11] W.D. Cornell, P. Cieplak, C.I. Bayly, I.R. Gould, K.M. Merz, D.M. Ferguson, D.C. Spellmeyer, T. Fox, J.W. Caldwell, and P.A. Kollman, J. Am. Chem. Soc., 117, 5179– 5197, 1995. [12] B.R. Brooks, R.E. Bruccoleri, B.D. Olafson, D.J. States, S. Swaminathan, and M. Karplus, J. Comp. Chem., 4, 187–217, 1983. [13] W.L. Jorgensen and J. Tirado-Rives, J. Am. Chem. Soc., 110, 1657–1666, 1988. [14] J.A. Grant, B.T. Pickup, and A. Nicholls, J. Comput. Chem., 22, 608–640, 2000. [15] D. Qiu, P.S. Shenkin, F.P. Hollinger, and W.C. Still, J. Phys. Chem. A, 101, 3005–3014, 1997. [16] V.S. Pande, I. Baker, J. Chapman, S. Elmer, S. Kaliq, S. Larson, Y.M. Rhee, M.R. Shirts, C. Snow, E.J. Sorin, and B. Zagrovic, Biopolymers, 68, 91–109, 2003. [17] V. Tsui and D.A. Case, Biopolymers, 56, 275–291, 2001. [18] R.L. Baldwin and G.D. Rose, TIBS, 24, 26–33, 1999. [19] R.L. Baldwin and G.D. Rose, TIBS, 24, 77–83, 1999. [20] M. Karplus and D.L. Weaver, Protein Sci., 3, 650–668, 1994. [21] R.V. Pappu and D.L. Weaver, Protein Sci., 7, 480–490, 1998. [22] R.E. Burton, J.K. Meyers, and T.G. Oas, Biochemistry, 37, 5337–5343, 1998. [23] V. Daggett, A. Li, L.S. Itzhaki, D.E. Otzen, and A.R. Fersht, J. Mol. Biol., 257, 430–440, 1996. [24] V. Daggett and M. Levitt, J. Mol. Biol., 232, 600–619, 1993. [25] V. Daggett and A.R. Fersht, Trends Biochem. Sci., 28, 18–25, 2003. [26] B. Zagrovic and V. Pande, J. Comput. Chem., 24, 1432–1436, 2003. [27] D. Chandler, J. Chem. Phys., 68, 2959–2970, 1978. [28] V.S. Pande and D.S. Rokhsar, Proc. Nat. Acad. Sci. U.S.A., 96, 9062–9067, 1999. [29] C. Snow, H. Nguyen, V.S. Pande, and M. Gruebele., Nature, 420, 102–106, 2002. [30] C. Snow, B. Zagrovic, and V.S. Pande, J. Am. Chem. Soc., 2002. [31] C. Dellago, P.G. Bolhuis, F.S. Csajka, and D. Chandler, J. Chem. Phys., 108, 1964– 1977, 1998. [32] D. Chandler, From the Lectures given at the Euroconference on Computer Simulations of Rare Events, Lerici, Villa Marigonla, Italy, 1997. [33] L.R. Pratt, J. Chem. Phys., 85, 5045–5048, 1986. [34] P.G. Bolhuis, Proc. Natl. Acad. Sci. U.S.A., 100, 12129–12134, 2003. [35] V.S. Pande, I. Baker, J. Chapman, S.P. Elmer, S. Khaliq, S.M. Larson, Y.M. Rhee, M.R. Shirts, C.D. Snow, E.J. Sorin, and B. Zagrovic, Biopolymers, 68, 91–109, 2003.

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[36] C.D. Snow, B. Zagrovic, and V.S. Pande, J. Am. Chem. Soc., 124, 14548–14549, 2002. [37] B. Zagrovic, E.J. Sorin, and V. Pande, J. Mol. Biol., 313, 151–169, 2001. [38] E. Paci, A. Cavalli, M. Vendruscolo, and A. Caflisch, Proc. Natl. Acad. Sci. U.S.A., 100, 8217–8222, 2003. [39] M. Shirts and V.S. Pande, Science, 290, 1903–1904, 2000.

6.1 POINT DEFECTS C.R.A. Catlow Royal Institution of Great Britain, London W1S 4BS, UK

1.

Introduction

Point defects are pervasive. They are present in all materials, as a result of the intrinsic thermodynamic equilibrium and of the inevitable levels of impurities. Indeed, elementary chemical thermodynamics shows that if a defect formation reaction (e.g., Frenkel or Schottky formation as discussed below) is associated with a free energy gD , then the equilibrium mole fraction, xD , of defects is given by:

gD xD = exp − , nkT

(1)

where n is the number of defects created in the defect formation process. The equation shows that xD > 0 for T > 0, and also demonstrates that properties dependent on defects will show “Arrhenius”-like temperature dependence. The three broad classes of point defects are vacancies (unoccupied lattice sites), interstitials (extra lattice or impurity atoms/ions at sites which are not normally occupied) or substitutionals (impurity atoms/ions at normal lattice sites). They may be created thermally (i.e., by the intrinsic thermodynamic equilibrium discussed above), chemically (owing to the presence of impurity species) and by irradiation or mechanical damage. In the case of thermally generated defects, there are two basic models of disorder referred to above. Frenkel disorder involves the displacement of lattice ions to interstitial sites and anion Frenkel disorder is the predominant mode of disorder in, e.g., CaF2 ; Schottky disorder involves the creation of vacancies in stoichiometric properties as in NaCl where Schottky pair formation (sodium and chlorine vacancies) are the dominant mode of disorder. Detailed discussions of point defects and defect-dependent properties are given in the monograph of Agullo-Lopez, Catlow and Townsend [1] in the proceedings of several NATO Advanced Studies Institute (e.g., [2, 3]. This 1851 S. Yip (ed.), Handbook of Materials Modeling, 1851–1854. c 2005 Springer. Printed in the Netherlands.

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chapter will focus on the calculation of point defect structures, energies and properties, with a strong emphasis on ionic materials where computational techniques have had a major impact.

2.

Defect Dependent Properties

Even small defect concentrations can have a major influence on key physical and chemical properties of materials. Here we introduce some of the most widely studied and significant aspects of the field. Atomic Transport including both diffusion and ionic conductivity is invariably affected by defect processes as both vacancy migration (in which neighboring atoms/ions jump into vacant sites) and interstitial migration affect transport of atomic charge. Conductivity and diffusion have been very extensively studied over the last 50 years, and calculations of defect formation and migration energies have played a crucial role in the development of the field as discussed later. Spectroscopic properties of solids may be drastically modified by defects which may provide localized hole or electron states within the energy band gap of the solid. The classic example is the “F center” in which one (or more) electron(s) are trapped at an anion vacancy site; and it has been known since the 1930s that the presence of such centers in, e.g., NaCl (induced by additive coloration, i.e., reaction of the crystal with metal vapor, or irradiation damage) impart an intense color to the crystal owing to optically induced excitations of the trapped electron. Substitutional impurities, especially transition metal ions can also provide absorption/emission centers in crystals, and indeed transition metal impurities are responsible for the colors of many gem-stones, e.g., ruby (Al2 O3 containing Cr3+ impurities). Spectroscopic properties of defective solids have been intensively investigated by theoretical/computational methods. An excellent account of the basic theory is given by Stoneham [4]. Equilibrium with the gas phase and nonstoichiometry. Nonstoichiometry is wide spread amongst the solid compounds of transition metals and lanthanide and actinide metals; examples are Fe1−x O, Ni1−x O, UO2+x , CeO2−x . In such compounds, nonstoichiometry can be understood in terms of variable cation valence, accompanied by defect formation. A simple example is Ni i−x O, where Ni2+ may be oxidized to Ni3+ – a process which is compensated by the formation of nickel vacancies. Indeed, oxidation of the crystal involves the following reaction: 3NiNi + 12 O2 → 2N•Ni + V||Ni + “NiO” where we have used the “Kroger-Vink” notion for defects, NiNi represents a || regular lattice Ni ion, Ni•Ni an oxidized cation and VNi a nickel vacancy.

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It is clear that the vacancy concentration equals the deviation from stoichiometry, x, in Nii−x O, and that it is controlled by the oxygen partial pressure, PO2 . Indeed, a simple mass action analysis shows that if the above equilibrium operates: x=

1 1/6 K PO2 , 4

where K is the equilibrium constant for the reaction. If, however, more complex processes are involved (e.g., defect clustering) then a different dependence of x on PO2 will be observed. Extensive analyses have been performed on the variation of x with PO2 for nonstoichiometric compounds with the aim of elucidating their defect structures (see, e.g., [5]). Nonstoichiometric compounds can in many cases accommodate very high levels of disorder; for example, in Fe1−x O materials with x = 0.3 can be prepared, and in UO2−i x , x varies from 0.0 to 0.25. Heavily disordered systems invariably contain defect clusters, and defect clustering is, in general, an important phenomenon in doped and nonstoichiometric solids. Examples will be given in the sections which follow, and reviews of earlier work can be found in the monograph edited by Sorensen [6].

3.

Defect Calculations

Attempts to calculate defect energies go back to the origins of the field with pioneering work of Mott in the 1930s and Coulson in the 1950s. Details will be given in the sections which follow; here we review general aspects of the different methods available which are as follows:

3.1.

Simple Cluster Calculations

Here a cluster is defined which contains the defect and surrounding coordination shells. The energy and structure of the cluster is calculated with and without the defect. The calculations are usually performed using an appropriate quantum mechanical method. The approach, of course, omits long-range effects of the surrounding lattice (which may, however, be relatively unimportant in covalent materials), and may also be problematic regarding the treatment of the “dangling bonds” of the atoms on the periphery of the cluster, where a common approach is to saturate by attaching hydrogen atoms. The method has, however, been widely used to study defects in semiconductors.

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C.R.A. Catlow

Embedded Cluster Methods

This approach attempts to remedy some of the deficiencies of simple cluster methods by embedding the cluster in a more approximate representation of the surrounding lattice. For example, if the cluster is described quantum mechanically, the embedding region may use interatomic potentials. Case must be taken in interfacing the two regions and in ensuring that the effects of the embedding region on the inner region containing the defect are accurately represented. An older, simpler variant of the approach in the Mott–Littleton method discussed in later sections, where the inner region is treated using interatomic potentials, and the outer region is a quasi-dielectric continuum. The method has enjoyed widespread success in modeling defects in ionic crystals.

3.3.

Periodic Methods

The approach here is particularly simple: a supercell is defined with the defect in the center and periodic boundary conditions are applied. The calculations are performed both with and without defects. The resulting defect energy and structure includes, of course, the effects of the interactions between the periodic images of the defects and the use of the method may be problematic, for charged defects (although there are prescriptions for overcoming this latter difficulty). The major advantage of the method is that it may use the extensive range of procedures and methods developed for modeling periodic solids. Further discussions of methods and applications of defect simulation will be given in the sections which follow.

References [1] F. Agullo-Lopez, C.R.A. Catlow, and P. Townsend, Point Defects in Materials, Academic Press, London, 1988. [2] F. B´eni`ere and C.R.A. Catlow (eds.), Mass Transport in Solids, NATO ASI Series, Plenum Press, vol. 97, 1983. [3] C.R.A. Catlow (ed.), Defects and Disorder in Crystalline and Amorphous Solids, NATO ASI Series C, vol. 418, Kluwer, Netherlands, 1993. [4] A.M. Stoneham, Theory of Defects in Solids, Clarendon Press, Oxford, 1976. [5] P. Kofstad, Non-Stoichiometry, Diffusion and Electrical Conductivity in NonStoichiometric Metal Oxides, John Wiley and Sons, 1972. [6] O.T. Sorensen, Non-Stoichiometric Oxides, Academic Press, New York, 1980.

6.2 POINT DEFECTS IN METALS Kai Nordlund and Robert Averback Accelerator Laboratory, P.O. Box 43 (Pietari Kalmin k. 2), 00014, University of Helsinki, Finland Department of Materials Science and Engineering, University of Illinois at Urbana-Champaign, Illinois, USA

Calculations of the properties of point defects in metals by computer simulations have the enormous advantages over other classes of materials since, to first order, charge exchange and angular forces can be neglected, thereby greatly reducing the computational effort. Moreover, the structure, concentration and diffusivity of both the vacancy and interstitial defects are known experimentally in a number of metals, so that a solid base of information is available to validate the simulation models employed. In this article we will first briefly summarize this base of experimental knowledge and then discuss current state of the art methods for calculating the properties of defects in metals, citing notable successes and failures. Each subsection is laid out discussing first the ground state energies of the defect structures, then the defect concentration (or entropy), and finally the migration properties of the defects. In the last section, we will briefly examine attempts to apply the concepts of defect structures in crystalline structures to metallic glasses.

1. 1.1.

Experimental Background Defect Structure

The structure of the single vacancy in metals is widely accepted to be the simplest possible one, that obtained when a single atom is removed from the lattice. In an elastic continuum, which works quite well for metals, the excess energy derives mostly from the creation of surface energy associated with the cavity, or breaking of bonds in a discrete lattice. Within this picture, the positive surface energy leads to a slight inward relaxation of surrounding atoms, 1855 S. Yip (ed.), Handbook of Materials Modeling, 1855–1876. c 2005 Springer. Printed in the Netherlands.

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giving rise to a negative vacancy relaxation volume and change in the lattice parameter, as generally observed experimentally [1, 2]. The structure of the interstitial has also been measured directly in several metals [1, 2]. The interstitials in metals with the FCC, BCC or HCP structures are generally so-called “split” structures where two atoms share a single lattice site (Fig. 1). The two atoms are symmetrically arranged on both sides of the ideal site, so that it is not possible to state which one of the two would be the extra atom. They strongly displace the surrounding atoms outwards, giving rise to a large positive relaxation volume. We can estimate the relaxation volume by simply noting that for close-packed hard spheres, the insertion of an atom increases the total volume by approximately two atomic volumes, one owing to the extra atom itself, and the other from breaking the closepacking arrangement. The crystal direction defined by the two atoms and the lattice site is used to characterize the defect, so that one can, for instance, refer to a 1 1 0 split interstitial. Frequently, these interstitials are called dumbbell interstitials. This curious name stems from illustrating the two atoms as two (a)

(b)

(c)

(d)

(e)

(f)

Figure 1. Some sample defect structures in a pure element. Atoms on the front side of the polygons are shown as open circles, and atoms on the back sides as dashed open circles. Atoms forming dumbbell interstitials are shown as solid circles. Vacancy positions are indicated as thin dashed open circles. (a) 1 0 0 dumbbell interstitial in an FCC metal. (b) 1 1 0 dumbbell interstitial in a BCC metal. (c) 0 0 0 1 dumbbell interstitial in an HCP metal. (d) Vacancy in an FCC metal. (e) Vacancy in a BCC metal. (f) Divacancy with atoms missing from second-nearest-neighbor sites in a BCC metal.

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1857

large spheres joined by a narrow bonding-rod, making the structure resemble a dumbbell weight-lifting device. Whether interstitials always have a split structure has long been questioned. In several BCC metals, calculations indicate that the energy of the so-called 1 1 1 crowdion structure is close or even lower than that of the 1 1 0 split structure [3, 4]. The extra atom in the crowdion structure creates a string-like defect along the 1 1 1 direction, encompassing some tens of atoms. Even when this structure does not represent the ground state, it may play a major role in interstitial migration; in the crowdion structure an interstitial can move several atomic distances very rapidly due to the collective and linear nature of the defect. The most obvious structure of the divacancy in the elastic continuum model is represented by two vacant nearest-neighbor sites, and indeed this is the structure experimentally observed in the BCC metal W [2], and it is believed to be the structure in most FCC metals as well [2]. Recent DFT calculations, however, suggest that this structure is not always the ground state, with the stable divacancy in Al being formed by two next-nearest-neighbor atoms [5]. In intermetallics, like ionic crystals, several additional point defect types arise. The designation of vacancies and interstitials must specify which constituent has been added or removed. A missing Al atom in NiAl, for example, can be denoted VAl and a Ni interstitial as INi . In addition, entirely new defect types can exist. An antisite defect represents an atom located on the wrong sublattice, e.g., a Ni atom on an Al site is denoted NiAl . A pair of adjacent antisites of opposite types is called an exchange defect. In addition, more complex point defect types can exist; examples of these are mentioned in Section 4.2.

1.2.

Concentration

The equilibrium concentration, c, of one defect type can be written [2, 6] as: c = g eS / k e−H f

f / kT

(1)

where k is the Boltzmann’s constant, H f the defect formation enthalpy, and S f the formation entropy. g is a geometrical factor which depends on the number of equivalent crystallographic orientations of the defect. For instance, the 1 0 0 dumbbell interstitial in FCC metals has the three equivalent orientations 1 0 0, 0 1 0, and 0 0 1 and hence g = 3. Since the g factor represents configurational entropy, it is commonly absorbed in the term S f . The type of the defect is conventionally given as a subscript, such that the monovacancy concentration is denoted cv or c1v , the interstitial formation entropy Sif , the divacancy formation f enthalpy H2v , and so on. The total equilibrium defect concentration and its temperature dependence has been measured experimentally for a number of metals using different techniques [2]. The predominant defect in all common transition metals in

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thermodynamic equilibrium is the vacancy. This was shown by Simmons and Balluffi [7] and has later been confirmed by independent experimental methods, such as positron annihilation spectroscopy which detects open volume in lattices. The reason vacancies dominate is easy to understand; most transition metals have atomically dense structures such as FCC, HCP and BCC, where it is very difficult to squeeze in an extra atom. The alkali metals appear to form an exception; their interatomic bonding distances are relatively large, and hence it has been suggested that it is also relatively easy to add atoms on interstitial sites [8]. While vacancies are the predominant defect in most metals, nonArrhenius diffusion behavior suggest that other defect types are also important [6, 9, 10]. Most often divacancies are assumed to contribute to diffusion at high temperatures [6, 9], although trivacancies and interstitials have also been considered. Since these other defects have much higher enthalpies of formation than does the vacancy, they can only be important if they have exceptionally high entropies of formation and low enthalpies of migration. As will be discussed below, the interstitial has an enthalpy of migration of ≈0.1 eV and an entropy of formation of ≈15kB [11, 12]. Simulation results indicating interstitials are present near the melting point are shown in Fig. 2.

Defect concentration

10⫺4

cv

10⫺5 10⫺6 c2v

10⫺7

ci

10⫺8 10⫺9 10⫺10

0.95

1.0

1.05

1.1 1.15 Tmelt/T

1.2

1.25

1.3

Figure 2. Defect concentrations in Cu around the melting temperature. The vacancy concentration is the experimental one, while the interstitial and divacancy concentrations are predictions from our computer simulations [39]. These results suggest that the interstitial concentration may in fact exceed the divacancy concentration at the highest temperatures.

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1.3.

1859

Defect Migration

The jump rate w of a single defect can be written [6] as: w = ν eS

m/ k

e−H

m / kT

(2)

ν is the vibrational frequency of the lattice, H m the migration enthalpy of the defect, and S m the migration entropy. The number of jumps due to all defects of the same type is the defect concentration c times the jump rate [6, 9], = Z cw

(3)

where Z is the number of directions available for the jump. If the defect migration occurs by a purely random walk process, the self-diffusion coefficient for a single defect type is given by the Einstein relationship [10, 13]: D = 16 r 2

(4)

where r is the jump distance in the lattice. If one measures the self-diffusion coefficient using, for example, tracer atoms, then one has to correct this equation for correlation effects. This can be accounted for by including a jump correlation factor f [6] to obtain the tracer self-diffusion equation, D = 16 r 2 f

(5)

The correlation factor is normally in the range 0.1–1; for vacancies on a FCC lattice, for example, f = 0.781. The migration mechanisms cannot be determined directly experimentally, but can for many simple defect types be deduced from simple geometrical arguments, treating the atoms as soft spheres or with a pair potential, see Fig. 3. Modern DFT calculations have, generally speaking, given defect migration mechanisms consistent with those obtained from the simple geometrical arguments [14, 15]. For instance, the vacancy (or in actuality one atom next to the vacancy) in FCC metals moves along the straight line connecting one lattice site with the next-nearest-neighbour site. The split interstitial migration is by necessity somewhat more complex, but occurs by one of the atoms A in the split configuration filling the original lattice site, while simultaneously the other atom B pushes away an atom C in a nearby site and forms a new split interstitial [14]. Thus, the split interstitial formed by atoms A and B becomes a new split interstitial formed by atoms B and C, see Fig. 3(b). The factors Z and f are known for a given defect structure in a given lattice. Hence the factors determining the defect migration S m and H m can be deduced by measuring the temperature-dependent self-diffusion of the material, if c and hence S f and H f are known. This procedure, however, is complicated by the curvature in the Arrhenius plots. If several defect types are present, they can contribute to the curvature of the self-diffusion coefficient D

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K. Norlund and R. Averback (a)

(b)

Figure 3. Migration of defects in an FCC metal. (a) Migration of a vacancy. The arrow indicates how an atom moves from one site to the next, and the atom is shown in the saddle point configuration in the middle of the line joining the two sites. The initial and final empty site are shown with thin dashed lines. (b) Migration of an interstitial. For clarity only atoms on the front side of the cube are shown. The initial positions of the two atoms forming the dumbbell are shown in black, and the final ones in a striped pattern. Arrows indicate the movement of the three atoms involved.

both via their concentration and the difference in the migration coefficients. Moreover, since self-diffusion is hard to measure at low temperatures, this method does not work well for defects with low defect migration energies. More typically, the migration enthalpy of defects H m has been determined using either quenching or irradiation experiments. In quenching experiments, for example, specimens are first quenched from high temperatures, to a temperature at which the defects are immobile. The specimen is then warmed to successively higher temperatures according to some annealing schedule, while monitoring some property that is proportional to the defect concentration. By following the rate at which the defects annihilate at their sinks, the migration enthalpy can be deduced, for details see Ref. [16]. For interstitial defects, which have immeasurably small concentrations at high temperatures, irradiation experiments are necessary. The irradiations are performed at low temperatures and these are followed by an annealing program as just described, see Fig. 4. The difficulty with irradiation experiments, however, is that interstitials and vacancies are created in pairs, and their initial locations are correlated. These seemingly simple considerations have led to much confusion in the field over the years (see, for example, the collection of articles in J. Nucl. Mater., 69–70, 1978). From these and other alternative measurement methods, it has become clear that defect migration in transition metals has some general trends. The interstitials are highly mobile, with activation energies typically in the range 0.05–0.5 eV. In Au stage I has not been observed, even for measurements well below 1 K [17, 18]. This has been interpreted to mean that the Au interstitial is

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1.0

Relative damage level

0.8

Stage I Stage II

0.6

Stage III 0.4 Stage IV 0.2

Stage V

0.0 10

2

5

10 2

2

5

10 3

Temperature (K) Figure 4. Annealing stages in Cu. After Ehrhart [2]. The fine structure in stage I is due to recombination of close, bound Frenkel pairs.

mobile at all temperatures due to a quantum mechanical tunneling. The vacancies have mobilities of the order of 1 eV, and thus are much less mobile than interstitials at low temperatures.

2. 2.1.

Methods of Defect Simulation Structure

The determination of the ground state structure of a point defect is made difficult by the need to solve the energy minimization problem in 3N dimensions, where N is the number of atoms in the simulation cell. Moreover, it is generally not possible to formally prove that a particular defect structure represents the ground state in any given model (it took close to 400 years even to prove Kepler’s conjecture, i.e., that the perfect FCC and HCP structures are the densest possible packing of hard spheres [19]). In practice, however, the number of reasonable structures that can be obtained in the common crystal structures, FCC, BCC and HCP, is small, since all of these structures are relatively close-packed.

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The computationally least demanding approach for finding the ground state is to use all of the high-symmetry defect positions in a lattice as initial trials for the defect structure. For each trial, the positions of the atoms surrounding the defect are then relaxed to the closest minimum. This is achieved efficiently using conjugate gradient (CG) energy minimization [20] or by carrying out a molecular dynamics (MD) simulation and scaling the temperature toward zero (MD methods are described elsewhere in this handbook series). A common trick for speeding up the latter method is to set the velocity of atoms to zero if the dot product of the velocity and force vectors is less than zero, i.e., the force is opposite to the velocity. In our experience these two methods are of comparable efficiency in small cells, while in large cells an adaptive CG method [21] is much faster than either plain CG or MD with the velocity–force trick. The lowest-energy state thus obtained yields the ground-state structure of the defect. The main drawback to this trial and error approach is that one is starting from educated guesses, and it is possible that some other state exists that has even lower energy. An approach to find the ground-state structure that circumvents this problem starts with a randomly chosen interstitial position (for interstitials inserting one extra atom in a random position in a periodic cell), and then carrying out an MD or MC simulated annealing run to find the ground state. In practice, this might be implemented by starting at half the melting temperature, simulating for 10 ps in this state, and then slowly cooling the system to 0 K. If this process always gives the same ground state for different starting positions and for a wide range of cooling rates, below some practical maximum rate, one can be quite confident that the true ground state has been obtained. Although simulated annealing is somewhat cumbersome, not using it may give wrong answers. For instance, the original Foiles embedded-atom method (EAM) potential for Pt [22] yields the tetrahedral position for the ground state of the interstitial, which would not be found by starting from the common (1 0 0) split interstitial and simply relaxing it to the closest minimum [23]. Also note that even for vacancies, just removing an atom may not result in the correct ground state of the model. Recent DFT simulations of graphite have revealed a Jahn–Teller effect, which breaks the vacancy symmetry and leads to a distorted vacancy structure [24]. In our experience, surprisingly long cooling times may be needed to obtain the true minimum; in our recent development of a bond-order potential for W we found that times of the order of 100 ps were needed to arrive at the correct ground-state structure of the interstitial. This makes it computationally very difficult to apply the simulation approach in DFT simulations, and in practice DFT methods remain limited to the first approach of testing a large number of high-symmetry structures. Another complication arises from finite-size effects. Since especially interstitial defects have a far extending strain field, a periodic simulation cell will

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always lead to artificial interaction between two defects due to their strain fields. While it is not possible to completely avoid size effects, the magnitude of the error can be checked by using different-sized or -shaped simulation cells, and different pressure relaxation approaches (fixed vs. pressure relaxed). Particular care must be exercised for calculations of extended defect structures, such as the crowdion interstitial. These defects can contain strings of some tens of atoms, and large cells are required to properly model them, sometimes with some thousands of atoms. Again this is particularly limiting for DFT methods, which usually are limited to a few hundreds of atoms. Once the ground-state structure has been found, the relaxation volume of the defect can be easily obtained by subtracting the volume of the perfect cell from that of the cell containing the defect, for cells relaxed to zero pressure. The formation volume f and relaxation volume, V are related by the expression: f = ±0 + V

(6)

where the plus sign is for vacancy formation and the minus sign for interstitials [2].

2.2.

Concentration

From Eq. (1), one can see that the equilibrium concentration of defects is determined by the formation enthalpy H f and formation entropy S f . At zero external pressure, and in the limit of low defect concentration, the formation energy E f of a defect can in a system with Na atom types be defined as [25]: E = E D − Q(E v − µe ) − f

Na

n i µi

(7)

i=1

where E D is the energy of the defect cell at 0 K, Q the charge of the defect, E v the valence band maximum, µe the chemical potential of the electrons, n i the number of atoms of each type i in the defect cell, and µi the chemical potential of atom type i. In monoatomic metal systems the charge state can be ignored and the other terms simplified to give the formation energy as:

Ef =

Ed Eu − Nd Nu

Nd

(8)

where E d and Nd are the potential energy and number of atoms in the defect cell, and E u and Nu the same in a defect-free cell. Thus, the procedure needed for determining E f , in practice, requires knowing the potential energy per atom E u /Nu in a defect-free cell, and then relaxing a simulation cell with a defect to

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zero temperature and pressure to determine E d . The only complications arise from possible finite-size effects, and the need to carry out the calculation of E d to high precision, since Eq. (8) involves the subtraction of two large, almost equal, numbers. The determination of the formation entropy S f is more difficult, since the entropy involves temperature effects and thus the simulations must be long to average out the thermal fluctuations. This can be especially cumbersome for small simulation cells, where the fluctuations are large. The conceptually simplest way of determining S f is to determine the defect concentration c by direct simulation, then use Eq. (1) to calculate S f once E f has been determined as explained above. The concentration can be simulated by placing a crystal in contact with a particle source/sink, e.g., a liquid, and simulating until the defect concentration equilibrates in the crystal part. Since c seldom exceeds ∼0.001 even at the melting temperature, one needs systems with at least tens of thousands of crystal atoms to observe a significant vacancy concentration. Moreover, since the crystal part needs to be large, one also needs long simulation times to allow for in-diffusion of the defect from the particle source. This makes the direct approach quite cumbersome computationally. Nevertheless, it does have the advantage that in principle one single ordinary N P T ensemble MD simulation is enough to determine c, and the method has been successfully employed, at least for vacancies in Cu [11]. Another approach is to use Frenkel–Ladd thermodynamic integration [26–28] to determine explicitly the entropy of a simulation cell with a defect, and then compare it with the entropy of a defect-free cell. This requires constructing a special interaction model where one can move smoothly between an Einstein harmonic solid and the “true” solid of interest, where “true” now signifies the solid modeled by a classical or quantum mechanical method. One needs to simulate a range of systems between the Einstein solid and the true solid. Integration of the obtained enthalpy data gives the free energy difference between the Einstein and true solid, which can be subtracted from the known free energy of an Einstein crystal to give the free energy and hence entropy of a defect. For defects, the reference Einstein lattice should be the fully relaxed defect lattice [26, 29]. The simulations that provide the enthalpies can be either MC or MD simulations of the N P T ensemble. Both previously described methods require long simulations to obtain good statistical accuracy, and are hence not conducive to DFT calculations or calculations using complicated classical potentials. For such methods, one still can determine the defect entropy in the harmonic approximation by explicitly evaluating the change in the vibrational modes (phonon spectrum) of the defect. This can be achieved by determining the eigenfrequencies by direct diagonalization of the force-constant matrix in systems with and without the defect [2, 28, 30–32]. Anharmonic effects and electronic contributions to the defect entropy are, of course, neglected by this procedure [33].

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2.3.

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Migration

Equation (2) can be put in the form: w = w0 e−H

m / kT

(9)

where the geometrical factor Z , lattice vibration rate ν and migration entropy S m have been embedded into the term w0 , which can be considered the attempt frequency of the jump. Since Z and ν are usually known for a given material, one can determine both S m and H m if the jump rate is known as a function of temperature. Thus a straightforward approach to determine the defect migration properties is to directly count defect jumps in an MD simulation. For simple point defects, such as the single vacancy, this can often be achieved by using some automated means for tracking the position of the defect as a function of time. The exact number of defect jumps can be then determined, yielding w for a given T . For more complicated defects, for example, even the interstitial, this becomes difficult since at high temperatures the “interstitialcy” is quite extended in space. In this case, it is easier to find the average square displacement of all atoms R 2 (t) in the system, then deduce the diffusion coefficient through the Einstein relation [13] which for three dimensions is R 2 (t) = 6 f Dt

(10)

and then w using Eqs. (2)–(4) for a single defect. Here, f is the correlation factor defined earlier. A completely different approach employs transition state theory to determine the migration rates. Within this approximation, which is generally valid for metals, one can determine both the migration enthalpy H m and prefactor w0 by nondynamic simulations, making this approach particularly attractive for time-constrained simulation methods such as DFT. To obtain H m , the height of the potential energy barrier separating a defect site with an adjacent site is first determined. For instance, if the migration path of a vacancy is a straight line connecting the two sites, this can be achieved simply by moving an atom stepwise along the straight line from an adjacent site to the vacancy site. For these calculations, the position of the moving atom is fixed at each step, while adjacent atoms are relaxed to the nearest potential energy minimum. The corners of the simulation cell are fixed to prevent motion of the whole cell. This yields an energy vs. distance curve, with H m given by the maximum height. If the migration mechanism is more complicated, a conceptually similar but more refined approach is required. Two recent popular methods are the nudged-elastic band and dimer methods [34, 35]. The attempt frequency w0 is determined by evaluating the eigenfrequencies of the ground state of the defect and migration saddle point state using the Vineyard approach [36].

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Accuracy of Defect Simulations

The accuracy of present defect simulations has been checked by comparisons with experiments in several systems. Due to the large number of studies on this topic, it is impossible to review the entire list of successes and failures in the field, but a few of the most studied cases illustrate the current state of the field.

3.1.

Structure

In FCC metals, both DFT methods and most modern EAM-like potentials reproduce well, at least qualitatively, the structure of the vacancy and interstitial [37–40]. They predict that the atoms surrounding the vacancy relax inwards slightly (leading to a small negative relaxation volume), and that the interstitial has the 0 0 1 dumbbell structure and a large positive relaxation volume, in agreement with experiments [2]. Similarly, the properties of vacancies on BCC and HCP lattices are also generally treated well [4, 41–44]. Predicting the properties of the BCC metal interstitial has proven to be more difficult, however, probably because of the more complex nature of the atomic interactions [45, 46], as well as magnetic effects. For instance, in Fe many classical models predict that the ground state of the interstitial is the 1 1 1 crowdion [47, 48], which disagrees with other classical potentials, DFT calculations and experimental information [32, 49–51]. On the other hand, recent DFT calculations indicate that the 1 1 1 dumbbell would be the stable interstitial in V [4].

3.2.

Concentration

The formation energy of vacancies E fv in metals is most often given quite accurately by EAM-like classical models [37, 38, 40, 43], and thus naturally also in more advanced DFT methods [15, 52–54]. This is, in part, because this quantity is often directly fitted during EAM potential construction, and in part, because the formation energy often is closely related to other basic properties of the material. For instance, in all common BCC metals E fv scales with the melting point of the material within an accuracy of about 10%. It is far more difficult to provide a general statement about the interstitial formation energies, since they are not very well known experimentally. Practically all models, however, consistently predict that E if is higher than E vf , in agreement with experimental observations. The defect formation entropy has been studied even less with simulations, due to the difficulties associated with carrying out the calculations, see above

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and Ref. [31]. The classical potential studies which have been carried out tend to predict low values, ∼1k for the vacancy formation entropy, which is in good agreement with experiments [11, 32]. The situation for the interstitial is particularly interesting, since the Granato model and recent experiments predict that (at least) the FCC metal interstitials should have very high values of Sif , ∼10k, due to the resonance modes associated with the dumbbell interstitial structure. This result has been supported by DFT and classical simulations giving a similar result [11, 14].

3.3.

Migration

The migration properties of metal vacancies have been studied both in classical and quantum mechanical frameworks [15, 38, 45, 55, 56]. The activation energy for vacancy migration is generally described satisfactorily both by classical and DFT models. Similarly to the case of the vacancy formation energy, this is largely due to the simple character of the migration and the fact that the migration energy is related to the elastic properties of the material [2]. For BCC metals it can be predicted quite well by Flynn’s simple analytical formula [57]. The interstitial migration is more complicated than the vacancy migration, and generally very rapid [2]. This is reproduced at least qualitatively by both EAM and DFT models, since the basic reason is related to the geometry of the defect migration path [4, 39, 49, 58]. One notable exception is the aforementioned quantum mechanical migration of the interstitial in gold, which cannot possibly be reproduced by conventional EAM or DFT models which always work in the Born–Oppenheimer approximation.

4.

Examples of Simulated Predictions

The results described in Section 3 are not all that interesting from a general scientific point of view, since they merely reproduce already known experimental results. The real value of simulations is making predictions of effects and properties that have not yet been experimentally measured, or can be measured only indirectly. We provide here a few examples of recent simulation results, which in our opinion are especially interesting in that they have wider implications.

4.1.

Temperature-dependent Defect Formation Energy

As mentioned in the beginning of this article, the curvature in Arrhenius plots of diffusion data is usually interpreted to be due to divacancies. Recently,

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an alternative explanation has been offered, one that does not require the presence of any other defect than the vacancy. Using DFT calculations Carling et al. report that the divacancy in Al is energetically unstable [30, 59]. Instead, they propose that the anharmonicity of the lattice vibrations leads to a temperaturedependent vacancy formation energy Hvf (T ). When Hvf (T ) is used in Eq. (1), a curved Arrhenius plot is obtained which agrees well with experiments. Later, Sandberg showed that this can also explain the nonArrhenius behaviour of the vacancy diffusion [60]. Later work on the same systems, however, indicates that even if the nearestneighbor divacancy is unstable, the second-nearest-neighbor one may be stable [5]. Also, Khellaf, Seeger and Emrick have argued that Carling’s result of a negative divacancy binding energy is simply in contradiction to experiments [61]. The analysis of the experiments, however, assumes the enthalpy of defect formation is temperature-independent, which Carling argues, it is not [59]. Hence the controversy can be regarded as unresolved at present.

4.2.

Defects in NiAl

Point defects in intermetallic compounds, particularly NiAl, have been studied intensively by computer simulations during the last 5 years [61–65]. These studies have revealed the surprisingly complex nature that point defects can have in intermetallics. The general rule that the simple vacancy is the predominant point defect at all temperatures in elemental metals, cannot be true in intermetallic compounds, owing to the need to preserve stoichiometry. For NiAl, the dominant thermal defect is found to be the so-called “triple (Ni) defect” [62, 63]. This defect is a divacancy consisting of one missing Ni and one missing Al atom, but reconstructed such that there are two vacancies on Ni sites, and between these vacancies is an Al antisite atom on a Ni site. Moreover, in nonstoichiometric alloys the situation becomes even more complicated. The ground state of a nonstoichiometric alloy can be viewed to have an intrinsic concentration of vacancy or antisite defects. For instance, Ni-rich NiAl has Ni antisite atoms on Al sites [62, 65]. These defects are called constitutional defects. At finite temperatures thermal activation can produce thermal defects, some of which may be produced by removing the constitutional defects. Such defects are called interbranch defects; for example, one constitutional Ni antisite can be replaced by two Al vacancies [62]. Since the thermal activation thus removes defects which are part of the alloy ground state, one arrives at the peculiar situation of a negative defect concentration, at least for one species of defect. These defects are also interesting since the entropy term becomes the dominant contribution in the Gibbs free energy [65].

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1869

The migration of atoms in NiAl is also interesting, as one has to distinguish between tracer diffusion and chemical diffusion (i.e., diffusion leading to a chemical composition gradient). For elemental metals tracer atoms are conventionally used to measure the diffusion. For an intermetallic, however, tracers may be misleading, since, for instance, a divacancy in a stoichiometric B2 structure lattice will contribute to tracer diffusion, but not to chemical diffusion [61]. Also interesting is that the vacancy cannot make an ordinary jump from one site to the nearest-neighbor site in the B2 structure of NiAl. Instead, it must always make a double jump to the next-nearest-neighbor site [64].

4.3.

Defects in Metallic Glasses

Hitherto we have only dealt with defects in crystalline metals. In crystalline materials the underlying lattice provides a straightforward means to characterize the structure of point defects, point defect clusters, and the Burgers vectors of dislocations. The same is not possible for metallic glasses, since lattice sites need not be conserved. Consequently, there are no experimental methods available to determine directly defect structures and properties. Positron annihilation does offer some help, however, in identifying vacancies [66]. As a consequence, very little is known about the nature of defects in metallic glasses, and indeed whether localized defects can be clearly defined. Most of what is known derives from MD computer simulations. Nevertheless, MD suffers from many of the same problems encountered by experiments in identifying the structure of defects. Also similar to experiments, most work has focused on the question of vacancies, since vacancies can be simply identified by a void volume. Vacancy-like defects in metallic glasses. The first MD simulation work probing the nature of defects in metallic glasses was performed by Bennett et al. [67] who examined the stability of vacancies in Lennard–Jones glasses. In these simulations, like many that followed, a glass was created by quenching a liquid from high temperatures to a temperature sufficiently low to suppress crystallization, using various stages of relaxation along the way. These early investigations showed that while vacancies created by removing an atom from the assembly were stable near 0 K, the localized excess volume quickly dissipated at temperatures quite low in comparison to the migration temperature of vacancies in the Lennard–Jones crystal. While these investigations provided an initial understanding of point defects in metallic glasses, they did not provide a systematic evaluation of the environment of the vacancy and its stability. In a simple Bravais lattice, all atoms have identical configurations, and thus statistical studies are unnecessary, but in a glass, all sites are unique and statistical investigations are necessary to examine for patterns of behavior.

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Limoge and Delaye [68–70] performed such a statistical investigation of vacancy-like entities in a Lennard–Jones glass. Like Bennett et al., they removed an atom from the glass and monitored the response. Their primary results are illustrated in Fig. 5 (a)–(c) where the local excess volume around the vacancy is plotted as a function of time. These volumes were obtained by forming Voronoi polyhedra for the atoms neighboring the removal site and calculating the average volume of these polyhedra. At 0 K three different histories are observed. As shown in case (a), the volume initially undergoes a brief relaxation, but then remains constant, denoting a stable vacancy-like entity. In contrast, case (b) shows a situation where the excess volume shrinks slowly with time and returns nearly to its initial value before creating the vacancy. Lastly, case (c) illustrates the case where the volume is at first stable, but then after some time relaxes quickly to its initial value. In this case, it was observed that a neighboring atom jumped into the vacancy, annihilating it, but creating a new vacancy at a neighboring site. In addition to characterizing the nature of the different types of vacancies, Delaye and Limoge calculated the thermodynamic and kinetic properties of these defects. In Fig. 6 the formation enthalpy is plotted vs. the local hydrostatic pressure. It is noteworthy that when the local pressure is low, suggestive of a crystal-like environment, the formation enthalpy is close to the value in the crystal, 0.088 eV [70]. Delaye and Limoge also observed that the 41.0 Case a)

40.5

Case b)

Volume (Å3)

40.0

Case c)

39.5 39.0 38.5 38.0 37.5 37.0

0

1000

2000

3000

Time

4000

(10⫺14

5000

6000

7000

s)

Figure 5. Local excess volume in a Lennard–Jones liquid as a function of time. The three lines correspond to three different cases. The data is from Ref. [70].

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0.14 0.12

Enthalpy (eV)

0.1 0.08 0.06 0.04 0.02 0.0 ⫺2

0

2 4 Hydrostatic pressure (kbar)

6

8

Figure 6. Formation enthalpy vs. the local hydrostatic pressure for vacancies in glasses. The data is from Ref. [68].

formation enthalpy increases linearly with formation volume, with a ratio of 0.14 eV per atomic volume. On the other hand, the formation entropy was found to decrease with increasing volume, reaching a value of 4–5 kB at the largest volumes [68]. These entropies are notably larger than in the crystal in which a value of ∼2.7kB [68] has been determined. Similar results regarding the formation of a stable point defect were recently observed in simulations of quenched amorphous copper using EAM potentials [71]. Calculations of system evolution following both the addition of an atom or the removal of an atom showed the same three types of system evolution: stable defect formation (with a positive formation energy and volume), localized defect annihilation and extended cooperative relaxation. The results are summarized in Table 1. The formation volumes and energies were considerably lower than the corresponding values for the crystalline case, which are 1.2 eV for a vacancy and 3.2 eV for the interstitial. The lower formation enthalpy of the interstitial is understood within the Granato model of glasses as a consequence of the reduced shear modulus (see Ref. [72] and below). Currently, there is still little agreement regarding the role of such “point defects” in diffusion in amorphous materials, as it seems that diffusion in the glassy state happens through a collective process [73]. However, the “interstitial-like” defects in the amorphous system caused large-scale cooperative motion as they

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Table 1. Formation volume V f and energy E f for stable point defects in quenched amorphous Cu, as described by our EAM potential Stable defect

Localized annihilation

Extended relaxation

Atom addition

V f (0 ) E f (eV)

0.3 ± 0.15 0.5 ± 0.25

0.0 ± 0.1 −0.1 ± 0.2

−0.7 ± 0.5 −1.0 ± 0.5

Atom removal

V f (0 ) E f (eV)

0.5 ± 0.2 0.7 ± 0.3

0.0 ± 0.1 0.1 ± 0.2

−0.4 ± 0.3 −0.7 ± 0.5

relaxed following insertion. This is consistent with simulations showing extended nonGaussian behavior during glass relaxation [74]. Granato interstitial model of a liquid. This chapter has dealt extensively with point defects in metals, and how they should be simulated on a computer. We have not mentioned application areas where point defects are important except in passing. Naturally point defects in metals do play a central role in many aspects of metals processing. We will finish this chapter, however, on a speculative note by mentioning a recent theory which proposes that point defects play a quite fundamental importance in the theory of liquids and glasses, i.e., the Granato model. This model of liquids and glasses essentially states that the structure of a liquid is, in fact, a crystal containing a few percent of interstitial defects and a glass is a frozen liquid. The basis for this model derives from two properties of interstitials in metals: (i) the entropy of formation of the interstitial is ∼15kB (see Section 1.2); (ii) the shear modulus of the crystal decreases rapidly with interstitial concentration. Since the formation enthalpy of the interstitial depends almost linearly on the shear modulus, every added interstitial reduces the energy required to add the next one. Granato has calculated the free energy of the solid as a function of interstitial concentration, and indeed finds that above the melting temperature, the crystal with a small concentration of interstitials (10−7 –10−8 ) is metastable with respect to the liquid, i.e., the crystal with a large number of interstitials. While defect models of melting have been proposed much earlier, these have mostly considered vacancy defects, presumably to account for the excess volume of the liquid. The entropy of a vacancy is, however, far too small to account for the entropy of melting. The interstitial, with its large entropy and large formation volume satisfies both criteria. While it is beyond the scope of this article to critically evaluate the model, we note that computer simulations are well suited to test the model predictions.

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[43] A.S. Goldstein and H. Jonsson, “An embedded atom method potential for the h.c.p. metal Zr,” Philos. Mag. B: Phys. Condens. Matt. Electr. Opt. Magn. Prop., 71, 1041– 1056, 1995. [44] Q.M. Hu, D.S. Xu, and D. Li, “First principles calculations for vacancy formation energy and solute-vacancy interaction energy in alpha-Ti,” Int. J. Mater. Prod. Technol., 622–627, 2001. [45] F. Willaime, A. Satta, M. Nastar, and O.L. Bacq, “Electronic structure calculations of vacancy parameters in transition metals: impact on the BCC self-diffusion anomaly,” Int. J. Quant. Chem., 77, 927–39, 2000. [46] W. Xu and J. Moriarty, “Atomistic simulation of ideal shear strength, point defects, and screw dislocations in bcc transition metals: Mo as a prototype,” Phys. Rev. B, 54, 6941, 1996. [47] Y.N. Osetsky, A. Serra, V. Priego, F. Gao, and D.J. Bacon, “Mobility of selfinterstitials in fcc and bcc metals,” In: Y. Mishin, G. Vogl, N. Cowern, R. Catlow, and D. Farkas (eds.), Diffusion-Mechanisms-in-Crystalline-Materials, MRS Symposium Proceedings MRS, Warrendale, pp. 49–58, 1998. [48] G. Simonelli, R. Pasianot, and E.J. Savino, “Self-interstitial configuration in B.C.C. metals. An analysis based on many-body potentials for Fe and Mo,” Phys. Stat. Sol. B, 217, 747–758, 2000. [49] B.D. Wirth, G.R. Odette, D. Maroudas, and G.E. Lucas, “Energetics of formation and migration of self-interstitials and self-interstitial clusters in alpha-iron,” J. Nucl. Mater., 244, 185–194, 1997. [50] Y. Kawazoe, K. Ohno, K. Shiga, H. Kamiyama, Z. Tang, M. Hasegawa, and H. Matsui, “How accurate the first-principles calculations can be applied to nuclear reactor materials research?” Nucl. Instr. Meth. Phys. Res. B, Beam Interactions Mater Atoms, 153, 77–86, 1999. [51] H. Bilger, V. Hivert, J. Verdone, J. Leveque, and J. Soulie, H. Bilger, In: Point defects in Iron, (ed.), International Conference on Vacancies and Interstitials in Metals, Kernforschunganlage Jülich, Jülich, p. 751–767, 1968. [52] T. Korhonen, First-principles Electronic Structure Calculations: Defects in Metals, Nitrides and Carbides, Ph.D. Thesis, Helsinki Univ. Technol, Espoo, Finland, 1996. [53] P.A. Korzhavyi, I.A. Abrikosov, B. Johansson, A.V. Ruban, and H.L. Skriver, “Firstprinciples calculations of the vacancy formation energy in transition and noble metals,” Phys. Rev. B: Condens. Matt., 59, 11693–11703, 1999. [54] T.R. Mattson and A.E. Mattson, “Calculating the vacancy formation energy in metals: Pt, Pd, and Mo,” Phys. Rev. B, 66, 214110, 2002. [55] M.J. Sabochick and S. Yip, “Migration energy calculations for small vacancy clusters in copper,” J. Phys. F: Met. Phys., 18, 1689–1701, 1988. [56] J.N. Adams, S.M. Foiles, and W.G. Wolfer, “Self-diffusion and impurity diffusion of fcc metals using the five-frequence model and the Embedded Atom Method,” J. Mater. Res., 4, 102, 1989. [57] P. Flynn, Point Defects and Diffusion, Clarendon Press, Oxford, UK, 1972. [58] S.M. Foiles, M.I. Baskes, and M.S. Daw, “Embedded-atom-method functions for the fcc metals Cu, Ag, Au, Ni, Pd, Pt, and their alloys,” Phys. Rev. B, 33, 7983, 1986; Erratum: ibid, Phys. Rev. B, 37, 10378, 1988. [59] K. Carling, G. Wahnström, T.R. Mattsson, N. Sandberg, and G. Grimvall, “Vacancy concentration in Al from combined first-principles and model potential calculations,” Phys. Rev. B, 67, 054101, 2003.

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[60] N. Sandberg, B. Magyari-Kope, and T.R. Mattsson, “Self-diffusion rates in Al from combined first-principles and model-potential calculations,” Phys. Rev. Lett., 89, 065901, 2002. [61] A. Khellaf, A. Seeger, and R.M. Emrick, “Quenching studies of lattice vacancies in high-purity aluminium,” Mater. Trans., 43, 186–189, 2002. [62] P.A. Korzhavyi, A.V. Ruban, A.Y. Lozovoi, Y.K. Vekilov, I.A. Abrikosov, and B. Johansson, “Constitutional and thermal point defects in B2 NiAl,” Phys. Rev. B, 61, 6003, 2000. [63] Y. Mishin, M.J. Mehl, and D.A. Papaconstantpoulos, “Embedded-atom potential for B2-NiAl,” Phys. Rev. B, 65, 224114, 2002. [64] Y. Mishin, A.Y. Lozovoi, and A. Alavi, “Evaluation of diffusion mechanisms in NiAl by embedded-atom and first-principles calculations,” Phys. Rev. B, 67, 014201, 2003. [65] A.Y. Lozovoi and Y. Mishin, “Point defects in NiAl: the effect of lattice vibrations,” Phys. Rev. B, 68, 184113, 2003. [66] X.D. Liu, J. Zhu, Z.Q. Hu, and J.T. Wang, “Investigation of defective structure of nanocrystalline Fe–Mo–Si–B alloys by the positron annihilation technique,” J. Mater. Sci. Lett., 12, 1826–1828, 1993. [67] C. Bennett, P. Chaudhari, V. Moruzzi, and P. Steinhardt, “On the stability of vacancy and vacancy clusters in amorphous solids,” Philos. Mag. A, 40, 485, 1979. [68] J.M. Delaye and Y. Limoge, “Molecular dynamics study of vacancy-like defects in a model glass: dynamical behavior and diffusion,” J. Phys. I, 3, 2079–2097, 1993. [69] J.M. Delaye and Y. Limoge, “Molecular dynamics study of vacancy-like defects in a model glass: static behaviour,” J. Phys. I, 3, 2063–2077, 1993. [70] Y. Limoge, “Microscopic and macroscopic properties of diffusion in metallic glasses,” Mater. Sci. Eng. A, 226–228, 228, 1997. [71] Y. Ashkenazy, R.S. Averback, and A. Granato, Point defects in supercooled amorphous Cu, to be published, 2004. [72] A.V. Granato, “Interstitialcy model for condensed matter states of face-centeredcubic metals,” Phys. Rev. Lett., 68, 974, 1992. [73] F. Faupel, W. Frank, M.-P. Macht, H. Mehrer, V. Naundorf, K. Rätzke, H.R. Schober, S.K. Sharma, and H. Teichler, “Diffusion in metallic glasses and supercooled melts,” Rev. Mod. Phys., 75, 237–280, 2003. [74] H.R. Schober, C. Oligschleger, and B.B. Laird, “Low-frequency vibrations and relaxations in glasses,” J. Non-Cryst. Solids, 156–158, 965–968, 1993. [75] G.E. Murch and I.V. Belova, “Chemical diffusion by vacancy pairs in intermetallic compounds with the B2 structure,” Phil. Mag. Lett., 80, 569–575, 2000.

6.3 DEFECTS AND IMPURITIES IN SEMICONDUCTORS Chris G. Van de Walle Materials Department, University of California, Santa Barbara, California, USA

Impurities are essential for giving semiconductors the properties that render them useful for electronic and optoelectronic devices. The intrinsic carrier concentrations in most semiconductors are quite low. Adding small amounts of impurities allows control of the conductivity of the semiconductor: shallow donors, such as phosphorous in silicon, produce n-type conductivity (carried by electrons), and shallow acceptors, such as boron in silicon, produce p-type conductivity (carried by holes). These doped layers and the junctions between them control carrier confinement, carrier flow, and ultimately the device characteristics. Commonly used semiconductors such as Si and GaAs can be doped both p-type and n-type. Constraints on doping still limit device performance, however. For instance, the shrinking size of Si field-effect transistors requires higher doping densities, with donors exhibiting deactivation when the doping increases above ∼3 ×1020 cm−3 . Doping problems have been more severe in wide-band-gap semiconductors such as ZnSe, GaN or ZnO, which typically exhibit unintentional n-type conductivity, and in which p-type conductivity has been difficult to achieve. Point defects (vacancies, self-interstitials, and antisites) have often been invoked to explain these difficulties. Computational studies have made important contributions to solving many of these problems. One application of the computational results is to provide a microscopic identification of a defect or impurity, by comparing the computed properties with experiment. For instance, atomic relaxations can be compared with EXAFS (extended X-ray absorption fine structure), hyperfine parameters, based on calculated wave functions, with EPR (electron paramagnetic resonance), and vibrational frequencies with infrared or Raman spectroscopy. In addition, the ability provided by first-principles calculations to calculate total energies enables us to attack the problems related to doping limitations. These limitations can be attributed to four causes: solubility; ionization energy; incorporation of impurities in undesired configurations; and 1877 S. Yip (ed.), Handbook of Materials Modeling, 1877–1888. c 2005 Springer. Printed in the Netherlands.

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compensation by native point defects or foreign impurities. Each of these issues can be addressed by performing total-energy calculations, as illustrated in the following sections. Unless otherwise specified I will use the word “defect” to denote both native point defects and impurities, since the methodologies are the same.

1.

Geometry

For most purposes, the key issue is to calculate the properties of a single, isolated point defect in an infinite solid. Interactions between defects may be important, but these can often be modeled once the properties of the isolated noninteracting species are known. There are three basic approaches for handling the geometry corresponding to a single defect: clusters, supercells, and Green’s functions. The cluster approach is based on the assumption that most of the essential physics is captured if the local environment of the defect is well described. It focuses on the interactions of the defect with the surrounding shells of host atoms, using a cluster geometry. The cluster should be large enough to provide a reasonable description of the band structure of the host, and to minimize interactions between the defect and the surface of the cluster. Passivation of the surface is required in order to suppress surface states that would otherwise dominate the results; hydrogen atoms are commonly used for this purpose. In a supercell geometry, the impurity is surrounded by a finite number of semiconductor atoms, and that whole structure is periodically repeated. This has the advantage of allowing the use of various techniques that require translational periodicity of the system. Supercells need to be large enough to provide adequate separation of the defects, which can be explicitly tested. Another major advantage is that the band structure of the host crystal is well described. Indeed, a calculation for a supercell that is simply filled with the host crystal, in the absence of any defect, simply produces the band structure of the host. This contrasts with cluster approaches, where even fairly large clusters still produce sizeable quantum-confinement effects that significantly affect the band structure. For the zinc-blende (ZB) structure, typical supercells contain 16, 32, 54, 64, 128, 216, or 256 atoms. The 16-, 54-, and 128-atom supercells are based on simply extending the basis vectors of the fcc lattice. The shape of these cells is ill suited to providing adequate separation between defects in all directions. The 64- and 216-atom supercells are based on enlarging the “conventional” 8-atom simple cubic cell. The 32- and 256-atom supercells, finally, have bcc symmetry and are most appropriate for separating defects in all spatial directions. Defect calculations should be carried out at the theoretical lattice constant, in order to avoid spurious elastic interactions with defects in neighboring supercells. Atomic relaxations of several shells of neighbors around

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the defect should be allowed. It may be necessary to break the symmetry in order to explore low-symmetry configurations of the defect. Another approach which provides a good desciption of the band structure of the host crystal is based on the Green’s function determined for the perfect crystal. This function is then used to calculate changes induced by the presence of the defect. The Green’s function approach is more cumbersome and less physically transparent than the supercell technique. Indeed, the supercell approach has become the method of choice for performing state-of-the-art defect calculations.

2.

Hamiltonian

Any Hamiltonian for a defect in a semiconductor must include terms that describe the interactions between the nuclei, the interactions of electrons with the nuclei, and the electron-electron interactions. The latter are the most difficult part of the problem. Historically, Hartree–Fock methods were the first to attack many-electron problems, with considerable success for atoms and molecules. The main problems are the neglect of correlation and the computational demands: ab initio Hartree–Fock methods can only be applied to systems with small numbers of atoms, because they require the evaluation of a large number of multicenter integrals. Quantum chemists have developed simpler semi-empirical methods that either neglect or approximate some of these integrals, but the accuracy and reliability of these methods is hard to assess. Tight-binding calculations have also often been used for defects. These methods take advantage of the fact that within a local basis set the Hamiltonian matrix elements rapidly decrease with increasing distance between the orbitals. Thus, instead of having to diagonalize the full Hamiltonian matrix, most of the matrix elements vanish and only a sparse matrix has to be diagonalized. Depending on how the remaining matrix elements are determined one can distinguish two main approaches: (i) Empirical tight-binding methods use parameters obtained from fitting a set of experimental or computed quantities, a procedure for which no consistent prescription exists. (ii) First-principles tight-binding methods, on the other hand, use local orbitals to explicitly calculate the Hamiltonian matrix elements [1]. The most rigorous calculations for defects in semiconductors are based on density-functional theory (DFT). DFT in the local density approximation (LDA) [2, 3] allows a description of the many-body electronic ground state in terms of single-particle equations and an effective potential. The effective potential consists of the ionic potential due to the atomic cores, the Hartree potential describing the electrostatic electron-electron interaction, and the exchange-correlation potential that takes into account the many-body effects. This approach has proven to describe with high accuracy such quantities as atomic geometries,

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charge densities, formation energies, etc. For calculations of defects and impurities in semiconductors use of the local density approximation seems to be well justified. The generalized gradient approximation offers few advantages, either for bulk properties or for formation energies of point defects [4, 5]. One shortcoming of DFT is its failure to produce accurate excited-states properties – the band gap is commonly underestimated. No method is currently available that goes beyond DFT and provides total-energy capability for the large supercell calculations required to investigate defects. Even methods aimed solely at calculating the band structure, such as quasiparticle calculations in the GW approximation [6] are currently prohibitively expensive for large cells. Defect calculations can, in principle, be performed in an all-electron approach, such as the FLAPW (full-potential augmented plane wave) or the fullpotential LMTO (linearized muffin-tin orbital) methods. Computationally, however, a pseudopotential approach is most efficient, particularly for the large-scale calculations required for defects. Most properties of molecules and solids are determined by the valence electrons, i.e., those electrons in outer shells which take part in the bonding between atoms. The core electrons can be removed from the problem by representing the ionic core (i.e., the nucleus plus inner shells of electrons) by a pseudopotential. State-of-the-art calculations employ nonlocal, norm-conserving pseudopotentials which are generated from atomic calculations and do not contain any fitting to experiment [7]. Such calculations can therefore be called “ab initio” or “first-principles”. A plane-wave basis set is most commonly used in the pseudopotential approach. Convergence as a function of plane-wave cutoff is straightforward to check. Integrations over the Brillouin zone are performed using the standard Monkhorst–Pack scheme [8] with a regularly spaced mesh of n × n × n points in the reciprocal unit cell shifted from the origin (to avoid picking up the point as one of the sampling points). Symmetry reduces this set to a set of points in the irreducible part of the Brillouin zone. Most problems require calculations not only of electronic wave functions, but also of atomic positions. An important advance in this respect was the development of the Car–Parrinello method [9] which allows simultaneous optimization of the electronic and atomic degrees of freedom. The ability to move atoms allows performing first-principles molecular dynamics, as well.

3.

Formation Energies

The formation energy is a key quantity determining the properties of a defect or impurity. Indeed, the equilibrium concentration of a defect is given by c = Nsites e−E f / k T

(1)

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where E f is the formation energy, Nsites is the number of sites on which the defect or impurity can be incorporated, k is the Boltzmann constant, and T the temperature. Equation (1) shows that defects with a high formation energy will occur in low concentrations. In principle, the free energy should be used in Eq. (1). Contributions from vibrational entropy are often neglected, however. Explicit calculations of entropies are quite demanding, and they also often cancel to some extent, for instance when solubilities are calculated. The formation energy is not a constant but depends on the growth conditions. We illustrate the key concepts with the example of a Mg acceptor (substituting on a Ga site) in GaN. Its formation energy is determined by the relative abundance of Mg, Ga, and N atoms, as expressed by the chemical potentials µMg , µGa and µN , respectively. If the Mg acceptor is charged (as is expected when it is electrically active), the formation energy also depends on the Fermi level (E F ), which acts as a reservoir for electrons. Forming a substitutional Mg acceptor requires the removal of one Ga atom and the addition of one Mg atom; the formation energy is therefore: − E f (GaN:Mg− Ga ) = E tot (GaN:MgGa ) − E tot (GaN, bulk) − µMg + µGa −[E F + E v + V ]. (2)

First-principles calculations allow explicit derivation of E tot (GaN:Mg− Ga ), the total energy of a system containing substitutional Mg on a Ga site. Similar expressions apply to other impurities and to the various native point defects. The chemical potentials depend on the experimental growth conditions, which can be either Ga-rich or N-rich. For the Ga-rich case, µGa = µGa[bulk] places an upper limit on µGa . Indeed, pushing µGa beyond this limit results in precipitation of bulk Ga, rather than growth of GaN. In equilibrium, µGa + µN = E tot [GaN], where E tot [GaN] is the total energy of a two-atom unit of bulk GaN; the upper limit on µGa therefore places a lower limit on µN . For the Nrich case, the upper limit on µN is given by µN = µN[N2 ] , i.e., the energy of N in an N2 molecule; this yields a lower limit on µGa . The total energy of GaN can also be expressed as E tot [GaN] = µGa[bulk] + µN[N2 ] + H f [GaN], where H f [GaN] is the enthalpy of formation, which is negative for a stable compound. We observe that the host chemical potentials thus vary over a range corresponding to the magnitude of the enthalpy of formation of the compound. The Mg atom in our example occurs in the negative charge state (i.e., it has donated a hole to the valence band). The electron that is placed on the Mg is taken out of a reservoir of electrons, the energy of which is the electron chemical potential or Fermi level, E F . E F is referenced to the valenceband maximum in the bulk. Due to the choice of this reference, Eq. (2) needs to explicitly contain a term representing the energy of the bulk valence-band maximum, E v , when expressing the formation energy of a charged state. However, we need to add a correction term, V , to align the reference potential in

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the defect calculation with that in the bulk, as discussed in Van de Walle and Neugebauer, 2004 [10]. Another issue regarding calculations for charged states is the treatment of the G = 0 term in the total energy of the supercell. This term would diverge for a charged system; we therefore assume the presence of a compensating uniform background (jellium) and evaluate the G = 0 term as if the system were neutral [11]. Correction terms have been proposed [12], but they tend to overestimate the effect, essentially because screening is more efficient than assumed in a simple dielectric model. The Fermi level E F is not an independent parameter, but is always determined by the condition of charge neutrality. In principle, equations such as Eq. (2) can be formulated for every native defect and impurity in the material; the complete problem (including free-carrier concentrations in valence and conduction bands) can then be solved selfconsistently, imposing charge neutrality. However, it is often instructive to plot formation energies as a function of E F in order to examine the behavior of defects and impurities when the doping level changes. An example of such a plot is given in Fig. 1. It shows results for defects and impurities relevant for p-type GaN. The zero of E F is located at the top of the 4

MgN

Formation energy (eV)

Mgi

Bei VN

2

MgGa H⫹

0

BeGa 0

1 EF (eV)

2

Figure 1. Formation energies as a function of Fermi level for point defects and impurities relevant for p-type GaN. Energies are shown for Mg and Be in different configurations (Ga-substitutional, N-substitutional, and interstitial configuration). Also included is the dominant native defect (VN ) and hydrogen, a foreign impurity. Nitrogen-rich conditions and equilibrium with Mg3 N2 or Be3 N2 are assumed.

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valence band. For ease of presentation, the chemical potentials are set equal to fixed values; however, a general case can always be addressed by referring back to Eq. (2). The fixed values correspond to N-rich conditions and to maximum incorporation of the impurities. For Mg, this is determined by equilibrium with Mg3 N2 . Indeed, increasing the chemical potential of Mg results in a lowering of the formation energy of MgGa [Eq. (2)] and an increase in its concentration [Eq. (1)]; however, at some point it becomes more favorable to form Mg3 N2 instead of incorporating Mg as an impurity, and this determines the solubility limit. For each defect in Fig. 1, only the line segment is shown that corresponds to the charge state that gives rise to the lowest energy at a particular value of E F . The slope of the line segment represents the charge state, and a kink (change in slope of the lines) therefore represents a change in the charge state of the defect [see Eq. (2)]. The Fermi-level position at which this change occurs corresponds to a transition level that can be experimentally measured.

4.

Doping-Limiting Mechanisms

In the introduction we mentioned that first-principles calculations can elucidate doping limitations and suggest ways to overcome them. We now illustrate this claim with the example of Fig. 1. (1) Solubility. A high free-carrier concentration obviously requires a high concentration of the dopant impurity. The solubility corresponds to the maximum concentration that the impurity can attain in the semiconductor, a quantity that depends on the growth temperature and on the abundance of the impurity as well as the host constituents in the growth environment. As discussed above, the solubility of Mg in GaN is limited by formation of Mg3 N2 . Figure 1 shows that, under comparable conditions, the formation energy of beryllium is markedly lower than that of magnesium, implying that Be has a higher solubility. (2) Ionization energy. The ionization energy of a dopant determines the fraction of dopants that contributes free carriers at a given temperature. A high ionization energy limits the doping efficiency. Ionization energies of acceptors in GaN are typically so large (around 200 meV) that at room temperature only about 1% of acceptor impurities are ionized. Ionization energies are mainly determined by intrinsic properties of the semiconductor, such as effective masses, dielectric constant, etc. Switching to a different acceptor therefore has no dramatic effect on the ionization energy. The ionization energy of a substitutional acceptor corresponds to the Fermi-level position where the neutral and negative charge states have equal energies, i.e., it is given by the kink in the

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curves for MgGa and BeGa in Fig. 1. Error bars on ionization energies are typically quite large. Still, Fig. 1 shows that the ionization energy of Be is slightly lower than that of Mg, suggesting that also from this point of view Be would be a superior acceptor. (3) Incorporation of impurities in other configurations. Most donor and acceptor impurities reside on substitutional sites, i.e., they replace one of the host atoms. In order for Mg in GaN to act as an acceptor, it needs to be incorporated on the gallium site. If Mg is located in other positions in the lattice, such as an interstitial position or substituting for a nitrogen atom, it actually behaves as a donor. For GaN doped with Mg, Fig. 1 shows that these other configurations (MgN and Mgi ) are energetically unfavorable, and hence will not form. The situation is different for Be: as shown in Fig. 1, the formation energy of beryllium interstitials is quite low, particularly when the Fermi level approaches the valence band. Self-compensation is therefore a serious risk in case of doping with Be. Another instance of impurities incorporating in undesirable configurations consists of the so-called DX centers. The prototype DX center is Si in AlGaAs. In GaAs and in AlGaAs with low Al content, Si behaves as a shallow donor. But when the Al content exceeds a critical value, Si behaves as a deep level. This has been explained in terms of Si moving off the substitutional site, towards an interstitial position [13]. It has been found that oxygen forms a DX center in AlGaN when the Al content exceeds about 30% [10]. (4) Compensation by native point defects. Native defects have frequently been invoked to explain doping problems in semiconductors. Recent studies have shown that this problem is not necessarily more severe in wide-band-gap semiconductors than in, e.g., GaAs. For GaN, compensation by vacancies can in some cases limit the doping level: gallium vacancies are acceptors and compensate n-type GaN, and nitrogen vacancies are donors and compensate p-type GaN (as illustrated in Fig. 1). Self-interstitials and antisites were found to be too high in energy in GaN to form in significant concentrations. Figure 1 also shows that the nitrogen vacancy has a transition level (between the 3+ and + charge states) at 0.5 eV above the valence band. Transfer of electrons from the conduction band (or from a shallow donor level) to this level of VN may therefore give rise to blue luminescence, roughly 0.5 eV below the band gap. Note that in order to calculate the transition energy more accurately the Franck–Condon shift should be taken into account. This is illustrated, for a general case, in Fig. 2. During the emission process the atomic configuration of the defect remains fixed – i.e., in the final state, the defect has undergone a change in charge state but still has the atomic configuration corresponding to the initial

energy

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q

X ⫹e

Ea

X

Ee

q ⫹1

Eg⫺ε(q /q⫹1)

Erel

zq⫹1

zq configuration coordinate z

Figure 2. Schematic configuration coordinate diagram illustrating the difference between thermal and optical ionization energies for a defect X. The curve for Xq is vertically displaced from that for Xq + 1 assuming the presence of an electron in the conduction band. E rel is the Franck–Condon shift, i.e., the relaxation energy that can be gained by relaxing from configuration z q (eguilibrium configuration for charge state q) to configuration z q + 1 (equilibrium configuration for charge state q + 1). E a is the absorption energy, E e the emission energy, ε(q/q + 1) the thermodynamic transition level, and E g the band gap.

charge state. Calculations can provide a complete configuration coordinate diagram showing the energy of the defect in different charge states as a function of the atomic configuration, from which the energies of all relevant transitions can be obtained. Figure 2 shows that the emission energy (e.g., in a photoluminescence experiment) is smaller than expected based on the position of the thermodynamic transition level ε(q/q + 1). (5) Compensation by foreign impurities. This source of compensation may seem rather obvious but it should be mentioned for completeness: for instance, when doping with acceptors in order to obtain p-type conductivity, impurities that act as donors should obviously be tightly controlled. Hydrogen, which is often unintentionally introduced during growth, is a prime example of such an impurity. In p-type GaN, H behaves as a donor (H+ ) (see Fig. 1); it thus compensates acceptors. A post-growth annealing step is required to render the acceptors electrically active. The presence of hydrogen during growth is actually beneficial. Indeed, as shown in Fig. 1 [14], the formation energy of hydrogen is actually lower than that of nitrogen vacancies. Hydrogen is therefore the preferred source of compensation, and suppresses the formation of VN . The presence of hydrogen during growth also shifts the Fermi level

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5.

Discussion

Strictly speaking, the relationship [Eq. (1)] between concentrations and formation energies only holds in thermodynamic equilibrium. Materials growth is obviously a non-equilibrium situation – however, many growth techniques are close enough to equilibrium to warrant the use of the equilibrium approach. It should be emphasized that not all aspects of the process need to be in equilibrium in order to justify the use of equilibrium expressions for defects and impurities. What is required is a sufficiently high mobility of the relevant impurities and point defects to allow them to equilibrate at the temperatures of interest. Computations can be used to assess the mobility of defects. The migration barrier of an impurity is the energy difference between the saddle point and the ground state. Identifying the saddle point can be accomplished, e.g., by using the nudged elastic band method [15]. It is also possible to map the complete total energy surface for an interstitial impurity moving through a solid [11]. Our discussion so far has focused on isolated point defects and impurities. Complexes between defects and/or impurities can also be important. It is often assumed that such complexes automatically form when two defects exhibit a positive binding energy. In fact, it can be shown that for the equilibrium concentration of complexes to exceed the concentration of either constituent the binding energy needs to be greater than the larger of the formation energies of the constituents [10]. We mentioned in the introduction that calculations of quantities that can be directly compared with experiment provide a powerful means of providing microscopic identification of a defect or impurity. The first-principles calculations explicitly produce wave functions; it is therefore possible to calculate hyperfine parameters. The wave functions have to be obtained from a spinpolarized calculation; i.e., spin-up and spin-down electrons have to be treated independently. It has been shown that it is important to take contributions to the spin density from all the occupied states into account. Hyperfine parameters are particularly sensitive to the wave functions in the core region. When using a pseudopotential approach, the wave function in the core region is replaced by a smooth pseudo-wave function. The information contained in the

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pseudo-wave function, in conjunction with information about wave functions in the free atom, is actually sufficient to calculate hyperfine parameters with a high degree of accuracy [16]. Defects or impurities often give rise to localized vibrational modes (LVM). Light impurities, in particular, exhibit distinct LVMs that are often well above the bulk phonon spectrum. The value of the observed frequency often provides some indication as to the chemical nature of the atoms involved in the bond, but a direct comparison with first-principles calculations has proven to be very valuable. Evaluating the vibrational frequency corresponding to a stretching or wagging mode of a particular bond can be accomplished by using calculated forces to construct a dynamical matrix. In the case of light impurities, where a large mass difference exists between the impurity and the surrounding atoms, it is often a good approximation to focus on the displacement of the light impurity alone, keeping all other atoms fixed. A fit to the calculated energies as a function of displacement then produces a force constant. This approach lends itself well to taking higher-order terms (anharmonic corrections) into account. In the case of an impurity such as hydrogen the anharmonic terms can be on the order of several 100 cm−1 , and therefore an accurate treatment is essential [17].

6.

Outlook

We have provided an overview of first-principles computational methods for calculating defects and impurities in semiconductors. The power of the approach was illustrated with examples for GaN, an important wide-band-gap semiconductor. However, the methodology is entirely general and can be applied to any material. First-principles calculations for defects and impurities in semiconductors are playing an increasingly important role in interpreting and guiding experiments. In fact, in a number of areas theory has led experiment. Examples include the prediction of the behavior of hydrogen and its interactions with dopant impurities, and the study of diffusion of point defects in GaN. New developments in methodology could make the approach even more powerful. The band-gap problem inherent in density-functional calculations limits the accuracy in some cases, and solving this problem is an important goal. Other developments are aimed at rendering the exploration of migration paths or of possible configurations for low-symmetry configurations less cumbersome. The latter capability would make it easier to study complexes between point defects and impurities. Another area of increasing interest is the interaction between point defects or impurities on the one hand and extended defects, interfaces or surfaces on the other hand. Just like point defects in the bulk play an important role in diffusion, point defects at surfaces determine atomic mobilities at surfaces,

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and hence play a decisive role in growth. Likewise, a full understanding of impurity incorporation requires comprehensive calculations of the behavior of impurities at and near the surface. Such first-principles calculations can then form the foundation for realistic simulations of the actual growth process.

References [1] M. Elstner, D. Porezag, G. Jungnickel et al., “Self-consistent-charge densityfunctional tight-binding method for simulations of complex materials properties,” Phys. Rev. B, 64, 7260–7268, 1998. [2] P. Hohenberg and W. Kohn, “Inhomogeneous electron gas,” Phys. Rev., 136, B864– B871, 1964. [3] W. Kohn and L.J. Sham, “Self-consistent equations including exchange and correlation effects,” Phys. Rev., 140, A1133–A1138, 1965. [4] C. Stampfl and C.G. Van de Walle, “Density-functional calculations for III-V nitrides using the local-density approximation and the generalized gradient approximation,” Phys. Rev. B, 59, 5521–5535, 1999. [5] C. Stampfl and C.G. Van de Walle, “Theoretical investigation of native defects, impurities, and complexes in aluminum nitride,” Phys. Rev. B, 65, 155212–1-10, 2002. [6] W.G. Aulbur, L. J¨onsson, and J.W. Wilkins, “Quasiparticle calculations in solids,” Solid State Physics, 54, 1–218, 2000. [7] D.R. Hamann, M. Schl¨uter, and C. Chiang, “Norm-conserving pseudopotentials,” Phys. Rev. Lett., 43, 1494–1497, 1979. [8] H.J. Monkhorst and J.D. Pack, “Special points for Brillouin-zone integrations,” Phys. Rev. B, 13, 5188–5192, 1976. [9] R. Car and M. Parrinello, “Unified approach for molecular dynamics and densityfunctional theory,” Phys. Rev. Lett., 55, 2471–2474, 1985. [10] C.G. Van de Walle and J. Neugebauer, “Applied Physics Review: First-principles calculations for defects and impurities: applications to III-nitrides,” J. Appl. Phys., 95, 3851–3879, 2004. [11] C.G. Van de Walle, P.J.H. Denteneer, Y. Bar-Yam, et al., “Theory of hydrogen diffusion and reactions in crystalline silicon,” Phys. Rev. B, 39, 10791–10808, 1989. [12] G. Makov and M.C. Payne, “Periodic boundary conditions in ab initio calculations,” Phys. Rev. B, 51, 4014–4022, 1995. [13] D.J. Chadi and K.J. Chang, “Theory of the atomic and electronic structure of DX centers in GaAs and Alx Ga1−x As alloys,” Phys. Rev. Lett., 61, 873, 1988. [14] J. Neugebauer and C.G. Van de Walle, “Theory of hydrogen in GaN,” In: N.H. Nickel (ed.), R.K. Willardson and E.R. Weber (treatise eds.), Hydrogen in Semiconductors II, Semiconductors and Semimetals, vol. 61. Academic Press, Boston, pp. 479–502, 1999. [15] H. J´onsson, G. Mills, and K.W. Jacobsen, “Nudged elastic band method for finding minimum energy paths of transitions,” In: B.J. Berne, G. Ciccotti, and D.F. Coker (eds.), Classical and Quantum Dynamics in Condensed Phase Simulations, World Scientific, Singapore, Chapter 16, 1998. [16] C.G. Van de Walle and P.E. Bl¨ochl, “First-principles calculations of hyperfine parameters,” Phys. Rev. B, 47, 4244–4255, 1993. [17] S. Limpijumnong, J.E. Northrup, and C.G. Van de Walle, “Identification of hydrogen configurations in p-type GaN through first-principles calculations of vibrational frequencies,” Phys. Rev. B, 68, 075206–1-14, 2003.

6.4 POINT DEFECTS IN SIMPLE IONIC SOLIDS John Corish Department of Chemistry, Trinity College, University of Dublin, Dublin 2, Ireland

1.

Nature, Occurrence and Modelling of Point Defects in Simple Ionic Solids

Apart from man’s innate need to model and the satisfaction that a good model can bring, the real purpose of scientific modelling is to increase our understanding of a system. More importantly, it provides the basis from which to move forward to understand more complex systems and to design such new systems for specific applications by making predictions about their properties. Simple ionic solids, such as the alkali and silver halides, some fluoritestructured crystals and binary oxides, provide the most accessible and well-developed testing grounds for the study, both experimental and theoretical, of point defects in crystalline materials. This is because defects in these crystals typically carry a charge different from those on the ions that comprise the normal components of the matrix. Their presence, nature, interactions and movements can therefore be rather easily quantitatively determined through the measurement of readily observed macroscopic properties such as ionic conductivity. These charges can be present whether the defects are intrinsic or extrinsic. Intrinsic defects, such as Schottky or Frenkel defects, are equilibrium thermodynamic defects and exist in all materials because the balance between the enthalpy required for their formation in a perfect lattice and the resulting increase in the entropy of the system gives rise to a minimum in the Gibbs free energy. There is a corresponding equilibrium intrinsic defect concentration at each temperature. Extrinsic defects involve ionic impurities, often aliovalent, that are either adventitiously present in the lattice or that are introduced purposely as dopants to confer particular desired properties. They can enter the lattice substitutionally, in which case they displace a regular ion and take up its position on the lattice, or they may occupy an interstitial position that is not normally occupied in the pure lattice. 1889 S. Yip (ed.), Handbook of Materials Modeling, 1889–1899. c 2005 Springer. Printed in the Netherlands.

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It is generally advantageous to consider the difference between the charge on a particular site and the charge that it would carry in the perfect crystal. This is termed the “virtual charge” and is the charge used in the Kroger– Vink notation system that is recommended to describe defects [1]. It is clear that the preservation of charge neutrality in a crystal requires that the overall sum of virtual charges be zero. This means that the total sum of the virtual charges on intrinsic defects must be neutral and that the introduction of aliovalent ions always requires the simultaneous formation of charge compensating defects. These constraints, coupled with the variety of extrinsic defects that can be realised, are again very useful in defining the range of permissible defect structures in simple ionic solids. The concentration of aliovalent extrinsic dopants that can enter simple ionic lattices varies from perhaps only a few hundred parts per million or even less for alkali halides to up to tens of percents in fluorite-structured crystals. In the latter each alternate cube is empty in the pristine lattice and is therefore available to accommodate chargecompensating fluoride ions when the cation is replaced by trivalent dopants such as the rare-earth cations. The most common classical ionic conductors like the alkali halides typically have much less than 1% intrinsic defects concentrations, even at very high temperatures approaching their melting points. However, other simple ionic solids contain defects such as vacancies, often in relatively large concentrations, as an integral part of the structure of a particular phase e.g., α-Ag2 S, in which only about two-thirds of the available cation sites are occupied. Such large concentrations of vacancies that do not require to be thermodynamically formed can render ionic transport very facile between them and so give rise to fast-ion conduction. The modelling of this type and of other fast-ion conducting materials will be discussed in Article 6.5 below.

2.

Modelling Techniques

Because of their simplicity and usefulness as a test bed for novel advances in modelling, virtually all simulation techniques have been applied, usually at an early stage in their development, to simple ionic solids. These systems have also been extensively studied experimentally so that reliable data are available with which the results of the simulations can be compared. A necessary prerequisite for the use of a defect modelling technique as a predictive tool is that it can first accurately calculate the macroscopic properties of the defect free material. It should also reproduce the energies for the formation of the basic defects, for their interactions and for their migrations, all in good agreement with their experimentally determined values. Applications of these techniques, particularly in inorganic crystallography, have been recently reviewed [2].

Point defects in simple ionic solids

2.1.

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Static Lattice Calculations

This is the technique that is most widely used in the systems of interest here. A foundation for the modelling of defects in ionic crystals was laid relatively early when Mott and Littleton [3]showed that a defect could be modelled successfully and its energy calculated by dividing the crystal into two regions. This two-region strategy overcomes the essentially infinite nature of the problem. In the inner region, that directly surrounding the defect, the interactions between the species are calculated explicitly, while the species in the outer region are treated as a continuum using macroscopic response functions. This strategy may be written mathematically by expressing the energy of the defective crystal as: E = E 1 (r) + E 2 (r, ξ ) + E 3 (ξ )

(1)

Here E 1 (r) is the energy of the inner region with r denoting the displacements as determined explicitly. E 2 (r, ξ ) is the interaction energy between the two regions and E 3 (ξ ) the energy or outer region with ξ being the displacements. If the inner region is sufficiently large it may be assumed that the outer region consists of perfect lattice with harmonic displacements. The calculations done by Mott and Littleton themselves involved only very small numbers of ions in the inner region but the method has since been incorporated, starting some 35 years later, into powerful simulation codes HADES [4]; CASCADE [5], GULP [6]. In common with others, these programmes can generate complete crystal structures from the specified lattice vectors and the contents of a unit cell. They then accept the detailed atomistic specifications for both simple and complex defects and use the two-region strategy, with sufficiently large numbers of ions in the inner region to ensure its convergence, to calculate the defect energies of the most relaxed defect configuration. Their output includes complete atomistic information on the detailed structure of the defect with the coordinates of the positions of the surrounding ions, and the energy of the system. Although the earlier programmes were restricted both in terms of the choice of simple interionic two-body potentials that could be used and of the crystal symmetries they could handle they were, in many applications, adequate for simple ionic solids. These static lattice programmes have, of course, been continuously developed and current codes provide for very wide choices of interaction potentials to represent the force fields in the materials and include the complete range of crystal symmetries. The programmes are relatively undemanding in terms of computational power and so the energies of large and complex defect structures can be calculated. In addition, quite sophisticated potentials, including treatment of the polarisation of the ions that is essential for reliable results, can be employed. Defect formation energies are determined by comparison with the energies, also calculated by the programme, for the perfect lattices. If migration

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pathways are assumed and calculations carried out for the saddle-point energy, the activation energy for a jumping ion can be also be determined by subtraction of the energy of the ground state from that calculated for the assumed saddle point. In such cases it is always advisable to search out for the lowest energy pathway by fixing the jumping ion at a variety of appropriate sites in the region of any assumed saddle points. The energies of association of complex defects are determined as the differences between the energy calculated for the complexes and the sum of the energies calculated for their isolated constituent parts. Static lattice calculations are essentially constant volume calculations and so provide a value for u v (0). The thermodynamic relationships between the results obtained and the corresponding experimental quantities, h p (T ), which are measured at constant pressure, have been set out by Catlow et al., [7]. The essential equations are: g p = fv

h p = uv − T

s p = uv −

∂V ∂T

∂V ∂T

p

p

∂ fv ∂V

∂ fv ∂V

(2)

(3) T

(4) T

Later work [8, 9] has shown that the reason that the internal energy of a defect at infinite zero can be identified with its enthalpy at a finite temperature is because two terms cancel each other to make h p (T ) ∼ u v (0). This is the comparison that is made between experimental and calculated values in normal practice.

2.2.

The Supercell Method

This technique can also be used to calculate defect energies by introducing the defect to the supercell to which periodic boundary conditions are to be applied. However there are some inherent difficulties because an artificial ordering is introduced for the defects and also because the presence of charges on the defects causes the Coulombic summation to diverge. The sum of the Coulombic energy requires a neutral cell with multiple defects that maintain charge neutrality and that these interact with each other and their images in the periodic summation. To reduce these effects a number of corrections have been suggested that allow the simulation of single charged point defects with subtraction of defect–defect interactions to yield a simulation that is very close to infinite dilution. The Ewald method used to sum the Coulombic terms assumes that the cell is charge neutral. However, a small modification [10] that assumes that there

Point defects in simple ionic solids

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is a counter charge distributed evenly throughout the cell and is the g = 0 term for the reciprocal space summation given by: −

π Q2 2V η

(5)

made it possible to consider charged cells. Here Q is the total charge on the cell, V is the cell volume and n is the Gaussian half-width used in the Ewald sum. In addition, the Coulombic interaction energy from defect–defect interactions can be calculated [11] as: defects ” i, j

qi q j 2r i j · ε R

(6)

where the ” indicates that the summation does not include pairs of defects within the unit cell, ri j is the distance between defects i and j and ε R is the static dielectric constant matrix. Parker and co-workers extended these extended these post-calculation corrections to all symmetries and, more importantly, corrected the energies and forces during the simulation, rather than applying them retrospectively. The forces due to defect–defect interactions were also calculated and subtracted and were therefore excluded from affecting the final configuration and energy. The techniques were implemented in the free energy minimisation code PARAPOCS [12, 13] and used to calculate the free energy of defect formation and migration in MgO. Watson et al. [14] later used the method to investigate defect enthalpy formation as a function of pressure in MgSiO3 perovskite, including defect volume. Such calculations are difficult using the Mott–Littleton approach.

2.3.

Molecular Dynamics

This technique is of rather limited use in the study simple ionic solids because it is not suited to the modelling of the hopping motion that is typical of ionic migration through these materials and which is too slow to yield results on a reasonable timescale. A related approach that has proved useful involves pushing ions from one site to another using forced molecular dynamics. A force is applied to an ion pushing it toward a vacant site. Molecular dynamics is then applied so that the system can relax around the moving ion. In this way a trace of the diffusion pathway and the value of the activation energy can be obtained. This approach has been tested on bulk MgO and has also been applied to diffusion in grain boundaries of MgO [15], NiO [16] and Al2 O3 .

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Electronic Structure Calculations

As with all quantum mechanical calculations those for defects in crystals are limited by the size of the system that can be handled. Periodic calculations result in defect–defect interactions and also exhibit problems with charge neutrality. These effects are more difficult to correct for than in force-field calculations because of the extra complication of the electronic rearrangement. However some basic defects in simple ionic materials, such as MgO and Li2 O, have been studied [17, 18].

2.5.

Calculation of Defect Entropies

Simulation techniques to calculate the entropies of defect formation have also been developed using both the two-region approach and the supercell method. The calculations are a natural follow-on from the calculation of defect energies and require the detailed relaxed positions that are calculated for the ions in the defective region. They then treat the ions in the defective lattice within the harmonic approximation and the calculation effectively estimates the effect of the defect on the lattice phonon spectrum. In the two-region strategy the ions in the outer region are not allowed to move while the ions in the inner region immediately about the defect may vibrate. Provided that sufficiently large numbers of ions are used the two techniques give similar answers [9, 19].

3.

Interionic Potentials

The static lattice codes now used to calculate defect properties are sufficiently well developed and flexible that the accuracy of the calculated energies for both defect formation and defect processes depend crucially on the quality of the interionic potentials used. This, in turn, depends on two factors. The first of these is the form of the potential and whether it can adequately represent the real forces that exist and act in the crystal. The second is the accuracy with which the parameters in that potential can be determined and their appropriateness to the calculation of defect properties. The potential most commonly used in simple ionic solids is the Buckingham potential which has the form: (r) = A exp(−r/ρ) − C/r 6

(7)

This is comprised of a repulsive term with a pre-exponential parameter, A, a hardness parameter, ρ, and an attractive term with a parameter, C. For a pure

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binary crystal three such interactions, cation–cation, cation–anion and anion– anion must be specified. The introduction of a dopant ion will necessitate at least two more analogous interactions, dopant–cation and dopant-anion and a third to represent dopant–dopant interactions if the dopant ions are sufficiently close to each other. For accurate calculations, it is also essential to take account in the simulation of the polarisability of the ions. This polarisability has been represented by a number of models of which the shell model of Dick and Overhauser [20] is the most successful and widely used. In this model the ion is represented by a spring-coupled core and a shell each of which carry charges: the sum of these charges is the overall charge on the ion. The cores and shells can move independently thus giving rise to the polarisation of the ions. The use of the shell model in the simulation of a binary crystal introduces four additional parameters, a shell charge, Y , and a core-shell coupling constant, k, for each ion, the values of which must also be determined in addition to those of the other parameters in the potential. The polarisability of the free ion, α, is related to Y and k by the equation: α = Y 2 /k

(8)

The parameters required for the interionic potentials in simple ionic solids are determined by two principal means. In the first of these they are evaluated through fitting of the potentials, using appropriate lattice dynamics and other equations, to available experimentally determined macroscopic properties of the crystal such as the lattice separation, cohesive energy and elastic and dielectric constants. Potentials for which the parameters have been determined in this way are called empirical potentials. Despite successful applications of such potentials for specific crystals, there are significant disadvantages to their more general use. Even for the most thoroughly experimentally investigated and simple crystals, the parameters evaluated in this way are underdetermined. Furthermore is it not always possible to discriminate between the extents of the contributions of each of the various ion-pair interactions to the overall measured properties. In addition, the values obtained for the parameters depend on properties measured when all the ions in the crystal are occupying equilibrium lattice sites and so they are unlikely to be completely appropriate to defect structures in which the constituents are displaced from such positions. This effect is further aggravated in calculations of the energies of defect processes during which ions may be expected to be substantially displaced from their equilibrium separations. The second method is to carry out electronic structure calculations at some suitable level of approximation to yield the energies of interaction for each ion pair in question as a function of their distance of separation. One such calculation used for simple ionic solids is the electron gas approximation [21] in which the crystal field effects corresponding to the environment in which the ion is placed in the crystal are sometimes included. The resulting

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energy-separation relationship can then be fitted to one of the analytical forms of potential available in the defect energy code. Alternatively the numerical values can be used directly by making use of splining techniques to provide values of the potential and its derivatives at separations intermediate between those at which its value was calculated. It is clear that neither of these methods is entirely satisfactory and that, in almost every case, some degree of model building is required to complete the potential. The parameters to describe the interaction between a particular pair of ions determined in one crystal may be transferred for use in another crystal in which the same two ions interact. However, care must be exercised to ensure that all the pair potentials assembled for use in a calculation were originally determined in the same way e.g., using an electronic structure calculation at the same level of approximation. Libraries of such interactions have been built up and can provide the parameters for the interactions when a new material is to be studied, for example the GULP programme includes a library of such potentials. The two-body potentials discussed here to date allow only central forces to be treated in the calculations. Whereas these are adequate for fully ionic or almost fully ionic materials they fail to properly represent the forces in crystals with deformable and highly polarisable ions in which non-central forces can play a significant role. Among the simple ionic solids the silver halides, both the chloride and the bromide, provide examples of materials that cannot be adequately represented by two-body potentials. The current defect codes include bond-bending terms usually of the form: i j k = 0.5 ki j k (ϑ − ϑ0 )2

(9)

so that energy is required to depart from the equilibrium bond angle, ϑ 0 .Such terms introduce directionality into the bonding and may be appropriate for use when modelling semi-covalent and covalent systems or molecular ions. ki j k is the bond-bending force constant between the bonds ik and ij. The codes also contain torsional terms, also called four-body potentials, to model systems that have a planar structure due, for example, to π -bonding. Bond-stretching terms to model bonded interactions, such as the hydroxide ion in ionic systems, and often written in the well-known Morse potential form, are also included. Further refinements include the extension of the shell model to include breathing shell and deformable shell models which can improve the representation of the polarisability of some ions. Because molecular dynamics codes are substantially more demanding of computational power it is usually necessary to carry out simulations using potentials that are simpler in form. However, it has proved possible to use full shell model potentials to calculate the diffusion energy barriers [22]. As has been mentioned above, molecular dynamics techniques are more suited to

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systems in which the energy barriers are small and in which there is a high concentration of defects, i.e., fast-ion conductors (Section X.Y).

4.

Discussion

Where adequate interionic potentials have been determined the energies calculated using static lattice techniques for the formation of point defects, for their interactions and for their migration in simple ionic crystals are generally in very good agreement with the analogous experimentally determined values. This is well demonstrated even in the early extensive compilations of calculated and experimental defect energies made by Corish et al [23, 24]. Indeed in the case of the most-studied of the alkali halides, in which Schottky defects predominate, such calculations were sufficiently accurate and reliable to show that the experimental values for the activation energies for anion vacancy migration, which were reported as being substantially larger than those for the corresponding cation vacancies, were not correct but rather resulted from an artefact of the non-linear fitting procedures used to determine the defect parameters from the conductivity data. In the identically structured silver halides, in which the dominant defect is cationic Frenkel, the use of the quasi-harmonic approximation (Eqs. (2)–(4) above) provided the answer to what had been a long-standing problem. The unusually very rapid increase in the conductivities of these materials at higher temperatures as they approach their melting points was explained when the Frenkel defect formation energy was shown to be temperature dependent [23]. More recently, DFT methods have been used to examine the defect formation energies as well as the very facile motion of the very polarisable and readily deformable silver ion through these crystals [24]. In simple ionic crystals simulations techniques have been particularly useful in enabling us to understand the interactions between defects, their aggregation into larger more complex defect clusters and the eventual formation of small domains with a different structure. In many cases these processes or the entities formed are not open to easy experimental identification or investigation. A particularly striking example is the work of Tomlinson et al. [25] on the rock-salt structured transition metal oxides. They calculated the free energies, and hence the equilibrium constants, for the aggregation of a range of complex defects that are formed when the valences of the cations increase in the non-stoichiometric materials. These equilibrium constants were then used in a mass action analysis to estimate values for the oxygen partial pressure as a function of the deviation from stoichiometry and the results were compared with experimental determinations to identify the nature of the defect clustering occurring in the material. In the rare-earth doped fluorites simulation studies were used to correlate the structures of the complex defects formed between

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several dopant ions and charge-compensating interstitial fluoride ions and the observed EXAFS spectra from the dopants [26].

5.

Summary

The study of point defects in simple ionic crystals has been a very productive test-bed for both experimental and theoretical investigations of defect chemistry and physics in solid materials. In terms of modelling the techniques and their embodiment into codes of ever-increasing flexibility, reliability and scope has depended on the availability of accurately known experimental data. The close interplay between calculation and experiment that has been achieved in these systems has been crucial to progress in both areas. It has been pursued vigorously and has now brought us to a very detailed understanding of the nature of defect formation and interaction and of the dynamics of defect processes. The codes that have been developed and tested as well as the techniques that have emerged to identify realistic and appropriate interionic potentials and to determine the values of the parameters for specific interactions have been invaluable in enabling progress towards the modelling of more technologically important materials where the force-fields that operate are more complex in nature. Current developments in quantum mechanical/molecular modelling techniques promise Mott–Littleton type quantum calculations for defects. Success in this endeavour will solve many of the difficulties that now surround the quantum mechanical investigation of defects and promise the prospect of calculating even more accurate defect energies.

References [1] F.A. Kr¨oger and H.J. Vink, “Relations between the concentrations of imperfections in crystalline solids,” Solid State Phys., 3, 307, 1956. [2] C.R.A. Catlow (ed.), “Computer Modelling in Inorganic Crystallography,” Academic Press, London, 1997. [3] N.F. Mott and M.J. Littleton, “Conduction in polar crystals. I. electrolytic conduction in solid salts,” Trans Faraday Soc., 34, 485, 1938. [4] M.J. Norgett, Harwell Report AERE-R 7650, AEA Technology, Harwell, Didcot, OX11. ORA, U.K., 1974. [5] M. Leslie, SERC, Daresbury Laboratory Report DL-SCI-TM3IT, CCL, Daresbury Laboratory, Warrington, WA4, 4AD, U.K., 1982. [6] J.D. Gale, General Utility Lattice Programme, Imperial College London, U.K, 1993. [7] C.R.A. Catlow, J. Corish, P.W.M.J. Jacobs et al., “The thermodynamics of characteristic defect parameters,” J. Phys. C., 14, L121, 1981. [8] M.J. Gillan, “The volume of formation of defects in ionic crystals,” Phil. Mag., 4, 301, 1981. [9] J.H. Harding, “Calculation of the free energy of defects in calcium fluoride,” Phys. Rev. B, 32, 6861, 1985.

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[10] M. Leslie and M.J. Gillan, “The energy and elastic dipole tensor of defects in ionic crystals calculated by the supercell method,”J. Phys. C – Solid State Phys. B, 973, 1985. [11] N.L. Allan, W.C. Mackrodt, and M. Leslie, “Calculated point defect entropies in MgO,” Advances in Ceramics, 23, 257, 1989. [12] S.C. Parker and G.D. Price, “Computer modelling of phase transitions in minerals,” Adv. Solid State Chem., 1, 295, 1989. [13] G.W. Watson, T. Tschaufeser, R.A. Jackson et al., “Modelling the crystal structures of inorganic solids using lattice energy and free-energy minimisation,” In: C.R.A. Catlow (ed.), Computer Modelling in Inorganic Crystallography Academic Press, London, 1997. [14] G.W. Watson, A. Wall, and S.C. Parker, “Atomistic simulation of the effect of temperature and pressure on point defect formation in MgSiO3 perovskite and the stability of CaSiO3 perovskite,”J. Phys. Condens. Matter, 12, 8427, 2000. [15] D.J. Harris, G.W. Watson, and S.C. Parker, “Vacancy migration at the {410}/[001] symmetric tilt grain boundary of MgO: an atomistic simulation study,” Phys. Rev. B, 56, 11477, 1997. [16] D.J. Harris, J.H. Harding, and G.W. Watson, “Computer simulation of the reactive element effect in NiO grain boundaries,” Acta Mater., 48, 3309, 2000. [17] A. Devita, M.J. Gillan, J.S. Lin et al., “Defect energies in MgO treated by 1st principles methods,” Phys. Rev. B, 46, 12964, 1992. [18] A. Devita, I. Manassidis, J.S. Lin et al., “The energetics of frenkel defects in Li2 O from 1st principles,” Europhys. Letts., 19, 605, 1992. [19] M.J. Gillan and P.W.M.J. Jacobs, “Entropy of a point defect in an ionic crystal,” Phys. Rev. B, 28, 759, 1983. [20] B.G. Dick and A.W. Overhauser, “Theory of the dielectric constants of alkali halide crystals,” Phys. Rev., 164, 90, 1964. [21] Y.S. Kim and R.G. Gordon, “Theory of binding of inorganic crystals: application to alkali-halides and alkaline-earth-dihalide crystals,” Phys. Rev. B, 9, 3548, 1974. [22] P.J.D. Lindan and M.J. Gillan, “Shell-model molecular dynamics simulation of superionic conduction in CaF2 ,” J. Phys –Cond. Matter, 5, 1019, 1993. [23] J. Corish, “Calculated and experimental defect parameters for Silver Halides,” J. Chem. Soc., Faraday Trans., 85, 437, 1989. [24] D.J. Wilson, S.A. French, and C.R.A. Catlow, “Computational studies of intrinsic defects in Silver Chloride,” Radiat. Eff. Defects Solids, 157, 857, 2002. [25] S. Tomlinson, C.R.A. Catlow, and J.H. Harding, “Computer modelling of the defect structure of non-stoichiometric binary transition metal oxides,” J. Phys. Chem. Solids, 51, 477, 1990. [26] C.R.A. Catlow, A.V. Chadwick, J. Corish et al., “Defect structure of doped CaF2 at high temperatures,” Phys. Rev. B, 39, 1897, 1989. [27] J. Corish and P.W.M.J. Jacobs, “Surface and defect properties of solids,” M.W. Roberts and J.M. Thomas (eds.), Specialist Periodical Reports, vol. 2, The Chemical Society, London, p. 160, 1973. [28] J. Corish, P.W.M. Jacobs, and S. Radhakrishna, “Surface and defect properties of solids,” M.W. Roberts and J.M. Thomas (eds.), Specialist Periodical Reports, vol. 6, The Chemical Society, London, p. 219, 1977.

6.5 FAST ION CONDUCTORS Alan V. Chadwick Functional Materials Group, School of Physical Sciences, University of Kent, Canterbury, Kent CT2 7NR, UK

1.

Introduction

Fast ion conductors, sometimes referred to as superionic conductors or solid electrolytes, are solids with ionic conductivities that are comparable to those found in molten salts and aqueous solutions of strong electrolytes, i.e., 10−2 – 10 S cm−1 . Such materials have been known of for a very long time and some typical examples of the conductivity are shown in Fig. 1, along with sodium chloride as the archetypal normal ionic solid. Faraday [1] first noted the high conductivity of solid lead fluoride (PbF2 ) and silver sulphide (Ag2 S) in the 1830s and silver iodide was known to be unusually high ionic conductor to the German physicists early in the 1900s. However, the materials were regarded as anomalous until the mid 1960s when they became the focus of intense interest to academics and technologists and they have remained at the forefront of materials research [2–4]. The academic aim is to understand the fundamental origin of fast ion behaviour and the technological goal is to utilize the properties in applications, particularly in energy applications such as the electrolyte membranes in solid-state batteries and fuel cells, and in electrochemical sensors. The last four decades has seen an expansion of the types of material that exhibit fast ion behaviour that now extends beyond simple binary ionic crystals to complex solids and even polymeric materials. Over this same period computer simulations of solids has also developed (in fact these methods and the interest in fast ion conductors were almost coincidental in their time of origin) and the techniques have played a key role in this area of research. The electrical conduction in fast ion conductors occurs via point defects and the modelling of these defects is covered in Article 6.4 in this Volume by

1901 S. Yip (ed.), Handbook of Materials Modeling, 1901–1914. c 2005 Springer. Printed in the Netherlands.

1902

A.V. Chadwick 2.0 1.0 0.0

log ó/S cm⫺1

⫺1.0

RbAg4I5

â-PbF2

⫺2.0 ⫺3.0 ⫺4.0 Ce1.8Gd0.2O1.9

⫺5.0 ⫺6.0

ä-Bi2O3

⫺7.0

NaCl

⫺8.0 0.8

1.2

1.6

Zr1.8Y0.2O1.9 2.0

2.4

Agl

NaSCN.P(EO)8 2.8

3.2

3.6

1000K/T Figure 1. The experimental conductivity plots for several fast ion conductors.

Corish. In an ionic solid the conductivity, o´ , can simply be expressed by the expression: σ=

i

n r |qr |µr

(1)

r=1

where n r is the number of defects of type r per unit volume, qr the effective charge and ìr the mobility of defects of type r. Equation 1 provides a simple, phenomenological interpretation of fast ion conduction. The unusually high o´ could arise from an unusually large n r and/or ìr . Most theoretical models have focused on a high concentration of defects as the cause of fast ion behaviour. In fact an early attempt at a general explanation of the phenomenon was the molten sub-lattice model, which assumed that a high concentration of defects on one of the sub-lattices led to it’s melting with concomitant liquidlike motion of the ions [5]. Although the focus still remains on an unusually high defect concentration this general model has been rejected and massive disorder is not a requirement; normal ionic conductors, like NaCl, typically have a site fraction of defects ∼10−3 and o´ ∼10−3 S cm−1 at the melting point and therefore defect site fractions of ∼1% would lead to anomalous behaviour.

Fast ion conductors

1903

Given the wide range of materials that exhibit fast ion conduction attention has turned to the development of models that are system specific. Detailed descriptions of the computer simulation methods are presented elsewhere in this Volume. Hence only a brief summary of the techniques is necessary here and this will be presented in the next section. This will be followed by the results that have been obtained for fast ion conductors. Given the large number of systems that are now known to exhibit the phenomenon the discussion will be restricted to the major material types. The technical details of the simulations, unless vital to the discussion, will not be presented but the focus will be on the major findings and the rˆole the simulations have played in understanding the experimental data and the insights they can provide. The final section will present a summary and a look at how simulations could develop in this field.

2.

Modelling Techniques

As mentioned earlier the studies of fast ion conductors and computer modelling share a common and interlinked history and most of the modelling techniques have been applied to these materials.

2.1.

Static Lattice Calculations

This technique, described by Corish [6] in this volume, has been widely used in the study of fast ion conductors. The two-region strategy of the Mott– Littleton procedure has been used as implemented in the codes HADES, CASCADE [8] and GULP [9]. A variety of interatomic potentials have been employed, ranging from simple point charge models to shell-model and ab initio potentials. In more recent work on complex systems DFT methods have been used to derive potentials using codes such as CASTEP [10]. In a similar manner to the study of normal ionic solids the codes have been used to calculate the energies of defect formation and migration, the energies of solution of impurities and the energies of defect clusters. In a few cases the calculation of the entropy has been included so that the free energies and hence absolute values of defect concentrations and diffusion coefficients could be obtained.

2.2.

Molecular Dynamics Simulations

Molecular dynamics (MD), simulations have been extensively used in the study of liquids, [11, 12] and have proved most informative of the methods applied to fast ion conductors. A variety of codes have been employed, the

1904

A.V. Chadwick

most recent being the DL POLY code from the Daresbury Laboratory [13]. The rapid transit of the ions in these materials, as in liquids, happens on a timescale that a significant number of displacements occur in the period of a typical calculation. Although heavily demanding on computing resources the number of ions in the simulation box, the length of time of the calculation, the complexity of the interatomic potential have all increased with the increase of computer power to the extent that modern calculations are directly comparable to experiment. The obvious advantage of this technique is that it produces parameters that are correlated to the measured experimental parameters without the need to invoke theoretical model. For example, the radial distribution functions (RDF) can be compared with diffraction or EXAFS data, and the mean square displacements can be compared with diffusion coefficients from tracer or NMR data. However, there is also additional mechanistic information in that the MD produces a detailed picture of the ion motion at the atomic scale. Hence it possible to decide whether the motion of the mobile ion is liquid like (i.e., the ion is continuous motion), or solid-like, (i.e., the ion jumps from site to site with the transit time much less than the site residence time), and, in the case of the latter process, if nature of the point defects involved and the degree of correlation between successive diffusive jumps.

2.3.

Monte Carlo Simulations

Monte Carlo (MC), modelling methods are well-suited to the investigation of highly disordered systems and have been used in the study of some classes of fast ion conductor where there are a range of defect sites and possible jump mechanisms for an ion. A typical use is to couple the calculation of jump activation energies for the different defect environments (for example, obtained from static lattice simulations) with an MC simulation. In a simulation box a defect and its jump direction is selected at random and using the knowledge of the environment the Boltzmann factor is obtained. A standard Metropolis sampling procedure is used to decide if a jump succeeds or fails. After several thousand moves, in which the configuration is continually updated, an average diffusion coefficient and corresponding activation energy can be obtained.

3.

Discussion

Fast ion conductors have been classified in a variety of ways all have which have their advantages. For example they have been classified on the nature of the mobile ion (i.e., H+ , Li+ , Na+ , O2− , etc.), or on the type of lattice structure (i.e., fluorite type, layered, tunnel, glasses, polymers, etc.), or in terms of their applications (i.e., batteries, fuel cells, sensors, etc.). The classification used

Fast ion conductors

1905

here is that developed by Catlow [14], as it is the most useful in terms of the mechanisms of the ion transport [15]. For each of these classes the rˆole computer simulations has played in understanding the structures and processes will be outlined.

3.1.

Solids with a Phase Transition

Many solids exhibit fast ion conduction after a phase transition, the low temperature behaviour being interpretable in terms of classical defect theories [16, 17]. Examples include AgI and crystals with the fluorite structure. In AgI ˆ to there is a first order transition at 146◦ C from the low temperature a-form the high temperature cubic a´ -form with an increase in o´ of two orders of magnitude, as shown in Fig. 1. The a´ -form has a bcc arrangement of the I− ions with the two Ag+ ions per unit cell distributed over 42 possible lattice sites which are occupied with the preference order 12d (tetrahedral) > 24h (trigo¯ space group. Conduction takes place by nal) > 6b (octahedral) in the Im3m + Ag ions moving between tetrahedral and trigonal sites and has been studied in MD simulations. The most extensively studied fast ion conductors are those with the fluorite structure [18–21] and a general feature of the structure is a broad thermal anomaly, often referred to as the Bredig transition, similar to a diffuse e¨ -type order-disorder transition at a temperature Tc , which is approximately 0.8 of the melting temperature. The changes in ionic conductivity are quite subtle, as indicated by the data for aˆ -PbF2 in Fig. 1. At low temperatures the conductivity is normal and can be interpreted with classical point defect models [22] however at the same temperature regime as the thermal anomaly the conductivity plot shows an increasing upward curvature which then plateaus around 1 S cm−1 in the fast ion region. Similar behaviour, though not extensively investigated is found in the rare-earth fluorides with the tysonite structure. There is a wealth of experimental data for the fluorite-structured materials, particularly the fluorides (CaF2 ,SrF2 ,BaF2 ) as they are one of the easiest systems to study (e.g., large single crystals are readily available, they are thermally stable, etc.). When this is coupled with the simplicity of the structure it is not surprising that these systems have been the subject of more computer modelling studies than any other system. Very early static lattice simulations using shell model potentials showed that the defects in the low temperature phase were Frenkel defects, anion vacancies and interstitials. The calculated formation and migration energies of these defects were in good agreement with experiment. Thus reliable potentials were available at the outset. More difficult was the successful modelling of the transition to the superionic phase, which is due to a rapid generation of Frenkel defects around Tc . This is believed to be due to a cooperative effect in which the energy of

1906

A.V. Chadwick

formation of the Frenkel defects lowers as the concentration of these defects increases, resulting in a rapid rise in their concentration as the temperature increase. An early explanation was that this was due to the formation of defect clusters, which are well documented in doped systems (as will be discussed in Section 3.2), but involving simply vacancies and interstitial anions. There was evidence from neutron scattering experiments for the existence of these clusters in the superionic phase and static lattice simulations showed that these clusters had significant binding energies [23]. However, it was the MD studies of the fluorides, notably by Gillan and co-workers [20] that led to a better understanding of these systems. These simulations used shell model potentials to treat the polarizability of the ions and were able to reproduce were able to reproduce the experimental data, notably the specific heat peak and the conductivity. The calculations also suggested and alternative explanation of the neutron data suggesting that the features observed were not necessarily due to defect clusters but arose from the dynamic nature of the system, i.e., a snapshot showed many anions were off the normal sites. In addition, the MD work showed that the motion of the anions was a solid state hopping process, the residence time on a lattice site being considerably longer than the transit time between sites, and the vacancy concentration was only the order a few per cent, consistent with conductivity data [22]. Work continues on these systems, with more experimental data and MD simulations using ab initio potentials to more effectively treat the ion polarizability [21, 24].

3.2.

Massively Disordered and Heavily Doped Solids

In a few materials there is massive disorder due to an exceptionally high defect concentration. An excellent example is bismuth oxide, Bi2 O3 , which undergoes a phase transition to a fluorite-structured a¨ -phase at high temperature, which has the highest known O2− ion conductivity. This is hardly surprising as the oxygen sublattice contains 25% vacancies and this is clearly different from the fluorite-structured materials described in the preceding section. The silver chalcogenides (e.g., Ag2 S, Ag2 Se and Ag2 Te) are similar to AgI in that there are more available cation sites than Ag+ ions and in these systems the motion of the cations appears to be truly liquid-like. The addition of aliovalent impurities to an ionic crystal is the traditional method of increasing the defect concentration [6, 16, 17]. However, in simple ionic solids the changes are limited by the extremely low solubilities of the impurities, typically less than 1 mole per cent. Exceptions to this general rule are open structures, particularly the fluorite structure which is capable of dissolving up to 50 mole per cent of cation impurities. These fall into this class of heavily doped systems and includes the technologically important fluorite-structured oxides zirconia (ZrO2 ) and ceria (CeO2 ). In fact, pure ZrO2 has a monoclinic

Fast ion conductors

1907

structure and is stabilised in the cubic, fluorite phase by doping with Y3+ or Ca2+ (hence the name cubic-stabilised zirconia, CSZ) and the creation of anion vacancies. Another group of compounds that can be usefully included in this class is the mixed fluorides, such as PbSnF4 and RbBiF4 , where the mixing of cations seems to generate defects and leads to fast fluoride conductivity at temperatures around 100◦ C [25]. A considerable research effort has focused on the fluorite structured oxides due to their technological importance, particularly for solid oxide fuel cells (SOFC). The effect of doping with lower valence cations will create chargecompensating anion vacancies and these vacancies are responsible for the high ionic conductivity. The doping can be expressed by the reaction (Krõger–Vink notation): 2T 3+ (dissolving in MO2 ) → 2DM + V•• O

(2)

where T is a trivalent cation (e.g., a rare earth or yttrium) and MO2 is the oxide, ceria or zirconia. However, it was found that the conductivity did not increase monotonically with dopant concentration but it peaked around 10 mole per cent and that the conductivity was dependent on the nature of the dopant. The origin of the effect was clear; as the concentration of the dopant increased there was an increase in the concentration of impurity-anion vacancy clusters (DM − V •• ) increased in size and complexity that “trapped” the vacancies and reduced their mobility. The binding in these clusters has been treated very successfully by static lattice calculations where the key factor is found to be elastic strain due to a mismatch of ion sizes. Therefore the best dopants, in terms of minimising the reduction in conductivity at high concentrations, are when they are the same size as the host cation. A detailed simulation of Y3+ doped ceria has been made in which the various clusters were modelled in a static lattice calculation and the transport of the vacancies by an MC simulation [26]. The qualitative agreement with the experimental data was excellent, showing the maximum in conductivity as a function of dopant concentration and the change in activation energy, as shown in Fig. 2. In addition, the simulation also showed that at high concentration the vacancies were still making jumps but they were mainly from site to site within the clusters.

3.3.

Layered and Tunnel Structures Solids

In this group of materials the ions move along rapid diffusion pathways that are defined by the crystal structure. The aˆ and aˆ -aluminas are the archetypal 2-D conducting materials of this class and were originally explored as membranes for the sodium-sulfur battery [2]. These compounds are nonstoichiometric sodium aluminates with the mobile Na+ ions located in the

1908

A.V. Chadwick

(a) ⫺1 calculated 833K

log ó/S cm⫺1

⫺3

experimental 833K

⫺5

calculated 455K

⫺7 experimental 455K ⫺9

0

0.02

0.04

0.06

0.08

0.1

fraction of anion sites vacant (b) 1.4 experimental 455K activation energy/eV

1.2

1 calculated 455K 0.8

0.6

0.4 0

0.05

0.1

0.15

fraction of anion sites vacant Figure 2. Calculated and experimental conductivity studies of CeO2 doped with Y2 O3 [26]. (a) conductivity plots and (b) Arrhenius energies.

Fast ion conductors

1909

planes between spinel blocks of alumina that are bridged by oxygen. The unit cell of the conduction plane is shown in Fig. 3. In this conduction plane the Na+ ions in the stoichiometric material occupy alternate sites on a hexagonal network, the so-called Beevers-Ross (BR) sites. However, most experimental samples contain between 15 and 30% excess Na+ and the nominal nonstoichiometric composition is (Na2 O)1+x Al22 O33 (where 0.15 < x < 0.3), with the additional Na+ and charge compensating additional oxygen ions located in the conduction plane. Both the structure and Na+ migration have been the subject of computer simulation and is summarised in the recent paper by Beckers et al. [27]. Early static lattice simulations showed that the stable position for excess Na+ ions is the anti-Beevers–Ross (aBR) site. However, recent diffraction studies have shown the preferred site is displaced from this position and is between the mid-oxygen (mO) and aBR site, the so-called A site. Cation migration proceeds by jumps between lattice sites and MD simulations have been particularly illuminating on the sequence of diffusive steps. The calculations show the basic process is a hopping process between BR and aBR sites,

aBR

aBR

O

mO

O

BR

BR mO

mO A

aBR

A aBR

Figure 3. The unit cell in the conduction plane of aˆ -alumina. Lattice sites are labelled BR (Beevers–Ross), aBR anti (Beevers–Ross), mO (mid-oxygen) and O (bridging oxygen).

1910

A.V. Chadwick

but the details depend on composition. In stoichiometric material all BR sites are occupied and all aBR sites are vacant, thus Na+ migration is via intermediate aBR defect sites and diffusion is slow and the conductivity is low. In the non-stoichiometric material there is a strong correlation between jumps, particularly at low temperatures; an aBR to BR jump is immediately followed by a BR to aBR jump and so on, creating trains of mobile ions. The calculations also show the role of the excess oxygen in interstitial sites is to stabilise the A site and their presence ‘blocks’ the motion of Na+ ions. There are other layered materials in which the conduction is 2D but on the whole they have not been thoroughly explored. A rare example of a simple binary 2D conductor is lithium nitride, Li3 N, which is the result of an unusual crystal structure. An example of a 3D ionic conductor in which the ions move through channels is the Na+ ion conductor Na3 Zr2 PSi2 O12 , which is now generally referred to as NASICON (Na superionic conductor). Like the aˆ -aluminas this is a ceramic material but has a Na+ higher conductivity at 300◦ C.

3.4.

Amorphous Solids

There are recognised applications for vitreous electrolytes and a number of glass systems do exhibit high ionic conductivities [28–30]. Glasses with conductivities of the order of 10−2 S cm−1 at room temperature are known and have been referred to as “superionic conducting glasses (SIG)” or “vitreous electrolytes”. The highly mobile ion is usually Li+ , Ag+ or F− and a variety of network forming materials have been identified as producing useful electrolytes, e.g., silicates, borates, phosphates and sulfides. To date, there have been relatively few computer simulation investigations of these systems however it appears that the fast ion migration occurs along preferential pathways in the glassy matrix.

3.5.

Solid Polymer Electrolytes

A dry number of polymers will dissolve ionic salts and yield films that have reasonable magnitudes of ionic conductivity. The most useful and widely investigated are based on high molecular weight polyethylene oxide (PEO; [–CH2 –CH2 –O–]n ) where the ether oxygen atoms are co-ordinated to the cation of the salt and effect solvation in the same manner as crown ethers [31]. The high conductivity is found above the melting point of PEO (∼65◦ C), when the material is amorphous and in an elastomer phase. Although the specific conductivity of these materials is only ∼10−3 S cm−1 (two orders of magnitude lower than the best fast ion conductors) they have the mechanical advantages

Fast ion conductors

1911

of being flexible and strong, compared to brittle ceramics, and are easily processed as very thin films. These materials have attracted major interest as the membranes for lithium ion batteries. The PEO based electrolytes have been extremely difficult systems to model due to both the dynamic nature of the system and the lack of long-range order. The similarity between the activation energies for ionic conduction and the reptation motion of the polymer backbone had led to the assumption that the two processes were inter-related [31]. However, there was a continuing debate concerning the nature of the mobile species; the relatively low dielectric constant led to the speculation that there was considerable clustering of the ions and that transport could involve ion dimers, trimers, etc. A recent computer simulation study the PEO-lithium triflate (Li+ CF3 SO− 3) has been particularly revealing [32]. Firstly, cubic amorphous cells were constructed using the amorphous cell module and force fields in the Insight code (Molecular Simulations Inc.). The salt was then introduced and the ionpolymer interactions were represented by potentials derived from high level ab initio calculations, which were validated by reference to crystallographic parameters. The system was equilibrated and the configurations used as input to the DL POLY MD code, with the NVT ensemble, periodic boundary conditions and the Ewald summation. There was reasonable quantitative agreement of the calculated diffusion coefficients with those obtained from NMR measurements, as shown in Fig. 4, and there was unique qualitative information on the system. RDFs were evaluated for the various pairs and there were strong

log D/m2 s⫺1

⫺9.5 Li⫹D calc CF3SO3⫺D calc Li⫹D NMR CF3SO3⫺D NMR

⫺10

⫺10.5

⫺11 1.8

1.9

2

2.1

2.2

2.3

2.4

2.5

2.6

1000K/T Figure 4. Comparison of the calculated and experimental diffusion coefficients of Li+ and CF3 SO− 3 ions in 8:1 PEO-lithium triflate polymer electrolyte [32].

1912

A.V. Chadwick

Li+ – Li+ and Li+ -triflate correlations at long distances, whereas ion-polymer correlations were weaker. The first shell of Li+ consisted of 2.5 triflate oxygens and 1.5 ether oxygens indicating that the ion is not located at specific crystallographic type site in the polymer structure. Molecular graphics showed that rather than isolated clusters of ions there was an extended network of ions which cross-linked the polymer chains and through which the diffusion occurred. There was little evidence of ion hopping and that the motion of ions in the network was faster than those close to the polymer. Local motion of the polymer chain was more rapid than the ion transport, but there was negligible diffusion of the chain, suggesting polymer reptation is important in the diffusion of the ions.

3.6.

Proton Conductors

The mechanisms that are thought to operate in these systems are different to those in other fast ion conductors and are taken as a separate class. In the case of hydrated salts like hydrogen uranyl phosphate, HUO2 PO4.4H2 O (HUP), the protons are believed to migrate by the classical Gr˝otthus mechanism used to explain proton mobility in water and involving a proton exchange between the water molecules [33]. A number of doped ceramic oxides with the perovskite structure, mainly cerates and zirconates, also exhibit high proton conductivity and the protons was believed to migrate by a tunnelling process [34]. The best example is Yb doped SrCeO3 , where the dopant is charge compensated by the creation of anion vacancies. Water molecules can be incorporated into these vacancies and generate hydroxyl ions according to the reaction (Kr˝oger–Vink notation): OO + VO + H2 O → 2(OH)O

(3)

Static lattice simulations have been used successfully in a conventional manner to study the solution of dopants in the perovskite oxides. However more revealing has been the simulation of the proton motion in calcium zirconate, CaZrO3 [35]. This involved an ab initio calculation of the electronic structure using a DFT approach in the CASTEP code [10] coupled with a classical MD simulation. The calculations showed that the proton moved via hops between oxygen anions. Thus the process was analogous to the Gr˝otthus mechanism and there was no evidence of the migration of hydroxyl ions. As in the Gr˝otthus mechanism the reorientation and alignment of the OH group towards the next oxygen anion was a rate-determining factor. This was consistent with the experimental observation that distorted perovskites have a lower conductivity than cubic perovskites.

Fast ion conductors

4.

1913

Conclusions

Fast ion conduction has provided challenges to conventional theories in terms of both defect structure and the mechanisms of ion migration. Computer simulation methods have provided a means of attacking these problems and have played a key role in developing the current level of understanding of the materials. Particularly successful have been the MD simulations. It is clear that there will be further developments with the expected increases in computing power. For example, this will allow better representation of the interatomic potentials with the greater use of ab initio methods, the use of larger simulation boxes and, in the case of MD calculations, longer run times. In terms of new areas that will develop it is reasonable to expect more work in nanocrystalline fast ion conductors. Enhanced diffusion is a feature of in nanocrystals and recent experimental work has shown that conductivity is increased when particle size or film of an ionic material is in the nanometre range [36]. It should soon be possible to explicitly simulate a nanocrystal of a few nanometre in diameter on the computer and explore the morphology and ion transport directly. Finally, it is worth mentioning developments in methodology. The common approaches used for fast ion conductors have been to construct a structure for the simulation based on crystallographic information and then introduce defects. Recently alternative approaches have been developed where the simulation involves some kind of structural evolution [37]. For example, in the study of films on substrates the technique can involve a MD simulation in which the system is melted or amorphised followed by a recrystallization. In these methods the system will evolve its own natural morphology and defects, such as grain boundaries, dislocations and point defects, are generated naturally in the simulation and depend solely on the interatomic potentials, rather than the intuitive input of the researcher. This can be very useful in exploring complex structures, such as the role of the substrate on film morphology. The methods should find applications in the study of fast ion conductors, particularly for nanocrystals and thin films, where the materials are being employed in technological applications.

References [1] M. Faraday, “Experimental researches in electricity,” J.M. Dent, London; Faraday’s Diaries 1820–1862, G. Bell, London, entries for 21st February, 1833 and 19th February, 1835, 1939. [2] S. Chandra, “Superionic solids,” North-Holland, Amsterdam, 1981. [3] A.M. Stoneham (ed.), “Ionic solids at high temperatures,” World Scientific, Singapore, 1989.

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[4] A. Laskar and S. Chandra (eds.), “Superionic solids and solid electrolytes,” Academic Press, New York, 1990. [5] S. Huberman, Phys. Rev. Letts., 32, 1000, 1974. [6] J. Corish, “Point defects in simple ionic solutions,” Article 6.4, this volume. [7] M.J. Norgett, Harwell Report AERE-R 7650, AEA Technology, Harwell, Didcot, OX11 0RA. U.K., 1974. [8] M. Leslie, SERC Daresbury Laboratory Report DL-SCI-TM3IT, CCLRC Daresbury Laboratory, Warrington WA4 4AD, U.K., 1982. [9] J.D. Gale, General Utility Lattice Programme, Imperial College, London, U.K., 1992. [10] M.D. Segall, P.J.D. Lindan, M.J. Probert C.J. Pickard, P.J. Hasnip, S.J. Clark, and M.C. Payne, J. Phys. Condens. Matter, 14, 2717, 2002. [11] M.P. Allen and D.J. Tyldesley, “Computer simulation of liquids,” OUP, Oxford, 1987. [12] S. Yip, this Volume, Chapter 6.5, 2004. [13] W. Smith and T.R. Forrester, J. Mol. Graph., 14, 136, 1996. [14] C.R.A. Catlow, J. Chem. Soc. Faraday Trans., 86, 1167, 1990. [15] A.V. Chadwick, in “Diffusion in materials – unsolved problems,” G. Murch (ed.), Transtech, Zurich, 1992. [16] A.B. Lidiard, Handbuch der Physik, XX, 157, 1957. [17] J. Corish and P.W.M. Jacobs, In: M.W. Roberts and J.M. Thomas (eds.) Surface and Defect Properties of Solids, The Chemical Society, London, vol. 2, p. 184, 1973. [18] C.R.A. Catlow, Comments in Solid State Phys., 9, 157, 1980. [19] A.V. Chadwick, Solid State Ionics, 8, 209, 1983. [20] M.J. Gillan, In: A.M. Stoneham (ed.), “Ionic Solids at High Temperatures,” World Scientific, Singapore, p. 169, 1989. [21] D.A. Keen, J. Phys.: Condens. Matter, 14, R819, 2002. [22] A. Azimi, V.M. Carr, and A.V. Chadwick et al., J. Phys. Chem. Solids, 45, 23, 1984. [23] A.R. Allnatt, A.V. Chadwick, and P.W.M. Jacobs, Proc. Roy. Soc., A410, 385, 1987. [24] M.J. Castiglione and P.A. Madden, J. Phys. Condens. Matter, 13, 9963, 2001. [25] J.M. Reau and J. Grannec, In: P. Hagenm˝uller (ed.), Inorganic Solid Fluorides, Academic Press, New York, p. 423, 1985. [26] G.E. Murch, C.R.A. Catlow, and A.D. Murray, Solid State Ionics, 18–19, 196, 1986. [27] J.V.L. Beckers, K.J. van der Bent, and S.W. de Leeuw, Solid State Ionics, 133, 217, 2000. [28] C.A. Angell, Solid State Ionics, 9–10, 3, 1983. [29] C.A. Angell, Solid State Ionics, 18–19, 72, 1986. [30] J.L. Soucquet, Solid State Ionics, 28–30, 693, 1988. [31] F.M. Gray, “Solid polymer electrolytes,” VCH, New York, 1991. [32] C.R.A. Catlow, A.V. Chadwick, and G. Morrison, Radia. Eff. Defects Solids, 156, 331, 2001. [33] A.N. Fitch, Mater. Sci. Forum, 39, 113, 1986. [34] H. Iwahara, H. Uchida, and K. Tanaka, Solid State Ionics, 9–10, 1021, 1983. [35] M.S. Islam, R.A. Davies, and J.D. Gale, Chem. Mater., 13, 2049, 2001. [36] N. Sata, K. Eberl, and K. Eberman et al., Nature, 408, 946, 2000. [37] D.C. Sayle and R.L. Johnston, Curr. Opin. Solid State Mater. Sci., 7, 3, 2003.

6.6 DEFECTS AND ION MIGRATION IN COMPLEX OXIDES M. Saiful Islam Chemistry Division, SBMS, University of Surrey, Guildford GU2 7XH, UK

1.

Introduction

Ionic or mixed conductivity in complex ternary oxides has attracted considerable attention owing to both the range of applications (e.g., fuel cells, oxygen generators, oxidation catalysts) and the fundamental fascination of fast oxygen transport in solid state ionics [1, 2]. In particular, the ABO3 perovskite structure has been dubbed an “inorganic chameleon” since it displays a rich diversity of chemical compositions and properties. For instance, the mixed conductor La1−x Srx MnO3 finds use as the cathode material in solid oxide fuel cells (SOFCs) and also exhibits colossal magnetoresistance (CMR), whereas Sr/Mg doped LaGaO3 shows superior oxygen ion conductivity relative to the conventional zirconia-based electrolyte at moderate temperatures. A range of perovskite-structured ceramics, particularly cerates (ACeO3 ) and zirconates (AZrO3 ), also exhibit proton conductivity with potential fuel cell and sensor applications. It has become increasingly clear that the investigation of defect phenomena and atomistic diffusion mechanisms in these complex oxides underpins both the fundamental understanding of macroscopic behaviour and the ability to predict their transport parameters. Computer modelling techniques are now well established tools in this field of solid state ionics, and have been applied successfully to studies of structures and dynamics of solids at the atomic level. A major theme of modelling work has been the strong interaction with experimental studies, which is evolving in the direction of increasingly complex systems. In general, three main classes of technique have been employed in the study of complex oxide materials: atomistic (static lattice) based on energy minimisation, molecular dynamics (MD) and quantum mechanical (ab initio) methods. Our focus here is to highlight the major findings of recent 1915 S. Yip (ed.), Handbook of Materials Modeling, 1915–1924. c 2005 Springer. Printed in the Netherlands.

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M.S. Islam

modelling studies since detailed descriptions of these computational methods are presented in Chapter 1 (Cohen), Chapter 2 (Gale; Catlow) and Chapter 6 (Corish) of this Volume. This review addresses recent trends in computational studies of the defect, dopant and ionic transport properties of topical perovskite oxide materials. In particular, we highlight contemporary work on different oxygen ion and proton-conducting perovskites (such as LaGaO3 and CaZrO3 ) to illustrate part of the wide scope of information that can be obtained.

2.

Dopants in LaMO3 Perovskites

The series of compounds based on LaMO3 (where, for example, M = Mn, Co, Ga) are some of the most fascinating members of the perovskite family, due to their applications in SOFCs, ceramic membranes and heterogeneous catalysis [1, 3, 4]. The addition of aliovalent cation dopants is crucial to the ionic (or mixed) conductivity in these oxides. These materials are typically acceptor-doped with divalent ions at the La3+ site, resulting in extrinsic oxygen vacancies at low vapour pressures. Considering Sr2+ substitution of La3+ as an example, this doping process can be represented by the following defect reaction: 1 × 1 •• 1 SrO + La× La + 2 OO = SrLa + 2 VO + 2 La2 O3

(1)

where, in Kroger–Vink notation, SrLa signifies a dopant substitutional and V•• O an oxygen vacancy. Atomistic simulations can been used to evaluate the energies of this “solution” reaction by combining appropriate defect and lattice energy terms. In this way, the modelling approach provides a useful systematic guide to the relative energies for different dopant species at the same site. The starting point is the modelling of the ABO3 perovskite structure, which is built upon a framework of corner-linked BO6 octahedra with the A cation in a 12-coordinate site; the orthorhombic phase can be considered as due purely to tilts of these octahedra from the ideal cubic configuration (shown in Fig. 1). The interatomic potential parameters were derived by empirical procedures (as discussed by Gale in Chapter 2) using their observed structures and crystal properties; these energy minimization methods produce good agreement between experimental and simulated structures [5], which provides a reliable starting point for the defect calculations. Detailed studies have calculated solution energies for a series of alkalineearth metal ions in the LaMO3 materials (M = Mn,Co,Ga). The defect calculations are based on well established Mott–Littleton methodology [6] embodied in the GULP code [7]. Figure 2 reveals that the lowest energy values are predicted for Sr and Ca at the La site. The favourable incorporation of these ions will ‘therefore’ enhance transport properties owing to the increase in the

Defects and ion migration in complex oxides

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(a)

(b)

Figure 1. rhombic.

Perovskite structure showing corner-linked MO6 octahedra: (a) cubic (b) ortho-

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5.0

LaGaO3

Solution energy (eV/dopant)

LaMnO3 4.0

LaCoO3 Ba

Mg

3.0

2.0

1.0 Ca 0.0

Sr 0.8

0.9

1.0

1.1

1.2

1.3

1.4

1.5

Dopant radius (Å)

Figure 2. Calculated energies of solution as a function of dopant ion radius for alkaline-earth cations substituting on the La site in LaMO3 perovskites.

concentration of mobile defects. These results accord well with experimental work in which Sr is the dopant commonly used to generate ionic (or mixed) conductivity in these perovskites, while Ca is often used to generate mixed valent Mn3+ /Mn4+ in the magnetoresistive (CMR) manganates. It is also apparent from Fig. 2 that a degree of correlation is found between the calculated solution energy and the size of the alkaline-earth dopant with minima near the radius of the host La3+ . However, ion size is not the sole factor as previous studies show that the solution energy for the alkali metal dopants with similar ionic radii are appreciably endothermic, in line with their observed low solubility.

3.

Oxygen Defect Migration in LaGaO3

The LaGaO3 material doped with Sr and Mg has attracted growing attention as a solid electrolyte competitive with Y/ZrO2 (and Gd/CeO2 ) due to its extremely high oxygen ion conductivity at lower operating temperatures [1, 3]. As discussed by Chadwick in Chapter 6, simulation methods have been able to investigate fundamental mechanistic problems of ion migration in complex ionic conductors.

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As part of an extensive survey of perovskite oxides, we first probed the energy profiles for oxygen ion migration by calculating the defect energy of the migrating ion along possible diffusion paths, and allowing relaxation of the lattice at each position. It should be noted that interstitial formation and migration are calculated to be highly unfavourable as expected for the closely packed ABO3 perovskite lattice. The simulations ‘therefore’ confirm the migration of oxygen ion vacancies as the lowest energy path, as well as predicting that any oxygen hyperstoichiometry will not involve interstitial defects. For the topical LaGaO3 oxygen ion conductor, the simulations find that the calculated migration energy (0.73 eV) is in good agreement with experimental activation energies of about 0.7 eV from high temperature dc conductivity [8] and 0.79 eV from SIMS data [9]. In terms of the precise diffusion mechanism, however, it has not been clear from experiment whether the migrating ion takes a direct linear path along the edge of the MO6 octahedron into a neighbouring vacancy. An important modelling result is that a small deviation from the direct path for vacancy migration is revealed, illustrated schematically in Fig. 3. The calculations ‘therefore’ indicate a curved route around the octahedron edge with the “saddle-point” away from the adjacent B site cation. Indeed, a recent neutron diffraction and scattering studies of doped LaGaO3 [10] provide evidence for our predicted curved pathway. In the saddle-point configuration, the migrating ion must pass through the opening of a triangle defined by two A site (La) ions and one B site (Ga) ion. The simulation approach is able to treat ionic polarizability and lattice relaxation, generating valuable information on local ion movements. From our analysis we find significant displacements (0.1 Å) of these cations away from the mobile oxygen ion. These results emphasise that neglecting lattice relaxation effects at the saddle-point may be a serious flaw in previous ion size approaches based on a rigid hard-sphere model, in which the ‘critical radius’ of the opening is derived. It is worth mentioning that in addition to dopant ion incorporation these modelling techniques have been used to investigate dopant–vacancy association where we find a minimum in the binding energy occurs for Sr2+ on La3+ in LaGaO3 , which would be beneficial to oxygen ion conductivity. Recent studies have also examined larger complex clusters (xMgGa (x/2)V•• O ) within 2D and 3D structures, which may be related to possible “nano-domain” formation at higher dopant regimes [11]. This work on nano-clusters is currently being extended to other materials.

4.

Proton Transport in AZrO3

In addition to oxygen ion conduction, perovskite oxides have received considerable attention as high temperature proton conductors with

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M

O

M

O

M

O

M

Figure 3.

Curved path for oxygen vacancy migration between adjacent anion sites in LaMO3 .

promising use in fuel cell and hydrogen sensor technologies [12–14]. Most attention has focused on A2+ B4+ O3 perovskites, particularly ACeO3 and AZrO3 . An important example is the development of a sensor for hydrogen in molten metal based upon doped CaZrO3 as the proton-conducting electrolyte. The CaZrO3 material is typically acceptor-doped with trivalent ions (e.g., In3+ ) at the Zr4+ site. When these perovskite oxides are exposed to water vapour, the oxygen vacancies are replaced by hydroxyl groups, described as follows: x • H2 O(g) + V•• O + OO → 2OHO

(2)

In an attempt to gain further insight into the mechanistic features of proton diffusion, we have focused on the orthorhombic phase of CaZrO3 comprised of O(1) and O(2) inequivalent oxygen sites; this study extends earlier simulation work on ideal cubic perovskites [15]. Here the DFT-pseudopotential approach has been utilised to perform ab initio dynamics calculations [16] using the

Defects and ion migration in complex oxides (i)

1921

(ii)

H

O

(iii)

Zr

Figure 4. Sequence of three snapshots from ab initio MD simulations showing inter-octahedra proton hopping in orthorhombic CaZrO3 . (The Ca ions are omitted for clarity).

CASTEP code [17], which essentially combines the solution of the electronic structure with classical molecular dynamics (MD) for the nuclei. Graphical analysis of the evolution of the system with time shows proton hopping events during the simulation run. Figure 4 presents “snapshots” of one of these proton hops between neighbouring O(1) oxygen ions illustrating both initial and barrier (transition) states. This confirms that proton conduction occurs via a simple transfer of a proton from one oxygen ion to the next (Gr¨otthuss mechanism). These simulations provide no evidence for the migration of hydroxyl ions (“vehicle” mechanism) or the existence of “free protons”. We also find rapid rotational and stretching motion of the O–H group, which allows the reorientation of the proton towards the next oxygen ion before the transfer process. Interestingly, our simulations reveal predominantly inter-octahedra proton hopping, rather than within octahedra, which is influenced by the [ZrO6 ] tilting within the orthorhombic structure of CaZrO3 . This work is consistent with the experimental observation that proton mobilities are lower in perovskite structures deviating strongly from cubic symmetry.

5.

Proton–Dopant Association in AZrO3

Proton conducting oxides are typically doped with aliovalent ions. However, there has been some debate as to whether there is any significant interaction between the dopant ion and the protonic defect (hydroxyl ion at oxygen site), which may lead to proton “trapping”. In an attempt to probe this question of proton-dopant association, DFT-based methods [17] have been

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used to examine defect pairs (OH•O MZr ) comprised of a hydroxyl ion and a neighbouring dopant substitutional (shown in Fig. 5). Attention was focused on three commonly used dopants in CaZrO3 , namely Sc3+ , Ga3+ and In3+ . The resulting binding energies (with respect to the two isolated defects) are in the range −0.2 to −0.3 eV which suggests that all the hydroxyl-dopant pairs are favourable configurations. Although there are no experimental data on CaZrO3 for direct comparison, the calculated values are in accord with proton “trapping” energies of about −0.2 and −0.4 eV for Sc-doped SrZrO3 and Yb-doped SrCeO3 respectively, derived from muon spin relaxation (µSR) and quasi-elastic neutron scattering (QENS) experiments [18]. These studies postulate that in the course of their diffusion, protons are temporarily trapped at single dopant ions. It is noted, however, that defect pairs do not necessarily preclude the presence of isolated protons and dopant ions, since clusters will be in equilibrium with single defects. This picture can be viewed as analogous

Figure 5. Dopant-OH pair at nearest-neighbour sites in the [Zr–O] plane in CaZrO3 .

Defects and ion migration in complex oxides

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to oxygen ion conductivity in fluorite oxides and the well-known importance of dopant–vacancy interactions [19].

6.

Concluding Remarks

Computational techniques now play an important role in contemporary studies of complex oxides such as perovskite-structured ionic conductors. Such modelling tools, acting as powerful “computational microscopes,” have been used here to provide deeper fundamental insight as to the defect and ion transport properties of some topical examples of complex oxides. Our simulations, using both energy minimization and quantum mechanical methods, have aimed to guide and stimulate further experimental work on these perovskite materials, relevant to their applications in solid oxide fuel cells, oxygen separation membranes and partial oxidation reactors. Future developments in the modelling of ternary oxides are likely to encompass the atomistic simulation of interfaces and nanocrystals, the greater use of shell model MD over longer time-scales, and the extension of quantum mechanical techniques to more complex oxide systems, with increasing emphasis on predictive calculations. These developments will be assisted by the constant growth in computer power, and will also draw on the strong interaction with complementary experimental techniques.

Acknowledgments The author is grateful for valuable discussions with C.R.A. Catlow, A.V. Chadwick, J.D. Gale, P.R. Slater and J.R. Tolchard. The work has been supported by the EPSRC and the Royal Society.

References [1] [2] [3] [4] [5] [6]

B.C.H. Steele, Solid State Ion., 134, 3, 2000. P. Knauth and H.L. Tuller, J. Am. Ceram. Soc., 85, 1654, 2002. T. Norby, J. Mater. Chem., 11, 11, 2001. S.W. Tao and J.T.S. Irvine, Chem. Record, 4, 83, 2004. M.S. Islam, J. Mater. Chem., 10, 1027, 2000. C.R.A. Catlow (ed.), Computer Modelling in Inorganic Crystallography, Academic Press, London, 1997. [7] J.D. Gale, J. Chem. Soc., Faraday Trans., 93, 629, 1997. [8] K. Huang, R.S. Tichy, and J.B. Goodenough, J. Am. Ceram. Soc., 81, 2565, 1998. [9] T. Ishihara, J.A. Kilner, M. Honda, and T. Takita, J. Am. Chem Soc., 119, 2747, 1997.

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[10] M. Yashima, K. Nomura, H. Kageyama, Y. Miyazaki, N. Chitose, and K. Adachi, Chem. Phys. Lett., 380, 391, 2003. [11] M.S. Islam and R.A. Davies, J. Mater. Chem., 14, 86, 2004. [12] H. Iwahara, H. Matsumoto, and K. Takeuchi, Solid State Ionics, 136–137, 133, 2000. [13] S.M. Haile, Acta Materalia, 51, 5981, 2003. [14] K.D. Kreuer, Ann. Rev. Mater. Res., 33, 333, 2003. [15] W. M¨unch, K.D. Kreuer, G. Seifert, and J. Maier, Solid State Ionics, 136–137, 183, 2000. [16] M.S. Islam, R.A. Davies, and J.D. Gale, Chem. Mater., 13, 2049, 2001. [17] M.C. Payne, M.P. Teter, D.C. Allan, T.A. Arias, and J.D. Joannopoulos, Rev. Mod. Phys., 64, 1045, 1992. [18] R. Hempelmann, M. Soetratmo, O. Hartmann, and R. Wappling, Solid State Ionics, 107, 269, 1998. [19] J.A. Kilner, Solid State Ionics, 129, 13, 2000.

6.7 INTRODUCTION: MODELING CRYSTAL INTERFACES Sidney Yip1 and Dieter Wolf2 1

Department of Nuclear Science and Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA 2 Materials Science Division, Argonne National Laboratory, Argonne, IL 60439, USA

Interfaces represent an integral part of the understanding, design, and processing of modern materials [1, 4–6]. Many phenomena and properties, ranging from electronic and optical to thermal and mechanical in nature, are known to be dominated by the presence of interfaces. The basic paradigm for understanding and controlling interfacial phenomena is the concept of structure–property correlation, well known in materials science. In the remaining sections of this chapter, we will discuss the application of this concept to grain boundaries. The reader should keep in mind that while the modeling of grain boundaries is a significant problem in and of itself, the modeling concepts and simulation methods developed and applied in this context, apply equally well to other types of interfacial materials. One way of classifying solid interfaces is shown in Fig. 1 where three types of interfacial systems are distinguished. The interfacial region (assumed to be infinite in the x–y plane) is embedded in the z direction between two perfect, semi-infinite bulk crystals. The lower and upper halves of this bicrystal generally consist of different materials, A and B. Only flat interfaces are considered in Fig. 1; however, from high-resolution electron microscopy we know that macroscopically curved interfaces are usually faceted on an atomic scale. The presence or absence of one or both of the bulk regions in Fig. 1 has a strong effect on the lattice parameters, and hence the physical properties in the interfacial region. By examining the different ways in which the interfacial region may or may not be sandwiched between the two bulk regions, three types of interfacial systems can be distinguished: (a) Bulk (or buried, internal) interfaces: Systems in which the interface region is surrounded by bulk material on both sides. Examples are interphase (for A =/ B) and grain (for A = B) boundaries. 1925 S. Yip (ed.), Handbook of Materials Modeling, 1925–1930. c 2005 Springer. Printed in the Netherlands.

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MATERIAL B

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Bulk Perfect Crystal 2

Interface Region 2

MATERIAL A

Interface Plane

z x

Interface Region 1

Bulk Perfect Crystal 1

y

Figure 1. Distinction of three types of interfacial systems. Depending on whether the system is embedded in bulk material on both sides of the interface, on only one side or not at all, we distinguish “bulk”, “epitaxial” and “thin-film” interfaces. A and B are generally different materials.

(b) Epitaxial interfaces: Systems with bulk material on only one side of the interface and a thin film on the other. For A = B, a bulk free surface is obtained. (c) Thin-film interfaces: Systems with both bulk regions removed. For A = B, a free-standing thin film, generally containing a grain boundary and bordered by free surfaces, is obtained. Strained-layer superlattices are included here for the case in which the structure of an A|B thin-film sandwich is periodic not only in the x–y plane but also in the z direction: . . . |A|B|A|B| . . . . Since no bulk embedding is left, the material near the interface would not “know” its bulk lattice parameter. Following this line of thinking one can go further and distinguish between coherent versus incoherent, or commensurate versus incommensurate interfaces (D. Wolf, in [1], Chapter 1). By grain boundaries we mean the regions between adjacent crystalline grains. Because the adjacent grains can have different shapes or orientations, atoms in the region of mismatch, the grain boundary, will be less well packed than those in the grain interior. This open structure means that the grain boundary region can act like a source or sink for defects, an easier path for atomic transport, or a more active site for mechanical deformation and even chemical

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reaction. The details of the structural openness and how the local atomic arrangements can affect the various physical properties of polycrystalline material are fundamental topics for modeling studies. In this sense the studies of crystal defects in the present chapter provide illustrations of the basic utility of materials theory and simulation that permeates throughout this volume. Many special interests in interface materials stem from their inherent inhomogeneity, i.e., the physical properties at or near an interface can differ dramatically from those of the nearby bulk material. For example, the thermal expansion, electrical resistivity or elastic response near an interface can be highly anisotropic in an otherwise isotropic material, and differ by orders of magnitude from those of the adjacent bulk regions. Typically these gradients extend over only a few atomic layers, so their experimental investigation requires techniques capable of atomic-level resolution and detection. For surfaces and thin films suitable characterization methods have been developed [2]. On the other hand, buried interfaces continue to be a major challenge, because the presence of the interfaces affects only a small fraction of the atoms (D. Seidman, in [1], Chapter 2). This inherent difficulty in the experimental investigation of buried interfaces actually presents an opportunity for atomic- and electronic-level calculations to contribute to our understanding of solid interfaces by means of simulations. Because the properties in the interfacial region are controlled by relatively few atoms, electronic- and atomic-level simulations can provide a close-up view of the most critical part of the material. The limitations of these simulations, our incomplete knowledge of electronic structure and interatomic interactions, the finite size of any simulation cell and its embedding in the surrounding material, and the finite duration of any simulation, are well known. Nonetheless, we now have the means to study the positions and movements of atoms, local stresses, etc., and relate this information to the physical properties of the material. For example, in molecular dynamics simulation it is as if one had an atomic-level camera with a field of view of about 100 Å at a speed of 1014 –1015 frames per second. The unique features of such simulations offer opportunities for a joint approach combining atomic-level experimental techniques with computer simulations. In atomistic simulations one has complete information about the model systems under study. This is a very significant advantage over experiments where either the microstructure of the sample has not been fully determined, or the phenomenon of interest cannot be measured in sufficient details. Because both structure and properties can be well characterized in simulation the results of such studies are particularly useful for establishing correlations. This feature will be illustrated repeatedly in the following sections. The layout of the remaining sections of this chapter follows the paradigm of structure-property correlation. We summarize in Chapter 6.8 the simulation concepts and methods that are relevant for the study of interfaces in

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general and grain boundaries in particular, making use of the treatments that can be found in Chapters 1 and 2. Even though grain boundaries have a finite depth, they are considered to be planar defects. As with all studies of extended defects in crystals, one begins with the structural aspects, such as classification of geometrical features in a lattice and considerations of atomic distributions and configurations, and then correlates various physical properties to the interfacial structure. From the modeling perspective, a pertinent question is which of the many possible structures that a grain boundary can have should one study. This is a nontrivial issue since in a simulation one must have a way of specifying the energy of the system in terms of the atomic coordinates, the information usually embodied in an interatomic potential, an energy function which depends on the positions of all the atoms in the system. Knowing the potential is generally not enough because border conditions for the simulation cell, which necessarily has to be finite, also must be specified. In Chapter 6.9 we will discuss static calculation of grain-boundary energy and the correlation of the results with the structure of the interface. This investigation illustrates how one can determine the stability of an interface, a procedure that can be applied to any system of atoms. To study finite-temperature behavior we will switch to molecular dynamics, in Chapter 6.10, in order to obtain the trajectories of the atoms as the system microstructure evolves in a thermal environment. We will consider results on grain growth and plastic deformation in this context. In the following section, Chapter 6.11, we go into even more fundamental details to probe an extreme form of thermal response, the crystal-to-liquid transition or melting. The question we ask is at the atomic level what are the mechanisms for a crystal lattice to collapse at a certain critical temperature. Another way to raise the issue is to ask about the thermodynamic significance of the melting point and its relation to the kinetics of melting. We will see that atomistic simulation can provide the mechanistic details to give us a fuller understanding of melting, and to address the thermodynamic and the kinetic aspects of the phenomena in a unifying manner. This kind of investigation can be extended in several directions, such as melting at a grain boundary and the connection with a related phenomenon, the crystal-to-amorphous transition or solid-state amorphization. In Chapter 6.12 we take up the elastic behavior of interface materials. Conceptually, crystals with interfaces are inherently inhomogeneous systems. We will find that their elastic behavior can be opposite to what is expected of homogeneous systems. For example, we see from a study of the elastic behavior of interfaces that the elastic moduli of a material do not necessarily soften when the material density decreases. Simulations reveal that the net elastic response of interfacial materials is the result of complex, highly nonlinear competition between interfacial structural disordering and consequent volume expansion. Volume expansion and the related increase in average

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interatomic distances usually give rise to elastic softening, whereas the structurally disordered interface region can cause either strengthening or softening [3]. The latter is readily seen via the underlying radial distribution function, as shown in Fig. 2. In spite of a decrease in the average density, indicated by the shift of the solid arrows towards larger distances, indicated by the open arrows, some atoms near an interface may be pushed closer together, causing elastic strengthening, while others are moved further apart, giving rise to a weakened elastic response. Because of anharmonicity, the former are weighed much more heavily than the latter, the net result being that some elastic moduli may actually strengthen even though the average density decreases. The final section of the chapter, Chapter 6.13, is a discussion of the role of grain boundaries in nanoscrystalline materials. Nanocrystals have emerged in recent years as materials systems with unique structures and properties. They are of interest not only for fundamental understanding, but also because they can be functionalized for exciting technological applications. From the standpoint of modeling material interfaces, nanocrystals constitute an increasingly important topic of study. Nanostructures may be viewed as the confluence of clusters of a few to tens of atoms and microstructural entities on the length scales of microns. Using atomistic simulation techniques one can probe the structural and thermodynamic features of nanocrystals and relate them to the

0.8

r 2 G (r)

0.6

0.4

0.2

0.0 0.6

0.8

1.0

1.2

1.4

1.6

r/a

Figure 2. Zero-temperature radial distribution function, G(r ), vs. distance r (in units of the lattice parameter a) for a superlattice of (001) twist grain boundaries in Cu (see Chapter 4) with six (001) planes between the interfaces [3]. The dashed lines delineate the peaks associated with the nearest, second-nearest, etc., neighbors in the fcc lattice. G(r ) is normalized such that the areas under the peaks correspond to the numbers of nearest (12), second-nearest (6), etc., neighbors; the peak centers are indicated by the open arrows. The solid arrows indicate the positions of the corresponding perfect-crystal (δ-function type) peaks. The difference between open and solid arrows signals a volume expansion of the multilayer over the perfect crystal.

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corresponding features characteristic of amorphous materials. Another aspect to be discussed is the how one can relate the behavior of nanocrystals to those of bicrystals and polycrystals. We have emphasized the modeling of crystal defects using the atomistic methods discussed in Chapter 2. It should be clear that the electronic structure methods discussed in Chapter 1 are also applicable so long as the number of atoms in the simulation are not so large as to preclude their use. There are significant connections between this chapter and the one following, Chapter 7, on microstructure, and also parts of Chapter 9 on soft matter.

Acknowledgments DW is supported by the US Department of Energy, BES Materials Sciences, under Contract W-31-109-Eng-38.

References [1] D. Wolf and S. Yip, (eds.), Materials Interfaces: Atomic-Level Structure and Properties, Chapman and Hall, London, 1992. [2] E. Meyer, S.P. Jarvis, and N.D. Spencer, “Scanning probe microscopy in materials science,” MRS Bull., 29, 443–445, 2004. [3] D. Wolf and J.F. Lutsko, Phys. Rev. Lett., 10, 1170, 1998. [4] A.P. Sutton and R.W. Balluffi, Interfaces in Crystalline Materials, Clarendon Press, Oxford, 1994. [5] D. Wolf and S. Yip, MRS Bull., 15, 21–23, 1990a. [6] D. Wolf and S. Yip, MRS Bull., 15, 23, 1990b.

6.8 ATOMISTIC METHODS FOR STRUCTURE–PROPERTY CORRELATIONS Sidney Yip Department of Nuclear Science and Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA

There is a general belief that physical properties of crystals can be classified, to a first approximation, according to its structure. The basis of this thinking is that there is close correlation between structure and the chemical bonding between atoms which in turn controls the properties [1]. Although it is not guaranteed to be always successful, this can be a good starting point toward the understanding of materials properties and behavior. In this section we discuss the use of atomistic techniques to study interfaces, primarily grain boundaries, in the context of structure–property correlation. As we will see, these methods are a subset of the multiscale techniques treated extensively in Chapters 1–4. Using grain boundary as a prototypical crystal defect, we examine how atomistic simulation techniques can be brought together to determine the physical properties of crystalline materials with well-characterized defect microstructure. This section also serves as an introduction to the subsequent sections which are concerned, in one way or another, with probing the structure and associated properties of grain boundaries. An integrated approach to establish the correlations between interfacial structure and the physical properties of grain boundaries has been proposed using intergranular fracture as a target application [2]. We adopt the same point of view here not so much to study intergranular fracture as to illustrate the capabilities of four related atomistic techniques, lattice statics (LS), lattice dynamics (LD), Monte Carlo (MC), and molecular dynamics (MD). A fundamental characteristic of all interfacial systems is that such systems are intrinsically inhomogeneous.. The presence of an interface means the

1931 S. Yip (ed.), Handbook of Materials Modeling, 1931–1951. c 2005 Springer. Printed in the Netherlands.

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immediate region surrounding the interface can have properties quite different from the bulk. It follows that any understanding of interfacial properties must explicitly take into account the local behavior of the interfacial region, for example, the local volume expansion at the boundaries or the local elastic constants. Because the interfacial region is usually only a few atomic spacing in extent, calculations of local properties necessarily involve details of displacements and forces at the molecular level. Such information is directly available from discrete-particle (atomistic) simulations in which the interfacial region is modeled as an assembly of particles interacting through specified interatomic potentials. In the case of intergranular fracture there are formidable difficulties in the experimental approach to structure–property correlations because the complexity and diversity of relevant phenomena involve the specification and determination of large numbers of parameters and variables. One can imagine such problems at every stage of an experiment, from sample preparation where one needs to control the interfacial structure, to sample characterization and measurement of details of the fracture process. On the other hand, these difficulties point to opportunities for the modeling approach. It is the inherent nature of atomistic calculations that one can specify interatomic structure and forces, follow the system evolution in dynamical details, and analyze the results with regard to various physical properties of interest. Simulation ‘therefore’ becomes a unique complement to experiment and the traditional theoretical methods of materials research. In assessing the capabilities of atomistic simulations, certain fundamental limitations of this approach also should be recognized at the outset. These are concerned with the availability of realistic interatomic potential models for the material system under study, and the finite system size and simulation duration that can be studied. The former has to be addressed in every individual study, since the significance (meaningfulness) of the results is always limited by the adequacy of the potential model adopted. With continuing research toward developing potentials for metals, ionics and semiconductors at first, and alloys and compounds more recently, and the increasing use of electronic-structure methods to produce databases for fitting and validation, the question of reliable potentials for atomistic simulation is not as critical as before. Nonetheless, the use of empirical potentials in atomistic simulation will always remain a compromise between tractability and predictive accuracy. The limitation of system size and simulation duration is an issue of computational resources. With computer power still increasing steadily, this is also becoming less serious in that what was not feasible only a year or two ago, is now within reach of current capabilities. It is important to recognize that development of ingenious boundary and initial conditions can be very effective in mitigating artifacts due to finite size and duration of simulation.

Atomistic methods for structure–property correlations

1.

1933

Linking Atomistic Techniques for Structure–Property Correlations

We begin with a brief review of the basic concepts in atomistic simulations. Consider a collection of N interacting atoms and denote their positions as {r N } ≡ (r 1 , r 2 , . . . , r N ), where r i is the position of atom i. We will call {r N }the atomic configuration or the system configuration. Suppose the system is in an initial configuration {r N }i which is not an equilibrium configuration, then the atoms will move under the effects of interatomic interactions once the system is allowed to evolve in time. At any given instant the state of the system is characterized by a potential energy, U ({r N }), which depends on the instantaneous positions of all the atoms, and a kinetic energy, K ({r N }), where {v N } is the set of atomic velocities (v 1 , v 2 , . . . , v N ). It is understood that the evolution of the N -particle system occurs as if the N atoms are actually part of a larger body, one of macroscopic dimensions, and a set of border conditions will be specified to describe the embedding of the simulation cell with N atoms in the larger medium. From a conceptual point of view three basic ingredients are required to predict the system evolution under the effects of interatomic forces. The interatomic potential has to be specified. In contrast to electronic-structure calculations, the potentials are assumed to depend only on the atom positions, with the electrons considered as an effective medium which mediates the atomatom coupling. Secondly, to account for interactions between the system and its environment border conditions have to be invoked. Such conditions may be a periodic extension of the simulation cell in three, two, or one dimension, for example. Or they may be a certain embedding of the system, such as enclosing the system in a linear elastic medium. Finally, an algorithm has to be used to determine the system response of interest, whether it is deterministic or stochastic, and whether the response is a static relaxation or dynamical evolution. The three aspects will be considered separately.

2.

Interatomic Interactions

The key to the computational efficiency of atomistic simulations lies in the description of the interaction between the atoms through a potential energy. The task of reducing the complex many-body problem of a system of interacting electrons and ions to a description involving only atomic coordinates is highly nontrivial and not unique. The successes of this approach which have been extensively documented thus far justify the belief that meaningful results can be obtained. Another practical motivation is that this is the only tractable way of investigating a large class of significant problems in materials research

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for which the use of first-principles methods is not yet feasible. The development of an empirical potential usually consists of two steps. Some analytical form that can be conceptually rationalized is first chosen for the potential energy function of the system of N atoms, U ({r N }), then the adjustable parameters in this function are determined by fitting to a database of properties, typically lattice structure and zero-temperature lattice parameters, elastic constants, the sublimation or vacancy formation energy, and others. The database used can be purely experimentally or theoretically determined, or a combination of the two. Current trend is to use only theoretical results. See Chapters 1 and 2 for extensive discussions. From a conceptual viewpoint one can discuss empirical potential models by writing the energy U ({r N }) of the system as an expansion in n, where n is the number of particles interacting with each other, U ({r N }) =

V1 (r i ) +

i

V2 (r i , r j ) +

V3 (r i , r j , r k ) + . . .

(1)

i< j
i< j

where V1 (r i ) is the one-body potential which depends only on the position of atom i, V2 (r i , r j ) is the two-body potential, and so on. The simplest possible representation of many-body interactions is the sum of two-body potentials. See the discussions on interatomic potentials in Chapters 2 and 9. For metals the two-body approximation is known to be inadequate. A more reasonable approach is to adopt a many-body potential in the following sense. One writes U ({r N }) =

i

Fi (ρi ) +

ϕ(r i j )

(2)

i< j

where Fi is the cohesive energy (attractive) contribution which depends on the electron density at atom i, ρi , and ϕ is the repulsive two-body interaction depending the separation between atoms i and j , r i j =r i −r j . See in particular Article 2.2. Much of the results on grain-boundary (GB) studies discussed in this chapter are based on the two-body potential model.

3.

Border Conditions

By its very nature an interfacial system is composed of two coupled regions, the interface and the surrounding bulk regions. Formulating proper border conditions is a compromise in trying to satisfy two requirements, (i) keeping the simulation cell size small without incurring artifacts, and (ii) treating the coupling to bulk surroundings realistically. This problem is discussed extensively in several subsequent sections in this chapter. The so-called Region I–Region II method has ben adopted in a number of calculations. In this case, the interface (GB) is embedded in two semi-infinite bulk ideal

Atomistic methods for structure–property correlations

1935

crystals, with periodic border conditions in the directions parallel to the interface. For an isolated interface, one which is embedded in two semi-infinite bulk ideal crystals, two-dimensional periodic border conditions (2-d PBC) in two directions parallel to the interface are clearly appropriate. This embedding is accomplished by surrounding the interface region, denoted as Region I in Fig. 1 by two semi-infinite bulk ideal crystals, denoted as Region II. With the same 2-d PBC applied in directions x and y to both Regions I and II, the system has no free surfaces. When thermal motions are considered in the calculation, the Region I–Region II treatment together with 2-d PBC is still appropriate. But now the periodicity enforced by the border conditions gives rise to a limitation on the dynamical processes that can be studied. As is well known in MD simulations, the system under investigation cannot propagate phonons with wavelengths greater than the dimensions of Region I. In strained-layer superlattice materials one has a periodic arrangement of interface planes in the direction of the interface-plane normal (z-direction). Such systems can be represented by periodically repeating the atom positions

REGION II

REGION I

REGION II

z x

y

Figure 1. Schematic of simulation cell showing the simulation region (Region I) and the border regions (Region II). The grain-boundary is a planar interface which is infinite in the x- and y-directions by virtue of the periodic border conditions. The border condition in the z-direction, normal to the interfacial plane, depends on the specification of Region II.

1936

S. Yip

in Region I in the direction normal to the interface, and by eliminating Region II. Instead of embedding a single interface in semi-infinite ideal crystals, the atoms near the surface of Region I are now surrounded by their own periodic images. The result is a three-dimensionally periodic arrangement of atoms (3-d PBC) in which the computational unit cell, Region I, now contains two identical interface planes. For studying an isolated interface, 3-d PBC have the undesirable feature that the two interface planes in the simulation cell can interact so that their mutual influence cannot be easily separated from the behavior of a single interface. A method for treating the z-borders at finite temperatures in a lessconstraining manner has been proposed [3]. Unlike a 3-d PBC this approach gives a simulation cell containing a single interface, and in contrast to the condition of a fixed z-border, it accommodates dimensional changes normal to the interface as well as translational motions parallel to the interface plane. This method also makes use of the Region I–Region II configuration shown in Fig. 1, where in Region I the particles are treated explicitly and Region II consists of two semi-infinite blocks of atoms held fixed at their ideal-crystal positions. The novel feature is that the rigid blocks are allowed to move by translations parallel to the interface plane, such motions being determined by the force exerted on the blocks across the Region I–Region II border. The blocks are also allowed translations in the z-direction; these movements are treated separately from the parallel translations and are governed by the pressure exerted on the blocks by Region I. At zero temperature and for finite temperature simulations, other borders have been used for different specific applications. For the simulation of equilibrium segregation, a 2-d PBC three-region simulation cell was used for the purpose of allowing the solute concentration at the interface to be calculated while keeping the solute concentration in the surrounding bulk region fixed (see below). For the simulation of crack-tip systems, different borders involving 1-d PBC which allow the external stress stress to be transmitted to the simulation cell have been proposed.

4.

Lattice Statics

The atomistic methods we will link together are all concerned with a common model system. We will briefly examine the physical basis of each method, indicate their complementarity, and consider how they can be used synergistically. The method of lattice statics enables one to determine the zero-temperature relaxed structure of the simulation system by minimization of the potential energy U ({r N }). It is widely used in problems dealing with the low-temperature structure and energetics of defects in crystals [4].

Atomistic methods for structure–property correlations

1937

The basis of the method is the expansion of U about a certain configuration {r oN }, U ({r N }) = U ({r oN }) +

∇ ri U (r − r oi )

i

1 ∇ ri ∇ r j U o :(r i − r oi )(r i − r oj ) + . . . + 2 i, j

(3)

where terms beyond the quadratic order in the displacement are notshown. To determine the equilibrium (relaxed) configuration, one can solve the equation given by the requirement that the force on each atom must vanish,

F i ({r oN }) = −∇ ri U ({r N })U ({r N })o = 0 which is the second term in Eq. (3). In lattice-statics calculation energy minimization is carried out by moving each atom in the direction of the force acting on it by a small amount. How small this should be is governed by the force constants, the third term in Eq. (3), or in simpler schemes, it is chosen arbitrarily. This process of relaxation is continued until all the forces are reduced below some prescribed value, then the system considered fully relaxed. The configuration {r oN } and the corresponding energy U ({r oN })thus obtained are the equilibrium structure and energy of the system at zero temperature. When the system contains an interface, it is not sufficient to relax only the atoms. The border conditions may need adjustment in response to the atomic relaxations at the interface, otherwise a residual strain can build up in the interfacial region.

5.

Lattice Dynamics

Although not generally considered as simulation in the sense of moving particles, this widely used method for calculating thermal properties of crystals can be computationally quite efficient [5]. In its simplest form, the harmonic approximation, the approach is most useful at low temperatures. However, with various modifications, known as quasiharmonic approximations, one can obtain useful results up to temperatures approaching the melting point in some cases [6]. The basis of the method is the expansion given in Eq. (3). With {r oN } taken to be the equilibrium configuration, the first term in Eq. (3) is the cohesive energy of the crystal and the second term vanishes. The harmonic approximation constitutes ignoring all terms cubic in the displacement and higher. The crystal lattice is thus treated as a system of coupled oscillators with force constants derived from the second derivatives of the potential energy. The combined use of lattice statics and lattice dynamics is quite clear. Starting with

1938

S. Yip

a model system with an initial configuration, one performs first energy minimization to obtain the relaxed configuration {r oN }and the energy Uo . This then provides the input structure for lattice dynamics calculation which involves the diagonalization of the dynamical matrix formed from the coefficients of the displacements in Eq. (3), giving the frequencies and eigenvectors of the various vibrational modes in the system. All the thermal properties of a harmonic solid can be calculated from the partition function once the normal modes are determined. There is some degree of freedom in choosing the unit cell for the formulation of the dynamical matrix; however, it is intrinsic in the method that the system be periodic.

6.

Molecular Dynamics

This method is the realization of the classical-dynamics description of a model system through the Newtonian equations of motion [7], d 2r i (t) = −∇ ri U ({r N (t)}), i = 1, . . . , N (4) dt 2 where mi the mass of atom i. Given the initial configuration of the atoms and the border conditions, these equations are integrated numerically to give the positions of the atoms at later times in incremental steps. The basic output of molecular dynamics (MD) is a set of particle trajectories, {r N (t)} = (r 1 (t), r 2 (t), . . . , r N (t)), the complete description of the model system as formulated in classical mechanics. Knowing how the system evolves in time one can proceed to determine all physical properties, equilibrium as well as dynamical, of interest. In molecular dynamics all the particles are displaced from one time step to the next in accordance with Eq. (3). Each particle therefore has an instantaneous velocity and kinetic energy. One can define an instantaneous temperature for the system as being proportional to the total kinetic energy. This quantity will fluctuate in time as the particles move through regions of different potential interaction. It is through these fluctuations that entropic effects into the simulation. Because of this property, MD is valid for classical systems at any temperature. In practice, the validity of the MD approach is limited to temperatures near or above the Debye temperature solid because it neglects the zero-point vibration effects of quantum mechanical nature. This neglect is not very serious since one can modify the MD procedure to take these effects into account. To the extent that the interatomic potential used holds for all interparticle separations, the simulation is valid for arbitrary deformation. It also follows that one can simulate the system behavior at any temperature desired, as well as the system response to a temperature change such as in a phase transition. mi

Atomistic methods for structure–property correlations

1939

Relative to lattice statics, MD can be regarded as a method for determining how a model system which is relaxed at zero temperature behaves at finite temperature and external stress. The effects of external stress can be treated either through the border conditions or modifications of Eq. (3) by introducing an appropriate Lagrangian [8]. By ‘behavior’ we mean here both the equilibrium properties such as thermal expansion, and the mechanical responses such as the elastic constants. Relative to lattice dynamics, MD provides a means to study anharmonic effects associated with large amplitudes of atomic displacements at elevated temperatures, arising from thermal activation or external stress. It is quite feasible for MD to verify the results of both lattice statics and lattice dynamics. The version of MD discussed here is based on classical mechanics, extensions to quantum MD are discussed in Chapter 2.

7.

Monte Carlo

This is a companion atomistic simulation method to MD in that both require the same input of potential energy, initial atomic configuration of the system, and border conditions [Binder, 1979]. In contrast to MD, particle displacements in MC are chosen by sampling from a prescribed probability distribution function. In the 3N -configurational space defined by the positions of the atoms, the system is represented by a phase point at any step during the simulation. As the system evolves, this point moves through the configurational space following a certain trajectory. Unlike MD, there is no velocity involved in the simulation. The system temperature is therefore predetermined by its appearance in the probability distribution and does not change for the entire simulation. Whereas the particle trajectory in MD is determined by Hamiltonian or Newtonian dynamics, the trajectory in MC is determined by stochastic dynamics. Stochastic simulations using a sequence of random (actually only pseudo-random) numbers, are sometimes also called Monte Carlo. In our usage, we will mean the method of simulation at finite temperature where the assignment of atom displacement is based on a procedure known as the Metropolis method, or any of its variations involving the concept of importance sampling. The Metropolis procedure is well-known for sampling the canonical distribution,

−U ({r N }) f ({r N }) ∝ exp kB T

(5)

where kB is the Boltzmann’s constant and T the system temperature. The result of a Monte Carlo simulation is a sequence of Ns particle configurations{r N } j , j = 1, . . . , Ns , where Ns specifies how many configurations one wishes to sample. This is the output corresponding to the particle trajectories, {r N (tk )},

1940

S. Yip

produced by MD, where k = 1, . . . , Nt is the time-step index, Nt being the number of time steps one wishes to simulate. Since all the equilibrium properties of interest are calculated as ensemble averages over an appropriate distribution of particle positions, they can be obtained using either MC or MD. The assumption is that the system is ergodic, for which ensemble and time averages are in principle equivalent. In practice this requires that the configuration steps and time steps are both large enough. It is generally believed that MD and MC, when both are properly carried out, will give the same description of the equilibrium properties of the model system, although it is quite rare that this correspondence is explicitly demonstrated in any study [20]. For dynamical properties, the two methods are not expected to give the same results; the time-dependent response given by the MD trajectories is regarded as the physically meaningful one in the sense of classical dynamics. On the other hand, MD is intrinsically bound to the microscopic time scale determined by the interatomic forces in condensed matter, typically a fraction of a picosecond. In the MC approach a step in the sequence is governed by the transition probability, which can be formulated to describe any kind of particle displacement of interest. Thus, the stochastic method is more flexible in terms of the time step of simulation; it can be used to treat kinetic phenomena which occur on longer time scales than the natural time scales of MD, such as impurity or solute segregation at an interface. From the standpoint of computational efficiency, one can compare the four techniques discussed in terms of the number of times the system energy and interatomic forces are evaluated in each method. In this respect, lattice dynamics is the most efficient since the dynamical matrix has to be evaluated only once. In lattice statics, energy minimization is achieved typically in a few hundred iterations or less. In MD, the force calculation is made every time step unless some bookkeeping device is introduced to reduce the frequency of updating the system configuration. Roughly speaking, the efficiency of MC is comparable to MD in calculations of equilibrium properties. Where MC can have a significant advantage is in the study of ‘slow’ timedependent phenomena; it can be the only viable means of atomistic simulation in special cases. Besides CPU-time efficiency, one should also consider storage requirements in comparing the different techniques. Here lattice dynamics is the least efficient since, for a system of N particles, it requires the storage of the dynamical matrix of order 3N × 3N . In practice this has limited LD to relatively small systems, containing typically 1000 atoms. Moreover, the CPU required for matrix diagonalization increases approximately as N 3, rendering the method inefficient for large systems. By contrast, MC simulation requires the least storage in that unlike MD and LS the interatomic forces are not required. Storage requirement for MC, LS and MD increases linearly with the number of particles.

Atomistic methods for structure–property correlations

1941

It should be evident that each of the four methods discussed provides unique capabilities for the study of structure–property correlations in interface materials. We have shown in Fig. 2 how they can be linked together synergistically. There are three levels of correlations. Beginning with a given interface geometry, specified by five macroscopic degrees of freedom, one prepares the initial atomic configuration, un-relaxed positions of all the atoms, by rotating two semi-infinite perfect-crystal blocks with respect to each other by an angle θ about the normal to the interface plane, n. ˆ At the second level, lattice statics methods are used to determine the relaxed structure and energy of the interfacial system, including the three microscopic degrees of freedom associated with translations parallel and perpendicular to the interface plane. Lastly, the relaxed structure is passed on to the third level at which finite temperature properties are investigated by means of LD, MD, or MC, including the effects of externally applied stress. Suppose this approach is applied to intergranular fracture. LS methods are well suited to explore zero-temperature correlation between the interface

INTERFACE GEOMETRY

â n1, â n2, θ; a 0(1)/a 0(2) CRYSTAL STRUCTURE(S)

PLANAR UNIT CELL

INTERATOMIC POTENTIAL(S)

LATTICE STATICS (LS)

2-d or 3-d PBCs

RELAXED STRUCTURE and ENERGY at T⫽0

LATTICE DYNAMICS (LD)

MOLECULAR DYNAMICS (MD)

MONTE CARLO (MC)

QUASI-HARMONIC and

ANHARMONIC EFFECTS

EQILIBRIUM PROPERTIES

HARMONIC ELASTIC and

'FAST' DYNAMICAL

'SLOW' KINETIC

THERMODYNAMIC PROPERTIES

PHENOMENA

PROCESSES

Figure 2. Methods for atomistic modeling of grain-boundary systems showing their interrelated roles in the study of structure-property correlations [2].

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geometry and the corresponding energy. Insights gained can be an invaluable guide to probe the effects of external stress and temperature, to unravel the correlation between the interface structure and its fracture properties. LD calculations are best suited to obtain local elastic constants which play a central role in the mechanical response of a stressed system. MC simulations constitute the most powerful approach to the determination of the equilibrium distribution of solutes in the interface region which are known to control the embrittlement behavior. The actual dynamical process of crack extension can be simulated by MD, thus leading to quantifiable insights that connects interfacial geometry and chemistry with fracture resistance.

8.

Structural Disorder at an Interface

We now consider basic atomic structural and mobility properties that have been studied by atomistic simulations. The most common form of structural disorder in a crystal lattice is caused by the thermal movements of the atoms, usually leading to thermal expansion. This homogeneous type of disorder, and the consequent volume increase, originate in the anharmonicity of the interatomic interactions. In the presence of planar defects, structural disorder occurs even at zero temperature. The effect is now localized, usually also in the form of volume expansion. Thus, volume expansion may be viewed as a measure of structural disorder in the system, homogeneous and inhomogeneous. The radial distribution function, r 2 g(r), is a useful tool for characterizing the effects of structural disorder. One can look for two effects of thermal disorder here. The δ-function like zero-temperature peaks associated with the shells of nearest, second nearest, and more distant neighbors are broadened, and because of the volume expansion the peak centers are shifted toward larger distances. As illustrated in Fig. 3, the structural disorder at a solid interface gives rise to the same two effects even at zero temperature. Figures 3(a) and 3(b) show the radial distribution function for atoms in the planes nearest and next-nearest to a high-angle twist boundary on the (100) plane of an fcc metal, respectively [9]. In the plane closest to the GB, the perfect crystal δ-function peaks have been replaced by a broad distribution, whereas in the secondclosest plane, the ideal-crystal peaks are largely recovered. This illustrates the highly localized nature of the structural disorder at the interface. Figure 3(a) also illustrates the cause for the local expansion at the GB. The distances to the left of the arrows represent atoms showed more closely together than in the perfect crystal; because of anharmonicity these atoms repel each other more strongly, thereby giving rise to a local expansion. A commonly used measure to characterize atomic mobility is the atomic mean-squared displacement (MSD). MSD is essentially time independent in

Atomistic methods for structure–property correlations (a)

0.6 Au (EAM) (100) θ ⫽ 43.60˚ (Σ 29) PLANE 1

0.5 0.4 r 2g(r)

1943

0.3 0.2 0.1 0.0 0.8

1.0

1.2

1.4

1.6

1.4

1.6

r/a

(b)

0.6 Au (EAM) (100) θ ⫽ 43.60˚ (Σ 29) PLANE 2

0.5

r 2g(r)

0.4 0.3 0.2 0.1 0.0

0.8

1.0

1.2 r/a

Figure 3. Radial distribution function, r 2 g(r ), for the two planes nearest to a (100)θ = 43.60◦ (29) grain boundary as described by an EAM potential for Au. Arrows indicate the corresponding perfect-crystal peak positions. While atoms in the plane nearest to the interface (a) are very strongly affected by the presence of the interface, the atoms in the second-nearest plane (b) have an environment much closer to that of an ideal crystal.

the solid, while in the liquid it increases approximately linearly with time, with a proportionality constant that is a direct measure of the liquid diffusion constant. To study the structural disorder at grain boundaries MSD is less useful than the magnitude of the static structure factor, S(k). We define S 2 (k) as 2 S (k) ≡ S(k) = 2

N 1 cos(k · r i ) N i=1

2

N 1 + sin(k · r i ) N i=1

2

(6)

where r i is the position of atom i. Because grain boundary is a planar defect, it is more appropriate to focus on the spatial ordering along the normal to the interface plane. We divide the simulation cell into slices along the z-direction

1944

S. Yip

(perpendicular to the interface plane), with each slice chosen to contain a single atomic plane in the crystal, and define the planar structure factor, S 2p (k), using in Eq. (6) only those atoms lying in a given lattice plane. For an ideal crystal at zero temperature, S 2p (k)is unity for any wave vector, k, which is a reciprocal lattice vector in the plane p. In the liquid state (without longrange order in plane p), S 2p (k)fluctuates near zero. As the two halves of a bicrystal are rotated with respect to each-other about the GB-plane normal, two different wave vector, k 1 and k 2 , are required, each corresponding to a principal direction in the related half. For a well-defined crystalline lattice plane, say in semicrystal 1, S 2p (k 1 ) then fluctuates near a finite value (∼1) appropriate for that temperature, whereas S 2p (k 2 ) ∼ 0. In the GB region, due to the local disorder, one expects somewhat lower values for S 2p (k 1 ). By monitoring S 2p (k 1 )and S 2p (k 2 )every slice may be characterized as (a) belonging to semicrystal 1 (for S 2p (k 1 )finite, S 2p (k 2 ) ∼ 0),(b) belonging to semi-crystal 2 (for S 2p (k 1 ) ∼ 0, S 2p (k 2 )finite) or (c) disordered or liquid (for S 2p (k 1 ) ∼ 0, S 2p (k 2 ) ∼0). See Chapters 6.9, 6.10 and 6.11 for further discussions of structural order parameter.

9.

Grain Boundary Sliding and Migration

When a bicrystal is brought into equilibrium at a finite temperature, thermal stresses can develop in the GB core and give rise to deformation or activated cooperative motions. Of particular interest are motions which result in a displacement of the interface with the GB structure remaining intact. One can imagine two modes of boundary displacements, a translation along the boundary plane by the upper half of the bicrystal relative to the lower half (sliding), and a movement of the boundary plane in a perpendicular direction (migration). It turns out that the individual atoms can move by relatively small distances and yet collectively they cause the boundary to slide and migrate. Because no large amplitude of atomic displacements are involved, these collective motions can be readily thermally activated and, therefore, observed in a dynamic simulation. Furthermore, it is reasonable to expect that sliding and migration will be coupled if the boundary is to continually reconstitute itself during its movements [10]. The coupled motions of sliding and migration, observed in two- and three-dimensional simulations, can be analyzed using the so-called DSC lattice. Figure 4 shows an example of the atomic displacements that have been observed in the bicrystal. One sees relative to the lower crystal the atoms in the upper crystal, except for the atoms in the transition region, all undergo a lateral displacement equal to one unit of the DSC lattice in the x-direction. The

Atomistic methods for structure–property correlations

1945

B

B'

A

A'

Figure 4. Atomic displacements in a molecular dynamics simulation of a symmetric tilt bicrystal which are associated with coupled GB sliding and migration [10]. As a result of these displacements which occurred under either thermal or shear activation, the boundary plane moved from position A to position B.

displacements of the atoms in the transition region are more complicated because they have to change allegiance from being part of the upper crystal to belonging to the lower crystal. In addition to thermal activation, boundary sliding and migration can be induced by external stress. This has been demonstrated in a MC study by applying a shear stress to the same two-dimensional bicrystal model discussed above [11]. With either thermal or stress activation, it was found that a threshold value exists below which boundary motion was not observed. See Article 6.7 for further discussions of sliding and migration.

10.

Atomic Mobility

The study of atomic migration by MD is appropriate provided the time required for migration is not greater than the time interval of simulation. One measure of the necessary level of diffusivity is given by the behavior of the mean squared displacement r 2 . If this displacement shows a linear increase with time over a period long compared to local fluctuations and correlations, then the motion can be considered as diffusive, and the diffusion coefficient obtained from the slope of the linear portion of r 2 . Alternatively, if the discrete jumps between lattice sites can be monitored, one can deduce the diffusion coefficient from the observation of several hundred jumps. In the study of liquid-state dynamics, the mean squared displacement procedure is routinely used to calculate the diffusion coefficient, which typically has values of order 10−5 cm2 /s.

1946

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Atomic diffusion along grain boundaries is an important metallurgical process for matter transport, especially at temperatures well below Tm where it may be orders of magnitude more rapid than bulk diffusion. It is generally believed that GB diffusion occurs via a point-defect mechanism [12]. A related question concerns the action of the grain boundary as a source or sink for defects, and, in the presence of impurities, the role of diffusion in grainboundary segregation. MD studies of GB diffusion have been carried out on a symmetrical tilt boundary in the fcc structure using the Lennard-Jones potential with 3-D PBC [13] and the Morse potential for Cu with a 2-D PBC and fixed z-borders [14], as well as in the bcc structure using anempirical potential for α − Fe with fixed z-borders [15]. Figure 5 shows the bcc bicrystal model in the form of a stack of ten layers of (001) atomic planes, each containing 40 atoms. After the system was relaxed, a vacancy was introduced into one of the sites in the GB core (labeled as A, B, C, or D). The different sites are clearly not equivalent, as can be seen by the corresponding vacancy formation energies calculated separately by LS methods. Long simulation runs were carried out out at several temperatures during which vacancy migration from one site to another was monitored. A typical jump sequence at 1500 K (observed melting point of iron is about 1800 K) is shown in Fig. 6. From this kind of data one can extract a vacancy migration energy by assuming an Arrhenius behavior for the jump frequency; a reasonable value of 0.51 eV was obtained in this case. In such simulations one also has the necessary details to investigate the relative frequency with which the vacancy visits the different sites, information that

(b)

5.00 a0

6.5 0a

0

(a)

Layer

1 10 9 8 7 6 5 4 3

6.32 a0

x [130] z [001] y [310] y [310] x [130]

2 1

Figure 5. Bicrystal model of a symmetric tilt boundary in an MD simulation of vacancy migration in bcc iron, (a) view showing the simulation cell, and (b) view of one of the (001) planes and the border regions, enclosed by dashed lines, containing fixed particles.

Atomistic methods for structure–property correlations

1947

Figure 6. A portion of the vacancy migration trajectory observed during MD simulation at an elevated temperature. Length scale along [001] has been expanded by a factor of 5. Sites in the GB core are labeled A, B, C, D with equivalent sites denoted by a prime. Three sequences are shown, vacancy migration predominantly in the GB plane (left sequency), migration involving an interstitial position I (middle sequence), and migration resulting in exchange of atoms at sites B and B (right sequence).

may be potentially useful for correlating the diffusion properties of a given GB with structural features of the boundary core. To obtain a diffusion coefficient from the simulation data one can use the jump-frequency information, or resort to a more fundamental quantity, the mean squared displacement (MSD), 1 r (t) = N 2

N

[r i (t) − r i (0)]

2

(7)

i=1

where on the right hand side means an average over different time origins t = 0 if one is dealing with a steady-state process, otherwise, the bracket notation should be ignored. The mean squared displacement ‘therefore’ is a measure of how far an atom in the system migrates during a time interval t. The way it is defined in Eq. (3) assumes that all the atoms in the system behave in the same

1948

S. Yip

2 MSD (IO⫺16 cm2)

C

B 1

A 0

0

100

200 TIME (psec)

300

400

Figure 7. Mean squared displacement of atoms observed in an MD study of GB diffusion [15], (A) atoms in the bulk region only, (B) all atoms in the simulation cell, and (C) atoms in the GB core only (sites A–D in Fig. 6).

way on average. If there is a particular atom or a small group of atoms one wishes to follow, then the factor (1/N ) and the summation over atoms would have to be adjusted accordingly. Figure 7 illustrates the difference between atomic mobility in the bulk crystal and that in the GB core. The grain-boundary coefficient, DG B , can be obtained from the definition D = r 2 (t)/6t, for t large compared to any local relaxation times. However, in this approach there is an ambiguity in deciding which atoms should be included in calculating the MSD. Clearly, only those atoms that could be considered as belonging to the GB core should be counted in determining DG B . The basic difficulty is that what is considered GB region is not precisely defined; in other words, we do not have a unique of defining the width of the GB. This problem becomes worse when the GB core becomes deformed and starts to migrate, which can happen at sufficiently high temperatures. See Article 6.7 for further discussion of GB diffusion.

11.

Grain Boundary Segregation

Grain-boundary segregation is known to strongly influence not only the mechanical properties of interface materials [16], but also the atomic mobility at the GBs [21]. By segregation we mean the migration of solute atoms from the bulk to the GB and the subsequent distribution of these particles. As discussed above MD is appropriate for the simulation of atomic migration,

Atomistic methods for structure–property correlations

1949

provided the diffusivity is such that significant motions can be captured during the interval of simulation. In polycrystals with micron grain size or larger, atomic diffusion in the bulk would be too slow for MD to follow. On the other hand, in nanocrystals the possibility is more promising [see Article 6.10]. For the equilibrium distribution of the segregants, the problem is basically one of determining the proper chemical composition of the GB region, and allowing for atomic and overall volume relaxations to relieve the lattice strain. Monte Carlo method [23] is well suited for this kind of study because it affords an efficient sampling of various lattice configurations in which pairs of atoms of different species are exchanged sequentially. If one wishes to perform only structural relaxation of a GB with a given impurity concentration and distribution, MD also can be used. The fundamental reason that MC results are the most physically meaningful is that entropy effects are taken into account so the free energy is minimized. There are different ways of performing the MC simulation depending on the ensemble distribution to be generated [17]. For problems where it is not necessary to let the total number of atoms fluctuate, it is convenient to work with either the isochoric canonical ensemble (NVT), where particle N , the system volume V , and temperature T are held constant, or the isobaric canonical ensemble (NPT). Since the number of segregants in the GB is not known a priori, a grand canonical ensemble, either (µV T ) or (µP T ), with µ being the chemical potential, should be used. However, determining the chemical potential, which is required in applying the grand-canonical-ensemble MC (GCEMC) method, is itself a challenging task. The problem of simulating equilibrium segregation begins with finding the concentration of each atom species at the interface subject to the condition that the interfacial region is in chemical equilibrium with the bulk. Chemical equilibrium means the two regions have the same chemical potential, which must be known before one can apply GCEMC. If the chemical potentials have not been determined, then one needs to first perform a simulation of the bulk system where the chemical potentials are adjusted to give the desired concentrations in the bulk. The effect of the GCEMC procedure then is to generate system configurations according to the distribution

[U ({r N }) − µA NA − µB NB ] f ({r N }, NA , NB ) ∝ exp − kB T

(8)

in the case of a binary system, with NA being the number of atoms of species A. Such a calculation is capable of giving the proper chemical composition at the interface while also taking care of the relaxation of lattice strain effects. We give an illustration of using MC to study the distribution of oversized impurities Bi in symmetrical and asymmetrical tilt boundaries in Cu [18]. Since these were rather early studies, very simple two-body interatomic potentials were employed, in particular the cross interactions between atoms of

1950

S. Yip

different species were fitted to the heat of mixing. A simulation cell consisting of three regions was set up in a manner similar to Fig. 1. In the interface region the atoms were allowed to exchange lattice sites as well as species identity. In the adjacent regions the atoms could exchange sites, but the average solute concentration was kept constant. The arrangement just described did not include local atomic relaxation effects since the atoms were always assigned to lattice sites. To study relaxation effects a different cell was used in which there was only the central region and all borders were 3-d periodic. In this case the impurities were randomly distributed initially, and atoms were allowed to exchange positions with one of their nearest neighbors; in addition they were allowed to move according to the usual Metropolis procedure. The simulation thus showed that segregation was localized in the first few layers near the GB, with increasing tendency to segregate as temperature was decreased. The net effect of a large solute atom was an overall outward movement of the boundary atoms. The amount and distribution of segregants were different for different GBs. In the case of the asymmetrical tilt boundary the extent of segregation was slightly higher and the segregant profile at the interface became asymmetrical. A final comment is that solute-atom segregation at internal interfaces is a problem where simulation has been integrated into interpretation of experiments [19].

12.

An Outlook

The correlation between the structure and property is a longstanding concept in materials science. Just as the study of materials has become a broad enterprise that encompasses many traditional disciplines, the concept of correlation has expanded from classical route of processing–properties– performance relations mostly for bulk materials to include both fundamentals and innovations in areas that may be called chem.-bio-nano [20]. The increased emphasis on functionality of materials will mean correspondingly more opportunities for multiscale modeling and simulation as discussed throughout this volume.

References [1] C. Kittel, Introduction to Solid State Physics, 3rd edn., John Wiley & Sons, New York, 1966. [2] S. Yip and D. Wolf, “Atomistic concepts for simulation of grain boundary fracture,” Mater. Sci. Forum, 46, 77–168, 1989. [3] J.F. Lutsko, D. Wolf, S. Yip, S.R. Phillpot, and T. Nguyen, “Molecular-dynamics method for the simulation of bulk-solid interfaces at high temperatures,” Phys. Rev. B, 38, 11572–11581, 1988.

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[4] J.R. Beeler, In: H. Herman (ed.), Advances in Materials Research, vol. 5, Wiley, New York, p. 295, 1970. [5] A.A. Maradudin, E.W. Montroll, G.H. Weiss, and I. Ipatova, Theory of Lattice Dynamics in the Harmonic Approximation, Academic, New York, 1971. [6] D.C. Wallace, Thermodynamics of Crystals, Wiley, New York, 1972. [7] M.P. Allen and D.J. Tildesley, Computer Simulation of Liquids, Clarendon, Oxford, 1987. [8] M. Parrinello and A. Rahman, “Polymorphic transitions in single crystals: a new molecular dynamics method,” J. Appl. Phys., 52, 7182–7190, 1981. [9] S.R. Phillpot, D. Wolf, and S. Yip, “Effects of atomic-level disorder at solid interfaces,” MRS Bull., XV, pp. 38–45, 1990. [10] G.H. Bishop, R.J. Harrison, T. Kwok, and S. Yip, “Simulation of grain boundaries at elevated temperature by computer molecular dynamics,” In: J.W. Christian, P. Haasen, T.B. Massalski (eds.), Progress in Materials Science, Chalmers Anniversary Volume, Pergamon, Oxford, pp. 49–95, 1981. [11] R. Najafabadi and S. Yip, Scripta Metall., 18, 159, 1984. [12] R.W. Balluffi, Metall. Trans. B, 13, 527, 1982, L. Peterson, Int. Metall. Rev., 28, 66, 1983. [13] G. Ciccotti, M. Guillope, and V. Pontikis, Phys. Rev. B, 27, 5576, 1983. [14] C. Nitta, “Computer simulation study of grain boundary diffusion in aluminum and aluminum-copper systems,” PhD Thesis, MIT, 1986. [15] T. Kwok, P.S. Ho, and S. Yip, “Molecular-dynamics studies of grain-boundary diffusion, II. Vacancy migration, diffusion mechanism, and kinetics,” Phys. Rev. B, 29, 5363–5371, 1984. [16] M.P. Seah, J. Phys., F10, 1043, 1985. [17] J. Ray, “Elastic constants and statistical ensembles in molecular dynamics,” Comput. Phys. Rep., 8, 109–151, 1988. [18] S. Foiles, Phys. Rev. B, 32, 7685, 1985. [19] S. Foiles and D. Seidman, “Solute–atom segregation at internal interfaces,” MRS Bull., XV, pp. 51–57, 1990. [20] M.C. Flemings and S. Suresh, “Materials education for the new century,” MRS Bull., November, 918–924, 2001. [21] W.G. Hoover and B.J. Alder, “Studies in molecular dynamics. IV. The pressure, collision rate, and their number dependence for hard disks,” J. Chem. Phys., 46, 686–691, 1967. [22] H. Gleiter and B. Chalmers, Progr. Mater. Sci., 16, 77, 1972. [23] K. Binder, Applications of the Monte Carlo Methods in Statistical Phys. (Springer Verlag, Berlin, 1979).

6.9 STRUCTURE AND ENERGY OF GRAIN BOUNDARIES Dieter Wolf Materials Science Division, Argonne National Laboratory, Argonne, IL 60439, USA

Grain boundaries (GBs) are internal interfaces formed when two crystals that are misoriented relative to each-other are brought into intimate contact. Together with the grain junctions (i.e., the lines and points were three or more GBs meet), GBs represent the elementary building blocks of polycrystalline microstructures. Their structure and physical behavior therefore controls many of the thermal, mechanical and electrical properties of polycrystalline solids. A central theme in GB research addresses this interrelation between GB structure and physical properties. In this context it is usually understood that the term “structure” includes both the GB geometry and the underlying GB atomic structure. While the GB geometry involves the eight crystallographic degrees of freedom of the interface (Section 1.), the atomic structure is determined by the physics of the material, i.e., by the nature of the interactions between the atoms, leading to phenomena such as misfit localization or the amorphous structure of high-energy GBs (Section 3.). At an intermediate level, the concept of the “atomic-level geometry” (Section 2.) provides a natural link between the macroscopic geometry and the atomic structure. Here we review the key concepts for the description of GB “structure” at each of these three levels, with particular emphasis on how they can be connected to certain key GB physical properties, such as the GB energy.

1.

Macroscopic Geometry

In addition to the crystal structure, the geometry of a bicrystalline GB is characterized by five “macroscopic” and three “translational” or “microscopic” degrees of freedom (DOFs). Their experimental characterization represents a major challenge. The translational DOFs are represented by the three components of a vector, T, associated with rigid-body translations parallel to 1953 S. Yip (ed.), Handbook of Materials Modeling, 1953–1983. c 2005 Springer. Printed in the Netherlands.

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(two DOFs) and perpendicular to the GB (one DOF). In principle, a sixth parameter, representing the position of the GB, should be added to the five macroscopic DOFs; however, for an immobile GB this DOF is irrelevant.

1.1.

Macroscopic Degrees of Freedom

The five macroscopic DOFs can be defined in different ways. The ultimate measure of any particular choice is its ability to expose GB physical properties in terms of the terminology thus introduced. Here we describe two methods for their definition (for details, see Ref. [1]). The traditional “CSL-misorientation” terminology focuses on the misorientation between two grains in terms of the coincident-site lattice [2–4]. While limited to commensurate interfaces (see Section 2.3.), this terminology is particularly useful for the description of dislocation boundaries (i.e., low-angle and vicinal GBs; see Section 3.2.). By contrast, the “interface-plane” terminology [1, 5] focuses on the GB plane and thus exposes the similarities between GBs, free surfaces and stacking faults. While not limited to commensurate GBs, it is particularly useful for the description of high-angle GBs (see Section 3.3.).

1.1.1. CSL-misorientation Method Within the framework of the CSL description of GBs, three DOFs are identified with the CSL rotation, R(nˆ CSL , φCSL ), about a rotation axis, nˆ CSL , by the angle φCSL ; the remaining two DOFs are assigned to the GB plane. A GB is then thought of as having been generated by a relative rotation, R, of two identical, interpenetrating crystal lattices such that a three-dimensional superlattice in common to the two is generated. The unit-cell volume of this superlattice (the CSL) is times larger than that of the two space lattices; the integer number may therefore be thought of as the inverse volume density of CSL sites obtained for a given rotation. Following generation of the CSL, a GB is created by choosing as the GB plane any plane of the CSL and subsequently removing corresponding halves from each crystal lattice. The five DOFs are thus defined as follows: {DOFs} = {ˆnCSL , φCSL , nˆ 1 }

(“CSL-misorientation method”),

(1)

where nˆ 1 represents the GB-plane normal in either of the two halves of the bicrystal (here chosen to be grain 1). Although redundant as it is uniquely determined by the GB misorientation, = {R(nˆ CSL , φCSL )} is usually added as a sixth parameter in Eq. (1). Due to the requirement that, at least prior to allowing for rigid-body translations, a superlattice in common to the two grains exists, the CSL method is limited to commensurate GBs.

Structure and energy of grain boundaries

1955

For dislocation boundaries the distinction between pure tilt (nˆ CSL ⊥nˆ 1 ), pure twist (nˆ CSL ||nˆ 1 ) and mixed (or “general”) GBs is extremely useful as it provides information about the types and planar densities of dislocations present in the GB atomic structure, with the tilt and twist components, respectively, defining the edge- and screw-dislocation content. For “general” GBs (i.e., those with five DOFs, thus having both twist and tilt components), the CSL rotation may be viewed as consisting of the tilt rotation, R(nˆ T , ψ), followed by a twist rotation, R(nˆ 1 , θ); i.e., R(nˆ CSL , φCSL ) = R(nˆ 1 , θ)R(nˆ T , ψ),

(2)

where nˆ T and ψ are the tilt axis and tilt angle. The three rotation matrices in Eq. (2) involve a total of nine parameters for an overall misorientation characterized by only the three DOFs in nˆ CSL and φCSL ; six relationships must therefore exist among nˆ CSL , φCSL , nˆ T , ψ, nˆ 1 and θ. Due to this myriad of parameters in the CSL terminology, the correct number of macroscopic DOFs of any given GB and its atomic-level geometry are not readily apparent. For example, the fact that asymmetric tilt boundaries (ATGBs) have only four DOFs and hence represent a subset of general boundaries and that, with only two DOFs, symmetric tilt boundaries (STGBs) are a subset of twist boundaries is not obvious. In fact, because the misorientation represents already three DOFs, it is difficult to envision any GB having only two DOFs. To illustrate these difficulties, we start with the conceptually very simple pure (or symmetric) twist boundary, for which nˆ CSL ≡ ± nˆ 1 (i.e., the CSL rotation axis is parallel to the GB plane normal). Equation (2) then becomes {DOFs} = {±nˆ 1 , φCSL , nˆ 1 }

(symmetric twist GB),

(4)

revealing that such a GB has only three DOFs. The fact that, compared to a general GB with five DOFs, an asymmetric tilt boundary (ATGB; see Fig. 1) has only four DOFs while a STGB has only two is not as readily apparent, however. Generally, for tilt boundaries nˆ CSL ≡ nˆ T and φCSL ≡ψ; Eq. (2) then yields {DOFs} = {ˆnT , ψ, nˆ 1 } (tilt GB).

(5)

From Eq. (5) it appears that a tilt boundary may have all five DOFs, similar to a general boundary (see Eq. (2)). However, because of the constraint that the tilt axis is perpendicular to the GB plane normal (i.e., nˆ T ⊥nˆ 1 or (nˆ T · nˆ 1 ) = 0), the number of independent variables in Eq. (5) is reduced from five to only four; an ATGB has therefore only four DOFs. In a symmetric GB, generally nˆ 2 = ± nˆ 1 ; in a symmetric tilt boundary, the normals nˆ 1 and nˆ 2 are related by the rotation: nˆ 2 = ±nˆ 1 = R(nˆ T , ψ)nˆ 1 .

(6)

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D. Wolf

(a)

(b)

(c)

â n

Ψ

â n

â n

Ψ

Ψ

â nT

Figure 1. Definition of the four macroscopic degrees of freedom of an asymmetric tilt boundary (ATGB) [1, 5].

This constraint further reduces the number of independent variables in Eq. (5) from four to only two. This little recognized fact means that an STGB has only two macroscopic DOFs, as do free surfaces and stacking faults. Within the CSL terminology this intimate geometrical relationship between these three rather different types of interfacial systems, each having only two DOFs, is far from obvious. Fortunately, however, within the interface-plane terminology, these similarities become readily obvious.

1.1.2. Interface-plane method GBs are planar defects on crystallographically well-defined lattice planes. Conceptually a bicrystal may therefore be thought of as having been formed by bringing two free surfaces, with normals nˆ 1 and nˆ 2 , into contact followed by a rotation about their common normal (see Fig. 2). The five macroscopic DOFs may then be simply defined as follows [5]: {DOFs} = {ˆn1 , nˆ 2 , θ} (“interface-plane method”).

(7)

ˆ The unit vectors nˆ 1 and nˆ 2 characterize the common GB-plane normal, (n), with respect to the two principal (say, cubic) coordinate systems, (x1 , y1 , z 1 ) and (x2 , y2 , z 2 ), in the two halves of the bicrystal, i.e., the normals of the two free surfaces brought into contact (see Fig. 2). Since each unit vector represents two DOFs, the GB plane characterized with respect to each of the two grains represents four DOFs. The only remaining DOF is the one associated with a “twist” rotation, i.e., a rotation about nˆ by the angle θ; any other rotation would change nˆ 1 and nˆ 2 .

Structure and energy of grain boundaries

1957

â n2 z2

θ

y2

x2

â n1 z1 y1

x1 Figure 2. Definition of the five macroscopic DOFs of a general (or “asymmetric twist”) GB within the interface-plane terminology [1, 5].

1.2.

Symmetric vs. Asymmetric Tilt and Twist GBs

By contrast with the CSL terminology, the five DOFs defined in Eq. (7) readily permit the correct number of macroscopic DOFs of any given GB to be identified and its atomic-level geometry to be visualized. By definition, the angle θ describes a twist rotation. We therefore define θ = 0◦ as the angle for which the GB is of a pure tilt type (i.e., an asymmetric tilt GB, ATGB), such that the GB atomic structure contains only edge dislocations (see Fig. 1(c)). For θ > 0, screw dislocations are introduced in addition to the edge dislocations already present for θ = 0◦ , leading to an increase in the GB planar unit-cell area. Hence, if the lattice planes associated with nˆ 1 and nˆ 2 are commensurate (see Section 2.3), the ATGB at θ = 0◦ has the smallest possible planar unit cell of any GB on the (nˆ 1 ,nˆ 2 ) GB plane.

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Because of the inversion symmetry of Bravais lattices, a twist rotation by θ = 180◦ produces another ATGB with an identical planar unit-cell area and shape (see Section 2.2); however, its atomic structure differs from that at θ =0◦ in that the planar stacking of lattice planes on one side of the GB is inverted with respect to that at θ = 0◦ [1,5,6]. The two ATGB configurations that exist on a given (nˆ 1 ,nˆ 2 ) GB plane are thus characterized by: {DOFs} = {ˆn1 , nˆ 2 , θ = 0◦ or θ = 180◦ } (asymmetric tilt GB).

(8)

It is then rather obvious that, with a fixed value of θ and hence only four DOFs, ATGBs represent a geometric subset of general (or “asymmetric twist”) GBs having five DOFs. Given the above definition of θ = 0◦ , the twist component, R(nˆ 1 , θ), of the general GB defined by (7) is also readily apparent. Its tilt component, R(nˆ T , ψ) (see Eq. (2)), is governed by the condition that nˆ T ⊥nˆ 1 , nˆ 2 (i.e., that the tilt axis lies in the GB plane; see also Fig. 2). Therefore nˆ T = [nˆ 1 × nˆ 2 ]/sin ψ,

(9)

with the tilt angle given by sin ψ = |[nˆ 1 × nˆ 2 ]|.

(10)

Equations (9) and (10) illustrate the fact that the tilt component of a general GB is solely determined by the four DOFs associated with the (nˆ 1 , nˆ 2 ) GB plane. For symmetric GBs, nˆ 2 = ± nˆ 1 , reducing the number of DOFs in Eq. (7) from five to three, according to (see Eq. (8)) {DOFs} = {ˆn1 , ±nˆ 1 , θ}

(symmetric GB).

(11)

θ = 0 is then conveniently chosen to represent the perfect crystal. As discussed in Section 2.2, in lattices with inversion symmetry θ = 180◦ yields the STGB configuration on a given lattice plane, characterized by the inversion of stacking at the GB. STGBs are therefore a subset of twist GBs, with only two DOFs since θ is fixed at 180◦ ; by contrast, in twist GBs θ represents a DOF.

2.

Atomic-level Geometry

To better visualize GB geometry, the interface-plane description can be taken all the way down to the level of the underlying Bravais and crystal lattices. This leads naturally to the atomic-level description of GB geometry and provides a useful link between the macroscopic geometry and GB atomic structure. In addition to the five DOFs, two useful quantities in this description are the size of the GB planar unit cell (for periodic GBs) and the interplanar spacing parallel to the GB.

Structure and energy of grain boundaries

2.1.

1959

Planar Stacking in Bravais Lattices

A three-dimensional Bravais lattice is usually defined by three vectors, a1 , a2 and a3 , such that any lattice point is given by r = la1 + ma2 + na3 (l, m, n = 0, ± 1..) (see Fig. 3(a)). To visualize the planar arrangement of ˆ a new Bravais sites perpendicular to some arbitrary (rational) direction, n, ˆ c2 (n) ˆ and c3 (n), ˆ can be defined “plane-based” set of Bravais vectors, c1 (n), ˆ and c2 (n) ˆ lie within the planes with normal nˆ while c3 (n) ˆ consuch that c1 (n) ˆ and c2 (n) ˆ thus define nects points within different planes (see Fig. 3(b)). c1 (n) the primitive planar unit cell of the nˆ planes while the out-of plane component, ˆ = d + e enables one to proceed from one lattice plane to the next. ˆ of c3 (n) d||n, As sketched in Fig. 4, the in-plane component, e, governs relative translations between neighboring planes, characterizing their overall, periodic stacking; by contrast, the inter-planar component, d, permits one to proceed from one plane to a neighboring one. To be specific, we consider cubic crystals in which nˆ = (hkl)/(h 2 + k 2 + 2 1/2 l ) may be expressed in terms of the Miller indices, (hkl). All relevant geometrical parameters may then be expressed in terms of (hkl) as well. For example, the interplanar spacing, |d| = d(hkl), is given by the well-known expression d(hkl) = αhkl a(h 2 + k 2 + l 2 )−1/2 ,

(αhkl = 0.5 or 1)

(12)

where a is the cubic lattice parameter and the value of αhkl depends on the combination of odd and even Miller indices. (For example, in the fcc lattice, αhkl = 1 if all (hkl) are odd but 0.5 otherwise.) The number of planes in the repeat stacking sequence, referred to as the stacking period P(hkl), is given by [1, 5, 6] P(hkl) = βhkl (h 2 + k 2 + 12 ),

(βhkl = 1 or 2), (b) z' || â n

(a)

z z'

(13)

a2

e

â n

d

c3

c2

y'

y

a3 x

a1 c1 x'

Figure 3. Two alternate methods for defining a three-dimensional Bravais lattice. (a) Conventional and (b) plane-based Bravais lattice.

1960

D. Wolf A

e B

d(hkl)

d C D E

Figure 4. Stacking of Bravais-lattice planes, schematically illustrated for a hypothetical five-plane stacking sequence, . . .|ABCDE|. . . [1, 5]. Table 1. Interplanar spacing, d(hkl), in units of the lattice parameter a (see Eq. (12)), for the most widely spaced planes in the fcc lattice. Also listed are the stacking period, P(hkl) (see Eq. (13)), and the values of (hkl) for an STGB on the (hkl) plane (see Eq. (18) below). [1] (hkl) 1 (1 1 1) 2 (1 0 0) 3 (1 1 0) 4 (1 1 3) 5 (3 3 1) 6 (2 1 0) 7 (1 1 2) 8 (1 1 5) 9 (5 1 3) 10 (2 2 1)

P(hkl)

d(hkl/a)

3 2 2 11 38 10 6 27 35 18

0.5774 0.5000 0.3535 0.3015 0.2294 0.2236 0.2041 0.1925 0.1690 0.1667

(hkl)

3 1 1 11 19 5 3 27 35 9

where, similar to αhkl , the value of βhkl depends on the combinations of odd and even (hkl). For example, the well-known . . . |ABC|ABC|. . . stacking of (111) planes in the fcc lattice is an example for P(hkl) = 3; the . . .|AB|AB|. . . stacking of the (100) and (110) planes is an example for P(hkl) = 2. Since each Bravais plane contains exactly one lattice site in the planar unit cell (see Figs. 3 and 4), its area, Amin (hkl), is related to d(hkl) via the unit-cell volume, : Amin (hkl)d(hkl) = .

(14)

The most widely spaced planes are therefore the ones with the smallest planar unit-cell areas; i.e., the densest planes of the crystal lattice. As an example, Table 1 lists the values of d(hkl) and P(hkl) for the 10 densest planes of the fcc lattice, in which = a 3 /4.

Structure and energy of grain boundaries

2.2.

1961

Symmetric Tilt Boundaries in Cubic Crystals

From a strictly geometrical viewpoint STGBs are fascinating, yet little understood objects since they are so closely related to free surfaces and stacking faults. These simplest of all crystalline interfaces share the common property that, apart from having different rigid-body translations, T, their geometry is fully characterized by only the two DOFs associated with the interface (hkl) plane. Consideration of their atomic-level geometry exposes this commonality (see Fig. 5). The atomic-level geometry of the STGB sketched in Fig. 5(c) is characterized by the familiar inversion at the GB of the stacking of the perfect-crystal lattice planes in Fig. 5(a) (see the shaded arrows on the right), a feature in common to all STGBs. In practice, the inverted configuration in Fig. 5(c) is unstable, giving rise to a rigid-body translation (Tx , Ty ) parallel to the GB accompanied by a volume expansion per unit GB area, Tz = δV , that minimizes the GB energy. A basic property of all lattices with inversion symmetry is that the stacking sequence in a given direction, hkl, may be inverted by a 180◦ twist rotation about hkl. In such lattices STGBs therefore represent special 180◦ twist boundaries [5, 6]. (a)

(b)

(c)

T⫽(Tx,Ty) A

E

B

D C B

e

C

d

D

A

d(hkl)⫹␦V

E

E

Ψ/2

E E

D

D

D

C

C

C

B

B

B

A

A Perfect crystal

Ψ/2 Ψ/2

A

Stacking Fault

Symmetrical Tilt ("Twin")

Figure 5. Geometrical relation between (a) planar stacking in the perfect crystal, (b) the stacking fault and (c) the STGB on the same (hkl) plane. (a) shows schematically two unit stacks of lattice planes, labeled |ABCDE|, in a (hypothetical) direction with a 5-plane stacking period (P(hkl) = 5), illustrating the constant in-plane and out-of-plane translations, e and d, as one proceeds from one plane to the next. In the stacking fault in (b), the upper grain has been merely translated parallel to the (hkl) interface plane. By contrast, in the STGB configuration in (c), the upper stack is inverted with respect to the lower one while preserving the shape and area of the perfect-crystal planar unit cell. In most STGBs, rigid-body translations parallel and perpendicular to the GB will destroy the mirror symmetry in (c) and give rise to a volume expansion per unit GB area, δV [1, 5].

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Apart from having a different rigid-body translation, the stacking fault on the (hkl) plane (sketched in Fig. 5(b)) differs from the STGB merely by the absence of this stacking inversion (i.e., θ = 0◦ ) while the free surface may be viewed as the Tz → ∞ limit. Obviously the STGB, stacking fault and free surface have equal planar unit-cell areas, with unit-cell dimensions that are identical to those of the perfect-crystal (hkl) plane in Fig. 5(a); this area is hence the smallest possible for any planar defect on the (hkl) plane. A special STGB translational state is the coherent-twin configuration in which the GB is not only a mirror plane (see Fig. 5(c)) but also an atom plane. In the fcc lattice there is only one such STGB, namely the (111) twin, with the stacking sequence, . . .|ABCBA|. . ., in which the C plane is the mirror plane (see Fig. 6). Starting from the . . .|ABC|ABC|. . . perfect-crystal stacking for P(111) = 3 (see Fig. 6(a)), a 180◦ twist rotation (or, because of symmetry, a 60◦ twist rotation) yields the inverted . . .|CBA|ABC|. . . STGB configuration in Fig. 6(b) which, after a rigid-body translation of the upper crystal relative to the lower one such that A → B (and hence B → C and C → A) results in the coherent twin configuration in Fig. 6(c). The fact that only (hkl) planes with P(hkl) = 3 can accommodate an STGB is also readily seen (see Fig. 7). For example, starting from the . . .|AB|AB|. . . perfect-crystal stacking for P(hkl) = 2 in (a) (e.g., for the (100) and (110) planes in the fcc lattice), a 180◦ twist rotation yields the inverted . . .|AB|BA|. . . “STGB” configuration in (b)) which, after a rigid-body translation such that B → A (and hence A → B), converts back to the perfect crystal (see Fig. 7(c)). Given that an STGB has only two DOFs, namely those associated with the GB (hkl) plane, all six CSL parameters in nˆ CSL ≡ nˆ T , φCSL ≡ ψ, nˆ 1 , and (a) <111>

(b) <111>

A B C

(c)

<111>

C B θ⫽ 60˚

A

A C T⫽(Tx,Ty)

B

A B C

A B C

A B C

Ideal crystal

"Unstable Twin"

"Coherent Twin"

Figure 6. Generation of the coherent-twin configuration in (c) by a 180◦ twist rotation of the perfect-crystal configuration in (a). The STGB configuration in (b) is energetically unstable, inducing the rigid-body translation that results in the coherent twin [1, 5].

Structure and energy of grain boundaries (a)

1963

(b)

(c)

â n

â n

â n

A B

B A

A B

θ⫽ 180˚

T⫽(Tx,Ty )

A B

A B

A B

Ideal crystal

"Unstable Twin"

"Coherent Twin"

Figure 7. Hypothetical generation of an STGB on a lattice plane with P(hkl) = 2. The 180◦ twist rotation of the perfect-crystal configuration in (a) results in the characteristic, inverted stacking sequence in the STGB configuration in (b). However, a simple rigid-body translation of this configuration parallel to the interface plane re-establishes the perfect-crystal translational state in (c) [1, 5].

(nˆ T , ψ) can be expressed in terms of the Miller indices [6]. However, applying Eqs. (9) and (10) to express for example nˆ T in terms of (hkl), at first sight it appears that STGBs have a vanishing tilt component because [nˆ 1 × nˆ 1 ] = 0. This apparent discrepancy originates from the fact that the underlying condition for symmetry, nˆ 2 = nˆ 1 , is too narrow as it should include all combinations of crystallographically equivalent planes, (hkl)(±h ± k ± l) (see Section 1.2). In cubic crystals an arbitrary combination of Miller indices, (±h, ±k, ±l), can always be converted, by a sequence of 90◦ rotations about 100 which transform the crystal into itself, e.g., into a form (h k ± l). The three DOFs of a symmetric GB in a cubic crystal are therefore given by [1] {D O Fs} = {(hkl), (hkl), θ} = {(hkl), (hk − l), 180◦ − θ}, (symmetric twist).

(15)

For θ = 180◦ , Eq. (15) describes the (one and only) STGB on the (hkl) plane. According to Eq. (15), it can be viewed either as a pure twist GB generated by a 180◦ twist rotation of two perfect-crystal stacks, (hkl)(hkl), or as a pure tilt boundary, starting from two already inverted stacks, (hkl)(hk − l), with no twist component at all [see also Fig. 1(b)]. In the latter case, Eqs. (5) and (6) readily yield its tilt component in terms of (hkl) [1, 6]: nˆ T = (h 2 + k 2 )−1/2 (−k h 0), sin ψ =

2 l(h 2 + k 2 )1/2 . h2 + k2 + l2

(16) (17)

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D. Wolf

Finally, may be expressed in terms of the Miller indices as well. As shown earlier [5, 6], = (hkl) is determined by the stacking period P(hkl). For an odd value of P(hkl), in the STGB configuration only two planes out of the 2P(hkl) planes in a double stack of (hkl) planes remain in perfect registry; i.e., = 2P(hkl)/2 = P(hkl). More generally, for both odd and even values of P(hkl), one can show that [5, 6] = δhkl P(hkl) = βhkl δhkl (h 2 + k 2 + l 2 ) = δhkl (h 2 + k 2 + l 2 ),

(18)

=βhkl δhkl =0.5 or 1, ensuring that is always where Eq. (13) was used and δhkl odd. Combining Eqs. (12) and (14), the STGB (or perfect-crystal) planar unitcell area, Amin (hkl), may be related directly to given by Eq. (18) as follows [5, 6]: −1/2 1/2 −1 Amin (hkl) = αhkl (δh kl) = εhkl 1/2 , (19) a a √ −1 where√εhkl = αhkl (δhkl )1/2 can assume the following four values: εhkl = 1, 2, 2 and 2 2. A small value of is therefore necessary but not sufficient for the STGB on the (hkl) plane to have a small planar unit-cell; the necessary and sufficient condition requires that for a particular (hkl) plane εhkl is small as well (for more on this distinction, see Section 3.1.).

2.3.

Commensurate Grain Boundaries

Within the CSL terminology, naturally only commensurate GBs are considered; i.e., those with special misorientations that result in a common superlattice, the CSL. Within the interface-plane terminology, a criterion for the commensurability of the lattice planes forming the GB is readily derived [1]. Two lattice planes, (h 1 k1l1 ) and (h 2 k2l2 ), are commensurate if under a general twist rotation a common planar unit cell exists which describes their periodic structures. This implies that n Amin (h 1 k1l1 ) = m Amin (h 2 k2l2 ), where n and m are positive integers. By definition, all symmetric GBs [for which (h 2 k2l2 ) = (h 1 k1 ± l1 )] are therefore commensurate. Using Eqs. (12) and (14), this condition can be written as follows: εh21 k1 l1 (h 21 + k12 + l12 ) m 2 = . εh21 k1 l1 (h 22 + k22 + l22 ) n 2

(20)

For two lattice planes to be commensurate, the ratio of the sum of the squares of the Miller indices therefore has to be itself a ratio, m 2 /n 2 , of squares of integers m and n (for details, see Ref. [1]). As an example, Table 2 lists pairs of cubic lattice planes with the smallest planar unit cells that are commensurate with the (111), (100) and (110) planes.

Structure and energy of grain boundaries

1965

Table 2. Cubic lattice planes with the smallest unit cells which are compatible with (111), (001) and (011) planes, respectively (see Eq. (20)) [1] No (h 2 k2l2 ) (h 1 k1l1 ) m 2 /n 2 (h 2 k2l2 ) (h 1 k1l1 ) m 2 /n 2 (h 2 k2l2 ) (h 1 k1l1 ) m 2 /n 2 1 2 3 4 5 6 7 8

(1 1 1) (1 1 1) (1 1 1) (1 1 1) (1 1 1) (1 1 1) (1 1 1) (1 1 1)

(1 1 1) (1 1 5) (1 5 7) (1 5 11) (1 11 11) (5 7 13) (1 1 19) (5 7 17)

1 9 25 49 81 81 121 121

(0 0 1) (0 0 1) (0 0 1) (0 0 1) (0 0 1) (0 0 1) (0 0 1) (0 0 1)

(0 0 1) (2 2 1) (4 3 0) (2 3 6) (1 4 8) (4 4 7) (6 7 7) (2 6 9)

1 9 25 49 81 81 121 121

3.

Grain-boundary Atomic Structure

(0 1 1) (0 1 1) (0 1 1) (0 1 1) (0 1 1) (0 1 1) (0 1 1) (0 1 1)

(0 1 1) (1 1 4) (0 7 1) (3 4 5) (1 4 9) (3 5 8) (4 5 11) (7 7 8)

1 9 25 49 81 81 121 121

The atomic structure of GBs is determined by the physics of the material. In accordance with their fundamentally different structures, we distinguish between three types of GBs. Special boundaries, involving the densest lattice planes and smallest planar unit cells, give rise to energy cusps in the phase space comprised of the five macroscopic DOFs. For small angular deviations from these, arrays of well-separated dislocations are formed. The resulting dislocation (i.e., low-angle or vicinal) GBs are to be distinguished from highangle GBs, in which the dislocation cores overlap completely.

3.1.

“Special” Boundaries

It has often been stated that the geometric reason for the existence of special GBs are the low values of associated with special misorientations. Much evidence to the contrary suggests, however, that a low value of does not guarantee an especially low GB energy. Instead, the main reasons for the appearance of energy cusps are the existence of (i) special GB planes, i.e., planes with large values of the interplanar spacing, d(hkl), parallel to the GB and (ii) GBs with especially small planar unit-cells (see, e.g., Refs. [5] and [7]). Similar to special (i.e., step-free or flat) free surfaces, special GBs contain no dislocations. The comparison of the structure of a stepped surface with that of a lowangle symmetric tilt boundary on the same lattice plane (see Fig. 8) demonstrates the close similarities between these simple types of interfaces. The steps in the free surface, with height h, are merely replaced by GB dislocations, with Burgers vector b.

1966

D. Wolf

(a)

∆Ψ ∆Ψ

b

δ

δ′

z

(b)

y

∆Ψ δ

h

x

â n cusp

δ′

â nv ∆Ψ

â nT

Figure 8. Step and dislocation structures of vicinal free surfaces and STGBs (schematic). Although both systems have only two DOFs (those associated with the interface plane), for vicinal interfaces the three-parameter description in terms of the tilt (or pole) axis, nˆ T , and angle, ψ, associated with the vicinal misorientation is more useful. nˆ cusp and nˆ v denote the normals of the “cusped” (or “special”, i.e., line-defect-free) and vicinal interface planes (see also Ref. [10]).

3.1.1. Special GB planes STGBs represent ideal model systems to deconvolute the distinct roles of (hkl) and d(hkl) in the appearance of special GBs since both parameters are governed by (hkl) [see Eqs. (12) and (18)]. Due to their close similarity, it is also instructive to compare STGBs with free surfaces. The appearance of GB dislocations or surface steps for small vicinal deviations from an energy cusp provides a fingerprint enabling the identification of special STGBs and surfaces. As illustrated in Fig. 8, in spite of having only two DOFs the dislocation and step structures of these vicinal interfaces are best characterized by the three CSL parameters associated with the tilt

Structure and energy of grain boundaries

1967

(100)

(113) (114) (116)

(112)

(111)

(331) (221) (332)

(110)

axis, nˆ T , and vicinal angle, ψ = arcos(nˆ v · nˆ cusp ), where nˆ cusp and nˆ v are the “special” and vicinal normals. As an example, Fig. 9 compares the simulated energies of fcc free surfaces and STGBs with a common 110 tilt axis plotted against the tilt angle, ψ = ψcusp + ψ. Given that each data point in Fig. 9 represents a different plane, the appearance of cusps indicates an extreme sensitivity of the surface and STGB energies to the interface plane. In fact, independent of the interatomic potential used [7] and in agreement with experiments [8], the STGBs exhibit cusps for the four fcc planes with the largest d(hkl) values (see Table 1). (We note that, with P(hkl) = 2, the (110) and (100) cusps at ψ = 0 and 180◦ correspond to perfect-crystal configurations; see Table 1 and Fig. 7.) That special free surfaces usually involve the two or three densest lattice planes has long been known (see, e.g., Ref. [9]); indeed, Fig. 9 reveals energy cusps associated with the (111) and (100) planes. All other STGBs and surfaces are vicinal to these special ones and it is the spacing between the line defects (i.e., ψ = ψ − ψcusp ) rather than d(hkl) that determines their energy [5].

1000

Σ9 Σ19 Σ3

Σ11

600

Σ9

<110 > Free surfaces Σ19

400 200

Σ11

Energy [mJ/m 2 ]

800

Σ3

0 Au (EAM)

<110 > STGBs

⫺200 0

30

60

90 Ψ

120

150

180

Figure 9. Simulated energies of fcc free surfaces and STGBs with a common 110 tilt axis plotted against the tilt angle, ψ [7]. For STGBs such plots were first obtained from both experiments and computer simulations by Hasson et al. [8].

1968

D. Wolf

In spite of their low values, remarkably absent in Fig. 9 are major cusps associated with the 3(112), 9(221), 9(114) and 11(332) STGBs. The reason is that these planes have rather small d(hkl) values, demonstrating that a low value alone is not sufficient to ensure a low STGB energy; a necessary and sufficient condition is that d(hkl) be large (see also Eq. (19)). The reason for the existence of special interface planes is readily understood in terms of a broken-bond model. The number of nearest-neighbor (nn) bonds within one of the densest planes is particularly large while the number of bonds per unit area across these planes is particularly small. Forming a free surface on such a plane therefore requires the smallest number of surface bonds to be broken; similarly, forming a GB on such a plane involves breaking, bending or stretching the smallest number of interplanar bonds per unit GB area. Another explanation, valid for GBs only, is that rigid-body translations parallel to the GB provide a particularly powerful relaxation mechanism if the GB area is very small [5].

3.1.2. Boundaries with small planar unit cells: STGB and ATGB cusps For STGBs a large value of d(hkl) is equivalent to a small GB planar unit-cell area, Amin (hkl) (see Eq. (19)). By contrast, in twist boundaries d and A are independent of each other. For twist angles other than the perfect crystal at θ = 0◦ and at the STGB at 180◦ , the GB area is given by A(hkl, θ) = (θ) Amin (hkl),

(21)

where the planar density of CSL sites, (θ) ≥ 1, indicates by how much A(hkl, θ) exceeds Amin . By definition, A(hkl, 0◦ ) = A(hkl,180◦ ) = Amin (hkl); i.e., (0) = (180◦ ) ≡ 1. For planes with P(hkl) ≤ 2, (θ) is identical to , the inverse volume density of CSL sites. For vicinal deviations from θ = 0◦ or 180◦ screw dislocations are formed; any twist GB on the (hkl) plane therefore has a larger GB area ((θ) > 1) than either the perfect crystal or the STGB. As seen in Fig. 10, this geometrical uniqueness gives rise to energy cusps at θ = 0◦ and 180◦ . The STGB on any given lattice plane is therefore special with respect to all other twist GBs on the same plane. While all STGBs are therefore special GBs with respect to vicinal twist deviations, some STGBs are clearly more special than others. As seen from Fig. 9, the STGBs on the four densest planes stand out in that they are special also with respect to vicinal tilt deviations; i.e., they represent cusps with respect to both tilt and twist vicinal deviations. Figure 10 also demonstrates that the large-d(hkl) criterion for low GB energy is valid even for high-angle twist GBs. An expression analogous to Eq. (21), in which A(hkl, θ) is replaced A(h 1 k1l1 , h 2 k2l2 , θ), is valid also for asymmetric GBs provided the two sets of planes are commensurate. θ = 0◦ and 180◦ then correspond to the two ATGB

Structure and energy of grain boundaries

GB ENERGY [mJ/m2]

1400

1969

(113)

1200 1000

(011)

800 (001) (m ⫽2)

600 400

(111) (m ⫽3)

200

Symm. twists, Cu(LJ)

0 0

30

60

90

120

150

180

mθ Figure 10. lattice [7].

Simulated energies of twist boundaries on the four densest planes of the fcc

configurations with the same, smallest planar GB area possible for this combination of planes, Amin (h 1 k1l1 ,h 2 k2l2 ), for which (0◦ ) = (180◦ ) ≡ 1. These two ATGB configurations differ from one-another merely by the inversion of stacking at the GB in one relative to the other. As in the symmetric case (Fig. 10), due to their small GB area, these geometrically unique GBs give rise to energy cusps [7]. ATGBs are therefore special with respect to asymmetric twist (or general) GBs involving the same combination of lattice planes. In summary, due to their small GB areas, all tilt GBs, symmetric or asymmetric, are special with respect to vicinal twist rotations. Moreover, similar to the STGBs, those ATGBs on GB planes with a large “effective” interplanar spacing, deff = [d(h 1 k1l1 ) = d(h 2 k2l2 )]/2, are special even with respect to vicinal tilt misorientations. In practice, deff is relatively large for any combination of lattice planes involving one of the densest planes on one side of the GB [7].

3.2.

Dislocation Boundaries

The CSL terminology in Eq. (1) is particularly useful for the description of dislocation GBs. For small vicinal deviations from a special GB, dislocations separated by strained perfect-crystal regions are formed to accommodate the misfit between the two grains. Depending on whether the vicinal deviation involves a pure tilt or twist rotation, the misfit dislocations are of pure edge or screw type; in general, however, they have mixed character.

1970

D. Wolf

3.2.1. Grain-boundary dislocations vs. surface steps Since their atomic structures are determined by the vicinal deviation, ψ, from a nearby “special” interface, it is instructive to compare the atomic structures of vicinal STGBs and free surfaces (Fig. 8). The main geometric difference between them is the replacement of the Burgers vector, b, in Fig. 8(a) by the step height, h ≡ |b| = b, in Fig. 8(b). The spacing, δ, between the steps, or between each of the two sets of edge dislocations, is given by the Frank formula, δ = b/sin ψ. Denoting the line energy per unit length by (δ), the interface energy may be written as follows [10]:

γ ( ψ) − γcusp cos ψ = n

(δ) sin ψ, b

(22)

where n = 1 for surfaces and n = 2 for STGBs. The mutual repulsion between steps, on one hand, and dislocations, on the other, due to their overlapping elastic strain fields may be incorporated by decomposing (δ) = core + el (δ) into its core and strain-field contributions 2 and assuming that only el depends on δ. For steps el (δ) = G s−s el /δ [11] d−d while for dislocations el (δ) = −G el ln(b/δ) [12] with the “elastic interacd−d tion strengths”, G s−s el and G el , governed by the elastic constants. Inserting these expressions into Eq. (22), for the surfaces we obtain [10] S S cos ψ = γcore ( ψ) + γelS ( ψ) γ S ( ψ) − γcusp

=

S core h

sin ψ +

G s−s el b3

while for the STGBs [10] γ

GB

( ψ) −

GB γcusp

sin3 ψ,

cos ψ =

GB γcore ( ψ)

−

G d−d el b3

+

γelGB ( ψ)

=2

GB core b

(23)

sin ψ

sin ψln sin ψ .

(24)

Here γcore and γel are the core and strain-field contributions to the interface energy. We note that Eq. (24) is valid for both tilt and twist boundaries. In the low-angle limit ( ψ → 0), Eq. (24) reduces to the famous expression,

γ

GB

( ψ) −

GB γcusp cos ψ

= 2 ψ

GB core b

−

(G d−d el /b) ln( ψ)

,

(25)

first derived by Read and Shockley [12], a derivation that also yields expressions for G d−d el for symmetric tilt and twist GBs in terms of the elastic moduli. The qualitatively different behaviors of STGBs and free surfaces for small vicinal angles ψ predicted by these expressions are clearly visible in Fig. 9.

Structure and energy of grain boundaries

1971

According to the Read–Shockley formula (25), for ψ → 0 the GB energy, γ GB ( ψ) ∼ γelGB ( ψ) ∼ ψln( ψ), decreases logarithmicly as it is dominated by the elastic strain-field term, giving rise to an infinite slope at ψ = 0. By contrast, due to the much weaker step-step interactions, for ψ → 0 the surface energy is dominated by the core term and therefore decreases sinusoidally, γ s ( ψ) ∼ γscore ( ψ) ∼ sin( ψ), with a finite slope at each cusp (see Eq. (23)). Combined with Eqs. (23) and (24), Fig. 8 demonstrates the little-known fact that cleavage decohesion of an STGB into free surfaces may be conceptualized as the reversible conversion of edge dislocations into surface steps; i.e., of long-ranged dislocation strain fields into short-ranged elastic strain fields near surface steps. The work of adhesion, Wad = 2γ s − γ GB , is therefore given by the difference between their respective core and elastic line energies [10].

3.2.2. The principle of misfit localization The formation of line defects may be viewed as a manifestation of a simple and rather general physical principle according to which a system responds to a structural inhomogeneity by localizing (“screening”) the disorder; i.e., a localized type of disorder is energetically preferred over a more spread-out, delocalized and inhomogeneous type of disorder. This fundamental principle originates from the anharmonicity present in virtually all interatomic interactions. Due to anharmonicity, interatomic distances shorter than the nearestneighbor distance cause a much greater increase in energy than the longer distances. By localizing the structural disorder, the number of these energetically most unfavored atoms is minimized while enabling the vast majority of atoms to relax into perfect crystal-like, elastically distorted local environments, thus minimizing the cost of structural disorder. For the case of vicinal GBs, this translates into the localization of the misfit within the highly defected dislocation cores separated by perfect-crystal-like regions.

3.3.

High-angle Boundaries

There has been much debate on the distinction between low-angle and high-angle GBs, in particular on whether a dislocation picture is appropriate at high angles and how the Read–Shockley model might possibly be extended to high-angle GBs. For example, in 1952 C. S. Smith stated [13]: “The success of dislocation theory simply and rationally applied to GBs is a great achievement . . .” and he suggested that “. . . there is a continuous transition from the pure dislocation boundary to whatever exists at higher angles . . .”. In support of this statement he suggested that the experimental evidence “. . . does not in any way preclude a nearly horizontal curve of energy vs. misorientation once

1972

D. Wolf

the maximum value has been achieved”, implying that the energy of highangle GBs is independent of the misorientation between the grains: “[Their structure] can undoubtedly be described formally in terms of dislocations, [although they are] clearly of a different type and the strain involved in the misfit has at least the appearance of being of shorter range”. That Smith’s intuition was correct is illustrated in Fig. 10 showing that, indeed, the energy increases monotonically with misorientation and levels off at a maximum value, suggesting that, indeed, there is a continuous transition from low-angle to high-angle GBs. Figure 11 shows a least-squares fit to the Read–Shockley expression (24) to the (100) data in Fig. 11, demonstrating that, in spite of the rather restrictive underlying assumptions, Eq. (24) (in GB /b and G d−d which core el /b are treated as adjustable parameters) represents the simulation data surprisingly well. Remarkably, the strain-field contribution, γelGB ( ψ), does not vanish at ∼ 15◦ where the dislocation cores start to overlap; rather, it goes through a maximum and then gradually decreases toward GB ( ψ), saturates simultaneously, zero. The sinusoidal core contribution, γcore however only as the angle ψ ≡ 2θ approaches 90◦ . By comparison, the total GB ( ψ) + γelGB ( ψ), converges much more rapidly, reaching GB energy, γcore a plateau already at ∼ 40◦ (see Fig. 10 for other GB planes). According to GB GB (90◦ ) = 2core /b; i.e., Eq. (24), the plateau energy is given by γ GB (90◦ ) ≡ γcore by the dislocation-core energy which depends on b and, hence, on the GB plane (see Fig. 11).

GB ENERGY [mJ/m 2 ]

800 (001) Twists Cu(LJ) CORE

600

400

200

STRAIN FIELD

0 0

15

30

45 2θ

60

75

90

Figure 11. Least-squares fit of the Read–Shockley expression (19) to the (100) twist-boundary data in Fig. 10 [7].

Structure and energy of grain boundaries

1973

As suggested by Smith [13], the above results demonstrate that, indeed, the energy of “true” high-angle GBs, i.e., those with a misorientation-independent energy, consists of both core and elastic strain-field contributions. Although for large angles the core energy clearly dominates, a residual strain energy arises from the elastic distortion of the dislocation cores.

3.3.1. Polyhedral-unit and broken-bond models That the atomic structure of high-angle GBs is composed of elastically distorted dislocation cores (“structural units”) is captured in the polyhedral-unit model [14, 15]. Clearly, the requirements of space filling at the GB and of the compatibility of the structural units with the adjoining grains cannot be satisfied simultaneously unless the polyhedra are elastically distorted in a manner that depends systematically on the misorientation. Unfortunately, because the model ignores these distortions and, moreover, offers no quantifiable measure of even the undistorted GB structure, it has virtually no predictive power as far as GB properties are concerned. The model can be at least partially quantified, by identifying the number of broken bonds per unit GB area associated with each type of structural unit [16]. Although elastic distortions of the units are thus neglected, it captures the broken-bond nature of the dislocation cores (i.e., the sinusoidal core contribuGB ( ψ), in Eq. (24) and Fig. 11) and hence becomes more quantitative tion, γcore as the elastic strain-field contribution, γelGB ( ψ), decreases towards zero for

ψ → 90◦ . In broken-bond models the atomic structure is generally quantified via the number of broken nearest-neighbor bonds per unit interface area. Given the rather weak elastic interactions between steps, such models work particularly well for surfaces because they quantify the dominating core contribution in Eq. (23). By contrast, these models are less quantitative for internal interfaces because they cannot capture the dominant elastic contribution due to the dislocations [16]. For example, whereas for small angles the GB energy is dominated by the (logarithmic) elastic contribution in Eq. (24), GB diffusion takes place within the cores (“pipe diffusion”) and is hence dominated by the (sinusoidal) core contribution. By contrast with the energy, GB diffusion can therefore be quantitatively predicted from a broken-bond description. In another group of models, known as hard-sphere models, the optimum translation parallel to the interface plane is assumed to be the one that minimizes the volume expansion at the GB. Although based on unrelaxed atomic structures, via the volume expansion at the GB these models provide at least a rough quantitative measure of the degree of GB structural disorder.

1974

D. Wolf

3.3.2. Short-range vs. long-range structural disorder In simulations, atomic-level GB structural disorder is best characterized in terms of the plane-by-plane radial distribution function, G(r), or its Fourier transform, S(k), averaged over the atoms in each of the lattice planes near the GB. The two structural measures are illustrated in Figs. 12 and 13 for the simulated (110) θ = 50.48◦ (11) twist GB and for the STGB on the (123) plane in fcc Pd. Figures 12(a) and (b) compare the G(r)’s for the atoms in the two center planes of each GB with that of bulk amorphous Pd (obtained by MD simulation of a rapidly quenched melt; see Ref. [17]). While the structure of the twist boundary in (a) is virtually indistinguishable from that of the bulk glass, with the characteristic split second peak indicated by the arrows, the more pronounced peak structure for the (123) STGB in (b) indicates a significant degree of residual crystallinity. These differences in the degree of long-range GB structural disorder can be quantified via the square of the planar structure factor, |S(kα )|2 (α = 1 or 2), where the wave vectors k1 and k2 lying within each lattice plane are chosen to be reciprocal lattice vectors in grains 1 and 2, respectively, with the smallest magnitude |kα |. Then, for a perfect-crystal plane at zero temperature belonging to grain 1, |S(k1 )|2 = 1 and |S(k2)|2 = 0 ; similarly, |S(k1 )|2 = 0 and |S(k2 )|2 = 1 for planes belonging to grain 2. At finite temperature, due to the vibration of the atoms the planar long-range order decreases by the Debye–Waller-factor. By contrast, in a liquid or an amorphous solid, due the absence of long-range order both |S(k1 )|2 and |S(k2 )|2 have near-zero values. The plane-by-plane structure factors for the two GBs are compared in Fig. 13. The center of the (110) twist GB is, indeed, highly disordered, as evidenced by the low values of |S(k)|2 ≈ 0.25 in the two center planes, indicating only a 25% residual crystallinity; by comparison, the (123) STGB is 94% crystalline even in the two planes immediately at the GB. In spite of these significant differences in the degree of long-range structural disorder, the two GBs are remarkably similar as far as short-range disorder is concerned, as evidenced by the rather similar shapes and widths of the nn peaks in G(r) in Fig. 12(a) and (b), indicating rather similar GB energies of 1027 mJ/m2 for the twist GB and 881 mJ/m2 for the STGB (see Ref. [17]). This comparison demonstrates that, in spite of fundamentally different long-range structures, the two GBs exhibit comparable degrees of short-range structural disorder, translating into similar energies. For GB properties dominated by the short-range disorder, such as the energy and self-diffusion behavior (and probably most other properties), the degree of long-range structural periodicity therefore appears to be of little importance (see also Ref. [17]). Hence, although the information in |S(k)| is equivalent to that in G(r), the

Structure and energy of grain boundaries

1975

(a) 8 Pd(EAM) T⫽0K

7 6

(110) Σ11 glass

G(r)

5 4 3 2 1 0 0

0.5

1 r [a 0 ]

1.5

2

(b) 8 Pd(EAM) T⫽0K

7 6

(123) STGB glass

G(r)

5 4 3 2 1 0 0

0.5

1 r [a 0 ]

1.5

2

Figure 12. Radial distribution function, G(r ), determined by molecular-dynamics simulation for the atoms in the two central planes of (a) the (110) θ = 50.48◦ (11) twist GB and (b) the STGB on the (123) plane in fcc Pd at zero temperature [22]. The dashed line indicates G(r ) for the glass obtained by quenching the melt to zero temperature.

latter provides a more useful structural measure as it is more directly related to GB properties. Finally, in a broken-bond model the nn peak in G(r) is replaced by the area under the peak. Because all the elastic information contained in the detailed peak shape is thus lost, strain-field effects due to line defects are thus ignored.

1976

D. Wolf 1

0.8

|S ( k )| 2

(123) STGB 0.6

0.4

(110) Σ11 twist Pd (EAM) T ⴝ0K

0.2

0 ⫺4

⫺2

0 z [a 0]

2

4

Figure 13. Square of the planar structure factor, |S(kα )|2 , in the two halves (α = 1 or 2) of the same two GBs considered in Fig. 12 [22].

G(r) therefore provides the most complete measure of GB structural disorder while a broken-bond model represents only a subset of this information.

3.3.3. Amorphous high-energy GBs The commonly observed existence of thin disordered intergranular films of nanometer thickness in ceramics, such as SiC and Si3 N4 , represents one of the most intriguing features in the atomic structures of GBs [18, 19]. Because these impurity-based films are usually formed during liquid-phase sintering, their presence at room temperature is usually thought to be a strictly kinetic, non-equilibrium phenomenon that has little to do with the structure of GBs in pure materials. However, recent simulations of pure materials raise the intriguing possibility that these films may be a manifestation of the thermodynamicequilibrium structure of high-energy GBs and thus may not require impurities for their stabilization. Early in the last century, based on the inherent low-temperature brittleness and high-temperature ductility of glasses, Rosenhain and his coworkers suggested that the GBs in a polycrystalline material represent an “amorphous cement” that holds the grains together (see, e.g., Refs. [20, 21]). However, based on observations in bicrystals and coarse-grained polycrystals of (a) good long-range structural periodicity within the GBs, (b) well-defined rigid-body

Structure and energy of grain boundaries

1977

translations and (c) the variation of the GB energy with misorientation, Rosenhain’s model has been thoroughly discredited. That Rosenhain appears to have been at least partially correct was suggested recently by extensive computer simulations of GBs in silicon, diamond and fcc palladium [22–24]. In complete qualitative agreement with the observations on the impurity-based amorphous intergranular films in ceramics, the simulations of impurity-free Si bicrystals revealed a common energy for all high-energy GBs and a universal, highly disordered (“confined amorphous”) structure of uniform thickness that is virtually indistinguishable from that of bulk amorphous Si. By contrast, low-energy GBs were found to exhibit very good crystalline order, also in agreement with the intergranular-film experiments. In these simulations, zero-temperature-relaxed GB structures, such as that in Fig. 14(a), were compared with the related high-temperature-equilibrated structure obtained either by high-temperature annealing or by growth from the melt containing well-oriented seeds followed by cooling to zero temperature. That the amorphous structure in Fig. 14(b) thus obtained has a significantly lower GB energy than the crystalline structure in (a) is demonstrated in Fig. 14(c), in which the plane-by-plane average energies per ion in excess of that in the perfect crystal are compared for the two GBs. The peak excess energy of 0.48 eV/atom for the crystalline structure in (a) far exceeds the average excess energy of bulk amorphous Si of 0.19 eV/atom (dashed line). By contrast, the most disordered plane of the amorphous GB in (b) exhibits an excess energy of only about 0.26 eV/atom, much closer to that of amorphous Si. More importantly, despite a broadening of the GB, the disordering results in an about 15% lower GB energy (given by the area under each peak). Having thus elucidated the thermodynamic and structural origins of the underlying driving forces, one might expect that all high-energy GBs should have similar, highly disordered “confined amorphous” structures. That this is, indeed, true is seen in Fig. 15 in which the energy profile of the 29 GB in Fig. 14(c) is compared with the profiles obtained for three other highangle twist boundaries on different Si planes: (100) φ = 31.89◦ ( 17), (110) φ = 44.00◦ (57) and (112) φ = 35.26◦ ( 35). Based on the above insights, one might expect that a low-energy GB is crystalline even after high-temperature equilibration. To test this hypothesis, Keblinski et al. [22] studied the (111), φ = 42.10◦ (31) twist GB which, like the 29 GB, is a high-angle boundary, however on the most widely-spaced (and hence, lowest-energy) plane in the diamond structure, with the rather low lattice-statics-relaxed energy of only 638 erg/cm2 . These simulations revealed that, indeed, no restructuring took place during the heat treatment and the GB energy remained totally unchanged. The projected “edge-on” zero-temperature structure of this GB shown in Fig. 16 indicates a highly ordered structure, which was confirmed by a quantitative characterization of the

1978

D. Wolf (a)

Crystalline (b)

Amorphous (c) 0.5 crystalline GB 0.4 bulk amorphous Si amorphous GB

0.3 0.2 0.1 0 ⫺16

⫺12

⫺8

⫺4

0 z [Å]

4

8

12

16

Excess energy (ev/atom)

Figure 14. (a) Crystalline, zero-temperature-relaxed structure and (b) amorphous, hightemperature-relaxed structure of the (100) θ = 43.60◦ (29) twist GB in Si. (c) shows the underlying plane-by-plane average-excess-energy-per-atom profiles for the two GBs by comparison with the average excess energy of bulk amorphous Si (dashed line) [22].

Structure and energy of grain boundaries ⫺3.95

1979

(100) Σ 29

bulk amorphous

(100) Σ 17 Energy [eV/atom]

⫺4.05

(110) Σ 57 (112) Σ 35

⫺4.15

⫺4.25

⫺4.35 ⫺2

⫺1

0 z [a0]

1

2

Figure 15. Energy profiles along the z direction. Figure shows data for four different highenergy GBs, (i) (100) φ = 43.60◦ (29) - solid squares, (ii) (100) φ = 31.89◦ (17) - open squares, (iii) (110) φ = 44.00◦ (57) - solid circles, (iv) (112) φ = 35.26◦ (35) – triangles [22].

structure via the corresponding bond-angle distribution function and a planeby-plane structure-factor analysis [22]. The atomic structure of the low-angle (100), φ = 10.60◦ (101) twist GB boundary on the (100) plane is also rather revealing. As seen in Fig. 17, the high-temperature annealed structure of this dislocation boundary on the same lattice plane as the high-angle GB in Fig. 14(b) consists of an array of highly disordered dislocation cores connected by more ordered, i.e., more perfect-crystal-like regions. Upon increasing the twist angle towards 45◦ , the continuously disordered structure in Fig. 14(b) is obviously obtained. The simulations of Keblinski et al. [22] also revealed that the “universal” GB energy of the structure in Fig. 14(b) obtained from the profiles in Figs. 14(c) and 15 is of similar magnitude as the excess energy of two noninteracting bulk crystal/amorphous interfaces brought into contact. The latter exhibit atomic structures, widths and energy profiles that are remarkably independent of the orientation of the crystalline substrate, an observation that may explain the practically uniform width of the constrained amorphous GB phase from one high-energy GB to another [18]. This comparison also demonstrates a slightly greater degree of confinement (i.e., a smaller width) of an amorphous GB coupled with a higher peak energy compared to a crystal/ amorphous interface; this provides further evidence for the stability of the

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z [a 0 ]

1

0

⫺1

⫺2 ⫺3

⫺2

⫺1

0 x [a 0 ]

1

2

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Figure 16. Zero-temperature structure obtained after high-temperature equilibration of the (111), φ = 42.10◦ (31) twist GB [22].

2

z [a 0 ]

1

0

⫺1

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⫺2

⫺1

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Figure 17. Zero-temperature structure obtained after high-temperature equilibration of the (100) φ = 10.60◦ (101) twist GB [22].

Structure and energy of grain boundaries

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amorphous GB phase against decomposition into two crystal/amorphous interfaces. These observations identify a simple criterion for the existence of an equilibrium disordered GB film: if atoms in an ordered GB have significantly higher energies than the atoms in the related bulk amorphous phase, the introduction of an amorphous film into the crystalline interface is energetically favorable. Although these results have yet to be confirmed experimentally for impurity-free GBs, they suggest that Rosenhain’s historic amorphous-cement model describes at least qualitatively the misorientation-independent structures of “true” high-angle GBs on high-energy GB planes (i.e., on non-special and non-vicinal planes; see also Fig. 11 and Section 3.2). These conclusions deduced from simulations of pure Si were later confirmed in similar studies of GBs in fcc Pd [17] (see also Figs. 12 and 13) and in diamond in which, however, structural disorder is partially replaced by coordination disorder [23].

4.

