Series in Soft Condensed Matter Vol.
UNDERSTANDING SOFT CONDENSED MATTER VIA MODELING AND COMPUTATION
3
All living organisms consist of soft matter. For this reason alone, it is important to be able to understand and predict the structural and dynamical properties of soft materials such as polymers, surfactants, colloids, granular matter and liquids crystals. To achieve a better understanding of soft matter, three different approaches have to be integrated: experiment, theory and simulation. This book focuses on the third approach — but always in the context of the other two.
World Scientific www.worldscientific.com 7642 hc
7642.Alvin.CC.indd 1
ISBN-13 978-981-4295-58-1 ISBN-10 981-4295-58-2
,!7IJ8B4-cjffib!
Hu Shi
UNDERSTANDING SOFT CONDENSED MATTER VIA MODELING AND COMPUTATION
Series in Soft Condensed Matter Vol.
3
Editors
Wenbing Hu • An-Chang Shi
UNDERSTANDING SOFT CONDENSED MATTER VIA MODELING AND COMPUTATION World Scientific
11/24/10 2:03 PM
UNDERSTANDING SOFT CONDENSED MATTER VIA MODELING AND COMPUTATION
Series in Soft Condensed Matter Vol.
UNDERSTANDING SOFT CONDENSED MATTER VIA MODELING AND COMPUTATION Editors
Wenbing Hu
Nanjing University, China
An-Chang Shi
McMaster University, Canada
World Scientific NEW JERSEY
•
LONDON
•
SINGAPORE
•
BEIJING
•
SHANGHAI
•
HONG KONG
•
TA I P E I
•
CHENNAI
3
SERIES IN SOFT CONDENSED MATTER Founding Advisor: Pierre-Gilles de Gennes (1932–2007) Nobel Prize in Physics 1991 Collège de France Paris, France
ISSN: 1793-737X
Series Editors: David Andelman Tel-Aviv University Tel-Aviv, Israel Günter Reiter Universität Freiburg Freiburg, Germany
Published: Vol. 1
Polymer Thin Films edited by Ophelia K. C. Tsui and Thomas P. Russell
Vol. 2
Polymers, Liquids and Colloids in Electric Fields: Interfacial Instabilities, Orientation and Phase Transitions edited by Yoav Tsori and Ullrich Steiner
Vol. 3
Understanding Soft Condensed Matter via Modeling and Computation edited by Wenbing Hu and An-Chang Shi
Alvin - Understanding Soft Condensed.pmd
2
9/29/2010, 11:40 AM
Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.
UNDERSTANDING SOFT CONDENSED MATTER VIA MODELING AND COMPUTATION Series in Soft Condensed Matter — Vol. 3 Copyright © 2011 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.
For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.
ISBN-13 978-981-4295-58-1 ISBN-10 981-4295-58-2
Printed in Singapore.
Alvin - Understanding Soft Condensed.pmd
1
9/29/2010, 11:40 AM
September 29, 2010
13:44
World Scientific Review Volume - 9in x 6in
Foreword
The study of Soft Condensed Matter has stimulated fruitful interactions between physicists, chemists, and engineers, and is now reaching out to biologists. A broad interdisciplinary community involving all these areas of science has emerged over the last 30 years, and with it our knowledge of Soft Condensed Matter has grown considerably with the active investigations of polymers, supramolecular assemblies of designed organic molecules, liquid crystals, colloids, lyotropic systems, emulsions, biopolymers and biomembranes, among others. Taking into account that research in Soft Condensed Matter involves ideas coming from physics, chemistry, materials science as well as biology, this series may form a bridge between all these disciplines with the aim to provide a comprehensive and substantial understanding of a broad spectrum of phenomena relevant to Soft Condensed Matter. The present Book Series, initiated by the late Pierre-Gilles de Gennes, comprises independent book volumes that touch on a wide and diverse range of topics of current interest and importance, covering a large number of diverse aspects, both theoretical and experimental, in all areas of Soft Condensed Matter. These volumes will be edited books on advanced topics with contributions by various authors and monographs in a lighter style, written by experts in the corresponding areas. The Book Series mainly addresses graduate students and junior researchers as an introduction to new fields, but it should also be useful to experienced people who want to obtain a general idea on a certain topic or may consider a change of their field of research. This Book Series aims to provide a comprehensive and instructive overview of all Soft Condensed Matter phenomena. The present volume of this Book Series, edited by Wenbing Hu and An-Chang Shi, provides a thorough set of examples for the power of modeling and computer simulations of Soft Condensed Matter. Each chapter addresses a different technique or system, but as an ensemble, all contributions provide evidence on how simulation approaches can complement v
00c-foreword
September 29, 2010
vi
13:44
World Scientific Review Volume - 9in x 6in
Foreword
theory and experiment, bridging the gap between atomistic models and coarse-grained ones. The diversity of the selected fields is representative to the large spectrum of Soft Condensed Matter. As special effort has been made to review important concepts and simulation techniques on an introductory level, we expect that this volume will serve as a useful guide to graduate students and young investigators who aim to get a broad overview on modeling approaches and simulation techniques used for Soft Condensed Matter. Within the next few years, our Series on Soft Condensed Matter will grow continuously and eventually cover the whole spectrum of phenomena in Soft Condensed Matter. We hope that many interested colleagues and scientists will participate and profit from these endeavours.
David Andelman and G¨ unter Reiter Series Editors
00c-foreword
September 29, 2010
13:45
World Scientific Review Volume - 9in x 6in
Preface
Soft condensed matter physics is concerned with the study of soft materials. Examples of soft materials are polymers, bio-macromolecules, liquid crystals, surfactant solutions, colloidal suspensions, and lipids. The territory of these materials extends to plastics, rubber, textiles, pharmaceuticals, foodstuffs and biomaterials, which are important and useful materials for our daily life and high-tech industries. Furthermore, soft materials provide the ground for the birth of new physics based on the theme of softness, malleability and fragility. At the extreme, all living organisms can be regarded as soft matter. For these reasons, it is important to be able to understand and predict the structural and dynamical properties of soft materials. The present volume of the book series in soft condensed matter focuses on the modeling and simulation of soft matter, as an essential investigation often concerted with the efforts of experiment and theory. One of the advantages of computer simulations is that they allow us to vary controlling parameters of microscopic models and to visualize physical processes in detail. This makes the simulation approach particularly valuable in the study of complex physiochemical systems such as soft condensed matter. Although simulations of a given physiochemical system are ideally carried out on atomistic models, using this approach for macroscopic systems sets the demand for us to track a huge number of particles, of the order of 1023 or so, which is a formidable task even for current computational abilities. In order to overcome this difficulty, a variety of coarse-grained models have been developed. The tasks of modeling and simulation are two-fold: to develop model parameters in the coarse-grained models arising from atomistic models, and to understand structure and dynamics of soft matter from simulations of the coarse-grained models. Both tasks require the development of sophisticated simulation methodologies and algorithms. This book collects a set of mini-reviews on a number of topics in soft condensed matter physics, emphasizing the advantages of simulation vii
00d-preface
September 29, 2010
viii
13:45
World Scientific Review Volume - 9in x 6in
Preface
approaches. Each mini-review demonstrates how a carefully chosen algorithm leads to the understanding of some key properties of soft matter. Because the book is aimed at non-specialists, topics in each chapter are placed in a broader context combining experimental and theoretical approaches while technical issues are discussed at an introductory level. The content of this book can be roughly divided into five parts, covering colloids, polymers, bio-macromolecules, liquid crystals and fluid droplets. Using these various soft matter systems as a platform, a number of simulation techniques are introduced, including molecular dynamics simulations, Monte Carlo simulations, self-consistent field theory, density functional theory, quantum chemistry modeling, and finite element method of fluid mechanics. The first part begins with a brief overview by Daan Frenkel on the role of computer simulations in the study of soft matter, in particular, of colloids. The rationale of this approach is based on the fact that the weakness of cohesive energies of soft matter allows the entropic forces to drive phase transitions. This intrinsic feature of soft matter can be captured by an extension of the classical density functional theory, which is introduced by Hartmut L¨ owen in the second chapter. The non-equilibrium problems of soft matter, like colloids, liquid crystals and polymer solutions can be tackled by the dynamic density functional theory derived from the Smoluchowski equation. The second part focuses on phase separation and crystallization in polymeric systems. Marcus M¨ uller first describes the modeling of liquid–liquid demixing in polymer blends and solutions, as well as simulations of interfacial properties, phase boundaries in confined geometries, and wetting. Then, Feng Qiu et al. present an introduction to the self-consistent-field theory (SCFT) for the study of the phase behavior of block copolymers, focusing on a new generic reciprocal-space method for the discovery of ordered block copolymer phases. Furthermore, Toshihiro Kawakatsu introduces the recent development of dynamic SCFT in the study of nonequilibrium behaviors of diblock copolymers with various geometries of microdomains. Turning to polymer crystallization, Takashi Yamamoto combines molecular dynamics and Monte Carlo simulations to study crystallization of helical polymers, using isotactic polypropylene as the model system to demonstrate the relationship between the molecular chirality and the symmetry of the lattice. Finally, Wenbing Hu presents a study of the interplay of liquid–liquid demixing and polymer crystallization in multi-component systems. Dynamic Monte Carlo simulations are performed
00d-preface
September 29, 2010
13:45
World Scientific Review Volume - 9in x 6in
Preface
00d-preface
ix
according to thermodynamic designs of interplay based on a mean-field theory. The third part of the book focuses on modeling and simulation of biologically important macromolecules containing hydrophobic interactions, hydrogen bonding and electrostatics. Chwen-Yang Shew and Kenichi Yoshikawa make a comparison between experiments and modeling on collapse transition of a giant DNA. The effects of chain rigidity, charge density, inter- and intra-molecular segregation, and temperature have been extensively studied. Jing Ma et al. present techniques combining quantum and molecular mechanics for the modeling of hydrogen bonding in solvated long oligomers and cyclic peptides. Naji et al. discuss the unusual features of electrostatic interactions between macromolecules. In the fourth part, Rauzah Hashim presents a survey of computer simulations of liquid crystals. Five classes of modeling have been identified, followed with applications of variants and hybrid models derived from these classes to elucidate the nature of mesophases and their relationships with molecular geometries and intermolecular forces. In the last part, James J. Feng et al. introduce finite element computations on the basis of diffuse-interface theory, demonstrating unusual behavior of interfaces between complex fluids. Two examples are partial coalescence of a droplet on a planar interface, and the self-assembly of droplets suspended in a nematic liquid crystal. We expect that this book will be beneficial to those readers working in the interdisciplinary area of soft matter physics and biophysics with some background in physics, chemistry, biological sciences and materials sciences. We believe that the selected contributions on the whole reflect the diversity of recent research activities in soft condensed matter, and provide useful guidance to graduate students and young investigators who are starting their research career in modeling and simulation of soft matter.
Editors Wenbing Hu Nanjing, Jiangsu Province, China An-Chang Shi Hamilton, Ontario, Canada
This page is intentionally left blank
September 29, 2010
13:45
World Scientific Review Volume - 9in x 6in
00e-contents
Contents
Foreword
v
Preface
vii
Introduction. Role of Modeling in Soft Matter Physics D. Frenkel 1.
Applications of Density Functional Theory in Soft Condensed Matter H. L¨ owen 1
2
3
4
Freezing of Spheres . . . . . . . . . . . . . . . . . . . . . . . 1.1 Phenomenological results . . . . . . . . . . . . . . . . 1.2 Independent treatment of the different phases . . . . 1.3 Unifying Microscopic theories . . . . . . . . . . . . . 1.4 Phase diagrams of simple potentials . . . . . . . . . . 1.5 Density Functional Theory (DFT) . . . . . . . . . . Brownian Dynamics . . . . . . . . . . . . . . . . . . . . . . 2.1 Brownian dynamics (BD) . . . . . . . . . . . . . . . 2.2 BD computer simulations . . . . . . . . . . . . . . . 2.3 Dynamical density functional theory (DDFT) . . . . 2.4 An example: Crystal growth at imposed nucleation clusters . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Hydrodynamic interactions . . . . . . . . . . . . . . Rod-Like Particlesx . . . . . . . . . . . . . . . . . . . . . . 3.1 Statistical mechanics of rod-like particles . . . . . . . 3.2 Simple models . . . . . . . . . . . . . . . . . . . . . . 3.3 Brownian dynamics of rod-like particles . . . . . . . 3.4 “Active” (self-propelled) Brownian particles . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . xi
1
9 11 11 12 13 14 17 23 23 27 27 29 31 33 33 38 40 42 42
September 29, 2010
13:45
World Scientific Review Volume - 9in x 6in
Contents
xii
2.
Polymer Phase Separation M. M¨ uller
47
1 2
47 56 56
3
3.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . Phase Behavior in the Bulk . . . . . . . . . . . . . . . . . . 2.1 Predictions of the mean-field theory . . . . . . . . . . 2.2 Estimating the Flory-Huggins parameter for simulation models . . . . . . . . . . . . . . . . . . . . 2.3 Simulation techniques for computing the bulk phase behavior . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Compressible mixtures . . . . . . . . . . . . . . . . . Outlook: Interfacial Properties, Phase Boundaries in Confined Geometry, and Wetting . . . . . . . . . . . . . . .
Self-Consistent Field Theory of Block Copolymers F. Qiu, A.-C. Shi and Y. Yang 1 2
3 4 5 4.
00e-contents
Introduction . . . . . . . . . . . . . . . . . . . . . Self-Consistent Field Theory of Block Copolymers 2.1 Polymer model and partition function . . . 2.2 Chain propagators . . . . . . . . . . . . . . 2.3 Self-consistent mean-field theory . . . . . . Reciprocal-Space Formulation . . . . . . . . . . . . Applications of the Reciprocal-Space Method . . . Summary . . . . . . . . . . . . . . . . . . . . . . .
3
63 70 74 85
. . . . . . . .
. . . . . . . .
. . . . . . . .
. . . . . . . .
. 85 . 90 . 90 . 92 . 93 . 95 . 100 . 101
Dynamic Self-Consistent Field Theories for Polymer Blends and Block Copolymers T. Kawakatsu 1 2
59
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . Basic Formalism for Dynamic SCF Theory . . . . . . . . . 2.1 Diffusion flux . . . . . . . . . . . . . . . . . . . . . . 2.2 Convection flux . . . . . . . . . . . . . . . . . . . . . 2.3 Flux induced by external fields . . . . . . . . . . . . Dynamic SCF Theories in Slow Diffusion Regime . . . . . . 3.1 Formulation . . . . . . . . . . . . . . . . . . . . . . . 3.2 Formation process of mesophases of block copolymer melt . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Dynamics and non-equilibrium domain structures in thin films and near solid surfaces . . . . . . . . . . .
105 105 107 108 110 111 112 112 113 115
September 29, 2010
13:45
World Scientific Review Volume - 9in x 6in
00e-contents
Contents
3.4
4
5 5.
. . 117 . . . . .
. . . . .
Molecular Dynamics in Crystallization of Helical Polymers: Crystal Ordering and Chirality Selection T. Yamamoto 1 2 3
4
5
6 6.
Structural phase transitions induced by external fields . . . . . . . . . . . . . . . . . . . . . . . . . Beyond Diffusion Dynamics — Hydrodynamics, Viscoelasticity and Hybrid Techniques . . . . . . . . . . 4.1 Hydrodynamic effects on domain formation . . . 4.2 Dynamic SCF with viscoelastic properties . . . . 4.3 Hybrid simulations with particles and fields . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . .
xiii
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . Our Strategies for Simulating Crystallization in Helical Polymers . . . . . . . . . . . . . . . . . . . . . . . . . . . . Molecular Models and Simulation Methods . . . . . . . . . 3.1 MD simulations . . . . . . . . . . . . . . . . . . . . . 3.2 MC simulation . . . . . . . . . . . . . . . . . . . . . Crystallization of the Bare Helix . . . . . . . . . . . . . . . 4.1 A primary nucleation of a single polymer in vacuo . 4.2 Crystallization of a single polymer on a growth front 4.3 Order-disorder transition and crystal chirality . . . . 4.4 Development of chiral crystal . . . . . . . . . . . . . Simulations for iPP, a Helical Polymer with Side Groups . . 5.1 Collapsing of a single iPP chain in vacuo . . . . . . . 5.2 Crystallization of a single chain with definite chiral recognition . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Crystallization and polymorph selection . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . .
Interplay of Liquid-Liquid Demixing and Polymer Crystallization W. Hu 1 2 3 4
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . Theoretical Model . . . . . . . . . . . . . . . . . . . . . . . Simulation Techniques . . . . . . . . . . . . . . . . . . . . . Results and Discussion . . . . . . . . . . . . . . . . . . . . . 4.1 Liquid-liquid demixing enhanced by crystallizability . 4.2 Crystal nucleation enhanced by prior L-L demixing .
121 121 124 127 129
133 134 136 138 138 140 142 142 147 150 156 163 164 166 170 175
179 179 182 184 186 186 188
September 29, 2010
13:45
World Scientific Review Volume - 9in x 6in
00e-contents
Contents
xiv
4.3
5 7.
Elucidation of Single Molecular Observation of a Giant DNA C.-Y. Shew and K. Yoshikawa
207
1 2
208
3
4
5
6
7
8
8.
Crystal nucleation enhanced by prior L-L demixing in the single-chain systems . . . . . . . . . . . . . . . . 199 Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
Polymer Physics Aspects of DNA Conformation . . . . . . Manipulation and Measurement DNA Conformation In Vitro . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Condensing agents . . . . . . . . . . . . . . . . . . . 2.2 Single molecular images . . . . . . . . . . . . . . . . 2.3 Limitation of traditional light scattering . . . . . . . All-or-none Conformation Transition of DNA . . . . . . . . 3.1 Discrete conformational transition of DNA . . . . . . 3.2 Chain stiffness and discrete conformational transition . . . . . . . . . . . . . . . . . . . . . . . . Dynamics of Conformational Relaxation . . . . . . . . . . . 4.1 Time dependent conformational behavior . . . . . . . 4.2 Folding and unfolding kinetics . . . . . . . . . . . . . Conformational Hysteresis . . . . . . . . . . . . . . . . . . . 5.1 Characterization of hysteresis . . . . . . . . . . . . . 5.2 Thermodynamics in conformational hysteresis . . . . 5.3 Hysteresis under mechanical forces . . . . . . . . . . Effect of Charge on DNA Conformation . . . . . . . . . . . 6.1 Conditions to induce intramolecular segregation . . . 6.2 Polyelectrolyte analogy . . . . . . . . . . . . . . . . . 6.3 Phase diagram of intermolecular and intramolecular segregation . . . . . . . . . . . . . . . . . . . . . . . Temperature Effect of DNA Conformation . . . . . . . . . . 7.1 Temperature induced conformational change . . . . . 7.2 Competition of smaller ions on compaction . . . . . . Applications . . . . . . . . . . . . . . . . . . . . . . . . . . Concluding Remarks . . . . . . . . . . . . . . . . . . . . . .
Theoretical Modeling of Hydrogen Bonding in Macromolecular Solutions: The Combination of Quantum Mechanics and Molecular Mechanics J. Ma, N. Jiang and H. Li 1
209 209 210 212 212 212 213 214 215 216 218 218 219 221 224 224 224 225 228 228 230 230 233
237
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 238
September 29, 2010
13:45
World Scientific Review Volume - 9in x 6in
Contents
2
3
4 9.
Fragmentation-Based QM/MM Simulations . . . . . . . . . 2.1 Solvent models . . . . . . . . . . . . . . . . . . . . . 2.2 Energy-based fragmentation QM . . . . . . . . . . . 2.3 Fragmentation QM/MM: Basic idea and formalisms 2.4 Fragmentation QM/MM simulations on poly(ethylene oxide) polyethylene . . . . . . . . . . . Simulations of Solvated Peptides Using Polarizable Force Field Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Fragmentation-based polarization model . . . . . . . 3.2 Configurations of solvated α-conotoxin GI and its analogues . . . . . . . . . . . . . . . . . . . . . . . . Concluding Remarks . . . . . . . . . . . . . . . . . . . . . .
00e-contents
xv
239 239 240 241 245 248 249 251 255
Exotic Electrostatics: Unusual Features of Electrostatic Interactions between Macroions A. Naji, M. Kanduˇc, R. R. Netz and R. Podgornik
265
1 2
266 268
3 4
5
6
7
8
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . Scenery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Colloids, polymers and membranes: The mesoscopic scale . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Charges: From industry to biology . . . . . . . . . . 2.3 Theoretical challenge and coarse-grained models . . . Length Scales in a Classical Charged System . . . . . . . . From Mean-Field to Strong Coupling Regime . . . . . . . . 4.1 Weak coupling or mean-field regime . . . . . . . . . . 4.2 Strong coupling regime . . . . . . . . . . . . . . . . . Interactions between Like-Charged Surfaces . . . . . . . . . 5.1 WC regime: Repulsion . . . . . . . . . . . . . . . . . 5.2 SC regime: Attraction . . . . . . . . . . . . . . . . . Counterions with Salt . . . . . . . . . . . . . . . . . . . . . 6.1 Functional integral formalism . . . . . . . . . . . . . 6.2 Dressed counterions . . . . . . . . . . . . . . . . . . . 6.3 WC dressed counterion theory . . . . . . . . . . . . . 6.4 SC dressed counterion theory . . . . . . . . . . . . . Counterions between Randomly Charged Surfaces . . . . . 7.1 General formalism: The replica method . . . . . . . . 7.2 Disorder effects in the WC regime . . . . . . . . . . . 7.3 Disorder effects in the SC regime . . . . . . . . . . . Lessons . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
268 269 271 272 273 273 274 276 277 278 279 279 281 282 282 286 287 288 288 291
September 29, 2010
13:45
World Scientific Review Volume - 9in x 6in
00e-contents
Contents
xvi
10. Computer Modeling of Liquid Crystals R. Hashim 1
2
3
297
Introduction . . . . . . . . . . . . . . . . . . . . . . 1.1 What is a liquid crystal? . . . . . . . . . . . . 1.2 Theoretical approach to understanding liquid crystals . . . . . . . . . . . . . . . . . . . . . Introduction to the Computer Simulation . . . . . . 2.1 Computer simulation techniques . . . . . . . . 2.2 Simulation limitations . . . . . . . . . . . . . Liquid Crystal Models . . . . . . . . . . . . . . . . . 3.1 The lattice class . . . . . . . . . . . . . . . . . 3.2 The Gay-Berne class . . . . . . . . . . . . . . 3.3 Full atomistic class . . . . . . . . . . . . . . . 3.4 Conclusion . . . . . . . . . . . . . . . . . . . .
. . . . 297 . . . . 298 . . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
11. Drop Dynamics in Complex Fluids J. J. Feng, X. Chen, P. Yue and C. Zhou 1 2
3
4 Index
Introduction . . . . . . . . . . . . . . . . . . . . . Partial Coalescence in Polymer Solutions . . . . . 2.1 Experimental observations . . . . . . . . . . 2.2 Numerical simulations . . . . . . . . . . . . Droplet Self-Assembly in Nematic Liquid Crystals 3.1 Experimental observations . . . . . . . . . . 3.2 Numerical simulations . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . .
301 304 305 306 307 308 317 325 331 339
. . . . . . . .
. . . . . . . .
. . . . . . . .
. . . . . . . .
. . . . . . . .
340 343 343 346 352 352 355 360 365
September 29, 2010
10:41
World Scientific Review Volume - 9in x 6in
01-introduction
Introduction Role of Modeling in Soft Matter Physics
Daan Frenkel Department of Chemistry, University of Cambridge, Lensfield Road, Cambridge CB2 1EW, UK
Writing an introduction to a book on Modeling Soft Matter is difficult because this is a field that arguably has links with more other disciplines than any other branch of physics or chemistry. In fact, the contents of this book illustrate this and the subjects covered range from polymer science and quantum mechanics to hydrodynamics and biophysics. There are, however, certain unifying features in soft-matter science. The first one is, not surprisingly, that soft matter is “soft”. In practice, this means that soft materials have elastic moduli that are a million to a billion times smaller than that of, say, steel. This softness can be understood on the basis of dimensional arguments. Elastic moduli have the dimension energy/length3 . In hard (covalently bonded) materials, the relevant energy scale is that of the chemical bond, which is typically expressed in electronvolts. The characteristic length scale is the interparticle spacing in a solid. This distance is of the order of a few ˚ A. Hence, typical elastic constants of such materials are of the order of 109 Pa. Now soft matter. Here the 1 relevant energy is the thermal energy, kB T, which is about 40 eV. The typical distances span a wide range: from several nanometers to microns. As a consequence, typical elastic constants should have values between 1 and 105 Pa. For modeling, the above observation is of crucial importance. First of all, the fact that these materials consist of building blocks that have linear dimensions that may be up to 10,000 times larger than an atom, means that fully atomistic modeling is almost always hopeless. A single colloid may contain some 1010 atoms. A colloidal suspension of only 100 colloids in a solvent will contain 1012 particles. Clearly, atomistic modeling is not an 1
September 29, 2010
2
10:41
World Scientific Review Volume - 9in x 6in
01-introduction
D. Frenkel
option. The only exception — and a partial one at that — is the behavior of particles or structures at the very lower end of this range of length scales: e.g. small mesogens or proteins. But even there it is rare to see simulations that contain more than a few proteins in water. The atomistic simulation of an entire living cell is, for the foreseeable future, utterly impossible. There is a second case where atomistic simulations of soft matter may work, namely in cases where the local properties of the macro-molecular units are important. For instance, one can study the structure of a polymer crystal with atomistic simulations, precisely because the molecules are so large: in the crystal there are few free end points and, as a consequence, the local structure does not “feel” the length of the building blocks, only their lateral extent. However, in the overwhelming majority of cases soft matter has to be modeled using a coarse-graining approach: polymers are modeled as chains of self-avoiding Kuhn segments and colloids are modeled as spheres (or rods, or platelets), to name but two examples. The energy-scale is equally important: it is the thermal energy and hence thermal fluctuations are always important in soft materials. This is where the science of soft matter starts to diverge from that of, say, granular materials. In the latter, the length scales are large (larger even than for soft materials) but thermal fluctuations play no role. Where is the boundary between colloids and granular materials? The behavior of colloids is governed by the laws of statistical mechanics. In equilibrium, colloidal suspensions occur in the phase with the lowest free energy, and the dynamics of colloids in equilibrium is due to thermal (“Brownian”) motion. In principle, this should also be true for grains of sand. But in practice, it is not. In order for grains to behave like colloids, they should be able to evolve due to Brownian motion. There are two reasons why grains of sand do not. First of all, on earth, all particles are subject to gravity. The probability to find a particle of mass m at a height h above the surface of the earth is given by the barometric height distribution P (h) = exp(−mgh/kB T ) ,
(1)
where m is the effective mass of the particle (in the case of a colloid: the mass of the particle minus the mass of the displaced solvent), T is the temperature and kB is Boltzmann’s constant. The average height of the particle above the bottom of the container is equal to < h >= kB T /(mg). For a 1 milligram grain of sand at room temperature, < h >= O(10−14 ) cm. This tells us something that we all know: the beach is not seething with thermal motion. One way to delimit the colloidal regime is to require that
September 29, 2010
10:41
World Scientific Review Volume - 9in x 6in
Role of Modeling in Soft Matter Physics
01-introduction
3
< h > is larger than the particle diameter. Suppose we have a spherical particle with diameter σ and (excess) mass density ρ, then our criterion implies π gρσ 4 = kB T . (2) 6 For a particle with an excess density of 1 g/cm3 , the above equality is satisfied for σ ≈ 1 µm. So, on earth anything larger than a few microns is dominated by gravity, not by thermal motion: grains of sand are not colloids. Naively one might think that in space, much larger particles could behave like colloids. Yet, even if we would launch a grain of sand in a spaceshuttle, that grain would still not behave like a colloid. The reason is the following: colloidal particles should be able to move due to diffusion (i.e. thermal motion). How long does it take for a particle to move a distance equal to its own diameter? In a time t, a particle typically diffuses a distance √ 2Dt. For a spherical particle, the diffusion constant is given by the StokesEinstein relation D = kT /(3πησ), where η is the viscosity of the solution. Hence, a particle diffuses a distance comparable to its own diameter in a time τ = O(ησ 3 /kT ) .
(3)
For a 1µm-colloid in water, this time is of the order of one second. For a 1 mm grain of sand, it is of the order of 30 years. Hence, even though grains of sand in zero-gravity may behave like colloids, they will not do so on a human time-scale. Clearly, what we define as a colloid, also depends on the observation time. Again, 1 micron comes out as a natural upper limit to the colloidal domain. Although soft materials all have length scales within a certain range, their properties may still be very different. This is particularly clear when we consider the role of entropy in the phase behavior of soft matter. As I will briefly discuss below, entropy is extremely important (often even dominant) in driving phase transitions in colloidal systems. However, in polymer mixtures, entropy is much less important. So much so, that a mixture of deuterated polymer and hydrogenated polymers can (under suitable conditions) demix spontaneously: the entropy of mixing is too small to compensate for the tiny difference in interaction energies of H and D polymers. 1 The reason is that the interaction energy of a polymer of type A in a melt of type B is proportional to the number of atoms in that polymer. Yet the mixing entropy NA polymers of type A in a melt of type B is proportional to NA , the number of polymers. Clearly, the number of polymers is much
September 29, 2010
4
10:41
World Scientific Review Volume - 9in x 6in
01-introduction
D. Frenkel
smaller than the number of atoms. In “hard matter” physics, the 3 He-4 He mixture is the only example of isotopes demixing — and that only at very low temperatures. Now let us consider the role of entropy in colloidal phase transitions. The second law of thermodynamics tells us that any spontaneous change in a closed system results in an increase of the entropy, S. In this sense, all spontaneous transformations of one phase into another are entropy driven. However, this is not what I mean with the term “entropy-driven phase transitions”. When discussing ordering transitions one usually considers a system that can exchange energy with its surroundings. In that case, the second law of thermodynamics implies that the system will tend to minimize its Helmholtz free energy F = E − T S, where E is the internal energy of the system and T the temperature. Clearly, a system at constant temperature can lower its free energy in two ways: either by increasing the entropy S, or by decreasing the internal energy E. In order to gain a better understanding of the factors that influence phase transitions, we must look at the statistical mechanical expressions for entropy. The simplest starting point is to use Boltzmann’s expression for the entropy of an isolated system of N particles in volume V at an energy E, S = kB ln Ω
(4)
where kB , the Boltzmann constant, is simply a constant of proportionality. Ω is the total number of (quantum) states that is accessible to the system. In the remainder of this paper, I shall choose my units such that kB =1. The usual interpretation of Eqn. 4 is that Ω, the number of accessible states of a system, is a measure for the “disorder” in that system. The larger the disorder, the larger the entropy. This interpretation of entropy suggests that a phase transition from a disordered to a more ordered phase can only take place if the loss in entropy is compensated by the decrease in internal energy. This statement is completely correct, provided that we use Eqn. 4 to define the amount of disorder in a system. However, we also have an intuitive idea of order and disorder: we consider crystalline solids “ordered”, and isotropic liquids “disordered”. This intuitive picture suggests that a spontaneous phase transition from the fluid to the crystalline state can only take place if the freezing lowers the internal energy of the system sufficiently to outweigh the loss in entropy: i.e. the ordering transition is “energy driven”. In many cases, this is precisely what happens. It would, however, be a mistake to assume that our intuitive definition of order always coincides with the one based on Eqn. 4.
September 29, 2010
10:41
World Scientific Review Volume - 9in x 6in
Role of Modeling in Soft Matter Physics
01-introduction
5
In order to observe “pure” entropic phase transitions, we should consider systems for which the internal energy is a function of the temperature, but not of the density. Using elementary statistical mechanics, it is easy to show that this condition is satisfied for a system of hard particles. Suspensions of (uncharged, refractive-index-matched) colloids behave effectively as systems of hard particles. Whenever these systems order at a fixed density and temperature, they can only do so by increasing their entropy (because, at constant temperature, their internal energy is fixed). Such systems are conveniently studied in computer simulations. But, increasingly, experimentalists — in particular, colloid scientists, have succeeded in making real systems that behave very nearly as ideal hard-core systems. Hence, the phase transitions discussed below can, and in many cases, do occur in nature. Below I list a few examples of entropic ordering in hard-core systems. But I stress that the list is far from complete. The earliest example of an entropy-driven ordering transition is described in a classic paper of Onsager,2 on the isotropic-nematic transition in a (three-dimensional) system of thin hard rods. Onsager showed that, on compression, a fluid of thin hard rods of length L and diameter D must undergo a transition from the isotropic fluid phase, where the molecules are translationally and orientationally disordered, to the nematic phase. In the latter phase, the molecules are translationally disordered, but their orientations are, on average, aligned. This transition takes place at a density such that (N/V )L2 D = O(1). Onsager considered the limit L/D → ∞. In this case, the phase transition of the hard-rod model can be found exactly (see e.g. Ref. 3). At first sight it may seem strange that the hard rod system can increase its entropy by going from a disordered fluid phase to an orientationally ordered phase. Indeed, due to the orientational ordering of the system, the orientational entropy of the system decreases. However, this loss in entropy is more than offset by the increase in translational entropy of the system: the available space for any one rod increases as the rods become more aligned. In fact, we shall see this mechanism returning time-and-again in ordering transitions of hard-core systems: the entropy decreases because the density is no longer uniform in orientation or position, but the entropy increases because the free-volume per particle is larger in the ordered than in the disordered phase. The most famous, and for a long time controversial, example of an entropy-driven ordering transition is the freezing transition in a system of hard spheres. This transition had been predicted by Kirkwood in the early fifties4 on basis of an approximate theoretical description of the hard-
September 29, 2010
6
10:41
World Scientific Review Volume - 9in x 6in
01-introduction
D. Frenkel
sphere model. As this prediction was quite counter-intuitive and not based on any rigorous theoretical results, it met with wide-spread skepticism until Alder and Wainwright5 and Wood and Jacobson6 performed numerical simulations of the hard-sphere system that showed direct evidence for this freezing transition. Even then, the acceptance of the idea that freezing could be an entropy driven transition, came only slowly.7 However, by now, the idea that hard spheres undergo a first-order freezing transition is generally accepted. And, although the hard-sphere model was originally devised as an idealized and highly unrealistic model of an atomic fluid, it is now realized that this model provides a good description of certain classes of colloidal systems. The next step in this sequence came in the mid-eighties when computer simulations8 showed that hard-core interactions alone could also explain the formation of more complex liquid crystals. In particular, it was found that a system of hard sphero-cylinders (i.e. cylinders with hemi-spherical caps) can form a smectic liquid crystal, in addition to the isotropic liquid, the nematic phase and the crystalline solid. In the smectic (A) phase, the molecules are orientationally ordered but, in addition, the translational symmetry is broken: the system exhibits a one-dimensional density-modulation. Subsequently, it was found that some hard-core models could also exhibit columnar ordering.9 In the latter case, the molecules assemble in liquidlike stacks, but these stacks order to form a two-dimensional crystal. In summary, hard-core interaction can induce orientational ordering and one-, two- and three-dimensional positional ordering. This is rather surprising because, in particular for the smectic and the columnar phase, it was generally believed that their formation required specific energetic interactions. A recent addition to the list of entropic phase transitions is, interestingly, the liquid-vapor transition. Since the seminal work of van der Waals, the standard picture of the liquid-vapor transition is that it is due to the competition between entropic repulsion (the gas molecules strive to maximize the volume that they can access) and energetic attraction (due to the so-called van der Waals dispersion forces). Hence, the condensation of a liquid seems to be the prototypical example of an energy-driven phase transition: the system “pays” for the loss in entropy by delivering the heat of condensation to the surroundings where it will increase the entropy. However, in soft matter things are, again, different. This is in particular true in the case of colloidal particles coated with very long polymers (e.g. DNA) that may form bridges between different particles. If the binding is very strong, there are no “dangling” polymers. For instance, in the case of hard
September 29, 2010
10:41
World Scientific Review Volume - 9in x 6in
Role of Modeling in Soft Matter Physics
01-introduction
7
colloids linked by DNA, the low density phase consists of colloidal “dimers” linked by a number of DNA strands.10 As there are no dangling bonds, the energy of this system is independent of density. Yet, beyond a certain density, the system undergoes a transition from a dilute vapor phase to a dense liquid phase. The reason is that there are many more ways of arranging the bonds in a dense phase than in the gas phase. Hence, the entropy of the network drives the condensation. I have mentioned these examples to illustrate that modeling soft materials is an exciting activity. Often, we observe/predict phenomena that have no counterpart in “hard matter”. These unusual properties of soft materials are not simply of academic interest. Increasingly, complex self-assembling materials are being produced with properties that rely on the “soft” nature of their building blocks. In fact, all living matter is an illustration of the amazing diversity of structures that can be formed by soft building blocks. In order to unravel the underlying design principles, simulation still has a long way to go. The chapters in this book discuss different examples of progress in this direction. The editors, Professors Wenbing Hu and An-Chang Shi have succeeded in collecting a number of very interesting aspects of soft-matter modeling. As the reader will see, in all its diversity, the book shows the underlying unity of the subject. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.
F. S. Bates and P. Wiltzius, J. Chem. Phys. 91, 3258 (1989). L. Onsager, Proc. NY. Acad. Sci. 51, 627 (1949). R. F. Kayser and H. J. Raveche, Phys. Rev. A17, 2067 (1978). J. E. Kirkwood, in Phase Transformations in Solids, R. Smoluchowski, J. E. Mayer and W. A. Weyl (eds.), Wiley, New York (1951), p. 67. B. J. Alder and T. E. Wainwright, J. Chem. Phys. 27, 1208 (1957). W. W. Wood and J. D. Jacobson, J. Chem. Phys. 27, 1207 (1957). The Many-Body Problem, J. K. Percus, editor, Interscience, New York, 1963. D. Frenkel, H. N. W. Lekkerkerker and A. Stroobants, Nature 332, 822 (1988). J. A. C. Veerman and D. Frenkel, Phys. Rev. A45, 5633 (1992). B. Bozorgui and D. Frenkel, Phys. Rev. Lett. 101, 045701 (2008).
This page is intentionally left blank
October 7, 2010
15:52
World Scientific Review Volume - 9in x 6in
Chapter 1 Applications of Density Functional Theory in Soft Condensed Matter Hartmut L¨ owen Heinrich-Heine University D¨ usseldorf, Germany Applications of classical density functional theory (DFT) to soft matter systems like colloids, liquid crystals and polymer solutions are discussed with a focus on the freezing transition and on nonequilibrium Brownian dynamics. First, after a brief reminder of equilibrium density functional theory, DFT is applied to the freezing transition of liquids into crystalline lattices. In particular, spherical particles with radially symmetric pair potentials will be treated (like hard spheres, the classical one-component plasma or Gaussian-core particles). Second, the DFT will be generalized towards Brownian dynamics in order to tackle nonequilibrium problems. After a general introduction to Brownian dynamics using the complementary Smoluchowski and Langevin pictures appropriate for the dynamics of colloidal suspensions, the dynamical density functional theory (DDFT) will be derived from the Smoluchowski equation. This will be done first for spherical particles (e.g. hard spheres or Gaussian-cores) without hydrodynamic interactions. Then we show how to incorporate hydrodynamic interactions between the colloidal particles into the DDFT framework and compare to Brownian dynamics computer simulations. Third orientational degrees of freedom (rod-like particles) will be considered as well. In the latter case, the stability of intermediate liquid crystalline phases (isotropic, nematic, smectic-A, plastic crystals etc) can be predicted. Finally, the corresponding dynamical extension of density functional theory towards orientational degrees of freedom is proposed and the collective behaviour of “active” (self-propelled) Brownian particles is briefly discussed.
Introduction Apart from their fundamental importance, soft matter materials (as colloidal dispersions, liquid crystals and polymer coils) represent excellent 9
02-chapter1
October 7, 2010
10
15:52
World Scientific Review Volume - 9in x 6in
H. L¨ owen
realizations of strongly interacting classical many-body systems. In this sense, soft matter provides us with model systems which in turn means that a reasonable (coarse-grained) modelling of soft matter is very realistic and can lead to valuable physical insight. This is one of the main reasons why theory, computer simulations and experiments go hand-in-hand in recent soft matter research. A classical many-body system is typically characterized by “effective” pairwise interaction forces [1], derived from an (optimal) effective pair potential V (r) where r denotes the interparticle distance [2]. Lots of theoretical and simulational efforts have been spent in the past to derive or to compute the effective interaction. Subsequently, once the effective pair interaction is known, equilibrium properties like structural correlations and phase transitions can be obtained by theory and computer simulation. This route can also be inversed: If for a particular shape of V (r) novel behaviour is predicted by theory or computer simulation, various soft matter systems (in particular mixtures) can be exploited to find a convenient realization of the prescribed potential V (r). While the concept of effective interactions is valid for static equilibrium quantities, it typically breaks down for dynamical correlations [3] and nonequilibrium situations [4]. The dynamics of colloidal particles and polymers in solution is Brownian due to the separation of time scales between the mesoscopic particles and the solvent molecules. The overdamped particle motion can be described using the Smoluchowski equation or the stochastically equivalent Langevin picture [5, 6]. A wealth of nonequilibrium effects have been discovered in these Brownian systems [7, 8]. In this book chapter we highlight the role of density functional theory in order to calculate phase diagrams in equilibrium and Brownian dynamics in nonequilibrium. Density functional theory represents a microscopic approach to many-body effects where the pair interaction potential V (r) is the only input. Computer simulations, on the other hand, are necessary to provide “benchmark” data to test the theory. For the topics considered here, these are mainly Monte-Carlo simulations for the equilibrium phase diagrams and Brownian dynamics simulations for nonequilibrium dynamics. In this contribution, we focus in particular on the freezing transition of simple systems, governed by a radially-symmetric pair potential (e.g. hard or soft spheres) and on orientational degrees of freedom relevant for e.g. rod-like systems. For both cases, phase diagrams and effects of nonequilibrium Brownian dynamics are described. Some examples from recent research in this field are used to illustrate the capacity of dynamical
02-chapter1
October 7, 2010
15:52
World Scientific Review Volume - 9in x 6in
Applications of Density Functional Theory in Soft Condensed Matter
02-chapter1
11
density functional theory as compared to Brownian dynamics computer simulations [9]. 1. Freezing of Spheres 1.1. Phenomenological results Experiments show that liquids freeze into periodic crystalline structures at low temperatures or high densities. In these states, the translational symmetry of the system is broken, i.e. the one-particle density N ρ(r) = δ(r − ri ) (1) i=1
is inhomogeneous. Here . . . denotes a canonical average, and ri (i = 1, . . . , N ) are the particle positions. As freezing is ubiquitous and general, it is one of the most important phase transition in nature. The basic question is: when does it happen? Answering this question is one of the central tasks of statistical physics. A full microscopic theory is highly desirable which uses the interaction forces as an input and predicts the thermodynamic conditions for freezing as an output. Since freezing is a collective effect, this is a very demanding task. Before turning to such a microscopic approach, let us first collect some empirical facts for freezing, as for more details and references, see [10–15]. We shall also summarize known phase behaviour for simple model potentials gained by computer simulations. i) Lindemann-criterion of melting If a = ρ−1/3 denotes a typical interparticle spacing (with the number density ρ being the number of particles per volume), then one can examine the root mean-square displacement u of a particle around a given crystalline i , i denoting a lattice site index, which is defined as lattice position R i )2 ri − R (2) u = ( This quantity can also be viewed as the spread of the inhomogeneous oneparticle density ρ(r) around a lattice position, see Figure 1. The Lindemann parameter L = u/a measures the fluctuations around the lattice positions in terms of the lattice constant. The traditional Lindemann rule of melting states that a solid melts if L ≈ 0.1. Computer simulations have confirmed this phenomenolgical rule where the actual value
October 7, 2010
15:52
World Scientific Review Volume - 9in x 6in
12
H. L¨ owen
ρ(r)
i
L
R1
R2
r
Fig. 1. One-dimensional sketch of the inhomogeneous one-particle density ρ( r ) in a 1 and R 2 . The spread of the density peak is crystalline solid with lattice points at R embodied in the Lindemann parameter L.
of L at melting varies between 0.129 for hard spheres and 0.18 for the onecomponent plasma. But it is always roughly one order of magnitude smaller than the lattice constant itself. ii) Hansen-Verlet rule of freezing Different to the Lindemann rule, the Hansen-Verlet rule starts from the liquid side of the freezing transition and states that the freezing occurs if the amplitude of the first peak in the liquid structure factor S(k) exceeds 2.85. Originally found for Lennard-Jones systems, this rule has been confirmed also for other interactions like hard-sphere, plasma and Yukawa pair potentials. Without any notable exception a value of 2.5-3.0 was found near equilibrium freezing. However, the peak can be much higher in a metastable glassy state. 1.2. Independent treatment of the different phases The simplest theoretical approach is to construct different theories for the different thermodynamic states (solid and liquid). In particular the internal energy of the solid phase can be accessed by a simple lattice sum of the given pair potentials. In particular, different candidate lattices can be assumed at fixed averaged density, and the one with minimal potential energy will be the stable one for temperature T = 0. Finite temperature corrections based on a harmonic phonon-picture can be added on top of that resulting in a (Helmholtz) free energy F of the solid state. Likewise the free energy of the liquid can be gained by using for instance liquid integral equation theories [16] where different closure schemes may be adopted. Combined with the free energy of the solid, a Mawell double-tangent construction for the isothermal free energy per particles
02-chapter1
October 7, 2010
15:52
World Scientific Review Volume - 9in x 6in
02-chapter1
Applications of Density Functional Theory in Soft Condensed Matter
13
F/N
ρe
ρs
ρ
Fig. 2. Sketch of the Maxwell double tangent construction to the free energy per particle in the liquid and solid phase resulting in the two coexistence densities ρ and ρs .
versus density ρ leads to the coexisting liquid and solid densities ρ and ρs , see Figure 2. The double tangent ensures the equality of the chemical potential and the pressure in the two phases. If this is repeated for various temperature, the full phase diagram emerges. In three spatial dimensions, freezing is typically a first order transition with a considerable density jump ∆ρ = ρs − ρ .
1.3. Unifying Microscopic theories Both from a fundamental and esthetic point of view, a unifying theory which treats both the liquid and the solid phase on the same footing is desirable. In the past decades, there have been considerable advances in this field. In three spatial dimensions, classical density functional theory (DFT) can be used to get a liquid-based description of the solid phase. Here the solid is viewed as a strongly inhomogeneous liquid with strong density peaks. Freezing in DFT is therefore a condensation of liquid density modes [14]. Conversely, in two spatial dimensions the Kosterlitz-Thouless approach is a solid-based approach which treats the liquid as a solid phase with an accumulation of defects. In the following we shall focus on the three-dimensional freezing and on density functional theory. We emphasize that a unifying treatment is mandatory for the description of solid-liquid interface and phenomena like crystal nucleation and growth out of an undercooled melt where indeed a single theory for both phases is needed.
October 7, 2010
15:52
World Scientific Review Volume - 9in x 6in
14
02-chapter1
H. L¨ owen
1.4. Phase diagrams of simple potentials Let us first summarize some familiar phase diagrams for various model pairwise interactions. These were obtained mainly by “exact” computer simulation of a many-body system [17] and therefore provide “benchmark” data for a microscopic theory. a) Hard spheres The simplest nontrivial interaction potential is that for hard spheres of diameter σ. The potential reads ∞, r ≤ σ V (r) = (3) 0, r>σ The internal energy is completely ideal U = 32 N kB T , i.e. the averaged potential energy is zero. Hence Helmholtz free energy F = U − T S scales with kB T alone (as kB T is the only energy scale for hard spheres). Therefore, for hard spheres, the entire thermodynamic behaviour is governed by entropy alone. This becomes different for other interactions which possess an explicit energy scale. This is the main reason why hard spheres are the most important models for freezing. From computer simulations, the hard sphere phase diagram is shown in Figure 3. The only parameter is the density which is conveniently scaled in terms of a volume or packing fraction η = πρσ3 /6. The quantity η measures the ratio of the volume occupied by all spheres to the total available volume V of the system. For η → 0 an ideal gas is recovered, while maximal packing fo hard spheres occurs for η = ηcp = 0.74 corresponding to stacked layers of triangular crystals. In fcc-crystal (ABC stacking)
fluid 0
ηfluid =0.494
fluid
ηsolid =0.545
ηCP =0.74
η
crystal
Fig. 3. Hard sphere freezing diagram versus packing fraction η. The intuitive picture of freezing is also shown: at high densities a fluid state involves blocked configurations and more configurations are achieved by a periodic packing.
October 7, 2010
15:52
World Scientific Review Volume - 9in x 6in
Applications of Density Functional Theory in Soft Condensed Matter
02-chapter1
15
between, there is a first order freezing transition with coexisting packing fractions ηell = 0.494 and ηs = 0, 545. The stable crystalline crystal is facecentred-cubic (fcc) which has an ABC stacking sequence. Interestingly, the freezing transition is driven by entropy. For η > ηs the solid state has a higher entropy than the fluid state clearly showing that entropy has nothing to do with structural order. More intuitively, a disordered fluid state at high densities implies jammed configurations, and much more configuration (i.e. higher entropy) is gained by taking as a reference configuration a solid and generating more configurations from slightly displaced particles (configurational entropy). b) Plasma The one-component plasma (OCP) with neutralizing background is defined by the pairwise Coulomb potential V (r) = V0 /r. By scaling the classical partition function, one can show that only the coupling param√ 3
4πρ
V
eter Γ = kB3T 0 determines the structure and phase behaviour. There is isochoric freezing from the fluid into a body-centered-crystal (bcc) at Γ = 178. c) Soft spheres Inverse power law potentials where V (r) = V0 (σ/r)n interpolate between the plasma (n = 1) and the hard sphere potential, formally obtained as n → ∞. Depending on n either bcc or fcc crystals are stable. d) Yukawa-system The Yukawa potential V (r) = V0 exp(−κr)/r applies e.g. to chargestabilized colloidal suspensions. Again κ interpolates between the OCP (κ = 0) and the hard-sphere-limit κ → ∞. The phase diagram depends λ 1 only on λ = κa (a = ρ− 3 ) and T˜ = kVB0T eλ and involves a fluid, a bcc solid and an fcc solid with a triple point as sketched in Figure 4. e) Lennard-Jones-system The Lennard-Jones the traditional model for rare gases, is potential, σ 6 σ 12 where is the energy and σ is the length given by V (r) = 4 r − r scale. This potential has a long-ranged attractive tail. Correspondingly it exhibits also a critical point separating a gas from a liquid and a triple point with gas-liquid-fcc solid coexistence.
October 7, 2010
15:52
World Scientific Review Volume - 9in x 6in
16
02-chapter1
H. L¨ owen
T
triple point 0.2 fluid fluid
fcc bcc bcc
0 plasma
2
4
6
λ
Fig. 4. Sketch of the Yukawa phase diagram in the plane spanned by λ and T˜. The special case λ = 0 is the one-component plasma.
f) Sticky hard spheres Sticky hard sphere possess a square-well attraction and are reasonable models for proteins. Here ∞ r ≤ σ V (r) = − σ ≤ r ≤ σ(1 + δ) (4) 0 elsewhere with a finite attraction range δσ and an attraction depth of − . The scaled range δ must be larger than about 0.25 in order to get a liquid-gas separation. For small δ < 0.05 there is a novel isostructural solid-solid-transition with a critical point [18]. g) Ultrasoft interactions Soft (floppy) objects like polymer coils have effective interactions which are even softer than the plasma, therefore these interactions are called “ultrasoft” [19]. A log-Yukawa interaction has been proposed for star polymers. Here the interaction diverges at the origin only logarithmically with distance, i.e. V (r) ∝ kB T ln(r/σ). The phase behaviour [20] involves fluid, bcc and fcc solids as well as body-centered-orthogonal and diamond lattices and exhibits reentrance effects.
October 7, 2010
15:52
World Scientific Review Volume - 9in x 6in
Applications of Density Functional Theory in Soft Condensed Matter
02-chapter1
17
h) Penetrable interactions One may even describe soft objects with pair potentials which are finite at the origin. Examples are Gaussian potentials which are a good model for linear polymer coils. The phase behaviour involves again the fluid, bcc-solid and fcc-solid phase with fluid reentrance [21]. Penetrable interactions with other shapes exhibit again also “exotic” solid phases and reentrance [22]. Finite potentials which have a negative Fourier transform exhibit cluster crystals [23] where a lattice points is occupied by more than one particle. To summarize: (1) Hard and “harsh” potentials freeze into fcc lattices. (2) Soft repulsive potentials with an at least 1r singularity for r 0 freeze into bcc lattices. (3) Ultrasoft v(r) ∼ −ln( σr ) and penetrable (V (r 0) = V (0) < ∞) potentials show besides fcc and bcc structures, more open “exotic” lattices and reentrance effects. (4) If the Fourier transform of V (r) has negative parts, a cluster crystal occurs. (5) Attractions lead to gas-liquid coexistence and isostructural solid-solid transition. In conclusion, various shapes of the pairwise interaction potential can lead to a rich phase behaviour and there is the theoretical challenge to construct a microscopic approach in order to predict and reproduce this complex phase behaviour. As will be discussed in the sequel, classical density functional theory for inhomogeneous fluids does provide such an approach. 1.5. Density Functional Theory (DFT) a) Basics The cornerstone of density functional theory (DFT) is an existence theorem combined with a basic variational principle [10]. In detail, there exists a unique grand-canonical free energy-density-functional Ω(T, µ, [ρ]), which gets minimal for the equilibrium density ρ0 (r) and then coincides with the real grandcanoncial free energy, i.e. δΩ(T, µ, [ρ]) = 0. (5) δρ(r) ρ( r )=ρ0 ( r)
October 7, 2010
18
15:52
World Scientific Review Volume - 9in x 6in
02-chapter1
H. L¨ owen
In particular DFT is also valid for systems which are inhomogeneous on a microscopic scale. In principle, all fluctuations are included in an external potential which breaks all symmetries. For interacting systems in 3d, however, Ω(T, µ, [ρ]) is not known. Fortunately, there are few exceptions where the density functional is known exactly. First, for low density, the ideal-gas-limit is reached and the density functional can be constructed analytically (see below). Next leading orders for finite densities can be incorporated via a virial expansion which is quadratic in the densities. Conversely, in the high-density-limit, the meanfield approximation (see below) becomes asymptocially exact for penetrable potentials. Indeed this approximation also works surprisingly well for finite densities beyond overlap. Furthermore, the density functional is exactly known (as so-called Percus-functional) in one spatial dimension for the Tonks gas (had rods on a line). However, the latter system does not exhibit freezing. Please note that the knowledge of a functional is much more than a bulk equation of state since it can be applied to any external potential Vext (r). In principle, the application of DFT to freezing works as follows: First one has to chose an approximation. Then the density field is parameterized with variational parameters. In the homogeneous gas and liquid bulk phase one takes ρ(r) = ρ¯
(6)
where ρ¯ is a variational parameter. On the other hand, for the solid, the Gaussian approximation of density peaks on the lattice positions is an excellent choice [24]. α −3/2 2 n exp −α r − R ρ(r) = (7) π n Here both the lattice structure and spacing as well as the width α are variational parameters. Finally, for a given chemical potential µ and temperature T , one has to minimize the functional Ω(T, µ, [ρ]) with respect to all variational parameters. As a result one obtains the phase diagram in the µT plane. The procedure itself is sketched close to the solid-liquid transition in Figures 5 and 6. A solid-liquid transition line in the µT plane is schematically shown in Figure 5 and we consider a path with fixed µ and increasing T crossing the solid-liquid transition at µ = µcoex and T = Tcoex . Coexistence implies that temperature T , chemical potential µ and pressure p are the
October 7, 2010
15:52
World Scientific Review Volume - 9in x 6in
02-chapter1
Applications of Density Functional Theory in Soft Condensed Matter
19
same in both phases. Since in the bulk p = −Ω/V (V denoting the system volume) coexistence means that at given µ and T , Ω/V has two minima with equal depth. A contour plot of the density functional in the space of variational parameters is shown in Figure 6 for three different temperature on the path shown in Figure 5. The liquid minimum occurs at zero α while the solid is characterized by a minimum at finite α. The global minimum is the stable phase and at coexistence, the two minima have equal depth. μ solid μ coex liquid
Tcoex
T
Fig. 5. Solid-liquid coexistence line in the µT plane. The path along which three state points are discussed in Figure 6 is indicated.
b) Approximations for the density functional Let us first recall the exact functional for the ideal gas where V (r) = 0. It reads as (8) Fid (T, [ρ]) = kB T d3 r ρ(r) ln(ρ(r)Λ3 ) − 1 and minimization
δFid = kB T ln(ρ(r)Λ3 ) + Vext (r) − µ δρ(r) 0
leads to the generalized barometric law 1 Vext (r) − µ ρ0 (r) = 3 exp − Λ kB T
(9)
(10)
for the inhomogeneous density. In the interacting case, V (r) = 0, one can split F(T, [ρ]) =: Fid (T, [ρ]) + Fexc (T, [ρ])
(11)
October 7, 2010
15:52
World Scientific Review Volume - 9in x 6in
20
02-chapter1
H. L¨ owen α
a)
T
solid
metastable liquid α
ρ
b)
at coexistence: μ=μ coex , T=Tcoex
αs solid
ρl
ρs
ρ
liquid (α=0) α
metastable solid
T>Tcoex
liquid
c)
ρ
Fig. 6. Contour plot of the grandcanonical free energy Ω(T, µ, ρ¯, α) for fixed T and µ as a function of two variational parameters α and the averaged density ρ¯. The latter is given by the lattice constant in the solid phase. a) with a stable solid phase, b) at solid-liquid coexistence, c) with a stable liquid phase.
which defines the excess free energy density functional Fexc (T, [ρ]). Approximations work on different levels. In the mean-field approximation, we set 1 d3 r d3 r V (|r − r |)ρ(r)ρ(r ) (12) Fexc (T, [ρ]) ≈ 2 In fact, the mean-field approximation (together with a correlational hole in the solid) yields freezing of the Gaussian potential [25] and is the correct starting point for cluster crystals for penetrable potentials [23]. The Ramakrishnan-Yussouff (RY) approximation is a perturbative treatment out of the bulk liquid which needs the bulk liquid direct correlation function c(2) (r, ρ¯, T ) as an input. A functional Taylor expansion
October 7, 2010
15:52
World Scientific Review Volume - 9in x 6in
Applications of Density Functional Theory in Soft Condensed Matter
02-chapter1
21
around a homogeneous reference density up to second order yields kB T Fexc (T, [ρ]) ∼ d3 r d3 r c(2) (|r − r |, ρ¯, T )(ρ(r) − ρ¯)(ρ(r ) − ρ¯) =− 2 (13) The RY approximation leads to freezing for hard spheres and was historically the first demonstration that freezing can be described within DFT. The RY functional can readily be generalized to soft interactions [26] (as the OCP) and gives reasonable results for freezing there (though it is better to improve the functional by including triplet correlations). A non-perturbative functional is based on Rosenfeld’s fundamental measure theory (FMT). This works, however, only for hard spheres. In FMT we have Fexc [ρ] = d3 r Φ[{nα (r)}] (14) kB T with (15) nα (r) = d3 r w(α) (r − r )ρ(r ) where the six weight function are given explicitly as w(2) (r) πσ 2 w(2) (r) w(1) (r) = 2πσ σ w(2) (r) = δ −r 2σ (3) −r w (r) = Θ 2 (V2 ) w (r) w(V1 ) (r) = 2πσ r σ w(V2 ) (r) = δ −r r 2 with σ denoting the hard sphere diameter and w(0) (r) =
Φ = Φ1 + Φ 2 + Φ 3 Φ1 = −n0 ln(1 − n3 ) n1 n2 − nv1 · nv2 Φ2 = 1 − n3 1 3 n − n2 (nv2 · nv2 ) Φ3 = 3 2 8π(1 − n3 )2
(16) (17) (18) (19) (20) (21)
(22) (23) (24) (25)
October 7, 2010
15:52
World Scientific Review Volume - 9in x 6in
22
02-chapter1
H. L¨ owen
This FMT functional yields the Percus-Yevick solution of the direct correlation function as an output. It furthermore survives the dimensional crossover [27]: If the three-dimensional hard sphere system is confined within a one-dimensional tube, the exact Percus functional is recovered. Moreover, in a spherical cavity which holds one or no particle at all, the exact functional is recovered. This helps to understand that the constraint packing argument of freezing is geometrically included in the FMT. In fact (also with a tensor modification in Φ3 [28]), the FMT gives excellent data for hard-sphere freezing [27], see Table 1 and the recent review by Roth [9]. Table 1. Coexisting number densities and solid Lindemann parameter at coexistence for the hard sphere systems. “Exact” computer simulation data are shown as well as DFT data using the Ramakrishnan-Yussouff (RY) or Rosenfeld’s fundamental measure theory. ρl σ 3 ρs σ 3 L (: Lindemann) computer simulations 0.94 1.04 0.129 RY 0.97 1.15 0.06 0.94 1.03 0.101 Rosenfeld
Last but not least we mention perturbation theories which can be used for attractive tails. The total potential V (r) is then split into a purely repulsive short-ranged part Vrep (r) and a longer-ranged attractive part Vattr (r) such that V (r) = Vrep (r)+Vattr (r). The repulsive part is treated as an effective hard core with an effective (temperature-dependent) Barker-Henderson diameter σ(T ) =
∞
dr
1 − e−βVrep (r)
(26)
0
and the attractive part is treated within mean-field approximation. Accordingly, the total excess free energy functional reads as HS (T, [ρ])σ=σ(T ) Fexc (T, [ρ]) ∼ = Fexc 1 d3 r d3 r ρ(r)ρ(r )Vattr (|r − r |) + 2
(27)
This procedure yields good phase diagrams for both Lennard-Jones potentials and sticky-hard-sphere systems including the isostructural solid-solid transition [18].
October 7, 2010
15:52
World Scientific Review Volume - 9in x 6in
Applications of Density Functional Theory in Soft Condensed Matter
02-chapter1
23
To summarize: (1) Rosenfeld’s FMT yields excellent data for hard sphere freezing. (2) The much less justified RY perturbative approach works in principle for softer repulsions. (3) The mean-field density functional approximation works for penetrable potentials. (4) Hard sphere pertubation theory yields stability of liquids and solid-solid isostructural transitions. 2. Brownian Dynamics 2.1. Brownian dynamics (BD) Colloidal particles are embedded in a molecular solvent and are therefore randomly kicked by the solvent molecules on timescales much smaller than the drift of the colloidal motion [5, 29]. Let us first discuss the Smoluchowski picture. Here the time-dependent density field is the central quantity. It should follow a simple deterministic diffusion equation. For noninteracting particles with an inhomogeneous time-dependent particle density ρ(r, t), Fick’s law states that the current density j(r, t) is r , t) j(r, t) = −D0 ∇ρ(
(28)
where D0 is a phenomenological diffusion coefficient. The continuity equation of particle conservation ∂ρ(r, t) + ∇ · j(r, t) = 0 ∂t then leads to the wellknown diffusion equation for ρ(r, t): ∂ρ(r, t) ≡ D0 ∆ρ(r, t) ∂t
(29)
(30)
ext (r) In the presence of an external potential Vext (r), the force F = −∇V acts on the particles and will induce a drift velocity vD resp. an additional current density jD = ρvD .
(31)
We now assume totally overdamped motion since inertia effects are small as the colloids are much bigger than the solvent molecules. This results in 1 F vD = = − ∇V r) (32) ext ( ξ ξ
October 7, 2010
15:52
World Scientific Review Volume - 9in x 6in
24
02-chapter1
H. L¨ owen
with ξ denoting a friction coefficient. For a sphere of radius R in a viscous solvent, ξ = 6πηs R, with ηs denoting the shear viscosity of the solvent (Stokes law). Now the total current density is r , t) − ρ(r, t) 1 ∇V ext (r) j = −D0 ∇ρ( (33) ξ In equilibrium, the one-particle density is a Boltzmann distribution ρ(r, t) ≡ ρ(1) (r) = ρ(0) (r) = A exp(−βVext (r))
(34)
Futhermore, in equilibrium, the total current has to vanish. Therefore, necessarily kB T (35) D0 = ξ + which is the so-called Stokes-Einstein relation. Hence j = − 1 (kB T ∇ρ ξ
ext ) and the continuity equation yields ρ∇V
∂ρ(r, t) 1 ext (r))) r , t)∇V = (kB T ∆ρ(r, t) + ∇(ρ( ∂t ξ
(36)
which is called Smoluchowski equation (for non-interacting particles). The same equation holds for the probability density w(r, t) to find a particle at position r for time t. For N non-interacting particles, 1 w(r, t) = ρ(r, t) , (37) N and the Smoluchowski equation reads: ∂w 1 ext )) = (kB T ∆w − ∇(w · ∇V (38) ∂t ξ Now we consider N interacting particles. Using a compact notation for the particle positions {xi } = {ri } = {x1 , x2 , x3 , x4 , x5 , x6 , · · · , x3N −2 , x3N −1 , x3N } r1
r2
(39)
rN
we assume a linear relation between acting forces on the particles and the resulting drift velocities. (The same compact notation is used for other multiple vectors.) The details of this relation embody the so-called hydrodynamic interations mediated between the colloidal particles by the solvent flow field induced by the moving colloidal particles. This linear relation is in general vi =
3N j=1
Lij ({xn })Fj
(40)
October 7, 2010
15:52
World Scientific Review Volume - 9in x 6in
Applications of Density Functional Theory in Soft Condensed Matter
02-chapter1
25
∂ where Fj = − ∂x Utot where Utot involves both the internal and the interacj tion potential energy and v is the drift velocity. The underlying assumption in (40) is that the hydrodynamic interactions act quasi-instantaneously. This is justified by the fact that the timescale upon which a shear perturbation is traveling through the suspension within an interparticle distance is much smaller than that of Brownian motion. The coefficients Lij constitute the so-called 3N ×3N mobility matrix and can in principle be obtained by solving the Navier-Stokes equations of N spheres with appropriate stick boundary conditions of the solvent flow field on the particle’s surfaces. In general, Lij depends on rN , and we postulate:
symmetry Lij = Lji positivity
Fi Fj Lij > 0 for all Fi,j = 0
(41)
(42)
ij
With w({ri }, t) denoting the probability density for interacting particles, the suitable generalization of the continuity equation is 3N ∂ ∂w =− (vtot,n w) ∂t ∂x n n=1
(43)
with vtot,n =
3N m=1
Lmn
∂ (kB T ln w + Utot ) ∂xm
(44)
which leads to the generalized Smoluchowski equation fo interacting particles. ∂w ˆ = Ow (45) ∂t with the Smoluchwoski operator ˆ= O
3N
∂ ∂ ∂Utot Lnm (kB T + ) ∂x ∂x ∂xm n m n,m=1
(46)
In many applications, hydrodynamic interactions are neglected. This means that the mobility matrix is constant and a diagonal Lnm =
1 δnm ξ
(47)
October 7, 2010
15:52
World Scientific Review Volume - 9in x 6in
26
02-chapter1
H. L¨ owen
‘Brownian motion’
Fig. 7.
Typical trajectory of a randomly kicked Brownian particle.
This assumption, however, is only true for small volume fraction of the colloidal particles. Complementary to the Smoluchowski approach which considers diffusion in phase space, stochastic trajectories in real-space are the basic ingredients for the Langevin picture. A typical “cuspy” Brownian trajectories of a colloidal particle is shown in Figure 7. First, we consider only one particle in an external potential Vext (r) with random force f(t). The stochastic differential equation for a single particle is completely overdamped ext (r) + f(t) ξr˙ = −∇V
(48)
where f(t) mimicks the random kicks of the solvent and is a Gaussian random variable which fulfills f(t) = 0
(49)
fi (t)fj (t ) = 2ξkB T δij δ(t − t )
(50)
One can show that this is stochastically equivalent to the Smoluchowski equation of non-interacting particles. For interacting particles, the Smoluchowski equation is obtained from the following Langevin equations [5]: x˙ n (t) =
3N m=1
Lnm (−
3N ∂Utotal ∂Lnm + fm (t)) + kB T ∂xm ∂xm m=1
(51)
with Gaussian random variables fm which fulfill fm (t) = 0
fm (t)fm (t ) =
2L−1 k T δ(t mm B
(52)
− t ).
(53)
October 7, 2010
15:52
World Scientific Review Volume - 9in x 6in
Applications of Density Functional Theory in Soft Condensed Matter
02-chapter1
27
2.2. BD computer simulations In Brownian Dynamic (BD) computer simulations we use the overdamped Langevin equation (48) with a single realization of the stochastic process. Applying this to a finite time step ∆t and integrating xn (t + ∆t) = xn (t) + ∆t ∆n =
Lnm (−
m=1
where
3N
t+∆t t
dt
∂Lnm ∂Utotal ) + ∆n + kB T ∆t ∂xm ∂xm m
Lnm fm (t )
is Gaussian distributed
m
and possesses the moments ∆n =
t+∆t t
dt
Lnm fm (t ) = 0
m
∆n ∆ = 2kB T ∆tLnn , n
which is the basic equation for BD computer simulations. For each step we need to generate a Gaussian random number and update the Langevin equation by calculating xn (t + ∆t). Although this algorithm is less efficient than MD/MC for equilibrium correlations, it is unavoidable for calculation of colloidal dynamics. 2.3. Dynamical density functional theory (DDFT) Here we derive a deterministic equation for the time-dependent one-particle density from the Smoluchowski equations [25]. We follow the idea of Archer and Evans [30]. First, we recall Smoluchowski equation for the N -particle density w(r1 , . . . , rN , t) ≡ w(rN , t) , rN = {r1 , . . . , rN } as N
∂w i+∇ i Utot (rN , t)]w i · [kB T ∇ ˆ =1 ∇ = Ow ∂t ξ i=1
(54)
with Utot (rN , t) =
N i=1
Vext (ri , t) +
N i,j=1 i<j
V (|ri − rj |)
(55)
October 7, 2010
15:52
World Scientific Review Volume - 9in x 6in
28
02-chapter1
H. L¨ owen
Here, hydrodynamic interactions have been neglected. Now the idea is to integrate out degrees of freedom. In fact, an integration yields 3 ρ(r1 , t) = N d r2 . . . d3 rN w(rN , t) (56) The 2-particle density is analogously obtained as ρ(2) (r1 , r2 , t) = N (N − 1) d3 r3 . . . d3 rN w(rN , t) Integrating the Smoluchowski equation with N ξ
(57)
d3 r2 . . . d3 rN yields
∂ 1 (ρ(r1 , t)∇ 1 Vext (r1 , t) ρ(r1 , t) = kB T ∆1 ρ(r1 , t) + ∇ ∂t 1 d3 r2 ρ(2) (r1 , r2 , t)∇ 1 V (|r1 − r2 |) +∇
(58)
r1 ,t) In equilibrium, necessarily ∂ρ( = 0 which implies ∂t r ) + ρ(r)∇V ext (r) + d3 r ρ(2) (r, r )∇V (|r − r |) kB T ∇ρ( 0=∇
(59) The constant must vanish for r → ∞ and is thus identical to zero. Therefore r ) + ρ(r)∇V ext (r) + d3 r ρ(2) (r, r )∇V (|r − r |) (60) 0 = kB T ∇ρ( This is also known as Yvon-Born-Green-hierarchy (YBG). In equilibrium, DFT implies: δF = µ − Vext (r) δρ(r) = kB T ln(Λ3 ρ(r)) +
(61) δFexc , since F = Fid + Fexc δρ(r )
(62)
We now apply the gradient which gives: ext (r) + kB T ∇ ln(Λ3 ρ(r)) + ∇ δFexc = 0 ∇V δρ(r) combined with YBG we obtain (|r − r |) = ρ(r)∇ · δFexc [ρ] d3 r ρ(2) (r, r )∇V δρ(r)
(63)
(64)
We postulate that this argument holds also in nonequilibrium. In doing so, non-equilibrium correlations are approximated by equilibrium ones at the same ρ(r, t) (via a suitable Vext (r) in equilibrium). Equivalently, one can say
October 7, 2010
15:52
World Scientific Review Volume - 9in x 6in
Applications of Density Functional Theory in Soft Condensed Matter
02-chapter1
29
that it is postulated that pair correlations decay much faster to their equilibrium one than the one-body density. Therefore the basic approximation of DDFT is sometimes called adiabatic approximation. (For an alternate derivation, see Marconi and Tarazona [31, 32] or Espa˜ nol and L¨ owen [33].) Hence: ξ
∂ρ(r, t) B T ∇ρ( r , t) + ρ(r, t)∇V ext (r, t) = ∇(k ∂t δFexc ) + ρ(r, t)∇ δρ(r, t)
(65)
or equivalently ξ
∂ρ(r, t) r , t)∇ δΩ[ρ] = ∇ρ( ∂t δρ(r, t)
(66)
which constitutes the basic equation of dynamical density functional theory (DDFT). The applications of DDFT are numerous. The dynamics of a strongly inhomogeneous Brownian fluid has found to be a good agreement with BD computer simulations. [34, 35] 2.4. An example: Crystal growth at imposed nucleation clusters The dynamical density functional theory can also be used for the dynamics of the crystal. This is demonstrated by an example in the following. A two-dimensional system with the repulsive inverse power-law potential V (r) = V0 /r3 corresponds to dipoles in a plane whose dipole moments are perpendicular to the plane. Two-dimensional systems are best realized for superparamagnetic colloids pending at an air-water interface in an external magnetic field [36, 37]. The RY functional gives a reasonable description for the two-dimensional crystallization [26]. The dynamical version can be applied to a situation where a cluster of 19 particles from a prescribed cut-out of a triangular lattice is imposed and offered as a nucleation seed to an undercooled fluid. This cluster is first fixed and then instantaneously released. Depending on the lattice constant of the prescribed nucleation cluster, there is either subsequent crystal growth or relaxation back to the undercooled fluid. Two examples are shown in Figure 8. The one-particle density field is shown for fixed time as a contour plot. In both examples, the crystalline seed is compressed relative to the
October 7, 2010
30
15:52
World Scientific Review Volume - 9in x 6in
H. L¨ owen
Fig. 8. Snapshots of the central region of the dimensionless density field ρ(r, t)/ρ of two colloidal clusters with strain parameters Aρ = 0.7 (left panel) and Aρ = 0.6 (right panel) at times t/τB = 0, 0.001, 0.1, 1.0 (from top to bottom; t/τB = 1.0 only for Aρ = 0.7). A is the area of the crystal unit cell. From [38].
stable bulk one. If the compression is significant but not too large (Figure 8, left panel), there is still crystal growth, but if the crystal is compressed further, the seed is too dissimilar to the bulk crystal as to initiate crystal growth (Figure 8, right panel). The same qualitative behaviour has been found in Brownian dynamics computer simulations [38]. This example shows that dynamical density functional theory represents a reliable microscopic approach to nonequilibrium phenomena like crystal growth in an external field [39]. A similar behaviour has been seen for
02-chapter1
October 7, 2010
15:52
World Scientific Review Volume - 9in x 6in
Applications of Density Functional Theory in Soft Condensed Matter
02-chapter1
31
growing crystal fronts [38, 40] and for vacancy dynamics in two-dimensional solids. 2.5. Hydrodynamic interactions Here we address the question how Lnm ({xj }) looks like explicitely. Solving the linearized Navier-Stokes equations with the appropiate stick boundary conditions on the particle surfaces, is a difficult problem. Furthermore it is problematic that i) Lnm ({xj }) is long-ranged in terms of distances between particles. ii) H.I. have many-body character, pair expansion only possible at low concentrations. iii) H.I. have quite different near-field behaviour. They are divergent lubrication terms. The linear relationship (40) can be rewritten as vn =
N
¯ F H nm m
(67)
m=1
¯ where each quantity H nm is a 3×3 matrix. In particular, we can discriminate the following cases: 1) No H.I. Hnm = ½ δnm ξ 2) Oseen-tensor ¯ =½ In the Oseen approximation, H nn ξ ¯ r − r ) for n = m ¯ H nm = H( n m
(68)
r
with the Oseen tensor ¯ r) = H(
1 1 , (½ + rˆ ⊗ rˆ) 8πηs r
rˆ =
r r
(69)
This is the leading far field term for two particles at large distance r. The symbol ⊗ denotes the dyadic product or tensor product. 3) Rotne-Prager-tensor In this approximation, the next leading correction is included. Hnn =
½ ξ
,
¯ (r − r ) Hnm = H RP n m
(70)
October 7, 2010
15:52
World Scientific Review Volume - 9in x 6in
32
02-chapter1
H. L¨ owen
with ¯ (r) = D0 H RP kB T
3 3 RH 1 RH [½ + rˆ ⊗ rˆ] + [1 − 3ˆ r ⊗ rˆ] 4 r 2 r3
(71)
Higher order expansions of higher order than r13 are possible. These include also terms of sphere rotation. Finally the triplet contribution can be estimated. The DDFT can be generalized to hydrodynamic interactions [41]. Again the starting point is the Smoluchowski equation which we now write in the form N Utot (rN , t) ∂w(rN , t) N ¯ = w(rN , t) (72) ∇i · Hij (r ) · ∇j + ∇j ∂t kB T i,j=1
We use the two particle approximation D 0 ¯ (rN ) ≈ ½δij + δij ω11 (ri − re ) + (1 − δij )ω12 (ri − re ) H ij kB T i=j
(73) on the level of the Rotne-Prager expression ω11 (r) = 0 3 σH 1 σH 3 σH 7 r ⊗ rˆ) + O(( ω12 (r) = (½ + rˆ ⊗ rˆ) + ( ) (1 − 3ˆ ) ) 8 r 16 r r where σH is the hydrodynamic diameter. Integrating Smoluchowski equation [25] then yields [41] kB T ∂ρ(r, t) δF [ρ] = ∇r · ρ(r, t)∇r D0 ∂t δρ(r, t) δF [ρ] + dr ρ(2) (r, r , t)ω11 (r − r ) · ∇r δρ(r, t) δF [ρ] + dr ρ(2) (r, r , t)ω12 (r − r ) · ∇r δρ(r, t)
(74) (75)
(76)
A possible closure is via the Ornstein-Zernike equation ρ(2) (r, r , t) = (1 + c(2) (r, r ))ρ(r, t)ρ(r , t) + ρ(r , t) dr ((ρ(2) (r, r , t) − ρ(r, t)ρ(r , t))c(2) (r , r )) (77)
October 7, 2010
15:52
World Scientific Review Volume - 9in x 6in
Applications of Density Functional Theory in Soft Condensed Matter
02-chapter1
33
with c(2) (r, r ) = −β
δ 2 Fexc [ρ] δρ(r, t)δρ(r , t)
(78)
In an easier attempt, one can approximate ρ(2) (r, r , t) ≈ ρ(r, t)ρ(r , t)g(|r − r |, ρ¯)
(79)
where ρ¯ is a suitable averaged density and g(r, ρ¯) is a pair distribution function in the equilibrium bulk fluid. Finally we describe an example which involves hard sphere colloids of diameter σ moving in a time-dependent oscillating radial-symmetric trapping potential Vext (r, t). The potential reads as Vext (r, t) = V1 r4 + V2 cos(ωt)r4
(80)
and corresponds to a shape which switches between stable and unstable situations at the origin r = 0 but is globally stable. The amplitudes are V1 = 10kB T /4096σ 4, V2 = kB T /σ 2 and the external switching frequency is ω = 4πD0 /σ2 . As a result, the density profile is picking up the external frequency ω and exhibits a breathing mode, i.e. it is periodically expanding and compressed again by the external potential (80). Time-dependent density profiles ρ(r, t) are presented in Figure 9 in the steady breathing state. Figures 9(a) and (b) show results for hydrodynamic interactions included on the Rotne-Prager level. DDFT data are in very good agreement with Brownian dynamics computer simulations which include hydrodynamic interactions on the same Rotne-Prager level. On the other hand, in Figure 9(c) and (d), hydrodynamic interactions are ignored. The density profiles are qualitatively different to that shown in Figures 9(a) and (b) but again DDFT data are in agreement with Brownian dynamics computer simulations. This demonstrates that DDFT is a reliable microscopic theory both if hydrodynamic interactions are included or ignored. 3. Rod-Like Particles 3.1. Statistical mechanics of rod-like particles Density functional theory can readily be extended to rod-like particles which possess an additional orientational degree of freedom described by a unit vector u ˆ. A configuration of N particles is now fully specified by the
October 7, 2010
15:52
World Scientific Review Volume - 9in x 6in
34
H. L¨ owen
Fig. 9. Steady-state DDFT (solid curves) and BD (noisy curves) results for the timedependent density profile ρ(r, t). In Fig. (a) and (b) hydrodynamic interactions are taken into account while in (c) and (d) they are neglected. (a) and (c) correspond to the expanding half period and (b) and (d) to the compressing half period, respectively. The profiles correspond to the following time sequence: t0 = 2.5τB , t1 = 2.6τB , t2 = 2.7τB , t3 = 2.75τB in (a) and (c), and t3 = 2.75τB , t4 = 2.85τB , t5 = 2.9τB and t6 = 3.0τB , τB = σ 2 /D0 . From Ref. [41].
set of positions of the center of masses and the corresponding orientations i, u {R ˆi , i = 1, . . . , N }, see Figure 10. Example for ansiotropic particles include (1) (2) (3) (4)
molecular dipolar fluids (e.g. H2 O molecule) rod-like colloids (e.g. tobacco-mosaic viruses) molecular fluids without dipole moment (apolar), (e.g. H2 molecule) plate-like objects (clays) The canonical partition function for rod-like particles now reads [42] 1 d3 R1 ... d3 RN d3 p1 ... d3 pN Z = 6N h N! 3 V V R3 R × d2 u1 ... d2 uN d3 L1 ... d3 LN e−βH (81) S2
S2
R3
R3
02-chapter1
October 7, 2010
15:52
World Scientific Review Volume - 9in x 6in
Applications of Density Functional Theory in Soft Condensed Matter
ûi
rod i
02-chapter1
35
orientation vector
Ri center-of-mass coordinate
platelet i ûi
Ri
i and the orientational unit vector u Fig. 10. Sketch of the center-of-mass position R ˆi for the ith particle both for a rod-like and plate-like particle.
with the Hamilton function H=
N 2 pi 1 ¯ −1 Li + L i (Θ) 2m 2 i=1
+
N
i, u Vext (R ˆi )
+
N 1 j, u v(Ri − R ˆi , u ˆj ) 2 i,j=1
(82)
i=1
which comprises the kinetic energy, the pair interaction energy and the ¯ is the inertia tensor and S the unit-sphere in 3d. external energy. Here Θ 2 (1) Again the central quantity is the one-particle density ρ0 (r, u ˆ) which is defined as N (1) ˆ) := δ(r − Ri )δ(ˆ u−u ˆi ) (83) ρ (r, u 0
i=1
Integrating the orientations over the unit sphere S2 results in the density of the center-of-masses 1 (1) d2 u ρ0 (r, u ˆ) (84) ρ0 (r) = 4π S2
On the other hand, the globally averaged orientational order is gained by integrating over the center-of-mass corrdinates and given by 1 (1) d3 r ρ0 (r, u ˆ) (85) f (ˆ u) = V V
October 7, 2010
15:52
World Scientific Review Volume - 9in x 6in
36
02-chapter1
H. L¨ owen
û1
1
2
r
û2
R2
R1
Fig. 11. Sketch of two interacting rods. The interaction potential depends on r = 1 − R 2 and u R ˆ1 , u ˆ2 .
In analogy to the isotropic case, one can define a pair correlation function N ! 1 − R i )δ(R 2 − R j )δ(ˆ δ(R u1 − u ˆj )δ(ˆ u2 − u ˆj ) 1, R 2, u g(R ˆ1 , u ˆ2 ) :=
i,j=1 i=j
1, u 2 , uˆ2 ) ρ0 (R ˆ1 )ρ0 (R (1)
(1)
(86)
Now different phases are conceivable which can be classified and distinguished by their one-particle density field. 1) Fluid (disordered) phase, isotropic phase Here the center-of-mass-positions and orientations are disordered: (1)
ˆ) = ρ0 = const ρ0 (r, u
(87)
2) Nematic phase Here, positions are disordered and orientations are ordered, i.e. (1)
ˆ) = ρf (ˆ u) ρ0 (r, u
,
u ˆ0 : nematic director
(88)
Typically the orientation is distributed around a nematic director uˆ0 . In order to quantify orientational order, it is convenient to introduce a nematic order parameter. In fact, this is defined via the second rank tensor N 1 3 1 ¯= ˆi − ½ (89) Q u ˆi ⊗ u N i=1 2 2 where the dyadic product is
uix uix uix uiy uix uiz ˆi = uiy uix uiy uiy uiy uiz u ˆi ⊗ u uiz uix uiz uiy uiz uiz
(90)
October 7, 2010
15:52
World Scientific Review Volume - 9in x 6in
Applications of Density Functional Theory in Soft Condensed Matter
02-chapter1
37
¯ is trace-less One can easily show that the tensor Q 1 Tr(3ˆ ui ⊗ u ˆi − ½) 2 1 = 3 · 1 − 3 = 0 2
¯= TrQ
(91) (92)
¯ is clearly symmetric and hence diagonalizable with three Furthermore Q eigenvalues λ1 ≥ λ2 ≥ λ3 where λ3 must be −λ2 −λ1 . The largest eigenvalue λ1 is called nematic order parameter S. The corresponding eigenvector is called nematic director. For perfect alignment along u0 we have ui ≡ u ˆ0 for all i. Then, S = 1. If the two lower eigenvalues are identical, λ2 = λ3 , the call it a uniaxial nematic phase. If λ2 = λ3 , the orientation is called biaxial. In the isotropic phase: S = 0. Orientational distributions are accessible experimentally, by e.g. birefringence. 3) Smectic-A phase The so-called smectic-A phase is positionally ordered along an orienation axis u ˆ0 . The associated one-particle density is periodic in the z-direction along uˆ0 : (1)
ˆ) = ρ(z, u ˆ) z-periodic ρ0 (r, u
(93)
4) Smectic-B phase The smectic-B phase is similar to the smectic-A phase but exhibits an inplane triangular lattice. 5) Columnar phase The columnar exhibits crystalline order perpendicular to the director uˆ0 but is disordered positionally along u0 . The one-particle density field thus reads as (1)
ˆ) ρ0 (r, uˆ) = ρ(x, y, u
(94)
6) Plastic crystal In a plastic crystal, the positions are ordered in all three spatial directions but the orientations are disordered. Therefore: (1)
ρ0 (r, uˆ) = f (r)
(95)
October 7, 2010
15:52
World Scientific Review Volume - 9in x 6in
38
02-chapter1
H. L¨ owen
7) Full crystalline phases Finally, in the full crystalline phase, positions and orientations are both ordered. The list (1)-(7) of liquid crystalline phases is by far not exhaustive. There are more “exotic” phases such as an AAA stacked phase, a smecticC with tilted rods, or cholesteric phases which possesses a helical twist with a pitch length lp . 3.2. Simple models Let us now discuss simple models for interactions between anisotropic rods. The phase behaviour of hard objects, as shown in Figure 12, is dominated by the shape. Temperature scales out in this case such that packing fraction alone (apart from the particle shape) is the only parameter. Hard spherocylinders have been studied, as well as hard platelets (“hard coins”) or thin needles which arise from spherocylinders in the limit of infinite aspect ratio L D → ∞. limit: L D h
L
D hard spherocylinders Fig. 12.
∞
hard coins
hard needles
Sketch of differently shaped hard bodies.
A) Analytical results by Onsager L In the limit p = D → ∞, a virial expansion up to second order is getting asymptotically exact. There is a isotropic-nematic transition which can be calculated analytically [43]. It is first order with a density jump. The scaled coexistence densities of the coexisting isotropic and nematic phases are
ρiso L2 D = 4.189 . . .
(96)
ρnem L2 D = 5.376 . . .
(97)
and At coexistence, the nematic order parameter in the nematic phase is S = 0.784. However, in practice, one needs pretty large aspect ratios (about p 200) in order to get reasonably into this Onsager limit.
