EDITOR IN CHIEF Rudy J. M. Konings European Commission, Joint Research Centre, Institute for Transuranium Elements, Karlsruhe, Germany
SECTION EDITORS Todd R. Allen Department of Engineering Physics, University of Wisconsin, Madison, WI, USA Roger E. Stoller Materials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, TN, USA Shinsuke Yamanaka Division of Sustainable Energy and Environmental Engineering, Graduate School of Engineering, Osaka University, Osaka, Japan
Elsevier Radarweg 29, PO Box 211, 1000 AE Amsterdam, The Netherlands The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, UK 225 Wyman Street, Waltham, MA 02451, USA Copyright © 2012 Elsevier Ltd. All rights reserved The following articles are US Government works in the public domain and not subject to copyright: Radiation Effects in UO2 TRISO-Coated Particle Fuel Performance Composite Fuel (cermet, cercer) Metal Fuel-Cladding Interaction No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means electronic, mechanical, photocopying, recording or otherwise without the prior written permission of the publisher Permissions may be sought directly from Elsevier’s Science & Technology Rights Department in Oxford, UK: phone (þ44) (0) 1865 843830; fax (þ44) (0) 1865 853333; email:
[email protected]. Alternatively you can submit your request online by visiting the Elsevier web site at http://elsevier.com/locate/permissions, and selecting Obtaining permission to use Elsevier material Notice No responsibility is assumed by the publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein, Because of rapid advances in the medical sciences, in particular, independent verification of diagnoses and drug dosages should be made British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Library of Congress Catalog Number: 2011929343 ISBN (print): 978-0-08-056027-4 For information on all Elsevier publications visit our website at books.elsevier.com Cover image courtesy of Professor David Sedmidubsky´, The Institute of Chemical Technology, Prague Printed and bound in Spain 12 13 14 15 16 10 9 8 7 6 5 4 3 2 1
Editorial : Gemma Mattingley Production: Nicky Carter
EDITORS BIOGRAPHIES Rudy Konings is currently head of the Materials Research Unit in the Institute for Transuranium Elements (ITU) of the Joint Research Centre of the European Commission. His research interests are nuclear reactor fuels and actinide materials, with particular emphasis on high temperature chemistry and thermodynamics. Before joining ITU, he worked on nuclear fuel-related issues at ECN (the Energy Research Centre of the Netherlands) and NRG (Nuclear Research and Consultancy Group) in the Netherlands. Rudy is editor of Journal of Nuclear Materials and is professor at the Delft University of Technology (Netherlands), where he holds the chair of ‘Chemistry of the nuclear fuel cycle.’
Roger Stoller is currently a Distinguished Research Staff Member in the Materials Science and Technology Division of the Oak Ridge National Laboratory and serves as the ORNL Program Manager for Fusion Reactor Materials for ORNL. He joined ORNL in 1984 and is actively involved in research on the effects of radiation on structural materials and fuels for nuclear energy systems. His primary expertise is in the area of computational modeling and simulation. He has authored or coauthored more than 100 publications and reports on the effects of radiation on materials, as well as edited the proceedings of several international conferences.
Todd Allen is an Associate Professor in the Department of Engineering Physics at the University of Wisconsin – Madison since 2003. Todd’s research expertise is in the area of materials-related issues in nuclear reactors, specifically radiation damage and corrosion. He is also the Scientific Director for the Advanced Test Reactor National Scientific User Facility as well as the Director for the Center for Material Science of Nuclear Fuel at the Idaho National Laboratory, positions he holds in conjunction with his faculty position at the University of Wisconsin.
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Editors Biographies
Shinsuke Yamanaka is a professor in Division of Sustainable Energy and Environmental Engineering, Graduate School of Engineering, Osaka University since 1998. He has studied the thermophysics and thermochemistry of nuclear fuel and materials. His research for the hydrogen behavior in LWR fuel cladding is notable among his achievements and he received the Young Scientist Awards (1980) and the Best Paper Awards (2004) from Japan Atomic Energy Society. Shinsuke is the program officer of Japan Science and Technology Agency since 2005 and the visiting professor of Fukui University since 2009, and he is also the associate dean of Graduate School of Engineering, Osaka University since 2011.
PREFACE There are essentially three primary energy sources for the billions of people living on the earth’s surface: the sun, radioactivity, and gravitation. The sun, an enormous nuclear fusion reactor, has transmitted energy to the earth for billions of years, sustaining photosynthesis, which in turn produces wood and other combustible resources (biomass), and the fossil fuels like coal, oil, and natural gas. The sun also provides the energy that steers the climate, the atmospheric circulations, and thus ‘fuelling’ wind mills, and it is at the origin of photovoltaic processes used to produce electricity. Radioactive decay of primarily uranium and thorium heats the earth underneath us and is the origin of geothermal energy. Hot springs have been used as a source of energy from the early days of humanity, although it took until the twentieth century for the potential of radioactivity by fission to be discovered. Gravitation, a non-nuclear source, has been long used to generate energy, primarily in hydropower and tidal power applications. Although nuclear processes are thus omnipresent, nuclear technology is relatively young. But from the moment scientists unraveled the secrets of the atom and its nucleus during the twentieth century, aided by developments in quantum mechanics, and obtained a fundamental understanding of nuclear fission and fusion, humanity has considered these nuclear processes as sources of almost unlimited (peaceful) energy. The first fission reactor was designed and constructed by Enrico Fermi in 1942 in Chicago, the CP1, based on the fission of uranium by neutron capture. After World War II, a rapid exploration of fission technology took place in the United States and the Union of Soviet Socialist Republics, and after the Atoms for Peace speech by Eisenhower at the United Nations Congress in 1954, also in Europe and Japan. A variety of nuclear fission reactors were explored for electricity generation and with them the fuel cycle. Moreover, the possibility of controlled fusion reactions has gained interest as a technology for producing energy from one of the most abundant elements on earth, hydrogen. The environment to which materials in nuclear reactors are exposed is one of extremes with respect to temperature and radiation. Fuel pins for nuclear reactors operate at temperatures above 1000 C in the center of the pellets, in fast reactor oxide fuels even above 2000 C, whereas the effects of the radiation (neutrons, alpha particles, recoil atoms, fission fragments) continuously damage the material. The cladding of the fuel and the structural and functional materials in the fission reactor core also operate in a strong radiation field, often in a dynamic corrosive environment of the coolant at elevated temperatures. Materials in fusion reactors are exposed to the fusion plasma and the highly energetic particles escaping from it. Furthermore, in this technology, the reactor core structures operate at high temperatures. Materials science for nuclear systems has, therefore, been strongly focussed on the development of radiation tolerant materials that can operate in a wide range of temperatures and in different chemical environments such as aqueous solutions, liquid metals, molten salts, or gases. The lifetime of the plant components is critical in many respects and thus strongly affects the safety as well as the economics of the technologies. With the need for efficiency and competitiveness in modern society, there is a strong incentive to improve reactor components or to deploy advanced materials that are continuously developed for improved performance. There are many examples of excellent achievements in this respect. For example, with the increase of the burnup of the fuel for fission reactors, motivated by improved economics and a more efficient use of resources, the Zircaloy cladding (a Zr–Sn alloy) of the fuel pins showed increased susceptibility to coolant corrosion, but within a relatively short period, a different zirconium-based alloy was developed, tested, qualified, and employed, which allowed reliable operation in the high burnup range.
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Nuclear technologies also produce waste. It is the moral obligation of the generations consuming the energy to implement an acceptable waste treatment and disposal strategy. The inherent complication of radioactivity, the decay that can span hundreds of thousands of years, amplifies the importance of extreme time periods in the issue of corrosion and radiation stability. The search for storage concepts that can guarantee the safe storage and isolation of radioactive waste is, therefore, another challenging task for materials science, requiring a close examination of natural (geological) materials and processes. The more than 50 years of research and development of fission and fusion reactors have undoubtedly demonstrated that the statement ‘technologies are enabled by materials’ is particularly true for nuclear technology. Although the nuclear field is typically known for its incremental progress, the challenges posed by the next generation of fission reactors (Generation IV) as well as the demonstration of fusion reactors will need breakthroughs to achieve their ambitious goals. This is being accompanied by an important change in materials science, with a shift of discovery through experiments to discovery through simulation. The progress in numerical simulation of the material evolution on a scientific and engineering scale is growing rapidly. Simulation techniques at the atomistic or meso scale (e.g., electronic structure calculations, molecular dynamics, kinetic Monte Carlo) are increasingly helping to unravel the complex processes occurring in materials under extreme conditions and to provide an insight into the causes and thus helping to design remedies. In this context, Comprehensive Nuclear Materials aims to provide fundamental information on the vast variety of materials employed in the broad field of nuclear technology. But to do justice to the comprehensiveness of the work, fundamental issues are also addressed in detail, as well as the basics of the emerging numerical simulation techniques. R.J.M. Konings European Commission, Joint Research Centre, Institute for Transuranium Elements, Karlsruhe, Germany T.R. Allen Department of Engineering Physics, Wisconsin University, Madison, WI, USA R. Stoller Materials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, TN, USA S. Yamanaka Division of Sustainable Energy and Environmental Engineering, Graduate School of Engineering, Osaka University, Osaka, Japan
FOREWORD ‘Nuclear materials’ denotes a field of great breadth and depth, whose topics address applications and facilities that depend upon nuclear reactions. The major topics within the field are devoted to the materials science and engineering surrounding fission and fusion reactions in energy conversion reactors. Most of the rest of the field is formed of the closely related materials science needed for the effects of energetic particles on the targets and other radiation areas of charged particle accelerators and plasma devices. A more complete but also more cumbersome descriptor thus would be ‘the science and engineering of materials for fission reactors, fusion reactors, and closely related topics.’ In these areas, the very existence of such technologies turns upon our capabilities to understand the physical behavior of materials. Performance of facilities and components to the demanding limits required is dictated by the capabilities of materials to withstand unique and aggressive environments. The unifying concept that runs through all aspects is the effect of radiation on materials. In this way, the main feature is somewhat analogous to the unifying concept of elevated temperature in that part of materials science and engineering termed ‘high-temperature materials.’ Nuclear materials came into existence in the 1950s and began to grow as an internationally recognized field of endeavor late in that decade. The beginning in this field has been attributed to presentations and discussions that occurred at the First and Second International Conferences on the Peaceful Uses of Atomic Energy, held in Geneva in 1955 and 1958. Journal of Nuclear Materials, which is the home journal for this area of materials science, was founded in 1959. The development of nuclear materials science and engineering took place in the same rapid growth time period as the parent field of materials science and engineering. And similarly to the parent field, nuclear materials draws together the formerly separate disciplines of metallurgy, solid-state physics, ceramics, and materials chemistry that were early devoted to nuclear applications. The small priesthood of first researchers in half a dozen countries has now grown to a cohort of thousands, whose home institutions are anchored in more than 40 nations. The prodigious work, ‘Comprehensive Nuclear Materials,’ captures the essence and the extensive scope of the field. It provides authoritative chapters that review the full range of endeavor. In the present day of glance and click ‘reading’ of short snippets from the internet, this is an old-fashioned book in the best sense of the word, which will be available in both electronic and printed form. All of the main segments of the field are covered, as well as most of the specialized areas and subtopics. With well over 100 chapters, the reader finds thorough coverage on topics ranging from fundamentals of atom movements after displacement by energetic particles to testing and engineering analysis methods of large components. All the materials classes that have main application in nuclear technologies are visited, and the most important of them are covered in exhaustive fashion. Authors of the chapters are practitioners who are at the highest level of achievement and knowledge in their respective areas. Many of these authors not only have lived through a substantial part of the history sketched above, but they themselves are the architects. Without those represented here in the author list, the field would certainly be a weaker reflection of itself. It is no small feat that so many of my distinguished colleagues could have been persuaded to join this collective endeavor and to make the real sacrifices entailed in such time-consuming work. I congratulate the Editor, Rudy Konings, and
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the Associate Editors, Roger Stoller, Todd Allen, and Shinsuke Yamanaka. This book will be an important asset to young researchers entering the field as well as a valuable resource to workers engaged in the enterprise at present. Dr. Louis K. Mansur Oak Ridge, Tennessee, USA
Permission Acknowledgments The following material is reproduced with kind permission of Cambridge University Press Figure 15 of Oxide Dispersion Strengthened Steels Figure 15 of Minerals and Natural Analogues Table 10 of Spent Fuel as Waste Material Figure 21b of Radiation-Induced Effects on Microstructure www.cambridge.org The following material is reproduced with kind permission of American Chemical Society Figure 2 of Molten Salt Reactor Fuel and Coolant Figure 22 of Molten Salt Reactor Fuel and Coolant Table 9 of Molten Salt Reactor Fuel and Coolant Figure 6 of Thermodynamic and Thermophysical Properties of the Actinide Nitrides www.acs.org The following material is reproduced with kind permission of Wiley Table 3 of Properties and Characteristics of SiC and SiC/SiC Composites Table 4 of Properties and Characteristics of SiC and SiC/SiC Composites Table 5 of Properties and Characteristics of SiC and SiC/SiC Composites Figure 5 of Advanced Concepts in TRISO Fuel Figure 6 of Advanced Concepts in TRISO Fuel Figure 30 of Material Performance in Supercritical Water Figure 32 of Material Performance in Supercritical Water Figure 19 of Tritium Barriers and Tritium Diffusion in Fusion Reactors Figure 9 of Waste Containers Figure 13 of Waste Containers Figure 21 of Waste Containers Figure 11 of Carbide Fuel Figure 12 of Carbide Fuel Figure 13 of Carbide Fuel Figure 4 of Thermodynamic and Thermophysical Properties of the Actinide Nitrides Figure 2 of The U–F system Figure 18 of Fundamental Point Defect Properties in Ceramics Table 1 of Fundamental Point Defect Properties in Ceramics Figure 17 of Radiation Effects in SiC and SiC-SiC Figure 21 of Radiation Effects in SiC and SiC-SiC Figure 6 of Radiation Damage in Austenitic Steels Figure 7 of Radiation Damage in Austenitic Steels Figure 17 of Ceramic Breeder Materials Figure 33a of Carbon as a Fusion Plasma-Facing Material Figure 34 of Carbon as a Fusion Plasma-Facing Material i
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Figure 39 of Carbon as a Fusion Plasma-Facing Material Figure 40 of Carbon as a Fusion Plasma-Facing Material Table 5 of Carbon as a Fusion Plasma-Facing Material www.wiley.com The following material is reproduced with kind permission of Springer Figure 4 of Neutron Reflector Materials (Be, Hydrides) Figure 6 of Neutron Reflector Materials (Be, Hydrides) Figure 1 of Properties and Characteristics of SiC and SiC/SiC Composites Figure 3 of Properties and Characteristics of SiC and SiC/SiC Composites Figure 4 of Properties and Characteristics of SiC and SiC/SiC Composites Figure 5 of Properties and Characteristics of SiC and SiC/SiC Composites Figure 6 of Properties and Characteristics of SiC and SiC/SiC Composites Figure 7 of Properties and Characteristics of SiC and SiC/SiC Composites Figure 8 of Properties and Characteristics of SiC and SiC/SiC Composites Figure 9 of Properties and Characteristics of SiC and SiC/SiC Composites Figure 10 of Properties and Characteristics of SiC and SiC/SiC Composites Figure 11 of Properties and Characteristics of SiC and SiC/SiC Composites Figure 12 of Properties and Characteristics of SiC and SiC/SiC Composites Figure 22d of Fission Product Chemistry in Oxide Fuels Figure 3 of Behavior of LWR Fuel During Loss-of-Coolant Accidents Figure 14a of Irradiation Assisted Stress Corrosion Cracking Figure 14b of Irradiation Assisted Stress Corrosion Cracking Figure 14c of Irradiation Assisted Stress Corrosion Cracking Figure 25a of Irradiation Assisted Stress Corrosion Cracking Figure 25b of Irradiation Assisted Stress Corrosion Cracking Figure 1 of Properties of Liquid Metal Coolants Figure 5b of Fast Spectrum Control Rod Materials Figure 3 of Oxide Fuel Performance Modeling and Simulations Figure 8 of Oxide Fuel Performance Modeling and Simulations Figure 10 of Oxide Fuel Performance Modeling and Simulations Figure 11 of Oxide Fuel Performance Modeling and Simulations Figure 14 of Oxide Fuel Performance Modeling and Simulations Figure 5 of Thermodynamic and Thermophysical Properties of the Actinide Nitrides Figure 51 of Phase Diagrams of Actinide Alloys Figure 6 of Thermodynamic and Thermophysical Properties of the Actinide Oxides Figure 7b of Thermodynamic and Thermophysical Properties of the Actinide Oxides Figure 9b of Thermodynamic and Thermophysical Properties of the Actinide Oxides Figure 35 of Thermodynamic and Thermophysical Properties of the Actinide Oxides Table 11 of Thermodynamic and Thermophysical Properties of the Actinide Oxides Table 13 of Thermodynamic and Thermophysical Properties of the Actinide Oxides Table 17 of Thermodynamic and Thermophysical Properties of the Actinide Oxides Figure 18 of Radiation Damage of Reactor Pressure Vessel Steels Figure 7 of Radiation Damage Using Ion Beams Figure 9b of Radiation Damage Using Ion Beams Figure 28 of Radiation Damage Using Ion Beams Figure 34 of Radiation Damage Using Ion Beams Figure 35 of Radiation Damage Using Ion Beams Figure 36d of Radiation Damage Using Ion Beams Figure 37 of Radiation Damage Using Ion Beams Table 3 of Radiation Damage Using Ion Beams
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Figure 5 of Radiation Effects in UO2 Figure 9a of Ab Initio Electronic Structure Calculations for Nuclear Materials Figure 9b of Ab Initio Electronic Structure Calculations for Nuclear Materials Figure 9c of Ab Initio Electronic Structure Calculations for Nuclear Materials Figure 10a of Ab Initio Electronic Structure Calculations for Nuclear Materials Figure 23 of Thermodynamic and Thermophysical Properties of the Actinide Carbides Figure 25 of Thermodynamic and Thermophysical Properties of the Actinide Carbides Figure 26 of Thermodynamic and Thermophysical Properties of the Actinide Carbides Figure 27 of Thermodynamic and Thermophysical Properties of the Actinide Carbides Figure 28a of Thermodynamic and Thermophysical Properties of the Actinide Carbides Figure 28b of Thermodynamic and Thermophysical Properties of the Actinide Carbides Figure 2 of Physical and Mechanical Properties of Copper and Copper Alloys Figure 5 of Physical and Mechanical Properties of Copper and Copper Alloys Figure 6 of The Actinides Elements: Properties and Characteristics Figure 10 of The Actinides Elements: Properties and Characteristics Figure 11 of The Actinides Elements: Properties and Characteristics Figure 12 of The Actinides Elements: Properties and Characteristics Figure 15 of The Actinides Elements: Properties and Characteristics Table 1 of The Actinides Elements: Properties and Characteristics Table 6 of The Actinides Elements: Properties and Characteristics Figure 25 of Fundamental Properties of Defects in Metals Table 1 of Fundamental Properties of Defects in Metals Table 7 of Fundamental Properties of Defects in Metals Table 8 of Fundamental Properties of Defects in Metals www.springer.com The following material is reproduced with kind permission of Taylor & Francis Figure 9 of Radiation-Induced Segregation Figure 6 of Radiation Effects in Zirconium Alloys Figure 1 of Dislocation Dynamics Figure 25 of Radiation Damage Using Ion Beams Figure 26 of Radiation Damage Using Ion Beams Figure 27 of Radiation Damage Using Ion Beams Figure 4 of Radiation-Induced Effects on Material Properties of Ceramics (Mechanical and Dimensional) Figure 7 of The Actinides Elements: Properties and Characteristics Figure 20 of The Actinides Elements: Properties and Characteristics Figure 18a of Primary Radiation Damage Formation Figure 18b of Primary Radiation Damage Formation Figure 18c of Primary Radiation Damage Formation Figure 18d of Primary Radiation Damage Formation Figure 18e of Primary Radiation Damage Formation Figure 18f of Primary Radiation Damage Formation Figure 1 of Radiation-Induced Effects on Microstructure Figure 27 of Radiation-Induced Effects on Microstructure Figure 5 of Performance of Aluminum in Research Reactors Figure 2 of Atomic-Level Dislocation Dynamics in Irradiated Metals Figure 3 of Atomic-Level Dislocation Dynamics in Irradiated Metals Figure 5 of Atomic-Level Dislocation Dynamics in Irradiated Metals Figure 10a of Atomic-Level Dislocation Dynamics in Irradiated Metals Figure 10b of Atomic-Level Dislocation Dynamics in Irradiated Metals Figure 10c of Atomic-Level Dislocation Dynamics in Irradiated Metals
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Figure 10d of Atomic-Level Dislocation Dynamics in Irradiated Metals Figure 12a of Atomic-Level Dislocation Dynamics in Irradiated Metals Figure 12b of Atomic-Level Dislocation Dynamics in Irradiated Metals Figure 12c of Atomic-Level Dislocation Dynamics in Irradiated Metals Figure 12d of Atomic-Level Dislocation Dynamics in Irradiated Metals Figure 16a of Atomic-Level Dislocation Dynamics in Irradiated Metals Figure 16b of Atomic-Level Dislocation Dynamics in Irradiated Metals Figure 16c of Atomic-Level Dislocation Dynamics in Irradiated Metals Figure 16d of Atomic-Level Dislocation Dynamics in Irradiated Metals Figure 16e of Atomic-Level Dislocation Dynamics in Irradiated Metals Figure 17a of Atomic-Level Dislocation Dynamics in Irradiated Metals Figure 17b of Atomic-Level Dislocation Dynamics in Irradiated Metals Figure 17c of Atomic-Level Dislocation Dynamics in Irradiated Metals Figure 17d of Atomic-Level Dislocation Dynamics in Irradiated Metals www.taylorandfrancisgroup.com
1.01
Fundamental Properties of Defects in Metals
W. G. Wolfer Ktech Corporation, Albuquerque, NM, USA; Sandia National Laboratories, Livermore, CA, USA
Published by Elsevier Ltd.
1.01.1 1.01.2 1.01.3 1.01.3.1 1.01.3.2 1.01.3.3 1.01.4 1.01.4.1 1.01.4.2 1.01.4.3 1.01.4.4 1.01.5 1.01.5.1 1.01.5.2 1.01.5.3 1.01.6 1.01.6.1 1.01.6.2 1.01.6.3 1.01.7 1.01.7.1 1.01.7.2 1.01.7.3 1.01.7.4 1.01.7.4.1 1.01.7.4.2 1.01.7.4.3 1.01.7.4.4 1.01.8 1.01.8.1 1.01.8.2 1.01.8.2.1 1.01.8.2.2 1.01.8.3 1.01.9 Appendix A A1 A2 Appendix B B1 B2 B3 B4 B5 References
Introduction The Displacement Energy Properties of Vacancies Vacancy Formation Vacancy Migration Activation Volume for Self-Diffusion Properties of Self-Interstitials Atomic Structure of Self-Interstitials Formation Energy of Self-Interstitials Relaxation Volume of Self-Interstitials Self-Interstitial Migration Interaction of Point Defects with Other Strain Fields The Misfit or Size Interaction The Diaelastic or Modulus Interaction The Image Interaction Anisotropic Diffusion in Strained Crystals of Cubic Symmetry Transition from Atomic to Continuum Diffusion Stress-Induced Anisotropic Diffusion in fcc Metals Diffusion in Nonuniform Stress Fields Local Thermodynamic Equilibrium at Sinks Introduction Edge Dislocations Dislocation Loops Voids and Bubbles Capillary approximation The mechanical concept of surface stress Surface stresses and bulk stresses for spherical cavities Chemical potential of vacancies at cavities Sink Strengths and Biases Effective Medium Approach Dislocation Sink Strength and Bias The solution of Ham Dislocation bias with size and modulus interactions Bias of Voids and Bubbles Conclusions and Outlook Elasticity Models: Defects at the Center of a Spherical Body An Effective Medium Approximation The Isotropic, Elastic Sphere with a Defect at Its Center Representation of Defects by Atomic Forces and by Multipole Tensors Kanzaki Forces Volume Change from Kanzaki Forces Connection of Kanzaki Forces with Transformation Strains Multipole Tensors for a Spherical Inclusion Multipole Tensors for a Plate-Like Inclusion
2 3 5 5 8 10 12 12 13 15 15 16 16 17 20 21 22 23 25 26 26 26 27 29 29 29 30 31 32 32 33 33 35 35 37 38 38 38 41 41 42 43 43 43 44
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Fundamental Properties of Defects in Metals
Abbreviations bcc CA CD dpa EAM fcc hcp INC IHG SIA Vac
Body-centered cubic Cavity model Center of dilatation model Displacements per atom Embedded atom method Face-centered cubic Hexagonal closed packed Inclusion model Inhomogeneity model Self-interstitial atom Vacancy
1.01.1 Introduction Several fundamental attributes and properties of crystal defects in metals play a crucial role in radiation effects and lead to continuous macroscopic changes of metals with radiation exposure. These attributes and properties will be the focus of this chapter. However, there are other fundamental properties of defects that are useful for diagnostic purposes to quantify their concentrations, characteristics, and interactions with each other. For example, crystal defects contribute to the electrical resistivity of metals, but electrical resistivity and its changes are of little interest in the design and operation of conventional nuclear reactors. What determines the selection of relevant properties can best be explained by following the fate of the two most important crystal defects created during the primary event of radiation damage, namely vacancies and self-interstitials. The primary event begins with an energetic particle, a neutron, a high-energy photon, or an energetic ion, colliding with a nucleus of a metal atom. When sufficient kinetic energy is transferred to this nucleus or metal atom, it is displaced from its crystal lattice site, leaving behind a vacant site or a vacancy. The recoiling metal atom may have acquired sufficient energy to displace other metal atoms, and they in turn can repeat such events, leading to a collision cascade. Every displaced metal atom leaves behind a vacancy, and every displaced atom will eventually dissipate its kinetic energy and come to rest within the crystal lattice as a self-interstitial defect. It is immediately obvious that the number of self-interstitials is exactly equal to the number of vacancies produced, and they form Frenkel pairs. The number of Frenkel pairs created is also referred to as the number of displacements, and their accumulated density is
expressed as the number of displacements per atom (dpa). When this number becomes one, then on average, each atom has been displaced once. At the elevated temperatures that exist in nuclear reactors, vacancies and self-interstitials diffuse through the crystal. As a result, they will encounter each other, either annihilating each other or forming vacancy and interstitial clusters. These events occur already in their nascent collision cascade, but if defects escape their collision cascade, they may encounter the defects created in other cascades. In addition, migrating vacancy defects and interstitial defects may also be captured at other extended defects, such as dislocations, cavities, grain boundaries and interface boundaries of precipitates and nonmetallic inclusions, such as oxide and carbide particles. The capture events at these defect sinks may be permanent, and the migrating defects are incorporated into the extended defects, or they may also be released again. However, regardless of the complex fate of each individual defect, one would expect that eventually the numbers of interstitials and vacancies that arrive at each sink would become equal, as they are produced in equal numbers as Frenkel pairs. Therefore, apart from statistical fluctuations of the sizes and positions of the extended defects, or the sinks, the microstructure of sinks should approach a steady state, and continuous irradiation should change the properties of metals no further. It came as a big surprise when radiation-induced void swelling was discovered with no indication of a saturation. In the meantime, it has become clear that the microstructure evolution of extended defects and the associated changes in macroscopic properties of metals in general is a continuing process with displacement damage. The fundamental reason is that the migration of defects, in particular that of self-interstitials and their clusters, is not entirely a random walk but is in subtle ways guided by the internal stress fields of extended defects, leading to a partial segregation of self-interstitials and vacancies to different types of sinks. Guided then by this fate of radiation-produced atomic defects in metals, the following topics are presented in this chapter: 1. The displacement energy required to create a Frenkel pair. 2. The energy stored within a Frenkel pair that consists of the formation enthalpies of the selfinterstitial and the vacancy. 3. The dimensional changes that a solid suffers when self-interstitial and vacancy defects are created,
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Fundamental Properties of Defects in Metals
4.
5.
6.
7.
and how these changes manifest themselves either externally or internally as changes in lattice parameter. These changes then define the formation and relaxation volumes of these defects and their dipole tensors. The regions occupied by the atomic defects within the crystal lattice possess a distorted, if not totally different, arrangement of atoms. As a result, these regions are endowed with different elastic properties, thereby changing the overall elastic constants of the defect-containing solid. This leads to the concept of elastic polarizability parameters for the atomic defects. Both the dipole tensors and elastic polarizabilities determine the strengths of interactions with both internal and external stress fields as well as their mutual interactions. When the stress fields vary, the gradients of the interactions impose drift forces on the diffusion migration of the atomic defects that influences their reaction rates with each other and with the sinks. At these sinks, vacancies can also be generated by thermal fluctuations and be released via diffusion to the crystal lattice. Each sink therefore possesses a vacancy chemical potential, and this potential determines both the nucleation of vacancy defect clusters and their subsequent growth to become another defect sink and part of the changing microstructure of extended defects.
The last two topics, 6 and 7, as well as topic 1, will be further elaborated in other chapters.
where mc2 is the rest energy of an electron and L ¼ 4 mM/(m þ M)2. The approximation on the right is adequate because the electron mass, m, is much smaller than the mass, M, of the recoiling atom. Changing the direction of the electron beam in relation to the orientation of single crystal film specimens, one finds that the threshold energy varies significantly. However, for polycrystalline samples, values averaged over all orientations are obtained, and these values are shown in Figure 1 for different metals as a function of their melting temperatures.1 First, we notice a trend that Td increases with the melting temperature, reflecting the fact that larger energies of cohesion or of bond strengths between atoms also lead to higher melting temperatures. We also display values of the formation energy of a Frenkel pair. Each value is the sum of the corresponding formation energies of a self-interstitial and a vacancy for a given metal. These energies are presented and further discussed below. The important point to be made here is that the displacement energy required to create a Frenkel pair is invariably larger than its formation energy. Clearly, an energy barrier exists for the recoil process, indicating that atoms adjacent to the one that is being displaced also receive some additional kinetic energy that is, however, below the displacement energy Td and is subsequently dissipated as heat. The displacement energies listed in Table 1 and shown in Figure 1 are averaged not only over crystal orientation but also over temperature for those metals
50 Displacement energy Td(eV) fcc Frenkel pair (eV) bcc Frenkel pair (eV)
Scattering of energetic particles from external sources, be they neutrons, electrons, ions, or photons, or emission of such particles from an atomic nucleus, imparts a recoil energy. When this recoil energy exceeds a critical value, called the threshold displacement energy, Td, Frenkel pairs can be formed. To measure this displacement energy, an electron beam is employed to produce the radiation damage in a thin film of the material, and its rise in electrical resistivity due to the Frenkel pairs is monitored. By reducing the energy of the electron beam, the resistivity rise is also reduced, and a threshold electron energy, Emin, can be found below which no Frenkel pairs are produced. The corresponding recoil energy is given by relativistic kinematics as 2mc 2 þ Emin m Emin ½1 E 4 1 þ Td ¼ LEmin min 2mc 2 þ LEmin 2mc 2 M
Displacement and Frenkel pair energies (eV)
1.01.2 The Displacement Energy 40
30
20
10
0
0
500
1000 1500 2000 2500 3000 3500 Melting temperature (K)
4000
Figure 1 Energies of displacement and energies of Frenkel pairs for elemental metals as a function of their melting temperatures.
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Fundamental Properties of Defects in Metals
Table 1 Element
Displacement and Frenkel pair energies of elemental metals Symbol
Z
Melt temp. ( K)
M
Td (eV)
Frenkel pair energy (eV) fcc
Silver Aluminum Gold Cadmium Cobalt Chromium Copper Iron Indium Iridium Magnesium Molybdinum Niobium Neodymium Nickel Lead Palladium Platinum Rhenium Tantalum Titanium Vanadium Tungsten Zinc Zirconium
Ag Al Au Cd Co Cr Cu Fe In Ir Mg Mo Nb Nd Ni Pb Pd Pt Re Ta Ti V W Zn Zr
47 13 79 48 27 24 29 26 49 77 12 42 41 60 28 82 46 78 75 73 22 23 74 30 40
107.9 26.98 197.0 112.4 58.94 52.01 63.54 55.85 114.8 192.2 24.32 95.95 92.91 144.3 58.71 207.2 106.4 195.1 186.2 181.0 47.90 50.95 183.9 65.38 91.22
fcc fcc fcc hcp hcp bcc fcc bcc tetragonal fcc hcp bcc bcc hcp fcc fcc fcc fcc hcp bcc hcp bcc bcc hcp hcp
1235 933.5 1337 594.2 1768 2180 1358 1811 429.8 2719 923.2 2896 2750 1289 1728 600.6 1828 2041 3458 3290 1941 2183 3695 692.7 2128
26.0 15.3 34.0 19.0 23.0 28.0 18.3 17.4 10.5 46.0 10.0 32.4 28.2 9.30 22.0 11.8 34.0 34.0 44.0 26.7 20.8 28.0 44.0 12.0 22.5
bcc
6.52 4.96 5.98 4.40
6.66
4.03
6.69 4.01 10.5 3.24 6.51 12.7 5.77 3.26 8.90
Source: Displacement energies from Jung, P. In Landolt-Bo¨rnstein; Springer-Verlag: Berlin, 1991; Vol. III/25, pp 8–11.
50 Td fcc Td bcc Td hcp and others Td, eV
40 Displacement energy (eV)
where the displacement energy has been measured as a function of irradiation temperature. For some materials, such as Cu, a significant decrease of the displacement energy with temperature has been found. However, a definitive explanation is still lacking. Close to the minimum electron energy for Frenkel pair production, the separation distance between the self-interstitial and its vacancy is small. Therefore, their mutual interaction will lead to their recombination. With increasing irradiation temperature, however, the self-interstitial may escape, and this would manifest itself as an apparent reduction in the displacement energy with increasing temperature. On the other hand, Jung2 has argued that the energy barrier involved in the creation of Frenkel pairs is directly dependent on the temperature in the following way. This energy barrier increases with the stiffness of the repulsive part of the interatomic potential; a measure for this stiffness is the bulk modulus. Indeed, as Figure 2 demonstrates, the displacement energy increases with the bulk modulus. Since the bulk modulus decreases with temperature, so will the displacement energy. The correlation of the displacement energy with the bulk modulus appears to be a somewhat better
30
20
10
0 0
50
100
150
200
250
300
350
400
Bulk modulus (GPa) Figure 2 Displacement energies for elemental metals as a function of their bulk modulus.
empirical relationship than the correlation with the melt temperature. However, one should not read too much into this, as the bulk modulus B, atomic
Fundamental Properties of Defects in Metals
volume O, and melt temperature of elemental metals approximately satisfy the rule BO 100kB Tm discovered by Leibfried3 and shown in Figure 3.
1.01.3 Properties of Vacancies 1.01.3.1
Vacancy Formation
The thermal vibration of atoms next to free surfaces, to grain boundaries, to the cores of dislocations, etc., make it possible for vacancies to be created and then diffuse into the crystal interior and establish an equilibrium thermal vacancy concentration of f EV TSVf eq ½2 CV ðT Þ ¼ exp kB T given in atomic fractions. Here, EVf is the vacancy formation enthalpy, and SVf is the vacancy formation entropy. The thermal vacancy concentration can be measured by several techniques as discussed in Damask and Dienes,4 Seeger and Mehrer,5 and Siegel,6 and values for EVf have been reviewed and tabulated by Ehrhart and Schultz;7 they are listed in Table 2. When these values for the metallic elements are plotted versus the melt temperature in Figure 4, an approximate correlation is obtained, namely EVf Tm =1067
½3
4000
Bulk mod.*atom. vol./(100 k)
3500 3000 2500
1500 1000 500 0
Using the Leibfried rule, a new approximate correlation emerges for the vacancy formation enthalpy that has become known as the cBO model8; the constant c is assumed to be independent of temperature and pressure. As seen from Figure 5, however, the experimental values for EVf correlate no better with BO than with the melting temperature. It is tempting to assume that a vacancy is just a void and its energy is simply equal to the surface area 4pR2 times the specific surface energy g0. Taking the atomic volume as the vacancy volume, that is, O ¼ 4pR3/3, we show in Figure 6 the measured vacancy formation enthalpies as a function of 4pR2g0, using for g0 the values9 at half the melting temperatures. It is seen that EVf is significantly less, by about a factor of two, compared to the surface energy of the vacancy void so obtained. Evidently, this simple approach does not take into account the fact that the atoms surrounding the vacancy void relax into new positions so as to reduce the vacancy volume VVf to something less than O. The difference VVrel ¼ VVf O
0
500
1000 1500 2000 2500 3000 3500 4000 Melt temperature (K)
Figure 3 Leibfried’s empirical rule between melting temperature and the product of bulk modulus and atomic volume.
½4
is referred to as the vacancy relaxation volume. The experimental value7 for the vacancy relaxation of Cu is 0.25O, which reduces the surface area of the vacancy void by a factor of only 0.825, but not by a factor of two. The difference between the observed vacancy formation enthalpy and the value from the simplistic surface model has recently been resolved. It will be shown in Section 1.01.7 that the specific surface energy is in fact a function of the elastic strain tangential to the surface, and when this surface strain relaxes, the surface energy is thereby reduced. At the same time, however, the surface relaxation creates a stress field in the surrounding crystal, and hence a strain energy. As a result, the energy of a void after relaxation is given by FC ½eðRÞ; e ¼ 4pR2 g½eðRÞ; e þ 8pR3 me2 ðRÞ
2000
5
½5
The first term is the surface free energy of a void with radius R, and it depends now on a specific surface energy that itself is a function of the surface strain e(R) and the intrinsic residual surface strain e* for a surface that is not relaxed. The second term is the strain energy of the surrounding crystal that depends on its shear modulus m. The strain dependence of the specific surface energy is given by g½e; e ¼ g0 þ 2ðmS þ lS Þð2e þ eÞe
½6
Here, g0 is the specific surface energy on a surface with no strains in the underlying bulk material.
6
Fundamental Properties of Defects in Metals
Table 2
Crystal and vacancy properties
Metal
Crystal
Melt temp. (K)
KO (eV)
Debye temp. (K)
EfV (eV)
Em v (eV)
g0 (J m2)
Ag Al Au Be Co Cr Cs Cu Fe Hf Ir K Li Mg Mn Mo Na Nb Nd Ni Os Pb Pd Pt Re Rh Ru Sb Sr Ta Ti Tl U V W Zn Zr
fcc fcc fcc hcp hcp bcc bcc fcc bcc hcp fcc bcc bcc hcp bcc bcc bcc bcc hcp fcc hcp fcc fcc fcc hcp fcc hcp rbh* fcc bcc hcp hcp bco** bcc bcc hcp hcp
1235 933.5 1337 1560 1768 2180 301.6 1358 1811 2506 2719 336.7 453.7 923 1519 2896 371 2750 1289 1728 3306 600.6 1828 2041 3458 2237 2607 904 1050 3290 1941 577.2 1408 2183 3695 693 2128
10.9 7.89 18.1 6.57 13.1 12.1 41.2 10.1 12.3 15.3 31.3 1.55 1.63 5.13 9.15 25.4 1.70 19.3 6.8 12.5 36.7 8.46 17.7 26.7 34 32.6 18.6 7.9
229.2 430.6 162.7
1.11 0.67 0.93 0.8
588.4
2.1
1.09 1.02 1.33 1.30 2.22 2.01
349.6 483.3
1.28 1.90
0.66 0.61 0.71 0.87 0.72 0.95 0.084 0.70 0.55
25.3 11.8 7.7 13 13.5 30.8 6.49 13.8
92.7 369.5
0.48 0.80
473.4 157.1 254.6
3.10 0.34 2.70
481.4
1.79
0.038 0.50 1.30 1.35 0.03 0.55 0.81 1.04
106.6 277.9
0.58 1.85 1.35 3.10 2.50
0.43 1.03 1.43 2.20 1.50
264.7
3.10
0.70
399.4 384.3
2.10 3.60 0.54
0.50 1.70 0.42 0.58
1.57 2.12 1.92 2.65 0.129 0.472 0.688 2.51 0.234 2.31 2.38 2.95 0.54 1.74 2.20 3.13 2.33 2.65 0.461 0.358 2.49 1.75 0.55 1.78 2.30 2.77 0.896 1.69
*
rbh: rhombohedral bco: body-centered orthorhombic
**
However, such a surface possesses an intrinsic, residual surface strain e*, because the interatomic bonding between surface atoms differs from that in the bulk, and for metals, the surface bond length would be shorter if the underlying bulk material would allow the surface to relax. Partial relaxation is possible for small voids as well as for nanosized objects. In addition to the different bond length at the surface, the elastic constants, mS and lS, are also different from the corresponding bulk elastic constants. However, they can be related by a surface layer thickness, h, to bulk elastic constants such that mS þ lS ¼ ðm þ lÞh ¼ mh=ð1 2nÞ
½7
where l is the Lame’s constant and n is Poisson’s ratio for the bulk solid. Computer simulations on
freestanding thin films have shown10 that the surface layer is effectively a monolayer, and h can be approximated by the Burgers vector b. For planar crystal surfaces, the residual surface strain parameter e* is found to be between 3 and 5%, depending on the surface orientation relative to the crystal lattice. On surfaces with high curvature, however, e* is expected to be larger. The relaxation of the void surface can now be obtained as follows. We seek the minimum of the void energy as defined by eqn [5] by solving @FC =@e ¼ 0. The result is eðRÞ ¼
ðmS þ lS Þe h e ¼ mR þ ðmS þ lS Þ ð1 2nÞR þ h
½8
and this relaxation strain changes the initially unrelaxed void volume
7
Fundamental Properties of Defects in Metals
4
4 bcc Hf/v, eV fcc Hf/v, eV hcp Hf/v, eV
3 2.5 2 1.5 1
0
2.5 2 1.5 1
0
1 2 3 4 5 Surface energy of a vacancy (eV)
6
Figure 6 Correlation between the surface energy of a vacancy void and the vacancy formation energy. 3.5 Vacancy formation energy and its bulk contribution (eV )
4 fcc Hf/v, eV bcc Hf/v, eV hcp Hf/v, eV
3.5 3 2.5 2 1.5 1 0.5 0
0
500 1000 1500 2000 2500 3000 3500 4000 Melting temperature (K)
Figure 4 Vacancy formation energies as a function of melting temperature.