Conclusions

In this chapter we have reviewed the key concepts for the description of GB “structure” at three distinct levels, with particular emphasis on how at each level the underlying terminology can be used to connect “structure” with GB physical properties. At the level of the macroscopic GB geometry, two complementary types of descriptions are useful. The coincident-site lattice terminology is particularly suited for the description of dislocation (i.e., low-angle or vicinal) GBs because the spacing between the dislocations is governed by the misorientation. By contrast, the interface-plane terminology is best suited for the description of high-angle GBs in which the crystallographic orientation of the GB plane is the key parameter; four out of the five macroscopic degrees of freedom are therefore assigned to the GB plane. The concept of the atomic-level GB geometry provides a useful link between the macroscopic geometry and GB atomic structure. The two key parameters connecting to GB properties are the spacing of the lattice planes parallel to the GB and the magnitude of the GB planar unit-cell area. The intuitive connection here is that the “special” combination of a large interplanar spacing with a small planar unit-cell area minimizes the degree of GB structural disordering, giving rise to “special” GB properties. Finally, the degree of atomic-level GB structural disordering is captured quantitatively by two complementary structural measures, the plane-by-plane radial distribution function, G(r), and the related planar structure factor, S(k). Because most GB properties are governed by the degree of short-range structural disorder, G(r) is clearly the more useful of the two measures; by contrast,

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S(k) captures the degree of long-range disorder, which appears to be relatively unimportant to most GB properties. This is the reason for the relative success of broken-bond models which utilize the information contained in the area under the nearest-neighbor peak in G(r). However, because they ignore the information on the elastic distortions of the bonds contained in the detailed peak shape, these models capture only the effects of the dislocation cores and are hence most suited for the description of high-angle GBs. A remaining challenge lies in the establishment of a closer connection between the highest-energy GBs in pure materials and the amorphous, impurity-based GB phases, for example, in ceramic materials. Computer simulations suggest that Rosenhain’s amorphous-cement idea must be at least partially correct, although the width of his “amorphous glue”, typically of less than 1 nm thickness, is clearly much smaller than what he anticipated. Experimental verification of the existence of such extremely thin amorphous GB films in the highest-energy GBs in single-phase materials would finally, after a century-long discussion, unify our understanding of GB structure. Such a unified picture would include both, low- and intermediate-energy GBs with well-defined rigid-body translations and an atomic structure and properties that depend strongly on the misorientation and the GB plane, and high-energy GBs (“true high-angle” or “Rosenhain boundaries”) with a universal, highly confined amorphous structure and misorientation-independent properties.

Acknowledgments This work was supported by the US Department of Energy, BES-Materials Science under contract W-31-109-Eng-38.

References [1] D. Wolf, Chapter 1 in Materials Interfaces, Atomic-level Structure and Properties, D. Wolf and S. Yip, (eds.), Chapman & Hall, New York, pp. 1–57, 1992. [2] D.G. Brandon, B. Ralph, S. Ranganathan, and M.S. Wald, Acta Metall., 12, 813, 1964. [3] W. Bollmann, Crystal Defects and Crystalline Interfaces, Springer, New York, 1970. [4] A.P. Sutton and R.W. Balluffi, Interfaces in Crystalline Materials, Clarendon Press, Oxford, 1995. [5] D. Wolf, J. Phys. Colloque, C4 46, C4–197, 1984. [6] D. Wolf and J.F. Lutsko, Z. Kristallographie, 189, 239, 1989. [7] D. Wolf and K.L. Merkle, Chapter 3 in Materials Interfaces, Atomic-level Structure and Properties, D. Wolf and S. Yip (eds.), Chapman & Hall, New York, pp. 87–150, 1992.

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[8] G. Hasson, J.-Y. Boos, I. Herbeuval, M. Biscondi, and C. Goux, Surf. Sci., 31, 115, 1972. [9] C. Herring, In: Structure and Properties of Solid Surfaces, R. Gomer and C.S. Smith (eds.), University of Chicago Press, Chicago, 1953. [10] D. Wolf and J. Jaszczak, Chapter 26 in Materials Interfaces, Atomic-level Structure and Properties, D. Wolf and S. Yip (eds.), Chapman & Hall, New York, pp. 662–690, 1992. [11] V.I. Marchenko and A.Y. Parshin, Sov. Phys.-JETP, 52, 129, 1980. [12] W.T. Read and W. Shockley, Phys. Rev., 78, 275, 1950. [13] C.S. Smith, In: Metal Interfaces, ASM, Cleveland, OH, p. 62, 1952. [14] M. Weins, H. Gleiter, and B. Chalmers, J. Appl. Phys., 42, 2639, 1971. [15] A.P. Sutton and V. Vitek, Phil. Trans. R. Soc. Lond. A, 309, 1, 1983. [16] D. Wolf, J. Appl. Phys., 68, 3221, 1990. [17] P. Keblinski, D. Wolf, S.R. Phillpot, and H. Gleiter, Phil. Mag. A, 79, 2735, 1999. [18] D.R. Clarke, In: L.C. Dufour, C. Monty, and G. Petot-Ervas (eds.), Surfaces and Interfaces of Ceramic Materials, Kluwer Academic Publishers, Boston, MA, 1989. [19] H.-J. Kleebe, M.K. Cinibulk, R.M. Cannon, and M. R¨uhle, J. Am. Ceram. Soc., 76, 1969, 1993. [20] W. Rosenhain and D. Ewen, J. Institute Metals, 10, 119, 1913. [21] K.T. Aust and B. Chalmers, Metal Interfaces, ASM, Cleveland, OH, 1952. [22] P. Keblinski, S.R. Phillpot, D. Wolf, and H. Gleiter, J. Am. Cer. Soc., 80, 717, 1997. [23] P. Keblinski, D. Wolf, S.R. Phillpot, and H. Gleiter, J. Mater Res., 13, 2077, 1999. [24] P. Keblinski, D. Wolf, S.R. Phillpot, and H. Gleiter, MRS Bull., 23(9), 36, 1998.

6.10 HIGH-TEMPERATURE STRUCTURE AND PROPERTIES OF GRAIN BOUNDARIES Dieter Wolf Materials Science Division, Argonne National Laboratory, Argonne, IL 60439, USA

The evolution of polycrystalline microstructures under the driving forces of temperature and stress, i.e., during grain growth and plastic deformation, is controlled by the high-temperature structure and dynamical behavior of grain boundaries (GBs). These microstructural processes, driven by temperature and/or stress, involve three distinct types of dynamical GB phenomena, namely GB diffusion, GB migration and GB sliding. All three are known to depend strongly on the GB energy. A better understanding of microstructural evolution in terms of the behavior of the individual interfaces therefore requires insight on the interrelation among GB structure, GB energy and these dynamical GB processes. A polycrystal generally contains GBs of different types covering a wide spectrum of energies and, hence, GB properties. As described in Chapter 6.9, much work of recent decades has suggested a distinction among three qualitatively different types of GBs: (a) special boundaries, (b) dislocation boundaries and (c) high-angle GBs [1]. Among these, the high-angle boundaries are the most important – yet least understood – ones because their high-temperature behavior is rate-controlling in many phenomena, such as grain growth and plastic deformation, particularly in nanocrystalline materials, i.e., polycrystals with a grain size of nanometer dimensions (see Chapter 6.13). The atomic structure of high-angle GBs is characterized by a complete overlap of the GB dislocation cores [2, 3], giving rise to a more or less continuously distributed type of structural disorder along the interface and a GB energy which is entirely independent of the GB misorientation (see, e.g., Figs. 10, 14(b) and 17 in Chapter 6.9). Whereas high-angle GBs on the 1985 S. Yip (ed.), Handbook of Materials Modeling, 1985–2008. c 2005 Springer. Printed in the Netherlands.

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lowest-index plane of a given crystal lattice, such as the (111) plane in fcc crystals, exhibit relatively little structural disorder – and hence a rather low GB energy (see, e.g., Fig. 16 in Chapter 6.9), boundaries on most other lattice planes are highly disordered, with a consequently very high energy (see Figs. 12(a) and (13) in Chapter 6.9). Although these high-energy, high-angle GBs may represent only a relatively small fraction of the GBs in a polycrystal, their very high mobility and diffusivity coupled with a rather low sliding resistance gives them a potentially rate-controlling role in many high-temperature properties of polycrystalline materials. A structural model that accounts for the highly non-perfect-crystal-like properties of these GBs, and one that connects also with other types of structural disorder, such as the amorphous state, is therefore clearly needed. Here we focus on recent molecular-dynamics (MD) simulations investigating the high-temperature structure and self-diffusion behavior of such highenergy, high-angle GBs. Many of the earlier simulations of high-temperature GB structure remain controversial (for a critical review, see [4]) in that they predict “premelting” at the GBs, i.e., GB disordering below the bulk melting point, Tm , with a width of the liquid-like GB layer that diverges as T → Tm , where the GB is completely wetted by the liquid. It was not until the simulations had provided a clearer understanding of the melting process itself [5, 6], of the related superheating limit [6, 7], and of the dependence of lowtemperature GB structure on the GB energy [3, 8], that a clearer picture of high-temperature GB structure and dynamical GB behavior started to develop. These simulations demonstrate that above the glass-transition temperature, Tg , the high-energy boundaries undergo a reversible structural and dynamical transition from a confined amorphous solid to a confined liquid. By contrast with the bulk glass transition, however, this equilibrium transition is continuous and thermally activated, starting at Tg and completed only at the melting point, Tm , at which the entire amorphous film is liquid. The coexistence of the confined amorphous and liquid phases in this two-phase region of less than 1 nm thickness has a profound effect on grain-boundary self-diffusion. The picture that emerges suggests that upon heating from zero temperature through the melting point, the highest-energy tilt and twist GBs in the microstructure undergo a reversible transition from a low-temperature, solid structure with a rather high, GB energy dependent activation energy, e.g., for GB diffusion, to a highly confined, liquid GB structure with a universal activation energy that is independent of the GB misorientation and related to selfdiffusion in the melt. This reversible structural transition should also manifest itself in an extremely high GB mobility and a low sliding resistance above the solid-to-liquid transition. It is this unique combination of high-temperature properties that appears to be the cause for the rate-controlling role of these GBs in grain growth and deformation of polycrystalline materials.

High-temperature structure and properties of grain boundaries

1.

1987

High-Energy, High-Angle Grain Boundaries in Silicon

As discussed in Chapter 6.9, recent simulations of silicon GBs equilibrated at high temperatures and subsequently cooled to zero temperature revealed a highly disordered equilibrium structure of uniform thickness and energy of all the high-energy boundaries while low-energy boundaries remained crystalline (see Figs. 14–17 in Chapter 6.9) [8, 9]. In Fig. 1(a) and (b) the local radial and bond-angle distribution functions for the atoms in the central two (100) planes of the (100) φ = 43.60 (29) twist boundary in Si determined by means of the Stillinger–Weber potential [9] are compared with the related distributions obtained for bulk amorphous Si. (The structure and energy of this GB are representative for those of other high-energy GBs [8].) The comparison in Fig. 1 reveals a virtually complete absence of crystalline order in the GB, in favor of short-range order that is strikingly similar to that seen in bulk amorphous Si. Qualitatively identical results were obtained via a tight-binding description of Si bonding [10] and the Tersoff potential [11], suggesting the robustness of these conclusions. Consistent with this, fully quantum-mechanical calculations for the (100) φ = 36.87◦ (5) twist GB in Ge yielded a disordered structure remarkably similar to that of amorphous Ge [12]. Although these simulation results have yet to be confirmed experimentally for impurityfree Si GBs, the misorientation-independent structures of these GBs seem to be what Rosenhain described in his historic “amorphous-cement” model (see also Chapter 6.9) [13]. Using the idea that liquid Si is significantly denser and better coordinated than either crystalline or amorphous Si, Keblinski et al. [14] investigated the possibility of an amorphous-to-liquid transition in high-energy GBs in Si. Their idea was that a transformation of an amorphous into a liquid-like GB structure should be detectable via a contraction at the GB, accompanied by an increase in the average nearest-neighbor (nn) coordination and changes in the nn bond-angles. Figure 2 compares the GB excess volume per unit GB area, δV (T ), between T = 0 K and Tm for two high-angle GBs: the high-energy (100) φ = 43.60◦ (29) twist GB with the amorphous structure shown in Figs. 1(a) and (b) and the (111) φ = 50.48◦ (31) twist GB whose structure remains crystalline all the way up to Tm [8]. It is interesting to note that at 0 K, the (100) GB is expanded relative to the perfect crystal whereas the (111) GB exhibits a volume contraction. The latter arises from the fact that all Si bonds point directly across the (111) GB and are therefore stretched during the 111 twist rotation, with a consequent contraction during relaxation so as to recover, as much as possible, perfect-crystal bond lengths. [8] More strikingly, however, as temperature increases the value of δV (T ) for the (111) GB remains unchanged; by

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(a)

10 bulk amorphous (100) Σ29

8

G(r)

6 4 2 0 0

0.3

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r [a 0 ] (b) 0.025 bulk amorphous (100) Σ29

P (cos θ)

0.02 0.015 0.01 0.005 0 ⫺1

⫺0.5

0

0.5

1

cos θ Figure 1. (a) Radial distribution function, G(r ), for the (100) φ = 43.60◦ (29) twist GB. The solid line shows G(r ) for bulk amorphous silicon obtained for the same interatomic potential. The atomic structure of this boundary is shown in Fig. 14(b) in Section 6.9. (b) Angular distribution function, P(cos θ), for the same GB. The solid line shows P(cos θ) for bulk amorphous silicon (redrawn from Keblinski et al., 1999).

High-temperature structure and properties of grain boundaries Tg

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Tm

0.01

ψV [a 0 ]

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(100) ⌺29

⫺0.01 ⫺0.02 (111) ⌺31

⫺0.03 ⫺0.04 0

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1000

1500

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T [K] Figure 2. Temperature dependence of the volume expansion per unit GB area, δV (in units of the zero-temperature lattice parameter, a0 ) for two high-angle GBs in silicon: the high-energy (100) φ = 43.60◦ (29) twist boundary and the low-energy (111) φ = 42.10◦ (31) twist GB. The bulk glass-transition temperature, Tg , and melting point, Tm , are indicated on the top [14].

contrast, at about 1000 K the (100) GB starts to contract continuously all the way up to Tm , indicating the formation of a liquid intergranular film. Finally, for T > Tm both systems melt by propagation of two crystal–liquid interfaces from the GBs until all crystalline material is consumed by the liquid [5–7]. Figure 3 demonstrates that the response of the (100) GB to temperature change is completely reversible, indicating the equilibrium nature of the amorphous-to-liquid transition [14]. Also, the fact that with increasing temperature, the contraction progresses continuously rather than occurs suddenly at a certain temperature, indicates that the structural transition involves a twophase mixture and progresses continuously between Tg and Tm . A detailed analysis of the underlying structural changes revealed a significant increase with increasing temperature in the average GB-atom coordination (from 4.21 at Tg to 4.55 at Tm ) accompanied by a decrease in the fraction of fourcoordinated GB atoms (from 0.78 at Tg to 0.49 at Tm ). Simultaneously, as shown in Fig. 4 the bond-angle distribution function changes from being similar to that of bulk amorphous Si with a broad peak near 110◦ indicating predominantly tetrahedral bonding (see Fig. 1b), to one that is similar to that of

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t [ns] Figure 3. Response to thermal cycling of the volume expansion per unit area, δV, normal to the (100) 29 twist GB. The plot illustrates the reversibility of the transition between the confined amorphous and liquid GB phases; t is the simulation time [14].

molten Si, with its peak shifted towards 90◦ , a characteristic of the almost six-coordinated high-temperature liquid [14]. The amorphous-to-liquid GB transition should profoundly affect the mechanism and activation energy for GB self-diffusion. Unfortunately, below Tg both lattice and GB diffusion are too slow to be observable on a typical MD time scale; in fact, even above Tg the atom mobility in the perfect-crystal regions surrounding the GB is negligible compared to that in the GB. The total observed mean-square-displacement (MSD) summed over all the atoms is therefore dominated by the in-plane (x − y) motions of the GB atoms, (x)2 + (y)2 . In analogy to the Gibbsian excess energy of the GB, this MSD normalized to the GB area represents the integrated, Gibbsian excess MSD of the GB atoms, from which the diffusion flux in the GB can be determined as follows: (x)2 + (y)2 , (1) 4t A where is the atomic volume and t the simulation time; DGB is the GB self-diffusion constant and δD is the “diffusional width” of the GB (to be DGB δD =

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0.02 GB (1600K)

bulk amorph. (900K)

P(ψ)

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bulk liquid (1600K) 0.01

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0 60

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ψ Figure 4. Comparison of the bond-angle distribution functions, P(θ), for the confined amorphous and confined liquid GB phases with those for bulk amorphous and supercooled liquid Si, respectively [14]. In perfect-crystal Si at T = 0 K, P(θ) exhibits a single, δ-function peak at the tetrahedral angle, θt = 109.47◦ .

distinguished from its “structural width”, δS ; see below). We note that DGB δD has the dimensions of (length)3 /time. As seen from the Arrhenius plot in Fig. 5, the simulated values of DGB δD (right scale) yield an activation energy of 1.4 ± 0.1 eV. By comparison, the self-diffusion constant of the bulk liquid, Dliq (left scale in Fig. 5), exhibits the much lower activation energy of 0.70 ± 0.05 eV. If one were to assume that δD is temperature independent, thus assigning the entire temperature dependence of DGB δD to that of DGB alone, one would be led to conclude that the rather different activation energies for DGB and Dliq dispel the notion that the GB is, indeed, liquid. However, by contrast with this interpretation, Fig. 6 reveals a strong, thermally activated increase of δD with increasing temperature, with an activation energy of 0.65 ± 0.05 eV [14]. These values of δD were determined from the expression [14] NGB , (2) A where NGB is the number of diffusing GB atoms with atomic volume . The structural GB width shown in Fig. 6 was defined [14] via the excess potential-energy profile of the GB, as the width-at-half-maximum of planar profiles like the ones shown in Fig. 7. According to Fig. 7, the excess-energy profiles at T = 0 and 900 K are practically indistinguishable. By contrast, the δD =

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1/T [eV⫺1] Figure 5. Arrhenius plot for self-diffusion in the (100) 29 Si GB (right scale) and in the bulk, supercooled Si melt (left scale) [14]. DGB δD is the diffusion flux in the GB with a “diffusional width” δD .

profile broadens above Tg , from approximately four (100) planes (or δS = a0 = 5.5Å) at low temperatures to approximately six planes at 1600 K (δS = 8.2Å), indicating that two initially crystalline planes have become liquid, in addition to the initially amorphous GB phase (see Fig. 14b in Chapter 6.9). The rather small value of δS = δD ( = 8.2 Å) near the melting point is noteworthy as it underscores the severe confinement of the liquid. To further elucidate this confinement, Keblinski et al. compared this value of the GB width with the width of a bulk crystal–liquid (100) interface in Si, δc−liq = 6 Å, at T = T m determined for the SW potential [14]. The fact that all the way up to the melting point 2δc−liq > δS = δD demonstrates the stability of the confined liquid against decomposition into two unbound crystal–liquid interfaces. The results in Figs. 5 and 6 for DGB δD , δD and Dliq suggest that the activation energy of 1.4 ± 0.1 eV for DGB δD in the 29 GB above Tg involves two distinct processes: first, the thermally activated formation of the confined liquid from the confined-amorphous phase, with an activation energy E F = 0.65 ± 0.05 eV (see Fig. 6) and second, the subsequent bulk-liquid like atom migration, with an activation energy of E liq = 0.70 ± 0.05 eV (see Fig. 5). It follows that above Tg , the mechanism for self-diffusion in the 29 GB should be rather similar to that in the bulk melt. As a consequence, at Tm

High-temperature structure and properties of grain boundaries

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structural width, ψS 10

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1/T [eV⫺1] Figure 6. Arrhenius plot for the ‘diffusional width’, δD , of the GB, indicating that the formation of the confined liquid in the confined-amorphous matrix is a thermally activated process. For comparison, the only weakly temperature dependent “structural width” of the GB, δS , is also shown [14].

the self-diffusion constants DGB and Dliq should be of comparable magnitude. In fact, the value of δD = 8.2 Å combined with the value for DGB δD = 1.43 × 10−12 cm3 /s obtained by extrapolation to Tm in Fig. 6, yields DGB = 2 × 10−5 cm2 /s, compared to the value Dliq = 6 × 10−5 cm2 /s [14]. Given the uncertainties in the underlying activation energies and the degree of arbitrariness in the definition of the GB width, this agreement is rather good.

2.

Evidence for Structural Transition in fcc-Metal Grain Boundaries

The above simulation results for silicon suggest the intriguing possibility that, in the spirit of Rosenhain’s amorphous-cement model [13], even in fcc metals the high-energy GBs might exhibit a universal, highly disordered structure that is liquid-like above a certain critical temperature and more or less disordered, but solid at lower temperatures. Unfortunately, compared to Si for which the coordination and density of the melt differ significantly from

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Figure 7. Plane-by-plane excess-potential-energy profiles for the (100) 29 GB at T = 0, 900 and 1600 K, indicating that the transition from a confined-amorphous to a confined-liquid GB structure is accompanied by an increase in the “structural width” of the GB, δS , defined as the width-at-half-maximum of such profiles [14].

those in the amorphous and crystalline phases, in fcc metals the structural fingerprint of such a transition is much more subtle because the coordination and density of the glass (producible by simulations only, by rapidly quenching the melt) are essentially the same as those of the melt. Recent self- and impurity-diffusion experiments by Budke et al. [15] on Cu 001 tilt GBs near the 5 (36.87◦ ) misorientation (see Fig. 8) revealed a crossover between a strongly misorientation-dependent low-temperature diffusion mechanism with a high activation energy and a high-temperature mechanism with a 60% lower activation energy that is independent of the GB misorientation. This transition was interpreted as a GB structural transition, from an ordered low-temperature GB structure, with well-defined atomic jump vectors, to a disordered high-temperature structure with randomly directed jump vectors [15]. Consistent with these experimental observations, recent MD simulations of GB migration (see Fig. 9a) and GB diffusion (see Fig. 9b) in a (100) φ = 43.60◦ (29) twist bicrystal described by a Lennard–Jones potential fitted to Cu suggest a crossover from a solid-like mechanism at low temperatures to a liquidlike mechanism at high temperatures. According to Fig. 9, in both types of

High-temperature structure and properties of grain boundaries

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Figure 8. Arrhenius plots for Au diffusion along [001] tilt GBs in Cu near the = 5 misorientation of θ = 36.87◦ [15].

GB processes this crossover occurs at a temperature of about T ∼ 750 K (or ∼ 0.62Tm ). Detailed analysis revealed a crossover in the underlying atomiclevel mechanisms, from ones based on solid-like atom hopping at low temperatures to ones involving liquid-like reshuffling of the GB atoms at high temperatures [16]. As seen in Figs. 9(a) and (b), this change in mechanism is accompanied by an about 50% decrease in the activation energies for both processes [16].

3.

Self-Diffusion in High-Energy fcc-Metal GBs

To systematically investigate the existence of a crossover from a solid lowtemperature to a liquid-like high-temperature GB structure, Keblinski et al. [17] have recently performed extensive simulations of self-diffusion in several well-chosen high-energy tilt and twist boundaries in Pd. These simulations not only confirm that, analogous to their Si work, a GB structural transition, indeed, takes place even in fcc-metal GBs, but also that (i) the transition, indeed, proceeds from a solid-like low-temperature to a liquid-like

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[eV⫺1]

Figure 9. Arrhenius plots for (a) the GB mobility, m (in units of 10−8 m4 /Js) and (b) the GB diffusivity, δ D GB (see Eq. (1)) for the (001) φ = 43.60◦ (29) twist GB in Cu as described by a Lennard–Jones potential [16].

high-temperature structure and (ii) consistent with the experiments [15], the transition temperature depends strongly on the GB energy. Consistent with their earlier study of Si GBs [14], at high-temperatures four different high-energy GBs (two tilt and two twist GBs; see Table 1) were found to exhibit very similar activation energies (see Fig. 10), with practically the same absolute values of the product of the GB diffusivity and the diffusional GB width, DGB δD (right-hand scale in Fig. 10). For comparison, Fig. 10 also shows the Arrhenius plot for self-diffusion in the bulk melt, Dliq (left-hand scale in Fig. 10). We note that the “universal” activation energy for

High-temperature structure and properties of grain boundaries

1997

Table 1. GBs chosen for the diffusion study in fcc Pd of Keblinski et al. [17], listed in order of decreasing GB energy. ( denotes the familiar inverse density of coincidence sites.) Also listed are the activation energies for GB self-diffusion in the high- and low-temperature regimes and the crossover temperature, Tc . Values of the square of the square of the planar structure factor, |S(k)|2 , are given for the most disordered plane in the center of each GB GB

Zero-temp. GB Zero-temp. Activation energy Activation energy energy (erg/cm2 ) |S(k)|2 high T (eV) low T (eV) Tc (K)

(110) φ = 50.48◦ (11) (113) φ = 67.11◦ (9) (310) (5) STGB (123) (7) STGB

1027 1021 952 881

0.25 0.28 0.82 0.94

0.61 ± 0.05 0.56 ± 0.05 0.65 ± 0.05 0.70 ± 0.10

Tm 1400 K

0.88 ± 0.05 1.50 ± 0.10

< 700 < 700 900 1300

1000 K

10⫺8

10⫺17

10⫺9

10⫺18 0.60eV (110) Σ11 twist (311) Σ9 twist (310) Σ5 sym. tilt

10⫺10

10⫺19

D GBψD [m3/s]

D liq [m2/s]

melt 0.41eV

10⫺20

10⫺11 5

6

7

8 9 1/kT [eV⫺1]

10

11

12

Figure 10. Comparison of self-diffusion constants in Pd grain boundaries and in the melt [17]. Right-hand scale: high-temperature regime (1000–1400 K) in the Arrhenius plots for the diffusion flux, DGB δD , for three high-energy GBs (see also Table 1). Left-hand scale: self-diffusion constant in the bulk, 3d periodic Pd melt supercooled all the way down to 1000 K. The melting point for the Pd embedded-atom-method potential is estimated to be about Tm ∼ 1500 K.

GB diffusion in this high-temperature regime (of E GB = 0.60 ± 0.05 eV) is distinctly higher than that associated with self-diffusion in the supercooled melt (of E melt = 0.41 ± 0.03 eV). On the other hand, E GB is significantly lower than that for self-diffusion in bulk perfect-crystal Pd via monovacancies (with a simulated activation energy of 2.41 eV [18]).

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To elucidate the difference between the activation energies for self-diffusion in these high-energy GBs and in the supercooled melt, we recall that GB diffusion involves the diffusion flux, DGB δD, rather than DGB alone. As first pointed out in the Si work (see Figs. 5 and 6) [14], only under the assumption that δD is temperature independent can the entire temperature dependence of DGB δD be assigned solely to that of DGB . To test this assumption, as in their earlier Si study Keblinski et al. [17] followed the diffusion of individual GB atoms over time. These Pd simulations [14, 17] also exposed the important distinction between the structural width of the GB, δ S (defined, for example, via the excess-energy profile across the GB), and the diffusional width of the GB (defined by the number of GB atoms actually responsible for the observed diffusion flux, i.e., for the excess mean-square-displacement of the GB atoms [14, 17]). As shown in Fig. 11, they observed that δS increases only slightly with increasing temperature (by about a factor of two between T = 0 K and T m ). However, the fraction of mobile GB atoms which governs δD was found to increase much more rapidly, involving a thermally-activated process as evidenced by the Arrhenius plot in Fig. 11 for the (110) 11 twist GB in Pd. Consistent with the Si simulations [14], the assumption of a temperatureindependent diffusional width is therefore inappropriate for fcc metals as well. Adding the activation energy of E δ = 0.22 ± 0.02 eV for δD (Fig. 11) go the value for diffusion in the melt, E melt = 0.41 ± 0.03 eV (Fig. 4), yields an 10

GB width [a 0 ]

structural width, δS

0.22 eV

1

diffusional width, ψD

0.1 8

9

10 1/kT

11

12

[eV⫺1]

Figure 11. Arrhenius plot for the “diffusional GB width”, δD , of the (110) φ = 50.48◦ (11) twist GB in Pd, indicating that the number of actually diffusing GB atoms is thermally activated. For comparison, the “structural GB width”, δS , with a much weaker temperature dependence, is also shown [17].

High-temperature structure and properties of grain boundaries

1999

activation energy of 0.63 ± 0.05 eV, which is in remarkable agreement with the “universal” high-temperature activation energy of ∼ 0.60 ± 0.05 for DGB δD (see Fig. 10 and Table 1). Similar to their Si work, Keblinski et al. therefore concluded that the high-temperature GB diffusion mechanism is liquidlike, involving two distinct processes: first the thermally activated formation of clusters of liquid from the solid GB, followed by liquid-like migration of the atoms within these clusters with an activation energy similar to that in the bulk melt; i.e., 0 δD0 exp[−(E δ + E melt )/kT ]. DGB δD = DGB

(3)

Some insight into the underlying self-diffusion mechanism can be gained from the analysis of the temporal evolution of the pattern of the diffusing GB atoms. Figs. 12(a) and (b) show two edge-on snapshots of the (110) 11 twist GB. (a)

(b)

Figure 12. Initial positions of those atoms that moved by at least one nearest-neighbor distance within 10 000 simulation time steps, projected onto the x − z plane for the (110) 11 twist boundary. (a) T = 1000 K; (b) T = 1400 K. The thin horizontal lines indicate the structural width of the GB defined in Eq. (3) and determined via the excess-energy profile of the GB (see also Fig. 11).

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The figure shows the initial positions of only those GB atoms that moved by at least one nearest-neighbor distance within a time interval of 10 000 time steps at (a) T = 1000 K and (b) T = 1400 K. The dashed horizontal lines indicate the structural width of the GB, δS , determined in simulations by the energy profile of the GB (see Fig. 13). A comparison between the two snapshots reveals (a)

4 (123) STGB

T⫽1400 K

bulk liquid

g(r)

3

2

1

0 0

0.5

1

1.5

2

r [a 0] (b)

8 7

(123) STGB

T⫽0 K

glass

6 g(r)

5 4 3 2 1 0 0

0.5

1 r [a 0]

1.5

2

Figure 13. Radial distribution functions, g(r ), for the atoms in the two central planes of the STGB on the (123) plane of Pd at (a) T = 1400 K compared to that of the bulk, supercooled melt and (b) T = 0 K compared to that of the bulk glass produced by MD simulation [17].

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2001

not only that δS increases somewhat with increasing temperature but, more importantly, that the fraction of mobile GB atoms increases significantly with increasing temperature. The diffusional width, δD , therefore increases much more rapidly with increasing temperature than the structural width, δS . That the high-temperature GB structure is, indeed, liquid-like was demonstrated [17] via the comparison of the radial distribution function for the bulk melt with the local radial distribution functions at T = 1400 K in Figs. 13(a) and 14(a) for the atoms in the centers of, respectively, the symmetric tilt boundary (STGB) on the (123) plane (Fig. 13a) and the (110) φ = 50.48◦ (11) twist GB (Fig. 14a). That the diffusion mechanism is, indeed, liquid-like in this high-temperature regime was shown directly by MD simulation, revealing a virtually complete absence of angular correlations between successive jump vectors of the diffusing atoms [17]. For comparison, Figs. 13(b) and 14(b) show the corresponding 0 K GB structures together with the radial distribution function of the bulk Pd glass (obtained by quench from the melt). By contrast with these high-temperature results, at lower temperatures the activation energy for GB self-diffusion can be significantly higher and strongly dependent on the GB energy. According to Fig. 15 and Table 1, the two STGBs exhibit a crossover in the diffusion behavior at different transition temperatures, Tc , from a high-temperature regime with the relatively low, universal activation energy to a low-temperature regime with a significantly higher activation energy that depends on the GB energy. For the case of the STGB on the (123) plane, Keblinski et al. demonstrated that, indeed, the diffusion mechanism in this low-temperature regime involves atom jump vectors in discrete directions, i.e., on a crystal lattice. The fact that the twist boundaries do not exhibit such a transition in Fig. 15 indicates that Tc is lower than 700 K, where the mean-square-displacement of the GB atoms became too small to be detectable by MD simulation [17]. On the other hand, that a structural transition must take place, for example, in the (110) 11 twist GB is seen from the comparison in Fig. 13(a) and (b) of the local radial distribution functions at 1400 and 0 K, respectively: the 0 K structure in (b) clearly exhibits the familiar split second peak of the amorphous solid (indicated by arrows).

4.

Short-Range vs. Long-Range Structural Disorder

The comparison between the (123) STGB and the (110) 11 twist GB in Figs. 13 and 14 reveals that, in spite of having rather similar GB energies (see Table 1) and similar, liquid-like high-temperature structures, the two GBs have very different low-temperature atomic structures: whereas the STGB in Fig. 13(b) is basically crystalline, the twist GB in Fig. 14(b) is essentially

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(a) 4 (110) Σ11

T⫽1400 K

bulk liquid

g(r)

3

2

1

0 0

0.5

1

1.5

2

r [a 0 ] (b) 8 7

(110) Σ11

T⫽0 K

glass

6

g(r)

5 4 3 2 1 0 0

0.5

1

1.5

2

r [a 0 ] Figure 14. Radial distribution functions, g(r ), for the atoms in the two central planes of the (110) φ = 50.48◦ (11) twist GB in Pd at (a) T = 1400 K compared to that of the bulk, supercooled melt and (b) T = 0 K compared to that of the bulk glass produced by MD simulation [17]. The arrows in (b) indicate the familiar split second peak associated with the glass.

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Tm 1400 K

1000 K

10⫺18

700 K (110) Γ11 twist (311) Γ9 twist

ψD GB [m3/s]

10⫺19

10⫺20 (123) sym. tilt 10⫺21 (310) sym. tilt 10⫺22 8

10

12 1/kT [1/eV]

14

16

Figure 15. Arrhenius plots for the diffusion flux, DGB δD , between 700 and 1400 K for all four Pd GBs considered in the study of Keblinski et al. (compare with Fig. 10) [17].

amorphous. This raises the question why, in spite of these qualitatively different 0 K atomic structures, the high-temperature behavior of these GBs is so similar. As demonstrated by Keblinski et al. [17], elucidation of the precise manner in which GB structure affects GB diffusion requires a better understanding of the interrelation between GB energy and atomic structure, with particular emphasis on the distinction between long-range and short-range GB structural disorder. Given plane-by-plane radial distribution functions, such as those in Figs. 13 and 14 in the center of each GB, the degree of long-range GB structural order can be quantified by the square of the planar structure factors, S(kα ), within each lattice plane in the two rotated crystals (α = 1 and 2) forming a given bicrystal [5, 6]. For each plane, S(kα ) is defined as the Fourier transform of the average g(r) for this particular plane. For a perfect-crystal plane at zero temperature, |S(k1 )|2 = 1 and |S(k2)|2 = 0 for planes belonging to the bulk of grain 1; similarly, |S(k1 )|2 = 0 and |S(k2 )|2 = 1 for planes belonging to the bulk of grain 2. At finite temperature, due to the vibration of the atoms, the long-range order within the planes decreases by the Debye–Wallerfactor. By contrast, in a liquid or an amorphous solid, due the absence of

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long-range order both |S(k1)|2 and |S(k2 )|2 fluctuate near zero at any temperature. The combination of |S(k1 )|2 and |S(k2 )|2 thus provides a convenient quantitative measure for the overall degree of crystallinity in each lattice plane [5, 6]. The zero-temperature plane-by-plane structure factors associated with the two GBs in Figs. 13 and 14 are compared in Fig. 16. Consistent with the g(r) in Fig. 14(b), the center of the (110) twist GB is, indeed, highly disordered at 0 K, as evidenced by the low values of |S(k)|2 = 0.25 in the two center planes, indicating only a 25% residual crystallinity. By comparison, the (123) STGB is 94% crystalline even in the two planes immediately at the GB. This qualitative difference is mostly due to the extremely high degree of structural periodicity exhibited by all STGBs: on any given lattice plane, the STGB has the smallest possible planar unit cell of any GB on that plane; by contrast, any twist boundary on the same plane has a significantly larger unit cell, i.e., less periodicity [19]. Whereas the degree of long-range GB structural disorder is governed by the higher-order peaks in g(r), the GB energy is dominated by the nn peak as it represents a weighted, highly damped sum over the peaks in g(r). Assuming, for example, that the interatomic interactions can be approximated by a pair

1

|S(k)|2

0.8

(123) STGB (110) Σ11 twist

0.6 0.4

Pd (EAM) T⫽0 K

0.2 0

⫺4

⫺2

0 z [a 0 ]

2

4

Figure 16. Square of the planar structure factors, |S(kα )|2 , in the two halves (α = 1 or 2) of the (123) STGB and the (110) φ = 50.48 (11) twist GB in Pd at zero temperature. The underlying radial distribution functions in the center of each GB are found in Figs. 6(b) and 7(b) [17].

High-temperature structure and properties of grain boundaries

2005

potential, the excess energy, γ , per unit GB area, A, of a bicrystal may be written as 1 γ= A

1 N N [gbi (rij ) − gid (rij )]V (rij ) , N i=1 i=1

(4)

where gbi (rij ) and gid (rij ) are the radial distribution functions of all N atoms in the bicrystal and the ideal reference crystal, respectively. Since in virtually all materials the interatomic interactions fall off rather quickly with decreasing separation between the atoms, the GB energy is controlled by the contribution in Eq. (4) from the nearest-neighbor (nn) peak, i.e., by the shape and width of the peak and the area under it. The magnitude of the GB energy is therefore a measure for the degree of short-range GB structural disorder that is very insensitive to any long-range structural order present in the GB. The fact that the (110) 11 twist GB and the (123) STGB have somewhat similar energies (of 1027 and 881 mJ/m2 ; see Table 1) therefore indicates comparable degrees of broadening of the nn peak in their related g(r)’s, as confirmed by the rather similar shapes and widths of the nn peaks in Figs. 13(b) and 14(b). Hence, although in principle the structural information contained in |S(k)|2 is equivalent to that in g(r), the GB energy represents a more direct predictor of GB properties, particularly of the diffusion behavior, than the magnitude of the planar structure factor. In fact, the success of brokenbond models for predicting surface and GB energies and physical properties [20] arises mostly from their focus on the nn peak in g(r), replacing its detailed shape by the area under the peak. Because all the elastic information contained in the detailed peak shape is thus lost, strain-field effects associated, for example, with line defects in vicinal interfaces are thus ignored. Nevertheless, the change in the area under the peak captures the total number of broken nn bonds, which is directly related to the excess energy of the average atom in the system. This comparison suggests that the zero-temperature GB energy is a much more reliable predictor of high-temperature GB behavior than the degree of long-range order present in the low-temperature GB structure. The above comparison also demonstrates that the degree of long-range structural disorder already present in the zero-temperature GB structure provides little indication as to whether or not the GB will undergo a solid-to-liquid transition at elevated temperatures. The fact that even the two tilt boundaries undergo the transition, in spite of being structurally much more ordered at low temperatures than the two twist boundaries, suggests that the degree of long-range order in the zero-temperature structure of the GB has little, if any, effect on the transition temperature, Tc . The decrease of Tc with increasing GB energy (see Table 1 and Fig. 15) therefore suggests that a higher degree of nn structural disorder present in the 0 K structure favors a transition into the liquid phase already at lower temperatures. In other words, it seems that

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a certain, critical amount of local GB structural disorder is required to trigger the transition to a liquid-like high-temperature structure. At finite temperatures the amount of GB structural disorder has two origins, one from the static disorder already present at 0 K and another, dynamical in nature, due to the thermal disorder. The above simulations suggest that the sum of both types of disorder in the GB region must reach a critical level in order to trigger the solid-to-liquid structural transition. This behavior is reminiscent of the Lindemann criterion for melting, requiring a certain amount of (vibrational) disordering in order to trigger the melting transition. The increase in the activation energy in the low-temperature regime with decreasing GB energy (see Table 1 and Fig. 15) can be explained similarly: irrespective of the detailed diffusion mechanism, any amount of static, shortrange structural disorder already present in the GB lowers the barrier for nn diffusion jumps of the atoms. Conversely, as evidenced by the perfect-crystal like activation energy for self-diffusion in low-energy GBs (such as the (111) twin), the absence of short-range GB structural disorder increases the saddlepoint energy for diffusion jumps, until a maximum activation energy is reached, namely the perfect-crystal value associated with zero GB energy.

5.

Conclusions

The picture that emerges from the above simulations of GB self-diffusion in Si and Pd bicrystals suggests that upon heating from zero temperature through the melting point, the highest-energy tilt and twist GBs in a polycrystalline microstructure undergo a reversible transition from a low-temperature, solid structure with a rather high, GB energy-dependent activation energy, to a highly confined liquid-like GB structure with a universal activation energy that is related to, and slightly higher than, that in the melt. In parallel, the diffusion mechanism changes from being solid-like to liquid-like; the latter is practically indistinguishable from that in the melt. The temperature at which the transition occurs increases with decreasing GB energy until it reaches the melting point. Lower-energy GBs, with less built-in short-range GB structural disorder, never reach the critical level in the overall (static and dynamic) structural disorder that is required to trigger the transition below Tm ; their diffusion behavior is therefore solid-like all the way up to Tm , with an activation energy that increases with decreasing GB energy until it reaches the perfect-crystal value. One should keep in mind that this classification of GB diffusion based on the GB energy applies only to high-angle GBs, i.e., boundaries with completely overlapping dislocation cores and hence a more or less homogeneously distributed type of structural disorder along the interface. This can clearly not be the whole picture, however. For example, in dislocation boundaries (i.e., low-angle or vicinal GBs) local structural disordering is distributed in a highly

High-temperature structure and properties of grain boundaries

2007

inhomogeneous manner: the disorder is localized in the dislocation cores and well separated by elastically strained perfect-crystal like regions. Two GBs with the same, relatively low energy, one a dislocation boundary and the other a “true” high-angle GB (with a homogeneously distributed type of GB structural disorder), will therefore exhibit rather different diffusion behaviors: the dislocation boundary will exhibit rather fast diffusion down the dislocation pipes but no diffusion at all in the perfect-crystal regions; by contrast, the diffusion in the low-energy, high-angle boundary will probably be much slower but homogeneously distributed throughout the GB.

Acknowledgments This work was supported by the US Department of Energy, BES-Materials Science under contract W-31-109-Eng-38.

References [1] D. Wolf, Grain boundaries: structure. In: R. Cahn, principal editor, The Encyclopedia of Materials Science and Technology, Pergamon Press, pp. 3597–3609, 2001. [2] W.T. Read and W. Shockley, Phys. Rev., 78, 275, 1950. [3] D. Wolf and K.L. Merkle, Chapter 3 in D. Wolf and S. Yip (eds.), Materials Interfaces, Atomic-level Structure and Properties, Chapman and Hall, pp. 87–150, 1992. [4] V. Pontikis, J. de Physique (Paris), 49, C5–327, 1988. [5] S.R. Phillpot, J.F. Lutsko, D. Wolf, and S. Yip, Phys. Rev. B, 40, 2831, 1989. [6] S.R. Phillpot, S. Yip and D. Wolf, Comput. in Phys., 3, 20, 1989. [7] D. Wolf, P. Okamoto, S. Yip, J.F. Lutsko, and M. Kluge, J. Mater. Res., 5, 286, 1990. [8] P. Keblinski, S.R. Phillpot, D. Wolf, and H. Gleiter, Phys. Rev. Lett., 77, 2965, 1997; see also J. Am. Ceram. Soc., 80, 717, 1997. [9] F.H. Stillinger and T.A. Weber, Phys. Rev. B, 31, 5262, 1985. [10] F. Cleri, P. Keblinski, L. Colombo, S.R. Phillpot, and D. Wolf, Phys. Rev. B, 57, 6247, 1998. [11] J. Tersoff, Phys. Rev. B, 38, 9902, 1988. [12] E. Tarnow, P. Dallot, P.D. Bristowe, J.D. Joannopoulos, G.P. Francis, and M.C. Payne, Phys. Rev. B, 42, 3644, 1990. [13] W. Rosenhain and D. Ewen, J. Inst. Metals, 10, 119, 1913; for a review of this work, see K.T. Aust and T. Chalmers in Metal Interfaces, ASM, Cleveland, OH, 1952. [14] P. Keblinski, S.R. Phillpot, D. Wolf, and H. Gleiter, Phil. Mag. Letters, 76, 143, 1999. [15] E. Budke, T. Surholt, S.I. Prokofjev, L. Shvindlerman, and C. Herzig, Acta Mater., 47, 385, 1999. [16] B. Schoenfelder, P. Keblinski, D. Wolf, and S.R. Phillpot, Intergranular and Interphase Boundaries in Materials, Trans. Tech. Publ., pp. 9–16, 1999.

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[17] P. Keblinski, D. Wolf, S.R. Phillpot, and H. Gleiter, Phil. Mag. A, 79, 2735, 1999. [18] J.B. Adams, S.M. Foiles, and W.G. Wolfer, J. Mater. Res., 4, 102, 1989. [19] D. Wolf, Chapter 1 In: D. Wolf and S. Yip (eds.), Materials Interfaces, Atomic-level Structure and Properties, Chapman and Hall, pp. 1–57, 1992. [20] D. Wolf and J. Jaszczak, Chapter 26 in D. Wolf and S. Yip (eds.), Materials Interfaces, Atomic-level Structure and Properties, Chapman and Hall, pp. 662–90, 1992.

6.11 CRYSTAL DISORDERING IN MELTING AND AMORPHIZATION Sidney Yip1 , Simon R. Phillpot2 , and Dieter Wolf3 1

Department of Nuclear Science and Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA 2 Department of Materials Science and Engineering, University of Florida, Gainesville, FL 32611, USA 3 Materials Science Division, Argonne National Laboratory, Argonne, IL 60439, USA

Among the structural phase transitions that evolve from an initially crystalline state, melting is the most common and most extensively studied. Another transformation that produces a disordered final state is solid-state amorphization. In this section the underlying thermodynamic and kinetic features of these two phenomena in a bulk lattice and at surfaces and grain boundaries will be discussed [1]. By focusing on the insights derived from molecular-dynamics simulations, we are led quite naturally to a view of structural disordering that unifies the crystal-to-liquid (C–L) and crystal-to-amorphous (C–A) transitions at high and low temperatures, respectively. On the one hand, a variety of models have been developed to describe melting in which intrinsic lattice defects, produced spontaneously within the crystal, play a central role [2]. On the other hand, it is known from experiments that melting generally occurs at extrinsic defects such as free surfaces, grain boundaries and voids [3]. From the standpoint of understanding the mechanisms of melting, the simulation technique of molecular-dynamics (MD) offers a way to follow the dynamics of the disordering transitions in molecular detail. In addition to offering precise control over specifying the initial atomic configuration of the system and the manner in which the sample is heated, MD simulation allows one to fully characterize the atomistic details associated with the onset of disorder. As we discuss here, MD simulations show that there are two distinct paths in melting. In the presence of an extrinsic defect, melting is a heterogenous process in which a small region of disorder is first nucleated at the defect and then propagates through the system. In the absence of extrinsic defects, melting is a homogenous process in 2009 S. Yip (ed.), Handbook of Materials Modeling, 2009–2023. c 2005 Springer. Printed in the Netherlands.

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which the crystal lattice becomes mechanically unstable to shear deformation. Heterogenous melting takes place at the thermodynamic melting point Tm , which can be independently determined by equating the free energies of the crystal and liquid. Homogenous melting, on the other hand, generally occurs at a higher temperature, Ts , which can be determined from an analysis of the elastic constants of the crystal [4]. (Additionally, entropy arguments may be used to predict crystalline instabilities at temperatures above the thermodynamic melting temperature.) If, however, surface melting is suppressed, then the solid can be substantially superheated, as has been observed in the case of crystalline spheres of silver coated with gold. In terms of the familiar thermodynamic phase diagram, conventional melting has a clear and simple representation. When the equation of state is projected onto the temperature-volume plane, the melting curve is seen to terminate at the triple point temperature Tt . Conceptually one may think of an extension of the melting curve below Tt as suggesting the existence of a metastable crystalline phase in which sublimation, a thermally activated process, is kinetically suppressed. Following this point of view one can construct an effective phase diagram which also describes solid-state amorphization in the sense of a combined representation of thermodynamics and kinetics.

1.

How Crystals Melt – Thermodynamics vs. Kinetics

In thermodynamics the melting point Tm is the temperature at which the crystal and the liquid phases coexist in equilibrium, with coexistence being governed by the equality of the Gibbs free energies, G, for the crystal and the liquid, G = E − T S + PV

(1)

where E is the internal energy, T, S, P, and V are the system temperature, entropy, pressure, and volume, respectively. For a material where the interatomic interactions are specified, the free energies can be evaluated using the atomistic simulation methods discussed in Chapter 2; in this way one can essentially predict the melting point from the intersection of the two free-energy curves. Notice, however, that while thermodynamics tells us when the system should melt, it says nothing about how the melting process should occur. To see the disordering of a crystal lattice actually taking place at the molecular level, a question of kinetics, it is appropriate to also turn to molecular-dynamics simulation. Consider the case of silicon where the free-energy calculation of Tm was made for a three-body potential for Si in the diamond cubic phase, giving a

Crystal disordering in melting and amorphization

2011

surprisingly accurate prediction of 1691 ± 30 K [5] (the experimental value was 1683 K). Given this result it would appear that using the same potential in a straightforward molecular dynamics simulation of crystal heating would reveal readily the kinetics of crystal melting. This, however, was not what happened [6, 7]. In heating a perfect lattice of Si atoms with periodic boundary conditions in incremental steps, it was observed that the crystal remained stable well past 1691 K, even up to 2200 K; finally, at T ∼ 2500 K the ordered lattice collapsed abruptly, the disordering of atomic positions seemed to occur everywhere in the system. The observation of structural collapse thus seemed to contradict the above definition of melting. Why did the crystal not melt at 1691 K as predicted by the free-energy calculations? What kind of transition took place at 2500 K when structural disordering set in? The answer to the first question was obtained by repeating the simulations with the periodic boundary condition turned off along one direction which resulted in introducing two free surfaces at the ends of the simulation cell. The results showed that in the temperature range above 1700 K, structural disordering appeared first in the free-surface region and then propagated into the bulk crystal, as shown in Fig. 1. By evaluating the static structure factor for each layer of atoms (see Chapter. 6.8) one could follow the melt–crystal interface and deduce the speed of the moving front v(T ), T being the temperature at which the system was being heated. Figure 2 shows the resulting variation of the speed of the interface with temperature. The significance of this apparently simple behavior lies in that by extrapolating the data to zero speed one obtains a temperature of 1710 K, essentially the melting temperature predicted by the free-energy calculations. With a bit of hindsight it became apparent that this way of analyzing the simulation results amounts to a practical method of implementing the thermodynamic definition of phase coexistence. That is, at the melting point the melt–crystal interface should not move in either direction. It is worth pointing out a general implication, namely, rather than trying to find a phase-transition point precisely it is often easier to approach it by extrapolation. Although not very elegant, this method is simple and reasonably robust [6, 7]. We determined the melting point of a crystal by using a free surface to nucleate the structural transition and then extrapolating the “melting speed” to zero. From the standpoint of a thermodynamic process, melting is seen to require the presence of a nucleation site, the free surface in the present discussion. In the absence of a nucleation site, the transition can be kinetically suppressed, as was the case in the simulations using fully periodic boundary conditions, forcing the system to go into a superheated state. Other defects could just as well serve as the necessary nucleation site. For example, a grain boundary or a 13-vacancy void have been found to lead to the same extrapolated melting temperature as obtained in the free surface simulations [8].

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(a)

S(K)

1.0 0.5 0.0 0

8

16 Atomic plane

24

32

0

8

16 Atomic plane

24

32

(b)

S(K)

1.0 0.5 0.0

Figure 1. Heterogenous thermodynamic melting of a silicon bicrystal containing in its center a (110) θ = 50.48◦ (11)twist boundary (GB). (The two semicrystals are pulled apart to facilitate visualization of the structural disorder.) After the bicrystal was heated from 1600 K (T < Tm ) to 2200 K (T > Tm ) over a period of 600 time steps (1000 steps corresponds to 1.15 ps of real time), the simulation time was set to t = 0. Shading of the atoms indicates nearest neighbor coordination, C, with white, gray and black circles denoting C = 4, 5 and 3, respectively. (a) After 2700 time steps, a number of planes on either side of the GB plane have melted. Nearzero values of the structure factors S(k1 ) and S(k2 ) show that a breakdown in long-range order in approximately seven (110) planes closest to the GB. By contrast, at t = 0 only a few atoms at the GB had coordination greater than four, and the structure-factor profiles showed a welldefined GB region consisting of about four (110) planes. (b) After 8100 time steps, melting has spread over half of the system, with loss of long-range in the 20 central planes.

Crystal disordering in melting and amorphization

2013

120 Si (SW)

Velocity (m/s)

100

(110) θ ⫽ 50.48˚ (Σ11)

80

60

40 Grain boundary Free surface

20

0 1700

1800

1900

2000

2100

2200

T(K)

Figure 2. Propagation velocity of the silicon solid–liquid interface as a function of temperature after nucleation from a grain boundary (solid squares with error bars) and from a free surface (open squares). The curve, representing a quadratic fit to the data points, extrapolates to zero velocity at T = 1710±30 K.

2.

Mechanical Melting via Elastic Instability

It still remains to clarify the nature of the abrupt structural disordering observed at 2500 K in the simulation runs with fully periodic boundary conditions. We know that in the absence of a nucleation site the crystal can be readily superheated to a metastable state. The upper limit of metastability is the mechanical stability limit of the lattice, the temperature at which the system would collapse without waiting for nucleation and growth of a liquid layer. This is what happened when the lattice disordered uniformly at 2500 K. We will henceforth refer to this homogenous process as mechanical melting in contrast to the heterogenous process of thermodynamic melting. Normally the former is not observed since the latter invariably occurs at a lower temperature. One can understand the nature of mechanical melting by recalling the simple criterion for crystal melting set forth by Born, in which melting is defined by the loss of shear rigidity [9]. Accordingly the melting point Tm is that temperature at which the shear modulus G vanishes, G(Tm ) = 0

(2)

In contrast to the thermodynamic definition based on free energies, this is a thermoelastic description based on elastic stability. Indeed Born extended

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Eq. (2) to a set of conditions governing the structural stability of a cubic lattice [10], C11 + 2C12 > 0,

C11 − C12 > 0,

C44 > 0

(3)

where C11 , C12 , and C44 (= G) are the three distinct elastic constants (in Voigt notation). The melting criterion, Eq. (2), was not successful in explaining the experimental results at the time; also it was not clear how it could account for the existence of a latent heat and a latent volume associated with a first-order thermodynamic phase transition. Given that molecular-dynamics allows one to both observe directly structural disordering and calculate the elastic constants, the validity of Eq. (2) can be tested. This was carried out by simulating isobaric heating of a perfect crystal with fully periodic boundary conditions. Such a study showed that an abrupt structural disordering was indeed triggered by the vanishing of a shear modulus, although it was C11 − C12 instead of C44 [11]. It was shown that the temperature at which the lattice collapsed was in good agreement with that predicted by the elastic stability conditions. Since the crystal analyzed was initially defect free, the structural disordering was unlikely to correspond to the free energy-based process which should occur at a lower temperature. The simulations performed employed an interatomic potential model for fcc Au (details of the potential are not important for the present discussion) and a cell containing 1372 atoms with deformable (Parinello–Rahman) periodic boundary conditions imposed at constant stress (see Chapter 2, Basic MD). A series of isostress–isothermal simulations (with velocity rescaling) were carried out over a range of temperatures. At each temperature the atomic trajectories generated were used to compute the elastic constants at the current state using fluctuation formulas [12]. Figure 3 shows the variation with temperature of the lattice strain a/ao along the three cubic symmetry directions. The slight increase with increasing temperature merely indicates the lattice is expanding normally with temperature. Also the results for the three directions are the same as they should be for a cubic crystal. At T = 1350 K one sees a sharp bifurcation in the lattice dimension where the system elongates in two directions while contracting in the third. This is a clear signal of symmetry breaking, from cubic to tetragonal. To see whether the simulation results are in agreement with the prediction based on Eq. (3) we show in Fig. 4 the variation of the elastic moduli with temperature, or equivalently with lattice strain (the one-to-one correspondence is seen in Fig. 3); the three moduli of interest are the bulk modulus BT = (C11 + 2C12 ) /3, tetragonal shear modulus G = (C11 − C12 )/2, and rhombohedral shear modulus G = C44 . On the basis of Fig. 4 one would predict the incipient instability to be the vanishing of G , occurring at the theoretical (predicted) lattice strain of (a/ao )th = 1.025. From the simulation at T = 1350 K the observed strain is (a/ao )obs = 1.024. Thus

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1.2

1.12

ai/a0

1.04

0.96 ax/a0 ay/a0

0.88

az/a0 0.8

0

200

400

600

800

1000 1200 1400 1600

T (K)

Figure 3. Temperature variation of lattice strain a/ao along the three directions of the cubic simulation cell in a series of isobaric (zero pressure) simulations of incremental heating of a perfect crystal of Au atoms, showing a sudden symmetry-breaking structural response at 1350 K. 0.5

1.8 1.6

BT

0.4

1.4 1.2

0.3

1.0 0.8

0.2

BT (Mbar)

G (Mbar)

G

0.6 0.4

0.1

0.2

G' 0.0 1.00

1.01

1.02

1.03

1.04

0.0 1.05

a/a0

Figure 4. Variation of elastic moduli BT , G, and G with lattice strain a/ao in the same isobaric, heating simulation as Fig. 3.

we can conclude that the vanishing of tetragonal shear is responsible for the structural bifurcation behavior. For more details of the system behavior at T = 1350 K, we show in Fig. 5 the time evolution of the lattice strain, the off-diagonal elements of the cell matrix H , and the system volume. It is clear that from Fig. 5(a) that the vanishing of G triggers both a shear deformation (cf. Fig. 5b) and a lattice decohesion (Fig. 5c), the latter providing the characteristic volume expansion

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1.2

a/a0

1.1

1.0

0.9

0.8

(b)

0

50

100 150 Time steps (*100)

200

0

50

100 150 Time steps (*100)

200

0

50

100 150 Time steps (*100)

200

0.4

Hij

0.2

0.0

⫺0.2 ⫺0.4

(c)

1.125

Ω/Ω0

1.110 1.095 1.080 1.065 1.050

Figure 5. Overall system response in time in the same simulation as Fig. 3 revealing further characteristic behavior beyond the onset of the triggering instability, (a) lattice strain a/ao along the initially cubic simulation cell, (b) off-diagonal elements of the cell matrix H, and (c) normalized system volume /o . Arrows indicate the onset of Born instability in (a), shear instability in (b), and lattice decohesion in (c).

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associated with melting. This sequence of behavior implies that the signature of a first-order transition, namely, latent volume change, is not necessarily associated with the incipient instability. Our results also provide evidence supporting Born’s picture of melting being driven by a thermoelastic instability, interpreted to involve a combination of loss of shear rigidity and vanishing of the compressibility [13]. Moreover, it is essential to recognize that this mechanism applies only to the process of mechanical instability (homogenous melting) of a crystal without defects, and not to the coexistence of solid and liquid phases at a specific temperature. Although our results for an fcc lattice with metallic interactions show that homogenous melting is triggered by G = 0 and not Eq. (2), nevertheless, they constitute clear-cut evidence that a shear instability is responsible for initiating the transition. The fact that simulation reveals a sequence of responses apparently linked to the competing modes of instabilities (cf. Fig. 5) implies that it is no longer necessary to explain all the known characteristic features of melting on the basis of the vanishing of a single modulus. In other words, independent of whether G = 0 is the initiating mechanism, the system will in any event undergo volume change and latent heat release in sufficiently rapid order (on the time scale of physical observation) that these processes are all identified as part of the melting phenomenon. Generalizing this observation further, one may entertain the notion of a hierarchy of interrelated stability catastrophes of different origins, elastic, thermodynamic, vibrational , and entropic [14]. It may be mentioned that generalizing the stability criteria to arbitrary external load has led to the identification of the elastic instability triggering a particular structural transition [15]. In hydrostatic compression of Si, the instability which causes the transition from diamond cubic to β-tin structure is the vanishing of G (P) = (C11 − C12 − 2P)/2. In contrast, compression of crystalline SiC in the zinc blende structure results in an amorphization transition associated with the vanishing of G(P) = C44 − P. For behavior under tension, crack nucleation in SiC and cavitation in a model binary intermetallic, both triggered by the spinodal instability, vanishing of BT (P)=(C11 +2C12 + P)/3, are results which are analogous to the observation reported here. Notice also that in the present study a crossover from spinodal to shear instability can take place at sufficiently high temperature [15].

3.

Premelting and Thermal Disordering at an Interface

The question of whether a liquid-like layer can form at a grain boundary at temperatures distinctly below the melting point Tm is of long-standing interest [16]. A number of molecular dynamics simulations have been performed to address this issue. The early studies had found significant structural

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disordering at T ≤ Tm which led to conflicting interpretations of the existence of ‘grain-boundary premelting’. A perceptive review of the factors that could account for the apparent discrepancy is a good reminder for students interested in atomistic simulations on the need to guard against numerical artifacts [17]. The present consensus concerning the high-temperature structural stability of grain boundaries is that there is no evidence for a premelting transition or a stable disorderd phase below the melting temperature of the bulk. This is consistent with the foregoing discussion of melting in Si where the same melting point was obtained from simulations of a single crystal with free surfaces and simulations of a bicrystal with no free surfaces. It is also consistent with a TEM study of aluminum bicrystals where no premelting was observed up to 0.999 Tm [18]. The surface-induced melting analysis discussed here has been applied to further investigate thermal structural disordering at a crystalline interface in aluminum [19]. Besides showing that melting is nucleated at the grain boundary in a similar manner as at the free surface, the results provided details of metastable behavior commencing at about 0.93 Tm . Three structural models were examined, a single crystal with 3D periodic boundary conditions (model A), a single crystal with 2D periodic boundary conditions and free surfaces along the third direction (model B), and a bicrystal with the same boundary conditions as model B (model C). The interatomic potential used was a many-body potential of the EAM-type for Al. From model A the mechanical melting temperature Ts was established to be 950 K. Using model B the thermodynamic melting point Tm was found by extrapolating to zero the propagation speed of the surface-nucleated disordered layer, the result was 865±15 K. Meanwhile, the bulk or ordered region of model B was observed to behave in the same manner as model A. (The experimental melting point of Al is 930 K; the difference between this and Tm is attributed to the interatomic interaction model used which was an EAM-type potential with no fitting to any thermal property.) With model C having two interfaces, the grain boundary and free surfaces, the analysis of interface-induced melting could be applied separately to the disordered layers nucleated in the vicinity of the interfaces to produce two extrapolated temperatures. Both extrapolations gave the same result for the thermodynamic melting point, a value of 865 K that was also obtained from model B. Concerning the nature of disordering in the free surface or grain boundary region at temperatures just below Tm , results on the energy–temperature variation and the re-emergence of structural order after a long run reveal the onset of metastable behavior. It is tempting to regard such data, along with the rapid growth of interfacial thickness as one approaches Tm , as supporting the theoretical prediction of a continuous process, in contrast to the first-order melting transition in the bulk. On the other hand, one should keep in mind that on the short time and distance scales of molecular dynamics simulation,

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kinetic-barrier effects associated with local energy minima could mask any “intrinsic metastability behavior”. Since the former are expected to be sensitive to system size, some assessment of their presence could be made by using larger simulation cells. To explicitly demonstrate metastability, one could examine potential-energy surfaces obtained by applying energy minimization to a sufficient number of system configurations generated during molecular dynamics simulation (see Chapter 2), or reaction pathway sampling methods (see Chapter 5). The concept of inherent structure [20] may well be relevant to the understanding of competing effects between ordering in the bulk and disordering at the interface.

4.

Parallels Between Melting and Amorphization

There are similarities in the underlying thermodynamic and kinetic features of structural disordering transitions to the liquid and the amorphous states that are worth exploring. We have already commented on such features in discussing the two types of melting transitions, the heterogenous process of nucleation and growth at defect sites and the homogenous process of lattice instability. It should not be surprising that similar distinctions can be made in the disordering transition where a destabilized lattice ends up in an amorphous state. The traditional methods of producing amorphous materials have been rapid solidification of a melt and quenching from the vapor state. More recent methods involving solid-state processes now include ion and electron irradiation, chemical interdiffusion, mechanical deformation, and pressure-induced amorphization [21]. Just as the discussion of mechanisms of melting continues to be a topic of current interest, the understanding of solid-state amorphization is by no means complete despite a considerable body of investigations [22]. Here we point out certain parallels between melting and amorphization by combining the mechanistic insights on melting derived from simulations with related observations from amorphization experiments [1]. Since melting occurs at elevated temperature, it is tempting to think that amorphization occurs more readily at low temperatures. Because solid-state amorphization experiments are usually carried out at temperatures where pointdefect mobility is limited, both heterogenous and homogenous transitions have been observed. In the case of beam irradiation, the amorphous phase was nucleated at defect sites when the temperature is close to a threshold value, whereas at lower temperatures homogenous transformation in the entire irradiated volume occurred. In experiments on hydrogen charging distinctly heterogenous or homogenous processes were observed to occur in regions that are, respectively, poor or rich in hydrogen. These observations show that the two mechanisms of melting occur also in amorphization.

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Besides temperature another state variable which comes into play is the elastic strain, or volume change (usually expansion), accompanying the onset of structural disorder. The existence of volume change associated with melting and amorphization is clearly seen in the decrease of the shear modulus C in both types of transitions. There is an additional effect of chemical disordering in the amorphization of alloys; however, this is not an issue in the case of hydrogenation experiments where little or no chemical disordering takes place. We can combine our considerations of temperature and volume expansion by giving an unified interpretation of melting and amorphization in terms of a thermodynamic phase diagram. In Fig. 6 we show the phase boundaries delineating the crystal, liquid and vapor states of an elemental substance in the temperature–volume plane. The condition for thermodynamic melting is expressed by the melting curve Tm (V ), which terminates at the triple-point temperature Tt , the lowest value at which the crystal can coexist with the liquid. The freezing curve Tf (V ) would lie more or less parallel to the melting curve, also terminating at Tt . For many materials the melting point Tm is only a little higher than the triple-point temperature (often by less than 1K). Thus any experiment performed at T < Tm is also performed at T < Tt , which

T

Tm(v ) Ts(V ) Ts C-L

Tm

L L-Vap

C

Vap

Tt

C-Vap

L VtC Vt

V

Figure 6. Schematic temperature–volume phase diagram of a monatomic substance showing the single-phase regions of crystal (C), liquid (L), and vapor (Vap), and the various two-phase regions. On the horizontal triple line, at temperature It , the crystal (at volume VtC ) and the liquid (at volume VtL ) coexist with the vapor. The points on the thermodynamic melting line, Tm (V ), and the freezing curve, Tf (V ), indicate conditions of ambient pressure. As discussed in Chapter 5, to a good approximation, the freezing curve and the mechanical-stability line, Ts (V ), coincide.

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calls into question the meaning of ‘surface premelting’ in the sense of equilibrium thermodynamics. Although the phase diagram indicates that below Tt C–L transition cannot occur as a equilibrium thermodynamic process, one nevertheless can define a critical volume Vs(T ), with T < Tt , at which the crystal becomes mechanically unstable upon uniform volume expansion, as shown in Fig. 7. In effect this extends the instability curve for mechanical melting to arbitrarily low temperatures. That this is still consistent with the vanishing of the shear modulus can be demonstrated by calculating C at a function of temperature down to zero temperature. The extended curve therefore represents the existence of a metastable crystalline phase in which the thermally activated process of sublimation is kinetically suppressed. In other words, we justify the crossing of the phase boundaries by restricting the interpretation to time scales short compared to the relevant kinetics. With sublimation kinetically suppressed, a similar extension of the thermodynamic melting curve Tm (V ) can be introduced, as also shown in Fig. 7. Notice that neither the triple line separating the C–L region from the C–Vap region nor the sublimation curve appear in this diagram. The region lying to the left of Tm (V ) now becomes the effective single-phase region for the crystalline state, and it is important to keep in mind that the region between the original sublimation curve and Tm (V ) defines a metastable over-expanded

T

L C-L L-vap C Tm' (V )

Ts' (V )

Vap

V

Figure 7. An effective T − V phase diagram showing the extensions of both the thermodynamic-melting and the mechanical-stability (or freezing) curves below the triple-point temperature. Along the extension of the former, an expanded crystal becomes unstable against the disordered phase by heterogenous amorphization; by contrast, along the extension of the mechanical-melting curve, homogenous amorphization can occur.

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solid. Similarly, the extended two-phase region below the triple-point temperature is where the over-expanded solid can coexist with the metastable supercooled liquid. The extended (effective) phase diagram applies equally well to solid-state amorphization. From this standpoint C–A transformation is viewed as an isothermal melting process driven by volum expansion at T < Tm . The analogy with C–L transition is that volume expansion produced by external forces at constant temperature plays the same role as thermal expansion during isobaric heating to melting. When the expansion is sufficient to cross Tm (V ), the crystal becomes unstable against structural disordering, and heterogenous amorphization may take place. If in addition to sublimation heterogenous amorphization is also kinetically suppressed, then the expansion may continue up to the curve Ts (V ), at which point homogenous amorphization triggered by the Born instability is expected to set in.

5.

Further Issues

With structural transitions affecting virtually all properties and behavior, it is unlikely all the issues relevant to a complete understanding of how a crystal lattice undergoes disordering will be resolved any time soon. We have addressed the thermodynamic meaning of melting as conventionally measured, which we now distinguish as thermodynamic melting, and demonstrate its relation to melting by an elastic instability, which we denote as mechanical melting. Whereas the former is governed by the kinetics of extrinsic defects, the latter is clearly a process intrinsic to all crystals. Further developments along the lines of our considerations are to be expected. The equivalence between the determination of melting by free energy calculations for a defectfree crystal and a liquid and by the method of defect-nucleation described above has been demonstrated in a much more extensive manner in studies that map out an melting curve of a noble-gas element, showing how well simulation can predict experiments [23]. Through molecular-dynamics simulation the connection between the elastic stability criterion and the empirical rule known as Lindemann’s law can be clarified, along with an investigation of surface melting [24]. As our understanding of melting mechanisms deepens, the issue of how quickly melting can occur becomes relevant. It is now feasible to observe melting on the time scale of electronic excitations by femtosecond laser spectroscopy. To interpret such measurements the studies described here would have to be extended to incorporate the role of electronic processes. A more complete understanding of structural disordering phenomena will then have to deal with a multiscale rate process.

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References [1] S.R. Phillpot, S. Yip, P.R. Okamoto, and D. Wolf, “Role of interfaces in melting and solid-state amorphization,” In: D. Wolf and S. Yip (eds.), Materials Interfaces, Chapman and Hall, London, pp. 228–254, 1992. [2] A.R. Ubbelohde, Molten State of Matter: Melting and Crystal Structure, Wiley, Chichester, 1978. [3] R.W. Cahn, “Melting and the surface,” Nature, 323, 668–669, 1986. [4] M. Born and K. Huang, Dynamical Theory of Crystal Lattices, Clarendon Press, Oxford, 1954. [5] J. Broughton and X.P. Li, “Phase diagram of silicon by molecular dynamics,” Phys. Rev. B, 35, 9120–9127, 1987. [6] S.P. Phillpot, J.F. Lutsko, D. Wolf, and S. Yip, “Molecular-dynamics simulation of lattice-defect-nucleated melting in silicon,” Phys. Rev. B, 40, 2831–2840, 1989. [7] S.R. Phillpot, S. Yip, and D. Wolf, “How crystals melt,” Comput. in Phys., 3, 20, 1989. [8] J.F. Lutsko, D. Wolf, S.R. Phillpot, and S. Yip, “Molecular-dynamics study of latticedefect-nucleated melting in metals using an embedded atom method potential,” Phys. Rev. B, 40, 2841–2855, 1989. [9] M. Born, J. Chem. Phys., 7, 591, 1939. [10] M. Born, Proc. Cambridge Philos. Soc.. 36, 160, 1940. [11] J. Wang, J. Li, S. Yip, D. Wolf, and S. Phillpot, “Unifying two criteria of born: elastic instability and melting of homogenous crystals,” Physica A, 240, 396–403, 1997. [12] J. Ray, “Elastic constants and statistical ensembles in molecular dynamics,” Comput. Phys. Rep., 8, 111–151, 1988. [13] L.L. Boyer, Phase Transitions, 5, 1, 1985. [14] J.L. Tallon, Crystal instability and melting, Nature, 342, 658–658, 1989. [15] J. Wang, J. Li, S. Yip, S. Phillpot, and D. Wolf, “Mechanical instabilities of homogenous crystals,” Phys. Rev. B, 52, 12627–12635, 1985. [16] H. Gleiter and B. Chalmers, High-Angle Boundaries, Pergamon, Oxford, p. 113, 1972. [17] V. Pontikis, “Grain-boundary structure and phase transformations–a critical review of computer-simulation studies and comparison with experiments,” J. de Phys., 49, C5, 327–338, 1988. [18] T.E. Hsieh and R.W. Balluffi, “Experimental study of grain-boundary melting in aluminum,” Acta Metall., 37, 1637–1644, 1989. [19] T. Nguyen, P.S. Ho, T. Kwok, C. Nitta, and S. Yip, “Thermal structural disorder and melting at a crystalline interface,” Phys. Rev. B, 10, 6050–6060, 1992. [20] F.H. Stillinger and T.A. Weber, “Computer simulation of local order in condensed phases of silicon,” Phys. Rev. B, 31, 5262–5271, 1985. [21] W.L. Johnson, “Thermodynamics and kinetic aspects of the crystal to glass TRna sition in metallic materials,” Prog. Mat. Sci., 30, 81–134, 1986. [22] P.R. Okamoto and M. Meshii, In: H. Wiedersich and M. Meshii (eds.), Science of Advanced Materials, ASM International, Metals Park, OH, p. 33, 1990. [23] M. de Koning, A. Antonelli, and S. Yip, “Single-simulation determination of phase boundaries: a dynamic clausius-clapeyron integration method,” J. Chem. Phys., 115, 11025–11035, 2001. [24] Z.H. Jin, P. Gumbsch, K. Lu, and E. Ma, “Melting mechanisms at he limit of superheating,” Phys. Rev. Lett., 87, 055703, 2001.

6.12 ELASTIC BEHAVIOR OF INTERFACES Dieter Wolf Materials Science Division, Argonne National Laboratory, Argonne, IL 60439

A large body of work on epitaxial thin films has focused on controlling interfacial structure, particularly on preventing formation of interface dislocations which can lead to diffusion and electrical breakdown in semiconductor devices [1]. However, in other types of interfacial materials, dislocations might actually be desirable for controlling or enhancing certain mechanical properties, such as toughness and ductility. In this section we illustrate – by means of atomistic computer simulations – the important role of the atomic structure and, in particular, misfit dislocations in the elastic behavior of metallic interface materials. In particular, we review atomic-level simulations that elucidate the causes of the anomalous elastic behavior of thin films and composition-modulated superlattice materials. (For earlier reviews, see Refs. [2] and [3].) The investigation of thin-film superlattices composed of grain boundaries (GBs) shows that the elastic anomalies are not necessarily an electronic but a structural interface effect that is intricately connected with the local atomic disorder at the interfaces [4]. The consequent predictions that (i) coherent strained-layer superlattices should show the smallest elastic anomalies and (ii) incoherent interfaces exhibit much larger anomalies are validated by simulations of dissimilar-material superlattices. We hope to demonstrate that such simulations can be an effective aid in tailoring the elastic behavior of composite materials because, by contrast with experiments, they allow one to systematically investigate simple, but well characterized model systems with increasing complexity. This unique capability of simulations has enabled elucidation of the underlying driving forces and, in particular, (i) deconvolution of the distinct effects due to the inhomogeneous atomic-level disorder localized at the interfaces from the consequent interfacestress-induced anisotropic lattice-parameter changes and (ii) separation of the homogeneous effects of thermal disordering from the inhomogeneous effects due to the interfaces. 2025 S. Yip (ed.), Handbook of Materials Modeling, 2025–2054. c 2005 Springer. Printed in the Netherlands.

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D. Wolf

This case study of the interfacial elastic behavior also demonstrates the intuitive insights into the physical behavior of inhomogeneous systems obtainable from atomistic simulations. This behavior is often opposite to our usual intuition gained from the investigation of homogeneous systems. The simulations discussed below suggest that the usual intuition that the elastic constants and moduli soften when the density of a material decreases (for example, by thermal expansion) does not necessarily apply to inhomogeneous systems. Instead, these simulations demonstrate that the net elastic response of an interface material is the result of a rather complex, highly non-linear competition between two influences, structural disordering and consequent volume expansion. Our focus is mostly on multilayered systems (see Fig. 1) [5]. The elastic behavior of multilayered materials is known to exhibit significant anomalies, in that some elastic constants and moduli are significantly strengthened (the so-called “supermodulus effect”) while others may actually be softened [6]. In the past, much controversy has focused on the actual magnitude of these anomalies; [7–11] however it now seems rather widely accepted that, although no longer thought to be quite as striking as suggested in the original paper [7], these anomalies (typically of the order of 10–50%) are significantly larger than predictions from continuum elasticity theory based on the underlying anisotropic lattice-parameter changes (typically of the order of a few n â

n â

(a)

(b)

Λ

z

coherent x

incoherent y

Figure 1. (a) Coherent or “strained-layer” and (b) incoherent thin-film superlattice (schematic). The lattice parameters parallel and perpendicular to the interfaces will adjust in response to interfacial stresses.

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percent) which usually accompany the elastic anomalies. The physical origin of these anomalies has also been the subject of much controversy, including suggestions that this behavior arises from (i) the modified net electronic structure of the multilayered material, (ii) the anisotropic – but homogeneous – state of strain of the two constituent films forming the multilayer, or (iii) the greatly disordered atomic structure of the inhomogeneous regions surrounding the interfaces (see Fig. 1) [6]. In this section present evidence that the anomalies are intimately connected with the detailed atomic structure of the interfaces, while the homogeneous strains and the related modification of the electronic structure appear to play only secondary roles. The fundamental importance of composition-modulated layered materials as model interfacial materials is due to the fact that, by controlling the composition modulation wavelength, (see Fig. 1), the fraction of atoms at or near the interfaces can be varied systematically. As a consequence, the physical response of these model systems consists of a tunable mixture of homogeneous and inhomogeneous effects: By decreasing gradually, more and more atoms in the system experience the presence of the interfaces, and their behavior resembles less and less that of a homogeneous system, thus gradually exposing the behavior characteristic of the inhomogeneous parts of the material. These characteristics consists mainly of a highly non-linear modification of the atomic structure and related elastic behavior with decreasing . In “real” multilayers the simultaneous presence of both structural and chemical disorder at the interfaces (the latter associated, for example, with interfacial reactions and segregation) gives rise to considerable difficulties in both, full characterization of the material and interpretation of any observed elastic anomalies in terms of the underlying atomic structure and composition of the interfaces. In simulations this difficulty can be avoided entirely by choosing model systems which are atomically and chemically flat, thus greatly simplifying the interpretation of the observed elastic behavior of these model systems, a simplification not easily possible in experiments. The outline of the paper follows a simple building-block concept in which the thin slab, delimited by flat surfaces and of variable thickness, is considered as the basic building block of the multilayered system (Fig. 1). Following a brief discussion of the computational approach in Section 1, we first consider the structure and elastic behavior of superlattices of GBs (see Fig. 2), thus avoiding any effects due to materials and interfacial chemistry (Section 2). These grain-boundary superlattices (GBSLs) represent ideal model systems for the investigation of the purely structural aspects in the anomalous elastic behavior, including the structure-property correlation. Next, by periodically stacking up thin films of different materials, the grain boundaries will be replaced with phase boundaries, thus modeling the more complex behavior of composition-modulated superlattices (Section 3). Finally, by comparing the temperature dependence of the elastic behavior of multilayers with that of a

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D. Wolf x, y

A

B

A

B z

Λ Figure 2. Schematic showing the periodic arrangement of thin slabs, A and B, to form a grainboundary superlattices (GBSL). A and B are identical materials of equal thickness, however rotated with respect to each other about the interface normal (||z) to form a periodic array of GBs. In a composition-modulated superlattice, A and B are different materials.

perfect crystal (Section 4), the homogeneous effects of temperature will be compared with the inhomogeneous effects due to the interfaces.

1.

Computational Approach

A hierarchy of atomistic simulation codes, schematically shown in Fig. 3, has been used to determine the structure and elastic behavior of multilayered systems. Following the choice of a suitable interatomic potential and the desired multilayer geometry, the system of atoms is first relaxed at zero temperature using an iterative energy-minimization algorithm (“lattice statics”). The periodic border conditions are chosen to be consistent with the geometry of the system. For example, three-dimensional (3D) periodic borders are used in the simulation of perfect crystals and superlattices, while 2D periodic border conditions are imposed in the plane of the interfaces for thin films and bicrystalline interfaces (i.e., individual grain or phase boundaries embedded between two bulk perfect crystals). The relaxation may be performed under conditions of either constant volume or constant stress, permitting one to elucidate the role of the anisotropic lattice-parameter changes in the elastic behavior. Following the complete relaxation of the system, the 6 × 6 elastic-constant and -compliance tensors at T = 0 are evaluated using a lattice-dynamics like method [12]. The elastic constants thus obtained can be tested and verified by a direct comparison with those extracted from stress-strain curves. Finally, the relaxed structures thus obtained can be used as input into molecular dynamics (MD) simulations to study the effects of temperature. A non-trivial conceptual problem in the evaluation of elastic constants for inhomogeneous systems arises from the internal relaxations that occur

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Hierachy of Codes Interatomic Potential(s)

Unrelaxed Structure

Force Relaxation ("Lattice statics")

Lattice Dynamics

Molecular Dynamics

Figure 3. Schematic representation of the hierarchy of computer codes used to atomistically simulate the structure and elastic behavior of interface materials.

following the application of an external strain or stress to the system. This relaxation effect, absent when homogeneously deforming, for example, a perfect monatomic cubic crystal, gives rise to a contribution to the zero-temperature elastic constants [12], in addition to the well-known Born term [13]. In MD simulations of elastic constants this relaxation contribution is part of the so-called fluctuation term [12,14], which for inhomogeneous systems does not vanish in the T → 0 limit. In order to elucidate the degree to which the simulated results depend on the potential and the material being simulated, the results obtained by means of two conceptually different fcc-metal potentials will be compared: a Lennard– Jones (LJ) pair potential fitted for Cu (with ε = 0.167 eV and σ = 2.315 Å) and an embedded-atom-method (EAM) many-body potential fitted for Au [15]. As discussed in detail elsewhere [16], the two types of potentials yield qualitatively the same behavior for most interfacially controlled materials properties, indicating that interfacial behavior is dominated by the (usually pair-wise) repulsive interactions in these potentials. Those properties that do seem to vary with the potential can usually be understood in terms of the rather different interfacial stresses that are generated with the two potentials [17]. The different magnitudes of stresses associated with the two potentials can be seen by considering the variation of the cohesive energy of the two potentials with

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D. Wolf Table 1. Zero-temperature elastic constants, selected moduli and Poisson ratios for perfect crystals of each potential in the principal cubic coordinate system. Units are 1012 dynes/cm2 , except for the Poisson ratios, which are dimensionless. Elastic property C11 = C22 = C33 C12 = C13 = C23 C44 = C55 = C66 Young’s modulus, Yz Biaxial modulus, Yb Poisson ratio, ν

LJ potential

EAM potential

1.808 1.016 1.016 1.076 1.681 0.360

1.807 1.571 0.440 0.346 0.647 0.465

the deviation of the lattice parameter from its equilibrium T = 0 value (see, e.g., Refs. [17] and [18]). The steeper slopes of the EAM cohesive-energy curve shows a greater resistance to straining, and is reflected in the larger interfacial stresses this potential generates [13, 14]. To avoid discontinuities in the energy and forces, both potentials were shifted smoothly to zero at their respective cutoff radius (Rc /a = 1.32 and 1.49 for the EAM and LJ potentials, respectively). The zero-temperature perfect-crystal lattice parameters, a, for these potentials were determined to be 4.0828 Å (EAM) and 3.6160 Å (LJ). The three elastic constants of the perfect fcc crystal at zero temperature in the principal cubic coordinate system (with the x, y, and z-axes parallel to the principal 001 directions) are summarized in Table 1 for the two potentials. We note that, because of the Cauchy relation, for an equilibrium pair potential, C12 = C44 , i.e., the LJ potential has only two independent elastic constants.

2.

Grain-boundary Superlattices (GBSLs)

As illustrated in Fig. 2, GBSLs represent idealized, somewhat hypothetical layered materials consisting of a periodic arrangement, . . . | A | B | A | B | . . . , of thin slabs A and B of equal thickness, /2. By contrast with a composition-modulated superlattice, however, A and B consist of the same material, thus avoiding any effects that might arise from materials chemistry. The GBSLs described below contain symmetrical twist boundaries on the three principal cubic lattice planes; i.e., the thin slabs A and B in Fig. 2 are merely rotated with respect to each-other about the interface normal by an angle θ (between A and B) and −θ (between B and A). Unencumbered by any effects due to interfacial chemistry, these idealized model systems were shown to capture the essential interfacial phenomena of inhomogeneous structural disorder coupled with anisotropic lattice-parameter changes [4, 19].

Elastic behavior of interfaces

2031

1400 Symm, Twists, Cu(LJ)

GB Energy [mJ/m2]

1200 1000

(011) (m⫽1)

800 600

(001) (m ⫽ 2)

400 (111) (m ⫽ 3)

200 0 0

30

60

90 120 mθ (deg)

150

180

Figure 4. Energies (in mJ/m2 ) of twist boundaries versus twist angle for the three densest planes in the fcc lattice using the LJ(Cu) potential; the EAM results are qualitatively identical (see also Section 6.9) [18]. The factor m is related to the rotation symmetries of the (111) (m = 3) and (001) (m = 2) planes.

For the (001) GBSLs considered below, the twist angle θ was chosen to be 36.87◦ (forming so-called 5 (001) twist GBs); for the (011) GBSLs, θ = 50.48◦ (11 (011)); finally, for the (111) GBSLs, θ = 21.79◦ (7 (111)). The choice of these twist angles was motivated by our desire to maximize the degree of structural disordering on each GB plane in order to achieve the largest possible elastic anomalies on that plane. As seen in Fig. 4, on their respective planes each of these particular twist angles (arrows) represents a high-angle twist GB (in which, by definition, the dislocation cores overlap and the energy is therefore independent of θ; see also Section 6.9) [18].

2.1.

Simulation Results

As shown in Fig. 5(a), both the (001) and (111) GBSLs show monotonic, isotropic contractions in the x-y plane with decreasing , while the (anisotropic) (011) plane shows an expansion in one direction and a contraction in the other; a similar behavior was observed for (011) free-standing thin films [20]. The accompanying Poisson expansions normal to the interfaces are shown in Fig. 5(b). Interestingly, analogous to the results obtained for free-standing thin films [20], the (001) GBSLs show the largest expansions normal and the greatest contractions parallel to the interfaces.

2032

D. Wolf (a)

0.01

∆a x /a, ∆a y /a

0.00 ⫺0.01

GBSL Au(EAM) (111) (001) (011) - ax

⫺0.02 ⫺0.03

(011) - ay ⫺0.04 0

2

4

6

8

10 12 14 16 18 20 22 Λ/a

(b) 1.10

GBSL Au(EAM)

∆a z /a

1.08

(111) (001)

1.06

(011)

1.04 1.02 1.00 0

2

4

6

8

10 12 14 16 18 20 22 Λ/a

Figure 5. (a) Change in the average lattice parameters parallel to the interfaces for GBSLs on the three principal fcc planes, as a function of the modulation wavelength , using the Au(EAM) potential. (b) Average lattice parameter perpendicular to the interfaces for the same three GBSLs [19].

In Ref. [20] it was shown that linear elasticity theory, using bulk elastic moduli and the bulk free-surface stress, could account for the lattice-parameter changes of the (001) and (111) free-standing thin films very well. Figure 6, showing az from Fig. 5(b) plotted as a function of the planar contraction ax of Fig. 5(a), illustrates that linear elasticity theory works equally well for the anisotropic lattice-parameter changes in the GBSLs on the (001) and (111)

Elastic behavior of interfaces

2033

0.05 ⫺1.738

GBSL Au(EAM)

∆a z /a

0.04

Column 2 Column 3

0.03 ⫺1.212 0.02 0.01 0.00 ⫺0.03

(001) (111) ⫺0.02

⫺0.01

0.00

⌬a x /a Figure 6. Poisson expansion, az , in the direction of the surface normal resulting from the stress-induced in-plane lattice-parameter changes, ax , for the GBSLs. The straight lines are predicted from linear-elasticity theory based on the surface stress [19, 20].

planes. The slopes of the solid lines in the Fig. 6 are governed by a combination of Poisson ratios [19, 20]. As a consequence of these lattice-parameter contractions and expansions, the average GB area, A, decreases while the average atomic volume, , increases with decreasing [19]. Based on the behavior of homogeneous systems, one might expect some strengthening of the in-plane elastic behavior combined with a softening of the out-of-plane response. However, as was observed for the thin films [20], some moduli are strengthened (the “supermodulus effect”) while some are softened, regardless of being in-plane or out-of-plane (see Fig. 7). (Notice that the moduli in the figures have been normalized to the corresponding → ∞ values [19], which are governed by the appropriate averages over two perfect fcc crystals rotated with respect to one another about the GB normals.) According to Fig. 7(a), the in-plane Young’s modulus, Yx , for the (111) GBSLs does, indeed, strengthen as one would expect from the related in-plane contraction; however, its out-of-plane Young’s modulus, Yz (Fig. 7(b)), strengthens as well with decreasing , despite the z expansion. On the other hand, in spite of their by-far largest in-plane contraction, the (001) GBSLs exhibit only a very small increase in Yx for the larger values of followed by a decrease; moreover, in spite of their by-far largest increase in az and the atomic volume , in the (001) GBSLs Yz nevertheless strengthens. This comparison demonstrates that, even based on a complete knowledge of the

2034

D. Wolf

Young's Modulus, Yx / Yx∞

(a)

1.2 1.1 1.0 0.9 0.8 0.7 (111) (001) (011) - Yx (011) - Yy

0.6 0.5 GBSL Au(EAM)

0.4 0.3 0

Young's Modulus, Yz / Yz∞

(b)

2

4

6

8

10 12 14 16 18 20 22 Λ/a

2.5 (111) (001)

2.0

(011) 1.5 1.0 0.5

GBSL Au(EAM)

0.0 0

2

4

6

8

10 12 14 16 18 20

22

Λ/a Figure 7. (a) in-plane and (b) out-of-plane Young’s moduli for fully-relaxed Au (EAM) GBSLs, normalized to the related bulk moduli. For symmetry reasons, for the (001) and (111) GBSLs Yx = Yy [19].

interface-stress-induced anisotropic lattice-parameter changes, it is impossible to predict the magnitude or even the sign of the observed elastic anomalies. It is striking that even when the x-y lattice-parameter changes are suppressed in the simulation, the Young’s moduli still show anomalous behavior, albeit less pronounced, pointing again to the structural disorder due to the interfaces as the cause for the elastic anomalies [19].

Elastic behavior of interfaces

2035

An important aspect of the elastic behavior of the GBSLs is the resistance to shear strains parallel to the interfaces. According to Fig. 8, the shear moduli for the (001) and (111) GBSLs (for which Gxz = Gyz ) exhibit a pronounced softening with decreasing . However, by contrast with the above Young’s moduli, the shear moduli are of the same magnitude for the different GB planes. Why, by contrast with Yx and Yz , the shear moduli are rather insensitive functions of the detailed atomic structure and energy of the GBs was discussed in detail in [21], where it was argued that all high-angle GBs should exhibit a greatly reduced shear resistance right at the GBs because in such boundaries virtually all correlation is lost between the atom positions on opposite sides of the interface. In fact, in a bicrystal study [22, 23] the degree of structural disorder was systematically varied by considering a range of (001) twist angles between 0◦ and 45◦ . In correspondence with the rapidly increasing GB energy (see Fig. 4), the related shear moduli were shown to decrease dramatically with increasing twist angle. We finally mention that, because of the virtually complete loss of stacking order across the interfaces, the modulus for shear parallel to the GBs is a rather insensitive function of the x-y contractions (see Fig. 9(b)), by contrast to the related Young’s moduli (see Fig. 9(a)). Figure 9(a) also emphasizes the point made earlier that even when lattice-parameter changes are suppressed in the simulation, the Young’s modulus exhibits anomalous behavior, albeit less pronounced. 0.6 (111) Σ7 (100) Σ5

∞ Gxz/Gxz

0.5

LJ (Cu)

0.4

0.3

0.2

0.1 0.0

5.0

10.0

15.0 Λ/a

20.0

25.0

30.0

Figure 8. Normalized moduli for shear parallel to the interface planes for the (001) and (111) GBSLs [21].

2036

D. Wolf

Young's Modulus, Yz /Yz

(a)

2.5 x-y contraction: on

(001) GBSL Au(EAM)

2.0

off

1.5

1.0

0.5 0

10

20 Λ/a

30

40

(b) 1.0 Shear Modulus, Gxz /Gxz

0.9

x-y contraction: on off

(001) GBSL Au(EAM)

0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0

10

20 Λ/a

30

40

Figure 9. Normalized (a) Young’s moduli, Yz , and (b) shear moduli, Gxz = Gyz , as obtained by the Au(EAM) potential, for (001) GBSLs with and without allowing for the stress-induced x-y lattice-parameter changes, as a function of .

2.2.

Role of the Atomic Structure of the Interfaces

We now investigate the relationship between the atomic structure of the interfaces and the elastic anomalies. The degree of structural disorder is best illustrated by radial distribution functions like the ones shown in Figs. 10(a)– (c) associated with high-angle GBs on the three principal fcc planes (see also

Elastic behavior of interfaces

2037

(a) 1.2 (111) θ ⫽ 17.90˚ (Σ ⫽ 31) PLANE 1

1.0

r 2 G(r)

0.8 0.6 0.4 0.2 0.0 0.6

0.8

1.0

1.2

1.4

1.6

1.2

1.4

1.6

r/a

(b)

1.5 (100) θ ⫽ 43.60˚ (Σ ⫽ 29) PLANE 1

r 2 G(r)

1.0

0.5

0.0 0.6

0.8

1.0 r/a

(c)

0.25 (110) θ ⫽ 50.48˚ (Σ ⫽ 11) PLANES 1 ⫹ 2

r 2 G(r)

0.20 0.15 0.10

0.05 0.00 0.6

0.8

1.0

1.2

1.4

1.6

r/a

Figure 10. Radial distribution functions, r 2 G(r ), for twist boundaries on the three densest planes of the fcc lattice at zero temperature. For the two densest planes (Figs. (a) and (b)) [24], only atoms in the planes immediately at the GB were included in G(r ); by contrast, because of the smaller spacing of the (110) planes and the larger degree of atomic-level “rumpling” in these planes, the two planes nearest to the GB were considered in the case of the (110) twist GB (Fig. (c)) [25]. The full arrows indicate the perfect-crystal δ-function peak positions at zero temperature.

2038

D. Wolf

Fig. 4 and Section 6.9). The comparison of the substantially broadened peaks with the corresponding zero-temperature δ-function peaks of heights 12, 6, 24, etc. at the nearest-neighbor (nn), 2nd nn, 3rd nn, etc. distances of 0.707a, a, 1.225a, etc. in the fcc lattice demonstrates the strongly defected local environments of the atoms near the GBs. Also, consistent with the related GB energies (see Fig. 4), the (110) boundaries show by far the greatest broadening, followed by the (001) and (111) GBs, the latter with the lowest energy. A detailed analysis given in Figs. 11(a)–(c) for the three planes nearest to an (001) high-angle twist boundary (labeled 29, with θ = 43.60◦ ) shows two effects [4]. First, as evidenced by the rapid recovery of sharp perfect-crystal peaks only three planes away from the GB (Fig. 11(c)), the structural disorder is highly localized at the GBs. Second, compared to the perfect crystal (solid arrows), the peak centers in the GBSLs (open arrows) are shifted slightly towards larger distances, by an amount approximately proportional to the corresponding volume expansion at the GBs [4]. The apparently paradoxical question is this: How can at least some elastic moduli of an interface material strengthen in spite of the decrease in its average density? Based on our usual intuition, gained from the study of homogeneous systems, one would expect all elastic moduli and constants to weaken upon expansion. As first pointed out in Ref. [4], although the overall volume of the system expands upon introduction of the interfaces (i.e., the average distance between the atoms increases), some atoms are in closer proximity to one-another, up to about 10%, than they are in the perfect crystal (see Figs. 10 and 11). These shorter distances are expected to strengthen the local elastic response whereas longer distances give rise to a softening, with the net effect apparently being a strengthening of some moduli. However, as illustrated above, the net outcome of this complex averaging process seems to depend strongly on the detailed atomic structure of the interfaces and on the particular elastic modulus considered. We finally consider the difference between the elastic constants and the related moduli, a rather fundamental distinction from both a conceptual and experimental viewpoint. When determining a modulus, an external stress is applied to the system and the ensuing strains are monitored; i.e., the stress is fixed and the strains are variables. In an elastic-constant measurement, by contrast, a strain is imposed on the system and the ensuing stresses are monitored. Hence, while a modulus describes the physical response of the system while permitting all lattice-parameter changes of the system in response to the applied stress to take place, an elastic constant describes the system response while all strains are fixed. The moduli are consequently given by the elastic compliances, thus representing combinations of elastic constants. Consequently, while the anomalies in the elastic constants may be rather small (see Fig. 12 for the (001) GBSLs), the anomalies in the related moduli may be much larger by comparison (see Fig. 7). It therefore appears that the reduced

Elastic behavior of interfaces

2039

(a) 1.5 (100) Σ 29 PLAN 1

G (r )

1.0

0.5

0.0 0.0

0.8

1.0

1.2

1.4

1.6

1.2

1.4

1.6

1.2

1.4

1.6

r

(b) 2

G (r )

(100) Σ 29 PLAN 2

1

0 0.6

0.8

1.0 r

(c) 5 (100) Σ 29 PLAN 3

4

G (r )

3 2 1 0 0.6

0.8

1.0 r

Figure 11. Plane-by-plane zero-temperature radial distribution functions for the three lattice planes closest to the (001) θ = 43.60 (29) symmetrical twist GB surrounded by bulk perfect crystals [4]. The full arrows indicate the perfect-crystal δ-function peak positions. The open arrows mark the average neighbor distance in each shell. The widths of these shells are indicated by the dashed lines. (We note that Fig. 2 in Section 6.7 represents the average G(r ) for a GBSL of (001) twist GBs with six (001) planes between the interfaces.)

2040

D. Wolf 1.4 (001) Cu(LJ)

Elastic Moduli

1.2 1.0 0.8

GBSL

0.6

Yz /Yz∞ Gx z /Yxz∞

0.4 Strained Perfect Crystal

0.2 0.0 0

10

20 Λ/ a

Yz /Yz∞ Gx z /Yxz∞ 30

40

Figure 12. Elastic constants (in 1012 dyn/cm2 ) for the (001) GBSLs whose Young’s moduli are shown in Figs. 7 and 8 [19, 26].

elastic constants reported in numerous experiments may not contradict experiments in which enhanced moduli were observed. The supermodulus effect may therefore be very aptly named since a “super elastic-constant” effect may not exist [26].

3.

Dissimilar-material Superlattices

Based on the interpretation of the elastic behavior of thin films [20] and GBSLs [4, 9, 21–23, 26] as structural interface effects, several predictions can be made regarding tailoring elastic behavior by controlling interface structure. Most importantly, one would expect that coherent interfaces (“perfect epitaxy”) should exhibit the smallest elastic anomalies. Moreover, similar to the increase in the elastic anomalies upon replacing (111) by (001) GBs or free surfaces by GBs, introduction of structural disorder into a relatively perfect interface, via misfit dislocations, should increase these anomalies significantly. In order to test these predictions, Jasczak and Wolf [27, 28] investigated the role of coherency in the elastic behavior of composition-modulated superlattices on the (001) plane of fcc metals. Again, in order to eliminate materials and interfacial chemistry as much as possible as a contributing factor, these simulations were performed using Lennard–Jones potentials with a 10% [27] and 20% lattice-parameter mismatch [28] but with the same cohesive energies. We here focus on the (001) superlattices with a 20% lattice-parameter

Elastic behavior of interfaces

2041

mismatch because the results are more dramatic than those obtained for a 10% mismatch [28]. The fully relaxed structures of the three types of dissimilar-material superlattices studied here, containing varying degrees of interfacial structural disorder, are shown in Figs. 13(a)–(c). The coherent superlattices (COHSLs), while strained due to the lattice-parameter mismatch, are highly ordered at the interfaces (Fig. 13(a)). In the incoherent superlattices shown in Fig. 13(b) (INCSLs), the mismatch strains are relieved via the introduction of a square network of misfit dislocations, with a consequent increase in the degree of interfacial structural disorder. In order to introduce even more structural disorder, the (001) planes in (b) were twisted relative to each other, thus introducing screw dislocations in addition to the misfit dislocations already there. In the superlattices shown in Fig. 13(c), a twist angle of ±16.26◦ between alternating A and B layers (Fig. 2) was chosen; because the resulting planar unit cell contains five times as many atoms as the INCSLs, these superlattices are designated 5 NCSLs. As expected, the incoherent superlattices exhibit greater elastic anomalies than the coherent superlattices for both the 10 and 20% lattice-parameter mismatch [27, 28]. For example, Fig. 14(a) illustrates the systematic increase in the anomalous stiffening of Yz with decreasing as the incoherency of the superlattices increases from COHSL, to INCSL, to 5 INCSL. Figure 14(b) shows a parallel decrease in the shear moduli, Gxz = G yz , most notably the virtual absence of any shear resistance in the incoherent superlattices [27, 28].

(a)

(b)

COHSL

(c)

INCSL

Σ5 INCSL

Figure 13. Illustration of the various degrees of structural disorder in the three latticemismatched superlattice types considered by Jasczak and Wolf [27, 28], with increasing degrees of interfacial structural disorder from (a) to (c). Shown are the fully relaxed atom positions in one plane of material A (solid circles) and one plane of material B (open circles) for superlattices containing eight planes of each material per modulation wavelength. (a) coherent superlattice (COHSL), (b) incoherent superlattice (INCSL) and (c) 5 incoherent superlattice (5 INCSL) [27, 28].

2042

D. Wolf (a) Σ 5 INCSL

1.3 Young's Modulus, Yz / Yz

INCSL 1.2

COHSL

1.1 1.0 0.9

0.8 0

2

4

6

8

10

12

14

16

18

20

(b)

Shear Modulus, G x z / G x z

1.2 1.0 0.8 COHSL

0.6

INCSL Σ 5 INCSL

0.4 0.2 0.0 0

2

4

6

8

10 12 Λ /a

14

16

18

20

Figure 14. (a) Young’s Modulus, Yz , and (b) shear modulus, Gxz , as a function of the modulation wavelength for the three superlattice types of Fig. 13. The normalization constants, which are the → ∞ values for the unstrained systems (i.e., the INCSLs), are 12 2 ∞ 12 2 Y∞ z = 0.793 × 10 dyn/cm and Gxz = 0.727 × 10 dyn/cm [27, 28].

Similar to the thin films [20] and GBSLs, the 5 INCSLs show a pronounced contraction parallel to the interfaces with a consequent Poisson expansion perpendicular to the interfaces. Jasczak and Wolf [27, 28] showed that the interface-stress-induced changes in lattice parameters cannot account for the anomalous elastic behavior. However, in complete analogy to the observations for the thin films and GBSLs, they do enhance the anomalies caused by the structural disorder. In order to differentiate between the effects of the interfaces and those caused by the changes in lattice-parameters, we compare the elastic response

Elastic behavior of interfaces

2043

of the 5 INCSLs with that of corresponding interface-free perfect-crystal reference systems with the same unit-cell size and shape as the 5 INCSLs themselves. The latter was determined as follows [27, 28]: The elastic constants of ideal crystals of A and B, however strained to the appropriate anisotropic lattice parameters of the 5 INCSLs, were individually determined and then averaged in the appropriate manner for superlattices [19, 29, 30] to give the “interface-free-superlattice” (IFSL) elastic constants and moduli. As shown in Figs. 15 and 16 for the Young’s and shear moduli, respectively, the IFSLs show a behavior that is very close to that of a homogeneous system that cannot even nearly account for either the anomalous strengthening of Yz nor the extreme softening of Gxz . In particular, although Yz does show some stiffening in the

Σ 5 INCSL Σ 5GBSL

1.3

Σ 5 in SLAB

Young's Modulus, Y z / Y z

Σ 5 SLAB Σ 5 IFSL

1.2

(001) 1.1

1.0

0.9 0

2

4

6

8

10 12 Λ /a

14

16 18

20

Figure 15. Comparison of the Young’s moduli, Yz , as a function of the modulation wavelength for the 5 INCSLs with those of the superlattices of the (001) θ = 36.87 (5) twist grain boundaries (5 GBSLs; see Section 2.1), thin slabs with and without a 5 grain boundary, and interface-free composition-modulated superlattices (IFSLs) with the lattice-parameter changes of the 5 INCSLs. (We note that all results were obtained for the LJ potential.) The normalization factors are 12 2 12 2 Y∞ z = 0.793 × 10 dyn/cm and 1.076 × 10 dyn/cm , while a¯ = 3.9776 Å and 3.6160 Å, respectively, for the composite systems (5 INCSL and IFSL) and for the monatomic systems (GBSLs and SLABs) [29, 30].

2044

D. Wolf 1.0

Shear Modulus, GXZ / GXZ

0.8

IFSL SLAB 5S in SLAB

0.6 (001) 0.4

0.2 5S GBSL 5S INCSL 0.0 0

2

4

6

8

10 12 14 16 18 20 22 Λ/a

Figure 16. Shear modulus Gxz as a function of for the same systems as in Fig. 15. The 12 2 12 2 normalization factors are G∞ xz = 0.727 × 10 dyn/cm and 1.016 × 10 dyn/cm , respectively, for the composite and the monatomic systems [29, 30].

IFSLs with decreasing (Fig. 15) while Gxz softens slightly (Fig. 16), the effects are very small compared to those in the 5 INCSLs. For comparison, Figs. 15 and 16 also show the LJ results obtained for the (001) GBSLs [19, 21] and slabs [20] and for bicrystalline (001) slabs containing a single 5 (001) twist GB in their center [26]. It is interesting to observe that, except for the appearance of a maximum in Yz and a minimum in Gxz , the 5 GBSL results agree very well with the dissimilar-materials 5 INCSL results, indicating that some saturation level is reached in the elastic behavior as the dislocation cores overlap completely, similar to the interface energies in Fig. 4. As the degree of interfacial structural disorder is systematically decreased – by going from the superlattice of 5 GBs, via a bicrystalline slab containing a single 5 GB to the single-crystal slab – the interfaceinduced stiffening of Yz and softening in Gxz disappear gradually, with the slab exhibiting even a softening in Yz . The main conclusion is therefore that increasing the degree of structural disorder in the superlattices, either by increasing the lattice-parameter mismatch or by introducing a relative rotation between the two materials (thus introducing screw dislocations), will dramatically enhance the small elastic anomalies present in the coherent system. That the transition from a coherent

Elastic behavior of interfaces

2045

to an incoherent interface structure is, indeed, associated with enhanced elastic anomalies has been verified experimentally [31, 32].

4.

Effects of Temperature

So far we have shown that by controlling the structural disorder due to the interfaces, one can “engineer” the elastic behavior of the material. While interfacial structural disorder is inhomogeneous, i.e., localized at the interfaces, one might ask whether homogeneous structural disorder has the same effect as inhomogeneous disorder. The best-known form of homogeneous disorder arises from the thermal movements of the atoms at finite temperature. Another way of formulating the same question is therefore to ask whether the temperature dependence of the elastic behavior can be understood in terms of the same underlying causes as the zero-temperature elastic behavior of interface materials, namely a competition between the strengthening due to the atomic-level disorder and the softening due to the consequent volume expansion. In an attempt to elucidate whether such a temperature-induced competition, indeed, exists, Jaszczak and Wolf [33] have performed extensive MD simulations of the thermo-elastic behavior of both perfect crystals and of the (001) GBSLs described above. By comparing the results of zero-stress simulations (in which volume expansion is permitted) with constant-volume simulations (in which thermal expansion is suppressed), they were able to deconvolute the homogeneous effects induced by thermal disorder from the consequent thermal expansion. Following the full equilibration of the system, the elastic constants and moduli were calculated using the stress-fluctuation formula [14, 34, 35] (for details, see Ref. [33]).

4.1.

Thermo-elastic Behavior of a Perfect fcc Crystal

It is well known that most materials soften thermally, typically by a factor of two between absolute zero and melting, as a consequence of their thermal expansion. If the thermal and interface-induced types of disorder had the same physical effects, one would expect that temperature should provide causes for both elastic softening and strengthening. More specifically, since the thermal fluctuations increase the degree of (homogeneous) structural disorder, as evidenced by the well-known broadening of the peaks in the radial distribution function (see Fig. 17), they should actually give rise to an elastic strengthening. However, as the temperature is increased, the peaks in Fig. 17 shift to larger separations due to the thermal expansion; thermal expansion, the “prize” of thermal disorder, thus provides a mechanism for elastic softening.

2046

D. Wolf

(r/a)2 g(r)

15

10

Zero Stress

Perfect Crystal 100K 400K 1000K

5

0 0.6

0.8

1.0

1.2

1.4

1.6

1.8

Figure 17. Pair distribution function of a perfect fcc crystal at various temperatures under zero stress. G(r ) is normalized such that at zero temperature the nearest-neighbor peak of the perfect crystal is a δ-function of height 12 [33].

According to Fig. 18(a), the elastic constants of a perfect crystal, indeed, soften dramatically (and approximately linearly) with increasing temperature if the thermal expansion is permitted. As illustrated in Fig. 18(b), however, if the thermal expansion is suppressed, even the perfect crystal stiffens elastically with increasing temperature. Although this stiffening, by about 8% in C11 and C44 at 1000 K (i.e., at about 80% of the melting temperature, Tm ∼1200 K, for this potential), is much less pronounced than the net softening when the volume expansion is permitted, this comparison illustrates that homogeneous (i.e., thermal) disorder in a perfect crystal affects the elastic behavior in much the same way as inhomogeneous (i.e., interfacial) disorder in interface materials.

4.2.

Thermo-elastic Behavior of (001) GBSLs

We now superimpose the homogeneous effects of thermal disorder on the already inhomogeneously disordered superlattices of grain boundaries investigated above. The peaks in G(r), already broadened even at zero temperature (see Figs. 10 and 11), can then be expected to broaden further with increasing temperature. By selectively fixing the temperature or the modulation wavelength, Jaszczak and Wolf [33] were able to deconvolute the homogeneous from the inhomogeneous effects: At a fixed temperature, i.e., for a fixed degree of thermal disorder, the amount of interfacial disorder in the GBSLs can be systematically varied by changing the modulation wavelength; conversely, by fixing , the amount of interfacial disorder can be fixed while the thermal disorder is varied with the temperature.

Elastic behavior of interfaces

2047

Elastic Constants

1.00

C11 / C110 C12 / C120

0.80

C14 / C140

0.60 Perfect Crystal Zero Stress 0.40 0

200

400

600

800

1000

800

1000

T (K) 1.08

Cjj(T) / C (T ⫽ 0)

C11 1.06

C12

1.04

C44

1.02 1.00 Perfect Crystal; Fixed Volume 0.98 0

200

400

600 T (K)

Figure 18. (a) Isothermal elastic constants as a function of temperature for a perfect fcc crystal (a) under zero external stress and (b) for constant T = 0 K volume (LJ potential). The elastic constants in (a) are in units of 1012 dyn/cm2 ; those in (b) are normalized to the zero-temperature, zero-stress perfect-crystal values given in Table 1. The solid lines represent least-squares fits to the data [33].

Figure 19 shows G(r) for a GBSL composed of four (001) planes between the GBs at T = 100 K and 400 K under zero external stress. (At higher temperatures the GBSLs tend to crystallize by grain-boundary migration to form a perfect crystal [33]). As is evident from Fig. 19, the peaks in the G(r) for the GBSLs both broaden and shift to larger separations with increasing temperature, in a fashion completely analogous to the perfect-crystal peaks in Fig. 17.

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D. Wolf 8 (001) GBSL 4 planes between GB's

(r/a) g(r)2

6

Zero Stress

100 K 400 K 4

2

0 0.6

0.8

1.0

1.2 r/a

1.4

1.6

1.8

Figure 19. Pair distribution function of a GBSL composed of four (001) planes between GBs at two temperatures under zero stress [33].

Elastic Constants

1.00

(001) GBSL 4 planes between GBs Zero Stress

0.90

0.80 C11 / C110 C33 / C330 0.70 0

100

200

300

400

T (K) Figure 20. Selected isothermal elastic constants as a function of temperature for a GBSL composed of four (001) planes between GBs under zero stress. Moduli for the GBSLs soften similarly. The GBSL values are normalized to the zero-temperature elastic constants for this particular GBSL: C011 = 1.788 × 1012 dyn/cm2 , C033 = 1.859 × 1012 dyn/cm2 [33].

Moreover, due to the interfaces the peaks are broader in the GBSLs than in the related perfect crystal [33]. Figure 20 shows the combined effects of the thermal disordering and consequent thermal expansion on the elastic properties of the highly inhomogeneous

Elastic behavior of interfaces

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GBSLs with four (001) planes between the GBs. Interestingly, in complete analogy to the perfect crystal (see Fig. 18(a)), the net effect is a nearly linear softening of the elastic constants and moduli with increasing temperature. Hence, despite their inherently inhomogeneous structure, the GBSLs behave homogeneously (i.e., perfect-crystal like) in response to homogeneous (thermal) disorder when the thermal expansion is allowed to take place; i.e., they elastically soften as their G(r) peaks broaden and shift to larger separations with increasing temperature. Also, the centers of the G(r) peaks for the GBSLs are at nearly the same positions as the peaks for a perfect crystal at the same temperature [33]. To demonstrate that this softening is mostly due to the thermal expansion of these highly inhomogeneous systems, in Fig. 21 the values of C33 obtained under conditions of constant volume and constant (zero) pressure, respectively, are compared for the same GBSLs. According to the figure, the role of thermal disorder alone (i.e., as the temperature is increased) is, indeed, a slight strengthening of C33 ; the Young’s modulus, Yz , strengthens similarly [33]. Moreover, similar to the behavior of the perfect crystal (Fig. 18), the enhancement of C33 and Yz over the zero-stress values at the same temperature is quite significant (by over 40% at 400 K). The thermal softening under zero stress is therefore predominantly due to the volume expansion. Finally, by changing the modulation wavelength, , the fraction of atoms “seeing” the interfaces can be systematically varied. The corresponding variation of the Young’s moduli on is shown in Fig. 22 for several temperatures. Even at non-zero temperatures, the GBSLs show the same generic

C33 (T) / C (T33 ⫽ 0)

1.00 Fixed Volume Zero Stress

0.90

0.80 (001) GBSL 4 planes between GBs 0.70 0

100

200

300

400

500

T (K) Figure 21. Comparison of the isothermal elastic constants, C33 , as a function of temperature of a GBSL with four (001) planes between GBs at fixed zero-temperature volume and at fixed zero external stress [33].

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D. Wolf (a)

1.3 (001) GBSL Zero Stress

Yz / Yz∞

1.2 1.1 1.0

T=0K T = 100 K T = 400 K

0.9 0.8 0

5

10

15

20

Λ/a (b) 1.0

Yx / Yx∞

0.9 (001) GBSL Zero Stress

0.8 0.7

T⫽0K T = 100 K T = 400 K

0.6 0.5 0

5

10

15

20

Λ/a Figure 22. Isothermal Young’s moduli (a) Yz and (b) Yx as a function of at T = 0, 100 and 400 K. The moduli are normalized to their T = 0 values in the → ∞ limit: 1.076 and 1.429 × 1012 dyn/cm2 , respectively [33].

behavior as a function of as they do at zero temperature. In particular, despite the large, anisotropic thermal expansion, there remains at small an anomalous enhancement of the Young’s modulus, Yz [Fig. 22(a)]. By contrast, the in-plane Young’s moduli, Yx = Yy in Fig. 22(b) show only softening with decreasing at both zero and non-zero temperatures. On the other hand, for a fixed value of , and therefore a fixed degree of interfacial structural disorder, the effect of increasing the temperature under zero stress is to soften all the moduli.

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We therefore conclude that the effect of thermal disorder and consequent volume expansion is largely to soften the elastic moduli by the same degree, independent of the amount of inhomogeneous structural disorder present (i.e., independent of ). The above investigation demonstrates that atomic-level structural disorder, be it homogeneous or inhomogeneous, can lead to elastic stiffening, provided that the related volume expansions do not dominate the elastic behavior and result in a softening. Whether the disorder or the consequent volume expansion will dominate the elastic response depends on the detailed nature of the disorder, the anisotropy of the volume expansion, and on the nature of the interatomic interactions. However, even using the simple Lennard–Jones potential, Jaszczak and Wolf [33] were able to demonstrate a full spectrum of behaviors, from stiffening to softening, by varying the degrees of homogeneous and inhomogeneous disorder and by controlling the resulting volume expansions.

5.

Conclusions

The above simulations illustrate the unique capabilities of atomic-level computer simulations to explore the physical origin of the anomalous elastic behavior of composition-modulated superlattice materials. These capabilities enable investigation of simple, but well characterized model systems which exhibit the same generic behavior as that observed experimentally in “real” composition-modulated alloys [6–11, 31, 32]. In particular, this behavior usually includes (i) a pronounced change in the lattice parameter in the plane of the interfaces accompanied by a related change in the normal direction (the Poisson effect) and (ii) a strengthening of some elastic moduli and constants accompanied by a softening of others. The most pronounced feature observed both experimentally and in the simulations is a strong variation of both the structure and elastic behavior as a function of the composition modulation wavelength, . This dependence expresses the fact that the average physical response of the material consists of a tunable mixture of homogeneous and inhomogeneous effects: By decreasing gradually, more and more atoms in the system experience the presence of the interfaces, and their behavior resembles less and less that of a homogeneous system, thus gradually exposing the behavior characteristic of the inhomogeneous parts of the system. By contrast with the typical experimental situation, in the simple model systems investigated above the atomic structure and any effects associated with materials and interfacial chemistry can be carefully controlled and systematically varied. Combined with the ability to apply external stress in simulations, these simulations were able to

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(a) deconvolute the distinct effects due to the inhomogeneous atomic disorder localized at the interfaces from the consequent interface-stressinduced anisotropic lattice-parameter changes, and (b) separate the homogeneous effects of thermal disordering from the inhomogeneous effects due to the interfaces. These simulations thus provide insight into the atomic-level phenomena and processes governing interfacial elasticity and expose the physical causes for the elastic anomalies of interface materials. Such atomic-level insights are difficult to obtain by experimental means alone or from theoretical methods based on continuum mechanics. They can provide a fundamental basis for the design of composite materials with desired elastic properties. The three major conclusions of this work may be summarized as follows. First, while the average interface-stress-induced structure of thin films and multilayers can be predicted from linear elasticity theory, their elastic behavior, showing some moduli to be hardened while others are softened, is considerably more complex. The latter is the result of a highly complex interplay between two competing causes, namely (i) the built-in structural disorder at the interfaces, as evidenced by a broadening of the radial distribution function, G(r), even at zero temperature, and (ii) the consequent anisotropic lattice-parameter changes, giving rise to a shift in the G(r) peaks. The broadening results in some atoms near the interfaces being shoved more closely together than in the perfect crystal, providing the ingredient necessary for elastic strengthening; the peak shifts, by contrast, usually soften the material. The key to elastic strengthening therefore lies in minimizing the volume expansion per degree of structural disordering. Second, based on the interpretation that the elastic anomalies arise from the interfacial structural disorder, several predictions have been made and verified by simulations. Most importantly, coherent (i.e., epitaxial) interfaces were shown to exhibit the smallest elastic anomalies; introduction of structural disorder via misfit dislocations increases these anomalies significantly, a prediction that was verified experimentally [31, 32]. Third, the effect of temperature on the elastic anomalies has been investigated. It was found that the elastic moduli of grain-boundary superlattices soften with increasing temperature as one would expect for a homogeneous system. Considering that the elastic anomalies arise from the inhomogeneous structural disorder localized at the interfaces, this result is somewhat a surprise. Ultimately the elastic anomalies of interfacial materials arise from a competition between structural disorder and the consequent (usually anisotropic) lattice-parameter and net volume change. This competition can be seen even in a perfect crystal at finite temperature: Increasing the temperature (i.e., broadening the radial distribution function) without permitting the crystal to expand actually strengthens the elastic constants and moduli. In superlattices, by

Elastic behavior of interfaces

2053

contrast, such a broadening in the radial distribution function is present even at zero temperature. This broadening may lead to an elastic strengthening perpendicular to the interfaces provided the related volume expansion, and the associated elastic softening, is not too large. Throughout these simulations the interfaces were assumed to be atomically flat and chemically sharp. By contrast to the typical experimental situation, any effects that might arise from chemical disordering at the interfaces have thus been avoided. Based on the insights gained on these simple model systems, one would expect that chemical disorder, as evidenced in a broadening of the partial radial distribution functions of the material, should play the same role in the anomalous elastic behavior of multilayers as does structural disorder.

Acknowledgment This work was supported by the U.S. Department of Energy, BES Materials Sciences, under Contract W-3l-l09-Eng-38.

References [1] R. Hull and J.C. Bean, Crit. Rev. Solid State and Mater. Sci., 17, 507, 1992. [2] D. Wolf and J.A. Jasczcak, Chapter 14 in Materials Interfaces: Atomic-Level Structure and Properties, D. Wolf and S. Yip (eds.), Chapman and Hall, London, pp. 364 ff, 1992. [3] D. Wolf and J.A. Jasczcak, J. Comput. Aided Mats. Design, 1, 111, 1993. [4] D. Wolf and J.F. Lutsko, Phys. Rev. Lett., 60, 1170, 1988. [5] See, for example, Chapter 1 in Materials Interfaces: Atomic-Level Structure and Properties, D. Wolf and S. Yip (eds.), Chapman and Hall, London, p. 1 ff, 1992. [6] For recent reviews, see R.G. Brandt, Mater. Sci. Eng. B, 6, 95, 1990; M. Grimsditch and I.K. Schuller, Chapter 13 in Materials Interfaces: Atomic-Level Structure and Properties, D. Wolf and S. Yip (eds.), Chapman and Hall, London, 1992, pp. 354 ff; M. Grimsditch in Topics in Applied Physics: Light Scattering in Solids V, M. Cardona and G. Guntherodt (eds.), Springer, Berlin, 1989, p. 285; B.Y. Yin and J.B. Ketterson, Adv. Phys., 38, 189, 1989. [7] W.M.C. Yang, T. Tsakalakos, and J.E. Hilliard, J. Appl. Phys., 48, 876, 1977. [8] A. Kueny, M. Grimsditch, K. Miyano et al., Phys. Rev. Lett., 48, 166, 1988. [9] U. Helmersson, S. Todorova, S.A. Barnett et al., J. Appl. Phys., 62, 491, 1987. [10] B.M. Davis, D.N. Seidman, A. Moreau et al., Phys. Rev. B, 43, 9304, 1991. [11] A. Fartash, E.E. Fullerton, I.K. Schuller et al., Phys. Rev. B, 44, 13760, 1991. [12] J.F. Lutsko, J. Appl. Phys., 65, 2991, 1989. [13] M. Born and K. Huang, “Dynamical theory of crystal lattices,” Clarendon Press, Oxford, 1954. [14] J. Ray and A. Rahman, J. Chem. Phys., 80, 4423, 1984 and Phys. Rev. B, 32, 733, 1985.

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[15] S.M. Foiles, M.I. Baskes, and M.S. Daw, Phys. Rev. B, 33, 7983, 1986. [16] D. Wolf, J.F. Lutsko, and M. Kluge, “Atomistic simulation in materials – beyond pair potentials,” V. Vitek and D. Srolovitz (eds.), Plenum Press, New York, p. 245, 1989. [17] D. Wolf, Surf. Sci., 226, 389, 1989; Phil. Mag. A, 63, 337, 1991. [18] D. Wolf and K.L. Merkle, Chapter 3, “Materials interfaces: atomic-level structure and properties,” D. Wolf and S. Yip (eds.), Chapman and Hall, London, pp. 87 ff, 1992. [19] D. Wolf and J.F. Lutsko, J. Mater. Res., 4, 1427, 1989. [20] D. Wolf, Appl. Phys. Lett., 58, 2081, 1991. [21] D. Wolf, Mater. Sci. Eng. A, 126, 1, 1990. [22] D. Wolf and M.D. Kluge, Scripta Metall. Mater., 24, 907, 1990. [23] M.D. Kluge, D. Wolf, J.F. Lutsko et al., J. Appl. Phys., 67, 2370, 1990. [24] D. Wolf, Acta Metall., 37, 1983, 1989. [25] D. Wolf, Acta Metall., 37, 2823, 1989. [26] D. Wolf and J.F. Lutsko, J. Appl. Phys., 66, 1961, 1989. [27] J.A. Jaszczak, S.R. Phillpot, and D. Wolf, J. Appl. Phys., 68, 4573, 1990. [28] J.A. Jaszczak and D. Wolf, J. Mater. Res., 6, 1207, 1991. [29] M.H. Grimsditch, Phys. Rev. B, 31, 6818, 1985. [30] M.H. Grimsditch and F. Nizzoli, Phys. Rev. B, 33, 5891, 1986. [31] G. Carlotti, D. Fioretto, G. Socino et al., J. Appl. Phys., 71, 4897, 1992. [32] E.E. Fullerton, I.K. Schuller, F.T. Parker et al., J. Appl. Phys., 73, 7370, 1993. [33] J.A. Jaszczak and D. Wolf, Phys. Rev. B, 46, 2473, 1992. [34] J. Ray, Comp. Phys. Reports, 8, 111, 1988. [35] D.R. Squire, A.C. Holt, and W.G. Hoover, Physica, 42, 388, 1969.

6.13 GRAIN BOUNDARIES IN NANOCRYSTALLINE MATERIALS Dieter Wolf Materials Science Division, Argonne National Laboratory, Argonne, IL 60439, USA

The rapidly developing ability to manipulate matter at the atomic level is making considerable impact in materials-science related technologies because it permits novel microstructures and metastable phases to be atomically designed, often resulting in unusual materials properties. The importance of metastable phases with novel microstructures, such as nanocrystalline materials, metallic glasses, multilayers and epitaxially-stabilized thin-film phases, is based primarily on their intrinsic inhomogeneity. This inhomogeneity originates from the atomic-level disorder generated by manipulating the microstructure, for example, via introduction of grain or phase boundaries or, in the case of amorphous materials, by creation of a frozen-liquid like structure. By design, these materials systematically exploit the altered, yet to-date usually unknown local physical behavior in the heavily defected, inhomogeneous structural environments experienced by a large fraction of the atoms. Atomiclevel computer simulations are uniquely suited to provide insight into what constitutes “typically inhomogeneous” behavior in heavily disordered materials because they provide information on local structures and properties. The nature of the grain boundaries (GBs) in nanocrystalline materials has been the subject of intense debate ever since the first ultrafine-grained polycrystals, with a typical grain size of 5–50 nm, were synthesized over a decade ago by consolidation of small clusters formed via gas condensation [1, 2]. For some time it had been thought that the atomic structures of the severely constrained GBs in NCMs differ fundamentally from those observed in coarsegrained or bicrystalline materials because, for a grain diameter of nanometer dimensions, a significant fraction of the atoms is situated in or near GBs and grain junctions [3, 4]. The suggestion of a “frozen-gas” like structure of the GBs [3, 4] was aimed at explaining the rather unusual diffraction results; such a model might also explain some of the unusual thermal properties reported for NCMs, such as their enhanced specific heat [5, 6] and lower Debye 2055 S. Yip (ed.), Handbook of Materials Modeling, 2055–2079. c 2005 Springer. Printed in the Netherlands.

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temperature [4, 7]. However, more recent experimental observations involving Raman spectroscopy [8], atomic-resolution TEM [9] and X-ray diffraction [10] indicate that the atomic structures of GBs in NCMs are, in fact, rather similar to those in coarse-grained polycrystalline or bicrystalline materials. Any attempt to elucidate the properties of NCMs requires, at the outset, a GB structural model that incorporates the severe microstructural constraints in NCMs and yet connects with the large body of knowledge on extended GBs in bicrystalline materials. Two key questions that have evolved from the experimental studies are: (1) What is the structural and thermodynamic relationship between nanocrystalline microstructures and amorphous solids? (2) To what extent can the atomic structure of the GBs in nanocrystalline materials be extrapolated from those of coarse-grained polycrystalline materials and bicrystals? Here we review the insights gained from atomic-level computer simulations on these two important questions. In Section 1, we discuss a simple model that is designed to capture both, the severe microstructural constraints associated with the small grain size and the physical inhomogeneity associated with the GBs. Simulations performed for this simple model system expose important parallels that exist in the dynamical properties of nanocrystalline materials and glasses. In Section 2, we review results obtained by a method for the computational synthesis of less idealized nanocrystalline microstructures, by growth from the melt into which randomly oriented crystal nuclei were inserted. This method enables a fuller comparison of the structures of the highly constrained GBs in nanocrystalline materials with those of entirely unconstrained, bicrystalline GBs. The simulations of GB diffusion creep reviewed in Section 3 represent an important test for the GB structural model suggested on the basis of the low-temperature observations discussed in Section 2. Finally, our most important conclusions are summarized in Section 4.

1.

A Simple Model

The problem of relating the structure and properties of the GBs in a polycrystal to those of corresponding bicrystalline GBs is extremely difficult as it requires three types of microstructural averages to be performed. These arise from the microstructural constraints present in a polycrystal and involve averages over (a) the five macroscopic degrees of freedom that each individual GB contributes to the total number of degrees of freedom of the polycrystal (three degrees associated with the misorientation between the grains and two characterizing the GB-plane normal; [12]), (b) the various grain shapes and (c) the distribution of grain sizes invariably present in polycrystals.

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In an early attempt to develop a structural model for nanocrystalline materials, Wolf et al. [12–14] presented a simple model that was tailored to capture the two essential structural features of NCMs, namely (i) the microstructural constraints associated with the finite grain size and (ii) the structural inhomogeneity due to the GBs and grain junctions. The question Wolf et al. [12] asked at the outset is this: As far as the total number of GB degrees of freedom, grain shapes and grain sizes are concerned, what is the conceptually simplest polycrystal that one can, at least in principle, construct? In other words: What is the smallest number of geometrically distinct types of GBs, grain shapes and grain sizes that a polycrystal has to contain and still be space filling? Figure 1 shows that it is geometrically possible to construct a space-filling, three-dimensional polycrystal with a uniform (and unique) rhombohedral grain shape in which all GBs are crystallographically equivalent; i.e., a monodisperse polycrystal with exactly the same number of macroscopic degrees of freedom (at most 5) as the corresponding bicrystal, the only difference being the finite, variable grain size. Having thus eliminated the distributions in the types of GBs and grain shapes, simulations of this model focus entirely on the effect of the grain size, i.e., on the role of the microstructural constraints, on the atomic structure and physical properties of a well-defined GB.

2

1 1

2 1

2 2

1 1

1 2

2 2

2 1 2

2

2 1

2

1

Figure 1. Idealized space-filling polycrystal model. The three-dimensionally periodic simulation cell shown here contains eight identically shaped rhombohedral grains delimited by two sets of crystallographically distinct surfaces (indicated by 1 or 2), forming a total of 24 crystallographically equivalent asymmetric tilt boundaries [12–14].

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Naturally, in the limit of infinite grain size, the model reproduces the corresponding bicrystal structure [12–14]. The simulation cell in Fig. 1 contains two sets of distinct rhombohedral grains, each set delimited by six crystallographically equivalent surfaces, {hkl}1 or {hkl}2, respectively, and therefore forming 24 crystallographically equivalent asymmetric tilt boundaries (for details see [12]). A series of four such model NCMs of increasing size (d = 8.2, 14.4, 20.6 and 26.9 Å, corresponding to 416, 2408, 7280 and 16 328 atoms in the simulation cell) were investigated. In these systems one rhombohedron was chosen to be bounded by {111} and the other by {115} planes; all the GBs are therefore asymmetric {115}{111} tilt boundaries. This particular GB, with an energy of 674 erg/cm2 in the bicrystal, is a reasonably representative of all types of asymmetric tilt boundaries [11].

1.1.

Structure

Lattice-statics simulations, in which the force on each atom and the external stresses on the simulation cell are iteratively reduced to zero by energy minimization, were used to determine the zero-temperature equilibrium structure of the NCMs. A Lennard–Jones potential was used to describe the interatomic interactions. The energy and length scales in this potential (ε = 0.167 eV, σ = 2.315 Å) were fitted to the melting point, Tm = 1356 K, and zero-temperature lattice parameter, a0 = 3.616 Å of Cu, with a cohesive energy, E id = −1.0378 eV/atom. Extensive comparisons with physically better justified many-body potentials have demonstrated that this simple potential represents face-centeredcubic (fcc) metals remarkably well [15]. This similarity is due to the fact that most interfacial phenomena are dominated by the short-range repulsions between the atoms (which are of a central-force type in both types of potentials). The energy distribution function in Fig. 2(a) for the largest of the three fully relaxed {115}{111} model NCMs shows three peaks, indicating three distinct types of crystal environments experienced by the atoms. (For comparison, Fig. 2(b) shows the distribution function for a Lennard–Jones glass produced by molecular-dynamics simulation of a 500-atom quenched liquid; see Section 1.2 below.) The lowest-energy peak in Fig. 2(a), centered at E id , corresponds to the grain interiors that, although elastically distorted, are essentially single crystalline. Detailed structural analysis showed the second peak to be due to the GBs while the third peak arises from the grain junctions (i.e., the lines where four grain edges and the points where eight grain corners meet). As the grain size is reduced, the area under the perfect-crystal peak decreases until it disappears completely for the smallest grain size, indicating overlapping GBs; simultaneously the areas under the two remaining peaks increase. Interestingly, the NCMs with the three largest grain diameters exhibit well-defined GBs with an atomic structure, energy, volume expansion and

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(a) 25 Polycrsytal (D ⫽ 20.6Å)

I

g (E ) [%]

20 15 10

II III

5 0 ⫺1.1

⫺1.0

⫺0.9

⫺0.8

⫺0.7

E [eV/atom] (b) 16 Glass

14

g (E ) [%]

12 10 8 6 4 2 0 ⫺1.1

⫺1.0

⫺0.9

⫺0.8

⫺0.7

E [eV/atom] Figure 2. Energy distribution functions, g(E), for (a) the fully relaxed {115}{111} model nanocrystalline material (see Fig. 1) with a grain size of 20.6 Å and (b) a Lennard–Jones glass produced by molecular-dynamics simulation of a 500-atom quenched liquid. The corresponding radial distribution functions are shown in Fig. 3 below. The arrows indicate the ideal-crystal cohesive energy, E id = −1.0378 eV/atom, for the Lennard–Jones potential used here [13].

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width (of about 1.5a0 ) which differ remarkably little from those of the corresponding bicrystalline asymmetric {115}{111} tilt boundary [14]. The radial distribution function for the third-largest grain size is shown in Fig. 3, indicating excellent crystallinity of the material; similar plots were obtained even for the two smallest grain sizes. Corresponding to the reduced density (of 94.2% of that of the perfect crystal), the mean peak positions are shifted slightly to larger values with respect to the δ-function peaks in the perfect fcc crystal at T = 0 K (situated at 0.707, 1.0, 1.225a0 , etc.). The radial distribution function for the NCM shown in Fig. 3 differs remarkably little from that obtained for the bicrystal (for the atoms within a distance of ±a0 from the GB; see also [14]). More importantly, the broadened peaks and the shift of the peak centers towards larger distances are generic features observed for virtually all GBs [15–17], suggesting that, as far as structural disorder is concerned, the particular GB considered in the model system in Fig. 1, is indeed reasonably representative.

0.6

Polycrystal (D ⫽ 20.6 Å) Glass

0.5

RDF

0.4

0.3

0.2

0.1

0 0.4

0.6

0.8

1.0 r [a ]

1.2

1.4

1.6

Figure 3. Radial distribution functions for the fully relaxed {115}{111} model nanocrystalline material (see Fig. 1) with a grain size of 20.6 Å and for a Lennard–Jones glass produced by molecular-dynamics simulation of a 500-atom quenched liquid [13].

Grain boundaries in nanocrystalline materials

1.2.

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Vibrational Behavior and Relationship to the Glass

Three types of experimental evidence suggest an intricate, possibly thermodynamic connection between the nanocrystalline and amorphous microstructures. However, partly due to different kinetic processes leading to one or the other microstructure and partly because the experiments were performed on different types of materials (such as pure metals, alloys and covalent semiconductors), a common thermodynamic framework connecting these observations has not been developed. (a) It is well established that mechanical alloying of two elements by ball milling (“mechanical attrition”) may lead to an amorphous phase [18], usually via a nanocrystalline precursor state with a lower limit in the grain size that depends on the concentration of the solid solution [19]. Also, the grain size in a series of nanocrystalline fcc metals produced by ball milling saturates to a minimum value which scales with the melting point of the elements [19]. Moreover, the excess energy due to the large number of GBs in the NCM has been shown to be of similar magnitude as the energy increase associated with solid-state amorphization [20]. (b) The crystallization of amorphous alloys provides a well-known method for the synthesis of fully dense, pore-free NCMs [21]. The NCMs thus obtained exhibit a smallest grain size below which crystallization does not occur [21, 22]. Conversely, the coexistence of nanocrystalline and amorphous phases observed, for example, in ball-milled silicon has been attributed to the existence of a “critical” grain size above which solid-state amorphization cannot take place [23]. (c) The synthesis of pure, submicron-grained polycrystalline metals by severe plastic deformation [24], although conceptually similar to ball milling, does not result in solid-state amorphization. While it is well known that pure metals cannot, in practice, be amorphized [25], the highly disordered non-equilibrium GBs thus obtained [24] – and their recovery to attain equilibrium structures upon annealing – suggest the existence of kinetically induced, locally disordered (and possibly amorphous) GB phases that disappear upon annealing. By contrast with the experiments, in computer simulations even singlecomponent materials can be amorphized, via very fast quenching of a liquid, i.e., orders of magnitude faster than in experiments. By capturing only the microstructural differences between the nanocrystalline and amorphous states, simulations of such simple systems provide an opportunity to compare these phases directly, unencumbered by the mostly kinetic effects of materials chemistry. The model of a fully dense, three-dimensional nanocrystalline material discussed above (see Fig. 1) is well-suited to elucidate how the low-temperature vibrational modes and the related thermal properties of this model NCM

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deviate from those of the perfect crystal and the glass. As seen in Fig. 3, the structures of the glass and the NCM obviously differ fundamentally as far as long-range order is concerned. However, it is well established that the broadening of the nearest-neighbor peak in the related radial distribution function represents a generic feature of all heavily disordered systems. Because many of their physical properties are governed by the short-range repulsions between the atoms, i.e., ion size, the presence or absence of long-range structural periodicity plays only a minor role in most of their properties. The similar nearest-neighbor peaks of the NCM and the glass (see Fig. 3) can therefore be expected to give rise to similarities, for example, in their phonon properties. Using the fully relaxed zero-temperature structures as starting points, Wolf et al. [13, 14] performed lattice-dynamics simulations to determine the phonon spectrum, from which the low-temperature thermodynamic and elastic properties of the material can be determined. Figure 4 compares the phonon density of states for the smallest {115}{111} model NCM with the phonon spectrum of a perfect fcc crystal and with that of the glass. Notice that, in agreement with

0.04

g (ψ)

0.03

0.02

0.01

0 0.0

2.0

4.0

6.0 ψ[THz]

8.0

10.0

12.0

Figure 4. Comparison of the vibrational density of states, g(ν), of the {115}{111} model nanocrystalline material with a grain size of 8.2 Å (solid line), the 500-atom glass (dash-dotted line) and a 500-atom perfect fcc crystal (dashed line); ν is the phonon frequency. The degree of broadening of g(ν) relative to the perfect crystal observed for the NCM decreases with increasing grain size as a more and more perfect crystal like microstructure is obtained [13].

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Raman [26] and neutron-scattering [27] experiments, the densities of state of both the NCM and the glass exhibit low- and high-frequency tails which are not present in the perfect crystal. These were shown [16] to originate from the broadening and shift in the peaks in the radial distribution function: distances shorter than in the perfect crystal give rise to local elastic stiffening – or higher phonon frequencies – while longer distances and the overall volume expansion cause elastic – or vibrational – softening. The net elastic and phonon responses of the system are therefore the result of a highly non-linear averaging process over these competing contributions [16, 17], seen here explicitly as tails in the phonon spectrum. Figure 5 shows the excess specific heat over that of the perfect crystal for the model NCMs and for the glass determined from the related frequency spectra. In the context of lattice-dynamics, it is well known [28] that only the lower-frequency modes in Fig. 4 should contribute significantly to the thermodynamic properties. This is confirmed by the complete disappearance of the specific-heat anomaly for the NCMs and the glass when the lower-frequency

0.3

Excess specific heat [k B ]

D ⫽ 8.2 Å D ⫽ 14.4 Å D ⫽ 20.6 Å 0.2

D ⫽ 43 Å (MD) Glass

0.1

0.0 0

100

200 T [K]

300

400

Figure 5. Comparison of the temperature and grain-size dependence of the excess specific heat (in units of Boltzmann’s constant, kB ) of the idealized {115}{111} model nanocrystalline material (see Fig. 1) with a grain size of 8.2 Å (open symbols) with that of the 500-atom glass (dash-dotted line) and of a 55 296-atom nanocrystalline material with randomly oriented grain boundaries and an average grain size of 43 Å (solid line) [13].

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modes were excluded; by contrast, omission of the high-frequency modes had no effect [14]. To test whether the {115}{111} asymmetrical tilt boundaries in our model NCMs are, indeed, reasonably representative of high-angle GBs in general, we have used molecular-dynamics (MD) simulations to crystallize from the melt a fully dense, three-dimensional NCM with random grain orientations and an average grain size of 43 Å [15]. Detailed structural characterization of the 55 296-atom system at zero temperature and stress gives a system-averaged theoretical density of 97.5% and a radial distribution function very similar to that of the model NCM in Fig. 2; most of the GBs were identified as “general” GBs, i.e., with random misorientations between the grains [15]. Subsequent lattice-dynamics simulations give the specific-heat anomaly shown also in Fig. 4 (solid line), with a maximum at virtually the same temperature as in the model NCMs, however, with a diminished peak height in accordance with the larger grain size [15]. These similarities in the specific-heat anomalies of NCMs and the glass, and their common origin in low-frequency phonon modes associated with the structural disorder in the material, strongly support the intuition that led to the design of the nanocrystalline model system in Fig. 1. Namely, at least as far as the low-temperature thermodynamic properties of NCMs are concerned, detailed microstructural averaging is not as important as the incorporation of the finite grain size and a realistic description of the structure and physics of the inhomogeneous regions. Lattice-dynamics calculations can also provide the free energy of the system. In Fig. 5, the temperature and grain-size dependence of the free energies of the {115}{111} model NCMs are compared with those of the glass and of the perfect crystal. Because of the use of the quantum-harmonic approximation, these results are strictly valid only at lower temperatures, typically up to about half the melting point [28]. According to Fig. 6, the glass and the NCMs exhibit a much higher free energy than the perfect crystal, as one would expect for these metastable phases. Interestingly, however, below a grain size of about 14 Å, the NCMs are unstable with respect to the glass. The grain size for which this free-energy based transition occurs is remarkably independent of the temperature; it can be expected to depend, however, on the particular GB incorporated into the mono-disperse model system in Fig. 1 and on the interatomic potential chosen for the simulations. Also, because the model NCM is so highly idealized, the actual grain size for which this transition occurs in a real material will be a function of the degree of microstructural averaging (i.e., on the distributions in the grain shapes, grain sizes and the types of GBs) and on the actual material considered, including the effects of interfacial chemistry. Irrespective of these factors, however, the existence of a reversible, free-energy-based transition between the NCM and the glass appears physically reasonable, given

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⫺0.85

F [eV/atom]

⫺1.00

⫺1.15 D = 8.2 Å D = 14.4 Å D = 20.6 Å Glass Perfect crystal

⫺1.30

⫺1.45 0

200

400

600

800

1000

1200

T [K] Figure 6. Comparison of the temperature and grain size dependence of the free energies of {115}{111} model nanocrystalline materials (open symbols) with those of the 500-atom glass (dashed line) and the perfect, 500-atom crystal (full symbols) [13, 14].

the common origin of the observed effects in the atomic-level structural disorder and in the related phonon spectra. The existence of such a transition has, indeed, been reported for nanocrystalline silicon if the grain size is reduced below about 20 Å [29]. Analogous to the crystal-to-liquid (“melting”) and crystal-to-glass (“solidstate amorphization”) transitions [30] (see also Chapter 6.11.), the existence of a free-energy based transition from the NCM to the glass below a critical minimum grain size can be expected to involve the nucleation of disorder at lattice defects and the subsequent growth of the amorphous phase into the grain interiors. To investigate the regions where the nucleation can actually occur, in Fig. 2(a) and (b) the zero-temperature energy distribution functions for the {115}{111} model NCM and for the glass are compared. As discussed in Section 1.1, the NCM exhibits three peaks, indicating three distinct types of crystal environments experienced by the atoms. The lowest-energy peak in Fig. 2(a) (near the perfect-crystal value indicated by the arrow) corresponds to the grain interiors which, although slightly distorted, are essentially single crystalline. This comparison suggests that nucleation of the amorphous phase is energetically possible at the grain junctions and the highest-energy GBs, which give rise to the two peaks in Fig. 2(a) with the highest energies.

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The above thermodynamics-based results have a number of important implications for the structure and properties of both nanocrystalline materials and glasses. (a) By analogy with the formation of a two-phase region upon melting, the smaller grains, with diameters below the critical size, can become amorphous while the larger grains remain crystalline, resulting in a two-phase microstructure seen, for example, in ball-milled Si [23]. Similarly, the observation that NCMs synthesized by crystallization of amorphous alloys exhibit a smallest grain size below which crystallization cannot occur [21, 22] follows naturally from Fig. 6. (b) The observation that solid-state amorphization via mechanical attrition proceeds via a nanocrystalline precursor state with a lower limit in the grain size [19] finds a natural explanation in terms of the above freeenergy plots, as does the observation of similar excess energies of the nanocrystalline precursor state and the amorphous phase [20]. (c) It appears that the synthesis of pure, submicron-grained polycrystalline metals by severe plastic deformation [24] does not result in solid-state amorphization because, according to Fig. 6, the grain size required for a nanocrystalline-to-amorphous transition to be possible is orders of magnitude too large. However, even for grains that are larger than the minimum critical size, one can expect that amorphous nuclei can kinetically be formed at the grain junctions and the highest-energy GBs (see Fig. 2a). Upon annealing, these thermodynamically unstable disordered regions disappear, as observed in the experiments [24]. Finally, one might speculate on how the transition from the glass to the NCM actually takes place at the atomic level; i.e., the mechanism by which the atoms reshuffle when the material transforms from a short-range ordered glass structure to a long-range ordered polycrystalline structure. Clearly, only minor changes are necessary in the nearest-neighbor environment of the atoms (see Fig. 3). A simple and rather natural way for establishing long-range order from the glass would be possible if the intermediate-range structure of the glass contained the “microstructural fingerprint” of the NCM, namely connected, less-well coordinated regions of lower density and higher energy density into which the better coordinated, more perfect crystal like “grains” are embedded. That intermediate-range density fluctuations lead to an inhomogeneous distribution of the free volume in liquids and glassy materials has, indeed, been reported from Fabry–Perot and Raman spectroscopy [31]. These fluctuations were interpreted in terms of a “microstructure” consisting of nanometer-sized clusters with different densities, dynamics, etc. In conclusion, in spite of its conceptual simplicity, the highly idealized model of a nanocrystalline material shown in Fig. 1 combines the severe microstructural constraints associated with a small grain size with a realistic

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treatment of the GBs, and thus provides insights not readily obtained from experiments. Most notably, simulations of this model demonstrate remarkably similar phonon spectra of the nanocrystalline and amorphous phases of the same material, giving rise also to similar thermodynamic properties at low temperatures, most notably an anomaly in their specific heats. The possibility of a reversible, free-energy based transition between the nanocrystalline material and the glass suggests that their atomic structures may share common elements; these may kinetically enable local amorphous-phase formation in NCMs and, conversely, give the glass an NCM-like intermediate-range structure.

2.

Molecular-Dynamics Synthesis of Nanocrystalline Model Microstructures

In addition to the distributions in the grain size and grain shapes, a coarsegrained polycrystal contains GBs with very differing structures and a wide spectrum of energies and properties. In close analogy to the classification of the structure of free surfaces, much work of recent years has suggested the usefulness of distinguishing among the following three different types of GBs: Special high-angle GBs (analogous to flat surfaces), dislocation boundaries (analogous to stepped surfaces), and general high-angle GBs (analogous to surfaces with overlapping steps). (For recent reviews, see Ref. [32] and Section 6.9.) The least understood of these are the general high-angle GBs, i.e., GBs, with a structure consisting of completely overlapping dislocation cores. Although in a coarse-grained polycrystal, this type usually represents only a relatively small fraction of the GBs, the pronounced properties of these GBs, particularly their high mobility and diffusivity coupled with a low sliding resistance and cohesion, can dominate the evolution of polycrystalline microstructures [33]. Because of the completely overlapping dislocation cores, the structural disorder in these GBs is distributed rather homogeneously, in a manner analogous to the surfaces of amorphous materials [34]. By contrast, similar to stepped surfaces, in dislocation boundaries the structural disorder is inhomogeneously distributed, consisting of well-defined, usually highly disordered dislocation cores separated by elastically distorted, perfect crystal like regions [35]. In an attempt to incorporate a distribution of GBs into nanocrystalline microstructures, Phillpot et al. [36, 37] used molecular-dynamics (MD) simulations to synthesize NCMs from a supercooled melt into which small crystalline seeds with more or less random orientations had been inserted. The subsequent crystal-growth simulation resulted in fully dense, impurity and porosity-free microstructures with fully equilibrated GBs that were subjected to full structural characterization.

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Figure 7 shows a cross-section through an fcc microstructure thus synthesized [36, 37]. The two crystallites in the lower half of the figure are of particular interest as their seeds were oriented so as to form a coherent twin boundary (i.e., the symmetric tilt boundary on the (111) plane of the fcc lattice). In its optimum translational state, this GB has mirror-plane symmetry and, hence, an extremely low energy (of 1 mJ/m2 for the Lennard–Jones potential used in these simulations; this potential is identical to that used in the studies described in Section 1) This energy is so low because only the third-nearest neighbors of the atoms at the GB are affected by the presence of the interface. However, in these simulations a small rigid-body translation away from the optimum, mirror-plane translational state was imposed on the original seeds; such translations might be present during the initial stages during the powder processing of nanocrystalline materials. The simulations revealed that this translation could not be optimized during the crystal-growth simulation, resulting in the highly 12

0

⫺12 ⫺12

0

12

Figure 7. Structural cross-sections of thickness 0.4a0 through the centers of four of the eight grains in the cubic, 3d periodic simulation cell. (a0 = 3.616 Å is the lattice parameter of the Lennard–Jones potential for Cu used in these simulations.) The different symbols denote different nearest neighbor miscoordinations, ranging between −3 (open squares), −2 (open triangles), −1 (open circles), 0 (small dots) and +1 (solid circles). (For more details, see Ref. [36].)

Grain boundaries in nanocrystalline materials

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disordered structure of this GB seen in Fig. 7, with the rather high energy of 701 mJ/m2 for the same Lennard–Jones potential. This high energy arises from the highly constrained nanocrystalline microstructure, in which the rigid-body translations of the grains parallel to the GB plane cannot be fully optimized, by contrast with an entirely unconstrained GB in a bicrystal. The GB structural disorder in this nanocrystalline microstructure was characterized in a rather simplistic way involving nearest-neighbor coordination, by simply determining the number of missing or extra nearest neighbors of each atom. In their first study of the deformation of nanocrystalline materials, Schiotz et al. [38] applied the much more powerful method of commonneighbor analysis (CNA) in which atoms are characterized as being either in a perfect crystal fcc environment, in an hcp environment (distinguished from fcc by the third neighbors) or miscoordinated (see Fig. 8). The extension of the work on fcc metals to silicon [39–41] further elucidated the connection between the GBs present in nanocrystalline and coarse-grained microstructures. The simulations of nanocrystalline Si [41], involving grain sizes of up to about 7 nm, revealed the presence of highly disordered GBs with a more or less uniform thickness. For comparison, extensive

Figure 8. Structural characterization of a deformed nanocrystalline microstructure using common neighbor analysis (CNA). The atoms with perfect fcc coordination are shown as white, those that are miscoordinated in the first neighbor shell are shown as dark. The atoms shown as gray are have perfect hcp coordination, which differs from fcc coordination only in the third neighbor shell [38].

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simulations of microstructurally unconstrained, bicrystalline Si GBs [39, 40] revealed a universal, highly disordered atomic structure of all the high-angle, high-energy GBs. Quantitative structural characterization in terms of the radial distribution function, g(r), revealed that this universal structure was virtually indistinguishable from that of both the GBs present in nanocrystalline Si with random grain orientations and bulk amorphous Si (compare with Fig. 1(c) for Pd) [39, 40]; by contrast, high-angle but low-energy bicrystalline GBs were found to exhibit good crystallinity (see Figs. 14 and 16 in Chapter 6.9). This work also revealed that the disordering of the high-energy GBs is driven by a lowering of the GB energy. It was therefore concluded that the existence of a highly disordered (“confined amorphous”) GB phase in NCMs as well as high-energy bicrystals represents a thermodynamic rather than kinetic effect. This work also demonstrated that nanocrystalline-Si microstructures with randomly oriented grains contain mostly high-energy, large-unit-cell or incommensurate GBs with atomic structures that are qualitatively identical to those of high-energy bicrystalline GBs [36–42] Similar simulations for nanocrystalline Pd [43] using the Pd (EAM) potential of Foiles and Adams [44] yielded qualitatively identical results although, by contrast with Si, fcc metals do not have a stable bulk amorphous phase in terms of which the degree of GB structural disorder can be quantified. As in the Si simulations [41], the three-dimensionally (3d) periodic, cubic simulation cell used in the Pd simulations contained four randomly oriented seed grains in an fcc arrangement; these seeds were embedded in the melt filling the rest of the cell. The fully dense microstructure thus obtained after the crystal-growth simulation consists of dodecahedral grains delimited by GBs, triple lines and higher-fold point junctions [43]. In order to generate a driving force for crystal growth, the melt containing the seeds was cooled down to 800 K (i.e., well below the melting point of Tm ∼ 1500 K [44, 45]). The entire system (melt plus seeds) was subsequently allowed to evolve freely at constant temperature and under zero pressure. To ensure that the microstructure thus obtained was indeed stable, prior to structural characterization the sample was subjected to thermal annealing at 600 K under zero external stress for 30 000 MD steps, followed again by cooling to 0 K. The final structure thus obtained, with a grain size of 8 nm, was found to be practically the same as the unannealed one, demonstrating that it is thermally stable against grain growth, at least on an MD time scale. To characterize the microstructures thus obtained, planar cuts of thickness a0 were made. Figure 9 shows gray-scale contours of equal energy per atom for a slice parallel to the microstructural (111) planes; due to the 3d periodicity imposed on the simulation cell, this cut slices through all four grain centers. Clearly all GBs (seen as dark regions) have roughly the same width, while the triple lines appear to be slightly wider. Although the grain interiors appear to be perfect-crystal like, close inspection reveals a number of (111) twins

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25 20 15 10

Y [a o]

5 0 ⫺5 ⫺10 ⫺15 ⫺20 ⫺25 ⫺25 ⫺20 ⫺15 ⫺10 ⫺5

3 nm 0 5 X [a o]

10

15

20

25

Figure 9. Energy per atom gray-scale contour plot for a (111) slice through an fcc Pd microstructure that contains the centers of all four grains of uniform grain shape and size. Dark regions indicate high excess energy (Periodic images of some of the grains are also shown [43]).

developed during the growth process. Their formation is not surprising due to their extremely low energy (of ∼ 3 mJ/m2 for this Pd potential); this low energy is also the reason that they are not visible in the energy contour plot [43]. The average structure of the material as characterized by the system averaged, overall radial distribution function, g(r), is shown in Fig. 10(a). The sharp crystalline peaks originate from the ordered grain interiors; the nonvanishing background between the peaks indicates the presence of structural disorder. To elucidate the origin of this non-crystalline signal, the local radial distribution functions associated with the GBs, triple lines and point junctions were determined by considering only those atoms in the system with the highest excess energies. According to Fig. 10(b), these local distribution functions reveal a complete absence of long-range order in the highly disordered GBs. In particular, the second crystalline peak has almost completely disappeared. Remarkably, this distribution function is virtually indistinguishable from the radial distribution function of the bulk Pd glass (dashed line; see also Fig. 12(a) in Chapter 6.9). A comparison of Fig. 10(b) with Fig. 12(a) in Chapter 6.9 reveals that the GBs in the nanocrystalline microstructure have a structure that is virtually

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D. Wolf (a) 25 nanocrystalline Pd d ⫽ 8 nm

20

g (r )

15 10 5 0 0

0.5

1

1.5

2

r [a 0] (b) 10 interfaces in nanocrystalline Pd 8

glass

g (r )

6 4 2 0 0

0.5

1 r [a 0]

1.5

2

Figure 10. (a) System-averaged radial distribution function for a Pd microstructure with a grain size of 8 nm (see also Fig. 9). (b) Local radial distribution function for the GB atoms; similar distributions were obtained for the atoms located in the line and point grain junctions. For comparison, g(r ) for the bulk Pd glass is also shown [43].

identical to that of the high-angle (110) twist GB, i.e., to the universal structure of the high-energy GBs in coarse-grained Pd. The fact that, with an energy of 1025 mJ/m2 the (110) GB is, indeed, representative of the high-energy GBs in the nanocrystalline material is supported by the histogram of GB energies in the nanocrystalline microstructure (see Fig. 11). The narrow spread of GB

Grain boundaries in nanocrystalline materials 8 nanocrystalline Pd

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high-energy GBs

4

3 twin GB

Frequency

6

2

(111) twist GBs

(100) twist GBs

0 0

500 1000 Energy [mJ/m2]

1500

Figure 11. Histogram of GB energies obtained for the 24 GBs in the microstructure in Fig. 9. The energies of several bicrystalline GBs are also indicated [43].

energies, with none of the 24 GBs in the system having an energy lower than 800 mJ/m2 or higher than 1300 mJ/m2 , originates from the random grain misorientations that give rise to GBs having both tilt and twist components, thus effectively inhibiting the formation of low-angle and “special” low-energy GBs (i.e., high-angle GBs on special, low-index lattice planes [32]). There is, however, no reason to assume that low-angle and special low-energy GBs do not exist in real materials, in which the misorientations between neighboring grains may not be random.

3.

Grain-Boundary Diffusion Creep in Nanocrystalline Pd

If the structure and properties of the high-energy GBs in a nanocrystalline microstructure are, indeed, identical to high-energy bicrystalline GBs, the simulation of GB diffusion creep in the idealized microstructures considered in Section 2 (see, e.g., Figs. 7 and 9) should provide an important test case. From experimental studies of coarse-grained polycrystals it is well known that Coble creep involves homogeneous grain elongation via GB diffusion, with a creep rate that is linear in the stress, proportional to d −3 , and governed by the activation energy of GB diffusion [49]. Due to this d −3 increase of the strain

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rate with decreasing grain size, d, according to the well-known Coble-creep formula [49], ε˙ = A

σ D δD DGB , kB T d 3

(1)

Coble-creep should be observable in nanocrystalline materials even during the short observation window of ∼10−9 s typically accessible by MD simulation. (In Eq. (1), δD DGB is the diffusion flux in the GBs with diffusion constant DGB , width δD and activation volume D ; kB T is the thermodynamic temperature, and A is a geometric constant depending on the grain shape.) To test the GB structural model described above, MD simulations of 3d periodic nanocrystalline Si [50] and Pd [51] microstructures were performed. So as to enable steady-state diffusion creep to be observed unencumbered by grain growth, model microstructures were tailored to have a uniform grain size and shape, with random grain orientations. As illustrated in Fig. 12 for the case of nanocrystalline Pd [51], these microstructures, indeed, exhibited steady-state diffusion creep that is homogenous, linear in the stress, and with a strain rate that agrees quantitatively with the Coble-creep formula in Eq. (1) [49]. The grain-size dependence of the creep rate, ε˙ , represents an important characteristic of the deformation mechanism. The observed linear relation 0.03 d ⫽7.6 nm; T ⫽1200 K

σ⫽

0.02

7

a; ε 4 Gp

a; ε

.3 Gp

ε

σ ⫽ 0.2

⫺1 ]

[s

2.

⫽1

0.

σ⫽0

0.01

0 6⫻1

⫺1 ] 07 [s

⫻1 ⫽ 8.7

7 ⫺1 ] ⫻10 [s

6.8 Gpa; ε ⫽

1 Gpa; ε ⫽ σ ⫽ 0.t[ps]

7 ⫺1 2.5⫻10 [s ]

0.00 0

50

100

t [ps] Figure 12. Total (elastic + plastic) strain vs. simulation time for a 7.6 nm grain size at 1200 K under different tensile stresses [51].

Grain boundaries in nanocrystalline materials

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between ε˙ and σ [51] suggests that, for a given grain size and temperature, the strain rates obtained for different stresses (Fig. 12) should – within the error bars – collapse into a single point, ε˙ /σ . The log–log plot of ε˙ /σ vs. the grain size, d, in Fig. 13 collects all the data points obtained at 1200 K. In this representation the strain rates, indeed, fall on a universal curve (dashed line), showing a d −3 dependence for the larger grain sizes. By contrast with Eq. (1), however, the d −2 dependence seen for the smallest grain sizes is a characteristic of Nabarro–Herring rather than Coble-creep. This apparent discrepancy is readily resolved by recognizing that in the small grain-size limit, Coble and Nabarro–Herring creep become essentially indistinguishable. In fact, Yamakov et al. [51] were able to show that the d −2 dependence follows naturally from Coble’s derivation as the limit in which δD /d ∼ 1. The activation energy represents an important fingerprint of the underlying deformation mechanism. The Arrhenius plot in Fig. 14 for the strain rates extracted from these simulations (see, e.g., Fig. 12) yields an activation energy of 0.61 ± 0.1 eV [51]. This value is in remarkable agreement with the universal high-temperature activation energy of 0.60 ± 0.1 eV for bicrystalline 101

ε/σ [S⫺1 Pa⫺1]

T ⫽1200 K

100 2 1 10⫺1

d ⫽3.8 nm d ⫽5.7 nm d ⫽7.6 nm d ⫽11.4 nm d ⫽15.2 nm

3 1

10⫺2 100

101

d [nm] Figure 13. Scaling plot of ε˙ /σ vs. d showing that all the data points obtained for different stresses at 1200 K collapse onto a single curve, thus indicating that the strain rate increases linearly with stress (see Eq. (1)). The dashed curve is merely a guide to the eye. The increase in error bars with decreasing grain size is due to the greater equilibrium fluctuations in the smaller systems [51].

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D. Wolf 109

d ⫽7.6 nm; σ ⫽ 0.4 GPa

1300 K ε [s⫺1]

1200 K 108

1100 K 0.61 eV 1000 K 900 K

107 8

10

12

14

1/ kB T [1/eV] Figure 14. Arrhenius plot for the strain rates at a stress of 0.4 GPa. The melting point for this potential is about 1500 K [51].

GBs determined in separate simulations of Pd self-diffusion in high-angle, high-energy GBs [45], i.e., for GB diffusion in the absence of applied stress and any microstructural constraints associated with the small grain size. This comparison reveals that GB diffusion is, indeed, the deformation-rate limiting process. Moreover, the fact that the activation energy for the creep in the nanocrystalline microstructure is the same as that for diffusion in high-energy, bicrystalline GBs confirms that the high-energy GBs predominantly present in nanocrystalline materials have a structure and properties that are virtually indistinguishable from those of the microstructurally unconstrained, extended GBs present in coarse-grained materials.

4.

Grain-Boundary Structural Model for Nanocrystalline Materials

The significant body of atomic-level simulations described above suggests important similarities and some differences between the GBs in coarse-grained polycrystalline and nanocrystalline microstructures. The most important differences seem to arise from the severe microstructural constraints present in

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nanocrystalline materials. As indicated by the highly disordered structure of the coherent twin boundary in Fig. 7, these constraints can have the effect of converting what would otherwise be a “special” (i.e., dislocation-free, lowenergy) GB into a highly disordered, high-energy interface. However, as evidenced by the observation in nanocrystalline Pd of perfectly ordered coherent twins transecting the grain interiors [43], the structures of “special” boundaries in a nanocrystalline microstructure may not be unique, but rather may depend on the nature of the microstructural confinement locally at the GB, and hence on the synthesis conditions and the degree of “thermal relaxation” of the material following its synthesis. Therefore, while nanocrystalline and coarse-grained polycrystalline microstructures appear to contain the same three types of GBs (see Section 2 above), the most important measure of the differences between them probably lies in their GB energy distribution functions. Whereas coarse-grained materials usually exhibit a broad distribution of GB energies, the severe microstructural constraints present in nanocrystalline microstructures seem to have the effect of significantly increasing the fraction of high-energy GBs at the expense of the special boundaries: The more severe the microstructural constraints become (e.g., by decreasing the grain size or by the use of a highly non-equilibrium synthesis route), the larger appears to be the fraction of the high-energy boundaries in the system. Valiev and coworkers [52, 53] have clearly demonstrated that in nanocrystalline metallic materials produced by severe plastic deformation, one observes highly non-equilibrium, high-angle GBs with a structure that facilitates significant GB sliding in the context of nanocrystalline superplasticity. Since, due to their high mobility and diffusivity coupled with a low sliding resistance and high GB energy, these boundaries dominate the evolution of these microstructures in response to stress and temperature, their effect should be particularly pronounced in nanocrystalline materials. The above simulations also suggest an intriguing structural and thermodynamic relation between nanocrystalline materials and bulk amorphous solids. Although the high-angle, high-energy GBs represent only a (synthesis and processing dependent) fraction of the GBs in a nanocrystalline microstructure, their structure and behavior strongly resembles Rosenhain’s historic “amorphouscement” model [46–48]. In fact, in the (hypothetical) limit in which all GBs in a nanocrystalline material are of this type, the two-phase microstructure consists of crystalline grains embedded in a glassy, intergranular, glue-like phase. In practice, this intergranular amorphous phase is disrupted by more inhomogeneously disordered dislocation boundaries and, perhaps, a few special boundaries. Given that in bicrystals this amorphous intergranular phase seems to be of thermodynamic, rather than purely kinetic origin, it appears that a rather intimate relation exists between nanocrystalline microstructures and the bulk amorphous phase. These observations are also consistent with the

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simulations of the phonon density of states and of the related free energy described in Section 1 [12–14], which demonstrated that below a certain critical grain size (of ∼ 1.5–2 nm) nanocrystalline microstructures are thermodynamically unstable with respect to the amorphous phase.

Acknowledgment This work was supported by the US Department of Energy, BES-Materials Science under contract W-31-109-Eng-38.

References [1] H. Gleiter, In: N. Hansen et al., Proceedings Second Risø International Symposium on Metallurgy and Materials Science, Roskilde, Denmark, p. 15, 1981. [2] R. Birringer et al., Phys. Lett. A, 102, 365, 1984. [3] X. Zhu et al., Phys. Rev. B, 35, 9085, 1987. [4] H. Gleiter, Prog. Mater. Sci., 33, 223, 1989. [5] J. Rupp and R. Birringer, Phys. Rev. B, 36, 7888, 1987; A. Tsch¨ope and R. Birringer, Acta Metall. Mater., 41, 2791, 1993, Phil. Mag. B, 68, 2223, 1993. [6] H.G. Klein, Diplom Thesis, Universit¨at des Saarlandes, November (unpublished), 1992. [7] J. Jiang et al., Solid State Commun., 80, 525, 1991; U. Herr et al., Appl. Phys. Lett., 50, 472, 1987. [8] C.A. Melendres et al., J. Mater. Res., 4, 1246, 1989. [9] G.J. Thomas et al., Scripta Metall., 24, 201, 1990. [10] J. Eastman et al., Phil. Mag. B, 66, 667, 1992. [11] See, for example, D. Wolf, In: D. Wolf and S. Yip (eds.), Materials Interfaces: Atomic-Level Structure and Properties, Chapman and Hall, p. 16, 1992. [12] D. Wolf, J. Wang, S.R. Phillpot, and H. Gleiter, Phys. Rev. Lett., 74, 4686, 1995. [13] D. Wolf, J. Wang, S.R. Phillpot, and H. Gleiter, Phys. Lett. A, 205, 274, 1995. [14] D. Wolf, J. Wang, S.R. Phillpot, and H. Gleiter, Phil. Mag. A, 73, 517. 1996. [15] D. Wolf and K.L. Merkle, Chapter 3 In: D. Wolf and S. Yip (eds.), Materials Interfaces: Atomic-Level Structure and Properties, Chapman and Hall, pp. 87–150, 1992. [16] D. Wolf and J. Lutsko, Phys. Rev. Lett. , 60, 1170, 1988. [17] D. Wolf and J. Jaszczak, J. Comput. Aided Mater. Des., 1, 111, 1993. [18] See, for example, P.J. Desre, Nanostruct. Mater., 4, 957, 1994. [19] J. Eckert, J.C. Holzer, C.E. Krill, and W.L. Johnson, J. Mater. Res., 7, 1751, 1992. [20] A.R. Yavari, Mater. Sci. Eng. A, 179/180, 20, 1994. [21] Y. Yoshizawa, S. Oguma, and K. Yamauchi, J. Appl. Phys., 64, 6044, 1988. [22] K. Lu, Phys. Rev. B, 51, 18, 1995. [23] T.D. Shen, C.C. Koch, T.L. McCormick, R.J. Nemanich, J.Y. Huang, and J.G. Huang, J. Mater. Res., 10, 139, 1995. [24] R.Z. Valiev, E.V. Kozlov, Y.F. Ivanov, J. Lian, A.A. Nazarov, and B. Baudelet, Acta Metall. Mater., 42, 2467, 1994. [25] Y.-W. Kim, H.-M. Lin, and T.F. Kelley, Acta Metall., 37, 247, 1989. [26] E.D. Obraztsova et al., Nanostruct. Mats., 6, 827, 1995. [27] J. Trampeneau, K. Bauszus, W. Petry, and U. Herr, Nanostruct. Mater., 6, 551, 1995.

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[28] See, e.g., A.A. Maradudin et al., In: H. Ehrenreich et al. (eds.), Solid State Physics, Suppl. 3, 1971. [29] S. Veprek, Z. Iqbal, H.R. Oswald, and A.P. Webb, J. Phys. C, 14, 295, 1981. [30] D. Wolf, P.R. Okamoto, S. Yip, J.F. Lutsko, and M. Kluge, J. Mater. Res., 5, 286, 1990. [31] R.W. Fischer, Physica A, 201, 183, 1993. [32] D. Wolf, “Grain boundaries: structure,” In: Robert Cahn, principal editor, The Encyclopedia of Materials, Science and Technology, Pergamon Press, 2001. [33] F.J. Humphreys and M. Hatherly, Recrystallization and Annealing Phenomena, Pergamon, 1995. [34] J.C.M. Li, J. Appl. Phys., 32, 525, 1961. [35] W.T. Read and W. Shockley, Phys. Rev., 78, 275, 1950. [36] S.R. Phillpot, D. Wolf, and H. Gleiter, J. Appl. Phys., 78, 847, 1995. [37] S.R. Phillpot, D. Wolf, and H. Gleiter, Scripta Metall. Mater., 33, 1245, 1995. [38] J. Schiotz, F.D. DiTolla, and K.W. Jacobsen, Nature, 391, 561, 1998. [39] P. Keblinski, S.R. Phillpot, D. Wolf, and H. Gleiter, Phys. Rev. Lett., 77, 2965, 1996. [40] P. Keblinski, S.R. Phillpot, D. Wolf, and H. Gleiter, J. Am. Ceram. Soc., 80, 717, 1997. [41] P. Keblinski, S.R. Phillpot, D. Wolf, and H. Gleiter, Acta Mater., 45, 987, 1997. [42] P. Keblinski, S.R. Phillpot, D. Wolf, and H. Gleiter, Phil. Magn. Lett., 76, 143, 1997. [43] P. Keblinski, D. Wolf, S.R. Phillpot, and H. Gleiter, Scripta Mater., 41, 631, 1999. [44] S.M. Foiles and J.B. Adams, Phys. Rev. B, 40, 5909, 1986. [45] P. Keblinski, D. Wolf, S.R. Phillpot, and H. Gleiter, Phil. Mag. A, 79, 2735, 1999. [46] W. Rosenhain and J.C.W. Humfrey, J. Iron Steel Inst., 87, 219, 1913. [47] W. Rosenhain and D. Ewen, J. Inst. Metals, 10, 119, 1913. [48] For an excellent description of Rosenhain’s amorphous-cement model, see K.T. Aust and B. Chalmers, in Metal Interfaces, ASM, p. 153, 1952. [49] R.L. Coble, J. Appl. Phys., 34, 1679, 1963. [50] P. Keblinski, D. Wolf, and H. Gleiter, Interface Sci., 6, 205, 1998. [51] V. Yamakov, D. Wolf, S.R. Phillpot, and H. Gleiter, Acta Mater., 50, 61, 2002. [52] R.Z. Valiev, R.K. Islamgaliev, and I.V. Alexandrov, Prog. Mater. Sci., 45, 103, 2000. [53] R.Z. Valiev, R. Islamgaliev, and N. Yunusova, Mater. Sci. Forum, 357–359, 449, 2001.

7.1 INTRODUCTION: MICROSTRUCTURE David J. Srolovitz1 and Long-Qing Chen2 1 Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, USA 2 Department of Materials Science and Engineering, Penn State University, University Park, PA 16802, USA

There is a very common observation that two nominally identical samples of a material may exhibit remarkably different properties. Such differences can often be traced back to how a material was synthesized or subsequently processed. If two materials have exactly the same composition, how can the properties be different? The answer is generally associated with heterogeneities in the material. These heterogeneities may be associated with spatial distributions of phases of different compositions and/or crystal structures, grains of different orientations, domains of different structural variants, domains of different electrical or magnetic polarizations, as well as of structural defects such as dislocations. Materials scientists routinely manipulate these inhomogeneities to optimize the properties of materials. Contrary to common usage, “microstructure” does not refer to the scale of the structure (e.g., the micrometer scale). Rather, “microstructure” describes the arrangements of the defects or the compositional inhomogeneity within a material. In this sense, microstructure exists on scales ranging from 10 s of atoms up to that above which the material behaves as a homogeneous continuum (often millimeters). What types of defects constitute the microstructure? These include, for example, dislocations, grain boundaries, interphase boundaries, domain walls, precipitates/inclusions, cracks, and surfaces. The shape of an individual extended defect is itself a type of microstructure since shape is simply a statement of the spatial arrangement of segments of the continuous defect. However, the term “morphology” is often used to describe this type of microstructure. In order to characterize a particular type of microstructure we must first choose a description. Practical descriptions of microstructure resolve scales that are fine on the scale of the spacing between defects (e.g., grain size), but 2083 S. Yip (ed.), Handbook of Materials Modeling, 2083–2086. c 2005 Springer. Printed in the Netherlands.

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not those that characterize the defect itself (the width of the grain boundary). Such descriptions may simply identify the location of a defect, but could also carry additional information, such as the Burgers vector of a dislocation. Generally, the second issue is to characterize the thermodynamics of the system. In cases where the defect only disrupts the crystal lattice on a scale which is very small compared to the spacing between defects, a thermodynamic description may only account for the total length or area of the defects and, possibly, be a function of the defect character (e.g., the change in crystal orientation upon crossing a boundary and/or the boundary inclination). However, in many cases, defects and volume elements in a microstructure interact with one another over very large distances. A classical case is in a dislocation microstructure. Other examples include coherent microstructures, ferroelectric domain structures, and ferromagnetic domain structures in which long-range elastic, electrostatic, and magnetic interactions exist. In such cases, a thermodynamic description may require summation or integration over the entire volume of the material. The third issue that we must decide is how to describe the temporal evolution of the microstructure. In most types of microstructure evolution problems, the defect dynamics are over-damped. In such cases, it is reasonable to assume that the defect velocity scales with the derivative of local chemical potential with respect to defect position (i.e., a thermodynamic force). In cases where the variables describing the defect are not conserved, this relation may simply be proportionality between velocity and thermodynamic force. In the conserved case, the rate of change of the microstructure descriptor will be proportional to the Laplacian of the thermodynamic force. In both cases, the proportionality constant is some type of mobility. Mobilities may be isotropic or anisotropic. Similarly, the rate of change of one variable may depend on gradients of the free energy with respect to non-conjugate variables; i.e., an Onsager form. There are times in which the simulator is faced with a situation in which the thermodynamic properties of the system are unavailable, but where experimental measurements of the defect velocity as a function of external variables have been made. In such situations, some form of front tracking method may be applied. The fourth issue (and often the most difficult one) is where do you get all of the parameters you need to describe the thermodynamics and kinetics. This issue is especially difficult in situations in which the key parameters are functions of many variables – e.g., the properties of grain boundaries depend on (at least) five distinct parameters (three to describe the misorientation between grains and two to specify the boundary plane). In cases where the parameter space is highly dimensional, we can (sometimes) use symmetry to reduce the volume of the space. However, in most cases, we employ the spherical chicken approximation. That is, we arbitrarily reduce the number of parameters we use until the solution can no longer describe the physics we care about

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(i.e., until you can no longer distinguish a chicken from a sphere). Let’s say that we have reduced the volume of the parameter space as much as possible, we will still be faced with the problem of choosing the parameters. In a multi-scale modeling framework, this is where we reach down to smaller scales. For example, many thermodynamic properties can be determined directly from first-principles calculations and kinetic parameters may be determined from atomistic simulations (perhaps combined with rate theory ideas). In many cases, these parameters may be extracted from experimental measurements (e.g., diffusivities) or combinations of experimental measurements and empirical calculations (e.g., CALPHAD data bases). Finally, the microstructure simulator will be faced with determining the properties of the microstructure that was found through the simulations (or from experiment). It is difficult to make general statements as to how to do this, since the procedures depend greatly on the properties of interest. However, this step is key to many activities because the ultimate goal is often to determine how to process a material to achieve a desired set of properties. In other cases, it is to determine the optimal microstructure for a particular set of objectives; this is so-called “materials by design.” In practice, many microstructure simulation activities stop short of this objective. Before leaving this general discussion of microstructure simulations, it is important to note that disparate microstructural features often interact in significant and surprising ways. For example, the evolution of the grain microstructure in a polycrystalline material is sensitive to the existence of precipitates (and the structure of their interfaces), the existence of domain structures within each grain, the evolution of the dislocation microstructure, and the distribution of solute within the material. Therefore, many of the most important issues in microstructure evolution can only be simulated through combinations of simulation methods. As such, microstructure evolution is itself a multi-physics multi-scale modeling activity. This chapter is organized as follows. We begin with a series of contributions on (7.2) phase field modeling and its application to (7.3) solidification, (7.4) precipitation, (7.5) ferroic domain structures and (7.6) grain growth. The phase field grain growth contribution is followed by a description of recrystallization using cellular automata (7.7). Next, (7.8) is a discussion of the coarsening of multi-phase systems using a front-tracking approach. Diffusion-controlled phase transformation modeling by kinetic Monte Carlo and microscopic kinetic simulations are the topics of the following two contributions (7.9 and 7.10). A series of four papers then addresses dislocation dynamics and/or dislocation microstructure using (7.11) front tracking, (7.12) phase field, (7.13) levelset methods and (7.14) coarse graining ideas. Then, a series of three papers describe the evolution of thin films during deposition using (7.15) level-set, (7.16) stochastic equations, and (7.17) Monte Carlo approaches. This chapter then concludes with discussions of (7.18) microstructure optimization and

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(7.19) microstructure characterization. While microstructures are being simulated today using too wide of an array of approaches to be adequately surveyed in a single chapter of a handbook, we have strived to represent many of the major classes of microstructure simulation methods, many different approaches to the same microstructure evolution problem and many uses of microstructure simulations.

7.2 PHASE-FIELD MODELING Alain Karma Northeastern University, Boston, MA, USA

The phase-field method is a powerful simulation tool to describe xxx the complex evolution of interfaces in a wide range of contexts without explicitly tracking these interfaces. Its main application to date has been to problems in materials science where the evolution of interfaces and defects in the interior or on the surface of a material has a profound impact on its behavior [8]. A partial list of applications to date in this general area includes alloy solidification [5], where models combine elements of the first phase-field models of the solidification of pure materials [9, 32] and the Cahn–Hilliard equation (7), solid-state precipitation [66], stress-driven interfacial instabilities [29, 41, 58], microstructural evolution in polycrystalline materials [17, 31, 36, 60], crystal nucleation [16], surface growth [13, 25, 44], thin film patterning [34], ferroelectric materials [57], dislocation dynamics [22, 49, 52, 55], and fracture [3, 11, 27, 56]. Interface tracking is avoided by making interfaces spatially diffuse with the help of order parameters that vary smoothly in space. Evolution equations for these order parameters are derived variationally from a Lyapounov functional that represents the total free-energy of the system. This theoretical construct provides great flexibility to model simultaneously various physical processes on different length and time scales within a single self-consistent set of coupled partial differential equations.

1.

Phase and Grain Boundaries

The formation of alloys microstructures during solidification, and the subsequent evolution of this microstructure in the solid-state, are controlled by the motion of boundaries between thermodynamically distinct phases, or between regions of the same crystalline phase with different orientations. The key ingredient of the phase-field approach is to make these interfaces spatially diffuse over a region of finite thickness. As a first example of a phase boundary, 2087 S. Yip (ed.), Handbook of Materials Modeling, 2087–2103. c 2005 Springer. Printed in the Netherlands.

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consider a pure crystal at equilibrium with its melt. The total free-energy of this system can be written in the phenomenological form

F[φ] =

κ 2 dV |∇φ| + h f (φ) ≡ 2

dV Fv ,

(1)

where the integral is over the entire volume of the two-phase system, assumed to be at the melting point. The free-energy density Fv is the sum of a gradient square term and a double-well potential f (φ) = φ 2 (1 − φ)2 with two minima that correspond to solid and liquid. In this context, φ can be interpreted physically as a phenomenological measure of crystalline order, which varies continuously through the diffuse interface region from φ = 1 in the crystal with perfect long range order to φ = 0 in the disordered liquid. Equivalently, φ can be seen as the envelope of the density wave that has a constant amplitude in the solid and decays into the liquid [40, 43], as illustrated by the dashed line in Fig. 1(a). For a flat interface in equilibrium, the phase-field φ only depends on the coordinate x that is normal to this interface. This equilibrium profile, φ0 (x), and the interface thickness are obtained by minimization of the total freeenergy, which is carried out by the standard calculus of variation. One replaces φ(x) = φ0 (x) + δφ(x) into Eq. (1) and expands the integrand to linear order in δφ, with the substitution dV = A dx, where A is the total interface 2 = (dφ/dx)2 ≡ φx2 . After integrating once by parts, and using area, and |∇φ| the fact that φx vanishes away from the interface, Eq. (1) can be written in the form δF = A

dx δφ(x)

δ F , δφ φ(x)=φ0 (x)

(a)

(2)

(b)

Crystal

ψ ψ

ψ

Grain 2

Grain 1

Melt

x

ψ

x

Figure 1. Schematic plots of crystalline order parameter φ through spatially diffuse interfaces: (a) solid–liquid interface, and (b) grain boundary. The other order parameters ρ in (a) and θ in (b) are the density and crystal orientation, respectively.

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where we have defined the functional derivative d ∂ Fv ∂ Fv δF ≡− , + δφ dx ∂φx ∂φ

(3)

For φ = φ0 (x) to minimize F, δ F must vanish for an arbitrary smooth variation δφ(x). This yields the equilibrium condition δ F/δφ = 0, and hence the equation −κ

d2 φ0 + h f (φ0 ) = 0, 2 dx

(4)

where f denotes the derivative of the double-well potential, f (φ0 ) = 2φ0 − 2 3 6φ √0 + 4φ0 . This equation√has the kink-shape solution φ0 (x) = (1 − tan h [x/ ( 2W )])/2, where W = κ/ h is a measure of the interface thickness. This solution interpolates smoothly between the values of φ in solid and liquid as shown in Fig. 1(a). In order to relate the diffuse interface model to a real physical system, one needs to calculate the excess free-energy of the solid–liquid interface, γsl, per unit area, which can be measured experimentally or computed from atomistic simulations [21]. This excess is the total free-energy of the two-phase system minus the total free-energy of a single phase system (either solid or liquid) occupying the same volume, divided by A, or here γsl =

+∞

dx −∞

2 κ φ0x + h f (φ0 ) 2

(5)

2 /2 and h f (φ0 ) yield equal By multiplying Eq. (4) by φ0x , one finds that κ φ0x contributions to this integral, such that Eq. (5) reduces to

γsl = W h

+∞

2 dy φ0y

(6)

−∞

where y = x/W and the integral are dimensionless. Consequently, γsl scales as the product of the interface thickness and the height of the double-well free-energy density. The crystalline nature of the solid is manifest in the fact that γsl varies with the orientation of the interface with respect to a fix set of crystal axes. The degree of variation depends generally on the atomic scale structure of the solid–liquid interface. This anisotropy can be incorporated into the phasefield model by letting the coefficient of the gradient square term in Eq. (1) be a function, κ(∇φ/| ∇φ|), of the direction normal to the interface [38, 63, is the unit normal pointing from solid to liquid. 65], where nˆ = −∇φ/| ∇φ| This approach has been applied to model dendritic evolution in materials with atomically rough interfaces and a smooth variation of γsl with orientation

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[6, 24], and in materials with faceted solid–liquid interfaces that have cusps in the γ -plot [10]. Grain boundaries in two dimensions have been modeled phenomenologically by coupling two order parameters [31, 36, 60]. The first, θ, measures the local crystal orientation; θ is constant in the interior of each grain and the jump θ across a grain boundary is the misorientation. The second, φ, is analogous to the order parameter used to describe the crsytal-melt interface. It differentiates between perfect crystalline order in the interior of a grain (φ = 1) and disordered material at the grain boundary (φ < 1). The two-order-parameter free-energy functional is written as [36, 60] F[φ, θ] =

dV

κ 2 + 2 H (φ)|∇θ| 2 , |∇φ| + h f (φ) + sg(φ)|∇θ| 2

(7)

in the third term inside the integrand where the singular dependence on |∇θ| is necessary to insure that the grain boundary is spatially localized [31]. This free-energy is rotationally invariant. Hence, both grain boundary migration and grain rotation can be modeled in dynamical applications of this model. One-dimensional stationary profiles of the two order parameters in the diffuse interface region, obtained from the equilibrium conditions δ F/δφ = δ F/δθ = 0, are shown schematically in Fig. 1(b). For =/ 0 (dashed lines), the profiles of both order parameters are smooth, which is desired in numerical applications of this model. For = 0 (solid lines), θ is a step function and φ has a cusp shape. While not suited for numerics, this limit is useful to gain insight into the static properties of the model. For the single-well function f (φ) = (1 − φ)2 /2, which penalizes energetically the formation of liquid at φ = 0, and g(φ) = φ 2 , the stationary φ-profile is φ0 (x) = 1 − (1 − φm ) exp (−|x|/W ),

(8)

where W = (κ/ h)1/2 is the interface thickness. This expression is easily seen to be solution of δ F/δφ =κd2 φ0 /dx 2 +h(1 − φ0 )=0 for x =/ 0, and the balance of the θ-jump and the jump of the first derivative of φ0 at x = 0 fixes φm = 1/(1 + W s θ/κ). This expression implies that disorder at the center of the grain boundary is larger (φm is smaller) for a larger misorientation. Furthermore, the grain boundary energy is given by γgb = sθ/(1 + W sθ/κ)2 ,

(9)

which is the sum of the contributions in Eq. (7) due to spatial disorder (φ variation) and the θ-jump across the boundary. For a different analytical form of g(φ), the model can also reproduce the form of the Read–Shockley energy,

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γgb ≈ −sθ ln θ, of a low angle tilt grain boundary composed of a stack of edge dislocations [31]. More importantly, both solid–liquid and grain boundaries can be described simultaneously in the model by choosing again f (φ) to be a double-well potential with minima at φ = 0 and φ = 1. In this case, grain boundary wetting can occur if, above a critical misorientation, the excess freeenergy of the crystal–crystal interface exceeds twice the excess free-energy of the solid–liquid interface. While the exact determination of this misorientation requires to repeat the analysis of the stationary φ-profile for the double-well potential [60], its scaling can be obtained √ by comparing γsl (Eq. (6)) and γgb (Eq. (9)), which yields θ ∼ W h/s ∼ κh/s. Both the crystal-melt interface and grain boundaries have also been simulated using a more microscopic “phase-field crystal” model [13] where the order parameter is the density ρ of the material. A typical profile of the density averaged spatially in the plane perpendicular to the normal to the solid–liquid interface is plotted schematically in Fig. 1(a), where each peak corresponds to a single plane of atoms. This approach is rooted in classical density functional theory [43], but does not constrain the density to be a sum of density waves corresponding to reciprocal lattice vectors of a perfect crystal. Hence, it can naturally describe dislocations and vacancies in the solid. Furthermore, because phonons are averaged out in this mean field approach, the evolution of crystalline defects can be followed on time scales several orders of magnitude larger than in molecular dynamics simulations.

2.

Cracks

The nucleation and propagation of crack surfaces has been a topic of long standing interest in the materials science and engineering community. To illustrate the extension of the phase-field method to fracture, let us consider the one-dimensional problem of a stationary infinitely long crack in an elastic solid, as illustrated in Fig. 2. The original solid is stretched along the x-axis by displacing its boundaries at x = ±W by ±. The crack splits this solid into two equal parts. In the traditional approach where the crack surfaces are treated as sharp boundaries, the standard displacement field u(x) of mass points measured from their original positions is simply u(x) = x /W for the stretched solid. After complete fracture, the displacement is a step function, u(x) = + for x > 0 and u(x) = − for x < 0, and is discontinuous at the origin, u(0± ) = ±. A phase-field model for cracks can be formulated by introducing a scalar field φ(x) which describes the state of the material [27]. The model retains the same parametrization of linear elasticity where u(x) measures the displacement of mass points from their original positions. Hence, φ measures the

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⫹1

Solid

Crack

Solid ψ

x

0

or x ⫹ u (x ) u /∆

⫺1 Figure 2. Schematic phase-field profiles vs. the material coordinate x (thick solid line) and vs. the spatial coordinate x + u(x) (dashed line), where u(x) (thin solid line) is the displacement of mass points with respect to their original positions in the unstretched material. The thick vertical solid lines denote the spatial locations of the two crack surfaces.

state of the material at a spatial location x + u(x). The unbroken solid, which behaves purely elastically, corresponds to φ = 1, whereas the fully broken material that cannot support stress corresponds to φ = 0. The total energy per unit area of crack surfaces is taken to be

E=

κ dx 2

dφ dx

2

µ + h f (φ) + g(φ) 2 − c2 , 2

(10)

where = du/dx is the strain, f (φ) = φ 2 (1 − φ)2 is the same double-well potential as before with minima at φ =1 and φ =0, µ is the elastic modulus, and c is the critical strain magnitude such that the unbroken (broken) state is energetically favored for | | < c (| | > c ). The function g(φ) is a monotonously increasing function of φ with limits g(0) = 0 and g(1) = 1, which controls the softening of the elastic energy at large strain. In equilibrium, the energy must be a minimum, which implies that δ E/δφ = 0 and δ E/δu = 0. The second condition is equivalent to uniform stress in the material. It implies that d(g(φ) )/dx = 0, and hence that = 0 /g(φ) where 0 is the value of the remanent strain in the bulk of the material far from the crack. The first condition δ E/δφ = 0, after the substitution = 0 /g(φ), can

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be written in the form of a one-dimensional mechanical problem of a rolling ball with coordinate φ and mass κ dVeff (φ) d2 φ , =− 2 dx dφ in an effective potential κ

(11)

µ 2 2 c g(φ) + 0 Veff (φ) = −h f (φ) + 2 g(φ)

(12)

This potential (Fig. 3) has a repulsive part because g(φ) vanishes for small φ. In this mechanical analog, the stationary phase-field profile φ(x) shown in Fig. 2 corresponds to the ball rolling down this potential, starting from φ = 1 at x = − W , to the turning point located close to φ = 0, and then back to φ = 1 at x = +W . This mechanical problem must be solved under the constraint that the

+Wintegral of the strain equals the total displacement of the fracture surfaces, −W dx 0 /g(φ)=2. An analysis of the solutions in the large system size limit [27] shows that the remanent strain is determined by the behavior of the function g(φ) for small φ. If this function has the form of a power law g ∼ φ 2+α

0.25 ε0 ⫽ 0.01 ε0 ⫽ 0.001 ε0 ⫽ 0.0001

Veff

0.15

0.05

⫺0.05

0

0.2

0.4

0.6

0.8

1

ψ Figure 3. Plots of the effective potential for different values of the remanent strain in the bulk material 0 for one-dimensional static fracture (µ = h = 1 and c = 1/2).

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near φ = 0, the result is 0 ∼ −(2+α)/α . Hence, as long as α is positive, 0 will vanish in the large system size limit such that the local contribution of the crack to the overall displacement is dominant compared to the bulk contribution, which scales √ as 0 W . In this limit, the width of φ-profile remains finite and scales ∼ κ/µ. The u-profile is also continuous in the diffuse interface region, but its width vanishes has an inverse power of the system size, such that the strain = du/dx becomes a Dirac delta function in the large system size limit. In addition, this analysis yields the expression for the surface energy [27] γ=

µ c2 κ

1

dφ 0

1 − g(φ) + 2

h f (φ) µ c2

(13)

In contrast to the interface energy for a phase boundary (Eq. (6)), γ for a crack remains finite when the height h of the double well potential vanishes. Therefore, the inclusion of this potential is not a prerequisite to model cracks within this model.

3.

Interface Dynamics

The preceding sections focused on flat static interfaces and their energies. This section examines the application of the phase-field method to simulate the motion of curved interfaces outside of equilibrium, when spatially inhomogeneous distributions of temperature, alloy concentration, or stress are present in the material. The effect of these inhomogeneities are straightforward to incorporate into the model by adding bulk internal energy and entropic contributions to the free-energy functional. Furthermore, the Ginzburg–Landau form [15, 18] of the equations is prescribed by conservation laws and by requiring that the total free-energy relaxes to a minimum. Three illustrative examples are considered: the solidification of a pure substance, the solidification of a binary alloy, and crack propagation. For the solidification of a pure melt [32], the total free-energy that includes the contribution due to the variation of the temperature field is a generalization of Eq. (1)

F[φ, T ] =

κ 2 α dV |∇φ| + h f (φ) + (T − Tm )2 , 2 2

(14)

which is minimum at the melting point T = Tm . Dynamical equations which guarantee that F decreases monotonically in time (dF/dt ≤ 0), and which

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conserve the total energy dV e in a closed system with no energy flux through the boundaries, are [32] δF ∂φ = −K φ , ∂t δφ δF ∂e = Ke ∇ 2 , ∂t δe

(15) (16)

where the energy density e = C(T − Tm ) − p(φ)L and φ are chosen as the independent field variables in Eq. (14), C is the specific heat per unit volume, L is the latent heat of melting per unit volume, and p(φ) is a function that increases monotonously with φ with limits p(0) = 0 and p(1) = 1. The energy equation (Eq. (16)) yields L ∂ p(φ) ∂T = D∇ 2 T + ∂t C ∂t

(17)

where we have defined the thermal diffusivity D = α K e /C 2 . This is the standard heat diffusion equation with an extra source term ∼ ∂ p/∂t corresponding to latent heat production. The equation of motion for the phase-field (Eq. (15)), in turn, gives K φ−1

∂φ = κ∇ 2 φ − h f (φ) − α(L/C) p (φ)(T − Tm ), ∂t

(18)

where the prime denotes differentiation with respect to φ. In the region near the interface, where T is locally constant, Eq. (18) implies that the phase change is driven isothermally by the modified double-well potential h f (φ) + α(L/c) p(φ)(T − Tm ). This potential has a “bias” introduced by the undercooling of the interface, which lowers the free-energy of the solid well relative to that of the liquid well. A one-dimensional analysis of this equation [9, 32] shows that the velocity of the interface is simply proportional to the undercooling, V = µsl (Tm − T ), where the interface kinetic coefficient µsl ∼ α K φ (κ/ h)1/2(L/c). Further refinement of this phase-field model [24] and algorithmic developments have made it possible to simulate dendritic evolution quantitatively both in a low undercooling regime where the scale of the diffusion field is much larger than the scale of the dendrite tip [45, 47, 48], and in the opposite limit of rapid solidification [6]. Parameter free results obtained for the latter case using anisotropic forms of γsl and µsl computed from atomistic simulations [20, 21] are compared with experimental data in Fig. 4. Next, let us consider the isothermal solidification of a binary alloy [5, 26, 30, 59, 61]. The total free-energy of the system can be written in the form

F[φ, c, T ] =

dV

κ 2 |∇φ| + f pure (φ, T ) + f solute(φ, c, T ) , 2

(19)

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A. Karma 100

80

V (m/s)

60

40

20

0 0

100

200 ∆T (K)

300

400

Figure 4. Example of application of the phase-field method to simulate the dendritic crystallization of deeply undercooled nickel [6]. A snapshot of the solid–liquid interface is shown for an undercooling of 87 K. The dendrite growth rate versus undercooling obtained from the simulations (filled triangles and solid line) is compared to two sets of experimental data from Ref. [37] (open squares) and Ref. [64] (open circles).

where c denotes the solute concentration defined as the mole fraction of B in a binary mixture of A and B molecules, f pure = h f (φ) + α(L/c) p(φ)(T − Tm ) is the double-well potential of the pure material, and f solute(φ, c, T ) is the contribution due to solute addition. Dynamical equations that relax the system to a free-energy minimum are δF ∂φ = −K φ , ∂t δφ ∂c δF , = ∇ · Kc∇ ∂t δc

(20) (21)

where Eq. (21) is equivalent to the mass continuity relation with µc ≡ δ F/δc c as the solute current density. identified as the chemical potential and −K c ∇µ The smooth spatial variation of φ in the diffuse interface can be exploited to interpolate between known forms of the free-energy densities in solid and liquid ( f s and f l , respectively), by writing f solute(φ, c, T ) = g(φ) f s (c, T ) + (1 − g(φ)) f l (c, T ),

(22)

where g(φ) has limits g(0) = 0 and g(1) = 1. For example, for a dilute alloy, f s,l = s,l c + (RTm /v 0 )(c ln c − c) where s,l c is the change of internal energy density due to solute addition in solid or liquid, and the second term is the standard entropy of mixing, where R is the gas constant and v 0 is the molar volume. This interpolation describes the thermodynamic properties of the diffuse interface region as an admixture of the thermodynamic properties of the bulk

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solid and liquid phases. The static phase-field and solute profiles through the interface are then obtained from the equilibrium conditions ∂φ/∂t = ∂c/∂t = 0. The limits of c in bulk solid (cs ) and liquid (cl ) are the same as the equilibrium values obtained by the standard common tangent construction of the alloy phase diagram. The method has been extended to non-isothermal conditions, multicomponent alloys, and polyphase transformations, as illustrated in Fig. 5 for the solidification of a ternary eutectic alloy. The first models of polyphase solidification used either the concentration field [23] or a second non-conserved order parameter [35, 62] to distinguish between the two solid phases in addition to the usual phase field that distinguishes between solid and liquid. The more recent multi-phase-field approach interprets the phase fields as local phase fractions and therefore assigns one field to each phase present [14, 42, 53, 54]. This approach provides a more general formulation of multi-phase solidification. The simplest nontrivial example of dynamic brittle fracture is antiplane shear (mode III) crack propagation where the displacement field u(x, y) perpendicular to the x–y plane is purely scalar. The total energy (defined here

Figure 5.

Phase-field simulation of two-phase cell formation in a ternary eutectic alloy [46].

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per unit length of the crack front) must now include both kinetic and elastic contributions to this energy, yielding the form

E=

dx dy

µ ρ 2 κ 2 u˙ + |∇φ| + h f (φ) + g(φ) | |2 − c2 , 2 2 2

(23)

is where dot denotes derivative with respect to time, ρ is the density, ≡ ∇u the strain and all the other functions and parameters are as previously defined. The dynamical equations of motion are then obtained variationally from this energy in the forms δE ∂φ = −χ ∂t δφ 2 δE ∂ u ρ 2 =− ∂t δu

(24) (25)

These equations describe both the microscopic physics of material failure and macroscopic linear elasticity. Figure 6 shows examples of cracks obtained in phase-field simulations of this model in a strip of width 2W with a fixed displacement u(x, ±W ) = ± at the strip edges. The stored elastic energy per unit area ahead of the crack tip is G = µ2 /W . The Griffith’s threshold for the onset fracture is well reproduced in this model [27]. This approach has been recently used to study instabilities of mode III [28] and mode I cracks [19]. (a)

(b)

(c)

Figure 6. Example of dynamic crack patterns for mode III brittle fracture [28] with increasing load from (a) to (c). Plots correspond to φ = 1/2 contours at different times.

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Discussion

The preceding examples illustrate the power of the phase-field method to simulate a host of complex interfacial pattern formation phenomena in materials. Making quantitative predictions on experimentally relevant length and time scales, however, remains a major challenge. This challenge stems from the fact that, in most applications, the interface thickness and the time scale of the phase field kinetics need to be chosen orders of magnitude larger than in a real material for the simulations to be feasible. Because of this constraint, phase-field results often depend on interface thickness and are only qualitative. Over the last decade, progress has been made in achieving quantitative simulations despite this constraint [12, 24, 26, 51, 66]. One important property of the phase-field model is that the interfacial energy (Eq. (6)) scales as W h. Hence, the correct magnitude of capillary effects can be modeled even with a thick interface by lowering the height h of the double-well potential. For alloys, the coupling of the phase field and solute profiles through the diffuse interface makes the interface energy dependent on interface thickness. This dependence, however, can be eliminated by a suitable choice of freeenergy density [12, 26]. More difficult to eliminate are nonequilibrium effects that become artificially magnified because of diffusive transport across a thick interface. These effects can compete with, or even supersede, capillary effects, and dramatically alter microstructural evolution. To illustrate these nonequilibrium effects, consider the solidification of a binary alloy. The effect best characterized experimentally and theoretically is solute trapping [1, 4], which is associated with a chemical potential jump across the interface. The magnitude of this effect scales with the interface thickness. Since W is orders of magnitude larger in simulations than in reality, solute trapping will prevail at growth speeds where it is completely negligible in a material. Additional effects modify the mass conservation condition at the interface cl (1 − k)Vn = −D

∂c + ··· ∂n

(26)

where cl is the concentration on the liquid side of the interface, k is the partition coefficient, Vn is the normal interface velocity, and “· · · ” is the sum of a correction ∼ cl (1 − k)W Vn κ, where κ is the local interface curvature, a correction ∼ W D∂ 2 cl /∂s 2 , corresponding to surface diffusion, and a correction ∼ kcl (1 − k)W Vn2 /D proportional to the chemical potential jump at the interface. All three corrections can be shown to originate physically from the surface excess of various quantities [12], such as the excess of solute illustrated in Fig. 7. These corrections are negligible in a real material. For this reason, they have not been traditionally considered in the standard free-boundary problem of alloy solidification. For a mesoscopic interface thickness, however, the

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c1

Solid

Liquid

cs 0 r Figure 7. Illustration of surface excess associated with a diffuse interface. The excess of solute is the integral, along the coordinate r normal to the interface, of the actual solute profile (thick solid line) minus its step profile idealization (thick dashed line) with the Gibbs dividing surface at r = 0. This excess is negative in the depicted example. The thin solid line depicts the phasefield profile. The use of a thick interface in simulations artificially magnifies the surface excess of several quantities and alters the results [12].

magnitude of these corrections becomes large. Thus, the phase-field model must be formulated to make these corrections vanish. Achieving this goal requires a detailed asymptotic analysis of the thin-interface limit of diffuse interface models [2, 12, 24, 26, 39]. This analysis provides the formal guide to formulate models free of these corrections. So far, however, progess has only been possible for dilute [12, 26] and eutectic alloys [14]. Thus, it is not yet clear whether or not it will always be possible to make phase-field models quantitative in more complicated applications.

5.

Outlook

The phase-field method has emerged as a powerful computational tool to model a wide range of interfacial pattern formation phenomena. The success of the approach can be judged by the rapidly growing list of fields in which it has been used from materials science to biology. It can also be judged by the wide range of scales that have been modeled from crystalline defects to nanostructures to microstructures. Like with any simulation method, however, obtaining quantitative results remains a major challenge. The core of this challenge is the disparity of length and time scales between phenomena on the

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scale of the diffuse interface and on the scale of energy or mass transport in the bulk material. For well-established problems like solidification, and a few others, quantitative simulations have been achieved in a few cases following two decades of research since the introduction of the first models. In more recent applications like fracture, with no clear separation between microscopic and macroscopic scales, results remain so far qualitative. In the future, one can expect phase-field simulations to be useful both to gain new qualitative insights into pattern formation mechanisms and to make quantitative predictions in mature applications.

Acknowledgments The author thanks the US Department of Energy and NASA for financial support.

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7.3 PHASE-FIELD MODELING OF SOLIDIFICATION Seong Gyoon Kim1 and Won Tae Kim2 1 Kunsan National University, Kunsan 573-701, Korea 2

Chongju University, Chongju 360-764, Korea

1.

Pattern Formation in Solidification and Classical Model

Pattern formation in solidification is one of the most well known freeboundary problems [1, 2]. During solidification, solute partitioning and release of the latent heat take place at the moving solid/liquid interface, resulting in a build-up of solute atoms and heat ahead of the interface. The diffusion field ahead of the interface tends to destabilize the plane-front interface. Conversely, the role of the solid/liquid interface energy, which tends to decrease by reducing the total interface area, is to stabilize the plane solid/liquid interface. Therefore the solidification pattern is determined by a balance between the destabilizing diffusion field effect and the stabilizing capillary effect. Anisotropy of interfacial energy or interface kinetics in a crystalline phase contributes to form an ordered pattern with a unique characteristic length scale rather than a fractal pattern. The key ingredients in pattern formation during solidification thus are contributions of diffusion field, interfacial energy and crystalline anisotropy [2]. The classical moving boundary problem for solidification of alloys assumes that the interface is mathematically sharp. The governing equations for isothermal alloy solidification [1] are given by ∂c L = DL ∇ 2cL ∂t ∂cS ∂c L V (ciL − ciS ) = D S − DL ∂n ∂n Hm 1 S i L i e βV + σ κ f c (cS ) = f c (c L ) = f c − e c L − ceS Tm ∂cS = DS ∇ 2 cS ; ∂t

2105 S. Yip (ed.), Handbook of Materials Modeling, 2105–2116. c 2005 Springer. Printed in the Netherlands.

(1) (2) (3)

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where c is composition and D is diffusivity. The subscripts S and L under c and D denote the values of solid and liquid phase, respectively. The superscripts i and e on composition denote the interfacial and equilibrium compositions, respectively. f cS and f cL are the chemical potentials of bulk solid and liquid, respectively, and f ce is the equilibrium chemical potential. Here the chemical potential denotes the difference between the chemical potentials of solute and solvent. β is the interface kinetics coefficient, V the interface velocity, Hm the latent heat of melting, Tm the melting point of the pure solvent, σ the interface energy, κ the interface curvature, ∂/∂t and ∂/∂n are the time and the interface normal derivatives, respectively. The solidification of pure substances involving the latent heat release at interface, instead of solute partitioning, can be described by the same set of Eqs. (1)–(3), which can be expressed by replacing variables: c → H/Hm , f c → T Hm /Tm , D → DT , where T , H and DT are temperature, enthalpy density and thermal diffusivity, respectively, with the same meanings for the superscripts and subscripts L, S and i.

2.

Phase-field Model

Many numerical approaches have been proposed to solve the Eqs. (1)– (3). These include direct front tracking methods and boundary integral formulations, where the interface is treated to be mathematically sharp. However these sharp interface approaches lead to significant difficulties due to the requirement of tracking interface position every time step, especially in handling topological changes in interface pattern or extending to 3D computation. An alternative technique for modeling the pattern formation in solidification is the phase-field model (PFM) [3, 4]. This approach adopts a diffuse interface model, where a solid phase changes gradually into a liquid phase across an interfacial region of a finite width. The phase state is defined by an order parameter φ as a function of position and time. The phase field φ takes on a constant value in each bulk phase, e.g., φ = 0 in liquid phase and φ = 1 in solid phase, and it changes smoothly from φ = 0 to φ = 1 across the interfacial region. Any point within the interfacial region is assumed to be a mixture of the solid and liquid phases, whose fractions are varying gradually from 0 to 1 across the transient region. All the thermodynamic and kinetic variables then are assumed to follow a mixture rule. A set of equations for PFM can be derived in a thermodynamically consistent way. Let us consider an isothermal solidification of an alloy. It is

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assumed that the total free energy of the system of volume is given by a functional

F=

[ 2 |∇φ|2 + ωg(φ) + f (φ, c, T )]d

(4)

During solidification, which is a non-equilibrium process, the system evolves toward a more stable state by reducing the total free energy. To decrease the total free energy, the first term (phase-field gradient energy) in the functional (4) makes the phase-field profile to spread out, i.e., to widen the transient region. The second term (double-well potential ωg(φ)) makes the bulk phases stable, i.e., to sharpen the transient region. The diffuse interface maintains a stable width by a balance between these two opposite effects. Once the stable diffuse interface is formed, the two terms start to cooperate to decrease the total volume (in 3D) or area (in 2D) of the diffuse interfacial region where|∇φ| and g(φ) are not vanishing. This is corresponding to the curvature effect in the classical sharp interface model. Thus the gradient energy and the doublewell potential play two-fold roles; formation of stable diffuse interface and incorporation of the curvature effect. √ As the result, the interface width scales as the ratio of the coefficients (/ √ ω), whereas the interface energy scales as the multiplication of them ( ω). The last term in the functional (4) is a thermodynamic potential assumed to follow a mixture rule f (φ, c, T ) = h(φ) f S (cS , T ) + [1 − h(φ)] f L (c L , T )

(5)

where c is the average composition of the mixture, cS and c L are the compositions of coexisting solid and liquid phases in the mixture, respectively, and f S and f L are the free energy densities of the solid and liquid phases, respectively. It is natural to take c(x) at a given point x to be c(x) = h(φ)cS (x) + [1 − h(φ)]c L (x)

(6)

The monotonic function h(φ) satisfying h(0) = 0 and h(1) = 1 has the meaning of solid fraction. One more restriction is required for h(φ) to ensure that the solid and liquid phases are stable or metastable(i.e., exhibit energy minima), the function ωg(φ) + f in the functional (4) must have local minima at φ = 0 and φ = 1. It then leads to the condition h (0) = h (1) = 0 which confines the phase change occurring within the interfacial region. Finally the anisotropy effect in interface energy can be incorporated into the functional (4) by allowing to depend on the local orientation of the phase-field gradient [5]. Note that all thermodynamic components controlling pattern formation during solidification are incorporated into a single functional (4). The kinetic components controlling pattern formation are incorporated into the dynamic equations of the phase and diffusion fields. In a solidifying

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system where its total free energy decreases monotonically with time, the total amount of solute is conserved, whereas the total volume of each phase is not conserved. Therefore the phase field and concentration are assumed to follow relaxation dynamics of δF ∂φ = −Mφ ∂t δφ

(7)

δF ∂c = −∇ Mc · ∇ ∂t δc

(8)

where Mφ and Mc are mobilities of the phase and concentration fields, respectively. From the variational derivatives of the functional (4), it follows 1 ∂φ = 2 ∇ 2 φ − ωg (φ) − f φ Mφ ∂t ∂c = ∇ Mc · ∇ f c ∂t

(9) (10)

where the subscripts in f denote the partial derivatives by the specific variables. Mc is related to the chemical diffusivity D(φ) by Mc = D/ f cc , where f cc ≡ ∂ 2 f /∂c2 , D(1) = D S and D(0) = D L . The PFM for isothermal solidification of alloys thus consists of Eqs. (9) and (10). To solve these equations, we need f φ , f c and f cc . For the given thermodynamic data f S (cS ) and f L (c L ) at a given temperature, the above functions are obtained by differentiating Eq. (5). For this differentiation, relationships between c(x), cS (x) and c L (x) are required. Two alternative ways have been proposed for these relationships [6]: equal composition condition; and equal chemical potential condition. In the former case, which has been widely adopted [3], it is assumed that cS (x) = c L (x) and so f c S (cS (x)) =/ f c L (c L (x)), resulting in c = cS = c L from Eq. (6). Under this condition, it is straightforward to find fφ , f c and fcc from Eq. (5). In the latter case, it is assumed that fc S (cS (x)) = f c L (c L (x)) and so cS (x) =/ c L (x), resulting in f c = f c S = f c L from Eqs. (5) and (6). Under this condition, f φ in the phase-field Eq. (9) is given by f φ = h (φ)[ f S (cS , T ) − f L (c L , T ) − (cS − c L ) f c ]

(11)

and the diffusion Eq. (10) can be modified into the form ∂c = ∇ · D(φ)[h(φ)∇cS + (1 − h(φ))∇c L ] (12) ∂t Note that this diffusion equation is derived in a thermodynamically consistent way, even though the same equation has been introduced in an ad hoc manner previously [3]. In case of nonisothermal solidification of alloys, the evolution equations for thermal, solutal and phase fields can also be derived in a thermodynamically

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consistent way, where positive entropy production is guaranteed [7]. The resulting evolution equations are dependent on the detailed form of the adopted entropy functional. With a simple form of the entropy functional, the thermal diffusion equation is given by ∂H = ∇k(φ) · ∇T (13) ∂t where H is the enthalpy density, k(φ) is the thermal conductivity, and the phase field and chemical diffusion equations remain identical with Eqs. (9) and (10). In the simplest case where the thermal conductivities and the specific heats of the liquid and solid are same and independent of temperature, the thermal diffusion equation can be written into ∂φ ∂T Hm h (φ) (14) − = ∇ DT · ∇T ∂t cp ∂t where c p is the specific heat.

3.

Sharp Interface Limit

Equations (9) and (10) in the PFM of alloy solidification can be mapped onto the classical free boundary problem, described in Eqs. (1)–(3). The relationships between the parameters in the phase-field equations and material’s parameters are obtained from the mapping procedure. It can be done at two different limit conditions: a sharp interface limit where the interface width 2ξ p is vanishingly small; and a thin interface limit where the interface width is finite, but much smaller than the characteristic length scales of diffusion field and the interface curvature. At first we deal with the sharp interface analysis. To find the interface width, consider an alloy system at equilibrium, with a 1D diffuse interface between solid (φ = 1 at x < − ξ p ) and liquid (φ = 0 at x > ξ p ) phases. Then the phase-field equation can be integrated and the equilibrium phase-field profile φ0 (x) [8] is the solution of

2 dφ0 2 = ωg(φ0 ) + Z (φ0 ) − Z (0) (15) 2 dx where Z (φ0 ) = f −c fc . The function Z (φ0 )−Z (0) in the right side of this equation has a double-well form under the equal composition condition, whereas it disappears under the equal chemical potential condition for alloys or the equal temperature condition for pure substances [6]. Integrating Eq. (15) again gives the interface width 2ξ p , corresponding to a length over which the phase field changes from φa to φb ; 2ξ p = √ 2

φb φa

√

dφ0 ωg(φ0 ) + Z (φ0 ) − Z (0)

(16)

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The interface energy is obtained by considering an equilibrium system with a cylindrical solid in liquid matrix, maintaining a diffuse interface between them. Integrating the phase-field equation in the cylindrical coordinate gives the chemical potential shift from the equilibrium value, which recovers the curvature effect in Eq. (3), if the interface energy σ is given by σ =

∞ 2 −∞

dφ0 dr

2

dr =

√

2

1

ωg(φ0 ) + Z (φ0 ) − Z (0) dφ0

(17)

0

The same expression for σ can be directly obtained from the excess free energy arising from the nonuniform phase-field distribution in the functional (4). In sharp interface limit, the interface width is vanishingly small, while the interface energy should remain finite. From Eqs. (16) and (17), it appears that the limit can be attained when → 0 and ω → ∞. This leads to ωg(φ0 ) Z (φ0 ) − Z (0) and then the interface width and the energy in the sharp interface limit are given by √ (18) 2ξ p = √ 2 2 ln 3 ω √ ω σ= √ (19) 3 2 when we used φa = 0.1, φb = 0.9 and g(φ) = φ 2 (1 − φ)2 . In sharp interface limit, Eq. (10) for chemical diffusion recovers not only the usual diffusion equations in bulk phases, but also the mass conservation condition at the interface. Similarly, the thermal diffusion Eq. (13) also reproduces the usual thermal diffusion equation in bulk phases and the energy balance condition at the interface. The remaining procedure is to find a relationship between the mobility Mφ and the kinetic coefficient β. Consider a moving plane-front interface with a steady velocity V . The 1D phase-field equation in a moving coordinate system can be integrated over the interfacial region, in which the chemical potential at the interface is regarded as a constant because its variation within the interfacial region can be ignored in the sharp interface limit. The integration yields a linear relationship between the interface velocity and the thermodynamic driving force, which recovers the kinetic effect in Eq. (3) if we put √ ωTm 1 (20) β= √ 3 2 Hm Mφ For given 2ξ p , σ and β, all the parameters , ω and Mφ in the phase-field Eq. (9) thus are determined from the three relationships (18)–(20). For the model of pure substances consisting of Eqs. (9) and (13), exactly same relationships between phase-field parameters and material’s parameters are maintained. When the phase-field parameters are determined with these equations,

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special care should be taken to avoid the interface width effect on the computational results. It is often computationally too stringent to choose 2ξ p small enough to resolve the desired sharp interface limit.

4.

Thin Interface Limit

Remarkable progress has recently been made to overcome the stringent restriction on interface width by using a thin-interface analysis of the PFM [5, 9]. This analysis maps the PFM with a finite interface width, 2ξ p , onto the classi˜ V and ξ p R, where D˜ cal moving boundary problem at the limit of ξ p D/ and R are the average diffusivity in the interfacial region and the local radius of interface curvature, respectively. Furthermore, this makes it possible to eliminate the interface kinetic effect by a specific choice of the phase-field mobility. The mapping of the thin interface PFM onto the classical moving boundary problem is based on the following two ideas. First, due to the finite interface width, there can exist anomalous effects in (1) interface energy, (2) diffusion in the interfacial region, (3) release of the latent heat and (4) solute partitioning. Crossing the interface with a finite width, 2ξ p , the anomalous effects vary sigmoidally and change their signs around the middle position of the interface. By specific choices of the functions in the PFM such as h(φ) and D(φ), these anomalous effects can be eliminated by summing them over the interface width. Second, the thermodynamic variables such as temperature T and chemical potential f c at the interface are not uniquely defined, but rather varying smoothly ˜ V is satisfied, within the finite interface width. When the condition ξ p D/ ˜ V are linhowever, their profiles at the solid and liquid sides of ξ p |x| D/ ear. Extrapolating two straight profiles into the interfacial region, we get a value of the thermodynamic variable at the intersection point. The value corresponds to that in sharp interface limit. In this way, we can find the unique thermodynamic driving force for the thin interface. First we deal with a symmetric model [5] for pure substances, where the specific heat, c p , and the thermal diffusivity, DT , are constant throughout the whole system. In this case, all the anomalous effects arising from the finite interface width are vanishing when φ0 (x) − 1/2 and h(φ0 (x)) are odd functions of x. Because the extra potential disappears in Eq. (15) for pure substances, usual choices of g(φ) and h(φ) satisfy these conditions, for example, g(φ) = φ 2 (1 − φ)2 and h(φ) = φ 3 (6φ 2 − 15φ + 10 ); furthermore the relationships (18) and (19) remain unchanged. The next step of the thin interface analysis is to find the linear relationship between the interface velocity and the thermodynamic driving force, which leads to √ ωTm Hm 1 √ J − (21) β= √ DT c p 2ω 3 2 Hm Mφ

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and J is a constant given by 1

J= 0

h p (φ)[1 − h d (φ)] √ dφ g(φ)

(22)

where the subscripts p and d under h(φ) are added to discriminate solid fractions from the phase-field and diffusion equations, respectively. The discrimination was made because a model with h p (φ) =/ h d (φ) can also be mapped onto the classical moving boundary problem, although both functions become identical when the model is derived from the functional (4). The second term in the right side of Eq. (21) is the correction from the finite √ interface width effect, which disappears in sharp interface limit 2ξ p ∼ / ω → 0. For given 2ξ p , σ and β, all the parameters , ω and Mφ in the phase-field Eq. (9) thus are determined from the three relationships (18)–(20) in thin interface limit. Note that Mφ can be determined at the vanishing interface kinetic condition by putting β = 0 in Eq. (21).

5.

One-sided Model

When the specific heat c p and thermal diffusivity DT in solid and liquid phases are different from each other, the thin interface analysis is more deliberate because one must take care of the anomalous effects associated with asymmetric functions of c p (φ) and DT (φ). There exists similar difficulties in the analysis for the PFM of alloys. The analysis requires additional care of the solute trapping arising from a finite relaxation time for solute partitioning in the interfacial region. The thin interface analysis, however, is still tractable for a one-sided system where the diffusion in solid phase is ignored, which is described below. When the interface width is finite, the interface width and energy are given by Eqs. (16) and (17), respectively. They are influenced by the extra potential Z (φ)− Z (0). The potential imposes a restriction on the interface width [8, 10], for a given interface energy. The restriction is often so tight that it prevents us from taking the merit of the thin interface analysis – enhancing the computational efficiency by increasing the interface width. For high computational efficiency, therefore, it is desirable to take the equal chemical potential condition instead of the equal composition condition, under which the extra potential Z (φ0 ) − Z (0) disappears [6, 10]. In a dilute solution, the equal chemical potential condition is reduced to a simple relationship cS (x)/c L (x) = ceS /ceL ≡ k,

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and the diffusion equation and the phase-field equation are as follows [9, 10]; c = [1 − (1 − k)h d (φ)]c L ≡ A(φ)c L

(23)

∂c = ∇ · D(φ)A( φ)∇c L ∂t

(24)

RT (1 − k) e 1 ∂φ = 2 ∇ 2 φ − ωg (φ) − (c L − c L )h p (φ) Mφ ∂t vm

(25)

where the last term in Eq. (25) is the dilute solution approximation of Eq. (11) and v m is the molar volume. The coefficient RT (1−k)/v m may be replaced by Hm /(m e Tm ), following the van’t Hoff relation, where m e is the equilibrium liquidus slope in the phase diagram. The mapping of Eqs. (24) and (25) in thin interface limit onto the classical moving boundary problem can be performed under the assumption of D S D L [9]. The following are the results obtained to remove anomalous interfacial effects in thin interface limit: Anomalous interface energy is vanishing if dφ0 (x)/dx is an even function of x, where the origin x = 0 is taken as the position with φ = 1/2. This is fulfilled by taking a symmetric potential, such as g(φ) = φ 2 (1−φ)2 . Anomalous solute partitioning is vanishing if h d (φ0 ) dφ0 /dx is an even function of x. This requirement is fulfilled when h d (φ0 (x)) is an even function of x because dφ0 (x)/dx also is an even function following the first condition. Usual choice for h d (φ) satisfies this condition, for example, h d (φ) = φ or h d (φ) = φ 3 (6φ 2 − 15φ + 10). Anomalous surface diffusion in the interfacial region is vanishing if D(φ(x))A(φ(x)) − D L /2 is an odd function of x, which can be fulfilled by putting D(φ) A(φ) = (1 − φ)D L . Also a condition for vanishing chemical potential jump is required at the imaginary sharp interface at x = 0. The chemical potential jump is directly related with the solute trapping effects arising from the finite interface width. Even though the solute trapping is one of the important physical phenomena in rapid solidification of alloys, it is negligible in normal slow solidification conditions. This often leads to a strong artificial solute trapping effect in such normal conditions, however, when a thick interface width is adopted for high efficiency in the phase-field computation. These artificial effects can be remedied by introducing an anti-trapping mass flux into the diffusion Eq. (24) [4], which is proportional to the interface velocity (∼ ∂φ/∂t) and directed toward the normal direction (∇φ/|∇φ|) to the interface. The modified diffusion equation then has the form

∂φ ∇φ ∂c = ∇ · D(φ) A(φ)∇c L + α(c L ) ∂t ∂t |∇φ|

(26)

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The coupling coefficient α(c L ) can be found from the condition for vanishing chemical potential jump; α(c L ) = √ (1 − k)c L (27) 2ω with the previous choices g(φ)=φ 2 (1−φ)2 , h d (φ)=φ and D(φ)A(φ)=(1−φ) D L . The linear relationship between the thermodynamic driving force and the interface velocity leads to a similar relationship between β and Mφ as for symmetric model, but with a replacement of Hm /(DT c p ) by m e ceL (1−k)/D L in the second term of the right side of Eq. (21).

6.

Multiphase and/or Multicomponent Models

The PFM explained above is for solidification of binary alloys into a single solid phase. Solidification of industrial alloys often involves more solid phases and/or more components. In multiphase systems, eutectic and peritectic solidification involving one liquid and two solid phases are of particular importance not only from engineering aspects, but also from scientific aspects because of their richness in interface patterns. Extending the number of phases for eutectic/peritectic solidification can be done by several ways; introducing three phase fields to denote each phase, introducing two phase fields where one is to distinguish between the solid and liquid phases and the other between two different solid phases, or coupling the PFM with the spinodal decomposition model where two solids phases are discriminated by two different compositions. Each approach has its own merits, yielding fruitful information for understanding pattern formation. For quantitative computation in real alloys with enhanced numerical efficiency, however, it is desirable for the models to have the following properties. First, thermodynamic and kinetic properties for three different interfaces in the system should be controlled independently. Second, the force balance at the triple interface junction should be maintained because it plays an essential role in pattern formation. Third, imposing the equal chemical potential condition is preferable because it significantly improves the numerical efficiency, as compared to the equal composition condition. Fourth, all the parameters should be determined to map the model onto the classical moving boundary problem of the eutectic/peritectic solidification. Such multiphase-field models are at the stage of development [10, 11]. The PFMs for binary alloys can be straightforwardly extended to the multicomponent system under the equal composition or equal chemical potential conditions. However, utilizing the advantage of the latter condition requires extra computation to find the compositions of coexisting solid and liquid phases having an equal chemical potential. If the thermodynamic database that are usually given by functions of the compositions are transformed into

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functions of the chemical potential as a preliminary step of computation, the extra cost may be significantly reduced. When the dilute solution approximation is adopted, in particular, the cost is negligible because the condition is reduced to the constant partition coefficients for a reference phase, e.g., liquid phase. Although multicomponent PFMs have been developed with the constant partition coefficients, the complete mapping of the models onto the classical sharp interface model has not yet been done. Presently, the multicomponent PFMs remain as tools for qualitative simulation.

7.

Simulations

The PFMs can be easily implemented into a numerical code by finite difference or finite element schemes, and various simulations for dendritic, eutectic, peritectic and monotectic solidifications have been performed. Examples of them can be found in [3]. The large disparity between the interface width, the microstructural scale and the diffusion boundary layer width hinders the simulation in physically relevant growth conditions. Therefore, early simulations have focused on the qualitative computations of the basic patterns. However, recent advances in hardware resources and the thin interface analysis greatly improved the computational power and efficiency in phasefield simulation. For modeling the free dendritic growth at low undercooling, where the diffusion field reaches far beyond the dendrite itself, computationally efficient methods such as adaptive mesh refining methods [12, 13] and the multi-scale hybrid method [14] of the finite difference scheme and the Monte Carlo scheme have been developed. Through a combination of such advances, not only qualitative but also quantitative phase-field simulation are possible in experimentally relevant growth conditions. The earliest quantitative phase-field simulation [5] was on the free dendritic growth of the symmetric model in 2D and 3D. This was the first numerical test of the microscopic solvability theory for free dendritic growth, which left little doubt about its validity. Quantitative 3D simulations of free dendritic growth in pure substance are further being refined to answer long-standing questions, for examples, the role of fluid flow for dendritic growth [3] and the origin of the abrupt changes of growth velocity and morphology in highly undercooled pure melt [4]. In spite of the variety of simulations for alloy solidification, the quantitative simulation for alloys have been limited. Recent advances in thin interface analysis for a one-sided model opened the window for quantitative calculations. One example is the 2D multiphase-field simulations of directional eutectic solidification in CBr4 −C2 Cl6 alloys [10]. The 2D experimental results of solidification in this alloy may be used for benchmarking the quantitative simulations because all the materials’ parameters were not only measured with reasonable accuracy, but also the various oscillatory/tilting

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instabilities occur with varying lamella spacing, growth velocity and composition. The 2D phase-field simulations of the eutectic solidification under real experimental conditions quantitatively reproduced all the lamella patterns and the morphological changes observed in experiments. In views of the recent success in the thin interface analysis and importance of the alloy solidification in both the engineering and scientific aspects, application of the one-sided PFM will soon be one of the most active fields in modeling alloy solidification. The quantitative application of PFMs to solidification of real alloys is hindered by the lack of information on thermo-physical properties such as interface energy, interface kinetic coefficient and their anisotropies. Combining the PFMs with atomistic modeling to determine these properties will provide powerful tools for studying the solidification behavior in real alloys.

References [1] J.S. Langer, “Instabilities and pattern formation in crystal growth,” Rev. Mod. Phys., 52, 1–28, 1980. [2] P. Meakin, “Fractals, scaling and growth far from equilibrium,” 1st edn., Cambridge Press, UK, 1998. [3] Boettinger, W.J. Warren, J.A., C. Beckermann, and A. Karma, “Phase-field simulation of solidification,” Annu. Rev. Mater. Res., 32, 163–194, 2002. [4] W.J. Hoyt, M. Asta, and A. Karma, “Atomistic and continuum modeling of dendritic solidification,” Mater. Sci. Eng. R, 41, 121–163, 2003. [5] A. Karma and W.-J. Rappel, “Quantitative phase-field modeling of dendritic growth in two and three dimension,” Phys. Rev. E, 57, 4323–4349, 1998. [6] S.G. Kim, W.T. Kim, and T. Suzuki, “Phase-field model for binary alloys,” Phys. Rev. E, 60, 7186–7197, 1999. [7] A.A. Wheeler, G.B. McFadden, and W.J. Boettinger, “Phase-field model for solidification of a eutectic alloy,” Proc. R. Soc. London. A, 452, 495–525, 1996. [8] S.G. Kim, W.T. Kim, and T. Suzuki, “Interfacial compositions of solid and liquid in a phase-field model with finite interface thickness for isothermal solidification in binary alloys,” Phys. Rev. E, 58, 3316–3323, 1998. [9] A. Karma, “Phase-field formulation for quantitative modeling of alloy solidification,” Phys. Rev. Lett., 87, 115701, 2001. [10] S.G. Kim, W.T. Kim, T. Suzuki, and M. Ode, “Phase-field modeling of eutectic solidification,” J. Cryst. Growth, 261, 135–158, 2004. [11] R.Folch and M. Plapp, “Toward a quantitative phase-field modeling of two-solid solidification,” Phys. Rev. E, 68, 010602, 2003. [12] N. Provatas, N. Goldenfeld, and J. Dantzig, “Adaptive mesh refinement computation of solidification microstructures using dynamic data structures,” J. Comp. Phys., 148, 265–290, 1999. [13] C.W. Lan, C.C. Liu, and C.M. Hsu, “An adaptive finite volume method for incompressible heat flow problem in solidification,” J. Comp. Phys., 178, 464–497, 2002. [14] M. Plapp and A. Karma, “Multiscale random-walk algorithm for simulating interfacial pattern formation,” Phys. Rev. Lett., 84, 1740–1743, 2000.

7.4 COHERENT PRECIPITATION – PHASE FIELD METHOD C. Shen and Y. Wang The Ohio State University, Columbus, Ohio, USA

Phase transformation is still the most efficient and effective way to produce various microstructures at mesoscales, and to control their evolution over time. In crystalline solids, phase transformations are usually accompanied by coherency strain generated by lattice misfit between coexisting phases. The coherency strain accommodation alters both thermodynamics and kinetics of the phase transformations and, in particular, produces various self-organized, quasi-periodical array of precipitates such as the tweed [1], twin [2], chessboard structures [3], and fascinating morphological patterns such as the stars, fans and windmill patters [4], to name a few (Fig. 1). These microstructures have puzzled materials researchers for decades. Incorporation of the strain energy in models of phase transformations not only allows for developing a fundamental understanding of the formation of these microstructures, but also provides the opportunity to engineer new microstructures of salient features for novel applications. Therefore, it is desirable to have a model that is able to predict the formation and time-evolution of coherent microstructural patterns. Yet coherent transformation in solid is the toughest nut to crack [5]. In a non-uniform (either compositionally or structurally) coherent solid where lattice planes are continuous on passing from one phase to another (Fig. 2), the lattice misfit between the adjacent non-uniform regions has to be accommodated by displacement of atoms from their regular positions along the boundaries. This sets up elastic strain fields within the solid. Being long-range and strongly anisotropic, the mechanical interactions among these strain fields are very different from the short-range chemical interactions. For example, the bulk chemical free energy and interfacial energy, both of which are associated with the short-range chemical interactions, depend solely on the volume fraction and the total area and inclination of interfaces of the precipitates, respectively. The elastic strain energy, on the other hand, depends on the size, shape, spatial orientation and mutual arrangement of the precipitates. When 2117 S. Yip (ed.), Handbook of Materials Modeling, 2117–2142. c 2005 Springer. Printed in the Netherlands.

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(a)

(b)

10µm (c)

(d)

30mm

50mm

70mm

50mm

70mm

Figure 1. Various strain-accommodating morphological patterns produced by coherent precipitation: (a) tweed, (b) twin, (c) chessboard structures, and (d) stars, fans and windmill patterns.

the elastic strain energy is included in the total free energy, every single precipitate (its size, shape and spatial position) contributes to the morphological changes of all other precipitates in the system through its influence on the stress field and the corresponding diffusion process. Therefore, many of the thermodynamic principles and rate equations obtained for incoherent precipitation may not be applicable anymore to coherent precipitation. A rigorous treatment of coherent precipitation requires a self-consistent description of microstructural evolution without any a priori assumptions about possible particle shapes and their spatial arrangements along a phase transformation path. The phase field method seems to satisfy this requirement. Over the past two decades, it has been demonstrated to have the ability to deal with arbitrary coherent microstructures produced by diffusional and

Coherent precipitation – phase field method (a)

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(b)

Figure 2. Schematic drawing of coherent interfaces (dashed lines). In (a) the precipitate (in grey) and the matrix have the same crystal structure but different lattice parameters while in (b) the precipitate has different crystal structure from the matrix.

displacive transformations with arbitrary transformation strains. Many complicated strain-induced morphological patterns such as those shown in Fig. 1 have been predicted (for recent reviews see [6–8]). A variety of new and intriguing kinetic phenomena underlying the development of these morphological patterns have been discovered, which include the correlated and collective nucleation [6, 7, 9, 10], local inverse coarsening, precipitate drifting and particle splitting [11–14]. These predictions have contributed significantly to our fundamental understanding of many experimental observations [15]. The purpose of this article is to provide an overview of the phase field method in the context of its applications to coherent transformations. We shall start with a discussion of the fundamentals of coherent precipitation, including how the coherency strain affects phase equilibrium (e.g., equilibrium compositions of coexisting phases and their equilibrium volume fractions), driving forces for nucleation, growth and coarsening, thermodynamic factors in diffusivity, and precipitate shape and spatial distribution. This will be followed by an introduction to microelasticity of an arbitrary coherent heterogeneous microstructure and its incorporation in the phase field method. Finally, implementation of the method in modeling coherent precipitations will be illustrated through three examples with progressively increasing complexity. For the purpose of simplicity and clarity, we limit our discussions to bulk materials of homogeneous modulus (i.e., all coexisting phases have the same elastic constants). Applications to more complicated problems such as small confining systems (such as thin films, multi-layers, and nano-particles) and elastically inhomogeneous systems will not be presented. For interested readers, these applications can be found in the references listed under Further Reading.

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Fundamentals of Coherent Precipitation

In depth coverage of this subject can be found in the monograph by Khachaturyan [16] and the book chapter by Johnson [17]. Below we discuss some of the basic concepts related to coherent precipitation. In a series of classical papers [18–21], Cahn laid the theoretical foundation for coherent transformations in crystalline solids. He distinguished the atomic misfit energy (part of the mixing energy of a solid solution) from the coherency elastic strain energy, and incorporated the latter into the total free energy to study coherent processes. He analyzed the effect of coherency strain energy on phase equilibrium, nucleation, and spinodal decomposition. Since the free energy is formulated within the framework of gradient thermodynamics [22], these studies are actually the earliest applications of the phase field method to coherent transformations.

1.1.

Atomic-misfit Energy and Coherency Strain Energy

A macroscopically stress-free solid solution with uniform composition can be in a “strained” state if the constituent atoms differ in size. The elastic energy associated with this microscopic origin is often referred to as the atomic-misfit energy in solid-solution theory [23]. It is the difference between the free energy of a real, homogeneous solution and the free energy of a hypothetical solution of the same system in which all the atoms have the same size. This atomicmisfit energy, even though mechanical in origin and long-range in character, is part of the physically measurable chemical free energy (e.g., free energy of mixing) and is included in thermodynamic databases in literature. The elastic energy associated with composition or structure non-uniformity (such as fluctuations and precipitates) in a coherent system is referred to as the coherency strain energy. The reference state for the measure of the coherency strain energy is a system of identical fluctuations or precipitate–matrix mixture, but with the fluctuations or precipitates/matrix separated into stressfree portions [21] (i.e., the incoherent counterpart). Since the coherency strain energy is in general a function of size, shape, spatial orientation and mutual arrangement of precipitates [16], it cannot be incorporated into the chemical free energy except for very special cases [18]. Thus the coherency strain energy is usually not included in the free energy from thermodynamic databases.

1.2.

Coherent and Incoherent Phase Diagrams

Different from the atomic misfit energy, the coherency strain energy is zero for homogeneous solid solutions and positive for any non-uniform coherent systems. It always promotes a homogeneous solid solution and suppresses

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phase separation. For a given system, the phase diagram determined by minimizing solely the bulk chemical free energy (including the contribution from the atomic misfit energy), or measured from a stage when precipitates already loose their coherency with the matrix, is referred to as incoherent phase diagram. Correspondingly the phase diagram determined by minimizing the sum of the bulk chemical free energy and the coherency strain energy, or measured from coherent stages of the system is referred to as coherent phase diagram. A coherent phase diagram, which is relevant to the study of coherent precipitation, could differ significantly from an incoherent one. This has been demonstrated clearly by Cahn [18] using an elastically isotropic system with a linear dependence of lattice parameter on composition. In this particular case the equilibrium compositions and volume fractions of coherent precipitates can be determined by the common-tangent rule with respect to the total bulk free energy (Fig. 3). Cahn showed that a coherent miscibility gap lies within an incoherent miscibility gap, with the differences in critical point and width of the miscibility gap determined by the amount of lattice misfit. In an elastically anisotropic system, the coherency strain energy becomes a function of precipitate size, shape and spatial location. In this case precipitates of different configurations will have different coherency strain energies, leading to a series of miscibility gaps lying within the incoherent one.

Incoherent free energy coherent free energy

c 1 c '1

c0

c

c'2 c 2

Figure 3. Incoherent (solid line) and coherent (dotted line) free energy as a function of composition for a regular solution that is elastically isotropic and its lattice parameter depends linearly on concentration. The equilibrium compositions in both cases (c1 ,c2 , c1 and c2 ) are determined by the common tangent construction. c0 is the average composition of the solid solution (after [21]).

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Coherent Precipitation

Precipitation involves typically phenomena of nucleation and growth of new phase particles out of a parent phase matrix, and subsequent coarsening of the resulting two-phase mixture. In the absence of coherency strain, nucleation is controlled by the interplay between the bulk chemical free energy and the interfacial energy, while growth and coarsening are dominated, respectively, by the bulk chemical free energy and the interfacial energy. For coherent precipitation, the coherency strain energy enters the driving forces for all three processes because it depends on both volume and morphology of the precipitates. In this case, nucleation is determined by the interplay among the bulk chemical free energy, the coherency strain energy, and the interfacial energy, while growth is dominated by the interplay between the chemical free energy and the coherency strain energy, and coarsening dominated by the interplay between the coherency strain energy and the interfacial energy. Therefore, many of the thermodynamic principles and rate equations derived for incoherent precipitation have to be modified for coherent processes. First of all, one has to pay attention to how the phase diagram and thermodynamic database for a given system were developed. For an incoherent phase diagram the thermodynamic data do not include the contribution from the coherency strain energy. In this case one needs to add the coherency strain energy to the chemical free energy from the database to obtain the total free energy for coherent transformations. However, if the phase diagram is determined for coherent precipitates and the thermodynamic database is developed by fitting the “chemical” free energy model to the coherent phase diagram, the “chemical” free energy already includes the coherency strain energy corresponding to the precipitate configuration encountered in the experiment. Adding again the coherency strain energy to such a “chemical” free energy will overestimate its contribution. Extra effort has to be made to formulate correctly the total free energy function in this case (see next section). Phase diagrams reported in literature are usually incoherent phase diagrams, but exceptions are not uncommon. For example, most existing Ni–Ni3 Al (γ /γ ) phase diagrams are actually coherent ones because the incoherent equilibrium between γ and γ are rarely observed in usual experiment [24]. To have an accurate chemical free energy model is essential for the construction of an accurate total free energy in the phase field method, which determines the coherent phase diagram and the driving forces for coherent precipitation. Even though the coherency strain energy always suppresses phase separation, reducing the driving force for nucleation and growth, coherent precipitation is still the preferred path at early stages of transformations in many material systems. This is because the nucleation barrier for a coherent precipitate is usually significantly lower than that for an incoherent precipitate because of the order-of-magnitude difference in interfacial energy between a

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coherent and an incoherent interface. Precipitates may loose their coherency at later stages when they grow to certain sizes; by then the strain-induced interactions among the coherent fluctuations and precipitates may have already fixed the spatial distribution of the precipitates. Therefore, developing any model for coherent precipitation has to start with coherent nucleation. Classical treatments of strain energy effect on nucleation (for reviews see, [5, 25, 26]) considered an isolated precipitate and calculate the strain energy per unit volume of the precipitate as a function of its shape. The strain energy was then added to the chemical free energy. In these approaches, the interaction of a nucleating particle with the existing microstructure was ignored. However, the strain fields associated with coherent particles interact strongly with each other in elastically anisotropic crystals. In this case the strain energy of a coherent precipitate depends not only on the strain field of its own but also on the strains due to all other particles in the system (for review, see [16]). This may have a profound influence on the nucleation process, e.g., making certain locations preferred nucleation sites [21]. In fact, many of the strain-induced morphological patterns observed (e.g., Fig. 1) may have been inherited from the nucleation stages and further developed during growth and coarsening. For example, the correlated (the position of a nucleus is determined by its interaction with the existing microstructure) [3, 6, 9] and collective (particles appear in groups) nucleation phenomena [10, 27] have been predicted for the formation of various self-organized, quasi-periodical morphological patterns as those shown in Fig. 1. Cahn [18, 21] analyzed coherent nucleation using the phase field method. He showed that one could derive analytical expressions for coherent interfacial energy, activation energy and critical size of a coherent nucleus for an elastically isotropic system. These expressions have exactly the same forms as those derived for incoherent precipitation, but with the chemical free energy replaced by a sum of the chemical free energy and the coherency strain energy. Although no solution is given for coherent nucleation in elastically anisotropic systems, Cahn illustrated qualitatively the effect of elastic interactions among coherent precipitates on the nucleation process in an elastically anisotropic cubic crystal. The driving force for nucleation reaches maximum at a nearby location in an elastically soft direction to an existing precipitate. In computer simulations using the phase field method, nucleation has been implemented in two ways: (a) solving numerically the stochastic phase field equations with the Langevin noise terms [6] and (b) stochastically seeding nuclei in an evolving microstructure according to the nucleation rates calculated as a function of local concentration and temperature [28] following the classical or non-classical nucleation theory. Recently the latter has been extended to coherent nucleation where the effect of elastic interaction of a nucleating particle with an existing microstructure is considered [29]. The Langevin approach is self-consistent with the phase field method but computationally

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intensive, because observation of nucleation requires sampling at very high frequency in the simulation. It has been applied successfully to the study of collective and correlated nucleation under site-saturation conditions [6, 9, 10, 27]. The explicit algorithm is computationally more efficient and has been applied successfully in concurrent nucleation and growth processes under either isothermal or continuous cooling conditions [28, 30]. Because the interfacial energy scales with surface area while the coherency strain energy scales with volume, the shape of a precipitate tends to be dominated by the interfacial energy when it is small and by the coherency strain energy when it grows to larger sizes. Therefore, shape transitions during growth and coarsening of coherent precipitates are expected. The long-range and highly anisotropic elastic interactions give rise to directionality in precipitate growth and coarsening, promoting spatial correlation among precipitates. Extensive discussions on these subjects can be found in the references listed in the Further Reading section. Indeed, significant shape transition (including splitting) and strong spatial alignment of precipitate have been observed (See reviews [6, 15, 31]). The shape transition of a growing particle may further induce growth instability, leading to faceted dendrite [32]. One of the major advantages of the phase field mode is that it describes growth and coarsening seamlessly in a single, self-consistent methodology. Incorporation of the coherency strain energy in the phase field model allows for capturing all possible microstructural features developing during growth and coarsening of coherent precipitates. For example, precipitate drifting, local inverse coarsening, and particle splitting have been predicted during growth and coarsening of coherent precipitates [11–14]. Incorporation of the coherency strain energy will also alter the thermodynamic factor in diffusivity, which is the second derivative of the total free energy with respect to concentration. Since atomic mobility rather than diffusivity is employed in the phase field model, the effect of coherency strain on the thermodynamic factor is included automatically. Note that the thermodynamic factor used in the calculation of atomic mobility from diffusivity should include the elastic energy contribution if the diffusivity was measured from a coherent system.

2. 2.1.

Theoretical Formulation Phase Field Microelasticity of Coherent Precipitation

In the phase field approach [7, 8], microstructural evolution during phase transformation is characterized self-consistently by the tempero-spatial evolution of a set of continuum order parameters or phase fields. One of the major

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advantages of the method is its ability to describe effectively and efficiently an arbitrary microstructure at mesoscale without exp-licitly tracking moving interfaces. In order to apply such a method to describe coherent transformations, one need to formulate the coherency strain energy as a functional of the phase fields without any a priori assumptions about possible particle shapes and their spatial arrangements along the transformation path. The theoretical treatment of such an elasticity problem was due to Khachaturyan and Shatalov [16, 33, 34] who derived a close form of the coherency strain energy for an arbitrary coherent multi-phase mixture in an elastically anisotropic crystal under the homogenous modulus assumption. The theory essentially solves the equation of mechanical equilibrium in the reciprocal space for the well-known virtual process by Eshelby [35, 36] (Fig. 4). The process consists of five steps: (1) isolate portions of a parent phase matrix; (2) transform the isolated portions into precipitate phases in a stress-free state (e.g., outside the parent phase matrix). The deformation involved in this step by assuming certain lattice correspondence between the precipitate and parent phases is defined as the stress-free transformation strain (SFTS) εi0j ; (3) apply an opposite stress −Ci j kl εkl0 to the precipitates to restore their original shapes and sizes; (4) placed them back into the spaces they occupied originally in the matrix; (5) allow both the precipitates and matrix to relax to minimize the elastic strain energy subject to the requirement of interface coherency. Step (1) is traditionally taken prior to the phase transformation. If the precipitates

(5)

(1)

(2)

ε0ij

(4) (3)

σij ⫽⫺Cijkl ε0kl

Figure 4. The Eshelby’s virtual procedures for calculating the coherency strain energy of coherent precipitates.

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differ in composition from the matrix the transformation in Step (2) will change the matrix composition as well because of mass conservation. To be consistent with the definition of the coherency strain energy given in Section 2, we may modify the Eshelby cycle as follows: (1 ) consider a coherent microstructure consisting of arbitrary concentration or structural non-uniformity produced along a phase transformation path; (2 ) decompose the microstructure into its incoherent counterpart (i.e., with all the microstructural constituents being in their stress-free states); (3 ) apply counter stress to force the lattices of all the constituents to be identical to nullify SFTS; (4 ) put them back together by re-stitching their corresponding lattice planes at interfaces; (5 ) let the system relax to minimize the elastic strain energy. The SFTS field associated with arbitrary compositional or structural inhomogeneities can be expressed either in terms of shape functions for sharpinterface approximation [16] of an arbitrary multi-phase mixture, or in terms of phase fields for diffuse-interface approximation of arbitrary concentration or structural non-uniformities: εi0j (x) =

N

εi00j ( p)φ p (x),

(1)

p=1

which is a linear superposition of all N types of non-uniformities with φ p (x) being the phase fields characterizing the p-th type non-uniformity and εi00j ( p) the corresponding SFTS measured from a given reference state. Note that εi00j ( p)(i, j = 1, 2, 3) depends on the lattice correspondence between the precipitate and parent phases. The calculation of εi00j ( p) is an important step towards formulating the coherency strain energy and it will be described in details later in several examples. The equilibrium elastic strain and hence the strain energy can be found from the condition of mechanical equilibrium [37] ∂σi j (x) + f i (x) = 0, ∂x j

(2)

subject to boundary conditions. Here σi j (x) is the ij component of the coherency stress at position x. f i (x) is a body force per unit volume exerted by, e.g., an external field. In Eq. (2) we have used the convention by Einstein where the repeated index j implies a summation over all its possible values. The boundary conditions include constraints on external surfaces and internal interfaces. At external surfaces the boundary conditions are determined by physical constraints on the macroscopic body of a sample, such as shape, surface traction, or a combination of the two. At internal interfaces,

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continuities of both displacement and coherency stress are required to ensure the coherency of the interfaces. The Green’s function solution of Eq. (2) under the homogeneous modulus assumption, gives the equilibrium elastic strain [16, 38]:

ei j (x) = ε¯ i j +

N 1 dg − [n j ki (n) + n i kj (n)]n l σkl00 ( p)φ˜ p (g) 2 (2π )3 p=1

−

N

εi00j ( p)φ p (x)

(3)

p=1

where ε¯ i j is a homogeneous strain that represents the macroscopic shape change of the material body, g is a vector in the reciprocal space and n ≡ g/g. [−1 (n)]ik ≡ Ci j kl n j n l is the inverse of the Green’s function in the reciprocal 00 ˜ space. σi00 j ( p) ≡ C i j kl εkl ( p), φ p (g) is the Fourier transform of φ p (x). – represents a principle value of the integral that excludes a small volume in the reciprocal space (2π )3 / V at g = 0, where V is the total volume of the system. The total coherency strain energy of the system at equilibrium is then readily obtained as

1 Ci j kl ei j (x)ekl (x)dx E = 2 N N V 1 Ci j kl εi00j ( p)εkl00 (q)φ p (x)φq (x)dx + Ci j kl ε¯ i j ε¯ kl = 2 p=1 q=1 2 el

− ε¯ i j

N

Ci j kl εkl00 ( p)φ p (x)dx

p=1

−

1 2

N N

00 ˜∗ ˜ − n i σi00 j ( p) j k (n)σkl (q)n l φ p (g)φq (g)

p=1 q=1

dg (2π )3

(4)

The asterisk in the last term stands for the complex conjugate. Equations (3) and (4) contain the homogeneous strain, ε¯ i j , which is suitable if the external boundary condition is given for a constrained macroscopic shape. Corresponding to the Eshelby circle aforementioned, the first term in the right-hand side of Eq. (4) is the energy required to “squeeze” the microstructural constituents to nullify the stress-free transformation strain in Step (3 ), and the remaining terms represent the energy reductions associated with relaxations of the “squeezed” state in Step (5 ). In particular, the second and third terms describe the homogeneous (macroscopic shape) relaxation and the fourth term

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describes the local heterogeneous relaxation. For a constrained stress condition at the external surface, ε¯ i j is determined by the minimization of the total elastic energy with respect to itself, which yields [38]. ε¯ i j =

appl Si j kl σkl

N 1 + εi00j ( p)φ p (x)dx V p=1

(5) appl

where Si j kl is the elastic compliance tensor and σi j is the applied stress that appl is related to the surface traction T and the surface normal s by Ti = σi j s j . Combining Eqs. (3)–(5) gives 1 E = 2 el

− −

N N

Ci j kl εi00j ( p)εkl00 (q)φ p (x)φq (x)dx

p=1 q=1

1 Ci j kl 2V 1 2

N

εi0j ( p)φ p (x)dx

p=1

N

εkl00 (q)φq (x )dx

q=1

N N p=1

appl − σi j

dg 00 ˜∗ ˜ − n i σi00 j ( p) j k (n)σkl (q)n l φ p (g)φq (g) (2π )3 q=1

N p=1

εi00j ( p)φ p (x)dx −

V appl appl Si j kl σi j σkl 2

(6)

The expression for the mixed constrained shape and surface traction boundary conditions can be derived in a similar way [38]. The above equations were derived under an assumption of constant Ci j kl , i.e., the homogeneous modulus assumption. In cases with spatially dependent Ci j kl the solution is found to be contained in an implicit equation and thus requires a suitable solver, such as an iteration method. Readers are referred to the recent development by Wang et al. [39]. Equations (2)–(6) provide the close forms of the coherency strain energy for a general elastically anisotropic system with arbitrary coherent precipitates described by the phase fields. In such formulations, the coherency strain energy can be added directly to the chemical free energy in the phase field method, because both of them are functionals of the same phase field variables. As mentioned earlier, the “chemical” free energy contains part of the coherency strain energy if it is obtained by fitting the free energy model to a coherent phase diagram. In order to avoid possible double counting, it is necessary to subtract this part of the coherency strain energy from Eq. (4) or (6). Therefore, it is useful to separate the coherency strain energy into self-energy

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and interaction-energy. Following the same treatment as that presented in the microscopic elasticity theory of solid solutions [16], we can rewrite Eq. (4) as el el E el = E sel f + E int , el E sel f =

N N 1 2 p=1 q=1

−¯εi j

N

Ci j kl εi00j ( p)εkl00 (q)φ p (x)φq (x)dx +

V Ci j kl ε¯ i j ε¯ kl 2

Ci j kl εkl00 ( p)φ p (x)dx

p=1

− el =− E int

1 2

N N p=1 q=1

dg Qδ pq − φ˜ p (g)φ˜ q∗ (g) , (2π )3

N N 1 00 − n i σi00 j ( p) j k (n)σkl (q)n l −Qδ pq 2 p=1 q=1

dg × φ˜ p (g)φ˜ q∗ (g) , (2π )3 00 00 00 where Q = n i σi00 j ( p) j k (n)σkl ( p)n l g is the average of n i σi j ( p) j k (n)σkl ( p)n l over the entire reciprocal space and δ pq is the Kronecker delta that equals el unity when p = q or zero otherwise. E sel f is configuration-independent and equals the elastic energy of placing a coherent precipitate of unit-volume multiplying the total volume of the precipitate (small as compared to the volume el of the system) into a uniform matrix. E int is configuration-dependent and contains the pair-wise interactions between precipitates and between volume elements within a finite precipitate. Since the self-energy depends only on the total volume of the precipitates and is independent of their morphology and spatial arrangement, it could be incorporated into and renormalizes the chemical free energy. Clearly, the self-energy should not be included in the calculation of the coherently strain energy if the “chemical” free energy of a system is obtained by fitting to a coherent phase diagram.