October 7, 2010
15:52
World Scientific Review Volume - 9in x 6in
02-chapter1
Applications of Density Functional Theory in Soft Condensed Matter
39
B) Computer simulations Full phase diagrams can be obtained by Monte Carlo computer simuations [44]. Hard spherocylinders show for various densities ρ and aspect ratios p the following stable phases: isotropic, plastic, ABC stacking, AAA stacking, smectic-A, and nematic phases. All those transitions are purely driven by entropy. Another anisotropic system are hard ellipsoids which are characterized by two different axial lengths a and b. The phase diagram involves isotropic, nematic fully crystalline and plastic crystalline phases [45]. There is a remarkable symmetry in x → x1 for the topology of the phase diagram where x = ab . x < 1 correspond to oblate and x > 1 to prolate objects. C) Density functional theory Again density functional theory tells us that there exists a unique grandcanonical free energy functional Ω(T, µ, [ρ(1) ]) (functional of the one-particle (1) density) which becomes minimal for the equilibrium density ρ0 (r, u ˆ) and then equals the real grand canonical free energy, i.e. δΩ(T, µ, [ρ(1) ]) =0 (98) δρ(1) (r, u ˆ) ρ(1) =ρ(1) (r,ˆu) 0
Here, the functional can be decomposed as follows 3 (1) Ω(T, µ, [ρ ]) = kB T d r d2 u ρ(1) (r, u ˆ)[ln(Λ3 ρ(1) (r, u ˆ)) − 1] + d3 rd2 u (Vext (r, u ˆ) − µ)ρ(1) (r, u ˆ) + Fexc (T, [ρ(1) ]) (99) The first term on the right hand side of equation (99) is the functional Fid [ρ(1) ] for ideal rotators. The excess part Fexc(T, [ρ(1) ]) is in general unknown and requires approximative treatments. For hard spherocylinders, Fexc (T, [ρ(1) ]) can be approximated by a smoothed density approximation (SMA) [46] yielding several stable liquidcrystalline phases, namely: isotropic, nematic, smectic-A and ABC crystalline. A modified weighted density approximation (MWDA) was subsequently proposed [47] which improves upon the SMA by exhibiting stable plastic crystalline and AAA crystals as well. An important recent progress was archieved by generalizing Rosenfeld’s fundamental measure theory from hard spheres to hard objects with any
October 7, 2010
15:52
40
World Scientific Review Volume - 9in x 6in
02-chapter1
H. L¨ owen
shape [48]. For spherocylinders the functional was worked out explicitly. This functional could be exploited also for attractions by employing a perturbation theory for the attractive parts in the potential. Finally, a meanfield density functional for rods with soft segments was proposed and studied [49]. 3.3. Brownian dynamics of rod-like particles In order to derive a dynamical density functional theory (DDFT) for rodlike particles one can start from the Smoluchowski equation for the full probability density distribution w(r1 , · · · , rN ; u1 , · · · , uN , t) of N rods with their corresponding center-of-mass positions rN = (r1 , · · · , rN ) and orienu1 , · · · , u ˆN ) which reads [6] tations uˆN = (ˆ ∂w ˆS w =O ∂t where the Smoluchowski operator is now given by N 1 N N ¯ u )· ∇ r · D(ˆ ˆS = ∇ O ∇ + U ( r , u ˆ , t) i ri r i kB T i i=1 ˆi · R ˆ i U (rN , u ˆi + 1 R ˆN , t) +Dr R kB T
(100)
(101)
ˆN , t) is the total potential energy. Here the rotation operator where U (rN , u ˆ ˆ i = u uˆi and the anisotropic translational diffusion i is defined as R ˆi × ∇ R tensor is given by ¯ u ) = D u ˆi ⊗ u ˆi + D⊥ (1 − u ˆi ⊗ u ˆi ) D(ˆ i
(102)
The two diffusion constants D and D⊥ , parallel and perpendicular to the orientations reflect the fact that the translational diffusion is anisotropic. For hard spherocylinders there are valid approximations for D and D⊥ [50]. Following the idea of Archer and Evans the can integrate 2Smolu [30] one chowski equation by now applying N d3 r2 · · · d3 rN d2 u1 · · · d un on both sides of Eqn (100). This results in [49]: (r, u F ˆ, t) 1 ∂ρ(r, u ˆ, t) ¯ u) · ∇ r · D(ˆ r Vext (r, u r ρ(r, u =∇ ρ(r, u ˆ, t) · ∇ ˆ, t) + ˆ, t) − ∂t kB T kB T 1 ˆ · Rρ( ˆ r, u r Vext (r, u + Dr R ˆ, t) − k 1 T T (r, u ˆ, t) + ˆ , t) ρ(r, u ˆ, t)∇ B kB T
(103)
October 7, 2010
15:52
World Scientific Review Volume - 9in x 6in
02-chapter1
Applications of Density Functional Theory in Soft Condensed Matter
41
with an average force 3 r v2 (r − r , uˆ, u ˆ, u ˆ , t)∇ ˆ ) F (r, u ˆ, t) = − d r d2 u ρ(2) (r, r , u
(104)
and average torque 3 ˆ 2 (r − r , u T (r, u ˆ, t) = − d r d2 u ρ(2) (r, r , u ˆ, u ˆ , t)Rv ˆ, u ˆ )
(105)
The two-particle density which is in general unknown can be approximated in equilibrium by using r δFexc (T, [ρ0 ]) ˆ)∇ F (r, u ˆ, t) = ρ0 (r, u δρ0 (r, u ˆ)
(106)
ˆ δFexc [ρ] T (r, u ˆ, t) = ρ0 (r, u ˆ)R δρ0 (r, u ˆ)
(107)
respectively
Similar as in the isotropic (spherical) case we now employ the “adiabatic” approximation. We assume that the pair correlations in nonequilibrium are the same as those for an equilibrium system with the same one-body density profile (established by a suitable Vext (r, u ˆ, t)). The resulting dynamical equation for the time-dependent one particle density ρ(r, u ˆ, t) is then given by Ref. [49]: ∂ρ(r, u ˆ, t) ˆ, t)] ¯ u) · ρ(r, u r · D(ˆ r δF [ρ(r, u =∇ ˆ, t)∇ kB T ∂t δρ(r, u ˆ, t) δF [ρ( r , u ˆ , t)] ˆ ˆ + Dr R ρ(r, u ˆ, t)R (108) δρ(r, u ˆ, t) with the equilibrium Helmholtz free energy density functional u ρ(r, u ˆ) ln(Λ3 ρ(r, u ˆ)) − 1 F[ρ] = kB T d3 r dˆ 3 + Fexc (T, [ρ]) + d r dˆ u ρ(r, u ˆ)Vext (r, u ˆ, t)
(109)
This sets the frame for dynamical density functional theory (DDFT) for rods. As for a special application of DDFT to dynamics in the confined isotropic phase we refer to [49] where the mean-field approximation for the functional was employed. More recent work has used the Rosenfeld functional for hard spherocylinders [48] for driven nematic phases [51, 52].
October 7, 2010
15:52
42
World Scientific Review Volume - 9in x 6in
02-chapter1
H. L¨ owen
3.4. “Active” (self-propelled) Brownian particles “Active” particles are self-propelled by their own intrinsic motor. In fact, apart from swimming bacteria, there are artificial microswimmers made by colloidal particles [53, 54]. Ignoring hydrodynamic interactions, these swimmers can simplest be modelled by rod-like particles which are driven by a constant force along their orientations; the force corresponds to an effective drift velocity and mimicks the actual propulsion mechanism. On top of the intrinsic propulsion, the particles feel Brownian noise of the solvent. The corresponding motion is intrinsically a nonequilibrium one and even the dynamics of a single Brownian swimmer was solved only recently [55, 56]. Starting from the Smoluchowski equation with an appropriate intrinsic drift term, a dynamical density functional theory can be derived using the same adiabatic approximation (64) as in the “passive” case. The resulting equation of motion for the one-particle density then [57] has an extra term on the right-hand side of Eqn. (108), namely ¯ u)ρ(r, u D(ˆ ˆ, t)ˆ u −F0 ∇
(110)
where F0 denotes the internal driving force. This represents a microscopic theory for concentrated “active” matter. For swimmers in a two-dimensional channel, the time-dependent density profiles were found to be in agreement with Brownian dynamics computer simulations [57] even if a crude Onsager-like density functional approximation [43] was used. Qualitatively, the transient formation of hedgehog-like clusters ear the channel boundaries was reproduced by the dynamical density functional theory. 4. Conclusions In conclusion, there is a variety of topologies for equilibrium phase diagrams, even for relatively simple radially-symmetric pair potentials V (r). Examples include freezing into crystalline lattices with unusual open structure and reentrant melting effects if the interaction is soft and isostructural solid-to-solid transition for system with short-ranged attractions. The effects are predicted by Monte Carlo computer simulations and can in principle be confirmed by density functional theory of freezing. Colloidal particles can be used as model systems to realize these interactions. Rod-like systems with anisotropic interactions, on the other hand, exhibit liquid crystalline phase with different degrees of orientational and
October 7, 2010
15:52
World Scientific Review Volume - 9in x 6in
Applications of Density Functional Theory in Soft Condensed Matter
02-chapter1
43
positional ordering. Again density functional theory of freezing can be formulated and be applied to predict the topology of the phase diagram. Rodlike colloidal particles represent ideal model systems to explore and test the phase behaviour experimentally. Effects of colloidal dynamics in a solvent both in equilibrium and nonequilibrium are conveniently simulated by Brownian dynamics computer simulations. The density functional theory can be extended towards dynamics, so-called dynamical density functional theory to tackle various nonequilibrium phenomena. This was demonstrated for crystal growth at imposed nucleation seeds and for the collective behaviour of “active” Brownian particles. In general, dynamical density functional theory is in good agreement with the simulations. This may be different for undamped Newtonian dynamics which occurs for example in molecular liquid crystals or in a complex plasma [58]. Acknowledgement I thank R. Evans, M. Rex, H. H. Wensink, S. van Teeffelen, U. Zimmermann, T. Glanz, M. Kohl and A. H¨ artel for many helpful suggestions. This work was supported by the DFG (SPP1296 and SFB TR6 (project D3)). References [1] A. F. Th¨ unemann, M. M¨ uller, H. Dautzenberg, J.-F. Joanny, H. L¨ owen, Adv. Polym. Sci. 166, pages 113-171 (2004), Springer Series. [2] H. L¨ owen, G. Kramposthuber, Europhys. Letters 23, 637-678 (1993). [3] G. A. Vliegenthart, P. van der Schoot, Europhys. Lett. 62, 600-606 (2003). [4] J. Dzubiella, H. L¨ owen, C. N. Likos, Phys. Rev. Letters 91, 248301 (2003). [5] M. Doi, S. F. Edwards, The Theory of Polymer Dynamics, Oxford Science Publications, Clarendon Press Oxford (1986). [6] J. K. G. Dhont, An Introduction to Dynamics of Colloids, Elsevier, Amsterdam, 1996. [7] H. L¨ owen, J. Phys.: Condensed Matter 13, R415-R432 (2001). [8] H. L¨ owen, Particle-resolved instabilities in colloidal dispersions, Soft Matter, 6, 3133-3142 (2010). [9] R. Roth, J. Phys.: Condensed Matter 22, 063102 (2010). [10] R. Evans, Advances in Physics 28, 143 (1979). [11] D. W. Oxtoby, in: Liquids, Freezing and Glass Transition, edited by J. P. Hansen, D. Levesque, J. Zinn-Justin, North Holland, Amsterdam, pages 145189, 1991. [12] H. L¨ owen, Physics Reports 237, 249 (1994).
October 7, 2010
44
15:52
World Scientific Review Volume - 9in x 6in
H. L¨ owen
[13] J.-L. Barrat, J.-P. Hansen, Basic Concepts for Simple and Complex Liquids, Cambridge University Press, 2003. [14] H. L¨ owen, J. Phys.: Condensed Matter 14, 11897 (2002). [15] P. Tarazona, J. A. Cuesta, Y. Martinez-Raton, Density functional Theories of Hard Particle Systems, Lect. Notes Phys. 753, Springer, Berlin, pages 247-341 (2008). [16] J. P. Hansen, I. McDonald, Theory of Simple Liquids. 3rd Edition, Elsevier, Amsterdam, Academic Press 2005. [17] M. P. Allen, D. J. Tildesley, Computer Simulation of Liquids, Oxford Science Publications, Clarendon Press, Oxford, 1987. [18] C. N. Likos, Z. T. N´emeth, H. L¨ owen, J. Phys.: Condensed Matter 6, 10965 (1994). [19] C. N. Likos, H. L¨ owen, M. Watzlawek, B. Abbas, O. Jucknischke, J. Allgaier, D. Richter, Phys. Rev. Letters 80, 4450 (1998). [20] M. Watzlawek, C. N. Likos, H. L¨ owen, Phys. Rev. Letters. 82, 5289 (1999). [21] A. Lang, C. N. Likos, M. Watzlawek, H. L¨ owen, J. Phys.: Condensed Matter 12, 5087 (2000). [22] D. Gottwald, C. N. Likos, G. Kahl, H. L¨ owen, Phys. Rev. Letters 92, 068301 (2004). [23] C. N. Likos, A. Lang, M. Watzlawek, H. L¨owen, Phys. Rev. E 63, 031206 (2001). [24] R. Ohnesorge, H. L¨ owen, H. Wagner, Europhys. Letters 22, 245 (1993). [25] A. J. Archer, Phys. Rev. E 72, 051501 (2005). [26] S. van Teeffelen, N. Hoffmann, C. N. Likos, H. L¨owen, Europhysics Letters 75, 583 (2006). [27] Y. Rosenfeld, M. Schmidt, H. L¨ owen, P. Tarazona, Phys. Rev. E 55, 4245 (1997). [28] P. Tarazona. Phys. Rev. Letters 84, 694 (2000). [29] P. N. Pusey, in: Liquids, Freezing and Glass Transition, edited by J. P. Hansen, D. Levesque, J. Zinn-Justin, North Holland, Amsterdam, pages 145189, 1991. [30] A. J. Archer, R. Evans, J. Chem. Phys. 121, 4246 (2004). [31] U. M. B. Marconi, P. Tarazona, J. Chem. Phys. 110, 8032 (1999). [32] U. M. B. Marconi, P. Tarazona, J. Phys.: Condensed Matter 12, A413 (2000). [33] P. Espa˜ nol, H. L¨ owen, J. Chem. Phys. 131, 244101 (2009). [34] J. E. Hug, F. van Swol, C. F. Zukoski, Langmuir 11, 111 (1995). [35] R. J. Hunter, Foundations of Colloid Science. 2nd ed. Oxford University Press, Oxford, 1989. [36] N. Hoffmann, F. Ebert, C. N. Likos, H. L¨ owen, G. Maret, Phys. Rev. Letters 97, 078301 (2006). [37] L. Assoud, F. Ebert, P. Keim, R. Messina, G. Maret, H. L¨ owen, Phys. Rev. Letters 102, 238301 (2009). [38] S. van Teeffelen, C. N. Likos, H. L¨ owen, Phys. Rev. Letters 100, 108302 (2008). [39] G. Kahl, H. L¨ owen, J. Phys.: Condensed Matter 21, 464101 (2009).
02-chapter1
October 7, 2010
15:52
World Scientific Review Volume - 9in x 6in
Applications of Density Functional Theory in Soft Condensed Matter
02-chapter1
45
[40] S. van Teeffelen, R. Backofen, H. L¨ owen, A. Voigt, Phys. Rev. E 79, 051404 (2009). [41] M. Rex, H. L¨ owen, Phys. Rev. Letters 101, 148302 (2008). [42] D. Frenkel, in: Liquids, Freezing and Glass Transition, edited by J. P. Hansen, D. Levesque, J. Zinn-Justin, North Holland, Amsterdam, pages 689756, 1991. [43] L. Onsager, Proc. New York Acad. Sci. 51, 627 (1949). [44] P. Bolhuis, D. Frenkel, J. Chem. Phys 106, 666 (1997). [45] D. Frenkel, B. M. Mulder, J. P. McTague, Phys. Rev. Letters 52, 287 (1984). [46] A. Poniewierski, R. Holyst, Phys. Rev. Letters 61, 2461 (1988). [47] H. Graf, H. L¨ owen, J. Phys.: Condensed Matter, 11, 1435 (1999). [48] H. Hansen-Goos, K. Mecke, Phys. Rev. Letters 102, 018302 (2009). [49] M. Rex, H. H. Wensink, H. L¨ owen, Phys. Rev. E 76, 021403 (2007). [50] H. L¨ owen, Phys. Rev. E 50, 1232 (1994). [51] A. H¨ artel, H. L¨ owen, J. Phys.: Condensed Matter 22, 104112 (2010). [52] A. H¨ artel, R. Blaak, H. L¨ owen, Towing, breathing, splitting and overtaking in driven liquid crystals, Phys. Rev. E 81, 051703(1-5) (2010). [53] R. Dreyfus, J. Baudry, M. L. Roper, M. Fermigier, H. A. Stone, J. Bibette, Nature 437, 862-865 (2005). [54] A. Erbe, M. Zientara, L. Baraban, C. Kreidler, P. Leiderer, J. Phys.: Condensed Matter 20, 404215 (2008). [55] S. van Teeffelen, H. L¨ owen, Phys. Rev. E 78, 020101 (2008). [56] B. ten Hagen, S. van Teeffelen, H. L¨ owen, Condensed Matter Physics 12, 725-738 (2009). [57] H. H. Wensink, H. L¨ owen, Phys. Rev. E 78, 031409 (2008). [58] G. E. Morfill, A. V. Ivlev, Rev. Mod. Phys. 81, 1353-1404 (2009).
This page is intentionally left blank
September 29, 2010
13:28
World Scientific Review Volume - 9in x 6in
Chapter 2 Polymer Phase Separation
Marcus M¨ uller Institut f¨ ur Theoretische Physik, Georg-August Universit¨ at, Friedrich-Hund-Platz 1, 37077 G¨ ottingen, Germany E-mail:
[email protected]
Computer simulations and numerical techniques for studying phase separation and interfacial properties of polymer blends and polymer-solvent mixtures are discussed and illustrated by a simulation study of coarsegrained models. The results of the simulation are quantitatively compared to the prediction of mean-field theory to identify the Flory-Huggins parameter. Different computational techniques for locating phase boundaries and free energies of interfaces between immiscible phases are described, and the relation between the bulk phase behavior, wetting and phase coexistence in confined geometry is explored.
1. Introduction Multicomponent mixtures comprising polymers find widespread use in industrial applications,1,2 e.g., purification, processing, or melt blending. The phase separation dictates the compositions of the coexisting phases, and the presence of interfaces between phases often dominate the material properties. Typically, additives of low molecular weight (e.g., solvents like hexadecane, toluene or carbon dioxide) can be dissolved in a polymer to some extent. These solvents are utilized to lower the glass transition temperature (i.e. plasticizer) and aid processability. If one increases the concentration further or changes the thermodynamic control variables — pressure or temperature — domains will form that are enriched in the polymer or preferentially contain solvent. For instance, this effect is utilized to fabricate 47
03-chapter2
September 29, 2010
48
13:28
World Scientific Review Volume - 9in x 6in
M. M¨ uller
polymer foams. A detailed knowledge of the phase behavior is crucial for understanding these processes. Blending polymers with different properties is a common strategy for producing materials3 with tailored properties that are difficult to achieve in a one-component material. A prototypical and industrially important example is the mixture of polystyrene (PS) with polybutadiene (PB). PS is a stiff and rather brittle material, which makes it inappropriate for many practical applications. A blend of it with the rubbery PB, however, meets the criterion of a tough material with a large stiffness. This rubber-toughened blend — high-impact PS — is fabricated in large amounts and used in a variety of applications (e.g., packaging materials). Unlike the situation in metallic alloys, however, different polymers of high molecular weight are typically immiscible but rather form domains, in which one or the other component is locally enriched. The size of the domains is kinetically determined and, typically, on the order of micrometers. Much effort has been targeted to tailor the property of the interfaces between the polymers, which, in turn, determine the properties of the material on macroscopic length scales. For instance, the lower the interfacial tension is, the smaller are the domains formed in shear flow (e.g., extruding the mixture).4 If the width of the interface is large enough for entanglements to form between different polymer species, the mechanical strength is greatly improved.5 Thus, the characterization, understanding and control of these properties on a microscopic level is of vital importance. Surface effects can dramatically alter the phase behavior compared to the bulk. The shift of the binodals, which characterize the composition of the two coexisting phases, upon confinement are illustrated in Fig. 1.7–9 At a solid substrate or the free surface of a polymer film to the vapor phase (vacuum), one component may enrich and form a wetting layer. The interplay between these surface effects and the interfaces in a phase-separated blend give rise to a complex phase behavior. These properties become increasingly important as the size of the confinement decreases and the sizes of the enrichment layers is comparable with the linear dimension of the confinement. For instance, in nano-fluidic devices the linear dimensions of channels may be on the same order than the size of a large macromolecule, or in a blend, which is loaded with a large volume fraction of nano-particles, the distance between the surfaces of nano-particles may shrink down to the molecules’ extensions. In this contribution, we will discuss simulation techniques to address the properties of phase-separating mixtures containing polymers. These com-
03-chapter2
September 29, 2010
13:28
World Scientific Review Volume - 9in x 6in
03-chapter2
Polymer Phase Separation
49
80
kBT/ε
60
bulk L=2.8Re sym L=2.8Re anti
40
20
0 0.0
0.2
0.4
0.6
0.8
Twet
1.0
φ Fig. 1. Phase diagrams of a confined, symmetric, binary polymer blend. The bulk phase behavior, binodals for symmetric (capillary condensation) and strictly anti-symmetric (interface localization-delocalization transition) preference of the confining surfaces are shown. The arrow marks the wetting transition temperature, Twet , of the semi-infinite system. The film thickness, L, in units of the molecule’s end-to-end distance, Re , is given in the key. From M¨ uller and Binder.6
plex multi-component materials posses interesting properties on widely different time, length and energy scales and, consequently, different numerical techniques have been employed to investigate them. Predicting the thermodynamics of a mixture of two polymers requires information about the structure on the level of atomistic constituents and their mutual interactions. Typically, in the molten state, chemical repeat units attract each other, which makes the polymers condense into a dense liquid with vanishingly low vapor pressure. The attraction per chemical repeat unit is on the order of the thermal energy scale, kB T , where T denotes the temperature and kB is Boltzmann’s constant. This cohesive energy can be obtained by convoluting the interaction potential and the pair correlation function, which measures the probability of finding two repeat units a distance r apart. Even for the pure component, it is a rather challenging task to determine both quantities with sufficient accuracy in order to predict, for instance, the melt density.
September 29, 2010
50
13:28
World Scientific Review Volume - 9in x 6in
M. M¨ uller
The free energy of mixing in a phase-separating mixture is on the order of kB T per molecule and thus, for a macromolecule of high molecular weight, this quantity only amounts to 10−3 − 10−5 kB T per chemical repeat unit. This tiny energy difference stems from the fact that like repeat units attract each other slightly more than unlike do. Moreover, minuscule changes in the packing, which are reflected in the pair correlation function, or changes of the molecular conformations upon mixing contribute to the thermodynamics of the blend. For most components of practical interest, neither the interaction potentials are known with sufficient accuracy nor it is possible to measure the free energy change upon mixing in simulations of atomistic resolution with the required precision. Fortunately, many properties of the phase behavior and the structure of interfaces between the coexisting phases in a polymer blend can be understood without resorting to an atomistic description. The large molecular weight imparts a large degree of universality on the structure and thermodynamics of phase-separating polymer systems, which makes them particularly amenable to a modeling on a coarse-grained level. Conceptually, these coarse-grained models are constructed by lumping a small number of chemical repeat units into an effective interaction center — denoted as segment in the following — and by calculating the effective interactions between these effective segments. The reduction of the number of degrees of freedom and the softness of the effective interactions on the coarse-grained level allow to investigate larger time and length scales by computer simulations. While this “systematic coarse-graining” procedure has successfully been applied to one-component systems,10,11 it meets significant challenges when applied to polymer blends: Besides the difficulties due to the lack of sufficiently accurate interaction potentials on the atomistic scale12 and the precision of the systematic coarse-graining procedure, the effective interactions on the coarse-grained scale are free energies rather than interaction energies. Generally, they are comprised of multi-body interactions instead of the pair-wise interactions between the constituents on the atomistic level, and they depend on the thermodynamic state (characterized by temperature and pressure), at which they have been derived.13,14 Given these difficulties of “deriving” the blend’s properties from the characteristics of the atomistic constituents, one often uses minimal, coarsegrained models15–17 that only incorporate the relevant interactions: connectivity of the repeat units along the macromolecule, repulsion between the unlike species, and near-incompressibility of the dense polymer liquid. The strength of each one of those interactions is characterized by a coarse-
03-chapter2
September 29, 2010
13:28
World Scientific Review Volume - 9in x 6in
Polymer Phase Separation
03-chapter2
51
grained parameter or “invariant”.18 The justification for utilizing these coarse-grained models is twofold: On the one hand, many polymer polymer blends exhibit common phase behavior and structure in experiments, although their interactions on the atomistic level significantly differ. On the other hand, the structure of long macromolecules can be described by renormalization group theory,19 which exploits the scale-invariant structure of the polymer coil on the length scale ranging from a few chemical repeat units up to the size of the entire molecule. de Gennes demonstrated20,21 that the behavior of long, flexible macromolecules is formally similar to the universal behavior at phase transitions of second order, and this analogy identifies the relevant interaction that a minimal, coarse-grained model has to incorporate to reproduce the universal features of polymer melts.a In a dense melt, linearly connecting chemical repeat units into a flexible molecule gives rise to random walk-like conformations on large length 2 increases linearly with scales, i.e., the mean-square end-to-end distance R e e, the number of repeat units. The distribution of the end-to-end vector, R 2 2 is Gaussian and characterized by zero mean and variance Re ≡ Re . The Gaussian chain conformations on large length scales are completely described by the coarse-grained parameter, Re . This invariant, coarse-grained parameter sets the length scale and, for instance, dictates the length of composition fluctuations in a blend or the thickness of enrichment layers at a surface. In a dense polymer liquid, density fluctuations are small and the system is characterized by a uniform number density, ρo , of segments. This density cannot be used as an invariant, coarse-grained parameter because it depends on the definition of an effective segment. One rather uses the ¯ = (ρo R3 /N )2 , where N is the number of segments per molecule. quantity N e ρo /N is the polymer number density, which is independent from how many effective segments are distributed along the molecule’s contour. In a dense ¯ ∝ N is often referred to as the invariant demelt, Re2 ∝ N and, thus, N gree of polymerization. It measures the number of neighboring molecules, which are contained inside the volume Re3 of a polymer coil. In a typical ¯ = 103 − 106 adopts large values, which indicates experimental system, N a In this chapter, we mainly focus on polymer blends and only briefly discuss polymersolvent systems. Both systems share many qualitative characteristics and many simulation techniques for polymer blends can straightforwardly be applied to polymer-solvent mixtures. Polymer-solvent mixtures, however, typically do not exhibit the same degree of universality as polymer blends, because the interface between polymer and solvent is narrow and there is no separation of scales between solvent and atomistic structure.
September 29, 2010
52
13:28
World Scientific Review Volume - 9in x 6in
M. M¨ uller
¯ is, the less that one molecule interacts with many neighbors. The larger N important become the long-ranged fluctuations. The strength of the repulsion between unlike molecules in a blend is characterized by χN , where χ denotes the Flory-Huggins parameter. This invariant, coarse-grained parameter sets the energy scale. It is defined by the free energy change upon mixing of a homogeneous system, which takes the form22,23 V ∆Fmix ¯ φ ln φ + (1 − φ) ln(1 − φ) + χN φ(1 − φ) (1) = N kB T Re3 where V is the volume of the system, and A and B denote the two species of homopolymers of the symmetric binary blend. 0 ≤ φA = φ ≤ 1 and φB = 1 − φA are the volume fractions (composition) of A- and B-segments, respectively, and both species are comprised of the same number of effective segments, N . If the constituents of the blend are structurally asymmetric, e.g., they differ in their chain extensions or segmental volumes, additional coarse-grained parameters are introduced to capture these asymmetries. The basic idea of minimal, coarse-grained models is that all systems (polymer blends in a laboratory or models utilized for computer simulations) with an identical set of invariant, coarse-grained parameters, Re , ¯ , and χN share the same equilibrium behavior. The information about N the detailed structure and interaction of the chemical repeat units on the atomistic levels is encoded in the small number of invariant, coarse-grained parameters and properties on larger length scales depend on the properties ¯ , and χN . Thus, minimal, coarseof chemical repeat units only via Re , N grained models are well suited to study the universal aspects of phase behavior, interface structure and wetting phenomena and are able to provide quantitative predictions about this class of materials on the length scale ranging from a few nanometers to microns. ¯ , and In addition to these invariant, coarse-grained parameters, Re , N χN , that describe the equilibrium behavior of a symmetric blend in the bulk, the model can be augmented by additional interactions. For example, if a polymer melt is in contact with a solid substrate, the density profile at the surface will rise to the bulk density of the melt over a short distance (e.g., smaller than nanometer). On this length scale, interactions on the atomistic scale between the polymer and the constituents of the solid substrate dictate the behavior and quantum chemical methods have to be employed to accurately predict them. A minimal, coarse-grained model, where each effective segment corresponds to a small number of chemical re-
03-chapter2
September 29, 2010
13:28
World Scientific Review Volume - 9in x 6in
Polymer Phase Separation
03-chapter2
53
peat units, is not capable of capturing the fine structural re-arrangements on the sub-nanometer scale. In order to describe the formation of enrichment layers or wetting phenomena — collective phenomena which occur on a length scale set by Re — the dimensionless surface tension, γRe2 /kB T , serves to encode the atomistic structure at the surface. Once the invariant, coarse-grained parameters have been identified, computer simulations of minimal, coarse-grained models can predict the structure and thermodynamics of multi-component systems containing polymers in the bulk and under confinement. A particular advantage of computer simulations is the fact that the parameters can be independently and systematically varied and a multitude of structural, thermodynamic and kinetic properties on different length scales is simultaneously accessible, e.g., the molecular conformation at the interface between coexisting phases in a blend and the excess free energy of the interface. Therefore, computer simulations can provide insights of properties on the nanoscale that are difficult to directly observe in experiment, and they can correlate them with macroscopic thermodynamical behavior and structure. This information obtained by computer simulation allows for a stringent test of theories and the comparison of different models that give rise to the same invariant, coarse-grained parameters can investigate, which properties are universal, i.e., independent from the microscopic interactions of the specific model. In order to exploit this potential, computational models and simulation techniques shall address some of the following challenges: • The invariant, coarse-grained parameters shall be accurately determined by computer simulations. While the extension, Re , of the macromolecules ¯ , can be straightforwardly measured in exand the polymer density, N periments or computed in simulations, the Flory-Huggins parameter, χ, is commonly defined by comparing the results of experiments or simulations to the prediction of the mean-field theory of a binary polymer blend (cf. Eq. (1)). If this comparison is performed in a regime of the phase diagram and for quantities, where the approximations of the meanfield theory are justified, consistent results are obtained. We shall discuss various possibilities of identifying the Flory-Huggins parameter in Sec. 2.2. • The invariant, coarse-grained parameters shall adopt values that are characteristic for experimental systems. If the minimal, coarse-grained model incorporates harsh, excluded volume interactions between the ef-
September 29, 2010
54
13:28
World Scientific Review Volume - 9in x 6in
M. M¨ uller
fective segments (e.g., the bond fluctuation model24,25 or a LennardJones bead-spring model26,27 ), the maximal density will be on the order of ρo b3 ∼ O(1), where b2 = Re2 /N denotes the statistical segment length. For flexible polymer models, b is comparable to the segment size. For larger densities, the liquid of segments will form a crystal or vitrify into a glass. In order to describe a polymer blend with an invariant degree of ¯ = 104 , which is quite typical for an experimental syspolymerization, N tem, the chain contour has to be comprised of many effective segments, N = N¯ /(ρo b3 ) ∼ O(104 ). This poses formidable computational problems: The relaxation time of chains with such a large value of N is enormous, τ ∼ Re2 /D ∼ N 3 , and a simulation box with√modest, linear dimension of ¯ (L/Re )3 N ∼ O(6 · 107 ), L = 4Re contains a huge number, ρo L3 = N of effective segments. Even on state-of-the-art supercomputers such calculations are not feasible. Thus, computer simulations of models with harsh, excluded volume interactions typically correspond to values of ¯ in the range 10 − 500, which is significantly smaller than experiN mental values. Therefore, long-ranged fluctuations are very important in these computational models; they are less important for experiments; and the mean-field theory, which ignores all fluctuations, corresponds ¯ → ∞. A low value of N ¯ is useful in order to highlight to the limit N the role of fluctuation effects and explore the limitations of mean-field theories. In order to achieve large values of N¯ in computational models, one can increase the polymer density ρo /N . A system of size L = 4Re and invari¯ = 104 , contains 6 400 polymer chains. If ant degree of polymerization, N one places N = 32 effective segments along the chain contour, which is just sufficient to approximate the random walk-like statistics of the chain conformations, then the system contains only 204 800 effective segments 3 ¯ but the segment density is ρo b = N /N ≈ 18 very large. Such values can only be achieved by replacing the excluded volume interaction of the segments by a soft repulsion without hard core. These soft potentials between effective segments on the coarse-grained scale arise naturally in the systematic coarse-graining procedure, where the degrees of freedom “inside an effective segment” are integrated out. Such soft potentials are typical for DPD models28 or models applied in conjunction with SingleChain-in-Mean-Field simulations.29–31 • Thermodynamic properties shall be accurately computed for a coarsegrained model. Since these thermodynamic characteristics involve the free energy, they are not simple functions of the segments’ coordinates,
03-chapter2
September 29, 2010
13:28
World Scientific Review Volume - 9in x 6in
Polymer Phase Separation
03-chapter2
55
and special simulation techniques have been devised to locate phase coexistence or compute interface free energies. Some of those techniques are described in Sec. 2.3. • On accessible, human time scales, equilibrium on macroscopic length scales often cannot be attained. For instance, the coarsening of the domain structure in a binary blend arrests, once the domains grow to a size on the order of micrometers, and one experimentally observes that the macroscopic structure depends on processing history. Unfortunately, the kinetics of phase separation exhibits less degree of universality than the equilibrium phase behavior. While the gross features have been classified by Hohenberg and Halperin,32 the application of minimal, coarse-grained models for investigating the kinetics of phase separation is still in its infancy. On the one hand, it is not obvious that different dynamic process can be described by a single rescaling factor of time, which relates the time scale of the minimal, coarse-grained model to the experiment. The largest single-chain relaxation time, τ = Re2 /D (with D being the self-diffusion coefficient of the molecules) is an appropriate invariant for symmetric blends but generalizations to blends with a strong dynamical asymmetry (e.g., where one component vitrifies in the course of phase separation) are a challenge. On the other hand, the softer interactions on the coarse-grained scale between the effective segments may alter the qualitative characteristics of the dynamics. For instance, the combination of finite bond length and harsh, excluded volume interactions on the atomistic scale ensures that chain molecules cannot cross through each other in the course of their motion. This incrossability condition leads to the reptation dynamics of long macromolecules in a melt,33 where the dynamics of fluctuations perpendicular to the chain contour is limited by the topological constraints to a tube around the instantaneous conformation. Long, flexible molecules slither like snakes along their contours. Fortunately, the√tube diameter, dT , does not introduce a ¯ with the constant of proportionality new length scale but dT ∝ Re / N being universal, i.e., independent from the details of the chemical repeat units. In coarse-grained models, which are able to address large values of N¯ by replacing harsh, excluded volume interactions between effective segments by soft repulsion, non-crossability of bonds between effective segments is not “automatically” guaranteed. It has to be enforced by explicitly checking that bonds do not cross in the course of the simulation,34 or they have to be incorporated in an effective manner by slip-links.35
September 29, 2010
13:28
World Scientific Review Volume - 9in x 6in
56
03-chapter2
M. M¨ uller
2. Phase Behavior in the Bulk 2.1. Predictions of the mean-field theory The excess free energy of mixing, Eq. (1), of a molecule in a symmetric polymer blend is comprised of two terms. The first one, stems from the combinatorial entropy of the mixture due to the indistinguishability of different A-molecules and different B-molecules. It is on the order of kB T per molecule. The second contribution describes the interactions between the different polymer species and the Flory-Huggins parameter, χ, quantifies the repulsion between segments of different type. Generically, this repulsion of coarse-grained segments is comprised of enthalpic and entropic contributions. This simple, analytical mean-field prediction lays at the the basis of self-consistent field (SCF) calculations16–18,36–39 for spatially inhomogeneous polymer systems in the framework of the Gaussian chain model. Its predictions have been abidingly employed to analyze experiments on the phase behavior and interfacial properties. Some of the results are compiled below: • The critical point of a symmetric blend is located at: N ∼ χMF c N = 2 Tc
and
φMF = c
1 , 2
(2)
This result quantifies that already minuscule repulsions between segments of different species, which are only on the order of 1/N , give rise to phase separation, and it explains why most pairs of polymers do phase separate. • In a symmetric blend, the binodals are given by the implicit form χN =
1 φ ln 2φ − 1 1 − φ
(3)
Due to the mean-field approximation, the binodal in the vicinity of the critical point are parabolic, while much flatter binodals are observed in simulations (cf. Fig. 1) and experiments, which are a hallmark of the Ising universality class. • Since it is difficult to measure the free energy of mixing in computer simulations, one often considers the relation between the exchange potential ∆µ ≡ µA − µB and the composition of the mixture, where µA and µB denote the chemical potentials per molecule. 1 ∆µ ∂∆F √ = = ln φ − ln(1 − φ) + χN (1 − 2φ) ¯ V /R3 ∂φ kB T kB T N e
(4)
September 29, 2010
13:28
World Scientific Review Volume - 9in x 6in
Polymer Phase Separation
03-chapter2
57
composition fluctuation loop
(b)
(a)
Fig. 2. (a) Snapshot of an interface between two coexisting phases in a binary polymer blend in the bond fluctuation model.24 Within this coarse-grained lattice model, effective segments are represented by cubes on a lattice. Each segment blocks the 8 corners from double occupancy and segments along the chain are connected by one of 108 different bond vectors. In the symmetric binary blend, segments interact via a square-well potential that is extended over the nearest 54 lattice sites. Like segments attract each other with strength , while unlike species repel each other with . (invariant polymerization ¯ = 91, incompatibility χN ≈ 17, linear box dimension L ≈ 7.5Re , or numindex N ber of effective segments N = 32, interaction /kB T = 0.1, monomer number density ρ0 = 1/16.0). (b) Cartoon of the molecular configuration illustrating loops of a chain into the domain of opposite type, fluctuations of the local interface position (capillary waves) and composition fluctuations in the bulk and the shrinking of the chains in the minority phase. From M¨ uller.17
• The experimentally accessible, collective structure factor, S(k), of composition fluctuations is given by: N SRPA (k)
=
1 1 + − 2χN φgA (k) (1 − φ)gB (k)
(5)
in the homogeneous phase, χN < 2. gA (k) = SA /N denotes the normalized single-chain structure factor of A-polymers and k the wave-vector of composition fluctuations. If the chains obey Gaussian statistics, it is given by the Debye-function: 12 −(kRe )2 /6 2 − 1 + (kR ) /6 (6) e g(k) = e (kRe )4 • The properties of the interface are dominated by excursions of Apolymers into the B-rich phase, and vice versa. This is illustrated by
September 29, 2010
58
13:28
World Scientific Review Volume - 9in x 6in
03-chapter2
M. M¨ uller
a snapshot of a simulation in Fig. 2. The energy cost of each loop is comparable to the thermal energy scale, kB T . Each monomer of a loop contributes to this energy an amount of the order χkB T , and, consequentially, the number of monomers per loop scales like 1/χ. The spatial extension of the loops determines the interfacial width. In the Gaussian chain model the conformational distribution is Gaussian on all length scales, √ and therefore the spatial extension of loops is of the order Re / χN . In the limit of strong segregation (SSL), this quantity sets the width of the interface between coexisting phases: wSSL 1 = √ Re 6χN
(7)
• Each monomer in the interfacial region contributes to the interface ten√ sion an amount χ, and the interface tension scales as χρo Re / χN . The result of the SCF theory (including prefactor of order unity) is:40 σ R2 χN √SSL e = (8) ¯ 6 N kB T ¯ , the FloryUnlike the other coarse-grained parameters, Re and N Huggins parameter is defined by comparing experimental or simulation data to the prediction of the mean-field theory. We emphasize, however, that these predictions are derived invoking a mean-field approximation (or even stronger assumptions) in order to obtain a tractable, analytical description. Of course, one should only use these mean-field predictions to identify invariant, coarse-grained parameters in the regime of validity of the mean-field approximation, i.e., the regime where long-ranged fluctuations are of negligible importance. In the hypothetical limit of large invariant de¯ → ∞, any of the above quantities can be utilized gree of polymerization, N to identify the Flory-Huggins parameter, χ. Unfortunately, large degrees of polymerization are often not accessible in computer simulations, fluctuations are important, and the extrapolation of simulation results to N¯ → ∞ can be difficult because the convergence towards the asymptotic behavior may set in only for large chain lengths, N . Moreover, the predictions above are affected to a different extent by the mean-field approximation. Thus, identifying χ via the comparison between simulation results and mean-field ¯ → ∞ or paying special attention to predictions without extrapolating to N the role of fluctuation effects might result in significant errors and different quantities might yield different, mutually inconsistent estimates.
September 29, 2010
13:28
World Scientific Review Volume - 9in x 6in
03-chapter2
4
3
MC (L=96)
6
2 0
0
3 0
0.5
1
φA
-1
ε=0.004kBT ε=0.007kBT 2.44(2φA-1)
-2 -3 0.0
0.2
0.4
φA
(a)
0.6
0.8
2
8
4
0
6 5
0
4
1
8
3 2
FH
1 0
1.0
~ χN
4
~ χN
1
FH 3D Ising (FSS fit)
4
N/S(k=0)
~ χkBT/ε
(kBTln[φA/φB]-∆µ)/2Nε
2
59
ln[φA/(1-φA)]/(2φA-1)
Polymer Phase Separation
0 0
0
0.2 0.4φ 0.6 0.8 A
MC
1
1 ~ χN=2Nzcε/kT with zc=2.44
2
(b)
Fig. 3. Estimation of the Flory-Huggins parameter for a symmetric polymer blend with chain lengths NA = NB = 64 obtained by semi-grandcanonical simulations of the bond fluctuation model. (a) First derivative of the Flory-Huggins free energy. The slope of the curve yields the effective coordination number zc = 2.44. The inset presents the value of the coordination number zc = χT /2 extracted from the collective structure factor. The horizontal line marks the estimate from the main panel. (b) Inverse maximum of the collective structure factor of composition fluctuations, N/S(k → 0), as a function of the incompatibility, χN . Symbols correspond to MC simulations for system size L = 96, the dashed curve presents the results of a finite size scaling analysis of the simulation data in the vicinity of the critical point, and the straight, solid line indicates the prediction of the Flory-Huggins theory. The critical incompatibility predicted by the Flory-Huggins ¯ ≈ 240) are theory and obtained from MC simulations of the bond fluctuation model (N indicated by arrows. The left inset compares the phase diagram obtained from simulations with the prediction of the Flory-Huggins theory. The right inset replots the composition of the coexistence compositions such that the mean-field theory predicts them to fall onto a straight line.
2.2. Estimating the Flory-Huggins parameter for simulation models A choice of possible identifications of the Flory-Huggins parameter is given below: (1) The relation between exchange potential ∆µ and composition φ¯A of the blend, cf. Eq. (4), or composition fluctuations/scattering, cf. Eq. (5), in the one-phase region far away from the critical point can be used to determine χ. (2) Measuring the composition of the two coexisting phases at large incompatibility and comparing the result to the prediction of the FloryHuggins theory, Eq. (3), one can estimate the χ-parameter. (3) The mean-field theory also relates the location of the critical point of demixing (cf. Eq. (2)) and scattering intensity of composition fluctuations to the Flory-Huggins parameter.
September 29, 2010
60
13:28
World Scientific Review Volume - 9in x 6in
M. M¨ uller
(4) The SCF theory makes explicit predictions for the width of the interface between coexisting phases (cf. Eq. (7)). Comparing experimental measurements or simulation results to those predictions also yields an estimate for χ. All estimates basically treat the Flory-Huggins parameter, χ, as a phenomenological quantity, determined by matching the properties of the blend to the predictions of the SCF theory and thereby absorbing all unknown information concerning blend miscibility.41 This procedure is well adopted to the concept of coarse-grained models, where the structure on the microscopic length scale determines the behavior on the large length scale only via a small number of parameters. The latter two approaches are, however, affected by long-wavelength fluctuations, namely composition fluctuations near critical points and capillary waves at interfaces (to be illustrated in Sec. 3). Measuring the dependence of the composition on the exchange potential is straightforward in the semi-grandcanonical ensemble, where the exchange potential ∆µ, is a control parameter and one observes the composition and its fluctuations in the simulation.42,43 The dependence φA (∆µ) (cf. Eq. (4)) and the composition fluctuations φ2A − φA 2 (cf. Eq. (5)) correspond to the first and second derivative of the Flory-Huggins free energy of mixing with respect to the composition, and therefore constitute a direct test of the Flory-Huggins free energy.22,23 Simulation data for the bond fluctuation model are presented in Fig. 3(a), where we plot the difference between the exchange chemical potential and the ideal gas contribution, (∆µ − kB T ln[φ¯A /φ¯B ])/2N , vs the composition φ¯A , where measures the repulsion between unlike segment species in this coarse-grained lattice model. In accord with the FloryHuggins estimate, Eq. (4), we observe a linear dependence on composition for all temperatures far above the demixing transition. This suggests that χ ∼ /kB T in our model at least for small values of , and we can identify the Flory-Huggins parameter, χ = 2zc /kB T with an effective coordination number zc = 2.44. In the inset of Fig. 3(a) we present the estimate for the effective coordination number via the inverse of the collective structure factor, Eq. (5). Within the statistical uncertainties, we again find that χ ∼ /kB T with a slight upwards parabolic correction for asymmetric compositions. This parabolic composition-dependence becomes stronger as we approach the critical point.44,45 Far above the critical temperature, however, the ratio
03-chapter2
September 29, 2010
13:28
World Scientific Review Volume - 9in x 6in
Polymer Phase Separation
03-chapter2
61
adopts the value χkB T / = 2zc ≈ 4.88 consistent with the results of the main panel. Even though composition fluctuations are directly related to the experimental procedure, we prefer to extract the Flory-Huggins parameter from the excess of the exchange potential, Eq. (4), in the simulations, because the translational contribution diverges only logarithmically for φA → 0 or 1. Also in experiments, strongly asymmetric compositions might induce inaccuracies46 when one utilizes the structure factor, Eq. (5). Moreover, the composition is a self-averaging quantity that can be obtained with high precision in the simulations while the error of composition fluctuations, S(k = 0), does not decrease with system size. In Fig. 3(b) we use this estimate of the Flory-Huggins parameter, χ = 2zc /kB T , to plot the inverse structure factor as a function of χN . Far above the critical temperature, χN χc N = 2, the data are consistently describable by the mean-field theory, i.e., for φA = 1/2 we find the simple relation, N/S(k = 0) = 4 − 2χN . This plot constitutes a common experimental procedure to measure the Flory-Huggins parameter. In the ultimate vicinity of the critical point, however, deviations from the SCF theory become notable. The inverse structure factor extracted from the MC simulations does not vanish at χSCF N = 2, and the mean-field theory c overestimates the true critical temperature by a factor 1.12 for the chain length, N = 64 utilized in the simulation. Note, that this value corresponds ¯ = 240. Similar to simulations44 to an invariant polymerization index, N 47–50 and experiments, we observe in Fig. 3(b) the crossover from the meanfield dependence to the Ising behavior: If we approach the critical point at the critical composition, φA = 1/2, we will not observe a linear dependence of the inverse scattering factor with /kB T up to the ultimate vicinity of the critical point, but we rather find 1/S(k → 0) ∼ ( /kB T − /kB Tc )γ , where γ ≈ 1.24 denotes the critical exponent of the order parameter fluctuations in the three dimensional Ising model. By virtue of the universality at second order phase transitions, a polymer mixture exhibits the same critical behavior as other mixtures characterized by a single scalar order parameter. The SCF theory, being a mean-field theory and assuming random mixing, cannot describe these critical, long-ranged composition fluctuations. The SCF theory predicts a parabolic shape of the binodal at the critB ical point, while simulations and experiments observe |φA coex − φcoex | ∼ β ( /kB Tc − /kB T ) , where the critical exponent of the order parameter takes the value of the three-dimensional Ising universality class, β ≈ 0.32.
September 29, 2010
62
13:28
World Scientific Review Volume - 9in x 6in
M. M¨ uller
The binodals are presented in the insets of Fig. 3(b) for N = 64 (and in Fig. 1 for N = 32). Note that the deviations are even more pronounced in thin films (see Fig. 1), where the binodals are characterized by the exponent β = 1/8 characteristic of the universality class of the two-dimensional Ising model. While the SCF theory fails in the ultimate vicinity of the critical point, at larger χN the binodals are well described by the SCF theory. We expect the region where long-wavelength composition fluctuations lead to a break-down of the SCF theory to become even smaller for larger invariant polymerization index, which is characteristic for experimental systems. ¯A )] φ In the left inset of Fig. 3(b) we plot the binodals in the form ln[φA2φ/(1− A −1 vs χN . Within the mean-field theory, all data fall onto a line with unit slope. This constitutes an alternative possibility to extract the Flory-Huggins parameter for a phase-separated blend. At low incompatibilities, in the ultimate vicinity of the critical point, we find deviations due to long-ranged fluctuations as discussed above. At very large incompatibilities, the FloryHuggins parameter becomes of the order unity for the short chain length considered in the simulations and, then, the repulsion between unlike segments is strong enough to change the structure and packing in the liquid of segments. To summarize, fitting the results of simulations to the predictions of the SCF theory far away from the critical point yields consistent results, i.e., all observations can be described by a Flory-Huggins parameter, which is proportional to the repulsion between unlike segments, /kB T for small values of . In the ultimate vicinity of the critical point, however, neglect of longranged composition fluctuations by the SCF theory affects the estimate of the Flory-Huggins parameter. Estimating the Flory-Huggins parameter from properties in the ultimate vicinity of the critical point leads to spurious effects due to fitting a mean-field theory to data which exhibit critical fluctuations according to the universality class of the three-dimensional Ising model. The estimate of the Flory-Huggins parameter does not only account for short-ranged correlations but also incorporates some effects of long-ranged critical fluctuations. If we defined the Flory-Huggins parameter via the critical properties, the Flory-Huggins parameter would not give an appropriate description further away from the critical point or for larger invariant degrees of polymerization because the effect of long-ranged composition fluctuations would become negligible but short-range correlations would remain important.
03-chapter2
September 29, 2010
13:28
World Scientific Review Volume - 9in x 6in
Polymer Phase Separation
03-chapter2
63
2.3. Simulation techniques for computing the bulk phase behavior The study of phase coexistence by computer simulation has played a central role in the study of condensed-matter systems. In order to obtain a very rough estimate of the properties of the coexisting phases without applying any special simulation technique, one can consider the coexistence between two domains in a simulation cell. By exchanging particles across the interface and adjusting the volume of the coexisting domains, one establishes chemical and mechanical equilibrium. The size of the simulation cell has to be chosen large enough for the volume fraction of the interfaces, which separate the two domains, be negligible and the properties in the interior of the domains be not influenced by the presence of the interfaces. Rather large system sizes are necessary in order to obtain accurate results, and the equilibration across the interface via the diffusive motion of the polymers is an extremely slow process. Even within the framework of computationally efficient, coarse-grained models, this approach is only a viable simulation strategy for small oligomers. In the late 80’s, the Gibbs ensemble simulation technique51 has been devised, which permits the direct calculation of phase coexistence properties. Instead of separating the two coexisting domains via interfaces, one places them in two, separate simulation cells. In each simulation cell, periodic boundary conditions are applied to mimic the behavior of an infinitely extended, bulk system. If the segregation between the phases is sufficiently strong, each cell will contain one phase but no interface. Since one studies the bulk properties of a phase, significantly smaller system size already yield accurate results. Chemical and mechanical equilibrium between the two simulation cells are established by exchanging particles and volume between the two cells. If the overall composition of the system, i.e., the volume-weighted average of the two simulation cells, is located somewhere inside the miscibility gap, one will observe phase coexistence and the two coexisting phases will form in the corresponding simulation cells. Thus, the knowledge of the coexistence pressure or chemical potential is not required, which makes this a rather robust method. A powerful and popular alternative are advanced thermodynamic integration methods that relate the two coexisting phases via a reversible path. In the context of a binary polymer blend, simulations in the semigrandcanonical ensemble in conjunction with reweighting techniques have proven particularly useful. In the semi-grandcanonical ensemble, which has
September 29, 2010
64
13:28
World Scientific Review Volume - 9in x 6in
03-chapter2
M. M¨ uller
been introduced by Sariban and Binder in the late 80’s,44,52 the temperature, T , the total number of polymers, nA + nB , and the volume, V , are A fluctuates and it is fixed. The composition of the mixture, φ ≡ nAn+n B controlled by the exchange chemical potential, ∆µ (see Eq. (4)). While we illustrate this technique for the specific example of a symmetric, binary polymer blend, the simulation strategy carries over to other systems by replacing the composition φ by the order parameter that distinguishes the two coexisting phase (e.g., the total number of particles in case of a liquid-vapor transition) and ∆µ by the thermodynamically conjugated variable (e.g., the chemical potential in case of a liquid-vapor transition). In order to realize such an ensemble in the framework of Monte Carlo simulations, two types of Monte Carlo moves have to be utilized — canonical moves, which alter the chain confirmations, and semi-grandcanonical moves, which relax the order parameter. In the case of a symmetric polymer blend, the latter moves merely consists in switching the labels A B at fixed conformation of the molecules. For structurally asymmetric mixtures generalizations can be devised for special cases.42,43,53,54 One advantage of the semi-grandcanonical ensemble consists in the fast relaxation of fluctuations of the order parameter. Rather than decaying via the slow diffusion of long macromolecules in a dense melt, composition fluctuations relax via relabeling. While those Monte Carlo moves do not mimic the kinetics of composition fluctuations in an experimental system, where the order parameter is locally conserved, they do allow for an extremely efficient sampling of configuration space. In the simulations, one monitors the probability distribution, P (φ), of the order parameter. In the vicinity of phase coexistence, this distribution is bimodal (see Fig. 4). Each peak corresponds to one of the coexisting phases. If one increases the exchange chemical potential, ∆µ, the weight of the peak, which corresponds to the A-rich phase increases. Two phases coexist, if at constant temperature and (exchange) chemical potential, they will result in the same (osmotic) pressure. The latter quantity is related to the (semi-)grandcanonical energy, G(∆µ, V, T ), via G(∆µ, V, T ) πV =− = ln Z dφ P (φ) kB T kB T A-rich peak
(9)
where Z(∆µ, V, T ) denotes the partition function of the (semi-) grandcanonical ensemble. The condition of equal pressure between coexisting phases leads to the equal weight rule: Two phases will coexist, if the (exchange) chemical potential is adjusted such that the weights of the two
September 29, 2010
13:28
World Scientific Review Volume - 9in x 6in
03-chapter2
Polymer Phase Separation
probability P(φ)
10
65
0
2
exp(2Lσ/kT)
10
-10
10
-20
0.0
0.2
0.4 0.6 composition φ
0.8
1.0
Fig. 4. Probability distribution of the composition at = 0.02kB T (i.e., T /Tcbulk = 0.7215) in a system of geometry 3.8Re ×3.8Re ×7.5Re with periodic boundary conditions in all three directions. The two peaks correspond to the A-rich and the B-rich phase as indicated by the snapshots. In the vicinity of φ = 1/2 the typical configuration consists in two domains separated by two AB interfaces of size L2 . The slab configuration — an A-rich domain that spans the simulation cell in the two shorter directions and separated by two interfaces from a B-rich domain — is stable over a larger range of composition and for symmetric composition, φ = 1/2, the two AB-interfaces are farther apart than in a cubic cell, thereby, reducing the interactions between the two interfaces. From M¨ uller and Binder.55
peaks in the probability distribution of the order parameter are equal, i.e. (10) dφ P (φ; ∆µcoex , V, T ) = dφ P (φ; ∆µcoex , V, T ) B-rich peak A-rich peak It has been demonstrated that away from the critical point, where the probability distribution exhibits a pronounced two-peak structure, this estimate for coexistence value of the (exchange) chemical potential, ∆µcoex , has only exponentially small finite size effects.56,57 This general strategy has become very popular because of two advanced simulation and analysis techniques: (i) weighted histogram analysis and (ii) multicanonical simulations. The first method, introduced by Ferrenberg and Swendsen in the late 80’s,58,59 provides a simple means for extrapolating data generated at one set of control parameters to nearby points in thermodynamic space. Specifically, for the example discussed above, there is no need to perform many different simulations for different values of the
September 29, 2010
13:28
World Scientific Review Volume - 9in x 6in
66
M. M¨ uller
exchange potential, ∆µ, in order to accurately locate phase coexistence according to the equal weight criterion (10), but if the simulation have been performed at value ∆µ and the corresponding probability distribution, P (φ; ∆µ, V, T ), has been sampled, the probability distribution at a slightly different value ∆µ is given by: A P (φ; ∆µ, V, T ) exp (∆µ k−∆µ)n BT (11) P (φ; ∆µ , V, T ) = 1 (∆µ −∆µ)nA dφ P (φ; ∆µ, V, T ) exp kB T 0 √ ¯ V /Re3 is the number of A-polymers in the simulation cell. where nA = φ N The same idea can be utilized to combine simulation data generated at different values of the control parameter, ∆µ. Consider a series of r semi-grandcanonical simulations at different ∆µi with i = 1, · · · , r. The histograms of the number of A-polymers are sorted and stored in the form, Hi (nA ) = H(nA , ∆µi , V, T ). Let #i = nA Hi (nA ) denote the number th of entries in the i histogram. Then, the results of the different simulations can be combined to yield an accurate estimate of the unnormalized probability distribution r ∆µnA H (n ) exp i=1 i A kB T P˜ (φ, ∆µ, V, T ) = (12) r ∆µi nA −Gi i=1 #i exp kB T where the value of the free energy, Gi , of the system at exchange chemical potential, ∆µi , has to be determined self-consistently by iterating the above equation and the normalization condition
√¯ NV −Gi = dφ P˜ (φ, ∆µi , V, T ) exp (13) kB T Re3 The second advance — multicanonical simulation technique — addresses the issue of efficiently sampling the bimodal probability distribution of the order parameter: In order to establish the equal weight criterion, the system must often tunnel between the two peaks of P (φ) in the course of the simulation. As can be seen in Fig. 4, the probability to find the system between the two peaks is excessively low. Sampling the states with the Boltzmann weight, P (φ), one will not encounter configurations with a φA
+φB coex
composition around coex 2 one of the two phases.
but the system will rather remain stuck in
03-chapter2
September 29, 2010
13:28
World Scientific Review Volume - 9in x 6in
Polymer Phase Separation
03-chapter2
67
In order to mitigate this problem, one can sample configurations {ri }nA with a weight
−E[{ri }nA ] + ∆µnA (14) Pmuca[{ri }nA ] ∼ exp × e−w(nA ) kB T The first factor characterizes the weight of a configuration comprised of nA polymers of type A and energy E in the semi-grandcanonical ensemble. The second, “re-weighting” factor modifies the Boltzmann weight and it only depends on the number, nA , but not on the detailed conformations of the polymers. Thus, if one averages any property over configurations with a fixed number of A-polymers, one will obtain a result that corresponds to the canonical ensemble. The probability, √ Pmuca (φ) of finding the system ¯ V /R3 , of A-polymers — i.e., a in a state with a fixed number, nA = φ N e canonical slice of the configuration space — will differ from the probability, P (φ), in the semi-grandcanonical ensemble. However, both quantities are related by the simple formula Pmuca (φ) ∼ P (φ)e−w(φ)
(15)
In order to sample the peaks of P (φ) and the region between them efficiently in a multicanonical simulation, Pmuca (φ) must be approximately flat, i.e., independent from φ.b Thus, we should choose the re-weighting factors such that P (φ) + const ≈ ln P (φ) (16) w(φ) = ln Pmuca (φ) Formally, these equations are valid for an arbitrary choice of the reweighting factors but an inappropriate, “bad” choice does not allow for a sufficient sampling of the interesting interval of nA -values. The problem of sampling the probability distribution, P (φ), is therefore shifted to that of obtaining appropriate re-weighting factors, w. An optimal choice of w(φ), however, requires a working estimate of the probability distribution, P . Several strategies have been devised to overcome this dilemma: (i) multicanonical recursion,61 (ii) Wang-Landau sampling,62 (iii) transition matrix Monte Carlo63 or (iv) successive umbrella sampling.64 We refer the reader to the literature65 for a thorough discussion of the advantages of the the different methods in terms of simplicity of implementation, efficient use of information, detailed-balance issues and convergence rate. In the following, we briefly describe the successive umbrella sampling technique, which b Alternatively, one can require that the number of transitions from the A-rich to the B-rich phase and vice versa be maximal.60
September 29, 2010
68
13:28
World Scientific Review Volume - 9in x 6in
03-chapter2
M. M¨ uller
is very easy to implement. The guiding idea of umbrella sampling66 is to divide the pertinent range of the order parameter, φ, into smaller windows and to investigate one small window after the other. If the windows are sufficiently narrow, the probability P (φ) will not substantially vary within a window, and a crude estimate of the re-weighting factors, e.g., w(φ) =constant within a window, is often sufficient to ensure sampling of all values of the order parameter within a window. In the limiting case, the windows just consist of two neighboring values of the order parameter, nA and nA + 1. A histogram Hi (nA ) monitors how often each state is visited in the + ith window, n− i ≤ nA ≤ ni . Care must be exercised at the boundaries of a window to fulfill detailed balance:67,68 If a semi-grandcanonical identity switch attempts to leave the window, it is rejected and H(window edge) is incremented by unity. Another question, which may arise from the discussion of the boundary is the optimum amount of overlap to minimize the uncertainty of the overall ratio. Here we choose the minimal overlap − of one state at the interval boundaries, i.e., n+ i = ni+1 . This is simple to implement and sufficient to match the probability distributions at their boundaries. A larger overlap may reduce the uncertainty but requires a higher computational effort. + th Let Hi− ≡ Hi (n− i ) and Hi+ ≡ Hi (ni ) denote the values of the i histogram at its left and right boundary, and Ri ≡ Hi+ /Hi− characterize their ratio. After a predetermined number of Monte Carlo steps per window, the (unnormalized) probability distribution can recursively be estimated according to: Hi (nA ) Hi (nA ) P (nA ) H0+ H1+ · ··· = Πi−1 k=1 Rk · − = H0− H1− Hi− Hi− P (n0 )
with Ri ≡
Hi+ , Hi− (17)
+ when nA ∈ [n− i : ni ]. The ratios in Eq. (17) correspond to the Boltzmann factor associated with the free energy difference across the order parameter interval i. We now consider how the overall error depends on the choice of the window size:64 For clarity, we assume that no sampling difficulties are encountered in our system, i.e., the order parameter in which we reweight (i.e., number of A-polymers) is suitable to flatten and overcome all barriers in the (multidimensional) free energy landscape and the restriction of fluctuations of the average order parameter by the window size does not impart sam-
September 29, 2010
13:28
World Scientific Review Volume - 9in x 6in
Polymer Phase Separation
03-chapter2
69
pling errors onto the simulation.c Under these circumstances, it has been suggested69,70 that small windows reduce computational effort by a factor of r, where r denotes the total number of windows into which the sampling range is subdivided: The time τ to obtain a predetermined standard deviation δ of the ratio Ri in a single window is proportional to the square of − 2 the window size, ∆i = n+ i − ni . With τ ∼ ∆i , we get a total computation 2 time (for all windows) of tcpu ∼ r∆i , implying that the overall error of the simulation is also δ. This contrasts the behavior of a single large window; r = 1 and ∆i = r∆i yield tcpu = (r∆i )2 = rtcpu , which suggests that a window size as small as possible be chosen.69,70 Since we are interested in localizing phase coexistence, however, the pertinent error is related to the free energy difference of the two end points of the order parameter interval, i.e., the peaks of the probability distribution that correspond to the distinct phases. Therefore, we have to account for error propagation in Eq. (17) and − we obtain for the error, ∆, of the ratio P (n+ r )/P (n1 ) r √ ) P (n+ r = ∆≡δ δRi2 ∼ O(δ r). (18) − P (n1 ) i=1 Due to error propagation across the windows, the error of each individual √ subinterval has to be smaller than the total error, ∆, by a factor of r. Thus, the time that must be spent in each window to achieve an error √ of δ = ∆/ r is rτ and the total simulational effort is r2 τ ∼ (r∆)2 ∼ rtcpu , which is identical to the time required for a single large window. This argument demonstrates that the statistical error for a given computational effort is independent from the window size, i.e., the number of intervals the range of order parameter is divided into. A more detailed analysis including possible systematic errors due to (i) estimating probability ratios and (ii) correlations between successive windows can be found in Ref. 64. The computational advantage from subdividing the range of order parameter into windows does not stem from an increased statistical accuracy due to the small correlation times within a window, but from the fact that successive umbrella sampling does not require an independent and comc Restricting
the window size limits order parameter fluctuations to far less than those explored in a semi-grandcanonical simulation and each sub-simulation resembles more closely a simulation in the canonical ensemble than in the semi-grandcanonical ensemble. We emphasize, however, that local composition (order parameter) fluctuations are not restricted and that, ideally, configurations with a fixed order parameter have identical statistical weight in the canonical ensemble, in the ensemble used in our simulation and in the semi-grandcanonical ensemble.
September 29, 2010
13:28
World Scientific Review Volume - 9in x 6in
70
M. M¨ uller
putationally costly generation of re-weighting factors. Making the window size small, we minimize the variation of the free energy across a single interval and achieve sufficiently uniform sampling without re-weighting factors. Therefore, successive umbrella sampling is as efficient as a multicanonical simulation with very good re-weighting factors, except that now w needs not be known beforehand. Moreover, the scheme is easy to implement on parallel computers and the range of order parameter can be easily enlarged by simply adding more windows. This computational scheme has successfully been applied to study phase equilibria and interface properties in polymersolvent mixtures,64,71 polymer-colloid mixtures72–75 and liquid crystals.76,77 2.4. Compressible mixtures The phase behavior of binary mixtures will become much more complex, if one relaxes the assumption of incompressibility. In this case, the free energy of mixing, ∆Fmix , depends on two order parameters — density and composition (ρ, φ) or the two densities of the species (ρA , ρB ). More importantly, the functional form of the free energy of mixing is much more complex, because both liquid-liquid demixing (as previously discussed for incompressible blends) and liquid-vapor phase separation are simultaneously possible. There is no single, coarse-grained parameter that characterizes the interaction between the molecules and dictates the phase behavior like the Flory-Huggins parameter in the incompressible limit. In fact, the interplay between liquid-liquid and liquid-vapor types of phase separation gives rise to six, qualitatively different types of phase behavior in the classification scheme of Konynenburg and Scott.78 Therefore, the phase behavior does depend on the detailed interactions of the computational model and a coarse-grained description has to be constructed with great care.71,79–83 Table 1. Critical points of the pure components from MC simulations and experiments. By comparing the critical temperatures from simulation and experiment, we identify CO2 = 4.201 · 10−21 J, HD = 5.787 · 10−21 J and CO2 /HD = 0.726. From the critical densities we derive σCO2 = 3.693 · 10−10 m, σHD = 4.523 · 10−10 m and σCO2 /σHD = 0.816. ρ refers to segment number densities. Tcrit ρcrit pcrit MC CO2 0.999 k 0.32 σ13 0.088 σ3 g EXP CO2 304 K 0.464 cm3 73.87 bar MC HD 1.725 k 0.27 σ13 0.022 σ3 g EXP HD 723 K 0.219 cm3 13.98 bar
03-chapter2
September 29, 2010
13:28
World Scientific Review Volume - 9in x 6in
Polymer Phase Separation
03-chapter2
71
To give a specific example, we shall discuss the mixture of hexadecane (HD), a short alkane, and carbon dioxide (CO2 ). Carbon dioxide has attracted much recent interest because it is environmentally benign (i.e., nonpoisonous) and small pressure changes give rise to pronounced changes of the physical properties in the supercritical region. The polymer is represented by a flexible, Lennard-Jones bead spring model. We utilize a chain length of N = 5 effective segments, which roughly mimics the geometrical shape of a hexadecane molecule. Comparing the two parameters of the Lennard-Jones potential, and σ, with experimental results for the critical temperature and pressure, we identify energy and length scales.71,81 Carbon dioxide is represented by a single Lennard-Jones interaction center; the interactions between CO2 -beads are adjusted to reproduce the experimental critical point. A modified Lorentz-Berthelot mixing rule is used to describe the interactions between hexadecane and CO2 beads:84,85 σHD + σCO2 √ σHD−CO2 = (19) and HD−CO2 = ξ HD CO2 2 where ξ describes the deviations from the standard mixing rule. The semi-grandcanonical simulation techniques can be straightforwardly transferred to compressible mixtures, where one controls the two chemical potentials of the individual species and the number of polymer molecules and carbon dioxide molecules fluctuates. In order to insert (or remove) a macromolecule into a dense liquid, the configurational bias algorithm86 is employed in order to generate trial configurations that do not strongly overlap with the other particles of the liquid. In the simulation, one monitors the joint probability distribution of the two order parameters, ρHD and ρCO2 , and adjusts via weighted histogram analysis the chemical potentials in order for the peaks in the ρHD -ρCO2 plane to have equal weight (cf. Eq. (10)). This procedure is illustrated in Fig. 5, where this two-dimensional probability distribution is depicted. The data are close to the critical point and the two peaks in the finite-sized system indicate the compositions of the two phase, which will coexist in the vicinity of the critical point. At low temperatures (top/left panel), the two phases are comprised of a dense carbon dioxide liquid and a liquid that contains both, polymer and CO2 , i.e., liquid-liquid phase separation occurs into a liquid solvent and a dense polymer-solvent mixture. Upon increasing the temperature, the character of the phase coexistence gradually changes and, at the highest temperature (bottom/right panel), the two coexisting phases correspond to a liquid and a vapor of polymer.
September 29, 2010
72
13:28
World Scientific Review Volume - 9in x 6in
M. M¨ uller
Fig. 5. Joint probability distribution of solvent and polymer density along the critical line in a finite-sized simulation cell for ξ = 0.886. The distributions correspond to temperatures T = 314K, 356K, 398K, 486K, 650K and 713K from up/left to down/bottom. For temperatures T = 314K and 356K the box size is L = 6.74σP while L = 9σP was employed for other temperatures. Reprinted with permission from Virnau et al.81 Copyright 2004 American Institute of Physics.
Fig. 6 shows projections of the phase diagram in the plane of temperature, T , and pressure, p, for two choices of ξ. The value ξ = 1 corresponds to the standard Berthelot rule, which describes the mixing of van-der-Waals interactions, while the value ξ = 0.886 matches the experimental behavior87,88 most closely. Solid lines represent the liquid-vapor phase equilibria of the two pure components that end in critical points marked by arrows. When a small amount of solvent is added to the pure polymer, the liquidvapor coexistence shifts and so does the critical point. The loci of critical points for the binary system form a critical line that is shown by the dashed line with squares for ξ = 1 and triangles for ξ = 0.886. In the former case — phase behavior of type I — the critical line connects the critical points of the two pure components and the two coexisting phases gradually change from vapor and solvent-rich liquid to vapor and polymer-rich liquid. This is not observed in experiments, where a phase diagram of type III is encountered. In a type-III diagram, the critical line that emerges from the liquid-vapor critical point of the polymer is not connected to the critical point of the pure solvent, but it gradually changes its character from a liquid-vapor co-
03-chapter2
September 29, 2010
13:28
World Scientific Review Volume - 9in x 6in
03-chapter2
Polymer Phase Separation
sim: lv coexist. [=1.000 [=0.886
300 pressure p [bar]
73
200
100 critical point CO2
0 200
300
critical point C16H34
400 500 600 temperature T [K]
700
800
Fig. 6. Projection of the global phase diagram for a compressible mixture of hexadecane, C16 H34 , and carbon dioxide, CO2 , into the temperature-pressure plane for two values of the mixing parameter, ξ = 1 (square) and ξ = 0.886 (triangle). Simulation results for the liquid-vapor coexistence of the pure components are shown by solid lines and end in critical points that are indicated by arrows. The line of critical points that emerges from the critical point of the less volatile polymer component is indicated by symbols. Reprinted with permission from Virnau et al.81 Copyright 2004 American Institute of Physics.
existence to a liquid-liquid coexistence between a dense solvent-liquid and a polymer-liquid. A minuscule change of the interaction between the different components from ξ = 1 to ξ = 0.886 qualitatively alters the type of the phase diagram. The reduction of the attractions between CO2 and hexadecane can be traced back to quadrupolar interactions between CO2 molecules.83,89 The well depth CO2 of the pure solvent parameterized the attraction between carbon dioxide molecules that is caused by the joined effect of both, van-der-Waals and quadrupole interactions. The Berthelot mixing rule implicitly stipulates that all attractions result from van-derWaals interactions and utilizes the strength CO2 , which only partially stems from dipolar interactions. If one used a coarse-grained representation of the carbon dioxide that takes due account of the quadrupolar interactions, one could predict the phase behavior without introducing a modification, ξ, of the mixing rule. This has been confirmed by recent simulations and greatly improves the predictive power of the coarse-grained modeling.83
September 29, 2010
13:28
World Scientific Review Volume - 9in x 6in
74
03-chapter2
M. M¨ uller
3. Outlook: Interfacial Properties, Phase Boundaries in Confined Geometry, and Wetting As illustrated in Fig. 4, the transition from the A-rich phase to the B-rich phase in the course of a semi-grandcanonical simulation of a symmetric, binary polymer blend occurs via configurations that are comprised of two domains — an A-rich and a B-rich domain — that are separated by two interfaces in a system with periodic boundary conditions. Thus, the simulation technique does not only accurately locate the bulk phase behavior but additionally yields information about the interfacial properties. The configurational snapshots in Fig. 4 suggest that the free energy barrier for tunneling between the two homogeneous phases is associated with the excess free energy of the two interfaces separating the A-rich and B-rich domains. This is further corroborated by the dependence of the free energy profile, F (φ) = −kB T ln P (φ), on the composition. Around the φA
+φB
composition of the rectilinear diameter, coex 2 coex = 0.5, the free energy profile is flat, i.e., one can change the composition and thereby the distance between the two AB-interfaces at no free energy costs. Therefore, the two interfaces do not interact, and the excess free energy is simply the sum of two, independent contributions.90 2 × L2 σAB is the interfacial tension and σAB
B P (φA coex )+P (φcoex ) kB T kB T ln L 2 = ln + O B φA 2L2 L2 coex +φcoex P 2
(20)
where L2 denotes the (smallest) cross-section of the simulation cell. In Fig. 4, we have utilized a cuboidal shape of the simulation cell of geometry L × L × 2L. This helps to increase the distance between the interfaces and thereby reduces their mutual interactions without increasing the free energy barrier between the two coexisting phases, which has to be compensated by the re-weighting technique. The results of such a free energy calculation are presented in Fig. 7. They illustrate the comparison with the prediction of Eq.(8), analytical predictions for corrections to the behavior in the strong segregation limit,40 and SCF calculations. The agreement between the mean-field prediction and the simulation data is quantitative without any adjustable parameter. Therefore, measuring the interfacial free energy in a phase-separated blend far below the critical point of demixing is a suitable method (see Sec. 2) for measuring the Flory-Huggins parameter, χ.
September 29, 2010
13:28
World Scientific Review Volume - 9in x 6in
03-chapter2
Polymer Phase Separation
75
1.0 0.8
VI/VSSL
0.6 0.4 0.2 0.0 0.0
N=16 N=32 N=64 N=32 (capillary waves) SCF 1-4ln2/FN 1-4ln2/FN+1.1842/(FN) 0.1
1/FN
3/2
0.2
0.3
√ ¯ χN/6. The symbols correFig. 7. Interface tension in units of σSSL R2e /kB T = N spond to MC simulations for three chain lengths as indicated in the key. The interfacial tension has been extracted from the probability distribution of the composition in the semi-grandcanonical ensemble, and the spectrum of capillary waves (for N = 32). Data are compared to SCF results and the exact asymptotic corrections in the limit χN → ∞ uller and de Pablo.65 by Semenov.40 From M¨
While excess properties of interfaces (like the interfacial free energy or the total interfacial excess of chain end segments, solvent particles, etc) can be straightforwardly obtained from such simulations, special care has to be exerted when comparing profiles across interfaces. While the meanfield theory makes rather detailed predictions about the spatial variation of properties (e.g., segment densities, orientations, etc) across the interface, the interface is assumed to be perfectly flat and long-ranged fluctuations of the local interface position, i.e., capillary waves,91–94 are ignored. If one averaged profiles across the total interfacial area, then these capillary waves would give rise to a system-size dependent broadening of the profiles. Therefore, the interfacial width, cf. Eq. (7), does not provide a reliable means of identifying the Flory-Huggins parameter. Fig. 8(a) presents a snapshot of the local interface position, u, in a binary polymer blends, where those capillary waves are clearly visible. Panel (b) depicts the variation of the local position for different incompatibilities. The larger the incompatibility between the two species is, the larger is the interfacial tension, σAB , and the stronger are capillary waves suppressed.
September 29, 2010
13:28
World Scientific Review Volume - 9in x 6in
76
03-chapter2
M. M¨ uller
0.15
0.10 P(u)
u(x,y)
5 60
0
ε = 0.02 ε = 0.024 ε = 0.03 ε = 0.04 ε = 0.05 ε = 0.07 ε = 0.1 ε = 0.146
0.05
-5 -10
40
x 20
20
0.00 −16
40
60
(a)
−8
0 u
y
8
16
(b)
Fig. 8. (a) Fluctuations of the local interface position in a binary polymer blend at = 0.03T (χN ≈ 5.1). (b) Distribution of the local interface positions as a function of the incompatibility /T . From Werner et al.95
In fact, this effect can be utilized to measure the interface tension (see Fig. 7) which yields consistent results. The semi-grandcanonical simulation strategy can also be employed to locate phase coexistence in confinement or to measure the free energy difference of the two coexisting phases with respect to a surface. To this end, periodic boundary conditions are only utilized in two directions and, in the third direction, surfaces are introduced, which preferentially attract the A-component of the mixture with an energy, wall . Using the semigrandcanonical reweighting technique in conjunction with re-weighting, the system tunnels between an A-rich phase in contact with the surface and a B-rich phase in contact with the surface. If the chemical exchange potential, ∆µ, is adjusted as to fulfill the equal weight criterion (cf. Eq. (10)), one accurately locates the phase transition in a slit-like pore. The results of such a procedure are depicted in Fig. 1 for symmetric surfaces, which both attract the A-species, and anti-symmetric surfaces, where the lower surface prefers A with the same strength as the upper surface prefers B. If the semi-grandcanonical simulation is performed at the bulk coexistence value of the chemical potential, the simulation measures the free energy difference between an A-rich bulk phase in contact with the surface and a B-rich phase in contact with the surface. Since the two phases coexist, the semi-grandcanonical free energies of the bulk regions are identical and the difference solely stems from the surface free energy difference, i.e., one computes L2 ∆γ. Knowing the interface tension, σAB , and the difference of
September 29, 2010
13:28
World Scientific Review Volume - 9in x 6in
03-chapter2
Polymer Phase Separation
0.05
1/2
)b(F/6) 'J=JWB-JWA JAB
0.03
1.0
WET
cos 4
J/kBT
0.04
0.02
77
0.5 H/kBT=0.05
0.01
NON-WET 0.00
0.02
0.04
0.0
H/kBT=0.025
0
0.06 H/kBT
0.05
Hwall/kBT 0.1
0.08
0.10
Fig. 9. Interface tension σAB and difference in surface tensions ∆γ = γW B − γW A as a function of inverse temperature /k B T obtained from simulations. Approximations for the interface tension σAB = bρo χ/6 in the strong segregation limit and ∆γ = 2Φdwall wall = wall /4 (dashed-dotted lines) are also shown. From Ref. 9. The inset shows the dependence of the contact angle on wall for the two temperatures, /kB T = 0.025 and 0.05. From M¨ uller and Binder.55
surface tensions, ∆γ, one can invoke Young’s equation96 σAB cos Θ = ∆γ
(21)
to accurately determine the contact angle, Θ, that an A-rich domain boundary makes with the surface. It turns out that utilizing Young’s equation is a very precise tool to determine the contact angle, which does not suffer from effects of finite domain size.7,9 The simulation techniques discussed provide a wealth of information about the phase behavior in the bulk and confinement and, simultaneously, yield excess properties of interfaces. In this contribution, we mainly focused on a simple system — a symmetric polymer blend — but the strategy carries over to other phase-separating systems, which simultaneously exhibit liquidliquid and liquid-vapor phase separation or liquid crystalline order. The two major advantages are that (i) the semi-grandcanonical scheme efficiently relaxes the order parameter of the transition, which would take a protracted time to relax in a polymer system if it were locally conserved and (ii) that a criterion of phase coexistence exists — the equal weight rule or sophisticated
September 29, 2010
13:28
78
World Scientific Review Volume - 9in x 6in
M. M¨ uller
finite-size scaling techniques in the vicinity of the critical point — which allows for an accurate computation of the properties of the coexisting phases from small systems. Given the importance of phase separation for practical applications, the wealth of new materials created by blending and altering the macromolecular architecture, and the necessity to predict it or extrapolate it into parameter regions, which are experimentally difficult or dangerous to access, computer simulations of atomistic and coarse-grained models will continue making contributions to this field of material science. References 1. R. Cahn, P. Haasen, and E. Kramer, Materials Science and Technology: A Comprehensive Treatment. 12, VCH, Weinheim, (1993). 2. F. Garbassi, M. Morra, and E. Occhiello, Polymer Surface: From Physics to Technology. pp. Wiley, Chichester, (2000). 3. D. R. Paul and J. W. Barlow, Polymer blends (or alloys), Journal of Macromolecular Science-Reviews In Macromolecular Chemistry. C18, 109–168, (1980). 4. G. Taylor, The viscosity of a fluid containing small drops of another fluid, Proc. R. Soc. London A. 138, 41–48, (1932). 5. C. Creton, E. J. Kramer, and G. Hadziioannou, Critical molecular-weight for block copolymer reinforcement of interfaces in a 2-phase polymer blend, Macromolecules. 24, 1846–1853, (1991). 6. 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, (2001). 7. M. M¨ uller and K. Binder, The interplay between wetting and phase behaviour in binary polymer films and wedges: Monte Carlo simulations and mean field calculations, J. Phys.: Condens. Matter. 17, S333–S361, (2005). 8. M. M¨ uller, K. Binder, and E. V. Albano, Non-monotonous crossover between capillary condensation and interface localisation/delocalisation transition in binary polymer blends, Europhys. Lett. 50, 724–730, (2000). 9. M. M¨ uller and K. Binder, Wetting and capillary condensation in symmetric polymer blends: a comparison between Monte Carlo simulations and selfconsistent field calculations, Macromolecules. 31, 8323–8346, (1998). 10. 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). 11. F. M¨ uller-Plathe, Coarse-graining in polymer simulation: From the atomistic to the mesoscopic scale and back, Chem. Phys. Chem. 3, 754–769, (2002).
03-chapter2
September 29, 2010
13:28
World Scientific Review Volume - 9in x 6in
Polymer Phase Separation
03-chapter2
79
12. D. Heine, D. T. Wu, J. G. Curro, and G. S. Grest, Role of intramolecular energy on polyolefin miscibility: Isotactic polypropylene/polyethylene blends, J. Chem. Phys. 118, 914–924, (2003). 13. P. G. Bolhuis, A. A. Louis, and J. P. Hansen, Many-body interactions and correlations in coarse-grained descriptions of polymer solutions, Phys. Rev. E. 6402, 021801, (2001). 14. A. A. Louis, Beware of density dependent pair potentials, J. Phys.: Condens. Matter. 14, 9187–9206, (2002). 15. K. Kremer and K. Binder, Monte Carlo simulations of lattice models for macromolecules, Computer Physics Reports. 7, 259–310, (1988). 16. M. M¨ uller and F. Schmid, Incorporating fluctuations and dynamics in selfconsistent field theories for polymer blends, Adv. Polym. Sci. 185, 1–58, (2005). 17. M. M¨ uller, Comparison of self-consistent field theory and Monte Carlo simulations, Soft Condensed Matter, G. Gompper and M. Schick (eds), Wiley VCH 1, 179–281, (2006). 18. E. Helfand, Theory of inhomogeneous polymers: Fundamentals of the Gaussian random–walk model, J. Chem. Phys. 62, 999–1005, (1975). 19. L. Sch¨ afer, Excluded volume effects in polymer solutions, Springer, Berlin. (1999). 20. P. G. de Gennes, Exponents for excluded volume problem as derived by Wilson method, Physics Letters A. A, 339, (1972). 21. P. de Gennes, Scaling Concepts in Polymer Physics, Cornell University Press, Ithaca and London. (1979). 22. P. J. Flory, Thermodynamics of high polymer solutions, J. Chem. Phys. 9, 660, (1941). 23. M. L. Huggins, Solutions of long chain compounds, J. Chem. Phys. 9, 440, (1941). 24. I. Carmesin and K. Kremer, The bond fluctuation method: a new effective algorithm for the dynamics of polymers in all spatial dimensions., Macromolecules. 21, 819–2823, (1988). 25. M. M¨ uller, Miscibility behavior and single chain properties in polymer blends: a bond fluctuation model study, Macromol. Theory Simul. 8, 343–374, (1999). 26. K. Kremer and G. S. Grest, Dynamics of entangled linear polymer melts: a molecular-dynamics simulation, J. Chem. Phys. 92, 5057–5086, (1990). 27. G. S. Grest, M. D. Lacasse, K. Kremer, and A. M. Gupta, Efficient continuum model for simulating polymer blends and copolymers, J. Chem. Phys. 105, 10583–10594, (1996). 28. R. D. Groot and P. B. Warren, Dissipative particle dynamics: Bridging the gap between atomisticand mesoscopic simulation, J. Chem. Phys. 107, 4423– 4435, (1997). 29. M. M¨ uller and G. D. Smith, Phase separation in binary mixtures containing polymers: a quantitative comparison of single-chain-in-mean-field simulations and computer simulations of the corresponding multichain systems, J. Polym. Sci. B: Polymer Physics. 43, 934–958, (2005).
September 29, 2010
80
13:28
World Scientific Review Volume - 9in x 6in
M. M¨ uller
30. K. C. Daoulas and M. M¨ uller, Single chain in mean field simulations: Quasiinstantaneous field approximation and quantitative comparison with Monte Carlo simulations, J. Chem. Phys. 125, 184904, (2006). 31. K. C. Daoulas, M. M¨ uller, J. J. de Pablo, P. F. Nealey, and G. D. Smith, Morphology of multi-component polymer systems: Single chain in mean field simulation studies, Soft Matter. 2, 573–583, (2006). 32. P. C. Hohenberg and B. I. Halperin, Theory of dynamic critical phenomena, Rev. Mod. Phys. 49, 435–479, (1977). 33. M. Doi and S. F. Edwards, The theory of polymer dynamics. Oxford University Press, (1986). 34. J. T. Padding and W. J. Briels, Time and length scales of polymer melts studied by coarse-grained molecular dynamics simulations, J. Chem. Phys. 117, 925–943, (2002). 35. A. E. Likhtman, Single-chain slip-link model of entangled polymers: Simultaneous description of neutron spin-echo, rheology, and diffusion, Macromolecules. 38, 6128–6139, (2005). 36. E. Helfand and Y. Tagami, Theory of interface between immiscible polymers, J. Chem. Phys. 57, 1812, (1972). 37. J. M. H. M. Scheutjens and G. J. Fleer, Statistical-theory of the adsorption of interacting chain molecules. 1. partition-function, segment density distribution, and adsorption-isotherms, J. Phys. Chem. 83, 1619–1635, (1979). 38. J. Noolandi and K. M. Hong, Interfacial properties of immiscible homopolymer blends in the presence of block copolymers, Macromolecules. 15, 482–492, (1982). 39. M. W. Matsen and M. Schick, Stable and unstable phases of a diblock copolymer melt, Phys. Rev. Lett. 72, 2660–2663, (1994). 40. A. N. Semenov, Theory of long-range interactions in polymer systems, J. Phys. (France) II. 6, 1759–1780, (1996). 41. J. Dudowicz, K. F. Freed, and J. F. Douglas, New patterns of polymer blend miscibility associated with monomer shape and size asymmetry, J. Chem. Phys. 116, 9983–9996, (2002). 42. M. M¨ uller and K. Binder, Computer-simulation of asymmetric polymer mixtures, Macromolecules. 28, 1825–1834, (1995). 43. M. M¨ uller, Effects of structural disparities in polymer blends – a Monte Carlo investigation, Macromolecules. 28, 6556–6564, (1995). 44. A. Sariban and K. Binder, Phase-separation of polymer mixtures in the presence of solvent, Macromolecules. 21, 711–726, (1988). 45. H. P. Deutsch and K. Binder, Critical-behavior and crossover scaling in symmetrical polymer mixtures – a Monte Carlo investigation, Macromolecules. 25, 6214–6230, (1992). 46. B. Crist, Thermodynamic interactions in isotope blends: Experiment and theory, Macromolecules. p. 5853, (1998). 47. F. S. Bates and G. H. Fredrickson, Block copolymer thermodynamics – theory and experiment, Ann. Rev. Phys. Chem. 41, 525–557, (1990).