Vacancy formation enthalpy (eV)
3
0.5
0.5 0
fcc Hf/v, eV bcc Hf/v, eV hcp Hf/v, eV
3.5 Vacancy formation enthalpy (eV)
Vacancy formation energy (eV)
3.5
0
5
10
15
20
25
30
35
40
Exp. value Computed values Bulk strain energy
3
2.5
Ni
2
1.5
1
0.5
0 –0.35
–0.3
Bulk modulus * atomic volume (eV)
4p 3 ½9 R 3 consisting of n aggregated vacancies, by the amount nO ¼
½10
Applying these equations to a vacancy, for which n ¼ 1 and R b, we obtain VVrel O
¼
3e 2ð1 nÞ
and for the vacancy formation energy
–0.2
–0.15
–0.1
–0.05
0
Vacancy relaxation volume
Figure 5 Vacancy formation energy versus the product of bulk modulus and atomic volume.
V rel ðRÞ ¼ 3nOeðRÞ
–0.25
½11
Figure 7 Vacancy formation energy and its dependence on the relaxation volume.
( EVf ¼ 4pR2
2 ) 2ð3 4nÞ VVrel g0 mb 9ð1 2nÞ O
rel 2 2 V þ mO V O 3
½12
This equation is evaluated for Ni and the results are shown in Figure 7 as a function of the vacancy relaxa tion volume VVrel =O. It is seen that relaxation volumes of 0.2 to 0.3 predict a vacancy formation energy comparable to the experimental value of 1.8 eV.
8
Fundamental Properties of Defects in Metals
Few experimentally determined values are available for the vacancy relaxation volume, and their accuracy is often in doubt. In contrast, vacancy formation energies are better known. Therefore, we use eqn [12] to determine the vacancy relaxation volumes from experimentally determined vacancy formation energies. The values so obtained are listed in Table 3, and for the few cases7 where this is possible, they are compared with the values reported from experiments. Computed values for the vacancy relaxation volumes are between 0.2O and 0.3O for both fcc and bcc metals. The low experimental values for Al, Fe, and Mo then appear suspect. The surface energy model employed here to derive eqn [12] is based on several approximations: isotropic, linear elasticity, a surface energy parameter, g0, that represents an average over different crystal orientations, and extrapolation of the energy of large voids to the energy of a vacancy. Nevertheless, this approximate model provides satisfactory results and captures an important connection between the vacancy relaxation volume and the vacancy formation energy that has also been noted in atomistic calculations. Finally, a few remarks about the vacancy formation entropy, SVf , are in order. It originates from the change in the vibrational frequencies of atoms surrounding the vacancy. Theoretical estimates based on empirical potentials provide values that range from 0.4k to about 3.0k, where k is the Boltzmann constant. As a result, the effect of the vacancy formation entropy on the magnitude of the thermal equilibrium vacancy coneq centration, CV , is of the same magnitude as the statistical uncertainty in the vacancy formation enthalpy. Table 3
1.01.3.2
Vacancy Migration
The atomistic process of vacancy migration consists of one atom next to the vacant site jumping into this site and leaving behind another vacant site. The jump is thermally activated, and transition state theory predicts a diffusion coefficient for vacancy migration in cubic crystals of the form DV ¼ nLV d02 expðSVm ÞexpðEVm =kB T Þ ¼ DV0 expðEVm =kB T Þ
½13
Here, nLV is an average frequency for lattice vibrations, d0 is the nearest neighbor distance between atoms, SVm is the vacancy migration entropy, and EVm is the energy for vacancy migration. It is in fact the energy of an activation barrier that the jumping atom must overcome, and when it temporarily occupies a position at the height of this barrier, the atomic configuration is referred to as the saddle point of the vacancy. It will be considered in greater detail momentarily. Values obtained for EVm from experimental measurements are shown in Figure 8 as a function of the melting point. While we notice again a trend similar to that for the vacancy formation energy, we find that EVm for fcc and bcc metals apparently follow different correlations. However, the correlation for bcc metals is rather poor, and it indicates that EVm may be related to fundamental properties of the metals other than the melting point. The saddle point configuration of the vacancy involves not just the displacement of the jumping atom but also the coordinated motion of other atoms that are nearest neighbors of the vacancy and of the jumping atom. These nearest neighbor atoms
Vacancy relaxation volumes for metals
Metal
g 0 (J m2)
m (GPa)
n
HfV (eV)
V rel V =V (model)
Ag Al Au Cu Ni Pb Pd Pt Cr a-Fe Mo Nb Ta V W
1.19 1.1 1.45 1.71 2.28 0.57 1.91 2.40 2.23 2.31 2.77 2.54 2.76 2.51 3.09
33.38 26.18 31.18 54.7 94.6 10.38 53.02 65.1 117.0 90.4 125.8 39.6 89.9 47.9 160.2
0.354 0.347 0.412 0.324 0.276 0.387 0.374 0.393 0.209 0.278 0.293 0.397 0.324 0.361 0.280
1.11 0.05 0.67 0.03 0.93 0.04 1.28 0.05 1.79 0.05 0.58 0.04 1.7, 1.85 1.35 0.05 2.0 0.3 1.4, 1.89 3.2 0.09 2.6, 3.07 2.2, 3.1 2.2 0.4 3.1, 4.1
0.247 0.005 0.311 0.003 0.262 0.003 0.259 0.005 0.236 0.004 0.282 0.005 0.239, 0.225 0.260 0.003 0.218 0.02 0.278, 0.245 0.191 0.004 0.284, 0.258 0.264, 0.228 0.298 0.028 0.201, 0.161
V rel V =V (experiment) 0.05, 0.38 0.15 to 0.5 0.25 0.2 0.24, 0.42 0.05 0.1
Fundamental Properties of Defects in Metals
lie at the corners of a rectangular plane as shown in Figure 9. As the jumping atom crosses this plane, they are displaced such as to open the channel. This coordinated motion can be viewed as a particular strain fluctuation and described in terms of phonon excitations. In this manner, Flynn11 has derived the following formula for the energy of vacancy migration in cubic crystals. EVm ¼
15C11 C44 ðC11 C12 Þa3 w 2½C11 ðC11 C12 Þ þ C44 ð5C11 3C12 Þ
½14
Here, a is the lattice parameter, C11, C12, and C44 are elastic moduli, and w is an empirical parameter that characterizes the shape of the activation barrier and can be determined by comparing experimental vacancy migration energies with values predicted by eqn [14]. Ehrhart et al.7,12 recommend that w ¼ 0.022 for fcc metals and w ¼ 0.020 for bcc metals.
Vacancy migration energy (eV)
2.5
In the derivation of Flynn,11 only the four nearest neighbor atoms are supposed to move, while all other atoms are assumed to remain in their normal lattice positions. On the other hand, Kornblit et al.13 treat the expansion of the diffusion channel as a quasistatic elastic deformation of the entire surrounding material. The extent of the expansion is such that the opened channel is equal to the cross-section of the jumping atom, and a linear anisotropic elasticity calculation is carried out by a variational method to determine the energy involved in the channel expansion. A vacancy migration energy is obtained for fcc metals of EVm ¼ 0:01727a3 C11
p02
p0 p1 p22 þ 29 p0 p2 þ 19 p22
½15
and the parameters pi will be defined momentarily. For bcc metals,14 the activation barrier consist of two peaks of equal height EVmax with a shallow valley in between with an elevation of EVmin , where
fcc Hm/v, eV hcp Hm/v, eV bcc Hm/v, eV
2
9
q0 q1 q22 q02 112 q0 p2 þ 883 q22
½16
s0 s1 s22 2 s0 0:29232s0 s2 þ 0:0413s22
½17
EVmax ¼ 0:003905a 3 C11 and
1.5
EVmin ¼ 0:002403a 3 C11 1
The parameters pi, qi, and si are linear functions of the elastic moduli with coefficients listed in Table 4. For example,
0.5
q1 ¼ 3:45C11 0:75C12 þ 4:35C44 0
0
500 1000 1500 2000 2500 3000 3500 4000 Melt temperature (K)
Figure 8 Vacancy migration energy as a function of melting temperature.
If the depth of the valley is greater than the thermal energy of the jumping atom, that is, greater than 32 kT , then it will be trapped and requires an additional activation to overcome the remaining barrier of
Table 4 expressions
Figure 9 Second nearest neighbor atom (blue) jumping through the ring of four next-nearest atoms (green) into adjacent vacancy in a fcc structure.
Coefficients
for
the
Kornblit
energy
Function
C11
C12
C44
p0 p1 p2 q0 q1 q2 s0 s1 s2
5.29833 0.86667 1.41903 6.36429 3.45 3.32143 3.62621 1.57190 1.46564
4.76499 0.3333 0.88570 3.66429 0.75 0.62143 2.88241 0.82810 0.72184
9.35238 1.9111 1.64444 12.92142 4.35 3.70714 11.30366 4.21984 3.68855
10
Fundamental Properties of Defects in Metals
max EV EVmin . As a result, Kornblit14 assumes that the
vacancy migration energy for bcc metals is given by
EVm ¼
EVmax ; max 2EV EVmin ;
if EVmax EVmin 32 kT if EVmax EVmin > 32 kT
½18
Using the formulae of Flynn and Kornblit, we compute the vacancy migration energies and compare them with experimental values in Figure 10. With a few exceptions, both the Flynn and the Kornblit values are in good agreement with the experimental results. The self-diffusion coefficient determines the transport of atoms through the crystal under conditions near the thermodynamic equilibrium, and it is defined as
m eq DSD ¼ DV CV ¼ nLV a 2 exp SVf þ SVm =k expðQSD =kT Þ ½19
0 expðQSD =kT Þ ¼ DSD
where the activation energy for self-diffusion is Q SD ¼ EVf þ EVm
½20
The most accurate measurements of diffusion coefficients are done with a radioactive tracer isotope of the metal under investigation, and in this case one obtains values for the tracer self-diffusion coeffiT ¼ fD cient DSD SD that involves the correlation factor f. For pure elemental metals of cubic structure, f is a constant and can be determined exactly by computation.15 For fcc crystals, f ¼ 0.78145, and for bcc crystals, f ¼ 0.72149.
Theoretical vacancy migration energy (eV)
2 Evm, Flynn, fcc Evm, Kornblit, fcc Evm, Flynn, bcc Evm, Kornblit, bcc
1.5
1
To determine the preexponential factor for selfdiffusion 0 ¼ nLV a 2 exp SVf þ SVm =k ½21 DSD requires the values for the entropy SVf þ SVm and for the attempt frequency nLV. Based on theoretical estimates, Seeger and Mehrer5 recommend a value of 2.5 k for the former. The atomic vibration of nearest neighbor atoms to the vacancy is treated within a sinusoidal potential energy profile that has a maximum height of EVm . For small-amplitude vibrations, the attempt frequency is then given by rffiffiffiffiffiffi 1 EVm for fcc and by nLV ¼ a M rffiffiffiffiffiffiffiffi 1 2EVm ½22 nLV ¼ for bcc a 3M crystals where M is the atomic mass. In contrast, Flynn11 assumes that the atomic vibrations can be derived from the Debye model for which the average vibration frequency is rffiffiffi 3 kYD ½23 nLV ¼ 5 h where YD is the Debye temperature and h is Planck’s constant. The calculated preexponential factors for some fcc metals according to the models by Seeger and Mehrer5 and by Flynn are listed in Table 5 together with experimental values. They are also shown in Figure 11. While the computed values are of the right order of magnitude, there exists no clear correlation between the experimental and theoretical values. In fact, the computed values change little from one metal to another, and the Flynn model predicts values about twice as large as the model by Seeger and Mehrer. Either model can therefore be used to provide a reasonable estimate of the preexponential factor where no experimental value is available. 1.01.3.3 Activation Volume for Self-Diffusion
0.5
When the crystal lattice is under pressure p, the selfdiffusion coefficient changes and is then given by 0
0
0.5
1
1.5
Experimental vacancy migration energy (eV) Figure 10 Comparison of computed vacancy migration energies according to models by Flynn and Kornblit with measured values.
2
0 DSD ðT ; pÞ ¼ DSD expðQSD =kT ÞexpðpVSD =kT Þ
½24
The activation volume VSD can be obtained experimentally by measuring the self-diffusion coefficient as a function of an externally applied pressure. Such measurements have been carried out only for a few
Fundamental Properties of Defects in Metals
11
Preexponentials for tracer self-diffusion
Metal
M
a (nm)
QD (K)
Em V (eV)
Experimental value
S&M
Flynn (m2 s1)
Ag Al Au Cu Ni Pb Pd Pt
107.9 26.98 197 63.54 58.71 207.2 106.4 195.1
0.409 0.405 0.408 0.361 0.352 0.495 0.389 0.392
229 430.6 162.7 349.6 481.4 106.6 278 240
0.66 0.61 0.71 0.70 1.04 0.43 1.0 1.4
4.5e6 4.7e6 3.5e6 1.6e5 9.2e5 6.65e5 2.1e5 6.0e6
3.00e6 5.69e6 2.29e6 3.55e6 4.39e6 2.11e6 3.53e6 3.11e6
5.90e6 1.08e6 4.16e6 7.02e6 9.18e6 4.01e6 6.46e6 5.66e6
bcc Fe
Seeger Mehrer Flynn
Experiment Brown and Ashby Wallace Wang et al.
Rb Na
10–5
Li K Cs Zn Tl Element
Theoretical preexponential (m2 s–1)
Table 5
10–6 10–6
Mg Cd Pt
10–5
0.0001
Experimental preexponential Do (m2 s–1)
Pb Ni
Figure 11 Comparison of preexponential factors for tracer self-diffusion as computed with two models and as measured.
fcc Fe Cu Al Ag
metals, and it has been found that the activation volumes have positive values. Therefore, self-diffusion decreases with applied pressure. However, it has been noticed that the self-diffusion coefficient at melting appears to be constant, and this can be explained by the fact that the melting temperature increases in general with pressure. It follows then from the condition d½ln DSD ðp; Tm ðpÞÞ=dpjp¼0 ¼ 0
that
Q dTm VSD ¼ 0 Tm dp p¼0
½25
where Tm0 is the melting temperature under ambient conditions. Brown and Ashby16 have used this relationship to evaluate the activation volumes for self-diffusion
0
0.2 0.4 0.6 0.8 1 Activation volume/atomic volume
1.2
Figure 12 Activation volumes of elements divided by their atomic volumes from experiments and from relationship [25], using the change of the melting temperature with pressure from different sources.
for a variety of metals. Using more recent values for the pressure derivative of the melting temperature by Wallace17 and Wang et al.,18 one obtains activation volumes as shown in Figure 12. They are in reasonably good agreement with the experimental values where they exist. With the exception of Pt, the predicted values are also similar, giving an activation volume of about 0.85O for fcc metals, 0.65O for hcp metals, and around 0.4O for bcc metals.
12
Fundamental Properties of Defects in Metals
Table 6
Activation volume for vacancy migration
Metal
VSD =O
VfV =V
Vm V =V
Ag Al Cu Ni Pb bcc Fe
0.872 0.835 0.895 0.841 0.791 0.655
0.753 0.689 0.741 0.764 0.718 0.722
0.119 0.146 0.154 0.077 0.073 0.067
The equilibrium vacancy concentration in a solid under pressure p is given by f E TSVf pV f eq exp V ½26 CV ðT Þ ¼ exp V kB T kB T where VVf is the vacancy formation volume. Since the self-diffusion coefficient is the product of the thermal vacancy concentration and the vacancy migration coefficient, the activation volume for self-diffusion is the sum of two contributions, namely VSD ¼ VVf þ VVm ¼ O þ VVrel þ VVm
½27
VVm
with being the activation volume for vacancy migration. If one takes the average of the predicted activation volumes shown in Figure 12, and the vacancy relaxation volumes from Table 3, one obtains values for VVm listed in Table 6 and also shown in Figure 12.
1.01.4 Properties of Self-Interstitials 1.01.4.1 Atomic Structure of Self-Interstitials The accommodation of an additional atom within a perfect crystal lattice remained a topic of lively debates at international conferences on radiation effects for many decades. The leading question was the configuration of this interstitial atom and its surrounding atoms. This scientific question has now been resolved, and there is general agreement that this additional atom, a self-interstitial, forms a pair with one atom from the perfect lattice in the form of a dumbbell. The configuration of these dumbbells can be illustrated well with hard spheres, that is, atoms that repel each other like marbles. Let us first consider the case of an fcc metal. In the perfect crystal, each atom is surrounded by 12 nearest neighbors that form a cage around it as shown on the left of Figure 13. When an extra atom is inserted in this cage, the two atoms in the center form a pair
Figure 13 An atom with its 12 nearest neighbors in the perfect fcc lattice, on the left, and a [001] self-interstitial dumbbell with the same nearest neighbors, on the right.
whose axis is aligned in a [001] direction. This [001] dumbbell constitutes the self-interstitial in the fcc lattice. The centers of the 12 nearest neighbor atoms are the apexes of a cubo-octahedron that encloses the single central atom in the perfect lattice, and it can be shown19 that the cubo-octahedron encloses a volume of VO ¼ 10O/3. However, around a self-interstitial dumbbell, this cubo-octahedron expands and distorts, and now it encloses a larger volume of V001 ¼ 4.435O. The volume expansion is the difference DV ¼ V001 V0 ¼ 1:10164 O
½28
which happens to be larger than one atomic volume. We shall see shortly that the volume expansion of the entire crystal is even larger due to the elastic strain field created by the self-interstitial that extends through the entire solid. We consider next the self-interstitial defect in a bcc metal. Here, each atom is surrounded in the perfect crystal by eight nearest neighbors as shown on the left of Figure 14. When an extra atom is inserted, it again forms a dumbbell configuration with another atom, and the dumbbell axis is now aligned in the [011] direction, as shown on the right of Figure 14. The cage formed by the eight nearest neighbor atoms becomes severely distorted. It is surprising, however, that the volume change of the cage is only DV ¼ 0:6418 O
½29
less than the volume of the inserted atom to create the self-interstitial in the bcc structure. The reason for this is that the bcc structure does not produce the most densely packed arrangement of atoms, and some of the empty space can accommodate the self-interstitial. In contrast, the fcc structure has in fact the densest arrangement of atoms, and disturbing it by inserting an extra atom only creates disorder and lower packing density. As already mentioned, the large inclusion volume DV of self-interstitials leads to a strain field
Fundamental Properties of Defects in Metals
Figure 14 On the left is the unit cell of the bcc crystal structure. The central atom shown darker is surrounded by eight nearest neighbors. On the right is the arrangement when a self-interstitial occupies the center of the cell.
throughout the surrounding crystal that causes changes in lattice parameter and that is the major source of the formation energy for self-interstitials. In order to determine this strain field, we treat in Appendix A the case of spherical defects in the center of a spherical solid with isotropic elastic properties. Although this represents a rather simplified model for self-interstitials, for vacancies, and for complex clusters of such defects, it is a very instructive model that captures many essential features. 1.01.4.2 Formation Energy of Self-Interstitials In contrast to the formation energy of vacancies, there exists no direct measurement for the formation energy of self-interstitials. We have mentioned in Section 1.01.2 that the displacement energy required to create a Frenkel pair is much larger than the combined formation energies of the vacancy and the self-interstitial. As pointed out, there exist a large energy barrier to create the Frenkel pair, namely the displacement energy Td, and this barrier is mainly associated with the insertion of the interstitial into the crystal lattice. However, although this barrier should be part of the energy to form a selfinterstitial, it is by convention not included. Rather, the formation energy of a self-interstitial is considered to be the increase of the internal energy of a crystal with this defect in comparison to the energy of the perfect crystal. In contrast, since vacancies can be created by thermal fluctuations at surfaces, grain boundaries, and dislocation cores by accepting an atom from an adjacent lattice site and leaving it vacant, no similar barrier exists. The activation energy for this process is simply the sum of the actual formation energy EVf and the migration energy EVm, that is, the energy for self-diffusion, QSD.
13
When Frenkel pairs are created by irradiation at cryogenic temperatures, self-interstitials and vacancies can be retained in the irradiated sample. Subsequent annealing of the sample and measuring the heat released as the defects migrate and then disappear provide an indirect method to measure the energies of Frenkel pairs. Subtracting from these calorimetric values, the vacancy formation energy should give the formation energy of self-interstitials. The values so obtained for Cu7 vary from 2.8 to 4.2 eV, demonstrating just how inaccurate calorimetric measurements are. Besides, measurements have only been attempted on two other metals, Al and Pt, with similar doubtful results. As a result, theoretical calculations or atomistic simulations provide perhaps more trustworthy results. For a theoretical evaluation of the formation energy, we can consider the self-interstitial as an inclusion (INC) as described in Appendix A. Accordingly, a volume O of one atom is enlarged by the amount DV, or in other words, is subject to the transformation strain eij ¼ dij
where 3 ¼ DV =O
½30
The energy associated with the formation of this inclusion is given in Table A2, and it can be written as 2K mO DV 2 ½31 U0 ¼ 3K þ 4m O This expression for the so-called dilatational strain energy provides a rough approximation to the formation energy of a self-interstitial in fcc metals when the above volume expansion results are used. However, as the nearest neighbor cells depicted in Figures 13 and 14 show, their distorted shapes cannot be adequately described with a radial expansion of the original cell in the ideal crystal as implied by eqn [31]. As the detailed analysis by Wolfer19 indicates, the [001] dumbbell interstitial in the fcc lattice does not change the cell dimension in the [001] direction. In fact, it shortens it slightly, implying that e33 ¼ 0.01005. The volume change is therefore due to the nearest neighbor atoms moving on average away from the dumbbell axis. This can be represented by the transformation strain components e11 ¼ e22 being determined by 2e11 þ e33 ¼
DV ¼ 1:10164 O
The transformation strain tensor for the [001] selfinterstitials in fcc crystals is then
14
Fundamental Properties of Defects in Metals 0
1 0:556 0 0 B C eij ¼ B 0:556 0 C @ 0 A 0 0 0:010 0 1 0 0:189 0 0 0:367 0 0 B C B B 0:189 0 ¼B 0:367 0 C @ 0 Aþ@ 0 0 0 0:377 0 0 0:367 ¼ dij þ~eij
The transformation strain tensor can again be separated into a dilatational and a shear part as 1 C C A ½32
and it can be divided into an dilatational part, dij, and a shear part, ~e ij , as shown. To find the transformation strain tensor for the [011] self-interstitial in bcc crystal, it is convenient to use a new coordinate system with the x3-axis as the dumbbell axis and the x1- and x2-axes emanating from the midpoint of the dumbbell axis and pointing toward the corner atoms. While the distance between these corner atoms and the central atom in the original bcc unit cell is the interatomic distance r0, in the distorted cell containing pffiffiffi the self-interstitial, their distance is reduced to 3r0 =2 as shown by Wolfer.19 As a result, pffiffiffi 3 1 0:134 e11 ¼ e22 ¼ 2 the remaining strain component is then determined by 2e11 þ e33 ¼
DV ¼ 0:6418 O
0
B eij ¼ B @ 0 0
U1 ¼
0:214 0
1
0
C B B 0 C Aþ@ 0:214
0:348
0
0
0:348
0
0
30.7 26.2 27.3 49.8 161.4 91.3 8.08 48.5 64.7 116.9 87.6 0.914 3.95 125.6 1.99 38.1 0.434 89.9 47.9 160.2
1.89 1.51 1.93 2.00 7.17 3.03 1.00 2.70 3.78 1.23 1.04 0.086 0.102 2.05 0.097 0.91 0.055 1.81 0.81 2.58
0.38 0.31 0.35 0.42 1.60 0.69 0.18 0.52 0.72 3.19 2.42 0.170 0.209 4.63 0.194 1.72 0.102 3.89 1.63 5.97
C 0 C A 0:696
2ð9K þ 8mÞmO I2 5ð3K þ 4mÞ
EIf U ¼ U0 þ U1
½34
½35
The dilatational and the shear strain energies for some elements are listed in Table 7 in the fourth and fifth column, respectively. For the fcc elements, the ratio of U1/U0 is about 0.2. In contrast, for the bcc elements (in italics) this ratio is about two.
dK/dP
dG/dP
Vrel/O Theory
102.3 76.1 170.7 137.7 354.7 183.7 44.7 192.7 283.0 161.9 167.7 3.30 12.1 261.7 6.90 172.3 2.20 225.0 155.7 311.0
1
and is equal to I2fcc ¼ 0:1068 and I2bcc ¼ 0:3633 for selfinterstitials in fcc and bcc metals, respectively. We consider now the total strain energy as a reasonable approximation for the formation energy of self-interstitials, namely
K (GPa)
Ag Al Au Cu Ir Ni Pb Pd Pt Cr Fe K Li Mo Na Nb Rb Ta V W
0
½33
where I2 ¼ ~e11~e22 ~e22~e33 ~e33~e11
Metal
U1 (eV)
0
The strain energy associated with the shear part can be shown (see Mura20) to be
Strain energies and relaxation volumes of self-interstitials U0 (eV)
0
¼ dij þ~eij
Table 7
G (GPa)
0:214
6.12 4.42 6.29 5.48 4.83 6.20 5.53 5.35 5.18 4.89 5.29 3.96 3.53 4.40 4.69 6.91 3.63 3.15 3.50 3.95
Elements in italics are bcc, all others are fcc. Experimental values from Ehrhart and Schultz.7
1.40 1.80 1.05 1.35 3.40 1.40 1.10 0.54 1.60 1.40 1.80 0.79 0.42 1.50 0.80 0.53 0.72 1.10 0.94 2.30
1.94 2.03 1.80 1.87 2.67 1.98 1.73 1.44 2.05 1.21 1.43 0.97 0.76 1.26 0.99 0.91 0.97 1.05 1.03 1.58
Experiment 1.9 0.4 1.55 0.3 1.8 1.86 0.3 1.1
1.1 0.2 1.1
Fundamental Properties of Defects in Metals
1.01.4.3 Relaxation Volume of Self-Interstitials If the elastic distortions associated with selfinterstitials could be adequately treated with linear elasticity theory, and if the repulsive interactions between the dumbbell atoms with their nearest neighbors were like that between hard spheres, then the volume change of a solid upon insertion of an interstitial atom would be equal to the volume change DV as derived above. This follows from the analysis of the inclusion in the center of a sphere given in Appendix A. From the results listed in Table A2, under column INC, we see that the volume change of the solid with a concentration S of inclusions is simply given by DV ¼ 3S V where 3 is the volume dilatation per inclusion as if it were not confined by the surrounding matrix. This remarkable result has been proven by Eshelby21 to be valid for any shape of the solid and any location of the inclusion within it, provided the inclusion and the solid can be treated as one linear elastic material. In other words, the elastic strains within the inclusion and within the matrix must be small. However, this is not the case for the elastic strains produced by self-interstitials. Here, the elastic strains are quite large. For example, the volume of the confined inclusion, also listed in Table A2 under column INC, is given by Du 3 3 1þn ¼ ¼ ¼ 3 u 1 þ o gE 3ð1 nÞ and so it is reduced to about 62% of the unconstrained volume for a Poisson’s ratio of n ¼ 0.3. This amounts to an elastic compression of 42% of the ‘volume’ of the self-interstitial in fcc materials, and 25% for the self-interstitial in bcc materials. Clearly, nonlinear elastic effects must be taken into account. Zener22 has found an elegant way to include the effects of nonlinear elasticity on volume changes produced by crystal defects such as self-interstitials and dislocations. If U represents the elastic strain energy of such defects evaluated within linear elasticity theory, and if one then considers the elastic constants in the formula for U to be in fact dependent on the pressure, then the additional volume change dV produced by the defects can be derived from the simple expression dV ¼
@U U @p K
½36
15
found by Schoeck.23 Its application to the strain energy of self-interstitials leads to the following result. 3K m0 =m þ 4mK 0 =K 1 U0 dV ¼ 3K þ 4m K 12ðK 0 m K m0 Þ m0 1 þ U1 ½37 þ ð3K þ 4mÞð9K þ 8mÞ m K Here, m0 and K 0 are the pressure derivatives of the shear and bulk modulus, respectively. The first term arising from the dilatational part of the strain energy was derived and evaluated earlier by Wolfer.19 It is the dominant term for the additional volume change for self-interstitials in fcc metals. Here, we evaluate both terms using the compilation of Guinan and Steinberg24 for the pressure derivatives of the elastic constants, and as listed in Table 7. The calculated relaxation volumes for selfinterstitials, VIrel ¼ DV þ dV
½38
are given in the eighth column of Table 7, and they can be compared with the available experimental values also listed. We shall see that the relaxation volume of selfinterstitials is of fundamental importance to explain and quantify void swelling in metals exposed to fast neutron and charged particle irradiations. 1.01.4.4
Self-Interstitial Migration
The dumbbell configuration of a self-interstitial gives it a certain orientation, namely the dumbbell axis, and upon migration this axis orientation may change. This is indeed the case for self-interstitials in fcc metals, as illustrated in Figure 15. Suppose that the initial location of the selfinterstitial is as shown on the left, and its axis is along [001]. A migration jump occurs by one atom of the dumbbell (here the purple one) pairing up with one nearest neighbor, while its former partner
Figure 15 Migration step of the self-interstitial in fcc metals.
16
Fundamental Properties of Defects in Metals
(the blue atom) occupies the available lattice site. Computer simulations of this migration process have shown25 that the orientation of the self-interstitial has rotated to a [010] orientation, and that this combined migration and rotation requires the least amount of thermal activation. Similar analysis for the migration of selfinterstitials in bcc metals has revealed that a rotation may or may not accompany the migration, and these two diffusion mechanisms are depicted in Figure 16. Which of these two possesses the lower activation energy depends on the metal, or on the interatomic potential employed for determining it. In general, however, the activation energies for self-interstitial migration are very low compared to the vacancy migration energy, and they can rarely be measured with any accuracy. Instead, in most cases only the Stage I annealing temperatures have been measured. In the associated experiments, specimens for a given metal are irradiated at such low temperatures that the Frenkel pairs are retained. Their concentration is correlated with the increase of the electrical resistivity. Subsequent annealing in stages then reveals when the resistivity declines again upon reaching a certain annealing temperature. The first annealing, Stage I, occurs when self-interstitials become mobile and in the process recombine with
vacancies, form clusters of self-interstitials, or are trapped at impurities. Table 8 lists the Stage I temperature,7 TIm , for pure metals as well as two alloys that represent ferritic and austenitic steels. For a few cases, an associated activation energy EIm is known, and in even fewer cases, a preexponential factor, D0I , has been estimated.
1.01.5 Interaction of Point Defects with Other Strain Fields 1.01.5.1
The Misfit or Size Interaction
Many different sources of strain fields may exist in real solids, and they can be superimposed linearly if they satisfy linear elasticity theory. If this is the case, we need to consider here only the interaction between one particular defect located at rd and an extraneous displacement field u0(r) that originates from some other source than the defect itself. In particular, it may be the field associated with external forces or deformations applied to the solid, or it may be the field generated by another defect in the solid. To find the interaction energy, we assume that the defect under consideration is modeled by applying a set of Kanzaki26 forces f(a)(R(a)), a ¼ 1, 2, . . ., z, at z atomic positions R(a) as described in greater detail
Figure 16 Two migration steps are favored by self-interstitials in bcc metals; the left is accompanied by a rotation, while the right maintains the dumbbell orientation.
Fundamental Properties of Defects in Metals Table 8 Annealing temperatures for Stage I and migration properties estimated from them for self-interstitials Metal
Stage I Tm I (K)
Cr Fe K Li Mo Na Nb Ta V W CrxFe1x Ag Al Au Cu Ir Ni Pb Pd Pt Rh Th Cr Fe Ni Be Cd Co Mg Re Sc Ti Zn Zr
36 23–144 <6 <6 3339 <6 5.5–6.0 4–6 3.8 11–38 100 15 37 <0.3 38 50 56 4 35 22 32 10 100–200 30–50 <3.6 45–50 13 90 105 120–130 13 102
Em I (eV)
DI0 (106 m2 s1)
0.25–0.32
0.054, 0.085 0.085, 0.088 0.112, 0.115
5.0
0.117
0.5–1.0
Pij ¼ OCijkl ekl W ¼ OCijkl ekl e0ij ¼ O ekl s0kl
ekl ¼
0.5–0.92
0.015 0.26, 0.30
If the displacement field varies slowly from one atom position to the next, we may employ the Taylor expansion for each displacement component,
ðÞ 0 ðrd ÞRj þ ½40 ui0 rd þ RðÞ ui0 ðrd Þ þ ui;j and obtain
¼1
ðÞ
0 ui;j ðrd Þ
z X ¼1
1 DV dkl 3 O
and
0.1–0.15 0.029 0.16
¼1
fi
½44
Finally, for the simplest model of a point defect as a misfitting spherical inclusion, as treated in Appendix A,
0.13
in Appendix B. We imagine that once applied, the extraneous source is switched on, whereupon the atoms are displaced by the field u0(r), and work is done by the Kanzaki forces of the amount z
X f ðÞ u0 rd þ RðÞ W ¼ ½39
z X
½43
and the interaction energy can be written as
0.14, 0.15 0.01 0.06–0.07
½42
where e0ij ðrd Þ is the extraneous strain field at the location of the defect. Point defects can also be modeled as inclusions, as we have seen, and they can be characterized by a transformation strain tensor, ekl. In Appendix B it is shown, with eqn [B25], that the dipole tensor is then given by
Source: Ehrhart, P.; Schultz, H. In Landolt-Bo¨rnstein; SpringerVerlag: Berlin, 1991; Vol. III/25.
W ui0 ðrd Þ
The first term vanishes because the sum of Kanzaki forces is zero, as pointed out in eqn [B4], and the second term can be expressed in terms of the dipole tensor defined in [B6]. Furthermore, since the dipole tensor is symmetric, we obtain finally for the interaction energy W1 Pij e0ij ðrd Þ
0.083
17
fi
ðÞ ðÞ Rj
½41
1 ½45 W ¼ V rel s0kk 3 In this last form, we used the notation for the relaxation volume for the misfit or transformation volume, that is, DV ¼ V rel, to emphasize the fact that the interaction energy of vacancies and self-interstitials does not depend on their formation volumes, but on their relaxation volumes. This dependence on the misfit volume or defect size gave this energy the name of misfit or size interaction. 1.01.5.2 The Diaelastic or Modulus Interaction The definition of the dipole tensor and of multipole tensors with Kanzaki forces assumes that they are applied to atoms in a perfect crystal and selected such that they produce a strain field that is identical to the actual strain field in a crystal with the defect present. In particular, the dipole tensor reproduces the long-range part of this real strain field, and it can be determined from the Huang scattering measurements of crystals containing the particular defects. The actual specification of the exact Kanzaki forces is therefore not necessary. However, if the crystal is
18
Fundamental Properties of Defects in Metals
under the influence of external loads, the Kanzaki forces may be different in the deformed reference crystal. Consider for example the case of a crystal with a vacancy and under external pressure. In the absence of pressure, the vacancy relaxation volume has a certain value. However, under pressure, the volume of the vacancy may change by a different amount than the average volume per atom, and therefore, the Kanzaki forces necessary to reproduce this additional change will have to change from their values in the crystal under no pressure. The change of the Kanzaki forces induced by the extraneous strain field may then also be viewed as a change in the dipole tensor by dPij . Assuming that this change is to first-order linear in the strains, dPij ¼ aijkl e0kl
½46
The tensor aijkl has been named diaelastic polarizability by Kro¨ner27 based on the analogy with diamagnetic materials. When the change of the dipole tensor is included in the derivation of the interaction energy performed in the previous section, an additional contribution arises, namely 1 W2 ¼ aijkl e0ij e0kl 2
½47
The factor of 1/2 appears here because when the extraneous strain field is switched on for the purpose of computing the work, the induced Kanzaki forces are also switched on. This additional contribution W2, the diaelastic interaction energy, is quadratic in the strains in contrast to the size interaction, eqn [44], which is linear in the strain field. A crystalline sample that contains an atomic fraction n of well-separated defects and is subject to external deformation will have an enthalpy of 1
n n f E Pij e0ij ½48 H ðcÞ ¼ Cijkl aijkl e0ij e0kl þ 2 O O per unit volume. It follows from this formula that the presence of defects changes the effective elastic constants of the sample by n ½49 DCijkl ¼ aijkl O Such changes have been measured in single crystal specimens of only a few metals that were irradiated at cryogenic temperatures by thermal neutrons or electrons. Significant reductions of the shear moduli C44 and C0 ¼ (C11–C12)/2 are observed from which the corresponding diaelastic shear polarizabilities listed
in Table 9 are derived.25,28 These values are per Frenkel pair, and hence each one is the sum of the shear polarizabilities of a self-interstitial and a vacancy. By annealing these samples and observing the recovery of the elastic constants to their original values, one can conclude that the shear polarizabilities of vacancies are small and that the overwhelming contribution to the values listed in Table 9 comes from isolated, single self-interstitials. The softening of the elastic region around the selfinterstitial to shear deformation is not intuitively obvious. However, the theoretical investigations by Dederichs and associates29 on the vibrational properties of point defects have provided a rather convincing series of results, both analytical and by computer simulations. According to these results, the selfinterstitial dumbbell axis is highly compressed, up to 0.6 of the normal interatomic distance between neighboring atoms. Therefore, the dumbbell axis can be easily tilted by shear of the surrounding lattice and thereby release some of this axial compression. The weak restoring forces associated with this tilt introduce low-frequency vibrational modes that are also responsible for the low migration energy of selfinterstitials in pure metals. Computer simulations carried out by Dederichs et al.30 with a Morse potential for Cu gave the results presented in Table 10. While the shear polarizabilities compare favorably with the experimental results for Cu listed in Table 9, the bulk polarizability in the last column of Table 10 is too large, and most likely of the wrong sign for the self-interstitial. The experimental results for Cu indicate that the bulk polarizability for the Frenkel pair is close to zero. Atomistic simulations
Table 9
Diaelastic shear polarizabilities per Frenkel pair
Metal
Al
Cu
Mo
a44 (eV) (a11a12)/2 (eV)
269 20 125 16
377 44 111 14
162 60 298 43
Table 10 Diaelastic polarizabilities from computer simulations of Cu Diaelastic polarizability
a44 (eV)
(a11a12)/2 (eV)
(a11þ2a12)/3 (eV)
Frenkel pair Self-interstitial Vacancy
481.6 443.9 37.7
109.5 77.7 31.8
117.7 90.3 27.4
Fundamental Properties of Defects in Metals
have also been reported by Ackland31 using an effective many-body potential. The predicted diaelastic polarizabilies all turn out to be of the opposite sign than those reported by Dederichs and those obtained from the experimental measurements. Furthermore, Ackland also reports that the simulation results are dependent on the size of the simulation cell, that is, on the number of atoms. Evidently, the predictions depend very sensitively on the type and the particular features of the interatomic potential as well as on the boundary conditions imposed by the periodicity of the simulation cell. The model of the inhomogeneous inclusion pioneered by Eshelby21 may be instructive to explain the diaelastic polarizabilities of vacancies and selfinterstitials. A defect is viewed as a region with elastic constants different from the surrounding elastic continuum. We suppose that this region occupies a spherical volume of NO, has isotropic elastic constants K* and G*, and is embedded in a medium with elastic constants K and G. Here, O is the volume per atom, N the number of atoms in the defect region, and K and G the bulk and shear modulus, respectively. As Eshelby21 has shown, when external loads are applied to this medium and they produce a strain field e0ij in the absence of the spherical inhomogeneity, then an interaction is induced upon forming it that is given by
1 W2 ¼ N O K A e0ii e0jj þ 2GB e~ij0 e~ij0 ½50 2 Here, 1 e~ij0 ¼ e0ij dij e0kk 3 is the deviatoric strain tensor, and A¼
3ðK K Þð1 nÞ ðK K Þð1 þ nÞ 3K ð1 nÞ
15ðG GÞð1 nÞ B¼ 2ðG G Þð4 5nÞ 15Gð1 nÞ
½51
½52 ½53
The Poisson’s ratio n in the above equations is that of the matrix. For an isotropic crystal, eqn [47] assumes the same form as eqn [50], and the diaelastic polarization tensor has then only two components. These can now be identified with the two coefficients in eqn [50] to give the bulk polarizability 1 aK ¼ N OK A ¼ ða11 þ 2a12 Þ 3 and the shear polarizability
½54
19
3 1 ½55 aG ¼ N OGB a44 þ ða11 a12 Þ 5 5 The approximation in the last equation is based on Voigt’s averaging of the shear moduli of cubic materials to obtain an isotropic value. Let us first apply the formulae [52] to [55] to a vacancy. It seems plausible to select N ¼ 1 and assume that K * ¼ G * ¼ 0. Then AV ¼
3ð1 nÞ 2ð1 2nÞ
½56
15ð1 nÞ ½57 7 5n With these expressions, it is easy to compute the bulk and shear polarizabilities for vacancies, and their values are listed in the second and third columns of Table 11. Next, we consider the bulk polarizability of selfinterstitials. The two atoms that form the dumbbell are under compression, and the local bulk modulus that controls their separation distance may be estimated as follows: BV ¼
K K þ
dK dp 0 dK Du0 Du ¼ K K dp dV dp O
½58
Here, Du0 is the volume compression of the dumbbell exerted on it by the surrounding material, and as shown in Section 1.01.4, it is given by Du0 ¼ Du DV ¼ ð1=gE 1ÞDV
½59
where DV is the relaxation volume of the selfinterstitial as evaluated for the linear elastic medium. As we have seen in Section 1.01.4, DV ¼ 1.10164O for fcc and DV ¼ 0.6418O for bcc crystals. With the relations [58] and [59] we obtain DV ðgE 1ÞdK dp O
AI ¼ DV ð1 1=gE ÞdK dp O þ gE
½60
and with it the bulk polarizability of self-interstitials as aKI ¼ 2OKAI
½61
Numerical values for it are listed in the third column of Table 11. We note that these values are negative, meaning that self-interstitials increase the effective bulk modulus of irradiated samples, and this is in contrast to the results from atomistic simulations by Dederichs et al.29 obtained with a Morse potential for Cu. To rationalize the shear polarizability of selfinterstitials, we recall that the crystal lattice that
20
Fundamental Properties of Defects in Metals
Table 11 Diaelastic polarizabilities in electron volts for vacancies and self-interstitials estimated with an Isotropic Inhomogeneity Model Metal
aKV
aG V
aKI
aG I
aG FP (from experiment)
dVI/V
Al Cu Ni Cr Fe Mo
38.12 31.19 31.46 24.71 30.02 65.20
6.02 6.86 12.00 17.45 12.43 23.41
15.93 18.09 27.47 16.80 15.06 27.41
264.8 301.8 527.4 174.5 124.3 234.1
211 19 271 32
0.93 0.77 0.88 0.57 0.79 0.62
216 53
The last column is the nonlinear elastic contribution to the relaxation volume of self-interstitials.
surrounds the dumbbell becomes significantly dilated due to nonlinear elastic effects. This additional dilatation, dV/O, has to be added to DV/O to obtain relaxation volumes that agree with experimental values. We repeat the values for dV/O in the last column of Table 11 as a reminder. As a result of this additional dilatation, the atomic structure adjacent to the dumbbell is more like that in the liquid phase, as it lost its rigidity with regard to shear. For this dilated region, consisting of NG atoms that include the two dumbbell atoms, we assume that its shear modulus G* ¼ 0. Then BI ¼ BV ¼
15ð1 nÞ 7 5n
and aG I ¼ NG OGBI
½62
If the dilated region extends out to the first, second, or third nearest neighbors, then NG ¼ 14, 20, or 44, respectively, for fcc crystals, and NG ¼ 10, 16, or 28 for bcc crystals. From these numbers we shall select those that enable us to predict a value for aG I that comes closest to the experimental value. Matching it for fcc Cu indicates that the dilated region reaches out to third nearest neighbors, and hence NG ¼ 44. However, the best match for bcc Mo is obtained with NG ¼ 10, a region that only includes the dumbbell and its first nearest neighbors. These respective values for NG are also adopted for the other fcc and bcc elements in Table 11, and the shear polarizabilities so obtained are listed in the fifth column. To compare these estimates with experimental results, the approximation given in eqn [55] is used with the data in Table 9 for the shear polarizabilities of Frenkel pairs. These isotropic averages are listed in the sixth column of Table 11, and they are to G be compared with aG V þ aI . It is seen that the
inhomogeneity model is quite successful in explaining the experimental results, in spite of its simplicity and lack of atomistic details. 1.01.5.3
The Image Interaction
This interaction arises not from the strain field of other defects or from applied loads but is caused by the changing strain field of the point defect itself as it approaches an interface or a free surface of the finite solid. We have shown in Section 1.01.4 that the strain energy associated with a point defect is given by 2 2K mO V rel 2mð1 þ nÞ ðV rel Þ2 ½63 ¼ U0 ¼ 3K þ 4m O 9ð1 nÞ O when the defect is in the center of a spherical body with isotropic elastic properties or when the defect is sufficiently far removed from the external surfaces of a finite solid. This strain energy has been obtained by integrating the strain energy density of the defect over the entire volume of the solid, and since this density diminishes as r6, where r is the distance from the defect center, it is concentrated around the defect. Nevertheless, close to a free surface, the strain field of the defect changes, and with it the strain energy. This change is referred to as the image interaction energy U im, and the actual strain energy of the defect becomes U ðhÞ ¼ U0 þ U im ðhÞ
½64
Here, h is the shortest distance to the free surface. The strain energy of the defect, U(h), changes with h for two reasons. First, as the defect approaches the surface, the integration volume over regions of high strain energy density diminishes, and second, the strain field around the defect becomes nonspherical and also smaller. The evaluation of both of these contributions requires advanced techniques for solving elasticity problems.