2.2.

Incorporation of Coherency Strain Energy into Phase Field Equations

The chemical free energy of a non-uniform system in the phase field approach is formulated as a functional of the field variable based on gradient thermodynamics [22]

F ch =

[ f (φ(x)) + κ|∇φ(x)|2 ]dx,

(7)

where the first term in the integrand is the local chemical free energy density that depends only on local values of the field, φ(x), while the second term is

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the gradient energy that accounts for contributions from spatial variation of φ(x). More complex system may require multiple phase fields, as will be seen in the examples given in the next section. For a coherent system, the total free energy is a sum of the chemical free energy, F ch , and the coherency strain energy, E el , F = F ch + E el ,

(8)

where the chemical free energy is usually measured from a stress-free reference state mentioned earlier, and the coherency strain energy contains both the self- and interaction-energy discussed above. The time evolution of the phase fields, and thus the coherent microstructure, is described by the Onsager-type kinetic equation that assumes a linear dependence of the rate of evolution, ∂φ/∂t, on the driving force, δ F/δφ, ∂φ(x, t) ˆ δ F + ξ(x, t), = −M ∂t δφ(x, t)

(9)

ˆ is a kinetic coefficient matrix and ξ is the Langevin where the operator M random force term that describes thermal fluctuation. The kinetic coefficient ˆ = M if the phase field is non-conserved and matrix is often simplified to M 2 ˆ M = − M∇ if the phase field is conserved, where M is a scalar. Note that the total free energy, F, is a functional of the spatial distribution of the phase field and the energy minimization is a variational process.

3.

Examples of Applications Cubic → Cubic Transformation

3.1.

For a simple cubic → cubic transformation the SFTS is dilatational. If we assume that the coherency strain is caused by concentration inhomogeneity, which is the case for most cubic alloys, the SFTS tensor becomes a function of concentration, e.g., εi0j = ε 0 (c)δi j . The compositional dependence of ε 0 (c) can be written in a Taylor series around the average composition of the parent phase matrix, c¯

dε 0 ε (c) = ε (c) ¯ + ¯ + ··· . (c − c) dc c=c¯ 0

0

(10)

¯ c), ¯ and By choosing a reference state at c(stress-free), ¯ ε 0 (c) = [a(c) − a(c)]/a( the leading term at the right hand side of Eq. (10) vanishes. The SFTS may be approximated by taking the first non-vanishing term

εi0j (x) =

1 da [c(x) − c]δ ¯ ij , a(c) ¯ dc c=c¯

(11)

Coherent precipitation – phase field method

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where we have added the explicit dependence of the SFTS on the spatial posi¯ tion x. Accordingly, εi00j = a −1 (c)(da/dc) c=c¯ δi j . With the stress-free condition for the external boundary applied in this and the subsequent examples, the coherency strain energy is reduced from Eq. (4) with substituting φ p by c(x) − c¯ to

1 V ¯ 2 dx + Ci j kl ε¯ i j ε¯ kl E el = Ci j kl εi00j εkl00 [c(x) − c] 2 2 −¯εi j Ci j kl εkl00

[c(x) − c]dx ¯

dg 1 00 ˜ c˜∗ (g) , − − n i σi00 j j k (n)σkl n l c(g) 2 (2π )3 ˜ is the Fourier where ε¯ i j is determined by the boundary condition and c(g) transform of c(x). The kinetics of coherent precipitates is then described by Eqs. (7)–(9). A typical example of such a cubic → cubic coherent transformation is the precipitation of an ordered intermetallic phase (γ -L12 (Ni3 Al)) from a disordered matrix (γ-fcc solid solution) in Ni–Al (Fig. 5). The coherency strain is caused by the difference in composition between γ and γ that modifies the lattice parameters of the two phases. Since the two-phase equilibrium is coherent equilibrium in the system, the coherency strain energy should include only the configuration-dependent part, as discussed earlier:

1 dg el 00 00 00 00 ˜ c˜∗ (g) . E = − − n i σi j j k (n)σkl n l − n i σi j j k (n)σkl n l c(g) g 2 (2π )3 Figure 6 shows the simulated microstructural evolution during coherent precipitation by the phase field method [40]. The chemical free energy is

(a)

(b)

Figure 5. Crystal structures of γ (fcc solid solution) (a) and γ (ordered L12 ) (b) phases in nickel-aluminum alloy. In (b) the solid circles indicate nickel atoms and the open circles indicate aluminum atoms.

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approximated by a Landau-type expansion polynomial, which provides appropriate descriptions of the equilibrium thermodynamic properties (such as equilibrium compositions and driving force) and reflects the symmetry relationship between the parent and product phases (for general discussion see [41, 42]. The elastic constants of the cubic crystal c11 (=C1111), c12 (=C1122), c44 (=C2323 ) are 231, 149, 117 GPa, respectively [43]. εi00j is chosen as 0.049δi j which corresponds to a SFTS of 0.56%. The simulation is performed on a 512 × 512 mesh with grid size of 1.7 nm. The starting microstructure is a homogeneous supersaturated solid solution of an average composition of 0.17at%Al. The nucleation processes in this and the subsequent examples was simulated by the Langevin noise terms described by ξ in Eq. (9). The noise terms were applied only for a short period of time at the beginning, corresponding to the site-saturation approximation. According to the group and subgroup relationship of crystal lattice symmetry of the parent and precipitate phases, three long-range order parameter fields were used in addition to the concentration field, which introduces automatically four anti-phase domains of the ordered γ phase. Periodical boundary conditions were employed. Because of the strong elastic anisotropy, the precipitates evolved into cuboidal shapes and align themselves into a quasi-periodical array, with both the interface inclination and spatial alignment along the elastically soft 100 directions. The simulated γ /γ microstructure agrees well with experimental observations (Fig. 6(b)). Through this example it becomes clear that the phase field method is able to handle high volume fractions of diffusionally and elastically interacting precipitates of complicated shapes and spatial distributions.

3.2.

Hexagonal → Orthorhombic Transformation

The hexagonal → orthorhombic transformation is a typical example of structural transformations with crystal lattice symmetry reduction. Different from a cubic → cubic transformation, there are several symmetry related orientation variants of the precipitate phase. Experimental observations [44–46] have shown remarkably similar morphological patterns formed by the low symmetry orthorhombic phase in different materials systems, indicating that accommodation of coherency strain among different orientation variants dominate the microstructural evolution during the precipitation reaction. In this example we present a generic transformation of a disordered hexagonal phase to an ordered orthorhombic phase with three lattice correspondence variants [27]. The atomic rearrangement during ordering occurs primarily on the (0001) plane of the parent hexagonal phase and, therefore, the essential features of the microstructural evolution can be well represented by ordering of the (0001) planes (Fig. 7) and effectively modeled in two-dimension.

Coherent precipitation – phase field method (a)

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(b)

0.2µm

Figure 6. (a) Simulated γ /γ microstructure by the phase field method. The lattice misfit is taken as (aγ − aγ ) / aγ ≈ 0.0056. (b) Experimental observation in Ni–Al–Mo alloy (Courtesy of M. F¨ahrmann). (a)

[010]O

[12 10]H

(b)

bO [12 10]H [100]O

aH

[100]O

bO

[12 10]H

Figure 7. Correspondence of the lattices of (a) the disordered hexagonal phase and (b) the ordered orthorhombic phase (with three orientation variants).

The lattice correspondence between the parent and product phases is shown in Fig. 7. For the first variant in Fig. 7(b) we have, 1 ¯¯ [2110]H 3 1 ¯ H [1120] 3

→ [100]O ,

→ 12 [110]O , [0001]H → [001]O , and the corresponding STFS tensor is 

 √ α 0 0 a b cO − cH − a − 3aH O H O 0 ,β = √ ,γ= , εi j =  0 β 0 , where α = aO cH 3aH 0 0 γ

where aH and cH are the lattice parameters of the hexagonal phase and aO , bO and cO are the lattice parameters of the orthorhombic phase. If we assume

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no volume change for the transformation and the lattice parameter difference between the hexagonal and orthorhombic phases along the c-axis is negligible, the SFTS is simplified to 



1 0 0 εi0j = ε 0  0 −1 0 , 0 0 0

(12)

where ε 0 = (aO − aH )/aO is the magnitude of the shear deformation. The three lattice correspondence variants of the orthorhombic phase are related by 120◦ rotation with respect to each other around the c-axis (Fig. 7b). The SFTS of the remaining two variants thus can be obtained by rotational operation (±120◦ around [100]0 ) on the strain tensor given in (Eq. (12). Furthermore, since the deformation along the c-axis is assumed zero, the SFTS of the three variants can be written as 2 × 2 tensors: √ 0 −1/2 3/2 00 0 1 00 0 , εi j (2) = ε √ , εi j (1) = ε 0 −1 3/2 1/2 √ −1/2 − 3/2 00 0 √ . (13) εi j (3) = ε 1/2 − 3/2 In the phase field method, the three variants are described by three longrange order (lro) parameters (η1 , η2 , η3 ), with each representing one variant. Since there is no composition change during the ordering reaction, the structural inhomogeneity is solely characterized by the lro parameters. Correspondingly, the chemical free energy is formulated as a Landau polynomial expansion with respect to the lro parameters. Substituting φ p by η2p ( p = 1, 2, 3) in Eq. (4) the elastic energy becomes, 3 3 1 Ci j kl εi00j ( p)εkl00 (q) E = 2 p=1 q=1 el

− ε¯ i j −

1 2

3

Ci j kl εkl00 ( p)

p=1 3 3

η2p (x)ηq2 (x)dx +

V Ci j kl ε¯ i j ε¯ kl 2

η2p (x)dx

00 2 2∗ − n i σi00 j ( p) j k (n)σkl (q)n l η p (g)ηq (g)

p=1 q=1

. dg (2π )3

Figure 8 shows the simulated microstructures by the phase field method [27]. The system was discretized into a 1024 × 1024 mesh with grid size 0.5 nm. The initial microstructure is a homogeneous hexagonal phase. Strong spatial correlation among the orthorhombic phase particles was developed during the nucleation (Fig. 8(a)). The subsequent growth and coarsening of the orthorhombic phase particles produced various special domain patterns

Coherent precipitation – phase field method (a)

(b)

t* = 20

2135 (c)

t* = 1000

t* = 3000

Figure 8. Microstructures obtained during hexagonal → orthorhombic ordering by 2D phase field simulation. Specific patterns (highlighted by circles, ellipses, and squares) are also found in experimental observations (Fig. 1d).

as a result of elastic strain accommodation among different orientation variants. These patterns show excellent agreements with experimental observations (Fig. 1(d)). Typical sizes of these configurations were also found in good agreement with the experimental observations. If the coherency strain energy was not considered, completely different domain pattern were observed. This indicates that the elastic strain accommodation among different orientation variants dominates the morphological pattern formation during the hexagonal → orthorhombic transformations. The coarsening kinetics of the domain structure deviates significantly from the one observed for an incoherent system [47].

3.3.

Cubic → Trigonal (ζ2 ) Martensitic Transformation in Polycrystalline Au–Cd Alloy

In the two examples presented above, single crystals with relative simple lattice rearrangements during precipitation are considered. In this example we present one of the most complicated cases that have been studied by the phase field method [48]. The trigonal lattice of the ζ2 martensite in Au–Cd can be visualized as a stretched cubic lattice in one of the body diagonal (i.e., [111]) directions. Four lattice correspondence variants are associated with the transformation, which correspond to the four 111 directions of the cube. In the phase field method, the spatial distribution of the four variants is characterized by four lro parameter fields and the chemical free energy is approximated by a Landau expansion polynomial with respect to the lro parameters. If we represent the trigonal phase in hexagonal indices, the lattice correspondence

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between the parent and product phases are [49]: ¯ ς , [121] ¯ β2 → [12 ¯ 10] ¯ ς , [111]β2 → [0001]ς , ¯ β2 → [21¯ 10] Variant 1: [211] 2 2 2 ¯ ¯ ¯ ¯ ¯ ¯ ¯ Variant 2: [121]β2 → [2110]ς2 , [211]β2 → [1210]ς2 , [111]β2 → [0001]ς2 , ¯ ς , [121] ¯ β2 → [12 ¯ 10] ¯ ς , [1¯ 11] ¯ β2 → [0001]ς , ¯ β2 → [21¯ 10] Variant 3: [211] 2 2 2 ¯ ¯ ¯ ¯ ¯ ¯ ¯ Variant 4: [121]β2 → [2110]ς , [211]β2 → [1210]ς , [111]β2 → [0001]ς , 2

2

2

Correspondingly, the SFTS for the four lattice correspondence variants are: 



α β β εi00j (1) =  β α β , β β α 





α −β −β β , εi00j (2) = −β α −β β α 





α −β β α β −β α −β , (14) εi00j (3) = −β α −β , εi00j (4) =  β β −β α −β −β α √ √ √ √ where α = ( 6ah + 3ch − 9ac )/9ac , β = (− 6ah + 2 3ch )/18ac , ac is the lattice parameter of the cubic parent phase, ah and ch are the lattice parameters of the trigonal phase represented in the hexagonal indices. The SFTS field that characterizes the structural inhomogeneity is a linear superposition of the SFTS of each variant, as given by Eq. (2). Thus the elastic energy (Eq. (4)) reduces to

4 4 1 00 − Ci j kl εi00j ( p)εkl00 (q) − n i σi00 E = j ( p) j k (n)σkl (q)n l 2 p=1 q=1 el

×η˜ p (g)η˜ q∗ (g)

dg . (2π )3

Figure 9(a) shows the 3D microstructure simulated in a 128 × 128 × 128 mesh for a single crystal. The grid size is 0.5 µm. The simulation started with a homogeneous cubic solid solution characterized by η1 (x) = η2 (x) = η3 (x) = η4 (x) = 0. The four orientation variants are represented by four shades of gray in the figure. The typical “herring-bone” feature of the microstructure formed by self-assembly of the four variants is readily seen, which agrees well with experimental observations (Fig. 9(c)). The treatment for a polycrystalline material may take the strain tensors in Eq. (14) as the ones in the local coordinate of each constituent single crystal grain. The SFTS expressed in the global coordinate thus requires applying a rotational operation 0,g

εi j (x) = Rik (x)R j l (x)εi0j (x),

(15)

where Ri j (x) is a 3×3 matrix that defines the orientation of the grain in the global coordinate, which has a constant value within a grain but differs from

Coherent precipitation – phase field method (a)

2137

(b)

(c)

Figure 9. Microstructures developed in a cubic → trigonal (ζ2 ) martensitic transformation in (a) single crystal and (b) polycrystal from 3D phase field simulations. The “herring-bone” structure observed in the simulation (a) agrees well with experiment observations (c).

one grain to another. The microstructure in Fig. 9(b) is obtained for a polycrystal with eight randomly oriented grains. The produced multi-domain structure is found to be quite different from the one obtained from the single crystal. Because of the constraint from neighboring randomly oriented grains, the martensitic transformation does not go to completion and the multi-domain structure is stable against further coarsening, which is in contrary to the case with single crystal where the martensitic transformation goes to completion and the multi-domain microstructure undergoes coarsening till a single domain state for the entire system is reached. This example demonstrates well the capability of the phase field method in predicting very complicated strain accommodating microstructural patterns produced by a coherent transformation in polycrystals.

4.

Summary

In this article we reviewed some of the fundamentals related to coherent transformations, the microelasticity theory of coherent precipitation and

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its implementation in the phase field method. Through three examples, the formulations of the stress-free transformation strain field associated with compositional or structural non-uniformity produced by diffusional and diffusionless transformations are discussed. For any given coherent transformations, if the lattice correspondence between the parent and product phases, their lattice parameters and elastic constants are known, the coherency strain energy can be formulated in a rather straightforward fashion as a functional of the same field variables chosen to characterize the microstructure in the phase field method. The flexibility of the method in treating various coherent precipitations involving simple and complex atomic rearrangements has been well demonstrated through these examples. The description of microstructures in terms of phase fields allows for complexities at a level close to that encountered in real materials. The evolution of the microstructures is treated in a self-consistent framework where the variational principle is applied to the total free energy of the system. It would not be surprising to see in the near future a significant increase in the attempts of exploring various kinds of complex coherent phenomena with phase field method owing to these benefits. The formulation of the chemical free energy for solid state phase transformations is not emphasized in this review, but can be found in other reviews (see e.g., [6–8]). The numerical techniques employed in current phase field modeling of coherent transformations involve uniform finite difference schemes, which pose serious limitations on length scales. As a physical model, the affordable system size that can be considered in a phase field simulation is limited by the thickness of the actual interfaces when real material parameters are used as inputs. In order to overcome this length scale limit, one has to either employ more efficient algorithms such as the adaptive [50] and wavelet method [51] that are currently under active development, or produce artificially diffuse interfaces at length scales of interest without altering the velocity of interface motion by modifying properly certain model parameters [52–55]. Since the close form of the coherency strain energy is given in the reciprocal space, Fourier transform is required in solving the partial differential equations, which may impose serious challenges to the adaptive or wavelet method. A common approach to scale up the length scale of phase field modeling of a coherent transformation is to increase the contribution of the coherency strain energy relative to the chemical free energy [40, 56]. While it seems to be a reasonable approach for qualitative studies, it may result in serious artifacts in quantitative studies. For example, it may produce artificially high strain-induced concentration non-uniformity which may affect the kinetics of nucleation, growth and coarsening. This issue has received increasing attentions as the phase field method is being applied to quantitative simulation studies.

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Further Reading

Monographs and Reviews on Coherent Phase Transformations 1. A.G. Khachaturyan, Theory of structural transformations in solids, John Wiley & Sons, New York, 1983. 2. Y. Wang, L.Q. Chen, and A.G. Khachaturyan, “Computer simulation of microstructure evolution in coherent solids,” Solid phase transformations, Warrendale, PA, TMS, 1994. 3. W.C. Johnson, “Influence of elastic stress on phase transformations,” In: H.I. Aaronson (ed.), Lectures on the theory of phase transformations, The Minerals, Metals & Materials Society, 35–134, 1999. 4. L.Q. Chen, “Phase field models for microstructure evolution,” Annu. Rev. Mater. Res., 32, 113–140, 2002. Articles on Elastically Inhomogeneous Solids and Thin Films 5. A.G. Khachaturyan, S. Semenovskaya, and T. Tsakalokos, “Elastic strain energy of inhomogeneous solids,” Phys. Rev. B, 52, 15909–15919, 1995. 6. S.Y. Hu and L.Q. Chen, “A phase-field model for evolving microstructures with strong elastic inhomogeneity,” Acta Mater., 49, 1879, 2001. 7. Y.U. Wang, Y.M. Jin, and A.G. Khachaturyan, “Phase field microelasticity theory and modeling of elastically and structurally inhomogeneous solid,” J. Appl. Phys., 92, 1351–1360, 2002. 8. Y.U. Wang, Y.M. Jin, and A.G. Khachaturyan, “Phase field microelasticity modeling of dislocation dynamics near free surface and in heteroepitaxial thin films,” Acta Mater., 51, 4209–4223, 2003.

References [1] L. Wang, D.E. Laughlin et al., “Magnetic domain structure of Fe-55 at %Pd alloy at different stages of atomic ordering,” J. Appl. Phys., 93, 7984–7986, 2003. [2] V.I. Syutkina and E.S. Jakovleva, Phys. Stat. Sol., 21, 465, 1967. [3] Y. Le Bouar and A. Loiseau, “Origin of the chessboard-like structures in decomposing alloys: Theoretical model and computer simulation,” Acta Mater., 46, 2777, 1998. [4] C. Manolikas and S. Amelinckx, “Phase-transitions in ferroelastic lead orthovanadate as observed by means of electron-microscopy and electron-diffraction 1. Static observations,” Phys. Stat. Sol., A60(2), 607–617, 1980. [5] K.C. Russell, “Introduction to: Coherent fluctuations and nucleation in isotropic solids by John W. Cahn,” In: W. Craig Carter and William C. Johnson (eds.), The selected works of John W. Cahn, Warrendale, Pennsylvania, The Minerals, Metals & Materials Society, 105–106, 1998. [6] Y. Wang, L.Q. Chen et al., “Computer simulation of microstructure evolution in coherent solids,” Solid → Solid Phase Transformations, Warrendale, PA, TMS, 1994.

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[7] Y. Wang and L. Chen, “Simulation of microstructural evolution using the phase field method,” In: E.N. Kaufman (Editor in chief) Methods in materials research, a current protocols, Unit 2a.3, John Wiley & Sons, Inc., 2000 [8] L.Q. Chen, “Phase field models for microstructure evolution,” Annu. Rev. Mater. Res., 32, 113–140, 2002. [9] Y. Wang, H.Y. Wang et al., “Microstructural development of coherent tetragonal precipitates in Mg-partially stabilized zirconia: a computer simulation,” J. Am. Ceram. Soc., 78, 657, 1995. [10] Y. Wang and A.G. Khachaturyan, “Three-dimensional field model and computer modeling of martensitic transformation,” Acta Metall. Mater., 45, 759, 1997. [11] Y. Wang, L.Q. Chen et al., “Particle translational motion and reverse coarsening phenomena in multiparticle systems induced by a long-range elastic interaction,” Phys. Rev. B, 46, 11194, 1992. [12] Y. Wang, L.Q. Chen et al., “Kinetics of strain-induced morphological transformation in cubic alloys with a miscibility gap,” Acta Metall. Mater., 41, 279, 1993. [13] D.Y. Li and L.Q. Chen, “Shape evolution and splitting of coherent particles under applied stresses,” Acta Mater., 47(1), 247–257, 1998. [14] J.D. Zhang, D.Y. Li et al., “Shape evolution and splitting of a single coherent precipitate,” Materials Research Society Symposium Proceedings, 1998. [15] M. Doi, “Coarsening behavior of coherent precipitates in elastically constrained systems – with particular emphasis on gamma-prime precipitates in nickel-base alloys,” Mater. Trans. Japan. Inst. Metals, 33, 637, 1992. [16] A.G. Khachaturyan, Theory of Structural Transformations in Solids, John Wiley & Sons, New York, 1983. [17] W.C. Johnson, “Influence of elastic stress on phase transformations,” In: H.I. Aaronson (ed.), Lectures on the Theory of Phase Transformations, The Minerals, Metals & Materials Society, 35–134, 1999. [18] J.W. Cahn, “Coherent fluctuations and nucleation in isotropic solids,” Acta Met., 10, 907–913, 1962. [19] J.W. Cahn, “On spinodal decomposition in cubic solids,” Acta Met., 10, 179, 1962. [20] J.W. Cahn, “Coherent two-phase equilibrium,” Acta Met., 14, 83, 1966. [21] J.W. Cahn, “Coherent stress in elastically anisotropic crystals and its effect on diffusional proecesses,” In: The Mechanism of Phase Transformations in Crystalline Solids, The Institute of Metals, London, 1, 1969. [22] J.W. Cahn and J.E. Hilliard, “Free energy of a nonuniform system. I. Interfacial free energy,” J. Chem. Phys., 28(2), 258–267, 1958. [23] J.W. Christian, The Theory of Transformations in Metals and Alloys, Pergamon Press, Oxford, 1975. [24] A.J. Ardell, “The Ni-Ni3 Al phase diagram: thermodynamic modelling and the requirements of coherent equilibrium,” Modell. Simul. Mater. Sci. Eng., 8, 277–286, 2000. [25] K.C. Russell, “Nucleation in solids,” In: Phase Transformations, ASM, Materials Park, OH, 219–268, 1970. [26] H.I. Aaronson and J.K. Lee, “The kinetic equations of solid → solid nucleation theory and comparisons with experimental observations,” In: H.I. Aaronson (ed.), Lectures on the Theory of Phase Transformation, TMS, 165–229, 1999. [27] Y.H. Wen, Y. Wang et al., “Phase-field simulation of domain structure evolution during a coherent hexagonal-to-orthorhombic transformation,” Phil. Mag. A, 80(9), 1967–1982, 2000.

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[28] J.P. Simmons, C. Shen et al., “Phase field modeling of simultaneous nucleation and growth by explicit incorporating nucleation events,” Scripta Mater., 43, 935–942, 2000. [29] C. Shen, J.P. Simmons et al., “Modeling nucleation during coherent transformations in crystalline solids,” (to be submitted), 2004. [30] Y.H. Wen and J.P. Simmons et al., “Phase-field modeling of bimodal particle size distributions during continuous cooling,” Acta Mater., 51(4), 1123–1132, 2003. [31] W.C. Johnson and P.W. Voorhees, Solid State Phenomena, 23–24, 87, 1992. [32] Y.S. Yoo, Ph.D. dissertation, Korea Advanced Institute of Science and Technology, Taejon, Korea, 1993. [33] A.G. Khachaturyan, “Some questions concerning the theory of phase transformations in solids,” Sov. Phys. Solid State, 8, 2163, 1967. [34] A.G. Khachaturyan and G.A. Shatalov, “Elastic interaction potential of defects in a crystal,” Sov. Phys. Solid State, 11, 118, 1969. [35] J.D. Eshelby, “The determination of the elastic field of an ellipsoidal inclusion, and related problems,” Proc. R. Soc. A, 241, 376–396, 1957. [36] J.D. Eshelby, “The elastic field outside an ellipsoidal inclusion,” Proc. R. Soc. A, 252, 561, 1959. [37] L.E. Malvern, Introduction to the Mechanics of a Continuous Medium, Prentice-Hall, Englewood Cliffs, 1969. [38] D.Y. Li and L.Q. Chen, “Shape of a rhombohedral coherent Ti11Ni14 precipitate in a cubic matrix and its growth and dissolution during constrained aging,” Acta Mater., 45(6), 2435–2442, 1997. [39] Y.U. Wang, Y.M. Jin et al., “Phase field microelasticity theory and modeling of elastically and structurally inhomogeneous solid,” J. Appl. Phys., 92(3), 1351–1360, 2002. [40] Y. Wang, D. Banerjee et al., “Field kinetic model and computer simulation of precipitation of L12 ordered intermetallics from fcc solid solution,” Acta Mater., 46(9), 2983–3001, 1998. [41] L.D. Landau and E.M. Lifshitz, Statistical Physics, Pergamon Press, Oxford, New York, 1980. [42] P. Tol`edano and V. Dimitriev, Reconstructive Phase Transitions : In Crystals and Quasicrystals, World Scientific, Singapore, River Edge, NJ, 1996. [43] H. Pottebohm, G. Neitze et al., “Elastic properties (the stiffness constants, the shear modulus and the dislocation line energy and tension) of Ni-Al solid-solutions and of the nimonic alloy pe16,” Mat. Sci. Eng., 60, 189, 1983. [44] J. Vicens and P. Delavignette, Phys. Stat. Sol., A33, 497, 1976. [45] R. Sinclair and J. Dutkiewicz, Acta Met., 25, 235, 1977. [46] L.A. Bendersky and W.J. Boettinger, “Transformation of bcc and B2 hightemperature phases to hcp and orthorhombic structures in the ti-al-nb system 2. Experimental tem study of microstructures,” J. Res. Natl. Inst. Stand. Technol., 98(5), 585–606, 1993. [47] Y.H. Wen, Y. Wang et al., “Coarsening dynamics of self-accommodating coherent patters,” Acta Mater., 50, 13–21, 2002. [48] Y.M. Jin, A. Artemev et al., “Three-dimensional phase field model of low-symmetry martensitic transformation in polycrystal: simulation of ζ2 martensite in aucd alloys,” Acta Mater., 49, 2309–2320, 2001. [49] S. Aoki, K. Morii et al., “Self-accommodation of ζ2 martensite in a Au-49.5%Cd alloy. Solid → Solid Phase Transformations, Warrendale, PA, TMS, 1994.

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[50] N. Provatas, N. Goldenfield et al., “Efficient computation of dendritic microstructures using adaptive mesh refinement,” Phys. Rev. Lett., 80, 3308–3311, 1998. [51] D. Wang and J. Pan, “A wavelet-galerkin scheme for the phase field model of microstructural evolution of materials,” Computat. Mat. Sci., 29, 221–242, 2004. [52] A. Karma and W.-J. Rappel, “Quantitative phase-field modeling of dendritic growth in two and three dimensions,” Phys. Rev. E, 57(4), 4323–4349, 1998. [53] K.R. Elder and M. Grant, “Sharp interface limits of phase-field models,” Phys. Rev. E, 64, 021604, 2001. [54] C. Shen, Q. Chen et al., “Increasing length scale of quantitative phase field modeling of growth-dominant or coarsening-dominant process,” Scripta Mater., 50, 1023– 1028, 2004. [55] C. Shen, Q. Chen et al., “Increasing length scale of quantitative phase field modeling of concurrent growth and coarsening processes,” Scripta Mater., 50, 1029–1034, 2004. [56] J.Z. Zhu, Z.K. Liu et al., “Linking phase-field model to calphad: Application to precipitate shape evolution in Ni-base alloys,” Scripta Mater., 46, 401–406, 2002.

7.5 FERROIC DOMAIN STRUCTURES USING GINZBURG–LANDAU METHODS Avadh Saxena and Turab Lookman Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM, USA

We present a strain-based formalism of domain wall formation and microstructure in ferroic materials within a Ginzburg–Landau framework. Certain components of the strain tensor serve as the order parameter for the transition. Elastic compatibility is explicitly included as an anisotropic, long-range interaction between the order parameter strain components. Our method is compared with the phase-field method and that used by the applied mathematics community. We consider representative free energies for a twodimensional triangle to rectangle transition and a three-dimensional cubic to tetragonal transition. We also provide illustrative simulation results for the two-dimensional case and compare the constitutive response of a polycrystal with that of a single crystal. Many minerals and materials of technological interest, in particular martensites [1] and shape memory alloys [2], undergo a structural phase transformation from one crystal symmetry to another crystal symmetry as the temperature or pressure is varied. If the two structures have a simple group–subgroup relationship then such a transformation is called displacive, e.g., cubic to tetragonal transformation in FePd. However, if the two structures do not have such a relationship then the tranformation is referred to as replacive or reconstructive [3, 4]. An example is the body-centered cubic (BCC) to hexagonal closepacked (HCP) transformation in titanium. Structural phase transitions in solids [5, 6] have aroused a great deal of interest over a century due to the crucial role they play in the fundamental understanding of physical concepts as well as due to their central importance in developing technologically useful properties. Both the diffusion-controlled replacive (or reconstructive) and the diffusionless displacive martensitic transformations have been studied although the former have received far more

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attention simply because their reaction kinetics is much more conducive to control and manipulation than the latter. We consider here a particular class of materials known as ferroelastic martensites. Ferroelastics are a subclass of materials known as ferroics [4], i.e., a non-zero tensor property appears below a phase transition. Some examples include ferromagnetic and ferroelectric materials. In some cases more than one ferroic property may coexist, e.g., magnetoelectrics. Such materials are called multi-ferroics. The term martensitic refers to a diffusionless first order phase transition which can be described in terms of one (or several successive) shear deformation(s) from a parent to a product phase [1]. The transition results in a characteristic lamellar microstructure due to transformation twinning. The morphology and kinetics of the transition are dominated by the strain energy. Ferroelasticity is defined by the existence of two or more stable orientation states of a crystal that correspond to different arrangements of the atoms, but are structurally identical or enantiomorphous [4, 5]. In addition, these orientation states are degenerate in energy in the absence of mechanical stress. Salient features of ferroelastic crystals include mechanical hysteresis and mechanically (reversibly) switchable domain patterns. Usually ferroelasticity occurs as a result of a phase transition from a non-ferroelastic high-symmetry “prototype” phase and is associated with the softening of an elastic modulus with decreasing temperature or increasing pressure in the prototype phase. Since the ferroelastic transition is normally weakly first order, or second order, it can be described to a good approximation by the Landau theory [7] with spontaneous strain as the order parameter. Depending on whether the spontaneous strain, which describes the deviation of a given ferroelastic orientation state from the prototype phase is the primary or a secondary order parameter, the low symmetry phase is called a proper or an improper ferroelastic, respectively. While martensites are proper ferroelastics, examples of improper ferroelastics include ferroelectrics and magnetoelastics. There is a small class of materials (either metals or alloy systems) which are both martensitic and ferroelastic and exhibit shape memory effect [2]. They are characterized by highly mobile twin boundaries and (often) show precursor structures (such as tweed and modulated phases) above the transition. Furthermore, these materials have small Bain strain, elastic shear modulus softening, and a weakly to moderately first order transition. Some examples include In1−x Tlx , FePd, CuZn, CuAlZn, CuAlNi, AgCd, AuCd, CuAuZn2 , NiTi and NiAl. In many of these transitions intra-unit cell distortion modes (or shuffles) can couple to the strain either as a primary or secondary order parameter. NiTi and titanium represent two such examples of technological importance. Additional examples include actinide alloys: UNb6 shape memory alloy and Ga-stabilized δ-Pu.

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Landau Theory

To understand the thermodynamics of the phase transformation and the phase diagram a free energy of the transformation is needed. This Landau free energy (LFE) is a symmetry allowed polynomial expansion in the order parameter that characterizes the transformation [7], e.g., strain tensor components and/or (intra-unit cell) shuffle modes. A minimization of this LFE with respect to the order parameter components leads to conditions that give the phase diagram. Derivatives of the LFE with respect to temperature, pressure and other relevant thermodynamic variables provide information about the specific heat, entropy, susceptibility, etc. To study domain walls between different orientational variants (i.e., twin boundaries) or diferent shuffle states (i.e., antiphase boundaries) symmetry allowed strain gradient terms or shuffle gradient terms must be added to the Landau free energy. These gradient terms are called Ginzburg terms and the augmented free energy is referred to as the Ginzburg–Landau (GLFE) free energy. Variation of the GLFE with respect to the order parameter components leads to (Euler–Lagrange) equations [8] whose solution leads to the microstruture. In two dimensions we define the symmetry-adapted dilatation (area change), deviatoric and shear strains [8, 9], respectively, as a function of the Lagrangian strain tensor components i j : 1 e1 = √ (x x + yy ), 2

1 e2 = √ (x x − yy ), 2

e3 = x y .

(1)

As an example, the Landau free energy for a triangular to (centered) rectangular transition is given by [10, 11] F(e2 , e3 ) =

A 2 B C A1 2 (e2 + e32 ) + (e23 − 3e2 e32 ) + (e22 + e32 )2 + e , 2 3 4 2 1

(2)

where A is the shear modulus, A1 is the bulk modulus, B and C are third and fourth order elastic constants, respectively. This free energy without the non-order parameter strain (e1 ) term below the transition temperature (Tc ) has three minima in (e2 , e3 ) corresponding to the three rectangular variants. Above Tc it has only one global minimum at e2 = e3 = 0 associated with the stable triangular lattice. Since the shear modulus softens (partially) above Tc , we have A = A0 (T − Tc ). In three dimensions we define symmetry-adapted strains as [8] 1 1 e1 = √ (x x + yy + zz ), e2 = √ (x x − yy ), 3 2 1 e3 = √ (x x + yy − 2zz ), e4 = x y , e5 = yz , 6

e6 = x z .

(3)

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As an example, the Landau part of the elastic free energy for a cubic to tetragonal transition in terms of the symmetry-adapted strain components is given by [8, 12, 13] F(e2 , e3 ) =

A 2 B C A1 2 (e + e32 ) + (e23 − 3e2 e32 ) + (e22 + e32 )2 + e 2 2 3 4 2 1 A4 2 + (e + e52 + e62 ), 2 4

(4)

where A1 , A and A4 are bulk, deviatoric and shear modulus, respectively, B and C denote third and fourth order elastic constants and (e2 , e3 ) are the order parameter deviatoric strain components. The non-order parameter dilatation (e1 ) and shear (e4 , e5 , e6 ) strains are included to harmonic order. For studying domain walls (i.e., twinning) and microstructure this free energy must be augmented [12] by symmetry allowed gradients of (e2 , e3 ). The plot of the free energy in Eq. (4) without the non-order parameter strain contributions (i.e., compression and shear terms) is identical to the two-dimensional case, Eq. (2), except that the three minima in this case correspond to the three tetragonal variants. The coefficients in the GLFE are determined from a combination of experimental structural (lattice parameter variation as a function of temperature or pressure), vibrational (e.g., phonon dispersion curves along different high symmetry directions) and thermodynamic data (entropy, specific heat, elastic constants, etc.). Where sufficient experimental data is not available, electronic structure calculations and molecular dynamics simulations (using appropriate atomistic potentials) can provide the relevant information to determine some or all of the coefficients in the GLFE. For simple phase transitions (e.g., two-dimensional square to rectangle [8, 9] or those involving only one component order parameter [14]) the GLFE can be written down by inspection (from the symmetry of the parent phase). However, in general the GLFE must be determined by group theoretic means which are now readily available for all 230 crystallographic space groups in three dimensions and (by projection) for all 17 space groups in two dimensions [14] (see the computer program ISOTROPY by Stokes and Hatch [15]).

2.

Microstructure

There are several different but related ways of modeling the microstructure in structural phase transformations: (i) GLFE based as described above [8], (ii) phase-field model in which strain variables are coupled in a symmetry allowed manner to the morphological variables [6], (iii) sharp interface models used by applied mathematicians [16, 17].

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The natural order parameters in the GLFE are strain tensor components. However, until recent years researchers have simulated the microstructure in displacement variables by rewriting the free energy in displacement variables [10, 13]. This procedure leads to the microstructure without providing direct physical insight into the evolution. A natural way to bring out the insight is to work in strain variables only. However, if the lattice integrity is maintained during the phase transformation, that is no dislocation (or topological defect) generation is allowed, then one must obey the St. Venant elastic compatibility constraints because various strain tensor components are derived from the displacement field and are not all independent. This can be achieved by minimizing the free energy with compatibility constraints treated with Lagrangian multipliers [9, 11]. This procedure leads to an anisotropic long-range interaction between the order parameter strain components. The interaction (or compatibility potential) provides direct insight into the domain wall orientations and various aspects of the microstructure in general. Mathematically, the elastic compatibility condition on the “geometrically linear” strain tensor is given by [18]: × (∇ × ) = 0. ∇ (5) which is one equation in two dimensions connecting the three components of the symmetric strain tensor: x x,yy + yy,x x = 2x y,x y . In three dimensions it is two sets of three equations each connecting the six components of the symmetric strain tensor ( yy,zz + zz,yy = 2 yz,yz and two permutations of x, y, z; x x,yz + yz,x x = x y,x z + x z,x y and two permutations of x, y, z). For periodic boundary conditions in Fourier space it becomes an algebraic equation which is then easy to incorporate as a constraint. For the free energy in Eq. (2), the Euler–Lagrange variation of [F−G] with respect to the non-O P strain, e1 is then [11, 14] δ(F c −G)/δe1 = 0, where G denotes the constraint equation, Eq. (5), is a Lagrange multiplier and F c = ( A1 /2)e12 is identically equal to k F c (k). The variation gives (in k space assuming periodic boundary conditions) (k x2 + k 2y )(k) . (6) A1 We then put e1 (k) back into the compatibility constraint condition, Eq. (5), and solve for the Lagrange multiplier (k). Thus e1 (k) is expressed in terms of e2 (k), e3 (k) and e1 (k) =

A1 F (k) = 2 c

2 (k 2 − k 2 )e2 2k x2 k 2y e3 x y + , k2 k2

(7)

(k)e (k) with l = 2, 3, which is used in a identically equal to (1/2) A1 U (k)e (static) free energy variation of the order parameter strains. The (static) “comˆ is independent of |k| and therefore only orientationally patibility kernel” U (k)

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→ U (k). ˆ In coordinate space this is an anisotropic long-range dependent: U (k) 2 (∼ 1/r ) potential mediating the elastic interactions of the primary order parameter strain. From these compatibility kernels one can obtain domain wall orientations, parent product interface (i.e., “habit plane”) orientations and local rotations [14] consistent with those obtained previously using macroscopic matching conditions and symmetry considerations [19, 20]. The concept of elastic compatibility in a single crystal can be readily generalized to polycrystals by defining the strain tensor components in a global frame of reference [21]. By adding a stress term (bilinear in strain) to the free energy one can compute the stress–strain constitutive response in the presence of microstructure for both single and polycrystals and compare the recoverable strain upon cycling. The grain rotation and grain boundaries play an important role when polycrystals are subject to external stress in the presence of a structural transition. Similarly, the calculation of the constitutive response can be generalized to improper ferroelastic materials such as those driven by shuffle modes, ferroelectrics and magnetoelastics.

3.

Dynamics and Simulations

The overdamped (or relaxational) dynamics can be used in simulations to obtain equilibrium microstructure e˙ = −1/ A δ(F + F c )/δe, where A is a friction coefficient and F c is the long-range contribution to the free energy due to elastic compatibility. However, if the evolution of an initial non-equilibrium structure to the equilibrium state is important, one can use inertial strain dynamics with appropriate dissipation terms included in the free energy. The strain dynamics for the order parameter strain tensor components εl is given by [11]

c2 2 δ(F + F c ) δ(R + R c ) + , ρ0 ¨l = l ∇ 4 δl δ ˙l

(8)

where ρ0 is a scaled mass density, cl is a symmetry-specific constant, R = ( A /2)˙εl2 is Rayleigh dissipation and R c is contribution to the dissipation due to the long-range elastic interaction. We replace the compressional free energy in Eq. (2) with the corresponding long-range elastic energy in the order parameter strains and include a gradient term FG = (K /2)[(∇e2 )2 + (∇e3 )2 ], where the gradient coefficient K determines the elastic domain wall energy and can be estimated from the phonon dispersion curves. Simulations performed with the full underdamped dynamics for the triangle to centered rectangular transition are depicted in Fig. 1. The equilibrium microstructure is essentially the same as that found from the overdamped dynamics. The three shades of gray represent the three rectangular variants (or orientations) in the martensite phase. A similar microstrucure has

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Figure 1. A simulated microstructure below the transition temperature for the triangle to rectangle transition. The three shades of gray represent the three rectangular variants.

been observed in lead orthovanadate Pb3 (VO4 )2 crystals [22]. This has also been simulated in the overdamped limit by phase–field [23] and displacement based simulations of Ginzburg–Landau models [10]. The 3D cubic to tetragonal transition (free energy in Eq. (4)) can be simulated either using the strain based formalism outlined here [12] or directly using the displacements [13]. In Fig. 2 we depict microstructure evolution for the cubic to tetragonal transition in FePd mimicked by a square to rectangle transition. To simulate mechanical loading of a polycrystal [21], an external tensile stress σ is applied quasi-statically, i.e., starting from the unstressed configuration of left panel (a), the applied stress σ is increased in steps of 5.13 MPa, after allowing the configurations relax for t ∗ = 25 time steps after each increment. The loading is continued till a maximum stress of σ = 200 MPa is reached in panel (e). Thereafter, the system is unloaded by

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Figure 2. Comparison of the constitutive response for a single crystal and a polycrystal for FePd parameters. The four right panels show the single crystal microstructure and the four left panels depict the polycrystal microstructure.

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decreasing σ to zero at the same rate at which it was loaded; see panel (g). Panel (c) relates to a stress level of σ = 46.15 MPa during the loading process. The favored (rectangular) variants have started to grow at the expense of the unfavored (differently oriented rectangular) variants. The orientation distribution does not change much. As the stress level is increased further, the favored variants grow. Even at the maximum stress of 200 MPa, some unfavored variants persist, as is clear from panel (e). We note that the grains with large misorientation with the loading direction rotate. Grains with lower misorientation do not undergo significant rotation. The mechanism of this rotation is the tendency of the system to maximize the transformation strain in the direction of loading so that the total free energy is minimized [21]. Within the grains that rotate, sub-grain bands are present which correspond to the unfavored strain variants that still survive. Panel (g) depicts the situation after unloading to σ = 0. Upon removing the load, a domain structure is nucleated again due to the local strains at the grain boundaries and the surviving unfavored variants in the loaded polycrystal configuration in panel (e). This domain structure is not the same as that prior to loading, see panel (a), and thus there is an underlying hysteresis. The unloaded configuration has non-zero average strain. This average strain is recovered by heating to the austenite phase, as per the shape memory effect [2]. Note also that the orientation distribution reverts to its preloading state as the grains rotate back when the load is removed. We compare the above mechanical behavior of the polycrystal to the corresponding single crystal. The recoverable strain for the polycrystal is smaller than that for the single crystal due to nucleation of domains at grain boundaries upon unloading. In addition, the transformation in the stress–strain curve for the polycrystal is not abrupt because the response of the polycrystal is averaged over all grain orientations.

4.

Comparison with Other Methods

We compare our approach that is based on the work of Barsch and Krumhansl [8] with two other methods that make use of Landau theory to model structural transformations. Here we provide a brief outline of the differences, the methods are compared and reviewed in detail in Ref. [24]. Khachaturyan and coworkers [6, 23, 25] have used a free energy in which a “structural” or “morphological” order parameter, η, is coupled to strains. This order parameter is akin to a “shuffle” order parameter [26] and the inhomogeneous strain contribution is evaluated using the method of Eshelby [6]. The strains are then effectively removed in favor of the η’s and the minimization is carried out for these variables. This approach (sometimes referred to as

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“phase-field”) applied to improper ferroelastics is essentially the same as our approach with minor differences in the way the inhomogeneous strain contribution is evaluated. However, for the proper ferroelastics that are driven by strain, rather than shuffle, essentially the same procedure is used with phasefield, that is, the minimization (through relaxation methods) is ultimately for the η’s, rather than the strains. In our approach, the non-linear free energy is written up front in terms of the relevant strain order parameters with the discrete symmetry of the transformation taken into account. Here terms that are gradients in strains, which provide the costs of creating domain walls, are also added according to the symmetries. The free energy is then minimized with respect to the strains. That the microstructure for proper ferroelastics obtained from either method would appear qualitatively similar is not surprising. Although the free energy minima or equilibrium states are the same from either procedure, differences in the details of the free energy landscape would be expected to exist. These could affect, for example, the microstructure associated with metastable states. Our method and that developed by the Applied Mechanics community [16, 17] share the common feature of minimizing a free energy written in terms of strains. The method is ideally suited for laminate microstructures with domain walls that are atomistically sharp. This sharp interface limit means that incoherent strains are incorporated through the use of the Hadamard jump condition [16, 17]. The method takes into account finite deformation and has served as an optimization procedure for obtaining static, equilibrium structures, given certain volume fractions of variants. Our approach differs in that we use a continuum formulation with interfaces that have finite width and therefore the incoherent strains are taken into account through the compatibility relation [9, 11]. In addition, we solve the full evolution equations so that we can study kinetics and the effects of inertia.

5.

Ferroic Transitions

Above we considered proper ferroelastic transitions. This method can be readily extended (including the Ginzburg–Landau free energy and elastic compatibility) to the study of improper ferroelastics (e.g., shuffle driven transitions such as in NiTi [26]), proper ferroelectrics such as BaTiO3 [27–29], improper ferroelectrics such as SrTiO3 [30] and magnetoelastics and magnetic shape memory alloys, e.g., Ni2 GaMn [31], by including symmetry allowed coupling between the shuffle modes (or polarization or magnetization) with the appropriate strain tensor components. However, now the elastic energy is considered only up to the harmonic order whereas the primary order parameter has anharmonic contributions. For example for a two-dimensional

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ferroelectric transition on a square lattice the Ginzburg–Landau free energy is given by [25, 32]: = α1 (Px2 + Py2 ) + α11 (Px4 + Py4 ) + α12 Px2 Py2 + α111 (Px6 + Py6 ) F( P) g1 2 g2 2 2 2 + Py,y ) + (Px,y + Py,x ) + α112 (Px2 Py4 + Px4 Py2 ) + (Px,x 2 2 1 1 1 + g3 Px,x Py,y + A1 e12 + A2 e22 + e32 + β1 e1 (Px2 + Py2 ) 2 2 2 2 2 + β2 e2 (Px − Py ) + β3 e3 Px Py , where Px and Py are the polarization components. The free energy for a twodimensional magnetoelastic transition is very similar with magnetization (m x , m y ) replacing the polarization (Px , Py ). For specific physical geometries the long-range electric (or magnetic) dipole interaction must be included. Certainly ferroelectric (and magnetoelastic) transitions can be modeled by phasefield [33] and other methods [34]. We have presented a strain-based formalism for the study of domain walls and microstructure in ferroic materials within a Ginzburg–Landau free energy framework with elastic compatibility constraint explicitly taken into account. The latter induces an anisotropic long-range interaction in the primary order parameter (strain in proper ferroelastics such as martensites and shape memory alloys [9, 11] or shuffle, polarization or magnetization in improper ferroelastics [28, 32]). We compared this method with the widely used phase-field method [6, 23, 25] and the formalism used by applied mathematics and mechanics community [16, 17, 34]. We also discussed the underdamped strain dynamics for the evolution of microstructure and compared the constitutive response of a single crystal with that of a polycrystal. Finally, we briefly mention four other related topics that can be modeled within the Ginzburg–Landau formalism. (i) Some martensites show strain modulation (or tweed precursors) above the martensitic phase transition. These are believed to be caused by disorder such as compositional fluctuations. They can be modeled and simulated by including symmetry allowed coupling of strain to compositional fluctuations in the free energy [9, 35, 36]. Similarly, symmetry allowed couplings of polarization (magnetization) with polar (magnetic) disorder can lead to polar [37] (magnetic [38]) tweed precursors. (ii) Some martensites exhibit supermodulated phases [39] (e.g., 5R, 7R, 9R) which can be modeled within the Landau theory in terms of a particular phonon softening [40] (and its harmonics) and coupling to the transformation shear. (iii) Elasticity at nanoscale can be different from macroscopic continuum elasticity. In this case one must go beyond the usual elastic tensor components and include intra-unit cell modes [41]. (iv) The results presented here are relevant for displacive transformations, i.e., when the parent and product crystal

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structures have a group-subgroup symmetry relationship. However, reconstructive transformations [3], e.g., BCC to HCP transitions, do not have a group– subgroup relationship. Nevertheless, the Ginzburg–Landau formalism can be generalized to these transformations [42]. Notions of a transcendental order parameter [3] and irreversibility [43] have also been invoked to model the reconstructive transformations.

Acknowledgments We acknowledge collaboration with R. Ahluwalia, K.H. Ahn, R.C. Albers, A.R. Bishop, T. Cast´an, D.M. Hatch, A. Planes, K.Ø. Rasmussen and S.R. Shenoy. This work was supported by the US Department of Energy.

References [1] Z. Nishiyama, Martensitic Transformations, Academic, New York, 1978. [2] K. Otsuka and C.M. Wayman (eds.), Shape Memory Materials, Cambridge University Press, Cambridge, 1998; MRS Bull., 27, 2002. [3] P. Tol´edano and V. Dimitriev, Reconstructive Phase Transitions, World Scientific, Singapore, 1996. [4] V.K. Wadhawan, Introduction to Ferroic Materials, Gordon and Breach, Amsterdam, 2000. [5] E.K.H. Salje, Phase Transformations in Ferroelastic and Co-elastic Solids, Cambridge University Press, Cambridge, UK, 1990. [6] A.G. Khachaturyan, Theory of Structural Transformations in Solids, Wiley, New York, 1983. [7] J.C. Tol´edano and P. Tol´edano, The Landau Theory of Phase Transitions, World Scientific, Singapore, 1987. [8] G.R. Barsch and J.A. Krumhansl, Phys. Rev. Lett., 53, 1069, 1984; G.R. Barsch and J.A. Krumhansl, Metallurg. Trans., A18, 761, 1988. [9] S.R. Shenoy, T. Lookman, A. Saxena, and A.R. Bishop, Phys. Rev. B, 60, R12537, 1999. [10] S.H. Curnoe and A.E. Jacobs, Phys. Rev. B, 63, 094110, 2001. [11] T. Lookman, S.R. Shenoy, K. Ø. Rasmussen, A. Saxena, and A.R. Bishop, Phys. Rev. B, 67, 024114, 2003. [12] K. Ø. Rasmussen, T. Lookman, A. Saxena, A.R. Bishop, R.C. Albers, and S.R. Shenoy, Phys. Rev. Lett., 87, 055704, 2001. [13] A.E. Jacobs, S.H. Curnoe, and R.C. Desai, Phys. Rev. B, 68, 224104, 2003. [14] D.M. Hatch, T. Lookman, A. Saxena, and S.R. Shenoy, Phys. Rev. B, 68, 104105, 2003. [15] H.T. Stokes and D.M. Hatch, Isotropy Subgroups of the 230 Crystallographic Space Groups, World Scientific, Singapore, 1988. (The software package ISOTROPY is available at http://www.physics.byu.edu/∼ stokesh/isotropy.html, ISOTROPY (1991)). [16] J.M. Ball and R.D. James, Arch. Rational Mech. Anal., 100, 13, 1987. [17] R.D. James and K.F. Hane, Acta Mater., 48, 197, 2000.

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[18] S.F. Borg, Fundamentals of Engineering Elasticity, World Scientific, Singapore, 1990; M. Baus and R. Lovett, Phys. Rev. Lett., 65, 1781, 1990; M. Baus and R. Lovett, Phys. Rev. A, 44, 1211, 1991. [19] J. Sapriel, Phys. Rev. B, 12, 5128, 1975. [20] C. Boulesteix, B. Yangui, M. Ben Salem, C. Manolikas, and S. Amelinckx, J. Phys., 47, 461, 1986. [21] R. Ahluwalia, T. Lookman, and A. Saxena, Phys. Rev. Lett., 91, 055501, 2003; R. Ahluwalia, T. Lookman, A. Saxena, and R.C. Albers, Acta Mater., 52, 209, 2004. [22] C. Manolikas and S. Amelinckx, Phys. Stat. Sol., (a) 60, 607, 1980; C. Manolikas and S. Amelinckx, Phys. Stat. Sol., 61, 179, 1980. [23] Y.H. Wen, Y.Z. Wang, and L.Q. Chen, Philos. Mag. A, 80, 1967, 2000. [24] T. Lookman, S.R. Shenoy, and A. Saxena, to be published. [25] H.L. Hu and L.Q. Chen, Mater. Sci. Eng., A238, 182, 1997. [26] G.R. Barsch, Mater. Sci. Forum, 327–328, 367, 2000. [27] W. Cao and L.E. Cross, Phys. Rev. B, 44, 5, 1991. [28] S. Nambu and D.A. Sagala, Phys. Rev. B, 50, 5838, 1994. [29] A.J . Bell, J. Appl. Phys., 89, 3907, 2001. [30] W. Cao and G.R. Barsch, Phys. Rev. B, 41, 4334, 1990. [31] A.N. Vasil’ev, A.D. Dozhko, V.V. Khovailo, I.E. Dikshtein, V.G. Shavrov, V.D. Buchelnikov, M. Matsumoto, S. Suzuki, T. Takagi, and J. Tani, Phys. Rev. B, 59, 1113, 1999. [32] R. Ahluwalia and W. Cao, Phys. Rev. B, 63, 012103, 2001. [33] Y.L. Li, S.Y. Hu, Z.K. Liu, and L.Q. Chen, Appl. Phys. Lett., 78, 3878, 2001. [34] Y.C. Shu and K. Bhattacharya, Phil. Mag. B, 81, 2021, 2001. [35] S. Kartha, J.A. Krumhansl, J.P. Sethna, and L.K. Wickham, Phys. Rev. B, 52, 803, 1995. [36] T. Cast´an, A. Planes, and A. Saxena, Phys. Rev. B, 67, 134113, 2003. [37] O. Tikhomirov, H. Jiang, and J. Levy, Phys. Rev. Lett., 89, 147601, 2002. [38] Y. Murakami, D. Shindo, K. Oikawa, R. Kainuma, and K. Ishida, Acta Mater., 50, 2173, 2002. [39] K. Otsuka, T. Ohba, M. Tokonami, and C.M. Wayman, Scr. Matallurg. Mater., 19, 1359, 1993. [40] R.J. Gooding and J.A. Krumhansl, Phys. Rev. B, 38, 1695, 1988; R.J. Gooding and J.A. Krumhansl, Phys. Rev. B, 39, 1535, 1989. [41] K.H. Ahn, T. Lookman, A. Saxena, and A.R. Bishop, Phys. Rev. B, 68, 092101, 2003. [42] D.M. Hatch, T. Lookman, A. Saxena, and H.T. Stokes, Phys. Rev. B, 64, 060104, 2001. [43] K. Bhattacharya, S. Conti, G. Zanzotto, and J. Zimmer, Nature, 428, 55, 2004.

7.6 PHASE-FIELD MODELING OF GRAIN GROWTH Carl E. Krill III Materials Division, University of Ulm, Albert-Einstein-Allee 47, D–89081 Ulm, Germany

When a polycrystalline material is held at elevated temperature, the boundaries between individual crystallites, or grains, can migrate, thus permitting some grains to grow at the expense of others. Planar sections taken through such a specimen reveal that the net result of this phenomenon of grain growth is a steady increase in the average grain size and, in many cases, the evolution toward a grain size distribution manifesting a characteristic shape independent of the state prior to annealing. Recognizing the tremendous importance of microstructure to the properties of polycrystalline samples, materials scientists have long struggled to develop a fundamental understanding of the microstructural evolution that occurs during materials processing. In general, this is an extraordinarily difficult task, given the structural variety of the various elements of microstructure, the topological complexities associated with their spatial arrangement and the range of length scales that they span. Even for single-phase samples containing no other defects besides grain boundaries, experimental and theoretical efforts have met with surprisingly limited success, with observations deviating significantly from the predictions of the best analytic models. Consequently, researchers are turning increasingly to computational methods for modeling microstructural evolution. Perhaps the most impressive evidence for the power of the computational approach is found in its application to single-phase grain growth, for which several successful simulation algorithms have been developed, including Monte Carlo Potts and cellular automata models (both discussed elsewhere in this chapter), and phase-field, front-tracking and vertex approaches. In particular, the phase-field models have proven to be especially versatile, lending themselves to the simulation of growth occurring not only in single-phase systems, but also in the presence of multiple phases or gradients of concentration, strain or temperature. It is no exaggeration to claim that these simulation techniques 2157 S. Yip (ed.), Handbook of Materials Modeling, 2157–2171. c 2005 Springer. Printed in the Netherlands.

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have revolutionized the study of grain growth, offering heretofore unavailable insight into the statistical properties of polycrystalline grain ensembles and the detailed nature of the microstructural evolution induced by grain boundary migration.

1.

Fundamentals of Grain Growth

From a thermodynamic standpoint, grain growth occurs in a polycrystalline sample because the network of grain boundaries is a source of excess energy with respect to the single-crystalline state. The interfacial excess free energy G int can be written as the product of the total grain boundary area AGB and the average excess energy per unit boundary area, γ: G tot = G bulk + G int = G X (T, P) + AGB γ (T, P),

(1)

where G X (T, P, . . .) denotes the free energy of the single-crystalline grain interiors at temperature T and pressure P. Because the specific grain boundary energy γ is a positive quantity, there is a thermodynamic driving force to reduce AGB or, owing to the inverse relationship between AGB and the average grain size R, to increase R. Consequently, grain boundaries tend to migrate such that smaller grains are eliminated in favor of larger ones, resulting in steady growth of the average grain size. The kinetics of this process of grain growth follow one of two qualitatively different pathways [1]: during so-called normal grain growth, the grain size distribution f (R, t) maintains a unimodal shape, shifting to larger R with increasing time t. In abnormal grain growth, on the other hand, only a subpopulation of grains in the sample coarsens, leading to the development of a bimodal size distribution. Although abnormal grain growth is far from rare, the factors responsible for its occurrence are poorly understood at best, depending strongly on properties specific to the sample in question [2]. In contrast, normal grain growth obeys two laws of apparently universal character: power-law evolution of the average grain size and the establishment of a quasistationary scaled grain size distribution [1, 3]. The first entails a relationship of the form Rm (t) − Rm (t0 ) = k (t − t0 ),

(2)

where k is a rate constant (with a strong dependence on temperature), and m denotes the growth exponent [Fig. 1(a)]. Experimentally, m is found to take on a value between 2 and 4, tending toward the lower end of this scale in materials of the highest purity annealed at temperatures near the melting point [2]. The second feature of normal grain growth encompasses the fact that, with increasing annealing time, f (R, t) evolves asymptotically toward a

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1.4 1.0 0.8 0.6 800˚C 750˚C 700˚C

0.4 0.2

f (R /, t )

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0.0 0

50

100

150

Annealing time (min)

200

0.0

0.5

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2.0

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R /

Figure 1. Normal grain growth in polycrystalline Fe. [Data obtained from Ref. [30].] (a) Plot of the average grain size as a function of time in samples annealed at the indicated temperatures. Dashed lines are fits of Eq. (2) with m = 2 (fit function modified slightly to take ‘size effect’ into account). (b) Self-similar evolution of the grain size distribution in the sample annealed at 800 ◦ C for the indicated times. Solid line is a least-squares fit of a lognormal function to the scaled distributions. Dashed line is the prediction of Hillert’s analytic model for grain growth in 3D.

time-invariant shape when plotted as a function of the normalized grain size R/R [Fig. 1(b)]; that is, f (R, t) −→ f˜(R/R),

(3)

with the quasistationary distribution f˜(R/R) generally taking on a lognormal shape [4]. Analytical efforts to explain the origin of Eqs. (2) and (3) generally begin with the assumption that the migration rate v GB of a given grain boundary is proportional to its local curvature, with the proportionality factor defining the grain boundary mobility M [5]. Hillert [6] derived a simple expression for the resulting growth kinetics of a single grain embedded in a polycrystalline matrix. Solving the Hillert model self-consistently for the entire ensemble of grains leads directly to a power-law growth equation with m = 2 and to selfscaling behavior of f (R, t), but the shape predicted for f˜(R/R)–plotted in Fig. 1(b)–has never been confirmed experimentally. This failure is typical of all analytic growth models, which, owing to their statistical mean-field nature, do not properly account for the influence of the grain boundary network’s local topology on the migration of individual boundaries. Computer simulations are able to circumvent this limitation, either by calculating values for v GB from instantaneous local boundary curvatures (cellular automata, vertex, front-tracking methods) or by determining the excess free energy stored in the grain boundary network and then allowing this energy to relax in a physically plausible manner (Monte Carlo, phase-field approaches) [7, 8].

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Phase-field Representation of Polycrystalline Microstructure

The phase-field model for simulating grain growth takes its cue from Eq. (1), expressing the total free energy Ftot as the sum of contributions arising from the grain interiors, Fbulk , and the grain boundary (interface) regions, Fint [9]: Ftot = Fbulk + Fint =

f bulk({φi }) + f int ({φi }, {∇φi }) dr.

(4)

Both Fbulk and Fint are specified as functionals of a set of phase fields {φi (r, t)} (also called order parameters), which are continuous functions defined for all times t at all points r in the simulation cell. The energy density f bulk describes the free energy per unit volume of the grain interior regions, whereas f int accounts for the free energy contributed by the grain boundaries. As discussed below, grain boundaries in the phase-field model have a finite (i.e., non-zero) thickness; therefore, the interfacial energy density f int –like f bulk–is an energy per unit volume and must be integrated over the entire volume of the simulation cell to recover the total interfacial energy. The function f bulk({φi }) can be constructed such that each of the phase fields φi takes on one of two constant values–such as zero or unity–in the interior region of each crystallite [9]. Only when a boundary between two crystallites is crossed do one or, generally, more order parameters change continuously from value to the other; consequently, grain boundaries are locations of large gradients in one or more φi , suggesting that the grain boundary energy term f int should be defined as a function of {∇φi }. The specific functional forms chosen for f bulk and f int, however, depend on considerations of computational efficiency, the physics underlying the growth model and, to a certain extent, personal taste. Over the past several years, two general approaches have emerged in the literature for simulating grain growth by means of Eq. (4).

2.1.

Discrete-orientation Models

In the discrete-orientation approach [10, 11], each order parameter φi is viewed as a continuous-valued component of a vector φ (r, t) = φ (r, t), φ2 1 (r, t), . . . , φ Q (r, t) specifying the local crystalline orientation throughout the simulation cell. Stipulating that the phase fields φi take on constant values of 0 or 1 within the interior of a grain, this model clearly allows at most 2 Q distinct grain orientations, with Q denoting the total number of phase fields. In the most common implementation of the discrete-orientation method, f bulk({φi }) is defined to have local minima when one and only one component of φ equals unity in a grain interior, thus reducing the total number of allowed

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orientations to Q. For example, in a simulation with Q = 4, a given grain might be represented by the contiguous set of points at which φ = (0, 0, 1, 0), and a neighboring grain by φ = (0, 1, 0, 0) [Fig. 2(a)]. As illustrated in Fig. 2(b), upon crossing from one grain to the other, φ2 changes continuously from 0 from 1 to 0; minimization of f int , which is defined to be proporto 1 and φ 3 Q tional to i=1 (∇φi )2 , leads to a smooth–rather than instantaneous–variation in the order-parameter values. The width of the resulting interfacial region is prevented from expanding without bound by the increase in f bulk that occurs when φ deviates from the orientations belonging to the set of local minima of f bulk. Thus, the mathematical representation of each grain boundary is determined by a competition between the bulk and interfacial components of Ftot – a common feature of phase-field representations of polycrystalline microstructures.

(a)

(b) (0, 0, 0, 1) 1

φ2

0

φ3

(0, 0, 1, 0)

(0, 1, 0, 0) (1, 0, 0, 0) (0, 0, 0, 1) (0, 0, 1, 0)

(c)

(d) φ1 θ 10˚

φ1 θ 18˚

φ1 θ 46˚ φ1 θ 72˚ φ1 θ 34˚

φ1 θ 27˚

crystal 50˚

0˚ 1

boundary

crystal θ

φ

0

Figure 2. Phase-field representations of polycrystalline microstructure. (a) Discreteorientation model: grain orientations are specified by a vector-valued phase field φ having four components in this example. (b) Smooth variation of φ2 and φ3 along the dashed arrow in (a). (c) Continuous-orientation model: grain orientations are specified by the angular order parameter θ, and local crystalline order by the value of φ. (d) Smooth variation of θ and φ along the dashed arrow in (c).

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Restricting the grains to a set of discrete orientations may simplify the task of constructing expressions for f bulk and f int in Eq. (4), but it also introduces some conceptual as well as practical limitations to the model. Clearly, it is unphysical for the free energy density of the grain interiors, f bulk, to favor specific grain orientations defined relative to a fixed reference frame, for the free energy of the bulk phase must be invariant with respect to rotation in laboratory coordinates [12]. Even more seriously, the energy barrier in f bulk that separates allowed orientations prohibits the rotation of individual grains during a simulation of grain growth. Since the rotation rate rises dramatically with decreasing grain size [13], grain rotations may be important to the growth process even when R is large, given that there is always a subpopulation of smaller grains losing volume to their growing nearest neighbors.

2.2.

Continuous-orientation Models

In an effort to avoid the undesirable consequences of a finite number of allowed grain orientations, a number of researchers have attempted to express Eq. (4) in terms of continuous, rather than discrete, grain orientations [14–16]. In two dimensions, the orientation of a given grain can be specified completely by a single continuous parameter θ representing, say, the angle between the normal to a particular set of atomic planes and a fixed direction in the laboratory reference frame [Fig. 2(c)]. In 3D, the same specification can be accomplished with three such angular fields. By choosing f bulk to be independent of the orientational order parameters, one ensures that grains are free to take on arbitrary orientations rather than only those corresponding to local minima of the bulk energy density. Because of this independence, however, there is no orientational energy penalty preventing grain boundaries from widening without bound during a growth simulation; thus, it is necessary to introduce an additional phase field that couples the width of the interfacial region to the value of f bulk. Generally, one defines an order parameter φ specifying the degree of crystallinity at each point in the simulation cell, with a value of unity signifying perfect crystalline order (such as obtains in the grain interior) and lower values (0 ≤ φ <1) denoting the disorder characteristic of the boundary core. As illustrated in Fig. 2(d), both φ and θ manifest gradients in the interfacial regions, but only the value of the orientational coordinate distinguishes a given grain from its neighbors. Despite the conceptual advantages enjoyed by continuous-orientation models over their discrete counterparts, the numerical implementation of the former has proven to be far more challenging, with significant difficulties arising from an unavoidable singularity in the expression for f int [12, 14]. Only recently have continuous-orientation simulations of 2D grain growth been

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reported [16], and the authors of this formalism anticipate having to overcome significant additional hurdles to extend it to 3D.

3.

Microstructural Evolution

In both the discrete and continuous-orientation models, the time evolution of the phase-field φi (r, t) is assumed to be governed by the Allen–Cahn equation for non-conserved order parameters [17]: ∂φi δ Ftot = −L i ({φ j }) , ∂t δφi

(5)

where the kinetic coefficients L i –in general, themselves functions of the phase fields–are related to the interface mobility. For example, in a discreteorientation model for ideal grain growth with Q allowed orientations, where Q f int = (κ/2) i=1 (∇φi )2 , and Ftot is given by Eq. (4) with L i ≡ L for all i, Eq. (5) takes on the form [10] ∂φi ∂ f bulk({φ j }) = −L + Lκ∇ 2 φi ∂t ∂φi

(i, j = 1, 2, . . . , Q).

(6)

This coupled set of differential equations can be discretized and solved numerically at each site of a grid spanning the simulation cell, thus generating the time development of the microstructure represented by the phase-fields. The resulting velocity of each grain boundary segment is exactly that expected for curvature-driven boundary migration, as can be demonstrated by applying Eq. (6) to the analytically solvable case of a spherical grain embedded in a homogeneous matrix: the calculated behavior is identical to the analytic solution, with the product Lκ of model parameters equalling the product Mγ of physical parameters [18]. Thus, the microstructural evolution calculated by the phase-field approach should indeed correspond to experimental findings for normal grain growth. Practical issues to consider when solving Eq. (5) numerically include the initial conditions of the calculation (i.e., the starting grain configuration), the boundary conditions of the simulation cell, the spacing of grid points and the numerical scheme chosen for solving the differential equations. The starting microstructure can be obtained by any method for dividing the simulation cell into a space-filling ensemble of grains, such as the Poisson–Voronoi tessellation, or one can begin with an ‘undercooled liquid,’ in which grains nucleate homogeneously and grow to impingement. In the latter case, once nucleation and growth has annihilated the last remnants of liquid phase, subsequent evolution of the microstructure can occur only by boundary migration and grain rotation, as in real polycrystalline materials. Usually, it is most convenient to assume periodic boundary conditions at the edges of the simulation cell, thus

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avoiding complications arising from surface effects. In contrast to the other grid-based method for simulating grain growth (Monte Carlo Potts model), the results of a phase-field simulation are insensitive to the symmetry chosen for the lattice of grid points, largely because the finite width of the boundary regions essentially ‘averages out’ the local symmetry of the grid. Thus, one is free to perform the calculation on a uniform lattice of grid points; however, care must be taken to ensure that the mesh size is small enough to include on the order of seven or more grid points within the thickness of a boundary, if boundary motion is to be independent of the detailed structure of the grid [19]. Finally, when solving Eq. (5) or (6), one must deal with the usual issues of stability and convergence encountered in any numerical solution of nonlinear differential equations. Surprisingly, even a simple numerical scheme like the explicit forward Euler method yields qualitatively correct behavior when applied to analytically solvable cases; however, if absolute rates of boundary migration are of interest, then it is advisable to implement more sophisticated techniques, like a semi-implicit Fourier scheme [20], although this inevitably entails additional computational complexity.

4.

“Ideal” Grain Growth

In a real polycrystalline material, the grain boundary mobility M and the specific grain boundary energy γ vary from grain boundary to grain boundary, depending on (i) the relative orientation of the crystalline regions on either side of the boundary (called the misorientation) and on (ii) the orientation of the boundary plane with respect to the adjacent crystal lattices (the boundary inclination) [2, 21]. For example, both M and γ manifest sharp minima or maxima (‘cusps’) at low-angle boundaries and at certain special misorientations [2, 22]. Analytic models for grain growth have traditionally ignored these dependencies, attributing the same values of M and γ to each grain boundary regardless of the nature of its misorientation or inclination. Grain growth occurring under this simplified scenario is sometimes considered to be “ideal,” and it is this case that, to date, has been investigated most extensively by computer simulation. Of the various phase-field approaches to simulating ideal grain growth, only the discrete-orientation models have been applied to 2D and 3D simulation cells containing a statistically significant number of grains (i.e., 103 ) [4, 18]. The coarsening observed during a phase-field simulation of normal grain growth in 3D is illustrated by the sequence of images in Fig. 3. The positions of grain boundaries can be established by locating the contours Q 2 of an appropriate function like ϕ(r, t) = i=1 φi (r, t), which, for the model employed in the calculations of Fig. 3, takes on a value of unity within individual grains and smaller values in the boundary regions. This allows

Phase-field modeling of grain growth

t 30.0 N 5901

t 100.0 N 2107

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t 300.0 N 696

t 800.0 N 216

Figure 3. Phase-field simulation of three-dimensional grain growth using a discreteorientation model [4]. The elapsed simulation time t and the number of grains N in the simulation cell are specified under each image.

topological properties like the volume, number of faces, number of edges, etc., of each grain in the simulation cell to be evaluated, thus enabling quantification of the evolution of both local and averaged topological properties. For example, in Fig. 4(a) the square of the average grain size, R2 , is plotted against the time t for the simulation of Fig. 3; the resulting straight line reveals that the simulated grain growth follows Eq. (2) with m = 2, consistent with the prediction of Hillert’s analytic model for ideal grain growth. Figure 4(b) illustrates that the grain size distribution evolves in a self-similar manner, as well, but the shape of the quasistationary distribution f˜(R/R) disagrees with the Hillert prediction. Finally, the average topological parameters of the 3D microstructures generated by phase-field simulation can be calculated and compared to measurements performed on real polycrystalline materials–an extraordinarily tedious task!–and to the results of other algorithms for simulating grain growth (Table 1).

5.

“Anisotropic” Grain Growth

With recent advances in computing power, it has become feasible to extend the phase-field approach to the “non-ideal” case in which the grain boundary mobility M and energy γ depend on misorientation and boundary inclination. Such a step is particularly straightforward for a continuous-orientation model, as the angular parameters needed to specify the orientation of each grain appear explicitly as phase fields in the free energy of Eq. (4). With discrete-orientation models, the misorientations and boundary inclinations must be calculated from the parameterization of allowed orientations represented by the array of values of the order parameters {φi }. In both cases, however, it is possible to introduce physically plausible expressions for the anisotropy of M and γ into the equations of motion for microstructural evolution [Eq. (5)].

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0 0

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t 200.0 N 5306 t 400.0 N 2534 t 800.0 N 1113 Hillert (3D)

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Figure 4. Quantitative analysis of microstructural evolution generated by a discreteorientation grain growth model in 3D, averaged over five simulation runs [4]. (a) Linear evolution of R2 , illustrating parabolic growth kinetics following an initial transient. (Dashed line is a guide to the eye.) (b) The grain size distribution f˜(R/R, t), plotted at various times t for the indicated number of grains N. The prediction of Hillert’s analytic model is included for comparison.

To date, all phase-field investigations of the influence of anisotropic M and γ on the kinetics of grain growth have been based on a discrete-orientation model [23, 24]. The results of these studies point to a qualitative difference in the consequences of the two types of anisotropy: whereas both mobility and

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Table 1. Topological parameters of 3D microstructures generated by various grain growth algorithms, measured in cellular materials, or modeled as tessellations of space. The quantity F denotes the average number of faces per cell (grain), and E F the average number of edges per face. See Refs. [4, 29] for references. Phase field Monte Carlo Surface Evolver Vertex Al98 Sn2 Fe Soap froth Tetrakaidecahedra Poisson–Voronoi

(a)

(b)

F

E F

13.7 13.7 13.5 13.8 13.9 13.4 13.4 14 15.535

5.12 5.05 5.05 5.01 5.14 5.11 5.11 5.143 5.228

(c)

Figure 5. Microstructures generated during a 2D phase-field simulation of grain growth, calculated assuming (a) isotropic grain boundary mobility M and energy γ; (b) anisotropic M and isotropic γ ; (c) anisotropic M and γ . Microstructures drawn such that darker boundaries correspond to larger misorientations. Only the introduction of misorientation-dependent grain boundary energies alters the topology of the microstructure. [After (Ref. [25]). Reprinted with permission.]

energy anisotropy affect the overall growth kinetics, only the latter alters the topology of the microstructure generated by grain growth (Fig. 5) [24]. This initially puzzling result can be understood as a consequence of the establishment of local equilibrium at grain boundary junctions: when both higher and lower-energy boundaries meet at a junction, equilibrium favors the lengthening of the lower-energy boundaries at the expense of the higher-energy ones, and it leads to the replacement of threefold-coordinated junctions (the only stable junctions in the isotropic case) with higher-order junctions [25]. Both of these effects have a major impact on the topology of the grain boundary network. In contrast, the introduction of a spectrum of boundary mobilities merely tends to increase or decrease the growth rate of R. Since control over microstructural topology is the primary goal of the processing of

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polycrystalline materials, the future improvement of anisotropic growth models will depend to a large extent on the development of more accurate treatments of energy anisotropy. Further extension of the phase-field approach to even more complex systems, such as those containing multiple phases or gradients of concentration or temperature, is likewise an active area of research [11, 16, 26].

6.

Comparison to other Grain Growth Simulation Algorithms

The phase-field method is just one of many approaches to simulating grain growth in two and three dimensions. Broadly speaking, the various computational algorithms can be divided into two classes: boundary-tracking models, in which the equations of motion are solved numerically for a set of points describing the network of grain boundaries in the simulation cell, thus permitting the positions of each boundary to be calculated as a function of time, and volumetric-relaxation models, in which local microstructural changes are calculated from an equation governing the evolution of the free energy of the overall system. Examples for the boundary-tracking approach include vertex, front-tracking, Surface Evolver and cellular automata models, whereas the primary representatives of the volumetric approach are the Monte Carlo Potts models and the phase-field method [3, 8]. The relative strengths and weaknesses of the two classes of algorithms are clearest for the conceptually and computationally demanding case of 3D growth. Computational considerations seem to favor the boundary-tracking approach, because the dimensionality of the grain boundary network is one less than that of the space in which it is embedded; thus, the computational resources required to simulate a given growth process are potentially far smaller than for a volumetric computation. Moreover, there is no intrinsic limit to the accuracy to which the boundary motion can be determined, as the boundary positions are not restricted to lie along the points of a discrete grid, as they are in the volumetric techniques. However, the topological richness of the grain growth process is the source of a fundamental weakness of boundary-tracking models, at least when applied to 3D growth, as determining the precise topological consequences of singular events like the disappearance of a grain is a currently unsolved problem in 3D. Volumetric-relaxation algorithms avoid this problem entirely, because in those models all topological changes occur naturally as a result of global energy minimization, not in a biased manner through the application of an ad hoc set of rules for local topological changes. The boundary positions need not be tracked explicitly in the volumetric approach, as they can always be determined from the instantaneous state of the simulation cell; however, this

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requires performing calculations throughout the cell volume. Consequently, the volumetric models are computationally tractable only on a discrete lattice, the symmetry and spacing of which can under certain circumstances influence the calculated growth kinetics. Most importantly, it is necessary to construct the energy function in such a manner that the energy minimization pathway yields physically plausible equations of motion for individual boundaries–i.e., curvature-driven migration, in the case of grain growth. For both the Monte Carlo Potts and the phase-field models, this has been verified by comparing the calculated shrinkage of a circular or spherical grain embedded in a homogeneous matrix against the analytic solution for curvature-driven boundary migration. In the case of ideal grain growth, boundary-tracking and volumetricrelaxation simulations yield essentially identical results for growth occurring in two dimensions [27, 28] as well as in three [4]. All calculations predict a value of m = 2 for the growth exponent in Eq. (2), and the simulated microstructures manifest nearly the same quasistationary grain size distribution (Fig. 6) and topological averages (Table 1). It is interesting to note that the simulated grain size distributions universally disagree with the prediction of Hillert’s analytic model, but–unlike in the experimental case illustrated in Fig. 1 (b)–this discrepancy cannot be blamed on the presence of impurities in the grain boundaries or on the existence of mobility and energy anisotropy. Although far less extensive comparisons have been carried out between various simulation algorithms for anisotropic grain growth, here, too, the initial results point to general agreement [25]. Thus, it appears that the choice of

phase field Monte Carlo Surface Evolver vertex Hillert (3D)

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0.8 0.6 0.4 0.2 0.0 0.0

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Figure 6. Comparison of quasistationary grain size distributions generated by various algorithms for simulating ideal grain growth in 3D [4]. The prediction of Hillert’s analytic model is included for comparison.

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an algorithm for simulating grain growth can safely be made on the basis of computational convenience for the specific problem at hand.

References [1] H.V. Atkinson, “Theories of normal grain growth in pure single phase systems,” Acta Metall., 36, 469–491, 1988. [2] F.J. Humphreys and M. Hatherly, Recrystallization and Related Annealing Phenomena, Pergamon Press, Oxford, 1996. [3] C.V. Thompson, “Grain growth and evolution of other cellular structures,” Solid State Phys., 55, 269–314, 2001. [4] C.E. Krill III and L.-Q. Chen, “Computer simulation of 3-D grain growth using a phase-field model,” Acta Mater., 50, 3057–3073, 2002. [5] J.E. Burke and D. Turnbull, “Recrystallization and grain growth,” Prog. Metal Phys., 3, 220–292, 1952. [6] M. Hillert, “On the theory of normal and abnormal grain growth,” Acta Metall., 13, 227–238, 1965. [7] H.J. Frost and C.V. Thompson, “Computer simulation of grain growth,” Curr. Op. Solid State Mater. Sci., 1, 361–368, 1996. [8] M.A. Miodownik, “A review of microstructural computer models used to simulate grain growth and recrystallisation in aluminium alloys,” J. Light Metals, 2, 125–135, 2002. [9] L.-Q. Chen, “Phase-field models for microstructure evolution,” Ann. Rev. Mater. Res., 32, 113–140, 2002. [10] L.-Q. Chen and W. Yang, “Computer simulation of the domain dynamics of a quenched system with a large number of nonconserved order parameters: The grain growth kinetics,” Phys. Rev. B, 50, 15752–15756, 1994. [11] I. Steinbach, F. Pezzolla, B. Nestler, M. Seeßelberg, R. Prieler, G.J. Schmitz, and J.L.L. Rezende, “A phase field concept for multiphase systems,” Physica D, 94, 135– 147, 1996. [12] R. Kobayashi, J.A. Warren, and W.C. Carter, “A continuum model of grain boundaries,” Physica D, 140, 141–150, 2000. [13] D. Moldovan, D. Wolf, and S.R. Phillpot, “Theory of diffusion-accommodated grain rotation in columnar polycrystalline microstructures,” Acta Mater., 49, 3521–3532, 2001. [14] R. Kobayashi, J.A. Warren, and W.C. Carter, “Vector-valued phase field model for crystallization and grain boundary formation,” Physica D, 119, 415–423, 1998. [15] M.T. Lusk, “A phase-field paradigm for grain growth and recrystallization,” Proc. R. Soc. London A, 455, 677–700, 1999. [16] J.A. Warren, R. Kobayashi, A.E. Lobkovsky, and W.C. Carter, “Extending phase field models of solidification to polycrystalline materials,” Acta Mater., 51, 6035–6058, 2003. [17] S.M. Allen and J.W. Cahn, “A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening,” Acta Metall., 27, 1085–1095, 1979. [18] D. Fan, and L.-Q. Chen, “Computer simulation of grain growth using a continuum field model,” Acta Mater., 45, 611–622, 1997. [19] D. Fan, L.-Q. Chen, and S.P. Chen, “Effect of grain boundary width on grain growth in a diffuse-interface field model,” Mater. Sci. Eng. A, A238, 78–84, 1997.

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[20] L.-Q. Chen and J. Shen, “Applications of semi-implicit Fourier-spectral method to phase field equations,” Comput. Phys. Commun., 108, 147–158, 1998. [21] G. Gottstein and L.S. Shvindlerman, Grain Boundary Migration in Metals: Thermodynamics, Kinetics, Applications, CRC Press, Boca Raton, FL, 1999. [22] D. Wolf and K.L. Merkle, “Correlation between the structure and energy of grain boundaries in metals,” In: D. Wolf and S. Yip (eds.), Materials Interfaces: AtomicLevel Structure and Properties, Chapter 3, pp. 87–150, Chapman & Hall, London, 1992. [23] A. Kazaryan, Y. Wang, S.A. Dregia, and B.R. Patton, “Generalized phase-field model for computer simulation of grain growth in anisotropic systems,” Phys. Rev. B, 61, 14275–14278, 2000. [24] A. Kazaryan, Y. Wang, S.A. Dregia, and B.R. Patton, “Grain growth in anisotropic systems: comparison of effects of energy and mobility,” Acta Mater., 50, 2491–2502, 2002. [25] M. Upmanyu, G.N. Hassold, A. Kazaryan, E.A. Holm, Y. Wang, B. Patton, and D.J. Srolovitz, “Boundary mobility and energy anisotropy effects on microstructural evolution during grain growth,” Interface Sci., 10, 201–216, 2002. [26] D. Fan and L.-Q. Chen, “Topological evolution during coupled grain growth and Ostwald ripening in volume-conserved 2-D two-phase polycrystals,” Acta Mater., 45, 4145–4154, 1997. [27] V. Tikare, E.A. Holm, D. Fan, and L.-Q. Chen, “Comparison of phase-field and Potts models for coarsening processes,” Acta Mater., 47, 363–371, 1999. [28] C. Maurice, “Numerical modelling of grain growth: Current status,” In: G. Gottstein, and D.A. Molodov (eds.), Recrystallization and Grain Growth, Vol.1, pp. 123–134, Springer-Verlag, Berlin, 2001. [29] K.M. D¨obrich, C. Rau, and C.E. Krill III, “Quantitative characterization of the threedimensional microstructure of polycrystalline Al-Sn using x-ray microtomography,” Metall. Mater. Trans. A, 35A, 1953–1961, 2004. [30] H. Hu, “Grain growth in zone-refined iron,” Can. Metall. Q., 13, 275–286, 1974.

7.7 RECRYSTALLIZATION SIMULATION BY USE OF CELLULAR AUTOMATA Dierk Raabe Max-Planck-Institut f¨ur Eisenforschung, Max-Planck-Str. 1, 40237 D¨usseldorf, Germany

1. 1.1.

Introduction to Cellular Automata Basic Setup of Cellular Automata

Cellular automata are algorithms that describe the discrete spatial and temporal evolution of complex systems by applying local (or sometimes longrange) deterministic or probabilistic transformation rules to the cells of a regular (or non-regular) lattice. The space variable in cellular automata usually stands for real space, but orientation space, momentum space, or wave vector space can be used as well. Cellular automata can have arbitrary dimensions. Space is defined on a regular array of lattice points which can be regarded as the nodes of a finite difference field. The lattice maps the elementary system entities that are regarded as relevant to the model under investigation. The individual lattice points can represent continuum volume units, atomic particles, lattice defects, or colors depending on the underlying model. The state of each lattice point is characterized in terms of a set of generalized state variables. These could be dimensionless numbers, particle densities, lattice defect quantities, crystal orientation, particle velocity, blood pressure, animal species or any other quantity the model requires. The actual values of these state variables are defined at each of the individual lattice points. Each point assumes one out of a finite set of possible discrete states. The opening state of the automaton which can be derived from experiment (for instance from a microtexture experiment) or theory (for instance from crystal plasticity finite element simulations) is defined by mapping the initial distribution of the values of the chosen state variables onto the lattice. 2173 S. Yip (ed.), Handbook of Materials Modeling, 2173–2203. c 2005 Springer. Printed in the Netherlands.

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The dynamical evolution of the automaton takes place through the application of deterministic or probabilistic transformation rules (also referred to as switching rules) that act on the state of each lattice point. These rules determine the state of a lattice point as a function of its previous state and the state of the neighboring sites. The number, arrangement, and range of the neighbor sites used by the transformation rule for calculating a state switch determines the range of the interaction and the local shape of the areas which evolve. Cellular automata work in discrete time steps. After each time interval the values of the state variables are updated for all lattice points in synchrony mapping the new (or unchanged) values assigned to them through the transformation rule. Owing to these features, cellular automata provide a discrete method of simulating the evolution of complex dynamical systems which contain large numbers of similar components on the basis of their local (or long-range) interactions. Cellular automata do not have restrictions in the type of elementary entities or transformation rules employed. They can map such different situations as the distribution of the values of state variables in a simple finite difference simulation, the colors in a blending algorithm, the elements of fuzzy sets, or elementary growth and decay processes of cells. For instance, the Pascal triangle which can be used to calculate higher-order binominal coefficients or the Fibonaccy numbers can be regarded as a one-dimensional cellular automaton where the value that is assigned to each site of a regular triangular lattice is calculated through the summation of the two numbers above it. In this case the entities of the automaton are dimensionless integer numbers and the transformation rule is a summation. Cellular automata were introduced by von Neumann [1] for the simulation of self-reproducing Turing automata and population evolution. In his early contributions von Neumann denoted them as cellular spaces. Other authors used notions like tessellation automata, homogeneous structures, tessellation structures, or iterative arrays. Later applications were particularly in the field of describing non-linear dynamic behavior of fluids and reaction-diffusion systems. During the last decade cellular automata increasingly gained momentum for the simulation of microstructure evolution in the materials sciences.

1.2.

Formal Description and Classes of Cellular Automata

The local interaction of neighboring lattice sites in a cellular automaton is specified through a set of transformation (switching) rules. While von Neumann’s original automata were designed with deterministic transformation rules probabilistic transformations are conceivable as well. The value of an arbitrary state variable ξ assigned to a particular lattice site at a time (t0 + t) is determined by its present state (t0 ) (or its last few states t0 , t0 − t, etc.) and the state of its neighbors [1–4].

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Considering the last two time steps for the evolution of a one-dimensional −t , cellular automaton, this can be put formally by writing ξ tj0 +t = f (ξ tj0−1 t0 −t t0 −t t0 t0 t0 t0 , ξ j +1 , ξ j −1, ξ j , ξ j +1 ) where ξ j indicates the value of the variable at a ξj time t0 at the node j . The positions ( j + 1) and ( j − 1) indicate the nodes in the immediate neighborhood of position j (for one-dimension). The function f specifies the set of transformation rules, for instance such as provided by standard discrete finite difference algorithms. If the state of the node depends only on its nearest neighbors (NN) the array is referred to as von Neumann neighboring (Fig. 1a). If both the NN and the next-nearest neighbors (NNN) determine the ensuing state of the node, the array is called Moore neighboring (Fig. 1b). Due to the discretization of space, the type of neighboring affects the local transformation rates and the evolving morphologies [1–4]. For the Moore and other extended configurations, which allows one to introduce a certain medium-range interaction among the sites the transformation rule can in one dimension and for interaction with the last −t −t −t , ξ jt0−n+1 , . . . , ξ tj0−1 , ξ tj0 −t , two time steps be rewritten as ξ tj0 +t = f ξ tj0−n

−t , ξ tj0−1 , ξ tj0 , ξ tj0+1 , . . . , ξ tj0+n−1 , ξ tj0+n where n indicates the range of the ξ tj0+1 transformation rule in units of lattice cells. Even for very simple automata there exists an enormous variety of possible transformation rules. If in a one-dimensional Boolean cellular automaton with von Neumann neighboring and reference to the preceding time step each node can have one of two possible ground states, say ξ j = 1 or ξ j = 0, the

(a)

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x (2,2) f {x (1,2), x (2,1), x (2,3), x (3,2)}

x (2,2) g {x (1,1), x (1,2), x (1,3), x (2,1), x (2,3), x (3,1), x (3,2), x (3,3)}

Figure 1. (a) Example of a 2D von Neumann configuration considering nearest neighbors. (b) Example of 2D Moore configuration considering both nearest and next-nearest neighbors.

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transformation rule assumes the form ξ jt0 +t = f ξ tj0−1 , ξ tj0 , ξ tj0+1 . This simple Boolean configuration defines 28 possible transformation rules. One of them has the form if if if if if if if if

t0 ξ = 1, jt0−1 ξ = 1, jt0−1 ξ = 1, jt0−1 ξ j −1 = 1, t0 ξ = 0, jt0−1 ξ = 0, jt0−1 ξ j −1 = 0, t0

ξ tj0 = 1, ξ tj0 = 1, ξ tj0 = 0, ξ tj0 = 0, ξ tj0 = 1, ξ tj0 = 1, ξ tj0 = 0, ξ j −1 = 0, ξ tj0 = 0,

ξ tj0+1 = 1 ξ tj0+1 = 0 ξ tj0+1 = 1 ξ tj0+1 = 0 ξ tj0+1 = 1 ξ tj0+1 = 0 ξ tj0+1 = 1 ξ tj0+1 = 0

then then then then then then then then

ξ tj0 +t ξ tj0 +t ξ tj0 +t ξ tj0 +t ξ tj0 +t ξ tj0 +t ξ tj0 +t ξ tj0 +t

=0 =1 =0 =1 =1 =0 =1 =0

(1, 1, 1) → 0 (1, 1, 0) → 1 (1, 0, 1) → 0 (1, 0, 0) → 1 (0, 1, 1) → 1 (0, 1, 0) → 0 (0, 0, 1) → 1 (0, 0, 0) → 0

This particular transformation rule can be encoded by (01011010)2 . Its digital description is of course only valid for a given arrangement of the corresponding basis. This order is commonly chosen as a decimal row with decreasing value, i.e. (1,1,1) translates to 111 (one hundred eleven), (1,1,0) to 110 (one hundred ten), and so on. Transforming the binary code into decimal numbers using 27

26

25

24

23

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20

0

1

0

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0

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leads to the decimal code number 9010 . The digital coding system is commonly used for compactly describing transformation rules for cellular automata in the literature [2–4]. In general terms the number of rules can be calculated by k (kn) , where k is the number of states for the cell and n is the number of neighbors including the core cell. For a two-dimensional automaton with Moore neighborhood and two possible cell states (i.e., k = 2, n = 9) 229 = 262 144 different transformation rules exist. If the state of a node is determined by the sum of the neighbor site values, the model is referred to as totalistic cellular automaton. If the state of a node has a separate dependence on the state itself and on the sum of the values taken by the variables of the neighbors, the model is referred to as outer totalistic cellular automaton. Cellular automata fall into four basic classes of behavior [2–4]. Class 1 cellular automata evolve for almost any initial configuration after a finite number of time steps to a homogeneous and unique state from which they do not evolve further. Cellular automata in this class exhibit the maximal possible order both at the global and local scale. The geometrical analogy for this class is a limit point in the corresponding phase space. Class 2 cellular automata

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usually create short period patterns that repeat periodically, typically recurring after small periods, or are stable. Local and global order exhibited is in such automata, although not maximal. Class 2 automata can be interpreted as filters, which derive the essence from discrete data sets for a given set of transformation rules. In phase space such systems form a limit cycle. Class 3 cellular automata lead from almost all possible initial states to aperiodic chaotic patterns. The statistical properties of these patterns and the statistical properties of the starting patterns are almost identical at least after a sufficient period of time. The patterns created by class 3 automata are usually self-similar fractal arrays. After sufficiently many time steps, the statistical properties of these patterns are typically the same for almost all initial configurations. Geometrically class 3 automata form so called strange attractors in phase space. Class 3 is the most frequent type of cellular automata. With increasing neighborhood and increasing number of possible cell states the probability to design a class 3 automaton increases for an arbitrary selected rule. Cellular automata in this class can exhibit maximal disorder on both global and local scales. Class 4 cellular automata yield stable, periodic, and propagating structures which can persist over arbitrary lengths of time. Some class 4 automata dissolve after a finite steps of time, i.e., the state of all cells becomes zero. In some class 4 a small set of stable periodic figures can occur (such as for instance in Conway’s game of life [5]). By properly arranging these propagating structures, final states with any cycle length may be obtained. Class 4 automata show a high degree of irreversibility in their time development. They usually reveal more complex behavior and very long transient lengths, having no direct analogue in the field of dynamical systems. The cellular automata in this class can exhibit significant local (not global) order. These introductory remarks show that the cellular automaton concept is defined in a very general and versatile way. Cellular automata can be regarded as a generalization of discrete calculation methods [1, 2]. Their flexibility is due to the fact that, besides the use of crisp mathematical expressions as variables and discretized differential equations as transformation rules, automata can incorporate practically any kind of element or rule that is deemed relevant.

2.

Application of Cellular Automata in Materials Science

Transforming the abstract rules and properties of general cellular automata into a materials-related simulation concept consists in mapping the values of relevant state variables onto the points of a cellular automaton lattice and using the local finite difference formulations of the partial differential equations of the underlying model as local transformation rules. The particular versatility of the cellular automaton approach for microstructure simulations particularly in the fields of recrystallization, grain growth, and phase transformation

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phenomena is due to its flexibility in considering a large variety of state variables and transformation laws. The design of such time and space discretized simulations of materials microstructures which track kinetics and energies in a local fashion are of interest for two reasons. First, from a fundamental point of view it is desirable to understand better the dynamics and the topology of microstructures that arise from the interaction of large numbers of lattice defects which are characterized by a wide spectrum of intrinsic properties and interactions in spatially heterogeneous materials. For instance, in the fields of recrystallization and grain growth the influence of local grain boundary characteristics (mobility, energy), local driving forces, and local crystallographic textures on the final microstructure is of particular interest. Second, from a practical point of view it is necessary to predict microstructure parameters such as grain size or texture which determine the mechanical and physical properties of real materials subjected to industrial processes on a phenomenological though sound physical basis. Apart from cellular automata a number of excellent models for discretely simulating recrystallization and grain growth phenomena have been suggested. They can be grouped as multi-state kinetic Potts Monte Carlo models, topological boundary dynamics and front–tracking models, and Ginzburg–Landau type phase field kinetic models (see overview in Ref. [6]). However, in comparison to these approaches the strength of scaleable kinetic cellular automata is that they combine the computational simplicity and scalability of a switching model with the physical stringency of a boundary dynamics model. Their objective lies in providing a numerically efficient and at the same time phenomenologically sound method of discretely simulating recrystallization and grain growth phenomena. As far as computational aspects are concerned, cellular automata can be designed to minimize calculation time and reduce code complexity in terms of storage and algorithm. As far as microstructure physics is concerned, they can be designed to provide kinetics, texture, and microstructure on a real space and time scale on the basis of realistic or experimental input data for microtexture, grain boundary characteristics, and local driving forces. The possible incorporation of realistic values particularly for grain boundary energies and mobilities deserves particular attention since such experimental data are increasingly available enabling one to make quantitative predictions. Cellular automaton simulations are often carried out at an elementary level using atoms, clusters of atoms, dislocation segments, or small crystalline or continuum elements as underlying units. It should be emphasized that particularly those variants that discretize and map microstructure in continuum space are not intrinsically calibrated by a characteristic physical length or time scale. This means that a cellular automaton simulation of continuum systems requires the definition of elementary units and transformation rules

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that adequately reflect the system behavior at the level addressed. If some of the transformation rules refer to different real time scales (e.g., recrystallization and recovery, bulk diffusion and grain boundary diffusion) it is essential to achieve a correct common scaling of the entire system. The requirement for an adjustment of time scaling among various rules is due to the fact that the transformation behavior of a cellular automaton is sometimes determined by non-coupled Boolean routines rather than by the exact local solutions of coupled differential equations. The same is true when underlying differential equations with entirely different time scales enter the formulation of a set of transformation rules. The scaling problem becomes particularly important in the simulation of non-linear systems (which applies for most microstructure based cellular automata). During the simulation it can be useful to refine or coarsen the scale according to the kinetics (time re-scaling) and spatial resolution (space re-scaling). Since the use of cellular automata is not confined to the microscopic regime, it provides a convenient numerical means for bridging various space and time scales in microstructure simulation. Important fields where microstructure based cellular automata have been successfully used in the materials sciences are primary static recrystallization and recovery [6–19], formation of dendritic grain structures in solidification processes [20–26], as well as related nucleation and coarsening phenomena [27–36]. In what follows this chapter is devoted to the simulation of primary static recrystallization. For studying related microstructural topics the reader is referred to the quotes given above.

3. 3.1.

Example of a Recrystallization Simulation by Use of a Probabilistic Cellular Automaton Lattice Structure and Transformation Rule

The model for the present recrystallization simulation is designed as a cellular automaton with a probabilistic transformation rule [16–18]. Independent variables are time t and space x = (x1 , x2 , x3 ). Space is discretized into an array of equally shaped cells (2D or 3D depending on input data). Each cell is characterized in terms of the dependent variables. These are scalar (mechanical, electromagnetic) and configurational (interfacial) contributions to the driving force and the crystal orientation g = g(ϕ1 , φ, ϕ2 ), where g is the rotation matrix and ϕ1 , φ, ϕ2 the Euler angles. The driving force is the negative change in Gibbs enthalpy G t per transformed cell. The starting data, i.e., the crystal orientation map and the spatial distribution of the driving force, can be provided by experiment, i.e., orientation imaging microscopy via electron back scatter diffraction or by simulation, e.g., a crystal plasticity finite element

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simulation. Grains or sub-grains are mapped as regions of identical crystal orientation, but the driving force may vary inside these areas. The kinetics of the automaton result from changes in the state of the cells (cell switches). They occur in accord with a switching rule (transformation rule) which determines the individual switching probability of each cell as a function of its previous state and the state of its neighbor cells. The switching rule is designed to map the phenomenology of primary static recrystallization in a physically sound manner. It reflects that the state of a non-recrystallized cell belonging to a deformed grain may change due to the expansion of a recrystallizing neighbor grain which grows according to the local driving force and boundary mobility. If such an expanding grain sweeps a non-recrystallized cell the stored dislocation energy of that cell drops to zero and a new orientation is assigned to it, namely that of the expanding neighbor grain. To put this formally, the switching rule is cast in a probabilistic form of a linearized symmetric rate equation, which describes grain boundary motion in terms of isotropic single-atom diffusion processes perpendicular through a homogeneous planar grain boundary segment under the influence of a decrease in Gibbs energy,

G − G t /2 G + G t /2 − exp − (1) x˙ = nνD λgb c exp kB T kB T where x˙ is the grain boundary velocity, νD the Debye frequency, λgb the jump width through the boundary, c the intrinsic concentration of grain boundary vacancies or shuffle sources, n the normal of the grain boundary segment, G the Gibbs enthalpy of motion through in the interface, G t the Gibbs enthalpy associated with the transformation, kB the Boltzmann constant, and T the absolute temperature. Replacing the jump width by the burgers vector and the Gibbs enthalpy terms by the total entropy, S, and total enthalpy, H , leads to a linearized form S H pV exp − (2) x˙ ≈ nνD b exp − kB kB T kB T where p is the driving force and V the atomic volume which is of the order of b3 (b is the magnitude of the Burgers vector). Summarizing these terms reproduces Turnbull’s rate expression

Q gb p (3) x˙ = n m p = n m 0 exp − kB T where m is the mobility. These equations provide a well known kinetic picture of grain boundary segment motion, where the atomistic processes (including thermal fluctuations, i.e., random thermal backward and forward jumps) are statistically described in terms of the pre-exponential factor of the mobility m 0 = m 0 (g,n) and of the activation energy of grain boundary mobility Q gb = Q gb (g, n).

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For dealing with competing switches affecting the same cell the deterministic rate equation can be replaced by a probabilistic analogue which allows one to calculate switching probabilities. For this purpose Eq. (3) is separated into a deterministic part, x˙ 0 , which depends weakly on temperature, and a probabilistic part, w, which depends strongly on temperature:

kB T m 0 pV Q gb exp − V kB T kB T pV Q gb w= exp − kB T kB T

x˙ = x˙ 0 w = n

with x˙ 0 = n

kB T m 0 , V (4)

The probability factor w represents the product of the linearized part pV /(kB T ) and the non–linearized part exp(−Q gb /(kB T )) of the original Boltzmann terms. According to this expression non-vanishing switching probabilities occur for cells which reveal neighbors with different orientation and a driving force which points in their direction. The automaton considers the first, second (2D), and third (3D) neighbor shell for the calculation of the total driving force acting on a cell. The local value of the switching probability depends on the crystallographic character of the boundary segment between such unlike cells.

3.2.

Scaling and Normalization

Microstructure based cellular automata are usually applied to starting data which have a spatial resolution far above the atomic scale. This means that the automaton lattice has a lateral scaling of λm b where λm is the scaling length of the cellular automaton lattice and b the Burgers vector. If a moving boundary segment sweeps a cell, the grain thus grows (or shrinks) by λ3m rather than b3 . Since the net velocity of a boundary segment must be independent of this scaling value of λm , an increase in jump width must lead to a corresponding decrease in the grid attack frequency, i.e., to an increase of the characteristic time step, and vice versa. For obtaining a scale-independent grain boundary velocity, the grid frequency must be chosen in a way to ensure that the attempted switch of a cell of length λm occurs with a frequency much below the atomic attack frequency which attempts to switch a cell of length b. This scaling condition which is prescribed by an external scaling length λm leads to the equation x˙ = x˙ 0 w = n (λm ν) w

with ν =

kB T m 0 V λm

(5)

where ν is the eigenfrequency of the chosen lattice characterized by the scaling length λm .

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The eigenfrequency represents the attack frequency for one particular grain boundary with constant mobility. In order to use a whole spectrum of mobilities and driving forces in one simulation it is necessary to normalize the eigenfrequency by a common grid attack frequency ν0 rendering it into

x˙ = x˙ 0 w = nλm ν0

ν w = xˆ˙ 0 ν0

ν w = xˆ˙ 0 wˆ ν0

(6)

The value of the attack frequency ν0 which is characteristic of the lattice can be calculated by the assumption that the maximum occurring switching probability cannot be larger than one

wˆ

max

max Q min m max gb 0 p = exp − min kB T λ m ν0

! ≤1

(7)

is the maximum occurring pre–exponential factor of the mobility, where m max 0 pmax the maximum possible driving force, ν0min the minimum allowed grid attack frequency, and Q min gb the minimum occurring activation energy. With wˆ max = 1 one obtains the normalization frequency as a function of the upper bound input data.

ν0min

max Q min m max gb 0 p = exp − λm kB T

(8)

This frequency and the local values of the mobility and the driving force lead to

wˆ

local

Q local m local plocal gb = 0 min exp − kB T λ m ν0

=

m local 0 m max 0

local

p pmax



exp−

min Q local gb − Q gb

kB T



m local plocal 0 = max max m0 p

(9) This expression is the central switching equation of the algorithm. One can interpret this equation also in terms of the local time t = λm /˙x which is required by a grain boundary with velocity x˙ to sweep an automaton cell of size λm

wˆ

local

=

m local plocal m max pmax

=

x˙ local x˙ max

t max = local t

(10)

Equation (9) shows that the local switching probability can be quantified by the ratio of the local and the maximum mobility m local/m max , which is a function of the grain boundary character and by the ratio of the local and the maximum driving pressure plocal / pmax . The probability of the fastest occurring

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local min boundary segment (characterized by m local = m max = pmax , Q local 0 0 , p gb = Q gb ) to realize a cell switch is equal to 1. Equation (9) shows that an increasing cell size does not influence the switching probability but only the time step elapsing during an attempted switch. This relationship is obvious since the volume to be swept becomes larger which requires more time. The characteristic time constant of the simulation t is 1/ν0min . While Eq. (8) allows one to calculate the switching probability of a cell as a function of its previous state and the state of the neighbor cells the actual decision about a cell switch is made by a Monte Carlo step. The use of random numbers ensures that all cell switches are sampled according to their proper statistical weight, i.e., according to the local driving force and mobility between cells. The simulation proceeds by calculating the individual local switching probabilities wˆ local for each cell and evaluating them using a Monte Carlo algorithm. This means that for each cell the calculated switching probability is compared to a randomly generated number r which lies between 0 and 1. The switch is accepted if the random number is equal or smaller than the calculated switching probability. Otherwise the switch is rejected.

  m local plocal   accept switch if r ≤    m max pmax 

random number r between 0 and 1

   m local plocal     reject switch if r >

m max pmax

(11) Except for the probabilistic evaluation of the analytically calculated transformation probabilities, the approach is entirely deterministic. Thermal fluctuations other than already included via Turnbull’s rate equation are not permitted. The use of realistic or even experimental input data for the grain boundaries enables one to make predictions on a real time and space scale. The switching rule is scalable to any mesh size and to any spectrum of boundary mobility and driving force data. The state update of all cells is made in synchrony.

3.3.

Simulation of Primary Static Recrystallization and Comparison to Avrami-type Kinetics

Figure 2 shows the kinetics and 3D microstructures of a recrystallizing aluminum single crystal. The initial deformed crystal had a uniform Goss orientation (011)[100] and a dislocation density of 1015 m−2 . The driving force was due to the stored elastic energy provided by the dislocations. In order to compare the predictions with analytical Avrami kinetics recovery

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100 recrystallized volume fraction [%]

90 80 70 60 50 40 30 20 10 0 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50 time [s] Figure 2. Kinetics and microstructure of recrystallization in a plastically strained aluminum single crystal. The deformed crystal had a (011)[100] orientation and a uniform dislocation density of 1015 m−2 . Simulation parameter: site saturated nucleation, lattice size: 10 × 10 × 10 × µ m3 , cell size: 0.1 µm, activation energy of large angle grain boundary mobility: 1.3 eV, pre–exponential factor of large angle boundary mobility: m 0 = 6.2 ×10−6 m3 /(N s), temperature: 800 K, time constant: 0.35 s.

and driving forces arising from local boundary curvature were not considered. The simulation used site saturated nucleation conditions, i.e., the nuclei were at t =0 s statistically distributed in physical space and orientation space. The grid size was 10 × 10 × 10 µm3 . The cell size was 0.1 µm. All grain boundaries had the same mobility using an activation energy of the grain boundary mobility of 1.3 eV and a pre–exponential factor of the boundary mobility of m 0 = 6.2 · 10−6 m3 /(N s) [37]. Small angle grain boundaries had a mobility of zero. The temperature was 800 K. The time constant of the simulation was 0.35 s. Figure 3 shows the kinetics for a number of 3D recrystallization simulations with site saturated nucleation conditions and identical mobility for all grain boundaries. The different curves correspond to different initial numbers of nuclei. The initial number of nuclei varied between 9624 (pseudo–nucleation energy of 3.2 eV) and 165 (pseudo–nucleation energy of 6.0 eV). The curves (Fig. 3a) all show a typical Avrami shape and the logarithmic plots (Fig. 3b)

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(a) recrystallized volume fraction [%]

100 3.2 eV (nucl.)

90 80 70 60 50

6.0 eV (nucl.)

40 30 20 10 0

0

5

10

15

20 25 30 annealing time [s]

35

40

45

(b) 1.5 1

In (In (1/(1 x )))

0.5 0 0.5 1 1.5 2 2.5 1.4

1.8

2.2

2.6 In(t )

3

3.4

3.8

Figure 3. Kinetics for various 3D recrystallization simulations with site saturated nucleation conditions and identical mobility for all grain boundaries. The different curves correspond to different initial numbers of nuclei. The initial number of nuclei varied between 9624 (pseudo– nucleation energy of 3.2 eV) and 165 (pseudo–nucleation energy of 6.0 eV). (a) Avrami diagrams. (b) Logarithmic diagrams showing Avrami exponents between 2.86 and 3.13.

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reveal Avrami exponents between 2.86 and 3.13 which is in very good accord with the analytical value of 3.0 for site saturated conditions. The simulations with a very high initial density of nuclei reveal a more pronounced deviation of the Avrami exponent with values around 2.7 during the beginning of recrystallization. This deviation from the analytical behavior is due to lattice effects: while the analytical derivation assumes a vanishing volume for newly formed nuclei the cellular automaton has to assign one lattice point to each new nucleus. Figure 4 shows the effect of grain boundary mobility on growth selection. While in Fig. 4a all boundaries had the same mobility, in Fig. 4b one grain boundary had a larger mobility than the others (activation energy of the mobility of 1.35 eV instead of 1.40 eV) and consequently grew much faster than the neighboring grains which finally ceased to grow. The grains in this simulation all grew into a heavily deformed single crystal. (a)

temporal evolution

deformed single crystal

growing nucleation front

(b)

temporal evolution

deformed single crystal

growing nucleation front

Figure 4. Effect of grain boundary mobility on growth selection. All grains grow into a deformed single crystal. (a) All grain boundaries have the same mobility. (b) One grain boundary has a larger mobility than the others (activation energy of the mobility of 1.35 eV instead of 1.40 eV) and grows faster than the neighboring grains.

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4.

4.1.

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Examples of Coupling Cellular Automata with Crystal Plasticity Finite Element Models for Predicting Recrystallization Motivation for Coupling Different Spatially Discrete Microstructure and Texture Simulation Methods

Simulation approaches such as the crystal plasticity finite element method or cellular automata are increasingly gaining momentum as tools for spatial and temporal discrete prediction methods for microstructures and textures. The major advantage of such approaches is that they consider material heterogeneity as opposed to classical statistical approaches which are based on the assumption of material homogeneity. Although the average behavior of materials during deformation and heat treatment can sometimes be sufficiently well described without considering local effects, prominent examples exist where substantial progress in understanding and tailoring material response can only be attained by taking material heterogeneity into account. For instance in the field of plasticity the quantitative investigation of ridging and roping or related surface defects observed in sheet metals requires knowledge about local effects such as the grain topology or the form and location of second phases. In the field of heat treatment, the origin of the Goss texture in transformer steels, the incipient stages of cube texture formation during primary recrystallization of aluminum, the reduction of the grain size in microalloyed low carbon steel sheets, and the development of strong {111}uvw textures in steels can hardly be predicted without incorporating local effects such as the orientation and location of recrystallization nuclei and the character and properties of the grain boundaries surrounding them. Although spatially discrete microstructure simulations have already profoundly enhanced our understanding of microstructure and texture evolution over the last decade, their potential is sometimes simply limited by an insufficient knowledge about the external boundary conditions which characterize the process and an insufficient knowledge about the internal starting conditions which are, to a large extent, inherited from the preceding process steps. It is thus an important goal to improve the incorporation of both types of information into such simulations. External boundary conditions prescribed by real industrial processes are often spatially non-homogeneous. They can be investigated using experiments or process simulations which consider spatial resolution. Spatial heterogeneities in the internal starting conditions, i.e., in the microstructure and texture, can be obtained from experiments or microstructure simulations which include spatial resolution.

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Coupling, Scaling and Boundary Conditions

In the present example the results obtained from a crystal plasticity finite element simulation were used to map a starting microstructure for a subsequent discrete recrystallization simulation carried out with a probabilistic cellular automaton. The finite element model was used to simulate a plane strain compression test conducted on aluminum with columnar grain structure to a total logarithmic strain of ε = –0.434. Details about the finite element model are given elsewhere [34, 35, 38, 39]. The values of the state variables (dislocation density, crystal orientation) given at the integration points of the finite element mesh were mapped on the regular lattice of a 2D cellular automaton. While the original finite element mesh consisted of 36 977 quadrilateral elements, the cellular automaton lattice consisted of 217 600 discrete points. The values of the state variables (dislocation density, crystal orientation) at each of the integration points were assigned to the new cellular automaton lattice points which fell within the Wigner–Seitz cell corresponding to that integration point. The Wigner–Seitz cells of the finite element mesh were constructed from cell walls which were the perpendicular bisecting planes of all lines connecting neighboring integration points, i.e., the integration points were in the centers of the Wigner–Seitz cells. In the present example the original size of the specimen which provided the input microstructure to the crystal plasticity finite element simulations gave a lattice point spacing of λm = 61.9 µm. The maximum driving force in the region arising from the stored dislocation density amounted to about 1 MPa. The temperature dependence of the shear modulus and of the Burgers vector was considered in the calculation of the driving force. The grain boundary mobility in the region was characterized by an activation energy of the grain boundary mobility of 1.46 eV and a pre-exponential factor of the grain boundary mobility of m0 = 8.3 × 10−3 m3 /(N s). Together with the scaling length λm = 61.9 µm these data were used for the calculation of the time step t = 1/ν0min and of the local switching probabilities wˆ local. The annealing temperature was 800 K. Large angle grain boundaries were characterized by an activation energy for the mobility of 1.3 eV. Small angle grain boundaries were assumed to be immobile.

4.3.

Nucleation Criterion

The nucleation process during primary static recrystallization has been explained for pure aluminum in terms of discontinuous subgrain growth [40]. According to this model nucleation takes place in areas which reveal high misorientations among neighboring subgrains and a high local driving force

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for curvature driven discontinuous subgrain coarsening. The present simulation approach works above the subgrain scale, i.e., it does not explicitly describe cell walls and subgrain coarsening phenomena. Instead, it incorporates nucleation on a more phenomenological basis using the kinetic and thermodynamic instability criteria known from classical recrystallization theory (see e.g., [40]). The kinetic instability criterion means that a successful nucleation process leads to the formation of a mobile large angle grain boundary which can sweep the surrounding deformed matrix. The thermodynamic instability criterion means that the stored energy changes across the newly formed large angle grain boundary providing a net driving force pushing it forward into the deformed matter. Nucleation in this simulation is performed in accord with these two aspects, i.e., potential nucleation sites must fulfill both, the kinetic and the thermodynamic instability criterion. The used nucleation model does not create any new orientations: at the beginning of the simulation the thermodynamic criterion, i.e., the local value of the dislocation density was first checked for all lattice points. If the dislocation density was larger than some critical value of its maximum value in the sample, the cell was spontaneously recrystallized without any orientation change, i.e., a dislocation density of zero was assigned to it and the original crystal orientation was preserved. In the next step the ordinary growth algorithm was started according to Eqs. (1)–(11), i.e., the kinetic conditions for nucleation were checked by calculating the misorientations among all spontaneously recrystallized cells (preserving their original crystal orientation) and their immediate neighborhood considering the first, second, and third neighbor shell. If any such pair of cells revealed a misorientation above 15◦ , the cell flip of the unrecrystallized cell was calculated according to its actual transformation probability, Eq. (8). In case of a successful cell flip the orientation of the first recrystallized neighbor cell was assigned to the flipped cell.

4.4.

Predictions and Interpretation

Figures 5–7 show simulated microstructures for site saturated spontaneous nucleation in all cells with a dislocation density larger than 50% of the maximum value (Fig. 5), larger than 60% of the maximum value (Fig. 6), and larger than 70% of the maximum value (Fig. 7). Each figure shows a set of four subsequent microstructures during recrystallization. The upper graphs in Figs. 5–7 show the evolution of the stored dislocation densities. The gray areas are recrystallized, i.e., the stored dislocation content of the affected cells was dropped to zero. The lower graphs represent the microtexture images where each color represents a specific crystal orientation.

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(a)

(b)

(c)

(d)

Figure 5. Consecutive stages of a 2D simulation of primary staticrecrystallization in a deformed aluminum polycrystal on the basis of crystal plasticity finite element starting data. The figure shows the change in dislocation density (top) and in microtexture (bottom) as a function of the annealing time during isothermal recrystallization. The texture is given in terms of the magnitude of the Rodriguez orientation vector using the cube component as reference. The gray areas in the upper figures indicate a stored dislocation density of zero, i.e., these areas are recrystallized. The fat white lines in both types of figures indicate grain boundaries with misorientations above 15◦ irrespective of the rotation axis. The thin green lines indicate misorientations between 5◦ and 15◦ irrespective of the rotation axis. The simulation parameters are: 800 K; thermodynamic instability criterion: site-saturated spontaneous nucleation in cells with at least 50% of the maximum occurring dislocation density (threshold value); kinetic instability criterion for further growth of such spontaneous nuclei: misorientation above 15◦ ; activation energy of the grain boundary mobility: 1.46 eV; pre-exponential factor of the grain boundary mobility: m0 = 8.3 × 10−3 m3 /(N s); mesh size of the cellular automaton grid (scaling length): λm = 61.9 µm.

The color level is determined as the magnitude of the Rodriguez orientation vector using the cube component as reference. The fat white lines in both types of figures indicate grain boundaries with misorientations above 15◦ irrespective of the rotation axis. The thin green lines indicate misorientations between 5◦ and 15◦ irrespective of the rotation axis.

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(a)

(b)

(c)

(d)

Figure 6. Parameters like in Fig. 5, but site-saturated spontaneousnucleation occurred in all cells with at least 60% of the maximum occurring dislocation density.

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(b)

(c)

(d)

Figure 7. Parameters like in Fig. 5, but site-saturated spontaneousnucleation occurred in all cells with at least 70% of the maximum occurring dislocation density.

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The incipient stages of recrystallization in Fig. 5 (cells with 50% of the maximum occurring dislocation density undergo spontaneous nucleation without orientation change) reveal that nucleation is concentrated in areas with large accumulated local dislocation densities. As a consequence the nuclei form clusters of similarly oriented new grains (e.g., Fig. 5a). Less deformed areas between the bands reveal a very small density of nuclei. Logically, the subsequent stages of recrystallization (Fig. 5 b–d) reveal that the nuclei do not sweep the surrounding deformation structure freely as described by Avrami– Johnson–Mehl theory but impinge upon each other and thus compete at an early stage of recrystallization. Figure 6 (using 60% of the maximum occurring dislocation density as threshold for spontaneous nucleation) also reveals strong nucleation clusters in areas with high dislocation densities. Owing to the higher threshold value for a spontaneous cell flip nucleation outside of the deformation bands occurs vary rarely. Similar observations hold for Fig. 7 (70% threshold value). It also shows an increasing grain size as a consequence of the reduced nucleation density. The deviation from Avrami–Johnson–Mehl type growth, i.e., the early impingement of neighboring crystals is also reflected by the overall kinetics which differ from the classical sigmoidal curve which is found for homogeneous nucleation conditions. Figure 8 shows the kinetics of recrystallization

100

recrystallized volume fraction [vol.%]

90 80 70 60 50 40 30 50% max. disloc. density

20

60% max. disloc. density

10

70% max. disloc. density

0 0

100

200

300

400

500

600

700

800

annealing time [s]

Figure 8. Kinetics of the recrystallization simulations shown in Figs. 5–7, annealing temperature: 800 K; scaling length λm = 61.9 µm.

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(for the simulations with different threshold dislocation densities for spontaneous nucleation, Figs. 5–7). Al curves reveal a very flat shape compared to the analytical model. The high offset value for the curve with 50% critical dislocation density is due to the small threshold value for a spontaneous initial cell flip. This means that 10% of all cells undergo initial site saturated nucleation. Figure 9 shows the corresponding Cahn–Hagel diagrams. It is found that the curves increasingly flatten and drop with an increasing threshold dislocation density for spontaneous recrystallization. It is an interesting observation in all three simulation series that in most cases where spontaneous nucleation took place in areas with large local dislocation densities, the kinetic instability criterion was usually also well enough fulfilled to enable further growth of these freshly recrystallized cells. In this context one should take notice of the fact that both instability criteria were treated entirely independent in this simulation. In other words only those spontaneously recrystallized cells which subsequently found a misorientation above 15◦ to at least one non-recrystallized neighbor cell were able to expand further. This makes the essential difference between a potential nucleus and a successful nucleus. Translating this observation into the initial deformation microstructure means that in the present example high dislocation densities

interface area between recrystallized and non-recrystallized matter devided by sample volume [cellsize1]

0.020 0.018 0.016 0.014 0.012 0.010 0.008 0.006 50% max. disloc. density

0.004

60% max. disloc. density

0.002

70% max. disloc. density

0.000 0

10

20

30

40

50

60

70

80

90

100

recrystallized volume fraction [%]

Figure 9. Simulated interface fractions between recrystallized and non-recrystallized material for the recrystallization simulations shown in Figs. 5–7, annealing temperature: 800 K; scaling length λm = 61.9 µm.

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2195

and large local lattice curvatures typically occurred in close neighborhood or even at the same sites. Another essential observation is that the nucleation clusters are particularly concentrated in macroscopical deformation bands which were formed as diagonal instabilities through the sample thickness. Generic intrinsic nucleation inside heavily deformed grains, however, occurs rarely. Only the simulation with a very small threshold value of only 50% of the maximum dislocation density as a precondition for a spontaneous energy drop shows some successful nucleation events outside the large bands. But even then nucleation is only successful at former grain boundaries where orientation changes occur naturally. Summarizing this argument means that there might be a transition from extrinsic nucleation such as inside bands or related large scale instabilities to intrinsic nucleation inside grains or close to existing grain boundaries. It is likely that both types of nucleation deserve separate attention. As far as the strong nucleation in macroscopic bands is concerned, future consideration should be placed on issues such as the influence of external friction conditions and sample geometry on nucleation. Both aspects strongly influence through thickness shear localization effects. Another result of relevance is the partial recovery of deformed material. Figures 5d, 6d, and 7d reveal small areas where moving large angle grain boundaries did not entirely sweep the deformed material. An analysis of the state variable values at these coordinates and of the grain boundaries involved substantiates that not insufficient driving forces but insufficient misorientations between the deformed and the recrystallized areas–entailing a drop in grain boundary mobility– were responsible for this effect. This mechanisms is referred to as orientation pinning.

4.5.

Simulation of Nucleation Topology within a Single Grain

Recent efforts in simulating recrystallization phenomena on the basis of crystal plasticity finite element or electron microscopy input data are increasingly devoted to tackling the question of nucleation. In this context it must be stated clearly that mesoscale cellular automata can neither directly map the physics of a nucleation event nor develop any novel theory for nucleation at the sub-grain level. However, cellular automata can predict the topological evolution and competition among growing nuclei during the incipient stages of recrystallization. The initial nucleation criterion itself must be incorporated in a phenomenological form. This section deals with such as an approach for investigating nucleation topology. The simulation was again started using a crystal plasticity finite

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element approach. The crystal plasticity model set-up consisted in a single aluminum grain with face centered cubic crystal structure and 12 {111}110 slip systems which was embedded in a plastic continuum which had the elasticplastic properties of an aluminum polycrystal with random texture. The crystallographic orientation of the aluminum grain in the center was ϕ1 = 32◦ , φ = 85◦ , ϕ2 = 85◦ . The entire aggregate was plane strain deformed to 50% thickness reduction (given as d/d0 , where d is the actual sample thickness and d0 its initial thickness). The resulting data (dislocation density, orientation distribution) were then used as input data for the ensuing cellular automaton recrystallization simulation. The distribution of the dislocation density taken from all integration points of the finite element simulation is given in Fig. 10. Nucleation was initiated as outlined in detail in Section 4.3, i.e., each lattice point which had a dislocation density above some critical value (500 × 1013 m−2 in the present case, see Fig. 10) of the maximum value in the sample was

25000

Λd /d 50% FCC, orentation ϕ1 32˚, φ85˚, ϕ285˚

22500

10000

20000

8000

frequency [1]

17500

6000

15000

4000

12500

2000

10000

0 350

400

450

500

550

600

650

700

7500 5000 2500 0 0

100

200

300 400 500 600 700 dislocation density [ 1013 m2 ]

800

900

1000

Figure 10. Distribution of the simulated dislocation density in a deformed aluminum grain embedded in a plastic aluminum continuum. The simulation was performed by using a crystal plasticity finite element approach. The set-up consisted of a single aluminum grain (orientation: ϕ1 = 32◦ , φ = 85◦ , ϕ2 =85◦ in Euler angles) with face centered cubic crystal structure and 12 {111}110 slip systems which was embedded in a plastic continuum which had the elasticplastic properties of an aluminum polycrystal with random texture. The sample was plane strain deformed to 50% thickness reduction. The resulting data (dislocation density, orientation distribution) were used as input data for a cellular automaton recrystallization simulation.

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spontaneously recrystallized without orientation change. In the ensuing step the growth algorithm was started according to Eqs. (1)–(11), i.e., a nucleus could only expand further if it was surrounded by lattice points of sufficient misorientation (above 15◦ ). In order to concentrate on recrystallization in the center grain the nuclei could not expand into the surrounding continuum material. Figures 11a–c show the change in dislocation density during recrystallization (Fig. 11a: 9% of the entire sample recrystallized, 32.1 s; Fig. 11b: 19% of the entire sample recrystallized, 45.0 s; Fig. 11c: 29.4% of the entire sample recrystallized, 56.3 s). The color scale marks the dislocation density of each lattice point in units of 1013 m−2 . The white areas are recrystallized. The surrounding blue area indicates the continuum material in which the grain is embedded (and into which recrystallization was not allowed to proceed). Figures 12a–c show the topology of the evolving nuclei without coloring the as-deformed volume. All recrystallized grains are colored indicating their crystal orientation. The non-recrystallized material and the continuum surrounding the grain are colored white. Figure 13 shows the volume fractions of the growing nuclei during recrystallization as a function of annealing time (800 K). The data reveal that two groups of nuclei occur. The first class of nuclei shows some growth in the beginning but no further expansion during the later stages of the anneal. The second class of nuclei shows strong and steady growth during the entire recrystallization time. One could refer to the first group as non-relevant nuclei while the second group could be termed relevant nuclei. The reasons of such a spread in the evolution of nucleation topology after their initial formation are nucleation clustering, orientation pinning, growth selection, or driving force selection phenomena. Nucleation clustering means that areas which reveal localization of strain and misorientation produce high local nucleation rates. This entails clusters of newly formed nuclei where competing crystals impinge on each other at an early stage of recrystallization so that only some of the newly formed grains of each cluster can expand further. Orientation pinning is an effect where not insufficient driving forces but insufficient misorientations between the deformed and the recrystallized areas – entailing a drop in grain boundary mobility – are responsible for the limitation of further growth. In other words some nuclei expand during growth into areas where the local misorientation drops below 15◦ . Growth selection is a phenomenon where some grains grow significantly faster than others due to a local advantage originating from higher grain boundary mobility such as shown in Fig. 4b. Typical examples are the 40◦ 111 rotation relationship in aluminum or the 27◦ 110 rotation relationship in iron–silicon which are known to have a growth advantage (e.g., Ref. [40]). Driving force selection is a phenomenon where some grains grow significantly faster than others due to a local advantage in driving force (shear bands, microbands, heavily deformed grain).

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(b)

(c)

Figure 11. Change in dislocation density during recrystallization (800 K).The color scale indicates the dislocation density of each lattice point in units of 1013 m−2 . The white areas are recrystallized. The surrounding blue area indicates the continuum material in which the grain is embedded. (a) 9% of the entire sample recrystallized, 32.1 s; (b) 19% of the entire sample recrystallized, 45.0 s; (c) 29.4% of the entire sample recrystallized, 56.3 s.

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(a)

(b)

(c)

Figure 12. Topology of the evolving nuclei of the microstructure given inFig. 11 without coloring the as-deformed volume. All newly recrystallized grains are colored indicating their crystal orientation. The non-recrystallized material and the continuum surrounding the grain are colored white. (a) 9% of the entire sample recrystallized, 32.1 s; (b) 19% of the entire sample recrystallized, 45.0 s; (c) 29.4% of the entire sample recrystallized, 56.3 s.

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D. Raabe 12000

volume of new grains [ cell3 ]

10000

8000

6000

4000

2000

0 0

10

20

30 annealing time [s]

40

60

80

Figure 13. Volume fractions of the growing nuclei in Fig. 11 during recrystallization as a function of annealing time (800 K).

5.

Conclusions and Outlook

A review was given about the fundamentals and some applications of cellular automata in the field of microstructure research. Special attention was placed on reviewing the fundmentals of mapping rate formulations for interfaces and driving forces on cellular grids. Some applications were discussed from the field of recrystallization theory. The future of the cellular automaton method in the field of mesoscale materials science lies most likely in the discrete simulation of equilibrium and non-equilibrium phase transformation phenomena. The particular advantage of automata in this context is their versatility with respect to the constitutive ingredients, to the consideration of local effects, and to the modification of the grid structure and the interaction rules. In the field of phase transformation simulations the constitutive ingredients are the thermodynamic input data and the kinetic coefficients. Both sets of input data are increasingly available from theory and experiment rendering cellular automaton simulations more and more realistic. The second advantage, i.e., the incorporation of local effects will improve our insight into cluster effects, such as arising from the spatial competition of expanding neighboring spheres already in the incipient stages of transformations. The third advantage, i.e., the flexibility of automata with respect to the grid structure and the interaction rules is probably the most

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important aspect for novel future applications. By introducing more global interaction rules (in addition to the local rules) and long-range or even statistical elements in addition to the local rules for the state update might establish cellular automata as a class of methods to solve some of the intricate scale problems that are often encountered in the materials sciences. It is conceivable that for certain mesoscale problems such as the simulation of transformation phenomena in heterogeneneous materials in dimensions far beyond the grain scale cellular automata can occupy a role between the discrete atomistic approaches and statistical Avrami-type approaches. The mayor drawback of the cellular automaton method in the field of transformation simulations is the absence of solid approaches for the treatment of nucleation phenomena. Although basic assumptions about nucelation sites, nucleation rates, and nucelation textures can often be included on an empirical basis as a function of the local values of the state variables, intrinsic physically based phenomenological concepts such as available to a certain extent in the Ginzburg–Landau framework (in case of the spinodal mechanism) are not yet available for automata. It might hence be beneficial in future work to combine Ginzburg–Landau-type phase field approaches with the cellular automaton method. For instance the (spinodal) nucleation phase could then be treated with a phase field method and the resulting microstructure could be further treated with a cellular automaton simulation.

References [1] J. von Neumann, “The general and logical theory of automata,” In: W. Aspray and A. Burks (eds.), Papers of John von Neumann on Computing and Computer Theory, vol. 12 in the Charles Babbage Institute Reprint Series for the History of Computing, MIT Press, Cambridge, 1987, 1963. [2] S. Wolfram, Theory and Applications of Cellular Automata, Advanced Series on Complex Systems, selected papers 1983–1986, vol. 1, World Scientific Publishing Co. Pte. Ltd, Singapore, 1986. [3] S. Wolfram, “Statistical mechanics of cellular automata,” Rev. Mod. Phys., 55, 601– 622, 1983. [4] M. Minsky, Computation: Finite and Infinite Machines, Prentice-Hall, Englewood Cliffs, NJ, 1967. [5] J.H. Conway, Regular Algebra and Finite Machines, Chapman & Hall, London, 1971. [6] D. Raabe, Computational Materials Science, Wiley-VCH, Weinheim, 1998. [7] H.W. Hesselbarth and I.R. G¨obel, “Simulation of recrystallization by cellular automata,” Acta Metall., 39, 2135–2144, 1991. [8] C.E. Pezzee and D.C. Dunand, “The impingement effect of an inert, immobile second phase on the recrystallization of a matrix,” Acta Metall., 42, 1509–1522, 1994. [9] R.K. Sheldon and D.C. Dunand, “Computer modeling of particle pushing and clustering during matrix crystallization,” Acta Mater., 44, 4571–4582, 1996. [10] C.H.J. Davies, “The effect of neighbourhood on the kinetics of a cellular automaton recrystallisation model,” Scripta Metall. et Mater., 33, 1139–1154, 1995.

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[11] V. Marx, D. Raabe, and G. Gottstein, “Simulation of the influence of recovery on the texture development in cold rolled BCC-alloys during annealing,” In: N. Hansen, D. Juul Jensen, Y.L. Liu, and B. Ralph (eds.), Proceedings 16th RISøInt. Sympos. on Mat. Science: Materials: Microstructural and Crystallographic Aspects of Recrystallization, RISø Nat. Lab, Roskilde, Denmark, pp. 461–466, 1995. [12] D. Raabe, “Cellular automata in materials science with particular reference to recrystallization simulation,” Ann. Rev. Mater. Res., 32, 53–76, 2002. [13] V. Marx, D. Raabe, O. Engler, and G. Gottstein, “Simulation of the texture evolution during annealing of cold rolled bcc and fcc metals using a cellular automaton approach,” Textures Microstruct., 28, 211–218, 1997. [14] V. Marx, F.R. Reher, and G. Gottstein, “Stimulation of primary recrystallization using a modified three-dimensional cellular automaton,” Acta Mater., 47, 1219–1230, 1998. [15] C.H.J. Davies, “Growth of nuclei in a cellular automaton simulation of recrystallisation,” Scripta Mater., 36, 35–46, 1997. [16] C.H.J. Davies and L. Hong, “Cellular automaton simulation of static recrystallization in cold-rolled AA1050,” Scripta Mater., 40, 1145–1152, 1999. [17] D. Raabe, “Introduction of a scaleable 3D cellular automaton with a probabilistic switching rule for the discrete mesoscale simulation of recrystallization phenomena,” Philos. Mag. A, 79, 2339–2358, 1999. [18] D. Raabe and R. Becker, “Coupling of a crystal plasticity finite element model with a probabilistic cellular automaton for simulating primary static recrystallization in aluminum,” Modell. Simul. Mater. Sci. Eng., 8, 445–462, 2000. [19] D.Raabe, “Yield surface simulation for partially recrystallized aluminum polycrystals on the basis of spatially discrete data,” Comput. Mater. Sci., 19, 13–26, 2000. [20] D. Raabe, F. Roters, and V. Marx, “Experimental investigation and numerical simulation of the correlation of recovery and texture in bcc metals and alloys,” Textures Microstruct., 26–27, 611–635, 1996. [21] M.B. Cortie, “Simulation of metal solidification using a cellular automaton,” Metall. Trans. B, 24, 1045–1052, 1993. [22] S.G.R. Brown, T. Williams, and JA. Spittle, “A cellular automaton model of the steady-state free growth of a non-isothermal dendrite,” Acta Metall., 42, 2893–2906, 1994. [23] C.A. Gandin and M. Rappaz, “A 3D cellular automaton algorithm for the prediction of dendritic grain growth,” Acta Metall., 45, 2187–2198, 1997. [24] C.A. Gandin, “Stochastic modeling of dendritic grain structures,” Adv. Eng. Mater., 3, 303–306, 2001. [25] C.A. Gandin, J.L. Desbiolles, and P.A. Thevoz, “Three-dimensional cellular automaton-finite element model for the prediction of solidification grain structures,” Metall. Mater. Trans. A, 30, 3153–3172, 1999. [26] J.A. Spittle and S.G.R. Brown, “A cellular automaton model of steady-state columnardendritic growth in binary alloys,” J. Mater. Sci., 30, 3989–3402, 1995. [27] S.G.R. Brown, G.P. Clarke, and A.J. Brooks, “Morphological variations produced by cellular automaton model of non-isothermal free dendritic growth,” Mater. Sci. Technol., 11, 370–382, 1995. [28] J.A. Spittle and S.G.R. Brown, “A 3D cellular automation model of coupled growth in two component systems,” Acta Metallurgica, 42, 1811–1820, 1994. [29] M. Kumar, R. Sasikumar, P. Nair, and R. Kesavan, “Competition between nucleation and early growth of ferrite from austenite-studies using cellular automaton simulations,” Acta Mater., 46, 6291–6304, 1998.

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[30] S.G.R. Brown, “Simulation of diffusional composite growth using the cellular automaton finite difference (CAFD) method,” J. Mater. Sci., 33, 4769–4782, 1998. [31] T. Yanagita, “Three-dimensional cellular automaton model of segregation of granular materials in a rotating cylinder,” Phys. Rev. Lett., 3488–3492, 1999. [32] E.M. Koltsova, I.S. Nenaglyadkin, A.Y. Kolosov, and V.A. Dovi, “Cellular automaton for description of crystal growth from the supersaturated unperturbed and agitated solutions,” Rus. J. Phys. Chem., 74, 85–91, 2000. [33] J. Geiger, A. Roosz, and P. Barkoczy, “Simulation of grain coarsening in two dimensions by cellular-automaton,” Acta Mater., 49, 623–629, 2001. [34] Y. Liu, T. Baudin, and R. Penelle, “Simulation of grain growth by cellular automata,” Scripta Mater., 34, 1679–1686, 1996. [35] T. Karapiperis, “Cellular automaton model of precipitation/sissolution coupled with solute transport,” J. Stat. Phys., 81, 165–174, 1995. [36] M.J. Young and C.H.J. Davies, “Cellular automaton modelling of precipitate coarsening,” Scripta Mater., 41, 697–708, 1999. [37] O. Kortluke, “A general cellular automaton model for surface reactions,” J. Phys. A, 31, 9185–9198, 1998. [38] G. Gottstein and L.S. Shvindlerman, Grain Boundary Migration in Metals– Thermodynamics, Kinetics, Applications, CRC Press, Boca Raton, 1999. [39] R.C. Becker, “Analysis of texture evoltuion in channel die compression-I. Effects of grain interaction,” Acta Metall. Mater., 39, 1211–1230, 1991. [40] R.C. Becker and S. Panchanadeeswaran, “Effects of grain interactions on deformation and local texture in polycrystals,” Acta Metall. Mater., 43,2701–2719, 1995. [41] F.J. Humphreys and M. Hatherly, Recrystallization and Related Annealing Phenomena, Pergamon Press, New York, 1995.

7.8 MODELING COARSENING DYNAMICS USING INTERFACE TRACKING METHODS John Lowengrub University of California, Irvine, California, USA

In this paper, we will discuss the current state-of-the-art in numerical models of coarsening dynamics using a front-tracking approach. We will focus on coarsening during diffusional phase transformations. Many important structural materials such as steels, aluminum and nickel-based alloys are products of such transformations. Diffusional transformations occur when the temperature of a uniform mixture of materials is lowered into a regime where the uniform mixture is unstable. The system responds by nucleating second phase precipitates (e.g., crystals) that then evolve diffusionally until the process either reaches equilibrium or is quenched by further reducing the temperature. The diffusional evolution consists of two phases – growth and coarsening. Growth occurs in response to a local supersaturation in the primary (matrix) phase and a local mass balance relation is satisfied at each precipitate interface. Coarsening occurs when a global mass balance is achieved and involves a dynamic rearrangement of the fixed total mass in the system so as to minimize a global energy. Typically, the global energy consists of the surface energy. If the transformation occurs between components in the solid state, there is also an elastic energy that arises due to the presence of a misfit stress between the precipitates and the matrix as their crystal structures are often slightly different. Diffusional phase transformations are responsible for producing the material microstructure, i.e., the detailed arrangement of distinct constituents at the microscopic level. The details of the microstructure greatly influence the material properties of the alloy (i.e., stiffness, strength, and toughness). In many alloys, an in situ coarsening process can occur at high temperatures in which a dispersion of very small precipitates evolves to a system consisting of a few very large precipitates in order to decrease the surface energy of the system. This coarsening severely degrades the properties of the alloy and can lead to in service failures. The details of this coarsening process depend strongly 2205 S. Yip (ed.), Handbook of Materials Modeling, 2205–2222. c 2005 Springer. Printed in the Netherlands.

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on the elastic properties and crystal structure of the alloy components. Thus, one of the goals of this line of research is to use elastic stress to control the evolution process so as to achieve desirable microstructures. Numerical simulations of coarsening two-phase microstructures have followed two directions – interface capturing and interface tracking. In capturing methods, the precipitate/matrix interfaces are implicitly determined through an auxiliary function that is introduced to delineate between the precipitate and matrix phases. Examples include phase-field and level-set methods. Typically, sharp interfaces are smoothed out and the elasticity and diffusion systems are replaced by mesoscopic approximations that mimic the true field equations together with interface jump conditions. These methods have the advantage that topological changes such as precipitate coalescence and splitting are easily described. A disadvantage of this approach is that the results can be sensitive to the parameters that determine the thickness of the interfacial regions and care needs to be taken reconcile the results using sharp interfaces and tracking methods. In interface tracking methods, which are the subject of this article, a specific mesh is introduced to approximate the interface. The evolution of the interface is tracked by explicitly evolving the interface mesh in time. Examples include boundary integral, immersed interface [1], ghost-fluid [2], front-tracking [3, 4]. In boundary integral, immersed interface and ghost-fluid methods, for example, the interfaces remain sharp and the true field equations and jump conditions are solved. These methods have the advantage that high order accurate solutions can be obtained. Thus, in addition to their intrinsic value, results from these algorithms can also be used as benchmarks to validate interface-capturing methods. Boundary integral methods have the additional advantage that the field equations and jump conditions are mapped to the precipitate/matrix interfaces thereby reducing the dimensionality of the problem. However, boundary integral methods typically apply only in the limited situation where the physical domains and parameters are piecewise homogeneous. The other tracking methods listed above do not suffer from this difficulty although they are generally not as accurate as boundary integral methods. A general disadvantage of the tracking approach is that ad-hoc cut-and-connect procedures are required to handle changes in interface topologies. In this article, we will focus primarily on a description of the state-of-theart in boundary integral methods.

1.

Coarsening

One of the central assumptions of mathematical models of coarsening is that the system evolves so as to decrease the total energy. This energy consists of an interfacial part, associated with the precipitate/matrix interfaces and a

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bulk part due to the elasticity of the constituent materials. In the absence of the elastic stress, precipitates tend to be roughly spherical and interfacial area is reduced by the diffusion of matter from regions of high interfacial curvature to regions of low curvature. During coarsening, this leads to a survival of the fattest since large precipitates grow at the expense of small ones. This coarsening process may severely degrade the properties of the alloy. In the early 1960s, an asymptotic theory, now referred to as the LSW theory, was developed by Lifshitz and Slyosov [5], and Wagner [6] to predict the temporal power law of precipitate growth and in particular the scaling at long times of the precipitate radius distribution. In this LSW theory, only surface energy is considered and it is found that the average precipitate radius R ∼ t 1/3 at long times. The LSW theory has two major restrictions, however. First, precipitates are assumed to be circular (spherical in 3-D) and second, the theory is valid only in the zero (precipitate) volume fraction limit. Extending the results of LSW to account for non-spherical precipitates, finite volume fractions and elastic interactions has been a subject of intense research interest and is one of the primary reasons for the development of accurate and efficient numerical methods to study microstructure evolution. See the recent reviews by Johnson and Voorhees [7], Voorhees [7] and Thornton et al. [8].

2.

Governing Equations

For the purposes of illustration, let us focus a two-phase microstructure in a binary alloy. We further assume that the matrix phase M extends to infinity (or in 2D may be contained in a large domain ∞ ), while the precipitate phase P consists of isolated particles occupying a finite volume. The interface between the two phases is a collection of closed surfaces . The evolution of the precipitate matrix interface is controlled by diffusion of matter across the interface. Assuming quasi-static diffusion, the normalized composition c is governed by Laplace’s equation c = 0

(1)

in both phases. The composition on a precipitate-matrix interface is given by the Gibbs–Thomson boundary condition [9] c = −(τ I + ∇n ∇n τ ) : K − Zg el − λVn ,

(2)

where τ = τ (n) is the non-dimensional anisotropic surface tension, n is the normal vector directed towards M , I is the identity tensor, (∇n ∇n τ )i j = ∂ 2 τ/ ∂n i ∂n j ,

K=−

L M N s1 s1 + √ (s1 s2 + s2 s1 ) + s2 s2 E F EG

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is the curvature tensor where s1 and s2 are tangent vectors to the interface and the definitions of L, E, M, G, F and N depend on the interface parametrization and can be found in standard differential geometry texts [10]. Note that ˆ = 2H where H is the mean curvature. In addition, Z characterizes the tr(E) relative strength of the surface and elastic energies, g el is the elastic energy density (defined below), Vn is the normal velocity of the precipitate/matrix interface and λis a non-dimensional linear kinetic coefficient. Roughly speaking, this boundary condition reflects the idea that changing the shape of a precipitate changes the energy of the system both through the additional surface energy, (τ I + ∇n ∇n τ ) : K, and also through the change in elastic energy of the system, Zg el . We note that the composition is normalized differently in the precipitate and matrix, so that the normalized composition is continuous across the interface; the actual dimensional composition undergoes a jump. The normal velocity is given by the flux balance Vn = k

∂c ∂c − , ∂n P ∂n M

(3)

and (∂c/∂n) P and (∂c/∂n) M denote the values of normal derivative of c evaluated on the precipitate side and the matrix side of the interface, respectively, and k is the ratio of thermal diffusivities. Two different far-field conditions for the diffusion problem can be posed. In the first, the mass flux J into the system is specified: 1 J= 4π

1 Vn d = 4π

∂∞

∂c d∂∞ , ∂n

(4)

where ∞ is a large domain containing all the precipitates. As a second, alternative boundary condition, the far-field composition c∞ is specified lim c(x) = c∞ .

|x|→∞

(5)

In 2D, the limit in Eq. (5) is taken only to ∂∞ since c diverges logarithmically at infinity (see the 2D Green’s function below). Since the elastic energy density g el appears in the Gibbs–Thomson boundary condition (1), one must solve for the elastic fields in the system before finding the diffusion fields. The elastic fields arise if there is a misfit strain, denoted by ε T between the precipitate and matrix. This misfit is taken into account through the constitutive relations between the stress σi j and strain εi j . These are σiPj = CiPj kl εlkP − εlkT in the precipitate and σiMj = CiMj kl εlkM in the matrix, where we have taken the matrix lattice as the reference. The superscripts P and M refer to the values in the precipitate and matrix respectively. The elastic stiffness tensor Ci j kl may be different in the matrix and precipitate (elastically inhomogeneous) and may also reflect different material symmetries of the two phases.

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The equations of elastic equilibrium require that σi j, j = 0,

(6)

in both phases (in the absence of body forces). We also assume the interfaces are coherent, so the displacement u (i.e., εi j =(u i, j +u j,i )/2) and the traction t (i.e., ti = σi j n j ) are continuous across them. For simplicity, we suppose that the far-field tractions and displacements vanish. Finally, the elastic energy density g el is given by 1 P P σi j εi j − εiTj − σiMj εiMj + σiMj εiMj − εiPj . (7) 2 Finally, the total energy of the system is the sum of the surface and elastic energies

g el =

Wtot = Ws + Wel . where

Ws =

(8) 

Z τ (n) d, and Wel =  2

σiPj εiPj − εiTj d+

P

 

σiMj εiMj d.

M

(9) For details on the isotropic formulation, derivation and nondimensionalization, see Li et al. [11], the review articles [8, 12] and the references therein.

3.

The Boundary Integral Formulation

We first consider the diffusion problem. If the interface kinetics λ > 0, then a single-layer potential can be used. That is, the composition is given by

c(x) =

σ (x ) G(x − x ) d(x ) + c¯∞ ,

(10)

where σ (x) is the single-layer potential, G(x)is the Green’s function (i.e., 2D: G(x) = (1/2π ) log |x|, 3D: G(x) = (1/4π |x|) and c¯∞ is a constant. Then, taking the limit as x → , and using Eq. (2), we get the Fredholm boundary integral equation

σ (x ) G(x − x ) d(x ) + λV n + c¯∞

(11)

where the normal velocity Vn is related to σ (x). In fact, if the ratio of diffusivities k = 1, then Vn = σ (x) and the equation is a 2nd kind Fredholm integral

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equation. See [13]. For simplicity, let us suppose this is the case. Then, if the flux is specified, Eq. (11) is solved together with Eq. (4) to determine Vn and c¯∞ . In 3D, if far-field condition (5) is imposed, then c¯∞ = c∞ . In 2D, if (5) is imposed, then another single layer potential must be introduced at the far-field boundary ∂∞ [14]. If the interface kinetics λ = 0, then a double-layer potential should be used: c(xi ) =

µi (x )

∂G np (xi − x )d(x ) + k=1 Ak G(xi − Sk ), ∂n

(12)

in each domain i where i = p, m, and n p is the number of precipitates and Sk is a point inside the kth precipitate. In the limit x → leads to the system of 2nd kind Fredholm equations

µi (x )

∂G µi np (xi − x ) d(x ) + k=1 Ak G(xi − Sk ) ± ∂n 2

= − (τ I + ∇n ∇n τ ) : K − Zg el ,

(13)

where the plus sign is taken when i = m [13]. The Ak are determined from the equations

µi (x) d(x ) = 0,

for i = 1, n p − 1,

and

n

p k=1 Ak = J.

The normal velocity Vn is obtained by taking the normal derivative of Eq. (12), taking care to treat the singularity of the Green’s function [13], and thus depends on µi (x ). Equation (13) is then solved together with the far-field conditions in either Eq. (4) or (5) to obtain µi (x ) and c¯∞ and Vn . We note that in 3D, we have recently found that a vector potential formulation [15] rather than a dipole formulation gives better numerical accuracy for computing Vn in this case (Pham, Lowengrub, Nie, Cristini, in preparation). Finally, once Vn is known, the interface is updated by n•

dx = Vn . dt

(14)

To actually solve the boundary integral equations, the elastic energy density g el must be determined first. This requires the solution of the elasticity equations. The boundary integral equations for the continuous displacement field u(x), and traction field t(x) on the interface involve Cauchy-principal-value

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integrals over the interface. The equations can, using a direct formulation, be written as

(u i (y) − u i (x))Ti Pj k (y − x)n k (y) d(y) −

=

ti (y)G iPj k (y − x) d(y)

tiT (y)G iPj k (y

− x) d(y),

(15)

and u i (x) −

(u i (y) − u i (x))Ti M j k (y − x)n k (y) d(y)

−

ti (y)G iMj k (y − x) d(y) = 0,

(16)

where Ti j k and G ikj are the Green’s functions associated with the traction T and displacement respectively and tiT = CiPj kp εkp n j is the misfit traction. For isotropic elasticity, the Green’s functions are given by the Kelvin solution. For general 3D anisotropic materials, the Green’s functions cannot be written explicitly and are formulated in terms of line integrals. In 2D, explicit formulas exist for the Green’s functions. See for example [16, 17]. From the components of the displacements and tractions, the elastic energy density g el can be calculated [12].

4.

Numerical Implementation

The numerical procedure to simulate the evolution is as follows. Given the precipitate shapes, the elasticity Eqs. (15) and (16) are solved and the elastic energy g el is determined. Then diffusion equation is solved, the normal velocity is calculated and the interfaces are advanced one step in time. Precipitates whose volume falls below a certain tolerance are removed from the simulation. In 2D, very efficient and spectrally numerical methods have been developed to solve this problem [12]. The integrals with smooth integrands are discretized with spectral accuracy using the trapezoid rule. The Cauchy principal value integrals are discretized with spectral accuracy using the alternating point trapezoid rule. The fast multipole method [18] is used to evaluate the discrete sums in O(N ) work where N is the total number of collocation points on all the interfaces. Further efficiency is gained by neglecting particle–particle interactions if the particles are well-separated. The iterative method GMRES is then used to solve the discrete nonsymmetric, non-definite elasticity and diffusion matrix systems. The surface tension introduces a severe third order time step constraint for stability: t ≤ Cs 3 where C is a constant and s is the minimum spacing in

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arclength along all the interfaces. To overcome this difficulty, Hou, Lowengrub and Shelley [12] performed a mathematical analysis of the equations of motion at small length-scales (the “small-scale decomposition”). This analysis shows that when the equations of motion are properly formulated, surface tension acts through a linear operator at small length-scales. This contribution, when combined with a special reference frame in which the collocation points remain equally spaced in arclength, can then be treated implicitly and efficiently in a time-integration scheme, and the high-order constraints removed. In 3D, efficient algorithms have been recently developed by Li et al. [11] and Cristini and Lowengrub [19]. In these approaches, the surfaces are discretized using an adaptive surface triangulated mesh [20]. As in 2D, the integral equations are solved using the collocation method and GMRES. In Li et al. [11], local quadratic Lagrange interpolation is used to represent field quantities (i.e., u, t, Vn , and the position of the interface x ) in triangle interiors. The normal vector is derived from the local coordinates using the Lagrange interpolants of the interface position. The curvature is determined by performing a local quadratic fit to the triangulated surface. This combination was found to yield the best accuracy for a given resolution. On mesh triangles where the integrand is singular, a nonlinear change of variables (Duffy’s transformation) is used to map the singular triangle to a unit square and to remove the 1/r divergence of the integrand. For triangles in a region close to the singular triangle, the integrand is nearly singular, and, so, each of these triangles is divided into four smaller triangles, and a high-order quadrature is used on each subtriangle individually. On all other mesh triangles, the highorder quadrature is used to approximate the integrals. In Cristini and Lowengrub [19], there are no effects of elasticity (Z = 0) the collocation method is used to solve the diffusion integral equation together with GMRES and the nonlinear Duffy transformation to remove the singularity of the integrand in the singular triangle. Away from the singular triangle, the trapezoid rule is used and no interpolations are used to represent the field quantities in triangle interiors. As in Li et al., the curvature is still determined by performing a local quadratic fit to the triangulated surface. In both Li et al., and Cristini and Lowengrub, a second-order Runge–Kutta method is used to advance the triangle nodes. The time-step size is proportional to the smallest diameter of the triangular elements raised to the 3/2 power:t = Ch 3/2 . This scaling is due to the fact that the adaptive mesh uniformly resolves the solid angle. Since the shape of the precipitate can change substantially during its evolution, one of the keys to the success of these algorithms is the use of the adaptive-mesh refinement algorithm developed originally by Cristini, Blawzdzieweicz, and Loewenberg [20]. In this algorithm, the solid angle is uniformly resolved throughout the simulation using the following local-mesh restructuring operations to achieve an optimal mesh density: grid equilibration,

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edge-swapping, and node addition and subtraction. This results in a density of node points that is proportional to the maximum of the curvature (in absolute value), so that grid points cluster in highly curved regions of the interface. Further, each of the mesh triangles is nearly equilateral. Finally, to further increase efficiency, a parallelization algorithm is implemented for the diffusion and elasticity solvers. The computational strategy for the parallelization is similar to the one designed for the microstructural evolution in 2D elastic media [12]. A new feature of the algorithm implemented by Li et al. is that the diffusion and elasticity matrices are also divided among the different processors in order to reduce the amount of memory required on each individual processor.

5.

Two-dimensional Results

The state-of-the-art in 2D simulations of purely diffusional evolution in the absence of elastic stress (Z = 0) is the work of [21]. In metallic alloy systems, this corresponds to simulating systems of very small precipitates where the surface energy dominates the elastic energy. Using the methods described above, Akaiwa and Meiron performed simulations containing over 5000 precipitates. Akaiwa and Meiron divided the computational domain into subdomains each containing 50–150 precipitates. Inside each sub-domain, the full diffusion field is computed. The influence of particles outside each subdomain is restricted to only involve those lying within a distance of 6–7 times the average precipitate radius from the sub-domain. This was found to give at most a 1% error in the diffusion field and significantly reduces the computational cost. In Fig. 1, two snapshots of a typical simulation are shown at the very late stages of coarsening. In this simulation, the precipitate area fraction is 0.5 and periodic boundary conditions are applied. In Fig. 1(left), there are approximately 130 precipitates remaining, while in Fig. 1(right) there are only approximately 70 precipitates left. Note that there is no discernible alignment of precipitates. Further, as the system coarsens, the typical shape of a precipitate shows significant deviation from a circle. The simulation results of Akaiwa and Meiron agree with the classical Lifshitz–Slyozov–Wagner (LSW) theory in which the average precipitate radius R is predicted to scale as R ∝ t 1/3 at large times t. It was found that certain statistics, such as the particle size distribution functions, are insensitive to the non-circular particle shapes at even at moderate volume fractions. Simulations were restricted to volume fractions less than 0.5 due to the large computational costs associated with refining the space and time scales to resolve particle-particle near contact interactions at larger volume fractions. The current state of the art in simulating diffusional evolution in homogeneous, anisotropic elastic media is the recent work of [22] who studied alloys

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Figure 1. The late stages of coarsening in the absence of elastic forces (Z = 0). Left: Moderate time; Right: Late time. After [21]. Reproduced with permission.

with cubic symmetry. In metallic alloys, such a system can be considered as a model for nickel–aluminum alloys. In the homogeneous case, one need not solve Eqs. (15)–(16). Instead, the derivatives of the displacement field and hence the elastic energy density g el due to a misfitting precipitate may be evaluated directly from the Green’s function tensor via the boundary integral [22]

u j,k (x) = Ci j + Ci j 22

gi j,k (x, x )n l (x )d(x ),

(17)

where the misfit is a unit dilatation and x is either in the matrix or precipitate and Ci j kl is the stiffness tensor. Using the methods described above together with a fast summation method to calculate the integral in Eq. (17), Akaiwa, Thornton and Voorhees, 2001 have performed simulations involving over 4000 precipitates. See Fig. 2 for results with isotropic surface tension and dilatational misfits. The value of Z is allowed to vary dynamically through an average precipitate radius. Thus, as precipitates coarsen and grow larger, Z increases correspondingly. The initial volume fraction of precipitates is 0.1. Thornton, Akaiwa and Voorhees find that the morphological evolution is significantly different in the presence of elastic stress. In particular, large-scale alignment of particles is seen in the 100 and 010 directions during the evolution process. In addition, there is significant shape dependence as nearly circular precipitates are seen at small Z and as Z increases, precipitates become squarish and then rectangular. It is found that in the elastically homogeneous system, elastic stress does not modify the 1/3 temporal exponent of the LSW coarsening law even though the precipitate morphologies are far from circular. Surprisingly, as long as the

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Figure 2. Coarsening in homogeneous, cubic elasticity. The volume fraction is 10%. The left column shows the computational domain, while the right column is scaled with the average particle size. After Thornton, Akaiwa and Voorhees, 2001. Reproduced with permission.

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shapes remain fourfold symmetric, the kinetics (coefficient of temporal factor) remains unchanged also. It is only when a majority of the particles have a two-fold rectangular shape that the coarsening kinetics changes [23]. The inhomogeneous elasticity problem is much more difficult to solve than the homogeneous problem because in the inhomogeneous case, the integral Eqs. (15)–(16) must be solved in order to obtain the inhomogeneous elastic fields and the elastic energy density g el . For this reason, the state of theory and simulations are less well-developed for the inhomogeneous case compared to the homogeneous problem. The current state-of-the-art in simulating microstructure evolution in inhomogeneous, anisotropic elastic media is the work of Leo, Lowengrub and Nie 2000. Although the system (15), (16) is a Fredholm equation of mixed type with smooth, logarithmic, and Cauchy-type kernels, it was shown by Leo, Lowengrub and Nie 2000, in the anisotropic case, that the system may be transformed directly to a second kind Fredholm system with smooth kernels. The transformation relies on an analysis of the equations at small spatial scales. Leo, Lowengrub and Nie, 2000 found that even small elastic inhomogeneities may have a strong effect on precipitate evolution in systems with small numbers of precipitates. For instance, in systems where the elastic constants of the precipitates are smaller than those of the matrix (soft precipitates), the precipitates move toward each other. In the opposite case (hard precipitates), the precipitates tend to repel one another. The rate of approach or repulsion depends on the amount of inhomogeneity. Anisotropic surface energy may either enhance or reduce this effect. The evolutions of two sample inhomogeneous systems in 2D are shown in Fig. 3. The solid curves correspond to Ni3 Al precipitates (soft, elastic constants less than the Ni matrix) and the dashed curves correspond to Ni3 Si precipitates (hard, elastic constants larger than the Ni matrix). In both cases, the matrix is Ni. Note that only the Ni3 Si precipitates are shown at time t = 20.09 for reasons explained below. From a macroscopic point of view, there seems to be little difference in the results of the two simulations over the times considered. The precipitates become squarish at very early times and there is only a small amount of particle translation. One can observe that the upper and lower two relatively large pairs of precipitates tend to align along the horizontal direction locally. The global alignment of all precipitates on the horizontal and vertical directions appears to occur on a longer time scale. On the time scale presented, the kinetics appears to be primarily driven by the surface energy which favors coarsening–the growth of large precipitates at the expense of the small precipitates to reduce the surface energy. Upon closer examination, differences between the simulations are observed. For example, consider the result at time t = 15.77 which is shown in Fig. 3. In the Ni3 Al case, the two upper precipitates attract one another and likely merge. In the Ni3 Si case, on the other hand, it does not appear that these two

Modeling coarsening dynamics using interface tracking methods t 0

t 2.5

t 5.0

t 15.0

t 15.77

t 20.09

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Figure 3. Evolution of 10 precipitates in a Ni matrix. Solid, Ni3 Al; dashed, Ni3 Si, Z=1. After Leo, Lowengrub and Nie 2000. Reproduced with permission.

precipitates will merge. This is consistent with the results of smaller precipitate simulations [24]. In addition, the interacting pairs of Ni3 Al precipitates tend to be “flatter” than their Ni3 Si counterparts. Also observe that the lower two precipitates in the Ni3 Al case attract one another. In the process, the lower right precipitate develops very high curvature (note its flat bottom) that ultimately prevents the simulation to be continued much beyond this time. This is why no Ni3 Al precipitates are shown in Fig. 3 at time t = 20.09. Finally, more work needs to be done in order to simulate larger inhomogeneous systems in order to reliably determine coarsening rate constants.

6.

Three-dimensional Results

Because of the difficulties in simulating the evolution of 2D surfaces in 3D, the simulation of microstructure evolution in 3D is much less developed than the 2D counterpart. Nevertheless, there has been promising recent work that is beginning to bridge the gap. The state-of-the-art in 3D boundary integral simulations is the work of Cristini and Lowengrub, 2004 and Li et al., 2003. Using the adaptive simulation algorithms described above, Cristini and Lowengrub, 2004 simulated the diffusional evolution of systems with a

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single precipitate growing under the influence of a driving force consisting of either an imposed far-field heat flux or a constant undercooling in the far-field. Under conditions of constant heat flux, Cristini and Lowengrub demonstrated that the Mullins–Sekerka instability can be suppressed and precipitates can be grown with compact shapes. An example simulation from Cristini and Lowengrub, 2004 is shown in Fig. 4. In this figure, the precipitate morphologies together with the shape factor δ/R are shown for precipitates grown under constant undercooling and constant flux conditions. R is the effective precipitate radius (i.e., radius of a (equivalent) sphere with the same volume enclosed) and δ/Rmeasures the shape deviation from the equivalent sphere. In Fig. 5, the coarsening of a system of 8 precipitates in 3D is shown in the absence of elastic effects (Z = 0), from Li, Lowengrub and Cristini, 2004. This adaptive simulation uses the algorithms described above and is performed an infinitely large domain. Because the precipitates are spaced relatively far from one another, there is little apparent deviation of the morphologies from spherical. However, this is not assumed or required by the algorithm. In Fig. 5, we see the classical survival of the fattest as mass is transferred from small precipitates to large ones. Work is ongoing to develop simulations at finite

Figure 4. Precipitate morphologies grown under constant undercooling and constant flux conditions. After Cristini and Lowengrub, 2004. Reproduced with permission.

Modeling coarsening dynamics using interface tracking methods t0

t 0.75

t 1.5

t 4.0

t 6.75

t 7.5

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Figure 5. The coarsening of a system of 8 precipitates in 3D in the absence of elastic effects (Z = 0). Figure courtesy of Li, Lowengrub and Cristini, 2004.

Figure 6. The evolution of a Ni3 Al precipitate in a Ni matrix (Z = 4). Left: early time. Right: late time (equilibrium). After Li et al., 2003. Reproduced with permission.

volume fractions of precipitate coarsening in periodic geometries [25] in order to determine statistically meaningful coarsening rate statistics. The current state-of-the-art in simulations of coarsening in 3D with elastic effects is the work of Li et al., 2003. To date, simulations have been performed with single precipitates. A sample simulation from Li et al., 2003 is shown in Fig. 6 for the evolution of a Ni3 Al precipitate in a Ni matrix with Z = 4. For this value of Z , and those above it (for Ni3 Al precipitates), there is a transition from cuboidal shapes to elongated shapes as seen in the figure. Such elongated

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Figure 7. Left and Middle: Growth shapes of a Ni3 Al precipitate in a Ni matrix. After Li et al., 2003. Reproduced with permission. A. Right: An experimental precipitate from a Ni-based superalloy after Yoo, Yoon and Henry, 1995. Reproduced with permission.

shapes are often seen in experiments. Finally, in Fig. 7, we present growth shapes (left and middle) of a Ni3 Al precipitate in a Ni matrix with Z = 4 under a driving force consisting of a constant flux of Al atoms [11]. In contrast to the precipitate in Fig. 6, under growth, the Ni3 Al precipitate retains its cuboidal shape although it develops concave faces. On the right, an image is shown from an experiment [26] showing Ni-based precipitates with concave faces similar to those observed in the simulation.

7.

Outlook

In this paper, we have presented a brief description of the state-of-theart in simulating microstructure evolution, and in particular coarsening, using boundary integral interface tracking methods. In general, the methods are quite well-developed in 2D. In particular, large-scale coarsening studies have been performed in the absence of elastic effects and when the elastic media is homogeneous and anisotropic. Although methods have been developed to study coarsening in fully inhomogeneous, anisotropic elastic media, so far the computational expense of the current methods have prevented large-scale studies to be performed. There have been exciting developments in 3D and although the state-ofthe-art in 3D simulations is still well behind those in 2D, this direction looks very promising for the future. This is also an important future direction as coarsening in metallic alloys, for example, is a fully 3D phenomenon. Efforts in this direction will have a significant potential payoff in that they will allow, for the first time, not only a rigorous check of the LSW coarsening kinetics in 3D but also will allow the effects of finite volume fraction and elastic forces on the coarsening kinetics to be assessed.

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References [1] Z. Li and R. Leveque, “Immersed interface methods for Stokes flow with elastic boundaries or surface tension,” SIAM J. Sci. Comput., 18, 709, 1997. [2] S. Osher and R. Fedkiw, “Level set methods: An overview and some recent results,” J. Comp. Phys., 169, 463, 2001. [3] J. Glimm, M.J. Graham, J. Grove et al., “Front tracking in two and three dimensions,” Comput. Math. Appl., 35, 1, 1998. [4] G. Tryggvason, B. Bunner, A. Esmaeeli et al., “A front tracking method for the computations of multiphase flow,” J. Comp. Phys., 169, 708, 2001. [5] I.M. Lifshitz and V.V. Slyozov, J. Phys. Chem. Solids, 19, 35, 1961. [6] C. Wagner, Z. Elektrochem., 65, 581, 1961. [7] W.C. Johnson and P.W. Voorhees, “Elastically-induced precipitate shape transitions in coherent solids,” Solid State Phenom, 23, 87, 1992. [8] K. Thornton, J. Agren, and P.W. Voorhees, “Modelling the evolution of phase boundaries in solids at the meso- and nano-scales,” Acta Mater., 51(3), 5675–5710, 2003. [9] C. Herring “Surface tension as a motivation for sintering,” In: W. E. Kingston, (ed.), The Physics of Powder Metallurgy, Mcgraw-Hill, p. 143, 1951. [10] M. Spivak, “A Comprehensive Introduction to Differential Geometry,” Vol. 4, Publish or Perish, 3rd edn., 1999. [11] Li Xiaofan, J.S. Lowengrub, Q. Nie et al., “Microstructure evolution in threedimensional inhomogeneous elastic media,” Metall. Mater. Trans. A, 34A, 1421, 2003. [12] T.Y. Hou, J.S. Lowengrub, and M.J. Shelley, “Boundary integral methods for multicomponent fluids and multiphase materials,” J. Comp. Phys., 169, 302–362, 2001. [13] S.G. Mikhlin, “Integral equations and their applications to certain problems in mechanics, mathematical physics, and technology,” Pergamon, 1957. [14] P.W. Voorhees, “Ostwald ripening of two phase mixtures,” Annu. Rev. Mater. Sci., 22, 197, 1992. [15] W.T. Scott, “The physics of electricity and magnetism,” Wiley, 1959. [16] A.E.H. Love, “A treatise on the mathematical theory of elasticity,” Dover, 1944. [17] T. Mura, “Micromechanics of defects in solids,” Martinus Nijhoff, 1982. [18] J. Carrier, L. Greengard, and V. Rokhlin, “A fast adaptive multipole algorithm,” SIAM J. Sci. Stat. Comput., 9, 669, 1988. [19] V. Cristini and J.S. Lowengrub, “Three-dimensional crystal growth II. Nonlinear simulation and control of the Mullins-Sekerka instability,” J. Crystal Growth, in press, 2004. [20] V. Cristini, J. Blawzdzieweicz, and M. Loewenberg, “An adaptive mesh algorithm for evolving surfaces: Simulations of drop breakup and coalescence,” J. Comp. Phys., 168, 445, 2001. [21] N. Akaiwa and D.I. Meiron, “Two-dimensional late-stage coarsening for nucleation and growth at high-area fractions,” Phys. Rev. E, 54, R13, 1996. [22] N. Akaiwa, K. Thornton, and P.W. Voorhees, “Large scale simulations of microstructure evolution in elastically stressed solids,” J. Comp. Phys., 173, 61–86, 2001. [23] K. Thornton, N. Akaiwa, and P.W. Voorhees, “Dynamics of late stage phase separation in crystalline solids,” Phys. Review Lett., 86(7), 1259–1262, 2001. [24] P.H. Leo, J.S. Lowengrub, and Q. Nie, “Microstructure evolution in inhomogeneous elastic media,” J. Comp. Phys., 157, 44, 2000.

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[25] Li Xiangrong, J.S. Lowengrub, and V. Cristini, “Direct numerical simulations of coarsening kinetics in three-dimensions,” In preparation, 2004. [26] Y.S. Yoo, D.Y. Yoon, and. M.F. Henry, “The effect of elastic misfit strain on the morphological evolution of γ -precipitates in a model Ni-base superalloy,” Metals Mater., 1, 47, 1995.

7.9 KINETIC MONTE CARLO METHOD TO MODEL DIFFUSION CONTROLLED PHASE TRANSFORMATIONS IN THE SOLID STATE Georges Martin1 and Fr´ed´eric Soisson2 1

´ Commissariat a` l’Energie Atomique, Cab. H.C., 33 rue de la F´ed´eration, 75752 Paris Cedex 15, France 2 CEA Saclay, DMN-SRMP, 91191 Gif-sur-Yuette, France

The classical theories of diffusion-controlled transformations in the solid state (precipitate-nucleation, -growth, -coarsening, order-disorder transformation, domain growth) imply several kinetic coefficients: diffusion coefficients (for the solute to cluster into nuclei, or to move from smaller to larger precipitates. . . ), transfer coefficients (for the solute to cross the interface in the case of interface-reaction controlled kinetics) and ordering kinetic coefficients. If we restrict to coherent phase transformations, i.e., transformations, which occur keeping the underlying lattice the same, all such events (diffusion, transfer, ordering) are nothing but jumps of atoms from site to site on the lattice. Recent progresses have made it possible to model, by various techniques, diffusion controlled phase transformations, in the solid state, starting from the jumps of atoms on the lattice. The purpose of the present chapter is to introduce one of the techniques, the Kinetic Monte Carlo method (KMC). While the atomistic theory of diffusion has blossomed in the second half of the 20th century [1], establishing the link between the diffusion coefficient and the jump frequencies of atoms, nothing as general and powerful occurred for phase transformations, because of the complexity of the latter at the atomic scale. A major exception is ordering kinetics (at least in the homogeneous case, i.e., avoiding the question of the formation of microstructures), which has been described by the atomistic based Path Probability Method [2]. In contrast, supercomputers made it possible to simulate the formation of microstructures by just letting the lattice sites occupancy change in course of time following a variety of rules: the Kinetic Ising model (KIM) in particular has been (and 2223 S. Yip (ed.), Handbook of Materials Modeling, 2223–2248. c 2005 Springer. Printed in the Netherlands.

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still is) extensively studied and is summarized in the appendix [3]; other models include “Diffusion Limited Aggregation”, etc. . . Such models stimulate a whole field of the statistical physics of non-equilibrium processes. However, we choose here a distinct point of view, closer to materials science. Indeed, a unique skill of metallurgists is to master the formation of a desired microstructure simply by well controlled heat treatments, i.e., by imposing a strictly defined thermal history to the alloy. Can we model diffusion controlled phase transformations at a level of sophistication capable of reproducing the expertise of metallurgists? Since Monte Carlo techniques were of common use in elucidating delicate points of the theory of diffusion in the solid state [4, 5], it has been quite natural to use the very same technique to simulate diffusion controlled coherent phase transformations. Doing so, one is certain to retain the full wealth that the intricacies of diffusion mechanisms might introduce in the kinetic pathways of phase transformations. In particular, the question of the time scale is a crucial one, since the success of a heat treatment in stabilizing a given microstructure, or in insuring the long-term integrity of that microstructure, is of key importance in Materials Science. In the following, we first recall the physical foundation of the expression for the atomic jump frequency, we then recall the connection between jump frequencies and kinetic coefficients describing phase transformation kinetics; the KMC technique is then introduced and typical results pertaining to metallurgy relevant issues are given in the last section.

1.

Jumps of Atoms in the Solid State

With a few exceptions, out of the scope of this introduction, atomic jump in solids is a thermally activated process. Whenever an atom jumps, say from site α to α , the configuration of the alloy changes from i to j . The probability per unit time, for the transition to occur, writes:

Wi, j = νi, j

Hi, j exp − kB T

(1)

In Eq. (1), ν i, j is the attempt frequency, kB is the Boltzmann’s constant, T is the temperature and Hi, j is the activation barrier for the transition between configurations i and j . According to the rate theory [6], the attempt frequency writes, in the (quasi-) harmonic approximation: 3N−3

νi, j = k=1 3N−4 k=1

νk νk

(2)

In Eq. (2), νk and νk are the vibration eigen-frequencies of the solid, respectively in the initial configuration, i, and at the saddle point between configurations i and j . Notice that for a solid with N atoms, the number of eigen modes

Diffusion controlled phase transformations in the solid state

2225

is 3N . However, the vibrations of the centre of mass (3 modes) are irrelevant in the diffusion process, hence the upper bound 3N −3 in the product at the numerator. At the saddle point position between configurations i and j , one of the modes is a translation rather than a vibration mode, hence the upper bound 3N −4 in the denominator. Therefore, provided we know the value of Hi, j and νi, j for each pair of configurations, i and j , we need to implement some algorithm which would propagate the system in its configuration space, as the jumps of atoms actually do in the real solid. Notice that the algorithm must be probabilistic since Wi, j in Eq. (1) is a jump probability per unit time. Before we discuss this algorithm, we give some more details on diffusion mechanisms in solids, since the latter deeply affect the values of Wi, j in Eq. (1). The most common diffusion mechanisms in crystalline solids are vacancy-, interstitial- and interstitialcy-diffusion [7]. Vacancies (a vacant lattice site) allow for the jumps of atoms from site to site on the lattice; in alloys, vacancy diffusion is responsible for the migration of solvent- and of substitutional solute- atoms. Therefore, the transition from configuration i to j implies that one atom and one (nearest neighbor) vacancy exchange their position. As a consequence, the higher the vacancy concentration, the more numerous are the configurations, which can be reached from configuration i: indeed, starting from configuration i, any jump of any vacancy destroys that configuration. Therefore the transformation rate depends both on the jump frequencies of vacancies, as given by Eq. (1), and on the concentration of vacancies in the solid. This fact is commonly taken advantage of, in practical metallurgy. At equilibrium, the vacancy concentration depends on the temperature, the pressure and, in alloys, of the chemical potential differences between the species:

Cve

gf = exp − v kB T

(3)

In Eq. (3), Cve = Nv /(N + Nv ), with N the number of atoms, and gvf is the free enthalpy of formation of the vacancy. At equilibrium, the probability for an atom to jump equals the product of the probability for a vacancy to be nearest neighbor of that atom (deduced from Eq. 3), times the jump frequency given by Eq. (1). In real materials, vacancies form and annihilate at lattice discontinuities (free surfaces, dislocation lines and other lattice defects). If, in course of the phase transformation the equilibrium vacancy concentration changes, e.g., because of vacancy trapping in one of the phases, it takes some time for the vacancy concentration to adjust to its equilibrium value. This point, of common use in practical metallurgy, is poorly known from the basic point of view [8] and will be discussed later. Interstitial diffusion occurs when an interstitial atom (like carbon or nitrogen in steels) jumps to a nearest neighbor unoccupied interstitial site.

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Interstitialcy diffusion mechanism implies that a substitutional atom is “pushed” into an interstitial position by a nearest neighbor interstitial atom, which itself, becomes a substitutional one. This mechanism prevails, in particular, in metals under irradiation, where the collisions of lattice atoms with the incident particles produce Frenkel pairs; a Frenkel pair is made of one vacancy and one dumb-bell interstitial (two atoms competing for one lattice site). The migration of the dumb-bell occurs by the interstitialcy mechanism. The concentration of dumb-bell interstitials results from the competition between the production of Frenkel pairs by nuclear collisions and of their annihilation either by recombination with vacancies or by elimination on some lattice discontinuity. The interstitialcy mechanism may also prevail in some ionic crystals, and in the diffusion of some gas atoms in metals.

2.

From Atomic Jumps to Diffusion and to the Kinetics of Phase Transformations

The link between the jump frequencies and the diffusion coefficients has been established in details in limiting cases [1]. The expressions are useful for adjusting the values of the jump frequencies to be used, to experimental data. As a matter of illustration, we give below some expressions for the vacancy diffusion mechanism in crystals with cubic symmetry (with a for the lattice parameter): – In a pure solvent, the tracer diffusion coefficient writes: D ∗ = a 2 f 0 W0 Cve ,

(4a)

with f 0 for the correlation factor (a purely geometrical factor) and W0 , the jump frequency of the vacancy in the pure metal. – In a dilute solution with Face Centered Cubic (FCC) lattice, with non interacting solutes, and assuming that the spectrum of the vacancy jump frequencies is limited to 5 distinct values (Wi , i = 0 to 4, for the vacancy jumps respectively in the solvent, around one solute atom, toward the solute, toward a solvent atom nearest neighbor of the solute, and away from the solute atom, see Fig. 1), the solute diffusion coefficient writes: W4 f 2 W2 , (4b) W3 where the correlation factor f 2 can be expressed as a function of the Wi ’s. In dilute solutions, the solvent- as well as the solute-diffusion coefficient depends linearly on the solute concentration, C, as: Dsolute = a 2 Cve

D ∗ (C) = D ∗ (0)(1 + bC). The expression of b is given in [1, 9].

(4c)

Diffusion controlled phase transformations in the solid state W3

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W1

W3 W2

W3

W1

W4 W0

Figure 1. The Five-frequency model in dilute FCC alloys: the five types of vacancy jumps are represented in a (111) plane (light gray: solvent atoms, dark gray: solute atom, open square: vacancies).

– In concentrated alloys, approximate expressions have been recently derived [10]. The atomistic foundation of the classical models of diffusion controlled coherent phase transformation is far less clear. For precipitation problems, two main techniques are of common use: the nucleation theory (and its atomistic variant sometimes named “cluster dynamics”) and Cahn–Hilliard diffusion equation [11]. In the nucleation theory, one defines the formation free energy (or enthalpy, if the transformation occurs under fixed pressure), F(R) of a nucleus with size R (volume vR 3 and interfacial area sR2 , v and s being geometric factors computed for the equilibrium shape): F(R) = δµvR 3 +σ sR 2 .

(5)

In Eq. (5), δµ and σ are respectively the gain of chemical potential on forming one unit volume of second phase, and the interfacial free energy (or free enthalpy) per unit area. If the solid solution is supersaturated, δµ is negative and F(R) first increases as a function of R, then goes through a maximum for the critical size R ∗ (R ∗ = (2s/3v) (σ/|δµ|)) and then decreases (Fig. 2). F(R) can be given a more precise form, in particular for small values of R. More details may be found in Perini et al. [12]. For the critical nucleus, F ∗ = F(R ∗ ) ≈ σ 3 /(δµ)2 .

(6)

F(R) can be seen as an energy hill which opposes the growth of sub-critical nuclei (R< R ∗ ) and which drives the growth of super-critical nuclei (R >R ∗ ). The higher the barrier, i.e., the larger F ∗ , the more difficult the nucleation is. F ∗ is very sensitive to the gain in chemical potential: the higher the supersaturation, the larger the gain, the shallower the barrier, and the easier the

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F (R )

F R

R Figure 2. Free energy change on forming a nucleus with radius R.

nucleation. F ∗ also strongly depends on the interfacial energy, a poorly known quantity, which, in principle depends on the temperature. With the above formalism, the nucleation rate (i.e., the number of supercritical nuclei which form per unit time in a unit volume) writes, under stationary conditions: F∗ ∗ (7a) Jsteady = β Z N0 exp − kB T with N0 for the number of lattice sites and Z for the Zeldovich’s constant:

1 Z= − 2π kT

∂2F ∂n 2

1/2

,

(7b)

n=n ∗

n for the number of solute atoms in a cluster and θ ∗ for the sticking rate of solute atoms on the critical nucleus. If the probability of encounter of one solute atom with one nucleus is diffusion controlled: β(R) = 4πDRC

(7c)

For a detailed discussion, see Waite [13]. In Eq. (7c), D is the solute diffusion coefficient in the (supersaturated) matrix with the solute concentration C. An interesting quantity is the incubation time for precipitation, τinc , i.e., the time below which the nucleation current is much smaller than Jsteady . The former writes: 1 (7d) τinc ∝ ∗ 2 β Z When the supersaturation is small and/or the interfacial energy is high, the incubation time gets very large. Also the incubation time is scaled to the diffusion coefficient of the solute.

Diffusion controlled phase transformations in the solid state

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The nucleation process can be described also by the technique named “cluster dynamics”. The microstructure is described, at any time, by the number density, ρn, of clusters made of n solute atoms. The latter varies in time as: dρn = − ρn (αn + βn ) + ρn+1 αn+1 + ρn−1 βn−1 dt

(8)

where α n and β n are respectively the rate of solute evaporation and sticking at a cluster of n solute atoms. Again, α n and β n can be expressed in terms of solute diffusion or transfer coefficients. At later stages, when the second phase precipitation has exhausted the solute supersaturation, Ostwald ripening takes place: because the chemical potential of the solute close to a precipitate increases with the curvature of the precipitate-matrix interface (δµ(R) = 2σ/R), the smaller precipitates dissolve to the benefit of the larger ones. According to Lifschitz and Slyosov and to Wagner [14], the mean precipitate volume increases linearly with time, or the mean radius (as well as the mean precipitate spacing) goes as: R(t) − R(0) = k t 1/3

(9a)

with k3 =

(8/9)Dσ Cs kB T

(9b)

In Eq. (9b), D is again the solute diffusion coefficient, Cs the solubility limit, and the atomic volume. The problem of multicomponent alloys has been addressed by several authors [15]. The above models do not actually generate a full microstructure: they give the size distribution of precipitates as a function of time, as well as the mean precipitate spacing, since the total amount of solute is conserved, provided that the precipitates do not change composition in the course of the phase separation process. The formation of a full microstructure (i.e., including the variability of precipitate shapes, the correlation in the positions of precipitates etc.) is best described by Cahn’s diffusion equation [16]. In the latter, the chemical potential, the gradient of which is the driving force for diffusion, includes an inhomogeneity term, i.e., is a function, at each point, both of the concentration and of the curvature of the concentration field. The diffusion coefficient was originally given the form due to Darken. Based on a simple model of Wi, j and a mean field approximation, an atomistic based expression of the mobility has been proposed, both for binary [17] and multicomponent alloys [18]. When precipitation occurs together with ordering, Cahn’s equation is complemented with an equation for the relaxation of the degree of order; the latter relaxation occurs at a rate proportional to the gain in free energy due to the onsite relaxation of the degree order. The rate constant is chosen arbitrarily [19]. Since in a crystalline

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sample the ordering reaction proceeds by the very same diffusion mechanism as the precipitation, both rate constants (for the concentration- and for the degree of order fields) should be expressed from the same set of Wi, j . This introduces some couplings, which have been ignored by classical theories [20]. As a summary, despite their efficiency, the theories of coherent phase separation introduce rate constants (diffusion coefficients, interfacial transfer coefficients, rate constants for ordering) the microscopic definition of which is not fully settled. The KMC technique offers a means to by-pass the above difficulties and to directly simulate the formation of a microstructure in an alloy where atoms jump with the frequencies defined by Eq. (1).

3.

Kinetic Monte Carlo Technique to Simulate Coherent Phase Transformations

The KMC technique can be implemented in various manners. The one we present here has a transparent physical meaning.

3.1.

Algorithm

Consider a computational cell with Ns sites, Na atoms and Nv = Ns − Na vacancies; each lattice site is linked to Z neighbor sites with which atoms may be exchanged (usually, but not necessarily, nearest neighbor sites). A configuration is defined by the labels of the sites occupied respectively by A, B, C, . . . atoms and by vacancies. Each configuration “i” can be escaped by Nch channels (Nch = Nv Z minus the number of vacancy–vacancy bounds if any), leading to Nch new configurations “ j1 ” to “ j Nch ”. The probability that the transition “i; jq ” occurs per unit time is given by Eq. (1) which can be computed a priori provided a model is chosen for Hi, j and νi, j . Since the configuration “i” may disappear by Nch independent channels, the probability for the configuration to disappear per unit time, Wiout , is the sum of the probabilities it decays by each channel (Wi, j q , q = 1 to Nch ), and the life time τ i of the configuration is the inverse of Wiout : 

τi = 

Nch

−1

Wi, jq 

(10a)

q=1

The probability that the configuration “ jq ” is reached among the Nch target configurations is simply given by: Wi, jq Pi jq = N = Wi, jq × τi (10b) ch

Wi, jq q=1

Diffusion controlled phase transformations in the solid state

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Assuming all possible values of Wi, jq are known (see below), the code proceeds as follows: Start at time t = 0 from the configuration “i 0 ”, set i = i 0 ;

1. Compute τi (Eq. (10a)) and the Nch values of Si,k = kq=1 Pi jq , k = 1 to Nch . 2. Generate a random number R on ]0; 1]. 3. Find the value of f to be given to k such that Si,k−1 < R ≤ Si,k . Choose f as the final configuration. 4. Increment the time by τi (t MC => t MC + τi ) and repeat the process from step 1, giving to i the value f .

3.2.

Models for the Transition Probabilities Wi , j (Eq. (1))

For a practical use of the above algorithm, we need a model for the transitions probabilities per unit time, Wi, j . In principle, at least, given an interatomic potential, all quantities appearing in Eqs. (1)–(3) can be computed for any pair of configurations, hence Wi, j . The computational cost for this is so high that most studies use simplified models for the parameters entering Eqs. (1)–(3); the values of the parameters are obtained by fitting appropriate quantities to available experimental data, such as phase boundaries and tie lines in the equilibrium phase diagram, vacancy formation energy and diffusion coefficients. We describe below the most commonly used models, starting from the simplest one. Model (a) The energy of any configuration is a sum of pair interactions ε with a finite range (nearest- or farther neighbors). The configurational energy is the sum of the contributions of two types of bounds: those which are modified by the jump, and those which are not. We name esp the contribution of the bounds created in the saddle point configuration. This model is illustrated in Fig. 3. The simplest version of this model is to assume that esp depends neither on the atomic species undergoing the jump, nor on the composition in the surrounding of the saddle point [17]. Model (b) Same as above, but with esp depending on the atomic species at the saddle point. This approximation turned out to be necessary to account for the contrast in diffusivities in the ternary Ni–Cr–Al [21]. Model (c) Same as above, but with esp written as a sum of pair interactions [22]. This turned out to provide an excellent fit to the activation barriers computed in Fe(Cu) form fully relaxed atomistic simulations based on an EAM potential. As shown on Fig. (4), the

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G. Martin and F. Soisson

non broken bonds

0

esp broken bonds

Saddle-Point position

∆Hi;j

Hj Hi

(

)

(

)

i

j

Figure 3. Computing the migration barrier between configurations i and j (Eq. (1)), from the contribution of broken- and restored bounds. 7.5 8 2

eFe(SP)

8.5

6

9

eCu(SP)

5 3 1

V

4

9.5 10

0

1

2

3

4

5

6

NCu(SP)

Figure 4. The six nearest-neighbors (labeled 1 to 6) of the saddle-point in the BCC lattice (left). Contribution to the configurational energy, of one Fe atom, eFe (SP), or one Cu atom, eCu (SP), at the saddle point, as a function of the number of Cu atoms nearest neighbor of the saddle point (right).

contribution to the energy of one Cu atom at the saddle point, eCu (SP), does not depend on the number of Cu atoms around the saddle point, while that of one Fe atom, eFe (SP), increases linearly with the latter. Model (d) The energy of each configuration is a sum of pair and multiple interactions [18]. Taking into account higher order interactions permits to reproduce phase diagrams beyond the regular solution

Diffusion controlled phase transformations in the solid state

2233

model. The attempt frequency (Eq. 2) was adjusted, based on an empirical correlation between the pre-exponential factor and the activation enthalpy. Complex experimental interdiffusion profiles in four components alloys (AgInCdSn) could be reproduced successfully. Multiplet interactions have been used in KMC to model phase separation and ordering in AlZr alloys [23]. Model (e) The energies of each configuration and at the saddle point, as well as the vibration frequency spectrum (entering Eq. (2)) are computed from a many body interaction potential [24]. The vibration frequency spectrum can be estimated either with Einstein’s model [25] or Debye approximation [26, 27]. The above list of approximations pertains to the vacancy diffusion mechanism. Fewer studies imply also interstitial diffusion, as carbon in iron, or dumbbell diffusion, in metals under irradiation, as will be seen in the next section. The models for the activation barrier are of model (b) described above.

3.3.

Physical Time and Vacancy Concentration

Consider the vacancy diffusion mechanism. If the simulation cell only contains one vacancy, the vacancy concentration is 1/Ns , often much larger than a typical equilibrium vacancy concentration Cve . From Eq. (10), we conclude that the time evolution in the cell is faster than the real one, by a factor equal to the vacancy supersaturation in the cell: (1/Ns )/Cve . The physical time, t is therefore longer than the Monte Carlo time, tMC , computed above: t = tMC /(Ns Cve )

(11)

Equation (11) works as long as the equilibrium vacancy concentration does not vary much in the course of the phase separation process, a point which we discuss now. Consider an alloy made of N A atoms A, N B atoms B on Ns lattice sites. For any atomic configuration of the alloy, there is an optimum number of lattice sites, Nse , that minimizes the configurational free energy; the vacancy concentration in equilibrium with that configuration is: Cve = (Nse − N A − N B )/Nse . For example assume that the configurations can be described by K types of sites onto which the vacancy is bounded by an energy E bk (k = 1 to K ), with k = 1 corresponding to sites surrounded by pure solvent (E b1 = 0). We name N1 , . . . , N K the respective numbers of such sites. The equilibrium concentrations of vacancies on the sites of type 1 to K are respectively:

e Cvk =

Nvk E f + E bk = exp − Nk + Nvk kB T

(12a)

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G. Martin and F. Soisson

In Eq. (11), E f is the formation energy of a vacancy in pure A. The total vacancy concentration, in equilibrium with the configuration as defined by N1 , . . . , N K is thus (in the limit of small vacancy concentrations):

e Nk Cvk e ≈ = Cv0 k Nk X k = Nk /N1

Cve

k

1+

X k exp(−E bk /kB T )

; 1 + k=2,K X k

k=2,K

(12b)

e is the equilibrium vacancy concentration in the pure solvent, In Eq. (12), Cv0 and X k depends on the advancement of the phase separation process: e.g., in the early stages of solute clustering, we expect the proportion of sites surrounded by a small number of solute atoms to decrease. The overall vacancy equilibrium concentration thus changes in time (Eq. (12b)), while it remains unaffected for each type of site (Eq. (12a)). Imposing a fixed number of vacancies in the simulation cell, creates the opposite situation: in the simulation, the overall vacancy concentration is kept constant, thus the vacancy concentration on each type of site must change in course of time: the kinetic pathway will be altered. This problem can be faced in various ways. We quote below two of them:

– Rescaling the time from an estimate of the free vacancy concentration, i.e., the concentration of those vacancies with no solute as neighbor [22]. The vacancy concentration in the solvent is estimated in the course of the simulation, at a certain time scale, t, from the fraction of the time, where the vacancy is surrounded by solvent atoms only. Each time interval t is rescaled by the vacancy super saturation, which prevails during that time interval. – Modeling a vacancy source (sink) in the simulation cell [28]: in real materials, vacancies are formed and destroyed at lattice discontinuities (extended defects), such as dislocation lines (more precisely jogs on the dislocation line), grain boundaries, incoherent interfaces and free surfaces. The simplest scheme is as follows: creating one vacancy implies that one atom on the lattice close to the extended defect jumps into the latter in such a way as to extend the lattice by one site; eliminating one vacancy implies that one atom at the extended defect jumps into the nearby vacancy. Vacancy formation and elimination are a few more channels by which a configuration may change. The transition frequencies are still given by Eq. (1) with appropriate activation barriers: Fig. 5 gives a generic energy diagram for the latter transitions. As shown by the above scheme, while the vacancy equilibrium concentration is dictated by the formation energy, E f , the time to adjust to a change in the equilibrium vacancy concentration implies the three parameters E f ,

Diffusion controlled phase transformations in the solid state

2235

Em

Ef

Figure 5. Configurational energy as a function of the position of the vacancy. When one vacancy is added to the crystal, the energy is increased by E f .

E m and δ. In other words, a given equilibrium concentration can be achieved either by frequent or by rare vacancy births and deaths. The consequences of this fact on the formation of metastable phases during alloy decomposition are not yet fully understood.

3.4.

Tools to Characterize the Results

The output of a KMC simulation is a string of atomistic configurations as a function of time. The latter can be observed by the eye (e.g., to recognize specific features in the shape of solute clusters); one can also measure various characteristics (short range order, cluster size distribution, cluster composition and type of ordering. . . ); one can simulate signals one would get from classical techniques such as small- or large-angle scattering, or use the very same tools as used in Atom Probe Field Ion Microscopy to process the very same data, namely the location of each type of atom. Some examples are given below.

3.5.

Comparison with the Kinetic Ising Model

The KIM, of common use in the Statistical Physics community, is summarized in the appendix. It is easily checked that the models presented above for the transition probabilities introduce new features, which are not addressed by the KIM. In particular, the only energetic parameter to appear in KIM is what is named, in the community of alloys thermodynamics, the ordering energy: ω = ε AB − (ε A A + ε B B )/2 (for the sake of simplicity, we restrict, here, to two

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G. Martin and F. Soisson

component alloys). While ω is indeed the only parameter to enter equilibrium thermodynamics, the models we introduced show that the kinetic pathways are affected by a second independent energetic parameter, the asymmetry between the cohesive energies of the pure elements: ε A A − ε B B . This point is discussed into details, by Ath`enes and coworkers [29–31]. Also, the description of the activated state between two configurations is more flexible in the present model as compared to KIM. For these reasons, the present model offers unique possibilities to study complex kinetic pathways, a common feature in real materials.

4.

Typical Results: What has been Learned

In the 70s the early KMC simulations have been devoted to the study of simple ordering and phase separation kinetics in binary systems with conserved or non-conserved order parameters. Based on the Kinetic Ising model and so called “Kawazaki dynamics” (direct exchange between nearest neighbor atoms, with a probability proportional to exp [−(Hfinal − Hinitial)/2kB T ]), with no point defects and no migration barriers, they could mainly reproduce some generic features of intermediate time behaviors, taking the number of Monte Carlo step as an estimate of physical time: the coarsening regime of precipitation with R − R0 ∝ t 1/3 ; the growth rate of ordered domains R − R0 ∝ t 1/2 , dynamical scaling laws, etc. [3, 32]. However, such models cannot reproduce important metallurgical features such as the role of distinct solute and solvent mobilities, of point defect trapping, or of correlations among successive atomic jumps etc. In the frame of the models (a)–(e) previously described, these features are mainly controlled by the asymmetry parameters for the stable configurationsp sp and saddle-point energies (respectively ε A A − ε B B , and e A − e B ). We give below typical results, which illustrate the sensitivity, to the above features, of the kinetic pathways of phase transformations.

4.1.

Diffusion in Ordered Phases

Since precipitates are often ordered phases, the ability of the transition probability models to well describe diffusion in ordered phases must be assessed. As an example, diffusion in B2 ordered phases presents specific features which have been related to the details of the diffusion mechanism: at a given composition, the Arrhenius plot displays a break at the order/disorder temperature and an upward curvature in the ordered phase; at a given temperature, the tracer diffusion coefficients are minimum close to the stoichiometric composition. The reason for that is as follows: starting from a perfectly

Diffusion controlled phase transformations in the solid state

2237

ordered B2 phase, any vacancy jump creates an antisite defect, so that the most probable next jump is the reverse one which annihilates the defect. As a consequence, it has been proposed that diffusion in B2 phases occurs via highly correlated vacancy jump sequences, such as the so-called 6-jump cycle (6JC) which corresponds to 6 effective vacancy jumps (resulting from many more jumps, most of them being canceled by opposite jumps). Based on the above “model (a)” for the jump frequency, Ath`enes’ KMC simulations [29] show that other mechanisms (e.g., the antisite-assisted 6JC) contribute to long-range diffusion, in addition to the classical 6JC (see Figure 6). Their relative importance increases with the asymmetry parameter u = ε A A − ε B B , which controls the respective vacancy concentrations on the two B2 sublattices and the relative mobilities of A and B atoms. Moreover while diffusion by 6JC only would implies a D ∗A /D ∗B ratio between 1/2 and 2, the newly discovered antisite-assisted cycles yield to a wider range, as observed experimentally in some B2 alloys, such as Co–Ga. Moreover, high asymmetry parameters produce an upward curvature of the Arrhenius plot in the B2 domain. Similar KMC model has been applied to the L12 ordered structures and successfully explains some particular diffusion properties in these phases [30].

4.2.

Simple Unmixing: Iron–Copper Alloys

Copper precipitation in α-Fe has been extensively studied with KMC: although pure copper has an FCC structure, experimental observations show that the first step of precipitation is indeed fully coherent, up to precipitate radii of the order of 2 nm, with a Cu BCC lattice parameter very close to that of iron. The composition of the small BCC copper clusters has long been debated: early atom probe or field ion microscopy studies or small angle neutron scattering experiments suggested that they might contain more than 50% (a)

(b) 0

4

3

1

4

3

1 5

5 2

0

6

2

6

Figure 6. Classical Six Jump Cycle (a) and Antisite assisted Six Jump Cycle (b) in B2 compounds [29].

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G. Martin and F. Soisson

iron, while others experimental techniques suggested pure copper clusters. Using the above simple “model (a)”, KMC suggest almost pure copper precipitates, but with very irregular shapes [33]: the significant iron content measured in some experiments could then be due to the contribution of atoms at the precipitate matrix interface if a simple smooth shape is attributed to the precipitate while the small Cu clusters have very irregular shapes. This explanation is in agreement with the most direct observations using a 3D atom probe [34]. The simulations have also shown that, with the parameter values we used, fast migration of small Cu clusters occurs: the latter induces direct coagulation between nuclei, yielding ramified precipitate morphologies. On the same Fe–Cu system, Le Bouar and Soisson [22] have used an EAM potential to parameterize the activation barriers in Eq. (1). In dilute alloys, the EAM computed energies of stable and saddle-point relaxed configurations, can be reproduced with pair interactions on a rigid lattice (including some vacancy-atom interactions). The saddle-point binding energies of Fe and Cu are shown in Fig. 4 and have already been discussed. Such a dependence of the SP binding energies does not modify the thermodynamic properties of the system (the solubility limit, the precipitation driving force, the interfacial energies, the vacancy concentrations in various phases do not depend on the SP properties) and it slightly affects the diffusion coefficients of Fe and Cu in pure iron. Nevertheless such details strongly affect the precipitation kinetic pathway, by changing the diffusion coefficients of small Cu clusters and thus the balance between the two possible growth mechanisms: classical emissionadsorption of single solute atoms and direct coagulation between precipitates. This is illustrated by Fig. 7, where two simulations of copper precipitation Fe on the are displayed: one which takes into account the dependence of esp Fe local atomic composition and one with a constant esp . In the second case small copper clusters (with typically less than 10 Cu atoms) are more mobile than in the first case, which results in an acceleration of the precipitation. Moreover, the nucleation regime in Fig. 7(b) almost vanishes, because two small clusters can merge as- or even more rapidly than a Cu monomer and a precipitate. The dashed line of Fig. 7 represents the results obtained with the empirical parameter values described in the previous paragraph [33]: as can be seen these results do not differ qualitatively from those obtained by Le Bouar et al. [22], so that the qualitative interpretation of the experimental observations is conserved. The competition between the classical solute emission–adsorption and direct precipitate coagulation mechanisms observed in dilute Fe–Cu alloys appears indeed to be quite general and to have important consequences on the whole kinetic pathway. First studies [35] focused on the role of the atomic jump mechanism (Kawasaki dynamics versus vacancy jump), but recent KMC simulations based on the transition probability models (a)–(c) above have shown that both single solute atom- and cluster-diffusion are observed when

Diffusion controlled phase transformations in the solid state

2239

t (year) 3

10

(a) 0,8

101

100

101

101

DSPE ISPE

0,6 Cu

10

2

0,4 0,2 0,0

(b)

1600 1200 Np(i 1) 800 400 0

(c) 102 101

100 104

105

106

107

108

109

1010

t (s)

Figure 7. Precipitation kinetics in a Fe-3at.%Cu alloy at T = 573 K [22]. Evolution of (a) the degree of the copper short-range order parameter, (b) the number of supercritical precipFe itates and (c) the averaged size of supercritical precipitates. Monte Carlo simulations with esp depending on the local atomic configuration (•) or not (♦). The dashed lines corresponds to the results of Soisson et al. [33].

vacancy diffusion is carefully modeled. Indeed the balance between both mechanisms is controlled by: – the asymmetry parameter which controls the relative vacancy concentrations in the various phases [31]. A vacancy trapping in the precipitates (e.g., in Fe–Cu alloys) or at the precipitate-matrix interface tends to favor direct coagulation, while if the vacancy concentration is higher in the matrix, as is the case for Co precipitation in Cu, [36], the migration of monomers and emission-adsorption of single solute atoms are dominant.

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G. Martin and F. Soisson

– the saddle-point energies which, together with the asymmetry parameter, control the correlation between successive vacancy jumps and the migration of solute clusters [22].

4.3.

Nucleation/Growth/Coarsening: Comparison with Classical Theories

The classical theories of nucleation, growth or coarsening, as well as the theory of spinodal decomposition in highly supersaturated solid solutions, can be assessed using KMC simulations [37]. For the nucleation regime, the thermodynamic and kinetic data involved in Eqs. (5)–(7) (the driving force for precipitation, δµ, the interfacial energy, σ , the adsorption rate β, etc.) can be computed from the atomistic parameters used in KMC (pair interaction, saddle-point binding energies, attempt frequencies): a direct assessment of the classical theories is thus possible. For low supersaturations and in cases where only the solute monomers are mobile, the incubation time and the steady–state nucleation rate measured in the KMC simulations are very close to those predicted by the classical theory of nucleation. On the contrary, when small solute clusters are mobile (keeping the overall solute diffusion coefficient the same), the classical theory strongly overestimates the incubation time and weakly underestimates the nucleation rate, as exemplified on Fig. 8.

100 10 5 4 1014

ψi Cv(s)

109

3

2.5

2.1

1012

1010

F

1011

W

1010 J 108

1012

106

13

10

1014 3 10

T = 0.5 Ω/2k b

st

T = 0.4 Ω/2k

104 102 1/(S0 (ln S0)3)

101

102

b

T = 0.3 Ω/2kb

0

0.5

1

1.5

2

1/ (ln S0)2

Figure 8. Incubation time and steady-state nucleation rate, in a binary model alloy A–B, as eq a function of supersaturation S0 = C 0B /C B (initial/equilibrium B concentration in the solid solution). Comparison of KMC (symbols) and Classical Theory of Nucleation (lines). On the left: the dotted lines refer to two classical expressions of the incubation time (Eq. (7d)), the plain line is obtained by numerical integration of Eq. (8); KMC with mobile monomers only, KMC with small mobile clusters. On the right: the dotted and plain lines refer to Eq. (7a) with respectively Z = 1 or Z from Eq. (7b); ♦, ◦ and refer to KMC with mobile monomers. For more details, see Ref. [37].

Diffusion controlled phase transformations in the solid state

2241

The above general argument has been assessed in the case of Al3 Zr and Al3 Sc precipitation in dilute aluminum alloys: the best estimates of the parameters suggest that diffusion of Zr and Sc in Al occurs by monomer migration [38]. When the precipitation driving force and interfacial energy are computed in the frame of the Cluster Variation Method, the classical theory of nucleation predicts nucleation rates in excellent agreement with the results of the KMC simulations, for various temperatures and supersaturations. Similarly, the distribution of cluster sizes in the solid solution ρn ∼ exp(−Fn /kB T ), with Fn given by the capillarity approximation (Eq. (5)) is well reproduced, even for very small precipitate sizes.

4.4.

Precipitation in Ordered Phases

The kinetic pathways become more complex when ordering occurs in addition to simple unmixing. Such kinetics have been explored by Ath`enes [39] in model BCC binary alloys, in which the phase diagram displays a tricritical point and a two-phase field (between a solute rich B2 ordered phase and a solute depleted A2 disordered phase). The simulation was able to reproduce qualitatively the main experimental features reported from transmission electron microscopy observations during the decomposition of Fe–Al solid solutions: (i) for small supersaturations, a nucleation-growth-coarsening sequence of small B2 ordered precipitates in the disordered matrix occurs; (ii) for higher supersaturations, a short range ordering starts before any modification of the composition field, followed by a congruent ordering with a very high density of antiphase boundaries (APB). In the center of the two phase field, this homogeneous state then decomposes by a thickening of the APBs which turns into the A2 phase. Close to the B2 phase boundary, the decomposition process also involves a nucleation of iron rich A2 precipitates inside the B2 phase. Varying the asymmetry parameter u mainly affects the time scale. However qualitative differences are observed, at very early stages, in the formation of ordered microstructures: if the value of u enhances preferentially the vacancy exchanges with the majority atoms (u > 0), ordering proceeds everywhere, in a diffuse manner; while if u favors vacancy exchanges with the solute atoms (u < 0), ordering proceeds locally by patches. This could explain the experimental observation of small B2 ordered domains in as-quenched Fe-Al alloys, in cases where phenomenological theories predict a congruent ordering [39]. Precipitation and ordering in Ni(Cr,Al) FCC alloys have been studied by Pareige et al. [21], with MC parameters fitted to thermodynamic and diffusion properties of Ni-rich solid solutions (Fig. 9a). For relatively small Cr and Al

2242

G. Martin and F. Soisson

<001>

30 nm

(a)

(b)

Figure 9. (a) Microstructure of a Ni-14.9at.%Cr-5.2at%Al alloy after a thermal ageing of 1 h at 600◦ C. Monte Carlo simulation (left) and 3D atom probe image (right). Each dot represents an Al atom (for the sake of clarity, Ni and Cr atoms are not represented). One observes the Al-rich 100 planes of γ precipitates, with an average diameter of 2 nm [21]. (b) Monte Carlo simulation of NbC precipitation in ferrite with transient precipitation of a metastable iron carbide, shown in faint in the snapshots at 1.5, 11 and 25 seconds [28].

Diffusion controlled phase transformations in the solid state

2243

contents, at 873 K, the phase transformation occurs in three stages: (i) a short range ordering of the FCC solid solution, with two kinds of ordering symmetry (a “Ni3 Cr” symmetry corresponding to the one observed at high temperature in binary Ni–Cr alloys, and an L12 symmetry) followed by a nucleation-growthcoarsening sequence, (ii) the formation of the Al-rich γ precipitates (with L12 structure), (iii) the growth and coarsening of the precipitates. In the γ phase Cr atoms substitute for both Al and Ni atoms, with a preference for the Al sublattice. The simulated kinetics of precipitation are in good agreement with 3D-atom probe observations during a thermal ageing of the same alloy, at the same temperature [21]. For higher Cr and Al contents, MC simulations predict an congruent L12 ordering (with many small antiphase domains) followed by the γ − γ decomposition, as in the A2/B2 case discussed above.

4.5.

Interstitial and Vacancy Diffusion in Parallel

Advanced high purity steels offer a field of application of KMC with practical relevance. In so called High-Strength Low-Alloy (HSLA) steels, Nb is used as a means to retain carbon in niobium carbide precipitates, out of solution in the BCC ferrite. The precipitation of NbC implies the migration, in the BCC Fe lattice, of both Nb, by vacancy mechanism, and C, by direct interstitial mechanism. At very early stages, the formation of coherent NbC clusters on the BCC iron lattice is documented from 3D atom probe observations. The very same Monte Carlo technique can be used [28]; the new feature is the large value of the number of channels by which a configuration can decay, because of the many a priori possible jumps of the numerous carbon atoms. This makes step 3 of the algorithm above, very time consuming. A proper grouping of the channels, as a function of their respective decay time, helps speeding up this step. Among several interesting features, KMC simulations revealed the possibility for NbC nucleation to be preceded by the formation of a transient iron carbide, due to the rapid diffusion of C atoms by comparison with Nb and Fe diffusion (Fig. 9b). This latter kinetic pathway is found to be sensitive to the ability of the microstructure to provide the proper equilibrium vacancy concentration during the precipitation process.

4.6.

Driven Alloys

KMC offers a unique tool to explore the stability and the evolution of the microstructure in “Driven Alloys”, i.e., alloys exposed to a steady flow of energy, such as alloys under irradiation, or ball milling, or cyclic loading. . . [40]. Atoms in such alloys, change position as a function of time because of two mechanisms acting in parallel: one of the thermal diffusion mechanisms as discussed above, on the one hand, and forced, or “ballistic jumps”

2244

G. Martin and F. Soisson

on the other hand. The latter occur with a frequency imposed by the coupling with the surrounding of the system: their frequency is proportional to some “forcing intensity” (e.g., the irradiation flux). This situation is reminiscent of the “Kinetic Ising Model with two competing dynamics”, much studied in the late 80s. However, one observes a strong sensitivity of the results to the details of the diffusion mechanism and of the ballistic jumps. The main results are : – a solubility limit which is a function both of the temperature and of the ratio of the frequencies of ballistic to thermally activated jumps (i.e., on the forcing intensity); – at given temperature and forcing intensity, the solubility limit may also depend on the number of ballistic jumps to occur at once (“cascade size effect”); – the “replacement distance”, i.e., the distance of ballistic jumps has a crucial effect on the phase diagrams as shown in Fig. 10. For appropriate replacement distances, self-patterning can occur, with a characteristic length, which depends on the forcing intensity and on the replacement distance [41]. What has been said of the solubility limit also applies to the kinetic pathways followed by the microstructure when the forcing conditions are changed. Such KMC studies and the associated theoretical work helped to understand, for alloys under irradiation, the respective effects of the time and space structure of the elementary excitation, of the dose rate and of the integrated dose (or “fluence”). (a)

(A) G 5 104 s1

(B) 103 s1

(b) 2

1 Patterning

Solid Solution

(C) 102 s1

(D) 1 s1

R (ann)

10

1

Macroscopic Phase Separation 102 101 100

101

102

103

104

105

(s1)

Figure 10. (a) Steady–state microstructures in KMC simulations of the phase separation in a binary alloy, for different ballistic jump frequencies . (b) Dynamical phase diagram showing the steady–state microstructure as a function of the forcing intensity and the replacement distance R [41].

Diffusion controlled phase transformations in the solid state

5.

2245

Conclusion and Future Trends

The above presentation is by no means exhaustive. It aimed mainly at showing the necessity to model carefully the diffusion mechanism, and the techniques to do so, in order to have a realistic kinetic pathway for solid state transformations. All the examples we gave are based on a rigid lattice description. The latter is correct as long as strain effects are not too large, as shown by the discussion of the Fe(Cu) alloy. Combining KMC for the configuration together with some technique to handle the relaxation of atomic positions is quite feasible, but for the time being requires a heavy computation cost if the details of the diffusion mechanism are to be retained. Interesting results have been obtained e.g., for the formation of strained hetero-epitaxial films [42]. A field of growing interest is the first principle determination of the parameters entering the transition probabilities. In view of the lack of experimental data for relevant systems, and of the fast improvement of such techniques, no doubt such calculations will be of extreme importance. Finally, at the atomic scale, all the transitions modeled so far are either thermally activated or forced at some imposed frequency. A field of practical interest is where “stick and slip” type processes are operating: such is the case in shear transformations, in coherency loss etc. Incorporating such processes in KMC treatment of phase transformations has not yet been attempted to our knowledge, and certainly deserves attention.

Acknowledgments We gratefully acknowledge many useful discussions with our colleagues at Saclay and at the Atom Probe Laboratory in the University of Rouen, as well as with Prs. Pascal Bellon (UICU) and David Seidman (NWU).

Appendix: The Kinetic Ising Model In the KIM, the kinetic version of the model proposed by Ising for magnetic

materials, the configurational Hamiltonian writes H = i=/ j Ji j σi σ j + i h i σi , with σ ι = ± 1, the spin at site i, Ji j , the interaction parameter between spins at sites i and j , and h i the external field on site i. The probability of a transition per unit time, between two configurations {σι } and {σι } is chosen as: W{σ },{σ } = w exp[−(H − H )/2kB T ], with w for the inverse time unit. Two models are studied:

KIM with conserved total spin, for which i σi = so that the configuration after the transition is obtained by permuting the spins on two (nearest neighbor) sites;

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G. Martin and F. Soisson

KIM with non-conserved total spin, for which the new configuration is obtained by flipping one spin on one given site. When treated by Monte Carlo technique, two types of algorithms are currently applied to KIM: Metropolis’ algorithm, where the final configuration is accepted with probability one if (H − H ) ≤ 0, and with probability exp[−(H − H )/kB T ] if (H − H ) > 0. Glauber’s the final configuration is accepted with proba algorithm, where bility 1/2 1 + tanh(−(H − H )/2kB T ) .

References [1] A.R. Allnatt and A.B. Lidiard, “Atomic transport in solids,” Cambridge University Press, Cambridge, 1994. [2] T. Morita, M. Suzuki, K. Wada, and M. Kaburagi, “Foundations and Applications of Cluster Variation Method and Path Probability Method,” Prog. Theor. Phys. Supp., 115, 1994. [3] K. Binder, “Applications of Monte Carlo methods to statistical physics,” Rep. Prog. Phys., 60, 1997. [4] Y. Limoge and J.-L. Bocquet, “Monte Carlo simulation in diffusion studies: time scale problems,” Acta Met., 36, 1717, 1988. [5] G.E. Murch and L. Zhang, “Monte Carlo simulations of diffusion in solids: some recent developments,” In: A.L. Laskar et al. (eds.), Diffusion in Materials, Kluwer Academic Publishers, Dordrecht, 1990. [6] C.P. Flynn, “Point defects and diffusion,” Clarendon Press, Oxford, 1972. [7] J. Philibert, “Atom movements, diffusion and mass transport in solids,” Les Editions de Physique, Les Ulis, 1991. [8] D.N. Seidman and R.W. Balluffi, “Dislocations as sources and sinks for point defects in metals,” In: R.R. Hasiguti (ed.), Lattice Defects and their Interactions, GordonBreach, New York, 1968. [9] J.-L. Bocquet, G. Brebec, and Y. Limoge, “Diffusion in metals and alloys,” In: R.W. Cahn and P. Haasen (eds.), Physical Metallurgy, North-Holland, Amsterdam, 1996. [10] M. Nastar, V.Y. Dobretsov, and G. Martin, “Self consistent formulation of configurational kinetics close to the equilibrium: the phenomenological coefficients for diffusion in crystalline solids,” Philos. Mag. A, 80, 155, 2000. [11] G. Martin, “The theories of unmixing kinetics of solids solutions,” In: Solid State Phase Transformation in Metals and Alloys, pp. 337–406. Les Editions de Physique, Orsay, 1978. [12] A. Perini, G. Jacucci, and G. Martin, “Interfacial contribution to cluster free energy,” Surf. Sci., 144, 53, 1984. [13] T.R. Waite, “Theoretical treatment of the kinetics of diffusion-limited reactions,” Phys. Rev., 107, 463–470, 1957. [14] I.M. Lifshitz and V.V. Slyosov, “The kinetics of precipitation from supersaturated solid solutions,” Phys. Chem. Solids, 19, 35, 1961. [15] C.J. Kuehmann and P.W. Voorhees, “Ostwald ripening in ternary alloys,” Metall. Mater Trans., 27A, 937–943, 1996.

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[16] J.W. Cahn, W. Craig Carter, and W.C. Johnson (eds.), The selected works of J.W. Cahn., TMS, Warrendale, 1998. [17] G. Martin, “Atomic mobility in Cahn’s diffusion model,” Phys. Rev. B, 41, 2279– 2283, 1990. [18] C. Desgranges, F. Defoort, S. Poissonnet, and G. Martin, “Interdiffusion in concentrated quartenary Ag–In–Cd–Sn alloys: modelling and measurements,” Defect Diffus. For., 143, 603–608, 1997. [19] S.M. Allen and J.W. Cahn, “A macroscopic theory for antiphase boundary motion and its application to antiphase domain coarsening,” Acta Metal., 27, 1085–1095, 1979. [20] P. Bellon and G. Martin, “Coupled relaxation of concentration and order fields in the linear regime,” Phys. Rev. B, 66, 184208, 2002. [21] C. Pareige, F. Soisson, G. Martin, and D. Blavette, “Ordering and phase separation in Ni–Cr–Al: Monte Carlo simulations vs Three-Dimensional atom probe,” Acta Mater., 47, 1889–1899, 1999. [22] Y. Le Bouar and F. Soisson, “Kinetic pathways from EAM potentials: influence of the activation barriers,” Phys. Rev. B, 65, 094103, 2002. [23] E. Clouet and N. Nastar, “Monte Carlo study of the precipitation of Al3 Zr in Al–Zr,” Proceedings of the Third International Alloy Conference, Lisbon, in press, 2002. [24] J.-L. Bocquet, “On the fly evaluation of diffusional parameters during a Monte Carlo simulation of diffusion in alloys: a challenge,” Defect Diffus. For., 203–205, 81–112, 2002. [25] R. LeSar, R. Najafabadi, and D.J. Srolovitz, “Finite-temperature defect properties from free-energy minimization,” Phys. Rev. Lett., 63, 624–627, 1989. [26] A.P. Sutton, “Temperature-dependent interatomic forces,” Philos. Mag., 60, 147– 159, 1989. [27] Y. Mishin, M.R. Sorensen, F. Arthur, and A.F. Voter, “Calculation of point-defect entropy in metals,” Philos. Mag. A, 81, 2591–2612, 2001. [28] D. Gendt, Cin´etiques de Pr´ecipitation du Carbure de Niobium dans la ferrite, CEA Report, 0429–3460, 2001. [29] M. Ath`enes, P. Bellon, and G. Martin, “Identification of novel diffusion cycles in B2 ordered phases by Monte Carlo simulations,” Philos. Mag. A, 76, 565–585, 1997. [30] M. Ath`enes and P. Bellon, “Antisite diffusion in the L12 ordered structure studied by Monte Carlo simulations,” Philos. Mag. A, 79, 2243–2257, 1999. [31] A. Ath`enes, P. Bellon, and G. Martin, “Effects of atomic mobilities on phase separation kinetics: a Monte Carlo study,” Acta Mater., 48, 2675, 2000. [32] R. Wagner and R. Kampmann, “Homogeneous second phase precipitation,” In: P. Haasen (ed.), Phase Transformations in Materials, VCH, Weinhem, 1991. [33] F. Soisson, A. Barbu, and G. Martin, “Monte Carlo simulations of copper precipitation in dilute iron-copper alloys during thermal ageing and under electron irradiation,” Acta Mater., 44, 3789, 1996. [34] P. Auger, P. Pareige, M. Akamatsu, and D. Blavette, “APFIM investigation of clustering in neutron irradiated Fe–Cu alloys and pressure vessel steels,” J. Nucl. Mater., 225, 225–230, 1995. [35] P. Fratzl and O. Penrose, “Kinetics of spinodal decomposition in the Ising model with vacancy diffusion,” Phys. Rev. B, 50, 3477–3480, 1994. [36] J.-M. Roussel and P. Bellon, “Vacancy-assisted phase separation with asymmetric atomic mobility: coarsening rates, precipitate composition and morphology,” Phys. Rev. B, 63, 184114, 2001. [37] F. Soisson and G. Martin, Phys. Rev. B, 62, 203, 2000.

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[38] E. Clouet, M. Nastar, and C. Sigli, “Nucleation of Al3 Zr and Al3 Sc in aluminiun alloys: from kinetic Monte Carlo simulations to classical theory,” Phys. Rev. B, 69, 064109, 2004. [39] M. Ath`enes, P. Bellon, G. Martin, and F. Haider, “A Monte Carlo study of B2 ordering and precipitation via vacancy mechanism in BCC lattices,” Acta Mater., 44, 4739–4748, 1996. [40] G. Martin and P. Bellon, “Driven alloys,” Solid State Phys., 50, 189, 1997. [41] R.A. Enrique and P. Bellon, “Compositional patterning in immiscible alloys driven by irradiation,” Phys. Rev. B, 63, 134111, 2001. [42] C.H. Lam, C.K. Lee, and L.M. Sander, “Competing roughening mechanisms in strained heteroepitaxy: a fast kinetic Monte Carlo study,” Phys. Rev. Lett., 89, 216102, 2002.

7.10 DIFFUSIONAL TRANSFORMATIONS: MICROSCOPIC KINETIC APPROACH I.R. Pankratov and V.G. Vaks Russian Research Centre, “Kurchatov Institute”, Moscow 123182, Russia

The term “diffusional transformations” is used for the phase transformations (PTs) of phase separation or ordering of alloys as these PTs are realized via atomic diffusion, i.e., by interchange of positions of different species atoms in the crystal lattice. Studies of kinetics of diffusional PTs attract interest from both fundamental and applied points of view. From the fundamental side, the creation and evolution of ordered domains or precipitates of a new phase provide classical examples of the self-organization phenomena being studied in many areas of physics and chemistry. From the applied side, the macroscopic properties of such alloys, such as their strength, plasticity, coercivity of ferromagnets, etc., depend crucially on their microstructure, in particular, on the distribution of antiphase or interphase boundaries separating the differently ordered domains or different phases, while this microstructure, in its turn, sharply depends on the thermal and mechanical history of an alloy, in particular, on the kinetic path taken during the PT. Therefore, the kinetics of diffusional PTs is also an important area of Materials Science. Theoretical treatments of these problems employ usually either Monte Carlo simulation or various phenomenological kinetic equations for the local concentrations and local order parameters. However, Monte Carlo studies in this field are difficult, and until now they provided limited information about the microstructural evolution. The phenomenological equations are more feasible, and they are widely used to describe the diffusional PTs, see, e.g., Turchi and Gonis [1], part I. However, a number of arbitrary assumptions are usually employed in such equations, and their validity region is often unclear [2]. Recently, the microscopic statistical approach has been suggested to treat the diffusional PTs [3–5]. It aims to develop the theoretical methods which can describe the non-equilibrium alloys as consistently and generally as the canonical Gibbs method describes the equilibrium systems. This approach was used for simulations of many different PTs. The simulations revealed a number 2249 S. Yip (ed.), Handbook of Materials Modeling, 2249–2268. c 2005 Springer. Printed in the Netherlands.

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of new and interesting microstructural effects, many of them agreeing well with experimental observations. Below we describe this approach.

1. 1.1.

Statistical Theory of Non-equilibrium Alloys Master Equation Approach: Basic Equations

A consistent microscopical description of non-equilibrium alloys can be based on the fundamental master equation for the probabilities of various atomic distributions over lattice sites [3, 4]. For definiteness, we consider a binary alloy Ac B1−c with c ≤ 0.5. Various distributions of atoms over lattice sites i are described by the sets of occupation numbers {n i } where the operator n i = n Ai is unity when the site i is occupied by atom A and zero otherwise. The interaction Hamiltonian H has the form H=

vi j ni n j +

vi j k ni n j nk + · · ·

(1)

i> j >k

i> j

where v i... j are effective interactions. The fundamental master equation for the probability P of finding the occupation number set {n i } = α is dP(α) = [W (α, β)P(β) − W (β, α)P(α)] ≡ Sˆ P dt β

(2)

where W (α, β) is the β → α transition probability per unit time. Adopting for this probability the conventional “thermally activated atomic exchange model”, we can express the transfer matrix Sˆ in Eq. (2) in terms of the probability WiAB j of an elementary inter-site exchange Ai B j : s ˆ in ˆ in WiAB j = n i n j ωi j exp[−β(E i j − E i j )] ≡ n i n j γi j exp(β E i j ).

(3)

Here n j = n B j = (1 − n j ); ωi j is the attempt frequency; β = 1/T is the reciprocal temperature; E isj is the saddle point energy; γi j is ωi j exp(−β E isj ); and Eˆ iinj is the initial (before the jump) configurational energy of jumping atoms. The most general expression for the probability P{n i } in (2) can be conveniently written in the “generalized Gibbs” form:

P{n i } = exp

β

+

i

λi n i − Q

.

(4)

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Here the parameters λi can be called the “site chemical potentials”; the “quasiHamiltonian” Q is an analogue of the hamiltonian H in (1); and the generalized grand-canonical potential = {λi , ai... j } is determined by the normalizing condition: Q=

ai j n i n j +

ai j k n i n j n k + · · ·

i> j >k

i> j

= −T ln Tr exp

β

λi n i − Q

(5)

i

where Tr (. . .) means the summation over all configurations {n i }. Multiplying Eq. (2) by operators n i , n i n j , etc., and summing over all configurations, we obtain the set of exact kinetic equations for averages gi j ...k = n i n j . . . n k , in particular, for the mean site occupation ci ≡ gi = n i where . . . means Tr (. . . )P: dgi... j ˆ = n i . . . n j S. (6) dt These equations enable us to derive an explicit expression for the free energy of a non-equilibrium state, F = F{ci , gi... j }, which obeys both the “generalized first” and the second law of thermodynamics: F = H + T ln P = + dF =

iα

λi =

λi dci +

∂F ∂ci

λi ci + H − Q

iα

(v i... j − ai... j ) dgi... j

i>... j

(v i... j − ai... j ) =

dF ≤ 0. dt

∂F ∂gi... j (7)

The stationary state (being not necessarily uniform) corresponds to the minimum of F with respect to its variables ci and gi... j provided the total number of atoms N A = i ci is fixed. Then the relations (7) yield the usual, Gibbs equilibrium equations: λi = µ = constant; ai... j = v i... j ,

or :

(8) Q = H.

(9)

Non-stationary atomic distributions arising under the usual conditions of diffusional PTs appear to obey the “quasi-equilibrium” relations which correspond to an approximate validity in the course of the evolution of the second

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equilibrium Eq. (9), while the site chemical potentials, generally, differ with each other [2]. Then the free energy F in (7) takes the form: F =+

λi ci

(10)

i

while the system of Eq. (6) is reduced to the “quasi-equilibrium” kinetic equation (QKE) for the mean occupations ci = ci (t) [3]:

β(λ j − λi ) dci Mi j 2 sinh . = dt 2 j

(11)

Here the quantities λ j are related to ci by the self-consistency equation:

ci = n i = Tr n i P{λ j }

(12)

while the “generalized mobility” Mi j for the pair interaction case, when the Hamiltonian (1) includes only the first term, can be written as [6]:

Mi j = γi j n i n j exp

     β λ + λ − (v + v + u + u )n   i j jk ik jk k  k ik   

2

  

. (13)

AB BB Here γi j , n i and v i j = ViAA j − 2Vi j + Vi j are the same as in Eqs. (3) and (1), AA BB while u i j = Vi j − Vi j is the so-called asymmetric potential. The description of the diffusional PTs in terms of the mean occupations ci given by Eqs. (11)–(13) seems to be sufficient for the most situations of practical interest, in particular, for the “mesoscopic” stages of such PTs when the local fluctuations of occupations are insignificant. At the same time, to treat the fluctuative phenomena, such as the formation and evolution of critical nuclei in metastable alloys, one should modify the QKE (11), for example, by an addition of some “Langevin-noise”-type terms [4].

1.2.

Kinetic Mean-field and Kinetic Cluster Approximations

To find explicit expressions for the functions F{ci }, λi {c j }, and Mi j {ck } in Eqs. (10)–(12), one should employ some approximate method of statistical physics. Several such methods have been developed [4]. For simplicity we consider the pair interaction model and write the interaction v i j in (1) as δi j,n v n where the symbol δi j,n is unity when sites i and j are nth neighbors in the lattice and zero otherwise, while v n is the interaction constant. Then the simplest,

Diffusional transformations: microscopic kinetic approach

2253

“kinetic mean-field” approximation (KMFA, or simply MFA) corresponds to the following expressions for , λi and Mi j : MFA =

T ln ci −

i

λMFA i

=T

ln (ci /ci )

1 δi j,n v n ci c j 2 i, j,n

+

(14)

δi j,n v n c j

j,n

MiMFA j

= γi j

ci ci c j cj

exp β

1/2

(u ik + u j k )ck )

.

(15)

k

Here ci is 1 − ci , while the free energy F is related to and λi by Eq. (10). For a more refined and usually more accurate, kinetic pair-cluster approximation (KPCA, or simply PCA), the expressions for and λi are more complex but still can be written analytically: PCA =

T ln ci +

i

λPCA i

= T ln (ci /ci ) +

1 δi j,n inj 2 i, j,n

(16)

ij

δi j,n λni .

j,n ij

Here inj = − T ln(1 − ci c j gni j ); λni = −T ln(1 − c j gni j ); and the function gni j is expressed via the Mayer function f n = exp (−βv n ) − 1 and the mean occupations ci and c j : gni j = Rni j

2 fn ij Rn

+ 1 + f n (ci + c j )

= [1 + (ci + c j ) f n ] − 4ci c j f n ( f n + 1) 2

1/2

(17) .

For the weak interaction, βv n 1, the function gni j becomes (−βv n ), inj − ij v i j ci c j , λni v n c j , and the PCA expressions (16) become the MFA ones (14). The MFA or the PCA is usually sufficient to describe the PTs between the disordered phases and/or the BCC-based ordered phases, such as the B2 and D03 phases. However, these simple methods are insufficient to describe the FCC-based L12 and L10 ordered alloys as strong many-particle correlations are characteristic of such systems. These alloys can be adequately described by the cluster variation method (CVM) which takes into account the correlations mentioned within at least 4-site tetrahedron cluster of nearest neighbors. However, the CVM is cumbersome, and it is difficult to use it for the non-homogeneous systems. At the same time, a simplified version of CVM, the tetrahedron cluster-field approximation (TCA), usually combines the high accuracy of CVM with great simplification of calculations [6].

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The TCA expressions for and λi can be written explicitly and are similar to those in Eq. (16), but to find the functions (ci ) and λi (c j ) explicitly one should solve the system of four algebraic equations for each tetrahedron cluster. In practice, these equations can easily be solved numerically using the conjugate gradients method [4, 7]. We can also use the PCA or the TCA methods to more accurately calculate the mobility Mi j in the expression (13) [4]. However, in this expression the above-mentioned correlations of atomic positions result only in some quantitative factors that weakly depend on the local composition and ordering and seem to be of little importance for the microstructural evolution. Therefore, the simple MFA expression (15) for Mi j was employed in the previous KTCAbased simulations of the L12 and L10 -type orderings [4, 7].

1.3.

Deformational Interactions in Dilute and Concentrated Alloys

The effective interaction v i... j in the Hamiltonian (1) includes the “chemic cal” contribution v i... j which describes the energy change under the substitution of some atoms A by atoms B in the rigid lattice, and the “deformational” d term v i... j due to the difference in the lattice deformation under such a substitution. The interaction v d includes the long-range elastic forces which can significantly affect the microstructural evolution, see, e.g., Turchi and Gonis [1]. A microscopical model to calculate the interaction v d in dilute alloys was suggested by Khachaturyan [8]. In the concentrated alloys, the deformational interaction can lead to some new effects, in particular, to the lattice symmetry change under PT, such as the tetragonal distortion under L10 ordering. Below we describe the generalization of the Khachaturyan’s model of deformational interactions to the case of a concentrated alloy [9]. Supposing a displacement uk of site k relative to its position Rk in the “average” crystal Ac B1−c to be small, we can write the alloy energy H as H = Hc {n i } −

u αk Fαk +

k

1 u αk u βl Aαk,βl 2 αk,βl

(18)

where α and β are Cartesian indices and both the Kanzaki force Fk and the force constant matrix Aαk,βl are some functions of occupation numbers n i . For the force constant matrix, the conventional average crystal approximation seems usually to be sufficient: Aαk,βl {n i } → Aαk,βl {c} ≡ A¯ αk,βl . The Kanzaki force Fαk can be written as a series in the occupation numbers n i : Fαk =

i

(1) f αk,i ni +

i> j

(2) f αk,i j ni n j + · · ·

(19)

Diffusional transformations: microscopic kinetic approach

2255

where the coefficients f (n) do not depend on n i . Minimizing the energy (18) with respect to displacements uk we obtain for the deformational Hamiltonian Hd: 1 Fαk ( A¯ −1 )αk,βl , Fβl (20) Hd = − 2 αk,βl where ( A¯ −1 )αk,βl means the matrix inverse to A¯ αk,βl which can be written ¯ explicitly using the Fourier transformation of the force constant matrix A(k). For the dilute alloys, one can retain in (19) only the first sum which corresponds to a pairwise H d by Khachaturyan [8]. The next terms in (19) lead to non-pairwise interactions which describe, in particular, the above-mentioned effects of a lattice symmetry change. To describe these effects, for example, for the case of the L10 ordering in the FCC lattice, we can retain in (19) only terms with f (1) and f (2) and estimate them from the experimental data about the concentration dilatation in the disordered phase and about the lattice parameter changes under the L12 and L10 orderings [6, 9].

1.4.

Vacancy-mediated Kinetics and Equivalence Theorem

In the most theoretical treatments of kinetics of diffusional PT, as well as in the previous part of this paper, the simplified direct exchange model was used which assumes direct exchange of positions between unlike neighboring atoms in an alloy. Actually, the exchange occurs between the main alloy component atom, e.g., A or B atom in an ABv alloy, and the neighboring vacancy “v”. As the vacancy concentration cv in alloys is actually quite small, cv 10−4 , employing the direct exchange model greatly simplifies the theoretical studies by reducing the computation times by several orders of magnitude. However, it is not clear a priori whether using the unrealistic direct exchange model results in some errors or missing some effects. In particular, a notable segregation of vacancies at interphase or antiphase boundaries was observed in a number of simulations, and the problem of possible influence of this segregation on the microstructural evolution was discussed by a number of authors. To clarify these problems, the statistical approach described above has been generalized to the vacancy-mediated kinetics case [5]. In particular, the QKE for an ABv alloy, instead of Eq. (11), takes the form of a set of equations for the A-atom and the vacancy mean occupations, cAi = ci and cvi :

dci Av = γi j Bi j eβ(λA j + λvi ) − eβ(λAi + λv j ) dt j

dcvi vA βλAi Bi j eβλv j γivB = j + γi j e dt j

(21)

− {i → j } .

(22)

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Here Bi j is an analogue of the second factor in Eq. (13), while λAi and λvi are the site chemical potentials for the A atom and the vacancy, respectively, in (14): which in the MFA have the form similar to λMFA i

λMFA Ai

= T ln

ci ci

+

v iAA j cj;

λMFA vi

j

= T ln

cvi ci

+

v ivA j cj

j

(23) where v ivA j is an effective interaction between a vacancy and an A atom. The main alloy components kinetics determined by the QKE (21) can usually be described in terms of a certain equivalent direct exchange model; this statement can be called “the equivalence theorem”. To prove it, we first note that the factor exp(βλvi ) in Eqs. (21) and (22) is proportional to the vacancy concentration cvi , which is illustrated by (22) and is actually a general relation of thermodynamics of dilute solutions. Thus the time derivatives of the mean occupations are proportional to the local vacancy concentration cvi or cv j , which is natural for the vacancy-mediated kinetics. As cvi is quite small, this implies that the main component relaxation times are by a factor 1/cvi larger than the time of the relaxation of vacancies to their “quasi-equilibrium” distribution cvi {ci } minimizing the free energy F{cvi , ci } at the given main component distribution {ci }. Therefore, neglecting the small correction of the relative order of cvi 1, we can find this “adiabatic” vacancy distribution cvi by equating the left-hand side of (22) to zero. Employing for simplicity the vB conventional nearest-neighbor vacancy exchange model: γivB j = δi j,1 γnn and vA vA γi j = δi j,1 γnn , we can solve this equation explicitly. The solution corresponds to the first term in square brackets in (22) to be constant not depending on the site number i, though it can, generally, depend on time: νi =

vB γnn exp(βλvi ) = ν(t) vB Av [γnn + γnn exp(βλρi )]c¯v

(24)

vB and the average concentration of vacancies c¯v where the common factor γnn are introduced for convenience. Relations (24) determine the adiabatic vacancy distribution cvi {ci } mentioned above. Substituting these relations into (21) we obtain the QKE for the main alloy component which has the “direct exchange” form (11) with an effective rate vA γieff j = γi j c¯v ν(t).

(25)

Physically, the opportunity to reduce the vacancy-mediated kinetics to the equivalent direct exchange kinetics is connected with the above-mentioned fact that in the course of the alloy evolution the vacancy distribution adiabatically fast follows that of the main components. Thus it is natural to believe that

Diffusional transformations: microscopic kinetic approach

2257

for the quasi-equilibrium stages of evolution under consideration such equivalence holds not only for the nearest-neighbor vacancy exchange model but is actually a general feature of any vacancy-mediated kinetics. In more detail, features of the vacancy-mediated kinetics for both the phase separation and the ordering case have been discussed by Belashchenko and Vaks [5] who used computer simulations based on Eqs. (21) and (22). The simulations confirmed the equivalence theorem for the “quasi-equilibrium” stages of evolution, t τAB , where τAB is the mean time needed for an exchange of neighboring A and B atoms. The function ν(t) in (24) was found to monotonously increase with the PT time t, and in the course of the PT this function slowly approaches its equilibrium value ν∞ . At the same time, at very early stages of PT, for times t less than the vacancy distribution equilibration time τve , the equivalence theorem does not hold as the spatial fluctuations in the initial vacancy distribution are here important. These fluctuations can lead, in particular, to a peculiar phenomenon of “localized ordering” observed by Allen and Cahn [10] in Fe–Al alloys. However, at later times t τve ∼ τAB · cv1/3 , the vacancy distribution equilibrates and the equivalence theorem holds.

2.

Applications of Statistical Approach for Simulation of Diffusional Transformations

Numerous applications of the above-described statistical methods for simulation of diffusional PTs are discussed and compared to experimental observations in reviews [4, 7]. Below we illustrate these applications with some examples.

2.1.

Methods of Simulation

Most of these simulations were based on the QKE (11). For the mobility Mi j in this equation, the MFA expression (15) with the “nearest-neighbor symmetric atomic exchange”, γi j = δi j,1 γnn and u i j = 0, was usually used. Vaks, Beiden and Dobretsov [11] also considered the effect of an asymmetric potential u i j =/ 0 on spinodal decomposition. For the site chemical potential λi in the disordered phase and in the BCC-based ordered phases, the MFA expression (14) was employed which is usually sufficient to describe PTs between these phases. The simulations of the L12 - and L10 -type orderings in FCC alloys were based on the KTCA expressions. Equations (11) were usually solved by the 4th-order Runge–Kutta method [12] with the dimensionless time variable t = tγnn and the variable time-step t . This time-step was chosen so that the maximum variation | ci | = |ci (t + t ) − ci (t )| for one time-step does not exceed 0.01. The typical t values were 0.01 − 0.1, depending on

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the evolution stage. For the PTs after a quench of a disordered alloy, the initial as-quenched distribution ci =c(Ri ) at t =0 was characterized by its mean value c and small random fluctuations δci ±0.01. The most of simulations were performed on 2D lattices with periodic boundary conditions as it enables us to study more sizable structures. However, some main conclusions were also verified by 3D simulations with periodic boundary conditions.

2.2.

Spinodal Decomposition of Disordered Alloys

Vaks, Beiden and Dobretsov [11] simulated spinodal decomposition (SD) of a disordered alloy after its quench into the spinodal instability area in the c, T plane. The interaction v i j = v(ri j ) = v(Ri − R j ) was assumed to be Gaussian and long-ranged: v(r)=−A exp (−r 2 /rv2 ) with rv2 a 2 and the constant A proportional to the critical temperature Tc . Some results of this simulation are presented in Figs. 1 and 2. The figures illustrate the transition from the initial stage of SD corresponding to the development of non-interacting Cahn’s concentration waves with growing amplitudes (see, e.g., [8]) to the next stages, first to the stage of non-linear interaction of concentration waves (Fig. 1), and then to the stage of interaction and fusion of new-formed precipitates via a peculiar “bridge” mechanism

(a)

(b)

Figure 1. Profiles of the concentration c(r) at spinodal decomposition for the 2D model described in the text at c = 0.35; T = T/Tc = 0.4, u i j = 0 , and the following values of the reduced time t = tγnn : (a) 5; and (b) 10. Distances at the horizontal axes are given in the interaction radius rv units.

Diffusional transformations: microscopic kinetic approach

2259

Figure 2. Distribution of c(r) for the same model as in Fig. 1 at the following t : (a) 20, (b) 120, (c) 130, (d) 140, (e) 160, (f) 180, (g) 200, and (h) 5000. The grey level linearly varies with c(r) for c between 0 and 1 from completely dark to completely bright.

illustrated by Fig. 2. This mechanism was discussed in detail by Vaks, Beiden and Dobretsov [11], while the microstructures shown in Fig. 2 reveal a striking similarity with those observed in the recent experimental studies of SD in some liquid mixtures [4].

2.3.

Kinetics of B2 and D03 -type Orderings

The B2 order corresponds to the splitting of the BCC lattice into two cubic sublattices, a and b, with the displacement vector rab = [1, 1, 1]a/2 and the mean occupations ca = c + η and cb = c − η where η is the order parameter. There are two types of antiphase ordered domain (APD) differing with the sign of η, and one type of antiphase boundary (APB) separating these APDs. The inhomogeneously ordered alloy states including APBs can be conveniently described in terms of the local order parameter ηi = η(Ri ) and the local concentration c¯i = c(Ri ) obtained by the averaging of mean occupations ci over site i and its nearest neighbors: c¯i =

1 1 1 1 ci + cj ηi = ci − c j exp(ik1 Ri ). (26) 2 z nn j =nn(i) 2 z nn j =nn(i)

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I.R. Pankratov and V.G. Vaks

Here index nn(i) means the summation over nearest-neighbors of site i; z nn is the number of such neighbors, i.e., 4 for the 2D square lattice and 8 for the 3D BCC lattice; and the superstructure vector k1 is (1, 1)2π/a or (1, 1, 1)2π/a for the 2D or 3D case, respectively. Dobretsov, Martin and Vaks [13] investigated kinetics of phase separation with B2 ordering using the KMFA-based 2D simulations on a square lattice of 128 × 128 sites and the Fe–Al-type interaction model. The simulations enabled one to specify the earlier phenomenological considerations [10] and to find a number of new effects. As an illustration, in Fig. 3 we show the evolution after a quench of an alloy from the disordered A2 phase to the two-phase state in which SD into the B2 and the A2 phases takes place. The volume ratio of these two phases in the final mixture is the same as that for the disordered “dark” and “bright” phases in Fig. 2, and so one might expect a similarity of microstructural evolution for these two transformations. However, the formation of numerous APBs at the initial, “congruent ordering” stage of PT A2 → A2 + B2 (which occurs at an approximately unchanged initial concentration c) ¯ and the subsequent “wetting” of these APBs by the A2 phase lead to significant structural differences with the SD into disordered phases. In particular, the concentration c(r) ¯ and the order parameter η(r) at the first stages of SD shown in Figs. 3(a)–3(c) form a “ridge-valley”-like pattern, rather

Figure 3. Temporal evolution of mean occupationals ci =c(ri ) for the Fe–Al-type alloy model under PT A2→A2+B2 at c = 0.175, T = 0.424, and the following t : (a) 50, (b) 100, (c) 200, (d) 1000, (e) 4000, and (f) 9000.

Diffusional transformations: microscopic kinetic approach

2261

than the “hill-like” pattern seen in Fig. 1. For the PT B2 → A2 + B2, the simulations reveal some peculiar microstructural effects in vicinity of initial APBs, the formation of wave-like distributions, “broken layers” of ordered and disordered domains parallel to the initial APB, and these results agree well with experimental observations for Fe–Al alloys [4, 10]. For the homogeneous D03 phase, the mean occupation ci can be written as ci = c + η exp(ik1 Ri ) + ζ [exp(ik2 Ri )sgn(η) + exp(−ik2 Ri )].

(27)

Here Ri is the BCC lattice vector of site i; k2 = [111]π/a is the D03 superstructure vectors, and η or ζ is the B2- or the D03 -type order parameter. Both η and ζ in (27) can be positive and negative, thus there are four types of ordered domain and two types of APB, which separate either the APDs differing in the sign of η (“η-APB”), or the APDs differing in the sign of ζ (“ζ -APB”). Using the relations analogous to (26), one can also define the local parameters ηi , ζi and c¯i , in particular, the local order parameter ηi2 used in Figs. 4 and 5: 

1  2 1 ci − cj + ηi2 = 16 z nn j =nn(i) z nnn

2

cj .

(28)

j =nnn(i)

Here nn(i) or nnn(i) means the summation over nearest or next-nearest neighbors of site i, and z nn or z nnn is the total number of such neighbors. The

a

b

c

d

e

f

Figure 4. Temporal evolution of model I for PT A2 → A2 + D03 at c = 0.187, T = T/Tc = 0.424, and the following t : (a) 10, (b) 30, (c) 100, (d) 500, (e) 1000, and (f) 2000. The grey level linearly varies with ηi2 defined by (28) between its minimum and maximum values from completely dark to completely bright.