03-chapter2
September 29, 2010
13:28
World Scientific Review Volume - 9in x 6in
Polymer Phase Separation
03-chapter2
81
48. F. S. Bates, M. Schultz, J. Rosedale, and K. Almdal, Correlation of binary polyolefin phase behavior with statistical segment length asymmetry, Macromolecules. 25, 5547, (1992). 49. M. D. Gehlsen and F. S. Bates, Conformational asymmetry in poly(vinylcyc1ohexane) containing diblock copolymers, Macromolecules. 27, 3611, (1994). 50. D. Schwahn, G. Meier, K. Mortensen, and S. Janssen, On the n-scaling of the Ginzburg number and the critical amplitudes in various compatible polymer blends, J. Phys. II (France). 4, 837, (1994). 51. A. Z. Panagiotopoulos, Direct determination of phase coexistence properties of fluids by Monte Carlo simulation in a new ensemble, Mol. Phys. 61, 813–826, (1987). 52. A. Sariban and K. Binder, Critical properties of the Flory-Huggins lattice model of polymer mixtures, J. Chem. Phys. 86, 5859–5873, (1987). 53. M. M¨ uller and K. Binder, An algorithm for the semi-grand-canonical simulation of asymmetric polymer mixtures, Comp. Phys. Comm. 84, 173–185, (1994). 54. M. M¨ uller and A. Werner, Interfaces between highly incompatible polymers of different stiffness: Monte Carlo simulations and self-consistent field calculations, J. Chem. Phys. 107, 10764–10776, (1997). 55. M. M¨ uller and K. Binder, The interplay between wetting and phase behaviour in binary polymer films and wedges: Monte Carlo simulations and mean field calculations, J. Phys. Condens. Matt. 17, S333–S361 (2005). 56. C. Borgs and R. Kotecky, J. Stat. Phys. 60, 79, (1990). 57. C. Borgs and S. Kappler, Phys. Lett. A. 171, 37, (1992). 58. A. M. Ferrenberg and R. H. Swendsen, New Monte Carlo technique for studying phase transitions, Phys. Rev. Lett. 61, 2635–2638, (1988). 59. A. M. Ferrenberg and R. H. Swendsen, Optimized Monte Carlo data analysis, Phys. Rev. Lett. 63, 1195–1198, (1989). 60. P. Dayal, S. Trebst, S. Wessel, D. Wurtz, M. Troyer, S. Sabhapandit, and S. N. Coppersmith, Performance limitations of flat-histogram methods, Phys. Rev. Lett. 92, 097201, (2004). 61. B. A. Berg, Multicanonical recursions, J. Stat. Phys. 82, 323, (1996). 62. 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). 63. G. R. Smith and A. D. Bruce, A study of the multi-canonical Monte Carlo method, J. Phys. A-Math. Gen. 28, 6623–6643, (1995). 64. P. Virnau and M. M¨ uller, Calculation of free energy through successive umbrella sampling, J. Chem. Phys. 120, 10925–10930, (2004). 65. M. M¨ uller and J. J. de Pablo, Simulation techniques for calculating free energies, Lec. Notes Phys. 703, 67–122, (2006). 66. J. P. Valleau, Thermodynamic-scaling methods in Monte Carlo and their application to phase equilibria, Adv. Chem. Phys. 105, 369–404, (1999). 67. B. J. Schulz, K. Binder, M. M¨ uller, and D. P. Landau, Avoiding boundary effects in Wang-Landau sampling, Phys. Rev. E. 67, 067102, (2003).
September 29, 2010
82
13:28
World Scientific Review Volume - 9in x 6in
M. M¨ uller
68. N. B. Wilding and M. M¨ uller, Accurate measurements of the chemicalpotential of polymeric systems by Monte Carlo simulation, J. Chem. Phys. 101, 4324–4330, (1994). 69. D. Chandler, Introduction to Modern Statistical Mechanics. (Oxford University Press, New York, 1987). 70. D. Frenkel and B. Smit, Understanding Molecular Simulation. (Academic Press, London, 2002). 71. K. Binder, M. M¨ uller, P. Virnau, and L. G. MacDowell, Polymer plus solvent systems: Phase diagrams, interface free energies, and nucleation, Adv. Polym. Sci. 173, 1–104, (2005). 72. R. L. C. Vink and J. Horbach, Grand canonical Monte Carlo simulation of a model colloid-polymer mixture: Coexistence line, critical behavior, and interfacial tension, J. Chem. Phys. 121, 3253–3258, (2004). 73. R. L. C. Vink, J. Horbach, and K. Binder, Capillary waves in a colloidpolymer interface, J. Chem. Phys. 122, 134905, (2005). 74. R. L. C. Vink, J. Horbach, and K. Binder, Critical phenomena in colloidpolymer mixtures: Interfacial tension, order parameter, susceptibility, and coexistence diameter, Phys. Rev. E. 71, 011401, (2005). 75. R. L. C. Vink and M. Schmidt, Simulation and theory of fluid demixing and interfacial tension of mixtures of colloids and nonideal polymers, Phys. Rev. E. 71, 051406, (2005). 76. R. L. C. Vink and T. Schilling, Interfacial tension of the isotropic-nematic interface in suspensions of soft spherocylinders, Phys. Rev. E. 71, 051716, (2005). 77. R. L. C. Vink, S. Wolfsheimer, and T. Schilling, Isotropic-nematic interfacial tension of hard and soft rods: Application of advanced grand canonical biased-sampling techniques, J. Chem. Phys. 123, 074901, (2005). 78. P. H. V. Konynenburg and R. L. Scott, Critical lines and phase equilibria in binary van der waals mixtures, Phil. Trans. R. Soc. A. 298, 496–540, (1980). 79. P. Virnau, M. M¨ uller, L. G. MacDowell, and K. Binder, Phase diagrams of hexadecane – CO2 mixtures from histogram-reweighting Monte Carlo, Comp. Phys. Comm. 147, 378–381, (2002). 80. M. M¨ uller, L. G. MacDowell, P. Virnau, and K. Binder, Interface properties and bubble nucleation in compressible mixtures containing polymers, J. Chem. Phys. 117, 5480–5496, (2002). 81. P. Virnau, M. M¨ uller, L. G. MacDowell, and K. Binder, Phase behavior of n-alkanes in supercritical solution: a Monte Carlo study, J. Chem. Phys. 121, 2169–2179, (2004). 82. L. Yelash, M. M¨ uller, W. Paul, and K. Binder, Artificial multiple criticality and phase equilibria: An investigation of the PC-SAFT approach, Phys. Chem. Chem. Phys. 7, 3728–3732, (2005). 83. B. M. Mognetti, L. Yelash, P. Virnau, W. Paul, K. Binder, M. Muller, and L. G. MacDowell, Efficient prediction of thermodynamic properties of quadrupolar fluids from simulation of a coarse-grained model: The case of carbon dioxide, J. Chem. Phys. 128, 104501, (2008). 84. H. A. Lorentz, Annln Phys. 12, 127, (1881).
03-chapter2
September 29, 2010
13:28
World Scientific Review Volume - 9in x 6in
Polymer Phase Separation
03-chapter2
83
85. D. C. Berthelot, r. hebd. Seanc. Acad Sci., Paris. 126, 1703, (1898). 86. J. I. Siepmann and D. Frenkel, Configurational-Bias Monte Carlo: a new sampling scheme for flexible chains, Mol. Phys. 75, 59–70, (1992). 87. G. Schneider, Z. Alwani, W. Heim, E. Horvath, and E. U. Franck, Phasengleichgewichte und kritische Erscheinungen in bin¨ aren Mischsystemen bis 1500 bar, Chem.-Ing.-Techn. 39, 649, (1967). 88. T. Charoensombut-Amon, R. J. Martin, and R. Kobayashi, Application of a generalized multiproperty apparatus to measure phase equilibrium and vapor phase densities of supercritical carbon dioxide in n-hexadecane systems up to 26 MPa, Fluid Phase Equilibria. 86, 89–104, (1986). 89. C. Mendui˜ na, C. McBride, and C. Vega, The second virial coefficient of quadrupolar two center Lennard–Jones models, Phys. Chem. Chem. Phys. 3, 1289–1296, (2001). 90. K. Binder, Monte Carlo calculation of the surface tension for two- and threedimensional lattice-gas models, Phys. Rev. A. 25, 1699–1709, (1982). 91. F. P. Buff, R. A. Lovett, and F. H. Stillinger, Interfacial density profile for fluids in critical region, Phys. Rev. Lett. 15, 621, (1965). 92. W. Helfrich, Elastic properties of lipid bilayers – theory and possible experiments, Zeitschrift Naturforschung C. 28, 693–703, (1973). 93. J. D. Weeks, Structure and thermodynamics of liquid-vapor interface, J. Chem. Phys. 67, 3106, (1977). 94. D. Bedeaux and J. D. Weeks, Correlation-functions in the capillary wave model of the liquid vapor interface, J. Chem. Phys. 82, 972, (1985). 95. 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). 96. T. Young, Philos. Trans. R. Soc. London. 5, 65, (1805).
This page is intentionally left blank
September 29, 2010
10:40
World Scientific Review Volume - 9in x 6in
Chapter 3 Self-Consistent Field Theory of Block Copolymers
Feng Qiu,∗ An-Chang Shi† and Yuliang Yang‡ Department of Macromolecular Science, Fudan University, Shanghai, China Department of Physics and Astronomy, McMaster University, Hamilton, Ontario L8S 4M1, Canada Self-consistent field theory (SCFT) is a powerful theoretical framework for the study of the phase behavior of polymeric systems, in particular, block copolymers. In this chapter the framework of SCFT is introduced, focusing on the reciprocal-space method to solve the SCFT equations for ordered phases of block copolymers. The approach utilizes Fourier representation of spatially varying functions. The SCFT equations are cast in terms of the Fourier coefficients. An advantage of the reciprocal-space approach is that the symmetry of the ordered phases can be easily and definitively deduced from the nonzero Fourier coefficients. The method has been used to successfully reproduced all the ordered phases observed in diblock copolymers and ABC linear triblock copolymers and predicted some previously unknown structures.
1. Introduction Block copolymers are macromolecules composed of two or more chemically distinct sub-chains (blocks) linked together by covalent bonds. Common examples of block copolymers are linear AB diblock copolymers, linear ABC triblock copolymers, and miktoarm (or star) ABC triblock copolymers. One intrigue property of block copolymers is that the competition of repulsion and connectivity between different blocks leads to the self-assemly of a rich array of ordered phases. The phase behavior of block copolymers has attracted tremendous attention in the past decades. In the content of ∗ Fudan
University University ‡ Fudan University † McMaster
85
04-chapter3
September 29, 2010
86
10:40
World Scientific Review Volume - 9in x 6in
F. Qiu, A.-C. Shi and Y. Yang
soft condensed matter physics, block copolymers have served as an ideal paradigm for the study of the self-assembly in soft materials. Due to a large number of theoretical and experimental studies over the last decades, remarkable progress on understanding the phase behavior of block copolymers has been achieved. It is well established that the self-assembly of block copolymers is governed by a delicate balance of the interaction energy and chain conformational entropy. The phase behavior of block copolymer melts is determined by several factors: the volume fraction and sequence of the chemically distinct blocks, the interaction strength between different components (usually quantified by the Flory-Huggins interaction parameters), the degree of polymerization of the macromolecules, an the topology of the block copolymers. For a given set of controlling parameters, block copolymers self-assemble to form a particular ordered phase so that the free energy of the system is minimized. Determination of the possible phases and transition sequences between these ordered phases has been the major focus of theoretical and experimental studies of block copolymers.1,2 For the simplest case of AB diblock copolymers, the equilibrium phases are largely governed by three parameters: the volume fraction of the A block, fA , the AB interaction parameter χ, and the degree of polymerization, N . Although the AB diblock copolymer is the simplest block copolymer, it exhibits a surprisingly large number of ordered phases: alternating lamellae (L), hexagonally packed cylinders (C), spheres packed in a bodycentered cubic or face-centered cubic lattice (S), double-gyroid network (G) and orthorhombic F ddd network (O 70 , which is a noncubic phase).3 It is important to point out that the determination of the complex networked phases is not a trivial task. The correct identifications of the two networked phases were only accomplished in recent years. For example, the first bicontinuous structure, the gyroid, was initially identified as a double diamond structure with the space group P3mm when it was first observed in experiment.4,5 The identification of this phase as the gyroid phase was made recently when high-resolution small angle X-ray scattering data become available.6 More complex block architectures, such as ABC triblock copolymers, offer opportunities to create a greater complexity of microphase morphologies. The richness of ordered structures in triblock copolymers stems from a greatly enlarged parameter space. At the mean-field level, there are at least five independent molecular parameters determining the phase behavior of ABC triblock copolymers: two independent compositions, i.e., volume fractions of A and B blocks (fA and fB ), and three integrated
04-chapter3
September 29, 2010
10:40
World Scientific Review Volume - 9in x 6in
Self-Consistent Field Theory of Block Copolymers
04-chapter3
87
interaction parameters (χAB N , χBC N and χAC N ). Furthermore, chain topology, i.e., linear triblock copolymer versus non-linear triblock copolymers (star-shaped, comb- and branched-type), and block sequence in linear triblock copolymers, can also play important roles in determining the equilibrium structures. Consequently, much more complex ordered morphologies can be obtained in ABC triblock copolymers. At the present time, more than three-dozen ordered structures have been identified for ABC triblock copolymers.3 A centre theme emerged from previous studies on block copolymer phase behavior is that the availability of block copolymers with different architectures will lead to complex phase behavior and diverse ordered phases. For a given block copolymer architecture and block type, the major challenge is to discover the possible ordered phases from this particular block copolymers. This challenge can only be met by a combination of experimental and theoretical approaches. Extensive experimental efforts provide valuable contributions to the understanding of the complex morphologies of different block copolymers. On the other hand, because varying copolymer compositions and monomer-monomer interactions involve the synthesis of a large number of block copolymers, experimental exploration of the phase space is a very costly and time consuming task. From this perspective, theory and simulation provide an ideal tool for the investigation of the phase behavior of complex block copolymers. In particular, theoretical studies can be used to obtain a good understanding of the detailed relationship between the mesoscopic structures and molecular parameters (block type, composition, architecture, and interactions). Over the last decades the self-consistent field theory (SCFT) of polymers has emerged as a powerful theoretical frameworks for the study of the statistical thermodynamics of block copolymers.7–9 The SCFT of polymers can be traced back to the work of Edwards in 1960s.10 Helfand adapted this theoretical framework to treat block copolymers in 1975.11 Because of the complexity of finding exact solutions of the SCFT equations, several approximate methods were developed. These approximation methods can be roughly divided into two classes, the weak segregation theory and the strong segregation theory. The weak-segregation theory (WST) proposed by Leibler in 198012 assumes that under weak-segregation limit condition, i.e., near order-disorder transition, the composition profile of an ordered structure is a small sinusoidal fluctuation in the disordered or homogeneous phase. Therefore the free energy of the system can be expressed as a Landau expansion based on the random phase approximation (RPA)
September 29, 2010
88
10:40
World Scientific Review Volume - 9in x 6in
F. Qiu, A.-C. Shi and Y. Yang
around the homogeneous phase. The weak segregation theory provides a good explanation for phase behaviors of block copolymers near ODT. On the other hand, the strong segregation theory (SST) proposed by Semenov in 198513 is based on the assumption that, at the strong segregation limit, the phase separation is almost complete. Therefore the free energy of an ordered structure can be separated into interfacial free energy and stretching free energy. Analytical expressions of these two contributions can be estimated for a given structure. The SST has been successful in constructing phase diagram of diblock copolymers, initially including equilibrium phases such as lamellae, sphere and cylinder. Complex phases, such as gyroid, perforated lamellae, double diamond, have been considered with the SST.14,15 The WST and SST approximations provide valuable insights into the self-assembly of block copolymers. On the other hand, most experimental systems correspond to intermediate degree of segregation, where the WST and SST become inaccurate. In this case numerical techniques become indispensible as they provide exact solutions to the SCFT equations. The initial attempt to obtain numerical solutions of SCFT equations was made by Helfand and coworkers.16 Subsequently Shull17 and Whitmore and coworkers18 constructed phase diagrams of block copolymer melts and solutions using approximate numerical techniques. The first exact threedimensional numerical solutions of the SCFT equations were obtained by Matsen and Schick using a spectral-method.19 The availability of the exact numerical solutions allowed precise calculations of free energies and phase diagrams for block copolymers. The main idea of the Matsen-Schick approach is to expand all spatially varying functions in terms of a finite set of basis functions determined from the symmetry of the ordered structure under consideration. With this powerful tool, an accurate phase diagram of diblock copolymer was successfully constructed.20 Subsequently, other systems of block copolymers, such as symmetric triblock copolymer melts21 and block copolymer blends,22 have also been investigated. Recently, Morse and coworkers have constructed phase diagrams of nonfrustrated linear ABC triblock copolymers by solving the SCFT equations using the spectral-method.23,24 Initially the reiprocal-space method utilizes basis functions which are determined from some assumed symmetries of the ordered phases. This restriction has been removed recently by Guo et al with a generic Fourier-space method for solving the SCFT equations.25 In this generic spectral-method, all spatially varying functions are expanded in terms of Fourier series. The Fourier series themselves are determined by discretizing the reciprocal space. This method can be viewed as a general-
04-chapter3
September 29, 2010
10:40
World Scientific Review Volume - 9in x 6in
Self-Consistent Field Theory of Block Copolymers
04-chapter3
89
ization of the Matsen-Schick method. The new method has been validated by applying it to AB diblock copolymers. Furthermore, the capability of the method to predict new phases has been demonstrated via the application to linear and star ABC triblock copolymer melts. Parallel to the spectral-method, tremendous progresses have been made for the development of real-space methods to solve the SCFT equations. For the purpose of exploring new phases, Drolet and Fredrickson26 proposed a combinatorial screening method, which is based on a direct implementation of SCFT equations in real space with an adaptive arbitrary cell. This approach has been applied to a number of block copolymer systems, leading to the construction of two-dimensional phase maps of linear and star ABC triblock copolymers.27 In a similar approach, Bohbot-Raviv and Wang28 proposed an efficient strategy to minimize the free energy function in an arbitrary unit cell with respect to the composition profile and the dimensions of the unit cell. These real-space strategies to solve the SCFT equations are greatly aided by the development of new numerical techniques, such as the split-step method proposed by Rasmussen et al,29 for solving the modified diffusion equation in SCFT. The split-step method is numerically stable and efficient. It has become the method of choice for real-space solutions of SCFT equations used in combinatorial screening of novel stable and metastable phases of block copolymers. The computational advantage of he real-space method has been demonstrated in a recent study by Cochran em et al,30 in which accurate phase diagram of AB diblock copolymers has been constructed up to χN = 100. A comparison of the efficiency of the real-space and reciprocal-space method is given by Matsen recently.31 The combination of experimental, theoretical and simulation studies has led to a good understanding of the phases and phase transitions of diblock copolymers and non-frustrated linear triblock copolymers. Despite all these progresses, predicting the possible ordered phases from a given complex block copolymer still presents a challenging task. From a theoretical point of view, the challenge is to find the possible solutions of the SCFT equations for a given block copolymer system. Progresses in real-space and reciprocal space methods to solve the SCFT equations have made it possible to predict the ordered phases of block copolymers, as demonstrated in many examples from AB diblock and ABC triblock copolymers. An effective strategy to construct phase diagrams of block copolymers is to divide the task into two steps. The first step is to obtain possible candidate ordered phases by combining the discovery power of the real-space and generic reciprocalspace methods. The symmetry of the candidate phases can then be used
September 29, 2010
90
10:40
World Scientific Review Volume - 9in x 6in
F. Qiu, A.-C. Shi and Y. Yang
as input to the more accurate Matsen-Schick method or the split-space real-space method to constructed the phase diagram. Because reviews of the real-space method9 and the Matsen-Schick method7,8 are available in the literature, we will focus on the generic reciprocal-space method in this chapter. 2. Self-Consistent Field Theory of Block Copolymers In this section, a brief description of self-consistent field theory for block copolymers is presented. It should be pointed out that several excellent reviews of self-consistent field theory have been published over the years.7–9,32,33 The readers are refereed to these sources for a systematic description of the self-consistent field theory framework. For simplicity, we will use AB diblock copolymer melts as an example first, followed by extensions to ABC star triblock copolymer melts. Generalization of the theory to other block copolymers or block copolymer blends is straightforward. 2.1. Polymer model and partition function The model system of an AB diblock copolymer melt consists of n block copolymer chains contained in a finite volume V . The copolymer chains are linear macromolecules formed from NA A-monomers connected with NB B-monomers (N = NA + NB is the degree of polymerization of the block copolymer). For simplicity, the A- and B-monomers are assumed to have the same volume v0 = 1/ρ0 , where the monomer density ρ0 , is defined as the number of monomers per unit volume. The conformation of the ith polymer chain is described by a space curve, Rα i (s), which specifies the spatial position of the s-th monomer at α-block (α = A or B for AB diblock copolymers). The monomer species can be described by a function γα (s), which is 1 when the s-th monomer belongs to the α-block, and 0 otherwise, i.e., γα (s) = δα,A for 0 < s < NA and γα (s) = δα,B for NA < s < N . It is P obvious that this function satisfies α γα (s) = 1 for all s. For a given chain configuration ({R(·)}), the α-monomer density at a spatial position r can be expressed as, n Z 1 X N ˆ φα (r) = dsδ(r − Ri (s))γα (s), (1) ρ0 i=1 0 where the hat on φα indicates that φα (r) is a functional of the chain conformation Ri (s). The probability distribution of a given chain conformation
04-chapter3
September 29, 2010
10:40
World Scientific Review Volume - 9in x 6in
04-chapter3
Self-Consistent Field Theory of Block Copolymers
91
is denoted by p0 (R(·)). For the Gaussian chains considered here, the probability distribution has the standard form " 2 # Z N dR(s) 3 ds , (2) p0 (R(·)) = A exp − 2 2b 0 ds R where A is a normalization constant such that D(R(·))p0 (R(·)) = 1. For AB diblock copolymers, the probability of a chain configuration {R(·)} (= {R1 (·), R2 (·), · · · , Rn (·)}) can be written in the form, n Y B P0 ({R(·)}) = p0 (Ri (s)) δ RA , (3) i (fA ) − Ri (fA i=1
B where the delta function δ RA i (fA ) − Ri (fA ensures that the A-block is linked to the B-block at segment position NA = fA N such that the polymers are diblock copolymers. The partition function of the AB diblock copolymer melts can be given in term of the functional integral over all the chain conformations, Z h i wn ˆ , (4) ZC = c D {R(·)} P0 ({R(·)}) δ (1 − φA − φB ) exp −βV (φ}) n! where wc is the partition function of a single diblock copolymer chain ˆ due to kinetic energy, and β = 1/kB T is the Boltzmann factor. V ({φ}) is the monomer-monomer interaction functional, which is often given by the Flory-Huggins expression, Z ˆ = χρ0 drφA (r)φB (r), V ({φ}) (5)
where χ is the Flory-Huggins parameter. The delta-function in the partition function is defined by, Y δ(1 − φA − φB ) = {δ (1 − φA (r) − φB (r))} , (6) r
which is introduced to ensure that the incompressibility condition at every spatial position r is satisfied. The partition function ZC in its current form presents a formidable many-body problem in statistical mechanics. In order to proceed, it is useful to insert an identity in terms of the delta-functions into the expression of the partition function. The delta-functions are in turn converted to functional integrals in terms of the auxiliary fields, Z 1 = Dφα δ φα − φˆα =
Z
Dφα
Z
+i∞
−i∞
Dωα exp
Z
drωα (r)δ φα (r) − φˆα (r)
.
(7)
September 29, 2010
10:40
World Scientific Review Volume - 9in x 6in
04-chapter3
F. Qiu, A.-C. Shi and Y. Yang
92
This identity allows one to replace the density operator φˆα (r) in the interˆ by the density field φα (r). The range of integral of action potential V ({φ}) the auxiliary field ωα (r) is along the imaginary axis in the complex plane from. Inserting the above identity into the partition function and after some simple manipulation, the partition function can be rewritten in terms of a functional integral Z Y ZC = [Dφα Dωα ] δ (1 − φA − φB ) exp [−βFC ({φ}, {ω})] , (8) α
where the “Hamiltonian” functional FC is given by ( ρ0 Rg3 −V ln QC ({ω}) FC ({φ}, {ω}) = N +
Z
"
dr χN φA (r)φB (r) −
X α
ωα (r)φα (r)
#)
.
(9)
Here the functional QC ({ω}) is the partition function of a single diblock copolymer chain in the external field ωα (r), " # Z XZ N 1 QC ({ω}) = D {R(·)} P0 ({R(·)}) exp − dsωα (R(s)γα (s) . V 0 α
(10)
2.2. Chain propagators The above expression of the single-chain partition function QC ({ω}) is in the form of a path integral. Computationally it is convenient to cast the single-chain partition function in terms of the chain propagators G(r, t|r0 , t0 ), which is the conditional probability distribution of monomer t (t0 ) at spatial position r (r0 ), given that the monomer t0 (t) is located at r0 (r) under an external field ωα (r), Z 1 dr1 dr2 dr3 G(r3 , 1|r2 , s)G(r2 , s|r1 , 0). (11) QC ({ω}) = V It can be proven that the chain propagator satisfies the following differential equations, ) ( X 2 2 ∂ 0 0 G(r, t|r , t ) = γα (t) σα ∇ − ωα (r) G(r, t|r0 , t0 ), (12) ∂t α
September 29, 2010
10:40
World Scientific Review Volume - 9in x 6in
Self-Consistent Field Theory of Block Copolymers
04-chapter3
93
with the initial conditions, G(r, t|r0 , t) = δ(r − r0 ). Here σα2 = N b2α /6 and bα is the statistical segment length of the α-block. For convenience, it is useful to introduce the end-integrated chain propagators. There are two types of end-integrated propagators, the forward end-integrated propagator q(r, s) and the backward end-integrated propagator q † (r, s), because the two ends of a block copolymer chain are not equivalent. These end-integrated propagators are defined by, Z q(r, s) = dr0 G(r, s|r0 , 0), (13) †
q (r, s) =
Z
dr0 G(r0 , 1|r, s).
(14)
Obviously, these two end-integrated propagators also satisfy the same differential equations, ( ) X 2 2 ∂ q(r, s) = γα (t) σα ∇ − ωα (r) q(r, s), (15) ∂s α ( ) X 2 2 † ∂ † q (r, s) = − γα (t) σα ∇ − ωα (r) q (r, s), ∂s α
(16)
but with different initial conditions q(r, 0) = 1 and q † (r, 1) = 1. In terms of these two new definitions, the single-chain partition function can be rewritten as Z 1 QC ({ω}) = drq(r, s)q † (r, s). (17) V 2.3. Self-consistent mean-field theory The thermodynamics of the diblock copolymer melts is completely determined by the partition function ZC . However, exact evaluation of the partition function is in general a formidable task. Numerous methods have been proposed to evaluate this partition function, including approximate analytical and numerical methods as well as numerical simulations of the field theory. Among the many approximation strategies, the mean-field approximation has provided extremely useful information about the phases and phase transitions of block copolymers. Technically, the mean-field approximation assumes that a single field configuration dominates the functional integral defining the partition function. This particular configuration is the one corresponding to the maximization of the integrand exp [−βF({φ}, {ω})]. Consequently, this field
September 29, 2010
94
10:40
World Scientific Review Volume - 9in x 6in
04-chapter3
F. Qiu, A.-C. Shi and Y. Yang
configuration is obtained by minimization of the free-energy functional F({φ}, {ω}) with respect to φα (r) and ωα (r), leading to a set of coupled SCFT equations, ωα (r) = χN φβ (r) + η(r) Z fα 1 φα (r) = dsqα (r, s)q † (r, fα − s), QC 0
(18) (19)
where the chain propagator qα (r), s) satisfies the modified diffusion equation in the mean field ωα (r), ∂ qα (r, s) = σα2 ∇2 − ωα (r) qα (r, s), ∂s
(20)
and η(r) is a Lagrangian multipler, or a pressure field, which ensures the incompressibility condition, φA (r) + φB (r) = 1. Because both φα (r) and ωα (r) are determined self-consistently from the above self-consistent equations, the mean-field approximation is referred as the self-consistent mean-field theory (SCMFT). In the literature, the SCMFT is often referred simply as the self-consistent field theory (SCFT). Within the mean-field approximation, the free energy per chain of the system is obtained by inserting the mean field solution into the free energy expression. It is interesting to examine the structure of the SCMFT for a diblock copolymer melt. Within the mean-field approximation, the parameters entering the theory are the combination χN , the block volume fraction fα , and the effective Kuhn lengths σα . The parameter χN characterizes the degree of segregation and it is controlled by temperature and the molecular weight. On the other hand, the polymer structure is characterized by the block composition fα and the block Kuhn lengths σα . Within the SCMFT, the thermodynamic properties of a diblock copolymer melt is completely specified by χN , fα and σα . The method to solve the SCMFT equations is conceptually straightforward. The first step is to make an initial guess of the mean fields ωα (r), which bears the symmetry of the ordered phase under investigation. The modified diffusion equations with the appropriate initial and boundary conditions are then solved to obtain the propagators, qα (r, s). These propagators can then be used to compute the mean field concentrations, φα (r). The next step is to adjust the mean fields ωα (r) according to an iterative procedure so as to satisfy the self-consistent equations and the incompressibility condition. For a given set of controlling parameters such as {χN, fα , σα }
September 29, 2010
10:40
World Scientific Review Volume - 9in x 6in
Self-Consistent Field Theory of Block Copolymers
04-chapter3
95
for a diblock copolymer melt, there are many solutions to the mean field equations, corresponding to different morphologies. The phase diagram is constructed by finding the structures with the lowest free energy density. Because of the complexity of the theory even within the mean-field approximation, analytic solutions of the mean-field theory can only be obtained under special conditions, such as for homogeneous phase. In order to obtain analytic solutions of the SCMFT equations, approximation methods are needed. In particular, approximation methods have been developed at the weak segregation limit (WSL)12 and at the strong segregation limit (SSL).13 The connection of the SCMFT and the WSL and SSL theories has been reviewed by Matsen.7 To obtain an exact solution, numeric techniques are required. Since the formulation of the SCFT for block copolymers by Helfand in 1975,11 a great effort has been devoted to the solution of the SCFT equations. The first efficient and accurate method to solve the self-consistent meanfield equations is the reciprocal-space method developed of Matsen and Schick,19 which is based the expansion in terms of plane wave like basis functions. Recently, with the availability of increasing computing power and new numerical techniques, real-space methods have been developed to the level that they can be used to explore the possible phases for a given block copolymer architecture.9 Mathematically, the structure of a block copolymer system is described by a density profile which is a spatially varying function determined by the minimization of the free energy function. For an ordered phase the density profile is a periodic function of the space. In the real-space formulation, the density profile is described by a real function φ(r) in space. Numerical methods in real-space start with discretizing the real-space into a lattice and the density profile is specified by its values at the lattice point. In the reciprocal-space formulation, the density presented as a Fourier R profile isik·r integral over the wavevector k, φ(r) = dkφ(k)e . Numerical methods in reciprocal-space start with the discretization of the reciprocal-space into a discrete lattice of wave vectors. In what follows the reciprocal-space formulation is given in detail. 3. Reciprocal-Space Formulation In the last section the SCFT framework of block copolymers is developed in real-sapce. Mathematically, any spatially varying functions can be represented in the Fourier space or the reciprocal-space. Therefore the SCFT
September 29, 2010
96
10:40
World Scientific Review Volume - 9in x 6in
04-chapter3
F. Qiu, A.-C. Shi and Y. Yang
can be cast in the reciprocal space. The advantage of the reciprocal space approach is that an ordered phase is naturally described by a set of basis functions which are periodic functions in space. Specifically, the functions of interest within the SCFT, such as the mean-field concentrations, the mean-field potentials, and the end-integrated propagators, are expanded in terms of Fourier series. The SCFT is then cast in terms of the expansion coefficients. The reformulation starts with expanding the functions of interest using the plane waves, eiG·r , as the basis functions. The wave vectors {G} are determined by specific discretization of the reciprocal space. One useful method to discretize the reciprocal space is to use periodic boundary conditions applied to the computation box with specific size and shape. Once the reciprocal vectors are specified, any spatially varying functions of interest, ψα (r), can be expressed in terms of Fourier series as X ψα (r) = ψα,j eiGj ·r . (21) j
Without loss of generality, the following discussions utilize a rectangular computation box. In this case the wave vector {G} is specified by three indices, {G} = 2π (h/Dx , k/Dy , l/Dz ), where Dx , Dy , Dz are the sizes of the rectangular box, and h, k, l are integers. A reciprocal vector can be represented by a set of integers (h, k, l). With these notations, the chain propagators qα (r, s), segments density φα (r), and the mean-field ωα (r) are specified as, X qα (r, s) = qα,j (s)eiGj ·r , (22) j
φα (r) =
X
φα,j eiGj ·r ,
(23)
ωα,j eiGj ·r .
(24)
j
ωα (r) =
X j
One of the important characteristics of the basis functions (fj (r) = eiGj ·r ) is that they are are orthonormal eigenfunctions of the Laplace operator, ∇2 fj (r) = −λj fj (r) with the corresponding eigenvalues λj = 2 2 2 2 4π (h/Dx ) + (k/Dy ) + (l/Dz ) . It is convenient to order the basis functions such that their eigenvalues λj form a non-decreasing series, therefore the first wave vector is G1 = 0. These basis functions are othronormal, Z 1 drfi (r)fi (r) = δij . (25) V
September 29, 2010
10:40
World Scientific Review Volume - 9in x 6in
Self-Consistent Field Theory of Block Copolymers
04-chapter3
97
In terms of the Fourier coefficients, the SCFT equations become, ωα,j = χN (φβ,j − fβ δi,1 ) + ηi Z 1 X X fα † φα,i = dsqα,j (s)qα,k (fα − s)δGi ,Gj +Gk , QC j 0
(26) (27)
k
where the single-chain partition function QC = qα,1 (1) and ηi is introduced to ensure the incompressibility condition φA,i + φB,i = δi,1 . The diffusion equations of the end-integrated propagators in reciprocal space are rewritten as, X ∂qα,i (s) =− Hα (Gi , Gj )qα,j (s) ∂s j
† X ∂qα,i (s) † = Hα (Gi , Gj )qα,j (s), ∂s j
where the “Hamiltonian” of the chain propagators is specified by X Hα (Gi , Gj ) = σα2 G2i δGi ,Gj + ωα,k (s)δGi ,Gj +Gk .
(28)
(29)
(30)
k
The initial conditions of the propagators are qA,i (0) = δGi ,0 , qB,i (fA ) = † † qA,i (fA ), qB,i (1) = δGi ,0 and qB,i (fB ) = qA,i (fB ). In the reciprocal-space, the modified diffusion equation becomes a set of coupled linear differential equation, whose solution can be found in terms of the transfer matrix Tα (s) = e−Hα s , X qα,i (s) = Tα,ij (s)qα,j (0). (31) j
Th transfer matrix can be expressed in terms of the eigenvalues and eigenvectors of the propagator Hamiltonian Hα . Because the Hamiltonian is a Hermitian matrix, diagonalization of matrix Hα can always be performed by a unitary transformation, leading to Hα = Uα Dα Uα† , where Dα is a diagonal matrix with the real eigenvalues dα,i of Hα as its matrix elements Dα,ij = dα,i δij . The columns of the unitary matrix Uα are the normalized eigenvectors of Hα . In terms of the diagonalized matrix the transfer matrix is given by Tα (s) = Uα e−Dα s Uα† , whose matrix elements are, X † Tα,ij (s) = Uα,ik e−dα,k s Uα,kj . (32) k
September 29, 2010
10:40
World Scientific Review Volume - 9in x 6in
04-chapter3
F. Qiu, A.-C. Shi and Y. Yang
98
The Fourier coefficients of the end-integrated propagators can now be written in terms of the eigenvalues and eigenfunctions for s < fA , X † qα,i (s) = UA,ij e−dA,j s UA,j1 (33) j
† qα,i (s)
=
XX j
† UA,ik e−dA,k (fA −s) UA,kj qB,j (fA ).
(34)
k
Similar expressions are obtained for s > fA , X † † qα,i (s) = UB,ij e−dB,j (1−s) UB,j1
(35)
j
qα,i (s) =
XX j
† UB,ik e−dB,k (s−fA ) UB,kj qA,j (fA ).
(36)
k
Using these expressions for the propagators, the density profiles and the single chain partition function QC can be obtained in terms of these eigenvalues and eigenvectors. Once the above equations are solved, the free energy per chain of the system is evaluated by, X FC = − ln [qB,1 (1)] − χN φA,i φB,i . (37) nkB T i
For a given set of molecular parameters, ordered phases are obtained by solving the SCFT equations. The minimizing the free energy can be further carried out with respect to the computation box parameters Dx , Dy and Dz . Once a solution is obtained, the space group of the resulting ordered structures can be determined by the wave vectors Gi with nonzero coefficients φα,i . Although the above formulae are derived for diblock copolymer melts, extension of extension to other block copolymers with varying architectures is straightforward. The following is a brief description of the extension to the case of ABC star triblock copolymers. The numerical procedure for the triblocks is very similar to that of the diblocks, with different initial conditions for the modified diffusion equations of chain propagators. We denote the junction point in star ABC triblock copolymers as segment position s = 0, and the three end-monomers along the A, B and C arms are assigned as s = fA N, fB N and fC N , respectively. Here N is the degree of polymerization of star polymer, and fA , fB , and fC (fA + fB + fC = 1) are the volume fractions of the three star chains. Starting from the initial conditions of the coefficients of the backward end-integrated chain propagators, † qα,i (fα ) = δi,1 ,
(38)
September 29, 2010
10:40
World Scientific Review Volume - 9in x 6in
Self-Consistent Field Theory of Block Copolymers
04-chapter3
99
and following the same procedures for solving the differential equations in reciprocal-space as diblock copolymers, the coefficients for the backward end-integrated chain propagators expanded in terms of the eigenvalues and eigenfunctions are obtained for 0 ≤ s ≤ fα as, X † † qα,i (s) = Uα,ij e−dα,j (fα −s) Uα,j1 . (39) j
For topological constraint that three chemically distinct chains are connected into a common junction at s = 0 in an ABC star triblock copolymer, the initial conditions qα,i (0) for solving the forward end-integrated chain propagators equations must be satisfied, XX † qA,i (0) = qB,j (0)qC,k (0)δGi ,Gj +Gk , (40) j
qB,i (0) =
j
qC,i (0) =
k
XX
(41)
† qA,j (0)qB,k (0)δGi ,Gj +Gk .
(42)
k
XX j
† qC,j (0)qA,k (0)δGi ,Gj +Gk
k
With these initial conditions, the “forward” chain propagators are found for 0 ≤ s ≤ fα as, XX † qα,i (s) = Uα,ik e−dα,k s Uα,kj qα,j (0). (43) j
k
With the solution of the chain propagators, the density profiles are obtained via. Z 1 X fα † φα,i = dsqα,j (s)qα,k (s)δGi ,Gj +Gk . (44) QC 0 j,k
P † The single chain partition function is QC = i,j qA,i (0)qA,j (0)δGi ,−Gj . Again all these quantities can be expressed in terms of the eigenvalues and eigenvectors of the chain Hamiltonian. The mean-fields are determined selfconsistently from, ωA,i = χAB N φB,i + χAC N φC,i + ηi ,
(45)
ωB,i = χAB N φA,i + χBC N φC,i + ηi ,
(46)
ωC,i = χAC N φA,i + χBC N φB,i + ηi .
(47)
where the field ηi is introduced to ensure the incompressibility of the system φA,i + φB,i + φC,i = δi,1 .
September 29, 2010
100
10:40
World Scientific Review Volume - 9in x 6in
F. Qiu, A.-C. Shi and Y. Yang
Once the SCFT equations are set up, their solutions are found by using an iterative procedure starting with the initial guess of the ω-fields. The chain propagators and density profiles are then computed for the given ω-field. These density profiles are used to constructed new ω-field. If the new ω-field agrees with the old one with a prescribed accuracy,9 the solution is considered to be converged. Otherwise a new ω-field is constructed by some mixing procedures and the new chain propagators and density are computed. This procedure continues till self-consistency is achieved. The accuracy of the numerical solutions depends on the number of reciprocal vectors used in the computation. Because many of the coefficients are zero when the wave vectors are from a generic set described above, the method is not designed to produce highly accurate solutions. Instead, it is a screening technique to obtain the symmetries of the possible ordered phases for given block copolymers. The knowledge of symmetry obtained from our method can be used as input to the more accurate and efficient method of Matsen and Schick. In this approach the basis functions are constructed choosing reciprocal lattice vectors according to the symmetry of ordered structures. In a general platform, the generic method and Matsen-Schick method can be used in tandem to produce accurate solutions for given block copolymers. 4. Applications of the Reciprocal-Space Method The reciprocal-space method of SCFT equations can be classified into two classes. The first class utilizes a restricted set of basis functions determined from a pre-assumed symmetry of the ordered phase, as initially proposed by Matsen and Schick.19 There are numerous applications of the Matsen-Schick method to a variety of block copolymer systems, starting with the seminal paper by Matsen and Schick19 on dliblock copolymers to the study of nonfrustrated ABC linear triblock copolymers by Morse and coworkers.23,24 The second class of the reciprocal-space method utilizes a generic set of plane waves as the basis functions as proposed by Guo et al.25 In this approach there is no restriction on the symmetry of the ordered phase. Instead, the symmetry of the ordered phase emerges from the solution of the SCFT equations. Specifically, the set of wave vectors corresponding to non-zero expansion coefficients are used to map out the symmetry of the solution. In this section a brief summary of the application of the generic reciprocal-space method is given.
04-chapter3
September 29, 2010
10:40
World Scientific Review Volume - 9in x 6in
Self-Consistent Field Theory of Block Copolymers
04-chapter3
101
In order to test the validity of the generic reciprocal-space approach for discovering equilibrium morphologies of block copolymers, the method has been applied to linear AB diblock copolymers. The three classic phases (lamellae, hexagonally packed cylinders and body-centered-cubic spheres) and some complex phases including the gyroid and O 77 structures have been successfully reproduced.25 The relative stability of these ordered phases is in agreement with the well-established phase diagram of diblock copolymers. A non-trivial application of the generic reciprocal-space method is the study of linear ABC triblock copolymers.25 A phase diagram for a model of the frustrated ABC triblock copolymers has been constructed. A number of new phases are predicted for the triblock copolymers. Furthermore, the method is applied to a realistic model of SEBM triblock copolymer, in which the fascinating KP phase was predicted to occur at the parameters that mostly match the experiment conditions. Recently the generic reciprocal-space method has been used to study the phases and phase transitions of ABC star triblock copolymers.34 The distinct feature of strongly segregated ABC star triblock copolymers is that their junction points must be on space curves, leading to the formation of two-dimensional packing patterns. A number of ordered two-dimensional patterns have been predicted by solving the SCFT equations using the generic reciprocal-space method. Phase diagrams are constructed for two typical ABC star triblock copolymers. The predicted phase diagrams are in good agreement with available experimental and simulations results. 5. Summary In this chapter the self-consistent field theory for block copolymers is reviewed. Although a diblock copolymer melt is used as a model system, it should be emphasized that the theoretical methods of deriving the SCFT and techniques of solving the SCFT equations are very flexible and versatile. Extension of the theory to more complex block copolymers is straightforward. The current formulation allows a systematic analysis of the thermodynamic properties of block copolymer systems. The equilibrium phases and phase diagrams can be obtained by solving the SCFT equations. We have emphasized the newly developed generic reciprocal-space method to solve the SCFT equations for complex ordered structures of block copolymers. The approach is based on a generic Fourier-space method. The essential idea is to expand all the spatial varying functions into a generic set of basis functions. The solutions of the SCFT equations can then be
September 29, 2010
10:40
102
World Scientific Review Volume - 9in x 6in
F. Qiu, A.-C. Shi and Y. Yang
obtained using any of the available numerical techniques, such as Matsen and Schick.19 The symmetry of the equilibrium phases emerges from the solutions. The space group of the ordered phases can be identified by mapping to scattering vectors. The essence of this reciptoal-space approach is to use the full power of the spectral method, while the symmetry of the ordered phases is not presumed. The generic reciprocal-space method has been applied to a number of block copolymers, including AB diblock copolymers and ABC linear and star triblock copolymers. The method is able to reproduce all known ordered phases of these block copolymers and prediction about a number of preciously unknown ordered phases has been made. Acknowledgments We would like to thank Dr. Guojie Zhang and Mr. Zhuojun Guo for their contributions in developing the generic reciprocal-space method. References 1. I. W. Hamley, The Physics of Block Copolymers. Oxford University Press c1998. 2. F. S. Bates and G. H. Fredrickson, Physics Today 52, 32 (1999). 3. A. J. Meuler, M. A. Hillmyer and F. S. Bates, Macromolecules 42, 7221 (2009). 4. E. L. Thomas, D. B. Alward, D. J. Kinning, D. C. Martin, D. L. Handin, L. J. Fetters, Macromolecules 19, 2197 (1987). 5. H. Hasegawa, H. Tanaka, K. Yamasaki, T. Hashimoto, Macromolecules 20, 1651 (1987). 6. D. A. Hajduk et al, Macromolecules 27, 4063 (1994). 7. M. W. Matsen, J. Phys.: Condens. Matter 14, R21 (2002). 8. Shi, A.-C. in Developments in Block Copolymer Science and technology, Edited by I. W. Hamley, Wiley & Sons 2004. 9. G. H. Fredrickson, The Equilibrium Theory of Inhomogeneous Polymers, Oxford University Press, New York, 2006. 10. S. Edwards, Proc. Phys. Soc. 85, 613 (1965). 11. E. Helfand, J. Chem. Phys. 62, 999 (1975); Macromolecules 8, 552 (1975). 12. L. Leibler, Macromolecules 13, 1602 (1980). 13. A. N. Semenov, Sov. Phys. JETP 61, 733 (1985). 14. A. E. Likhtman and A. N. Semenov, Macromolecules 27, 3101 1994. 15. P. D. Olmsted and S. T. Milner, Macromolecules 31, 4011 1998. 16. E. Helfand and Z. R. Wasserman, Macromolecules 9, 879 (1976). 17. K. R. Shull, Macromolecules 25, 2122 (1992).
04-chapter3
September 29, 2010
10:40
World Scientific Review Volume - 9in x 6in
Self-Consistent Field Theory of Block Copolymers
04-chapter3
103
18. J. D. Vavasour and M. W. Whitmore, Macromolecules 25, 5477 (1992); M. Banaszak and M. W. Whitmore, Macromolecules 25, 3406 (1992). 19. M. W. Matsen and M. Schick, Phys. Rev. Lett. 72, 2660 (1994). 20. M. W. Matsen and F. S. Bates, Macromolecules 29, 1091 (1996). 21. M. W. Matsen, J. Chem. Phys., 108, 785 (1998). 22. M. W. Matsen, Macromolecules 28, 5765 (1995); 36, 9647 (2003). 23. C. A. Tyler and D. C. Morse, Phys. Rev. Lett. 94, 208302 (2005). 24. C. A. Tyler, J. Qin, F. S. Bates and D. C. Morse, Macromolecules 40, 4654 (2007). 25. Z. J. Guo, G. J. Zhang, F. Qiu, H. D. Zhang, Y. L. Yang and A.-C. Shi, Phys. Rev. Lett. 101, 028301 (2008). 26. F. Drolet and G.H. Fredrickson, Phys. Rev. Lett. 83, 4317 (1999). 27. P. Tang, F. Qiu, H. D. Zhang and Y. L. Yang, Phys. Rev. E 69 031803 (2004); J. Phys. Chem. B 108, 8434 (2004). 28. Y. Bohbot-Raviv and Z.-G. Wang, Phys. Rev. Lett. 85, 3428 (2000). 29. G. Tzeremes, K. O. Rasmussen, T. Lookman and A. Saxena, Phys. Rev. E 65, 041806 (2002); K. O. Rasmussen and G. Kalosakas, J. Polym. Sci.: Part B: Polym. Phys. 40, 1777 (2002). 30. E. W. Cochran, C. J. Garcia-Cervera and G. H. Fredrickson, Macromolecules 39, 2449 (2006); 39, 4264 (2006). 31. M. W. Matsen, Eur. Phys. J. E, in press (2009). 32. M. W. Whitmore and J. D. Vavasour, Acta. Polymerica 46, 341 (1990). 33. F. Schmid, J. Phys.: Condens. Matter 10, 8105 (1998). 34. G. J. Zhang, F. Qiu, H. D. Zhang, Y. L. Yang and A.-C. Shi, Macromolecules 43, 2981 (2009).
This page is intentionally left blank
September 29, 2010
10:42
World Scientific Review Volume - 9in x 6in
Chapter 4 Dynamic Self-Consistent Field Theories for Polymer Blends and Block Copolymers Toshihiro Kawakatsu Department of Physics, Tohoku University, Sendai 980-8578, Japan E-mail:
[email protected] Understanding the behavior of the phase separated domain structures and rheological properties of multi-component polymeric systems require detailed information on the dynamics of domains and that of conformations of constituent polymer chains. Self-consistent field (SCF) theory is a useful tool to treat such a problem because the conformation entropy of polymer chains in inhomogeneous systems can be evaluated quantitatively using this theory. However, when we turn our attention to the dynamic properties in a non-equilibrium state, the basic assumption of the SCF theory, i.e. the assumption of equilibrium chain conformation, breaks down. In order to avoid such a difficulty, dynamic SCF theories were developed. In this chapter, we give a brief review of the recent developments of dynamic SCF theories, and discuss where the cutting-edge of this theory is.
1. Introduction One of the characteristic features of multi-phase polymeric materials, such as polymer blends or block copolymers, is the formation of mesophases where phase-separated domains are formed on the length scale of the gyration radius of the constituent polymer chains (i.e. so-called microphase separation). It is important to recognize that the physical properties of these polymeric materials very often depend on the structure and dynamics of the domains in mesophases.1–3 Typical examples are the self-assembling processes of polymer blends and block copolymers, where the final domain structure is often strongly affected by the kinetic pathway that the system traces from the initial state to the final state. Of course, the deformation of constituent chains due to the flow or to the deformation of the domains also contribute to this dynamics of mesophases. 105
05-chapter4
September 29, 2010
106
10:42
World Scientific Review Volume - 9in x 6in
T. Kawakatsu
For the study of equilibrium domain structures and equilibrium chain conformations in mesophases, there is a well-established technique called (static) self-consistent field (SCF) theory1,4–8 Using this static SCF theory, one can predict the equilibrium domain structures in mesophases of polymeric systems not only qualitatively but also quantitatively. On the other hand, the techniques to study dynamic phenomena in polymeric systems on the length scale of the domains in the mesophases are still limited to Monte Carlo or molecular dynamics simulation techniques on molecular models, which require a large amount of computational cost. In order to improve the computational efficiency of the dynamic simulations, it is essential to construct a model that is based on the coarse-grained picture like the SCF theory. The dynamics of polymeric flows are roughly categorized into two regimes. One is the slow flow regime where only the phase separated domains are deformed but the chain conformations are still in equilibrium. In such a regime, we can still rely on the static SCF theory by assuming that the chain conformations are always in local equilibrium and the chains diffuse according to the gradient of the thermodynamic forces (Diffusion regime). The other extreme is the fast flow regime where the chain conformation are deformed considerably from its equilibrium one due to the external forces or by migration of the domains (Viscoelastic regime).5,9 There have been considerable works for the dynamic extensions of the SCF theory to the diffusion regime, and various simulation techniques and knowledge have been accumulated.3,5,10 For example, dynamics of microphase separation of diblock copolymer melts,10,11 phase behavior of complex polymeric systems,12,13 dynamics of block copolymers under external constraints and under external fields14–22 Contrary to the situation for these dynamic SCF theories in the diffusion regime, studies for the viscoelastic regime have not yet been well developed from the viewpoint of the SCF theory. Actually, there have been several trials to study such viscoelastic regime using dynamic SCF theories. Typical examples are studies on long-range hydrodynamic interactions,23–28 effects of long-range kinetic correlations in mesophases,29,30 chain dynamics in dilute polymer solutions (Rouse dynamics),31–33 the chain dynamics in dense polymer systems (reptation dynamics),34 and development of hybrid approaches between SCF theory and particles.27,35,36 Despite these investigations, the state of the art of the dynamic SCF theories for viscoelastic regime is still in a developing stage.
05-chapter4
September 29, 2010
10:42
World Scientific Review Volume - 9in x 6in
Dynamics Self-Consistent Field Theories for Polymers
05-chapter4
107
In this short review, we try to figure out the current status of this developing dynamic SCF theories for both diffusion regime and for viscoelastic regime. This article is organized as follows. In the next section, we give a brief summary of the theoretical formulations of the dynamic SCF theories. In section 3, we review typical studies on the domain motion in the diffusion dynamics regime, where dynamics of polymer blends and block copolymer systems are discussed. Then, in section 4, several recent trials to extend the dynamic SCF to viscoelastic regime. Finally, in section 5, we give a conclusion. 2. Basic Formalism for Dynamic SCF Theory For a simple illustration of the dynamic SCF theories, let us consider multicomponent polymer blends (including block copolymers as components). The essence of SCF theory for polymers is the coarse-graining, where microscopic details of the polymer chains are averaged out and only several essential physical properties survive. In this coarse-grained level, the polymer chain conformations are well described by the Gaussian chain statistics, where the conformation entropy of the chains is evaluated using the path integrals.5 In the static SCF theories,1,4–8 the important coarse-grained variables that characterize the state of the system are the density distributions of individual type of segments. Let us denote the density distribution of Ktype segments as φK (r), where K is the index for the segment species. The calculation of the static SCF theory is performed so that the total free energy of the system is minimized with respect to these density distributions of segments. On the other hand, when we turn our attention to dynamic processes, we immediately recognize that the segment density distributions φK (r) are not the only relevant field variables to specify the state of the system. Extra variables we need are, the local flow velocities of each components and the stress field. In general, time evolution of the density field φK (r, t), which is now also a function of time, is described by the following continuity equation because it is a conserved quantity; ∂ φK (r, t) + ∇ · jK (r, t) = 0, ∂t where jK (r, t) is the flux for the K-type segments.
(1)
September 29, 2010
10:42
World Scientific Review Volume - 9in x 6in
108
05-chapter4
T. Kawakatsu
The flux jK (r, t) is decomposed into three contributions, i.e. diffusion flux, convection flux and the flux induced by the external field such as an electric field, as conv ext jK (r, t) = jdiff K (r, t) + jK (r, t) + jK (r, t).
(2)
In the following, we will discuss these contributions to the dynamics of the segment density separately. 2.1. Diffusion flux The first term on the right-hand side of Eq. (2) accounts for the diffusion flux. The simplest and the most popular choice is the following Fick’s law for linear diffusion driven by the gradient of the local chemical potential µK (r, t): LKK ∇µK (r, t), (3) jdiff K (r, t) = − K
where the coefficients LKK are the Onsager kinetic coefficients, which is in general a function of the local density φK (r). The following approximation for this kinetic coefficient is often adopted; (0)
LKK = LK φK (r)δKK , (0)
(4)
where LK are constants and δKK is Kronecker’s delta symbol. This simple form of the kinetic coefficient is accepted because in most of the cases, the off-diagonal elements of the kinetic coefficient matrix, LKK (K = K ), are negligible compared to the diagonal elements LKK , which means that the effects of the dynamical cross coupling between different types of fluxes is small. In Eq. (4), ∇µK (r) corresponds to a thermodynamic force acting on the individual segment of K-type, which originates from the inhomogeneity in the segment density distributions. As the polymer flow is very viscous, each segment moves with a limiting velocity driven by this thermodynamic force (over damped case). Then, the diffusion flux is a product of the drift velocity of a single segment and the total numbers of segments per volume, which results in the expression described above. In the dynamic SCF theory, the system is in general non-equilibrium. Therefore, the chemical potential µK in Eq. (3) or Eq. (4) is also nonequilibrium. Then, how to evaluate this chemical potential is a big problem.
September 29, 2010
10:42
World Scientific Review Volume - 9in x 6in
Dynamics Self-Consistent Field Theories for Polymers
05-chapter4
109
If the motion of the segment density distributions is so slow that the chain conformations are always relaxed under the constraint of the instantaneous distributions of the segment densities φK (r, t) at time t, we can use local equilibrium assumption as in the static SCF theory and can evaluate the chemical potential µK (r, t). Only difference from the static SCF is that the segment density distributions φK (r) are not determined by the SCF calculation but they are regarded as constraints imposed to the SCF calculation. Then, these density distributions φK (r) are determined by integrating the equation of motion, Eq. (1), rather than to be determined in SCF calculation. In this case, the SCF scheme is modified so that the density distribution of K-type segments calculated in the SCF scheme at every time step, denoted as ϕK (r), should coincide with the given density field φK (r). This can be done by solving the following set of SCF equations:5 b2 ∂ QK (r, s) = K ∇2 QK (r, s) − βVK (r)QK (r, s) ∂s 6 ϕK (r) = CK dsQK (r, s)Q†K (r, NK − s) VK (r) = χKK ϕK (r) + λK (r),
(5)
K
where QK (r, s) and Q†K (r, s) are forward and backward path integrals (or sometimes called end point distributions) for the K-type polymer chains, bK is the size of the statistical segment, VK (r) is the self-consistent mean field potential, ϕK (r) is the segment density distribution calculated by the SCF scheme, CK is a normalization constant, χKK is the χ-parameter between K and K type segment pair, and λK (r) is the Lagrange multiplier that constrains the segment density distribution calculated by this SCF scheme, ϕK (r), to the given density distribution φK (r). Once the above SCF scheme is solved (usually an iteration procedure is used), the chemical potential µK (r) can be obtained by the following relation: µK (r) = −λK (r).
(6)
This relation is understood in the following way. As we discussed above, the SCF calculation generates an equilibrium state under the constraint ϕK (r) = φK (r) by using the Lagrange multiplier λK (r). If this Lagrange multiplier is absent, the density field φK (r) will move according to the thermodynamic driving force due to the chemical potential µK (r). Thus,
September 29, 2010
10:42
World Scientific Review Volume - 9in x 6in
110
05-chapter4
T. Kawakatsu
the Lagrange multiplier λK (r, t) should cancel this thermodynamic driving force so that the system becomes static (local equilibrium). This balance between µK (r) and λK (r) gives Eq. (6). 2.2. Convection flux The convection flux in Eq. (2) is driven by the local convection flow field v(r, t), and is given by jconv K (r, t) = v(r, t)φK (r, t).
(7)
This flow velocity v(r) is determined by the following conservation equations for the mass and the momentum: ∇ · v(r, t) = 0,
(8)
and
∂ v(r, t) = ∇ · Π(r, t) + ρK(r, t), (9) ∂t where ρ is the total density of the polymeric fluid that is assumed to be constant due to incompressibility condition, Π(r, t) is the stress tensor and K(r, t) is the body force which will be defined below. The stress tensor Π(r) expresses the viscoelastic properties of the polymeric fluid. In general, it is described as: ρ
Π(r) = −p(r)1 + σ(r),
(10)
where p(r) is the local pressure that plays the role of Lagrange multiplier for the incompressibility condition for total density, 1 is the unit tensor, and σ(r) is the shear stress tensor that in general depends on the history of the flow. For polymeric flow, the inertia term ρvv that appears in the stress tensor for low molecular weight fluids is usually neglected because of the high viscosity. The shear stress σ(r) characterizes the viscoelastic properties of the fluid. The relation between the shear stress σ at time t and the history of the velocity gradient tensor κ ≡ (∇v)T is called the constitutive equation. The simplest one is the Newtonian fluid,24 for which the constitutive equation is given by σ(r) = η(r)κ(r) = η(r)(∇v)T ,
(11)
where η(r) is the viscosity that in general depends on the local composition. For more general case with viscoelastic properties, one has to assume a model constitutive equation, such as Maxwell model, Rouse model, or reptation model.5,9 These models will be discussed in section 4.2 later.
September 29, 2010
10:42
World Scientific Review Volume - 9in x 6in
Dynamics Self-Consistent Field Theories for Polymers
05-chapter4
111
The body force K(r) in Eq. (9) accounts for the effects from the inhomogeneity in the segment density distribution (interfaces), and is given by23,24 K(r) = −
φK (r)∇µK (r).
(12)
K
Contrary to the static SCF theories discussed in the previous chapters, dynamic SCF simulation is usually done in the real space because the main target system of dynamic SCF is deformed domain structures that have no crystalline symmetry, for which spectral method is no longer useful.5,37 2.3. Flux induced by external fields Imposing an external field to the polymeric system also induces a nontrivial polymeric flow. A typical example is the electric field, which is often used to make the phase separated domains in a thin film align along the direction of the electric field.38 Let us consider a situation where an electric field is imposed in the zdirection of an A/B binary polymer blend or an A-B block copolymer melt. Assuming slow spatial variation of the local compositions φK (r), we can derive following approximated expression for the flux due to this electric field39 jext K (r, t) = α {ez · ∇φK (r)} ez ,
(13)
where α is a constant that measures the effect of the electric field, and ez is a unit vector in z-direction. This expression means that the imposed electric field induces an anisotropic diffusion in its direction. The parameter α depends on both the electrostatic nature of the material and the strength of the electric field. It is given by α=
0 21 2 E vL,
¯ 0
(14)
where E0 is the amplitude of the electric field, v is the volume of a single polymer chain, L is the Onsager kinetic coefficient that appeared in Eq. (3) where we assumed LKK = LδKK for simplicity, 0 is the dielectric constants of vacuum, and 1 and ¯ are defined by the dependence of the local dielectric constant (r) on the local composition φA (r) as
(r) ≈ ¯ + 1 (φA (r) − φ¯A ), where φ¯A is the average composition of the A-component.
September 29, 2010
10:42
112
World Scientific Review Volume - 9in x 6in
05-chapter4
T. Kawakatsu
3. Dynamic SCF Theories in Slow Diffusion Regime 3.1. Formulation When there is mesoscopic phase separated domains in the system, such as the microphase separation of block copolymers, these domains dominate the rheological properties of the total system in the low frequency regime or long time regime. In this case, we can adopt quasi-static approximation where the chain conformation is always equilibrium under the constraint imposed by the profile of the segment density distribution φK (r, t). If we further neglect the flow field, Eq. (2) is simplified to (0)
jK (r, t) = −LK φK (r)∇µK (r, t),
(15)
where we adopted the simplified form of the Onsager kinetic coefficients given in Eq. (4). In Eq. (15), the connectivity of polymer chain is taken into account into the SCF calculation through the chemical potential {µK }. This diffusion dynamics takes only the slowest Rouse mode of the constituent chains into account.40 If we apply this diffusion type model to polymer blends, the dominant contribution to the dynamics of whole system is the motion of the interfaces that separate the domains. However, for immiscible polymer blends, such domains continuously grow and result in a macrophase separation. In such a late stage of the macrophase separation, the gyration radius of constituent polymers are negligibly small compared to the domain size, and the effect of the conformation entropy of the polymer chains becomes less and less important. This is why SCF theory was not extensively applied to dynamics of macrophase separation of polymer blends. For such macrophase separations, the so-called Ginzburg-Landau (GL) type coarse-grained theory is often used.5,41 Situation changes drastically when we consider block copolymer melts or blends of homopolymers and block copolymers. In these systems, the domains do not grow to a macroscopic scale because of the connectivity of the subchains of the block copolymer or of the surfactant nature of the block copolymer. The former limits the domain size and the latter diminishes the driving force for the domain coarsening. As a result, the system shows a microphase separation on the length scale of the block copolymer chain. (The domain size is proportional to N 1/2 in the vicinity of the critical point and to N 2/3 well inside the coexistence region for a melt of a block copolymer with chain length N .5 ) Therefore, block copolymer systems are very suitable targets of the dynamic SCF theory, where the conformation entropy plays an important role.
September 29, 2010
10:42
World Scientific Review Volume - 9in x 6in
Dynamics Self-Consistent Field Theories for Polymers
05-chapter4
113
Another suitable target of dynamic SCF technique is a thin film of polymeric fluid. If the thickness of the film is of the same order as the gyration radius of the polymer chain, the polymers undergo considerable deformation due to the constraint imposed by the wall of the film. These systems can also be simulated with dynamic SCF theories. 3.2. Formation process of mesophases of block copolymer melt The first simulation based on the diffusion type dynamic SCF theory was reported by Fraaije on the formation process of lamellar phase of a symmetric diblock copolymer melt using a 2-dimensional system,10 and later they extended this simulation to 3-dimenional systems.11 Figure 1 is a series of snapshot pictures obtained by a 3-dimenisonal simulation on formation of lamellar phase of an A8 B8 symmetric diblock copolymer melt with χN = 16.0.11 The initial state is a uniform phase of the block copolymer quenched from a high-temperature disordered state. They simulated this process based on Eqs. (1) and (2), where jdiff is given by Eq. (15) and other fluxes are neglected. In this simulation, the authors assumed that diffusion dynamics, where the kinetic coefficients LK were (0) (0) given by LK = LK φK (r) with constant LK . (a)
(b)
(c)
(d)
Fig. 1. (a)–(c) Formation of a lamellar phase of an A8 B8 symmetric diblock copolymer melt from its initial uniform state with χN = 16.0. (d) An isosurface representation of the data is shown in (d). Reprinted with permission from Ref. 11.
September 29, 2010
10:42
114
World Scientific Review Volume - 9in x 6in
T. Kawakatsu
(a)
(b)
(c)
Fig. 2. (a) Dynamic SCF simulation on phase separation of a mixture of long and short A-B type block copolymers. Upper figures show the density distributions of long block copolymer, while the lower figures show the density distributions of A-segments of both long and short block copolymers. (b) Domain structure obtained by a 3-dimensional simulation on the same system. (c) A TEM image of the corresponding experimental system. Reprinted with permission from Ref. 13.
Although there exists GL type theory for block copolymer melts42 based on coarse-grained free energy model,43 dynamic SCF has several advantages. For example, the chain architecture can easily be introduced, and the simulation can be done even for larger value of χN , for which the GL free energy is no longer valid. One of the early simulations in this direction was performed on Pluronics, which are A-B-A type triblock copolymers made of two poly(propylene oxide) blocks and a poly(ethylene oxide) middle block.12 This triblock copolymer is a typical polymeric surfactant, and is widely used in chemical industries. Dynamic SCF simulations were performed for Pluronic/water solutions by changing the composition of the solution.12 It was shown that the simulations reproduce most of the phase diagram of the experimental system without assuming any crystal symmetries of the domain structures. Dynamic SCF simulations can also predict the domain structures and phase diagram of more complex systems. Figure 2 shows a phase separation dynamics of a mixture of a long and a short block copolymers of the same kind.13 This simulation was intended to reproduce the behavior of an experimentally investigated systems.44 Therefore the parameters such as the χ-parameters, chain lengths and the molecular structures of the block copolymers are all taken from the experimentally obtained data.44 Due to this reason, the system is a mixture of a slightly asymmetric A − B type diblock copolymers with different chain lengths. In Figs. 2(a) and (b), one
05-chapter4
September 29, 2010
10:42
World Scientific Review Volume - 9in x 6in
Dynamics Self-Consistent Field Theories for Polymers
05-chapter4
115
can confirm that the lamellar domains are generated in a matrix of the disordered phase. Such a structure is similar to that observed in real experiment shown in Fig. 2(c). We also confirmed that the experimentally determined phase diagram is very well reproduced by the dynamic SCF simulations. It should be noted that the generated lamellar domains both in the experiment and in the dynamic SCF simulation are irregular ones. This means that the system is not yet reached the true equilibrium state. For such a non-equilibrium state without crystal symmetry, the dynamic SCF simulation in real space is a powerful tool. 3.3. Dynamics and non-equilibrium domain structures in thin films and near solid surfaces Another interesting target of dynamic SCF technique where the diffusion dynamics is dominant is the phase separation dynamics and irregular domain structures in thin films and near solid surfaces. Typical examples are surface-induced phase separation and selective adsorption of polymers to a solid or free surface. Application of dynamic SCF technique to these problems was done by Hasegawa and Doi.14 They investigated the adsorption of homopolymers to an attractive solid surface from a semidilute polymer solution using 1dimensional dynamic SCF technique. Using the path integrals calculated in the SCF simulation, one can extract information on the polymer chain conformations. As is shown in Fig. 3(a), they calculated the average lengths of tail, train, and loop parts in the adsorbed polymers. (The definitions of these terminologies are given in Fig. 3(b).) From these data, one can (a)
(b)
average length
10
tail
5 train
tail loop
loop 0
train
10-2 10-1 100 101 102 time
Fig. 3. (a) Time evolution of average lengths of tail, train and loop parts of adsorbed homopolymers. Reprinted with permission from Ref. 14. (b) Schematic explanation for the tail, train and loop parts.
September 29, 2010
10:42
World Scientific Review Volume - 9in x 6in
116
T. Kawakatsu
(a)
(b)
(c)
(d)
Fig. 4. Mesophases in thin block copolymer layer obtained by experiments ((a)-(c)) and dynamic SCF simulations ((d)). Reprinted with permission from Ref. 15.
understand that the adsorption process consists of two stages. In the first stage, segments that were close to the surface in the initial time are adsorbed rather independently. This is indicated by the emergence of long tails and loops. Then, using the adsorbed segment as a nucleus, connected segments to this adsorbed segment are successively adsorbed onto the solid surface. This leads to a formation of long trains, which replaces the initially generated tails and loops. Another example is the mesophase formation in thin films of block copolymers placed between two parallel solid plates.15 The data shown in Fig. 4 are comparison between experimental observation on SBS block copolymer film and corresponding dynamic SCF simulation of a thin film of block copolymer A3 B12 A3 and χN = 117, where the film thickness is gradually changed so that the various mesophases can be observed simultaneously. Not only traditional spherical and cylindrical phases, one can also observe perforated lamellar domains which is not stable in a 3-dimensional bulk system. Dynamic SCF technique is also applicable to a free surface between a polymer film and the air.16 In such a system, domains formed by the phase separation near the free surface induce deformation of this surface due to the difference in the interfacial tension. In Fig. 5, we show the domain structures and the surface roughening of an interface between A/B polymer blend film and the air obtained by dynamic SCF simulation. In
05-chapter4
September 29, 2010
10:42
World Scientific Review Volume - 9in x 6in
Dynamics Self-Consistent Field Theories for Polymers
(a)
05-chapter4
117
(c)
(b)
Fig. 5. Formation of domains at the free surface of an A/B polymer blend film obtained by dynamic SCF simulations. (a) Segment density distributions and (b) deformation of free surface. (c) Phase diagram for the surface roughening. χAB and χBV are interaction parameter between A and B segments and that between A-segment and the void (air) segment. Reprinted with permission from Ref. 16.
this simulation, the air phase is modeled by an artificial “void” phase that has a very strong repulsive interaction with the polymer segments. Using a series of simulations, we can determine the phase diagram of the surface roughening as a function of the interaction parameters between A and B segments and that between A segment and the void (Fig. 5(c)). 3.4. Structural phase transitions induced by external fields When a mesophase of a polymeric system is subjected to a weak external field, such as an electric field or an external flow, in response to this external field, the phase separated domains are deformed. If the strength of the external field exceeds certain threshold value, a structural phase transition will take place. Here we discuss several examples of such a structural phase transition induced by external fields. The first example is concerned with the gyroid phase of diblock copolymers. Gyroid phase is composed of bicontinuous networks of two A-domains and two B-domains, each of which consists of 3-fold junctions.17 Figure 6 shows results of dynamic SCF simulations on the effect of an external shear flow on gyroid phase of an A7 −B13 diblock copolymer melt at χN = 20 done by Honda and Kawakatsu.17 The shear flow direction is in
September 29, 2010
10:42
World Scientific Review Volume - 9in x 6in
118
T. Kawakatsu
(a)
(b)
(d)
(e)
(c)
(f)
Fig. 6. (a)–(f) Structural phase transition from a gyroid phase to a cylindrical phase induced by an external shear flow. Reprinted with permission from Ref. 17.
the [1 1 1] direction of the Gyroid structure (chosen as x-direction) and its velocity gradient in the [1 1 ¯2] direction (chosen as y-direction), where the shear rate γ˙ ≡ ∂vx /∂y is γ˙ = 0.001 (in unit of 1/L, where L is the kinetic coefficient which is assumed to be the same for A and B components). In this simulation, the effect of the flow is introduced by assuming v(r) in Eq. (7) as v(x, y, z) = (γy, ˙ 0, 0). (An example of the treatment of full hydrodynamics is discussed in section 4.1.) A coexistence of the gyroid phase and the cylindrical phase is observed in the late stage. This means that the transition proceeds as nucleation and growth process. For this coexistence, it is essential that the size of the simulation box is adjusted so that the free energy density is minimized with respect to the system box size at every time step. (We call this technique as “system size optimization”.) Otherwise the coexistence is smothered by the constraint of the constant system size.17 A similar strategy of adjusting the system size was used by Li et al. for 2-dimensional models of sheared micro phases of linear and star triblock copolymers.18 On the other hand, Fig. 7 shows the results of dynamic SCF simulations on a similar structural transition of gyroid phase but induced by an external static electric field imposed in the [1 1 1] direction of the gyroid structure.19
05-chapter4
September 29, 2010
10:42
World Scientific Review Volume - 9in x 6in
Dynamics Self-Consistent Field Theories for Polymers
(a)
(d)
(b)
(e)
05-chapter4
119
(c)
(f)
Fig. 7. (a)–(f) Structural phase transition from gyroid phase to cylindrical phase induced by an external electric field. Reprinted with permission from Ref. 19.
The effect of the electric field is taken into account using Eq. (13). Differently from the case with shear flow (Fig. 6), we observe uniform transition from the initial gyroid structure to the final cylindrical structure. We confirmed that this uniform nature of the intermediate structure is not due to the effect of the system size optimization19 as in the case of the shear flow. The main difference between the shear flow case and this electric field case is the fact that the external field has two characteristic directions in the shear flow case, i.e. the flow direction and the direction of the velocity gradient, while the electric field has only one characteristic direction. This difference affects the kinetic pathways from the initial gyroid phase to the final cylindrical phase because in shear flow case, anisotropy in the domain structure is easily be generated due to the bi-directional nature of the shear flow. The effects of the external electric field is especially important when one try to align the micro domains in a thin film of block copolymer to a desired direction. It is usually not easy for an electric field to penetrate the bulk specimen of a polymeric material due to the screening effects by the dielectric property. However, by using a ultra thin film of block copolymers, such an alignment can be achieved.38 The effects of an external electric field on block copolymer thin film were extensively studied by Knoll et al.39 They studied the alignment of the cylinder domains in a thin film both by experiment and by dynamic
September 29, 2010
10:42
World Scientific Review Volume - 9in x 6in
120
T. Kawakatsu
(b)
(a)
(i)
(ii)
(i)
(ii)
(iii)
(iv)
(iii)
(iv)
(v) Fig. 8. (a) (i)–(v) Time evolution of the domain structures upon structural phase transition from initial perforated lamellar phase to cylindrical phase in a thin film of A-B diblock copolymer induced by an external electric field in the normal direction of the thin film. (b) Variation of the shapes of the final cylinders by changing the strength of the electric field. From (i) to (iv), the electric field becomes stronger. Reprinted with permission from Ref. 20.
SCF simulation for χN = 19.0 (weak segregation regime). They observed splitting of cylinders into small spheres due to the imposed electric field. Then, these spheres reconnect in a different way so that new cylinders that are parallel to the electric field are generated. This is the scenario of domain reorientation in weak segregation regime. If the system is in the strong segregation regime (larger χN ), rotation of the grains of ordered mesophase is expected instead of splitting and reconnection of domains. In this case, the domain rotation is achieved by migration of defects in the ordered domains. As we have discussed in section 3.3, block copolymers can show exotic domain structures, such as perforated lamellar domains, which are no longer stable in a 3-dimensional bulk system.15 Thus, it is interesting to study the stability of such exotic phase in thin film under the influence of an electric field. Ly et al. performed a series of dynamic SCF simulations on thin films of an A-B diblock copolymer20 or of an A-B-A triblock copolymer21 under an electric field. Figure 8 shows results of a dynamic SCF simulation on perforated lamellar phase in a thin film of A-B diblock copolymer under an electric field. The electric field is imposed in the normal direction of the film, in which direction the final cylinders are aligned.
05-chapter4
September 29, 2010
10:42
World Scientific Review Volume - 9in x 6in
Dynamics Self-Consistent Field Theories for Polymers
05-chapter4
121
In this case, the kinetic pathway is rather complex. First, the narrow arms of the perforated lamellar domains shrink. Then, these arms are disconnected to form cylinder domains perpendicular to the film. The shape of the final cylindrical domains are determined by the competition between the effect of the electric field and the effect of the wall which prefers one of the two segment species. In this simulation, the interaction parameters between the polymer segments and the solid walls are chosen so that the cylinder-forming block dislikes the wall. Therefore the top and the bottom of the cylinders are shrinking when the electric field is weak (Figure 8(b)(i)(ii)). However, as we increase the strength of the electric field, the cylinder tend to be straight (Figure 8(b)(iii)-(vi)). 4. Beyond Diffusion Dynamics — Hydrodynamics, Viscoelasticity and Hybrid Techniques In the previous section, we have discussed the dynamic phenomena where the diffusion process and the deformation of domains are dominant. However, if the system is strongly and rapidly deformed by an external field or by a rapid migration of the interfaces, the characteristic time scale of the deformation becomes shorter than the characteristic relaxation time of the chain conformation. In such a case, the assumption of diffusion dynamics breaks down, and we have to introduce more microscopic dynamic processes into the model. In this section, we review the recent developments of the extensions of dynamic SCF theory to several interesting situations. 4.1. Hydrodynamic effects on domain formation Even though a polymer melt or a polymer blend is so viscous, hydrodynamic effects cannot be neglected in the long time behavior.45 This is because the hydrodynamic interaction is long ranged and does not have characteristic length scale (it decays as inverse distance). For example, in the case of the macrophase separation of a polymer blend, the hydrodynamic interaction may be negligible in the early stage of the phase separation. However, its importance gradually increases as the domains grow, and finally it dominates the dynamics of the domain coarsening. This property has been well verified experimentally46 and by simulations on time-dependent GL model.47,48 Even for the diffusion type dynamics, the dynamic SCF simulation requires a massy computation. The introduction of hydrodynamics requires an additional heavy computational cost, which is necessary to solve the
September 29, 2010
10:42
World Scientific Review Volume - 9in x 6in
122
05-chapter4
T. Kawakatsu
Simple diffusion Hydrodynamics
Fig. 9. Time evolution of the free energy of an A8 B8 diblock copolymer melt that undergoes microphase separation from an initial uniform state. Two model systems, i.e. simple diffusion dynamics system (upper curve) and hydrodynamic system where Darcy’s law is used (lower curve), are compared. Reprinted with permission from Ref. 23.
time evolution equation for the Navier-Stokes equation for the velocity field, Eqs. (9)-(11): ρ
∂ v(r, t) = −∇p(r) + ∇ · [η(r)∇v(r)] + ρK(r, t), ∂t
(16)
where the pressure p(r) is determined so that the incompressibility condition Eq. (8) is always satisfied. In order to avoid the heavy computational cost associated with the numerical solution of Eq. (16), Maurits, et al. used an assumption of Darcy’s law23 1 (17) ζv(r, t) = K(r, t), ρ where ζ = 6πη0 /ξ 2 is the friction constant of an object that has a characteristic size ξ. This assumption means that the segment is assumed to be a moving object in a static medium with viscosity η0 and is always moving with the termination velocity driven by the body force K(r). In Fig. 9, we show a comparison of the time evolution of the free energy between the two model systems, one is the simple diffusion system and the other is the hydrodynamic system where Darcy’s law is used.23 It is obvious that the hydrodynamic effects accelerates the phase separation. This acceleration originates from the alignment of domains induced by the long-range hydrodynamic interaction.
September 29, 2010
10:42
World Scientific Review Volume - 9in x 6in
Dynamics Self-Consistent Field Theories for Polymers
05-chapter4
123
(a)
(b)
Fig. 10. (a) Dynamic SCF simulation with hydrodynamic interaction is applied to polymeric flows in micro channels with various shapes. (b) Dynamic SCF with hydrodynamics is applied to polymeric flows with filler particles. Here the filler particles are treated as discrete molecules while the polymers are described by dynamic SCF theory. Reprinted with permission from (a) Ref. 25 and (b) Ref. 27, respectively.
On the other hand, dynamic SCF simulations based on the full hydrodynamic equation, Eq. (16), were performed recently.24–26 Honda and Kawakatsu investigated the effects of the hydrodynamic interaction on the formation process of mesophases of block copolymers.24 They simulated the formation processes of lamellar, cylinder, spherical and gyroid structure from initial disordered phase. Comparison between the two cases with and without the hydrodynamic interactions revealed that the well-ordered lamellar, cylindrical and spherical domains can only be formed for the cases with hydrodynamic interaction. This result corresponds to the observation by Maurits et al.23 However, even with the hydrodynamic interaction, well-ordered gyroid phase was not obtained. The main difference between the gyroid phase and the other phases is the connectivity of the domains. As the gyroid domains are interconnected, the flow field induced by the interfacial motion cannot propagate to distant points and cannot induce long-range dynamic/static correlations between domains. Hall et al. used the dynamic SCF with hydrodynamic interaction for the study of polymeric flows in microscopic channels with arbitrary shapes25 (Figure 10(a)) and those containing filler particles27 (Figure 10(b)). For the latter case of the simulation with fillers, they used a hybrid approach where the polymer flow is described by continuous field while the filler particles are described by discrete molecules,28,49–51 which will be discussed in section 4.3.
September 29, 2010
10:42
World Scientific Review Volume - 9in x 6in
124
05-chapter4
T. Kawakatsu
As shown in Fig. 10, the dynamic SCF simulations give us an understanding of micro-fluidics and composite flows, and offers a way to design high-functional materials and micro machines. 4.2. Dynamic SCF with viscoelastic properties It is well known that polymeric materials show viscoelastic properties. The viscoelastic property of a fluid is described by the relation between the stress tensor σ(r, t) and the velocity gradient tensor κ(r, t) ≡ (∇v)T as5,9 σ(r, t) = S [{κ(r, t )}; t ≤ t}] ,
(18)
where the right-hand side means that the stress at time t depends on the history of the deformation in the past. The relation Eq. (18) is called the constitutive equation. A simple example of the constitutive equation is the following one for a linear viscoelastic fluid: t G(t − t ) κ(t ) + κT (t ) dt , (19) σ(t) = −∞
where G(t) is called the stress relaxation function, and we assumed that the fluid is uniform and isotropic. The non-Markovian nature of the constitutive equation Eq. (18) or Eq. (19) means that the fluid has a memory. In the case of multi-component polymeric fluids, this memory is stored in the fluid in the form of (a) deformation of the domains, and (b) deformation of the polymer chains or polymer networks. The former (factor (a)) is already taken into account in the dynamic SCF theory using the diffusion dynamics. However, the latter (factor (b)) is not accounted for by the diffusion dynamics or by hydrodynamics discussed so far. There are two typical chain dynamics in polymer solutions and melts. One is the Rouse dynamics and the other is the reptation dynamics.5,9 The Rouse dynamics is valid for dilute polymer solutions without hydrodynamic interaction or for short time dynamics of dense polymer systems. In a dilute polymer solution, a chain undergoes a Brownian motion under the constraint of the chain connectivity. This constraint leads to the Rouse dynamics. On the other hand, reptation dynamics explains the dynamics for overall time regime of dense polymer melts and blends. In dense polymer systems, a polymer chain moves as if it is confined in a tube formed by the steric hindrance by the surrounding chains. This chain dynamics is called reptation dynamics.
September 29, 2010
10:42
World Scientific Review Volume - 9in x 6in
Dynamics Self-Consistent Field Theories for Polymers
05-chapter4
125
Because SCF theory can quantitatively evaluate a change in the polymer conformation, it should basically be possible to take the viscoelastic properties into SCF theory. However, both static and dynamic SCF theories assume that the chain conformation is always equilibrium, which contradicts with the chain conformation in deformed viscoelastic polymeric fluids. 4.2.1. Non-local kinetic coefficients with chain deformations Early works intended to overcome this difficulty in the framework of dynamic GL and SCF theories were done by introducing non-local kinetic coefficient.29,30,52 Instead of the time evolution equations with local kinetic coefficients, Eqs. (1)-(4), Kawasaki and Sekimoto proposed a time evolution equation with non-local kinetic coefficients for diffusion flux as ∂ φK (r, t) = dr LKK (r, r )∇ µK (r , t), (20) ∂t K
where LKK (r, r ) is the non-local kinetic coefficient which depends on the instantaneous deformed conformations of the chains.52 The non-locality comes from the dynamical correlations between segments at distant points r and r that are connected by polymer chains, which move in Rouse dynamics or in reptation dynamics.5,9 This non-local kinetic coefficient was applied to formation of lamellar phase of a block copolymer melt using dynamic GL and SCF simulations with reptation dynamics in Ref. 29 and with Rouse dynamics in Ref. 30, respectively. As is shown in Fig. 11, these simulations with non-local kinetic coefficient lead to an acceleration of the coarsening compared to that with local kinetic coefficient. This acceleration is a result of an apparent longrange transport of segments. 4.2.2. Full dynamic SCF theory with Rouse dynamics More direct way of introducing non-equilibrium chain conformation into dynamic SCF theory is to calculate the time evolution of the probability distribution of each bond vector connecting the segments along the chain. If we can calculate this probability distribution of bond vectors, we can obtain the instantaneous stress tensor σ(r, t) as a sum of the contributions from each bond vectors. Then, we can substitute the result into Eq. (10) and solve Eq. (9). Maurits et al. introduced Smoluchowski equation, which is the time evolution equation of the probability distribution of bond vector, into dynamic
September 29, 2010
10:42
World Scientific Review Volume - 9in x 6in
126
T. Kawakatsu
(a)
(b)
Fig. 11. Time evolutions of inverse of the average domain size of lamellar formation of a symmetric diblock copolymer. Results of two models with local (broken curve) and non-local kinetic coefficients (solid curve) based on reptation dynamics are compared. The ratio between the gyration radius of the chain Rg and the interfacial thickness ξ is chosen as (a) Rg /ξ = 2.0 and (b) Rg /ξ = 5.0, respectively. Reprinted with permission from Ref. 29.
(a)
(b)
Fig. 12. A comparison of domain structures of diblock copolymer melt under a shear flow between dynamic SCF simulations with (a) simple diffusion dynamics and with (b) Rouse dynamics is shown. Reprinted with permission from Ref. 31.