21
Fundamental Properties of Defects in Metals
where h is the distance from the center of the defect to the surface. Equation [65] clearly demonstrates that the strain energy of the defect decreases as it approaches the surface. The minimum distance h0 is obviously that for which U(h) ¼ 0, and it is given by
ð1 þ nÞ 1=3 r0 ½66 h0 ¼ 4 Another case for the image interaction has been solved by Moon and Pao,32 namely when a point defect approaches either a spherical void of radius R or, when inside a solid sphere of radius R, approaches its outer surface. For a defect in a sphere, its strain energy changes with its distance r from the center of the sphere according to 1 þ n r0 3 US ðr Þ ¼ U0 1 R 4 % 1 X ðn þ 1Þðn þ 1Þð2n þ 1Þð2n þ 3Þ r 2n ½67 n2 þ ð1 þ 2nÞn þ 1 þ n R n¼0 while the strain energy of a defect at a distance r from the void center is given by 1 þ n r0 3 Uv ðr Þ ¼ U0 1 R 4 % 1 X nðn 1Þð2n 1Þð2n þ 1Þ R 2nþ2 ½68 n2 þ ð1 2nÞn þ 1 n r n¼2 Again, at a distance of closest approach to the void, h0(R), the strain energy of the defect vanishes. The numerical solutions of US(R þ h0) ¼ 0 and of UV(Rh0) ¼ 0 gives the results for h0/r0 shown in Figure 17. There is a modest dependence on the radius of curvature of the surface. Approximately, however, the defect strain energy becomes zero about halfway between the top and first subsurface atomic layer, assuming that r0 is equal to the atomic radius.
1.01.6 Anisotropic Diffusion in Strained Crystals of Cubic Symmetry The diffusion of the point defects created by the irradiation and their subsequent absorption at
0.85
Closest distance/atomic radius
Eshelby21 has shown that the strain energy of a defect, modeled as a misfitting inclusion of radius r0, in an elastically isotropic half-space, is given by ð1 þ nÞ r03 ½65 U ðhÞ ¼ U0 1 4 h3
Sphere Halfspace Void
0.8
0.75
0.7
0.65 Ni Poisson's ratio = 0.287
0.6
0.55
1
10 100 Surface radius/atomic radius
1000
Figure 17 Distance to surface where the defect strain energy disappears.
dislocations and interfaces in the material is the most essential process that restores the material to its almost normal state. The adjective of ‘almost normal’ is anything but a casual remark here, but it hints at some subtle effects arising in connection with the long-range diffusion that constitute the root cause for the gradual changes that take place in crystalline materials exposed to continuous irradiation at elevated temperatures. If these effects were absent, then a steady state would be reached in the material subject to continuous irradiation at a constant rate and temperature in which the rate of defect generation would be balanced by their absorption at sinks, meaning the above-mentioned dislocations and interfaces. As vacancies and self-interstitials are created as Frenkel pairs in equal numbers, they would also be absorbed in equal numbers at these sinks. At this point, the microstructure of these sinks would also be in a steady, unchanging state. While this steady state would be different from the initial microstructure or the one reached at the same temperature but in the absence of irradiation, it would correspond to material properties that reached constant values. The subtle effects alluded to in the above remarks arise from the interactions of the point defects with strain fields created both internally by the sinks and externally by applied loads and pressures on the materials that constitute the reactor components. The internal strain fields from sinks give rise to long-range forces that render the diffusion migration nonrandom, while the external strains induce anisotropic diffusion throughout the entire material. In the
22
Fundamental Properties of Defects in Metals
next section, we derive the diffusion equations for cubic materials to clearly expose these two fundamental effects. 1.01.6.1 Transition from Atomic to Continuum Diffusion During the migration of a point defect through the crystal lattice, it traverses an energy landscape that is schematically shown in Figure 18. The energy minima are the stable configurations where the defect energy is equal to E f(r), the formation energy, but modified by the interactions with internal and external strain fields, which in general vary with the defect location r. In order to move to the adjacent energy minimum, the defect has to be thermally activated over the saddle point that has an energy E S ðrÞ ¼ E f ðrÞ þ E m ðrÞ
½69
where E m ðrÞ is the migration energy. As the properties of the point defect, such as its dipole tensor and its diaelastic polarizability, are not necessarily the same in the saddle point configurations as in the stable configuration, the interactions with the strain fields are different, and the envelope of the saddle point energies follows a different curve than the envelope of the stable configuration energies, as indicated in Figure 18. For a self-interstitial, we
Potential profile Envelope for formation energies Envelope for saddle points
must also consider the different orientations that it may have in its stable configuration. Accordingly, let Cm ðr; t Þ be the concentration of point defects at the location r and at time t with an orientation m. For instance, the point defect could be the selfinterstitial in an fcc crystal, in which case, there are three possible orientations for the dumbbell axis and m may assume the three values 1, 2, or 3 if the axis is aligned in the x1, x2, or x3 direction, respectively. The elementary process of diffusion consists now of a single jump to one adjacent site at r þ R, where R is one of the possible jump vectors. The rate of change with time of the concentration Cm ðr; t Þ is now given by @Cm X ¼ Cn ðr R; t ÞLnm ðr R j RÞ @t R;n X Cm ðr; t ÞLmn ðr j RÞ ½70 R;n
Here, the first term sums up all jumps from neighboring sites to site r thereby leading to an increase of Cm ðr; t Þ, while the second term adds up all the jumps (really the probabilities of jumps) out of the site r. The frequency (or better the probability) of a particular jump from r to r þ R while changing the orientation from m to n is denoted by Lmn ðr j RÞ. The eqn [70] applies now to each of the possible orientations, and it appears that this leads to as many diffusion equations as there are possible orientations, and these equations may be coupled if the defect can change its orientation between jumps. To circumvent this complication, one considers an ensemble of identical systems, all having identical microstructures, and identical internal and external stress fields. The ensemble average of the defect concentration at each site, denoted simply as C(r,t) without a subscript, is now assumed to be the thermodynamic average such that expðbEmf Þ Cm ðr; t Þ ¼ Cðr; t ÞP expðbEnf Þ n
¼ Cðr; t ÞexpðbEmf Þ=N ðrÞ
r
r + R/2
r+R
Figure 18 Schematic of the potential energy profile for a migrating defect.
½71
where the normalization factor N only depends on the location r as do the energies for the stable defect configurations. Substituting this into eqn [70] on both sides constitutes another assumption. To see this, suppose that the defect concentrations Cn ðr R; t Þ on all neighbor sites happen, at the particular instance t, to be aligned in one direction. Since their new
Fundamental Properties of Defects in Metals
alignments after the jump to site r is correlated with the jump vector R and the previous orientation, the added defect population does not possess the equilibrium distribution of eqn [71]. However, Kronmu¨ller et al.33 argue that after several subsequent jumps of defects from the neighbors to this site r, the earlier deviation from the equilibrium distribution will have died out. Thus, introducing the thermodynamic averages on both sides of the eqn [70] is a plausible approximation. To proceed further requires a more specific form of the jump probability. For a jump from the site r to r þ R, it is assumed that
1 S Lmn ðr j RÞ ¼ ðL0 =nÞexp bEmn ðr þ RÞ þ bEmf ðrÞ ½72 2
where L0 is a constant to be defined later and n is the number of possible jump vectors. It is further assumed that the saddle point is halfway between r and r þ R, although different locations between r and r þ R have no bearing on the final results. The saddle point energy is then affected by the strain fields at the location r þ R/2. For a reverse jump of a defect initially at r þ R with orientation n to the site r and with a new orientation m, the same saddle point energy needs to be overcome. Hence, S S
Enm at the same location. Emn When eqns [71] and [72] are inserted in eqn [70], and then the latter is summed over all orientations m, one obtains @C ¼ fCðr R; t Þ Cðr; t Þg @t
½73
with
L0 X 1 S Cðr; t Þ ¼ Cðr; tÞ exp bEmn ðr þ RÞ N ðrÞn R;m;n 2
½74
This latter function may be viewed as an analytic function of r, since the saddle point energies vary with strains that are obtained from continuum elasticity theory, and they are by definition analytic functions of r. Expanding the first term in eqn [73] into a Taylor series up to second order, and then reverting back to the real defect concentration C(r,t), one arrives at the diffusion equation @C @ @ Dij ðrÞCðr; t Þ Fi ðrÞCðr; t Þ ½75 ¼ @xi @xj @t @xi 2
with the diffusion tensor defined as Dij ðrÞ ¼
L0 X 1 f S Xi Xj exp bEmn ðr þ RÞ þ bE ðrÞ ½76 2n R;m;n 2
23
and a drift force as Fi ðrÞ ¼
L0 X 1 f S Xi exp bEmn ðr þ RÞ þ bE ðrÞ n R;m;n 2
½77
The components of the jump vector R are denoted by capital letters Xi, while the components of the location vector r are given by the lower case letters xi. The normalization factor N(r) is replaced in the eqns [76] and [77] with an exponential function of the average defect formation energy according to h i h i X f exp bEmf ðrÞ ½78 exp bE ðrÞ ¼ N ðrÞ ¼ m
It is important to emphasize, as Dederichs and Schro¨der34 first did, that the above Taylor expansion does not remove the dependence of the saddle point energy on the jump direction R. To what degree it still depends on the jump direction is a function of the crystal lattice and magnitudes of the elastic strains. 1.01.6.2 Stress-Induced Anisotropic Diffusion in fcc Metals To evaluate the diffusion tensor and the drift force, we consider here the diffusion of self-interstitials and vacancies in fcc crystals, and begin first with the ideal case of a crystal free of any stresses other than those produced by the migrating defect itself. The jump vectors for both self-interstitials and vacancies coincide with the n ¼ 12 nearest neighbor locations of the defect in its stable configuration. Table 12 lists the components Xi of these jump vectors, and they are divided into three groups according to the orientations of the dumbbell axes before and after the jump. These are indicated in the first row of Table 12 by the indices 1, 2, or 3 when the dumbbell axis is aligned with the x1, x2, or x3 crystal coordinate direction, respectively. In the absence of stress, all saddle point energies f are equal, say ES, and E S E ¼ E m is just the defect migration energy. It is then a simple matter to Table pffiffiffi 12 d0 = 2.
X1 X2 X3
Components of the jump vectors R in units of 1$2
2$3
3$1
1 1 1 1 1 1 1 1 0000
0000 1 1 1 1 1 1 1 1
1 1 1 1 0000 1 1 1 1
d0 is the nearest neighbor distance between atoms.
Fundamental Properties of Defects in Metals
perform the summations in the definitions of the diffusion tensor and the drift force to show that 1 ½79 Dij ¼ L0 d02 expðbE m Þdij ¼ D0 ðT Þdij 6 and Fi ¼ 0. Next, let us consider the case of an applied spatially uniform stress field. According to Section 1.01.5, eqn [44], the interaction energy to linear order in the applied stress will change the energy of the defect in its stable configurations to Emf ¼ E f þ Wmf ¼ E f Oeijm s0ij
½80
and in its saddle point configurations to S S Emn ¼ E S þ Wmn ¼ E S Oeijmn s0ij
½81
Here, eijm is the transformation strain tensor of the defect in its stable configuration with orientation m. We had specified this tensor for self-interstitials in Section 1.01.4 for an orientation in the [001] direction, that is, for m ¼ 3, although the nonlinear contributions were not included. Below, it will be given with these contributions for self-interstitials in Cu and for the same orientation. The corresponding transformation strain tensors for the other two orientations, for m ¼ 1, 2, can be obtained from eij3 by appropriate coordinate rotations. Similarly, eijmn is the transformation strain tensor of the defect under consideration in its saddle point configuration while changing its orientation from m to n. Specifying it for one particular jump is sufficient to obtain the transformation strain tensors for all other jumps by appropriate coordinate rotations. The inspection of the jump directions for a particular pair mn reveals that there are equal but opposite jump directions with the same saddle point interaction energy; the only difference for such a pair of jump directions is that the components of R and –R have equal and opposite signs. As a result, the drift force F vanishes again. However, for the diffusion tensor, the two opposite jump directions make positive and equal contribution. Suppose, the applied stress is uniaxial with the only nonvanishing component s033 ¼ s. When the crystal is oriented such that this uniaxial stress is perpendicular to a (001) plane, then Chan et al.35 have shown that the diffusion within the (001) plane and perpendicular to it are given by ð001Þ
D11
ð001Þ
¼ D22
¼ D0
S s þ exp½bOðe S þ e S Þs=2 3 exp½bOe33 11 22 f s þ exp½bOe f s 2 2 exp½bOe11 33
S þ e S Þs=2 3exp½bOðe11 ð001Þ 22 D33 ¼ D0 f s þ exp½bOe f s 2exp½bOe11 33
respectively.
½82
When the uniaxial stress is perpendicular to a (111) crystal plane, then the diffusion tensor in the reference frame of the stress tensor has the components35 ð111Þ
D11
ð111Þ
¼ D22 ¼ D0
ð111Þ
D33
¼ D0
S þ e S Þs=3 þ exp½bOð2e S þ e S Þs=3 3exp½bOð2e22 33 11 33 f þ e f þ e f Þs=3 4exp½bOðe11 22 33 S þ e S Þs=3 exp½bOð2e11 33
½83
f þ e f þ e f Þs=3 exp½bOðe11 22 33
In order to obtain the transformation strains for the saddle point configurations, Chan, Averback, and Ashkenazy35 carried out molecular dynamics simulations of diffusion as a function of the applied stress. Fitting their results to the above equations enabled them to determine the tensors eijS and eijf . Their principal values are listed in Table 13 for self-interstitials and vacancies in Cu. The ratio of the diffusion coefficients in the plane perpendicular to the uniaxial stress, namely D11 =D0, is shown in Figures 19 and 20 as solid curves, while Table 13 Principal transformation strain components for self-interstitial atoms and vacancies in Cu
SIA, stable configuration SIA, saddle point Vacancy, stable configuration Vacancy, saddle point
e11
e22
e33
0.66 0.73 0.08 0.64
0.66 0.32 0.08 0.46
0.48 0.75 0.08 1.26
1.6 Stress in [001] SIA in (100) SIA out (100) Vac in (100) Vac out (100)
1.4
Diffusion ratio
24
1.2
1
0.8
0.6
–0.4
–0.2 0 0.2 Uniaxial stress (GPa)
0.4
Figure 19 Change of the diffusion coefficients within and perpendicular to (001) crystal planes when a uniaxial stress is applied.
Fundamental Properties of Defects in Metals
@spq L0 X mn Xi Xj Xk epq @xk 4n R;m;n h i f S exp bEmn ðrÞ þ bE ðrÞ
1.6
Dij ðrÞ þ bO
Stress in [111] SIA in (111) SIA out (111) Vac in (111) Vac out (111)
Diffusion ratio
1.4
1.2
1
0.8
0.6
–0.4
–0.2
0
25
0.2
0.4
Uniaxial stress (GPa) Figure 20 Change of the diffusion coefficients within and perpendicular to (111) crystal planes when a uniaxial stress is applied.
diffusion parallel to the stress, D33 =D0 , is shown as dashed curves. The enhancement of diffusion by tensile (positive) stresses in the (001) planes is much larger for vacancies than for self-interstitials. However, when tensile stress is applied to (111) crystal planes, diffusion in these planes is reduced for self-interstitials but remains almost unaltered for vacancies. 1.01.6.3 Diffusion in Nonuniform Stress Fields Internal stress fields from dislocations and other defects such as precipitates are spatially varying, that is, they are functions of the location r of the migrating point defect. However, the stresses sij (r) may be viewed as continuous functions except at material interfaces such as grain boundaries and free surfaces, and their differences between adjacent lattice sites may be approximated by 1 1 @sij ðrÞ ½84 sij r þ R sij ðrÞ Xk @xk 2 2 where a summation over the repeated index k is implied. The saddle point energy as given in eqn [81] is now replaced by @spq 1 S mn mn 1 Emn r þ R ¼ E S Oepq spq ðrÞ Oepq ½85 Xk 2 2 @xk
When this is inserted into eqn [76], a new diffusion tensor is given obtained that, to linear order in the stress gradients, is given by
½86
Here, the first term is just the diffusion tensor obtained for a uniform stress field, but now with the stress replaced by the local stress field at r. The second term is a correction linear in the stress gradient. However, this term vanishes for the following reasons. We have seen that for fcc crystals there are pairs of opposite jump directions that have identical transformation mn , but they differ only in the signs of the strain values epq jump vector components Xk. Hence, these opposite pairs of jump directions cancel each other’s contribution to the sum in eqn [86]. As a result, there is no correction to the diffusion tensor that is linear in stress gradients. Spatially varying stress field, however, produce a drift force. Inserting eqn [84] into the formula [77] leads to @spq L0 X mn Xi Xk epq Fi ¼ bO @xk 2n R;m;n h i f S ½87 exp bEmn ðrÞ þ bE ðrÞ The factor with the sum has a remarkable resemblance with the expression for the diffusion tensor. Indeed, it is straightforward to show that Fi ðrÞ ¼
@ f @ Dik ðrÞ Dik ðrÞbOepq spq ðrÞ ½88 @xk @xk
Since the average energy of the point defect in its stable configuration in the presence of a stress field is given by F
f E ðrÞ ¼ E f Oepq spq ðrÞ
½89
we may also write eqn [88] as h i @ n h i o f f Fi ðrÞ ¼ exp bE ðrÞ exp bE ðrÞ Dik ðrÞ ½90 @xk
In this last expression, the product of the two functions in the curly brackets depends now only on the saddle point energies, and this shows that it is the spatial dependence of the saddle point energy only that gives rise to a drift force, while the variation of the defect formation energy is not contributing to the drift force. The general formulae [88] or [90] reveal that stressinduced diffusion anisotropy affects the direction of the drift force such that it is in general not collinear with the stress-induced interaction force.
26
Fundamental Properties of Defects in Metals
1.01.7 Local Thermodynamic Equilibrium at Sinks 1.01.7.1
Introduction
Global thermodynamic equilibrium of a crystalline solid is nearly impossible to realize. Such a system would have to be a single crystal free of any defects except for a small concentration of vacancies. This concentration depends on the temperature. If the temperature is changed, these vacancies could not spontaneously appear and disappear within the perfect crystal lattice. The thermal fluctuations are too low in energy to nucleate a Frenkel pair, that is, a vacancy and a self-interstitial. However, at surfaces of this perfect solid, vacancies can form without the creation of a self-interstitial. Of course, surfaces are in reality extended defects just as grain boundaries are in real crystals. The latter contain also dislocations, which are lattice imperfections formed during the solidification of the crystal from its melt, and when thermal stresses are relaxed by plastic deformation. The conclusion one reaches is that the extended crystal defects, namely grain boundaries, dislocation cores, incoherent interfaces, and surfaces, are the places where vacancies are produced by thermal fluctuations and where they also disappear. It is at these places, called sinks (or sources), where a local equilibrium concentration of vacancies is established and maintained. The vacancy concentration in the remaining, perfect part of the crystal, adjusts by diffusion to the local equilibrium concentrations at the sinks. However, these local equilibrium concentrations are not necessarily all equal. In fact, they can differ depending on the type of sink, the local state of stress, and the local composition in the case of alloys. Here, we shall derive local equilibrium vacancy concentrations for some of the important types of sinks that are typically found in irradiated metals and alloys. The procedure we will employ is the same for all sink types, namely we will consider the reaction of species indicated by square brackets
procedure is a thermal vacancy concentration that is in local thermodynamic equilibrium with the particular sink, and this concentration defines a boundary condition for the diffusion of vacancy into or out of the sink or source, respectively. Henceforth, sink and source are considered synonymous. 1.01.7.2
Edge Dislocations
Edge dislocations contain jogs as illustrated by Figure 21. Here, the edge dislocation is the termination of an atomic plane, and the last row of atoms, rendered in a darker color, delineates the dislocation line. The step in this line forms a jog. The two atomic planes next to the terminated atomic plane are shown in the upper part with perfect occupancies of all atomic positions, except that they bend along the dislocation line to become neighboring planes above the dislocation. A vacancy may now form by an atom adjacent to the jog jumping to the terminating plane and thereby moving the jog by one atomic distance, as illustrated in the lower part of Figure 21. The vacancy created
Nv ½V þ ½S $ ðNv þ 1Þ½V þ ½S 0 in which a sink of type [S] surrounded by Nv vacancies [V] will emit one more vacancy and change to a different sink [S 0 ]. The two sinks [S] and [S 0 ] do not differ from each other in type, but may differ in their internal energies. The change in Gibbs free energy, DGðT Þ, for this reaction will be equal to zero when local thermodynamic equilibrium exists around the sink. The following specific example will clarify the meaning of the procedure. The outcome of this
Figure 21 Pure edge dislocation with a jog, before (upper part) and after (lower part) the formation of a vacancy in the adjacent lattice plane.
Fundamental Properties of Defects in Metals
may then diffuse away and into the surrounding crystal. Conversely, a vacancy that migrates to this jog will take the place of the atom next to the jog. As a result, the jog is simply displaced in the opposite direction. The absorption or emission of a vacancy restores the configuration of the edge dislocation. The absorption or emission of a vacancy from a perfect core site of an edge dislocation is also possible. However, the formation of a double jog would involve a large energy change, and for this reason, vacancy emission and absorption is unlikely for any site along the edge dislocation other than at already existing jogs. Assuming local thermodynamic equilibrium for the region around a jog, the thermal vacancy concentration can be found as follows. We imagine that we create a vacancy in the surrounding lattice and add the extra atom to the jog. The total change in Gibbs free energy is then given by f TS f Þ þ kT ‘n W ðN þ 1Þ kT ‘n W ðN Þ ½91 DG ¼ ðEV V V V
since we added the Gibbs free energy ðEVf TSVf Þ to the crystal to create the vacancy but gained some configurational entropy of the amount ðN þ N V þ 1Þ!N V ! DS ¼ k ‘n W ðN V þ 1Þ k ‘n W ðN V Þ ¼ k ‘n ðN V þ 1Þ!ðN þ N V Þ! N þNV þ1 N k ‘n V ¼ k ‘nCV ¼ k ‘n N NV þ1
½92
Here, N is the number of lattice sites per unit volume and N V the number of vacancies, the latter being much smaller than N. The creation of an additional vacancy in the vicinity of the dislocation jog occurs as a result of thermal fluctuation, and the Gibbs free energy change is on average equal to zero for this process. The condition DG ¼ 0 defines the vacancy concentration in local thermodynamic equilibrium with the edge dislocation, and one obtains eq
CV ¼ e SV =k e EV =kT f
f
½93
Two minor but nevertheless important energy contributions have been neglected in the above derivation. First, we did not consider the interaction of the vacancy with the stress field of the dislocation. Second, we also neglected any volume change of the crystal with the formation or absorption of a vacancy. A volume change will give rise to a work term if an external stress field soij is present in addition to the dislocation stress field. The first contribution can be taken into account by adding to the vacancy formation energy the
27
interaction energy EI ðro ; ’Þ, which was discussed in Section 1.01.5 and is further elaborated in Section 1.01.8. This energy is evaluated at the core radius rC, and it varies with the polar angle ’. To evaluate the second contribution, we note that if vacancies were persistently absorbed at an edge dislocation, it would eventually eliminate the incomplete crystal plane of atoms. The crystal volume would therefore contract in the direction perpendicular to this incomplete plane by the amount of the Burgers vector component bi in this direction. If ti ¼ soij bj =b is the force per unit area on the incomplete plane as exerted by the external stress field soij , then the work done against the external loads is
dA bi ti ¼ dA bi soij bj =b ½94 when the area dA is eliminated from the incomplete plane. The negative sign turns into a positive one if vacancies are emitted from the dislocation and if it climbs to expand the area of the incomplete plane. For one vacancy emitted and hence one atom added to the incomplete plane, dAb ¼ O, and the energy gained from the work expended by the external loads is ½95 O=b 2 soij bi bj þ VVR soH ¼ Oso? þ VVR soH The second term is due to the fact that the crystal does not expand by the entire atomic volume but is reduced by the vacancy relaxation volume. Note that VVR has in general a negative value, and that it results in an isotropic volume change; hence, the hydrostatic stress performs the work. As a result of these two additional contributions one finds that the local equilibrium vacancy concentration at a dislocation is EI ðrC ; ’Þ VVR soH Oso? eq þ ½96 þ CV? ðro ; ’Þ ¼ CV exp kT kT kT
We see that the interaction energy EI ðrC ; ’Þ gives a spatial dependence to the local equilibrium vacancy concentration. In addition, external forces on the crystal or stresses produced at the location of the dislocation by other distant defects will further change the local equilibrium concentration. 1.01.7.3
Dislocation Loops
For dislocation loops, the equilibrium vacancy concentration can be found by a similar analysis. However, as point defects are absorbed or emitted by the loop, it shrinks or expands, thereby changing its strain energy.
28
Fundamental Properties of Defects in Metals
Let us first consider an interstitial loop in a crystal subject to a stress soij . The force exerted on the loop plane is ti ¼ soij nj , where n is a normal vector to the loop plane. Upon forming the loop in the crystal, the atomic planes adjacent to the loop platelet are displaced by the Burgers vector b. Therefore, the work done by the external stresses is pR2 bi soij nj , where pR2 is the area of the loop assumed to be circular in shape. Since the separation of the two atomic planes next to the loop platelet is n b ¼ ni bi , the volume of the inserted platelet is pR2 ni bi ¼ N O, where N is the number of atoms in the loop. The work done by the external forces can then be written as soIL O ¼ bi soij nj O=ðni bi Þ
½97
Note, when a vacancy loop is formed, the crystal contracts and the work performed by external forces is just the opposite, namely soVL O ¼ bi soij nj O=ðni bi Þ
½98
Let us return to the case of an interstitial loop and consider the change in loop energy when a vacancy is emitted. This change is ELoop ðN þ 1Þ ELoop ðN Þ ffi dELoop =dN , where the number N of interstitials is related to the loop radius as pR2 ni bi ¼ N O. This change may be considered as a vacancy chemical potential associated with the prismatic interstitial loop of radius R, and it may be computed according to the equation Loop
DmV
ðRÞ ¼
dELoop dELoop O ¼ dN 2pðni bi ÞR dR
½99
Following our procedure now, we consider the total change in Gibbs free energy when a vacancy is created in the crystal lattice next to the dislocation core and the extra atom is attached to the loop platelet. This change is DG ¼
EVf
þ
TSVf
Loop DmV
EI ðrC ; ’Þ þ
kT ‘n CVIL
OsoIL
þ
VVR soH ½100
and it must be equal to zero when local thermodynamic equilibrium prevails. This condition then determines the local vacancy concentration as
( ) Loop o R IL ðr ; ’Þ ¼ C eq exp EI ðrC ; ’Þ þ OsIL þ VV DmV CV o V kT kT kT ( ) Loop ? exp DmV ½101 ffi CV kT
The local equilibrium vacancy concentration at an interstitial-type loop differs from the value at a
straight edge dislocation mainly by an exponential factor containing the vacancy chemical potential change. In fact, it is lower than for a straight edge dislocation, as will be shown below. For a vacancy-type loop, the sign in this exponential factor is positive, and the local equilibrium vacancy concentration is enhanced compared to the straight edge dislocation according to ( ) Loop Dm V ½102 CVVL ffi CV? exp kT To quantify how different these local vacancy concentrations can be from the true equilibrium concentration, we consider the case of perfect prismatic loops. The Burgers vector b in this case is parallel to the normal vector n, and nibi ¼ b. The energy is given by36,37
mb 2 n
rC r C o R K 1 E 1 þ pR2 gSF 1n R R pffiffiffi mb 2 8R R ‘n ffi 2 þ pR2 gSF ½103 2ð1 nÞ rC
ELoop ðRÞ ¼
where K and E are the complete elliptic integrals. For large loop radii R much larger than the core radius rC, the approximation given in the second part of the above equation can be used. The second term in addition to the strain energy represents the energy of the stacking fault. Note that the shear modulus m is not to be confused with the vacancy chemical potential DmV . Taking the derivative of the above equation with respect to N gives the vacancy chemical potential difference as Loop
DmV
ðRÞ ¼
mO b d r r R K 1 C E 1 C R R 2pð1 nÞ R dR O þ gSF b
½104
Inserting this result into eqns [101] and [102] gives the ratio of the vacancy concentration near the dislocation core of prismatic dislocation loops relative to the equilibrium concentration [93]. As an example, these ratios are evaluated for Ni and at a temperature of 773 K and shown in Figure 22. Both the exact expression for the loop energy and the logarithmic one in eqn [103] are employed to demonstrate that the latter provides an excellent approximation. However, for small prismatic loops, both expressions for the loop energy become questionable, and atomistic calculations are necessary to obtain energies of small, plate-like clusters of self-interstitials or of vacancies.
Fundamental Properties of Defects in Metals
The growth of the cavity volume by O implies that the gas inside expands and thereby performs the work pO. The overall change in Gibbs free energy is then
104 1000
Equilibrium vacancy ratio
At vacancy-type loops 100
2g0 O pO kT ‘nCVO R Under local thermodynamic equilibrium, DG ¼ 0, and we find that 2g0 O eq p ½108 CVO ¼ CV exp R kT F F DG ¼ E1V þ TS1V þ
10 1 0.1 0.01 At interstitial-type loops 0.001
0.0001 10 Loop radius, R/b
100
Figure 22 The ratio of vacancy equilibrium concentrations at prismatic dislocation loops to that at straight edge dislocations. The results are for Ni at T = 772 K, for a stacking fault energy of 0.11 J m2. Solid curves employ the exact loop energy, and dashed curves the logarithmic approximation.
1.01.7.4
29
Voids and Bubbles
1.01.7.4.1 Capillary approximation
Consider a spherical cavity of radius R with an internal gas pressure of p. To obtain the local equilibrium vacancy concentration, we take one atom from the cavity surface and place it in one of the vacancies in the crystal lattice next to the cavity surface. The cavity volume is thereby increased by one atomic volume, O, and its surface free energy is changed by DFS ðnÞ ¼ FS ðn þ 1Þ FS ðnÞ h i ¼ ð4pÞ1=3 ð3OnÞ2=3 ð1 þ 1=nÞ2=3 1 g0
½105
Here, g0 is the specific surface energy per unit area, assumed to be a constant, and n is the number of vacancies contained in the cavity. This number is related to the unrelaxed cavity volume by 4p 3 R ¼ nO 3
½106
When the cavity is large, that is, n 1, we can use the approximation
1 2=3 2 1þ ffi 1þ n 3n
DFS ffi 2g0 ð4p=3nÞ
2=3
ð3OÞ
2g ffi 0O R
This constitutes the well-known capillary approximation, and it is often interpreted as the combined mechanical effect of surface tension and pressure. However, the above derivation shows that the surface tension 2g0/R is not a mechanical force, but a thermodynamic or chemical force. The assumption that the surface tension acts like a mechanical pressure and generates a stress field whose radial component satisfies the boundary condition 2g0 p ½110 R is incorrect. However, to clarify this point, we need to first introduce the correct definition of a surface stress. srr ðRÞ ¼
1.01.7.4.2 The mechanical concept of surface stress
Surface areas in solids can be changed by elastic deformation, and this leads to the concept of surface stresses, gab, and surface strains, eSab . Here the lower indices designate two orthogonal coordinate directions, x1 and x2, which are tangential to the surface. Greek indices are used here for surface quantities, and they assume the values of 1 or 2. In contrast, Latin indices are employed for vector- and tensor-valued quantities within the bulk material, and they assume values of 1, 2, or 3. The definition of the surface stress in a Lagrangian reference frame is according to Cahn38 gab ¼
and then obtain 1=3
The chemical potential of vacancies at cavities is then 2g0 eq p O ½109 mCV ðRÞ ¼ kT ‘n½CVC =CV ¼ R
½107
@g @eSab
½111
Here, the specific surface free energy g is assumed to depend now on the elastic strain components tangential to the surface. We note that eqn [111] differs from
30
Fundamental Properties of Defects in Metals
the Shuttleworth39 equation. The latter is inconsistent with continuum mechanics as has been argued by Gutman.40 In contrast, the surface elasticity theory of Gurtin and Murdoch41,42 is compatible with the elasticity theory of bulk solids, and the two are connected by appropriate boundary conditions as described below. The theory of Gurtin and Murdoch requires that the surface energy be a function of the surface strains. As found from recent ab initio calculations and from atomistic calculations with empirical interatomic potentials, the dependence of the specific surface energy g on surface strains can be written as 1 gðeSab Þ ¼ gð0Þ þ Gabgd eSab eSgd þ Oðe3 Þ 2
½112
for small surface strains. Here, Gabgd is the surface elastic modulus tensor, and summation is implied over repeated indices. The surface elastic constants have a dimension of N m1 while bulk elastic constants have the dimension of N m2. Since the surface is attached to a bulk solid, surface strains eSab and the bulk strains eij are connected. Bulk strains are defined relative to a stress-free reference configuration of the bulk solid. However, a stress-free reference state for its surfaces may not coincide with the stress-free reference configuration for the bulk solid. Ab initio calculations on slabs43–47 of metals have in fact revealed that surface layers on stress-free bulk materials possess in general a positive residual strain, eab . The surfaces of metals are naturally stretched and will contract if they can deform the underlying solid. When the solid is elastically strained, the total surface strains become eSab ¼ eab þ eab
½113
where the strains eab are identical to the bulk strains eij at the surface. The surface stresses relative to the stress-free reference configuration of the bulk are now defined as gab
@g ¼ ¼ gab þ Gabgd egd @eab
½114
The first term gives the residual or intrinsic surface stresses, and they are connected to the residual surface strains by gab
¼
Gabgd egd
½115
This mechanical theory of surface stresses, as outlined above, has been developed previously in its full mathematical rigor by Gurtin and Murdoch for general finite strains.41 The above linearized version for
small strains has also been presented by these authors, and they have shown42 that for isotropic materials, the residual surface strains are eab ¼ e dab
½116
and the surface elastic modulus tensor Gabgd ¼ mS ðdag dbd þ dad dbg Þ þ lS dab dgd
½117
contains two surface elastic constants mS and lS. Typically, e* is between 0.01 and 0.1, and the surface stretch modulus is on the order of 2mMd/(1–2nM), where mM and nM are the elastic constants of the bulk material and d is the thickness of the surface layer, about one interatomic distance. With the above equations, the specific surface energy expansion, eqn [112], can be written as gðeab Þ ¼ g0 þ 2ðmS þ lS Þe eaa þ lS eab eab 1 ½118 þ lS eaa ebb 2 for isotropic materials. The energy associated with the residual surface stresses gab is included in the term g0. It represents the surface energy of a planar surface on a stress-free solid. 1.01.7.4.3 Surface stresses and bulk stresses for spherical cavities
The stretched surface of a cavity can relieve its residual strain e* to some degree by reducing the surface area at the expense of creating stresses in the surrounding material. In the absence of externally applied stresses, this creates a spherically symmetric deformation field that can be derived from a radial displacement function uðr Þ ¼ A=r 2
½119
where r is the distance from the cavity center. The following bulk strains and stresses can then be obtained: eyy ¼ e’’ ¼ A=r 3 err ¼ 2A=r 3 ; 3 srr ¼ 4mM A=r ; syy ¼ s’’ ¼ 2mM A=r 3
½120
The surface stresses, on the other hand, are determined from g ¼ gyy ¼ g’’ ¼ 2ðmS þ lS Þðe þ A=R3 Þ
½121
The constant A is chosen such as to satisfy the boundary condition for the radial bulk stress at the cavity surface: srr ðRÞ ¼
2g p R
½122
Fundamental Properties of Defects in Metals
This boundary condition replaces the incorrect one stated in eqn [110]. The cavity must always satisfy this mechanical equilibrium condition expressed by eqn [122], regardless of whether the thermodynamic equilibrium condition is satisfied or not. In other words, thermodynamic equilibrium and mechanical equilibrium obey two different and separate conditions. From eqns [121] and [122] it follows that the surface strains eyy and e’’ are both equal to
¼
A pR 4ðmS þ lS Þe ¼ R3 4mM R þ 4ðmS þ lS Þ pR 2g 4mM R þ 4ðmS þ lS Þ
½123
Using this result we can determine from eqn [118] the surface energy of the cavity as a function of its radius R: gðeðRÞÞ ¼ g0 þ 2ðmS þ lS Þ½2e þ eðRÞeðRÞ ½124 Associated with the surface strain e(R) is a stress and a strain field in the surrounding material given by eqn [120]. It gives rise to the strain energy ððð 1 sij eij d 3 r ¼ 8pR3 mM e2 ðRÞ ½125 U ðRÞ ¼ 2 The reference cavity radius R defined by eqn [106] undergoes a small change as the surface strains adjust to their mechanical equilibrium values given by eqn [120]. As a result, the change in cavity volume, its relaxation volume, is DV R ðRÞ ¼ 4pR2 uðRÞ ¼ 4pR3 eðRÞ
½126
When gas is present in the cavity at a pressure p, it performs the work pDVR when the surface relaxes. Therefore, the total free energy associated with the creation of a cavity is FC ðRÞ ¼ 4pR2 gðeðRÞÞ þ 8pR3 mM e2 ðRÞ 4pR3 peðRÞ
biaxial surface stretch modulus 2(mS þ lS). As mentioned above, the latter has the dimension of N m1, and we may then relate it to the corresponding bulk modulus 2mM/(1–2nM) by multiplying the latter with a surface layer thickness parameter h. The surface energy g0 has been determined both experimentally and from ab initio calculations and can be considered as known. The surface layer has been determined by Hamilton and Wolfer10 from atomistic simulations on Cu thin films to be one monolayer thick; hence d ¼ b. A value for the residual surface strain parameter e* has been chosen in Section 1.01.3.1 such that it reproduces the relaxation volume of a vacancy according to eqn [11]. What if one selects the same value for voids containing n vacancies? The relative relaxation volume, R =ðnOÞ ¼ 3eðRðnÞÞ, can now be that is, the ratio VnV computed with eqn [123] and the results are shown in Figure 23 by the solid curve. As it must, for n ¼ 1 it reproduces the vacancy relaxation volume of 0.25O. In addition, it also agrees with the overall trend of the atomistic results of Shimomura.48 Of course, the atomistic results for small vacancy cluster vary in a discontinuous manner with the cluster size. The surface stress model gives not only a reasonable approximation to these atomistic results, but also a valid extrapolation to relaxation volumes of large voids. The chemical potential of vacancies for voids can now be computed with eqn [127] as FC(R(n þ 1)) FC(R(n)). Figure 24 shows the results for Ni as the solid curve. The vacancy chemical potentials for
½127
FC(R(N)) replaces now the surface free energy FS(N) used in eqn [105] to arrive at the cavity surface tension 2g0/R. The latter is now given by FC(N þ 1) – FC(N). It will be evaluated in the next section and compared with 2g0/R. 1.01.7.4.4 Chemical potential of vacancies at cavities
The free energy of a void or bubble, according to eqn [127], depends now on three surface parameters instead of just one as in eqn [105]: it depends on the surface energy g0 of a planar surface, on the residual surface strain e* for such a planar surface, and on the
0 EAM, Shimomura Surface stress model
–0.05 Relaxation volume/void volume
eðRÞ ¼
31
Cu
Planar surface energy: 1.71 J m–2
–0.1
–0.15
–0.2
–0.25
–0.3
–0.35 1
10 100 1000 Number of vacancies in void
104
Figure 23 The relaxation volume of voids in Cu according to atomistic simulations by Shimamura48 and according to the surface stress model.