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b

c

d

e

f

Figure 5. As Fig. 4, but for model II and PT A2 → A2 + B2 at c = 0.325, T = 0.424.

distribution of ηi2 is similar to that observed in the transmission electron microscopy (TEM) images with the reflection vector k1 [14]. To study kinetics of D03 ordering, Belashchenko, Samolyuk and Vaks [15] simulated PTs A2 → D03 , A2 → A2 + D03 , A2 → B2 + D03 and D03 → B2 + D03 using the Fe-Al-type interaction models. They also considered two more models, I and II, in which the deformational interaction v d was taken into account for the PT A2 → A2 + D03 and A2 → A2 + B2, respectively. The simulations reveal a number of microstructural features related to the “multivariance” of the D03 orderings. Some of them are illustrated in Figs. 4 and 5 where the PT A2 → A2 + D03 for model I is compared to the PT A2 → A2 + B2 for model II. The first stage of both PTs corresponds to congruent ordering at approximately unchanged initial concentration. Frame 4a illustrates the transient state in which only the B2 ordered APDs (“η-APDs”) are present. Frame 4b shows the formation of the D03 -ordered APDs (“ζ -APDs”) within initial η-APDs, and these ζ -APDs are much more regular-shaped than the η-APDs in frame 5b. Frames 4b–4d also illustrate wetting of both the η-APBs and ζ -APBs by the disordered A2 phase. Later on the deformational interaction tends to align the ordered precipitates along elastically soft (100) directions, and frame 4f shows an array of approximately rectangular D03 -ordered precipitates, unlike rod-like structures seen in frame 5f. The microstructure in frame 4f is similar to those observed for the PT A2 → A2 + D03 in alloys Fe– Ga, while the microstructure in frame 5f is similar to those observed for the PT B2 → B2 + D03 in alloys Fe–Si. The latter similarity reflects the topological equivalence of the A2 → A2 + B2 and B2 → B2 + D03 PTs [4].

Diffusional transformations: microscopic kinetic approach

2.4.

2263

Kinetics of L12 and L10 -type Orderings

For the FCC-based L12 - or L10 -ordered structures, the occupation ci of the FCC lattice site Ri is described by three order parameters ηα corresponding to three superstructure vectors kα : ci = c + η1 exp(ik1 Ri ) + η2 exp(ik2 Ri ) + η3 exp(ik3 Ri ) k1 = (1, 0, 0)2π/a k2 = (0, 1, 0)2π/a k3 = (0, 0, 1)2π/a

(29)

where a is the FCC lattice constant. For the cubic L12 structure |η1 | = |η2 | = |η3 |, η1 η2 η3 > 0, and four types of ordered domain are possible. In the L10 phase with the tetragonal axis α, a single nonzero parameter ηα is present which is either positive or negative. Thus six types of ordered domain are possible with two types of APB. The APB separating two APDs with the same tetragonal axis can be for brevity called the “shift-APB”, and that separating the APDs with perpendicular tetragonal axes can be called the “flip-APB”. The inhomogeneously ordered alloy states can be described by the local 2 similar to those in Eqs. (26) and (30), and by quantities ηi2 parameters ηαi characterizing the total degree of the local order: 

2

1  1 2 = ci + c j exp(ikα Ri j ) ; ηαi 16 4 j =nn(i)

2 2 2 ηi2 = η1i + η2i + η3i

(30)

where R j i is R j − Ri . Belashchenko et al. [6, 9] simulated PTs A1 → L12 , A1 → A1 + L12 , and A1 → L10 after a quench of an alloy from the disordered FCC phase A1. The simulations were performed in FCC simulation boxes of sizes Vb = L 2 × H , and the value H = 1 (in the lattice constant a units) corresponds to quasi-2D simulation when the simulation box contains two atomic planes. A number of different models have been considered: the short-range-interaction models 1, 2, and 3; the intermediate-range-interaction model 4 with v n estimated from the experimental data for Ni–Al alloys; and the extended-interaction model 5. In studies of PTs A1 → L10 , the models 1 –5 were also considered in which the deformational interaction v d was added to the “chemical” interactions v n of models 1–5. This v d was found with the use of Eq. (20) and the experimental data for Co–Pt alloys. The simulations revealed many interesting microstructural features for both the L12 and L10 -type orderings. It was found, in particular, that the character of the microstructural evolution strongly depends on the type of the interaction v i j , particularly on its interaction range rint , as well as on temperature T and the degree of non-stoichiometry δc which is (c − 0.25) for the L12 phase, and (c − 0.5) for the L10 phase. With increasing rint , T , or δc, the microstructures become more isotropic and the APBs become more diffuse and mobile. At the same time, for the short-range-interaction systems at not-high T and small δc,

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the microstructures are highly anisotropic while the most of APBs are thin and low-mobile. Figures 6 and 7 illustrate these features for the L12 -type orderings. Figure 6 shows the evolution under the A1 → L12 PT for the intermediate-interactionrange model 4 at non-stoichiometric c = 0.22. We see that the distribution of APBs is virtually isotropic. The main evolution mechanism is the growth of larger domains at the expense of smaller ones which is also typical for the simple B2 ordering. At the same time, one more mechanism, the fusion of in-phase domains, is also important for the multivariant orderings under consideration. For the later stages of evolution, Fig. 6 also reveals many approximately equiangular triple junctions of APDs with angles 120◦ ; it agrees with TEM observations for Cu–Pd alloys [14]. Kinetics of the A1 → L12 PT for the short-range-interaction system is illustrated in Fig. 7. The distribution of APBs here reveals a high anisotropy, a tendency to the formation of thin “conservative” APBs with (100)-type orientation. One also observes many “step-like” APBs with the conservative segments; the triple junctions of APBs with one non-conservative APBs and two conservative APBs; and the “quadruple” junctions of APDs. All these features were

Figure 6. Temporal evolution of model 4 under PT A1 → L12 for the simulation box size Vb = 1282 × 1 at c = 0.22, T = 0.685 and the following t : (a) 5; (b) 50; (c) 120; (d) 125; 2 + η2 + η2 between its mini(e) 140; and (f) 250. The grey level linearly varies with ηi2 = η1i 2i 3i mum and maximum values from completely dark to completely bright. The symbol A, B, C or D indicates the type of the ordered domain, and the thick arrow indicate the fusion-of-domain process.

Diffusional transformations: microscopic kinetic approach a

b

c

d

e

f

2265

Figure 7. As Fig. 6, but for model 1 and Vb = 642 × 1 at c = 0.25, T = 0.57 and the following t : (a) 2, (b) 3, (c) 20, (d) 100, (e) 177 and (f) 350.

observed in the electron microscopy studies of Cu3 Au alloys [14]. Figure 7 also illustrates the peculiar kinetic processes related to conservative APBs and discussed by Vaks [4, 7]. The L10 structure, unlike the cubic L12 structure, is tetragonal and has a tetragonal distortion . Depending on the importance of this distortion, the evolution in the course of the A1 → L10 PT can be divided into three stages. I. The initial stage when the L10 -ordered APDs are quite small, their tetragonal distortion is insignificant, and all six types of APD are present in the same proportion. II. The intermediate stage when the tetragonal distortion of APDs leads to some predominance of the (110)-type orientations of flip-APBs and to decreasing of the portion of APDs with the unfavorable orientation (001). III. The final, “twin” stage when the well-defined twin bands delimited by the flip-APBs with (110)-type orientation are formed. Each band includes only two types of APD with the same tetragonal axis, and these axes in the adjacent bands are “twin” related, i.e., have alternate (100) and (010) orientations. The thermodynamic driving force for the (110)-type orientation of flipAPBs is the gain in the elastic energy: at other orientations this energy increases proportionally to the volume of the adjacent APDs [8].

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The simulations of PTs A1 → L10 [9] revealed a number of peculiar microstructural features for each of the stages mentioned above. Figures 8 and 9 illustrate some of these features. Frame 8a corresponds to stage I ; frames 8b–8c, to stage II; and frames 8d–8f and 9a–9d, to stage III. The following processes and configurations are seen to be characteristic of both the stage I and stage II: (1) The abundant processes of fusion of in-phase domains which are among the main mechanisms of domain growth at these stages. (2) Peculiar long-living configurations, the quadruple junctions of APDs (4-junctions) of the type A1 A2 A1 A3 where A2 and A3 can correspond to any two of four types of APD different from A1 and A1 . (3) Many processes of “splitting” of a shiftAPB into two flip-APBs which leads either to the fusion of in-phase domains or to the formation of a 4-junction. For the final, “nearly equilibrium” twin stage, Figs. 8f and 9a–9d demonstrate a peculiar alignment of shift-APBs: within a (100)-oriented twin band in a (110)-type polytwin the APBs tend to align normally to some direction n = (cos α, sin α, 0) characterized by a “tilting” angle α which is mainly

Figure 8. Temporal evolution of model 4 under PT A1 → L10 for Vb = 1282 × 1 at c = 0.5, T = 0.67, and the following t : (a) 10; (b) 20; (c) 50; (d) 400; (e) 750; and (f) 1100. ¯ B or B ¯ and C or C¯ indicates an APD with the tetragonality axis along The symbol A or A, (100), (010) and (001), respectively. The thick, the thin and the single arrow indicates the fusion-of-domain process, the quadruple junction of APDs, and the splitting APB process, respectively.

Diffusional transformations: microscopic kinetic approach

2267

Figure 9. As Fig. 8, but for model 2 at the following values of c, T , and t : (a) c = 0.5, T = 0.77, and t = 350; (b) c = 0.5, T = 0.95, and t = 300; (c) c = 0.46, T = 0.77, and t = 350; and (d) c = 0.44, T = 0.77, and t = 300.

determined by the type of chemical interaction. For the short-range interaction systems this angle is close to zero, in agreement with observations for CuAu. For the intermediate-interaction-range systems, the scale of α is illus-trated by Fig. 8f, and the alignment of APBs shown in this figure is very similar to that observed for a Co0.4 Pt0.6 alloy [4]. Figure 9 also illustrates sharp changes of the alignment type under variation of temperature T and non-stoichiometry δc, including the “faceting-tilting”-type morphological transitions.

3.

Outlook

For the last decade the statistical theory of diffusional PTs has been formulated in terms of both approximate and exact kinetic equations and was applied to studies of many concrete problems. These applications yielded numerous new results, many of them agreeing well with experimental observations. Many predictions of this theory are still awaiting experimental verification. At the same time, there remain a number of further problems in this approach to be solved, such as the elaboration of a microscopical “phase-fieldtype” approach suitable for treatments of sizeable and complex structures [2]; the consistent treatment of fluctuative effects, including the problem of nucleation of embryos of a new phase within the metastable one, and others. Some of these problems are now underway, and for the nearest future one can expect a further progress in that field.

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References [1] P.E.A. Turchi and A. Gonis (eds.), “Phase transformations and evolution in materials,” TMS, Warrendale, 2000. [2] I.R. Pankratov and V.G. Vaks, “Generalized Ginzburg–Landau functionals for alloys: general equations and comparison to the phase-field method,” Phys. Rev. B, 68, 134208 (in press), 2003. [3] V.G. Vaks, “Master equation approach to the configurational kinetics of nonequilibrium alloys: exact relations, H-theorem and cluster approximations,” JETP Lett., 78, 168–178, 1996. [4] V.G. Vaks, “Kinetics of phase separation and orderings in alloys,” Physics Reports, 391, 157–242, 2004. [5] K.D. Belashchenko and V.G. Vaks, “Master equation approach to configurational kinetics of alloys via vacancy exchange mechanism: general relations and features of microstructural evolution,” J. Phys. Condensed Matter, 10, 1965–1983, 1998. [6] K.D. Belashchenko, V. Yu. Dobretsov, I.R. Pankratov et al., “The kinetic clusterfield method and its application to studes of L12 -type orderings in alloys,” J. Phys. Condens. Matter, 11, 10593–10620, 1999. [7] V.G. Vaks, “Kinetics of L12 -type and L10 -type orderings in alloys,” JETP Lett., 78, 168–178, 2003. [8] A.G. Khachaturyan, “Theory of structural phase transformations in solids,” Wiley, New York, 1983. [9] K.D. Belashchenko, I.R. Pankratov, G.D. Samolyuk et al., “Kinetics of formation of twinned structures under L10 -type orderings in alloys,” J. Phys. Condens. Matter, 14, 565–589, 2002. [10] S.M. Allen and J.W. Cahn, “Mechanisms of phase transformations within the miscibility gap of Fe-rich Fe-Al alloys,” Acta Metall., 24, 425–437, 1976. [11] V.G. Vaks, S.V. Beiden, V. Dobretsov, and Yu., “Mean-field equations for configurational kinetics of alloys at arbitrary degree of nonequilibrium,” JETP Lett., 61, 68–73, 1995. [12] G. Korn and T. Korn, “Mathematical handbook for scientists and engineers,” McGraw-Hill, New York, 1961. [13] V. Yu. Dobretsov, V.G. Vaks, and G. Martin, “Kinetic features of phase separation under alloy ordering,” Phys. Rev. B, 54, 3227–3239, 1996. [14] A. Loiseau, C. Ricolleau, L. Potez, and F. Ducastelle, “Order and disorder at interfaces in alloys,” In: W.C. Johnson, J.M. Howe, D.E. Mc Laughlin, and W.A. Soffa (eds.), Solid–Solid Phase Transformations, pp. 385–400, TMS, Warrendale, 1994. [15] K.D. Belashchenko, G.D. Samolyuk, and V.G. Vaks, “Kinetic features of alloy ordering with many types of ordered domain: D03 -type ordering,” J. Phys. Condens. Matter, 10, 10567–10592, 1999.

7.11 MODELING THE DYNAMICS OF DISLOCATION ENSEMBLES Nasr M. Ghoniem Department of Mechanical and Aerospace Engineering, University of California, Los Angeles, CA 90095-1597, USA

1.

Introduction

A fundamental description of plastic deformation is under development by several research groups as a result of dissatisfaction with the limitations of continuum plasticity theory. The reliability of continuum plasticity descriptions is dependent on the accuracy and range of available experimental data. Under complex loading situations, however, the database is often hard to establish. Moreover, the lack of a characteristic length scale in continuum plasticity makes it difficult to predict the occurrence of critical localized deformation zones. It is widely appreciated that plastic strain is fundamentally heterogenous, displaying high strains concentrated in small material volumes, with virtually undeformed regions in-between. Experimental observations consistently show that plastic deformation is internally heterogeneous at a number of length scales [1–3]. Depending on the deformation mode, heterogeneous dislocation structures appear with definitive wavelengths. It is common to observe persistent slip bands (PSBs), shear bands, dislocation pile ups, dislocation cells and sub grains. However, a satisfactory description of realistic dislocation patterning and strain localization has been rather elusive. Since dislocations are the basic carriers of plasticity, the fundamental physics of plastic deformation must be described in terms of the behavior of dislocation ensembles. Moreover, the deformation of thin films and nanolayered materials is controlled by the motion and interactions of dislocations. For all these reasons, there has been significant recent interest in the development of robust computational methods to describe the collective motion of dislocation ensembles. Studies of the mechanical behavior of materials at a length scale larger than what can be handled by direct atomistic simulations, and smaller than what allows macroscopic continuum averaging represent particular difficulties. Two 2269 S. Yip (ed.), Handbook of Materials Modeling, 2269–2286. c 2005 Springer. Printed in the Netherlands.

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complimentary approaches have been advanced to model the mechanical behavior in this meso length scale. The first approach, commonly known as dislocation dynamics (DD), was initially motivated by the need to understand the origins of heterogeneous plasticity and pattern formation. In its early versions, the collective behavior of dislocation ensembles was determined by direct numerical simulations of the interactions between infinitely long, straight dislocations [3–9]. Recently, several research groups extended the DD methodology to the more physical, yet considerably more complex 3D simulations. Generally, coarse resolution is obtained by the Lattice Method, developed by Kubin et al. [10] and Moulin et al. [11], where straight dislocation segments (either pure screw or edge in the earliest versions, or of a mixed character in more recent versions) are allowed to jump on specific lattice sites and orientations. Straight dislocation segments of mixed character in the The Force Method, developed by Hirth et al. [12] and Zbib et al. [13] are moved in a rigid body fashion along the normal to their mid-points, but they are not tied to an underlying spatial lattice or grid. The advantage of this method is that the explicit information on the elastic field is not necessary, since closed-form solutions for the interaction forces are directly used. The Differential Stress Method developed by Schwarz and Tersoff [14] and Schwarz [15] is based on calculations of the stress field of a differential straight line element on the dislocation. Using numerical integration, Peach–Koehler forces on all other segments are determined. The Brown procedure [16] is then utilized to remove the singularities associated with the self-force calculation. The method of The Phase Field Microelasticity [17–19] is of a different nature. It is based on Khachaturyan–Shatalov (KS) reciprocal space theory of the strain in an arbitrary elastically homogeneous system of misfitting coherent inclusions embedded into the parent phase. Thus, consideration of individual segments of all dislocation lines is not required. Instead, the temporal and spatial evolution of several density function profiles (fields) are obtained by solving continuum equations in Fourier space. The second approach to mechanical models at the mesoscale has been based on statistical mechanics methods [20–24]. In these developments, evolution equations for statistical averages (and possibly for higher moments) are to be solved for a complete description of the deformation problem. We focus here on the most recent formulations of 3D DD, following the work of Ghoniem et al. We review here the most recent developments in computational DD for the direct numerical simulation of the interaction and evolution of complex, 3D dislocation ensembles. The treatment is based on the parametric dislocation dynamics (PDD), developed by Ghoniem et al. In Section 2, we describe the geometry of dislocation loops with curved, smooth, continuous parametric segments. The stress field of ensembles of such curved dislocation loops is then developed in Section 3. Equations of motion for dislocation loops

Modeling the dynamics of dislocation ensembles

2271

are derived on the basis of irreversible thermodynamics, where the time rate of change of generalized coordinates will be given in Section 4. Extensions of these methods to anisotropic materials and multi-layered thin films are discussed in Section 5. Applications of the parametric dislocation dynamics methods are given in Section 6, and a discussion of future directions is finally outlined in Section 7.

2.

Computational Geometry of Dislocation Loops

Assume that the dislocation line is segmented into (n s ) arbitrary curved segments, labeled (1 ≤ i ≤ n s ). For each segment, we define rˆ (ω)=P(ω) as the position vector for any point on the segment, T(ω) = T t as the tangent vector to the dislocation line, and N(ω) = N n as the normal vector at any point (see Fig. 1). The space curve is then completely described by the parameter ω, if one defines certain relationships which determine rˆ (ω). Note that the position of any other point in the medium (Q) is denoted by its vector r, and that the vector connecting the source point P to the field point is R, thus R = r − rˆ . In the following developments, we restrict the parameter 0 ≤ ω ≤ 1, although we map it later on the interval −1 ≤ ωˆ ≤ 1, and ωˆ = 2ω − 1 in the numerical quadrature implementation of the method. To specify a parametric form for rˆ (ω), we will now choose a set of gen( j) eralized coordinates qi for each segment ( j ), which can be quite general. If one defines a set of basis functions C i (ω), where ω is a parameter, and allows

g3 ⫽ b冫冩 b 冩 P ω⫽ 0

R

g2 ⫽ t

g2 ⫽ e r

Q

ω⫽ 1

1z

1x

Figure 1. segment.

1y

Differential geometry representation of a generalparametric curved dislocation

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N.M. Ghoniem

for index sums to extend also over the basis set (i = 1, 2, . . . , I ), the equation of the segment can be written as ( j) rˆ ( j ) (ω) = qi Ci (ω)

2.1.

(1)

Linear Parametric Segments

The shape functions of linear segments Ci (ω), and their derivatives Ci,ω take the form: C1 = 1 − ω, C2 = ω and C1,ω = −1, C2,ω = 1. Thus, the available degrees of freedom for a free, or unconnected linear segment ( j ) are just the position vectors of the beginning ( j ) and end ( j + 1) nodes. ( j)

( j)

q1k = Pk

2.2.

and

( j)

( j +1)

q2k = Pk

(2)

Cubic Spline Parametric Segments

For cubic spline segments, we use the following set of shape functions, their parametric derivatives, and their associated degrees of freedom, respectively: C1 = 2ω3 − 3ω2 + 1, C2 = −2ω3 + 3ω2 , C3 = ω3 − 2ω2 + ω, and C4 = ω3 − ω2 C1,ω = 6ω2 − 6ω, C2,ω = −6ω2 + 6ω2 , C3,ω = 3ω2 − 4ω + 1, and C4,ω = 3ω2 − 2ω ( j) q1k

=

( j) Pk ,

( j) q2k

=

( j +1) Pk ,

( j) q3k

=

( j) Tk ,

and

( j) q4k

=

( j +1) Tk

(3) (4) (5)

Extensions of these methods to other parametric shape functions, such as circular, elliptic, helical, and composite quintic space curves are discussed by Ghoniem et al. [25]. Forces and energies of dislocation segments are given per unit length of the curved dislocation line. Also, line integrals of the elastic field variables are carried over differential line elements. Thus, if we express the Cartesian ( j) ( j) ( j) differential in the parametric form: dk = rˆk, ω dω = qsk Cs, ω dω. The arc length differential for segment j is then given by

( j)

( j ) 1/2

| d( j ) | = dk dk

( j)

( j)

( j ) ( j ) 1/2

= rˆk, ω rˆk, ω

= q pk C p, ω qsk Cs, ω

1/2

dω

dω

(6) (7)

Modeling the dynamics of dislocation ensembles

3.

2273

Elastic Field Variables as Fast Sums

3.1.

Formulation

In materials that can be approximated as infinite and elastically isotropic, the displacement vector u, strain ε and stress σ tensor fields of a closed dislocation loop are given by deWit [26] ui = −

εi j =

σi j

bi 4π

1 8π

=

Ak dlk +

C

ikl bl R, pp +

C

1 kmn bn R,mi dlk 1−ν

−

1 j kl bi R,l + ikl b j R,l − ikl bl R, j − j kl bl R,i , pp 2

×

kmn bn R,mi j dlk 1−ν

C

µ 4π

1 8π

(8)

C

(9)

1 1 R,mpp j mn dli + imn dl j + kmn 2 1−ν

× R,i j m − δi j R, ppm dlk

(10)

where µ and ν are the shear modulus and Poisson’s ratio, respectively, b is Burgers vector of Cartesian components bi , and the vector potential Ak (R) = i j k X i s j /[R(R+R· s)] satisfies the differential equation: pik Ak, p (R) = X i R −3 , where s is an arbitrary unit vector. The radius vector R connects a source point on the loop to a field point, as shown in Fig. 1, with Cartesian components Ri , successive partial derivatives R,i j k... , and magnitude R. The line integrals are carried along the closed contor C defining the dislocation loop, of differential arc length dl of components dlk . Also, the interaction energy between two closed loops with Burgers vectors b1 and b2 , respectively, can be written as µb1i b2 j EI = − 8π

R,kk C (1) C (2)

2ν dl2 j dl1i + dl2i dl1 j 1−ν

2 + (R,i j − δi j R,ll )dl2k dl1k 1−ν

(11)

The higher order derivatives of the radius vector, R,i j and R,i j k are components of second and third order Cartesian tensors that are explicitly known [27]. The dislocation segment in Fig. 1 is fully determined as an affine mapping on the scalar interval ∈ [0, 1], if we introduce the tangent vector T,

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the unit tangent vector t, the unit radius vector e, and the vector potential A, as follows T=

dl , dω

t=

T , |T|

e=

R , R

A=

e×s R(1 + e · s)

Let the Cartesian orthonormal basis set be denoted by 1 ≡ {1x , 1 y , 1z }, I = 1 ⊗ 1 as the second order unit tensor, and ⊗ denotes tensor product. Now define the three vectors (g1 = e, g2 = t, g3 = b/|b|) as a covariant basis set for the curvilinear segment, and their contravariant reciprocals as: gi · g j = δ ij , where δ ij is the mixed Kronecker delta and V = (g1 × g2 ) · g3 the volume spanned by the vector basis, as shown in Fig. 1. When the previous relationships are substituted into the differential forms of Eqs. (8), (10), with V1 = (s × g1 ) · g2 , and s an arbitrary unit vector, we obtain the differential relationships (see Ref. [27] for details)

|b||T|V (1 − ν)V1 / V du = g3 + (1 − 2ν)g1 + g1 dω 8π(1 − ν)R 1 + s · g1 V |T| d 1 1 =− −ν g ⊗ g + g ⊗ g 1 1 dω 8π(1 − ν)R 2

+ (1 − ν) g3 ⊗ g3 + g3 ⊗ g3 + (3g1 ⊗ g1 − I) µV |T| dσ 1 1 = g ⊗ g + g ⊗ g 1 1 dω 4π(1 − ν)R 2

+ (1 − ν) g2 ⊗ g2 + g2 ⊗ g2 − (3g1 ⊗ g1 + I)

µ|T1 ||b1 ||T2 ||b2 | d2 E I =− (1 − ν) g2I · g3I g2II · g3II dω1 dω2 4π(1 − ν)R

+ 2ν g2II · g3I

+ g3I · g1

g2I · g3II − g2I · g2II

g3II · g1

g3I · g3II

µ|T1 ||T2 ||b|2 d2 E S =− (1 + ν) g3 · g2I g3 · g2II dω1 dω2 8π R (1 − ν)

− 1 + (g3 · g1 )2

g2I · g2II

(12)

The superscripts I and II in the energy equations are for loops I and II , respectively, and g1 is the unit vector along the line connecting two interacting points on the loops. The self energy is obtained by taking the limit of 1/2 the interaction energy of two identical loops, separated by the core distance. Note that the interaction energy of prismatic loops would be simple, because g3 · g2 = 0. The field equations are affine transformation mappings of the scalar interval neighborhood dω to the vector (du) and second order tensor (d, dσ)

Modeling the dynamics of dislocation ensembles

2275

neighborhoods, respectively. The maps are given by covariant, contravariant and mixed vector, and tensor functions.

3.2.

Analytical Solutions

In some simple geometry of Volterra-type dislocations, special relations between b, e, and t can be obtained, and the entire dislocation line can also be described by one single parameter. In such cases, one can obtain the elastic field by proper choice of the coordinate system, followed by straight-forward integration. Solution variables for the stress fields of infinitely-long pure and edge dislocations are given in Table 1, while those for the stress field along the 1z -direction for circular prismatic and shear loops are shown in Table 2. Note that for the case of a pure screw dislocation, one has to consider the product of V and the contravariant vectors together, since V = 0. When the parametric equations are integrated over z from −∞ to +∞ for the straight dislocations, and over θ from 0 to 2π for circular dislocations, one obtains the entire stress field in dyadic notation as: 1. Infinitely-long screw dislocation µb − sin θ 1x ⊗ 1z + cos θ 1 y ⊗ 1z + cos θ 1z ⊗ 1 y 2πr − sin θ 1z ⊗ 1x }

σ=

(13)

Table 1. Variables for screw and edge dislocations Screw dislocation

Edge dislocation

g2

1 (r cos θ1x + r sin θ1 y + z1z ) R 1z

1 (r cos θ1x + r sin θ1 y + z1z ) R 1z

g3

1z

1x

g1

0

1 1y V

g1

g2 V g3 V T R V

r

r

r 2 + z2 r 2 + z2

dz 1z dω

(− sin θ1x + cos θ1 y ) V (sin θ1x − cos θ1 y ) V

r 2 + z2 r

r 2 + z2

dz 1z dω

0

r 2 + z2

1

r 2 + z2

r sin θ r 2 + z2

(−z1 y + r sin θ1z ) (sin θ1x − cos θ1 y )

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N.M. Ghoniem

Table 2. Variables for circular shear and prismatic loops Shear loop 1

Prismatic loop

g2

− sin θ1x + cos θ1 y

− sin θ1x + cos θ1 y

g3

1x

1z

g1

−

r 2 + z2

g2 V g3 V

T R V

(r cos θ1x + r sin θ1 y + z1z )

cos θ 1y V 1

(−z1 y + r sin θ1z )

(−z cos θ1x − z sin θ1 y

r 2 + z2 1 r 2 + z2

−r sin θ

+ r 1z )

dθ dθ 1x + r cos θ 1y dω dω

1

g1

r 2 + z2

1 (cos θ1x + sin θ1 y ) V r (− sin θ1x + cos θ1 y ) V r 2 + z2 1 (−z cos θ1x − z sin θ1 y V r 2 + z2 + r 1z ) −r sin θ

r 2 + z2

(r cos θ1x + r sin θ1 y + z1z )

dθ dθ 1x + r cos θ 1y dω dω

r 2 + z2

z cos θ − r 2 + z2

r

r 2 + z2

2. Infinitely-long edge dislocation µb sin θ(2 + cos 2θ )1x ⊗1x − (sin θ cos 2θ )1 y ⊗1 y 2π(1 − ν)r + (2ν sin θ)1z ⊗ 1z − (cos θ cos 2θ)(1x ⊗ 1 y + 1 y ⊗ 1x ) (14)

σ=−

3. Circular shear loop (evaluated on the 1z -axis)

σ=

µbr 2 2 2 2 (ν − 2)(r + z ) + 3z 4(1 − ν)(r 2 + z 2 )5/2 × 1 x ⊗ 1z + 1z ⊗ 1 x

(15)

4. Circular prismatic loop (evaluated on the 1z -axis)

σ=

µbr 2 (2(1 − ν)(r 2 + z 2 ) − 3r 2 ) 4(1 − ν)(r 2 + z 2 )5/2

× 1x ⊗ 1x + 1 y ⊗ 1 y − 2(4z 2 + r 2 ) 1z ⊗ 1z

(16)

As an application of the method in calculations of self- and interaction energy between dislocations, we consider here two simple cases. First, the

Modeling the dynamics of dislocation ensembles

2277

interaction energy between two parallel screw dislocations of length L and with a minimum distance ρ between them is obtained by making the following substitutions in Eq. (12) g2I = g2II = g3I = g3II = 1z ,

|T| =

dl = 1, dz

z2 − z1 1z · g1 = 2 ρ + (z 2 − z 1 )2

where z 1 and z 2 are distances along 1z on dislocations 1 and 2, respectively, connected along the unit vector g1 . The resulting scalar differential equation for the interaction energy is d2 E I µb2 =− dz 1 dz 2 4π(1 − ν)

ν (z 2 − z 1 )2 − 2 ρ 2 + (z 2 − z 1 )2 [ρ + (z 2 − z 1 )2 ] 3/2

(17) Integration of Eq. (17) over a finite length L yields identical results to those obtained by deWit [26] and by application of the more standard Blin formula [28]. Second, the interaction energy between two coaxial prismatic circular dislocations with equal radius can be easily obtained by the following substitutions g3I = g3II = 1z , g2I = − sin ϕ1 1x + cos ϕ1 1 y , g2II = − sin ϕ2 1x + cos ϕ2 1 y ϕ1 − ϕ2 2 z 1z · g2I = 0, R 2 = z 2 + (2ρ sin ) , 1z · g1 = 2 R Integration over the variables ϕ1 and ϕ2 from (0 − 2π ) yields the interaction energy.

4.

Dislocation Loop Motion

Consider the virtual motion of a dislocation loop. The mechanical power during this motion is composed of two parts: (1) change in the elastic energy stored in the medium upon loop motion under the influence of its own stress (i.e., the change in the loop self-energy), (2) the work done on moving the loop as a result of the action of external and internal stresses, excluding the stress contribution of the loop itself. These two components constitute the Peach– Koehler work [29]. The main idea of DD is to derive approximate equations of motion from the principle of Virtual Power Dissipation of the second law of thermodynamics Ghoniem et al. [27]. Once the parametric curve for the dislocation segment is mapped onto the scalar interval {ω ∈ [0, 1]}, the stress field everywhere is obtained as a fast numerical quadrature sum [30]. The Peach– Koehler force exerted on any other dislocation segment can be obtained from the total stress field (external and internal) at the segment as [30]. F P K = σ · b × t.

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N.M. Ghoniem

The total self-energy of the dislocation loop is determined by double line integrals. However, Gavazza and Barnett [31] have shown that the first variation in the self-energy of the loop can be written as a single line integral, and that the majority of the contribution is governed by the local line curvature. Based on these methods for evaluations of the interaction and self-forces, the weak variational form of the governing equation of motion of a single dislocation loop was developed by Ghoniem et al. [25] as

Fkt − Bαk Vα δrk |ds| = 0

(18)

Here, Fkt are the components of the resultant force, consisting of the Peach– Koehler force F P K (generated by the sum of the external and internal stress fields), the self-force Fs , and the Osmotic force F O (in case climb is also considered [25]). The resistivity matrix (inverse mobility) is Bαk , Vα are the velocity vector components, and the line integral is carried along the arc length of the dislocation ds. To simplify the problem, let us define the following dimensionless parameters r r∗ = , a

f∗ =

F , µa

t∗ =

µt B

Here, a is lattice constant, and t is time. Hence Eq. (18) can be rewritten in dimensionless matrix form as dr∗ ∗ ∗ ∗ δr f − ∗ ds = 0 (19) dt ∗

Here, f∗ = [ f 1∗ , f 2∗ , f 3∗ ] and r∗ = [r1∗ , r2∗ , r3∗ ] , which are all dependent on the dimensionless time t ∗ . Following Ghoniem et al. [25], a closed dislocation loop can be divided into Ns segments. In each segment j , we can choose a set of generalized coordinates qm at the two ends, thus allowing parametrization of the form r∗ = CQ

(20)

Here, C = [C1 (ω), C2 (ω), . . . , Cm (ω)], Ci (ω), (i = 1, 2, . . . , m) are shape functions dependent on the parameter (0 ≤ ω ≤ 1) and Q = [q1 , q2 , . . . , qm ] , qi are a set of generalized coordinates. Substituting Eq. (20) into Eq. (19), we obtain Ns j =1

Let,

δQ

dQ C f − C C ∗ |ds| = 0 dt ∗

j

fj = j

C f∗ |ds| ,

kj = j

C C |ds|

(21)

Modeling the dynamics of dislocation ensembles

2279

Following a similar procedure to the FEM, we assemble the EOM for all contiguous segments in global matrices and vectors, as F=

Ns j =1

fj,

K=

Ns

kj

j =1

then, from Eq. (21) we get, dQ =F (22) dt ∗ The solution of the set of ordinary differential Eq. (22) describes the motion of an ensemble of dislocation loops as an evolutionary dynamical system. However, additional protocols or algorithms are used to treat: (1) strong dislocation interactions (e.g., junctions or tight dipoles), (2) dislocation generation and annihilation, (3) adaptive meshing as dictated by large curvature variations [25]. In the The Parametric Method [25, 27, 32, 33] presented above, the dislocation loop can be geometrically represented as a continuous (to second derivative) composite space curve. This has two advantages: (1) there is no abrupt variation or singularities associated with the self-force at the joining nodes in between segments, (2) very drastic variations in dislocation curvature can be easily handled without excessive re-meshing. K

5.

Dislocation Dynamics in Anisotropic Crystals

Extension of the PDD to anisotropic linearly elastic crystals follows the same procedure described above, with the exception of two aspects [34]. First, calculations of the elastic field, and hence forces on dislocations, is computationally more demanding. Second, the dislocation self-force is obtained from non-local line integrals. Thus PDD simulations in anisotropic materials are about an order of magnitude slower than in isotropic materials. Mura [35] derived a line integral expression for the elastic distortion of a dislocation loop, as u i, j (x)= ∈ j nk C pqmn bm

G ip,q (x − x )νk dl(x ),

(23)

L

where νk is the unit tangent vector of the dislocation loop line L, dl is the dislocation line element, ∈ j nh is the permutation tensor, Ci j kl is the fourth order elastic constants tensor, G i j ,l (x − x ) = ∂ G i j (x − x )/∂ xl , and G i j (x − x ) are the Green’s tensor functions, which correspond to displacement component along the xi -direction at point x due to a unit point force in the x j -direction applied at point x in an infinite medium.

2280

N.M. Ghoniem

The elastic distortion formula (23) involves derivatives of the Green’s functions, which need special consideration. For general anisotropic solids, analytical expressions for G i j,k are not available. However, these functions can be expressed in an integral form (see, e.g., Refs. [36–39]), as G i j ,k (x − x ) =

1 2 8π |r|2

Ck

¯ −1 (k) ¯ − r¯k Ni j (k)D

¯ j m (k)D ¯ −2 (k) ¯ dφ + k¯k Clpmq (¯r p k¯q + k¯ p r¯q )Nil (k)N (24) where r = x − x , r¯ = r/|r|, k¯ is the unit vector on the plane normal to r, the integral is taken around the unit circle Ck on the plane normal to r, Ni j (k) and D(k) are the adjoint matrix and the determinant of the second order tensor Cikj l kk kl , respectively. The in-plane self-force at the point P on the loop is also obtained in a manner similar to the external Peach–Koehler force, with an additional contribution from stretching the dislocation line upon a virtual infinitesimal motion [40] F S = κ E(t) − b · σ¯ S · n

(25)

where E(t) is the pre-logarithmic energy factor for an infinite straight dislocation parallel to t: E(t) = 12 b · (t) · n, with (t) being the stress tensor of an infinite straight dislocation along the loop’s tangent at P. σ S is self stress tensor due to the dislocation L, and σ¯ = 12 [σ S (P + m) + σ S (P − m)] is the average self-stress at P, κ is the in-plane curvature at P, and = |b|/2. Barnett [40] and Gavazza and Barnett [31] analyzed the structure of the self-force as a sum 8 S − J (L , P) + Fcore (26) F = κ E(t) − κ E(t) + E (t) ln κ where the second and third terms are line tension contributions, which usually account for the main part of the self-force, while J (L , P) is a non-local contribution from other parts of the loop, and Fcore is due to the contribution to the self-energy from the dislocation core.

6.

Selected Applications

Figure 2 shows the results of computer simulations of plastic deformation in single crystal copper (approximated as elastically isotropic) at a constant strain rate of 100 s−1 . The initial dislocation density of ρ = 2 × 1013 m−2 has been divided into 300 complete loops. Each loop contains a random number

Modeling the dynamics of dislocation ensembles

2281

Figure 2. Results of computer simulations for dislocation microstructure deformation in copper deformed to increasing levels of strain (shown next to each microstructure).

of initially straight glide and superjog segments. When a generated or expanding loop intersects the simulation volume of 2.2 µm side length, the segments that lie outside the simulation boundary are periodically mapped inside the simulation volume to preserve translational strain invariance, without loss of dislocation lines. The number of nodes on each loop starts at five, and is then increased adaptively proportional to the loop length, with a maximum number of 20 nodes per loop. The total number of Degrees of Freedom (DOF) starts at 6000, and is increased to 24 000 by the end of the calculation. However, the number of interacting DOF is determined by a nearest neighbor criterion, within a distance of 400a (where a is the lattice constant), and is based on a binary tree search. The dislocation microstructure is shown in Fig. 2 at different total strain. It is observed that fine slip lines that nucleate at low strains evolve into more pronounced slip bundles at higher strains. The slip bundles are well-separated in space forming a regular pattern with a wavelength of approximately one micron. Conjugate slip is also observed, leading to the formation of dislocation junction bundles and stabilization of a cellular structures. Next, we consider the dynamic process of dislocation dipole formation in anisotropic single crystals. To measure the degree of deviation from elastic isotropy, we use the anisotropy ratio A, defined in the usual manner: A = 2C44 /(C11 − C12 ) [28]. For an isotropic crystal, A = 1. Figure 3(a) shows the configurations (2D projected on the (111)-plane) of two pinned dislocation segments, lying on parallel (111)-planes. The two dislocation segments are

2282

N.M. Ghoniem (a) 300

A⫽1 A⫽2 A ⫽ 0.5

200

[⫺1 ⫺1 2]

b Stable dipole

100

0

⫺500

0

500

[⫺1 1 0]

(b) 0.4 A⫽1 A⫽1

0.35

A⫽2 A⫽1

τ/µ (%)

0.3 A ⫽ 0.5

0.25

0.2

0.15 Backward break up Forward break up Infinite dipole

0.1

0.05

0.04

0.08

0.12

a/h

Figure 3. Evolution of dislocation dipoles without applied loading (a) and dipole break up shear stress (b).

¯ initially straight, parallel, and along [110], but of opposite line directions, ¯ have the same Burgers vector b = 1/2[101], and are pinned √ at both ends. Their 3a, L : d : h = 800 : glide planes are separated by h. In this figure, h = 25 √ 300 : 25 3, with L and d being the length of the initial dislocation segments and the horizontal distance between them, respectively. Without the application of any external loading, the two lines attract one another, and form an equilibrium state of a finite-size dipole. The dynamic shape of the segments during the dipole formation is seen to be dependent on the anisotropy ratio A, while the final configuration appears to be insensitive to A. Under external loading, the dipole may be unzipped, if applied forces overcome binding forces between dipole arms. The forces (resolved shear stresses τ , divided by µ = (C11 − C12 )/2) to break up the dipoles are shown in Fig. 3(b). It can be seen that the break up stress is inversely proportional to the separation distance h, consistent with the results of infinite-size dipoles. It is easier to break up dipoles in crystals with smaller A-ratios (e.g., some BCC crystals). It is also noted that two ways to break up dipoles are possible: in backward direction (where the self-force assists the breakup), or forward direction (where the

Modeling the dynamics of dislocation ensembles

2283

self-force opposes the breakup). For a finite length dipole, the backward break up is obviously easier than the forward one, due to the effects of self forces induced by the two curved dipole arms, as can be seen in Fig. 3(b). As a final application, we consider dislocation motion in multi-layer anisotropic thin films. It has been experimentally shown that the strength of multilayer thin films is increased as the layer thickness is decreased, and that maximum strength is achieved for layer thickness on the order of 10–50 nm. Recently, Ghoniem and Han [41] developed a new computational method for the simulation of dislocation ensemble interactions with interfaces in anisotropic, nanolaminate superlattices. Earlier techniques in this area use cumbersome and inaccurate numerical resolution by superposition of a regular elastic field obtained from a finite element, boundary element, surface dislocation or point force distributions to determine the interaction forces between 3D dislocation loops and interfaces. The method developed by Ghoniem and Han [41] utilizes two-dimensional Fourier Transforms to solve the full elasticity problem in the direction transverse to interfaces, and then by numerical inversion, obtain the solution for 3D dislocation loops of arbitrary complex geometry. Figure 4 shows a comparison between the numerical simulations (stars) for the critical yield strength of a Cu/Ni superlattice, compared to Freund’s analytical solution (red solid line) and the experimental data of the Los Alamos group (solid triangles). The saturation of the nanolayered system strength (and hardness) with a nanolayer thickness less than 10–50 nm is a result of dislocations overcoming the interface Koehler barrier and loss of dislocation confinement within the soft Cu layer.

4.0 Freund critical stress Experiment (Misra, et al.,1998) Simulation, image force

Critical yield stress (GPa)

3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0 1

Figure 4.

10 Cu layer thickness h (nm)

100

Dependence of a Cu/Ni superlattice strength onthe thickness of the Cu layer [41].

2284

7.

N.M. Ghoniem

Future Outlook

As a result of increased computing power, new mathematical formulations, and more advanced computational methodologies, tremendous progress in modeling the evolution of complex 3D dislocation ensembles has been recently realized. The appeal of computational dislocation dynamics lies in the fact that it offers the promise of predicting the dislocation microstructure evolution without ad hoc assumptions, and on sound physical grounds. At this stage of development, many physically-observed features of plasticity and fracture at the nano- and micro-scales have been faithfully reproduced by computer simulations. Moreover, computer simulations of the mechanical properties of thin films are at an advanced stage now that they could be predictive without ambiguous assumptions. Such simulations may become very soon standard and readily available for materials design, even before experiments are performed. On the other hand, modeling the constitutive behavior of polycrystalline metals and alloys with DD computer simulations is still evolving and will require significant additional developments of new methodologies. With continued interest by the scientific community in achieving this goal, future efforts may well lead to new generations of software, capable of materials design for prescribed (within physical constraints) strength and ductility targets.

Acknowledgments Research is supported by the US National Science Foundation (NSF), grant #DMR-0113555, and the Air Force Office of Scientific Research (AFOSR), grant #F49620-03-1-0031 at UCLA.

References [1] H. Mughrabi, “Dislocation wall and cell structures and long-range internal-stresses in deformed metal crystals,” Acta Met., 31, 1367, 1983. [2] H. Mughrabi, “A 2-parameter description of heterogeneous dislocation distributions in deformed metal crystals,” Mat. Sci. & Eng., 85, 15, 1987. [3] R. Amodeo and N.M. Ghoniem, “A review of experimental observations and theoretical models of dislocation cells,” Res. Mech., 23, 137, 1988. [4] J. Lepinoux and L.P. Kubin, “The dynamic organization of dislocation structures: a simulation,” Scripta Met., 21(6), 833, 1987. [5] N.M. Ghoniem and R.J. Amodeo, “Computer simulation of dislocation pattern formation,” Sol. St. Phen., 3&4, 377, 1988. [6] A.N. Guluoglu, D.J. Srolovitz, R. LeSar, and R.S. Lomdahl, “Dislocation distributions in two dimensions,” Scripta Met., 23, 1347, 1989.

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[7] N.M. Ghoniem and R.J. Amodeo, “Numerical simulation of dislocation patterns during plastic deformation,” In: D. Walgreaf and N. Ghoniem (eds.), Patterns, Defects and Material Instabilities, Kluwer Academic Publishers, Dordrecht, p. 303, 1990. [8] R.J. Amodeo and N.M. Ghoniem, “Dislocation dynamics I: a proposed methodology for deformation micromechanics,” Phys. Rev., 41, 6958, 1990a. [9] R.J. Amodeo and N.M. Ghoniem, “Dislocation dynamics II: applications to the formation of persistent slip bands, planar arrays, and dislocation cells,” Phy. Rev., 41, 6968, 1990b. [10] L.P. Kubin, G. Canova, M. Condat, B. Devincre, V. Pontikis, and Y. Brechet, “Dislocation microstructures and plastic flow: a 3D simulation,” Diffusion and Defect Data–Solid State Data, Part B (Solid State Phenomena), 23–24, 455, 1992. [11] A. Moulin, M. Condat, and L.P. Kubin, “Simulation of frank-read sources in silicon,” Acta Mater., 45(6), 2339–2348, 1997. [12] J.P. Hirth, M. Rhee, and H. Zbib, “Modeling of deformation by a 3D simulation of multi pole, curved dislocations,” J. Comp.-Aided Mat. Des., 3, 164, 1996. [13] R.M. Zbib, M. Rhee, and J.P. Hirth, “On plastic deformation and the dynamics of 3D dislocations,” Int. J. Mech. Sci., 40(2–3), 113, 1998. [14] K.V. Schwarz and J. Tersoff, “Interaction of threading and misfit dislocations in a strained epitaxial layer,” Appl. Phys. Lett., 69(9), 1220, 1996. [15] K.W. Schwarz, “Interaction of dislocations on crossed glide planes in a strained epitaxial layer,” Phys. Rev. Lett., 78(25), 4785, 1997. [16] L.M. Brown, “A proof of lothe’s theorem,” Phil. Mag., 15, 363–370, 1967. [17] A.G. Khachaturyan, “The science of alloys for the 21st century: a hume-rothery symposium celebration,” In: E. Turchi and a. G.A. Shull, R.D. (eds.), Proc. Symp. TMS, TMS, 2000. [18] Y.U. Wang, Y.M. Jin, A.M. Cuitino, and A.G. Khachaturyan, “Presented at the international conference, Dislocations 2000, the National Institute of Standards and Technology,” Gaithersburg, p. 107, 2000. [19] Y. Wang, Y. Jin, A.M. Cuitino, and A.G. Khachaturyan, “Nanoscale phase field microelasticity theory of dislocations: model and 3D simulations,” Acta Mat., 49, 1847, 2001. [20] D. Walgraef and C. Aifantis, “On the formation and stability of dislocation patterns. I. one-dimensional considerations,” Int. J. Engg. Sci., 23(12), 1351–1358, 1985. [21] J. Kratochvil and N. Saxlo`va, “Sweeping mechanism of dislocation patternformation,” Scripta Metall. Mater., 26, 113–116, 1992. [22] P. H¨ahner, K. Bay, and M. Zaiser, “Fractal dislocation patterning during plastic deformation,” Phys. Rev. Lett., 81(12), 2470, 1998. [23] M. Zaiser, M. Avlonitis, and E.C. Aifantis, “Stochastic and deterministic aspects of strain localization during cyclic plastic deformation,” Acta Mat., 46(12), 4143, 1998. [24] A. El-Azab, “Statistical mechanics treatment of the evolution of dislocation distributions in single crystals,” Phys. Rev. B, 61, 11956–11966, 2000. [25] N.M. Ghoniem, S.-H. Tong, and L.Z. Sun, “Parametric dislocation dynamics: a thermodynamics-based approach to investigations of mesoscopic plastic deformation,” Phys. Rev., 61(2), 913–927, 2000. [26] R. deWit, “The continuum theory of stationary dislocations,” In: F. Seitz and D. Turnbull (eds.), Sol. State Phys., 10, Academic Press, 1960. [27] N.M. Ghoniem, J. Huang, and Z. Wang, “Affine covariant-contravariant vector forms for the elastic field of parametric dislocations in isotropic crystals,” Phil. Mag. Lett., 82(2), 55–63, 2001.

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[28] J. Hirth and J. Lothe, Theory of Dislocations, 2nd edn, McGraw–Hill, New York, 1982. [29] M.O. Peach and J.S. Koehler, “The forces exerted on dislocations and the stress fields produced by them,” Phys. Rev., 80, 436, 1950. [30] N.M. Ghoniem and L.Z. Sun, “Fast sum method for the elastic field of 3-D dislocation ensembles,” Phys. Rev. B, 60(1), 128–140, 1999. [31] S. Gavazza and D. Barnett, “The self-force on a planar dislocation loop in an anisotropic linear-elastic medium,” J. Mech. Phys. Solids, 24, 171–185, 1976. [32] R.V. Kukta and L.B. Freund, “Three-dimensional numerical simulation of interacting dislocations in a strained epitaxial surface layer,” In: V. Bulatov, T. Diaz de la Rubia, R. Phillips, E. Kaxiras, and N. Ghoniem (eds.), Multiscale Modelling of Materials, Materials Research Society, Boston, Massachusetts, USA, 1998. [33] N.M. Ghoniem, “Curved parametric segments for the stress field of 3-D dislocation loops,” Transactions of ASME. J. Engrg. Mat. & Tech., 121(2), 136, 1999. [34] X. Han, N.M. Ghoniem, and Z. Wang, “Parametric dislocation dynamics of anisotropic crystalline materials,” Phil. Mag. A., 83(31–34), 3705–3721, 2003. [35] T. Mura, “Continuous distribution of moving dislocations,” Phil. Mag., 8, 843–857, 1963. [36] D. Barnett, “The precise evaluation of derivatives of the anisotropic elastic green’s functions,” Phys. Status Solidi (b), 49, 741–748, 1972. [37] J. Willis, “The interaction of gas bubbles in an anisotropic elastic solid,” J. Mech. Phys. Solids, 23, 129–138, 1975. [38] D. Bacon, D. Barnett, and R. Scattergodd, “Anisotropic continuum theory of lattice defects,” In: C.J.M.T. Chalmers, B (ed.), Progress in Materials Science, vol. 23, Pergamon Press, Great Britain, pp. 51–262, 1980. [39] T. Mura, Micromechanics of Defects in Solids, Martinus Nijhoff, Dordrecht, 1987. [40] D. Barnett, “The singular nature of the self-stress field of a plane dislocation loop in an anisotropic elastic medium,” Phys. Status Solidi (a), 38, 637–646, 1976. [41] X. Han and N.M. Ghoniem, “Stress field and interaction forces of dislocations in anisotropic multilayer thin films,” Phil. Mag., in press, 2005.

7.12 DISLOCATION DYNAMICS – PHASE FIELD Yu U. Wang,1 Yongmei M. Jin,2 and Armen G. Khachaturyan2 1 Department of Materials Science and Engineering, Virginia Tech., Blacksburg, VA 24061, USA 2 Department of Ceramic and Materials Engineering, Rutgers University, 607 Taylor Road, Piscataway, NJ 08854, USA

Dislocation, as an important category of crystal defects, is defined as a one-dimensional line (curvilinear in general) defect. It not only severely distorts the atomic arrangement in a region (called core) around the mathematical line describing its geometrical configuration, but also in a less severe manner (elastically) distorts the lattice beyond its core region. Dislocation core structure is studied by using the methods and models of atomistic scale (see Chapter 2). The long-range strain and stress fields generated by dislocation are well described by linear elasticity theory. In the elasticity theory of dislocations, dislocation is defined as a line around which a line integral of the elastic displacement yields a non-zero vector (Burgers vector). The elastic fields, displacement, strain and stress, of an arbitrarily curved dislocation are known in the form of line integrals. For complex dislocation configurations, the exact elasticity solution is quite difficult. A conventional alternative is to approximate a curved dislocation by a series of straight line segments or spline fitted curved segments. This involves explicit tracking of each segment of the dislocation ensemble (see “Dislocation Dynamics – Tracking Methods” by Ghoniem). In a finite body, the strains and stresses depend on the external surface. For general surface geometries, the elastic fields of dislocations are difficult to determine. In this article we discuss an alternative to the front-tracking methods in modeling dislocation dynamics. This is the structure density phase field method, which is a more general version of the phase field method used to describe solidification process. Instead of explicitly tracking the dislocation lines, the phase field method describes the slipped (plastically deformed by shear) and unslipped regions in a crystal by using field variables (structure density functions or, less accurately but more conventionally called, phase 2287 S. Yip (ed.), Handbook of Materials Modeling, 2287–2305. c 2005 Springer. Printed in the Netherlands.

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fields). Dislocations are the boundaries between the regions of different degrees of slipping. One of the advantages of the phase field approach is that it treats the system with arbitrarily complex microstructures as a whole and automatically describes the evolution events producing changes of the microstructure topology (e.g., nucleation, multiplication, annihilation and reaction of dislocations) without explicitly tracking the moving segments. Therefore, it is easy for numerical implementation even in three-dimension (a front-tracking scheme often results in difficult and untidy numerical algorithm). No ad hoc assumptions are required on evolution path. The micromechanics theory proposed by Khachaturyan and Shatalov (KS) [1–3] and recently further developed by Wang, Jin and Khachaturyan (WJK) in a series of works [4–9] is formulated in such a form that it is easily incorporated in the phase field theory. It allows one to determine the elastic interactions at each step of the dislocation dynamics. In the case of elastically homogeneous systems, the exact elasticity solution for an arbitrary dislocation configuration can be formulated as a closed-form functional of the Fourier transforms of the phase fields describing the dislocation microstructure irrespective of its geometrical complexity (the number of the phase fields is equal to the number of operative slip systems that is determined by the crystallography instead of by a concrete dislocation microstructure). This fact makes it easy to achieve high computational efficiency by using Fast Fourier Transform technique, which is also suitable for parallel computing. The Fourier space solution is formulated in terms of arbitrary elastic modulus tensor. This means that the solution for dislocations in single crystal of elastic anisotropy practically does not impose more difficulty. By simply introducing a grain rotation matrix function that describes the geometry and orientation of each grain and the entire multi-grain structure, the phase field method is readily extended to model dislocation dynamics in polycrystal composed of elastically isotropic grains. If the grains are elastically anisotropic, their misorientation makes the polycrystal an elastically inhomogeneous body. The limitation of grain elastic isotropy could be lifted without serious complication of the theory and model by an introduction of additional virtual misfit strain field. This field acting in the equivalent system with the homogeneous modulus produces the same mechanical effect as that produced by elastic modulus heterogeneity. The introduction of the virtual misfit strain greatly simplifies a treatment of elastically inhomogeneous system of arbitrary complexity, in particular, a body with voids, cracks, and free surfaces. The structural density phase field model of multi-crack evolution can be developed in the formalism similar to the phase field model of multidislocation dynamics. This development of the theory has been an extension of the corresponding phase field theories of diffusional and displacive phase transformations (e.g., decomposition, ordering, martensitic transformation, etc.). All these structure density field theories are conceptually similar and

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are formulated in the similar theoretical and computational framework. The latter facilitates an integration of multi-physics such as dislocations, cracks and phase transformations into one unified structure density field model, where multiple processes are described by simultaneous evolution of various relaxing density fields. Such a unified model would be highly desirable for simulations of complex materials behaviors. The following sections will discuss the basic ingredients of the phase field model of dislocation dynamics. Single crystalline system is considered first, followed by the extension to polycrystal composed of elastically isotropic grains. Finite body with free surfaces is discussed next. The phase field model of cracks, in many respects, is similar to the dislocation model. It is also discussed. The article concludes with a brief outlook on the structural density field models for integration of multiple solid-state physical phenomena and connections between mesoscale phase field modeling and atomistic as well as continuum models.

1.

Dislocation Loop as Thin Platelet Misfitting Inclusion

Consider a simple two-dimensional lattice of circles representing atoms, as shown in Fig. 1(a). Imagine that we cut and remove from the lattice a thin platelet consisting of two monolayers indicated by shaded circles, deform it by gliding the top layer with respect to the bottom layer by one interatomic distance, as shown in Fig. 1(b), then reinsert the deformed thin platelet back into the original lattice, and allow the whole lattice to relax and reach mechanical equilibrium. In doing so, we create an edge dislocation that is located at (a)

(c)

(d)

(b)

Figure 1. Illustration of dislocations as thin platelet misfitting inclusions. (a) A 2D lattice. (b) A thin platelet misfitting inclusion generated by transformation. (c) Bragg–Nye bubble model of an edge dislocation in mechanical equilibrium (after Ref. [10], reproduced with permission). (d) Continuum presentation of the dislocation line ABC ending on the crystal surface at points A and C and a dislocation loop by the thin platelet misfitting inclusions (after Ref. [4], reproduced with permission). b is the Burgers vector, d is the thickness of the inclusion equal to the interplanar distance of the slip plane, and n is the unit vector normal to the inclusion habit plane coinciding with the slip plane.

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the edge of the thin platelet. The equilibrium state of such a lattice is demonstrated in Fig. 1(c), which shows the Bragg–Nye bubble model of an edge dislocation [10]. In the continuum elasticity theory of dislocations, dislocation loop can be created in the same way by transforming thin platelet in the matrix of untransformed solid. Consider an arbitrary-shaped plate-like misfitting inclusion, whose habit plane (interface between inclusion and matrix) coincides with slip plane, as shown in Fig. 1(d). The misfit strain (also called stress-free transformation strain or eigenstrain describing the homogeneous deformation of the transformed stress-free state) of the platelet is a dyadic prod inclusion under = b n + b n 2d, where b is a Burgers vector, n is the normal and d uct, εidis i j j i j is the platelet thickness equal to the interplanar distance of the slip plane. Such a misfitting thin platelet generates stress that is exactly the same as generated by a dislocation loop of Burgers vector b encircling the platelet [2]. This fact, as will be shown in next two sections, greatly facilitates the description of dislocation microstructure and the solution of the elasticity problem, which is the basis of the WJK phase field microelasticity (PFM) theory of dislocations [4]. This theory was extended by Shen and Wang [11] and WJK [7, 9, 12]. In fact, the dislocation-associated misfit strain εidis j characterizes the plastic strain of the transformed (plastically deformed) platelet inclusion.

2.

Structure Density Field Description of Dislocation Ensemble

As discussed above, by treating dislocation loops as thin platelet misfitting inclusions, instead of describing dislocations by lines, we describe the transformed regions in the untransformed matrix. The transformed regions are the regions that have been plastically deformed by slipping. Dislocations correspond to the boundaries separating the regions of different degrees of slipping. In this description, we track a spatial and temporal evolution of the dislocationassociated misfit strain (plastic strain), which is the structure density field. This field describes the evolution of individual dislocations in an arbitrary ensemble. For an arbitrary dislocation ensemble involving all operative slip systems, the total dislocation-associated misfit strain εidis j (r) is the sum over all slip planes numbered by α: εidis j (r) =

1 α

2

bi (α, r) H j (α) + b j (α, r) Hi (α) ,

(1)

where b(α, r) is the slip displacement vector, H(α) = n(α)/d(α) is the reciprocal lattice vector of the slip plane α, n(α) and d(α) are the normal and interplanar distance, respectively, of the slip plane α. Therefore, a set of

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vector fields, {b(α, r)}, completely characterizes the dislocation configuration. Slipped (plastically deformed) regions are the ones where b(α, r) =/ 0. The vector b(α, r) can be expressed as a sum of the slip displacement vectors numbered by m α corresponding to the operative slip modes within the same slip plane α: b (α, r) =

b (α, m α , r).

(2)

mα

It is convenient to present each field b(α, m α , r) in terms of an order parameter η (α, m α , r) through the following relation b (α, m α , r) = b (α, m α ) η (α, m α , r),

(3)

where η (α, m α , r) is a scalar field, and b (α, m α ) is the corresponding elementary Burgers vector of the slip mode m α in the slip plane α. Thus, an arbitrary dislocation configuration involving all possible slip systems is completely characterized by a set of order parameter fields (phase fields), {η(α, m α , r)}. The number of the fields is equal to the number of the operative slip systems that is determined by the crystallography rather than a concrete dislocation configuration. For example, face-centered cubic (fcc) crystal has four {111} slip planes (α=1, 2, 3, 4) and three 110 slip modes in each slip plane (m α =1, 2, 3), thus has 12 slip systems. A total number of 12 phase fields are used to characterize an arbitrary dislocation ensemble in a fcc crystal if all possible slip systems are involved. An in-depth discussion on the choice of Phase Fields (dislocation density fields) is presented in Ref. [12]. It is noteworthy that the structural density phase field (order parameter) here has the physical meaning of structure (dislocation) density, which is more general than the order parameter used in the phase field model of solidification that assumes 1 in solid and 0 in liquid.

3.

Phase Field Microelasticity Theory

As discussed in the preceding section, the micromechanics of an arbitrary dislocation ensemble involving all operative slip systems is characterized by the dislocation-associated misfit strain εidis j (r) defined in Eq. (1). Substituting Eqs. (2) and (3) into Eq. (1) expresses εidis j (r) as a linear function of a set of phase fields, {η (α, m α , r)}: εidis j (r) =

1 α

mα

2

bi (α, m α ) H j (α) + b j (α, m α ) Hi (α) η (α, m α , r). (4)

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The elastic (strain) energy generated by such a dislocation ensemble is

E

elast

= V

1 Ci j kl εi j (r) − εidis εkl (r) − εkldis (r) d 3r, j (r) 2

(5)

where Ci j kl is elastic modulus tensor, V is body volume, and εi j (r) is the equilibrium strain that minimizes the elastic energy (5) under the compatibility (continuity) condition. The exact elastic energy E elast can be expressed as closed-form functional of εidis j (r). This is obtained by using the KS theory developed for arbitrary multi-phase and multi-domain misfitting inclusions in the homogeneous anisotropic elastic modulus case. The total elastic energy for an arbitrary multidislocation ensemble described by a set of phase fields {η(α, m α , r)} in an appl elastically homogeneous anisotropic body under applied stress σi j is E elast =

1 d 3k α,m α β,m β

−

α,m α

−

2

−

(2π )

3

K α, m α , β, m β , e

∗

×η˜ (α, m α , k) η˜ β, m β , k appl σi j

bi (α, m α ) H j (α)

η (α, m α , r) d 3r

V

V −1 appl appl C σ σkl , 2 i j kl i j

(6)

where η˜ (α, m α , k) = V η (α, m α , r) e−ik·r d 3r is the Fourier transform of η(α, m α , r), the superscript asterisk (*) indicates complex conjugate, e = k/k is

a unit directional vector in the reciprocal (Fourier) space, and the integral as a principal value excluding the point – in the reciprocal space is evaluated k = 0. The scalar function K α, m α , β, m β , e is defined as

K α, m α , β, m β , e = Ci j kl −em Ci j mn np (e) Cklpq eq × bi (α, m α ) H j (α) bk β, m β Hl (β),

(7)

where i j (e) is the Green function tensor inverse to the tensor −1 i j (e) = Cikj l ek el . The elastic energy (6) is a closed-form functional of η(α, m α , r) and their Fourier transform η˜ (α, m α , k) irrespective of dislocation geometrical complexity. This fact makes it easy to achieve high computational efficiency in solving elasticity problem of dislocations. In computer simulations, elasticity solution is obtained numerically. The fields η˜ (α, m α , k) are evaluated by using fast Fourier transform technique, which is also suitable for parallel computing. Since the functional (6) is formulated for arbitrary elastic modulus tensor Ci j kl , a consideration of elastic anisotropy does not impose more difficulty. In fact, in simulations the function K(α, m α , β, m β , e) defined in Eq. (7)

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needs to be evaluated only once and stored in computer memory. Therefore, elastic anisotropy practically does not affect computational efficiency. The elastic energy E elast consists of dislocation self-energy and interaction appl energy as well as the energy generated by the applied stress σi j and the (potential) energy associated with the external loading device. The elastic energy is calculated by using the linear elasticity theory. Equation (6) provides the exact solution for the long-range elastic interactions between individual dislocations in an arbitrary configuration, which is the same as described by the Peach–Koehler equation.

4.

Crystalline Energy and Gradient Energy

In the phase field model, individual dislocations of an arbitrary configuration are completely described by a set of phase fields, {η(α, m α , r)}. For perfect dislocations, each slip displacement vector b(α, m α , r) should relax to a discrete set of values that are multiples of the corresponding elementary Burgers vector b(α, m α ). Thus according to Eq. (3), the order parameter η(α, m α , r) should relax to integer values. The elementary Burgers vectors b(α, m α ) correspond to the shortest crystal lattice translations in the slip planes. For partial dislocations, b(α, m α , r) do not correspond to crystal lattice translations, and η(α, m α , r) may assume non-integer values. The integers η(α, m α , r) are equal to the number of perfect dislocations with Burgers vector b(α, m α ) sweeping through the point r. The sign of the integer determines the slip direction with respect to b(α, m α ). The above-discussed behavior of η(α, m α , r) is automatically achieved by a choice of the Landau-type coarse-grained “chemical” free energy functional of a set of phase fields {η(α, m α , r)}. In the case of dislocations, this free energy is the crystalline energy that reflects the periodic properties of the host crystal lattice:

E

cryst

=

f cryst ({η (α, m α , r)})d 3r,

(8)

V

which should be minimized at {η(α, m α )} equal to integers. The integrand f cryst ({η(α, m α )}) is a periodical function of all parameters {η(α, m α )} with periods equal to any integers. This property follows from the fact that the Burgers vectors b(α, m α , r) in Eq. (3) corresponding to the integers η(α, m α , r) are lattice translation vectors that do not change the crystal lattice. The crystalline energy characterizes an interplanar potential during a homogeneous gliding of one atomic plane above another atomic plane by a slip displacement vector b(α). In the case of one slip mode, say (α1 , m α1 ), the

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local specific crystalline energy function f cryst ({η(α, m α )}) can be reduced to the simplest form by keeping the first non-vanishing term of its Fourier series: f cryst [b(α1 , m α1 )η (α1 , m α1 ) , 0, . . . , 0] = A sin2 π η (α1 , m α1 ),

(9)

where A is a positive constant providing the shear modulus at small strain limit. Its general behavior is schematically illustrated in Fig. 2(a). Any deviation of the slip displacement vector b from the lattice translation vectors is penalized by the crystalline energy. In the case where all slip modes are operative, the general expression of the multi-periodical function f cryst({η(α, m α )}) can also be presented as a Fourier series summed over the reciprocal lattice vectors of the host lattice, which reflects the symmetry of the crystal lattice (see, for detailed discussion, Refs. [4, 9, 11, 12]). The energy E cryst characterizes an interplanar potential of a homogeneous slipping. If the interplanar slipping is inhomogeneous, correction should be made to the crystalline energy (8). This is done by gradient energy E grad that characterizes the energy contribution associated with the inhomogeneity of the slip displacement. For one dislocation loop characterized by the phase field η(α1 , m α1 , r), as shown in Fig. 3(a) where η = 1 inside the disc domain describing the slipped region and 0 outside, E grad is formulated as

E

grad

= V

1 β [n (α1 ) × ∇η (α1 , m α1 , r)]2 d 3r, 2

(10)

where β is a positive coefficient, and ∇ is the gradient operator. As shown in Fig. 3(a), the term n (α) × ∇η (α, m α , r) defines the dislocation sense at point r. The gradient energy (10) is proportional to the dislocation loop perimeter and vanishes over the slip plane. For an arbitrary dislocation configuration characterized by a set of phase fields {η(α, m α )}, the general form of the gradient energy is

E

grad

=

ϕi j (r) d 3r,

(11)

V

(a)

(b) f(b)

f(h)

2γ/d b0

2b0

b

h hc

Figure 2. Schematic illustration of the general behavior of Landau-type coarse-grained “chemical” energy function for (a) dislocation (crystalline energy) and (b) crack (cohesion energy).

Dislocation dynamics – phase field (a)

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Figure 3. (a) A thin platelet domain describing the slipped region. The term n×∇η (r) defines the dislocation sense along the dislocation line (plate edge) and vanishes over the slip plane (plate surface). (b) Schematic of a polycrystal model. Each grain has a different orientation described by its rotation matrix Qi . The rotation matrix function Q (r) completely describes the geometry and orientation of each grain and the entire multi-grain structure.

where the argument of the integrand, ϕi j (r), is defined as ϕi j (r) =

α

[H(α) × ∇η(α, m α , r)]i b j (α, m α ).

(12)

mα

The choice of the tensor ϕi j (r) is dictated by the physical requirements that (i) the gradient energy is proportional to the dislocation length and vanishes over the slip planes and (ii) the gradient energy depends on the total Burgers vector of the dislocation. Following the Landau theory, we can approximate the function ϕi j (r) by the Taylor expansion, which reflects the symmetry of the crystal lattice. As discussed in the preceding section, the elastic energy of dislocations is calculated by using the linear elasticity theory. The nonlinear effects associated with dislocation cores are described in the phase field model by both the crystalline energy E cryst and the gradient energy E grad , which produce significant contributions only near dislocation cores. More detailed discussion on the crystalline and gradient energies is presented in Refs. [4, 9, 11, 12].

5.

Time-dependent Ginzburg–Landau Kinetic Equation

The total energy of a dislocation system is the sum of elastic energy (6), crystalline energy (8) and gradient energy (11): E = E elast + E cryst + E grad ,

(13)

which is a functional of a set of phase fields {η(α, m α , r)}. The temporalspatial dependence of η (α, m α , r, t ) describes the collective motion of the dislocation ensemble. The evolution of η (α, m α , r, t ) is characterized by a

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phenomenological kinetic equation, which is the time-dependent Ginzburg– Landau equation: δE ∂η(α, m α , r, t) = −L + ξ(α, m α , r, t), (14) ∂t δη(α, m α , r, t ) where L is the kinetic coefficient characterizing dislocation mobility, E is the total system energy (13), and ξ(α, m α , r, t) is the Langevin Gaussian noise term reproducing the effect of thermal fluctuations (an in-depth discussion on the invariant form of the time-dependent Ginzburg–Landau kinetic equation is presented in Ref. [12]). A numerical solution η(α, m α , r, t) of the kinetic Eq. (14) automatically takes into account the dislocation multiplication, annihilation, interaction and reaction without ad hoc assumptions. Figure 4 shows one example of the PFM simulation of self-multiplying and self-organizing dislocations during plastic deformation of single crystal ([4]; more simulations are presented therein, and also in Ref. [13] on dislocations in polycrystal, Ref. [11] on network formation, Ref. [14] on solute–dislocation interaction, and Ref. [15] on alloy hardening). The kinetic Eq. (14) is based on the assumption that the relaxation rate of a field is proportional to the thermodynamic driving force. Note that Eq. (14) assumes a linear dependence between dislocation glide velocity v and local resolved shear stress τ along the Burgers vector, i.e., v = mτ b, where m is a constant. In fact, ∂η/∂t = −Lδ E elast/δη −Lδ(E cryst + E grad )/δη, where the first term of the right-hand side gives the linear dependence (L/d) σi j n j bi with σi j being local stress. The second term provides the effect of lattice friction on dislocation motion. It is worth noting that the WJK theory is an interpolational theory providing a bridge between the high and low spatial resolutions. In the high resolution

Figure 4. PFM simulation of stress–strain curve and the corresponding 3D dislocation microstructures during plastic deformation of fcc single crystal under uniaxial loading (after Ref. [4], reproduced with permission).

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limit, it is a 3D generalization of the Peierls–Nabarro (PN) theory [16] to arbitrary dislocation configuration: the WJK theory reproduces the results of the PN theory in a particular case considered in this theory, i.e., a 2D model with a straight single dislocation. The gradient energy (11) is one ingredient that the PN theory lacks. As discussed in the preceding section, the gradient term is necessary as an energy correction associated with slip inhomogeneity and, together with the crystalline energy, describes the core radius and nonlinear core energy. As the PN theory, the WJK theory is applicable in the atomic resolution as well. However, to make the PN and WJK theories fully consistent with atomic resolution modeling, instead of continuum Green function, the atomistic Green function of the crystal lattice statics should be used [2]. To obtain the atomic resolution in the computer simulations, the computational grid sites should be the crystal lattice sites. Another option is to use subatomic scale phase field model where density function models individual atoms [17]. In the low resolution, the WJK theory gives a natural transition to the continuum dislocation theory where local dislocation density εidis j (r), which is related to the dislocation density fields η(α, m α , r) by Eq. (4), is smeared over volume elements corresponding to a computational grid cell, where the grid size l is much larger than the crystal lattice parameter. Then the reciprocal lattice vectors should be defined as H (α) = n (α)/l. In such situations, individual dislocation’s position is uncertain within one grid cell. The dislocation core width, which is the order of crystal lattice parameter, is too small to be resolved by the low resolution computational grids. To effectively eliminate the inaccuracy associated with the Burgers vector relaxation (the core effect) to the dislocation interaction energies at distances exceeding a computational grid length, a non-linear relation between the slip displacement vector b(α, m α , r) and the order parameter η(α, m α , r), rather than the linear relation (3), should be used in the low resolution cases. One simple example of such non-linear relation is [14]:

b(α, m α , r) = b(α, m α ) η(α, m α , r) −

1 2π

sin 2π η (α, m α , r) ,

(15)

which shrinks the effective radius of the dislocation core to improve the accuracy in the mesoscale diffuse-interface modeling. If the resolution of the simulation is microscopic, the use of the non-linear relation becomes unnecessary and the linear dependence (3) of the Burgers vector on the order parameter should be used.

6.

Dislocation Dynamics in Polycrystals

Equation (4) completely characterizes the dislocation configuration in a single crystal, where the elementary Burgers vectors b(α, m α ) and reciprocal

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lattice vectors H(α) are defined in the coordinate system related to the crystallographic axes of crystal. However, it should be modified to characterize a dislocation microstructure in a polycrystal. In the same global coordinate system the components of the vectors b(α, m α ) and H(α) will have different values in different grains because of the mutual rotations of crystallographic axes of grains. In the latter case, we have to describe the orientation of each grain in the polycrystal. To do this, we introduce a static rotation matrix function Q i j (r) that is constant within each grain but assumes different constant values in different grains [13]. In fact, Q i j (r) describes the geometry and orientation of each grain and the entire multi-grain structure, as shown in Fig. 3(b). Then the misfit strain εidis j (r) of a dislocation microstructure in a polycrystal is given by εidis j (r) =

1 α

mα

2

Q ik (r) Q j l (r) [bk (α, m α ) Hl (α)

+ bl (α, m α ) Hk (α)] η (α, m α , r).

(16)

For a single crystal, Q i j (r) = δi j and Eq. (6) is reduced to Eq. (4). Therefore, a dislocation microstructure consisting of all possible slip systems in both single crystal and polycrystal can be completely described by a set of phase fields {η(α, m α , r)}. The elastic energy E elast is still determined by Eq. (6) if the polycrystal is composed of elastically isotropic grains, since the KS theory is applicable to elastically homogeneous body. Otherwise if the grains are elastically anisotropic, their mutual rotations would make the polycrystal an elastically inhomogeneous body. The limitation of grain elastic isotropy could be lifted without serious complication of the theory and computational model by using the PFM theory of elastically inhomogeneous solid [6]. A special case of this theory, viz., a discontinuous body with voids, cracks and free surfaces, will be discussed in the following sections. With the simple modification (16), the above-discussed theory is applicable to dislocation dynamics in polycrystal composed of elastically isotropic grains. Simulation examples are presented in Ref. [13].

7.

Free Surfaces and Heteroepitaxial Thin Films

Free surface is one common type of defects that is shared by all real materials. The stress field is significantly modified near free surfaces (the so-called image force effect). This produces important effects on dislocation dynamics. It is generally a difficult task to calculate the image force corrections to stress field and elastic energy for an arbitrary dislocation configuration in the vicinity of arbitrary-shaped free surfaces. To address this problem, the WJK theory has been extended to deal with finite systems with arbitrary-shaped free

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surfaces based on the theory of a stressed discontinuous body with arbitraryshaped voids and free surfaces [5]. The latter provides an effective method to solve the elasticity problem without sacrificing accuracy. In this section, we discuss the applications of the phase field dislocation dynamics to a system with free surfaces. We first discuss a recently established variational principle that makes this extension possible. A body containing voids is no longer continuous. The elasticity problem for this discontinuous body under applied stress can be solved by using the (r), located following variational principle [5]: if a virtual misfit strain, εivirtual j within the domains of equivalent continuous body minimizes its elastic energy, the generated strain and elastic energy of this equivalent continuous body are the equilibrium strain and elastic energy of original discontinuous body with voids. This variational principle is equally applicable to the cases of voids within a solid and a finite body with arbitrary-shaped free surfaces. The latter can be considered as the body fully “immersed into a void”, where the vacuum around the body can be regarded as the domain defined in the vari(r) ational principle. The position, shape and size of the domains with εivirtual j coincide with those of the voids and surrounding vacuum. Together with the (r), generates externally applied stress, the strain energy minimizer, εivirtual j the stress that vanishes within the domains. The latter allows one to remove the domains without disturbing the strain field and thus return to the initial externally loaded discontinuous body. This variational principle enables one to reduce the elasticity problem of a stressed discontinuous elastically anisotropic body to a much simpler equivalent problem of the continuous body. The above-discussed variational principle leads to the method of determination of the virtual misfit strain εivirtual (r) through a numerical minimization j elast , for the equivalent continuous body with of the strain energy functional, E equiv (r) under external stress. The explicit form of this functional of εivirtual εivirtual (r) j j is given by the KS theory. We may employ a Ginzburg–Landau type equation for energy minimization, which is similar to Eq. (14): elast δ E equiv ∂εivirtual (rd , t) j , (17) = −K i j kl virtual ∂t δεkl (rd , t) where K i j kl is “kinetic” coefficient, t is “time”, and rd represents the points inside the void domains. The “kinetic” Eq. (17) leads to a steady-state solution (r) that is the energy minimizer and generates vanishing stress in the εivirtual j void domains. Equation (17) provides a general approach to determining 3D elastic field, displacement and elastic energy of an arbitrary finite multi-void system in an elastically anisotropic body under applied stress. In particular, it can be used to calculate elasticity solution for a body with mixed-mode cracks of arbitrary configuration, which enables us to develop a phase field model of cracks, as discussed in next section.

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The system with free surface is also structurally inhomogeneous if defects generate a crystal lattice misfit. In the case of dislocations in a heteroepitaxial film, the structural inhomogeneity is characterized by dislocation-associated epitax misfit strain εidis (r) associated with j (r) as well as epitaxial misfit strain εi j crystal lattice misfit between film and substrate. The effective misfit strain εieffect (r) of equivalent system is a sum as j epitax

εieffect (r) = εi j j

virtual (r) + εidis (r). j (r) + εi j

(18)

elast of equivalent system is expressed in terms of The elastic energy E equiv effect εi j (r). For a given dislocation microstructure characterized by εidis j (r), the virtual virtual misfit strain εi j (r) can be determined by using Eq. (17), which has to be solved only at points rd inside the domains corresponding to vacuum (r) generates vanishing stress in around the body. As discussed above, εivirtual j the vacuum domains. Since the whole equivalent system (regions corresponding to vacuum and film/substrate) is in elastic equilibrium, the vanishing stress in the vacuum region automatically satisfies free surface boundary condition. The total energy of a dislocation ensemble near free surfaces is also given elast . Since the role of virby Eq. (13), where the elastic energy is given by E equiv virtual tual misfit strain εi j (r) is just to satisfy the free surface boundary condition, it does not enter crystalline energy (8) or gradient energy (11). As discussed above, the dislocation-associated misfit strain εidis j (r) is a function of a set of phase fields {η (α, m α , r)} given by Eq. (4). Since the epitaxy misfit strain epitax εi j (r) is a static field describing heteroepitaxial structure, the total energy is

a functional of two sets of evolving fields, i.e., E {η (α, m α , r)}, εivirtual (r) . j Following Wang et al. [5], the evolution of dislocations in a heteroepitaxial film is characterized by simultaneous solutions of Eqs. (14) and (17), which is driven by an epitaxial stress relaxation under influence of image forces near free surfaces. Figure 5 shows one example of the PFM simulation of

Figure 5. PFM simulation of motion of a threading dislocation and formation of misfit dislocation at film/substrate interface during stress relaxation in heteroepitaxial film. The numbers indicate the time sequence of dislocation configurations (after Ref. [5], reproduced with permission).

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misfit dislocation formation through threading dislocation motion in epitaxial film [5].

8.

Phase Field Model of Cracks

According to the variational principle discussed in the preceding section, the effect of voids can be fully reproduced by an appropriately chosen virtual misfit strain εivirtual (r) defined inside the domains corresponding to the voids. In j particular, the domains corresponding to cracks are thin platelets of interplanar thickness. To model moving cracks, which can spontaneously nucleate, propagate and coalesce, the virtual misfit strain εivirtual (r) is no longer constrained inside fixed j domains and is allowed to evolve driven by a reduction of total system free energy. In this formalism, εivirtual (r) describes evolving cracks: regions where j virtual εi j (r) =/ 0 are the laminar domains describing cracks. The crack-associated virtual misfit strain is also a dyadic product, εicrack = j (h i n j + h j n i )/2d, where n is the normal and d is the interplanar distance of the cleavage plane, and h(r) is the crack opening vector. As in the phase field model of dislocations, individual cracks of an arbitrary configuration are completely described by a set of fields, {h(α, r)}, where α numbers operative cleavage planes [15]. The total number of the fields is determined by the crystallography rather than a concrete crack configuration. For an arbitrary crack configuration in a polycrystal involving all operative cleavage planes, the total virtual misfit strain is expressed as a function of the fields h(α, r): εicrack (r) = j

1 α

2

Q ik (r) Q j l (r) [h k (α, r) Hl (α) + h l (α, r) Hk (α)], (19)

where H (α) = n (α)/d(α) is the reciprocal lattice vector of the cleavage plane α, and Q i j (r) is the grain rotation matrix field function that describes polycrystalline structure. Under stress, the opposite surfaces of cracks undergo opening displacements h(α, r). For given crack configuration, h(α, r) are a priori unknown and vary under varying stress. The crack-associated virtual misfit strain εicrack (r) j defined in Eq. (18), and thus the fields h(α, r), can be obtained through a numerical minimization procedure similar to that in Eq. (17), where the elaselast tic energy E equiv of such a crack system is also given by the KS elastic energy functional in terms of εicrack (r). j

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The non-linear effect of cohesive forces resisting crack-opening is described by the Landau-type coarse-grained “chemical” energy, which in the case of cracks is the cohesion energy,

f cohes [{h (α, r)}]d 3r,

E cohes =

(20)

V

whose integrand is a function of a set of fields {h(α, r)}. The specific cohesion energy f cohes (h) characterizes an energy that is required to provide a separation of two pieces of crystals by the distance h cut along the cleavage plane. From a microscopic point of view, the energy f cohes (h) is the atomistic energy required for a continuous breaking of atomic bonds across cleavage plane and thus creating two free surfaces during a process of crack formation. A specific approximation of this function similar to the one first proposed by Orowan is formulated by Wang et al. [5]. The general behavior of specific cohesion energy is schematically illustrated in Fig. 2(b), which introduces crack tip cohesive force acting in small crack tip zones. The cohesion energy E cohes defined in Eq. (20) describes a homogeneous separation where both boundaries of crack-opening are kept flat and parallel to cleavage plane. The energy correction associated with the effect of crack surface curvature is taken into account by the gradient energy

E

grad

=

φi j (r) d 3r,

V

(21)

where the argument of the integrand φi j (r) is defined as φi j (r) =

[H (α) × ∇]i h j (α, r),

(22)

α

which is similar to the tensor ϕi j (r) defined in Eq. (12) in the case of dislocations. The choice of the tensor φi j (r) is dictated by similar physical requirement, i.e., the gradient energy is significant only near crack tip where the surface curvature is big and is proportional to the crack front length while vanishes at flat surfaces of homogeneous opening. Following the Landau theory approach, we can also approximate the function φi j (r) by the Taylor expansion, which reflects the symmetry of the crystal lattice (see, for detailed discussion, Refs. [5, 9, 12]). The total free energy of the crack system characterized by the fields h(α, r) (r)), cohesion energy (20) and is the sum of elastic energy (in terms of εicrack j gradient energy (21): E = E elast + E cohes + E grad ,

(23)

which is a functional of a set of fields, {h(α, r)}. The temporal-spatial dependences of h(α, r, t) describe the collective motion of the crack ensemble.

Dislocation dynamics – phase field (a)

2303 (b)

Figure 6. PFM simulation of crack propagation during cleavage fracture in a 2D polycrystal composed of elastically isotropic grains (after Ref. [5], reproduced with permission). Different grain orientations are shown in gray scales.

The evolution of h(α, r, t) is obtained as a solution of the time-dependent Ginzburg–Landau kinetic equation: δE ∂h i (α, r, t) = −L i j + ξi (α, r, t), ∂t δh j (α, r, t)

(24)

where L i j is the kinetic coefficient characterizing crack propagation mobility, E is the system free energy (23), and ξi (α, r, t) is the Gaussian noise term reproducing the effect of thermal fluctuations. As shown by Wang et al. [5], a numerical solution h(α, r, t) of kinetic Eq. (24) automatically takes into account crack evolution without ad hoc assumption on possible path. Figure 6 shows one example of the PFM simulation of self-propagating crack during cleavage fracture in polycrystal [5].

9.

Multi-physics and Multi-scales

This article discusses the recent developments of the phase field theory and models of structurally inhomogeneous systems and their applications to modeling of the multi-dislocation dynamics and multi-crack evolution. The phase field approach can be used to simulate diffusional and displacive phase transformations (see “Phase Field Method–General Description and Computational Issues” by Karma and Chen, “Coherent Precipitation–Phase Field” by Wang, “Ferroic Domain Structures/Martensite” by Saxena and Chen, and the references therein), dislocation dynamics during plastic deformation and cracks development during fracture, as well as dislocation dynamics and morphology evolution [7, 8] of the heteroepitaxial thin films driven by the relaxation of epitaxial stress. These computational models are formulated in the same PFM formalism of the structure density dynamics. The difference between them is only in the analytical form of the Landau-type coarse-grained energy reflecting the physical nature and invariancy properties of the structural heterogeneities. This common analytical framework makes it easy to integrate the models of

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physically different processes into one unified structure density dynamics model. A cost of this would be just an increase in the number of evolving fields. A use of such a unified model allows one to address problems of arbitrary multi-mode microstructure evolution in complex materials systems. In particular, it enables one to investigate structure–property relationships of structurally inhomogeneous materials in situations where the structural heterogeneities of different kinds, which determine the mechanical properties of these materials, simultaneously evolve. The PFM theories and models presented in this article show that while challenges remain, significant advances have been achieved in integrating multiple physical phenomena for simulation of complex materials behavior. The second issue of equal importance is to bridge multiple length and time scales in materials modeling and simulation. Since the PFM approach is based on continuum theory, the PFM simulation is performed at mesoscale from a few nanometers to hundreds of micrometers. The PFM theory can also be applied to the atomic scales, in which case the role of structure density Fields is played by the occupation probabilities of the crystal lattice sites [18]. Recently the Phase Field model has been further extended to the subatomic scale where the field is the subatomic scale continuum density describing individual atoms [17]. The latter model bridges the molecular dynamics approach and the phase field theories discussed in this article. At an intermediate length scale, the mesoscale PFM theory and modeling bridge the gap between the modeling of atomistic level physical processes and macroscopic level material behaviors. The input information to the mesoscale modeling is the macroscopic material constants such as crystallographic data, elastic moduli, bulk chemical energy, interfacial energy, equilibrium composition, domain wall mobility, diffusivity, etc., which could be obtained via either atomistic calculations (first principle, molecular dynamics) or experimental measurements or both. Its output could be directly used to formulate the continuum constitutive relations for macroscopic materials theory and modeling. In particular, the PFM theory and models require a determination of the functional forms of Landau-type energy for different physical processes. This could be obtained through atomistic scale calculations. Incorporation of the results of atomistic simulations into the mesoscale PFM theories is a feasible way for multi-scale modeling.

References [1] A.G. Khachaturyan, Fiz. Tverd. Tela, 8, 2710 (1967. Sov. Phys. Solid State, 8, 2163), 1966. [2] A.G. Khachaturyan, Theory of Structural Transformations in Solids, John Wiley & Sons, New York, 1983. [3] A.G. Khachaturyan and G.A. Shatalov, Sov. Phys. JETP, 29, 557, 1969.

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[4] Y.U. Wang, Y.M. Jin, A.M. Cuiti˜no, and A.G. Khachaturyan, Acta Mater., 49, 1847, 2001. [5] Y.U. Wang, Y.M. Jin, and A.G. Khachaturyan, J. Appl. Phys., 91, 6435, 2002a. [6] Y.U. Wang, Y.M. Jin, and A.G. Khachaturyan, J. Appl. Phys., 92, 1351, 2002b. [7] Y.U. Wang, Y.M. Jin, and A.G. Khachaturyan, Acta Mater., 51, 4209, 2003. [8] Y.U. Wang, Y.M. Jin, and A.G. Khachaturyan, Acta Mater., 52, 81, 2004. [9] Y.U. Wang, Y.M. Jin, and A.G. Khachaturyan, “Mesoscale modeling of mobile crystal defects – dislocations, cracks and surface roughening: phase field microelasticity approach,” accepted to Phil. Mag., 2005a. [10] W.L. Bragg and J.F. Nye, Proc. R. Soc. Lond. A, 190, 474, 1947. [11] C. Shen and Y. Wang, Acta Mater., 51, 2595, 2003. [12] Y.U. Wang, Y.M. Jin, and A.G. Khachaturyan, “Structure density field theory and model of dislocation dynamics,” unpublished, 2005b. [13] Y.M. Jin and A.G. Khachaturyan, Phil. Mag. Lett., 81, 607, 2001. [14] S.Y. Hu, Y.L. Li, Y.X. Zheng, and L.Q. Chen, Int. J. of Plast., 20, 403, 2004. [15] D. Rodney, Y. Le Bouar, and A. Finel, Acta Mater., 51, 17, 2003. [16] F.R.N. Nabarro, Proc. Phys. Soc. Lond., 59, 256, 1947. [17] K.R. Elder and M. Grant, “Modeling elastic and plastic deformations in nonequilibrium processing using phase field crystals,” unpublished, 2003. [18] L.Q. Chen and A.G. Khachaturyan, Acta Metall. Mater., 39, 2533, 1991.

7.13 LEVEL SET DISLOCATION DYNAMICS METHOD Yang Xiang1 and David J. Srolovitz2 1

Department of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong 2 Princeton Materials Institute and Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, New Jersey 08544, USA

1.

Introduction

Although dislocation theory had its origins in the early years of the last century and has been an active area of investigation ever since (see [1–3]), our ability to describe the evolution of dislocation microstructures has been limited by the inherent complexity and anisotropy of the problem. This complexity has several contributing features. The interactions between dislocations are extraordinarily long-ranged and depend on the relative positions of all dislocation segments and the orientation of their Burgers vectors and line orientation. Dislocation mobility depends on the orientations of the Burgers vector and line direction with respect to the crystal structure. A description of the dislocation structure within a solid is further complicated by such topological events as annihilation, multiplication and reaction. As a result, analytical descriptions of dislocation structure have been limited to a small number of the simplest geometrical configurations. More recently, several dislocation dynamics simulation methods have been developed that account for complex dislocation geometries and/or the motion of multiple, interacting dislocations. The first class of these dislocation dynamics simulation methods is based upon front tracking methods. Three-dimensional simulations based upon these methods were first performed by Kubin et al. [4, 5] and later augmented by other researchers [6–11]. In these simulation methods, dislocation lines are discretized into individual segments. During the simulations, each segment is tracked and the forces on each segment from all other segments are calculated at each time increment (usually through the Peach–Koehler formula 2307 S. Yip (ed.), Handbook of Materials Modeling, 2307–2323. c 2005 Springer. Printed in the Netherlands.

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[12]). Three-dimensional front tracking methods made it possible to simulate dislocations motion with a degree of reality heretofore not possible. Such methods require, however, large computational investments because they track each segment of each dislocation line and calculate the force on each segment due to all other segments at each time increment. Moreover, special rules are needed to describe the topological changes that occur when segments of the same or different dislocations annihilate or merge [8, 9, 11]. Another class of dislocation dynamics models employs a phase field description of dislocations, as proposed by Khachaturyan, et al. [13, 14]. In their phase field model, density functions are used to model the evolution of a three-dimensional dislocation system. Dislocation loops are described as the perimeters of thin platelets determined by density functions. Since this method is based upon the evolution of a field in the full dimensions of the space, there is no need to track individual dislocation line segments and topological changes occur automatically. However, contributions to the energy that are normally not present in dislocation theory must be included within the phase field model to keep the dislocation core from expanding. In addition, dislocation climb is not easily incorporated into this type of model. Recently, a three-dimensional level set method for dislocation dynamics has been proposed [15, 16]. In this method, dislocation lines in three dimensions are represented as the intersection of zero levels (or zero contors) of two three-dimensional scalar functions (see [17–19] for a description of the level set method). The two three-dimensional level set functions are evolved using a velocity field extended smoothly from the velocity of the dislocation lines. The evolution of the dislocation lines is implicitly determined by the evolution of the two level set functions. Linear elasticity theory is used to compute the stress field generated by solved using a fast Fourier transform (FFT) method, assuming periodic boundary conditions. Since the level set method does not track individual dislocation line segments, it easily handles topological changes associated with dislocation multiplication and annihilation. This level set method for dislocation dynamics is capable of simulating the threedimensional motion of dislocations, naturally accounting for dislocation glide, cross-slip and climb through the choice of the ratio of the glide and climb mobilities. Unlike previous field-based methods [13, 14], no unconventional contributions to the system energy are required to keep the dislocation core localized. Numerical implementation of the level set method is through simple and accurate finite difference schemes on uniform grids. Results of simulation examples using this method agree very well with the theoretic predictions and the results obtained using other methods [15]. This method has also been used to simulate the dislocation-particle bypass mechanisms [16]. Here we shall review this level set dislocation dynamics method and present some of the simulation results in [15, 16].

Level set dislocation dynamics method

2.

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Continuum Dislocation Theory

We first briefly review the aspects of the continuum theory of dislocations that are relevant to the development of the level set description of dislocation dynamics. More complete descriptions of the continuum theory of dislocations can be found in, e.g., [2, 3, 20, 21]. Dislocations are line defects in crystals for which the elastic displacement vector satisfies

du = b,

(1)

L

where L is any contor enclosing the dislocation line with Burgers vector b and u is the elastic displacement vector. We can rewrite Eq. (1) in terms of the distortion tensor w, wi j = ∂u j /∂ xi for i, j = 1, 2, 3, as ∇ × w = ξ δ(γ ) ⊗ b,

(2)

where ξ is the unit vector tangent to the dislocation line, δ(γ ) is the two dimensional delta function in the plane perpendicular to the dislocation and is zero everywhere except on the dislocation, the operator ⊗ implies the tensor product of two vectors. While the Burgers vector is constant along any individual dislocation line, different dislocation lines may have different Burgers vectors. Equation (2) is valid only for dislocations with the same Burgers vector. In crystalline materials, the number of possible Burgers vectors, N , is finite (e.g., typically N = 12 for a FCC metal). Equation (2) may be extended to account for all possible Burgers vectors: ∇×w=

N

ξi δ(γi ) ⊗ bi

(3)

i=1

where γi represents all of the dislocations with Burgers vector bi , and ξi is the tangent to dislocation line i. Next, we consider the tensors describing the strain and stress within the body containing the dislocations. The strain tensor is defined as i j = 12 (wi j + w j i )

(4)

for i, j = 1, 2, 3. The stress tensor σ is determined from the strain tensor by the linear elastic constitutive equations (Hooke’s law) σi j =

3 k,l=1

Ci j kl kl

(5)

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for i, j = 1, 2, 3, where {Ci j kl } is the elastic constant tensor. For an isotropic medium, the constitutive equations can be written as 2ν (6) (11 + 22 + 33 )δi j 1 − 2ν for i, j = 1, 2, 3, where G is the shear modulus, ν is the Poisson ratio, and δi j is equal to 1 if i = j and is equal to 0, otherwise. In the absence of body forces, the equilibrium equation is simply σi j = 2Gi j + G

∇ · σ = 0.

(7)

Finally, the stress and strain tensors associated with a dislocation can be found by combining Eqs. (2), (4), (5) and (7). Dislocations can be driven by stresses within the body. The driving force for dislocation motion, referred to as the Peach–Koehler force, is f = σ tot · b × ξ,

(8)

where the total stress field σ includes the applied stress σ self-stress σ obtained by solving Eqs. (2), (4), (5) and (7): tot

σ tot = σ + σ appl.

appl

and the (9)

Dislocation migration can, at low velocities, be thought of as purely dissipative, such that the local dislocation velocity can be written as v = M · f,

(10)

where M is the mobility tensor. The interpretation of the mobility tensor M is deferred to the next section.

3.

The Level Set Dislocation Dynamics Method

The level set framework was devised by Osher and Sethian [17] in 1987 and and has been successfully applied to a wide range of physical and computer graphics problems [18, 19]. In this section, we present the level set approach to dislocation dynamics. More details and applications of this method can be found in [15, 16]. A level set is defined as a surface on which the level set function has a particular constant value. Therefore, an arbitrary scalar level set function can be used to describe a surface in three dimensional space, a line in two dimensional space, etc. In the level set method for dislocation dynamics, a dislocation in three dimensional space γ (t) is represented by the intersection of the zero levels of two level set functions φ(x, y, z, t) and ψ(x, y, z, t) defined in the three-dimensional space, i.e., where φ(x, y, z, t) = ψ(x, y, z, t) = 0,

(11)

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2311

see Fig. 1. The evolution of the dislocation is described by φt + v · ∇φ = 0 ψt + v · ∇ψ = 0

(12)

where v is the velocity of the dislocation extended smoothly to the threedimensional space, as described below. The reason this system of partial differential equations gives the correct motion of the dislocation can be understood in the following way. Assume that the dislocation γ (s, t), described in parametric form using the variable s, is given by φ(γ (s, t), t) = 0 ψ(γ (s, t), t) = 0,

(13)

where t is time. The derivative of Eq. (13) with respect to t gives ∇φ(γ (s, t), t) · γt (s, t) + φt (γ (s, t), t) = 0 ∇ψ(γ (s, t), t) · γt (s, t) + ψt (γ (s, t), t) = 0.

(14)

Comparing this result with Eq. (12) shows that γt (s, t) = v,

(15)

which means the velocity of the dislocation is equal to v, as required. The velocity field of a dislocation is computed from the stress field using Eqs. (8), (9) and (10). The self-stress field is obtained by solving the elasticity equations: (2), (4), (5) and (7). The unit vector locally tangent to the dislocation line, ξ , in Eqs. (2) and (8), is calculated from the level set functions φ and ψ using ξ=

∇φ × ∇ψ . |∇φ × ∇ψ|

(16)

ψ(x,y,z) ⫽ 0

ψ(x,y,z) ⫽ 0

Figure 1. A dislocation in three-dimensional space γ (t) is the intersection of the zero levels of the two level set functions φ(x, y, z, t) and ψ(x, y, z, t).

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The self-stress obtained by solving the elasticity equations (2), (4), (5) and (7) is singular on the dislocation line. This singularity is artificial because of the discreteness of the atomic lattice and non-linearities in the stress–strain relation not included in the linear elastic formulation. This non-linear region corresponds to the dislocation core. One approach to handling this problem is to use a smeared delta function instead of the exact delta function in Eq. (2) near each point on the dislocation line. The smeared delta function, like the exact one, is defined in the plane perpendicular to the dislocation line, and the vector ξ is defined everywhere in this plane to be the dislocation line tangent vector. This smeared delta function can be considered to be the distribution of the Burgers vector in the plane perpendicular to the dislocation line. The width of the smeared delta function is the diameter of the core region of the dislocation line. We use this approach to treat the dislocation core and its smeared delta function description. More precisely, the smeared delta function in Eq. (2) is given by δ(γ ) = δ(φ)δ(ψ),

(17)

where the delta functions on the right-hand-side are one-dimensional smeared delta functions δ(x) =

  1 1 + cos π x

2  0

− ≤ x ≤

,

(18)

otherwise

and scales the distance over which the delta function is smeared. The level set functions φ and ψ are usually chosen to be signed distance functions to their zero levels (i.e., the magnitude of the function is the distance from the closest point on the surface and the sign changes as we cross the zero level) and their zero levels are kept perpendicular to each other. A procedure called reinitialization is used to retain these properties of φ and ψ during their temporal evolution (see the next section for details). Therefore the delta function defined by (17) is a two-dimensional smeared delta function in the plane perpendicular to the dislocation line. Moreover, the size and the shape of the core region do not change during the evolution of the system. We now define the mobility tensor M. A dislocation line can glide conservatively (i.e., without diffusion) only in the plane containing both its tangent vector and the Burgers vector (i.e., the slip plane). A screw segment on a dislocation line can move in any plane containing the dislocation segment, since the tangent vector and Burgers vector are parallel. The switching of a screw segment from one slip plane to another is known as cross-slip. At high temperatures, non-screw segments of a dislocation can also move out of the slip plane by a non-conservative (i.e., diffusive) process; i.e., climb. The

Level set dislocation dynamics method

2313

following form of the mobility tensor satisfies these constraints:

M=

m g (I − n ⊗ n) + m c n ⊗ n

non-screw (ξ not parallel to b)

mgI

screw (ξ parallel to b)

,

(19) where n=

ξ ×b |ξ × b|

(20)

is the unit vector normal to the slip plane (i.e., the plane that contains the tangent vector ξ of the dislocation and its Burgers vector b), I is the identity matrix, I − n ⊗ n is the orthogonal matrix that projects vectors onto the plane with normal vector n, m g is the mobility constant for dislocation glide and m c is the mobility constant for dislocation climb. Typically, mc 1. (21) 0≤ mg The mobility tensor M, defined above, can account for the relatively high glide mobility and slow climb mobility. The present method is equally applicable to all crystal systems and all crystal orientations through appropriate choice of the Burgers vector and the mobility tensor (which can be rotated into any arbitrary orientation). In the present model, the dislocation can slip on all mathematical slip planes (i.e., planes containing the Burgers vector and line direction) and are not constrained to a particular set of crystal plane {hkl}, although it would be relatively simple to impose this constraint. Finally, while we implicitly assume that the glide mobilities of screw and non-screw segments are identical, this restriction is also easily relaxed. For simplicity, we restrict our description of the problem throughout rest of this discussion to the case of isotropic elasticity. While anisotropy will not cause any essential difficulties in the model, the added complexity clouds the description of the method. If we further assume periodic boundary conditions, the stress field can be solved analytically from the elasticity system (2), (4), (6) and (7) in Fourier space. The formulation can be found in [15]. A necessary condition for the elasticity system to have a periodic solution is that the total Burgers vector is equal to zero in the simulation cell. If the total Burgers vector is not equal to zero, the stress is equal to a periodic function plus a linear function in x, y and z [22, 23]. In this case, we also use the above mentioned expression for the stress field, as it only gives the periodic part of that field. This is consistent with the approach suggested by Bulatov et al. for computing periodic image interactions in the front tracking method [22, 23]. The above description of the method can only be applied to the case where all dislocations have the same Burgers vector b. For a more general case,

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where dislocation lines have different Burgers vectors, we would use different level set functions φi and ψi for each of the unique set of Burgers vectors bi , i = 1, 2, . . . , N , where N is the total number of the possible Burgers vectors, and use Eq. (3) instead of Eq. (2) in the elasticity equations.

4. 4.1.

Numerical Implementation Computing the Elastic Fields and the Dislocation Velocity

We solve the elasticity equations associated with the dislocations (2), (4), (6) and (7) using the FFT approach. The first step is to compute the dislocation tangent vector ξ δ(γ ) from the level set functions φ and ψ. The delta function δ(γ ) is computed using Eq. (17) with core radius = 3dx, where dx is the spacing of the numerical grid. The tangent vector ξ is computed using a regularized form of Eq. (16) (to avoid division by zero), i.e., ∇φ × ∇ψ , ξ= |∇φ × ∇ψ|2 + dx 2

(22)

as is standard in level set methods. The gradients of φ and ψ in Eq. (22) are computed using the third order weighted essentially nonoscillatory (WENO) method [24]. Since (WENO) derivatives are one-sided, we switch sides after several time steps to reduce the error caused by asymmetry. After we obtain the stress field, we compute the velocity field using Eqs. (8)–(10). We now use central differencing to compute the gradients of φ and ψ in (22) to get the tangent vector ξ in Eqs. (8) and (20). The mobility tensor in Eq. (10) is computed using Eqs. (19) and (20). We also regularize the denominator in Eq. (20) to avoid division by zero, as we did in Eq. (22). For the mobility tensor (19), we use the mobility for a screw dislocation when |ξ × b| < 0.1 and use the mobility for a non-screw dislocation otherwise.

4.2.

Numerical Implementation of the Level Set Method

4.2.1. Solving the evolution equations The level set evolution equations are commonly solved using high order essentially nonoscillatory (ENO) or WENO methods for the spatial discretization [17, 25, 24] and total variation diminishing (TVD) Runge–Kutta methods for the time discretization [26, 27]. Here we compute the spatial upwind derivatives using the third order WENO method [24] and use the fourth order TVD Runge–Kutta [27] to solve the temporal evolution equations (12).

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4.2.2. Reinitialization In level set methods for three-dimensional curves, the desired level set functions φ and ψ are signed distance functions to their zero levels (i.e., the value at each point in the scalar field is equal to the distance from the closest point on the zero level contor surface with a positive value on one side of the zero level and a minus sign on the other). Ideally, the zero level surfaces of these two functions should be perpendicular to each other. Initially, we choose φ and ψ to be such signed distance functions. However, there is no guarantee that the level set functions will always remain orthogonal signed distance functions during their evolution. This has the potential for causing large numerical errors. Standard level set techniques are used to reconstruct new level set functions from old ones with the dislocations unchanged. The resultant new level set functions are signed distance functions and their zero levels are perpendicular to each other. It has been shown [28, 29, 18, 30] that this procedure does not change the evolution of the lines represented by the intersection of the two level set functions, which are the dislocations here. (1) Signed Distance Functions To obtain a new signed distance function φ˜ from φ, we solve the following evolution equation to steady state [29] φ˜ ˜ − 1) = 0 (|∇ φ| φ˜t +

˜ 2 dx 2 . φ˜ 2 + |∇ φ| ˜ φ(t = 0) = φ

(23)

The new signed distance function ψ˜ from the level set function ψ can be found similarly. We solve for the steady state solutions to these equations using fourth order TVD Runge Kutta [27] in time and Godunov’s scheme [25, 31] combined with third order WENO [24] in space. We iterate these equations several steps of the fourth order TVD Runge Kutta method [27] using a time increment equal to half of the Courant-Friedrichs-Levy (CFL) number (i.e., the numerical stability limit). We solve for the new level set functions φ˜ and ψ˜ at each time step for use in solving the evolution equation (12). (2) Perpendicular Zero Levels Theoretically, the following equation resets the zero level of φ perpendicular to that of ψ [18, 30] ψ

∇ψ · ∇ φ˜ = 0 φ˜t +

2 2 2 ψ + |∇ψ| dx |∇ψ|2 + dx 2 . (24) ˜ φ(t = 0) = φ We solve for the steady state solution to this equation using fourth order TVD Runge Kutta [27] in time and third order WENO [24] for the upwind one˜ The gradient of ψ in the equation is computed using sided derivatives of φ.

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the average of the third order WENO [24] derivatives on both sides. We iterate this equation several steps of the fourth order TVD Runge–Kutta method given in [27] using a time increment of half of the CFL number. We reset the zero level of ψ perpendicular to that of φ similarly. We perform this perpendicular resetting procedure once every few time steps in the integration of the level set evolution equations (Eq. (12)).

4.2.3. Visualization The plotting of the dislocation line configurations is complicated by the fact that the dislocation lines are determined implicitly by the two level set functions. We use the following plotting method, described in more detail in [18]. Each cube in the grid is divided into six tetrahedra. Inside each tetrahedron, the level set functions φ and ψ are approximated by linear functions. The intersection of the zero levels of the two linear functions is a line segment inside the tetrahedron if the intersection is not empty (i.e., we need only compute the two ending points of the line segment on the tetrahedron surface), see Fig. 2. The union of all of these segments is the dislocation configuration.

4.2.4. Velocity interpolation and extension We use a smeared delta function (rather than an exact delta function) to compute the self-stress of the dislocations in order to smooth the singularity in the dislocation self-stress. The region near the dislocations where the smeared delta function is non-zero is the core region of the dislocations. The size of the core region is set by the discretization of space rather than by the physical

A

E G B

D F C

Figure 2. A cube in the grid, a tetrahedron A BC D and a dislocation line segment E F inside the tetrahedron. Point G is on the segment E F and the length of CG is the distance from the grid point C to the segment E F.

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core size. The leading order of the self-stress near the dislocations, when using a smeared delta function, is of the order 1/, where is the dislocation core size. This O(1/) self-stress near the dislocations does not contribute to the motion of the dislocations. We remove this contribution to the self-stress by a procedure which we call velocity interpolation and extension. We first interpolate the velocity on the dislocation line and then extend the interpolated value to the whole space using the fast sweeping method [32–36]. In the velocity interpolation, we use a method similar to that used in the plotting of dislocation lines. For any grid point, the dislocation line segments in nearby cubes can be found by the plotting method. The distance from this grid point to the dislocation line is the minimum distance to any dislocation segment. The remainder of the procedure is most simply described by consideration of the example in Fig. 2. The distance from the grid point of interest, point C for example, to the dislocation line is the distance from C to the segment E F. We locate a point G on the segment E F such that the length of C G is the minimum distance from C to E F. We know the velocity on the grid points of the cube in Fig. 2. We compute the velocity on the points E and F by trilinear interpolation of the velocity on these grid points. Then, we compute the velocity on the point G using a linear interpolation of the velocity on E and F. The velocity of point C is approximated as that on grid point G. To extend the velocities calculated at grid points neighboring the dislocation lines to the whole space, we employ the fast sweeping method [32–36]. The fast sweeping method is an algorithm for obtaining the distance function d(x) to the dislocations at all gridpoints from the distance values at gridpoints neighboring the dislocations (obtained as described above). This involves solving |∇d(x)| = 1

(25)

using the Godunov scheme with Gauss-Seidel iterations [35, 36]. Velocity extension is incorporated into this algorithm by updating the velocity v = (v 1 , v 2 , v 3 ) at each gridpoint after the distance function is determined such that the velocity is constant in the directions normal to the dislocations (the gradient directions of the distance function). This involves solving equations ∇v i (x) · ∇d(x) = 0,

(26)

for i = 1, 2, 3 simultaneously d(x) [32–34].

4.2.5. Initialization Initially, we choose the level set functions φ and ψ such that (1) the intersection of their zero levels gives the initial configuration of the dislocation lines; (2) φ and ψ are signed distance functions to their zero levels, respectively; and (3) the zero levels of φ and ψ are perpendicular to each other.

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Though we solve the elasticity equations assuming periodicity, the level set functions are not necessarily periodic and may be defined in a region smaller than the periodic simulation box.

5.

Applications

Figures 3–10 show several applications of the level set method for dislocation dynamics, described above. Additional simulation details and results can be found in [15, 16]. The simulations were performed within simulation cells that were l × l × l (where l = 2) in arbitrary units. The simulation cell is discretized into 64 × 64 × 64 grid points (For Fig. 6, the simulation cell is 2l ×2l ×l discretized into 128×128×64 grid points). We set the Poisson ratio ν = 1/3 and the climb mobility m c = 0, except in Figs. 3 and 4. The simulations described in Fig. 3, performed with these parameters, required less than five hours on a personal computer with a 450 MHz Pentium II microprocessor. 1 0.8 0.6 0.4 0.2

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Figure 3. A prismatic loop shrinking under its self-stress by climb. The Burgers vector b is pointing out of the paper. The loop is plotted at uniform time intervals starting with the outermost circle. The loop eventually disappears. (a)

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Figure 4. An initially circular glide loop in the x y plane, with a Burgers vector b in the x direction, expanding under a complex applied stress (σx z , σx y =/ 0) with mobility ratios m c /m g of (a) 0, (b) 0.25, (c) 0.5, (d) 0.75, and (e) 1.0. The loop is plotted at regular intervals in time.

Level set dislocation dynamics method

2319

The computational efficiency is independent of the absolute value of the glide mobility or the absolute value of the grid spacing. Figure 3 shows a prismatic loop (Burgers vector perpendicular to the plane containing the loop) shrinking under its self-stress by climb (the climb mobility m c > 0). The simulation result agrees with the well-known fact that the leading order shrinking force in this case is proportional to the curvature of the loop. Figure 4 shows an initially circular glide loop expanding under a complex applied stress with mobility ratios m c /m g of 0, 0.25, 0.5, 0.75, and 1.0. The applied stress generates a finite force on all the dislocation segments that tends to move them out of the initial slip plane. However, if the climb mobility m c = 0, only the screw segments move out of the slip plane; the non-screw segments cannot because the mobility in such direction is zero (Fig. 4(a)). If the climb mobility m c > 0, both the screw and non-screw segments move out of the slip plane (Fig. 4(b)–(e)). Figure 5 shows the intersection of two initially straight screw dislocations with different Burgers vectors. One dislocation is driven by an applied stress towards the other and then cuts through it. Two pairs of level set functions are used and the elastic fields are described using Eq. (3) instead of Eq. (2). Figure 6 shows the simulation of the Frank-Read source. Initially the dislocation segment is an edge segment. It bends out under an applied stress and generates a new loop outside. The initial configuration in this simulation is a rectangular loop. Of its four segments, two opposite ones are operating as the

b2

z

b1 y x

Figure 5. Intersection of two initially straight screw dislocations with Burgers vectors b1 and b2 . Dislocation 1 is driven in the direction of the −x axis by the applied stress σ yz .

Figure 6. Simulation of the Frank-Read source. Initially the dislocation segment is an edge segment in the x y plane (the z axis is pointing out of the paper). The Burgers vector is parallel to the x axis and a stress σx z is applied. The configuration in the slip plane is plotted at different time during the evolution.

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Y. Xiang and D.J. Srolovitz

Frank-Read source in the plane perpendicular to the initial loop and the other two are fixed. Figure 7 shows an edge dislocation bypassing a linear array of impenetrable particles, leaving Orowan loops [37] around the particles behind. The dislocation moves towards the particles under an applied stress. The glide plane of the dislocation intersects the centers of the particles (the particles are coplanar). The impenetrable particles are assumed to exert a strong short-range repulsive force on dislocations, see [15] for details. Figure 8 shows a screw dislocation bypassing an impenetrable particle by a combination of Orowan looping [37] and cross-slipping [38]. The dislocation moves towards the particle under an applied stress. It leaves two loops behind on the two sides of the particle. The plane in which the screw dislocation would glide in the absence of the particle is above the particle center. Figure 9 shows an edge dislocation bypassing a misfitting spherical particle by cross-slip [38], where the slip plane of the dislocation is above the particle center. The misfit > 0. The dislocation moves towards the particles under an applied stress. Two loops are left behind: one is behind the particle and the other is around the particle. They have the same Burgers vector but opposite line directions. The stress fields generated by a (dilatational) misfitting spherical particle (isotropic elasticity) were given by Eshelby [39]. Figure 10 shows the critical stress for an edge dislocation to bypass co-planar impenetrable particles by the Orowan mechanism. The stress is 3

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Figure 7. An edge dislocation bypassing a linear array of impenetrable particles, leaving Orowan loops [37] around the particles behind. The Burgers vector b is in the x direction. The applied stress σx z =/ 0, where the z direction is pointing out of the paper.

z b y

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Figure 8. A screw dislocation bypassing an impenetrable particle by a combination of Orowan looping [37] and cross-slipping [38]. The Burgers vector b is in the y direction, the applied stress is σ yz =/ 0, and the plane in which the screw dislocation would glide in the absence of the particle is above the particle center (in the +z direction).

Level set dislocation dynamics method

2321

Figure 9. An edge dislocation bypassing a misfitting spherical particle by cross-slip [38], where the slip plane of the dislocation is above the particle center. The Burgers vector b is in the x direction, the applied stress is σx z =/ 0. The misfit > 0.

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plotted in the unit (Gb/L) against log(D1 /r0 ), where G is the shear modulus, b is the length of the Burgers vector, L is the inter-particle distance, D is the diameter of the particle, D1 is the harmonic mean of L and D, and r0 is the inner cut-off radius, associated with the dislocation core. The data points represent the simulation results and the straight line is the best fit to our data using the classic equation (Gb/2π L) log(D1 /r0 ) [37, 40, 41]. It shows a good agreement between the simulation results using the level set method and the theoretical estimates.

References [1] V. Volterra, Ann. Ec. Norm., 24, 401, 1905. [2] F.R.N. Nabarro, Theory of Crystal Dislocations, Clarendon Press, Oxford, England, 1967. [3] J.P. Hirth and J. Lothe, Theory of Dislocations, 2nd edition, John Wiley, New York, 1982. [4] L.P. Kubin and G.R. Canova, In: U. Messerschmidt et al. (eds.), Electron Microscopy in Plasticity and Fracture Research of Materials, Akademie Verlag, Berlin, p. 23, 1990. [5] L.P. Kubin, G. Canova, M. Condat, B. Devincre, V. Pontikis, and Y. Brechet, Solid State Phenomena, 23/24, 455, 1992. [6] H.M. Zbib, M. Rhee, and J.P. Hirth, Int. J. Mech. Sci., 40, 113, 1998. [7] M. Rhee, H.M. Zbib, J.P. Hirth, H. Huang, and T. de la Rubia, Modelling Simul. Mater. Sci. Eng., 6, 467, 1998. [8] K.W. Schwarz, J. Appl. Phys., 85, 108, 1999. [9] N.M. Ghoniem, S.H. Tong, and L.Z. Sun, Phys. Rev. B, 61, 913, 2000. [10] B. Devincre, L.P. Kubin, C. Lemarchand, and R. Madec, Mat. Sci. Eng. A-Struct., 309, 211, 2001. [11] D. Weygand, L.H. Friedman, E. Van der Giessen, and A. Needleman, Modelling Simul. Mater. Sci. Eng., 10, 437, 2002. [12] M. Peach and J.S. Koehler, Phys. Rev., 80, 436, 1950. [13] A.G. Khachaturyan, In: E.A. Turchi, R.D. Shull, and A. Gonis (eds.), Science of Alloys for the 21st Century, TMS Proceedings of a Hume-Rothery Symposium, TMS, p. 293, 2000. [14] Y.U. Wang, Y.M. Jin, A.M. Cuitino, and A.G. Khachaturyan, Acta Mater., 49, 1847, 2001. [15] Y. Xiang, L.T. Cheng, D.J. Srolovitz, and W. E, Acta Mater., 51, 5499, 2003. [16] Y. Xiang, D.J. Srolovitz, L.T. Cheng, and W. E, Acta Mater., 52, 1745, 2004. [17] S. Osher and J.A. Sethian, J. Comput. Phys., 79, 12, 1988. [18] P. Burchard, L.T. Cheng, B. Merriman, and S. Osher, J. Comput. Phys., 170, 720, 2001. [19] S. Osher and R.P. Fedkiw, J. Comput. Phys., 169, 463, 2001. [20] R.W. Lardner, Mathematical Theory of Dislocations and Fracture, University of Toronto Press, Toronto and Buffalo, 1974. [21] L.D. Landau and E.M. Lifshitz, Theory of Elasticity, 3rd edn., Pergamon Press, New York, 1986. [22] V.V. Bulatov, M. Rhee, and W. Cai, In: L. Kubin, et al. (eds.), Multiscale Modeling of Materials – 2000, Materials Research Society, Warrendale, PA, 2001.

Level set dislocation dynamics method [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41]

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W. Cai, V.V. Bulatov, J. Chang, J. Li, and S. Yip, Phil. Mag., 83, 539, 2003. G.S. Jiang and D. Peng, SIAM J. Sci. Comput., 21, 2126, 2000. S. Osher and C.W. Shu, SIAM J. Numer. Anal., 28, 907, 1991. C.W. Shu and S. Osher, J. Comput. Phys., 77, 439, 1988. R.J. Spiteri and S.J. Ruuth, SIAM J. Numer. Anal., 40, 469, 2002. M. Sussman, P. Smereka, and S. Osher, J. Comput. Phys., 114, 146, 1994. D. Peng, B. Merriman, S. Osher, H.K. Zhao, and M. Kang, J. Comput. Phys., 155, 410, 1999. S. Osher, L.T. Cheng, M. Kang, H. Shim, and Y.H.R. Tsai, J. Comput. Phys., 179, 622, 2002. M. Bardi and S. Osher, SIAM J. Math. Anal., 22, 344, 1991. H.K. Zhao, T. Chan, B. Merriman, and S. Osher, J. Comput. Phys., 127, 179, 1996. S. Chen, M. Merriman, S. Osher, and P. Smereka, J. Comput. Phys., 135, 8, 1997. D. Adalsteinsson and J.A. Sethian, J. Comput. Phys., 148, 2, 1999. Y.H.R. Tsai, L.T. Cheng, S. Osher, and H.K. Zhao, SIAM J. Numer. Anal., 41, 673, 2003. H.K. Zhao, Math Comp., to appear. E. Orowan, In: Symposium on Internal Stress in Metals and Alloys, London: The Institute of Metals, p. 451, 1948. P.B. Hirsch, J. Inst. Met., 86, 13, 1957. J.D. Eshelby, In: F. Seitz and D. Turnbull, (ed.), Solid State Physics, vol. 3, Academic Press, New York, 1956. M.F. Ashby, Acta Metall., 14, 679, 1966. D.J. Bacon, U.F. Kocks, and R.O. Scattergood, Phil. Mag., 28, 1241, 1973.

7.14 COARSE-GRAINING METHODOLOGIES FOR DISLOCATION ENERGETICS AND DYNAMICS J.M. Rickman1 and R. LeSar2 1

Department of Materials Science and Engineering, Lehigh University, Bethlehem, PA 18015, USA 2 Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA

1.

Introduction

Recent computational advances have permitted mesoscale simulations, wherein individual dislocations are the objects of interest, of systems containing on the order of 106 dislocation [1–4]. While such simulations are beginning to to elucidate important energetic and dynamical features, it is worth noting that the large-scale deformation response in, for example, wellworked metals having dislocation densities ranging between 1010 −1014 /m2 can be accurately described by a relatively small number of macrovariables. This reduction in the number of degrees of freedom required to characterize plastic deformation implies that a homogenization, or coarse-graining, of variables is appropriate over some range of length and time scales. Indeed, there is experimental evidence that, at least in some cases, the mechanical response of materials depends most strongly on the macroscopic density of dislocations [5] while, in others, the gross substructural details may also be of importance. A successful, coarse-grained theory of dislocation behavior requires the identification of the fundamental homogenized variables from among the myriad of dislocation coordinates as well as the time scale for overdamped defect motion. Unfortunately, there has been, to date, little effort to devise workable coarse-graining strategies that properly reflect the long-ranged nature of dislocation–dislocation interactions. Thus, in this topical article, we review salient work in this area, highlighting the observation that seemingly unrelated problems are, in fact, part of a unified picture of coarse-grained dislocation behavior that is now emerging. More specifically, a prescription is given for identifying a relevant macrovariable set that describes a collection of mutually interacting dislocations. This set follows from a real-space 2325 S. Yip (ed.), Handbook of Materials Modeling, 2325–2335. c 2005 Springer. Printed in the Netherlands.

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analysis involving the subdivision of a defected system into volume elements and subsequent multipole expansions of the dislocation density. It is found that the associated multipolar energy expansion converges quickly (i.e., usually at dipole or quadrupole order) for well-separated elements. Having formulated an energy functional for the macrovariables, the basic ingredients of temporal coarse-graining schemes are then outlined to describe dislocation–dislocation interactions at finite temperature. Finally, we suggest dynamical models to describe the time evolution of the coarse macrovariables. This article is organized as follows. In Section 2 we outline spatial coarsegraining strategies that permit one to link mesoscale dislocation energetics and dynamics with the continuum. In Section 3 we review some temporal coarsegraining procedures that make it possible to reduce the number of macrovariables needed in a description of thermally induced kinks and jogs on dislocation lines. Section 4 contains a summary of the paper and a discussion of coarse-grained dynamics.

2.

Spatial Coarse-Graining Strategies

A homogenized description of the energetics of a collection of dislocations in, for example, a well-worked metal is complicated by the long-ranged, anisotropic nature of dislocation–dislocation interactions. Such interactions lead to the formation of patterns at multiple length scales as dislocations polygonize to lower the energy of the system [6, 7]. This tendency to form dislocation walls can be quantified via the calculation of an orientationally weighted pair correlation function [8, 9] from a large-scale, two-dimensional mesoscale simulation of edge dislocations, as shown in Fig. 1. As is evident from the figure, both 45◦ and 90◦ walls are dominant (with other orientations also represented), consistent with the propensity to form dislocation dipoles with these relative orientations. Thus, a successful coarse-graining strategy must preserve the essential features of these dislocation structures while reducing systematically the number of degrees of freedom necessary for an accurate description. There are different, although complementary, avenues to pursue in formulating a self-consistent, real-space numerical coarse-graining strategy in which length scales shorter than some prescribed cutoff are eliminated from the problem. One such approach involves subdividing the system into equally sized blocks and then, after integrating out information on scales less than the block size, inferring the corresponding coarse-grained free energy from probability histograms compiled during finite-temperature mesoscale simulations [10–12]. In this context, each block contains many dislocations, and so the free energy extracted from histograms will be a function of a block-averaged dislocation density. This method is motivated by Monte Carlo coarse-graining (MCCG) studies of spin system and can be readily applied, for example, to a

Coarse-graining methodologies for dislocation energetics

2327

Figure 1. An angular pair-correlation function. In the white (black) region there is a relatively high probability of finding a dislocation with positive (negative) Burgers vector, given that a dislocation with positive Burgers vector is located at the origin. From [9].

two-dimensional dislocation system, modeled as a “vector” lattice gas, once the long-ranged nature of the dislocation–dislocation interaction is taken into account. Unfortunately, however, the energy scale associated with dislocation interactions is typically much greater than kB T , where kB is Boltzmann’s constant and T is the temperature, and therefore the finite-temperature sampling inherent in the MCCG technique is not well-suited to the current problem. To develop a more useful technique that reflects the many frustrated, lowenergy states relevant here, consider first the ingredients of a coarse-graining strategy based on continuous dislocation theory. The theory of continuous dislocations follows from the introduction of a coarse-graining volume over which the dislocation density is averaged. The dislocation density is a tensor r ), where k indicates the component of field defined at r with components ρki ( the line direction and i indicates the component of the Burgers vector. In this development it is generally assumed that is large relative to the dislocation spacing, yet small relative to the system size [13]. However, the exact meaning of this averaging prescription is unclear, and it is not obvious at what scales a continuum theory should hold. In particular, if one takes the above assumption that a continuum theory holds for length scales much greater than the typical dislocation spacing, then the applicability of the method is restricted to

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scales much greater than the dislocation structures known to be important for materials response [14]. Clearly, if the goal is to apply this theory at smaller length scales so as to capture substructures relevant to mechanical response, then one must build the ability to represent such substructures into the formalism. As previous work focused on characterizing these dislocation structures (calculated from two-dimensional simulations) through the use of pair correlation functions [8, 9], we outline here an extension to the continuous dislocation theory that incorporates important spatial correlations. The starting point for this development is the work of Kosevich, who showed that the interaction energy of systems of dislocations (in an isotropic linear elastic medium) can be written in terms of Kr¨oner’s incompatibility tensor [15]. From that form one can derive an energy expression in terms of the dislocation density tensor [16] µ EI = 16π

ipl j mn R,mp ( r , r )

r )ρin ( r ) + δi j ρkl ( r )ρkn (r ) + × ρ j l (

2ν ρil ( r )ρ j n ( r ) d r d r , 1−ν (1)

where the integrals are over the entire system, δi j is the Kronecker delta, and repeated indices are summed. The notation a,i denotes the derivative of a with respect to xi .R,mk indicates the derivative ∂ 2 | r − r | /∂ xm ∂ xk . It should be noted here that the energy expression in Eq. (l) includes very limited information about dislocation structures at scales smaller than the averaging volume. Here we summarize results from one approach to incorporate the effects of lower-scale structures, with a more complete derivation given elsewhere [17]. The basic plan is to divide space into small averaging volumes, calculate the local multipole moments of the dislocation microstructure (as described next), and then to write the energy as an expansion over the multipoles. Consider a small region of space with volume containing n distinct dislocation loops, not necessarily entirely contained within . We can define a set of moment densities of the distribution of loops in as [17] = ρl() j ρl() jα = ···

n 1 (q) b q=1 j

1

n q=1

,

(q)

(q)

dll ,

(2)

C(q),

bj

C(q),

(q)

rα(q) dll ,

(3)

Coarse-graining methodologies for dislocation energetics

2329

where b is the Burgers vector and the notation (C (q) , ) indicates that we integrate over those parts of dislocation line q that lie within the volume . () Here ρl() j is the dislocation density tensor and ρl j α is the dislocation dipole moment tensor for volume . Higher-order moments can also be constructed. Consider next two regions in space denoted by A and B. We can write the interaction energy between the dislocations in the two regions as sums of pair interactions or, equivalently, as line integrals over the dislocation loops [18, 19]. Now, if the volumes are well separated, then the interaction energy can be written as a multipole expansion [17]. Upon truncating this expansion at zeroth order (i.e., the “charge–charge” term) one finds (o) = E AB

µ 8π

A B

×

ipl j mn R,mp

B ) (A ) ρ ( j l ρin

+

(A ) δi j ρkl(B ) ρkn

2ν (B ) (A ) ρ + ρ jn d rA d rB , (4) 1 − ν il

where R connects the centers of the two regions. Summing the interactions between all regions of space and then taking the limit that the averaging volumes A and B go to differential volume elements, the Kosevich form for continuous dislocations in Eq. (l) is recovered and the dislocation density tensor approaches asymptotically the continuous result. Corrections to the Kosevich form associated with a finite averaging volume can now be obtained by including higher-order moments in the expansion. For example, the first-order term (“charge–dipole”) has the form (dipole−charge)

EI

=

µ 16π

ipl j mn R,mpα

ρ j l ( r )ρinα (r )

2ν ρil ( r )ρknα (r ) + r )ρ j nα (r ) + δi j ρkl ( 1−ν

− ρ j lα ( r )ρin (r ) + δi j ρklα ( r )ρkn (r ) 2ν ρil,α ( r )ρ j n (r ) + 1−ν

d r dr

(5)

where R,mpα is the next higher-order derivative of R [17]. We note that inclusion of terms that depend on the local dipole are equivalent to gradient corrections to the Kosevich form. The expression in Eq. (5) (and higher-order terms) can be used as a basis for a continuous dislocation theory with local structure by including the dipole (and higher) dislocation moment tensors as descriptors. For a systematic analysis of the terms in a dislocation multipolar energy expansion and their dependence on coarse-grained cell size, the reader is referred to a review elsewhere [20].

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3.

J.M. Rickman and R. LeSar

Temporal Coarse-graining – Finite-temperature Effects

At finite temperatures dislocation lines may be perturbed by thermally induced kinks and jogs. While such perturbations are inherent in 3D mesoscale dislocation dynamics simulations at elevated temperatures, it is of interest here to explore methods to integrate out these modes to arrive at a simpler description of dislocation interactions. For example, motivated by calculations of the fluctuation-induced coupling of dipolar chains in electrorheological fluids and flux lines in superconductors [21], one can determine the interaction free energy between fluctuating dislocation lines that are in contact with a thermal bath and thereby deduce the effective force between dislocations. Indeed, the impact of temperature-induced fluctuations on the interaction of two (initially) parallel screw dislocations was the focus of a recent paper [16]. In this work it was assumed that perturbations in the dislocation lines that arise from thermal fluctuations in the medium can be viewed as a superposition of modes having screw, edge and mixed character. The impact of these fluctuations on the force between the dislocations at times greater than the those associated with the period of a fluctuation was then examined by integrating out the vibrational modes of the dislocation lines. The procedure employed was similar to that used to construct quaisharmonic models of solids in which vibrational atomic displacements are eliminated in favor of their corresponding frequency spectrum in the canonical partition function [22]. In both cases the resulting free energy then depends on a small set of coarse-grained variables. To see how a finite-temperature force may be constructed, consider a prototypical system in which harmonic perturbations are added to two straight screw dislocation lines without changing the Burgers vector, which remains along the z (i.e., x3 ) axis. We describe those fluctuations by parameterizing the line position in the x1 −x2 plane with a Fourier series with r = xˆ1 F (x3 ) + xˆ2 F⊥ (x3 ) where Fκ (x3 ) =

n max

C+,n,κ einκ π x3 /L + C−,n,κ e−inκ π x3 /L ,

(6)

n κ=1

κ is either ⊥ or , L is a maximal length characterizing the system, and n max is related to a minimum characteristic length. An expression for the dislocation density tensor (ρi j ( r )) in the form of the expansion in Eq. (6) can be written in terms of Dirac delta functions indicating the line position. The next step in the analysis is to calculate the Fourier transform of the dislocation density for the perturbed dislocation lines. While it is possible to write these densities in terms of infinite series expansions, it is more useful here to restrict attention to the lowest-order terms in the fluctuation amplitudes that are excited at low temperatures. Having determined the dislocation density tensor, the aim is then to calculate the interaction energy between two

Coarse-graining methodologies for dislocation energetics

2331

perturbed dislocation lines. This energy will, in turn, determine the corresponding Boltzmann weight for the fluctuating pair of lines and, hence, the equilibrium statistical mechanics of this system. The interaction energy can be obtained from an expression for the total energy, E, based on ideas from continuous dislocation theory [23]. For this purpose it is again convenient to write the Kosevich energy functional, this time as an integral in reciprocal space, [13, 15] as E[ρ] ¯ =

1 2

d3 k ρ˜i j (k) ρ˜kl (−k), K i j kl (k) (2π)3

(7)

where the integration is over reciprocal space (tilde denoting a Fourier transform), the kernel (without core energy contributions)

K i j kl

µ 2ν Ci j Ckl , = 2 Q ik Q j l + Cil Ckj + k 1−ν

(8)

and Q¯ and C¯ are longitudinal and transverse projection operators, respectively. (The energetics of the disordered core regions near each line can be incorporated, at least approximately, by the inclusion of a phenomenological energy penalty term in the kernel above.) The Helmholtz free energy and, therefore, the associated finite-temperature k ) for forces can be obtained by first constructing the partition function Z (k, ˆ ¯ the system of two perturbed screw dislocations with associated k = i k + jˆk¯⊥ and k = iˆ k¯ + jˆk¯ ⊥ . This is accomplished by considering the change in energy,

e(a), associated with fluctuations on the (initially straight) dislocations and noting that it can be written as a sum of contributions, ( e) and ( e)⊥ , corresponding to in-plane and transverse fluctuation modes. One then finds that the factorized partition function k ) = N Z (k,

= Z⊥ Z,

−L( e) dω exp kB T

−L( e)⊥ dω⊥ exp kB T

(9)

where N is a normalization factor and ω is the eight-dimensional configuration space described by the complex fluctuation amplitudes. The Helmholtz free energy associated with the interactions between the fluctuating screws is then given by A = −kB T ln(Z ) = −kB T {ln(Z ) + ln(Z ⊥ )}.

(10)

In our earlier work [16] we gave analytic expressions for both Z ⊥ and Z . Upon integrating A over all possible perturbation wavevectors one finally

2332

J.M. Rickman and R. LeSar 0.002 0.0015

b2/kBT

0.001 0.0005 0 ⫺0.0005 ⫺0.001 ⫺0.0015 ⫺0.002 20

22

24

26

28 a*

30

32

34

36

Figure 2. The contributions to the normalized force versus normalized separation for two perturbed dislocations. The parallel (perpendicular) contribution is denoted by triangles (circles). From [16].

arrives at the total free energy, now a function of coarse-grained variables (i.e., the average line locations.) From the development above it is clear that the average force between the dislocations is obtained by differentiating the total free energy with respect to the line separation a. For the purposes of illustration it is convenient to decompose this force into a sum of components both parallel and perpendicular to a line joining the dislocations. For concreteness, we evaluate the resulting force for dislocations embedded in copper and having the same properties. The maximum size of the system is taken to be L = 200b, where b is the magnitude of the Burgers vector of a dislocation. As can be seen from Fig. 2, a plot of the normalized force contributions versus normalized separation a ∗ (a ∗ = a/b), the parallel (perpendicular) contribution to the force is repulsive (attractive), both components being of similar magnitude. Further analysis indicates that the net thermal force at a temperature of 600 K at a separation of a ∗ = 22 is approximately 1.3 × 10−4 J/m 2 for b = 2.56 Å. This thermal force is approximately 1000 smaller in magnitude than the direct (Peach–Koehler) force for the same separation.

4.

Discussion

Several applications of spatial and temporal coarse graining to systems containing large numbers of dislocations have been outlined here. A common

Coarse-graining methodologies for dislocation energetics

2333

theme linking these strategies is the classification of relevant state variables and the subsequent elimination of a subset of degrees of freedom (via averaging, etc.) in favor of those associated with a coarser description. For example, in the case of the straight screw dislocations interacting with a thermal bath (see Section 3), the vibrational modes of the dislocation lines can be identified as “fast” variables that can be integrated out of the problem, with the resultant free energy based on the long-time, average location of these lines. Furthermore, the spatial coarse graining schemes proposed above involve the identification of a dislocation density, based on localized collections of dislocations, and the separation of interaction length scales (i.e., in terms of a multipolar decomposition and associated gradient expansions) with the aim of developing a model based solely on the dislocation density and other macrovariables. It remains to link coarse-grained dislocation energetics with the corresponding dynamics. While the history of the theory of dislocation dynamics goes back to the early work of Frank [24], Eshelby [25], Mura [26] and others, who deduced the inertial response for isolated edge and screw dislocations in an elastically isotropic medium, we note that the formulation of equations of motion for an ensemble of mutually interacting dislocations at finite temperature is an ongoing enterprise that presents numerous challenges. We therefore merely outline promising approaches here. The construction of a kinetic model is, perhaps, best motivated by earlier work in the field of critical dynamics [27, 28]. More specifically, in this approach, one formulates a set of differential equations that reflect any conservation laws that constrain the evolution of the variables (e.g., conservation of Burgers vector in the absence of sources). Different workers have employed variations of this formalism in dislocation dynamics simulations. For example, in early work in this area, Holt [29] postulated a dissipative equation of motion for the scalar dislocation density, subject to the constraint of conservation of Burgers vector, with a driving force given by gradients of fluctuations in the dislocation interaction energy. Rickman and Vinals [30], following an earlier statistical-mechanical treatment of free dislocation loops [13] and by hydrodynamic descriptions of condensed systems, considered a dynamics akin to a noise-free Model B [28] to track the time evolution of the dislocation density tensor in an elastically isotropic medium. Equations of motion for dislocation densities have also been advanced by Marchetti and Saunders [31] in a description of a viscoelastic medium containing unbound dislocations, by Haataja et al. [32] in a continuum model of misfitting heteroepitaxial films and, recently, by Khachaturyan and coworkers [33–35] in several phase-field simulations. The elegant approach of this group is, however, an alternative formulation of overdamped discrete dislocation models, as opposed to a spatially coarse-grained description. As indicated above, work in this area continues, with some current efforts directed at incorporating dislocation substructural information in the dynamics.

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Acknowledgments J.M. Rickman would like to thank the National Science Foundation for its support under grant number DMR-9975384. The work of R. LeSar was performed under the auspices of the United States Department of Energy (US DOE under Contract No. W-7405-ENG-36) and was supported by the Office of Science/Office of Basic Energy Sciences/Division of Materials Science of the US DOE.

References [1] E. Van der Giessen and A. Needleman, “Micromechanics simulations of fracture,” Ann. Rev. Mater. Res., 32, 141, 2002. [2] R. Madec, B. Devincre, and L. Kubin, “Simulation of dislocation patterns in multislip,” Scripta Mater., 47, 689–695, 2002. [3] M. Rhee, D.H. Lassila, V.V. Bulatov, L. Hsiung, and T.D. de la Rubia, “Dislocation multiplication in BCC molybdenum: a dislocation dynamics simulation,” Phil. Mag. Lett., 81, 595, 2001. [4] M. Koslowski, A.M. Cuitino, and M. Ortiz, “A phase-field theory of dislocation dynamics, strain hardening and hysteresis in ductile single crystals,” J. Mech. Phys. Solids, 50, 2597, 2002. [5] A. Turner and B. Hasegawa, “Mechanical testing for deformation model development,” ASTM, 761, 1982. [6] J.P. Hirth and J. Lothe, Theory of Dislocations, Krieger, Malabar, Florida, 1982. [7] D.A. Hughes, D.C. Chrzan, Q. Liu, and N. Hansen, “Scaling of misorientation angle distributions,” Phys. Rev. Lett., 81, 4664–4667, 1998. [8] A. Gulluoglu, D.J. Srolovitz, R. LeSar, and P.S. Lomdahl, “Dislocation distributions in two dimensions,” Scripta Metall., 23, 1347–1352, 1989. [9] H.Y. Wang, R. LeSar, and J.M. Rickman, “Analysis of dislocation microstructures: impact of force truncation and slip systems,” Phil. Mag. A, 78, 1195–1213, 1998. [10] K. Binder, “Critical properties from Monte Carlo coarse graining and renormalization,” Phys. Rev. Lett., 47, 693–696, 1981. [11] K. Kaski, K. Binder, and J.D. Gunton, “Study of cell distribution functions of the three-dimensional ising model,” Phys. Rev. B, 29, 3996–4009, 1984. [12] M.E. Gracheva, J.M. Rickman, and J.D. Gunton, “Coarse-grained Ginzburg-Landau free energy for Lennard–Jones systems,” J. Chem. Phys., 113, 3525–3529, 2000. [13] D.R. Nelson and J. Toner, “Bond-orientational order, dislocation loops and melting of solids and smectic–a liquid crystals,” Phys. Rev. B, 24, 363–387, 1981. [14] U.F. Kocks, A.S. Argon, and M.F. Ashby, Thermodynamics and Kinetics of Slip, Prog. Mat. Sci., 19, 1975. [15] A.M. Kosevich, In: F.R.N. Nabarro (ed.), Dislocations in Solids, New York, p. 37, 1979. [16] J.M. Rickman and R. LeSar, “Dislocation interactions at finite temperature,” Phys. Rev. B, 64, 094106, 2001. [17] R. LeSar and J.M. Rickman, Phys. Rev. B, 65, 144110, 2002. [18] N.M. Ghoniem and L.Z. Sun, “Fast-sum method for the elastic field of three-dimensional dislocation ensembles,” Phys. Rev. B, 60, 128, 1999.

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[19] R. de Wit, Solid State Phys., 10, 249, 1960. [20] R. LeSar and J.M. Rickman, “Coarse-grained descriptions of dislocation behavior,” to be published in Phil. Mag., 83, 3809–3827, 2003. [21] T.C. Halsey and W. Toor, “Fluctuation-induced couplings between defect lines or particle chains,” J. Stat. Phys., 61, 1257–1281, 1990. [22] J.M. Rickman and D.J. Srolovitz, “A modified local harmonic model for solids,” Phil. Mag. A, 67, 1081–1094, 1993. [23] E. Kr¨oner, Kontinuumstheorie der Versetzungen and Eigenspannungen, Ergeb. Angew. Math. 5 (Springer-Verlag, Berlin 1958). English translation: Continuum Theory of Dislocations and Self-Stresses, translated by I. Raasch and C.S. Hartley, (United States Office of Naval Research), 1970. [24] F.C. Frank, “On the equations of motion of crystal dislocations,” Proc. Phys. Soc., 62A, 131–134, 1949. [25] J.D. Eshelby, “Supersonic dislocations and dislocations in dispersive media,” Proc. Phys. Soc., B69, 1013–1019, 1956. [26] T. Mura, “Continuous distribution of dislocations,” Phil. Mag., 8, 843–857, 1963. [27] J.D. Gunton and M. Droz, “Introduction to the theory of metastable and unstable states,” Springer-Verlag, New York, pp. 34–42, 1983. [28] P.C. Hohenberg and B.I. Halperin, “Theory of dynamic critical phenomena,” in Rev. Mod. Phys., 49, 435–479, 1977. [29] D.L. Holt, “Dislocation cell formation in metals,” J. Appl. Phys., 41, 3197 1970. [30] J.M. Rickman and Jorge Vinals, “Modeling of dislocation structures in materials,” Phil. Mag. A, 75, 1251, 1997. [31] M.C. Marchetti and K. Saunders, “Viscoelasticity from a microscopic model of dislocation dynamics,” Phys. Rev. B 66, 224113, 2002. [32] M. Haataja, J. Miiller, A.D. Rutenberg, and M. Grant, “Dislocations and morphological instabilities: continuum modeling of misfitting heteroepitaxial films,” Phys. Rev. B, 65, 165414, 2002. [33] Y.U. Wang, Y.M. Jin, A.M. Cuitino, and A.G. Khachaturyan, “Nanoscale phase field microelasticity theory of dislocations: model and 3D simulations,” Acta Mater., 49, 1847–1857, 2001. [34] Y.M. Jin and A.G. Khachaturyan, “Phase field microelasticity theory of dislocation dynamics in a polycrystal: model and three-dimensional simulations,” Phil. Mag. Lett., 81, 607–616, 2001. [35] S.Y. Hu and L.-Q. Chen, “Solute segregation and coherent nucleation and growth near a dislocation – a phase-field model for integrating defect and phase microstructures,” Acta Mater., 49, 463–472, 2001.

7.15 LEVEL SET METHODS FOR SIMULATION OF THIN FILM GROWTH Russel Caflisch and Christian Ratsch University of California at Los Angeles, Los Angeles, CA, USA

The level set method is a general approach to numerical computation for the motion of interfaces. Epitaxial growth of a thin film can be described by the evolution of island boundaries and step edges, so that the level set method is applicable to simulation of thin film growth. In layer-by-layer growth, for example, this includes motion of the island boundaries, merger or breakup of islands, and creation of new islands. A system of size 100 × 100 nm may involve hundreds or even thousands of islands. Because it does not require smoothing and or discretization of individual island boundaries, the level set method can accurately and efficiently simulate the dynamics of a system of this size. Moreover, because it does not resolve individual hopping events on the terraces or island boundaries, the level set method can take longer time steps than those of an atomistic method such as kinetic Monte Carlo (KMC). Thus the level set approach can simulate some systems that are computationally intractable for KMC.

1.

The Level Set Method

The level set method is a numerical technique for computing interface motion in continuum models, first introduced by [11]. It provides a simple, accurate way of computing complex interface motion, including merger and pinchoff. This method enables calculations of interface dynamics that are beyond the capabilities of traditional analytical and numerical methods. For general references on level set methods, see the books [12, 21]. The essential idea of the method is to represent the interface as a level set of a smooth function, φ(x) – for example the set of points where φ = 0. For numerical purposes, the interface velocity is smoothly extended to all points x of the domain, as v(x). Then, the interface motion is captured simply by 2337 S. Yip (ed.), Handbook of Materials Modeling, 2337–2350. c 2005 Springer. Printed in the Netherlands.

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R. Caflisch and C. Ratsch

convecting the values of the smooth function φ with the smooth velocity field v. Numerically, this is accomplished by solving the convection equation ∂φ + v · ∇φ = 0 ∂t

(1)

on a fixed, regular spatial grid. The main advantage of this approach is that interface merger or pinch off is captured without special programming logic. The merger of two disjoint level sets into one occurs naturally as this equation is solved, through smooth changes in the function φ(x, t). For example, two disjoint interface loops would be represented by a φ with two smooth humps, and their merging into a single loop is represented by the two humps of φ smoothly coming together to form a single hump. Pinch off is the reverse process. In particular, the method does not involve smoothing out of the interface. The normal component of the velocity v = n · v contains all the physical information of the simulated system, where n is the outward normal of the moving boundary and v · ∇ϕ = v|∇ϕ|. Another advantage of the method is that the local interface geometry – normal direction, n, and curvature, κ – can be easily computed in terms of partial derivatives of φ. Specifically, −∇φ |∇φ| κ =∇·n n=

(2) (3)

provide the normal direction and curvature at points on the interface.

2.

Epitaxial Growth

Epitaxy is the growth of a thin film on a substrate in which the crystal properties of the film are inherited from those of the substrate. Since an epitaxial film can (at least in principle) grow as a single crystal without grain boundaries or other defects, this method produces crystals of the highest quality. In spite of its ideal properties, epitaxial growth is still challenging to mathematically model and numerically simulate because of the wide range of length and time scales that it encompasses, from the atomistic scale of Ångstroms and picoseconds to the continuum scale of microns and seconds. The geometry of an epitaxial surface consists of step edges and island boundaries, across which the height of the surface increases by one crystal layer, and adatoms which are weakly bound to the surface. Epitaxial growth involves deposition, diffusion and attachment of adatoms on the surface. Deposition is from an external source, such as a molecular beam. The principal dimensionless parameter (for growth at low temperature) is the ratio D/(a 4 F),

Level set methods for simulation of thin film growth

2339

in which a is the lattice constant and D and F are the adatom diffusion coefficient and deposition flux. It is conventional to refer to this parameter as D/F, with the understanding that the lattice constant serves as the unit of length. Typical values for D/F are in the range of 104 –108 . The models that are typically used to describe epitaxial growth include the following: Molecular dynamics (MD) consists of Newton’s equations for the motion of atoms on an energy landscape. A typical Kinetic Monte Carlo (KMC) method simulates the dynamics of the epitaxial surface through the hopping of adatoms along the surface. The hopping rate comes from an Arrhenius rate of the form e−E/kB T in which E is the energy barrier for going from the initial to the final position of the hopping atom. Island dynamics and level set methods, the subject of this article, describe the surface through continuum scaling in the lateral directions but atomistic discreteness in the growth direction. Continuum equations approximate the surface using a smooth height function h = h(x, y, t), obtained by coarse graining in all directions. Rate equations describe the surface through a set of bulk variables without spatial dependence. Within the level set approach, the union of all boundaries of islands of height k + 1, can be represented by the level set ϕ = k, for each k. For example, the boundaries of islands in the submonolayer regime then correspond to the set of curves ϕ = 0. A schematic representation of this idea is given in Fig. 1, where two islands on a substrate are shown. Growth of these islands is described by a smooth evolution of the function ϕ (cf. Figs. 1 (a) and (b)). (a) ϕ⫽0

(b) ϕ ⫽0

(c) ϕ⫽0

(d)

ϕ ⫽1 ϕ ⫽0

Figure 1. A schematic representation of the level-set formalism. Shown are island morphologies (left side), and the level-set function ϕ (right side) that represents this morphology.

2340

R. Caflisch and C. Ratsch

The boundary curve (t) generally has several disjoint pieces that may evolve so as to merge (Fig. 1(c)) or split. Validation of the level set method will be detailed in this article by comparison to results from an atomistic KMC model. The KMC model employed is a simple cubic pair-bond solid-on-solid (SOS) model [24]. In this model, atoms are randomly deposited at a deposition rate F. Any surface atom is allowed to move to its nearest neighbor site at a rate that is determined by r = r0 exp{−(E S + n E N )/kB T }, where r0 is a prefactor which is chosen to be 1013 s−1 , kB is the Boltzmann constant, and T is the surface temperature. E S and E N represent the surface and nearest neighbor bond energies, and n is the number of nearest neighbors. In addition, the KMC simulations include fast edge diffusion, where singly bonded step edge atoms diffuse along the step edge of an island with a rate Dedge , to suppress roughness along the island boundaries.

3.

Island Dynamics

Burton, Cabrera and Frank [5] developed the first detailed theoretical description for epitaxial growth. In this “BCF” model, the adatom density solves a diffusion equation with an equilibrium boundary condition (ρ = ρeq ), and step edges (or island boundaries) move at a velocity determined from the diffusive flux to the boundary. Modifications of this theory were made, for example in [9], to include line tension, edge diffusion and nonequilibrium effects. These are “island dynamics” models, since they describe an epitaxial surface by the location and evolution of the island boundaries and step edges. They employ a mixture of coarse graining and atomistic discreteness, since island boundaries are represented as smooth curves that signify an atomistic change in crystal height. Adatom diffusion on the epitaxial surface is described by a diffusion equation of the form 2dNnuc (4) ∂t ρ − D∇ 2 ρ = F − dt in which the last term represents loss of adatoms due to nucleation and desorption from the epitaxial surface has been neglected. Attachment of adatoms to the step edges and the resulting motion of the step edges are described by boundary conditions at an island boundary (or step-edge) for the diffusion equation and a formula for the step-edge velocity v. For the boundary conditions and velocity, several different models are used. The simplest of these is ρ = ρ∗ v=D

∂ρ ∂n

(5)

Level set methods for simulation of thin film growth

2341

in which the brackets indicate the difference between the value on the upper side of the boundary and the lower side. Two choices for ρ∗ are ρ∗ = 0, which corresponds to irreversible aggregation in which all adatoms that hit the boundary stick to it irreversibly, and ρ∗ = ρeq for reversible aggregation. For the latter case, ρeq is the adatom density for which there is local equilibrium between the step and the terrace [5]. Line tension and edge diffusion can be included in the boundary conditions and interface velocity as in ∂ρ = DT (ρ± − ρ∗ ) − µκ, ∂n ±

(6) µ κss , v = DT n · [∇ρ] + βρ∗ ss + DE in which κ is curvature, s is the variable along the boundary, and D E is the coefficient for diffusion along and detachment from the boundary. Snapshots of the results from a typical level-set simulation are shown in Fig. 2. Shown is the level-set function (a) and the corresponding adatom concentration (b) obtained from solving the diffusion Eq. (4). The island boundaries that correspond to the integer levels of panel (a) are shown in (c). Dashed (solid) lines represent the boundaries of islands of height 1. Comparison of panels (a) and (b) illustrates that ρ is indeed zero at the island boundaries (where ϕ takes an integer value). Numerical details on implementation of the level set method for thin film growth are provided in [7]. The figures in this article are taken from [17] and [15].

4.

Nucleation and Submonolayer Growth

For the case of irreversible aggregation, a dimer (consisting of two atoms) is the smallest stable island, and the nucleation rate is dNnuc = Dσ1ρ 2 , (7) dt where · denotes the spatial average of ρ(x, t)2 and σ1 =

4π ln[(1/α)ρD/F]

(8)

is the adatom capture number as derived in [4]. The parameter α reflects the island shape, and α 1 for compact islands. Expression (7) for the nucleation rate implies that the time of a nucleation event is chosen deterministically. Whenever Nnuc L 2 passes the next integer value (L is the system size), a new island is nucleated. Numerically, this is realized by raising the level-set function to the next level at a number of grid points chosen to represent a dimer.

2342

R. Caflisch and C. Ratsch (a)

2.5 2 1.5 1 0.5 0 90

90 60

60 30

30 0

0

(b) 5

z 10 5 4 3 2 1 0 90

90 60

90

60 30

30 0 0

(c)

Figure 2. Snapshots of a typical level-set simulation. Shown are a 3D view of the level-set function (a) and the corresponding adatom concentration (b). The island boundaries as determined from the integer levels in (a) are shown in (c), where dashed (solid) lines correspond to islands of height 1 (2).

Level set methods for simulation of thin film growth

2343

The choice of the location of the new island is determined by probabilistic choice with spatial density proportional to the nucleation rate ρ 2 . This probabilistic choice constitutes an atomistic fluctuation that must be retained in the level set model for faithful simulation of the epitaxial morphology. For growth with compact islands, computational tests have shown additional atomistic fluctuations can be omitted [18]. Additions to the basic level set method, such as finite lattice constant effects and edge diffusion, are easily included [17]. The level set method with these corrections is in excellent agreement with the results of KMC simulations. For example, Fig. 3 shows the scaled island size distribution (ISD)

s ns = 2 g , sav sav

(9)

where n s is the density of islands of size s, sav is the average island size, and g(x) is a scaling function. The top panel of Fig. 3 is for irreversible attachment; the other two panels include reversibility that will be discussed below. All three panels show excellent agreement between the results from level set simulations, KMC and experiment.

5.

Multilayer Growth

In ideal layer-by-layer growth, a layer is completed before nucleation of a new layer starts. In this case, growth on subsequent layers would essentially be identical to growth on previous layers. In reality, however, nucleation on higher layers starts before the previous layer has been completed and the surface starts to roughen. This roughening transition depends on the growth conditions (i.e., temperature and deposition flux) and the material system (i.e., the value of the microscopic parameters). At the same time, the average lateral feature size increases in higher layers, which we will refer to as coarsening of the surface. These features of multilayer growth and the effectiveness of the level set method in reproducing them is illustrated in Fig. 4 that shows the island number density N as a function of time for two different values of D/F from both a level set simulation and from KMC. The results show near perfect agreement. The KMC results were obtained with a value for the edge diffusion that is 1/100 of the terrace diffusion constants. The island density decreases as the film height increases which implies that the film coarsens. The surface roughness w is defined as w 2 = (h i − h)2 ,

(10)

where the index i labels the lattice site. Figure 5 shows the increase of surface roughness for various different values of the edge diffusion, which implies that

2344

R. Caflisch and C. Ratsch 1.4

n s s av 2/ψ

1.2

KMC

1.0

LS

0.8

Exp

0.6 0.4 0.2 0.0 1.4 1.2

n s s av 2/ψ

1.0 0.8 0.6 0.4 0.2 0.0 1.4 1.2

n s s av 2/ψ

1.0 0.8 0.6 0.4 0.2 0.0 0

1

2

3

s /s av

Figure 3. The island size distribution, as given by KMC (squares) and LS (circles) methods, in comparison with STM experiments(triangles) on Fe/Fe(001) [23]. The reversibility increases from top to bottom.

Level set methods for simulation of thin film growth

2345

0.0015 KMC Levelset

N

0.001

0.0005

0

N

0.002

0.001

0

0

2

4 6 Coverage (ML)

8

Figure 4. Island densities N on each layer for D/F =106 (lower panel) and D/F =107 (upper panel) obtained with the level-set method and KMC simulations. For each data set there are 10 curves in the plot, corresponding to the 10 layers.

edge diffusion contributes to roughening, as also observed in KMC studies. It suggests that faster edge diffusion leads to more compact island shapes, and as a result the residence time of an atom on top of compact islands is extended. This promotes nucleation at earlier times on top of higher layers, and thus enhanced roughening. Effects of edge diffusion were included in these simulations through a term of the form κ − κ rather than κss as in (6).

6.

Reversibility

The simulation results presented above have been for the case of irreversible aggregation. If aggregation is reversible the KMC method must simulate a large number of events that do not affect the time-average of the system: Atoms detach from existing islands, diffuse on the terrace for a short period of time and reattach to the same island most of the time. These processes can slow down KMC simulations significantly. On the other hand, in a level set simulation these events can directly be replaced by their time average

2346

R. Caflisch and C. Ratsch D edge 0 D edge 10 D edge 20 D edge 50 D edge 100

0.7

Roughness

0.6

0.5

0.4

0.3

0.2 0

5

10

15

Coverage (ML) Figure 5. Time evolution of the surface roughness w for different values of edge diffusion Dedge .

and therefore the simulation only needs to include detachment events that do not lead to a subsequent reattachment, making the level set method much faster than KMC. Reversibility does not necessarily depend only on purely local conditions (e.g., local bond strength) but often on more global quantities such as strain or chemical environment. To include these kind of effects is a rather hard task in a KMC simulation but can be quite naturally included in a mean field picture. Reversibility can be included in the level set method using the boundary conditions (5) with ρ∗ = ρeq in which ρeq depends on the local environment of the island, in particular the edge atom density [6]. For islands consisting of only of a few atoms, however, the stochastic nature of detachment becomes relevant and is included through random detachment and breakup for small islands, as detailed in [14]. Figure 3 shows that the level set method with reversibility reproduces nicely the trends in the scaled ISD found in the KMC simulations and experiment. In particular, the scaled ISD depends only on the degree of reversibility, and it narrows and sharpens in agreement with the earlier prediction of [19].

Level set methods for simulation of thin film growth 1.4

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1.3 1.2

1.2

1.1

log R

1 |

1

1.2

1.4

1

ψ 0.085 ψ 0.16

0.8

0.6 0.5

0

0.5 log t

1

1.5

Figure 6. Time dependence (in seconds) of the average island radius R¯ (in units of the lattice constant) for two different coverages on a log–log plot. The straight lines have slope 1/3, which was the theoretical prediction.

In [15], the level set method with reversibility was used to determine the long time asymptotics of Ostwald ripening. A similar computation was performed in [8]. Figure 6 shows that the average island size R¯ grows as t 1/3 , which was an earlier theoretical prediction. Because reversibility greatly increases the number of hopping events and thus lowers the time step for an atomistic computation, KMC simulations have been unable to reach this asymptotic regime. The longer time steps in the level set simulation give it a significant advantage over KMC for this problem.

7.

Hybrid Methods and Additional Applications

As described above, the level set method does not include island boundary roughness or fractal island shapes, which can be significant in some applications. One way of including boundary roughness is by including additional state variables φ for the density of edge atoms and k for the density of kinks along an island boundary or step edge. A detailed step edge model was derived

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R. Caflisch and C. Ratsch

in [6] and used in determination of ρeq for the level set method with reversibility. While adequate for simulating reversibility, this approach will not extend to fractal island shapes. A promising alternative is a hybrid method that combines island dynamics with KMC; e.g., the adatom density is evolved through diffusion of a continuum density function, but attachment at island boundaries is performed by Monte Carlo [20]. In a different approach [10], where diffusion is described and the adatom density is evolved by explicit solution of the master equation, the atoms are resolved explicitly only once they attach to an island boundary. While this methods do not use a level set method, it is sufficiently similar to the method discussed here to warrant mention in this discussion. Level set methods have been used for a number of thin film growth problems that are related to the applications described above. In [22] a level set method was used to describe spiral growth in epitaxy. A general level set approach to material processing problems, including etching, deposition and lithography, was developed in [1], [2] and [3]. A similar method was used in [13] for deposition in trenches and vias.

8.

Outlook

The simulations described above have established the validity of the level set method for simulation of epitaxial growth. Moreover, the level set method makes possible simulations that would be intractable for atomistic methods such as KMC. This method can now be used with confidence in many applications that include epitaxy along with additional phenomena and physics. Examples that seem promising for future developments include strain, faceting and surface chemistry: Elastic strain is generated in heteroepitaxial growth due to lattice mismatch between the substrate and the film. It modifies the material properties and surface morphology, leading to many interesting growth phenomena such as quantum dot formation. Strained growth could be simulated by combining an elasticity solver with the level set method, and this would have significant advantages over KMC simulations for strained growth. Faceting occurs in many epitaxial systems, e.g., corrugated surfaces and quantum dots, and can be an important factor in the energy balance that determines the kinetic pathways for growth and structure. The coexistence of different facets can be represented in a level set formulation using two level set functions, one for crystal height and the second to mark the boundaries between adjacent facets [16]. Determination of the velocity for a facet boundary, as well for the nucleation of new facets, should be performed using energetic arguments. Similarly, surface chemistry such as the effects of different surface reconstructions could in principle be represented using two level set functions.

Level set methods for simulation of thin film growth

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References [1] D. Adalsteinsson and J.A. Sethian, “A level set approach to a unified model for etching, deposition, and lithography 1. Algorithms and two-dimensional simulations,” J. Comp. Phys., 120, 128–144, 1995. [2] D. Adalsteinsson and J.A. Sethian, “A level set approach to a unified model for etching, deposition, and lithography. 2. 3-dimensional simulations,” J. Comp. Phys., 122, 348–366, 1995. [3] D. Adalsteinsson and J.A. Sethian, “A level set approach to a unified model for etching, deposition, and lithography. 3. Redeposition, reemission, surface diffusion, and complex simulations,” J. Comp. Phys., 138, 193–223, 1997. [4] G.S. Bales and D.C. Chrzan, “Dynamics of irreversible island growth during submonolayer epitaxy,” Phys. Rev. B, 50, 6057–6067, 1994. [5] W.K. Burton, N. Cabrera, and F.C. Frank, “The growth of crystals and the equilibrium structure of their surfaces,” Phil. Trans. Roy. Soc. London Ser. A, 243, 299–358, 1951. [6] R.E. Caflisch, W.E, M. Gyure, B. Merriman, and C. Ratsch, “Kinetic model for a step edge in epitaxial growth,” Phys. Rev. E, 59, 6879–87, 1999. [7] S. Chen, M. Kang, B. Merriman, R.E. Caflisch, C. Ratsch, R. Fedkiw, M.F. Gyure, and S. Osher, “Level set method for thin film epitaxial growth,” J. Comp. Phys., 167, 475–500, 2001. [8] D.L. Chopp. “A level-set method for simulating island coarsening,” J. Comp. Phys., 162, 104–122, 2000. [9] B. Li and R.E. Caflisch, “Analysis of island dynamics in epitaxial growth,” Multiscale Model. Sim., 1, 150–171, 2002. [10] L. Mandreoli, J. Neugebauer, R. Kunert, and E. Sch¨oll, “Adatom density kinetic Monte Carlo: A hybrid approach to perform epitaxial growth simulations,” Phys. Rev. B, 68, 155429, 2003. [11] S. Osher and J.A. Sethian, “Front propagation with curvature dependent speed: Algorithms based on Hamilton-Jacobi formulations,” J. Comp. Phys., 79, 12–49, 1988. [12] S.J. Osher and R.P. Fedkiw, Level Set Methods and Dynamic Implicit Surfaces, Springer Verlag, New York, 2002. [13] P.L. O’Sullivan, F.H. Baumann, G.H. Gilmer, J.D. Torre, C.S. Shin, I. Petrov, and T.Y. Lee, “Continuum model of thin film deposition incorporating finite atomic length scales,” J. Appl. Phys., 92, 3487–3494, 2002. [14] M. Petersen, C. Ratsch, R.E. Caflisch, and A. Zangwill, “Level set approach to reversible epitaxial growth,” Phys. Rev. E, 64, #061602, U231–U236, 2001. [15] M. Petersen, A. Zangwill, and C. Ratsch, “Homoepitaxial Ostwald ripening,” Surf. Sci., 536, 55–60, 2003. [16] C. Ratsch, C. Anderson, R.E. Caflisch, L. Feigenbaum, D. Shaevitz, M. Sheffler, and C. Tiee, “Multiple domain dynamics simulated with coupled level sets,” Appl. Math. Lett., 16, 1165–1170, 2003. [17] C. Ratsch, M.F. Gyure, R.E. Caflisch, F. Gibou, M. Petersen, M. Kang, J. Garcia, and D.D. Vvedensky, “Level-set method for island dynamics in epitaxial growth,” Phys. Rev. B, 65, #195403, U697–U709, 2002. [18] C. Ratsch, M.F. Gyure, S. Chen, M. Kang, and D.D. Vvedensky, “Fluctuations and scaling in aggregation phenomena,” Phys. Rev. B, 61, 10598–10601, 2000. [19] C. Ratsch, P. Smilauer, A. Zangwill, and D.D. Vvedensky, “Submonolyaer epitaxy without a critical nucleus,” Surf. Sci., 329, L599–L604, 1995.

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[20] G. Russo, L. Sander, and P. Smereka, “A hybrid Monte Carlo method for surface growth simulations,” preprint, 2003. [21] J.A. Sethian. Level Set Methods and Fast Marching Methods: Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision, and Materials Science, Cambridge U. Press, Cambridge, 1999. [22] P. Smereka, “Spiral crystal growth,” Physica D, 138:282–301, 2000. [23] J.A. Stroscio and D.T. Pierce, “Scaling of diffusion-mediated island growth in ironon-iron homoepitaxy,” Phys. Rev. B, 49:8522–8525, 1994. [24] D.D. Vvedensky, “Atomistic modeling of epitaxial growth: comparisons between lattice models and experiment,” Comp. Materials Sci., 6:182–187, 1996.

7.16 STOCHASTIC EQUATIONS FOR THIN FILM MORPHOLOGY Dimitri D. Vvedensky Imperial College, London, United Kingdom

Many physical phenomena can be modeled as particles on a lattice that interact according to a set of prescribed rules. Such systems are called “lattice gases”. Examples include the non-equilibrium statistical mechanics of driven systems [1, 2], cellular automata [3, 4], and interface fluctuations of growing surfaces [5, 6]. The dynamics of lattice gases are generated by transition rates for site occupancies that are determined by the occupancies of neighboring sites at the preceding time step. This provides the basis for a multi-scale approach to non-equilibrium systems in that atomistic processes are expressed as transition rates in a master equation, while a partial differential equation, derived from this master equation, embodies the macroscopic evolution of the coarse-grained system. There are many advantages to a continuum representation of the dynamics of a lattice system: (i) the vast analytic methodology available for identifying asymptotic scaling regimes and performing stability analyses; (ii) extensive libraries of numerical methods for integrating deterministic and stochastic differential equations; (iii) the extraction of macroscopic properties by coarsegraining the microscopic equations of motion, which, in particular, enables (iv) the discrimination between inherently atomistic effects from those that find a natural expression in a coarse-grained framework; (v) the more readily discernible qualitative behavior of a lattice model from a continuum representation than from its transition rules, which (vi) helps to establish connections between different models and thereby facilitate the transferal of concepts and methods across disciplines; and (vii) the ability to examine the effect of apparently minor modifications to the transition rules on the coarse-grained evolution which, in turn, facilitates the systematic reduction of full models to their essential components.

2351 S. Yip (ed.), Handbook of Materials Modeling, 2351–2361. c 2005 Springer. Printed in the Netherlands.

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1.

D.D. Vvedensky

Master Equation

The following discussion is confined to one-dimensional systems to demonstrate the essential elements of the methodology without the formal complications introduced by higher dimensional lattices. Every site i of the lattice has a column of h i atoms, so every configuration H is specified completely by the array H = {h 1 , h 2 , . . .}. The system evolves from an initial configuration according to transition rules that describe processes such as particle deposition and relaxation, surface diffusion, and desorption. The probability P(H, t) of configuration H at time t is a solution of the master equation [7], ∂P [W (H − r; r)P(H − r, t) − W (H; r)P(H, t)], = ∂t r

(1)

where W (H; r) is the transition rate from H to H + r, r = {r1 , r2 , . . .} is the array of all jump lengths ri , and the summation over r is the joint summation over all the ri . For particle deposition, H and H + r differ by the addition of one particle to a single column. In the simplest case, random deposition, the deposition site is chosen randomly and the transition rate is W (H; r) =

1 δ(ri , 1) δ(r j , 0), τ0 i j= /i

(2)

where τ0−1 is the deposition rate and δ(i, j ) is the Kronecker delta. A particle may also relax immediately upon arrival on the substrate to a nearby site within a fixed range according to some criterion. The two most common relaxation rules are based on identifying the local height minimum, which leads to the Edwards–Wilkinson equation, and the local coordination maximum, i.e., the site with greatest number of lateral nearest neighbors, which is known as the Wolf-Villain model [5]. If the search range extends only to nearest neighbors, the transition rate becomes W (H; r) =

1 (1) wi δ(ri , 1) δ(r j , 0) + wi(2) δ(ri−1 , 1) δ(r j , 0) τ0 i j= /i j= / i−1

+ wi(3)δ(ri+1 , 1)

δ(r j , 0) ,

(3)

j= / i+1

where the wi(k) embody the rules that determine the final deposition site. The sum rule wi(1) + wi(2) + wi(3) = 1

(4)

Stochastic equations for thin film morphology

2353

expresses the requirement that the deposition rate per site is τ0−1 . The transition rate for the hopping of a particle from a site i to a site j is W (H; r) = k0

wi j δ(ri , −1)δ(r j , 1)

δ(rk , 0),

(5)

k= / i, j

ij

where k0 is the hopping rate and the wi j contain the hopping rules. Typically, hopping is considered between nearest neighbors ( j = i ± 1).

2.

Lattice Langevin Equation

Master equations provide the same statistical information as kinetic Monte Carlo (KMC) simulations [8] and so are not generally amenable to an analytic solution. Accordingly, we will use a Kramers–Moyal–Van Kampen expansion [7] of the master equation to obtain an equation of motion that is a more manageable starting point for detailed analysis. This requires expanding the first term on the right-hand side of Eq. (1) which, in turn, relies on two criteria. The first is that W is a sharply peaked function of r in that there is a quantity δ > 0 such that W (H; r) ≈ 0 for |r| > δ. For the transition rates in Eqs. (2), (3) and (5), this “small jump” condition is fulfilled because the difference between successive configurations is at most a single unit on one site (for deposition) or two sites (for hopping). The second condition is that W is a slowly varying function of H, i.e., W (H + H; r) ≈ W (H; r)

for

|H| < δ.

(6)

In most growth models, the transition rules are based on comparing neighboring column heights to determine, for examine, local height minima or coordination maxima, as discussed above. Thus, an arbitrarily small change in the height of a particular column can lead to an abrupt change in the transition rate at a site, in clear violation of Eq. (6). Nevertheless, this condition can be accommodated by replacing the unit jumps in Eqs. (2), (3) and (5) with rescaled jumps of size −1 , where is a “largeness” parameter that controls the magnitude of the intrinsic fluctuations. The time is then rescaled as t → τ = t/ to preserve the original transition rates. The transformed master equation reads ∂P = ∂τ

(H − r; r)P(H − r, t) − W (H; r)P(H, t) dr, W

(7)

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D.D. Vvedensky

corresponding to those in Eqs. (2), (3) and (5) are where the transition rates W given by (H; r) = τ −1 W

δ ri −

i

(H; r) = τ −1 W

i

wi(1)δ

+ wi(2) δ ri−1 −

+ wi(3) δ ri+1 − (H; r) = W

ij

1 δ(r j ), j =/ i

(8)

1 ri − δ(r j ) j =/ i

1 1

δ(r j )

j= / i−1

δ(r j ) ,

j= / i+1

1 1 wi j δ r i + δ rj −

(9) δ(rk ),

(10)

k= / i, j

in which δ(x) is the Dirac δ-function. The central quantities for extracting a Langevin equation from the master : equation in Eq. (7) are the moments of W K i(1) (H) = K i(2) j (H) =

(H; r)dr ∼ O(1), ri W

(11)

(H; r)dr ∼ O(−1 ), ri r j W

(12)

and, in general, K (n) ∼ O(1−n ). With these orderings in , a limit theorem due to Kurtz [9] states that, as → ∞, the solution of the master equation (1) is approximated, with an error of O(ln / ), by that of the Langevin equation dh i (13) = K i(1) (H) + ηi , dτ where the ηi are Gaussian noises that have zero mean, ηi (τ ) = 0, and covariance ηi (τ )η j (τ ) = K i(2) j (H)δ(τ − τ ).

(14)

The solutions of this stochastic equation of motion are statistically equivalent to those of the master equation (1).

3.

The Edwards–Wilkinson Model

There are several applications of the Langevin equation (13). If the occupancy of only a single site is changed with each transition, the correlation

Stochastic equations for thin film morphology

2355

matrix in Eq. (14) is site-diagonal, in which case the numerical integration of Eq. (13) provides a practical alternative to KMC simulations. More important for our purposes, however, is that this equation can be used as a starting point for coarse-graining to extract the macroscopic properties produced by the transition rules. We consider the Edwards–Wilkinson model as an example. The Edwards–Wilkinson model [10], originally proposed as a continuum equation for sedimentation, is one of the standard models used to investigate morphological evolution during surface growth. There are several atomistic realizations of this model, but all are based on identifying the minimum height or heights near a randomly chosen site. In the version we study here, a particle incident on a site remains there only if its height is less than or equal to that of both nearest neighbors. If only one nearest neighbor column is lower than that of the original site, deposition is onto that site. However, if both nearest neighbor columns are lower than that of the original site, the deposition site is chosen randomly between the two. The transition rates in Eq. (3) are obtained by applying these relaxation rules to local height configurations. These configurations can be tabulated by using the step function

θ(x) =

1 if x ≥ 0 0 if x < 0

(15)

to express the pertinent relative heights between nearest neighbors as an identity:

θ(h i−1 − h i ) + (h i−1 − h i ) θ(h i+1 − h i ) + (h i+1 − h i ) = 1, (16)

where (h i − h j ) = 1 − θ(h i − h j ). The expansion of this equation produces four configurations, which are shown in Fig. 1 together with the deposition ( j) rules described above. Each of these is assigned to one of the wi , so the sum rule in Eq. (4) is satisfied by construction, and we obtain the following expressions: wi(1) = θ(h i−1 − h i )θ(h i+1 − h i ),

wi(2) = θ(h i+1 − h i ) 1 − θ(h i−1 − h i ) +

× 1 − θ(h i+1 − h i ) , wi(3) = θ(h i−1 − h i ) 1 − θ(h i+1 − h i ) +

× 1 − θ(h i+1 − h i ) .

1 2

1 2

1 − θ(h i−1 − h i ) 1 − θ(h i−1 − h i )

(17)

The lattice Langevin equation for the Edwards–Wilkinson model is, therefore, from Eq. (13), given by 1 (1) dh i (2) (3) = wi + wi+1 + wi−1 + ηi , dτ τ0

(18)

2356

D.D. Vvedensky (a)

(b)

(c)

(d)

Figure 1. The relaxation rules of the Edwards–Wilkinson model. The rule in (a) corresponds (1) (2) (3) to wi , those in (b) and (d) to wi , and those in (c) and (d) to wi . The broken lines indicates sites where greater heights do not affect the deposition site.

where the ηi have mean zero and covariance ηi (τ )η j (τ ) =

1 (1) (2) (3) wi + wi+1 + wi−1 δi j δ(τ − τ ). τ0

(19)

The statistical equivalence of solutions of this Langevin equation and those of the master equation, as determined by KMC simulations, can be demonstrated by examining correlation functions of the heights. One such quantity is the surface roughness, defined as the root-mean-square of the heights,

W (L , t) = h 2 (t) − h(t)2

1/2

,

(20)

where h k (t) = L −1 i h ki (t) for k = 1, 2, and L is the length of the substrate. For sufficiently long times and large substrate sizes, W is observed to conform to the dynamical scaling hypothesis [5], W (L , t) ∼ L α f (t/L z ), where f (x) ∼ x β for x 1 and f (x) → constant for x 1, α is the roughness exponent, z = α/β is the dynamic exponent, and β is the growth exponent. The comparison of W (L , t) obtained from KMC simulations with that computed from the Langevin equation in (18) is shown in Fig. 2 for systems

Stochastic equations for thin film morphology

2357

(a)

W(lattice units)

L⫽100

Ω⫽1 Ω⫽2

100

Ω⫽50 KMC

100

101

t(ML)

102

(b)

W(lattice units)

L⫽1000

Ω⫽1 Ω⫽20

100

KMC 100

101

102 t(ML)

103

Figure 2. Surface roughness obtained from the lattice Langevin Eq. (18) and KMC simulations for systems of size L = 100 and 1000 for the indicated values of . Data sets for L = 100 were averaged over 200 independent realizations. Those for L = 1000 were obtained from a single realization. The time is measured in units of monolayers (ML) deposited. Figure courtesy of A.L.-S. Chua.

2358

D.D. Vvedensky

of lengths L = 100 and 1000, each for several values of . Most apparent is that the roughness increases with time, a phenomenon known as “kinetic roughening” [5], prior to a system-size-dependent saturation. The roughness obtained from the Langevin equation is greater than that of the KMC simulation at all times, but with the difference decreasing with increasing . The greater roughness is due, in large part, to the noise in Eq. (19): the variance includes information about nearest-neighbors, but the noise is uncorrelated between sites. Thus, as the lattice is scanned, the uncorrelated noise produces a larger variance in the heights than the simulations. But even apart from the rougher growth front the discrepancies for smaller are appreciable. For L = 100 and = 1, 2, the saturation of the roughness is delayed to later times and the slope prior to saturation differs markedly from that of the KMC simulation. There are remnants of these discrepancies for L = 1000, though the slope of the roughness does approach the correct value at sufficiently long times even for = 1.

4.

Coarse-grained Equations of Motion

The non-analyticity of the step functions in Eq. (17), which reflects the threshold character of the relative column heights on neighboring sites, presents a major obstacle to coarse graining the lattice Langevin equation in Eq. (18), as well as those corresponding to other growth models [11, 12]. To address this problem, we begin by observing that θ(x) is required only at the discrete values h k±1 − h k = n, where n is an integer. Thus, we are free to interpolate between these points at our convenience. Accordingly, we use the following representation of θ(x) [13]:

θ(x) = lim+ →0

e(x+1)/ + 1 ln e x/ + 1

.

(21)

For finite , the right-hand side of this expression is a smooth function that represents a regularization of the step function (Fig. 3). This regularization can be expanded as a Taylor series in x and, to second order, we obtain θ(x) = A +

B2x 2 Cx3 Bx − − + ··· , 2 8 62

(22)

where A = ln

1 (1 2

+ e1/ ) ,

B=

e1/ − 1 , e1/ + 1

C=

e1/ (e1/ − 1) . (e1/ )3

As → 0, A → 1 − ln 2 + · · · , B → 1, and C → 0.

(23)

Stochastic equations for thin film morphology

2359

1 ∆⫽1

0.8 ∆⫽0.5

θ (x)

0.6

∆⫽0.25

0.4 0.2 0 ⫺2

⫺1

0

1

x

Figure 3. The regularization in (21) showing how, with decreasing , the step function (shown emboldened) is recovered.

We now introduce the coarse-grained space and time variables x = i and t = z τ/τ0 , where z is to be determined and parametrizes the extent of the coarse-graining, with = 1 corresponding to a smoothed lattice model (with no coarse-graining) and → 0 corresponding to the continuum limit. The coarse-grained height function u is u(x, t) =

α

τ hi − , τ0

(24)

where α is to be determined and τ/τ0 is the average growth rate. Upon applying these transformations and the expansion in Eq. (22) to Eqs. (18) and (19), we obtain the following leading terms in the equation of motion:

z−α

∂u ∂ 2u ∂ 4u ∂ 2 ∂u = ν 2−α 2 + K 4−α 4 + λ1 4−2α 2 ∂t ∂x ∂x ∂x ∂x 3 ∂ ∂u + λ2 4−3α + · · · + (1+z)/2ξ, ∂x ∂x

2

(25)

where ν = B,

K=

1 (4 − 3A), 12

λ1 =

B2 B2 − (1 − A), 8 8

λ2 = −

C , 3 (26)

and ξ is a Gaussian noise with mean zero and covariance ξ(x, t)ξ(x , t ) = δ(x − x )δ(t − t ).

(27)

2360

D.D. Vvedensky

The most direct approach to the continuum limit is obtained by requiring (i) that the coefficients of u t , u x x , and ξ have the same scale in and (ii) that these are the dominant terms as → 0. The first of these necessitates setting z = 2 and α = 1/2. To satisfy condition (ii), we first write = δ . A lower bound of the scale of the nth order term in the expansion in Eq. (25) can be estimated from Eq. (22) as

1−n

∂h ∂x

n

1

∼ n(1−α)−(n−1)δ = 2 n−(n−1)δ .

(28)

This yields the condition δ < 1/2, and satisfies condition (ii) for λ1 and λ2 as well. Thus, in the limit → 0, we obtain the Edwards–Wilkinson equation: ∂u ∂ 2 u = + ξ. ∂t ∂ x 2

(29)

The method used to obtain this equation can be applied to other models and in higher spatial dimensions. There have been several simulation studies of the Edwards–Wilkinson [14] and Wolf–Villain [15, 16] models that suggest intriguing and unexpected behavior that is not present for one-dimensional substrates. Taking a broader perspective, if a direct coarse-graining transformation is not suitable, our method can be used to generate an equation of motion as the initial condition for a subsequent renormalization group analysis. This will provide the basis for an understanding of continuum growth models as the natural expression of particular atomistic processes.

5.

Outlook

There are many phenomena in science and engineering that involve a disparity of length and time scales [17]. As a concrete example from materials science, the formation of dislocations within a material (atomic-scale) and their mobility across grain boundaries of the microstructure (“mesoscopic” scale) are important factors for the deformation behavior of the material (macroscopic scale). A complete understanding of mechanical properties thus requires theoretical and computational tools that range from the atomic-scale detail of density functional methods to the more coarse-grained picture provided by continuum elasticity theory. One approach to addressing such problems is a systematic analytic and/or numerical coarse-graining of the equations of motion for one range of length and time scales to obtain equations of motion that are valid over much longer length and time scales. A number of approaches in this direction has already been taken. Since driven lattice models are simple examples of atomic-scale systems, the approach described here may serve as a paradigm for such efforts.

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References [1] C. Godr`eche (ed.), Solids far from Equilibrium, Cambridge University Press, Cambridge, England, 1992. [2] H.J. Jensen, Self-Organized Criticality: Emergent Complex Behavior in Physical and Biological Systems, Cambridge University Press, Cambridge, England, 2000. [3] S. Wolfram (ed.), Theory and Applications of Cellular Automata, World Scientific, Singapore, 1986. [4] G.D. Doolen (ed.), Lattice Gas: Theory Application, and Hardware, MIT Press, Cambridge, MA, 1991. [5] A.-L. Barab´asi and H.E. Stanley, Fractal Concepts in Surface Growth, Cambridge University Press, Cambridge, England, 1995. [6] J. Krug, “Origins of scale invariance in growth processes,” Adv. Phys., 46, 139–282, 1997. [7] N.G. Van Kampen, Stochastic Processes in Physics and Chemistry, North-Holland, Amsterdam, 1981. [8] M.E.J. Newman and G.T. Barkema, Monte Carlo Methods in Statistical Physics, Oxford University Press, Oxford, England, 1999. [9] R.F. Fox and J. Keizer, “Amplification of intrinsic fluctuations by chaotic dynamics in physical systems,” Phys. Rev. A, 43, 1709–1720, 1991. [10] S.F. Edwards and D.R. Wilkinson, “The surface statistics of a granular aggregate,” Proc. R. Soc. London Ser. A, 381, 17–31, 1982. [11] D.D. Vvedensky, A. Zangwill, C.N. Luse, and M.R. Wilby, “Stochastic equations of motion for epitaxial growth,” Phys. Rev. E, 48, 852–862, 1993. [12] M. Pˇredota and M. Kotrla, “Stochastic equations for simple discrete models of epitaxial growth,” Phys. Rev. E, 54, 3933–3942, 1996. [13] D.D. Vvedensky, “Edwards–Wilkinson equation from lattice transition rules,” Phys. Rev. E, 67, 025102(R), 2003. [14] S. Pal, D.P. Landau, and K. Binder, “Dynamical scaling of surface growth in simple lattice models,” Phys. Rev. E, 68, 021601, 2003. ˇ [15] M. Kotrla and P. Smilauer, “Nonuniversality in models of epitaxial growth,” Phys. Rev. B, 53, 13777–13792, 1996. [16] S. Das Sarma, P.P. Chatraphorn, and Z. Toroczkai, “Universality class of discrete solid-on-solid limited mobility nonequilibrium growth models for kinetic surface roughening,” Phys. Rev. E, 65, 036144, 2002. [17] D.D. Vvedensky, “Multiscale modelling of nanostructures,” J. Phys.: Condens. Matter, 16, R1537–R1576, 2004.

7.17 MONTE CARLO METHODS FOR SIMULATING THIN FILM DEPOSITION Corbett Battaile Sandia National Laboratories, Albuquerque, NM, USA

1.

Introduction

Thin solid films are used in a wide range of technologies. In many cases, strict control over the microscopic deposition behavior is critical to the performance of the film. For example, today’s commercial microelectronic devices contain structures that are only a few microns in size, and emerging microsystems technologies demand stringent control over dimensional tolerances. In addition, internal and surface microstructures can greatly influence thermal, mechanical, optical, electronic, and many other material properties. Thus it is important to understand and control the fundamental processes that govern thin film deposition at the nano- and micro-scale. This challenge can only be met by applying different tools to explore the various aspects of thin film deposition. Advances in computational capabilities over recent decades have allowed computer simulation in particular to play an invaluable role in uncovering atomic- and microstructure-scale deposition and growth behavior. Ab initio [1] and molecular dynamics (MD) calculations [2, 3] can reveal the energetics and dynamics of processes involving individual atoms and molecules in very fine temporal and spatial resolution. This information provides the fundamentals – the “unit processes” – that work in concert to deposit a solid film. The environmental conditions in the deposition chamber are commonly simulated using either the basic processing parameters directly (e.g., temperature and flux for simple physical vapor deposition systems); or continuum transport/reaction models [4] or direct simulation Monte Carlo methods [5] for more complex chemically active environments. These methods offer a wealth of information about the conditions inside a deposition chamber, but perhaps most important to the modeling of film growth itself are the fluxes and identities of species arriving at the deposition surface. All of 2363 S. Yip (ed.), Handbook of Materials Modeling, 2363–2377. c 2005 Springer. Printed in the Netherlands.

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this information, including atomic-scale information about unit processes and chamber-scale information about surface fluxes and chemistry, must be used to construct a comprehensive model of deposition. Many methods have been used to model film growth. These range from one-dimensional solutions of coupled rate equations, which usually provide only growth rate information; to time-intensive MD simulations of the arrival and incorporation of many atoms at the growth surface, which yield detailed structural and energetic information at the atomic scale. This chapter addresses an intermediate approach, namely kinetic Monte Carlo (KMC) [6], that has been widely and successfully used to model a variety deposition systems. The present discussion is restricted to latticed-based KMC approaches, i.e., those that employ a discrete (lattice) representation of the material, which can provide a wealth of structural information about the deposited material. In addition, the underlying KMC foundation allows the treatment of problems spanning many time and length scales, depending primarily on the nature of the input kinetic data. These kinetic data are often derived using transition state information from experiments or from atomistic simulations. The growth model is often coupled to information about the growth environment such as temperature, pressure, vapor composition, and flux, and these data can be measured experimentally or computed using reactive transport models. The following discussion begins with a brief theoretical background of the Monte Carlo (MC) method in the context of thin film deposition, then continues with a discussion of its implementation, and concludes with an overview of both historical and current applications of KMC (and related variants) to the modeling of thin film growth. The intent is to instill in the reader a basic understanding of the foundations and implementation of the MC method in the context of thin film deposition simulations, and to provide a starting point in their exploration of this broad and rich topic.

2.

The Monte Carlo Method

Many collective phenomena in nature are essentially deterministic. For example, a ball thrown repeatedly with a specific initial velocity (in the absence of wind, altitudinal air density variations, and other complicating factors) will follow virtually the same trajectory each time. Other behaviors appear stochastic, as evidenced by the seemingly random behavior of a pachinko ball. Nanonscopically (i.e., on the time and length scale of atomic motion), most processes behave stochastically rather than deterministically. The vibrations of an atom or molecule as it explores the energetic landscape near the potential energy minimum created by the interactions with its environment are, for all practical purposes, random, i.e., stochastic. When that atom is in the vicinity of others, e.g., in a solid or liquid, the energetic landscape is very

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complex and consists of many potential energy minima separated by energy barriers (i.e., maxima). Given enough time, a vibrating atom will eventually happen to “hop over” one of these barriers and “fall into” an adjacent potential energy “well.” In doing so, the atom has transitioned from one state (i.e., energy) to a new one. The energetics of such a transition are depicted in Fig. 1, where the states are described by their free energies (i.e., both enthalpic and entropic contributions). These concepts apply not only to vibrating atoms but also to the fundamental transitions of any system that has energy minima in configurational space. Transition state theory describes the frequency of any transition that can be described energetically by a curve like the one in Fig. 1. Although a detailed account of transition state theory is beyond the scope of this chapter, suffice it to say that the average rate of transitioning from State A to State B is described by the rate constant

E , kA→B = A exp − kT

(1)

where A is the frequency with which the system attempts the transition, E is the activation barrier, k is Boltzmann’s constant equal to 1.3806503 × 10−23 J K−1 = 8.617269 × 10−5 eV K−1 , and T is the temperature. Likewise, the

Energy

E

A ∆G

B Reaction coordinate Figure 1.

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average rate of the reverse transition from State B to State A is described by the rate constant E − G , (2) kA←B = A exp − kT where G is the change in free energy on transitioning from State A to B (notice from Fig. 1 that G is negative), and the reuse of the symbol, A, implies that the attempt frequencies for the forward (A → B) and reverse (A ← B) transitions are assumed equal. (The rate constants are obviously in the same units as the attempt frequency. If these units are not those of an absolute rate, i.e., sec−1 , then the rate constant can be converted into an absolute rate by multiplying by the appropriate quantity, e.g., concentrations in the case of chemical reactions.) Whereas Eqs. (1) and (2) describe the average rates for the transitions in Fig. 1, the actual rates for each instance of a particular transition will vary because the processes are stochastic. The state of the system will vary (apparently) randomly inside the energy well at State A until, by chance, the system happens to make an excursion that reaches the activated state, at which point (according to transition state theory) the system has a 50% chance of returning to State A and a 50% chance of transitioning into State B. The Monte Carlo (MC) method, named after the casinos in the Principality of Monaco (an independent sovereign state located between the foot of the Southern Alps and the Mediterranean Sea) is ideally suited to modeling not only realistic instantiations of individual state transitions (provided the relevant kinetic parameters are known) but also time- and ensemble-averages of complex and collective phenomena. The MC method is essentially an efficient method for numerically estimating complex and/or multidimensional integrals [7]. It is commonly used to find a system’s equilibrium configuration via energy minimization. Early MC algorithms involved choosing system configurations at random, and weighting each according to its potential energy via the Boltzmann equation, E (3) P = exp − kT where P is the weight (i.e., the probability the configuration would actually be realized). The configuration with the most weight corresponds to equilibrium. Metropolis et al. [7] improved on this scheme with an algorithm that, instead of choosing configurations randomly and Boltzmann-weighting them, chooses configurations with the Boltzmann probability in Eq. (3) and weighting them equally. In this manner, the model system wastes less time in configurations that are highly unlikely to exist. Bortz et al. [8] introduced yet another rephrasing of the MC method, and termed the new algorithm the N-Fold Way (NFW). This algorithm always accepts the chosen changes to the system’s configuration, and shifts the stochastic component of the computation into the time

Monte Carlo methods for simulating thin film deposition

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incrementation (which can thereby vary at each MC step). Independent discoveries of essentially the same algorithm were presented shortly thereafter by Gillespie [9], and more recently by Voter [10]. The NFW is only applicable in situations where the Boltzmann probability is nonzero only for a finite and enumerable set of configurational transitions. So, for example, it cannot be used (without adaptation of either the algorithm or the model system) to find the equilibrium positions of atoms in a liquid, since the phase space representing these positional configurations is continuous and thus contains a virtually infinite number of possible transitions. Both the Metropolis algorithm (in its kinetic variation, described below) and the NFW can treat kinetic phenomena, but the NFW is better suited to generating physically realistic temporal sequences of configurational transitions [6] provided the rates of all possible state transitions are known a priori. To illustrate the concepts behind these techniques, it is useful to consider a simple example. Imagine a system that can exist in one of three states: A, B, or C. All the possible transitions for this system are therefore A ↔ B ↔ C. When the system is in State A, it can undergo only one transition, i.e., conversion to State B. When in State C, the system is only eligible for conversion to State B. When in State B, the system can either convert to State A, or convert to State C. Assume that the energetics of the transition paths are described by Fig. 2. The symbol *IJ denotes the activated state for the transition between

AB

CB Energy

EAB

ECB C

A

∆GCB

∆GAB

B Reaction coordinate Figure 2.

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States I and J, E I J is the activation barrier encountered upon the transition from State I to State J, and G IJ is the difference in the free energies between States I and J. (Note that both G AB and G CB are negative because the free energy decreases upon transitioning from State A to B, and from State C to B.) The lowest-energy state is B, the highest is C, and A is intermediate. Simply by examining Fig. 2, it is clear that the thermodynamic equilibrium for this system is State B. However, the kinetic properties of the system depend on the transition rates, which in turn depend not only on the energies but also on the attempt frequencies. If the attempt frequencies of all four transitions are equal, then the state with the maximum residence time (in steady state) would certainly be State B, and that with the minimum time would be State C. Otherwise the residence properties might be quite different. As aforementioned, the Metropolis algorithm proceeds by choosing configurations at random, and accepting or rejecting them based on the change to the system energy that is incurred by changing the system’s configuration. So, in the present example, such an algorithm would randomly choose one of the three states – A, B, or C – and accept or reject the chosen state with a probability based on the energy difference between it and the previous state. Specifically, the probability of accepting a new State J when the system is in State I is PI→J =

 G IJ   exp − kT  

G IJ > 0

.

(4)

G IJ ≤ 0

1

This so-called thermodynamic Metropolis MC approach clearly utilizes only the states’ energy differences, and does not account for the properties of the activated states or the dynamics that lead to transition attempts. As such, it can reveal the equilibrium state of the system, but provides no information about the kinetics of the system’s evolution. However, the same algorithm can be adapted into a kinetic Metropolis MC scheme in order to capture kinetic information. This is accomplished by introducing time into the approach, and by using the transition rate information from Eqs. (1) and (2). Specifically, the rate constants for the “forward” transition, I→J, and the “backward,” I←J, are E IJ and (5) kI→J = AIJ exp − kT

kI←J = AJI exp −

E IJ − G IJ . kT

(6)

Assuming that the rate and the rate constant are the same, the probability of accepting a new State J when the system is in an “adjacent” State I is PI→J = kI→J t,

(7)

Monte Carlo methods for simulating thin film deposition

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where t is a constant time increment that is chosen a priori to accommodate the fastest transitions in the problem. Generally, t is chosen to be near 0.5/kmax . Thus, at each step in a kinetic Metropolis MC calculation, a transition is chosen at random from those that are possible given the state of the system, the chosen transition is realized with a probability according to Eq. (7), and the time is incremented by the constant t. Notice that while the thermodynamic Metropolis scheme allows the system to change its configuration to a state that is not directly accessible (e.g., A→C), the kinetic Metropolis approach considers only transitions between accessible states (i.e., the transitions A ↔ C in Fig. 2 would be forbidden). Similarly, the NFW deals only with accessible transitions, but unlike the kinetic Metropolis formulation, the NFW realizes state transitions with unit probability. Specifically, at each step in an NFW computation, a transition is chosen at random from those that are possible given the state of the system. The probability of choosing a particular transition depends on its relative rate. As such, i−1 j =1

kj < ζ ≤

M

kj,

(8)

j =i

where j merely indexes each transition, i denotes the chosen transition, ζ is a random number between zero (inclusive) and one (exclusive) such that ζ ∈ [0, 1), and is the sum of the rates of all the transitions that are possible given the state of the system. (Recall that the transition rates are equal to the rate constants in the present example, as aforementioned.) The chosen transition is always realized, and the time is incremented by ln (ξ ) , (9) t = − where ξ is another random number between zero and one (exclusive of both bounds) such that ξ ∈ (0,1). On closer inspection, it is apparent that the NFW is simply a rearrangement of the kinetic Metropolis MC algorithm [8]. Consider a system in some arbitrary State I. Assume that the system can exist in multiple states, so that the system will eventually (at non-zero temperature) transition out of State I. Because the transitioning process is stochastic, the time that the system spends in State I will vary each time it visits that state. (This “fact” is evident in the kinetic MC algorithms discussed above.) Let P− (dt) denote the probability that the system remains in State I for a time of at least dt, and P+ (dt) be the probability that the system leaves State I before dt has elapsed, where dt = 0 refers to the moment that the system entered State I. Since the system has no other choices but to either stay in State I during dt or leave State I sometime during dt, it is clear that P− (dt) + P+ (dt) = 1.

(10)

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Consider some value of time, t =/ dt, where again t =0 refers to the moment that the system entered State I. Multiplying Eq. (10) by P− (t) yields P− (dt) P− (t) + P+ (dt) P− (t) = P− (t).

(11)

Notice that P− (dt)P− (t) = P− (t)P− (dt), and is simply the probability that the system is still in State I after t and also after the following dt, i.e., it is the probability that the system remains in State I for at least a time of t + dt. Therefore, P− (t + dt) + P+ (dt) P− (t) = P− (t).

(12)

Also notice that P+ (dt) = dt,

(13)

where is the average number of transitions from State I per unit time, i.e., the sum of the rates of all the transitions that the system can make from State I. Substituting Eqs. (12) and (13) into Eq. (11) yields P− (t + dt) + P− (t) dt = P− (t).

(14)

Rearranging Eq. (14) produces P− (t + dt) − P− (t) = − P− (t). dt In the limit that dt → 0, Eq. (15) becomes

(15)

dP−

= − P− (t). dt t

(16)

Integrating Eq. (16), and realizing that P− (0) = 1, yields

ln P− (t) = −t, hence

(17)

ln P− (t) . (18) t = − Let t∗ be the average residence time for State I. On each visit that the system makes to State I, it remains there for a different amount of time, and the associated residence probabilities for each visit follow a uniform random distribution such that P− (t∗ ) ∈ (0,1). Therefore, the individual residence times from visit to visit follow a distribution of the form ln (ξ ) , (19) t∗ = − where ξ is a random number such that ξ ∈ (0,1), and thus Eq. (9) is obtained. Clearly the time that elapses between one transition and the next is stochastic

Monte Carlo methods for simulating thin film deposition

2371

and is a function only of the sum of the rates of all available transitions. When in any given state, the probability that the system will actually make a particular transition, provided it is accessible, is equal to the rate of the transition relative to the sum of the rates of all accessible transitions, as described in Eq. (8).

3.

Implementing the N-Fold Way

One can readily see the utility of the NFW for simulating the fundamental processes involved in thin film deposition. Simply put, one need only apply the algorithms described above, illustrated for the idealized system in Fig. 2, onto each fundamental location on the model deposition surface. For example, consider the simple two-dimensional surface in Fig. 3a. Assume that each square represents a fundamental unit of the solid structure (e.g., an atom), that there is a flux of material toward the substrate, and that gray denotes a static substrate. If the incoming material is appropriate for coherent epitaxy on the substrate, then the evolution of the surface in Fig. 3a will begin by the attachment of material to the surface, i.e., the filling of one of the sites above the surface denoted by dotted squares in Fig. 3b, by a unit of incoming material. Consider only one of these candidate sites, e.g., the one labeled d in Fig. 3c. Site d represents a subsystem of the entire surface, and that subsystem is in a particular state whose configuration is defined by the “empty” site just above the surface. Site d can transition into another state, namely one in which the site contains a deposited unit of material, as depicted in Fig. 3d. This local transition occurs just as described in the simple example for Fig. 2 above. In fact, the evolution of the entire surface can be modeled by collectively considering the local transitions of each fundamental unit (i.e., site) in the system. Consider the behavior of the entire system from the initial configuration shown in Fig. 3c. Each site above the surface can be filled by incoming

(a)

Flux

(b)

(c)

(d)

Figure 3. Deposition of a single practicle onto a simple two-dimensional substrate. Gray squares are substrate sites, white dotted squares are sites into which particles can potentially deposit, and black squares are deposited particles.

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material. The NFW algorithm suggests that the time that passes before a particular site, e.g., Site a transitions is ta = −

X ln (ξ ), F

(20)

where X is the aerial density of surface sites in units of length−2 and F is the deposition flux in length−2 sec−1 (taking into account such factors as the sticking probability), so that F/X is the average rate at which material can deposit into Site a. But how much time passes before any of the sites makes a transition? In other words, how long will the system remain in the configuration of Fig. 3c? By way of analogy, consider rolling six-sided dice. If only one die is rolled, the chance that a particular side faces upwards (after the die comes to rest) is 1/6. So the chance of rolling “a three” is 1/6, as is the chance of rolling a five, etc. If three dice are rolled, the chance that at least one of them shows a three is 3/6 = 1/2. Thus, since there are seven sites in Fig. 3c that can accept incoming material, the probability that at least one of them will transition during some small time increment is seven times the probability that a specific isolated site will transition in the same increment. Because more probable events obviously occur more often, i.e., require less time, then the time that passes before the entire system leaves the configuration in Fig. 3c, i.e., the time it takes for a unit of material to deposit somewhere on the surface, is t =

1X td =− ln (ξ ). 7 7F

(21)

Notice that 7F/X is simply the sum of the rates of all the per-site transitions that can occur in the entire system, i.e., the system’s activity, and thus it is clear that the general form of Eq. (21) is Eq. (9). As described above, the NFW algorithm prescribes that the choice of transition at each time step be randomized, with the probability of choosing a particular transition proportional to its relative rate. Since one of only seven transitions can occur on the surface in Fig. 3c, and each has the same rate, then the selection of a transition from the configuration in Fig. 3c involves simply choosing at random one of the seven sites marked with dotted outlines. Duplicating Fig. 3c as Fig. 4a, and assuming that Site d is randomly selected to transition, then the configuration after the first time step is that in Fig. 4b. If the per-site flux (i.e., F/ X ) is 1 sec−1 , then the time increment that elapses before the first transition is dictated by Eq. (9) to be t1 = −

ln (ξ1 ) sec . 7

(22)

A random number of ξ1 = 0.631935 yields a time increment for the first step of t1 = 0.065567 sec for a total time after the first step of, obviously, t1 = 0.065567 sec (where the starting time is naturally t0 = 0 sec).

Monte Carlo methods for simulating thin film deposition

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Assume that the deposited material at Site d can either diffuse to Site c, diffuse to Site e, or desorb. Assume further that the temperature is T = 1160 K such that kT =0.1 eV; the attempt frequency and activation barrier for diffusion are A D = 1x104 sec−1 and E D = 0.70 eV, respectively; and those for desorption are A R = 1x104 sec−1 and E R = 0.85 eV. Then the per-site rate of diffusion is approximately 9 sec−1 , and that of desorption is 2 sec−1 . The set of transitions available to the configuration in Fig. 4b includes seven deposition events at the dotted sites, two diffusion events, and one desorption event. To illustrate the process of choosing one of these ten transitions in the NFW algorithm, it is useful to visualize them on a graph. Figure 5b shows the ten possible transitions on a line plot, with the width of each corresponding to its relative rate. A transition can be selected in accord with Eq. (8) simply by generating a random number ζ2 ∈ [0,1), plotting it on the graph in Fig. 5b, and selecting the appropriate transition. (Figure 5a depicts the same type of plot for the configuration in Fig. 4a, assuming a value of ζ1 = 0.500923.) For example, if a random number of ζ2 = 0.652493 is generated, then the black atom at Site d in Fig. 4b would diffuse into Site e yielding the configuration in Fig. 4c. Since the activity of the system in Fig. 4b is = 27 sec −1 , a random number of ξ2 = 0.548193 yields a time increment for the second step of t2 = 0.022264 sec yielding a time value of t2 = 0.087831 sec. (Notice that when fast transitions are available to the system, as in Fig. 5b, the activity of the system increases and the time resolution in the NFW becomes finer to accommodate the fast processes.) By repeating this recipe, the evolution of the surface from its initial state in Fig. 4a can be simulated, as shown in Figs. 4 and 5. The random numbers (for transition selection) corresponding to the system’s evolution from Fig. 4c are ζ3 = 0.132087, ζ4 = 0.327872, and ζ5 = 0.891473, and the simulation time would be calculated as prescribed above. This NFW approach can be straightforwardly extended into three dimensions, and all manner of complex, collective, and environment- and structure-dependent transitions can be modeled provided their rates are known.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 4. Possible configurations for the first few steps of deposition onto a simple twodimensional substrate. Gray squares are substrate sites, white dotted squares are sites into which particles can potentially deposit, and black squares are deposited particles.

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Figure 5. Lists of transitions for use in the NFW algorithm applied to the surface evolution depicted in Fig. 4. The numerals at the upper right of each plot indicate the total rate in sec−1 , those below each plot demark relative rates, and the letters above each plot denote transition classes and locations. The letter F corresponds to particle deposition, D to diffusion, and R to desorption. Lowercase italic letters correspond to the site labels, in Fig. 4, and the notation i ⇒ j indicates diffusion of the particle at site i into site j . The thick gray lines below each plot mark the locations of the random numbers used to select a transition from each configuration.

4.

Historical Perspective

Thousands of papers have been published on Monte Carlo simulations of thin film deposition. They encompass a wide range of thin film applications

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and employ a variety of methods. This section contains a brief overview of some selected examples. No attempt is made here to provide a comprehensive review; instead, the goal is to present selected sources for further exploration. Some of the earliest applications of MC to the computer simulation of deposition used simple models of deposition on an idealized surface. One of the first of these is attributed to Young and Schubert [11], who simulated the multilayer adsorption (without desorption or migration) of tungsten onto a tungsten substrate. Chernov and Lewis [12] performed MC calculations of kink migration during binary alloy deposition using a 1000-particle linear chain in one dimension, a 99×49 square grid in two dimensions (where the grid represented a cross-section of the film), and a 99×49×32 cubic grid in three dimensions. Gordon [13] simulated the monolayer adsorption and desorption of particles onto a 10×10 grid of sites with hexagonal close-packing (where the grid represented a plan view of the film). Abraham and White [14] considered the monolayer adsorption, desorption, and migration of atoms onto a 10×10 square grid (again in plan view), with atomic transitions modeled using a scheme that resembles the NFW. (Notice that Abraham’s and White’s publication appeared five years before the first publication of the NFW algorithm.) Leamy and Jackson [15], and Gilmer and Bennema [16], used the solid–on-solid (SOS) model [17–19] to analyze the roughness of the solid– vapor interface on a three-dimensional film represented by a 20×20 square grid. The SOS model represents the film by columns of atoms (or larger solid quanta) so that no subsurface voids or vacancies can exist. One major advantage of this approach is that the three-dimensional film can be represented digitally by a two-dimensional matrix of integers that describe the height of the film at each location on the free surface. Their approach was later extended [20] to alleviate the restrictions of the SOS model so that the structure and properties of the diffuse solid-vapor interface could be examined. Over the years, KMC methods have been applied to a wide range of materials and deposition technologies. These include materials such as simple metals, alloys, semiconductors, oxides, diamond, nanotubes, and quasicrystals; and technologies like molecular beam epitaxy, physical vapor deposition, chemical vapor deposition, electrodeposition, ion beam assisted deposition, and laser assisted deposition. Because of their relative simplicity, lattice KMC models were used in many of the computational deposition studies performed to date. However, MC methods can also be applied to model systems where the basic structural units (e.g., atoms) do not adhere to prescribed lattice positions. For example, continuous-space MC methods [21] allow particles to assume any position inside the computational domain. The motion of the particles is generally simulated by attempting small displacements, computing the associated energy changes via an interparticle potential, and applying the MC algorithms described above to accept or reject the attempted displacements. Alternatively, MC methods can be combined with other techniques within

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the same simulation framework to create a hybrid approach [22]. Common applications of these hybrids involve relaxing atomic positions near the surface, usually by means of energy minimization or molecular dynamics, and performing the MC calculations at off-lattice locations that are identified as potential transition sites on the relaxed structure.

5.

Summary

The preceding discussion should demonstrate clearly that the topic of MC deposition simulations is broad and rich. Unfortunately, a comprehensive review of existing methods and past research is beyond the scope of this article, and the reader is referred to the works mentioned herein and to the numerous reviews on the subject [23–30] for further study. As the techniques for applying MC methods to the study of thin film deposition continue to mature, novel approaches and previously inaccessible technologies will emerge. Hybrid MC methods seem particularly promising, as they allow for a physically based description of the fundamental surface structure, can allow for the real-time calculation of transition rates via physically accurate methods, and are able to access spatial and temporal scales that are well beyond the reach of more fundamental approaches. Whatever the future holds, it is certain that our ability to study thin film processing using computer simulations will continue to evolve and improve, yielding otherwise unobtainable insights into the physics and phenomenology of deposition, and that MC methods will play a crucial role in that process.

References [1] J. Fritsch and U. Schr¨oder, “Density functional calculation of semiconductor surface phonons,” Phys. Lett. C – Phys. Rep., 309, 209–331, 1999. [2] M.P. Allen, “Computer simulation of liquids,” Oxford University Press, Oxford, 1989. [3] J.M. Haile, “Molecular dynamics simulation: elementary methods,” John Wiley and Sons, New York, 1992. [4] C.K. Harris, D. Roekaerts, F.J.J. Fosendal, F.G.J. Buitendijk, P. Daskopoulos, A.J.N. Vreenegoor, and H. Wang, “Computational fluid dynamics for chemical reactor engineering,” Chem. Eng. Sci., 51, 1569–1594, 1996. [5] G.A. Bird, “Molecular gas dynamics and the direct simulation of gas flows,” Oxford University Press, Oxford, 1994. [6] K.A. Fichthorn and W.H. Weinberg, “Theoretical foundations of dynamical Monte Carlo simulations,” J. Chem. Phys., 95, 1090–1096, 1991. [7] N. Metropolis, A.W. Rosenbluth, M.N. Rosenbluth, A.H. Teller, and E. Teller, “Equation of state calculations by fast computing machines,” J. Chem. Phys., 21, 1087–1092, 1953.

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[8] A.B. Bortz, M.H. Kalos, and J.L. Lebowitz, “A new algorithm for Monte Carlo simulation of ising spin systems,” J. Comp. Phys., 17, 10–18, 1975. [9] D.T. Gillespie, “Exact stochastic simulation of coupled chemical reactions,” J. Phys. Chem., 81, 2340–2361, 1977. [10] A.F. Voter, “Clasically exact overlayer dynamics: diffusion of rhodium clusters on Rh(100),” Phys. Rev., B, 34, 6819–6829, 1986. [11] R.D. Young and D.C. Schubert, “Condensation of tungsten on tungsten in atomic detail – Monte Carlo and statistical calculations vs experiment,” J. Chem. Phys., 42, 3943–3950, 1965. [12] A.A. Chernov and J. Lewis, “Computer model of crystallization of binary systems – kinetic phase transitions,” J. Phys. Chem. Solids, 28, 2185–2198, 1967. [13] R. Gordon, “Adsorption isotherms of lattice gases by computer simulation,” J. Chem. Phys., 48, 1408–1409, 1968. [14] F.F. Abraham and G.W. White “Computer simulation of vapor deposition on twodimensional lattices,” J. Appl. Phys., 41, 1841–1849, 1970. [15] H.J. Leamy, and K.A. Jackson, “Roughness of crystal–vapor interface,” J. Appl. Phys., 42, 2121–2127, 1971. [16] G.H. Gilmer and P. Bennema, “Simulation of crystal-growth with surface diffusion,” J. Appl. Phys., 43, 1347–1360, 1972. [17] T.L. Hill, “Statistical mechanics of multimolecular adsorption 3: introductory treatment of horizontal interactions – Capillary condensation and hysteresis,” J. Chem. Phys., 15, 767–777, 1947. [18] W.K. Burton, N. Cabrera, and F.C. Frank, “The growth of crystals and the equilibrium structure of their surfaces,” Phil. Trans. Roy. Soc. A, 243, 299–358, 1951. [19] D.E. Temkin, “Crystallization processes,” Consultant Bureau, New York, 1966. [20] H.J. Leamy, G.H. Gilmer, K.A. Jackson, and P. Bennema, “Lattice–gas interface structure: a Monte Carlo simulation,” Phys. Rev. Lett., 30, 601–603, 1973. [21] B.W. Dodson and P.A. Taylor, “Monte Carlo simulation of continuous-space crystal growth,” Phys. Rev. B, 34, 2112–2115, 1986. [22] M.D. Rouhani, A.M. Gu´e, M. Sahlaoui, and D. Est`eve, “Strained semiconductor structures: simulation of the first stages of the growth,” Surf. Sci., 276, 109–121, 1992. [23] K. Binder, “Monte Carlo methods in statistical physics,” Springer-Verlag, Berlin, 1986. [24] T. Kawamura, “Monte Carlo simulation of thin-film growth on si surfaces,” Prog. Surf. Sci., 44, 67–99, 1993. [25] J. Lapujoulade, “The roughening of metal surfaces,” Surf. Sci. Rep., 20, 195–249, 1994. [26] M. Kotrla, “Numerical simulations in the theory of crystal growth,” Comp. Phys. Comm., 97, 82–100, 1996. [27] G.H. Gilmer, H. Huang, and C. Roland, “Thin film deposition: fundamentals and modeling,” Comp. Mat. Sci., 12, 354–380, 1998. [28] M. Itoh, “Atomic-scale homoepitaxial growth simulations of reconstructed III–V surfaces,” Prog. Surf. Sci., 66, 53–153, 2001. [29] H.N.G. Wadley, A.X. Zhou, R.A. Johnson, and M. Neurock, “Mechanisms, models, and methods of vapor deposition,” Prog. Mat. Sci., 46, 329–377, 2001. [30] C.C. Battaile, and D.J. Srolovitz, “Kinetic Monte Carlo simulation of chemical vapor deposition,” Ann. Rev. Mat. Res., 32, 297–319, 2002.

7.18 MICROSTRUCTURE OPTIMIZATION S. Torquato Department of Chemistry, PRISM, and Program in Applied & Computational Mathematics Princeton University, Princeton, NJ 08544, USA

1.

Introduction

An important goal of materials science is to have exquisite knowledge of structure-property relations in order to design material microstructures with desired properties and performance characteristics. Although this objective has been achieved in certain cases through trial and error, a systematic means of doing so is currently lacking. For certain physical phenomena at specific length scales, the governing equations are known and the only barrier to achieving the aforementioned goal is the development of appropriate methods to attack the problem. Optimization methods provide a systematic means of designing materials with tailored properties for a specific application. This article focuses on two optimization techniques: (1) the topology optimization procedure used to design composite or porous media, and (2) stochastic optimization methods employed to reconstruct or construct material microstructures.

2.

Topology Optimization

A promising method for the systematic design of composite microstructures with desirable macroscopic properties is the topology optimization method. The topology optimization method was developed almost two decades ago by Bendsøe and Kikuchi [1] for the design of mechanical structures. It is now also being used in smart and passive material design, mechanism design, microelectro-mechanical systems (MEMS) design, target optimization, multifunctional optimization, and other design problems [2–7]. Consider a two-phase composite material consisting of a phase with a property K 1 and volume fraction φ1 and another phase with a property K 2 and 2379 S. Yip (ed.), Handbook of Materials Modeling, 2379–2396. c 2005 Springer. Printed in the Netherlands.

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volume fraction φ2 (= 1 − φ1 ). The property K i is perfectly general: it may represent a transport, mechanical or electromagnetic property, or properties associated with coupled phenomena, such as piezoelectricity or thermoelectricity. For steady-state situations, the generalized flux F(r) at some local position r in the composite obeys the following conservation law in the phases: ∇ · F(r) = 0.

(1)

In the case of electrical conduction and elasticity, F represents the current density and stress tensor, respectively. The local constitutive law relates F to a generalized gradient G, which in the special case of a linear relationship is given by F(r) = K (r)G(r),

(2)

where K (r) is the local property. In the case of electrical conduction, relation (2) is just Ohm’s law, and K and G are the conductivity and electric field, respectively. For elastic solids, relation (2) is Hooke’s law, and K and G are the stiffness tensor and strain field, respectively. For piezoelectricity, F is the stress tensor, K embodies the compliance and piezoelectric coefficients, and G embodies both the electric field and strain tensor. The generalized gradient G must also satisfy a governing differential equation. For example, in the case of electrical conduction, G must be curl free. One must also specify the appropriate boundary conditions at the two-phase interface. One can show that the effective properties are found by homogenizing (averaging) the aforementioned local fields [8, 9]. In the case of linear material, the effective property K e is given by F(r) = K e G(r),

(3)

where angular brackets denote a volume average and/or an ensemble average. For additional details, the reader is referred to the other article (“Theory of Random Heterogeneous Materials”) by the author in this encyclopedia.

2.1.

Problem Statement

The basic topology optimization problem can be stated as follows: distribute a given amount of material in a design domain such that an objective function is extremized [1, 2, 4, 7]. The design domain is the periodic base cell and is initialized by discretizing it into a large number of finite elements (see Fig. 2) under periodic boundary conditions. The problem consists in finding the optimal distribution of the phases (solid, fluid, or void), such that the objective function is minimized. The objective function can be any combination of the individual components of the relevant effective property tensor

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subject to certain constraints [2, 7]. For target optimization [5] and multifunctional optimal design [6], the objective function can be appropriately modified, as described below. In the most general situation, it is desired to design a composite material with N different effective properties, which we denote by K e(1) , K e(2) , . . . , K e(N) , given the individual properties of the phases. In principle, one wants to know the region (set) in the multidimensional space of effective properties in which all composites must lie (see Fig. 1). The size and shape of this region depends on how much information about the microstructure is specified and on the prescribed phase properties. One could begin by making an initial guess for the distribution of the two phases among the elements, solve for the local fields using finite elements and then evolve the microstructure to the targeted properties. However, even for a small number of elements, this integer-type optimization problem becomes a huge and intractable combinatorial problem. For example, for a small design problem with N = 100, the number of different distributions of the three material phases would be astronomically large (3100 = 5 · 1047 ). As each function evaluation requires a full finite element analysis, it is hopeless to solve the optimization problem using random search methods such as, genetic algorithms or simulated annealing methods, which use a large number of function evaluations and do not make use of sensitivity information. Following the idea of standard topology optimization procedures, the problem is therefore relaxed by allowing the material at a given point to be a gray-scale mixture of the two phases. This makes it possible to find sensitivities with respect to design changes, which in turn allows one to use linear programming methods to solve the optimization problem. The optimization

Property Ke(2)

All composites

Property Ke(1) Figure 1. Schematic illustrating the allowable region in which all composites with specified phase properties must lie for the case of two different effective properties.

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procedure solves a sequence of finite element problems followed by changes in material type (density) of each of the finite elements, based on sensitivities of the obj-ective function and constraints with respect to design changes. At the end of the optimization procedure, however, we desire to have a design where each element is either phase 1 or phase 2 material (Fig. 2). This is achieved by imposing a penalization for grey phases at the final stages of the simulation. In the relaxed system, let xi ∈ [0, 1] be the local density of the ith element, so that when xi = 0, the element corresponds to phase 1 and when xi = 1, the element corresponds to phase 2. Let x (xi ,i = 1, . . . , n) be the vector of design variables which satisfies the constraint for the fixed volume fraction φ2 = xi . For any x, the local fields are computed using the finite element method and the effective property K e (K ;x), which is a function of the material property K and x, is obtained by the homogenization of the local fields. The optimization problem is specified as follows: Minimize : subject to :

= K e (x) n 1 xi = φ2 n i=1

(4)

0 ≤ xi ≤ 1, i = 1, . . . , n and prescribed symmetries. The objective function K e (x) is generally nonlinear. To solve this problem, the objective function is linearized, enabling one to take advantage of powerful sequential linear programming techniques. Specifically, the objective function is expanded in Taylor series for a given microstructure x0 : K e (X0 ) + ∇ K e · x, Design domain (base cell)

Phase 1:

(5) Periodic material structure

Phase 2:

Figure 2. Design domain and discretization for a two-phase, three-dimensional topology optimization problem. Each cube represents one finite element, which can consist of either phase 1 material or phase 2 material.

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where x = x − x0 is the vector of density changes. In each iteration, the microstructure evolves to the optimal state by determining the small change x. One can use the simplex method [2] or the interior-point method [5] to minimize the linearized objective function in Eq. (5). In each iteration, the homogenization step to obtain the effective property K e (K ; x0 ) is carried out numerically via the finite-element method on the given configuration x0 . Derivatives of the objective function (∇ K e ) are calculated by a sensitivity analysis which requires one finite element calculation for each iteration. One can use the topology optimization to design at will composite microstructures with targeted effective properties under required constraints [5]. The objective function for such a target optimization problem has been chosen to be given by a least-square form involving the effective property K e (x) at any point in the simulation and a target effective property K 0 : = [K e (x) − K 0 ]2 .

(6)

The method can also be employed for multifunctional optimization problems. The objective function in this instance has been chosen to be a weighted average of each of the effective properties [6].

2.2.

Illustrative Examples

The topology optimization procedure has been employed to design composite materials with extremal properties [2, 3, 10], targeted properties [5, 11], and multifunctional properties [6]. To illustrate the power of the method, we briefly describe microstructure designs in which thermal expansion and piezoelectric behaviors are optimized, the effective conductivity achieves a targeted value, and the thermal conduction demands compete with the electrical conduction demands. Materials with extreme or unusual thermal expansion behavior are of interest from both a technological and fundamental standpoint. Zero thermal expansion materials are needed in structures subject to temperature changes such as space structures, bridges and piping systems. Materials with large thermal displacement or force can be employed as “thermal” actuators. A negative thermal expansion material has the counterintuitive property of contracting upon heating. A fastener made of a negative thermal expansion material, upon heating, can be inserted easily into a hole. Upon cooling, it will expand, fitting tightly into the hole. All three types of expansion behavior have been designed [2]. In the negative expansion case, one must consider a three-phase material: a high expansion material, a low expansion material, and a void region. Figure 3 shows the two-dimensional optimal design that was found; the main mechanism behind the negative expansion behavior is the reentrant cell

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Figure 3. Optimal microstructure for minimization of effective thermal expansion coefficient [2]. White regions denote void, black regions consist of low expansion material and cross-hatched regions consist of high expansion material.

structure having bimaterial components which bend (into the void space) and cause large deformation when heated. In the case of piezoelectricity, actuators that maximize the delivered force or displacement can be designed. Moreover, one can design piezocomposites (consisting of an array of parallel piezoceramic rods embedded in a polymer matrix) that maximize the sensitivity to acoustic fields. The topology optimization method has been used to design piezocomposites with optimal performance characteristics for hydrophone applications [3]. When designing for maximum hydrostatic charge coefficient, the optimal transversally isotropic matrix material has negative Poisson’s ratio in certain directions. This matrix material itself turns out be a composite, namely, a special porous solid. Using an autocad file of the three-dimensional matrix material structure and a stereolithography technique, such negative Poisson’s ratio materials have actually been fabricated [3]. For the case of a two-phase, two-dimensional, isotropic composite, the popular effective-medium approximation (EMA) formula for the effective electrical conductivity σ e is given by

φ1

σe − σ1 σe − σ2 + φ2 = 0, σe + σ1 σe + σ2

(7)

where φi and σi are the volume fraction and conductivity of phase i, respectively. Milton [12] showed that the EMA expression is exactly realized by granular aggregates of the two phases such that spherical grains (in any dimension) of comparable size are well separated with self-similarity on all length scales. This is why the EMA formula breaks down when applied to dispersions of identical circular inclusions. An interesting question is the following: Can the EMA formula be realized by simple structures with a single

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length scale? Using the target optimization formulation in which the target effective conductivity σ0 is given by the EMA function (7), Torquato et al. [6] found a class of periodic, single-scale dispersions that achieve it at a given phase conductivity ratio for a two-phase, two-dimensional composite over all volume fractions. Moreover, to an excellent approximation (but not exactly), the same structures realize the EMA for almost the entire range of phase conductivities and volume fractions. The inclusion shapes are given analytically by the generalized hypocycloid, which in general has a non-smooth interface (see Fig. 4). Minimal surfaces necessarily have zero mean curvature, i.e., the sum of the principal curvatures at each point on the surface is zero. Particularly fascinating are minimal surfaces that are triply periodic because they arise in a variety of systems, including block copolymers, nanocomposites, micellar materials, and lipid-water systems [6]. These two-phase composites are bicontinuous in the sense that the surface (two-phase interface) divides space into two disjoint

␾2 ⫽ 0.001

␾2 ⫽ 0.05

␾2 ⫽ 0.089

␾2 ⫽ 0.3

␾2 ⫽ 0.5

␾2 ⫽ 0.7

␾2 ⫽ 0.911

␾2 ⫽ 0.95

␾2 ⫽ 0.999

Figure 4. Unit cells of generalized hypocycloidal inclusions in a matrix that realize the EMA relation (1) for selected values of the volume fraction in the range 0 < φ2 < 1. Phases 1 and 2 are the white and black phase, respectively.

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but intertwining phases that are simultaneously continuous. This topological feature of bicontinuity is rare in two dimensions and therefore virtually unique to three dimensions [8]. Using multifunctional optimization [6], it has been discovered that triply periodic two-phase bicontinuous composites with interfaces that are the Schwartz P and D minimal surfaces (see Fig. 5) are not only geometrically extremal but extremal for simultaneous transport of heat and electricity. More specifically, these are the optimal structures when a weighted sum of the effective thermal and electrical conductivities ( = λe + σ e ) is maximized for the case in which phase 1 is a good thermal conductor but poor electrical conductor and phase 2 is a poor thermal conductor but good electrical conductor with φ1 = φ2 = 1/2. The demand that this sum is maximized sets up a competition between the two effective transport properties, and this demand is met by the Schwartz P and D structures. By mathematical analogy, the optimality of these bicontinuous composites applies to any of the pair of the following scalar effective properties: electrical conductivity, thermal conductivity, dielectric constant, magnetic permeability, and diffusion coefficient. It will be of interest to investigate whether the optimal structures when φ1 =/ φ2 are bicontinuous structures with interfaces of constant mean curvature, which would become minimal surfaces at the point φ1 = φ2 = 1/2. The topological property of bicontinuity of these structures suggests that they would be mechanically stiff even if one of the phases is a compliant solid or a liquid, provided that the other phase is a relatively stiff material. Indeed, it has recently been shown that the Schwartz P and D structures are extremal when a competition is set up between the bulk modulus and electrical (or thermal) conductivity of the composite [13].

Figure 5. Unit cells of two different minimal surfaces with a resolution of 64 × 64 × 64. Left panel: Schwartz simple cubic surface. Right panel: Schwartz diamond surface.

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3.

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Reconstruction Techniques

The reconstruction of realizations of disordered materials, such as liquids, glasses, and random heterogeneous materials, from a knowledge of limited microstructural information (lower-order correlation functions) is an intriguing inverse problem. Clearly, one can never reconstruct the original material perfectly in the infinite-system limit, i.e., such reconstructions are nonunique. Thus, the objective here is not the same as that of data decompression algorithms that efficiently restore complete information, such as the gray scale of every pixel in an image. The generation of realizations of random media with specified lower-order correlation functions can: 1. shed light on the nature of the information contained in the various correlation functions that are employed; 2. ascertain whether the standard two-point correlation function, accessible experimentally via scattering, can accurately reproduce the material and, if not, what additional information is required to do so; 3. identify the class of microstructures that have exactly the same lowerorder correlation functions but widely different effective properties; 4. probe the interesting issue of nonuniqueness of the generated realizations; 5. construct structures that correspond to specified correlation functions and categorize classes of random media; 6. provide guidance in ascertaining the mathematical properties that physically realizable correlation functions must possess [14]; and 7. attempt three-dimensional reconstructions from slices or micrographs of the sample: a poor man’s X-ray microtomography experiment. The first reconstruction procedures applied to heterogeneous materials were based on thresholding Gaussian random fields. This approach to reconstruct random media originated with Joshi [15], and was extended by Adler [16] and Roberts and Teubner [17]. This method is currently limited to the standard two-point correlation function, and is not suitable for extension to non-Gaussian statistics.

3.1.

Optimization Problem

It has recently been suggested that reconstruction problems can posed as optimization problems [18, 19]. A set of target correlation functions are prescribed based upon experiments, theoretical models, or some ansatz. Starting from some initial realization of the random medium, the method proceeds to find a realization by evolving the microstructure such that the calculated correlation functions best match the target functions. This is achieved by minimizing

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an error based upon the distance between the target and calculated correlation functions. The medium can be a dispersion of particles [18] or, more generally, a digitized image of a disordered material [19]. For simplicity, we will introduce the problem for the case of digitized heterogeneous media here and consider only a single two-point correlation function for statistically isotropic two-phase media (the generalization to multiple correlation functions is straightforward [18, 19]). It is desired to generate realizations of two-phase isotropic media that have a target two-point correlation function f 2 (r) associated with phase i, where r is the distance between the two points and i = 1 or 2. Let fˆ2 (r) be the corresponding function of the reconstructed digitized system (with periodic boundary conditions) at some time step. It is this system that we will attempt to evolve towards f 2 (r) from an initial guess of the system realization. Again, for simplicity, we define a fictitious “energy” (or norm-2 error) E at any particular stage as E=

[ fˆ2 (r) − f 2 (r)]2 ,

(8)

r

where the sum is over all discrete values of r. Potential candidates for the correlation functions [8] include: (1) the standard two-point probability function S2(r), lineal path function L(z), pore-size density function P(δ), and twopoint cluster function C2 (r). For statistically isotropic materials, S2 (r) gives the probability of finding the end points of a line segment of length r in one of the phases (say phase 1) when randomly tossed into the system, whereas L(z) provides the probability of finding the entire line segment of length r in phase 1 (or 2) when randomly tossed into the system.

3.2.

Solution of Optimization Problem

The aforementioned optimization problem is very difficult to solve due to the complicated nature of the objective function, which involves complex microstructural information in the form of correlation functions of the material, and due to the combinatorial nature of the feasible set. Standard mathematical programming techniques are therefore most likely inefficient and likely to get trapped in local minima. In fact, the complexity and generality of the reconstruction problem makes it difficult to devise deterministic algorithms of wide applicability. One therefore often resorts to heuristic techniques for global optimization, in particular, the simulated annealing method. Simulated annealing has been applied successfully to many difficult combinatorial problems, including NP-hard ones such as the “traveling salesman” problem. The utility of the simulated annealing method stems from its simplicity in that it only requires “black-box” cost function evaluations, and in its physically designed ability to escape local minima via accepting locally

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unfavorable configurations. In its simplest form, the states of two selected pixels of different phases are interchanged, automatically preserving the volume fraction of both phases. The change in the error (or “energy”) E = E − E between the two successive states is computed. This phase interchange is then accepted with some probability p(E) that depends on E. One reasonable choice is the Metropolis acceptance rule, i.e.,

p(E) =

1, E ≤ 0, exp(−E/T ), E > 0,

(9)

where T is a fictitious “temperature”. The concept of finding the lowest error (lowest energy) state by simulated annealing is based on a well-known physical fact: If a system is heated to a high temperature T and then slowly cooled down to absolute zero, the system equilibrates to its ground state. We note that there are various ways of appreciably reducing computational time. For example, computational cost can be significantly lowered by using other stochastic optimization schemes such as the “Great Deluge” algorithm, which can be adjusted to accept only “downhill” energy changes, and the “threshold acceptance” algorithm [20]. Further savings can be attained by developing strategies that exploit the fact that pixel interchanges are local and thus one can reuse the correlation functions measured in the previous time step instead of recomputing them fully at any step [19]. Additional cost savings have been achieved by interchanging pixels only at the two-phase interface [8].

3.3.

Illustrative Examples

Lower-order correlation functions generally do not contain complete information and thus cannot be expected to yield perfect reconstructions. Of course, the judicious use of combinations of lower-order correlation functions can yield more accurate reconstructions than any single function alone. Yeong and Torquato [19, 21] clearly showed that the two-point function S2 alone is not sufficient to reconstruct accurately random media. By also incorporating the lineal-path function L, they were able to obtain better reconstructions. They studied one-, two- and three-dimensional digitized isotropic media. Each simulation began with an initial configuration of pixels (white for phase 1 and black for phase 2) in the random checkerboard arrangement at a prescribed volume fraction. A two-dimensional example illustrating the insufficiency of S2 in reconstructions is a target system of overlapping disks at a disk volume fraction of φ 2 = 0.5; see Fig. 6(a). Reconstructions that accurately match S2 alone, L alone, and both S2 and L are shown in Fig. 6. The S2-reconstruction is not very accurate; the cluster sizes are too large, and the system actually percolates.

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(b)

(c)

(d)

Figure 6. (a) Target system: a realization of random overlapping disks. System size = 400 × 400 pixels, disk diameter = 31 pixels, and volume fraction φ2 = 0.5. (b) S2 -reconstruction. (c) Corresponding L-reconstruction. (d) Corresponding hybrid (S2 + L)-reconstruction.

(Note that overlapping disks percolate at a disk area fraction of φ2 ≈ 0.68 [8]). The L-reconstruction does a better job than the S2 -reconstruction in capturing the clustering behavior. However, the hybrid (S2 + L)-reconstruction is the best. The optimization method can be used in the construction mode to find the specific structures that realize a specified set of correlation functions. An interesting question in this regard is the following: Is any correlation function physically realizable or must the function satisfy certain conditions? It turns out that not all correlation functions are physically realizable. For example, what are the existence conditions for a valid (i.e., physically realizable) auto covariance function χ(r) ≡ S2(r)−φ12 for statistically homogeneous twophase media? It is well known that there are certain nonnegativity conditions involving the spectral representation of the auto covariance χ(r) that must be obeyed [14]. However, it is not well known that these nonnegativity conditions are necessary but not sufficient conditions that a valid auto covariance χ(r) of a statistically homogeneous two-phase random medium (i.e., a binary stochastic spatial process) must meet. Some of these “binary” conditions are described by Torquato [8] but the complete characterization is a very difficult problem. Suffice it to say that that the algorithm in the construction mode can be used

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to provide guidance on the development of the mathematical conditions that a valid auto covariance χ(r) must obey. Cule and Torquato [20] considered the construction of realizations having the following autocovariance function: sin(qr) S2(r) − φ12 , = e−r/a φ1 φ2 qr

(10)

where q = 2π/b and the positive parameter b is a characteristic length that controls oscillations in the term sin(qr)/(qr), which also decays with increasing r. This function possesses phase-inversion symmetry [8] and exhibits a considerable degree of short-range order; it generalizes the purely exponentiallydecaying function studied by Debye, et al. [22]. This function satisfies the nonnegativity condition on the spectral function but may not satisfy the “binary” conditions, depending on the values of a, b, and φ1 [14]. Two structures possessing the correlation function (10) are shown in Fig. 7 for φ2 = 0.2 and 0.5, in which a = 32 pixels and b = 8 pixels. For these sets of parameters, all of the aforementioned necessary conditions on the function are met. At φ2 = 0.2, the system resembles a dilute particle suspension with “particle” diameters of order b. At φ2 = 0.5, the resulting pattern is labyrinthine such that the characteristic sizes of the “patches” and “walls” are of order a and b, respectively. Note that S2(r) was sampled in all directions during the annealing process. In all of the previous two-dimensional examples, however, both S2 and L were sampled along two orthogonal directions to save computational time. This time-saving step should be implemented only for isotropic media, provided that there is no appreciable short-range order; otherwise, it leads to unwanted anisotropy [20, 23]. However, this artificial anisotropy can be reduced by optimizing along additional selected directions [24].

␾2 ⫽ 0.2

␾2 ⫽ 0.5

Figure 7. Structures corresponding to the target correlation function given by (10) for φ2 = 0.2 and 0.5. Here a = 32 pixels and b = 8 pixels.

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To what extent can information extracted from two-dimensional cuts through a three-dimensional isotropic medium, such as S2 and L , be employed to reproduce intrinsic three-dimensional information, such as connectedness? This question was studied for the aforementioned Fontainebleau sandstone for which we know the full three-dimensional structure via X-ray microtomography [21]. The three-dimensional reconstruction that results by using a single slice of the sample and matching both S2 and L is shown in Fig. 8. The reconstructions reproduce accurately certain three-dimensional properties of the pore space, such as the pore-size functions, the mean survival time of a Brownian particle, and the fluid permeability. The degree of connectedness of the pore space also compares remarkably well with the actual sandstone, although this is not always the case [25]. As noted earlier, the aforementioned algorithm was originally applied to reconstruct realizations of many-particle systems [18]. The hard-sphere system in which pairs of particles only interact with an infinite repulsion when they overlap is one of the simplest interacting particle systems [8]. Importantly, the impenetrability constraint does not uniquely specify the statistical ensemble. The hard-sphere system can be in thermal equilibrium or in one of the infinitely many nonequilibrium states, such as the random sequential addition (or adsorption) (RSA) process that is produced by randomly, irreversibly, and sequentially placing nonoverlapping objects into a volume [8]. While particles in equilibrium have thermal motion such that they sample the configuration space uniformly, particles in an RSA process do not sample the configuration space uniformly, since their positions are forever “frozen” (i.e., do not diffuse) after they have been placed into the system.

Figure 8. Hybrid reconstruction of a sandstone (described in Ref. [8]) using both S 2 and L obtained from a single “slice”. System size is 128 × 128 × 128 pixels. Left panel: Pore space is white and opaque, and the grain phase is black and transparent. Right panel: 3D perspective of the surface cuts.

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The geometrical blocking effects and the irreversible nature of the process results in structures that are distinctly different from corresponding equilibrium configurations, except for low densities. The saturation limit (the final state of this process whereby no particles can be added) occurs at a particle volume fraction φ2 ≈ 0.55 in two dimensions [8]. The reconstruction of the two-dimensional RSA disk system in which the target correlation function is the well-known radial distribution function (RDF) g(r). In two dimensions, the quantity ρ2πrg(r) dr gives the average number of particle centers in an annulus of thickness dr at a radial distance of r from the center of a particle (where ρ is the number density). The RDF is of central importance in the study of equilibrium liquids in which the particles interact with pairwise-additive forces since all of the thermodynamics (a)

(b)

Figure 9. (a) A portion of a sample RSA system at φ2 = 0.543. (b) A portion of the reconstructed RSA system at φ2 = 0.543.

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Figure 10. Configurations of 289 particles for φ2 = 0.2 in two dimensions. The equilibrium hard-disk system (left) shows more clustering and larger pores than the annealed step-function system (right).

can be expressed in terms of the RDF. The RDF can be ascertained from scattering experiments, which makes it a likely candidate for the reconstruction of a real system. The initial configuration was 5000 disks in equilibrium. Figure 9 shows a realization of the the RSA system at φ2 = 0.543 in (a), and the reconstructed system. As a quantitative comparison of how the original and reconstructed systems matched, it was found that the corresponding pore-size distribution functions [8] were similar. This conclusion gives one confidence that a reasonable facsimile of the actual structure can be produced from the RDF for a class of many-particle systems in which there is not significant clustering of the particles. For the elementary unit step-function g2 , previous work [26] indicated that this function was achievable by hard-sphere configurations up to a terminal covering fraction of particle exclusion diameters equal to 2−d in d dimensions. To test whether the unit step g2 is actually achievable by hard spheres for nonzero densities, the aforementioned stochastic optimization procedure was applied in the construction mode. Calculations for d = 1 and 2 confirmed that the step-function g2 is indeed realizable up to the terminal density [27]. Figure 10 compares an equilibrium hard-disk configuration at φ2 = 0.2 to a corresponding annealed step-function system.

4.

Summary

The fundamental understanding of the microstructure/properties connection is the key to designing new materials with the tailored properties for

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specific applications. Optimization methods combined with novel synthesis and fabrication techniques provide a means of accomplishing this goal systematically and could make optimal design of real materials a reality in the future. The topology optimization technique and the stochastic reconstruction (construction) method address only a small subset of optimization issues of importance in materials science, but the results that are beginning to emerge from these relatively new methods bode well for progress in the future.

References [1] M.P. Bendsøe and N. Kikuchi, “Generating optimal topologies in structural design using a homogenization method,” Comput. Methods Appl. Mech. Eng., 71, 197–224, 1988. [2] O. Sigmund and S. Torquato, “Design of materials with extreme thermal expansion using a three-phase topology optimization method,” J. Mech. Phys. Solids, 45, 1037–1067, 1997. [3] O. Sigmund, S. Torquato, and I.A. Aksay, “On the design of 1-3 piezocomposites using topology optimization,” J. Mater. Res., 13, 1038–1048, 1998. [4] M.P. Bendsøe, Optimization of Structural Topology, Shape and Material, SpringerVerlag, Berlin, 1995. [5] S. Hyun and S. Torquato, “Designing composite microstructures with targeted properties,” J. Mater. Res., 16, 280–285, 2001. [6] S. Torquato, S. Hyun, and A. Donev, “Multifunctional composites: optimizing microstructures for simultaneous transport of heat and electricity,” Phys. Rev. Lett., 89, 266601, 1–4, 2002. [7] M.P. Bendsøe and O. Sigmund, Topology Optimization, Springer-Verlag, Berlin, 2003. [8] S. Torquato, Random Heterogeneous Materials: Microstructure and Macroscopic Properties, Springer-Verlag, New York, 2002. [9] G.W. Milton, The Theory of Composites, Cambridge University Press, Cambridge, England, 2002. [10] U.D. Larsen, O. Sigmund, and S. Bouwstra, “Design and fabrication of compliant mechanisms and material structures with negative Poisson’s ratio,” J. Microelectromechanical Systems, 6(2), 99–106, 1997. [11] S. Torquato and S. Hyun, “Effective-medium approximation for composite media: realizable single-scale dispersions,” J. Appl. Phys., 89, 1725–1729, 2001. [12] G.W. Milton, “Multicomponent composites, electrical networks and new types of continued fractions. I and II,” Commun. Math. Phys., 111, 281–372, 1987. [13] S. Torquato and A. Donev, “Minimal surfaces and multifunctionality,” Proc. R. Soc. Lond. A, 460, 1849–1856, 2004. [14] S. Torquato, “Exact conditions on physically realizable correlation functions of random media,” J. Chem. Phys., 111, 8832–8837, 1999. [15] M.Y. Joshi, A Class of Stochastic Models for Porous Media, Ph.D. thesis, University of Kansas, Lawrence, 1974. [16] P.M. Adler, Porous Media – Geometry and Transports, Butterworth-Heinemann, Boston, 1992. [17] A.P. Roberts and M. Teubner, “Transport properties of heterogeneous materials derived from Gaussian random fields: bounds and simulation,” Phys. Rev. E, 51, 4141–4154, 1995.

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[18] M.D. Rintoul and S. Torquato, “Reconstruction of the structure of dispersions,” J. Colloid Interface Sci., 186, 467–476, 1997. [19] C.L.Y. Yeong and S. Torquato, “Reconstructing random media,” Phys. Rev. E, 57, 495–506, 1998a. [20] D. Cule and S. Torquato, “Generating random media from limited microstructural information via stochastic optimization,” J. Appl. Phys., 86, 3428–3437, 1999. [21] C.L.Y. Yeong and S. Torquato, “Reconstructing random media: II. Three-dimensional media from two-dimensional cuts,” Phys. Rev. E, 58, 224–233, 1998b. [22] P. Debye, H.R. Anderson, and H. Brumberger, “Scattering by an inhomogeneous solid. II. The correlation function and its applications,” J. Appl. Phys., 28, 679–683, 1957. [23] C. Manwart and R. Hilfer, “Reconstruction of random media using Monte Carlo methods,” Phys. Rev. E, 59, 5596–5599, 1999. [24] N. Sheehan and S. Torquato, “Generating microstructures with specified correlation function,” J. Appl. Phys., 89, 53–60, 2001. [25] C. Manwart, S. Torquato, and R. Hilfer, “Stochastic reconstruction of sandstones,” Phys. Rev. E, 62, 893–899, 2000. [26] F.H. Stillinger, S. Torquato, J.M. Eroles, and T.M. Truskett, “Iso-g (2) processes in equilibrium statistical mechanics,” J. Phys. Chem. B, 105, 6592–6597, 2001. [27] J.R. Crawford, S. Torquato, and F.H. Stillinger, “Aspects of correlation function realizability,” J. Chem. Phys., 2003.

7.19 MICROSTRUCTURAL CHARACTERIZATION ASSOCIATED WITH SOLID–SOLID TRANSFORMATIONS J.M. Rickman1 and K. Barmak2 1

Department of Materials Science and Engineering, Lehigh University, Bethlehem, PA 18015, USA 2 Department of Materials Science and Engineering, Carnegie Mellon University, Pittsburgh, PA 15213, USA

1.

Introduction

Materials scientists have long been interested in the characterization of complex poly-crystalline systems, as embodied in the distribution of grain size and shape, and have sought to link microstructural features with observed mechanical, electronic and magnetic properties [1]. The importance of detailed microstructural characterization is underscored by systems of limited spatial dimensionality, with length scales of the order of nanometers to microns, as their reliability and performance are greatly influenced by specific microstructural features rather than by average, bulk properties [2]. For example, the functionalities of many electronic devices depend critically on the microstructure of thin metallic films via the film deposition process and the occurrence of reactive phase formation at metallic contacts. Various tools are available for quantitative microstructural characterization. Most notably, microstructural analyses often employ stereological techniques [1] and the related formalism of stochastic geometry [3] to interrogate grain populations and to deduce plausible growth scenarios that led to the observed grain morphologies. In this effort computer simulation is especially valuable, permitting one to implement various growth assumptions and to generate a large number of microstructures for subsequent analysis. The acquisition of comparable grain size and shape information from experimental images is, however, often problematic given difficulties inherent in grain recognition. The case of polycrystalline thin films is illustrative here. In these systems transmission 2397 S. Yip (ed.), Handbook of Materials Modeling, 2397–2408. c 2005 Springer. Printed in the Netherlands.

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electron microscopy (TEM) is necessary to resolve pertinent microstructural features. Unfortunately, complex contrast variations peculiar to TEM images plague grain recognition and therefore image interpretation. As a result, the tedious, state-of-the-art analysis, until quite recently [4, 6], involved human intervention to trace grain boundaries and to collect grain statistics. In this topical article we review methods for quantitative microstructural analysis, focusing first on systems that evolve from nucleation and subsequent growth processes. As indicated above, computer simulation of these processes provides considerable insight into the link between initial conditions and product microstructures, and so we will highlight some recent work in this area of evolving, first-order phase transformations in the absence of grain growth (i.e., coarsening). The analysis here will involve several important descriptors that are sensitive to different microstructural details and that can be used to infer the conditions that led to a given structure. Finally, we conclude with a discussion of new, automated image processing techniques that permit one to acquire information on large grain populations and to make useful comparisons of the associated grain-size distributions with those advanced in theoretical investigations of grain growth [6–8].

2.

Phase Transformations

Computer simulations are particularly useful for investigating the impact of nucleation conditions on product grain morphology resulting from a firstorder phase transformation [9, 3]. Several schemes for modeling such transformations have been discussed in the literature [10, 11], and it is generally possible to use them to describe a variety of nucleation scenarios, including those involving site saturation (e.g., a burst) and a constant nucleation rate. To illustrate a typical microstructural analysis, consider the constant radial growth to impingement of product grains originating from a burst of nuclei that are randomly distributed in two dimensions. The resulting microstructures consists of a set of Voronoi grains that tile the system, as shown in Fig. 1.

2.1.

Grain Area Distribution

Our characterization of this microstructure begins with the compilation of ¯ where the bar a frequency histogram of normalized grain areas, A = A/ A, denotes a microstructural average. The corresponding probability distribution P( A ), as obtained for a relatively large grain population (∼106 grains) is

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Figure 1. A fully coalesced product microstructure produced by a burst of nuclei that subsequently grow at a constant radial rate until impingement.

shown in Fig. 2. While no exact analytical form for this distribution is known, approximate expressions based on the gamma distribution P γ ( A ) =

1 (A )α−1 exp( A ) β α (α)

(1)

follow from stochastic geometry [12, 13], where α and β are parameters such that α = 1/β. For the Voronoi structure β is then the variance, as obtained analytically by Gilbert [14]. As can be seen from Fig. 2, the agreement between the simulated and approximate distributions is excellent. As P( A ) is a quantity of central importance in most microstructural analyses, it is of interest to determine whether it can be used to deduce, a posteriori, nucleation conditions. For this purpose, consider next the product microstructure resulting from nuclei that are randomly distributed on an underlying microstructure. A systematic analysis of such structures follows from a comparison of the relevant length scales here, namely the average underlying cell diameter, lu , and the average internuclear separation along the boundary, lb . For this discussion it is convenient to define a relative measure of these length scales r = lb /lu , and so one would intuitively expect that in the limit r > 1 (r < 1) the product microstructure comprises largely equiaxed (elongated) grains. Several product microstructures corresponding to different values of r, shown in Fig. 3, confirm these expectations.

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Figure 2. The corresponding probability distribution, P(A ), of normalized grain areas, A , and an approximate representation, P γ (A ) (solid line), based on the gamma distribution. Note the excellent agreement between the simulated and approximate distributions.

Figure 3. Product microstructures corresponding, from left to right, to r < 1, r ∼ 1, and r > 1. Note that in the limit r > 1(r < 1) the product microstructure comprises largely equiaxed (elongated) grains.