SCF theory.31 When they calculated the shear stress under an external flow filed, they approximated the Rouse dynamics of a connected bonds along the chain by a set of independently moving dumbbells. By applying a shear flow on a diblock copolymer melt using both simple diffusion dynamics and Rouse dynamics, they observed that the response of the domains to the external flow is slowed down when they introduce the Rouse dynamics (Figure 12). Since these works, several groups extended dynamic SCF mainly with Rouse dynamics.32,33
05-chapter4
September 29, 2010
10:42
World Scientific Review Volume - 9in x 6in
Dynamics Self-Consistent Field Theories for Polymers
(a)
05-chapter4
127
(b)
Fig. 13. Example of dynamic SCF simulation with reptation dynamics. (a) Two parallel plates with grafted polymer brushes. (b) Time evolution of the stress when a start-up flow is imposed. Inset figures are the conformations of a single chain before (t = 0.0) and after (t = 1.0) the shear deformation is imposed. Reprinted with permission from Ref. 34.
4.2.3. Full dynamic SCF theory with reptation dynamics As the main target of dynamic SCF theory is dense polymer systems, reptation dynamics that accounts for the entanglements between chains is more suitable than the Rouse dynamics discussed in section 4.2.2. Shima et al. introduced the reptation dynamics into dynamic SCF theory.34 Similar to the model by Maurits et al., they also used the Smoluchowski equation for the distribution function of set of bond vectors along the chain. The time evolution of this distribution function of set of bond vectors are modeled using the reptation dynamics, which gives correct constitutive equation for dense polymer melts.5,9 As an example, they applied this dynamic SCF technique to a system of two parallel solid plates onto which polymer brushes are grafted as is shown in Fig. 13(a). Figure 13(b) shows the time evolution of the stress when a steady shear deformation is suddenly imposed on the system at t = 0.0 (so-called “start-up deformation”). This simulation data clearly shows that this dynamic SCF technique with reptation dynamics can successfully reproduce the stress overshoot and the relaxation to final steady state. The inset figures show the conformation change of individual polymer chain, which cannot be realized by dynamic SCF technique with simple diffusion dynamics. 4.3. Hybrid simulations with particles and fields One of the difficulties of SCF theory is its use of the coarse-grained Gaussian chain model. As this model is based on a very simple picture of bead-spring
September 29, 2010
128
10:42
World Scientific Review Volume - 9in x 6in
T. Kawakatsu
model, where spherical beads are connected by harmonic bonds,5 most of the microscopic chemical details are neglected except for a few molecular features, such as chain architecture (branched molecular structure or block structure), interaction parameters (χ-parameters), and local composition of the components. The same is true for dynamic properties. Except for the viscoelastic simulations discussed in section 4.2, microscopic chain dynamics is always neglected in standard dynamic SCF theories. One of the strategies to go beyond these limitations is a hybrid approach where microscopic molecular model is combined with the coarse-grained field approach.27,28,49–51 In these references, relatively small molecules, such as surfactant molecules or filler particles, compared to the phase separated domains are modeled using discrete molecules, while the phase separating binary mixtures or block copolymers are modeled using field variables. However, for the present purpose of introducing chemical details or microscopic dynamics of polymer chains into the field model, it is essential to use molecular model of polymer chains. Saphiannikova et al. combined bead-spring chains with self-consistent potential to simulate Rouse dynamics of polymer brushes.35 Here, the beadspring chains are used as a method to calculate non-equilibrium chain conformations, which replaces the path integrals in dynamic SCF theory. They succeeded in simulating sheared brushes in non-entangled Rouse regime. Although this hybrid simulation is proposed to simulate viscoelastic behavior, it can also be used to introduce chemical details into the coarse-grained SCF simulations. Recently, Milano and Kawakatsu introduced a similar technique to account for the chemical details of the constituent polymer chains into the SCF model.36 They combined discrete polymer chains with self-consistent potential field. The polymer chain is modeled by a united-atom model,5 where not only the bond potential between adjacent monomers along the chain but also bending and torsion potentials of the backbone of the chain are included. In this hybrid approach, the non-bonded monomer-monomer interactions such as the Lenard-Jones potential, are replaced by the selfconsistent field as is done in the standard SCF simulation. For the polymer chains, they used MD simulations, while the self-consistent potential is solved on a spatial grid. Using this hybrid particle-filed MD approach, one can treat the microscopic detail of the polymers while keeping the computational cost considerably smaller than the full MD simulations. In Fig. 14, we show a result of this hybrid particle-field MD approach on lamellar-forming block copolymer melt.36 Although this simulation uses
05-chapter4
September 29, 2010
10:42
World Scientific Review Volume - 9in x 6in
Dynamics Self-Consistent Field Theories for Polymers
05-chapter4
129
me Fig. 14. Time evolution of the self-consistent potential of a lamellar-forming diblock copolymer melt simulated with a hybrid particle-field MD approach. Reprinted with permission from Ref. 36.
a simple bead-spring model of block copolymer chains, the method can also be applied to complex molecules, such as lipid molecules that form bilayer membranes. 5. Conclusion As is well-known, the static SCF theory is a powerful tool to predict equilibrium domain structures and phase diagrams of polymeric systems. With this technique, one can predict the domain structures and the phase diagram of an unknown system based on the molecular architecture and some other physical parameters of the system. Despite such a success of static SCF theories, its application to dynamic problems were rather limited. In this chapter, we have reviewed recent developments of dynamic SCF theories starting from early works on diffusion dynamics to recent developments on hydrodynamic and viscoelastic polymeric flows. Both of these dynamic SCF theories are still under development, and we expect a rapid and extensive growth of the research field. Acknowledgments The author thanks A.-C. Shi, W. Hu and D. Andelman for their encouragements and help in preparing this article. He also thanks T. Honda, A.V.Zvelindovsky, D.Q. Ly, H. Morita, X.-F. Yuan, M. Doi, G. Milano,
September 29, 2010
10:42
130
World Scientific Review Volume - 9in x 6in
T. Kawakatsu
T. Shima, H. Kuni, Y. Okabe, T. Hashimoto, M. Takenaka, D. Yamaguchi, K. Furuichi, and C. Nonomura for collaborations on dynamic self-consistent field theory. Many of the studies cited in this article were performed using SUSHI code in OCTA software system, which was produced under the national project on nano-structured polymeric materials, which has been entrusted to the Japan Chemical Innovation Institute by the New Energy and Industrial Technology Development Organization (NEDO) under METI’s Program for the Scientific Technology Development for Industries that Creates New Industries. Discussions and collaborations at the project were very useful. This study was also supported by Grant-in-Aid for Scientific Research on Priority Area “Soft Matter Physics” from the Ministry of Education, Culture, Sports, Science, and Technology of Japan. References 1. The chapter by A.-C. Shi in this book. 2. G. Gompper and M. Schick, eds., Soft Matter, Volume 1, (Wiley-VCH, Weinheim, 2006). 3. A. V. Zvelindovsky, ed., Nanostructured Soft Matter, (Springer-Verlag, Berlin, 2007). 4. J. Fleer, M. A. C. Stuart, J. M. H. M. Scheutjens, and B. V. T. Cosgrove, Polymers at Interfaces, (Chapman and Hall, London, 1993). 5. T. Kawakatsu, Statistical Physics of Polymers — An Introduction, (SpringerVerlag, Berlin, 2004). 6. G. H. Fredrickson, The Equilibrium Theory of Inhomogeneous Polymers, (Oxford University Press, Oxford, 2006). 7. M. W. Matsen, in Ref 2, pp. 87-178. 8. M. M¨ uller, in Ref 2, pp. 179-281. 9. M. Doi and S. F. Edwards, The Theory of Polymer Dynamics, (Oxford University Press, Oxford, 1986). 10. J. G. E. M. Fraaije, J. Chem. Phys., 99, 9202(1993). 11. J. G. E. M. Fraaije, B. A. C. van Vlimmeren, N. M. Maurits, M. Postma, O. A. Evers, C. Hoffmann, P. Altevogt, and G. Goldbeck-Wood, J. Chem. Phys., 106, 4260(1997). 12. B. A. C. van Vlimmeren, N. M. Maurits, A. V. Zvelindovsky, G. J. A. Sevink, and J. G. E. M. Fraaije, Macromolecules, 32, 646(1999). 13. H. Morita, T. Kawakatsu, M. Doi, D. Yamaguchi, M. Takenaka, and T. Hashimoto, Macromolecules, 35, 7473(2002). 14. R. Hasegawa and M. Doi, Macromolecules, 30, 3086(1997). 15. A. Knoll, A. Horvat, K. S. Lyakhova, G. Krausch, G. J. A. Sevink, A. V. Zvelindovsky, and R. Magerle, Phys. Rev. Lett., 89, 035501(2002). 16. H. Morita, T. Kawakatsu, and M. Doi, Macromolecules, 34, 8777(2001). 17. T. Honda and T. Kawakatsu, Macromolecules, 39, 2340(2006).
05-chapter4
September 29, 2010
10:42
World Scientific Review Volume - 9in x 6in
Dynamics Self-Consistent Field Theories for Polymers
05-chapter4
131
18. X. Li, P. Tang, H. Zhang, F. Qiu, and Y. Yang, J. Chem. Phys., 128, 114901(2008). 19. D. Q. Ly, T. Honda, T. Kawakatsu, and A. V. Zvelindovsky, Macromolecules, 40, 2928(2007). 20. D. Q. Ly, T. Honda, T. Kawakatsu, and A. V. Zvelindovsky, Macromolecules, 41, 4501(2008). 21. D. Q. Ly, T. Honda, T. Kawakatsu, and A. V. Zvelindovsky, Soft Matter, 5, 4814(2009). 22. K. S. Lyakhova, A. V. Zvelindovsky, and G. J. A. Sevink, Macromolecules, 39, 3024(2006). 23. N. M. Maurits, A. V. Zvelindovsky, G. J. A. Sevink, B. A. C. van Vlimmeren, and J. G. E. M. Fraaije, J. Chem. Phys., 108, 9150(1998). 24. T. Honda and T. Kawakatsu, J. Chem. Phys., 129, 114904(2008). 25. D. M. Hall, T. Lookman, G. H. Fredrickson, and S. Banerjee, Phys. Rev. Lett., 97, 114501(2006). 26. D. M. Hall, T. Lookman, G. H. Fredrickson, and S. Banerjee, J. Compt. Phys, 224, 681(2007). 27. D. M. Hall, T. Lookman, and S. Babnerjee, Chem. Eng. Sci., 64, 4754 (2009). 28. R. B. Thompson, V. V. Ginzburg, M. W. Matsen, and A. C. Balazs, Macromolecules, 35, 1060(2002). 29. T. Kawakatsu, Phys. Rev. E, 56, 3240(1997); Phys. Rev. E, 57, 6214 (1998). 30. N. M. Maurits and J. G. E. M. Fraaije, J. Chem. Phys., 107, 5879(1997). 31. N. M. Maurits, A. V. Zvelindovsky, and J. G. E. M. Fraaije, J. Chem. Phys., 109, 11032(1998). 32. B. Narayanan, V. A. Pryamitsyn, and V. Ganesan, Macromolecules, 37, 10180(2004). 33. T. S. Lo, M. Mihajlovic, Y. Shnidman, W. Li, and D. Gersappe, Phys. Rev. E, 72, 040801(2005). 34. T. Shima, H. Kuni, Y. Okabe, M. Doi, X.-F. Yuan, and T. Kawakatsu, Macromolecules, 36, 9199(2003). 35. M. G. Saphiannikova, V. A. Pryamitsyn, and T. Cosgrove, Macromolecule, 31, 6662(1998). 36. G. Milano and T. Kawakatsu, J. Chem. Phys., 130, 214106(2009). 37. T. Honda and T. Kawakatsu, in Ref 3, pp. 461-493. 38. T. Xu, Y. Zhu, S. P. Gido, and T. P. Russell, Macromolecules, 37, 2625(2004). 39. T. Xu, A. V. Zvelindovsky, G. J. A. Sevink, K. S. Lyakhova, H. Jinnai, and T. P. Russell, Macromolecules, 38, 10788(2005). 40. K. Furuichi, C. Nonomura, T. Kawakatsu, and M. Doi, J. Chem. Phys., 117, 9959(2002). 41. A. Onuki, Phase Transition Dynamics, (Cambridge University Press, Cambridge, 2002). 42. M. Bahiana and Y. Oono, Phys. Rev. A, 41, 6763(1990). 43. T. Ohta and K. Kawasaki, Macromolecules, 19, 2621(1986). 44. D. Yamaguchi and T. Hashimoto, Macromolecules, 34, 6495(2001).
September 29, 2010
132
10:42
World Scientific Review Volume - 9in x 6in
T. Kawakatsu
45. E. D. Siggia, Phys. Rev. A, 20, 595(1979). 46. P. Guenoun, R. Gastaud, F. Perrot, and D. Beysens, Phys. Rev. A, 36, 4876(1986). 47. T. Koga and K. Kawasaki, Physica A, 196, 389(1993). 48. A. Shinozaki and Y. Oono, Phys. Rev. E, 48, 2622(1993). 49. V. V. Ginzburg, F. Qiu, M. Paniconi, G. Peng, D. Jasnow, and A. C. Balazs, Phys. Rev. Lett., 82, 4026(1999). 50. T. Kawakatsu, K. Kawasaki, M. Furusaka, H. Okabayashi, and T. Kanaya, J. Chem. Phys., 99, 8200(1993). 51. T. Kawakatsu and K. Kawasaki, Physica A, 167, 690(1990). 52. K. Kawasaki and K. Sekimoto, Physica A, 143, 361(1988).
05-chapter4
Chapter 5 Molecular Dynamics in Crystallization of Helical Polymers: Crystal Ordering and Chirality Selection
Takashi Yamamoto Department of Physics and Informatics, Yamaguchi University, Yamaguchi 753-8512, Japan E-mail:
[email protected]
Simulating crystallization in helical polymers is a big challenge. We here report on our recent efforts toward better understanding of crystallization in helical polymers. We adopt a combination of molecular dynamics and Monte Carlo (MC) methods. We first categorize helical polymers into two types: simple bare helix and general helix. With a slight modification of the united atom model for polyethylene we build our bare helix, while realistic model of isotacticpolypropylene (iPP) is taken as a representative of the general helix. We find that a single isolated chain of the bare helix shows rather fast ordering into chain-folded conformations. Though our bare helix is confirmed to have a chiral low temperature phase, chirality selection during crystallization is not found to be pronounced. On the other hand, crystallization in iPP is extremely sluggish. By restricting the spatial dimension of the system, however, we can observe iPP molecules crystallize rather quickly. Quite remarkable is that iPP molecules show very clear recognition of chain chirality during crystallization. By use of off-lattice MC method, we also study formation of crystal polymorphs in iPP, where we find that the crystal chirality is closely related with the symmetry of the lattice.
133
134
T. Yamamoto
1. Introduction Understanding molecular mechanisms of crystallization is of great importance in science and technologies of polymers, and enormous efforts have been made since the early days of polymer science.1-3 However, it is still an intriguing subject to understand and control crystallization for the design of favorable microscopic and macroscopic properties.4-6 It is a great academic issue how polymers with huge internal degrees of freedom perform a remarkable feat of crystallization into neat chain folded structures.7,8 The molecular process of polymer crystallization has also a great relevance to that of folding in proteins which is now a central subject in molecular biology. A consistent molecular scenario of polymer crystallization was first proposed by Lauritzen and Hoffman, who succeeded in giving very simple explanations for various characteristics of polymer crystallization.9 In the Lauritzen-Hoffman (LH) theory, the polymer molecules were modeled as structureless strings and the specific molecular characteristics were renormalized into a small number of parameters such as surface free energies, enthalpy of melting, etc. The great success of the theory strongly suggested the universality in crystallization of polymers having different chemical structures. Recent experimental efforts as well as vast knowledge accumulated so far have revealed many aspects of polymer crystallization that cannot be readily comprehended by the LH-theory.6,7 In the LH-theory, elementary molecular processes were greatly simplified to make analytical treatments tractable without definite experimental evidence, in many cases. Various molecular level scenarios more detailed have hitherto been put forward,7,10-14 but they have yielded macroscopic predictions similar to those derived from the LH-theory. The difficulty in directly observing crystallizing molecules has long been a greatest obstacle in clarifying detailed molecular processes in polymer crystallization. Molecular simulations of polymer crystallization are recently emerging as unique and promising tools to break through the difficulty. In fact, many pioneering studies have been reported on the primary nucleation,15-22 the steady-state growth of chain-folded lamellae,23-33 and the crystallization under flow or large deformation,34-38
Molecular Dynamics in Crystallization of Helical Polymers
135
lamellae,23-33 and the crystallization under flow or large deformation,34-38 though most of these are treating simple polyethylene-like polymers. The polyethylene-like polymers are appropriate for molecular simulation studies, since they show rather fast ordering. We hereafter
Fig. 1. Model step-up from (a) the linear polymer, (b) planar-zigzag polymer, the usual united atom model of polyethylene, (c) simplified helical polymer having approximate 4/1 helix, and (d) realistic united atom model of iPP having pendant methyl units.
call the polyethylene-like polymers linear polymers against helical polymers which are the focus of this review (Fig. 1). The helical structures are quite common in synthetic as well as natural polymers: slightly twisted polytetrafluoroethylene (PTFE), approximate 9/5 helix of polyoxymethylene (POM), regular 3/1 helix of isotacticpolypropylene (iPP) (Fig. 1), not to mention the double helix of DNA. The molecular process of crystallization in helical polymers necessitates sophisticated intramolecular and intermolecular cooperation and will be much more complex than that in linear polymers. Crystallization in helical polymers has another interesting aspect, the molecular chirality recognition during crystallization.39-41 Let us consider a helical polymer, such as iPP, which equally takes both the right-handed (R-handed) and the left-handed (L-handed) helices. It is evident that the crystallizing molecules select their chirality according to the structural symmetry of the growing crystal, a sort of molecular recognition. However, basic molecular mechanisms of structure formation with strict
136
T. Yamamoto
chiral selection have not received sufficient attention. This will be again due to experimental difficulties in observing detailed molecular process of crystallization. In this paper, we will review our recent computer simulation studies on the crystallization of helical polymers with or without chirality selection. 2. Our Strategies for Simulating Crystallization in Helical Polymers Crystallization in helical polymers involves complicated chain dynamics: the intramolecular ordering into specific helical conformations and the interchain ordering to meet required crystal symmetry. Characteristic time-scales of the crystallization are expected much longer than those of simple linear polymers. Though several attempts to simulate crystallization in realistic or coarse-grained models have been reported, they could not reproduce well-ordered crystalline state.42-48 Even a single molecule ordering in vacuo was found to be very sluggish in comparison with simple polyethylene-like polymers. It appears that crystallization in helical polymers is too slow to be investigated by straightforward molecular simulations. In order to cope with the difficulty, we took a stepwise approach.45,49,50 Let us remember various types of helical polymers, from those of very simple architectures to more complex ones. Of various helical polymers, we readily notice two types. One is the simple “bare” helix such as PTFE or POM, which is essentially made of backbone groups only and has smooth molecular contour. Their helical conformations are well-determined by the basic intramolecular potentials such as the torsion angle potential. The other is the general helix having large side groups such as iPP or polypeptides. Their helical conformations are also sensitive to the detailed interactions among the side groups and the backbone atoms. The molecules have rugged molecular contours and their favorable ways of chain packing in the crystals are much more sophisticated. In the former type bare helices, crystallization will be faster since the geometrical requirements (or the entropic barriers) in placing these nearly cylindrical chains in crystal register are much smaller. For example, PTFE will readily crystallize at relatively low undercooling.
Molecular Dynamics in Crystallization of Helical Polymers
137
Though the incipient crystals are known to be disordered with respect to the dispositions of the R- and the L-handed helices, they quickly transform to the ordered crystalline form through solid-solid transitions around room temperature. The tolerance in the R-L selection during crystallization as well as larger molecular mobility is expected to facilitate crystallization. On the contrary, considerable geometrical constraints are imposed on helical polymers with large side groups. Crystallization would be seriously slowed down by the presence of larger internal degrees of freedom that must be properly adjusted during crystallization. We here discuss crystallization in these two types of helical polymers separately. Since the former type bare helices crystallize faster, we can treat larger system in three dimensional (3D) space. It will be shown that our model bare helix crystallizes directly from 3D random coil states, but with less clear chirality selection. As a latter type polymer having large side groups we consider iPP. We will see that crystallization of iPP is much more sluggish and the similar approaches as the bare helices are difficult. Therefore we consider a single iPP chain confined within
Fig. 2. A model iPP chain confined within a slit and within a tube. The width of the slit is given as Ly = 0.656 nm, while those of the tube are Ly = 0.656 nm and Lz = 1.124 nm.
138
T. Yamamoto
2D slit and 1D tube (Fig. 2). We focus attention to the adsorption onto a chiral crystalline substrate and the crystallization with definite chirality selection. In addition to the molecular dynamics (MD) method, we also utilize Monte Carlo (MC) simulations of the packing of rigid helices in studying crystal structures, solid-solid phase transitions, and crystallization. 3. Molecular Models and Simulation Methods 3.1 MD simulations49,50 The bare helical polymer and oligomer we consider have POM-like conformation of hypothetical 4/1 helix of polyethylene-like backbone51 except its torsion angle preference to gauche; the torsion potential has deep minima at gauche positions and therefore the chains take approximate 4/1-helical conformation. The molecule has a bond stretching potential 1 U b (r) = k b (r − r0 ) 2 2
(1)
where kb and r0 are the force constant and the average C–C bond length, respectively. A potential for the C–C–C bond angle is given by
θ
θ
1 U angle (θ ) = k θ (cos θ − cos θ 0 ) 2 2
(2)
where kθ and 0 are the bending stiffness and the equilibrium bond angle. We consider a torsion angle potential that has deep minima at the gauche positions, 4
U torsion (τ ) = k τ ∑ a n cos n τ
(3)
n =0
The deep gauche minima and a high trans barrier are necessary to make the 4/1-helical conformation stable. There is still some freedom at hand in choosing the torsion barriers at cis position. Though higher cis barrier makes the helix reversals more sluggish and the crystallization much slower, we here assume the high cis barrier, except when considering oligomeric system, by referring to the literature data for
Molecular Dynamics in Crystallization of Helical Polymers
139
Table 1. Parameter values used for the bare 4/1-helix.
Parameters
Values
mass (CH2)
1.4Χ10-2 kg mol-1
r0 kb θ0
0.152 nm 3.46107 J nm-2 mol-1 70.5°
kθ
5.0Χ105 J mol-1
kτ
1.18Χ104 J mol-1
a0 a1 a2 a3 a4 ε σ cutoff
1.0 2.63657 -1.07025 -3.6976 1.9801 500 J mol-1 0.38 nm 2.50σ
POM.52 We assume usual Lennard-Jones potential, between atoms more than three bonds apart along the chain and between atoms of different chains. For a reason described later, we consider two parameter values for , 60 K (0.5 kcal/mol) for the polymer and 120 K (1.0 kcal/mol) for the oligomer of the bare helix. The values of the parameters used in this work are given in Table 1. The bare helix, though it has the torsion energy minima at the gauche positions, actually takes slightly unwound 18/5 helical conformation, which is due to van der Waals repulsion between 1-5 carbon atoms along the chain. We take iPP as a representative helical polymer with large side groups. The intermolecular and intramolecular potentials used are the “transferrable potentials for phase equilibria” (TraPPE) given by Martin and Siepmann.53 In order to keep the isotactic structure of the polymer,
ε
140
T. Yamamoto
we imposed the phantom bond-length constraints with RATTLE algorithm for the atom pairs between the nearest CH3 and CH2, and between CH2 and CH2, the lengths of which were those of the ground state 3/1 helix of iPP chain. The confinement of the iPP chain within the 2D slit and the 1D tube were made by placing repulsive walls of the following potential form, 1 σ wall 10 1 σ wall 4 6 4ε − + ε wall d < σ wall Uwall(d) = wall 5 d 2 d 5 0 d > σwall
(4)
where d is the distance of each atom from the wall; soft repulsion The wall = 0.468 nm and wall = 0.0198 kcal/mol is here assumed. widths of the slit and the tube were made very thin of the order of the lattice constants of iPP crystals (Fig. 2). All MD simulations reported in this work are performed using an MD program, COGNAC, in recently developed simulation platform for soft materials, OCTA.54 Most of the MD simulations are carried out under constant NVT ensemble with periodic boundary conditions, where the temperature is controlled with the conventional Nose’-Hoover method. Systems of many oligomers of the bare helix, on the contrary, are studied under constant NPT ensemble also with periodic boundary conditions by Andersen-Nose-Hoover algorithm.
σ
ε
3.2 MC simulation50 When we study crystal structure or crystallization of polymers, it is often useful to disintegrate the polymer chain into a collection of rigid stems having crystalline conformation. In the LH-theory, for example, stepwise dispositions of the stems on the growth surface are considered as an elementary molecular process of polymer crystallization, where internal degrees of freedom of the chain are not explicitly taken into consideration. In many cases, the crystal structure of polymer is determined as the lowest free energy packing mode of the stems of given conformation. Then the problem is essentially equivalent to that of searching for the 2D magnetic spin configurations, except that the
Molecular Dynamics in Crystallization of Helical Polymers
141
Fig. 3. Comparison of (a) a spin system and (b) a crystal system of chain molecules.
polymer chains in the crystal have much larger number of possible states: chain rotation, chain translation, helical sense, etc (Fig. 3). We here consider 2D lattice and off-lattice models to investigate crystal-crystal transitions and crystallization. In case of lattice simulations, where the chains are placed on a lattice of given symmetry and lattice constants, all the possible interchain energies can be calculated and stored in a large interaction table, and every MC move are efficiently performed by only looking up the table.55,56 In case of off-lattice simulations, on the contrary, the relative positions of the chain stems are variable and the interchain energy cannot be practically stored in moderate sized table. Therefore complicated interchain energies must be calculated at every MC move making the simulation very time-consuming. The rigid conformation of the POM-like bare helix is constructed by use of equilibrium bond-length and the bond-angle adopted in our MD simulation, and the dihedral angles are assumed 105° to give approximate 18/5 helix. Following three kinds of freedom are allotted every 1° interval to each rigid chain on the lattice: 360 orientations around the chain axis ( = 1°), 10 translations Z along the chain axis ( z = 0.00986 nm), and the two helical senses (either R- or L-hand).
τ
Δ
Δθ
θ
142
T. Yamamoto
The interchain interaction energy is calculated from van der Waals energy eq. (4) between all atom-atom pairs of different chains. The conformation of rigid iPP is modeled as the perfect 3/1 helix of propylene monomers: all the C-C bond length and the C-C-C bond angle are 0.152 nm and 113°, respectively. Crystallization of the rigid iPP by use of the off-lattice model is treated by taking into account translational motions of the chains perpendicular to their axes, in addition to the four kinds of lattice moves, chain rotation, z-translation, and inversions of helical sense and direction of the methyl groups. The necessary parameters in calculating the interchain energies are obtained from Ref. 53. In the present MC simulations, we always considered infinitely long crystal stems to eliminate the stem end effects. Our interchain energies are thus exactly linear functions of the stem length.
4. Crystallization of the Bare Helix 4.1. A primary nucleation of a single polymer in vacuo49 If a helical polymer is flexible enough and helix reversal defects can sweep fast along the chain, the polymer chain would readily collapse and form chain-folded conformations. We first survey the conformational changes of the bare helix of 500 atom length by a rapid stepwise cooling of 60 K/0.8 ns. Figure 4 shows that the molecule indeed quickly collapsed and formed a compact globule. It is readily noticed that the degree of conformational order, the number helical segments and their lengths, grows markedly with decreasing temperature. However, the chain does not form well-ordered crystalline conformation but leaves many helix reversal defects even at the lowest temperature simulated. The chain is considered to be quenched into a metastable state of partial order (Fig. 4(d)). Shown in Fig. 4(e) is a degree of conformational fluctuation, the root mean square displacement of atoms in a short period of 8 ps, vs. temperature. By lowering the temperature, the conformational fluctuation decreases monotonically until a break around Tf @ 300 K below which the degree of fluctuation points toward the origin at 0 K.
Molecular Dynamics in Crystallization of Helical Polymers
143
Fig. 4. Snapshots (a)–(d) show the trajectory of the bare 4/1 helix of 500 atoms during stepwise cooling (60 K/0.8 ns) from 840 K down to 120 K. The helical segments, the sequences of gauche+ or gauche- bonds, which are longer than six bonds are depicted by cylinders; right- and left-handed ones in different colors. Each picture is the last snapshot at temperature, (a) 540 K, (b) 420 K, (c) 300 K, and (d) 180 K. Also shown in (e) is the average displacement of constituent atoms within 8 ps, which represents the degree of dynamic motions of atoms. An apparent freezing or glass transition is observed around Tf = 300 K.
During the stepwise cooling, the radius of gyration constantly shrinks (Fig. 5(a)) and the nonbonded energy decreases (Fig. 5(b)) until they come to show slight changes around Tf in their slopes and fluctuations. Both the global order parameter of bond-orientation Gbare (Fig. 5(c)) and the average helix length < l > (Fig. 5(d)) sharply increase around Tf . The global order parameter for the bare helix Gbare is here defined as
(
G bare = 3cos 2 θ ij − 1)/2
)
(5)
where θij is the angle between the vector connecting the i-th and the (i+4)-th atoms and that connecting the j-th and the ( j+4)-th atoms, where the average is taken over all possible pairs of vectors, and the average helix length < l > is the lengths of the successive gauche+ or gauche− bonds, where the gauche+ and the gauche− are here defined
144
T. Yamamoto
loosely as torsion angles between 60° < τ < 180° and −180° < τ < −60°, respectively. A little bit strange is that < l >, an intramolecular order parameter, grows earlier than Gbare. How do the intermolecular order and intramolecular order cooperate during crystallization?
Fig. 5. Changes in (a) the radius of gyration, (b) the nonbonded energy, (c) the global order parameter, and (c) the average helical length during the stepwise cooling 60 K/ 0.8 ns of the bare 4/1 helix of 500 atoms. In the abscissas in Figs. 4(a), both the time scale (lower) and the corresponding approximate temperature scale (above) are given; note that the temperature is changed stepwise and does not precisely correspond to the time scale. All the quantities, except the average helix length discussed later, show appreciable changes around 300 K (or 7.5 ns).
In order to elaborate on this point, we adopted much slower cooling and heating scheme 50 K/2 ns for twice as long chain C1000, and monitored the global order parameter and the average helix lengths (Fig. 6).
Molecular Dynamics in Crystallization of Helical Polymers
145
During cooling, < l > started growing around Ts @ 400 K while Gbare slightly retarded in increasing until around 350K, but both growth ceased around Te @ 250 K; the final degree of order is still limited to Gbare ≅ 0.1 and < l > ≅ 5. During the subsequent heating on the other hand, < l > started decreasing around 350 K while Gbare showed sudden drop between 350 K and 400 K after showing large fluctuations. In the development and the loss of crystalline order during cooling and heating, respectively, the appearance of the global order slightly lags behind that of the growth of the intramolecular order. These may be either due to a very small crystallite size of the chain or due to very imperfect ordering attained within the time scale considered. Between Ts and Te, both Gbare and < l > show thermal hysteresis; larger hysteresis in the former will come from its long-range order nature, while smaller hysteresis in < l > will be due to its short-range nature which enable quicker response to the temperature change.
Fig. 6. Changes in (a) the global order parameter and (b) the average helix length, in the longer bare 4/1 helix of 1000 atoms during much slower (50 K/2 ns) stepwise cooling (solid line) and heating (dash). Two circles indicate the approximate temperature ranges where the ordering starts and ends, respectively.
Rapid cooling made the chain freeze into a rather poorly ordered state. Crystallization rate of the bare helix, though it will be much faster than
146
T. Yamamoto 0.5
(a)
(b) ave rage helix length (bond)
0.3
0.2
0.1
0 0 –2500
100
non–bonded energy (reduced)
t ime (ns )
(c)
100
200
3500
3000
–3500 2500
100
5
time (ns)
–3000
–4000 0
10
0
200
torsion energy (reduced)
global order parameter
0.4
(d) gg-
g-
t
g+ g+ g+
200
time (ns)
Fig. 7. Changes with time in (a) the global order parameter, (b) the average helical length
(solid) and (dotted), and (c) the nonbonded (dash) and the torsion (solid line) energies, during long-time annealing at 300 K. A snapshot (d) obtained after 200 ns of annealing indicates that the crystalline stems are often mixture of the R- and the L-handed helical segments with characteristic helix reversal defects made of an intervening trans bond.
iPP, must be slow due to the presence of helix reversal defects. Annealing or isothermal crystallization for much longer times is expected, and indeed we found, to give much higher ordered state. Figure 7 shows the late stage of development, by annealing up to 240 ns, seen in Gbare and the average helix lengths and < lL> for the R-handed and the L-handed helices, together with the torsion and nonbonding energy changes. Very conspicuous is a sharp increase in Gbare leading to a much larger value of about 0.35 (Fig. 7(a)), while both and
Molecular Dynamics in Crystallization of Helical Polymers
147
increase gradually (Fig. 7(b)). It is clear that the global order showed steep increase within 30 ns but with many helix reversal defects left within the crystallite. Figure 7(d) shows a typical snapshot of the chain conformation, where the ordered helical segments are drawn with cylinders. Development of parallel helical segments is seen but with many helix reversal defects as well as disordered fold conformations; removal of the R-L defect is seen to be much slower (Fig. 7(b)). Such characteristic ordering by the annealing is also evident in the intermolecular and intramolecular energies (Fig. 7(c)); while the nonbonding energy decreases only slightly indicating small alteration in the inter-atomic contacts, the intramolecular torsion energy decreases slowly but markedly within 100ns showing considerable decrease in the helix reversal defects. Crystallization of a single chain in free space, a simple model of primary nucleation, was thus found to be very slow in the helical polymer. Probably due to its small size, the intermolecular and the intramolecular order showed slightly different rate processes. The global ordering slightly retarded in the early stage but it attained higher order at the late stage, while the intramolecular ordering is faster in the early stage but gets slower in the late stage due to the sluggishness in removing the helix reversal defects from the crystallites. Unexpected is that the final chain folded crystallite of the helical polymer did not show marked chirality but consisted of a mixture of the R-handed and L-handed stems. Crystallization at lower temperatures is expected to yield chiral crystallites, but the rate of crystallization would be much slower.
4.2 Crystallization of a single polymer on a growth front49 When we study crystal growth, we must treat many chain systems. In our former studies of crystallization in simple polyethylene-like polymers, we first studied a single chain in vacuo crystallizing onto growth surface, and then we proceeded to 3D crystal growth from the melt of hundreds of chains. Here again, we begin with studying a single bare helix placed on the growth front. We first build a crystal
148
T. Yamamoto
substrate of the same helical oligomers C100 all R-handed. The substrate comprises two layers, ten chains in each layer, which were equilibrated to have hexagonal close-packed structure (Fig. 8). Upon the substrate, we place a single bare helix C500 and switch on the interactions between the substrate and the polymer; we assumed a periodic boundary in the x-y directions (Fig. 8). Crystallization of the chain during fast stepwise cooling of 60 K/0.4 ns was then monitored, where all the substrate atoms were fixed. At high temperature of 800 K, the chain is a random coil weakly adsorbed to the substrate. With decreasing temperature the polymer comes to attach to and spread on the substrate. The polymer is nearly fully adsorbed around 540 K, but the appreciable ordering sets in a little later around 500 K (Fig. 8).
y
x Fig. 8. Adsorption and ordering of the bare 4/1 helix of 500 atoms onto the crystal substrate made of the same chains single sensed; the figures are snapshots at 540 K, 480 K, and 420 K, from the left. Given above are side views along the chain axis direction of the substrate, and below are top views of the growth surface; black and gray indicate the opposite-handed helical segments. Despite rather rapid cooling 60 K/0.4 ns, fast adsorption onto the substrate followed by quick alignment parallel to the substrate chains was observed. The chiral order in the adsorbed helical stems was not attained even after extremely long annealing at various temperatures.
Molecular Dynamics in Crystallization of Helical Polymers
149
Fig. 9. (a) The degree of adsorption, defined as the number of atoms within the first adsorption layer, and (b) the global order (solid line) and the average helix length (dash) vs. time (and temperature) during stepwise cooling 60 K/ 0.4 ns. A very sharp onset of ordering takes place around 3 ns (400 K) with both order parameters showing quite cooperative increases.
The substrate attraction is generally very strong in vacuo. The polymer chain is firmly arrested by the substrate, and most of the polymer atoms are within the first adsorption layer when the crystallization sets in. In Fig. 9 are shown (a) the degree of chain adsorption, the fraction of atoms forming the first adsorption layer, and (b) the global order parameter Gbare and the averaged helix length < l >. In contrast to the previous case of primary nucleation in vacuo, the intermolecular order Gbare and the intramolecular order < l > show very sharp and cooperative increases around 400 K. The highly cooperative behavior supports the conventional picture of polymer crystal growth, where no specific intramolecular ordering precedes crystal growth. Furthermore, crystallization on the substrate is very fast with Gbare attaining a value of 0.5 within only 4 ns. However, the chain did not form a single cluster. It is suggested that the interchain cooperation among the crystalline stems and the substrate chains is not efficient enough to give a single chiral crystallite.
150
T. Yamamoto
4.3 Order-disorder transition and crystal chirality50
Fig. 10. (a) Two dimensional on-lattice model for the MC simulation, where the black and white circles represent the R-handed and the L-handed chains, respectively. Two types of the boundary conditions are adopted: the periodic boundary condition and the free boundary condition. (b) The torsion angle potential used for the present 18/5-helix oligomer. The helix parameters that describe the lowest-energy conformation of the chain are the same as before: turn/atom φ = 100°, z translation/atom z = 0.0987 nm, helix radius R = 0.0756 nm, and the corresponding internal parameters are C–C bond length r0 = 0.152 nm, C–C–C bond angle q0 = 109.5°, and torsion angle measure from the trans state t = 105°.
Δ
Δ
The absence of marked chirality selection during crystallization poses a question whether the present bare helix has a chiral crystalline phase or not. We therefore examined the lowest free energy structure of the crystal for the bare helix by an on-lattice MC simulation for the rigid chains. We arranged 1500 chains, 30×50, into the two dimensional hexagonal form, the lattice constant of which was estimated from the MD simulation. Since crystal surfaces are the forefront of crystal growth, we are also interested in the structure of the crystal surfaces. We therefore use a lattice system that is bounded by the periodic boundary in one direction and the free boundary in the other direction
Molecular Dynamics in Crystallization of Helical Polymers
151
(Fig. 10(a)). First we prepare a crystal fully disordered with respect to the helical sense and the chain orientation/translation around/along the axis. The crystal is then equilibrated at various temperatures. Figure 11 shows the total equilibrium lattice energy versus reduced temperature T/T0, where T0 = /kB; data for three simulations with different lattice constant are shown. For any choice of the lattice constant, the energy shows a stepwise increase corresponding to the onset of order-disorder transition; the smaller lattice constant gives the stronger interchain interactions resulting in the higher transition temperature. We here focus on the case of a = 0.497 nm which shows good temperature correspondence with later MD simulations. For ease of comparison with later MD simulations, we here consider the rigid chains of 18 atom length C18, one fiber period of the 18/5 helix.
ε
Fig. 11. Total interchain van der Waals energy plotted vs. reduced temperature T/T0, T0 = e/kB. The lattice energies for three independent runs with different hexagonal lattice parameters are plotted.
152
T. Yamamoto
The transition accompanies disordering in the chain chirality as well as in chain rotation and translation, but we here discuss the arrangement of the R- and L-handed helices vs. temperature (Fig. 12). At temperatures below T/T0 @ 4, the crystal makes a single domain of either the R-handed or the L-handed helices only. With increasing temperature, the chiral order of the crystal is gradually degraded until it is completely lost around T/T0 = 4.5. Here we define a parameter Mbulk to describe the long-range-order of the crystal chirality,
M bulk = N R − N L /(N R + N L )
(6)
where NR and NL are the numbers of the R- and the L-handed chains in the crystal. The order parameter, when plotted vs. temperature, shows a rather sharp transition around T/T0 = 4.5 (Fig. 13), and the crystal is evidently chiral below the transition temperature.
●
○
Fig. 12. Changes in the arrangement of the R-handed ( ) and the L-handed ( ) chains with temperature T/T0 at (a) 1.67, (b) 3.33, (c) 4.0, (d) 4.17, (e) 4.33, (f) 4.5, (g) 4.67, (h) 4.83, (i) 5.0, and (j) 6.67.
Molecular Dynamics in Crystallization of Helical Polymers
Fig. 13. energy ( Mbulk ( Msurface (
Changes in the van der Waals
▲), the long-range order parameter ● ), and the order at the surface ○).
153
Fig. 14. Chain configurations during the stepwise heating 12 K/0.4 ns, showing onset of helical disordering and melting; molecules of different helical senses are depicted in different colors (brightness). The temperature T/T0 of each snapshot is (a) 3.7, (b) 3.9, (c) 4.1, (d) 4.3, (e) 4.5, (f) 4.7, (g) 4.9, (h) 5.1, and (i) 5.3.
Each molecule within the crystal has six neighbors, while the molecules at the outermost free surface have only four neighbors. A simple mean field argument for the Ising ferromagnet tells that the order-disorder transition temperature Tc is around zJ /kB, where z is the number of the nearest neighbor spins and J is the strength of the spin-spin coupling. This suggests that the transition occurs at much lower temperatures near the crystal surface where the number of neighboring chains is smaller. By using the order parameter Msurface, which is again defined as eq. (6) but counting the chains on the surface layers only, we can see that the free surface shows disordering at considerably lower temperature than the inner bulk region (Fig. 13). At the growth front or in small crystal nucleus, the chirality selection must be more tolerant and blurs the chirality recognition on the growth surface.
154
T. Yamamoto
The MC prediction about the solid-solid phase transition in the present bare helix was also confirmed by MD simulations for a system of the bare oligomeric chains C20. As will be explained later we here assume a higher van der Waals parameter = 120 K and use a lower cis energy barrier (Fig. 10(b)). A fully ordered crystal which comprises the R-handed chains only is first relaxed at T/T0 = 3.5 for 0.4 ns, and then we monitored structural changes during stepwise heating 12 K/0.4 ns. Figure 14 shows the structural changes with temperature viewed along the chain axis. The crystal is fully ordered well below T/T0 = 4.0, but appreciable number of helix reversals appear around T/T0 = 3.9 until a fully disordered state of equal population of chains of both hands is attained around T/T0 = 4.5. These changes exactly correspond to those predicted by the MC simulation.
ε
Fig. 15. Changes in (a) the number of the R-handed and the L-handed bonds, and the degree of single sense (DSS), and (b) the nonbonded energies, both in the ordered monolayer (lower curve) and in the stacked lamellae (upper curve), during the stepwise heating of 12 K/0.4 ns. As the abscissa is plotted the time (below) and the corresponding average temperature (above).
155
Molecular Dynamics in Crystallization of Helical Polymers
The numbers of the R-handed and the L-handed bonds in the system are plotted vs. temperature in Fig. 15(a). The initial chiral crystal of the R-handed chains transforms to a racemic one within the temperature range of 4.0 < T/T0 < 4.5. The structural change is also reflected in the nonbonding energy (the lower curve in Fig. 15(b)); an appreciable energy jump of about 2.5% at T/T0 = 4.0 agrees well with the corresponding increase of about 3% in our previous MC calculation (Fig. 11). Much larger energy increase around T/T0 = 5.3 in Fig. 15(b) is evidently due to melting. The value of Tm has large uncertainty; higher values of Tm are usually due to superheat originating in the difficulty of forming melt nuclei in perfect crystals. Also shown in Fig. 15(b) is the corresponding energy changes in a system of melt-crystallized lamellae containing many crystal defects; the melting point in this system was around T/T0 = 4.9 but the solid-solid transition was again around T/T0 = 4.0. We also confirmed that the observed solid-solid transition is a reversible phase transition. Conformation of the helix reversal defect and its translational motion along the chain are of considerable interest. A typical series of conformation of a chain with the moving defect are shown in Fig. 16. The helix reversal defect is found to be properly accommodated within the crystal lattice if the opposite handed helical segments are connected by an intervening trans or skew bond. Also found is that the helix reversal defects quickly traverse the full length of the chain. The conformational path of the traveling defect is delineated by the time trajectories of torsion angles of the related bonds (Fig. 16(b)). The bonds far from the defect keep the gauche states showing only small fluctuations around = ±105°, while the bonds near the defect experience large alterations, from one gauche position to the other surmounting the high energy barrier at trans. The defect translation along the chain is mainly achieved by the gauche bonds successively traversing the trans state; however, bonds sometimes go over the cis barrier. The MC and MD simulations have thus confirmed the presence of a chiral crystal phase at low temperatures. In the MC simulation, the only parameter that determines the transition temperature is the interchain energy which is proportional to the van der Waals parameter ; the transition temperature TOD scales linearly with as TOD @ 4 /kB.
τ
ε
ε
ε
156
T. Yamamoto
a
b
c
d
e
L-hand, gauche L-hand, trans
f
g
h
i
R-hand, gauche R-hand, trans
Fig. 16. A typical molecular trajectory of the traveling helix reversal defect observed at T/T0 = 3.9 and times t of (a) 16 ps, (b) 48 ps, (c) 52 ps, (d) 56 ps, (e) 60 ps, (f) 64 ps, (g) 68 ps, (h) 72 ps, and (i) 116 ps. The L-handed and the R-handed helical bonds are drawn in light blue and red, while the trans bonds with positive and negative torsion angles are in blue and yellow.
On the other hand, the melting point Tm of the crystal estimated by the MD simulation was Tm @ 5 /kB ( = 120 K) and Tm @ 6.8 /kB ( = 60 K, not shown here). Usual choice of = 60 K gives Tm @ 400 K and TOD @ 240 K, while larger value of = 120 K gives Tm @ 600 K and TOD @ 480 K. Direct observation by MD simulation of the growth of chiral crystal requires thermal energy that facilitates frequent helix reversals. Therefore in this and next sections, we adopted the higher value of = 120 K to make the chiral phase stable at higher temperatures; also we adopted a lower cis barrier than before in order to accelerate crystallization.
ε ε
ε
ε ε
ε
ε
4.4 Development of chiral crystal50 In above discussions, the presence of chiral low temperature phase below T/T0 ≅ 4 has been confirmed for the present oligomeric bare helix, and
Molecular Dynamics in Crystallization of Helical Polymers
157
isothermal crystallization below this temperature is expected to give chiral crystals. First we prepared an isotropic melt fully relaxed at T/T0 = 6.5. The melt was then quenched down to various temperatures Tc. Figure 17 indicates the non-bonding energy changes, which clearly show the onset of crystallization after various induction periods. The crystal nucleation is a statistical event, but we can conclude from repeated runs that the induction period is longer at higher Tc or smaller undercooling; an extreme case is the crystallization at T/T0 = 4.0 where
Fig. 17. Changes in the nonbonded energy vs. time during the isothermal crystallization at different crystallization temperatures, (a) T/T0 = 3.5, (b) T/T0 = 3.75, (c) T/T0 = 3.95, and (d) T/T0 = 4.0. Remarkably long induction time is noticed at the highest crystallization temperature.
we had to wait more than 40 ns before the onset of crystallization. Final configurations obtained at various Tc are given in Fig. 18. Well-developed parallel lamellae are generated at higher temperatures, while small crystallites of different chain orientations fill up the MD cell at the lowest temperature Tc /T0 = 3.5. At the highest temperature Tc /T0 = 4.0 the system is only partially crystalline.
158
T. Yamamoto
Fig. 18. Typical final configuration of chains attained at different crystallization temperatures: (a) T/T0 = 3.5, (b) T/T0 = 3.75, (c) T/T0 = 3.95, and (d) T/T0 = 4.0. The R-handed and the L-handed chains are depicted in red and light blue.
In order to examine the molecular mechanism of crystallization in helix forming chains, we define the degree of helicity DH as the averaged fraction of the R-handed plus the L-handed bonds in the system,
DH = ( N RB + N LB ) / 17
(7)
all chains
where NRB and NLB are, respectively, the number of the R-handed bonds (60° < < 180°) and the L-handed bonds (−180° < < −60°) out of 17 bonds of a chain. We also use the degree of single sense DSS
τ
τ
DSS = N RB − N LB /(N RB + N LB )
all chains
(8)
Molecular Dynamics in Crystallization of Helical Polymers
159
which takes one when all the chains become perfect single-sensed helix without helix reversal defects. In addition to these, we again consider a global order in oligomer orientations. We introduce the average director vector n for each oligomer by use of the cord vectors bi = ri+4 − ri , connecting every four carbon atoms ri and ri+4, M
n = ∑ bi / M
(9)
i =1
where M is the number of such pairs within the chain. The global order parameter Goligomer is then defined just like eq. (6) from the cosines of the angles qij between the vectors ni , nj of all pairs of oligomers,
(
G oligomer = 3 cos 2 θij − 1) / 2
)
all chain pairs
(10)
We also use a short range order in the oligomer orientation Soligomer defined through a similar equation except that the average is taken over pairs of chains whose centers of mass are within 1 nm,
(
Soligomer = 3 cos 2 θij − 1) / 2
)
local chain pairs
(11)
By using these parameters, we now discuss the characteristics of crystallization at various temperatures (Fig. 19). At lower temperatures, crystallization starts soon after the quench (Figs. 19(a), (b)), while long induction periods are noticed at higher temperatures (Figs. 19(c), (d)). In every case, all order parameters show quite cooperative increases. The large value of Goligomer observed at T/T0 = 3.75 reflects the development of parallel crystalline domains, while the small Goligomer at the largest undercooling T/T0 = 3.5 is due to the polycrystalline structure. Slightly smaller values of Goligomer at Tc /T0 = 3.95 and 4.0 are evidently due to the presence of amorphous chains. Quite conspicuous in Fig. 19 is a large fluctuation in Soligomer during the induction periods at higher temperatures; local ordered clusters of chains are created and annihilated repeatedly in the undercooled melt. When deeply quenched, on the other hand, the local order once created is not readily eliminated due to smaller thermal fluctuation and the Soligomer makes constant increases.
160
T. Yamamoto
●
○
▲
△
Fig. 19. The development of the order parameters, DH ( ), DSS( ), S ( ), G( ), vs. time during the isothermal crystallization: (a) at T/T0 = 3.5, (b) at T/T0 = 3.75, (c) at T/T0 = 3.95, and (d) at T/T0 = 4.0.
It is suggestive that at lower temperature T/T0 = 3.75 the local order parameters DSS and Soligomer show gradual increases before an appreciable increase in the global order Goligomer, while at higher temperatures they do not show any appreciable increases during the long induction period. Structure of the undercooled melt is an intriguing problem and has been discussed in recent simulations of primary nucleation. The primary nucleation is generally very slow in the usual time scale of MD simulation, and we are compelled to study highly undercooled melt where nucleation is much more frequent. Present simulation suggests that we need special precautions when we discuss
Molecular Dynamics in Crystallization of Helical Polymers
161
the structure of undercooled melt. As far as the degree of undercooling is not so large, the melt structure and the crystallization mode indicate simple first order nature without any sign of preordering, and Goligomer and Soligomer as well as other order parameters show highly cooperative behavior. Now look at the emergence and development of the crystalline order. In order to demarcate the crystalline regions, we again use the average chain directors {n} and the center of mass positions {R} of the oligomers. If the two chains have the center of mass positions and the directors that satisfy the following conditions, we consider these two chains are crystalline neighbors: (1) the chains are well paralleled with the angle between the chain directors smaller Fig. 20. Geometrical parameters to define dij^ and dij// than 20° and (2) the chains are closer enough for the vector connecting the two centers of mass to satisfy the following conditions (Fig. 20):
d ⊥ ij < 1.5σ, d // ij < 1.2σ
(12)
If a chain has at least three crystalline neighbors, then we consider the chain belongs to a crystalline domain. Figure 21 shows the snapshots of the initial stage of crystallization around t = 16 ns at Tc /T0 = 3.95, where the nonbonding energy showed clear drop in Fig. 17; we here depicted the chains forming the crystalline domains only. A small cluster of the chains first appears, and then follows the steady growth. Actually small embryos are created and annihilated repeatedly before t = 16 ns without growing into larger crystallites, but some crystal embryos eventually grow into stable crystallites. During the growth of the crystallites, the incoming chains adhere onto the crystal surface, especially to the kink sites (Fig. 22). The molecular mechanism of crystal growth in the helical oligomer is thus similar to that in a simple linear polymer reported in our previous papers. Though crystallization
162
T. Yamamoto
Fig. 21. Appearance of the crystal nucleus and the growth of the crystallites at T/T0 = 3.95 and at times t = 16 + t (ns): (a) t = 0.16, (b) t = 0.32, (c) t = 0.48, (d) t = 0.64, (e) t = 0.80, and (f) t = 0.96. Four equivalent images of the neighboring MC cells are shown, where the R-handed and the L-handed chains are depicted in red and light blue.
△
△
△
△
△
△
△
was conducted below the solid-solid transition temperature, we could not notice prominent chiral selections during crystal nucleation and initial crystal growth; the helical chains adhered to the crystal surfaces by selecting their helical senses apparently at random. Our MC simulation described earlier suggested that the chirality selection at the crystal surfaces was rather tolerant. This can be a possible origin of the appearance of racemic crystals in the early stage of crystallization. As described before, the MC and MD simulations verified the presence of chiral ordered phase at lower temperatures, and therefore crystallization below the order-disorder transition temperature should have led to the development of chiral crystallites. However the chiral disorder introduced in the initial stage of crystallization appear frozen within the conventional MD time scale, especially at lower temperatures. Though the solid-solid phase transformation is generally slow, we already confirmed the transition is reversible. In case of crystallization from the melt also, though the chirality recognition in the early stages of
Molecular Dynamics in Crystallization of Helical Polymers
163
crystallization is not readily noticed, we found that the final crystal obtained is made of chiral domains. The domains are however mainly formed through the solid state transition (a lower path for bare helix in Fig. 23).
Fig. 22. The surface deposition processes of the chains during the crystal growth at T/T0 = 3.95 and at times t = 16 + t (ns): (a) t = 1.6, (b) t = 1.76, (c) t = 1.92, and (d) t = 2.08; the solid circles represent the positions where new chains will be deposited within the next 0.16 ns, and the dotted circles are the newly deposited chains.
△
△
△
△
△
5. Simulations for iPP, a Helical Polymer with Side Groups We have explained in the previous section that the bare helical polymers crystallize relatively fast, but the chirality selection was less efficient than expected. Now we consider crystallization in the general helical polymers by taking iPP as an example.
164
T. Yamamoto
Fig. 23. Two possible pathways from random coiled to ordered crystal. Many helical polymers directly transform from the random coil to the ordered crystal, where the crystalline order will be attained on or near the growth front of the crystal (upper path). Some polymers, however, have highly disordered crystalline states, though the states may be transient, below the melting points. In this case, the polymers first crystallize into the disordered crystals, which afterwards transform to the ordered crystalline phases through solid-solid transformations (lower path).
5.1 Collapsing of a single iPP chain in vacuo49 Crystallization of helical polymers was shown to be rather slow even in case of simple bare helical polymer. Helical polymers with large side groups will be much more sluggish in crystallization. Molecular simulations of crystallization in iPP by use of realistic molecular models were already attempted with limited success, where only local incomplete ordering has been observed. It seems that the iPP molecules must follow much more complex paths than the bare helix. As an illustrative example of the slow dynamics, our simulation of total 50 ns during a stepwise cooling of 10 K/1 ns from 700 K down to 200 K is shown in Fig. 24. Though the radius of gyration Rg decreases
Molecular Dynamics in Crystallization of Helical Polymers
165
Fig. 24. Collapse of the iPP chain of 100 monomers into a noncrystalline state during very slow stepwise cooling (10 K/1 ns). The radius of gyration (filled circle) shows constant shrinkage of the chain, while the global order parameter (open circle) does not indicate any sign of ordering. Typical snapshots obtained at 700 K and 200 K are inserted.
continuously, the global order parameter GiPP does not increase at all; the parameter GiPP is again defined by eq. (5). Corresponding vectors are those connecting the i-th and the (i+6)-th backbone atoms taking the 3/1 helix of the iPP molecule. Actually the number of helical segments, the colored atoms in the insets, is increasing with decreasing temperature, but long successions of the same handed helical segments are hardly observed. The free energy barrier for creating single-sensed helical conformations must be much larger than those of the bare helix. We suspect that the molecular motions necessary to make ordered crystalline
166
T. Yamamoto
states are also suppressed due to rugged molecular contours, specific intermolecular interactions, and chain entanglements.
5.2 Crystallization of a single chain with definite chiral recognition49 In order to accelerate ordering, we hereafter consider a single short chain geometrically confined within a 2D slit or a 1D tube, where the chain is flattened or stretched before the onset of crystallization. In addition, we have learned that the polymer crystallization is greatly accelerated if the molecule is placed on a crystalline substrate. We here use fully ordered crystalline substrates placed within the 2D slit and the 1D tube. The substrate is made of a single-sensed L-handed helix of iPP of infinite length, where the periodic boundary condition is assumed in the x-axis direction (Fig. 2); the orientation of the 3/1 helix around the chain axis of the substrate iPP is so selected that the one surface of the triangular prism is facing upward (Fig. 25). Within the slit, we placed a molecule of 100 monomer length. Figure 25 shows typical snapshots of chains crystallizing during slow cooling, 10 K/1 ns, from 700 K to 200 K. At higher temperatures the randomly coiled chain is wobbling on the substrate. With decreasing temperature, the chain starts to attach to and stretch on the substrate. Finally below 490 K, the chain is fully stretched and makes perfectly ordered R-handed helix. Very remarkable is that the iPP chain always selects its helical sense to be just opposite to that of the substrate chain. Such conformational ordering with chirality recognition is well-described by use of the fraction of monomer units with definite handedness NR and NL for the R-handed or the L-handed ones (Fig. 26); the monomer unit is called the R-handed or the L-handed when the pair of bonds, CH–CH2 and CH2–CH, has trans-gauche+ or gauche−-trans, respectively; here the trans and the gauche bonds are respectively defined as the bonds whose torsion angles satisfy | τ | < 30° and 90° < |τ| < 150°.
167
Molecular Dynamics in Crystallization of Helical Polymers
(a)
(b)
(c) (d)
Fig. 25. The molecule of iPP confined within a 2D slit (see inset), and the molecular process of crystallization onto the substrate made of the left handed iPP; the substrate iPP chain has the inverted triangle orientation with one face of the triangular prism looking upward. During the stepwise cooling (10 K/1 ns) from 700 K to 200 K, the chain ordering is accompanied with strict selection of the helical sense. The snapshots are at (a) 560 K, (b) 530 K, (c) 510 K, and (d) 490 K.
Above 600 K, both the R- and the L-handed units have nearly the same fraction of about 20% with remaining 60% monomer units not showing definite handedness. The chain conformation begins to fluctuate greatly around 600 K, and around 530 K a sharp transition into the ordered R-handed helix takes place. The substrate of ordered L-handed helix compels the crystallizing chain to take the opposite R-handed helical conformation. The random coil chain in the 2D space showed partial attachment first and the attached segments sequentially transformed into the ordered helix through strict chiral recognition process. This adsorption and ordering process would be a simple molecular model in crystallization from the isotropic melt or solution.
168
T. Yamamoto
Fig. 26 The process of conformational ordering with marked selection of the helical sense is represented by use of the number of R- ( ) and L-( ) handed monomer units and their sum ( ) vs. temperature for the chain, (a) confined within a narrow slit (Fig. 25) and (b) confined within a narrow tube (Fig. 27); in both cases the substrate is the left-handed helix of the same iPP.
○
▲
▽
When the chain is highly stretched initially, on the other hand, like in crystallization under flow or from oriented melt, what modifications would occur in its crystallization and helix recognition? Figure 27 shows an iPP chain confined within a tube together with the L-handed crystal substrate of iPP. The chain is stretched well from the beginning,
(a) L
(b)
R
(c)
L Fig. 27. Trajectories of the chain confined within a tube during the stepwise cooling (10 K/1 ns) from 700 K to 200 K at (a) 650 K, (b) 500 K, and (c) 400 K. The chain is forced to stretch along the direction of the tube, such as a chain in a stretched amorphous state or that under elongational flow.
Molecular Dynamics in Crystallization of Helical Polymers
169
though the conformation is highly disordered with very small fraction of the helical monomer units (Fig. 26(b)). With decreasing temperature, the fraction of the R-handed monomer units increases. The symmetry breaking transition with the spontaneous selection of R- and L-handed units starts at a high temperature of 600 K. The earlier onset of helical sense recognition is undoubtedly due to the forced adsorption of the chain to the substrate. We have demonstrated the clear selection of the helical hand during crystallization onto the substrate of given chirality. The pair of opposite handed chains was found to have lower interchain energy than the same handed pairs. We will expect proper rewinding of helices of unfavorable helical pairs in the crystal, whereby the helix reversal defects must propagate along the crystalline stems. We studied the motion of the helix reversal defects within the same tube as Fig. 27. On the chiral L-handed substrate, we placed a chain made of three helical blocks, L-R-L, having
Fig. 28. Motion of the helix reversal defects along the chain at (a) 300 K and (b) 400 K. The iPP chain placed on the chiral (L-handed) substrate has initially tri-block structure of the L-R-L helical segments. The L- and the R-bonds of the chain are drawn blue and red, respectively. The abscissa represents the bond number (position) and the ordinate is time. The initial chain conformation and the helix reversal defect positions (circles) are shown on top of each graph.
170
T. Yamamoto
two helix reversal points. Figure 28 shows the distribution of the R and L handed monomer units, each drawn in red and blue, as a function of time. The boundaries of the two colors at each time correspond to the L-R and R-L reversal points. At lower temperature of 300 K, the helix-reversal defects are almost immobile and the chain keeps its initial overall conformation. At higher temperature of 400 K, however, the blue area decreases quickly indicating that the helical tri-block conformation L-R-L is transforming into a dominantly R-handed one. Thus the solid state process of the helix rewinding in iPP is also expected to play an important role in the ordering of the iPP crystals especially at higher temperatures.
5.3 Crystallization and polymorph selection57 Crystallization of iPP in free 3D space or even on the 2D substrate is expected to be quite complex and very slow. Highly cooperative process in the intermolecular and intramolecular degrees of freedom will provide very narrow free-energy paths, which makes the direct atomistic scale MD very difficult. A possible alternative approach is to investigate a collection of rigid crystalline stems of iPP by ignoring all intramolecular degrees of freedom. Here we further assume that every rigid stem is oriented to the same direction. Crystallization in this system can then be envisaged as the aggregation of stems and their ordering with respect to their positions, orientations around the axes, their helical senses, and directions of methyl groups. The present MC simulation method for iPP is similar to those adopted in our on-lattice simulations for PTFE,55 iPP,56 and for the bare helix,50 except that we here adopt the off-lattice model. With such simplified model, we can study the growth of various crystalline forms if iPP, such as the α-form, the β-form, and the smectic mesophase. Some time ago we reported on an MC simulation of the crystal structures of iPP56, where we found that the chiralβ-form or the racemic α-form crystal spontaneously forms when the symmetry of the 2D crystal lattice is given either hexagonal or rectangular. In that work, the
Molecular Dynamics in Crystallization of Helical Polymers
171
crystal lattice was given beforehand, and the lattice model was available where the interchain energies were calculated and tabulated. When we consider crystallization from the melt (or more precisely from the nematic state) of random chain dispositions however, we lose the lattice framework to be referred. We here consider an off-lattice model of 120 stems of iPP. By use of the conventional Metropolis method for the system of rigid chains, we study the crystallization and selection of the crystal polymorph with definite chirality. From a random state with respect to the chain positions and orientations around the axis as well as helical sense and methyl direction, we performed a cooling run of 10K/105 MCS and observed spontaneous nucleation and growth of the crystallites (Fig. 29); the chains are here confined within a cylindrical area to avoid the chains to disperse at higher temperatures. We can readily notice the emergence of the α-from nuclei around Tstart ≅ 210 K, where the R- and the L-handed helices are arranged in order. The nuclei grow by further cooling until they collide, coalesce,
α
Fig. 29. (a) Emergence of -form cluster of iPP during slow cooling, and (b) lattice energy vs. temperature. Right and left handed helixes are drawn as filled and open triangles.
172
T. Yamamoto
number of α –chain
and finally form a single crystalline domain. From the decrease in the nonbonding energy (Fig. 29(b)), dominant ordering is seen to occur between Tstart and Tend ≅ 180 K with residual crystal perfection processes below Tend. The structure obtained is just the α-form with approximate lattice constants a ≅ 0.62 nm, b ≅ 2.06 nm which are slightly smaller than experimental values; slightly smaller values will be due to the united atom model for methyl and methylene groups. We counted the number N α of chains which have local α-form structure, two same sensed neighbors and three 60 opposite sensed neighbors. Figure 30 shows that the 50 local order parameter sharply increases around 40 Tstart and come to take a 30 constant value around Tend. Such simple modeling 20 of crystallization is also useful in studying a frozen 10 disordered crystal, socalled smectic mesophase, 0 by deep quench. Figure 100 150 200 31 shows the crystalline temperature (K) states obtained after Fig. 30. Development of local order during 5×105 MCS by quenching slow cooling. The local order is represented from 500 K to various Tc. by the number of chains that has local -form At 150 K, slightly below environment Nα. Tend, the growth of the small α-form clusters is noticed. At lower temperatures, however, the ordered clusters become smaller and smaller. The number of the chains having the local α-form neighbors N α is considerably smaller than those given in Fig. 30. The loss of the local α-form order is characterized well by use of the average separations between the neighboring chains. We here introduce the parameters from the average interchain separations of the neighboring chains closer than 0.7 nm: the average separation between
α
Molecular Dynamics in Crystallization of Helical Polymers
173
the same-sensed chains dSS, between the opposite-sensed chains dOS, and the parameter AS to represent the asymmetry of the chain packing,
AS = d SS − d OS /d OS
(13)
Fig. 31. (a) Appearance of the frozen “smectic” structure in iPP by rapid quenching from 500 K to the respective temperatures indicated, and (b) the degree of disorder vs. temperature represented by the average interchain spacing between same-sensed chains dss ( ) and between opposite-sensed chains dos ( ) and the asymmetry parameter As ( ). Representation of right- and left-handed helix is the same as Fig. 29.
●
○
△
The average separation between the same-sensed chains dSS slightly decreases by deep quench, while that between the opposite-sensed chains dOS increases markedly, and the asymmetry AS between the same-sensed and the opposite-sensed chains is gradually lost (Fig. 31(b)). The chains in the highly quenched state have thus disordered chain-orientations with distorted hexagonal arrangements. However, the preference for the local R-L pairing is still quite evident (Fig. 31(a)). Both gradual cooling and rapid quenching resulted in the racemic mixture of the R- and the L-handed chains. In search for the origin of chiral β-from crystal, we attempted many off-lattice simulations but we could not find out proper conditions for the appearance of the β-form.
174
T. Yamamoto
In our previous on-lattice simulation, we found that the imposed hexagonal lattice gave rise to the chiral β-form. Here we show the reverse is also true, that is the imposed crystal chirality gives the hexagonal lattice. We here consider an initial high temperature state that is similar to those adopted before (Figs. 29 and 31) except that all
β
Fig. 32. Emergence of the -form structure of iPP during slow cooling from 500 K. (a) the snapshots showing rapid ordering into the chiral hexagonal form, and (b) total interchain lattice energy vs. temperature.
the chains are of the same hand. From this initial state, we attempted similar off-lattice simulations as before, with an additional constraint that the helix inversion is prohibited thereby preserving crystal chirality. Structural changes by gradual cooling of 10 K/105 MCS are given in Fig. 32. Quite remarkable is that chains quickly form a hexagonal structure; the hexagonal chain packing is evident at temperatures as high as 220 K. The lattice energy also shows a rapid convergence to the equilibrium value, which shows constant decrease with decreasing temperature. Such behavior is quite in contrast to Fig. 29 during the growth of the α-form crystals. The present model for the β-form growth involves no helix inversions, which assumption must have greatly accelerated crystallization. From the Fourier analysis of the chain
Molecular Dynamics in Crystallization of Helical Polymers
175
orientations, we also confirmed that the structure has a characteristic super-structure that nicely corresponds to the incommensurate chain packing proposed from electron diffraction experiments.58,59
6. Conclusions In this chapter we reviewed our recent efforts to understand molecular mechanisms of crystallization in helical polymers with special emphasis on the microscopic process of chirality recognition. Since the straightforward approach to the problem was difficult, we adopted a strategy of categorizing helical polymers into two distinct types and used the combination of MD and MC methods. We found that the isolated bare helix froze into partially ordered state by rapid quenching but it remarkable improved the conformational order and transformed into the chain-folded structure by long annealing. On the contrary the bare helix crystallized much faster on the crystal substrate through highly cooperative intermolecular and intramolecular motions. We also studied the chirality selection process by use of oligomers of the bare helix. We confirmed that our bare helix has a chiral low temperature phase. However, even when crystallization was conducted below the order-disorder transition temperature, marked chirality selection was not readily noticed in the early stage of crystallization, but the final chiral structure was attained through solid-solid phase transition at the late stage of crystal growth or after completion of the crystal. We also observed frequent occurrence of helical sense inversions and the propagation of the helix reversal defects in the crystal. We studied the crystallization of iPP in free 3D space but found it extremely sluggish. By restricting the spatial dimension of the system into 2D slit or 1D tube, we observed that the molecule of iPP crystallizes very quickly on the crystal substrate. Quite remarkable was an observation that the molecule of iPP efficiently recognizes the helical sense of the substrate chain and strictly selects right chirality during crystallization. Propagation of the helix reversal defects in iPP was also observed, which clearly indicated marked preference of the pairing of opposite handed helixes.
176
T. Yamamoto
Finally we studied spontaneous formation of various polymorphs of iPP by use of the off-lattice MC method for the rigid chains. Gradual cooling from disordered state always gave rise to the α-form , while the rapid quenching into much lower temperatures yielded the emergence of the disordered crystalline phase that corresponds to the smectic form. Conditions for the appearance of the β-form were not clarified, but we confirmed that the crystal chirality and the lattice symmetry are closely related; the chiral β-form is to the hexagonal lattice, while the racemic α-form is to the rectangular orthorhombic lattice.
References 1. F. C. Frank, Faraday Discuss. Chem. Soc. 68, 7 (1979). 2. B. Wunderlich, Macromolecular Physics (Academic, New York, 1976), Vols. 1–3. 3. L. Mandelkern, Crystallization of Polymers (Cambridge University Press, Edinburgh, 2002), Vols. 1–2. 4. J. Macromol. Sci., Phys. 42, 403 (2003), special issue on the International Symposium on Polymer Crystallization. 5. Polymer 46, 8661 (2005), special issue of polymer to honor the memory of John D. Hoffman. 6. Interphases and Mesophase in Polymer Crystallization, Advances in Polymer Science Vols. 180, 181, and 191, edited by G. Allegra (Springer, Berlin, 2005). 7. K. Armistead and K. G. Goldbeck-Wood, Adv. Polym. Sci. 100, 219 (1992). 8. M. Muthukumar, Adv. Chem. Phys. 128, 1 (2004). 9. J. D. Hoffman and R. L. Miller, Polymer 38, 3135 (1997). 10. D. M. Sadler and G. M. Gilmer, Polymer 25, 1446 (1984). 11. J. P. K. Doye and D. Frenkel, Phys. Rev. Lett. 81, 2160 (1998). 12. J. J. Point, Macromolecules 12, 770 (1979). 13. M. Hikosaka, Polymer 31, 458 (1990). 14. G. Strobl, Eur. Phys. J. E 3, 165 (2000). 15. T. A. Kavassalis and P. R. Sundararajan, Macromolecules 26, 4144 (1993). 16. C. Liu and M. Muthukumar, J. Chem. Phys. 109, 2536 (1998). 17. P. Welch and M. Muthukumar, Phys. Rev. Lett. 87, 218302 (2001). 18. H. Meyer and F. J. Mueller-Plathe, J. Chem. Phys. 115, 7807 (2001). 19. H. Meyer and F. J. Mueller-Plathe, Macromolecules 35, 1241 (2002). 20. T. Miura, R. Kishi, M. Mikami, and Y. Tanabe, Phys. Rev. E 63, 061807 (2001). 21. R. H. Gee and L. E. Fried, J. Chem. Phys. 118, 3827 (2003). 22. N. Lacevic, L. E. Fried, and R. H. Gee, J. Chem. Phys. 128, 014903 (2008). 23. T. Yamamoto, J. Chem. Phys. 107, 2653 (1997). 24. C. M. Chen and P. G. Higgs, J. Chem. Phys. 108, 4305 (1998).
Molecular Dynamics in Crystallization of Helical Polymers 25. 26. 27. 28. 29. 30. 31. 32. 33.
34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52. 53. 54. 55.
177
T. Yamamoto, J. Chem. Phys. 89, 2356 (1998). J. P. Doye and D. Frenkel, J. Chem. Phys. 109, 10033 (1998). J. P. Doye and D. Frenkel, J. Chem. Phys. 110, 2692 (1999). T. Yamamoto, J. Chem. Phys. 115, 8675 (2001). N. Waheed, M. S. Lavine, and G. Rutledge, J. Chem. Phys. 116, 2301 (2002). W. Hu, D. Frenkel, and V. B. F. Mathot, Macromolecules 36, 549 (2003). T. Yamamoto, Polymer 45, 1357 (2004). N. Waheed, M. J. Ko, and G. C. Rutledge, Polymer 46, 8689 (2005) T. Yamamoto in Interphases and Mesophases in Polymer Crystallization III, Advances in Polymer Science Vol. 191, edited by G. Allegra (Springer, New York, 2006). A. Koyama, T. Yamamoto, K. Fukao, and Y. Miyamoto, Phys. Rev. E 65, 050801 (2002). W. Hu, D. Frenkel, and V. B. F. Mathot, Macromolecules, 35, 7172 (2002). M. S. Lavine, N. Waheed, G. C. Rutledge, Polymer, 44, 1771 (2003) I. Dukovski and M. Muthukumar, J. Chem. Phys. 118, 6648 (2003) M. J. Ko, N. Waheed, M. S. Lavine, and G. C. Rutledge, J. Chem. Phys. 121, 2823 (2004). I. Weissbuch, L. Addadi, M. Lahav, and L. Leiserowitz, Science 253, 637 (1991). L. Cartier, T. Okihara, Y. Ikada, H. Tsuji, J. Puiggali, and B. Lotz, Polymer 41, 8909 (2000). C. Mathieu, W. Stocker, A. Thierry, J. C. Wittman, and B. Lotz, Polymer 42, 7033 (2001). P. Choi, H. P. Blom, T. A. Kavassalis, and A. Rudin, Macromolecules 28, 8247 (1995). K. Nagarajan and A. S. Myerson, Cryst. Growth Des. 1, 131 (2001). G. Xu, and W. L. Mattice, Polymer 43, 7007 (2002) T. Yamamoto, N. Orimi, N. Urakami, and K. Sawada, Faraday Discuss. 128, 75 (2005). X. Chen, R. Ozisik, S. K. Kumar, J. Polym. Sci. Part B, 44, 3453 (2006). X. Chen, R. Ozisik, S. K. Kumar, and P. Choi, J. Polym. Sci. Part B, 45, 3349 (2007) S.A. Sabeur, F. Hamdache, and F. Schmid, Phys. Rev. E77, 020802 (2008). T. Yamamoto and K. Sawada J. Chem. Phys. 123, 234906 (2005). T. Yamamoto, J. Chem. Phys. 125, 064902 (2006). D. Rigby and R. Roe, J. Chem. Phys. 87, 7285 (1987). R. Aich and P. C. Haegele, Prog. Colloid Polym. Sci. 71, 86 (1985). M. G. Martin and J. I. Siepmann, J. Phys. Chem. B 103, 4508 (1999). Integrated simulation system for polymeric material design OCTA; http://octa.jp.index.html. T. Yamamoto, J. Polym. Sci., Polym. Phys. Ed. 23, 771 (1985).
178
T. Yamamoto
56. M. Hirose, T. Yamamoto, and M. Naiki, Comput. Theor. Polym. Sci. 10, 345 (2000). 57. T. Yamamoto, European Discussion Meeting EDM 2007 on polymer crystallization L-5 (2007). 58. S. V. Meille, D. R. Ferro, S. Bruckner, A. J. Lovinger, and F. J. Padden, Macromolecules, 27, 2615 (1994). 59. B. Lotz, J. C. Wittmann, and A. J. Lovinger, Polymer 37, 4979 (1996).
Chapter 6 Interplay of Liquid-Liquid Demixing and Polymer Crystallization
Wenbing Hu School of Chemistry and Chemical Engineering, Nanjing University 210093 Nanjing, PR China E-mail: [email protected] This mini-review made a survey on recent dynamic Monte Carlo simulations of lattice polymers performing interplay of liquid-liquid (L-L) demixing and crystallization in polymer blends and solutions. The thermodynamic conditions for interplay were designed by the developed mean-field lattice statistical theory. The simulation results were separated into three categories: L-L demixing enhanced by the component-selective crystallizability, crystal nucleation enhanced by prior L-L demixing, and enhanced crystal nucleation in single-chain systems. The implications of these results in real polymer systems have been discussed. Some perspectives on the self-assembly of multicomponent copolymers were provided at the end.
1. Introduction Polymeric materials formed commonly of multiple components are processed with two basic types of phase transitions: polymer crystallization and liquid-liquid (L-L) demixing. Conventional investigations on polymer phase transitions focus only on one type of phase transitions, and often overlook their interplay. The interplay of phase transitions actually offers us an ‘emergence’ point of view (meaning integral larger than simple addition) to understand the complex behaviors of polymers in phase transitions. Such behaviors are responsible for the formation of versatile hierarchical multi-phase structures of complex polymeric systems.
179
180
W.-B. Hu
In polymer solutions, when both L-L demixing and crystallization occur in the same temperature windows, their phase diagrams will intercept with each other, making a monotectic triple point. At this triple point, the diluted phase, the concentrated phase and the crystalline phase coexist in equilibrium. Early in 1945, Richards et al. made the first phase diagrams of polyethylene solutions demonstrating both L-L demixing and polymer crystallization.1,2 Since then, the full sets of phase diagrams in polyethylene solutions have been well studied.3,4 Flory and his collaborators measured the same sets of phase diagrams for poly-(N,N’sebacoylpiperazine) solutions.5,6 Deberdt and Berghmans also measured the full sets of phase diagrams of syndiotactic polystyrene solutions.7-9 L-L demixing conventionally occurs before crystallization, even at the triple point where they share the same equilibrium transition temperatures.10 Schaaf et al. studied the crystal morphology produced by segregated polymers in solutions.11 The porous structure formed during the early stage of demixing can be frozen by subsequent crystallization.12 Such a thermally stimulated process will provide us a useful way to control the pore sizes in thermo-reversible gels and membranes.13-15 Keller made a review on the role of such an interplay of phase transitions in the sol-gel transition.16 Guenet discussed the importance of phase diagrams in understanding the interplay of phase transitions.17 For polyolefin binary blends, the full sets of phase diagrams were measured by Wang et al. in 2002.18,19 Zhang et al. further found that crystal nucleation could be enhanced in the immiscible polymer blends.20-23 Recently, Cheng made an overview on the above experimental progress in his book.24 Theoretical investigations on multi-component systems of polymers took the lattice model for the prototypical polymers, considering the fact that the volume exclusion of molecules plays a dominant role in determining the combinatorial structures of mixtures.25 In the classical lattice statistical theory, the conformational entropy of polymers together with the combinatorial entropy of multi-components have been regarded as the main contributions in the entropy of mixing/demixing of polymer solutions, first by Meyer26 in 1939 and then by Huggins27 and Flory28 in 1942. The thermodynamic results of the lattice theory have been widely applied into polymer blends,29-32 polymer networks,33 and diblock copolymers,34 etc. On the other hand, the excess energy for the formation of a pair of mutual interactions between monomers and solvent, denoted as B, has been regarded as the molecular driving force for L-L demixing.
Interplay of Liquid-Liquid Demixing and Polymer Crystallization
181
Such a contribution in enthalpy was successfully calculated under a mean-field assumption35 and be further refined by series expansions for a better accuracy in the description of real polymer solutions.36-38 Flory developed the classical lattice statistical theory for semiflexible polymer solutions, by introducing the bending energy at each bond connection along the lattice polymer chain, denoted as EC.39 He regarded such a local intra-molecular chain-rigidity as the sole driving force for polymer crystallization, and supposed that in analogy to the Onsager’s entropy-driving scheme for lyotropic liquid-crystalline order,40,41 the packing of rigid rods due to the inflexibility of polymers at low temperatures would favor parallel packing between polymers. Such an entropic driving force actually had not been estimated in his theory. He omitted the minor excess energy for compact packing of crystalline polymers, simply because the lattice model could not directly handle the volume contraction in company with crystalline packing between polymers. Such a compromise is, however, unacceptable in the common sense, for crystallization is conventionally driven by the intention of compact packing between molecules. In analogy to the Maier-Saupe enthalpy-driving scheme for thermotropic liquid-crystalline order,42 Hu proposed the excess energy for parallel packing of two lattice bonds to represent the compact packing tendency of crystalline polymers, denoted as EP.43 The volume contraction accompanied with ordering is assumed to disperse right away without changing the relative positions of monomers in the local lattice. The contribution of such an excess energy can be calculated under the mean-field assumption as well.44,45 Therefore here, by combining both the energy parameter B driving L-L demixing and the energy parameter EP driving polymer crystallization, we can use the developed mean-field lattice theory to design the thermodynamic conditions for the interplay of phase transitions in multi-component polymer systems. The theoretical designs of interplay were realized in the dynamic Monte Carlo (MC) simulations of lattice polymers under the same protocols of thermodynamic and molecular parameters. Molecular simulations provide vivid pictures of the kinetic competitions between phase transitions, without the mean-field assumption. In this review, we first introduce the thermodynamic model to calculate phase diagrams, followed with simulation techniques. Then, we present the simulation results in three separate categories: L-L demixing enhanced by crystallizability, crystal nucleation enhanced by prior L-L
182
W.-B. Hu
demixing, and enhanced crystal nucleation in the single-chain systems. The paper ends up with some perspectives for further investigations of this issue. 2. Theoretical Model In the condensed-matter physics, the lattice model has been regarded as the most fundamental tool in the study of phase transitions. The application of the mean-field assumption in the lattice model allows us to calculate separately the combinatorial energy and entropy. As mentioned in the above section, the lattice model for describing polymer solutions have been substantially developed with the consideration of parallelpacking energy. The developed mean-field lattice theory assumes that n2 polymer chains, each having r monomers, and n1 solvent sites are randomly mixed in a regular lattice with the coordination number q (the total volume n = rn2 + n1). The partition function for such a homogeneous polymer solution is thus given by44,45
n Z = n1
n1
n2
n
n q 2 −n2 ( r −1) n2 ( r −2 ) n2 ( r −1) n2r e zc zp z m , (1) n2 2
where
E z c = 1 + (q − 2) exp − c , kT
q − 2 2n2 (r − 1) E P z p = exp− 1− kT , qn 2 n (q − 2) B . z m = exp − 1 kT n EC represents the conformational energy for each collinear connection of consecutive bonds along the chain, reflecting semi-flexibility of polymers; EP is the packing energy for two parallel bonds, reflecting crystallizability of polymers; B is the net energy for each contact of monomer and solvent, reflecting their compatibility; k is the Boltzmann’s constant, and T the temperature.
183
Interplay of Liquid-Liquid Demixing and Polymer Crystallization
The mixing free energy of polymer solutions can thus be obtained as 2 ∆f mix φ B 2 1 E = φ1 ln φ1 + 2 ln φ 2 + φ1φ 2 (q − 2) + 1 − 1 − P , (2) r kT q r kT nkT where φ1 and φ2 are the volume fractions of solvent and polymers, respectively. From this expression, the phase diagrams for L-L demixing can be calculated according to the equivalence of chemical potentials between two coexisting phases, either for solvent or for solute. On the other hand, the phase diagrams for liquid-solid coexistence, e.g. the melting-point lines of polymer crystals can be calculated according to the equivalence of chemical potentials between solutions and crystals. The crystalline state of polymers appears as very similar with the ground state, in which polymer chains are fully extended in an ordered and phase-separated state; so its chemical potential is near zero. We then simply solve the equation that the chemical potential of solution equals to zero. In the above calculations, two phase transitions are assumed to be independent with each other because the free energy of homogeneous solutions has been applied. In reality, however, two phase transitions coexist, and the prior one may change the homogeneous state of polymer solutions. The latter situation can be studied in the present molecular simulations under the protocols of the same thermodynamic and structural parameters. In polymer blends, when the above solvent is replaced by noncrystallizable polymers (meaning EP = 0) with the same chain lengths, the partition function for a homogeneous symmetric blend is updated as45
n Z = n1 where
n1
n2
n q n2 2
n1 + n2
zc
( n1 + n2 )( r − 2 )
e −( n1 + n2 )( r −1) z p
E z c = 1 + (q − 2) exp − c , kT q − 2 2n 2 (r − 1) E P z p = exp− 1− kT , qn 2 n r (q − 2) B z m = exp − 1 . kT n
n2 ( r −1)
zm
n2 r
, (3)
184
W.-B. Hu
Similarly, the mixing free energy for the above polymer blends can be obtained as 2 ∆f mix φ1 φ2 B 2 1 E P , (4) = ln φ1 + ln φ 2 + φ1φ 2 (q − 2) + 1 − 1 − r r kT q r kT nkT
The phase diagrams for L-L demixing and polymer crystallization in the above polymer blends can be calculated as well. The thermodynamic conditions for their interplay can thus be designed, and the interplay will be performed in molecular simulations of lattice polymers under the protocols of the same parameters as the theoretical designs.
3. Simulation Techniques Molecular simulations have become a powerful tool in the study of polymer phase transitions. There mainly exist two types of simulation algorithms, i.e. molecular dynamics and MC simulations. Molecular dynamics simulations provide quite realistic information with heavy consideration of chemical details, but under the expense of efficiency and sample sizes, although the most powerful computation facility can be used. MC simulations sacrifice the structural and dynamical details, but win back efficiency and sample sizes. The latter is important for comparisons with thermodynamic predictions that are commonly under the assumption of infinitely large systems. In this sense, we employ MC simulations of lattice polymers by assuming each polymer chain consecutively occupying the lattice sites with its chain length r, and the phantom solvent occupying those single sites. To mimic the Brownian motion of polymers with volume exclusion, we allow polymer chains to move randomly on the lattice space without double occupation or bond crossing. The move called conventionally as micro-relaxation of polymer chains is driven by thermal fluctuations that make local deviations from the equilibrium states. The micro-relaxation model of lattice polymers considers two basic modes of local relaxation of polymer chains, i.e. crankshaft rotation and sliding diffusion, which provide high efficiency to generate new bond orientations inside polymer chains.45,46 In the cubic lattice, the bonds can be either along the lattice axis or along the face and body diagonals, so the coordination number is as high as q = 6 + 12 + 8 = 26. Such a big
Interplay of Liquid-Liquid Demixing and Polymer Crystallization
185
number could minimize the effect of discrete space on phase transitions. Periodic boundary conditions were applied to a lattice box with definite sizes to open a small window of the spatial image for an infinitely large system, under the assumption that the sizes are large enough to neglect those long-distance correlations between boundaries. The unit of time evolution can be defined as MC cycle, which is the total amount of trial moves equivalent to the amount of the monomers in the sample systems.