32
Fundamental Properties of Defects in Metals
The task is then to divide the solid into cells, each containing one individual sink, and to solve in each diffusion equations of the following type
2.5
Ni Vacancy chemical potential (eV)
2
r j ¼ P recombination
Capillary approx. Surface stress model EAM simulations 1.5
Planar surface energy:
2.38 J m–2
1
0.5
0 1
10 Void radius, R/b
Figure 24 Vacancy chemical potentials for voids in Ni.
voids are significantly lower than the capillary approximation predicts with a fixed surface energy (dashed curve). The chemical potentials from atomistic simulations of voids in Ni have been obtained by Adams and Wolfer49 using the Ni-EAM potential of Foiles et al.50 These results converge to those predicted with the surface stress model.
1.01.8 Sink Strengths and Biases 1.01.8.1
Effective Medium Approach
The fate of the radiation-produced atomic defects, namely self-interstitials and vacancies, is mainly determined at elevated temperatures by their diffusion from the places where they were created to the sinks where they are absorbed or annihilated, as in the encounter of an interstitial and a vacancy. As there are many sinks within each grain of an irradiated material, the spatial distribution of the atomic defects requires the solution of a very complex diffusion problem. Clearly, some approximations must be sought to arrive at an acceptable solution. First, we assume that the rate of defect generation, that is, the rate of displacements, is constant, and the defect concentrations between the sinks no longer changes with time or adjust rapidly when the number of sinks and their arrangement changes. Then within the regions between sinks, the diffusion fluxes are stationary, that is, @ ji ¼ Dij ðrÞCðrÞ þ Fi ðrÞCðrÞ @xj is independent of time.
½128
½129
for each mobile defect. Here, P is the rate of defect production per unit volume generated by the radiation, and the other terms represent the rates of defect disappearance by recombination with other migrating defects. On the outer boundary of this cell, the defect concentration C must then match the concentration in adjacent cells occupied by other sinks, and its gradient must vanish. This cellular approach has been pioneered by Bullough and collaborators,51 but the drift term, the second term in eqn [128], is omitted when solving the diffusion equation. Its effect is subsequently taken into account by changing the actual sink boundary into another, effective boundary at which the interaction energy between the sink and the approaching defect becomes of the order of kT, k being the Boltzmann constant. An alternate approach52,53 is to view a particular sink as embedded in an effective medium that maintains an average concentration of mobile defects at a distance far from this particular sink, and to neglect production and losses of mobile defects nearby. In this approximation, the r.h.s. of eqn [129] is set to zero, and the outer boundary condition far from the sink is that C approaches a constant value C that remains to be determined later from average rate equations. The diffusion equation is now solved with and without the drift term and the resulting defect current to the sink is evaluated. The ratio of the currents with and without the drift defines the sink bias factor. It is thereby possible to define unambiguously bias factors for each type of sink, and with these determine the net bias for a given microstructure. It is this embedding approach that we follow here to evaluate the bias of a sink. The first attempt to determine the dislocation bias by solving the diffusion equation with drift appears to have been made by Foreman.54 He employed a cellular approach retaining only the defect production term. Furthermore, anticipating small bias values, the drift term was treated by perturbation theory and numerous approximations were introduced in the derivation. The intent was to obtain rough estimates; nevertheless, Foreman concluded that the bias was larger than the empirical estimate. Shortly thereafter, Heald55 employed the embedding approach and used the solution of Ham56 in the form presented by Margvalashvili and Saralidze.57 We shall return to this solution below, as it is the only analytical one
Fundamental Properties of Defects in Metals
known. However, this solution applies only when the mobile defect is modeled as a center of dilatation (CD). For more general interactions, Wolfer and Askin58 developed a rigorous perturbation theory that includes also the interaction induced by externally applied stresses. The latter was shown to result in radiation-induced creep. They also compared the bias obtained from the perturbation theory carried out to second order with the bias from Ham’s solution, and demonstrated that both results agree only for weak centers of dilatation. Vacancies can be considered as such weak centers, but interstitials cannot. As a result, perturbation theory to second order is in general insufficient to evaluate the dislocation bias. 1.01.8.2
Dislocation Sink Strength and Bias
In order to obtain the sink strength, one first solves the steady state diffusion equation r j ¼ r2 ðDCÞ ¼ 0
per unit length of the dislocation. Here, R is an outer cut-off radius taken as half the average distance between dislocations, and rd is the dislocation core radius. It is assumed that at this radius the defect concentration becomes equal to the local, thermal equilibrium concentration. The total defect current to all dislocations is then proportional to the ½131
where r is the dislocation density. When the drift term is now included, the defect current changes to J ¼ ZJ0
1.01.8.2.1 The solution of Ham
One is for the diffusion to an edge dislocation when the interaction energy is given by eqn [45] and the stress field is that in an isotropic material. In this case W1 ðr ; ’Þ ¼
mb 1 þ n rel sin’ V 3p 1 n r
½133
expressed in polar coordinates (r, ’). The solution of the diffusion equation with drift determined by the size interaction energy [133] can be obtained in terms of products of modified Bessel functions with cosine functions, Kn cosðn’Þ and In cosðn’Þ,56 and the edge dislocation bias factor is then obtained 57 in the form
Kn ðrC =RÞ Kn ðrC =rd Þ 1 In ðrC =RÞ In ðrC =rd Þ n¼0
K0 ðrC =RÞ K0 ðrC =rd Þ 1 ‘nðR=r0 Þ ½134 I0 ðrC =RÞ I0 ðrC =rd Þ
Zedge ¼ ‘nðR=r0 Þ
1 X
ð2 dn0 Þ
where the capture radius is defined as
without a drift term. For the case of a straight dislocation, the solution depends only on the radial direction in a cylindrical coordinate system with the dislocation as the axis. The total current of defects is then given by 2p DC DC eq ½130 J0 ¼ ‘nðR=rd Þ
2pr Dislocation sink strength ¼ ‘nðR=rd Þ
33
ð1 þ nÞb mV rel rC ¼ 6pð1 nÞ kT
The series converges very rapidly for R rC, and it is sufficient, as the numerical evaluation shows, to retain only the zeroth order term. As before, 2R is the average distance between dislocations. If this distance becomes small as in a dense dislocation cell wall with narrow dislocation multipoles, the longrange stress fields of individual edge dislocations cancel each other, and the net interaction energy of eqn [133] is no longer valid. For example, consider an edge dislocation dipole as shown in Figure 25. The interaction energy with the migrating defects is now given by E1 ðx; yÞ ¼
mb 1 þ n rel V 3p 1 n " yþh ðx þ hÞ2 þ ðy þ hÞ2
yh ðx hÞ2 þ ðy hÞ2
½132
Z is called the bias factor, and there are as many such factors as there are diffusing defects and different types of sinks. The complexity of the interaction of a migrating defect with the strain field of the sink makes it difficult to find an analytical solution to the diffusion equation with drift. However, there exist a few important solutions.
½135
r
h
a
Figure 25 Edge dislocation dipole.
# ½136
34
Fundamental Properties of Defects in Metals
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi At distances r ¼ x 2 þ y 2 >> h, this interaction energy becomes mb 1 þ n rel 3=2 sin2a W1 ðr ; aÞ V 2 h 2 ½137 3p 1 n r when expressed in the polar coordinates indicated in Figure 25. Comparing this with the interaction energy for a singular edge dislocation, eqn [133], we see that interaction with an edge dislocation dipole falls off as r–2, and it has an angular periodicity of twice that for the single edge dislocation. The solution of the diffusion equation can again be constructed in terms of products of cosine functions, cos(2na), and the modified Bessel functions, but now of a different argument, namely 2rCrD/r2, where rD ¼ 21/2h is the radius of the dipole. If we take this radius to be the sink radius, then the bias factor for the dipole is Zdipole ¼ ‘nðR=rD Þ
1 K0 ð2rC rD =R2 Þ K0 ð2rC =rD Þ ½138 I0 ð2rC rD =R2 Þ I0 ð2rC =rD Þ
Again, just as for the bias of a single edge dislocation, eqn [134], we find when evaluating eqn [138] numerically that the first term, as shown, in this series already provides accurate results. The important material parameter that determines the bias factor of edge dislocation is the capture radius defined in eqn [135]. At this radius, and at polar angles perpendicular to the direction of the Burgers vector, the interaction energy is of the magnitude
p 1 ½139 W1 rC ; ¼ kT 2 2 We may then attach the following physical meaning to this capture radius: when a defect reaches the dislocation at this distance from its core, and the interaction is attractive, meaning negative, it is inevitably pulled into the core, and when it is repulsive, that is, positive, the defect is definitively repelled.
Table 14 lists values for both the interstitial and vacancy capture radii evaluated at half the melting points for various metals, and the bias factors for rd ¼ 2b and R/b ¼ 2250, corresponding to a dislocation density of 1012 m2. The vacancy capture radii are small for all metals, and about equal to what one would expect for the sum of the dislocation core radius plus the point defect radius, namely rd ’ 1–2b. In contrast, the interstitial capture radii are significantly larger, in particular for fcc metals when compared with bcc metals. The evaluation of eqn [134] gives the solid curve displayed in Figure 26. As already mentioned above, terms in the sum with n 1 contribute less than 0.00025 to the dislocation bias factors. In addition, for large values of R/rC, the term that depends on rC/r0 can be neglected, and the series expansions can be used for the modified Bessel functions K0 and I0. As a result, one then obtains the asymptotic approximation59 Zedge
‘nðR=r0 Þ ‘nð2R=rC Þ g
½140
where g ¼ 0.577216 is the Euler constant. This approximation is also shown in Figure 26, and it is seen that it coincides with the exact results for rC/b 6. However, for rC/b 2, eqn [140] gives incorrect bias factors less than one. For small values of rC/b, Wolfer and Ashkin58 have obtained from perturbation theory the following expression: Zedge 1 þ
½rC =ð2r0 Þ2 Oð½rC =ð2r0 Þ4 Þ ‘nðR=r0 Þ
½141
As indicated, extension of this perturbation theory to higher orders shows that an alternating series is obtained with poor convergence. This then suggests to seek a Pade approximation that may extend the usefulness of eqn [141]. For example,58
Table 14 Capture radii and bias factors for interstitials and vacancies evaluated at half the melting points and with the size interaction only Element
Vrel I =V
Vrel V =V
rC =b SIA
rC =b vacancies
ZdI
ZdV
Net bias ¼ ZdI =ZdV 1
Al Cr Cu Fe Mo Nb Ni Ta V W
1.9 1.21 1.55 1.1 1.1 0.76 1.8 1.05 0.25 1.03 0.92
0.31 0.22 0.25 0.27 0.19 0.27 0.20 6.43 0.30 0.18
14.32 8.31 13.46 8.69 10.62 5.08 14.77 1.56 5.06 14.91
2.36 1.11 1.80 1.94 1.84 1.24 1.64 1.177 1.47 1.70
1.358 1.229 1.342 1.239 1.284 1.135 1.366 1.020 1.134 1.369
1.043 1.011 1.026 1.030 1.027 1.013 1.022 0.154 1.020 1.024
0.302 0.216 0.308 0.208 0.250 0.120 0.337 0.112 0.337
Fundamental Properties of Defects in Metals
where Ei is the exponential integral function. Evaluation of eqn [144] gives the curve labeled as ‘Average’ in Figure 26. The angular average approximation [144] compares well with the exact result for large values of rC/b, that is, for interstitial capture radii, but slightly over-predicts vacancy bias factors.
Edge dislocation bias factor
1.5 Ham’s solution Asymptote Pade approx. 3.0 Angular average
1.4
35
1.3
1.01.8.2.2 Dislocation bias with size and modulus interactions 1.2
1.1
1 0
5 10 15 Capture radius/Burgers vector
20
The modulus interaction has been discussed in Section 1.01.5.3. Treating both the material as well as the defect inclusion as elastically isotropic, the modulus interaction depends on two diaelastic polarizabilities, aK and aG, for which values are provided in Table 11. For this isotropic case, the modulus interaction for edge dislocations is58 W2 ðr ; ’Þ ¼ ðA0 þ A2 cos2’Þðb=r Þ2
Figure 26 Edge dislocation bias factors based on Ham’s solution and various approximations to it.
with A0 ¼
2
Zedge ¼ 1 þ
½rC =ð2r0 Þ ‘nðR=r0 Þ þ m½rC =ð2r0 Þ2
½142
with m ¼ 3 fits the exact results for 0 rC/b 6 as seen in Figure 26. Very accurate results for the bias factors of edge dislocations can therefore be produced with eqn [142] for small capture radii and with eqn [141] for large capture radii. The two approximations transition extremely well at rC/b ¼ 6. These approximations suggest a way to proceed when the interaction energy assumes a more complicated form than in eqn [133]. It is therefore tempting to see if an angular average of the size interaction energy, eqn [134], could be used to evaluate at least approximately the bias factors. Obviously, the angular average of sinj is zero. The diffusion flux will wind around the dislocation as it approaches the core in order to avoid regions where the interaction energy W1 becomes strongly repulsive. Therefore, an average should only be taken over the angular range where W1 is attractive, that is, negative. So, if we replace W1(r, ’) in eqn [134] with W1(r, p/2)/2 in the case of interstitials and with W1(r, 3p/2)/2 in the case of vacancies and evaluate the equation Zd Ð R rd
‘nðR=rd Þ exp½bW1 ðr Þd ‘nðr Þ
½143
‘nðR=r0 Þ EiðrC =rd Þ EiðrC =RÞ
½144
aK ð1 2nÞ2 þ 43am ð1 n þ n2 Þ ½4pð1 nÞ2
½146
and A2 ¼
ðaK 23am Þð1 2nÞ2 þ 4am nð1 nÞ ½4pð1 nÞ2
½147
The perturbation theory of Wolfer and Ashkin58 with the sum of the size interaction [130] and the modulus interaction [145] gives the result
1 2A0 edge 2 ½148 1þ rC þ Z kT ð2rd Þ2 ‘nðR=r0 Þ This suggests that an effective capture radius can be defined as rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2A0 rC;eff ¼ rC2 þ ½149 kT and replacing rC with it in eqn [144] yields bias factors that include both the effects of size and modulus interactions. Using the values given in Table 11 for the diaelastic polarizabilities, effective capture radii and new bias factors are obtained and presented in Table 15. Comparing these results with the corresponding ones in Table 14 shows that the modulus interaction contributes to the net bias about 25% for fcc metals, and about 10–15% for bcc metals. 1.01.8.3
we obtain58 Zd
½145
Bias of Voids and Bubbles
In elastically isotropic materials, the interaction of vacancies as well as interstitials with spherical cavities depends only on the radial distance r to the
36
Fundamental Properties of Defects in Metals
Table 15
Capture radii and bias factors of edge dislocations with size and modulus interactions
Element
rC, eff /b SIA
rC, eff /b vacancy
ZdI
ZdV
Net bias ¼ ZdI =ZdV 1
Al Cu Ni Cr Fe Mo
20.38 18.30 20.09 10.47 10.75 12.73
3.55 2.89 2.98 2.69 3.12 3.27
1.70 1.66 1.69 1.46 1.47 1.53
1.21 1.18 1.18 1.17 1.19 1.20
0.405 0.408 0.434 0.256 0.240 0.276
cavity center. Denoting by WS(r) this interaction energy at the saddle point configurations, the radial defect flux that includes the drift can be written as jr ¼ exp½bW S ðr Þ
d DCðr Þexp½bW S ðr Þ ½150 dr
The defect current J through each concentric sphere around the cavity is a constant, that is, 4pr2jr ¼ J. Integration of [150] then leads to DCðr Þexp½bW S ðr Þ ¼DCðRÞexp½bW S ðRÞ ðr J R
dr exp½bW S ðrÞ r2
Matching this solution to the boundary conditions at r ! 1, where Cðr Þ ! C, and at r ¼ R, where CðRÞ ¼ C 0 is the defect concentration in local equilibrium with the cavity, gives the defect current 4pRðDC DC 0 Þ J ¼ Ð R=ðRþhÞ exp½bW S ðR=r Þd ðR=r Þ 0 ¼ 4pRZ0 ðDC DC 0 Þ
ð1 þ nÞ2 mðV rel Þ2 U im ðr ; RÞ ¼ 36pð1 nÞR3
2gðRÞ 2me ¼ 1 þ ð1 2nÞðR=bÞ R
½154
where e* is the residual surface strain of a planar surface. It is shown in Section 1.01.3.1, that this residual surface strain parameter can be related to the relaxation volume of the vacancy, and according to eqn [11] 2 e ¼ ð1 nÞVVrel =O 3
½155
With eqns [154] and [155], the modulus interaction then takes the final form rel 2 2 V 2 1n W2 ðR=r Þ ¼ aG V O 3 1 þ ð1 2nÞðR=bÞ
The evaluation of the void bias factor "ð Z ðRÞ ¼ 0
R=ðRþhÞ
exp½bW ðR=r Þd ðR=r Þ S
½156
#1 ½157
0
1 X nðn 1Þð2n 1Þð2n þ 1Þ R 2n þ 2 n2 þ ð1 2nÞn þ 1 n
The strain field around the cavity is caused by both a pressure p when gas resides in the cavity, and by the surface stress g(R). Note that this surface stress is not equal to the surface energy or a constant as usually assumed. As shown in Section 1.01.7.4.3, it changes with the cavity radius and the gas pressure p. For the case of voids, when p ¼ 0, the surface tension is found to be
½151
The integration extends up to the distance h(R) from the cavity surface where the strain energy of the defect becomes zero, as discussed in Section 1.01.5.3 and shown in Figure 17. The interaction energy W S(R/r) consists of two parts, the image interaction and a modulus interaction. The former has been discussed in Section 1.01.5.3, and it has been given by Moon and Pao32 in the form
n¼2
Next, the modulus interaction with the strain field of the cavity is60
3aG 2gðRÞ 2 R 6 W2 ðR=r Þ ¼ 2 p ½153 8m R r
r
½152
The series converges slowly as r approaches the cavity radius R. However, when R/r 0.99, no more than about 1000 terms are required to obtain accurate results.
requires numerical integration. As an example, the bias factors for self-interstitials and vacancies are computed for Ni, at a temperature of 773 K, and using the defect parameters in their stable configurations as given in Tables 2, 7, and 11. The results are presented in Figure 27. The solid curves are obtained when both the image and the modulus interactions are included, while the dashed curves neglect the contribution of
Fundamental Properties of Defects in Metals
2.2
SIA total bias SIA image bias Vac. total bias Vac. image bias
Void bias factors
2
1.8 Ni 1.6
1.4
1.2
1 1
10 Void radius, R/b
100
Figure 27 Bias factors for voids in nickel as a function of the void radius in units of the Burgers vector.
the modulus interaction to the void bias factors. It is seen that the modulus interaction contributes little to the void bias for vacancies, while it enhances the void bias for interstitials significantly.
1.01.9 Conclusions and Outlook Several past decades of intense research have resulted in a good understanding of the fundamental properties of vacancies and self-interstitials in pure metals. We have reviewed this understanding from the following point of view: how do these fundamental properties affect radiation damage at elevated temperatures that exist in nuclear reactors. The key parameters that emerge from this perspective are the displacement energy of Frenkel pairs, the formation and migration energies of vacancies and selfinterstitials, and their relaxation volumes and elastic polarizabilities. While the physical basis for these key parameters is understood, obtaining precise values for them by experimental and theoretical means remains a formidable challenge. In particular, this is true for the type of alloys that are used for components in the core of nuclear reactors. In general, these are complex alloys. For example, the austenitic stainless steels are composed of major alloy constituents, namely iron, nickel, and chromium, and many minor alloy elements such as molybdenum, titanium, manganese, carbon, and silicon. Therefore, many different types of vacancies and self-interstitials can exist in
37
these alloys with potentially different properties. It is unlikely that all these different properties can actually be measured. Rather, only effective average properties, such as the self-diffusion coefficient, can be determined experimentally, and theoretical models must be employed to relate effective properties to the individual properties of the different vacancies. At the present time, electronic structure method still require further development before effective properties of defects in complex alloys can be calculated. In fact, only recently has it become possible to calculate, for example, accurate values for the formation energies of mono- and divacancies in pure metals. Two advances have been responsible for the progress. First, density functional theory requires different implementations when applied to bulk and to surface properties of metals. The uniform electron gas that serves as a starting point for the electron density functional in the bulk interior of solids is not suitable to formulate the corresponding functional for the ‘electron edge gas.’ As shown by Kohn and Mattsson,61,62 functionals must be developed that join the edge and the interior bulk regions, and Armiento and Mattsson63,64 have proposed and tested functionals for these two regions and how to join them. When applied to vacancies in metals,65,66 formation energies are predicted that are in much better agreement with experimental results. But there are also differences. For example, Carling et al.65 obtain a repulsive (positive) binding energy for divacancies in Al. However, this result may be due to the limited number of atoms employed in the calculations and the periodic boundary conditions. This brings us to the second advance that has recently been made, namely the implementation of an orbital-free electron density functional theory based on finite-element methods.67 With this approach, much larger systems containing effectively millions of atoms can be treated; these systems are truly finite, and realistic boundary conditions can be applied to them. Figures 28 and 29 reproduced from Gavini et al.67 reveal a surprisingly large effect of the system size, that is, the effective number of atoms in a finite crystal into which one or two vacancies have been introduced. As demonstrated by these results, in order to obtain defect properties that are independent of the size of a finite system requires thousands of atoms as well as their full relaxation. In other words, electronic structure calculations need to be combined with continuum elasticity descriptions to predict radiation effects and to develop better alloys for nuclear power generation.
38
Fundamental Properties of Defects in Metals
clusters of several hundreds of atomic defects, and the volume fraction they occupy ranges from a part per million to perhaps a tenth of a percent. So if r is the typical radius of a defect under consideration and 2R the average distance between them, then the defect volume fraction
0.9
Vacancy formation energy (eV)
Unrelaxed Relaxed 0.85
S ¼ ð=RÞ3
0.8
0.75
0.7
1
10
100 1000 104 Number of atoms
105
106
Figure 28 Vacancy formation energy for Al as a function of system size, and with and without relaxing the atomic positions when removing a central atom to create a vacancy. Reproduced from Gavini, V.; Bhattacharya, K.; Ortiz, M. Mech. J. Phys. Solids 2007, 55, 697.
Di-vacancy binding energy in Al (eV)
0.1 <110> <100>
0.05 0 –0.05 –0.1 –0.15 –0.2 –0.25
1
10
100 1000 104 Number of atoms
105
106
Figure 29 Binding energy of a divacancy in Al as a function of system size and orientation. Reproduced from Gavini, V.; Bhattacharya, K.; Ortiz, M. Mech. J. Phys. Solids 2007, 55, 697.
½A1
is a small number. As a result, each defect is surrounded by a cell that contains no other defect, and the solid can be viewed as composed of these cells. If the solid as a whole is free of external stresses on its macroscopic surface, then the normal stress component averaged over the surface of each cell is also zero, since on the cell boundary, the stress fields of two neighboring defects overlap and cancel on average. In the final step in this cellular approach, we approximate the typical cell by a spherical region with radius R with a defect of radius r at its center. It is this effective medium approximation that elevates the analysis of a sphere with a defect at its center to more than just an academic exercise, and its results are more generally valid for defects in solids of finite extent. Atomistic computer simulations using either semi-empirical potentials or first-principle methods are usually carried out in a periodically repeated cell that contains a finite number of atoms. However, in this case the medium is in fact infinite, and the results are different from those obtained for a finite crystal on whose external surface the stress field created by the defect satisfies the boundary condition of zero tractions, that is, the vanishing of the stress component normal to the external surface. Satisfying this boundary condition is different from having a defect stress field that approaches zero at infinity. This difference is often attributed to the so-called image stresses that exist in a finite crystal, but not in an infinite one. The model of a defect at the center of a finite sphere of radius R captures this difference, and as we shall see, this difference becomes independent of the radius R as it approaches infinity but such that S, the defect concentration, remains small but constant.
Appendix A Elasticity Models: Defects at the Center of a Spherical Body
A2 The Isotropic, Elastic Sphere with a Defect at Its Center
A1
Consider then a small defect with a radius r situated at the center of an elastically isotropic sphere of radius R. The defect is modeled in four different ways:
An Effective Medium Approximation
The crystal defects that we consider here are mainly of microscopic size ranging from single atoms to
Fundamental Properties of Defects in Metals
As a center of dilatation (CD): the defect is created by displacing the inner radius r of the surrounding matrix by an amount uðrÞ ¼ cr
½A2
39
partial differentiation. Hooke’s law then connects the stresses with the elastic part of the strains, which for the isotropic case reads as sij ¼ mðui;j þ uj ;i Þ þ ldij uk;k 2meij ldij ekk
½A5
Here, the parameter c is referred to as the strength of the defect. As a cavity (CA): the inner surface is loaded by a pressure p, giving rise to a radial stress component in the surrounding medium, which has the boundary value
Here, repeated indices imply a summation and a comma before an index, say j, means a partial derivative with regard to xj . The mechanical equilibrium equations are sij ; j ¼ 0, or after substitutions of Hooke’s law
sr ðrÞ ¼ p
ðm þ lÞuj ; ji þ mui; jj ¼ 0
½A3
The pressure p can also represent radial forces that the defect exerts on its nearest neighbor atoms of the surrounding matrix, and they are then referred to as Kanzaki forces. As an inclusion (INC): the material within the defect region is subject to a transformation strain eij as if it had endured a phase transformation. We shall treat here only the simple case of an isotropic transformation strain eij ¼ dij where dij is the Kronecker matrix. As an inhomogeneity (IHG): In addition to a transformation strain, the defect region has also acquired different elastic constants from the surrounding matrix. The boundary conditions at the interface between the defect and its surrounding matrix are that the radial displacement and the radial stress component must be continuous. These different models of defects correspond in essence to the three possible boundary conditions on the defect–matrix interface, namely the Dirichlet boundary condition for the CD, the von Neuman condition for the CA, and the mixed one for the INC and IHG. One could also consider these three different boundary conditions for the external surface. However, prescribing a value for the displacement of the external surface or a mixed boundary condition implies the presence of yet another surrounding medium, in which the defect-containing sphere would be embedded. Here, these cases will not be considered. We then restrict the following treatment to the boundary condition of a free external surface for which sr ðRÞ ¼ 0
½A4
The solution of the elasticity problems associated with all defect models is rather elementary and can be found in many textbooks. However, as a way to introduce our notation, we sketch the procedure. The deformation is described by a displacement r Þ, from which the strain tensor is obtained by field ui ð~
½A6
We assumed here that the transformation strains eij are constant within the defect region. For a spherically symmetric problem, the displacement field possesses only one radial component u, which satisfies the differential equation d 1 d 2 uÞ ¼0 ½A7 ðr ðl þ 2mÞ dr r 2 dr The solution to this equation is uðr Þ ¼ Ar þ B=r 2
½A8
with two unknown constants A and B for each region, for the matrix that surrounds the defect and for the defect region itself. However, for the defect region B ¼ 0. The remaining constants are to be determined by the boundary conditions. For later use we give the radial stress component sr ¼ ð2m þ 3lÞðA Þ 4mB=r 3 ¼ 3K ðA Þ 4mB=r 3
½A9
for the case of an isotropically transformed inclusion, that is, when eij ¼ dij . Furthermore, instead of the Lame’s constants m and l, we introduced the bulk modulus K and the shear modulus m. The following ratio involving the two elastic constants will appear often, and we reserve the symbol o¼
4m 2ð1 2nÞ ¼ ¼ gE 1 3K 1þn
½A10
for it, where n is Poison’s ratio and gE is the Eshelby factor. The hydrostatic stress is related to the elastic dilatation or the lattice parameter change by the equation 1 Da ½A11 sH ¼ sii ¼ 3K ðA Þ ¼ 3K 3 a We see that for a defect in the center of an isotropic sphere the lattice parameter changes are uniform throughout the matrix.
40
Fundamental Properties of Defects in Metals
The density of strain energy, on the other hand, is strongly peaked near the defect, as can be seen from the formula 1 9 fðr Þ ¼ sij ui; j ¼ K ðA Þ2 þ 6mB 2 =r 6 2 2
½A12
Integrating this function over the entire sphere gives the strain energy associated with the defect, ððð ðR 1 sij ui; j dV ¼ 4p fðr Þr 2 dr U0 ¼ ½A13 2 0 Finally, let us state the formulae for volume changes. The external volume change is computed from the equation DV ¼ 4pR2 uðRÞ ¼ 4pR3 ðA þ B=R3 Þ or
DV ¼ 3ðA þ B=R3 Þ V
½A14
average by < > brackets. However, if the defect region consists of a very different crystal structure or atoms with much lower form factors, then the defect region does not contribute to the diffraction pattern, and the lattice parameter change is due to the matrix only. When the defect region is excluded, the lattice parameter change is indicated by ( ) parentheses. We obtain the coefficients A and B by solving the equations for the boundary conditions. For example, for the case of an inhomogeneity, the radial stress components vanish on the external surface, sr ðRÞ ¼ 3KA 4mB=R3 ¼ 0
½A16
On the interface between the inclusion region and the matrix, that is, at r ¼ r, the displacements and the radial stress components must be continuous, or AI r ¼ Ar þ B=r2
½A17
In a similar fashion, one defines the change of the defect volume due to the constraints imposed by the surrounding matrix, as
and
Du ½A15 ¼ 3ðA þ B=r3 Þ u A last point must be made regarding the measurement of the lattice parameter change. If the atoms in the defect region contribute to the diffraction pattern just as the matrix atoms, we must calculate the appropriate average lattice parameter change over the entire volume V. We denote this
Here, the subscript ‘I’ designates a parameter for the inclusion region, while parameters for the matrix are without a subscript. Solution of the three eqns [A17]-[A19] determines the three parameters A, B, and AI listed in the first three rows and the last column of Table A1. We note that the stress within the inhomogeneous inclusion is purely hydrostatic (because BI ¼ 0), and the results do not depend on
3KI ðAI Þ ¼ 3KA 4mB=r3
½A18
Table A1 The solutions for the integration constants A and B, and volume changes, lattice parameter changes, and defect strain energies that derive from them Defect type
CD
CA
INC
IHG
A for matrix
coS 1 þ oS
pS 3Kð1 SÞ
oS 1þo
oS 1 þ oS þ koð1 SÞ
B for matrix
cr3 1 þ oS
pr3 4mð1 SÞ
r3 1þo
r3 1 þ oS þ koð1 SÞ
ð1 þ oSÞ 1þo 3ð1 þ oSÞ 1þo
ð1 þ oSÞ 1 þ oS þ koð1 SÞ 3ð1 þ oSÞ 1 þ oS þ koð1 SÞ 3ð1 þ oSÞ 1 þ oS þ koð1 SÞ ð1 þ oÞS 1 þ oS þ koð1 SÞ
AI for defect Du of defect u DV of solid V Da for solid a Da for matrix a Defect energy U0
NA
NA 3c
3pð1 þ oSÞ 4mð1 SÞ
1þo 3c S 1 þ oS 1þo c S 1 þ oS
3pð1 þ oÞ S 4mð1 SÞ pS 3K
3S
ð1 SÞo S 1 þ oS oþS 9c2 uk 1S
pS 3K
oð1 SÞS 1þo 1S 62 um 1þo
c
3p2 uð1 þ oSÞ 8mð1 SÞ
S
oð1 SÞS 1 þ oS þ koð1 SÞ 62 umð1 SÞ 1 þ oS þ koð1 SÞ
41
Fundamental Properties of Defects in Metals
Table A2
Simplified expressions obtained from the general ones in Table 2 for small defect concentrations, S << 1
Defect type
CD
A for matrix
coS
B for matrix
cr3
A for defect Du of defect u DV of solid V Da for solid a Da for matrix a
NA
CA pS 3K pr3 4m NA 3p 4m
3c
3pS ð1 þ oÞ 4m pS 3K pS 3K 3p2 u 8m
3cSð1 þ oÞ cSð1 þ oÞ cSo 12mc2 u
Defect energy U0
the shear modulus of the inhomogeneity, but only on the ratio k ¼ K =KI
½A19
For the inclusion (INC), its bulk modulus is the same as for the matrix. Therefore, its parameters follow from those for the IHG by setting k ¼ 1. With the parameters A and B determined, it is straightforward to evaluate volume changes, lattice parameter changes, and the defect strain energy U0. Table A1 contains a tabulation of all these parameters for the four different defect types, and for all possible values of the defect volume fraction S. For application to defect concentrations in irradiated materials, it may be assumed that S<<1, and the expressions in Table A1 can be simplified to those shown in Table A2.
Appendix B Representation of Defects by Atomic Forces and by Multipole Tensors B1 Kanzaki Forces In a perfect crystal, the equilibrium positions of all atoms are such that they exert no net forces on each other. However, when the crystal is subject to either external forces or internal forces originating from crystal defects, mutual interaction forces arise. For example, the atoms surrounding a vacancy move to new positions in response to the missing interaction
INC
IHG
oS 1þo r3 1þo 1þo 3 1þo
oS 1 þ ko r3 1 þ ko 1 þ ko 3 1 þ ko 1þo 3S 1 þ ko 1þo S 1 þ ko oS 1 þ ko 6m2 u 1 þ ko
3S S oS 1þo 6m2 u 1þo
forces from the atom that would normally occupy the vacant site. One may imagine that these missing forces are applied to atoms in a perfect crystal and given such magnitudes and directions that they produce the same strain and stress fields as exist in a crystal with a real vacancy. These fictitious forces that are applied to a perfect crystal are known by the name of their inventor,26 as the Kanzaki forces. For a localized defect in an elastic medium, the elastic strain or stress field can be generated with a finite number of point forces f(a) acting at the lattice sites r0 þ R(a), where r0 is the location of the center of the defect region and R(a) is the lattice vector from this center to the adjacent atom on which the force is to be applied. In harmonic crystal lattices, or equivalently, in solids that deform according to linear elasticity theory, the displacement field created by all these point forces is then given by ul ðrÞ ¼
Z X
ðaÞ
Gij ðr; r0 þ RðaÞ Þf j
½B1
a
where Gij is either the lattice Green’s function when the solid is described by a harmonic crystal lattice or the elastic Green’s function when it is described as an elastic continuum. For distances jr r0 j >> RðaÞ , we can expand the elastic Green’s function into a Taylor series around the point r0. Using the notation Gij ; k0 ¼
@ Gij @xk0
½B2
42
Fundamental Properties of Defects in Metals
one obtains ui ðrÞ ¼ Gij ðr; r0 Þ
X
ðaÞ fj
þ Gij ;k0 ðr; r0 Þ
X
a
ðaÞ ðaÞ Rk f j
X
ðaÞ ðaÞ ðaÞ Rk Rl fj
þ ½B3
a
The set of z forces must of course be selfequilibrating so as to not impose any net force or net force moment on the solid medium. Hence, z X
ðaÞ
fj
¼0
ðaÞ
ðaÞ
ðaÞ
f j Rk f k Rj
¼0
½B5
a¼1
As a consequence of eqn [B3], the first term in the multipole expansion (B3) vanishes. The next term contains the first moment of the forces, which is called the dipole tensor of the defect, and is denoted by z X
½B11
Gij1;k0 ¼ Gij1;k
½B12
The multipole expansion for the displacement field of a point defect can then finally be written as
and ðaÞ
gij ðoÞ jr r 0 j
where o is the solid angle of the unit vector parallel to (r-r0 ). It is then permissible to change the differentiations with respect to r0 to differentiations with respect to r, taking into account changes in sign. For example,
½B4
a¼1
Pjk ¼
Gij1 ¼
a
1 þ Gij ;k0 l 0 ðr; r0 Þ 2!
z
X
In an infinite medium, the elastic Green’s function has the form
ðaÞ
ðaÞ
f j Rk ¼ Pkj
½B6
1 ui ðrÞ ¼ Gij1;k ðr r0 ÞPjk þ Gij1;kl ðr r0 ÞPjkl 2! 1 Gij1;klm ðr r0 ÞPjklm þ ½B13 3! Using eqn [B11], we find that the first term falls off as 1/r2, the second as 1/r3, etc. B2
Volume Change from Kanzaki Forces
a¼1
and as indicated, it is a symmetric tensor by virtue of eqn [B5]. The second moment of the forces Pjkl ¼
z X
ðaÞ ðaÞ ðaÞ f j Rk Rl
½B7
a¼1
is called the quadrupole tensor, and the third moment Pjklm ¼
z X
ðaÞ
ðaÞ
ðaÞ
f j Rk Rl RðaÞ m
½B8
a¼1
the octupole tensor. In terms of these multipole tensors, the displacement field of the defect region can be written in the series expansion ui ðrÞ ¼ Gij ;k0 ðr; r0 ÞPjk þ Gij ;k0 l 0 ðr; r0 ÞPjkl þ
½B9
If the crystal lattice defect has certain symmetries, some terms in this multipole expansion may not be present. For example, if the defect possesses an inversion symmetry, then for each force there exists an equal but opposite force at a position equal to but opposite to the position vector belonging to the mirror force. For such a defect, n
P jkl......: ¼ ð1Þ Pjkl......: n indices
and all multipole tensors of odd rank vanish.
½B10
The local volume change at any point in a solid due to elastic strains is given by 1 sii 3K and the total volume change is then ððð 1 sii ðrÞd3 r DV ¼ 3K e11 þ e22 þ e33 ¼ eii ¼
½B14
½B15
We now transform this volume integral by using the following identity: sii ¼ sim dmi ¼ ðsim xi Þ;m sim;m xi ¼ ðsim xi Þ;m þfi xi
½B16
Here, f (r) represents the distribution of internal forces that enter into the equilibrium equations that the stresses must satisfy, namely sim;m þ fi ¼ 0
With the formula [B15], the total volume change can be written as the sum of two terms. The first one contains as its integrant the divergence of the vector field sim xi , and it can therefore be converted with the Gauss theorem into a surface integral. As a result DV ¼
ððð 1 xi fi d 3 r ∯ xi sim nm dS þ 3K
½B17
The surface integration is over the surface tractions sim nm , where n is the surface normal vector. If no
Fundamental Properties of Defects in Metals
external loads are applied on the surface of the solid, then the surface tractions vanish, and the first term in [B16] is zero. When the internal body forces are Kanzaki forces, then fi ðrÞ ¼
z X a¼1
fi
ðaÞ
ðaÞ
dðr RðaÞ Þ xi ¼ Ri
½B18
and using (B6) one obtains DV ¼
1 ðP11 þ P22 þ P33 Þ 3K
½B19
It is immediately obvious from this derivation that when more than one point defect is present in the solid, the volume change is simply the sum of individual traces of their dipole tensors divided by three times the bulk modulus of the solid. B3 Connection of Kanzaki Forces with Transformation Strains As we have shown in Appendix A, lattice imperfections, such as a self-interstitials, a small precipitate, or small dislocation loops can also be modeled as an inclusion. These can be created by transforming a region O in the perfect crystal to a different crystal structure. If the transformation strain is emn ðRÞ, then the displacement field outside this region is given by ð ½B20 ui ðrÞ ¼ Cjkmn d3 RGij :k0 ðr; r0 þ RÞemn ðRÞ O
where G is now the elastic Green’s function. Again, far from the defect region where jr r0 j >> jRj, we may employ a Taylor series expansion for the Green’s function, and we end up with the multipole expansion of eqn [B9]. The multipole tensors are now given by ð ½B21 Pjk ¼ Cjkmn d3 R emn ðRÞ O
ð Pjkl ¼ Cjkmn d3 R Rl emn ðRÞ
½B22
ð Pjklp ¼ Cjkmn d3 R Rl Rp emn ðRÞ
B4 Multipole Tensors for a Spherical Inclusion 3 Suppose that in a spherical domain O ¼ 4p 3 a the transformation strain is uniform while it vanishes outside this domain. Then, in the above eqns [B20] to [B22], the transformation strain tensor can be taken outside the integral, and it remains to solve integrals of the following type: þ ða ð 3 ^ R ^ ^ d RRa Rb .. .R2n ¼ dR R2þ2n doR R a b ... R2n 0
O
þ 4pa 3þ2n ^ R ^ ^ doR R ¼ ½B24 a b ... R2n 2n þ 3 ^ ¼ R =jRj is the a-th component of the unit Here, R a a vector of R, and the remaining integral in eqn [B24] is over the surface of the unit sphere. The value for these surface integrals can be found from the general formula
þ X 1 1 ^_ 2n ¼ doR R^_ a R^_ b :... R dab dcd ...d2n1;2n 4p ð2n þ 1Þ!!