Upon examining the probability distributions for large collections of grains with these values of r (see Fig. 4), it is evident that, upon decreasing r, the distribution shifts to the left and a greater number of both relatively small and large grains is created. A more detailed analysis of these distributions demonstrates, again, that the gamma distribution is a good approximation in many cases, and a calculation of lower-order moments reveals a scaling regime for intermediate values of r [9]. Despite these features, it is found that, in general, P( A ) lacks the requisite sensitivity to variations in r needed for an unambiguous identification of nucleation conditions. As an alternative to the grain-area distribution, one can obtain descriptors that focus on the spatial distribution of the nucleation sites themselves, regarded here as microstructural generators. The utility of such descriptors depends, of course, on the ability to extract from a product microstructure the spatial distribution of these generators. As a reverse Monte Carlo method was recently devised to accomplish this task in some circumstances [3], we merely

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Figure 4. The probability distribution P(A ) for different values of the ratio of length scales r . Although there is a discernible shift in curve position and attendant change in curve shape upon changing r, the distribution is not very sensitive to these changes.

outline here the use of one such descriptor. Now, from the theory of point processes one can define a neighbor distribution wk (r), the generalization of a waiting-time distribution to space, that measures the probability of finding the k-th neighbor at a distance r (not to be confused with the dimensionless microstructural parameter defined above) away from a given nucleus [15]. Consider then the first moment of this distribution rk for the kth neighbor. For points randomly distributed in d dimensions one finds that

1 d 1+ rk = √ 1/d π(λd ) 2

1/d

(k + 1/d) , (k)

(2)

where λd is the d-dimensional volume density of points. Thus, the dependence of rk on k is a signature of the effective spatial dimensionality of the site-saturated nucleation process. Figure 5 shows the dependence of the normalized first moment on k for several cases of catalytic nucleation on an underlying microstructure, each corresponding to a different value of ζ = 1/r. An interpretation of Fig. 5 follows upon examining Fig. 6, the latter showing the dependence of the moment on k for the small and large ζ along with the predicted results for strictly oneand two-dimensional random distributions of nuclei. For low linear nucleation densities (e.g., ζ = 0.1) the underlying structure is effectively unimportant and so rk follows the theoretical two dimensional random curve for small to intermediate k. By contrast, at high nucleation densities, nuclei have many neighbors along a given edge and so rk initially exhibits pseudo-one-dimensional behavior. As more distant neighbors are considered, rk is consistent with

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Figure 5. The first moment of the neighbor distribution, rk , as a function of neighbor number k for different values of ζ = 1/r.

two-dimensional random behavior as these neighbors are now on other boundary segments distributed throughout the system. With this information it is possible to infer different nucleation scenarios from rk vs. k [3]. Finally, it is worth noting that other, related descriptors are useful in distinguishing different nucleation conditions. For example, as is often done in atomistic simulation, one can calculate the pair correlation function, g(r), for the nuclei. The results of such a calculation are presented in Fig. 7 for nucleation on the corners of an underlying grain structure. A measure of the nonrandomness of this spatial distribution of nuclei at a particular r is given by g(r) − 1. Thus, g(r) is a sensitive measure of deviations from randomness, and has been employed to investigate spatial correlations among nuclei formed at underlying grain junctions [3, 16].

3.

Image Processing and Grain-size Distributions

As indicated above, the acquisition of statistically significant grain size and shape information from experimental micrographs is difficult owing to

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Figure 6. The dependence of rk on k for the small and large ζ along with the predicted results for strictly one- and two-dimensional random distributions of nuclei.

problems associated with grain recognition. Nevertheless, it is essential to obtain such information to enable meaningful comparisons with simulated structures and to investigate various nucleation and growth scenarios. With this in mind, we outline below recent progress toward automated analysis of experimental micrographs. In this short review, our focus will be on assessing models of grain growth (i.e., coarsening) in thin films that describe microstructural kinetics after transformed grains have grown to impingement. The grain size of thin films is known to scale with the thickness of the film. Thus, for films with thicknesses of 1 nm to 1 µm it is necessary to employ transmission electron microscopy (TEM) to image the film grain structure. Although the grain structure of these film is easily discernable by eye from TEM micrographs, the image contrast is often quite complex. Such image contrast arises primarily from variations in the diffraction condition that result from: (1) changes in crystal orientation as a grain boundary is traversed, (2) localized straining of the lattice, and (3) long-wavelength bending of the sample. The latter two sources of contrast cannot be easily deconvoluted from the first, and, as a result, conventional image processing algorithms have been of limited utility in thin film grain structure analysis.

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Figure 7. The pair correlation function g(r ) versus internuclear separation, r , for nucleation on the corners of an underlying grain structure. A measure of the nonrandomness of this spatial distribution of nuclei at a particular r is given by g(r ) − 1.

Recently we have developed and used a practical, automated scheme for the acquisition of microstructural information from a statistically significant population of grains imaged by TEM [4]. Our overall philosophy for automated detection of grain boundaries is to first optimize the microscope operating conditions and the resultant image, and to then eliminate as much as possible false features in the processed images, even sometimes at the expense of real microstructural features. The true information deleted in this manner is recovered by optimally combining processed images of the same field of view taken at different sample tilts. The new algorithms developed to independently process the images taken at different samples tilts are automated thresholding and three filters for removal of (i) short, disconnected pixel segments, (ii) excessively connected or “tangled” lines, and (iii) isolated clusters. The segment and tangle filters employ a length scale specified by the user that is estimated as the length, in

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pixels, of the shortest real grain boundary segment in the TEM image. These newly developed filters are used in combination with other existing image processing routines including, for example, gray scale and binary operations such as the median noise filter, the Sobel and Kirsch edge detection filters, dilation and erosion operations, skeletonization, and opening and closing operations to generate the binary image seen Fig. 8. The images at different sample tilts are then combined to generate a single processed image that can then be analyzed using available software (e.g., NIH image, Rasband, US National Institutes of Health, or Scion Image, http://www.scioncorp.com.) Additional details of our automated image processing methodology can be found elsewhere [4, 5]. The experimentally determined grain size data for 8185 Al grains obtained using our automated methodology is shown in Fig. 9. The figure also shows three continuous probability density functions, corresponding to the lognormal (l), gamma (g), and Rayleigh (r) distributions, respectively, that have been fitted to the experimental data. The functional forms of these distributions are given by pl (d) =

1 2 2 exp −(1n(d) − α) /2β , d(2πβ 2 )1/2

(3)

pg (d) =

d α−1 exp(−d/β), β α (α)

(4)

pr (d) =

αd exp −d 2 /4β , β

(5)

where α and β are fitting parameters that are different for each distribution and, in the case of the Rayleigh distribution, normalization requires that α = 1/2. In these expressions, d represents an equivalent circular grain diameter, i.e., the diameter of a circle with equal area to that of the grain. The figure clearly demonstrates that the Rayleigh density is a poor representation of the experimental data, while both the lognormal and gamma densities fall within the error of the experimental distribution. It should be emphasized that large data sets, acquired here via automated methodologies, are needed to examine quantitatively the predictions of various grain growth models.

4.

Conclusions

Various techniques for the analysis of microstructures generated both experimentally and by computer simulation were described above. In the case

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Figure 8. A bright-field scanning transmission electron micrograph and processed images of a 100 nm thick Al film. (B1–D1) Results from conventional image processing, after (B1) median noise filter and Sobel edge detection operator, (C1) dilation, skeletonization. (B2–D2) Results from using a combination of new and conventional image processing operations, after (B2) hybrid median noise filter and Kirsch edge detection filter, (C2) dilation, skeletonization, segment filter and tangle filter, and (D2) cluster filter and final consolidation. Note that conventional image processing results in a number of false grains.

of computer simulation the focus was on developing descriptors that can be used to infer nucleation and growth conditions associated with a first-order phase transformation from a final, coalesced product microstructure. We also describe a methodology for the automated analysis of experimental TEM

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Figure 9. Fig. 3 (a) Lognormal, (b) Gamma, and (c) Rayleigh distributions fitted to experimental grain size data comprising 8185 Al grains in a thin film. Error bars represent a 95% confidence level.

micrographs. The purpose of such an analysis is to obtain statistically significant size and shape data for a large grain population. Finally, we use the information from the automated analysis to assess the validity of different grain growth models.

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Acknowledgments The authors are grateful for support under DMR-9256332, DMR-9713439 and DMR-9996315.

References [1] E.E. Underwood, Quantitative Stereology, Addison-Wesley, Reading, Massachusetts, 1970. [2] J. Harper and K. Rodbell, J. Vac. Sci. Technol. B, 15, 763, 1997. [3] W.S. Tong, J.M. Rickman, and K. Barmak, Acta Mater., 47, 435, 1999. [4] D.T. Carpenter, J.M. Rickman, and K. Barmak, J. Appl. Phys., 84, 5843, 1998. [5] D.T. Carpenter, J.R. Codner, K. Barmak, and J.M. Rickman, Mater. Lett., 41, 296, 1999. [6] N. Louat, Acta Metall., 22, 721, 1974. [7] P. Feltham, Acta Metall., 5, 97, 1957. [8] W.W. Mullins, Acta Mater., 46, 6219, 1998. [9] W.S. Tong, J.M. Rickman, and K. Barmak, “Impact of boundary nucleation on product grain size distribution,” J. Mater. Res., 12, 1501, 1997. [10] K.W. Mahin, K. Hanson, and J.W. Morris, Jr., Acta Metall., 28, 443, 1980. [11] H.J. Frost and C.V. Thompson, Acta Metall., 35, 529, 1987. [12] T. Kiang, Z. Astrophys, 48, 433, 1966. [13] D. Weaire, J.P. Kermode, and J. Wejchert, Phil. Mag. B, 53, L101–105, 1986. [14] E.N. Gilbert, Ann. Math. Stat., 33, 958, 1962. [15] N.G. van Kampen, Stochastic Processes in Physics and Chemistry, North-Holland, New York, 1992. [16] D. Stoyan and H. Stoyan, Appl. Stoch. Mod. Data Anal., 6, 13, 1990.

8.1 MESOSCALE MODELS OF FLUID DYNAMICS Bruce M. Boghosian1 and Nicolas G. Hadjiconstantinou2 1 Department of Mathematics, Tufts University, Bromfield-Pearson Hall, Medford, MA 02155, USA 2 Department of Mechanical Engineering, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA

During the last half century, enormous progress has been made in the field of computational materials modeling, to the extent that in many cases computational approaches are used in a predictive fashion. Despite this progress, modeling of general hydrodynamic behavior remains a challenging task. One of the main challenges stems from the fact that hydrodynamics manifests itself over a very wide range of length and time scales. On one end of the spectrum, one finds the fluid’s “internal” scale characteristic of its molecular structure (in the absence of quantum effects, which we omit in this chapter). On the other end, the “outer” scale is set by the characteristic sizes of the problem’s domain. The resulting scale separation or lack thereof as well as the existence of intermediate scales are key to determining the optimal approach. Successful treatments require a judicious choice of the level of description which is a delicate balancing act between the conflicting requirements of fidelity and manageable computational cost: a coarse description typically requires models for underlying processes occuring at smaller length and time scales; on the other hand, a fine-scale model will incur a significantly larger computational cost. When no molecular or intermediate length scales are important, e.g., for simple fluids, modeling the fluid at the outer scale and as a continuum results in the most efficient approach. The most well known example of these “continuum” approaches is the Navier–Stokes description of a viscous fluid. Continuum hydrodynamic descriptions are typically derived from conservation laws which require transport models before they can be solved. The resulting mathematical model is in the form of partial differential equations. A variety of methods have been developed for the solution of these, including finitedifference, finite-element, finite-volume, and spectral-element methods, such 2411 S. Yip (ed.), Handbook of Materials Modeling, 2411–2414. c 2005 Springer. Printed in the Netherlands.

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as are described in Ref. [1]. All of these methods require that the physical domain is discretized using a mesh, the generation of which can be fairly involved, depending on the complexity of the problem. More recent efforts have culminated in the development of meshless methods for solving partial differential equations, an exposition of which can be found in Ref. [2]. In certain circumstances, the separation between the molecular and macroscopic scales of length and time is lost. This happens locally in, inter alia, liquid droplet coalescence, amphiphilic membranes and monolayers, contactline dynamics in immiscible fluids, and shock formation. It may also happen globally, for example if ultra-high frequency waves are excited in a fluid. In these cases, one is forced to use a particulate description, the most well known of which is Molecular Dynamics (MD) in which particle orbits are tracked numerically. An extensive description of MD can be found in Chapter 2 [3], while a discussion of its applications to hydrodynamics can be found in Ref. [4]. The Navier–Stokes equations on one hand and MD on the other, represent two extreme possibilities. Typical problems of interest, and in particular of practical interest involving complex fluids and inhomogeneities, are significantly more complex leading to a wide range of intermediate scales that need to be addressed. For the foreseeable future, MD can be applied only to very small systems and for very short periods of time due to the computational cost associated with this approach. The principal purpose of this Chapter is to describe numerous intermediate or “mesoscale” approaches between these extremes, which attempt to coarse-grain the particulate description to varying degrees to address modeling needs. An example of a mesoscale approach can be found in descriptions of a dilute gas, in which particles travel in straight line orbits for the great majority of the time. In this situation, calculating trajectories between collisions in an exact fashion is unnecessary and therefore inefficient. A particularly ingenious method, known as Direct Simulation Monte Carlo (DSMC) takes advantage of this observation to split particle motion into successive collisionless advection and collision events. The collisionless advection occurs in steps on the order of a collision time, in contrast to MD which may require on the order of 102 time steps per collision time; likewise, collision events are processed in a stochastic manner in DSMC, in contrast to MD which tracks detailed trajectories of colliding particles. The result of this coarse graining is a description which is many orders of magnitude more computationally efficient than MD, but sacrifices atomic-level detail and precise representation of interparticle correlations. The method is described in Ref. [5]. An extension of this method, called Direct Simulation Automata (DSA), includes multiparticle collisions that make it suitable for the description of liquids and complex fluids; this is described in Ref. [6]. For a wider range of materials, including dense liquids and complex fluids, thermal fluctuations and viscous dissipation are among the essential emergent

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properties captured by MD. For example, these are the principle ingredients of a Langevin description of a colloidal suspension. In a physical system, with microscopically reversible particle orbits, these quantities are related by the Fluctuation–Dissipation Theorem of statistical physics. Dissipative Particle Dynamics (DPD) takes advantage of this to include these ingredients in a physically realistic fashion. It modifies the conservative forces of an MD simulation by introducing effective potentials, as well as fluctuating and frictional forces that represent the degrees of freedom that are lost by coarse-graining. The result is a description that is orders of magnitude more computationally efficient than MD, but which sacrifices the precise treatment of correlations and fluctuations, and requires the use of effective potentials. The DPD model is described in Ref. [7]. If one is willing to dispense with all representation of thermal fluctuations and multiparticle correlations, one may retain only the single-particle distribution function, as for example in the Boltzmann equation of kinetic theory. It was discovered in the late 1980’s that the Navier–Stokes limit of the Boltzmann description is surprizingly robust with respect to radical discretizations of the velocity space. In particular, it is possible to adopt a velocity space that consists only of a small discrete set of velocities, coincident with lattice vectors of a particular lattice. For certain choices of lattice and of collision operator, the resulting Boltzmann equation, which describes the transport of particles on a lattice with collisions localized to lattice sites, can be rigorously shown to give rise to Navier–Stokes behavior. These lattice Boltzmann models are described in Ref. [8]. Since their discovery, they have been extended to deal with compressible flow, adaptive mesh refinement on structured and unstructured grids, multiphase flow, and complex fluids. In a number of situations, the hydrodynamics of certain problems evolve in a wide range of length and time scales. If this range of scales is sufficiently wide such that no single description can be used, hybrid methods which combine more than one description can be used. The motivation for hybrid methods stems from the fact that, invariably, the “higher fidelity” description is also more computationally expensive and thus it becomes advantageous to limit its use only in the regions in which it is necessary. Clearly, hybrid methods in this respect make sense only when the “higher fidelity” description is required in small regions of space. Although hybrid methods coupling any of the methods described in this chapter can be envisioned, currently most effort has been focused towards the development of Navier–Stokes/MD and Navier–Stokes/DSMC hybrids. These are described in detail in Ref. [9]. The list of topics chosen for inclusion in this chapter is representative but not exhaustive. In particular, space limitations have precluded us from including much interesting and excellent work in the area of mesh genration, adaptive mesh refinement, and boundary element methods for the Navier–Stokes equations. Also missing are descriptions of certain mesoscale methods, such

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as lattice-gas automata and smoothed-particle hydrodynamics. Nevertheless, we feel that the topics included provide a representative cross section of this fast developing and exciting area of materials modeling research.

References [1] S. Sherwin and J. Peiro, “Finite difference, finite element and finite volume methods for partial differential equations,” Article 8.2, this volume. [2] G. Li, X. Jin, and N.R. Aluru, “Meshless methods for numerical solution of partialdifferential equations,” Article 8.3, this volume. [3] J. Li, “Basic molecular dynamics,” Article 2.8, this volume. [4] J. Koplik and J.R. Banavar, “Continuum deductions from molecular hydrodynamics,” Ann. Rev. Fluid Mech., 27, 257–292, 1995. [5] F.J. Alexander, “The direct simulation Monte Carlo method: going beyond continuum hydrodynamics,” Article 8.7, this volume. [6] T. Sakai and P.V. Coveney, “Discrete simulation automata: mesoscopic fluid models endowed with thermal fluctuations,” Article 8.5, this volume. [7] P. Espa˜nol, “Dissipative particle dynamics,” Article 8.6, this volume. [8] S. Succi, W.E, and E. Kaxiras, “Lattice Boltzmann methods for multiscale fluid problems,” Article 8.4, this volume. [9] H.S. Wijesinghe and N.G. Hadjiconstantinou, “Hybrid atomistic-continuum formulations for multiscale hydrodynamics,” Article 8.8, this volume.

8.2 FINITE DIFFERENCE, FINITE ELEMENT AND FINITE VOLUME METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS Joaquim Peir´o and Spencer Sherwin Department of Aeronautics, Imperial College, London, UK

There are three important steps in the computational modelling of any physical process: (i) problem definition, (ii) mathematical model, and (iii) computer simulation. The first natural step is to define an idealization of our problem of interest in terms of a set of relevant quantities which we would like to measure. In defining this idealization we expect to obtain a well-posed problem, this is one that has a unique solution for a given set of parameters. It might not always be possible to guarantee the fidelity of the idealization since, in some instances, the physical process is not totally understood. An example is the complex environment within a nuclear reactor where obtaining measurements is difficult. The second step of the modeling process is to represent our idealization of the physical reality by a mathematical model: the governing equations of the problem. These are available for many physical phenomena. For example, in fluid dynamics the Navier–Stokes equations are considered to be an accurate representation of the fluid motion. Analogously, the equations of elasticity in structural mechanics govern the deformation of a solid object due to applied external forces. These are complex general equations that are very difficult to solve both analytically and computationally. Therefore, we need to introduce simplifying assumptions to reduce the complexity of the mathematical model and make it amenable to either exact or numerical solution. For example, the irrotational (without vorticity) flow of an incompressible fluid is accurately represented by the Navier–Stokes equations but, if the effects of fluid viscosity are small, then Laplace’s equation of potential flow is a far more efficient description of the problem. 2415 S. Yip (ed.), Handbook of Materials Modeling, 2415–2446. c 2005 Springer. Printed in the Netherlands.

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After the selection of an appropriate mathematical model, together with suitable boundary and initial conditions, we can proceed to its solution. In this chapter we will consider the numerical solution of mathematical problems which are described by partial differential equations (PDEs). The three classical choices for the numerical solution of PDEs are the finite difference method (FDM), the finite element method (FEM) and the finite volume method (FVM). The FDM is the oldest and is based upon the application of a local Taylor expansion to approximate the differential equations. The FDM uses a topologically square network of lines to construct the discretization of the PDE. This is a potential bottleneck of the method when handling complex geometries in multiple dimensions. This issue motivated the use of an integral form of the PDEs and subsequently the development of the finite element and finite volume techniques. To provide a short introduction to these techniques we shall consider each type of discretization as applied to one-dimensional PDEs. This will not allow us to illustrate the geometric flexibility of the FEM and the FVM to their full extent, but we will be able to demonstrate some of the similarities between the methods and thereby highlight some of the relative advantages and disadvantages of each approach. For a more detailed understanding of the approaches we refer the reader to the section on suggested reading at the end of the chapter. The section is structured as follows. We start by introducing the concept of conservation laws and their differential representation as PDEs and the alternative integral forms. We next discusses the classification of partial differential equations: elliptic, parabolic, and hyperbolic. This classification is important since the type of PDE dictates the form of boundary and initial conditions required for the problem to be well-posed. It also permits in some cases, e.g., in hyperbolic equations, to identify suitable schemes to discretise the differential operators. The three types of discretisation: FDM, FEM and FVM are then discussed and applied to different types of PDEs. We then end our overview by discussing the numerical difficulties which can arise in the numerical solution of the different types of PDEs using the FDM and providing an introduction to the assessment of the stability of numerical schemes using a Fourier or Von Neumann analysis. Finally we note that, given the scientific background of the authors, the presentation has a bias towards fluid dynamics. However, we stress that the fundamental concepts presented in this chapter are generally applicable to continuum mechanics, both solids and fluids.

1.

Conservation Laws: Integral and Differential Forms

The governing equations of continuum mechanics representing the kinematic and mechanical behaviour of general bodies are commonly referred

Numerical methods for partial differential equations

2417

to as conservation laws. These are derived by invoking the conservation of mass and energy and the momentum equation (Newton’s law). Whilst they are equally applicable to solids and fluids, their differing behaviour is accounted for through the use of a different constitutive equation. The general principle behind the derivation of conservation laws is that the rate of change of u(x, t) within a volume V plus the flux of u through the boundary A is equal to the rate of production of u denoted by S(u, x, t). This can be written as d dt

u(x, t) dV +

V

f(u) · n dA −

A

S(u, x, t) dV = 0

(1)

V

which is referred to as the integral form of the conservation law. For a fixed (independent of t) volume and, under suitable conditions of smoothness of the intervening quantities, we can apply Gauss’ theorem

∇ · f dV =

V

f · n dA

A

to obtain

V

∂u + ∇ · f (u) − S dV = 0. ∂t

(2)

For the integral expression to be zero for any volume V , the integrand must be zero. This results in the strong or differential form of the equation ∂u + ∇ · f (u) − S = 0. ∂t

(3)

An alternative integral form can be obtained by the method of weighted residuals. Multiplying Eq. (3) by a weight function w(x) and integrating over the volume V we obtain V

∂u + ∇ · f (u) − S w(x) dV = 0. ∂t

(4)

If Eq. (4) is satisfied for any weight function w(x), then Eq. (4) is equivalent to the differential form (3). The smoothness requirements on f can be relaxed by applying the Gauss’ theorem to Eq. (4) to obtain V

∂u − S w(x) − f (u) · ∇w(x) dV + ∂t

f · n w(x) dA = 0.

A

(5) This is known as the weak form of the conservation law.

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J. Peir´o and S. Sherwin

Although the above formulation is more commonly used in fluid mechanics, similar formulations are also applied in structural mechanics. For instance, the well-known principle of virtual work for the static equilibrium of a body [1], is given by δW =

(∇ σ + f ) · δv dV = 0

V

where δW denotes the virtual work done by an arbitrary virtual velocity δv, σ is the stress tensor and f denotes the body force. The similarity with the method of weighted residuals (4) is evident.

2.

Model Equations and their Classification

In the following we will restrict ourselves to the analysis of one-dimensional conservation laws representing the transport of a scalar variable u(x, t) defined in the domain = {x, t : 0 ≤ x ≤ 1, 0 ≤ t ≤ T }. The convection–diffusionreaction equation is given by ∂u ∂u ∂ au − b −r u =s (6) L(u) = + ∂t ∂x ∂x together with appropriate boundary conditions at x = 0 and x = 1 to make the problem well-posed. In the above equation L(u) simply represents a linear differential operator. This equation can be recast in the form (3) with f (u) = au − ∂u/∂ x and S(u) = s + ru. It is linear if the coefficients a, b, r and s are functions of x and t, and non-linear if any of them depends on the solution, u. In what follows, we will use for convenience the convention that the presence of a subscript x or t under a expression indicates a derivative or partial derivative with respect to this variable, for example du ∂u (x); u t (x, t) = (x, t); dx ∂t Using this notation, Eq. (6) is re-written as u x (x) =

u x x (x, t) =

∂ 2u (x, t). ∂x2

u t + (au − bu x )x − ru = s.

2.1.

Elliptic Equations

The steady-state solution of Eq. (6) when advection and source terms are neglected, i.e., a=0 and s =0, is a function of x only and satisfies the Helmholtz equation (bu x )x + ru = 0.

(7)

Numerical methods for partial differential equations

2419

This equation is elliptic and its solution depends on two families of integration constants that are fixed by prescribing boundary conditions at the ends of the domain. One can either prescribe Dirichlet boundary conditions at both ends, e.g., u(0) = α0 and u(1) = α1 , or substitute one of them (or both if r=/ 0) by a Neumann boundary condition, e.g., u x (0) = g. Here α0 , α1 and g are known constant values. We note that if we introduce a perturbation into a Dirichlet boundary condition, e.g., u(0) = α0 + , we will observe an instantaneous modification to the solution throughout the domain. This is indicative of the elliptic nature of the problem.

2.2.

Parabolic Equations

Taking a = 0, r = 0 and s = 0 in our model, Eq. (6) leads to the heat or diffusion equation u t − (b u x )x = 0,

(8)

which is parabolic. In addition to appropriate boundary conditions of the form used for elliptic equations, we also require an initial condition at t = 0 of the form u(x, 0) = u 0 (x) where u 0 is a given function. If b is constant, this equation admits solutions of the form u(x, t) = Aeβt sin kx if β + k 2 b = 0. A notable feature of the solution is that it decays when b is positive as the exponent β < 0. The rate of decay is a function of b. The more diffusive the equation (i.e., larger b) the faster the decay of the solution is. In general the solution can be made up of many sine waves of different frequencies, i.e., a Fourier expansion of the form u(x, t) = Aeβt sin k x u(x, t) =

Am eβm t sin km x,

m

where Am and km represent the amplitude and the frequency of a Fourier mode, respectively. The decay of the solution depends on the Fourier contents of the initial data since βm = −km2 b. High frequencies decay at a faster rate than the low frequencies which physically means that the solution is being smoothed. This is illustrated in Fig. 1 which shows the time evolution of u(x, t) for an initial condition u 0 (x) = 20x for 0 ≤ x ≤ 1/2 and u 0 (x) = 20(1 − x) for 1/2 ≤ x ≤ 1. The solution shows a rapid smoothing of the slope discontinuity of the initial condition at x = 1/2. The presence of a positive diffusion (b > 0) physically results in a smoothing of the solution which stabilizes it. On the other hand, negative diffusion (b < 0) is de-stabilizing but most physical problems have positive diffusion.

2420

J. Peir´o and S. Sherwin u(x)

11 10

t0

9

t T

8

t 2T

7

t 3T t 4T

6 5

t 5T t 6T

4 3 2 1 0 0.0

0.5

1.0

x Figure 1. Rate of decay of the solution to the diffusion equation.

2.3.

Hyperbolic Equations

A classic example of hyperbolic equation is the linear advection equation u t + a u x = 0,

(9)

where a represents a constant velocity. The above equation is also clearly equivalent to Eq. (6) with b = r = s = 0. This hyperbolic equation also requires an initial condition, u(x, 0) = u 0 (x). The question of what boundary conditions are appropriate for this equation can be more easily be answered after considering its solution. It is easy to verify by substitution in (9) that the solution is given by u(x, t) = u 0 (x − at). This describes the propagation of the quantity u(x, t) moving with speed “a” in the x-direction as depicted in Fig. 2. The solution is constant along the characteristic line x − at = c with u(x, t) = u 0 (c). From the knowledge of the solution, we can appreciate that for a > 0 a boundary condition should be prescribed at x = 0, (e.g., u(0) = α0 ) where information is being fed into the solution domain. The value of the solution at x = 1 is determined by the initial conditions or the boundary condition at x = 0 and cannot, therefore, be prescribed. This simple argument shows that, in a hyperbolic problem, the selection of appropriate conditions at a boundary point depends on the solution at that point. If the velocity is negative, the previous treatment of the boundary conditions is reversed.

Numerical methods for partial differential equations

2421 Characteristic x at c

u (x,t ) t

x

u (x,0 )

x Figure 2. Solution of the linear advection equation.

The propagation velocity can also be a function of space, i.e., a = a(x) or even the same as the quantity being propagated, i.e., a = u(x, t). The choice a = u(x, t) leads to the non-linear inviscid Burgers’ equation u t + u u x = 0.

(10)

An analogous analysis to that used for the advection equation shows that u(x, t) is constant if we are moving with a local velocity also given by u(x, t). This means that some regions of the solution advance faster than other regions leading to the formation of sharp gradients. This is illustrated in Fig. 3. The initial velocity is represented by a triangular “zig-zag” wave. Peaks and troughs in the solution will advance, in opposite directions, with maximum speed. This will eventually lead to an overlap as depicted by the dotted line in Fig. 3. This results in a non-uniqueness of the solution which is obviously non-physical and to resolve this problem we must allow for the formation and propagation of discontinuities when two characteristics intersect (see Ref. [2] for further details).

3.

Numerical Schemes

There are many situations where obtaining an exact solution of a PDE is not possible and we have to resort to approximations in which the infinite set of values in the continuous solution is represented by a finite set of values referred to as the discrete solution. For simplicity we consider first the case of a function of one variable u(x). Given a set of points xi ; i = 1, . . . , N in the domain of definition of u(x), as

2422

J. Peir´o and S. Sherwin t

t3

t2

u u>0

t1

t 0

x

u 0 Figure 3. Formation of discontinuities in the Burgers’ equation.

ui ui 1

ui 1

x1

x i 1

xi

xi 1

xn

x

Ωi

xi

1 2

xi 12

Figure 4. Discretization of the domain.

shown in Fig. 4, the numerical solution that we are seeking is represented by a discrete set of function values {u 1 , . . . , u N } that approximate u at these points, i.e., u i ≈ u(xi ); i = 1, . . . , N . In what follows, and unless otherwise stated, we will assume that the points are equally spaced along the domain with a constant distance x = xi+1 − xi ; i = 1, . . . , N − 1. This way we will write u i+1 ≈ u(xi+1 ) = u(xi + x). This partition of the domain into smaller subdomains is referred to as a mesh or grid.

Numerical methods for partial differential equations

3.1.

2423

The Finite Difference Method (FDM)

This method is used to obtain numerical approximations of PDEs written in the strong form (3). The derivative of u(x) with respect to x can be defined as u(xi + x) − u(xi ) x→0 x u(xi ) − u(xi − x) = lim x→0 x u(xi + x) − u(xi − x) . = lim x→0 2x

u x |i = u x (xi ) = lim

(11)

All these expressions are mathematically equivalent, i.e., the approximation converges to the derivative as x → 0. If x is small but finite, the various terms in Eq. (11) can be used to obtain approximations of the derivate u x of the form u i+1 − u i x u i − u i−1 u x |i ≈ x u i+1 − u i−1 . u x |i ≈ 2x

u x |i ≈

(12) (13) (14)

The expressions (12)–(14) are referred to as forward, backward and centred finite difference approximations of u x |i , respectively. Obviously these approximations of the derivative are different.

3.1.1. Errors in the FDM The analysis of these approximations is performed by using Taylor expansions around the point xi . For instance, an approximation to u i+1 using n + 1 terms of a Taylor expansion around xi is given by

u i+1

x 2 dn u x n = u i + u x |i x + u x x |i + · · · + n 2 dx i n! dn+1 u ∗ x n+1 + n+1 (x ) . dx (n + 1)!

(15)

The underlined term is called the remainder with xi ≤ x ∗ ≤ xi+1 , and represents the error in the approximation if only the first n terms in the expansion are kept. Although the expression (15) is exact, the position x ∗ is unknown.

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J. Peir´o and S. Sherwin

To illustrate how this can be used to analyse finite difference approximations, consider the case of the forward difference approximation (12) and use the expansion (15) with n = 1 (two terms) to obtain x u i+1 − u i = u x |i + u x x (x ∗ ). x 2 We can now write the approximation of the derivative as u i+1 − u i + T x where T is given by u x |i =

(16)

(17)

x u x x (x ∗ ). (18) 2 The term T is referred to as the truncation error and is defined as the difference between the exact value and its numerical approximation. This term depends on x but also on u and its derivatives. For instance, if u(x) is a linear function then the finite difference approximation is exact and T = 0 since the second derivative is zero in (18). The order of a finite difference approximation is defined as the power p such that limx→0 (T /x p ) = γ =/ 0, where γ is a finite value. This is often written as T = O(x p ). For instance, for the forward difference approximation (12), we have T = O(x) and it is said to be first-order accurate ( p = 1). If we apply this method to the backward and centred finite difference approximations (13) and (14), respectively, we find that, for constant x, their errors are T = −

x u i − u i−1 + u x x (x ∗ ) ⇒ T = O(x) x 2 x 2 u i+1 − u i−1 − u x x x (x ) ⇒ T = O(x 2 ) u x |i = 2x 12 u x |i =

(19) (20)

with xi−1 ≤ x ∗ ≤ xi and xi−1 ≤ x ≤ xi+1 for Eqs. (19) and (20), respectively. This analysis is confirmed by the numerical results presented in Fig. 5 that displays, in logarithmic axes, the exact and truncation errors against x for the backward and the centred finite differences. Their respective truncation errors T are given by (19) and (20) calculated here, for lack of a better value, with x ∗ = x = xi . The exact error is calculated as the difference between the exact value of the derivative and its finite difference approximation. The slope of the lines are consistent with the order of the truncation error, i.e., 1:1 for the backward difference and 1:2 for the centred difference. The discrepancies between the exact and the numerical results for the smallest values of x are due to the use of finite precision computer arithmetic or round-off error. This issue and its implications are discussed in more detail in numerical analysis textbooks as in Ref. [3].

Numerical methods for partial differential equations

2425

1e 00 Backward FD Total Error Backward FD Truncation Error Centred FD Total Error Centred FD Truncation Error

1e 02 1e 04

1

1e 06

ε

1 2

1e 08

1 1e 10 1e 12 1e 14 1e 00

1e 02

1e 04

1e 06

1e 08

1e 10

1e 12

∆x

Figure 5.

Truncation and rounding errors in the finite difference approximation of derivatives.

3.1.2. Derivation of approximations using Taylor expansions The procedure described in the previous section can be easily transformed into a general method for deriving finite difference schemes. In general, we can obtain approximations to higher order derivatives by selecting an appropriate number of interpolation points that permits us to eliminate the highest term of the truncation error from the Taylor expansions. We will illustrate this with some examples. A more general description of this derivation can be found in Hirsch (1988). A second-order accurate finite difference approximation of the derivative at xi can be derived by considering the values of u at three points: xi−1 , xi and xi+1 . The approximation is constructed as a weighted average of these values {u i−1 , u i , u i+1 } such as u x |i ≈

αu i+1 + βu i + γ u i−1 . x

(21)

Using Taylor expansions around xi we can write x 2 x 3 u x x |i + u x x x |i + · · · 2 6 x 2 x 3 u x x |i − u x x x |i + · · · = u i − x u x |i + 2 6

u i+1 = u i + x u x |i +

(22)

u i−1

(23)

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J. Peir´o and S. Sherwin

Putting (22), (23) into (21) we get 1 αu i+1 + βu i + γ u i−1 = (α + β + γ ) u i + (α − γ ) u x |i x x x x 2 u x x |i + (α − γ ) u x x x |i + (α + γ ) 2 6 x 3 u x x x x |i + O(x 4 ) + (α + γ ) (24) 12 We require three independent conditions to calculate the three unknowns α, β and γ . To determine these we impose that the expression (24) is consistent with increasing orders of accuracy. If the solution is constant, the left-hand side of (24) should be zero. This requires the coefficient of (1/x)u i to be zero and therefore α+β +γ = 0. If the solution is linear, we must have α−γ =1 to match u x |i . Finally whilst the first two conditions are necessary for consistency of the approximation in this case we are free to choose the third condition. We can therefore select the coefficient of (x/2) u x x |i to be zero to improve the accuracy, which means α + γ = 0. Solving these three equations we find the values α = 1/2, β = 0 and γ = −(1/2) and recover the second-order accurate centred formula u x |i =

u i+1 − u i−1 + O(x 2 ). 2x

Other approximations can be obtained by selecting a different set of points, for instance, we could have also chosen three points on the side of xi , e.g., u i , u i−1 , u i−2 . The corresponding approximation is known as a one-sided formula. This is sometimes useful to impose boundary conditions on u x at the ends of the mesh.

3.1.3. Higher-order derivatives In general, we can derive an approximation of the second derivative using the Taylor expansion 1 1 αu i+1 + βu i + γ u i−1 u x |i = (α + β + γ ) 2 u i + (α − γ ) 2 x x x 1 x u x x x |i + (α + γ ) u x x |i + (α − γ ) 2 6 x 2 u x x x x |i + O(x 4 ). + (α + γ ) 12

(25)

Numerical methods for partial differential equations

2427

Using similar arguments to those of the previous section we impose 

α + β + γ = 0 α−γ =0 ⇒ α = γ = 1, β = −2.  α+γ =2

(26)

The first and second conditions require that there are no u or u x terms on the right-hand side of Eq. (25) whilst the third conditon ensures that the righthand side approximates the left-hand side as x tens to zero. The solution of Eq. (26) lead us to the second-order centred approximation u i+1 − 2u i + u i−1 + O(x 2 ). (27) x 2 The last term in the Taylor expansion (α − γ )xu x x x |i /6 has the same coefficient as the u x terms and cancels out to make the approximation second-order accurate. This cancellation does not occur if the points in the mesh are not equally spaced. The derivation of a general three point finite difference approximation with unevenly spaced points can also be obtained through Taylor series. We leave this as an exercise for the reader and proceed in the next section to derive a general form using an alternative method. u x x |i =

3.1.4. Finite differences through polynomial interpolation In this section we seek to approximate the values of u(x) and its derivatives by a polynomial P(x) at a given point xi . As way of an example we will derive similar expressions to the centred differences presented previously by considering an approximation involving the set of points {xi−1 , xi , xi+1 } and the corresponding values {u i−1 , u i , u i+1 }. The polynomial of minimum degree that satisfies P(xi−1 ) = u i−1 , P(xi ) = u i and P(xi+1 ) = u i+1 is the quadratic Lagrange polynomial (x − xi )(x − xi+1 ) (x − xi−1 )(x − xi+1 ) + ui (xi−1 − xi )(xi−1 − xi+1 ) (xi − xi−1 )(xi − xi+1 ) (x − xi−1 )(x − xi ) . + u i+1 (xi+1 − xi−1 )(xi+1 − xi )

P(x) = u i−1

(28)

We can now obtain an approximation of the derivative, u x |i ≈ Px (xi ) as (xi − xi+1 ) (xi − xi−1 ) + (xi − xi+1 ) + ui (xi−1 − xi )(xi−1 − xi+1 ) (xi − xi−1 )(xi − xi+1 ) (xi − xi−1 ) . (29) + u i+1 (xi+1 − xi−1 )(xi+1 − xi )

Px (xi ) = u i−1

If we take xi − xi−1 = xi+1 − xi = x, we recover the second-order accurate finite difference approximation (14) which is consistent with a quadratic

2428

J. Peir´o and S. Sherwin

interpolation. Similarly, for the second derivative we have Px x (xi ) =

2u i−1 2u i + (xi−1 − xi )(xi−1 − xi+1 ) (xi − xi−1 )(xi − xi+1 ) 2u i+1 + (xi+1 − xi−1 )(xi+1 − xi )

(30)

and, again, this approximation leads to the second-order centred finite difference (27) for a constant x. This result is general and the approximation via finite differences can be interpreted as a form of Lagrangian polynomial interpolation. The order of the interpolated polynomial is also the order of accuracy of the finite diference approximation using the same set of points. This is also consistent with the interpretation of a Taylor expansion as an interpolating polynomial.

3.1.5. Finite difference solution of PDEs We consider the FDM approximation to the solution of the elliptic equation u x x = s(x) in the region = {x : 0 ≤ x ≤ 1}. Discretizing the region using N points with constant mesh spacing x = (1/N − 1) or xi = (i − 1/N − 1), we consider two cases with different sets of boundary conditions: 1. u(0) = α1 and u(1) = α2 , and 2. u(0) = α1 and u x (1) = g. In both cases we adopt a centred finite approximation in the interior points of the form u i+1 − 2u i + u i−1 = si ; x 2

i = 2, . . . , N − 1.

(31)

The treatment of the first case is straightforward as the boundary conditions are easily specified as u 1 = α1 and u N = α2 . These two conditions together with the N − 2 equations (31) result in the linear system of N equations with N unknowns represented by            

1 0 ... 1 −2 1 0 ... 0 1 −2 1 0 ... .. .. .. . . . 0 ... 0 1 −2 1 0 ... 0 1 −2 0 ... 0

0 0 0



u1 u2 u3 .. .

         u 0    N−2  1  u N−1

1

uN





α1 x 2 s2 x 2 s3 .. .

          =       x 2 s N−2     x 2 s N−1

α2

      .     

Numerical methods for partial differential equations

2429

This matrix system can be written in abridged form as Au = s. The matrix A is non-singular and admits a unique solution u. This is the case for most discretizations of well-posed elliptic equations. In the second case the boundary condition u(0) = α1 is treated in the same way by setting u 1 = α1 . The treatment of the Neumann boundary condition u x (1) = g requires a more careful consideration. One possibility is to use a one-sided approximation of u x (1) to obtain u x (1) ≈

u N − u N−1 = g. x

(32)

This expression is only first-order accurate and thus inconsistent with the approximation used at the interior points. Given that the PDE is elliptic, this error could potentially reduce the global accuracy of the solution. The alternative is to use a second-order centred approximation u x (1) ≈

u N+1 − u N−1 = g. x

(33)

Here the value u N+1 is not available since it is not part of our discrete set of values but we could use the finite difference approximation at x N given by u N+1 − 2u N + u N−1 = sN x 2 and include the Neumann boundary condition (33) to obtain 1 u N − u N−1 = (gx − s N x 2 ). 2

(34)

It is easy to verify that the introduction of either of the Neumann boundary conditions (32) or (34) leads to non-symmetric matrices.

3.2.

Time Integration

In this section we address the problem of solving time-dependent PDEs in which the solution is a function of space and time u(x, t). Consider for instance the heat equation u t − bu x x = s(x)

in

= {x, t : 0 ≤ x ≤ 1, 0 ≤ t ≤ T }

with an initial condition u(x, 0) = u 0 (x) and time-dependent boundary conditions u(0, t) = α1 (t) and u(1, t) = α2 (t), where α1 and α2 are known

2430

J. Peir´o and S. Sherwin

functions of t. Assume, as before, a mesh or spatial discretization of the domain {x1 , . . . , x N }.

3.2.1. Method of lines In this technique we assign to our mesh a set of values that are functions of time u i (t) = u(xi , t); i = 1, . . . , N . Applying a centred discretization to the spatial derivative of u leads to a system of ordinary differential equations (ODEs) in the variable t given by b du i {u i−1 (t) − 2u i (t) + u i+1 (t)} + si ; = dt x 2

i = 2, . . . , N − 1

with u 1 = α1 (t) and u N = α2 (t). This can be written as 

u2





−2 1





 u2    u3        ..    .  +       u N−2     u N−1 1 −2

 u3   1 −2 1  b  d   ..   .. .. ..  . =  . . .  x 2  dt   u N−2   1 −2 1

u N−1



bα1 (t) s2 + x 2 s3 .. .



        s N−2  bα2 (t) 

s N−1 +

x 2

or in matrix form as du (t) = A u(t) + s(t). (35) dt Equation (35) is referred to as the semi-discrete form or the method of lines. This system can be solved by any method for the integration of initial-value problems [3]. The numerical stability of time integration schemes depends on the eigenvalues of the matrix A which results from the space discretization. For this example, the eigenvalues vary between 0 and −(4α/x 2 ) and this could make the system very stiff, i.e. with large differences in eigenvalues, as x → 0.

3.2.2. Finite differences in time The method of finite differences can be applied to time-dependent problems by considering an independent discretization of the solution u(x, t) in space and time. In addition to the spatial discretization {x1 , . . . , x N }, the discretization in time is represented by a sequence of times t 0 = 0 < · · · < t n < · · · < T . For simplicity we will assume constant intervals x and t in space and time, respectively. The discrete solution at a point will be represented by

Numerical methods for partial differential equations

2431

u ni ≈ u(xi , t n ) and the finite difference approximation of the time derivative follows the procedures previously described. For example, the forward difference in time is given by u t (x, t n ) ≈

u(x, t n+1 ) − u(x, t n ) t

and the backward difference in time is u t (x, t n+1 ) ≈

u(x, t n+1 ) − u(x, t n ) t

both of which are first-order accurate, i.e. T = O(t). Returning to the heat equation u t − bu x x = 0 and using a centred approximation in space, different schemes can be devised depending on the time at which the equation is discretized. For instance, the use of forward differences in time leads to − u ni u n+1 b n i u i−1 − 2u ni + u ni+1 . = 2 t x

(36)

This scheme is explicit as the values of the solution at time t n+1 are obtained directly from the corresponding (known) values at time t n . If we use backward differences in time, the resulting scheme is − u ni u n+1 b n+1 i n+1 n+1 = u − 2u + u i−1 i i+1 . t x 2

(37)

Here to obtain the values at t n+1 we must solve a tri-diagonal system of equations. This type of schemes are referred to as implicit schemes. The higher cost of the implicit schemes is compensated by a greater numerical stability with respect to the explicit schemes which are stable in general only for some combinations of x and t.

3.3.

Discretizations Based on the Integral Form

The FDM uses the strong or differential form of the governing equations. In the following, we introduce two alternative methods that use their integral form counterparts: the finite element and the finite volume methods. The use of integral formulations is advantageous as it provides a more natural treatment of Neumann boundary conditions as well as that of discontinuous source terms due to their reduced requirements on the regularity or smoothness of the solution. Moreover, they are better suited than the FDM to deal with complex geometries in multi-dimensional problems as the integral formulations do not rely in any special mesh structure.

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J. Peir´o and S. Sherwin

These methods use the integral form of the equation as the starting point of the discretization process. For example, if the strong form of the PDE is L(u) = s, the integral from is given by 1

L(u)w(x) dx =

0

1

sw(x) dx

(38)

0

where the choice of the weight function w(x) defines the type of scheme.

3.3.1. The finite element method (FEM) Here we discretize the region of interest = {x : 0 ≤ x ≤ 1} into N − 1 subdomains or elements i = {x : xi−1 ≤ x ≤ xi } and assume that the approximate solution is represented by u δ (x, t) =

N

u i (t)Ni (x)

i=1

where the set of functions Ni (x) is known as the expansion basis. Its support is defined as the set of points where Ni (x)=/ 0. If the support of Ni (x) is the whole interval, the method is called a spectral method. In the following we will use expansion bases with compact support which are piecewise continuous polynomials within each element as shown in Fig. 6. The global shape functions Ni (x) can be split within an element into two local contributions of the form shown in Fig. 7. These individual functions are referred to as the shape functions or trial functions.

3.3.2. Galerkin FEM In the Galerkin FEM method we set the weight function w(x) in Eq. (38) to be the same as the basis function Ni (x), i.e., w(x) = Ni (x). Consider again the elliptic equation L(u) = u x x = s(x) in the region with boundary conditions u(0) = α and u x (1) = g. Equation (38) becomes 1

w(x)u x x dx =

0

1

w(x)s(x) dx.

0

At this stage, it is convenient to integrate the left-hand side by parts to get the weak form −

1 0

wx u x dx + w(1) u x (1) − w(0) u x (0) =

1 0

w(x) s(x) dx.

(39)

Numerical methods for partial differential equations ui 1

u1

2433

ui ui 1 Ωi

x1

xi 1

xi

uN x i 1

xN

x

u1 x 1

x

.. . Ni (x)

ui x

1 x

.. . uN x

1 x

Figure 6. A piecewise linear approximation u δ (x, t) =

N

i=1 u i (t)Ni (x).

ui ui 1 Ωi

xi

ui

x i 1

Ni 1

Ni 1

xi

ui 1

x i 1

x

1

x i 1

Figure 7. Finite element expansion bases.

This is a common technique in the FEM because it reduces the smoothness requirements on u and it also makes the matrix of the discretized system symmetric. In two and three dimensions we would use Gauss’ divergence theorem to obtain a similar result. The application of the boundary conditions in the FEM deserves attention. The imposition of the Neumann boundary condition u x (1) = g is straightforward, we simply substitute the value in Eq. (39). This is a very natural way of imposing Neumann boundary conditions which also leads to symmetric

2434

J. Peir´o and S. Sherwin

matrices, unlike the FDM. The Dirichlet boundary condition u(0) = α can be applied by imposing u 1 = α and requiring that w(0) = 0. In general, we will impose that the weight functions w(x) are zero at the Dirichlet boundaries. N δ Letting u(x) ≈ u (x) = j =1 u j N j (x) and w(x) = Ni (x) then Eq. (39) becomes −

1

N dNi dN j uj (x) (x) dx = dx dx j =1

0

1

Ni (x) s(x) dx

(40)

0

for i =2, . . . , N . This represents a linear system of N − 1 equations with N − 1 unknowns: {u 2 , . . . , u N }. Let us proceed to calculate the integral terms corresponding to the i-th equation. We calculate the integrals in Eq. (40) as sums of integrals over the elements i . The basis functions have compact support, as shown in Fig. 6. Their value and their derivatives are different from zero only on the elements containing the node i, i.e.,  x − xi−1   xi−1 < x < xi    x i−1 Ni (x) =   xi+1 − x   xi < x < xi+1  xi  1   xi−1 < x < xi    x i−1

dNi (x) =  dx  −1    xi < x < xi+1 xi with xi−1 = xi − xi−1 and xi = xi+1 − xi . This means that the only integrals different from zero in (40) are xi

−

x i−1

dNi dNi−1 dNi + ui u i−1 dx dx dx

xi

=

Ni s dx +

x i−1

−

x i+1

xi

x i+1

Ni s dx

(41)

xi

The right-hand side of this equation expressed as xi

F= x i−1

x − xi−1 s(x) dx + xi−1

x i+1

xi

xi+1 − x s(x) dx xi

can be evaluated using a simple integration rule like the trapezium rule x i+1

xi

g(x) dx ≈

dNi dNi dNi+1 + u i+1 ui dx dx dx dx

g(xi ) + g(xi+1 ) xi 2

Numerical methods for partial differential equations and it becomes

F=

2435

xi xi−1 si . + 2 2

Performing the required operations in the left-hand side of Eq. (41) and including the calculated valued of F leads to the FEM discrete form of the equation as −

u i+1 − u i xi−1 + xi u i − u i−1 + = si . xi−1 xi 2

Here if we assume that xi−1 = xi = x then the equispaced approximation becomes u i+1 − 2u i + u i−1 = x si x which is identical to the finite difference formula. We note, however, that the general FE formulation did not require the assumption of an equispaced mesh. In general the evaluation of the integral terms in this formulation is more efficiently implemented by considering most operations in a standard element st = {−1 ≤ x ≤ 1} where a mapping is applied from the element i to the standard element st . For more details on the general formulation see Ref. [4].

3.3.3. Finite volume method (FVM) The integral form of the one-dimensional linear advection equation is given by Eq. (1) with f (u) = au and S = 0. Here the region of integration is taken to be a control volume i , associated with the point of coordinate xi , represented by xi− 1 ≤ x ≤ xi+ 1 , following the notation of Fig. 4, and the integral form is 2 2 written as x i+ 1

x i+ 1

2

u t dx +

x i− 1

2

f x (u) dx = 0.

(42)

x i− 1

2

2

This expression could also been obtained from the weighted residual form (4) by selecting a weight w(x) such that w(x) = 1 for xi− 1 ≤ x ≤ xi+ 1 and 2 2 w(x) = 0 elsewhere. The last term in Eq. (42) can be evaluated analytically to obtain x i+ 1

2

f x (u) dx = f u i+(1/2) − f u i−(1/2)

x i− 1 2

2436

J. Peir´o and S. Sherwin

and if we approximate the first integral using the mid-point rule we finally have the semi-discrete form

u t |i xi+ 1 − xi− 1 + f u i+ 1 − f u i− 1 = 0. 2

2

2

2

This approach produces a conservative scheme if the flux on the boundary of one cell equals the flux on the boundary of the adjacent cell. Conservative schemes are popular for the discretization of hyperbolic equations since, if they converge, they can be proven (Lax-Wendroff theorem) to converge to a weak solution of the conservation law.

3.3.4. Comparison of FVM and FDM To complete our comparison of the different techniques we consider the FVM discretization of the elliptic equation u x x = s. The FVM integral form of this equation over a control volume i = {xi− 1 ≤ x ≤ xi+ 1 } is 2

x i+ 1

2

x i+ 1

2

2

u x x dx = x i− 1

s dx. x i− 1

2

2

Evaluating exactly the left-hand side and approximating the right-hand side by the mid-point rule we obtain

u x xi+ 1 − u x xi− 1 = xi+ 1 − xi− 1 2

2

2

2

si .

(43)

If we approximate u(x) as a linear function between the mesh points i − 1 and i, we have u i − u i−1 u i+1 − u i , u x |i+ 1 ≈ , u x |i− 1 ≈ 2 2 xi − xi−1 xi+1 − xi and introducing these approximations into Eq. (43) we now have u i − u i−1 u i+1 − u i − = (xi+ 1 − xi− 1 ) si . 2 2 xi+1 − xi xi − xi−1 If the mesh is equispaced then this equation reduces to u i+1 − 2u i + u i−1 = x si , x which is the same as the FDM and FEM on an equispaced mesh. Once again we see the similarities that exist between these methods although some assumptions in the construction of the FVM have been made. FEM and FVM allow a more general approach to non-equispaced meshes (although this can also be done in the FDM). In two and three dimensions, curvature is more naturally dealt with in the FVM and FEM due to the integral nature of the equations used.

Numerical methods for partial differential equations

4.

2437

High Order Discretizations: Spectral Element/ p-Type Finite Elements

All of the approximations methods we have discussed this far have dealt with what is typically known as the h-type approximation. If h = x denotes the size of a finite difference spacing or finite elemental regions then convergence of the discrete approximation to the PDE is achieved by letting h → 0. An alternative method is to leave the mesh spacing fixed but to increase the polynomial order of the local approximation which is typically denoted by p or the p-type extension. We have already seen that higher order finite difference approximations can be derived by fitting polynomials through more grid points. The drawback of this approach is that the finite difference stencil gets larger as the order of the polynomial approximation increases. This can lead to difficulties when enforcing boundary conditions particularly in multiple dimensions. An alternative approach to deriving high-order finite differences is to use compact finite differences where a Pad´e approximation is used to approximate the derivatives. When using the finite element method in an integral formulation, it is possible to develop a compact high-order discretization by applying higher order polynomial expansions within every elemental region. So instead of using just a linear element in each piecewise approximation of Fig. 6 we can use a polynomial of order p. This technique is commonly known as p-type finite element in structural mechanics or the spectral element method in fluid mechanics. The choice of the polynomial has a strong influence on the numerical conditioning of the approximation and we note that the choice of an equi-spaced Lagrange polynomial is particularly bad for p > 5. The two most commonly used polynomial expansions are Lagrange polynomial based on the Gauss–Lobatto–Legendre quadratures points or the integral of the Legendre polynomials in combination with the linear finite element expansion. These two polynomial expansions are shown in Fig. 8. Although this method is more (a)

(b) 1

1

1

0

0

0

0

1

1

1

1

1

1

1

1

0

0

0

0

1

1

1

1

1

1

1

1

0

0

0

0

1

1

1

1

1

Figure 8. Shape of the fifth order ( p = 5) polynomial expansions typically used in (a) spectral element and (b) p-type finite element methods.

2438

J. Peir´o and S. Sherwin

involved to implement, the advantage is that for a smooth problem (i.e., one where the derivatives of the solution are well behaved) the computational cost increases algebraically whilst the error decreases exponentially fast. Further details on these methods can be found in Refs. [5, 6].

5.

Numerical Difficulties

The discretization of linear elliptic equations with either FD, FE or FV methods leads to non-singular systems of equations that can easily solved by standard methods of solution. This is not the case for time-dependent problems where numerical errors may grow unbounded for some discretization. This is perhaps better illustrated with some examples. Consider the parabolic problem represented by the diffusion equation u t − u x x = 0 with boundary conditions u(0) = u(1) = 0 solved using the scheme (36) with b = 1 and x = 0.1. The results obtained with t = 0.004 and 0.008 are depicted in Figs. 9(a) and (b), respectively. The numerical solution (b) corresponding to t = 0.008 is clearly unstable. A similar situation occurs in hyperbolic problems. Consider the onedimensional linear advection equation u t + au x = 0; with a > 0 and various explicit approximations, for instance the backward in space, or upwind, scheme is − u ni u n+1 u n − u ni−1 i +a i = 0 ⇒ u n+1 = (1 − σ )u ni + σ u ni−1 , i t x the forward in space, or downwind, scheme is u n − u ni − u ni u n+1 i + a i+1 =0 t x

⇒

(a)

u n+1 = (1 + σ )u ni − σ u ni+1 , i

(44)

(45)

(b)

0.3

0.3

t0.20 t0.24 t0.28 t0.32

0.2

t0.20 t0.24 t0.28 t0.32

0.2

0.1 u(x,t)

u(x,t)

0.1

0

0

0.1

0.1

0.2

0

0.2

0.4

0.6 x

0.8

1

0.2

0

0.2

0.4

0.6

0.8

1

x

Figure 9. Solution to the diffusion equation u t + u x x = 0 using a forward in time and centred in space finite difference discretization with x = 0.1 and (a) t = 0.004, and (b) t = 0.008. The numerical solution in (b) is clearly unstable.

Numerical methods for partial differential equations

2439

u(x,t)

 0   

u(x, 0) =

1 + 5x

 1 − 5x  

0

x ≤ −0.2 −0.2 ≤ x ≤ 0 0 ≤ x ≤ 0.2 x ≥ 0.2

a 1.0 0.0 0.2

0.2

x

Figure 10. A triangular wave as initial condition for the advection equation.

and, finally, the centred in space is given by u n − u ni−1 u n+1 − u ni i + a i+1 =0 t 2x

⇒

= u ni − u n+1 i

σ n (u − u ni−1 ) 2 i+1 (46)

where σ = (at/x) is known as the Courant number. We will see later that this number plays an important role in the stability of hyperbolic equations. Let us obtain the solution of u t + au x = 0 for all these schemes with the initial condition given in Fig. 10. As also indicated in Fig. 10, the exact solution is the propagation of this wave form to the right at a velocity a. Now we consider the solution of the three schemes at two different Courant numbers given by σ = 0.5 and 1.5. The results are presented in Fig. 11 and we observe that only the upwinded scheme when σ ≤ 1 gives a stable, although diffusive, solution. The centred scheme when σ = 0.5 appears almost stable but the oscillations grow in time leading to an unstable solution.

6.

Analysis of Numerical Schemes

We have seen that different parameters, such as the Courant number, can effect the stability of a numerical scheme. We would now like to set up a more rigorous framework to analyse a numerical scheme and we introduce the concepts of consistency, stability and Convergence of a numerical scheme.

6.1.

Consistency

A numerical scheme is consistent if the discrete numerical equation tends to the exact differential equation as the mesh size (represented by x and t) tends to zero.

2440

J. Peir´o and S. Sherwin 3

1 0.9

2 0.8 0.7

1

u(x,t)

u(x,t)

0.6 0.5

0

0.4 1

0.3 0.2

2

0.1 0

1

0.8

0.6

0.4

0.2

0

0.2

σ 0.5

0.4

0.6

0.8

3 1

1

1

0.8

0.6

0.4

0.2

0

0.2

0.4

0.6

0.8

1

0.8

0.6

0.4

0.2

0 x

0.2

0.4

0.6

0.8

1

0.4

0.6

0.8

1

σ 1.5

30

20

2

10 1 0

u(x,t)

0 u(x,t)

10 1

20

2

3

30

1

0.8

0.6

0.4

0.2

0 x

0.2

0.4

0.6

0.8

1

40 1

σ 1.5

σ 0.5 3

1.2 1

2

0.8 1

0

0.4 u(x,t)

u(x,t)

0.6

0.2

1

0 2 0.2 3

0.4 0.6 1

0.8

0.6

0.4

0.2

0

0.2

0.4

0.6

0.8

1

4 1

x

σ 0.5

0.8

0.6

0.4

0.2

0 x

0.2

σ 1.5

Figure 11. Numerical solution of the advection equation u t + au x = 0. Dashed lines: initial condition. Dotted lines: exact solution. Solid line: numerical solution.

Consider the centred in space and forward in time finite diference approximation to the linear advection equation u t + au x = 0 given by Eq. (46). Let us , u ni+1 and u ni−1 around (xi , t n ) as consider Taylor expansions of u n+1 i = u ni + t u t |ni + u n+1 i

t 2 u t t |ni + · · · 2

Numerical methods for partial differential equations

2441

x 2 x 3 u x x |ni + u x x x |ni + · · · 2 6 x 2 x 3 u ni−1 = u ni − x u x |ni + u x x |ni − u x x x |ni + · · · 2 6 Substituting these expansions into Eq. (46) and suitably re-arranging the terms we find that − u ni u n+1 u n − u ni−1 i + a i+1 − (u t + au x )|ni = T (47) t 2x where T is known as the truncation error of the approximation and is given by u ni+1 = u ni + x u x |ni +

t x 2 u t t |ni + au x x x |ni + O(t 2 , x 4 ). 2 6 The left-hand side of this equation will tend to zero as t and x tend to zero. This means that the numerical scheme (46) tends to the exact equation at point xi and time level t n and therefore this approximation is consistent. T =

6.2.

Stability

We have seen in the previous numerical examples that errors in numerical solutions can grow uncontrolled and render the solution meaningless. It is therefore sensible to require that the solution is stable, this is that the difference between the computed solution and the exact solution of the discrete equation should remain bounded as n → ∞ for a given x.

6.2.1. The Courant–Friedrichs–Lewy (CFL) condition This is a necessary condition for stability of explicit schemes devised by Courant, Friedrichs and Lewy in 1928. Recalling the theory of characteristics for hyperbolic systems, the domain of dependence of a PDE is the portion of the domain that influences the solution at a given point. For a scalar conservation law, it is the characteristic passing through the point, for instance, the line P Q in Fig. 12. The domain of dependence of a FD scheme is the set of points that affect the approximate solution at a given point. For the upwind scheme, the numerical domain of dependence is shown as a shaded region in Fig. 12. The CFL criterion states that a necessary condition for an explicit FD scheme to solve a hyperbolic PDE to be stable is that, for each mesh point, the domain of dependence of the FD approximation contains the domain of dependence of the PDE.

2442

J. Peir´o and S. Sherwin

(a)

(b) t

t

∆x

∆x

a∆t

Characteristic P

P

a∆t

∆t

∆t

x Q

x Q

Figure 12. Solution of the advection equation by the upwind scheme. Physical and numerical domains of dependence: (a) σ = (at/x) > 1, (b) σ ≤ 1.

For a Courant number σ = (at/x) greater than 1, changes at Q will affect values at P but the FD approximation cannot account for this. The CFL condition is necessary for stability of explicit schemes but it is not sufficient. For instance, in the previous schemes we have that the upwind FD scheme is stable if the CFL condition σ ≤ 1 is imposed. The downwind FD scheme does not satisfy the CFL condition and is unstable. However, the centred FD scheme is unstable even if σ ≤ 1.

6.2.2. Von Neumann (or Fourier) analysis of stability The stability of FD schemes for hyperbolic and parabolic PDEs can be analysed by the von Neumann or Fourier method. The idea behind the method is the following. As discussed previously the analytical solutions of the model diffusion equation u t − b u x x = 0 can be found in the form u(x, t) =

∞

eβm t e I km x

m=−∞

if βm + b km2 = 0. This solution involves a Fourier series in space and an expocomnential decay in time since βm ≤ 0 for b > 0. Here we have included the√ I km x = cos km x + I sin km x with I = −1, plex version of the Fourier series, e because this simplifies considerably later algebraic manipulations. To analyze the growth of different Fourier modes as they evolve under the numerical scheme we can consider each frequency separately, namely u(x, t) = eβm t e I km x .

Numerical methods for partial differential equations

2443

A discrete version of this equation is u ni = u(xi , t n ) = eβm t e I km xi . We can take, without loss of generality, xi = ix and t n = nt to obtain n

n

u ni = eβm nt e I km ix = eβm t e I km ix . The term e I km ix = cos(km ix) + I sin(km ix) is bounded and, therefore, any growth in the numerical solution will arise from the term G = eβm t , known as the amplification factor. Therefore the numerical method will be stable, or the numerical solution u ni bounded as n → ∞, if |G| ≤ 1 for solutions of the form u ni = G n e I km ix . We will now proceed to analyse, using the von Neummann method, the stability of some of the schemes discussed in the previous sections. Example 1 Consider the explicit scheme (36) for the diffusion equation u t − bu x x = 0 expressed here as u n+1 = λu ni−1 + (1 − 2λ)u ni + λu ni+1 ; i

λ=

bt . x 2

We assume u ni = G n e I km ix and substitute in the equation to get G = 1 + 2λ [cos(km x) − 1] . Stability requires |G| ≤ 1. Using −2 ≤ cos(km x) − 1 ≤ 0 we get 1 − 4λ ≤ G ≤ 1 and to satisfy the left inequality we impose −1 ≤ 1 − 4λ ≤ G

=⇒

1 λ≤ . 2

This means that for a given grid size x the maximum allowable timestep is t = (x 2 /2b). Example 2 Consider the implicit scheme (37) for the diffusion equation u t − bu x x = 0 expressed here as n+1 n + λu n+1 λu n+1 i−1 + −(1 + 2λ)u i i+1 = −u i ;

λ=

bt . x 2

The amplification factor is now G=

1 1 + λ(2 − cos βm )

and we have |G| < 1 for any βm if λ > 0. This scheme is therefore unconditionally stable for any x and t. This is obtained at the expense of solving a linear system of equations. However, there will still be restrictions on x

2444

J. Peir´o and S. Sherwin

and t based on the accuracy of the solution. The choice between an explicit or an implicit method is not always obvious and should be done based on the computer cost for achieving the required accuracy in a given problem. Example 3 Consider the upwind scheme for the linear advection equation u t + au x = 0 with a > 0 given by = (1 − σ )u ni + σ u ni−1 ; u n+1 i

σ=

at . x

Let us denote βm = km x and introduce the discrete Fourier expression in the upwind scheme to obtain G = (1 − σ ) + σ e−Iβm The stability condition requires |G| ≤ 1. Recall that G is a complex number G = ξ + I η so ξ = 1 − σ + σ cos βm ;

η = −σ sin βm

This represents a circle of radius σ centred at 1 − σ . The stability condition requires the locus of the points (ξ, η) to be interior to a unit circle ξ 2 + η2 ≤ 1. If σ < 0 the origin is outside the unit circle, 1 − σ > 1, and the scheme is unstable. If σ > 1 the back of the locus is outside the unit circle 1 − 2σ < 1 and it is also unstable. Therefore, for stability we require 0 ≤ σ ≤ 1, see Fig. 13. Example 4 The forward in time, centred in space scheme for the advection equation is given by = u ni − u n+1 i

σ n (u − u ni−1 ); 2 i+1

σ=

at . x

η

1 σ

1

G σ

ξ

Figure 13. Stability region of the upwind scheme.

Numerical methods for partial differential equations

2445

The introduction of the discrete Fourier solution leads to σ G = 1 − (e Iβm − e−Iβm ) = 1 − I σ sin βm 2 Here we have |G|2 = 1 + σ 2 sin2 βm > 1 always for σ =/ 0 and it is therefore unstable. We will require a different time integration scheme to make it stable.

6.3.

Convergence: Lax Equivalence Theorem

A scheme is said to be convergent if the difference between the computed solution and the exact solution of the PDE, i.e. the error E in = u ni − u(xi , t n ), vanishes as the mesh size is decreased. This is written as lim

x,t →0

|E in | = 0

for fixed values of xi and t n . This is the fundamental property to be sought from a numerical scheme but it is difficult to verify directly. On the other hand, consistency and stability are easily checked as shown in the previous sections. The main result that permits the assessment of the convergence of a scheme from the requirements of consistency and stability is the equivalence theorem of Lax stated here without proof: Stability is the necessary and sufficient condition for a consistent linear FD approximation to a well-posed linear initial-value problem to be convergent.

7.

Suggestions for Further Reading

The basics of the FDM are presented a very accessible form in Ref. [7]. More modern references are Refs. [8, 9]. An elementary introduction to the FVM can be consulted in the book by Versteeg and Malalasekera [10]. An in-depth treatment of the topic with an emphasis on hyperbolic problems can be found in the book by Leveque [2]. Two well established general references for the FEM are the books of Hughes [4] and Zienkiewicz and Taylor [11]. A presentation from the point of view of structural analysis can be consulted in Cook et al. [11] The application of p-type finite element for structural mechanics is dealt with in the book of Szabo and Babu˘ska [5]. The treatment of both p-type and spectral element methods in fluid mechanics can be found in the book by Karniadakis and Sherwin [6]. A comprehensive reference covering both FDM, FVM and FEM for fluid dynamics is the book by Hirsch [13]. These topics are also presented using a more mathematical perspective in the classical book by Quarteroni and Valli [14].

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References [1] J. Bonet and R. Wood, Nonlinear Continuum Mechanics for Finite Element Analysis. Cambridge University Press, 1997. [2] R. Leveque, Finite Volume Methods for Hyperbolic Problems, Cambridge University Press, 2002. [3] W. Cheney and D. Kincaid, Numerical Mathematics and Computing, 4th edn., Brooks/Cole Publishing Co., 1999. [4] T. Hughes, The Finite Element Method: Linear Static and Dynamic Finite Element Analysis, Dover Publishers, 2000. [5] B. Szabo and I. Babu˘ska, Finite Element Analysis, Wiley, 1991. [6] G.E. Karniadakis and S. Sherwin, Spectral/hp Element Methods for CFD, Oxford University Press, 1999. [7] G. Smith, Numerical Solution of Partial Differential Equations: Finite Diference Methods, Oxford University Press, 1985. [8] K. Morton and D. Mayers, Numerical Solution of Partial Differential Equations, Cambridge University Press, 1994. [9] J. Thomas, Numerical Partial Differential Equations: Finite Difference Methods, Springer-Verlag, 1995. [10] H. Versteeg and W. Malalasekera, An Introduction to Computational Fluid Dynamics. The Finite Volume Method, Longman Scientific & Technical, 1995. [11] O. Zienkiewicz and R. Taylor, The Finite Element Method: The Basis, vol. 1, Butterworth and Heinemann, 2000. [12] R. Cook, D. Malkus, and M. Plesha, Concepts and Applications of Finite Element Analysis, Wiley, 2001. [13] C. Hirsch, Numerical Computation of Internal and External Flows, vol. 1, Wiley, 1988. [14] A. Quarteroni and A. Valli, Numerical Approximation of Partial Differential Equations, Springer-Verlag, 1994.

8.3 MESHLESS METHODS FOR NUMERICAL SOLUTION OF PARTIAL DIFFERENTIAL EQUATIONS Gang Li∗ , Xiaozhong Jin† , and N.R. Aluru‡ Beckman Institute for Advanced Science and Technology, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA

A popular research topic in numerical methods recently has been the development of meshless methods as alternatives to the traditional finite element, finite volume, and finite difference methods. The traditional methods all require some connectivity knowledge a priori, such as the generation of a mesh, whereas the aim of meshless methods is to sprinkle only a set of points or nodes covering the computational domain, with no connectivity information required among the set of points. Multiphysics and multiscale analysis, which is a common requirement for microsystem technologies such as MEMS and Bio-MEMS, is radically simplified by meshless techniques as we deal with only nodes or points instead of a mesh. Meshless techniques are also appealing because of their potential in adaptive techniques, where a user can simply add more points in a particular region to obtain more accurate results. Extensive research has been conducted in the area of meshless methods in recent years (see [1–3] for an overview). Broadly defined, meshless methods contain two key steps: construction of meshless approximation functions and their derivatives and meshless discretization of the governing partial-differential equations. Least-squares [4–6, 8–13], kernel based [14–18] and radial basis function [19–23] approaches are three techniques that have gained considerable attention for construction of meshless approximation functions (see [26] for a detailed discussion on least-squares and kernel approximations). The meshless discretization of the partial-differential equations can be categorized into three classes: cell integration [5, 6, 12, 15, 16], local point integration [9, 24, 25], and point collocation [8, 10, 11, 17, 18, 20, 21]. Another class of important meshless methods are developed for boundaryonly analysis of partial differential equations. Boundary integral formulations 2447 S. Yip (ed.), Handbook of Materials Modeling, 2447–2474. c 2005 Springer. Printed in the Netherlands.

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[27], especially when combined with fast algorithms based on multipole expansions [28], Fast Fourier Transform (FFT) [29] and singular value decomposition (SVD) [30, 31], are powerful computational techniques for rapid analysis of exterior problems. Recently, several meshless methods for boundary-only analysis have been proposed in the literature. Some of the methods include the boundary node method [32, 33], the hybrid boundary node method [34] and the boundary knot method [35]. The boundary node method is a combined boundary integral/meshless approach for boundary only analysis of partial differential equations. A key difficulty in the boundary node method is the construction of interpolation functions using moving least-squares methods. For 2-D problems, where the boundary is 1-D, Cartesian coordinates cannot be used to construct interpolation functions (see [36] for a more detailed discussion). Instead, a cyclic coordinate is used in the moving least-squares approach to construct interpolation functions. For 3-D problems, where the boundary is 2-D, curvilinear coordinates are used to construct interpolation functions. The definition of these coordinates is not trivial for complex geometries. Recently, we have introduced a boundary cloud method (BCM) [36, 37], which is also a combined boundary-integral/scattered point approach for boundary only analysis of partial differential equations. The boundary cloud method employs a Hermite-type or a varying polynomial basis least-squares approach to construct interpolation functions to enable the direct use of Cartesian coordinates. Due to the length restriction, boundary-only methods are not discussed in this article. This paper summarizes the key developments in meshless methods and their implementation for interior problems. This material should serve as a starting point for the reader to venture into more advanced topics in meshless methods. The rest of the article is organized as follows: In Section 1, we introduce the general numerical procedures for solving partial differential equations. Meshless approximation and discretization approaches are discussed in Sections 2 and 3, respectively. Section 4 provides a brief summary of some existing meshless methods. The solution of an elasticity problem by using the finite cloud method is presented in Section 5. Section 6 concludes the article.

1.

Steps for Solving Partial Differential Equations: An Example

Typically, the physical behavior of an object or a system is described mathematically by partial differential equations. For example, as shown in Fig. 1, an irregular shaped 2-D plate is subjected to certain conditions of heat transfer: it has a temperature distribution of g(x, y) on the left part on its boundary (denoted as u ) and a heat flux distribution of h(x, y) on the remaining part of the boundary (denoted as q ). At steady state, the temperature at any point on

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u g(x,y)

Ω

Γu

∇2 u 0

Γq

u,n h(x,y) Figure 1. Heat conduction within a plate.

the plate is described by the steady-state heat conduction equation, i.e., ∇ 2u = 0

(1)

where u is the temperature. The temperature and the flux prescribed on the boundary are defined as boundary conditions. The prescribed temperature is called the Dirichlet or an essential boundary condition, i.e., u = g(x, y) on u

(2)

and the prescribed flux is called the Neumann or anatural boundary condition, i.e., ∂u = h(x, y) on q (3) ∂n where n is the outward normal to the boundary. The governing equations along with the Dirichlet and/or Neumann boundary conditions permit a unique temperature field on the plate. There are various numerical techniques available to solve the simple example considered above. Finite difference method (FDM) [38], finite element method (FEM) [39] and boundary element method (BEM) [27, 40] are the most popular methods for solving PDEs. Recently, meshless methods have been proposed and they have been successfully applied to solve many physical problems. Although the FDM, FEM, BEM and meshless methods are different in many aspects, all these methods contain three common steps: 1. Discretization of the domain 2. Approximation of the unknown function 3. Discretization of the governing equation and the boundary conditions.

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In the first step, a meshing process is often required for conventional methods such as finite element and boundary element methods. For objects with complex geometry, the meshing step could be complicated and time consuming. The key idea in meshless methods is to eliminate the meshing process to improve the efficiency. Many authors have shown that this can be done through meshless approximation and meshless discretization of the governing equation and the boundary conditions.

2.

Meshless Approximation

In meshless methods, as shown in Fig. 2, a physical domain is represented by a set of points. The points can be either structured or scattered as long as they cover the physical domain. An unknown function such as the temperature field in the domain is defined by the governing equation along with the appropriate boundary conditions. To obtain the solution numerically, one first needs to approximate the unknown function (e.g., temperature) at any location in the domain. There are several approaches for constructing the meshless approximation functions as will be discussed in the following sections.

2.1.

Weighted Least-squares Approximations

Assume we have a 2-D domain and denote the unknown function as u(x, y). In a weighted moving least-squares (MLS) approximation [41], the unknown function can be approximated by u a (x, y) =

m

a j (x, y) p j (x, y)

(4)

j =1

z

approximated unknown funtion x

weighting function

y support domain

Figure 2. Meshless approximation.

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2451

where a j (x, y) are the unknown coefficients, p j (x, y) are the basis functions and m is the number of basis functions. Polynomials are often used as the basis functions. For example, typical 2-D basis functions are given by linear basis: p(x, y) = [1 x y]T qudratic basis: p(x, y) = [1 x y x 2 x y y 2 ]T cubic basis: p(x, y) = [1 x y x 2 x y y 2 x 3 x 2 y x y 2 y 3 ]T

m=3 m=6 m = 10

(5)

The basic idea in weighted least-squares method is to minimize the weighted error between the approximation and the exact function. The weighted error is defined as E(u) =

NP i=1

=

NP

wi (x, y) u a (xi , yi ) − u i

2

 2 m wi (x, y)  a j (x, y) p j (xi , yi ) − u i 

i=1

(6)

j =1

where NP is the number of points, wi (x, y) is the weighting function centered at the point (x, y) and evaluated at the point (xi , yi ). If the weighting function is a constant, the weighted least-squares approach reduces to the classical least-squares approach. The weighting function is used in meshless methods for two reasons: first is to assign the relative importance of the error as a function of distance from the point (x, y); second, by choosing weighting functions whose value will vanish outside certain region, the approximation becomes local. The region where a weighting function has a non-zero value is called a support, a cloud or a domain of influence. The center point (x, y) is called a star point. As shown in Fig. 2, a typical weighting function is bellshaped. Several popular weighting functions used in meshless methods are listed below [1, 17, 42]:  2 3 2/3 − 4r + 4r

r ≤ 1/2 4/3 − 4r + 4r 2 − 4/3r 3 1/2 ≤ r ≤ 1  r >1 0 2 3 4 1 − 6r + 8r − 3r r ≤1 quartic spline: wi (r ) = r >1 0 2 2 e−(r/c) − e−(rmax /c) 0 ≤ r ≤ rmax Gaussian: wi (r) = 1 − e−(rmax /c)2 wi (r) 0 ≤ r ≤ rmax Modified Gaussian: wi (r) = 1 − wi (r) +

cubic spline:

wi (r) =

(7)

where r = r/rmax , r is the distance from the point (x, y) to the point (xi , yi ), i.e., r = |x − xi | = (x − xi )2 + (y − yi )2 and rmax is the radius of the support and c is called the dilation parameter which controls the sharpness of the weighting function. Typical value of c is between rmax /2 and rmax /3. The shape

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of the support, which defines the region where the weighting function is nonzero, can be arbitrary. The parameter in the modified Gaussian weighting is a small number to prevent the weighting function from being singular at the center. Multidimensional weighting functions can be constructed as products of one-dimensional weighting functions. For example, it is possible to define the 2-D weighting function as the product of two 1-D weighting functions in each direction, i.e., wi (x, y) = w(x − xi , y − yi ) = w(x − xi )w(y − yi )

(8)

In this case, the shape of the support/cloud is rectangular. The support size of the weighing function associated with a node i is selected to satisfy the following considerations [43]: 1. The support size should be large enough to cover a sufficient number of points and these points should occupy all the four quadrants of the star point (for boundary star points, the quadrants outside the domain are not considered). 2. The support size should be small enough to provide adequate local character to the approximation. Algorithm 1 gives a procedure for determining the support size for a given point i. Note that several other algorithms [8, 42, 44] are available for determining the support size. However, determining an “optimal” support size for a set of scattered points in meshless methods is still an open research topic. Algorithm 1 The implementation of determining support size rmax for a given point i 1: Select the nearest N E points in the domain (N E is typically several times of m). 2: For each selected point (x j , y j ), j = 1, 2, . . . , N E , compute the distance

3: 4: 5: 6:

from the point i, ρi j = (xi − x j )2 + (yi − y j )2 . Sort nodes in order of increasing ρi j and designate the first m nodes of the sort to a list. Draw a ray from the point i to each of the node in the list. If the angle between any two consecutive rays is greater than 90o , add the next node from the sort to the list and go to 4, if not, go to 6. Set rmax = Max(ρi j ) and multiply rmax by a scaling factor αs . The value of the scaling factor is provided by user.

Once the weighting function is selected, the unknown coefficients are computed by minimizing the weighted error (Eq. (6)) ∂E =0 ∂a j

j = 1, 2, . . . , m

(9)

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For a point (x, y), Eq. (9) leads to a linear system, which in matrix form is BW B T a = BW u

(10)

where a is the m × 1 coefficient vector, u is an NP × 1 unknown vector, B is an m × NP matrix,    

B=

p1 (x1 , y1 ) p2 (x1 , y1 ) .. .

p1 (x2 , y2 ) p2 (x2 , y2 ) .. .

pm (x1 , y1 )

··· ··· .. .

pm (x2 , y2 ) · · ·

p1 (x NP , y NP ) p2 (x NP , y NP ) .. .



   W =     

0 .. .

0 ··· w(x − x2 , y − y2 ) · · · .. .. . .

0

0

···

  , 

(11)

pm (x NP , y NP )

W is an NP × NP diagonal matrix defined as w(x − x1 ,  y−y ) 1 





     0   ..   .  w(x − x NP , 

0

(12)

y − y NP )

Rewriting M(x, y) = BW B T

(13)

C(x, y) = BW

(14)

and where the matrix M(x, y) of size m × m is called the moment matrix and from Eqs. (10), (13), and (14), the unknown coefficients can be written as a = M −1 Cu

(15)

Therefore, the approximation of the unknown function is given by u a (x, y) = pT (M −1 C)u

(16)

One can write Eq. (16) in short form as u a (x, y) = N(x, y)u =

NP

Ni (x, y)u i

(17)

i=1

Note that typically u i =/ u a (xi , yi ). In the moving least-squares method, the unknown coefficients a(x, y) are functions of (x, y). The approximation of the first derivatives of the unknown function is given by

T −1 (M −1 C) + pT (M −1 C ,k ) u u a,k (x, y) = p,k ,k C + M

= N ,k (x, y)u

(18)

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where k = 1 is the x-derivative or k = 2 is the y-derivative. One alternative to the moving least-squares approximation is the fixed least-squares (FLS) approximation [10, 13]. In FLS, the unknown function u(x, y) is approximated by u a (x, y) =

m

a j p j (x, y)

(19)

j =1

Note that a j in Eq. (19) is not a function of (x, y), i.e., the coefficients a j , j = 1, 2, . . . , m are constants for a given support or cloud. The weighting matrix W in the fixed least-squares approximation is 

w(x K − x1 ,  y −y ) 1  K

   W =     

0 .. . 0



0 ··· w(x K − x2 , y K − y2 ) · · · .. .. . . 0

···

     0   ..   .  w(x K − x NP , 

0

(20)

y K − y NP )

where (x K , y K ) is the center of the weighting function. Note that (x K , y K ) can be arbitrary and consequently the interpolation functions can be multivalued (see [18] for details). A unique set of interpolation functions can be constructed by fixing (x K , y K ) at the center point (x, y), i.e., when computing Ni (x, y), i = 1, 2, . . . , NP and its derivatives, the center of the weighting function is always fixed at (x, y). Therefore, it is clear that the moment matrix M and matrix C are not functions of (x, y) and the derivatives of the function are given by T u a,k (x, y) = p,k (M −1 C)u

k ∈ {1, 2}

(21)

Comparing Eqs. (18) and (21), it is easily shown that the cost of computing the derivatives in FLS is much less than that in MLS. However, it is reported in literature [6] that the approximated derivatives obtained from FLS may be less accurate. Algorithm 2 gives the procedure for computing the moving leastsquares approximation. In Algorithm 2, N C is the number of points in a cloud.

2.2.

Kernel Approximations

Consider again an arbitary 2-D domain, as shown in Fig. (2), and assume the domain is discretized into NP points or nodes. Then, for each node an approximation function is generated by constructing a cloud about that node (also referred to as a star node). A support/cloud is constructed by centering a kernel (i.e., the weighting function in the case of weighted least-squares

Meshless methods for numerical solution

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Algorithm 2 The implementation of moving least-squares approximation 1: Discretize the domain into NP points to cover the entire domain and its boundary . 2: for each point in the domain, (x j , y j ), do 3: Center the weighting function at the point. 4: Search the nearby domain and determine the support size to get N C points in the cloud by using Algorithm 1. 5: Compute the matrices M, C and their derivatives. 6: Compute the approximation function Ni (x j , y j ), i = 1, 2, . . . , N C and its derivatives by using Eqs. (16,18). 7: end for approximation) about the star point. The kernel is non-zero at the star point and at few other nodes that are in the vicinity of the star point. Two types of the kernel approximations can be considered: the reproducing kernel [15] and the fixed kernel [18]. In a 2-D reproducing kernel approach, the approximation u a (x, y) to the unknown function u(x, y) is given by u (x, y) =

a

C (x, y, s, t)w(x − s, y − t)u(s, t)ds dt

(22)

where w is the kernel function centered at (x, y). Typical kernel functions are given by Eq. (7). C (x, y, s, t) is the correction function which is given by C (x, y, s, t) = pT (x − s, y − t)c(x, y)

(23)

pT ={p1 , p2 , . . . , pm } is an m ×1 vector of basis functions. In two dimensions, a quadratic polynomial basis vector is given by

pT = 1, x − s, y − t, (x − s)2 , (x − s)(y − t), (y − t)2

m = 6 (24)

c(x, y) is an m × 1 vector of unknown correction function coefficients. The correction function coefficients are computed by satisfying the consistency conditions, i.e.,

pT (x − s, y − t)c(x, y)w(x − s, y − t) pi (s, t)ds dt = pi (x, y)

i = 1, 2, . . . , m

(25)

In discrete form, Eq. (25) can be written as NP

pT (x − x I , y − y I )c(x, y)w(x − x I , y − y I ) pi (x I , y I )VI

I =1

= pi (x, y)

i = 1, 2, . . . , m

(26)

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where NP is the number of points in the domain and VI is the nodal volume of node I . Typically a unit nodal volume of the nodes is assumed (see [18] for a discussion on nodal volumes). Equation (26) can be written in a matrix form as M c(x, y) = p(x, y)

(27)

where M is the m × m moment matrix and is a function of (x, y). The entries in the moment matrix are given by Mij =

NP

p j (x − x I , y − y I )w(x − x I , y − y I ) pi (x I , y I )VI

(28)

I =1

From Eq. (27), the unknown correction function coefficients are computed as c(x, y) = M −1 (x, y) p(x, y)

(29)

Substituting the correction function coefficients into Eq. (23) and employing a discrete approximation for Eq. (22), we obtain u a (x, y) =

NP

pT (x, y)M −T (x, y) p(x − x I , y − y I )

I =1

×w(x − x I , y − y I )VI uˆ I =

NP

N I (x, y)uˆ I

(30)

I =1

where uˆ I is the nodal parameter for node I , and N I (x, y) is the reproducing kernel meshless interpolation function. The first derivatives of the correction function coefficients can be computed from Eq. (27) M ,k (x, y)c(x, y) + M(x, y)c,k (x, y) = p,k (x, y)

(31)

c,k = M −1 ( p,k − M ,k c)

(32)

where k = 1 (for x-derivative) or k = 2 (for y-derivative). Thus, the first derivatives of the approximation can be written as

u a (x, y)

,k

= =

NP

(cT ),k pw + cT p,k w + cT pw,k VI uˆ I

I =1 NP

N I,k (x, y)uˆ I

(33)

I =1

Similarly, the second derivatives of the correction function coefficients are given by M ,mn (x, y)c(x, y) + M ,m (x, y)c,n (x, y) + M ,n (x, y)c,m (x, y) + M(x, y)c,mn (x, y) = p,mn (x, y)

(34)

c,mn = M −1 ( p,mn − M ,mn c − M ,m c,n − M ,n c,m )

(35)

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where m, n = x or y, and

u a (x, y)

,mn

=

NP

(cT ),mn pw + cT p,mn w + cT pw,mn + (cT ),m p,n w

I =1

+ (cT ),m pw,n + (cT ),n pw,m + (cT ),n p,m w

+ cT p,m w,n + cT p,n w,m VI uˆ I =

NP

N I,mn (x, y)uˆ I

(36)

I =1

The other major type of the kernel approximation is the fixed-kernel approximation. In a fixed-kernel approximation, the unknown function u(x, y) is approximated by

u (x, y) = C (x, y, x K − s, y K − t)w(x K − s, y K − t)u(s, t)ds dt (37) a

Note that in the fixed-kernel approximation, the center of the kernel is fixed at (x K , y K ) for a given cloud. Following the same procedure as in the reproducing kernel approximation, one can obtain the discrete form of the fixed kernel approximation u a (x, y) =

NP

pT (x, y)M −T (x K , y K ) p(x K − x I , y K − y I )

I =1

× w(x K − x I , y K − y I )VI uˆ I =

NP

N I (x, y)uˆ I

(38)

I =1

Since (x K , y K ) can be arbitrary in Eq. (38), the interpolation functions obtained by Eq. (38) are multivalued. A unique set of interpolation functions can be constructed by computing N I (x K , y K ), I = 1, 2, . . . , NP, when the kernel is centered at (x K , y K ) (see [18] for more details). Equation (38) shows that only the leading polynomial basis vector is a function of (x, y). Therefore, the derivatives of the interpolation functions can be computed simply by differentiating the polynomial basis vector in Eq. (38). For example, the first and second x derivatives are computed as:

N I ,x (x, y) = 0 1 0 2x y 0 M −T p(x K − x I , y K − y I ) ×w(x K − x I , y K − y I )VI

(39)

N I ,x x (x, y) = [0 0 0 2 0 0] M −T p(x K − x I , y K − y I ) ×w(x K − x I , y K − y I )VI

(40)

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It has been proved in [26] that, if the nodal volume is taken to be 1 for each node, the reproducing kernel approximation is mathematically equivalent to the moving least-squares approximation, and the fixed kernel approximation is equivalent to the fixed least-squares approximation. The algorithm to construct the approximation functions by using the fixed-kernel approximation method is given by Algorithm 3 The implementation of fixed-kernel approximation 1: Allocate NP points to cover the domain and its boundary . 2: for each point in the domain, (x j , y j ), do 3: Center the weighting function at the point. 4: Determine the support size to get N C points in the cloud by using Algorithm 1. 5: Compute the moment matrix M and the basis vector p(x, y). 6: Solve M c = p 7: Compute the approximation function N I (x j , y j ) I = 1, 2, . . . , N C and its derivatives by using Eqs. (38)–(40). 8: end for

2.3.

Radial Basis Approximation

In a radial basis meshless approximation, the approximation of an unknown function u(x, y) is written as a linear combination of NP radial functions [19], u a (x, y) =

NP

α j φ(x, y, x j , y j )

(41)

j =1

where NP is the number of points in the domain, φ is the radial basis function and α j , j = 1, 2, . . . , NP are the unknown coefficients. The unknown coefficients α1 , . . . , α NP can be computed by solving the governing equation by using either a collocation or a Galerkin method, which we will discuss in the following sections. The partial derivatives of the approximation function in a multidimensional space can be calculated as NP ∂ k φ(x, y, x j , y j ) ∂ k u a (x, y) = α j ∂ x a ∂ yb ∂ x a ∂ yb j =1

(42)

where a, b ∈ 0, 1, 2 and k = a + b. The multiquadrics [19–21] and thin-plate spline functions [45] are among the most popular radial basis functions. The multiquadrics radial basis function is given by φ(x, y, x j , y j ) = (x, x j ) = (r j ) = (r 2j + c2j )0.5

(43)

Meshless methods for numerical solution

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where r j = ||x − x j || is the Euclidian norm and c j is a constant. The value of c controls the shape of the basis function. The reciprocal multiquadrics radial basis function has the form (r) =

1 (r 2 + c2 )0.5

(44)

The thin-plate spline radial basis function is given by (r) = r 2m log r

(45)

where m is the order of the thin-plate spline. To avoid the singularity of the interpolation system, a polynomial function is often added to the approximation Eq. (41) [46]. The modified approximation is given by u a (x, y) =

NP

α j φ(x, y, x j , y j ) +

j =1

m

βi pi (x, y)

(46)

i=1

along with m additional constraints NP

α j pi (x j , y j ) = 0 i = 1, . . . , m

(47)

j =1

where βi , i = 1, 2, . . . , m are the unknown coefficients and p(x) are the polynomial basis functions as defined in Eq. (5). Equations (46) and (47) lead to a positive definite linear system which is gauranteed to be nonsingular. The radial basis function approximation shown above is global since the radial basis function are non-zero everywhere in the domain. It is required to solve a dense linear system to solve the unknown coefficients. The computational cost could be very high when the domain contains a large number of points. Recently, compactly supported radial basis functions have been proposed and applied to solve PDEs with largely reduced computational cost. For more details on compactly supported RBFs, please refer to [23].

3.

Discretization

As shown in Eqs. (17), (18), (30), (33), (36), (38) and (41), although each approximation method has a different way of computing the approximation functions, all the methods presented in previous sections represent u(x, y) in the same general form as u (x, y) = a

NP I =1

N I (x, y)uˆ I

(48)

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and the approximation of the derivtaives can also be written in the general form given by NP k ∂ N I (x, y) ∂ k u a (x, y) = uˆ I a b ∂x ∂y ∂ x a ∂ yb I =1

(49)

where a, b ∈ 0, 1, 2 and k = a + b. After the approximation functions are constructed, the next step is to compute the unknown coefficients in Eq. (48) by discretizing the governing equations. The meshless discretization techniques can be broadly classified into three categories: (1) point collocation; (2) cell integration and (3) local domain integration.

3.1.

Point Collocation

Point collocation is the simplest and the easiest way to discretize the governing equations. In a point collocation approach, the governing equations for a physical problem can be written in the following general form L (u(x, y)) = f (x, y) in G (u(x, y)) = g(x, y) on g H (u(x, y)) = h(x, y) on h

(50) (51) (52)

where is the domain, g is the portion of the boundary on which Dirichlet boundary conditions are specified, h is the portion of the boundary on which Neumann boundary conditions are specified and L , G and H are the differential, Dirichlet and Neumann operators, respectively. The boundary of the domain is given by = g ∪ h . After the meshless approximation functions are constructed, for each interior node, the point collocation technique simply substitutes the approximated unknown into the governing equations. For nodes with prescribed boundary conditions the approximate solution or the derivative of the approximate solution are substituted into the given Dirichlet and Neumann-type boundary conditions, respectively. Therefore, the discretized governing equations are given by L (u a ) = f (x, y) for points in G (u a ) = g(x, y) for points on g H (u a ) = h(x, y) for points on h

(53) (54) (55)

The point collocation approach gives rise to a linear system of equations of the form, K uˆ = F

(56)

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The solution of Eq. (56) provides the nodal parameters at the nodes. Once the nodal parameters are computed, the unknown solution at each node can be computed from Eq. (48). Let’s revisit the heat condution problem presented in Section 2 as an example. The governing equation is the steady-state heat conduction along with the appropriate boundary conditions stated in Eqs. (1)–(3). As shown in Fig. 3(a), the points are distributed over the domain and the boundary. Using the meshless approximation functions, the nodal temperature can be expressed by Eq. (48). If a node i is an interior node, the governing equation is satisfied, i.e.,

∇2

NP

N I (xi , yi )uˆ I

I =1

=

NP

(∇ 2 N I (xi , yi ))uˆ I = 0

(57)

I =1

If a node j is a boundary node with a Dirichlet boundary condition, we have NP

N I (x j , y j )uˆ I = g(x j , y j )

(58)

I =1

and if a node q is a boundary node with a Neumann boundary condition (heat flux at the boundary) ∂(

NP I =1

NP N I (xq , yq )uˆ I ) ∂(N I (xq , yq )) = uˆ I = h(xq , yq ) ∂n ∂n I =1

(59)

Assuming that there are ni interior points, nd Dirichlet boundary points, and nn Neumann boundary nodes (NP = ni + nd + nn) in the domain, the final

(a)

Governing equation

(b)

(c) background cells Ωs Γs

Dirichlet boundary condition

Neumann boundary condition

Ls Γsq

Figure 3. Meshlessdiscretization: (a) point collocation. (b) cell integration. (c) local domain integration.

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linear system takes the form 

∇ 2 N1 (x1 )  ∇ 2 N1 (x2 )  .  ..

  ∇ 2 N (x )  1 ni   N1 (xni+1 ) . . .   N1 (xni+nd )  ∂(N (x 1 ni+nd+1 )   ∂n .  ..   ∂(N (x )) 1

∂n

NP

∇ 2 N2 (x1 ) ∇ 2 N2 (x2 ) .. .

∇ 2 N2 (xni ) N2 (xni+1 ) .. .

··· ··· .. . ··· ··· .. .

N2 (xni+nd ) · · · ∂(N2 (xni+nd+1 )) ··· ∂n .. .. . . ∂(N2 (x NP )) ··· ∂n



  0           0           .   .      .    2    ∇ N NP (xni )      u ˆ 0 1         N NP (xni+1 )         g(x u ˆ )  2 ni+1 ..  . = .  .   . .   .  .          N NP (xni+nd )       ) u ˆ g(x NP ni+nd   ∂(N NP (xni+nd+1 ))         ∂n ) h(x   ni+nd+1    ..      .   . .      .      ∂(N NP (x NP ))

∇ 2 N NP (x1 ) ∇ 2 N NP (x2 ) .. .

∂n

h(x NP )

(60) where xni denotes the coordinates of node ni. Equation (60) can be solved ˆ The nodal temperature can be computed by to obtain the nodal parameters u. using Eq. (48). Algorithm 4 summarizes the key steps involved in the implementation of a point collocation method for linear problems. The point collocation steps are the same for nonlinear problems. However, a linear system such as Eq. (60) cannot be directly obtained by substituting the approximated unknown into the governing equation and the boundary conditions. A Newton’s method can be used to solve the discretized nonlinear system (please refer to [47] for detail). The point collocation method provides a simple, efficient and flexible meshless method for interior domain numerical analysis. Many meshless methods, such as the finite point method [10], the finite cloud method [18] and the h–p meshless cloud method [8], employ the point collocation technique to discretize the governing equation. However, there are several issues one needs to pay attention to improve the robustness of the point collocation method: 1. Ensuring the quality of clouds: We have found that, for scattered point distributions, the quality of the clouds is directly related to the numerical error in the solution. When the point distribution is highly scattered, it is likely that certain stability conditions, namely the positivity conditions (see [42] for details), could be violated for certain clouds. For this reason, the modified Gaussian, cubic or quartic inverse distance functions [42] are better choices for the kernel/weighting function in point collocation. In [42], we have proposed quantitative criteria to measure the cloud quality and approaches to ensure the satisfaction of the positivity conditions for 1-D and 2-D problems. However, for really bad point distributions, it could be difficult to satisfy the positivity conditions and modification of the point distribution may be necessary.

Meshless methods for numerical solution

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Algorithm 4 Implementation of a point collocation technique for numerical solutions of PDEs 1: Compute the meshless approximations for the unknown solution 2: for each point in the domain do 3: if the node is in the interior of the domain then 4: substitute the approximation of the solution into the governing equation 5: else if the node is on the Dirichlet boundary then 6: substitute the approximation of the solution into the Dirichlet boundary condition 7: else if the node is on the Neumann boundary then 8: substitute the approximation of the solution into the Neumann boundary condition 9: end if 10: assemble the corresponding row of Eq. (60) 11: end for 12: Solve Eq. (60) to obtain the nodal parameters 13: Compute the solution by using Eq. (48)

2. Improving the accuracy for high aspect-ratio clouds: Like the conventional finite difference and finite element methods, large error could occur with the collocation meshless methods when the point distribution has a high aspect ratio (i.e. anisotropic cloud). Further investigation is needed to deal with the high aspect ratio problem.

3.2.

Cell Integration

Another approach to discretize the governing equation is the Galerkin method. The Galerkin approach is based on the weak form of the governing equations. The weak form can be obtained by minimizing the weighted residual of the governing equation. For the heat condution problem, a weak form of the governing equation can be written as

w ∇ u d + 2

v (u − g(x, y)) d = 0

(61)

u

where w and v are the test functions for the governing equation and the Dirichlet boundary condition, respectively. Note that the second integral in Eq. (61) is used to enforce the Dirichlet boundary condition. By applying the

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divergence theorem and imposing the natural boundary condition, Eq. (61) can b written as u

∂u wd + ∂n

q

∂u wd − ∂n

u ,i w,i d +

v (u − g(x, y)) d = 0

u

(62) The approximation for the unknown function is given by the meshless approximation (Eq. (48)) and the normal derivative of the unknown function can be computed by !

NP ∂ NI ∂ NI ∂u a = nx + n y uˆ I ∂n ∂x ∂y I =1

(63)

Denoting ∂ NI ∂ NI nx + ny ∂x ∂y

I =

(64)

The normal derivative of the unknown function can be rewritten as N ∂u = I uˆ I ∂n I =1

(65)

We choose the test functions w and v by w= v=

NP I =1 NP

N I uˆ I

(66)

I uˆ I

(67)

I =1

Subtituting the approximations into the weak form, we obtain NP I =1

" NP

uˆ I

N I,i N J,i duˆ J −

J =1

=

NP I =1

uˆ I

" q

NP J =1

u

N I h(x, y) d −

N I J d uˆ J − u

NP J =1

#

I g(x, y) d

#

I N J d uˆ J

u

(68)

Meshless methods for numerical solution

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Equation (68) can be simplified as   NP    N I,i N J,i d − N I J d − I N J d  uˆ J J =1

=

u

N I h(x, y) d −

q

u

I g(x, y) d

(69)

u

In matrix form

K − G − G T uˆ = h − g

(70)

where the entries of the coefficient matrix and the right hand side vector are given by

K IJ =

N I,i N J,i d

(71)

N I J d

(72)

N I h(x, y)d

(73)

I g(x, y)d

(74)

G IJ = u

hI = q

gI = u

As shown in Eqs. (71)–(74), the entries in the matrices and the right hand side vector are integrals over the domain or over the boundary. Since there is no mesh available to compute the various integrals, one approach is to use a background cell structure as shown in Fig. 3(b). The integrations are computed by appropriately summing over the cells and using Gauss quadrature in each cell. The implementation of cell integration is summarized in Algorithm 5. In a cell integration approach, the approximation order is reduced, i.e., for a second order PDE, there is no need to compute the second derivatives of the approximation functions. However, the cell integration approach requires background cells and the treatment of the boundary cells is not straightforward. Element-free Galerkin method [6], partition of unity finite element method [12], diffuse element method [5] and reproducing kernel particle method [15] are among the meshless methods using cell integration technique for discretizating the governing equation.

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Algorithm 5 Implementation of cell integration technique [48] 1: Compute the meshless approximations 2: Generate the background cells which cover the domain. 3: for each cell C i do 4: for each quadrature points x Q in the cell do 5: if the quadrature point is inside the physical domain then 6: Check all nodes in the cell Ci and surrounding cells to determine the nodes x I in the domain of influence of x Q 7: if x I − x Q does not intersect the boundary segment then 8: Compute the N I (x Q ) and N I,i (x Q ) at the quadrature point. 9: Evaluate contributions to the integrals. 10: Assemble contributions to the coefficient matrix. 11: end if 12: end if 13: end for 14: end for 15: Solve Eq. (77) to obtain the nodal parameters 16: Compute the solution by using Eq. (48)

3.3.

Local Domain Integration

Another method for discretizing the governing equation is based on the concept of local domain integration [9]. In the local domain integration method, the global domain is covered by local subdomains, as shown in Fig. 3(c). The local domains can be of arbitrary shape (typically circles or squares are convenient for integration) and can overlap with each other. In the heat conduction example, for a given node, a generalized local weak form over the node’s subdomain s can be written as s

v ∇ u d − α 2

v u − u b d = 0

(75)

su

where su = ∂s ∩ u is the intersection of the boundary of s and the global Dirichlet boundary. For nodes near or on the global boundary, ∂s = s + Łs . s is a part of the local domain boundary which is also located on the global boundary. Łs is the remaining part of the local boundary which is inside the global domain. α 1 is a penalty parameter used to impose the Dirichlet boundary conditions.

Meshless methods for numerical solution

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By applying the divergence theorem and imposing the Neumann boundary condition, for any local domain s , we obtain the local weak form

∂u v d + ∂n

Ls

−

su

∂u v d + ∂n

u ,k v ,k d − α

s

h(x, y)v d sq

v (u − g(x, y)) d = 0

(76)

su

in which sq is the intersection of the boundary of s and the global Neumann boundary. For a sub-domain located entirely within the global domain, there is no intersection between ∂s and , the integrals over su and sq vanish. In order to simplify the above equation, one can deliberately select a test function v such that it vanishes over ∂s . This can be easily accomplished by using the weighting function in the meshless approximations as also the test function, with the support of the weighting function set to be the size of the corresponding local domain s . In this way, the test function vanishes on the boundary of the local domain. By substituting the test function and the meshless approximation of the unknown (Eq. (48)) into the local domain weak form (Eq. (76)), we obain the matrix form K uˆ = f

(77)

where

Ki j = si

N j,k v i,k d + α

sui

N j v i d −

N j,n v i d

(78)

sui

and

fi = sqi

h(x, y)v i d + α

g(x, y)v i d

(79)

sui

where si , sui and sqi are the domain and boundary for the local domain i. The integrations in Eqs. (78) and (79) can be computed within each local domain by using Gauss quadrature. The implementation of the local integration can be carried out as summarized in Algorithm 6. Meshless methods based on local domain integration include the meshless local Petrov–Galerkin method [9] and the method of finite spheres [24].

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Algorithm 6 Implementation of the local domain integration technique 1: Compute the meshless approximations for the unknown solution 2: for each node (x i , yi ) do 3: Determine the local sub-domain s and its corresponding local boundary ∂s 4: Determine Gaussian quadrature points x Q in s and on ∂s 5: for each quadrature points x Q in the local domain do 6: Compute the Ni (x Q ) and Ni, j (x Q ) at the quadrature point x Q . 7: Evaluate contributions to the integrals. 8: Assemble contributions to the coefficient matrix. 9: end for 10: end for 11: Solve Eq. (77) to obtain the nodal parameters 12: Compute the solution by using Eq. (48)

4.

Summary of Meshless Methods

In this paper, we have introduced several approaches to construct the meshless approximations and three approaches to discretize the governing equations. Many meshless methods published in the literature can be viewed as different combinations of the approximation and discretization approaches introduced in the previous sections. Table 1 lists the popular methods with their approximation and discretization components.

Table 1. The catagory of meshless methods Point collocation

Cell integration Galerkin

Local domain integration Galerkin

Moving leastSquares

Finite point method [10]

Element-free Galerkin method [6], partition of unity finite element method [12]

Meshless local Petrov-Galerkin method [3], method of finite spheres [24]

Fixed leastsquares

Geleralized finite difference method [7] h − p meshless cloud method [8], finite point method [10]

Diffuse element method [5]

Reproducing

Finite cloud method [18]

Repeoducing kernel

kernel

particle method [15]

Fixed kernel

Finite cloud method [18]

Radial basis

Many

Many

Meshless methods for numerical solution

5.

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Example: Finite Cloud Method for Solving Linear Elasticity Problems

As shown in Fig. 4, an elastic plate containing three holes and a notch is subjected to a uniform pressure at its right edge [49]. We solve this problem by using the finite cloud method to demonstrate the effectiveness of the meshless method. To show the accuracy of the solution, the problem is solved by both the finite element method by using ANSYS and the finite cloud method. We construct the FCM discretizations by employing the same set of FEM nodes. For two-dimensional elasticity, there are two unknowns associated with each node in the domain, namely the displacements in the x and y directions. The governing equations assuming zero body force, can be rewritten as the Navier–Cauchy equations of elasticity  1 ∂   ∇ 2u +    1 − 2ν ∂ x      ∇ 2v +

∂ 1 1 − 2ν ∂ y

!

∂u ∂v + ∂x ∂y ∂u ∂v + ∂x ∂y

=0 (80)

!

=0

with

ν =

  

ν

 

ν 1+ν

for plane strain (81) for plane stress

where ν is the Poisson’s ratio. In this paper we consider the plane stress situation. In the finite cloud method, the first step is to construct the fixed kernel approximation for the displacements u and v by using Algorithm 3. In this example, the cloud size is set for each node to cover 25 neighboring nodes. 200

100

150 q

75

30

5

100

30

75

5

E 20

250 120

75 55

100

95

Thickness 1

115

Figure 4. Plate with holes.

υ 0.3 q 1.0

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The 2-D version of the modified Gaussian weighting function (Eq. (7)) is used as the kernel. After the approximation funcitons are computed, a point collocation approach is used for discretizing the governing equation and the boundary conditions by using Algorithm 4 to obtain the solution of the displacements. Figure 5 shows the deformed shape obtained by the FEM code ANSYS. The FEM mesh consists of 4474 nodes. All the 4474 ANSYS nodes are taken as the points in the FCM simulation. The deformed shapes obtained by FCM are shown in Fig. 6. The results obtained from the FEM and FCM agree with each other quite well and the difference of the maximum displacement is within 1%. Figure 7 shows a quantitative comparison of the computed σx x stress on the surfaces of the holes obtained from the two methods. The results

FEM solution

Figure 5. Deformed shape obtained by the finite element method.

400 FCM solution 350 300 250 200 150 100 50 0 0

100

200

300

400

500

Figure 6. Deformed shapeobtained by the finite cloud method.

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5 FEM (ANSYS) FCM 4

3

σxx

θ 2

1

0

1

0

10

20

30

40

50

60

70

80

90

100

θ (degree)

Figure 7. Results comparionfor σx x at the lower left circular boundary.

show very good agreement and demonstrate that the FCM approach provides accurate results for problems with complex geometries.

Remarks: 1. The construction of approximation functions is more expensive in meshless methods compared to the cost associated with construction of interpolation functions in FEM. The integration cost in Galerkin meshless methods is more expensive. Galerkin meshless methods can be a few times slower (typically more than five times) than FEM [25]. 2. Collocation meshless methods are much faster since no numerical integrations are involved. However, they may need more points and their robustness needs to be addressed [42]. 3. Meshless methods introduce a lot of flexibility. One needs to sprinkle only a set of points or nodes covering the computational domain as shown in Fig. 6, with no connectivity information required among the set of points. This property is very appealing because of its potential in adaptive techniques, where a user can simply add more points in a particular region to obtain more accurate results.

References [1] T. Belytschko, Y. Krongauz, D. Organ, M. Fleming, and P. Krysl, “Meshless methods: an overview and recent developments,” Comput. Methods Appl. Mech. Engrg., 139, 3–47, 1996.

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[2] S. Li and W.K. Liu, “Meshfree and particle methods and their applications,” Appl. Mech. Rev., 55, 1–34, 2002. [3] S.N. Atluri, The Meshless Local Petrov–Galerkin (MLPG) Method, Tech Science Press, 2002. [4] P. Lancaster and K. Salkauskas, “Surface generated by moving least squares methods,” Math. Comput., 37, 141–158, 1981. [5] B. Nayroles, G. Touzot, and P. Villon, “Generalizing the finite element method: diffuse approximation and diffuse elements,” Comput. Mech., 10, 307–318, 1992. [6] T. Belytschko, Y.Y. Lu, and L. Gu, “Element free galerkin methods,” Int. J. Numer. Methods Eng., 37, 229–256, 1994. [7] T.J. Liszka and J. Orkisz, “The finite difference method at arbitrary irregular grids and its application in applied mechanics,” Comput. Struct., 11, 83–95, 1980. [8] T.J. Liszka, C.A. Duarte, and W.W. Tworzydlo, “hp-meshless cloud method,” Comput. Methods Appl. Mech. Eng., 139, 263–288, 1996. [9] S.N. Atluri and T. Zhu, “A new meshless local Petrov–Galerkin (MLPG) approach in computational mechanics,” Comput. Mech., 22, 117–127, 1998. [10] E. O˜nate, S. Idelsohn, O.C. Zienkiewicz, and R.L. Taylor, “A finite point method in computational mechanics. Applications to convective transport and fluid flow,” Int. J. Numer. Methods Eng., 39, 3839–3866, 1996. [11] E. O˜nate, S. Idelsohn, O.C. Zienkiewicz, R.L. Taylor, and C. Sacco, “A stabilized finite point method for analysis of fluid mechanics problems,” Comput. Methods Appl. Mech. Eng., 139, 315–346, 1996. [12] I. Babuska and J.M. Melenk, “The partition of unity method,” Int. J. Numer. Meth. Eng., 40, 727–758, 1997. [13] P. Breitkopf, A. Rassineux, G. Touzot, and P. Villon, “Explicit form and efficient computation of MLS shape functions and their derivatives,” Int. J. Numer. Methods Eng., 48(3), 451–466, 2000. [14] J.J. Monaghan, “Smoothed particle hydrodynamics,”Annu. Rev. Astron. Astrophys., 30, 543–574, 1992. [15] W.K. Liu, S. Jun, S. Li, J. Adee, and T. Belytschko, “Reproducing Kernel particle methods for structural dynamics,” Int. J. Numer. Methods Eng., 38, 1655–1679, 1995. [16] J.-S. Chen, C. Pan, C. Wu, and W.K. Liu, “Reproducing Kernel particle methods for large deformation analysis of non-linear structures,” Comput. Methods Appl. Mech. Eng., 139, 195–227, 1996. [17] N.R. Aluru, “A point collocation method based on reproducing Kernel approximations,” Int. J. Numer. Methods Eng., 47, 1083–1121, 2000. [18] N.R. Aluru and G. Li, “Finite cloud method: a true meshless technique based on a fixed reproducing Kernel approximation,” Int. J. Numer. Methods Eng., 50(10), 10, 2373–2410, 2001. [19] R.L. Hardy, “Multiquadric equations for topography and other irregular surfaces,” J. Geophys. Res., 176, 1905–1915, 1971. [20] E.J. Kansa, “Multiquadrics – a scattered data approximation scheme with applications to computational fluid dynamics – I, surface approximations and partial derivative estimates,” Comp. Math. Appl., 19, 127–145, 1990. [21] E.J. Kansa, “Multiquadrics – a scattered data approximation scheme with applications to computational fluid dynamics – II, solutions to parabolic, hyperbolic and elliptic partial differential equations,” Comp. Math. Appl., 19, 147–161, 1990. [22] M.A. Golberg and C.S. Chen, “A bibliography on radial basis function approximation,” Boundary Elements Comm., 7, 155–163, 1996.

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[23] H. Wendland, “Piecewise polynomial, positive definite and compactly supported radial functions of minial degree,” Adv. Comput. Math., 4, 389–396, 1995. [24] S. De and K.J. Bathe, “The method of finite spheres,” Comput. Mech., 25, 329–345, 2000. [25] S. De and K.J. Bathe, “Towards an efficient meshless computational technique: the method of finite spheres,” Eng. Comput., 18, 170–192, 2001. [26] X. Jin, G. Li, and N.R. Aluru, “On the equivalence between least-squares and Kernel approximation in meshless methods,” CMES: Comput. Model. Eng. Sci., 2(4), 447– 462, 2001. [27] J.H. Kane, Boundary Element Analysis in Engineering Continuum Mechanics, Prentice-Hall, 1994. [28] L. Greengard and V. Rokhlin, “A fast algorithm for particle simulations,” J. Comput. Phys., 73(2), 325–348, 1987. [29] J.R. Phillips and J.K. White, “A precorrected-FFT method for electrostatic analysis of complicated 3-D structures,” IEEE Transact. on Comput.-Aided Des. of Integrated Circuits Sys., 16(10), 1059–1072, 1997. [30] S. Kapur and D.E. Long, “I E S 3 : a fast integral equation solver for efficient 3-dimensional extraction,” IEEE Computer Aided Design, 1997, Digest of Technical Papers 1997, IEE/ACM International Conference, 448–455, 1997. [31] V. Shrivastava and N.R. Aluru, “A fast boundary cloud method for exterior 2-D electrostatics,” Int. J. Numer. Methods Eng., 56(2), 239–260, 2003. [32] Y.X. Mukherjee and S. Mukherjee, “The boundary node method for potential problems,” Int. J. Numer. Methods Eng., 40, 797–815, 1997. [33] M.K. Chati and S. Mukherjee, “The boundary node method for three-dimensional problems in potential theory,” Int. J. Numer. Methods Eng., 47, 1523–1547, 2000. [34] J. Zhang, Z. Yao, and H. Li, “A hybrid boundary node method,” Int. J. Numer. Methods Eng., 53(4), 751–763, 2002. [35] W. Chen, “Symmetric boundary knot method,” Eng. Anal. Boundary Elements, 26(6), 489–494, 2002. [36] G. Li and N.R. Aluru, “Boundary cloud method: a combined scattered point/boundary integral approach for boundary-only analysis,” Comput. Methods Appl. Mech. Eng., 191, (21–22), 2337–2370, 2002. [37] G. Li and N.R. Aluru, “A boundary cloud method with a cloud-by-cloud polynomial basis,” Eng. Anal. Boundary Elements, 27(1), 57–71, 2003. [38] G.E. Forsythe and W.R. Wasow, Finite Difference Methods for Partial Differential Equations, Wiley, 1960. [39] T.J.R. Hughes, The Finite Element Method, Prentice-Hall, 1987. [40] C.A. Brebbia and J. Dominguez, Boundary Elements An Introductory Course, McGraw-Hill, 1989. [41] K. Salkauskas and P. Lancaster, Curve and Surface Fitting, Elsevier, 1986. [42] X. Jin, G. Li, and N.R. Aluru, “Positivity conditions in meshless collocation methods,” Comput. Methods Appl. Mech. Eng., 193, 1171–1202, 2004. [43] W.K. Liu, S. Li, and T. Belytschko, “Moving least-square reproducing kernel methods (I) methodology and convergence,” Comput. Methods Appl. Mech. Eng., 143, 113– 154, 1997. [44] P.S. Jensen, “Finite difference techniques for variable grids,” Comput. Struct., 2, 17– 29, 1972. [45] G.E. Fasshauer, “Solving differential equations with radial basis functions: multilevel methods and smoothing,” Adv. Comput. Math., 11, 139–159, 1999.

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8.4 LATTICE BOLTZMANN METHODS FOR MULTISCALE FLUID PROBLEMS Sauro Succi1, Weinan E2 , and Efthimios Kaxiras3 1

Istituto Applicazioni Calcolo, National Research Council, viale del Policlinico, 137, 00161, Rome, Italy 2 Department of Mathematics, Princeton University, Princeton, NJ 08544-1000, USA 3 Department of Physics, Harvard University, Cambridge, MA 02138, USA

1.

Introduction

Complex interdisciplinary phenomena, such as drug design, crackpropagation, heterogeneous catalysis, turbulent combustion and many others, raise a growing demand of simulational methods capable of handling the simultaneous interaction of multiple space and time scales. Computational schemes aimed at such type of complex applications often involve multiple levels of physical and mathematical description, and are consequently referred to as to multiphysics methods [1–3]. The opportunity for multiphysics methods arises whenever single-level methods, say molecular dynamics and partial differential equations of continuum mechanics, expand their range of scales to the point where overlap becomes possible. In order to realize this multiphysics potential specific efforts must be directed towards the development of robust and efficient interfaces dealing with “hand-shaking” regions where the exchange of information between the different schemes takes place. Two-level schemes combing atomistic and continuum methods for crack propagation in solids or strong shock fronts in rarefied gases have made their appearance in the early 90s. More recently, three-level schemes for crack dynamics, combining finite-element treatment of continuum mechanics far away from the crack with molecular dynamics treatment of atomic motion in the near-crack region and a quantum mechanical description of bond-snapping in the crack tip have been demonstrated. These methods represent concrete instances of composite algorithms which put in place seamless interfaces between the different mathematical models associated with different physical levels of description, say continuum and atomistic. An alternative approach is to explore methods 2475 S. Yip (ed.), Handbook of Materials Modeling, 2475–2486. c 2005 Springer. Printed in the Netherlands.

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that can host multiple levels of description, say atomistic, kinetic, and fluid, within the same mathematical framework. A potential candidate is the lattice Boltzmann equation (LBE) method. The LBE is a minimal form of Boltzmann kinetic equation in which all details of molecular motion are removed except those that are strictly needed to recover hydrodynamic behavior at the macroscopic scale (mass-momentum and energy conservation) [4, 5]. The result is an elegant and simple equation for the discrete distribution function f i ( x , t) describing the probability to find a particle at lattice site x at time t with speed v. LBE has potential to combine the power of continuum methods with the geometrical flexibility of atomistic methods. However, as multidisciplinary problems of increasing complexity are tackled, it is evident that significant upgrades are called for, both in terms of extending the range of scales accessible by LBE itself and in terms of coupling LBE downwards/upwards with micro/macroscopic methods. In the sequel, we shall offer a cursory view of both these research directions. Before proceeding further, a short review of the basic ideas behind LBE theory is in order.

2.

Lattice Boltzmann Scheme: Basic Theory

The lattice Boltzmann equation is based on the idea of moving pseudoparticles along prescribed directions on a discrete lattice (the discrete particle speeds define the lattice connectivity). At each lattice site, these pseudoparticles undergo collisional events designed in such a way as to conserve the basic mass, momentum and energy principles which lie at the heart of fluid behavior. Historically, LBE was generated in response to the major problems of its ancestor, the lattice gas cellular automaton, namely statistical noise, high viscosity, and exponential complexity of the collision operator with increasing number of speeds [6, 7]. A few years later, its mathematical connections with model kinetic equations of continuum theory have also been clarified [8]. The most popular, although not necessarily the most efficient, form of lattice Boltzmann equation (Lattice BGK, for Bhatnagar, Gross, Krook) reads as follows [9]

x + ci t, t + t) − f i ( x , t) − ωt f i − f ie ( x , t) + Fi t, f i ( →

(1)

where f i ( x , t) = f ( x , v = ci , t), i = 1,b, is the discrete one-body distribution function moving along the lattice direction defined by discrete speed ci. At the left hand side, we recognize the streaming operator of the Boltzmann equation, ∂t f + v · ∇ f, advanced in discrete time from t to t + t, along the characteristics xi = ci t. The right hand side represents the collisional operator in the form of single-time relaxation to the local equilibrium f ie · Finally, the effect of an external force, Fi , is also included. In order to recover fluid-dynamic

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behavior, the set of discrete speeds must guarantee the basic symmetries of fluid equations, namely mass, momentum and energy conservation, as well as rotational invariance. Only a limited subclass of lattices qualifies. A popular choice in three-dimensional space is the nineteen-speed lattice, consisting of one speed-zero (c = 0) particle sitting on the center of the cell, six speed-one (c = 1) particles√connecting to the face centers of the cell, and twelve particles with speed c = 2, connecting the center of the cell with edge centers. The local equilibrium is usually taken in the form of a quadratic expansion of a Maxwellian





uu · ci ci − cs2 I u · ci e  , f i = ρωi 1 + 2 + cs 2cs4

(2)

i /ρ the flow speed. Here where ρ = i f i the fluid density, and u = = i fi c 2 cs is the lattice sound speed defined by the condition cs I = i ωi ci ci , where I denotes the unit tensor. Finally, ωi is a set of lattice-dependent weights normalized to unity. For athermal flows, the lattice sound speed is a constant of order one (cs2 = 1/3 for the 19-speed lattice of Fig. 1). Local equilibria obey the following conservation relations (mass made unity for convenience):

f ie = ρ,

(3)

f ie ci = ρ u,

(4)

i

i

f ie ci ci = ρ uu + cs2 I .

i

Figure 1. The D3Q19 lattice.

(5)

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Using linear transport theory, in the limit of long-wavelengths as compared to particle mean free path, (small-Knudsen number) and low fluid speed as compared to the sound speed (low-Mach number), the fluid density and speed are shown to obey the Navier-Stokes equations for a quasi-incompressible fluid (with no external force for simplicity) ∂t ρ + divρ u = 0,

(6)

u + ( u )T + λdiv uI , ∂t ρ u + divρ uu = − ∇ P + div µ(

(7)

where P = pcs2 is the fluid pressure, and µ = ρu is the dynamic viscosity, and λ is the bulk viscosity (this latter term can be neglected to all practical purposes since we deal with quasi-incompressible fluids). Note that, according to the above relation, the LBE fluid obeys an ideal equation of state, as it belongs to a system of molecules with no potential energy. Potential energy effects can be introduced via a self-consistent force Fi , but in this work we shall not deal with such non-ideal gas aspects. The kinematic viscosity of the LBE fluid turns out to be:

ν=

cs2

1 τ − t 2

x 2 . t

(8)

The term τ ≡ 1/ω is the relaxation time around local equilibria, while the factor –1/2 is a genuine lattice effect which stems from second order spatial derivatives in the Taylor expansion of the discrete streaming operator. It is fortunate that such a purely numerical effect can be reabsorbed into a physical (negative) viscosity. In particular, by choosing ωt = 2 − , very small viscosities of order O() (in lattice units) can be achieved, corresponding to the very challenging regime of fluid turbulence [10]. Main assets of LBE are: • • • •

mathematical simplicity; physical flexibility; easy implementation of complex boundary conditions; excellent amenability to parallel processing.

Mathematical simplicity is related to the fact that, at variance with the Navier-Stokes equations in which non-linearity and non-locality are lumped into a single term, u∇ u, in LBE the non-local term (streaming) is linear and the non-linear term (the local equilibrium) is local. This disentangling proves beneficial from both the analytical and computational point of views. Physical flexibility relates to opportunity of accomodating additional physics via generalizations of the local equilibria and/or the external source Fi , such as to include the effects of additional fields interacting with the fluid.

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Easy implementation of complex boundary conditions results from the fact that the most common hydrodynamic boundary conditions, such as prescribed speed at solid boundaries, or prescribed pressure at fluid oulets, can be imposed in terms of elementary mechanical operations on the discrete distributions. However, in the presence of curved boundaries, i.e., boundaries which do not fit into the lattice sites, the boundary procedure may become considerably more involved. This represents one of the most active research topic in the field. It must be pointed out that in addition to fluid density and pressure, LBE also carries along the momentum flux tensor, whose equilibrium part corresponds to the fluid pressure. As a result, LBE does not need to solve the Poisson problem to compute the pressure distribution corresponding to a given flow configuration. This is a significant advantage as compared to explicit finite-difference schemes for incompressible flows. The price to pay is an extra-amount of information as compared to a hydrodynamic approach. For instance, in two dimensions, the most popular LBE requires nine populations (one rest particle, four nearest-neighbors and four next-to-nearest neighbors) to be contrasted with only three hydrodynamic fields (density, two velocity components). On the other hand, since LBE populations always stream “upwind” (from x to x + ci t, only one time level needs to be stored, which saves a factor two over hydrodynamic representations. As per efficiency on parallel computers, the key is again the locality of the collision operator which can be advanced concurrently at each lattice site independently of all others. Owing to these highlights, LBE has been used for more than 10 years for the simulation of a large variety of flows, including flows in porous media, turbulence, and complex flows with phase transitions, to name but a few. Multiscale applications, on the other hand, have appeared only recently, as we shall discuss in the sequel.

3.

Multiscale Lattice Boltzmann

Multiscale versions of LBE were first proposed by Filippova and Haenel [11] in the form of a LBE working on locally embedded grids, namely regular grids in which the lattice spacing is locally refined or coarsened, typically in steps of two for practical purposes. The same option was available since even longer in commercial versions of LB methods [12]. In the sequel, we shall briefly outline the main elements of multiscale LBE theory on locally embedded cartesian grids.

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3.1.

Basics of the Multiscale LB Method

The starting point of multiscale LBE theory is the lattice BGK equation (1). Grid-refinement is performed by introducing an n-times finer grid with spacing: δx =

x t , δt = , n n

The kinematic viscosity on the coarse lattice is given by Eq. (8) from which we see that in order to achieve the same viscosity on both coarse and fine grids, the relaxation parameter in the fine grid has to be rescaled as follows

τn = nτ1 1 −

n − 1 t/2 , n τ1

(9)

where rn and τ1 ≡ τ are the relaxation parameters on the n times-refined and on the original coarse grids, respectively n = 2l after l levels of grid-refinement). Next, we need to set up the interface conditions controlling the exchange of information between the coarse and fine grids. The guiding requirement is the continuity of hydrodynamic quantities (density, flow speed) and of their fluxes. Since hydrodynamic quantities are microscopically conserved, the corresponding interface conditions simply consists in setting the local equilibria in the fine grid equal to those in the coarse one. The fluxes, however, do not correspond to any microscopic invariant, and consequently their continuity implies requirements on the non-equilibrium component of the discrete distribution function. Therefore, the first step of the interface procedure consists in splitting the discrete distribution function into an equilibrium and non-equilibrium components: f i = f ie + f ine .

(10)

Upon expanding the left hand side of the LBE equation (1) to first order in at, the non-equilibrium component reads as

f ine = −τ [∂t + cia ∂a ] f ie + O K n 2 ,

(11)

where the latin index a runs over spatial dimensions and repeated indices are summed upon. This is second-order accurate in the Knudsen number K n = x/L, where L is a typical macroscopic scale of the flow. In the low-frequency limit t/τ ∼ K n 2 , the time derivative can be neglected, and by combining the above relation with continuity of the hydrodynamic variables at the interface between the two grids, one obtains the following scaling relations between the coarse and fine grid populations

f i = F˜ie + F˜i − F˜ie −1 ,

Fi = f ie + ( f i − f ie ) ,

(12) (13)

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where capital means coarse-grid, prime means post-collision, and tilde stands for interpolation from the coarse grid. In the above,

=n

(τ1 − t) · (τn − t)

The basic one-step algorithm reads as follows: 1. Advance (Stream, and Collide) F on the coarse grain grid. 2. For all subcycles k = 0, l, . . . , n − 1 do: a. Interpolate F on the interface coarse-to-fine grid. b. Scale F to f via (12) on the interface coarse-to-fine grid. c. Advance (Stream and Collide) f on the fine-grain grid. 3. Scale back f to F via (13) on the interface of the fine-to-coarse grid. Step 1 applies to all nodes in the coarse grid, bulk and interface, Steps 2a and 2b apply to interface nodes which belong only to the fine grid, Step 2c applies to bulk nodes of the fine grid, and Step 3 applies to interface nodes which belong to both coarse and fine grids. It is noted that becomes singular at τn = t, corresponding to n = (t/2)/(τ1 − t/2) = cs2 t/2ν (see Eq. (8)). For high-Reynolds applications, in which v is of the order of ∼10−3 or less (in units of the original lattice), the above singularity is of no practical concern, for it would be met only after hundred levels refinement. For low-Reynolds flow applications, however, this flaw needs to be cured. To this purpose, a more general approach that avoids the singularity has been recently developed by Dupuis [13]. These authors show that by defining the scale transformations between the coarse and fine grain populations before they collide, the singularity disappears ( = n(τ1 )/(τn )). In practice, this means that, at variance with Filippova’s model, the collision operator is applied also to the interface nodes which belong to the fine grid only.

4.

Multiscale LBE Applications

To date, Multiscale LBEs have been applied mainly to macroscopic turbulent flows [14, 15]. Here, however, we focus our attention to microscale problems of more direct relevance to material science applications.

4.1.

Microscale Flows with Chemical Reactions

The LBE couples easily to finite difference/volume methods for continuum parial differential equations. Distinctive features of LBE in this context

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are: (1) Use of very-small time-steps, (2) Geometrical flexibility. Item (1) refers to the fact that since LBE is an explicit method ticking at the particle speed, not the fluid one, it advances in much smaller time-steps than usual fluid-dynamics methods, typically a factor ten. (The flip side, is that a large number of time-step is required in long-time evolutions.) As an example, take a millimetric flow with, say 100 grid points per side, yielding a mesh spacing dx =10 µm. Assuming a sound speed of the order of 300 m/s, we obtain a timestep of the order of dt= 30 ns. Such a small time-step permits to handle relatively fast reactions without going to implicit time stepping, thus avoiding the solution of large systems of algebraic equations. Item (2) is especially suited to heterogeneous catalysis since the simplicity of particle trajectories permits to describe fairly irregular geometries and boundary conditions. Because of these two points, LBE is currently being used to simulate reactive flows over microscopically corrugated surfaces, an application of great interest for the design of chemical traps, catalytic converters and related devices [16, 17] (Fig. 2). These problems are genuinely multiphysics, since they involve a series of hydrodynamic and chemical time-scales. The major control parameters are the Reynolds number Re = U d/ν, the Peclet number Pe = Ud/D, and the Damkohler number Da = d 2 /Dτc . In the above, U and d are typical flow speed and size, D is the mass diffusivity of the chemical species and τc is a typical chemical reaction time-scale. Depending on various physical and geometrical parameters, a wide separation of these time-scales can arise. In general, the LBE time-step is sufficiently small to resolve all the relevant time-scales.

Figure 2. A multiscale computation of a flow in a microscopic restriction of a catalytic converter. Local flow gradients may lead to significant enhancements of the fluid-wall mass transfer, with corresponding effects on the chemical reactivity of the device. Note that three levels of refinement are used.

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Whenever faster time-scales develop, e.g., fast chemical reactions, the chemical processes are sub-cycled, i.e., advanced in multiple steps each with the smallest time-scale, until completion of a single LBE step [18].

4.2.

Nanoscale Flows

When the size of the micro/nanoscopic flow becomes comparable to the molecular mean free path, the Knudsen number is no longer small, and the whole fluid picture becomes questionable. A fundamental question then arises as to whether LBE can be more than a “Navier-Stokes solver in disguise”, namely capture genuinely kinetic information not available at the fluid-dynamic level. Mathematically, this possibility stems from the fact that – as already observed – discrete populations f i consistently outnumber the set of hydrodynamic observables, so that the excess-variables are potentially available to carry non-hydrodynamic information. This would represent a very significant advance, for it would show that LBE can be used as a tool for computational kinetic theory, beyond fluid dynamics. Nonetheless, a few numerical simulations of LBE microflows in microscopic electro-mechanical systems (MEMS) seem to indicate that standard LBE can capture some genuinely kinetic features of rarefied gas dynamics, such as slip motion at solid walls [19]. LBE schemes for nanoflow applications will certainly require new types of boundary conditions. A simple way to accomodate slip motion within LBE is to allow a fraction of LBE particles to be elastically reflected at the wall. A typical slip-boundary condition for, say, southeast propagating molecules entering the fluid domain from the north wall, y = d, would read as follows (lattice spacing made unity for simplicity): f se (x, d) = (1 − r) fne (x − 1, d − 1) + r fnw (x + 1, d − 1). Here r is a bounce-back coefficient in the range 0 < r < 1, and subscripts se, ne stand for south-east and north-east propagation, respectively [20]. It is easily seen that the special case r = 1 corresponds to a complete bounceback along the incoming direction, a simple option to implement zero fluid speed at the wall. More general conditions, borrowed from “diffusive” boundary conditions used in rarefied gas dynamics for the solution of the “true” Boltzmann equation have also been developed [21]. Much remains to be done to show that existing LBE models, extended with appropriate boundary conditions, can solve non-hydrodynamic flow regimes. This is especially true if thermal effects must be taken into account, as it is often the case in nanoflows applications.

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Even if the use of LBE stand-alone turned out to be unviable, one could still think of coupling LBE with truly microscopic methods, such as direct simulation or kinetic Monte Carlo [22, 23]. A potential advantage of coupling LBE instead of Navier-Stokes solvers to atomistic, or kinetic Monte Carlo, descriptions of atomistic flows is that the shear tensor Sab =

ν(∂a u b + ∂b u a ) 2

(14)

can be computed locally as 1

Sab = µ ( f i − f ie )(cia cib − cs2 δab ) 2 i

(15)

with no need of taking spatial derivatives (a delicate, and often error-prone, task at solid interfaces). Moreover, while the expression (14) is only valid in the limit of small Knudsen number, no such restriction applies to the kinetic expression (15). Both aspects could significantly enhance the scope of sampling procedures converting fluid-kinetic information (the discrete populations) into atomistic information (the particles coordinates and momenta) and vice versa, at fluid–solid interfaces [24]. This type of coupling procedures represent one of the most exciting frontiers for multiscale LBE applications at the interface between fluid dynamics and material science [25].

5.

Future Prospects

LBE has already made proof of significant versatility in addressing a wide range of problems involving complex fluid motion at disparate scales. Much remains to be done to further boost the power of the LB method towards multiphysics applications of increasing complexity. Important topics for future research are: • robust interface conditions for strongly non-equilibrium flows; • locally adaptive LBEs on unstructured, possibly moving, grids; • acceleration strategies for long-time and steady-state calculations. Finally, the development of a solid mathematical framework identifying the general conditions for the validity (what can go wrong and why!) of multiscale LBE techniques is also in great demand [26]. There are good reasons to believe that further upgrades of the LBE technique, as indicated above, hopefully stimulated by enhanced communication with allied sectors of computational physics, will make multiphysics LBE applications flourish in the near future.

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References [1] M. Seel, “Modelling of solid rocket fuel: from quantum chemistry to fluid dynamic simulations,” Comput. Phys., 5, 460–469, 1991. [2] W. Hoover, A.J. de Groot, and C. Hoover, “Massively parallel computer simulation of plane-strain elastic–plastic flow via non-equilibrium molecular dynamics and Lagrangian continuum mechanics,” Comput. Phys., 6(2), 155–162, 1992. [3] F.F. Abraham, J. Broughton, N. Bernstein, and E. Kaxiras, “Spanning the length scales in dynamic simulation,” Comput. Phys., 12(6), 538–546, 1998. [4] R. Benzi, S. Succi, and M. Vergassola, “The lattice Boltzmann equation: theory and applications,” Phys. Rep., 222, 145–197, 1992. [5] S. Succi, “The lattice Boltzmann equation for fluid dynamics and beyond,” Oxford University Press, Oxford, 2001. [6] G. McNamara and G. Zanetti, “Use of the Boltzmann equation to simulate lattice gas automata,” Phys. Rev. Lett., 61, 2332–2335, 1988. [7] F. Higuera, S. Succi, and R. Benzi, “Lattice gas dynamics with enhanced collisions,” Europhys. Lett., 9, 345–349, 1989. [8] X. He and L.S. Luo, “A priori derivation of the lattice Boltzmann equation,” Phys. Rev. E, 55, R6333–R6336, 1997. [9] Y.H. Qian, D. d’Humieres, and P. Lallemand, “Lattice BGK models for the Navier– Stokes equation,” Europhys. Lett., 17, 479–484, 1992. [10] S. Succi, I.V. Karlin, and H. Chen, “Role of the H theorem in lattice Boltzmann hydrodynamic simulations,” Rev. Mod. Phys., 74, 1203–1220, 2002. [11] O. Filippova and D. H¨anel, “Grid-refinement for lattice BGK models,” J. Comput. Phys., 147, 219–228, 1998. [12] H. Chen, C. Teixeira, and K. Molvig, “Realization of fluid boundary conditions via discrete Boltzmann dynamic,” Int. J. Mod. Phys. C, 9, 1281–1292, 1998. [13] A. Dupuis, “From a lattice Boltzmann model to a parallel and reusable implementation of a virtual river,” PhD Thesis n. 3356, University of Geneva, 2002. [14] O. Fippova, S. Succi, F.D. Mazzocco, C. Arrighetti, G. Bella, and D. Haenel, “Multiscale lattice Boltzmann schemes with turbulence modeling,” J. Comp. Phys., 170, 812–829, 2001. [15] S. Chen, S. Kandasamy, S. Orszag, R. Shock, S. Succi, and V. Yakhot, “Extended Boltzmann kinetic equation for turbulent flows,” Science, 301, 633–636, 2003. [16] A. Gabrielli, S. Succi, and E. Kaxiras, “A lattice Boltzmann study of reactive microflows,” Comput. Phys. Commun., 147, 516–521, 2002. [17] S. Succi, G. Smith, O. Filippova, and E. Kaxiras, “Applying the Lattice Boltzmann equation to multiscale fluid problems,” Comput. Sci. Eng., 3(6), 26–37, 2001. [18] M. Adamo, M. Bernaschi, and S. Succi, “Multi-representation techniques for multiscale simulation: reactive microflows in a catalytic converter,” Mol. Simul., 25(1–2), 13–26, 2000. [19] X.B. Nie, S. Chen, and G. Doolen, “Lattice Boltzmann simulations of fluid flows in MEMS,” J. Stat. Phys., 107, 279–289, 2002. [20] S. Succi, “Mesoscopic modeling of slip motion at fluid–solid interfaces with heterogeneus catalysis,” Phys. Rev. Lett., 89(6), 064502, 2002. [21] S. Ansumali and I.V. Karlin, “Kinetic boundary conditions in the lattice Boltzmann method,” Phys. Rev. E, 66, 026311–17, 2002. [22] M. Silverberg, A. Ben-Shaul, and F. Rebentrost, “On the effects of adsorbate aggregation on the kinetics of surface-reactions,” J. Chem. Phys., 83, 6501–6513, 1985.

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[23] T.P. Schulze, P. Smereka, and Weinan E, “Coupling kinetic Monte Carlo and continuum models with application to epitaxial growth,” J. Comput. Phys., 189, 197–211, 2003. [24] W. Cai, M. de Koning, V.V. Bulatov, and S. Yip, “Minimizing boundary reflections in coupled-domain simulations,” Phys. Rev. Lett., 85, 3213–3216, 2000. [25] D. Raabe, “Overview of the lattice Boltzmann method for nano and microscale fluid dynamics in material science and engineering,” Model. Simul. Mat. Sci. Eng., 12(6), R13–R14, 2004. [26] W. E, B. Engquist, Z.Y. Huang, “Heterogeneous multiscale method: a general methodology for multiscale modeling,” Phys. Rev. B, 67(9), 092101, 2003.

8.5 DISCRETE SIMULATION AUTOMATA: MESOSCOPIC FLUID MODELS ENDOWED WITH THERMAL FLUCTUATIONS Tomonori Sakai1 and Peter V. Coveney2,∗ 1 Centre for Computational Science, Queen Mary, University of London, Mile End Road, London E1 4NS, UK 2 Centre for Computational Science, Department of Chemistry, University College London, 20 Gordon Street, London WC1H 0AJ, UK

1.

Introduction

Until recently, theoretical hydrodynamics has largely dealt with relatively simple fluids which admit or are assumed to have an explicit macroscopic description. It has been highly successful in describing the physics of such fluids by analyses based on the Navier-Stokes equations, the classical equations of fluid dynamics which describe the motion of fluids, and usually predicated on a continuum hypothesis, namely that matter is infinitely divisible [1]. On the other hand, many real fluids encountered in our daily lives, in industrial, biochemical, and other fields are complex fluids made of molecules whose individual structures are themselves complicated. Their behavior is characterized by the presence of several important length and time scales. It must surely be among the more important and exciting research topics of hydrodynamics in the 21st century to properly understand the physics of such complex fluids. Examples of complex fluids are widespread – surfactants, inks, paints, shampoos, milk, blood, liquid crystals, and so on. Typically, such fluids are comprised of molecules and/or supramolecular components which have a non-trivial internal structure. Such microscopic and/or mesoscopic structures lead to a rich variety of unique rheological characteristics which not only make the study of complex fluids interesting but in many cases also enhance our quality of life.

* Corresponding author: P.V. Coveney, Email address: P.V. [email protected]

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In order to investigate and model the behavior of complex fluids, conventional continuum fluid methods based on the governing macroscopic fluid dynamical equations are somewhat inadequate. The continuous, uniform, and isotropic assumptions on which the macroscopic equations depend are not guaranteed to hold in such fluids where complex and time-evolving mesoscopic structures, such as interfaces, are present. As noted above, complex fluids are ones in which several length and time scales may be of importance in governing the large scale dynamical properties, but these micro and mesoscales are completely omitted in macroscopic continuum fluid dynamics, where empirical constitutive relations are instead shoe-horned into the Navier–Stokes equations. On the other hand, fully atomistic approaches based on molecular dynamics [2], which are the exact antithesis of conventional continuum methods, are in most cases not viable due to their vast computational cost. Thus, simulations which provide us with physically meaningful hydrodynamic results are out of reach of present day molecular dynamics and will not be accessible within the near future. Mesoscopic models are good candidates for mitigating problems with both conventional continuum methods and fully atomistic approaches. Spatially and temporally discrete lattice gas automata (LGA)[3] and lattice Boltzmann (LB) [4–7] methods have proven to be of considerable applicability to complex fluids, including multi-phase [8, 9] and amphiphilic [8, 9] fluids, solid–fluid suspensions [10], and the effect of convection–diffusion on growth processes [11]. These methods have also been successfully applied to flow in complex geometries, in particular to flow in porous media, an outstanding contemporary scientific challenge that plays an essential role in many technological, environmental, and biological fields [12–16]. Another important advantage of LGA and LB is that they are ideally suited for high performance parallel computing due to the inherent spatial locality of the updating rules in their dynamical time-stepping algorithms [17]. However, lattice-based models have certain well-known disadvantages associated with their spatially discrete nature [4, 7]. Here, we describe another mesoscopic model worthy of study. The method, which we call discrete simulation automata (DSA), is a spatially continuous but still temporally discrete version of the conventional spatio-temporally discrete lattice gas method, whose prototype was proposed by Malevanets and Kapral [18]. Since the particles now move in continuous space, DSA has the advantage of eliminating the spatial anisotropy that plagues conventional lattice gases, while also providing conservation of energy which enables one to deal with thermohydrodynamic problems not easily accessible by conventional lattice methods. We have coined the name DSA by analogy with the direct simulation Monte Carlo (DSMC) method [24] to which it is closely related, as we discuss further in Section 2. Some authors have referred to this method as a “realcoded lattice gas” [19–22]. Others have used the terms “Malevanets–Kapral

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method” “stochastic rotation method” or “multiple particle collision dynamics”. We have proposed the term DSA which we hope will be widely adopted in order to avoid further confusion [23]. The remainder of our paper is structured as follows. Starting from a review of single-phase DSA in Section 2, Section 3 describes how DSA can deal with binary immiscible fluids. In Section 4, we describe the application of DSA to amphiphilic fluids. Two of the latest developments of DSA, flow in porous media and a parallel implementation, are discussed in Section 5. Section 6 concludes our paper with a summary of the method.

2.

The Basic DSA Model and its Physical Properties

DSA are based on a microscopic, bottom-up approach and are comprised of cartesian cells between which massive point particles with a certain mass move. For a single component DSA fluid, state variables evolve by a twostep dynamical process: particle propagation and multi-particle collision. Each particle changes its location in the propagation process r = r + v

(1)

and its velocity in a collision process v = V + σ (v − V ),

(2)

where V is the mean velocity of all particles within a cell in which the collision occurs and σ is a random rotation, the same for all particles in one cell but differing between cells. In these equations, primes denote post-collision values and the mass as of all the particles are set to unity for convenience. This collision operation is equivalent to that in the direct simulation Monte Carlo (DSMC) method [24], except that pairwise collisions in DSMC are replaced by multi-particle collisions. The loss of molecular detail is an unavoidable consequence of the DSA algorithms as with other mesoscale modeling methods; however, these details are not required in order to describe the universal properties of fluid flow. Evidently, the use of multi-particle collisions allows DSA to deal readily with phenomena on mesoscopic and macroscopic scales which would be much more costly to handle using DSMC. Mass, momentum and energy are locally and hence globally conserved during the collision process. The velocity distribution of DSA particles corresponds to that of a Maxwellian when the system has relaxed to an equilibrium state [18]. We can thus define a parameter which may be regarded as a measure of average kinetic energy of the particles; this is the temperature T . For example, T = 1.0 specifies a state when each cartesian velocity component for the particles is described by a Maxwell distribution, whose variance is equal to one lattice unit (i.e., one DSA cell length).

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The existence of an H-theorem has been established using a reduced one-particle distribution function [18]. By applying a Chapman–Enskog asymptotic expansion to the reduced distribution function, the Navier–Stokes equations can be derived, as in the case of LGA [3] and LB [5]. When σ rotates v − V (see Eq. (2)) by a random angle in each cell, the fluid viscosity in DSA is written as ν=

1 ρ + 1 − e−ρ +T , 12 2(ρ − 1 + e−ρ )

(3)

where ρ is the number density of particles.

3. 3.1.

DSA Models of Interacting Particles Binary Immiscible Fluids

DSA have been extended to model binary immiscible fluids by introducing the notion of “color”, in both two and three dimensions [20]. Individual particles are assigned color variables, e.g., red or blue, and “color charges” which act rather like electrostatic charges. This notion of “color” was first introduced by Rothman and Keller [8]. With the color charge Cn of the nth particle given by

Cn =

+1

red particle,

−1

blue particle,

(4)

there is an attractive force between particles of the same color and a repulsive force between particles of different colors. To quantify this interaction, we define the color flux vector N( r) Q(r) = Cn (vn − V (r)), (5) n=1

where the sum is over all particles, and the color field vector F(r) =

i

wi

N( r ) Ri i Cn , |Ri | n

(6)

where the first and the second sums are over all nearest neighbor cells and all particles, respectively. N (r) is the number of particles in the local cell, vn the velocity of the nth particle, and V (r) the mean velocity of particles in a cell. The weighting factors are defined as wi = 1/|Ri |, where Ri = r − r i and r i is the location of the centre of ith nearest neighbor cell. The range of the index i differs according to the definition of the neighbors. With two- and threedimensional Moore neighbors, for example, i would range from 0 to 7 and 0 to

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26, respectively. One can model the phase separation kinetics of an immiscible binary fluid by choosing a rotation angle for each collision process such that the color flux vector points in the same direction as the color field vector after the collision. The model exhibits complete phase separation in both two [20, 22] and three [22] dimensions and has been verified by investigating domain growth laws and the resultant surface tension between two immiscible fluids [21], see Figs. 1–3. Although the precise location of the spinodal temperature has not thus far been investigated within DSA, we have confirmed that all binary immiscible fluid simulations presented in this review operate below it.

Initial

50 steps

500 steps

1000 steps

Figure 1. Two-phase separation in a binary immiscible DSA simulation [22]. Randomly distributed particles of two different colors (dark grey for water, light grey for oil) in the initial state segregate from each other, until two macroscopic domains are formed. The system size is 32 × 32 × 32, and the number density of both water and oil particles is 5.0. 1.2 1.1

T ⫽ 2.0

1 0.9 0.8 ∆P

0.7 0.6

T ⫽ 1.0

0.5 0.4 0.3

T ⫽ 0.5

0.2 0.1 0

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

1/ R

Figure 2. Verification of Laplace’s law for two-dimensional DSA [21]. The pressure difference between inside and outside of a droplet of radius R, P = Pin − Pout , was measured in a system of size 4R × 4R(R = 16, 32, 64, 128), averaged over 10 000 time-steps. The error bars are smaller than the symbols. T is the “temperature” which can be regarded as the indicator of averaged kinetic energy of particles and is defined by T = kT ∗ /m (k is Boltzmann’s constant, T ∗ the absolute temperature, and m the mass of the particles).

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3.6 ⴛ t⫺0.5

0.6

7.8 ⴛ t⫺0.67

0.5

0.4

0.3

0.2

100 Time step

200

300

Figure 3. Temporal evolution of the characteristic wave number [25] in two-dimensional DSA simulations of binary phase separation, averaged over seven independent runs [21]. The domain growth is characterized with two distinct rates, namely, a slow growth rate R ∼ t 1/2 in the initial stage and a fast growth rate R ∼ t 2/3 at later times.

3.2.

Ternary Amphiphilic Fluids

A typical surfactant molecule has a hydrophilic head and a hydrophobic tail. Within DSA this structure is described by introducing a dumbbell-shaped particle in both two [21] and three [22] dimensions. Figure 4 is a schematic description of the two-dimensional particle model. A and B correspond to the hydrophilic head and the hydrophobic tail. G is the centre of mass of the surfactant particle. Color charges Cphi and Cpho are assigned to A and B, respectively. If we take the other DSA particles to be water particles whose color charges are

l phi

F (r)

A C phi θ

l pho

G

x

B C pho Figure 4. The schematic description of the two-dimensional surfactant model. A and B with color charges Cphi and Cpho correspond to the hydrophilic head and the hydrophobic tail, respectively. The mass of the surfactant particle is assumed to be concentrated at G, the centre of mass of the dumbbell particle.

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positive, Cphi and Cpho should be set as Cphi > 0 and Cpho < 0. The attractive interaction between A and water particles and the repulsive interaction between A and oil particles and conversely for B are described in a similar way to those in the binary immiscible DSA. For simplicity, the mass of the surfactant particle is assumed to be concentrated at the centre of mass. This assumption provides the model with great simplicity especially in describing the rotational motion of surfactant particles, while adequately retaining the ability to reproduce essential properties of surfactant solutions. Since there is no need to consider the rotational motions of the surfactant particle explicitly, its degrees of freedom are reduced to only three, that is, its location, orientation angle, and translational velocity. Calculations of the color flux F(r) and the color field Q(r) resemble those in the binary immiscible DSA. For the calculation of F(r), we use Eq. (5), without taking the contributions of surfactant particles into account. Note that motions of A and B only result in suppressing the tendency of F(r) and Q(r) to overlap each other, because they would not influence the “non-color” momentum exchanges. Q(r) is determined by considering both the distribution and the structure of surfactant particles. When a surfactant particle is located at r G with an orientation angle θ (see Fig. 4), A and B ends of the particle are located at

rA =

rB =

r Ax r Ay

rBx rBy

=

=

rG x rGy rG x rGy

+

−

cos θ sin θ cos θ sin θ

· lphi ,

(7)

· lpho .

(8)

In these equations, lphi and lpho are the distance between G and the hydrophilic end (A in Fig. 4), and the distance between G and the hydrophobic end (B in Fig. 4), respectively. We then add the color charge Cphi and Cpho to cells located at r A and r B , which corresponds to modifying Eq. (4) into  +1   

red particle, −1 blue particle, Cn =  hydrophilic head, C   phi Cpho hydrophobic tail.

(9)

After calculating the color flux and the color field in each cell, a rotation angle is chosen using the same method as for binary immsicible DSA fluids, namely, the color flux vector overlaps the color field vector. Finally, the orientation angle θ of each surfactant particle, after the momentum exchange, is set in such a way that it overlaps with the color field, which can be expressed as:

cos θ sin θ

=

F(r) . |F(r)|

(10)

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Both two- and three-dimensional versions of this model have been derived in this way [21, 22]. Using this model, the formation of spherical micelles, water-in-oil and oil-in-water droplet microemulsion phases, and water/oil/ surfactant sponge phase in both two [21] and three [22] dimensions have been reported (see Figs. 5 and 6). Suppression of phase separation and resultant domain growth, the lowering of interfacial tension between two immiscible fluids, and the connection between the mesoscopic model parameters and the macroscopic surfactant phase behavior have been studied within the model in both two and three dimensions [21, 22]. These studies have been primarily qualitative in nature, and correspond to some of the early papers published on ternary amphiphilic fluids using LGA [26, 27] and LB [17, 28] methods. Much more extensive work on the

Initial

After 250 steps

After 2500 steps

After 10000 steps

Figure 5. A two-dimensional DSA simulation of a sponge phase in a ternary amphiphilic fluid starting from a random initial condition [21]. Surfactant is visible at the interface between oil (dark grey) and water (light grey) regions. The system size is 64×64, the number density of DS A particles 10, the concentration ratio of water/oil/surfactant 1 : 1 : 1, the temperature of the system 0.2, color charges for hydrophilic and hydrophobic end groups Cphi = 10.0, Cpho = − 10.0.

Figure 6. The formation of spherical micelles in aqueous solvent [22]. The system size is 32 × 32 × 32, the concentration of surfactant particles is 10%.

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quantitative aspects of self-assembly kinetics has already been published using these two techniques.

4.

Some Recent Developments and Applications

DSA is currently attracting growing attention; the most recent published works using the method include the modeling of colloids [29], a detailed quantitative analysis of single-phase fluid behavior [30], and studies on the theoretical and numerically determined viscosity [31, 32]. Here we describe our own latest developments, concerning flow in porous media and parallel implementation.

4.1.

Flow in Porous Media

Within DSA, updating the state in porous media simulations requires close attention to be paid to the propagation process. This is due to the fact that particles are allowed to assume velocities of arbitrary directions and magnitudes: it frequently happens that a particle penetrates unphysically through an obstacle and reaches a fluid area on another side. It is thus not enough to know only the information about the starting and ending sites of the moving particles, as is done in LGA and LB studies, but rather their entire trajectories need to be investigated. We detect whether a particle hits an obstacle or not in the following way. First, we look at the cell containing r =r +v. If the cell is inside an obstacle the particle move is rejected and bounce-back boundary conditions are applied to update the particle velocity in the cell. When the cell is within a pore region, we extract a rectangular set of cells where the cells including r and r face each other on the diagonal line, as shown in Fig. 7. From this set of cells we further extract cells which intersect the trajectory of the particle. In order to do this, every cell C j in the “box” shown in Fig. 7 except those containing r and r , is investigated by taking the cross product v × c j k , where c j k denotes the position vector of four points of a cell C j and k = 1, 2, 3 and 4. If the v × c j k s for all k have the same sign, this means that the whole of C j is located on either side of v, that is, it does not intersect v and there is no need to check whether the site is inside a pore or the solid matrix. Otherwise, C j intersects v and the move is rejected if the site is inside the solid matrix, see Fig. 7. Using this method we have simulated single phase and binary immiscible fluid flow in two-dimensional porous media [23]. Good linear force–flux relationships were observed in single phase fluid flows, as is expected from Darcy’s law. In binary immiscible fluid flows, our findings are in good

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T. Sakai and P.V. Coveney DSA cells

r'

r

"box"

Solid matrices

Figure 7. Scheme for detecting particles’ collisions with obstacles within the discrete simulation automata model [23]. Assume a particle moves from r to r = r + v (v is the velocity of the particle) in the current time-step. This particle obviously collides with the obstacle which is colored gray. However, the collision cannot be detected if we only take into account the information on r which is within the pore region. The whole trajectory of the particle must be investigated to accurately detect the collisions. In order to do this, we first extract all – in this case twelve – cells comprising the “box”, a rectangular set of cells where the cells including r and r are aligned with each other on a diagonal line. Secondly, from within the box, we further extract cells which overlap with the trajectory of the particle. The six cells comprising the region bordered with slashed lines are such cells in this case. These cells except those which include r and r are finally checked to establish whether they are part of an obstacle or a pore region.

agreement with previous studies using LGA [12–14]: a well defined linear force–flux relationship was obtained only when the forcing exceeded specified thresholds. We also found a one-to-one correspondence between these thresholds and the interfacial tension between the two fluids, which supports the interpretation from previous LGA studies that the existence of these thresholds is due to the presence of capillary effects within the pore space. In the study [23], we assumed that the binary immiscible fluids are uncoupled. However, a more general force–flux relationship allows for the fluids to be coupled and there have been a few studies of two-phase flow taking such coupling into account [12–14, 33, 34]. Within LGA, using the gravitational

Discrete simulation automata

2497 υj

j

i

υi Figure 8. Occluded particles [23]: some particles can be assigned to a pore completely surrounded by solid matrices at the initial state, like particle i. Other particles can be occluded in a depression on the surface of an obstacle, like particle j . By imposing the gravitational force on such particles, they will gain kinetic energy limitlessly because their energy cannot be dissipated through interactions with other particles.

forcing method, it is possible to apply the forcing to only one species of fluid and discuss similarities with the Onsager relations [12, 13]. In our DSA study, we have used pressure forcing [23] and thus have not been able to investigate the effect of the coupling of the two immiscible fluids. The difficulty in implementing gravitational forcing within DSA is partly due to the local heating effects caused by occluded particles which are trapped within pores and will gain kinetic energy in an unbounded manner by the gravitational force imposed on them at every time step; see Fig. 8.

4.2.

Parallel Implementation

For large scale simulations in three dimensions, the computational cost of DSA is high, as with LGA and LB methods. Due to the spatially local updating rules, however, all basic routines in DSA algorithms are parallelizable. Good

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computer performance can thus be expected given an efficient parallel implementation. We have parallelized our DSA codes in two and three dimensions, written in C++ and named DSA2D and DSA3D, respectively, by spatially decomposing the system and implementing the MPI libraries [35]. It is in the propagation process that the MPI library functions are mainly used. There are two key features which are worth pointing out here. First, in the propagation process, information on the particles which exit each domain is stored in arrays which are then handed over to a send function MPI_Isend. The size of the arrays depends on temperature and the direction of the target domain. Second, as the number of particles within a domain fluctuates, 10 – 20% of the memory allocated for particles in the domain is used as an absorber. (Particles are allocated at an initial stage up to 80 – 90% of the total capacity.) Figures 9 and 10 show the parallel performance in two and three dimensions, respectively. Although DSA2D scales superlinearly across all processor counts, DSA3D scales well only with a large number of CPU (DSA3D’s propagation() routine even slows down with increasing processor counts for certain sets of parameters). The difference here is due to the way the system is spatially decomposed: DSA2D has been domain decomposed in one direction whereas DSA3D has been decomposed in three directions. In order to realise good scalability in three dimensions for our current parallel

50 45 collision()

40 35 Performance

TOTAL

30 IDEAL

25 propagation()

20 15 10 5 0

0

5

10

15

20

25

30

35

Number of CPUs

Figure 9. Scalability of two-dimensional DSA (DSA2D) for single-phase fluids on SGI Origin 3000 (400 MHz MIPS R12000) processors. “Performance” (vertical axis) means “speed-up”, which is relative to the number of processors. The overall performance is indeed superlinear.

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(a) 60 IDEAL

50 Performance

collision() 40 30

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Number of CPUs

(c) 60 IDEAL

Performance

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collision()

40 TOTAL 30 propagation() 20 10 0

0

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30

40

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60

Number of CPUs

Figure 10. Parallel performance of DSA3D for single-phase fluids of varying system sizes: (A) 643 ; (B) 1283 ; (C) 2563 , on SGI Origin 3000 (400 MHz MIPS R12000) processors. “Performance” (vertical axis) means “speed-up”, which is relative to the number of processors. For 643 and 1283 systems the performance of the propagation process actually decreases when the number of CPUs becomes large.

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implementation, a large system and a large number of CPUs are required. The present parallel implementation should be regarded only as preliminary; further optimization may be expected to result in better overall performance.

5.

Summary

Discrete simulation automata (DSA) represent a mesoscopic fluid simulation method which, in common with lattice gas automata (LGA) and the lattice Boltzmann (LB) methods, has several advantages over conventional continuum fluid dynamics. Beyond LGA and LB’s beneficial aspects, DSA’s most eminent characteristic is that a temperature can be defined very naturally. It is thus a promising candidate to deal with complex fluids where fluctuations can often play an essential role in determining macroscopic behavior. There remain, however, some drawbacks to the DSA technique. The existence of particles with continuously valued velocities coupled to the intrinsic temporal discreteness of the model leads to some problems in handling wall boundary collisions, including trapping of particles with increasing energy in certain flow regimes, which do not arise with LGA and LB methods. Nonetheless, DSA appears to be a promising technique for the study of numerous complex fluids. We have reviewed a few examples here, including immiscible fluids, amphiphilic fluids, and flow in porous media. Most of these studies have so far not reached an equivalent maturity and quantitative level to that of LGA and LB publications. DSA is amenable to fairly straightforward parallel implementation. We therefore expect to see further fruitful explorations of complex fluid dynamics using DSA in the future.

References [1] G.K. Batchelor, An Introduction to Fluid Dynamics, Cambridge University Press, Cambridge, 1967. [2] D.C. Rapaport, The Art of Molecular Dynamics Simulation, Cambridge University Press, Cambridge, 1995. [3] U. Frisch, B. Hasslacher, and Y. Pomeau, Phys. Rev. Lett., 56, 1505, 1986. [4] S. Succi, The Lattice Boltzmann Equation, Oxford University Press, Oxford, 2001. [5] R. Benzi, S. Succy, and M. Vergassola, Phys. Rep., 222, 145, 1992. [6] S. Chen, Z. Wang, X. Shan, and G. Doolen, J. Stat. Phys., 68, 379, 1992. [7] D.H. Rothman and S. Zaleski, Lattice Gas Cellular Automata, Cambridge University Press, Cambridge, 1997. [8] D.H. Rothman, and J. Keller, J. Stat. Phys., 52, 1119, 1988. [9] D. Grunau, S. Chen, and K. Eggert, Phys. Fluids A, 5, 2557, 1993. [10] A.J.C. Ladd, J. Fluid Mech., 271, 285, 1994. [11] J.A. Kaandorp, C. Lowe, D. Frenkel, and P.M.A. Sloot, Phys. Rev. Lett., 77, 2328, 1996.

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[12] P.V. Coveney, J.-B. Maillet, J.L. Wilson, P.W. Fowler, O. Al-Mushadani, and B.M. Boghosian, Int. J. Mod. Phys. C, 9, 1479, 1998. [13] J.-B. Maillet and P.V. Coveney, Phys. Rev. E, 62, 2898, 2000. [14] P.J. Love, J.-B. Maillet, and P.V. Coveney, Phys. Rev. E, 64, 061302, 2001. [15] N.S. Martys and H. Chen, Phys. Rev. E, 53, 743, 1996. [16] A. Koponen, D. Kandhai, E. Hellen, M. Alava, A. Hoekstra, M. Kataja, K. Niskanen, P. Sloot, and J. Timonen, Phys. Rev. Lett., 80, 716, 1998. [17] P.J. Love, M. Nekovee, P.V. Coveney, J. Chin, N. Gonzalez-Segredo and J.M.R. Martin, Comput. Phys. Commun., 153, 340, 2003. [18] A. Malevanets and R. Kapral, J. Chem. Phys., 110, 8605, 1999. [19] Y. Hashimoto, Y. Chen, and H. Ohashi, Int. J. Mod. Phys. C, 9(8), 1479, 1998. [20] Y. Hashimoto, Y. Chen, and H. Ohashi, Comput. Phys. Commun., 129, 56, 2000. [21] T. Sakai, Y. Chen, and H. Ohashi, Phys. Rev. E, 65, 031503, 2002. [22] T. Sakai, Y. Chen, and H. Ohashi, J. Coll. and Surf., 201, 297, 2002. [23] T. Sakai and P.V. Coveney, “Single phase and binary immiscible fluid flow in two-dimensional porous media using discrete simulation automata,” 2002 (preprint). [24] G.A. Bird, Molecular Gas Dynamics and the Direct Simulation of Gas Flows, Clarendon, Oxford, 1994. [25] T. Kawakatsu, K. Kawasaki, M. Furusaka, H. Okabayashi and T. Kanaya, J. Chem. Phys., 99, 8200, 1993. [26] B.M. Boghosian, P.V. Coveney, and A.N. Emerton, Proc. R. Soc. A, 452, 1221, 1996. [27] B.M. Boghosian, P.V. Coveney, and P.J. Love, Proc. R. Soc. A, 456, 1431, 2000. [28] H. Chen, B.M. Boghosian, P.V. Coveney, and M. Nekovee, Proc. R. Soc. A, 456, 2043, 2000. [29] S.H. Lee and R. Kapral, Physica A, 298, 56, 2001. [30] A. Lamura and G. Gompper, Eur. Phys. J.E, 9, 477, 2002. [31] T. Ihle and D.M. Kroll, Phys. Rev. E, 63, 020201(R), 2001. [32] A. Lamura, G. Gompper, T. Ihle, and D. M. Kroll, Europhys. Lett., 56, 319, 2001. [33] C. Zarcone and R. Lenormand, C.R. Acad. Sci. Paris, 318, 1429, 1994. [34] J.F. Olson and D.H. Rothman, J. Fluid Mech., 341, 343, 1997. [35] http://www-unix.mcs.anl.gov/mpi/index.html

8.6 DISSIPATIVE PARTICLE DYNAMICS Pep Espa˜nol Dept. Física Fundamental, Universidad Nacional de Educaci´on a Distancia, Aptdo. 60141, E-28080 Madrid, Spain

1.

The Original DPD Model

In order to simulate a complex fluid like a polymeric or colloidal fluid, a molecular dynamics simulation is not very useful. The long time and space scales involved in the mesoscopic dynamics of large macromolecules or colloidal particles as compared with molecular scales imply to follow an exceedingly large number of molecules during exceedingly large times. On the other hand, at these long scales, molecular details only show up in a rather coarse form, and the question arises if it is possible to deal with coarse-grained entities that reproduce the mesoscopic dynamics correctly. Dissipative particle dynamics (DPD) is a fruitful modeling attempt in that direction. DPD is a stochastic particle model that was introduced originally as an off-lattice version of Lattice gas automata (LGA) in order to avoid its lattice artifacts [1]. The method was put in a proper statistical mechanics context a few years later [2] and the number of applications since then is growing steadily. The original DPD model consists of a collection of soft repelling frictional and noisy balls. From a physical point of view, each dissipative particle is regarded not as a single molecule of the fluid but rather as a collection of molecules that move in a coherent fashion. In that respect, DPD can be understood as a coarse-graining of molecular dynamics. There are three types of forces between dissipative particles. The first type is a conservative force deriving from a soft potential that tries to capture the effects of the “pressure” between different particles. The second type of force is a friction force between the particles that wants to describe the viscous resistance in a real fluid. This force tries to reduce velocity differences between dissipative particles. Finally, there is a stochastic force that describe the degrees of freedom that have been eliminated from the description in the coarse-graining process. 2503 S. Yip (ed.), Handbook of Materials Modeling, 2503–2512. c 2005 Springer. Printed in the Netherlands.

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This stochastic force will be responsible for the Brownian motion of polymer and colloidal particles simulated with DPD. The postulated stochastic differential equations (SDEs) that define the DPD model are [2] dri = vi dt FCi j (ri j )dt − γ ω(ri j )(ei j ·vi j )ei j dt m i dvi = j= /i

+σ

(1)

j= /i

ω

1/2

(ri j )ei j dWi j

j= /i

Here, ri , vi are the position and velocity of the dissipative particles, m i is the mass of particle i, FCi j is the conservative repulsive force between dissipative particles i, j , ri j = ri −r j , vi j = vi −v j , and the unit vector from the j th particle to the ith particle is ei j = (ri − r j )/ri j with ri j = |ri − r j |. The friction coefficient γ governs the overall magnitude of the dissipative force, and σ is a noise amplitude that governs the intensity of the stochastic forces. The weight function ω(r) provides the range of interaction for the dissipative particles and renders the model local in the sense that the particles interact only with their neighbors. A usual selection for the weight function in the DPD literature is a linear function with the shape of a Mexican hat, but there is no special reason for such a selection. Finally, dWi j = dW j i are independent increments of the Wiener process that satisfy the Itˆo calculus rule dWi j dWi j = (δii δ j j + δi j δ j i ) dt. There are several remarkable features of the above SDEs. They are translationally, rotationallyand Galilean invariant. Most importantly, total momentum is conserved, d( i pi )/dt = 0, because the three types of forces satisfy Newton’s Third Law. Therefore, the DPD model captures the essentials of mass and momentum conservation which are responsible for the hydrodynamic behavior of a fluid at large scales [3, 4]. Despite its appearance as Langevin equations, Eq. (2) is quite different from the ones used in Brownian Dynamics simulations. In the Brownian Dynamics method, total momentum of the particles is not conserved and only mass diffusion can be studied. The above SDE are mathematically equivalent to a Fokker–Planck equation (FPE) that governs the time-dependent probability distribution ρ(r, v; t) of positions and velocities of the particles. The explicit form of the FPE can be found in Ref. [2]. Under the assumption that the noise amplitude and the friction coefficient are related by the fluctuation–dissipation relation σ = (2kB T γ )1/2, the equilibrium distribution ρ eq of the FPE has the familiar form

1 1 ρ (r, v) = exp − Z kB T eq

m i v i2 i

2

+ V (r)

(2)

where V is the potential function that gives rise to the conservative forces FC , kB is Boltzmann’s constant, T is the equilibrium temperature and Z is the normalizing partition function.

Dissipative particle dynamics

2.

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DPD Simulations of Complex Fluids

One of the most attractive features of the model is its enormous versatility in order to construct simple models for complex fluids. In DPD, the Newtonian fluid is made “complex” by adding additional interactions between the fluid particles. Just by changing the conservative interactions between the fluid particles, one can easily construct polymers, colloids, amphiphiles, and mixtures. Given the simplicity of modeling of mesostructures, DPD appears as a competitive technique in the field of complex fluids. We review now some of the applications of DPD to the simulation of different complex fluids systems (see also Ref. [5]). Colloidal particles are constructed by freezing fluid particles inside certain region, typically spheres or ellipsoids, and moving those particles as a rigid body. The idea was pioneered by Koelman and Hoogerbrugge [6] and has been explored in more detail by Boek et al. [7]. The simulation results for shear thinning curves of spherical particles compare very well with experimental results for volume fractions below 30%. At higher volume fractions somewhat inconsistent results are obtained, which can be attributed to several factors. The colloidal particles modeled in this way are to certain degree “soft balls” that can interpenetrate leading to unphysical interactions. At high volume fractions solvent particles are expelled from the region in between two colloidal particles. Again, this misrepresents the hydrodynamic interaction, which is mostly due to lubrication forces [8]. Depletion forces appear [9, 10] which are unphysical and due solely to the discrete representation of the continuum solvent. It seems that a judicious selection of lubrication forces that would take into account the effects of the solvent when no dissipative particle exist in between two colloidal particles can eventually solve this problem. Finally, we note that DPD can resolve the time scales of fluid momentum transport on the length scale of the colloidal particles or their typical interdistances. These scales are probed experimentaly by diffusive wave spectroscopy [11]. Polymer molecules are constructed in DPD through the linkage of several dissipative particles with springs (either Hookean or FENE [12]). Dilute polymer solutions are modeled by a set of polymer molecules interacting with a sea of fluid particles. The solvent quality can be varied by fine tuning the solvent–solvent and solvent–monomer conservative interactions. In this way, a collapse transition has been observed in passing from a good solvent to a poor solvent [13]. Static scaling results for the radius of gyration and relaxation time with the number of beads are consistent with the Rouse/Zimm models [14]. The model displays hydrodynamic interactions and excluded volume interactions, depending on solvent quality. Rheological properties have been also studied showing a good agreement with known kinetic theory results [15, 16]. Polymer solutions confined between walls have also been modeled showing anisotropic relaxation in nanoscale gaps [17]. Polymer melts have been

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simulated showing that the static scaling and rheology correspond to the Rouse theory, as a result of screening of hydrodynamic and excluded volume interactions in the melt [14]. The model is unable to simulate entanglements due to the soft interactions between beads that allow polymer crossing [14], although this effect can be partially controlled by suitably adjusting the length and intensity of the springs. At this point, DPD appears as a well benchmarked model for the simulation of polymer systems. Nevertheless, there is still not a direct connection between the model parameters used in DPD and actual molecular parameters like molecular weight, torsion potentials, etc. Immiscible fluid mixtures are modeled in DPD by assuming two types of particles [18]. Unequal particles repel each other more strongly than equal particles thus favoring phase separation. Starting from random initial conditions representing a high temperature miscible phase suddenly quenched, the domain growth has been investigated [19, 20]. Although lattice Boltzmann simulations allow to explore larger time scales than DPD [21], the simplicity of DPD modeling allows one to generalize easily to more complex systems in a way that lattice Boltzmann cannot. For example, mixtures of homopolymer melts have been modeled with DPD [22]. Surface tension measurements allow for a mapping of the model to the Flory–Huggins theory [22]. In this way, thermodynamic information has been used to fix the model parameters of DPD. A recent more detailed analysis of this procedure has been presented in Refs. [23, 24], where a calculation of the phase diagram of monomer and polymer mixtures of DPD particles allowed to discuss the connection of the repulsion parameter difference and the Flory–Huggins parameter χ. Another successful application of DPD has been the simulation of microphase separation of diblock copolymers [25], that has allowed to discuss the pathway to equilibrium. This pathway is strongly affected by hydrodynamics [26]. In a similar way, simulations of rigid DPD dimers in a solution of solvent monomers has allowed to study the growth of amphiphilic mesophases and its behavior under shear [27] and the self-assembly of model membranes [28]. DPD has also been applied to other complex situations like the dynamics of a drop at a liquid–solid interface [29], flow and rheology in the presence of polymers grafted to walls [30], vesicle formation of amphiphilic molecules [31] and polyelectrolytes [32].

3.

Thermodynamically Consistent DPD Model

Despite its successes, the DPD model suffers from several conceptual shortcomings that originate from the oversimplification of the so-formulated

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dissipative particles as representing mesoscopic portions of fluid. There are several issues in the original model that are unsatisfactory. For example, even though the macroscopic behavior of the model is hydrodynamic [3], it is not possible to relate in a simple direct way the viscosity of the fluid with the model parameters. Only after a recourse to the methods of kinetic theory can one estimate what input values for the friction coefficient should be imposed to obtain a given viscosity [4]. Another problem with the original model is that the conservative forces fix the thermodynamic behavior of the fluid [22]. The pressure equation of state, for example, is an outcome of the simulation, not an input. The model is isothermal and not able to study energy transport. There are no rules for specifying the range and shape of the weight functions that affect both, thermodynamic and transport properties. Perhaps the biggest problem of the model is the unclear physical length and time scales that are actually simulated. How big is a dissipative particle is not known from the model parameters. DPD appeared as a quick way of getting hydrodynamics suitable for “mesoscales”. Of course, the fact that there exists a well-defined Hamiltonian with a proper equilibrium ensemble, still makes the DPD model useful, at least as a thermostating device that respect hydrodynamics. In particular, when considering models of coarse-grained complex molecules (like amphiphiles or macromolecules) DPD as it was originally formulated can be very useful, despite the fact that an explicit correspondence between molecular parameters and DPD parameters are not known. However, the above-mentioned problems render DPD as a poor tool for the simulation of Newtonian fluids at mesoscopic scales. One needs to simulate a Newtonian fluid when dealing with colloidal suspension, dilute polymeric suspensions or mixtures of Newtonian fluids. In these cases, one should use better models that are thermodynamically consistent. These models consider each dissipative particle as a fluid particle, this is, a small moving thermodynamic system with proper thermodynamic variables. The idea of introducing an internal energy variable in the DPD model was developed in Refs. [33, 34] in order to obtain an energy conserving DPD model. Yet, it is necessary to introduce a second thermodynamic variable to have a full thermodynamic description. This variable is the volume of the fluid particles. There have been also attempts to introduce a volume variable in the isothermal DPD model [35, 36], but a full non-isothermal and thermodynamically consistent model has only appeared recently [37]. One way to define the volume is with the help of a bell-shaped weight function W (r) of finite range h normalized to unity.We introduce the density of every fluid particle through the relation di = j W (ri j ). Clearly, if around particle i there are many particles j , the density di defined above will be large. One associates a volume V i = di−1 to the fluid particle. Another possibility for defining the volume of each fluid particle relies on the Voronoi tessellation [38–40].

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P. Espa˜nol

The equations for the evolution of the position, velocity, and entropy of each fluid particle in the thermodynamically consistent DPD model are [37] r˙ i = vi m v˙ i =

j

Ti S˙i = −2κ

Pj 5η Fi j Pi + F r − vi j + ei j ei j ·vi j + F˜ i i j i j 2 2 3 j di d j di dj

Fi j j

di d j

Ti j +

5η Fi j 2 vi j + (ei j ·vi j )2 + Ti J˜i 6 j di d j

(3)

Here, Pi , Ti are the pressure and temperature of the fluid particle i, given in terms of equations of state, and Ti j = Ti − T j . We have introduced the function F(r) through ∇W (r) = −rF(r) and F˜ i , J˜i are suitable stochastic forces that obey the fluctuation–dissipation theorem [37]. Some small terms have been neglected in Eq. (3) for the sake of presentation. It can be shown that the above model conserves mass, momentum and energy and that the total entropy is a non-decreasing function of time, rendering the model consistent with the Laws of Thermodynamics. What are the similarities and differences between the thermodynamically consistent DPD model in Eq. (3) and the original DPD model of Eq. (2)? As in DPD, now particles of constant mass m move according to their velocities and exert forces of finite range to each other of different nature. The conservative forces of DPD are now replaced by a repulsive force directed along the line joining the particles that has a magnitude given by the pressure Pi and densities of the particles. Because the pressure Pi depends on the density, these type of force is not pair-wise but multibody [35]. The friction forces still depend on velocity differences between neighbor particles, but there is an additional term directly proportional to vi j . This new bit is necessary in order to have a faithful representation of the second space derivative terms that appear in the continuum equations of hydrodynamics [41]. In other words, it can be shown that, when thermal fluctuations can be neglected, Eq. (3) is a Lagrangian discretization of the continuum equations for hydrodynamics. Note that the friction coefficient is now given by the actual viscosity η of the fluid to be modeled and not an arbitrary tuning parameter. Finally, there is an additional dynamic equation for the entropy Si of the fluid particles. The terms in the entropy equation have a simple meaning as heat conduction and viscous heating. The heat conduction term tries to reduce temperature differences between particles by suitable energy exchange [42], whereas the viscous heating term proportional to the square of the velocities ensures that the kinetic energy dissipated by the friction forces is transformed into internal energy of the fluid particles. The model solves all the conceptual problems of DPD mentioned in the beginning of this section. In particular, the pressure and any other thermodynamic information is introduced as an input. The conservative forces of the

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original model become physically sounded pressure forces. Arbitrary equations of state and, in particular, of the van der Waals type can be used to study liquid–vapor coexistence in dynamic situations. Energy is conserved and we can study transport of energy in the system. The Second Law is satisfied. The transport coefficients are input of the model. The range functions of DPD enter in a very specific form, both in the conservative part of the dynamics through the density and pressure and in the dissipative part through the function Fi j . The particles have a physical size given by its physical volume and it is possible to specify the physical scale being simulated. The concept of resolution enters into play in the sense that one has to use many fluid particles per relevant length scale in order to recover the continuum results. Therefore, for resolving micron-sized objects one has to use very small fluid particles, whereas for resolving meter-sized objects large fluid particles are sufficient. In the model, it turns out that the amplitude of the thermal fluctuations scales with the square root of the volume of the fluid particles, in accordance with the usual notions of equilibrium statistical mechanics. Therefore, we expect that thermal fluctuations can be neglected in a simulation of meter-sized objects, but they are essential in the simulation of colloidal particles. This natural switching off thermal fluctuations with size is absent in the original DPD model. The model in Eq. (3) (without thermal fluctuations) is actually a version of the smoothed particle hydrodynamics (SPH) model, which is a Lagrangian particle method introduced by Lucy [43] and Monaghan [44] in the 70s in order to solve hydrodynamic problems in astrophysical contexts. Generalizations of SPH in order to include viscosity and thermal conduction and address laboratory scale situations like viscous flow and thermal convection have been presented only quite recently [42, 45, 46]. In order to formulate the thermodynamically consistent DPD model in Eq. (3), we have resorted to the GENERIC framework, which is a very elegant and useful way of writing dynamic equations that, by structure, are thermodynamically consistent [47]. It is possible to derive new fluid particle models based on both, the SPH methodology for discretizing continuum equations, and the GENERIC framework to ensure thermodynamic consistency. Continuum models for complex fluids typically involve additional structural or internal variables that are coupled with the conventional hydrodynamic variables. The coupling renders the behavior of the fluid non-Newtonian and complex. For example, polymer melts are characterized by additional conformation tensors, colloidal suspensions can be described by further concentration fields, mixtures are characterized by several density fields (one for each chemical species), emulsions are described with the amount and orientation of interface, etc. All these continuum models rely on the hypothesis of local equilibrium and, therefore, the fluid particles are regarded as thermodynamic subsystems. The physical picture that emerges from these fluid particles is that they represent “large” portions of the fluid and therefore, the scale of these

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fluid particles is supramolecular. This allows one to study large time scales. The price, of course, is the need for a deep understanding of the physics at this more coarse-grained level. In order to model polymer solutions, for example, ten Bosch [48] has associated to each dissipative particle an elongation vector representing the average elongation of polymer molecules. Although the ten Bosch model has all the problems of the original DPD model, it can be cast into a thermodynamically consistent model for non-isothermal dilute polymer solutions [49]. Another example where the strategy of internal variables can be successful is in the simulation of chemically reacting mixtures. Chemically reacting mixtures are not easily implemented with the usual approach taken by DPD in order to model mixtures. In DPD, mixtures are represented by “red” and “blue” particles. It is not trivial to specify a chemical reaction in which, for example, two red particles react with a blue particle to form a “green” particle. In this case, it is better to start from the well-established continuum equations for chemical reactions [41]. The fluid particles in the model have as additional variable the fraction of component red and blue inside the fluid particle. These two examples show how one can address viscoelastic flow problems and chemical reacting fluids with a simple methodology that involves fluid particles with internal variables. The idea can, of course, be applied to other complex fluids where the continuum equations are known.

Acknowledgments This work has been partially supported by the project BFM2001-0290 of the Spanish Ministerio de Ciencia y Tecnología.

References [1] P.J. Hoogerbrugge and J.M.V.A. Koelman, “Simulating microscopic hydrodynamics phenomena with dissipative particle dynamics,” Europhys. Lett., 19(3), 155–160, 1992. [2] P. Espa˜nol and P. Warren, “Statistical mechanics of dissipative particle dynamics,” Europhys. Lett., 30, 191, 1995. [3] P. Espa˜nol, “Hydrodynamics from dissipative particle dynamics,” Phys. Rev. E, 52, 1734, 1995. [4] C. Marsh, G. Backx, and M.H. Ernst, “Static and dynamic properties of dissipative particle dynamics,” Phys. Rev. E, 56, 1976, 1997. [5] P.B. Warren, “Dissipative particle dynamics,” Curr. Opinion Colloid Interface Sci., 3, 620, 1998. [6] J.M.V.A. Koelman and P.J. Hoogerbrugge, “Dynamic simulations of hard-sphere suspensions under steady shear,” Europhys. Lett., 21, 363–368, 1993.

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[7] E.S. Boek, P.V. Coveney, H.N.W. Lekkerkerker, and P. van der Schoot, “Simulating the rheology of dense colloidal suspensions using dissipative particle dynamics,” Phys. Rev. E, 55(3), 3124–3133, 1997. [8] J.R. Melrose, J.H. van Vliet, and R.C. Ball, “Continuous shear thickening and colloid surfaces,” Phys. Rev. Lett., 77, 4660, 1996. [9] E.S. Boek and P. van der Schoot, “Resolution effects in dissipative particle dynamics simulations,” Int. J. Mod. Phys. C, 9, 1307, 1997. [10] M. Whittle and E. Dickinson, “On simulating colloids by dissipative particle dynamics: issues and complications,” J. Colloid Interface Sci., 242, 106, 2001. [11] M. Kao, A. Yodh, and D.J. Pine, “Observation of brownian motion on the time scale of hydrodynamic interactions,” Phys. Rev. Lett., 70, 242, 1993. [12] A.G. Schlijper, P.J. Hoogerbrugge, and C.W. Manke, “Computer simulation of dilute polymer solutions with dissipative particle dynamics,” J. Rheol., 39(3), 567–579, 1995. [13] Y. Kong, C.W. Manke, W.G. Madden, and A.G. Schlijper, “Effect of solvent qualityon the conformation and relaxation of polymers via dissipative particle dynamics,” J. Chem. Phys., 107, 592, 1997. [14] N.A. Spenley, “Scaling laws for polymers in dissipative particle dynamics,” Mol. Simul., 49, 534, 2000. [15] Y. Kong, C.W. Manke, W.G. Madden, and A.G. Schlijper, “Modeling the rheology of polymer solutions by dissipative particle dynamics,” Tribol. Lett., 3, 133, 1997. [16] A.G. Schlijper, C.W. Manke, W. GH, and Y. Kong, “Computer simulation of non-Newtonian fluid rheology,” Int. J. Mod. Phys. C, 8(4), 919–929, 1997. [17] Y. Kong, C.W. Manke, W.G. Madden, and A.G. Schlijper, “Simulation of a confined polymer on solution using the dissipative particle dynamics method,” Int. J. Thermophys., 15, 1093, 1994. [18] P.V. Coveney and K. Novik, “Computer simulations of domain growth and phase separation in two-dimensional binary immiscible fluids using dissipative particle dynamics,” Phys. Rev. E, 54, 5134, 1996. [19] S.I. Jury, P. Bladon, S. Krishna, and M.E. Cates, “Test of dynamical scaling in threedimensional spinodal decomposition,” Phys. Rev. E, 59, R2535, 1999. [20] K.E. Novik and P.V. Coveney, “Spinodal decomposition off of-critical quenches with a viscous phase using dissipative particle dynamics in two and three spatial dimensions,” Phys. Rev. E, 61, 435, 2000. [21] V.M. Kendon, J.-C. Desplat, P. Bladon, and M.E. Cates, “Test of dynamical scaling in three-dimensional spinodal decomposition,” Phys. Rev. Lett., 83, 576, 1999. [22] R.D. Groot and P.B. Warren, “Dissipative particle dynamics: bridging the gap between atomistic and mesoscopic simulation,” J. Chem. Phys., 107, 4423, 1997. [23] S.M. Willemsen, T.J.H. Vlugt, H.C.J. Hoefsloot, and B. Smit, “Combining dissipative particle dynamics and Monte Carlo techniques,” J. Comput. Phys., 147, 50, 1998. [24] C.M. Wijmans, B. Smit, and R.D. Groot, “Phase behavior of monomeric mixtures and polymer solutions with soft interaction potential,” J. Chem. Phys., 114, 7644, 2001. [25] R.D. Groot and T.J. Madden, “Dynamic simulation of diblock copolymer microphase separation,” J. Chem. Phys., 108, 8713, 1997. [26] R.D. Groot, T.J. Madden, and D.J. Tildesley, “On the role of hydrodynamic interactions in block copolymer microphase separation,” J. Chem. Phys., 110, 9739, 1999.

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[27] S. Jury, P. Bladon, M. Cates, S. Krishna, M. Hagen, N. Ruddock, and P.B. Warren, “Simulation of amphiphilic mesophases using dissipative particle dynamics,” Phys. Chem. Chem. Phys., 1, 2051, 1999. [28] M. Venturoli and B. Smit, “Simulating the self-assembly of model membranes,” Phys. Chem. Commun., 10, 1, 1999. [29] J.L. Jones, M. Lal, N. Ruddock, and N.A. Spenley, “Dynamics of a drop at a liquid/solid interface in simple shear fields: a mesoscopic simulation study,” Faraday Discuss., 112, 129, 1999. [30] P. Malfreyt and D.J. Tildesley, “Dissipative particle dynamics of grafted polymer chains between two walls,” Langmuir, 16, 4732, 2000. [31] S. Ymamoto, Y. Maruyama, and S. Hyodo, “Dissipative particle dynamics study of spontaneous vesicle formation of amphiphilic molecules,” J. Chem. Phys., 116(13), 5842, 2003. [32] R.D. Groot, “Electrostatic interactions in dissipative particle dynamics – simulation of polyelectrlytes and anionic surfactants,” J. Chem. Phys., 118, 11265, 2003. [33] J. Bonet-Aval´os and A.D. Mackie, “Dissipative particle dynamics with energy conservation,” Europhys. Lett., 40, 141, 1997. [34] P. Espa˜nol, “Dissipative particle dynamics with energy conservation,” Europhys. Lett., 40, 631, 1997. [35] I. Pagonabarraga and D. Frenkel, “Dissipative particle dynamics for interacting systems,” J. Chem. Phys., 115, 5015, 2001. [36] S.Y. Trofimov, E.L.F. Nies, and M.A.J. Michels, “Thermodynamic consistency in dissipative particle dynamics simulations of strongly nonideal liquids and liquid mixtures,” J. Chem. Phys., 117, 9383, 2002. [37] P. Espa˜nol and M. Revenga, “Smoothed dissipative particle dynamics,” Phys. Rev. E, 67, 026705, 2003. [38] E.G. Flekkøy, P.V. Coveney, and G. DeFabritiis, “Foundations of dissipative particle dynamics,” Phys. Rev. E, 62, 2140, 2000. [39] M. Serrano and P. Espa˜nol, “Thermodynamically consistent mesoscopic fluid particle model,” Phys. Rev. E, 64, 046115, 2001. [40] M. Serrano, G. DeFabritiis, P. Espa˜nol, E.G. Flekkoy, and P.V. Coveney, “Mesoscopic dynamics of voronoi fluid particles,” J. Phys. A: Math. Gen., 35, 1605–1625, 2002. [41] S.R. de Groot and P. Mazur, Non-equilibrium Thermodynamics, North Holland, Amsterdam, 1964. [42] P.W. Cleary and J.J. Monaghan, “Conduction modelling using smoothed particle hydrodynamics,” J. Comput. Phys., 148, 227, 1999. [43] L.B. Lucy, “A numerical testing of the fission hypothesis,” Astron. J., 82, 1013, 1977. [44] J.J. Monaghan, “Smoothed particle hydrodynamics,” Annu. Rev. Astron. Astrophys., 30, 543–574, 1992. [45] H. Takeda, S.M. Miyama, and M. Sekiya, “Numerical simulation of viscous flow by smoothed particle hydrodynamics,” Prog. Theor. Phys., 92, 939, 1994. [46] O. Kum, W.G. Hoover, and H.A. Posch, “Viscous conducting flows with smoothparticle applied mechanics,” Phys. Rev. E, 52, 4899, 1995. ¨ [47] H.C. Ottinger and M. Grmela, “Dynamics and thermodynamics of complex fluids. II. Ilustrations of a general formalism,” Phys. Rev. E, 56, 6633, 1997. [48] B.I.M. ten Bosch, “On an extension of dissipative particle dynamics for viscoelastic flow modelling,” J. Non-Newtonian Fluid Mech., 83, 231, 1999. [49] M. Ellero, P. Espa˜nol, and E.G. Flekkøy, “Thermodynamically consistent fluid particle model for viscoelastic flows,” Phys. Rev. E, 68, 041504, 2003.

8.7 THE DIRECT SIMULATION MONTE CARLO METHOD: GOING BEYOND CONTINUUM HYDRODYNAMICS Francis J. Alexander Los Alamos National Laboratory, Los Alamos, NM, USA

The Direct Simulation Monte Carlo method is a stochastic, particle-based algorithm for solving kinetic theory’s Boltzmann equation. Materials can be modeled at a variety of scales. At the quantum level, for example, time-dependent density functional theory or quantum Monte Carlo may be used. At the atomistic level, typically molecular dynamics is used, while at the continuum level, partial differential equations describe the evolution of conserved quantities and slow variables. Between the atomistic and continuum descriptions lives is the kinetic level. The ability to model at this level is crucial for electron and phonon transport in materials. For classical fluids, especially gases in certain regimes, modeling at this level is required. This article addresses computer simulations at the kinetic level.

1.

Direct Simulation Monte Carlo

The equations of continuum hydrodynamics, such as Euler and NavierStokes, model fluids under a variety of conditions. From capillary flow, to river flow, to the flow of galactic matter, these equations describe the dynamics of fluids over a wide range of space and time scales. However, these equations do not apply in important situations such as gas flow in nanoscale channels and flight in rarefied atmospheric conditions. Because these flows may be collisionless, or nonequilibrium or have sharp gradients, they require a finer-grained description than that provided by hydrodynamics. In these situations, the single particle distribution function, f (r, v, t) is used. Here, f is the number density of atoms or molecules in an infinitesimal six-dimensional volume of phase space, centered at location r and with 2513 S. Yip (ed.), Handbook of Materials Modeling, 2513–2522. c 2005 Springer. Printed in the Netherlands.

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velocity v. For dilute gases, Boltzmann was the first to determine how this distribution changes in time. His insight led to the equation that bears his name [1]: ∂ f (r, v, t) ∂t

=

+v·

dv1

∂ f (r, v, t) ∂r

+

F ∂ f (r, v, t) · m ∂v

d ( f (r, v , t) f (r, v1 , t) − f (r, v, t) f (r, v1 , t))|v − v1 |σ (v − v1 ).

(1)

The Boltzmann equation for hard spheres (1) accounts for all of the processes which change the particle distribution function. The advection term, v · (∂ f /∂r), accounts for the change in f due to particles’ velocities carrying them into and out of a given region of space around r. The force term, (F/m) · (∂ f /∂v) accounts for the change in f due to forces acting on particles of mass m to carry them into and out of a given region of velocity space around v. Terms on the right hand side represent the changes due to collisions. The first term on the right accounts for particles at r, with velocities v1 and v1 which, upon collision, are scattered into a small volume of velocity phase-space around v. The second term accounts for particles at r which, upon collision, are scattered out of this region of velocity space. The collision rate is given by σ and is a function of relative velocities. Though it provides the level of detail necessary to describe many important flows, the Boltzmann equation (1) has several features which make solving it extremely difficult. First, it is a nonlinear integro-differential equation. Only in special cases has it been amenable to exact analytic solution. Second, the Boltzmann equation lives in infinite dimensional phase space. Thus, the methods which work so well for partial differential equations cannot be used. As a result, approximate numerical methods are required. Monte Carlo methods are ideally suited for such high dimensional problems. In the early 1960s, Graeme Bird developed a Monte Carlo technique to solve the Boltzmann equation. This method, now known as Direct Simulation Monte Carlo (DSMC), has been extraordinarily successful in aerospace applications and is also gaining popularity with computational scientists in many fields. A brief outline of DSMC is given here. For more comprehensive descriptions, see Refs. [2–4]. The DSMC method solves the Boltzmann equation by using a representative sample of particles drawn from the actual single particle distribution function. Each DSMC particle represents Ne molecules in the original physical system. For flows of practical interest, typically Ne 1. This approach allows the modeling of extremely large systems while using a computationally tractable number of particles Ntot ≤ 108 , instead of a macroscopic number, 1023 .

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A DSMC simulation is set up in the following way. First the spatial domain, boundary conditions and initial conditions of the simulation are specified. The domain is then partitioned into cells, typically, though not always, of uniform size. These cells are later used in the collision phase of the algorithm. Particles are placed according to a density distribution specified by the initial conditions. To guarantee accuracy, the number of particles used in the simulation should not be too small, i.e., not fewer than about 20 particles per cell [5]. Along with its spatial location ri , each particle is also initialized with a velocity vi . If the system is in equilibrium, this velocity distribution is Maxwellian. However, the velocity distribution can be set to accomodate any flow. The state of the DSMC system is given by the positions and velocities of the particles, {ri , vi }, for i = 1, . . . , N . The DSMC method simulates the dynamics of the single particle distribution using a two-step, splitting algorithm. These steps are advection and collision and model two physical processes at work in the Boltzmann equation. Advection models the free streaming between collisions, and the collision step models the two-body collisions. Each advection–collision step simulates a time t.

2.

Advection Phase

During the advection phase, all particles’ positions are changed from ri to ri + vi t. When a particle strikes a boundary or interface, it responds according to the appropriate boundary condition. The time of the collision is then determined by tracing the straight line trajectory from the initial location ri to the point of impact, rw . The time of flight from the particle’s initial position ˆ i · n), ˆ where nˆ is the unit norto the point of impact is tw = (rw − ri ) · n/(v mal to the surface. After striking the surface, the particle rebounds with a new velocity. This velocity depends on the boundary conditions. The particle then propagates freely for the remaining time t − tw . If, in the remaining time, the same particle again strikes a wall, this process is repeated until all of the time in that step has been exhausted. DSMC can model several types of boundaries (for example, specular surfaces, periodic boundaries, and thermal walls). Upon striking a specular surface, the component of a particle’s velocity normal to the surface is reversed. If a particle should strike a perfect thermal wall at temperature Tw , then all three components of the velocity are reset according to a biasedMaxwellian distribution. The resulting component normal to the wall is distributed as m 2 P⊥ (v ⊥ ) = v ⊥ e−mv ⊥ /2kTw . (2) kTw

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The individual parallel components are distributed as

P (v ) =

2 m e−mv /2kTw , 2πkTw

(3)

where Tw is the wall temperature, m is the particle’s mass and k is Boltzmann’s constant. Along with the tangential velocity component generated by thermal equilibration with the wall, an additional velocity is required to account for any translational motion of the wall. The distribution (3) is given in the rest frame of the wall. Assume the x and y axes are parallel to the wall. If the wall is moving in the lab frame, for example in the x-direction with velocity u w , then u w is added to the x-component of velocity for particles scattering off the wall. The components of the velocity of a particle leaving a thermal wall are then

vx =

vy =

v⊥ =

kTw RG + u w m

(4)

kTw R m G

(5)

−

2kTw ln R m

(6)

where R is a uniformly distributed random number in [0,1) and RG , RG are Gaussian distributed random numbers with zero mean and unit variance. For most engineering applications, gas-surface scattering is far more complicated. Nevertheless, these scattering rates can usually be effectively modeled in the gas-surface scattering part of the algorithm [6].

3.

Collision Phase

Interparticle collisions are addressed independently from the advection phase. For this to be accurate, the interaction potential between molecules must be short-range. While many short-range interaction models exist, the DSMC algorithm is formulated in this article for a dilute gas of hard sphere particles with diameter σ . When required for specific engineering applications, more realistic representations of the molecular interaction may be used [2, 8]. During the collision phase, some of the particles are selected at random to undergo collisions with each other. The selection process is determined by classical kinetic theory. While there are many ways to accomplish this, a simple and effective method is to sort the particles into spatial cells, the size of which should be less than a mean free path. Only particles in the same cell

The direct simulation monte carlo method

2517

are allowed to collide. As with the particles themselves, the collisons are only statistical surrogates of the actual collisions that would occur in the system. At each time-step, and within each cell, sets of collisions then are generated. All pairs of particles in a cell are eligible to become collision partners. This eligibility is independent of the particles’ positions within the cell. Only the magnitude of the relative velocity between particles is used to determine their collision probability. Even particles that are moving away from each other may collide. The collision probability for a pair of hard spheres, i and j , is given by Pcoll (i, j ) =

2|vi − v j | Nc (Nc − 1)|v rel |

(7)

where |v rel | is the mean magnitude of the relative velocities of all pairs of particles in the cell, and Nc is the number of particles in the cell. To implement this in an efficient manner, a pair of potential collision partners, i and j , is selected at random from the particles within the cell. The pair collides if |vi − v j | > r, v r,max

(8)

where v r,max is the maximum relative speed in the cell and r is a uniform random variable chosen from the interval [0, 1). (Rather than determining v r,max exactly each time step, it is sufficient to simply update it everytime a relative velocity is actually calculated.) If the pair does not collide,then another pair is selected and the process repeats until the required number of pairs Mcoll (explained below) in the cell have been handled. If the pair does collide, then the new velocities of the particles are determined by the following procedure, and the process repeats until all collisions have been processed. In an elastic hard sphere collision, linear momentum and energy are conserved. These conserved quantities fix the magnitude of the relative velocity and center of mass velocity v r = |vi − v j | = |vi − vj | = v r ,

(9)

and vcm = 12 (vi + v j ) = 12 (vi + vj ) = vcm ,

(10)

where vi and vj are the post-collision velocities. In three dimensions, Eqs. (9) and (10) constrain four of the six degrees of freedom. The two remaining degrees of freedom are chosen at random. These correspond to the azimuthal and polar angles, θ and φ, for the post-collision relative velocity. vr = v r [(sin θ cos φ) xˆ + (sin θ sin φ) yˆ + cos θ zˆ ].

(11)

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F.J. Alexander

For the hard sphere model, these angles are uniformly distributed over the unit sphere. Specifically, the azimuthal angle φ is uniformly distributed between 0 and 2π, and the angle θ has the following distribution P(θ) dθ = 12 sin θ dθ.

(12)

Since only sin θ and cos θ are required, it is convenient to change variables from θ to ζ = cos θ.Then ζ is chosen uniformly from [−1, 1], and setting cos θ = ζ and sin θ = 1 − ζ 2 . These values are used in (11). The post-collision velocities are then given by vi = vcm + 12 vr , vj = vcm − 12 vr .

(13)

The mean number of collisions that take place in a cell during a time-step is given by Nc (Nc − 1)πσ 2 v r Ne t , (14) 2Vc where Vc is the volume of the cell, and v r is the average relative velocity in the cell. To avoid the costly computation of v r , and since the ratio of total accepted to total candidates is v r Mcoll = . (15) Mcand v r,max Using (14) and (15) Mcoll =

Nc (Nc − 1)πσ 2 v r,max Ne t , (16) 2Vc produces the number of candidate pairs to select over a time step t. Note that Mcoll will, on average, equal the acceptance probability (8) multiplied by (16) and is independent of v r,max . Setting v r,max too high still processes the same number of collisions on the average, but the program is inefficient because the acceptance probability is low. This procedure selects collision pairs according to (7). Even if the value of v r,max is overestimated, the method is still correct, though less efficient because too many potential collisions are rejected. A better option is to make a guess which slightly overestimates v r,max [7]. To maintain accuracy while using the two-step, advection–collision algorithm, t should only be a fraction of the mean free time. If too large a time-step is used, then particles move too far between collisions. On the other hand, if the spatial cells are too large, then collisions can occur between particles which are “too far” from each other. Time steps beyond a mean free time and spatial cells larger than a mean free path have the effect of artificially enhancing transport coefficients such as viscosity and thermal conductivity [17, 18]. Mcand =

The direct simulation monte carlo method

4.

2519

Data Analysis

In DSMC, as with other stochastic methods, most quantities of interest are computed as averages. For example, the instantaneous, fluctuating mass ˜ t), and energy density e(r, density, ρ(r, ˜ t), momentum density, p(r, ˜ t) are given by   ˜ t)   ρ(r,





1  m  ˜ t) = . p(r, mvi   Vs 1 2 i e(r, ˜ t) m|vi | 2

(17)

The sum is over particles over a volume of space surrounding r. Because it contains details of the single particle distribution function, DSMC can provide far more information than what is contained in the hydrodynamic variables above. However, this extra information comes at a price. As with other Monte Carlo-based methods, DSMC suffers from errors√due to the finite number of particles used. Convergence is typically O(1/ N ). These errors can be reduced by using more particles in the simulation, but for some systems, that can be prohibitive. For a detailed discussion on the statistical errors in DSMC and the techniques to estimate them in a variety of flow situations, refer to the recent work of Hadjiconstantinou et al. [9]. To reduce the fluctuations in the average quantities, a large number of particles is used, or, in the case of time-independent flows, statistics are gathered over a long run after the system has reached its steady state. For timedependent problems, a statistical ensemble of realizations of the simulation is used. Physical quantities of interest can be obtained from these averages. From the description of the algorithm above it should be clear that DSMC is computationally very expensive and should not be used in situations when Navier-Stokes or Euler PDE solvers apply. To check if DSMC is necessary, one should determine the Knudsen number K n. This dimensionless parameter is defined as K n = λ/L, where L is the characteristic length scale of the physical system, and λ is the molecular mean free path (i.e., the average distance between successive collisions of a given molecule). While there is no clear dividing line, a useful rule of thumb is that DSMC should be used when K n > 1/10. For a dilute gas, the mean free path λ is given by λ= √

1 2 πσ 2 n

,

(18)

where n is the number density, and σ is the effective diameter of the molecule. Air at atmospheric pressure has λ ≈ 50 nm. In the upper atmosphere, however, (e.g., > 100 km altitude), the mean free path is several meters. The Knudsen number for air flow through a nanoscale channel or around a meter scale space vehicle can therefore easily exceed K n ≈ 1. For these cases

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F.J. Alexander

continuum hydrodynamics is not an option and DSMC should be used. Other, more detailed criteria can also be used [7].

5.

Discussion

Despite obvious similarities, key differences exist between DSMC and molecular dynamics. In molecular dynamics, the trajectory of every particle in the gas is computed from Newton’s equations, given an empirically determined interparticle potential. Each MD particle represents one atom or molecule. In DSMC, each particle represents Ne atoms or molecules, where Ne is on the order of 1/20 of the number of atoms/molecules in a cubic mean free path. Using MD to simulate one cubic micron of air at standard temperature and pressure MD requires integrating Newton’s equations for approximately 1010 molecules for 104 time steps to model one mean free time. With DSMC, only 105 particles and approximately 10 time steps are required. The DSMC method is therefore an efficient alternative for simulating a dilute gas. The method can be viewed as a simplified molecular dynamics (though DSMC is several orders of magnitude faster). DSMC can also be considered a Monte Carlo method for solving the time-dependent nonlinear Boltzmann equation. Instead of exactly calculating collisions as in molecular dynamics, the DSMC method generates collisions stochastically with scattering rates and post-collision velocity distributions determined from the kinetic theory of a dilute gas. Although DSMC simulations are not correct at the length scale of an atomic diameter, they are accurate at scales smaller than a mean free path. However, if more detail is required, then MD is the best option.

6.

Outlook

Though it originated in the aerospace community, since the mid-1980s DSMC has been used in a variety of other areas which demand a kinetic level formulation. These include the study of nonequilibrium fluctuations [10], nanoscale fluid dynamics [11] and granular gases [13]. Originally, DSMC was confined to to dilute gases. Several advances, however, such as the consistent Boltzmann algorithm (CBA) [8] and Enskog simulation Monte Carlo (ESMC) [12] have extended DSMC’s reach to nonideal, dense gases. Among other areas, CBA has found applications in heavy ion dynamics [14]. Similar methods also are used in transport theories of condensed matter physics [15]. While the DSMC method has been quite successful in these applications, only within the last decade has it been put on a firm mathematical foundation.

The direct simulation monte carlo method

2521

Wagner [16], for example, proved that the method, in the limit of infinite particle number, has a deterministic evolution which solves an equation “close” to the Boltzmann equation. Subsequent work has shown that DSMC and its variants converge to a variety of kinetic equations. Other analytical work has determined the error incurred in DSMC by the use of a space and time discretization [17, 18]. Efforts have been made to improve the computational efficiency of DSMC for flows in which some spatial regions are hydrodynamic and others kinetic. Pareschi and Caflisch [19] have developed an implicit DSMC method which seamlessly interpolates between the kinetic and hydrodynamic scales. Another hybrid approach optimizes performance by using DSMC where required and then using Navier-Stokes or Euler in regions where allowed. The two methods are then coupled across an interface to provide information to each other [20, 21]. This is currently a rapidly growing field.

Acknowledgments This document was prepared at LANL under the auspices of the Department of Energy LA-UR 03-7358.

References [1] C. Cercignani, The Boltzmann Equation and its Applications, Springer, New York, 1988. [2] G.A. Bird, Molecular Gas Dynamics, Clarendon, Oxford, 1976; G.A. Bird, Molecular Gas Dynamics and the Direct Simulation of Gas Flows, Clarendon, Oxford, 1994. [3] A.L. Garcia, Numerical Methods for Physics, Prentice Hall, Englewood Cliffs, 1994. [4] E.S. Oran, C.K. Oh, and B.Z. Cybyk, Annu. Rev. Fluid Mech., 30, 403, 1998. [5] M. Fallavollita, D. Baganoff, and J. McDonald, J. Comput. Phys., 109, 30, 1993; G. Chen and I. Boyd, J. Comput. Phys., 126, 434, 1996. [6] A.L. Garcia and F. Baras, Proceedings of the Third Workshop on Modeling of Chemical Reaction Systems, Heidelberg, 1997 (CD-ROM only). [7] I. Boyd, G. Chen, and G. Candler, Phys. Fluids, 7, 210, 1995. [8] F. Alexander, A.L. Garcia, and B. Alder, Phys. Rev. Lett., 74, 5212, 1995; F. Alexander, A.L. Garcia, and B. Alder, in 25 Years of Non-Equilibrium Statistical Mechanics, J.J. Brey, J. Marco, J.M. Rubi, and M. San Miguel (eds.), Springer, Berlin, 1995; A. Frezzotti, A particle scheme for the numerical solution of the Enskog equation, Phys. Fluids, 9(5), 1329–1335, 1997. [9] N. Hadjiconstantinou, A. Garcia, M. Bazant, and G. He, J. Comput. Phys., 187, 274–297, 2003. [10] F. Baras, M.M. Mansour, A.L. Garcia, and M. Mareschal, J. Comput. Phys., 119, 94, 1995. [11] F.J. Alexander, A.L. Garcia, and B.J. Alder, Phys. Fluids, 6, 3854, 1994. [12] J.M. Montanero and A. Santos, Phys. Rev. E, 54, 438, 1996; J. M. Montanero and A. Santos, Phys. Fluids, 9, 2057, 1997.

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[13] H.J. Herrmann and S. Luding, Continuum Mechanics and Thermodynamics, 10, 189, 1998; J. Javier Brey, F. Moreno, R. García-Rojo, and M.J. Ruiz-Montero, “Hydrodynamic Maxwell demon in granular systems,” Phys. Rev. E, 65, 011305, 2002. [14] G. Kortemeyer, F. Daffin, and W. Bauer, Phys. Lett. B, 374, 25, 1996. [15] C. Jacoboni and L. Reggiani, Rev. Mod. Phys., 55, 645, 1983. [16] W. Wagner, J. Stat. Phys., 66, 1011, 1992. [17] F.J. Alexander, A.L. Garcia, and B.J. Alder, Phys. Fluids, 10, 1540, 1998; Phys. Fluids, 12, 731, 2000. [18] N.G. Hadjiconstantinou, Phys. Fluids, 12, 2634, 2000. [19] L. Pareschi and R.E. Caflisch, J. Comput. Phys., 154, 90, 1999. [20] H.S. Wijesinghe and N.G. Hadjiconstantinou, “Hybrid atomistic-continuum formulations for multiscale hydrodynamics,” Article 8.8, this volume. [21] A.L. Garcia, J.B. Bell, W.Y. Crutchfield, and B.J. Alder, J. Comput. Phys., 154, 134, 1999.

8.8 HYBRID ATOMISTIC–CONTINUUM FORMULATIONS FOR MULTISCALE HYDRODYNAMICS Hettithanthrige S. Wijesinghe and Nicolas G. Hadjiconstantinou Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA

Hybrid atomistic-continuum formulations allow the simulation of complex hydrodynamic phenomena at the nano and micro scales without the prohibitive cost of a fully atomistic approach. Hybrid formulations typically employ a domain decomposition strategy whereby the atomistic model is limited to regions of the flow field where required and the continuum model is implemented side-by-side in the remainder of the domain within a single computational framework. This strategy assumes that non-continuum phenomena are localized and that coupling of the two descriptions can be achieved in a spatial region where both formulations are valid. In this article we review hybrid atomistic-continuum methods for multiscale hydrodynamic applications. Both liquid and gas formulations are considered. The choice of coupling method and its relation to the fluid physics as well as the differences between incompressible and compressible hybrid methods are discussed using illustrative examples.

1.

Background

While the fabrication of MEMS devices has received much attention, transport mechanisms at the nano and micro scale environment are currently poorly understood. Furthermore, efficient and accurate design capabilities for nano and micro engineering components are also somewhat limited since design tools based on continuum formulations are increasingly reaching their limit of applicability. 2523 S. Yip (ed.), Handbook of Materials Modeling, 2523–2551. c 2005 Springer. Printed in the Netherlands.

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H.S. Wijesinghe and N.G. Hadjiconstantinou

For gases, deviation from the classical Navier–Stokes behavior is typically quantified by the Knudsen number, K n = λ/L where λ is the atomistic mean free path ( = 4.9 × 10−8 m for air) and L is a characteristic dimension. The Navier–Stokes formulation is found to be invalid for K n 0.1. Ducts of width 100 nm or less which are common in N/MEMS correspond to Knudsen numbers of order 1 or above [1]. The Knudsen number for Helium leak detection devices and mass spectrometers can reach values of up to 200 [2]. Also material processing applications such as chemical vapor deposition and molecular beam epitaxy involve high Knudsen number flow regimes [3]. The Navier–Stokes description also deteriorates in the presence of sharp gradients. One example comes from Navier–Stokes formulations for high Mach number shock waves which are known to generate spurious post-shock oscillations [4, 5]. In such cases, a Knudsen number can be defined using the characteristic length scale of the gradient. A significant challenge therefore exists to develop accurate and efficient design tools for flow modeling at the nano and micro scales. Liquids in nanoscale geometries or under high stress and liquids at material interfaces may also exhibit deviation from Navier–Stokes behavior [6]. Examples of problems which require modeling at the atomistic scale include the moving contact-line problem between two immiscible liquids [6], corner singularities, the breakup and merging of droplets [7], dynamic melting processes [8], crystal growth from a liquid phase and polymer/colloid wetting near surfaces. Accurate modeling of wetting phenomena is of particular concern in predicting microchannel flows. While great accuracy can be obtained by an atomistic formulation over a broader range of length scales, a substantial computational overhead is associated with this approach. To mitigate this cost, “hybrid” atomisticcontinuum simulations have been proposed as a novel approach to model hydrodynamic flows across multiple length and time scales. These hybrid approaches limit atomistic models to regions of the flow field where needed, and allow continuum models to be implemented in the remainder of the domain within a single computational framework. A hybrid method therefore allows the simulation of complex hydrodynamic phenomena which require modeling at the microscale without the prohibitive cost of a fully atomistic calculation. In what follows we provide an overview of this rapidly expanding field and discuss recent developments. We begin by discussing the challenges associated with hybrid formulations, namely the choice of the coupling method and the imposition of boundary conditions on atomistic simulations. We then illustrate hybrid methods for incompressible and compressible flows by describing recent archetypal approaches. Finally we discuss the effect of statistical fluctuations in the context of developing robust criteria for adaptive placement of the atomistic description.

Hybrid atomistic–continuum formulations

2.

2525

Challenges

Over the years a fair number of hybrid simulation frameworks have been proposed leading to some confusion over the relative merits and applicability of each approach. Original hybrid methods focused on dilute gases [9–12], which are arguably easier to deal with within a hybrid framework than dense fluids, mainly because boundary condition imposition is significantly easier in gases. The first hybrid methods for dense fluids appeared a few years later [13–16]. These initial attempts have led to a better understanding of the challenges associated with hybrid methods. Coupling the continuum and atomistic formulations requires a region of space where information exchange takes place between the two descriptions. This information exchange between the two subdomains is typically in the form of state variables and/or hydrodynamic fluxes, with the latter typically measured across the matching interface. This process may be viewed as a boundary condition exchange between subdomains. In some cases transfer of information is facilitated by an overlap region. The transfer of information from the atomistic subdomain to the continuum subdomain is fairly straightforward. A hydrodynamic field can be obtained from atomistic simulation data through averaging (see for example the article “The Direct Simulation Monte Carlo: going beyond continuum hydrodynamics” in the Handbook). The relative error due to statistical sampling in atomistic hydrodynamic formulations was also recently characterized [17]. Imposition of the latter data as boundary conditions on the continuum method is also well understood and depends on the numerical method used (see article “Finite Difference, Finite Element and Finite Volume Methods for Partial Differential Equations” of the Handbook). As discussed below, the most challenging aspect of the information exchange lies in imposing the hydrodynamic field obtained from the continuum subdomain onto the atomistic description, a process which is not well defined in the absence of the complete distribution function (hydrodynamic fields correspond to the first few moments of the distribution function). Thus to a large extent, the two major issues in developing a hybrid method is the choice of a coupling method and the imposition of boundary conditions on the atomistic simulation. Generally speaking, these two can be viewed as decoupled. The coupling technique can be developed on the basis of matching two compatible and equivalent over some region of space hydrodynamic descriptions and can thus be borrowed from the already existing and extensive continuum-based numerical methods literature. Boundary condition imposition can be posed as a general problem of imposing “macroscopic” boundary conditions on an atomistic simulation. In our opinion this is a very challenging problem that has not been, in general, resolved to date completely satisfactorily. Boundary condition imposition on the atomistic subdomain is discussed shortly.

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3. 3.1.

H.S. Wijesinghe and N.G. Hadjiconstantinou

Developing a Hybrid Method The Choice of Coupling Method

Coupling a continuum to an atomistic description is meaningful in a region where both can be presumed valid. In choosing a coupling method it is therefore convenient to draw upon the wealth of experience and large cadre of coupling methods nearly 50 years of continuum computational fluid dynamics have brought us. Coupling methods for the compressible and incompressible formulations generally differ, since the two correspond to two different physical and mathematical limits. Faithful to their mathematical formulations, the compressible formulation lends itself naturally to time-explicit flux-based coupling while incompressible formulations are typically coupled using either state properties (Dirichlet) or gradient information (Neumann). Given that the two formulations have different limits of applicability and physical regimes in which each is significantly more efficient than the other, care must be exercised when selecting the ingredients of the hybrid method. In other words, the choice of a coupling method and continuum subdomain formulation needs to be based on the degree to which compressibility effects are important in the problem of interest, and not on a preset notion that a particular coupling method is more appropriate than all others. The latter approach was recently pursued in a variety of studies which enforce the use of the compressible formulation to steady and essentially incompressible problems to achieve coupling by time-explicit flux matching. This approach is not recommended. On the contrary, for an efficient simulation method, similarly to the case of continuum solution methods, it is important to allow the flow physics to dictate the appropriate formulation, while the numerical implementation is chosen to cater to the particular requirements of the latter. Below, we expand on some of the considerations which influence the choice of coupling method under the assumption that the hybrid method is applied to problems of practical interest and therefore the continuum subdomain is appropriately large. Our discussion focuses on timescale considerations that are more complex but equally important to limitations resulting from lengthscale considerations, such as the size of the atomistic region(s). It is well known that the timestep for explicit integration of the compressible Navier–Stokes formulation, τc , scales with the physical timestep of the problem, τx , according to [18] M τx (1) 1+ M where M is the Mach number. As the Mach number becomes small, we are faced with the classical stiffness problem whereby the numerical efficiency of the solution method suffers [18] due to disparity of the time scales in the τc ≤

Hybrid atomistic–continuum formulations

2527

system of governing equations. For this reason, when the Mach number is small, the incompressible formulation is used which allows integration at the physical timestep τx . In the hybrid case matters are complicated by the introduction of the atomistic integration timestep, τm , which is at most of the order of τc in gases (if the discretization scale is O(λ)) and in most cases significantly smaller. Thus as the global domain of interest grows, the total integration time grows, and transient calculations in which the atomistic subdomain is explicitly integrated in time become more computationally expensive and eventually infeasible. The severity of this problem increases with decreasing Mach number and makes unsteady incompressible problems very computationally expensive. New integrative frameworks which coarse grain the time integration of the atomistic subdomain are therefore required. Fortunately, for low speed steady problems implicit (iterative) methods exist which provide solutions without the need for explicit integration of the atomistic subdomain to the global problem steady state. One such implicit method is discussed in this review; it is known as the Schwarz method. This method decouples the global evolution timescale from the atomistic evolution timescale (and timestep) by achieving convergence to the global problem steady state through an iteration between steady state solutions of the continuum and atomistic subdomains. Since the atomistic subdomain is small, explicit integration to its steady state is feasible. Although the steady assumption may appear restrictive, it is interesting to note that the majority of both compressible and incompressible test problems solved to date have been steady. A variety of other iterative methods may be suitable if they provide for timescale decoupling. The choice of the Schwarz coupling method using state variables versus a flux matching approach was motivated by the fact (as explained below) that state variables suffer from smaller statistical noise and are thus easier to prescribe on a continuum formulation. The above observations do not preclude the use of the compressible formulation in the continuum subdomain for low speed flows. In fact, preconditioning techniques which allow the use of the compressible formulation at very low Mach numbers have been developed [18]. Such a formulation can, in principle, be used to solve for the continuum subdomain while this is being coupled to the atomistic subdomain via an implicit (eg., Schwarz) iteration. What should be avoided is a time-explicit compressible flux-matching coupling procedure for solving essentially incompressible steady state problems. The issues discussed above have not been very apparent to date because in typical test problems published so far, the continuum and atomistic subdomains are of the same size (and, of course, small). In this case the large cost of the atomistic subdomain masks the cost of the continuum subdomain and also typical evolution timescales (or times to steady state) are small. It should not be forgotten, however, that hybrid methods make sense when the continuum subdomain is significantly larger than the atomistic subdomain.

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H.S. Wijesinghe and N.G. Hadjiconstantinou

The stiffness resulting from a small timestep in the atomistic subdomain may be remedied by implicit timestepping methods [19]. However, flux-based coupling additionally suffers from adverse signal to noise ratios in connection with the averaging required for imposition of boundary conditions from the atomistic subdomain to the continuum subdomain. In the case of an ideal gas it has been shown for low speed flows [17] that for the same number of samples, flux (shear stress, heat flux) averaging exhibits relative noise, E f , which scales as Ef ∝

E sv Kn

(2)

where E sv is the relative noise in √ the corresponding state variable (velocity, temperature) which varies as 1/ (number of samples). Here K n is based on the characteristic lengthscale of the transport gradients. Since, by assumption, in the matching region a continuuum description is appropriate, we expect K n = λ/L 1. It thus appears that flux coupling will be significantly disadvantaged in this case, since 1/K n 2 times the number of samples required by state-variable averaging is required to achieve comparable noise levels in the matching region. Statistical noise has important implications on hybrid methods which will be discussed throughout this paper. The effect of statistical noise becomes of critical importance in unsteady incompressible flows which are discussed later.

4.

Boundary Condition Imposition

Consider the boundary, ∂ of the atomistic region on which we wish to impose a set of hydrodynamic (macroscopic) boundary conditions. Typical implementations require the use of particle reservoirs R (see Fig. 1) in which particle dynamics may be altered in such a way that the desired boundary conditions appear on ∂; the hope is that the influence of the perturbed dynamics in the reservoir regions decays sufficiently fast and does not propagate into the region of interest, that is, the relaxation distance both for the velocity distribution function and the fluid structure is small compared to the characteristic scale of . Since ∂ represents the boundary with the continuum region, R extends into the continuum subdomain. Knowledge of the continuum solution in R is typically used to aid imposition of the above on ∂. In a dilute gas, the non-equilibrium distribution function in the continuum limit has been characterized [20] and is known as the Chapman–Enskog distribution. Use of this distribution to impose boundary conditions on atomistic simulations of dilute gases results in a robust, accurate and theoretically elegant approach. Typical implementations [21] require the use of particle generation and initialization within R. Particles that move into within the

Hybrid atomistic–continuum formulations

2529 ∂Ω

Ω

R

Figure 1. Continuum to atomistic boundary condition imposition using reservoirs.

simulation timestep are added to the simulation whereas particles remaining in R are discarded. For liquids, both the particle velocity and the fluid structure distribution functions are important and need to be imposed. Unfortunately no theoretical results for these distributions exist. A related issue is that of domain termination; due to particle interactions, , or in the presence of a reservoir R, needs to be terminated in a way that does not have significant effect on the fluid state inside of . As a result, researchers have experimented with possible methods to impose boundary conditions. It is now known that similarly to a dilute gas, use of a Maxwell–Boltzmann distribution for particle velocities leads to slip [14]. In [22] a Chapman–Enskog distribution is used to impose boundary conditions to generate a liquid shear flow. In this approach, particles crossing ∂ acquire velocities that are drawn from a Chapman–Enskog distribution parametrized by the local values of the required velocity and stress boundary condition. Although this approach was only tested for a Couette flow, it appears to give reasonable results (within atomistic fluctuations). Since in Couette flow no flow normal to ∂ exists, ∂ can be used as symmetry boundary separating two back-to-back shear flows; this sidesteps the issue of domain termination. Boundary conditions on MD simulations can also be imposed through the method of constraint dynamics [13]. Although the approach in [13] did not allow hydrodynamic fluxes across the matching interface, the latter feature can be integrated into this approach with a suitable domain termination. In a different approach [16] external forces are used to impose boundary conditions. More specifically, the authors apply an external field with a magnitude such that the total force on the fluid particles in R is the one required by momentum conservation as required by the coupling procedure.

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The outer boundary of the reservoir region is terminated by using a force preventing particles from leaving the domain and an ad-hoc weighting factor for the force distribution on particles. This weighting factor diverges as particles approach the edge of R. This prevents their escape and also ensures new particles introduced in R move towards . Particles introduced into the reservoir are given velocities drawn from a Maxwell–Boltzmann distribution, while a Langevin thermostat keeps the temperature constant. The method appears to be successful although the non-unique choice of force fields and Maxwell– Boltzmann distribution makes it not very theoretically pleasing. It is also not clear what the effect of these forces are on the local fluid state (it is well known that even in a dilute gas gravity driven flow [23] exhibits significant deviations from Navier–Stokes behavior) but this effect is probably small since force fields are only acting in the reservoir region. The above approach was refined [24] by using a version of the Usher algorithm to insert particles in the energy landscape such that they have the desired specific energy, which is beneficial to imposing a desired energy current while eliminating the risk of particle overlap at some computational cost. This approach uses a Maxwell– Boltzmann distribution, however, for the initial velocities of the inserted particles. Temperature gradients are imposed by a small number of thermostats placed in the direction of the gradient. Although no proof exists that the disturbance to the particle dynamics is small, it appears that this technique is successful at imposing boundary conditions with moderate error [24]. A method for terminating incompressible molecular dynamics simulations with small effect on particle dynamics has been suggested and used [14]. This simply involves making the reservoir region fully periodic. In this manner, the boundary conditions on ∂ also impose a boundary value problem on R, where the inflow to is the outflow from R. As R becomes bigger, the gradients in R become smaller and thus the flowfield in R will have a small effect on the solution in . The disadvantage of this method is the number of particles that are needed to fill R as this grows, especially in high dimensions. We believe that significant contributions can still be made by developing methods to impose boundary conditions in hydrodynamically consistent and, most importantly, rigorous approaches.

4.1.

Particle Generation in Dilute Gases Using the Chapman–Enskog Velocity Distribution Function

In the case of dilute gases the atomistic structure is not important and the gas is characterized by the single-particle distribution function. This relative simplicity has led to solutions of the governing Boltzmann equation [25, 26] in the Navier–Stokes limit. The resultant Chapman–Enskog solution [20, 25] can be used to impose boundary conditions in a robust and rigorous manner.

Hybrid atomistic–continuum formulations

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In what follows we illustrate this procedure using the direct simulation Monte Carlo (DSMC) as our dilute gas simulation method. DSMC is an efficient method for the simulation of dilute gases which solves the Boltzmann equation [27] using a splitting approach. The time evolution of the system is approximated by a sequence of discrete timesteps, t, in which particles undergo, successively, collisionless advection and collisions. An appropriate number of collisions are performed between randomly chosen particle pairs within small cells of linear size x. DSMC is discussed further in the article, “The Direct Simulation Monte Carlo Method: going beyond continuum hydrodynamics” of the Handbook. In a typical hybrid implementation, particles are created in a reservoir region in which the continuum field to be imposed is known. Correct imposition of boundary conditions requires generation of particles with the correct single particle distribution function which includes the local value of the number density [21]. Current implementations [21, 28, 29] show that linear interpolation of the density gradient within the reservoir region provides sufficient accuracy. Generation of particles according to a linear density gradient can be achieved with a variety of methods, including acceptance-rejection schemes. In the next section we outline an approach for generation of particle velocities from a Chapman–Enskog distribution parametrized by the required flow variables. After particles are created in the reservoir they move for a single DSMC timestep. Particles that enter the atomistic region are incorporated into the standard convection/collision routines of the DSMC algorithm. Particles that remain in the reservoir are discarded. Particles that leave the atomistic region are also deleted from the computation.

4.2.

Generation of Particle Velocities Using the Chapman–Enskog Velocity Distribution

The Chapman–Enskog velocity distribution function f (C) can be written as [30], f (C) = f 0 (C)(C)

(3)

where, C = C/(2kT /m) 1 ∈ f 0 (C) = 3/2 e−C π and,

1/2

is the normalized thermal velocity, (4)

2 2 C −1 (C) = 1 + (qx Cx + q y C y + qz Cz ) 5 − 2(τx y Cx C y + τx z Cx Cz + τ yz C y Cz ) − τx x (Cx2 − Cz2 ) − τ yy (C y2 − Cz2 )

(5)

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with,

2m 1/2 ∂ T kT ∂ xi µ ∂v i ∂v j 2 ∂v k τi j = + − δi, j P ∂x j ∂ xi 3 ∂ xk qi = −

κ P

(6) (7)

where qi and τi j are the dimensionless heat flux vector and stress tensor respectively with µ, κ, P and v = (v x , v y , v z ) being the viscosity, thermal conductivity, pressure and mean fluid velocity. An “Acceptance–Rejection” scheme [30, 31] can be utilized to generate Chapman–Enskog distribution velocities. In this scheme an amplitude parameter A = 1 + 30B is first chosen where B = max(|τi j |, |qi |). Next a trial velocity Ctry is drawn from the Maxwell–Boltzmann equilibrium distribution function f 0 given by Eq. (4). Note f 0 is a normal (Gaussian) distribution that can be generated using standard numerical techniques [32]. The trial velocity Ctry is accepted if it satisfies AR ≤ (Ctry ) where R is a uniform deviate in [0, 1). Otherwise a new trial velocity Ctry is drawn. The final particle velocity is given by C = (2kT /m)1/2 Ctry + v

5.

(8)

Incompressible Formulations

Although in some cases compressibility may be important, a large number of applications are typically characterized by flows where use of the incompressible formulation results in a significantly more efficient approach [18]. As explained earlier, our definition of incompressible formulation is based on the flow physics and not on the numerical method used. Although in our example implementation below we have used a finite element discretization based on the incompressible formulation, we believe that a preconditioned compressible formulation [18] could also be used to solve the continuum subdomain problem if it could be successfully matched to the atomistic solution through a coupling method which takes into account the incompressible nature of the (low speed) problem to provide solution matching consistent with the flow physics. From the variety of methods proposed to date, it is becoming increasingly clear that almost any continuum–continuum coupling method can be used so long as it is properly extended to account for boundary condition imposition. The challenge thus lies more in choosing a method that best matches the physics of the problem of interest (as explained above) rather than developing general methods for large classes of problems. Below we illustrate a hybrid implementation appropriate for incompressible steady flow using the Schwarz alternating coupling method.

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Before we proceed with our example, a subtle numerical issue associated with the incompressible formulation should be discussed. Due to inherent statistical fluctuations, boundary conditions obtained from the atomistic subdomain may lead to mass conservation discrepancies. Although this phenomenon is an artifact of the finite sampling, in the sense that if a sufficiently large (infinite) number of samples are taken the mean field obtained from the atomistic simulation should be appropriately incompressible, it is sufficient to cause a numerical instability in the continuum calculation. A simple correction that can be applied consists of removing the discrepancy in mass flux equally across all normal velocity components of the atomistic boundary data that are to be imposed on the continuum subdomain. If 1 is the portion of the continuum subdomain φ that receives boundary data from the atomistic subdomain (φ ⊇ 1 ), n is the unit outward normal vector to φ, and d S is a differential element of φ, the correction to the atomistic data on 1 , v1 , can be written as

(v1 .n)corrected = v1 .n −

φ

vφ .ndS

1

dS

(9)

Tests with various problems [14, 15, 28] indicate that this simple approach is successful at removing the numerical instability.

5.1.

The Schwarz Alternating Method for Steady Flows

The Schwarz method was originally proposed for molecular dynamicscontinuum methods [14, 15], but it is equally applicable to DSMC-continuum hybrid methods [28, 33]. This approach was chosen because of its ability to couple different descriptions through Dirichlet boundary conditions (easier to impose on liquid atomistic simulations compared to flux conditions, because fluxes are non-local in liquid systems), and its ability to reach the solution steady state in an implicit manner which requires only steady solutions from each subdomain. The importance of the latter characteristic cannot be overemphasized; the implicit convergence in time through steady solutions guarantees timescale decoupling that is necessary for the solution of macroscopic problems; the integration of atomistic trajectories at the atomistic timestep for total times corresponding to macroscopic evolution times is, and will for a long time be, infeasible, while integration of the small molecular region to its steady state solution is feasible. A continuum–continuum domain decomposition can be used to illustrate the Schwarz alternating method as shown graphically in Figs. 2–4 (adapted from [34]) to solve for the velocity in a simple, one-dimensional problem, a pressure driven Poiseuille flow. Starting with a zero guess for the solution in domain 2, the first steady solution in domain 1 can be obtained. This provides the first boundary condition for a steady solution in domain 2 (Fig. 2). The

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H.S. Wijesinghe and N.G. Hadjiconstantinou

Domain 1 Domain 2 Overlap Region

Wall

Wall First BC for domain 2 First BC for domain 1

Domain 1 first iteration a x

b L

Figure 2. Schematic illustrating the Schwarz alternating method for Poiseuille flow. Solution at the first Schwarz iteration. Adapted from [34].

Figure 3. Schematic illustrating the Schwarz alternating method for Poiseuille flow. Solution at the second Schwarz iteration. Adapted from [34].

Hybrid atomistic–continuum formulations

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Figure 4. Schematic illustrating the Schwarz alternating method for Poiseuille flow. Solution at the third Schwarz iteration. Adapted from [34].

new solution in domain 2 provides an updated second boundary condition for domain 1 (Fig. 3). This process is repeated until the two solutions are identical in the overlap region. As seen in Fig. 4 the solution across the complete domain rapidly approaches the steady state solution. This method is guaranteed to converge for elliptic problems [35]. The Schwarz method was recently applied [33] to the simulation of flow through micromachined filters. These filters have passages that are sufficiently small that require an atomistic description for the simulation of the flow through them. Depending on the geometry and number of filter stages the authors have reported computational savings ranging from 2 to 100.

5.2.

Driven Cavity Test Problem

In this section we solve the steady driven cavity problem using the Schwarz alternating method. The driven cavity problem is used here as a test problem for verification and illustration purposes. In fact, although wall effects might be important in small scale flows, and a hybrid method which treats only the regions close to the walls using the molecular approach may be an interesting problem, the formulation chosen here is such that no molecular effects are present. This is achieved by placing the molecular description in the center of the computational domain such that it is not in contact with any of the system

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H.S. Wijesinghe and N.G. Hadjiconstantinou

walls (see Fig. 5). The rationale is that the hybrid solution of this problem should reproduce the full Navier–Stokes solution and thus the latter can be used as a benchmark result. In our formulation the continuum subdomain is described by the Navier– Stokes equations solved by finite element discretization. Standard Dirichlet velocity boundary conditions for a driven cavity problem were applied on the system walls which in this implemetation are captured by the continuum subdomain; the horizontal velocity component on the left, right and lower walls were held at zero, while on the upper wall it was set to 50 m/s. The vertical velocity component on all boundaries was set to zero. Boundary conditions from the atomistic domain are imposed on nodes that have been centered on DSMC cells (see Fig. 6). The pressure is scaled by setting the middle node on the lower boundary at atmospheric pressure (1.013×105 Pa). Despite the relatively high flow velocity, the flow is essentially incompressible and isothermal.

Figure 5. Continuum and atomistic subdomains for Schwarz coupling for the twodimensional driven cavity problem.

Hybrid atomistic–continuum formulations

2537

Figure 6. Boundary condition exchange. Only the bottom left corner of the matching region is shown for clarity. Particles are created with probability density proportional to the local number density.

The imposition of boundary conditions on the atomistic subdomain is facilitated by a particle reservoir as shown in Fig. 6. Note that in this particular implementation the reservoir region serves also as part of the overlap region, thus reducing the overall cost of the molecular description. Particles are created at locations x, y within the reservoir with velocities C x , C y drawn from a Chapman–Enskog velocity distribution. The Chapman Enskog distribution is generated, as explained above, by using the mean and gradient of velocities from the continuum solution; the number and spatial distribution of particles in the reservoir are chosen according to the overlying continuum cell mean density and density gradients. The rapid convergence of the Schwarz approach is demonstrated in Fig. 7. The continuum numerical solution is reached to within ±10% at the 3rd Schwarz iteration and to within ±2% at the 10th Schwarz iteration. The error estimate which includes the effects of statistical noise [17] and discretization error due to finite timestep and cell size is approximately 2.5%. Similar convergence is also observed for the velocity field in the vertical direction. The close agreement with the fully continuum results indicates that the Chapman–Enskog procedure is not only theoretically appropriate but also robust. Despite a Reynolds number of Re ≈ 1, the Schwarz method converges

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H.S. Wijesinghe and N.G. Hadjiconstantinou

Figure 7. Convergence of the horizontal velocity component along the Y = 0.425 × 10−6 m plane with successive Schwarz iterations.

with negligible error. This is in agreement with findings [36] which have recently shown that the Schwarz method is expected to converge for Re ∼ O(1).

5.3.

Unsteady Formulations

Unsteady incompressible calculations are particularly challenging for two reasons. First, due to the low flow speeds associated with them and the associated large number of samples required, the computational cost of the atomistic subdomain simulation rises sharply. Second, because of Eq. (1) and the fact that τm τc (typically), explicit time integration to the time of interest is very expensive. Approaches which use explicit time coupling based on compressible fluxmatching schemes have been proposed for these flows but it is not at all clear that these approaches provide the best solution. First, they suffer from signal to noise problems more than state-variable based methods. Second, integration of the continuum subdomain using the compressible formulation for an incompressible problem becomes both expensive and inaccurate [18]. On the other hand, iterative methods require a number of re-evaluations of the molecular

Hybrid atomistic–continuum formulations

2539

solution to achieve convergence. This is an additional computational cost that is not shared by the time-explicit coupling and leads to a situation whereby (for incompressible unsteady problems) the choice between a time-explicit fluxmatching coupling formulation or an iterative (Schwarz-type) coupling formulation is not clear and may be problem dependent. An alternative approach would be the adaptation of non-iterative continuum-continuum coupling techniques which take into account the incompressible nature of the problem and avoid the use of flux matching, such as the coupling approach presented in O’Connell and Thompson [13]. We should also recall that from Eq. (1), unless time coarse-graining techniques are developed, large, low-speed, unsteady problems are currently too expensive to be feasible by any method.

6.

Compressible Formulations

As discussed above, consideration of the compressible equations of motion leads to hybrid methods which differ significantly from their incompressible counterparts. The hyperbolic nature of compressible flows means that steady state formulations typically do not offer a significant computational advantage, and as a result, explicit time integration is the preferred solution approach and flux matching is the preferred coupling method. Given that the characteristic evolution time, τh , scales with the system size, the largest problem that can be captured by a hybrid method is limited by the separation of scales between the atomistic integration time and τh . Local mesh refinement techniques [21, 29] minimize the regions of space that need to be integrated at small CFL timesteps (due to a fine mesh), such as the regions adjoining the atomistic subdomain. Implicit timestepping methods [19] can also be used to speed up the time integration of the continuum subdomain. Unfortunately, although both approaches enhance the computational efficiency of the continuum sub-problem, they do not alleviate the issues arising from the disparity between the atomistic timestep and the total integration time. Compressible hybrid continuum-DSMC approaches are popular because compressible behavior is often observed in gases. In these methods, locally refining the continuum solution cells to the size of DSMC cells leads to a particularly seamless hybrid formulation in which DSMC cells differ from the neighboring continuum cells only by the fact that they are inherently fluctuating. The DSMC timestep required for accurate solutions [37–39] is very similar to the CFL timestep of a compressible formulation, and thus a finite volume formulation can be used to couple the two descriptions (for finite volume methods see the article, “Finite Difference, Finite Element and Finite Volume Methods for Partial Differential Equations” in the Handbook). In such a method [9, 10, 40] the flux of mass, momentum and energy from DSMC

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H.S. Wijesinghe and N.G. Hadjiconstantinou

to the continuum domain can be used directly for finite volume integration. Going from the continuum solution to DSMC requires the use of reservoirs. A DSMC reservoir extending into the continuum subdomain is attached at the end of the DSMC subdomain and initialized using the overlying continuum field properties. Flux of mass, momentum and energy is then provided by the particles entering the DSMC subdomain from the reservoir. The particles leaving the DSMC subdomain to the reservoir are discarded (after their contribution to mass, momentum and energy flux to the continuum is recorded). Another characteristic inherent to compressible formulations is the possibility of describing parts of the domain by the Euler equations of motion [29]. In that case, consistent coupling to the atomistic formulation can be performed using a Maxwell–Boltzmann distribution [21]. It has been shown [41] that explicit time-dependent flux-based formulations preserve the fluctuating nature of the atomistic description within the atomistic regions but the fluctuation amplitude decays rapidly within the continuum regions; correct fluctuation spectra can be obtained in the entire domain by solving a fluctuating hydrodynamics formulation [42] in the continuum subdomain. Below we discuss a particular hybrid implementation to illustrate atomisticcontinuum coupling in the compressible limit. We would like to emphasize again that a variety of methods can be used, although the compressible formulation is particularly well suited to flux matching. The method illustrated here is an extended version of the original Adaptive Mesh and Algorithm Refinement (AMAR) method [21]. This method was chosen since it is both the current state of the art in compressible fully adaptive hybrid methods and since it also illustrates how existing continuum multiscale techniques can be used directly for atomistic-continuum coupling with minimum modification.

6.1.

Fully Adaptive Mesh and Algorithm Refinement for a Dilute Gas

The compressible adaptive mesh and algorithm refinement formulation of Garcia et al., [21], referred to as AMAR, pioneered the use of mesh refinement as a natural framework for the introduction of the atomistic description in a hybrid formulation. In AMAR the typical continuum mesh refinement capabilities are supplemented by an algorithmic refinement (continuum to atomistic) based on continuum breakdown criteria. This seamless transition is both theoretically and practically very appealing. In what follows we briefly discuss a recently developed [29] fully adaptive AMAR method. In this method DSMC provides an atomistic description of the

Hybrid atomistic–continuum formulations

2541

flow while the compressible two-fluid Euler equations serve as the continuumscale model. Consider the Euler equations in conservative integral form d dt where

U dV +

φ

F · nˆ dS = 0

(10)

∂φ



    U=   

ρ px py pz e ρc





    ;   

    x F =   

ρu x ρu 2x + P ρu x u y ρu x u z (e + P)u x ρcu x

        

(11)

Only the x-direction component of the flux terms are listed here; other directions are similar. A two-species gas is assumed with the mass concentrations being c and (1 − c). Discrete time integration is achieved by using a finite volume approximation to Eq. (10). This yields a conservative finite difference expression with Unij k appearing as a cell-centered quantity at each x,n+1/2 time level and Fi+1/2, j,k located at faces between cells at half-time levels. A second-order version of an unsplit Godunov scheme is used to approximate the fluxes [43–45]. Time stepping on an AMR grid hierarchy involves interleaving time steps on individual levels [46]. Each level has its own spatial grid resolution and timestep (typically constrained by a CFL condition). The key to achieving a conservative AMR algorithm is to define a discretization for Eq. (10) that holds on every region of the grid hierarchy. In particular, the discrete cell volume integrals of U and the discrete cell face integrals of F must match on the locally-refined AMR grid. Thus, integration of a level involves two steps: solution advance and solution synchronization with other levels. Synchronizing the solution across levels assumes that fine grid values are more accurate than coarse grid values. So, coarse values of U are replaced by suitable cell volume averages of finer U data where levels overlap, and discrete fine flux integrals replace coarse fluxes at coarse-fine grid boundaries. Although the solution is computed differently in overlapping cells on different levels as each level is advanced initially, the synchronization procedure enforces conservation over the entire AMR grid hierarchy.

6.2.

Details of Coupling

During time integration of continuum grid levels, fluxes computed at each cell face are used to advance the solution U (Fig. 8b). Continuum values are

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H.S. Wijesinghe and N.G. Hadjiconstantinou

(a)

(b)

(c)

(d)

(e)

(f)

Figure 8. Outline of AMAR hybrid: (a) Beginning of a time step; (b) Advance the continuum grid; (c) Create buffer particles; (d) Advance DSMC particles; (e) Refluxing; (f) Reset overlying continuum grid. Adapted from [29].

advanced using a time increment tc appropriate for each level, including those that overlay the DSMC region. When the particle level is integrated, it is advanced to the new time on the finest continuum level using a sequence of particle time steps, tp . The relative magnitude of tp to the finest continuum grid tc depends on the finest continuum grid spacing x (typically a few λ) and the particle mean collision time. Euler solution information is passed to the particles via buffer (reservoir) cells surrounding the DSMC region. At the beginning of each DSMC integration step, particles are created in the buffer cells using the continuum hydrodynamic values (ρ, u, T ) and their gradients (Fig. 8c) in a manner analogous to the incompressible case discussed above and the guidelines of the section on particle generation in dilute gases. Since the continuum solution is advanced first, these values are time interpolated between continuum time steps for the sequence of DSMC time steps needed to reach the new continuum solution time. DSMC buffer cells are one mean free path wide; thus, the time step tp is constrained so that it is extremely improbable that a particle will travel further than one mean free path in a single time step. The particle velocities are drawn from an appropriate distribution for the continuum solver, such as the Chapman–Enskog distribution when coupling to a Navier–Stokes description and a Maxwell–Boltzmann when coupling to an Euler description. During each DSMC time integration step, all particles are moved, including those in the buffer regions (Fig. 8d). A particle that crosses the interface

Hybrid atomistic–continuum formulations

2543

between continuum and DSMC regions will eventually contribute to the flux at the corresponding continuum cell face during the synchronization of the DSMC level with the finest continuum level. After moving particles, those residing in buffer regions are discarded. Then, collisions among the remaining particles are evaluated and new particle velocities are computed. After the DSMC region has advanced over an entire continuum grid timestep, the continuum and DSMC solutions are synchronized in a manner analogous to the AMR level synchronization process described earlier. First, the continuum values in each cell overlaying the DSMC region interior are set to the conservative averages of data from the particles within the continuum grid cell region (Fig. 8e). Second, the continuum solution in cells adjacent to the DSMC region is recomputed using a “refluxing” process (Fig. 8f). That is, a flux correction is computed using a space and time integral of particle flux data, δF = −AFn+(1/2) +

Fp.

(12)

particles

The sum represents the flux of the conserved quantities carried by particles passing through the continuum cell face during the DSMC updates. Finally, Un+1 = Un+1 +

tc δF xyz

(13)

is used to update the conserved quantities on the continuum grid where Un+1 is the coarse grid solution before computing the flux correction. Note, multiple DSMC parallelepiped regions (i.e., patches) are coupled by copying particles from patch interiors to buffer regions of adjacent DSMC patches (see Fig. 9). That is, particles in the interior of one patch supply boundary values (by acting as a reservoir) for adjacent particle patches. After copying particles into buffer regions, each DSMC patch may be integrated independently, in the same fashion that different patches in a conventional AMR problems are treated after exchanging boundary data. In summary, the coupling between the continuum and DSMC methods is performed in three operations. First, continuum solution values are interpolated to create particles in DSMC buffer cells before each DSMC step. Second, conserved quantities in each continuum cell overlaying the DSMC region are replaced by averages over particles in the same region. Third, fluxes recorded when particles cross the DSMC interface are used to correct the continuum solution in cells adjacent to the DSMC region. This coupling procedure makes the DSMC region appear as any other level in the AMR grid hierarchy. Figure 10 shows the adaptive tracking of a shockwave of Mach number 10 used as a validation test for this method. Density gradient based mesh refinement ensures the DSMC region tracks the shock front accurately. Furthermore, as shown in Fig. 11 the density profile of the shock wave remains smooth and

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Figure 9. Multiple DSMC regions are coupled by copying particles from one DSMC region (upper left) to the buffer region of an adjacent DSMC region (lower right). After copying, regions are integrated independently over the same time increment. Adapted from Wijesinghe et al. [29].

Figure 10. Moving Mach 10 shock wave though Argon. The AMAR algorithm tracks the shock by adaptively moving the DSMC region with the shock front. Note, dark Euler region shading corresponds to density = 0.00178 g/cm3 , light Euler region shading corresponds to density = 0.00691 g/cm3 .

Hybrid atomistic–continuum formulations

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Figure 11. Moving Mach 10 shock wave though Argon. The AMAR profile (dots) is compared with the analytical time evolution of the initial discontinuity (lines). τm is the mean collision time.

is devoid of oscillations that are known to plague traditional shock capturing schemes [4, 5]. Further details of the implementation using the Structured Adaptive Mesh Refinement Application Infrastructure (SAMRAI) developed at Lawrence Livermore National Laboratory [47] can be found in [29].

7.