Fig. 1. Schematic presentation of the micro-relaxation model of lattice polymers. The solid circles represent monomers connected by solid bonds, the dashed lines are new bonds produced in a step of model evolution, and the arrows are the directions of monomer jumping.46
Each step of micro-relaxation will be checked by the conventional algorithm for the importance sampling called as the Metropolis sampling method. In this method, we allow micro-relaxation with a probability [1, exp(−E/kT)], which means the probability as one when the total potential energy change E is negative, and the probability proportional to exp(−E/kT) when E positive. The total potential energy change counts the net energies for mixing, bending and non-parallel packing during each step of micro-relaxation, as giving by (cEC + pEP + bB)/kT, where c, p and b are the net numbers of collinear connections of bonds along a
186
W.-B. Hu
given chain, of crystallizable bonds which are packed in parallel, and of mixed pairs of lattice sites occupied by different components, respectively. In practice, we usually set EC/kT as the reduced system temperature, B/EC as the driving force for L-L demixing, and EP/EC as the driving force for polymer crystallization. In the following, we will specify three kinds of interplay in the multicomponent polymer systems and perform MC simulations to observe phase transitions under the theoretically designed conditions. The quantities that monitor phase transitions will be introduced in details. 4. Results and Discussion 4.1. Liquid-liquid demixing enhanced by crystallizability In polymer blends, according to eq 4, the mixing entropy is close to zero due to the applied large chain lengths, so the miscibility of polymer blends is quite sensitive to the mixing enthalpy, mainly contributed by B, but also by EP. In the cases of polymer blends with similar chemistry but different crystallizability, for instance, in stereo-chemical compositions like the blend of isotactic polypropylene (PP) and atactic PP, their mixing energy B closes to zero. But since isotactic sequences are regular and crystallizable, and atactic sequences are non-crystallizable, like the blends we have discussed in eq 4, their difference in crystallizability leads to a positive mixing free energy, which implies incompatibility in the melt blend.47,48 Such a story indeed exists in the blend of isotactic and syndiotactic PP,49 as well as in the blend of syndiotactic and atactic PP.50,51 Decreasing the chain lengths can enhance the mixing entropy and thus the compatibility of the above polymer blends, as observed in experiments.50,52 Such L-L demixing driven by component-selective crystallizability has been demonstrated by MC simulations.53,54 Furthermore, theoretical treatment makes a mean-field assumption, which neglects thermal fluctuations of parallel packing in the melt. Near the equilibrium melting point, the tendency for parallel packing in the melt will be even stronger due to thermal fluctuations. According to eq 4, any enhancement in the EP contribution will enhance L-L demixing too. Figure 2 shows the theoretical design for various B/EC values in symmetric polymer blends with chain lengths 32 and one component crystallizable (EP/EC = 1) in 323 lattice space. The solid curves are theoretical binodals for L-L demixing, and the dashed curves are theoretical liquid-solid coexistence curves. In this theoretical design,
Interplay of Liquid-Liquid Demixing and Polymer Crystallization
187
with the decrease of B/EC values, the L-L demixing curves are approaching to the curves of equilibrium melting points. One can see that with B/EC = 0, L-L demixng occurs prior to crystallization. Such demixing is purely driven by the component-selective crystallizability as discussed in the above paragraph. We know that the data points in Fig. 2 are obtained from MC simulations under the same thermodynamic conditions but without the mean-field assumption. When the phase diagrams of demixing are far away from the phase diagrams of crystallization, the simulation results agree quite well with the theoretical predictions including the case of B/EC = 0. However, when the demixing occurs near the equilibrium melting point, the simulation results shift to higher temperatures, implying enhanced demixing. Such an enhancement in demixing can be attributed to the enhanced thermal fluctuations of EP interactions near the melting point, as captured only in molecular simulations without the theoretical mean-field assumption.54 20
Temperature (units of EC/k)
0.01 16
0 12
-0.01 8
-0.02 4 0.0
0.2
0.4
0.6
0.8
1.0
Volume fraction of the crystallizable component Fig. 2. L-L binodals (solid curves) and liquid-solid coexistence lines (dashed curves) of symmetric polymer blends with one component crystallizable (EP/EC = 1) and variable B/EC values as denoted nearby to the solid curves, predicted by the mean-field lattice theory. The data points (B/EC = 0.01 crosses, 0 squares, −0.01 triangles, and −0.02 spheres) are “real” binodals obtained from simulations of lattice polymer blends. Each data point was averaged over five individual simulation results.54
188
W.-B. Hu
4.2. Crystal nucleation enhanced by prior L-L demixing We exchange the counterparts in the above interplay and now focus our attentions on crystal nucleation. We will see the similar performance, i.e. crystal nucleation can be enhanced by the prior demixing. On the one hand, if demixing occurs in the vicinity of melting point, concentrating polymer solutions may shift up the melting point, which implies larger supercooling in the bulk polymer phase than in the previous solutions under the constant crystallization temperatures; so crystal nucleation in the bulk polymer phase will be enhanced. On the other hand, if demixing occurs far above the melting point, the interfacial zone may contain the melting point even higher than the segregated bulk polymer phase, and hence crystal nucleation favors to occur at interfaces.
Temperature (units of Ec/k)
6
3rd 2nd 1st
5
4
1st. Ep/Ec=1.0, B/Ec=0.092
3 2nd. Ep/Ec=1.064, B/Ec=0.05 2 3rd. Ep/Ec=1.14, B/Ec=0 1 0.0
0.2
0.4
0.6
0.8
1.0
Polymer volume fraction Fig. 3. Liquid-solid coexistence curves (solid lines) and L-L binodals (dashed lines) for lattice chains, each with 32 units, in a 643 cubic lattice and with variable sets of interaction parameters as denoted. The curves are calculated from the mean-field theory. The arrow indicates the subsequent performance in simulations.55
We firstly look at the effects in the bulk phase. Figure 3 demonstrates the theoretical design for three series of polymer solutions: all three have the same melting points at the polymer volume concentration 0.15, but various critical points for L-L demixing with different depths below the melting points.55 If we have three samples of homogeneous polymer
189
Interplay of Liquid-Liquid Demixing and Polymer Crystallization
solutions with the volume concentration 0.15 and quench these samples down to the temperature denoted in Fig. 3, the time-evolution curves of crystallinity for these three samples reveal very different crystallization rates, as shown in Fig. 4. This is interesting because all three samples have the same melting points and the same crystallization temperatures, implying the same supposed supercoolings or thermodynamic driving forces. 1.0
Disorder parameter
0.9 0.8 3rd 0.7 0.6 2nd
0.5 1st 0.4 0.3
0
10
20
30
40
50
Time evolution /10000 MC cycles Fig. 4. Time evolution curves of disorder parameters for the sample systems in Fig. 3 quenched from the infinite temperature to 2.857 EC/k. The lines are drawn on the data points with 500 MC cycle intervals. The disorder parameter is defined by the mean fraction of non-collinear connections along the chains, which gives zero for fully extended chains and 0.96 for random coils.55
The difference in crystallization rates can be attributed to prior L-L demixing that diverts the course of thermodynamic conditions in the first and second samples. One segregated phase increases polymer concentration and hence it may shift up the melting point, implying larger supercooling or stronger thermodynamic driving forces for polymer crystallization. This is the reason why the first sample exhibits the highest crystallization rate. The second sample increases a relatively less concentration in the segregated phase, and thus exhibits the second crystallization rate. The third sample performs typically simple crystallization with a significant incubation period for crystal nucleation.
190
W.-B. Hu
(a)
(b) Fig. 5. Snapshots of resulted crystalline states after isothermal crystallization shown in Fig. 4. The linear size of cubic box is 64 with periodic boundary conditions. All the bonds of 32-mer chains are drawn in cylinders. (a) The 1st sample, (b) the 2nd sample, and (c) the 3rd sample.55
Interplay of Liquid-Liquid Demixing and Polymer Crystallization
191
(c)
The resulted morphologies of crystallites are also very different among three sample systems, as demonstrated in the snapshots shown in Fig. 5. The first sample exhibits spinodal decomposition upon L-L demixing, which generates tremendous amount of small concentrated domains well distributed in the space. Each segregated domain makes a spatial confinement to subsequent fast crystallization of internal polymers. So we observed many small crystallites well distributed in the space. The third sample makes only one nucleus and thus yields a single lamellar crystal. The second sample is just in the middle case between two extremities. Homogeneous crystal nucleation generates a large amount of crystal nuclei at low temperatures. By counting the amount of crystallites under various temperatures, we could find the onset temperature of homogeneous nucleation in the time window of our simulations. Figure 6 demonstrates comparisons of theoretical phase diagrams with the onsets of homogeneous crystal nucleation in simulations.56 The energy parameters for polymer solutions are variable in our observations. One can see that below the theoretical spinodal, the onsets of crystal nucleation appear horizontal. In this region, the thermodynamic driving forces for crystal nucleation are decided by the same concentrations of
192
W.-B. Hu
the segregated polymer phases. This result implies that prior L-L demixing dominates crystal nucleation in the bulk polymer phase, or in other words, crystal nucleation of polymers is triggered by spinodal decomposition upon prior demixing.56 The spinodal decomposition occurs in the thermodynamically instable region of homogeneous solutions (not in the metastable regions cut between binodal and spinodal curves), which fits into the very short time window of our molecular simulations. Above the crossing point, the onsets of crystal nucleation increase with the increase of polymer concentrations, following the theoretical dependence of equilibrium melting points on polymer concentrations.
Temperature (units of Ec/k)
4.0
3.5
3.0
2.5
2.0
C1
C1
C2
C1
C2
C3 C2
1.5 0.0
0.2
0.4
0.6
0.8
Polymer volume fraction Fig. 6. Kinetic boundaries of polymer crystal nucleation for three types of 128-mer solutions labeled with squares (C1: EP/EC = 1.0 and B/EC = 0.076), triangles (C2: EP/EC = 1.072 and B/EC = 0.03), and spheres (C3: EP/EC = 1.275 and B/EC = −0.1), respectively. The dashed and the dotted lines are the theoretical binodals and spinodals of liquid-liquid demixing, respectively. The data points are the onset temperatures of homogeneous crystal nucleation obtained through counting the number of crystallites after isothermal crystallization at various temperatures, and the error bars are the possible reading ranges due to the curvature in the connection of crystallite populations. The straight segments connecting the same groups of data points are drawn to guide the eyes.56
Interplay of Liquid-Liquid Demixing and Polymer Crystallization
193
Melting temperatures (units of EC/k)
10 0.5
9 8
0.3
7 6
0.1 0
5 -0.1
4 0.0
0.2
0.4
0.6
0.8
1.0
Volume fractions of crystallizable polymers Fig. 7. Theoretical liquid-solid coexistence curves of 16-mers in the homogeneous phase of binary symmetric blends with various B/EC values as denoted nearby, calculated from the developed mean-field lattice theory with EP/EC = 1 for the crystallizable species. Theoretical L-L binodals are far above these curves, which make these regions practically forbidden for bulk polymers but not for those at interfaces.57
Next, we look at the effects of interfaces between the immiscible components in polymer blends and solutions. In experiments of polymer blends, Zhang et al have found that with the increase of time evolution of prior L-L demixing at a high temperature, crystal nucleation at a low temperature could be depressed.20-23 It is well-known that Oswald ripening minimizes the content of interfaces during the long-term evolution of L-L demixing. In this sense, the decrease of interfaces could be responsible for the depression of crystal nucleation. Indeed, crystal nucleation may favor to occur at such interfaces. If we look at the theoretical phase diagrams of immiscible polymer blends, for instance, shown in Fig. 7, the melting points go upwards upon dilution.57 This implies that those polymers forced to be diluted at interfaces could contain a higher melting point than the bulk polymers, while the latter can never reach the most part of these melting points due to prior L-L demixing occurring at higher temperatures. This theoretical consideration means that crystal nucleation should be enhanced near the interfaces of immiscible polymer blends and thus provides an interpretation to the experimental observations.
194
W.-B. Hu
Fig. 8. Snapshot of the symmetric binary blend of 16-mers at the temperature of 3.95 EC/k and the time period of 22,000 MC cycles after having been annealed at a slightly higher temperature of 4.0 EC/k for a time period of 200,000 MC cycles. EP/EC = 1, and B/EC = 0.5. The bonds of non-crystallizable component are shown as tiny dark cylinders, while the bonds of crystalline component containing more than 11 parallel neighbors of same species are drawn in tiny bright cylinders. All other bonds of crystallizable species are not shown for clarity.57
The above principle can be verified by MC simulations of parallel situations.57 To this end, two parallel interfaces are established by guiding L-L demixing in the lattice box, as shown in the snapshot in Fig. 8. At a temperature above the occurrence of irreversible crystal nucleation, the distribution of locations for the largest crystallites along the direction Z normal to the interfaces has been calculated. According to the classical nucleation theory, the largest crystallites contain the highest priority to pass through the size threshold upon thermal fluctuations. So the properties of the largest crystallites represent those critical nuclei. Figure 9 shows the distributions of these probabilities to find the largest crystallites upon thermal fluctuations at a high temperature, for the immiscible sample systems with variable B/EC values.
Interplay of Liquid-Liquid Demixing and Polymer Crystallization
195
700 0.0 0.1 0.3 0.5 0.7
Populations of clusters
600 500 400 300 200 100 0 0
5
10 15 20 25 30 35 40 45 50 55 60 65
Z locations of the largest clusters Fig. 9. Distributions of the largest crystalline clusters with their mass centers as a function of the distance along the direction Z normal to the blend interface for variable B/EC values as labeled, at the temperature 4.0 EC/k after having annealed the sample for 200,000 MC cycles with EP/EC = 1 and B/EC = 0.5 for the completion of L-L demixing. The largest cluster was defined as the cluster containing the maximum number of parallel bonds of crystallizable species, and its location was represented by the Z position of its center of mass. 10,000 observations were collected with 20 MC cycle intervals, and the mass centers of crystallizable phase were calibrated back to the same locations to eliminate blurring of distributions due to the floating of phase boundaries.57
One can see from Fig. 9 that when B > 0 the distributions exhibit double peaks. The double peak implies the preference to generate crystal nuclei near interfaces. The occurrence of such a phenomenon appears quite a sudden, as we can also see from the size distributions of the largest crystalline clusters, shown in Fig. 10. Just above B = 0, all the positive B values give the similar size distributions. According to the above theoretical principle, larger B values should make stronger enhancement of crystal nucleation. The similarity of those results above B = 0 can be attributed to the fact that with the increase of B values, the interfaces become thinner. Such a size effect will, to some extent, compensate the thermodynamic effect, and thus makes the similar results for various positive B values.
196
W.-B. Hu 300 0.0 0.1 0.3 0.5 0.7
Population of clusters
250 200 150 100 50 0 50
100
150
200
250
Sizes of the largest clusters Fig. 10. Size distributions of the largest crystalline clusters with variable B/EC values as labeled. The cluster size was defined by the total amount of bonds connected by parallel neighbors of the crystallizable species. The data were collected under the same conditions described in Fig. 9.57
Polymers segregated from solutions exhibit something different from the above immiscible polymer blends.58 Figure 11 summarizes the theoretical phase diagrams for immiscible polymer solutions. The melting points are going not just upwards, but rather, first downwards and then upwards. The first going-down can be attributed to the strong contribution of mixing entropy in the mixing free energy. As can be seen in Fig. 11, L-L binodals intercept with those solidification lines at temperatures and concentrations both increasing with B/EC values. This tendency of interceptions implies that crystal nucleation will be enhanced in the bulk phase with the increase of B values. However, assuming the typical polymer concentration as 0.5 in the interfacial region, only when B/EC values are large enough (the solvent qualities are poor enough), the melting points at the assumed concentration on interfaces can be higher than those in the bulk phases, as indicated by the dotted straight lines in Fig. 11.
Interplay of Liquid-Liquid Demixing and Polymer Crystallization
197
5.7
Temperatures (units of EC/k)
5.6 5.5 5.4
0.5
5.3 5.2
0.4
5.1 5.0
0.3
0.5
4.9 4.8
0.2
0.4
4.7 4.6 4.5 0.5
0.3
0.2 0.6
0.7
0.8
0.9
1.0
Volume fractions of polymers
Fig. 11. Theoretical phase diagrams of polymer solutions for 128-mers with variable B/EC values as denoted and EP/EC = 1. The dashed curves are for L-L binodals, and the solid curves are for liquid-solid coexistence curves. The straight dotted lines indicate the melting points in the concentrated polymer phases at T = 4.5 EC/k and make comparisons with the melting points at volume fraction 0.5 assumed typically for interfacial polymers.58
The above principle has also been verified in MC simulations of parallel situations.58 Two parallel interfaces were established under the thermodynamic conditions for L-L demixing, as shown in the snapshot in Fig. 12. The distributions of the largest crystalline clusters on their locations have also been calculated with various B/EC values, and the results are summarized in Fig. 13. One can see that the double peaks occur only when B/EC values become larger than 0.3, coincident with the theoretical prediction in Fig. 11. As shown in Fig. 14, the size distributions now shift in gradual with the increase of B values, contributed mainly by the bulk phase of concentrated polymers due to their higher melting points. The present results imply that only when the solvent becomes poor enough, can the interface enhance crystal nucleation. This situation is something different from immiscible polymer blends.
198
W.-B. Hu
Fig. 12. Snapshot of the phase-separated polymer solutions for 128-mers with the volume fraction 0.5 under the conditions of EP/EC = 0, B/EC = 0.1 and T = 1.5 EC/k in the 643 lattice box. Polymer bonds are drawn in dark tiny cylinders in the middle of Z, and the single solvent sites are drawn in bright tiny spheres in two side-regions of Z.58
Probablity of the largest clusters
0.05
0.04
0.03
0.5 0.4 0.3 0.2
0.02
0.01
0.00 0
5
10 15 20 25 30 35 40 45 50 55 60 65
Z locations of the largest clusters
Fig. 13. Distributions of locations for the largest crystallites calculated from about 30,000 events, while each event is collected with an interval of 200 MC cycles in the evolving 128-mer solutions established in Fig. 12, under variable B/EC values as denoted with EP/EC = 1 and T = 4.5 EC/k. Two arrows are indicating the double peak. The segments are drawn to guide the eyes.58
Interplay of Liquid-Liquid Demixing and Polymer Crystallization
199
Probability of the largest crystallite
0.05
0.5 0.4 0.3 0.2
0.04
0.03
0.02
0.01
0.00 0
50
100
150
200
250
300
Size of the largest crystallite
Fig. 14. Size distributions of the largest crystallites calculated under the conditions same as in Fig. 13. The segments are drawn to guide the eyes.58
4.3. Crystal nucleation enhanced by prior L-L demixing in the single-chain systems When polymer concentration becomes small enough, single polymer coils suspend in the solvent, and we can regard each chain as an isolated system. The single chain can perform L-L demixing called as collapse transition.46 The single chain can also perform crystallization to maximize parallel packing of the bonds and meanwhile to minimize the surface free energy, giving rise to chain folding.59 Single-chain crystallization plays a role of prototype in the intramolecular nucleation model for polymer crystallization.60,61 Moreover, interplay of phase transitions in such a single-chain system shows of significant importance in the understanding of polymer background for protein folding. L-L demixing corresponds to the hydrophobic collapse transition of proteins, while crystallization with chain folding corresponds to the beta-sheet formation or to the core formation towards the native tertiary structure of proteins. It is well known that from the coil state to the native state, proteins have to find their fast path. Such a path of fast transformation may look like a funnel, i.e. the first step like the hydrophobic collapse transition of a molten-globule state to reach the funnel neck, followed with the accelerated chain-folding of the native state.62 Daggett and Fersht have described such a fast path as the unified condensationnucleation model.63 The physical background of this fast path is similar with above crystal nucleation enhanced by prior L-L demixing.
200
W.-B. Hu
(a)
(b) Fig. 15. Snapshots of single 512-mers with EC = 0 in (a) the coil state with B/EP = −0.1 and T = 2.174 EP/k; (b) the collapsed state with B/EP = 0.1 and T = 3.289 EP/k; (c) the crystalline state with B/EP = 0.1 and T = 2.289 EP/k.62
Interplay of Liquid-Liquid Demixing and Polymer Crystallization
201
(c)
Such a physical picture has been verified by MC simulations of single lattice polymer systems.64 The snapshots shown in Fig. 15 demonstrates the coil state, the molten-globule state and the crystalline state of single 512-mers under the various thermodynamic conditions selected from phase diagrams. The phase diagrams for single chain system cannot be directly calculated from the mean-field lattice theory because the concentration in the single-chain systems is far from homogeneous. However, the developed lattice theory can calculated the critical temperatures for L-L demixing and the melting points at the critical concentrations at various B/EP values (here assumes EC = 0 for simplicity). Such theoretical phase diagrams show almost parallel shifts from the simulation results and very similar crossing-over point critical for exhibiting the collapse transition, as shown in Fig. 16. The onset temperatures for crystallization and collapse transitions on cooling make the phase diagrams in simulations. Since the single chains are small systems, current computation power allowed us to calculate the free energy curves upon melting under variable thermodynamic conditions. To this end, we used the trick of biased sampling with parallel tempering in computation.64 Figure 17 summarizes the free energy curves at the equilibrium melting points where the ordered and disordered states share the same levels of local
202
W.-B. Hu
free energy minimum. The equilibrium melting points are very close to those simulation results obtained from cooling process shown in Fig. 16. The heights of free energy barriers under the equilibrium conditions reflect the difficulty of crystal nucleation, which offer us a way to evaluate the kinetics of phase transitions. Figure 18 compares the variations of equilibrium heights of free energy barriers with the phase diagrams. One can see that the free energy barriers drop down around the triple point, implying crystal nucleation accelerated by the occurrence of prior collapse transition. Such an interplay of phase transitions in polymer solutions provides a polymer background to the fast path of protein folding described by the funnel model. 16
Temperature (units of EP/k)
14 12 10 8 6 4 2 0 -0.1
b a
c
0.0
0.1
0.2
0.3
B/EP
Fig. 16. Tentative phase diagrams of a single 512-mer as obtained from dynamic MC simulations. The data points correspond to cooling runs performed at different B/EP values. The cooling procedure involved a stepwise increase in the value of EP/(kT), starting from the infinite temperature limit. During cooling, the step size in EP/(kT) was chosen to be equal to 0.01. After every step, the chains were equilibrated during one million MC cycles. The filled circles denote the onset of crystallization while the filled triangles denote the onset of the collapse transition. The estimates of the freezing transition based on free-energy calculations are indicated by filled pentagons. The dashed curve denotes the mean-field prediction for the critical temperatures of L-L demixing, and the solid curve refers to the mean-field prediction for freezing at the critical demixing concentration. The arrows next to the letters a, b, and c indicate the state points of the snapshots in Fig. 15.64
Interplay of Liquid-Liquid Demixing and Polymer Crystallization 70
B/Ep, T /Ep/k -0.1, 2.198 0, 2.755 0.04, 2.976 0.1, 3.289 0.3, 3.683
60 50 40
∆F /(kT)
203
30 20 10 0
0
100
200
300
400
500
Molten units Fig. 17. Free-energy curves (∆F) of single 512-mers between the stable ordered and disordered states at the temperatures where the free-energy difference between two local minima vanishes. In the thermodynamic limit, this corresponds to the melting temperature. The B/EP values corresponding to the different curves are indicated in the figure.64 140
Height of ∆F/kT
10
120 100
Tcol
8
80
6
60
Tcry
4
40 2 -0.1
0.0
0.1
0.2
0.3
Height of free-energy barrier
Temperature (units of EP/k)
12
20
B/Ep Fig. 18. Comparison of the free-energy barriers at equilibrium melting temperatures for intramolecular crystallization on the parameter B/EP, with phase diagrams same as in Fig. 16. As can be seen from the figure, the barrier for nucleation drops sharply upon approach to the point where the molten globule intervenes between the coil state and the crystalline state.64
204
W.-B. Hu
5. Perspectives The multi-component systems performing interplay of phase transitions are not limited with the above-discussed intermolecular multiplicity, but also include the intramolecular multiplicity; for instance, the block copolymers, the graft or branch copolymers, the star-shaped copolymers, and even the statistical copolymers. Those biological macromolecules often contain the intramolecular multiplicity of components. In these cases, L-L demixing is restricted in the macromolecular scale, which gives rise to arrays of microdomains with specific geometries. Such a process is often called as self-assembly. Crystallization after such microphase separation meets two interplay effects: on the one hand, the interfaces with nano-scale spacing make confinement to polymer crystallization; on the other hand, crystallization may break out of the microdomains and thus causes destruction of the pattern with nanoresolutions. There have been extensive investigations of molecular simulations on the above-mentioned intramolecular interplay of phase transitions in diblock copolymers. We expect another mini-review focusing on diblock copolymers in near future. This does not mean that the interplay in other kinds of copolymers can be neglected, although they have received little investigations up to now. The molecular simulations of other copolymers will be on the march and demonstrate again the power of modeling. Acknowledgments This work was supported by the National Natural Science Foundation of China (NSFC Grant No. 20674036 and 20825415). References H. C. Raine, R. B. Richards and H. Ryder, Trans. Faraday Soc, 41, 56 (1945). R. B. Richards, Trans. Faraday Soc., 42, 10 (1946). X. W. He, J. Herz, and J. M. Guenet, Macromolecules, 20, 2003(1987). L. Aerts, H. Berghmans, and R. Koningsveld, Makromol. Chem., 194, 2697(1993). P. J. Flory, L. Mandelkern and H. K. Hall, J. Am. Chem. Soc., 73, 2532 (1951). P. J. Flory, Principles of Polymer Chemistry (Cornell University Press, Ithaca, 1953). 7. F. Deberdt and H. Berghmans, Polymer, 34, 2192(1993). 8. F. Deberdt and H. Berghmans, Polymer, 35, 1694(1994).
1. 2. 3. 4. 5. 6.
Interplay of Liquid-Liquid Demixing and Polymer Crystallization 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42.
205
H. Berghmans and F. Deberdt, Phil. Trans., 348, 117(1994). R. Koningsveld, Ph.D. thesis, University of Leiden, 1967. P. Schaaf, B. Lotz and J. C. Wittmann, Polymer, 28,193(1987). N. Inaba, K. Sato, S. Suzuki, and T. Hashimoto, Macromolecules, 19, 1690(1986). H. K. Lee, A. S. Myerson, and K. Levon, Macromolecules, 25, 4002(1992). P. D. Graham and A. J. McHugh, Macromolecules, 31, 2565(1998). H. Berghmans, R. De Cooman, J. De Rudder, and R. Koningsveld, Polymer, 39, 4621(1998). A. Keller, Faraday Discuss., 101, 1(1995). J. M. Guenet, Thermochim. Acta, 284, 67(1996). H. Wang, K. Shimizu, H. Kim, E. K. Hobbie, Z. G. Wang, C. C. Han, J. Chem. Phys., 116, 7311(2002). G. Matsuba, K. Shimizu, H.Wang, Z. G.Wang, and C. C. Han, Polymer, 44, 7459 (2003). X. H. Zhang, Z. G. Wang, M. Muthukumar, C. C. Han, Macromol. Rapid. Commun., 26, 1285(2005). X. H. Zhang, Z. G. Wang, X. Dong, D. J. Wang, C. C. Han, J. Chem. Phys., 125, 024907(2006). X. H. Zhang, Z. G. Wang, R. Y. Zhang, C. C. Han, Macromolecules, 39, 9285(2006). Y. H. Niu, Z. G. Wang, C. A. Orta, D. H. Xu, H. Wang, K. Shimizu et al., Polymer, 48, 6668(2007). S. Z. D. Cheng. Phase Transitions in Polymers (Elsevier, Amsterdam, 2008). J. S. Rowlinson, Faraday Disc. Chem. Soc., 49, 30(1970). K. H. Meyer, Z. Phys. Chem. B, 44, 383(1939) M. L. Huggins, Ann. NY Acad. Sci., 43, 1(1942) P. J. Flory, J. Chem. Phys., 10, 51(1942). R. L. Scott, J. Chem. Phys., 17, 279(1949). H. Tompa, Trans. Faraday Soc., 45, 1142(1949). I. C. Sanchez and M. T. Stone, in Polymer Blends: Formulation & Performance, edited by D. R. Paul and C. B. Bucknall (Wiley, New York, 2000), Vol. 1, Chap. 2. K. F. Freed and J. Dudowicz, Adv. Polym. Sci., 183, 63(2005). P. J. Flory, J. Rehner Jr, J. Chem. Phys., 11, 521(1943). L. Leibler, Macromolecules, 13,1602(1980). E. A. Guggenheim, Mixtures (Clarendon, Oxford, 1952). R. Koningsveld and L. A. Kleintjens, Macromolecules, 4, 637 (1971). M. G. Bawendi and K. F. Freed, J. Chem. Phys., 88, 2741(1988). D. Buta, K. F. Freed, and I. Szleifer, J. Chem. Phys., 112, 6040(2000). P. J. Flory, Proc. R. Soc. London, Ser. A, 234, 60(1956). L. Onsager, Ann. NY Acad. Sci., 51, 627(1949). A. Ishihara, J. Chem. Phys., 19, 1142(1951). W. Maier and A. Y. Saupe, Z. Naturforsch., 14A, 882 (1959); 15A, 287(1959).
206 43. 44. 45. 46. 47. 48. 49. 50. 51. 52. 53. 54. 55. 56. 57. 58. 59. 60. 61.
62. 63. 64.
W.-B. Hu W.-B. Hu, J. Chem. Phys., 113, 3901(2000). W. B. Hu, D. Frenkel, V. B. F. Mathot, J. Chem. Phys., 118, 10343(2003). W. B. Hu and D. Frenkel, Adv. Polym. Sci., 191,1(2005). W. B. Hu, J. Chem. Phys., 109, 3686(1998). R. Silvestri and P. Sgarzi, Polymer, 39, 5871(1998). Z. G. Wang, R. A. Phillips, and B. S. Hsiao, J. Polym. Sci., Part B: Polym. Phys., 38, 2580(2000). R. Thomann, J. Kressler, S. Setz, C. Wang, and R. Muelhaupt, Polymer, 37, 2627 (1996). R.-D. Maier, R. Thomann, J. Kressler, R. Muelhaupt, and B. Rudolf, J. Polym. Sci., Part B: Polym. Phys., 35, 1135(1997). T. Haliloglu and W. L. Mattice, J. Chem. Phys., 111, 4327(1999). D. Lohse, Polym. Eng. Sci., 26, 1500(1986). W.-B. Hu and V. B. F. Mathot, J. Chem. Phys., 119, 10953(2003). Y. Ma, W.-B. Hu, H. Wang, Phys. Rev. E, 76, 031801 (2007). W. B. Hu and D. Frenkel, Macromolecules, 37, 4336(2004). L. Y. Zha and W. B. Hu, J. Phys. Chem. B, 111, 11373(2007). Y. Ma, L. Y. Zha, W.-B. Hu, G. Reiter, C. C. Han, Phys. Rev. E, 77, 061801(2008). L. Y. Zha and W. B. Hu, Polymer, 50, 3828(2009). W.-B. Hu, D. Frenkel, V. B. F. Mathot, J. Chem. Phys., 118, 3455(2003). W.-B. Hu, D. Frenkel, V. B. F. Mathot, Macromolecules, 36, 8178(2003). W.-B. Hu, in Lecture Notes in Physics Vol. 714: Progress in Understanding of Polymer Crystallization, Ed. by Reiter, G. and Strobl, G. (Springer-Verlag: Berlin, Heidelberg, 2007), p 47. K. A. Dill and H. S. Chan, Nat. Struct. Biol., 4, 10(1997). V. Daggett and A. R. Fersht, Trends Biochem. Sci., 28, 18(2003). W.-B. Hu and D. Frenkel, J. Phys. Chem., B, 110, 3734(2006).
Chapter 7 Elucidation of Single Molecular Observation of a Giant DNA
Chwen-Yang Shew Department of Chemistry, Graduate Center and College of Staten Island, City University of New York, Staten Island, NY 10314, USA E-mail: [email protected] Kenichi Yoshikawa Department of Physics, Graduate School of Science, Kyoto University, Kyoto, 606-8502, Japan E-mail: [email protected] Single molecular images detail the conformational behavior of a giant DNA, important for both life science and polymer physics. Mediated by condensing agents, DNA forms compact structure, like the coilglobule transition of a single polymer molecule. Through adjusting condensing agent concentration, DNA undergoes a conformational transition, as in the first-order phase transition, with coexistent elongated and compact states. The conformation of a DNA is governed by its two intrinsic properties: semflexibility and charge. In contrast to flexible chains, semiflexible chains exhibit many unique properties, including discrete conformational transition, highly ordered toroid structure in compact state, and conformational hysteresis. The origin of conformational hysteresis can be understood from kinetics and thermodynamics. Moreover, the charge of DNA contributes to intramolecular segregation, with both compact and coiled segments coexistent within a singe chain. This is different from intermolecular segregation in which each DNA takes either coiled or compact structure. Meanwhile, with cationic condensing agents, an increase of temperature enhances the folding of DNA due to ion exchange among counterions. The recent experiment on DNA gels is also discussed. This review aims at elucidating the observed conformational behaviors from theoretical standpoint. 207
208
C.-Y. Shew and K. Yoshikawa
1. Polymer Physics Aspect of DNA Conformation DNA is one of the most studied molecules because of its active role in life science.1-5 As a matter of fact, the number of cells in a human body is in the order of 1014, and DNA is located in the nucleus of each cell.6 The total length of a completely stretched DNA molecule in human cells can reach 2 m long or so.7 How can the tiny cell8 at around 10-20 µm accommodate such a long molecule? Despite its gigantic length, the diameter of a double helix DNA is indeed only about 2 nm, far smaller than a cell.4 To further address DNA compaction in a cell, one needs to revisit DNA from the standpoint of polymer physics. DNA can be classified as semiflexible chain and/or polyelectrolyte. The semiflexibility arises from the local rigidity of double helix structure with a persistence length around 50 nm, and the charge9,10 is due to the highly ionizable phosphate groups. These two intrinsic properties tend to elongate the chain molecule, and impose hurdles to contract a long chain DNA. Odijk11 and Skolnick and Fixman12 suggest that the total persistence length lp of DNA is the sum of intrinsic persistence length (lp0) and electrostatic persistence length (lpe). l p = l p0 + l pe
(1)
where lpe = lBλ2/4κ2 without considering counterion condensation; lB is the Bjerrum length, lB = e2/4πεε0kBT (≈ 0.7 Å at T = 25 oC in pure water with dielectric constant ε ≈ 78); λ is mean line charge density along the chain; κ (in nm) is the inverse Debye screening length (= 8πl B IN A with I = ionic strength in mM; NA = Avogadro’s number). In Eq. (1), lpe depends on monomer charge and salt concentration. In nature, histone proteins with charge opposite to DNA assist contraction of a giant DNA within a human cell. There are multiple classes of histone proteins, but roughly, they create two major functions in DNA compaction: the first type keeps DNA winding around adsorbed histones to form nucleosomes; the other type serves as linkers to bridge nucleosomes within a DNA.13 Contraction of a DNA molecule is a kind of coil-globule transition in polymer physics. Such a transition is induced by changing solvent quality from good to poor solvents. In good
Elucidation of Single Molecular Observation of a Giant DNA
209
solvents, the effective interaction between monomers is repulsive, whereas in poor solvents, it becomes attractive. At Θ−solvent, the effective interaction of monomers is essentially zero (ideal chain).14 In the next section, the brief overview of the experimental technique developed to measure DNA conformation will be given. The rest of sections divide discussions into DNA as semiflexible chains in Sections 3, 4 and 5 as well as DNA as polyelectrolytes in Sections 6 and 7. Extension of previous studies to a new system regarding DNA gels will be briefly reviewed in Section 8 with conclusion remarks to follow. 2. Manipulation and Measurement DNA Conformation In Vitro 2.1 Condensing agents Water is a good solvent for a DNA owing to the strong dipole-ion interaction between solvent and the ionic groups of DNA. To reduce the solvent quality of a DNA solution, it is common to mix the solution with condensing agents,2 such as high valent metal ions or polycations, to neutralize the DNA charges with great extent and induce effective attraction among monomer units. Mixing DNA with condensing agent is a common laboratory procedure in purification of DNA by precipitating out DNA molecules in the solution of high enough DNA concentration.2 While the DNA concentration is relatively low, precipitation becomes less significant. Instead, compaction of an individual DNA molecule occurs, where the compact DNA behaves like a charged colloid because of the surviving negative charge on the surface. With the addition of excess condensing agent, the compact DNA molecules form aggregate/precipitate. High valent cations are different from histones in chromosomes because their sizes are far too small for a DNA to wind around them. Besides multivalent metal ions and polycations, a broad range of condensing agents have been identified, including surfactants, charged lipids and even neutral polyethylene glycol (PEG). DNA compaction in high PEG concentration is attributed to crowding effect, refraining DNA from elongating in solution.1-4
210
C.-Y. Shew and K. Yoshikawa
2.2 Single molecular images A systematic study of the conformational behavior of DNA is advanced through using single molecular imaging method that rendered a thorough study on the unique conformational behavior of a giant DNA in vitro.1-5 Single molecular images of DNA are obtained by using a high sensitive fluorescence microscope. Since no significant fluorescence emitted from DNA, fluorescent chromophores are introduced to bind with DNA. The recent reviews have systematically investigated each type of chromophore, and concluded that DAPI is most advantageous to facilitate the observation of the conformation transition of DNA. Actually, DAPI is a minor groove binder with DNA. Even though it slightly changes the persistent length of DNA, there is no perceivable influence on contour length as well as no hindrance on formation of compact DNA. It is noted that as opposed to DAPI, the chromophores, acting as intercalators, often retard DNA compaction.2 Due to limited spatial resolution, DNA fluorescence images are subjected to the blurred effect at around 0.3 µm. Actually, the naturally occurring DNA, for example T4- and λ−DNA, have contour lengths in the order of a few tens of micrometer. These giant DNA molecules become ideal systems for conformational study through this optical method, like “live simulations” without using computers.1-4 Molecular images also render the statistical histogram of molecular sizes. Figure 1 displays the single molecular images of T4DNA (three-dimension representation of the fluorescence intensity from a DNA), the mean longaxis length L against spermidine (spd) (+3) concentrations, and the histogram of L. In the control experiment without spd, DNA is elongated and exhibits coiled morphology, whereas in higher spd concentrations, DNA collapses and forms compact structure. In Fig. 1, the mean L first decreases slightly but continuously for spd below 140 µM. In the medium spd concentrations between 140 and 160 µM, two distinct morphologies emerge simultaneously: elongated and compact ones with different mean L. After spd is increased to 160 µM, only one conformational state exists, and the chain size becomes leveled off rapidly. The corresponding histograms of chain size (or L) display the
Elucidation of Single Molecular Observation of a Giant DNA
211
Fig. 1. (Left) Histogram of long-axis length L of T4DNA for different concentrations of spermidine (+3), spd. (Right top) Variation of average L as a function of spd concentration for elongated-coil and folded-compact states, where grey band corresponds to the coexistence region. (Right bottom) Examples of the fluorescence image for the coil and compact states. Modified based on the data reported in Ref. 15.
single modal distribution for both low and high spd concentrations, but the bimodal profile appears when these two conformational states coexit.2 Single molecular images enable us to discern the DNA molecules freely moving in solution from those adsorbed onto a substrate. In the
212
C.-Y. Shew and K. Yoshikawa
former case, DNA undergoes Brownian motion with fuzzy fluorescence images, while those fixed on the substrate exhibit sharp images. 2.3 Limitation of traditional light scattering Prior to single molecular imaging method, scattering method was widely used to study chain conformation. However, scattering method failed to discern the region where the elongated and compact states coexist due to distinct effective segmental densities between elongated state (very low scattering intensity) and compact state (high scattering intensity). Namely, the average physico-chemical properties of DNA in solution seem continuous through the observation by light scattering. On the contrary, single molecular images are based on the real-time observation of each molecule, which are not subjected to ensemble average.15 3. All-or-none Conformation Transition of DNA A systematic study has shown that the coil-globule transition of a neutral polymer, such as polystyrene in hexane, is continuous.16 In this section, the effect of condensing agents on DNA conformation and the role of DNA semiflexibility will be discussed. 3.1 Discrete conformational transition of DNA Figure 1 discloses the fundamental physics to elucidate the nature of the conformational transition of DNA.15 From the histogram of the long-axis length L of T4DNA, two distinct conformational states, coil and globule, are characterized through bimodal chain size distribution, indicating two coexistent conformational states when the spd concentration is between 140 and 160 µM. The molecular volume of the coil state is estimated to be 104–105 greater than that of the compact state. This all-or-none transition is very different from the continuous conformational transition observed in a neutral polymer. Namely, the long chain DNA undergoes a discrete conformational transition, accompanied with a finite region of coexistent compact and elongated conformations. The discrete transition
Elucidation of Single Molecular Observation of a Giant DNA
213
is an indication of the first-order phase transition under the criterion of symmetry argument by Landau.1-4 For a giant DNA, the conformational transition is more like liquidsolid or vapor-solid transition of pure substances. Figure 2 shows the morphologies of the globule state of DNA under spermidine (+3), consisting of highly ordered rod (left) and toroid structure (right) similar to crystal states. It has been claimed that the cations must at least be trivalent to become an effective condensing agent, which can neutralize more than 90% of charged groups in a DNA,2 following the framework of the counterion condensation theory.17-18 This argument implies that “DNA condensation” is a continuous process. However, the new experimental evidence indicates that divalent amines also induce compaction.19 This finding clarifies the concept of DNA condensation. The onset of DNA condensation should mean the appearance of single DNA compaction, and the thus obtained compact DNA is essentially neutral except some residual negative charges on its surface.
Fig. 2. Transmission electron micrographs of rod morphology (left) and toroid structure (right) in the compact state of λ-DNA induced by spermidine (+3) from Ref. 2. The side bar is 0.1 µm. The toroid structure is somewhat larger than the typical dimension (ca. 80 nm in diameter), when it is formed at high spermidine concentration.
3.2 Chain stiffness and discrete conformational transition Computer simulations have been applied to understand the influences of chain stiffness on the all-or-none conformational transition, with monomer charge excluded to model high salt solution in experiment. In the simulation, an increase of condensing agent lowers solvent quality and increases the attraction between monomers. From such a simple approximation, it becomes possible to deduce the intrinsic characteristic
214
C.-Y. Shew and K. Yoshikawa
of a semiflexible chain after the grasp of the essential property of a giant DNA molecule, i.e., a linear molecule with a long contour length and an enough number of Kuhn segments. Figure 3 is the phase diagram of the amount of condensing agent (or effective monomer-monomer interaction) vs. chain stiffness obtained from Monte Carlo simulation with multi-Canonical method. For flexible chains, the coil-globule transition is continuous, and the globule state is liquid-like, lack of ordered structure. For stiffer chains, the discrete transition is observed, along with formation of the highly ordered toroid structure after chain compaction. The role of semiflexibility is clearly illustrated in the calculation. Note that the rod-like morphology may arise from the kinetics reason, other than a thermodynamically stable state. The above simulation results agree well with experimental observations, and have stimulated extensive studies on the properties of a semiflexible chain since then.20
Fig. 3. Simulated phase diagram for polymer chain as a function of chain stiffness and condensing agent concentration from Ref. 20.
4. Dynamics of Conformational Relaxation Time-evolution single molecular fluorescent images allow us to acquire the real-time conformational relaxation of a DNA. Since most of experiments were conducted under high salt solutions, semiflexibility is a
Elucidation of Single Molecular Observation of a Giant DNA
215
key factor to govern the conformational behavior of DNA and its chain dynamics. Sections 4.1 and 4.2 review experimental measurement and theoretical analysis, respectively.
Fig. 4. Time dependent fluorescence intensity and long-axis length L, along with timeelapse snapshots, for the folding (top) and unfolding (bottom) processes of T4DNA in PEG solution from Ref. 21.
4.1 Time dependent conformational behavior The observation of the chain dynamics of DNA has been systematically carried out. Figure 4 is the time dependent fluorescence intensity and long-axis length L, together with snapshots, for the folding (top) and unfolding (bottom) processes of T4DNA by a sudden increase and decrease of PEG concentration, respectively. The folding process is
216
C.-Y. Shew and K. Yoshikawa
usually initiated from either end of a DNA chain as a nucleation site. Then, the chain segments “condense” on the nucleation site with a constant rate (top middle image), similar to crystal nucleation growth. The unfolding process undertakes a different pathway, which is slow in the beginning and is accelerated after surpassing the initial stage.21 The chain compaction has also been studied by using dynamics simulations for a semiflexible chain with sticky monomers to model the poor solvent condition. Figure 5 shows the time evolution of compaction from a coil to a toroid or a rod.5 The time dependent snapshots capture the feature of experimental observations, and enable us to visualize the pathways of the folding process.
Fig. 5. Dynamics simulation for the folding process of a semiflexible chain model from coil to toroid (top) and rod structure (bottom) from Ref. 5.
4.2 Folding and unfolding kinetics A critical analysis for the folding and unfolding kinetics of a semiflexible chain has been carried out based on the de Gennes’s argument, in which the rate of heat dissipation of a process is equal to the rate of the change of free energy,14 that is, dQ dF ~− dt dt
(2)
Elucidation of Single Molecular Observation of a Giant DNA
217
Figure 6 is the schematic representations of the folding and unfolding pathways. In folding, the nucleation-and-growth mechanism dominates the process, whereas for the unfolding process, three steps are involved, including swelling, disentangle, and relaxation. In compaction, DNA nucleates at one end, and starts to fold into the toroidal segment.22 If the heat dissipation to ambient solvents can be neglected, the heat produced by the free energy change due to compaction is dissipated to the coil segment. By equating the rate of free energy change and heat dissipation, the equation of motion at the limit RL >> R can be obtained as follows.22 εl p ln(l p / a) dRL (3) ~− ηL2 dt Note all the parameters in Eq. (3) are defined in Fig. 6. On r.h.s of the above equation, all parameters are time-independent, meaning that the contraction of chain size is linearly proportional to time t. Such a result agrees with the experimental observation in Fig. 4.
Fig. 6. Schematics of the folding (top) and unfolding (bottom) pathways for a polymer of contour length Lc (=l1 + l2) and diameter a from Ref. 22, where l1 and l2 are the contour lengths of the coil and tororid segments; R and r are the diameters of toroid and tube, respectively; RL is the long-axis length of coil (≈ l2); R⊥ is the lateral fluctuation.
As the toroid is changed from poor solvents to good solvents, the compact structure is expected to swell till the point all monomer units are frozen due to entanglement. The subsequent relaxation process is disentanglement, of which free energy change is governed by the change
218
C.-Y. Shew and K. Yoshikawa
of elastic energy, and its heat is dissipated into the inside of each blob with segments (~ L/lp). The balance of energy and heat results in RL t ~ ( + 1)1/ 8 τ2 R1
(4)
where R1 ~ L1/3a 2/3 and τ2 ~ ηL2/3a16/3/(Tlp3 ln(lp/a)). For the long enough time scale, RL ~ t1/8. In the final step, the heat dissipation is expected to take place at the level of a segment, and from the analysis of equation of motion, it can be shown that RL t ~ ( + 1)1/ 4 τ3 R2
(5)
where R2 ~ (Llp)1/2 and τ3 ~ ηL2lp/(Tln(lp/a)). This leads to the power law RL ~ t1/4.22 These scaling relations can be further tested in the future computer simulation. From the standpoint of experiment, the pronounced time-scale difference between these steps (τ3/τ2 ~ L4/3lp4) may suggest different techniques required to discern τ2 and τ3. 5. Conformational Hysteresis Due to the nature of the first-order phase transition, hysteresis may occur in the conformational transition of DNA. Sec 5.1 summarizes the experimental measurement of the hysteresis loop for the folding and unfolding processes. This kinetic phenomenon can be rationalized from the aspect of thermodynamics given in Section 5.2. In Section 5.3, the mechanical force induced conformational hysteresis will be reviewed. 5.1 Characterization of hysteresis The conformational hysteresis is characterized by using T4DNA of contour length around 50 µm mixed with PEG under the NaCl concentration of 300 mM to largely screen out the electrostatic interaction within a giant DNA. The DNA solution is kept as low as 0.2 µM to reduce intermolecular interaction. The forward and backward titration are then conducted to induce the folding and unfolding processes of T4DNA by increasing and decreasing PEG concentration,
Elucidation of Single Molecular Observation of a Giant DNA
219
respectively. Figure 7 shows histograms of the long-axis length of DNA and variation of the fraction of elongated state of DNA with PEG concentration for the folding and unfolding processes, as marked, after incubating for 60 and 1000 minutes. For both incubation times, the folding and unfolding processes form hysteresis loops, suggesting that these two processes undertake different pathways. The folding process is kinetically slower than that of the unfolding process because of the longer incubation time needed to induce the process.23
Fig. 7. Histograms of the long-axis length and fraction of coiled state of T4DNA in different PEG concentrations for the folding and unfolding processes, induced by titration and backtitration of DNA solution with PEG from Ref. 23.
5.2 Thermodynamics in conformational hysteresis Monte Carlo simulations are resorted to examine the role of chain semiflexibility on the hysteresis loop formation by using a bead-spring chain model. Two parameters are introduced in the model: bending energy to adjust semiflexibility and monomer-monomer attraction via attractive Yukawa potential. In the simulation, the monomers undergo random walk, equivalent to their Brownian motion. Figure 8 plots the
220
C.-Y. Shew and K. Yoshikawa
Fig. 8. Variation of simulated mean squared radius of gyration, divided by a factor Γ, with ε (strength of monomer-monomer interaction) and some snapshots for κ = 2 (Γ = 1) and 10 (Γ = 10), where κ is the strength of bending energy, when N = 8, and for κ = 20 (Γ = 100) when N = 32, as marked, under the folding (solid symbols) and unfolding (open symbols) processes; lines provide as a visual guide, and the factor Γ is introduced to collapse all the curves in the same plot. (Ref. 23)
simulated radius gyration (chain size) as a function of the strength of monomer-monomer attraction for various chain stiffness, along the folding and unfolding processes. In the simulation, the hysteresis loop emerges only for stiff chains, but not for flexible chains. Within the hysteresis loop, the elongated and compact chain conformations coexist in a range of ε. To determine their relative thermodynamic stability, the Density of State Monte Carlo (DOSMC) simulation is applied. The energy distribution of elongated and compact conformational states within the hysteresis loop obtained from random walk Monte Carlo simulation is compared with the calculation of DOSMC. The DOSMC reproduces the energy distribution corresponding to compact state, meaning that the compact conformation in the hysteresis loop is more stable. The DOSMC also shows that the number of density of states at lower energy decreases for a stiffer chain but it increases as the monomer-monomer attraction is increased. These results explain why the elongated DNA chain, thermodynamically less stable,
Elucidation of Single Molecular Observation of a Giant DNA
221
persists until the concentration of condensing agents becomes high enough. An increase of chain stiffness attenuates the number of chain conformations, including most of intermediate conformations required to make a transition to compact state. As a result, the folding process is impeded, in particular, when the intermediate conformations have a higher energy. On the contrary, an increase of the concentration of condensing agent enhances monomer-monomer attraction, which facilitates the folding process by increasing the number of possible chain conformations.23 As for the unfolding process, it has been shown that the nucleation process is not involved, and the unfolding rating is determined by different mechanisms as discussed in Sec. 4.2.
Fig. 9. End-to-end distance dependent force response in flexible (A) and semiflexible (B) chains, and the pathways for the unloading/loading processes of a flexible chain in (C) and a semiflexible chain in (D), obtained from the simulation in Ref. 24.
5.3 Hysteresis under mechanical forces The hysteresis behavior for the folding and unfolding cycle of a semiflexible chain under mechanical forces has been recently investigated through Langevin dynamics simulation by letting one chain end fixed and moving the other end at constant velocity.24 Figure 9
222
C.-Y. Shew and K. Yoshikawa
shows the simulation results for the chain size (characterized by end-toend distance) dependent force response for flexible and semiflexible chains along with their pathways for the unloading/loading processes. For flexible chains, the unloading/loading cycle is near reversible, and the two processes take the same pathway with a globule embedded within the chain molecule as the intermediate state. For semiflexible chains, two distinct force response curves appear for the unloading/ loading processes, and the hysteresis is clearly observed. In the loading process, the chain is gradually adsorbed into the cylindrical segment, but in the unloading process, the intermediate toroid segment is formed before the elongated chain structure is reached. A further calculation shows that the hysteresis area of a semiflexible chain tends to be larger than that of a flexible chain under a broad range of external forces. Generally, the hysteresis area decreases and gradually diminishes as the operation is done in an infinitesimally small fashion, but for semiflexible chains, the possible high free energy barrier between the cylinder and toroid states may hinder the closure of hysteresis. Owing to the distinct intermediate states for the unloading and loading processes, it requires at least two order parameters in the free energy expression. A possible free energy form is given by µ µ 1 1 (6) F ( X , Y ) = − 1 X 2 − 2 Y 2 − αXY + X 4 + Y 4 − h(t ) X 2
2
4
4
where X and Y are two order parameters, such as monomer density and the orientational order of a polymer; µ1 and µ2 are positive coefficients; α is the coupling constant between X and Y. µ1 and µ2 may be interpreted as the gradient to regulate the change of X and Y, respectively. h(t) can be viewed as time dependent end-to-end distance. With the free energy, the kinetic equations based on Langevin equation for X and Y are respectively given by ∂F dX = − L1 + ξ1 = L1 (− X 3 + µ1 X + αY + h(t )) + ξ1 dt ∂X
(7)
∂F dY = − L2 + ξ 2 = L2 (−Y 3 + µ 2Y + αX ) + ξ 2 dt ∂Y
(8)
Elucidation of Single Molecular Observation of a Giant DNA
223
where L1 and L2 are the Onsager coefficients (L1 = L2 = 1 in Ref. 24); ξ1 and ξ2 are the Gaussian random forces. In the calculation, it assumes that the force is changed linearly in time, and h(t) = vt + h0 where v (=1/λ) is the velocity to drag one chain end and h0 is the initial value (e.g., end-toend distance). Also, µ1 is chosen be to 100 and α = 5.0. By adjusting µ2, the dynamical behavior of flexible chain and semiflexible chain is reproduced. In Fig. 10, the hysteresis area is plotted against inverse velocity λ. µ2 = 0.1 and 100 qualitatively agree with the behavior of flexible chains and semiflexible chains in the simulations, respectively.24 Also, for the zero external force (h(t) = h0), the hysteresis for flexible chains completely diminishes, but it persists in semiflexible chains. Such a result is consistent with the work introduced in Section 5.2.
Fig. 10. End-to-end distance dependent force response in flexible (A) and semiflexible (B) chains, and the pathways for unloading/loading processes for a flexible chain in (C) and a semiflexible chain in (D), obtained from the simulation in Ref. 24.
Fig. 11. Schematic diagram of the possible pathways to form intermolecular and intramolecular segregation from Ref. 25 by increasing condensing agents from left to right, and a fluorescence image of intramolecular segregation of DNA from Ref. 26.
224
C.-Y. Shew and K. Yoshikawa
6. Effect of Charge on DNA Conformation The discrete conformational transition of DNA induces the coexistence of coiled and globular conformational states. This process is known as intermolecular segregation. However, intramolecular segregation has also been identified, containing coexistent coiled and globular segments within a single giant DNA. Figure 11 is the schematic plot of intermolecular and intramolecular segregation25 together with a fluorescence image for the intramolecular segregation of DNA.26
Fig. 12. AFM images of PVA in different pH values, as marked, with collapsed globules at pH = 10, and pearl necklace structure at pH = 4.0 and 4.9, from Ref. 27.
6.1 Conditions to induce intramolecular segregation The intramolecular segregation can be induced by charged condensing agents, for instance cationic PEG-A1, and the image in Fig. 11 indicates that the morphology of DNA undertakes the form of ring-on-a-string, and the ring structure is essentially an ordered toroid.26 The total net charge of DNA is crucial for the formation of intramolecular segregation. 6.2 Polyelectrolyte analogy Recently, Kirwan et al prepared poly(vinyl amine) (PVA) solution of different pH to adjust the total charge of polymers.27 These ionic polymers are then adsorbed rapidly onto a substrate to freeze their chain
Elucidation of Single Molecular Observation of a Giant DNA
225
conformations, and AFM is used to probe the morphology of polyelectrolyte chains. They found that PVA makes a transition from a collapsed chain at high pH to an elongated chain at lower pH, as shown in Fig. 12. At high pH, PVA is essentially in poor solvents, without charges, due to its hydrophobic backbone. A reduction of pH increases the total charge on the chain through protonation, and turns the polymer chain into a polyelectrolyte but the poor solvent condition is maintained by its backbone. The pearl necklace chain structure, similar to intramolecular segregation in Fig. 11, is found at pH = 4.0 and 4.9 with collapsed globules connected by coiled chain segments. Unlike the toroid rings in the intramolecular segregated DNA, these intramolecular globules are disordered. A further decrease of pH transforms the pearl necklace chain into an elongated coil,27 suggesting the conformation depends on the net charge of a polyion under poor solvents. 6.3 Phase diagram of intermolecular and intramolecular segregation In addition to its semiflexibilty, the giant DNA exhibits the behavior of ionic polymers, and its charge may affect local chain stiffness. Experimentally, the condensing agents of DNA induce intermolecular and intramolecular segregation by regulating the total net charge of DNA, as shown in Fig. 12. For intermolecular segregation, a giant DNA forms either elongated or compact conformation, and such a transition is known as the first-order phase transition. For intramolecular segregation, elongated and compact chain segments coexist within a chain molecule. The challenge lies in how to correlate these two conformational transitions in the same phase diagram. Motivated by experiments, the DiMarzio theory is extended to elucidate the inter- and intra-molecular phase segregation of a single flexible polyelectrolyte in dilute salt-free solutions.25 The theory formulates the single chain free energy in terms of chain size R or swelling ratio α (= R/R0 where R0 is the size of an ideal chain), with solvent quality and electrostatic interaction incorporated. Thus, the size distribution of a polymer chain W(R) takes the following form25,28 W ( R) = [ R 2 exp(−
3R 2 jb3 Nb3 3ΓN 2 )][∏ nj−=10 (1 − 3 )][exp{− χ (1 − 3 )][exp(− )] 2 R R R
(9)
226
C.-Y. Shew and K. Yoshikawa
where χ is the Flory-Huggins χ-parameter to depict solvent qualities; Γ is the interaction strength of Coulomb interaction within a chain (Γ = lB f 2/3, where lB is the Bjerrum length and f is the monomer charge fraction). In the free energy expression, the first argument depicts the elasticity of a flexible chain. The second argument is the excluded volume interaction. The third term takes the polymer-solvent interaction into account. The forth term considers the electrostatic repulsion of a salt-free polyelectrolyte chain. DiMarzio has suggested that the free energy expression should be F/kBT = −ln[RW(R)].29-30 In experiment, this free energy is corresponding to the conditional probability to depict the statistical distribution of chain size in histogram (~ exp(–F(R)/kBT)). At the long chain limit, this theory yields the same formalism obtained from the more sophisticated polyelectrolyte theory based on Edwards Hamiltonian,31-34 which reads
α 5 −α 3 =
α −3 3
1 − ( χ − ) N 1/ 2 + (ΓN 3 / 2 )α 2 . 2
(10)
Equation (10) generates the correct power laws for the conformational behavior of a neutral and an ionic chain, as summarized in Table 1. Table 1. Predicted power laws from Eq. (10). Polymer type Neutral
Solvent condition Good
Power Laws R2 ~ N6/5
References 29
R2 ~ N
29
Poor
R2 ~ N2/3
29
Good
R2 ~ Γ2/3N2
28
Θ
Polyelectrolyte 28
Θ R2 ~ Γ2/3N2
Note: The conformational behavior of a polyelectrolyte in poor solvents is discussed in context.
The above mean-field model reveals the impact of polymer charge on inter- and intra-molecular segregation. The theory predicts the first-order phase transition in poor solvents for an ionic polymer, with continuous and discontinuous transitions separated by a critical point. The critical
Elucidation of Single Molecular Observation of a Giant DNA
227
constants follow the power laws: (χ* − 1/2) ~ N −1/2 and Γ* ~ N −3/2. These scaling relations indicate that for a long enough chain, a small monomer charge and the solvent quality slightly below the Θ-solvent would trigger the first order conformational transition. Figure 13 plots the calculated phase diagram for Coulomb interaction strength Γ against swelling ratio α for two different χ. As Γ is increased, the globular ionic chain starts to swell continuously prior to approaching the critical point. Under the discontinuous transition, the polyelectrolyte chain exhibits coexistent elongated and collapsed conformational states of equal free energy, same as the inter-molecular phase segregation. It can be shown that the intramolecular segregation is attainable from Maxwell construction,25 which is also relevant to Rayleigh splitting, like a highly charged water droplet splitting into smaller droplets to reduce its electrostatic repulsion.35 This Rayleigh splitting has been attributed to the formation of pearl necklace chain for a polyelectrolyte in poor solvents.36 The charge needed to induce intramolecular segregation is smaller than that of intermolecular segregation for a given chain length. Moreover, the difference between intermolecular and intramoelcular segregation
Fig. 13. Comparison of phase boundary obtained from the Maxwell construction method (broken line), and the equal free energy method (dotted line) for N = 1000. χ = 0.58 and 0.62 exemplify the calculated critical charge fraction for spontaneous Rayleigh splitting through Maxwell construction, denoted by solid lines, and the two bistable conformations of equal free energy connected by dotted broken lines. This figure is from Ref. 25.
228
C.-Y. Shew and K. Yoshikawa
becomes more pronounced for increasing χ, meaning that by decreasing the solvent quality, it may enhance intramolecular segregation. Also, the findings show that for a longer chain, intramolecular segregated conformations differ significantly from intermolecular segregated conformations, but the difference becomes insignificant for shorter chains. 7. Temperature Effect of DNA Conformation In this section, the thermodynamics of the DNA conformational transition is further investigated by examining the temperature effect. The temperature dependent free energy study gives rise to the entropy of the system. For the conformational transition of DNA, the entropy can be divided into conformational entropy as well as the translational entropy of condensing agents and small ions. The entropy of water is not considered because no experiment evidence is found to link the conformational transition with ambient water at the present stage. 7.1 Temperature induced conformational change The temperature dependent conformational behavior of DNA is greatly sensitive to the choice of condensing agents. For neutral condensing agents, such as PEG, an increase of temperature impedes the chain compaction, whereas for multivalent cations, such as spermine (+4), the DNA chain tends to compact by increasing temperature.1 It is noticeable that the temperature dependent behavior is related to the electrostatic interaction of DNA. This can be shown by adding excess salts into the DNA solution at a given temperature. In the case of neutral condensing agents, an increase of salt concentration enhances chain compaction, but the opposite behavior is found in the presence of multivalent cations. The mechanism to contract a DNA through neutral condensing agents can be attributed to the excluded volume interaction. An increase of salt concentration screens out the monomer charge on the chain, and facilitates chain compaction.1 As for multivalent cations, ion exchange is essential to induce chain contraction as will be discussed below.
Elucidation of Single Molecular Observation of a Giant DNA
229
Recently, Saito et al carried out the temperature dependent experiment for T4DNA in the mixture of spermine (+4) and monovalent cation, with concentration equal to 1.5 mM and 10 mM, respectively.37 From the histogram, the fractions of the elongated (Pcoil) and compact states (Pcompact) are determined. The free energy difference between these two states can be computed from the following expression. ln(Pcoil /Pcompact) = ∆G/T = ∆H/kBT – ∆S/kB
(11)
The transition temperature Tc is defined as the temperature when ∆G = 0 or Pcoil = Pcompact, with equal conditional probability for both conformational states. Figure 14 displays the temperature dependent van ’t Hoff plot, and ln(Pcoil/Pcompact) is linear against 1/T, indicating that ∆H and ∆S are near constant in the experimental temperature range. As a result, the solution enthalpy change per DNA from the coil state to the compact state (∆H) is estimated to be around 30 kBTc, and ∆S = (Scompact − Scoil) = +30 kB. The experimental ∆S also suggests that formation of compact state is favorable at higher temperatures. This result is somehow surprising by considering the high elastic entropy penalty due to chain compaction, and its cause will be given in the next section.
Fig. 14. Temperature dependent histogram of the long-axis length L, and the van Hoff plot to determine ln(Pcoil /Pcompact) against the inverse of temperature from Ref. 37.
230
C.-Y. Shew and K. Yoshikawa
7.2 Competition of smaller ions on compaction The chain elasticity of the compact state is entropically less favorable for the highly ordered chain conformation (with entropy −1000 kB less than the coiled state). Therefore, the driving force for compaction may arise from smaller ions, including cationic condensing agent and salt in solution. These ions contribute to free energy in translational entropy and electrostatic energy. The translation entropy is crucial to compensate the high elastic energy and to stabilize the compact state. With all these factors, the mean field free energy of the system is given by F/kBT = F(elastic) + F(translation) + F(electrostatic), and the total entropy change is given by ∆S = ∆S(elastic) + ∆S(translation). The change of the translational entropy due to compaction (∆S(translation) = Scompact(translation) – Scoil(translation)) is suggested to be ec i v i ec i v i 2 − Q ∑ θ icoil log coil ) . (12) S (translation) = Q ∑ θ icompact log compact θi θi i = +4 , +1, −1 i = +4, +1, −1 where Q is the total charge number of T4DNA (c.a., 3.32 × 103 e), ci is the concentration of ionic species i (spermine, cation or anion), vi is the effective volume of ionic species i, and θi is the fraction of ionic species i bound to each DNA charge site. The translational entropy is computed by assuming that in the compact state, the DNA is totally neutralized by spermines, and the mononvalent ions are freely dispersed in solution. θicoil is then obtained by minimizing the free energy, and the translational entropy change due to ion exchange is estimated to be +1000 kB. To this end, the free energy is lowered because ion exchange contributes to the stronger electrostatic attraction between spermine and DNA as well as an increase of the translation entropy from monovalent ions. 8. Applications Single molecular images reveal the fundamental physics of the conformational behavior of DNA, and are utilized as a tool for a direct examination on the role of DNA in life science. Several recent reviews have concluded that elongation of a DNA chain is relevant to the
Elucidation of Single Molecular Observation of a Giant DNA
231
Fig. 15. Variation of the volume swelling ratio of DNA gels and single DNA with spermidine concentration from Ref. 41.
biological activity in vivo.1-4 Here, an extension of the current study of DNA to the more complex DNA gels is discussed. Gels are ubiquitous in industrial materials and in nature, for example, superabsorbent gels38 and gelation of egg proteins.39 The superabosrbent gels are widely used in sanitary products, such as diapers, based on crosslinking of polyelectrolytes, such as polymethacrylic acid. While dissolving in water, polyelectrolyte gels tend to swell and trap water, due to electrostatic repulsion among ionized groups. In fact, semiflexible DNA can also be crosslinked by introducing ethylene glycol diglycidyl ether (DGDE) as crosslinking reagents under basic solution.40 Mayama et al apply single molecular images to measure the swelling ratio of a single T4DNA and a DNA gel.41 The swelling ratio (V/V0) is defined as the volume ratio of a T4DNA chain (a T4DNA gel) and an ideal chain (or an ideal gel in absence of effective monomer-monomer interaction). Figure 15 compares the swelling ratio of a single DNA and a DNA gel as a function of spermidine (+3) concentration. For single
232
C.-Y. Shew and K. Yoshikawa
DNA molecules, the swelling ratio is near constant at low spermindine concentrations. When the spermidine concentration is as high as 0.12 to 0.25 mM, the two coexistent conformational states emerge. At a higher concentration of spermidine, the chain collapses into compact structure. In contrast, the DNA gel shows a transition between the elongated and collapsed state at [spermidine+3] = 3.2 mM without the coexistent region. In addition, the molecular volume (V) of a DNA chain or a DNA gel is measured. The change of the swelling ratio of volume (V/V0) between the expanded and collapsed states of a DNA gel is significantly smaller than its counterpart of a single DNA molecule. The virial expansion of free energy is used to discern the fundamental difference between a single DNA and a DNA gel41: F = F(elastic) + B ρ2V + C ρ3V + …. where F(elastic) is the chain elastic energy; ρ is the monomer density; B is the second virial coefficient; C is the third virial coefficient; V is the volume occupied by a polymer chain or a gel particle. The elastic energy is same for both cases, i.e., F(elastic) = α2 + 1/α2 where α is the swelling ratio and V/V0 = α3. For single chains, the second virial coefficient can be divided into steric and attractive interactions between segments (or B = B0 + Batt). The steric interaction accounts for the intrinsic volume of a semiflexible segment, and the segment-segment attraction is determined by the degree of segment alignment. For gels, the disturbance of chain alignment is needed in the second virial coefficient, (i.e., B = B0 + Batt + Brep). For two perpendicular segments, their attraction is reduced due to the spatial hindrance to pack with other segments. With the same third virial coefficient expression and overall molecular excluded volume, the free energy of a DNA chain and a DNA gel can be calculated. Figure 16 compares the free energy and size distribution of a DNA chain and a DNA gel with 560 segments for different effective third coefficients (C’= Batt + C), as marked. The two local minima in the free energy show a smaller separation for gels and the discrete transition occurs at a much poorer solvent condition (c.f. a single chain), consistent with Fig 15. The absence of the coexistence region for gels may be due to the high free energy barrier between the expanded and collapsed states of a gel.
Elucidation of Single Molecular Observation of a Giant DNA
233
Fig. 16. Comparison of free energy and distribution of swelling ratio α for DNA gels and single DNA with spermidine concentration from Ref. 41.
9. Concluding Remarks Single molecular images shed new lights on the conformational behavior of DNA, and led to the remarkable finding regarding the discrete coil-globule transition of DNA (as in first-order phase transition). Such a result has opened up a new interdisciplinary research area. The semiflexibility of DNA is the key on its discrete conformational transition, and the formation of the highly ordered toroid compact structure. Dynamically, DNA undergoes different pathways for the folding and unfolding processes, whereas a flexible chain takes a reversible path. During folding, the nucleation-growth mechanism at chain ends dominates, and the number of compact segments is increased with time. For the unfolding process, it takes multiple steps to expand. Owing to different pathways, the folding and unfolding processes form a hysteresis loop. Computer simulations confirm the role of chain stiffness on the conformational hysteresis. Also, the hysteresis is also observed under external forces for the loading and unloading processes. The simulations show that these processes undertake distinct intermediate states for conformational transition. To depict the chain dynamics under external forces, the free energy of a semiflexible chain needs two order parameters, corresponding to two intermediate states.
234
C.-Y. Shew and K. Yoshikawa
Besides its semiflexibility, DNA is also a polyelectrolyte. With condensing agents, a DNA exhibits intermolecular segregation, with distinct coil and compact structure for each molecule. However, some cationic condensing agents induce intramolecular segregation with coexistent compact toroidal and coil segments within a single DNA, taking the rings-on-a-string structure. Such a structure is similar to the pearl necklace chain of a flexible polyelectrolyte in poor solvents except that the pearls are composed of liquid-like collapsed globules. A simple mean field theory is able to correlate intermolecular and intramolecular segregation on the phase diagram. The total charge of an ionic polymer needed to induce intramolecular segregation is different from that of intermolecular segregation. Furthermore, the charge of DNA explains its temperature dependent behavior. Under high valent condensing agents and monovalent ions, an increase of temperature enhances compaction due to the ion exchange of monovalent ions and condensing agents, by which it increases electrostatic attraction between DNA and charged condensing agent, and the translational entropy of monvalent ions. Despite enormous studies, the accurate free energy expression of a single DNA in solution remains unavailable. The more sophisticated field theory has been developed,42,43 but it is difficult to incorporate the excluded volume interaction within a chain. Hence, the mean field theory remains a feasible way to elucidate the conformational behavior of DNA. Furthermore, the future model should be able to combine both charge and semiflexibility for elucidating their interplay in a single giant DNA. Acknowledgments CYS thanks partial supports from the City University of New York PSCCUNY grants and NYSTAR grant for the Center of Engineered Polymeric Materials. KY is partially supported by Japan Society for the Promotion of Science (JSPS) under Grant-in Aid for Creative Scientific Research (Project No. 18GS0421).
Elucidation of Single Molecular Observation of a Giant DNA
235
References 1. K. Yoshikawa, Adv. Drug Delivery Rev. 52, 235-244 (2001). 2. K. Yoshikawa and Y. Yoshikawa, in Pharmaceutical Perspectives of Nucleic AcidBased Therapeutics (R. I. Mahato, Ed.), Taylor & Francis, London, 2002. 3. K. Yoshikawa. J. Biol. Phys. 31, 243 (2005). 4. K. Yoshikawa and Y. Yoshikawa in Non-virial Gene Therapy Gene Design and Delivery, Springer-Verlag (2005). 5. A. Zinchenko, D. Baigl and K. Yoshikawa, in Polymeric Nanostructures and their Applications, Volume 1, Chapter 17, American Scientific Publishers, 2007. 6. C. L. Sears, Anaerobe 5, 247 (2005). 7. McGraw Hill Encyclopedia of Science and Technology New York: McGraw Hill, 1997. 8. A. Maton, D. Lahart, J. Hopkins, M. Q. Warner, S. Johnson, and J. D. Wright Cells Building Blocks of Life. New Jersey: Prentice Hall, 1997. 9. G. S. Manning, Biophys. J. 91, 3607 (2006). 10. A. A. Kornyshev, D. J. Lee, S. Leikin, A. Wynveen, Rev. Modern Phys. 79, 943 (2007) 11. T. Odijk, J. Polym. Sci. 15, 477 (1977). 12. J. Skolnick and M. Fixman, Macromolecules 10, 944 (1977). 13. C. Redona, D. Pilcha, E. Rogakoub, O. Sedelnikovaa, K. Newrocka, and W. Bonnera, Current Opinion in Genetics & Development 12, 162 (2002). 14. P.-G. de Gennes, Scaling Concepts in Polymer Physics, Cornell University Press, Ithaca, NY, 1979. 15. K. Yoshikawa, M. Takahashi, V. V. Vasilevskaya, and A. R. Khokhlov, Phys. Rev. Lett. 76, 3029 (1996). 16. C. Williams, F. Brochard, and H. L. Frisch, Ann. Rev. Phys. Chem. 32, 433 1981. 17. G. S. Manning, J. Chem. Phys. 51, 924 (1969). 18. G. S. Manning, J. Chem. Phys. 51, 934 (1969). 19. Y. Yoshikawa and K. Yoshikawa, FEBS Lett. 361, 277 (1995). 20. H. Noguchi and K. Yoshikawa, Chem. Phys. Lett. 278, 184 (1997). 21. K. Yoshikawa and Y. Matsuzawa, J. Am. Chem. Soc. 118, 929 (1996). 22. N. Yoshinaga, Progress of Theoretical Physics Supplement, 161, 397 (2006). 23. C.-Y. Shew, Y. Higuchi and K. Yoshikawa,” J. Chem. Phys. 127 085103 (2007). 24. N. Yoshinaga, K. Yoshikawa, T. Ohta, Eur. Phys. J. E 17, 485 (2005). 25. C.-Y. Shew and K. Yoshikawa, J. Chem. Phys. 126 144913 (2007). 26. T. Iwaki and K. Yoshikawa, Europhys. Lett. 68, 113 (2004). 27. L. J. Kirwan, G. Papastavrou, and M. Borkovec, Nano Lett. 4, 149 (2004). 28. C.-Y. Shew and A. Yethiraj, J. Chem. Phys. 110, 676 (1999). 29. E. A. DiMarzio, Macromolecules 17, 969 (1984). 30. E. A. DiMarzio and C. M. Guttman, J. Phys. Chem. 93, 7004 (1989).
236 31. 32. 33. 34. 35. 36. 37. 38.
39. 40. 41. 42. 43.
C.-Y. Shew and K. Yoshikawa B.-Y. Ha and D. Thirumalai, Phys. Rev. A 46, R3012 (1992). Y. Chen and A. L. Kholodenko, J. Chem. Phys. 86, 1540 (1987). M. Muthukumar, J. Chem. Phys. 86, 7230 (1987). A. Dua and T. A. Vilgis, Europhys. Lett. 71, 49 (2005). Y. Kantor and M. Kardar, Phys. Rev. E 51, 1299 (1995). A. V. Dobrynin, M. Rubinstein and S. P. Obukhov, Macromolecules 29, 2974 (1996). T. Saito, T. Iwaki, and K. Yoshikawa, Europhys. Lett. 71, 304 (2005). Fredric L. Buchholz and Nicholas A. Peppas, Superabsorbent polymers: Science and Technology, ACS Symposium Series, American Chemical Society Publication (1994). P. W. Gossett, S. S. H. Rizvi, R. C. Baker, Food Technology 38, 67 (1984). T. Amiya and T. Tanaka, Macromolecules 20, 1162 (1987). H. Mayama, T. Nakai, T. Takushi, K. Yoshikawa, J. Chem. Phys. 127, 34901 (2007). Y. Ishimoto and N. Kikuchi, J. Chem. Phys. 128, 134906 (2008). K. Alim and E. Frey, Phys. Rev. Lett. 99, 198102 (2007).
September 29, 2010
10:44
World Scientific Review Volume - 9in x 6in
Chapter 8 Theoretical Modeling of Hydrogen Bonding in Macromolecular Solutions: The Combination of Quantum Mechanics and Molecular Mechanics
Jing Ma,∗ Nan Jiang and Hui Li School of Chemistry and Chemical Engineering, Institute of Theoretical and Computational Chemistry, Key Laboratory of Mesoscopic Chemistry of MOE, Nanjing University, Nanjing, 210093, People’s Republic of China E-mail: ∗ [email protected]
Hydrogen bonding interaction takes an important position in solutions. The non-classic nature of hydrogen bonding requires the resourcedemanding quantum mechanical (QM) calculations. The molecular mechanics (MM) method, with much lower computational load, is applicable to the large-sized system. The combination of QM and MM is an efficient way in the treatment of solution. Taking advantage of the low-cost energy-based fragmentation QM approach (in which the macromolecule is divided into several subsystems, and QM calculation is carried out on each subsystem that is embedded in the environment of background charges of distant parts), the fragmentation-based QM/MM and polarization models have been implemented for the modeling of macromolecule in aqueous solutions, respectively. Within the framework of the fragmentation-based QM/MM hybrid model, the solute is treated by the fragmentation QM calculation while the numerous solvent molecules are described by MM. In the polarization model, the polarizability is considered by allowing the partial charges and fragment-centered dipole moments to be variables, with values coming from the energy-based fragmentation QM calculations. Applications of these two methods to the solvated long oligomers and cyclic peptides have demonstrated that the hydrogen bonding interaction affects the dynamic change in chain conformations of backbone.
237
09-chapter8
September 29, 2010
10:44
238
World Scientific Review Volume - 9in x 6in
J. Ma, N. Jiang and H. Li
1. Introduction The hydrogen bonding interactions are ubiquitous in numerous chemical and biochemical systems. A typical example is the various alignments of hydrogen bonds between water molecules in gas, liquid, and solid phases, giving rise to unique physical properties (such as high boiling point, melting point and viscosity in liquid water). For macromolecules in solutions, the hydrogen bonding interactions are complicated by the coexistence of the intramolecular and intermolecular hydrogen bonds. Hydrogen bonds, which fluctuate with time and depend on the temperature, also play an important role in determining the dynamical conformational transitions of proteins. Because of their strength and directionality, the cooperative hydrogen bonding interaction is also the driving force of self-assembling of polymer chains with tunable functions and applications. Theoretical study of these macromolecular systems is highly desired to gain insights into the nature of the hydrogen bonding. Theoretical study of hydrogen bonding has been a topic of long-standing interest.1–10 The hydrogen bond is characterized by its strength and directionality. It usually consists of a proton donor (C-H, O-H, N-H, F-H) and a basic acceptor (O, N) atom. The hydrogen bonding is mainly electrostatic with the strength weaker than the normal covalent bond, but stronger than the van der Waals force. The difficulty in theoretical study of hydrogen bonding lies in such an ambiguity in the noncovalent bonding nature and huge computational costs demanded by the many-body problem. The continuous improvement of theoretical methods and computational techniques is making them efficient tools for the study of hydrogen bonding systems. Ab intio quantum mechanics (QM) methods have been employed to investigate the electronic structures of hydrogen bonding systems at various levels of accuracy. The description of intermolecular hydrogen bonding interaction requires highly sophisticated electron correlation methods, such as the second order Møller-Plesset perturbation (MP2) and coupled cluster (CC) theory. For history and details, readers are referred to pertinent books and review articles that have been devoted to this subject.1–15 However, the exponential scale of computational cost with the system size (called O(Nbα ), where α > 3 and Nb is the number of basis sets) in conventional QM methods prohibits their applications to the description of hydrogen bonds in solvated macromolecules. Another thread of theoretical methods that has enjoyed the extensive applications to large molecular systems is the molecular mechanics
09-chapter8
September 29, 2010
10:44
World Scientific Review Volume - 9in x 6in
Theoretical Modeling of Hydrogen Bonding in Macromolecular Solutions
09-chapter8
239
(MM).16–23 This kind of methods is also called the force field (FF) methods, since the energy of a system is expressed as a function of the nuclear coordinates, in terms of a classical physical picture of the intra-molecular (bond stretching, angle bending, and backbone torsion) forces and nonbonded (van der Waals and electrostatic) interactions within the systems. Although MM calculations are fast with much lower computational scale (O(Na2 ), Na is the number of atoms, smaller than Nb ) than that of QM computation, the ignorance of electronic motions in MM makes it difficult to describe the non-classic hydrogen bonding. Most force fields attempt to reproduce the hydrogen bonding on the van der Waals (Lennard-Jones 6-12) and electrostatic (Coulombic interaction) terms with refitted parameters. There are also some efforts in modeling the hydrogen bond by using an explicit hydrogen-bonding term, such as the 10-12 Lennard-Jones potential24 or direction-dependent 6-4 function.25 The accuracy of MM largely depends on the selection and transferability of force field parameters. Thus, it is necessary to study the hydrogen bonding in macromolecule solutions at ab initio QM level. An efficient way to solve this problem is taking advantage of both QM and MM methods in simulations. This review will concentrate on theoretical simulations of hydrogen bonding of macromolecules in aqueous solution by using the combination of QM and MM calculations. The rest of this chapter is organized as follows. In Sec. 2, a fragmentation-based QM/MM hybrid method is introduced with an emphasis on the treatment of inter-molecular hydrogen bonding between the long oligomers and solvent water molecules. Section 3 presents another method which supplements the polarization effects in the force field calculations. The applications of that polarizable force field model are also illustrated by aqueous solvated cyclic peptides. Finally, concluding remarks are made in the last section. 2. Fragmentation-Based QM/MM Simulations 2.1. Solvent models For simulations of solutions, there are three types of solvent models, implicit continuum dielectric approach, explicit solvent model, and the combined continuum/explicit strategy.26–29 In the continuum dielectric approach, the effect of solvent molecules is mimicked in terms of a continuous dielectric constant, ε.30 Although the continuum solvent model is the most computationally economical way to treat solvent effect, some short-range interac-
September 29, 2010
240
10:44
World Scientific Review Volume - 9in x 6in
J. Ma, N. Jiang and H. Li
tions such as hydrogen bonding cannot be described well. To another end, the explicit solvent model takes the specific short-range interactions into considerations by including the geometrical structures of individual solvent molecules.31–35 However, it is impossible to use explicit model (with thousands of solvent molecules) to include long range interactions in full QM calculation due to tremendous computational cost. The MM is then adopted in explicit solvent model to investigate the microscopic packing configurations of solution. As mentioned before, the force field calculations are difficult to give reasonable description of hydrogen bonds. Recently, a combined discrete/continuum strategy36–49 has also been employed in order to describe both specific and bulk effects in solutions. Another efficient way to describe the dilute solutions is the hybrid QM/MM method,50 in which the solute is treated as QM part, and the solvent molecules are taken as MM part. Recently, enormous works using QM/MM method surged out.51–75 The size of QM part in QM/MM, however, is still restrained to the medium-sized solute molecule in conventional QM methods. Therefore, there is an urgent call for the low-cost QM approach in the study of large-sized solute. 2.2. Energy-based fragmentation QM The difficulty associated with the large-sized molecules in standard electronic structure methods, such as Hartree-Fock (HF) and density-functional theory (DFT), lies in the high computational scaling of the construction and diagonalization of Fock matrix. Especially, Coulomb and exchange terms are two computationally expensive parts involved in the formation of Fock matrix. The fast multipole method76,77 and its later developments78–82 as well as some similar fast formulations83–86 achieve the linear scaling (O(N )) in the calculation of Coulomb integrals. Several other strategies, such as the near-field-exchange method,87 the order N exchange method,88 the LinK scheme89 were proposed to form the exact exchange matrix, realizing O(N ) or near-O(N ). There were also several attempts to reduce the computational cost of Fock matrix diagonalization.90–94 There has been a surge of interest in developing another category of linear scaling methods termed fragment-based approach. The progress has been summarized in a number of recent reviews.95–97 The basic idea is to divide a macromolecule (or molecular cluster) into a series of subsystems and obtain properties of the whole molecule from a combination of those of subsystems in a proper way. The success of fragmentation-based methods lies
09-chapter8
September 29, 2010
10:44
World Scientific Review Volume - 9in x 6in
Theoretical Modeling of Hydrogen Bonding in Macromolecular Solutions
09-chapter8
241
in the locality and transferability of properties of a subsystem in molecules. Among them, the energy-based approaches, independently proposed by Li et al.98–103 and Collins’ group,104,105 can be easily implemented at various theoretical levels and applied to geometry optimization and calculations of various properties of large-sized systems.98–108 Since the energy-based fragmentation method only requires a standard QM calculation on each subsystem, the problem of high scaling, O(Nbα ) (α ≥ 3), of conventional QM methods is significantly alleviated. Furthermore, by introducing the background point charges on the distant parts to mimic the electrostatic and polarization effects in the calculation of each fragment100,109–115 the fragment-based methods can give satisfactory descriptions for many properties, such as atomic charge, dipole moment, and static polarizability. Within the framework of the energy-based fragmentation method (as illustrated in Fig. 1),100 the total QM energy of the system can be expressed as N sub N atom N sub qa qb F rag ˜ EQM = Ci Ei − Ci − 1 , (1) rab i=1
i
a>b
˜i and Ci are the energy (includwhere Nsub is the number of subsystems, E ing the self energy of the point charges) and coefficient of i-th subsystem, respectively. The details were given in related references.98–103 In fact, the introduction of background point charges in the fragmentation QM model to consider the long-range electrostatics interaction has the same idea as electrostatics embedding model65,116 in QM/MM method. In this way, the polarization effect from MM part can be partially considered. 2.3. Fragmentation QM/MM: Basic idea and formalisms In the conventional QM/MM, the solute is treated as QM core, while the surrounding solvent molecules is considered as MM part, as shown in Fig. 2(a). In order to study dilute solutions of macromolecules, we have implemented a hybrid method by combining energy-based fragmentation QM calculations101 with MM, which was called fragmentation QM/MM (Fig. 2(b)). First of all, a long oligomer is partitioned into fragments in the QM part, as illustrated in Fig. 2(b). As in the electrostatics embedding65,116 model, the partial point charges of MM part are included in Hamiltonian of QM part when the fragmentation QM calculation100 is performed. The difference between the traditional QM/MM (Fig. 2(a)) and fragmentation
September 29, 2010
10:44
World Scientific Review Volume - 9in x 6in
242
09-chapter8
J. Ma, N. Jiang and H. Li
...
…
i-1
i-th
i+1
subsystem
subsystem
subsystem
…
…
~
Calculation of E i
Background charges
i-th
Background
subsystem
charges
Fig. 1. A schematic illustration for the energy-based fragmentation method, in which a macromolecule (or molecular cluster) is divided into a series of subsystems. In the calculation of the energy or properties of each fragment, a standard QM computation is carried out in atmosphere of the background point charges on distant parts from the central subsystem.
Fig. 2. A schematic description of (a) conventional and (b) fragmentation QM/MM models.