½B23
½B25
The sum extends over all possible combinations of the indices, and hence it contains (2n-1)!! terms. The double factorial is defined as ð2n þ 1Þ!! ¼ 1 3 5 ð2n þ 1Þ From these relations, one then finds the following multipole tensors for a spherical inclusion: Pjk ¼ OCjkmn emn a2 djk Ppq 5 4 a Pjkpqrs ¼ ðdjk dpq þ djp dkp þ djq dkp ÞPrs ½B26 35 We see that all multipoles tensors of higher rank than two are given in terms of the dipole tensor, and all tensors with an odd rank are zero. When the transformation strain is that associated with an isotropic expansion, then 1 DV ½B27 dmn emn ¼ 3 O For cubic crystals, the dipole tensor is in this case Pjkpq ¼
Pjk ¼ K DV djk
O
43
½B28
where K is the bulk modulus. This equation agrees with eqn [B19].
O
The characterization of atomic defects by multipole tensors then encompasses their description by either Kanzaki forces or by transformation strains. These tensors, in particular the dipole tensor Pjk, serve as more general parameter for their properties.
B5 Multipole Tensors for a Plate-Like Inclusion Self-interstitials may aggregate into planar clusters with their dumbbell axes aligned in parallel. We may
44
Fundamental Properties of Defects in Metals
view them as plate-like inclusions of thickness h and with a normal vector n. If we transform this plate by displacing one of its faces by b relative to the other face, then the transformation displacement field throughout the unconstrained plate is xj ½B29 uiT ¼ bi nj h where x j are the components of the position vector. The transformation strain is obtained by differentiation and found to be 1 T 1 þ ujT;i ¼ ðbi nj þ bj ni Þ ½B30 eij ¼ ui;j 2 2h Inserting these transformation strains into eqn [B21], we find for this plate-like inclusion the dipole tensor Pjk ¼ Cjkmn bm nn A
½B31
where A is the area of the plate. Note that this result is independent of the thickness h and of the shape of the plate. However, to evaluate the higher order multipole tensors, one must specify the shape of the plate. Let us consider a circular plate of radius c, vanishing thickness h!0, and with its normal unit vector n pointing in the x3-direction. Then, all tensors Pjkpq. . . vanish that have one or more indices that are equal to 3. For all other cases, the tensor components can be obtained with the formula ðð
the one for a circular dislocation loop by Kroupa36 reveals that the dipole approximation gives an accurate representation for the dislocation loop for distances r 2c.
References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.
c 2ðMþN þ1Þ ð2M 1Þ!!ð2N 1Þ!! x12M x22N dx1 dx2 ¼ 2p ½B32 2ðM þ N þ 1Þ 2ðMþN Þ ðM þ N Þ
14. 15.
All multipole tensors of odd rank vanish, and the remaining can again be expressed in terms of the dipole tensor. For example, the quadrupole tensor can be written as
16. 17. 18.
Pjkpq ¼ Pjk Qpq where Q
pq
0 1 c2@ ¼ 0 4 0
0 1 0
1 0 0 1 A 0 A¼ @ 0 4p 0 0
½B33 0 1 0
1 0 0A 0
½B34
For an infinite isotropic material, the far-field displacement components of this circular platelet are given by ui ðrÞ ffi Gij1;k ðrÞPjk ffi
A 1 1 2n ½ni bk xk þ bi nk xk xi bk nk 8pð1 nÞ r 2 r 3 ½B35 þ 3 xi bk nk bl nl r
where the distance vector r is from the center of the platelet. A comparison of this displacement field with
19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34.
Jung, P. In Landolt-Bo¨rnstein; Springer-Verlag: Berlin, 1991; Vol. III/25. Jung, P. Radiat. Effects 1981, 59, 103–104. Leibfried, G. Z. Phys. 1950, 127, 344–356. Damask, A. C.; Dienes, G. J. Point Defects in Metals; Gordon and Breach: New York, 1963. Seeger, A.; Mehrer, H. In Vacancies and Interstitials in Metals; Seeger, A., Schumacher, D., Schilling, W., Diehl, J., Eds.; North-Holland: Amsterdam, 1970; pp 1–58. Siegel, R. W. J. Nucl. Mater. 1978, 69&70, 117–146. Ehrhart, P.; Schultz, H. In Landolt-Bo¨rnstein; Springer-Verlag: Berlin, 1991; Vol. III/25. Varotsos, P. A.; Alexopoulos, K. D. Thermodynamics of Point Defects and Their Relation to Bulk Properties; North-Holland: Amsterdam, 1986. Tyson, W. R.; Miller, W. A. Surf. Sci. 1977, 62, 267–276. Hamilton, J. C.; Wolfer, W. G. Surf. Sci. 2009, 603, 1284–1291. Flynn, C. P. Phys. Rev. 1968, 171, 682–698; Phys. Rev. 1969, 179, 920. Ehrhart, P.; Robrock, K. H.; Schober, H. R. In Physics of Radiation Effects in Crystals; Johnson, R. A., Orlov, A. N., Eds.; Elsevier: Amsterdam, 1986; p 3. Kornblit, L.; Pelleg, J.; Rabinovitch, A. Phys. Rev. 1977, 16, 1164–1167. Kornblit, L. Phys. Rev. 1978, B17, 575–582. Flynn, C. P. Point Defects and Diffusion; Oxford University Press: London, 1972; Chapter 6. Brown, A. M.; Ashby, M. F. Acta Met. 1980, 28, 1085–1101. Wallace, D. Proc. Roy. Soc. Lond A 1991, 433, 615–630. Wang, Z.; Lazor, P.; Saxena, S. K. Physica B 2001, 293, 408–416. Wolfer, W. G. J. Phys. F Met. Phys. 1982, 12, 425–433. Mura, T. Micromechanics of Defects in Solids; Martinus Nijhoff: The Hague, 1982, Chapter 2.13. Eshelby, J. D. Progress in Solid Mechanics; Sneddon, I. N., Hill, R., Eds.; North-Holland: Amsterdam, 1961; Vol. II, Chapter 3. Zener, C. Trans. AIME 1942, 147, 361. Schoeck, G. Phys. Stat. Sol. B 1979, 94, 147–151. Guinan, M. W.; Steinberg, D. J. J. Phys. Chem. Solids 1974, 35, 1501–1512. Schilling, W. J. Nucl. Mater. 1978, 69&70, 465–489. Kanzaki, H. J. Phys, J. Chem. Solids 1957, 2, 24–36. Kro¨ner, E. Phys. Kond. Mat. 1964, 2, 262. Robrock, K. H. Mechanical Relaxation of Interstitials in Irradiated Metals; Springer-Verlag: Berlin, 1990. Dederichs, P. H.; Lehmann, C.; Schober, H. R.; Scholz, A.; Zeller, R. J. Nucl. Mater. 1978, 69&70, 176–199. Dederichs, P. H.; Lehmann, C.; Scholz, H. R. Z. Physik B 1975, 20, 155–163. Ackland, G. J. J. Nucl. Mater. 1988, 152, 53–63. Moon, F. C.; Pao, Y. H. J. Appl. Phys. 1967, 38, 595–601. Kronmu¨ller, H.; Frank, W.; Hornung, W. Phys. Stat. Sol. B 1971, 46, 165–176. Dederichs, P. H.; Schroeder, K. Phys. Rev. 1978, 17, 2524–2536.
Fundamental Properties of Defects in Metals 35. Chan, W. L.; Averback, R. S.; Ashkenazy, Y. J. Appl. Phys. 2008, 104, 023502. 36. Kroupa, F. Czech. J. Phys. 1960, B10, 284–293. 37. Mura, T. In Advance Materials Research; Herman, H., Ed.; Wiley: New York, 1968; Vol. 3, pp 1–108. 38. Cahn, J. W. Acta Met. 1980, 28, 1333–1338. 39. Shuttleworth, R. Proc. Phys. Soc. Lond. A 1950, 63, 444–457. 40. Gutman, E. M. J. Phys. Condens. Matter. 1995, 7, L663–L667. 41. Gurtin, M. E.; Murdoch, A. I. Arch. Rat. Mech. Anal. 1975, 57, 291–323; 1975, 59, 389–390. 42. Gurtin, M. E.; Murdoch, A. I.; Int. J. Solids Struct. 1978, 14, 431–440. 43. Needs, R. J. Phys. Rev. Lett. 1987, 58, 53–56. 44. Needs, R. J.; Mansfield, M. J. Phys. Condens. Matter. 1989, 1, 7555–7563. 45. Needs, R. J.; Godfrey, M. J.; Mansfield, M. Surface Sci. 1991, 242, 215–221. 46. Feibelman, P. J. Phys. Rev. B 1994, 50, 1908–1911. 47. Medasani, B.; Young, H. P.; Vasiliev, I. Phys. Rev. B 2007, 75, 235436. 48. Shimomura, Y. Mater. Chem. Phys. 1997, 50, 136–151. 49. Adams, J. B.; Wolfer, W. G. J. Nucl. Mater. 1989, 166, 235–242. 50. Foiles, S. M.; Baskes, M. I.; Daw, M. S. Phys. Rev. B 1986, 33, 7983–7991. 51. Bullough, R.; Eyre, B. L.; Perrin, R. C. Nucl. Appl. Technol. 1970, 9, 346.
52. 53. 54.
55. 56. 57. 58. 59. 60. 61. 62. 63. 64. 65. 66. 67.
45
Harkness, S. D.; Li, Che-Yu Met. Trans. 1971, 2, 1457–1470. Mansur, L. K. Nucl. Technol. 1978, 40, 5–34. Foreman, A. J. E. The preferential trapping of interstitials at dislocations as a mechanism for void growth during irradiation damage, Harwell report AERE-R-7629; May, 1974. Heald, P. T. Phil. Mag. 1975, 31, 551–558. Ham, F. S. J. Appl. Phys. 1959, 30, 915–926. Margvalashvili, I. G.; Saralidze, Z. K. Sov. Phys. Solid State 1974, 15, 1774–1776. Wolfer, W. G.; Ashkin, M. J. Appl. Phys. 1976, 47, 791–800. Wolfer, W. G. J. Comput. Aided Mater. Des. 2007, 14, 403–417. Wolfer, W. G.; Ashkin, M. Scripta Met. 1973, 7, 1175–1180. Kohn, W.; Mattsson, A. E. Phys. Rev. Lett. 1998, 81, 3487. Mattsson, A. E.; Kohn, W. J. Chem. Phys. 2001, 115, 3441. Armiento, R.; Mattsson, A. E. Phys. Rev. 2002, B66, 165117. Armiento, R.; Mattsson, A. E. Phys. Rev. 2005, B72, 085108. Carling, K.; Wahnstro¨m, G.; Mattsson, T. R.; Mattsson, A. E.; Sandberg, N.; Grimvall, G. Phys. Rev. Lett. 2000, 85, 3862. Mattsson, A. E.; Armiento, R.; Schultz, P. A.; Mattsson, T. R. Phys. Rev. 2006, B73, 195123. Gavini, V.; Bhattacharya, K.; Ortiz, M. Mech. J. Phys. Solids 2007, 55, 697.
1.02
Fundamental Point Defect Properties in Ceramics
A. Chroneos University of Cambridge, Cambridge, UK
M. J. D. Rushton and R. W. Grimes Imperial College of Science, London, UK
ß 2012 Elsevier Ltd. All rights reserved.
1.02.1 1.02.2 1.02.2.1 1.02.2.2 1.02.2.3 1.02.2.4 1.02.3 1.02.3.1 1.02.3.2 1.02.3.3 1.02.3.4 1.02.3.5 1.02.3.6 1.02.3.7 1.02.4 1.02.4.1 1.02.4.2 1.02.4.3 1.02.4.4 1.02.5 1.02.6 1.02.6.1 1.02.6.2 1.02.7 References
Introduction Intrinsic Point Defects in Ionic Materials Point Defects Compared to Defects of Greater Spatial Extent Intrinsic Disorder Reactions Concentration of Intrinsic Defects Kro¨ger–Vink Notation Defect Reactions Intrinsic Defect Concentrations Effect of Doping on Defect Concentrations Decrease of Intrinsic Defect Concentration Through Doping Defect Associations Nonstoichiometry Lattice Response to a Defect Defect Cluster Structures Electronic Defects Formation Concentration of Intrinsic Electrons and Holes Band Gaps Excited States The Brouwer Diagram Transport Through Ceramic Materials Diffusion Mechanisms Diffusion Coefficient Summary
1.02.1 Introduction The mechanical and electronic properties of crystalline ceramics are dependent on the point defects that they contain, and as a consequence, it is necessary to understand their structures, energies, and concentration defects and their interactions.1,2 In terms of their crystallography, it is often convenient to characterize ceramic materials by their anion and cation sublattices. Such models lead to some obvious expectations. It might, for example, be energetically unfavorable for an anion to occupy a site in the cation sublattice and vice versa. This is because it would lead to anions having nearest neighbor anions with a substantial electrostatic energy penalty. Further, there should exist an equilibrium between the concentration of intrinsic defects (such as lattice vacancies), extrinsic defects (i.e., dopants), and
47 48 48 48 49 50 51 51 52 52 52 53 55 56 57 57 57 58 58 60 61 61 63 63 64
electronic defects in order to maintain charge neutrality.1,2 Such constraints on the types and concentrations of point defects are the focus of this chapter. In the first section, we consider the intrinsic point defects in ionic materials. This is followed by a discussion of the defect reactions describing the effect of doping, defect cluster formation, and nonstoichiometry. Thereafter, we consider the importance of electronic defects and their influence on ceramic properties. In the final section, we examine solidstate diffusion in ceramic materials. Examples are used throughout to illustrate the extent and range of the point defects and associated processes occurring in ceramics. The subsequent chapters (see Chapter 1.03, Radiation-Induced Effects on Microstructure and Chapter 1.06, The Effects of Helium in Irradiated Structural Alloys) will deal with defects 47
48
Fundamental Point Defect Properties in Ceramics
of greater spatial extent, such as dislocations and grain boundaries, in greater detail; here, however, we begin by comparing them with point defects.
1.02.2 Intrinsic Point Defects in Ionic Materials 1.02.2.1 Point Defects Compared to Defects of Greater Spatial Extent In crystallography, we learn that the atoms and ions of inorganic materials are, with the exception of glasses, arranged in well-defined planes and rows.3 This is, however, an idealized representation. In reality, crystals incorporate many types of imperfections or defects. These can be categorized into three types depending on their dimensional extent in the crystal: 1. Point defects, which include missing atoms (i.e., vacancies), incorrectly positioned atoms (e.g., interstitials), and chemically inappropriate atoms (dopants). Point defects may exist as single species or as small clusters consisting of a number of species. 2. Line defects or dislocations, which extend through the crystal in a line or chain. The dislocation line has a central core of atoms, which are located well away from the usual crystallographic sites (in ceramics, this extends, in cylindrical terms, to a nanometer or so). Most dislocations are of edge, screw, or mixed type.4 3. Planar defects, which extend in two dimensions and are atomic in only one direction. Many different types exist, the most common of which is the grain boundary. Other common types include stacking faults, inversion domains, and twins.1,2 The defect types described above are the chemical or simple structural models for the extent of defects. It is critical to bear in mind that all defect types, in all materials, may exert an influence via an elastic strain field that extends well beyond the chemical extent of the defect (i.e., beyond the atoms replaced or removed). This is because the lattice atoms surrounding the defect have had their bonds disrupted. Consequently, these atoms will accommodate the existence of the defect by moving slightly from their perfect lattice positions. These movements in the positions of the neighboring atoms are referred to as lattice relaxation. As a result of the elastic strain and electrostatic potential (if the defect is not charge-neutral), defects can affect the mechanical properties of the lattice. In addition, defects have a chemical effect, changing the
oxidation/reduction properties. Defects also provide mechanisms that support or impede the movement of ions through the lattice. Finally, defects alter the way in which electrons interact with the lattice, as they can alter the potential energy profile of the lattice (whether or not the defect is charged). For example, this may lead to the trapping of electrons. Also, because dopant ions will have a different electronic configuration from that of the host atom, defects may donate an electron to a conduction band, resulting in n-type conduction, or a defect may introduce a hole into the electronic structure, resulting in p-type conductivity. 1.02.2.2
Intrinsic Disorder Reactions
A number of different point defects can form in all ceramics, but their concentration and distributions are interrelated. In the event of the production of a vacancy by the displacement of a lattice atom, this released atom can be either contained within the crystal lattice as an interstitial species (forming a Frenkel pair), or it can migrate to the surface to form part of a new crystal layer (resulting in a Schottky reaction). Figure 1 represents a Frenkel pair: both cations and anions can undergo this type of disorder reaction, resulting in cation Frenkel and anion Frenkel pairs, respectively. In ceramic materials, both the vacancy and interstitial defects are usually charged, but the overall reaction is charge-neutral. The energy necessary for this reaction to proceed is the energy to create one vacancy by removing an ion from the crystal to infinity plus the energy to create one interstitial ion by taking an ion from infinity and placing it into the crystal. The implication of removing and taking ions from infinity implies that the two species are infinitely separated in the crystal (unlike the two species shown in Figure 1). As separated species, these are defects at infinite dilution, a Vacancy
Interstitial Figure 1 Schematic representation of a Frenkel pair in a binary crystal lattice.
Fundamental Point Defect Properties in Ceramics
well-defined thermodynamic limit. As the two defects are charged, they will interact if not infinitely separated, a point we will return to later. As with the Frenkel reaction, the Schottky disorder reaction must be charge-neutral. Here, only vacancies are created, but in a stoichiometric ratio. Thus, for a material of stoichiometry AB, one A vacancy and one B vacancy are created. The displaced ions are removed to create a new piece of lattice. It is important to realize that we are dealing with an equilibrium process for the whole crystal. Thus, as the temperature changes, many thousands of vacancies are created/destroyed and new material containing many thousands of ions is formed. Thus, it is not simply that one new molecule is formed, but there is also an increase in the volume of the crystal, which is why the lattice energy is part of the Schottky reaction. The energy for the Schottky disorder reaction to proceed in an AB material is the energy to create one A-site vacancy by removing an ion from the crystal to infinity plus the energy to create one B-site vacancy by removing an ion from the crystal to infinity, plus the lattice energy associated with one unit of the AB compound. For example in Al2O3, the energy would be that associated with the sum of two Al vacancies plus three O vacancies plus the lattice energy of one Al2O3 formula unit. Again, the vacancy species are assumed to be effectively infinitely separated. In a crystalline material with more than one type of atom, each species usually occupies its own sublattice. If two different species are swapped, this produces an antisite pair (see Figure 2). For example, in an AB compound, one A atom is swapped with the B atom. While this would be of high energy for an AO compound, where A and O are of opposite charge (e.g., Mg2þ and O2), in an ABO3 material where A and B may have similar or even identical positive charges, antisite energies can be small.
Antisite pair
49
In general, the energies needed to form each type of disorder, in a given material, are different. Therefore, only one type of intrinsic disorder dominates: this is often described as the intrinsic disorder of the material. If one intrinsic process is of much lower energy than the others, it will dominate the equilibrium: this is useful when investigating other defect processes, as we will see later. In most metals and metal alloys, Schottky disorder dominates because of the closely packed nature of their crystal structure. In ceramics, both Schottky and Frenkel disorders are possible; for example, in NaCl and MgO (both having the rock salt structure), Schottky disorder dominates, but in CaF2 and UO2 (both having the fluorite structure), anion Frenkel disorder is predominant, while antisite disorder is observed to dominate in MgAl2O4 spinel. In Al2O3, the situation is too close to call and it is not clear whether Frenkel or Schottky disorder dominates.5 1.02.2.3
Concentration of Intrinsic Defects
We start with the assumption that, for a given set of ions, their crystal structure represents the most stable arrangement of those ions in space. Thus, there is an enthalpy cost to form atomic defects: energy is expended in forming the defects. How then do defects form? The answer is related to free energy considerations; that is, the increase in the enthalpy of the system can be balanced with a corresponding increase in the entropy and more particularly, the configurational entropy. Point defects in a crystal can therefore be described as entropically stabilized, and as such they are equilibrium defects (dislocations and grain boundaries, on the other hand, are not equilibrium defects). If the enthalpy of forming n Schottky pairs in an AB material is n Dh, the vibrational entropy is nT Ds, where T is temperature (in K) so that n Dgf ¼ n Dh þ nT Ds, and the change in the entropy associated with this reaction is DSc; the change in the free energy (DG) of the system (if we ignore pressure volume term effects) is DG ¼ nDgf T DSc If we assume that the entropy is all associated with configuration DSc ¼ klnO
Figure 2 Schematic representation of an antisite pair in a binary crystal lattice.
where k is Boltzmann’s constant and O is the number of distinct ways that n Schottky pairs can be arranged in the crystal. If we assume that there are N ‘A’ lattice sites (in defect chemistry terms, a lattice
Fundamental Point Defect Properties in Ceramics
site means a position in the crystal that an ion will usually occupy in that crystal structure), the number of ways, OA, of arranging n A-site vacancies is N! OA ¼ n!ðN nÞ! As we have n B-site vacancies to distribute over N, B-lattice sites, the total number of configurations is the product of OA and OA: N! N! DSC ¼ k ln n!ðN nÞ! n!ðN nÞ! N! ¼ 2k ln n!ðN nÞ! where N and n are large, as they are when dealing with crystals, we can invoke Stirling’s approximation, which states that ln(M!) ¼ M ln(M) M. Thus, DSC ¼ 2k½N lnðN Þ ðN nÞlnðN nÞ nlnðnÞ Therefore, DG¼nDgf 2kT ½N lnðN ÞðN nÞlnðN nÞnlnðnÞ N N n þnln ¼nDgf 2kT N ln N n n To find the equilibrium number of defects, we need to find the minimum of DG with respect to n (see Figure 3). That is @DG N n ¼0¼Dgf 2kT ln @n T ;P n Assuming that the number of defects is small in comparison to the number of available lattice sites, then N n N:
Energy
nΔgf
n Dgf Dh Ds ¼ exp exp ¼ exp N 2kT 2k 2kT Usually, we assume that the energy associated with the change in vibrational entropy is negligible so that the concentration of defects (n/N) is dominated by the enthalpy of reaction: n Dh ½n ¼ ¼ exp N 2kT However, this is not always a valid assumption and care must be taken. When defect concentrations are measured experimentally, they are presented on an Arrhenius plot of ln(concentration) versus 1/T, which yields straight lines with slopes that are proportional to the disorder enthalpy (see Figure 4). 1.02.2.4
Kro¨ger–Vink Notation
It is usual for defects in ceramic materials to be described using a short hand notation after Kro¨ger and Vink.6 In this, the defect is described by its chemical formula. Thus, a sodium ion would be described as Na, whatever its position in whatever lattice. A vacancy is designated as ‘V.’ The description is made with respect to the position within the lattice that the defect occupies. For example, a vacant Mg site is designated by VMg and an Na substituted at an Mg site is designated by NaMg. Interstitial ions are represented by ‘i’ so that an interstitial fluorine ion in any lattice would be Fi. The charge on an ion is described with respect to the site that the ion occupies. Thus, an Na ion (which has formal charge þ) sitting on an Mg site in MgO (which expects to be occupied by a 2þ ion) has one too few þ charges; it has a relative charge of 1 which is designated as a vertical dash, meaning that it is written as Na0Mg . An Al3þ ion at an Mg site in
ΔG Defect concentration
−TΔSc Figure 3 Relationship of terms contributing to the defect-free energy.
In [n]
50
1/T Figure 4 Disorder enthalpy is proportional to the gradient of a ln [n] versus 1/T graph.
Fundamental Point Defect Properties in Ceramics h
MgO has too high a charge. Positive excess charge relative to a site is designated with a dot, thusAlMg . Similarly, a vacant Mg site in MgO is designated by V00Mg and an interstitial Mg ion in MgO byMg i . Finally, a neutral charge is indicated by a cross ‘,’ so that an Mg ion at an Mg site in MgO is Mg Mg . Ions such as Fe may assume more than one oxidation state. Therefore, in MgO, we might find both Fe2þ and Fe3þ ions on Mg sites, that is, Fe Mg andFeMg . It is also possible to encounter bound defect pairs or clusters. These are indicated using brackets and an indication of the overall cluster charge; for example, an Fe3þ ion bound to an Naþ ion, both substituted at magnen o sium sites, would be FeMg : Na0Mg . These cases are summarized in Figure 5. Finally, defect concentrations are indicated using square brackets. Thus, the concentration of Fe3þ ions substituted at magnesium sites in MgO would be
FeMg
i
. When we consider the role of hole and electron species, these are represented as h and e0 , respectively.
1.02.3 Defect Reactions 1.02.3.1
Intrinsic Defect Concentrations
Introducing a doping agent to a crystal lattice can have a significant effect on the defect concentration within the material. As such, doping represents a powerful tool in the engineering of the properties of ceramic materials. The concept of a solid solution, in which solute atoms are dispersed within a diluent matrix, is used in many branches of materials science. In many respects, the doped lattice can be viewed as a solid solution in which the point defects are dissolved in the host
• = Positive charge
Defect species Element label, V for a vacancy, h for hole or e for electron
S
Charge
ı = Negative charge ´ = Neutral
Site
Subscript denotes species in the nondefective lattice at which defect currently sits. Interstitial defects are represented by letter ‘i’
Examples Vacancy on an oxygen site with an effective 2+ charge
••
Vo
ıı
Oxygen interstitial with an effective 2- charge
Oi
•
AIMg
Substitutional defect in which an aluminum atom is situated on a magnesium site and has an effective 1+ charge
ı {FeMg:NaMg }´
Neutral defect cluster containing: Fe on Mg site (1+ charge) and Na on Mg site (1- charge). Braces indicate defect association
•
´
´
MgMg + Oo
ıı
VMg + Vo¨ + MgO
Figure 5 Overview of Kro¨ger–Vink notation.
51
Defect equation showing Schottky defect formation in MgO
52
Fundamental Point Defect Properties in Ceramics
lattice. The critical issue with such a view is in defining the chemical potential of an element. This is straightforward for the dopant species but is less clear when the species is a vacancy. This is circumvented by defining a virtual chemical potential, which allows us to write equations similar to those that describe chemical reactions. Within these defect equations, it is critical that mass, charge, and site ratio are all conserved. Using Kro¨ger–Vink notation, we can describe the formation of Schottky defects in MgO thus: Null ! V00Mg þ V O or MgMg þ OO
! V00Mg þ V O þ MgO Note that in this case, the equation balances in terms of charge, chemistry, and site. As this is a reaction, it may be described by a reaction constant ‘K,’ which is related to defect concentrations by h i KS ¼ V00Mg V O In the case of the pure MgO Schottky reaction, charge neutrality dictates that h i V00Mg ¼ V O So, if Dh is the enthalpy of the Schottky reaction, if we use our previous definition of concentration, n Dh ¼ exp N 2kT h i Dh V00Mg ¼ V ¼ exp O 2kT 1.02.3.2 Effect of Doping on Defect Concentrations Similar reactions can be written for extrinsic defects via the solution energy. For example, the solution of CoO in MgO, where the Co ion has a charge of 2þ and is therefore isovalent to the host lattice ion, CoO ! Co Mg þ OO þ MgO
Ksolution ¼
h i Co Mg OO ½CoO
Dhsol exp kT
where Dhsol is the solution enthalpy. As the concentration of CoO in CoO ¼ 1, h i Dhsol exp Co O Mg O kT Consider the solution of Al2O3 in MgO. In this case, the Al ions have a higher charge and are termed aliovalent. These ions must be charge-compensated by other defects, for example,
00 Al2 O3 ! 2AlMg þ 3O O þ VMg
Then,
h i2 h i 00 AlMg O O VMg ½Al2 O3
Dhsol ¼ exp kT h
i
As electronegativity dictates that AlMg ¼ 2 follows that h i pffiffiffi Dhsol 3 AlMg ¼ 2 exp 3kT
h
V00Mg
i
, it
In general, the law of mass action6 states that for a reaction aA þ bB $ cC þ dD ½C c ½Dd DG ¼ K ¼ exp reaction kT ½Aa ½B b 1.02.3.3 Decrease of Intrinsic Defect Concentration Through Doping While considering the intrinsic defect reaction forh MgO, i we wrote the defect reaction K S V00Mg VO . This implies that there is equilibrium between these two defect concentrations. Assuming that the enthalpy of the Schottky reaction, Dh ¼ 7.7 eV, h i 7:7 00 VMg VO ¼ exp kT Now, consider the effect that solution of 10 ppm Al2O3 has on MgO. The solution reaction implies that a concentration of 5 ppm of V00Mgh hasi been introduced into the lattice, that is, V00Mg ¼ 5 106. Therefore, 7:7 VO ¼ 2 105 exp kT Thus, at 1000 C, the VO ¼ 6.4 10–26 compared to an oxygen vacancy concentration in pure MgO of 5.66 10–16. The introduction of the extrinsic defects has therefore lowered the oxygen vacancy concentration by 10 orders of magnitude! 1.02.3.4
Defect Associations
So far, we have assumed that when we form a set of defects through some interaction, although the defects reside in the same lattice, somehow they do not interact with one another to any significant extent. They are termed noninteracting. This is valid in the dilute limit approximation; however, as defect concentrations increase, defects tend to form
Fundamental Point Defect Properties in Ceramics
into pairs, triplets, or possibly even larger clusters. Take, for example, the solution of Al2O3 into MgO. 00 Al2 O3 ! 2AlMg þ 3O O þ VMg þ 3MgO
When the concentration is great enough, n o0 AlMg þ V00Mg ! AlMg : V00Mg
the scope of this chapter.7 Therefore, to illustrate the types of relationships n that canoxoccur, we use the 00 cluster resulting example of the binary Ti Mg : VMg from TiO2 solution in MgO via 00 TiO2 ! Ti Mg þ 2OO þ VMg þ 2MgO
and
n o 00 00 Ti þ V ! Ti : V Mg Mg Mg Mg
As seen for solution energies, using the enthalpy associated with this pair cluster formation (the binding energy Dhbind), the reaction can be analyzed using mass action: hn o0 i AlMg : V00Mg Dh ih i exp bind Kbinding pair ¼ h kT V00 Al
and the electroneutrality condition h i h i V00Mg ¼ Ti Mg
But since,
yields
Mg
Mg
h i2 h i Dhsol AlMg V00Mg ¼ exp kT we have the relationship hn
AlMg
o0 ih i Dhbind þ Dhsol 00 : VMg AlMg ¼ exp kT
which describes the solution process n o0 00 Al2 O3 ! AlMg þ 3O þ Al : V O Mg Mg Further, itn is possible tooform a neutral triplet defect , which has a binding cluster, AlMg : V== Mg : AlMg bind with respect to isolated defects enthalpy of DE so that hn o i AlMg : V00Mg : AlMg Kbinding triplet ¼ h i2 h i AlMg V00Mg Dhbind exp kT which leads to hn o i Dhbind þ Dhsol AlMg : V00Mg : AlMg ¼ exp kT We now investigate the relative significance of defect clusters over isolated defects as a function of temperature for a fixed dopant concentration. For most systems, there are a great number of possible isolated and cluster defects, and the equilibria between them quickly become very complex. Solving such equilibria requires iterative procedures that are beyond
53
Then, using h ih i hn o i 00 00 Ti Ti Mg VMg Kbinding pair ¼ Mg : VMg
hn
00 Ti Mg : VMg
o i
h i2 ¼ Ti Mg Kbinding pair
a
If the concentration of titanium ions on magnesium sites is x so that Mg(1–2x)TixO is the formula of the material, then, hn o i h i 00 Ti : V ¼ x Ti b Mg Mg Mg Substituting b into a yields the quadratic equation, h i2 h i þ Ti Kbinding pair Ti Mg Mg x ¼ 0 Solvingh thisi in the usual manner allows us to deteras a function of Kbinding energy. If we mine Ti Mg now assume that Dhbind ¼ 1 eV (a typical binding energy between charged pairs of defects in oxide ceramics), relationships between the concentration of the clusters and the isolated substitutional ions can be determined as a function of either total dopant concentration, x, or temperature. These are shown in Figure 6, which assumes a fixed temperature of 1000 K and Figure 7, which assumes a fixed value of x ¼ 1 106 and a range of temperature from 500 to 2000 K. 1.02.3.5
Nonstoichiometry
Although some materials such as MgO maintain the ratio between Mg and O very close to the stoichiometric ratio 1:1, other crystal structures such as FeO can tolerate large nonstoichiometries; Fe1xO with 0.05 x 0.15. The extent of the deviation from stoichiometry depends on how easily the host ions can assume charge states other than those associated with
54
Fundamental Point Defect Properties in Ceramics
1⫻10-2
Isolated favored
-4
Cluster favored
1⫻10
Concentration
1⫻10-6 1⫻10-8 1⫻10-10 1⫻10-12
Cluster ⬘⬘ ´ [{TiMg:VMg }] ••
-14
1⫻10
Isolated
1⫻10-16
·· ] [TiMg
1⫻10-18 1⫻10-12
1⫻10-11
1⫻10-10
1⫻10-9
1⫻10-8
1⫻10-7
1⫻10-6
1⫻10-5
1⫻10-4
X Figure 6 Cluster and isolated defect concentration as a function of x at a temperature of 1000 K.
1⫻10−4
Isolated favored
Cluster favored 1⫻10−5
Concentration
1⫻10−6
1⫻10−7
1⫻10−8
Cluster ·· :V ⬘⬘ }⫻] [{TiMg Mg
1⫻10−9
1⫻10−10
Isolated [Ti ·· ] Mg
1⫻10−11
600
800
1000
1200
1400
1600
1800
2000
T (K) Figure 7 Cluster and isolated defect concentration for x ¼ 1 10–6 as a function of temperature.
the host material, Fe3þ in the last case. Usually, this is dependent on how easily the cation can be oxidized or reduced. Associated with these reactions is the removal or introduction of oxygen from the atmosphere. For example, the reduction reaction follows 1 ! O2 ðgÞ þ V O þ 2e 2 where e represents a spare electron, which will reside somewhere in the lattice. For example, in CeO2, the electron is localized on a cation site forming a Ce3þ O O
ion. This is usually written as Ce0Ce . Similarly, the oxidation reaction is 1 O2 ðgÞ ! O O þ 2h 2 where h represents a hole, that is, where an electron has been removed because the new oxygen species requires a charge of 2. Thus in CoO, for example, 1 00 O2 ðgÞ þ 2Co Co ! OO þ 2CoCo þ VCo 2
Fundamental Point Defect Properties in Ceramics
If the enthalpy for the oxidation reaction is DhOX, 2 00 CoCo VCo DhOX ¼ exp kT ðPO2 Þ1=2 ðPO2 Þis the partial pressure of oxygen, that is, the concentration of oxygen in the atmosphere. Since electroneutrality gives us CoCo ¼ 2 V00Co 00 VCo ðPO2 Þ1=6 Now, if the majority of cobalt vacancies are associated with a single charge-compensating Co3þ species, that is, we have some defect clustering, then the oxidation reaction will be 0 1 00 O2 ðgÞ þ 2Co Co ! OO þ CoCo þ CoCo : VCo 2 and
h 0 i CoCo CoCo : V00Co ðPO2 Þ1=2
¼ exp
DhOX kT
which, given that
h 0 i CoCo ¼ CoCo : V00Co
h
i
implies that CoCo : V00Co 0 is proportional to ðPO2 Þ1=4 . Similar relations can be formulated for even larger clusters. The defect concentration can be determined by measuring the self-diffusion coefficient. When this is related to the oxygen partial pressure on a lnPO2 versus ln CoCo : V00Co graph, the slope shows how the material behaves. For CoO, the experimentally determined slope is 1=4, showing that the cation vacancy is predominantly associated with a single charge-compensating defect.8
manifested as a volume change arising as a result of the way in which the lattice responds, that is, how the lattice ions relax around the defect. For example, vacancies in ionic materials usually result in positive defect volumes. Consider the example of a vacancy in MgO. The nearest neighbor cations are displaced outward, away from the vacant site, causing an increase in volume, whereas the second neighbor anions move inward, albeit to a lesser extent (see Figure 8). What drives these ion relaxations is the change in the Coulombic interactions due to defect formation. We say that the oxygen vacancy carries an effective positive charge because an O2 has been removed and thus, an electrostatic attraction between O2 and Mg2þ is removed. As ionic forces are balanced in a crystal, the outer O2 ions now attract the Mg2þ away from the V O defect site. In covalent materials, vacant sites result in atomic relaxations that are due to the formation of an incomplete complement of bonds, often termed ‘dangling bonds.’ In this case, the net result can be different from those in ionic solids and by way of an example, in silicon, a vacancy results in a volume decrease. On the other hand, an arsenic substitutional atom causes an increase in volume.10 Finally, in a material such as ZrN or TiN, which exhibits both covalent and metallic bonding, the volume of a nitrogen vacancy is practically zero.11 Clearly, the overall response of the lattice can be rather complicated. However, the defect volume can be determined fairly easily by applying the relationship dfV uP ¼ KT V dV T
Mg O
1.02.3.6
55
O
Lattice Response to a Defect
To formulate quantitative or often even qualitative models for defect processes in materials, it is essential that lattice relaxation be effected. Without lattice relaxation, the total energies calculated for defect reactions would be so great that we would have to conclude that no point defects would ever form in the material.9 Each defect has an associated defect volume. That is, each defect, when introduced into the lattice, causes a distortion in its surroundings, which is
Vo··
Mg
O
Mg
O Mg
Figure 8 Schematic of the lattice relaxations around an oxygen vacancy in MgO.
56
Fundamental Point Defect Properties in Ceramics
where uP is the defect volume (in A˚3); KT, the isothermal compressibility (in eV/A˚3); V, the volume of the unit cell (in A˚3); and fV, the Helmholz free energy of formation of the defect (in eV). Finally, defect associations can also (but not necessarily) have a significant effect on defect volumes for a given solution reaction. For example, for the Al2O3 solution in MgO if we assume isolated AlMg and V00Mg hashthe least i effect on lattice parameter as a function of AlMg whereas the formation of neutral AlMg : V00Mg has the greatest effect (in fact ten times the reduction in lattice parameter).12
3rd
1st
2nd
M3+
Defect Cluster Structures
So far, we have ignored possible geometric preferences between the constituent defects of a defect cluster. Of course, for oppositely charged defects, electrostatic considerations would drive the defects to sit as close as possible to one another, which would be described as a nearest neighbor configuration. However, as we saw in the previous section, defects can cause considerable lattice strain. Consequently, the most stable defect configuration will be dictated by a balance between electrostatic and strain effects. To illustrate cluster geometry preference, we will consider simple defect pairs in the fluorite lattice, specifically in cubic ZrO2. These are formed between a trivalent ion, M3þ, that has substituted for a tetravalent lattice ion (i.e.,M0Zr ) and its partially chargecompensating oxygen vacancy (i.e., V O ). This doping process produces a technologically important fast ion-conducting system, with oxygen ion transport via oxygen vacancy migration.2,13 The lowest energy solution reaction that gives rise to the constituent isolated defects14 is 0 M2 O3 þ 2Zr Zr ! 2MZr þ VO þ 2ZrO2
with the pair cluster formation following: 0 M0Zr þ V O ! MZr þ VO Figure 9 shows the options for the pair cluster geometry, in which, if we fix the trivalent substitutional ion at the bottom left-hand corner, the associated oxygen vacancy can occupy the first near neighbor, the second (or next) near neighbor, or the third near neighbor position. Defect energy calculations have been used to predict the binding energy of the pair cluster as a function of the ionic radius15 of the trivalent substitutional
Figure 9 First, second, and third neighbor oxygen ion sites with respect to a substitutional ion (M3þ).
0.8
AI Cr
(M⬘zr: V•o•)•binding energy (eV)
1.02.3.7
Ga Fe
0.6
Ce
0.4 Yb
Y
Gd
La
Sm
Sc In 0.2
0.0
-0.2 0.7
0.8
0.9
1.0
1.1
1.2
Cation radius (Å)
Figure 10 Binding energies of M3þ dopant cations to an oxygen vacancy: ▪ a first configuration; second configuration, and ▼ third configuration. Open symbols represent calculations that required stabilization to retain the desired configuration. Reproduced from Zacate, M. O.; Minervini, L.; Bradfield, D. J.; Grimes, R. W.; Sickafus, K. E. Solid State Ionics 2000, 128, 243.
ion.14 These suggest (see Figure 10) that there is a change in preference from the near neighbor configuration to the second neighbor configuration as the ionic radius of the substitutional ion increases. The change occurs close to the Sc3þ ion. Furthermore, the binding energy of the near neighbor cluster falls as a function of radius; conversely, the binding energy of the second neighbor cluster increases. Consequently, the change in preference occurs at a minimum in binding energy. The third neighbor cluster is largely independent of ionic radius. Interestingly, the minimum coincides with a maximum in the ionic conductivity, perhaps because the trapping of the oxygen vacancies as they move through the lattice is at a minimum.14
Fundamental Point Defect Properties in Ceramics
The change in preference for the oxygen vacancy to reside in a first or second neighbor site is a consequence of the balance of two factors: first, the Coulombic attraction between the vacancy and the dopant substitutional ion, which always favors the first neighbor site, and is largely independent of ionic radius, and second, the relaxation of the lattice, a crystallographic effect that always favors the second neighbor position. This is because, in the second neighbor configuration, the Zr4þ ion adjacent to the oxygen vacancy can relax away from the effectively positive vacancy without moving away from the effectively negative substitutional ion. Nevertheless, lattice relaxation in the first neighbor configuration contributes an important energy term. However, in the first neighbor configuration, the relaxation of oxygen ions is greatly hindered by the presence of larger trivalent cations, while small trivalent ions provide more space for relaxation. Thus, the relaxation preference for the second neighbor site increases in magnitude as the ionic radius increases and consequently, the second neighbor configuration becomes more stable compared to the first.14 This example shows that even in a simple system such as a fluorite, which has a simple defect cluster, the factors that are involved in determining the cluster geometry become highly complex. Even so, we have so far only considered structural defects. Next, we investigate the properties of electronic defects.