Refinement Criteria

The AMAR scheme discussed above allows grid and algorithm refinement based on any combination of flow variables and their gradients. Density gradient based refinement has has been found to be generally robust and reliable. However, refinement may be triggered by any number of user defined criteria. For example, concentration gradients or concentration values within some interval are also effective refinement criteria especially for multispecies flows. In the AMAR formulation, refinement is triggered by spatial gradients exceeding user defined tolerances. This approach follows from the continuum breakdown parameter method [48]. Due to spontaneous stochastic fluctuations in atomistic computations, it is important to track gradients in a manner that does not allow the fluctuations

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H.S. Wijesinghe and N.G. Hadjiconstantinou

to trigger unnecessary refinement and excessively large atomistic regions. Let us consider a dilute gas for simplicity and the gas density as an example. For an ideal gas under equilibrium conditions, the number of particles in a given volume is Poisson distributed; the standard deviation in the normalized density gradient perceived by the calculation at cell i is dρ/dx 2 ≈

ρ

Ni+1 − Ni 2 =

x Ni

√

2 √ x N

(14)

where N is the number of particles in a cell where macroscopic properties are defined. The use of equilibrium fluctuations is sufficiently accurate as long as the deviation from equilibrium is not too large [17]. The fluid density fluctuation can thus only be reduced by increasing the number of simulation particles. This has consequences for the use of density gradient tolerances Rρ , the value of which, as a result, must be based on the number of particles used in the atomistic subdomain. Let us illustrate this through an example. Consider the domain geometry shown in Fig. 12 where an atomistic region is in contact with a continuum region. Let the gas be in equilibrium. As stated above, the effect of nonequilibrium on fluctuations is small. In this problem, grid refinement occurs when the density gradient at the interface between two descriptions exceeds a normalized threshold,

2λ dρ Rρ < ρ dx

(15)

After such a “trigger” event the atomistic region grows by a single continuum cell width. Lets us assume that we would like to estimate the value of

Figure 12. 3D AMAR computational domain for investigation of tolerance parameter variation with number of particles in DSMC cells. From [29].

Hybrid atomistic–continuum formulations

2547

refinement threshold such that a given trigger rate, say 5–10%, is achieved. The interpretation of this trigger rate, is that there is a probability of 5–10% of observing spurious growth of the atomistic subdomain due to density fluctuations. Following [29] we show how the trigger rate can be related to the number of particles per cell used in the calculation. For the geometry considered in the above test problem, each continuum cell consists of 8 DSMC cells and hence effectively the contribution of 8 × N particles is averaged to determine the density gradient between continuum cells. Applying Eq. (14) to these continuum cells we obtain, dρ/dx 2 ≈ σ=

ρ

c

1 √

2x N

(16)

Note that we are assuming that the fluctuation of the continuum cells across from the atomistic-continuum interface is approximately the same as that in the atomistic region. This was shown to be the case for the diffusion equation and a random walk model [41], and has been verified for the Euler–DSMC system [29] (see Fig. 14). This allows the use of Eq. (14) that was derived assuming 2 atomistic cells. Note that the observed trigger event is a composite of a large number of possible density gradient fluctuations that could exceed Rρ ; gradients across all possible nearest neighbor cells, next-to-nearest neighbor cells and diagonally-nearest neighbor cells are all individually evaluated by the refinement routines and checked against Rρ . For a 10% trigger rate (or equivalent probability of trigger) the probability of an individual cell having a density fluctuation exceeding Rρ can be estimated as O(0.1/100) by observing that, 1. since the trigger event is rare, probabilities can be approximated as additive, 2. for the geometry considered, there are ≈ 300 nearest neighbor, nextnearest neighbor and diagonal cells that can trigger refinement and 3. the rapid decay of the Gaussian distribution ensures the decreasing probability (O(0.1/100) ∼ O(0.001)) of a single event does not significantly alter the corresponding confidence interval and thus an exact enumeration of all possible trigger pairs with correct weighting factors is not necessary. Our probability estimate at O(0.001) suggests that our confidence interval is 3σ − 4σ . This is verified in Fig. 13. Smaller trigger rates can be achieved by increasing Rρ , that is, by increasing the number of particles per cell.

2548

Figure 13.

H.S. Wijesinghe and N.G. Hadjiconstantinou

Variation of density gradient tolerance with number of DSMC particles. From [29].

Figure 14. Average density for stationary fluid Euler–DSMC hybrid simulation with 80 particles per cubic mean free path. Errorbars give one standard deviation over 10 samples. From [29].

Hybrid atomistic–continuum formulations

8.

2549

Outlook

Although hybrid methods provide significant savings by limiting atomistic solutions only to the regions where they are needed, solution of timeevolving problems which span a large range of timescales is still not possible if the atomistic subdomain, however small, needs to be integrated for the total time of interest. New frameworks are therefore required which allow timescale decoupling or coarse grained time evolution of atomistic simulations. Significant computational savings can be obtained by using the incompressible formulation, when appropriate, for steady problems. Neglect of these simplifications can lead to a problem that is simply intractable when the continuum subdomain is appropriately large. It is interesting to note that, when a hybrid method was used to solve a problem of practical interest [33] while providing computational savings, the Schwarz method was preferred because it provides a steady solution framework with timescale decoupling. For dilute gases the Chapman–Enskog distribution provides a robust and accurate method for imposing boundary conditions. Further work is required for the development of similar frameworks for liquids.

Acknowledgments The authors wish to thank R. Hornung and A.L. Garcia for help with the computations and valuable comments and discussions, and A.T. Patera and B.J. Alder for helpful comments and discussions. This work was supported in part by the Center for Computational Engineering, and the Center for Advanced Scientific Computing, Lawrence Livermore National Laboratory, US Department of Energy, W-7405-ENG-48. The authors also acknowledge the financial support from the University of Singapore through the Singapore-MIT alliance.

References [1] A. Beskok and G.E. Karniadakis, “A model for flows in channels, pipes and ducts at micro and nano scales,” Microscale Thermophys. Eng., 3, 43–77, 1999. [2] S.A. Tison, “Experimental data and theoretical modeling of gas flows through metal capillary leaks,” Vacuum, 44, 1171–1175, 1993. [3] D.G. Coronell and K.F. Jensen, “Analysis of transition regime flows in low pressure CVD reactors using the direct simulation Monte Carlo method,” J. Electrochem. Soc., 139, 2264–2273, 1992. [4] M. Arora and P. L. Roe, “On postshock oscillations due to shock capturing schemes in unsteady flows,” J. Comput. Phys., 130, 25, 1997. [5] P.R. Woodward and P. Colella, “The numerical simulation of two-dimensional fluid flow with strong shocks,” J. Comput. Phys., 54, 115, 1984.

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[6] J. Koplik and J.R. Banavar, “Continuum deductions from molecular hydrodynamics,” Annu. Rev. Fluid Mech., 27, 257–292, 1995. [7] M.P. Brenner, X.D. Shi, and S.R. Nagel, “Iterated instabilities during droplet fission,” Phys. Rev. Lett., 73, 3391–3394, 1994. [8] P.A. Thompson and M.O. Robbins, “Origin of stick–slip motion in boundary lubrication,” Science, 250, 792–794, 1990. [9] D.C. Wadsworth and D.A. Erwin, “One-dimensional hybrid continuum/particle simulation approach for rarefied hypersonic flows,” AIAA Paper 90-1690, 1990. [10] D.C. Wadsworth and D.A. Erwin, “Two-dimensional hybrid continuum/particle simulation approach for rarefied hypersonic flows,” AIAA Paper 92-2975, 1992. [11] J. Eggers and A. Beylich, “New algorithms for application in the direct simulation Monte Carlo method,” Prog. Astronaut. Aeron., 159, 166–173, 1994. [12] D. Hash and H. Hassan, “A hybrid DSMC–Navier Stokes solver,” AIAA Paper 95-0410, 1995. [13] S.T. O’Connell and P. Thompson, “Molecular dynamics-continuum hybrid computations: A tool for studying complex fluid flows,” Phys. Rev. E, 52, R5792–R5795, 1995. [14] N.G. Hadjiconstantinou and A.T. Patera, “Heterogeneous atomistic-continuum representations for dense fluid systems,” Int. J. Mod. Phys. C, 8, 967–976, 1997. [15] N.G. Hadjiconstantinou, “Hybrid atomistic-continuum formulations and the moving contact-line problem,” J. Comput. Phys., 154, 245–265, 1999. [16] E.G. Flekkoy, G. Wagner, and J. Feder, “Hybrid model for combined particle and continuum dynamics,” Europhys. Lett., 52, 271–276, 2000. [17] N.G. Hadjiconstantinou, A.L. Garcia, M.Z. Bazant, and G.He, “Statistical error in particle simulations of hydrodynamic phenomena,” J. Comput. Phys., 187, 274–297, 2003. [18] P. Wesseling, Principles of Computational Fluid Dynamics, Springer, 2001. [19] X. Yuan and H. Daiguji, “A specially combined lower-upper factored implicit scheme for three dimensional compressible Navier-Stokes equations,” Comput. Fluids, 30, 339–363, 2001. [20] S. Chapman and T.G. Cowling, The Mathematical Theory of Non-uniform Gases, Cambridge University Press, 1970. [21] A.L. Garcia, J.B. Bell, W.Y. Crutchfield et al., “Adaptive mesh and algorithm refinement using direct simulation Monte Carlo,” J. Comput. Phys., 54, 134, 1999. [22] J. Li, D. Liao and S. Yip, “Nearly exact solution for coupled continuum/MD fluid simulation,” J. Comput. Aided Mater. Design, 6, 95–102, 1999. [23] M.M. Mansour, F. Baras, and A.L. Garcia, “On the validity of hydrodynamics in plane poiseuille flows,” Physica A, 240, 255–267, 1997. [24] R. Delgado–Buscalioni and P.V. Coveney, “Continuum–particle hybrid coupling for mass, momentum and energy transfers in unsteady fluid flow,” Phys. Rev. E, 67(4), 2003. [25] C. Cercignani, The Boltzmann Equation and its Applications, Springer-Verlag, New York, 1988. [26] G.A. Bird, Molecular Gas Dynamics and the Direct Simulation of Gas Flows, Clarendon Press, Oxford, 1994. [27] W. Wagner, “A convergence proof for bird’s direct simulation Monte Carlo method for the Boltzmann equation,” J. Statist. Phys., 66, 1011, 1992. [28] H.S. Wijesinghe and N.G. Hadjiconstantinou, “A hybrid continuum-atomistic scheme for viscous incompressible flow,” In: Proceedings of the 23th International Symposium on Rarefied Gas Dynamics, 907–914, Whistler, British Columbia, 2002.

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[29] H.S. Wijesinghe, R. Hornung, A.L. Garcia et al., “Three–dimensional hybrid continuum–atomistic simulations for multiscale hydrodynamics,” ASME J. Fluids Eng., 126, 768–777, 2004. [30] A.L. Garcia and B.J. Alder, “Generation of the Chapman Enskog distribution,” J. Comput. Phys., 140, 66, 1998. [31] L. Devroye, “Non-uniform random variate generation,” In: A.L. Garcia (ed.), Numerical Methods for Physics, Prentice Hall, New Jersey, 1986. [32] W.H. Press, B.P. Flannery, S.A. Teukolsky, and W.A. Vetterling, Numerical Recipes in Fortran, Cambridge University Press, 1992. [33] O. Aktas and N.R. Aluru, “A combined continuum/DSMC Technique for multiscale analysis of microfluidic filters,” J. Comput. Phys., 178, 342–372, 2002. [34] N. G. Hadjiconstantinou, Hybrid Atomistic-Continuum Formulations and the Moving Contact Line Problem, Phd Thesis edn., Mechanical Engineering Department, Massachusetts Institute of Technology, Cambridge, Massachusetts, 1998. [35] P.L. Lions, “On the Schwarz alternating method,” I. In: R. Glowinski, G. Golub, G. Meurant, and J. Periaux (eds.), First International Symposium on Domain Decomposition Methods for Partial Differential Equations, pp. 1–42, SIAM, Philadelphia, 1988. [36] S.H. Liu, “On Schwarz alternating methods for the incompressible Navier–Stokes equations,” SIAM J. Sci. Comput., 22(6), 1974–1986, 2001. [37] F.J. Alexander, A.L. Garcia, and B.J. Alder, “Cell size dependence of transport coefficients in stochastic particle algorithms,” Phys. Fluids, 10, 1540, 1998. [38] N.G. Hadjiconstantinou, “Analysis of discretization in the direct simulation Monte Carlo,” Phys. Fluids, 12, 2634–2638, 2000. [39] A.L. Garcia and W. Wagner, “Time step truncation error in direct simulation Monte Carlo,” Phys. Fluids, 12, 2621–2633, 2000. [40] R. Roveda, D.B. Goldstein, and P.L. Varghese, “Hybrid Euler/direct simulation Monte Carlo calculation of unsteady slit flow,” J. Spacecraft and Rockets, 37(6), 753–760, 2000. [41] F.J. Alexander, A.L. Garcia, and D. Tartakovsky, “Algorithm refinement for stochastic partial diffential equations: I. Linear diffusion,” J. Comput. Phys., 182(1), 47–66, 2002. [42] L.D. Landau and E.M. Lifshitz, Statistical Mechanics Part 2, Pergamon Press, Oxford, 1980. [43] P. Colella, “A direct Eulerian (MUSCL) scheme for gas dynamics,” SIAM J. Sci. Statist. Comput., 6, 104–117, 1985. [44] P. Colella and H.M. Glaz, “Efficient solution algorithms for the riemann problem for real gases,” J. Comput. Phys., 59, 264–289, 1985. [45] J. Saltzman, “An unsplit 3D upwind method for hyperbolic conservation laws,” J. Comput. Phys., 115, 153, 1994. [46] M. Berger and P. Colella, “Local adaptive mesh refinement for shock hydrodynamics,” J. Comput. Phys., 82, 64, 1989. [47] CASC, “Structured adaptive mesh refinement application infrastructure,” http://www.llnl.gov/CASC/, 2000. [48] G.A. Bird, “Breakdown of translational and rotational equilibrium in gaseous expansions,” Am. Inst. Aeronautics and Astronaut. J., 8, 1998, 1970.

9.1 POLYMERS AND SOFT MATTER L. Mahadevan1 and Gregory C. Rutledge2 1

Division of Engineering and Applied Sciences, Department of Organismic and Evolutionary Biology, Department of Systems Biology, Harvard University Cambridge, MA 02138, USA 2 Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA

1.

Introduction

Within the context of this Handbook, the combined areas ofpolymers and soft matter encompasses a vast range of complex materials, including both synthetic and natural polymers, many biological materials, and complex fluids such as colloids and viscoelastic media. What distinguishes these materials from most of those considered in other chapters of this Handbook is the macromolecular or supermolecular nature of the basic components of the material. In addition to the usual atomic level interactions responsible for chemically specific material behavior, as is found in all materials, these macromolecular and supermolecular objects exhibit topological features that lead to new, larger scale, collective nonlinear and nonequilibrium behaviors that are not seen in the constituents. As a consequence, these materials are typically characterized by a broad range of both length and time scales over which phenomena of both scientific and engineering interest can arise. In polymers, for instance, the organic nature of the molecules is responsible for both strong (intramolecular, covalent) and weak (intermolecular, van der Waals) interactions, as well as interactions of intermediate strength such as hydrogen bonds that are common in macromolecules of biological interest. In addition, however, the long chain nature of the molecule introduces a distinction between dynamics that occur along the chain or normal to it; one consequence of this is the observation of certain generic behaviors such as the “slithering snake” motion, or reptation, in polymer dynamics. It is often the very ability of polymers and soft matter to exhibit both atomic (or molecular) and macro- (or super-) molecular behavior that makes them so interesting and powerful as a class of materials and as building blocks for living systems. 2555 S. Yip (ed.), Handbook of Materials Modeling, 2555–2559. c 2005 Springer. Printed in the Netherlands.

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Nevertheless, polymers and soft matter are, at their most basic level, collections of atomic and subatomic particles, like any other class of materials. They exhibit both liquid-like and crystalline (or at least semi-crystalline) order in their condensed forms. For polymers, vitrification and the glassy state are particularly important, as both the vitrification temperature and the kinetics of vitrification are strong functions of the inverse of molecular weight. For the most part, the methods developed for atomic and electronic level modeling described in the earlier chapters of this Handbook are equally applicable, at least in principle, to the descriptive modeling of polymers and soft matter. Electronic structure calculations, atomistic scale molecular dynamics and Monte Carlo simulations, coarse-grained and mesoscale models such as Lattice Boltzmann and Dissipative Particle Dynamics all have a role to play in modeling of polymers and soft matter. As materials, these interesting solids and fluids exhibit crystal plasticity, amorphous component viscoelasticity, rugged energy landscapes, and fascinating phase transitions. Indeed, block copolymers consisting of two or more covalently-joined but relatively incompatible chemical segments, and the competition they represent between intermolecular interactions and topological constraints, give rise to the rich field of microphase separation, with all its associated issues and opportunities regarding manipulation of microstructure, size and symmetry. It has not been our objective in assembling the contributions to this chapter to repeat any of the basic elements of modeling that have been developed to describe materials at any of these particular length and time scales, or strategies for generating thermodynamics information relevant to ensembles, phase transitions, etc. Rather, in recognition of those features which make polymers and soft matter distinct and novel with respect to their atomic or monomolecular counterparts, we have attempted to assemble a collection of contributions which highlight these features, and which describe methods developed specifically to handle the particular problems and complexities of dimensionality, time and length scale which are unique to this class of materials. With this in mind, the following sections in this chapter should be understood as extensions and revisions of what has gone before. We begin with a discussion of interatomic potentials specific to organic materials typical of synthetic and natural polymers and other soft matter. Accurate force fields lie at the heart of any molecular simulation intended to describe a particular material. Over the years, numerous apparently dissimilar force fields for organic materials have been proposed. However, certain motifs consistently reappear in such force fields, and common pitfalls in parameterization and guidelines for application of such force fields can be identified. These are discussed in the contribution by Smith. The recognition that one of the defining features of macromolecules is their very large conformation space motivated the relatively early development by Volkenstein in the late 1950s of the concept of rotational isomeric

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states for each of the rotatable bonds along the backbone (i.e., the topologically connected dimension) of molecular chains. This essential discretization of conformation space allowed the development by Flory and others of what is now known as the rotational isomeric states (RIS) method, discussed in the section by Mattice. This method for evaluation of conformational averages is unique to polymers and provides an important alternative to the sampling strategies embodied by molecular dynamics and Monte Carlo simulation. What RIS gives up in assuming a simplified, discrete set of allowed rotational states for each bond, it more than makes up for in its computational efficiency and rigorous representation of contributions from all allowed conformers to the partition function and resulting conformational averages. The issues in sampling of phase space using molecular dynamics or Monte Carlo simulations for chain models are discussed by Mavrantzas. Molecular dynamics is of course applicable to the study of polymers and soft matter, but the broad range of length and, in particular, time scales alluded to earlier as being a consequence of the macromolecular and/or supermolecular nature of such matter, render this method of limited utility for many of the most interesting and unique behaviors in this class of materials. For this reason, Monte Carlo simulation has come to play a particularly important role in the modeling of polymers and soft matter. At the expense of detailed dynamics, the state of the art in Monte Carlo simulations of chain molecules and aggregates has advanced through the development of new sampling schemes that permit drastic, sometimes seemingly unphysical, moves through phase space. These moves are designed with both intermolecular interactions and intramolecular topology in mind. Without them, full equilibration and accurate simulation of complex materials are all but impossible. An alternative approach to accessing the long length and time scales of interest in polymers and soft matter is to coarse-grain the model description, gaining computational efficiency at the price of atomic scale detail. Such methods are useful for studying the generic, or universal, properties of polymers and aggregates. In the field of polymers and soft matter, lattice models have long been employed for rendering such coarse-grained models. The Bond Fluctuation Model, in particular, is typical of this class of methods and has enjoyed widespread application, due at least in part to the delicate compromise it achieves between the complexity of conformation space and the simplification inherent in rendering on a lattice. Importantly, it does so while retaining the essential topological connectivity. These methods are discussed by M¨uller and provide a link to continuum-based methods. Continuum based methods start to become relevant when the number of particles involved is very large and one is interested in long wavelength, long time modes, as is typical of hydrodynamics. The dimensionalities of both the “material” component and the embedding component, or matrix, play important roles in determining the behavior of mesophases such as suspensions, colloids

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and membranes. The article by Sierou provides an introduction to Stokesian dynamics, a molecular dynamics-like method for simulating the multi-phase behavior of particles suspended in a fluid. The particles are treated in a discrete sense, while the surrounding fluid is treated using a continuum approximation and is thus valid when the particle size is much larger than that of the molecules of the solvent. By accounting for Brownian motion, Stokesian dynamics provides a generalization of Brownian dynamics, treated by Doyle and Underhill in the next section, wherein the many-body contribution from hydrodynamics is accounted for properly. It thus paves the road for a study of the equilibrium and non-equilibrium rheology of colloids and other complex multiphase fluids. Moving up in dimensionality from particles to chains, the section by Doyle and Underhill discusses Brownian dynamics simulation of long chain polymers. The topological connectivity of these polymers implies a separation in time and energy scales for deformations tangential to and normal to the backbone. Coarse-grained models that account for this separation of scales range from bead-spring models to continuum semi-flexible polymers. While these models have been corroborated with each other and with simple experiments involving single molecules, the next frontier is clearly the use of these dynamical methods to probe the behavior of polymer solutions, a subject that still merits much attention. Next, Powers looks at the 2D generalization of polymers, i.e., membranes, which are assemblies of lipid molecules that are fluid-like in the plane but have an elastic response to bending out of the plane. In contrast to the previous sections, the focus here is on the continuum and statistical mechanics of these membranes using analytical tools via a coarse-grained free energy written in terms of the basic broken-symmetries of the system. Once again the role of non-equilibrium dynamics comes up in the example of active membranes. The last section in this chapter offers a union of the molecular and continuum perspectives, in some sense, to address problems such as molecular structure-mediated microphase formation. Here again continuum models based on density fields and free energy functionals are most appropriate. It is a relatively recent development, however, that such models have been used as a starting point for computer simulations. The Field Theoretic Simulation method developed by Frederickson and co-workers does just this, and is discussed by Ganesan and Frederickson in this chapter. They provide a prescription by which a molecular model can be recast as a density field with its projected Hamiltonian, and then present appropriate methods for discretizing and sampling phase space during the simulation. Thus, polymers and soft matter are in some sense no different than hard matter, in that their constituents are atomic in nature. Yet they are distinguished by the predominance of weak interactions comparable to the thermal fluctuations, which makes them amenable to change. Looking to the future, the wide

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variety of phases and broken symmetries that they embody is nowhere more abundant than in living systems that operate far from equilibrium and are eternally mutable. From a materials perspective, polymers and soft matter offer opportunities to mimic and understand nature in ways that we are only just beginning to appreciate. It is our hope that the sections in this chapter offer a glimpse of the techniques that one may use and the questions that motivate them.

9.2 ATOMISTIC POTENTIALS FOR POLYMERS AND ORGANIC MATERIALS Grant D. Smith Department of Materials Science and Engineering, Department of Chemical Engineering, University of Utah, Salt Lake City, Utah, USA

Accurate representation of the potential energy lies at the heart of all simulations of real materials. Accurate potentials are required for molecular simulations to accurately predict the behavior and properties of materials, and even qualitative conclusions drawn from simulations employing inaccurate or unvalidated potentials are problematic. Various forms of classical potentials (force fields) for polymers and organic materials can be found in the literature [1–3]. The most appropriate form of the potential depends largely upon the properties of interest to the simulator. When interest lies in reproducing the static, thermodynamic and dynamic (transport and relaxational) properties of non-reactive organic materials, the potential must accurately represent the molecular geometry, nonbonded interactions, and conformational energetics of the materials of interest. The relatively simple representation of the classical potential energy discussed below has been found to work remarkable well for these properties. More complicated potentials that can handle chemical reactions [4] or are designed to very accurately reproduce vibrational spectra [5] can be found in the literature. The form of the force field considered here has the advantages of being more easily parameterized than more complicated forms. Parameterization of even simple potentials is a challenging task, however, as discussed below.

1.

Form of the Potential

The classical force field represents the total potential energy of an ensemble of atoms V ( r ) with positions given by the vector r as a sum of nonbonded

2561 S. Yip (ed.), Handbook of Materials Modeling, 2561–2573. c 2005 Springer. Printed in the Netherlands.

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interactions V N B ( r ) and energy contributions due to all bond, valence bend, and dihedral interactions: V ( r ) = V nb ( r) +

V bond(ri j ) +

bonds

V bend (θi j k ) +

bends

V tors (ϕi j kl )

dihedrals

(1) The various interactions are illustrated in Fig. 1. The dihedral term also includes four-center improper torsion or out-of-plane bending interactions that occur at sp2 hybridized centers. r ) consists of a sum of the twoCommonly, the nonbonded energy V N B ( body repulsion and dispersion energy terms between atoms i and j represented by the Buckingham (exponential-6) potential, the energy due to the interactions between fixed partial atomic or ionic charges (Coulomb interaction), and the energy due to many-body polarization effects: r ) = V pol ( r) + V nb (

N 1 Ci j qi q j Ai j exp(−Bi j ri j ) − 6 + 2 i, j =1 4π ε0ri j ri j

(2)

The generic behavior of the dispersion/repulsion energy for an atomic pair is shown in Fig. 2. The dispersion interactions are weak compared to repulsion, but are longer range, resulting in an attractive well with well depth ε at an interatomic separation of σ ∗ . The separation where the net potential is zero, σ , is often used to define the atomic diameter. In addition to the exponential-6 dihedral twist

intramolecular nonbonded intermolecular nonbonded

bond stretch

valence angle bend

Figure 1. Schematic representation of intramolecular bonded and nonbonded (intramolecular and intermolecular) interactions in a typical polymer.

V DIS-REP(r)

Atomistic potentials for polymers and organic materials

0

2563

ε σ σ∗

r Figure 2. Schematic representation of the dispersion/repulsion potential between two atoms as a function of separation.

form, the Lennard–Jones form of the dispersion–repulsion interaction, 

Ai j Ci j σ V D I S−R E P (ri j ) = 12 − 6 = 4ε  ri j ri j ri j

12

 12 6  σ ∗ σ ∗  = ε −2

ri j

ri j

−

σ ri j

6  

(3)

is commonly used, although this form tends to yield a poorer (too stiff) description of repulsion. The relationship between the well depth and atomic diameter and the dispersion–repulsion parameters is particularly simple for the Lennard–Jones potential (ε = C 2 /4A, σ = (A/C)1/6 , σ ∗ = 21/6 σ ), allowing the dispersion–repulsion interaction to be expressed in terms of these parameters, as shown in Eq. (3). Nonbonded interactions are typically included between all atoms of different molecules and between atoms of the same molecule separated by more than two bonds (see Fig. 1). It is not uncommon, however, to scale intramolecular nonbonded interactions between atoms separated by three bonds. Care must therefore be taken in implimenting a potential that the 1–4 intramolecular nonbonded interactions are correctly treated. Repulsion parameters have the shortest range and typically become negligible at 1.5 σ. Dispersion parameters are longer range than the repulsion parameters requiring cutoff distances of 2.5 σ. The Coulomb term is long-range, necessitating use of special summing methods [6, 7]. While dispersion interactions are typically weaker and are shorter range than Coulomb interactions, they are always attractive, independent of the configuration of the molecules, and typically make the dominate contribution to the cohesive energy even in highly polar polymers and organic materials.

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A further complication arises in cases where many-body dipole polarization needs to be taken into account explicitly. The potential energy due to dipole polarization is not pair-wise additive and is given by a sum of the interaction energy between the induced dipoles µi and the electric field Ei0 at atom i generated by the permanent charges in the system (qi ), the interaction energy between the induced dipoles and the energy required to induce the dipole moments [7] V

pol

(r) = −

N i=1

N N 1 µ i • µ i 0 µ • Ei − µ i • T ij • µ j + 2 i, j 2α i i=1

(4)

tot where µ i = αi E tot i , αi is the isotropic atomic polarizability, E i is the total electrostatic field at the atomic site i due to permanent charges and induced dipoles, and the second order dipole tensor is given by 1 1 T i j = ∇i ∇ j = 4π ε0ri j 4π ε0ri3j

3 ri j ri j −1 ri2j

(5)

where ri j is the vector from atom i to atom j . Because of the expense involved in simulations with explicit inclusion of many-body dipole polarization, it may be desirable to utilize a two-body approximation for these interactions [8]. The contributions due to bonded interactions are represented as

ri j − ri0j V bond (ri j ) = 12 kibond j

2

θi j k − θi0j k V bend (θi j k ) = 12 kibend jk V tors(ϕi j kl ) = V tors(ϕi j kl ) =

1 2

(6) 2

= 12 k bend cos θi j k − cos θi0j k ijk

kitors j kl (n) 1 − cos nϕi j kl

n 1 oop k 2 i j kl

φi j k

2

2

(7)

or (8)

Here, ri0j is an equilibrium bond length and θi0j l is an equilibrium valence bend oop bend tors angle while kibond j , ki j k , ki j kl (n) and ki j kl are the bond, bend, torsion and outof-plane bending force constants, respectively. Note that the forms given for the valence bend interaction are entirely equivalent for sp2 and sp3 bonding geometries for reasonably stiff bends at reasonable temperatures, with k = k/sin2 θ 0 . The indices indicate which (bonded) atoms that are involved in the interaction. These geometric parameters and force constants, combined with the nonbonded parameters qi , αi , Ai j , Bi j and Ci j , constitute the classical force field for a particular material. In contrasting the form of the potential for polymers and organics with potentials for other materials, the nature of bonding in organic materials becomes manifestly apparent. In organic materials the relatively strong covalent bonds and valence bends serve primarily to define the geometry of the

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2565

molecule. Much weaker/softer intramolecular degrees of freedom, namely torsions, and intermolecular nonbonded interactions, primarily determine the thermodynamic and dynamic properties of polymers and large organic molecules. Hence relatively weak (and consequently difficult to parameterize) torsional and repulsion/dispersion parameters must be determined with great accuracy in potentials for polymers and organics. However, this separation of scales of interaction strengths (strong intramolecular covalent bonding, weak intermolecular bonding) has the advantage of allowing many-body interactions, which often must be treated through explicit many-body nonbonded terms in simulations of other classes of materials, to be treated much more efficiently as separate intramolecular bonded interactions in organic materials.

2.

Existing Potentials

By far the most convenient way to obtain a force field is to utilize an extant one. In general, force fields can be divided into three categories: (a) force fields parametrized based upon a broad training set of molecules such as small organic molecules, peptides, or amino acids including AMBER [1], COMPASS [9], OPLS-AA [3] and CHARMM [10]; (b) generic potentials such as DREIDING [11] and UNIVERSAL [12] that are not parameterized to reproduce properties of any particular set of molecules; and (c) specialized force fields carefully parametrized to reproduce properties of a specific compound. A procedure for parameterizing the latter class of potential is described below. A summary of the data used in the parametrization of some of the most common force fields is presented in Table 1. Parametrized force fields (AMBER, OPLS and CHARMM) can work well within the class of molecules they have been parametrized upon. However, when the force field parameters are utilized for compounds similar to those in the original training set but not contained in the training set significant errors can appear and the quality of force field predictions is often no better than that of the generic force fields [13]. Similar behavior is expected when parameterized force fields transferred to new classes of compounds. Therefore, in choosing a potential, both the quality of the potential and the transferability of the potential need to be considered. The quality of a potential can be estimated by examining the quality and quantity of data used in its parameterization. For example, AMBER ff99 (Table 1) uses a much higher level of quantum chemistry calculation for determination of dihedral parameters than the early AMBER ff94. The ability of the force fields to describe the molecular and condensed-phase properties of the training set is another indicator of the force field quality. The issue of transferability of a potential is faced when a high-quality force field, adequately validated for compounds similar to the one of interest, is used in modeling

[N/A, N/A, experiment]

X-ray structure, IR, Raman

various experimental sources, QC

peptides, nucleic acids, organics

polarization

bond/bend

torsion

training set

organic liquids

QC

AMBER[ff94] with some values from CHARMM

peptides

Microwave and electron diffraction, QC

IR, Raman, microwave and electron diffraction, X-ray crystal data, QC

N/A

QC, experimental dipoles

PVT, H vap

QC

electrostatic N/A

PVT, H vap , crystal structures, QC

PVT, H vap

PVT, H vap

repulsion/dispersion

CHARMM

OPLS-AA

AMBER[ff94, ff99, ff02]

Interactions

Table 1. Summary of the primary data used in parameterization of popular force fields

generic

Generic based on hybridization

Generic

N/A

predictive method

Crystal structures and sublimation energies

DREIDING

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related compounds not in the training set, or in modeling entirely new classes of materials. Transferability varies tremendously upon the potential function parameter, with some parameters being in general quite transferable between similar compounds and others being much less so.

3.

Sources of Data for Force Field Parametrization

In order to judge the quality of existing force fields for a compound of interest, or to undertake the demanding but often inevitable task of parameterizing or partially parameterizing a new force field, one requires data against which the force field parameters (or subset thereof) can be tested and if necessary, fit. As can be seen in Table 1, there are two primary sources for such data: experiment and ab initio quantum chemistry calculations. Experimentally measured structural, thermodynamic and dynamic data for condensed phases (liquid and/or crystal) of the material of interest or closely related compounds are particularly useful in force field parameterization and validation. Highlevel quantum chemistry calculations are the best source of molecular level information for force field parameterization. While such calculations are not yet possible on high polymers and very large organic molecules, they are feasible on small molecules representative of polymer repeat units and oligomers, fragments of large molecules, as well as molecular clusters that reproduce interactions between segments of polymers or organic molecules or the interaction of a these with surfaces, solvents, ions, etc. These calculations can provide the molecular geometries, partial charges, polarizabilities, conformational energy surface, and intermolecular nonbonded interactions critical for accurate prediction of structural, thermodynamic and dynamic properties of polymers. Of key importance in utilizing quantum chemistry calculations for force field parameterization is use of an adequate level of theory and the choice of the basis set. As a rule of thumb, augmented correlation-consistent polarizable basis sets (e.g., aug-cc-pVDZ) utilizing DFT geometries (e.g., B3LYP) and correlated (MP2) energies work quite well, often providing molecular dipole moments within a few percent of experimental values, conformer energies within ±0.3 kcal/mol, rotational energy barriers between conformations within ± 0.5 kcal/mol, and intermolecular binding energies after basis set superposition error (BSSE) correction within 0.1–1 kcal/mol. However, whenever force field parameterization for any new class of molecule for which extensive quantum chemistry studies do not exist is undertaken, a comprehensive study of the influence of basis set and electron correlation on molecular geometries, conformational energies, cluster energies, dipole moments, molecular polarizabilities and electrostatic potential is warranted.

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Determining Potential Function Needs

In examining candidate potentials for a material, one should ascertain whether they have been parameterized for the material of interest or for closely related materials. One should also determine what data (quantum chemistry and experimental) were used in the parametrization, the quality of the data employed, and how well the potential reproduces the “training set” data. Finally, what if any validation steps that have been carried by the originators of the potential or by others who have utilized the potential should be determined. Next, one should determine what force field parameters are missing or may need reparameterization for the material of interest. The parameters that have most limited transferability from the training set to related compounds and hence are most likely to need parameterization are partial charges and dihedral parameters. Other parameters that may need parameterization in order of decreasing probability (increasing transferability) are equilibrium bond lengths and angles, bond, bend and improper torsion force constants, dispersion/repulsion parameters and atomic polarizabilities (for many-body polarizable potentials). A general procedure for systematic parameterization and validation of potential functions suitable for any polymer, organic compound or solution is provide below. Detailed derivations of quantum-chemistry based potentials for organic compounds and polymers can be found in the literature [9, 14].

5.

Establishing the Quantum Chemistry Data Set

Once it has been determined that parameterization or partial parameterization of a potential function is needed, it is necessary to determine the set of model molecules to be utilized in the potential function parameterization. If dispersion/repulsion parameters are needed, this may include molecular complexes containing the intermolecular interactions of interest. For smaller organic molecules, the entire molecule should be included in the data set. For polymers and larger organic molecules, oligomers/fragments containing all single conformations and conformational pairs extant in the polymer/large organic should be included. A search for existing quantum chemistry studies of these and related molecules should be conducted before beginning quantum chemistry calculations. When a new class of material (one for which extensive quantum chemistry studies have not yet been conducted) is being investigated, the influence of basis set and level of theory should be systematically investigated. Comparison with experiment (binding energies, molecular geometry, conformational energies, etc.) can help establish what level of theory is adequate.

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Once the level of theory is established, all important conformers and rotational energy barriers for the model molecule(s) in the data set should be found, as well as dipole moments and electrostatic potential for the lowest energy conformers. BSSE corrected binding energies for important configurations of molecular clusters should also be determined if parameterization of dispersion/repulsion interactions is required. These data provide the basis for parameterization of the potential as described briefly below.

6. 6.1.

Potential Function Parameterization and Validation Partial Charges

Most organic molecules are sufficiently polar that Coulomb interactions must be accurately represented. Often it is sufficient to treat Coulomb interactions with fixed partial atomic charges (Eq. (2)) and neglect explicit inclusion of many-body dipolar polarizability. The primary exception occurs when small ionic species are present. In such cases the force field needs to be augmented with additional terms describing polarization of a molecule (Eq. (4)). When needed, atomic polarizabilities can be determined straightforwardly from quantum chemistry [14, 15]. In parameterization of partial atomic charges, one attempts to reproduce the molecular dipole moment and electrostatic potential in the vicinity of model molecules as determined from high-level quantum chemistry calculations with a set of partial charges of the various atoms. Fig. 3 illustrates the quality of agreement that can be achieved in representing the electrostatic potential with partial atomic charges.

6.2.

Dispersion and Repulsion Interactions

Carrying out quantum chemistry studies of molecular clusters of sufficient accuracy to allow for final determination of dispersion parameters is very computationally expensive. Fortunately repulsion and dispersion parameters are highly transferable. Therefore, it is expedient to utilize literature values for repulsion and dispersion parameters where high-quality, validated values exist. Where necessary BSSE corrected Hartree–Fock binding energies of molecular clusters can be used establish repulsion parameters and initial values for dispersion parameters can be determined from fitting to correlated binding energies [14, 15]. Regardless of the source of data utilized to parameterize dispersion interactions (experimental thermodynamic or structural data, quantum chemistry data on molecular clusters, or direct use of existing parameters)

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G.D. Smith

5 5

-30 -20 0 0

-10

Figure 3. Electrostatic potential in the plane of a 1,2-dimethoxyethane molecule from ab initio electronic structure calculations (QC) and from partial atomic charges (FF) parameterized to reproduce the potential. Energy contours are in kcal/mol.

it may be necessary to make (hopefully) minor empirical adjustments (as large as ±10%) to the dispersion parameters so as to yield highly accurate thermodynamic properties for the material of interest. This can be accomplished by carrying out simulations of model molecules and comparing predicted thermodynamic properties (density, heat of vaporization, thermal expansion, compressibility) with experiment and adjusting dispersion parameters as needed to improve agreement.

6.3.

Bond and Bend Interactions

The covalent bond and valence bend force constants are also highly transferable between related compounds. As long as the dihedral potential (see

Atomistic potentials for polymers and organic materials

2571

below) is parameterized with the chosen bond and bend force constants, the particular (reasonable) values of the force constants will not strongly influence structural, thermodynamic, or dynamic properties of the material. It is therefore recommended that stretching bending force constants be taken from the literature where available. When not available, stretching and bending force constants can be taken directly from quantum chemistry normal mode frequencies determined for representative model molecules with appropriate scaling of the force constants.

6.4.

Molecular Geometry

The molecular geometry can strongly influence static, thermodynamic and dynamic properties and needs to be accurately reproduced. Therefore, accurate representation of bond lengths and angles is important. Equilibrium bond lengths and bond angles can be adjusted so as to accurate reproduce the bond lengths and bond angles of model compounds determined from high-level quantum chemistry.

6.5.

Dihedral Potential

It is crucial that the conformational energies, specifically the relative energies of important conformations and the rotational energy barriers between them, be accurately represented for polymers and conformationally flexible organic compounds. As a minimum a force field must be able to reproduce the relative energies of the important conformations of single dihedrals and dihedral pairs (dyad) in model molecules. The conformational energies and rotational energy barriers obtained from quantum chemistry for model molecules are quite sensitive to the level of theory utilized, both basis set size and electron correlation. Fortunately, it is typically not necessary to conduct geometry optimizations with electron correlation—for many compounds SCF or DFT geometries are sufficient. Unfortunately, relative conformational energies and rotational energy barriers obtained at the SCF and DFT level are usually not sufficient accurate, necessitating the calculation of MP2 energies at SCF or DFT geometries. In fitting the dihedral potential, it is sometimes possible to utilize only 1, 2 and 3-fold dihedral terms (n = 1–3 in Eq. (8)). However, it is often necessary to up to 6-fold dihedral terms to obtain a good representation of the conformational energy surface. One must be cognizant of possible artifacts (e.g., spurious minima and conformational energy barriers) that can be introduced into the conformational energy surface when higher-fold terms (n > 3) with large amplitudes are utilized. Fig. 4 show the quality of agreement for conformational energies between quantum chemistry and molecule

2572

G.D. Smith 7

conformational energy (kcal/mol)

QC

6

FF

5 4 3 2 1 0 0

60

120

180

240

300

360

β dihedral angle Figure 4. The relative conformational energy for rotation about the β-dihedral in 1,5hexadiene from ab initio electronic structure calculations (QC) and a force field parameterized to reproduce the conformational energy surface (FF).

mechanics that is possible with a 1-3 fold potential for model molecules for poly(butadiene).

6.6.

Validation of the Potential

As a final step, the potential, regardless of its source, should be validated through extensive comparison of structural, thermodynamic and dynamic properties obtained from simulations of the material of interest, closely related materials, and model compounds used in the parameterization, with available experimental data. The importance of potential function validation in simulation of real materials cannot be overemphasized.

References [1] W.D. Cornell et al., “A second generation force field for simulations of proteins, nucleic acids, and organic molecules,” J. Am. Chem. Soc., 117, 5179–5197, 1995. [2] J.W. Ponder and D.A. Case, “Force fields for protein simulation,” Adv. Prot. Chem., 66, 27–85, 2003. [3] W.L. Jorgensen, D.S. Maxwell, and J. Tirado-Rives, “Development and testing of the OPLS all-atom force field on conformational energetics and properties of organic luquids,” J. Am. Chem. Soc., 118, 11225–11236, 1996. [4] D.W. Brenner, “Empirical potential for hydrocarbons for use in simulating the chemical vapor deposition of diamond films,” Phys. Rev. B, 42, 9458–9471, 1990.

Atomistic potentials for polymers and organic materials

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[5] T.A. Halgren, “Merck molecular force field. III. Molecular geometries and vibrational frequencies for MMFF94,” J. Comput. Chem., 17, 553–586, 1996. [6] A. Toukmaji, C. Sagui, J. Board, and T.Darden, “Efficient particle-mesh ewald based approach to fixed and induced dipolar interactions,” J. Chem. Phys., 113, 10912– 10927, 2000. [7] T.M. Nymand and P. Linse, “Ewald summation and reaction field methods for potentials with atomic charges, dipoles, and polarizabilities,” J. Chem. Phys., 112, 6152–6160, 2000. [8] O. Borodin, G.D. Smith, and R. Douglas, “Force field development and MD simulations of poly(ethylene oxide)/LiBF4 polymer electrolytes,” J. Phys. Chem. B, 108, 6824–6837, 2003. [9] H. Sun, “COMPASS: An ab initio force-field optimized for condensed-phase applications-overview with details on alkane and benzene compounds,” J. Phys. Chem. B, 102, 7338–7364, 1998. [10] A.D. MacKerell et al., “All-atom empirical potential for molecular modeling and dynamics studies of proteins,” J. Phys. Chem. B, 102, 3586–3616, 1998. [11] S.L. Mayo, B.D. Olafson, and W.A. Goddard, III, “DREIDING: A generic force field for molecular simulations,” J. Phys. Chem., 94, 8897–8909, 1990. [12] A.K. Rapp´e, C.J. Casewit, K.S. Colwell, W.A. Goddard, and W.M. Skiff, “UFF, a full periodic table force field for molecular mechanics and molecular dynamics simulations,” J. Am. Chem. Soc., 114, 10024–10035, 1992. [13] F. Sato, S. Hojo, and H. Sun, “On the transferability of force field parameters-with an ab initio force field developed for sulfonamides,” J. Phys. Chem. A., 107, 248–257, 2003. [14] O. Borodin and G.D. Smith, “Molecular modeling of poly(ethylene oxide) melts and poly(ethylene oxide)-based polymer electrolytes,” In: L. Curtiss and M. Gordon, (eds.), Methods and Applications in Computational Materials Chemistry, Kluwer Academic Publishers, 35–90, 2004. [15] O. Borodin and G.D. Smith, “Development of the quantum chemistry force fields for poly(ethylene oxide) with many-body polarization interactions,” J. Phys. Chem. B, 108, 6801–6812, 2003.

9.3 ROTATIONAL ISOMERIC STATE METHODS Wayne L. Mattice Department of Polymer Science, The University of Akron, Akron, OH 44325-3909

At very small degree of polymerization, x, the conformation-dependent physical properties of a chain are easily evaluated by discrete enumeration of all allowed conformations. Each conformation can be characterized in terms of bond lengths, l, bond angles, θ, torsion angles, φ, and conformational energy, E. The rapid increase in conformations as x → ∞ prohibits discrete enumeration when the chain reaches a degree of polymerization associated with a high polymer. This difficulty is overcome with the rotational isomeric state (RIS) model. This model provides a tractable method for computation of average conformation-dependent physical properties of polymers, based on the knowledge of the properties of the members of the homologous series with very small values of x. The physical property most commonly computed with the RIS method is the mean square unperturbed end-to-end distance, r 2 0 . Zero as a subscript denotes the unperturbed state, where the properties of the chain are controlled completely by the short-range interactions that are present at very small values of x. This assumption is appropriate for the polymer in its melt, which is a condition of immense importance both for modeling studies and for the use of polymers in reality. The assumption also applies in dilute solution in a solvent, where the excluded volume effect is nil [1]. The second virial coefficient for the osmotic pressure is zero in this special solvent. In good solvents, where the second virial coefficient is positive, the mean square end-to-end distance is larger than r 2 0 , due to the expansion of the chain produced by the excluded volume effect. The excluded volume effect is not incorporated in the usual applications of the RIS model. The first use of the RIS method was reported over five decades ago, well before the widespread availability of fast computers [2]. Given this date of origin of the method, it is not surprising that the correct numerical evaluation of a RIS model requires very little computer time, in comparison with newer 2575 S. Yip (ed.), Handbook of Materials Modeling, 2575–2582. c 2005 Springer. Printed in the Netherlands.

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W.L. Mattice

simulation methods that were developed after fast computers populated nearly every desktop.

1.

Information Required for Calculation of r 2 0

The essential features of the RIS method are well illustrated by the classic calculation of r 2 0 for a long unperturbed polyethylene chain, using as input the properties of n-butane and n-pentane [3]. This illustration identifies the information that is required from the small molecules, and shows how that information is incorporated into the model in order to calculate r 2 0 for a very long chain. The information required for a successful RIS treatment of polyethylene is summarized in Table 1. From n-butane we obtain the values for the length of the C–C bond, l = 0.154 nm, and the C–C–C bond angle, 112◦ . The internal C–C bond is subject to a symmetric torsion potential with three preferred conformations, ν = 3, denoted by trans(t), gauche+ (g + ), and gauche− (g − ). When φ is defined to be zero in the cis state, the torsion angles are 180◦ and ± (60◦ + φ), with the value of φ being about 7.5◦ . The g states are higher in energy that the t state by E σ = E g − E t = 2.1 kJ/mol. This first-order (dependence on a single torsion angle) interaction energy specifies a temperature-dependent statistical weight of σ = exp (−E σ /RT) for a g state relative to a t state. The input from n-butane would be sufficient for the RIS model if the bonds in polyethylene were independent of one another. However, independence of bonds in not observed in polyethylene or in most other polymers. Information about the pair-wise interdependence of the bonds comes from the next higher alkane in the homologous series. Specifically it is from the examination of the energies of the four conformations of n-pentane in which both internal C–C bonds adopt g states. If the two bonds were independent, the four gg states would have the same conformational energy, and that energy would be higher Table 1. Input from small n-alkanes to the RIS model for polyethylene Alkane Butane

Pentane

Information

Symbol

Value for polyethylene

C–C bond length C–C–C bond angle Number of rotational isomeric states Torsion angles

l θ

0.154 nm 112◦

ν φ

First-order interaction energy Second-order interaction energy

Eσ = E g – Et

3 180◦ and ± (60◦ + φ), φ = 7.5◦ 2.1 kJ/mol

E ω = E g + g− − E g+g+

8.4 kJ/mol

Rotational isomeric state methods

2577

by 2Eσ than the conformational energy in the tt state. This expectation is realized if both g states are of the same sign. However, if they are of opposite sign, a strong repulsive interaction of the pendant methyl groups causes the energy to be higher than the energy of the tt conformation by 2Eσ + 8.4 kJ/mol. This important extra energy, denoted E ω , is termed a second-order interaction because it depends on two torsion angles. Examination of the remaining conformations of n-pentane reveals no other important second-order interactions. Third- and higher-order interactions can be incorporated in the model, but often they are unnecessary. Polyethylene is an example of a chain where the performance of the model is not improved by the incorporation of thirdorder interactions. Third-order interactions occur between the methyl groups in n-hexane. Their interaction is prohibitively repulsive when the intervening C–C bonds are all in g states that alternate in sign. However, the g + g − g + conformation of n-hexane is severely penalized by the second-order interactions described in the previous paragraph. Penalizing it further by specifically incorporating the third-order interaction has a trivial effect on numerical results calculated from the model. Therefore the simpler approach, based on first- and second-order interactions only, is the one usually adopted. All of the information in Table 1 is used in the calculation of r 2 0 for a long unperturbed polyethylene chain via the RIS method. Initially the thermodynamic (or energetic) and structural (bond lengths, bond angles, torsion angles) contributions are considered separately. Then these two pieces of the problem are combined for the final answer.

2.

Thermodynamic (energetic) Information: The Conformational Partition Function

The thermodynamic information appears in the conformational partition function, Z, which is the sum of the statistical weights for all ν (n−2) conformations for an unperturbed chain with n bonds. The first- and second-order interactions from Table 1 are counted correctly in an expression for Z that uses a statistical weight matrix, Ui , for each bond. Z = U1 U2 . . . Un

(1)

For internal bonds, Ui is a ν × ν matrix, with rows indexed by the state at bond i − 1, and columns indexed in the same order by the state at bond i. Each column contains the first-order statistical weight appropriate for the conformation that indexes that column, and each element contains the second-order statistical weight appropriate for the pair of states defined by that row and

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W.L. Mattice

column. If the order of indexing is t, g + , g − , Ui is specified by Eq. (2) for 1 < i < n. 



1 σ σ Ui =  1 σ σ ω , 1 σω σ

1
(2)

The terminal row and column vectors in Eq. (1) are U1 = [1 0 0] and Un = [1 1 1]T , where T as a superscript denotes the transpose. The Z for n-butane and n-pentane are 1 + 2σ and 1 + 4σ + 2σ 2 (1 + ω), respectively. Equation (1) continues to correctly count the contributions of σ and ω at all higher n.

3.

Geometric Information: r 2

The geometric information is utilized first for a single, arbitrarily chosen, conformation of the chain. At this stage of the development of the model, the energy of this conformation is not relevant. The squared end-to-end distance of a specified conformation of a chain is often written in terms of the bond vectors, li , as shown in Eq. (3).

r =r·r= 2

li

·

i

i

li

=

li2 + 2

i

li · l j

(3)

1≤i< j ≤n

Evaluation of the last term in this equation requires knowledge of the angles between all pairs of bond vectors. The RIS method uses an alternative (but completely equivalent) formulation of r 2 in which every bond vector is written in its own coordinate system as li = [li 0 0]T . The angles between the bond vectors are treated with transformation matrices, defined so that Ti li + 1 expresses bond vector i + 1 in the local coordinate system of bond i. The local coordinate system for bond i is usually defined so that the x axis runs along this bond, the y axis is in the plane of bonds i and i − 1, with a positive projection on bond i − 1, and the z-axis completes a right-handed Cartesian coordinate system. With this definition, Ti is given by Eq. (4). 



− cos θ sin θ 0 Ti =  − sin θ cos φ − cos θ cos φ − sin φ  − sin θ sin φ − cos θ sin φ cos φ

(4)

Equation (4) is written using the convention that φ = 0 in the cis state. With transformation matrices defined in this manner, r 2 can be written as shown in Eq. (5). r2 =

i

li2 + 2

1≤i< j ≤n

liT Ti Ti+1 . . . T j −1 l j

(5)

Rotational isomeric state methods

2579

The most important difference in the two formulations is that every bond vector is expressed in its own local coordinate system as [li 0 0]T in Eq. (5), but in Eq. (3) none of the bond vectors can be written until their orientation in a common coordinate system has been established. The expression in Eq. (5) can be cast into a matrix form that is similar in structure to Eq. (1). The desired property, r 2 , is obtained as a serial product of n matrices, and all necessary information about bond i is found in the ith matrix. r 2 = F1 F2 . . . Fn

(6)

The internal Fi are 5 × 5 matrices that are more conveniently written in block form as 3 × 3 matrices. 



1 2liT Ti li2 Ti li , Fi =  0 0 0 1

1
(7)

The F1 and Fn in Eq. (6) are the first row and the last column, respectively, of the matrix in Eq. (7).

4.

Combination of Matrix Expressions for Z and r 2

The desired average of r 2 can be written as a sum over the κ conformations of the chain.

r2

0

=

κ

pκ rκ2

(8)

Here rκ2 is obtained from Eq. (6) with the assignment of the n −2 torsion angles that are found in conformation κ. The information required for the normalized probability of this conformation, pκ , is present in Z. It is the statistical weight for conformation κ divided by Z. The desired statistical weight is the product of σ , raised to a power given by the number of g states in conformation κ, and ω, raised to a power given by the number of adjacent pairs of bonds in g states of opposite sign in that conformation. The proper number of factors of σ and ω, and the sum over all κ conformations, are obtained with another serial product of n matrices. r 2 0 = Z −1 G1 G2 . . . Gn

(9)

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W.L. Mattice

The internal Gi matrices in Eq. (9) are obtained by expansion of each element of Ui by the appropriate form for Fi . This form for Fi must use the torsion angle in Ti that is appropriate for the state indexed by that column of Ui . Using Ft , Fg+ , and Fg− to denote these forms of the Fi defined in Eq. (7), the internal Gi for Eq. (9) can be written in block form as ν × ν matrices. 

Ft Gi =  Ft Ft



σ Fg+ σ Fg− σ Fg+ σ ωFg− , σ ωFg+ σ Fg−

1
(10)

When written out element by element, the dimensions are 5ν × 5ν. The terminal Gi in Eq. (9) are given by G1 = [F1 0 0] and Gn = [Fn Fn Fn ]T . Numerical evaluation of Eq. (9) is fast, even for very long chains, because computers can rapidly perform the matrix multiplication that is required. Numerical results are usually reported as the dimensionless characteristic ratio, Cn , in which r 2 0 has been normalized by the value expected for a random flight chain with the same n and l. Thus Cn = r 2 0 /nl 2 . The values of r 2 0 increase without limit as n increases, but Cn approaches an asymptotic limit as n → ∞. The approach to this limit is linear in 1/n at large n. For this reason, C∞ is easily obtained by linear extrapolation to 1/n = 0 of a plot of Cn vs. 1/n. A short FORTRAN program that exploits this property of Cn can be found in Appendix C of reference 4. Table 2 presents a comparison of the behavior of the RIS method with simpler analytical descriptions of C∞ . The freely jointed chain has C∞ = 1 at all temperatures. Fixing the bond angle yields a temperature-independent value given by (1 − cos θ)(1 + cos θ)−1 . Introduction of the symmetric torsion, via C∞ = [(1 − cos θ)(1 + cos θ)−1 ][(1−cosφ)(1 + cosφ)−1 ], produces a further increase in C∞ and correctly predicts the negative temperature coefficient of the mean square dimensions for unperturbed polyethylene chains. The value of C∞ remains too small, however. Introduction of the pair-wise interdependence of the bonds, achieved with the RIS method, is required if the computed C∞ is to be in agreement with experiment.

Table 2. C∞ for unperturbed polyethylene at 413 K, as evaluated by four methods Method

Information used for r 2 0

C∞

Freely jointed chain Freely rotating chain Symmetric hindered rotation, independent bonds RIS model, including bond interdependence

n, l n, l, θ

1 2.20

∂ ln C∞ / ∂ T 0 0

n, l, θ, ν, φ, E σ

3.91

−0.0011 K−1

n, l, θ, ν, φ, E σ , E ω

7.95

−0.0010 K−1

Rotational isomeric state methods

5.

2581

Other Common Uses of the RIS Method

Other conformation-dependent properties of the unperturbed polyethylene chain can also be evaluated by simple modification of the method described above for r 2 0 . Equation (1) for Z is still pertinent, and we also retain the form of Eq. (9). The only important change is alteration of Eq. (7) so that the Fi are the ones appropriate for the new property of interest. For example, the average of the end-to-end vector is obtained through replacement of Eq. (7) with the following expression:

Fi =

Ti 0

li , 1

1
(11)

The terminal matrices in the form of Eq. (6) that is appropriate for r instead of r 2 are F1 = [T1 l1 ] and Fn = [ln 1]T . Elaboration of this approach to other types of conformation-dependent properties may require the introduction of additional information into the Fi . For example, calculation of the mean square dipole moment for a polar chain such as polyoxyethylene or poly(vinyl chloride) requires introduction into Fi of the dipole moment for that bond, mi . In addition to mi , the anisotropic part of the polarizability tensor for the bond is also required in the calculation of the molar Kerr constant. Additional properties, such as the macrocyclization equilibrium constant and the stereochemical composition of a vinyl polymer after epimerization to stereochemical equilibrium, are also accessible. The construction of a RIS model takes its simplest form for a molecule such as polyethylene in which all of the bonds are identical. The method is not restricted to such chains, however. The bonds can be of different types, as they are in polyoxyethylene, and the bonds do not all need to have the same number of rotational isomeric states. Differences in the numbers of rotational isomeric states merely produce rectangular Ui , with the number of rows given by ν at bond i − 1, and the number of columns given by ν at bond i, as is seen in polycarbonate. Third order interactions can be incorporated if the dimensions of Ui are expanded to νi−2 νi−1 × νi−1 νi . Manipulation of Z yield probabilities for the local conformations that can be useful for comparison with experimental data, such as average bond conformations deduced from NMR spectra. This information can also be used to control the behavior of coarse-grained chains so that the chain, and all of its subchains, have distributions functions for their end-to-end distances that match those for the real polymer that the coarse-grained chain represents. These probabilities also provide the basis for Monte Carlo calculations that efficiently generate representative chains, such that the probability of the generation of a chain is directly proportional to its statistical weight. This procedure can be used to efficiently generate data such as the distribution function for the end-to-end distance.

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Flory’s classic book [5] and review [6] are required reading for new users of the RIS model. The subject was updated 25 years after the first publication of Flory’s book, in another book that also includes problems, many with answers, that may facilitate self-instruction in the use of the method [4]. Detailed RIS models have been devised for an enormous number of polymers. The RIS models that appear in the literature through the mid-1990s have been tabulated in an extensive review [7].

Acknowledgment Preparation of this manuscript was supported by NSF DMR 00-98321.

References [1] P.J. Flory, Principles of Polymer Chemistry, Cornell University Press, Ithaca, New York, 1953. [2] M.V. Volkenstein, Dokl. Akad. Nauk SSSR, 78, 879, 1951. [3] A. Abe, R.L. Jernigan, and P.J. Flory, J. Am. Chem. Soc., 88, 631, 1966. [4] W.L. Mattice and U.W. Suter, “Conformational theory of large molecules. The rotational isomeric state model in macromolecular systems,” Wiley-Interscience, New York, 1994. [5] P.J. Flory, “Statistical mechanics of chain molecules,” Wiley-Interscience, New York, 1969; reprinted with the same title by Hanser, München, 1989. [6] P.J. Flory, Macromolecules, 7, 381, 1974. [7] M. Rehahn, W.L. Mattice, and U.W. Suter, Adv. Polym. Sci., 131/132, 1, 1997.

9.4 MONTE CARLO SIMULATION OF CHAIN MOLECULES V.G. Mavrantzas Department of Chemical Engineering, University of Patras, Patras, GR 26500, Greece

Molecular simulations differ from other forms of numerical computation in that the computer with which the calculations are carried out is not merely a machine but the virtual laboratory in which the system is studied. In such a “laboratory”, understanding is achieved by constructing first a theoretical model of molecular behavior able to reproduce and predict experimental observations and solving it using a suitable algorithm or a computer program. Molecular dynamics and Monte Carlo are two such methods that provide exact results to statistical mechanics problems (for the given molecular model) in preference to approximate solutions. Monte Carlo, in particular, has developed to a powerful tool for simulating the properties of complex systems such as chain molecules, because of its capability to accelerate system equilibration through the implementation of large or unphysical moves that do not require the system to follow the natural trajectory.

1.

The Monte Carlo Method

The Monte Carlo (MC) is a computing method for simulating the properties of matter that relies on probabilities. Like molecular dynamics (MD), it aims at providing exact solutions to statistical mechanical problems through a rigorous calculation of the potential energy of interaction; simultaneously, many of the assumptions invoked in analytical or approximate approaches to the same problems are avoided. In contrast to MD, however, where the atoms are moved according to the inter- and intra-molecular forces derived from the potential function by solving Newton’s equations of motion, MC is a stochastic method: it relies on transition probabilities between different states of the simulated system [1, 2]. These transitions are traced through a scheme that 2583 S. Yip (ed.), Handbook of Materials Modeling, 2583–2597. c 2005 Springer. Printed in the Netherlands.

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involves in general three (3) steps: (a) generation of an initial configuration, (b) trial of a randomly generated system configuration, and (c) evaluation of an “acceptance criterion” for the trial configuration and comparison to a random number to decide whether the trial configuration will be accepted or rejected. The acceptance criterion is usually formulated in terms of the potential energy change between trial (new) and existing (old) states and some other properties of the new and old configurations. To accelerate sampling in a MC process, it is important to sample preferentially those states that make the most significant contributions to the configurational properties of the system. This is achieved by the technique of “importance sampling” [1]. According to this, the simulation proceeds by generating a Markov chain of states, i.e., a sequence of states in which the outcome of a trial state depends only on the state that immediately precedes it. Such random states are chosen from a certain distribution, ρ X (Γ), where X denotes the macroscopic constraints of the statistical ensemble in which the simulation is carried out and Γ the phase space. This allows function evaluations to be concentrated in the region of space that makes important contributions to ensemble averages such as the energy. In MC (see details in Chapter 2: Basic MC, Volume 1 of the Handbook), two states Γm and Γn are linked by the element (mn) of a transition matrix π which gives the probability of going from state m to state n. To generate the phase space trajectory, Metropolis et al. [3] suggested the following rule for constructing the matrix π:

πmn =

  amn

if ρn ≥ ρm m =/ n

 amn ρn ρ

if ρn ≺ ρm m =/ n

m

(1)

In Eq. (1), a is a symmetric (amn = anm ) stochastic matrix designed to take the system from state m into any of its neighboring states nwith equal probability and is often called the underlying matrix of the Markov chain. According to the Metropolis rule, then, the probability of accepting the new state is in general:

U Pacc = min 1, exp − kB T

(2)

where U = Unew − Uold denotes the difference in potential energy between new and old states, kB is Boltzmann’s constant and T the temperature. Equation (2) guarantees that energetically more “favorable” states are accepted preferentially. By construction, there is considerable freedom in choosing a, the only requirement being that anm = amn . For example, instead of one (as in the original MC algorithms) several or all atoms of the system may be simultaneously moved. Moreover, one can devise totally unphysical ways for moving atoms

Monte Carlo simulation of chain molecules

2585

in space that substantially depart the system from its natural trajectory. These allow moving through configuration space much more efficiently than by MD. For systems of chain molecules (e.g., synthetic polymers, highly branched macromolecules and biopolymers), this is of paramount importance. Chain systems present considerable difficulties for molecular simulation relative to either atoms or short polyatomic molecules due to the wide spectrum of time and length scales characterizing their dynamics and structure. The computational difficulties are more acute for the dynamic methods, such as MD. These methods are plagued by the problem of long relaxation times and their applicability is restricted to short chain-length systems. Of course, one can think of a domain decomposition method and algorithm execution on a parallel computer for decreasing the length scale of MD. However, one is still faced with the problem of long relaxation times, since the time scale that can be spanned by a brute-force dynamic method today falls short of the longest relaxation time of real-life chain systems [4]. In this direction, MC methods can play a key role: through the design of clever (sometimes “unphysical”, from the point of view of true dynamics) moves for generating new, trial configurations, they can accelerate system equilibration by many orders of magnitude more efficiently than MD, for the same model of molecular geometry and interatomic potential.

2.

Simple Monte Carlo Moves

For systems of simple molecules, trial configurations in a MC process can typically be generated by displacing, exchanging, removing or adding a molecule [2]. For systems of chain molecules, the situation is more complex. Excluded volume interactions among polymer segments, connectivity of atoms along chains, and conformational stiffness of chain backbones make it very difficult to sample configuration space efficiently. Thus, the earliest MC simulations were conducted using lattice models, initially on single chains and later on multi-chain systems [5]. An early “unphysical” MC move that proved particularly useful in simulations of dense polymer systems was “reptation” [6]. Reptation is a “slithering snake” move that deletes a segment from one end of a randomly selected chain and appends it to the other end; through this, the chain “slides” along its contour by one segment. Other simple MC moves include [7]: (a) end-mer rotation, (b) libration or flip, (c) configurational bias, (d) concerted rotation, (e) generalized reptation and (f) parallel rotation [8]. Configurational bias (CB), in particular, has found extensive applications in simulations of phase equilibria formulated initially in the form of the Gibbs ensemble MC method [9] and more recently in the form of an expanded grand canonical ensemble [10]; the latter formulation

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alleviates problems associated with the insertion/deletion or exchange of large chain molecules in dense systems. From the above MC moves, reptation, configurational bias, end-mer rotation and generalized reptation operate only at chain ends. Due to this, when used with realistic continuum models, they benefit from the excess free volume available near chain ends and can significantly enhance system equilibration. However, their effectiveness degrades in longer-chain systems where chain ends are scarce. As a result, only chains up to 70 units long can be simulated with the cocktail of simple MC moves mentioned above.

3.

Complex MC Moves: Variable Connectivity and Extended Configurational Bias MC Methods

To simulate longer-chain systems, moves capable of inducing drastic reconfiguration of large internal sections along the chain are also needed. Such moves were first introduced in the 1980s in the form of “chain breaking” or “pseudokinetic” MC algorithms [11]. These algorithms alter the connectivity among polymer segments in the lattice model at the expense of introducing some polydispersity (i.e., a distribution of chain lengths) in the polymer. Small alterations in chain connectivity are very desirable in a MC simulation because they substantially enhance the efficiency with which the long-range structural features of the system (chain end-to-end vectors, radii of gyration, etc.) are sampled. Guided by these “chain-breaking” lattice MC algorithms, variable connectivity MC methods of continuous-space models were designed in the mid 1990’s for simulations of chain systems. The first such method to be developed was end-bridging [12]. End-bridging (EB) effects a change in chain connectivity by constructing a trimer bridge between a chain end and an interior segment of another chain. Simultaneously, one of the trimers adjacent to the bridged interior atom is excised; this ensures that the total polymer mass remains constant. EB is schematically shown in Fig. 1. EB as well as the rest of the variable connectivity MC moves for chain molecules developed in the ensuing years were founded around the geometric problem of trimer bridging; this is mathematically formulated as follows: Given two dimers in space, connect them with a trimer such that the resulting heptamer has prescribed bond lengths and angles [7]. The problem has so far been addressed by two different methods [13, 14]. Other inter- or intra-molecular variable connectivity moves include intramolecular rebridging or concerted rotation, directed internal bridging, directed end-bridging, fusion and scission, and self-end-bridging. These moves belong to the class of chain connectivity-altering MC methods, whose introduction and

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Figure 1. Schematic of the end-bridging (EB) move. The attacking chain is denoted as ich and the victim chain as jch. Trimer ( ja , jb , jc ) is to be excised from the victim chain. ( ja , jb , jc ) is the trimer bridging end i of the attacking chain to internal mer j of the victim chain. The two new chains are labeled ich and j ch .

application in simulations of chain molecules have revolutionized the study the conformational and thermodynamic properties of these systems [7]. To comply with the condition of microscopic reversibility in a variable connectivity MC move such as EB, care must be taken to: (a) evaluate all possible geometric solutions to the trimer bridging problem associated with the move, (b) incorporate appropriate Jacobians of the transformations (involved in the solution of the geometric problem) into the acceptance criteria, and (c) calculate the attempt probabilities for both the forward and reverse problems. A typical acceptance criterion of an EB move, for example, reads:

Pacc = min 1,

Pselect (new → old) J (new) exp (−U (new)/kb T ) Pselect (old → new)J (old) exp (−U (old)/kb T )

(3)

where J and Pselect denote the Jacobian of transformation and attempt probability of the corresponding move (old → new or new → old). To deal with polydispersity effects, simulations with chain connectivityaltering MC algorithms are carried out in a semi-grand canonical ensemble, in which a spectrum µ∗ of chemical potentials is employed to control the chain length distribution. Such a semigrand ensemble is denoted as Nch nPT µ∗ , since the following variables are kept constant: the pressure P, the temperature T ,

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the total number of chains Nch , the total number of mers n, and the spectrum of relative chemical potentials µ∗ of all chain species in the chain except two, which are taken as reference species. Expressions for µ∗ that generate the most important chain length distributions have been derived by Pant and Theodorou [12] for bulk systems, and Daoulas et al. [15] for chains at the interface with a substrate. Complex MC moves which do not effect alterations in chain connectivity and, still, enable equilibration of rather long chain length systems have also been formulated. These moves were first introduced by Escobedo and de Pablo [16] as extensions of the original continuum confgurational bias (CCB) method. Today, they have developed to what are known as internal configurational bias (ICB) and self-adapting fixed end point configurational bias (SAFE-CB) methods which, by combining the philosophies of Con-Rot and CB methods, enable sampling of arbitrarily long internal chain sections [17, 18]. Generalized CB algorithms have proven very useful in simulations of cross-linked-network structures, but most importantly in designing simulation strategies based on density-of-states sampling methods [19–21].

4.

Advanced Monte Carlo Moves: Simulation of Non–linear Chain Systems

The class of variable connectivity MC moves was initially developed for simulating model systems of linear chains and was heavily dependent on the presence of chain ends. MC moves that are independent of chain ends were for the first time proposed by Balijepalli and Rutledge [22]. They are the intrachain and inter-chain concerted rotation moves developed as generalizations of the ConRot move for simulating the morphology and elasticity [23] of semicrystalline polymer interphases consisting of chain segments whose terminus is fixed to the crystal surface. Through a novel design by Karayiannis et al. [24], these moves have now developed to what is known as the double bridging (DB) and intramolecular double rebridging (IDR). The two moves are generalizations of the EB move, since they involve the construction of two trimer bridges (instead of one); they are schematically shown in Figs. 2(a) and (b). DB and IDR are applicable to a variety of systems such as nonlinear chain architectures, long-chain branched macromolecules, cyclic peptides, grafted polymers, chains with stiff backbones, and infinite-length chain molecules (see Fig. 3); the simulation of all these systems is almost impossible with MD or other MC methods. A typical example is that of the long-chain branched molecules [see, for example, Fig. 4]. By properly re-designing the two moves to allow for bridgings between: (a) the main backbones of two different chains, (b) the branches of two different chains, and (c) the branches of the same

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Figure 2a. Schematic of the double bridging (DB) move. (a): Local configuration of the two chains, ich and jch, prior to the DB move. Trimer ( ja , jb , jc ) is to be excised from jch and trimer (i a , i b , i c ) from ich. (b): Local configurations of the two new chains after the DB move. Trimer ( ja , jb , jc ) connects atoms i and j in ich’. Trimer (i a , i b , i c ) connects atoms i 2 and j2 in j ch (after Karayiannis et al. 2002).

chain, and by introducing special moves (such as the H-BR and the double ConRot) to effect atom displacements at the junction points, a novel MC algorithm arises capable of simulating H-shaped, comb and star-like chain molecules [25]. In addition to developing new MC algorithms capable of more efficient sampling of long chain systems, key to balancing CPU time and code performance for a given system is the choice of the optimal (or near-optimal) mix of moves. For a given application, with what appears to be a host or a cocktail of moves, choosing or identifying the most efficient mix can substantially affect code performance. In searching for optimal move mixes, one can profit by simple scaling arguments that consider polymer chains as random coils. Such an analysis was carried out, for example, by Mavrantzas et al. [13] who quantified the performance of EB in simulations of bulk polymers. For polydisperse, linear-chain systems, a mix of moves which results in nearly optimal code performance includes: 5% reptations, 5% end-mer rotations, 5% CCBs, 5% flips, 30% ConRots, 48% EBs and 2% volume fluctuations. This mix combines the higher acceptance ratio of the simpler moves with the power of the more complex (less frequently accepted, though) EB move. Such a scheme remains near-optimum for systems with a polydispersity index

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Figure 2b. Schematic of the intramolecular double rebridging (IDR) move. Top: Local configuration of the chain prior to the IDR move. The attack shown by the dark gray arrow is combined either with attack a or with attack b represented by the light gray arrows. Bottom: trial configurations of the chain after both a and b attacks have been completed (after Karayiannis et al. 2002).

higher than 1.1. At lower polydispersities or for strictly monodisperse systems, the EB moves should be replaced by DB’s and IDR’s at equal proportions. Including DB’s and IDR’s (in favor of ConRot or EB moves) could accelerate system equilibration also: (a) In cases where one or both chain ends are permanently fixed at a surface; typical examples include systems of end-grafted polymer ends (polymer brushes) or semicrystalline interphases. (b) In systems of non-linear polymer architectures; typical examples are the long-chain branched and the cyclic molecules, and (c) In systems of oriented chains such as in the presence of a deforming tensorial field [26]. For chains bearing short, frequently-spaced branches along their backbone, code performance could be enhanced by re-growing the short branches with extended CB moves. For

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Figure 3. Application of the DB and/or IDR moves to MC simulations of chain molecules with a variety of chemical architectures. (a): H-shaped molecules, (b): cyclic molecules and (c): grafted molecules (after Karayiannis et al. 2002).

systems consisting of different chain species (such as bi-disperse melts, mixtures of end-grafted and free chains or mixtures of linear and non-linear molecules), configurational sampling can be accelerated by allowing the available variable connectivity moves to also operate on pairs of chains that belong to different species.

5.

Applications

An important test of the ability of any atomistic simulation method to describe liquids in a realistic manner is the calculation of atomic structure. This is usually quantified by calculating the pair radial distribution function g(r) describing the spatial correlations between two atoms at separation distance r in the liquid. For chain molecules, the (total) pair distribution function has contributions from both intra- and inter-molecular correlations: g tot (r) = g(r) +

w(r) ρN

(4)

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Scheme 2

(a) (a)

(b)

(c)

(b)

Figure 4. Schematic representation of possible combinations of the DB and IDR moves to be used in MC simulations of non-linear, H-shaped chain molecules. Scheme 1(a)–(c): bridges between the main backbones or the branches of two different molecules. Scheme 2(a)–(b): the H-BR move designed to effect displacements of the branch points (after Karayiannis et al. 2003).

where w(r) is the intra-chain pair density function and g(r) the intermolecular pair distribution function. In Eq. (4), ρ = Nch /V is the chain number (or molecular) density and N the number of mers per chain. The total pair distribution function is of great significance because its Fourier transform gives the static structure factor S(k), which is experimentally measured by X-ray diffraction: S(k) − 1 = ρ N

∞

4πr 2 0

sin(kr) tot g (r) − 1 dr kr

(5)

Figure 5 demonstrates the unique ability of the variable connectivity MC algorithms to reliably simulate the atomic structure of a long-chain linear polyethylene (PE) melt, by comparing simulated against experimentally measured X-ray patterns. The agreement is really excellent. The MC simulation has been carried out in a cubic box characterized by periodic boundary conditions in all three dimensions with a 25-chain C500 strictly monodisperse PE melt at T = 450 K and P = 1 atm, using the united atom model [24]. The superiority of MC algorithms based on chain connectivity-altering moves relative to the conventional MD method in efficiently simulating systems of chain molecules is documented in Fig. 6. The figure compares the rate with which the melt long-length scale characteristics is relaxed with the two methods in two systems, representative of a linear and a non-linear chain

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2.0 simulation results Xⴚray diffraction data

1.5

S(k)ⴚ1

1.0 0.5 0.0 ⴚ0.5 ⴚ1.0

0

2

4

6

8

10

12

14

k (A⫺1) Figure 5. Simulated (at T = 450 K) and experimental (at T = 430 K) X-ray diffraction patterns of linear polyethylene (P = 1 atm). The simulations have been executed with the variable connectivity MC algorithms discussed in the main text (after Karayiannis et al. 2002).

architecture, respectively. The first is a linear, strictly monodisperse C500 PE melt and the second a non-linear H-shaped PE melt containing on the average 300 carbon atoms on the backbone and 50 carbon atoms on each one of its four branches denoted as PEH(50)2 (300)(50)2 . As a measure of method efficiency, we use the rate of decay of the end-to-end vector orientational autocorrelation function with CPU time. Figure 6 shows MC to be orders of magnitude more efficient than MD: Even if the length scale of the MD simulation is decreased with a domain decomposition method and execution on a 16-node parallel computer, MC still remains the method of choice. Figure 7 shows a typical snapshot of a 4-chain PEH(70)2 (400)(70)2 ) melt before and after the MC simulation with the advanced moves discussed above. Within about 2 × 106 CPU s on a dual 2.8 GHz Xeon system, the bad initial configuration (characterized by unusually large voids and an unphysical distribution of mass in the simulation box) is equilibrated to a structure fully representative of that of the real melt (characterized by a uniform distribution of the polymer mass all over the box and excellent statistics of chain dimensions). The new variable-connectivity MC moves can also be designed for effective vector implementation through domain decomposition methods. This has

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1.0

0.8

(d)

0.6

(a)

(c)

0.4

0.2 (e)

(b) 0.0 0.0

0.5

1.0 CPU TIME (106 S)

1.5

2.0

Figure 6. Decay of the chain end-to-end vector orientational autocorrelation function in MC and MD simulations of linear and H-shaped PE melts: (a) A monodisperse C500 PE melt simulated with brute force MD. (b) The same system simulated with the variable connectivity MC moves. (c) The same system simulated with a parallel MD code on a 16-node Beowulf cluster. (d) A PEH(50)2 (300)(50)2 PE melt simulated with brute force MD. (e) The same system simulated with the new chain connectivity-altering MC moves.

Figure 7. Typical atomistic snapshots of the simulated PEH(70)2 (400)(70)2 system before (a) and after (b) the simulation with the variable connectivity MC algorithms (T = 450 K, P = 1 atm). Shown in black and dark gray are atoms belonging to the backbone and the four branches of an arbitrarily selected H-shaped molecule (after Karayiannis et al. 2003).

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allowed simulating by MC the equilibrium atomic structure of a molten linear PE melt with an average chain length of 6000 backbone carbon atoms (C6000); this is typical of the commercial grades that are widely used for injection molded articles [27].

6.

Outlook

Through the design of efficient (mostly “unphysical”) moves, MC has developed to a principal research tool for simulating chain systems of a variety of chemical architectures. Polyolefin and polydiene melts, polymers grafted on substrates, cyclic peptides, oriented systems, as well as binary systems composed of chemically similar macromolecules are all amenable to MC simulation in continuous space and full atomistic detail. Of course, being a stochastic method, MC cannot unfortunately provide any direct dynamic information. Due to its dynamical nature, the basic technique of MD remains the only molecular simulation method that can give real-time information about the system evolution. MC can, however, be used to calculate dynamic properties indirectly. This can be achieved either in the context of non-equilibrium thermodynamics or by combining MC with MD simulations in a hierarchical scheme. In the former case, MC can be used to address questions related to the viscoleastic features of the melt [26, 28]. For example [26], it can address questions related to: (a) the dependence of the entropy of the deformed chain system on its average conformation or (b) the functional form of its effective spring constant. In the latter case, model configurations thoroughly equilibrated with MC can serve as starting points for executing MD simulations in order to estimate the spectrum of relaxation times of the chain system and its frictional and other dynamic properties [4]. Designing MC algorithms for simulations of beyond-equilibrium systems or combining MC with MD in the context of coarse-grained methodologies constitute one of the most active research areas in the field of molecular simulation. These two directions offer a more promising avenue for probing both structure and dynamics in chain systems than direct MD. The simulation of polymers (either purely amorphous or semi-crystalline) beyond equilibrium, the prediction of mixing thermodynamics in polymer blends, the derivation of “coarse-grained” model representations for complex chain molecules (such as polyimides and polyesters) from atomistic information, and the simulation of the structural and conformational properties of peptide molecules constitute only a small part in the list of materials problems to be investigated by MC in the coming years. As large-scale scientific computing becomes more accessible and computing performance grows each passing day, so the popularity and usefulness of MC will continue to increase.

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References [1] M.P. Allen and D.J. Tildesley, Computer Simulation of Liquids, Clarendon Press, Oxford, 1987. [2] R.J. Sadus, Molecular Simulation of Fluids: Theory, Algorithms and ObjectOrientation, Elsevier, Amsterdam, 1999. [3] N. Metropolis, A.W. Rosenbluth, M.N. Rosenbluth, A.H. Teller, and E. Teller, “Equation of state calculations by fast computing machines,” J. Chem. Phys., 21, 1087–1092, 1953. [4] V.A. Harmandaris and V.G. Mavrantzas, “Molecular dynamics simulations of polymers.” In: M. Kotelyanski and D.N. Theodorou (eds.), Simu. Meth. for Poly., Marcel Dekker, New York, pp. 177–222, 2004. [5] Z. Alexandrowicz and Y. Accad, “Monte Carlo of chains with excluded volume: distribution of intersegmental distances,” J. Chem. Phys., 54, 5338–5345, 1971. [6] M. Vacatello, G. Avitabile, P. Corradini, and A. Tuzi, “A computer model of molecular arrangement in a n-paraffinic liquid,” J. Chem. Phys., 73, 548–552, 1980. [7] D.N. Theodorou, “Variable-connectivity Monte Carlo algorithms for the atomistic simulation of long-chain polymer systems,” In: P. Nielaba, M. Mareschal, G. Ciccotti (eds.), Bridging Time Scales: Molecular Simulations for the Next Decade, Springer Verlag, Berlin, pp. 64–127, 2002. [8] S. Santos, U.M. Suter, M. M¨uller, and J. Nievergelt, “A novel parallel-rotation algorithm for atomistic Monte Carlo simulation of dense polymer systems,” J. Chem. Phys., 114, 9772–9779, 2001. [9] Panagiotopoulos, “Direct determination of fluid phase equilibria in the Gibbs ensemble: a review,” Mol. Phys., 9, 1–23, 1992. [10] A.P. Lyubartsev, A.A. Martsinovski, S.V. Shevkunov, and P.N. VorontsovVelyaminov, “New approach to Monte Carlo calculation of the free energy: method of expanded ensembles,” J. Chem. Phys., 96, 1776–1783, 1991. [11] O.F. Olaj and W. Lantschbauer, “Simulation of chain arrangement in bulk polymer .1. Chain dimensions and distribution of the end-to-end distance,” Makromol. Chem.Rapid Commun., 3, 847–858, 1982. [12] P.V.K. Pant and D.N. Theodorou, “Variable connectivity method for the atomistic Monte Carlo simulation of polydisperse polymer melts,” Macromolecules, 28, 7224–7234, 1995. [13] V.G. Mavrantzas, T.D. Boone, E. Zervopoulou, D.N. Theodorou, “End-bridging Monte Carlo: a fast algorithm for atomistic simulation of condensed phases of long polymer chains,” Macromolecules, 32, 5072–5096, 1999. [14] M.G. Wu and M.W. Deem, “Efficient Monte Carlo for cyclic peptides,” Mol. Phys., 97, 559–580, 1999. [15] K.Ch. Daoulas, A.F. Terzis and V.G. Mavrantzas, “Variable connectivity methods for the atomistic Monte Carlo simulation of inhomogeneous and/or anisotropic polymer systems of precisely defined chain length distribution: tuning the spectrum of chain relative chemical potentials,” Macromolecules, 36, 6674–6682, 2003. [16] F.A. Escobedo and J.J. de Pablo, “Extended continuum configurational bias Monte Carlo methods for simulation of flexible molecules,” J. Chem. Phys., 102, 2636– 2652, 1994. [17] A. Uhlherr, “Monte Carlo conformational sampling of the internal degrees of freedom of chain molecules,” Macromolecules, 33, 1351–1360, 2000.

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[18] C.D. Wick and J.I. Siepmann, “Self-adapting fixed end-point configurational-bias Monte Carlo method for the regrowth of interior segments of chain molecules with strong intramolecular interactions,” Macromolecules, 33, 7207–7218, 2000. [19] F. Wang and D.P. Landau, “Efficient, multiple-range random walk algorithm to calculate the density of states,” Phys. Rev. Lett., 86, 2050–2053, 2001. [20] T.S. Jain and J.J. de Pablo, “A biased Monte Carlo technique for calculation of density of states of polymer films,” J. Chem. Phys., 116, 7238–7243, 2002. [21] M.S. Shell, P.G. Debenedetti, and A.Z. Panagiotopoulos, “An improved Monte Carlo method for direct calculation of the density of states,” J. Chem. Phys., 119, 9406– 9411, 2003. [22] S. Balijepalli and G.C. Rutledge, “Simulation study of semi-crystalline polymer interphases,” Macromol. Symp., 133, 71–99, 1998. [23] P.J. in ’t Veld and G.C. Rutledge, “Temperature-dependent elasticity of a semicrystalline interphase composed of freely rotating chains,” Macromolecules, 36, 7358– 7365, 2003. [24] N.Ch. Karayiannis, V.G. Mavrantzas, and D.N. Theodorou, “A novel Monte Carlo scheme for the rapid equilibration of atomistic model polymer systems of precisely defined molecular architecture,” Phys. Rev. Lett., 88, 105503: 1–4, 2002. [25] N.Ch. Karayiannis, A.E. Giannousaki, and V.G. Mavrantzas, “An advanced Monte Carlo method for the equilibration of model long-chain branched polymers with a well-defined molecular architecture: detailed atomistic simulation of an H-shaped polyethylene melt,” J. Chem. Phys., 118, 2451–2454, 2003. ¨ [26] V.G. Mavrantzas and H-Ch. Ottinger, “Atomistic Monte Carlo simulations of polymer melt elasticity: their nonequilibrium thermodynamics GENERIC formulation in a generalized canonical ensemble,” Macromolecules, 35, 960–975, 2002. [27] A. Uhlherr, S.J. Leak, N.E. Adam, P.E. Nyberg, M. Doxastakis, V.G. Mavrantzas, and D.N. Theodorou, “Large scale atomistic polymer simulations using Monte Carlo methods for parallel vector processes,” Comp. Phys. Commun., 144, 1–22, 2002. ¨ [28] H-Ch. Ottinger, Beyond Equilibrium Thermodynamics, Wiley, New York, 2004.

9.5 THE BOND FLUCTUATION MODEL AND OTHER LATTICE MODELS Marcus M¨uller Department of Physics, University of Wisconsin, Madison, WI 53706-1390

Lattice models constitute a class of coarse-grained representations of polymeric materials. They have enjoyed a longstanding tradition for investigating the universal behavior of long chain molecules by computer simulations and enumeration techniques. A coarse-grained representation is often necessary to investigate properties on large time- and length scales. First, some justification for using lattice models will be given and the benefits and limitations will be discussed. Then, the bond fluctuation model by Carmesin and Kremer [1] is placed into the context of other lattice models and compared to continuum models. Some specific techniques for measuring the pressure in lattice models will be described. The bond fluctuation model has been employed in more than 100 simulation studies in the last decade and only few selected applications can be mentioned.

1.

Coarse Graining and Universal Behavior

Lattice models are well-suited to investigate universal behavior. Long chain molecules share many common mesoscopic characteristics which are independent of the atomistic structure of the chemical repeat units. For instance, in solutions or melts the self-similar structure at large length scales is only characterized by a single length scale, the chain’s end-to-end distance R. This independence of the qualitative behavior on chemical details is born out by many experimental observations. Therefore, one can use a coarse-grained description, which represents a small number of chemical repeat units by an effective segment. There are few examples of explicit coarse-graining [2] from a specific material onto a lattice model, often this mapping is only invoked conceptually. In this case, the question which interactions are necessary to bring about the observed universal behavior is of great interest in itself [3]. The concept of universality is 2599 S. Yip (ed.), Handbook of Materials Modeling, 2599–2606. c 2005 Springer. Printed in the Netherlands.

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not restricted to one-component polymer systems, but it is also observed, for instance, in amphiphilic systems, i.e., diblock copolymers in the molten state, amphiphilic polymers in aqueous solution, and biological liquids self-assemble into spatially periodic structures on the length scale of the molecule’s extension. Although the systems differ strongly in their microscopic interactions they share many qualitative features of their phase behavior [4]. For the self-similar structure of polymers in solutions (and melts) there exists a formal justification of coarse-grained models: de Gennes [5] related the structure of a polymer chain in a good solvent to a field theory of a n-component vector model in the limit n → 0. This class of models exhibits a continuous phase transition and the properties close to this critical point have been investigated extensively with renormalization group calculations. The inverse chain length plays the role of the distance from the critical point of the n = 0 component vector model. As in the theory of critical phenomena, the behavior in the vicinity of this critical point (i.e., 1/N 1) is governed by a universal scaling behavior, which is brought about by only a few relevant interactions. This fact justifies the use of highly coarse-grained models that incorporate only two relevant interactions: connectivity along the chain and binary segmental interactions.

2.

Lattice Models and Computational Techniques

Lattice models of polymer solutions are a particularly simple and computationally efficient realization, and therefore they have attracted abiding interest both for single chain simulations as well as for simulations of polymer solutions and melts. In simple lattice models, a small group of atomistic repeat units is represented by a site on a simple cubic lattice. Also other lattice structures have been considered, e.g., face-centered cubic, square or triangular. Segments along a polymer occupy neighboring lattice sites and multiple occupation of lattice sites is forbidden (excluded volume). The latter constraint corresponds to the repulsive binary interaction under good solvent conditions. Isolated chains on the lattice adopt configurations of self-avoiding walks. Lattice models and algorithms have been devised for multi-chain systems. Some analytical theories (e.g., Flory–Huggins theory of polymer mixtures or Flory equation of state) or enumeration techniques are particularly clearly formulated for lattice models and can be directly investigated by Monte Carlo simulations. Compared to coarse-grained models in continuum space (e.g., bead-spring type models), lattice models are computationally more efficient. The structure of the underlying lattice allows for a fast rejection of forbidden configurations and additional program optimizations. For instance, there is only a (small) number of allowed bond vectors in a lattice model and excluded volume constraints can be checked very efficiently. In a generic off-lattice model, there are many bond vectors with a very large energy and the excluded volume constraint is often modeled as a

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large, but not infinite energy of overlap. Even though configurations with a large repulsive energy in the off-lattice model will have only a negligible statistical weight, they cannot be rejected upfront and the whole energy change of a move has to be calculated to accept or reject the move. Therefore, lattice models are particularly suitable for investigating phenomena on mesoscopic length and time scales, which pose large computational challenges that cannot yet be addressed with atomistic or off-lattice models [4]. There are also disadvantages of lattice models: only the configurational part of the partition function can be investigated. There are no forces or momenta. Therefore, one cannot obtain the pressure via the virial expression and one cannot use molecular dynamics simulations to study the ballistic motion of segments at short time scales or hydrodynamic behavior at large time scales. This holds a fortiori for non-equilibrium situations, e.g., shear flow, etc. Random local displacements of segments can at most mimic a purely diffusive dynamics. Moreover, simulations at constant pressure or in the Gibbs ensemble are difficult. The equilibration of dense multi-chain systems is a challenging problem for computer simulations [6, 7], and lattice models have been a testing ground for many algorithms. Algorithms can be particularly easily formulated on a lattice (e.g., configurational bias Monte Carlo) and efficiently implemented. Some methods are tailored to isolated chains or very dilute systems (e.g., the pivot algorithm [8, 9]); other methods provide an effective relaxation of the overall chain dimensions in dense systems (e.g., configurational bias Monte Carlo [10, 11]). Special techniques have been devised to calculate the pressure in lattice models. Dickman [12] proposed a method to measure the pressure by calculating the free energy change associated with moving a hard wall by one lattice unit. This method is very efficient if one takes due account of the increase of the density that occurs upon compression [13]. An alternative method has been devised for systems with periodic boundary conditions, where one inserts or removes a whole slice of the lattice and regrows the chains which are affected by this Monte Carlo move via configurational bias [14]. This scheme has been applied in lattice Gibbs ensemble Monte Carlo simulations [15], but it is a tour de force and the acceptance rate and complexity depends on the system size. If one is interested in the equation of state it might be more convenient to obtain the pressure by integrating the excess chemical potential.

3.

The Bond Fluctuation Model and Selected Applications

Although simple lattice models reproduce the universal features of polymer solutions and melts, it is difficult to incorporate some qualitative details

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of molecular architecture. The simple lattice model allows only for two bond angles which makes the investigation of orientational effects prone to lattice artifacts, e.g., there is a strong cubic anisotropy and the lattice structure tends to stabilize liquid crystalline phases. Moreover, particles in real fluids arrange to form neighboring shells. This local packing structure of the fluid does not affect the universal scaling behavior but it is pertinent to relating coarse-grained effective interactions to underlying microscopic potentials. Since the vacancies on the lattice and the polymer segments have the same size, packing effects in the density correlation function are largely absent. More sophisticated lattice models, in which monomers are represented by extended objects (e.g., a whole unit cube) on the lattice, have been explored.∗ These models exhibit packing effects, albeit much weaker than bead-spring type models in continuum space. A large number of bond vectors and angles (see also Ref. [16]) results in a better approximation of isotropic space while still retaining the computational advantages of lattice models. They also allow for a diffusive dynamics of the polymers on the lattice which consists of random local displacements of the monomers. Moreover, the bond vectors can be chosen such that the excluded volume constraint prevents bonds from crossing through each other in the course of these local displacements. This non-crossability takes account of topological effects which are important for the dynamical properties of linear chains and influence the conformational statistics of ring polymers (i.e., to avoid topological interactions rings collapse in a concentrated solution) and networks. The bond fluctuation model can be formulated in two [1] and three [17] spatial dimensions. In three dimensions a monomer blocks all eight corners of a unit cell of a simple cubic lattice from further occupancy. This represents the segmental excluded volume. Monomers along polymer chain √ a√ √ are connected via one of 108 bond vectors of length 2, 5, 6, 3, and 10 in units of the lattice spacing. This realizes the connectivity on the monomers along a chain. The basic Monte Carlo move consists in randomly selecting a monomer and attempting to displace it by one lattice unit in one randomly chosen direction. While this move mimics a diffusive dynamics, other Monte Carlo moves (e.g., slithering snake, configuration bias Monte Carlo or semi-grandcanonical identity switches) have been applied to investigate thermodynamic equilibrium properties. In addition to its computational efficiency, one of the key advantages of the bond fluctuation model is the knowledge of a variety of different quantities. The density and chain length dependence of many static and dynamic properties of the basic model are compiled in Refs. [18, 19]. Upon increasing

* The use of extended particles on a lattice has also been applied to fluids (cf. A.Z. Panagiotopoulos, “On the equivalence of continuum and lattice models for fluids,” J. Chem. Phys., 112, 7132–7137, 2000.)

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the density from a dilute solution to a melt one observes a cross-over from self-avoiding to Gaussian chain statistics and from (renormalized) Rouse to reptation-like dynamics for very long chain lengths. Inter- and intramolecular paircorrelation functions in melts and semi-dilute solutions have been investigated. Different chain topologies have been studied: ring polymers [20], polymer networks [21, 22], end-tethered chains [23, 24] as well as equilibrium polymers [25]. Various additional interactions have been incorporated, e.g., bond-length potentials, chain stiffness, and binary interactions: attractive interactions between monomers result in a collapse of an isolated chain from a self-avoiding walk (for T > Tθ ) to a dense globule (for T < Tθ ). At the θ-temperature Tθ the chain conformations are Gaussian. A multi-chain system separates into liquid and vapor below the θ-temperature. The scaling of the critical temperature and density with chain length has attracted much interest [26]. At very low temperatures the liquid that coexists with the vapor becomes very dense and the lattice structure becomes important. Using a bond-length potential, Baschnagel [27] investigated the glass transition. The bond potential favors extended bonds which block lattice sites and effectively decrease the free volume. At low temperatures or high densities, the competition between extending bonds and dense packing frustrates the systems and leads to a glassy arrest of the dynamics, which has been investigated both in the bulk and in thin films [28]. Two component mixtures have been modeled by incorporating a shortranged repulsion between unlike segments. From the structure of the polymer fluid and the binary repulsion, the Flory–Huggins parameter can be extracted [29]. This makes the bond fluctuation model an ideal testing bed for comparing simulation results quantitatively to predictions of the self-consistent field theory for spatially inhomogeneous systems. Reister et al. [30] compared the dynamics of phase separation to different versions of dynamic mean field theory. The phase diagram of diblock copolymers in confinement [31], and reactive compatibilization [32] has been investigated, and also the phase behavior of more complex architectures, e.g., triblock-copolymers [33], is accessible. Due to extended structure, a polymer interacts with many neighbors and mean field predictions are often accurate. There are however caveats: capillary waves broaden interface profiles [34], they renormalize the interaction between an interface and a boundary (wall) [35, 36], and they give rise to a microemulsion in homopolymer/diblock-copolymer mixtures [15, 37]. The phase diagram of random copolymers [38] in three spatial dimensions and the scaling of the critical temperature in two-dimensional homopolymer blends [39] exhibits large deviations from mean field predictions. This list of selected applications of the bond fluctuation model illustrates the versatility and range of applications of coarse-grained lattice models. By virtue of their computational efficiency and the availability of information on

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a wide variety of properties, coarse-grained lattice models will continue to be important for tackling computationally challenging questions about the structure and the thermodynamics of polymer systems and for evaluating the approximations invoked in analytical approaches.

References [1] I. Carmesin and K. Kremer, “The bond fluctuation method – a new effective algorithm for the dynamics of polymers in all spatial dimensions,” Macromolecules, 21, 2819–2823, 1988. [2] J. Baschnagel, K. Binder, P. Doruker, A.A. Gusev, O. Hahn, K. Kremer, W.L. Mattice, F. M¨uller-Plathe, M. Murat, W. Paul, S. Santos, U.W. Suter, and V. Tries “Bridging the gap between atomistic and coarse-grained models of polymers: status and perspectives,” Adv. Polym. Sci., 152, 41–156, 2000. [3] M. M¨uller, “Mesoscopic and continuum models. In: J.H. Moore and J.H. Spencer (eds.), Encyclopedia of Physical Chemistry and Chemical Physics, vol II, IOP, Bristol, pp. 2087–2110, 2001. [4] M. M¨uller, K. Katsov, and M. Schick, “Coarse grained models and collective phenomena in membranes: computer simulation of membrane fusion,” J. Polym. Sci. B, Polym.Phys., 41, 1441–1450, 2003 (highlight article). [5] P.G. de Gennes, Scaling Concepts in Polymer Physics, Cornell University Press, Ithaca, 1979. [6] K. Kremer and K. Binder, “Monte Carlo simulations of lattice models for macromolecules,” Comput. Phys. Rep., 7, 259–310, 1988. [7] J.J. de Pablo and F.A. Escobedo, “Monte Carlo methods for polymeric systems,” Adv. Chem. Phys., 105, 337–367, 1999. [8] N. Madras and A.D. Sokal, “The pivot algorithm – a highly efficient Monte Carlo method for the self-avoiding walk,” J. Stat. Phys., 50, 109–186, 1988. [9] A.D. Sokal, “Monte Carlo methods for the self-avoiding walk,” In: K. Binder (ed.), Monte Carlo and Molecular Dynamics Simulations in Polymer Science, Oxford University Press, New York, p. 47, 1995. [10] J.I. Siepmann and D. Frenkel, “Configurational bias Monte Carlo – a new sampling scheme for flexible chains,” Mol. Phys., 75, 59–70, 1992. [11] D. Frenkel and B. Smit, “Understanding molecular simulations: from algorithms to applications,” 2nd edn., Academic Press, Boston, 2001. [12] R. Dickman, “New simulation method for the equation of state of lattice chains,” J. Chem. Phys., 87, 2246–2248, 1987. [13] M.R. Stukan, V.A. Ivanov, M. M¨uller, W. Paul, and K. Binder, “Finite size effects in pressure measurements for Monte Carlo simulations of lattice polymer systems,” J. Chem. Phys., 117, 9934–9941, 2002. [14] A.D. Mackie, A.Z. Panagiotopoulos, D. Frenkel, and S.K. Kumar, “Constantpressure Monte Carlo simulations for lattice models,” Europhys. Lett., 27, 549–544, 1994. [15] A. Poncela, A.M. Rubio, and J.J. Freire, “Gibbs ensemble simulations of a symmetric mixtures composed of the homopolymers AA and BB and their symmetric diblock copolymer,” J. Chem. Phys., 118, 425–433, 2003. [16] J.S. Shaffer, “Effects of chain topology on polymer dynamics: bulk melts,” J. Chem. Phys., 101, 4205–4213, 1994.

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[17] H.-P. Deutsch and K. Binder, “Interdiffusion and self-diffusion in polymer mixtures: a Monte Carlo study,” J. Chem. Phys., 94, 2294–2304, 1991. [18] W. Paul, K. Binder, D.W. Heermann, and K. Kremer, “Crossover scaling in semidilute polymer solutions: a Monte Carlo test,” J. Phys. II, 1, 37–60, 1991. [19] W. Paul, K. Binder, D.W. Heermann, and K. Kremer, “Dynamics of polymer solutions and melts – reptation predictions and scaling of relaxation times,” J. Chem. Phys., 95, 7726–7740, 1991. [20] M. M¨uller, J.P. Wittmer, and M.E. Cates, “Topological effects in ring polymers: a computer simulation study,” Phys. Rev. E, 53, 5063–5074, 1996. [21] J.U. Sommer and S. Lay, “Topological structure and nonaffine swelling of bimodal polymer networks,” Macromolecules, 25, 9832–9843, 2002. [22] Z. Chen, C. Cohen, and F.A. Escobedo, “Monte Carlo simulation of the effect of entanglements on the swelling and deformation behavior of end-linked polymeric networks,” Macromolecules, 25, 3296–3305, 2002. [23] J. Wittmer, A. Johner, J.F. Joanny, and K. Binder, “Chain desorption from a semidilute polymer brush – a Monte Carlo simulation,” J. Chem. Phys., 101, 4379–4390, 1994. [24] P.Y. Lai and K. Binder, “Structure and dynamics of grafted polymer layers: a Monte Carlo simulation,” J. Chem. Phys., 95, 9288–9299, 1991. [25] J.P. Wittmer, A. Milchev, and M.E. Cates, “Dynamical Monte Carlo study of equilibrium polymers: static properties,” J. Chem. Phys., 109, 834–845, 1998. [26] N.B. Wilding, M. M¨uller, and K. Binder, “Chain length dependence of the polymersolvent critical point parameters,” J. Chem. Phys., 105, 802–809, 1996. [27] J. Baschnagel, “Analysis of the incoherent intermediate scattering function in the framework of the idealized mode-coupling theory – a Monte Carlo study for polymer melts,” Phys. Rev. B, 49, 135–146, 1994. [28] J. Baschnagel and K. Binder, “On the influence of hard walls on the structuralproperties in polymer glass simulation,” Macromolecules, 28, 6808–6818, 1995. [29] M. M¨uller, “Miscibility behavior and single chain properties in polymer blends: a bond fluctuation model study,” Macromolecules Theory Simul., 8, 343–374, 1999 (feature article). [30] E. Reister, M. M¨uller, and K. Binder, “Spinodal decomposition in a binary polymer mixture: dynamic self-consistent field theory and Monte Carlo simulations,” Phys. Rev. E, 64, 041804/1–17, 2001. [31] K. Binder and M. M¨uller, “Monte Carlo simulation of block copolymers,” Curr. Opin. Colloid Interface Sci., 5, 315–323, 2001. [32] M. M¨uller, “Reactions at polymer interfaces: a Monte Carlo simulation,” Macromolecules 30, 6353–6357, 1997. [33] G. Szamel and M. M¨uller, “Thin films of asymmetric triblock copolymers: a Monte Carlo study,” J. Chem. Phys., 118, 905–913, 2003. [34] A. Werner, F. Schmid, M. M¨uller, and K. Binder, “Intrinsic profiles and capillary waves at homopolymer interfaces: a Monte Carlo study,” Phys. Rev. E, 59, 728–738, 1999. [35] M. M¨uller and K. Binder, “Wetting and capillary condensation in symmetric polymer blends: a comparison between Monte Carlo simulations and self-consistent field calculations,” Macromolecules, 31, 8323–8346, 1998. [36] M. M¨uller and K. Binder, “Interface localization–delocalization transition in a symmetric polymer blend: a finite size scaling Monte Carlo study,” Phys. Rev. E, 63, 021602/1–16, 2001.

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[37] M. M¨uller and M. Schick, “Bulk and interfacial thermodynamics of a symmetric, ternary homopolymer–copolymer mixture: a Monte Carlo study,” J. Chem. Phys., 105, 8885–8901, 1996. [38] J. Houdayer and M. M¨uller, “Deviations from the mean field predictions for the phase behavior of random copolymers,” Europhys. Lett., 58, 660–665, 2002. [39] A. Cavallo, M. M¨uller, and K. Binder, “Anomalous scaling of the critical temperature of unmixing with chain length for two-dimensional polymer blends,” Europhys. Lett., 61, 214–220, 2003.

9.6 STOKESIAN DYNAMICS SIMULATIONS FOR PARTICLE LADEN FLOWS Asimina Sierou University of Cambridge, Cambridge, UK

Stokesian Dynamics is a molecular-dynamics-like method for simulating the behavior of many particles suspended in a fluid. The method treats the suspended particles in a discrete sense while the continuum approximation remains valid for the surrounding fluid, i.e., the suspended particles are generally assumed to be significantly larger than the molecules of the solvent. The particles then interact through hydrodynamic forces transmitted via the continuum fluid, and when the particle Reynolds number is small, these forces are determined through the linear Stokes equations (hence the name of the method). In addition, the method can also resolve non-hydrodynamic forces, such as Brownian forces, arising from the fluctuating motion of the fluid, and interparticle or external forces. Stokesian Dynamics can thus be applied to a variety of problems, including sedimentation, diffusion and rheology, and it aims to provide the same level of understanding for multiphase particulate systems as molecular dynamics does for statistical properties of matter.

1.

Equations of Motion

For N rigid particles of radius a suspended in an incompressible Newtonian fluid of viscosity η and density ρ, the motion of the fluid is governed by the Navier–Stokes equations, while the motion of the particles is described by the coupled equation of motion m·

dU = FH + FP + FB, dt

(1)

which simply states that the mass times the acceleration equals the sum of the forces. In this equation, m is a generalized mass/moment-of-inertia matrix, U is the particle translational/rotational velocity vector of dimension 6N , 2607 S. Yip (ed.), Handbook of Materials Modeling, 2607–2617. c 2005 Springer. Printed in the Netherlands.

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and the 6N vector F represents the hydrodynamic (F H ), Brownian (F B ) and arbitrary interparticle or external forces/torques (F P ) acting on the particles. The method then aims to evaluate each force acting on the particles as a known function of the velocities and positions of all other particles, and then integrate Eq. (1) in time to follow the dynamic evolution of the suspension microstructure.

2.

Hydrodynamic Forces

When the particle Reynolds number is small, the hydrodynamic force exerted on the particles should scale linearly with the particles’ velocity (due to the linearity of Stokes equations). For particles in a suspension undergoing a bulk linear shear flow (e.g., simple shear) this hydrodynamic force can be written as (see Ref. [1], and references therein) F H = −R FU · (U − u∞ ) + R F E : E ∞ ,

(2)

where u∞ is the imposed bulk flow evaluated at the particle center, xp . For ˙ · x p and Γ ˙ = E ∞ + Ω∞ is the velocity gradient a simple shear flow u∞ = Γ tensor of the bulk flow (split into a symmetric and antisymmetric part). The resistance tensors R FU and R F E give the hydrodynamic force/torque on the particles due to their motion relative to the fluid and due to an imposed flow, respectively. Note that these resistance tensors can only depend on the positions of the particles (xp ) and not on their velocities, due to the linearity of the problem. Similarly, resistance tensors which relate the particle stresslet S (the symmetric first moment of the force density on a particle) to the velocity and rate of strain can be defined and thus a “grand resistance” matrix constructed

R=

R FU R SU

RF E RS E

(3)

with

F U − u∞ = −R · . S −E ∞

(4)

The inverse of the grand resistance matrix is the grand mobility matrix, M and it gives the particle velocities and rate of strain in terms of the total forces/torques and stresslets. Note that in this notation, RSE is a fourth-order tensor, while RFU a second-order one (see also Refs. [2, 3] for a more detailed introduction on the resistance and mobility formulations). The calculation of the resistance tensors is trivial for the case of a single particle (where, for example, RFU = 6πηa I for a particle of radius a,and the hydrodynamic drag simply corresponds to the Stokes drag), and exact analytical expressions are

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known for the case of two particles [3]. Stokesian Dynamics extends these results and makes an accurate approximation for the resistance and mobility tensors for the case of an arbitrary number of particles N . This procedure is briefly outlined below. Initially the force density on the surface of each particle is expanded in a series of moments around the center of the particle. (The zeroth moment is simply the net force acting on a particle, while the first moment is split into an antisymmetric torque and a symmetric stresslet.) Then the i-component of the velocity at any position x in the fluid can be simply expressed as u i (x) − u ∞ i (x) =

G i j (x, x np ) F jn ,

(5)

n

where F jn is the j -component of the hydrodynamic force/torque/stresslet (or even higher moment) on each particle n, and G i j corresponds to an appropriate solution function for the Stokes equations, which again can only depend on the configuration of the particles (see Refs. [1, 4, 5] for details on the functional form of G i j and issues of convergence, periodicity and infinite systems). To determine the motion of a particle immersed in a flow field given by Eq. (5) the method makes use of the Fax´en formulae for spheres [6] which relate the force on a particle n to the velocity of the particle Uin and the fluid velocity at the particle center u i (xpn )

Fin

=

−6πηa(Uin

−

n u∞ i (x p ))

a2 + 6πηα 1 + ∇ 2 u i (x np ) 6

(6)

(similar expressions can be written for the torque and stresslet as a function of the angular velocities and rate of strain). Through the combination of Eqs. (5) and (6), the method eliminates the fluid velocity u i (x np ) and a mobility matrix can thus be constructed relating the velocity of each particle Uin to the force moments on all particles. This approach, however, becomes computationally prohibitive when a pair of particles is close to contact. The hydrodynamic forces then diverge as −1 and ln −1 , where is the surface-to-surface distance of two particles (non-dimensionalized by the particle radius) and a large number of moments is required in order to capture the surface force density correctly. To circumvent this problem, higher order moments in the expansion are neglected and their effect is instead approximated by an added lubricationtype force between particles. Since these lubrication forces are dominated by interactions between contact points rather than boundary surfaces, they can be approximated as two-body interactions and thus simply added to the far-field solution in a pairwise-additive manner. The method then can be used to simulate a realistic number of particles by decomposing the hydrodynamic interactions into a far-field and a near-field part, thus maintaining both a relatively small number of parameters (moments) and high accuracy for dense systems.

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The algorithm, however, remains computationally expensive, as it requires O(N 2 ) operations for the construction of the mobility matrix (Eq. (5) applied for all N particles) and additionally an O(N 3 ) inversion of the resulting matrix (since for many problems the calculation of the resistance matrix is also required). Faster versions of Stokesian dynamics have thus been developed [7] which utilize fast Fourier transform techniques in combination with iterative inversions to reduce the cost of the hydrodynamic calculations to a O(N ln N ) scaling.

3.

Brownian Forces

When the suspended particles are sufficiently small the fluctuating forces they receive from the fluid influence their dynamics and give rise to the familiar phenomenon of Brownian motion. Stokesian Dynamics successfully accounts for these Brownian forces, in combination with an accurate manybody description of the hydrodynamic forces, and is thus an extension to “Brownian Dynamics” [8], a method commonly used to examine the properties of colloidal systems, where the many-particle hydrodynamics are ignored. The Brownian force F B arising from the thermal fluctuations of the fluid is a Gaussian stochastic variable defined by

F B = 0 and

F B (0)F B (t) = 2kT R FU δ(t),

(7)

where the angle brackets denote an ensemble average, k is Boltzmann’s constant, T is the absolute temperature and δ(t) the delta function. The correlation of the Brownian forces at times 0 and t is a direct consequence of the fluctuation–dissipation theorem for the N -particle system (see also Ref. [9] for an introduction to colloidal dynamics). Once the hydrodynamic resistance matrix is known, it is straightforward (although often computationally expensive) to calculate the “random” component of the displacement. Equation (1) can be integrated in time (assuming that the configuration of the particles does not change significantly during the time scale of the Brownian motion, i.e., during the time required for a particle’s momentum to relax after a Brownian pulse) leading to [8] X B = kT ∇ · R−1 FU t + X(t)

(8)

with X = 0,

X(t)X(t) = 2kT R−1 FU t.

(9)

Here X B is the change in the position of the particle associated with the Brownian motion and X(t) is a Gaussian random displacement with zero mean and covariance given by the inverse of the resistance tensor. (Note that

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due to the spatial variation of the resistance matrix and the simple forward time-stepping scheme, the random walk has a mean drift with a mean velocity kT ∇ · R−1 FU in addition to any systematic velocity.) The motion and rheological properties arising from the Brownian forces can then be combined with the hydrodynamic contributions described in Section 2, to give an accurate description of hydrodynamically interacting colloidal suspensions. As was the case with the hydrodynamic force, new versions of Stokesian Dynamics [10] have utilized iterative techniques and Chebyshev polynomial approximations (for calculating the square root of matrices) to reduce the computational cost to O(N 1.25 ln N ) and thus make possible the simulation of larger systems.

4.

Random Hard Sphere Dispersions

Stokesian Dynamics has been used successfully to calculate the transport properties of a random dispersion of hard spheres (see Refs. [7, 11, 12] for a full review). Figure 1 shows the high-frequency dynamic viscosity η∞ 8 7 6 5 4 3

2

N ⫽ 125 N ⫽ 343 N ⫽ 512 N ⫽ 1000 N ⫽ 2000 Ladd (1990) Vander Werff et al. (1989) Shikata & Pearson (1994)

10 η∞' /η

9 8 7 6 5 4 3

2

1

9 8

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

φ Figure 1. High-frequency dynamic viscosity vs. volume fraction. The experiments are for sterically stabilized silica particles which behave to a very good approximation as hard spheres.

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(non-dimensionalized by the solvent viscosity η) for hard spheres determined by Stokesian Dynamics simulation and compared with experiment. The highfrequency dynamic viscosity is the dissipative part of the stress measured in a very small amplitude and high frequency oscillatory shear flow (where the distribution of the solid particles is unaffected by shear flow and it still corresponds to the equilibrium hard-sphere structure). In simulations, it is completely determined by the hydrodynamic interactions and is easily extracted from the particle contribution to the bulk stress, P

P = n SH = −n R SU · (U − u∞ ) − R S E : E ∞

(10)

where n is the number density of particles, the non-zero particle velocities U are defined by Eqs. (1) and (2), and the resulting stress is ensemble-averaged over a number of equilibrium configurations. (New implementations of the method, which do not calculate the resistance tensors explicitly, obtain the far-field contribution to the hydrodynamic stress through an iterative inversion of an equivalent mobility matrix.) The results demonstrate that the hydrodynamic interactions are calculated accurately by Stokesian Dynamics for all volume fractions, capturing the singular nature of the viscosity as random close packing is approached.

5.

Rheology of Non-Colloidal Particles

The rheological properties of suspensions can be calculated with a dynamic simulation of a sheared system. In the absence of Brownian motion it has been demonstrated [13, 14] that the presence of a repulsive interparticle force is always necessary to prevent the system from forming infinite clusters. A pair-wise repulsive force of the form F Pαβ = F0

τ e−τ e(αβ) 1 − e−τ

(11)

is commonly used [1]. In this equation, F Pαβ is the force exerted on sphere α by sphere β, F0 represents its magnitude, τ −1 its range, is the dimensionless spacing between particle surfaces, and e(αβ) the unit vector connecting the centers of the two spheres. Since now there are two driving forces affecting the motion of the particles, a non-dimensional parameter γ = 6πηa 2 γ˙ /F0 is defined, describing the ratio of the shear rate to the magnitude of the ˙ is the magnitude of the velocity gradient interparticle forces (where γ˙ = | | tensor). The particle contribution to the stress is now given by [1]

P = −n R SU · (U − u∞ ) + n R S E : E ∞ − n x p F P ,

(12)

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3

2

10 ηr

9 8 7 6

Stokesian Dynamics N ⫽ 512, ψ⫽ 1000, ∗ ⫽ 1000 Experiments Zarraga et al. (2000) Pätzold (1980) Gadala-Maria (1979) Rutgers (1962)

5 4 3

2

1 0.0

0.1

0.2

0.3

0.4

0.5

φ Figure 2. fraction.

Steady-shear viscosity for non-colloidal particles as a function of the volume

where n is the particle number density, and the ensemble average denotes a number of steady-state configurations, or equivalently a time average over a large strain. The suspension relative viscosity is presented in Fig. 2 for a given magnitude of the interparticle force and compared with a number of experimental results. The steady-shear viscosity is generally larger than the highfrequency dynamic viscosity, since under steady-shear particle clusters form, which increase the total stress. The agreement between simulation and experiment is very good for low to moderate volume fractions, while for large volume fractions sizeable discrepancies exist, not only between the experimental and simulation results, but also between different sets of experiments. This is attributed to the fact that the exact form and magnitude of the interparticle force (e.g., surface roughness, particle material) can have a profound effect on the rheology, especially for large volume fractions, where the total stress is dominated by the near-particle interactions [15].

6.

Rheology of Colloidal Particles

Stokesian dynamics has also been used extensively to study the behavior of colloidal systems, where in addition to the hydrodynamic forces, Brownian forces are also important. For such systems the action of an external driving

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force such as shear disturbs their equilibrium microstructure and strongly affects the resulting behavior. Figure 3 shows the steady-shear viscosity ηr as a function of the P´eclet number for a volume fraction of φ = 0.45. The P´eclet number measures the relative importance of shear to thermal forces and is given by Pe = 6πηa 3 γ˙ /(kT ). The stress for such systems is still given by Eq. (10), where now the particle velocities include both a purely hydrodynamic and a Brownian contribution. It is thus instructive to decompose the total stress into two similar contributions: a hydrodynamic one, given by Eq. (10) with the particle velocities coming from only the deterministic displacements, and a Brownian one, given by [16]:

SB = −kT ∇ · R SU · R−1 FU

.

(13)

The overall viscosity appears to have a non-monotonic dependence on the shear rate, decreasing as a function of the shear rate (shear-thinning behavior) for small to moderate P´eclet numbers, but increasing as a function of the shear rate at high P´eclet numbers (shear-thickening behavior). The same transitional behavior has also been seen in experimental studies; as a measure of the quantitative predictive ability of Stokesian Dynamics, Fig. 4 shows the simulated steady-shear viscosity for a number of volume fractions compared with 16 14 12 10 ηr

8 6 φ ⫽ 0.45, N ⫽ 27 ηT

4

ηH

2

ηB

0 10⫺3

10⫺2

10⫺1

100

101

102

103

104

Pe

Figure 3. The steady-shear viscosity and its different contributions determined by Stokesian Dynamics for a system of N = 27 and φ = 0.45.

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35 Stokesian Dynamics φ ⫽0.316

30

van der Werff & de K ruif (1989) φ ⫽0.316

φ ⫽0.419

φ ⫽0.419

φ ⫽0.470 φ ⫽0.490

φ ⫽0.488

φ ⫽0.470

25

D' Haene et al. (1993) φ ⫽0.276 φ⫽0.389 φ ⫽0.460 φ ⫽0.484

20 ηr 15

10

5

0

10⫺3

10⫺2

10⫺1

100

101

102

103

104

Pe

Figure 4. The steady-shear viscosity (normalized by the solvent viscosity) as a function of the Peclet number Pe for different volume fractions.

experiments. Despite the different types and sizes of particles the agreement between the two sets of experiments and the simulation results is satisfactory. This complex behavior can be further clarified by analyzing the two contributions to the stress separately (see also Ref. [17]). The Brownian contribution to the stress is a result of the flow-induced deformation of the equilibrium structure (particles are trying to diffuse against the flow towards their unstressed/equilibrium configuration and the resulting stress is proportional to this deformation). The Brownian viscosity (stress normalized by ηγ˙ ) then scales simply as the deformation over the P´eclet number. As the P´eclet number increases, however, it has been suggested that the particle motion can no longer keep up with the flow and the deformation of the microstructure saturates, leading to a Brownian viscosity which is a decreasing function of the P´eclet number (as shown in Fig. 3). The hydrodynamic stress, on the other hand, remains constant for small values of the P´eclet number, since, due to the Brownian forces, the particles are still well-dispersed; its value is thus very close to the high-frequency dynamic viscosity. As the shear-rate is increased further, however, particle clusters form (since the shear-flow pushes particles together and the Brownian forces are no longer strong enough to disperse them) and strong lubrication stresses are manifested, leading to an increase in the total particle stress.

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Discussion

The Stokesian Dynamics method provides a rigorous procedure for the simulation of hydrodynamically interacting particles and suspensions. The results discussed above demonstrated the ability of the method to make accurate predictions for a number of rheological properties, and shed light on the physical mechanisms dictating the particles’ motion. However, they only represent a small fraction of the problems to which the method has been applied, or could be applied in the future. The combination of simulation with experiment and theory has allowed for a very detailed picture to emerge, describing the structure and rheology of colloidal and non-colloidal systems. The system microstructure, expressed through the pair-distribution function, has been studied extensively [1, 15, 17] and the observed anisotropy in the microstructure has been correlated with the suspension non-Newtonian behavior (i.e., normal stress differences). The method has also been expanded to the study of bounded systems and pressure-driven flows and has been instrumental in developing macroscopic models for such systems (see Ref. [18]). The diffusive motion of the particles, and in particular the effect of shear on diffusion, has also been the subject of recent studies; such studies have demonstrated the ability of Stokesian Dynamics (and simulation in general) to investigate effects which are often hard to measure experimentally.

8.

Outlook

Stokesian Dynamics, and faster O(N ln N ) implementations inspired by it, have been used extensively over the last decade and have increased our understanding and ability to predict suspension behavior. Although the method has so far mainly been used for monodisperse, hard-sphere systems, it is relatively straightforward to extend it to deformable drops, to non-spherical particles, or polydisperse systems. The method has thus opened up opportunities for the study of a much larger class of new problems and flows, which hopefully will provide further insight into the physics of complex fluids.

References [1] J.F. Brady and G. Bossis, “Stokesian dynamics,” Annu. Rev. Fluid Mech., 20, 111–157, 1988. [2] J. Happel and H. Brenner, Low Reynolds Number Hydrodynamics, Prentice Hall, New York, 1965. [3] S. Kim and S.J. Karilla, Microhydrodynamics: Principles and Selected Applications, Butterworths, London, 1991.

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[4] L.J. Durlofsky, J.F Brady, and G. Bossis, “Dynamic simulations of hydrodynamically interacting particles,” Fluid Mech., 180, 21–49, 1987. [5] J.F. Brady, R.J. Phillips, J.C. Lester, and G. Bossis, dynamic simulation of hydrodynamically interacting suspensions,” J. Fluid Mech., 195, 257–280, 1988. [6] G.K. Batchelor and J.T. Green, “The hydrodynamic interaction of two small freely moving spheres in a linear flow field,” J. Fluid Mech., 56, 375–400, 1972. [7] A. Sierou and J.F. Brady, “Accelerated Stokesian dynamics simulations,” J. Fluid Mech., 448, 115–146, 2001. [8] D.L. Ermak and J.A. McCammon, “Brownian dynamics with hydrodynamic interactions,” J. Chem. Phys., 69, 1352–1360, 1978. [9] W.B. Russel, D.A. Saville, and W.R. Schowalter, Colloidal Dispersions, Cambridge University Press, Cambridge, 1989. [10] A.J. Banchio and J.F. Brady, “Accelerated Stokesian dynamics: Brownian motion,” J. Chem. Phys., 118, 10323–10332, 2003. [11] R.J. Phillips, J.F. Brady, and G. Bossis, “Hydrodynamic transport properties of hard-sphere dispersions. I. Suspensions of freely mobile particles,” Phys. Fluids, 31, 3462–3472, 1988. [12] A.J.C. Ladd, “Hydrodynamic transport coefficients of random dispersions of hard spheres,” J. Chem. Phys., 93, 3484–3494, 1990. [13] J.R. Melrose and R.C. Ball, “The pathological behavior of sheared hard-spheres with hydrodynamic interactions,” Europhys. Lett., 32, 535–546, 1995. [14] D.I. Dratler and W.R. Schowalter, “Dynamic simulation of suspensions of non-Brownian hard spheres,” J. Fluid Mech., 325, 53–77, 1996. [15] J.F. Brady and J.F. Morris, “Microstructure of strongly sheared suspensions and its impact on rheology and diffusion,” J. Fluid Mech., 348, 103–139, 1997. [16] J.F. Brady, “The rheological behavior of concentrated colloidal dispersions,” J. Chem. Phys., 99, 567–581, 1993. [17] D.R. Foss and J.R. Brady, “Structure, diffusion and rheology of Brownian suspensions by Stokesian dynamics simulations,” J. Fluid Mech., 407, 167–200, 2000. [18] P.R. Nott and J.F. Brady, “Pressure driven flow of suspensions: simulation and theory,” J. Fluid Mech., 275, 157–199, 1994.

9.7 BROWNIAN DYNAMICS SIMULATIONS OF POLYMERS AND SOFT MATTER Patrick S. Doyle and Patrick T. Underhill Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, MA, USA

The Brownian dynamics (BD) simulation technique is a mesoscopic method in which explicit solvent molecules are replaced instead by a stochastic force. The technique takes advantage of the fact that there is a large separation in time scales between the rapid motion of solvent molecules and the more sluggish motion of polymers or colloids. The ability to coarse-grain out these fast modes of the solvent allows one to simulate much larger time scales than in a molecular dynamics simulation. At the core of a Brownian dynamics simulation is a stochastic differential equation which is integrated forward in time to create trajectories of molecules. Time enters naturally into the scheme allowing for the study of the temporal evolution and dynamics of complex fluids (e.g., polymers, large proteins, DNA molecules and colloidal solutions). Hydrodynamic and body forces, such as magnetic or electric fields, can be added in a straightforward way. Brownian dynamics simulations are particularly well suited to studying the structure and rheology of complex fluids in hydrodynamic flows and other nonequilibrium situations.

1.

Basic Brownian Dynamics

The technique of Brownian dynamics is used to simulate the dynamics of particles that undergo Brownian motion. Because of the small mass of these particles, it is common to neglect inertia. Using Newton’s Second Law for particle i, Ftot i = m i ai , the neglect of inertia means that the total force is always approximately zero. The total force on a particle is composed of a drag force Fdi from the particle moving through the viscous solvent, a Brownian force FBi

2619 S. Yip (ed.), Handbook of Materials Modeling, 2619–2630. c 2005 Springer. Printed in the Netherlands.

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due to random collisions of the solvent with the particle, and all nonhydrodynamic forces Fnh i d B nh Ftot i = Fi + Fi + Fi 0 .

(1)

This total nonhydrodynamic force includes any external body forces, any spring forces, and any excluded volume interactions. In creeping flow and neglecting hydrodynamic interactions (free-draining), the drag force is taken as Stokes drag on a sphere dri − u∞ (ri ) (2) Fdi = −ζ dt where ζ is the drag coefficient and u∞ (ri ) is the unperturbed velocity of the solvent evaluated at the position of the particle. The differential equation governing the motion of the particle then becomes dri 1 nh = u∞ (ri ) + Fi ({r j }) + FBi (t) (3) dt ζ and is commonly called a Langevin equation. Note that the nonhydrodynamic force depends on the set of all particle positions {r j }. This is a stochastic differential equation because the Brownian force is taken from a random distribution. In order for the dynamics to satisfy the fluctuation–dissipation theorem, the expectation values of the Brownian force are FBi (t) = 0

(4)

FBi (t)FBj (t ) = 2kB T ζ δi j δ(t − t )δ

(5)

where kB is Boltzmann’s constant, T is the absolute temperature, δi j is the Kronecker delta, δ(t − t ) is the Dirac delta function, and δ is the unit secondorder tensor. Equations (3)–(5) are equivalent to the Fokker–Plank equation description, which is a diffusion equation for the phase space probability density [1]. Having developed the governing stochastic differential equation, one performs a BD simulation by integrating this equation forward in time. The stochastic nature means that one must produce many independent trajectories that are averaged together, producing the time-evolution of an ensembleaveraged property. The repetition of many independent trajectories is a time-consuming but necessary part to follow the time-evolution of a property. However, to calculate a steady-state property, one uses the ergodic hypothesis to time-average a single trajectory.

2.

Hydrodynamic Interactions

As a particle moves along its trajectory, it exerts a force on the solvent which changes the velocity field from its undisturbed value. The disturbance

Brownian dynamics simulations of polymers and soft matter

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velocity changes the viscous drag force exerted on the other particles. This interaction between particles mediated by the solvent is called hydrodynamic interaction (HI). The hydrodynamic interactions are included in Brownian dynamics through the use of an interaction tensor Ωi j included as part of the diffusion tensor Di j [1]. The two tensors are related by Di j (ri , r j ) =

kB T δi j δ + ζ Ωi j (ri , r j ) ζ

(6)

where Ωii = 0. The stochastic differential equation including HI then becomes N N √ dri 1 Di j (ri , r j ) · Fnh ({r }) + 2 Bi j ({rk }) · n j (t) = u∞ (ri ) + k j dt kB T j =1 j =1

(7) where n j are random vectors with expectation values n j (t) = 0

(8)

ni (t)n j (t ) = δi j δ(t − t )δ

(9)

and the weighting factors Bi j must be calculated from the diffusion tensor in order to satisfy the fluctuation–dissipation theorem Di j (ri , r j ) =

N

Bip ({rk }) · BTjp ({rk }).

(10)

p=1

This can be inverted to calculate Bi j by Cholosky decomposition. However, a more efficient method has been developed by Fixman [2] and implemented in BD simulations. The interaction tensor in an unbounded solvent is taken as the Rotne– Prager–Yamakawa tensor, which is a regularized version of the Oseen–Burgers tensor. The Oseen–Burgers tensor represents the disturbance due to a pointforce in creeping flow. However, it results in a nonpositive-definite diffusion tensor if particle separations are comparable to the particle radius. The Rotne– Prager–Yamakawa tensor modifies the small separation disturbance such that the diffusion tensor is always positive-definite.

3.

Polymer Models used in Brownian Dynamics

The choice of polymer model is intrinsically a modeling decision which depends upon the real polymer one wants to model and the level of fine-scale molecular detail one needs to retain or can computationally afford to simulate. Polymers can be broadly separated into flexible and semiflexible chains. The

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flexibility of a chain is determined by the ratio L/lp , where lp is the persistence length and L the contour length of the chain. Flexible chains have L/lp 1 and semiflexible chains have L/lp ∼ 1. The most common coarse-grained models for flexible polymers are the freely jointed bead-rod and bead-spring chains. The polymer is modeled as a series of beads connected by either rods or springs, as shown in Fig. 1. The frictional drag on the chain is distributed at bead centers. The term “freely jointed” implies that there is no energetic penalty to rotating a spring or rod about a bead center. The spirit of these mesoscopic models is to coarse-grain out molecular details smaller than the finest length scale in the given model (rod or spring). We consider first the flexible bead-rod chain. Physically, the rod in a beadrod chain corresponds to a Kuhn length lk (twice the value of the persistence length). Mathematically, the rods act as a constraint on the system which ensures that adjacent beads in the chain are maintained at a constant separation at all times. How one achieves this constraint is important as there is a subtle difference between a completely rigid constraint and the approximation of that constraint using a very (infinitely) stiff potential [3]. For example, the equilibrium distribution of a bead-rod chain using stiff constraints yields a random walk configuration while imposing rigid constraints gives rise to correlations in the rod directions, even in the absence of bending forces. Physically, one would argue that the random walk configuration is more realistic. However, rigid constraints are attractive from a computational standpoint as they freeze out the usually uninteresting rapid bond fluctuations. In practice, one chooses to simulate the stiff system, but does so by imposing rigid constraints (which introduces a new tension force Ftens ) and adding a corrective pseudopotential metric force Fmet which makes the system equivalent to imposing an infinitely stiff

Figure 1. Schematic of the canonical polymer model used in Brownian dynamics simulations: a bead-spring chain. The continuous curve is the “real” polymer that the bead-spring chain is meant to represent.

Brownian dynamics simulations of polymers and soft matter

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potential. A recent detailed review of constrained Brownian motion and the implementation of bead-rod BD algorithms can be found in Morse [3]. With current computers typically one can only simulate chains with up to ∼100−200 Kuhn steps which corresponds to a low molecular weight polystyrene polymer ( ∼ 75 000−150 000 Da). Large flexible polymers are more commonly modeled as a series of Ns springs connected by beads. Each spring models a portion of the full chain and has a contour length L s = L/Ns . The spring represents the entropic restoring force associated with stretching a subsection of the chain. The entropic force can be calculated starting from a fine-scale micromechanical model (e.g., freely jointed bead-rod chain) using equilibrium statistical mechanics and calculating the extension of a chain when subject to a constant force. The force-extension response of a freely joined chain is exactly described by the inverse Langevin function which is closely approximated by the more convenient FENE force law 3kB T r/L s (11) F FENE (r) = lk [1 − (r/L s )2 ] where r is the spring extension and L s is the spring length when fully stretched [4]. A slightly more accurate approximation to the inverse Langevin function can be obtained using a Pad´e approximation (see for example Ref. [5]). Most flexible synthetic polymers (polystyrene, poly(ethylene oxide), etc.) and single-stranded DNA have significant bond rotations and should be modeled using the FENE force law. However, many biopolymers resist local bond torsion (e.g., duplex DNA) and are more accurately described by a wormlike chain model which for L s /lp 1 is well approximated by the Marko–Siggia spring law [6] F

wlc

kB T (r) = lp

r 1 1− 4 Ls

−2

r 1 − + . 4 Ls

(12)

It is important to note some limitations and assumptions when using common spring force laws such as Eqs. (11) and (12). These relations are derived assuming a large number of persistence or Kuhn lengths are contained in the spring. More refined calculations show, however, that corrections to the spring laws must be made when the spring has a contour length less than ∼20lk . Furthermore, if the underlying micromechanical model is freely jointed at each node connecting the springs (e.g., freely jointed chains), then spring force laws can be derived for springs containing an arbitrarily small number of Kuhn lengths which will exactly reproduce the force extension response of the micromechanical model. The use of equilibrium statistical mechanics to derive spring forces also raises a subtle point when using a bead-spring model in flow or other nonequilibrium situations. In such cases, it is implicitly assumed that the springs are deformed slowly enough such that the micromechanical model

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P.S. Doyle and P.T. Underhill

describing the spring would be able to fully sample its configuration space; in a sense achieving a local equilibration in phase space. Lastly, it is important to note that parameters in the spring force are directly related to the physical polymer system being modeled (e.g., a certain subsection of a polymer chain of length L s with a given persistence length lp ). An extensive discussion of these and other issues on the coarse-graining polymers into bead-spring chains is given in Underhill and Doyle [7]. The model for a semiflexible polymer is based on the premise that the polymer can be described as a homogeneous, isotropic elastic filament. The energy associated with

bending a continuous filament described by a curve r(s) is U bend = κ/2 ds |∂2 r/∂s 2 |2 , where s is a contour distance along the chain. The bending rigidity κ is related to the persistence length by lp =κ/kB T . The chain is typically coarse-grained to a series of beads connected by rods which have a bending energy given by

U

bend

κ N−1 =− u j · u j −1 a j =2

(13)

where a is the length of the connecting rod and u j is the unit vector directed = − ∂U bend/∂ri . This descripfrom bead j to j + 1. The bending force is Fbend i tion of a semiflexible polymer is commonly referred to as the wormlike chain or Kratky–Porod model [8]. Systems with constraints (rods between beads) which ensure local inextensibility of a chain are computationally expensive and are sometimes replaced by stiff Fraenkel springs of the form Fsi = H (|ri+1 − ri | − a)ui , where H is the spring constant and a is the equilibrium length of the spring. Pasquali and coworkers have shown that rigid constraints and stiff Fraenkel springs give similar results for the collapse of DNA in poor solvent [9]. They found typical time savings in using stiff springs over rigid constraints are on the order of 10–50-fold. Most researchers use stiff springs only when simulating a polymer which is at or near equilibrium. Detailed comparisons between Fraenkel springs and constraints have not yet been performed for polymers in flow or under large tensions. In general, it is recommended to use rigid rods (with the corrective metric force discussed previously) unless one is certain stiff springs introduce no computational artifacts.

4.

Numerical Algorithms

We have developed the stochastic differential equation and discussed the types of models and corresponding forces involved. To solve for the trajectory,

Brownian dynamics simulations of polymers and soft matter

2625

one must integrate forward in time. A simple explicit Euler time integration scheme is widely used

ri (t + δt) ri (t) +

dri (t) t. dt

(14)

In this discrete version, the random vectors have an expectation value ni (t)n j (t ) = δi j δt t δ/t.

(15)

If the forces are steep, this would require very small time steps. This is particularly important for bead-spring chains with finitely extensible springs such as the FENE or Marko–Siggia force law. The above algorithm allows for the possibility of a spring being stretched beyond its fully extended length. A semi-implicit predictor corrector method including HI has been developed that prevents this overstretching of springs [5]. The realization of a trajectory requires the sampling of random vectors with the appropriate expectation values. The calculation of these “random” numbers is a significant computational cost of BD simulations. Because of this cost, it is better to use uniform random numbers than Gaussian distributed. ¨ These random number generators should be used with care. Ottinger [1] reviews important aspects of random number generators. The use of random variables in the simulation and a finite number of trajectories in the ensemble means that there is intrinsic statistical noise to −1/2 the method. The size of this error is proportional to NT , where NT is the number of independent trajectories. An important technique for the reduction of this error by reducing the proportionality factor (not by increasing NT ) is called variance reduction. The exact way of performing variance reduction depends on the system of study. Two important types of variance reduction are importance sampling and subtracting off a control variable [1]. Another issue to be noted for the implementation of BD algorithms concerns the neglect of mass, which is a singular limit. This limit is discussed in detail by Grassia et al. [10]. We note here that one consequence of this singular limit is a drift that results if the diffusivity depends on position. To correct for this drift, a term with the gradient of the diffusivity must be added to the algorithm. However, if a higher order scheme is used such as a midpoint method or predictor corrector method, the extra term should not be added.

5.

Computing Stress

The rheology of most complex fluids is not described well by simple constitutive relations. Central to the study of rheology then is the calculation of the bulk stress tensor σ in a simulation. The bulk stress of the mixture is a linear combination of the solvent contribution σs (typically a simple

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P.S. Doyle and P.T. Underhill

Newtonian fluid) and the polymer/colloid contribution σp . The exact form of the stress tensor depends upon the details of the microstructural model. However, the most common model for a polymer or colloid is a collection of discrete beads (with positions ri ) for which the particle contribution to the bulk stress is given by the Kramers–Kirkwood expression 1 σ =− V p

N

ri Fhi

(16)

i=1

where Fhi is the total hydrodynamic force exerted on bead i, V is the volume of the simulation box and · · · denotes an ensemble average. Equation (16) is a quite general result and is still applicable for systems with constraints (e.g., bead-rod chains) and when hydrodynamic interactions are taken into account [3, 4]. While it is important to establish Eq. (16) as a rather general form for the stress tensor, it useful to consider the specific form of the tensor for restricted classes of models and other commonly encountered cases. Consider first the canonical model employed in Brownian dynamics for a polymer: a series of N beads connected by springs. The polymer contribution to the stress is then p

σ = np

N

Fsi

+

Fother i

Ri

+ n p N kB T δ

(17)

i=1

where Ri is the position of a bead relative to the center of mass of the chain, n p is the number density of polymer molecules, Fsi the net force by springs is the sum of all the nonhydrodynamic and nonspring on bead i and Fother i forces exerted on bead i, such as bead–bead excluded volume interactions or bending forces. Equation (17) is valid both when hydrodynamic interactions are accounted for or neglected. The last term in Eq. (17) is the familiar ideal gas-like internal energy contribution from each bead which is a direct result of applying the usual “equilibration in momentum space” and assuming an isotropic friction tensor for a bead [4]. There can be computational advantages to using certain formulas over others to calculate the stress tensor in particular simulations. To gain some appreciation of this consider the steady-state rheology of a bead-rod (or beadspring) chain in a simple shear flow u ∞ x (r) = γ˙ r y and neglect hydrodynamic interactions between beads. Using the Giesekus stress formula the polymer contribution to the shear viscosity (ηp = −σxpy /γ˙ ) can be written npζ d η =− 2γ˙ dt p

N i=1

Rxi R iy

n pζ + 2

N

R iy R iy

.

(18)

i=1

However, because in this example we are only interested in the stress of the system at steady-state we can immediately set the first term on the right side

Brownian dynamics simulations of polymers and soft matter

2627

of Eq. (18) to zero and eliminate it from our simulation code. Not only is this a great simplification, but now one has a simple expression which gives physical insight into the connection between polymer viscosity and molecular configurations. For this example, the effective width of the chain in the shear-gradient direction is directly related to the contribution to the viscosity. Furthermore, simplifying the formula for the stress tensor can aid in the development of scaling relations which generalize the trends [11]. An extensive discussion of the various forms of the stress tensor can be found in Bird et al. [4]. We note that extra care must be taken when handling polymers containing constraint forces. In this case, a brute force calculation of the stress using Eq. (16) will contain some contributions which fluctuate about zero but are quite large (order t −1/2 ). Various efficient algorithms have been developed specifically for this case which essentially filter out these uninteresting “noise” terms [3, 11].

6.

Polymer Solutions in Hydrodynamic Flows

The ability to predict the dynamics and rheology of dilute polymer solutions in simple linear flows is an important benchmark by which researchers evaluate BD simulations and mesoscopic polymer models. In recent years, it has become possible to directly observe single polymer molecule dynamics in flow using double stranded DNA [12]. DNA is a unique polymer in that its force-extension relation (entropic spring-like force) has been well characterized and for large DNA is accurately described by the Marko–Siggia force law. To date most experiments have been performed with a particular commercially available DNA molecule, λ-DNA, which when stained with typical fluorescent dyes contains approximately 400 persistence lengths. Figure 2 shows a comparison between BD simulations using a free-draining (no HI) bead-spring chain (spring law given by Eq. (12)) for free and tethered λ-DNA in simple linear flows. Excellent quantitative agreement between free-draining BD simulations and experiments of λ-DNA in shear, elongational and mixed flows has been attained (see for example Refs. [13, 14]). The ability to neglect HI in the BD simulations and obtain quantitative agreement with experiments is because λ-DNA contains a modest number of persistence lengths and so even when HI is included there is little difference in the drag on a coiled and fully extended chain. RecentexperimentsusingextremelylongDNA(genomiclength, ∼20 000 persistence lengths) have demonstrated conformational hysteresis in extensional flows [15]. The authors show the qualitative nature of this behavior is captured only if HI is included in BD simulations. As it stands now, quantitative description of most trends in the stretching of DNA with lengths up to ∼1000 persistence lengths can be captured without including HI in a simulation. Further simulations need to be done to

2628

P.S. Doyle and P.T. Underhill 1.0 Mean fractional extension

Free extensional flow 0.8 Tethered shear flow

0.6 0.4

Free shear flow 0.2 0.0 0

20

40

60

80

100

Wi Figure 2. Comparison of Brownian dynamics simulations (lines) to single molecule DNA experiments (symbols). Mean fractional extension (stretch of the DNA scaled by the contour length of the molecule) vs. Weissenberg number (W i) is shown for tethered and free DNA in linear hydrodynamic flows. W i is the product of the shear rate (or elongation rate) and the longest relaxation time of the DNA. Figure adapted from Doyle et al. [13].

quantitatively test HI when comparing to larger genomic length DNA. Furthermore, when polymers are placed near hard surfaces the hydrodynamic interaction between segments will change due to satisfying the no-slip boundary condition at the surface. These wall effects can be taken into account by numerically solving for the Green’s function for creeping flow in an arbitrary geometry [16]. These modified hydrodynamic interactions are needed in a BD simulation to predict the correct trend of shear-induced migration in channels. However, it has been shown that quantitative agreement between free-draining BD simulations and experiments for the stretch and fluctuations of tethered λ-DNA is obtained if the drag coefficient is merely adjusted [13]. While the ability to observe single polymer molecules is very powerful, a measure of the polymer contribution to the stress is not attained in such experiments. Recently, it has become possible to accurately measure transient extensional stresses of high molecular weight polymer solutions using filament stretching rheometers [17]. The stresses developed in these flows are challenging to model because the elongational nature of the flow leads to large deformation of the initially coiled polymer and brings with this a large contribution by the polymer to solution viscosity. Hsieh and Larson have performed the most detailed study of the role in HI in BD simulations of polystyrene and DNA in extensional flows [5]. A method was developed to determine the HI

Brownian dynamics simulations of polymers and soft matter

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parameters in the model which on one hand will keep the number of springs to a modest number while on the other hand will match the relaxation time or diffusivity of the experimental polymer system and the drag on the fully stretch polymer (estimated from Batchelor’s formula). They confirmed that including HI has little effect for λ-DNA. Hsieh and Larson find that it is necessary to include HI in order to quantitatively match stress–strain behavior up to strains of ∼6 in filament stretching experiments of polystyrene with a molecular weight of 2 million ( ∼5400 persistence lengths). However, the simulations do not properly predict the experimental values for the high strain plateau in the stress.

7.

Outlook

Brownian dynamics is a powerful technique to simulate nonequilibrium dynamics of polymers and other soft matter. Efficient and stable algorithms have been developed which allow for the simulation of a wide class of polymer models ranging from flexible bead-spring chains to semiflexible bead-rod filaments. Furthermore, the strengths and deficiencies of springs representing a small number of persistence lengths is now well understood, at least for unconfined polymers. Quantitative comparisons of BD simulations to the rheology and dynamics of dilute polymer solutions show that our understanding of the importance and correct implementation of hydrodynamic interactions into a simulation is continuing to evolve. With the growing importance of processing biological and other complex fluids in micro and even nanochannels, the BD simulation technique will continue to advance to properly treat molecules in tight spaces. Further quantitative comparisons to single molecule DNA experiments, both in ideal bulk flows and in microfluidic devices, will be critical in helping us to evaluate the state of the art in Brownian dynamics simulations.

References ¨ [1] H.C. Ottinger, Stochastic Processes in Polymeric Fluids, Springer-Verlag, New York, 1996. [2] M. Fixman, “Construction of Langevin forces in the simulation of hydrodynamic interaction,” Macromolecules, 19, 1204, 1986. [3] D.C. Morse, “Theory of constrained Brownian motion,” Adv. Chem. Phys., 128, 65, 2004. [4] R.B. Bird, C.F. Curtiss, R.C. Armstrong, and O. Hassager, Dynamics of Polymeric Liquids, Volume 2, 2nd edn., Wiley, New York, 1987. [5] C.-C. Hsieh, L. Li, and R.G. Larson, “Modeling hydrodynamic interaction in Brownian dynamics: simulations of extensional flows of dilute solutions of DNA and polystyrene,” J. Non-Newtonian Fluid Mech., 113, 147, 2003.

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[6] E. Marko and E.D. Siggia, “Stretching DNA,” Macromolecules, 28, 8759, 1995. [7] P.T. Underhill and P.S. Doyle, “On the coarse-graining of polymers into bead-spring chains,” J. Non-Newtonian Fluid Mech., 122, 3, 2004. [8] H. Yamakawa, Helical Wormlike Chains in Polymer Solutions, Springer, Berlin, 1997. [9] A. Montesi, M. Pasquali, and F.C. MacKintosh, “Collapse of a semiflexible polymer in poor solvent,” Phys. Rev. E, 69, 021916, 2004. [10] P.S. Grassia, E.J. Hinch, and L.C. Nitsche, “Computer simulations of Brownian motion of complex systems,” J. Fluid Mech., 282, 373, 1995. [11] P.S. Doyle, A.P. Gast, and E.S.G. Shaqfeh, “Dynamic simulation of freely draining flexible polymers in steady linear flows,” J. Fluid Mech., 334, 251, 1997. [12] S. Chu, “Biology and polymer physics at the single molecule level,” Phil. Trans. R. Soc. Lond. A, 361, 689, 2003. [13] P.S. Doyle, B.Ladoux, and J.L. Viovy, “Dynamics of a tethered polymer in shear flow,” Phys. Rev. Lett., 84, 4769, 2000. [14] R.G. Larson, H. Hu, D.E. Smith, and S. Chu, “Brownian dynamics simulations of DNA molecules in an extensional flow field,” J. Rheol., 43, 267, 1999. [15] C.M. Schroeder, H.P. Babcock, E.S.G. Shaqfeh, and S. Chu, “Observation of polymer hysteresis in extensional flow,” Science, 3001, 1515, 2003. [16] R.M. Jendrejack, D.C. Schwartz, J.J. de Pablo, and M.D. Graham, “Shearinduced migration in flowing polymer solutions: simulation of long-chain DNA in microchannels,” J. Chem. Phys., 120, 2513, 2004. [17] G.H. McKinley and T. Sridhar, “Filament stretching rheometry of complex fluids,” Annu. Rev. Fluid Mech., 34, 375, 2002.

9.8 MECHANICS OF LIPID BILAYER MEMBRANES Thomas R. Powers Division of Engineering, Brown University, Providence, RI, USA

All cells have membranes. The plasma membrane encapsulates the cell’s interior, acting as a barrier against the outside world. In cells with nuclei (eukaryotic cells), membranes also form internal compartments (organelles) which carry out specialized tasks, such as protein modification and sorting in the case of the Golgi apparatus, and ATP production in the case of mitochondria. The main components of membranes are lipids and proteins. The proteins can be channels, carriers, receptors, catalysts, signaling molecules, or structural elements, and typically contribute a substantial fraction of the total membrane dry weight. The equilibrium properties of pure lipid membranes are relatively well-understood, and will be the main focus of this article. The framework of elasticity theory and statistical mechanics that we will develop will serve as the foundation for understanding biological phenomena such as the nonequilibrium behavior of membranes laden with ion pumps, the role of membrane elasticity in ion channel gating, and the dynamics of vesicle fission and fusion. Understanding the mechanics of lipid membranes is also important for drug encapsulation and delivery. Lipid molecules are amphiphilic, with a charged or polar hydrophilic head group, and one or two hydrophobic hydrocarbon tails. In water, lipid molecules can self-assemble into sheet-like bilayer structures, with the hydrocarbon tails sandwiched between the hydrophilic head groups [1]. Free edges that expose the hydrophobic core are energetically unfavorable, and tend to cause the membrane to close up into a vesicle. The membrane structures formed by lipid molecules range in size from the 50 nm-diameter vesicles used for internal transport in eukaryotic cells to the 100 µm-diameter giant vesicles studied in vitro as model systems. The discrepancy between the size of such vesicles and the small size of the individual molecules justifies a coarse-grained approach in which the membrane is treated as a continuum. In this approach, 2631 S. Yip (ed.), Handbook of Materials Modeling, 2631–2643. c 2005 Springer. Printed in the Netherlands.

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most of the details of the chemistry or structure of the molecules are unimportant. The diffusion constant of lipid molecules in a lipid bilayer membrane at room temperature is roughly D ≈ 10−8 cm2 s−1 , much smaller than that of a small molecule in water, but still large enough that a single lipid molecule will diffuse to the other side of a 1 µm-diameter vesicle in about 1 s. Furthermore, above the chain-melting temperature [1], there is no crystalline order in the plane of the monolayers. Thus, lipid membranes are two-dimensional fluids. The hydrophobic nature of the hydrocarbon chains has several important consequences. For lipid molecules with two chains, the solubility in water is low, implying a constant total number of lipid molecules in a vesicle. Also, flip-flop of molecules from one monolayer to another is relatively rare since the polar or charged heads would have to pass through the hydrophobic inner region. Thus to a good approximation, the number of lipid molecules in each monolayer is also constant. Despite the hydrophobic chains, lipid membranes are relatively permeable to water. However, the permeability of ions is small; thus, changes in vesicle volume are resisted by osmotic pressure.

1.

Equilibrium Vesicle Shapes

Vesicles exhibit a rich variety of shapes (see Fig. 1): prolate and oblate surfaces of revolution, cup-shaped stomatocytes, pear shapes, budding shapes,

(a)

(b)

(c)

(d)

(e)

(f)

Figure 1. Sketches of observed vesicle shapes (not to scale): (a) prolate, (b) oblate, (c) stomatocyte, (d) pear-shaped, (e) budding, (f) nonaxisymmetric. Shapes (a)–(e) are surfaces of revolution with a vertical symmetry axis; shape (f) has a threefold symmetry axis normal to the plane of the paper.

Mechanics of lipid bilayer membranes

2633

and nonaxisymmetric “starfish”. In this section we will review the theory used to calculate these shapes (for more comprehensive reviews, see Refs. [2, 3]). In particular, we will see that accounting properly for the bilayer structure is crucial for generating the complete set of shapes. Unlike the interface between oil and water, lipid membranes have a resistance to bending. To develop the elastic theory of membranes, we exploit the small thickness of a membrane and take a purely two-dimensional approach. We begin by ignoring the bilayer structure and consider the minimal model in which the membrane is a monolayer. Once we see how the minimal model fails to account for all of the observed vesicle shapes, we will generalize the model to include bilayer structure. For a fluid membrane, there is no reference configuation for the positions of the constituent molecules (there is a reference density), and the bending energy depends on the current shape only. Therefore, the bending energy must depend on purely geometric quantities, such as curvature. To define the curvature at a point p of a surface, construct the tangent plane at p and measure the height h of the surface above the tangent plane as a function of local Cartesian coordinates with origin p (Fig. 2). To leading order in x and y, the height will be a quadratic form, h(u) = ij Lij (u)uiuj , where u = (x, y). The invariants of the matrix L (the trace and the determinant) are independent of the choice of coordinate system and depend on shape only. Thus, we define the mean curvature H ≡ trL/2 and the Gaussian curvature K ≡ det L; see Ref. [4]. Note that 2H = 1/R1 + 1/R2 and K = 1/(R1 R2 ), where the principal curvatures 1/R1 and 1/R2 are the eigenvalues of L . For a sphere of radius R, R1 = R2 = R. For a cylinder of radius R, R1 = ∞ and R2 =R. Note that we must choose a convention to define the sign of H. The direction of positive height could be along either the inward or outward surface normal ˆ For a symmetric monolayer (or bilayer) with the same solvent on either side, n. there is no reason to prefer one choice over the other. The Gaussian curvature, on the other hand, is independent of the choice of normal. Thus, the energy ˆ H −→ −H, K −→ K . must have the symmetry nˆ −→ −n,

p x

y h(x,y)

Figure 2. Height h of the surface above the tangent plane at p.

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T.R. Powers

We can think of H as a measure of strain due to bending. For small curvature (d/R1 1 and d/R2 1, where d is the membrane thickness), the leading terms of the bending energy are F=

κ 2

(2H )2 dA +

κG 2

K dA.

(1)

Although the energy is quadratic in the “strains”, it depends nonlinearly on the membrane shape r(u). The Gaussian curvature term is usually disregarded, since the Gauss–Bonnet theorem implies that it is insensitive to continuous changes of shape for a closed vesicle [4]. If we relax the bilayer symmetry requirement, for example because the solution inside the vesicle is different from the solution outside, then we have the spontaneous curvature model [5, 6]: FSC = (κ/2) (2H − c0 )2 dA. We will set c0 = 0 for simplicity, but density-dependent spontaneous curvature terms will arise in our discussions of bilayer structure and active membranes. In real cell membranes, the two monolayers have different compositions, and generally c0 =/ 0. The deformation energy of a fluid membrane is similar to that of a solid plate. But there is a crucial difference between plates and membranes. It costs energy to shear a flat plate in its plane. On the other hand, no energy is required to shear a flat membrane in its plane, since this deformation rearranges the molecules without stretching any bonds. (Note that we assume the shear occurs so slowly that viscous effects are small.) For a solid plate, the ratio of stretching energy to bending energy diverges as the thickness vanishes [15]. To see why bending is easier than stretching, consider a thin solid plate bent into the shape of a section of a cylinder with small curvature 1/R (Fig. 3). The difference in length between the inside and outside faces of the plate is proportional to d/R, where d is the plate thickness. These small strains can add up to a deflection which is large compared to the plate thickness. But to get a similar magnitude deflection from pure stretching requires strains which are independent of the thickness and much larger. The result is that the lowest energy configurations accessible to a thin solid plate are those in which there is no stretching. In the limit of vanishing thickness, we can replace the high penalty for stretching with a constraint which forces any deformation to be isometric, in which the distance between any two nearby points on the surface is fixed. This effective stiffness is known as geometric rigidity. For example, it is easy to bend a sheet of paper into a cylinder or cone shape, but it is impossible to smoothly shape the paper into a spherical cap. However, a flat fluid membrane could be shaped into a spherical cap without stretching – the lipid molecules flow to adjust to the new configuration. Therefore, the fluid nature of membranes implies that the geometric constraint in the limit of vanishing thickness is weaker than the constraint for solid plates; the total area is fixed,

Mechanics of lipid bilayer membranes

2635

(a)

R

d

inner

δ

outer

(b)

δ

L Figure 3. (a) The outer face of the bent plate is longer than the inner face by δd/R; therefore, the strain vanishes as d/R → 0. (b) The strain in the stretched plate is δ/L , independent of d.

but the deformation need not be isometric. This weaker constraint leads to a rich morphology of vesicle shapes. Membranes in solution undulate constantly due to Brownian motion. Although these thermal fluctuations are always present, they are usually not large, and they are often either disregarded or treated as a perturbation in discussions of vesicle shape. In this section, we use the equipartition theorem to estimate the amplitude of the fluctuations. Consider a membrane patch which is almost flat, parametrized in the Monge representation, r(u)=(x, y, h(x, y)), where h(x, y) is theheight above the xy-plane. Then the bending energy (1) reduces to F ≈ k/2 (∇ 2 h)2 dx dy. Assuming that the patch is square with side length L, and imposing periodic boundary conditions for simplicity, we write the height as a Fourier series, h(u)= q exp(iq · u)h q /L 2 , where the sum over q = (2πm/L , 2πn/L) runs over all the integers m and n less than a cutoff m max . The cutoff corresponds to the shortest excitable wavelength, which is comparable to the thickness of the membrane. Applying the equipartition theorem (each mode of a quadratic energy function contributes kB T /2 to the average energy [3]) to the bending energy yields the mean-square-amplitude

h q h q = L 2 δq+q ,0

kB T q4.

(2)

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T.R. Powers

The mean square δn · δn of the deviation δn ≡ nˆ − zˆ of the unit surface normal nˆ from zˆ is a measure of the size of the fluctuations. Since the membrane is almost flat, δn ≈ −∇h, and [7]

δn · δn =

1 q,q

L

q · q h q h q ≈ 2

d2 q kB T . (2π)2 κq 2

(3)

The integral in (3) is cut off at small wavenumbers by the lateral size of the membrane, and at large wavenumbers by a molecular scale a, such as the size of the molecules or the thickness of the membrane. Approximating the sum as an integral and supposing the wavevectors range from qmin ≈ 2π/L to qmax ≈ 2π/a leads to δn · δn ≈ kB T /(2πκ) ln(L/a). Thus, the magnitude of the fluctuations of the normal become comparable to the magnitude of the normal itself when L ≈ ξP , where the persistence length ξP ≈ a exp(2πκ/kB T ). For a ≈ 1 nm and κ ≈ 10kB T , the persistence length is enormous, and thermal fluctuations may be safely ignored in calculations of vesicle shapes. Perhaps the simplest approach to computing vesicle shapes is to treat the membrane as a monolayer with constant area and enclosed volume, and a resistance to bending. Minimizing the bending energy subject to these constraints leads to the Euler–Lagrange equation

p + 2σ H − 2κ 2H (H 2 − K ) − ∇ 2 H = 0,

(4)

where ∇ 2 is the Laplacian on the surface, and σ and p are the Lagrange multipliers for the constraints of fixed area and volume, respectively [2]. The first two terms of (4) correspond to the Young–Laplace law for an interface, and the rest are the contributions from the bending stresses. Note that the bending terms vanish for a perfectly spherical vesicle, since the bending energy of a sphere of any radius is 8πκ. More generally, if r(u) describes the shape of a vesicle, and if λ is a positive, constant scale factor, then the new shape λr(u) has the same bending energy. We can use this scale-invariance to reduce the two-parameter family of shapes (one for every area A and volume V ) to a one-parameter family. Thus, to find the critical points of F for area λ2 A and volume λ3 V , use the stationary solutions of F + σ A + pV and rescale the shapes is to use the all lengths by λ. A convenient way to parameterize √ reduced volume υ ≡ 3V /(4π R03 ), where R0 ≡ A/(4π) is the radius of the sphere with area A. Since the sphere has the maximum volume for a given area, 0 ≤ υ ≤ 1. Equivalently, since the sphere has minimum area for a given volume, any υ < 1 corresponds to a shape with excess area over the sphere that encloses the same volume. Numerical solution of the Euler–Lagrange equations (4) yields prolate, oblate, and stomatocyte shapes only, with the oblate shapes occuring only in a narrow range of υ. There are no budding shapes, pear shapes, or nonaxisymmetric shapes [2]. Thus, the minimal model captures some of the

Mechanics of lipid bilayer membranes

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features of the experimentally measured phase diagram of vesicle shape, but fails to predict some of the observed shapes. The shortcomings of the minimal model arise because it disregards an important physical effect. Bending a bilayer necessarily stretches and compresses the two monolayers, whereas bending a liquid monolayer does not stretch its midplane. For example, consider an element of area dA ≡ n · ∂r/∂u 1 × ∂r/∂u 2 on the midplane of the bilayer. By projecting all the points in dA along the positive and negative normal, we define the regions dA± on each leaf of the bilayer. If 2d is the distance between the two leaves, then a short calculation shows dA± = dA[1 ± 2d H + O(d 2 K )]. This formula is useful for writing ± relative the number densities φ ± of each monolayer in terms of densities φproj ± ± ± to the middle surface of the bilayer: φ dA = φproj dA. If φ0 is the equilibrium density on each monolayer, then the projected dimensionless density is ± /φ0 −1), and relative and total dimensionless projected densities are ρ ± = (φproj + ρ ≡ (ρ −ρ − )/2 and ρ¯ ≡ (ρ + −ρ − )/2. Note that under the bilayer symmetry ˆ H −→ −H, ρ −→ −ρ, and ρ¯ −→ ρ. operation, nˆ −→ −n, ¯ Thus, bilayer symmetry forbids a term in the elastic energy density of the form ρ¯ H, but allows ρ H. The total elastic energy is F=

κ 2

dA(2H )2 + km

dA (ρ − 2dH )2 + ρ −2 ,

(5)

where we have assumed that each monolayer has the same area compression modulus km [2]. The reader should check that (5) correctly gives no stretching or compression cost when the bilayer is bent with curvature H but each monolayer has been allowed to relax to its equilibrium density φ ± = φ0 . This relaxation always possible in a vesicle where the number of molecules is not ± in each leaf of the bilayer is fixed. N ± = dA φproj To reduce the energy density of (5) to a function of shape only, minimize over ρ and ρ¯ subject to the constraint of fixed N ± . Just as for the minimal model, the penalty for changing the total area is much larger than the cost of bending, and we may regard the area as constant. However, the coupling of curvature and density-difference leads to a new elastic term F=

κ 2

dA (2H )2 +

κβ π ( A − A0 )2 , 8Ad 2

(6)

where β = 2km d 2 /(πκ), A = 4d dAH, and A0 = (N + − N − )/φ0 . This model is known as the “area-difference elasticity” model. The parameter β is often regarded as an independent parameter, since its value depends on the model for stretching elasticity. If β → ∞, then (6) becomes the “bilayer couple model”, in which there is a constraint on the integrated mean curvature. Both of these models have an additional parameter, A0 , relative to the single parameter υ of the minimal model. Thus, the phase diagram is now

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two-dimensional, and the upshot is that these models can predict the observed shapes. The reader is referred to Refs. [2, 8] for the details.

2.

Entropic Elasticity

In this section and the next, we consider situations in which membrane stretching cannot be ignored. First we review Evans and Rawicz’s classic measurement of tension vs. apparent areal strain for a spherical vesicle [9]. We saw earlier that collisions of solvent molecules with a membrane excite undulations. If the membrane is stretched across a frame, these undulations will tend to contract the frame, leading to tension. The actual area of the membrane will be greater than the area of the frame. Likewise, a fluctuating vesicle will have an actual area which is larger then its apparent area, the area of the average vesicle shape. Applying tension will reduce the amplitude of the undulations, pulling some of the excess area out of the undulations and increasing the apparent area. Eventually, most of the undulations are smoothed out, and increasing the tension will stretch the membrane like an elastic material. To measure the relation between the tension and the excess area, Evans and Rawicz aspirated a vesicle onto a pipet. At low suction pressures, the vesicle shape fluctuates. With increased suction pressure, the outer portion of the vesicle becomes spherical, and the fluctuations are no longer optically visible. Nevertheless, as the suction pressure increases further, significant excess area is pulled from the microscopic fluctuations, which is reflected by an increase in the length of the vesicle in the pipet (Fig. 4). As the apparent area increases, the tension increases. The fluid nature of the membrane implies that the tension is uniform over the entire vesicle (the experimentalists take care that the membrane does not adhere to the pipet). There are two spherical portions where the Young–Laplace law [(Eq. (4) with κ = 0] applies: the main

R0 P1

Rp P0

Figure 4. Vesicle aspirated onto a pipet; p = p2 − p1 .

Mechanics of lipid bilayer membranes

2639

body of the vesicle with radius R0 , and the little spherical cap with radius RP inside the pipet. Eliminating the pressure inside the vesicle from these relations yields the tension τ=

p Rp , 2(1 − Rp /R0 )

(7)

where p is the pressure difference between the solvent and the pipet. To determine the apparent areal strain, define α ≡ (A − A0 )/ A0 , where A and A0 are the final and initial apparent areas of vesicle, respectively. The strain α is the (dimensionless) difference in the excess areas stored in the undulations at tensions τ0 and τ, where τ0 is the tension corresponding to the area A0 . Typically A0 is the apparent area at the lowest suction that holds the vesicle onto the pipet. If R˜ 0 is the final vesicle radius and R0 is the initial vesicle radius, then α A0 =2π Rp +4π( R˜ 02 − R02 ). Using volume conservation, /R0 1, R p /R0 1, and /R0 1, it follows that 1 α= 2

Rp R0

2

−

Rp R0

3

. Rp

(8)

Thus, α and τ may be determined by measuring R0 and as p is varied. There are two regimes. For α less than a few percent, the resistance to stretching is purely entropic, and the tension is related nonlinearly to strain: τ ∝ exp α. For greater strains, most of the thermal undulations have been pulled out, and the intrinsic membrane elasticity leads to a linear (Hooke’s law) response. To understand these two regimes, we present the self-consistent approach of Helfrich and Servuss [10] (see also Ref. [3]). The problem is considerably simplified if we consider a membrane attached to a square frame, and allow the membrane area (but not the area of the frame) to fluctuate. In this approach, the “tension” τ plays the role of a chemical potential per area. Our goal is to compute the excess area hidden in the fluctuations as a function of τ Up to an additive constant, the free energy to quadratic order is τ κ 2 2 2 d u(∇ h) + d2 u(∇h)2 , (9) F≈ 2 2 where 2 the second term arises from the expansion of the membrane area, Am = d u 1 + (∇h)2 . The average excess area hidden in the undulations is

Am − L 2 ≈

1 2

d2 u (∇h)2 ,

(10)

or, using the equipartition theorem [now with |h q |2 /L 2 = kB T /(κq 4 + τ q 2 )]

Am − L 2 1 = 2 L 4π

π/L

kB T q dq , κq 2 + τ

(11)

2640

T.R. Powers

where ∼ π/a is the short-wavelength cutoff, and π/L is the long-wavelength cutoff. Evaluating the integrals leads to the dimensionless excess area

Am − L 2 kB T κ 2 + τ = ln . L2 8πκ κ(π/L)2 + τ

(12)

To relate this calculation to the experiment of Evans and Rawicz, define α F to be the difference in the areas hidden in the fluctuations for zero tension and tension τ : α F ≡ ( Am τ = 0 − Am τ ) /L 2 . Except for geometrical factors which arise because α F applies to a planar geometry and α applies to a spherical geometry, α F and α both describe precisely the same difference in excess area (assuming τ0 ≈ 0). Thus, we identify α = α F , and let L 2 = A, the apparent area. In the low-tension regime, τ κ 2 , which means that drops out of the difference Am τ = 0 − Am τ . Finally, we combine the intrinsic membrane elasticity with the entropic elasticity by assuming these two effects can be modeled as springs in series

kB T τA α≈ ln 1 + 2 8πκ κπ

+

τ , KA

(13)

where K A is the intrinsic stretch modulus. For realistic values of tension, area, and bending stiffness, it turns out that τ A/(κ π2 ) 1, and the logarithmic term dominates at low tension. Thus, τ ∝ exp (α) at low tension. Evans and Rawicz determined the bending stiffness κ and the stretch modulus K A by fitting (13) to their measure of τ vs. α.

3.

Active Membranes

Our discussion so far has been confined to membranes in thermal equilibrium, or “passive membranes”. However, since living systems are by definition out of equilibrium, it is desirable to extend our discussion to nonequilibrium processes. This area is largely unexplored. In this section, we review recent work on the effects of ion pumps on membrane fluctuations [11]. The fluctuations of red blood cell membranes have been observed for many years, and have been interpreted using the equilibrium theory discussed in the Section 2 [12]. However, recent observations have cast doubt on the purely thermal origin of these fluctuations and shown them to depend on the concentration of ATP [13], the energy source for many of the active processes in the cell. The fluctuations increase with ATP concentration, and for a given ATP concentration, decrease as the solvent viscosity increases. When ATP is depleted, the fluctuations become independent of viscosity. Enhanced fluctuations due to non-thermal effects have also been seen in an artificial system

Mechanics of lipid bilayer membranes

2641

consisting of a lipid vesicle studded with bacteriorhodopsin (BR) [14]. BR is a light-driven proton pump purified from the purple membrane of the saltloving bacterium Halobacterium salinurum. Manneville et al. aspirated the vesicle onto a pipet and measured the tension versus the areal strain to find that fluctuations were enhanced when the BR was “on” (illuminated by light of the proper wavelength). To begin to understand these phenomena, we outline a simple theory for an almost flat monolayer membrane embedded with diffusing pumps. When a pump transfers an ion across the membrane, the pump exerts a force on the membrane. We will neglect the random fluctuations of the force over time, and suppose each pump exerts a constant force Fp ≈ 10kB T /d, where 10kB T is the free energy given up by ATP hydrolysis at physiological concentrations, and d is the membrane thickness. Each pump acts in only one direction, and cannot flip across the membrane to change its orientation. Assume there is an equal number of upward and downward-directed pumps, with number densities n + and n − , respectively, so that n + − n − = 0. Just as in our discussion of bilayer structure, it is convenient to introduce the dimensionless total density φ ≡ (n + + n − )/n 0 and density difference ψ ≡ (n + − n − )/n 0 , where n 0 = n + + n − . Bilayer symmetry allows a coupling between the density difference and the mean curvature but implies that φ is decoupled from the other ˆ leading to a quadratic energy fields (φ → φ and ψ → −ψ under nˆ → −n), similar to (5) F=

1 2

d2 u κ(∇ 2 h)2 + τ (∇h)2 + Bψ 2 − 2κc0 ψ∇ 2 h ,

(14)

where B is a compression or osmotic modulus. At low pump density, B ∼ n 0 kB T, since the compression modulus of an ideal gas is the pressure. When the pumps are on, they consume ATP and the equipartition theorem cannot be used to determine the variance of h; instead, we must solve the dynamical equations of motion. The details of this calculation are technical and we refer the reader to the review of (Ramaswamy and Rao, [11]); here we just summarize the main points. At typical membrane length and time scales, inertia is unimportant, and the equation for the height field amounts to a balance of elastic, viscous, and pump forces. The elastic forces per unit area can be deduced from the elastic energy (14). To determine the pump force per unit area f pumps , let Fp be the force exerted by an isolated pump when H = 0. The symmetry nˆ → −nˆ allows a coupling between density difference and mean curvature f pumps = (n + − n − )Fp + (n + + n − ) 2 H Fp ,

(15)

where 2 is a coupling constant with units of length. Finally, the simplest model for the viscous forces is to disregard the hydrodynamics of the solvent and use a local model, analogous to the Rouse model for polymer dynamics.

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T.R. Powers

In the membrane version of this model, the local membrane velocity is proportional to the elastic and pump forces per unit length, with the permeation constant µp acting as the constant of proportionality. The dynamics for h is completed with a random force density representing the effects of Brownian motion, leading to a Langevin equation [7]. The equation of motion for ψ is the diffusion equation modified with terms corresponding to the coupling of density and curvature in (14), and again with a random forcing term. Solving the coupled equations for the variance of h and computing the areal strain for small tensions leads to α ≡ α(τ0 ) − α(τ ) ≈ (kB Teff /κeff ) ln(τ/τ0 ), with kB Teff = kB T +

κeff µp Fa . D

(16)

In (16), D is the diffusion constant of the pumps, κeff = κ − (κc0 )2 /B, and

¯ = 2 + κc0 /B. In this model, the fluctuations are enhanced by the pumps, but have the same dependence on tension as in passive membranes. Note that increasing the solvent viscosity should decrease µp and thus α, in accord with the observations of Tuvia et al., [13]. Increasing the membrane viscosity should decrease D, enhancing the nonequilibrium fluctuations.

4.

Outlook

The good agreement between theoretical predictions and experimental observations of vesicle shapes demonstrates that the equilibrium elastic behavior of lipid bilayer membranes is well-understood. Likewise, we have seen that quantitative understanding of entropic tension has progressed to the point that measurements of thermally generated tension may be used as a tool for probing new phenomena such as the fluctuations of active membranes. These fluctuations are but one example requiring a deeper understanding of the dynamics of membranes, which is likely to be the main front for future developments.

Acknowledgment Preparation of this review was supported in part by Grant No. CMS0093658 from the National Science Foundation.

References [1] J. Israelachvili, Intermolecular and Surface Forces, 2nd edn. Academic Press, London, 1992.

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[2] U. Seifert, Configurations of fluid membranes and vesicles, Adv. Phys., 46, 13, 1997. [3] D. Boal, Mechanics of the Cell, Cambridge University Press, Cambridge, 2002. [4] R. Kamien, “The geometry of soft materials: a primer,” Rev. Mod. Phys., 74, 953, 2002. [5] P. Canham, “The minimum energy of bending as a possible explanation of the biconcave shape of the human red blood cell,” J. Theor. Biol., 26, 61, 1970. [6] W. Helfrich, “Elastic properties of lipid bilayers: theory and possible experiments,” Z. Naturforsh., 28c, 693, 1973. [7] P. Chaikin and T. Lubensky, Principles of Condensed Matter Physics, Cambridge University Press, Cambridge, 1995. [8] H.-G. D¨obereiner, E. Evans, M. Kraus, U. Seifert, and M. Wortis, “Mapping vesicle shapes into the phase diagram: a comparison of experiment and theory,” Phys. Rev. E, 55, 4458, 1997. [9] E. Evans and W. Rawicz, “Entropy-driven tension and bending elasticity in condensed-fluid membranes,” Phys. Rev. Lett., 64, 2094, 1990. [10] W. Helfrich and R.-M. Servuss, “Undulations, steric interaction and cohesion of fluid membranes,” Nuovo Cimento, D3, 137, 1984. [11] S. Ramaswamy and M. Rao, “The physics of active membranes,” C.R. Acad. Sci. IV Paris, 2, 817, 2001. [12] F. Brochard and J. Lennon, “Frequency spectrum of the flicker phenomenon in erythrocytes,” J. Phys. (Paris), 36, 1035, 1975. [13] S. Tuvia, A. Almagor, A. Bitler, S. Levin, R. Korenstein, and S. Yedgar, “Cell membrane fluctuations are regulated by medium viscosity: evidence for a metabolic driving force,” Proc. Natl. Acad. Sci. USA, 94, 5045, 1997. [14] J.-B. Manneville, P. Bassereau, S. Ramaswamy, and J. Prost, “Active membrane fluctuations studied by micopipet aspiration,” Phys. Rev., E64, 021908, 2001. [15] J. Rayleigh, The Theory of Sound, vol. I, 2nd edn. Dover Publications, New York, 1945.

9.9 FIELD-THEORETIC SIMULATIONS Venkat Ganesan1 and Glenn H. Fredrickson2 1 Department of Chemical Engineering, The University of Texas at Austin, Austin, TX, USA 2

Department of Chemical Engineering & Materials, The University of California at Santa, Barbara Santa Barbara, CA, USA

The science and engineering of materials is entering a new era of so-called “designer materials”, wherein, based upon the properties required for a particular application, a material is designed by exploiting the self-assembly of appropriately chosen molecular constituents [1]. The desirable and marketable properties of such materials, which include plastic alloys, block and graft copolymers, and polyelectrolyte solutions, complexes, and gels, depend critically on the ability to control and manipulate morphology by adjusting a combination of molecular and macroscopic variables. For example, styrene– butadiene block copolymers can be devised that serve either as rigid, tough, transparent thermoplastics or as soft, flexible, thermoplastic elastomers, by appropriate control of copolymer architecture and styrene/butadiene ratio. In this case, the property profiles are intimately connected to the extent and type of nanoscale self-assembly that is established within the material. One of the main challenges confronting the successful design of nano-structured polymers is the development of a basic understanding of the relationship between the molecular details of the polymer formulation and the morphology that is achieved. Unfortunately, such relationships are still mainly determined by trial and error experimentation. A purely experimental-based program in pursuit of this objective proves cumbersome – primarily, due to the broad parameter space accessible at the time of synthesis and formulation. Consequently, there is a significant motivation for the development of computational tools that can enable a rational exploration of the parameter space. Atomistically faithful computer simulations of self-assembly in dense phases of soft materials prove to be difficult or impossible for many systems of practical interest [2, 3]. Such methods typically involve building classical descriptions (with atomic resolution) of a complex fluid. Interactions in such models are

2645 S. Yip (ed.), Handbook of Materials Modeling, 2645–2656. c 2005 Springer. Printed in the Netherlands.

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described by some combination of bonded and non-bonded potential functions, typically parameterized at the two-body and/or three-body level. Determining the equilibrium and non-equilibrium properties involves carrying out a computer simulation, usually by employing Monte Carlo (MC) or molecular dynamics (MD) techniques. The major drawback of such atomistic methods is that except in rare instances, it is very difficult to equilibrate sufficiently large systems at realistic densities in order to extract meaningful information about structure and thermodynamics. Such limitations become particularly significant in the context of modeling inhomogeneous self-assembled phases, wherein the length scale of the self-assembled morphology corresponds to many atomic lengths. Consequently, most modern computer simulation methods for self-assembly in complex fluids have focused on coarsegrained, particle-based methods, i.e., particle based simulations (PBS) [4–6], where atoms or groups of atoms are lumped into larger “particles”. This could simply amount to a “united atom” approach where, e.g., each CH2 unit in a polyethylene chain is replaced by a single effective particle. Interactions in such a model are then effective interactions between lumped CH2 particles and standard MC or MD simulation methods can be employed. Often even more extensive coarse-graining is carried out. For example, bead-spring polymer chains are often employed in which each bead can represent the force center associated with 10 or more backbone atoms. Despite the great success and impact of such approaches, a difficulty is that there is no unique coarse-graining procedure and that the effective interactions between beads (particles) are often difficult to parameterize accurately. Moreover, PBS methods remain expensive to simulate, especially at melt densities and for heterogeneous systems that exhibit nanoscale or macroscale phase separation [7]. Alternative methods such as dissipative particle dynamics (DPD) [8] speed up the simulations by introducing soft inter-particle potentials, but at the cost of producing artificially high compressibilities, loss of topological constraints between chains, and often also a loss of connection to the chemical details of the underlying complex fluid. In the above modeling strategies, the fundamental degrees of freedom to be sampled in a computer simulation are the generalized coordinates (including bond and torsional angles) associated with the atoms or particles. An alternative approach for computing equilibrium properties is the focus of this review and is termed as field-theoretic simulations (FTS). Field theory based approaches have long been used as the basis for approximate analytical calculations on a variety of complex fluid systems including polymer solutions, melts, blends, and copolymers [9–11]. Further, lattice gauge simulation methods [12, 13] have also applied to field theories in nuclear, high energy, and “hard” condensed matter physics. However, it is only very recently that such field-theory models have considered as the starting point for a computer

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simulation strategy for polymers or other soft condensed matter systems [14, 15]. In the field-theoretic framework one integrates out the particle coordinates in the partition function, replacing them instead with functional integrals over smoothed, coarse-grained density fields and/or chemical potential fields that are conjugate to such density fields. The degrees of freedom in the FTS scheme are the values of the density and the chemical potential fields at different positions in the simulation domain. In order to represent these fields with a finite number of degrees of freedom, a variety of techniques are available including spectral methods, finite differences, and finite elements. These techniques are used extensively for the numerical solution of partial differential equations, e.g., in computational fluid mechanics [16], and a vast literature exists for efficiently and accurately representing fields of physical interest. Subsequently, an appropriate sampling strategy is used to generate different configurations of the fields (“ensembles”) with a thermodynamically consistent statistical weight, allowing one to calculate the experimental observables and the equilibrium thermodynamic properties of the material through averages over such fluctuating fields. It is hard to delineate precisely the systems for which field-theory based approaches should work more effectively than PBS and vice versa. Smallmolecule fluids and complex fluids that are characterized by harsh repulsions at small separations (like for instance, suspensions of colloidal particles, liquid crystals, etc.) possess rich liquid structure at small length scales that plays an important role in determining the thermodynamic properties of such materials [17]. Field-theoretic approaches would typically require a high spatial resolution (and correspondingly larger computational effort) to capture such short range effects, and hence such materials are better simulated by PBS methods. On the other hand, PBS methods are expensive to simulate, especially at high densities, and for heterogeneous systems that exhibit nanoscale or macroscale phase separation. In contrast, field-theories and the simulations of such theories work best for situations where the number densities of the particles are quite large so that the fluctuations in a coarse-grained description are kept at a modest level. Consequently, field-theoretic approaches offer an attractive platform (from the standpoint of computational effort) to simulate inhomogeneous polymers in situations where the short-range details turn out to be unimportant in determining the macroscale behavior. Examples include the cases of concentrated polymer solutions and blends, multiblock copolymers, charge-stabilized colloidal suspensions, etc. In the following, we take up the simple case of a fluid of colloidal particles to illustrate the two steps involved in developing and implementing the FTS approach. Many details, which have been omitted in the development to preserve brevity, can be found in Ref. [15]. We conclude with an overview of some recent applications and some potential future directions.

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V. Ganesan and G.H. Fredrickson

Step 1: Developing the Field Theory

The first step of FTS requires the development of a model of the classical/complex fluid whose thermodynamic properties are desired. In principle, such a model could be constructed at the atomistic scale (with interactions determined from quantum chemical calculations) or at a coarse-grained PBS scale with effective interactions between the lumped “particles”. In some instances the microscopic model could have particles smeared into fields, as in polymeric fluids where chains are often represented as flexible space curves. As an illustrative example, we consider the case of a fluid of colloidal particles that are interacting by means of a specified general pairwise interaction potential v(r). For a collection of n such particles in a volume V , the configurational partition function can be written as Z∝

drn exp −

β v(|r j − rk |) , 2 j =/ k k

(1)

where rn ≡ (r1 , r2 , . . . , rn ) and r j represents the coordinates of the particle j . In the above, β ≡ (kB T )−1 , where kB represents the Boltzmann constant and T the temperature. We note that the above framework also constitutes the starting point of PBS strategies. In such approaches, different configurations are characterized by different positions of the particles (or molecules), and the corresponding simulation strategies focus on generating ensembles of such configurations with the statistical weight: exp − β/2 j =/ k k v(|r j − rk |) [18]. The starting point for the FTS method smears out the coordinates of the particles, describing the thermodynamics of the system by means of a coarsegrained density field ρ(r). The appropriate statistical weights for such a description is obtained through two steps: (i) The first step is to specify the relationship between the microscopic density field and the positions of the particles. It is possible to postulate a variety of physically reasonable prescriptions for smearing out the positions of the particles into a density field. In the present context, we consider the simple case of “point” particles where we can introduce a microscopic density field ρ(r) ˆ ≡ ni=1 δ(r − ri ). We can rewrite the above partition function in terms of ρ(r) ˆ as Z∝

drn exp −

β 2

dr

dr ρ(r)v(|r ˆ − r |)ρ(r ˆ ) ,

(2)

where an unimportant self-energy term has been omitted. (ii) The second step involves “projecting” the above microscopic description to an effective

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“Hamiltonian” H [ρ] that is a functional of the real density fields ρ(r): exp(−β H [ρ]) ∝

drn

δ[ρ(r) − ρ(r)] ˆ

r

β dr dr ρ(r)v(|r ˆ − r |)ρ(r ˆ ) . (3) 2 The above step can be simply understood as enumerating the microscopic configurations for which ρ(r) ˆ = ρ(r), to it the weight exp(−β √ H [ρ]). and ascribing Dw exp[i dr w(ρ − ρ)], ˆ where i = −1 and Using the identity δ(ρ − ρ) ˆ = Dw denotes a functional integral over a scalar “chemical potential” field w(r), we obtain × exp −

exp(−β H [ρ]) ∝

Dw

drn

× exp i ×

drw(ρ − ρ) ˆ −

β 2

dr dr ρ(r)v(|r − r |)ρ(r ) .

(4)

After a few simple mathematical manipulations we obtain exp(−β H [ρ]) ∝ where Hˆ [ρ, w]=−i and Q[iw] = V −1

Dw exp(− Hˆ [ρ, w]),

dr wρ+

β 2

(5)

dr drρ(r)v(|r−r |)ρ(r )−n ln Q[iw], (6)

dr exp[−iw(r)].

(7)

Note that the above mathematical transformations have transformed the integrals over the coordinates of the particles rn to a functional integral over a real chemical potential field w(r). This step has also had the effect of eliminating all the particle–particle interactions, instead replacing them with the interaction between the particles and the fluctuating field w(r). As a result the coordinate integrals dr j are identical for each particle, allowing us to replace the n integrals by (Q[iw])n , where Q[iw] represents the single particle partition function in a purely imaginary potential iw(r) [9, 15]. For the case of flexible polymers, Eq. (7) is replaced by an expression related to the solution of a modified diffusion equation describing the conformations of a polymer experiencing the potential iw(r). The formulation embodied in Eq. (5) represents the thermodynamic description at the scale of coarse-grained density fields ρ(r). Indeed, the partition function Z can be equivalently expressed as Z∝

Dρ

Dw exp(− Hˆ [ρ, w]).

(8)

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Thermodynamic observables like the average densities, density correlations and osmotic pressures can be computed as appropriate ensemble averages over the different configurations of the density fields ρ(r). For instance, the density–density correlations ρ(r)ρ(r ) can be expressed as: ρ(r)ρ(r ) = Z −1

Dρ

Dw ρ(r)ρ(r ) exp(− Hˆ [ρ, w]).

(9)

The above approach can also be easily generalized to describe the thermodynamics of multicomponent systems in terms of the density fields of each of the components. As might be evident, the introduction of additional density fields would also necessitate the introduction of a conjugate chemical potential field for each such density [15, 19]. More recent applications have also generalized the above description to incorporate additional coarse-grained variables like stress fields to describe the thermodynamics of deformed systems [20].

2.

Step 2: Discretizing and Sampling the Field-Theory

The transformation of Eq. (1) into the field-theoretic formulation of Eqs. (5), (8) and (9) would seem to be a step in the wrong direction, since we now are faced with the task of evaluating infinite-dimensional functional integrals over the fields ρ(r) and w(r). However, upon discretization, the functional integrals reduce to finite-dimensional integrals that can be tackled with stochastic simulation methods. In such methods, large numbers of configurations of the fields ρ(r) and w(r) are generated with a probability weight exp(− Hˆ [ρ, w]), and the field configurations are used to evaluate thermodynamic averages like Eq. (9). The latter constitutes the basis of FTS, where the space (i.e., the simulation domain) is explicitly discretized and different configurations of the density and potential fields (as specified by their values at the different points in the domain) are generated with a weight exp(− Hˆ [ρ, w]) by an appropriate sampling scheme [15, 19]. The appropriate scheme for discretizing the spatial domain is a highly flexible component of FTS and can be accomplished by conventional finite difference or finite element representations of the fields. Spectral and pseudo-spectral techniques are particularly attractive for this purpose [21, 22]. Modern adaptive, unstructured finite element methods [16] could also be applied. On the other hand, the appropriate sampling scheme for generating the density and the potential fields involves some subtle issues which are discussed below. Conventional simulation strategies such as Monte Carlo schemes and molecular dynamics methods are well tuned to generating configurations with a given ˆ positive definite weight [18]. However, the effective “Hamiltonians” H √[ρ, w] which accompany FTS are complex (due to the appearance of i = −1 in Eq. (6)), despite the fact that the fields ρ and w and the partition function Z are

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real. If Hˆ is decomposed as Hˆ R + i Hˆ I , then it follows that Z can be expressed as Z = Dρ Dw exp(− Hˆ R ) cos( Hˆ I ), where the integrand is explicitly real, but is not positive semi-definite due to the phase factor cos(HI ). To overcome this subtle, albeit important feature of the FTS weights, two modified simulation strategies have been developed.

2.1.

Steepest Descent Sampling

This method [15, 23] tackles the issue of positive definiteness by generating the different configurations of ρ and w fields with a modified positive semidefinite weight, viz., exp(− Hˆ R ). Averages of observables are then computed by using an explicit phase factor, cos( Hˆ I )

φ([ρ]) =

φ([ρ]) cos[ Hˆ I ]R , cos[ Hˆ I ]R

(10)

where the averages · · · R are now over the potential fields generated with a weight exp(− Hˆ R ). While this approach is simple, in practice it is not very useful. The integration path, which in this case is the real axis for the density and potential fields, is not a constant phase or steepest-descent(ascent) path, whence the phase factor oscillates in sign from state to state along a Monte Carlo trajectory. As a result, it proves very difficult to accurately compute the right hand side of expressions such as Eq. (10). The steepest-descent (SD) simulation strategy adds one more step to the above idea where such oscillations are minimized by deforming the path of integration onto a new path that passes through the relevant saddle point and is also a constant phase path to quadratic order near the saddle point [24]. In this manner, provided that fluctuations about the relevant saddle point are weak, the phase factor cos( Hˆ I ) is kept at (an almost) constant value, thereby damping the spurious oscillations encountered in evaluating the above averages. In situations of strong fluctuations, a global stationary phase technique has recently become available [25].

2.2.

Complex Langevin Sampling

The Complex Langevin (CL) method was originally developed by Klauder [26] and Parisi [27] as a strategy for sampling quantum field theories on a lattice and for simulating more general types of lattice gauge theories with complex actions. The basic idea behind the CL method is to stochastically sample the relevant fields, i.e., ρ(r) and w(r), not just along the real axis, but

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in the entire complex plane of ρ =ρR +iρI and w =wR +iwI . For any observable φ, the expectation (average) value can be expressed as φ([ρ]) = Z

−1

D[ρR ]

D[wR ] exp(− Hˆ [ρR , wR ])φ([ρR ]).

(11)

In the CL method, the strategy is to instead express such an observable as φ([ρ]) =

D[ρR ]

D[ρI ]

D[wR ]

D[wI ] P[ρR , ρI , wR , wI ]

× φ([ρR + iρI ]),

(12)

where the complex weight Z −1 exp(− Hˆ ) has been replaced by a real, positive definite statistical weight P[ρR , ρI , wR , wI ]. The statistical weight P[ρR , ρI , wR , wI ] is generated as the steady-state distribution of a stochastic “complex Langevin” dynamics, which gives the probability of observing the field configurations w = wR + iwI and ρ = ρR + iρI at time t. The extension of the fields to the complex plane has a practical cost in that the number of configurational degrees of freedom in a simulation is doubled. However, applications of this approach have demonstrated that such an increase in the number of degrees of freedom is more than offset by the efficiency of this approach in generating statistically relevant field configurations.

2.3.

Saddle-points and Self-consistent Field Theories

An important feature of the field theory models [11, 28], described in the previous sections is the identification of stationary field configurations that correspond to extrema of the complex effective Hamiltonian Hˆ . Such configuration(s) can correspond to a local minimum, maximum, or (usually) a saddle point in the field configuration space. While the above two simulation strategies focus on generating large numbers of configurations with appropriate weights, the extremum field configurations correspond to dominating (in most cases) configurations, and in many a cases provide a wealth of information about the thermodynamic properties of the material. In the specific context of polymer models, the solutions corresponding to inhomogeneous saddle points are identical to those obtained using a mean-field theory known as selfconsistent field theory (SCFT). Such theories have been applied with great success in the analysis of the excluded volume effect, the characteristics of polymer–polymer interfaces and polymer brushes, the self-assembly features of block copolymer mesophases, polymer blends, thin films of polymers, block copolymer–nanoparticle blends, etc. [29]. New pseudo-spectral algorithms for efficient numerical implementation of SCFT will undoubtedly further extend the impact of the saddle point approach [21, 22].

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Outlook

In the present review we have described the idea behind the field-theoretic computer simulation (FTS) tools for analyzing the equilibrium structure and thermodynamics of both simple and complex fluids. The preceding sections illustrated how field theory models can be formulated starting from conventional particle-based models of fluids at the atomistic, mesoscopic, or macroscopic scales. Thus, it is possible to connect the potential parameters used in traditional MD or MC simulations to the parameters in field theory models amenable to study by the methods described here. Further, field theory models are commonly used in analytical studies aimed at extracting “universal” features of the structure and thermodynamics of polymers and complex fluids [10]. FTS methods thus enable numerical studies of field theory models in parameter ranges or situations where approximate analytical tools are inadequate or fail. Finally, experimental studies of complex fluids are often interpreted in the context of parameters (e.g., Flory χ parameters) and predictions derived from field theory models [30, 31]. As a result, it is often more straightforward to connect experimental data to results from a FTS simulation than to numerical data from particle-based MD or MC simulations. It is still too early in the development of FTS methods to predict their competitiveness against other theoretical and computational techniques. Nevertheless, the following problems appear particularly promising in the context of applying the FTS approach: • Concentrated polymer systems, especially multiphase blends and copolymer melts are ideally suited for study by FTS methods, especially when the atomic-scale structure is not of interest or relevant to mesoscopic/ macroscopic self-assembly behavior. We list two such examples where FTS has been applied: – Order–disorder transitions of block copolymers [15, 19]. FTS simulations have been used to explore quantitatively the effect of fluctuations and the polymerization index N of the polymer on shifting the order–disorder transition (ODT) in two-dimensional, symmetric diblock copolymer melts. The results of these studies matched quantitatively with approximate analytical calculations, but also extended the predictions to regimes where such calculations were not applicable. It is important to emphasize that such results would be very difficult to obtain by means of a conventional particle-based simulation of a highly incompressible block copolymer melt. – Polymeric microemulsion phases [23]. A variant of the steepest descent Monte Carlo method was used to analyze a field theory model of a ternary blend of AB diblock copolymers with A and B homopolymers. These studies found a shift in the line of order–disorder

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V. Ganesan and G.H. Fredrickson transitions from their mean-field values, as well as strong signatures of the existence of a polymeric bicontinuous microemulsion phase in the vicinity of the mean-field Lifshitz critical point. These results matched qualitatively with a series of experiments conducted with various three-dimensional realizations of this model system. – Confined polymer solutions [32]. FTS simulations using the CL technique were used to explore the structure of semi-dilute and concentrated polymer solutions confined to a slit geometry. The crossover behavior between the semi-dilute and concentrated regimes was explicitly accessed and the role of concentration fluctuations on the structure and thermodynamics was examined.

• Systems with soft, long-ranged interactions such as electrolyte solutions, polyelectrolytes, block co-polyelectrolytes, etc. may prove to be easier to study using FTS techniques. A long-ranged Coulomb interaction, v(|r − r |) ∼ |r − r |−1 , can be transformed into a short-ranged interaction in the field-theoretic framework by the Hubbard–Stratonovich transformation. Thus, computationally expensive approaches to treat long-range forces, for example Ewald summations, can be avoided in the FTS approach. • The determination of potentials of mean force between colloidal particles can be conveniently addressed by FTS techniques. The stability of colloidal (and nanoparticle) suspensions with surface charges, grafted polymers, free polymers, counterions, and salts can in principle be investigated by the methods described here. • The dynamical and rheological properties of complex fluids are currently best addressed with particle-based simulation methods. Recently, two field theory approaches have been advanced that appear promising for describing the dynamics and rheology of complex fluids. While the first approach generalizes the coarse-graining methodology to include new variables characterizing nonequilibrium states, the second approach combines particle-based simulations with the field-theoretic ideas to effect dynamical simulations in dense systems. With further development, either or both these methods could become the appropriate generalization of FTS for addressing nonequilibrium phenomena in complex fluids [20, 33].

References [1] R.A. Ball, Made to Measure, Princeton University Press, Princeton, 1997. [2] F. Muller-Plathe, “Combining quantum chemistry and molecular simulation,” Adv. Quantum Chem., 28, 81–87, 1997. [3] J. Bascnagel, K. Binder, P. Doruker, A.A. Gusev, O. Hahn, K. Kremer, W.L. Mattice, F. Muller-Plathe, M. Murat, W. Paul, S. Santos, U.W. Suter, and V. Tries, “Bridging

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[4] [5] [6] [7] [8] [9] [10] [11] [12]

[13] [14]

[15] [16] [17] [18] [19] [20] [21]

[22]

[23] [24] [25]

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the gap between atomistic and coarse-grained models of polymers: status and perspectives,” Adv. Pol. Sci., 152, 41–156, 2000. K. Binder (ed.), Monte Carlo and Molecular Dynamics Simulations in Polymer Science, Oxford University Press, New York, 1995. K. Binder and W. Paul, “Monte Carlo simulations of polymer dynamics: recent advances,” J. Polym. Sci., Part B – Polym. Phys., 35, 1–31, 1997. K. Kremer and F. Muller-Plathe, “Multiscale problems in polymer science: simulation approaches,” MRS Bull., 26, 205–210, 2001. G.S. Grest, M.D. Lacasse, and M. Murat, “Molecular-dynamics simulations of polymer surfaces and interfaces,” MRS Bull., 22, 27–31, 1997. R.D. Groot and P.B. Warren, “Dissipative particle dynamics: bridging the gap between atomistic and mesoscopic simulation,” J. Chem. Phys., 107, 4423–4435, 1997. E. Helfand, “Theory of inhomogeneous polymers – fundamentals of Gaussian random-walk model,” J. Chem. Phys., 62, 999, 1975. M. Doi and S.F. Edwards, The Theory of Polymer Dynamics, Oxford University, Press, New York, 1986. K. Freed, Renormalization Group Theory of Macromolecules, Wiley, New York, 1987. G.G. Batrouni, G.R. Katz, A.S. Kronfeld, G.P. Lepage, B. Svetitsky, and K.G. Wilson, “Langevin simulations of lattice-field theories,” Phys. Rev. D, 32, 2736, 1985. A.D. Kennedy, “The Hybrid Monte Carlo algorithm on parallel computers,” Parallel Comput., 25, 1311, 1999. P. Altevogt, O.A. Evers, J. Fraaije, N.M. Maurits, and B.A.C. van Vlimmeren, “The MesoDyn project: software for mesoscale chemical engineering,” Theochem – J. Mol. Struct., 463, 139–143, 1999. G.H. Fredrickson, V. Ganesan, and F. Drolet, “Field-theoretic computer simulation methods for polymers and complex fluids,” Macromolecules, 35, 16–39, 2002. T.J. Chung, Computational Fluid Dynamics, Cambridge, University Press, Cambridge, 2002. J.-P. Hansen and I.R. McDonald, Theory of Simple Liquids, Academic Press, New York, 1986. M.P. Allen and D.J. Tildesley, Computer Simulation of Liquids, Oxford University, Press, New York, 1987 V. Ganesan and G.H. Fredrickson, “Field-theoretic polymer simulations,” Europhys. Lett., 55, 814–820, 2001. G.H. Fredrickson, “Dynamics and rheology of inhomogeneous polymeric fluids: a complex Langevin approach,” J. Chem. Phys., 117, 6810–6820, 2002. G. Tzeremes, K.O. Rasmussen, T. Lookman, and A. Saxena, “Efficient computation of the structural phase behavior of block copolymers,” Phys. Rev. E., 65, 041806, 2002. S.W. Sides and G.H. Fredrickson, “Parallel algorithm for numerical self-consistent field theory simulations of block copolymer structure,” Polymer, 44, 5859–5866, 2003. D. Duechs, V. Ganesan, F. Schmid, and G.H. Fredrickson, “Fluctuation effects in ternary AB + A + B polymeric emulsions,” Macromolecules, 36, 9237, 2003. C.M. Bender and S.A. Orszag, Advanced Mathematical Methods for Scientists and Engineers, McGraw-Hill, New York, 1978. A.G. Moreira, S.A. Baeurle, and G.H. Fredrickson, “Global stationary phase and the sign problem,” Phys. Rev. Lett., 91, 150201, 2003.

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[26] J.R. Klauder, “Coherent-state Langevin-equations for canonical quantum-systems with applications to the quantized hall effect,” Phys. Rev. A, 29, 2036, 1984. [27] G. Parisi, “On complex probabilities,” Phys. Lett. B, 131, 393, 1983. [28] M.W. Matsen and M. Schick, “Stable and unstable phases of a diblock copolymer melt.” Phys. Revi. Lett., 72, 2660–2663, 1994. [29] M.W. Matsen, “The standard gaussian model for block copolymer melts,” J. Phys. Cond,. Matter, 14, R21–R47, 2002. [30] F.S. Bates and G.H. Fredrickson, “Block copolymer thermodynamics – Theory and Experiment,” Annu. Rev. Phys. Chem., 41, 525–557, 1990. [31] F.S. Bates, “Polymer–polymer phase behavior,” Science, 251, 898–905, 1991. [32] A. Alexander-Katz, A.G. Moreira, and G.H. Fredrickson, “Field-theoretic simulations of confined polymer solutions,” J. Chem. Phys., 118, 9030–9036, 2003. [33] V. Pryamitsyn and V. Ganesan, “Dynamical mean-field theory for inhomogeneous polymeric systems,” J. Chem. Phys., 118, 4345–4348, 2003.

Perspective 1 PROGRESS IN UNIFYING CONDENSED MATTER THEORY Duane C. Wallace Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA

Over the years, experimentalists have uncovered an array of fascinating properties of condensed matter. The initial response of theorists has been to treat each new property as a separate problem to be solved, with the result that condensed matter theory presents the appearance of “a different Hamiltonian for every problem.” But there has always been an undercurrent of work aimed at searching out common ground, and of reconstructing disparate theories from a more universal basis. Ultimately, the same wavefunctions and energy levels should explain all the properties of a given material in a given state. Here we sketch development along this line, and note how it is useful in continuing research. In qualitative terms, the physical nature of condensed matter is well understood. In a single isolated atom, the negatively charged electrons are bound by the Coulomb force to the positively charged nucleus. If the Coulomb force were unopposed, the electron cloud would shrink to the size of the nucleus. But the electron wavefunctions have to be mutually orthogonal solutions of the Shrödinger equation, and since electrons have only two orthogonal spin states, their spatial wavefunctions must oscillate to achieve orthogonality in a many-electron system, and this oscillation produces positive kinetic energy. Hence, the balance between negative Coulomb energy and positive kinetic energy determines the binding energy of the atom. The same principle operates when several atoms are brought together to form a molecule, or when many atoms are brought together to form condensed matter. Here the atomic electron clouds can deform so as to move electron density into regions between nearest-neighbor nuclei, thus lowering the negative Coulomb energy of the condensed system, but this process is opposed by the positive kinetic energy that results from the increased localization of the electrons. Hence the equilibrium configuration of the condensed system, and its binding energy relative to 2659 S. Yip (ed.), Handbook of Materials Modeling, 2659–2661. c 2005 Springer. Printed in the Netherlands.

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free atoms, is again controlled by the balance between Coulomb attraction and kinetic-energy repulsion of the electrons. From this qualitative picture, a broad part of condensed matter theory works out as follows [1]: we can calculate all the equilibrium properties if we know the energy of the electronic groundstate as function of the nuclear positions, and specifically for metals, if we also know the density of excited electronic energy levels. To illustrate the breadth and accuracy of current theory, let us consider the material class composed of metallic elements in the crystalline state, where density functional theory provides the required electronic energy information. By calculating the groundstate energy for various crystal structures, we can account for the observed stable crystal structure, its equilibrium lattice parameters, and its binding energy relative to free atoms. By calculating the increase in groundstate energy when nuclei are displaced from equilibrium, we obtain the elastic constants and phonon frequencies. With these energy levels, and statistical mechanics theory, we understand the separate phonon and electron contributions to thermal properties, such as specific heat and thermal expansion. The theory also reproduces observed crystal–crystal phase transitions, both compression induced transitions, and the theoretically subtle temperature induced transitions. With few exceptions, all these properties can be calculated to an accuracy on the order of 1%, or a few percent, for all the elemental crystals, leaving no doubt that the theory is correct in detail. Extending our view, we see a wide variety of material types and material properties. For insulating crystals, both elemental and polyatomic, density functional theory is still reliable, and from it we can again calculate equilibrium crystal properties to high accuracy. The rare gases are a special case, since their binding is extremely weak and is not accurately given by density functional theory, but once their interatomic forces are constructed, by the appropriate theory, all crystal properties can be calculated to high accuracy. Alloys are polyatomic metals, for which the atomic disorder presents difficulties for electronic structure theory, but these difficulties are currently being overcome. For the liquid state, the nuclear motion potential is given by precisely the same theory as for the crystal, but the problem has always been the complicated shape of this potential surface. The liquid problem is finally being solved, with development of an accurate theory for monatomic liquids, and with help from computer simulations. Similar techniques are being applied to the study of glasses and polymers. Much of modern condensed matter research is devoted to properties more delicate than those mentioned above, such as magnetism, ferroelectricity, and correlated electron effects including superconductivity, with the goal of making qualitative theories more accurate, and this inevitably calls for making theory more fundamental, hence more universal. Finally, we note that nonequilibrium properties can be understood from the same physical nature of condensed matter, but nonequilibrium theory

Progress in unifying condensed matter theory

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is more complicated because it requires a higher resolution of the nuclear and electronic motion. Our discussion leads to an important point about how we do research. Research is, after all, an enterprise in problem solving. The common situation calls for us to answer questions about a specific property of a given material, when the current understanding of this property for this material is insufficient for the purpose. The most advantageous way to proceed is to examine theory at its most fundamental and most universal level, to identify the physical processes which contribute to the property in question, in materials where it is well understood. This does not mean one cannot make approximations, since often they are unavoidable, but it is only by knowing the correct physics that one can make correct approximations. Suppose, for example, one needs to examine effects associated with thermal expansion of a polymer. The physical process involved is the same as for an insulating crystal, but the polymer is much more complicated owing to its greatly reduced symmetry. Consider the polymer nuclei, having equilibrium positions and normal modes of vibration about equilibrium. Under a small increase in the system volume, the equilibrium positions change, with a corresponding increase in the system potential energy, and the vibrational modes also change, with a corresponding decrease in their frequencies. With the decrease in frequencies the nuclei have larger vibrational amplitudes, hence they cover more space, or have more entropy, and the competition between higher potential energy and higher entropy determines the increase in volume which will accompany an increase in temperature. The equations expressing this picture are essentially exact, and the potential energy and frequencies involved are defined through the electronic structure. What makes the problem difficult is what also makes it interesting: we must find the variation with temperature of the entire atomic microstructure of the polymer. This microstructural change will then influence a host of other equilibrium and nonequilibrium properties.

Reference [1] D.C. Wallace, Statistical Physics of Crystals and Liquids, World Scientific, New Jersey.

Perspective 2 THE FUTURE OF SIMULATIONS IN MATERIALS SCIENCE D.P. Landau Center for Simulational Physics, The University of Georgia, Athens, GA 30602

The early part of the 21st century is rapidly developing into the era of man-made “materials by design.” The full realization of this situation requires the requisite understanding of the microscopic origins of diverse phenomena and the subsequent incorporation of this knowledge into the process which leads to the production of new materials. The optimum approach to scientific research now often requires the interplay between theory, experiment, and simulation as shown schematically in Fig. 1 below. Of course, each vertex of this triangular array represents a spectrum of different methods, some of which are more sensitive than others. From the perspective of computer simulations the situation has been brightening rapidly with the passage of time. Nevertheless, even with the dramatic increase in algorithmic sophistication and computer speed that has occurred during the past several decades, the treatment of models with large numbers of atoms is quite difficult. Moreover, because of the need to examine diverse, imperfect systems over wide ranges of temperature and other applied “fields,” it is quite likely that fully classical simulations employing approximate potentials will continue to play an exceedingly important role for a number of years to come. Heretofore, these potentials have been empirical; but because of the limitations of many such potentials, the interplay between quantum studies of smaller systems and the extraction of more fundamentally based and quantitatively accurate effective interactions for classical simulations will become increasingly important. We should thus expect to see electronic structure methods used to provide information that is used to help parameterize interactions that are then used as input to molecular dynamics and Monte Carlo simulations. Certainly, with the increased emphasis on systems at the nanoscale, there will be a class of simulations for which it will be possible to fully examine models containing a number of atoms that approximates that, which is present in the physical system under consideration. In other cases, it will be necessary 2663 S. Yip (ed.), Handbook of Materials Modeling, 2663–2666. c 2005 Springer. Printed in the Netherlands.

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Simulation

Nature

Theory

Experiment

Figure 1. Schematic representation of modern approach to scientific research in physics/ materials science.

to develop multiscale methods that will span length and time scales within a single coherent effort. As an example of the need for information over wide ranges of length and time scales, in Fig. 2 we portray the approximate length and times that are important for the study of fracture along with methods that are suitable to each range [1]. While this picture applies to one specific class of problems in material science, there are certainly many others for which the same degree of complexity exists. One pronounced feature of recent developments in computer simulations in statistical physics has been that new algorithms have succeeded in pushing back the frontier of accessible time scales quite substantially. This allowed advances to be made near phase transitions where very high resolution has been needed. Initially most of these new approaches have been applied to simple systems with discrete variables, but later methods were devised to allow them to be applied to models with continuous degrees of freedom. As an example, we can consider studies of the phase transitions in magnetic systems such as the Ising or XY-models. The combination of new “cluster flipping” simulation algorithms [2, 3] and new histogram reweighting analysis techniques [4] has permitted the extraction of extremely accurate values of critical temperatures (relative errors in the 6th significant digit) and critical exponents [5–7]. This lesson should be remembered when more complex models of real materials are being investigated, and brute force alone should not be relied on.

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Figure 2. Schematic view of the multiscale nature of fracture and the approaches that apply in different regions of length and time (from Ref. [1]).

A completely different problem centered about the study of crack propagation in silicon. Here, processes are occurring on quite disparate length and time scales. In a pioneering study Broughton et al. [8] used a tight binding method to examine the immediate region of the crack tip, classical molecular dynamics to simulate the region around the crack (excluding the crack tip), and a finite element method to treat the system at scales that are large compared to atomic dimensions. The “handshaking” between the different regions was an essential part of the calculation to allow the passage of energy between regions in a continuous fashion, and the entire algorithm was coded for a parallel platform. Although the implantation was, for an “ideal” problem of brittle cleavage, it was an early indication of the manner in which significant problems in materials science will be treated in the future. Although the algorithms, in particular the “handshaking” techniques, will require further developments, the authors aptly describe the study as the “beginnings of computational atomistic engineering.”

References [1] D.P. Landau, F.F. Abraham, G.G. Batrouni, J.M. Carlson, J.R. Chelikowsky, D.D. Koelling, S.G. Louie, C. Mailhiot, A.C. Switendick, P.R. Taylor, A.R. Williams, and

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[2] [3] [4] [5] [6] [7]

[8]

D.P. Landau B.L. Holian, Computational and Theoretical Techniques for Materials Science, NRL Strategic Series, National Academy Press, Washington, DC, 1995. R.H. Swendsen and J.-S. Wang, “Nonuniversal critical dynamics in Monte Carlo simulations,” Phys. Rev. Lett., 58, 86, 1987. U. Wolff, “Collective Monte Carlo updating for spin systems,” Phys. Rev. Lett., 62, 361, 1989. A.M. Ferrenberg and R.H. Swendsen, “New Monte Carlo technique for studying phase transitions,” Phys. Rev. Lett., 61, 2635, 1988. A.M. Ferrenberg and D.P. Landau, “Critical behaviour of the three dimensional Ising model: a high resolution Monte Carlo study,” Phys. Rev. B, 44, 5081, 1991. H.W.J. Blöte, E. Luijten, and J.R. Heringa, “Ising universality in three dimensions,” J. Phys. A, 28, 6289, 1995. K. Chen, A.M. Ferrenberg, and D.P. Landau, “Static behavior of three dimensional classical Heisenberg models: a high resolution Monte Carlo study,” Phys. Rev. B, 48, 239, 1993. J.Q. Broughton, F.F. Abraham, N. Bernstein, and E. Kaxiras, “Concurrent coupling with length scales,” Phys. Rev. B, 60, 2391, 1999.

Perspective 3 MATERIALS BY DESIGN Gregory B. Olson Department of Materials Science and Engineering, Northwestern University, Evanston, IL

As a new millennium unfolds, a Science Age of three centuries draws to a close, replaced by a Technology Age based not in scientific discovery but in a revolution in engineering design led by U.S. industry. The resulting New Economy, which we now strive to sustain, is based in technology not found in a laboratory, but deliberately created from the human mind in response to perceived needs. While we tend to be nostalgic about exploration ages, at this point in history humankind not only enjoys an unprecedented ability to create wealth from thought, but holds all the tools for a much-needed transformation from mere technology to responsible technology. At the strategic level, this revolution is founded in new systems-based design methodologies that accelerate the total product development cycle while achieving new levels of product reliability. At the tactical level, it integrates a new understanding of human team creativity with new opportunities in information technology to create powerful computational tools tailored to strategic needs. Meanwhile, our “modern” research universities we inherited from the Cold War have continued to train explorers for a bygone era. Dominated by a culture of reductionist analysis, university engineering has for the past half century been replaced by engineering science, leaving industry on its own to advance engineering practice, and leaving the teaching of modern engineering to business schools and corporate universities. While this cultural incongruity has affected all fields of engineering, no field has suffered more damage than the materials profession. Left on its own, the industrial practice of materials development has languished in a slow and costly empirical discovery process that cannot keep pace with the compressed product development cycle, leaving no chance for participation in concurrent engineering. Despite a widespread deeply-held goal of scientific engineering from within the community, the academic materials enterprise has been diverted under external forces through 2667 S. Yip (ed.), Handbook of Materials Modeling, 2667–2669. c 2005 Springer. Printed in the Netherlands.

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funding policies toward reductionism and the pursuit of novelty, leading to highly dissipative random-walk exploration that yields much paper but no materials. Against this background, the multi-institutional steel research group (SRG) was founded in 1985 to build within a modern systems engineering framework the methods, tools and databases to support the rapid computational design of materials, using high-performance steels as a first example. Treating materials as dynamic multilevel-structured systems, integration of process/structure/property/performance relations has generated a hierarchy of design models. The methods, tools and models, and their successful application in the thermodynamics-based parametric design of new alloys, are described in detail elsewhere [1–3]. Transfer of this technology to the commercial sector has been led by the university spinoff company QuesTek Innovations, commercializing both the design technology and the first cyber-materials emerging from it, and a range of small businesses now furnish software tools and supporting databases as surveyed in a recent National Academy study [4]. Credibility of computational materials design based on the SRG/QuesTek success has helped to bring about major initiatives supported by the Defense Advanced Research Projects Agency (DARPA) and other DoD agencies that are facilitating a much needed transformation of the materials profession. Of particular note is the recently completed 3-year DARPA-AIM initiative in Accelerated Insertion of Materials, expanding the scope of computational materials design to accelerate the full development and qualification cycle. With both structural composites and aeroturbine disc superalloys as motivating use cases, a combination of integrated high-fidelity simulation and focused strategic testing demonstrated both accelerated process optimization at the component level, and the efficient prediction of part-to-part property variation in manufacturing for accurate forecast of minimum design allowables. A particularly historic achievement was the demonstration of enhanced performance in a subscale turbine disc for which the disc designer was enabled to continuously vary materials processing in component design. The industry-led AIM initiative also set a new standard for extracting useful work from academic research. This experience in turn has had a significant positive impact on DoD universitycentered programs, notably including the ongoing ONR Grand Challenge in Naval Materials by Design and the AFOSR-MEANS (Materials Engineering for Affordable New Systems) initiatives. Addressing the early stages of computational materials design, these initiatives have notably integrated emerging quantum engineering tools for both the accelerated assessment of fundamental databases and the predictive control of crucial interfacial phenomena. This significant investment in modern materials engineering has enabled a new learning environment on our campuses that has gone beyond traditional graduate education to invigorate undergraduate design, dominating the TMS national undergraduate design competition, and allowing upper-level materials students

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in multidisciplinary projects to participate in a new form of concurrent materials engineering that did not exist 3 years ago. The new AIM paradigm of industry-led university engineering projects supported by mission-driven agencies promises a new rationalized research infrastructure meeting the needs of our times, where science toys can be intelligently fashioned into purposeful tools of engineering optimization, enabling materials modeling and simulation to reach their full potential for deliberate value creation. If indeed it ushers a new era where the physical sciences support engineering as well as the life sciences have supported medicine, we can expect to be as successful and as valued by society as we bring materials into this new Design Age.

References [1] G.B. Olson, “Science of steel,” In: G.B. Olson, M. Azrin, and E.S. Wright (eds.), Innovations in Ultrahigh-Strength Steel Technology, Sagamore Army Materials Research Conference Proceedings: 34th, 3–66, 1990. [2] G.B. Olson, “Computational design of hierarchically structured materials,” Science, 277(5330), 1237–1242, 1997. [3] G.B. Olson, “Designing a new material world,” Science, vol. 288, 12 May, 993–998, 2000. [4] National Research Council Report, Accelerating Technology Transition, The National Academies Press, Washington D.C., 2004.

Perspective 4 MODELING AT THE SPEED OF LIGHT J.D. Joannopoulos Francis Wright Davis Professor of Physics, Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA

For over half a century, semiconductor physics has played a vital role in almost every aspect of modern technology. Advances in this field have allowed scientists to adapt the conducting properties of materials and have led to the transistor revolution in electronics. New research suggests that we may now be able to initiate a similar revolution by adapting the properties of light. The key in achieving this goal lies in the use of a new class of materials called photonic crystals [1–3]. The basic idea is to design materials that can affect the properties of photonmodes (or just photons, for brevity) in much the same way that ordinary semiconductor crystals affect the properties of electrons. That this is feasible, becomes clear if one considers that Maxwell’s equations for linear materials in the frequency-domain, and in the absence of external currents and sources, can be cast in a form that is reminiscent of the Schroedinger equation, namely

1 ω2 ∇× H(r) = 2 H(r), (1) ε(r) c where H(r) is the magnetic field subject to the transversality constraint ∇ · H(r) = 0. Equation (1) represents a linear Hermitian eigenvalue problem whose solutions are determined entirely by the properties of the macroscopic dielectric function ε(r) . Therefore, if one were to construct a crystal consisting of a periodic array of macroscopic metallic or uniform-dielectric “atoms”, the photons in this crystal could be described in terms of a bandstructure, as in the case of electrons. Of particular interest is a photonic crystal whose bandstructure possesses a complete photonic band gap (PBG). A PBG defines a range of frequencies for which light is forbidden to exist inside the crystal, regardless of its direction of propagation. Forbidden, that is, unless there is a defect in the otherwise perfect crystal. A defect could lead to one or more localized photon states in ∇×

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the gap, whose shapes and properties would be dictated by the nature of the defect. A point defect could act like a microcavity to confine light and a line defect could act as a linear waveguide to guide light. Thus, deliberately designed structural defects are good things in photonic crystals, providing a new mechanism for molding and controlling the properties of light. Therein lies the exciting potential of photonic crystals. Moreover, given a new mechanism one might expect to enable photon phenomena that have never been possible before. Recent examples include theoretical predictions of negative refraction [4], novel Cerenkov radiation [5], reversed Doppler shifts [5] and anomalous dispersion relations [6]. A very significant and attractive difference between photonic crystals and electronic semiconductor crystals is the former’s inherent ability to provide complete tunability. A defect in a photonic crystal could, in principle, be designed to be of any size, shape or form and could be chosen to have any of a wide variety of dielectric constants. Thus, defect states in the gap could be tuned to any frequency and spatial extent of design interest. But, in addition to tuning the frequency, one also has some control over the symmetry of the localized photon state. For example, the very specific symmetry associated with each photon mode in a photonic crystal microcavity [3] translates into an effective orbital angular momentum for each photon, which can exist in addition to the intrinsic spin angular momentum [8]. This is a very intriguing notion that can have spectacular consequences in the selection rules of electronic transition rates in quantum well or quantum well structures. In addition to changing the symmetry of a photon state, one can also change the density of states in order to affect spontaneous emission. The rate of spontaneous emission of a given initial state is proportional to the density of final photon states available at the transition frequency. Hence, by operating within the gap of a finite photonic crystal and thus eliminating nearly all photon modes at the transition frequency, the emission rate will be greatly reduced. Conversely, by operating at a point-defect resonance, the emission rate could be dramatically enhanced due to the large increase in the density of final states. In this way one could improve the efficiency of current lasers, or even design lasers that operate in frequency ranges yet unachieved. The similarity of Maxwell’s equation (1) to Schroedinger’s equation implies that computational techniques used to study electrons in solids, such as the conjugate gradients approach, may also be used to study photon modes in photonic crystals [9–11]. The main differences are that electrons are described by a complex scalar field and strongly interact with each other whereas the photons are described by a real vector field and do not interact with each other. Solution of the photon-equations is thus only a single-particle problem and leads, to all intents and purposes, to an “exact” description of their properties.

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To solve Maxwell’s equations for periodic dielectric media in the frequencydomain one typically begins by expanding the fields in plane waves. As in the case of electrons, the use of a plane wave basis set has a number of extremely desirable consequences. First, the transversality constraint is easily satisfied. Second, the set is complete and orthonormal. Third, finite sets can be systematically improved in a straightforward manner. Finally, a priori knowledge of the field distribution is not required for the generation of the set. The chief difficulty, however, in using plane waves is that huge numbers of plane waves are typically required in order to describe the sudden changes in dielectric constant inherent in a photonic crystal structure. This problem can easily be overcome by a better treatment of the boundaries between the dielectric media. In particular, construction of a dielectric tensor [10] to interpolate in the boundary regimes allows the proper screening of photons with different polarization and leads to a rapid convergence of all eigenmodes by over an order of magnitude as compared to a scalar dielectric. A variational functional can then be constructed from Eq. (1) whose iterative minimization leads to the required stationary solutions [3]. To solve Maxwell’s equations for periodic dielectric media in the timedomain, one typically employs Yee-lattice finite difference time domain (FDTD) methods [12, 13] that can include periodic as well as absorbing boundary conditions. Such computations are extremely useful for making direct comparisons with experiments by modeling the experimental setup and measurement process. Moreover, complex frequency dependent dielectric functions, as well as non-linear response, can be straightforwardly implemented in this approach. Future computational work will involve enabling the treatment of photons together with electrons and phonons within a single framework. This capability will be critical for enabling future numerical studies of photonic crystals where material gain, acousto-optic response, or thermal response are of interest. In conclusion, theory and computation play a particularly important role in the field of photonic crystals. As opposed to electrons, where the exchange– correlation functional is only known approximately, photons in linear material systems do not interact. And in non-linear materials systems this interaction is well known. Thus, Maxwell’s equations can be solved numerically to any desired degree of precision or accuracy. This represents one of the few cases in science where numerical experiments can be as accurate as laboratory experiments! Future directions of research in this field will involve the design and realization of novel microdevices and device-components whose sizes are on the order of the wavelength of light of interest. This capability would make it possible, eventually, to integrate a large number and variety of optical devices on a single chip just as is now done for electronic devices. Moreover, the search for unusual or anomalous photonic phenomena associated with photonic crystal systems will continue both theoretically and

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experimentally, with ultimate goal the identification of novel and useful photon functionalities.

References [1] E. Yablonivitch, Phys. Rev. Lett., 58, 2509, 1987. [2] S. John, Phys. Rev. Lett., 58, 2486, 1987. [3] J.D. Joannopoulos, R.D. Meade, and J.N. Winn, Photonic Crystals, Princeton, New York, 1995. [4] C. Luo, S. Johnson, and J.D. Joannopoulos, Appl. Phys. Lett., 81, 2352, 2002. [5] C. Luo, M. Ibanescu, S. Johnson, and J.D. Joannopoulos, Science, 299, 368, 2003. [6] E. Reed, M. Soljacic, and J.D. Joannopoulos, Phys. Rev. Lett., 91, 133901, 2003. [7] M. Ibanescu, S.G. Johnson, D. Roundy, C. Luo, Y. Fink, and J.D. Joannopoulos, Phys. Rev. Lett., 92, 063903, 2004. [8] J.D. Joannopoulos, P.R. Villeneuve, and S. Fan, Nature, 386, 143, 1997. [9] K. Ho, C. Chan, and C. Soukoulis, Phys. Rev. Lett., 65, 3125, 1990. [10] R. Meade, K. Brommer, A. Rappe, and J. Joannopoulos, Phys. Rev. Rapid Comm. B, 44, 13772, 1991; Erratum: Phys. Rev. B, 55, 15942, 1997. [11] S. Johnson and J.D. Joannopoulos, Opt. Express, 8, 173, 2001. [12] K.S. Yee, IEEE Trans. Ant. Prop. AP, 14, 302, 1966. [13] J.P. Berenger, J. Comput. Phys., 114, 185, 1994.

Perspective 5 MODELING SOFT MATTER Kurt Kremer MPI for Polymer Research, 55021 Mainz, Germany

Soft matter science or soft materials science is a relatively new term for the science of a huge class of rather different materials such as colloids, polymers (of synthetic or biological origin), membranes, complex molecular assemblies, complex fluids, etc. and combinations thereof. While many of these systems are contained in or are even the essential part of everyday products (“simple” plastics such as yoghurt cups, plastic bags, CDs, many car parts; gels and networks such as rubber, many low fat foods, “gummi” bears; colloidal systems such as milk, mayonnaise, paints, almost all cosmetics or body care products, the border lines between the different applications and systems are of course not sharp) or as biological molecules or assemblies (DNA, proteins, membranes and cytoskeleton, etc.) are central to our existence, others are basic ingredients of current and future high tech products (polymers with specific optical or electronic properties, conducting macromolecules, functional materials). Though the motivation is different in life science rather than in materials science biomolecular simulations, the basic structure of the problems faced in the two fields is very similar. Often combinations of the above-mentioned materials are employed both in actual research as well as technology. Though rather different from the beginning and thus asking for rather different modeling tools, there is one unifying aspect, which makes it very reasonable to treat such systems from a common point of view. Compared to “hard matter” the characteristic energy density is much smaller. While the typical energy of a chemical bond (C–C bond) is about 3 × 10−19 J ≈ 80 kB T the non-bonded interactions are of the order of kB T and allow for the strong fluctuations even though the molecular connectivity is never affected. kB is Boltzmann’s constant and T the temperature. Two typical examples might illustrate this in comparison to a prototypical conventional crystal. To give a very rough and simple estimate one can compare a typical crystal to soft matter. Taking 10 nearest neighbors (8 for bcc, 12 for fcc/hcp) in a crystal and an interaction energy between 1 . . . 2 kB T 2675 S. Yip (ed.), Handbook of Materials Modeling, 2675–2686. c 2005 Springer. Printed in the Netherlands.

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one reaches a typical energy density of a crystal made up of atoms of about E/ V ≈ 5–10 kB T /Å3 . Comparing this to a typical polymer melt, where the strand–strand distance is about 3–6 Å and a chain typically has of the order of six neighbors we arrive at (depending on the persistence length) not more than 0.3–0.1 kB T /Å3 . Thus, since to a first approximation the energy density corresponds to the elastic constants, polymeric systems are at least 10–100 times softer than classical crystals. For typical colloidal crystals, the situation is even much more dramatic due to the typical size of the unit cells (often around 103 Å) leading to a factor of up to 109 compared to conventional crystals. This is the reason why many colloidal systems can be “shear melted” just by turning the containers by hand. As a consequence the thermal energy kB T is not a “small energy” for the systems any more, but rather defines the essential energy scale. This means that entropy, which typically is of the order of kB T per degree of freedom, plays a crucial role. Especially in the case of macromolecules, this mainly means intramolecular entropy, which for a linear polymer of length N is of order kB T 0(N ). As an immediate consequence it is clear that typical quantum chemical approaches cannot be sufficient to characterize a material and even be less sufficient to properly predict/interpret macroscopic properties. How these different contributions influence each other can most easily be seen in Fig. 1. Any study of electronically excited states, reactions, etc. in principle requires quantum mechanical calculations. Also specific interactions might require this approach. On a next level typically all atom or united atom force field simulations are performed. It should, however, be kept in mind that because of the many hydrogen atoms in typical soft matter systems solving classical equations of motion might cause severe problems. There is no universal force field. In both regimes the energies of local bond lengths, bond angles and torsions dominate the properties. Special difficulties here pose non-bonded interactions as will be discussed below. If the view is more coarsened the detailed atomistic picture is replaced by a more or less flexible path in space in the case of polymers. On that level the many possible conformations, i.e., the resulting intra-chain entropy, govern the overall conformation of the chains. The rather delicate interplay of entropic and energetic contributions give raise to the huge variety of properties one encounters in soft matter. Any modeling attempt has to keep this in mind. How this affects the properties can be seen in two rather simple cases. Need less to say that in most experimental cases the situation is more complicated. The first example is the miscibility of different polymers in a melt. Figure 2 illustrates this in a cartoon like manner. Consider a mixture of two different polymers of type A and B. In a homogenous mixture the average number of AA, BB as well AB nearest neighbor contacts is of order 0(N ) as is the intra-chain entropy. To a first approximation the intra-chain entropy does not change in a pure A or a pure B melt. Thus, just as in a mixture of small molecules an effective interaction energy

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Figure 1. Sketch of the different time and length scales in a typical soft matter system such as polymers.

of order E ≈ 0 (kB T ) per molecule, which is a polymer in our case, is sufficient to drive the phase separation. For polymers this means E ∼ N (εAB – 0.5 (εAA + εBB )), where the ε denote the specific interactions in a lattice model for example, due to the macromolecular structure of the molecules. Compared to small molecules the driving pair wise nearest neighbor energy difference at the phase separation point is proportional to N −1 , which eventually vanishes for larger chains. This is generic and independent of the systems, while the actual phase separation temperature (the “prefactor”) for a given N depends on chemical details. This generic behavior was impressively demonstrated by computer simulations on lattice models and by experiments studying the phase separation of protonated and partially deuterated polystyrene [1, 2]. The prediction of the actual value of critical interaction certainly is a challenge, which at least in the limiting case of large N will remain a challenge. In a similar way dynamical properties are governed by a combination of generic aspects originating from the connectivity and the non-crossability of the chains as well as local interaction parameters, which determine the bead friction and the local packing. For polymer chains exceeding the entanglement molecular weight the reptation model, which roughly assumes a motion

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Figure 2. Cartoon of the origin of the very poor miscibility of different polymers.

of the polymer beads along the coarse-grained contour of the chains (“reptating chains”) holds. Within this concept one can write for the melt viscosity η = AN 3.4 , N again being the chain length and A the prefactor where everything is included which is difficult to determine and which is (to a first order) independent of N . Now the viscosity can be modified by orders of magnitude in two ways, varying N or A. E.g., changing N by a factor of two changes η by about factor of 23.4 ≈ 10. In a similar fashion one can vary A by slightly changing the chemical structure or simply changing the temperature. For BPA-Polycarbonate, the classical CD material, a shift in the process temperature from 500 K down to 470 K also increases η by roughly a factor of 10 (the glass transition temperature for BPA-PC is around TG ≈ 420 K). Again one encounters two equally important ways to manipulate the system properties, one based on universal aspects and the other based on local details. These were two almost trivial examples, which illustrate the interplay of contributions originating from different length scales. Of course, the above discussion was over simplified. Actual theoretical, experimental, or technical problems often do not allow for such a well-defined separation of scales. On the other hand in many cases investigations on one level of description have been and still are of high relevance and are still pursued at a high level. Thus they are a very active research topic in their own right. In the following I would like to shortly discuss a few examples illustrating the different directions.

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The attempts to understand soft matter systems theoretically follow a long tradition. Beginning with the early work of Flory simplified models were studied, which were able to explain many generic/universal properties but failed to provide a solid basis for a theoretical understanding. It was then up to the seminal works by Edwards and deGennes to provide a link between statistical mechanics of phase transitions (critical phenomena) and polymer chain conformations. This link to modern concepts of theoretical physics not only provided a huge momentum to the field but also marked a starting point for statistical mechanics computer simulations applied to soft matter problems on a larger scale. Already for the simplest problems, computer simulations play a crucial role, i.e., the problem of an isolated self-avoiding walk cannot be solved exactly in three dimensions and until today the best data result from very extensive computer simulations. In a similar fashion basic features, such as the non-crossability of the chains are hard to deal with analytically and can only be included properly within a simulation approach. In this context, highly optimized and highly simplified models were and are still employed very extensively, and contributed significantly to our present knowledge. Typical other examples are phase separation studies of polymer mixtures [1], wetting phenomena in polymers, generic aspects of the glass transition of polymeric systems, dynamics of long and short chain melts [3, 4], hydrodynamic interactions in polymer solutions [5], and increasingly often many component complex fluids. These are just a few examples out of the huge literature, which occurred over the last 30 years. Increasing computer power, but especially ever improved and optimized models and algorithms allowed for this success and are still central to many research projects. For melts typical current examples are the dynamics of polymeric melts under shear or mixtures of linear polymers and polymers including branches [3]. To study such systems on a simple bead spring model level (the chains are represented by a string of beads and springs) is a challenging problem for modern super computers and simultaneously of highest technical relevance. Figure 3 shows an example of the determination of the backbone of the reptation tube in a melt of linear polymers. Other fields concern the coupling of the polymeric degrees of freedom to the hydrodynamic interaction in a solution [5] or related questions for colloidal systems. In both cases very recently developed algorithms allow for a first glance at such problems. From a statistical mechanics point of view the investigation of multi-component systems is just at the beginning. A typical example is the mixture of polystyrene and carbon dioxide which turns out to give rise to a very complicated phase diagram [6]. This is a special polymer solvent system, where the solvent itself is not as usual in such studies an inert species providing a background for a polymeric solute, but rather undergoes a liquid gas transition itself. The most prominent example, the above mentioned CO2 and polystyrene, is used to produce Styrofoam. For systems of this kind still many qualitative aspects of the phase diagrams are not known and before

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Figure 3. Original polymer melts (left) and network of the backbone of the tube of the red chain with the chains causing the confinement. All other chains are shown as very thin lines only [4]. From this the melt plateau modulus can be determined.

going into too many specific details they have to be established. On a more detailed level, namely force field simulations employing models for polymers, membranes or even proteins, all atoms are treated explicitly and Newton’s equations of motion are solved numerically. The problems in determining such a force field have already been mentioned before. But even then such calculations can only provide insight on a very short time scale or for rather small systems. Equilibrating such systems and deducing quantitative results for a macroscopic quantity still is a severe problem [7, 8] (see also the contribution by D.N. Theodorou). On the other hand it is even questionable whether runs on very long time scales, i.e., the time scale on which a long polymer chain would move its own diameter in space, on such a level would be very useful, because the enormous amount of data has to analyzed and structured. Therefore, it is useful to study such problems on a lower level of detail. Another field, which nowadays still is mostly confined to rather coarse-grained and simple models are polyelectrolytes in solution, due to the long range nature of the interaction combined with a typically slow relaxation [9]. While these individual studies provide important insight in either atomistic details or generic aspects such a separation quite often is not possible or a link between different scales within a hierarchical simulation scheme is needed. This is the center part of what is nowadays called “multi-scale” or “scale bridging” modeling. For many future applications and especially for a close link to experiment it is absolutely crucial to establish this bridge. To illustrate this I would again like to mention two examples. If one wants to study the statics and dynamics of a long chain polymer melt a highly simplified model is needed in order to reach the necessary time scales for the diffusion of the chains or for stress relaxation (cf. Fig. 3). This even nowadays requires highly optimized programs and top of the line super computers.

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However, to treat a specific system enough details of the atomistic structure have to be carried along. One way would be to run two separate studies, one all atom simulation for very short times to determine the time scaling from small length scale mean square displacements and one simulation of a more coarse-grained model. This implies to link these to sets of data, which in that case is quite difficult as both a time and a length scale mapping are needed. This is the traditional way which still dominates the literature but also poses some conceptual problems. An alternative way is to start from an all atom description and derive from that directly a coarse-grained model, which is efficient enough to study long time dynamics. Since this fixes the length scaling, the time mapping by comparing runs from the micro and meso level becomes unique and truly quantitative time dependent “measurement” becomes possible. This approach has another advantage, if one is interested in the atomistic structure of melts. By employing an inverse mapping back to the atomistic model one can actually use the simulation on the coarse-grained level to efficiently equilibrate all atom models of huge size. The strategy of this way of performing a hierarchical or multiscale modeling is illustrated in Fig. 4. However, in some cases this sequential ansatz of working on different levels is not sufficient. Consider the situation of selective adsorption of parts of a macromolecule at a surface. Due to the constraints coming from the connectivity within the molecule and the structure of the surface it can only bind to the surface in a rather specific configuration. This requires a significant amount

Figure 4. Cartoon like illustration of the hierarchical modeling scheme.

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of detail for this problem. On the other hand, the whole rest of the molecule, located in the polymer matrix, where it is not in contact to the surface, has to be equilibrated as well. For the latter more coarse-grained models are sufficient and also needed due to the CPU time requirements. Such a problem (e.g., polycarbonate at a nickel surface) led us to develop a dual scale simulation where the level of description along the backbone varied [10, 11]. In that case the bond angles close to the chain ends had to be considered within a simple bead spring model with atomistic resolution. The outcome of such a study as well as the effect of different chain ends on the morphology of short chain melts are illustrated in Fig. 5. Typical for many problems is that regions in space or periods of time where higher resolution is needed can be identified. Often these regions can change significantly with time. For not too large systems it might be advantageous to switch completely between two levels of representation (e.g., fully atomistic vs. fully coarse-grained) depending on some intrinsic system parameters. For this reason the above mentioned dual–resolution approach (where two levels of description are applied simultaneously at different parts of a given system) and uniform-resolution approaches (where either one or the other level is used to describe the entire system, given a suitable mapping/back – mapping procedure) are being developed. Alternatively, when there is no exchange of

Figure 5. Illustration of the multiscale modeling approach for specific surface interactions of a polymer with a surface [10, 11]. Depending on the specific interactions, in that case of the (non) phenolic chain ends with a Ni surface, the melt morphology of short chains close to the surface can be significantly different.

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particles/atoms regions of space with different resolutions are studied. An example for this is the so-called “QM/MM” or quantum/classical hybrid approach, which is successfully applied in biochemistry and catalysis research. A part of the system is treated by electronic structure methods while being coupled to a force field which represents the environment (e.g., in a protein) at the level of classical mechanics [12]. At the classical atomistic (microscopic) level the starting point usually is a parameterized force field mostly based on experimental input. Different classical multi-level and coarse graining techniques are currently developed within the realm of materials science/statistical physics. Still an open issue is to connect the quantum level via the microscopic up to the mesoscopic description quite independently from the specific problem at hand. Though this might be already complicated, this for many important questions is not sufficient. An even deeper challenge is the development of adaptive schemes, which allow for an adjustment of the description locally in space and/or in time as required, thus coupling different levels of description in a dynamical manner. Many important questions in materials science, soft matter and biophysical chemistry require an adaptive multi-scale approach for a deep overall understanding. Typical problems, which would benefit significantly from such a development, are, how does the atomistic structure of a functionalized polymer grafted onto a surface affect the materials properties of a composite system, how does the topological structure of a (block co-)polymer melt affect the overall rheological properties, how does the atomistic structure of photoactive molecules affect the optical properties of a cross–linked film of such chromophores or how do concentration fluctuations and chain conformations of liquids with polymer additives influence the turbulent drag reduction by such additives? In particular, local electronic properties are often crucially linked dynamically to global conformational properties. These are just a few typical challenges for modeling of soft matter. The above examples illustrate how phenomena on different length and time scales are linked to each other. In this context modern experimental work poses another additional challenge as well as opportunity. The characteristic size of experimental systems is constantly downsizing. Often structures and assemblies on the nanometer or micrometer scale are studied. Simultaneously simulations become more and more efficient due to both, hardware and software development. Eventually both will meet. In this regime it is essential that simulations will be able to tackle whole systems and not small parts, where one tries to eliminate surface effects by periodic boundary conditions. Such systems can be characterized in many ways. However, one important characteristic is that the distinction between bulk and surface contributions to the free energy does not make any sense anymore, they are both (if separable) of the same order. Since such systems will still contain many thousands of atoms

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and can be of high structural complexity (e.g., ion conducting columns of dendritic molecules, decorated micelles, etc.) a hierarchical modeling scheme as discussed above will be essential. If at hand and versatile enough such a development could open up a path to a new and improved interaction with experiments in nanotechnology, physical chemistry as well as biophysics. However, there is still a long way to go. As this book shows, many of the questions/problems discussed are subject to very active research in many laboratories throughout the world. In the soft matter field we still face many challenges, which will keep researchers busy for many years. Here I name just a few, which certainly reflect my own experience, but also my own taste of important problems: – Thorough studies of generic phase diagrams of many component soft matter systems (solvent–polymers–colloids, etc.) – Dynamics (equilibrium AND non-equilibrium) of complex branched systems and mixtures of linear and branched polymers in melt and solution – Multi-scale simulations of ordered structures such as hierarchical assemblies or proteins – Adaptive simulation schemes for a wide class of systems, including the quantum level simulations. – Simulations of large systems of charged macromolecules, statics and dynamics, with explicit ions taking dielectric contrast explicitly into account. These are just a few examples and this list could easily be extended significantly. In conclusion, the development of adaptive multilevel simulation methods is an, if not the most important challenge in multiscale modeling of complex systems, which is shared by such diverse fields as materials, life sciences, and beyond. Thus, adaptive multilevel techniques should connect the atomistic description based on quantum mechanics via the microscopic to the mesoscopic level and eventually include also the macroscopic view.

References [1] [2] [3] [4] [5] [6]

H.P. Deutsch and K. Binder, J. Phys. (France) II, 3, 1049, 1993. D.J. Londono et al., Macromolecules, 27, 2864, 1994. T.C.B. McLeish, Adv. Phys., 5, 1379, 2002. R. Everaers et al., Science, 303, 823, 2004. P. Ahlrichs, R. Everaers, and B. D¨unweg, Phys. Rev. E, 64, 040501, (R), 2001. K. Binder et al., Polymer + solvent systems: phase diagrams, interface free energies, and nucleation, In: C. Holm and K. Kremer (eds.), 2004.

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[7] D.R. Heine, G.S. Grest, and J.G. Curro, “Structure of polymer melts and blends: comparison of integral Equation theory and computer simulation,” In: C. Holm and K. Kremer (eds.), 2004. [8] D.N. Theodorou, Mol. Phys., 102, 147, 2004. [9] C. Holm et al., “Polyelectrolytes with defined molecular architecture II,” In: M. Schmidt (ed.), Adv. Pol. Sci., vol. 166, p. 67, 2004. [10] C.F. Abrams, L. Delle Site, and K. Kremer, Phys. Rev., E67, 021807, 2003. [11] L. Delle Site, S. Leon, and K. Kremer, JACS, 126, 2944, 2004. [12] U.F. R¨ohrig, I. Frank, J. Hutter, A. Laio, J. VandeVondele, and U. Rothlisberger, Chem. Phys. Chem., 4, 1177, 2003.

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Suggested General Reading: M. Doi and S.F. Edwards, The Theory of Polymer Dynamics, Clarendon Press, Oxford, 1986. P.G. deGennes, Scaling Concepts in Polymer Physics, Cornell Univ. Press, Ithaca, NY, 1979. Computational Soft Matter: From Synthetic Polymers to Proteins, N. Attig, K. Binder, H. Grubm¨uller, V.K. Kremer (eds.), NIC Series 23 J¨ulich 2004. D. Frenkel and B. Smit, Understanding Molecular Simulations: From Algorithms to Applications, Academic Press, San Diego, 2002. J. Baschnagel et al., Bridging the Gap between Atomistic and Coarse Grained Models of Polymers: Status and Perspectives, Adv. Pol. Sc., 152, 2000. K. Kremer and F. M¨uller-Plathe, Multiscale Problems in Polymer Science: Simulation Approaches, MRS Bulletin, 26, 205, 2001. C.F. Abrams, L. Delle Site, and K. Kremer, in: Bridging Time Scales: Molecular Simulations for the Next Decade (P. Nielaba, M. Mareschal, and G. Ciccotti (eds.)), Multiscale Computer Simulations for Polymeric Materials in Bulk and Near Surface, Proceedings Simu Conference, Konstanz, August 2002, 143 (Springer, Berlin - Heidelberg, 2002). K. Kremer, Multiscale Aspects of Polymer Simulations, in: Multiscale Modelling and Simulation, Lecture Notes in Computational Science and Engineering, S. Attinger and P. Koumoutsakos (eds.), Springer Verlag, 2004. C. Holm, K. Kremer (eds.), Advanced Computer Simulation Approaches for Soft Matter Sciences I Adv. Pol. Sc. 179, 2004.

Perspective 6 DROWNING IN DATA – A VIEWPOINT ON STRATEGIES FOR DOING SCIENCE WITH SIMULATIONS Dierk Raabe Max-Planck-Institut für Eisenforschung, Düsseldorf, Germany

1.

Introduction

Computational materials scientists are nowadays capable of producing an enormous wealth of simulation data. When analyzing such predictions the challenge often consists in extracting meaningful observations from them, and, wherever possible, to discover general and representative principles behind the often-huge data sets. Only the capability of condensing large data sets into the discovery of new microstructure principles renders materials simulations into computational materials science. This chapter is devoted to this topic. The following sections present some strategies for filtering new observations from materials simulations.

2.

Microstructure or the Hunt for Mechanisms

While the evolutionary direction of microstructure is prescribed by thermodynamics, its actual evolution path is selected by kinetics. It is this strong influence of thermodynamic non-equilibrium mechanisms that entails the large variety and complexity of microstructures typically encountered in materials. It is an essential observation that it is not those microstructures that are close to equilibrium, but often those that are in a highly non-equilibrium state that provide advantageous and desired material properties. Following Haasen [1] microstructure can be understood as the sum of all thermodynamic nonequilibrium lattice defects on a space scale that ranges from Angstroms (point defects) to meters (sample surface), Figs. 1, 2. Its temporal evolution ranges from picoseconds (dynamics of atoms) to years (fatigue, creep, corrosion, 2687 S. Yip (ed.), Handbook of Materials Modeling, 2687–2693. c 2005 Springer. Printed in the Netherlands.

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Figure 1. Example of relevant scales occurring in automotive crash simulations [2, 3].

Figure 2. Example of some relevant scales in polymer mechanics [4].

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diffusion). Haasens’s definition clearly underlines that microstructure does not mean micrometer, but nonequilibrium. Some of the size- and time-scale hierarchy classifications typically suggested for materials group microstructure research into macroscale, mesoscale, microscale, and nanoscale. They take a somewhat different perspective in which they refer to the real length scale of microstructures (often ignoring the intrinsic time scales which are more relevant when it comes to achieving relevant integration times). This might oversimplify the situation and suggest that we can linearly isolate the different space scales from each other. In other words, the classification of microstructures into a scale sequence merely reflects a spatial rather than a crisp physical classification. For instance, small defects, such as dopants, can have a larger influence on strength or conductivity than large defects such as precipitates. Or, think of the highly complex phenomenon of shear banding. These can be initiated not only by interactions among dislocations or between dislocations and point defects but as well by macroscopic stress concentrations introduced by the local sample shape, surface topology, and contact situation. However, if we accept that everything is connected with everything and that linear scale separation could blur the view on important scale-crossing mechanisms what is the consequence of this insight? One clear answer to that is: we do what materials scientists always did – we look for mechanisms. Let us be more precise. While former generations of materials researchers often focused on mechanisms or effects that pertain to single lattice defects and less complex mechanisms not amenable to basic analytical theory or experiments available in those times, present materials researchers have three basic advantages for identifying new mechanisms. First, ground state and molecular dynamics simulations have matured to a level at which we can exploit them to discover possible mechanisms at high resolution and reliability [5]. This means that materials theory is standing – in terms of the addressed time and space scale – for the first time on robust quantitative grounds allowing us insights we could not get before. It is hardly necessary to mention the obvious benefits arising from increased computer power in this context. Second, experimental techniques have been improved to such a level that – although sometimes only with enormous efforts – new theoretical findings can be critically scrutinized by experiment (e.g., microscopy, nanomechanics, diffraction techniques). Third, due to advances in both, theory and experiment more complex, self-organizing, critical, and collective non-linear mechanisms can be elucidated which cannot be understood by studying only one single defect or one single length scale. All these comments can be condensed to the statement, that microstructure simulation – as far as a fundamental understanding is concerned – consists in the hunt for key mechanisms. Only after identifying those we can (and should) make scale classifications and decide how to integrate them into macroscopic

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constitutive concepts or subject them to further detailed investigation. In other words the mechanisms that govern microstructure kinetics do not know about scales. It was often found in materials science that – once a basic new mechanism or effect was discovered – a subsequent avalanche of basic and also phenomenological work followed opening the path to new materials, new processes, new products, and sometimes even new industries. Well-known examples are the dislocation concept, transistors, aluminium reduction, austenitic stainless steels, superconductivity, or precipitation hardening. The identification of key mechanisms, therefore, has a bottleneck function in microstructure research and computational materials science plays a key role in it. This applies particularly when closed analytical expressions cannot be formulated and when the investigated problem is not easily accessible to experiments.

3.

Drowned by Data – Handling and Analyzing Simulation Data

A very good multi-particle simulation nowadays faces the same problem as a good experiment, namely, the handling, analysis, and interpretation of the resulting huge data sets [6]. Let us take for a moment the position of a quantum-Laplace-daemon and assume we can solve the Schr¨odinger equation for 1023 particles over a period of time, which covers significant processes in microstructure. What had we learned at the end? The answer to that is: not much more than from an equivalent experiment with high lateral and temporal resolution. The major common task of both operations would be to filter, analyze, and understand what we simulated or measured, respectively. We must not forget that the basic aim of most scientific initiatives consists in obtaining a general understanding of principles, which govern processes and states we observe in nature. This means that the mere mapping and reproduction of 1023 sets of single data packages (e.g., 1023 times the positions and momentum of all particles as a function of time) can only build a quantitative bridge to a basic understanding, but it cannot replace it. However, the advantages of this quantitative bridge built by the Laplacian super-simulation introduced above are at hand. First, it would give us a complete and well documented history record of all particles over the simulated period, i.e., more details than from any experiment. Second, once a simulation procedure is working properly it is often a small effort to apply it to other cases. Third, simulations have the capability to predict rather than only to describe. Fourth, usually in simulations the boundary conditions are exactly known (because they are mathematically imposed) as opposed to experimentation where they are typically not so well known. Therefore my initial statement about the concern of being drowned by simulation data aims at

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encouraging the computational materials science community to better cultivate an expertise of discovering basic mechanisms and microstructural principles behind simulations rather than getting lost in the details.