QM/MM model (Fig. 2(b)) lies in that the subsystem, constructed by the fragment (black region) and its nearby fragments (green regions), is calculated by QM, and the distant QM fragments (yellow regions) and MM parts are considered as background point charges in the QM calculation of each subsystem. The fragmentation QM/MM Hamiltonian can be expressed as F rag ET OL = EQM + EMM + EQM−MM
(2)
F rag is the energy of QM part by using the energy-based fragmenwhere EQM tation method (given in Eq. (1)), EMM is the energy of MM part treated by force field. For the studied aqueous solutions of oligomer (in which no covalent bond exists between the QM and MM parts), EQM−MM is the interaction between QM part and MM part, consisting of electrostatics
September 29, 2010
10:44
World Scientific Review Volume - 9in x 6in
Theoretical Modeling of Hydrogen Bonding in Macromolecular Solutions
09-chapter8
243
elec V dW (EQM−MM ) and van de Waals (EQM−MM ) interactions. Thus, Eq. (2) can be rewritten as F rag elec V dW + EMM + EQM−MM + EQM−MM ET OL = EQM
(3)
In combination with the MM part in the fragmentation QM/MM ˜ , the energy of the i-th subsystem embedded in method, we can obtain E i background point charges on those distant atoms in QM part and solvent molecules in MM part. Finally, the total energy is obtained N sub QM+MM N sub qa qb ˜ Ci Ei − Ci − 1 ET OL = rab i=1 i=1 a>b
−
MM a>b
qa qb V dW + EMM + EQM−MM rab
(4)
where in the second term, the electrostatic interaction between atomic charges covers the whole system including both QM and MM parts. The electrostatic interaction within the MM part is also counted in EMM and should be removed, as shown in the third term. It should be mentioned that the fragmentation QM/MM scheme will become more complicate when the QM and MM parts are linked by some covalent bonds. The formalism introduced above shows some advantages of fragmentation QM/MM method. The polarization of solvents is included in QM/MM method by updating the solvent atomic charge in QM Hamiltonian in the electrostatics embedding model. It can be also anticipated that the fragmentation-based QM/MM method is applicable to the hydrogen bonding interaction in solutions. As addressed in the Introduction, within the conventional MM, the two-body nonbonded interaction potential, which was introduced by the modified VdW and electrostatics interactions, is hard to give reasonable descriptions for the many-body characteristics of hydrogen bonds. It will be demonstrated in the following subsection that the important features of hydrogen bonding interactions, such as orientations and saturations, are well described in QM/MM simulations. The application of fragmentation QM/MM in geometry optimization or the molecular dynamics (MD) simulations needs the information of force (energy gradient). The forces in fragmentation QM/MM model are divided into two groups, forces involved in QM and MM parts, respectively. Forces to QM part can be derived as fαQM = gα −
∂ V dW E ∂rα QM−MM
(5)
September 29, 2010
10:44
World Scientific Review Volume - 9in x 6in
244
09-chapter8
J. Ma, N. Jiang and H. Li
where gα is the force to α-th atoms, which are obtained from the fragmentation QM calculation, and rα is the position of the α-th atom in QM part. By using QM Natural Population Analysis (NPA)117 charges, the force to MM part is approximately written as faMM ≈ −
∂ elec V dW EQM−MM + EMM + EQM−MM ∂ra
(6)
elec where EQM−MM is the electrostatic interaction between QM and MM parts, calculated by using point charges, and ra is the position of the a-th atom within MM part. For dilute solution of a long oligomer with thousands of atoms, the limited-memory and nearly linear scaling Broyden-Flether-GoldfarbShanno method118,119 is adopted in geometry optimization. Due to the approximations made in the force calculation, the forces do not correspond to the exact gradients of the potential energy surface. This may be not a serious problem when the gradients are large. However, when the system gets close to a local minimum, where the forces are very small, the fragmentation QM/MM forces may have difficulties in getting converged. It should be also mentioned that for such large systems containing thousands of atoms and innumerable local minima, it is meaningless to locate an exact minimum. A loose criterion of convergence of maximum force less than 0.002 Hartree/Angstrom was suggested, and the energy change was set to be smaller than 10−4 Hartree (0.063 kcal/mol). The accuracy of gradient increases with the increase in the size of fragment, and more strict criteria could be used. On the basis of the fragmentation QM/MM forces, the BornOppenheimer molecular dynamics (BOMD) simulations were implemented. The approximations made in calculations of forces give rise to slight errors of pairwise forces of QM/MM method. Although these errors do not affect geometry optimization significantly, they can be accumulated and cause problems in Hamiltonian energy conservation in molecular dynamics simulations, as noticed in other methods.153 Some techniques, such as smooth functions for modifying the forces and energies,121–123 have been used to improve the energy conservation in QM/MM MD simulations. In the following MD simulations, the Nose-Hoover-Chain (NHC)124,125 method was applied for controlling temperature in the BOMD simulations in canonical (NVT) ensemble, by which the energy drift is adsorbed by the heat bath. NHC method was implemented in Verlocity-Verlet125 method to control the temperature at 300.0 K.
September 29, 2010
10:44
World Scientific Review Volume - 9in x 6in
Theoretical Modeling of Hydrogen Bonding in Macromolecular Solutions
09-chapter8
245
2.4. Fragmentation QM/MM simulations on poly(ethylene oxide) polyethylene In this subsection, some examples are presented for applications of fragmentation QM/MM to geometry optimizations and BOMD. Two sets of oligomers with increasing chain length were selected. Poly(ethylene oxide), PEO, is a kind of water-soluble polymer bearing extensive industrial applications,126 while the hydrophobic polyethylene (PE) is widely used in plastic industry.127 Several MM-based molecular dynamics and Monte Carlo simulations have been carried out to investigate physical characteristics of PEO128–134 and PE135–142 in various media. Recently, the fragmentation QM/MM method143 was applied to study chain conformations of PEO and PE oligomers in aqueous solutions as well as hydrogen bonding interactions between PEO and water molecules, in comparison with the conventional QM/MM method. Different lengths of PEO [H(CH2 CH2 O)n H, n = 6, 8, 10, 15, 20] and PE [H(CH2 CH2 )n H, n = 9, 12, 15, 21, 30] have been studied. PEO10 and PE9 are chosen to make comparison between the fragmentation-based QM/MM method and conventional QM/MM. Every three successive heavy atoms along the main chains were selected as a fragment. Then the nearestneighboring fragments of this central fragment were included to form a work subsystem for QM calculations. Although the accuracy of the fragmentation-based QM calculations can be further improved by using larger subsystems, the computational time will increase accordingly. The target of modeling solutions of long oligomers with up to 30 repeat units (n = 30) rendered us to adopt the modest fragmentation scheme. Helix and linear conformations were used as the initial geometries for PEO and PE, respectively. Orthorhombic boxes of solvent with 12.0 ˚ A buffers were added around a solute molecule to build the solvent shells. Single point charge144 (SPC) model was employed for water solvents. 12-6 Lennard-Jones potential was used for describing the van de Waals interactions. Parameters of consistent valence force field (CVFF)145 were employed. To validate the results of fragmentation QM/MM method, the conventional QM/MM was also carried out by using Gaussian03 package146 and the same force field parameters. In order to save CPU time, HF/3-21G level was used for all the geometry optimizations and molecular dynamics simulations. Figure 3 shows the optimized structures of PEO10 and PE9 , along with the adjacent water molecules in the first solvation shell. The average bond lengths and bond
September 29, 2010
246
10:44
World Scientific Review Volume - 9in x 6in
J. Ma, N. Jiang and H. Li
Fig. 3. Comparison of the first solvation shells (r < 3.0 ˚ A) of PEO10 (left) and PE9 (right) solutions between the fragmentation and conventional QM/MM optimizations. (Figure redrawn from H. Li, W. Li, S. Li and J. Ma, J. Phys. Chem. B 112, 7061 (2008).)
angles obtained by the fragmentation-based QM/MM method are consistent with the conventional method. Compared with the PEO oligomer in gas phase, the C-O bond length in aqueous solution increases obviously because of the polarization effect, but the covalent C-C bonds are hardly affected by the adjacent water molecules. The hydrogen bonding interactions between water molecules and the oxygen atoms in PEO make the C-O bonds weaker than those in the gas phase and crystal phase. Hydrogen bonding plays an important role in the chain configurations in aqueous PEO solutions. The hydrogen bond criterion, the O• • •H distance of being less than 3.0 ˚ A and the O-H• • •O (or C-H• • •O) angle larger than 110◦ , was adopted, as suggested in other works.147 Significant O-H• • •O hydrogen bonds have been observed with short O• • •H distance, and the coordinate number of the O-H• • •O hydrogen bond is one or two. PEO oxygen/water hydrogen (OPEO-HW ), PEO hydrogen/water oxygen (HPEO -OW ) and PE hydrogen/water oxygen (HPE -OW ) intermolecular pair distribution functions are shown in Fig. 4(a). The fragmentation and conventional QM/MM methods gave the nearly identical curves for OPEO HW , HPEO -OW and HPE -OW pair distribution functions. In the OPEO HW pair distribution function, the first aqueous shell of oxygen atoms in PEO is distinctly denoted by the peak at 1.7 ˚ A. In contrast to the strong O-H•••O hydrogen bonds, C-H•••O hydrogen bonds are absent in the MD simulations under 300 K, as shown by the flat shape of HPE -OW pair distribution curves without any specific peaks (Fig. 4(a)). The distributions of
09-chapter8
September 29, 2010
10:44
World Scientific Review Volume - 9in x 6in
Theoretical Modeling of Hydrogen Bonding in Macromolecular Solutions
09-chapter8
247
Fig. 4. (a) Pair distribution functions and (b) distributions of O-H • • • OPEO and CH ••• OW angles in the vicinity of solutes of PEO10 and PE9 solutions in simulations by both fragmentation and conventional QM/MM methods at 300K, respectively. (Figure redrawn from H. Li, W. Li, S. Li and J. Ma, J. Phys. Chem. B 112, 7061 (2008).)
O-H• • •O (when H • • • O < 2.5 ˚ A) and C-H• • •O (when H • • • O < 3.0 ˚ A) angles in the vicinity of solute molecules were also shown in Fig. 4(b). The sharp peak at 170◦ in the distribution curve of O-H• • •O angles again displays the specific O-H• • •O hydrogen bonds in aqueous PEO10 solution.
September 29, 2010
248
10:44
World Scientific Review Volume - 9in x 6in
J. Ma, N. Jiang and H. Li
It was demonstrated that pair distribution functions of OPEO -HW in aqueous PEO solutions with increasing chain length (n = 6 ∼ 20) are almost the same, indicating little length dependence on the intermolecular hydrogen bonding interaction in PEO solutions.143 In comparison with conventional QM/MM methods, the fragmentationbased method combines advantages of both QM/MM method and molecular fragmentation method. With this method, it is also possible to perform ab initio MD simulations of large systems such as polymers, biomolecules, and even bulk materials. 3. Simulations of Solvated Peptides Using Polarizable Force Field Model It is well-recognized that the conventional FF models cannot properly describe the polarization effects in solutions. The electrostatic potential is simply expressed in a sum of pairwise Coulombic interactions between the atom-centered point charges with the pre-determined values.148–158 A straightforward way to explicitly include polarization effects is the employment of induced dipoles to treat the local change in charge density around an atom.159–167 This scheme can be further improved by using higher order multipole expansion.168–173 There are also some other models which take the combination of the fixed point charges/dipoles and inducible dipoles.174–180 Similar to inducible point charge models, Drude oscillator model181–192 adds some massless charged particles onto the polarizable atoms to model the electronic induction. Belonging to another category, fluctuating charge (FQ) methods193–221 describe the polarization response to the movement of charge density from one atom (or a bond) to another one. The values of the atomic charges are taken as dynamical variables, which are derived from electronegativity equalization.222–232 Friesner et al. further introduced site-centered inducible dipoles into the electrostatic potential.202–207 Pater et al. embedded fluctuating charges and dipoles into CHARMM.209–211 A variable internal dielectric model has also been employed to implicitly consider the polarization effects in proteins with charged side chain.233 Other efforts have been invested to model polarization by modifying atomic charges.234–236 Another type of polarization models is built on the basis of quantum mechanical (QM) calculations.237–240 Inspired by the successful application of the linear-scaling fragment-based methods in macromolecules, we fused the energy-based fragmentation calculation into the polarizable force field.241
09-chapter8
September 29, 2010
10:44
World Scientific Review Volume - 9in x 6in
Theoretical Modeling of Hydrogen Bonding in Macromolecular Solutions
09-chapter8
249
3.1. Fragmentation-based polarization model The polarization model can be implemented in both the FF03152 and OPLS153–158 with the potential function shown in Eq. (7) Utotal = Unon−elec + Uelec = Kb (b − beq )2 + Kθ (θ − θeq )2 bonds
angles
Vn [1 + cos(nφ − γ)] 2 dihedrals Aij Bij + Uelec + 12 − R6 Rij ij i<j +
(7)
where all the bond stretching, angle bending, torsional rotating, and Lennard-Jones parameters are taken from the FF03152 (or OPLS153–158 ) force field. Various polarizable force field methods differ in the definition of the electrostatic term (Uelec ). In comparison with the conventional MM models, the fragmentation-based polarizable model has two distinct features in electrostatic contribution, as schematically shown in Fig. 5. (i) In the calculation of the Coulombic interactions (Eq. (8)) between atomic partial qq , the atomic partial charges are obtained directly from the charges, Uelec linear-scaling QM calculations. qi qj qq Uelec = k (8) Rij i<j This has some similarity with the OPLS/CM1A force field,242 which incorporates partial atomic charges obtained from semiempirical AM1 calculations. Truhlar et al. also embedded the polarizable partial charges at different levels243–245 in the self-consistent reaction field model for various solutions. (ii) The fragment-centered dipoles are employed in the description of polarization (Fig. 5). The dipole-dipole interactions between the µµ , is then obtained from fragment dipoles, Uelec
rag IJ · µF rag 1 µF · RIJ · R µµ F rag F rag I J Uelec = k · µJ −3 µ I (9) IJ |3 IJ |2 | R | R I<J where qi and qj are the atom-centered partial charges of atoms i and j, rag rag and µF are the fragment dipoles centered in fragments I and J, µF I J respectively.
September 29, 2010
10:44
World Scientific Review Volume - 9in x 6in
250
09-chapter8
J. Ma, N. Jiang and H. Li
Polarization model: Utotal = Unon-elec + Uelec or atom-centered charge-charge interactions
fragment-centered dipole-dipole interactions FragI
qi at atom i
OI
qj at atom j
i, j = 1, 2,
(a)
, Natom
QM Atomic charges
(b)
OJ
I, J = 1, 2,
FragJ
, Mfrag
QM Fragment dipoles
Linear-scaling fragmentation QM calculation Fig. 5. The polarization model with the electrostatic term obtained from either (a) atom-centered charge-charge or (b) fragment-centered dipole-dipole interactions. The OI and OJ represent the geometric centers of the Ith and Jth fragments, respectively. (Figure taken from N. Jiang and J. Ma, J. Phys. Chem. A 112, 9854 (2008))
It has already been emphasized that the polarization is a group property and the reduction of the number of polarizable sites is an efficient way for treating polarization.246 The representation of peptide groups by dipolar structural units was demonstrated to give rapid evaluation of electrostatic interactions.247 The introduction of fragment dipole moments in the calcuµµ lation of dipole-dipole interactions (Uelec ) does save much computational time in comparison with those using the atom-centered dipole moments. The geometry optimization and MD simulations can be implemented on the basis of the polarization model. In the following subsection, an application of the fragmentation-based polarization model to the aqueous solvated cyclic peptide is given.
September 29, 2010
10:44
World Scientific Review Volume - 9in x 6in
Theoretical Modeling of Hydrogen Bonding in Macromolecular Solutions
09-chapter8
251
3.2. Configurations of solvated α-conotoxin GI and its analogues The polarization model was employed to investigate the conformational diversity and unfolding process of α-conotoxin GI (PDB248 code: 1not), a valuable probe of nicotinic acetylcholine receptors (nAChRs) and ion channels,249 and its two single-disulfide analogues, Cono-1 and Cono-2 (Fig. 6). α-Conotoxin GI is a cyclic peptide containing 13 residues with the sequence of
ECCNPACGHYSC-NH2 , as drawn in Fig. 6(a). Two disulfide bonds, Cys2-Cys7 and Cys3-Cys13, make the protein folding with diverse secondary structures: random coil (Glu1-Cys2-Cys3-Asn4 and Ser12-Cys13), α-helix (Pro5-Ala6-Cys7), and β-turn (Gly8-Arg9-His10-Tyr11). It has been shown that the polarizable FF based on the fragmentation QM calculations can reasonably describe not only the relative energies of conformers but also the structural and dynamical properties of the studied peptides.241 The polarizability was explicitly described by allowing the partial charges or fragment dipole moments to be variables, with values coming from the linear-scaling energy-based molecular fragmentation calculations at B3LYP/6-31G(d) level. The fragmentation procedure follows: (i) decompose a protein into several fragments, according to its residue units or secondary structures. The important intramolecular hydrogen bonding interactions are considered in the QM treatment of each fragment. (ii) The nearest neighboring residues attached to each fragment are included in the subsystem to mimic the influence of environment on the cutting bonds. Electrostatic interactions coming from the distant atoms outside each subsystem are approximated by the background point charges, as done in other works.100,109–115 The B3LYP/6-31G(d) charges or fragment-centered dipole moments were subsequently used in calculations of electrostatic terms in Amber FF03 or OPLS force fields. Comparison was made among three types of FF models: (i) the standard FF03 with the predetermined partial charges; (ii) FF03 with the dynamically variable NBO and ESP charges taken from the linearscaling DFT (B3LYP) calculations, designated FF03-qq(NBO) and FF03qq(ESP), respectively; and (iii) FF03 combined with the fragment dipoles
September 29, 2010
252
10:44
World Scientific Review Volume - 9in x 6in
J. Ma, N. Jiang and H. Li
Cys2 Cys7
Cys13
Cys3 Glu1-Cys2-Cys3-Asn4-Pro5-Ala6-Cys7-Gly8-Arg9-His10-Tyr11-Ser12-Cys13-NH2
(a) α-Conotoxin GI Cys2 Cys7 Glu1-Cys2-Ala3-Asn4-Pro5-Ala6-Cys7-Gly8-Arg9-His10-Tyr11-Ser12-Ala13-NH2
(b) Cono-1
Cys13 Cys3 Glu1-Ala2-Cys3-Asn4-Pro5-Ala6-Ala7-Gly8-Arg9-His10-Tyr11-Ser12-Cys13-NH2
(c) Cono-2 Fig. 6. Amino acid sequences and cysteine frameworks for α-conotoxin GI and its singledisulfide analogues, Cono-1 and Cono-2. NH2 denotes an amidated C-terminus. (Figure taken from N. Jiang and J. Ma, J. Phys. Chem. A 112, 9854 (2008).)
calculated from NBO and ESP charges, designated FF03-µµ(NBO) and FF03-µµ(ESP), respectively. A scaling of 0.1 was introduced for the van der Waals interactions, as suggested by others.239 It is very difficult for traditional force fields to reasonably describe the intramolecular hydrogen bonds, where polarization effects are particularly important. As displayed in Table 1, the accuracy of the polarization model (with average deviations of 0.093 ˚ A and 0.173 ˚ A for fragmentation FF03-qq and FF03-µµ force fields, respectively) from experiments is superior to that of the standard FF03 (average deviation: 0.448 ˚ A). The accuracy is closely related to the proper description of hydrogen bonding interactions in each subsystem.
09-chapter8
September 29, 2010
10:44
World Scientific Review Volume - 9in x 6in
09-chapter8
Theoretical Modeling of Hydrogen Bonding in Macromolecular Solutions
253
Table 1. Intramolecular hydrogen bonds (˚ A) of solvated α-conotoxin GI calculated from the standard FF03 and polarizable force fields, with the deviations from x-ray structure given in parentheses. Res1 -NH· · · O=C-Res2 Res1 Asn 4
(δ1 N)
Ser 12
Cys 7
Cly 8 His 10
Tyr 11
Cys 2
FF03
Res2 Glu 1 Cys 2
Asn 4
Pro 5 Cys 7
Gly 8
Ser 12
Mean deviation
Polarization modela FF03-qq
X-ray structureb
FF03-µµ
4.926
2.728
2.803
(2.209)
(0.011)
(0.086)
2.962
2.778
3.136
(0.073)
(0.111)
(0.247)
3.077
2.825
3.008
(0.192)
(0.060)
(0.123)
3.259
3.256
3.443
(0.170)
(0.167)
(0.354)
3.180
2.893
2.974
(0.312)
(0.025)
(0.106)
3.091
3.074
3.195
(0.168)
(0.151)
(0.272)
2.801
2.659
2.819
(0.013)
(0.129)
(0.031)
(0.448)
(0.093)
(0.173)
2.717 2.889
2.885
3.089 2.868
2.923
2.788
a In
Note: the polarization model, the atomic charges and fragment dipoles are calculated at fragmentation B3LYP/6-31G(d) level. b The experimental crystal results come from L. W. Guddat, J. A. Martin, L. Shan, A. B. Edmundson and W. R. Gray, Biochemistry 35, 11329 (1996).
The simulated structure of water solvated α-conotoxin GI from the FF03-based polarization model is compared with the x-ray crystal and NMR structures in Fig. 7(a). One can see that the polarization model can well reproduce the experimental structures, except in the terminal residue of Glu 1. The apparent difference in local conformations of terminal residues between the crystal (or NMR) structure and simulation has also been found in the water solvated BPTI206 and α-conotoxin MI.250 The terminal residues usually bear both negative and positive charge centers (e.g., COO− and NH+ 3 in Glu 1 of α-conotoxin GI), giving rise to significant hydrogen bonding between the end residues and solvent water. The water molecules move much faster relative to the backbone atoms, hence the terminal residues change their local conformations accordingly from time to time. This implies local structures of the protein terminals are very
September 29, 2010
10:44
World Scientific Review Volume - 9in x 6in
254
J. Ma, N. Jiang and H. Li
1.75 Å .. ....
1.75 Å
....
..
g(r)
Glu1
Tyr11 1.75 Å 1.85 Å
RO···H (Å) (a)
(b)
Fig. 7. (a) The structure of solvated α-conotoxin GI calculated from the polarization model. (b) Radial distribution functions (RDFs) for distance from water H atoms and carbonyl O (of Tyr 11) and carboxyl O (of Glu1), labeled ROTyr11···HWat (top) and ROGlu1···HWat (bottom), respectively. The molecular dynamics simulations were carried out by using standard FF03 (in dotted line) and the polarization model (in solid line), respectively. The fused NBO charges (at B3LYP/6-31G(d) level) in the polarizable FF03 model are updated every 2 ps.
sensitive to the solvent environment, making the characterization of their solution conformations a big challenge in both experiments and simulations. Figure 7(b) displays the solute-solvent radial distribution functions (RDF), illustrated by distributions of the distance from the water H atoms to carbonyl O (of residue Tyr11) and carboxyl O (of residue Glu1) atoms, labeled ROT yr11 ···Hwat and ROGlu1 ···Hwat , respectively. Both the standard and polarizable FF03 RDFs exhibit the first peak around 1.8 ˚ A, indicating a typical hydrogen bonding interaction exists between Tyr 11 and Glu 1 residues and water molecules. As shown in Fig. 7, the polarization model predicts a slightly shorter OTyr · · · Hwat distance than the standard FF03 model. The present polarization model was also employed to study the conformational dynamics of flexible proteins, such as the single-disulfide analogues of α-conotoxin GI, Cono-1 and Cono-2 (Fig. 6). The NMR experiment has shown that both Cono-1 and Cono-2 are considerably less structured than the native α-conotoxin GI.251 Some representative MD snapshots in the frameworks of both FF03 and FF03-qq(NBO) force fields are shown in Fig. 8. The conformational change displayed by the polarization model is more evident than that shown by the traditional FF03 force field. The MD
09-chapter8
September 29, 2010
10:44
World Scientific Review Volume - 9in x 6in
09-chapter8
Theoretical Modeling of Hydrogen Bonding in Macromolecular Solutions
Cono-1
Cono-2
FF03
FF03
FF03-qq (NBO)
FF03-qq (NBO) 200
400
600
800
Time (ps) (a)
1000
255
200
400
600
800
1000
Time (ps) (b)
Fig. 8. Snapshots of representative unfolding trajectories for (a) Cono-1 and (b) Cono-2 at several MD steps using standard FF03 (top) and polarizable FF03-qq(NBO) (bottom) at 300 K, respectively. The backbones of α-conotoxin GI analogues are shown in ribbon representations and the disulfide bonds are given in CPK forms (Figure adopted from N. Jiang and J. Ma, J. Phys. Chem. A 112, 9854 (2008).)
simulations using the polarizable force field indicate that both Cono-1 and Cono-2 unfold gradually in the absence of a disulfide bond, in good agreement with experiments.251 Therefore, the formation of the second disulfide bond is indispensable to compact the backbone of peptide into the native folding. It is also revealed that the polarization effect is important in simulations of the folding/unfolding process of solvated proteins. 4. Concluding Remarks This article has focused on the molecular simulations of oligomers or peptides in aqueous solution in combination with linear-scaling energy-based fragmentation QM calculations. Fragmentation QM/MM method has been developed in the electrostatics embedding QM/MM model, where the low-cost energy-based fragmentation calculations are employed for QM part. The intermolecular hydrogen bonding and chain configurations of oligomers in aqueous solution obtained from the fragmentation QM/MM simulations are consistent with the conventional QM/MM method. The fragmentation QM/MM MD is applicable to the increasingly long oligomers, such as PEOn (n = 6 ∼ 20) and PEn (n = 9 ∼ 30), exhibiting the length dependence of chain conformations and dynamics of aqueous solvated oligomers. A polarizable force field has also been implemented on the basis of linear-scaling molecular fragmentation approach. In comparison with other
September 29, 2010
10:44
256
World Scientific Review Volume - 9in x 6in
J. Ma, N. Jiang and H. Li
FF methods, the fragmentation-based polarization model has two features. Firstly, the partial charges and dipole moments are directly obtained from the energy-based fragmentation QM calculations, without the need of parametrization. Secondly, fragment-centered dipole moments are used to calculate the dipole-dipole interactions. The introduction of fragmentbased dipole-dipole interactions in calculations of electrostatic interactions can save much computational time in comparison with those using the atom-centered charge-charge interactions. The polarization model has been applied to the simulations of the solution conformation of α-conotoxin GI and its two single-disulfide analogues, from which the importance of polarization effect was shown in the folding/unfolding process of solvated proteins. It should be mentioned that the description of the hydrogen bonding interaction in condense phase is still a big challenge in the field of theoretical and computational chemistry. The basic idea underlying the fragment-based method may be also applicable to reduce the complexity in treating various biological or self-assembling systems. Acknowledgments This work was supported by the China NSF (No.20825312), the Chinese Ministry of Education (NCET-05-0442), and Fok Ying Tong Education Foundation (Grant No. 111013), and the National Basic Research Program (Grant No. 2004CB719901). References 1. G. C. Pimentel and A. L. McClellan, The Hydrogen Bond (Freeman, San Francisco, 1960). 2. L. Pauling, The Nature of the Chemical Bond, 3rd edn. (Cornell University Press, Ithaca, 1963). 3. W. C. Hamilton and J. A. Ibers, Hydrogen Bonding in Solids (Benjamin, New York, 1968). 4. G. A. Jeffrey and W. Saenger, Hydrogen Bonding in Biological Structures (Springer, Berlin, 1991). 5. S. Scheiner, Hydrogen Bonding: A Theoretical Perspective (Oxford University Press, New York, 1997). 6. G. A. Jefferey, An Introduction to Hydrogen Bonding (Oxford University Press, New York, 1997). 7. G. R. Desiraju and T. Steiner, The Weak Hydrogen Bond in Structural Chemistry and Biology (Oxford University Press, New York, 1999).
09-chapter8
September 29, 2010
10:44
World Scientific Review Volume - 9in x 6in
Theoretical Modeling of Hydrogen Bonding in Macromolecular Solutions
8. 9. 10. 11. 12. 13. 14. 15. 16. 17.
18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34.
35. 36. 37. 38. 39. 40. 41.
09-chapter8
257
A. E. Reed, L. A. Curtiss and F. Weinhold, Chem. Rev. 88,899 (1988). G. R. Desiraju, Acc. Chem. Res. 35, 565 (2002). T. Steiner, Angew. Chem. Int. Ed. 41, 48 (2002). A. S. N. Murthy and C. N. R. Rao, J. Mol. Struct. 6, 253 (1970). P. A. Kollman and L. C. Allen, Chem. Rev. 72, 283 (1972). K. Morokuma, Acc. Chem. Res. 10, 294 (1977). I. N. Levine, Physical Chemistry; 3rd ed. (McGraw-Hill, New York, 1988). M. Meot-Ner (Mautner), Chem. Rev. 105, 213 (2005). A. R. Leach, Molecular Modelling: Principles and Applications (Person Education EMA, Prentice-Hall, 2001). P. A. Kollman, I. Massova, C. Reyes, B. Kuhn, S.Huo,L. Chong, M. Lee, T. Lee, Y. Duan, W. Wang, O. Donini, P. Cieplak, J. Srinivasan, D. A. Case, and T. E. Cheatham, Acc. Chem. Res. 33, 889 (2000). M. Zimmer, Chem. Rev. 8, 2629 (1995). J. Aaqvist and A. Warshel, Chem. Rev. 93, 2523 (1993). P. Kollman, Chem. Rev. 93, 2395 (1993). C. Brooks, and D. Case, Chem. Rev. 93, 2487 (1993). J. E. Eksterowicz and K. N. Houk, Chem. Rev. 93, 2439 (1993). C. E. Dykstra, Chem. Rev. 93, 2339 (1993). A. Vedani, J. Comput. Chem. 9, 269 (1988). P. J. Goodford, J. Med. Chem. 28, 849 (1985). M. Orozco and F. J. Luque, Chem. Rev. 100, 4187 (2000) and references therein. C. J. Cramer and D. G. Truhlar, Chem. Rev. 99, 2161 (1999) and references therein. N. S. Hush and J. R. Reimers, Chem. Rev. 100, 775 (2000) and references therein. J. Tomasi, B. Mennucci and R. Cammi, Chem. Rev. 105, 2999 (2005). J. Tomasi and M. Persico, Chem. Rev. 94, 2027 (1994) and references therein. L. Meneses, P. Fuentealba, R. Contreras, Chem. Phys. Lett. 433, 54 (2006). K. Kitaura, K. Morokuma, Int. J. Quantum Chem. 10, 325 (1976). A. Pullman, in The New World of Quantum Chemistry, Ed. B. Pullman, R. Parr (Reidel:Dordrecht, The Netherlands, 1976), p. 149. A. Pullman, in Quantum Theory of Chemical Reactions, Ed. R. Daudel, A. Pullman, L. Salem and A. Veillard, (Reidel: Dordrecht, The Netherlands, 1981), p. 1. C. L. Mois´es, R. R. Daniel, and J. A. Santaballa, Chem. Phys. Lett. 417, 28 (2006). C. Alem´ an, Chem. Phys. 244, 151 (1999). C. Alem´ an and S. E. Galembeck, Chem. Phys. 232, 151 (1998). C. Alem´ an, Chem. Phys. Lett. 302, 461 (1999). C. Alem´ an, Chem. Phys. 253, 13 (2000). B. Mennucci and J. M. Mart´ınez, J. Phys. Chem. B 109, 9818 (2005). B. Mennucci and J. M. Mart´ınez, J. Phys. Chem. B 109, 9830 (2005).
September 29, 2010
258
10:44
World Scientific Review Volume - 9in x 6in
J. Ma, N. Jiang and H. Li
42. B. Mennucci, J. M. Mart´ınez and J. Tomasi, J. Phys. Chem. A 105, 7287 (2001). 43. K. S. Ekanayake and P. R. Lebreton, J. Comput. Chem. 27, 277 (2006). 44. S. Kong and J. D. Evanseck, J. Am. Chem. Soc. 122, 10418 (2000). 45. G. S. M. Kiruba and M. W. Wong, J. Org. Chem. 68, 2874 (2003). 46. P. Bandyopadhyay, and M. S. Gordon, J. Chem. Phys. 113, 1104 (2000). 47. R. Improta and V. Barone, J. Am. Chem. Soc. 126, 14320 (2006). 48. S. Meng, J. Ma and Y. Jiang, J. Phys. Chem. B 111, 4128 (2007). 49. S. Meng and J. Ma, J. Phys. Chem. B 112, 4313 (2008). 50. A. Warshel and M. Levitt, J. Mol. Biol. 103, 227 (1976). 51. U. C. Singh and P. A. Kollman, P. A. J. Comput.Chem. 7, 718 (1986). 52. M. J. S. Dewar, E. G. Zoebisch, E. A. Healy and J. J. P. Stewart, J. Am. Chem. Soc. 107, 3902 (1985). 53. M. J. Field, P. A. Bash and M. Karplus, J. Comput. Chem. 11, 700 (1990). 54. J. Gao and X Xia, Science 258, 631 (1992). 55. R. V. Stanton, D. S. Hartsough and K. M. Jr. Merz, Jr. J. Phys. Chem. 97, 11868 (1993). 56. V. Thery, D. Rinaldi and J.-L. Rivail, J. Comput. Chem. 15, 269 (1994). 57. F. Maseras and K. Morokuma, J. Comput. Chem. 16, 1170 (1995). 58. M. A. Thompson, J. Phys. Chem. 99, 4794 (1995). 59. K. P. Eurenius, D. C. Chatfield, B. R. Brooks and M. Hodoscek, Int. J. Quantum Chem. 60, 1189 (1996). 60. D. Bakowies and W. Thiel, J. Phys. Chem. 100, 10580 (1996). 61. I. B. Bersuker, M. K. Leong, J. E. Boggs and R. S. Pearlman, Int. J. Quantum Chem. 63, 1051 (1997). 62. J. Gao, P. Amara, C. Alhambra, and M. J. Field, J. Phys.Chem. A 102, 4714 (1998). 63. P. D. Lyne, M. Hodoscek and M. Karplus, M. J. Phys. Chem. A 103, 3462 (1999). 64. M. Eichinger, P. Tavan, J. Hutter and M. Parrinello, J. Chem. Phys. 110, 10452 (1999). 65. Y. Zhang, T.-S. Lee and W. Yang, J. Chem. Phys. 110, 46 (1999). 66. Y. Zhang, H. Liu and W. Yang, J. Chem. Phys. 112, 3483 (2000). 67. M. J. Field, M. Albe, C. Bret, F. Proust-De Martin and A. Thomas, J. Comput. Chem. 21, 1088 (2000). 68. D. A. Yarne, M. E. Tuckerman and G. J. Martyna, J. Chem. Phys. 115, 3531 (2001). 69. A. Laio, J. VandeVondele and U. Rothlisberger, J. Chem. Phys. 116, 6941 (2002). 70. A. Crespo, D. A. Scherlis, M. A. Marti, P. Ordejon, A. E. Roitberg and D. A. Estrin, J. Phys. Chem. B 107, 13728 (2003). 71. P. Sherwood, A. H. de Vries, M. F. Guest, G. Schreckenbach, C. Richard A. Catlow, S. A. French, A. A. Sokol, S. T. Bromley, W. Thiel, A. J. Turner, S. Billeter, F. Terstegen, S. Thiel, J. Kendrick, S. C. Rogers, J. Casci, M. Watson, F. King, E. Karlsen, M. Sjøvoll, A. Fahmi, A. Sch¨ afer and C. Lennartz, J. Mol. Struct. 632, 1 (2003).
09-chapter8
September 29, 2010
10:44
World Scientific Review Volume - 9in x 6in
Theoretical Modeling of Hydrogen Bonding in Macromolecular Solutions
09-chapter8
259
72. N. Ferre and M. Olivucci, J. Am. Chem. Soc. 125, 6868 (2003). 73. M. J. Loferer, H. H. Loeffler and K. R. Liedl, J. Comput. Chem. 24, 1240 (2003). 74. K. Nam, J. Gao and D. M. York, J. Chem. Theory Comput. 1, 2 (2005). 75. J. Gao, S. Ma, D. T. Major, K. Nam, J. Pu and D. G. Truhlar, Chem. Rev.106, 3188 (2006). 76. L. Greengard and V. Rokhlin, J. Comput. Phys. 60, 187 (1985). 77. L. Greengard and V. Rokhlin, J. Comput. Phys. 73, 325 (1987). 78. C. A. White and M. Head-Gordon, J. Chem. Phys. 101, 6593 (1994). 79. M. C. Strain, G. E. Scuseria and M. J. Frisch, Science 271, 51 (1996). 80. C. A. White, B. G. Johnson, P. M. W. Gill and M. Head-Gordon, Chem.Phys. Lett. 230, 8 (1994). 81. K. N. Kudin and G. E. Scuseria, Phys. Rev. B 61, 16440 (2000). 82. R. Kutteh, E. Apr` a and J. Nichols, Chem. Phys. Lett. 238, 173 (1995). 83. J. M. P´erez-Jord´ a and W. Yang, Chem. Phys. Lett. 247, 484 (1995). 84. M. Challacombe, E. Schwegler and J. Alml¨ of, J. Chem. Phys. 104, 4685 (1996). 85. J. M. P´erez-Jord´ a and W. Yang, J. Chem. Phys. 104, 8003 (1996). 86. J. M. P´erez-Jord´ a and W. Yang, J. Chem. Phys. 107, 1218 (1997). 87. J. C. Burant and G. E. Scuseria, J. Chem. Phys. 105, 8969 (1996). 88. E. Schwegler and M. Challacombe, J. Chem. Phys. 105, 2726 (1996). 89. C. Ochsenfeld, C. A. White and M. Head-Gordon, J. Chem. Phys. 109, 1663 (1998). 90. M. S. Daw, Phys. Rev. B 47, 10895 (1993). 91. X. Li, J. M. Millam, G. E. Scuseria, M. J. Frisch and H. B. Schlegel, J. Chem. Phys. 119, 7651 (2003). 92. S. Baroni and P. Giannozzi, Europhys. Lett. 17, 547 (1992). 93. E. Hern´ andez, M. J. Gillan and C. M. Goringe, Phys. Rev. B 53, 7147 (1996). 94. X.-P. Li, R. W. Nunes and D. Vanderbilt, Phys. Rev. B 47, 10891 (1993). 95. S. Li and W. Li, Annu. Rep. Prog. Chem., Sect. C: Phys. Chem. 104, 256 (2008). 96. W. Li, H. Dong and S. Li, in Frontiers in Quantum Systems in Chemistry and Physics, Eds. S. Wilson, P. J. Grout, G. Delgado-Barrio, J. Maruani and P. Piecuch (Springer, Berlin, 2008). 97. H. M. Netzloff and M. A. Collins, J. Chem. Phys. 127, 134113 (2007). 98. S. Li, W. Li and T. Fang, J. Am. Chem. Soc. 127, 7215 (2005). 99. W. Li, T. Fang and S. Li, J. Chem. Phys. 124, 154102 (2006). 100. W. Li, S. Li and Y. Jiang, J. Phys. Chem. A 111, 2193 (2007). 101. S. Li, W. Li, T. Fang, J. Ma and Y. Jiang, Low Scaling Quantum Chemical (LSQC) program; ver. 1.1; Nanjing University: Nanjing, 2006. 102. H. Cao, T. Fang, S. Li and J. Ma, Macromolecules. 40, 4363 (2007). 103. W. Hua, T. Fang, W. Li, J. G. Yu and S. Li, J. Phys. Chem. A 112, 10864 (2008). 104. V. Deev and M. A. Collins, J. Chem. Phys. 122, 154102 (2005). 105. M. A. Collins and V. A. Deev, J. Chem. Phys. 125, 104104 (2006).
September 29, 2010
260
10:44
World Scientific Review Volume - 9in x 6in
J. Ma, N. Jiang and H. Li
106. S. R. Gadre, R. N. Shirsat and A. C. Limaye, J. Phys. Chem. 98, 9165 (1994). 107. K. Babu and S. R. Gadre, J. Comput. Chem. 24, 484 (2003). 108. V. Ganesh, R. K. Dongare, D. P. Balanarayan and S. R. Gadre, J. Chem. Phys. 125, 104109 (2006). 109. T. E. Exner and P. G. Mezey, J. Phys. Chem. A 108, 4301 (2004). 110. N. Jiang, J. Ma and Y. Jiang, J. Chem. Phys. 124, 114112 (2006). 111. S. Morita and S. Sakai, J. Comput. Chem. 22, 1107 (2001). 112. S. Sakai and S. Morita, J. Phys. Chem. A 109, 8424 (2005). 113. E. E. Dahlke and D. G. Truhlar, J. Chem. Theory Comput. 3, 46 (2007). 114. E. E. Dahlke and D. G. Truhlar, J. Chem. Theory Comput. 4, 1 (2008). 115. E. E. Dahlke, H. R. Leverentz and D. G. Truhlar, J. Chem. Theory Comput. 4, 33 (2008). 116. P. Sherwood, in Modern and Algorithm of Quantum chemistry, 2nd ed., Ed. J. Grotendorst, (NIC: J¨ ulich, 2000) Vol. 3, p. 285. 117. J. P. Foster and F. Weinhold, J. Am. Chem. Soc. 102, 7211 (1980). 118. J. Nocedal, Math. Comput. 35, 773 (1980). 119. D. C. Liu and J. Nocedal, Math. Prog. 45, 503 (1989). 120. M. P.Allen and D. J. Tildesley, in Computer Simulation of Liquids (Clarendon Press: Oxford, 1987) p. 99. 121. T. Kerdcharoen, K. R. Liedl and B. M. Rode, Chem. Phys. 211, 313 (1996). 122. K. Morokuma and T. Kerdcharoen, Chem. Phys. Lett. 355, 257 (2002). 123. T. S. Hofer, A. B. Pribil, B. R. Randolf and B. M. Rode, J. Am. Chem. Soc. 127, 14231 (2005). 124. G. J. Martyna, M. L. Klein and M. Tuckerman, J. Chem. Phys. 97, 2635 (1992). 125. S. Jang and G. A. Voth, J. Chem. Phys. 107, 9514 (1997). 126. R. W. Body and V. L. Kyllingstad, in Encyclopedia of Polymer Science and Engineering, 2nd ed.(Wiley-Interscience: New York, 1986) vol. 6, p. 225. 127. S. L. Aggarwal and O. J. Sweeting, Chem. Rev.57, 665 (1957). 128. D. Rigby, H. Sun and B. E. Eichinger, Polym. Int. 44, 311 (1997). 129. G. D. Smith, D. Bedrov, O. Borodin, Phys. Rev. Lett. 85, 5583(2000). 130. P. Ahlstr¨ om, O. Borodin, G. Wahnstr¨ om, E. J. W. Wensink, P. Carlsson and G. D. Smith, J. Chem. Phys. 112, 10669 (2000). 131. G. D. Smith, D. Bedrov and O. Borodin, J. Am. Chem. Soc. 122, 9548 (2000). 132. D. Bedrov and G. D. Smith, Langmuir 22, 6189 (2006). 133. Y. Sasanuma, H. Ohta, I. Touma, H. Matoba, Y. Hayashi and A. Kaito, Macromolecules 35, 3748 (2002). 134. L. J. A. Siqueira and M. C. C. Ribeiro, J. Chem. Phys. 125, 214903 (2006). 135. S. Fujiwara and T. Sato, J. Chem. Phys. 107, 613 (1997). 136. V. A. Harmandaris, V. G. Mavrantzas and D. N. Theodorou, Macromolecules 31, 7934 (1998). 137. Y. Jin and R. H. Boyd, J. Chem. Phys. 108, 9912 (1998). 138. C. Liu and M. Muthukumar, J. Chem. Phys. 109, 2536 (1998). 139. M. P¨ utz, K. Kremer and G. S. Grest, Europhys. Lett. 49, 735 (2000).
09-chapter8
September 29, 2010
10:44
World Scientific Review Volume - 9in x 6in
Theoretical Modeling of Hydrogen Bonding in Macromolecular Solutions
09-chapter8
261
140. S. J. V. Frankland, A. Caglar, D. W. Brenner and M. Griebel, J. Phys. Chem. B 106, 3046 (2002). 141. J. T. Hirvi and T. A. Pakkanen, J. Chem. Phys. 125, 144712 (2006). 142. V. Knyazev, J. Phys. Chem. A 111, 3875 (2007). 143. H. Li, W. Li, S. Li and J. Ma, J. Phys. Chem. B 112, 7061 (2008). 144. H. J. Berendsen, J. P. M. Postma, W. F. van Gunsteren and J. Hermans, in Intermolecular Forces; Ed. B. Pullman, (Reidel: Dordrecht, The Netherland, 1981), p. 331. 145. P. Dauber-Osguthorpe, V. A. Roberts, D. J. Osguthorpe, J. Wolff, M. Genest and A. T. Hagler, Proteins: Struct., Funct., Genet. 4,31 (1988). 146. M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria, M. A. Robb, J. R. Cheeseman, J. A. Montgomery Jr, T. Vreven, K. N. Kudin, J. C. Burant, J. M. Millam, S. S. Iyengar, J. Tomasi, V. Barone, B. Mennucci, M. Cossi, G. Scalmani, N. Rega, G. A. Petersson, H. Nakatsuji, M. Hada, M. Ehara, K. Toyota, R. Fukuda, J. Hasegawa, M. Ishida, T. Nakajima, Y. Honda, O. Kitao, H. Nakai, M. Klene, X. Li, J. E. Knox, H. P. Hratchian, J. B. Cross, V. Bakken, C. Adamo, J. Jaramillo, R. Gomperts, R. E. Stratmann, O. Yazyev, A. J. Austin, R. Cammi, C. Pomelli, J. W. Ochterski, P. Y. Ayala, K. Morokuma, G. A. Voth, P. Salvador, J. J. Dannenberg, V. G. Zakrzewski, S. Dapprich, A. D. Daniels, M. C. Strain, O. Farkas, D. K. Malick, A. D. Rabuck, K. Raghavachari, J. B. Foresman, J. V. Ortiz, Q. Cui, A. G. Baboul, S. Clifford, J. Cioslowski, B. B. Stefanov, G. Liu, A. Liashenko, P. Piskorz, I. Komaromi, R. L. Martin, D. J. Fox, T. Keith, M. A. Al-Laham, C. Y. Peng, A. Nanayakkara, M. Challacombe, P. M. W. Gill, B. Johnson, W. Chen, M. W. Wong, C. Gonzalez and J. A. Pople, Gaussian 03 ; Revision D. 01; Gaussian, Inc.: Wallingford, CT, 2004. 147. T. Steiner, Angew. Chem. Int. Ed. 41, 48 (2002). 148. S. J. Weiner, P. A. Kollman, D. A. Case, U. C. Singh, C. Ghio, G. Alagona, S., Jr. Profeta and P. Weiner, J. Am. Chem. Soc. 106, 765 (1984). 149. S. J. Weiner, P. A. Kollman, D. T. Nguyen and D. A. Case, J. Comput. Chem. 7, 230 (1986). 150. F. A. Momany and R. Rone, J. Comput. Chem. 13, 888 (1992). 151. 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 (1995). 152. Y. Duan, C. Wu, S. Chowdhury, M. C. Lee, G. Xiong, W. Zhang, R. Yang, P. Cieplak, R. Luo, T. Lee, J. Caldwell, J. Wang and P. Kollman, J. Comput. Chem. 24, 1999 (2003). 153. W. L. Jorgensen and J. Tirado-Rives, J. Am. Chem. Soc. 110, 1657 (1988). 154. W. L. Jorgensen, D. S. Maxwell and J. Tirado-Rives, J. Am. Chem. Soc. 118, 11225 (1996). 155. N. A. McDonald and W. L. Jorgensen, J. Phys. Chem. B 102, 8049 (1998). 156. W. L. Jorgensen and N. A. McDonald, J. Mol. Struct. (THEOCHEM) 424, 145 (1998). 157. M. L. P. Price, D. Ostrovsky and W. L. Jorgensen, J. Comput. Chem. 22, 1340 (2001).
September 29, 2010
262
158. 159. 160. 161. 162. 163. 164. 165. 166. 167.
168. 169. 170. 171. 172. 173. 174. 175. 176. 177. 178. 179. 180. 181. 182. 183. 184. 185. 186. 187. 188. 189. 190.
10:44
World Scientific Review Volume - 9in x 6in
J. Ma, N. Jiang and H. Li
G. A. Kaminski and R. A.Friesner, J. Phys. Chem. B 105, 6474 (2001). F. H. Stillinger and C. W. David, J. Chem. Phys. 69, 1473 (1978). T. P. Lybrand and P. A. Kollman, J. Chem. Phys. 83, 2923 (1985). J. A. C. Rullmann and P. Th. van Duijnen, Mol. Phys. 63, 451 (1988). M. Sprik and M. L. Klein, J. Chem. Phys. 89, 7556 (1988). P. Cieplak, P. Kollman and T. Lybrand, J. Chem. Phys. 92, 6755 (1990). A. Wallqvist and B. J. Berne, J. Phys. Chem. 97, 13841 (1993). J. W. Caldwell and P. A. Kollman, J. Phys. Chem. 99, 6208 (1995). W. Xie, J. Pu, A. D. MacKerell Jr. and J. Gao, J. Chem. Theory Comput. 3, 1878 (2007). ´ I. Soteras, C. Curutchet, A. Bidon-Chanal, F. Dehez, J. G. Angy´ an, M. Orozco, C. Chipot and F. J. Luque, J. Chem.Theory Comput. 3, 1901 (2007). A. J. Stone, Chem. Phys. Lett. 83, 233 (1981). D. S. Kosov and P. L. A. Popelier, J. Phys. Chem. A 104, 7339 (2000). P. L. A. Popelier, L. Joubert and D. S. Kosov, J. Phys. Chem. A 105, 8254 (2001). P. L. A. Popelier and D. S. Kosov, J. Chem. Phys. 114, 6539 (2001). C. Sagui, L. G. Pedersen and T. A. Darden, J. Chem. Phys. 120, 73 (2004). A. J. Stone, J. Chem. Theory Comput. 1, 1128 (2005). R. D. Mountain, J. Chem. Phys. 103, 3084 (1995). D. N. Bernardo, Y. Ding, K. Krogh-Jespersen and R. M. Levy, J. Phys. Chem. 98, 4180 (1994). Y. Ding, D. N. Bernardo, K. Krogh-Jespersen and R. M. Levy, J. Phys. Chem. 99, 11575 (1995). D. E. Smith and L. X. Dang, J. Chem. Phys. 100, 3757 (1994). J. Gao, D. Habibollazadeh and L. Shao, J. Phys. Chem. 99, 16460 (1995). P. Cieplak, J. Caldwell and P. Kollman, J. Comput. Chem. 22, 1048 (2001). G. A. Kaminski, R. A. Friesner and R. Zhou, J. Comput. Chem. 24, 267 (2003). P. Drude, C. R. Mann and R. A. Millikan, The Theory of Optics (Longmans, Green, and Co., New York, 1902). J. S. Høye and G. Stell, J. Chem. Phys. 73, 461 (1980). J. Cao and B. J. Berne, J. Chem. Phys. 99, 6998 (1993). F. Lado, J. Chem. Phys. 106, 4707 (1997). G. Lamoureux and B. Roux, J. Chem. Phys. 119, 3025 (2003). G. Archontis, E. Leontidis and G. Andreous, J. Phys. Chem. B 109, 17957 (2005). I. V. Vorobyov, V. M. Anisimov and A. D. MacKerell Jr., J. Phys. Chem. B 109, 18988 (2005). E. Harder, V. M. Anisimov, I. V. Vorobyov, P. E. M. Lopes, S. Y. Noskov, A. D. MacKerell Jr. and B. Roux, J. Chem. Theory Comput. 2, 1587 (2006). G. Lamoureux, E .Harder, I. V. Vorobyov, B. Roux and A. D. MacKerell Jr., Chem. Phys. Lett. 418, 245 (2006). I. V. Vorobyov, V. M. Anisimov, S. Greene, R. M. Venable, A. Moser, R. W. Pastor and A. D. MacKerell Jr., J. Chem. Theory Comput. 3, 1120 (2007).
09-chapter8
September 29, 2010
10:44
World Scientific Review Volume - 9in x 6in
Theoretical Modeling of Hydrogen Bonding in Macromolecular Solutions
09-chapter8
263
191. V. M. Anisimov, I. V. Vorobyov, V. Roux and A. D. MacKerell Jr., J. Chem. Theory Comput. 3, 1927 (2007). 192. D. P. Geerke and W. F. van Gunsteren, J. Chem. Theory Comput. 3, 2128 (2007). 193. S. W. Rick, S. J. Stuart and B. J. Berne, J. Chem. Phys. 101, 6141 (1994). 194. S. W. Rick, S. J. Stuart, J. S. Bader and B. J. Berne, J. Mol. Liq. 65-66, 31 (1995). 195. S. W. Rick and B. J. Berne, J. Am. Chem. Soc. 118, 672 (1996). 196. S. W. Rick and B. J. Berne, J. Phys. Chem. B 101, 10488 (1997). 197. S. W. Rick, J. Chem. Phys. 114, 2276 (2001). 198. L. R. Olano and S. W. Rick, J. Comput. Chem. 26, 699 (2005). 199. S. J. Stuart, B. J. Berne, J. Phys. Chem. 100, 11934 (1996). 200. Y.-P. Liu, K. Kim, B. J. Berne, R. A. Friesner and S. W. Rick, J. Chem. Phys. 108, 4739 (1998). 201. J. L. Banks, G. A. Kaminski, R. Zhou, D. T. Mainz, B. J. Berne and R. A. Friesner, J. Chem. Phys. 110, 741 (1999). 202. H. A. Stern, G. A. Kaminski, J. L. Banks, R. Zhou, B. J. Berne and R. A. Friesner, J. Phys. Chem. B 103, 4730 (1999). 203. H. A. Stern, F. Rittner, B. J. Berne and R. A. Friesner, J. Chem. Phys. 115, 2237 (2001). 204. G. A. Kaminski, H. A. Stern, B. J. Berne, R. A. Friesner, Y. X. Cao, R. B. Murphy, R. Zhou and T. A. Halgren, J. Comput. Chem. 23, 1515 (2002). 205. G. A. Kaminski, H. A. Stern, B. J. Berne and R. A. Friesner, J. Phys. Chem. A 108, 621 (2004). 206. E. Harder, B. Kim, R. A. Friesner and B. J. Berne, J. Chem. Theory Comput. 1, 169 (2005). 207. J. R. Maple, Y. Cao, W. Damm, T. A. Halgren, G. A. Kaminski, L. Y. Zhang and R. A. Friesner, J. Chem. Theory Comput. 1, 694 (2005). 208. N. Yoshii, S. Miura and S. Okazaki, Chem. Phys. Lett. 345, 195 (2001). 209. S. Patel and C. L. Brooks III., J. Comput. Chem. 25, 1 (2004). 210. S. Patel, A. D. MacKerell Jr and C. L. Brooks III., J. Comput. Chem. 25, 1504 (2004). 211. S. Patel and C. L. Brooks III., J. Chem. Phys. 123, 164502 (2005). 212. C. S. Wang and Z. Z. Yang, J. Chem. Phys. 110, 6189 (1999). 213. Z. Z. Yang, Y. Wu and D. X. Zhao, J. Chem. Phys. 120, 2541 (2004). 214. Z. Z. Yang and Q. Zhang, J. Comput. Chem. 27, 1 (2005). 215. Z. Z. Yang and P. Qian, J. Chem. Phys. 125, 064311 (2006). 216. E. Llanta, K. Ando and R. Rey, J. Phys. Chem. B 105, 7783 (2001). 217. E. Llanta and R. Rey, Chem. Phys. Lett. 340, 173 (2001). 218. K. Ando, J. Chem. Phys. 115, 5228 (2001). 219. R. Chelli, S. Ciabatti, G. Cardini, R. Righini and P. Procacci, J. Chem. Phys. 111, 4218 (1999). 220. R. Chelli and P. Procacci, J. Chem. Phys. 117, 9175 (2002). 221. R. Chelli, R. Righini, S. Califano and P. Procacci, J. Mol. Liq. 96, 87 (2002). 222. W. J. Mortier, K. van Genechten and J. Gasteiger, J. Am. Chem. Soc. 107, 829 (1985).
September 29, 2010
264
10:44
World Scientific Review Volume - 9in x 6in
J. Ma, N. Jiang and H. Li
223. W. J. Mortier, S.K. Ghosh and S. Shankar, J. Am. Chem. Soc. 108, 4315 (1986). 224. K. T. No, J. A. Grant and H. A. Scheraga, J. Phys. Chem. 94, 4732 (1990). 225. A. K. Rapp´e and W. A. Goddard, J. Phys. Chem. 95, 3358 (1991). 226. B. G. Baekelandt, W. J. Mortier, J. L. Lievens and R. A. Schoonheydt, J. Am. Chem. Soc. 113, 6730 (1991). 227. J. Cioslowski and B. B. Stefanov, J. Chem. Phys. 99, 5151 (1993). 228. S. K. Ghosh, Int. J. Quantum Chem. 49, 239 (1994). 229. F. De Proft, W. Langenaeker and P. Geerlings, J. Mol. Struct. (THEOCHEM) 339, 45 (1995). 230. D. M. York and W. Yang, J. Chem. Phys. 104, 159 (1996). 231. Z. Z. Yang and C. S. Wang, J. Phys. Chem. A 101, 6315 (1997). 232. Z. Z. Yang and B. Q. Cui, J. Chem. Theory Comput. 3, 1561 (2007). 233. K. Zhu, M. R. Shirts and R. A. Friesner, J. Chem. Theory Comput. 3, 2108 (2007). 234. P. J. Winn, G. G. Ferenczy, C. A. Reynolds, J. Comput. Chem. 20, 704 (1999). 235. G. G. Ferenczy, C. A. Reynolds, J. Phys. Chem. A 105, 11470 (2001). 236. C. Curutchet, J. M. Bofill, B. Hern´ andez, M. Orozco and F. J. Luque, J. Comput. Chem. 24, 1263 (2003). 237. W. Xie and J. Gao,. J. Chem. Theory Comput. 3, 1890 (2007). ´ 238. F. Dehez, J. G. Angy´ an, I. S. Guti´errez, F. J. Luque, K. Schulten and C. Chipot, J. Chem. Theory Comput. 3, 1914 (2007). 239. S. Nakagawa, P. Mark and H. ˚ Agren, J. Chem. Theory Comput. 3, 1947 (2007). 240. N. Gresh, G. A. Cisneros, T. A. Darden and J.-P. Piquemal, J. Chem. Theory Comput. 3, 1960 (2007). 241. N. Jiang and J. Ma, J. Phys. Chem. A 112, 9854 (2008). 242. W. L. Jorgensen, K. P. Jensen and A. N. Alexandrova, J. Chem. Theory Comput. 3, 1987 (2007). 243. A. V. Marenich, R. M. Olson, C. P. Kelly, C. J. Cramer and D. G. Truhlar, J. Chem. Theory Comput. 3, 2011 (2007). 244. R. M. Olson, A. V. Marenich, C. J. Cramer and D. G. Truhlar, J. Chem. Theory Comput. 3, 2046 (2007). 245. A. V. Marenich, R. M. Olson, A. C. Chamberlin, C. J. Cramer and D. G. Truhlar, J. Chem. Theory Comput. 3, 2055 (2007). 246. K. Palmo and S. Krimm, J. Chem. Theory Comput. 3, 2120 (2007). 247. C. Niedermeier and P. Tavan, J. Chem. Phys. 101, 734 (1994). 248. H. M. Berman, J. Westbrook, Z. Feng, G. Gilliland, T. N. Bhat, H. Weissig, I. N. Shindyalov and P. E. Bourne, Nucleic Acid Res. 28, 235 (2000). 249. L. W. Guddat, J. A. Martin, L. Shan, A. B. Edmundson and W. R. Gray, Biochemistry 35, 11329 (1996). 250. H. Gouda, K. Yamazaki, J. Hasegawa, Y. Kobayashi, Y. Nishiuchi, S. Sakakibara and S. Hirono, Biochim. Biophys. Acta 1343, 327 (1997). 251. A. Kaerner and D. L. Rabenstein, Biochemistry 38, 5459 (1999).
09-chapter8
September 29, 2010
10:45
World Scientific Review Volume - 9in x 6in
Chapter 9 Exotic Electrostatics: Unusual Features of Electrostatic Interactions between Macroions Ali Naji Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge CB3 0WA, United Kingdom Matej Kanduˇc Department of Theoretical Physics, J. Stefan Institute, SI-1000 Ljubljana, Slovenia Roland R. Netz Department of Physics, Technical University of Munich, James Franck Strasse, D-85748 Garching, Germany Rudolf Podgornik Department of Theoretical Physics, J. Stefan Institute, SI-1000 Ljubljana, Slovenia and Department of Physics, Faculty of Mathematics and Physics and Institute of Biophysics, Medical Faculty, University of Ljubljana, SI-1000 Ljubljana, Slovenia We present an overview of our understanding of electrostatic interactions between charged macromolecular surfaces mediated by mobile counterand coions. The dichotomy between the weak and the strong coupling regimes is described in detail and the way they engender repulsive and attractive interactions between nominally equally charged macroions. We also introduce the concept of dressed counterions in the case of many-component Coulomb fluids that are partially weakly and partially strongly coupled to local electrostatic fields leading to non-monotonic interactions between equally charged macroions. The effect of quenched surface charge disorder on the counterion-mediated electrostatic interactions is analyzed within the same conceptual framework and shown to
265
10-chapter9
September 29, 2010
10:45
266
World Scientific Review Volume - 9in x 6in
A. Naji et al.
lead to unexpected and extraordinary electrostatic interactions between randomly charged surfaces with equal mean surface charge densities or even between effectively neutral macroion surfaces. As a result, these recent developments challenge some cherished notions of pop culture.
1. Introduction The nature of electrostatic interactions between charges started to be studied intensively in the second half of the 18th century.1 Benjamin Franklin first inferred from an observation that surprisingly there is no force on a charge inside a charged sphere, an observation later repeated by Joseph Priestley (1767). John Robison (1769) determined that the electrostatic force falls off with (almost) the second power of separation between charges. Based on the work of Charles A. de Coulomb (1777) who is a co-inventor of the torsion balance for measuring the force of magnetic and electrical attraction, Henry Cavendish measured directly the interactions between charges (1779) but did not publish his results. They were eventually published by William Thomson (Lord Kelvin) one hundred years after the original discovery (1879). From hard science electrostatic interaction penetrated the pop culture in general and today everybody knows that opposites attract and likes repel! The exact form of the electrostatic interaction is since known to be given by the Coulomb’s law which states that the interaction potential between two charges e1 and e2 in vacuo located at r and r respectively, can be written in the standard form (in SI units) as e1 e 2 , (1) V (r, r ) = 4πε0 |r − r | were ε0 is the permittivity of vacuum. Electrostatic interaction is the fundamental interaction in molecular world giving rise to short range intramolecular bonds as well as longer ranged interactions between molecules and their aggregates.2 While atomic bonds are obtained by combining the Coulomb potential with the quantum of action, longer ranged colloidal and nanoscale electrostatic interactions are obtained by combining the Coulomb potential with the thermal energy kB T . Almost exactly a hundred years ago Gouy3 and Chapman4 were the first to combine thermal energy and Coulomb interactions into a statistical theory of Coulomb fluids basing their approach on what became later known as the Poisson-Boltzmann (PB) equation. Coulomb fluid in general is an assembly of (variously) charged particles in thermal equilibrium. This
10-chapter9
September 29, 2010
10:45
World Scientific Review Volume - 9in x 6in
Exotic Electrostatics of Charged Macromolecular Systems
10-chapter9
267
mean-field approach to Coulomb fluids was gradually elaborated in uncountably many ways starting from the seminal work of Debye and H¨ uckel5 and finally codified as a cornerstone of the fundamental DLVO theory of colloidal interactions by Derjaguin and Landau6 as well as Verwey and Overbeek,7 where it figures as the repulsive electrostatic part of the total disjoining pressure acting between macromolecular bodies.8,9 The attractive component in this case is provided solely by the van der Waals interactions that have their origin in the quantum and thermal fluctuations of electromagnetic fields.10 One could claim that modern formulation of statistical mechanics of Coulomb fluids starts with the work of Edwards and Lenard11 where the grand-canonical partition function of a Coulomb fluid was written in the form of a functional integral over fluctuating electrostatic fields. While ˇ s12 realized that the collective description based on the Podgornik and Zekˇ mean-field PB theory arises from the saddle-point approximation to this functional integral in the case of a weak coupling regime corresponding, for instance, to small external charges, Netz showed13,14 that another approximation valid in the regime of large external charges leads to a completely different fixed point, formulated in terms of a single-particle description of the same system. This so-called strong coupling regime has been the focus of various theoretical studies over the past decade.15–19 The two approximations were shown to correspond to extremal values of a single electrostatic coupling parameter and to consistently bracket all available simulation results on statistical properties of non-homogeneous Coulomb fluids,15,16 i.e., Coulomb fluids confined between charged boundaries. In what follows we shall describe the main consequences of this weakstrong coupling dichotomy especially as they transpire in the case of interactions between charged (planar) macromolecular surfaces across a Coulomb fluid comprising mobile counter- and coions. We shall see that the counterion-mediated electrostatic interactions are strictly repulsive in symmetric external charge configurations for the case of weak coupling but can turn attractive for strongly coupled surfaces. In the case of many-component systems, composed typically of a weakly coupled univalent salt and strongly coupled polyvalent counterions, these interactions show a subtle interplay between repulsions and attractions for nominally equally charged surfaces. On the other hand, the presence of quenched charge disorder on bounding surfaces can lead to pronounced electrostatic attraction. These exotic features only arise in special cases of strongly coupled systems and can not be rationalized within the standard mean-field PB approach.
September 29, 2010
10:45
268
World Scientific Review Volume - 9in x 6in
A. Naji et al.
2. Scenery Electric charges and electrostatic interactions are ubiquitous in soft matter and biological systems.20,21 Soft materials are typically composed of macromolecules such as polymers, colloids and proteins which often acquire surface charges when dissolved in a polar solvent like water. This is usually due to dissociation of surface chemical groups, which leaves permanent charges on macromolecular surfaces and releases oppositely charged microscopic counterions into the solution. Soft materials are easily deformed or re-arranged by interaction potentials comparable in magnitude to thermal energy. It thus becomes clear that electrostatic interactions, that are typically long-ranged and strong, constitute a prominent factor in determining the behavior and properties of soft materials. This makes charged materials central to many technological applications and on the other hand, a challenging subject for fundamental research in inter-disciplinary sciences.2 In what follows, we briefly review a few examples to demonstrate the diversity of phenomena associated with charged soft matter systems. 2.1. Colloids, polymers and membranes: The mesoscopic scale Colloids are abundant in nature and industry: smoke, fog, milk, paint and ink are only a few examples of colloidal systems. They comprise tiny solid or liquid particles that are suspended in another medium such as air or another liquid. An important factor, which makes colloidal solutions in many ways different from molecular or simple electrolyte solutions (such as sugar or salt solution), is the large asymmetry in size and mass between the colloidal particles and solvent molecules (or microscopic ions): colloids are mesoscopic or even nanoscopic particles with sizes in the range of a few nanometers to microns that are indeed made of many atoms, but not yet sufficiently many to make them behave like macroscopic bodies. In colloidal dispersions, the total area that is in contact with solvent is tremendously large: for nanometer-sized colloids, nearly half of the atoms are at the surface whereas for macroscopic bodies this ratio tends to zero. This is even more true for extended quasi-two-dimensional macromolecular aggregates such as lipid membranes and surfactant-covered interfaces that can carry a substantial amount of charged molecular moieties.21 Therefore, contrary to the typical situation in the macroscopic world, the physics of meso- and nanoscopic particles are dominated by “surface” properties and interactions.7–9
10-chapter9
September 29, 2010
10:45
World Scientific Review Volume - 9in x 6in
Exotic Electrostatics of Charged Macromolecular Systems
10-chapter9
269
Fig. 1. Schematic view of a charged solution consisting of charged colloids (dark spheres), charged polymer chains and microscopic neutralizing counterions (small black spheres). Colloids may be covered by charged polymer brushes, which generate an additional repulsive interaction between them.
Another relevant mesoscopic or macromolecular system are polymers (with everyday-life examples like chewing gum, dough or egg white), in which many repeating subunits (monomers) are chemically connected to form a flexible chain. Flexible polymers are distinguished by their many degrees of freedom associated with conformational rearrangements of monomers that are easily excited by thermal energy at room temperature leading to a diverse phase behavior spanning extended and strongly entangled polymers such as DNA in solution, all the way to collapsed polymer chains organizing into compact globular states as in the case of proteins. Depending on their chemical structure, polymer chains can have a large mechanical stiffness as well, behaving like rigid rods at small length scales, or can be substantially charged giving rise to polyelectrolytes, in both cases playing an important role in biological processes occurring in the cell.22 2.2. Charges: From industry to biology In the mesoscopic world, only electromagnetic interactions that emerge in a variety of forms, are important; they are capable of overcoming thermal fluctuations — which are characterized by an energy scale of about 0.025eV at room temperature8 — and thus enable formation of stable condensed phases for soft complex materials.
September 29, 2010
270
10:45
World Scientific Review Volume - 9in x 6in
A. Naji et al.
In general, colloids dissolved in an aqueous solvent attract each other due to ubiquitous van der Waals dispersion forces that result from induced charges on their surface.2 As a result, colloidal particles tend to form large aggregates that typically sediment and destroy the dispersion. In many applications (for example in food emulsions such as milk), however, stability of a colloidal dispersion is a desirable property. One way to stabilize dispersions against aggregation is to generate long-range repulsive interactions between colloidal particles by charging their surfaces, which leads to the DLVO mechanism for the stability of colloidal dispersions.6,7 Another method of stabilization is to end-graft polymer chains (or polymer brushes) to the particle surfaces.23 For this task, charged polymers are ideal since they swell substantially in aqueous solutions and inhibit close contact between colloids (see Fig. 1). This latter mechanism has the advantage that it is highly stable against the addition of electrolyte or salt ions.24 Charged polymers, or polyelectrolytes, play a significant role in the production of cheap, non-toxic and environmentally friendly materials.25,26 In contrast to water-insoluble hydrocarbon chains, polyelectrolytes typically show high affinity for water and heavy metal ions, which makes them useful in applications such as super-absorbing diapers, waste water purifiers and washing detergents and their additives. In biology electrostatic effects between charged polymers such as DNA and RNA are ubiquitous.20 DNA, for instance, is a long biomolecule with a total length of about two meters in human cells, bearing one elementary charge per 1.7˚ A, which for human DNA adds up to 1010 elementary charges overall! Yet the DNA is densely packed inside the cell nucleus with a diameter of less than a few microns. In eucaryotic cells, this storage process involves a hierarchical structure on the lowest level of which short segments of DNA are tightly wrapped around positively charged histone protein complexes of a few nanometers in diameter.22 This protein-DNA complexation is believed to be governed by electrostatic interactions.15,27,28 Electrostatic effects also play a key role in complexes of DNA with cationic lipids,29–31 which are promising synthetically based non-viral carriers of DNA for gene therapy.32 Another example (which is closely related to the results presented later in this chapter) is the DNA condensation,20,33,34 in which electrostatic effects enter in a counter-intuitive fashion: here like-charged segments of DNA strongly attract each other! In the in vitro experiments,35 the condensation of DNA is realized using bacteriophages, which consist of a rigid
10-chapter9
September 29, 2010
10:45
World Scientific Review Volume - 9in x 6in
Exotic Electrostatics of Charged Macromolecular Systems
10-chapter9
271
shell (the capsid) that accommodates a single molecule of viral DNA. These viruses can inject their DNA into a cell or a lipid vesicle. As a result, large lengths of DNA (up to a hundred microns) can be fitted and condensed into a tightly packed, circumferentially wound torus with a diameter of about a hundred nanometers. This packaging process, which works against the Coulomb self-repulsion and the conformational entropy of the DNA chain, is facilitated and depends upon the presence of high-valency counterions in the medium. Similarly, other highly charged polymers, such as negatively charged F-actin and microtubules can aggregate into closely packed rod-like bundles when small amounts of polyvalent cations are added to the solution.36–38 It turns out that, in general, when particles are strongly charged, the role of electrostatic interactions dramatically changes:15 here electrostatic interactions themselves can trigger the destabilization of charged solutions by mediating attractive like-charge interactions! 2.3. Theoretical challenge and coarse-grained models From a theoretical point of view, charged systems pose a many-body problem: macroions, such as charged colloids and polymers, and other charged macromolecular surfaces, such as lipid membranes and surfactant layers, are always surrounded by counterions, and also in general by coions. These particles form loosely bound ionic clouds around macroions and tend to screen their charges. In particular, counterions that are attracted towards macromolecular surfaces, predominantly determine the static and dynamic properties of macromolecular solutions. Understanding the interactions between macromolecules across an ionic medium thus requires an understanding of the counterionic clouds first. In the most common theoretical approaches known also as primitive models, the molecular nature of the solvent is neglected and is represented by a continuum dielectric medium. In reality, the solvent structure is locally perturbed around particles, which can give rise to additional short-ranged solvent-induced interactions.8,39,40 On the other hand, the microscopic features of the macroions are taken into account using coarse-grained models that incorporate only a few effective parameters such as an effective surface charge density. In most cases, the specific effects associated with ions39 as well as the image charge effects due to dielectric inhomogeneities are also neglected. These models therefore represent a crude simplification of reality, yet given those simplifications, they can still lead to a systematic and clear understanding of electrostatic effects.
September 29, 2010
272
10:45
World Scientific Review Volume - 9in x 6in
10-chapter9
A. Naji et al.
Here we shall first begin by adopting such a simple model for the interaction between charged macromolecular surfaces in the presence of counterions only, but then examine the effects due to the additional salt41 and the heterogeneous or disordered distribution of surface charges42–45 in more detail. For simplicity, we shall also focus only on the case of planar charged surfaces, appropriate for the case of charged membranes, solid substrates, or large colloids. Other factors such as non-planar geometry of charged surfaces,46–48 image charges,43,48–50 dissimilar surfaces51 or multipolar structure of counterions52 have been analyzed within the same context. 3. Length Scales in a Classical Charged System Consider a system of fixed charged objects with uniform surface charge density −σs e0 (with e0 being the elementary charge) that are surrounded by their neutralizing counterions of charge valency +q in a solvent of dielectric constant ε at temperature T .a One of the basic characteristic length scales in a charged system is the Bjerrum length 53 B = e20 /(4πεε0 kB T ),
(2)
which is set by the ratio between the thermal energy, kB T , and the Coulomb interaction energy between two elementary charges at separation r, i.e., V /(kB T ) = B /r. The Bjerrum length thus measures the distance at which two elementary charges interact with an energy equal to kB T . In water and A. For counterions of charge at room temperature (ε = 80), one has B 7.1˚ valency +q, the Bjerrum length may be redefined as q 2 B . Other length scales may be identified by considering the specific form of the charge distribution and geometry of macroions. For uniformly charged planar surfaces (Fig. 2), one can define another key length scale by comparing the thermal energy with the energy scale of the counterion-wall attraction, i.e., u/(kB T ) = z/µ, where z is the distance from the wall and µ = 1/(2πqB σs )
(3)
is known as the Gouy-Chapman (GC) length.3,4 The GC length measures the distance at which the thermal energy equals the counterion-wall electrostatic interaction energy. It also gives a measure of the thickness of the counterion layer at a charged wall as we shall see later. a We conventionally assume that macroions are negatively charged and counterions are positively charged, thus σs and q are both positive by definition.
September 29, 2010
10:45
World Scientific Review Volume - 9in x 6in
Exotic Electrostatics of Charged Macromolecular Systems
(a)
10-chapter9
273
(b)
Fig. 2. Schematic view of the structure of a layer of counterions at an oppositely charged surface for small (a) and large (b) value of the coupling parameter.
For planar systems where no other length scales are present, it follows that only the dimensionless ratio between the above two length scales matters, i.e., Ξ = q2 B /µ = 2πq 3 2B σs .
(4)
This parameter is known as the electrostatic coupling parameter.13,14 4. From Mean-Field to Strong Coupling Regime For small coupling parameters Ξ 1, equation (4) shows that the GC length is relatively large, which indicates that counterions form a loosely bound cloud at an oppositely charged wall (Fig. 2(a)). For large coupling parameter Ξ 1, in contrast, the GC length is relatively small and counterions are strongly attracted toward the wall (Fig. 2(b)). Further insight may be obtained by considering the typical distance between counterions at a charged surface. For counterions residing near the surface, the local electroneutrality condition implies a typical lateral separation of (5) a⊥ ∼ q/σs , since each counterion neutralizes the charge of an area that scales as a2⊥ ∼ q/σs . Counterion spacing a⊥ is not an independent length scale and may be written in the units of the GC length as √ (a⊥ /µ) ∼ Ξ. (6) 4.1. Weak coupling or mean-field regime In the regime where Ξ 1, equation (6) shows that the lateral separation of counterions near the surface is small compared with the typical layer
September 29, 2010
10:45
World Scientific Review Volume - 9in x 6in
274
10-chapter9
A. Naji et al.
thickness, µ, which further indicates that the counterions tend to form a diffuse fluid-like layer at the surface (Fig. 2(a)).b This regime is dominated by collective mean-field-like effects, i.e., counterions become uncorrelated from each other in a statistical sense as each counterion in the layer interacts with a diffuse cloud of many other counterions. Therefore, Ξ 1 identifies the weak coupling (WC) or mean-field regime, which is relevant to systems with weakly charged surfaces, low valency counterions and/or high temperature.15 Formally, one can employ a mean-field approximation in order to describe the system in the WC regime by neglecting all inter-particle correlations on the leading order. The mean-field approximation is exact in the limit Ξ → 054 and leads to the so-called Poisson-Boltzmann (PB) equation for the mean-field electrostatic potential ψ(r),7,8 i.e. −εε0 ∇2 ψ = ρ0 (r) + qe0 n0 Ω(r) exp(−βqe0 ψ),
(7)
where β = 1/(kB T ). The first term on the r.h.s. represents the charge distribution due to fixed external charges (macroions), ρ0 (r), and the second term is the mean-field number density of counterions nPB (r) = n0 Ω(r) exp(−βqe0 ψ), where n0 is a normalization prefactor. The “blip” function Ω(r) is equal to one in the region accessible to counterions and zero elsewhere. For point-like counterions at a single uniformly charged wall, the PB equation yields the well-known algebraically decaying density profile7,8 1 nPB (z) = 2πB σs2 (z/µ + 1)2
z > 0,
(8)
where z is the distance from the wall. Note that the density profile is normalizable to the total number of counterions in order to ensure electroneutrality, and that the GC length, µ, equals the height of a layer containing half of the counterions, thus giving a measure of the typical layer thickness. The contact density of counterions nPB (z = 0) = 2πB σs2 turns out to be an exact result within the present model and remains valid beyond the mean-field theory.55 4.2. Strong coupling regime In the strong coupling (SC) regime Ξ 1, equation (6) shows that the lateral separation of counterions becomes larger than the GC length, thus b A more accurate estimate of the typical distance, a, between counterions in an extended three-dimensional layer gives a/µ ∼ Ξ1/3 .13
September 29, 2010
10:45
World Scientific Review Volume - 9in x 6in
Exotic Electrostatics of Charged Macromolecular Systems
10-chapter9
275
indicating that the counterions tend to form a quasi-2D layer at the surface (Fig. 2(b)). Such a layer is dominated by strong mutual repulsions between counterions as can be seen by considering the effective 2D plasma parameter56 Γ ≡ q 2 B /a⊥ ∼ Ξ1/2 , which gives the ratio between Coulombic inter-particle repulsions and the thermal energy. For elevated Ξ, Coulombic repulsions tend to freeze out lateral fluctuations of counterions on the surface, leading to strong correlations and a trend toward crystallization in the ionic structure.17,18 Individual counterions thus become isolated in relatively large correlation holes of size a⊥ from which neighboring counterions are statistically depleted. The Wigner crystallization of the 2D onecomponent plasma is known to occur for Γ > Γc 125,56 which corresponds to the range of coupling parameters Ξ > Ξc 3.1 × 104 .14 For Ξ 1, the PB description completely breaks down, nonetheless, one can obtain a simple analytical theory by employing a virial and 1/Ξ expansion to the leading order, which is known as the strong coupling theory.13 The SC theory turns out to contain contributions that involve only singleparticle interaction energies between counterions and the fixed macroion surface charges. For instance, the SC density profile of counterions at a single charged wall comes exclusively from the vertical degree of freedom, z, through which single isolated counterions are coupled to the wall with the interaction potential u/(kB T ) = z/µ. Hence using the Boltzmann weight, one finds the (laterally averaged) density profile nSC (z) = n0 exp(−z/µ),
(9)
where the prefactor (contact density) is again found from the electroneutrality condition to be n0 = 2πB σs2 . Unlike in the WC case, the SC density profile decays exponentially away from the charged wall. Moreover, the average distance of counterions is finite and equal to the GC length, z SC = µ, reflecting again the quasi-2D structure of the layer. Formally, the single-particle SC theory is exact in the asymptotic limit of an infinitely large correlation hole size, a⊥ /µ → ∞, or simply Ξ → ∞. However, its validity holds in a wider range of system parameters as is evident from comparison with computer simulations.13,14 For instance, for a finite coupling parameter Ξ, the SC density profile (9) still holds approximately at distances z < a⊥ , which yields the criterion √ z/µ < Ξ, (10) identifying the limits of applicability of the SC theory. At larger distances z > a⊥ , multi-particle interactions become increasingly more important and the mean-field-like features eventually dominate for z a⊥ .13,14
September 29, 2010
10:45
World Scientific Review Volume - 9in x 6in
276
10-chapter9
A. Naji et al. Table 1.
Typical values of physical parameters for some realistic systems. σs (e0 /nm2 ) q µ(˚ A) Ξ
charged membranes
∼1
1 2
2.2 1.1
3.1 24.8
3
0.7
83.7
DNA
0.9
1 (Na+ ) 2 (Mn2+ )
2.4 1.2
2.8 22.4
3 (spd3+ )
0.8
75.6
4 (sp4+ )
0.6
179 85
highly charged colloids (surfactant micelles)
∼1
3
0.7
weakly charged colloids
∼ 0.1
1
∼2
∼ 0.1
(polystyrene particles)
In brief, thus, one can identify two asymptotic regimes of weak and strong coupling based on the value of the electrostatic coupling parameter, where a charged system may be studied by means of two limiting theories, namely, the WC (mean-field) and the SC theory. In Table 1, we present illustrative examples of the parameter values (surface charge density σs , counterion valency q, GC length µ, and the coupling parameter Ξ in water and at room temperature) from a few realistic weakly and strongly coupled systems. Note that a typical coupling strength of Ξ ∼ 102 (or larger) already falls within the SC regime and a value of Ξ ∼ 1 (or smaller) typically falls well inside the WC regime.13–16 5. Interactions between Like-Charged Surfaces Macroions in solution are often like-charged and thus repel each other by their bare Coulomb interaction. The overall interaction is however different from this bare interaction due to the presence of counterions, which can mediate both repulsive and attractive effective forces. Obviously, the counterion-mediated interactions depend strongly on the distribution of counterions around macroions. In order to demonstrate the underlying physical picture, we shall focus on the interaction between two identical planar like-charged walls of uniform surface charge density −σs e0 at separation D, where q-valent counterions fill only the space between the walls, Fig. 3 (the dielectric constant is assumed to be uniform in space). In this system, an extra length scale is set by the wall separation, D. Two limiting regimes of repulsion and attraction may
September 29, 2010
10:45
World Scientific Review Volume - 9in x 6in
Exotic Electrostatics of Charged Macromolecular Systems
(a)
10-chapter9
277
(b)
Fig. 3. Schematic representation of the WC (a) and SC (b) interaction regimes for two like-charged walls.
be distinguished qualitatively by comparing D with other length scales of the system as follows. 5.1. WC regime: Repulsion Let us first consider the WC limit, Ξ 1, in the regime where the wall separation, D, is large compared with all other length scales (Fig. 3(a)). In this case, counterions form a diffuse layer at each wall, but due to the large separation, the system is approximately decoupled into two nearly neutral sub-systems, each consisting of a charged wall and its counterion cloud. The effective pressure acting between the walls is dominated by the osmotic pressure of counterions across the mid-plane, which is positive and thus corresponds to an effective repulsion between the walls. The scaling behavior of this effective repulsion follows by noting that the midplane osmotic pressure can be estimated from the local counterion density, nmid , and by using the ideal-gas equation of state as βP ∼ nmid . Thus, according to Eq. (8), the interaction pressure is expected to decay as ∼ D−2 . It turns out that the PB pressure obtained in the limit Ξ → 0 coincides exactly with the mid-plane osmotic pressure of counterions as discussed above and may be expressed as7,8,13 βPPB (D) = Λ, 2πB σs2
(11)
where Λ is obtained from Λ1/2 tan[Λ1/2 (D/2µ)] = 1. From here one can obtain the large-separation D/µ 1 behavior βPPB (D) πµ 2 . (12) 2πB σs2 D
September 29, 2010
10:45
World Scientific Review Volume - 9in x 6in
278
10-chapter9
A. Naji et al.
5.2. SC regime: Attraction Now let us consider the SC limit, Ξ 1, in the regime where D is smaller than the lateral spacing between counterion, a⊥ D (Fig. 3(b)). Since counterions are highly separated from each other, the two opposite layers of counterions tend to form an inter-locking pattern at small separations. The system may be thought of as a collection of laterally frozen “correlation cells”, each consisting of a single counterion sandwiched between two opposing sections of the walls with lateral size of about a⊥ . Since a⊥ D, the effective pressure between the walls is dominated by the contribution from single counterions fluctuating in single correlation cells. The electrostatic energy of the system per cell is the sum of the bare interactions between the two surfaces with each other and with the single counterion, which–using the electroneutrality condition per cell and the fact that the wall separation is small–follows as uelec/(kB T ) 2πB σs2 D per unit area. This energetic contribution gives an attractive pressure as βPelec −2πB σs2 between the walls. On the other hand, the entropic contribution due to counterion confinement is of the order Sci ∼ kB ln D (per cell), which generates a repulsive component. The total pressure between strongly coupled walls is then obtained by combining these two effects as 2µ βPSC (D) = −1 + . 2 2πB σs D
(13)
This expression clearly predicts a closely packed bound state for the likecharged walls with an equilibrium surface-surface separation, D∗ , equal to twice the GC length, i.e., D ∗ = 2µ. The like-charged walls therefore attract each other for D > D∗ and repel at smaller separations. The analytical expression (13) is indeed an exact result for planar walls in the limit Ξ → ∞.13,14 It turns out that in a system with finite coupling parameter, Ξ, the asymptotic (Ξ → ∞) SC results still hold approximately as long as the surface separation, D, is smaller than the typical lateral distance between counterions, a⊥ , i.e., for D < a⊥ . This condition in fact yields a simple and generic criterion identifying the regime where the SC attraction is expected to emerge between two like-charged macroions. It was originally suggested by Rouzina and Bloomfield17 and verified and generalized later using extensive analytical and numerical methods.13–16,41,46–49,51 For larger inter-surface separations, D > a⊥ , the mean-field features become increasingly more important and the strength of attraction reduces. Eventually at very large D, the interaction becomes repulsive.13,14
September 29, 2010
10:45
World Scientific Review Volume - 9in x 6in
Exotic Electrostatics of Charged Macromolecular Systems
10-chapter9
279
6. Counterions with Salt The SC theory was so far designed exclusively for counterions-only systems, i.e., Coulomb fluids composed of only counterions in the absence of any salt ions.13 Though an approximation of this type can be used to describe situations where a large amount of polyvalent counterions dominate the system, it has to be amended in the general case in order to deal with the complexity of real systems that always contain some amounts of simple salt.8 An experimentally oft-encountered situation would be a system composed of fixed surface charges with polyvalent counterions bathed in a solution of univalent salt.57,58 This situation leads to a difficult problem of asymmetric aqueous electrolytes where different components of the Coulomb fluid are differently coupled to local electrostatic fields.59 Polyvalent counterions are coupled strongly, whereas univalent salt ions are coupled weakly. In this case no single approximation scheme that would treat all the charged components on the same level would be expected to work. Whereas the SC framework would certainly work for the polyvalent counterions, it would fail for the univalent salt. The converse is true for the WC framework. One is thus faced with a problem since no single approximation scheme appears to be valid in any range of coupling parameters. One can nevertheless build a theoretical framework that allows to selectively use different approximation schemes for different components of the asymmetric Coulomb fluid. This combined WC-SC approach appears to bring forth all the salient features of these asymmetric systems at high electrostatic couplings.41 6.1. Functional integral formalism Our arguments until now were strictly intuitive. A formal theory can be developed exactly in terms of the functional integral representation of the classical partition function of the Coulomb fluid along the lines first introduced by Edwards and Lenard.11–13 Assume first that the system is composed of charged macromolecules with fixed charge density ρ0 (r), mobile polyvalent counterions and an additional univalent salt. The total electrostatic interaction energy of a given configuration of the system can be written as W =
1 2
ρ(r)v(r, r )ρ(r ) dr dr ,
(14)
September 29, 2010
10:45
World Scientific Review Volume - 9in x 6in
280
10-chapter9
A. Naji et al.
where v(r, r ) is the Coulomb kernel given by v(r, r ) = 1/(4πεε0 |r − r |), and ρ(r) is the total charge density ρ(r) = ρ0 (r) + qe0 δ(Rci− r) + e0 δ(R+ e0 δ(R− i − r) − i − r), (15) i
i
i
− where Rci , R+ i and Ri are the positions of the polyvalent counterions, univalent cations (salt counterions) and univalent anions (salt coions), respectively.c Assuming again that the system is composed of two apposed planar surfaces at z = ±D/2 with the surface charge density −σs , we have ρ0 (r) = −σs e0 δ(z − D/2) + δ(z + D/2) . (16)
The salt ions are assumed to be present in all regions in space, whereas the counterions are assumed to be present only in the inter-surface region |z| < D/2 as specified by the geometry “blip” function Ω(r) (Section 4.1). We then follow the standard procedure by introducing a fluctuating local potential, φ, via the Hubbard-Stratonovich transformation, which leads to the following exact functional integral representation for the grandcanonical partition function12,13 Z = Dφ e−βH[φ] , (17) where the field-functional Hamiltonian reads 1 −1 (18) H[φ] = φ(r)v (r, r )φ(r ) dr dr + i ρ0 (r)φ(r) dr 2 Λc Λ+ Λ− e−iβqe0 φ(r) Ω(r) dr − e−iβe0 φ(r) dr − eiβe0 φ(r) dr, − β β β and Λc and Λ± represent the fugacities of polyvalent counterions and salt ions and v −1 (r, r ) = −εε0 ∇2 δ(r − r ) is the inverse Coulomb kernel. The special case of counterions-only system, as analyzed in the previous section, is recovered by setting Λ± = 0. We shall assume that salt ions are in equilibrium with a bulk reservoir containing equal concentration nb of both positive and negative ions, which implies Λ+ = Λ− ≡ nb . One can thus introduce the Debye-H¨ uckel (DH) screening parameter κ (inverse “screening length”) as κ2 = 8πB nb . The polyvalent counterions shall be treated here within the canonical ensemble d , i.e., their number in the slit is assumed to be fixed and equal to N . The number of counterions may be expressed via the dimensionless parameter η = N q/(2σs S), c We d see
may refer to the q-valency (polyvalent) counterions simply as “counterions”. Ref.41 for a grand-canonical description of polyvalent counterions.