1.02.4 Electronic Defects 1.02.4.1
Formation
Electronic defects are formed when single or small groups of atoms in a crystal have their electronic structure changed (e.g., electrons removed, added, or excited). In particular, they are formed when an electron is excited from its ground state configuration into a higher energy state. Most often this involves a valence electron, although electrons from inner orbits can also be excited if sufficient energy is available. In either case, the state left by this transition, which is no longer occupied by an electron, is usually termed a hole. These defects can be generated thermally, optically, by radiation or through ion beam damage. The excited electron component may be localized on a single atomic site and if the electron is transferred to another center, it is represented as a change in the ionization state of the ion or atom to which it is localized. This is sometimes described as a small polaron or trapped electron. Such electronic defects
57
might migrate through the lattice via an activated hopping process. An example of a small polaron electron is a Ce3þ ion in CeO2x.16 Alternatively, the excited electron may be delocalized so that it moves freely through the crystal. In this case, the electron occupies a conduction band state, which is formed by the superposition of atomic wave functions from many atoms. This is the case with most semiconductor materials. Similarly, the hole may also be localized to one atomic center and be represented as a change in the ionization state of the ion or atom. Holes may also move via an activated hopping process. An example is a Co3þ ion in Co1xO. Similarly, the hole may also be delocalized. Intermediate situations may occur with the hole or electron being localized to a small number of atoms or ions (known as a large polaron) or a specific type of hole state associated with a particular chemical bond. The relationship between doping and its influence on electronic defects is of great technological importance in the field of semiconducting materials. For example, doping silicon with defect concentrations in the order of parts per million is sufficient for most microelectronic applications. Incorporation of a phosphorous atom in silicon results in a shallow state below the conduction band that will easily donate an electron to the conduction band. The remaining four valence electrons of the phosphorous dopant will form sp3 hybrid bonds with the four neighboring tetrahedral silicon atoms. Recently, it has been suggested that the state from which the electron is removed is associated with the dopant species and the four silicon atoms surrounding it; in other words, it is associated with a cluster.17 1.02.4.2 Concentration of Intrinsic Electrons and Holes Under equilibrium conditions, the number of electronic defects of energy E is given by,2,3 n ðE Þ ¼ N ðE Þ F ðE Þ where N(E) is the volume density of electronic levels that have energy E (known as the density of states) and F(E) is the probability that a given level is occupied, called the Fermi–Dirac distribution function. N(E) is a function of energy. It is the maximum density of electrons of energy E allowed (per unit volume of crystal) by the Pauli exclusion principle. For a semiconductor, this has an approximately parabolic behavior close to the band edges (i.e., N ðE Þ E 1=2 , refer to Figure 11).
58
Fundamental Point Defect Properties in Ceramics
F(E)
E
1
Nc(E)
T=0
Ec Ef T⫽0
Ev Nv(E)
E
Ef N(E)
Figure 12 Variation of the Fermi probability function with respect to the electron energy.
Figure 11 Schematic representation of the density of states function N(E). Eg > kT
To determine Nc, the effective conduction band density of states, we need to integrate2,3 ð Nc ðE ÞdE 2pm e kT 3=2 1019 cm3 Nc ¼ 2 h2 m e
where is the effective mass of an electron in the conduction band. Similarly, the effective valence band density of states is given by2,3 2pm h kT 3=2 Nv ¼ 2 1019 cm3 h2 where m h is the effective mass of a hole in the valence band. Note that m e and m h are between two and ten times greater than the mass of a free electron. Also, per volume, these densities are approximately four orders of magnitude less than the typical atom density in a solid. The Fermi–Dirac distribution function is given by2,3 F ðE Þ ¼
Ec Ef Ec
E
1 f 1 þ exp EE kT
At 0 K, this implies that all energy levels are occupied up to Ef, the Fermi energy. This is a step function. The Fermi–Dirac function with respect to the energy is represented in Figure 12. At the Fermi energy, F(E) is 1=2. Above 0 K, some energy levels above Ef are occupied. This implies that some levels below Ef are empty.
Eg >> kT
Eg
Ev
Ev
Metal
Eg upto 1.5 eV ˜
Eg > 3.5 eV
Intrinsic semiconductor
Insulator
Figure 13 Characteristic electron energy band levels for a metal, an intrinsic semiconductor and an insulator, where Ec is the bottom of the conduction band, Ev is the top of the valence band, Eg is the band gap, and Ef is the Fermi level.
1.02.4.3
Band Gaps
Materials can be classified based on the occupancy of the energy bands (Figure 13). In an insulator or a semiconductor, an energy band gap, Eg, is between the filled valence band, Ev, and the unoccupied (at 0 K) conduction band. In metals, the conduction band is partially filled (refer to Figure 13). Typical semiconductors have band gaps up to 1.5 eV; when the band gap exceeds 3.5 eV, the material is considered to be an insulator. Table 1 reports the band gaps of some important semiconductors (Ge, Si, GaAs, and SiC) and insulators (UO2, MgO, MgAl2O4, and Al2O3). 1.02.4.4
Excited States
The definition of an electronic defect is effectively ‘a deviation from the ground state electronic
Fundamental Point Defect Properties in Ceramics
Table 1 insulators
Band gaps of important semiconductors and
Material
Band gap (eV)
Ge Si GaAs SiC UO2 MgO MgAl2O4 Al2O3
0.66 1.11 1.43 2.9 5.2 7.8 7.8 8.8
3s
2p
Ground state
Excited state
Figure 14 The 2p ! 3s excitation of an oxygen ion in MgO.
Source: Chiang, Y.-M.; Birnie, D.; Kingery, W. D. Physical Ceramics; Wiley: New York, 1997.
Excited state
Energy
configuration.’ The defects discussed in Section 1.02.4.2. were holes and electrons. Here, we consider defects in which the excited species is localized around the atom by which it was excited. If an electron is excited into a higher lying orbital, there must be a difference between the angular momentum of the ground state and the excited state to accommodate the angular momentum of the photon that has been absorbed during the excitation process (conservation of angular momentum). For example, if the ground state is a singlet, then the excited state may be a triplet. A simple example would be 2p ! 3s excitation of an oxygen ion in MgO (Figure 14). Notice how the energy levels in Figure 14 alter their energies between the ground state and excited states. Therefore, in this case, it is not correct to estimate the energy difference between the ground state and excited states based on the knowledge of only the ground state energy configuration. If the excitation energy is calculated based on the ground state ion positions, it is known as the Franck– Condon vertical transition. When a photon is absorbed, the energy can be equal to this transition. However, the electron in the higher orbital will cause the forces between the ions to be altered. Consequently, the ions in the lattice will change their positions slightly, that is, relaxation will occur. Such relaxation processes are known as nonradiative, that is light is not emitted. Notice that the total energy of the system in the excited state decreases. However, if the triplet excited state now decays back to the singlet ground state (a process known as luminescence,18 see Figure 15), locally the ions are no longer in their optimum positions for the ground state. That is, the relaxed system in the ground state has become higher. The difference between the excitation energy
59
Excitation energy Luminescence Ground state Relaxation
Figure 15 The process of luminescence.
O2−
e h Mg2+
Figure 16 A schematic representation of an exciton in MgO.
and the luminescence energy is known as the Stokes shift.18 Figure 16 represents an example of an excited state electron in MgO, known as a self-trapped exciton.19 The model uses the idea that an exciton is composed of a hole species and an excite electron. Notice that the excited electron has an orbit that is between the hole and its nearest neighboring cations. Thus, the hole is shielded from the cations. This means that the cations do not relax to the extent
60
Fundamental Point Defect Properties in Ceramics
1.02.5 The Brouwer Diagram
Li e
h Cl2-
Cl-
Figure 17 A model for the exciton in alkali halides. The exciton is composed of a hole shared between two halide ions (Vk center) and an excited electron (the so-called Vk þ e model). Interestingly, the two halide ions that comprise the Vk center are not displaced equally from their original lattice positions. Reproduced from Shluger, A. L.; Harker, A. H.; Grimes, R. W.; Catlow, C. R. A. Phil. Trans. R. Soc. Lond. A 1992, 341, 221.
they would if there was a bare hole (the small relaxations are indicated by the arrows). Experimentally, the excitation energy in MgO is 7.65 eV, and the luminescence is 6.95 eV, which yields a small Stokes shift of only 0.7 eV.20 In comparison, a model for the exciton in alkali halides is shown in Figure 17. In this case, the exciton is composed of a Vk center (a hole shared between two halide ions) and an excited electron (the so-called Vk þ e model). However, it is to be noted that the two halide ions that comprise the Vk center are not displaced equally from their original lattice positions. In fact, one of the halide ions is essentially still on its lattice site, while the other is almost in an interstitial site. As calculations suggest that the hole is about 80% localized on this interstitial halide ion, it is almost an interstitial atom known as an H-center. Also, the electron is shifted away from the hole center and is sited almost completely in the empty halide site (called an F-center). As such, the model is almost a Frenkel pair plus an electron localized at a halide vacancy (the so-called F–H pair model). Whichever model is nearest to reality, Vk þ e or F–H pair, it is clear that there is considerable lattice relaxation. This is reflected in the large Stokes shift. In LiCl, the optical excitation energy is 8.67 eV and the p-luminescence energy is only 4.18 eV, leading to a Stokes shift of 4.49 eV.21
Thus far, we have considered both structural and electronic defects. In addition, we have derived the relationship between oxygen vacancies and the oxygen partial pressure, PO2 , which gives rise to nonstoichiometry. It should therefore not come as any surprise that we now consider the equilibrium between isolated structural defects, electronic defects, and PO2 . Of course, we have also considered the equilibrium that exists between isolated structural defects and defect clusters, but defect clusters will not be considered in the present context. Nevertheless, defect clustering does play an important role in the equilibrium between electronic and structural defects and cannot, in a research context, be ignored. In solving defect equilibria in previous sections, we have generally ignored the role that minority defects might have. For example, when considering Schottky disorder in MgO, which we know from experiments is the dominant defect formation process, the effect that oxygen interstitials might have was not taken into account.2 This is certainly reasonable within the context of determining the oxygen vacancy concentration of MgO. The oxygen vacancy concentration is the important parameter to know when predictions of the oxygen diffusivity in MgO are required. However, minority defects may well play an important role in other physical processes. For example, the electrical conductivity or resistivity will depend on the hole or electron concentration; these may be minority defects compared to oxygen vacancies, but understanding them is nevertheless crucial. Thus, we must be concerned with four different defect processes2 simultaneously: 1. The dominant intrinsic structural disorder process (e.g., Schottky or Frenkel). 2. The intrinsic electronic disorder reaction. 3. The REDOX reaction. 4. Dopant and impurity effects. Again we begin by considering MgO.6 If we ignore impurity effects, the three reactions are2 00 Mg Mg þ OO ! VMg þ VO þ MgO
Null ! e0 þ h 1 0 O O ! O2 þ VO þ 2e 2
h i KS ¼ V00Mg V O
Kele ¼ ½e0 ½h 2 1=2 Kredox ¼ PO2 ½e0 V O
Fundamental Point Defect Properties in Ceramics
Conversely at high PO2 , both oxygen vacancies and their charge-compensating electrons must have relatively low concentrations and therefore, the electroneutrality condition becomes dominated by the h i V00Mg and ½h defects so that2 h i ½h ¼ V00Mg Between these two regimes, the Brouwer approximation depends on whether structural or electronic defects dominate. In the case of MgO, we know that Schottky disorder dominates over electronic disorder (as it is a good insulator) and therefore, at intermediate values of PO2 , the appropriate electroneutrality condition is h 00 i VO ¼ VMg If the electronic disorder was dominant, this last reaction would be replaced by ½e0 ¼ ½h We are now in a position to be able to construct a Brouwer diagram, which is usually in the form of ln (defect concentration) versus lnPO2 for various defect components at a constant temperature. In the case of MgO, as indicated above, the diagram will clearly
· = 2[V ⬘⬘ ] [h] M
[e⬘]
1/2
[V··o ]
[VM⬘⬘ ] (VM⬘⬘ Vo·· )*
Ks
[V·· o]
[VM⬘⬘ ] 1/2
Ki
· 1/2 [VM⬘⬘ ] [h]µp o 2
[h]·
Stoichiometric crystal
To make the problem more tractable, we now introduce the Brouwer approximations, which simplify the form of the electroneutrality condition. These effectively concern the availability of defects via the partial pressure of oxygen. For example, if the PO2 is very low, the REDOX equilibrium will reaction 0 and ½ e concentrations are require that the V O relatively high so that these are the dominant positive and negative defect concentrations. Therefore, for low PO2,2 ½e0 ¼ 2 V O
Neutrality condition ⬘⬘ [V·o·] = [VM ] 2[V··o ] = [e⬘]
log concentration
These equations contain six unknown quantities: four are defect concentrations, the other two variables are the PO2 and the temperature, which are experimental variables and are thus given. Of course, we must know the enthalpies of the defect reactions. Nevertheless, to solve these equations simultaneously, we need a further relationship. This is provided by the electroneutrality condition, which, for MgO states that2 h i 2 V00Mg þ ½e0 ¼ 2 V O þ ½h
61
[e⬘]
1/2
log po2
Figure 18 The Brouwer diagram for MgO. Reproduced from Chiang, Y.-M.; Birnie, D.; Kingery, W. D. Physical Ceramics; Wiley: New York, 1997.
have three regimes corresponding to the three Brouwer conditions (refer to Figure 18 and Chiang et al.2).
1.02.6 Transport Through Ceramic Materials 1.02.6.1
Diffusion Mechanisms
Diffusion in ceramic materials is a process enabled by defects and controlled by their concentrations. Owing to the existence of separate sublattices, cation and anion diffusion is restricted to taking place separately (i.e., without exchange of anions and cations), which is one of the main differences with respect to diffusion in other materials.22 Therefore, mechanistically, diffusion theory is applied in ceramics by considering the anion and cation sublattices separately. Interestingly, it has recently been suggested23 that where there is more than one cation sublattice, cations can move on an alternate sublattice through the formation of cation antisite defects. Finally, it can be the case that ion transport in one of the sublattices is more pronounced. For example, in oxygen fast ion conductors, oxygen self-diffusion is faster than cation diffusion by orders of magnitude.24–26
62
Fundamental Point Defect Properties in Ceramics
Transport in crystalline materials requires the motion of atoms away from their equilibrium positions and, therefore, the role of point defects is significant.22 For example, vacancies provide the space into which neighboring atoms in the lattice can jump,27–29 although it is often the interstitial defects that provide the transport mechanism.22 Diffusion mechanisms refer to the way an atom can move from one position in the lattice to another, generally through an activated process that sees the ion move over an energy barrier. The beginning and end points to each jump may be symmetrically identical, providing a contiguous pathway through the crystal, but this need not be so. In some cases, the contiguous migration pathway may involve a number of nonidentical steps. Nevertheless, in most materials, the motion of an atom is restricted to a few paths. There are three main mechanisms that are relevant to most ceramic systems: the interstitial, the vacancy, and the interstitialcy mechanism. However, for completeness, we will also briefly describe the
collective and the interstitial–substitutional exchange mechanisms, which may be encountered in other classes of materials.22 In the interstitial mechanism, atoms at interstitial sites initially migrate by jumping from one interstitial site to a neighboring one (Figure 19). At the completion of a single jump, there is no permanent displacement of the other ions, although, of course, in the process of diffusion, the extent of lattice relaxation is likely to have become greater to facilitate the saddle point configuration. In principle, it is a simple mechanism as it does not require the existence of defects other than the interstitial ion, although it is possible that transient defects are produced if the lattice relaxation is great enough in the course of the jump. Interstitial diffusion is not common in ceramic materials but does occur if the interstitial species is small. In the vacancy mechanism, a host or substitutional impurity atom diffuses by jumping to a neighboring vacancy (Figure 20). Vacancy-mediated diffusion is common in a number of systems (particularly ceramics with higher atomic density where interstitial defect energies are high). For example, the vacancy mechanism is important for the diffusion of substitutional impurities, for self-diffusion and the transport of n-type dopants in germanium,30,31 and for oxygen self-diffusion in a number of hypostoichiometric perovskite and fluorite-related systems.32 In the vacancy mechanism, the interaction, attractive or repulsive, between the species that undergo transport and the vacancy can be very important. Of course, the vacancy mechanism requires the presence of lattice vacancies and therefore, their concentration in the lattice will influence the kinetics.23 In the interstitialcy mechanism, an interstitial atom displaces an atom from its normal substitutional site (Figure 21). The displaced atom, in turn, moves
Interstitial
(i)
(ii)
Figure 19 The interstitial mechanism of diffusion. The red and blue atoms are lattice species.
Vacancy
(i)
(ii)
(iii)
Figure 20 The vacancy mechanism of diffusion. The red and blue ions are lattice species.
Fundamental Point Defect Properties in Ceramics
1.02.6.2
63
Diffusion Coefficient
The temperature dependence of the diffusion coefficient has an Arrhenius form: Ha D ¼ D0 exp kT
(i)
(ii)
Figure 21 The interstitialcy mechanism of diffusion. The red and blue ions are lattice species, the blue ion with the red perimeter is initially an interstitial species but becomes a lattice species.
to an interstitial site. This mechanism is important for the diffusion of dopants such as boron in silicon.33 In hyperstoichiometric oxides, such as La2NiO4þd, it was recently predicted that oxygen diffuses predominantly via an interstitialcy mechanism.26 Collective mechanisms involve the simultaneous transport of a number of atoms. They can be found in ion-conducting oxide glasses22 and have been predicted during the annealing of radiation damage.34 Finally, in the interstitial–substitutional exchange mechanism, the impurities can occupy both substitutional and interstitial sites.22 One possibility for the interstitial atom is to migrate in the lattice until it encounters a vacant site, which it then occupies to become a substitutional impurity (dissociative mechanism).22 Another possibility for the impurity interstitial atom is to migrate in the lattice until it displaces an atom from its normal crystallographic site, thus forming a substitutional impurity and a host interstitial atom (kick-out mechanism). The interstitial–substitutional mechanism has been encountered in zinc diffusion in silicon and gallium arsenide.22 Naturally, there are potential energy barriers hindering the motion of atoms in the lattice. The activation energy associated with the barriers may be overcome by providing thermal energy to the system. The jump frequency o of a defect is given by3 DGm o ¼ n exp kT where DGm is the free energy required to transport the defect from an initial equilibrium position to a saddle point and n is the vibrational frequency. In real materials, the atomic transport may be locally affected by interactions with other defects especially if the defect concentration is high.35–37
where Ha is the activation enthalpy of diffusion, and D0 is the diffusion prefactor that contains all entropy terms and is related to the attempt frequency for migration. When diffusion involves only an interstitial migrating from one interstitial site to an adjacent interstitial site, the activation enthalpy of diffusion is composed mainly of the migration enthalpy. In comparison, for vacancy-mediated diffusion, dopants are trapped in substitutional positions and form a cluster with one or more vacancies. In such a situation, diffusion requires the formation of the cluster that assists in diffusion, migration of the cluster, and finally, the dissociation of the cluster. It is common for experimental studies referring to vacancy-mediated diffusion to refer to the activation enthalpy of diffusion. The activation enthalpy is the sum of the formation enthalpy and the migration enthalpy. The formation energy represents the energetic cost to construct a defect in the lattice (which may well require a complete Frenkel or Schottky process to occur). The formation energy of a defect, Ef (defect), is defined by X Ef ðdefectÞ ¼ E ðdefectÞ þ qme nj mj j
where Ef (defect) is the total energy of the supercell containing the defect; q, the charge state of the defect; me, the electron chemical potential with respect to the top of the valence band of the pure material; nj, the number of atoms of type j; and mj, the chemical potential of atoms of type j. It should be noted that in this definition, contributions of entropy and phonons have been neglected. The migration energy is the energy barrier between an initial state and a final state of the diffusion process. For a system with a complex potential energy landscape, there are a number of different paths that need to be considered.
1.02.7 Summary Point defects are ubiquitous: as intrinsic species, they are a consequence of equilibrium, but usually they are far more numerous incorporated as extrinsic species formed as a consequence of fabrication conditions. Slow kinetics mean that impurities are trapped
64
Fundamental Point Defect Properties in Ceramics
in ceramic materials, typically once temperatures drop below 800 K, although this value is quite material-dependent. The intentional incorporation of dopants into a crystal lattice can be used to fundamentally alter a whole range of processes: this includes the transport of ions, electrons, and holes. As a result, diffusion rates and electrical conductivity can be manipulated to increase or decrease by many orders of magnitude.1,2 Other mechanical or radiation tolerance-related properties can also be changed radically. This chapter has provided the framework for understanding the properties of point defects. In particular, the understanding of the concentration of equilibrium-intrinsic species, dopant ions and their interdependence, defect association to form clusters and nonstoichiometry. In each case, these defects alter the lattice surrounding them, with atoms being shifted from their perfect lattice positions in response to the specific defect type. Electronic defects have been described: not only electrons and holes formed by doping, but also states formed by excitation. Structural defects and electronic defects are considered together through Brouwer diagrams. Finally, we have also considered the transport of ions through the lattice via different processes, all of which require the formation of point defects.
14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32.
33.
References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.
Kingery, W. D.; Bowen, H. K.; Uhlmann, D. R. Introduction to Ceramics; Wiley: New York, 1976. Chiang, Y. M.; Birnie, D.; Kingery, W. D. Physical Ceramics: Principles for Ceramic Science and Engineering; MIT Press: Cambridge, 1997. Kittel, C. Introduction to Solid State Physics; Wiley: New York, 1996. Hull, D.; Bacon, D. J. Introduction to Dislocations, 4th ed.; Butterworth-Heinemann: Oxford, 2001. Harding, J. H.; Atkinson, K. J. W.; Grimes, R. W. J. Am. Ceram. Soc. 2003, 86, 554. Kro¨ger F. A.; Vink, H. J. In Solid State Physics; Seitz, F., Turnbull, D., Eds.; Academic Press: New York, 1956; Vol. 3, p 307. Ball, J. A.; Pirzada, M.; Grimes, R. W.; Zacate, M. O.; Price, D. W.; Uberuaga, B. P. J. Phys. Condens. Matter 2005, 17, 7621. Chen, W. K.; Peterson, N. L. J. Phys. Chem. Solids 1980, 41, 647. See papers in the special issue of Faraday Transactions II. Mol. Chem. Phys. 1989, 85(5), 335–579. Chroneos, A.; Grimes, R. W.; Tsamis, C. Mat. Sci. Semicond. Process. 2006, 9, 536. Ashley, N. J.; Grimes, R. W.; McClellan, K. J. J. Mat. Sci. 2007, 42, 1884. Vyas, S.; Grimes, R. W.; Binks, D. J.; Rey, F. J. Phys. Chem. Solids 1997, 58, 1619. Goodenough, J. B. Nature 2000, 404, 821.
34. 35. 36. 37.
Zacate, M. O.; Minervini, L.; Bradfield, D. J.; Grimes, R. W.; Sickafus, K. E. Solid State Ionics 2000, 128, 243. Shannon, R. D. Acta Cryst. 1976, A32, 751. Tuller, H. L.; Nowick, A. S. J. Phys. Chem. Solids 1977, 38, 859. Schwingenschlogl, U.; Chroneos, A.; Schuster, C.; Grimes, R. W. Appl. Phys. Lett. 2010, 96, 242107. Stoneham, A. M. Theory of Defects in Solids; Clarendon: Oxford, 1975. Shluger, A. L.; Grimes, R. W.; Catlow, C. R. A.; Itoh, N. J. Phys. Condens. Matter 1991, 3, 8027. Rachko, Z. A.; Valbis, T. A. Phys. Stat. Sol. B 1979, 93, 161. Shluger, A. L.; Harker, A. H.; Grimes, R. W.; Catlow, C. R. A. Phil. Trans. R. Soc. Lond. A 1992, 341, 221. Mehrer, H. Diffusion in Solids; Springer: Berlin Heidelberg, 2007. Murphy, S. T.; Uberuaga, B. P.; Ball, J. A.; et al. Solid State Ionics 2009, 180, 1. Rupasov, D.; Chroneos, A.; Parfitt, D.; et al. Phys. Rev. B 2009, 79, 172102. Miyoshi, S.; Martin, M. Phys. Chem. Chem. Phys. 2009, 11, 3063. Chroneos, A.; Parfitt, D.; Kilner, J. A.; Grimes, R. W. J. Mater. Chem. 2010, 20, 266. Bracht, H.; Nicols, S. P.; Walukiewicz, W.; Silveira, J. P.; Briones, F.; Haller, E. E. Nature (London) 2000, 408, 69. Weiler, D.; Mehrer, H. Philos. Mag. A 1984, 49, 309. Chroneos, A.; Bracht, H. J. Appl. Phys. 2008, 104, 093714. Brotzmann, S.; Bracht, H. J. Appl. Phys. 2008, 103, 033508. Chroneos, A.; Bracht, H.; Grimes, R. W.; Uberuaga, B. P. Appl. Phys. Lett. 2008, 92, 172103. Kilner, J. A.; Irvine, J. T. S. In Handbook of Fuel Cells – Advances in Electrocatalysis, Materials, Diagnostics and Durability; Vielstich, W., Gasteiger, H. A., Yokokawa, H., Eds.; John Wiley & Sons: Chichester, 2009; Vol. 5. Sadigh, B.; Lenosky, T. J.; Theiss, S. K.; Caturla, M. J.; de la Rubia, T. D.; Foad, M. A. Phys. Rev. Lett. 1999, 83, 4341. Uberuaga, B. P.; Smith, R.; Henkelman, G. Phys. Rev. B 2005, 71, 104102. Brotzmann, S.; Bracht, H.; Lundsgaard Hansen, J.; et al. Phys. Rev. B 2008, 77, 235207. Chroneos, A.; Grimes, R. W.; Uberuaga, B. P.; Bracht, H. Phys. Rev. B 2008, 77, 235208. Bernardi, F.; dos Santos, J. H. R.; Behar, M. Phys. Rev. B 2007, 76, 033201.
Further Reading Agullo-Lopez, F.; Catlow, C. R. A.; Townsend, P. D. Point Defects in Materials; Academic Press: San Diego, 1988. Greenwood, N. N. Ionic Crystals Lattice Defects and Nonstoichiometry; Butterworth: London, 1970. Kofstad, P. Nonstoichiometry, Diffusion and Electrical Conductivity in Binary Metal Oxides; Wiley: New York, 1972. Schmalzried, H. Solid State Reactions; Academic Press: New York, 1974. Stoneham, A. M. Theory of Defects in Solids: Electronic Structure of Defects in Insulators and Semiconductors; Oxford University Press: Oxford, 2001. Tilley, R. J. D. Defect Crystal Chemistry and its Applications; Blackie & Son: Glasgow, 1987. Van Gool, W. Principles of Defect Chemistry of Crystalline Solids; Academic Press: New York, 1966.
1.03
Radiation-Induced Effects on Microstructure*
S. J. Zinkle Oak Ridge National Laboratory, Oak Ridge, TN, USA
ß 2012 Elsevier Ltd. All rights reserved.
1.03.1 1.03.2 1.03.3 1.03.3.1 1.03.3.2 1.03.3.3 1.03.3.3.1 1.03.3.3.2 1.03.3.3.3 1.03.3.3.4 1.03.3.3.5 1.03.3.4 1.03.3.5 1.03.3.6 1.03.3.7 1.03.3.8 1.03.3.9 1.03.4 1.03.4.1 1.03.4.2 1.03.4.3 1.03.4.4 1.03.4.5 1.03.4.6 1.03.5 1.03.5.1 1.03.5.2 1.03.5.3 1.03.5.4 1.03.5.5 1.03.5.6 1.03.5.7 1.03.6 References
Introduction Overview of Defect Cluster Geometries in Irradiated Materials Influence of Experimental Conditions on Irradiated Microstructure Irradiation Dose Role of Primary Knock-on Atom (PKA) Spectra Role of Irradiation Temperature Very low temperature regime: immobile SIAs (T< Stage I) Low temperature regime: mobile SIAs, immobile vacancies (Stage I
Stage V) Very high temperature regime: He cavities (T >> Stage V) Role of Atomic Weight Role of Crystal Structure Role of Atomic Bonding Role of Dose Rate Role of Ionizing Radiation Role of Solute Segregation and Precipitation Overview of Key Radiation-Induced Property Degradation Phenomena Radiation-Induced Amorphization Radiation-Induced Hardening Thermal and Electrical Conductivity Degradation Radiation-Induced Segregation and Precipitation Dimensional Instabilities: Irradiation Growth, Creep, and Swelling High Temperature Embrittlement Examples of Radiation-Induced Microstructural Changes Dislocation Loop Formation Network Dislocation Formation Stacking Fault Tetrahedra Dislocation Channeling and Flow Localization Crystalline to Amorphous Phase Transitions Radiation-Induced Precipitation Cavity Formation Summary
Abbreviations appm Atomic parts per million bcc Body-centered cubic dpa Displacements per atom
*Prepared for the Oak Ridge National Laboratory under Contract No. DE-AC05-000R22725
fcc HCP MD PKA RIS SIA SFT
66 66 67 67 68 70 71 72 72 74 77 77 78 78 79 80 82 83 83 84 85 85 86 87 88 88 89 89 90 91 91 92 93 93
Face-centered cubic Hexagonal close packed Molecular dynamics Primary knock-on atom Radiation induced segregation Self-interstitial atom Stacking fault tetrahedron
65
66 TEM TM
Radiation-Induced Effects on Microstructure
Transmission electron microscope Melting temperature
1.03.1 Introduction Irradiation of materials with particles that are sufficiently energetic to create atomic displacements can induce significant microstructural alteration, ranging from crystalline-to-amorphous phase transitions to the generation of large concentrations of point defect or solute aggregates in crystalline lattices. These microstructural changes typically cause significant changes in the physical and mechanical properties of the irradiated material. A variety of advanced microstructural characterization tools are available to examine the microstructural changes induced by particle irradiation, including electron microscopy, atom probe field ion microscopy, X-ray scattering and spectrometry, Rutherford backscattering spectrometry, nuclear reaction analysis, and neutron scattering and spectrometry.1,2 Numerous reviews, which summarize the microstructural changes in materials associated with electron3–6 and heavy ion or neutron4,7–20 irradiation, have been published. These reviews have focused on pure metals5–10,12–14,16,19 as well as model alloys,3,9,13,14 steels,11,20 and ceramic3,4,15,17,18 materials. In this chapter, the commonly observed defect cluster morphologies produced by particle irradiation are summarized and an overview is presented on some of the key physical parameters that have a major influence on microstructural evolution of irradiated materials. The relationship between microstructural changes and evolution of physical and mechanical properties is then summarized, with particular emphasis on eight key radiation-induced property degradation phenomena. Typical examples of irradiated microstructures of metals and ceramic materials are presented. Radiation-induced changes in the microstructure of organic materials such as polymers are not discussed in this overview.
1.03.2 Overview of Defect Cluster Geometries in Irradiated Materials A wide range of defect cluster morphologies can be created by particle irradiation.8,21,22 The thermodynamic stability of these defect cluster geometries is dependent on the host material and defect cluster size as well as the potential presence of impurities. There are four common geometric configurations for
clusters of vacancies and self-interstitial atoms (SIAs): two planar dislocation loop configurations (faulted and perfect loops) that occur for both vacancies and SIAs, and two three-dimensional configurations that occur only for vacancy clusters (the stacking fault tetrahedron, SFT, and cavities). The faulted loop (also called Frank loop) is most easily visualized as either insertion or removal of a layer of atoms, creating a corresponding extrinsic or intrinsic stacking fault associated with condensation of a planar monolayer of vacancies and SIAs, respectively. The faulted loop generally forms on close packed planes, i.e., {111} habit planes with a Burgers vector of b ¼ 1/3h111i for face-centered cubic (fcc) materials, {110} habit planes with b ¼ 1/2h110i for body-centered cubic (bcc) metals, and f1010g habit planes with b ¼ a/2 h1010i for hexagonal close packed (HCP) metals.23 Faulted loops with b ¼ a/2 [0001] on the (0001) basal plane are also observed in many irradiated HCP materials. All of these faulted loops are immobile (sessile). The high stacking fault energy of bcc metals inhibits faulted loop nucleation and growth, and favors formation of perfect loops. There have been several observations of faulted loops consisting of multiple atomic layers.8,21 The perfect loop in fcc materials is typically created from initially formed faulted loops by nucleation of an a/6h112i Shockley partial dislocation that sweeps across the surface of the faulted loop and thereby restores perfect stacking order by this atomic shear of one layer of atoms. The resultant Burgers vector in fcc materials is a/2h110i, maintaining the {111} loop habit planes. After unfaulting, rotation on the glide cylinder gradually changes the habit plane of the fcc perfect loop from {111} to {110} to create a pure edge loop geometry. After the loop rotates to the {110} habit plane, the perfect loop is glissile. Experimental studies of irradiated fcc materials typically observe perfect loops on either {111} or {110} habit planes (or both), depending on the stage of the glide cylinder rotation process. The glissile perfect loop configurations for bcc materials consist of b ¼ a/2h111i loops on {111} habit planes and b ¼ ah100i loops on {100} habit planes. The typical corresponding HCP perfect loop configuration is b ¼ a/3 h1120i on f1120g prismatic habit planes. SFTs are only observed in close-packed cubic structures (i.e. fcc materials). The classic Silcox– Hirsch24 mechanism for SFT formation is based on dissociation of b ¼ 1/3h111i faulted loops into a/6h110i stair rod and a/6h121i Shockley partial dislocations on the acute intersecting {111} planes. Interaction between the climbing Shockley partials creates a/6h011i stair rod dislocations along the
Radiation-Induced Effects on Microstructure
67
0.8 0.7
Faulted loop SFT
Clean void
0.6 Energy per vacancy (eV)
Perfect loop
Copper Stacking fault = 0.055 J m–2 energy Surface energy = 1.7 J m–2
0.5 0.4 0.3 0.2
R L = 1.3 nm R V = 0.7 nm
0.1 0.0 10
L T = 3.6 nm
100
1000 Number of vacancies
10 000
100 000
Figure 1 Comparison of calculated size-dependent energies for different vacancy cluster geometries in pure copper. Reproduced from Zinkle, S. J.; Seitzman, L. E.; Wolfer, W. G., Philos. Mag. A 1987, 55(1), 111–125.
tetrahedron edges. The Silcox–Hirsch mechanism has been verified during in situ transmission electron microscope (TEM) observation of vacancy loops in quenched gold.25 Evidence from molecular dynamics (MD) simulations26–29 and TEM observations12,19,30–32 during in situ or postirradiation studies indicate that SFT formation can occur directly within the vacancyrich cascade core during the ‘thermal spike’ phase of energetic displacement cascades. There is an important distinction between the definitions for the terms void, bubble, and cavity, all of which describe a three-dimensional vacancy cluster that is roughly spherical in shape. Void refers to an object whose stability is not dependent on the presence of internal pressurization from a gaseous species such as helium. Bubbles are defined as pressurized cavities. The term cavity can be used to refer to either voids or bubbles and is often used as a generic term for both cases. In many cases, voids exhibit facets (e.g. truncated octahedron for fcc metals) that correspond with closepacked planes of the host lattice, whereas bubbles are generally spherical in shape. However, the absence of facets cannot be used as conclusive evidence to discriminate between a void and a bubble. Figure 1 shows the calculated energy for different vacancy geometries in pure fcc copper.22 The SFT is calculated to be the most energetically favorable configuration in copper for small sizes (up to about 4 nm edge lengths). Faulted loops are calculated to be stable at intermediate sizes, and perfect loops are calculated to be most stable at larger sizes. In practice,
many metastable defect cluster geometries may occur. For example, it is well established that the transition from faulted to perfect loops is typically triggered by localized stress such as physical impingement of adjoining loops, and not simply by loop energies; the activation energy barrier for unfaulting may be on the order of 1 eVatom 1.8 Similarly, large activation energy barriers exist for the conversion between planar loops and voids.33
1.03.3 Influence of Experimental Conditions on Irradiated Microstructure 1.03.3.1
Irradiation Dose
As discussed in Chapter 1.01, Fundamental Properties of Defects in Metals; Chapter 1.02, Fundamental Point Defect Properties in Ceramics; and Chapter 1.11, Primary Radiation Damage Formation, the international standardized displacement per atom (dpa) unit for radiation damage34 is a useful parameter for comparing displacement damage levels in a variety of irradiation environments. The calculated damage level is directly proportional to the product of the fluence and the average kinetic energy transferred to the host lattice atoms (damage energy). The effective damage cross-sections for 1 MeV particles incident on copper range from 30 barns (1 barn ¼ 1 1024 cm2) for electrons35 to 600 barns for neutrons36 and 2 109 barns for Cu ions.37
68
Radiation-Induced Effects on Microstructure
The dpa unit is remarkably effective in correlating the initial damage production levels over a wide range of materials and irradiating particles and is the singular most important parameter for quantifying radiation effects in materials. Numerous aspects of microstructural evolution are qualitatively equivalent on a dpa basis for materials irradiated in widely different irradiation environments. However, the dpa unit does not accurately capture some of the complex differences in primary damage production for energetic displacement cascade conditions compared to isolated Frenkel pair production.38 For example, defect production at cryogenic temperatures (where long-range defect migration and annihilation does not occur) for neutron and heavy ion-irradiated materials is about 20–30% of the calculated dpa value due to athermal in-cascade recombination processes.38,39 In addition, the accumulated damage, as evident in the form of point defect clusters or other microstructural features, typically exhibits a complex nonlinear relationship with irradiation dose that depends on irradiation temperature and several other factors. The impact of other experimental variables on the dosedependent damage accumulation behavior is discussed in Sections 1.03.3.2–1.03.3.9. 1.03.3.2 Role of Primary Knock-on Atom (PKA) Spectra Displacement damage can occur in materials when the energy transferred to lattice atoms exceeds a critical value known as the threshold displacement energy (Ed), which has a typical value of 30–50 eV.8,18,40 Figure 2 shows an example of the effect of bombarding energy on the microstructure of CeO2 during electron irradiation near room temperature.41 The loop density increases rapidly with increasing energy
200 keV
500 keV
above 200 keV, suggesting that 200 keVelectrons transfer elastic energy that is slightly above the threshold displacement energy. High-resolution microstructural analysis determined that the dislocation loops were associated with aggregation of oxygen ions only (i.e., no Ce displacement damage) for electron energies up to 1250 keV, whereas perfect interstitial-type dislocation loops were formed for electron energies of 1500 keV and higher. Therefore, the corresponding displacement energies in CeO2 are 30 and 50 eV for the O and Ce sublattices, respectively.41 A wide range of PKA energies can be achieved during irradiation, depending on the type and energy of irradiating particle. For example, the average PKA energies transferred to a Cu lattice for 1 MeV electrons, protons, Ne ions, Xe ions, and neutrons are 25 eV, 0.5 keV, 9 keV, 50 keV, and 45 keV, respectively.42 Irradiation of materials with electrons and light ions introduces predominantly isolated SIAs and vacancies (together known as Frenkel pairs) and small clusters of these point defects, because of the low average recoil atom energies of 0.1–1 keV. Conversely, energetic neutron or heavy ion irradiations produce energetic displacement cascades that can lead to direct formation of defect clusters within isolated displacement cascades due to more energetic average recoil atom energies that exceed 10 keV. Figure 3 compares the weighted PKA energy values for several irradiation species.40,42 These differences in PKA energy produce significant changes in primary damage state that can have a pronounced effect on the microstructural evolution observed during irradiation. As briefly mentioned in Section 1.03.3.1, the defect production efficiency per dpa determined from electrical resistivity measurements during irradiation near absolute zero and MD simulation studies is significantly lower (by about
750 keV
1000 keV
1250 keV
100 nm Figure 2 Bright-field images of interstitial-type nonstoichiometric dislocation loops formed in CeO2 during 200–1250 keV electron irradiation to a fluence of 3 1026 electrons per square meter at room temperature. The beam direction is along [011] and the diffraction vector is g ¼ 111. Reproduced from Yasunaga, K.; Yasuda, K.; Matsumura, S.; Sonoda, T. Nucl. Instrum. Methods Phys. Res. B 2008, 266(12–13), 2877–2881.