4.

Scaling, Coarse Graining, and Renormalization in Computational Materials Science

When realizing that quantum mechanics is not capable of directly treating 1023 particles the question arises how macroscopic material properties of microstructured samples can nonetheless be recovered from first principles. Numerous methods have been suggested to tackle such scale problems. They can be classified into two basic groups, namely, multi-scale and scale-bridging methods [2]. The first set of approaches (multiscale) consists in repeatedly including in simulation parameters or rules that were obtained from simulations at a respective smaller scale. For instance, the interatomic potentials of a material can be approximated using local density functional theory. This result can enter molecular dynamics simulations by using it for the design of embedded atom or tight-binding potentials. The molecular dynamics code could now be used, say for the simulation of a dislocation reaction. Reaction rules and resulting force fields of reaction products obtained from such predictions could be part of a subsequent elastic discrete dislocation dynamics simulation. The results obtained from this simulation could be used to derive the elements of a phenomenological hardening matrix formulation, and so on. Scale bridging methods take a somewhat different approach at scaling. They try to identify in phenomenological macroscopic constitutive laws those few key parameters, which mainly map the atomic scale physical nature of the investigated material and try to skip some of the regimes between the atomic and the macroscopic scale. Both approaches suffer from the disadvantage that they do not follow some basic and general transformation or scaling rules but require instead complete heuristical and empirical guidance. This means that both, multi-scale and scale-bridging methods must be directed by well-trained intuition and experience. An underlying and commonly agreed methodology of extracting meaningful information from different scales and implementing them into another does simply not exist. A similar but less arbitrary approach to this scale problem might lie in introducing suitable averaging methods, which are capable of reformulating forces among lattice defects accurately in terms of a new model with renormalized interactions obtained form an appropriate coarse graining algorithm [7–9]. Such a method could render some of the multi-scale and scale-bridging

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efforts more systematic, general, and reproducible. The basic idea of coarse graining and renormalization group theory consists in identifying a set of transformations (group) that translate characteristic properties of a system from one resolution to another. This procedure is as a rule not symmetric (semi-group). The transformations are conducted in such a way that they preserve the fundamental energetics of the system such as for instance the value of the partition function. For understanding the principle of renormalization let us consider an Ising lattice model, which is defined by some spatial distribution of the (Boolean) spins and its corresponding partition function. The Ising model can be reformulated by applying Kadanoff’s original real space coarse graining or block spin approach [7]. This algorithm works by summarizing a small subset of neighboring lattice spins and replacing them by one single new lattice point containing one new single lattice spin. The value of the new spin can be derived according to a decimation or majority rule. The technique can be applied to the system in a repeated fashion until the system remains invariant under further transformation. The art of rendering these repeated transformations a meaningful operation lies in how the value of the partition function or some other property of significance is preserved throughout the repeated renormalizations. The effect of successive scale transformations is to generate a flow diagram in the corresponding parameter space. The state flow occurs in our present example as a gradual change of the renormalized coupling constant. Reaching finally scale invariance is equivalent to transforming the system into a fixed point. These are points in parameter space that correspond to a state where the system is self-similar, i.e., it remains unchanged upon further coarse graining and transformation. The system properties at fixed points are, therefore, truly fractal. Each fixed point has a surrounding region in parameter space where all initial states finally end at the fixed point upon renormalization, i.e., they are attracted by this point. The surface where such competing areas abut is referred to as a critical surface. It separates the regions of the phase diagram that scale toward different single-component limits. The approach outlined in this section is called direct or real-space renormalization. For states on the critical surface, all scales of length coexist. The characteristic length for the system goes to infinity, becoming arbitrarily large with respect to atomic-scale lengths. For magnetic phase transitions, the measure of the diverging length scale is the correlation length. For percolation, the diverging length scale is set by the connectivity length of magnetic clusters. The presence of a diverging length scale is what makes it possible to apply to percolation the technique of real-space renormalization. Although the idea of applying the principles of renormalization group theory to microstructures might seem appealing at first sight it is essential to underline that there exists no general unified method of coarse graining microstructure problems. Approaches in this context must therefore also first take

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a heuristic view at scaling and carefully check which material property might be useful to assume the position of a preserved function when translating the system to the respective coarser scale. Another challenge consists in identifying appropriate methods to describe the in-grain and grain-to-grain behavior of the system clusters obtained by coarse graining. On the other hand, coarse graining and renormalization group theory might offer an elegant opportunity to eliminate empiricism encountered in some of the scaling approaches used in microstructure simulations [10–12]. Another advantage of applying renormalization group theory to microstructure ensembles might be to elucidate universal scaling laws, which are common also to other materials problems.

References [1] P. Haasen, Physikalische Metallkunde, Springer-Verlag, Berlin, Heidelberg, 1984. [2] D. Raabe, Computational Materials Science, Wiley-VCH, Weinheim, 1998. [3] R.W. Cahn, The Coming of Materials Science, Pergamon Press, Amsterdam, New York, 2001. [4] D. Raabe, “Mesoscale simulation of spherulite growth during polymer crystallization by use of a cellular automaton,” Acta Mater., in press, 2004. [5] M.P. Allen and D.J. Tildesley, Computer Simulation of Liquids, Oxford Science, 1987. [6] H.O. Kirchner, L.P. Kubin, and V. Pontikis, Proceedings of NATO ASI on Computer Simulation in Materials Science. NATO Advanced Science Institutes Series, Series E: Applied Sciences, vol. 308, Kluwer Academic in cooperation with NATO Science Division, 1996. [7] J. Cardy, Scaling and Renormalization in Statistical Physics, Cambridge, University Press, Cambridge, 1996. [8] D. Raabe, “Challenges in computational materials science,” Adv. Mater., 14, 639– 650, 2002. [9] D. Raabe, “Don’t Trust your Simulation – Computational materials science on its way to maturity?” Adv. Eng. Mater., 4, 255–267, 2002. [10] K. Binder, Monte-Carlo Methods in Statistical Physics, Springer-Verlag, New York, 1986. [11] H.E. Stanley, Phase Transitions and Critical Phenomena, Oxford University Press, London, 1971. [12] J.M. Yeomans, Statistical Mechanics of Phase Transitions, Clarendon, Oxford, 1992.

Perspective 7 DANGERS OF “COMMON KNOWLEDGE” IN MATERIALS SIMULATIONS Vasily V. Bulatov Lawrence Livermore National Laboratory, University of California, Livermore, CA 94550

For someone entering the field of materials simulations, it may be difficult to navigate through the maze of various ideas and concepts existing in the literature and to make one’s own judgment about their validity and certainty. Monographs and chapter books make it easier for a beginner to prepare for reading the literature describing the state of the art. Yet, even while reading a textbook, a novice may get the discomforting feeling of “not digging” a certain statement. If and when this happens, the first urge is usually to reread the passage and think harder and, if that fails, to re-view the preceding discussion trying to pay more specific attention to the facts and logic behind the elusive idea. Then, depending on one’s patience, it may become necessary to read other texts or talk to more experienced people. But what if all of this fails to clarify the point in question? What if the misunderstanding persists through the years and continues to nag even after most of the other, initially difficult, ideas happily find their proper place in one’s mind. It is quite natural then to begin to doubt oneself: why does no one else have this difficulty? Is it only I who is stupid? Eventually, the feeling of desperation subsides, often replaced by a conditional acceptance: “I don’t dig it but I can live with it”. Having changed my research area several times by now – from nuclear theory to statistical physics of polymers to physics of dislocations – I have had my share of such moments of desperation. My most recent foray into computer simulations of dislocations began in 1993 when Ali Argon and Sid Yip suggested for me to work on kink mechanisms of dislocation mobility in silicon. My prior exposure to dislocation theory consisted of a half-lecture in an undergraduate course on solid-state theory. The only thing I carried away with me from that lecture was that dislocation theory is something exceptionally boring and involves a lot of tedious tensor algebra. And that, however boring, the theory of dislocation is a very well established chapter of solid-state physics. It 2695 S. Yip (ed.), Handbook of Materials Modeling, 2695–2700. c 2005 Springer. Printed in the Netherlands.

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is with this pre-conception that I started reading the literature on dislocations in 1993. Now, having read most of the existing textbooks and hundreds if not thousands of papers on the subject, I feel humbled and humiliated by a vast variety of ideas and concepts existing in this fascinating field of study. And, surely, though still a relative novice I have accumulated my share of qualms most of which I learned to get over with. Yet some of the nagging issues appeared more troubling than others precisely because they seemed so simple and basic. Below I give a brief account of several such troubling statements and argue, at the risk of sounding contrarian, that some of the common understanding presented in the literature may be suspect. First, I will describe “a saga of misconceptions” around one technical issue in dislocation simulations, namely the use of periodic boundary conditions in dislocation dynamics simulations. Second, I will discuss one very basic idea concerned with the tendency of dislocations to glide on the most widely spaced crystallographic planes. Finally, I will speculate on several other bits of common knowledge that grow increasingly suspicious in my mind the more I think about them. The first issue may appear trivial but, mysteriously, remained unresolved until the year 2000. It concerns an alleged impossibility of using periodic boundary conditions (PBC) in 3D dislocation dynamics simulations. The following excerpts from the literature speak for themselves. “In order to avoid artifacts due to PBC, free surfaces are set” (1992). “In 3D, periodic boundary conditions should be implemented by mapping each set of slip planes on itself, which may be problematic. . . ” (1992). “PBC can be used in 2D but not in 3D if continuity of each dislocation line is to be maintained across the boundary” (1996). “The building of PBC at the boundaries of the simulation box is an unsolved problem. Indeed, the Born von Karman periodic condition used at the atomic scale can not be geometrically reproduced since we are considering linear defects moving in more than three glide systems and inside a 3-D space volume” (1996). “However, 3D periodicity is geometrically impossible since continuity of the dislocation lines can never be maintained” (1999). Although this list is certainly incomplete, it is nevertheless representative of the thinking at the time among the developers of a relatively new method of dislocation line dynamics. Rather than trying to examine what were the perceived difficulties∗ in using PBC in 3D line dynamics, it is curious to observe that the tone of the statements quoted above grew progressively more assertive to outright aggressive. The tone changed from doubts to absolute certainty that PBC in 3D line dynamics are not usable, and the notion continued to reinforce itself over time. Unfortunately, this misunderstanding was not * There were none. In fact, after PBC were “acquitted” in 2000 [1], in most cases it took just a few extra lines of code to modify the existing 3D algorithms to handle PBC gracefully. In a few weeks after the first use of PBC in 3D dislocation dynamics was demonstrated, essentially every group actively working in this area has implemented the simple trick.

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entirely harmless: having been convinced that PBC are impossible to use, several groups pursued alternative ideas for boundary conditions for dislocation line dynamics in the bulk. These alternative developments led to some rather complicated boundary conditions full of their own technical problems and artifacts and are now essentially abandoned in favor of PBC. The most amusing aspect of this story was that, all at the same time, there were plenty of people (myself included) who knew, from experience, that PBC are not only possible but also very simple to use when one is concerned with the line objects. Needless to say, I had many an agonizing thought about this issue while reading the literature on dislocation dynamics. I now turn to another notion that appears well entrenched in the dislocation physics community. Unlike the technical issue discussed above, this one is much more basic and appears very early in the textbooks. Give or take a few words, it is usually stated approximately like this: “Dislocations tend to exist and glide in those crystallographic planes that are most widely separated from each other”. Over some 10 years of reading the literature, I have seen quite a few such statements. Granted, exceptions to this rule are known and the notion itself is not pushed as aggressively among the specialists. Yet, the logic behind it is flawed and any reliance on such a geometric rule can lead to misunderstanding. Two factors are likely to have contributed to this preconceived notion. One is that the thinking about dislocations is still dominated by the observations of their behavior in FCC metals where dislocations indeed move in the widely spaced {111} planes. The other possible culprit is the Peierls–Nabarro model that, in its formulation, invokes the notion of inter-planar sliding: it appears logical that coupling between the wide spaced planes and, hence, the resistance to dislocation motion are the weakest. To see that this notion is dubious at best it suffices to consider dislocation glide in BCC metals. It is straightforward to show, by direct atomistic simulations, that Peierls stress of 1/2 111 dislocations in BCC crystals depends only weakly on the selection of glide plane of the 111 zone [2]. Instead, within one and the same plane, Peierls stress may vary by orders of magnitude as a function of dislocation character angle [3]. In particular, in the {110} planes, Peierls stress varies from 2 GPa for screw dislocations to below 20 MPa for edge dislocations. It appears that, the single geometrical parameter of the lattice that matters most is the period of the Peierls potential in the direction perpendicular to the dislocation line: the longer the period the higher is the Peierls stress. This period is the same as the spacing between the atomic rows parallel to the line direction and depends not only on the character angle in the geometric glide plane but also on the glide plane itself. Our yet unpublished results suggest that, even among edge dislocations, Peierls stress may vary from several hundreds of MPa to barely detectable levels of several MPa, depending on the selected glide plane [2]. Coming back to dislocations in FCC metals, the fact that dislocations glide (mostly) in the {111} planes is because these are the

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planes where dislocations dissociate into Shockley partials. This dissociation confines dislocations to gliding in the {111} planes. Thus, it is not the fact that {111} planes are most widely spaced but the fact that they are the only ones with low energy stacking faults that defines this characteristic behavior. In pure Al, where the stacking fault energy is relatively high, dissociation is suppressed and glide on planes other than {111} is observed [4]. Surely, one can counter this argument by saying that low energy stacking faults are more likely to exist between widely spaced planes. This conjecture is probably acceptable as a tendency, but not as a rule. Having described two examples of the existing “common knowledge”, I would like to name a few other common perceptions with respect to which my initial doubts are now giving way to a sense of “growing discontent”. I realize that such a speculative argument is quite risky and may eventually fly back in my face at some later time. However my sense of self-preservation is outweighed by a desire to call attention to possible misunderstandings and to persuade others to take a harder look at the issues at hand. 1. The traditional meaning of the Peierls stress does not seem to hold up against the recent results of direct MD simulations of screw dislocation motion in BCC metals. Simulation data from our group [5] as well as several unpublished observations [6, 7] suggest that screw dislocations in BCC metals are able to move under stress well below the (statically computed) Peierls threshold even at temperatures as low as 10 K. Furthermore, the ability of screw dislocations to move appears to depend strongly on the length of the simulated dislocation segment. 2. First suggested by P. Hirsch, the idea that a three-way non-planar splitting makes it difficult for screw dislocations in BCC metals to glide became rather pervasive in the literature. Yet, there is an emerging new consensus, based on recent atomistic simulations, that the high resistance to glide of screw dislocations in BCC metals is not a consequence of the notorious three-way splitting of the screw dislocation core [8, 9]. It appears that, as long as the screws do not experience a planar splitting, their lattice resistance is bound to be high regardless of other details of the core structure. 3. The well-known Friedel–Escaig (FE) mechanism of cross-slip in FCC metals has been recently confirmed in a series of large-scale atomistic simulations [10, 11]. There was little doubt, from the outset, that FE is the mechanism by which dislocation motion in FCC metals becomes 3D. However, the setup of atomistic simulations reported in [10, 11] made it virtually impossible to find any other cross-slip mechanism but FE. In particular, the choice of the method for finding the transition pathway together with the choice of the initial and final states along the sampling path, have restricted the search to FE-like transitions. Is this yet another

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case of self-reinforcing misconception? Notably, our recent simulations [12] suggest that cross-slip in FCC metals may proceed via a different mechanism suggested earlier by Fleischer [13]. 4. One finds numerous references to cross-slip from dislocation pile-ups as the mechanism underlying the transition from stage II to stage III hardening in FCC metals. Having examined much of the literature on this subject, including the original papers by Seeger, I have been frustrated by my inability to understand this mechanism and to paint a clear picture in my mind how exactly cross-slip from pile-ups defines this transition. The only weak consolation I had was that many other researchers, especially outside of Europe, have found this equally difficult to do. My frustration culminated in searching for and proposing an alternative mechanism of this transition [14, 15]. I am nearly certain that other researchers can offer more alternative ideas. In this short remark, I described my personal views on several aspects of dislocation theory. My intention was to share a few misgivings about some of the seemingly well-established knowledge in the field of dislocation physics. This is my current area of research and, obviously, the only one I can draw from for experience. Yet, given the vast amount of literature on dislocation physics, I will not be surprised to eventually find that other researchers have thought and even, possibly, stated their reservations concerning the same or related issues. In any case, my only and rather evident conclusion is that it is a good practice to remain agnostic about the ideas and concepts prevalent in any field of research, materials modeling included. Others may have their own “skeletons in the closet” waiting to get out in the open. This brief article was my way of doing just this.

References [1] V.V. Bulatov, W. Cai, and M. Rhee, “Periodic boundary conditions for dislocation dynamics simulations in three dimensions,” L.P. Kubin, J.L. Bassani, K. Cho et al., (eds.), Mat. Res. Soc. Symp., vol. 653, 2001. [2] V.V. Bulatov, W. Cai, and C. Krenn, unpublished. [3] J.P. Chang, W. Cai, V.V. Bulatov, and S. Yip, “Molecular dynamics simulations of motion of edge and screw dislocations in a metal,” Computat. Mater. Sci., 23, 111, 2002. [4] J.P. Hirth and J. Lothe, Theory of Dislocations, 2nd edn., Wiley, New Yourk, p. 272, 1982. [5] J. Marian, W. Cai, and V.V. Bulatov, “Dynamic transitions from smooth to rough to twinning in dislocation motion,” Nat. Mater., 3, 158–163, 2004. [6] V. Pontikis, private communication. [7] Yu. Osetsky, private communication.

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[8] S. Ismail-Beigi and T.A. Arias, “Ab initio study of screw dislocations in Mo and Ta: a new picture of plasticity in bcc transition metals,” Phys. Rev. Lett., 84, 1499, 2000. [9] C. Woodward and S.I. Rao, “Flexible ab initio boundary conditions: simulating isolated dislocations in bcc Mo and Ta,” Phys. Rev. Lett., 88, 216402, 2002. [10] T. Rasmussen, K.W. Jacobsen, T. Leffers, O.B. Pedersen, S.G. Srinivasan, and H. J´onsson, “Atomistic determination of cross-slip pathway and energetics,” Phys. Rev. Lett., 79, 3676–3679 1997. [11] T. Rasmussen, K.W. Jacobsen, T. Leffers, and O.B. Pedersen, “Simulations of the atomic structure, energetics, and cross slip of screw dislocations in copper,” Phys. Rev. B, 56, 2977, 1997. [12] W. Cai, V.V. Bulatov, J.P. Chang, J. Li, and S. Yip, “Dislocation core effects on mobility,” In: F.R.N. Nabarro and J.P. Hirth (eds.), Dislocations in Solids, Elsevier, Amsterdam, vol. 12, chap. 12, p. 17, 2004. [13] R.L. Fleischer, “Cross slip of extended dislocations,” Acta Metall., 7, 134, 1959. [14] V.V. Bulatov, “Unlocking dislocation secrets – challenges in theory and simulations of crystal plasticity,” In: J.V. Carstensen, T. Leffers, T. Lorentzen et al. (eds.), Modelling of Structure and Mechanics of Materials from Microscale to Product, RISO National Laboratory, Roskilde Denmark, pp. 39–60, 1998. [15] V.V. Bulatov, “Connecting the micro to the mesoscale: review and specific examples,” In: J. Lepinoix et al. (eds.), Multiscale Phenomena in Plasticity, Kluwer Academic Publishers, Netherlands, pp. 259–269, 2000.

Perspective 8 QUANTUM SIMULATIONS AS A TOOL FOR PREDICTIVE NANOSCIENCE Giulia Galli and François Gygi Lawrence Livermore National Laboratory, CA, USA

In the last two decades, the coming of age of first principles theories of condensed and molecular systems, and the continuous increase in computer power have positioned physicists to address anew the complexity of matter at the microscopic level. Theoretical and algorithmic developments in ab initio molecular dynamics [1] and quantum Monte Carlo methods [2], together with optimized codes running on high-performance computers, have allowed many properties of matter to be inferred from the fundamental laws of quantum mechanics, without input from experiment. In particular, quantum simulations are playing an increasingly important role in understanding and controlling matter at the nanoscale and in predicting with controllable, quantitative accuracy the novel and complex properties of nanomaterials. In the next few years, we expect quantum simulations to acquire a central role in nanoscience, as further theoretical and algorithmic developments will allow one to simulate a wide variety of alternative nanostructures with specific, targeted properties. In turn this will open the possibility of designing optimized materials entirely from first principles. Although the full accomplishment of this modeling revolution will be years in the making, its unprecedented benefits are already becoming clear. Indeed, ab initio simulations are providing key contributions to the understanding of a rapidly growing body of measurements at the nanoscale. A microscopic, fundamental understanding is very much in demand as such experimental investigations are often controversial and they cannot be explained on the basis of simple models. Quantum simulations provide simultaneous access to numerous physical properties (e.g., electronic, thermal and vibrational), and they allow one to investigate properties which are not yet accessible to experiment. A notable example is represented by microscopic models of the structure of surfaces at the nanoscale, which cannot yet be characterized experimentally due to the lack of appropriate imaging techniques. The characterization 2701 S. Yip (ed.), Handbook of Materials Modeling, 2701–2706. c 2005 Springer. Printed in the Netherlands.

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of nanoscale surfaces and interfaces is of paramount importance to predict the function of nanomaterials and eventually their assembly into macroscopic solids: the surface to bulk ratio is very large in any nanostructure and key chemical reactions determining the properties of nanomaterials are usually occurring at surfaces and interfaces. In Fig. 1 we show some examples of prediction of nanostructure properties recently obtained by quantum simulations. The first illustrates a prediction of the existence of a new class of nanoparticles (bucky diamonds) [3] which has been experimentally verified; the second shows how quantum simulations can be used to understand the complex interplay between quantum confinement effects and surface properties (in simple Si dots) [4] and to predict yet unexplored solvation effects on Si clusters. The third example shows how accurate calculations for CdSe nanoparticles can help interpret and better understand a set of apparently well established experimental data, and provide atomistic models which open the way to complex nanomaterials growth studies [5]. One common and important point shown by all of these examples is the unique ability of quantum simulations to separate different physical effects and assess their quantitative relevance in determining various properties (e.g., the

Figure 1. Example of nanostructures investigated with first principles, state-of-the-art calculations: carbon nanoparticles, in particular the structure of bucky-dimamond which was predicted by ab initio molecular dynamics simulations (left hand side [3]); surface reconstructions of hydrogenated silicon nanoparticles, simulated by ab initio MD with electronic band gap computed by highly accurate quantum Monte Carlo techniques (bottom right [4]) and self-healing of CdSe dots ([upper right [5]).

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relative importance of quantum confinement and surface structure in determining stability and optical gaps of semiconductor nanoparticles, or the relative importance of thermal disorder and solvation effects in determining electronic properties). This ability to discern between various physical effects is a unique feature of quantum simulations and it is an essential prerequisite to the development of materials design tools. The left hand side of Fig.1 shows structural models of bare nanodiamonds as obtained using ab initio molecular dynamics (MD) simulations [3]. These calculations have shown that, in the 1–4 nm size range, nanodiamond has a fullerene-like surface and, unlike silicon and germanium, exhibits very weak quantum confinement effects. These carbon nanoparticles characterized by a diamond core and a fullerene-like surface reconstruction have been called bucky diamonds. The proposed microscopic structure of bucky diamonds has been experimentally verified by a series of X-ray absorption and EELS measurements. In addition, ab initio calculations of bare and hydrogenated nanodiamonds have shown that at about 3 nm, and in a broad range of pressure and temperature, particles with bare, reconstructed surfaces become thermodynamically more stable than those with hydrogenated surfaces. These findings provided an explanation of the size distribution of extra-terrestrial nanodiamond found in meteorites and in outer space (e.g., proto-planetary nebulae) and of terrestrial nanodiamond found in ultradispersed and ultra-crystalline diamond films. Carbon is unique among group IV elements in exhibiting very weak quantum confinement effects at the nanoscale. Both Si and Ge are known to present stronger quantum confinement (below 5–6 nm), although experimentally it has been difficult to understand the interplay between mere size reduction of the crystalline nanoparticle core and surface reconstruction effects. Using a combination of quantum Monte Carlo (QMC) and ab initio MD techniques, the relative stability of Si nanoparticles (up to 2 nm) with reconstructed and unreconstructed surfaces has been predicted. Interestingly, these simulations have permitted to identify reconstructions which are unique to the highly curved surfaces of nanostructured materials and could not be guessed by a simple knowledge of the structure of solid surfaces. In addition, a clear connection between structure and function has been established: for example, calculations have shown that reconstructions of surface steps dramatically reduce the optical gap (right hand side of Fig.1) of hydrogenated Si dots, and decrease excitonic lifetimes, by localizing the band edge electronic states on the surface of the clusters. These predictions provided an explanation of both measured photoluminescence spectra of colloidally synthesized nanoparticles and observed deep gap levels in porous silicon. While surface reconstruction and some surface passivation (e.g., by oxygen) have been found to greatly influence the optical properties of Si dots, ab initio MD simulations of solvation of oxygenated Si clusters in water have shown no observable impact of the solvent.

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This is in contrast with blue shifts observed for several organic molecules in polar solvents and indicates that the vacuum optical properties of Si dots are preserved in the presence of water. This information is extremely important for possible applications of Si dots as sensors in aqueous environments. The need for quantum simulations in investigations of group IV nanostructures is apparent, given current experimental difficulties encountered in synthesizing and characterizing most of these nanoparticles. In addition, these simulations play a very important role in understanding other systems (such as, e.g., CdSe dots) which are better characterized experimentally. For example, ab initio calculations of the structural and electronic properties of CdSe nanoparticles [5] (right hand side of Fig.1) have shown significant geometrical rearrangements of the nanoparticle surface while the wurtzite core is maintained. Remarkably these reconstructions, which are very different from those of group IV dots, are similar in vacuo and in the presence of ligands used in colloidal synthesis. Surface rearrangements lead to the opening of an optical gap even in the absence of passivating ligands, thus “self-healing” the surface electronic structure. These calculations provided microscopic models which open the way to study the growth of both spheres and wires and eventually the surface functionalization of CdSe nanostructures. The nanostructures illustrated in Fig. 1 all contain between 100 and 500 atoms and they are representative of what can be dealt with today with ab initio MD and QMC tools. At present, state-of-the-art ab initio molecular dynamics can treat systems with a few hundred atoms (200–500 depending on the number of electrons and the accuracy required to describe the electronic wave-functions) and simulation times of 10–100 ps (depending on the size of the systems involved). State-of-the-art Quantum Monte Carlo (QMC) codes using newly developed linear scaling algorithms [6] can now enable the calculation of the energy and optical gaps of sp-bonded systems with up to 100–300 atoms, as illustrated above in the case of Si clusters. We estimate that in the next few years, algorithmic developments (e.g., linear scaling methods [7]) along with an anticipated surge in computational power will enable ab initio simulations of systems comprising 3000–4000 atoms for several picoseconds, as well as of systems comprising 200–300 atoms in the nanosecond range. In addition, nearly linear scaling QMC calculations [6] of systems containing several thousand atoms, will be made possible with unprecedented levels of accuracy. This will permit realistic simulations of organic/inorganic interfaces found in nanoscale devices for bio-detection, transport properties of single-molecule electronic devices and semiconductor nanowires, the properties of magnetic systems at the nanoscale and in general of advanced materials. Finally, the application of ab initio MD and QMC techniques is extending beyond the traditional fields of condensed matter physics and physical chemistry into biochemistry and biology. In the next decade we expect

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quantum simulations to effectively enter the realm of biology and to tackle problems such as microscopic modeling of DNA repair mechanisms and drug/ DNA interactions. In particular nearly exact QMC results may provide invaluable theoretical benchmarks that help overcome some of the current limitations of experimental biology. Although promising, quantum simulations still require improvements, in order to provide tools that theoreticians and experimentalists alike can use to design new materials, and many challenging problems remain to be solved. Besides the clear need for theoretical and algorithmic developments and complex code optimizations to adapt to new and changing platform architectures, new strategies need to be developed to best use these techniques in a way fully complementary to experiment. In particular, novel approaches to analyze, store and use data obtained from quantum simulations (including visualization tools and simulation data bases) need to be established. Progress in all of these areas will bring quantum simulations to be robust predictive tools for the design of new materials with targeted properties. Large increases in computer power – together with efficient coupled classical/ quantum-mechanical techniques (e.g., classical and ab initio molecular dynamics and quantum Monte Carlo) – will enable the design of new materials at the nanoscale, by generating a vast amount of accurate data to be used in configurational-expansion searches. The creation of easily accessible libraries of ab initio data for quantum design of materials will then allow one to predict systems with desired properties and quantities which are amenable to experimental validation. Many of the next-generation technologies will benefit from an ab initio computational design process, including optoelectronic materials, energy and information storage, detection of biological and chemical contaminants and spintronic devices.

Acknowledgment This work was performed under the auspices of the U.S. Department of Energy by University of California, Lawrence Livermore National Laboratory under Contract W-7405-Eng-48.

References [1] R. Car and M. Parrinello, Phys.Rev. Lett., 55, 2471, 1985. [2] D.M. Ceperley and B. Alder, Phys. Rev. Lett., 45, 566, 1980. [3] J-Y. Raty, G. Galli, A. Van Buuren, C. Boestedt, and L. Terminello, Phys. Rev. Lett., 90, 037401, 2003; J-Y. Raty and G. Galli, Nature Mat., 2, 792, 2003.

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[4] A. Puzder, A.J. Williamson, F. Reboredo, and G. Galli, Phys. Rev. Lett., 91, 157405, 2003. [5] A. Puzder, A.J. Williamson, F. Gygi and G. Galli, Phys. Rev. Lett., 92, 217401, 2004. [6] A. Williamson, R.Q. Hood, and J.C. Grossman, Phys. Rev. Lett., 87, 246406, 2001. [7] J.-L. Fattebert and J. Bernholc, Phys. Rev. B, 62, 1713, 2000; J.-L. Fattebert and F. Gygi, Comp. Phys. Comm., 162, 24, 2004.

Perspective 9 A PERSPECTIVE OF MATERIALS MODELING William A. Goddard III Materials and Process Simulation Center, California Institute of Technology, Pasadena, California 91125, USA

1.

The Vision and Opportunity of de novo Multi Paradigm and Multiscale Materials Modeling

The impossible combinations of materials properties required for essential industrial applications have made the present paradigm of empirically based experimental synthesis and characterization increasingly untenable. Since all properties of all materials are in principle describable by quantum mechanics (QM), one could in principle replace current empirical methods used to model materials properties by first principles or de novo computational design of materials and devices. This would revolutionalize materials technologies, with rapid computational design, followed by synthesis and experimental characterization only for materials and designs predicted to be optimum. From good candidate materials and processes, one could iterate between theory and experiment to optimize materials. The problem is that direct de novo applications of QM are practical for systems with ∼102 atoms whereas the materials designer deals with systems of ∼1022 atoms. The solution to this problem is to factor the problems into several overlapping scales each of which can achieve a scale factor of ∼ 104 . By adjusting the parameters of each scale to match the results of the finer scale, it is becoming possible to achieve de novo simulations on practical devices with just ∼ 5 levels. This would allow accurate predictions of the properties for novel materials never previously synthesized and would allow the intrinsic bounds on properties to be established so that one does not waste time on impossible challenges. The levels in this hierarchy of methods (theory, algorithms, and software) must overlap as in the Figure so that results of the finer level can be used to determine the parameters and models suitable at the coarse level while retaining the accuracy of the finer level. This will allow a designer to modify in real time 2707 S. Yip (ed.), Handbook of Materials Modeling, 2707–2711. c 2005 Springer. Printed in the Netherlands.

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time scale

De novo Hierarchical Strategy for Predicting Real Optimized Materials Design process: stage 1 Target properties at the macroscale determine the desired behavior at smaller scales FF/MD

QM

Software integration MACRO MESO

Design process: stage 2 High quality multi-scale modeling of promising materials for accurate predictions of materials properties

length

the choice of materials, microstructures, and assembly and obtain rapid feedback on the target properties until attaining the designed performance. Despite enormous progress, there remain enormous gaps in our ability to use theory and computation to address with reliability the prediction of optimized materials. Indeed other articles in this handbook address some of the innovative ideas for extending current theory, algorithms, and computational methods to solve the remaining problems. What is clear now is that the progress over the last 40 years has brought us to the brink of a whole new age in materials science in which theory and simulation are trusted to do the broad elements of designing bold new materials and the optimized solutions are refined experimentally to obtain the bringing to practice of these designs. Indeed the chapters in this Handbook provide details of the progress that has been made in various parts of this problem. The methods involved include: (a) First Principles Quantum Mechanics (QM) calculations to solve the Schr¨odinger equation, HØ = −i ∂Ø/∂t to predict the electronic, vibrational, excitation, and reactive properties of molecules, surfaces, and solids. The elements at this level are the wavefunctions Ø describing the electrons. Ø is a function of the coordinates of the N electrons in the system and of the N atom positions. There are various flavors of QM theory as discussed below. However it is not necessary to input any data about the system in order to predict the structures and properties. This allows us to predict the properties of novel materials never before synthesized or characterized. Unfortunately practical use of these methods is limited to ∼102 atoms. (b) Force Fields (FF) to describe the potential energy of the system in terms of the positions of the atoms, V (Ri , i = 1, . . . , N), where Ri is the 3D

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vector describing the location of atom i. Thus the electronic information in the QM description is captured in a far simpler form in terms of the atom locations (with the electron coordinates averaged out). Traditional FF were designed to describe the molecule or solid near equilibrium, making them useful for describing the structure and vibrational levels, but not the reactions or chemistry. (c) Reactive force fields (ReaxFF). A major recent breakthrough is the ReaxFF force field [1–3] that provides an accurate description of reactive processes (including barriers) with a classical FF, allowing simulation of complex materials and processes involving 1000’s of atoms (including chemical reactions, charge transfer, polarizabilities, and mechanical properties for metals, oxides, organics, and their interfaces). This is an essential part of the Hierarchy of methods discussed above. The parameters for ReaxFF are obtained entirely from QM. Critical elements of ReaxFF are: a) a general model for electrostatics involving self-consistent charge transfer and atomic polarizability in which the charge distribution is determined from the instantaneous environment of each atom and applied to all pairs [Goddard et al., 2002, Zhang et al., 2003, Rapp´e and Goddard, 1991)]. b) Valence terms are based on partial bond orders allowing proper bond dissociation. c) Pauli Repulsion and dispersion is applied to all pairs (no exclusions). ReaxFF accurately describes various metals, oxides, and covalent systems and has been used for simulating shock-induced decomposition in RDX [Strachan et al., 2003]. We are excited that this will complement the methods discussed in this handbook by providing the chemical reaction input for many important materials problems. (d) Molecular Dynamics (MD) simulations. Given the FF we can describe the dynamics of the system in terms of Newton’s equations F = Ma or −∇V = M ∂ 2 R∂t2 properly modified to take into account temperature and pressure. With non-reactive FF, MD simulations are practical for systems with 105 – 107 atoms. For ReaxFF they are practical for ∼105 atoms. This allows Large-scale ab initio-based Force Fields (FF) on systems with millions of particles to predict properties (yield strength, Peierls stresses, elasticity, electrical and thermal conductivity, dielectric loss, ferroelectric domain boundaries) relevant for materials design (phase fields, reaction models) to be input into mesoscale and continuum models. (e) Mesoscale Dynamics (MesoD) (Kinetic Monte Carlo, Phase Fields, Chemical Kinetics) of heterogeneous interfaces and structures, using parameters from the MD and predicting critical performance properties for continuum modeling. Here some cases may involve coarse grain

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descriptions in which beads represent collections of atoms. In other cases (say plasticity of metals) the mesoscale may involve only defects such as dislocations, with the atoms completely gone (of course that atoms were there at the MD level to obtain the Peierls stresses, kink energies and other quantities needed to describe plasticity. (f) Macroscale and Continuum Modeling of real devices to consider packaging of the components, effects of environment. A model here is NEMO 3D [Klimeck, 2002] used successfully in the semiconductor industry to predict quantum transport in devices. To obtain first principles-based results for macroscale systems, we must ensure that each scale of simulation overlaps sufficiently with the finer description so that all input parameters and constitutive laws at each level of theory can be determined from more fundamental theory. Equally important we must ensure that these relations are invertible so that the results of coarse level simulations can be used to suggest the best choices for finer level parameters, which can be used to suggest new choices of composition and structure. Validation and error propagation. Essential to design of materials is estimating the reliability (error bars) in the calculated results. Unfortunately progress here has been slow with much to do in establishing methods to estimate likely uncertainties by comparing to finer more accurate theory and by validating against available experimental data. Often in applications it is expedient to use less accurate levels than the best available to obtain answers quickly, which requires that the errors for the various level of theory be estimated and propagated forward so that the designer can know what confidence to have in modifying compositions and materials to optimize performance. Of course to be useful to designers, the simulation software must integrate the various computational methods (QM, FF, MD, MM, and FE) into an integrated framework so that they can focus on the design issues. To provide this framework we developed the Computational Materials Design Facility (CMDF) to allow the multiscale software to be accessed transparently with data automatically passed sequentially from one generation to the next for optimization and utilization of parameters.

Applications A recent application using multiscale methods to design a new material is Deng [8] in which a new previously unsynthesized material capable of reversible binding of H2 up to 6% by weight (the DOT goal for 2010) designed using FF developed from QM and using grand canonical MD to predict the pressure-temperature loading, including a viable synthetic strategy.

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Other recent examples use a multiscale strategy to predict the 3D structures of membrane bond proteins and to validate the structures by predicting the binding sites for agonists and antagonists [9, 10]. A recent application of the multiscale strategy (first principles QM through modeling of the stress-strain behavior as a function of temperature and strain rate is given in Cuitino et al. [11]. An implementation of the multiscale approach to rapid throughput screening of catalysts is given in Muller et al. [12].

References [1] A.C.T. van Duin, S. Dasgupta, F. Lorant et al., “ ReaxFF: A reactive force field for hydrocarbons,” J. Phys. Chem. A, 105, 9396–9409, 2001. [2] A.C.T. van Duin, A. Strachan et al., “ReaxFF sio reactive force field for silicon and silicon oxide systems,” J. Phys. Chem. A, 107, 3803–3811, 2003. [3] A. Strachan, A.C.T. van Duin, D. Chakraborty et al., “Shock waves in high-energy materials: the initial chemical events in nitramine RDX,” Phys. Rev. Let., 91(9): art. No. 098301, 2003. [4] W.A. Goddard III, Q. Zhang, M. Uludogan et al., “The ReaxFF polarizable reactive force fields for molecular dynamics simulation of ferroelectrics,” R.E. Cohen and T. Egami (eds.), Fundamental Physics of Ferroelectrics, 45–55, 2002. [5] Q. Zhang, T. Cagin, A. van Duin, et al., “Adhesion and nonwetting-wetting transition in the Al/alpha-Al2O3 interface,” Phys. Rev. B, 69(4): art. No. 045423, 2004. [6] A.K. Rapp´e and W.A. Goddard, “Charge equilibration for molecular dynamics simulations,” J. Phys. Chem., 95, 3358–3363, 1991. [7] G. Klimeck, F. Oyafuso, T.B. Boykin, R.C. Bowen, and P.V. Allmen, Comput. Modeling Eng. Sci., 3, 5, 601–642, 2002. [8] W.Q. Deng, X. Xu, and W.A. Goddard, “New alkali doped pillared carbon materials designed to achieve practical reversible hydrogen storage for transportation,” Phys. Rev. Let., 92(16): art. No. 166103, 2004. [9] M. Yashar, S. Kalani, N. Vaidehi et al., “The predicted 3D structure of the human D2 dopamine receptor and the binding site and binding affinities for agonists and antagonists,” PNAS, 101(11), 3815–3820, 2004. [10] P.L. Freddolino, M.Y.S. Kalani, N. Vaidehi et al., “Predicted 3D structure for the human beta 2 adrenergic receptor and its binding site for agonists and antagonists,” PNAS, 101(9), 2736–2741, 2004. [11] A.M. Cuitino, L. Stainier, G. Wang et al., “A multiscale approach for modeling crystalline solids,” J. Comput. Aided Mater. Des., 8, 127–149, 2001. [12] R.P. Muller, D.M. Philipp, and W.A. Goddard III, “Quantum mechanical – rapid prototyping applied to methane activation,” Top. Catal., 23, 81–98, 2003.

Perspective 10 AN APPLICATION ORIENTED VIEW ON MATERIALS MODELING Peter Gumbsch Institut f¨ur Zuverl¨assigkeit von Bauteilen und Systemen izbs, Universit¨at Karlsruhe (TH), Kaiserstr. 12, 76131Karlsruhe, Germany and Fraunhofer Institut f¨ur Werkstoffmechanik IWM, W¨ohlerstr. 11, 79194, Freiburg, Germany

Modeling and simulation has become a major part of Materials Science and Engineering in academia as well as in industrial research and development. Materials oriented modeling and simulation, however, is not a well established monolithic area. It encompasses all the tools which physicists, chemists, mechanical engineers and materials scientists have developed over the years to describe materials as such or their behavior. One usually distinguishes three distinct areas connected with materials modeling: materials modeling, process simulation and component simulation. Materials modeling is directed towards the simulation of the materials itself, its evolution, changes in its internal structure and its properties. Almost all contributions in this volume are directly oriented towards this part of materials modeling. The final goal of all materials modeling is better understanding of materials behavior and the development of simple models which accurately describe it. These models are the basis for all other materials oriented modeling and simulation. Process simulation is aiming at the description of the synthesis of a material and the further processing into a component. It takes into account the specific processing parameters and is often used to optimize individual processing steps. Component simulation is aiming at the characterization and evaluation of the properties of components or entire devices. This includes their behavior in service, possible changes during service and their lifetime. In industry, computer simulation is used to reduce the number of cycles of trial and error during development and to transfer these cycles from the lab to the computer. Before the insertion of products into the market, computer simulation is also used to reduce the costly testing of prototypes to a few well selected cases as in the case of crash tests of cars. 2713 S. Yip (ed.), Handbook of Materials Modeling, 2713–2718. c 2005 Springer. Printed in the Netherlands.

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Recent developments in process simulation aim at not only considering a single process but the chain of consecutive processing steps and the influence of one on the other. The need for the latter comes from the increasing demand for higher precision which requires inhomogenities in the material to be correctly considered. Such inhomogenities are usually associated with alterations in the microstructure of the material, which in turn have their origin in locally different processing conditions in the previous processing step. Microstructural information is therefore explicitly or implicitly transferred when transferring locally different processing conditions. The need to couple simulations of the individual steps of a process chain also arises because potential improvements in complex production processes can only be exploited if the entire process chain is considered as a whole. Whatever the particular reason for the modeling of the entire process chain, the value added is due to the use of information which has already been acquired at a previous step. Making this information useful is highly non trivial. In practical applications it often constitutes a mayor effort since it requires that implicitly available information (e.g., the cooling rate in casting) must be explicitly transformed to the relevant characteristics of the microstructure (e.g., the distribution of grain sizes and precipitates). The importance and the potential of the simulation of process chains is illustrated with the example of the producing of a geometrically complex aluminum oxide ceramic component. (Details can be found in [1] and references therein.) The geometry of the component is predetermined from the design. The essential process steps are the filling of the powder into the die, the pressing of the green body and the sintering of the ceramic. The difficulty lies within sintering, the last processing step, during which the component shrinks and bends and often leaves locally varying remaining porosity. Differences in density in the green body as a consequence of the pressing, shown in Fig. 1, are responsible for this. The changes in geometry during sintering have to be

Figure 1. Simulation of the pressing process of an alumina ceramic and the resulting density distribution in the green body [4].

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anticipated and counteracted through alterations of the die and or the pressing route. Such alterations can ideally be simulated. The density distribution after pressing can directly be transferred to the simulation of sintering and the outcome of this simulation can directly be compared to the specifications. Particularly valuable in this case is the possibility to invert the order of these simulations. This allows questions to be addressed like whether and how the desired design can be achieved with an existing tool by changes to the pressing only. The experience with simulations of powder metallurgical processing steps is extremely positive. The results match experimental findings to a very good degree. Remaining differences can often be traced back to inadequate knowledge of the first process step, the filling of the die, which may give an inhomogeneous density distribution before pressing. The simulation of filling processes of granular media is intensively worked on [2]. It makes use of discrete (atomistic) simulation methods similar to the large scale MD methods described in this volume. Recent developments in the component simulation aim at accounting for actual conditions in service, instead of the often much simplified testing conditions. They also aim at taking into account local inhomogenities in the material, which of course result from the processing of the material. A particularly memorable example of locally different materials properties due to the processing history is known to everyone who ever tried to straighten a crooked nail. Similar experience can be gained if one tries to bend a paper clip back to a straight wire. One experiences local differences in strain hardening which make it quite difficult to bend the clip back at precisely the location where it had previously been bent. This is shown in Fig. 2. In practical application such local differences in strain hardening also lead to locally different elastic stresses in a component during forming, which after removal of the forming tool lead to very different spring back behavior. The difficulty in correctly simulating the spring back behavior originates from the difficulties in correctly describing the flow stress and the work hardening. Often the flow stress may be significantly lower when reversing the direction of deformation. Changing the mode of deformation can lead to even more complex conditions and parts of the hardening can also relax with time (dependent on temperature). Altogether this results in a huge parameter space which can hardly be covered experimentally. In practical application one partly circumvents this problem by fitting advanced material laws to few experiments, specifically designed to test different fields in parameter space (e.g., [3]). A strict reference to the microstructure of the material can however not be drawn today. Materials modeling has seen enormous advances in multiscale simulation in the recent years. To simulate materials properties and to develop materials models often demands access to very different time and length scales. Sticking

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Figure 2. Bending a copper paper clip back usually does not lead to a straight wire due to local differences in hardening.

with the example of plastic deformation, one first has to consider the grain structure with which the dislocations interact. Typical grain sizes are millimeters and below. Dependent on the type of deformation the dislocations arrange in sub-grain structures of micrometer dimensions; and the development of these dislocation structures is substantially determined by cutting processes of the dislocations which occur at an atomic scale. All these different processes can not be described with one particular simulation tool. Thus different methods are established at different scales from crystal plasticity via dislocation dynamics simulation to atomistic methods all described in this volume. Until recently, all these methods were safely separated in space and time but the enormous advances in computing power and the development of dedicated hybrid simulation methods which couple several of the methods helped to close these gaps. The first atomistic simulations of dislocation intersections became possible five years ago [4, 5]. Of current interest are atomistic simulations that realistically describe the interaction between dislocations and grain boundaries [6, 7] and details of dislocation-dislocation interactions and their consequences for strain hardening [8, 9]. Simulating the evolution of a multitude of dislocations is still a formidable task. However, recent advances in the three dimensional discrete dislocation simulations now enable the simulation of simple strain paths and small deformations. Soon these simulations will go to large enough deformations in non-trivial geometries and thereby enable statistical

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analyses along the lines of recent two dimensional studies [10] and of dislocation density evolution for more complicated strain paths. Such statistical methods could in principle directly help to determine the flow stress of a material in classical engineering simulations. They will certainly aid the development of dislocation field theories which will enable one to do so. These topics are currently under investigation for example in the research network SizeDepEn [11]. Even if the emphasis of this commentary is on aspects of the mechanical properties of materials, multiscale simulation is not limited to that. Various other examples can for example be found in reference [12]. A common trend of all the different aspects of computational materials science discussed so far is the attempt to include microstructural aspects of the material. Advanced materials modeling must in this respect provide the basis for the other disciplines. The drive may however come from the users in the component and process simulation for whom a physically based integral materials simulation is the goal, which will give them a qualitative new basis for product design. So far the development of a new component is performed sequentially thru the design, the processing and the component evaluation phases as schematically displayed in Fig. 3. Starting with a proposal for component design, a first simple component evaluation can be done using standard component simulation tools. Next the possibilities for processing are evaluated using process simulation tools. Accurate evaluation and simulation of the properties of the component and detailed assessment with respect to the specifications are however only possible after the production of a few prototypes, which have the precise microstructure that results from the particular processing steps chosen to produce the component. If intolerable differences to the specifications occur, the entire design cycle has to be repeated. In general, an inversion of the design process is not possible today. Consequently, questions like: “What type of microstructure is needed in which part of the component to fulfill specifications?” and “How is the processing to be done to reach this microstructure?” can not be answered today. Component design on the basis of an integral materials simulation which uses transferable microstructure-based materials models at all steps will in the future allow precisely that. Besides qualitatively and quantitatively improved simulations in the specific steps, integral materials simulation will at least partly allow to invert the design steps and to back calculate how processing needs to be done to reach the desired properties. In full generality the picture drawn right now is still futuristic. In particular cases, however, and this includes several of the aforementioned examples, this scenario is reacheable in the near future if the still separated disciplines of materials, process and component simulation find together on common subjects.

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Figure 3. Component design has so far relied on a sequence of steps from the geometrical design and specification to assessment of the processing steps and component evaluation. Each of these steps is aided by simulation. Today, integral modeling of the genesis of a component on a common simulation basis is not yet available. Integral materials simulation which takes into consideration the microstructure of the material in the individual processing steps and carries it over to component simulation will enable much more precise simulation and will help to partly reverse the design process.

References [1] T. Kraft and H. Riedel, Powder Metall., 45, 227–231, 2002. [2] H.J. Herrmann, J.-P. Hovi, and S. Luding (eds.), Physics of Dry Granular Media, Kluwer Academic Publishers, Netherlands, 1998. [3] A.A. Krasowsky et al., “Computational fluid and solid mechanics,” K.J. Bathe (ed.), Elsevier Science, New York, pp. 403–406, 2003. [4] S.J. Zhou et al., Sci., 279, 1525–1527, 1998. [5] P. Gumbsch, Sci., 279, 1489–1490, 1998. [6] J. Schiotz and K.W. Jakobsen, Sci., 301, 1357–1359, 2003. [7] H. van Swygenhoven et al., Nat. Mater., 3, 399–403, 2004. [8] R. Madec et al., Sci., 301, 2003. [9] P. Gumbsch, Sci., 301, 1857–1858, 2003. [10] M. Zaiser, M.-C. Miguel and I. Groma, Phys. Rev. B, 64, 224102, 1–9, 2001. [11] www.sizedepen.net [12] MRS Bulletin, 26/3, 169–221, 2001.

Perspective 11 THE ROLE OF THEORY AND MODELING IN THE DEVELOPMENT OF MATERIALS FOR FUSION ENERGY Nasr M. Ghoniem Mechanical and Aerospace Engineering Department, University of California, Los Angeles, CA 90095-1597, USA

The environmental and operational conditions of First Wall/ Blanket (FW/B) structural materials in fusion energy systems are undoubtedly amongst the harshest in any technological application. These materials must operate reliably for extended periods of times without maintenance or repair. They must withstand the assaults of high particle and heat fluxes, as well as significant thermal and mechanical forces. Rival conditions have not been experienced in other technologies, with possible exceptions in aerospace and defense applications. Moreover, the most significant dilemma here is that the actual operational environment cannot be experimentally established today, with all of the synergistic considerations of neutron spectrum, radiation dose, heat and particle flux, and gigantic FW/B module sizes. Because of these considerations, we may rely on a purely empirical and incremental boot-strapping approach (as in most human developments so far), or an approach based on data generation from non prototypical setups (e.g., small samples, fission spectra, ion irradiation, etc.), or a theoretical/computational methodology. The first approach would have been the most direct had it not been for the unacceptable risks in the construction of successively larger and more powerful fusion machines, learning from one how to do it better for the next. The last approach (theory and modeling alone) is not a very viable option, because we are not now in a position to predict materials behavior in all its aspects from purely theoretical grounds. The empirical, extrapolative approach has also proved itself to be very costly, because we cannot practically cover all types of material compositions, sizes, neutron spectra, temperatures, irradiation times, fluxes, etc. Major efforts had to be scrapped because of our inability to encompass all of these variations simultaneously. While all three approaches must be considered for the 2719 S. Yip (ed.), Handbook of Materials Modeling, 2719–2729. c 2005 Springer. Printed in the Netherlands.

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development of fusion materials, the multi-scale materials modeling (MMM) framework that we propose here can provide tremendous advantages if coupled with experimental verification at every relevant length scale. A wide range of structural materials has been considered over the past 25–30 years for fusion energy applications [1]. This list includes conventional materials (e.g., austenitic stainless steel), low-activation structural materials (ferritic/martensitic steels, V-4Cr-4Ti, and SiC/SiC composites), oxide dispersion strengthened (ODS) ferritic steels, conventional high temperature refractory alloys (Nb, Ta, Cr, Mo, W alloys), titanium alloys, Ni-based super alloys, ordered intermetallics (TiAl, Fe3 Al, etc.), high-strength, high-conductivity copper alloys, and various composite materials (C/C, metal-matrix composites, etc.). Numerous factors must be considered in the selection of structural materials, including material availability, cost, fabricability, joining technology, unirradiated mechanical and thermophysical properties, radiation effects (degradation of properties), chemical compatibility and corrosion issues, safety and waste disposal aspects (decay heat, etc.), nuclear properties (impact on tritium breeding ratio, solute burnup, etc.). Strong emphasis has been placed within the past 10–15 years on the development of three reduced-activation structural materials: ferritic/martensitic steels containing 8–12%Cr, vanadium base alloys (e.g., V-4Cr-4Ti), and SiC/SiC composites. Recently there also has been increasing interest in reduced-activation ODS ferritic steels. Additional alloys of interest for fusion applications include copper alloys (CuCrZr, Cu–NiBe, dispersion-strengthened copper), tantalumbase alloys (e.g., Ta-8W–2Hf), niobium alloys (Nb–1Zr), molybdenum, and tungsten alloys. In the following, we give a brief analysis of the most limiting mechanical properties based on our earlier work [1].

1.

Lower Operating Temperature Limits

The lower temperature limits for FW/B structural materials (i.e., excluding copper alloys) are strongly influenced by radiation effects. For body-centered cubic (BCC) materials such as ferritic-martensitic steels and the refractory alloys, radiation hardening at low temperatures can lead to a large increase in the Ductile-To-Brittle-Transition-Temperature (DBTT)[2, 3]. For SiC/SiC composites, the main concerns at low temperatures are radiation-induced amorphization (with an accompanying volumetric swelling of ∼11%) [4] and radiation-induced degradation of thermal conductivity. The radiation hardening in BCC alloys at low temperatures (0.3TM ) is generally pronounced, even for doses as low as ∼1 dpa [3]. The amount of radiation hardening typically decreases rapidly with irradiation temperature above 0.3 TM , and radiationinduced increase in the DBTT may be anticipated to be acceptable at temperatures above ∼0.3TM . A Ludwig–Davidenkov relationship between hardening

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and embrittlement was used to estimate the DBTT shift with increased irradiation dose. In this model, brittle behavior occurs when the temperature dependent yield strength exceeds the cleavage stress. It is worth noting that operation at lower temperatures (i.e., within the embrittlement temperature regime) may be allowed for some low-stress fusion structural applications (depending on the value of the operational stress intensity factor relative to the fracture toughness). Numerous studies have been performed to determine the radiation hardening and embrittlement behavior of ferritic-martensitic steels. The hardening and DBTT shift are dependent on the detailed composition of the alloy. For example, the radiation resistance of Fe-9Cr-2WVTa alloys appears to be superior (less radiation hardening) to that of Fe-9Cr-1MoVNb. The radiation hardening and DBTT shift appear to approach saturation values following low temperature irradiation to doses above 1–5 dpa, although additional high-dose studies are needed to confirm this apparent saturation behavior. At higher doses under fusion conditions, the effects of He bubble accumulation on radiation hardening and DBTT need to be addressed. Experimental observa√ tions revealed brittle behavior (K I C ∼30 MPa- m) in V-(4–5)%Cr-(4–5)%Ti specimens irradiated and tested at temperatures below 400 ◦ C. From a comparison of the yield strength and Charpy impact data of unirradiated and irradiated V-(4–5)%Cr-(4–5)%Ti alloys, brittle fracture occurs when the tensile strength is higher than 700 MPa. Therefore, 400 ◦ C may be adopted as the minimum operating temperature for V-(4–5)%Cr-(4–5)%Ti alloys in fusion reactor structural applications[5]. Further work is needed to assess the impact (if any) of fusion-relevant He generation rates on the radiation hardening and embrittlement behavior of vanadium alloys. Very little information is available on the mechanical properties of irradiated W alloys. Tensile elongation of ∼ 0 have been obtained for W irradiated at relatively low temperatures of 400 and 500 ◦ C (0.18–0.21 TM ) and fluences of 0.5−1.5×1026 n/m2 (≺2 dpa in tungsten) [6]. Severe embrittlement (DBTT ≥ 900 ◦ C) was observed in un-notched bend bars of W and W-10%Re irradiated at 300 ◦ C to a fluence of 0.5 × 1026 n/m2 (≺ 1 dpa). Since mechanical properties data are not available for pure tungsten or its alloys irradiated at high temperatures, an accurate estimate of the DBTT versus irradiation temperature cannot be made. The minimum operating temperature which avoids severe radiation hardening embrittlement is expected to be 900 ± 100 ◦ C.

2.

Upper Operating Temperature Limits

The upper temperature limit for structural materials in fusion reactors may be controlled by four different mechanisms (in addition to safety considerations): Thermal creep, high temperature helium embrittlement, void swelling, and

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compatibility: corrosion issues. Void swelling is not anticipated to be significant in ferritic-martensitic steel [7] or V–Cr–Ti alloys [8] up to damage levels in excess of 100 dpa, although swelling data with fusion-relevant He:dpa generation rates are needed to confirm this expectation and to determine the lifetime dose associated with void swelling. The existing fission reactor database on high temperature (Mo, W, Ta) refractory alloys (e.g., [6]) indicates low swelling (≺2%) for doses up to 10 dpa or higher. Radiation-enhanced recrystallization (potentially important for stress-relieved Mo and W alloys) and radiation creep effects (due to a lack of data for the refractory alloys and SiC) need to be investigated. Void swelling is considered to be of particular importance for SiC (and also Cu alloys, which were shown to be unattractive fusion structural materials [1]. An adequate experimental database exists for thermal creep of ferriticmartensitic steels [7] and the high temperature (Mo, W, Nb, Ta) refractory alloys [10]. Oxide-dispersion-strengthened ferritic steels offer significantly higher thermal creep resistance compared to ferritic-martensitic steels [11], with a steady-state creep rate at 800 ◦ C as low as 3 × 10−10 s−1 for an applied stress of 140 MPa. The V-4Cr-4Ti creep data suggest that the upper temperature limit lies between 700 and 750 ◦ C, although strengthening effects associated with the pickup of 200–500 ppm oxygen during testing still need to be examined. The predicted thermal creep temperature limit for advanced crystalline SiC-based fibers is above 1000 ◦ C [12]. One convenient method to determine the dominant creep process for a given stress and temperature is to construct an Ashby deformation map. Using the established constitutive equations for grain boundary sliding (Coble creep), dislocation creep (power law creep) and self-diffusion (Nabarro–Herring) creep, the dominant deformation-mode regimes can be established [1].

3.

Operating Temperature Windows

Figure 1 summarizes the operating temperature windows (based on thermal creep and radiation damage considerations) for nine structural materials considered by Zinkle and Ghoniem [1]. The temperature limits for Type 316 austenitic stainless steel are also included for sake of comparison. In this figure, the light shaded regions on either side of the dark horizontal bands are an indication of the uncertainties in the temperature limits. Helium embrittlement may cause a reduction in the upper temperature limit, but sufficient data under fusion-relevant conditions are not available for any of the candidate materials. Due to a high density of matrix sinks, ferritic/martensitic steel appears to be very resistant to helium embrittlement [13]. An analysis of He diffusion kinetics in vanadium alloys predicted that helium embrittlement would

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Figure 1. Operating temperature windows (based on radiation damage and thermal creep considerations) for refractory alloys, Fe-(8-9%)Cr ferritic-martensitic steel, Fe-13%Cr oxide dispersion strengthened ferritic steel, Type 316 austenitic stainless steel, solutionized and aged Cu-2%Ni-0.3%Be, and SiC/SiC composites. The light shaded bands on either side of the dark bands represent the uncertainties in the minimum and maximum temperature limits.

be significant at temperatures above 700 ◦ C [14]. The lower temperature limits in Fig. (1) for the refractory alloys and ferritic:martensitic steel are based on fracture toughness embrittlement associated with low temperature neutron √ irradiation. An arbitrary fracture toughness limit of 30 MPa- m was used as the criterion for radiation embrittlement. Further work is needed to determine the minimum operating temperature limit for oxide dispersion strengthened (ODS) ferritic steel. The value of 290 ± 40 ◦ C used in Fig. (1) was based on results for HT-9 (Fe-12Cr ferritic steel). The minimum operating temperature for SiC/SiC was based on radiation-induced thermal conductivity degradation, whereas the minimum temperature limit for CuNiBe was simply chosen to be near room temperature. The low temperature fracture toughness radiation embrittlement is not sufficiently severe to preclude using copper alloys near room temperature [15], although there will be a significant reduction in strain hardening capacity as measured by the uniform elongation in a tensile test. The high temperature limit was based on thermal creep for all of the materials except SiC and CuNiBe. Due to a lack of long-term (10 000 h), low-stress creep data for several of the alloy systems, a Stage II creep deformation limit of 1% in 1000 h for an applied stress of 150 MPa was used as an arbitrary criterion for determining the upper temperature limit associated with thermal creep. Further creep data are needed to establish the temperature limits for longer times and lower stresses in several of the candidate materials.

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Main Challenges

4.1.

Deformation and Fracture

Problems of deformation and fracture stem from several phenomena that can render structural material brittle either at low or at high temperature. The interplay between these phenomena is complex resulting in deformation and fracture properties to depend on many intrinsic and extrinsic variables. We must therefore take into account environmental variables, when we develop a database for materials properties, which can be functions of (T, dpa, dpa rate, He, H, stress, etc.). Since properties involve many mechanisms at multiscale levels, and sometimes differences between large rates result in macroscopic changes (e.g., void swelling), we need to develop physically based properties models. We also need a modeling-experiment integration strategy for validation of models at different scales. Models that are to be developed can be hierarchical, starting from the atomic information all the way to the prediction of constitutive relationships and macroscopic fracture. The main crosscutting issues in deformation and fracture are: 1. Irradiation effects on stress-strain, constitutive laws, and consequences of flow localization; 2. Validity & physical basis of the Master–Curve (MC) for predicting the ductile-to-brittle-transition; 3. Embrittlement – MC shifts due to hardening & He effects 4. Model-based designs for high performance alloys; 5. Irradiation effects on constitutive properties: J2 laws linked to microstructure evolution; 6. Development of plasticity models for constitutive properties, for example bridging between Dislocation Dynamics, crystal plasticity, and polycrystalline plasticity; 7. Understanding flow localization and ductility loss of irradiated materials; 8. The apparent universality of the MC shape, and the physical basis for this universality. 9. Effects of helium on GB, and how that influences shifts in DBTT; 10. Model-based design of alloys; for example including a high density of nano-clusters to trap helium in high-pressure bubbles and thus preventing them from going to grain boundaries.

4.2.

Helium Effects

Several methods of modeling helium effects on irradiated materials have been developed over the past two decades. Atomistic MD simulations are

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now being used more extensively to determine the energetics of binding and migration of various helium-vacancy complexes. Information on the fraction of residual defects has also been obtained from cascade simulations. Such atomic level information is passed on to mesoscale simulations of microstructure evolution based on reaction rate theory. Most of these simulations have assumed that the microstructure is spatially homogeneous in space and time. However, some of these assumptions have been relaxed, such as the effects of cascades on point defect diffusion, formation of microstructure patterns, etc. One of the key advantages of rate theory is that the results of simulations can be directly compared to experiments, while the key parameters are obtained from either experiments or atomistic simulations. For example, KMC simulations can now be used to solve complex point defect diffusion problems in the stress field of dislocations, and thus derive more realistic values for the dislocation bias factors. At the same time, large systems of equations describing the nucleation and growth of void and bubble populations can be solved with current day large-scale computers, thus providing more accurate descriptions of nucleation and growth. This level of detailed rate theory modeling is essential, because experiments show that several phenomena are influenced by helium in a complex fashion, for example, the swelling rate is not a monotone function of the helium-to-dpa ratio. Likewise, the effects of small helium concentrations on grain boundary fracture depend on many details of the microstructure, while the effects of helium bubbles on hardening or embrittlement at low temperature is not yet clear.

4.3.

Radiation Stability of Alloys

Real alloys are made of major and minor components. While the fate of minor elements under irradiation can be handled, in principle at least, with the same tools as point defects (i.e., book keeping of the mean or local concentration as a function of time), such is not the case for major alloy components. In particular, the cluster dynamics technique (rate theory) fails, because of percolation problems. A small community works at developing a theoretical framework to assess the stability of stationary phases under irradiation. At the present time, it is acknowledged that the latter stability depends altogether on the temperature, composition, irradiation flux and “cascade size”. This implies that both the spatial extension of the cascade and the number of replacements per cascade are important factors. For the overall approach to be justified, the evolution of the precipitate population under irradiation must be fast compared to that of the defect sink structure (dislocation network, defect aggregates of various forms): such is indeed the case since the latter evolves at a rate proportional to the small difference between the vacancy and the interstitial fluxes

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at sinks, two large quantities; on the contrary, the precipitates grow or shrink because of the coupling of the solute flux with the two above fluxes, an additive process.

5.

Modeling Research Needs

5.1.

Interatomic Potentials for Radiation Damage

The crucial properties for radiation damage simulations are: (1) (2) (3) (4) (5)

Point defect formation, migration and interaction energies; Elastic constant anisotropy; Grain boundary energetics; Dislocation structure and response to stress; Alloy phase stability.

None of these properties are automatically correct as a result of the physical basis of potentials. With pair-wise interactions, some are necessarily wrong. With many-body potentials (used here as a generic term covering glue, Finnis– Sinclair, embedded atom, modified embedded atom and effective medium theory potentials) many can be fitted provided “correct” values are available. These types of potentials have been the “state of the art” for twenty years. Historically, there has been an insufficient database for robust potential fitting. Not all this data is available experimentally for parameterization and verification of potentials. Recent renewed interest in interatomic potentials is based on the ability of ab initio calculations to provide this missing data – with teraflop machines verification of predictions is finally possible. Where tested against new data, existing potentials have generally proved disappointing. Some common problems include poor interstitial formation energies, the energy difference between configurations too small and no satisfactory description of the austenitic-ferritic transition. Some of these problems can be traced to problems in parameterization of the potentials and have been addressed in recent work by simple reparameterization. Others, such as the absence of a physically sensible treatment of magnetization, point to more fundamental problems in the many-body potential concept. The majority of the effort in potentials for metallic phases has focused on elemental materials. Potentials for multi-component systems have been developed in isolated cases, but the predictive capability of these potentials is typically disappointing. Reasonable models for the mission-critical helium impurities exist, the inertness of helium making its behavior in MD somewhat insensitive to parameterization. There are two challenges in the development of potentials in alloy systems. First, there is generally much less data available though this can be rectified through the use of ab initio methods.

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Second, the appropriate functional forms are not as well developed. Most potentials are based on simple pictures of bonding. In alloy systems, the nature of the bonding is inherently more complex suggesting that more sophisticated potentials are needed to describe the energetics reliably. Non-metallic impurities (carbon, phosphorus) are more problematic.

5.2.

Dislocation Interactions & Dynamics

One of the critical problems for the development of radiation-resistant structural materials is the embrittlement, loss of ductility and plastic flow localization. Modeling the interaction between dislocations and radiationinduced obstacles is providing great insights into the physics of this problem, and will eventually lead to the design of radiation-resistant structural alloys. Models of dislocation-defect interactions are pursues at two levels: (1) the atomistic level, where MD simulations are playing significant role; and (2) the mesoscopic level, where DD simulations are providing insights into largerscale behavior. Both types of models are complementary, and provide direct information for experimental validation on the effects of irradiation on hardening, yield drop, and plastic flow localization, etc. Atomic scale models are used to “inform” DD models on the details of dislocation-defect interactions. Presently, MD models can simulate 1–10 million atoms on a routine basis. Both static and dynamic simulations are used. For static simulations, fixed displacement boundary conditions are applied, and conjugate gradient minimization is used. On the other hand, Newtonian equations of motion are used for dynamic simulations, and either force or velocity conditions are applied on boundary atoms. Atomistic simulations have shown the range where elasticity estimates are valid for dislocation-defect interactions, and where they break down due to new mechanisms. For example, the interaction of dislocations with small precipitates can result in local phase transitions and an associated energy cost that cannot be predicted from DD models. Also, it has been shown that dislocation-void interaction leads to dislocation climb, and the formation of a dislocation dipole before the dislocation completes cutting through the void completely. These effects are all of an atomic nature, and the information should be passed on to DD simulations. A number of challenges remain in the area of dislocation-defect interactions, as described below: (1) The strain rates in MD simulations are far in excess of experimentally achievable rates, and methods to incorporate slow rate events due to temperature or force field fluctuations have not yet been developed. (2) The information passing between MD and DD is not systematic yet. For example, the “angle” between dislocation arms before it leaves the

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(3)

(4)

(5)

(6)

(7)

N.M. Ghoniem obstacle is often used in DD simulations as a measure of obstacle strength. However, the definition of this angle in both MD and experiments is problematic for a variety of reasons. Force-displacemt information will be necessary. Methods for incorporating lower length scale microstructure effects into DD simulations are not well developed. For example, we do not have information on obstacle dynamics, solute effects, dislocations near cracks, dislocation nucleation, etc. The size of atomistic simulations is very small, and cannot deal with complex dislocation structures. Methods for reducing the degrees of freedom are needed. The boundary conditions used in MD simulations are either periodic, fixed, or represented by elastic Green’s functions. General methods for embedding MD simulations into the continuum are in an early stage of development. DD codes are limited to small size crystals. To improve their speed and range of applicability, new methods of designing these codes on massively parallel computers are needed. The connection between DD and macroscopic plasticity has not yet been made through “coarse graining” and a systematic reduction of the degrees of freedom. Development of this area is essential to the prediction of constitutive relations and macroscopic plastic deformation.

References [1] S. Zinkle, N. Ghoniem, “Operating temperature windows for fusion reactor structural materials,” Fusion Engineering and Design, 2000. [2] R. Klueh and D. Alexander, “Embrittlement of crmo steels after low fluence irradiation in hfir,” J. Nucl. Mater., 218, 151, 1995. [3] M. Rieth, B. Dafferner and H.-D. Rohrig, “Embrittlement behaviour of different international low activation alloys agter neutron irradiation,” J. Nucl. Mater., 258– 263, 1147, 1998. [4] L. Snead, S. Zinkle, J. Hay, and M. Osborne, “Amorphization of sic under ion and neutron irradiation,” Nucl. Instr. Methods B, 141, 123, 1998. [5] S. Zinkle et al., “Research and dvelopment on vanadium alloys for fusion applications,” J. Nucl. Mater., 258–263, 205, 1998. [6] F. Wiffen, “Effects of irradiation on properties of refractory alloys with emphasis on space power reactor applications,” Proc. Symp. on Refractory Alloy Technology for Space Nuclear Power Applications, CONF-8308130, Oak Ridge National Lab, p. 254, 1984. [7] D. Gelles, “Microstructural examination of commerical ferritic alloys at dpa,” J. Nucl. Mater., 233–237, 293, 1996. [8] B. Loomis and D. Smith “Vanadium alloys for structural applications in fusion systems: a review of vanadium alloy mechanical and physical properties,” J. Nucl. Mater., 191–194, 84, 1992.

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[9] K. Shiba, A. Hishinuma, A. Tohyama and K. Masamura, “Properties of low activation ferritic steel f82h iea heat-interim report of iea round-robin tests (1),” Japan Atomic Energy Research Institute Report JAERITech. [10] H. McCoy, “Creep properties of selected refractory alloys for potential space nuclear power applications,” Oak Ridge National Lab Report ORNL:TM-10127. [11] P. Maziasz et al., “New ods ferritic alloys with dramatically improved hightemperature strength,” J. Nucl. Mater., Proc. 9th Int. Conf. Fusion Reactor Materials, 1999. [12] G. Youngblood, R. Jones, G. Morscher and A. Kohyama, “Creep behavior for advanced polycrystalline sic fibers,” in Fusion Materials Semiann,” Prog. Report for period ending June 30 1997, DOE:ER-0313:22, Oak Ridge National Lab, p. 81, 1997. [13] H. Schroeder and H. Ullmaier, “Helium and hydrogen effects on the embrittlement of iron and nickel-based alloys,” J. Nucl. Mater., 179–181, 118, 1991. [14] A. Ryazanow, V. Manichev, and van W. Witzenburg, “Influence of helium and impurities on the tensile properties of irradiated vanadium alloys,” J Nucl. Mater., 227–263, 304, 1996. [15] D. Alexander, S. Zinkle and A. Rowcliffe “Fracture toughness of copper-base alloys for fusion energy applications,” J. Nucl. Matter, 271&272, 429, 1999.

Perspective 12 WHERE ARE THE GAPS? Marshall Stoneham Centre for Materials Research, and London Centre for Nanotechnology, Department of Physics and Astronomy, University College London, Gower Street, London WC1E 6BT, UK

Reading a Handbook like this gives a vivid picture of the enormous vigour and power of materials modelling. One is tempted to believe that we can answer all the questions materials technology might pose. Even if that were partly true, we should be identifying just what we do not know how to do. Some gaps will be depend on new hardware and software, especially when modelling quantum systems. Some gaps will be recognised only after some social or technological change has brought them into focus. Among the developments likely to stimulate innovation could be novel nanoelectronics, or the fields where physics meets biology. Still further gaps exist because we have been slaves to fashion, and have been drawn away from unpopular (roughly translating as “too difficult”) fields; examples might include excited state spectroscopy, or electrical breakdown. Much pioneering work on materials modelling was based on very simple potentials and non-self-consistent electronic structure. Today, potentials are sophisticated and accurate, and self-consistent electronic structure for molecular dynamics is routine for adiabatic energy surfaces. This has given us confidence. But can almost any process be modelled with success? Sadly, this is not so. There are at least three groups of challenges. One first challenge concerns electronic excited states and other non-equilibrium systems. Predicting the crystal structure of lowest energy is not the major issue in modelling, and is rarely a performance characteristic by itself. For many systems, the key characteristics are those of metastable forms (e.g., most steels, diamond), not the state of lowest free energy. Predicting which crystal structure will result from a particular growth process can be very hard indeed, especially for larger organic molecules. A second challenge concerns timescales, from femtoseconds to tens of years, and length scales, and the link between microscopic (atomistic) and mesoscopic (microstructural) scales for hierarchical phenomena. 2731 S. Yip (ed.), Handbook of Materials Modeling, 2731–2736. c 2005 Springer. Printed in the Netherlands.

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A third challenge lies in understanding the quantum physics of highly correlated systems, for which intuition based on the human scale is not a safe guide.

1.

Special Excited States

Why bother with excited states? They offer new ways to control systems, especially at the nanoscale, ranging from materials modification to quantum coherence [1]. In modifying materials, the basic solid-state processes rely on energy localisation, whether ionic or electronic. If atoms are to be displaced, some minimum energy is needed, and this energy must be associated with specific local atomic motions. Self-trapping, the immobilisation of excitations by local lattice deformation, is often the key. The amount of energy available for a process is crucial, yet current methods are disappointing in predicting available energies. Often, electronic excited states have features qualitatively different from the ground state, notably their equilibrium geometry and their degeneracy or near-degeneracy. Their excess energy can be dissipated by radiative or non-radiative transitions. Charge localisation is another key process. It is less necessary for materials modification, but offers one simple means of guiding energy localisation. Again, current methods for predicting self-trapping disappoint: the local density approximation (LDA) often fails to predict it, even when experiment is unambiguous, and Hartree–Fock methods (HF) predict localisation when they should not. Energy transfer is a third process. The atomic displacements do not need to occur at the sites originally excited, but can occur at more distant sites. A common example, if not yet fully understood, is photosynthesis. Fourthly, there is energy storage. Energy sinks can delay damage processes, and sometimes change their character. Even conduction electrons in a metal can store energy for long enough to affect outcomes. Finally, there are the effects of charge transfer and space charge. Charge buildup can be important for indirect reasons, e.g., because a macroscopic field can influence subsequent damage events.

2.

Away from Equilibrium and the Steady State

In a semiconductor, it is natural to assume that carrier injection will be followed by various processes leading to equilibrium or to some driven steady state. But when an exciton is created, or a muon implanted, these species have a finite lifetime. The exciton or the muon, can decay before it reaches its most stable state. So experiments may measure the properties of metastable states. This should be good news, in that the variety of excited states includes a wealth of different behaviours. There may be opportunities to manipulate

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and exploit the states and the branchings between competing excited state processes. The consequences might range from solid state reactions (such as the photographic process) and enhanced diffusion in semiconductors to minimal invasive dentistry. In radiation damage, it is a convenient metaphor to say that the centre of a collision cascade becomes liquid, since the energy deposited can be tens of eV per atom. How good is the “liquid” description? How well does it describe the more complex, but important, cases like Si, where solid Si is a semiconductor and liquid Si is a metal? Can one pretend that the system is neutral everywhere, even though electrons will have been scattered out of the central “liquid” zone? If there is a transiently positive core region, how soon is neutrality restored? And how quickly and where will the electrons deposit their energy? The Fermi level is not defined, so it is hard to know how to decide what to do about charge states. Life processes, as Schr¨odinger noted, are inherently non-equilibrium. There are many striking phenomena that are hard to mimic, let alone model. For example, is Davydov’s model of efficient energy transfer along the α-helix of a protein by a soliton mechanism correct [2]? And how are we to relate the biologist’s views of force [3] to the physicist’s clear ideas on forces as derivatives of potentials, acting at well-defined sites? Modelling biosystems, going beyond mimicry to understanding, is an open field.

3.

Beyond the Adiabatic Approximation

Even the commonest non-radiative transitions, the non-adiabatic processes by which many excited systems revert to the ground state, have special difficulties [1]. The normal input to such transition rates involved two components. One is an electronic matrix element of the non-adiabatic part of the Hamiltonian, and this needs wavefunctions. There, known (and subtle) difficulties have been identified, but there are few attempts to do state-of-the-art calculations. The other component concerns the quantum treatment of nuclear motion, going beyond the usual classical dynamics with quantum treatments of electrons. Unfortunately, the usual Fermi Golden Rule, or something broadly equivalent, does not suffice to predict the evolution of a system by self-consistent molecular dynamics. This is not a problem of the harmonic approximation, or of unusual adiabatic dynamics, such as solitons in conducting polymers, all of which can be handled. The non-adiabatic aspects are tricky, elusive in a convenient form. What is needed here is a means to avoid the Fermi Golden Rule. Various tools, like frozen Gaussian methods or energy surface hopping, offer ways forward, but there is little theory that addresses the remarkable experiments on transient excited state phenomena in halides and oxides, for example.

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Situations involving multiple energy surfaces are also challenging. It is only too tempting to treat a Jahn–Teller system as if there were a single energy surface. Yet this would be wrong dynamically, and would also omit extra excitations, like orbitons in coupled Jahn–Teller systems. Such coupled systems can exhibit other ideas hard to simulate, such as frustration, or distinctions between real and quasi spins when the interactions between component magnetic moments are significant.

4.

Full Quantum Treatments, Including Predictive Decoherence

Serious predictions of electronic structure inevitably involve quantum electrons, usually through the h¯ in kinetic energy terms, through quantum statistics, or through the exclusion principle. Quantum information processing, and the even more demanding ideas of quantum computing, are widely discussed [4] Quantum tunnelling has become a common phenomenon, although its modelling is often primitive, intended more to make use of simple analytical models than to represent the system accurately. When the adiabatic approximation is not enough, quantum nuclear motion is needed, as well as that of electrons. In some highly-correlated systems, and in superconductivity, the quantum effects become both more sophisticated and less intuitive. Indeed, it is not clear what practical limits exist to the modelling of quantum systems. Some of the most interesting cases involve entanglement, the quantum dance of one electron with another, and the way that is reduced by decoherence, the quantum analogue of classical dissipation. The most interesting aspects of quantum behaviour can also take one away from equilibrium phenomena. Quantum statistics, of course, is at its most useful at or close to equilibrium. Quantum entanglement, and the manipulations of quantum information processing, need the avoidance of decoherence, the quantum analogue of dissipation, and this can often be achieved in highly non-equilibrium situations.

5.

Open Systems, and Interfaces between Unlike Materials

Many important interfaces involve dissimilar materials: a metal and an oxide, a polymer and an indium tin oxide electrode, an adhesive and wood, a biomaterial and blood. Some of the challenges are to understand mechanical properties, like adhesion and friction. Sometimes simple ideas work well, like the image interaction picture of non-reactive metal/oxide adhesion, and can give at least a framework for discussion of complex interfaces. Other issues concern the processes of transferring charge (electrons, protons, etc.) across the interface. It is not simple to match absolute energies on the two

Where are the gaps?

2735

sides of most interfaces. Ther are often significant dipole moments, for instance. Tunnelling rates and injection phenomena need a dynamic description of screening. Electron emission needs to recognise the long-range electric field. Charge transfer, whether through a blocking electrode or through an ohmic contact, usually often implies an open system, whether it be a biosystem and its environment, or a nanocomponent within a device. In such cases, the boundary conditions demand subtle treatment. Moreover, the dynamics of screening can take one well away from the usual classical electrostatic descriptions.

6.

Mesoscopic Issues where Rare Events Dominate

Condensed matter processes exhibit enormous range. Some are fast processes: electronic relaxation in metals (fs); photochemistry and fast nonradiative processes (0.1–1 ps), and allowed radiative transitions (say 10 ns). Others are slow processes: spin-forbidden transitions (ms) and diffusioncontrolled processes (0.1 s to geological times). For a steel component in a nuclear reactor, the fastest processes in collision cascades take only a few femtoseconds, and most of the action is over in a few picoseconds. The consequences of these cascades and subsequent slower diffusion processes on the ductility of the steel can grow in importance over 30 years, i.e., the billion second timescale, and may affect dramatically reactor economics. In some cases, energy densities can be extremely large, perhaps tens of eV per atom. Radiationinduced processes are hierarchical, as for polymers, for which scission or cross-linking affect subsequent damage. Such cross-linking can generate diamond-like carbon, and this can be modelled. Key features of radiation behaviour are radiochemical yields (what happens for every 100 eV put in?), gas production, and the influence of interstitial molecular oxygen. Embrittlement and tendency to fracture, plus a reduced electrical breakdown performance, are critical for applications like insulation in fission reactors. Yet fracture and breakdown predictions are classic examples of behaviour for which phenomenological behaviour is well documented, but reliable predictions, even of behaviour statistics, barely attempted. When there are hierarchical processes, one common symptom is non-Debye relaxation behaviour (stretched exponential, “universal response”, etc.). The differences can be more dramatic, of course, with qualitative changes in some cases. Clearly, fracture is qualitatively distinct from modest shifts in elastic constants. And, if one is dealing with fracture, or electrical breakdown, for which extremal statistics apply, how do you ensure your model has the right Weibull parameters? Focussing one’s model on the appropriate scale seems much more effective [5]. Whilst multiscale modelling is not uncommon, there should be real doubts about the wisdom of using an all-embracing code that

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incorporates sophisticated calculations at several levels, e.g., finite difference macroscopic, atomistic, and electronic structure. One problem is making sure the ideas are right at each stage: in a silicon MEMS device, for instance, is the atomistic treatment of the silicon to have priority over treatment of thermal oxide?

7.

Posing the Problem: Making Contact with Reality

For all the early days of materials modelling, there was an implicit assumption that either the computer hardware or the software was the most serious limitation. This no longer seems to be the case. Posing the problem effectively, and interpreting the results properly, were supposed straightforward. The emphasis has changed. Major developments in hardware and software are welcome, but the serious limits need brainware and experience, rather than computer power or software [6]. Can we frame the problem in a way that could be modelled at the right level of detail, precision and sophistication? Do we understanding enough of the modelling output to give adequate answers to key questions? Have we the underlying science to recognise the limitations of the models used and be aware of the value of the answer assessed? Do we have enough confidence to use our models to take unpopular decisions? Could we tell an influential person that their favourite idea can’t work? Anyone who has worked in a technology-based organisation will know that new and unexpected situations arise with remarkable regularity. Sometimes these are problems, sometimes opportunities. Usually, the window of opportunity is short, since timescales are fixed by non-scientific constraints. Posing the problem in a soluble form can be the biggest challenge. It can also be rewarding. As thermodynamicist J Willard Gibbs acknowledged, thermodynamics owed more to the steam engine than the steam engine owed to thermodynamics.

References [1] N. Itoh and A.M. Stoneham, Materials Modification by Electronic Excitation, Cambridge University Press, Cambridge, 2001. [2] P.-A. Lindgard and A.M. Stoneham, “Self trapping, biomolecules and free-electron lasers,” J. Phys. Cond. Matter, 15, V5-V9, 2003. [3] M.E. Fisher and A.B. Kolmeiski, Proc. Nat. Acad. Sci., 96, 6597–6602, 1999. [4] C.P. Williams and S.H. Clearwater, Ultimate Zero and One, Copernicus, New York, 2000. [5] A.M. Stoneham and J.H. Harding, “Not too big, not too small: the appropriate scale,” Nature Materials, 2, 77–83, 2003. [6] A.M. Stoneham, A. Howe, and T. Chart, “Predictive Materials Modelling,” UK Department of Trade and Industry/Office of Science and Technology Foresight Report DTI/Pub5344/02/01/NP, URN 01/630, 2001.

Perspective 13 BRIDGING THE GAP BETWEEN QUANTUM MECHANICS AND LARGE-SCALE ATOMISTIC SIMULATION John A. Moriarty Lawrence Livermore National Laboratory, University of California, Livermore, CA 94551-0808

The prospect of modeling across disparate length and time scales to achieve a predictive multiscale description of real materials properties has attracted widespread research interest in the last decade [1]. To be sure, the challenges in such multiscale modeling are many, and in demanding cases, such as mechanical properties or dynamic phase transitions, multiple bridges extending from the atomic level all the way to the continuum level must be built. Although often overlooked in this process, one of the most fundamental and important problems in multiscale modeling is that of bridging the gap between first-principles quantum mechanics, from which true predictive power for real materials emanates, and the large-scale atomistic simulation of thousands or millions of atoms, which is usually essential to describe the complex atomic processes that link to higher length and time scales. For example, to model single-crystal plasticity at micron length scales via dislocation-dynamics simulations that evolve the detailed dislocation microstructure requires accurate large-scale atomistic information on the mobility and interaction of individual dislocations. Similarly, modeling the kinetics of structural phase transitions requires linking accurate large-scale atomistic information on nucleation processes with higher length and time scale growth processes.

1.

Electronic-atomic Gap

As indicated in Fig. 1, there currently exists a wide spectrum of atomic-scale simulation methods in condensed-matter and materials physics, extending from essentially exact quantum-mechanical techniques to classical descriptions with totally empirical force laws. All of these methods fall into one of two distinct 2737 S. Yip (ed.), Handbook of Materials Modeling, 2737–2747. c 2005 Springer. Printed in the Netherlands.

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J.A. Moriarty Material-dependent gap

Electronic: electron + ion motion Many-electron Self-consistent …… states mean field Correlated electron theory

Density Functional Theory (DFT)

Atomic: ion motion only Coarse-grained No electronic …… electronic structure structure Quantum based potentials

Empirical potentials

Exact quantum mechanics

“ ab initio”

Total empirical description QMC DMFT ...

QMD PP FP-LMTO ...

Quantum simulation

1⫺10

GPT MGPT BOP ...

EAM FS ...

“empirical”

Atomistic simulation

10⫺102

102⫺106

104⫺108

Number of atoms Figure 1. Representative sample of the wide spectrum of electronic and atomistic simulation approaches used in condensed-matter and materials physics and the material-dependent gap separating them.

categories, which are separated by a material-dependent gap. On one side of this gap are electronic methods based on direct quantum-mechanical treatments. These include quantum simulations that attempt to treat electron and ion motion on an equal footing, solving quantum-mechanical equations on the fly for both the electronic states of the system and the forces on the individual ions. In principle, such methods can provide a highly accurate description of the system and are chemically very robust, but they come at the price of being severely limited in the size and duration of the simulation. Typically, even efficient mean-field methods such as quantum molecular dynamics (QMD) [2, 3] can at best treat a hundred or so atoms for a few picoseconds of time. On the other side of the gap are methods used in atomistic simulations that treat only the ion motion, solving classical Newtonian equations of motion with the forces derived from explicit interatomic potentials, which may or may not be encoded with detailed quantum information about the electronic structure. For the simplest short-range empirical potentials, tens or hundreds of millions of atoms can be so simulated with molecular dynamics (MD) for time durations extending to tens or hundreds of nanoseconds. But this computational robustness often comes at the price of losing any connection to the underlying electronic structure of the material. For studying generic phenomena in simple systems this may not be a major drawback, but more generally, for the predictive multiscale modeling of real complex materials that we envision here, the retention of adequate quantum information is essential.

Gap between quantum mechanics and large-scale atomistic simulation 2739 In practice, both quantum and atomistic simulations may be approached at many different levels of approximation. Many-body quantum-mechanical methods attempt to treat the full many-electron states of the system and provide a general means of addressing the fundamental issue of electron correlation. The valence electrons in the vast majority of systems of practical interest, however, including most metals and semiconductors, effectively exhibit only weak electron correlation. For such systems accurate total energies and forces can be achieved through self-consistent, mean-field electronic-structure methods based on modern density functional theory (DFT) [4, 5]. Indeed, today DFT-based electronic-structure and quantum simulation methods are usually described as “ab initio,” even though significant approximations from exact quantum mechanics are involved. Nonetheless, for weakly correlated systems first-principles DFT methods are quantitatively predictive and rely on only the barest input information: the atomic numbers and masses of the material constituents. Within this category of computational method are all-electron, full-potential (e.g., FP–LMTO) techniques as well as pseudopotential (PP) techniques, which treat only valence electrons and are normally essential in QMD simulations. In addition to such direct computational approaches, simplified representations of DFT are also possible via orbital-basis-state approaches using plane waves, localized atomic orbitals or a hybrid combination of the two. Such simplification provides a useful starting point to “coursegrain” the electronic structure and actually bridge the electronic-atomic gap. The interatomic potentials used in atomistic simulations have likewise been developed to many different levels of sophistication and quantum compatibility. One basic consequence of the quantum-mechanical nature of electronic bonding in both metals and semiconductors is that the total energy of the system, E tot (R1 , . . . , R N ), is inherently a many-body functional of its atomic coordinates Ri and must contain terms beyond radial-force pair interactions. This requirement is directly manifest in a number of measurable properties including the elastic moduli, where in metals it is well known that the Cauchy relations implied by pure pair potentials are not satisfied in general (e.g., C12 =/ C44 in cubic metals) and in semiconductors non-radial forces are needed even to stabilize the basic diamond structure. Most modern empirical potentials satisfy this requirement by adding a more general functional to a pairpotential contribution in E tot . In some cases, this additional contribution is inspired by specific quantum-mechanical considerations, but typically arbitrary short-ranged functional forms are still maintained in both pair and nonpair terms. In addition to such empirical potentials, however, there are also more rigorous quantum-based interatomic potentials (QBIPs) – potentials that are actually derived in whole or in part from the underlying quantum mechanics by suitably course graining of the electronic structure. The gap between electronic and atomistic methods and between QMD and MD/QBIP simulations is then directly related to the additional approximations entailed in such

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course graining. The size of the gap and our ability to bridge it is dependent both on the complexity of the material in question and the complexity of its environment. For some materials and/or some environments this gap is actually relatively small and can be readily bridged, such as is the case for bulk simple metals. For other materials and/or environments, however, the gap is larger and bridging it is still a forefront challenge of current research. This is the case, for example, with directionally-bonded transition metals and semiconductors as well as for chemically reactive surfaces. Nonetheless, significant progress has been made in the last decade or so in many of these latter areas, inspired in part by the demands of multiscale modeling.

2.

Quantum-based Interatomic Potentials

In the case of simple sp-bonded metals (e.g., Na, Mg, Al), a rigorous formulation of QBIPs in the bulk material has been available since the 1960s in terms of pseudopotential perturbation theory [6]. In this approach, it is recognized that the electronic structure of such materials is nearly free electron in character and that the electron-ion interaction can be represented by a weak nonlocal pseudopotential. Using an orbital basis of plane waves, one can develop E tot to second-order in the pseudopotential and express the result explicitly in the real-space form 1 E tot (R1 , . . . , R N ) = N E vol () + v 2 (Ri j ; ), (1) 2 i, j where is the atomic volume of the metal, E vol is a collective volume term that satisfies the many-body requirement for E tot , and v 2 is a volumedependent, but structure-independent and transferable, radial-force pair potential. The volume dependence of v 2 is a consequence of the self-consistent electron screening, which also gives rise to long-range Friedel oscillations in the potential tail. By the 1970s and 80s, first-principles DFT-based implementations of this approach had already become well developed [7–9]. This method is particularly effective in dealing with bulk structural properties, including phonons and elasticity, solid phase transitions, liquid structure and dynamics, and melting. This approach has also been readily and successfully extended to compounds and alloys as well as to high pressures, where atomic volume is a very compatible environmental variable. In the 1980s, interest in simulating materials properties beyond the bulk environment and particularly at surfaces led to the development of alternative “glue” models for simple metals [10]. These include the radial-force potential models obtained from the embedded-atom method (EAM) [11] and from effective medium theory (EMT) [12]. The total-energy functional in the EAM or EMT is inspired by the DFT notion that the total energy of a system is

Gap between quantum mechanics and large-scale atomistic simulation 2741 a functional of its electron density and is assumed to take the form of an attractive embedding contribution balanced by a repulsive pair-potential contribution: 1 F(n¯ i ) + v 2 (Ri j ), (2) E tot (R1 , . . . , R N ) = 2 i, j i where F is a nonlinear function of the average electron density n¯ i on the site i. The embedding contribution correctly accounts for the increased bond strength at the surface relative to the bulk, although Eq. (2) itself is an ansatz and cannot be directly derived from quantum mechanics. In the EMT, the ingredients of this equation are evaluated from first-principles DFT considerations within a well-defined prescription starting from the embedding of an atom in a free-electron gas. In the EAM, on the other hand, these ingredients are all treated empirically with convenient parameterized analytic forms chosen for F, n¯ i and v 2 to maximize flexibility and achieve high computational speed. Alternatively, F, n¯ i and v 2 have also been spline-fit to larger databases that also include DFT energies and forces. Regardless of how they are parameterized, the EAM and EMT models are most appropriate for simple sp-bonded metals and series-end transition metals (e.g., Cu), where a description of the bonding in terms of radial forces is reasonable, and to such systems these models been extensively applied in the past twenty years. A general approach to QBIPs that does allow one to go beyond radialforce interactions in a rigorous way makes use of a local-orbital, tight-binding (TB) representation of the electronic structure [13]. In principle, if an accurate TB representation can be found, one can then develop a QBIP model based on an expansion of the total energy in terms of moments of the local electronic density of states, µ2 , µ3 , µ4 , . . . . In practice, the notion of casting potentials in terms of moments has been used both to develop empirical models as well as in full quantum-mechanical derivations. Such approaches have been mainly directed at the d states in central transition metals (e.g., Mo) and at the s and p states in covalently bonded semiconductors (e.g., Si). The simplest empirical scheme in this category is the second-moment, radialforce Finnis–Sinclair (FS) model [14]. This model is formally similar to the “glue” models discussed above with an assumed embedding function in the form F(µ2 ) ∝ (µ2 )1/2 , where the second moment µ2 is treated empirically as a short-ranged radial function about each atomic site. The FS model has been mostly applied to central bcc transition metals, although at this level of treatment there are in fact no angular-force terms to accommodate the d-state directional bonding. In this regard, empirical fourth-moment schemes for transition metals have also been developed that do implicitly include angularas well as radial-force contributions. The most complete and fundamental TB approach to QBIPs, however, is the bond-order-potential (BOP) model of Pettifor [13], which is based on an explicit expansion of the total energy within

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TB theory and has been considerably developed and applied over the last ten years [15]. In all the TB schemes, an empirical repulsive pair-potential contribution is included in the total energy, as in Eq. (2), and parameterization of the local-orbital matrix elements defining the moments is required. In the BOP model, the embedding energy of Eq. (2) is directly replaced by the full TB bond energy derived from quantum mechanics for the dominant bonding electrons, e.g., the d electrons in transition metals. In such a case an additional empirical environmental energy correction term to the pair potential is also added to E tot to account for the fact that local s and p orbitals in a TB representation of the sp electrons are effectively environmentally dependent. Another general approach to QBIPs in metals involves combining a plane-wave-based pseudopotential treatment for s and p electrons with a localorbital-based tight-binding treatment for d electrons, allowing application to both simple and transition metals. The primary example of this approach is first-principles generalized pseudopotential theory (GPT), which has been rigorously developed from DFT quantum mechanics [16]. In the GPT applied to transition metals, a mixed basis of plane waves and localized d-state orbitals is used to self-consistently expand the electron density and total energy of the system in terms of weak sp pseudopotential, d-d tight-binding, and sp-d hybridization matrix elements, which in turn are all directly calculable from first principles. The GPT total-energy expansion has been carried out to the level of four-ion interactions and formally generalizes Eq. (1). In a bulk elemental transition metal, one obtains the explicit real-space form E tot (R1 , . . . , R N ) = N E vol () + +

1 1 v 2 (i j ; ) + v 3 (i j k; ) 2 i, j 6 i, j,k

1 v 4 (i j kl; ). 24 i, j,k,l

(3)

The leading volume term in this expansion, E vol , as well as the two-, three-, and four-ion interatomic potentials, v 2 , v 3 , and v 4 , are as in Eq. (1) volume dependent, but structure independent quantities and thus transferable to all bulk ion configurations, either ordered or disordered. This includes all structural phases as well as the deformed solid and the imperfect bulk solid with either point or extended defects present. The angular-force multi-ion potentials v 3 and v 4 in Eq. (3) reflect contributions from partially-filled d bands and are generally important for central transition metals. In the full ab initio GPT, however, these potentials are long-ranged, nonanalytic and multidimensional functions, so that v 3 and v 4 cannot be readily tabulated for application purposes. This has led to the development of a simplified and complementary model GPT or MGPT applicable to central transition metals [17]. Within the MGPT, the multi-ion potentials v 3 and v 4 are systematically approximated by introduing canonical d bands and other simplifications to

Gap between quantum mechanics and large-scale atomistic simulation 2743 achieve short-ranged, analytic forms, which can then be applied to large-scale atomistic simulations. To compensate for the approximations introduced into the MGPT, a limited amount of parameterization is allowed in which the coefficients of the modeled potential terms are constrained by either DFT or experimental data. In practice, the ab inito GPT and the MGPT potentials have complementary ranges of application. The ab initio GPT is most effective in situations where the total-energy expansion (3) can be truncated at the pair-potential level, as in Eq. (1), since tabulation and interpolation of a nonanalytic pair potential v 2 (r, ) represents no computational barrier for atomistic simulations. Thus ab initio GPT applications include simple metals and series-end transition metals as well as appropriate binary alloys, including the transitionmetal aluminides [18]. The primary application range for the MGPT, on the other hand, is the bcc transition metals (e.g., Ta, Mo). Both GPT and MGPT potentials have been implemented in atomistic simulations and applied to a wide range of bulk structural, thermodynamic, defect and mechanical properties at both ambient and extreme conditions of temperature and pressure [17]. Extension of the bulk GPT and MGPT potentials to highly nonbulk situations, such as surfaces, voids and clusters, is also possible through appropriate environmental modulation [19]. This refinement has only been studied in detail, however, in the cases of free surfaces, where environmental corrections have been shown to be very important (∼ 50 − 70%), and for vacancies, where such corrections have been confirmed to be negligible (∼1 − 2%).

3.

Outlook

There are still many remaining challenges in bridging the gap between quantum and atomistic simulations. One inherent advantage that QBIPs have over empirical potentials in this quest is that they are systematically improvable in a manner consistent with quantum mechanics. In spite of the significant progress made over the past forty years, the collective amount of time and energy spent to date on developing QBIPs has actually been very small compared to that spent on developing advanced electronic-structure methods and quantum simulations themselves. As a result, this is still a young research field that has very much room to grow. Below we discuss three general areas where there would seem to be great opportunities for major progress over the next decade.

3.1.

Improved Accuracy and Computational Speed

In first-principles QBIPs such as GPT, the main additional approximation beyond DFT is the truncation of the total-energy expansion at finite order.

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For both simple and transition metals, it is now computationally feasible to extend this expansion to higher order as needed. For transition metals in particular, it should be possible to extend Eq. (3), both in the GPT and MGPT representations, to include five- and six-ion d-state interactions. This would improve the description of certain structural properties, such as the hcp-fcc energy difference and corresponding stacking faults, and would enable accurate applications both to the left and right of the central bcc metals. Moreover, in the context of semi-empirical QBIPs such as MGPT or BOP, it should be possible to eliminate or improve many of the secondary approximations that are currently used. In the MGPT, for example, it has recently been possible to formulate a more general matrix version of the theory that allows one to go beyond simple canonical d bands and hence achieve a more accurate representation of the electronic structure. Also, in both MGPT and BOP current applications to transition metals, explicit sp-d hybridization contributions have been dropped for convenience, but these should be included in the future, especially for non-central transition metals. The balancing consideration to increased accuracy is, of course, computational speed. One enormously appealing aspect of empirical EAM potentials is their extreme speed, which can be up to six orders of magnitude faster than first-principles DFT electronic-structure or QMD methods and up to two orders of magnitude faster than QBIPs. While angular-force QBIPs will be inherently slower than radial-force EAM potentials, a reasonable goal would be to come within one order of magnitude of their speed, at least at some basic level of operation. In the case of MGPT, recent algorithm improvements have increased computational speed dramatically by up to a factor of six and put us at or close to that goal. Adding higher-order interactions and/or sp-d hybridization will, of course, work to reverse that gain, but inevitably one must think in terms of having QBIPs at various levels of approximation and match the level and speed with the intended application.

3.2.

Treatment of Complex Systems and Complex Environments

The general application area of intermetallic compounds and alloys would seem to a potentially ideal one for QBIPs since in general they are much better positioned to handle chemical and structural complexity than empirical potentials. In this regard, BOP and GPT treatments of transition-metal intermetallics have already had significant success and these applications are expected to continue and grow in the future. Future MGPT and hydrid MGPT/GPT treatments also look very promising for transition-metal rich systems. Similarly, another fruitful application area is expected to be high-pressure physics. Here

Gap between quantum mechanics and large-scale atomistic simulation 2745 volume-dependent GPT and MGPT treatments to date have proven to be successful, both in terms of predicting new high-pressure phases and in describing how materials properties scale with pressure for an existing phase. More generally, QBIPs seem to be well suited to deal with changes in structural stability under pressure and the prediction of the thermodynamics and mechanical properties of new encountered phases. The application of QBIPs to non-bulk environments, such as free surfaces, voids and clusters, would also seem to be a potentially important area, especially for doing large-scale simulations involving growth and interaction that are beyond the reach of QMD. In this case, however, significantly more developmental work may be needed, since applications to date have been limited and bulk assumptions and approximations often require modification near surfaces. Yet another promising application area for QBIPs is that of f -electron metals. The BOP and GPT/MGPT transition-metal methodologies are readily adaptable from d to f electrons, at least in the weak correlation DFT regime. Initial MGPT applications in this area look promising, but an added challenge is the structural complexity of some f - electron phases. Treating strongly correlated systems poses another new challenge that will require going beyond DFT to correlated-electron theories such dynamical mean field theory (DMFT) and re-building the bridge to QBIPs from that starting point. This looks possible but no work has yet been done in this area.

3.3.

Temperature-dependent QBIPs and Direct Linkage with QMD

One interesting possible use of QMD simulations is to interface them with MD/QBIP simulations both to extend the time scale of the QMD simulations and to develop improved QBIPs at finite temperature. This linkage has actually been tried with empirical potentials in a few cases, but mostly as an interpolation mechanism and without any regard as to whether or not the potentials had physical meaning at the conditions used. For d- and f -electron metals, however, the concept of temperature-dependent potentials is actually a very important one. In such materials there are large electron-thermal effects at temperatures as low as melt arising from the high density of electronic states at the Fermi level. These effects can have a dramatic impact on high-temperature properties including the melt curve itself. Electron-thermal effects are normally treated separately from the more familiar ion-thermal effects that are associated with QBIPs constructed at zero temperature. It should be possible, however, to capture the coupled electron plus ion thermal effects simultaneously and self-consistently by building QBIPs on the basis of the total electron free energy at finite temperature. In principle, this can be done without resort

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to QMD simulations, but adding the possibility of refining such temperaturedependent potentials by matching QMD and MD/QBIP simulations on the fly might substantially improve the accuracy of such potentials.

Acknowledgment This work was performed under the auspices of the U.S. Department of Energy by the University of California Lawrence Livermore National Laboratory under contract number W-7405-ENG-48.

References [1] J.A. Moriarty, V. Vitek, V.V. Bulatov, and S. Yip, “Atomistic simulation of dislocations and defects,” J. Comput.-Aided Mater. Desi., 9, 99–132, 2002. [2] R. Car and M. Parrinello, “Unified approach for molecular dynamics and densityfunctional theory,” Phys. Rev. Lett., 55, 2471–2474, 1985. [3] M.C. Payne, M.P. Teter, D.C. Allan, T.A. Arias, and J.D. Joannopoulos, “Iterative minimization techniques for ab initio total-energy calculations: molecular dynamics and conjugate gradients,” Rev. Mod. Phys., 64, 1045–1097, 1992. [4] P. Hohenberg and W. Kohn, “Inhomogeneous electron gas,” Phys. Rev., 136, B864– B871, 1964. [5] W. Kohn and L.J. Sham, “Self-consistent equations including exchange and correlation effects,” Phys. Rev., 140, A1133–A1138, 1965. [6] W.A. Harrison, “Pseudopotentials in the theory of metals,” Benjamin, Reading, 1966. [7] L. Dagens, M. Rasolt, and R. Taylor, “Charge densities and interionic potentials in simple metals: Nonlinear effects II,” Phys. Rev. B, 11, 2726–2734, 1975. [8] A.K. McMahan and J.A. Moriarty, “Structural phase stability in third-period simple metals,” Phys. Rev. B, 27, 3235–3251, 1983. [9] J. Hafner, “From Hamiltonians to phase diagrams,” Springer-Verlag, Berlin, 1987. [10] R.M. Nieminen, M.J. Puska, and M.J. Manninen (eds.), “Many-atom interactions in solids,” Springer-Verlag, Berlin, 1990. [11] M.S. Daw, S.M. Foiles, and M.I. Baskes, “The embedded atom method: a review of theory and applications,” Mat. Sci. Rep., 9, 251–310, 1993. [12] K.W. Jacobsen, J.K. Norskov, and M.J. Puska, “Interatomic interactions in the effective-medium theory,” Phys. Rev. B, 35, 7423–7442, 1987. [13] D.G. Pettifor, “Bonding and structure of molecules and solids,” Oxford University Press, Oxford, 1995. [14] M.W. Finnis and J.E. Sinclair, “A simple N-body potential for transition metals,” Philos. Mag. A, 50, 45–55, 1984. [15] M. Mrovec, D. Nguyen-Manh, D.G. Pettifor, and V. Vitek, “Bond-order potential for molybdenum: application to dislocation behavior,” Phys. Rev. B, 69, 94115–94130, 2004. [16] J.A. Moriarty, “Density-functional formulation of the generalized pseudopotential theory. III. Transition-metal interatomic potentials,” Phys. Rev. B, 38, 3199–3230, 1988.

Gap between quantum mechanics and large-scale atomistic simulation 2747 [17] J.A. Moriarty, J.F. Belak, R.E. Rudd, P. S¨oderlind, F.H. Streitz, and L.H. Yang, “Quantum-based atomistic simulation of materials properties in transition metals,” J. Phys.: Condens. Matter, 14, 2825–2857, 2002. [18] J.A. Moriarty and M. Widom, “First-principles interatomic potentials for transitionmetal aluminides: theory and trends across the 3d series,” Phys. Rev. B, 56, 7905– 7917, 1997. [19] J.A. Moriarty and R. Phillips, “First-principles interatomic potentials for transitionmetal surfaces,” Phys. Rev. Lett., 66, 3036–3039, 1991.

Perspective 14 BRIDGING THE GAP BETWEEN ATOMISTICS AND STRUCTURAL ENGINEERING J.S. Langer Department of Physics, University of California, Santa Barbara, CA 93106-9530, USA

When Sid Yip asked me to write a commentary for this section of the handbook, I promptly reminded him that I am a co-author of a longer article in the section on mathematical methods. I told him that my article on amorphous plasticity, written with Michael Falk and Leonid Pechenik, already is more of a departure from conventional ideas than may be appropriate for a book like this one, which should serve as a reliable reference for years into the future; and I asked whether I really ought to be given yet more space for expressing my opinions. Sid insisted that I should write the commentary anyway. So here are some remarks about one of the topics of interest in this book, the search for predictive models of deformation and failure of solids, and the role of nonequilibrium physics in this effort. Like many of my colleagues, I am impatient about the slow rate of progress in theoretical solid mechanics. We theorists have been given great opportunities. Remarkable developments in instrumentation and computation have advanced our knowledge about the atomic-scale behavior of solids far beyond what most of us could have imagined a decade or so ago; and yet it seems to me that our ability to bring that knowledge to bear on practical problems has not kept pace. I blame ourselves – the theorists – for this state of affairs. We have not been quick enough to explore new concepts that might move us from atomistic models and numerical simulations to engineering practice. To bridge this gap between atomistics and structural engineering, it seems almost trivially obvious that we need new phenomenologies. Our goal must be to develop predictive, quantitative and tractable descriptions of an enormously wide range of complex materials and processes. First-principles theories may be necessary to get us started, but they do not take us far enough by themselves, especially if they require that each physical situation be treated separately 2749 S. Yip (ed.), Handbook of Materials Modeling, 2749–2756. c 2005 Springer. Printed in the Netherlands.

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as in molecular dynamics simulations or even the most powerful multi-scale analyses. Moving across these length and time scales, learning how the smallscale phenomena fit together to produce complex, larger scale behaviors, is every bit as important and challenging a research goal as is atomic-scale investigation. Phenomenological research of the kind needed here requires physical insight to extract the general principles from the less relevant details and, therefore, it necessarily involves a substantial amount of guesswork. The classic phenomenological models that are most relevant to deformation and failure in solids are Hookean elasticity for static stress analysis and the Navier–Stokes equation for fluid dynamics. In both of those cases, the atomicscale theories can in principle be used to compute constitutive parameters such as elastic constants or viscosities; and those first-principles calculations are very valuable in themselves because they tell us about the limits of validity of the phenomenological descriptions. Much of the value of phenomenology, however, lies in the fact that it is usually easier and more reliable to obtain the constitutive parameters experimentally. The challenge is to make sure that the phenomenological framework truly captures the essential features of the systems that we need to describe. In elasticity and fluid dynamics, the essential ingredients are Newton’s laws of motion plus continuity and symmetry criteria. Those must be the basic ingredients of a theory of solid plasticity, but they are not sufficient. The Navier–Stokes analogy is particularly relevant to my argument because I want to talk mostly about noncrystalline materials. Deformable amorphous solids are very similar to fluids in all but a few, albeit very important, respects. Although their molecular structures look very much like those of fluids, they support shear stresses, they exhibit stress-driven transitions between jammed and flowing states, and they even exhibit memory effects. Nevertheless, because of their molecular-level similarities, I see no fundamental reason why amorphous solids should not be amenable to a level of analysis roughly similar to that which we use for fluids. Moreover, I suspect that, once we have found a useful way of describing the dynamics of deformation in amorphous solids, we shall be well on our way to a useful description of polycrystalline materials as well. When I make remarks like these in public lectures, I am invariably accused (not always politely) of ignoring the huge body of literature and tradition in plasticity theory. Indeed, conventional approaches to plasticity have been extremely successful in conventional engineering applications; but many of the problems that these theories must now confront – in biological materials, for example – are distinctly unconventional. Textbook treatments of plasticity generally appear in two different forms, one based on the usual formulation of elasticity supplemented by phenomenological stress-strain relations and plastic yield criteria, and another liquid-like approach that focuses on rheological relations between stresses and strain rates, usually with no reference to yield

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stresses or the irreversible deformations that occur in low-stress, non-flowing regimes. The hydrodynamic analogy, however, tells us that we should not try to separate these two kinds of descriptions. Deformation or failure in a bounded material under loading is usually a localized phenomenon; plastic flow occurs where the stresses are large while, elsewhere, the stresses are relaxed and the material behaves nearly elastically. The material may harden in some places and soften in others. To be truly useful, therefore, our phenomenological equations of motion must incorporate all of those behaviors, and they must do so in a natural and relatively simple way. What new concepts and analytic tools will be needed in order to develop a unified theory of this kind? How might those techniques differ from the ones we have been using in the past? I shall try to start answering these questions by pointing to some puzzles and internal inconsistencies that persist in the conventional theories. These puzzles include the question of how breaking stresses can penetrate plastic zones near crack tips, and the possibly related question of why brittle fracture becomes dynamically unstable at high speeds. I know of no convincing solution to either of those problems, certainly not for the noncrystalline materials in which the definitive experiments have been performed; and the fact that these apparently simple problems have remained unsolved for such a long time is, by itself, enough to convince me that there is something seriously missing in our theories. Here, however, I would like to focus on a few more basic questions that I think lead us to a better understanding of what the missing ingedients might be. The most elementary and familiar of these questions is: What are the fundamental distinctions between brittle and ductile behaviors? A brittle solid breaks when subjected to a large enough stress, whereas a ductile material deforms plastically. Remarkably, we do not yet have a deep understanding of the distinction between these two behaviors. Conventional theories of crystalline solids say that dislocations form and move more easily through ductile materials than brittle ones, thus allowing deformation to occur in one case and fracture in the other. But the same behaviors occur in amorphous solids; thus the dislocation mechanism cannot be the essential ingredient of all theories. Moreover, the brittleness or ductility of some materials depends upon the speed of loading, which implies that a proper description of deformation and fracture must be dynamic; that is, it must be expressed in the form of equations of motion rather than the conventional static or quasistatic formulations. A second question that I find especially revealing is the following: What is the origin of memory effects in plasticity? Standard, hysteretic, stress-strain curves for deformable solids tell us that these materials – even the simplest amorphous ones – have rudimentary memories. For example, they “remember” the direction in which they most recently have been deformed. When unloaded and then reloaded in the original direction, they harden and respond elastically, whereas, when loaded in the opposite direction, they deform

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plastically. The conventional way of dealing with such behavior is to specify rules for how the response to an applied stress is determined by the history of prior loading; but such rules provide little insight about the nature of a theory based more directly on atomic mechanisms. A much better way to deal with memory effects is to introduce internal state variables that carry information about prior history and determine the current response of the system to applied forces. The trick is to identify the relevant variables. I am coming to believe that this is one of the main points at which a gap opens between atomistic understanding and engineering practice. All too often, for example, the plastic strain itself is used as such a state variable. This procedure has its roots in the conventional Lagrangian formulation of solid mechanics in which deformations of a material are described by displacements relative to fixed reference states. When applied to materials undergoing irreversible plastic deformations, such a procedure violates basic principles of nonequilibrium physics because, if taken literally, it implies that a material somehow must remember its configurations at times arbitrarily far in the past. That cannot be possible for an amorphous solid any more than it is for a liquid, where it is well understood that only displacement rates, and not the displacements themselves, may appear in equations of motion. When a solid undergoes a sequence of loadings and unloadings, bendings and stretchings, the displacement of an element of material from its original position cannot possibly be a physically meaningful quantity, thus it cannot be a sensible way of characterizing the internal state of the system. Nevertheless, the use of the total plastic strain, for example as a “hardening parameter,” appears frequently in the literature on plasticity. What, then, are the appropriate state variables for amorphous solids? My proposed answer to this question starts with the “flow-defect” or “sheartransformation-zone” (STZ) picture of Cohen et al. [1–4], in which plastic deformation occurs only at localized sites where molecules undergo irreversible rearrangements in response to applied stresses. Falk, Pechenik and I, in our paper in this volume (here denoted "FLP"), present a critical analysis of those earlier STZ theories, which I will not repeat in any detail here. On the plus side, these theories nicely satisfy my criteria for sensible phenomenological approaches. Their central ingredient is an internal state variable, i.e., the density of zones, and they generally postulate equations of motion for this density. On the other hand, they make a crucial assumption with which I disagree – that the plastic flow is equal simply to the density of zones multiplied by a stressand temperature-dependent Eyring rate factor. The interesting behavior, then, is contained in various assumptions about rates of annihilation and creation of zones as functions of temperature and strain rate. Such theories can do reasonably well in accounting for some rheological and calorimetric behaviors of, say, metallic or polymeric glasses. They do not predict yield stresses, however,

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nor can they convincingly account for the wide range of dynamic behavior observed recently by Johnson and coworkers in bulk metallic glasses. In the work described in FLP and in earlier papers [5–8], we have extended and modified the original STZ theories in two ways. First, instead of assuming that the STZ’s are structureless objects, we have modeled them as two-state systems; that is, we have assumed that they transform back and forth between two different orientations in response to applied stresses. This two-state picture is inspired by molecular-dynamics simulations [5]. Among other implications, it tells us that we must supplement the scalar density by a tensorial quantity that carries information about the orientations of the zones, and that the actual plastic flow will depend upon those orientations. The appearance of this internal state variable produces entirely new dynamical properties, specifically, jamming behavior when the zones are all aligned with the stress and cannot transform further in the same direction, and a yield stress at which the jammed state starts to flow. Although the STZ’s are structural irregularities that live in an unperturbed material for very long times, they are ephemeral in the sense that they are created and annihilated during irreversible deformations. These annihilation and creation terms play the same roles as those that appear in the original STZ theories. It is here that we have made the second of our basic changes using a combination of phenomenological guesswork and the constraints imposed by symmetry and the second law of thermodynamics. In particular, we have argued that the simplest possible creation rate is proportional to the rate at which energy is dissipated during plastic deformation, which is necessarily a non-negative scalar quantity. This phenomenological assumption leads us to a rate factor that is substantially different from earlier versions, and which seems free from unphysical features. As described in more detail in FLP, our equations of motion for the STZ populations are best expressed in terms of two dimensionless state variables: , a scalar field proportional to the density of STZ’s, and i j , a traceless symmetric tensor field that describes the local orientation of the zones. The full theory is necessarily expressed in Eulerian coordinates, as in fluid dynamics. It consists of equations of motion for and i j supplemented by the usual acceleration equation relating the vector flow field v i to the divergence of the stress σi j , and an equation expressing the rate-of-deformation tensor as the sum of elastic and plastic parts. Because these equations refer to solids rather than liquids, they are necessarily more complicated than Navier–Stokes; but they are capable of serving similar purposes. The most unconventional result of this theory is the way in which the yield stress emerges. The equations of motion for an isotropic system have two kinds of steady state solutions at fixed applied stress. One of these solutions is jammed, i.e., non-flowing, and the other is unjammed. The jammed

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solution is dynamically stable below some stress (a function of the material parameters) and is unstable above that stress. Conversely, the unjammed solution is unstable below this “yield” stress and stable above it. Thus the conventional yielding criterion is replaced by an exchange of dynamic stability between two branches of steady-state solutions of a set of coupled, nonlinear differential equations. The physical interpretation of this situation is that, at smaller stresses, the two-state STZ’s become saturated in the direction of the stress – the magnitude of the orientational bias i j reaches a stress-dependent maximum – and the motion stops. At larger stresses, jammed zones are annihilated and unjammed ones created fast enough to sustain steady-state plastic flow. The resulting dynamic version of an STZ theory reproduces a wide range of the phenomena observed in plastically deforming materials. Depending upon the choice of just a small number of material parameters and initial conditions, theoretical sress-strain curves may exhibit work hardening, strain softening (for annealed samples with low initial densities of STZ’s), strain recovery following unloading, Bauschinger effects, necking instabilities, and the like. With the addition of thermal fluctuations that cause spontaneous relaxation of the STZ state variables, the theory quantitatively explains the experimentally observed transition between Newtonian viscosity at small loading to superplastic flow at larger stresses as a transition from thermally assisted creep at small stress to plastic flow at the STZ yield stress. There are also some interesting shortcomings. In the form described in FLP, the theory does not predict the results of calorimetric measurements. Also, like almost all other theories of plasticity, this version of the STZ theory lacks an intrinsic length scale. The theory does show signs of shear banding instabilities; but a complete theory of shear banding will have to predict both the width of the bands and the thickness of the transition region between flowing and jammed material. I shall conclude my remarks by suggesting that these shortcomings may be associated with a second theoretical gap in solid mechanics. I am thinking of the largely unexplored possibility that the statistical physics of a nonequilibrium system such as a deforming solid, even when it is deforming very slowly, may be qualitatively different from that of a system in thermal and mechanical equilibrium. Specifically, I want to raise the possibility that, during irreversible plastic deformation, the slow, configurational degrees of freedom associated with molecular rearrangements may fall out of thermal equilibrium with the fast, vibrational degrees of freedom that couple strongly to a thermal reservoir. The statistical properties of both of these kinds of degrees of freedom may be described by “temperatures”; and the two temperatures may be quite different from one another under nonequilibrium conditions. This kind of effective temperature is formally similar to the free volume introduced by Spaepen and others, and can be used in much the same ways. The important difference is

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that the effective temperature is a measure of the configurational disorder in the system. Like ordinary temperature, it is an intensive quantity, and does not necessarily carry any implication of volume changes. It does, however, have clear thermodynamic consequences. Evidence for the existence of well defined effective disorder temperatures is emerging from recent studies of granular materials, foams, and related systems [9–13]. (So far, most of these studies are based on numerical simulations.) The elementary components of such systems, such as sand grains, are much too large for the ambient temperature to be relevant to their motions. Nevertheless, the use of fluctuation-dissipation relations in conjunction with measurements of diffusion constants, viscosities, stress fluctuations and the like, yield estimates of effective temperatures that are nonzero and remarkably consistent with one another. How might the addition of an effective disorder temperature resolve the remaining shortcomings of the STZ equations? In principle, this concept should be closely related to the mechanism that we have postulated for the annihilation and creation of STZ’s. That is, the rate of energy dissipation associated with plastic deformation must also be the heat source for the effective temperature. There also must be a cooling mechanism by which the effective temperature decreases in the absence of driving forces. The latter two effects, which couple the effective temperature to the ordinary bath temperature, should determine the calorimetric properties of the material. Finally, it will be important that this disorder temperature diffuses from hotter to cooler regions of the material; but its diffusion constant must naturally be very much smaller than that for ordinary temperature because the associated molecular rearrangements are very much slower than thermal vibrations. Higher effective disorder temperatures imply a higher density of STZ’s and thus a higher plastic strain rate at fixed stress, which in turn implies nonlinear amplification of the plastic response to driving forces. Preliminary investigations indicate that the resulting theory predicts experimentally interesting behavior of this kind. The effective thermal enhancement of plastic flow also appears to imply a picture of shear banding in which the flowing material inside the band is “hotter” than the jammed material outside, and the thickness of the boundary between the two regions is determined by the length scale contained in the effective – not the ordinary thermal – diffusion constant. In summary, I think that a dynamic version of the STZ theory has a good chance of closing the gap between atomistics and engineering applications. Essential elements of the theory are the identification of physically meaningful state variables, the choice of rate factors that are consistent with basic principles of nonequilibrium physics, and – perhaps – an effective disorder temperature to account for the fact that the configurational degrees of freedom may fall out of equilibrium with the heat bath in systems undergoing irreversible deformations.

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References [1] D. Turnbull and M. Cohen, J. Chem. Phys., 52, 3038, 1970. [2] F. Spaepen, Acta Metall., 25(4), 407, 1977. [3] F. Spaepen and A. Taub, In: R. Balian and M. Kleman (ed.), Physics of Defects, Les Houches Lectures, North Holland, Amsterdam, p. 133, 1981. [4] A.S. Argon, Acta Metall., 27, 47, 1979. [5] M.L. Falk and J.S. Langer, Phys. Rev. E, 57, 7192, 1998. [6] L.O. Eastgate, J.S. Langer, and L. Pechenik, Phys. Rev. Lett., 90, 045506, 2003. [7] J.S. Langer and L. Pechenik, Phys. Rev. E, 2003. [8] M.L. Falk, J.S. Langer, and L. Pechenik, Phys. Rev. E, 70, 011507, 2004. [9] I.K. Ono, C.S. O’Hern, D.J. Durian, S.A. Langer, A. Liu, and S.R. Nagel, Phys. Rev. Lett., 89, 095703, 2002. [10] L. Cugliandolo, J. Kurchan, and L. Peliti, Phys. Rev. E, 55, 3898, 1997. [11] P. Sollich, F. Lequeux, P. Hebraud, and M. Cates, Phys. Rev. Lett., 78, 2020, 1997. [12] L. Berthier and J.-L. Barrat, Phys. Rev. Lett., 89, 095702, 2002. [13] D.J. Lacks, Phys. Rev. E, 66, 051202, 2002.

Perspective 15 MULTISCALE MODELING OF POLYMERS Doros N. Theodorou School of Chemical Engineering, National Technical University of Athens, 9 Heroon Polytechniou Street, Zografou Campus, 157 80 Athens, Greece

Meeting today’s technological challenges calls for a quantitative understanding of structure-property-processing-performance relations in materials. Developing precisely this understanding constitutes the main objective of materials modeling and simulation. Along with novel experimental techniques, which probe matter at an increasingly finer scale, and new screening strategies, such as high-throughput experimentation, modeling has become an indispensable tool in the development of new materials and products. A challenge faced by materials modelers, which is especially serious in the case of polymeric materials, is that structure and dynamics are characterized by extremely broad spectra of length and time scales, ranging from tenths of nanometers to centimeters and from femtoseconds to years [1]. It is by now generally accepted that the successful solution of design problems involving polymers calls for hierarchical, or multiscale, modeling or simulation involving a judicious combination of atomistic (<10 nm), mesoscopic (10–1000 nm) and macroscopic methods. A rough sketch of how multiscale modeling approaches can be developed for polymeric materials is given in Fig. 1. Quantum mechanical methods can take us from chemical constitution to the bonded geometry, to electronic properties, as well as to potentials describing the interactions between building blocks in a material. Using molecular geometry and potentials as input, statistical mechanicsbased theories and molecular simulations can provide estimates of macroscopic thermal, mechanical, rheological, electrical, optical, interfacial, permeability, and other properties, as well as information on equations of state and constitutive relations governing equilibrium and nonequilibrium behavior. In addition, molecular simulations provide a wealth of detailed information on molecular organization and motion and their consequences for properties; this information can be validated against, and used to interpret contemporary microscopic measurements and is of great value for materials design. Atomistic simulations are 2757 S. Yip (ed.), Handbook of Materials Modeling, 2757–2761. c 2005 Springer. Printed in the Netherlands.

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Figure 1. Some methods and interconnections involved in multiscale modeling of polymeric materials [1].

quite limited in terms of the length scales (typically on the order of 10 nm) and time scales (typically on the order of 10–100 ns for molecular dynamics (MD)) they can address. To deal with longer time- and length scale phenomena, one can resort to mesoscopic simulations. These employ coarse-grained representations of the material, cast in terms of fewer degrees of freedom. They can use as input information derived from atomistic simulations. For example, atomistically calculated chain dimensions, bulk densities, and cohesive energy densities can be used to estimate the radii of gyration, bending energies, and interaction parameters invoked by self-consistent field (SCF) theories or dynamic density functional theories of inhomogeneous polymers; rate constants for elementary jumps, calculated by atomistic transition-state theory (TST), can be fed to kinetic Monte–Carlo (KMC) simulations to track diffusion in glassy polymers over micro- or millisecond time scales; and potentials of mean force between coarse-grained moieties, computed atomistically, can be used within Brownian dynamics or dissipative particle dynamics (DPD) simulations. Nonequilibrium mesoscopic simulations are particularly helpful for addressing how the processing conditions imposed on a material affect its morphology and microstructure. Finally, macroscopic calculations, based on the continuum engineering sciences, can derive input from all previous levels of modeling to predict product performance under specific application conditions and address materials and product design issues. Multiscale modeling approaches may be either sequential, involving series of simulations at progressively coarser or finer levels, or parallel, involving simultaneous simulations of phenomena at different length and time scales

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and passage of information between the different simulations. An example of a parallel approach is the simultaneous simulation of plastic deformation in a polymeric glass by continuum finite elements and atomistic molecular dynamics [2]; the atomistic simulation is carried out in a small volume embedded in the continuously represented material and provides the fundamental constitutive relation for the continuum deformation, which is tracked with finite element methods. Efficient sampling of configuration space according to the probability density of specific equilibrium ensembles is a prerequisite for the reliable calculation of thermodynamic properties of melts and solutions. Recent breakthroughs in the design of Monte–Carlo (MC) algorithms for polymers have dramatically enhanced our ability to sample configuration space, even for systems of long, entangled chains, where the longest relaxation time scales with the 3.4 power of the chain length for linear architectures. Examples of such algorithms are Configurational Bias MC, which offers itself for phase equilibrium calculations [3], Concerted Rotation MC, the connectivity-altering End Bridging and Double Bridging MC [4] and various combinations of the above with each other and with parallel tempering and expanded ensemble schemes. New density of states MC algorithms [5] open another promising avenue for bold sampling of configuration space; they have already been applied to single biological macromolecules with considerable success. For polymers of complex chemical constitution, a promising strategy for sampling configurations involves coarse-graining atomistic models into models cast in terms of fewer degrees of freedom and therefore governed by smoother potentials of mean force; equilibrating at the coarse-grained level; and reverse mapping back to the atomistic representation. Approaches for doing this using both lattice-based [6, 7] and continuous-space [8–10] models are under development. The ability to equilibrate long-chain polymer models at all length scales has opened up promising avenues for predicting polymer melt viscoelastic properties, which are of key importance in processing operations. Recent accomplishments include the computation of segmental friction factors, entanglement tube diameters and zero-shear viscosities by mapping atomistic MD trajectories onto the Rouse and reptation models [1] and the determination of entanglement structures and plateau moduli through direct topological analysis of well-equilibrated melt configurations [11]. Entanglement network-based KMC approaches have already proven useful for the prediction of rheological [12] and mechanical [13] properties from the architecture and length distribution of chains in the bulk and at interfaces. Thus, recent developments generate excellent prospects for connecting these properties rigorously all the way through to atomic-level chemical constitution. Modeling methods have already emerged into valuable tools for addressing molecular-level design issues. Examples include the computational

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investigation of permeability properties of glassy and rubbery polymeric materials for industrial gas separations and packaging applications, based on MD simulations, TST analysis and KMC; the prediction of stable and metastable equilibrium morphologies resulting from self-organization in multicomponent block-copolymer based systems, including self-adhesive materials, based on field-theoretic methods [14]; the development of new design principles for self-healing nanocomposites or nanocomposites with controlled optical properties based on a combination of SCF and density functional theory; calculations of phase and microphase separation phenomena under conditions of flow in polymer, block copolymer, and surfactant solutions through dynamic density functional theory [15] and DPD simulations [16]; and computation of intercalation and exfoliation phenomena in clay-filled nanocomposites through coarse-grained MD. There is still much to be done in establishing rigorous quantitative links between the various levels of simulation invoked in the applications mentioned above. Projection strategies, whereby one can pass from detailed atomistic to coarse-grained descriptions of thermodynamics and dynamics cast in terms of a small number of coarse-grained degrees of freedom or order parameters, and sampling strategies, whereby one can generate ensembles of detailed configurations consistent with a given coarse-grained description, are objects of active current research. There are several frontier problems involving polymeric materials, where methodological breakthroughs in multiscale modeling would be highly desirable. One such problem is polymer crystallization. Prediction of the correct crystal structure from chemical constitution requires the use of refined atomistic models; yet the long time scales of nucleation phenomena and the intricate, hierarchical, processing history-dependent semicrystalline morphologies (e.g., spherulites, axialites) obtained from polymer melts can only be addressed at a mesoscopic level. At the latter level, recent coarse-grained MD simulations [17] appear promising for understanding, at least qualitatively, nucleation and growth, melting and crystallization phenomena. Another frontier problem is the generation of polymer glasses with a formation history that is both realistic and well-defined. MD cooling from the melt is restricted to cooling rates in excess of 109 K s−1 , which are many orders of magnitude higher than those encountered in most applications. On the other hand, generating molecular packings at glassy densities through energy minimization and MD techniques may give satisfactory results for many purposes, but is difficult to map onto a well-defined, experimentally realizable vitrification history. Given the intense interest in and research talent devoted to multiscale modeling of polymers worldwide, there is every indication that challenges such are these will soon be overcome, and that computer-aided molecular design will steadily gain ground in future materials science and engineering.

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References [1] D.N. Theodorou, “Understanding and predicting structure-property relations in polymeric materials through molecular simulations,” Mol. Phys., 102, 147–166, 2004. [2] J.S. Yang, W.H. Jo, S. Santos, and U.W. Suter, “Plastic deformation in bisphenol-Apolycarbonate: applying an atomistic-continuum model,” In: M. Kotelyanskii, D.N. Theodorou (eds.), Simulation Methods for Polymers, Marcel Dekker, New York, 2004. [3] T.S. Jain and J.J. de Pablo, “Configurational bias techniques for simulation of complex fluids,” In: M. Kotelyanskii and D.N. Theodorou (eds.), Simulation Methods for Polymers, Marcel Dekker, New York, 2004. [4] D.N. Theodorou, “Variable connectivity monte carlo algorithms for the atomistic simulation of long-chain polymer systems,” In: P. Nielaba, M. Mareschal, and G. Ciccotti (eds.), Bridging Time Scales : Molecular Simulations for the Next Decade, Springer-Verlag, Berlin, pp. 69–128, 2002. [5] F.G. Wang and D.P. Landau, “Efficient, multiple-range random walk algorithm to calculate the density of states,” Phys. Rev. Lett., 86, 2050–2053, 2001. [6] J. Baschnagel, K. Binder, and W. Paul, “On the construction of coarse-grained models for linear flexible polymer chains: distribution functions for groups of consecutive monomers,” J. Chem. Phys., 95, 6014–6025, 1991. [7] R.F. Rapold and W.L. Mattice, “Introduction of short and long range energies to simulate real chains on the 2nd lattice,” Macromolecules, 29, 2457–2466, 1996. [8] W. Tch¨op, K. Kremer, J. Batoulis, T. B¨urger, and O. Hahn, “Simulation of polymer melts I. Coarse-graining procedures for polycarbonates,” Acta Polymerica, 41, 61–74 1998a. [9] W. Tch¨op, K. Kremer, O. Hahn, J. Batoulis, and T. B¨urger, “Simulation of polymer melts II. From coarse-grained models back to atomistic description,” Acta Polymerica 41, 75–79, 1998b. [10] D. Reith, M. P¨utz, and F. M¨uller-Plathe, “Deriving effective mesoscale potentials from atomistic simulations,” J. Comput. Chem., 24, 1624–1636, 2003. [11] R. Everaers, S.K. Sukumaran, G.S. Grest et al. “Rheology and microscopic topology of entangled polymeric liquids,” Science, 202, 823–826, 2004. [12] M. Doi and J. Takimoto, “Molecular modeling of entanglement,” Philos. T. Roy. Soc. A, 361, 641–50, 2003. [13] A.F. Terzis, D.N. Theodorou, and A. Stroeks, “Entanglement network of the polypropylene/polyamide interface 3. Deformation to fracture,” Macromolecules, 35, 508–521, 2002. [14] G.H. Fredrickson, V. Ganesan, and F. Drolet, “Field-theoretic computer simulation methods for polymers and complex fluids,” Macromolecules, 35, 16–39, 2002. [15] A.V. Zvelindovsky, G.J.A. Sevink, and J.G.E.M. Fraaije, “Dynamic mean-field DFT approach to morphology development,” In: M. Kotelyanskii and D.N. Theodorou (eds.), Simulation Methods for Polymers, Marcel Dekker, New York, 2004. [16] W.K. den Otter and J.H.R. Clarke, “Simulation of polymers by dissipative particle dynamics,” In: M. Kotelyanskii and D.N. Theodorou (eds.), Simulation Methods for Polymers, Marcel Dekker, New York, 2004. [17] H. Meyer and F. M¨uller-Plathe, “Formation of chain-folded structures in supercooled polymer melts examined by MD simulations,” Macromolecules, 35, 1241–1252, 2002.

Perspective 16 HYBRID ATOMISTIC MODELLING OF MATERIALS PROCESSES Mike Payne,1 G´abor Cs´anyi,2 and Alessandro De Vita3 1 Cavendish Laboratory, University of Cambridge, UK 2 Cavendish Laboratory, University of Cambridge, UK 3

King’s College London, UK, Center for Nanostructured, Materials (CENMAT) and DEMOCRITOS National Simulation Center, Trieste, Italy

Hybrid atomistic modelling schemes aim to combine the lengthscale and timescale capabilities of simulations performed using empirical potentials with the accuracy of first principles calculations. However, there are considerable challenges to developing a hybrid atomistic modelling scheme that can describe materials processes. In this article, we outline some of these challenges and describe a scheme we have developed that overcomes some of these.

1.

Materials Simulations

Atomistic simulations using empirical potentials have gained an increasingly important role in the understanding of materials processes. However, empirical potentials are unable to capture some of the most basic features of the formation and breaking of atomic bonds, which can only be described accurately using first principles quantum mechanical techniques. First principles density functional calculations can presently be performed for system sizes up to a thousand atoms. Such calculations provide the capability of predicting many materials properties, one example being the prediction of superhard materials [1]. One obstacle to further progress in first principles simulations is that the computational time for conventional density functional theory calculations scales as the cube of the number of atoms in the system. We expect that over the next five years these approaches will be replaced with linear scaling techniques which will make it possible to perform quantum mechanical calculations on systems containing many tens of thousands of atoms for static problems, and thousands of atoms for dynamical simulations. 2763 S. Yip (ed.), Handbook of Materials Modeling, 2763–2771. c 2005 Springer. Printed in the Netherlands.

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However, even these system sizes are far too small to study materials processes such as crack propagation which involve very strong coupling between atomistic effects, such as the breaking of a bond at the crack tip, and the long range elastic fields surrounding the crack. This coupling between short range quantum mechanical processes and long range elastic fields is common to many problems and appears to be beyond any conceivable future capability of first principles calculations alone, at least within a reasonable cost.

2.

Hybrid Modelling Schemes

Hybrid modelling schemes provide a natural approach for dealing with the demands of including both short range chemistry and long range elasticity in a single simulation. A general hybrid modelling approach is schematically illustrated in Fig. 1, in this case applied to a system containing two cracks. Over most of the system the distortion of the material due to the elastic fields is extremely small and can be accurately described by a continuum description of the material or using a simple atomistic model based on empirical potentials. In Fig. 1, we actually show both descriptions with the regions described using empirical potentials closer to the crack tips and a continuum region further away from the crack tips, where the distortion of the materials is smaller.

Empirical atomistic

Quantum

Continuum

Figure 1. Schematic illustration of a hybrid modelling scheme.

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Irrespective of the description of the material away from the crack tips, the region in the vicinity of each crack tip, where the bond breaking processes will occur, must be described quantum mechanically. For the purposes of this article we shall concentrate primarily on purely atomistic hybrid modelling schemes in which a region described quantum mechanically is combined with a region described using empirical interatomic potentials. Many approaches are being developed which embed a region described quantum mechanically within a region described empirically. One class consists of the so-called QM/MM techniques, which are being developed primarily for the study of biological systems [2]. In these approaches the active site of a protein would be described quantum mechanically while the remainder of the system is described using one of the standard potentials developed specifically for proteins.

3.

Challenges of Materials Simulations

There are many ways that the results of first principles calculations may be used in simulations of materials processes on lengthscales and time-scales that are far beyond those accessible to first principles techniques. One approach is to calculate some of the parameters that are relevant to the materials processes, such as surface energies or activation barriers, using first principles calculations and then using these parameters in the large scale simulations for instance in Bulatov et al.’s modelling of dislocation motion in silicon [3]. It has also become quite common to use the results of first principles calculations to construct the empirical potentials that will be used in the large scale atomistic simulations of materials processes, so that almost all modern potentials are fitted to reproduce data which at least in part is calculated. One can perform the first principles calculations on atomic configurations that are similar to those that will be encountered in the materials process so that the resulting empirical potential is more likely to give an accurate description of the process being studied. Such approaches to studying materials processes are hierarchical approaches, in which simulations for larger lengthscales and/or longer timescales are to be performed using parameters obtained from more accurate simulations performed over smaller lengthscales and timescales. While this approach can be successful in many cases, as mentioned previously, the hierarchical approach breaks down when there is a complex interplay between phenomena at different lengthscales, as in the case of crack propagation. Such problems need large numbers of atoms to correctly describe the long range elastic fields but also require quantum mechanical accuracy to correctly describe the bond breaking processes, which however take place as a direct consequence of the near-singular stress enhancement at the crack tip. For these problems it appears that the only way of modelling these processes is to use hybrid schemes.

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The application of hybrid modeling techniques to study materials processes is much more challenging than their application to biological systems. The active site of a protein does not move its position within the protein and so the region to be treated quantum mechanically within a QM/MM simulation does not change with time. Unfortunately, as illustrated schematically in Fig. 2, this is not true in the case of most materials processes and so any hybrid modeling scheme for such systems must be able adjust the region treated quantum mechanically in response to the dynamical evolution of the system. A further challenge, also illustrated schematically in Fig. 2, is that it is often not clear which regions of the material should be treated quantum mechanically since one cannot know in advance where or when a new crack may be initiated in the material in response to the continuous changes that take place in the material during the simulation. What these simplistic illustrations do not make clear is that, realistically, both of these issues must be addressed within the hybrid modeling scheme. It would be totally impossible to run a huge simulation of a complex materials process if one had to study the system after each step in the simulation in order to decide which regions should be treated quantum mechanically at the next step. However, choosing which atoms must be treated quantum mechanically is totally beyond the capability of virtually every hybrid modeling scheme currently being used. It is, however, within the capability of a scheme originally proposed by De Vita and Car [4].

Empirical atomistic

Quantum

?

Continuum

Figure 2. Challenges for hybrid modelling schemes for materials simulations. These schemes must be able to follow dynamically evolving systems and to identify which regions must be treated quantum mechanically.

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The Learn-on-the-fly Scheme

The scheme proposed by De Vita and Car takes a very different approach to the problem of hybrid modelling. In contrast to other schemes in which atoms are either in a quantum mechanical region or a region described by empirical potentials, in their scheme all the atoms in the system are described using empirical potentials. However, the parameters in the empirical potential are updated on an atom by atom basis where and when necessary during the simulation. A schematic illustration of this scheme is shown in Fig. 3. The updating of the interatomic potentials is carried out by performing accurate quantum mechanical calculations to determine the correct forces on the atoms in each critical region of the material and then updating the parameters of the potentials for all the atoms in this region to reproduce these quantum mechanical forces. As the parameters are updated “on-the-fly” we have named this approach “Learn-on-the Fly” or “LOTF”. In the LOTF approach every atom in the system can have a different interatomic potential as the parameters in the potential are assigned on an atom by atom basis. Furthermore, the potential of each atom can vary with time. For instance, in the case of crack propagation an atom would initially be described by a simple potential that worked well for the bulk material. While the atom was in the vicinity of the crack tip, the atomic potential would be updated regularly so that the potential is able to accurately describe the behaviour of this atom in its highly distorted and rapidly

Continuum Empirical atomistic

Atoms represented by empirical potentials with parameters determined by by quantum mechanical calculations

Figure 3. The Learn-on-the fly scheme.

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varying environment. Finally, when the crack has passed the atom, the empirical potential evolves to accurately describe an atom on the surface of the material and at this point the potential no longer needs to be routinely updated until the next major event takes place in the vicinity of this atom. The real test of the LOTF scheme is whether it can reproduce the results of an accurate quantum mechanical calculation despite using empirical interatomic potentials. To test this point we have calculated the self-diffusion rates for a vacancy in silicon using systems containing 64 atoms. This size is small enough for us to perform one set of simulations using tight-binding calculations for the entire system. We also performed a simulation using the Stillinger–Weber potential [5]. Finally we performed a simulation using LOTF with a Stillinger–Weber type potential, where the potential parameters were fit to the forces calculated using the tight binding method for a number atoms in the immediate vicinity of the defect. The results of these calculations are shown in Fig. 4. In each case the LOTF scheme gives values for the selfdiffusion constants which are consistent with those generated when the tight

4 LOTF SW TB

4.2

log10 D [cm2/sec]

4.4 4.6 4.8 5 5.2 5.4 5.6 5.8 6 0.65

0.7

0.75

0.8

0.85 0.9 1000/T [K]

0.95

1

1.05

Figure 4. Self diffusivity of a vacancy in a 64 atom bulk silicon cell as a function of inverse temperature. Diamonds and squares show the fully quantum mechanical (tight binding) and fully classical (Stillinger-Weber) results, respectively, and triangles (and dashed line) represent the results of the hybrid simulation, which agree with the tight binding reference within statistical error.

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binding scheme is applied to the entire system. It can also be seen that the diffusion constants calculated using the fixed original Stillinger–Weber potential are incorrect. This figure proves the ability of LOTF to inherit the accuracy of the quantum mechanical method used to generate the accurate forces on the atoms. This ability has been reproduced in every test of the method we have performed to date. It should be emphasized that LOTF will work with any form of empirical potential, even extremely unphysical forms. However, if very unphysical forms are used then the parameters will have to be updated more frequently in order to continue to reproduce the results of the underlying quantum mechanical calculations. To perform a LOTF simulation you set up the initial configuration of the system to be studied, choose the empirical potential to be used and the quantum mechanical scheme to be used to compute the accurate forces. There is, in principle, no reason why only one potential or quantum mechanical scheme has to be used. It could be more than one and it could even change during the simulation. Then one selects the criteria that will be used to select the atoms whose potentials will be updated. During the simulation, when these criteria are met, the forces on these atoms and their neighbours are calculated quantum mechanically and the parameters describing the empirical potentials of these atoms adjusted to reproduce the quantum mechanical forces on these atoms. We have found that it is important to use time averaged atomic positions when selecting these atoms. If one selects on the basis of instantaneous atomic configurations an enormous computational effort is expended following large thermal fluctuations on individual atoms that do not actually affect the materials process being studied. The selection criteria to be applied could include changes in interatomic distance above a chosen threshold, or changes in number of nearest neighbours within a specific distance. One of the challenges of successfully applying LOTF to any particular problem will be the choice of suitable criteria that will correctly identify the critical atomistic processes that give rise to the materials property one wishes to study. However, we emphasize that once these criteria are chosen, the simulation can proceed to run with no further user intervention. The major motivation for developing hybrid modelling schemes has generally been the wish to simulate ever larger systems. The same motivation lies behind attempts to develop linear scaling first principles calculations. However, increasing system size brings a further set of difficulties to simulation. These are primarily associated with the increased size and complexity of the phase space associated with the system. In general, the larger a physical system the larger the number of local energy minima in the potential energy surface and hence the slower the physical processes that occur within it. Without techniques for vastly accelerating the search through phase space, larger systems will require intractably longer simulations. Until recently, this problem was addressed by very few researchers but this situation now appears to

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be changing. LOTF does not directly address this problem of the complexity of the phase space of large systems. However, it is worth pointing out that the scheme allows simulations with quantum mechanical accuracy to be performed for simulation times which are normally only accessible to empirical simulations, routinely many nanoseconds or even microseconds. In contrast, due to fundamental limitations of first principles molecular dynamics approaches, hybrid techniques which explicitly include a large quantum mechanical region may be restricted to the picosecond timescale.

5.

Outlook

LOTF appears to deal successfully with the hybrid modelling problem at the atomistic scale. But, as can be seen in Fig. 1, there is the further problem of matching the atomistic region to the continuum. In fact, this problem was addressed more than ten years ago by Kohlhoff, Gumbsch and Fischmeister in their simulations of crack propagation in metals [6]. Their method has yet to be tested on a system containing a number of interacting defects. At present, our LOTF scheme can only deal with short range interatomic potentials. We still need to extend the technique to deal with systems in which there are long range electrostatic interactions, but this could be done similarly to most current QM/MM schemes. All the LOTF simulations performed to date have used very simple empirical potentials. We are currently investigating the best way to augment much more complicated classical potentials. This will be crucial for applying LOTF to biological systems. It would seem that changing the parameters of such elaborate potentials is technically difficult, and a better route is to retain a very simple functional form, and tune its parameters to reproduce the forces of the empirical potential in most regions, and of the quantum mechanical model in the activated regions. This would make the scheme even more general than described above, capable of using a range of “black box” potentials (of varying degrees of accuracy) each used in their assigned region of the system.

References [1] A.Y. Liu and M.L. Cohen, “Prediction of new low compressibility solids,” Sci., 245, 841–2, 1989. [2] J. Gao, “Methods and applications of combined quantum mechanical and molecular mechanical potentials,” In: K.B. Lipkowitz and D.B. Boyd (eds.), Reviews in Comp. Chem., VCH Publishers, New York, vol. 7, pp. 119–185, 1995. [3] V.V. Bulatov, J.F. Justo and W. Cai et al., “Parameter–free modelling of dislocation motion: the case of silicon,” Philos. Mag. A, 81, 1257–81, 2001.

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[4] A. De Vita and R. Car, “A novel scheme for accurate MD simulations of large systems,” Mat. Res. Soc. Symp. Proc., 491, 473–480, 1998. [5] F.H. Stillinger and T.A. Weber, “Computer simulation of local order in condensed phases of silicon,” Phys. Rev. B, 31, 5262–5271, 1985. [6] S. Kohlhoff, P. Gumbsch and H.F. Fischmeister, “Crack propagation in BCC crystals studied with a combined finite-element and atomistic model,” Phil. Mag., A64, 851– 78, 1991.

Perspective 17 THE FLUCTUATION THEOREM AND ITS IMPLICATIONS FOR MATERIALS PROCESSING AND MODELING Denis J. Evans Research School of Chemistry, Australian National University, Canberra, ACT, Australia

Thermodynamics describes the framework within which all macroscopic processes operate. Until the discovery of the Fluctuation Theorem [1], there was no equivalent framework for small (nano) systems observed for short times. The Fluctuation Theorem provides a generalisation of the Second Law of thermodynamics, that applies to finite systems observed over finite times. The Second Law of thermodynamics states that for all macroscopic processes the total entropy of the Universe can only increase. The Fluctuation Theorem says that for finite systems, the probability ratio that for a finite time the entropy decreases rather than increases, vanishes exponentially with system size and observation time. Thus in the so-called “thermodynamic limit”, the entropy can only increase and we obtain the Second Law. The Fluctuation Theorem places limits on the operation of nanomachines and biological processes taking place in small organelles. The Theorem states that as “engines” are made ever smaller, the probability that they will operate thermodynamically in reverse, increases exponentially with the size of the system and the duration of operation. The Fluctuation Theorem also resolves another paradox. All the laws of mechanics (quantum or classical) are time reversible. If you look at the motion of the planets of the solar system, orbiting the sun, then if all the motions are reversed in time, the resulting motion is still a valid solution of the laws of mechanics. However when the system involves billions upon billions of interacting molecules say in a glass of water or a waterfall, thermodynamics says the motion can only occur in the direction which increases the total entropy! This is in spite of the fact that even for large systems (like a glass of water or a waterfall), the laws of mechanics are, as always, completely time reversible. The first satisfactory mathematical proof of a Fluctuation Theorem was given 2773 S. Yip (ed.), Handbook of Materials Modeling, 2773–2776. c 2005 Springer. Printed in the Netherlands.

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by [2]. This proof assumes the initial distribution of microstates is known and computes the relative probabilities of changes in entropy by explicitly using the time reversal symmetry of the equations of motion and the assumption of causality, namely that probabilities of observing final microstates can be computed from the probabilities of observing the initial states from which the final states are generated.

1.

The Fluctuation Theorem(s) in More Detail

Because the Fluctuation Theorem (FT) deals with fluctuations we expect that there will be different versions of the FT for systems at constant energy, constant temperature, constant volume constant pressure etc. A recent experiment performed to confirm the FT [3], serves as an instructive example. A single transparent colloid particle was trapped in the harmonic force field of a focused laser beam – a so-called optical trap. The colloid particle is in rather obvious contact with a heat bath: namely the surrounding water solvent. Wang et al, allowed the system to come to equilibrium, then at an arbitrary zero time, the optical trap was suddenly translated at fixed velocity relative to the solvent. The resulting transient motion of the colloid particle was monitored as the particle responded to the sudden motion of the trap. This same experiment was repeated several hundreds of times. The transient FT for this system considers the quantity, ¯ t = (tkB T )−1

t 0

ds vopt • Fopt(s)

(1)

where Fopt(t) is the optical force on the trapped particle,vopt is the (constant) velocity of the optical trap and kB T is Boltzmann’s constant times the abso¯ t is a lute temperature of the solvent. From its definition (1), we can see that quantity that is recognizable as the time average of rate of entropy absorption by the solvent. In thermodynamic terms it is the time average of the work divided by the temperature of the solvent. For thermostatted dissipative systems, ¯ t , is always something that is recognizable as a rate the argument of the FT , of entropy absorption [4]. However for more general systems the argument of the FT, the dissipation function, is not directly recognizable as an entropy production/absorption. The general expression for the dissipation function that is valid for arbitrary ergodically consistent combinations of initial ensemble and dynamics can be found in Evans and Searles, 2002 [4]. For the experiment of Wang et al., the transient FT makes a very simple prediction for the ratio of probabilities of observing complementary values for ¯ t, the time averaged entropy prduction, ¯ t = A) Pr( = exp( At). (2) ¯ t = −A) Pr(

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The original paper of Wang et al. [3] experimentally confirmed the integrated form of (2), namely, ¯ t < 0) Pr( ¯ t t) = exp(− . ¯ t >0 ¯ t > 0) Pr(

(3)

A direct check of the validity of the transient FT itself was recently performed for a simpler experiment. In this experiment, an equilibrium ensemble of trapped particles was subject to a step function change in the strength of the optical trap. In this case the dissipation function appearing in the transient FT is not directly related to entropy production. Instead the dissipation function ¯ t is ¯ t = (tkB T )−1 (k0 − k1 )

0

t

•

ds r(s) • r(s)

= (2tkB T )−1 (k0 − k1 )(r(t)2 − r(0)2 )

(4)

where k0 , k1 are the initial and final values of the spring constant for the optical trap and and r(t) is the vector position of the colloid particle relative to the centre of the optical trap. From its definition (4) we can see the dissipation function is like entropy production in that its ensemble average is always expected to be positive. For this system the transient FT states that, ¯ t = A) Pr( = exp( At). ¯ Pr(t = −A)

(5)

This prediction was confirmed in the experiments of Carberry, et al., 2004 [5]. A recent review of the theoretical status of the FT has been published by Evans and Searles 2002 [4]. Fluctuation Theorems are extraordinarily general. There are stochastic versions of the FT. The FT is completely consistent with Langevin dynamics [6]. There are quantum versions of the FT [7, 8]. This theorem is obviously very general. It can be used to derive Green–Kubo relations for linear transport coefficients and the Fluctuation Dissipation Theorem [9]. However, it is more general than either of these two relations since the FT applies to the nonlinear regime far from equilibrium where Green– Kubo and Fluctuation Dissipation relations fail. The transient FT can be applied to nonequilibrium paths that connect two equilibrium states. When this is done the FT can be used to derive new expressions for free energy differences between equilibrium states in terms of sums over all nonequilibrium path integrals which connect those two equilibrium states [10–14]. We expect that over the next decade further extensions of the Fluctuation Theorem will be explored. The practical utility of these nonequilibrium free energy expressions is yet to be properly assessed. However the mere fact that equilibrium free energy differences can be related to nonequilibrium path integrals is quite surprising.

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So one hundred years after Boltzmann, I think we can finally say to our students that we understand how macroscopic irreversibility arises from reversible microscopic dynamics.

References [1] D.J. Evans, E.G.D. Cohen, and G.P. Morriss, “Probability of second law violations in shearing steady states,” Phys. Rev. Lett., 71, 2401–2402, 1993. [2] D.J. Evans and D.J. Searles, “Equilibrium microstates which generate second law violating steady states,” Phys. Rev. E, 50, 1645–1648, 1994. [3] G.M. Wang, E.M. Sevick, E. Mittag et al., “Experimental demonstration of violations of the second law of thermodynamics for small systems and short time scales,” Phys. Rev. Lett., 89, 050601–4, 2002. [4] D.J. Evans and D.J. Searles, “The Fluctuation theorem,” Adv. In Phys., 51, 1529– 1585, 2002. [5] D.M. Carberry et al., “Fluctuations and irreversibility: An experimental demonstration of a second-law-like theorem using a colloidal particle held in an optical trap,” Phys. Rev. Lett., 92, 140601–4, 2004. [6] J.C. Reid et al., “Reversibility in non-equilibrium trajectories of an optically trapped particle,” Phys. Rev. E, 70, 016111–9, 2004. [7] J. Kurchan, “A quantum fluctuation theorem,” arXiv:cond-mat/0007360, 1–8, 2001. [8] T. Monnai and S. Tasaki, “Quantum correction of fluctuation theorem,” J.Phys.A Math.Gen., 37, L75–L79, 2004. [9] D.J. Evans, D.J. Searles, and L. Rondoni, “On the application of the gallavotti-cohen fluctuation relation to thermostatted steady states near equilibrium,” arXiv:condmat/0312353, 1–40, 2004. [10] C. Jarzynski, “Nonequilibrium equality for free energy differences,” Phys. Rev. Lett., 78, 2690–2693, 1997a. [11] C. Jarzynski, “Equilibrium free energy differences from nonequilibrium measurements: A master equation approach,” Phys. Rev. E, 56, 5018–5035, 1997b. [12] G.E. Crooks, “Entropy production fluctuation theorem and nonequilibrium work relation for free energy differences,” Phys. Rev. E, 60, 2721–2726, 1999. [13] G.E. Crooks, “Path-ensemble averages in systems driven far from equilibrium,” Phys. Rev. E, 61, 2361–2366, 2000. [14] D.J. Evans, “A non-equilibrium free energy theorem for deterministic systems,” Mol. Phys., 101, 1551–1554, 2003.

Perspective 18 THE LIMITS OF STRENGTH J.W. Morris, Jr. Department of Materials Science and Engineering, University of California, Berkeley

In the usual case the strength of a crystalline material is determined by the motion of defects such as dislocations or cracks that are present within it. Materials scientists control strength by modifying the microstructure of the material to eliminate defects or flaws and inhibit the motion of dislocations. There is, however, an ultimate limit to the strength that can be obtained in this way. The mechanical stresses that are not relieved by plastic deformation or fracture are supported by elastic deformation, which is, essentially, the stretching of the interatomic bonds. These bonds have finite strength. There is a value of the stress at which bonding itself becomes unstable and the material must fracture or deform, whatever its microstructure. This elastic instability sets an upper bound on mechanical strength that cannot be exceeded, however creative a scientist may be. There are several practical reasons to be interested in the limit of elastic stability [1]. First, elastic instability defines the ideal strength [2–4], and it is useful to know the highest strength a particular material could possibly have. This is particularly true in times when new concepts in materials, such as nanomaterials, have led to optimistic predictions that have not always been vetted against the limits nature has set. Second, the elastic limit is reached, or, at least, closely approached in a number of experimental situations. A familiar example is deformation via stress-induced phase transformations, as in certain austenitic steels. However, even normal, ductile metals seem to approach the limit of strength in nanoindentation experiments, and stronger alloys may also do so in the region of stress concentration ahead of a sharp crack. Third, elastic instability is one of the few problems in solid mechanics that can actually be solved ab initio. Existing pseudopotential codes are capable of following elastic deformation to the point of instability with reasonable accuracy, and a number of calculations have been done [1, 5–9]. Fourth, as we shall see, the 2777 S. Yip (ed.), Handbook of Materials Modeling, 2777–2785. c 2005 Springer. Printed in the Netherlands.

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theoretical study of behavior at the limit of strength can provide new insight into mechanical behavior in more common situations. From the perspective of understanding deformation, it is interesting that a wide variety of mechanical phenomena that are ordinarily attributed to the specific behavior of dislocations would also be found in a defect-free world. In the limited space available here we discuss three examples. (1) In a defectfree world, the common bcc metals would cleave on {100} and exhibit “pencil glide” in 111. Most would be brittle at low temperature, as they are. (2) The common fcc metals would glide in {111} and would not cleave under simple tensile loads. They would be ductile at low temperature, as most of them are. (3) The maximum values of the nanohardness of simple metals would be very nearly what they are.

1.

Methodology

The calculation of ideal strength is based on modern methods that make it possible to compute the energy of a crystalline configuration of atoms with rather good accuracy. A number of computational techniques are available, almost all of them based on density functional theory. The Vienna Ab-Initio Simulation Package (VASP) [10] provides a set of readily available tools to accomplish these calculations, and other packages are also available. In most cases the local density approximation (LDA) to density functional theory yields good results. In certain circumstances (e.g., Fe) the generalized gradient approximation (GGA) gives significantly more accurate results [11]. The relative quality of the two approximations can be judged by comparing experimental data to theoretical predictions. In these methods Schrodinger’s equation is solved within a single particle approximation. It is usually sufficient to employ pseudopotentials for the atomic cores and a plane wave expansion of the wave functions (this is the approach used in VASP). This technique works reasonably well in situations for which the core states are not strongly affected by the imposed strains. However, under severe compressive stresses, for example, all electron methods, such as that employed in WIEN97, provide a better description of the solid’s properties. The elastic stress within a solid is related to the derivative of the energy with respect to the elastic strain. While there is some subtlety involved in doing this (there is no unique definition of the strain), it is usually sufficient to define the strain from the displacements of the crystal lattice points from a reference configuration [11, 12]. One can then compute the energy as a function of strain by incrementally displacing the atom positions within the unit cell to increment the strain along the desired path, and evaluate the associated stresses from the energy increment. The ultimate strength is associated with

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the maximum in the stress, which occurs at (or near) the inflection point of a plot of energy against strain. The precise way in which the stress-strain relation is evaluated depends on the specific problem one is trying to solve. For example, one is often interested in the ideal strength under simple tension along a particular crystal axis, or in simple shear on a particular crystallographic plane. In these cases the only nonzero stress is the stress that is conjugate to the strain of interest: the uniaxial tensile stress in the case of simple tension, the conjugate shear stress in the case of simple shear. In these cases the atom positions must be adjusted after each increment to the strain so that all other stresses are relaxed to zero. Methods for doing this in a variety of load geometries and crystal types are described in the references cited at the end of this article. We shall now describe the results of some of these calculations with an emphasis on the physical insight they provide.

2. 2.1.

BCC Metals Ideal Strength in Tension

Computations of the ideal tensile strengths of unconstrained bcc metals show that they are weakest when pulled in a 100 direction. The majority of those we have investigated fail in tension (cleave) on {100} planes, both in theory and in experiment. Ab initio calculations [1, 11–15] give an ideal tensile strength of about 30 GPa for W (0.07E100 ), 29 GPa for Mo (0. 078E100 ) and 13 GPa for Fe (0.087E100 ) where E100 is the tensile modulus in the 100 direction, and the stresses are computed at 0K. There is a simple crystallographic argument that explains both the cleavage plane and the ideal tensile strength (Fig. 1). A relaxed tensile strain along

Figure 1. The Bain strain connecting the bcc and fcc structures. If bcc is pulled in tension on [001] while contracting along [100] and [010] it generates an fcc crystal as shown.

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100 carries the bcc structure into an fcc structure with the same volume at a tensile strain of 0.26 (the “Bain strain”). By symmetry, both structures are unstressed, so the tensile stress must pass through at least one maximum along the transformation path, at a critical stain much less than 0.26. If we fit the stress-strain curve to a sinusoid that has the correct modulus at low strain, the tensile strength in 100 is given by

σm ∼

eB E100 = 0.08E100 π

(1)

where eB is the Bain strain and E100 is Young’s modulus for 100 tension. The same reasoning explains why the ideal strength increases when the normal tension is supplemented by a hydrostatic tension, as it is, for example, near the tip of a crack. Hydrostatic tension expands the unit cell, which increases the Bain strain and raises the stress at instability. Ab initio calculations for Fe [16] show that the ideal tensile strength increases by almost 50% when the tensile stress is supplemented by a hydrostatic tension that is equal in magnitude. The element Nb is anomalous among the bcc metals we have studied [13, 17]. While the ideal tensile strength of Nb is lowest for 100 loading, as in the other bcc’s, the failure mode is not in tension across the {100} planes, but in shear on the 111{112} system. After some significant tensile strain, Nb deviates from the tetragonal, Bain strain path onto an orthorhombic strain path that is characterized by unequal contractions in the 100 directions perpendicular to the axis of load. The eventual failure is in shear, rather than tension. This preference for shear failure is preserved, though with a smaller margin, when hydrostatic tension is superiomposed. The results suggest that Nb may not exhibit the ductile-brittle transition that is typical of bcc metals. The experimental evidence is unclear [13].

2.2.

Ideal Strength in Shear

The ideal shear strengths of bcc metals also reflects their symmetry. Calculations of the ideal strength of bcc W [12] in relaxed shear in the 111 direction on the {110}, {112} and {123} planes give almost identical values, τm ∼ 17.7 GPa ∼ 0.11G111 , where G111 is the shear modulus for shear in the 111 direction. In all three cases the shear strain at instability is about 0.17. Calculations for Fe [11] and Mo [13] give very similar results, with τm ∼ 7.2 GPa (0.11G111 ) for Fe, ∼15.8 GPa ( 0.12G111 for Mo). Nb is, again, unusual; the ideal shear strength in the 111{112} system is anomonously large, 6.4 GPa (0.15G111 ), and is still larger for the common alternative systems [13].

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The symmetry rule that governs the shear strength of typical bcc’s is illustrated in Fig. 2 [1, 12]. Essentially, a shear in the 111 direction tilts planes that are perpendicular to the 111 axis. If we allow relaxation of the atom positions in these planes, they come into an atomic registry that changes the crystal symmetry at a shear strain of ∼ 0.34, irrespective of the plane of tilt. This common stress-free, saddle-point structure is body-centered tetragonal. There is a maximum in the shear strength at about half the saddle-point shear, at e ∼ 0.17. If we fit the stress-strain relation with a sinusoid that gives the correct modulus in the elastic limit, we obtain

τm ∼ 0.11G111

(2)

in good agreement with the ab initio calculations. A number of bcc metals have similar strengths for slip in the 111 direction on various planes, a phenomenon that is known as “pencil glide” and is attributed to the peculiarities of dislocation glide in bcc. These calculations show that defect-free bcc crystals would tend to behave in a very similar way. The balance between the shear and tensile strengths suggested by Eqs. (1) and (2) is such that, in a defect-free world, the common bcc metals would cleave if loaded along 100, but not if loaded in other directions. Taking W as an example, a uniaxial load along 100 would reach the ideal cleavage strength, ∼ 30 GPa, when the shear stress in the most favorable slip system was only around 14 GPa, below the ideal shear strength. However, a uniaxial load along 111 or 110 would cause the shear strength to be exceeded before the tensile stress in 100 reached the ideal value. A single crystal would be ductile or brittle, depending on the direction of the load.

Figure 2. Instability in shear in [111], [111] shear tilts planes of atoms (equilateral triangles perpendicular to [111]) until they come into registry, as at right, creating a new symmetry.

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FCC Metals

3.1.

Ideal Strength in Tension

The tensile strength of defect-free fcc crystals could, in theory, also be governed by the Bain strain [18]. As illustrated in Fig. 3, an fcc crystal can be converted into bcc by straining in tension in the [110] direction. The tensile strain required to reach bcc is, in fact, relatively small, so the estimated strength would be small as well. However, reaching bcc from fcc with a [110] pull requires very substantial relaxations in the perpendicular directions. The ¯ crystal must expand along [110] and contract to an even greater degree along [001]. These large relaxations are inconsistent with the Poisson contractions of typical fcc metals in the linear elastic limit. Fcc metals do not start out along this deformation path when pulled along [110] and, apparently, never find it. Nonetheless, the 110 directions are the weak directions for tension in all of the fcc metals that have been studied to date: Al, Cu, Ir and Pd [19]. If the fcc crystal is pulled quasistatically to failure under uniaxial tension, the failure mode at the elastic limit is not a tensile failure across the perpendicular {110} plane, but rather a shear failure (the “flip strain”). In this deformation mode the 110 tensile direction is stretched while the perpendicular 100 direction contracts, with the ultimate consequence that the two directions are interchanged. It can be shown that the failure mode is, in fact, a failure in shear in the 112{111} system, which is the normal mode of shear failure in an fcc crystal. This result suggests a simple explanation for the fact that fcc crystals do not exhibit a conventional ductile-brittle transition on cooling; the inherent failure mechanism is in shear. Those fcc metals that do become brittle at low temperature, such as Ir and nitrided austenitic steels, do so only after [110] [1⫺10]

[001]

[110]

[1⫺10]

[001]

Figure 3. Bain strain of an fcc lattice through tension along [110]. The crystal must expand ¯ equally along [110] and contract dramatically along [001]. The alternative is the “flip strain” shown at right.

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significant plastic deformation; their “cleavage” is by decohesion on slip or twin planes. If we assume deformation at constant volume with a sinusoidal stress-strain curve, the “flip” instability occurs at an engineering strain of 0.08 and a stress of σm = 0.05E110 , where E110 is Young’s modulus for tension in the 110 direction. The most recent ab initio calculations of the ideal tensile strengths of fcc metals under quasistatic loading give the following values: for Al, σm = 5.2 GPa = 0.07E110 , for Cu, σm = 6.2 GPa = 0.05E110 , for Pd, σm = 5.2 GPa = 0.06 E110 , and for Ir, σm = 36 GPa = 0.06E110 [19]. Note that the exceptional strength of Ir is a consequence of its large elastic modulus, and is in no way anomalous. The anomaly is the high dimensionless strength of Al. However, recent research has shown that the actual strength of Al is determined by a phonon instability that intrudes slightly before the elastic instability [20], decreasing the ideal strength. No similar instability has been found in other fcc’s.

3.2.

The Ideal Strength in Shear

The mode of failure of fcc crystals in shear is conventional, though the behavior of at least some fcc’s, Al in particular, is not. The weak directions in shear are 112 directions in {111} planes, as one would expect from a rigidball model of the close-packed fcc structure. A sinusoidal model of that failure mode predicts a shear strength, τm ∼ 0.085G111 . The ideal shear behavior of Al and Cu [20, 21] make an interesting comparison. While the deformation of Cu remains nearly planar in the {111} shear plane as the instability is approached, Al expands significantly perpendicular to the shear plane. The consequence is that while Cu has an ideal strength near the estimate, τm = 2.7 GPa ∼ 0.11G 111 , the calculated shear strength of Al is much larger, both in absolute magnitude and in dimensionless terms: τm ∼ 3.4 GPa = 0.15G 111 . (This number is a bit of an overestimate, since Al experiences a phonon instability before reaching peak strength [20], but is qualitatively correct. It is significantly higher than the number reported in earlier work [8], which used a less accurate pseudopotential.) Comparing the ideal tensile and shear strengths of Al and Cu leads to a curious result: at the limit of strength, Cu is stronger than Al in tension, though weaker in shear. This is true despite the fact that the failure mode is precisely the same in the two cases: a shear instability in the system 112{111}. The reason is the perpendicular expansion of Al during shear. Tension in the 110 direction imposes a tension across the {111} shear plane, assisting the normal displacement of {111}planes and lowering the stress required for Al to fail in shear.

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Nanoindentation

A nanoindentation test is, essentially, a microhardness test done with a nanotipped indenter. Until the substrate yields, the deformation field of the indenter should be approximately Hertzian, which makes it possible to use the data to infer the stresses and strains at which yielding occurred. Moreover, since the maximum shear in a Hertzian strain field is well beneath the surface, nanoindentation tests can sample defect-free volumes, and may, therefore, test the ideal strength. Surprisingly, the shear strengths inferred from recent nanoindentation tests substantially exceed the computed ideal strengths. Thus, Bahr, et al. [22] report data showing shear stresses as high as 28 GPa in W prior to yielding, well beyond the value (18 GPa) that corresponds to the ideal strength on any of the common slip planes. Nix [23] reported preliminary Mo data giving a maximum strength of 23 GPa, compared to the theoretical shear strength of 15.6 GPa. The discrepancy between these values is almost entirely removed if one makes two corrections [24]. First, the Hertzian stress field is modified by nonlinearity as the ideal strength is approached. Finite-element calculations using a sinusoidal stress-strain relation show that the Hertzian stress field is correct except in the immediate vicinity of the maximum shear stress, even when the maximum shear stress approaches the ideal strength. However, the value of the maximum shear stress is significantly decreased, to τm ∼ 0.69 τH , where τH is the Hertzian value. Second, the triaxiality of the stress field near the point of maximum shear increases the ideal shear strength. When these (and a couple of other, minor corrections) are made the maximum shear strengths that can be inferred from nanoindentation experiments on W (22.8–24.0 GPa) and Mo (16.0–16.8 GPa) are very close to the theoretical values of the ideal strength (W = 22.1–23.3 GPa; Mo = 17.6–18.8 GPa), as they should be. This result suggests that nanoindentation may provide a viable means for measuring ideal strength.

References [1] J.W. Morris, Jr., C.R. Krenn, D. Roundy et al., In: P.E. Turchi and A. Gonis (eds.), Phase Transformations and Evolution in Materials, TMS, Warrendale, PA, pp. 187– 208, 2000. [2] R. Hill and F. Milstein, Phys. Rev. B, 15, 3087–3097, 1977. [3] J. Wang, J. Li, S. Yip et al., Phys. Rev. B, 52, 12, 627–635, 1995. [4] J.W. Morris, Jr. and C.R. Krenn, Phil. Mag. A, 80, 2827–2840, 2000. [5] A.T. Paxton, P. Gumbsch, and M. Methfessel, Phil. Mag. Lett., 63, 267–274, 1991. [6] W. Xu and J.A. Moriarty, Phys. Rev. B, 54, 6941–6951, 1996. [7] M. Sob, L.G. Wang, and V. Vitek, Mat. Sci. Eng., A234–236, 1075-1078, 1997.

The limits of strength [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24]

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D. Roundy, C.R. Krenn, M.L. Cohen et al., Phys. Rev. Lett., 82, 2713–2716 1999. S. Ogata, J. Li, and S. Yip, Phys. Rev. B, in press, 2004. G. Kresse and J. Hafner, J. Phys. Condens. Matter, 6, 8245, 1994. D.M. Clatterbuck, D.C. Chrzan, and J.W. Morris, Jr., Acta Mater., 51, 2271–2283, 2003. D. Roundy, C.R. Krenn, M.L. Cohen et al., Phil. Mag. A, 81, 1725–1747, 2001. W. Luo, D. Roundy, M. L. Cohen et al., Phys. Rev. B, 66, 94110, 2002. D.M. Clatterbuck, D.C. Chrzan, and J.W. Morris, Jr., Phil. Mag. Lett., 82, 141–147, 2002. M. Friak, M. Sob, and V. Vitek, Proc. Int. Conf. Juniormat 2000, Brno Univ. Technology, Brno, 2001. D.M. Clatterbuck, D.C. Chrzan, and J.W. Morris, Jr., Scripta Mat., 49, 1007, 2003. C.R. Krenn, D. Roundy, J.W. Morris, Jr. et al., Mat. Sci. Eng. A, A319–321, 111–114, 2001. J.W. Morris, Jr., C.R. Krenn, D. Roundy, and M.L. Cohen, Mat. Sci. Eng. A, 309– 310, 121–124, 2001. J.W. Morris, Jr., D.M. Clatterbuck, D.C. Chrzan et al., Mat. Sci. Forum, 426–432, 4429–4434, 2003. D.M. Clatterbuck, C.R. Krenn, M.L. Cohen et al., Phys. Rev Lett., 91, 135501, 2003. S. Ogata, J. Li, and S. Yip, Science, 298, 807, 2002. D.F. Bahr, D.E. Kramer, and W.W. Gerberich, Acta Mater., 46, 3605–3617, 1998. W.D. Nix, Dept. Materials Science, Stanford Univ., Private Communication, 1999. C.R. Krenn, D. Roundy, M.L. Cohen et al., Phys. Rev. B, 65, 13411–13416, 2002.

Perspective 19 SIMULATIONS OF INTERFACES BETWEEN COEXISTING PHASES: WHAT DO THEY TELL US? Kurt Binder Institut fuer Physik, Johannes Gutenberg Universitaet Mainz, Staudinger Weg 7, 55099 Mainz

Interfaces between coexisting phases are ubiquitous in the physics and chemistry of condensed matter: Bloch walls in ferromagnets; antiphase domain boundaries in ordered binary (AB) or ternary alloys; the surface of a liquid droplet against its vapor; boundaries between A-rich regions and B-rich regions in fluid binary mixtures, etc. These interfaces control material properties in many ways (e.g., when a fluid polymer mixture is frozen-in to form an amorphous material, the mechanical strength of this macromolecular glass is controlled by the extent that A-polymers are entangled with B-polymers across the interface, etc.). For a detailed understanding of the properties of such interfaces, one must consider their structure from the scale of “chemical bonds” between atoms in the interfacial region up to much larger, mesoscopic, scales (e.g., when one tries to measure a concentration profile across an interface between coexisting phases in a partially unmixed polymer blend by suitable depth profiling methods, typical results for the interfacial width are of the order of 50 nm [1]). So from the point of view of simulations, one deals with a multiscale problem [2]. However, the situation is even worse: interfaces may exhibit longwavelength excitations, such as capillary waves [3], which lead to the effect that many properties associated with interfaces do depend on the geometry used for the experiment [1] or the simulation [4]. This fact causes a strong danger so that the properties observed in the simulation (or experiment, respectively) are mis-interpreted [2]. For simplicity, we shall consider in the following only interfaces in fluid systems (the gas liquid-interface or the interface in a fluid binary mixture, respectively); although many considerations can be carried over immediately 2787 S. Yip (ed.), Handbook of Materials Modeling, 2787–2791. c 2005 Springer. Printed in the Netherlands.

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to interfaces in solid phases (e.g., Bloch walls in antiferromagnets, antiphase domain boundaries in ordered alloys with negligible mismatch between the lattice parameters of the constituent atomic species, etc.), there are many cases in solids where additional complications arise due to long-range elastic interactions, and it is clear that the latter effects are very important (consider e.g., “coherent” vs. “incoherent” precipitation: in the latter case elastic strains destroy the coherence of the lattice structure between a precipitated grain and the surrounding matirx). These complications will stay beyond the considerations of the present note, however. Now the standard concept to describe an interfacial profile (i.e., a density or concentration variable c(z) observed as a function of the coordinate z across the interface) is the concept of an “intrinsic interfacial profile”. This concept dates back to van der Waals in the 19th century (and to Cahn and Hilliard in the late 50s of the 20th century, as far as ordinary binary mixtures are concerned, and to deGennes, Joanny and Leibler in the late 70s for polymer blends). The result for the interfacial profile near the critical point of the fluid (or fluid binary mixture, respectively) can be cast into the familiar tanh-form

c(z) =

z − z0 1 c1 + c2 + (c2 − c1 ) tanh 2 w0

.

(1)

Here c1 , c2 are the concentrations (or densities, respectively) of the two coexisting phases very far away from the interface that is located at z = z 0 . The variation from c1 to c2 extends essentially over the distance z = w0 around z 0 , the “intrinsic width” w0 . Near the critical point, w0 is just twice the correlation length of the order parameter fluctuations (remember that the order parameter is the concentration (density) difference c2 − c1 , for the considered systems). Near the critical point, the correlation length is large, particularly for polymer blends (where the scale for this length is set by the gyration radius of the coils, see e.g., [5]). Also away from the critical point, Eq. (1) often is obtained, at least as a good approximation, though the required theories are more complicated than the Ginzburg–Landau type mean field theory required to derive Eq. (1) near the critical point [5]. However, these theories (such as the density functional theories for fluids, or the self-consistent field theories for polymeric systems in a sense also have a mean field character, and all ignore lateral fluctuations in the (x, y) directions parallel to the interface. For very large length scales of an interface in a fluid system, these lateral fluctuations are nothing but the well-known capillary waves [3] (for simplicity, effects of gravity are ignored here: this is anyway a very good approximation for polymer mixtures, but also reasonable for many after systems). One can show from the equipartition theorem that the thermally averaged mean square amplitude of interfacial height fluctuations due to the capillary waves with wavelenghths 2 /q scales like the inverse of the square of the wavenumber q. This causes a long wavelength instability of the interface: on a lateral length

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scale L parallel to the interface the mean square height scales proportional to the logarithm of L, ln (L/B), where the length B is a short-wavelength cutoff needed in the theory. In fact, on scales of the order of atomic distances (and near the critical point, already on scales of the order of the correlation length), the picture of a sharp interface locally displaced by capillary waves makes no sense anymore. However, a good understanding is lacking of what cutoff length B is [2–4, 6]. And when one considers the convolution of the intrinsic profile with this capillary wave broadening, there is no unique way to separate in the resulting dependence of the total width w the “intrinsic width” w0 from the term involving ln (B) [2, 4, 6]. These considerations have indeed dramatic consequences on simulations (and experiments [1]), as shown in [4]. Simulations intended to study interfacial profiles typically use one of the following two geometries: 1. A single interface can be stabilized in the geometry of a thin film of thickness D between two walls (of lateral linear dimensions L ). The nature of the walls is chosen such that they distinguish between the phases: in a binary mixture, one wall prefers A, the other wall prefers B; for a gas-liquid interface, one wall has a purely repulsive interaction between the wall and the fluid particles (prefering the gas phase), the other wall exerts an attractive interaction (prefering the liquid phase). In the lateral directions, periodic boundary conditions are used. The disadvantage of this geometry, of course, is that D must be rather large, to avoid a too strong perturbance of the interfacial profile due to the forces from the walls (there must be room on both sides of the interface for the bulk phases unperturbed by the walls). 2. One chooses periodic boundary conditions in all directions, and D considerably larger than L, and creates via suitable initial conditions a slab configuration, e.g., a liquid separated by two parallel interfaces from the gas (e.g., [7]). D must be large enough, so that any interactions between the two interfaces safely are avoided. In addition, L should not be too small either, because otherwise the average orientation of the interfaces around the z-directions also would fluctuate, and hence typically only part of the capillary wave spectrum is suppressed by the periodic boundary condition. In view of these problems, it is not clear to what extent observations of interfacial widths [7] should be compared with experiments, in particular when a careful study of the effects of varying both linear dimensions has not been made. A caveat needs also to be made with respect to the interfacial free energy f, which usually is computed from the profile of the anisotropy of the pressure tensor across the interface [3], since this profile is size-dependent as well. A variant of this geometry is used for symmetric systems (e.g., Isinglattice gas models, where there is a symmetry between particles and holes or

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“spin up” and “spin down” phases), there one can simulate a single interface but dispense of walls in favour of an “antiperiodic” boundary condition (crossing the boundaries in z-direction the sign of the spin is flipped) [8]. For a symmetric binary mixture, this means A turns into B (and vice versa) when a boundary in z-direction is crossed [6]. For this geometry (as well as for the above alluded “slab geometry” with two parallel interfaces) one has the problem already alluded to above, that the mean squared interfacial width varies with ln (L), and a unique identification of the intrinsic width (and intrinsic profile) is not possible. However, for the geometry with real walls (above choice No. 1) there is an additional very strong effect of the linear dimension D : for L , and shortrange forces due to the walls, the mean square width of the interface varies linearly with D (while it varies only logarithmically with D for long-range wall forces that decay with a power law with the distance from the wall and hence stabilize better the position of the interface in the middle of the thin film). This very strong size effect results from a kind of “soft mode”, the local position of the interface can be pushed away from the center of the thin film with very little energy cost, and in thermal equilibrium these “cheap” fluctuations cause a very strong broadening of the interfacial profile. These effects have been seen in simulations of Ising models and of polymer mixtures [2] as well as in experiments [1]. We conclude these comments by noting that finite size effects, on the other hand, are useful for the estimation of interfacial free energies f {e.g., [8,9]}. Choosing a geometry with periodic boundary conditions in all spatial directions and the (semi-) grandcanonical ensemble, the distribution of the order parameter is sampled. While its maxima correspond to the pure phases, its minimum corresponds to the slab configuration mentioned above. To be able to sample the minimum accurately, “multicanonical Monte Carlo” or “umbrella sampling techniques” are needed. From the ratio of the logarithms of the probabilities in the maximum and in the minimum, one can extract the free energy excess due to the interfaces, which is 2 times the area of a single interface times f . Checking that the result indeed scales linearly with the interface area proves that the asymptotic regime has indeed been reached. This “first principles” method avoids the need for taking any “measurements” at the interfaces or even to observe them explicitly in the simulations! There is now ample evidence for the reliability of this approach [6].

References [1] T. Kerle, J. Klein, and K. Binder, “Effects of finite thickness on interfacial widths in confined thin films of coexisting phases,” Eu. Phys. J. B7, 401–410, 1999.

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[2] K. Binder, “Simulations of interfaces between coexisting phases: multiscale aspects,” In: Multiscale Computational Methods in Chemistry and Physics, pp. 207–220, IOS Press, Amsterdam, 2001. [3] J.S. Rowlinson and B. Widom, “Molecular theory of capillarity,” Clarendon, Oxford, 1982. [4] A. Werner, F. Schmid, M. Mueller, and K. Binder, “Anomalous size-dependence of interfacial profiles between coexisting phases of polymer mixtures in thin film geometry: a Monte Carlo simulation,” J. Chem. Phys., 107, 8175–8188, 1997. [5] K. Binder, “Phase transitions in polymer blends and block copolymer melts: some recent developments,” Adv. Polym. Sci., 112, 181–299, 1994. [6] A. Werner, F. Schmid, M. Mueller, and K. Binder, “Intrinsic profiles and capillary waves at homopolymer interfaces: a Monte Carlo study,” Phys. Rev. E, 59, 728–738, 1999. [7] J. Alejandre, D.J. Tildesley, and G.A. Chapela, “Molecular dynamics simulation of the orthobaric densities and surface tension of water,” J. Chem. Phys., 102, 4754– 4583, 1995. [8] K. Binder, “Monte Carlo simulations of surfaces and interfaces in materials,” In: A. Gonis, P.A. Turchi, and J. Kudrnovsky (eds.), Stab. Mater., pp. 3–37, Plenum, New York, 1996. [9] M. Mueller, K. Binder, and W. Oed, “Structural and thermodynamic properties of interfaces between coexisting phases in polymer blends: a Monte Carlo investigation,” J. Chem. Soc. Faraday Trans., 91, 2369–2379, 1995.

Perspective 20 HOW FAST CAN CRACKS MOVE? Farid F. Abraham IBM Almaden Research Center, San Jose, California

1.

Molecular Dynamics Experiments

With the present-day supercomputers, simulation is becoming a very powerful tool for providing important insights into the nature of materials failure. Atomistic simulations yield “ab initio” information about materials deformation at length and time scales unattainable by experimental measurement and unpredictable by continuum elasticity theory. Using our “computational microscope,” we can see what is happening at the atomic scale. Our simulation tool is computational molecular dynamics [2], and it is very easy to describe. Molecular dynamics predicts the motion of a large number of atoms governed by their mutual interatomic interaction, and it requires the numerical integration of the equations of motion, “force equals mass times acceleration or F = ma.” We learn in beginning physics that the dynamics of two atoms can be solved exactly. Beyond two atoms, this is impossible except for a few very special cases, and we must resort to numerical methods. A simulation study is defined by a model created to incorporate the important features of the physical system of interest. These features may be external forces, initial conditions, boundary conditions, and the choice of the interatomic force law. In the present simulations, we adopt simple interatomic force laws since we wish to investigate the generic features of a particular many-body problem common to a large class of real physical systems and not governed by the particular complexities of a unique molecular interaction. It is very important to emphasize that this is a conscious choice since it is not uncommon to hear others object that one is not studying “real” materials when using simple potentials. Feynman summarized this viewpoint well on page two, volume I of his famous three volume series Feynman’s Lectures In Physics [3]: “If in some cataclysm all scientific knowledge were to be destroyed and only one sentence passed on to the next generation of creatures, what statement would contain the most information in the fewest words? I believe it is 2793 S. Yip (ed.), Handbook of Materials Modeling, 2793–2804. c 2005 Springer. Printed in the Netherlands.

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the atomic hypothesis that all things are made of atoms – little particles that move around in perpetual motion, attracting each other when they are a little distance apart, but repelling upon being squeezed into one another. In that one sentence, you will see there is an enormous amount of information about the world, if just a little imagination and thinking are applied”. A simple interatomic potential may be thought of as a “model potential,” and the model potentials for the present studies are harmonic and anharmonic springs and the Lennard–Jones 12:6 potential. Complaints of model approximations are not new. In his book entitled The New Science of Strong Materials, Gordon comments on Griffith’s desire to have a simpler experimental material that would have an uncomplicated brittle fracture [4]. He writes, “In those days, models were all very well in the wind tunnel for aerodynamic experiments but, damn it, who ever heard of a model material?” In the mid 1960s, a few hundred atoms could be treated. In 1984, we reached 100,000 atoms. Before that time, computational scientists were concerned that the speed of scientific computers could not go much beyond 4 Gigaflops, or 4 billion arithmetic operations per second and that this plateau would be reached by the year 2000! That became forgotten history with the introduction of concurrent computing. A modern parallel computer is made up of several (tens, hundreds or thousands) small computers working simultaneously on different portions of the same problem and sharing information by communicating with one another. The communication is done through message passing procedures. The present record is well over a few tens of Teraflops for optimized performance. Moore’s Law states that computer speed doubles every one and one-half years. For 35 years, that translates into a computer speed increase of ten million. This is exactly the increase in the number of atoms that we could simulate over the last 35 years.

2.

Supersonic Crack Propagation in Brittle Fracture

Our simulation study addresses the important question “how fast can cracks propagate?” In this study, we used system sizes of about twenty million atoms, very large by present-day standards but modest compared to our second study on ductile failure, which will follow [1]. Based on our current simulation model, we develop our earlier studies on transonic crack propagation in linear materials and supersonic crack propagation in nonlinear solids. Our new finding centers on a bilayer solid which behaves under large strain like an interface crack between a soft (linear) material and a stiff (nonlinear) material. In this mixed case, we observe that the initial mother crack propagating at the Rayleigh sound speed gives birth to a transonic daughter crack. Then, quite unexpectedly, we observe the birth of a supersonic granddaughter crack.

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We verify that the crack behavior is dominated by the local (nonlinear) wave speeds, which can be in excess of the conventional sound speeds of a solid. In this problem, there are three important wave speeds in the solid. In order of increasing magnitude, they are the Rayleigh wave speed, or the speed of sound on a solid surface, the shear (transverse) wave speed and the longitudinal wave speed. Predictions of continuum mechanics [5] suggest that a brittle crack cannot propagate faster than the Rayleigh wave speed. For a mode I (tensile) crack, the energy release rate vanishes for all crack velocities in excess of the Rayleigh wave speed, implying that the crack cannot propagate at a velocity greater than the Rayleigh wave speed. A mode II (shear) crack behaves similarly to a mode I crack in the subsonic velocity range; i.e., the energy release rate monotonically decreases to zero at the Rayleigh wave speed and remains zero between the Rayleigh and shear wave speeds. However, the predictions for the two loading modes differ for crack velocities greater than the shear wave speed. While the energy release rate remains zero for a mode I crack, it is positive for a mode II crack over the entire range of intersonic velocities. From these theoretical solutions, it has been concluded that a mode I crack’s limiting speed is clearly the Rayleigh speed. The same conclusion has also been suggested for a mode II crack’s limiting speed because the “forbidden velocity zone” between the Rayleigh and shear wave speeds acts as an impenetrable barrier for the shear crack to go beyond the Rayleigh wave speed. The first direct experimental observation of cracks moving faster than the shear wave speed has been reported by Rosakis et al., [6]. They investigated shear dominated crack growth along weak planes in a brittle polyester resin under dynamic loading. Around the same time, we performed 2D molecular dynamics simulations of crack propagation along a weak interface joining two strong crystals [7]. We assumed that the interatomic forces are harmonic except for those pairs of atoms with a separation cutting the centerline of the simulation slab. For these pairs, the interatomic potential is taken to be a simple model potential that allows the atomic bonds to break. Our simulations demonstrated intersonic crack propagation and the existence of a mother-daughter crack mechanism for a subsonic shear crack to jump over the forbidden velocity zone. We have since discovered that a crack cannot only travel supersonically [8] but that there exists a mother-daughter-granddaughter crack mechanism in bilayer slabs. The classical theories of fracture [5] are largely based on linear elastic solutions to the stress fields near cracks. An implicit assumption in such theories is that the dynamic behavior of cracks is determined by the linear elastic properties of a material. We have found [9] that the MD simulation results for harmonic atomic forces are indeed well interpreted by elasticity theories. However, the effects of anharmonic material properties on dynamic behaviours of cracks are not clearly understood. This is partly due to the

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general difficulties in obtaining non-linear elastic solutions to dynamic crack problems. Molecular dynamics (MD) simulations can be easily adapted to the anharmonic case so that non-linear effects can be thoroughly investigated. We now discuss the anharmonic simulations.

3.

The Computer Experiments Setup

We consider a strongly non-linear elastic solid described by a tethered Lennard–Jones potential where the compressive part of this potential is identical to that of the usual Lennard–Jones 12:6 function and the tensile part is the reflection of the compressive part with respect to the potential minimum. An FCC crystal formed by this potential exhibits a strongly non-linear stress-strain behaviour resulting in elastic stiffening and an increase of the elastic modulus with strain, as shown in Fig. 1. Note that the elastic modulus increases by a factor of 10 at 13% of elastic strain, indicating that the material properties of such a solid is strongly non-linear in the hyperelastic regime. We performed 3D MD simulations of two face-center-cubic (fcc) crystals joined by a weak interface. In this study, we used system sizes of about twenty million atoms, though a billion atoms were used in a preliminary simulation [8]. For comparison, we consider the anharmonic tethered potential together with the harmonic potential where the spring constant is equal to the

3500 anhrarmonic potential modulus

2500

1500

500 0

0.05

0.1 strain

0.15

0.2

Figure 1. Variation of elastic modulus as a function of strain under uniaxial stretching in the [110] direction of a fcc crystal formed by the anharmonic (tethered repulsive LJ) potential.

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tethered potential at equilibrium. The interatomic force bonding the two crystals is given by the Lennard–Jones 12:6 potential. The simulation results are expressed in reduced units: lengths are scaled by the value of the interatomic separation where the LJ potential is zero and energies are scaled by the depth of the minimum of the LJ potential. Atoms bond only with their original nearest neighbors. Hence, rebonding of displaced atoms due to applied loading does not occur. For the adjoining two crystal slabs, we consider the following three simulation cases: (1) Harmonic case: both crystal slabs are characterized by the harmonic potential. This is used as a control for the anharmonic studies; (2) Anharmonic case: both crystal slabs are characterized by the anharmonic potential; (3) Mixed case: one crystal slab is characterized by the anharmonic potential, and the other is characterized by the harmonic potential. In each case, a shear crack lies along the (110) plane and oriented toward the [110] direction. The crack front is parallel to the [001] direction. The applied loading is dominated by shear. In order to interpret the results of MD simulation, we need the following wave speeds in the fcc crystals formed by harmonic and/or anharmonic potentials: the conventional longitudinal and shear wave speeds in the harmonic and anharmonic crystals in the direction of crack propagation; the longitudinal and shear wave speeds under applied loading in the anharmonic crystal in the direction of crack propagation; the local longitudinal and shear wave speeds near the crack tip in the anharmonic crystal in the direction of crack propagation. The calculations of these wave speeds will be presented in detail in a forthcoming paper. We give the results in Table 1. We add some comments with regard to the concept of local wave speeds near the crack tip, which play a critically important role in explaining our simulation results. Conceptually, it is clear that the fracture process in brittle solids involves breaking of atomic bonds and is intrinsically a highly non-linear process. The anharmonic material properties of solids near the cohesive strength of atomic bonds would in general be quite different from the harmonic properties. In an earlier attempt to explain why the highest crack velocities recorded for mode I crack propagation in a homogeneous body are significantly lower than the Rayleigh wave speed, Broberg [5, 10] has suggested that the reason could be due to some kind of a “local” Rayleigh wave speed in the highly strained region near the crack tip, rather than the Rayleigh wave speed in the

Table 1. Calculated wave speeds for the harmonic wave speeds, bulk anharmonic wave speeds due to applied loading and local wave speeds Longitudinal wave Shear wave

Bulk harmonic

Bulk anharmonic

Local

9.49 4.24

10.36 5.95

13.4 10.4

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undisturbed material. Since the local strain is an extremely strong function of position from the crack tip, one might think that the local wave speed should depend on distance from the crack tip and cannot have one single value. It was not until 30 years after Broberg’s suggestion that this issue was finally quantitatively studied by Gao [11] who made use of the Barenblatt cohesive model of a mode I crack and, for the first time, defined the local wave speed unambiguously as the wave speed at the location where stress is exactly equal to the cohesive strength of the material, i.e., the true point of “fracture nucleation”. The local wave speed characterizes how fast elastic energy is transported near the region of bond breaking in front of a crack tip. For example, for a mode I crack√in a homogeneous isotropic elastic, the local wave speed is calculated to be σmax /ρ [11] where σmax is the cohesive strength and ρ is the density of the undisturbed materials. The cohesive strength is typically around 1/10 of the shear modulus, suggesting that the local wave speed for a mode I crack is approximately 1/3 of the shear wave speed. Interestingly, experiments [12] and MD simulations [13] show that mode I cracks exhibit a dynamic instability at 30% of the shear wave speed which suggests a possible dependence on the local wave speed [11]. We note that the local wave speeds differ from the conventional wave speeds both qualitatively and quantitatively. In the present study, the focus is shifted to study the effect of stiffening anharmonic behaviors of materials (as in many polymers) on a mode II crack propagating along a weak interface described by the Lennard–Jones potential. The solid itself is described by the tethered Lennard–Jones potential. We follow the same approach used in [11] and define the local wave speed as the wave speed of the solid at the location adjacent to the interface where the shear stress is exactly equal to the cohesive strength of the interface. Note that in this case the local wave speed depends on both the interface cohesive strength and the non-linear elastic properties of the solid.

4.

The Computer Experiments Results & Discussion

Figure 2 presents distance-time histories of a crack moving in the three different slab configurations: two weakly bonded harmonic crystals (designated “harmonic”); two weakly bonded anharmonic crystals (designated “anharmonic”); and a harmonic crystal weakly bonded to an anharmonic crystal (designated “mixed”). The cracks begin their motion when the Griffith criterion is satisfied. The respective dip-spike regions for each history represent the birth of a new crack. For the harmonic and anharmonic simulations (see Figs. 3 and 4), we observe one such region representing the birth of a daughter crack, the former traveling at the longitudinal sound speed for the harmonic solid and the latter achieving a supersonic sound speed of Mach 1.6. For the mixed simulation (see Fig. 5), we see two such dip-spike regions where a daughter crack

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speed

30

20 anharmonic 10 harmonic 100

mixed 200

time Figure 2. The space-time history of the crack tip for the three different simulations described in the text (reduced units are used).

Figure 3. Snapshot pictures of a crack traveling in the harmonic slab, where the progression in time is from the top to bottom. The boxed snapshot pictures represent a progression in time from the top to bottom.

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Figure 4. Snapshot pictures of a crack traveling in the anharmonic slab, where the progression in time is from the top to bottom. The boxed snapshot pictures represent a progression in time from the top to bottom.

is born from the mother crack, which then gives birth to a granddaughter crack at a later time. In Figs. 3 and 4, the harmonic and anharmonic simulations are shown, respectively. The boxed snapshot pictures in each figure represent a progression in time from the top to bottom. In the top images, the mode II daughter cracks are born. The early-time occurrence of the Mach cone attached to the crack tip is evident in the anharmonic slab; In middle image of Fig. 3, the crack in the harmonic slab has a single Mach cone and a circular stress-wave halo, which indicates a crack speed equal to the longitudinal sound speed of the linear solid. This is in contrast to the middle image of Fig. 4 where the two Mach cones for the crack in the anharmonic slab signify that the crack is traveling supersonically. For the bottom images of the two figures, we conclude that the crack in the anharmonic slab wins. In Fig. 5, snapshot pictures of the crack traveling in the mixed slab are presented, where the progression in time is from the top to bottom. The material properties for the harmonic and anharmonic regions are labeled, and the different sound waves associated with the crack’s dynamics are denoted. The sequence shows the following progression of events: (1) the daughter

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Figure 5. Snapshot pictures of a crack traveling in the mixed slab, where the progression in time is from the top to bottom. The sound waves associated with the crack’s dynamics and the material properties of the harmonic and anharmonic regions of the mixed slab are labeled.

crack is born; (2) the daughter crack is traveling at the longitudinal sound speed of the harmonic slab; (3) the granddaughter crack is born; (4) the granddaughter crack speeds ahead to Mach 1.6, matching the crack speed in the anharmonic slab. We have shown that the behavior of the crack in the harmonic crystal is controlled by the conventional elastic wave speeds [7]. In contrast, the crack behavior in the anharmonic crystal is controlled by the local wave speeds, which play an important role in the dynamic behavior of crack propagation. A similar dependence has been identified in a related phenomenon where hyperelasticity plays an important role in whether a solid will undergo brittle fracture or ductile failure at the crack tip [14]. The local wave speed represents the non-linear, hyperelastic, material properties near cohesive failure of atomic bonds while the conventional elastic (shear and longitudinal) wave speeds represent the material properties under infinitesimal deformation and can differ significantly from the harmonic wave speeds. The crack propagation speeds observed in molecular dynamics simulations are tabulated in Table 2 for comparison.

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F.F. Abraham Table 2. Observed speeds at which a daughter crack or a granddaughter crack nucleates and the limiting speeds for the harmonic, anharmonic and mixed simulation cases Daughter crack Grand daughter Limiting speed

Harmonic

Anharmonic

Mixed

3.5 – 9.4

4.1 – 15

4.1 10.35 15

Comparing the MD simulation results with the calculated wave speeds for harmonic and anharmonic crystals, we reach the following conclusions. The harmonic case is consistent with the linear elastic theory of intersonic crack propagation. We have previously discussed this case in detail for the 2D harmonic MD simulation [7]. The same theory is found to apply for the present 3D case. Essentially, the initial crack starts to propagate when the Griffith criterion is satisfied. Near the Rayleigh wave speed, the crack encounters a velocity barrier and a vanishing of stress singularity at the crack tip; i.e., both stress intensity factor and energy release rate vanish at the Rayleigh wave speed. This velocity barrier is overcome by the nucleation of a daughter crack at a distance ahead of the mother crack. This distance corresponds to the shear wave front at which a peak of shear stress increases to a critical magnitude to cause cohesive failure of the interface. The daughter crack’s speed is only limited by the longitudinal wave speed. Comparing Tables 1 and 2, we see that the daughter crack indeed nucleates at the Rayleigh wave speed and the limiting speed agrees very well with the longitudinal wave speed for the harmonic crystal. The mother-daughter mechanism described above is consistent with the Burridge–Andrew model of intersonic crack propagation [15]. In the anharmonic case, the nucleation of the daughter crack is consistent with linear elastic theory. The mother crack initiates according to the Griffith criterion and achieves a limiting velocity equal to the Rayleigh wave speed. At this point, it is necessary to nucleate a daughter crack to break the velocity barrier. The limiting speed of the daughter crack is more than 50% higher than the longitudinal wave speed and cannot be explained by the linear theory of intersonic fracture. In comparison, the calculated local wave speed is approximately equal to (only 10% lower than) the observed limiting speed. In calculating the local wave speeds, we have ignored the large gradient of deformation field near the crack tip. In view of this simplification, we conclude that the local wave speed provides a reasonable explanation of the observed limited crack speed. In the mixed case, the nucleation of the daughter crack still occurs at the Rayleigh wave speed for reasons discussed above. The daughter crack breaks the velocity barrier at the Rayleigh speed and propagates near the longitudinal wave speed. This behavior is similar to the harmonic case. However, at a

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velocity of 10.35, we observe a granddaughter crack forming ahead of the daughter crack. The critical speed at which this transition occurs is very close to the local shear wave speed in the anharmonic crystal. The granddaughter crack rapidly accelerates toward the local longitudinal wave speed of the stretched non-linear solid. It is tempting to conclude that the nucleation of the granddaughter crack is controlled by the local Rayleigh wave speed magnified by the local stress concentration, although this issue is worth further investigation. In summary, we conclude that the local wave speeds play a dominant role in the behavior of cracks in the anharmonic crystals. The mixed case behaves somewhat like an interface crack between a soft material and a stiff material. Although the harmonic and anharmonic crystals have identical material properties under infinitesimal deformation, the local material properties near the crack tip are resembled by a bimaterial. Rosakis et al. [16] have previously studied crack propagation along an interface between PMMA (soft) and Al (hard) and found that the crack speed can significantly exceed the longitudinal wave speed of PMMA. Our present study shows that the crack behavior is dominated by the local (nonlinear) wave speeds. This is not only of theoretical interest, but also of practical importance. It is known that many polymeric materials, especially rubbers, increase their modulus significantly when stretched. The underlying physical mechanism is that initial elasticity in rubbers is due to entropic effects. When stretched to large deformation, the polymeric chains are straightened and covalent atomic bonds eventually dominate their hyperelastic response. In such solids, the elastic modulus increases with strain and the local wave speeds near a crack tip would be larger than the linear elastic wave speeds. The dynamic behavior of cracks in such solids should propagate at a speed exceeding the conventional wave speeds. Another point worth commenting here is the remarkable success of continuum theories in predicting the behaviors obtained by atomistic simulations. We have found previously [9] that the MD simulation results for shear crack propagation along a weak interface in harmonic solids are well interpreted by elasticity theories. In particular, calculations based on linear elasticity were able to predict the time and location of the daughter crack as well as the initiation time of the mother crack. Our atomistic simulation of the mother-daughter crack mechanism for intersonic shear crack is consistent with the continuum mechanics based discovery made earlier by Burridge and Andrew [15]. An important message of the present study is that the nonlinear continuum theory of local wave speeds is capable of predicting crack velocities in strongly nonlinear solids. Indeed, it is quite remarkable that the dynamic behavior of cracks may retain its basic nature over such a wide range of length scales, from atomistic calculations using interatomic potentials all the way up to macroscopic laboratory experiments and continuum elasticity treatments. This bridging of length scales in dynamic materials failure should be of great interest to the

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general scientific audience since it points out the fantastic power of continuum mechanics.

References [1] F.F. Abraham, R. Walkup, H. Gao et al., Proc. Na. Acad. Sci., 99, 5777, 2002. [2] M.P. Allen and D.J. Tildesley, “Computer simulation of liquids,” Clarendon Press, Oxford, 1987. [3] R. Feynman, R. Leighton, and M. Sands, “The Feynman lectures on physics,” Addison–Wesley, Redwood City, 1963. [4] J.E. Gordon, “The new science of strong materials or why you don’t fall though the floor,” Princeton University Press, Princeton, 1988. [5] L.B. Freund, “Dynamical fracture mechanics,” Cambridge Univ. Press, New York, 1990; B. Broberg, Cracks and Fracture, Academic Press, San Diego, 1999. [6] A.J. Rosakis, O. Samudrala, and D. Coker, Science, 284, 1337, 1999. [7] F.F. Abraham and H. Gao, Phys. Rev. Lett., 84, 3113, 2000. [8] F.F. Abraham, J. Mech. Phys. Solids, 49, 2095, 2001. [9] H. Gao, Y. Huang, and F.F. Abraham, J. Mech. Phys. Solids, 49, 2113, 2001. [10] B. Broberg, J. Appl. Mech., 546, 1964. [11] H. Gao, J. Mech. Phys. Solids, 44, 1453, 1996. [12] J. Fineberg, S.P. Gross, M. Marder, and H.L. Swinney, Phys. Rev. Lett., 67, 457, 1991; Phys. Rev. B, 45, 5146, 1992, The dynamic instability is generally known as the “mirror-mist-hackle” instability. A quantitative study of the onset of this instability was not done previously. [13] F.F. Abraham, D. Brodbeck, R. Rafey et al., Phys. Rev. Lett., 73, 272, 1994; J. Mech. Phys. Solids, 45, 1595, 1997. [14] F.F. Abraham, Phys. Rev. Lett., 77, 869, 1996; F.F. Abraham, D. Schneider, B. Land et al., J. Mech. Phys. Solids, 45, 1461, 1997. [15] R. Burridge, Geophys. J. Roy. Astron. Soc., 35, 439, 1973; D.J. Andrews, J. Geophys. Res., 81, 5679, 1976. [16] A.J. Rosakis, O. Samudrala, R.P. Singh et al., J. Mech. Phys. Solids, 46, 1789, 1998.

Perspective 21 LATTICE GAS AUTOMATON METHODS Jean Pierre Boon Center for Nonlinear Phenomena and Complex Systems, Universit´e Libre de Bruxelles, 1050-Bruxelles, Belgium

When one is interested in studying the dynamical behavior of fluid systems starting at the microscopic level, a logical approach is to begin with a molecular dynamics description of the interactions between the constituting particles. This approach quite often turns into a formidable task when the fluid evolves into a non-linear regime where chaos, turbulence, or reactive processes take place. But one may question whether a ‘realistic’ description of the microscopic dynamics is indispensable to gain insight on the underlying mechanisms of large scale non-linear phenomena. Around 1985, a considerable simplification was introduced [1] when pioneering studies established theoretically and computationally the feasibility of simulating fluid dynamics via a microscopic approach based on a new paradigm: a virtual simplified micro-world is constructed as an automaton universe based not on a realistic description of interacting particles, but merely on the laws of symmetry and of invariance of macroscopic physics. Suppose we implement point-like particles on a regular lattice where they move from node to node at each time step and undergo collisions when their trajectories meet at the same node. As the system evolves, we observe its collective dynamics by looking at the lattice from a distance. And the remarkable fact is that, if the collisions occur according to some simple logical rules (satisfying fundamental conservations) and if the lattice has the proper symmetry, this Lattice Gas Automaton shows global behavior very similar to that of a real fluid. So we can infer that, despite its simplicity at the microscopic scale, the lattice gas automaton (LGA) should contain, at the elementary level, the essentials that are responsible for the emergence of complex behavior, and thereby can help us understand the basic mechanisms where from complexity builds up. The LGA consists of a set of particles moving on a regular d-dimensional lattice L at discrete time steps, t = nt, with n an integer. The lattice is 2805 S. Yip (ed.), Handbook of Materials Modeling, 2805–2809. c 2005 Springer. Printed in the Netherlands.

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composed of V nodes labeled by the d-dimensional position vectors r ∈ L. Associated to each node there are b channels (labeled by indices i, j, . . . , running from 1 to b). At a given time t, a channel can be either occupied by one particle or empty, so that the occupation variable n i (r, t) = 1 or 0. When channel i at node r is occupied, then the particle at the specified node r has velocity ci . The set of allowed velocities is such that the condition r + ci t ∈ L is fulfilled. The “exclusion principle” requirement that the maximum occupation be of one particle per channel allows for a representation of the automaton configuration in terms of a set of bits {n i (r, t)}; r ∈ L, i = {1, b}. The evolution rules are thus simply logical operations over sets of bits. The time-evolution of the automaton takes place in two stages: propagation and collision. In the propagation phase, particles are moved according to their velocity vector, and in the (local) collision phase, the particles occupying a given node are redistributed amongst the channels associated to that node. So the microscopic evolution equation of the LGA reads n i (r + ci t, t + t) = n i (r, t) + i ({n j (r, t)}),

(1)

where i ({n j }) represents the colision term which depends on all channel occupations at node r. By performing an ensemble average (denoted by angular brackets) over an arbitrary distribution of initial occupations, one obtains a hierarchy of coupled equations for the successive n-body distribution functions. This hierarchy can be truncated to yield the Lattice Boltzmann equation for the single particle distribution function f i (r, t) = n i (r, t): ({ f j (r, t)}), f i (r + ci t, t + t) − f i (r, t) = Boltz i

(2)

The l.h.s. can easily be recognized as the discrete version of the l.h.s. of the classical Boltzmann equation for continuous sytems, and the r.h.s. denotes the collision term where the precollisional uncorrelated state ansatz has been used to factorize the b-particle distribution function. The Lattice Boltzmann Eq. (2) is one of the most important results in LGA theory. It can be used as the starting point for the derivation (via multi-scale analysis) of the macroscopic equations describing the long wave-length behavior of the lattice gas. The LGA macroscopic equations are found to exhibit the same structure as the classical hydrodynamic equations, and under the incompressibility condition, one retrieves the Navier–Stokes equations for non-thermal fluids. Another important feature of the Lattice Boltzmann equation is that it can be used as an efficient and powerful simulation algorihtm. In practice one usually prefers to use a simplified equation where the collision term is approximated by a single relaxation time process inspired by the Bhatnagar–Gross– Krook model, known in its lattice version as the LBGK equation: f i (r + ci t, t + t) − f i (r, t) = −

1 leq f i (r, t) − f i (r, t) , τ

(3)

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where the r.h.s. is proportional to the deviation from the local equilibrium distribution function. There is a wealth of applications of the lattice gas methods which have established their validity and their usefulness. LGA simulations, based on Eq. (1), are most valuable for fundamental problems in statistical mechanics such as for instance the study of fluctuation correlations in equilibrium and non-equilibrium sytems [2, 3]. As an example, Fig. 1 shows the trajectories of tracer particles suspended in a Kolmogorov flow (above the critical Reynolds number) produced by a lattice gas automaton and where from turbulent diffusion was analyzed [4]. Simulations of more direct practical interest such as for instance profile optimization in car design or turbulent drag problems are most efficiently treated with the lattice Boltzmann method, in particular using the LBGK model. The examples given in Figs. 2 and 3 illustrate the method for the study of viscous fingering in Hele–Shaw geometry, showing the effect of reactivity between the two fluids as a determinant factor in the dynamics of the moving interface [5]. Applications of the LGA approach and of the lattice Boltzmann equation cover a wide variety of theoretical and practical problems ranging from the

Figure 1. Lattice gas simulation of the Kolmogorov flow: the tracer trajectories reflect the topology of the A BC flow in the regime beyond the critical Reynolds number (Re = 2.5× Rec ).

Figure 2. Lattice Boltzmann (LBGK) simulation of viscous fingering in miscible fluids.

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Figure 3. Lattice Boltzmann (LBGK) simulation of viscous fingering showing the interface sharpening effect of a reactive process between the two fluids (compare with Fig. 2).

dynamics of thermal fluctuations and quantum lattice gas automata to multiphase flow, complex fluids, reactive sytems and inhomogeneous turbulence. The list of selected references given below should guide the reader interested in specific topics related to the subject: Statistical Mechanics of Lattice Gas Automata [2], Lattice Gas Automaton applications [3], Reactive Lattice Gas Automata [6], Quantum Lattice Gas Automata [7], Complex fluids [8], Lattice Boltzmann method [9], and recent developments [10].

References [1] U. Frisch, B. Haslacher, and Y. Pomeau, “Lattice gas automata for the Navier–Stokes equation,” Phys. Rev. Lett., 56, 1505–1508, 1986. [2] J.P. Rivet and J.P. Boon, Lattice Gas Hydrodynamics, Cambridge, Cambridge University Press, 2001. [3] D. Rothman and S. Zaleski, Lattice Gas Cellular Automata, Cambridge, Cambridge University Press, 1997. [4] J.P. Boon, D. Hanon, and E. Vanden Eijnden, “Lattice gas automaton approach to turbulent diffusion,” Chaos, Solitons and Fractals, 11, 187–192, 2000. [5] P. Grosfils and J.P. Boon, “Viscous fingering in miscible, immiscible, and reactive fluids,” J. Modern Phys. B, (to appear), 2002. [6] J.P. Boon, D. Dab, R. Kapral, and A. Lawniczak, “Lattice gas automata for reactive systems,” Phys. Rep., 273(2), 55–148, 1996. [7] D. Meyer, “From quantum cellular automata to quantum lattice gases,” J. Statist. Phys., 85, 551–574, 1996.

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[8] B.M. Boghosian, P.V. Coveney, and A.N. Emerton, “A lattice gas model of microemulsions,” Proc. Soc., A452, 1221–1250, 1996. [9] S. Succi, The Lattice Boltzmann Equation, Oxford, Clarendon Press, 2001. [10] P.V. Coveney and S. Succi (eds.), “Discrete modeling and simulation of fluid dynamics,” Philos. Trans. R. Soci., 360, 291–573, 2002.

Perspective 22 MULTI-SCALE MODELING OF HYPERSONIC GAS FLOW Iain D. Boyd University of Michigan, Ann Arbor, MI, USA

On March 27, 2004, NASA successfully flew the X-43A hypersonic test flight vehicle at a velocity of 5000 mph to break the aeronautics speed record that had stood for over 35 years. The final flight of the X-43A on November 16, 2004 further increased the speed record to 6,600 mph which is almost ten times the speed of sound. The very high speed attainable by hypersonic airplanes could revolutionize air travel by dramatically reducing inter-continental flight times. For example, a hypersonic flight from New York to Sydney, Australia, a distance of 10,000 miles, would take less than 2 h. Reusable hypersonic vehicles are also being researched to significantly reduce the cost of access to space. Computer modeling of the gas flows around hypersonic vehicles will play a critical part in their development. This article discusses the conditions that can prevail in certain hypersonic gas flows that require a multi-scale modeling approach.

1.

Hypersonic Flight

Hypersonic flight is one in which the vehicle velocity is much greater than the speed of sound. For the remainder of this article, we will use the Galilean transformation that allows us to consider the hypersonic flow of air around a fixed vehicle as being the same as when a hypersonic vehicle flies through stationary air. In gas dynamics, the Mach number is the ratio of the flow velocity to the speed of sound. Although there is not a fixed definition, hypersonic flow is generally considered to involve a Mach number greater than five. The fastest commercial passenger airplane ever developed is the Concorde that cruised at a Mach number of about two. The fastest military aircraft ever developed is the SR-71 Blackbird that cruised at a Mach number of three. Important examples of truly hypersonic vehicles include the X-15 powered by rockets and flown 2811 S. Yip (ed.), Handbook of Materials Modeling, 2811–2818. c 2005 Springer. Printed in the Netherlands.

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in the 1960s, the recently flown X-43A powered by a scramjet that reached a peak Mach number of about ten, and the space shuttle that re-enters the atmosphere at about Mach 25. Note, however, that the space shuttle during its hypersonic re-entry is a glider and does not fly under its own power. Flights of hypersonic passenger aircraft will not happen any time in the near future as there are many difficult technology issues faced in the development of such vehicles. Some of the most important of these issues concern the basic aerodynamics of the vehicle. Because hypersonic vehicles fly at very high speed, they generate a lot of aerodynamic drag. For cruise conditions, the drag must be overcome by the propulsion system and so directly affects the efficiency of the vehicle. Note that lack of economic viability for flight at Mach 2 for the Concorde was the primary reason for the end of commercial service of that aircraft. On hypersonic vehicles, the problem is even more acute calling for minimization of vehicle drag to the greatest extent possible. Also, at hypersonic speed, the high velocity is converted into high gas temperature at the vehicle surface requiring development of a thermal protection system (TPS) that may consist of special surface materials and active cooling strategies.

2.

Need for Computer Simulation of Hypersonic Flows

The development of hypersonic vehicles relies heavily on computational modeling. This is in part because laboratory experiments are both technically challenging and cost a lot of money. In order to generate in the laboratory an air flow at Mach 12 representative of a hypersonic flight condition, the air must be compressed to about 100 times atmospheric pressure and heated to a temperature of 7500 K. Expanding this hot, compressed gas through the hypersonic nozzle of a wind-tunnel will typically produce test times of less than one second making measurements difficult. Flight development programs for hypersonic vehicles such as the X-43A are very rare due to the orders of magnitude increase in cost compared to laboratory testing. By comparison, computer simulation techniques offer the potential to provide useful results to aid in the design of hypersonic vehicles in a fraction of the time and money required for laboratory and flight investigations. However, computer simulations are only useful if they are demonstrated to provide accurate results through direct comparison with measured data. Thus, laboratory testing remains an extremely important component in hypersonic gas dynamics research. Finally, similar to the development of any air vehicle, flight tests will always be required to address issues that cannot be foreseen in simulation and laboratory investigations.

Multi-scale modeling of hypersonic gas flow

3.

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Modeling Hypersonic Gas Dynamics

The hypersonic flow of air over a vehicle shape involves a number of complex gas dynamic phenomena. The air becomes compressed as it is decelerated by the presence of the vehicle surfaces. The hypersonic character of the free stream results in an extremely compact region of strong compression called a shock wave. For hypersonic vehicle flight conditions, the shock wave typically has a thickness much less than 1 mm over which gas properties such as pressure, temperature, and density are increased by substantial factors. Downstream of the shock wave, the high pressure and high temperature gas may undergo excitation of vibrational energy modes of the air molecules and the molecules may even begin to react chemically. As the gas finally approaches the surface of the vehicle, it encounters a relatively cold surface that causes the density to rise further in another thin layer of fluid called the boundary layer. It is the properties of the gas in this final layer of fluid immediately adjacent to the vehicle surface that determine the drag force and heat transfer to the vehicle. The most fundamental model of dilute gas dynamics is the Boltzmann equation that describes the evolution of the velocity distribution function of molecules. By taking moments of the Boltzmann equation, Maxwell derived the equation of change [1]: ∂ (n W ) ∂ (n ci W ) = [W ] + ∂t ∂ xi

(1)

in which W (ci ) is some molecular property that depends on the random velocity ci in the xi direction, indicates the average over the velocity distribution function otherwise called the moment, and [ ] indicates the change due to collisions. For example, if W is the mass of a molecule, then in the absence of chemical reactions the mass of the particle is conserved in a collision, and Eq. (1) becomes the well-known continuity equation of continuum gas dynamics. In addition, if equilibrium is assumed, and W is set to the momentum vector and the energy of a molecule, Eq. (1) provides a set of five continuum transport equations that is often called the Euler equations. This equation set corresponds to the case where the velocity distribution function everywhere in the flow is of the equilibrium Maxwellian form. To derive higher order sets of transport equations from Maxwell’s equation of change, it must be noted from Eq. (1) that the temporal derivative of any moment W depends on the divergence of the next higher velocity moment. This problem is addressed by one of two methods. In the Chapman–Enskog approach, a specific form of the velocity distribution function is assumed for flows perturbed slightly from the equilibrium state. In Grad’s method of moments, specific relations are assumed between the second and fourth order velocity moments. Setting W = mci c j and W = mci c j ck in Eq. (1), using either the Chapman–Enskog or

2814

I.D. Boyd

the Grad methods, leads to a set of 20 equations (the 20-moment equations) consisting of the five Euler equations, five further equations involving the shear stress tensor τ i j , and ten further equations involving the symmetric heat flux tensor Q i j k . For modeling hypersonic gas flow (and many other flows involving viscosity and thermal conductivity effects), a set of transport equations called the Navier–Stokes (NS) equations is widely employed. The NS equations are obtained from the 20-moment equations by replacing the heat flux tensor with a heat flux vector, qi . As discussed above, one of the methods for deriving transport equation sets from Maxwell’s equation of change is the Chapman–Enskog approach in which the following form for the velocity distribution function is assumed explicitly:

f (C) = f 0 (C) (C),

f 0 (C)dC =

1

c (2kT /m)1/2

(2)

e−C dC 2

π 3/2

C=

(3)

(C) = 1 + qi∗ Ci ( 25 C 2 − 1) − τi∗j Ci C j qi∗

κ =− P

2m kT

1/2

∂T , ∂ xi

τi∗j

µ = P

∂ci ∂c j 2 ∂ck + − δi j ∂x j ∂ xi 3 ∂ xk

(4)

(5)

where f o is the equilibrium Maxwellian form of the distribution. When the Chapman–Enskog parameter, , is slightly perturbed from unity, then the 20moment (or perhaps the Navier–Stokes) equations are valid. When is sufficiently far from unity then these equation sets can be expected to fail, and a more detailed approach is required. Examination of Eqs. (4) and (5) indicates that will become large when gradients in temperature and/or velocity become significant. This is exactly the situation in the critical regions of the shock waves and boundary layers created around hypersonic vehicles. In these regions, the characteristic length scales of the flow based on flow field gradients are so steep that the Chapman–Enskog distribution function cannot accurately describe the strong non-equilibrium behavior. Thus, higher order terms must be included in the perturbation to the Maxwellian distribution. In a continuum sense, these additional terms introduce higher order derivatives into the conservation equations. The next higher set of conservation equations produced in this manner is called the Burnett equations. Research on numerical solution of the Burnett equations has met with mixed success. The mathematical properties of the full set of Burnett equations presents several difficulties in terms of numerical solution. Attempts to simplify the equations by eliminating “difficult” terms have resulted in unfortunate side effects such as a failure to obey the second law of thermodynamics. In addition, development of the

Multi-scale modeling of hypersonic gas flow

2815

associated boundary conditions required for these higher order equation sets presents problems of its own. An alternative approach to the numerical simulation of the strongly nonequilibrium conditions experienced in hypersonic shock waves and boundary layers is the direct simulation Monte Carlo (DSMC) method [2]. In the DSMC technique, model particles are used to simulate the motions and collisions of real molecules. In a single iteration of the DSMC technique, the model particles are first moved over the distance given by the product of their velocity vector and a time-step t that is smaller than the mean free time, the average time that elapses between successive collisions of a molecule. After all particles are moved, any boundary interactions are processed, such as collision of particles with a solid wall. Next, the particles are binned into computational cells. The size of each cell should be less than the local mean free path, λ, that is the average distance a molecule travels between successive collisions. Within each of the cells, a number of particles are paired together, and collision probabilities determined for each pair to determine which particles actually collision. For the pairs that collide, collision mechanics is applied that conserve linear momentum and energy. Finally, the particle properties in each cell are sampled to determine time averaged, macroscopic properties such as density, flow velocity, temperature, and pressure. The DSMC technique has been successfully applied to a variety of nonequilibrium gas flow systems including hypersonic aerothermodynamics, spacecraft propulsion systems, materials processing, micro-scale gas flows, and space physics. Since the DSMC technique requires that the dimensions of its computational cells are of the order of a local mean free path (less than 1 mm under hypersonic vehicle flight conditions), the method becomes prohibitively expensive for continuum flows.

4.

The Multi-scale Hybrid Simulation Approach

The above discussion indicates that, in aiming to develop a computer model of the gas flow around hypersonic vehicles, we are faced with a dilemma. Solution of the sets of continuum conservation equations (such as the Navier– Stokes equations) using standard methods from computational fluid dynamics (CFD) [3], is relatively efficient numerically but may be physically inaccurate in the important regions of the shock waves and boundary layers where flow field gradients become very steep. The direct simulation Monte Carlo method is physically valid for the entire flow field, but is so numerically expensive as to make this approach impossible. One way of considering this problem is from a multi-scale perspective. Specifically, the continuum CFD approach is physically valid except in small thin regions where the scale length changes so

2816

I.D. Boyd

dramatically that sub-scale physical processes become important. Since these sub-scale, molecular level processes can be accurately simulated using the DSMC technique, a multi-algorithm approach to the problem presents itself as a natural way to proceed. This is a common solution technique in many areas of physics and engineering for multi-scale phenomena. Thus, a hybrid method that uses a CFD approach for most of the flow field, and the DSMC technique only in the high-gradient, sub-scale regions, appears to offer the potential to combine the physical accuracy and numerical efficiency of each of the component methods. There are two key issues in the development of a hybrid CFD-DSMC method: (1) how to determine the location of the domain boundaries between the CFD and DSMC regions; and (2) how to communicate information back and forward between continuum and discrete representations of the same gas flow. The first issue is resolved using continuum breakdown parameters that assess the physical accuracy of the continuum conservation equations. There are a number of such parameters in the literature [4, 5], and the most successful of these are defined in terms of flow field gradients similar to those that appear in the Chapman–Enskog distribution. The second issue is usually resolved by computing fluxes of conserved properties (mass, momentum, and energy) across the CFD/DSMC domain interfaces. This can either be achieved directly using a finite volume CFD formulation, or by evaluating the fluxes from primitive variables such as density, velocity, and temperature.

5.

Illustrative Results

The primary objective of a hybrid CFD-DSMC simulation is to improve the physical accuracy of a pure CFD solution without paying the enormous computational penalty of a pure DSMC solution. A common approach is to first obtain a pure CFD solution and to then use it to initialise a hybrid simulation. Figures 1 and 2 show some illustrative results for Mach 12 flow of molecular nitrogen over a blunt cone from a hybrid simulation performed in this manner. In Fig. 1, the decomposition of the flow domain between the CFD and DSMC methods is shown. A breakdown parameter based on flow field gradients is employed [4] and shows, as expected, that the DSMC technique is required in the bow shock wave and in the boundary layer next to the body surface. Figure 2 shows comparisons of three different numerical solutions along the axis of the flow (the stagnation streamline). Assuming that the pure DSMC solution is more physically accurate than the pure CFD solution, what we would like to see from the hybrid simulation is that it starts from the initial CFD solution, and moves it to the pure DSMC solution. This is exactly the behavior achieved in the results shown in Fig. 2.

Multi-scale modeling of hypersonic gas flow

2817

Figure 1. CFD (white) and DSMC (grey) domains for Mach 12 flow over a cone.

Figure 2. Numerical solutions of temperature along the flow axis.

6.

Outlook

The results shown in the figures represent a solid foundation for the further development of hybrid CFD-DSMC methods for hypersonic gas flows. The physical modeling capabilities of the hybrid methods need to be extended beyond the perfect gas description that is presently employed. Specifically,

2818

I.D. Boyd

models for vibrational excitation and chemical reactions must be developed. In addition, the numerical performance of the hybrid scheme must be improved dramatically. In the results shown here, the cost of the hybrid simulation was greater than that of the pure DSMC solution! Therefore, it is expected that refinement of hybrid CFD-DSMC hybrid methods will continue for several years. The development of such methods is intellectually challenging, and the need for accurate simulations of these complex hypersonic flows provides plenty of practical motivation.

References [1] T.I. Gombosi, Gaskinetic Theory, Cambridge University Press, Cambridge, 1994. [2] G.A. Bird, Molecular Gas Dynamics and the Direct Simulation of Gas Flows, Oxford University Press, New York, 1994. [3] J.C. Tannehill, D.A. Anderson, and R.H. Pletcher, Computational Fluid Mechanics and Heat Transfer, Taylor and Francis, Philadelphia, 1997. [4] W.-L. Wang and I.D. Boyd, “Predicting continuum breakdown in hypersonic viscous flows,” Phys. Fluids, 15, 91–100, 2003. [5] A.L. Garcia, J.B. Bell, W.Y. Crutchfield, and B.J. Alder, “Adaptive mesh and algorithm refinement using direct simulation Monte Carlo,” J. Comp. Phys., 154, 134– 155, 1999.

Perspective 23 COMMENTARY ON LIQUID SIMULATIONS AND INDUSTRIAL APPLICATIONS Raymond D. Mountain Physical and Chemical Properties Division, Chemical Science and Technology Laboratory, National Institute of Standards and Technology, Gaithersburg, MD 20899-8380, USA

Molecular dynamics and Monte Carlo simulations have proven to be invaluable tools in the study of liquids. Statistical mechanics tells us what to calculate to obtain liquid state properties (equation of state, specific heat, and transport coefficients for example) from a model of the interaction between molecules. Simulations are the tools needed to make those calculations for physically realistic models. While the impact of simulations on the investigation of fluid properties and structure cannot be overstated, there has been very little success in the use of these methods to predict liquid properties outside the range where model potentials have been fit to some property. One result of this is that industrial modelers, who are called upon to estimate properties for various compositions, temperatures, and pressures of liquids, have not found simulation methods to be of much value for their work. Some of the reasons for the lack of use of molecular level simulations in the industrial sector will be examined here. These reasons point to opportunities and challenges for research that could lead to industrially useful molecular simulation methods and practices. A situation where simulations could be industrially useful involve conditions where data are scarce and measurements would be difficult, expensive, or hazardous. The solubility of oxygen in a combustible liquid at elevated temperature is an example. Another would involve compounds that are expensive to prepare. Specific properties that have industrial interest and where simulations could be useful are the vapor pressure, solubility of gases in liquids, and the thermal conductivity and shear viscosity of liquids. Simulation methods exist for determining these properties, although they are not easily used by non-experts.

2819 S. Yip (ed.), Handbook of Materials Modeling, 2819–2821. c 2005 Springer. Printed in the Netherlands.

2820

R.D. Mountain

The ingredients that go into a simulation are model interaction potentials and state conditions. For this discussion, boundary conditions, initial conditions, and simulation software will not be discussed. At present, almost all model intermolecular potential functions are empirical or semiempirical constructs that have parameters adjusted so that some set of calculated properties match the experimental values of those properties over some range of state conditions. The accuracy of property predictions outside the range where the fit was made is unpredictable. This is not a satisfactory state of affairs as uncertainty estimates are essential for industrial purposes. The way a molecule–molecule interaction is represented can be quite artibrary. For example, interaction sites may be placed on atomic sites or are they may be placed elsewhere. Often, the location of “atomic sites” differs from the location of the sites on the isolated molecule. A related issue is the placement of partial charges to mimic the charge distribution in a molecule. When a mean-field approach to induced polarization is employed, the criteria for the placement and magnitude of the charges is not governed by unambiguous physical considerations. Since the electrostatic interactions are generally much stronger than dispersion interactions, small changes in the magnitude and/or position of charges can have significant consequences. How to improve the predictive ability of molecular simulations for thermodynamic and transport properties is an open topic. At the root of the problem is the quantum mechanical nature of the potentials that are the result of effectively replacing electronic degrees of freedom by potentials. This suggests that at least some of the electronic degrees of freedom should be explicitly included in the simulation. How to do this in a way that is computationally tractable for system sizes of interest, for simulation intervals of sufficient length that statistically adequate samples are obtained, and that include the necessary physics so that the arbitrariness is reduced to acceptable levels is not yet understood, although there are continuing research efforts. One approach is to use high level quantum chemistry methods to calculate directly the energy of interaction between pairs of molecules and then to map the results onto functions that represent the energy of interaction for various separations and orientations of a pair of molecules. This can result in an accurate representation of the energy of a pair of molecules, but may be quite complicated and may require considerable computational effort if used in a molecular dynamics or Monte Carlo simulation. Also, significant many body effects associated with induced polarization require separate calculations. Extending this approach to explicitly consider three or more molecules simultaneously is beyond current capabilities for all but very simple molecules. A second scheme is to use quantum mechanics to calculate energies and forces as needed during a simulation. These methods are currently limited to using low level methods such as density functional based quantum methods. Even so, the computational load is heavy, the number of molecules is restricted

Commentary on liquid simulations and industrial applications

2821

to 100 or so, and and accuracy of predicted properties relative to experimental values is often no better than that obtained from empirical models. The challenge and opportunity is to find and develop ways to incorporate the relevant quantum mechanical interactions into forms that are computationally manageable for molecular simulations. A second reason why industrial modelers have not embraced molecular simulations is that the computational time required to obtain statistically significant results, particularly for the viscosity and the thermal conductivity, can be quite long. This is the case when either the equilibrium time correlation function/Einstein relation approach is used or a nonequilibrium externally applied gradient is imposed on the fluid. The challenge and opportunity is to develop alternative, efficient ways to extract transport coefficients with known uncertainties from simulations.

Perspective 24 COMPUTER SIMULATIONS OF SUPERCOOLED LIQUIDS AND GLASSES Walter Kob Laboratoire des Verres, Universit´e Montpellier 2, 34095 Montpellier, France

Glasses are materials that are ubiquitous in our daily life. We find them in such diverse items as window pans, optical fibers, computer chips, ceramics, all of which are oxide glasses, as well as in food, foams, polymers, gels, which are mainly of organic nature. Roughly speaking glasses are solid materials that have no translational or orientational order on the scale beyond O(10) diameters of the constituent particles (atoms, colloids, . . .) [1]. Note that these materials are not necessarily homogeneous since, e.g., alkali-glasses such as Na2 O-SiO2 show (disordered!) structural features on the length scale of 6–10 Å (compare to the interatomic distance of 1–2 Å) and gels can have structural inhomogeneities that extend up to macroscopic length scales. Apart from their relevance as a technological material, glasses, or more general glass-forming liquids, are also an important subject of fundamental research since many of their unique properties are not understood well (or at all) from a microscopic point of view and it is a great challenge to close this gap in our knowledge. For example, why does the addition of 100 ppm of water change the viscosity of silica, SiO2 , by 2–3 orders of magnitude, a question that is important, e.g., to understand the flow of magmatic material? What are the mechanisms that make a glass “age”, i.e., lead to a change in its properties with time, a phenomenon that is highly important to understand material properties on long time scales? What are the “best” compositions to obtain metallic glasses with a prescribed mechanical/electric/magnetic/. . . property? The most pertinent question is however an embarrassingly simple one: What is a glass? To understand this question we can consider the temperature dependence of the viscosity η of a glass-forming liquid, i.e., of a liquid that can be supercooled by a significant fraction of its melting temperature. 2823 S. Yip (ed.), Handbook of Materials Modeling, 2823–2828. c 2005 Springer. Printed in the Netherlands.

2824

W. Kob

Experimentally it is found that η shows a very strong T -dependence in that it increases from values on the order of 10−2 Pa · s– 1014 Pa · s (these values are typical for atomic and molecular liquids). This is demonstrated in Fig. 1 where we show an Arrhenius plot of η(T ) for various glass-forming liquids. Note that in order to take into account the different nature of the liquids, and hence the different relevant temperature scales, we have normalized the temperature axis by Tg , the temperature at which the viscosity of the liquid takes the value 1012 Pa · s, i.e., a value that corresponds to a typical relaxation time of 1–100 s. From this plot, we thus see that a change in temperature by a factor of two leads to an increase of the viscosity by about 10–15 decades! If the temperature is lowered a bit further, the viscosity, and hence the relaxation times, increases even more and the material does not flow anymore on human time scales, i.e., it has become a glass. The reason for the dramatic slowing down of the dynamics of glass-forming liquids is presently not really known. Although there are reliable theoretical approaches, such as the

Figure 1. Temperature dependence of the viscosity of various glass-forming liquids as a function of Tg /T , where Tg is the glass transition temperature. Adapted from Angell et al. [2] (reproduced with permission).

Computer simulations of supercooled liquids and glasses

2825

so-called “mode-coupling theory of the glass transition” [3], that are able to rationalize in a semi-quantitative and sometimes even quantitative way the observed slowing down, these approaches usually work only in a relatively narrow temperature range, i.e., in a T -window in which η is on the order of 10−1 −104 Pa · s and thus changes by “only” 3–5 decades. Finding a reliable theoretical description that is able to describe the slowing-down in the whole T -range is one of the most important issues in the field. Despite the many experiments that have been made to determine the properties of glass-forming systems [4] and hence to shed some light on the reason for the occurrence of the glass transition, one is currently still far away to have an answer. The problem is that in experiments, the microscopic information that is needed to identify the mechanism for the slowing down, namely the trajectories of the particles as well as their statistical properties, is not really available. Therefore, also in the last twenty years there have been large efforts to use computer simulations to study glass-forming systems [5]. Roughly speaking these numerical efforts can be divided into two large, not completely disjoint, groups. Simulations that aim to increase our understanding of a (real) specific material (or a narrow class of materials) such as silica, ortho-terphenyl, polyethylene, etc. Since the properties of the materials are often very specific, it is necessary to use force fields that are highly accurate. Although for the case of crystalline systems accurate effective force fields are often available, this is unfortunately not the case for supercooled liquids and therefore the results obtained with these types of interactions must always be looked at with some caution. The alternative is to use ab initio calculations in which the forces between the ions are determined directly from the electronic degrees of freedom [6]. Although the so obtained interactions are very accurate, the price one has to pay is an increase of about a factor 106 in the computational burden. Therefore the current state of the art ab initio simulations are done on systems with only a few hundred particles and cover a time scale of just a few tens of a pico-second. The second large group of simulations of glass-forming materials are numerical investigations aimed to understand the more universal aspects of these systems. Therefore one uses models that are relatively simple, such as Lennard–Jones particles, lattice gases, or spin models, i.e., systems in which the positions of the “particles” are frozen on a lattice and only their orientational degrees of freedom are relevant. Due to their simplicity these models are very useful to obtain results with a good statistics and to study the dynamical behavior of the system at relatively long times, which presently means O(109 ) time steps. Although making 109 time steps for a system size of O(1000) particles is presently still a huge numerical effort, it is still significantly less than the number of steps that one needs in order to reach macroscopic time scales. For example, in the case of silica a typical time step is around 1 fs and hence 109 steps correspond to only 1 µs! Thus the last 6–8 decades in η of the curves

2826

W. Kob

shown in Fig. 1 are presently not accessible in (equilibrium!) simulations and thus this technique is presently not able to make a significant contribution to our understanding of these highly supercooled liquids. Note that here we emphasize the notion of “equilibrium”. Of course it is possible to quench the system rapidly to a relatively low temperature, e.g., by coupling it to a heat bath, and then to start a microcanonical or canonical simulation. However, since the system is now out of equilibrium the measured properties might be, and usually are, very different from the ones found for the same system at this temperature but in equilibrium Bouchaud 1998 [10]. Note that in principle one is of course not forced to use these 109 time steps to do a “realistic dynamics”, i.e., one in which Newton’s equations of motions are solved by means of the standard algorithm, such as the one of Verlet. Instead it would be perfectly reasonable to use a cleverly designed Monte Carlo scheme which allows to equilibrate the system at relatively low temperatures and then to use the so generated equilibrium configurations as a starting point for a “conventional” simulation, i.e., a dynamics in which the particles follow Newton’s equations of motion. This approach is, e.g., used very successfully in the context of second order phase transitions, another physical situation where the dynamics becomes exceedingly slow (critical slowing down). However, whereas in this latter case there are indeed efficient algorithms, such as the one by Swendsen and Wang [7] that allows to equilibrate the system even very close to the critical point, this is not the case for glass-forming liquids, despite strong efforts to find such accelerating schemes. Nevertheless, some progress has already been made [8] and therefore it can be hoped that in the not too distant future it will indeed be possible to equilibrate glass-forming systems even at very low temperatures. (Note that since from a mathematical point of view equilibrating a glass-forming liquid has many similarities with more formal optimization problems, such as the traveling salesman problem (in both cases one searches for low lying minima of a complicated function, the potential energy/cost function), in recent years, there have been quite a few very fruitful exchanges between these two communities that have lead to an improvement of the algorithms [8].) Once such a method has been found, it will be possible to investigate on the microscopic level the vibrational and relaxation dynamics of such systems, and hence to make a much closer connection to experimental data and thus to allow to interpret it in a more reliable way and to make new predictions on the structure and dynamics of glass-forming systems. On the other hand such simulations will also give the possibility to obtain a better theoretical (analytical) description of the dynamics of supercooled liquids. As a possible example we return to the above mentioned mode-coupling theory. In this theory, one discusses the temperature dependence of the so-called intermediate scattering function F(q, t), a space-time correlation function that is directly measurable

Computer simulations of supercooled liquids and glasses

2827

in light or neutron scattering experiments and hence is of great theoretical and practical importance [9]. This function is defined given by F(q, t) =

N N 1 exp(iq · (r j (t) − rk (0)), N j =1 k=1

(1)

where r j (t) is the location of particle j at time t and q is a wave-vector. Using a procedure that is called “Zwanzig-Mori projection operator formalism” it is possible to derive exact equations of motion for F(q, t) and one finds that they have the form [9]: ¨ F(q, t) +

2q F(q, t)

+

2q

t

M(q, t − t )F(q, t )dt = 0.

(2)

0

Here the (squared) frequency 2q is given by q 2 k B T /(m S(q)), where m is the mass of the particles and S(q) is the static structure factor. The mode-coupling theory now makes the approximation that the so-called “memory function” M(q, t) is a bi-linear product of F(q , t) with coefficients that depend only on S(q ) [3]. As mentioned above, the theory works very well at temperatures corresponding to low and intermediate values of the viscosity. However, at lower temperatures the approximation seems not to be accurate any more but for the moment it is not clear at all how it can be improved. Thus to advance, it will be necessary to determine the memory function and to see why the approximation is no longer good. This can be done, e.g., by making large scale molecular dynamics simulations in order to determine F(q, t) with high precision. Using a simple Laplace transform it is then possible [9] to invert Eq. (2) in order to express M(q, t) as a function of F(q, t) and hence to see how the mode-coupling approximations can be improved. This procedure would thus allow to make finally one important step forward to our understanding of the relaxation dynamics of deeply supercooled liquids, a field that has not seen much theoretical progress since about 20 years, i.e., after the mode-coupling theory has been proposed. Thus it is evident that for the next few years the big challenges in simulations of glass-forming liquids and glasses are to obtain potentials that are sufficiently accurate to describe these disordered structures on a quantitative level and to develop new simulation algorithms that allow to equilibrate these systems also at low temperatures. Once these two goals are attained it will be possible to make on the one hand qualitative and quantitative predictions on the properties of glass-forming materials and on the other hand allow to make an important step forward in our understanding of the nature of the glassy state.

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References [1] J. Zarzycki (ed.), Materials Science and Technology, Vol. 9: Glasses and Amorphous Materials, VCH Publ., Weinheim, 1991. [2] C.A. Angell, P.H. Poole, and J. Shao, “Glass-forming liquids, anomalous liquids, and polyamorphism in liquids and biopolymers,” Nuovo Cim. D, 16, 993–1025, 1994. [3] W. G¨otze and L. Sj¨ogren, “Relaxation processes in supercooled liquids,” Rep. Prog. Phys., 55, 241–376, 1992. [4] K.L. Ngai, G. Floudas, A.K. Rizos, and E. Riande (eds.), “Relaxation in complex systems,” J. Non-Cryst. Solids, 307–310, 1–1080, 2002. [5] W. Kob, “Supercooled liquids, the glass transition, and computer simulations,” In: J.-L. Barrat, M. Feigelman, J. Kurchan, and J. Dalibard (eds.), Lecture Notes for Slow Relaxations and Nonequilibrium Dynamics in Condensed Matter, Les Houches July, 1–25, 2002; Les Houches Session LXXVII. Springer, Berlin, pp. 199–270, 2003. [6] M.E. Tuckerman, “Ab Initio, molecular dynamics: basic concepts, current trends and novel applications,” J. Phys. Condens. Matter., 14, R1297–R1355, 2002. [7] D.P. Landau and K. Binder, Monte Carlo Simulations in Statistical Physics, Cambridge University, Cambridge, 2000. [8] Y. Okamoto, “Generalized-ensemble algorithms: enhanced sampling techniques for Monte Carlo and molecular dynamics simulations,” J. Mol. Graphics Modelling, 22, 425–439, 2004. [9] U. Balucani and M. Zoppi, Dynamics of the Liquid State, Oxford University Press, Oxford, 1994. [10] J.-P. Bouchaud, L.F. Cugliandolo, J. Kurchan, and M. M´ezard, “Out of equilibrium dynamics in spin glasses and other glassy systems,” In: A.P. Young (ed.), Spin Glasses and Random Fields, World Scientific, Singapore, pp. 161–224, 1998.

Perspective 25 INTERPLAY BETWEEN MATERIALS THEORY AND HIGH-PRESSURE EXPERIMENTS Raymond Jeanloz University of California, Berkeley, CA, USA

High-pressure experiments have played an important role in materials physics, both as a route for synthesizing new materials and as a means of validating theory. These roles are complementary, and have proven to be remarkably synergistic. Perhaps the most famous example of materials synthesis under pressure is that of diamond, the high-pressure form of carbon that is produced in the Earth’s mantle and – since the early 1950s – in the laboratory; industrial diamonds have subsequently found a wide range of applications [1, 2]. Focusing on a property rather than a particular material, superconductivity illustrates an important characteristic that has been systematically pursued in the laboratory. Not only do the examples discovered under pressure almost double the number of superconductors documented among elements (Fig. 1)[3], but the highesttemperature superconducting transition found to date has been measured at high pressure [4]. Pressure is also significant for condensed-matter theory, which is founded on quantum mechanics but involves important approximations in actual implementation (e.g., of exchange and correlation effects; applicability of one-electron and density-functional approaches; separation of electronic and vibrational degrees of freedom; and considerations of electron-spin and of relativistic effects). The best way to evaluate the reliability of theory is to have it predict the stability and properties of new materials, or of known materials under new conditions. If one changes composition to form a new material, that also changes the approximations embedded in the potential term of the Schr¨odinger equation, however; if one changes temperature, new complications arise as thermal excitations have to be reliably accounted for in the theory. 2829 S. Yip (ed.), Handbook of Materials Modeling, 2829–2835. c 2005 Springer. Printed in the Netherlands.

2830

R. Jeanloz Superconductivity of Elements He

H

Li

Be

17

0.03

Na

Mg

K Rb

T

P⫽0

TC in K for bulk sample

B

C

N

T

P>0

maximum reported TC in K

Ca

Sc

Ti

V

Cr

15

0.3

0.5

5.4

3

Sr

Y

Zr

Nb

Mo

Tc

Ru

4

3

0.6

9.3

0.9

8.2

0.5 10

Mn

Fe

Co

Ni

Cu

2 Rh

Pd

Ag

⫺6

Cs

Ba

Hf

Ta

W

Re

Os

Ir

1.7

5

0.4

4.5

0.01

1.7

0.7

0.1

Fr

Ra

Rf

Db

Sg

Bh

Hs

Mt

Ds

Rg

La

Ce

Pr

Nd

Pm

Sm

Eu

Gd

Tb

6

1.8

Ac

Th 1.4

Pt

Au

O

F

Ne

Cl

Ar Kr

0.6

11 Al

Si

P

S

1.2

8

20

17

Zn

Ga

Ge

As

Se

Br

0.9

1.1

5.5

2.7

11

1.5

Cd

In

Sn

Sb

Te

I

0.5

3.4

3.7

3.6

7.5

1.2

Po

At

Hg

Tl

Pb

Bi

4.2

2.4

7.2

7

Dy

Ho

Er

Tm

Yb

Xe

Rn

Lu 1.2

Pa 1.4

U 1.3

Np

Pu

Am 0.6

Cm

Bk

Cf

Es

Fm

Md

No

Lr

Figure 1. Periodic table identifying elements that have been found to be superconductors at high pressures (shaded), and those elements known as superconductors at zero pressure, with transition temperatures shown for bulk samples at zero applied magnetic field [3].

These and other means of perturbing the system can be experimentally applied and theoretically modeled, but none is as straightforward for theory as considering the effect of pressure. Quite simply, the inter-atomic spacing is changed, and the quantum mechanical problem is now solved (and optimized) again in order to predict the properties of matter under new conditions. Experiments then verify the predictions, or not.

1.

Effect of Pressure on Materials Chemistry

As shown by many theoretical studies, pressure can dramatically change the properties of materials. One of the earliest examples is the 1935 prediction by E. Wigner and H. B. Huntington that hydrogen – normally a transparent, electrically insulating gas at ambient conditions – would become a metal at high pressures [5]. Ironically, this example has become notorious because, though there is no doubt that hydrogen becomes metallic at high enough pressures and temperatures, the conditions of metallization and the detailed processes involved remain controversial. It is unclear whether the metallization of

Interplay between materials theory and high-pressure experiments

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hydrogen can be “exactly solved” even with present-day quantum mechanical methods. That the periodic table of the elements is fundamentally altered under pressure is suggested by the following reasoning. The PdV pressure–volume work induced by compression to 1 million times atmospheric pressure (1 Mbar or 100 GPa), conditions readily achieved using either static (diamond-anvil cell) or dynamic (shock-wave) methods, amounts to an internal-energy change of order eV (∼105 J/mol of atoms). This is comparable to the energies of valence-shell electrons, and thus to chemical-bonding energies [6, 7]. In other words, the thermodynamic perturbation caused by Megabar pressures is enough to alter chemical bonding. Numerous experiments document major changes in chemical properties under pressure. Oxygen becomes chalcogenide-like, for example, transforming from a transparent, electrically insulating gas to an opaque metal by about

300

250

Pressure (GPa)

CsI-Xe CONVERGENCE 200

150

Insulator-Metal Transition 100

50

Structural Transitions 0 0

10

20

30

40

50

60

Volume (A3/atom) Figure 2. Crystalline phases and equations of state of Xe (open symbols; face-centered cubic structure on extreme right) and CsI (closed symbols; CsCl structure on right) at low pressures converge at pressures above 50 GPa (hexagonal close-packed structure on left), and both materials are metallic at pressures above 120 GPa [7].