(19)
September 29, 2010
10:45
World Scientific Review Volume - 9in x 6in
10-chapter9
Exotic Electrostatics of Charged Macromolecular Systems
281
where S is the area of the interacting surfaces. The case η = 0 represents a system with salt only, and η = 1 is the case where the total charge due to counterions exactly compensates the surface charge. Note that η can take any non-negative value when salt ions are present. This is because salt ions turn the long-range Coulomb potential into a short-range DH potential (see below) and can thus ensure the electroneutrality condition themselves. 6.2. Dressed counterions Assuming that the system is highly asymmetric q 1, one can formulate an approximate theory in order to evaluate the partition function (17) analytically by acknowledging the fact that the polyvalent counterions are strongly coupled while the simple salt ions are weakly coupled to the fluctuating electrostatic fields. This leads to a mixed WC-SC evaluation of the partition function.41 The salt terms (the last two terms) in Eq. (18) can be combined into cos βe0 φ(r) and in a highly asymmetric system this can be expanded up to the quadratic order in the fluctuating potential. Thus up to an irrelevant constant we remain with an effective field Hamiltonian of the form 1 −1 Heff [φ] = φ(r)vDH (r, r )φ(r ) dr dr + i ρ0 (r)φ(r) dr 2 Λc (20) e−iβqe0 φ(r) Ω(r) dr. − β This procedure therefore yields an effective Hamiltonian for a “counterionsonly” system but with the proviso that the inverse Coulomb kernel is replaced by the standard inverse DH kernel
−1 (r, r ) = −εε0 (∇2 − κ2 )δ(r − r ), vDH
with vDH (r, r ) =
e−κ|r−r | . 4πεε0 |r − r | (21)
We have thus effectively integrated out the salt degrees of freedom leading to a renormalized interaction potential between all the remaining charge species of the screened DH form. One can thus drop any reference to explicit salt ions and infer the thermodynamic properties of the original system by analyzing it as a system composed of dressed counterions and fixed external charges interacting via a screened DH pair potential. In the SC limit for the polyvalent counterions we term this approximation scheme as the SC dressed counterion theory. Our SC analysis thus proceeds in the same way as for the counterions-only systems13 except that the interactions between the charges are now of a dressed form.
September 29, 2010
282
10:45
World Scientific Review Volume - 9in x 6in
10-chapter9
A. Naji et al.
We note that any Bjerrum pairing53,60 or even electrostatic collapse of the salt or formation of salt-counterion complexes61 is beyond the framework developed here. 6.3. WC dressed counterion theory We again focus on a system composed of two plane-parallel surfaces defined via Eq. (16). In the WC limit (for both the counterions as well as the salt ions), the functional integral derived in the previous section is dominated by the contribution from the saddle-point solution φSP . This subsequently leads to the mean-field equation for the real-valued mean-field potential ψ = iφSP , i.e., −εε0 (∇2 ψ − κ2 ψ) = ρ0 (r) + qe0 Λc Ω(r) e−βqe0 ψ ,
(22)
which, in rescaled units w = βqe0 ψ and by virtue of the lateral symmetry for planar surfaces w = w(z), may be written as w = κ2 w − C e−w
|z| < D/2.
(23)
The constant C can be evaluated when one stipulates the fixed amount of counterions. Outside the slit |z| > D/2, the mean-field equation has the standard DH form w = κ2 w, which yields w(z) = w0 exp(±κz). The interaction pressure, P , between the bounding surfaces is given by the difference of the ion concentrations at the mid-plane (z = 0), where the mean electric field vanishes, and the bulk concentration, i.e., βP = n+ (0) + n− (0) + nc (0) − 2nb , which leads to the dimensionless expression 1 1 βP (D) = (κµ)2 w2 (0) + C e−w(0) . 2πB σs2 4 2
(24)
As evident from the above equation, the pressure can never be negative and the effective interaction is thus always repulsive within this type of meanfield approach.62 The canonical mean-field equation (23) can be solved numerically and the results can be used to evaluate the pressure (24) as a function of the inter-surface separation (Fig. 4(a), dashed line). 6.4. SC dressed counterion theory The analysis of the dressed counterions system in the SC limit is very similar to the traditional SC approach in the case of counterions only.13 We proceed by expanding the grand-canonical partition function associated with the dressed counterion approximation, Eq. (20), to the first order in counterion
September 29, 2010
10:45
World Scientific Review Volume - 9in x 6in
10-chapter9
Exotic Electrostatics of Charged Macromolecular Systems
283
4
η = 5.0
3
~ P
PB SC Ξ = 50 Ξ = 100 Ξ = 500
κµ = 0.4
2 1 0 -1 0
10
20
30
D/µ
40
(a)
(b)
Fig. 4. a) Rescaled interaction pressure P˜ ≡ βP (D)/(2πB σs2 ) between two like-charged surfaces as a function of the rescaled inter-surface distance. b) Van der Waals type iso-inverse screening length curves are shown along with the corresponding Maxwell construction. The inverse screening length is varied in the range from κµ = 0.4 to κµ = 0.56 in intervals of 0.027 (from bottom to top). Adapted from Ref. 41.
fugacity, Λc . We then perform an inverse Legendre transformatione in order to obtain the canonical SC free energy.13,46 We thus find (25) FN = U0 − N kB T ln e−βu(r) dr, where the first term is the screened interaction energy of fixed charges U0 =
1 2
ρ0 (r)vDH (r, r )ρ0 (r ) dr dr ,
(26)
and the term in the exponent is the single-particle interaction energy of the dressed counterions with fixed macroion charges u(r) = qe0 ρ0 (r )vDH (r, r ) dr . (27) The SC attraction between like-charged macroions stems from the second term in Eq. (25), which contains the counterion-induced effects.13,15,16,46 For the planar system under consideration (Eq. (16)), the above quantities may be evaluated explicitly41 and we find the SC free energy as βFN /N = e This
1 e−κD − ln I(D), 2κµη
(28)
is achieved by mapping the fugacity to the number of counterions, N , via the relation Λc ∂ ln Z/∂Λc = N .
September 29, 2010
284
10:45
World Scientific Review Volume - 9in x 6in
10-chapter9
A. Naji et al.
where we have introduced D/2 2 I(D) = e−κD/2 cosh κz dz. exp κµ 0
(29)
The first term in Eq. (28) corresponds to the usual salt-mediated repulsive DH interaction between the two surfaces, and the second one is the contribution of counterions, which is proportional to η, Eq. (19), on the single-particle SC level. The dimensionless pressure acting between the surfaces can be obtained from the free energy via the standard thermodynamic relation P = −∂(FN /S)/∂D, thus yielding βP (D) I (D) , = e−κD + 2η 2 2πB σs I(D)
(30)
where the prime denotes the derivative with respect to the argument. The analytical SC pressure (30) is shown in Fig. 4(a) as a function of the inter-surface distance (solid curve) along with the results from MonteCarlo simulations41 (symbols) for a few different screening parameters. We also show in the figure the WC pressure from Eq. (24) (dashed curve). First note that the simulation results are bracketed by the two limiting analytical theories of WC and SC within the dressed counterion scheme and thus agree with the general feature obtained before13–16 that the WC and SC limits in fact establish the upper and lower bounds for the interaction pressure between charged surfaces. For both small and large separations the interaction pressure becomes positive (repulsive), whereas for a sufficiently large coupling parameter,41 an effective inter-surface attraction can emerge at intermediate separations between the walls. For κD 1, the SC pressure reduces to βP (D) 2µη + (1 − 2η), 2πB σs2 D
(31)
which to the leading order corresponds to the ideal-gas osmotic pressure of counterions squeezed between the two surfaces. For κD 1, the pressure behaves as βP (D) 2µη 4η , − 2πB σs2 D (κD)2
(32)
which indicates that at large separations the counterions again behave as an ideal gas on the leading order as all electrostatic interactions are effectively screened out and hence only the repulsive osmotic contribution remains.
September 29, 2010
10:45
World Scientific Review Volume - 9in x 6in
Exotic Electrostatics of Charged Macromolecular Systems
10-chapter9
285
The interaction pressure in the canonical ensemble thus always possesses repulsive branches at small and large separations and can show nonmonotonic behavior in between. In fact, for certain values of the parameters the interaction pressure shows a van der Waals-like loop which could suggest a coexistence regime between two different “phases”. This loop is obtained for certain iso-ionic strength curves. From a thermodynamic perspective one thus has a coexistence between a dense phase, identified with a small intersurface separation, at equilibrium with an expanded phase with a larger inter-surface separation. Such van der Waals-like coexistence between interacting charged surfaces has been seen in other contexts before63 and can be demonstrated by means of a Maxwell construction analysis as shown in Fig. 4(b). In an experiment such as osmotic stress experiments and surface force experiments8 one can only probe stable equilibrium states of the system implying interaction pressure vs. separation curves that are in agreement with the appropriate Maxwell construction. The binodal or the coexistence curve, which delimits the region in the pressure-separation plots where a Maxwell construction is feasible (red curve in Fig. 4(b)), ends at a critical point corresponding to a critical amount of salt above which the interaction pressure remains purely repulsive. For the case with η = 1, we find the critical point as (P˜c = 0.092, Dc /µ = 6.14, κc µ = 0.546). It is interesting to note that this type of interaction pressure equilibria corresponding to abrupt transitions from one equilibrium separation to a different one have been observed in experiments with strongly charged macromolecules in the presence of polyvalent counterions and univalent salt. A typical example would be the osmotic stress experiments on DNA in the presence of trivalent CoHex counterions and 0.25M NaCl salt, that show abrupt transitions in osmotic pressure for intermediate CoHex3+ concentrations from one repulsive osmotic pressure branch to another one.57 Similar features are discerned even for a divalent counterion Mn2+ at various concentrations or temperatures.58 The agreement between the SC dressed counterion theory and simulations becomes better as the coupling parameter Ξ becomes larger. The agreement is also better for a smaller fraction of counterions, η, in the slit. Using a similar argument as in the counterions-only case in Section that the theory is expected to hold at small separations 5.2,13–16 we find given by D/µ Ξ/η. Thus, for η < 1 (i.e., when the amount of the bare charge due counterions is less than the bare fixed charge on the macroions), the SC dressed counterion theory is expected to hold in a wider range of
September 29, 2010
286
10:45
World Scientific Review Volume - 9in x 6in
A. Naji et al.
separations as compared with the original counterions-only SC theory.13 At very large separations, where most of the electrostatics is screened out, the interaction between counterions becomes negligible and the SC theory of dressed counterions retains its validity again. This result is a consequence of the dressed counterion theory and is not obtained in the standard SC theory with counterions only. Thus, the SC dressed counterion theory captures the physics both at large and small separations but would require improvements at intermediate separations. It should be noted that the validity of the DH-type linearization that we have used to derive the dressed interaction potentials is also limited by stipulating that the dimensionless DH potential itself is always small enough. This leads to the condition that κ 2πB σs , or κµ 1/q, which turns out to cover a whole range of realistic parameter values.41 7. Counterions between Randomly Charged Surfaces The assumption of homogeneity of surface charges is in general quite severe and there are well known cases where this assumption is not realistic at all. Random polyelectrolytes and polyampholytes present one such case.64–66 There the sequence of charges can be distributed along the polymer backbone in a disordered manner where the disorder distribution may be of a quenched type. The Coulomb (self-)interactions of such polyelectrolytes are distinct and different from homogeneously charged polymers. A case even closer to the present line of reasoning are investigations of interactions between solid surfaces in the presence of charged surfactants. The aggregation of surfactants on crystalline hydrophobic substrates in aqueous solutions can sometimes show structures consistent with highly inhomogeneous and disordered surface charge distributions.67 Similar interfacial structures are seen for interacting hydrophilic mica surfaces in the presence of cetyl-trimethyl-amonium bromide (CTAB) or other surfactantcoated surfaces.68 The surfaces appear to be covered by a mosaic of positively and negatively charged regions and experience a strong, long-ranged attraction, which is comparable in magnitude to that between hydrophobic surfaces, and is orders of magnitude larger than the expected Lifshitz-van der Waals forces.68 The patterning of interacting surfaces described above is highly disordered, depends on the method of preparation and has basic implications also for the forces that act between other types of hydrophilic surfaces with mixed charges.
10-chapter9
September 29, 2010
10:45
World Scientific Review Volume - 9in x 6in
Exotic Electrostatics of Charged Macromolecular Systems
10-chapter9
287
It thus seem appropriate to investigate the effect of quenched disordered charge distribution on the interactions between macroions in ionic solutions.f As a particular case, we shall again focus on the effective interaction between two randomly charged planar surfaces across a one-component Coulomb fluid.42 7.1. General formalism: The replica method The partition function of a Coulomb fluid in the presence of an external fixed charge distribution ρ0 (r), can be again written in the form of a functional integral over the fluctuating electrostatic field φ(r) as given in Eq. (17). However, the fixed charges are now assumed to be randomly distributed on macromolecular surfaces. Thus ρ0 (r) is represented by a probability distribution, which is assumed to be Gaussian with no spatial correlations, i.e. 2 1 −1 P[ρ0 (r)] = const. × e− 2 dr g (r) (ρ0 (r) − ρ¯0 (r)) , (33) where ρ¯0 (r) is the mean value and g(r) the width or variance of the charge disorder distribution. For clarity, we shall also focus on the counterions-only case by setting Λ± = 0 in Eq. (18). The average over quenched charge disorder is now obtained by applying the standard Edwards-Anderson replica ansatz69,70 in the form Zn − 1 , (34) n→0 n where the disorder average is defined as (. . . ) = Dρ0 (· · · ) P[ρ0 (r)]. The Gaussian integrals involved in Eq. (34) can be evaluated straightforwardly and the final form of the replicated partition function follows as42
n Zn = Dφa e−βS[φa (r)] , (35) F = −kB T ln Z = −kB T lim
a=1
with S[φa (r)] =
1 φa (r)Dab (r, r )φb (r) dr dr + i ρ¯0 (r) φa (r) dr 2 a a,b Λc Ω(r) e−iβqe0 φa (r) dr, (36) − β a
f See Refs.44,45 for an analysis of the effects due to annealed and partially annealed charge disorder.
September 29, 2010
10:45
World Scientific Review Volume - 9in x 6in
288
10-chapter9
A. Naji et al.
where a, b = 1, . . . , n are the replica indices and Dab (r, r ) = v −1 (r, r )δab + βg(r)δ(r − r ).
(37)
The expression (35) together with Eq. (34) represents the starting formulation for the free energy in the presence of quenched charge disorder. This free energy can only be evaluated approximately. Thus, in order to proceed we shall combine the methods developed for the one-component (counterions-only) Coulomb fluid without disorder13 and modify them in order to incorporate appropriately the disorder effects. We shall start with the WC limit giving rise to the corresponding mean-field theory and then proceed to the SC limit. 7.2. Disorder effects in the WC regime In the WC limit Ξ → 0, one may proceed by employing a saddle-point analysis of the functional integral (35), just as in the case with no disorder in Section 6.3. It is easy to show that the real-valued mean-field replica potential ψa is governed by the following equation, −εε0 ∇ ψa (r) + βg(r) 2
n
ψb (r) = ρ¯0 (r) + qe0 Λc Ω(r) e−βqe0 ψa .
(38)
b=1
In the replica formulation we have to take the limit n → 0, which further more implies that limn→0 b ψb (r) → 0. It is thus evident that in the limit n → 0, the contributions from the disorder vanish and, because the index a becomes irrelevant, one recovers the standard PB equation (7). Therefore, the quenched charge disorder effects completely vanish in the WC limit.42,71 The above result is a consequence of the mean-field approximation and holds in the limit Ξ → 0 even if the system is generalized to contain additional ionic species or dielectric discontinuities at the bounding surfaces. The quenched charge disorder however turns out to play a significant role in dielectrically inhomogeneous systems when electrostatic field fluctuations are taken into account. It can be shown to lead to an additional attractive or repulsive contribution to the total free energy (depending on the dielectric mismatch and the salt screening in the system) even when the surfaces are assumed to be net-neutral.43,44 7.3. Disorder effects in the SC regime The partition function (35) can be calculated in the SC limit via a virial expansion up to the first nontrivial leading order in powers of the fugacity
September 29, 2010
10:45
World Scientific Review Volume - 9in x 6in
10-chapter9
Exotic Electrostatics of Charged Macromolecular Systems
289
as noted before. The canonical SC free energy of the system then follows from Eq. (34) by using a standard Legendre transform42 as 1 1 ρ¯0 (r)v(r, r )¯ FN = ρ0 (r ) dr dr + Tr g(r)v(r, r ) 2 2 −N kB T ln Ω(R) e−βu(R) dR. (39) The first term in Eq. (39) is nothing but the direct Coulomb interaction between the mean charge densities ρ¯0 (r) of the fixed charged surfaces (macroions). The second term is an additive contribution from the charge disorder, which becomes important only in dielectrically inhomogeneous systems43,44 and will be irrelevant in the present study. The third term, however, embodies the disorder effects on the SC level in the presence of a Coulomb fluid. It depends on the single-particle interaction potential β ρ0 (r ) dr − (qe0 )2 g(r )v 2 (r , R) dr , (40) u(R) = qe0 v(r , R)¯ 2 where the second term comes from the disorder variance g(r) and exhibits a non-trivial dependence on the Coulomb kernel v(r, r ) = 1/(4πεε0|r − r |). Assuming again that our system is composed of two planar surfaces located at z = ±D/2 with statistically identical random charge distributions, we write the mean density and variance of the disordered surface charge as ρ¯0 (r) = −σs e0 δ(z−D/2)+δ(z+D/2) , g(r) = ge20 δ(z−D/2)+δ(z+D/2) . (41) The electroneutrality again stipulates that 2σs S = N q. The geometry function Ω(R) is the same as before. All the terms in the expression for the free energy, Eq. (39), can be computed explicitly. At the end we obtain a surprisingly simple expression βFN D = + (χ − 1) ln D. N 2µ
(42)
Here we have introduced the dimensionless disorder coupling parameter χ = 2πq 2 2B g,
(43)
which is very similar to the electrostatic coupling parameter, Ξ, in Eq. (4), except that it is defined based on the disorder variance g and scales with the counterion valency as q2 instead of q 3 . The free energy (42) is plotted in Fig. 5(a) for different values of the disorder coupling parameter. Note that the disorder leads to a long-range attractive contribution, which is additive in the SC free energy and has a logarithmic dependence on the
September 29, 2010
10:45
World Scientific Review Volume - 9in x 6in
290
10-chapter9
A. Naji et al.
2
µ
β
1
/*D
µ
0
0
1
χ
2
3
Fig. 5. Left: Rescaled SC free energy, Eq. (42), of two charged walls bearing quenched charge disorder as a function of the rescaled inter-surface distance D/µ for χ = 0, 1 and 2 (bottom line, middle line, top line). Right: Rescaled equilibrium distance D∗ /µ as a function of the disorder coupling parameter χ. Adapted from Ref. 42.
separation, i.e., χ ln D. It thus appears that the quenched charge disorder and the counterions confinement entropy, i.e., the − ln D term in Eq. (42), in some sense counteract one another. Evaluating the interaction pressure from the free energy, Eq. (42), we find P (D) = PSC (D) + Pdisorder (D), where the first term is the standard SC pressure,13 Eq. (13), and the second term is the additive contribution from the disorder
βPdisorder (D) 2µ . (44) = −χ 2πB σs2 D We can then derive the equilibrium distance D∗ between the two surfaces, corresponding to zero interaction pressure, as D∗ = 2 (1 − χ) µ.
(45)
In the undisordered case, χ = 0, this reduces to the known result D∗ = 2µ,13 which corresponds to a stable bound state for the two surfaces at a separation equal to twice the GC length. However, as χ is increased, the equilibrium separation decreases and vanishes at the critical value χc = 1 and remains at zero thereafter. This behavior has all the features of a secondorder, quenched-disorder-induced collapse transition with an unusual value of the critical exponent (see Fig. 5(b)). Note also that for χ = 1 the interaction pressure between the surfaces is obviously constant in the whole range of separations D right down to zero as the counterions confinement entropy is completely wiped out by the charge disorder contribution.
September 29, 2010
10:45
World Scientific Review Volume - 9in x 6in
Exotic Electrostatics of Charged Macromolecular Systems
10-chapter9
291
8. Lessons The two limiting laws, i.e., the WC and the SC limits for Coulomb fluids that we explored above, are valid in disjoint regions of the parameter space. While the WC limit is valid for sufficiently small macroion surface charge densities, low counterion valencies, high medium dielectric constant and/or high temperatures, the SC limit becomes valid for respectively opposite parameter values. The two together bracket the region of all possible behaviors of Coulomb fluids confined between charged boundaries, a view that was completely corroborated by extensive simulation studies. The parameter space in between these limiting values can be analyzed by approximate methods72–74 but is most often accessible solely via computer simulations.13–16,41,46–49,51,72,74–76 Exact solutions for the whole range of coupling parameters are unfortunately available only in one dimension.77 The WC-SC paradigm has been tested extensively13–16,41,46–49,51,72–74,77 and fits computer simulations quantitatively correctly in the respective regimes of validity, thus providing a unifying conceptual framework of the behavior of Coulomb fluids. Though we have shown that in the important limit of SC the much cherished and widespread PB approach does not work, formally its applicability can be systematically extended by perturbative corrections in the local potential fluctuations and correlations12–14,51,54,78–81 along the lines of the standard approach used in the mean-field context.82 This kind of fix is nevertheless severely limited83 since the perturbative expansion is only weakly convergent13–16 and higher-order corrections beyond the first-loop Gaussian term are very complicated and difficult to carry through.80,81 Such perturbative corrections offer in effect only a relatively insignificant improvement over the PB approximation13–16,51 and can not predict phenomena such as like-charge attraction.75 The emerging world of Coulomb interactions reviewed above is indeed fascinating. While the counterion-mediated electrostatic interactions between equally charged surfaces are always repulsive on the WC level, the SC regime offers a much richer framework with plethora of new phenomena. Interactions between equally charged macromolecular surfaces can be attractive for strongly coupled counterions, or non-monotonic– showing repulsion at small and large as well as attraction at intermediate separations–for strongly coupled counterions in the presence of weakly coupled simple salt. While charge disorder on macromolecular surfaces has no effect on the WC level, it can quite unexpectedly lead to strong electrostatic
September 29, 2010
10:45
292
World Scientific Review Volume - 9in x 6in
A. Naji et al.
attractions between randomly charged surfaces on the SC level. In view of these developments, the commonly held pop culture wisdom that likes repel and opposites attract should thus be substantially amended! Acknowledgments Rudolf Podgornik would like to acknowledge the financial support by the Slovenian Research Agency under contract Nr. P1-0055 (Biophysics of Polymers, Membranes, Gels, Colloids and Cells) and under contract Nr. J1-0908 (Dispersion force nanoactuators). Matej Kanduˇc would like to acknowledge the financial support by the Slovenian Research Agency under the young researcher grant. Ali Naji is supported by a Newton International Fellowship from the Royal Society, the Royal Academy of Engineering, and the British Academy. References 1. E. T. Whittaker, A History of the Theories of Aether and Electricity: The Classical Theories, the Modern Theories 1900-1926 (Two Volumes Bound As One, Dover Classics Edition, 1990). 2. R. French et al., Rev. Mod. Phys. 82, 1887 (2010). 3. G. Gouy, J. Phys. (Paris) 9, 457 (1910). 4. D.L. Chapman, Phil. Mag. 25, 475 (1913). 5. P. Debye, E. H¨ uckel, Physik. Z. 24, 185 (1923). 6. B.V. Derjaguin, L. Landau, Acta Physicochim. USSR 14, 633 (1941). 7. E.J. Verwey, J.T.G. Overbeek, Theory of the Stability of Lyophobic Colloids (Elsevier, Amsterdam, 1948). 8. J. Israelachvili, Intermolecular and Surface Forces (Academic Press, London, 1991). 9. R.J. Hunter, Foundations of Colloidal Science (Clarendon, Oxford, first edition 1987, second edition 2001). 10. V.A. Parsegian, Van der Waals Forces (Cambridge University Press, Cambridge, 2005). 11. S. F. Edwards, A. Lenard, J. Math. Phys. 3, 778 (1962). ˇ s, J. Chem. Soc., Faraday Trans 2 5, 611 (1988); R. 12. R. Podgornik, B. Zekˇ Podgornik, J. Phys. A 23, 275 (1990); J. Chem. Phys. 91, 5840 (1989). 13. R.R. Netz, Eur. Phys. J. E 5, 557 (2001). 14. A.G. Moreira, R.R. Netz, Europhys. Lett. 52, 705 (2000); Phys. Rev. Lett. 87, 078301 (2001); Eur. Phys. J. E 8, 33 (2002). 15. H. Boroudjerdi, Y.-W. Kim, A. Naji, R.R. Netz, X. Schlagberger, A. Serr, Physics Reports 416, 129 (2005). 16. A. Naji, S. Jungblut, A.G. Moreira, R.R. Netz, Physica A 352, 131 (2005). 17. I. Rouzina, V.A. Bloomfield, J. Phys. Chem. 100, 9977 (1996).
10-chapter9
September 29, 2010
10:45
World Scientific Review Volume - 9in x 6in
Exotic Electrostatics of Charged Macromolecular Systems
10-chapter9
293
18. A.Yu. Grosberg, T.T. Nguyen, B.I. Shklovskii, Rev. Mod. Phys. 74, 329 (2002). 19. Y. Levin, Rep. Prog. Phys. 65, 1577 (2002). 20. C. Holm, P. Kekicheff, R. Podgornik (Eds.), Electrostatic Effects in Soft Matter and Biophysics (Kluwer Academic, Dordrecht, 2001). 21. W. C. K. Poon, D. Andelman (Eds.), Soft condensed matter physics in molecular and cell biology (Taylor & Francis, New York, London, 2006). 22. B. Alberts, A. Johnson, J. Lewis, M. Raff, K. Roberts, P. Walter, Molecular Biology of the Cell (Garland Science, New York, 2002). 23. D.H. Napper, Polymeric Stabilization of Colloidal Dispersions (Academic Press, London, 1983). 24. P. Pincus, Macromolecules 24, 2912 (1991). 25. H. Dautzenberg, W. Jaeger, B.P.J. K¨ otz, C. Seidel, D. Stscherbina, Polyelectrolytes: Formation, characterization and application (Hanser Publishers, New York, 1994). 26. F. Oosawa, Polyelectrolytes (Marcel Dekker, New York, 1971). 27. K.K. Kunze, R.R. Netz, Phys. Rev. Lett. 85, 4389 (2000). 28. H. Boroudjerdi, R.R. Netz, Europhys. Lett. 64, 413 (2003); J. Phys.: Condens. Matter 17, S1137 (2005); Europhys. Lett. 71, 1022 (2005). 29. J.O. R¨ adler, I. Koltover, T. Salditt, C.R. Safinya, Science 275, 810 (1997). 30. J.O. R¨ adler, I. Koltover, A. Jamieson, T. Salditt, C.R. Safinya, Langmuir 14, 4272 (1998). 31. I. Koltover, T. Salditt, J.O. R¨ adler, C.R. Safinya, Science 281, 78 (1998). 32. R.G. Crystal, Science 270, 404 (1995). 33. W.M. Gelbart, R.F. Bruinsma, P.A. Pincus, V.A. Parsegian, Physics Today (September 2000), pp. 38-44. 34. V.A. Bloomfield, Biopolymers 31, 1471 (1991); Curr. Opin. Struct. Biol. 6, 334 (1996). 35. O. Lambert, L. Plan¸con, J.L. Rigaud, L. Letellier, Mol. Microbiol. 30, 761 (1998). 36. J.X. Tang, S. Wong, P.T. Tran, P.A. Janmey, Ber. Bunsenges. Phys. Chem. 100, 796 (1996). 37. J.X. Tang, T. Ito, T. Tao, P. Traub, P.A. Janmey, Biochemistry 36, 12600 (1997). 38. D. J. Needleman, M. A. Ojeda-Lopez, U. Raviv, H. P. Miller, L. Wilson, C. R. Safinya, Proc. Natl. Acad. Sci. USA 101, 16099 (2004). 39. D. Ben-Yaakov, D. Andelman, D. Harries, R. Podgornik, J. Phys.: Condens. Matter 21 424106 (2009). 40. Y. Burak, D. Andelman, Phys. Rev. E 62, 5296 (2000); J. Chem. Phys. 114, 3271 (2001) 41. M. Kanduˇc, A. Naji, J. Forsman, R. Podgornik, J. Chem. Phys. 132, 124701 (2010). 42. A. Naji, R. Podgornik, Phys. Rev. E 72, 041402 (2005). 43. R. Podgornik, A. Naji, Europhys. Lett. 74, 712 (2006). 44. A. Naji, D.S. Dean, J. Sarabadani, R. Horgan, R. Podgornik, Phys. Rev. Lett. 104, 060601 (2010).
September 29, 2010
294
10:45
World Scientific Review Volume - 9in x 6in
A. Naji et al.
45. Y.Sh. Mamasakhlisov, A. Naji, R. Podgornik, J. Stat. Phys. 133, 659 (2008). 46. A. Naji, R. R. Netz, Eur. Phys. J. E 13, 43 (2004); A. Naji, A. Arnold, C. Holm, R. R. Netz, Europhys. Lett. 67, 130 (2004). 47. A. Naji, R. R. Netz, Phys. Rev. Lett. 95, 185703 (2005); Phys. Rev. E 73, 056105 (2006). 48. M. Kanduˇc, J. Dobnikar, R. Podgornik, Soft Matter 5, 868 (2009); M. Kanduˇc, A. Naji, R. Podgornik, J. Chem. Phys. 132, 224703 (2010). 49. Y.S. Jho, M. Kanduˇc, A. Naji, R. Podgornik, M.W. Kim, P.A. Pincus, Phys. Rev. Lett. 101, 188101 (2008). 50. M. Kanduˇc, R. Podgornik, Eur. Phys. J. E 23, 265 (2007). 51. M. Kanduˇc, M. Trulsson, A. Naji, Y. Burak, J. Forsman, R. Podgornik, Phys. Rev. E 78, 061105 (2008). 52. M. Kanduˇc, A. Naji, Y. S. Jho, P. A. Pincus, R. Podgornik, J. Phys.: Condens. Matter 21, 424103 (2009). 53. N. Bjerrum, Kgl. Danske Videnskab. Selskab. Mat.-fys. Medd. 7, 1 (1926). 54. R.R. Netz, H. Orland, Eur. Phys. J. E 1, 203 (2000). 55. H. Wennerstr¨ om, B. J¨ onsson, P. Linse, J. Chem. Phys. 76, 4665 (1982). 56. M. Baus, J. Hansen, Phys. Rep. 59, 1 (1980). 57. D.C. Rau, V. A. Parsegian, Biophys J 61, 246 (1992). 58. D.C. Rau, V. A. Parsegian, Biophys J 61, 260 (1992). 59. O. Punkkinen, A. Naji, R. Podgornik, I. Vattulainen, P.-L. Hansen, Europhys. Lett. 82, 48001 (2008). 60. J. Zwanikken, R. van Roij, J. Phys.: Condens. Matter 21, 424102 (2009). 61. M. E. Fisher, Y. Levin, Phys. Rev. Lett. 71, 3826 (1993). 62. J.C. Neu, Phys. Rev. Lett. 82, 1072 (1999); J.E. Sader, D.Y.C. Chan, J. Colloid Interface Sci. 213, 268 (1999); Langmuir 16, 324 (2000); E. Trizac, J.-L. Raimbault, Phys. Rev. E 60, 6530 (1999); E. Trizac, Phys. Rev. E 62, R1465 (2000). 63. D. Harries, R. Podgornik, V.A. Parsegian, E. Mar-Or, D. Andelman, J. Chem. Phys. 124, 224702 (2006). 64. I. Borukhov, D. Andelman, H. Orland, Eur. Phys. J. B 5, 869 (1998). 65. Y. Kantor, M. Kardar, Europhys. Lett. 28, 169 (1994); 14, 421 (1991). 66. Y. Kantor, M. Kardar, D. Ertas, Physica A 249, 301 (1998). 67. S. Manne, J.P. Cleveland, H.E. Gaub, G.D. Stucky, P.K. Hansma, Langmuir 10, 4409 (1994); S. Manne, H.E. Gaub, Science 270, 1480 (1995). 68. E.E. Meyer, Q. Lin, T. Hassenkam, E. Oroudjev, J.N. Israelachvili, Proc. Natl. Acad. Sci. USA 102, 6839 (2005); S. Perkin, N. Kampf, J. Klein, Phys. Rev. Lett. 96, 038301 (2006); J. Phys. Chem. B 109, 3832 (2005); E.E. Meyer, K.J. Rosenberg, J. Israelachvili, Proc. Natl. Acad. Sci. USA 103, 15739 (2006). 69. J.W. Negele, H. Orland, Quantum Many-Particle Systems (Perseus Books Group, New York, 1998). 70. V. Dotsenko, Introduction to the Replica Theory of Disordered Statistical Systems (Cambridge University Press, New York, 2001). 71. C. Fleck, R.R. Netz, Europhys. Lett. 70, 341 (2005). 72. J. Forsman, J. Phys. Chem. B 108, 9236 (2004).
10-chapter9
September 29, 2010
10:45
World Scientific Review Volume - 9in x 6in
Exotic Electrostatics of Charged Macromolecular Systems
10-chapter9
295
73. Y. Burak, D. Andelman, H. Orland, Phys. Rev. E 70, 016102 (2004); C. D. Santangelo, Phys. Rev. E 73, 041512 (2006); M. M. Hatlo, L. Lue, Soft Matter 5, 125 (2009). 74. Y.-G. Chen, J. D. Weeks, Proc. Natl. Acad. Sci. 103, 7560 (2006); J. M. Rodgers, C. Kaur, Y.-G. Chen, J. D. Weeks, Phys. Rev. Lett. 97, 097801 (2006). 75. L. Guldbrand, B. J¨ onsson, H. Wennerstr¨ om, P. Linse, J. Chem. Phys. 80, 2221 (1984); D. Bratko, B. J¨ onsson, H. Wennerstr¨ om, Chem. Phys. Lett. 128, 449 (1986); J. P. Valleau, R. Ivkov, G. M. Torrie, J. Chem. Phys. 95, 520 (1991); R. Kjellander, T. ˚ Akesson, B. J¨ onsson, S. Marˇcelja, J. Chem. Phys. 97, 1424 (1992). 76. M. Trulsson, B. J¨ onsson, T. ˚ Akesson, J. Forsman, C. Labbez, Phys. Rev. Lett. 97, 068302 (2006); Langmuir 23, 11562 (2007). 77. D.S. Dean, R.R. Horgan, D. Sentenac, J. Stat. Phys. 90, 899 (1998); D.S. Dean, R.R. Horgan, A. Naji, R. Podgornik, J. Chem. Phys. 130, 094504 (2009). 78. P. Attard, J. Mitchell, B. W. Ninham, J. Chem. Phys. 88, 4987 (1988). 79. B.-Y. Ha, A. J. Liu, Phys. Rev. Lett. 79, 1289 (1997); P. A. Pincus, S.A. Safran, Europhys. Lett. 42, 103 (1998); M. Kardar, R. Golestanian, Rev. Mod. Phys. 71, 1233 (1999); R. R. Netz, H. Orland, Eur. Phys. J. E 1, 203 (2000); B.-Y. Ha, Phys. Rev. E 64, 031507 (2001); A. W. C. Lau, P. Pincus, Phys. Rev. E 66, 041501 (2002). 80. R. Podgornik, V. A. Parsegian, Phys. Rev. Lett. 80, 1560 (1997). 81. D. S. Dean, R. R. Horgan, J. Phys. C. 17, 3473, (2005); Phys. Rev. E 69, 061603 (2004); ibid 70, 011101 (2004); ibid 68, 061106 (2003). ˇ 82. P. Ziherl, R. Podgornik, S. Zumer, Chem. Phys. Lett. 295, 99 (1998). 83. Y. Levin, J. J. Arenzon, J. F. Stilck, Phys. Rev. Lett. 83, 2680 (1999).
This page is intentionally left blank
September 29, 2010
10:46
World Scientific Review Volume - 9in x 6in
Chapter 10 Computer Modeling of Liquid Crystals
Rauzah Hashim Department of Chemistry, University of Malaya, 50603 Kuala Lumpur, Malaysia E-mail: [email protected] This chapter outlines the methodologies and models which are commonly used in the simulation of liquid crystals. The approach in the simulation of liquid crystals has always been to understand the nature of the phase and to relate this to fundamental molecular features such as geometry and intermolecular forces, before important properties related to certain applications are elucidated. Hence, preceding the description of the main “molecular-based” models for liquid crystals, a general but brief outline of the nature of liquid crystals and their historical development is given. Three main model classes, namely the coarse-grained single-site lattice and Gay-Berne models and the full atomistic model will be described here where for each a brief review will be given followed by assessment of its application in describing the phase phenomena with an emphasis on understanding the molecular organization in liquid crystal phases and the prediction of their bulk properties. Variants and hybrid models derived from these classes and their applications are given.
1. Introduction Puzzled by an unusual double melting behavior of cholesteryl benzoate, but confident of its purity, Friedrich Reinitzer, an Austrian botanist working in the Institute of Plant Physiology at the University of Prague in 1888, contacted a German physicist, Otto Lehmann, who inspected this material under the polarizing microscope. Lehmann, an expert in optics, was surprised when he observed a unique crystalline property for a system that flowed. He suggested that this material possessed a special flowing crystalline nature and called it by various terms such as “fliessender Kristalle” (flowing crystal) or “fl¨ ussige Kristalle” which means liquid crystal, a term 297
11-chapter10
September 29, 2010
298
10:46
World Scientific Review Volume - 9in x 6in
R. Hashim
that is commonly used today, although some have argued for its change since it is clearly a contradiction.1 Although this event is always taken to mark the discovery of liquid crystalline phenomena, there were many reports on strange optical behavior observed near the melting point much earlier, like that observed by Willhelm Heintz (1850) on melting stearin from a solid to a cloudy liquid, which first changed to an opaque and finally to a clear liquid. Similarly, Rudolf Virchow, Ch. Freiherr von. Mettenheimer and Gabriel Gustav Valentin also reported strange optical behavior when a solution of nerve fiber was observed under polarized light. These early discoveries, we now know, are examples of lyotropic liquid crystal, which are much more pervasive in living systems. The lack of understanding on the fundamental molecular physics, combined with the fact that these scientists were mostly biologists, quite unconcerned with such pursuits, they glossed over the importance of the “two melting points” behavior and the strange crystal-like observation, unlike Lehmann and Renitzer, who postulated some explanation on the nature of these phase transitions and the related optical properties of the phases. Liquid crystal science is very much interdisciplinary and its understanding must necessarily be traceable from function or property down to the molecular level, both energetic and structure; hence this important conclusion sets the theme of this chapter based on those foundations. 1.1. What is a liquid crystal? A liquid crystal is a phase that exists in between a solid and a liquid, hence the term “mesophase” is often used to describe it; while “mesogen” refers to the material forming the phase. Mesogenic molecules shaped like rods or plates or some other anisotropic form, encourage build up of long range correlations between molecular axes, which are then aligned collectively along a certain preferred direction called the director. The minimum necessary requirement for the formation of a liquid crystal phase is that the molecular shape must deviate from the spherical symmetry, that is must be anisotropic. The directors within the liquid crystal can be manipulated with external perturbations such as mechanical, magnetic or electric forces because of the strong coupling to the field. Liquid crystals can be formed by the effect of heat, hence, are temperature sensitive; they turn into a solid when cooled, and into an isotropic liquid when heated. Pure materials forming this type of liquid crystal are classified as thermotropic.
11-chapter10
September 29, 2010
10:46
World Scientific Review Volume - 9in x 6in
Computer Modeling of Liquid Crystals
11-chapter10
299
Liquid crystals can also be formed by dissolving certain molecules called amphiphiles in specific solvents such as water. These amphiphiles usually contain two parts: one part tends to associate with the solvent (eg. water), while the other part avoids the solvent. The ordering or self-assembly of amphiphiles is due to the micro-phase separation between these two parts and their association (love/hate) with the solvents. Of course, other state variables (such as temperature and composition) also play an important role in governing the phase formation. Another class of liquid crystals is termed amphitropic, which forms liquid crystals by both thermal and solvent effects;2 these form thermotropic phases (in pure form) and lyotropic phases when dissolved in a solvent. Some examples of the different classes of mesogenic molecules are given in Fig. 1.
Fig. 1. Examples of mesogenic molecules which form liquid crystal phases by (a) thermal effect, that is, thermotropic; (b) solvent/composition effect, lyotropic; and (c) both thermotropic and lyotropic effects, amphitropic mesogen.
There are two main types of molecular organization within the mesophase and one of these is called the nematic phase (N), typified by the presence of one orientational director, which shows the long range order of the mesogenic molecules in a certain direction within the mesophase. Another type of common organization is the smectic phase (Sm), where in addition to the orientational order there is also a positional ordering which takes the defining form of a layer structure. Both lyotropic and thermotropic liquid crystals may exhibit some of these nematic and smectic liquid crystal phases, but different terminologies are used to denote them. For example,
September 29, 2010
300
10:46
World Scientific Review Volume - 9in x 6in
R. Hashim
the smectic phase is often called, more descriptively, a lamellar phase in lyotropic studies. The nematic phase has mechanical properties, for example, flow rheology, which closely resembles that of the liquid phase, since the molecular centers of mass are arranged randomly. Hence, the molecules move freely in a liquid phase, while retaining some degree of orientational ordering. This feature of thermotropic mesophase is exploited to make optical switching devices such as the widely used liquid crystal display. If the nematic phase contains chiral molecules, they arrange themselves into a strongly twisted structure, in which the director adopts a helical arrangement; this reflects visible light selectively, at different wavelengths depending on the temperature. These materials are often used in temperature sensors or thermometers. This phase was used to be called the cholesteric, but is now known more correctly as a chiral nematic (N*). The symmetry of the nematic phase is usually assumed to be uniaxial. However, most of the molecules forming the nematic phase deviate from this ideal symmetry; in other word these are biaxial. Therefore, by the symmetry argument it should be possible to observe a biaxial nematic phase. This phase was predicted by theoreticians more than thirty years ago; since then experimentalists have been attempting to make the materials which may form this phase. Interestingly, despite many claims,3 the experimental observations of a thermotropic biaxial nematic (NB ), remain somewhat debatable. On the other hand, its lyotropic equivalent had been reported conclusively by Yu and Saupe much earlier in 1980.4 A typical example of lyotropic nematic is a mixture of potassium laurate-1-decanol-water. This system gives three nematic phases, namely the nematic rod (NR ) containing mainly rod-like micelles, nematic disc (ND ), contains mainly disc-like micelle and the biaxial nematic phase, NB . Both NR and ND are uniaxial nematic. The presence of positional order, that is the layered structure, makes the smectic (and the lamellar) phase more closely related to the solid phase. Within the smectic layers, the molecular centers of mass are randomly distributed and hence these molecules diffuse freely, but interlayer exchanges are usually less likely. Within the layer, the molecules tend to arrange themselves in more or less the same direction orientationally along the layer normal. This type of liquid crystal layer organization is called the smectic A phase (SmA). There are other smectic structures, with different degrees of ordering such as the smectic C (SmC), which is like a smectic A but having the director within the layer tilted with respect to the layer normal. Another common layered smectic structure is the smectic B phase (SmB), which is
11-chapter10
September 29, 2010
10:46
World Scientific Review Volume - 9in x 6in
Computer Modeling of Liquid Crystals
11-chapter10
301
similar to smectic A, but having bond orientational ordering within the layer, whereby the molecular centers of mass arrange themselves hexagonally. Figure 2 shows rather simplified orientational arrangements for some of these phases.
Fig. 2. Typical liquid crystal molecular organizations; solid lines represent the directors. The common notations for isotropic (Iso), nematic (N), chiral nematic (N∗ ), smectic A (SmA), smectic C (SmC), crystal (Cr).
1.2. Theoretical approach to understanding liquid crystals Development of molecular theories for liquid crystals can be very challenging since, like all fluid phases, their densities are high and there are strong attractive and repulsive interactions which make many-body interactions relevant. In addition, these interactions contain anisotropic terms which are essential for the formation of liquid crystals but complicate the analysis further. Despite that, over the 100 years since the discovery of liquid crystals, there have been a number of fairly simple theories developed, at least to predict the general behavior of the phase transitions as well as the extent of the defining order. 1.2.1. The order parameter Central to the study of liquid crystals is the description of the order characterizing the phase. To make this description useful, it is necessary to identify the nature of the order, which may be positional or orientational. Here we consider the simplest case of the nematic phase with orientational order only; the common orientational order parameter is defined using the ensemble average: P2 (cos2 θ) = 3/2 cos2 θ − 1/2 (1) In this equation, P2 (cos2 θ) is the second rank Legendre polynomial ; θ is the angle between the molecular symmetry axis and the director, which is
September 29, 2010
10:46
World Scientific Review Volume - 9in x 6in
302
11-chapter10
R. Hashim
defined as the symmetry axis of the phase. It is also the symmetry axis for all its properties, accordingly represents the optical axis. This functional definition is convenient, since for a completely random distribution of molecules, or for an isotropic sample, P2 = 0, given that 2 cos (θ) is 1/3, whereas for a perfectly aligned sample P2 = 1. Experimentally, for a typical liquid crystal sample, P2 is of the order of 0.3 to 0.8, and decreases as the temperature is raised until it reaches the phase transition, when the system changes from a liquid crystal phase into the isotropic phase, where the order parameter drops discontinuously to zero. The order parameter can be measured experimentally in a number of ways through techniques such as diamagnetism, birefringence, Raman scattering, NMR and EPR spectroscopy. For example, from NMR experiments in an anisotropic medium such as a liquid crystal,5 the direct nuclear spin-spin dipolar coupling is non-zero and may be related to the order parameters. A more complete description of the orientational order requires higher rank Legendre polynomials. In the case of the uniaxial nematic phase, only even members, such as P4 (cos2 θ), are relevant since the odd terms are zero due to the presence of a mirror plane symmetry element. These higher rank order parameters are usually difficult to measure but can yield additional information on the nature of the phase. Other basis functions from the family of Legendre functions, such as the modified spherical harmonics and Wigner rotation matrices, may be used to characterize the order of a liquid crystal, and these are essential when the molecules or the phase deviate from the uniaxial symmetry.6 1.2.2. Liquid crystal theories One of the earliest and most successful theories developed for nematic liquid crystals is the Maier-Saupe theory.7 The original derivation assumes the anisotropic dispersion energy to be dominant and so uses a simple attractive interaction potential, Uij , between a pair of molecules, i and j, based on the relative orientation θij namely: Uij = −(rij )P2 (cos θij )
(2)
where (rij ) is the strength interaction function depending on the distance rij between the molecules. This function is also defined to be positive, so that the molecules tend to align with their symmetry axes parallel. Using the statistical mechanical treatment to obtain the average behavior and applying the molecular field approximation, the single particle potential of
September 29, 2010
10:46
World Scientific Review Volume - 9in x 6in
Computer Modeling of Liquid Crystals
11-chapter10
303
mean torque for one molecule with its interactions averaged over all other particles is, Uij = − P 2 P2 (cos θij )
(3)
In equation (3), is the average strength interaction that depends on the volume of the sample but is taken to be independent of temperature. This potential of mean torque is the anisotropic interaction which stabilizes the alignment of a molecule with respect to the director. The use of this potential enables the prediction of the thermotropic nematic phase transitions with results which are comparable to those found experimentally. In view of its simplicity, this theory is surprisingly successful, although it tends to over-predict slightly the transitional properties, such as the order NI parameter P 2 = 0.43 and the entropy ∆SN I /R = 0.42, while experimentally, these are 0.36 and 0.19 respectively, for a nematogen like PAA (4,4’-dimethoxyazoxybenzene). As we shall see later, results from computer simulation have been used to explain the observed discrepancy between theory and experiment. Several reasons have been proposed for the differences, including the use of the simple dispersive interaction energy and the molecular field treatment itself, which is an over-simplification that neglects any near neighbor interactions. Numerous extensions have been proposed to improve the theoretical approach to address these two issues. For example, an extended form of the pair interaction potential has been applied to include the fourth rank interaction,8 and the inclusion of terms that allow for the deviation from the molecular symmetry of the molecules. Despite these well-known drawbacks, the averaging approach using the molecular field formalism has been widely applied with relative success to other liquid crystal systems including those for biaxial nematics, chiral nematics, smectics and even for mixtures.9 Complementary to the Maier-Saupe theory is the hard particle theoretical approach which has been equally successful in describing the liquid crystal phase transition and this was originally proposed by Onsager in 1949, where he proved by using statistical mechanic, rod-like molecules display a nematic phase at a certain density.10 He started by considering the interaction of a pair of hard particles, simply defined as, ∞ if i and j overlapped Uij = U (rij , Ωi , Ωj ) = (4) 0 otherwise where rij , is the vector connecting the centers of particle i and j; Ωi and Ωj are the sets of Euler angles (α, β, γ) describing the orientation of each
September 29, 2010
304
10:46
World Scientific Review Volume - 9in x 6in
R. Hashim
particle in a space-fixed coordinate system. The free energy of this system may be estimated, using the cluster virial expansion approximation.11 In essence, Onsager predicted the density-driven nematic phase transition by simply considering the volume excluded from the center-of-mass of one idealized cylinder as it approaches another. In this formalism, when two hard particles (cylinders or ellipsoids) approach each other along the parallel direction, there will be a minimum excluded volume. On the other hand, if the direction of approach is at some angle, there will be a large excluded volume around the cylinder, where the approaching second cylinder cannot enter due to hard particle repulsion. The pair interaction potential depends on the angle of approach causing the net positional entropy function to decrease when the particles approach each other. The fundamental insight here is that, while parallel arrangements of anisotropic objects lead to a decrease in orientational entropy, there is an increase in positional entropy. Therefore, in some cases, a lesser positional order will be entropically favorable and so drive the increase in orientational order. Onsager’s approach deals with a low density system and unsuitable for real mesogen, whose aspect ratio is. Subsequently, many improvements have been proposed, for example using the scaled particle theory.12,13 There are other theoretical developments to describe the behavior of liquid crystals, such as the Landau-de Gennes theory,14 which uses the phenomenal approach. The two outlined above are molecular based to serve as a supplement to complement the rest of the content in this review on computer simulation. 2. Introduction to the Computer Simulation Despite the sophistication of current scientific equipment to study Nature, it is still desirable to perform a computer simulation (a virtual experiment), because it allows scientists to explore and manipulate precisely the realm of molecules and atoms in a way which experimental techniques could not do satisfactorily. In liquid crystal studies, there are many properties such as the ordering tensor or the phase diagram which are difficult to resolve experimentally, but these may be computed quite easily in a virtual experiment. In computer simulations, a designed interaction between particles (or a model) may be defined with parameters chosen at the simulator’s will. This model is then simulated using a set of physical rules to obtain the specific predictions of the properties. The level of complexity is determined by the
11-chapter10
September 29, 2010
10:46
World Scientific Review Volume - 9in x 6in
Computer Modeling of Liquid Crystals
11-chapter10
305
simulator, who would consider many issues, including technical limitations. Most important of these is the reason he performs the simulation in the first place. For example, in molecular studies, even though an accurate model based on quantum interaction is now available, and it is possible to compare directly to the behavior of the real system of a specific chemical nature, it may be unnecessary and wasteful to do so, if one is interested to know only the generic nature of the model or to discriminate the good from the bad theory; all of which could have been discerned using a simplified model that contains the essential physics of the system. Often, a simulation technique is chosen to test an idea or a theory. Based on a particular model, a theory may be tested by the simulation. Likewise, a model may be tested by comparing the simulation results with those from an experiment. All simulators are guided by the same basic approach, where the suitability, advantages and disadvantages of a chosen model and technique, will be argued along with the question “why did the simulator perform his simulation?” 2.1. Computer simulation techniques Two most commonly applied simulation techniques are the Monte-Carlo (MC) and molecular dynamics (MD). For the general techniques in MC and MD, there are many excellent text books and reviews,15,16 available for reference. In this section only their brief essentials are given, while more specific issues of the methods related to the model will be described separately later. The earliest MC simulation of the fluid phase was performed by Metropolis et al. in 1953,17 at the Los Alamos National Laboratories on MANIAC which was the second computer in the world after ENIAC. The Metropolis MC, as it is often called, is a technique based on random numbers to evaluate the multi-dimension integration needed to evaluate the statistical averages. This stochastic method was first developed by von Neumann in 1940s but is now widely applied in many statistical problems not limited to science but also include engineering, economics and even commerce. Its application in the study of condensed phases is more prominent and due to the robustness of the method it remains relevant until now.18 In the molecular simulation, the basic tenet of the techniques comprises the generation of a list of random configurations of the system that is Markovian by a bias importance sampling, where a configuration is selected with a certain probability related to the Boltzmann distribution. Thus, from these biased configurations, any statistical average or the thermodynamics
September 29, 2010
10:46
World Scientific Review Volume - 9in x 6in
306
R. Hashim
property of the system is calculated after the system has been suitably equilibrated. In contrast to the random MC method, molecular dynamics (MD) simulation follows step wisely the real time dynamics of the particles using Newton’s equations of motion; this makes an MD simulation to be more physical than an MC. The first MD was carried out for a system of hard spheres by Alder and Wainwright.19 Compared to MC, the implementation of MD simulation codes is more complicated, but the underlying physics is relatively simple and involves setting the equation of motion for N-particle system and solving these numerically. Of course, MD has added advantage since dynamical properties such as diffusion coefficients and viscosities may be calculated in addition to the usual thermodynamic behavior. 2.2. Simulation limitations A major criticism faced by a numerical simulation on a molecular model came from the smallness of the sample size (N ), limited by both the CPU power and memory storage. Nevertheless, early simulations achieved some degree of success even with N sometimes less than 100 particles. The important issue in simulation is to mimic the properties of a real system, which is defined mostly by the bulk particles and not those on the surface. Thus, it is sensible to consider a model containing only bulk particles and eliminate the surface ones completely. This is usually achieved by applying the periodic boundary condition (PBC), which makes the surface particles to become like bulk. PBC is implemented by surrounding the central N-particle box with its replicas infinitely or as many as is desirable for the system under study. Nevertheless the application of PBC may give rise to some unwanted periodical correlations for certain measured properties, especially those which are length dependent. This correlation is minimized if a free surface is used. Therefore, periodic and free boundary conditions define the upper and lower limits to those affected properties and the difference between them is minimized as the number of particles increases. As we shall see later, other boundary constraints may be applied for different cases. If the pair potential is long ranged, such as those involving electrostatic interactions, PBC is applied together with the adoption of the minimum image convention, whereby the distance between two particles i and j is taken to be either the distance between particle, i and j or the distance between, i and the nearest image of j, whichever is nearer.
11-chapter10
September 29, 2010
10:46
World Scientific Review Volume - 9in x 6in
Computer Modeling of Liquid Crystals
11-chapter10
307
Imposing these conditions on the simulation of the small sized system usually gives a reasonable representation of their bulk behavior. Furthermore, these are only weakly dependent on N , except when the system approaches the phase transition, during which by the strict statistical thermodynamic definition, there is no true phase transition if N is finite. In addition, if N is small and confined to a periodic box, density fluctuations typical of the critical region may not be observed, any study on the spatial fluctuation with wavelength greater than the dimension of the box is impossible. Another worrisome problem for the small size simulation is, the encounter of a quasi-ergodic behavior,15 where the system found itself locked in a meta-stable state indefinitely. This may be avoided if the simulation starts from a suitable initial configuration, so that it will reach equilibrium in a reasonable number of steps. However, it must be stressed, the end results of the simulation should be independent of the initial configuration chosen since the statistical ensemble average definition assumes the system to be ergodic. It is also important to conduct the simulation carefully near the critical region, by extending the number of steps in the equilibration and production stages, where fluctuations in the system cause slow equilibration for some properties. 3. Liquid Crystal Models The simulation methodologies and models developed for simple liquids were later applied to other complex phases including liquid crystal in the 1970s, which happened to coincide with a renewed interest in liquid crystals due to their potential as materials for display application. The rapid growth in both science and technology had called for a more systematic approach to understand their fundamentals, which may be achieved through simulation and theory. Thus, effective simulation models were developed for liquid crystal phases, where useful properties related to applications were calculated, such as order parameters, elastic constants and rotational viscosities. In addition, these simulations also provided a fair amount of understanding on the role of different forces governing the stability of these phases. Liquid crystals, which include other soft matter such as polymer, micelles, bilayers and membranes, now expand into the boundary of different ensembles such as peptides and proteins. This cross discipline junction is rich in opportunities and stimulates exciting challenges for designing new liquid crystal models, measuring more realistic properties and developing
September 29, 2010
10:46
308
World Scientific Review Volume - 9in x 6in
R. Hashim
much improved techniques. Thus, some current issues include, the simulation of novel phases; the interaction of mesogens with surfaces to the formation of defects; the simulation of macromolecular liquid crystal species and the development of atomistic force field and polarizable potentials; linking coarse-grained simulation to continuum models and theory, etc. Of the numerous models based on molecular interaction for mesogenic molecules, four broad classes have been identified namely the generic type, including the lattice, hard-particle, Gay-Berne as well as that of the atomistic force-field, since they are relatively versatile to represent a rich spectrum of liquid crystal phases while simple enough to track down their molecular origin. Limited by space only three of these will be selected here, namely the lattice, Gay-Berne and atomistic force-field. The discussion for each model class begins by defining the potential function where the salient features of the molecular interaction are given; followed by some selected properties for them before outlining a few variants to the models and their applications wherever possible. 3.1. The lattice class We shall discuss the lattice class first, since historically it was the first successful liquid crystal simulation ever performed using the classical spinmodel proposed by Lebwohl and Lasher (LL).20 In the original LL-model, spins (represented by unit vectors) were confined on the sites of a lattice (Fig. 3), using a simple cube with six nearest neighbors. In later simulation work, other box geometries such as the face centered cube had been used as well.21 Each spin, traditionally, represents a molecule which interact with the neighboring spins (vectors) through an interaction potential given by, ˆi ·w ˆ j )2 − 1/2 , Uij = −P2 (cos θij ) = − 3/2 cos2 θij − 1/2 = − 3/2(w (5) where defines the strength of interaction between two particles i and j, while θij is the angle between them. In equation (5), P2 (cos θ) is the second rank Legendre polynomial. As far as coding is concerned, it is easier to express this anisotropic pair interaction using the unit vectors of these ˆ j . Thus, each particle’s orientation is stored as rotating spins w ˆ i and w three components of the unit vector, w. ˆ There were many LL-simulations 22 performed using MD techniques, but by and large, MC Metropolis is the more popular technique amongst the LL simulators, and so is used in most examples here.
11-chapter10
September 29, 2010
10:46
World Scientific Review Volume - 9in x 6in
Computer Modeling of Liquid Crystals
11-chapter10
309
Fig. 3. The lattice class: the ellipsoid is a cartoon representation of the particle whose direction is given by the unit vectors and is the angle between the symmetry axes of particles i and j. Here the particles are placed on a 2D simple cubic lattice.
3.1.1. General techniques In an N V T ensemble, since the LL-lattice model has no translational motion and does not depend on the lattice spacing, the random MC changes are only those from the spinning vectors, brought about by applying the Barker and Watts procedure.23 This is implemented quite simply, by choosing randomly one of the three Cartesian axes (x, y, z) and followed by selecting one of the particles in the list (1, 2, . . . N ) either randomly or sequentially. Then, a trial move is attempted by rotating about the chosen axis (say x) by a random amount δψ = ∆(ξ − 0.5), which is proportional to the random number ξ (where 0 ≤ ξ ≤ 1) and ∆ is the maximum rotation parameter whose value is chosen so that the ratio of the accepted and rejected moves is roughly equal to ensure the simulation reaches equilibrium. Other types of “change” may be considered for some modified LLmodels. For example, in the simulation of an N P T ensemble, where instead of volume, the pressure is kept constant; in such a case, the box dimension, hence, the volume is allowed to change randomly.24 Or when considering a multi-component system of a LL-model nematic mixture, and if mixing is allowed, in order to observe the phase separation phenomena, a particle exchange is defined as a “change”.25 Whatever ingenious procedure devised for changing the configuration, hence, the energy, during each MC-step, microscopic reversibility must be ensured, where the transition probabilities of forward and backward moves are equal; failing which, the whole MC procedure will be invalid. A simulation is usually divided into the equilibration and production stages. A typical equilibration stage may contain a few hundred thousand
September 29, 2010
310
10:46
World Scientific Review Volume - 9in x 6in
R. Hashim
to a few million cycles (a cycle is defined as one complete sweep of N particle changes), while that of a production run would contain many more cycles than this. During equilibration, a select number of properties (including energy) are measured to check the condition for equilibrium, since some properties converge at different rates. But these measurements do not constitute the final averages, which are only evaluated at the end of the production stage. Usually, the final ensemble averages are harvested from macrostep sub-averages, and within a macrostep, these properties of interest are calculated during every MC step. Most LL-lattice simulations calculate a set of standard properties to monitor the changes within the phase and the onset of the phase transition, including the internal energy per particle and the heat capacity and various correlation functions. Although the average long range orientational order parameter is usually defined as P2 = P2 (w ˆ ·n ˆ ), this quantity is calculated differently since the director n ˆ fluctuates within the laboratory frame during the course of the simulation; but n ˆ may be located by evaluating the Q-tensor, a 3 × 3 matrix, whose element is defined as, (6) Qa,b = 3la lb − δab /2 where la is the direction cosine of the molecular symmetry axis with the laboratory a-axis.15 The largest eigenvalue of Q gives the order parameter, P 2 and the components of the eigenvector associated with this, defines ˆ within the laboratory frame. In principle, Q may be deterthe director n mined and diagonalised after every MC-step to calculate P 2 step-wisely, from which its final average value may be obtained. But in practice, this is neither economical nor necessary. Usually the Q-tensor is first averaged over many configurations, the number being restricted only to ensure the director is not fluctuating significantly. This averaged Q is then diagonalised, where its largest eigenvalue gives P 2 for a particular macrostep. This process is repeated for subsequent macrosteps to determine the subaverages of P 2 before the final average order parameter for the production run is calculated. 3.1.2. Limitations and advantages of the lattice class The size of the sample or the number of particles (N ) is related to the lattice type; thus, for a simple cube (SC) and face-centered cube (FCC), N is simply n3 and 4n3 respectively, where n is the box dimension. In most LL-lattice simulations, the range of the interaction is limited to the nearest
11-chapter10
September 29, 2010
10:46
World Scientific Review Volume - 9in x 6in
Computer Modeling of Liquid Crystals
11-chapter10
311
neighbors; accordingly the implementation of PBC is simplified since the central box is surrounded by only one adjacent site interaction from an image box on each face (of the cube). A lookup neighborhood table, which defines the neighbors of every particle, can be prepared only once at the beginning of the simulation run. For example, for a SC lattice, every particle is associated with six nearest neighbors, while those for FCC lattice are twelve. If a nematic binary mixture is considered, solute and solvent neighborhood tables will also be prepared and these tables must be updated after an effective particle exchange.25 Since the beginning, LL model simulations were conducted using N values which typically ranged from a few hundred to a few thousand; this value now has increased and may easily exceed tens of thousands.26 With domain decomposition MC technique implemented on the parallel computers and Condor (a specialized workload management system) in distributed Grid environment, simulations with several million particles of are possible.27–29 Such large scale calculations allow detailed examinations on bulk properties and field responses. It is quite ironic that a model without any translational motion is used to represent a complex liquid crystal state. Nevertheless, it has been shown that the translational degrees of freedom do not affect the long range orientational properties of the liquid crystal phase.30 Moreover, its proven popularity outweighs all criticisms on the lack of translational degree of freedom and most importantly, as we shall see in the following sections, this simple model produces properties which are comparable with those of theory and experiment. 3.1.3. Some results and applications The purpose of LL-model simulations20 was partly to verify and validate the molecular field approximation in the Maier-Saupe theory,7 in addition to testing how well a simple lattice model could represent a liquid crystal phase. Surprisingly, for such a simple model, it exhibits a first order phase transition between orientational ordered (nematic) and disordered (isotropic) states as a function of temperature and has a small entropy change associated with the phase transition (as with real liquid crystals). Furthermore, the simulation demonstrates a relatively lower order parameter, P 2 , compared to that predicted by the theory. This implies the molecular field formalism has overestimated slightly the nature of nematic order due the neglect of the short range correlation, which is unimportant except close to the phase transition.8 While there is a certain universality in
September 29, 2010
312
10:46
World Scientific Review Volume - 9in x 6in
R. Hashim
behavior found experimentally with those predicted by the mean field and LL model, the latter could not account for the variety of N-I phase transitions, each unique to the type of molecules of the real systems. This has prompted many improvements to the pair potential, for example, by adding to it a fourth rank interaction via the Legendre polynomial, P4 . Within the potential function of this extended model, there is an adjustable parameter, λ, which may be chosen to tune the results to improve the agreement with the real data.31 A creative variety of modifications has been made to the simple LLmodel to explore some hypothetical ordered liquid phases. For example, by adding a first order term to the pair-potential, a ferroelectric phase may be simulated,32 and the effect of molecular asymmetry may be assessed. While the simulation results agreed qualitatively with the molecular field predictions and the first order term was found to stabilize a low temperature ferroelectric nematic, experimentally, this novel phase has never been observed. An LL-like model has also been used to investigate the hypothetical thermotropic biaxial nematic, first suggested by Fresier.33 Many simulations were conducted using a biaxial potential interaction, which assumed the particle’s geometry to be orthorhombic or “lath-like”.34 This model was found to be too simple and insufficient to be implemented in designing a real mesogen that will exhibit a thermotropic biaxial nematic phase. Nevertheless, the simulations on this highly simplified model for a thermotropic biaxial nematic, served to characterize the phase, in particular its orientational order. Moreover, these early speculative simulation results are useful to other designers,3 and led them to a more guided investigation in search of this elusive phase either experimentally or using other much improved models such as those formed by molecules with monoclinic symmetry, which has been developed recently by theory,35 and simulated using an MC technique.36 Table 1 lists a selection of these potential models and some of their transitional results. Of course, in this table, only the results of MS and LL models are comparable. The others have been included to demonstrate its adaptive nature so much so many more sophisticated LL-like variant models have been developed. In the next section, two of these will be described in some detail. The LL-model has also been altered by adding another scalar invariant term as originally suggested by Nehring-Saupe to include additional anisotropy effects.38 In recent years, even the cubic phase has been simulated using the LL-cubatic model,39 where the overall order within the lattice is isotropic but each of the three component axes is highly ordered.
11-chapter10
September 29, 2010
10:46
World Scientific Review Volume - 9in x 6in
11-chapter10
Computer Modeling of Liquid Crystals
313
Table 1. Potential functions of LL and related models, Maier-Saupe theory and real system 4,4’-dimethoxyazoxybenzene (PAA) at constant pressure and their typical transitional properties. ∗ = k T / TNI B
NI
Model
Potential function/system
MS (theory)
Ui (θ) = −P 2 P (cos θ)
1.322
0.429
LL Real system
Uij = −ij P (cos θ) 4,4’-dimethoxyazoxybenzene (PAA) at constant pressure Uij = −ij P2 (cos θij ) + λP4 (cos θij ) Uij = − [P2 (cos θij ) + ξP1 (cos θij )]
1.127 409 K
0.33 0.39
1.205(λ = 0.2) 1.14(ξ = −2)
0.36 0.37
2.238
0.3
LL+P4 LL+ferro Nehring-Saupe
Uij = −Uij =
−ij /r 6 (3ai aj
2
− bij )
P2
Ref 7 20 37 31 32 38
3.1.4. LL-chiral nematic model The chiral nematic is characterized uniquely by its twisted director. In a lattice system the director is twisting layer by layer and the helix axis set along the lattice vector. Hence, a LL-like model had been developed to simulate a chiral nematic using a pair potential function based on the electric multipole expansion suggested by van der Meer,40 ˆ j )2 − K (ˆ ˆ j ) · ˆrij (ˆ ˆj ) ui · u Uij = −J (ˆ uj · u ui × u (7) 4 3 ˆ j ) i − M (ˆ ˆ j ) · ˆrij (ˆ ˆj) ui × u uj · u −L(ˆ ui · u ˆ i and u ˆ j are the unit vectors along the symmetry axes of molecules, where u i and j, respectively. rij is the intermolecular unit vector, which joins the centers of molecules i and j. J and K are scalar quantities, which depend on the intermolecular separation; if positively defined, they will enhance the corresponding terms to favor the parallel alignment of these molecules as in the nematic phase. While K and M are pseudo-scalar parameters related to the twisting function of the molecular orientation and vanish if the molecules are achiral. Using only the first two terms of the pair potential in equation (7) would give a chiral phase with a helical pitch independent of temperature, hence, higher order terms were included which made the pitch temperature dependent.41–43 A subtle feature of the lattice simulation for the chiral nematic phase concerns the boundary condition. While the usual PBC was adopted for both x and y axes, for the z axis, either twisted41 or self-determined boundary condition need to be used.42,43 A self-determined boundary is applied using commensurable periodic images generated first by a translation; this
September 29, 2010
10:46
314
World Scientific Review Volume - 9in x 6in
R. Hashim
Fig. 4. Self-determined boundary condition for the simulation of LL-chiral nematic model.42 Reproduced by permission of The Royal Society of Chemistry.
is followed by a rotation of all molecules around the z axis with a twist angle Φ as sketched in Fig. 5. A trial configuration was generated by choosing randomly a particle, i, whose new trial orientation was obtained by a rotation ˆ i around the z axis by an angle ϕ = (ξ−0.5) ϕ0 , where ξ of the unit vector, u is a random number (0 < ξ < 1) and ϕ0 is an adjustable parameter related to the optimum acceptance to rejection ratio subject to the Metropolis test in the MC steps.42 The usual set of properties were calculated to locate the phase transitions were calculated including the energy, specific heat, second and fourth rank order parameters. In addition, the equilibrium pitch was determined and found to be independent of the system size studied. The overall results were found to be in agreement with the molecular field theory when higher order terms are taken into account.41–43 3.1.5. LL-mixture models In early days of simulation, computer resources were scarce. In order to maximize and economize the utilization of CPU time, simulators developed many strategies, from performing routine code optimization to simply being very cautious and frugal in running the simulation to avoid CPU wastage. For example, what was originally an accidental mistake made by a student
11-chapter10
September 29, 2010
10:46
World Scientific Review Volume - 9in x 6in
Computer Modeling of Liquid Crystals
11-chapter10
315
in applying the wrong input interaction parameter to a particular set of simulation runs were saved to be a model LL-like nematic binary mixture.44 And this is now said to be the first of its kind ever performed,45 containing two different nematogens with different values, fixed on the lattice sites. For this mixture system, there were three possible interaction strength parameters, namely, that for the pure solvent-solvent (AA ), the solute-solute (BB ) and for a mixed solvent-solute interaction, which is √ simply taken as the geometric mean of the first two, thus AB = A B . The aim of this binary LL-mixture simulation was to study the effect of different interaction strengths on the phase transition motivated by the fact that LCD materials are mixtures. Since then, many other complex binary and ternary mixtures have been investigated; for some, complete phase diagrams have been obtained with different strength interaction parameters. In one simulation of a binary system of nematic phase the temperature variations of the order parameter P 2 were obtained for the different strength parameters.46 For one type, is set to one, while the other varies from 0.1 to 0.9. For those whose coupling constants vary between 0.1 and 0.5, a double-stepped behavior was observed, which may be related to the phase separation of the two molecular types. These results also imply the ordering depression within the nematic phase is not linear with respect to the decreasing trend in the coupling constant of the second molecule. As expected, the transition temperature shifts downwards with the trend.46 For LCD applications, liquid crystal materials used must be of high purity. The presence of impurities causes many complications. One of such problems is that of a spherical solute, which will depress the transition temperature of the nematic phase. Encouraged by such issues, a LL-mixture model was developed, where some of the sites contain isotropic or spherical particles, which in lattice system, may be easily represented by vacancies.25 In this simulation, the vacancies were allowed to exchange positions with the nematogenic rods, allowing the phase separation into coexisting isotropic and nematic phases to be observed, confirming the results from the molecular field theory for a similar potential model.47–50 The theory gives a phase diagram which contains three regions, nematic (N), isotropic (I) and biphasic (N+I). Moreover, it also predicts for some concentration, a re-entrant into the biphasic region at lower temperature may be observed. Recently, the simulations of this model were revisited with much larger system sizes and treated more generally as a lattice-gas extension, with anisotropic term P2 in three dimensions using the grand cannonical
September 29, 2010
316
10:46
World Scientific Review Volume - 9in x 6in
11-chapter10
R. Hashim
Fig. 5. LL-model mixture of a nematic mixture with spherical particles (vacancies), at 50% composition; (a) T*=0.55, in the two phase co-existence state (nematic and isotropic) and (b) T*= 0.75, in the isotropic phase.50
ensemble.48,49 The pair potential has a more general form of Uij = −si sj ij [λ + P2 (cos θij )]
(8)
where, apart from the usual definitions of the terms used previously, si = 0 or 1 depending on whether the site contains a rod (s=1) or a sphere/vacancy (s=0). In here, λ governs the strength of the isotropic term, where a small value means the model behaves like the original model,25 while a large λ indicates the liquid-vapor coexistence curve occurs at relatively high temperatures. In these simulations, the phase boundaries were calculated and compared directly with those predicted by the theory.47–50 Figure 5 shows the LL-mixture model for 50% composition of vacancies at two temperatures, T*=0.55 corresponding to the two phase coexistence region (nematic+isotropic) and T*=0.75, where the system is completely in the isotropic phase.50 In the two phase region, the rod-like particles are separated, but not completely, from the vacancies which are represented as white spheres. The effect of PBC is also very visible in this lattice case, (see Fig. 5(a)). 3.1.6. Final comment on LL-like models LL-like model is simple and cheap; thus its kinds invariably take the leading role in breaking new investigative grounds for liquid crystal computerassisted research. These few examples demonstrate their simplicity, adaptability and they may reasonably be compared with both theoretical predictions and observed phenomena. The simplicity of the potential model means it is expandable with the growing resources and this factor makes the model popular and highly versatile for the study of liquid crystals.
September 29, 2010
10:46
World Scientific Review Volume - 9in x 6in
Computer Modeling of Liquid Crystals
11-chapter10
317
Complexity may be added to the model itself or to the problems. For example, the simulation of LL-model with applied field performed previously,51 has now been expanded using large scale model and special boundary conditions to simulate macroscopic functions such as LCDs switching.52 Another large scale LL-model has been used to study the factors controlling the surface defects by coating a LL-nematic around a colloidal particle.53 The other major advantage of the lattice model, which contributes to its popularity, is that it cannot freeze. This means that the nematic phase can always be observed, unlike the real system, where freezing can impede the observation of NU and NB phases. 3.2. The Gay-Berne class Two simple model classes of an extreme nature used to represent liquid crystals are the soft attractive LL and its variants (which we have discussed above) and the purely repulsive kind i.e. the hard-particle model class which have been omitted in this chapter, but is discussed elsewhere.54 Even though, relatively speaking, they are easily adaptable and successful, neither of these models could claim to be “realistic”. enough, since a real molecular system must necessarily contain both contributions from the soft attractive forces, due to the distortion in the electronic charge distribution (from such effects as electrostatic and dispersion), and the repulsive force, due to the violation of Pauli’s exclusion principle as a result of electron overcrowding.55 Therefore, a realistic pair potential model must contain these two contributions, like that for an atom or particle with the spherical symmetry, as demonstrated by the famous Lennard-Jones (LJ) 12-6 potential, ULJ (rij ) = 4 (σ/rij )12 − (σ/rij )6 (9) where σ is the spherical contact distance and is the strength of the attractive interaction and always defined to be positive; both parameters are usually determined empirically. Prompted by the success of the LJ interaction potential, Corner proposed a similar site-site pair potential to describe the anisotropic interaction between molecules, where σ and in equation (9) took a functional form, which vary with the molecular orientations as well as that of the intermolecular distance, rij .56 Although, the proposed analytical functions for such dependencies were found unsuitable, Berne and Pechukas re-discovered the idea and suggested the hard Gaussian overlapped (HGO) for two prolate ellipsoids, from which
September 29, 2010
10:46
World Scientific Review Volume - 9in x 6in
318
11-chapter10
R. Hashim
they obtained the angular dependent functions for the inter-molecular contact and interaction strength.57 Later, Gay and Berne considered four LJ sites in the HGO treatment and obtained the highly successful Gay-Berne (GB) potential model.58 Their most significant contribution to the model development was the functional form proposed for (see equation (14)); while the parameterization of the potential model is secondary. This model has a form similar to the LJ potential,59 simply written as, ˆ j r) = UGB = 4 (ˆ ˆ j ˆr) R−12 − R−6 . U (ˆ ui u ui u (10) This potential assumes that the molecules are cylindrically symmetric so ˆ . In this equathat their orientations are represented by the unit vectors, u tion, the ˆ denotes a unit vector. Since it is also a single-site model, the potential depends on the intermolecular vector, r separating two molecules. R is defined as, ˆ j ˆr) + σ0 ] /σ0 . R = [r − σ (ˆ ui u
(11)
Equation (11) contains, the orientation-dependent contact function,
−1/2 ˆj) (ˆ ui · ˆ r)2 + (ˆ uj · ˆ r)2 − 2χ(ˆ ui · ˆ r)(ˆ uj · ˆ r)(ˆ ui · u ˆ ˆ 1 − χ σ(ˆ ui uj r) = σ0 , ˆ j )2 1 − χ2 (ˆ ui · u
(12) where χ, the shape anisotropy parameter is, χ = (κ2 − 1)/(κ2 + 1),
(13)
and is related to the length-to-breadth ratio κ = σe /σs , where σe is the endto-end contact distance between two GB particles, while σs is that for the side-by-side. The angular dependence of the strength parameter function is given as,
where
and
ˆ j ˆr) = 1 − χ ui u 2 (ˆ
ˆ j ˆr) = 0 ν1 (ˆ ˆ j ) µ2 (ˆ ˆ j ˆr) , ui u ui u (ˆ ui u
(14)
−1/2 ˆ j )2 ˆ j ) = 1 − χ2 (ˆ 1 (ˆ ui u ui · u
(15)
ˆj ) uj · ˆr)2 − 2χ (ˆ ui · ˆr)(ˆ uj · ˆr)(ˆ ui · u (ˆ ui · ˆr)2 + (ˆ 2 ˆj ) 1 − χ (ˆ ui · u
(16)
χ is analogous to χ, equation (13) and is the anisotropy parameter related to the well-depth, χ = (κ1/µ − 1)/(κ1/µ + 1).
(17)
September 29, 2010
10:46
World Scientific Review Volume - 9in x 6in
Computer Modeling of Liquid Crystals
11-chapter10
319
Fig. 6. (a) Two GB molecules. (b) UGB as a function of the intermolecular distance, r ∗ , for side-by-side, end-to-end and T-configurations for the GB(3, 5, 2, 1), dashed lines and GB(3, 5, 1, 3), solid lines.61 Reproduced by permission of The Royal Society of Chemistry.
The quantity κ provides a measure of the interaction strength anisotropy of the well-depth and is defined by s /e , where the well-depth for the sideby-side configuration is s and that for the end-to-end is e . Four parameters define the details of the GB potential, namely, the exponents µ and ν, which are both adjustable, κ and κ ; hence the model is often written as GB(κ, κ , µ, ν).60 In the simulation, σ0 and 0 are used to scale distance and energy. Figures 6(a) show two GB molecules, while Fig. 6(b) plots UGB as a function of intermolecular distance r to show how the energy varies for different angular approaches.61 3.2.1. Some results and applications The most common parameterization set that give stable Iso, N, SmB and Cr phases for this potential is the (3, 5, 2,1),62 which also represents a uniaxial ellipsoidal shape GB-molecule of the calamitic type. This potential function is shown by the dotted lines in Fig. 6(b). It is related to the length-to-breadth ratio or κ = σe /σs = 3 and the well-depth anisotropy κ = e /s = 5, while the exponent energy parameters µ and ν in equation (14), take the values of 2 and 1, respectively. The latter two parameters
September 29, 2010
10:46
World Scientific Review Volume - 9in x 6in
320
11-chapter10
R. Hashim
Table 2. Various GB models for elongated rod-like molecules simulated at different conditions and the resultant phases observed. Here Iso, N, Sm indicate isotropic, nematic and smectic phases.61 Reproduced by permission of The Royal Society of Chemistry. GB(κ, κ , µ, ν)
Phase observed
Simulation method and comments
(3,5,2,1)
Iso, N, SmB
MD MD, viscosity determined MD, N = 1024, 2048, elastic constants determined
(3,5,1,2)
Iso, N, Sm
(3,5,1,3)
Iso, N, SmA, SmB
(3.0,4,2,1); (3.2,4,2,1); (3.6,4,2,1); (3.8,4,2,1); (3,4,2,1) (3,1,1,3); (3,1.25,1,3); (3,2.5,1,3); (3,5,1,3); (3,10,1,3); (3,25,1,3) (4.4,20.0,1,1)
Iso, N, SmA, SmB
Iso, N, SmA, SmB
Iso, N, SmA, SmB
Ref 51,52,59 72 64,73
MC, N = 256
74
MC (N V T ), N = 1000 MD N = 8000, elastic constants determined MD N = 8000, pretransitional effects studied MD, N = 65,536, defects
63
Effect of changing the elongation, χ MD (N V T ), N = 254 MC (N P T ), N = 600 Effect of changing the elongation χ or well-depth anisotropy MD, N = 254-864
Nematic is only observed at high pressure MC (N P T ), N = 200; MD (N V T ), N = 16,000
64 71 65
66
75
60
control the interaction strength or the well-depth of the interacting GB particles. Hence, bigger values will deepen the well depth, as shown in (Fig. 6(b)) for example, when ν is increased from 1 (dotted lines) to 3 (solid lines) for both the end-to-end and side-by-side configuration.61 On the other hand, if µ is increased to 3, as in GB(3,5,3,1), the nematic range is enhanced.63,64 As it turns out, the set (κ, κ , 0, 0) reduces the model to a soft ellipsoid while that of (0, 0, µ, ν), to a spherical LJ. Table 2 summarizes the details of a selection of GB models simulated using MC and MD techniques under different thermodynamic conditions (N V T and N P T ),61 and the many reasons and motivation for the studies, from testing the model parameterization,65,66 to fitting GB models to real
September 29, 2010
10:46
World Scientific Review Volume - 9in x 6in
Computer Modeling of Liquid Crystals
11-chapter10
321
Fig. 7. A model of GB discs using the parameter set of σf /σe = 0.345, e /f = 1/9, µ = 1 and ν = 2 gives a range of phases (a) columnar, (b) nematic and (c) isotropic.79 Reproduced by permission of The Royal Society of Chemistry and the PCCP Owner Societies.
mesogens.61 Besides obtaining the orientational and positional orders, characteristic of these phases, the simulations also calculated other relevant physical observables, including a variety of correlation functions,62 viscosities,67 elastic constants,64,68,73 thermal conductivities and diffusion coefficients,69 all with the aim of phase identification and making some comparison with the real systems. GB model has also been used to test assumptions used in theory, such as the molecular field approximation,70as well as extracting information difficult to measure experimentally.71 Although, the parameter sets chosen in the previous discussion are for the GB rod-like shape, changing these values slightly will produce a GBdiscotic molecule.76–79 For example, by defining new contact distance and strength parameters such as σf and f for the face-to-face configuration and selecting the following values: σf /σe = 0.345, e /f = 1/9, µ = 1 and ν = 2. Such parameter sets will give a rich discotic polymorphism from isotropic, nematic and rectangular columnar phases (see Fig. 7). To get a hexagonal columnar phase,77 a slightly different value of µ and ν, should be used to lower the well-depths of the face-to-face and side-by-side configurations of the GB-disc.61 For these GB discotic phases, the usual properties ranging from the order parameters, radial distributions, thermodynamic quantities and elastic constants have been calculated.76–79 Although, the calamitic GB found a wide range of applications, the discotic GB is also interesting for columnar material applications; for example, to model energy transfer in columnar phases.80
September 29, 2010
10:46
World Scientific Review Volume - 9in x 6in
322
R. Hashim
Even though the anisotropic dispersion interaction (through induced dipole), is the minimum requirement necessary for the formation of a liquid crystal phase, many mesogenic molecules actually contain permanent molecular dipole and charges which further enhanced the stability of the phase. Since the simple GB model assumes only dispersion energy, permanent dipoles may be added to the GB potential to increase its anisotropic long-range interaction, while specifying its location and strength, may tune the model to imitate a specific system. The added dipole stabilizes the mesophase depending on its position, orientation and strength; for example, a terminal axial dipole shifts the N-I transition to higher temperatures and a central axial dipole stabilizes the smectic.82 Similarly, a quadrapolar term may also be added to the potential, and if placed transversely, the smectic transition temperature rises, while a larger magnitude favors the cubic smectic arrangement.83,84 In another MC study of a quadrupolar GB-disc, for a pure system, the quadrupole interaction destabilizes the columnar phase. In contrast, the binary mixtures of these were found to give a liquid crystal phase at much higher temperatures and may even give a columnar phase, when the components have opposite signs to the quadrupole terms.81 This observation is consistent with that found experimentally for the chemically induced phases formed by certain multiynes with 2,4,7-trinitrofluorenone.85
3.2.2. Gay-Berne variant models The GB model is remarkably more versatile and tunable than LL and hardparticle classes. It has an LJ type interaction, off-lattice, but still has only a single site interaction, which makes it computationally cheaper than the full atomistic force field model (see later). However, the GB models described so far are for molecules, whose shape approximate to that of an ellipsoid, both for calamatic and discotic with a uniaxial symmetry. This limiting feature has motivated GB variant models to be developed since most liquid crystal molecules deviate somewhat from an ellipsoidal shape and uniaxial symmetry. The most obvious variation to be included in a “perfect”. ellipsoidal model would be a deviation from the uniaxial symmetry. Thus, a biaxial GB model has been developed, where there are three semi-axes for the ellipsoid, from which two relative biaxiality parameters, λσ and λe , may be defined and these are related to the biaxial shape function, σ, and the biaxial interaction strength function, .78
11-chapter10
September 29, 2010
10:46
World Scientific Review Volume - 9in x 6in
Computer Modeling of Liquid Crystals
11-chapter10
323
A number of GB biaxial model variety (using different parameter sets) have been developed and simulated, and recently one has been used to study the response of the biaxial nematic bulk director to an applied external field (electric or magnetic) in an attempt to measure its switching time, which may have significant implications for the electro-optic and display industries.86 In this study, a particular GB model, described previously,78 was used with N = 8192 and biaxiality parameters, λσ = 0.216 and λ = −0.060. A series of MD director switching experiments were performed and they found that the secondary director switches faster than the principal director, and above certain threshold field, reorganization of local domains is not through a collective concerted reorientation but one which disrupts the nematic and biaxial order.86 Another variant model is the GB dimer,59 which consists of two GBellipsoids connected together with some flexibility at the joint. While the model parameters are fairly straight forward, the joint dynamics require a slightly different treatment. Liquid crystal dimers such as α,ω-bis (4,4.cyanobiphenyloxy) alkanes form novel materials and this was first described by Vorl¨ ander,87 who reported its special property which depends on both the spacer length and the parity of the number of atoms. Figure 8(a) shows the GB dimer’s structure linked together through a bond whose length is 2σ0 /3 and makes an angle θ0 with respect to the two molecular symmetry axes. If the ground state of the dimer is assumed to be trans- and the two mesogenic groups are co-planar, then, the torsional angle φ is zero. Thus the combined energetic contour of a typical GB(3,5,2,1) is given as in Fig. 8(b). The torsional energy and bend-bonding energy are given respectively as follows, Utors (φ) =
5
cn cosn φ , Ubend (θ) = (kbend /2)(θ − θ0 )2 .