Radiation-Induced Effects on Microstructure
69
1.00
0.80
Copper
Protons
W (T)
0.60
Kr Ne
0.40
0.20
0.00 101
Neutrons
102
103
104
105
106
T (eV) Figure 3 Weighted average recoil atom energy for 1 MeV particles in copper, plotted as a function of recoil energy (T). Reproduced from Averback, R. S. J. Nucl. Mater. 1994, 216, 49–62.
a factor of 3–4) for energetic displacement cascade conditions compared to isolated Frenkel pair conditions, due to pronounced in-cascade recombination and clustering processes.38,39 MD computer simulations43–46 and in situ or postirradiation thin foil experimental studies13,14,47,48 (where interaction between different displacement damage events is minimal due to the strong influence of the surface as a point defect sink) have found that defect clusters visible by transmission electron microscopy (TEM) can be produced directly in displacement cascades if the average PKA energy exceeds 5–10 keV. Irradiations with particles having significantly lower PKA energies typically produce isolated Frenkel pairs and submicroscopic defect clusters that can nucleate and coarsen via diffusional processes. The microstructural evolution of an irradiated material is controlled by different kinetic equations if initial defect clustering occurs directly within the displacement cascade (0.1–1 ps timescale) versus three-dimensional random walk diffusion to produce defect cluster nucleation and growth, particularly if some of the in-cascade created defect clusters exhibit one-dimensional glide.49–52 As discussed in Chapter 1.13, Radiation Damage Theory, this can produce significant differences in the microstructural evolution for features such as voids and dislocation loops. Figure 4 compares the microstructure produced in copper following irradiation near 200 C with fission neutrons53 and 1 MeV electrons.54,55 Vacancies and SIAs are fully mobile in copper at this temperature. The 1 MeV electron produces a steady flux of point defects that leads to the
(a)
(b)
200 nm
200 nm
Figure 4 Comparison of the microstructure of copper irradiated near 200 C with (a) fission neutrons (reproduced from Zinkle, S. J.; Sindelar, R. L. J. Nucl. Mater. 1988, 155–157, 1196–1200) and (b) 1 MeV electrons (modified from Kiritani, M. Ultramicroscopy 1991, 39(1–4), 180–186; Kiritani, M.; Takata, H. J. Nucl. Mater. 1978, 69–70, 277–309).
creation of a moderate density of large faulted interstitial loops. On the other hand, the creation of SFTs and small dislocation loops directly in fission neutron displacement cascades creates a high density (2 1023 m3) of small defect clusters, and the high point defect sink strength associated with these defect clusters inhibits the growth of dislocation loops. As shown in Figure 4, the net result is a dramatic qualitative and quantitative
70
Radiation-Induced Effects on Microstructure
difference in the irradiated microstructure due to differences in the PKA spectrum. Electron microscopy48,56 and binary collision48,57 and MD simulation45 studies have found that irradiation with PKA energies above a critical materialdependent value of 10–50 keV results in formation of multiple subcascades (rather than an everincreasing single cascade size), with the size of the largest subcascades being qualitatively similar to an isolated cascade at a PKA energy near the critical value. Figure 5 compares MD simulations of the peak displacement configurations of PKAs in iron with energies ranging from 1 to 50 keV.58 At low PKA energies, the size of the displacement cascade increases monotonically with PKA energy. When the PKA energy in Fe exceeds a critical value of 10 keV, multiple subcascades begin to appear, with the largest subcascade having a size comparable to the 10 keV cascades. The number of subcascades increases with increasing PKA energy, reaching 5 subcascades for a PKA energy of 50 keV in Fe. A fortunate consequence of subcascade formation is that fission reactor irradiations (1 MeV neutrons) can be used for initial radiation damage screening studies of potential future fusion reactor (14 MeV neutrons) materials, since both would have comparable primary damage subcascade structures.59,60 Further details on the effect of PKA spectrum on primary damage formation
Y
10 keV
1 keV
are given in Chapter 1.11, Primary Radiation Damage Formation. 1.03.3.3
Role of Irradiation Temperature
Irradiation temperature typically invokes a very large influence on the microstructural evolution of irradiated materials. There are several major temperature regimes delineated by the onset of migration of point defects. Early experimental studies used isochronal annealing electrical resistivity measurements on metals irradiated near absolute zero temperature to identify five major defect recovery stages.61–64 Figure 6 shows the five major defect recovery stages for copper irradiated with electrons at 4 K.65 The quantitative magnitude of the defect recovery in each of the stages generally depends on material, purity, PKA spectrum, and dose. Based on the currently accepted one-interstitial model, Stage I corresponds to the onset of long-range SIA migration. Stage I often consists of several visible substages that have been associated with close-pair (correlated) recombination of Frenkel defects from the same displacement event and long range uncorrelated recombination of defects from different primary displacement events. Stage II involves migration of small SIA clusters and SIA-impurity complexes. Stage III corresponds to the onset of vacancy motion. Stage IV involves migration of vacancy–impurity clusters, and Stage V corresponds to thermal dissociation of sessile vacancy clusters. It should be noted that the specific recovery stage temperature depends on the annealing time (typically 10 or 15 min in the resistivity studies), and therefore needs to be adjusted to lower values when considering the onset temperatures for defect migration in typical
Induced resistivity
50 keV
Z
I
II
III IV V
X
Figure 5 Comparison of the molecular dynamics simulations of 1–50 keV PKA displacement cascades in iron. PKA energies of 1 (red), 10 (green), and 50 (blue) keV for times corresponding to the transient peak number of displaced atoms are shown. The length of the Z (horizontal) dimension of the simulation box is 170 lattice parameters (49 nm). Adapted from Stoller, R. E., Oak Ridge National Lab, Private communication, 2010.
20
50
100
200
500
Temperature (K) Figure 6 Electrical resistivity defect recovery stages for copper following electron irradiation at 4 K. Reproduced from Agullo-Lopez, F.; Catlow, C. R. A.; Townsend, P. D., Point Defects in Materials. Academic Press: San Diego, CA, 1988; p 445.
Radiation-Induced Effects on Microstructure
Table 1
71
Summary of defect recovery stage temperatures for materials8,18,63,66–69
Material
Melting temperature (K)
Crystal structure
Stage I (K)
Stage III (K)
Stage V (K)
Pb Al Ag Au Cu Ni Pd Pt Rh SiC a-Fe Cr V Nb Mo Ta W Cd Zn Mg Ti Be Co Zr Re Al2O3
601 933 1233 1337 1357 1726 1825 2045 2236 3103 1809 2130 2175 2740 2890 3287 3680 594 693 922 1043 1560 1768 2125 3453 2324
fcc fcc fcc fcc fcc fcc fcc fcc fcc cubic bcc bcc bcc bcc bcc bcc bcc HCP HCP HCP HCP HCP HCP HCP HCP HCP
5 35 35 <4 50 60 50 30 32 220 (C) 450 (Si) 110 40 <6 5 40 10 30 <4 18 13 120 45 55 150 100 150
150 220 240 290 270 350 350 350 500 1400 (C) 1050 (Si) 230 380 220 230 470 270 650 120 125 130 250 280 310 270 630 850
300 540 530 550
1180
Source: Eyre, B. L. J. Phys. F 1973, 3(2), 422–470. Zinkle, S. J.; Kinoshita, C. J. Nucl. Mater. 1997, 251, 200–217. Schilling, W.; Ehrhart, P.; Sonnenberg, K. In Fundamental Aspects of Radiation Damage in Metals, CONF-751006-P1; Robinson, M. T.; Young, F. W., Jr., Eds. National Tech. Inform. Service: Springfield, VA, 1975; Vol. I, pp 470–492. Hautojarvi, P.; Pollanen, L.; Vehanen, A.; Yli-Kauppila, J. J. Nucl. Mater. 1983, 114(2–3), 250–259. Lefevre, J.; Costantini, J. M.; Esnouf, S.; Petite, G. J. Appl. Phys. 2009, 106(8), 083509. Schultz, H. Mater. Sci. Eng. A 1991, 141, 149–167. Xu, Q.; Yoshiie, T.; Mori, H. J. Nucl. Mater. 2002, 307–311(2), 886–890. Young, F. W., Jr. J. Nucl. Mater. 1978, 69/70, 310. Hoffmann, A.; Willmeroth, A.; Vianden, R. Z. Phys. B 1986 62, 335. Takamura, S.; Kobiyama, M. Rad. Eff. Def. Sol. 1980, 49(4), 247. Kobiyama, M.; Takamura, S. Rad. Eff. Def. Sol. 1985, 84(3&4), 161.
neutron irradiation experiments that may occur over time scales of months or years. Table 1 provides a summary of defect recovery stage temperatures for several fcc, bcc, and HCP materials.8,18,63,66–69 Although there is a general correlation of the recovery temperatures with melting temperature, Table 1 shows there are several significant exceptions. For example, Pt has one of the lowest Stage I temperatures among fcc metals despite having a very high melting temperature. Similarly, Cr has a much higher Stage III temperature than V or Nb that have higher melting points. As illustrated later in this chapter, the microstructures of different materials with the same crystal structure and irradiated within the same recovery stage temperature regime are generally qualitatively similar.
Several analytic kinetic rate theory models have been developed to express the dose dependence of defect cluster accumulation in materials at different temperature regimes.6,70–72 In the following, summaries are provided on the experimental microstructural observations for five key irradiation temperature regimes. 1.03.3.3.1 Very low temperature regime: immobile SIAs (T< Stage I)
At very low temperatures where defect migration does not occur, defect accumulation is typically proportional to dose until the defect concentration approaches the level where defects created in displacement events begin to overlap and annihilate preexisting defects created earlier in the irradiation
72
Radiation-Induced Effects on Microstructure
Relative defect concentration (N/Nmax)
0.03 ZnO: Ar (200 keV)
0.02
0–0.04 mm 0.04–0.08 mm
0.01
0.06–0.08 mm 0.11–0.15 mm
0.00 0.00
0.05
0.10
0.15
0.20
Displacement damage dose (dpa) Figure 7 Defect concentration normalized to the total atom concentration Nmax at four different depths in ZnO irradiated with 200 keV Ar ions at 15 K as determined by Rutherford backscattering spectrometry. Reproduced from Wendler, E.; Bilani, O.; Ga¨rtner, K.; et al. Nucl. Instrum. Methods Phys. Res. B 2009, 267(16), 2708–2711.
exposure. The defect accumulation kinetics73 can be described by N ¼ Nmax[1 exp(Aft)], where the parameter A is determined by the spontaneous recombination volume for point defects or the cascade overlap annihilation volume for defect clusters and ft is the product of the irradiation flux and time. Due to the lack of defect mobility, defect clusters resolvable by TEM are usually not visible in this irradiation temperature regime unless they are created directly in displacement cascades by energetic PKAs.74 Saturation in the defect concentration typically occurs after 0.1 dpa as monitored by atomic disorder,75–77 electrical resistivity,78–82 and dimensional change.83–85 Due to the large increase in free energy associated with lattice disordering and defect accumulation, amorphization typically occurs in this temperature regime in many ceramics15,85,86 and ordered metallic alloys87,88 for doses above 0.1– 0.5 dpa. Figure 7 shows an example of the dosedependent defect concentration in ion-irradiated ZnO at 15 K as determined by Rutherford backscattering spectrometry.89 1.03.3.3.2 Low temperature regime: mobile SIAs, immobile vacancies (Stage I
Between recovery Stage I and Stage III, the SIA point defects and small SIA clusters have sufficient mobility to migrate and form visible dislocation loops as well as recombine with sessile monovacancies and vacancy clusters. The defect accumulation in this temperature regime is initially linear with dose when the defect concentration is too low for
uncorrelated recombination to be a significant contribution, but then transitions to a square root dependence at an intermediate dose in pure materials when interaction between defects from different PKA events becomes important.6,70–72,90 The critical dose for this kinetic transition is dependent on the concentration of other defect sinks in the lattice (dislocations, grain boundaries, precipitates, etc.). The high sink strength associated with the immobile vacancies limits the growth rate (i.e., size) of the SIA loops for doses above 0.1 dpa, and the observable defect cluster size and density typically approach a constant value at higher doses. Figure 8 shows an example of the microstructure of AlN following ion irradiation at 80 K (mobile SIAs, immobile vacancies) to a damage level of about 5 dpa.91 The microstructure consists of small (<5 nm diameter) interstitial dislocation loops. 1.03.3.3.3 Medium temperature regime: mobile SIAs and vacancies (Stage III
At temperatures where both SIAs and vacancies are mobile, the defect cluster evolution is complex due to the wide range of defect cluster geometries that can be nucleated.8,47,92,93 The predominant visible features in this temperature regime are vacancy and interstitial loops and SFTs for irradiated fcc materials and vacancy and interstitial loops and voids for irradiated bcc materials. For medium- to highatomic number fcc metals exposed to energetic displacement cascades (e.g., fast neutron and heavy ion irradiation), most of the vacancies are tied up in
Radiation-Induced Effects on Microstructure
73
Tirr = 0.2–0.3TM
Defect cluster density (m-3)
1024
50 nm Figure 8 Weak beam microstructure of dislocation loops in AlN after 2 MeV Si ion irradiation to 5 dpa at 80 K. The TEM figure is based on irradiated specimens described in Zinkle et al.91
sessile vacancy clusters (SFTs, vacancy loops) that are formed directly in the displacement cascades. As a consequence, the majority of observed dislocation loops in fcc metals in this temperature and PKA regime are extrinsic (interstitial type), and void nucleation and growth is strongly suppressed. For bcc metals, the amount of in-cascade clustering into sessile defect clusters is less pronounced, and therefore, vacancy loop and void swelling are observed in addition to interstitial dislocation loop evolution. Due to the typical high sink strength of interstitial clusters in this temperature regime, the magnitude of void swelling is generally very small (<1% for doses up to 10 dpa or higher). The loop density and nature in bcc metals is strongly dependent on impurity content in this temperature regime.5,8,55 For example, the loop concentration in molybdenum irradiated with fission neutrons at 200 C is much higher in low-purity Mo with 99% of the loops identified as interstitial type, whereas 90% of the loops were identified to be vacancy type in high-purity Mo irradiated under the same conditions.8 The dose dependence of defect cluster accumulation in this temperature regime is dependent on the material and defect cluster type. For dislocation loops and SFTs in fcc metals, the defect accumulation is initially linear and may exhibit an extended intermediate regime with square root kinetics before reaching a maximum concentration level. The maximum defect cluster density is largely determined by displacement cascade annihilation of preexisting defect clusters. In fcc metals, the defect cluster
Copper
n = 1/2
1023
Nickel
1022 n=1
1021
1020 -5 10
10-4
10-3 10-2 Damage level (dpa)
10-1
100
Figure 9 Defect cluster density in neutron-irradiated copper and nickel following fission reactor, 14 MeV, and spallation neutron irradiation near room temperature, as measured by TEM. Depending on the purity of the nominally high-purity copper, the defect cluster accumulation at intermediate doses (0.001–0.01 dpa) either exhibits a continuation of linear kinetics or switches to a square root accumulation behavior. Based on data reported by Zinkle94 and Hashimoto et al.95,96
density may approach 1024 m3, which corresponds to a defect cluster spacing of less than 10 nm and is approximately equal to the maximum diameter of subcascades during the collisional phase in neutronirradiated metals. As with irradiation near recovery Stage II, the critical dose for transition in defect cluster accumulation kinetics is dependent on the overall defect sink strength. With continued irradiation, the loops may unfault and evolve into network dislocations, particularly if external stress is applied. Figure 9 summarizes the dose-dependent defect cluster densities in neutron-irradiated copper and nickel.94–96 In both of these materials, the predominant visible defect cluster was the SFT over the entire investigated dose and temperature regime. Depending on the purity of the copper investigated, the transition from linear to square root accumulation behavior may or may not be evident (cf. the differing behavior for Cu in Figure 9). The visible defect cluster density in irradiated copper reaches a constant saturation value (attributed to displacement cascade overlap with preexisting clusters) for damage levels above 0.1 dpa. The lower visible defect cluster density in Ni compared to Cu at doses up to 1 dpa has been attributed to a longer thermal spike lifetime of the Cu displacement cascades due to inefficient
74
Radiation-Induced Effects on Microstructure
coupling between electrons and phonons (thereby promoting more complete vacancy and interstitial clustering within the displacement cascade).97,98 Figure 10 compares the defect cluster accumulation behavior for two fcc metals (Cu, Ni) and two bcc metals (Fe, Mo) following fission neutron irradiation near room temperature.30,95,96,99–101 For all four materials, the increase in visible defect cluster density is initially proportional to dose. The visible defect cluster density is highest in Cu over the 1025
Defect cluster density (m-3)
Tirr = 295–340 K 1024
Cu Ni
1023
Mo
1022
1.03.3.3.4 High temperature regime: mobile defects and vacancy loop dissociation (T>Stage V)
Fe 1021
1020 0.0001
investigated damage range of 104–1 dpa. The irradiated Fe has the lowest visible density at low doses, whereas Ni and Mo have comparable visible cluster densities. At doses above 0.01 dpa, the visible loop density in Mo decreases due to loop coalescence in connection with the formation of aligned ‘rafts’ of loops. Partial formation of aligned loop rafts has also been observed in neutron-irradiated Fe for doses near 0.8 dpa, as shown in Figure 11.100 The individual loops within the raft aggregations in neutronirradiated Fe exhibited the same Burgers vector. The maximum visible cluster density in the fcc metals is about one order of magnitude higher than in the bcc metals (due in part to loop coalescence associated with raft formation). Positron annihilation spectroscopy analyses suggest that submicroscopic cavities are present in the two irradiated bcc metals, with cavity densities that are about two orders of magnitude higher than the visible loop densities.99–102
0.1 0.001 0.01 Displacement damage (dpa)
1
Figure 10 Defect cluster density in copper, nickel, molybdenum, and nickel following fission reactor and 14-MeV neutron irradiation near room temperature, as measured by TEM. Based on data reported by Kiritani30, Hashimoto et al.95,96, Eldrup et al. 99, Zinkle and Singh100, and Li et al.101
The typical microstructural features that appear during irradiation at temperatures above recovery Stage V include dislocation loops (vacancy and interstitial type), network dislocations, and cavities. SFTs are thermally unstable in this temperature regime and therefore only SFTs created in the latter stages of the irradiation exposure are visible during postirradiation examination.94 A variety of precipitates may also be nucleated in irradiated alloys.11,103–106 Defect cluster accumulation in this temperature regime exhibits
B = 111 g = 110
g = 110
(a)
50 nm
(b)
100 nm
Figure 11 Examples of aligned rafts of dislocation loops in iron following fission neutron irradiation to 0.8 dpa at 60 C. The microstructure in thin (a) and thick (b) foil regions are shown. Reproduced from Zinkle, S. J.; Singh, B. N. J. Nucl. Mater. 2006, 351, 269–284.
Radiation-Induced Effects on Microstructure
several different trends. The visible SIA clusters evolve from a low density of small loops to a saturation density of larger loops after damage levels of 1–10 dpa. Upon continued irradiation, a moderate density of network dislocations is created due to loop unfaulting and coalescence. The dislocation loop and network dislocation density monotonically decrease with increasing temperature above recovery Stage V,20,107 whereas the density of precipitates (if present) can either increase or decrease with increasing temperature. The major microstructural difference from lower temperature irradiations in most materials is the emergence of significant levels of cavity swelling. After an initial transient regime associated with cavity nucleation, a prolonged linear accumulation of vacancies into voids is typically observed.108,109 The cavity density monotonically decreases with increasing temperature in this temperature regime.20,107,110 Figure 12 summarizes the densities of voids and helium bubbles (associated with n,a transmutations) in austenitic stainless steel as a function of fission reactor irradiation temperature for damage rates near 1 106 dpa s1.20 The bubble and void densities exhibit similar temperature dependences in fission reactor-irradiated austenitic stainless steel, with the bubble density approximately one order of magnitude higher than the void density between 400 and 650 C. For neutron-
75
irradiated copper and Cu–B alloys, the bubble density is similarly observed to be about one order of magnitude larger than the void density for temperatures between 200 and 400 C.107,110 At higher temperatures, the void density in copper decreases rapidly and becomes several orders of magnitude smaller than the bubble density. The results from several studies suggest that the lower temperature limits for formation of visible voids111–113 and helium bubbles53 can each be reduced by 100 C or more when the damage rate is decreased to 109–108 dpa s1, due to enhanced thermal annealing of sessile vacancy clusters during the time to achieve a given dose. Dose rate effects are discussed further in Section 1.03.3.7. The void swelling regime for fcc materials typically extends from 0.35 to 0.6TM, where TM is the melting temperature, with maximum swelling occurring near 0.4–0.45TM for typical fission reactor neutron damage rates of 106 dpa s1.92,114 Figure 13 summarizes the temperature-dependent void swelling for neutron-irradiated copper.110 The results for a neutron-irradiated Cu–B alloy, where 100 atomic parts per million (appm) He was produced during the 1 dpa irradiation due to thermal neutron transmutation reactions with the B solute, are also shown in this figure.107 For both materials the onset
1024
Bubbles
1023
304 SS 0.5 dpa per 10 yr.
Cavity density (m-3)
1022
1021
Voids
1020
SA CW Maziasz (1985) Maziasz (1991) Hamada et al. (1989) Zinkle and Sindelar53 Brager and Straalsund (1973) Farrell and Packan (1982) Bagley et al. (1971)
1019
1018
0
100
200
300 400 Temperature (⬚C)
500
600
700
Figure 12 Effect of neutron irradiation temperature on the cavity density observed in austenitic stainless steels for damage rates near 1 10 6 dpa s 1 (except the labeled data point at 120 C which had a damage rate of 10 9 dpa s 1). Reproduced from Zinkle, S. J.; Maziasz, P. J.; Stoller, R. E. J. Nucl. Mater. 1993, 206, 266–286.
76
Radiation-Induced Effects on Microstructure
0.7 Cu-100 appm 10B 0.6
Density change (%)
0.5
0.4 Pure Cu 0.3
0.2
0.1
0 150
200
250
300
350
400
450
500
550
Irradiation temperature (⬚C) Figure 13 Temperature-dependent void swelling behavior in neutron-irradiated copper and Cu–B alloy after fission neutron irradiation to a dose near 1.1 dpa. Adapted from Zinkle, S. J.; Farrell, K.; Kanazawa, H. J. Nucl. Mater. 1991, 179–181, 994–997; Zinkle, S. J.; Farrell, K. J. Nucl. Mater. 1989, 168, 262–267.
8
0.35 TM
0.55 TM 50
7 51 6 Swelling (%)
of swelling occurs at temperatures near 180 C, which corresponds to recovery Stage V in Cu for the 2 107dpa s1 damage rates in this experiment. The swelling in Cu was negligible for temperatures above 500 C, and maximum swelling was observed near 300 C. The lower temperature limit for swelling in fcc materials is typically controlled by the high point defect sink strength of sessile defect clusters below recovery Stage V. The upper temperature limit is controlled by thermal stability of voids and a reduction in the vacancy supersaturation relative to the equilibrium vacancy concentration. As noted by Singh and Evans,92 the temperature dependence of the void swelling behavior of bcc and fcc metals can be significantly different. In particular, due to the lower amount of in-cascade formation of large sessile vacancy clusters in medium-mass bcc metals compared to fcc metals, the recovery Stage V is much less pronounced in bcc metals. The presence of a high concentration of mobile vacancies at temperatures below recovery Stage V (and a concomitant reduction in the density of sessile vacancy-type defect cluster sinks) allows void swelling to occur in bcc metals for temperatures above recovery Stage III (onset of long-range vacancy migration). Figure 14 compares the temperature dependence of the void swelling behavior of Ni (fcc) and Fe (bcc) after high dose neutron irradiation.115 Whereas the peak
57
Ni
5 4 3
70 dpa
53
58 dpa
Fe
2 1
51 36
0 300
57
38
400
50 500
25 600
24 dpa
700
Temperature (⬚C)
Figure 14 Comparison of the temperature-dependent void swelling behavior in Fe and Ni, based on data reported by Budylkin et al.115
swelling after 50 dpa in neutron-irradiated Ni occurred near 0.45TM, the peak swelling in Fe occurred at the lowest investigated temperature of 0.35TM. Several other bcc metals including Mo, W, Nb, and Ta exhibit void formation for irradiation temperatures as low as 0.2TM, which is approaching the upper limit of recovery Stage III.92 It is worth noting the peak swelling temperature for neutron-irradiated bcc metals Mo and Nb–1Zr after exposures of 50 dpa
Radiation-Induced Effects on Microstructure
occur near 0.3–0.35TM,116,117 which is much lower than the 0.4–0.45TM peak swelling temperature observed for fcc metals. 1.03.3.3.5 Very high temperature regime: He cavities (T >> Stage V)
Irradiation at temperatures near or above 0.5TM typically results in only minor microstructural changes due to the strong influence of thermodynamic equilibrium processes, unless significant amounts of impurity atoms such as helium are introduced by nuclear transmutation reactions or by accelerator implantation. When helium is present, cavities are nucleated in the grain interior and along grain boundaries. The cavity size increases and the density decreases rapidly with increasing temperature. Figure 15 compares the helium cavity density for various implantation and neutron irradiation conditions in austenitic stainless 1024 Ih Ih + Rh
1023
Bubbles
1022
Ca (m-3)
Ic+A
Voids Rh (n)
1021
1020
1019
1018
8
10
12
14
steels as a function of temperature.118,119 The temperature dependence of the cavity density is distinguished by two different regimes. At very high temperatures, the cavity density is controlled by gas dissociation mechanisms with a corresponding high activation energy, and at lower temperatures by gas or bubble diffusion kinetics.118 The cavity density decreases by nearly two orders of magnitude for every 100 K increase in irradiation temperature in this very high temperature regime. The helium cavity densities in materials irradiated at low temperatures (near room temperature) and then annealed at high temperature are typically much higher than in materials irradiated at high temperature, due to excessive cavity nucleation that occurs at low temperature. In the absence of applied stress, the helium-filled cavities tend to nucleate rather homogeneously in the grain interiors and along grain boundaries. If the helium generation and displacement damage occurs in the presence of an applied tensile stress, the helium cavities are preferentially nucleated along grain boundaries and may cause grain boundary embrittlement.120 1.03.3.4
Rh (n)
16
18
Figure 15 Temperature dependence of observed cavity densities in commercial austenitic steels during He implantation or neutron irradiation at elevated temperatures (Ih and Rh, respectively). The dashed lines denote the densities of voids during neutron irradiation (Rh(n)) and bubbles during implantation near room temperature followed by high temperature annealing (IcþA). Adapted from Singh, B. N.; Trinkaus, H. J. Nucl. Mater. 1992, 186, 153–165; Trinkaus, H.; Singh, B. N. J. Nucl. Mater. 2003, 323 (2–3), 229–242.
77
Role of Atomic Weight
Materials with low atomic weight, such as aluminum, exhibit more spatially diffuse displacement cascades than high atomic weight materials due to the increase in nuclear and electronic stopping power with increasing atomic weight. For example, the calculated average vacancy concentration in Au displacement cascades is about two to three times higher than in Al cascades for a wide range of PKA energies.57 This increased energy density and compactness in the spatial extent of displacement cascades can produce enhanced clustering of point defects within the energetic displacement cascades of high atomic weight materials. Electrical resistivity isochronal annealing studies of fission neutron-irradiated metals have confirmed that the amount of defect recovery during Stage I annealing decreases with increasing atomic weight,79 which is an indication of enhanced SIA clustering within the displacement cascades. The importance of atomic weight on defect clustering depends on the material-specific critical energy for subcascade formation compared to the average PKA energy. For example, in the fcc noble metal series Cu, Ag, Au, the subcascade formation energy increases slightly with mass (10, 13, and 14 keV, respectively), and very little qualitative difference exists in the defect cluster accumulation behavior of these three materials.13,56 In general, there is not a universal relation
78
Radiation-Induced Effects on Microstructure
between atomic weight and microstructural parameters such as overall defect production,121 defect cluster yield,122,123 or visible defect cluster size.56 1.03.3.5
Fe
Role of Crystal Structure
MD simulations23 predict the absolute level of defect production is not strongly affected by crystal structure. Conversely, electrical resistivity studies of fission neutron-irradiated metals suggest that the overall defect production is highest in HCP metals, intermediate in bcc metals, and lowest in fcc metals,121 which suggests that the anisotropic nature of HCP crystals might inhibit defect recombination within displacement cascades. TEM measurements of defect cluster yield (number of visible cascades per incident ion) in ion-irradiated metals have found that the relatively few visible defect clusters are formed directly in displacement cascades in bcc metals,122 whereas cluster formation is relatively efficient in fcc metals and variable behavior is observed for HCP metals.123 Faulted dislocation loops are often observed in irradiated fcc and HCP metals, but due to their high stacking fault energies most studies on irradiated bcc metals have only observed perfect loops.8,16,21,47,124 Since perfect loops are glissile, this can lead to more efficient sweeping up of radiation defects and accelerate the development of dislocation loop rafts or network dislocation structures in bcc materials. Figure 16 shows examples of the dislocation loop microstructures in bcc, fcc, and HCP metals with similar atomic weight following electron irradiation at temperatures above recovery Stage III.47 All of the loops are interstitial type with comparable size for the same irradiation dose. However, significant differences exist in the loop configurations, in particular habit planes and faulted (Ni, Zn) versus perfect (Fe) loops. One significant aspect of loop formation in HCP materials is that differential loop evolution on basal and prism planes can lead to significant anisotropic growth.125–129 In general, defect accumulation in the form of void swelling is significantly lower in bcc materials compared to fcc materials, although there are notable exceptions where very high swelling rates (approaching 3% per dpa)130,131 have been observed in some bcc alloys. Pronounced elastic and point defect diffusion anisotropy128 can also suppress void swelling in HCP materials, although high swelling has been observed in some HCP materials such as graphite.132 It has long been recognized that ferritic/martensitic steels exhibit significantly lower void swelling than
200 nm
Ni
200 nm
Zn
200 nm Figure 16 Dislocation loop microstructures in Fe, Ni, and Zn following electron irradiation at temperatures above recovery Stage III. The loops in Fe were perfect and located on (100) planes, and the loops in Ni and Zn were faulted and located on {111} and (0001) planes, respectively. Reproduced from Kiritani, M. J. Nucl. Mater. 2000, 276(1–3), 41–49.
austenitic stainless steels.109,133,134Figure 17 compares the microstructure of austenitic stainless steel and 9%Cr ferritic/martensitic steel after dual beam ion irradiation at 650 C to 50 dpa and 260 appm He.135 Substantial void formation is evident in the Type 316 austenitic stainless steel, whereas cavity swelling is very limited in the 9%Cr ferritic/martensitic steel for the same irradiation conditions. Several mechanisms have been proposed to explain the lower swelling in ferritic/martensitic steel, including lower dislocation bias for SIA absorption, larger critical radii for conversion of helium bubbles to voids, and higher point defect sink strength. 1.03.3.6
Role of Atomic Bonding
Atomic bonding (i.e., metallic, ionic, covalent, and polar covalent) is a potential factor to consider when comparing the microstructural evolution between metals and nonmetals, or between different nonmetallic materials that may have varying amounts of directional covalent or ionic bonds. For example, several authors have proposed an empirical atomic bonding
Radiation-Induced Effects on Microstructure
SA 316 LN
79
9Cr–2WVTa
500 nm
Figure 17 Comparison of the microstructure of Type 316 LN austenitic stainless steel and 9%Cr–2%WVTa ferritic/ martensitic steel after dual beam ion irradiation at 650 C to 50 dpa and 260 appm He. Reproduced from Kim, I.-S.; Hunn, J. D.; Hashimoto, N.; Larson, D. L.; Maziasz, P. J.; Miyahara, K.; Lee, E. H. J. Nucl. Mater. 2000, 280(3), 264–274.
1017 W Fe Density of loops (cm-3)
criterion to correlate the amorphization susceptibility of nonmetallic materials.136,137 Materials with ionicity parameters above 0.5 appear to have enhanced resistance to irradiation-induced amorphization. However, there are numerous materials which do not follow this correlation,86,138,139 and a variety of alternative mechanisms have been proposed86–88,138–141 to explain resistance to amorphization. Atomic bonding can directly or indirectly influence point defect migration and annihilation mechanisms (e.g., introduction of recombination barriers), and thereby influence the overall microstructural evolution.
1016
Au,150 K
Cu Au 1015
1.03.3.7
Role of Dose Rate
The damage accumulation is independent of dose rate at very low temperatures, where point defect migration does not occur. However, at elevated temperatures (above recovery Stage I) the damage rate can have a significant influence on the damage accumulation. Simple elevated temperature kinetic models for defect accumulation72,142–144 predict a transition from linear to square root dependence on the irradiation fluence when the radiation-induced defect cluster density becomes comparable to the density of preexisting point defect sinks such as line dislocations, precipitates, and grain boundaries. Similar square root flux dependence is predicted from more comprehensive kinetic rate theory models6,70,71,145 for irradiation temperatures between recovery Stage II and IV. Electron microscopy analyses of electron5 and neutron146 irradiation experiments performed above recovery Stage I have reported defect cluster densities that exhibit square root dependence on irradiation flux or fluence.
Mo
1018 1019 Irradiation intensity (electrons cm-2 s-1)
1020
Figure 18 Effect of irradiation flux on the density of interstitial dislocation loops in several fcc and bcc metals during electron irradiation near room temperature or at cryogenic temperature (above recovery Stage I). Reproduced from Kiritani, M. In Fundamental Aspects of Radiation Damage in Metals, CONF-751006-P2; Robinson, M. T.; Young, F. W., Jr., Eds. National Tech. Inform. Service: Springfield, VA, 1975; Vol. II, pp 695–714.
Figure 18 summarizes the square root dose rate dependence for dislocation loop densities at intermediate temperatures in several electron-irradiated pure metals.5 Similarly, the predicted critical dose to achieve amorphization is independent of dose rate below
80
Radiation-Induced Effects on Microstructure
at temperatures as low as 280–300 C, which is significantly lower than the 400 C lower limit for void swelling observed during fission reactor irradiations near 106 dpa s1 (cf. Figure 12).
Dcrit-Do (dpa) = (AF F)e(-Em/2kT) 0.80 Tirr = 380 K Flux exponent, F = -0.34
0.70
Dcrit-Do (dpa)
0.60
1.03.3.8
Tirr = 360 K F = -0.22
0.50
0.40
Tirr = 340 K F = -0.135
0.30
0.20 10-6
10-5 0.0001 Dose rate (dpa s-1)
0.001
Figure 19 Effect of dose rate (F) on the critical dose (Dcrit) to induce complete amorphization in 6H–SiC single crystals during 2 MeV Si ion irradiation. The dose D0 corresponds to the amorphization dose at very low temperatures, where all defects are immobile. The equation at the top of the figure is the prediction from a model (ref. 147) for the dose dependence of amorphization on dose rate, point defect migration energy (Em) and irradiation temperature (T). The parameter F describes the dose rate power law dependence and k is Boltzmann’s constant. Based on data reported by Snead et al.148
recovery Stage I and depends on the inverse square root of dose rate for temperatures above recovery Stage I.147 Experimental studies have confirmed that the threshold dose to achieve amorphization in ion-irradiated SiC is nearly independent of dose rate below 350 K (corresponding to recovery Stage I) and approaches an inverse square root flux dependence for irradiation temperatures above 380 K, as shown in Figure 19.148 In the void swelling149–151 and high temperature helium embrittlement119,152,153 regimes, damage rate effects are very important considerations due to the competition between defect production and thermal annealing processes. Experimental studies using ion irradiation (103 dpa s1) and neutron irradiation (106 dpa s1) damage rates have observed that the peak void swelling regime is typically shifted to higher temperatures by about 100–150 C for the high-dose rate irradiations compared to test reactor neutron irradiation conditions.114,154–158 Similarly, the minimum and maximum temperature for measureable void swelling increase with increasing dose rate. For example, recent low dose rate neutron irradiation studies111–113 performed near 109–108 dpa s1 have observed void swelling in austenitic stainless steel
Role of Ionizing Radiation
Due to relatively large concentrations of conduction electrons, materials with metallic bonding typically do not exhibit sensitivity to ionizing radiation. On the other hand, semiconductor and insulating materials can be strongly affected by ionizing radiation by various mechanisms that lead to either enhanced or suppressed defect accumulation.159 Some materials such as alkali halides, quartz, and organic materials, are susceptible to displacement damage from radiolysis reactions.65,160–163 In materials that are not susceptible to radiolysis, significant effects from ionizing radiation can still occur via modifications in point defect migration behavior. Substantial reductions in point defect migration energies due to ionization effects have been predicted, and significant microstructural changes attributed to ionization effects have been observed in several semiconductors and inorganic insulator materials.18,159,164–169 The effect of ionizing radiation can be particularly strong for electron or light ion beam irradiations of certain ceramic materials since the amount of ionization per unit displacement damage is high for these irradiation species; the ionization effect per dpa is typically less pronounced for heavy ion, neutron, or dual ion beam irradiation. Figure 20 summarizes the effect of variations in the ratio of ionizing to displacive radiation (achieved by varying the ion beam mass) on the dislocation loops density and size in several oxide ceramics.94,169,170 The loop density decreases rapidly when the ratio of ionizing to displacive radiation (depicted in Figure 20 as electron-hole pairs per dpa) exceeds a materialdependent critical value, and the corresponding loop size simultaneously increases rapidly. Numerous microstructural changes emerge in materials irradiated with so-called swift heavy ions that produce localized intense energy deposition in their ion tracks. Defect production along the ion tracks is observed above a material-dependent threshold value for the electronic stopping power with typical values of 1–50 keV nm1.159,171–175 The microstructural changes are manifested in several ways, including dislocation loop punching,176 creation of amorphous tracks with typical diameters of a few nm,159,173,174,177–180 atomic disordering,176,181,182 crystalline phase transformations,171 destruction of preexisting small dislocation
Radiation-Induced Effects on Microstructure
50 650 ⬚C
Fe Al
1023
C
10-6 to 10-4 dpa s-1
AI Fe
MgAl2O4
Mg
1021
Loop diameter (nm)
Loop density (m-3)
Mg
Zr
Al2O3
Fe Mg
H
1020
He
MgO 1019 1018 1017 10
H (d~200 nm)
40
He
1022
650 ⬚C 10-7 to 10-4 dpa s-1
81
Zr
30
He
AI C
C
He
He
Al2O3 20 AI He
10
H
H
MgO
H Fe Mg
H
MgAl2O4
Fe Mg
C
H
0 10
100 1000 104 Electron–hole pairs/dpa ratio
100 1000 104 Electron–hole pairs/dpa ratio
105
Figure 20 Effect of variations in ionizing to displacive radiation on the dislocation loop density and size in ion-irradiated MgO, Al2O3, and MgAl2O4. Adapted from Zinkle, S. J. Radiat. Eff. Defects Solids 1999, 148, 447–477; Zinkle, S. J. J. Nucl. Mater. 1995, 219, 113–127; Zinkle, S. J. In Microstructure Evolution During Irradiation; Robertson, I. M., Was, G. S., Hobbs, L. W., Diaz de la Rubia, T., Eds. Materials Research Society: Pittsburgh, PA, 1997; Vol. 439, pp 667–678.
(a)
20 nm
(b)
50 nm
Figure 21 Plan view microstructure of disordered ion tracks in MgAl2O4 irradiated 430 MeV Kr ions at room temperature to a fluence of 6 1015 ions per square meter (isolated ion track regime) under (a) weak dynamical bright field and (b) g ¼ h222i centered dark field imaging conditions (tilted 10 to facilitate viewing of the longitudinal aspects of the ion tracks). High-resolution TEM and diffraction analyses indicate disordering of octahedral cations (but no amorphization) within the individual ion tracks. Adapted from Zinkle, S. J.; Skuratov, V. A. Nucl. Instrum. Methods B 1998, 141(1–4), 737–746; Zinkle, S. J.; Matzke, H.; Skuratov, V. A. In Microstructural Processes During Irradiation; Zinkle, S. J., Lucas, G. E., Ewing, R. C., Williams, J. S., Eds. Materials Research Society: Warrendale, PA, 1999; Vol. 540, pp 299–304.
loops,176 and formation of nanoscale hillocks and surrounding valleys183,184 at free surfaces. Annealing of point defects occurs for irradiation conditions below the material-dependent threshold electronic stopping power for track creation,159,180,185,186 whereas defect production occurs above the stopping power threshold.159,171,173,175,178,180,183,185,186 The swift heavy ion annealing and defect production phenomena are observed in both metals and alloys171,175,183,185,186 as well as nonmetals.159,172,173,178–180,187–190 Defect production by swift heavy ions is of importance for
understanding the radiation resistance of current and potential fission reactor fuel systems, including the mechanisms responsible for the finely polygonized rim effect188,191 in UO2 and radiation stability of inert matrix fuel forms.182,189,191 The swift heavy ion defect production mechanism is generally attributed to thermal spike178,192 and self-trapped exciton187 effects. Figure 21 shows examples of the plan view (i.e. along the direction of the ion beam) microstructure of disordered ion tracks in MgAl2O4 irradiated with swift heavy ions.176,182
82
Radiation-Induced Effects on Microstructure
1.03.3.9 Role of Solute Segregation and Precipitation Solute atoms of importance include elements originally added to the material during fabrication and species produced by nuclear transmutation reactions (e.g., He and H, and a range of other elements). Solute atoms may exhibit preferential coupling with point defects created during irradiation, leading to either enhancement or depletion of solutes at point defect sink structures such as dislocations, grain boundaries, preexisting precipitates, and voids.193–198 The solute-defect coupling can modify the kinetics for point defect diffusion, and the resultant solute enrichment or depletion may sufficiently modify the local composition to induce the formation of new phases. There are three general categories of precipitation associated with radiation-induced segregation processes103,199: radiation-induced (phases that form due to irradiation-induced nonequilibrium solute segregation and dissolve during postirradiation annealing), radiation-enhanced (precipitate formation accelerated or occurring at lower temperatures due to irradiation, but are thermally stable after formation), and radiation-modified (different chemical composition of precipitates compared to thermodynamically stable composition). In some materials,
radiation-retarded precipitation (phase formation shifted to higher temperatures or longer exposure times) has been reported.200 A phenomenon that is uniquely associated with ion irradiation is the potential for the ions from the irradiating beam to modify the microstructural evolution by perturbing the relative balance of SIAs compared to vacancies flowing to defect sinks. The injected ions act as a source of additional interstitial atoms and can significantly suppress void nucleation and growth.149,154,201,202 The peak concentration of the injected ions occurs near the displacement damage peak for ion irradiation, and therefore considerable care must be exercised when evaluating the void swelling data obtained near the peak damage region in ion-irradiated materials.154,201,202 Figure 22 shows an example of the dramatic changes in microstructure that can occur in the injected ion region.203 In this example, void formation in ion-irradiated nickel at 400 C is completely suppressed in the regions with the injected interstitials and the void microstructure is replaced with an aligned array of small interstitialtype dislocation loops. Numerous studies have observed that the precipitation behavior during irradiation can strongly influence microstructural evolution, for example, the swelling behavior of austenitic stainless steels.103,106,204–206
[111]
[020]
Incident ions
a 0
b
c 1
2 Depth (µm)
a
b 3
a) Voids b) Random loops and voids c) Ordered loops
Figure 22 Cross-section TEM microstructure of nickel irradiated at 400 C with 14 MeV Cu ions to a fluence of 5 1020 ions m2 which produced a peak damage level of about 55 dpa at a depth near 2 mm. Void formation is completely suppressed in the injected interstitial regime (1.3–2.8 mm) and the void microstructure is replaced with an array of small interstitial-type dislocation loops aligned along {100} planes. Reproduced from Whitley, J. B. Depth dependent damage in heavy ion irradiated nickel. University of Wisconsin, Madison, 1978.