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100 GPa (and a superconductor at somewhat higher pressures) [7]. Perhaps more remarkable, the “inert” gas xenon becomes a close-packed metal by 100 GPa, with a crystal structure and equation of state matching those of the iso-electronic compound CsI (Fig. 2). Indeed, the salt (with a gap between valence– and conduction–electron energy bands exceeding 6 eV at zero pressure) becomes metallic by 100 GPa, and then superconducting by 200 GPa. Could it be that xenon will also become a superconducting metal at high pressures? Experiments have, so far, failed to confirm this.∗

2.

Comparison of Theory with Experiment

One of the earliest high-pressure successes of modern electronic bandstructure calculations was the prediction that potassium would transform from an alkali to a transition metal at high pressure [8], and could therefore alloy with iron under compression.† Spectroscopic and other measurements confirmed that K does take on a transition-metal character by 25 GPa [9], and potassium has subsequently been alloyed with nickel and with iron at pressures above 25–30 GPa [10]. Quantitative predictions of structural transitions are notoriously difficult because the energy differences between bulk crystalline phases are often quite small: comparable, in many cases, to the energy resolution of calculations until the 1970s. By the 1980s, however, the feasibility of theoretically deriving transition pressures was demonstrated through studies on Si. Superconductivity was then predicted for some of the high-pressure phases of silicon, and experimentally confirmed shortly thereafter [11]. The role of theory relative to high-pressure experiment underwent a quiet but significant change in the 1990s, with predictions of phase stability being made well before they were experimentally studied, and not just for elements but also for compounds exhibiting more complex structures and variations in bonding. A subtle structural transition (from corundum to Rh2 O3 II structures) was predicted for Al2 O3 in 1987–1994 for example, and only experimentally confirmed in 1997 [12].

∗ The failure to find superconductivity in xenon is plausibly due to the difficulty of experiments in the

“multi-Megabar” pressure range, but might also be due to subtle effects being unexpectedly important. For example, the slightly different masses of Cs and I result in the compound having twice as many vibrational mode-branches as the element, with optic as well as acoustic modes, and the possible effect of this in xenon could be explored through studies of a solid solution consisting of a crystallographically ordered 50:50 mixture of 126 and 132 isotopes of Xe. † The most abundant naturally occurring radioactive isotope with a half life in the billion-year range, 40 K, is an important source of heat for planets. If it alloys with iron at high-pressures, it could also be an important source of heat for the Earth’s iron-rich core and its dynamo-generated magnetic field.

Interplay between materials theory and high-pressure experiments

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At the same time, molecular dynamics, with interatomic forces derived from first-principles (quantum mechanical) calculations, was being developed to make predictions at high temperatures as well as high pressures. That crystalline CH4 methane would polymerize and ultimately form diamond at elevated pressures and temperatures was thus predicted, and subsequently observed experimentally [13]. By the end of the 20th century, it was becoming evident to high-pressure experimentalists that they had to pay attention to theory, not only to help identify interesting problems (determine which combination of elements to study, at what conditions and with what transformations or other phenomena to expect), but also to help interpret and go beyond the experimental measurements. At the highest pressures, many observations are indirect or require considerable interpretation, and theory can play a crucial role in validating and quantifying these interpretations. Also, to the degree that measured properties agree with theoretical predictions, one can have some confidence in the theoretical values for properties that cannot be measured (for example, elastic shear moduli are generally more difficult to measure under pressure than compressional moduli, so theory can be used to estimate the former if it has been shown to yield good estimates for the latter). Although it is useful to validate theory by demonstrating good agreement with measurement, comparisons between the two are especially powerful when disagreements are uncovered, because this is when the limitations in current modeling approaches or approximations are revealed. For example, the molecular-dynamics prediction that pressures of 100–300 GPa would polymerize methane and then transform it to diamond is experimentally found to be too high by approximately one order of magnitude. Such quantitative discrepancies provide important information about how theory can be improved.

3.

Future Directions for Materials Modeling

Theoretical modeling is pursued in order to develop a fundamental understanding of material properties, including what controls those properties and how those properties can be tuned in practice. An important goal of theory is thus to be able to take a desired property, such as superconductivity or high elastic modulus, and deduce the range of materials that would have these properties as well as predicting the conditions under which the relevant phases are thermodynamically stable. Extension beyond classical thermodynamic equilibrium (the material’s ground-state) is important in at least two regards. First, non-equilibrium properties such as strength and absorption or reflectance spectra (i.e., excited electronic or vibrational states) are of great practical interest. There is again much

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opportunity for comparing theoretical predictions against experimental observations, in particular with pressure as a variable over which to make the comparisons (as a function of compression for a given phase, as well as across pressure-induced phase transformations). Second, once materials having particular desirable properties have been identified, it becomes all-important to consider synthesis routes going beyond the bulk equilibrium of classical thermodynamics. After all, the value of a material is directly proportional both to the nature of its properties and to the ease with which it is synthesized. Carbon is a case in point, with formation by chemical vapor deposition (CVD) having revolutionized the utility of synthetic diamond. High-pressure experiments thus helped establish the equilibrium properties and conditions for synthesis of diamond, but it is the low-pressure method of growth that has taken over as one of the most effective means of making high-quality samples [14]. Indeed, the toughness properties of CVD diamond can be tuned, and this in turn will prove useful for future high-pressure experiments as the diamonds are used as anvils. The ultimate hope is that by fine-tuning theoretical modeling of materials, it will become possible to predict optimal routes for synthesis. Understanding the microscopic details of how materials are nucleated or can grow, and not only under conditions of bulk thermodynamic equilibrium, can then provide the practical answer to those needing a particular set of material properties for a given application: from a prediction of materials that have the required properties to a recipe for the optimal synthesis route. At that point, high-pressure experiments may no longer have their current utility for the study of materials.

References [1] K. Nassau and J. Nassau, J. Crystal Growth, 46, 157–172; F.P. Bundy and R.C. DeVries, “Diamond: high-pressure synthesis,” In: Encyclopedia of Materials: Science and Technology, Elsevier, Amsterdam, 2001. [2] R.M. Hazen, The Diamond Makers, Cambridge University Press, Cambridge, 1999. [3] C. Buzea and K. Robbie, Supercond. Sci. Technol., 18, R1–R8, 2005. [4] L. Gao, Y.Y. Xue, F. Chen, Q. Xiong, R.L. Meng, D. Ramirez, C.W. Chu, J.H. Eggert, and H.K. Mao, Phys. Rev. B, 50, 4260–4263, 1994. [5] E. Wigner and H.B. Huntington, J. Chem. Phys., 3, 764–770, 1935. [6] R. Jeanloz, Ann. Rev. Phys. Chem., 40, 237–259, 1989. [7] R.J. Hemley and N.W. Aschroft, Phys. Today, 51, 26, 1998; R.J. Hemley, Ann. Rev. Phys. Chem., 51, 763–800, 2000. [8] M.S.T. Bukowinski, Geophys. Res. Lett., 3, 491–503, 1976. [9] K. Takemura and K. Syassen, Phys. Rev. B, 28, 1193–1196, 1983. [10] L.J. Parker, T. Atou, and J.V. Badding, Science, 273, 95–97, 1996; L.J. Parker, M. Hasegawa, T. Atou, and J.V. Badding, Eur. J. Solid-State Inorg. Chem., 34, 693–704, 1997; K.K.M. Lee and R. Jeanloz, Geophys. Res. Lett., 30,

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[11] [12]

[13]

[14]

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doi:10.1029/2003GL018515, 2003; K.K.M. Lee, G. Steinle-Neumann, and R. Jeanloz, Geophys. Res. Lett., 31, doi:10.1029/2004GL019839, 2004. D. Erskine, P.Y. Yu, K.J. Chang, and M.L. Cohen, Phys. Rev. Lett., 57, 2741–2744, 1986. R.E. Cohen, Geophys. Res. Lett., 14, 37, 1987; H. Cynn, D.G. Isaak, R.E. Cohen, M.F. Nicol, and O.L. Anderson, Am. Mineral., 75, 439, 1990; F.C. Marton and R.E. Cohen, Am. Mineral., 79, 789, 1994; K.T. Thompson, R.M. Wentzcovitch, and M.S.T. Bukowinski, Science, 274, 1880, 1996; N. Funamori and R. Jeanloz, Science, 278, 1109–1111, 1997. F. Ancilotto, G.L. Chiarotti, S. Scandolo, and E. Tosatti, Science, 275, 1288–1290, 1997; L.R. Benedetti, J.H. Nguyen, W.A. Caldwell, H. Liu, M. Kruger, and R. Jeanloz, Science, 286, 100–102, 1999. C.S. Yan, K.K. Mao, W. Li, J. Qian, Y. Zhao, and R.J. Hemley, Phys. Sta. Sol.(a), 201, R25–R27, 2004.

Perspective 27 ATOMISTIC SIMULATION OF FERROELECTRIC DOMAIN WALLS I-Wei Chen Department of Materials Science and Engineering, University of Pennsylvania, Philadelphia, PA 19104-6282, USA

1.

Introduction

Atomistic simulation of moving discommensurations is most useful when atomic details have a strong influence on the outcome. A classical example is the problem of dislocation core and movement in which the nature of atomic bonding has a direct effect on the configuration of the dislocation core, which in turn affects the manner the dislocation moves [1]. The final macroscopic characteristics of plastic deformation are intimately related to these details, of which a complete understanding can only come from the atomistic simulation of both the static and the (activated) non-equilibrium configurations of dislocations. The extension to the interface problem is encountered in martensitic transformations, involving either the parent/product interface or the product/product interface, which we call variant interface [2]. Variant interfaces are mobile and may be regarded as a group of dislocations, and like dislocations they move under a stress. The problem is relevant to shape memory alloys in which the stress and rate dependence of shape change is ultimately controlled by the atomic configurations at variant interfaces during their movement, even though the final (presumably equilibrium) multi-variant configurations are dictated by crystallography and elastic energy minimization. As a further extension to the interface problem, domain walls in ferroelectric crystals are like variant interfaces in that, crystallographically, both belong to the class of twin boundaries, but domain walls are complicated by electrostatic considerations [3, 4]. Since the origin of ferroelectricity lies in the instability of covalent bonding, the atomic configurations of domain walls are necessarily sensitive to both covalent bonding and long-range elastic and electrostatic interactions. Such complexity makes the problem a good candidate for atomistic simulation studies from which one may hope to better understand domain switching 2843 S. Yip (ed.), Handbook of Materials Modeling, 2843–2847. c 2005 Springer. Printed in the Netherlands.

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characteristics such as coercivity and fatigue, which are important for ferroelectric applications. To date, however, there have been very few atomistic simulation studies on domain walls, none being relevant to domain wall movement. Nevertheless, the recent development of phenomenological but realistic potentials for Pb(Zr,Ti)O3 (PZT) [5] makes such studies entirely feasible within today’s computational resources. The purpose of this commentary is to outline the basic domain wall problem to stimulate the community to undertake atomistic simulations.

2.

Synopsis

Some general considerations based on our theoretical understanding of dislocation and ferroelectricity provide the following synopsis. A domain wall separates two domains of different polarization vectors. Like twinning, ferroelectric distortion causes internal strains that vary from domain to domain, albeit the net shear component of such strains can be greatly reduced by alternating the shear direction. Therefore, the simplest domain wall (called 180◦ domain wall) has opposite polarization on the two sides, and it has no elastic distortion at all. All the non-180◦ domain walls, however, are like twin boundaries and have some short-range strains associated with them. Therefore, they interact with stresses, including self-stress of the walls and internal stresses due to impurities and heterogeneities. A stress-free domain wall, in turn, is electrically neutral, which requires the normal component of the polarization vector to be continuous across the domain wall. This is illustrated for a tetragonal crystal (e.g., BaTiO3 ) for both 180◦ (BB’ in Fig. 1) and 90◦ (AA’ in Fig. 1) domain walls. This neutrality condition is not satisfied if the domain wall is aligned in other directions. (For example, BB” in Fig. 1 has a A'

B'

B"

A' D C

A

B

A Figure 1.

B

Atomistic simulation of ferroelectric domain walls

2845

negatively charged kink at CD.) Therefore, curved or kinked domain walls are generally charged. Such is the case at the edge of a lens-shaped domain where two domain walls bend to meet each other. Also, when a domain wall moves, it prefers to first form charged kinks (or ledges in three dimensions) as shown in Fig. 1, then propagate the kinks sidewise, rather than moving the entire domain wall forward all at once. (The electrostatic forces on the charged kinks always move them in a way to enlarge the side where the polarization vector is electrically favored.) These kinks and ledges are structurally similar to those found on moving dislocations, twin boundaries and variant interfaces, and their formation determines the activation energy for the wall movement. Since all moving walls are charged and all stationary non-180◦ walls are strained, they will attract or repel defects either electrically or elastically. Oxygen vacancies, which are positively charged and elastically soft, are known to be a potent defect that strongly interacts with domain walls and greatly influences the ferroelectric behavior of BaTiO3 and PZT. Second phases can similarly interact with domain walls and modify ferroelectric behavior.

3.

Stationary Domain Wall and Wulff Plot

The above synopsis makes evident the path to follow in the atomistic simulation of stationary and moving domain walls. In the following, we outline the case primarily for (strain-free) 180◦ domain walls to focus on the electrostatic aspect that is unique to the domain wall problem. To begin with, we recognize that there is a strong anisotropy due to both crystallography and charge. Take the case of BaTiO3 for example, in which the polarization is along 001. Then all the (hk0) domain walls are neutral, so their different domain wall energies are due to crystallographic anisotropy. On the other hand, the (hkl) domain walls are charged, so with increasing l there is a rapid rise in energy from the electrostatic contribution. The Wulff plot of the domain wall is therefore a highly elongated rod around the four-fold symmetry axis 001. Also of interest is the second derivative of the domain wall energy which determines the torque on a curved domain wall. These aspects, as well as the atomic positions and polarization distribution across the domain wall, can be readily studied by atomistic simulation. Our current understanding is that domain walls are relatively sharp, across which a near discontinuity of polarization vector exists. In this respect, ferroelectric domain walls are different from magnetic domain walls which can be quite diffuse. The sharpness of ferroelectric domain walls is due to the strong preference for polarization to align in certain crystallographic orientations and with a constant magnitude, given by the minimum of the “doublewell”. Ionic displacements at intermediate positions and with intermediate orientations may simply have too high an energy to be allowed even in the

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transition layer at the domain wall. This should be verified by atomistic simulation, however, and any deviation from the double-well minimum will modify the domain wall energy due to the so-called “gradient energy” term in the Ginsburg Landau equation. Moreover, the possibility of a more extended domain wall cannot be ruled out in some ferroelectric crystals. In BCC metals, extended dislocation cores with three-fold symmetry have been seen in both electron microscopy and atomistic simulations.

4.

Activation and Domain Wall Movement

As a first step to understand domain wall movement, we consider a translation that brings a domain wall initially to a higher energy state, then passing a maximum, before finally returning to the equilibrium state after one unit cell displacement. The periodic energy landscape along such translation can be studied by atomistic simulation, and its maximum slope gives the theoretical coercive field (at 0 K) of a flat domain wall. This energy landscape corresponds to the Peierls (lattice) energy for dislocations, and the theoretical coercive field corresponds to the Peierls stress. Lattice periodicity similarly manifests itself in energy profiles of a kinked/ledged domain wall when the entire wall is translated forward or when the kink/ledge is translated sideway. Polarization distribution across the translated wall/kink/ledge is again of interest. Both an electric and a stress field can motivate domain wall movement, but the energetics of domain wall movement can be studied without applying an external field. For a flat wall, the energy barrier for first forming a bulge, then enlarging the bulge sideway, is lower than the “Perierls” energy for translation. This bulge has a thickness of a unit cell, is electrically neutral as a whole, but is locally charged on some portions of its perimeter. The total energy of a bulged domain wall as a function of the size and shape of the bulge can be studied by atomistic simulation, which determines the activation barrier (and the coercive field at 0 K) for the flat wall. We expect the shape of the bulge to be highly anisotropic in order to minimize the domain wall energy of the charged perimeter and to keep the oppositely charged segments as far apart as possible. For the latter reason, the shape optimization of a bulge (of a constant area) involves non-local interaction and does not immediately follows from the Wulff plot. In addition, the incline of the perimeter may not be vertical but, instead, may be quite gradual especially when the wall energy of the segment is low. Therefore, atomistic simulation is again useful for elucidating these details. Of course, the simulation can also be repeated under an applied field. Since the activation nature implies that the wall movement is a dissipative, frictional process even at 0 K, a finite wall velocity obtains under an applied field once it exceeds the coercive field.

Atomistic simulation of ferroelectric domain walls

5.

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Extension to Finite Temperature and non-180◦ Domain Walls

Clearly, the simulation studies outline above can be extended to finite temperature. Information of domain wall entropy, roughening, and thermally activated movement can then be analyzed. Because of the non-local electrostatic interaction (and more generally, elastic interaction for non-180◦ walls) and the strong anisotropy of domain wall energy, the size of the simulated volume should be sufficiently large to avoid artifacts that either suppress or promote a particular mode of fluctuation. This size limitation is the main concern for atomistic domain wall simulations but it should be within reach of today’s computational resources. We have thus far emphasized the electrostatic aspect which is unique to the domain wall problem and is appropriate for the 180◦ domain walls, which only respond to the electrical field. The insensitivity to the stress field makes 180◦ domain walls less likely trapped and more mobile. In a real ferroelectric crystal, however, other domain walls are also present and they may well control the ferroelectric properties. For these domain walls, incorporation of both electrostatic and elastic interactions is obviously needed. Two sets of scaling parameters are therefore necessary. For electrostatic interaction, it is polarization; for elastic interaction, it is shear modulus and strain, which scales as (polarization)2 . Since the polarization vanishes at the Curie temperature whereas the shear modulus thermally softens only gradually, the temperature dependence in the domain wall problem is much stronger than in the dislocation problem, where only the shear modulus matters while strain remains constant. The additional influence of crystal chemistry and crystallography also need to be considered. This can be best incorporated into the simulation by parameterizing the semi-empirical potentials employed to allow simple interpretation in terms of charge, covalency, elasticity and polarization. No additional computational difficulty is foreseen with these studies.

References [1] J.P. Hirth and J. Lothe, Theory of Dislocations, 2nd edn., John Wiley & Sons, New York, 1982. [2] Y-H. Chiao and I-W. Chen, “Martensitic growth in ZrO2 –an in situ, small particle TEM study of a single-interface transformation,” Acta Metall., 38, 1163–1174, 1990. [3] F. Jona and G. Shirane, Ferroelectric Crystals, Pergamon Press, New York, 1962. [4] M.E. Lines and A.M. Glass, Principles and Applications of Ferroelectrics and Related Materials, Clarendon, Oxford, 1977. [5] I, Grinberg, V.R. Cooper, and A.M. Rappe, “Relationship between local structure and phase transitions of a disordered solid solution,” Nature, 419, 6910, 909–911, 2002.

Perspective 28 MEASUREMENTS OF INTERFACIAL CURVATURES AND CHARACTERIZATION OF BICONTINUOUS MORPHOLOGIES Sow-Hsin Chen Department of Nuclear Engineering, MIT, Cambridge, MA 02139, USA

1.

Introduction

Studies of bicontinuous and interpenetrating domain structures developed in the process of spinodal decomposition (SD) of binary mixtures of molecular fluids, binary alloys, and polymer blends have been an attractive research theme over the past several decades [1]. Complex structures can be observed in the phase separation process induced by a temperature quench from the stable one-phase region into the unstable two-phase spinodal region. The fact that the dynamical processes in complex fluids such as polymer blends, glasses, and gels are extremely slow due to their long characteristic relaxation times allows one to study their morphologies with better accuracy than those of simple liquid mixtures [2]. Similar kinds of bicontinuous structures have been observed in the water/oil/ surfactant three-component microemulsion system in the one-phase region close to the three-phase boundary (often called the “fish tail” in microemulsion literature because of the similarity in shape of the phase boundary of this region to a fish tail) and in the vicinity of hydrophile-lipophile balance temperature [3, 4]. It is interesting and physically relevant to examine the common and universal features of these bicontinous structures observed in phase-separated polymer blends [5] and in microemulsions [6]. During the course of the past three decades our comprehension of amphiphile solutions has been profoundly transformed [7]. A major conceptual force behind these developments has been the realization that the bulk phases of amphiphiles in solution could be visualized as systems of interfacial films and 2849 S. Yip (ed.), Handbook of Materials Modeling, 2849–2863. c 2005 Springer. Printed in the Netherlands.

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could therefore be successfully described in terms of their properties. Thus, the structural and thermodynamic features of micellar or vesicular solutions, of microemulsions, and of lyotropic liquid crystals, began to be progressively rationalized in terms of the properties of fluctuating surfaces [8]. Because interfaces formed by amphiphiles have often very small or vanishing interfacial tensions the dominant contribution to their free energy is its bending elastic energy. The latter, in turn, is given by the following expression,

FH =

1 dS κ (H − c0 )2 + κ K 2

(1)

where dS is the element of the surface area, H and K, are respectively, the mean and the Gaussian curvatures of the surface that represents the interface, and c0 , κ and κ are, respectively, the spontaneous curvature, the bending rigidity and the saddle-splay constants. FH is known as the Helfrich free energy [9] in the literature. Thus it is seen from the above formula, that the basic physical variables describing the elastic properties of an interface are the mean and Gaussian curvatures, much the same as those describing the mechanical properties of an interacting particle system are the kinetic and the potential energies. It follows that a statistical mechanical description of an interface must involve the average mean and Gaussian curvatures over the whole interface. It is then clear that to devise a method to measure and compute these average curvatures for physically relevant interfacial systems is utmost important. At any point on the oil–water interface (with the surfactant mono-layer in-between), the interface may be characterized by its two principal radii of curvature R1 and R2 , which by convention are positive when the interface is curved towards the oil phase and are negative when it is curved towards the water phase. One then defines a mean curvature by H =1/2 ((1/R1 ) + (1/R2 )), and a Gaussian curvature by K = 1/(R1 R2 ) at that point. The geometry of the surface is completely specified by giving a pair of values (H, K) at every point on the surface. The first practical method for measuring the average mean curvature H was proposed and implemented by Lee and Chen [10]. The system they studied was a three-component non-ionic microemulsion system composed of water/octane/ C10 E4 (tetraethylene glycol monodecyl ether) with equal volume fractions of water and oil. The phase diagram in the temperature- φs (volume fraction of the surfactant) plane shows that the hydrophile–lipophile balance (HLB) temperature is about 24 ◦ C and the one-phase microemulsion begins to form at the minimum volume fraction, φs = 0.10. Hence at φs = 0.20, if one varies the temperature from 15◦ to 30 ◦ C, the one-phase microemulsion should transform from a water-in-oil droplet microemulsion to an oil-in-water microemulsion through an intermediate state of lamellar microemulsion. In this process the average mean curvature H should switch sign from an initial

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positive value to a final negative value through an intermediate value of zero at the HLB temperature. They indeed found that the average mean curvature varies linearly in this temperature interval starting with a value H = 0.008 Å−1 at 15 ◦ C and ends up with a value H = −0.006 Å−1 at 30 ◦ C. The method was based on the measurements of three separate specific interfacial areas using a contrast variation technique in a small-angle neutron scattering (SANS) experiment. Because the surfactant layer in-between the water and oil phases of a microemulsion has a finite thickness d, the water–surfactant interfacial area AW and the oil–surfactant interfacial area AO are generally not equal if the film is curved. This can most easily seen by considering the case of a spherical shell; the inner surface has less area than the outer surface and this difference is related to the thickness of the shell and the radius of the sphere. The exact geometrical relationships between AW , AO and the surface area AS measured at the midpoints of the surfactant molecules to the average curvatures H and K are given by:

Aw = As

d2 K , 1 + d H + 4

Ao = As

d2 K 1 − d H + 4

(2) The average mean curvature can thus be calculated from thetwo measured areas AW and AO by a formulaH = ( Aw − AO ) / ( Aw + Ao ) /d. With this method, errors involved in the specific interfacial areas measurements were large enough that the average Gaussian curvature could not be deduced with enough accuracy. In order to measure the average Gaussian curvature Chen et al. proposed another method. This is called a Clipped Random Wave (CRW) model [6]. Cahn in 1965 proposed a scheme for generating a three-dimensional morphology of a phase separated A–B alloy by clipping a Gaussian random field generated by superposing many isotropically propagating sinusoidal waves with random phases [11]. The Gaussian random field can be normalized in such a way that it fluctuates continuously between −1 and +1. One can then realizes the bicontinuous two-phase morphology by clipping the continuous random process, namely, by assigning say 0 to all negative signals representing A, and +1 to all positive signals representing B. In this scheme, the essential features of the morphology depends only on the “spectral density function” (SDF), which is the three-dimensional Fourier transform of the twopoint correlation function g(r) of the Gaussian random field. The SDF gives the distribution of the magnitudes of the propagation wave vectors of the sinusoidal waves. Berk in 1987 further developed the idea of Cahn mathematically for the purpose of analyzing scattering data [12]. In particular, he derived an important relation connecting the two-point correlation function and the Debye correlation function which determines the scattering intensity. In his

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original paper, however, Berk discussed only a SDF which is a delta function. This resulted in generating a morphology which is only partially disordered and is not realistic. Chen, Chang and Strey [13] later point out that a broader but peaked SDF is necessary to generate a more disordered morphology which will reproduce small-angle neutron scattering (SANS) intensity distributions. In particular, it was recognized that the SDF should be chosen in such a way that it has the correct second moment to ensure the agreement with the measured finite interfacial area per unit volume (the specific interface area S/V).

2.

Theory of Scattering From Bicontinuous Porous Materials in Bulk Contrast

The intensity distribution of SANS from an isotropic, disordered twocomponent porous material can be calculated generally from a Debye correlation function (r) by the following formula [14]: I (Q)= < η >

∞

2

dr4πr 2 j0 (Qr)(r)

(3)

0

whereη2 = ϕ1 ϕ2 (ρ1 − ρ2 )2 , is the mean square fluctuation of the local scattering length density, ϕ1 and ϕ2 the volume fractions of components 1 and 2, ρ1 and ρ2 the corresponding scattering length densities. There are two physical boundary conditions that the Debye correlation function, a function of a scalar distance between the two points under consideration, must satisfy: it is normalized to unity at the origin, r = 0, and it should decay to zero at infinity. The most important property of the Debye correlation function for the case with a sharp boundary between two regions having different scattering length densities, ρ1 and ρ2 , is that it has a linear and a cubic terms in the small r expansion of the form: (r → 0) = 1 − ar + br 3 + . . . 1 S b = 1− r 1 − r2 + . . . 4ϕ1 ϕ2 V a

(4)

where a = (S/ V )/4ϕ1 ϕ2 is a factor proportional to the total interfacial area per unit volume of the sample and the ratio of the coefficient of the cubic term to the linear term has been given by Kirste and Porod [15] in terms of curvatures as: b 1 2 1 K . = H − a 8 24

(5)

Measurements and characterization of bicontinuous morphologies

3.

2853

Clipped Random Wave (CRW) Model for Computation of Interfacial Curvatures

In the clipped random wave model of Berk [12], an order parameter field ψ( r ), which for example, can be a quantity proportional to the volume fraction of the majority phase in the system, is constructed by superposition of a large number N of cosine waves with random phases:

ψ( r) =

N 2 cos(ki · r + ϕi ) N i=1

(6)

where directions of the wave vector ki are assumed to be distributed isotropically over a unit sphere and the phase, ϕi distributed randomly over an interval (0, 2π ). In constructing the sum, the magnitude of the ki vector is sampled from a scalar distribution function f (k) called the “spectral density function”. This order parameter field is certainly a Gaussian random field due to the central limit theorem in statistics. The statistical properties of a Gaussian random field is completely characterized by giving its spectral density function f (k) or the corresponding two-point correlation function g(r). We define the two-point correlation funcri )ψ( r2 ) and the associated SDF f (k) by a Fourier tion by g(|ri − r2 |) = ψ( transform relation: g(|ri − r2 |) =

∞

4π k 2 j0 (k | ri − r2 |) f (k) dk

(7)

0

This continuous random process ψ( r ), varying between +1 and −1, having a mean square value of unity (by its definition Eq. 6), is then clipped and transformed into a discrete, two-state discrete random process. By clipping we mean assigning a constant value +1 to the function whenever the Gaussian random field at that point is above a certain “clipping level” called α, and a constant 0 whenever its value is below α. This transformation can be defined as follows: 1, when ψ( r) ≥ α r )) = (8) ζ( r ) = α (ψ( 0, otherwise where α is a step function. Then the Debye correlation function can be written as a function of the discrete random variable ς (r) in the form: ζ(0)ζ( r ) − ζ 2 (9) ζ − ζ 2 where the quantities on the right-hand side are given by Teubner [16] as: ( r) =

1 1 ζ = − √ 2 2π

0 α

exp(−x 2 /2)dx

(10)

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S.H. Chen

1 ζ(0)ζ( r ) = ζ − 2π

cos−1 (g(r))

α2 exp − 1 + cos θ

0

dθ.

(11)

The average value of the clipped Gaussian random filed, ζ , and 1 − ζ can be interpreted as volume fractions of the majority and minority phases, ϕ1 and ϕ2 , respectively. Using (10) and ζ = ϕ1 , Eq. (9) can be rewritten as 1 ( r) = 1 − 2π ϕ1 (1 − ϕ1 )

cos−1 (g(r))

0

α2 exp − dθ 1 + cos θ

(12)

For a small α, meaning a slight deviation from an isometric case, Eq. (12) can be approximated as:

( r) ∼ = 1−

cos−1 (g(r)) 1 cos−1 (g(r)) − α 2 tan 2π ϕ1 (1 − ϕ1 ) 2

(13)

where the volume fraction ϕ1 can be approximated as α 1 ϕ1 ∼ = −√ . 2 2π

(14)

For an isometric (ϕ1 = ϕ2 = 1/2) microemulsion α = 0, and Eq. (12) reduces to a simple form 2 sin−1 (g(r)) (15) π which is a well-known result given by Berk [12]. From the relation in Eq. (7), one has the small r expansion of g(r) of the form ( r) =

∞

g(r) = 0

1 1 4 4 k r + . . . f (k)dk 4π k 2 1 − k 2r 2 + 6 120

=1−

1 4 4 1 2 2 k r + k r + ... 6 120

(16)

where we used the normalization condition g(0) = 1, and k 2 and k 4 denote the second and fourth moment of the spectral density function. Note that this expansion has a quadratic term followed by a quartic term. Using the result of Eq. (16) in Eq. (12), we obtain a small r expansion of the Debye correlation function the form: 1/2 1 2 √ k2 e−α /2r 2π 3ϕ1 ϕ2 1 2 2 1 k4 − k (α − 1) r 2 × 1− 40 k 2 72

B (r → 0) = 1 −

(17)

Measurements and characterization of bicontinuous morphologies

2855

Comparing this with Eq. (4), we arrive at a useful relation connecting the second moment of the spectral density function to the fundamental quantity of a porous material, the interfacial area per unit volume, 2 2 1/2 −α 2 /2 S = √ k e . V π 3

(18)

This relation also implies that one of the basic requirements for the physically acceptable spectral function is that the second moment be finite. We next consider a random surface generated by the level set ψ( r ) = ψ(x, y, z) = α

(19)

where the function ψ(x, y, z) is defined by Eq. (6). Teubner [16] has proved a remarkable theorem that for this random surface the average mean, Gaussian and mean square curvatures are given by:

K =

(20)

1 2 2 k α −1 6

(21)

H2 =

where

π 2 k 6

α H = 2

1 2 2 k α + v2 6

(22)

6 k4 v = 2 − 1 5 k2 2

(23)

So the second requirement of the physically acceptable spectral function is that it has a finite fourth moment also. Since for a bicontinuous microemulsion the level surface defined by Eq. (19) is approximately the mid-plane passing through the surfactant monolayer in a bulk contrast experiment, the average mean, Gaussian and square mean curvatures of the surfactant monolayer can be computed once a physically acceptable SDF can be found. Choice of the SDF can be based on a criterion that when it is substituted into equations Eqs. 3, 7 and 12, it would give an intensity distribution which agrees with SANS data in an absolute scale. A suitable form of the SDF has been proposed by Chen and Choi [17], which is an inverse eighth order polynomial in k and which contains three parameters a, b, and c. The first two parameters a and b have their approximate correspondences in the approximate theory of Teubner and Strey [18]. In the T-S theory, the Debye correlation function is given by: TS (r) = e−r/ξ

sin(2πr/d) (2πr/d)

(24)

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S.H. Chen

The correspondences are: a ≈ 2π/d and b ≈ 1/ξ , where d is the interdomain (water–water or oil–oil) repeat distance and ξ the decay length of the local order [19, 20]. The parameter c controls the transition from the main Q peak to the large Q behavior of the scattering intensity distribution, the existence of which is essential for the good agreement of the theory and experiment for the entire range of Q. This agreement is important if the correct surface to volume ratio of the system is to be incorporated into the theory. We thus choose [17] a SDF for which both the second and fourth moments exist. This function is an inverse eighth order polynomial containing three parameters, which are the minimal set for the physical situation under study: f (k) =

bc(a 2 + (b + c)2 )2 /(b + c)π 2 (k 2 + c2 )2 (k 4 + 2(b2 − a 2 )k 2 + (a 2 + b2 )2 )

(25)

√ This spectral density function has a peak approximately at kmax ≈ a 2 −b2 . From Eq. (25), the second and fourth moments of the spectral density function are given by:

c(a 2 + b2 + bc) (b + c)

c(a 4 + 2a 2 b2 + b4 + 4a 2 bc + 4b3 c + 4b2 c2 + bc3 ) k4 = (b + c) k2 =

(26)

The two-point correlation function g(r) can be given in an analytical form as well, which has only even powers of r in the small r expansion. From this two-point correlation function, the Debye correlation function can be calculated analytically using Eq. (12), once the clipping level (i.e., the volume fraction ϕ1 ) is specified. Explicit expressions for the curvatures are given in terms of the three parameters and the clipping level α (determined by the volume fraction of the majority phase as calculated by Eq. (10)) as:

π c(a 2 + b2 + bc) 6 (b + c) 2 2 1 c(a + b + bc) K = − (1 − α 2 ) 6 (b + c)

a 4 + 2a 2 b2 + b4 + 4a 2 bc + 4b3 c + 4b2 c2 + bc3 H 2 = K + 5(a 2 + b2 + bc) α H = 2

(27) (28) (29)

This model can be used to fit all the bulk contrast data nicely to obtain parameters a, b and c. One can then use Eqs. (27)–(29) to calculate the respective curvatures.

Measurements and characterization of bicontinuous morphologies

4.

2857

Examples of Small-angle Neutron Scattering (SANS) Data Analysis Using CRW Model

Figure 1a shows a series of scattering intensities I (Q) measured from an AOT microemulsion system at points in the one-phase region close to the lamellar phase boundary. The symbols are experimental data and solid lines are theoretical fits using Eq. (7), (15) and (25) in Eq. (3) plus an incoherent background. These curves show good agreement between the theory and experimental data over the entire range of Q. The fitted parameters a, b, and c are listed in Table 1. As the surfactant volume fraction increases a and c increase rapidly while b increases slowly. Considering the relations d ≈ 2π/a and ξ ≈ 1/b, the inter-domain distance d and the decay length ξ decreases with the surfactant volume fraction. This makes sense because we create more surfaces per unit volumes as the number of surfactant molecules increase and thus the inter-domain distance should decrease. Considering the ratio ξ /d, one sees that a bicontinuous microemulsion becomes more disordered at smaller surfactant volume fractions. The average Gaussian and square mean curvatures calculated by using Eqs. (28) and (29), respectively, are given in Fig. 2a. The solid line is a fit using a phenomenological parabolic equation K = co + c1 (φs − φo )2 (a)

(30)

(b) 104

103

102

102 101

Theory

100

0.08 0.11 0.14 0.17 0.20

I(Q) (cm⫺1)

I(Q) (cm⫺1)

103

φs

10⫺1 0.01

Theory octane decane dodecane tetradecane

100 0.1

Q (Å⫺1)

101

10⫺2

10⫺1

100

Q (Å⫺1)

Figure 1. (a) Analyses of the bulk contrast scattering intensities of isometric AOT-based microemulsions as a function ofsurfactant volume fraction. The scattering intensities were measured at points close to the lamellar phase boundary, where the average mean curvature is nearly zero; (b) analyses of the bulk contrast scattering intensities of isometric Ci E j -based microemulsion systems taken at the fish tail of its phase diagram.

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S.H. Chen

Table 1. Fitted parameters a, b, c and the calculated interfacial curvatures for an isometric (equal oil and water volume fractions,H = 0) AOT/decane/water (NaCl) microemulsion system in the one-phase region along the lamellar phase boundary. ϕs is volume fraction of the surfactant AOT φs

a (Å−1 )

b (Å−1 )

c (Å−1 )

0.05 0.08 0.11 0.14 0.17 0.20

0.00613 0.01317 0.02139 0.03005 0.03939 0.04968

0.01104 0.01198 0.01120 0.01236 0.01346 0.01459

0.01710 0.04682 0.09114 0.1096 0.1530 0.2096

2 η

backgrd

25.92 13.07 10.86 9.33 9.30 9.00

0.385 0.383 0.393 0.382 0.352 0.321

K

H 2

NaCl

–0.353 –1.165 –2.382 –3.610 –5809 –8.943

1.62 6.35 16.78 21.65 37.67 64.85

0.49 0.46 0.42 0.36 0.32 0.26

(1020 cm −4 ) (cm −1 ) (10−4 Å−2 ) (10−4 Å−2 ) (wt%)

and the dashed line is a fit using a similar equation

H 2 = c0 + c1 (φs − φo )2 .

(31)

Results of the fit are c0 = −0.434 × 10−4 Å−2 ; c1 = −310.2 × 10−4 Å−2 ; φ0 = 0.036 and c0 = 2.35 × 10−4 Å−2 ; c1 = 2530 × 10−4 Å−2 ; φ0 = 0.046. In both cases, φ0 turns out to be very close to the volume fraction at the fish tail of the phase diagram.We can say that c0 and c0 are the average Gaussian curvature and the average square mean curvature at the fish tail which is the lowest volume fraction of the surfactant where the one-phase microemulsion first forms. Magnitudes of average Gaussian curvatures obtained in this experiment are comparable to that obtained for a C10 E 4 /D2 O/Octane microemulsion near the fish tail before [10]. The quadratic dependence of the average Gaussian curvature on the volume fraction of surfactant is also understandable. According to Eqs. (18) and (21), the magnitude of the average Gaussian curvature is proportional to the square of the interfacial area per unit volume. The interfacial area is in turn proportional to the amount of surfactant added to the system. Since the average mean curvature is expected to be zero for microemulsions that we studied at these phase points, the average square mean curvature is the variance of the fluctuation of the mean curvature. One also observes that the variance of the fluctuation increases quadratically as the volume fraction increases. This is reasonable because as more surfactant is added to the system, more interfacial area is created. The surfactant film has to bend around to accommodate itself in the liquid. Figure 1b shows scattering intensities and their analyses for a series of isometric microemulsion systems composed of C8 E 3 /D2 O/n-alkane at their respective fish tail in the phase diagram. The theory again agrees with experiments uniformly well and the extracted curvatures are plotted as symbols in Fig. 2b. The fish tail is a very special point in the phase diagram where the system has the lowest specific internal surface area. Figure 2b shows results

Measurements and characterization of bicontinuous morphologies

⫺2

60

⫺4

50

40

⫺6

30

⫺8 ⫺10 0

0.05

0.1

φs

(a)

0.15

0.2

120 100 80

⫺6

⫺8

60

⫺10

20

⫺12

10

⫺14

0

⫺16

40

(10⫺4 Å⫺2)

⫺4

0

70

(10⫺4 Å⫺2)

(10⫺4 Å⫺2)

⫺2

80

(10⫺4 Å⫺2)

0

2859

20

0

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

φs

(b)

Figure 2. (a) Average Gaussian and square mean curvatures as a function of surfactant volume fraction obtained from analyses of a series of isometric AOT-based microemulsions. The corresponding SANS intensities are those shown in Fig. 1a. Solid lines are parabolic fits based on Eqs. (30) and (31). Average Gaussian and square mean curvatures as a function of surfactant volume fraction at the fish tail obtained from analyses of a series of isometric Ci E j -based microemulsions. The corresponding SANS intensities are those shown in Fig. 1b. Solid lines are the parabolic fits.

of curvatures we extracted from 18 generic non-ionic microemulsion systems [21]. Again, the parabolic dependence of the curvatures on the surfactant volume fraction is evident. Figure 3 shows the level surface as defined by Eq. (19) for the system C8 E3 /D2 O/tetradecane at its fish tail. The three parameters in the spectral density function needed for the three-dimensional reconstruction comes from analysis of the fourth intensity curve from the top shown in Fig. 1b. It is clear from the figure the geometry of the level surface is locally hyperbolic giving rise to a negative Gaussian curvature. In Fig. 4 we show scattering intensity distributions and their analyses for three non-isometric microemulsions. This new microemulsion system is composed of F8 H16 /H2 O/perfluorooctane [22]. We use an abbreviation F8 H16 ≡ CF3 (CF2 )7 −COO(CH2 CH2 O)7.2 CH3 . The analyses are successful and we are able to extract both the average mean and Gaussian curvatures given in Table 2. It is interesting to ask the following question at this point: is CRW method of analysis applicable to bicontinuous structures generated by a late stage spinodal decomposition of an isometric system such as a binary alloy or a polymer blend? Answer is partially yes. In Fig. 5 we show a light scattering intensity distribution [5] of a 50–50 polymer blend of near critical mixture of perdeuterated polybutadiene (dPB) and polyisoprene (PI) at the late stage spinodal decomposition. The analysis result is shown as the solid line. The plot

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S.H. Chen

Figure 3. ‘The level surface as defined by Eq. (19) for the system C8 E3 /D2 O/tetradecane at its fish tail. It is clear from the picture that the surface is locally hyperbolic having a negative average Gaussian curvature. The size of cube is 240 × 240 × 240µm3 .

I(Q) (cm -1 )

102

101

The ory 100 α⫽0.0530 α⫽0.2534 ⫺1

10

10⫺2

α⫽0.4438

10⫺2

10⫺2 ⫺1

Q (Å )

Figure 4. Analyses of scattering intensity distributions for threenon-isometric microemulsions composed of F8 H16 /H2 O/perfluorooctane [22]. The curvatures obtained are given in Table 2.

Measurements and characterization of bicontinuous morphologies

2861

Table 2. Average mean and Gaussian curvatures extracted from non-isometric microemulsions α (clipping level) 0.0530 0.2544 0.4438

φ1 (volume fraction)

H in 10−3 Å−1

K in 10−4 Å−2

0.48 0.40 0.33

0.86 4.80 9.90

−3.35 −4.27 −5.05

is presented in a scaling form because the characteristic length scale of the microstructure is of the order of 10 µm. Agreement between the experiment and the theory is excellent for large Q portion of the curve but is only moderately satisfactory for smaller Q region. The following results are obtained:

α = 0, η2 = 6.836 × 1012 cm−4 a = 5.36 × 10−5 Å−1 ; b = 1.04 × 10−5 Å−1 ; K = −1.99 × 10−9 Å−2 ≈ 0.2 µm−2

c = 87.2 × 10−5 Å−1 (32)

101

Q3maxI(Q)/<η2>

100

10⫺1

10⫺2

Experiment Theory

10⫺3

100

101

Q /Qm ax Figure 5. Analysis of a light scattering intensity distribution [5] of a 50–50 (isometric) polymer blend of near critical mixture of perdeuterated polybutadiene (dPB) and polyisoprene (PI) at its late stage spinodal decomposition. The analysis results are given in Eq. (32).

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S.H. Chen

Conclusion

We have outlined a method, called the Clipped Random Wave model, by which the interfacial curvatures of a porous material can be measured through a scattering experiment. This method also allows one to reconstruct a three dimensional interfacial structure of the porous medium from which the morphology of the two-phase bicontinuous microstructure can be visualized. We illustrate this method using SANS data taken from bicontinuous microemulsion samples having bulk contrast. Specifically, we can extract the average mean, Gaussian and square mean curvatures of the internal interfaces. The method relies on a choice of a physically reasonable SDF of the underlying Gaussian random field which upon clipping generates a bicontinuous structure that produces the given scattering intensity pattern. A physically reasonable SDF of the Gaussian random field should have finite lower order moments to ensure a finite internal surface area per unit volume and a finite average square mean curvature. The method has been put to test in three cases: a nearly isometric AOT/D2 O(NaCl)/decane microemulsion system as a function of surfactant volume fraction; Several isometric microemulsions made of Ci E j /D2 O/alkane at their respective fish tails in the phase diagrams; and several non-isometric microemulsions made of F8 H16 /H2 O/perfluorooctane. Theoretical intensities calculated by the CRW model can be made to agree with the scattering data in an absolute scale by adjusting the three length scale parameters, 1/a,1/b,1/c, in the SDF. The commonly used two-parameter Teubner–Strey model for the Debye correlation function can also be used to fit the scattering data in the small Q region. But the fit is distinctly worse compared to our three parameter theory for Q region after the peak.The parameter a can be approximately identified as a = 2π/d and b = 1/ξ of the corresponding parameters in the T-S theory. The third parameter c is necessary for ensuring a smooth transition of scattering intensity from small Q behavior near the peak to large Q behavior dominated by the Porod’s law. We can thus say that three length scales are necessary for a complete description of the microstructure of bicontinuous microemulsions. The CRW model with an appropriate choice of the SDF is also shown to be a useful tool for quantitative analyses of small angle scattering data from microphase-separated system such as a symmetric binary mixture of polymers at late stage of spinodal decomposition. It not only gives information on curvatures of the interface, which cannot be obtained by other means, but can also generate the morphology of 3-d microstructure as shown in Fig. 3.

Acknowledgment This research is supported by a grant from Materials Science Program of US Department of Energy. We are grateful to the Intense Pulse Neutron Source

Measurements and characterization of bicontinuous morphologies

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Division of Argonne National Laboratory for neutron beam time at the SAND Low-Angle Diffractometer.

References [1] J.D. Gunton, M. San Miguel and P. Sahni, In: C. Domb and J.L. Lebowitz (eds.), Phase Transitions and Critical Phenomena, Academic Press, New York, 1983, p. 269. [2] T. Hashimoto, Phase Transitions, 1988, 12, 47; Materials Science and Technology, Vol 12, Structure and Properties of Polymers, VCH, Weinheim, 1993. [3] W. Jahn and R. Strey, J. Phys. Chem., 92, 2294, 1988. [4] S.H. Chen, S.L. Chang, and R. Strey, J. Chem. Phys., 93, 1907, 1990. [5] H. Jinnai, T. Hashimoto, D.D. Lee, and S.H. Chen, Macromolecules, 30, 130, 1997. [6] S.H. Chen, D.D. Lee, K. Kimishima, H. Jinnai, and T. Hashimoto, Phys. Rev. E, 54, 6526, 1996. [7] W.M. Gelbart, A. Ben-Shaul and D. Roux (eds.), Micelles, Membranes, Microemulsions, and Monolayers, Springer, New York, 1994. [8] S.A. Safran, Statistical Thermodynamics of Surfaces, Interfaces, and Membranes, Addison-Wesley, 1994. [9] W. Helfrich and Z. Naturforsch, 28c, 693, 1973. [10] D.D. Lee and S.H. Chen, Phys. Rev. Lett., 73, 106, 1994. [11] J.W. Cahn, J. Chem. Phys., 42, 93, 1965. [12] N.F. Berk, Phys. Rev. Lett., 73, 106, 1987. [13] S.H. Chen, S. L. Chang, and R. Strey, J. Appl. Crystallogr., 24, 721, 1991. [14] P. Debye, H.R. Anderson, Jr. and H. Brumberger, J. Appl. Phys., 28, 679–683, 1957. [15] R. Kirste, Von & G. Porod, Kolloid-Z&Z.f.Polym., 184, 1–7, 1962. [16] M. Teubner, Europhys. Lett., 14(5), 403–408, 1991. [17] S.H. Chen and S.M. Choi, J. Appl. Cryst., 30, 755–760, 1997. [18] M. Teubner and R. Strey, J. Chem. Phys., 87, 3195–3200, 1987. [19] S.H. Chen, S.L. Chang, and R. Strey, Prog. Colloid Polymer. Sci., 81, 30–35, 1990. [20] S.H. Chen and S.L. Chang, and R. Strey, J. Appl. Cryst., 24, 721–731, 1991. [21] S.-M Choi, S.H. Chen, T. Sottmann, and R. Strey, “The existence of three length scales and their relation to the interfacial curvatures in bicontinuous microemulsions,” Physica A, 304, 85–92, 2002. [22] P. LoNostro, S.M. Choi, C.Y. Ku, and S.H. Chen “Fluorinated microemulsions–a study of the phase behavior and structure by SANS,” J. Phys. Chem., 103, 5347– 5352, 1999.

Perspective 29 PLASTICITY AT THE ATOMIC SCALE: PARAMETRIC, ATOMISTIC, AND ELECTRONIC STRUCTURE METHODS Christopher Woodward Northwestern University, Evanston, Illinois, USA

Over the last hundred years our evolving comprehension of deformation has been based on the discovery and understanding of the line defects (dislocations) that control plasticity. While our ability to directly model various defects has improved over this time, a great deal has been learned about deformation processes using parametric approaches. A natural extension of analytic models, parametric studies are sometimes overlooked in our search for the most accurate computational representation of the mechanism that controls a given materials property. However, parametric strategies has been broadly employed in the materials community. For example, we have used parametric approaches to study the influence of micro-structural properties on the strengthening mechanisms in model Ni-based super alloys and the influence of chemistry on high temperature strengthening in Ti–Al alloys [1, 2]. Also, current dislocation dynamics calculations can be viewed as a template for parametric studies of the role of various defect-defect interactions and how they control macroscopic behavior. The deformation behavior of the bcc transition metals provides an excellent historical example of this type of work and how it influenced our understanding of these materials. In the early 1920s as materials scientists were exploring the deformation behavior of various simple metals. Schmid proposed the seemingly reasonable premise that: Plastic flow will occur when the shear stress resolved on a particular slip system reaches a critical value, the critical resolved shear stress (CRSS), which will be independent of slip system and the sense of slip. Further, the CRSS should not be influenced by other components of the applied stress tensor. Since that time many examples of violations to Schmid’s law have been documented. In fact for materials slated for high-temperature structural 2865 S. Yip (ed.), Handbook of Materials Modeling, 2865–2869. c 2005 Springer. Printed in the Netherlands.

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C. Woodward

applications (refractory metals and intermetallics) violations to Schmid’s law seem to be the rule rather than the exception. In these materials edge dislocations tend to be very mobile and deformation is limited by the lattice friction stress (Peierls stress) of screw dislocations. These materials also exhibit large Peierls stresses compared to the simple fcc metals, particularly at low temperatures. This and the significant deviations from Schmids law led Mitchel and co-workers [3] to propose that in the bcc transition metals this behavior was produced by dislocations with non-planar dislocation centers (cores) that made planar glide difficult. Realizing that the differences in geometry for fcc and bcc metals may explain the gross differences in the plasticity of these materials, several groups in the early 1960s developed parametric atomistic potentials for the bcc metals. One set of parametric potentials were based on variations of a generalized stacking fault energy [4, 5]. This is defined as the energy as a function of shear for two blocks of material along the screw direction where the plane of shear is on the assumed glide plane. Structural parameters (lattice and elastic constants) were held fixed for the suite of potentials, but no effort was made to model a particular material. These simple studies showed that a range of non-planar dislocation cores could be obtained for screw dislocations in the bcc metals. Even though these studies were hampered by the simple forms of inter-atomic potentials for many years the qualitative parametric description of the screw dislocations was found to be satisfactory. The large lattice friction stresses, and some of the deviations from Schmids law could be understood in terms of the derived non-planar dislocation core structures. In the last 15 years, as confidence grew in new atomistic potential schemes such as the Embedded Atom Method, and the Model Generalized Pseudopotential Theory several groups produced carefully designed atomistic potentials for specific bcc transition metals [6–9]. These methods produced core structures consistent with previous, parametric studies, lending credence to the idea that the local geometry determines the shape and mobility of dislocations on the atomic scale. Specifically the studies showed that the predicted dislocation cores for the group V and VI bcc transition metals spread into three {110} planes and the shapes of the cores fell into two distinct classes (Fig. 1). The group V metals exhibited a core spread symmetrically about a central point (Fig. 1a) while the core for the group VI metals spreads asymmetrically about this central point (Fig. 1b). Differences in the macroscopic plasticity of the group V and VI transition metals, for example higher sensitivity to non-glide stresses propensity in the group VI metals, were linked to the differences in the equilibrium core structures. Also, atomistic studies showed that details of the generalized stacking fault determined the differences in equilibrium shape of the dislocations [10]. Only recently has it been possible to actually model these dislocation cores using electronic structure methods based on Density Functional Theory (DFT).

2867 <110>

Plasticity at the atomic scale

<112> <111>

(a)

(b)

Figure 1. Schematic of the strain field near the dislocation cores found using atomistic potentials (before 2001) for the group V and group VI bcc transition metals.

The geometry of an isolated dislocation is problematic for current large-scale electronic structure methods, so two approaches have been taken to accommodate these boundary conditions. The group at Wright Patterson Air Force Base employed a flexible boundary condition method that self-consistently couples the local strain field produced by the dislocation core to the long range elastic field [11]. This technique is an extension of a method originally proposed by Sinclair and allows the dislocation to be contained in a very small simulation cell [12, 13]. Other groups have used simulation cells with dislocation dipoles carefully arranged to minimize the total stress field produced by the dislocation array [14, 15]. Surprisingly all the electronic structure calculations show Ta (group V) and Mo (group VI) transition metals to have a dislocation core consistent with Fig. 1a. Frederiksen and Jacobsen also find the same result for the screw dislocation in bcc Fe (group VIII). They also find that the generalized stacking fault calculated using DFT for all the group V and VI bcc metals are consistent with a core spread symmetrically about a central point (Fig. 1a). Our calculations also show the atomic scale response of the screw dislocations to applied stress in Mo and Ta is similar to that found using atomistic potentials. Thus the differences in macroscopic plasticity of these two materials is probably not linked to the shape of the equilibrium cores, but is more likely tied to anisotropies in the local bonding of the dislocation core. While the atomistic methods are correctly incorporating a great deal of this information content, predicting differences between elemental metals may require improved interaction models or ab-initio methods.

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Electronic structure calculations indicate that one extreme of the parametric model (Fig. 1b) is not realized in the bcc transition metals. This is not a failure of the previous methods; it just indicates that this level of detail resides at the electronic level and must be modeled appropriately if we are to understand the behavior of a particular material. The question remains; can we find examples of the core shown in Fig. 1b. in nature and how will this configuration affect plasticity. Thus the original parametric investigations, based on the geometry and reasonable values for the physical parameters, had great value in breaking into a new area of materials science. These configurations may be realized in other bcc metals, or perhaps in studies of interstitial-dislocation interactions. Simple parametric-models will continue to play and important role extending our understanding of materials behavior, don’t overlook this strategy when approaching a new area in computational materials science.

References [1] S.I. Rao, T.A. Parthasarathy, D.M. Dimiduk, and P.M. Hazzledine, “Discrete dislocation simulation of precipitate hardening in superalloys,” Phil. Mag., in press, 2004. [2] C. Woodward, and J.M. MacLaren, “Planar fault energies and sessile dislocation configurations in substitutionally disordered Ti-Al with Nb and Cr ternary additions,” Phil. Mag., A74, 337, 1996. [3] T.E. Mitchell, R.A. Foxall, and P.B. Hirsch, “Work hardening in Niobium Single Crystals,” Phil. Mag., 8, 1895, 1963. [4] V. Vitek, R.C. Perrin, and D.K. Bowen, “The core structure of a/2111 screw dislocations in BCC crystals,” Phil. Mag., 21, 1049, 1970. [5] M.S. Duesbery, V. Vitek, and D.K. Bowen, “The effect of shear stress on the screw dislocation core structure in BCC cubic lattices,” Proc. Roy. Soc. Lond., A332, 85– 111, 1973. [6] G.J. Ackland and V. Vitek, “Many body potentials and atomic scale relaxations in nobel metals alloys,” Phys. Rev., B 41, 10324, 1990. [7] D. Farkas and P.L. Rodriguez, “Embedded atom study of dislocation cores structure in Fe,” Scripta Metall. Mater., 30, 921, 1994. [8] W. Xu and J.A. Moriarty, “Atomistic simulation of ideal shear strength, point defects, and screw dislocations in BCC transition metals: Mo and a prototype,” Phys. Rev., B 54, 6941, 1996. [9] L.H. Yang, P. Soderlind, and J.A. Moriarty, “Accurate atomistic simulation of (a/2)111, screw dislocations and other defects in BCC tantalum” Phil. Mag., A81, 1355–85, 2001. [10] M.S. Duesbery and V. Vitek, “Plastic anisotrophy in BCC transition metals,” Acta Mater., 46, 1481–1492, 1998. [11] C. Woodward and S.I. Rao, “Flexible ab-initio boundary conditions: Simulating isolated dislocations in BCC Mo and Ta,” Phys. Rev. Lett., 88, 216402-1-4, 2002. [12] J.E. Sinclair, P.C. Gehlen, R.G. Hoagland et al., “Flexible boundary conditions and nonlinear geometric effects an atomic dislocation modeling,” J. Appl. Phys., 3890– 3897, 1978.

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[13] S.I. Rao, C. Hernandez, J.P. Simmons et al., “Green function boundary conditions in 2-d and 3-d atomistic simulations of dislocations,” Philos. Mag. A77, 231, 1998. [14] S. Ismail-Beigi and T.A. Arias, “Ab-initio study of screw dislocations in Mo and Ta: A new picture of plasticity in BCC transition metals,”Phys. Rev. Lett., 84, 1499– 1503, 2000. [15] S.L. Frederiksen and K. Jacobsen, “Density functional theory studies of screw dislocation core structures in BCC metals,” Phil. Mag., 83, 365–375, 2003.

Perspective 30 A PERSPECTIVE ON DISLOCATION DYNAMICS Nasr M. Ghoniem Mechanical and Aerospace Engineering Department, University of California, Los Angeles, CA 90095-1597, USA

A fundamental description of plastic deformation has been recently pursued in many parts of the world as a result of dissatisfaction with the limitations of continuum plasticity theory. Although continuum models of plastic deformation are extensively used in engineering practice, their range of application is limited by the underlying database. The reliability of continuum plasticity descriptions is dependent on the accuracy and range of available experimental data. Under complex loading situations, however, the database is often hard to establish. Moreover, the lack of a characteristic length scale in continuum plasticity makes it difficult to predict the occurrence of critical localized deformation zones. Although homogenization methods have played a significant role in determining the elastic properties of new materials from their constituents (e.g., composite materials), the same methods have failed to describe plasticity. It is widely appreciated that plastic strain is fundamentally heterogenous, displaying high strains concentrated in small material volumes, with virtually undeformed regions in-between. Experimental observations consistently show that plastic deformation is heterogeneous at all length-scales. Depending on the deformation mode, heterogeneous dislocation structures appear with definitive wavelengths. A satisfactory description of realistic dislocation patterning and strain localization has been rather elusive. Attempts aimed at this question have been based on statistical mechanics, reaction-diffusion dynamics, or the theory of phase transitions. Much of the efforts have aimed at clarifying the fundamental origins of inhomogeneous plastic deformation. On the other hand, engineering descriptions of plasticity have relied on experimentally verified constitutive equations. At the macroscopic level, shear bands are known to localize plastic strain, leading to material failure. At smaller length scales, dislocation distributions are mostly heterogeneous in deformed materials, leading to the formation of 2871 S. Yip (ed.), Handbook of Materials Modeling, 2871–2877. c 2005 Springer. Printed in the Netherlands.

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a number of strain patterns. Generally, dislocation patterns are thought to be associated with energy minimization of the deforming material, and manifest themselves as regions of high dislocation density separated by zones of virtually undeformed material. Dislocation-rich regions are zones of facilitated deformation, while dislocation poor regions are hard spots in the material, where plastic deformation does not occur. Dislocation structures, such as Persistent slip Bands (PSB’s), planar arrays, dislocation cells, and subgrains, are experimentally observed in metals under both cyclic and steady deformation conditions. Persistent slip bands are formed under cyclic deformation conditions, and have been mostly observed in copper and copper alloys. They appear as sets of parallel walls composed of dislocation dipoles, separated by dislocation-free regions. The length dimension of the wall is orthogonal to the direction of dislocation glide. Dislocation planar arrays are formed under monotonic stress deformation conditions, and are composed of parallel sets of dislocation dipoles. While PSB’s are found to be aligned in planes with normal parallel to the direction of the critical resolved shear stress, planar arrays are aligned in the perpendicular direction. Dislocation cell structures, on the other hand, are honeycomb configurations in which the walls have high dislocation density, while the cell interiors have low dislocation density. Cells can be formed under both monotonic and cyclic deformation conditions. However, dislocation cells under cyclic deformation tend to appear after many cycles. Direct experimental observations of these structures have been reported for many materials. Two of the most fascinating features of micro-scale plasticity are the spontaneous formation of dislocation patterns, and the highly intermittent and spatially localized nature of plastic flow. Dislocation patterns consist of alternating dislocation rich and dislocation poor regions usually in the µm range (e.g., dislocation cells, sub-grains, bundles, veins, walls, and channels). On the other hand, the local values of strain rates associated with intermittent dislocation avalanches are estimated to be on the order of 1–10 million times greater than externally imposed strain rates. Understanding the collective behavior of defects is important because it provides a fundamental understanding of failure phenomena (e.g., fatigue and fracture). It will also shed light on the physics of elf-organization and the behavior of critical-state systems (e.g., avalanches, percolation, etc.) Because the internal geometry of deforming crystals is very complex, a physically-based description of plastic deformation can be very challenging. The topological complexity is manifest in the existence of dislocation structures within otherwise perfect atomic arrangements. Dislocation loops delineate regions where large atomic displacements are encountered. As a result, longrange elastic fields are set up in response to such large, localized atomic displacements. As the external load is maintained, the material deforms

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plastically by generating more dislocations. Thus, macroscopically observed plastic deformation is a consequence of dislocation generation and motion. A closer examination of atomic positions associated with dislocations shows that large displacements are confined only to a small region around the dislocation line (i.e., the dislocation core). The majority of the displacement field can be conveniently described as elastic deformation. Even though one utilizes the concept of dislocation distributions to account for large displacements close to dislocation lines, a physically-based plasticity theory can paradoxically be based on the theory of elasticity! Studies of the mechanical behavior of materials at a length scale larger than what can be handled by direct atomistic simulations, and smaller than what allows macroscopic continuum averaging represent particular difficulties. When the mechanical behavior is dominated by microstructure heterogeneity, the mechanics problem can be greatly simplified if all atomic degrees of freedom were adiabatically eliminated, and only those associated with defects are retained. Because the motion of all atoms in the material is not relevant, and only atoms around defects determine the mechanical properties, one can just follow material regions around defects. Since the density of defects is many orders of magnitude smaller than the atomic density, two useful results emerge. First, defect interactions can be accurately described by long-range elastic forces transmitted through the atomic lattice. Second, the number of degrees of freedom required to describe their topological evolution is many orders of magnitude smaller than those associated with atoms. These observations have been instrumental in the emergence of meso-mechanics on the basis of defect interactions by Eshelby, Kr¨oner, Kossevich, Mura and others. Thanks to many computational advances during the past two decades, the field has steadily moved from conceptual theory to practical applications. While early research in defect mechanics focused on the nature of the elastic field arising from defects in materials, recent computational modelling has shifted the emphasis on defect ensemble evolution. Although the theoretical foundations of dislocation theory are wellestablished, efficient computational methods are still in a state of development. Other than a few cases of perfect symmetry and special conditions, the elastic field of 3-D dislocations of arbitrary geometry is not analytically available. The field of dislocation ensembles is likewise analytically unattainable. A relatively recent approach to investigating the fundamental aspects of plastic deformation is based on direct numerical simulation of the interaction and motion of dislocations. This approach, which is commonly known as dislocation dynamics (DD), was first introduced for 2-D straight, infinitely long dislocation distributions, and then later for complex 3-D microstructure. In DD simulations of plastic deformation, the computational effort per time-step is proportional to the square of the number of interacting segments, because

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of the long-range stress field associated with dislocation lines. The computational requirements for 3-D simulations of plastic deformation of even single crystals are thus very challenging. The study of dislocation configurations at short-range can be quite complex, because of large deformations and reconfiguration of dislocation lines during their interaction. Thus, adaptive grid generation methods and more refined treatments of self-forces have been found to be necessary. In some special cases, however, simpler topological configurations are encountered. For example, long straight dislocation segments are experimentally observed in materials with high Peierls potential barriers (e.g., covalent materials), or when large mobility differences between screw and edge components exist (e.g., some bcc crystals at low temperature). Under conditions conducive to glide of small prismatic loops on glide cylinders, or the uniform expansion of nearly circular loops, changes in the loop shape is nearly minimal during its motion. Also, helical loops of nearly constant radius are sometimes observed in quenched or irradiated materials under the influence of point defect fluxes. It is clear that, depending on the particular application and physical situation, one would be interested in a flexible method which can capture the essential physics at a reasonable computational cost. A consequence of the longrange nature of the dislocation elastic field is that the computational effort per time step is proportional to the square of the number of interacting segments. It is therefore advantageous to reduce the number of interacting segments within a given computer simulation, or to develop more efficient approaches to computations of the long range field. While continuum approaches to constitutive models are limited to the underlying experimental data-base, DD methods offer new directions for modeling microstructure evolution from fundamental principles. The limitation to the method presented here is mainly computational, and much effort is needed to overcome several difficulties. First, the length and time scales represented by the present method are still short of many experimental observations, and methods of rigorous extensions are still needed. Second, the boundary conditions of real crystals are more complicated, especially when external and internal surfaces are to be accounted for. Thus, the present approach does not take into account large lattice rotations, and finite deformation of the underlying crystal, which may be important for explanation of certain scale effects on plastic deformation. And finally, a much expanded effort is needed to bridge the gap between atomistic calculations of dislocation properties on the one hand, and continuum mechanics formulations on the other. Nevertheless, with all of these limitations, the DD approach is worth pursuing, because it opens up new possibilities for linking the fundamental nature of the microstructure with realistic material deformation conditions. It can thus provide an additional tool to both theoretical and experimental investigations of plasticity and failure of materials.

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Two main approaches have been advanced to model the mechanical behavior in this meso length scale. The first is based on statistical mechanics methods. In these developments, evolution equations for statistical averages (and possibly for higher moments) are to be solved for a complete description of the deformation problem. The main challenge in this regard is that, unlike the situation encountered in the development of the kinetic theory of gases, the topology of interacting dislocations within the system must be included. The second approach, commonly known as Dislocation Dynamics (DD), was initially motivated by the need to understand the origins of heterogeneous plasticity and pattern formation. An early variant of this approach (the cellular automata) was first developed by Lepinoux and Kubin [1], and that was followed by the proposal of DD [2–4]. In these early efforts, dislocation ensembles were modelled as infinitely long and straight in an isotropic infinite elastic medium. The method was further expanded by a number of researchers, with applications demonstrating simplified features of deformation microstructure. Since it was first introduced in the mid-eighties independently by Lepinoux and Kubin, and by Ghoniem and Amodeo, Dislocation Dynamics (DD) has now become an important computer simulation tool for the description of plastic deformation at the micro- and meso-scales (i.e., the size range of a fraction of a micron to tens of microns). The method is based on a hierarchy of approximations that enable the solution of relevant problems with today’s computational resources. In its early versions, the collective behavior of dislocation ensembles was determined by direct numerical simulations of the interactions between infinitely long, straight dislocations [5]. Recently, several research groups extended the DD methodology to the more physical, yet considerably more complex 3-D simulations. The method can be traced back to the concepts of internal stress fields and configurational forces. The more recent development of 3-D lattice dislocation dynamics by Kubin and co-workers has resulted in greater confidence in the ability of DD to simulate more complex deformation microstructure [6–8]. More rigorous formulations of 3-D DD have contributed to its rapid development and applications in many systems [9–15]. We can classify the computational methods of DD into the following categories: 1. The Parametric Method: The dislocation loop can be geometrically represented as a continuous (to second derivative) composite space curve. This has two advantages: (1) there is no abrupt variation or singularities associated with the self-force at the joining nodes in between segments, (2) very drastic variations in dislocation curvature can be easily handled without excessive re-meshing. Other approximation methods have been developed by a number of groups. These approaches differ mainly in the representation of dislocation loop geometry, the manner by which the elastic field and self energies are calculated, and some additional details

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3.

4.

5.

N.M. Ghoniem related to how boundary and interface conditions are handled. The suitability of each method is determined by the required level of accuracy and resolution in a given application. dislocation loops are divided into contiguous segments represented by parametric space curves. The Lattice Method: Straight dislocation segments (either pure screw or edge in the earliest versions , or of a mixed character in more recent versions) are allowed to jump on specific lattice sites and orientations. The method is computationally fast, but gives coarse resolution of dislocation interactions. The Force Method: Straight dislocation segments of mixed character in the are moved in a rigid body fashion along the normal to their midpoints, but they are not tied to an underlying spatial lattice or grid. The advantage of this method is that the explicit information on the elastic field is not necessary, since closed-form solutions for the interaction forces are directly used. The Differential Stress Method: This is based on calculations of the stress field of a differential straight line element on the dislocation. Using numerical integration, Peach–Koehler forces on all other segments are determined. The Brown procedure [16] is then utilized to remove the singularities associated with the self force calculation. The Phase Field Microelasticity Method: This method is based on the reciprocal space theory of the strain in an arbitrary elastically homogeneous system of misfitting coherent inclusions embedded into the parent phase . Thus, consideration of individual segments of all dislocation lines is not required. Instead, the temporal and spatial evolution of several density function profiles (fields) are obtained by solving continuum equations in Fourier space [17].

References [1] J. Lepinoux and L.P. Kubin, “The dynamic organization of dislocation structures: a simulations,” Scripta Met., 21(6), 833, 1987. [2] N.M. Ghoniem and R.J. Amodeo, “Computer simulation of dislocation pattern formation,” Solid State Phenomena, 3 & 4, 377, 1988. [3] R.J. Amodeo and N.M. Ghoniem, “Dislocation dynamics I: a proposed methodology for deformation micromechanics,” Phys. Rev., 41, 6958, 1990b. [4] R.J. Amodeo and N.M. Ghoniem, “Dislocation dynamics II: applications to the formation of persistent slip bands, planar arrays, and dislocation cells,” Phys. Rev., 41, 6968, 1990a. [5] H.Y. Wang and R. LeSar, “O(N) algorithm for dislocation dynamics,” Phil. Mag. A, 71, 1, 149, 1995. [6] L.P. Kubin, G. Canova, M. Condat, B. Devincre, V. Pontikis, and Y. Brechet, “Dislocation microstructures and plastic flow: a 3D simulation,” Diffussion and Defect Data – Solid State Data, Part B (Solid State Phenomena), 23–24, 455, 1992.

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[7] G. Canova, Y. Brechet, and L.P. Kubin, “3D dislocation simulation of plastic instabilities by work-softening in alloys,” In: S.I. Anderson et al., Modeling of Plastic Deformation and its Engineering Applications, RISØ National Laboratory, Roskilde, Denmark, 1992. [8] B. Devincre and L.P. Kubin, “Simulations of forest interactions and strain hardening in fcc crystals,” Mod. Sim. Mater. Sci. Eng., 2(3A), 559, 1994. [9] J.P. Hirth, M. Rhee, and H. Zbib, “Modeling of deformation by a 3D simulation of multipole, curved dislocations,” J. Comp.-Aided Mat. Design, 3, 164, 1996. [10] K.V. Schwarz and J. Tersoff, “Interaction of threading and misfit dislocations in a strained epitaxial layer,” App. Phys. Lett., 69(9), 1220, 1996. [11] R.M. Zbib, M. Rhee, and J.P. Hirth, “On plastic deformation and the dynamics of 3D dislocations,” In: J. Mech. Sci., 40(2–3), 113, 1998. [12] M. Rhee, H.M. Zbib, J.P. Hirth, H. Huang, and T. de la Rubia, “Models for long/ short-range interactions and cross slip in 3D dislocation simulation of bcc single crystals,” Mod. Sim. Mater. Sci. Eng., 6(4), 467, 1998. [13] N.M. Ghoniem and L.Z. Sun, “Fast sum method for the elastic field of 3-D dislocation ensembles,” Phy. Rev. B, 60(1), 128–140, 1999. [14] N.M. Ghoniem, S.-H. Tong, and L.Z. Sun, “Parametric dislocation dynamics: a thermodynamics-based approach to investiagations of mesoscopic plastic deformation,” Phys. Rev., 61(2), 913–927. [15] N.M. Ghoniem, J. Huang, and Z. Wang, “Affine covariant-contravariant vector forms for the elastic field of parametric dislocations in isotropic crystals,” Phil. Mag. Lett., 82(2), 55–63, 2001. [16] L.M. Brown, “A proof of Lothe’s theorem,” Phil. Mag., 15, 363–370, 1967. [17] Y. Wang, Y. Jin, A. Cuitino, and A.G. Khachaturyan, “Nanoscale phase field microelasticity theory of dislocations: model and 3D simulations,” Acta Mat., 49, 1847, 2001.

Perspective 31 DISLOCATION-PRESSURE INTERACTIONS J.P. Hirth Ohio State and Washington State Universities, 114 E. Ramsey Canyon Rd., Hereford, AZ 85615, USA

1.

Introduction

There are various ways in which a dislocation can interact with isostatic stress, with most effects being important at elevated stress levels. Some of these that have received extensive attention are the effects of isostatic stress on elastic constants [1–3]; the direct influence of pressure, or the indirect effect via the stress normal to the glide plane, on the Peierls stress, or more specifically the kink formation energy in many metals and alloys [4, 5]; the influence of the degree and symmetry of core splitting of screw dislocations in bcc metals [6–8] as well as some semiconducting crystals and intermetallic compounds [8, 9]; the implementation of core-splitting effects in terms of the general stress tensor [10, 11]; the indirect effect on operative slip systems that modifies geometric hardening [12]; and weak effects on factors such as lattice parameter, vibrational frequency and diffusivity. However, little or no consideration has been given to the coupling of isostatic stress with the nonlinear, long-range, elastic field of the dislocation. Here, we briefly treat this effect and indicate how it can be incorporated into constitutive models for dislocation motion.

2.

Origin of the Coupling

The nonlinear elastic theory of dislocations has been developed, in the perturbation sense of including third and fourth order elastic constants, for both the isotropic and anisotropic elastic cases [1, 13, 14]. These treatments predict, for example, that there is a biaxial dilatation associated with a dislocation, and X-ray diffraction verifies the presence of the dilatation [1]: of course, in low symmetry crystals such as triclinic, there is a dilatational response to 2879 S. Yip (ed.), Handbook of Materials Modeling, 2879–2882. c 2005 Springer. Printed in the Netherlands.

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any applied stress so local dilatation should exist near a dislocation.. However, the perturbation models are not accurate in predicting the magnitude of the dilatation, although they are useful in giving the dominant terms for the near core region once the core fields are known. The dilatation is associated with highly nonlinear core effects and can be estimated from atomistic simulations of the near-core region [15] or from an atomistic simulation involving the variation of strain energy with stress in a local near-core region [16]. The core field entails sets of line-force dipoles without moment [15], but at long-range converges to a biaxial dilatation, so a consideration of the latter suffices at long range. A list of typical biaxial dilatations determined in such a manner is given by Puls [17]. Typically, the biaxial dilatational area, δa, including image relaxation at free surfaces, is of the order of 1 to 1.5 atomic areas per plane cut by the dislocation. This corresponds to a volume change per unit length δv/L = δa.

3.

Influence on Dislocation Motion

We consider local coordinates xi fixed on the dislocation with x3 // ξ , the sense vector of the dislocation. The dilatation will then interact with the biaxial stress σ = (σ11 +σ22 )/2. In most practical situations σ will correspond to the isoaxial stress σ I = − p = (σ11 + σ22 + σ33 )/3, where p is the isostatic pressure. We consider this correspondence to apply, although the more specific σ case can easily be introduced into the results. The local coupling will then produce a positive interaction energy per unit length given by W p = pδa.

4.

Systems Deforming by Kink Motion

As a first application, we treat kink pair formation on screw dislocations in bcc metals. For Fe a typical kink formation energy is 0.8 eV [7], δa = 0.62 b2 [18], and kinks should be inclined at an angle φ ∼45◦ to the Peierls valley. Here b is the length of the Burgers vector of a 12 [111] dislocation. These values lead to an energy contribution from the pressure interaction of W p λ=7.7 ( p/µ) eV, normalized to the shear modulus µ, where λ is the kink length. Thus at ( p/µ) = 0.01, the contribution of W p λ is about ten percent of Wk . The value ( p/µ) = 0.01 is readily attained in shock loading and it is of the order of pressures achieved in constrained high pressure tensile testing devices. This value is also typical of the isostatic stress σI = − p achieved in uniaxial tensile tests of heavily drawn steel wire [19]. In general, the kink formation energy can be written W f = Wk + Wh + W p λ +

Wσ 2

(1)

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Entropy terms would also enter the kink free energy [20]. Here, Wk is the kink formation energy in the absence of stress, Wh is the contribution arising from core splitting and the influence of the applied stress tensor [10], and Wσ is the work done by the effective shear stress resolved on the glide plane as a kink pair is created. The numerical example indicates that W p λ is significant and should be included in the analysis of kink formation at high stress levels. Less obviously, there is an indirect effect of pressure on kink energy. When W p is appreciable, there is a tendency to increase φ from its zero-stress value in order to minimize the length λ that interacts with p: that is the minimum free energy kink will become sharper. This means that Wk will increase relative to the zero-stress value. The effect is second-order relative to the direct effect just discussed, but it can be important at very high stress levels. For fcc metals, the isostatic stress interaction term is roughly W p λ ≈ 5 ( p/µ) eV. This is similar to the value for bcc metals. Thus for fcc metals, the kink mechanism is only operative at temperatures typically well below room temperature, and there the interaction again can be important under shock loading. The effect is negligible for bulk metals under uniaxial tension because of the relatively low values of the flow stress for fcc metals. However, it could also be important for in-situ composites [21] and thin multiplayer structures [22], where stresses also reach values where ( p/µ) ∼0.01. When the interaction is important, the influence on φ should be marked because of its initially low value ∼ 5◦ [20]. Other systems with large kink formation energies should behave analogously to the bcc screw case. Intermetallic compounds [9] and ceramic crystals such as alumina [23] should exhibit such behavior.

5.

Systems Deforming by Dislocation Bowout and Breakaway

In fcc metals at room temperature and in most bcc metals above a critical temperature that exceeds room temperature, deformation involves locally bowed out segments that breakaway from obstacles by cutting or bypassing them. The barrier to bowout is the increase in line length of the bowing dislocation. The exact line energy is a complicated function of segment length, dislocation character and local surrounding segment configurations [20]. For our purposes, it suffices to use the simple line tension approximation. The increase in energy per unit length is then W = S L L = (µb2 /2)L where S L is the constant line tension and L is the increase in line length. With the pressure interaction present, W p directly adds to S L . Thus, the total line tension is S = S L + W p = µb2 /2 + pδa

(2)

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And the increase in energy with an increase of line length is W = SL. The importance of the pressure interaction term can be determined by direct substitution.

6.

Summary

Dislocations have a long-range dilatational strain field arising from large nonlinearities in the core. The field interacts with isostatic stresses. The resultant interaction energy can be an important contribution to kink formation energy and to dislocation line tension at high pressures.

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23]

A. Seeger and P. Haasen, Philos. Mag., 3, 470, 1958. D.J. Steinberg, S.G. Cochran, and J. Guinan, J. Appl. Phys., 51, 1498, 1980. L.H. Yang, P. S¨oderlind, and J.A. Moriarty, Philos. Mag., A81, 1355, 2001. J.A. Moriarty, Phys. Rev., B49, 12431, 1994. J.A. Moriarty, V. Vitek, V.V. Bulatov et al., J. Computer-Aid. Mater. Des., 9, 99, 2002. V. Vitek, Cryst. Lattice Defects, 5, 1, 1974. M.S. Duesbery, Acta Metall., 31, 1747, 1983. W. Cai, V.V. Bulatov, J. Chang et al., In: F.R.N. Nabarro and J.P. Hirth (eds.), Dislocations in Solids, vol. 12, N. Holland, Amsterdam, 2004. V. Paidar, D.P. Pope, and V. Vitek, Acta Metall., 32, 435, 1984. Q. Qin and J.L. Bassani, J. Mech. Phys. Solids, 40, 835, 1992. M.S. Duesbery and V. Vitek, Acta Metall. Mater., 46, 1481, 1998. R.J. Asaro and J.R. Rice, J. Mech. Phys. Solids, 25, 309, 1977. J.R. Willis, Int. J. Engin. Sci., 5, 171, 1967. C. Teodosiu, Elastic Models of Crystal Defects, Springer-Verlag, Berlin, p. 208, 1982. R.G. Hoagland, J.P. Hirth, and P.C. Gehlen, Philos. Mag., 34, 413, 1976. R.G. Hoagland, M.S. Daw, and J.P. Hirth, J. Mater. Sci., 6, 2565, 1991. M.P. Puls, Dislocation Modeling of Physical Systems, M.F. Ashby et al. (eds.), Pergamon, Oxford, p. 249, 1981. B.L. Adams, J.P. Hirth, P.C. Gehlen, et al., J. Phys. F, 7, 2021, 1977. J.D. Embury, A.S. Keh, and R.M. Fisher, Metall. Trans., 12, 478, 1967. J.P. Hirth and J. Lothe, Theory of dislocations, Kliewer, Melbourne, FL, 1992. K. Han, J.D. Embury, J.J. Petrovic et al., Acta Mater., 46, 4691, 1998. A. Misra and H. Kung, Adv. Engin. Mater., 3, 217, 2001. T.E. Mitchell, P. Peralta, and J.P. Hirth, Acta Mater., 47, 3687, 1999.

Perspective 32 DISLOCATION CORES AND UNCONVENTIONAL PROPERTIES OF PLASTIC BEHAVIOR V. Vitek Department of Materials Science and Engineering, University of Pennsylvania, Philadelphia, PA 19104

1.

Concept of Dislocations

Dislocations are line defects found in all crystalline materials and their motion produces plastic flow. The notion of dislocations has two starting points. First, the dislocation was introduced as an elastic singularity by considering the deformation of a body occupying a multiply connected region of space. Secondly, dislocations were introduced into crystal physics when analyzing the large discrepancy between the theoretical and experimental strength of crystals. These two approaches are intertwined since the crystal dislocations are sources of long-ranged elastic stresses and strains that can be examined in the continuum framework. In fact, the bulk of the dislocation theory employs the continuum elasticity when analyzing a broad variety of dislocation phenomena encountered in plastically deforming crystals [1–4]. From the continuum point of view dislocations are line singularities invoking long-ranged elastic stress and strain fields that decrease as d −1 , where d is the distance from the dislocation line. Formally, both the stress and strain diverge at the dislocation line and the corresponding strain energy would also diverge. However, physically this means that there is a region, centered at the dislocation line, in which the linear elasticity does not apply. This region, the dimensions of which are of the order of the lattice spacing, is called the dislocation core. The properties of the core region and its impact on dislocation motion and thus on plastic yielding, can only be fully understood when the atomic structure is adequately accounted for. In general, when a dislocation glides its core undergoes changes that are the source of an intrinsic lattice friction. This 2883 S. Yip (ed.), Handbook of Materials Modeling, 2883–2896. c 2005 Springer. Printed in the Netherlands.

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friction is periodic with the period of the crystallographic direction in which the dislocation moves. The applied stress needed to overcome this friction at 0 K temperature is called the Peierls stress and the corresponding periodic energy barrier is called the Peierls barrier. In this Commentary we first focus on the dominant aspects of dislocation cores, in particular their symmetries and the form of spreading of atomic displacements in the core region. Specifically, the core displacements can be either confined to a given crystallographic plane or extended three-dimensionally. The latter cores, encountered in a broad variety of materials, are responsible for unconventional aspects of the plastic deformation such as the break-down of the Schmid law, anomalous dependence of the yield stress on temperature, unusually strong orientation and temperature dependence of the yield and flow stress, orientation dependence of the ductility and brittleness, strong strain rate sensitivity etc., [5–10]. Using body-centered-cubic (bcc) metals as an example, we discuss the core features that have to be captured in any theoretical description of the dislocation motion and related plastic properties. Finally, we demonstrate by a brief overview that analogous core phenomena are encountered in many other materials than bcc metals. However, prior to the discussion of atomic structures of dislocations, we reflect on the most important “core phenomenon”, dislocation dissociation into partial dislocations separated by metastable stacking-fault like planar defects.

2.

Dislocation Dissociation and Stacking Faults

A vital characteristic of dislocations in crystalline materials is their possible dissociation into partial dislocations with Burgers vectors smaller than the lattice vector. Such dislocation splitting can occur if the displacements corresponding to the Burgers vectors of the partials lead to the formation of metastable planar faults which then connect the partials. The reason for the splitting is, of course, the decrease of the dislocation energy when it is divided into dislocations with smaller Burgers vectors. A well-known example is splitting of 1/2110 dislocations in fcc materials into two Shockley partials with the Burgers vectors of the type 1/6112 on {111} planes. The planar fault formed by the 1/6112 displacement is an intrinsic stacking fault. In general, stacking-fault-like defects that include not only stacking faults but also other planar defects, such as antiphase domain boundaries and complex stacking faults encountered in ordered alloys and compounds, can be very conveniently analyzed using the notion of γ-surfaces, first employed in investigation of possible stacking faults in bcc metals [11]. To introduce the idea of a γ-surface, we first define a generalized stacking-fault: Imagine that the crystal is cut along a given crystallographic plane and the upper part displaced , parallel to the plane of the cut, as with respect to the lower part by a vector u

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shown in Fig. 1. The fault created in this way is called the generalized stackingfault and it is not in general metastable. The energy of such fault, γ ( u), can be evaluated using atomistic and/or density functional theory (DFT) methods; relaxation perpendicular to the fault has to be carried in such calculations. Re within the repeat cell of the given peating this procedure for various vectors u crystal plane, an energy-displacement surface can be constructed, commonly called the γ-surface. The local minima on this surface determine the displacement vectors of all possible metastable stacking-fault-like defects, and the values of γ at these minima are the energies of these faults. Many such calculations were performed employing a broad variety of descriptions of interatomic interactions. They became particularly popular with the advent of the DFT based methods since γ-surface calculations are much less computation intensive than studies of dislocations while providing information that is often sufficient for understanding the atomic level properties of dislocations. Examples are recent calculation of γ-surfaces in bcc transition metals [12], aluminum [13], silicon [13, 14], graphite [15], TiAl [16, 17] and MoSi2 [18, 19]. Symmetry arguments can be utilized to assess the general shape of γ-surfaces. If a mirror plane of the perfect lattice perpendicular to the plane of a generalized stacking-fault passes through the point corresponding to a , the first derivative of the γ-surface along the normal to this displacement u mirror plane vanishes owing to the mirror symmetry. This implies that the

Upper part of the crystal

u Generalized stacking fault

Lower part of the crystal Figure 1. Formation of the generalized stacking fault.

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γ-surface will possess extrema (minima, maxima or inflexions) for those displacements, for which there are at least two non-parallel mirror planes of the perfect lattice perpendicular to the fault. Whether any of the extrema correspond to minima, and thus to metastable faults, can often be determined by considering the change in the nearest neighbor configuration produced by the corresponding displacement. Hence, the symmetry-dictated metastable stacking-fault-like defects can be ascertained on a crystal plane by analyzing its symmetry. Such faults are then common to all materials with a given crystal structure. The intrinsic stacking faults in fcc crystals are symmetry dictated since three mirror planes of the {101} type pass through the points that correspond to the displacements 1/6112. However, other minima than those associated with symmetry-dictated extrema may exist in any particular material. These can not be anticipated on crystallographic grounds but their existence depends on the details of atomic interactions and they can only be revealed by calculations of the γ-surface. The primary significance of the dislocation dissociation is that it determines uniquely the slip planes, identified with the planes of splitting and corresponding stacking-fault-like defects and, consequently, the operative slip systems. The fact that {111} planes are the slip planes in fcc materials is a typical example. However, the core of individual partial dislocations may still spread spatially and introduce effects similar to those found in undissociated dislocations with non-planar cores. In summary, when analyzing dislocation core structure, possible splitting into well-defined partials separated by metastable stacking fault-like defects has to be considered first. If such splitting cannot occur, either because no metastable stacking fault-like defects exist or their energy is so high that the splitting is not favored, the core of total dislocations has to be studied. In the opposite case, investigation of the cores of the corresponding partials needs to be performed.

3.

Dislocation Cores

In general, the dislocation core structure can be described in terms of the relative displacements of atoms in the core region. These displacements are usually not distributed isotropically but are confined to certain crystallographic planes. Two distinct types of dislocation cores have been found, depending on the mode of the distribution of significant atomic displacements. When the core displacements are confined to a single crystallographic plane the core is planar∗ . In metallic materials dislocations with such cores usually glide

∗ A more complex planar core, called zonal, may be spread into several adjacent parallel crystallographic

planes of the same type [20].

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easily in the planes of the core spreading and their Peierls stress is commonly low. However, the Peierls stress can be high even for planar cores in covalently bonded solids owing to the need to break the covalent bonds during the dislocation glide. In contrast, if the core displacements spread into several nonparallel planes of the zone of the dislocation line, the core is non-planar. The glide planes of dislocations with such cores are often not defined uniquely, the corresponding Peierls stress is high and well below the melting temperature the dislocation glide is enabled by thermal activation over the Peierls barriers.

3.1.

Planar Cores

Only full-scale atomistic calculations can reveal all the details of the core structure for both planar and non-planar cores. However, for planar cores semiatomistic models of the Peierls-Nabarro type are capable to describe the cores with high precision [3]. In such models the core is regarded as a continuous distribution of dislocations in the plane of the core spreading. If we choose the coordinate system in the plane of the core spreading such that the axes x1 and x2 are parallel and perpendicular to the dislocation line, respectively, the corresponding density of the continuously distributed dislocations has two components ρα = ∂ u α /.∂ x2 (α = 1, 2), where u α is the α component of the dis in the x1 , x2 plane. The displacement u increases graduplacement vector u +∞ ally in the direction x2 from zero to the Burgers vector so that −∞ ρα dx2 = bα , where bα is the corresponding component of the Burgers vector. In the continuum approximation the elastic energy of such dislocation distribution can be expressed as the interaction energy of the dislocations within this distribution: E el =

+∞ +∞ 2

K αβ ρα ρβ ln x2 − x2 dx2 dx2

(1)

α,β=1−∞ −∞

where K αβ are constants depending on the elastic moduli and orientation of the dislocation line. On the atomic scale the displacement across the plane of the core spreading causes a disregistry that leads to an energy increase. The produces locally a generalized stacking-fault and in the local displacement u approximation the energy associated with the disregistry can be approximated as +∞

γ ( u) dx2

Eγ =

(2)

−∞

where γ ( u) is the energy of the corresponding γ -surface for the displacement . The continuous distribution of dislocations describing the core structure is u

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then found by the functional minimization of the total energy E tot = E el + E γ [21]∗ . with respect to the displacement u The salient feature of this model is that the energy E tot does not depend on the position of the core and thus when the dislocation moves no energy change would be found. Nevertheless, the Peierls stress can be evaluated by re-introducing the discrete structure of the lattice by placing the atoms of this obtained from lattice into the positions determined by the displacement field u the model. The Peierls stress and the Peierls barrier can then be evaluated by gradually moving this displacement distribution along the slip plane by one period along the glide plane. This is the approach adopted originally by Nabarro [22]. The result is a rather low Peierls stress in the case of planar cores. This is seen from the analytical expression obtained for the original model with sinusoidal restoring force when the Peierls stress is τ P = 2µ/α exp(−4π ζ /b); µ is the shear modulus and α a factor of the order of one and ζ is the width of the core. This stress is very small even when the core is very narrow; for example, if ζ = b, τ P ≈ 10−5 µ. Comparisons with full-scale atomistic calculations have shown a good agreement with this approximate approach in a number of cases, in particular when the cores are wide. A variety of improvements have been suggested recently in which the atomic structure is taken into account explicitly in the model rather than a posteriori [13, 23, 24]. However, the model generally performs poorly for cores the width of which is comparable with the lattice spacing and when covalent bonds dominate crystal bonding. In the former case the continuum description of the core is poor since it is applied to distances smaller than the lattice spacing and in the latter case possible breaking and/or formation of covalent bonds is not included into the model.

3.2.

Non-planar Cores

The non-planar cores can be divided into two classes: cross slip and climb cores. In the former case the core displacements lie in the planes of the core spreading while in the latter case they possess components perpendicular to these planes. Climb cores are less common and are usually formed at high temperatures by a climb process. The best known example of the cross-slip core is the core of 1/2111 screw dislocations in bcc metals. Atomistic studies of dislocations are the principal source of our understanding of the structure of such cores since direct observations are mostly outside the limits of experimental techniques. Two alternate structures of the core of the 1/2[111] ∗ If the displacement vector is all the time parallel to the Burgers vector, so that only the component u in ∂γ / 0 for any finite value of u, the Euler equation corresponding this direction needs to be considered, and ∂u = to the condition δ E tot = 0 leads to the well-known Peierls equation [21].

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(a)

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(b)

Figure 2. Two alternate core structures of the 1/2[111] screw dislocation depicted using differential displacement maps. (a) Calculated using Finnis–Sinclair type central force potential for Mo [25], (b) Calculated using a bond-order potential for Mo [27]. The atomic arrangement is shown in the projection perpendicular to the direction of the dislocation line ([111]) and circles represent atoms within one period. An arrow between them represents the [111] (screw) component of the relative displacement of the neighboring atoms produced by the dislocation. The length of the arrows is proportional to the magnitude of these components. The arrows, which indicate out-of-plane displacements, are always drawn along the line connecting neighboring atoms and their length is normalized such that it is equal to the separation of these atoms in the projection when the magnitude of their relative displacement is equal to |1/6 [111]|.

screw dislocation found by atomistic modeling are presented in Fig. 2. The core ¯ ¯ and (110) ¯ in Fig. 2a is spread asymmetrically into the (101), (011) planes that ¯ diad; belong to the [111] zone and is not invariant with respect to the [101] another energetically equivalent configuration related by this symmetry operation exists and this core is called degenerate or polarized. The core in Fig. 2b ¯ diad and it is called non-degenerate or is invariant with respect to the [101] non-polarized. The core structures shown in Figs. 2a and b were found by atomistic calculations employing central-force many-body potentials [25] and a tight binding and/or density functional theory based approaches [26, 27], respectively. This example demonstrates that the structure of the core is not determined solely by the crystal structure but may vary from material to material with the same crystal structure. The non-planar dislocation cores are the more common the more complex is the crystal structure and for this reason these cores are more prevalent than planar cores. In this respect, fcc materials (and also hcp materials with basal slip) in which the dislocations possess planar cores, are a special case rather than a prototype for more complex structures [5]∗ . ∗ Additional complexities of the dislocation core structures arise in covalent crystals where the breaking and/or readjustment of the bonds in the core region may be responsible for a high lattice friction stress, and in ionic solids where the cores can be charged which then strongly affects the dislocation mobility. Such dislocation cores affect not only the plastic behavior but also electronic and/or optical properties of covalently bonded semiconductors and ionically bonded ceramic materials.

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V. Vitek

Glide of Dislocations with Non-planar Cores and Breakdown of the Schmid Law

The Peierls stress of dislocations with non-planar cores is typically at least an order of magnitude higher than that of dislocations with planar cores. Furthermore, the movement of such dislocations is frequently affected not just by the shear stress parallel to the Burgers vector in the slip plane but by other stress components. Thus the deformation behavior of materials with non-planar dislocation cores may be very complex, often displaying unusual orientation dependencies and breakdown of the Schmid law. It has been known ever since the first studies of the plastic deformation of bcc metals that the Schmid law does not apply [7, 28] and, indeed, already the early atomistic calculations showed that the non-planar core has to transform prior to the dislocation motion and this transformation may be influenced by shear stresses in the slip direction acting in planes other than the slip plane [29]. Furthermore, more recent calculations revealed that such transformation may also be affected by shear stresses in the direction perpendicular to the Burgers vector [25, 30, 31]. This is well demonstrated by calculating the Peierls stress as a function of the shear stresses perpendicular to the Burgers vector while keeping the plane of the maximum resolved shear stress parallel to the Burgers vector (MRSSP) fixed. As suggested by Ito and Vitek [25], this can be achieved by applying the stress tensor with the components σ11 = −τ2 , σ22 = τ2 , σ33 = σ12 = σ13 = σ23 = 0 in the right-handed coordinate system with the x1 axis in the MRSSP, x2 axis perpendicular to the MRSSP and x3 axis parallel to [111], together with the shear stress parallel to the Burgers vector. Figure 3 shows the calculated dependence of the Peierls ¯ stress, τM , for the core shown in Fig. 2b on τ2 when the MRSSP is the (101) plane. It is important to note that changing the sign of τ2 corresponds to the rotation of the coordinates by 180◦ around the [111] axis. However, the core structure is not invariant with respect to this transformation and thus the effect of τ2 upon the dislocation behavior will, in general, be different for positive and negative values, as observed. The values of τM corresponding to tensile and compressive loadings along ¯ [238] and [012] axes are also shown in Fig. 3; for each uniaxial loading the value of τ2 is uniquely related to the loading stress at which the disloca¯ tion started to move. For these axes the (101) plane is the MRSSP and if there were no effect of shear stresses perpendicular to the Burgers vector the value of τM would be the same for both axes as well as for tension and compression. Obviously, this is not the case and the magnitude of τM depends on the shear stresses perpendicular to the slip direction, described by τ2 , that are different for different uniaxial loading. Moreover, for large negative τ2 the ¯ ¯ plane although the shear slip plane changes from the (101) plane to (011)

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τM C44

τ2/C44 Figure 3. Dependence of the Peierls stress, τM , on τ2 that defines the shear stress perpendic¯ ular to the Burgers vector, when the MRSSP is the (101) plane.

stress in the direction of the Burgers vector is lower in this plane. Apparently, the shear stress perpendicular to the Burgers vector alters the core such that ¯ plane is preferred. transformation into the (011) It should be emphasized at this point that the phenomena discussed above have been observed for both polarized and non-polarized cores [25]; similarly the polarity has not been found to influence significantly the magnitude of the Peierls stress. This is in contrast with the recent suggestions that the polarity plays a considerable role in the glide of dislocations with non-planar cores [32]. The reason is that when a stress is applied the symmetry of the nonpolarized core is broken and becomes the same as that of the polarized one. This has recently been discussed in detail by Vitek [33]. The results of atomistic calculations suggest that the motion of the 1/2[111] screw dislocation is governed by four distinct shear stress components when ¯ it glides in the (101) plane. First two are the Schmid stress, i.e., the stress in ¯ the direction of the Burgers vector in the slip plane (101), and the shear stress parallel to the Burgers vector in another {110} plane of the [111] zone. The other two are shear stresses perpendicular to the Burgers vector acting in two different {110} planes of the [111] zone. As discussed in detail in [34, 35], the effects of non-glide stresses may enter the flow rules for a single crystal

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loaded by a stress tensor σ by introducing for a slip system α an effective yield stress, τ ∗α = τ α +

3 η=1

aηα τηα

(3)

Here τ α = nα ·σ·mα is the Schmid stress; nα is the slip plane normal and mα the slip direction. τηα = nαη ·σ·mαη are the non-glide shear stresses; nαη is the normal to the {110} plane η, and mαη is the direction in this plane, either parallel or perpendicular to the Burgers vector, depending on which type of the shear stress is considered. At the onset of the plastic flow this effective yield stress has to attain a critical value τcr∗α . The coefficients aηα and the value of τcr∗α are determined so as to fit the results of atomistic calculations of the dislocation motion. This criterion reduces to the Schmid law if there is no influence of non-glide stresses and all aηα = 0. All the above results of atomistic studies relate to the glide of straight dislocations at 0 K but at finite temperatures the dislocation motion is aided by thermal activation and a generally accepted mechanism of the dislocation motion involves formation and extension of pairs of kinks. Assuming the usual rate theory, the dislocation velocity is then

U − W v ∝ exp − kB T

(4)

where kB is the Boltzman constant, T temperature, U = Uactivated − Uground the difference in the energy between the activated state and the state prior to the activation and W = τ α · b · A is the work done by the shear stress parallel to the Burgers vector (Schmid stress) during the activation process; b is the magnitude of the Burgers vector and A the area swept by the dislocation segment during activation. This mechanism of the thermally activated overcoming of the Peierls barriers has been employed in the framework of the dislocation theory by a number of authors, starting with the classical paper of Dorn and Rajnak [36] [37–39]. In these developments U had either been taken as constant or considered as a function of the Schmid stress τ α for a given slip system α. The same applies in the recent atomistic studies of kinks [40–43]. However, to include fully the effects of non-glide stresses at finite temperatures it has to be recognized that both Uactivated and Uground are functions of the applied stress tensor, so that U = f (σi j ), where σi j includes both glide and non-glide stress components affecting the dislocation motion. In principle, this could be achieved by molecular dynamics simulations of the formation of kink pairs at finite temperatures with applied stresses involving both glide and non-glide components. However, such calculations are feasible only for very

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high stresses and thus very large strain rates even when using the most powerful supercomputers [44]. The reason is that at usual strain rates the frequency of the formation of kink pairs is many orders of magnitude lower than the frequency of atomic vibration. Hence, the yet unsolved task is to develop a mesoscopic theory of the formation of kink pairs that will utilize the results of atomistic studies of the motion of straight dislocations to establish the dependence of U on all components of the stress tensor that govern the dislocation glide. A possible approach is to consider that U is a function of the effective yield stress τ ∗α such that U = 0 when τcr∗α = τcr∗α .

4.

Conclusions: Generality of the Core Effects

The non-planar dislocation cores, the example of which is the 1/2111 screw dislocation in bcc metals, are common in many materials. In hexagonal crystals such cores are not found when the slip is confined to basal planes but they are common in the case of prismatic and/or pyramidal slip [45] where they are responsible for strong temperature dependence of the yield stress. Similarly, non-planar dislocation cores have been identified in many intermetallic compounds. Examples are screw dislocations in NiAl [46], TiAl [47, 48] and MoSi2 [19, 49] where they are, presumably, responsible for an unusually large orientation dependence of the yield stress [50]. The transformation of 110 superdislocations in Ni3 Al from the planar glissile form to non-planar sessile form is responsible for the anomalous increase of the flow stress with temperature [10, 51]. However, non-planar cores leading to unusual temperature, strain rate and orientation dependencies have not been found only in metallic materials but also in materials such as olivine [52], sapphire [53] and anthracene [54]. Recently, an unusual ductile-to-brittle transition has been observed in the perovskite SrTiO3 that deforms plastically at room temperature but becomes brittle at about 1000 K [55]. This unusual behavior is most likely associated with formation of non-planar dislocation cores at high temperatures associated with local changes in stoichiometry [56]. In covalently bonded semiconductors it is both the atomic and electronic effects which affect significantly the Peierls barrier [57] that may lead to very complex mechanisms of formation of kinks [58]. In all these cases atomic level modeling is capable to identify the essential features of dislocation cores and identify the stress components governing the dislocation motion. The development of mesoscopic models of the thermally activated dislocation motion that incorporate the results of atomic level calculations is then the next essential step. However, at present such studies are either very rudimental or non-existent and this is, therefore, an open avenue for further research linking atomic level, nano-scale and macroscale deformation properties.

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[47] R. Porizek, S. Znam, D. Nguyen-Manh, V. Vitek, and D.G. Pettifor, “Atomistic studies of dislocation glide in y-TiAl,” Defect Properties and Related Phenomena in Intermetallic Alloys, E.P. George, H. Inui, M.J. Mills, and G. Eggeler (eds.), Pittsburgh: Materials Research Society, vol. 753, p. BB4.3.1–BB4.3.6, 2003. [48] C. Woodward and S.I. Rao, “Ab initio simulation of a/2110 screw dislocations in γ -TiAl,” Philos. Mag., 84, 401–414, 2003. [49] M.I. Baskes and R.G. Hoagland, “Dislocation core structures and mobilities in MoSi2 ,” Acta Mater., 49, 2357–2364, 2001. [50] K. Ito, H. Inui, Y. Shirai, and M. Yamaguchi, “Plastic deformation of MoSi2 single crystals,” Philos. Mag. A, 72, 1075–1097, 1995. [51] T. Kruml, E. Conforto, B. LoPiccolo, D. Caillard, and J.L. Martin, “From dislocation cores to strength and work-hardening: A study of binary Ni3 Al,” Acta Mater., 50, 5091–5101, 2002. [52] J.-P. Poirier, and B. Vergobbi, “Unusual deformation properties of olivire,” Physics of the Earth and Planetary Interiors, 16, 370–382, 1978. [53] J. Chang, C.T. Bodur, and A.S. Argon, “Pyramidal edge dislocation cores in sapphire,” Philos. Mag. Lett., 83, 659–666, 2003. [54] N. Ide, I. Okada, and K. Kojima, “Computer simulation of core structure and Peierls stress of dislocations in anthracene crystals,” J. Phys.: Condens. Matter, 5, 3151–3162, 1993. [55] P. Gumbsch, S. TaeriBaghbadrani, D. Brunner et al., “Plasticity and an inverse brittle-to-ductile transition in strontium titanate,” Phys. Rev. Lett., 8708, 5505+, 2001. [56] Z.L. Zhang, W. Sigle, W. Kurtz et al., “Electronic and atomic structure of a dissociated dislocation in SrTiO3 ,” Phys. Rev. B, 6621, 4112–4118, 2002a; “Atomic and electronic characterization of the a[100] dislocation core in SrTiO3 ,” Phys. Rev. B, 6609, 4108–4115, 2002b. [57] J.F. Justo, A. Antonelli, and A. Fazzio, “The energetics of dislocation cores in semiconductors and their role in dislocation mobility,” Physica B, 302, 398–402, 2001. [58] V.V. Bulatov, J.F. Justo, W. Cai et al., “Parameter-free modelling of dislocation motion: the case of silicon,” Philos. Mag. A, 81, 1257–1281, 2001.

Perspective 33 3-D MESOSCALE PLASTICITY AND ITS CONNECTIONS TO OTHER SCALES Ladislas P. Kubin LEM, CNRS-ONERA, 29 Av. de la Division Leclerc, BP 72, 92322 Chatillon Cedex, France

1.

Multiscale Analysis

There is a dislocation theory but there is no dislocation theory of plasticity. The reasons for this situation are multiple. By definition, a dislocation is a linear defect that ensures compatibility between slipped and unslipped parts of a crystal. The motion of this defect propagates microscopic shears and is responsible for the plastic (i.e., permanent) deformation of crystalline materials. A dislocation line can be viewed in two different manners. From an atomistic viewpoint, it consists of a highly distorted region, the core, which surrounds the geometrical line of the defect and has a diameter of a few lattice spacings. In the continuum, a dislocation consists of a singularity line to which are associated long range stress and strain fields. The line energy of a dislocation is mainly located outside the core region and can conveniently be calculated by elasticity theory. This allows treating all the elementary dislocation properties that derive from their self and interaction energies to any desired degree of accuracy. However, the properties of dislocation cores also govern important properties, like dislocation mobility or the selection of the dislocation slip planes. Although our current knowledge of core properties has significantly improved in the past years, it is still far from being just satisfactory. Thus, dislocation theory has not yet been able to bridge the gap between atomic scale properties and mesoscale properties (the mesoscale is understood here as the scale of the defect microstructure). At the mesoscale, the treatment of dislocation ensembles and the spontaneous emergence of dislocation patterns poses several types of problems. It is now understood that plasticity is a dissipative process far from thermal equilibrium. In such conditions, which are beyond the reach of thermodynamics, dislocation patterning is now modeled as a purely dynamic phenomenon. 2897 S. Yip (ed.), Handbook of Materials Modeling, 2897–2901. c 2005 Springer. Printed in the Netherlands.

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Nevertheless, there is no common agreement on the mechanisms that trigger the formation of self-organized dislocation microstructures, or on the way the latter influence the mechanical response. Finally, in many important situations, of fundamental or applied relevance, a connection has to be established with continuum formulations. Indeed, only the latter can compute, through a proper treatment of boundary conditions, the complex field of a dislocated crystal in a specimen with finite dimensions. The objective of what is called multiscale analysis is to reach, through a combination of simulation and modeling, a physically-based picture of the plasticity of crystalline solids. Within this goal, 3-D dislocation dynamics (DD) simulations have been developed in the last decade as a complement to more ancient and well established simulation methods that exist in the domains of solid state physics or solid mechanics.

2.

Connection to the Continuum

At the mesoscale, dislocation core properties can only be incorporated by imposing “local rules”, which translate atomic scale properties into a continuum formulation. Indeed, in the absence of these rules, DD simulations make little sense, especially if they cannot treat thermally activated processes. Since atomistic simulations provide everything but analytical models, one has to make use of intermediate models or approaches to implement this connection. The most important core mechanisms are related to changes in core structure and energy under stress. They involve small energy changes, in the eV range, and are, therefore, assisted by thermal fluctuations. Thus, they can be described with the help of rate equations, for instance semi-empirical Arrhenius forms, forms derived from elastic models for the core structure or Monte– Carlo simulations. This information passing procedure between the atomic and mesoscopic scales presents a major advantage, that of decoupling the time and length scales between the two types of simulations. As thermally activated events are slow events, with a time scale of the order of a fraction of a second in conditions of conventional deformation testing, they cannot be fully treated by molecular dynamics (MD) simulations. The search for saddle point configurations is then performed with the help of various types of energy minimization techniques. In this domain, progress has been slow but continuous, taking advantage of the now possible comparison between ab initio calculations and quantum-based potentials. The most important results recorded in the past years are concerned with the lattice friction (also called Peierls stress), especially in bcc metals. In silicon, the absence of consistency between experiment, continuum modeling and atomistic modeling is still a bit confusing. Studies of the cross-slip mechanism in fcc crystals have confirmed the physical picture yielded by early

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elastic models, but have stopped short of providing stress-dependent activation energies and scaling laws for fcc metals. The question of solute hardening, notably in the presence of lattice friction, is still an open domain. Studies on fast moving dislocations by MD simulations have a potential in relation to dynamic deformation tests (i.e., shock loading). A critical area of investigation is concerned with the study of athermal or weakly thermally assisted events, of which the most important one is dislocation generation in dislocation-free volumes, through either homogeneous or heterogeneous processes (occurring at surfaces, interfaces, voids, crack tips. . .). At present, the lack of local rules in this domain constitutes a serious handicap to mesoscale simulations. The investigation of fundamental dislocation mechanisms at atomic scale and the completion of dislocation theory is, indeed, a challenging task. Although this exercise may seem less seducing than the production of colorful but complex events, it is critical for the progress of multiscale analyses.

3.

DD Simulations: Limitations and Prespective

As exemplified by the variety of solutions adopted by the groups working in the field, implementing the elastic properties of dislocations in DD simulations is no longer a challenge. The microstructure may include defects other than dislocations, e.g., small clusters, precipitates, or grain boundaries. This entails the definition of additional local rules of topological nature, for instance telling dislocations in which conditions they can penetrate precipitates or grain boundaries. However, the elastic theory of dislocations is not providing tractable solutions for the treatment of complex stress fields originating from the conditions of compatible deformation at interfaces. As a result, DD simulations are essentially suited to studies on dislocation dynamics and collective behavior under uniform applied stresses, in crystals of infinite dimensions. Several technical factors (time step, number of interacting segments, dimension of the simulated volume, applied strain rate, available computing power) contribute to the maximum strain that can be achieved by DD simulations and, most of all, to the accuracy of the results. In spite of these limitations, the domain accessible to DD simulations is enormous. Globally, these simulations can be divided into “mass” simulations and “model” simulations, dedicated to the study of one or a few elementary mechanisms. Both are complementary and, as always, cannot be carried out without reference to experiment and current theoretical modeling. A nonexhaustive list of the main topics under investigation or that can be treated in the near future is as follows. – Single crystal studies, particularly in the absence of lattice friction. Such investigations can be very useful to clear up a number of controversies

2900

–

–

–

–

–

4.

L.P. Kubin accumulated over the years on dislocation pattern formation and strain hardening in monotonic and cyclic deformation. In materials with high lattice friction, especially covalent or iono-covalent compounds, similar investigations are feasible provided that enough information is available to construct local rules. Several models, deterministic or stochastic, some of them constructed along the lines of continuum dislocation theory, describe the collective behavior of dislocations in crystals. These models can benefit from a comparison with simulation data. Mechanical properties at medium temperatures (T > 0.3−0.4 Tm , where Tm is the melting temperature) have not been studied up to now by DD simulations. Any attempt to go beyond the current phenomenology of creep mechanisms at intermediate temperatures would be highly welcome. Another topic of practical interest, which has so far been treated only in 2-D, is the strength of alloys containing coherent or semi-coherent particles in the presence of cross-slip or climb. A major unknown in plasticity theory is concerned with the composition rules for obstacles of different strengths and of same or different nature. Only semi-empirical estimates exist in this domain, and mesoscale simulations could be of a great help to initiate modeling in a few simple cases. The interaction of dislocations with small clusters, in practice small prismatic loops, is involved in two important traditional domains: one is the instabilities and strain localizations observed during the plastic deformation of irradiated materials. The other is dislocation patterning in cyclic deformation. The investigation of size effects is another very seducing topic, although it has to be approached with care in the absence of a rigorous solution for the local fields. The influence of reduced dimensionality on line tension effects and dislocation mean-free paths is now being investigated. It implies defining new local rules, for instance for the interactions between dislocations and interfaces.

Connection to Continuum Mechanical Aspects

Simulations that simultaneously treat discrete dislocations and solve the boundary value problem have an enormous potential. In fundamental terms, they allow introducing length scales into the continuum framework. They are particularly suited for the study of size effects in nanostructured materials, of plasticity under strain gradients, of materials with complex microstructure, of complex modes of loading, static or dynamic, and fracture processes.

3-D mesoscale plasticity and its connections to other scales

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Two methods are being developed to establish this connection. One is based on the combination of DD simulations, which treat the plastic strains resulting from dislocation motion, with finite element (FE) codes, which solve the elastic fields. The existing solutions only differ in the distribution of tasks between the two codes. A variant consists in applying the boundary conditions via a Green function methods when the latter can be computed. The phase field method, which was initially developed to simultaneously treat stress and concentration fields, is now being applied to dislocation problems. At present, it is perhaps too early to assess the potential of this method. Its major advantage resides in the fact that it incorporates a “chemical” dimension that was up to now missing in mesoscale simulations. Another type of connection is possible, which is much less heavy. It simply consists in constructing physically based constitutive formulations for the evolution of dislocation densities and implementing them in a FE code that cares of the boundary value problem. Although this procedure is not really new, it takes advantage of the recent development of mesoscale simulations, for checking or developing existing models, and of continuum crystal plasticity codes, i.e., codes that include the slip geometry. This methodology is also very attractive for two reasons: it allows making use of a substantial existing body of knowledge on constitutive formulations, and it yields access to large strain behavior. How far it can be developed will depend on its ability to account for the spatial organization of the microstructures.

5.

Conclusion

At present, the development of DD simulations is governed in the one hand by the input information generated at atomic scale and, on the other hand, by the strain limit fixed by the available computing power. Within these bounds, DD simulations are still far from having exhausted their potential. The connection to continuum mechanical aspects constitutes the final step toward a physically-based modeling of plasticity. It will probably take a major importance in the coming years. Indeed, it is essential to keep in mind that this has been from the beginning the main objective of dislocation theory. Mesoscale simulations perhaps still have to confirm that they can go beyond providing textbook illustrations and are able to bring major useful information to materials scientists. This objective will, however, not be reached without a strong synergy with theoretical modeling. Current simulations are now able to reproduce known patterns of behavior, but reproducing does not necessarily means understanding.

Perspective 34 SIMULATING FLUID AND SOLID PARTICLES AND CONTINUA WITH SPH AND SPAM Wm.G. Hoover Department of Applied Science, University of California at Davis/Livermore and Lawrence Livermore National Laboratory, Livermore, California, 94551-7808

1.

SPH and SPAM

In my own research career I have been primarily interested in the statistical mechanics of nonequilibrium systems of particles, stressing new techniques for undertaking and understanding computer simulations [1, 2]. The blind alleys toward which molecular dynamics naturally leads provide a compensating appreciation of continuum mechanics, with its length scale more appropriate to everyday experiences. For me, it was a pleasure to learn that the two approaches, microsopic and macroscopic, can be usefully combined, using ideas due to Lucy and Monaghan. About 25 years ago these men independently introduced a new numerical method for simulating continuum problems. They used particles to represent extended macroscopic material elements. Though their method is not at all restricted to fluids, and certainly not to water, Monaghan has consistently adopted the name “smooth-particle hydrodynamics” for it. In the work I have carried out during the last decade I have used instead the alternative “Smooth particle applied mechanics,” (SPAM) for short, thinking it more apt, particularly for solid-phase applications. A Google search of the internet turns up hundreds of references to sph and SPAM. The interested reader could begin with my website at http:// williamhoover.info or could work backward from Carol’s and my review [3]. SPAM defines the local average value (at any location r) of any particle quantity Fi as a quotient of weighted sums:

F(r) ≡ i

Fi m i w(|r − ri| ) . i m i w(|r − ri| ) 2903

S. Yip (ed.), Handbook of Materials Modeling, 2903–2906. c 2005 Springer. Printed in the Netherlands.

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All particles lying within a distance h of the location r (h is the “range” of w) are to be included in the sums. The “weight function” w(|r| < h) is short-ranged and must be continuously twice differentiable, resembling a foreshortened Gaussian function. Lucy’s choice for w is the simplest polynomial satisfying these conditions: w(x < 1) ∝ (1 − x)3 (1 + 3x) = 1 − 6x 2 + 8x 3 − 3x 4 ;

x ≡ |r|/ h.

The implied multiplicative proportionality constant depends upon the dimensionality of the problem (1, 2, or 3) and is to be chosen such that the spatial integral of w is unity. In applying SPAM tofluids and solids a fixed mass m i is associated with the ith particle so that i m i w(|r − ri |) is the mass density ρ(r) while the density at the location of the ith particle is a sum over nearby particle pairs (and including the term i = j ): ρi ≡

m j w(|ri − r j |).

j

The related identities, F(r)ρ(r) ≡

Fi m i w(|r − ri |) −→

i

∇r [F(r)ρ(r)] ≡

Fi m i ∇r w(|r − ri |),

i

show off the method’s key advantage. First-order or second-order spatial derivatives (of velocity, temperature, stress, . . . ) can all be evaluated from simple sums involving w and w . Only the polynomial w is affected by the gradient operation. A consistent application of these ideas leads to new particle forms for the continuum equations of motion,

r¨i = υi = −m

(P/ρ )i + (P/ρ ) j · ∇i wi j , 2

2

i

and for the particle representations of the continuum energy equation,

e˙i ≡ −

(m/2)[(P/ρ 2 )i + (P/ρ 2 ) j ] :(υ j − υ˙ i ) ∇i wi j

j

−

m[(Q/ρ )i + (Q/ρ ) j ]·∇i wi j . 2

2

j

Note that the special case P ∝ ρ 2 gives particle trajectories isomorphic to those found with molecular dynamics (with w playing the rˆole of a pair potential).

Simulating fluid and solid particles and continua with SPH and SPAM 2905 In a SPAM particle simulation the pressure tensor P and the heat-flux vector Q follow from the underlying continuum constitutive relations (such as Newtonian viscosity and Fourier conductivity). A SPAM simulation proceeds by solving the differential equations for all the particles’ {˙r , v, ˙ e} ˙ subject to assumed initial and boundary conditions. (It must be emphasized that developing properly-formed algorithms for boundary conditions is a high art form.) There is a tremendous literature on particular fluid and solid applications, including the effects of analogs of the artificial viscosity and artificial heat conductivity encountered in the more usual simulations using regular meshes. An interesting and simple problem, with philosophical connections to fundamental statistical mechanics, is Gibbs’ (time-reversible!) expansion problem, in which a fluid expands (irreversibly!) to fill a larger container [4, 5]. See Fig. 1. Smooth particles have an advantage, for multiscale physics, in that their length scale h is arbitrary (it can also be a function of time or direction in space), so that “big” zones can be linked to “smaller” ones, even to zones of molecular dimensions. The particles also promote rezoning in a natural way Because their basic attributes include mass, momentum, and energy, particles

Figure 1. Fourfold free expansion of a 2D gas using smooth-particle applied mechanics. The particles themselves are shown, as well as are contours of density and kinetic energy density, computed using the smooth-particle weight functions. The separation between the white and black regions in the density and kinetic energy plots is the contour characterizing the corresponding mean value. The simulations show rapid equilibration, on a timescale of a few sound traversal times. In the Figure τ is the sound-traversal time for the (spatially-periodic) system.

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can easily be combined or subdivided in such a way as to conserve all three variables. Hybrid atomistic-continuum simulations based on these ideas are another useful application area.

References [1] W.G. Hoover, “Computational statistical mechanics,” Elsevier, New York, 1991. [2] Wm.G. Hoover, “Time reversibility, computer simulation, and chaos,” World Scientific, Singapore, 1999 and 2001. [3] Wm.G. Hoover and C.G. Hoover, “Links between microscopic and macroscopic fluid mechanics,” Mol. Phys., 101, 1559–1573, 2003. [4] Wm.G. Hoover and H.A. Posch, “Entropy increase in confined free expansions via molecular dynamics and smooth particle applied mechanics,” Phys. Rev. E, 59, 1770–1776, 1999. [5] Wm.G. Hoover, H.A. Posch, V.M. Castillo, and C.G. Hoover, “Computer simulation of irrversible expansions via molecular dynamics, smooth particle applied mechanics, eulerian, and Lagrangian continuum mechanics,” J. Stat. Phys., 100, 313–326, 2000.

Perspective 35 MODELING OF COMPLEX POLYMERS AND PROCESSES Tadeusz Pakula Max Planck Institute for Polymer Research, Mainz, Germany and Department of Molecular Physics, Technical University, Lodz, Poland

1.

Can Complex Macromolecular Architectures Lead to New Properties?

Creating new macromolecular architectures with increasing complexity can constitute a challenge for synthetic chemists but can additionally be justified if it would result in new material properties. For example, joining monomeric units into linear polymer chains, results in a dramatic change of properties of products with respect to properties of substrates. Whereas, a monomer in bulk can usually be only liquid-like or solid (e.g., glassy), the polymer can additionally exhibit a rubbery state with properties, which make these materials extraordinary for a large number of applications. This new state is due to the very slow relaxation of polymer chains in comparison with a fast motion of the monomers, especially, when the chains become so long that they can entangle in a bulk melt. The dynamic mechanical characteristics indicate a single relaxation in the monomer system (Fig. 1a) in contrast to the two characteristic relaxations in the polymer (Fig. 1b). The rubbery state of the polymer extends in the time scale between the segmental (monomer) and the chain relaxation times and is controlled by a number of parameters related to the polymer structure. The most important among these parameters is the chain length determining the ratio of the two relaxation rates. In the rubbery state, the material is much softer than in the solid state. If expressed by the real part of the modulus, the typical glassy state elasticity is of the order of 109 Pa and higher, whereas the rubber like elasticity in bulk polymers is of the order of 105 –106 Pa. It has been demonstrated, recently, that some highly branched macromolecular structures can lead to considerably different properties than these obtained 2907 S. Yip (ed.), Handbook of Materials Modeling, 2907–2915. c 2005 Springer. Printed in the Netherlands.

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(a)

109

107

liquid

106

glassy state

105

T = 233K

(c)

104 103 102 108

109

1010 1011 1012 1013 1014

ω [rad/s.]

(b)

109

G', G" [Pa]

109

Isobutylene MW=186

G', G" [Pa]

G', G" [Pa]

108

polymeric rubber

107

106

side chain relaxation

103

105

103

101

G' G''

brush relaxation 10

linear PnBA

⫺4

⫺2

10

0

10

2

10

4

10

⫺7

10

⫺4

10

T = 254K ⫺1

102

105

ω [rad/s.]

T = 254K

10

segmental relaxation

gel

super-soft rubber

6

10

ω [rad/s]

Figure 1. A comparison of mechanical behavior of bulk systems with various molecular complexity: (a) low molecular liquid, (b) linear entangled polymer and (c) molecular brush with PnBA side chains.

by linking monomeric units into linear chains [1, 2]. Examples consider dynamic behavior of multiarm stars in the melt [3], the melts of brush-like macromolecules [2] and hairy micelles dispersed in linear polymer matrices. In all these systems, the more complex structures extend the spectrum of the relaxations by a third well distinguishable process with the longest relaxation time, which is interpreted as related to slow structural rearrangements in the structured system. When the new process is slow enough, it can create a new elastic plateau with the plateau modulus (102 –103 Pa) by orders of magnitude lower than that characteristic for the conventional polymeric rubbery state (Fig. 1c). Appropriate cross-linking of such structured systems based on macromolecules with complex architectures can lead to bulk super soft elastomers for which we expect a broad range of application possibilities [2].

2.

Where are the Main Obstacles? What is Predictable?

In contrast to the situation we have in the case of linear polymers, an understanding, a theoretical description and consequently a predictability of

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behavior of the bulk systems with macromolecules having complex architecture are in a stage in which the problems are more or less recognized but because of the increased complexity the methods and solution are missing. Tackling the problems of complexity in macromolecular systems in order to provide new ways advancing our possibilities in predicting controlling and analyzing the complex macromolecular systems is, therefore, required. This concerns both natural and synthetic systems and consideration of the later along the whole pathway from substrates to products i.e., from monomers to bulk materials. The macromolecular chemistry is receiving recently a special attention because of development of new synthetic methods and skills which make possible creation of macromolecules with a complexity and precision comparable with that met in biological systems. This includes macromolecules having complex architectures such as multiarm stars, hyperbranched polymers, dendrimers, comb-like or dendronized polymers, rings, catenanes as well as various variants of these topologies made additionally more complex by various distributions of interacting fragments such as functional groups, incompatible blocks or intramolecular composition distributions of chemically different units (Fig. 2). Synthesis of such complex macromolecules requires well controlled processes for which a detailed understanding and possibility of description or modeling is of crucial importance. Existing theoretical methods of description of synthetic processes in macromolecular systems are not sufficiently developed. The problems result from neglecting that for bond formation a presence homopolymers linear

cyclic

star

comb

microgel

network

block copolymers

Figure 2. Some examples of complex macromolecular architectures differing by topology and intramolecular composition distributions.

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of reacting sites and catalysts within the reaction volume is necessary. This condition can only be reasonably considered in spatial models of reacting systems in which substrates and products coexist in space interacting mutually under reaction progress dependent conditions. Therefore, one of the important challenges for the future should be a development of effective tools for modeling of complex synthetic processes in 3D space under controlled conditions. A lot of systems have been developed in which a specific molecular architecture leads to self-organization of molecules to various supramolecular structures. In spite of a considerable progress in this field still many questions such as, what and how to synthesize in order to achieve desired functions or properties, remain open. Also in natural systems the mechanisms of adopting certain structural states and fulfilling certain functions are for most systems not understood. In all these systems, complexity and related physical effects play an important role and can not be considered within the existing theoretical tools. Experimental analysis of such processes and of products is difficult and in many cases leads to misleading results because of unknown effects of the complex macromolecules on the measured quantities (e.g., size exclusion chromatography). On the other hand, it can be expected that appropriate modeling can provide information concerning kinetics of such processes and properties of the products. They can be characterized, for example, by molecular weights, molecular weight distributions, compositions in the case of copolymers as well as composition heterogeneities and distributions, etc. When the complex molecules are obtained, the questions arise: which properties they have, which functions they can fulfill or what they are good for? The further aim of our efforts should, therefore, concern an analysis of structure and dynamics of the complex macromolecules in order to answer the above questions. The analysis should consider single complex macromolecules, spatially confined aggregates, as well as, bulk systems in which specific spatial arrangements, self-organization and formation of heterophases influence the behavior. The later structures constitute usually hierarchical supramolecular architectures in which the structural and dynamic complexity requires new methodological developments in order to be analyzed. Such structures can extend over many orders of magnitude in the size scale. The broad size range involves a broad variety of related relaxations which contribute to the dynamics extending over an extremely broad time range. The dynamic spectrum of such materials becomes very important for understanding the correlation between parameters of the molecular and supramolecular structures on one hand, and their specific functions or macroscopic properties on the other hand [2]. Analysis of these correlations appears to be extremely difficult because of complexity of the systems. Experimentally, it usually requires application of many techniques for characterization of both the structure

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and the dynamics. It often, however, leaves many questions open because of limitations in resolution, accuracy, sensitivity or selectivity of the experimental methods. The systems are usually to complex to be considered by existing theoretical tools. A hope is in computer modeling which due to rapid software and hardware developments becomes a valuable research tool providing a lot of details concerning structure and dynamics in such complex macromolecular systems [1].

3.

Some Perspectives for a Progress

The other important scientific objective of the future efforts should, therefore, be a methodological improvement in solving problems which appear in complex systems of a large number of molecular units interacting strongly or subjected to strong external constraints. Two aspects are considered as crucial for understanding such systems: hierarchy and cooperativity. We can take advantage of recent developments made in modeling of dynamics in complex molecular systems. A new liquid model – the Dynamic Lattice Liquid model [4] – could be used. The model seems to solve the long standing problem of cooperativity in dense complex molecular or macromolecular systems. It provides a microscopic picture of cooperative molecular rearrangements (cooperative loops) resulting from system continuity under conditions of excluded volume and dense packing of molecules (Fig. 3). The rearrangements are considered as taking place in systems with fluctuating density and with rates dependent on thermal activation barriers which depend on and fluctuate with the local density (intermolecular distances). Dependencies of this kind

single molecule trajectory

collective rearrangement

Figure 3. Illustration of a dynamic simplification of a rearrangement of beads in a dense system with all lattice sites occupied. The rearrangement consists in cooperative displacement of system elements along closed trajectories within which each element replaces one nearest neighbor.

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may include chemical specificity of systems. It has been shown, that the model is able to reproduce the extreme cases of behavior of so called fragile and strong liquids, as well as, various cases filling the gap between these extremes [4]. It reproduces also effects of pressure on the molecular dynamics. The model considering a detailed microscopic behavior become a basis for parallel computer simulation algorithms and in this form can be applied for both liquids and polymers of various complexities [1, 4, 5]. It constitutes new chances for very efficient molecular modeling. Computer modeling is nowadays one of the most important sources of information about details concerning behavior of complex molecular and macromolecular systems at the molecular or atomic scale. Main obstacles in exploring possibilities in this field concern limitations in computational speed and in related limits of spatial resolution of considered models. Possibilities to overcome these problems are both in development of appropriate efficient software, as well as, in development of suitable super fast hardware. Recent modeling techniques allow simulating systems over time intervals not exceeding six orders of magnitude and with spatial resolution still far from reaching two orders of magnitude in the 1D size scale. In most problems concerning material development on nano- and micro-meter size scales, as well as, in biological systems on the size scale of cells, the requirements for the spatial resolution of models are higher than the resolutions accessible recently in modeling of molecular or macromolecular dynamics and organization. Tests of the models based on cooperative rearrangements on the sequentially working computing hardware indicted already a high potential of this type of models for predicting behavior of the complex molecular systems (CMA algorithms) [1]. Based on this experience, we suggest a development and construction of a multipurpose fast logical unit which can serve as a basic element of a parallel computing system realizing the 3D architecture and the DLL dynamics. A building block of such DLL system will be a single chip that integrates multiple processors, each representing single site in the model, and a connectivity and communication logic corresponding to the DLL architecture. This should allow to build a full system by replicating such a chip. A prototype of such a system should allow estimations concerning finite size limits of the DLL machine and its computational efficiency, as well as the technological problems related to energy supply, thermal stabilization and spatial requirements and so on. Schematic illustration of the pathway between the DLL model and the very innovative and technologically pretentious computing system is shown in Fig. 4. Realization of such a system is of a great challenge and when successful would constitute a unique example of a modeling system having the architecture directly corresponding to the structure of modeled system. This would allow using of highest level of parallelism of computation and logic in a similar way as in real systems. To our knowledge such computing systems do not

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DYNAMIC LATTICE LIQUID

Processor controlling single lattice site

3D network of logical units corresponding to structure of modeled system CDLL

Basic Cell of the DLL

1 000 000

100

100

DLL system 100 CDLL

Figure 4. Illustration of a pathway between the DLL model and a DLL massively parallel computing system.

exist yet. It is worth while to mention that the systems will constitute a 3D network with 12 coordinated knots (sites) which probably will make the system suitable for modeling of complex networks such as brain system as well. We postulate future research to be related both to computational modeling of processes in complex macromolecular systems but embedded in, supported by and confronted with experimental investigations of related systems. Many, more and more complex macromolecular systems are synthesized nowadays with arguments that they potentially can show interesting properties. This, however, can only be tested after often very difficult and time consuming synthetic and characterization efforts. Some modeling examples demonstrate, however, another access to this kind of tests providing molecular modeling tools having predictive potential both in the synthetic and in characterization areas [1, 2]. The main properties towards which the predictive recognition postulated here is directed are the mechanical properties dependent on the macromolecular dynamics specific for the complex macromolecular systems. It has been recently discovered (MPG and CMU) that such systems can exhibit super soft elastic states, the presence of which is controlled by the details of the complex macromolecular architecture and related dynamics [2]. An improvement of modeling methods should provide answers to questions which structures

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under which conditions can lead to such states which can become a basis of useful applications. On the other hand, the development discussed should be considered as giving bases for much broader further investigations of other properties, other systems and other processes.

4.

Conclusions

The models based on the cooperative dynamics allow to expect that a real qualitative improvement in the field of macromolecular modeling can be achieved. Main advantages of these methods are: physically reasonable simplifications of the dynamics and structures, applicability to dense systems, flexibility in representation of various molecular topologies and high computational efficiency [1]. The cooperativity understood as concerted action of a number of system elements which results in realization of effects remaining strongly restricted or impossible to achieve in actions of individual elements is responsible for the high efficiency of the related computational methods. The simplicity of the cooperative phenomena in the suggested models lies in recognition of the essential conditions under which in a many body system the cooperative actions can take place [4]. The concepts of the DLL model can directly serve as basis for a parallel special purpose computer system realizing super fast modeling which can open new possibilities to improve spatial resolution of models by decoupling it from limitations imposed by computational speed. The building block of such a system will be a chip that integrates multiple processors each corresponding to the elemental site and the communication logic corresponding to the architecture of the DLL system. We consider it as a perspective which might be very helpful from the point of view of the material science and technology and which may allow in future simulations of systems comparable in sizes and complexity with biological cells, still represented with molecular resolution.

References [1] T. Pakula, “Simulations on the completely occupied lattice,” In: Simulation Methods for Polymers, M.J. Kotelyanskii and D.N. Theodorou, Marcel-Dekker, New York, Ch. 5. pp. 147–176, 2004. [2] T. Pakula, P. Minkin, and K. Matyjaszewski, “Polymers, particles, and surfaces with hairy coatings: Synthesis, structure, dynamics, and resulting properties,” In: K. Matyjaszewski (ed.), Advances in Controlled/Living Radical Polymerization, ACS Symp. Series 854, pp. 366–382, 2003. [3] T. Pakula, D. Vlassopoulos, G. Fytas, and J. Roovers, “Structure and dynamics of melts of multiarm polymer stars,” Macromolecules, 31, 8931–8940, 1998.

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[4] T. Pakula, “Collective dynamics in simple and supercooled polymer liquids,” Molecular Liquids, 86, 109–121, 2000. [5] P. Polanowski and T. Pakula, “Studies of mobility, interdiffusion, and self-diffusion in two-component mixtures using the dynamic lattice liquid model,” J. Chem. Phys., 118, 11139–11146, 2003.

Perspective 36 LIQUID AND GLASSY WATER: TWO MATERIALS OF INTERDISCIPLINARY INTEREST H. Eugene Stanley Center for Polymer Studies and Department of Physics Boston, University, Boston, MA 02215, USA

1.

Puzzling Behavior of Liquid Water

We can superheat water above its boiling temperature and supercool it below its freezing temperature, down to approximately −40 ◦ C, below which water inevitably crystallizes. In this deeply supercooled region, strange things happen: response functions and transport functions appear as if they might diverge to infinity at a temperature of about −45 ◦ C. These experiments were pioneered by Angell and co-workers over the past 30 years [1–4]. Down in the glassy region of water, additional strange things happen, e.g., there is not just one glassy phase [1]. Rather, just as there is more than one polymorph of crystalline water, so also there appears to be more than one polyamorph of glassy water. The first clear indication of this was a discovery of Mishima in 1985: at low pressure there is one form, called low-density amorphous (LDA) ice [5], while at high pressure Mishima discovered a new form, called highdensity amorphous (HDA) ice [6]. The volume discontinuity separating these two phases is comparable to the volume discontinuity separating low-density and high-density polymorphs of crystalline ice, 25–35 percent [7, 8]. In 1992, Poole and co-workers hypothesized that the first-order transition line separating two glassy states of water does not terminate when it reaches the no-man’s land (the region of the phase diagram where only crystalline ice is found experimentally), but extends into it [9]. If experiments could avoid the no-man’s land connecting the supercooled liquid with the glass, then the LDA–HDA first order transition line would continue into the liquid phase. This first-order liquid-liquid (LL) phase transition line separates two phases of liquid—high-density liquid (HDL) and low-density liquid (LDL)—which 2917 S. Yip (ed.), Handbook of Materials Modeling, 2917–2922. c 2005 Springer. Printed in the Netherlands.

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are the precise analogs of the two amorphous solids LDA and HDA. Like essentially all first-order transition lines, the LL transition line between noncrystalline phases must terminate in a critical point. Above the critical point is an analytic extension of the LL phase transition line. Called the Widom line, this extension exhibits apparent singularities—i.e., if the system approaches the Widom line, then thermodynamic response functions appear to diverge to infinity until the system is extremely close, when the functions will round off and ultimately remain finite—as seen in adiabatic compressibility data [10].

2.

Plausibility Arguments

That a LL phase transition exists is at least plausible. Liquid water is a tetra-hedral liquid, and two water tetrahedra can approach each other together in many different ways. One way is coplanar, as in ordinary hexagonal ice Ih —creating a “static heterogeneity” with a local density not far from that of ordinary ice, about 0.9 g/cm3 . A second way is altogether different: one of the two tetrahedra is rotated 90◦ , resulting in a closer distance where the minimum of potential energy occurs, and hence a static heterogeneity with a local density substantially larger (by about 30%) than that of ordinary ice [11]. In fact, this rotated configuration occurs in solid crystalline water (“ice VI”), which occurs at very high pressure. In liquids close to the freezing temperature, there are heterogeneities with local order resembling that of the nearby crystalline phases. Not surprisingly, then, in water at low pressure, there are more heterogeneities that have icelike entropy (local order) and specific volume, while at high pressure there are more heterogeneities that have a Ice VI-like entropy and specific volume. The potential that represents the relative orientations of two water tetrahedra has two wells: a deeper, “high-volume, low-entropy” well corresponding to LDL and a shallower, “low-volume, high-entropy” well corresponding to HDL. Note that LDL has a higher specific volume and a lower entropy. Therefore when water cools, each molecule must decide how to partition itself between these two minima. The specific volume fluctuations increase because of these two possibilities. The entropy fluctuations also increase, and the cross-fluctuations of volume and entropy have a negative contribution—i.e., high volume corresponds to low entropy so that the coefficient of thermal expansion, proportional to these cross fluctuations, can become negative. The possibility that these static heterogeneities gradually shift their balance between low-density and high-density as pressure increases is plausible, but need not correspond to a genuine phase transition. There is no inherent reason why these heterogeneities need to “condense” into a phase, and the first guess might be that they do not condense – what is now called

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the singularity free hypothesis [12, 13]. However if we reason by analogy with the gas-liquid transition, then there is one reason to believe that they will condense. This is related to the fact that a permanent gas is impossible so long as there is a weak attraction, no matter how weak. If such a weak attraction has an energy scale e, at low enough temperature T the ratio /T will become large enough to influence the Boltzmann factor sufficiently that the system will condense. For example, if a a lattice-gas fluid (ex: single-well potential) with = 1 condenses below Tc = 1, then if = 0.001 one anticipates condensation below Tc ≈ 0.001. For this reason, one anticipates that at low enough temperature the one-component liquid would condense into a low density liquid corresponding to a deeper potential well.

3.

Simulations

That the static heterogeneities should condense at sufficiently low temperature is found in simulations using a wide range of molecular potentials, ranging from “overstructured” potentials such as ST2 to “understructured” potentials like SPC/E. Recent work has focused on the newest of all water potentials, Tip5p [14]. Regardless of potential used, all results seem to be consistent with the LL phase transition hypothesis [2].

4.

Experiments

Experimental data are consistent with the LL phase transition hypothesis. The volume fluctuations are proportional to the compressibility, and this compressibility is a spectacularly anomalous function. Below 46 ◦ C, the compressibility start to increase as the temperature is lowered. This phenomenon is no longer counterintuitive if the double-well potential is correct. Similarly, below 35 ◦ C the entropy fluctuations, which correspond to the specific heat, start to increase. Finally, consider the coefficient of thermal expansion, which is proportional to the product of the entropy and volume fluctuations. This is positive in a typical liquid because large entropy and large volume go together, but for water this cross-correlation function has a negative contribution—and as we lower the temperature this contribution gets larger and larger until we reach 4 ◦ C, at which point the coefficient of thermal expansion passes through zero. The experimental work of Angell and collaborators shows apparent singularities when experimental data are extrapolated into the No-Man’s Land. Mishima measured the metastable phase transition lines of ice polymorphs and found that the slopes of these lines exhibited sharp kinks in the vicinity of the hypothesized liquid-liquid phase transition line as predicted by extrapolation

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[15, 16]. The nature of these kinks can be explained if we take into account that an ice polymorph must melt into a metastable liquid before it can recrystallize into a different polymorph. By the Clausius–Clapeyron relation the slope of that first-order metastable melting line must be equal to the ratio of the entropy change divided by the volume change of the two phases that coexist. In one phase the coexistence is always the high-pressure polymorph of ice. The other phase is either HDL or LDL. The volumes and entropies of those two liquids are different, and therefore as the first-order solid-liquid phase transition line crosses the hypothesized LL phase transition line, the slope changes. The Gibbs potential of two phases that coexist along a first-order transition line must be equal. We already know the Gibbs potential of all the polymorphs of ice, so we know, experimentally, the Gibbs potential of the LDL and the HDL. From the Gibbs potential of any substance one can obtain, by differentiation, the volume. Thus, if we know the Gibbs potential as a function of temperature and pressure, we know the volume as a function of temperature and pressure—which is called the equation of state. In this way Mishima and Stanley were able to find an experimental equation of state for water deep inside the no-man’s land. This is, of course, not quite the same as actually measuring the densities of two liquids coexisting at the LL phase transition line since the Mishima experiments concerned metastable melting lines in which the Gibbs potentials of the two phases are not necessarily equal to each other. Very recently, Reichert and collaborators [17] discovered experimentally a HDL under conditions outside the no-man’s land. They studied the thin, quasi-liquid layer between ice Ih and a solid substrate (amorphous SiO2 ). Using a clever experimental technique at Grenoble, they were able to measure the density and found 1.17 g/cm3 , the density of HDA.

5.

Discussion

The LL phase transition hypothesis does not fully answer the question “what matters?”—i.e., it does not tell us which liquids should exhibit LL phase transitions and which should not. It has been conjectured that it is local tetrahedral geometry of water that matters, since a tetrahedral local geometry leads to static heterogeneities which lead to a LL phase transition [18]. But then what about other tetrahedral liquids? Phosphorus [19] and SiO2 [20] have a local tetrahedral geometry, and experimental evidence supports the LL phase transition hypothesis. There is recent evidence for a LL phase transition in silicon, which is also a tetrahedral liquid [21]. It has been argued that LL phase transitions are associated with liquids possessing a line in the T-P phase diagram at which the density achieves a maximum [22–24].

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In summary, the presence of a local tetrahedral geometry leads to two distinct forms of static heterogeneities (or “local order”) differing in specific volume and entropy, with the specific volume and entropy anticorrelated. This fact gives rise to anomalous fluctuations in compressibility, specific heat, and the coefficient of thermal expansion. The hypothesis that at low enough temperatures these small regions of local order condense into two separate phases (LDA and HDA) is supported by simulations, but remains an open question experimentally.

References [1] C.A. Angell, “Amorphous water,” Ann. Rev. Chem., 55, 559–583, 2004. [2] P.G. Debenedetti, “Supercooled and glassy water,” J. Phys.: Condens. Matter, 15, R1669–R1726, 2003. [3] P.G. Debenedetti and H.E. Stanley, “The physics of supercooled and glassy water,” Physics Today, 56[6], 40–46, 2003. [4] O. Mishima and H.E. Stanley, “The relationship between liquid, supercooled, and glassy water,” Nature, 396, 329–335, 1998. [5] P. Briigeller and E. Mayer, “Complete vitrification in pure liquid water and dilute aqueous solutions,” Nature, 288, 569–571, 1980. [6] O. Mishima, L.D. Calvert, and E. Whalley, “An apparently first-order transition between two amorphous phases of ice induced by pressure,” Nature, 314, 76–78, 1985. [7] O. Mishima, “Reversible first-order transition between two H2 O amorphs at −0.2 GPa and 135 K,” J. Chem. Phys., 100, 5910–5912, 1994. [8] O. Mishima, “Relationship between melting and amorphization of ice,” Nature, 384, 546–549, 1996. [9] P.H. Poole, F. Sciortino, U. Essmann, and H.E. Stanley, “Phase behaviour of metastable water,” Nature, 360, 324–328, 1992. [10] E. Trinh and R.E. Apfel, “Sound velocity of supercooled water down to −33 ◦ C using acoustic levitation,” J. Chem. Phys., 72, 6731–6735, 1980. [11] M. Canpolat, F.W. Starr, M.R. Sadr-Lahijany et al., “Local structural heterogeneities in liquid water under pressure,” Chem. Phys. Lett., 294, 9–12, 1998. [12] H.E. Stanley and J. Teixeira, “Interpretation of the unusual behavior of H2 O and D2 O at low temperatures: tests of a percolation model,” J. Chem. Phys., 73, 3404– 3422, 1980. [13] S. Sastry, P. Debenedetti, F. Sciortino, and H.E. Stanley, “Singularity-free interpretation of the thermodynamics of supercooled water,” Phys. Rev. E, 53, 6144–6154, 1996. [14] M. Yamada, S. Mossa, H.E. Stanley, and F. Sciortino, “Interplay between timetemperature-transformation and the liquid-liquid phase transition in water,” Phys. Rev. Lett., 88, 195701, 2002. [15] O. Mishima and H.E. Stanley, “Decompression-induced melting of ice IV and the liquid-liquid transition in water,” Nature, 392, 164–168, 1998. [16] O. Mishima and Y. Suzuki, “Vitrification of emulsified liquid water under pressure,” J. Chem. Phys., 115, 4199–4202, 2001. [17] S. Engemann, H. Reichert, H. Dosch, J. Bilgram, V. Honkimaki, and A. Snigirev, “Interfacial melting of ice in contact with SiO2 , Phys. Rev. Lett., 92, 205 701, 2004.

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[18] H.E. Stanley, S.V. Buldyrev, N. Giovambattista, E. La Nave, A. Scala, F. Sciortino, and F.W. Starr, “Statistical physics and liquid water: ‘what matters’,” Physica A, 306, 230–242, 2002. [19] Y. Katayama, T. Mizutani, W. Utsumi, O. Shimomura, M. Yamakata, and K.-i. Funakoshi, “A first-order liquid-liquid phase transition in phosphorus,” Nature, 403, 170–173, 2000. [20] P.H. Poole, M. Hemmati, and C.A. Angell, “Comparison of thermodynamic properties of simulated liquid silica and water,” Phys. Rev. Lett., 79, 2281–2284, 1997. [21] S. Sastry and C.A. Angell, “Liquid–liquid phase transition in supercooled silicon,” Nature Materials, 2, 739–743, 2003. [22] F. Sciortino, E. La Nave, and P. Tartaglia, “Physics of the liquid–liquid critical point,” Phys. Rev. Lett., 91, 155701, 2003. [23] G. Franzese and H.E. Stanley, “A theory for discriminating the mechanism responsible for the water density anomaly,” Physica A, 314, 508–513, 2002. [24] G. Franzese, G. Malescio, A. Skibinsky, S.V. Buldyrev, and H.E. Stanley, “Generic mechanism for generating a liquid-liquid phase transition,” Nature, 409, 692–695, 2001.

Perspective 37 MATERIAL SCIENCE OF CARBON Wesley P. Hoffman Air Force Research Laboratory, Edwards, CA, USA

Carbon is a ubiquitous material that is essential for the functioning of modern society. Because carbon can exist in a multitude of forms, it can be tailored to possess practically any property that might be required for a specific application. The list of applications is very extensive and includes: aircraft brakes, electrodes, high temperature molds, rocket nozzles and exit cones, tires, ink, nuclear reactors and fuel particles, filters, prosthetics, batteries and fuel cells, airplanes, and sporting equipment. The different forms of carbon arise from the fact that carbon exists in three very different crystalline forms (allotropes) with a variety of crystallite sizes, different degrees of purity and density, as well as various degrees of crystalline perfection. These allotropes are possible because carbon has four valence electrons and is able to form different kinds of bonds with other carbon atoms. For example, diamond with a covalently-bonded face-centered cubic structure can exist as a naturally–formed single crystal as large as 200 g. Diamond, the hardest material known to man, is also able to be made synthetically by a variety of processes. For example, high pressure anvils can be utilized to produce relatively small single crystals while a vapor-phase process such as chemical vapor deposition (CVD) is employed to deposit crystalline and amorphous coatings having grain sizes on the micron scale, and a variety of degrees of crystallite orientations. Synthetic diamonds in all forms are used as hard scratch-resistant coatings and tool coatings for grinding, cutting, drilling and wire drawing. Other applications include heat sinks and optical windows among others. The most abundant forms of carbon exist as various forms of the allotrope hexagonal graphite. The perfect crystalline structure of graphite is a hexagonal layered structure in which the atoms in each layer are covalently-bonded while the graphene layers are held together by weaker Van Der Waals forces. This difference in bonding is what is responsible for the great anisotropy in mechanical, thermal, electrical and electronic properties. 2923 S. Yip (ed.), Handbook of Materials Modeling, 2923–2928. c 2005 Springer. Printed in the Netherlands.

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0.2456 nm

0.3354 nm

Diamond Structure

Hexagonal Graphite Structure

With the relative recent discovery of the nano-forms of carbon in the last two decades, the range of properties that carbon can possess and the gamut of potential applications has greatly increased. The linear portions of these molecules are simply rolled up graphene layers, while the curved portions consist of graphite hexagons in contact with pentagons. Although commercial applications for both bucky balls and carbon nanotubes are not well defined at the time of this writing, the high aspect ratio of nanotubes along with the fact that they are stronger ( 63 GPa ) and stiffer (∼1000 GPa ) than any other known material, means that the potential for these materials is great.

BUCKYBALL

NANOTUBE

Adding to the complexity of understanding and modeling carbon is the fact that, in its various forms, it rarely exists in a perfect single crystalline state. For example, perfect graphitic structure only exists in the various forms of natural graphite flakes and graphitizable carbons, which are carbons formed from a gas or liquid phase process. An example of a graphitizable carbon would be highly oriented pyrolytic graphite (HOPG), which is formed by depositing one atom at a time on a surface utilizing the pyrolysis of a hydrocarbon, such as methane or propylene. This deposited material is then graphitized employing both thermal and mechanical stress. In the overwhelming number of applications single crystal graphite is not employed. Rather a carbon with some degree of graphite structure is utilized. Excluding crystallite imperfection, such as, vacancies, interstitials,

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substitutions, twin planes, etc., the form of carbon that most closely approaches single crystal graphite is turbostratic graphite. This form of carbon looks very similar to graphite except that, although there may be some degree of perfection within the planes, the adjacent planes are out of registry with one another. That is, in the hexagonal graphite structure, there is an atom in each adjacent plane that sits directly over the center of the hexagonal ring. In turbostratic graphite, the adjacent planes are shifted with respect to one another and are out of registry. This results in an increase in the interlayer spacing, which can increase from 0.3354 nm to more than 0.345 nm. If the value exceeds this, the structure exfoliates. Heating to temperatures in excess of 2800 K provides energy for mobility and can convert turbostratic graphite to single crystal graphite in a process called graphitization. As stated above, a carbon material that goes through either a gas phase or liquid phase process in its conversion to carbon (carbonization process) can be convert to a graphitic structure employing time and temperature in the range from 2000–2800 K. This means that materials formed in the gas phase like carbon black and pyrolytic carbon can be converted to a graphitic structure. Cokes and carbon fibers fabricated from petroleum, coal tar, or mesophase pitch are also graphitizable. On the other hand, chars formed directly from organic materials, such as wood and bone used for activated carbon, PAN fibers formed from polyacrylonitrile, and vitreous carbons formed from polymers, such as, phenolic or phenyl-formaldehyde, are amorphous and not graphitizable because they maintain the same rigid non-aligned structure that they possessed before carbonization. In addition to the degree of crystalline order, the properties of carbons are also determined by the crystallite size and orientation of polycrystalline carbons. The largest carbon parts that are manufactured are electrodes for the steel and aluminum industries. These electrodes can weigh more than 3 tons and are fabricated by extruding a mixture of fine petroleum coke and coal tar pitch. The extrusion process causes some preferential alignment of the crystallites and baking to 2800 K produces a polycrystalline graphite part that has high strength and conductivity. To make isotropic graphite, fine grain coke and petroleum pitch are isostatically pressed. By proper selection of precursor material and processing condition, a carbon with practically any property can be produced. For example carbons can be hard (chars) or soft (blacks), strong (PAN fibers) or weak ( aerogel), r as well as anisotropic (HOPG) or stiff (pitch fibers) or flexible (Graphoil), isotropic (polycrystalline graphite). In addition, porosity, lubricity, hydrophobicity, hydrophilicity, thermal conductivity and surface area can be varied over a wide range. For example surface area can vary from 0.5 m2 /g for a fiber to >2000 m2 /g for an activated carbon while thermal conductivity can range from 0.001 W/(m-K) for an amorphous carbon foam to 1100 W/(m-K) for a pitch fiber, and 3000 W/(m-K) for diamond.

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Carbon materials have been studied and produced for thousands of years. (The Chinese used lampblack 5000 years ago.) While much is understood about carbon, there are some very important areas in which there is still a lack of understanding. These areas fall generally into the production of graphitic material and the oxidation protection of carbon and graphite materials. A better understanding of the science of carbon formation will allow increased performance at reduced cost, while effective oxidation protection at ultra high temperature will enable a whole range of new technology. Currently, the highest performance and highest cost form of carbon are carbon-carbon composites, which are truly a unique class of materials. These composites, which are stronger and stiffer than steel as well as being lighter than aluminum, are currently used principally in high performancehigh value applications in the aerospace and astronautics industries. The highest volume of carbon-carbon is used as brake rotors and stators for military and commercial aircraft because it has high thermal conductivity, good frictional properties, and low wear. Astronautic applications include, rocket nozzles, exit cones, and nose tips for solid rocket boosters as well as leading edges and engine inlets for hypersonic vehicles. In these applications carbon-carbon’s high strength and stiffness as well as its thermal shock resistant are keys to its success. For re-useable hypersonic vehicles, the fact that carbon does not go through phase changes like some ceramics and in fact its mechanical properties actually increase with temperature make this a very valuable material. For satellite applications carbon-carbon’s high specific strength and stiffness as well as its near zero thermal expansion make it an ideal material for large structures that require dimensional stability as they circle the earth. Carbon–carbon composites are fabricated through a multi-step process. First, carbon fibers, which carry the mechanical load, are woven, braided, felted, or filament wound into a preform which has the shape of the desired part. The perform is then densified with a carbon matrix, which fills the space between the fibers and distributes the load among the fibers. It is this densification process that is not well understood. Unlike the manufacture of fiberglass, in the formation of carbon-carbon composites the matrix precursor is not just cured but must be converted into carbon. The conversion of polymers to produce a char, as well as the conversion of a hydrocarbon gas to produce a graphitizable matrix, is fairly well understood. What is not clear is the process of converting a pitch-based material to a high quality graphitizable matrix. That is, for example, starting with a petroleum or coal tar pitch, heat soaking converts this isotropic mixture of polyaromatics into anisotropic liquid crystalline spherical droplets called mesophase spherules. These spherules ultimately coalesce to form a continuous second phase which ultimately pyrolyzes to form a graphitic structure. Although this process has been observed microscopically, little is known about the process, or even exactly what mesophase is or what molecular precursors are needed to form

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mesophase. Little is known because it has proved impossible to accurate analyze mesophase precursors, mesophase itself, or the intermediates between mesophase and graphite. What is needed is a model that can predict the growth of single aromatic rings through the mesophase intermediate and into an ordered graphitic structure. Moreover, it is well known that during pyrolysis, mesophase converts into a matrix that is very anisotropic. The formation of onion-like “sheaths” takes place on the surface of individual carbon fibers in the carbon composite. As these sheaths grow outward from the fiber surfaces, they ultimately collide forming point defects called disclinations. This behavior has a pronounced effect on both the chemical and physical properties of the carbon-carbon composite. A model that describes this matrix contribution to the composite’s properties would be of great benefit both for understanding experimental data, such as thermal conductivity and mechanical properties, as well as for prediction of composite behavior. Although carbon–carbon composites possess high strength, stiffness, and thermal shock resistance making them an excellent high temperature structural material, their Achilles heel is oxidation. Above 670 K, carbon oxidizes. This means that if it is unprotected, it can not be used for long term high temperature applications. Today it is used for short term applications, such as, aircraft brakes as well as rocket nozzles, exit cone, and nose tips. To be used in long term applications such as reusable hypersonic vehicles, it must be protected. Currently, an adequate protection system at ultra high temperatures (>2700 K) does not exist. There is a tremendous need and payoff for a non-structural barrier that can keep oxygen from reaching a carbon surface at 2700 K. Efforts to fabricate these coatings have been unsuccessful. What is probably required for success is some sort of novel functionally graded coating which will require modeling material properties as well as thermal stresses. Finally, the rather recent discovery of Buckyballs and nanotubes, which are 3D analogs of hexagonal graphite, have also rekindled the need for models of the intercalation of graphite. This process occurs when various elements, such as lithium, sodium, or bromine, “sandwich” themselves between graphene planes. This greatly alters the thermal and electrical properties of the graphite. Similar behavior is beginning to be demonstrated in Buckyballs and nanotubes. The need for models based on computational chemistry would be of great help in this area. For additional information on carbon, there are many good reference works. Most focus on specific forms of carbon such as carbon blacks, active carbons, fiber, composites, intercalation compounds, nuclear graphites, etc. For general topics the handbook by Pierson [1] is a good text. A reference work covering scores of subjects in great detail is the 28 Volume Chemistry and Physics of Carbon [2] which has had three Editors over the last 40 years.

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References [1] H.O. Pierson, “Handbook of carbon, graphite, diamond and fullerenes – properties, processing and applications,” Noyes Punlications, Park Ridge, 1993. [2] P.L. Walker Jr., P. Thrower, and L.R. Radovic (eds.), “Chemistry and physics of carbon,” vol. 1–28, 1965–2004, Marcel Decker, New York, 2001.

Perspective 38 CONCURRENT LIFETIME-DESIGN OF EMERGING HIGH TEMPERATURE MATERIALS AND COMPONENTS Ronald J. Kerans Air Force Research Laboratory, Materials and Manufacturing Directorate, Wright-Patterson Air Force Base, Ohio, USA

We ask a great deal of structural components used at high temperatures. In addition to performing their primary load bearing function, most of them are subjected to sizable thermal stresses and aggressive atmospheres. Their microstructures and phase compositions, and hence their properties, evolve throughout their lifetime. Many are used in regimes where significant local creep deformation accumulates, so their shapes and residual stress states change also. Finally, many of them are used in situations where failure is extremely undesirable. Failure of a power generation turbine or an aircraft engine carries substantial fiscal costs, and the latter has potential for tragic human costs. The customers – all of us – have very low tolerance for other than extreme reliability. This is a challenging environment in which to introduce new materials. It is an equally challenging environment in which to substitute computation for time-proven testing and design techniques. The penalties for mistakes are extreme and the development costs are correspondingly large. On the other hand, the high stakes involved provide strong motivation to use computation to improve the ability to predict component end-of-life and thereby avoid both catastrophic failures and expensive premature retirement. Likewise, the long and costly development / certification cycles for introducing even evolutionary changes in materials provide a substantial benefit for substituting computation for experiment. Fortunately the fundamental understanding and many of the multi-scale modeling tools will be similar for both tasks.

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Current Efforts

In many ways, the need for computational infrastructure for both design and life management is most pronounced for new materials systems for which both design and life limiting behavior differ from materials now in service. However, the breadth and depth of understanding of behavior, substantial existing databases, and cost benefit associated with fielded systems substantially argues for application to existing and evolutionary systems. The computational tools required for either task are numerous, as are the materials science aspects that will require enhanced understanding; hence a comprehensive, coordinated attack on either is a significant undertaking. In fact, a number of efforts have been coordinated under the umbrellas of three substantial activities which have begun addressing both design [1–3] and life management [4, 5]. The long-range scope of the design-related modeling effort extends from atomic level to component design via finite element analysis (FEA). The life management effort covers a broad span in degradation and failure behavior, plus goals of comparable complexity in logging and predicting the effects of actual use history and in sensing the actual state of the system in situ.

2.

Emerging Materials

While current efforts boldly attempt to devise entirely new approaches to design and life management, the focus is quite logically on current or evolutionary variations of current materials. However, the real impact of the approach may be most profound when applied to the introduction of new materials, such as low-ductility intermetallics, ceramic composites and perhaps hybrids combining two or more materials systems. These systems almost “demand” a more robust computational approach to design. Consider that while developmental ceramic composites are made in sheet form with uniform properties, many actual components will have fiber and void volume fractions that vary strongly with location in the part. The magnitude and anisotropy of the properties of the material will be a local feature of the component. In this sense, it is almost impossible to separate the material from the component; optimal design will require that the material and component be dealt with concurrently. While these issues apply to all fibrous composites, they will apply especially to ceramic composites. Organic-matrix composites are often used for large sheet-like components such as wing skins, whereas ceramic composites will largely be used for the smaller, more complex shapes of high temperature hardware. In almost all aspects, ceramic composites represent a significant departure from conventional practice. Optimizing the distributed damage mechanisms

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that provide macroscopic fracture toughness is an exercise in carefully designed fracture processes. In conventional materials, the design work is mostly oriented towards avoiding fracture entirely, or at least designing for minimum growth rates. The task of promoting local fracture and designing crack paths is relatively new territory [6]. A consequent significant difference between ceramic composites and other systems is the relationship between properties and damage. Local deformation in metals, for example, does not radically change properties whereas comparable deformation in a composite may significantly change the local properties with the largest changes possibly being in an unloaded direction. The local nature of properties both imposes obligation and provides opportunity to the designer. The idea is to design the local material/properties to best suit the requirements of the component. In many cases this will be best achieved not by fastening sub components together, but by way of designed architectures and blending materials systems within the component [7, 8]. Such hybrids might comprise oxide composites at the hot surface blending to nonoxide reinforcements in mid-temperature regions, blending to metal matrices in warm regions and finally to monolithic metals for cool attachments. The idea of hybrids is a natural extension of local design.

3.

Processing

Further complicating the local property situation is the certainty that the properties will also be a consequence of local processing. For most materials systems, processes will need to vary with the component. For example, the microstructures of both wrought and cast metal parts vary with location and depend on the geometry of the part. Ceramic composites are by their very nature particularly difficult to process. The basic goal is to consolidate a very refractory matrix material that is constrained by an equally refractory fiber structure, without damaging the fibers. The degree of densification tends to vary with local fiber fraction and most processes will always yield significant void space that varies with location. If the void space can’t be eliminated, then the objective should be to control it to be distributed in the most benign arrangement. Uniform fine porosity is typically more desirable than large pores, which are more desirable than large cracks. The challenge is three-fold; to design what can be processed reliably, to process what was designed, and to model the performance and life of what actually exists. This will require the ability to know and model the effects of such things as actual fiber locations in real preforms and components. Irrespective of the material system, modeling that represents real processes will be essential.

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Lifetime Design

Materials that operate at very high temperatures will evolve over time; no component in use is in a truly stable state. While this fact is not currently neglected entirely, there is opportunity to develop a much higher level of sophistication. The long-range goal should be to establish materials design and modeling capabilities that allow design of materials and components to achieve an optimum blend of properties over their lifetimes and reliable lifetimes in the specific application. This optimization will require: in-depth knowledge of, and the ability to simulate, the service environments; and the ability to predict the evolution of microstructures and damage and the consequent effects on properties and life limiting behavior. The ability to use this knowledge to predict the correct initial state of the materials for optimum lifetime performance implies a very high level of sophistication. The ability to achieve the goal of concurrent lifetime-designed, life-managed materials/components is completely dependent upon conquering what may be an equally challenging task: acquiring knowledge of the actual use environment to a far greater depth than is currently possible. It requires knowing well the actual conditions, not just nominal conditions, and the impact that has on each particular component, which may be as challenging as the preceding items.

5.

Models

The highest-level models may be similar to current design Finite Element Analysis. Much of the design portion of the work might best be done through inverse analysis. That is, the design analysis would determine what the component “wants” for properties and design the material to best suit that. In any event, the more basic analysis will need to be integrated such that it is invisible to the designer [3]. The shortest-scale models will depend somewhat on the materials systems and the nature of the component. In at least some cases, there will be a continuing need to deal with quantum-level modeling. An example of an atomic level issue in a metal alloy might be the introduction of a minor alloying element for the purpose of slowing diffusive microstructural changes. The modification could easily introduce unintended effects on boundary strength, or perhaps subtly affect stability of undesirable minor phases that could be problematic late in service life. Evaluation for all such effects requires very basic modeling, or, as in the current system, extensive prior knowledge. For ceramic composites, it has thus far seemed unnecessary to consider scales below the micromechanics level for design purposes; that is, the smallest dimension would be the approximate 200 nm thickness of the fiber coating.

Design of emerging high temperature materials and components

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It has been assumed that all constituents remain perfectly elastic (brittle) even at quite high temperatures, and that composite performance is not directly affected by subtleties at scales below that of the smallest constituent. However, recent basic work on deformation of monazite and scheelite fiber-coating materials indicates a large role of plasticity in the function of the coatings [9–11]. Further evidence confirms that the extent of plasticity will be temperature dependent, though considerably less so than for most metals. This will add a significant temperature dependent component to composite behavior, and a high level of modeling fidelity may require dealing with ductility via dislocations and deformation twins in the 200 nm coating. Moreover, it is clear that there are significant corrosion effects on fibers that remain uncharacterized, much less understood [12, 13]. These will have profound effects on lifetime properties, and surely other such effects lurk in the details. The degree to which we see simplicity in the modeling of ceramic composites is probably more a reflection of our depth of understanding of the materials than of their actual complexity in long-term service. In any event, very basic modeling will be necessary, but not nearly sufficient. Computational concurrent design will require integrated modeling on a wide variety of time and length scales, integration of materials disciplines, and integration of materials and design disciplines. An elegant and readily usable solution to multi-scale integration will be essential to an infrastructure that has real impact [3].

6.

Outlook

The ultimate goal could be: (1). concurrent material/component design for the best balance of properties over the lifetime of the component; (2). a reliable lifetime with graceful, detectable failure processes; (3). the ability to determine the current state of the component and predict its remaining life; (4). the power to do much of the preceding computationally with an economical set of physical experiments to confirm the results to high reliability. The achievement of this goal is dependent as much upon progress in the science of materials as upon progress in the science of modeling. It is probably safe to say that all current models are situation specific. They do not describe the actual physics; they describe an approximation of the actual physics. Each one describes a facet of the situation wonderfully well in certain circumstances and is completely wrong in others. Integrating them and their successors into an infrastructure that will apply them properly to this nearly infinitely faceted, multi-scale problem is a formidable challenge. There is the attendant danger that each implementation will be so material-system specific that applying the approach to a new system will require rebuilding the entire infrastructure. While such an outcome might still yield significant benefit, much of the

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savings would be lost. The resources currently spent building temporarily useful databases would largely be spent building temporarily useful models. Nevertheless, it seems self-evident that it is the logical goal of materials and computational science. It is a goal with tremendous benefit in both monetary and human terms and that is wonderfully engaging in addition. Though when it will be achieved is uncertain, we can look forward to the day when intrinsic failures are an artifact of the past.

References [1] D.M. Dimiduk and P.L. Martin, et al., “Accelerated insertion of materials: the challenges of gamma alloys are really not unique,” Gamma Titanium Alumindes, H.C.Y-W. Kim and A.H. Rosenberger, TMS, Warrendale, PA, 15–28, 2003. [2] D.M. Dimiduk and T.A. Parthasarathy et al., “Structural alloy performance prediction for accelerated use: evolving computational materials science & multiscale modeling,” 2nd International Conference on Multiscale Materials Modeling, University of California Los Angeles, Los Angeles, CA USA, American Scientific Publishers, 2004. [3] D.M. Dimiduk and T.A. Parthasarathy et al., “Predicting the microstructuredependent mechanical performance of materials for early-stage design,” NUMIFORM 2004, Columbus, OH, USA, 2004. [4] J.M. Larsen and B. Rasmussen et al., “The engine rotor life extension (ERLE) initiative and its contributions to increased life and reduced maintenance cost,” 6th National Turbine Engine High Cycle Fatigue (HCF) Conference, Jacksonville, FL, USA, 2001. [5] L. Christodoulou and J.M. Larsen, “Using materials prognosis to maximize the utilization potential of complex mechanical systems,” JOM(March), 15–19, 2004. [6] R.J. Kerans and R.S. Hay et al., “Interface design for oxidation resistant ceramic composites,” J. Am. Ceram. Soc., 85(11), 2599–632, 2002. [7] G. Jefferson and T.A. Parthasarathy et al., “Materials design of hybrid ceramic composites for hot structures,” Proceedings of the 35th International SAMPE Technical Conference, Dayton, OH, SAMPE, 2003. [8] T.A. Parthasarathy and R.J. Kerans et al., “Reduction of thermal gradient-induced stresses in composites using mixed fibers,” J. Amer. Cer. Soc., 87(4), 617–625, 2004. [9] R.S. Hay, “(120) and (122) monazite deformation twins,” Acta Mater., 51(18), 5255– 5262, 2003. [10] R.S. Hay and D.B. Marshall, “Monazite deformation twins,” Acta Mater., 51(18), 5235–5254, 2003. [11] R.S. Hay, “Climb dissociation of dislocations in monazite at low temperature,” J. Am. Ceram. Soc., 87(6), 1149–1152, 2004. [12] E.E. Boakye, R.S. Hay et al., “Monazite coatings on fibers: II, coating without strength degradation,” J. Am. Ceram. Soc., 84(12), 2793–2801, 2001. [13] R.S. Hay and E.E. Boakye, “Monazite coatings on fibers: I, effect of temperature and alumina-doping on coated fiber tensile strength,” J. Am. Ceram. Soc., 84(12), 2783–2792, 2001.

Perspective 39 TOWARDS A COHERENT TREATMENT OF THE SELF-CONSISTENCY AND THE ENVIRONMENT-DEPENDENCY IN A SEMI-EMPIRICAL HAMILTONIAN FOR MATERIALS SIMULATION S.Y. Wu, C.S. Jayanthi, C. Leahy, and M. Yu Department of Physics, University of Louisville, Louisville, KY 40292

The construction of semi-empirical Hamiltonians for materials that have the predictive power is an urgent task in materials simulation. This task is necessitated by the bottleneck encountered in using density functional theory (DFT)-based molecular dynamics (MD) schemes for the determination of structural properties of materials. Although DFT/MD schemes are expected to have predictive power, they can only be applied to systems of about a few hundreds of atoms at the moment. MD schemes based on tight-binding (TB) Hamiltonians, on the other hand, are much faster and applicable to larger systems. However, the conventional TB Hamiltonians include only two-center interactions and they do not have the framework to allow the self-consistent determination of the charge redistribution. Therefore, in the strictest sense, they can only be used to provide explanation for system-specific experimental results. Specifically, their transferability is limited and they do not have predictive power. To overcome the size limitation of DFT/MD schemes on the one hand and the lack of transferability of the conventional two-center TB Hamiltonians on the other, there exists an urgent need for the development of semi-empirical Hamiltonians for materials that are transferable and hence, have predictive power. The key ingredient to the development of semi-empirical Hamiltonians for materials that have predictive power is a reliable and efficient scheme to mimic the effect of screening by electrons when atoms are brought together to form a stable aggregate. Such an ingredient requires the construction of

2935 S. Yip (ed.), Handbook of Materials Modeling, 2935–2942. c 2005 Springer. Printed in the Netherlands.

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the semi-empirical Hamiltonian based on a framework that allows a coherent treatment of the self-consistent (SC) determination of charge redistribution and environment-dependent (ED) multi-center interactions. Various schemes can be found in the literature in recent years that are designed to improve the transferability of TB Hamiltonians by including the self-consistency and/or the environment-dependency. Among these are methods that can also be conveniently implemented in MD schemes because the atomic forces can be readily calculated. They include methods whose emphasis is placed on a phenomenological description of the environment-dependency [1, 2] and two similar methods whose frameworks take into account the self-consistency as well as the environment-dependency [3, 4]. The latter two approaches can be construed as the expansion of the DFT-total energy in terms of the charge density fluctuations about some reference density. To the second order in the density fluctuations, the total energy is approximated as the sum of a band structure term, a short-range repulsive term akin to that in the conventional two-center TB Hamiltonian, and a term representing the Coulomb interaction between charge fluctuations. The charge fluctuations in this approach are self-consistently determined by solving an eigenvalue equation with the two-center Hamiltonian modified by a term that depends on the charge redistribution. In this framework, the Hamiltonian does contain the features of self-consistency in the charge redistribution and the environment-dependency for systems with charge fluctuations. The environment-dependent feature, however, disappears when systems under consideration do not involve charge fluctuations, e.g., periodic systems with one atomic species per primitive unit cell. But the environment-dependent multi-center interactions are key features in a realistic modeling of the screening effect of the electrons in an aggregate of atoms, including extended periodic systems. This deficiency in properly mimicking the screening of the electrons can be critical in the development of a truly transferable Hamiltonian. Thus the development of semi-empirical Hamiltonians for materials with predictive power requires the treatment of the self-consistency as well as the environment-dependency on equal footing. We have recently developed a scheme for the construction of semi-empirical Hamiltonians for materials within the framework of linear combination of atomic orbitals (LCAO) that allows a coherent treatment of the SC determination of the charge redistribution and the environment-dependent (ED) multi-center interactions in a transparent manner [5]. In this scheme, we set up the framework of the semi-empirical Hamiltonian in accordance with the Hamiltonian of the many-atom aggregate. The salient feature of the resulting semi-empirical Hamiltonian, referred as the SCED/LCAO Hamiltonian, is that it has the flexibility to allow the database to provide the necessary ingredients for fitting parameters to capture the effect of electron screening.

Semi-empirical Hamiltonian for materials simulation

2937

The Hamiltonian of a many-atom aggregate may be written as H =−

h¯ 2 l

2m

∇l2 +

ν( rl − Ri ) +

l,i

l,l

Z i Z j e2 e2 + 4π ε0rll 4π ε0 Ri j i, j

(1)

energy where ν( rl − Ri ) is the potential between an electron at rl and the ion at Ri , rll = rl − rl , Ri j = Ri − R j , and Z i corresponds to the number of valence electrons associated with the ion at site Ri . Within the one-particle approximation in the framework of LCAO, the on-site (diagonal) element of the Hamiltonian can be written as 0 inter + u intra Hiα,iα = εiα iα + u iα + v iα

(2)

0 denotes the sum of the kinetic energy and the energy of interaction where εiα with its own ionic core of an electron in the orbital iα. The terms u intra iα and inter u iα are the energies of interaction of the electron in orbital iα with other electrons associated with the same site i and with other electrons in orbital jβ ( j =/ i), respectively. The term v iα represents the interaction energy between the electron in orbital α at site i and the ions at the other sites. In our scheme, the terms in Eq. (2) are represented by 0 = εiα − Z i U εiα

(3)

u intra iα = Ni U

(4)

and u inter iα + v iα =

[Nk VN (Rik ) − Z k VZ (Rik )]

(5)

k= /i

where εiα may be construed as the energy of the orbital α for the isolated atom at i, Z i the number of positive charges carried by the ion at i (also the number of valence electrons associated with the isolated atom at i), Ni the number of valence electrons associated with the atom at i when the atom is in the aggregate, U , a Hubbard-like term, the effective energy of electronelectron interaction for electrons associated with the atom at site i, VN (Rik ) the effective energy of electron-electron interaction for electrons associated with different atoms (atoms i and k), and Z k VZ (Rik ) the effective energy of interaction between an electron associated with an atom at i and an ion at site k.

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Following the same reasoning, we can set up the off-diagonal matrix element Hiα, jβ ( j =/ i) as

Hiα, jβ

1 = K (Ri j )(ε iα + εjβ ) + [(Ni − Z i ) + (N j − Z j )]U 2 +

(Nk VN (Rik ) − Z k VZ (Rik ))

k= /i

+

(Nk VN (R j k ) − Z k VZ (R j k ))

Siα, jβ (Ri j )

(6)

k= /j

Thus, in addition to the conventional two-center hopping-like first term, Eq. (6) also includes both intra- and inter-electron-electron interaction terms as well as environment-dependent multi-center (three-center explicitly and four-center implicitly) interactions. In its broadest sense, the first term in Eq. (6) corresponds to the Wolfsberg– Helmholtz relation in the extended H¨uckel theory. In our approach, K is treated as a function of Ri j rather than a constant parameter to ensure a reliable description of the dependence of the two-center term on Ri j in the off-diagonal Hamiltonian matrix element. The overlap matrix elements Siα, jβ (Ri j ) are expressed in terms of Si j,τ , with τ denoting, for example, molecular orbitals ssσ, spσ, ppσ, and ppπ in a sp3 configuration. They are expected to be shortranged function of Ri j . Equations (2) through (6) completely define the recipe for constructing semi-empirical SCED-LCAO Hamiltonians for materials in terms of parameters and parameterized functions. An examination of Eqs. (2)–(6) clearly indicates that the presence of Ni , the charge distribution at site i, in the Hamiltonian provides the framework for a self-consistent determination of the charge distribution. From Eqs. (5) and (6), it can be seen that the environmentdependent multi-center interactions are critically dependent on VN (Rik ) and VZ (Rik ), in particular their difference VN (R ik ) = VN (Rik ) − VZ (Rik ). As both VN (Rik ) and VZ (Rik ) must approach E 0 Rik for Rik beyond a few nearest neighbor separations, VN (Rik ) is expected to be a short ranged function of Rik . The parameters, including those characterizing the parameterized functions, are to be optimized with respect to a judiciously chosen database for a particular material. In our approach, εiα may be chosen according to its estimated value based on the orbital iα, or treated as a parameter of optimization. The quantity U will be treated as a parameter of optimization while VZ (Rik ) and VN (Rik ) will be treated as parameterized functions to be optimized. The parameterized function VZ (Rik ) is modeled as the energy of the effective interaction per ionic charge between an ion at site k and an electron associated with the atom at site i. VN (Rik ) is then modeled in terms of VZ (Rik ) and the short-range function VN .

Semi-empirical Hamiltonian for materials simulation

2939

The recognition of the difference between VN (Rik ) and VZ (Rik ) in the SCED/LCAO Hamiltonian assures that the environment-dependent feature will not disappear even for systems with no charge redistribution. The presence of the environment-dependent terms in the SCED/LCAO Hamiltonian for systems with no on-site charge redistribution affects the distribution of the electrons among the orbitals even though the total charge associated with a given site is not changed. Therefore, the effect of the environment-dependency will be reflected in the band structure energy through the solution to the general eigenvalue equation corresponding to the SCED/LCAO Hamiltonian as well as the total energy. This feature, together with the self-consistency in the determination of the charge redistribution, provides the flexibility for the SCED/LCAO Hamiltonian to mimic the effect of electron screening. According to the strategy given above, the framework of the proposed semi-empirical SCED-LCAO Hamiltonian will allow the self-consistent determination of the electron distribution at site i. The inclusion of environmentdependent multi-center interactions will provide the proposed Hamiltonian with the flexibility of treating the screening effect associated with electrons which is important for the structural stability of narrow band solids such as d-band transition metals, while at the same time, handling the effect of charge redistribution for systems with reduced symmetry on equal footing. Furthermore, as described above, the Hamiltonian is set up in such a way that the physics underlying each term in the Hamiltonian is transparent. Therefore, it will be convenient to trace the underlying physics for properties of a system under consideration when such a Hamiltonian is used to investigate a manyatom aggregate and predict its properties. The salient feature of our strategy is that, with the incorporation of all the relevant terms discussed previously, there is no intrinsic bias towards ionic, covalent, or metallic bonding for the proposed Hamiltonian. The construction of the SCED/LCAO Hamiltonian depends critically on the database. If one can judiciously compile a systematic and reliable database, the scheme has the flexibility to allow the database to properly model the screening effect of the electrons in an atomic aggregate. Thus the strategy represents an approach that provides the appropriate conceptual framework to allow the chemical trend in a given atomic aggregate to determine the structural as well as electronic properties of condensed matter systems. The total energy of the system consistent with the Hamiltonian described by Eqs. (2)–(6) is given by E tot = E B S − E corr + E ion−ion

(7)

where E B S is the band-structure energy and is obtained by solving the general eigenvalue equation corresponding to the SCED/LCAO Hamiltonian, E corr is the correction to the double counting of the electron-electron interactions between the valence electrons in the band-structure energy calculation, and

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E ion−ion is the repulsive interaction between ions. Based on Eqs. (2)–(6), Eq. (7) can be rewritten as E tot = E B S + +

1 2 1 (Z i − Ni2 )U − Ni Nk VN (Rik ) 2 i 2 i,k(i =/ k)

1 Z i Z k VC 2 i,k(i =/ k)

(8)

with VC =

e2 E0 = 4π ε0 Rik Rik

(9)

For the MD simulation, the forces acting on the atoms in the atomic aggregate must be calculated at each MD step. The calculation of the band structure contribution to atomic forces can be carried out by the Hellmann–Feynman theory. With the presence of terms involving Ni and Nk in the SCED-LCAO Hamiltonian (see Eqs. (5) and (6)), terms such as ∇k Ni where ∇k refers to the gradient with respect to Rk will appear in the electronic contribution to the atomic forces. However, these terms are canceled exactly by terms arising from the gradients of the second and the third terms in the total energy expression (Eq. (8)). Thus terms involving ∇k Ni will not contribute to the calculation of atomic forces. This fact greatly simplifies the calculation of atomic forces needed in the MD simulations. In other words, if one disregards the extra time due to the self-consistency requirement, the calculation of atomic forces based on the SCED–LCAO Hamiltonian is not anymore difficult compared with conventional TB approaches. We have tested the SCED/LCAO Hamiltonian by investigating a variety of different structures of silicon (Si), including the bulk phase diagrams of Si, the equilibrium structure of an intermediate-size Si71 cluster, the reconstruction of the Si(100) surface, and the energy landscape for a Si monomer adsorbed on the reconstructed Si(111)-7×7 surface [5]. In all the cases studied, the results have demonstrated the robustness of the SCED/LCAO Hamiltonian. For example, results showing the binding energy vs relative atomic volume curves for the diamond, the simple cubic (sc), the body centered cubic (bcc), and the face centered cubic (fcc) phases of silicon, obtained by using the SCED–LCAO Hamiltonian constructed for Si with our scheme, are presented in Fig. 1. Also shown in Fig. 1 are the corresponding curves obtained using three existing traditional (two-center and non-self consistent) non-orthogonal tight binding (NOTB) Hamiltonians [6–8], and two more recently developed non-self consistent but environment-dependent Hamiltonians [1, 2]. All the curves (solid) are compared with the results obtained by DFT–LDA calculations [9].

Semi-empirical Hamiltonian for materials simulation

2941

1

fcc DFT LCAO

bcc

0.4

sc

0.2 cdia 0 0.4

0.6

0.8

1

1.2

0.6

0.8

1

1.2

1.4

0.4

0.6

0.8

1

1.2

1.4

0.4

0.6

0.8

1

1.2

1.4

0.4

0.6

0.8

1

1.2

1.4

0.4

0.6

0.8

1

1.2

1.4

1

0.6

NRL

0.8

Wang et al.

Frauenheim et al.

binding energy per atom

0.6

Bernstein & Kaxiras

Menon & Subbaswamy

SCED

0.8

0.4

0.2 0 0.4

1.4

relative atomic volume Figure 1. The binding energy versus relative atomic volume curves for the diamond (cdia), the simple cubic (sc), the body centerd cubic (bcc), and the face centered cubic (fcc) phases of silicon. Top-left panel: SCED-LCAO Hamiltonian. Top-central panel: [7]; Top-right panel: [8]; Bottom-left panel: [6]; Bottom-central panel: [1]; Bottom-right panel: [2]. All the curves (solid) are compared with the result obtained by a DFT–LDA calculation [9].

It can be seen that while the results obtained by all the existing Hamiltonians fail for the high pressure phases, those obtained using Hamiltonians with environment-dependent terms give much better agreement for those phases. This is an indication of the importance of the inclusion of the environmentdependent effects in the Hamiltonian, even for single-element extended crystalline phases where there is no charge redistribution. However, the most striking message conveyed by Fig. 1 is how well our result compares with the DFT–LDA results for all the extended crystalline phases, both at low as well as high pressures. It indicates that the SCED/LCAO Hamiltonian has the capacity and the flexibility of capturing the environment-dependent screening effect under various local configurations. The framework of the SCED/LCAO Hamiltonian outlined in Eqs. (2)–(6) is very flexible. It can be conveniently extended to include the spin-polarized effect and to construct SCED/LCAO Hamiltonians for heterogeneous systems in terms of parameters of SCED/LCAO Hamiltonians of their constituent elemental systems. Work along these lines is in progress.

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Acknowledgment This work was supported by NSF (DMR-0112824 and ECS-0224114) and DOE (DE-FG02-00ER4582).

References [1] C.Z. Wang, B.C. Pan, and K.M. Ho, “An environment-dependent tight-binding potential for Si,” J. Phys.: Condens. Matter., 11, 2043–2049, 1999. [2] D.A. Papaconstantopoulos, M.J. Mehl, S.C. Erwin et al., “Tight-binding Hamiltonians for carbon and silicon,” In: P.E.A. Turchi, A. Gonis and L. Colombo (eds.), TightBinding Approach to Computational Materials Science. MRS Symposia Proceedings No. 491, Materials Research Society, Pittsburg, pp. 221–230, 1998. [3] K. Esfarjani and Y. Kawazoe, “Self-consistent tight-binding formalism for charged systems,” J. Phys.: Condens. Matter., 10, 8257–8267, 1998. [4] Th. Frauenheim, M. Seifert, M. Elsterner et al., “A self-consistent charge densityfunctional based tight-binding method for predictive materials simulations in physics, chemistry, and biology,” Phys. Stat. Sol. (b), 217, 41–62, 2000. [5] C. Leahy, M. Yu, C.S. Jayanthi, and S.Y. Wu, “Self-consistent and environmentdependent Hamiltonians for materials simulations: case studies on Si structures,” http://xxx.lanl.gov/list/cond-mat/0402544, 2004. [6] Th. Frauenheim, F. Weich, Th. K¨ohler et al., “Density-functional-based construction of transferable nonorthogonal tight-binding potentials for Si and SiH,” Phys. Rev. B, 52, 11492–11501, 1995. [7] M. Menon and K.R. Subbaswamy, “Nonorthogonal tight-binding moleculardynamics scheme for Si with improved transferability,” Phys. Rev. B, 55, 9231–9234, 1997. [8] N. Bernstein and E. Kaxiras, “Nonorthogonal tight-binding Hamiltonians for defects and interfaces,” Phys. Rev. B, 56, 10488–10496, 1997. [9] M.T. Yin and M.L. Cohen, “Theory of static structural properties, crystal stability, and phase transformations: applications to Si and Ge,” Phys. Rev. B, 26, 5668–5687, 1982.

INDEX OF CONTRIBUTORS (Article/Perspective numbers are given in bold.) Abraham, F.F. P20, 2793 Alexander, F.J. 8.7, 2513 Aluru, N.R. 8.3, 2447 Angelis, F. de 4, 59 Artacho, E. 1.5, 77 Asta, M. 1.16, 349 Averback, R. 6.2, 1855 Bammann, D.J. 3.2, 1077 Barmak, K. 7.19, 2397 Baroni, S. 1.1, 195 Bartlett, R.J. 1.3, 27 Battaile, C. 7.17, 2363 Bazant, M.Z. 4.1, 1217; 4.1, 1417 Bernstein, N. 2.24, 855 Binder, K. P19, 2787 Blöchl, P.E. 1.6, 93 Boghosian, B.M. 8.1, 2411 Boon, J P. P21, 2805 Boyd, I.D. P22, 2811 Bulatov, V.V. P7, 2695 Caflisch, R. 7.15, 2337 Cai, W. 2.21, 813 Car, R. 4, 59 Carloni, P. 1.13, 259 Carter, E.A. 1.8, 137 Catlow, C.R.A. 6.1, 1851; 2.7, 547 Ceder, G. 1.17, 367; 1.18, 395 Chadwick, A.V. 6.5, 1901 Chan, H S. 5.16, 1823 Chelikowsky, J.R. 1.7, 121 Chen, I-W. P27, 2843 Chen, L-Q. 7.1, 2083 Chen, S-H. P28, 2849 Chipot, C. 2.26, 929 Ciccotti, G. 2.17, 745; 5.4, 1597 Cohen, M.L. 1.2, 13 Corish, J. 6.4, 1889 Coveney, P.V. 8.5, 2487 Crocombette, J-P. 2.28, 987 Crowdy, D. 4.1, 1417

Csányi, G. P16, 2763 Cuong, N.N. 4.15, 1529 de Koning, M. 2.15, 707 De Vita, A. P16, 2763 Dellago, C. 5.3, 1585 Doll, J.D. 5.2, 1573 Doyle, P.S. 9.7, 2619 Eggers, J. 4.9, 1403 Español, P. 8.6, 2503 Evans, D.J. P17, 2773 Evans, J.W. 5.12, 1753 Falk, M.L. 4.3, 1281 Farkas, D. 2.23, 839 Först, C.J. 1.6, 93 Fredrickson, G H. 9.9, 2645 Frenkel, D. 2.14, 683 Gale, J.D. 2.3, 479; 1.5, 77 Galli, G. P8, 2701 Ganesan, V. 9.9, 2645 García, A. 1.5, 77 Gear, C.W. 4.11, 1453 Germann, T.C. 2.11, 629 Geva, E. 5.9, 1691 Ghoniem, N M. 7.11, 2269; P11, 2719; P30, 2871 Giannozzi, P. 1.1, 195; 4, 59 Gillespie, D.T. 5.11, 1735 Gilmer, G. 2.1, 613 Goddard, W.A. III. P9, 2707 Gro, A. 5.1, 1713 Gumbsch, P. P10, 2713 Gygi, F. P8, 2701 Hadjiconstantinou, N.G. 8.1, 2411; 8.8, 2523 Hirth, J.P. P31, 2879 Ho, K.M. 1.15, 307 Hoffman, W.P. P37, 2923 Hoover, W.G. P34, 2903 Horstemeyer, M.F. 3.1, 1071; 3.5, 1133 Hou, T.Y. 4.14, 1507 Huang, H. 2.3, 1039

2943

2944 Hummer, G. 4.11, 1453 Islam, M.S. 6.6, 1915 Jang, S. 5.9, 1691 Jayanthi, C.S. P39, 2935 Jeanloz, R. P25, 2829 Jensen, P. 5.13, 1769 Jin, X. 8.3, 2447 Jin, Y.M. 7.12, 2287 Joannopoulos, J.D. P4, 2671 Junquera, J. 1.5, 77 Justo, J.F. 2.4, 499 Kaburaki, H. 2.18, 763 Kalia, R.K. 2.25, 875 Kapral, R. 2.17, 745; 5.4, 1597 Karma, A. 7.2, 2087 Kästner, J. 1.6, 93 Katsoulakis, M.A. 4.12, 1477 Kaxiras, E. 2.1, 451; 8.4, 2475 Kerans, R.J. P38, 2929 Kevrekidis, I.G. 4.11, 1453 Khachaturyan, A.G. 7.12, 2287 Khraishi, T.A. 3.3, 1097 Kim, S G. 7.3, 2105 Kim, W.T. 7.3, 2105 Klein, M.L. 2.26, 929 Kob, W. P24, 2823 Kofke, D.A. 2.14, 683 Korkin, A. 1.3, 27 Kremer, K. P5, 2675 Krill, C.E., III. 7.6, 2157 Kubin, L.P. P33, 2897 Landau, D.P. P2, 2663 Langer, J.S. 4.3, 1281; P14, 2749 Leahy, C. P39, 2935 LeSar, R. 7.14, 2325 Li, G. 8.3, 2447 Li, J. 2.8, 565; 2.19, 773; 2.31, 1051 Li, X. 4.13, 1491 Lignères, V.L. 1.8, 137 Lookman, T. 7.5, 2143 Louie, S.G. 1.11, 215 Lowengrub, J. 7.8, 2205 Lu, G. 2.2, 793 MacKerell, A.D., Jr. 2.5, 509 Magistrato, A. 1.13, 259 Mahadevan, L. 9.1, 2555 Margetis, D. 4.8, 1389 Marin, E.B. 3.5, 1133 Maroudas, D. 4.1, 1217 Martin, G. 7.9, 2223 Martin, R.M. 1.5, 77 Marzari, N. 1.1, 9; 1.4, 59 Mattice, W.L. 9.3, 2575 Mavrantzas, V.G. 9.4, 2583 McDowell, D.L. 3.6, 1151; 3.9, 1193

Index of contributors Mehl, M.J. 1.14, 275 Metiu, H. 5.1, 1567 Miller, R.E. 2.13, 663 Milstein, F. 4.2, 1223 Mishin, Y. 2.2, 459 Montalenti, F. 2.11, 629 Morgan, D. 1.18, 395 Moriarty, J.A. P13, 2737 Morris, J.W., Jr. P18, 2777 Mountain, R.D. P23, 2819 Müller, M. 9.5, 2599 Nakano, A. 2.25, 875 Needleman, A. 3.4, 1115 Nitzan, A. 5.7, 1635 Nordlund, K. 6.2, 1855 Odette, R.G. 2.29, 999 Ogata, S. 1.2, 439 Olson, G.B. P3, 2667 Ordejón, P. 1.5, 77 Pakula, T. P35, 2907 Pande, V. 5.17, 1837 Pankratov, I.R. 7.1, 2249 Papaconstantopoulos, D.A. 1.14, 275 Pask, J.E. 1.19, 423 Patera, A.T. 4.15, 1529 Payne, M. P16, 2763 Pechenik, L. 4.3, 1281 Peiró, J. 8.2, 2415 Phillpot, S.R. 2.6, 527; 6.11, 2009 Potirniche, G.P. 3.5, 1133 Powers, T.R. 9.8, 2631 Raabe, D. 7.7, 2173 Radhakrishnan, R. 5.5, 1613 Ratsch, C. 7.15, 2337 Ray, J.R. 2.16, 729 Reinhard, W.P. 2.15, 707 Reuter, K. 1.9, 149 Rickman, J.M. 7.14, 2325; 7.19, 2397 Rubio, A. 1.11, 215 Rudd, R.E. 2.12, 649 Rutledge, G.C. 9.1, 2555 Sakai, T. 8.5, 2487 Sánchez-Portal, D. 1.5, 77 Sauer, J. 1.12, 241 Saxena, A. 7.5, 2143 Scheffler, M. 1.9, 149 Schulten, K. 5.15, 1797 Schwartz, S.D. 5.8, 1673 Selinger, R.L.B. 2.23, 839 Sepliarsky, M. 2.6, 527 Sergi, A. 2.17, 745; 5.4, 1597 Sethian, J.A. 4.6, 1359 Shelley, M.J. 4.7, 1371 Shen, C. 7.4, 2117 Sherwin, S. 8.2, 2415

Index of contributors Sierka, M. 1.12, 241 Sierou, A. 9.6, 2607 Smith, G.D. 9.2, 2561 Soisson, F. 7.9, 2223 Soler, J.M. 1.5, 77 Sornette, D. 4.4, 1313 Srolovitz, D.J. 7.1, 2083; 7.13, 2307 Stachiotti, M.G. 2.6, 527 Stampfl, C. 1.9, 149 Stanley, E.H. P36, 2917 Sterne, P.A. 1.19, 423 Stone, H.A. 4.8, 1389 Stoneham, M. P12, 2731 Succi, S. 8.4, 2475 Tadmor, E.B. 2.13, 663 Tajkhorshid, E. 5.15, 1797 Tang, M. 2.22, 827 Tarek, M. 2.26, 929 Taylor, DeCarlos E. 1.3, 27 Theodorou, D.N. P15, 2757 Thompson, C.V. P26, 2837 Tornberg, A.K. 4.7, 1371 Torquato, S. 4.5, 1333; 7.18, 2379 Trout, B.L. 5.5, 1613 Tuckerman, M.E. 2.9, 589 Uberuaga, B.P. 5.6, 1627; 2.11, 629 Underhill, P.T. 9.7, 2619 Vaks, V.G. 7.1, 2249 Van de Walle, A. 1.16, 349 Van de Walle, C.G. 6.3, 1877

2945 Van der Giessen, E. 3.4, 1115 Van der Ven, A. 1.17, 367 Vashishta, P. 2.25, 875 Veroy, K. 4.15, 1529 Vitek, V. P32, 2883 Vlachos, D.G. 4.12, 1477 Voter, A.F. 2.11, 629; 5.6, 1627 Voth, G.A. 5.9, 1691 Voyiadjis, G.Z. 3.8, 1183 Vvedensky, D.D. 7.16, 2351 Wahnström, G. 5.14, 1787 Wallace, D.C. P1, 2659 Wang, C.Z. 1.15, 307 Wang, Y. 7.4, 2117 Wang, Y.U. 7.12, 2287 Weinan, E. 4.13, 1491; 8.4, 2475 Wijesinghe, H S. 8.8, 2523 Wirth, B.D. 2.29, 999 Wolf, D. 6.7, 1925; 6.9, 1953; 6.1, 1985; 6.11, 2009; 6.12, 2025; 6.13, 2055 Woo, C.H. 2.27, 959 Woodward, C. P29, 2865 Wu, S.Y. P39, 2935 Xiang, Y. 7.13, 2307 Yip, S. 2.1, 451, 613; 6.7, 1925; 6.8, 1931; 6.11, 2009 Yu, M. P39, 2935 Zbib, H.M. 3.3, 1097 Zhu, F. 5.15, 1797 Zikry, M. 3.7, 1171

INDEX OF KEYWORDS ab-initio 1877, 2671, 2687, 2865 ab initio calculations 13, 349, 423, 2823 ab initio molecular dynamics 9, 59, 93, 195, 259, 349, 2701 ab initio potentials 1901 ab initio pseudopotential 121 abnormal grain growth 2157 accelerated molecular dynamics 629 acceptors 1877 acoustic emissions 1313 activated processes 1613 activation barrier 1281, 2223 activation energy 773, 1985, 2055 activation free energy 259 activation volume 1281 active sites 241 adaptive simulation 2675 adatom diffusion 613, 2337 adiabatic energy surfaces 2731 adsorption resonances 1713 aggregation 1769 AgI 1901 AIM 2667 Al(Zr,Sc) 2223 alkali metals 1223 all-atom-models 2675 all-electron potential 121 Allen-Cahn equation 2157 alumina 479 aluminum in melting 2009 Alzheimer’s disease 259 AMBER 509, 2561 amorphization 987, 2009 amorphous 1953, 1985 amorphous carbon 307 amorphous cement model 1953, 1985 amorphous GaAs static structure factor 875 amorphous polymers 1281 amorphous solid water 2917 amorphous solids 1901, 2055, 2749 amorphous structure of high-energy grain boundaries 1953, 1985

amphiphilic fluids 2411, 2487 Andrade law 1313 angular-dependent forces 459 angular-force multi-ion potential 2737 anharmonic 349 anharmonic correction 1877 anisotropic diffusion 959 anisotropic grain growth 2157 anisotropic long-range interaction 2143 anisotropic media 729 anisotropy 2173 annealing stages 1855 annealing temperature 613 anticancer drugs 259 antifreeze proteins 1613 antiphase domain boundaries 2787 antisymmetrically coupled environmental modes 1673 AOT microemulsion 2849 a posteriori error estimation 1529 aquaporins (AQPs) 1797 argon 1635 Arrhenius 1635 Arrhenius dynamics 1477 Arrhenius factor 1735 Arrhenius plot 1985 a-SiO2 structure factor 875 asymmetric GB 1953 asymmetric tilt GB (ATGB) 1953 asymmetrical grain boundaries 1931 asymmetry parameter 2223 ATGB cusp 1953 atomic configuration 613 atomic displacement 349 atomic displacements at interfaces 1931, 2055 atomic force field 1837 atomic hypothesis 451 atomic jump 2223 atomic jump frequency 1787 atomic-level geometry of grain boundaries 1953

2947

2948 atomic mechanism [of diffusion] 1787 atomic misfit energy 2117 atomic pseudopotential wave functions 121 atomic simulations 929 atomic structure 1953, 1985 atomic transport 1851 atomistics 649 atomistic/continuum coupling 663 atomistic description 2523 atomistic (modeling) 2757, 2763 atomistic picture 959 atomistic simulation 451, 793, 1837, 1931, 2737 atomistic simulation of interfaces 1925 atomistic simulation of melting 2009 atomistic thermodynamics 149 automatic model adaptation 663 Bain strain 2777 Bain transformation 1223 ballistic jumps 2223 bands 13 bandgap problem 215 barrier crossing 1635 basin constrained molecular dynamics 629 basis function 2447 basis-sets 93 bcc lattice 827, 1953 bcc metals 2777, 2865, 2883 Beeman algorithm 565 Beevers–Ross site 1901 bending elasticity 2631 Bethe–Salpeter equation 215 Bhatnagar–Gross–Krook model 2805 bias potential 629 bias sampling 613, 1613 biased random walk 1635 bicontinuous morphologies 2849 bicrystal 1953, 1985 bicrystal melting 2009 bifurcation 1223 biharmonic functions 1417 bilayers 2631 bimolecular modeling 259 bimolecular reactions 1735 binary collision approximation 987 binary mixture 2787 binding energy 1877, 2659 biomolecules 1613 biosystems 2731 Bloch walls 2787 Bloch-periodic boundary conditions 423 Bloch’s theorem 423 block copolymers 2599 Blue Moon ensemble 1597

Index of keywords boiling point 395 Boltzmann equation 2411, 2513 Boltzmann factor 613 Boltzmann H theorem 2487 bonds 13 bond-angle distribution function 1985 bond breaking 2763 bond fluctuation model 2599 bond-order 499 bond-order potentials 2737 bond orientational order parameter 1613 border conditions 1931 Born effective charges 195, 479 Born stability criteria 2009 Born-Oppenheimer approximation 59, 195 Born–von K´arm´an model 349 Bortz algorithm 1753 boundary 1491 boundary conditions 1931, 2475 boundary integral methods 2205 boundary layer 1389 boundary-tracking 2157 Bravais lattice 1953 Bredig transition 1901 brittle 855 brittle fracture 1417 brittle-to-ductile transition 839, 773, 855 broken-bond model 1953 Brønsted acidic sites 241 Brownian dynamics 649, 2757, 2607, 2619 buckyball 2923 bulk free surface 2025 bulk interfaces 1925, 2025 bulk melting point 1985 bulk modulus 855, 2009 Burgers vector 2307 C60 307, 1627 calorimetric 1823 canonical d-bands 2737 canonical ensemble 613 capillary waves 2787 carbon 855, 2923 carbon nanotubes 215 carbon-carbon composite 2923 carboxylate bridged binuclear motif 259, 1877 Car–Parrinello molecular dynamics 59, 259, 1877 cascade 987 CASCADE 1889, 1901 cascade damage 959 CASTEP 1901 catalysis 241, 395, 1753 Cauchy relation 2025

Index of keywords Cauchy–Born rule 663 cellular automata 2351 cellular automata 2687 central limit theorem 1635 central symmetry parameter 1051 centroid molecular dynamics 1691 ceramic composites 527, 875, 2929 certification 1529 CGMD 649 chain entropy 2675 Chapman Kolmogorov equation 1635 Chapman–Enskog analysis 2475, 2487 charge localisation 2731 charge transfer 2731 charged state 1877 CHARMM 509, 2561 chemical-bonding changes 2829 chemical Fokker–Planck equation 1735 chemical Langevin equation 1735 chemical master equation 1735 chemical potential 349, 407, 707, 1389, 1877, 2645 chemical rates 59, 149, 1573, 1585 chemically reacting flows 2475 chemically synthesized quantum dot structures 875 cisplatin-DNA adduct 259 classical limit 349 classical molecular dynamics 59, 349, 451 classical potentials – DFT comparisons 27 clathrate hydrates 1613 cleavage anisotropy 855 climb 2307 clipped random wave model 2849 closure 1477 closure on demand 1453 cluster 241, 349, 1851, 1877 cluster dynamics 2223 cluster expansion 349, 367 cluster migration 241, 2223 coarse bifurcation (algorithms/computations) 1453 coarse grained models 2675 coarse projective integration 1453 coarse time stepper(s) 1453 coarse-grain models 929 coarse-grained molecular dynamics 649 coarse-grained Monte Carlo 1477 coarse-grained polycrystal 2055 coarse-grained statistical model 349 coarse-grained stochastic models 1477 coarse-graining 649, 1477, 1613, 2083, 2325, 2351 2503, 2599, 2645, 2687, 2757 coarsening 2117, 2205 coherency strain energy 2117

2949 coherent 2025 coherent interfaces 1925 coherent phase diagram 2117 coherent potential approximation 349 coherent precipitation 2117 coherent treatment 2935 coherent-twin boundary 1953, 2055 cohesive zone model 855 coincident-site lattice (CSL) 1953 collective damage 1313 collective diffusion model 1797 colloidal fluids 2503, 2607, 2645 combined quantum mechanics–interatomic potential functions method 241 commensurate GB 1953 commensurate interfaces 1925, 2025 committor(s) 1585 COMPASS 2561 compensation 1877 complex fluids 1371, 2411, 2487, 2503, 2675 complex Langevin sampling 2645 complex systems 1217, 1453, 1475 component simulation 2713 composite 2173, 2379 composition-modulated superlattice 2025 compressibility 745 compressibility in melting 2009 computational efficiency 2907 computational fluid dynamics 2811 computational materials design facility 2707 computational nanoscience 2701 computer-aided analysis 1453 concerted rotation 2583 concerted rotation Monte Carlo 2757 concurrent 2929 concurrent material/component design 2929 concurrent multiscale simulation 649 condensed matter 137, 2659 condensed phase systems 1597 conditional convergence 813 conditional probability 1635 conditional reaction probability 1735 conductivity 1333, 1877, 1901 configuration 349 configuration coordinate diagram 1877 configuration phase space 613 configurational and displacive states 349 configurational arrangement 349 configurational bias 2583 configurational bias Monte Carlo 2757 configurational disorder 367 confined liquid 1985 conformal map 1417 conformational partition function 2575

2950 conjugate gradients 2671 conjugated polymers 215 conservation laws 1491 consistency 2415 constant pressure molecular dynamics 589 constant-coverage algorithms 1753 constitutive equations 1071 constitutive formulations 2897 constitutive law 745 constitutive theories 1281 constrained equations 745 constrained equilibrium 149 constrained phase space flow 745 continuity equation 745 continuous Markov process 1735 continuous-orientation model 2157 continuous-time random-walk (CTRW) model 1797 continuum 2173 continuum approach [to modeling diffusion] 1787 continuum description 2523 continuum limit 2351 continuum mechanics 1529, 2903 convergence 2415 convex hull 349 cooperative rearrangement 2907 cooperativity 1823, 2907 coordinate scaling 707 coordination number 1051 copolymers 2645 copper 629, 1223 copper100 629 core energy 813 core properties 793 core-shell nanoparticles 875 correlated events 629 correlation factor 1855 correlation function 1333, 1635, 1673, correlation length 2787 correlation time 1635 correspondence relation 649 coupling method 2523 covalent bonding 499 covalent solids 451 cracks 839, 2287 crack dynamics 2475 crack propagation 2087, 2763 crack propagation in amorphous SiO2 875 crack tip 773 crack tip plasticity 839 creep 959, 1313, 2719 critical behavior 1613 critical point 2787

Index of keywords criticality 1313 cross-slip 793, 2307, 2897 cross-validation 395 crowdion 1855 crown thioether complexes of rhenium and technetium 259 crystal growth 613, 629 crystal interface 1925 crystal plasticity code 2897 crystal structure 395, 423, 2829 crystal symmetry 1223 crystal-growth simulation 2055 crystallography 2173 crystal-to-amorphous transition 2009 C-S bond cleavage 259 CSL misorientation 1953 curvature 1039, 2837 curvature-driven grain growth 2157 cusped orientation 1953 CVD diamond 2829 cyclic plasticity 1193 DAD effect 959 damage 1071, 1183 Damkohler number 2475 data management and mining 875 dislocation dynamics simulations 2897 deposition 1039 dissipative particle dynamics (DPD) 2503 de novo 2707 deacylation of peptide 259 Debye correlation function 2849 decoherence 2731 defects 241,1855, 1877, 1915, 2269, 2737, 2871 defect annealing 1855 defect calculations 547 defect concentration 1855 defect-dependent properties 1851 defect diffusion 613, 1855 defect entropies 1889 defect-induced melting 2009 defects in amorphous materials 1855 defects in intermetallics 1855 deformation 1281, 2173 deformation gradient tensor 439, 1133 deformed materials 1217 degrees of freedom (DOF) of a grain boundary 1953 deNOx process 241 density field 2645 density functional theory (DFT) 9, 59, 93, 121, 137, 149, 195, 349, 423, 439, 451, 855, 1613, 1877, 1889, 2737

Index of keywords density of CSL sites 1953 density of states 683, 1823 deposition 2363 deprotonation energy 241 design 2667 detailed balance 613, 1477, 1585, 1753 deterministic dynamics 613 diamond 2923 diamond-anvil cell 2829 diamond nanoparticle 307 diamond structure 855 diamond surface 307 diamond-to-graphite transition 307 dielectric breakdown 1417 dielectric tensor 195 differential displacement map 773 diffraction 1713 diffuse interface 2087, 2117 diffusion 629, 1627, 2117 diffusion coefficient 1635, 1691, 1901, 1931, 2055 diffusion controlled reactions 1635 diffusion equation 1635, 1787 diffusion mechanism 2223 diffusion permeability 1797 diffusional phase-transition 2205 diffusional width 1985 diffusion-limited aggregation 1417 diiron proteins 259 dimanganese proteins 259 dimensional analysis 565 dimer method 629, 1627 dimer-TAD 629, 1627 diomimetic compound of methane monooxygenase 259 direct correlation function 1613 direct force method 349 direct simulation Monte Carlo (DSMC) 2411, 2811, 2487, 2523 directed end-bridging 2583 directed internal bridging 2583 disclinations 2923 discrete simulation automata (DSA) 2411, 2487 discrete-orientation model 2157 discretization 2415 dislocation 793, 813, 839, 855, 1077, 1098, 1115, 2083, 2269, 2843, 2865, 2871 dislocation core energy 773 dislocation density tensor 2325 dislocation dipole 2325 dislocation dynamics 813, 827, 2083, 2269, 2307, 2325, 2871 dislocation dynamics simulation, 3D 2695 dislocation GB 1953, 1985

2951 dislocation generation 2897 dislocation microstructure 827 dislocation nucleation 773 dislocation patterns 2897 dislocation theory 2695 dislocation-pressure interaction 2879 disorder at interfaces 1931 disorder temperature 2749 disordered crystalline alloy 349 disordered materials 349, 1217 dissipation 1151, 1281, 1635, 2773 dissipative particle dynamics (DPD) 2411, 2757 dissociation 1713 distributed computing 1837 distributed damage 2929 distribution of grain sizes 2837 distribution of nucleation sites 2397 dividing surface 629 DL POLY code 1901 DNA 77, 2619 DNA binding 259 domain 2083, 2843 domain wall, 90◦ 2843 domain wall, 180◦ 2843 doping 1877 double bridging 2583 double bridging Monte Carlo 2757 DREIDING 2561 drift 1635 drift-diffusion process 1635 driven alloys 2223 DSC lattice 1931 d-state directional bonding 2737 ductile 855 due ferro 1 (DF1) 259 dumbbell 1855 dynamic density functional theories 2757 dynamic fracture in n-Si3 N4 875 dynamic heterogeneities 2917 dynamic lattice liquid 2907 dynamical matrix 195, 349, 459, 649 dynamical scaling hypoethesis 2351 dynamics 1491, 2083 EAM potential 1931, 2055 Earth’s interior 2829 edge 1098, 1115, 2307 edge diffusion 2337 EDTB potential for molybdenum 307 EDTB potential for Si 307 Edwards–Wilkinson equation 2351 effective cluster interaction 349 effective Hamiltonian 349, 2645 effective lifetime 959

2952 effective medium theory 2737 effective potential 499, 1823 effective temperature 1281, 2749 eigenvalue 349 eigenvector following [method] 1573 Einstein crystal 349 elastic anomalies at interfaces 2025 elastic compatibility 2143 elastic constants 275, 459, 479, 729, 773, 1223, 1333, 2025 elastic effect 349 elastic energy 2117 elastic instability 2009, 2777 elastic stability 1223 elastic stiffness 773 elastic wave 649 elastically-mediated interaction 349 elasticity 649, 1371, 1529 elastodiffusion 959 electro-migration 1417 electronegativity equalisation 479 electron-hole interaction 215 electronic 2829 electronic band energy 349 electronic chemical potential 349 electronic density of states 349 electronic entropy 307, 349 electronic excitation 349 electronic excited states 2731 electronic free energy 349 electronic structure 13, 93, 137, 149, 349, 423, 1889 electronic structure, coarse grained 2737 electrostatic effect 349, 423, 479 element modeling 649 elliptic equations 2415 embedded atom method (EAM) 459, 1223, 1953, 1985, 2025, 2737 embrittlement 2719 empirical potential 499, 509, 855 empirical valence bond (EVB) method 241 end bridging Monte Carlo 2757 end-bridging 2583 end-of-life 2929 end-to-end distance 2575 end-to-end vector 2575 energy barrier 1585 energy cusp 1953, 2055 energy density functionals 137 energy function 509 energy localisation 2731 energy minimization 855, 1931 energy release rate 855 energy transfer 1713, 2731 ensemble 729, 2645

Index of keywords ensemble average 349, 707, 763 entropic elasticity 2631 entropy 349, 707, 1931 entropy production 707 environment dependence 2935 environmental effects on fracture 875 environment-dependent tight-binding (EDTB) potential 307 enzyme 1613 epitaxial growth 2337 equation of continuity 763 equation of state 2599 equation-free modeling 1453 equilibrium ensembles 589 equilibrium Monte Carlo 613 equilibrium properties 613 equilibrium volume 349 ergodic hypothesis 613 ergodic systems 1931 error bounds 1529 Eshelby’s virtual procedures 2117 Euler–Lagrange equation 649 evaporation 1769 evaporation–condensation 1389 evolution 1823, 2083 Ewald method 479, 1889 exact enumeration 1823 EXAFS 1901 excess energy 1985 excess energy profile 1985 exchange process 1627 exchange-correlation potential 1877 exciton 2731 excitonic effects 215 explicit solvent model 1837 explicit tau-leaping simulation procedure 1735 external free surface 1953 facets 1389 fast ion conductors 1901 fast marching methods 1359 fast multipole method 875 fast sweeping method 2307 fatigue 1071, 1193 fcc lattice 1953 fcc metals 2777 Fe(Cu) 2223 Fe(Nb,C) 2223 f-electron metals 2737 femtosecond laser pulse 307 FENE 2619 Fermi distribution 349 Fermi level 349 fermi surface 275

Index of keywords ferroelectrics 349, 527, 2843 Feynman propagators 1673 fiber-bundle models 1313 film growth 629 finite difference 2415 finite difference method 423 finite difference time domain 2671 finite element method 121, 423, 649, 663, 1529, 2173, 2415, 2663 finite size effects 2787 finite temperature 349, 1491 finite volume 2415 finite-time singularity 1313 Finnis–Sinclair model 2737 first generation radiopharmaceutical compounds 259 first principles 149, 349, 367, 423, 1877, 2707, 2865 first wall 2719 first-principles density functional theory 275 fitting 479 ‘flip’ instability 2777 Flory–Huggins theory 2599 flow defect 2749 flow in porous media 1507, 2487 fluctuation theorem 2773 fluctuation-dissipation theorem 1635, 2411 fluctuations 1635, 1735 fluid binary mixture 2787 fluid dynamics 2411 fluid flow 1507, 1529 fluid permeability 1333 fluorite structure 1901 flux 1635, 1787 flux autocorrelation function 1673 Fokker–Planck equation 1635, 2503 folding temperature 1823 force constant 349, 1877 force field 2561, 2707 forced oscillator model 1713 formation energies 1877 formation enthalpy 1855 formation entropy 1855 formation volume 1855 Fourier transforms 121, 423 Fourier’s law of thermal conduction 763 fractal 1417 fracture 839, 855, 1071, 1171, 2793 Frank–Read source 827, 2307 free energy 613, 683, 707, 1585, 1613 free volume 1281, 1953, 1985 free-energy perturbation 683 freezing 1613 Frenkel defects 1901 friction coefficient 1635

2953 front-tracking 2837 fuel cells 1915 functional integral 2645 funnel sampling 683 fusion energy 2719 fusion reactors 999 GaAs phonon dispersion 875 gamma-surface 855, 2883 Gaussians 121 Gaussian (“normal”) distribution 1635 Gaussian chains 2599 Gaussian curvature 2849 Gaussian Markovian stochastic process 1635 Gaussian random field 2849 Gaussian random force 1635 Gaussian stochastic process(es) 1635 Gaussian variable 1635 Gaussian white noise 1735 general grain boundary 1953, 2055 generalized configurational bias 2583 generalized gradient approximation 439, 1877 generalized Langevin equation 1673 generalized pseudopotential theory 2737 generalized stacking fault (GSF) energy 793 GENERIC framework 2503 genetic algorithm 307, 547, 1613 geometric glide plane 2695 germanium 855 grain boundary sliding and migration 1931 ghost forces 663 Gibbs–Duhem integration 683 Gibbs ensemble 683 Gibbs–Feynman identity 707 Gibbs free energy 2009 Ginzburg-Landau framework 2143 glass transition 1281, 1985 glass transition temperature 1823 glasses 2823 glide 1098, 1115, 2307 global climatic changes 1613 global minimum 613 global structure optimization 307 go model / go-like 1823 GPT 2737 gradient thermodynamics 2117 grain area distributions 2397 grain boundary (GB) 547, 1039, 1925, 1931, 1953, 1985, 2025, 2055, 2083 grain boundary area 2157 grain boundary diffusion 1953, 1985, 2055 grain boundary diffusion creep 1985, 2055

2954 grain boundary energy 1953, 1985, 2025, 2055, 2157 grain boundary fracture 1953 grain boundary inclination 2157 grain boundary migration 1985 grain boundary misorientation 2157 grain boundary mobility 1985, 2157 grain boundary self-diffusion 1985 grain boundary sliding 2055 grain boundary superlattice (GBSL) 2025 grain boundary width 2157 grain growth 1985, 2837 grain growth exponent 2157 grain junction 1953, 1985, 2055 grain shape 2055 grain size 2055, 2157 Granato model 1855 grand canonical ensemble 1931 graphite 2923 gray scale operations and filters 2397 Green’s function 1877 Green-Kubo formula 745, 763 Green’s function 2117 grid 2415 grid computing 875 grid methods 121 Griffith criterion 855 GROMOS 509 Grotthus mechanism 1901 ground state 349 growth 1769, 2117 Gr¨uneisen parameter 349 GULP 1889, 1901 GW approximation 215 GW bandgap 215 GW-BSE approach 215 HADES 1889, 1901 Hamiltonian 2935 Hamiltonian matrix 121, 307 Hamilton’s equations of motion 763 handshake regions 875 hardening 1281 hardness 855 hard-sphere model 1953 harmonic approximation 349 harmonic functions 1417 harmonic transition state theory 629 Hartree potential 121 Hartree–Fock 1877 heat capacity 1823 heat flux 763 heat of formation 275 Hele–Shaw 1417 Helfrich free energy 2849

Index of keywords Hellmann–Feynman theorem 59, 195 Helmholtz 1529 Helmholtz inequality 707 Herring relation 2055 heterogeneous and homogeneous melting 2009 heterogeneous catalysis 149 heterogeneous materials 1217, 1333 heterogeneous microstructure development 959 heterogeneous multiscale method 1491 heteropolymer 1823 hexatic phase 1613 hierarchical modeling 649 hierarchical processes 2731 high friction limit 1635 high pressure 2737, 2829 high temperature limit 349 high-angle GB 1953, 1985, 2055 high-energy GB 1953, 1985, 2055 higher order finite difference expansions 121 high-temperature relaxed structure 1953, 1985 Hillert model 2157 histogram reweighting 683 histograms 1613 holonomic constraints 745 homogeneous 1613 homogeneous system 1635, 2025 homogenization 1333, 2325, 2379 HP model 1823 hybrids 2929 hybrid algorithms 1753 hybrid Car–Parrinello/MM molecular dynamics 259 hybrid energy 349 hybrid FE/MD scheme 875 hybrid FE/MD/QM scheme 875 hybrid formulation 2523 hybrid MD/QM scheme 875 hybrid multiscale modeling 649, 2763 hybrid quantum mechanics/molecular mechanics methods 241 hydrocarbons conversion 241 hydrodynamic interaction 2607, 2619 hydrodynamics 1403, 2513, 2523 hydrogen 1877 hydrogen bonding 1613 hydrogen tunneling 259 hydrophobic 1613 hydrostatic (compression/expansion) 1223, 2009 hyperbolic equations 2415 hyperdynamics 629

Index of keywords hyperfine parameters 1877 hypersonic flows 2811 ideal strength 439, 855, 2777 image analysis 2397 image force effect 2287 immiscible fluids 2487, 2503 implicit solvent model 1837 importance sampling 349, 613, 2583 impurities 1877 In melts 1985 InAs/GaAs square nanomesa 875 incoherent 2025 incubation 1193 incubation time for nucleation 2223 indented surface of Si3 N4 875 industrial applications 2819 industrial applications of materials modeling 2713 information theory 1477 infrared spectroscopy 195 infrequent event system 629, 1627 inherent structure(s) 1573 inhomogeneous 1613, 2025 instability 1371 integral materials simulation 2713 interatomic force constants 195 interatomic potential 241, 451, 459, 479, 499, 1901, 1931, 2731, 2763 interconversion of atoms 349 inter-diffusion 367 interface 1953 interface capturing 2205 interface fluctuations 2351 interface-plane method 1953 interface tracking 2205 interfacial 2025 interfacial curvatures 2849 interfacial dynamics 1217 interfacial energy 2157 interfacial fracture and dislocation emission 875 interfacial free energies 2787 interfacial profile 2787 interfacial width 2787 intergranular fracture 1931 interionic potentials 1889 intermediate scattering function 2823 intermetallic compounds and alloys 2737 intermetallics 2883 internal interface 1953 internal state variables 1077, 1183 internal variables 1151 interplanar spacing 1953 interstitial 2223

2955 interstitial diffusion 367 interstitial position 1877 interstitialcy 2223 intrinsic interfacial profile 2787 intrinsic profile 2787 intrinsic width 2787 invariance 1223 invariant volume element 589 inversion symmetry 1953 ion core 121, 613 ion implantation 649 ion polarizability 1901 ion transport mechanism 1901 ionic materials 479 ionization energy 1877 ion-pair potentials 241 IR spectra 259 irradiation 987 irradiation growth 959 irreversibility paradox 2773 irreversible aggregation 2337 irreversible process 707 irreversible work 707 Ising 2687 Ising models 2787 Ising-type models 1477 island dynamics 2337 island number density 2337 isostress molecular dynamics 1223 iterative diagonalization methods 121 IVR 1673 Jahn–Teller effect 2731 jammed state 1281 Jarzynski average 707 jump frequency 2223 jump Markov process 1735 JWKB approximation 1673 kinematics 1183 kinetic 137 kinetic equation 2249 kinetic Ising model 2223 kinetic Monte Carlo 149, 613, 629, 1627, 1753, 1769, 2083, 2223, 2351, 2363, 2757 kinetic pathway 2223 kinetic theory 2411, 2475, 2513 kinetics 959, 2083 kinetics (of reactions) 1585 kinetics of melting 2009 kink-pair mechanism 827 kink-pressure 2879 Kleinman–Bylander form 121 Knudsen number 2475, 2513, 2523

2956 Kohn–Sham eigenvalues 215 Kohn–Sham equation 121 Kolmogorov flow 2805 Kosevich energy functional 2325 k-point sampling 439 Kramer–Moyal–van Kampen expansion 2351 Kroger–Vink notation 1889 Kubo–Green 367 Lagrange multipliers 745 Lagrangian formulations 1281 Landau free energy 1613 Landau, Ginzburg, De Gennes 1613 Landau-type coarse-grained “chemical” free energy 2287 Langevin equation 649, 2351, 2411, 1635 Langevin leaping 1735 Langevin noise 2117 Langevin update formula 1735 LAPW 93 large strain mechanical response 1223 large-scale deformation 2749 laser ablation 307 latent heat and volume 2009 lateral interactions 149 lattice Boltzmann equation 2411, 2475, 2805 lattice dynamics 649, 1931, 2025, 2055 lattice friction 2897 lattice gas 1753, 2351 lattice gas automata 2805 lattice gas Hamiltonians 149 lattice Green’s function 649 lattice model 2599 lattice Monte Carlo 613 lattice protein 1823 lattice site 349 lattice statics 1223, 1931, 2025 lattice strain in melting 2009 lattice symmetry 349 lattice trapping 855 lattice vibration 349 layer-by-layer growth 2337 layered and tunnel structures 1901 leap condition 1735 leap-frog algorithm 565 Lee–Edwards boundary condition 745 legacy codes 1453 length and time scales 2663 length scales 1077 Lennard Jones potential 565, 1953, 1985, 2025 level set methods 1359, 2337, 2083, 2307 level surface 2849

Index of keywords life management 2929 life-managed 2929 limitations of atomistic simulations 451 Lindemann criterion 1985 linear inequality 349 linear programming 349 linear regression 395 linear response theory 195, 349, 707, 745 linear scaling 77, 649 line–tension–pressure interaction 2879 link atoms 241 Liouville equation 745, 763 Liouville operator 589 lipid bilayers 929 lipid membrane structure 929 liquid 349 liquid simulations 2819 liquid–liquid phase transition 2917 liquid–solid interface 2009 LMTO 93 load balancing 875 local density approximation 121, 439, 1877 local expansion at interfaces 1931 local processing 2929 local property 2929 local pseudopotentials 137 local rules 2897 localization 1713 localized basis sets 423 localized orbitals 77 localized vibrational modes 1877 lognormal 2397 long range elastic fields 2763 long-range interactions 241 long-range interatomic interaction 349 long-range order parameter 2117 long-range structural order 1953, 1985 look-up tables 1753 low viscosity simulation 1837 low-angle GB 1953, 1985 low-dimensional continuum models 2631 low energy 1953, 2055 low-energy GB 1985 lubrication forces 2607 macromolecular architectures 2907 macroscale 1071 macroscopic 1953 macroscopic (modeling) 2757 magnetism 275 manganese catalase 259 many-body expansion 499 many-body perturbation theory 215 many-electron Green’s function 215 mapping 1039

Index of keywords Markov chain 2583 Markov process 1477, 1735 Markovian stochastic processes 1635 martensitic phase transformation 349, 2117 mass matrix 649 mass-action equations 1735 master equation 613, 1281, 1635, 1753, 2351 materials by design 2667 materials failure 2793 materials for fission and 999 materials modeling 1217, 2707 materials processing 1529 materials simulation 2935 mathematical methods 1217 matrix-free (numerical methods) 1453 Maxwell’s equations 2671 mean curvature 2849 mean field theory 2787 mean free path 2513 mean occupation 2249 mean square curvatures 2849 mean-field methods 349 mean-squared displacement 1931, 1985, 2055 measure 745 mechanical deformation 439 mechanical embedding 241 mechanical properties 2173, 2737 mechanism 1613, 1985, 2687 melting 2009 melting at interfaces 1931 melting point 349 membrane dynamics 929 membrane transport 929 membranes 2631, 2675 memory kernel 649 memory time 1635 Mermin order parameter 1613 mesh 2415 meshless method 2447 mesophase 2923 mesoscale 1071 mesoscale phenomena 2411, 2487 mesoscopic (modeling) 2731, 2749, 2757 metallic glasses 1281, 2749 metallic hydrogen 2829 metallization 2829 metals 451, 2173 metastable phase 2009 metastable phase diagram 349 metastable states 1613 methane monooxygenase 259 Metropolis 2363

2957 Metropolis algorithm 349, 613, 1585, 1823, 1931, 2583 Metropolis-type dynamics 1477 MgO 1627 MGPT 2737 microemulsion 2645, 2849 micro–macro hybrid methods 2903 microphotonics 2671 microporous materials 547 microscopic 1953 microscopic electro-mechanical systems (MEMS) 2475 microscopic reversibility 613 microscopic state 349 microstructural evolution 2087 microstructurally small crack 1193 microstructure 999, 1193, 1333, 1371, 2083, 2117, 2269, 2379, 2687, 2871 microstructure evolution 2157, 2249 microstructure, ferroic 2143 migration barrier 1877 migration enthalpy 1855 migration entropy 1855 minimum energy path 773 mirror symmetry 1953 misfit 2307 misfit energy 793 misfit localization 1953 misfit strain 2287 mixed basis 2737 mixed quantum classical [time evolution methodologies] 1673 MMFF 509 mobility 1877, 2307 mobility constant 745 mode-coupling theory 2823 model order reduction 1529 modeling 2173, 2269, 2871 modeling of excited states 259 modeling of interfaces 1925 modified embedded atom method 459, 855 modulation wavelength 2025 molecular assemblies 2675 molecular complexity 2907 molecular dynamics (MD) 275, 451, 527, 565, 629, 649, 707, 729, 813, 839, 855, 1039, 1491, 1613, 1627, 1735, 1787, 1837, 1877, 1889, 1901, 1931, 2523, 2663, 2737, 2793, 2411, 2475, 2487, 2503, 2707 molecular dynamics and Monte Carlo comparison 613 molecular dynamics simulation 349, 509, 929, 1585, 1597, 1797

2958 molecular dynamics simulation of melting 2009 molecular mechanics 2763 molecular mechanics force fields 241 molecular modeling 509 molecular statics 839 molten sub-lattice model 1901 monazite 2929 mon-equilibrium molecular dynamics 745 Monte Carlo 451, 707, 729, 1039, 1931, 2663, 2687 Monte Carlo [path] 1613 Monte Carlo [procedure] 1585 Monte Carlo simulation 349, 1901, 2583, 2599 morphological and microstructural evolution 1217 morphological evolution 2351 morphology 2083 Morse model 1223 Mott–Littleton 1889 Mott–Littleton procedure 1901 multicanonical Monte Carlo 2787 multicomponent alloy 349 multifunctionality 2379 multilayer growth 2337 multimillion atom molecular dynamics simulations 875 multi-paradigm 2707 multiphysics 2475 multiplicity 349 multipole expansion 2325 multiscale 2523, 2707 multiscale analysis 1491, 1507 multiscale computation 1507 multiscale gas dynamics 2811 multiscale modeling 149, 793, 999, 1217, 2737, 2657 multiscale modeling of polymers 2757 multiscale visualization 875 Nabarro–Herring creep 2055 nanoclusters 547 nanocrystalline 839 nanocrystalline material 1985, 2055 nanocrystals 1925 nanodiamond 307 nano-forms of carbon 2923 nanoindentation 2777 nanoindentation of silicon nitride 875 nanomechanics 649 nanophase SiC 875 nanophotonics 2671 nanostructured a-SiO2 875 nanostructured ceramics 875

Index of keywords nanostructured materials 875 nanostructured Si3 N4 875 nanostructures 215 nanostructures in solution 2701 nanotube 307, 2923, 1797 native point defects 1877 native-centric 1823 natural convection 1529 Navier–Stokes 1529 Navier–Stokes equations 2411, 2475, 2805 NDDO 27 neighbor distribution 2397 neighbor list 565 nematic order parameter 1613 NEMS 649 neural network 395 neutral net 1823 Newtonian equations of motion 1931 Newtonian fluids 2503 n-fold way 2363 Ni(Cr,Al) 2223 nickel 1223 NMR 1901 NMR spectra 259 nonbonded interaction 2561 non-collinear magnetization 275 non-Debye relaxation 2731 non-equilibrium alloy 2249 non-equilibrium free-energy calculation 707 non-equilibrium processes 1753 non-equilibrium states 745 non-equilibrium work 683 non-Hamiltonian dynamical systems 589 non-Hamiltonian system 745 non-linear perturbations 745 nonlocal pseudopotentials 423 nonlocality 121 non-Markovian 1635 non-Markovian dynamics 1635 non-Newtonian fluids 2503 nonorthogonal tight-binding 307 nonperiodic systems 121 non-planar core 2883 non-reactive collisions 1735 nonstoichiometry 1851 norm conserving 121 normal modes 195, 349, 1635 normal random variable 1735 Nose–Hoover chains 589 nucleation 1098, 1115, 1613, 2117, 2223, 2337 nucleation and growth 2397 nucleation barrier 2223 nucleation of voids and dislocation loops 959

Index of keywords nucleation rate 2223 nucleation sites in melting 2009 nudged elastic band 773 numerical analysis 1453 O(N) multiresolution algorithms 875 observable variables 1151 occupation variable 349 offline–online procedures 1529 Omori law 1313 one-dimensional diffusion 959 ONIOM method 241 Onsager’s regression hypothesis 707, 745 on-the-fly algorithms 1753 on-the-fly kinetic Monte Carlo 1627 open systems 2731 operator resummation 1673 OPLS 509, 2561 optical properties 215 optical properties of silicon dots 2701 optimization 613, 2083, 2379 orbital-free density functional theory 137 order parameter 683, 1585, 1613 order-disorder transition 2645 ordered phase 349 ordering 2223, 2249 order-N 77, 565 orientation variants 2117 Orowan looping 2307 orthogonal tight-binding 307 osmotic permeability 1797 Ostwald ripening 2337 output bounds 1529 overdamped limit 1635 overlap matrix 307 oxidation of alkanes 259 oxidation of aluminium nanoparticles 875 oxygen diffusion 1915 oxygen molecule 121 pair correlation function 2325, 2397, 2583 pair potentials 499 parabolic equations 2415 parallel (multiscale modeling approaches) 2757 parallel calculations 423, 2487 parallel molecular dynamics 875 parallel replica dynamics 629, 1627 parallel tempering 1613 parallel-replica 629, 1627 parametrized problems 1529 parent lattice 349 partial charge 2561 partial differential equations 1529, 2415, 2447

2959 partial least squares 395 partial occupation 349 particle and radiation transport 613 particle-based 2513 particle bypass 2307 particle-laden flows 2607 particle tracking 451, 613 particle trajectories 613 partition function 349, 707, 2645 path integral molecular dynamics 259 path-integral propagator 1691 path integral quantum transition state theory 1713 path sampling 1837 pattern formation 2117 pattern recognition 1613 Peach–Koehler force 1098, 1115 Peclet number 2475, 2607 Peierls stress 793, 813, 2865, 2883 Peierls–Nabarro model 793, 2287, 2695 ‘pencil glide’ 2777 peptide bundle, 4-α-helix 259 perfect crystal 2025 periodic boundary condition 241, 423, 565, 773, 813, 1051, 2695 perovskite oxides 527, 1915 Pettifor map 395 phase behavior 527 phase coexistence 2009 phase diagram 149, 349 phase equilibria 683 phase field 2083, 2105, 2117, 2157, 2287, 2837 phase-field models 2087 phase-field simulation 2157 phase separation 2249 phase-separated polymer blends 2849 phase space 349, 683, 1635 phase space flow 745 phase space sampling 613 phase stability 349 phase transformation 1223, 2117 phase transitions 149, 1613, 2829 phonons 195, 1713 phonon density of states 349 phonon dispersion 459 phonon frequencies 275, 349 Photobacterium leiognathi 1597 photonic band gap 2671 photonic crystals 2671 physical properties 2379 π-back donation 259 planar core 2883 planar stacking 1953 planar structure factor 1953, 1985

2960 plane waves 59, 121, 195 plane-wave basis 93 plane-wave method 423 plastic deformation 663, 793, 1985, 2055, 2713 plastic spin 1133 plasticity 1071, 1151, 1281, 2173, 2269, 2749, 2793, 2865, 2871, 2929 plateau value 1573 point charge model 1901 point collocation 2447 point defects 1851, 1855 Poisson–Boltzmann 1837 Poisson effect 2025 Poisson process 1797 Poisson random numbers 1735 Poisson random variable 1735 polarization 479, 2561 polycarbonate 2675 polycrystals 1953, 1985 polycrystalline 839, 1039, 2837 polydispersity 2583 polyethylene 2575 polyhedral unit model 1953 polymer blends 2599, 2645, 2787 polymer glass 2599 polymer melts 2599 polymer mixtures 2787 polymer solutions 2599, 2645 polymeric fluids 2503 polymers 2173, 2675 potential cutoff 565 potential energy function 509, 2561 potential energy surface 149, 1713 potential of mean force 2645, 2757 Potts model 2687, 2837 powder metallurgical processing 2713 power laws 1313 precipitation 2117 predictive simulations 27 predictor–corrector algorithm 565 premelting 1985 premelting at surface 2009 pressure distribution in a Si/Si3 N4 nanopixel 875 principal component analysis 395 probabilistic clustering 1573 problem posing 2731 process simulation 2713 processing 2173 processing–structure–property relationships 395 production bias 959 projection strategies (in coarse-graining) 2757

Index of keywords projector augmented wave method (PAW) 93 propensity function 1735 protein data bank 1051 protein folding 1837 protein-based nanostructures 875 proton conductors 1901 proton exclusion 1797 proton mobility in zeolites 241 proton transport 1915 prototype sequence 1823 pseudopotential approximation 439 pseudopotential model 1223 pseudopotential perturbation theory 2737 pseudopotentials 13, 59, 93, 121, 195, 423, 1877 Pt chemical shift 259 Pulay forces 59, 121, 195 pyrolsis 2923 pyrosilicic acid 27 QM/MM 259, 241, 2675, 2763 QM-Pot method 241 quadrature domains 1417 quantitative structure-activity relationships 395 quantitative structure–property relationships 395 quantum-based interatomic potentials 2737 quantum dots 121, 649 quantum effects 1713 quantum entanglement 2731 quantum information processing 2731 quantum Kramers [approach] 1673 quantum mechanical rate constants 1691 quantum mechanics 27, 349, 2707, 2763 quantum molecular dynamics 2737 quantum Monte Carlo 2701 quantum reflection 1713 quantum rods and tetrapods 875 quantum simulations 451, 2701 quantum statistics 2731 quantum tunnelling 2731 quasi-atomic minimal-basis-sets orbitals (QUAMBOs) 307 quasicontinuum 649, 663, 1491 quasi-elastic neutron scattering 1787 quasiharmonic approximation 349, 1931 quasiparticle calculations 1877 quasiparticle excitations 215 quasiparticle lifetime 215

Index of keywords radial distribution function 1901, 1953, 1985 radial distribution function at interfaces 1925, 2025 radiation damage 613, 649, 999, 1627 radiation effects 999, 2719 radiopharmaceutical compounds 259 Raman spectra 195, 259 random deposition 2351 random-dissipative forces 649 random force 1635, 1673 random numbers 613, 1635 random sampling 613 random walk 1585, 1635, 1787 rare event 1585 rare reactive events 1597 rare transitions 1613 ratcheting 1193 rate catalog 1627 rate dependent 1133 rate equations 2337 rate independent 1133 rate table 1627 Rayleigh and Gamma probability densities 2397 Rayleigh scattering 649 Rayleigh–Taylor 1417 Rayleigh wave speed 855 reaction coordinates 1585, 1635, 1597 reaction force field 241 reaction mechanism 1585 reaction of H2 O molecules at a silicon crack tip 875 reaction rate constants 1585, 1691, 1735 reaction rate equations 1735 reaction time 1585 reactive collisions 1735 reactive flux correlation function 1597 reactive force fields (ReaxFF) 2707 Read–Shockley model 1953 real space 121 real space methods 121, 423 real-time computation 1529 realizations (stochastic trajectories) 1635 reciprocal space 121 reduced basis 1529 reduced coordinates 1051 reduced descriptions 1635 reduced dimensional systems 215 reduced dynamics 1635 reduced unit system 565 re-emission 1359 reference system 349 refinement 2447, 2523 reflection coefficient 649

2961 refractory metals 2865 relative fluctuations 1735 relaxation 1953 relaxation time 1635 relaxation volume 1855 renormalization 2687 renormalization group 1613, 2351 reptation 1901, 2583, 2599, 2675 reversible aggregation 2337 reversible glass transition in high-energy grain boundaries 1985 reversible process 707 reversible work 707, 1585 Reynolds number 2475 rezoning 2903 rheology 2607 Rice criterion 855 rigid-body translation 1953, 2055 ROCKS 259 rolling 2173 rotation 1953 rotational excitation 1713 rotational isomeric state 2575 roughening transition 1389 Rouse dynamics 2599 rupture prediction 1313 saddle points 1585, 1877, 2223 sampling strategies (in coarse-graining) 2757 scalable data-compression scheme 875 scalable simulation 875 scale-bridging 2687 scale-bridging simulation 2675 scaling 2687 scaling properties of fracture surfaces 875 scattering 649, 1713 scheelite 2929 Schmid law 2883 screened Coulomb interaction 215 screw dislocation 1098, 1115, 2307 screw dislocation in bcc molybdenum 307 second law 2773 sedimentation 2607 segregation 1931 self-avoiding walks 2599 self-consistency 2935 self-consistency cycle 121 self-consistent field (SCF) 2645, 2757 self-diffusion 367, 1985 self end-bridging 2583 self-energy operator 215 self-limiting growth 875 self-organization 2117 self-similarity 1403

2962 self-trapping 2731 semiclassical mechanics 1673 semiconductor etching and deposition 1359 semiconductor growth 149 semiconductor nanostructures 2701 semiconductors 855, 1877, 2737 semi-discrete variational Peierls–Nabarro model 793 semi-empirical 2935 semi-empirical potentials 839 semi-empirical quantum chemistry 27 sequential (multiscale modeling approaches) 2757 sequential multiscale modeling 649 shape function 649 shape selectivity 241 shape transition 2117 shear deformation 2009 shear failure 2777 shear instability 2009 shear localization 2749 shear modulus 855, 2009 shear-rate dependence 745 shear softening 1281 shear strength 2777 shear stress 1223 shear-transformation zone 2749 shear viscosity 745 shell model 527, 1889 shock waves 2829 shooting and shifting [algorithms] 1585 short-range interactions 2575 short-range order 349, 1953, 1985 Si/Si3 N4 interface 875 Si3 N4 phonon density of states 875 SIESTA 77 silica stress and strain curves 27 silicon 499, 613, 855, 2009 silicon (110) 855 silicon (111) 855 silicon cluster 307 silicon nitride 855 silver 629 simple ionic solids 349, 547, 613, 1889 simulated annealing 613 simulations 275, 729, 2173, 2363 simulations of nanostructured materials 875 single crystal plasticity 827 single exponential kinetics 1837 single molecule spectroscopy 1635 single particle distribution function 2513 single-file water transport 1797 singularities 1403 sink competition 959 sintering of n-SiC 875

Index of keywords sintering of silicon nitride nanoclusters 875 size effects 2897 Slater–Koster theory 307 slip 793 Smoluchowski equation 1635 smooth particles 2903 smoothed particle hydrodynamics 2503 soft matter 2675 soft matter properties 2675 soft mode 2787 solid 349 solid electrolytes 1901 solid oxide fuel cells 1901 solid polymer electrolytes 1901 solid with a phase transition 1901 solidification 2087, 2105 solid–liquid interface 2009 solid–liquid phase boundary 349 solid-on-solid model 2337 solid-state amorphization 2055 solubility 1877 solute 349 solvent 349, 1613 solvent model 1837 Sommerfeld model 349 SPAM 2903 spatial decomposition 875 special GB 1953 special grain-boundary plane 1953 specific reaction probability rate constant 1735 spectral density 1673 spectral density function 2849 spectroscopic properties 1851 sph 2903 spherical chicken 2083 spinodal decomposition 2487 splitting algorithm 2513 static structure factor 1931 stability 2415 stacking fault 1953 stacking period 1953 stacking sequence 1953 standard model 13 state-change vector 1735 static heterogeneities 2917 static lattice 547 static lattice calculations 1889, 1901 stationary 1635 statistical mechanics 149, 729 statistical sampling 613 statistical weight matrix 2575 statistics of point processes 2397 steepest descent sampling 2645 Steinhardt order parameter 1613

Index of keywords steps 1389 STGB cusp 1953 sticking 1713 stiffness 1735 stiffness matrix 649 Stillinger–Weber potential 499, 855 stochastic 1635, 1735, 2513 stochastic differential equations 2503, 2619 stochastic dynamics 613 stochastic effects 959 stochastic equations of motion 1635, 2083 stochastic mesoscopic models 1477 stochastic PDE 1477 stochastic simulation algorithm 1735 stochastic time evolution 1635 stochastic trajectory 1635 stochastic variable 1635 Stokesian dynamics 2607 strain 773 strain faceting 2337 strength 2777 stress 439, 773, 2205 stress-free transformation strain 2117 stress intensity factor 839 stress–strain curve 827 stress tensor 459, 2619 structural 2737 structural components 2929 structural disorder 1953, 1985 structural instability 2009 structural materials 2719 structural phase transitions 195, 1985, 2009 structural properties 2737 structural relaxation 1931 structural transformation in GaAs nanocrystals 875 structural width 1985 structure 13, 349 structure map 395 structure–property correlations 1925, 1931, 1953, 2025, 2055, 2675 submonolayer growth 2337 substitutional diffusion 367 subtraction scheme 241 subtraction technique 745 super soft elastomers 2907 supercell 565, 813, 1051, 1877, 1889 superconductivity 2829 supercooled liquids 2009, 2823 supercooled melt 1985 supercooled water 2917 superfunnel 1823 superhard materials 2829 superheating 2009 superheating limit 1985

2963 superionic conductors 1901 superlattice 2025 super-plasticity 1281 surface 855, 1953 surface chemistry 2337 surface diffusion 629, 1389, 1627 surface exchange mechanism 1627 surface growth 629, 2351 surface hopping 1673 surface roughness 2337, 2351 surface simulations 547 surface steps 1953 surface structure 149 surface tension 1403 surface-state excitons 215 suspensions 1371, 2607 Suzuki–Yoshida factorization scheme swelling 2719 switched Hamiltonian 349 switching parameter 707 symmetric GB 1953 symmetric tilt GB 1953 symmetry breaking 2009 symplectic integration 565, 589 system instability 959 systems theory 1453

589

tau leaping 1735 temperature accelerated dynamics 629, 1627 temperature-dependent elastic constants and moduli 2025 temperature-dependent potentials 2737 tensile strength 2777 tensor 1183 Tersoff model 499 tertiary creep 1313 tetrahedral order parameter 1613 texture 1039, 1133, 2173, 2837 theoretical strength 1223 thermal activation 1223 thermal conduction 763 thermal conductivity 2819 thermal defect 349 thermal expansion 349, 459, 2659 thermal properties 707, 2659 thermally activated processes 2897 thermodynamics 1151, 2083, 2773 thermodynamic and mechanical melting 2009 thermodynamic driving force 707 thermodynamic factor 367 thermodynamic integration 259, 349, 683, 707, 1855 thermodynamic limit 1735

2964 thermodynamic path 707 thermodynamic phase diagram 2009 thermodynamic stability 349 thermo-elastic behavior of interfaces 2025 thermo-mechanical behavior 2055 thermostatted molecular dynamics 589 theta state 2575 thin films 1039, 2363, 2837 thin films, structure and elastic behavior of 2025 thin film growth 1753, 2337 thin-film interfaces 1925 threshold displacement energy 987 threshold effects 1713 tight-binding 275, 855, 1877, 2737 tight-binding Hamiltonian 307, 451 tight-binding model for carbon 307 tight-binding molecular dynamics 307 tilt 1953 tilt axis 1953 tilt boundary structures in Si 307 tilt GB 1953, 2055 time average 763 time correlation function 763, 1635 time dependent DFT (TDDFT) 259 time evolution of dislocation motion 875 time integration 2415 time-dependent Ginzbur–Landau equation 2287 time-dependent rate coefficient 1597 time–temperature–transformation (TTT) diagrams 395 topology 2157 total energy 423 total-energy calculation 349 total energy functional 93 total energy surface 1877 tracer 1901 tracer (self-diffusion) coefficient 1787 trajectory in phase space 763 transfer Hamiltonian 27 transitions 2143 transition level 1877 transition metals 2737 transition metals containing zeolites 241 transition path ensemble 1585 transition path sampling 1585 transition rates 613, 1635 transition state 1585, 1613, 1635 transition state ensemble 1585 transition state rate 149,1635 transition-state theory 629, 1573, 1627, 1635, 1673, 2757 translational 1953 translational order parameter 1613

Index of keywords transport coefficient 745, 763 trapping constant 1333 trimolecular reactions 1735 triple junctions 2055, 2157 triple lines 2055 Trotter theorem 589 Tsallis statistics 1613 TVD Runge–Kutta 2307 twin boundary 2843 twinning plane 1953 twist 1953 twist GB 1953, 2055 two-center approximation 307 two-phase flow 1403 two-phase microstructure 2205 two-phase region 1985 two-state model 2025 two-state systems 1281 ultrasoft pseudopotential 59, 195 umbrella sampling 613, 683, 1613, 2787 uniaxial loading 1223 unimolecular reactions 1735 unique properties of molecular dynamics and Monte Carlo 451 unit-cell area 1953 unit-cell volume 1953 UNIVERSAL 2561 unmixing 2223 unperturbed state 2575 unstable mode 349 unstable stacking energy 855 unstructured mesh 649 upscaling 1507 vacancies 1901, 1985, 2223 vacancy formation energy 275 vacancy source, sink 2223 valence interaction 2561 variable charge MD simulation 875 variable connectivity Monte Carlo 2583 variance 1635 variance reduction 2619 variational transition state theory 1635 velocity gap 855 velocity Verlet algorithm 565 velocity Verlet time integration 649 Verlet algorithm 565 vertex tracking 2837 vesicles 2631 vibrational energy relaxation rate constants 1691 vibrational entropy 349 vibrational free energy 349 vicinal GB 1953

Index of keywords vicinal surface 1953 virtual environment 875 viscosity 2675, 2815 viscous fingering 1417, 2805 viscous sintering 1417 visibility 1359 visualization 451, 1051 visualization algorithms 875 void coalescence 1171 void growth 1171 void nucleation 1171 volumetric relaxation 2157 von Mises strain 1051 Voronoi construction 839 Voronoi diagram 2503 waiting-time distribution 2397 water 1613, 2917 water channels 1797 water nucleophilic attack 259 water/silica interactions 27

2965 water/water interactions 27 wave reflection 649 weight function 2415 weighting function 2447 WENO 2307 width 1953, 1985 Wolf–Villain model 2351 wormlike chain 2619 wrapper, computational 1453 yield stress 827, 1281, 2749 yield surface 1077, 2173 Zeldovich factor 2223 zeolite catalysts 241 zero-point vibrations 1931 zero-temperature relaxed structure 1985 Zwanzig Hamiltonian 1673

1953,

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