(18)
n=0
The equations of motion are treated in a manner similar to that for polyatomic molecules ensuring the bond lengths within the molecule are conserved. For the flexible dimer, the force and torque which acted on the centre of mass and the symmetry axis of a GB group must be replaced with those forces acting on two centers displaced from the molecular center of mass.88 This simple dimeric GB model had been simulated to mimic a typical dimeric molecule, such as an α, ω-bis(4-cyanobiphenyl-4’-yloxy) alkane,90 where, for example, the N-I transition temperature of the real material, which was found to be roughly about 440 K had been used as the scaling constant. The dimeric GB mesogen was found to reduced the
September 29, 2010
324
10:46
World Scientific Review Volume - 9in x 6in
R. Hashim
Fig. 8. (a) The structure of a GB-model dimer. (b) The energy contour of the dimer.59 G. R. Luckhurst, Liquid Crystals 32, 1335 (2005). Reprinted by permission of Taylor & Francis Ltd. http://www.informaworld.com (c) A typical dimeric molecule α, ω-bis(4-cyanobiphenyl-4’-yloxy) alkanes.
stability of the crystal phase and enhanced the liquid crystal phase when compared with the monomeric GB under the same condition.59 Quite recently, another GB variant potential was proposed by Zewdie,91 who used Corner’s original idea about the orientation dependence of the contact and interaction strength functions, but expanded these instead in terms of the orthogonal basis set S-functions.55,92 Expanding these equations in terms of the S-functions, may give any desired shape provided sufficient terms are included; but for the potential to be useful and not too demanding, computationally, only a minimum set of these terms should be used. Zewdie determined some coefficients for a combination of shapes typically for a particle with the length-to-breadth ratio of 3:1.91 This special GB variant potential is also called the CornerS model. Using some of these coefficients, a simulation of the CornerS spherocylinder (k = 3) model had been performed by MC-N V T technique.59 The simulation results showed that the CornerS spherocylinder gave the usual phases such as crystal, smectic B, smectic A and isotropic (see Figs. 9(b), 9(c)). On revisiting the simulation, it was found that the smectic B claimed previously was more likely a crystalline phase.59 In contrast to the finding made by Bolhuis and Frenkel,89 for a purely hard spherocylinder (HSC) in which an aspect ratio of k = 4.1 was necessary for the formation of a smectic A, for the corresponding soft spherocylinder
11-chapter10
September 29, 2010
10:46
World Scientific Review Volume - 9in x 6in
Computer Modeling of Liquid Crystals
11-chapter10
325
Fig. 9. CornerS spherocylinder model (k=3) in three phases (a) crystal/B (b) nematic and (c) isotropic.59 G. R. Luckhurst, Liquid Crystals 32, 1335 (2005). Reprinted by permission of Taylor & Francis Ltd. http://www.informaworld.com
of the CornerS potential, the inclusion of an attractive force enhances the side-by-side interaction and so stabilize the smectic phase with a shorter aspect ratio k = 3. It turns out that if a sphere is added to the center of the spherocylinder, a peculiar shape (Fig. 9(c)) model is produced, which gives a crystal nematic and isotropic phases. Distortion of the spherocylinder by the embedded sphere introduces a packing constraint disfavoring the sideby-side arrangement leading to a shape quadrupolar effect and destabilizes the smectic phase.59 The approach of CornerS potential to fit almost any shape makes it possible now to develop a non-centrosymmetric single-site model. This has been applied to a combination of sphere and ellipsoid to model a pear shaped molecule and to study the flexoelectric effect.93,94 3.3. Full atomistic class Although simulation of atomistic models is well-established as a tool for studying solids, liquids and gases, this model class is less popular to study the liquid crystal properties compared to the generic models; despite the sensitivity of many mesogenic properties towards molecular details. For example, the range of a liquid crystal phase is controlled not only by the aspect ratio of mesogenic rigid section, but also by the alkyl chain flexibility, which is often ignored in generic models. However, limited by economy due to the overwhelming computational requirement for an atomistic simulation, liquid crystal simulations usually adopt a simplified design strategy, using a potential model generalized over shape (geometry) and attractive interactions. But this approach is changing in recent years; the growing CPU power, the widespread availability of general purpose force field soft-
September 29, 2010
10:46
World Scientific Review Volume - 9in x 6in
326
11-chapter10
R. Hashim
ware and the pure challenge from such a task, have gradually drawn more attention and interest to the atomistic simulation of liquid crystals. Atomistic simulations have different aims and challenges to those introduced before and one of these must be to reproduce the exact properties of a real molecular system since it uses the most exact “realistic” (ab initio) model. Unfortunately, this is still a formidable task, since typically one liquid crystal molecule contains more than 30 atoms, which makes the number of corresponding interaction sites so enormous, even for a simulation containing a few hundred molecules. Under such circumstance, some approximations have to be applied to reduce this number, for example, using the united atom approach especially for hydrogen, which is normally fused to the atom it is bonded to. Before the turn of the century, reliable force fields to reproduce liquid crystal properties were lacking, unlike those for biomolecular systems which are readily available from software packages such as AMBER and CHARMM. Only recently in 2002, did Cheung et al.95 develop a liquid crystal force field based on Jorgenson’s method,96 which employed OPLS (for united atoms) and OPLS-AA (all atoms) within the harmonic AMBER-type force field. This force field is sufficiently flexible to model any calamitic liquid crystal mesogen,28 and typically, it takes the usual form of equation (19), which contains contributions from the bond stretching (Ustretch ), bending (Ubend ), torsion (Utor ), non-bonding (ULJ ) and electrostatic interactions (Uelec ) as in equations (20)-(24): Uf f = Ustretch + Ubend + Utor + ULJ + Uelec ,
1 Ustrecth = kl (l − leq )2 , 2 bend
1 Ubend = kθ (θ − θeq )2 , 2
(19) (20) (21)
angle
Utor =
1 dih
ULJ =
i,j
Uelec =
i,j
2
ki (l + cos(iτ + δi )),
4ij
σij rij
1 qi qj . 4π0 rij
12
−
σij rij
(22) 6 ,
(23) (24)
The parameters within these equations, kl , kθ and ki may be determined either experimentally or now from high quality ab initio calculations, using for example, the Gaussian software package. Similarly, high
September 29, 2010
10:46
World Scientific Review Volume - 9in x 6in
Computer Modeling of Liquid Crystals
11-chapter10
327
level quantum mechanical techniques may also be used to determine the partial charges which are used in the electrostatic component modeled through a Coulomb potential (equation (24)).97 Lennard-Jones 12-6 potential (equation (23)) exclude the 1-2 and 1-3 non-bonded terms while all 1-4 interactions were scaled. The standard Lorentz-Berthelot mixing rules are applied for the Lennard-Jones parameters and thus 1i2j = (1i 2j )1/2 and σ1i2j = (σ1i + σ2j )/2. The potential defined in equation (23) is only an approximation to represent the non-bonding interaction for atoms in the fluid phase; improvements on both the dispersion contribution as well as the treatment of many-body interactions have been variously suggested.95 Over the years, sample sizes have varied from as low as N ≤ 100,98–100 to only a few hundred.101,102 The largest reported value of N is about 1000, but in this “considerably” large scale atomistic simulation, no attempt was made to compute the temperature dependency or to identify the phase transition.103 Given the monumental task of computing the interaction force field potential for the number of effective particles in the past, the MD observation window usually ranged from a few hundred picoseconds (ps) to 1-2 nanoseconds (ns). For solving the equations of motion, in this case, a shorter time-step, ∆t, is used and it ranges between 1-3 femtoseconds (fs). The short time-step is necessary to deal with the fast intramolecular motions. Lacking the knowledge of what the realistic values for N and the MD time-window were, those pioneering atomistic simulations inevitably produced poor quality results as described in a few examples below.
Fig. 10.
4-n-pentyl-4.-cyanobiphenyl (5CB), and its liquid crystal transitions.99
The first and preliminary atomistic molecular dynamics simulation of the nematic phase of 4-pentyl-4’-cyanobiphenyl (5CB) (see Fig. 10) was conducted by Picken et al.,99 using only 64 molecules containing 19 atom types of N, C, CH, CH2 and CH3 . Some of these were united atoms (CH, CH2 and CH3 ) which reduced the total number of interacting sites in this system to be 1216(= 19×64). The bond stretching, bending, torsion parameterizations were taken from experimental data, while those of the partial charges came from ab initio calculations.
September 29, 2010
10:46
World Scientific Review Volume - 9in x 6in
328
R. Hashim
This simulation was performed to study the effect of molecular dipole moment on the phase behavior; hence, two runs were conducted for charged and uncharged systems, observed under an MD time-window of about 60 ps each and applying an MD time-step of 0.004 ps using the leap-frog algorithm. Even though the MD-run was short by today’s standard, on the Control-Data Cyber 170/760, where it was running, one simulation step took about 3-10 CPU seconds. Consequently, a 60 ps run was equivalent to more than 11 CPU hours. For such a short run-duration on this computer, the systems were insufficiently equilibrated since many properties, like the order parameters, were found to be still decreasing even though the temperature and pressure were found to be stable. Nevertheless, from this preliminary work, they concluded that the presence of molecular charge affects the orientational distribution function and in qualitative agreement with experimental measurements of the orientational order parameters. Furthermore, the authors also cautioned that these results were qualitative, but acceptable for comparison purposes since both the simulations (with and without charge) were conducted under similar conditions. Many other atomistic simulations have been performed since then, such as that by Wilson and Allen for the system with N = 128 of the trans-4(trans-4-n-pentylcyclohexyl) cyclohexylcarbonitrile (CCH5).104 In this report, the authors claimed, an MD time frame of 1 ns was sufficient to stabilize the nematic order. But later in 1999, one of these authors (MRW) refuted the claim and admitted that a much longer time frame (> 5 ns) is needed to observe the equilibrium order.105 Table 3. The NI transition temperatures for a series of amino-cinnamate system.106 n
TNI (K)
< P2 >iso
Texp (K)
0 1 2
540 430 465
0.13 0.19 0.09
553.3 433.7 460.9
Recently, a set of three homologous series of 98 amino-cinnamate molecules were simulated for over 50 ns, where Berardi et al claimed the time to equilibrate the orientational order within the nematic phase must be greater than 10 ns.106 This simulation demonstrated the classic oddeven effect, which was initially described by Gray,107 and reproduced the experimental TNI for the odd and even series (see Fig. 11 and Table 3).
11-chapter10
September 29, 2010
10:46
World Scientific Review Volume - 9in x 6in
Computer Modeling of Liquid Crystals
11-chapter10
329
Fig. 11. A series of amino-cinnamate system.106 R. Berardi, L. Muccioli, C. Zannoni, Chem. Phys. Chem. 5, 104 (2004). Copyright Wiley-VCH Verlag GmbH Co. KGaA. Reproduced with permission.
From the extensive calculations performed on this system, they were able to offer reasons for the observed odd-even effect. Firstly, they found no significant increase in the molecular elongation ratio when passing from an even to odd number of the methylene group, and secondly, for the n=1, the terminal ring does not have a preferential alignment with respect to the phase director. Cheung et al., (2004), simulated a system of 4-(trans-4-n-pentylcyclohexyl) benzonitrile molecules (PCH5) (see Fig. 12) with the aim of understanding the influence of the quadrupolar interaction on its flexoelectric coefficient. The results from this study, where N = 216 (or equivalent to several thousand atomic interaction sites), showed the correct temperature dependency of the orientational order parameter in a nematic phase (Table 4). Although at the temperature of 330K, the simulated system remained nematic, due to possibly, an over prediction of the system density.102
Fig. 12.
4-(trans-4-n-pentylcyclohexyl) benzonitrile (PCH5).
September 29, 2010
10:46
World Scientific Review Volume - 9in x 6in
330
R. Hashim Table 4. Order parameters as a function of temperature for 4-(trans-4-n-pentylcyclohexyl) benzonitrile (PCH5).102 Reused with permission from David L. Cheung, J. Chem. Phys., 121, 9131 (2004). Copyright 2004, American Institute of Physics. T/K 300 310 320 330
< P2 >simulation
< P2 >experiment
± ± ± ±
0.63 0.58 0.50 0.00
0.68 0.65 0.55 0.51
0.02 0.01 0.03 0.04
The cyanobiphenyl family (CB) is the most thoroughly studied atomistic system using many different combinations of force field and ab initio parameterization strategies.45 These undertakings may appear collectively a major effort, but partly due to the computational limitations, no agreement has yet been achieved, even on which is the most realistic parameter set to represent the nCB family. Hence, the qualitative and quantitative agreements with the behaviour of the real systems remain inconclusive. Nevertheless, in a limited sense, the nCB atomistic simulations have been successfully applied to a variety of investigations including that on the intramolecular structure,105 Kirkwood correlation factors,108 rotational viscosity,95 and flexoelectric coefficients.106 The parameterization has also been used to study a preconstructed structure such as the smectic phase of free-standing film of 4-octyl-4’-cyanobiphenyl (8CB).109,110 It will not be too long before the full atomistic force field model be used in a predictive sense to forecast the liquid crystal phase transition and phase sequences. Recently, Tiberio et al.111 suggested an improved strategy to develop the force field for the nCB (n=4-8) with 250 molecules using the united atoms approach; many physical properties including the nematic-transition temperature were determined to within 4K of accuracy. Currently, although atomistic simulations are restricted to a few hundred molecules and lengthy simulations are needed to see the equilibrium liquid crystal behavior, such difficulties may be overcome by suitably combining both the Gay-Berne (GB) and molecular force field to produce a hybrid model usually referred as GBMOL, (see equation (25)), which has now been developed and applied to polymeric materials and to measure the helical twisting power.112 UGBMOL = Ustretch +Ubend +Utor +ULJ +Uelec +ULJ−GB +UGB−GB (25)
11-chapter10
September 29, 2010
10:46
World Scientific Review Volume - 9in x 6in
Computer Modeling of Liquid Crystals
11-chapter10
331
In summary, we have seen that atomistic simulation (including the united atom approach and the hybrid model) is beginning to compete in popularity with the generic models, due to not only large scale CPUs accessibility but also the increase in the confidence level as its methodology matures. 3.4. Conclusion This short chapter serves only as an introductory review on the molecular simulation of liquid crystals. Three model classes have been chosen for discussion, namely the lattice, Gay-Berne and atomistic; the first and second are also classified as generic models. Perhaps it must be emphasized here that none of these models is better than the others, as each has its own strengths and weaknesses. Moreover, the decision as to which model or technique used must depend heavily on the questions which the simulators wish to address, apart from the availability of computer resources. These issues must be clear from the beginning. Models from the generic class are cheaper to use and allow for larger system size and longer simulation runs. The lattice models, for example, have been extremely useful in theoretical investigations especially with respect to the molecular field theory. It has been used to study the precise nature and the thermodynamics of the N-I transition and even make many predictions of novel phases, such as the biaxial nematic or cubatic phases. The GB class, although is classified as a soft model, contains a hard repulsive component which makes the overall potential model more realistic and popular amongst liquid crystal simulators. Using this model class, they were able to optimize the fullest by balancing between the problem demands and computer resources. Numerous GB investigations include: mapping out the phase diagram for several parameterizations, addition of dipole and quadrupole interactions or chirality function; all with the aim of studying the influence of structure and electronic interactions on liquid crystal properties. The recently proposed CornerS model has features such as highly tunable, versatile and a single site; hence cheap and therefore will offer to study a wider spectrum of phase phenomena, uncovered previously by the other generic models such as the study of flexoelectricity. The simplicity and cost effectiveness of some of these generic models allow them to be used in function modeling such as the LCD switching, which is normally undertaken by a continuum model, whose observation time window is much longer and for a larger system size.
September 29, 2010
10:46
World Scientific Review Volume - 9in x 6in
332
11-chapter10
R. Hashim
Table 5. The types of simulation model and related processes which may be studied and the minimum simulation time window for each (extract from Ref. 28). Process
Bond motion Intermolecular motion Diffusion Director Alignment Smectics hline LC polymer Function
Simulation Approach
Time-Window
10−15 s 10−14 s 10−9 s 10−8 s 10−7 s > 10−7 s
Quantum
Atomistic
X X
X X X X X X X
Coarse Continuum Grain
X X X X X
X X
As cheap computer resources and reliable force fields become widely available, liquid crystal simulation advance gradually towards atom-based model. Despite the demand for larger system size for accuracy, the cluster computers or shared resources offer some hope towards solving such a colossal effort. For an MD simulation, further efficiency may be achieved by the use of an efficient integrator such as the symplectic multi-time step to reduce CPUs requirement per MD-step. These new technologies will definitely boost the capability of atomistic simulation of liquid crystals in the foreseeable future. The continuum and quantum descriptions of liquid crystal modeling have been omitted from this chapter since the former will be covered by other contributors; while the latter is somewhat not directly relevant to issues in self-assembly, although obviously it defines the molecular interaction. In Table 5, these simulation approaches are listed together with those from the atomistic and coarse grain, where within each one, the relevant processes and the minimum time-window required to make simulation observations are summarized. Obviously, the bond and intramolecular motions, which take place within the femtoseconds region are best studied by using full ab initio quantum and atomistic approaches. While for much slower processes such as diffusion, director alignment and smectic phase formation, the coarse grain models will be more suitable since these require much longer observation times greater than nanoseconds. The continuum simulation on the other hand is best applied for much bigger systems and application functions, but being macroscopic, loses all the molecular descriptions at the microscopic level which defeats the ultimate dream — for
September 29, 2010
10:46
World Scientific Review Volume - 9in x 6in
Computer Modeling of Liquid Crystals
11-chapter10
333
an in-silico liquid crystal simulation to bridge the microscopic world to the macroscopic world — from molecule to phase to function. This dream is not impossible, as computer power advances, its realization is slowly emerging from the atomistic, coarse grain (or even hybrid) approaches, which are marked by grey X’s in Table 5. Although, some may argue, this brute approach is not optimum, since similar results may be achieved from a combination of different methods; for example, using the molecular simulations to derive various properties relevant to applications and feeding these results to the continuum simulations.
Acknowledgments I would like to express my gratitude to G.R. Luckhurst for reading the draft manuscript and giving his valuable comments. I am thanking many friends who have allowed me to use their work and sent me the reprints, preprints and unpublished materials. Further, I acknowledge various grant agencies and the University of Malaya for supporting my scientific work.
References 1. T. J. Sluckin, D. A. Dunmur, H. Stegmeyer, Eds., Crystals that flow: Classic papers from the history of liquid crystals, (Taylor and Francis, London, 2004). 2. M. Bar´ on, (coordinator), Definitions of Basic Terms relating to Low-MolarMass and Polymer Liquid Crystals. Pure Appl. Chem. IUPAC (London, 2001), vol. 73. 3. G. R. Luckhurst, Nature 430, 413 (Jul 22, 2004). 4. L. J. Yu, A. Saupe, Phys. Rev. Lett. 45, 1000 (1980). 5. C. Zannoni, in Nuclear magnetic resonance of liquid crystals, J. W. Emsley, Ed. (Reidel, Dordrecht, Berlin, 1985), pp. 1-34. 6. C. Zannoni, in The Molecular Physics of Liquid Crystals, G. R. Luckhurst, G. W. Gray, Eds. (Academic Press, New York, 1979), pp. 51-83. 7. W. Maier, A. Saupe, Z. Nuturforsch 13, 564 (1958). 8. G. R. Luckhurst, in The Molecular Physics of Liquid Crystals, G. R. Luckhurst, G. W. Gray, Eds. (Academic Press, New York, 1979), pp. 85-119. 9. (See for example) G. R. Luckhurst, C. Zannoni, Nature 267, 412 (1977). 10. L. Onsager,Ann. N.Y. Acad. Sci. 51, 627 (1949). 11. (See for example) N. V. Madhusudana, Bulletin of Materials Science 3, 119 (1981). 12. M. A. Cotter, D. E. Martire, J. Chem. Phys. 52, 1902 (1970). 13. S. Singh, Physics Reports 324, 107 (2000).
September 29, 2010
334
10:46
World Scientific Review Volume - 9in x 6in
R. Hashim
14. P. G. de Gennes, J. Prost, The Physics of Liquid Crystals (Oxford University Press, ed. 2, 1993). 15. (See for example) C. Zannoni, in The Molecular Physics of Liquid Crystals, G. R. Luckhurst, G. W. Gray, Eds. (Academic Press, 1979), pp. 191-220. 16. (a) A. R. Leach, Molecular modeling: principles and applications. (Addison Wesley Longman, 1996); (b) M M. P. Allen, John von Neumann Institute for Computing. J¨ ulich, NIC Series (2004), vol. 23. 17. N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, E. Teller, J. Chem. Phys. 21, 1087 (1953). 18. M. P. Allen, D. J. Tildesley, Computer Simulation of Liquids. (Oxford University Press, 1987). 19. B. J. Alder, T. E. Wainwright, J. Chem. Phys. 31, 459 (1959). 20. P. A. Lebwohl, G. Lasher, Phys. Rev. A 6, 426 (1972). 21. C. Chiccoli, P. Pasini, F. Semeria, C. Zannoni, Phys. Lett. A 176, 428 (1993). 22. (See for example) C. Zannoni, M. Guerra, Molec. Phys. 44, 849 (1981). 23. J. A. Barker, R. O. Watts, Chem. Phys. Lett. 3, 144 (1976). 24. S. Romano, in Physics of liquid crystalline materials, I.-C. Khoo, F. Simoni, Eds. (Gordon and Breach, 1991), pp. 65. 25. R. Hashim, G. R. Luckhurst, S. Romano, Proc. R. Soc. Lond. A 429, 323 (1990). 26. P. Pasini, G. Skacej, C. Zannoni, Chem. Phys. Lett. 413, 463 (Sep 26, 2005). 27. S. Boschi, M. P. Brunelli, C. Zannoni, C. Chiccoli, P. Pasini, Int. J. Mod. Phys. C 8, 547 (1997). 28. M. R. Wilson, International Reviews in Physical Chemistry 24, 421 (Dec 21, 2005). 29. C. Chiccoli, P. Pasini, F. Semeria, C. Zannoni, Int. J. Mod. Phys. C 18, 1725 (2007). 30. G. R. Luckhurst, S. Romano, Proc. R. Soc. Lond. A 373, 111 (1980). 31. G. J. Fuller, G. R. Luckhurst, C. Zannoni, Chem. Phys. 92, 105 (1985). 32. (a) F. Biscarini, C. Chiccoli, P. Pasini, C. Zannoni, Molec. Phys.73, 439 (1991). (b) T. J. Krieger, H. M. James, J. Chem. Phys. 22, 796 (1954). 33. M. J. Freiser, Phys. Rev.Lett. 24, 1041 (1970). 34. (See for example) (a) G. R. Luckhurst, S. Romano, Molec. Phys. 40, 129 (1980); (b) S. Romano, Physics Lett. A 333, 110 (2004). 35. G. R. Luckhurst, S. Naemura, T. J. Sluckin, T. to and S. Turzi, to be submitted. 36. Rauzah Hashim, G. Luckhurst and H. S. Nguan, Poster Presentation, 23rd International Liquid Crystals Conference, 11–16 July 2010, Krakow, Poland. 37. V. Vill, LiqCryst 4.7-Database of liquid crystalline compounds for personal computers, (Fujitsu Kyushu Systems and LCI, 2008). 38. R. Hashim, S. Romano, Int. J. Mod. Phys. B 13, 3879 (1999). 39. (a) S. Romano, Phys. Rev. E 74, 011704 (2006); (b) S. Romano, Phys. Rev. E 77, (Feb, 2008). 40. B. W. V. D. Meer, G. Vertogen, in The Molecular Physics of Liquid Crystals, G. R. Luckhurst, G. W. Gray, Eds. (Academic Press, London, 1979).
11-chapter10
September 29, 2010
10:46
World Scientific Review Volume - 9in x 6in
Computer Modeling of Liquid Crystals
11-chapter10
335
41. G. R. Luckhurst, S. Romano, H. B. Zewdie, J. Chem. Soc., Faraday Trans. 92, 1781 (1996). 42. R. Memmer, F. Janssen, J. Chem. Soc.: Faraday Trans. 94, 267 (1998). 43. R. Memmer, F. Janssen, Liquid Crystals 24, 805 (1998). 44. R. Hashim, G. R. Luckhurst, S. Romano, Liquid Crystals 1, 133 (1986). 45. C. M. Care, D. J. Cleaver, Rep. Prog. Phys. 68, 2665 (2005). 46. V. Yarmolenko, Lattice modelling of liquid crystal mixtures, (Ph.D. Thesis, Sheffield Hallam, University, 2003). 47. (a) R. L. Humphries, G. R. Luckhurst, Proc. R. Soc. Lond. A 352, 41 (1976) (b) P. Pallfy-Muhoray, D. A. Dunmur, W. H. Miller, D. A. Balzarini, in Liquid Crystals and Ordered Fluids, A. C. Griffin, J. F. Johnson, Eds. (Plenum Press, 1984). 48. M. A. Bates, Phys. Rev. E 64, 051702 (2001) 49. M. A. Bates, Phys. Rev. E 65, 041706 (2002). 50. M. A. Bates, N. Halim, R. Hashim, G. R. Luckhurst, S. Romano, and S. M. Zain (Unpublished) 51. G. R. Luckhurst, P. Simpson, C. Zannoni, Chem. Phys. Lett. 78, 429 (1981). 52. E. Berggren, C. Zannoni, C. Chiccoli, P. Pasini, F. Semeria, Int. J. Mod. Phys. C 6, 135 (1995). 53. G. Skacej, C. Zannoni, Phys. Rev. Lett. 100, (May 16, 2008). 54. M. P. Allen, G. T. Evans, D. Frenkel, B. M. Mulder, Adv. Chem. Phys. 86, 1 (1993). 55. A. J. Stone, in The Molecular Physics of Liquid Crystals, G. R. Luckhurst, G. W. Gray, Eds. (Academic Press, 1979), p. 31-49. 56. J. Corner, Proc. R. Soc. Lond. A 192, 275 (1948). 57. B. J. Berne, P. Pechukas, J. Chem. Phys. 56, 4213 (1972). 58. J. G. Gay, B. J. Berne, J. Chem. Phys. 74, 3316 (1981). 59. G. R. Luckhurst, Liquid Crystals 32, 1335 (2005). 60. M. A. Bates, G. R. Luckhurst, J. Chem. Phys. 110, 7087 (1999). 61. C. Zannoni, J.Mater. Chem 11, 2637 (2001). 62. (a) M. K. Chalam, K. E. Gubbins, E. de Miguel, L. F. Rull, Molecular Simulation 7, 357 (1991) (b) E. de Miguel, L. F. Rull, M. K. Chalam, K. E. Gubbins, Molec. Phys.74, 405 (1991). 63. R. Berardi, A. P. J. Emerson, C. Zannoni, J. Chem. Soc., Faraday Trans 89, 4069 (1993). 64. M. P. Allen, M. A. Warren, M. R. Wilson, A. Sauron and W. Smith, J. Chem. Phys. 105, 2850 (1996). 65. J. L. Billeter, A. M. Smondyrev, G. B. Loriot, R. A. Pelcovits, Phys. Rev. E 60, 6831 (1999). 66. J. T. Brown, M. P. Allen, E. Mart´in del R´io, E. D. Miguel, Phys. Rev. E 57, 6685 (1998) and (b) J. T. Brown, M. P. Allen and M. Warren, J. Phys. Condens. Matter 8, 9433 (1996). 67. (a) L. Bennett, S. Hess, Phys. Rev. E 60, 5561 (1999) (b) S. Hess, in Advances in the computer simulations of liquid crystals, P. Pasini and C. Zannoni, Eds. (Kluwer, Dordrecht, 2000) and (c) S. Cozzini, L. F. Rull, G. Ciccotti, G. V. Paolini, Physica A 240, 173 (1997).
September 29, 2010
336
10:46
World Scientific Review Volume - 9in x 6in
R. Hashim
68. J. Stelzer, L. Longa, H.-R. Trebin, J. Chem. Phys. 103, 3098 (1995). 69. S. Sarman, D. J. Evans, J. Chem. Phys. 99, 620 (1993). 70. A. P. J. Emerson, R. Hashim, G. R. Luckhurst, Molec. Phys. 76, 241 (1992). 71. M. P. Allen, M. A. Warren, Phys. Rev. Lett. 78, 1291 (1997). 72. A. M. Smondyrev, G. B. Loriot, R. A. Pelcovits, Phys. Rev. Lett. 75, 2340 (1995). 73. J. Stelzer, H.-R. Trebin, L. Longa, J. Chem. Phys. 107, 1295 (1997). 74. G. R. Luckhurst, R. A. Stephens, R. W. Phippen, Liquid Crystals 8, 451 (1990). 75. E. de Miguel, E. M. del Ro, J. T. Brown, M. Allen, P., J. Chem. Phys. 105, 4234 (1996). 76. A. P. J. Emerson, G. R. Luckhurst, S. G. Whatling, Molec. Phys. 82, 113 (1994) 77. M. A. Bates, G. R. Luckhurst, J. Chem. Phys. 104, 6696 (1996). 78. R. Berardi, C. Zannoni, J. Chem. Phys. 113, 5971 (2000). 79. R. Berardi, S. Orlandi, C. Zannoni, PCCP 2, 2933 (2000). 80. C. Bacchiocchi, C. Zannoni, Phys. Rev. E 58, 3237 (1998). 81. M. A. Bates, G. R. Luckhusrt, Liquid Crystals 24, 229 (1998). 82. K. Satoh, S. Mita, S. Kondo, Chem. Phys. Lett. 255, 99 (1996). 83. M. P. Neal, A. J. Parker, Chem. Phys. Lett. 294, 277 (1998). 84. M. M. P. Neal, A. J. Parker, Mol. Cryst. Liq. Cryst. 330, 1809 (1999). 85. K. Praefcke, D. Singer, M. Langner, B. Kohne, M. Ebert, A. Liebmann, and J. H. Wendorff, Mol. Cryst. Liq. Cryst., 215, 121 (1992). 86. R. Berardi, L. Muccioli, C. Zannoni, J. Chem. Phys. 128, 024905 (2008). 87. D. Vorl¨ ander, Z. Phys. Chem. 126, 449, (1927). 88. (a) D. J. Adams, G. R. Luckhurst, R. W. Phippen, Molec. Phys.61, 1575 (1987) and (b) G. R. Luckhurst, R. A. Stephens, R. W. Phippen, Liquid Crystals 8, 451 (1990). 89. P. Bolhuis, D. Frenkel, J. Chem. Phys. 106, 666 (1997). 90. J. W. Emsley, G. R. Luckhurst, G. N. Shilstone, I. Sage, Mol. Cryst. Liq. Cryst. 102, 223 (1984). 91. H. Zewdie, J. Chem. Phys. 108, 2117 (1998). 92. L. Blum, A. J. Torruella, J. Chem. Phys. 56, 303 (1972). 93. J. Stelzer, R. Berardi, C. Zannoni, Chem. Phys. Lett. 299, 9 (1999). 94. J J. L. Billeter, R. A. Pelcovits, Liquid Crystals 9, 1151 (2000). 95. D. L. Cheung, S. J. Clark, M. R. Wilson, Phys. Rev. E 65, 051709 (2002). 96. W. L. Jorgensen, J. Tirado-Rives, Journal of the American Chemical Society 110, 1657 (1988). 97. R. J. Woods, M. Khalil, W. Pell, S. H. Moffat, and V. H. Smith, J. Comput. Chem. 11, 297 (1990). 98. S. S. Patnaik, S. J. Plimpton, R. Pachter, W. W. Adams, Liquid Crystals 19, 213 (1995). 99. S. J. Picken, W. F. van Gunsteren, P. T. van Duijnen and W. H. de Jeu, Poster Presentation, at the 10th International Liquid Crystal Conference York (1984).
11-chapter10
September 29, 2010
10:46
World Scientific Review Volume - 9in x 6in
Computer Modeling of Liquid Crystals
11-chapter10
337
100. S. J. Picken, W. F. van Gunsteren, P. T. van Duijnen and W. H. de Jeu, Liquid Crystals 6, 357 (1989). 101. S. Y. Yakovenko, A. A. Muravski, F. Eikelschulte, A. Geiger, Liquid Crystals 24, 657 (1998). 102. D. L. Cheung, S. J. Clark, and M. R. Wilson, J. Chem. Phys. 121, 9131 (2004). 103. Z. Q. Wang, J. A. Lupo, S. Patnaik, R. Pachter, Comput. Theor. Polym. Sci 11, 375 (2001) 104. M. R. Wilson, M. P. Allen, Mol. Cryst. Liq. Cryst. 198, 465 (1991). 105. M. R. Wilson, Structure and Bonding: Liquid Crystals. M. Mingos, Ed. (Springer-Verlag, Heidelberg, 1999) 106. R. Berardi, L. Muccioli, C. Zannoni, Chem. Phys. Chem. 5, 104 (Jan 23, 2004). 107. G. W. Gray, in The Molecular Physics of Liquid Crystals, G. R. Luckhurst and G. W. Gray, Eds. (Academic Press, 1979), pp. 1-28. 108. M. J. Cook, M. R. Wilson, Liquid Crystals 27, 1573 (2000). 109. N. Matsuhashi, M. Kimura, T. Akahane, M. Yoshida, Azojomo, Journal of Material Online 3, (2007). 110. N. Matsuhashi, Y. Okumoto, M. Kimura and T. Akahane, Japan. J. Appl. Phys. 44, 582 (2005). 111. G. Tiberio, L. Muccioli, R. Berardi and C. Zannoni Chem. Phys. Chem. 9, 1.13 (2008). 112. (a) M. R. Wilson, GBMOL: A replicated data molecular dynamics program to simulate the combinations of Gay-Berne and Lennard-Jones sites. (University of Durham, 1996) and (b) M. R. Wilson, J. M. Ilnytskyi, L. M. Stimson and Z. E. Hughes in Computer Simulations of liquid crystals and polymers, P. Pasini, C. Zannoni and S. Z˘ umer, Eds. (Kluwer 2004), pp. 57.
This page is intentionally left blank
September 29, 2010
10:47
World Scientific Review Volume - 9in x 6in
Chapter 11 Drop Dynamics in Complex Fluids
James J. Feng Department of Chemical and Biological Engineering and Department of Mathematics, University of British Columbia, Vancouver, BC V6T 1Z3, Canada E-mail: [email protected] Xiaopeng Chen School of Mechanics, Civil Engineering and Architecture, Northwestern Polytechnical University, Xi’an, Shaanxi, P. R. China 710072 Pengtao Yue Department of Mathematics, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061-0123, USA Chunfeng Zhou Department of Chemical Engineering and Materials Science, University of Minnesota, Minneapolis, MN 55455-0132, USA This chapter describes the unusual behavior of interfaces between complex fluids through two examples: the partial coalescence between a drop and a planar interface, and the interaction and self-assembly of droplets suspended in a nematic liquid crystal. The main message is that coupling among 3 disparate length scales — the microscopic scale of molecular and supramolecular configuration, the mesoscopic scale of the interfaces, and the macroscopic scale of hydrodynamics — produces interfacial dynamics that may differ markedly from that in Newtonian fluids. A diffuse-interface theory provides a convenient framework for describing two-phase complex fluids, and finite-element computations reproduce the main features of the experimental observations, and reveal the underlying physical mechanisms.
339
12-chapter11
September 29, 2010
10:47
340
World Scientific Review Volume - 9in x 6in
J. J. Feng et al.
1. Introduction The terms complex fluids and soft matter seem to both refer to materials with unusual mechanical behavior that are intermediate, in some sense, between ordinary fluids and solids. Examples range from common household materials such as tooth paste and hair gel to micro-engineered highperformance composite materials, from synthetic polymers to biological tissues and blood. From a mechanical viewpoint, these materials are distinguished by a microstructure that evolves as a result of deformation, which in return modifies the rheology of the material and thus its flow behavior. For example, polymer solutions exhibit viscoelasticity, a combination of viscosity and elasticity, and peculiar behavior such as rod-climbing.1 The elasticity stems from the polymer chains — the microstructure for polymer solutions — being stretched and oriented by flow and deformation. In technological applications, complex fluids are often used not in the pure form, but in mixtures with another material. An example is polymer blends made of two immiscible polymers which possess, ideally, desirable properties of each of the components. Other examples of complex fluid mixtures include thermoplastic foam and oil-water emulsions. The overall rheology of such materials depends critically on the shape, size and orientation of the interfaces between the components. With the advent of micro-engineering and nano-technology, the characteristic length scale of the processing flow becomes ever smaller. As a result, the interfaces exert ever increasing influences on the overall properties of the mixture relative to the bulk properties. This has motivated much of the recent research on interfacial dynamics in complex fluids. From a more fundamental point of view, such problems are intriguing because the interfacial behavior of complex fluids can differ markedly from that of Newtonian fluids. As an example, consider a thin filament of Newtonian fluid such as water suspended in an oil bath or in air. Because of surface tension, the thread will quickly break up into droplets of more or less uniform radius. The process, known as the Rayleigh instability,2 starts with the appearance of capillary waves on the thread, which quickly grow in amplitude and cause pinch-off at the nodes of the waveform. If the filament is a viscoelastic polymer solution, the appearance of capillary waves is followed by thinning of the thread but not rapid breakup. Instead, smaller beads form between the larger ones as the thread continues to thin. In the end, several generations of beads coexist in a spectacular beads-on-a-string morphology (Fig. 1), which persists for a long time before breakup.3 The
12-chapter11
September 29, 2010
10:47
World Scientific Review Volume - 9in x 6in
Drop Dynamics in Complex Fluids
12-chapter11
341
Fig. 1. Evolution of a thread of polymer solution (0.2% of poly(ethylene oxide) in ethylene glycol and water). The images of the thread represent advance in time from left to right, with neighboring images 25 ms apart. The initial filament diameter is c approximately 40 µm. Adapted from Oliveira and McKinley3 with permission, 2005 American Institute of Physics.
longevity of the thread is due to the long polymer chains in the solution. The thinning of the thread stretches and aligns the chains, which drastically increases the filament’s elongational viscosity, namely its resistance to further thinning and breakup. As a second example, consider a drop of nematic liquid crystal suspended in an isotropic medium with matching density.4 If the drop was an isotropic liquid such as water, naturally it would assume a spherical shape so as to minimize its surface area. The liquid crystal is made of elongated molecules that, in the nematic state, are more or less aligned with each other. The average orientation at any spatial point can be indicated by a unit vector called the director. Furthermore, they prefer to orient in a specific angle relative to the drop surface, a tendency commonly called anchoring. For the materials in this example, the anchoring is tangential to the interface. The drop then takes on an elongated lemon shape (Fig. 2). This is the result of minimizing the system’s free energy with respect to competing mechanisms. The spherical shape has minimum surface area. But if the anchoring is respected on the drop surface, the interior of the drop will have severe distortions of molecular orientation. The lemon shape emerges as the minimizer for the sum of the interfacial energy, anchoring energy and internal distortional energy. Incidentally, the same argument explains similar lemon shapes if the drop itself is isotropic but the surrounding medium is nematic.4,5
September 29, 2010
342
10:47
World Scientific Review Volume - 9in x 6in
J. J. Feng et al.
Fig. 2. Non-spherical shape of a liquid crystal drop in an isotropic medium. The image on the right is the micrograph from a polarizing microscope while the one on the left is a sketch of the director orientation field. The length of the drop is about 40 µm. Adapted c from Nastishin et al.4 with permission, 2005 The American Physical Society.
These relatively simple examples highlight the fact that when complex fluids are in contact, their interface often behave anomalously. Ultimately this is because the microstructures in the two bulk phases affect the interfacial behavior. In a broader context, the dynamics of a two-phase complex fluid mixture depends on the coupling among 3 length scales: (i) microstructural configuration, such as the conformation of the polymer chains and the molecular orientation of a liquid crystal; (ii) interfacial morphology; (iii) macroscopic hydrodynamic flow. The coupling between (i) and (iii) is the main subject matter of theoretical rheology.6 For example, a flow tends to stretch and align the polymer chains. This microstructural change in turn modifies the rheology of the fluid and its resistance to the deformation. Hence the microstructure modifies the flow in return. Similarly, droplet size distribution in an oil-in-water emulsion determines its flow behavior, e.g. viscosity. But shearing of the emulsion as a whole changes the drop size distribution via breakup and coalescence. This features the coupling between (ii) and (iii), which has been studied extensively for Newtonian fluids in two-phase fluid dynamics. This chapter introduces the reader to the fascinating dynamics of multiphase complex fluids that involve all three length scales. Given the wide range of complex fluids and flow situations, many physical mechanisms may be important.7 In this chapter, we shall focus on drop dynamics in two types of two-phase complex fluids: one involving a Newtonian fluid and a viscoelastic polymeric liquid, and the other involving a nematic liquid crystal and a Newtonian fluid. For the former, we will analyze an intriguing phenomenon known as partial coalescence, while
12-chapter11
September 29, 2010
10:47
World Scientific Review Volume - 9in x 6in
Drop Dynamics in Complex Fluids
12-chapter11
343
Fig. 3. A cycle in the partial coalescence cascade for an ethanol drop in air. The initial drop diameter is roughly 1.5 mm, and the frames are taken 0, 0.7, 1.8, 3.5, 5 and 50 ms after the initial contact. The grid on the drop comes from background lighting. From c Thoroddsen9 with permission, 2006 Nature Publishing Group.
for the latter, we will investigate the self-assembly of Newtonian droplets suspended in a nematic liquid crystal. Obviously, this is not a comprehensive review that covers all known facets of drop dynamics in complex fluids. Rather, our objectives are (i) to give the reader a flavor of the unusual interfacial dynamics that may be encountered in complex fluids, and (ii) to introduce a theoretical model and numerical methods for solving such problems. 2. Partial Coalescence in Polymer Solutions 2.1. Experimental observations Fill a beaker with a liquid, which naturally forms a flat, stationary interface with air. Now gently deposit a drop of the same liquid on the interface. A great deal of surface energy can be saved if the drop coalesces with its homophase below. But to make contact with the liquid below, the drop must first press and squeeze out the cushion of air beneath it. Depending on the drop size and the density of the liquid, the film drainage can take an appreciable length of time.8 Eventually the air layer gets thin enough that van der Waals forces cause it to rupture, and the drop starts to coalesce with the liquid layer below.
September 29, 2010
10:47
World Scientific Review Volume - 9in x 6in
344
12-chapter11
J. J. Feng et al.
0.8 viscous 0.6
inertiocapillary
gravity
ζ
0.4
0.2
0 10-4
10-3
10-2
10-1
100
101
Bo Fig. 4. The daughter-to-mother drop size ratio ζ as a function of the mother drop size, represented by the Bond number Bo = ∆ρgD2 /σ, ∆ρ being the density difference, g the gravitational acceleration, D the mother drop diameter, and σ the interfacial tension. ζ = 0 signifies complete coalescence. Different symbols denote different fluid pairs, and for each pair, ζ is taken through the steps of the partial coalescence cascade. Adapted c American Institute of Physics. from Chen et al.8 with permission, 2006
Under favorable conditions, the coalescence is not completed at once. Instead, a smaller daughter drop is left on the interface (Fig. 3). It then repeats the film drainage process until coalescence, which may leave a still smaller daughter drop on the interface. Such a cascade of partial coalescence has been documented with the aid of high-speed video by a number of groups, both for air-liquid systems10–12 and for liquid-liquid systems.8,13–17 It has been further discovered that the partial coalescence occurs only for an intermediate range of drop sizes. Drops too large or too small will merge entirely with the underlying liquid in one shot (Fig. 4). Chen et al.8 explored the same process when the drop (and the underlying liquid) is a viscoelastic polymer solution. The non-Newtonian rheology, as it turns out, tends to suppress partial coalescence; under conditions that would have led to partial coalescence for Newtonian fluids, now the coalescence is completed at once (Fig. 5). Furthermore, if the fluid surrounding the drop is viscoelastic, partial coalescence also tends to be suppressed, although the effect is weaker than if the drop phase is polymeric.
September 29, 2010
10:47
World Scientific Review Volume - 9in x 6in
Drop Dynamics in Complex Fluids
12-chapter11
345
Fig. 5. Suppression of partial coalescence by viscoelasticity. The drop fluid is a 0.18% solution of poly(ethylene oxide) in water, and the surrounding liquid is decane. The initial drop diameter D = 1.8 mm, and the numbers in the frames indicate the advance in time scaled by the capillary time tc = (ρD3 /σ)1/2 . From Chen et al.8 with permission, c 2006 American Institute of Physics.
This effect can be understood by contrasting the Newtonian and viscoelastic scenarios depicted in Figs. 3 and 5. The initial rupture of the cushioning fluid sends a capillary wave up the drop. Simultaneously, a capillary wave also ripples out on the interface but this is not directly related to our argument. In time, the drop turns into an elongated column; see the fourth frame of Fig. 3 and the third frame of Fig. 5. This process is driven by surface energy being turned first into kinetic energy in the wave motion, and then into gravitational potential energy of the column. Up to this point, the Newtonian and viscoelastic behaviors are qualitatively the same. But a viscous liquid column longer than its circumference is subject to the Rayleigh instability as mentioned in the Introduction. Thus, the column in Fig. 3 forms a neck that pinches off in time, producing the daughter drop and partial coalescence. For the viscoelastic column of Fig. 5, on the other hand, the same stabilizing effect as evident in the beads-on-a-string formation of Fig. 1 comes into play. A neck forms and turns into a thin filament, as indicated in Frame 5 of Fig. 5, which persists without breaking up. In time the drop fluid drains down the filament and the shrinking
September 29, 2010
10:47
World Scientific Review Volume - 9in x 6in
346
12-chapter11
J. J. Feng et al.
drop falls and merges into the underlying polymer solution. To explain the inverted case of a Newtonian drop in a polymeric upper liquid, Chen et al.8 observed the breakup of a Newtonian filament in a polymer solution, and found that it is subject to the same type of stabilization with the formation of beads on the thinning thread. To understand why partial coalescence appears only for an intermediate range of drop sizes, one needs to examine the time scales of two competing mechanisms. If it takes longer to drain the drop through the neck than for the neck to pinch off, the latter occurs and leaves a daughter drop behind. Conversely, if the drop drains into the lower layer before the neck pinches off, complete coalescence results. For drops that are too large, gravity dominates and the drop practically collapses into the bottom layer in one shot. For drops that are too small, viscosity dominates and slows down the thinning of the neck so complete merging takes place. In the intermediate range, a balance between capillarity and inertia produces a self-similar regime in which the drop diameter shrinks approximately by one half through each cycle of the cascade.10,16 Chen et al.8 have documented how viscoelasticity in either component narrows down the range of drop sizes for partial coalescence. To reflect on an argument advanced in the Introduction, the suppression of partial coalescence by viscoelasticity provides an example that clearly demonstrates how bulk rheology qualitatively modifies the behavior of the interface. In this case, the rheology is manifested by “strain hardening”, namely a steep increase in elongational viscosity during straining. Its molecular origin is the stretching and alignment of polymer chains by the elongational flow. While the explanation given above sounds reasonable, it needs to be confirmed by a quantitative study of the fluid mechanical process. This will be accomplished by numerical computations in the following.
2.2. Numerical simulations Laminar flows of incompressible Newtonian fluids are governed by the Navier-Stokes equations: ρ
∇ · v = 0, ∂v + v · ∇v = −∇p + ∇ · τ , ∂t τ = µ[∇v + (∇v)T ],
(1) (2) (3)
September 29, 2010
10:47
World Scientific Review Volume - 9in x 6in
12-chapter11
Drop Dynamics in Complex Fluids
347
where v and p are the velocity and pressure in the fluid, ρ and µ are the density and viscosity. The stress tensor τ is related to the velocity gradient linearly for a Newtonian viscous fluid. Computational fluid dynamics is concerned mostly with solving this set of partial differential equations subject to proper boundary conditions. If the fluid is non-Newtonian, τ depends on the fluid deformation in a more complex way. For viscoelastic liquids, in particular, τ depends not only on the deformation at the current moment, but also the history of deformation. Hence comes the idea of “memory” and elasticity. For a systematic exposition on rheology and constitutive modeling, the reader may consult several monographs.1,6,7,18 For our purpose, it suffices to give the Giesekus equation as an example of viscoelastic constitutive equations: τ = τ p + τ s,
(4)
τ s = µs [∇v + (∇v) ], λH τ p + λH τ p(1) + α τ p · τ p = µp [∇v + (∇v)T ], µp T
(5) (6)
where the total stress tensor τ is the sum of a Newtonian solvent contribution τ s and a polymer stress τ p . In the partial differential equation governing τ p , the subscript (1) denotes the so-called upper-convected derivative: τ p(1) =
∂τ p + v · ∇τ p − (∇v)T · τ p − τ p · ∇v. ∂t
(7)
λH and α are respectively the relaxation time and mobility parameter of the model, and µs and µp are the solvent and polymer viscosities. The numerical solution of non-Newtonian flows is a great deal more complex than that of Newtonian flows, and interested readers are referred to Owens and Phillips.19 If one deals with flow of two immiscible components, the situation is also more complex because now the fluid domain contains internal boundaries that deform and move. The motion of the interfaces is most conveniently described in Lagrangian terms, while the Navier-Stokes equations are customarily solved in an Eulerian framework. The crux is in reconciling these two viewpoints. A conceptually straightforward treatment is to lay grid points on the internal boundaries, which will track the interfacial motion in every time step. The governing equations are solved for each component, with matching boundary conditions on the interface. This typically calls for a moving grid and periodic remeshing and interpolation.20 Besides the computational overhead, a limitation of these interface-tracking methods
September 29, 2010
348
10:47
World Scientific Review Volume - 9in x 6in
J. J. Feng et al.
is the appearance of singularities when interfaces merge and rupture. An alternative is the interface-capturing methods, which introduce an auxiliary scalar field to demarcate the two components and indicate the location of the interface.21 The scalar field evolves according to a postulated convection or convection-diffusion equation. Thus, the computational burden of tracking the interface is replaced by that of solving an additional evolution equation. The main attractions are (i) now the governing equations can be solved over the entire domain on an Eulerian grid, with no need to match the boundary conditions on the interface; (ii) topological changes of the interface, such as rupture, can be handled more naturally. The most widely used interface-tracking methods include the volume-of-fluid method, the level-set method and the diffuse-interface method. The computations to be described here are based on the diffuse-interface method.22 We imagine the two components, although nominally immiscible, nevertheless mix to a slight degree over a narrow interfacial region. We define a phase-field variable φ as a scaled “concentration” such that φ = −1 in fluid bulk A and φ = +1 in B, and the interface is traced out by the level curve of φ = 0. We further require the evolution of φ be governed by the Cahn-Hilliard equation:23,24 φ(φ2 − 1) ∂φ 2 2 , (8) + v · ∇φ = γλ∇ −∇ φ + ∂t 2 where γ is the Cahn-Hilliard mobility parameter, λ is the density of the mixing energy between the two components, and is the capillary width. This is a phenomenological model based on the ideas that the evolution of φ is governed by a mixing energy in the diffuse interface, and that the species diffuse across the interfaces as driven by gradients of the chemical potential.25 The equilibrium φ profile is a minimizer of the mixing energy, and has the characteristic sigmoidal shape sketched in Fig. 6. The interfacial tension can be shown to be related to the Cahn-Hilliard model parameters as √ 2 2λ σ= , (9) 3 and the interfacial thickness, say defined between φ = ±0.9, is roughly 5. Feng et al.22 have given a general review of the theoretical background of the model, as well as its computational implementation for simulating twophase flows. We only mention that for the problems at hand, this formalism enjoys the advantage that the moving interface and viscoelastic rheology can be handled in a unified framework based on the free energy.
12-chapter11
September 29, 2010
10:47
World Scientific Review Volume - 9in x 6in
12-chapter11
Drop Dynamics in Complex Fluids
349
φ +1
Bulk fluid B
x Diffuse interface
Bulk fluid A
Fig. 6.
-1
Sketch of the phase-field profile across the diffuse interface.
The numerical solution of the continuity and momentum equations (1), (2), the constitutive equations (4)–(6) and the Cahn-Hilliard equation (8) is a daunting task. Beside the complexity of the equations, the numerical challenge lies in resolving the thin interfacial profile adequately so the interfacial tension can be computed accurately. Over the past few years, we have developed a finite-element package AMPHI that uses finite-elements with a novel adaptive meshing scheme. The scope of this chapter does not allow a detailed discussion of the algorithm, for which we refer the reader to Yue et al.26 In the following, we focus on using this methodology to simulate the partial coalescence phenomenon in Newtonian and viscoelastic liquids. Axisymmetry is always assumed in the geometric setup of the simulation. As a validation of the diffuse-interface model and the numerical algorithm, we first tried to reproduce the partial coalescence experiment for Newtonian liquids, with a water drop above a decane-water interface.27 The calculation uses the true densities and viscosities of water and decane, and proper values for λ and to produce the correct interface tension σ. Figure 7 gives a frame-by-frame comparison between experiment and simulation for one cycle of partial coalescence. The experimental snapshots are separated by a fixed interval of 542 µs, and the time below each gives the time for the numerical picture that best matches the experimental snapshot. Therefore, following the initial rupture at t = 0, the numerical simulation accurately reproduces the progress of the coalescence, from the propagation
September 29, 2010
350
10:47
World Scientific Review Volume - 9in x 6in
J. J. Feng et al.
Fig. 7. Frame-by-frame comparison between observation and computation of one cycle of partial coalescence. Streamlines in the computational plot indicate the local flow field. c American Institute of Physics. From Yue et al.27 with permission, 2006
of the capillary wave up the drop (a–e) to the formation of a liquid column (e–g), and finally to the formation of a neck (g–i). The next 2 frames, however, covers the pinchoff of the neck that generates the daughter drop. This is when the numerical simulation fails to track the progression in the correct time. Instead, the pinchoff occurs some 40% faster numerically than in reality. After the daughter drop is formed (k), the simulation again captures the real event precisely.
12-chapter11
September 29, 2010
10:47
World Scientific Review Volume - 9in x 6in
Drop Dynamics in Complex Fluids
12-chapter11
351
Fig. 8. Suppression of partial coalescence by viscoelasticity. The drop and lower fluid is a viscoelastic Giesekus fluid, while the upper layer is a Newtonian fluid. From left to right, the snapshots are at dimensionless times t = 0.740, 0.837, 0.934, 1.04 and 1.19, c scaled by the capillary time tc . From Yue et al.27 with permission, 2006 American Institute of Physics.
The discrepancy for frames (j) and (k) highlights a fundamental limitation to the diffuse-interface method. In reality, interfaces between smallmolecule liquids are nanometers thick. If the drop itself is millimeters in diameter, there is a gap of 106 in length scales that needs to be bridged. For the current generation of computational power and algorithms, it is impossible to fully resolve the minute details within the interfacial profile, as a diffuse-interface method must, while covering the drop-scale hydrodynamics at the same time. Thus, the interfacial thickness in the simulations, given by , is typically much larger than the real values. Experience shows that once the ratio Cn = /D, known as the Cahn number, falls below a threshold on the order of 10−2 , the numerical result becomes independent of Cn in most situations.25,26 This is when the sharp interface limit is reached, a necessary condition for the model predictions to be physically meaningful. The pinchoff in Fig. 7 is an exception to this rule in that the physical length scale of interest, say the neck radius, tends to zero. Thus, the Cn = 5×10−3 used, sufficiently small up to frame (i), eventually becomes too large as the neck pinches off. Two thicker interfaces start to overlap and interact sooner than thinner ones, and so the numerical simulation proceeds faster than reality. Thus, the diffuse-interface model can be a powerful and in some sense unique tool for computing interfacial flows, but one has to be aware of its inherent limitations. That partial coalescence occurs only for a range of drop sizes, being suppressed by gravity and viscosity on either end of the range, has also been reproduced numerically.27 Turning now to the role of viscoelasticity in suppressing partial coalescence, we use the Giesekus constitutive equation (6), with model parameters fitted roughly to the rheology of the polymer solutions. Figure 8 shows the evolution of the interfaces for a polymer drop surrounded by a Newtonian oil. The neck forms but the thin thread persists without breaking, as seen experimentally in Fig. 5. Note the close proximity of the dimensionless times
September 29, 2010
352
10:47
World Scientific Review Volume - 9in x 6in
J. J. Feng et al.
Fig. 9. Flow and stress fields near the neck for the viscoelastic drop in the second frame of Fig. 8, t = 0.837. The left half shows contours of the vertical velocity scaled by D/tc . The right half shows contours of the polymer stress component τpyy scaled by σ/D. c From Yue et al.27 with permission, 2006 American Institute of Physics.
between the experiment and the simulation. To confirm the explanation proposed before, we plot the flow and polymer stress fields when the neck is at its thinnest (Fig. 9). It is evident that the strong polymer tensile stress, due to the strain-hardening rheology, resists continued stretching and thinning of the neck, and suppresses partial coalescence. A similar scenario occurs when the viscoelasticity occurs in the surrounding fluid.27 There the polymer stress is activated indirectly through the no-slip boundary condition on the interface; pinchoff of the neck requires stretching of the surrounding fluid as well, which is resisted by the polymer stress. 3. Droplet Self-Assembly in Nematic Liquid Crystals 3.1. Experimental observations More than a decade ago, Poulin and coworkers28 reported that water droplets suspended in a nematic liquid crystal (LC) organize themselves into a chain, with a more or less constant spacing between neighboring droplets. This was later confirmed by spectacular pictures of parallel chains that form by self-assembly of silicone oil droplets in a nematic medium29
12-chapter11
September 29, 2010
10:47
World Scientific Review Volume - 9in x 6in
Drop Dynamics in Complex Fluids
(a)
12-chapter11
353
(b)
Fig. 10. (a) Silicone oil droplets, about 2 µm in size, arrange themselves into roughly equally spaced lines along the direction of the nematic director (indicated by the black c arrow). Adapted from Loudet et al.29 with permission, 2000 Macmillan Magazines Ltd. (b) Chains of silica particles, 2.32 µm in diameter, are coaxed into forming a regular 2D crystal in a nematic LC. From left to right, the frames at advancing times show the interaction between the single particle and the chains, which is discussed later in the text. c by the American Association for Adapted from Musevic et al.30 with permission, 2006 the Advancement of Science.
(Fig. 10(a)). More recently, 2D colloidal crystals have been made via selfassembly of colloidal particles in a nematic LC30 (Fig. 10(b)). The key to such pattern formation is topological defects created by droplets or particles inserted into an otherwise uniformly oriented LC. Nematic LC consists of anisotropic — rod-like or disc-like — molecules that are more or less aligned with each other.31 This produces a local average orientation that is conventionally marked by a unit vector n called the director. There is also long-range order in nematic LC in that any spatial distortion of the n field, in terms of the gradient ∇n, is resisted by distortional elasticity. Besides this bulk elasticity, LC molecules also have a preferred anchoring direction on surfaces, the most common ones being homeotropic (perpendicular) and planar (tangential) anchoring. Thus, when a water droplet with homeotropic anchoring is inserted into an LC single crystal, the radial n field near the droplet comes into conflict with the uniform far field. This conflict is resolved by nucleating a topological defect. For homeotropic anchoring, two potential configurations are depicted in Fig. 11 with either a point defect or a line defect. These are aptly called the satellite point defect and the Saturn ring defect, respectively. Both numerical computations and direct observations have confirmed their existence, and their stability relative to each other has been investigated extensively.32–36 For planar anchoring, surface defects called boojums prevail,33 but these are not relevant to the self-assembly of interest here. Poulin and coworkers28,29 observed that water and silicone-oil droplets exhibit homeotropic anchoring on the interface, and that when droplets
September 29, 2010
10:47
World Scientific Review Volume - 9in x 6in
354
J. J. Feng et al.
(a)
(b)
Fig. 11. Sketch of the satellite point defect (a) and Saturn ring defect (b) near a droplet with homeotropic anchoring. The solid lines trace out the director orientation.
line up into regular chains, each is always accompanied by a satellite point defect. They proposed an explanation for the self-assembly based on an analogy to electric dipoles. The droplet and its satellite point defect form a dipole, say pointing from the drop center to the defect. When two droplets are nearby, they interact through the equivalent of dipole-dipole attraction in electrostatics. Of course, the physical origin of the attraction is different here: when two droplets have their satellite defects on their line of centers and on the same side, elastic distortional energy is reduced when the two move toward each other until an optimal center-to-center separation of roughly 2.6a, a being the drop radius.28 Thus, droplets form a line with uniform spacing between neighbors along the direction of the background nematic director. The same analogy also explains the repulsion between two chains in parallel, with their dipoles pointing in the same direction. Thus, the parallel chains maintain a more or less equal distance from each other in Fig. 10(a). Furthermore, two chains with their dipolar directions opposite to each other (anti-parallel) should attract each other, and this explains the formation of the regular 2D arrays in Fig. 10(b) consisting of chains of alternating dipole directions. The top row of images in Fig. 10(b) shows a single droplet, with a dipole parallel to the nearby chain, being pushed away. In the bottom row, a single particle with a dipole anti-parallel to the nearby chain is attracted toward it. But the drop-defect pair can be seen as a dipole only when viewed from a large distance, i.e., when neighboring drops are far apart. When their separation is on the order of the drop diameter, the interaction is
12-chapter11
September 29, 2010
10:47
World Scientific Review Volume - 9in x 6in
Drop Dynamics in Complex Fluids
12-chapter11
355
necessarily more complex. To gain a more direct understanding of the physics underlying the self-assembly, we have carried out numerical computations to probe both the pairwise interaction between two droplets and the dynamic process of self-assembly. 3.2. Numerical simulations The simulations are based on the same diffuse-interface model explained above. However, the microstructure of the nematic LC, namely the molecular orientation represented by the director n, leads to a more complex mathematical model. The LC itself possesses a solid-liquid duality; it flows like a liquid but resists orientational distortion and transmits torque like an elastic solid. The former is characterized by 6 anisotropic Leslie viscosities and the latter by a linear Frank elasticity theory. The two are integrated in the Ericksen-Leslie theory for LC hydrodynamics,31 which will be used in our computations. Since we have a diffuse interface between the isotropic and nematic liquids, the anchoring condition requires a careful treatment. Besides, the defect normally presents a singularity in the n field, and is regularized here using a variable order parameter near the defect core. Details of the derivation can be found in the literature.5,25,37,38 For a mixture of a nematic (marked by φ = 1) and a Newtonian viscous liquid (φ = −1), the governing equations are: ρ
∇ · v = 0, ∂v + v · ∇v = −∇p + ∇ · σ, ∂t φ(φ2 − 1) ∂φ , + v · ∇φ = γλ∇2 −∇2 φ + ∂t 2 h = γ1 N + γ2 D · n.
(10) (11) (12) (13)
Equation (13) governs the evolution of the n field subject to elastic and viscous torques. The molecular field vector h represents the elastic torque: 1+φ 1 + φ (n · n − 1)n h=K ∇· − g, (14) ∇n − 2 2 δ2 where K is the bulk elastic constant, δ is a small defect core size used for regularizing the defect, and g = A[(∇φ·∇φ)n−(n·∇φ)∇φ] for homeotropic anchoring, A being the anchoring energy. The time derivative of n is em− 12 [(∇v)T − ∇v] · n, which is the rotation bedded in the vector N = dn dt of n with respect to the background flow field. D = 12 [∇v + (∇v)T ] is the
September 29, 2010
10:47
356
World Scientific Review Volume - 9in x 6in
J. J. Feng et al.
strain rate tensor, and γ1 and γ2 are viscosity coefficients. The stress tensor 1+φ 1+φ (∇n) · (∇n)T − G + σ + (1 − φ)µD, (15) 2 2 with G = A[(n · n)∇φ − (n · ∇φ)n]∇φ for homeotropic anchoring, and µ being the viscosity of the Newtonian component. σ is the Leslie viscous stress in the nematic phase39 σ = −λ∇φ∇φ − K
σ = α1 D : nnnn + α2 nN + α3 N n + α4 D + α5 nn · D + α6 D · nn, (16) where α1 to α6 are the Leslie viscous coefficients observing the Onsager relationship:31 α2 + α3 = α6 − α5 . The coefficients γ1 = α3 − α2 and γ2 = α3 + α2 . The numerical algorithm is essentially the same as used in Sec. 2.2, except that now the point defect, as well as the interface, requires locally refined grid for accurate resolution of the sharp gradients n. For details see Yue et al.26 and Zhou et al.38 In all the simulations to be discussed, we use a relatively large interfacial tension so that the drops remain essentially spherical in all times. This is true in the experimental observations, and also simplifies the analysis of the results by excluding the effect of drop deformation, which was essentially absent in the experimental systems. For an example of LC anchoring coupling with drop deformation, see Zhou et al.5 First, we place two droplets in the so-called parallel configuration, with their “dipoles” in the same direction and along their line of centers. The computational domain is axisymmetric in this case. Under the effect of distortional elasticity, the droplets start to move toward each other (Fig. 12(a)). From this motion, one can estimate the elastic driving force on each droplet from the Stokes drag by neglecting inertia and assuming that the drop is in force balance during the approach. This estimate turns out to be slightly below the true force computed from integrating the stress over the interface. Both forces are compared with prior theoretical and experimental results in Fig. 12(b). The long-range attraction manifests a R−4 scaling, as is expected from the attraction between two electric dipoles given by Lubensky et al.33 As the separation decreases, however, the attraction force decreases sharply, falling below the dipole formula, and approaches zero toward R = 2.45a. If the two droplets were initially placed closer than the equilibrium separation, they separate until R = 2.45a. The two experimental data sets indicate that the F ∝ R−4 power-law persists to smaller separations. But the equilibrium separation corresponding to F = 0 is in close agreement between computation and observations.
12-chapter11
September 29, 2010
10:47
World Scientific Review Volume - 9in x 6in
Drop Dynamics in Complex Fluids
(a)
12-chapter11
357
(b)
Fig. 12. Attraction between two droplets bearing satellite point defects placed in the parallel configuration. (a) The distance between the centers of the droplets decreasing as the two droplets attract each other. The two insets show the birefringence patterns around the defects computed from the director field. The point defect sits at the tip of the two bright lobes. (b) The attraction force as a function of the droplets’ separation, compared with the dipole formula33 and experimental data.40,41 Note that both axes are in logarithmic scale and the long-range attraction exhibits a F ∼ R−4 power law. c Adapted from Zhou et al.38 with permission, 2008 by the American Chemical Society.
The dynamic simulation of droplet interaction thus confirms the idea that the long-range attraction between droplets in the parallel configuration resembles dipolar attraction. For smaller separations, the idea of dipoledipole attraction no longer applies, and it certainly cannot account for the equilibrium separation and the repulsion between droplets that are too close to each other. This is where the dynamic computation provides results and insight that cannot come from the heuristic argument. Going beyond pairwise attraction, Zhou et al.38 have confirmed that a group of droplets in a 2D domain indeed form a chain along the undisturbed nematic director. To probe sidewise interactions between chains of droplets, we arrange two chains in initial configurations with their “dipoles” either in the same direction (parallel) or reversed (anti-parallel). These computations are done in a 2D planar domain. Figure 13 demonstrates that the parallel chains repel each other, while the anti-parallel chains attract each other. These trends are in qualitative agreement with the experimental observations in Fig. 10(b). The anti-parallel chains eventually approach an equilibrium separation of 2.44a, which is close to the observed value of 2.31a by Musevic et al.30 despite the two-dimensionality of the computations.
September 29, 2010
10:47
World Scientific Review Volume - 9in x 6in
358
J. J. Feng et al.
(a)
(b)
Fig. 13. Lateral interaction between chains of droplets. (a) The chains in the parallel configuration, with their dipoles in the same direction, repel each other. (b) The chains in the anti-parallel configuration, with their dipoles in the opposite direction, attract each other. The grayscale contours are for (n2x − 1/2)2 , nx being the horizontal component of the director, such that white indicates a vertical or horizontal n and black means a c by the American Chemical 45◦ tilt. Adapted from Zhou et al.38 with permission, 2008 Society.
The interaction can be understood from the director patterns. For the parallel configuration sketched in Fig. 14(a), the director n is forced to make a sharp upward turn between the chains. This is reflected in Fig. 13(a) by the narrow vertical dark and bright bands between the chains. The closer the two chains, the more severe the spatial distortion in between and the greater the elastic energy. Thus, the two chains repel each other. Note, however, the unexpected break of symmetry in the dynamic simulation of Fig. 13(a). The bottom droplet on the right chain has rotated its dipole clockwise so as to point its satellite defect toward the bottom droplet of the left chain. This created an attraction similar to Fig. 12 and pulls the two bottom droplets toward each other. In the anti-parallel configuration of Fig. 14(b), the director field merges smoothly between the two chains, with n assuming a relatively uniform, horizontal orientation. This n field has lower distortional energy than if the chains are far apart, which explains the attraction between the chains. If they are too close together, however, the radial n field surrounding each droplet comes into direct conflict, resulting in greater distortion and higher energy. The minimumenergy state is the equilibrium with a separation of 2.44a between the chains.
12-chapter11
September 29, 2010
10:47
World Scientific Review Volume - 9in x 6in
Drop Dynamics in Complex Fluids
(a)
12-chapter11
359
(b)
Fig. 14. Sketch of the director field around two chains of droplets. (a) The parallel configuration with the same dipolar orientation for the chains. (b) The anti-parallel configuration with opposite dipolar orientations.
Finally, we place 8 identical droplets in random positions in a doubly periodic 2D domain, and observe their self-assembly (Fig. 15). The dropdrop interaction is dominated by longitudinal dipole-like attractions (when nearby droplets have their satellite defects on their line of centers) and sideways repulsions (when their dipoles are in the parallel configuration). For example, drops 3, 5 and 7 initially move away from each other sideways. Then drop 5 is attracted by drop 6, and is eventually pulled in between drops 3 and 6 to form a diagonal chain. Drops 1, 4 and 2 seem to form a vertical chain by themselves. Since the domain is doubly periodic, there is no prescribed background nematic orientation to which a chain may align. Thus, the two chains spontaneously assume different angles. Conceivably, they will eventually line up into a single long chain, although our domain would not be large enough to accommodate that. Despite the small number of droplets in the simulation, the dynamic scenario of self-assembly exhibits the main features observed in reality (cf. Fig. 10(a)), and confirms that pairwise attraction (longitudinal) and repulsion (lateral) are the dominant mechanisms at play.
September 29, 2010
10:47
World Scientific Review Volume - 9in x 6in
360
J. J. Feng et al.
(a) t= 6.38
(b) t= 17.2
(c) t= 33.1
(d) t= 547.8
Fig. 15. Self-assembly of 8 drops in a doubly periodic domain. Time is made dimensionc less by ηa2 /K. Adapted from Zhou et al.38 with permission, 2008 by the American Chemical Society.
4. Summary To recapitulate, we have described two intriguing phenomena in drop dynamics involving polymer solutions and nematic liquid crystals, and demonstrated how numerical computation based on a diffuse-interface model can provide explanations to the experimental observations. The main insights from this chapter are: (1) The interface between complex fluids may exhibit behavior qualitatively different from that for Newtonian fluids. (2) The cause of such behavior should be sought from the coupling among microstructural conformation, interfacial morphology and large-scale fluid flow.
12-chapter11
September 29, 2010
10:47
World Scientific Review Volume - 9in x 6in
Drop Dynamics in Complex Fluids
12-chapter11
361
(3) Numerical computation is a powerful tool that can be used to gain an understanding of the physical mechanisms that would otherwise be difficult to obtain. The above insights, although derived from two specific examples, is relevant to all two-component complex fluids having internal boundaries. In the case of drop-interface partial coalescence, it is the strain-hardening rheology of the polymeric component, a direct manifestation of molecular alignment and stretching, that suppresses partial coalescence under suitable conditions. In the case of droplets self-assembling in a nematic suspending medium, it is the distortional elasticity in the bulk and anchoring on the interfaces, both originating from the molecular orientation, that drive the maneuver of the droplets relative to each other. As mentioned at the beginning, the purpose of the chapter is not to provide a comprehensive summary of the literature on drop dynamics in complex fluids. Instead, we use concrete examples to illustrate the rich dynamics of interfaces between complex fluids, and demonstrate a theoretical model and numerical technique that are well suited for this type of problems. We hope that the chapter has aroused the readers’ interest in such problems, and given them the basic ideas for formulating a research approach to similar systems in their own work. Acknowledgments We thank Siddharth Khullar, Chun Liu, Shreyas Mandre and Jie Shen for discussions. Financial support has come from the Petroleum Research Fund, the Canada Research Chair program, NSERC, CFI and NSFC (Grant Nos. 50390095, 20674051). Part of this work was presented at the Nanoscale Interfacial Phenomena in Complex Fluids program at the Kavli Institute for Theoretical Physics in Beijing, June 2008. References 1. R. B. Bird, R. C. Armstrong, and O. Hassager, Dynamics of Polymeric Liquids, Vol. 1. Fluid Mechanics. (Wiley, New York, 1987). 2. P. G. Drazin and W. H. Reid, Hydrodynamic Stability. (Cambridge University Press, New York, 1981). 3. M. S. N. Oliveira and G. H. McKinley, Iterated stretching and multiple beadson-a-string phenomena in dilute solutions of highly extensible flexible polymers, Phys. Fluids. 17, 071704, (2005).
September 29, 2010
362
10:47
World Scientific Review Volume - 9in x 6in
J. J. Feng et al.
4. Y. A. Nastishin, H. Liu, T. Schneider, V. Nazarenko, R. Vasyuta, S. V. Shiyanovskii, and O. D. Lavrentovich, Optical characterization of the nematic lyotropic chromonic liquid crystals: Light absorption, birefringence, and scalar order parameter, Phys. Rev. E. 72, 041711, (2005). 5. C. Zhou, P. Yue, J. J. Feng, C. Liu, and J. Shen, Heart-shaped bubbles rising in anisotropic liquids, Phys. Fluids. 19, 041703, (2007). 6. R. B. Bird, C. F. Curtiss, R. C. Armstrong, and O. Hassager, Dynamics of Polymeric Liquids, Vol. 2. Kinetic Theory. (Wiley, New York, 1987). 7. R. G. Larson, The Structure and Rheology of Complex Fluids. (Oxford, New York, 1999). 8. X. Chen, S. Mandre, and J. J. Feng, An experimental study of the coalescence between a drop and an interface in newtonian and polymeric liquids, Phys. Fluids. 18, 092103, (2006). 9. S. T. Thoroddsen, Droplet genealogy, Nature Phys. 2, 223–224, (2006). 10. S. T. Thoroddsen and K. Takehara, The coalescence cascade of a drop, Phys. Fluids. 12(6), 1265–1267, (2000). 11. E. M. Honey and H. P. Kavehpour, Astonishing life of a coalescing drop on a free surface, Phys. Rev. E. 73, 027301, (2006). 12. F. Blanchette and T. P. Bigioni, Partial coalescence of drops at liquid interfaces, Nature Phys. 2, 254–257, (2006). 13. G. E. Charles and S. G. Mason, The mechanism of partial coalescence of liquid drops at liquid/liquid interfaces, J. Colloid Sci. 15, 105–122, (1960). 14. Z. Mohamed-Kassim and E. K. Longmire, Drop coalescence through a liquid/liquid interface, Phys. Fluids. 16, 2170–2181, (2004). 15. A. D. Nikolov and D. T. Wasan, Effects of surfactant on multiple stepwise coalescence of single drops at liquid-liquid interfaces, Ind. Eng. Chem. Res. 34, 3653–3661, (1995). 16. X. Chen, S. Mandre, and J. Feng, Partial coalescence between a drop and a liquid-liquid interface, Phys. Fluids. 18, 051705, (2006). 17. T. Gilet, K. Mulleners, J. P. Lecomte, N. Vandewalle, and S. Dorbolo, Critical parameters for the partial coalescence of a droplet, Phys. Rev. E. 75, 036303, (2007). 18. R. I. Tanner, Engineering Rheology. (Oxford, New York, 2000). 19. R. G. Owens and T. N. Phillips, Computational Rheology. (Imperial College Press, London, 2002). 20. H. H. Hu, N. A. Patankar, and M. Y. Zhu, Direct numerical simulations of fluid-solid systems using the arbitrary Lagrangian-Eulerian technique, J. Comput. Phys. 169, 427–462, (2001). 21. J. A. Sethian and P. Smereka, Level set methods for fluid interfaces, Ann. Rev. Fluid Mech. 35, 341–372, (2003). 22. J. J. Feng, C. Liu, J. Shen, and P. Yue. An energetic variational formulation with phase field methods for interfacial dynamics of complex fluids: advantages and challenges. In eds. M.-C. T. Calderer and E. M. Terentjev, Modeling of Soft Matter, pp. pp. 1–26. Springer, New York, (2005). 23. J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system. I. interfacial free energy, J. Chem. Phys. 28, 258–267, (1958).
12-chapter11
September 29, 2010
10:47
World Scientific Review Volume - 9in x 6in
Drop Dynamics in Complex Fluids
12-chapter11
363
24. J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system. III. nucleation in a two-component incompressible fluid, J. Chem. Phys. 31, 688–699, (1959). 25. P. Yue, J. J. Feng, C. Liu, and J. Shen, A diffuse-interface method for simulating two-phase flows of complex fluids, J. Fluid Mech. 515, 293–317, (2004). 26. (a) P. Yue, C. Zhou, J. J. Feng, C. F. Ollivier-Gooch, and H. H. Hu, Phasefield simulations of interfacial dynamics in viscoelastic fluids using finite elements with adaptive meshing, J. Comput. Phys. 219, 47–67, (2006); (b) C. Zhou, P. Yue, J. J. Feng, C. F. Ollivier-Gooch, and H. H. Hu, 3D phase-field simulations of interfacial dynamics in Newtonian and viscoelastic fluids, J. Comput. Phys. 229, 498–511, (2010). 27. P. Yue, C. Zhou, and J. J. Feng, A computational study of the coalescence between a drop and an interface in Newtonian and viscoelastic fluids, Phys. Fluids 18, 102102, (2006). 28. P. Poulin, H. Stark, T. C. Lubensky, and D. A. Weitz, Novel colloidal interactions in anisotropic fluids, Science 275, 1770–1773, (1997). 29. J. C. Loudet, P. Barois, and P. Poulin, Colloidal ordering from phase separation in a liquid crystalline continuous phase, Nature. 407, 611–613, (2000). 30. I. Musevic, M. Skarabot, U. Tkalec, M. Ravnik, and S. Zumer, Twodimensional nematic colloidal crystals self-assembled by topological defects, Science. 313, 954–958, (2006). 31. P. G. de Gennes and J. Prost, The Physics of Liquid Crystals. (Oxford, New York, 1993). 32. E. M. Terentjev, Disclination loops, standing alone and around solid particles, in nematic liquid crystals, Phys. Rev. E. 51, 1330–1337, (1995). 33. T. C. Lubensky, D. Pettey, N. Currier, and H. Stark, Topological defects and interactions in nematic emulsions, Phys. Rev. E. 57, 610–625, (1998). 34. J. J. Feng and C. Zhou, Orientational defects near colloidal particles in a nematic liquid crystal, J. Colloid Interface Sci. 269, 72–78, (2004). 35. Y. Gu and N. L. Abbott, Observation of Saturn-ring defects around solid microspheres in nematic liquid crystals, Phys. Rev. Lett. 85, 4719–4722, (2000). 36. S. Khullar, C. Zhou, and J. J. Feng, Dynamic evolution of topological defects around drops and bubbles rising in a nematic liquid crystal, Phys. Rev. Lett. 99, 237802, (2007). 37. C. Zhou, P. Yue, and J. J. Feng, The rise of Newtonian drops in a nematic liquid crystal, J. Fluid Mech. 593, 385–404, (2007). 38. C. Zhou, P. Yue, and J. J. Feng, Dynamic simulation of droplet interaction and self-assembly in a nematic liquid crystal, Langmuir. 24, 3099–3110, (2008). 39. F. M. Leslie, Some constitutive equations for liquid crystals, Arch. Rational Mech. Anal. 28, 265–283, (1968). 40. P. Poulin, V. Cabuil, and D. A. Weitz, Direct measurement of colloidal forces in an anisotropic solvent, Phys. Rev. Lett. 79, 4862–4865, (1997). 41. C. M. No¨el, G. Bossis, A.-M. Chaze, F. Giulieri, and S. Lacis, Measurement of elastic forces between iron colloidal particles in a nematic liquid crystal, Phys. Rev. Lett. 96, 217801, (2006).
This page is intentionally left blank
September 29, 2010
13:27
World Scientific Review Volume - 9in x 6in
Index
Active particles, 42 adiabatic approximation, 29 amphitropic, 299
developed mean-field lattice theory, 182 diffuse-interface method, 348 DiMarzio theory, 225 disorder coupling parameter, 289 DNA condensation, 213, 270 dressed counterions, 281 Dynamic SCF theory, 107 Dynamical density functional theory, 27
Barker and Watts procedure, 309 Bjerrum length, 208, 272 Blending polymers, 48 Block copolymers, 85 boojums prevail, 353 Brownian dynamics, 23 Cahn number, 351 Cahn-Hilliard equation, 348 Chain propagators, 92 chiral crystal, 156 chiral recognition, 166 cholesteric, 300 coarse-grained, 271 coarse-graining, 2, 50 colloidal dispersions, 268 colloidal suspensions, 2 Columnar phase, 37 columnar phases, 321 commensurable periodic images, 313 condensing agent, 209 conformational hysteresis, 219 constitutive equation, 110 CornerS model, 324 Coulomb’s law, 266
Edwards-Anderson replica ansatz, 287 electrostatic coupling parameter, 273 electrostatic interactions, 250 entropy-driven phase transitions, 4 Fick’s law, 108 Flory-Huggins parameter, 56 fragmentation-based methods, 240 Functional integral, 279 functional integral, 91, 92 fundamental measure theory, 21 Gaussian chains, 91 Gay-Berne (GB) potential model, 318 Gibbs ensemble simulation technique, 63 Giesekus equation, 347 Gouy-Chapman (GC) length, 272 gyroid phase, 117
DAPI, 210 Darcy’s law, 122 Debye-H¨ uckel (DH) screening parameter, 280 Density functional theory, 17
Hansen-Verlet rule of freezing, 12 Hard spheres, 14 harmonic AMBER-type force field, 326 365
13-index
September 29, 2010
13:27
World Scientific Review Volume - 9in x 6in
366
histone proteins, 208 Hydrodynamic interactions, 31 hydrogen bonding, 238 interface-capturing methods, 348 interface-tracking methods, 347 intramolecular segregation, 224 isotropic phase, 36 Langevin dynamics simulation, 221 lattice model polymer solutions, 182 Lauritzen-Hoffman (LH) theory, 134 Lebwohl–Lasher model, 308 Lennard-Jones system, 317 Lennard-Jones-system, 15 Like-charged surfaces, 276 Lindemann-criterion of melting, 11 Liquid crystals, 297 lyotropic phases, 299 macroions, 271 Maier-Saupe theory, 302 Maxwell construction, 285 Maxwell double tangent construction, 13 mean-field approximation, 93 mesophase, 298 molecular dynamics, 138 molecular mechanics, 238 molten-globule state, 199 monotectic triple point, 180 Monte Carlo (MC) simulations, 181 Monte-Carlo, 305 Navier-Stokes equation, 122 Navier-Stokes equations, 346 nematic phase, 36, 299 nematogen, 303 Non-local kinetic coefficients, 125 odd-even effect, 328 Onsager kinetic coefficients, 108 orientational order parameter, 301 Ornstein-Zernike equation, 32 Oseen-tensor, 31
Index
partial coalescence, 342 path integrals, 109 Penetrable interactions, 17 perforated lamellar phase, 120 periodic boundary condition, 306 persistence length, 208 phase diagrams for L-L demixing, 183 phase diagrams for liquid-solid coexistence, 183 Plasma, 15 Plastic crystal, 37 Poisson-Boltzmann (PB) equation, 274 Polarizable force field model, 248 polarizable model, 249 polyelectrolytes, 270 primitive models, 271 protein folding, 199 protein-DNA complexation, 270 Q-tensor, 310 quantum mechanics (QM) methods, 238 Ramakrishnan-Yussouff (RY) approximation, 20 Randomly charged surfaces, 286 Rayleigh splitting, 227 Reciprocal-space method, 100 reciprocal-space method, 95 reptation dynamics, 127 rod-like particles, 33 Rotne-Prager-tensor, 31 satellite point defect, 353 Saturn ring defect, 353 self-consistent field theory, 87 self-consistent mean-field theory, 94 Self-propelled particles, 42 shape anisotropy parameter, 318 Single molecular images, 210 single-disulfide analogues, 251, 252 smectic phase, 299 Smectic-A phase, 37 Smoluchowski equation, 24
13-index
September 29, 2010
13:27
World Scientific Review Volume - 9in x 6in
Index
soft matter, 1, 9, 268 Soft spheres, 15 Sticky hard spheres, 16 Stokes-Einstein relation, 24 Strong coupling, 274 strong segregation theory, 88 system size optimization, 118 thermotropic, 298 toroidal segment, 217 total persistence length, 208
13-index
367
Ultrasoft interactions, 16 van ’t Hoff, 229 velocity gradient tensor, 124 weak coupling, 274 weak segregation theory, 87 wetting, 48, 53 Yukawa-system, 15 Yvon-Born-Green-hierarchy, 28
Series in Soft Condensed Matter Vol.
UNDERSTANDING SOFT CONDENSED MATTER VIA MODELING AND COMPUTATION
3
All living organisms consist of soft matter. For this reason alone, it is important to be able to understand and predict the structural and dynamical properties of soft materials such as polymers, surfactants, colloids, granular matter and liquids crystals. To achieve a better understanding of soft matter, three different approaches have to be integrated: experiment, theory and simulation. This book focuses on the third approach — but always in the context of the other two.
World Scientific www.worldscientific.com 7642 hc
7642.Alvin.CC.indd 1
ISBN-13 978-981-4295-58-1 ISBN-10 981-4295-58-2
,!7IJ8B4-cjffib!
Hu Shi
UNDERSTANDING SOFT CONDENSED MATTER VIA MODELING AND COMPUTATION
Series in Soft Condensed Matter Vol.
3
Editors
Wenbing Hu • An-Chang Shi
UNDERSTANDING SOFT CONDENSED MATTER VIA MODELING AND COMPUTATION World Scientific
11/24/10 2:03 PM