Radiation-Induced Effects on Microstructure
83
70 dpa
40 nm
200 nm 0.4 dpa/0.2 appm He/675 ⬚C
109 dpa/2000 appm He/675 ⬚C
Figure 23 Comparison of the cavity microstructures for a pure Fe–13Cr–15Ni austenitic alloy (left panel) and the same alloy with P, Si, Ti, and C additions that produced dense radiation-induced phosphide precipitation (center and right panels) following dual beam Ni þ He irradiation at 675 C. The irradiation conditions were 0.4 dpa and 0.2 appm He for the left panel (70 dpa and 35 appm for the inset figure), and 109 dpa and 2000 appm He for the other two figures. Reproduced from Mansur, L. K.; Lee, E. H. J. Nucl. Mater. 1991, 179–181, 105–110.
In extreme cases, large-scale phase transformations can occur such as the g (austenite, fcc) to a (ferritic, bcc) transformation in austenitic stainless steel following high dose neutron irradiation.106,207 Depending on the type of precipitation, either enhanced or suppressed swelling can occur. Void swelling enhancement has generally been attributed to a point defect collector mechanism and typically occurs for moderate densities of relative coarse precipitates such as G phase in austenitic stainless steels, whereas void swelling suppression is generally observed for high densities of finely dispersed precipitates and is usually attributed to high sink strength effects.103,151,208 Figure 22 shows an example of the strong void swelling suppression associated with formation of radiation-induced Si- and Ti-rich phosphide precipitates compared to a simple Fe–Cr–Ni ternary austenitic alloy.208 Similarly, the He/dpa ratio can influence the types and magnitude of point defect clusters and precipitation due to modifications in the point defect evolution under irradiation (Figure 23).106
1.03.4 Overview of Key RadiationInduced Property Degradation Phenomena There are eight major property changes that may occur in irradiated materials due to a variety of microstructural changes. Listed in order of increasing temperature where the effects are typically dominant, these
phenomena are radiation-induced amorphization, radiation hardening (often accompanied by loss of tensile elongation and reduction in fracture toughness), decrease in thermal and electrical conductivity, mechanical property or corrosion degradation due to radiation-induced segregation and precipitation, dimensional instabilities due to three distinct phenomena (anisotropic irradiation growth, irradiation creep, void swelling), and high temperature embrittlement of grain boundaries due to helium accumulation. The microstructural origins associated with these eight degradation processes are summarized in the following sections, and more detailed descriptions of the property degradations in metals and nonmetals are given in accompanying chapters in this Comprehensive. The radiation doses at which these phenomena emerge to become of practical engineering significance are generally dependent on irradiation temperature, PKA energy, and material. 1.03.4.1
Radiation-Induced Amorphization
At very low temperatures where motion of SIAs or SIA clusters is limited, a crystalline to amorphous phase transition can be induced. The phase transition usually produces large swelling (5–30%) and decreases in elastic moduli.15,91,182,209 This phase transition typically occurs for damage levels of 0.1–1 dpa at low temperatures and has been attributed to several mechanisms including direct amorphization within collision cascades, and an increase in
Radiation-Induced Effects on Microstructure
Dose (dpa)
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CuTi
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2 e
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Inui et al. (1990) Zinkle and Snead305 Weber and Wang (1995)
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1 2 MeV electrons 0.56 MeV Si+ 1.5 MeV Xe+
0.1
600
SiC
0
100 200 300 400 500 Irradiation temperature (K)
Figure 24 Effect of irradiating particle on the temperature-dependent dose for amorphization in irradiated CuTi214 and SiC.139 In both plots, filled symbols denote complete amorphization and open symbols denote amorphization did not occur.
1.03.4.2
Radiation-Induced Hardening
Irradiation of metals and alloys at temperatures below recovery Stage V typically produces pronounced radiation hardening, as discussed in Chapter 1.04, Effect of Radiation on Strength and Ductility of Metals and Alloys. The matrix hardening is typically accompanied by reduction in tensile elongation and in many cases lower fracture toughness.215–222 The uniform elongation measured in tensile tests for
1025 Copper
1024
Defect cluster density (m-3)
the crystalline free energy due to point defect accumulation and disordering processes.86,147,210–213 The dose dependence for accumulation of the amorphous volume fraction is significantly different for the direct impact mechanism compared to point defect accumulation or multiple overlap mechanisms.213 As the irradiation temperature is raised to values where long range SIA and SIA cluster migration occurs, point defect diffusion to reduce the increase in free energy occurs and the dose to induce amorphization typically increases rapidly with increasing temperature until a temperature is reached where it is not possible to induce amorphization. In many cases, the critical temperature for amorphization increases with increasing PKA energy. Figure 24 compares the effect of PKA energy on the temperaturedependent dose for complete amorphization for an intermetallic alloy214 and a ceramic139 material. In both materials, for all types of irradiating particles, the critical dose for amorphization increases rapidly when the irradiation temperature exceeds a critical value. The critical temperature for amorphization is significantly higher for heavy ion irradiation conditions compared to electron irradiation conditions.
1023 316 SS or 304 SS
V-4Cr-4Ti
1022 Zinkle et al.304 Maziasz (1992) PCA Yoshida et al (1992) Zinkle and Sindelar (1993) Horiki and Kiritani (1994) Horiki and Kiritani (1996) Rice and Zinkle224
1021
1020
0
100
200
300
400
500
Temperature (⬚C) Figure 25 Comparison of the temperature-dependent defect cluster densities in neutron-irradiated Cu, austenitic stainless steel, and V–4Cr–4Ti. Based on data reported by Rice and Zinkle224 and Rowcliffe et al.225
metals and alloys irradiated in this temperature regime usually decreases to <1% for damage levels above 0.1–1 dpa, which may require use of more conservative engineering design rules for the allowable stress of structural materials.223 The hardening is largely due to the creation of high densities of sessile defect clusters, which act as obstacles to dislocation motion in the matrix. The defect cluster densities decrease rapidly with increasing temperature above recovery Stage V. Figure 25 compares the temperature-dependent defect cluster densities224,225
Radiation-Induced Effects on Microstructure
observed in neutron-irradiated Cu, austenitic stainless steel, and V–4Cr–4Ti. Stage Vannealing of defect clusters is evident for temperatures above 150, 200, and 275 C for Cu, stainless steel, and V–4Cr–4Ti, respectively. The mechanical properties in irradiated nonmetals at temperatures below recovery Stage V exhibit variable behavior, with observations of increased hardness,226,227 unchanged strength,228 and decreased hardness or flexural strength.229–232 1.03.4.3 Thermal and Electrical Conductivity Degradation Thermal and electrical conductivity degradation can occur over a wide range of irradiation temperatures. For pure metals, there are two primary contributions: electron scattering from point defects (and associated defect clusters) and nuclear transmutation solute atoms. The conductivity degradation associated with radiation defects usually amounts to less than 1% change except in the case of high void swelling conditions.233–236 Conversely, the conductivity degradation associated with neutron-induced transmutation products tends to monotonically increase with increasing dose and typically becomes larger than the radiation defect contribution for doses above 1 dpa. Thermal conductivity degradation much greater than 10% can occur in high-conductivity metals and ceramics.235,237 The conductivity degradation in irradiated alloys can be complex due to shortrange ordering and precipitation phenomena,238 with the possibility for either increased or decreased conductivity compared to the unirradiated condition. For nonmetallic irradiated materials, the electrical conductivity during irradiation typically experiences a transient increase due to excitation of valence electrons into the valence band by ionizing radiation.239–243 The thermal conductivity of irradiated nonmetals is typically degraded by displacement damage due to phonon scattering by point defects and defect clusters.237,243–246
85
phenomena in irradiated ferritic and austenitic steels at elevated temperatures for doses above about 10 dpa,11,20,104,106,200,204 and in irradiated reactor pressure vessel steels at low dose rates for damage levels above 0.001–0.01 dpa.247,248 Some general aspects of radiation-induced and -enhanced solute segregation and precipitation were described previously in Section 1.03.3.9. The solute segregation and precipitation associated with irradiation can lead to several deleterious effects including property degradation due to grain boundary or matrix embrittlement224,247,249–252 and enhanced susceptibility for localized corrosion or stress corrosion cracking.253–256 Solute segregation and precipitation can lead to either enhanced or suppressed void swelling behavior.149,257,258 For austenitic stainless steel, undesirable precipitate phases that generally are associated with high void swelling include the radiation-induced phases M6Ni16Si7 (G), Ni3Si (g0 ), MP, M2P, and M3P, and the radiation-modified phases M6C, Laves, and M2P.200 The undesirable radiation-induced and -modified phases generally are associated with undersized misfits with the lattice, which tends to preferentially attract SIAs and thereby enhance the interstitial bias effect. Figure 26 shows an example of enlarged cavity formation in association with G phase precipitates in neutronirradiated austenitic stainless steel.106 Potentially desirable radiation-enhanced and -modified phases (when present in the form of finely dispersed precipitates) include M6C, Laves, M23C6, MC, s, and w.200
G-phase
1.03.4.4 Radiation-Induced Segregation and Precipitation At intermediate temperatures where SIAs and vacancies are mobile, significant solute segregation to point defect sinks can occur. This can lead to precipitation of new phases due to the local enrichment or depletion of solute. These radiation-induced or -enhanced precipitation reactions typically become predominant
50 nm Figure 26 Enlarged cavity formation in association with G phase (Mn6Ni16Si7) precipitates in Ti-modified ‘prime candidate alloy’ austenitic stainless steel following mixed-spectrum fission reactor irradiation at 500 C to 11 dpa that generated 200 appm He. Reproduced from Maziasz, P. J. J. Nucl. Mater. 1989, 169, 95–115.
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1.03.4.5 Dimensional Instabilities: Irradiation Growth, Creep, and Swelling Irradiation growth (due to anisotropic nucleation and growth of dislocation loops on different habit planes) can be of significant practical concern at intermediate temperatures in anisotropic materials such as Zr alloys, Be, BeO, Al2O3, uranium, and graphite.40,126,259–261 Anisotropic growth in individual grains in polycrystalline materials can produce large grain boundary stresses, leading to loss of strength and grain boundary fracture in some materials. Figure 27 shows the large anisotropy in measured lattice parameter change in the basal and prism planes for BeO irradiated near room temperature.262 For neutron fluences above 2 1020 cm2 (0.2 dpa) with a c-axis expansion >0.5% and an a-axis expansion near 0.1%, a rapid decrease in flexural strength was observed.262,263 In materials with highly textured grains, unacceptable anisotropic growth at the macroscopic level can occur. One engineering solution is to use processing techniques to produce randomly aligned, small grain-sized materials. Irradiation creep occurs in the presence of applied stress, due to biased absorption of point defects at cavities and along specific dislocation orientations relative to the applied stress.264 Irradiation creep produces dimensional expansion that acts in addition to normal thermal creep mechanisms and is most
prominent at temperatures from recovery Stage III up to temperatures where thermal creep deformation becomes rapid (typically above 0.5TM). The magnitude of steady-state irradiation creep is proportional to the applied stress level and dose, and consists of a creep compliance term and a void swelling term. The magnitude of typical irradiation creep compliance coefficients260,265,266 for fcc and bcc metals is 0.5–1 1012 Pa1 dpa1. The irradiation creep compliance for ferritic/martensitic steels appears to be about one-half of that for austenitic steels.109 Accelerated irradiation creep due to differential absorption of point defects at low temperatures (e.g. below recovery Stage V) or at low doses can produce creep deformation rates that are up to 10–100 times larger than the steady-state irradiation creep rates.267,268 Volumetric swelling from void formation occurs at temperatures above recovery Stage V in fcc and HCP materials (and above Stage III for bcc materials), and typically exhibits a linear increase with dose after an initial transient regime. As summarized in Figure 28 the dose-dependent swelling in fast fission reactor-irradiated austenitic stainless steel progresses
510 ⬚C
80 1%/dpa 538 ⬚C
3.5 Hickman 1966
482 ⬚C
60
2.5 c-parameter a-parameter
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40 427 ⬚C
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20
1
454 ⬚C
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0.2%/dpa
0
0 0
12 2 4 6 8 10 Fast neutron fluence, 1020 n cm-2 (E > 1 MeV)
Figure 27 Effect of fission neutron irradiation near 75 C on the measured lattice parameter changes for BeO. Adapted from Hickman, B. S. In Studies in Radiation Effects, Series A: Physical and Chemical; Dienes, G. J., Ed. Gordon and Breach: New York, 1966; Vol. 1, pp 72–158.
0
10 20 30 ´ 1022 Neutron fluence, n cm-2 (E > 0.1 MeV)
Figure 28 Summary of dose-dependent swelling behavior in 20% cold-worked Type 316 austenitic stainless steel due to fast fission reactor irradiation. Reproduced from Garner, F. A.; Toloczko, M. B.; Sencer, B. H. J. Nucl. Mater. 2000, 276, 123–142.
Radiation-Induced Effects on Microstructure
at a swelling rate of 1%/dpa without evidence for saturation up to swelling levels approaching 100%.109 Similar high swelling levels without evidence of saturation have been observed in pure copper108 and some simple bcc alloys.131 Volumetric swelling levels in structural materials in excess of 5% are difficult to accommodate by engineering design,269 and additional embrittlement mechanisms may appear in austenitic stainless steel for swelling levels above 10% including void channeling and loss of ductility.270,271 Therefore, there is strong motivation to design structural materials that are resistant to void swelling by introducing a high matrix density of point defect sinks or other techniques. In general, the amount of void swelling is lower in bcc materials compared to fcc materials.50,92,109 For example, the observed void swelling in many ferritic/martensitic steels is <2% after fission neutron damage levels of 50 dpa or higher, whereas the void swelling in simple austenitic stainless steels may be 30% or higher.109 The superior swelling resistance in ferritic/martensitic steels is largely due to a higher transient dose before onset of steady-state swelling, along with a lower steady-state swelling rate. For many HCP materials, the amount of void swelling is relatively small compared to fcc materials due to anisotropic point defect migration that tends to promote defect recombination.128 However, the potential for anisotropic swelling associated with cavity formation in HCP materials may induce large stresses and potential cracking at grain boundaries.263,272,273 Figure 29 shows an example of aligned cavity formation and grain boundary separation in Al2O3 following fast fission reactor irradiation.272 1.03.4.6
High Temperature Embrittlement
High temperature helium embrittlement occurs at elevated temperatures (typically near or above 0.5TM) when sufficient levels of helium are produced by nuclear transmutation reactions and mechanical stress is applied during irradiation. Intergranular fracture is induced by the transformation of grain boundary bubbles to voids, leading to breakaway growth, cavity coalescence, and rupture in the presence of mechanical stress.120,152,153,274–277 The application of tensile stress during high temperature irradiation induces migration of the helium to the grain boundaries, where large cavities can be formed.120 In the absence of applied stress, the helium bubbles are distributed throughout the material. The observed tensile ductility due to helium embrittlement decreases with decreasing strain rate120,278 and decreasing
87
100 nm
Figure 29 Aligned cavity formation and grain boundary separation in Al2O3 following fast fission reactor irradiation to 12 dpa at 1100 K. Reproduced from Clinard, F. W., Jr.; Hurley, G. F.; Hobbs, L. W. J. Nucl. Mater. 1982, 108–109, 655–670.
stress120 (opposite of the behavior observed in many unirradiated metals and alloys), pointing out the importance of exposure time at elevated temperature on helium embrittlement. Figure 30 shows examples of the grain boundary microstructures of an Fe–Cr–Ni ternary alloy preimplanted with 160 appm He during annealing at 750 C with and without applied tensile stress.279 Cavity formation along the grain boundary is very limited in the absence of applied stress for annealing times up to 60 h, whereas pronounced grain boundary cavity swelling occurs for annealing times as short as 8 h when 20 MPa stress is applied. Evidence for high temperature helium embrittlement has been observed during tensile and creep testing of austenitic stainless steel at temperatures above 550 C (0.45–0.5TM) when the helium concentration exceeds 30 appm.255,265,277,280,281 Austenitic stainless steels containing fine dispersions of precipitates exhibit better resistance to helium embrittlement than simple Fe–Cr–Ni alloys, and microstructural investigations suggest that helium trapping at grain interior locations (thereby impeding the flow of helium to grain boundaries) is an important factor.152,277,282–284 It has been observed that ferritic/martensitic steels exhibit better resistance to grain boundary helium cavity formation and growth compared to austenitic stainless steels.274,285–287 This has been attributed to several potential factors, including efficient trapping
88
Radiation-Induced Effects on Microstructure
YE-11612
YE-11611
YE-11567
Triple grain junction
G.B.
0 MPa
(a)
Matrix
(b)
(c) 0.1 µm
8h YE-11561
18 h
60 h YE-11559
YE-11560
Matrix
19.6 MPa
G.B.
Triple grain junction
(a)
(b)
(c)
Figure 30 Effect of exposure time and applied stress during annealing at 750 C on the formation of grain boundary cavities in Fe–17Cr–17Ni austenitic alloy preimplanted with 160 appm helium. Reproduced from Braski, D. N.; Schroeder, H.; Ullmaier, H. J. Nucl. Mater. 1979, 83, 265–277.
of helium in the ferritic steel grain interior by precipitates and other features, a potentially larger critical radius for conversion of helium bubbles to voids in ferritic steel, and lower matrix strength for ferritic steel compared to austenitic steel.119,274,286,288 The helium bubble densities observed in model Fe–Cr ferritic alloys and commercial ferritic steels following high temperature implantation are comparable to that observed in austenitic steels.118 Relatively good resistance to helium embrittlement compared to austenitic stainless steel has been observed in other bcc metals such as Nb and Nb–1Zr (no severe embrittlement observed for He concentrations up to 100–500 appm),289–291 whereas simple fcc metals such as pure copper are readily susceptible to helium embrittlement even at relatively high (tensile) strain rates at temperatures near 0.5TM for He concentrations of 100–330 appm.292,293
1.03.5 Examples of RadiationInduced Microstructural Changes 1.03.5.1
Dislocation Loop Formation
A common feature in many irradiated metals and nonmetals at temperatures between recovery Stage III and Stage V is dislocation loop formation (either perfect or faulted), with typical loop diameters ranging from 2 to 100 nm. Both vacancy (intrinsic) and interstitial (extrinsic) loops are frequently observed in irradiated materials. The dislocation loop shape is frequently circular (in order to minimize dislocation line length), but rhombus, square, hexagonal, or other shapes have been observed in some materials due to elastic energy considerations.21Figure 31 shows an example of circular faulted interstitial-type dislocation loop formation in MgAl2O4 due to ion irradiation at 650 C. The parallel fringes visible in the loop
Radiation-Induced Effects on Microstructure
89
202
50 nm Figure 31 Faulted interstitial-type dislocation loop formation in MgAl2O4 irradiated with 2 MeV Al þ ions at 650 C to 14 dpa. The image was taken with a beam direction near [101] using weak beam dark field (g, 6g), g ¼ 202 diffraction imaging conditions (data from S. J. Zinkle, unpublished research).
interiors are a signature of the stacking fault and are visible in TEM by selecting the appropriate diffraction imaging conditions. Faulted loop formation is energetically unfavorable in most bcc materials due to their high stacking fault energies, although there is some evidence for formation of small faulted loops in some cases.224 Experimental studies using energetic ion beams at cryogenic temperatures (where long range point defect migration does not occur) have obtained convincing evidence for direct formation of visible defect clusters directly within displacement cascades above a threshold energy value.294 Dislocation loop formation is usually randomly distributed on the relevant habit planes, with no pronounced spatial correlation. In some cases where mechanical or radiation-induced stresses are present, significant anisotropy occurs regarding the habit planes for loop formation.295,296 Within a limited temperature and damage rate regime, the dislocation loop microstructure in some materials also exhibits a tendency to self-organize into aligned walls.297–299 Figure 32 shows an example of well-developed defect cluster patterning in pure copper following proton irradiation to 2 dpa.298 The defect clusters within the walls consist of SFTs and small dislocation loops. 1.03.5.2
Network Dislocation Formation
Network dislocation structures are routinely observed in metals5,8,200 and ceramics300,301 irradiated at
500 nm Figure 32 Defect cluster patterning into aligned {001} walls in single crystal copper irradiated with protons at 100 C to 2 dpa. Reproduced from Ja¨ger, W.; Trinkaus, H. J. Nucl. Mater. 1993, 205, 394–410.
temperatures above recovery Stage I to temperatures in excess of recovery Stage V. During prolonged irradiation, the microstructural evolution typically involves formation and growth of faulted dislocation loops, loop unfaulting to create perfect dislocation loops, and then loop interaction/impingement to form network dislocation structures. The network dislocations are typically randomly distributed and are often heavily jogged as opposed to the relatively straight dislocations found in unirradiated metals. Figure 33 shows a typical network dislocation microstructure for irradiated copper.302 The quantitative value of the dislocation density can vary significantly among different materials within the same crystal structure. For example, typical network dislocation densities in irradiated metals at temperatures between recovery Stages III and V range from 0.01 to –0.1 1014 m2 for Cu302–304 to 1–10 1014 m2 for pure Ni304 and austenitic stainless steel.20 1.03.5.3
Stacking Fault Tetrahedra
Irradiation of fcc metals under energetic displacement cascade conditions induces the formation of stacking fault tetrahedra. Figure 34 shows an example of the formation of small dislocation loops and SFTs (triangle-shaped projected images) in copper due to irradiation with 750 MeV protons (2.5 MeV average PKA energy) at 90 C to 0.7 dpa.302
90
Radiation-Induced Effects on Microstructure
gold,307,310,311 palladium,310,312 and austenitic stainless steels.53,96,307,312,313 Evidence from thin film and low-dose irradiation studies using ion beams or other energetic displacement cascade conditions suggests that SFTs can be formed directly in displacement cascades when the PKA energy exceeds a threshold value of 5–10 keV, in agreement with molecular dynamics simulations.26,29 There are also several observations of SFT formation in some fcc metals due to point defect nucleation and growth during electron irradiation.299,305 The results from irradiations performed under energetic displacement cascade conditions at temperatures near recovery Stage I suggest that SFTs are not visible, perhaps due to insufficient rearrangement of the vacancy-rich core within the rapidly quenched displacement cascade.74,305
0.5 µm Figure 33 Dislocation microstructure of pure copper following irradiation with 750 MeV protons at 200 C to 2 dpa. Reproduced from Zinkle, S. J.; Horsewell, A.; Singh, B. N.; Sommer, W. F. J. Nucl. Mater. 1994, 212–215, 132–138.
20 nm Figure 34 Weak beam dark field (g, 4g), g ¼ 002 microstructure of pure copper following irradiation with 750 MeV protons at 90 C to 0.7 dpa. The TEM figure is based on irradiated specimens described in Zinkle et al.302 The SFTs are visible as small triangle-shaped defects since the electron beam direction was near [110] (data from S. J. Zinkle, unpublished research).
The SFTs are thermally stable up to recovery Stage V. SFTs have been observed in numerous irradiated fcc metals, including aluminum,305 copper,12,53,146,302,306,307 nickel,304,307–309 silver,306,307
1.03.5.4 Dislocation Channeling and Flow Localization Mechanical deformation of metals and alloys after irradiation at temperatures below recovery Stage V produces deformation microstructures that typically evolve from predominantly dislocation cell microstructures in the unirradiated and low-dose irradiated conditions to a variety of localized deformation microstructures above a threshold damage level including twinning, planar dislocation deformation, and formation of dislocation channels.314–316 Formation of cleared dislocation channels has been suggested to be the cause of low uniform elongations observed in tensile tests of metals and alloys irradiated at temperatures below recovery Stage V,221,317 and dislocation channeling is frequently observed following deformation of irradiated materials that exhibit low uniform elongation.95,96,100,312,316,318–321 An alternative mechanism for the low uniform elongations in irradiated materials, based on a material-specific threshold stress for plastic instability, has also been proposed.216,322–324 The spacing between dislocation channels is typically on the order of 1 mm, and the width of the individual channels ranges from 20 to 200 nm. Localized deformation visible as surface slip steps in irradiated copper following tensile straining has been directly correlated with cleared dislocation channels.325 The matrix regions between the cleared channels do not exhibit evidence of substantial dislocation activity, suggesting that all of the dislocation motion associated with deformation is restricted to the dislocation channel regions. Figure 35 shows an example of cleared dislocation channels observed in austenitic stainless steel following fission neutron
91
10 nm 1 µm Figure 35 Cleared dislocation channels observed in Type 316 austenitic stainless steel following fission neutron irradiation to 0.78 dpa near 80 C and subsequent uniaxial tensile deformation to 32% strain. The electron beam direction was near [110]. Reproduced from Byun, T. S.; Hashimoto, N.; Farrell, K.; Lee, E. H. J. Nucl. Mater. 2006, 349, 251–264.
irradiation to 0.78 dpa near 80 C and subsequent uniaxial tensile deformation to 32% strain.326 The mechanisms responsible for annihilation of SFTs by gliding dislocations within the dislocation channel include stress-induced collapse to triangle loops, multiple shear, partial annihilation with a remnant apex, collapse to a triangle loop or complete annihilation with multiple super jogs, and complete annihilation by screw dislocations followed by cross slip.327–329 Computer simulations of dislocation loop interactions with gliding dislocations suggest multiple potential mechanisms that could lead to defectcleared dislocation channels, including absorption, unfaulting, and shear of the loops.330–333 Detailed experimental confirmation of these annihilation mechanisms is still needed. 1.03.5.5 Crystalline to Amorphous Phase Transitions Radiation-induced amorphization can proceed by several different mechanisms, including direct impact amorphization and gradual accumulation of lattice defects and chemical disorder that eventually causes destabilization of the crystalline matrix.213 Figure 36 shows an example of the microstructure near the crystalline to amorphous transition dose in ion-irradiated SiC, where the amorphization is induced by gradual buildup of radiation defects.209
00
01
Radiation-Induced Effects on Microstructure
Figure 36 High-resolution transmission electron microscopy image of single crystal 6H–SiC following 0.56 MeV Si ion irradiation at 60 C to a damage level of 2.6 dpa. Reproduced from Snead, L. L.; Zinkle, S. J.; Hay, J. C.; Osborne, M. C. Nucl. Instrum. Methods B 1998, 141, 123–132.
At intermediate doses, amorphous islands gradually emerge from the initially crystalline matrix in SiC irradiated at low temperatures. Direct amorphization within individual displacement cascades has been observed in several intermetallic,334 semiconductor,12,335 and ceramic insulator15,336,337 materials. In many other materials, extensive chemical disordering from displacement cascades or point defects precedes amorphization.3,76,87 The chemical disordering can be monitored either on the nanoscale dimensions (e.g., due to individual displacement cascades) by techniques such as transmission electron microscopy,12,338 or an integrated average value by various techniques including X-ray diffraction, TEM, and Rutherford backscattering spectrometry.76,339 As previously noted in Section 1.03.3.8, the intense ionization associated with swift heavy irradiation can lead to amorphization either directly within ion tracks, or by a cumulative process involving chemical disordering before amorphization due to multiple overlapping ion tracks. 1.03.5.6
Radiation-Induced Precipitation
As previously outlined in Sections 1.03.3.9 and 1.03.4, phase changes associated with irradiation can be manifested in a variety of geometries, including randomly distributed matrix or grain boundary precipitates, continuous grain boundary films, precipitatefree zones near grain boundaries or other point defect sinks, spatially ordered arrays of precipitates, large-scale (>100 nm) phase transformations, and
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Radiation-Induced Effects on Microstructure
350 ⬚C
250 nm Figure 37 Radiation-induced precipitates on {001} habit planes observed next to a grain boundary in V–4Cr–4Ti following neutron irradiation to 0.1 dpa at 505 C. The fringe contrast in the precipitate interior is due to the a/3h001i displacement vector of the precipitates relative to the vanadium alloy matrix. The beam direction was {111} and the diffraction vector was g ¼ 011. Reproduced from Rice, P. M.; Zinkle, S. J. J. Nucl. Mater. 1998, 258–263, 1414–1419.
dissolution or growth of thermally stable precipitates. Preferential coupling of solute atoms to point defect fluxes can lead to modifications in the chemistry of precipitates as well as nucleation of phases that would not be stable under thermal equilibrium conditions. Figure 37 shows an example of radiation-induced platelet precipitates observed in the grain interiors of V–4Cr–4Ti following neutron irradiation to 0.1 dpa at 505 C.224 A precipitate-free zone is observed adjacent to the grain boundary in this figure. 1.03.5.7
Cavity Formation
Several different cavity geometries are created in irradiated materials. For helium-filled bubbles, the cavity shape is typically spherical. For voids, faceted cavities with faces corresponding to low-index crystallographic planes are often created, e.g. truncated {111} octahedra or {001} cubes in fcc materials, truncated {110} dodecahedra or {001} cubes in bcc materials, and more complex shapes in HCP materials,21,22,340–342 although nearly spherical shapes are also sometimes observed for voids. When helium is generated during irradiation (due to neutroninduced transmutation reactions, etc.), a bimodal cavity distribution is usually observed with the small cavities corresponding to helium-filled bubbles and the large cavities corresponding to underpressurized voids. The critical radius transition between bubbles and voids is determined by a balance between dislocation bias-induced vacancy influx and pressure-modified thermal emission of vacancies.120,151,208,274,343 Figure 38
100 nm Figure 38 Voids and small helium-filled bubbles in a copper–boron alloy following fission neutron irradiation to 1.2 dpa at 350 C. Reproduced from Zinkle, S. J.; Farrell, K.; Kanazawa, H. J. Nucl. Mater. 1991, 179–181, 994–997.
shows an example of large faceted voids and small helium-filled spherical bubbles in a neutronirradiated copper–boron alloy.107 The visible cavity density usually increases rapidly at low doses, and approaches a constant value for damage levels above 1–50 dpa. The void size tends to increase continuously with increasing dose. Well-developed periodic void lattices have been observed in several irradiated materials.297,344,345 Void lattice formation has most frequently been observed in bcc materials, but periodically aligned void structures have also been observed in HCP272,346,347 and fcc303,348–351 materials. Aligned voids have been observed in both metals and ceramic insulators. The aligned cavities in HCP materials are usually manifested as one- or two-dimensional arrays perpendicular or parallel to the basal plane, respectively.297,346 The void lattices in bcc and fcc materials adopts the same three-dimensional crystallographic symmetry as the host lattice.297 The swelling levels in bcc metals with well-developed void lattices are typically a few percent, which has led to hypotheses that void lattice formation may coincide with a cessation in steadystate swelling.117,352 The saturation in void swelling is associated with achieving a constant average void size. Figure 39 shows an example of a well-developed bcc void lattice in ion-irradiated Nb–1Zr.353 In the study by Loomis et al. it was reported that void lattice formation did not occur unless a threshold level of oxygen was present (60–2700 appm oxygen, depending on the irradiated material).
Radiation-Induced Effects on Microstructure
110 100 nm
110
Figure 39 Void lattice formation in Nb–1Zr containing 2700 appm oxygen following irradiation with 3.1 MeV V þ ions to 50 dpa at 780 C. Reproduced from Loomis, B. A.; Gerber, S. B.; Taylor, A. J. Nucl. Mater. 1977, 68, 19–31.
1.03.6 Summary Radiation-induced microstructural modifications can create large changes in the physical and mechanical properties of materials, as detailed in accompanying chapters in this book. The two most important extrinsic variables that influence microstructural evolution under irradiation are the radiation damage level and temperature. Many similarities are observed for diverse materials and irradiation spectra if the comparisons are performed at comparable damage levels and defect mobility regimes (defect recovery stages). The PKA energy often exerts a significant influence on the microstructural evolution, in particular by inducing direct cascade amorphization or creation of defect clusters within displacement cascades when the PKA energy exceeds a threshold energy value. Numerous other parameters such as dose rate, crystal structure, and atomic weight typically exert less pronounced influence on microstructural evolution, although very large qualitative and quantitative effects can be observed under some circumstances.
Acknowledgments Much of the author’s work discussed in this chapter was sponsored by the U.S. Department of Energy, Office of Fusion Energy Sciences.
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1.04 Effect of Radiation on Strength and Ductility of Metals and Alloys M. L. Grossbeck University of Tennessee, Knoxville, TN, USA
ß 2012 Elsevier Ltd. All rights reserved.
1.04.1 1.04.2 1.04.3 1.04.4 1.04.5 1.04.6 1.04.7 1.04.7.1 1.04.7.2 1.04.7.3 1.04.8 1.04.8.1 1.04.8.2 1.04.9 1.04.10 References
Introduction Mechanisms of Irradiation Hardening Tensile Behavior Effects of Neutron Spectrum Tensile Ductility Effect of Test Temperature Ferritic–Martensitic Alloys Introduction Tensile Behavior Helium Effects Refractory Metals Tensile Behavior Helium Effects Amorphous Metals Conclusions
Abbreviations A1 ANN appm ASTM ATR bcc BR2 CW DBTT dpa EBR-II fcc FFTF HFBR HFIR HFR JPCA
Lowest equilibrium temperature at which the austenite phase exists in steel Annealed Atomic parts per million ASTM International Advanced Test Reactor, Idaho Falls, ID, USA Body-centered cubic Belgian Reactor-2, Mol, Belgium Cold worked Ductile-brittle transition temperature Displacements per atom Experimental Breeder Reactor-II, Idaho Falls, ID, USA Face-centered cubic Fast Flux Test Facility, Richland, WA, USA High Flux Beam Reactor, Brookhaven, Upton, NY, USA High Flux Isotope Reactor, Oak Ridge, TN, USA High Flux Reactor, Petten, The Netherlands Japanese Prime Candidate Alloy
99 100 101 102 103 107 108 108 108 113 116 116 118 119 120 121
LMFBR Liquid Metal Fast Breeder Reactor LWR Light Water Reactor ORR Oak Ridge Research Reactor, Oak Ridge, TN, USA PCA Prime candidate alloy, adopted by the US Fusion Program in mid-1970s ppm Parts per million Unirr Unirradiated
1.04.1 Introduction The most commonly considered mechanical properties of metals and alloys include strength, ductility, fatigue, fatigue crack growth, thermal and irradiation creep, and fracture toughness. All these properties are important in the design of a structure that is to experience an irradiation environment. While determining the mechanical properties of irradiated materials, tensile properties, typically yield strength, ultimate tensile strength, uniform elongation, total elongation, and reduction of area are the most commonly considered because they are usually the simplest and the least costly to measure. In addition, the tensile properties can be used as an indicator of the other mechanical 99
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Effect of Radiation on Strength and Ductility of Metals and Alloys
properties. Space in a reactor or in an accelerator target is often so limited that the larger specimens required for fatigue and fracture toughness testing are not practical; consequently, the number of specimens that can be irradiated is so small that a meaningful test matrix is not possible. Shear punch testing of 3-mm diameter disks, typically used as transmission microscopy specimens, was developed to address the problem of irradiation space. Although much information can be obtained from shear punch testing, the tensile test remains the most reliable indicator of strength and ductility. For these reasons, the tensile test is usually the first mechanical test used in determining the irradiated properties of new materials. This chapter addresses the tensile strength and ductility of alloys.
1.04.2 Mechanisms of Irradiation Hardening Irradiation introduces obstacles to dislocation motion, which results in plastic deformation, in the form of defects resulting from atomic displacement and from transmutation products. Small Frank loops and defect clusters, known as black dots, large Frank loops (about an order of magnitude larger), precipitates, and cavities (either voids or bubbles) contribute to hardening in an irradiated alloy. Frank loops unfault and eventually contribute to the network dislocation density. Precipitates are certainly present in the unirradiated alloy, but additional precipitation results from the segregation of elements during irradiation and from the irradiation-induced changes that shift the thermodynamic stability of phases. Transmutation production of new elements in the alloy can also result in the formation of new precipitates. The production of insoluble species, most importantly helium, also results in precipitation, especially in the form of bubbles. Defects are divided into two classes: long range and short range. Short-range obstacles are defined as those that influence moving dislocations only on the same slip plane as opposed to long-range obstacles, which impede dislocation motion on slip planes not containing the obstacle.1 Coherent precipitates and large loops are long-range obstacles, but for this analysis, only network dislocations will be considered as long-range obstacles, a reasonable simplification from observations. As recommended by Bement,2 the contributions from short-range obstacles are added directly, DFTS ¼ DFLR þ DFSR
½1
where the quantities in eqn [1] are total stress, longrange contribution to stress, and short-range contribution to stress. The contributions from the short-range obstacles are added in quadrature as follows3: ðDFSR Þ2 ¼ ðDFSMloop Þ2 þ ðDFLGLoop Þ2 þ ðDFPRECIP Þ2 þ ðDFCAVITY Þ2
½2
where the term on the left represents the contribution from all short-range obstacles, and the terms on the right represent the stress contributions from small loops, large loops, precipitates, and cavities, either voids or bubbles. The contribution to hardening by network dislocations may be expressed by pffiffiffiffiffi ½3 tnet ¼ aGb rd where tnet is the increment in shear stress, G is the shear modulus, b is the Burgers vector, and rd is the dislocation density. The constant a is dependent upon the geometry of the dislocation configuration and is usually determined experimentally. However, Taylor has calculated a to be between 0.15 and 0.3,4 and Seeger has determined the value to be 0.2, incorporating the assumption of a random distribution of dislocation directions.5 Short-range defects such as small and large Frank loops and precipitates are treated as hard impenetrable obstacles where dislocations bow around them by the Orowan mechanism. The stress increment is expressed by pffiffiffiffiffiffiffiffiffiffiffi Dt ¼ Gb Nd =b ½4 where N is the defect density and d is the diameter. The constant b ranges between 2 and 4 as suggested by Bement2 or 6 as suggested by Olander.6 Voids and bubbles are also treated as hard obstacles using the same expression. Precipitates and bubbles have been observed in austenitic stainless steels to nucleate and grow together.7 In this case, the bubbles and precipitates are considered as one obstacle where the hardening increment is calculated assuming rod geometry using a treatment by Kelly expressed by8: pffiffiffiffiffiffi pffiffiffi 6d 0:16Gb Nd pffiffipffiffiffiffiffiffi ln Bubble-precip ¼ ½5 6 3b 1 Nd 3
where the parameters are the same as for eqn [4]. From the previous discussion, it can be inferred that because the nature of the irradiation-induced defects determines the degree of hardening, and because the nature, size, and density of defects is a strong function of temperature, radiation strengthening will be a strong function of irradiation temperature. Figure 1 illustrates
Effect of Radiation on Strength and Ductility of Metals and Alloys
101
Relative contribution to strength
1 Black dots
0.8
Frank loops
0.6
Bubbles-precipitates Network dislocation
0.4 0.2 0 0
100
200 Temperature (⬚C)
300
400
Figure 1 Relative contribution to strengthening from irradiation-induced defects in the austenitic stainless steel, PCA, irradiated to 7 dpa in the Oak Ridge Research Reactor. Reproduced from Grossbeck, M. L.; Maziasz, P. J.; Rowcliffe, A. F. J. Nucl. Mater. 1992, 191–194, 808.
160
Irrad. temp. » Test Temp. Strain rate ~ 4 ´ 10-5 S-1
1000
140
900
371 ⬚C
Yield strength (MPa)
800
Test Symbol Temp. (⬚C)
427 ⬚C
700
371 427 483 538 593 649 704 760 816
483 ⬚C
600 500 538 ⬚C 593 ⬚C
400 300 760 ⬚C
100
816 ⬚C
0
100 80 60
649 ⬚C
200
0
1
2
120
Yield strength (ksi)
1100
40
704 ⬚C 20
3
4
5
6
7
8
9
10
12
0
Neutron fluence (n cm−2 ) (E > 0.1 MeV)
Figure 2 Yield strength of 20% cold-worked type 316 stainless steel irradiated in the EBR-II. Reproduced from Fish, R. L.; Cannon, N. S.; Wire, G. L. In Effects of Radiation on Structural Materials; Sprague, J. A., Dramer, K., Eds.; ASTM: Philadelphia, PA, 1979; ASTM STP 683, p 450. Reprinted, with permission, from Effects of Radiation on Structural Materials, copyright ASTM International, West Conshohocken, PA.
strengthening from individual types of defects as a function of irradiation temperature for the austenitic stainless steel PCA.7 As can be seen from Figure 1, the black dot damage characteristic of low temperatures vanishes at temperatures over 300 C as Frank loops emerge. Bubbles and precipitates also become major contributors to hardening above 200 C.
1.04.3 Tensile Behavior Tensile behavior is determined by the irradiationinduced defect structure previously discussed. Austenitic stainless steels will again be used for the example since they are typical of fcc alloys and in
many respects to other alloys (see Chapter 2.09, Properties of Austenitic Steels for Nuclear Reactor Applications and Chapter 4.02, Radiation Damage in Austenitic Steels). The behavior of other example classes of alloys will be discussed in later sections of this chapter. The tensile behavior characteristic of austenitic stainless steels is shown in Figure 2, where yield strength is plotted as a function of fluence and displacement level.9 Saturation in strength is clear with the saturation time becoming shorter as irradiation temperature is increased. At temperatures above about 500 C, saturation is evident, but in this case, strength decreases. This decrease is a result of recovery of the coldworked microstructure of the 20% cold-worked type 316 stainless steel presented in Figure 2. Figure 3
102
Effect of Radiation on Strength and Ductility of Metals and Alloys
850
600
Ultimate tensile strength (UTS)
650 ⬚C
20% cold-worked
400
800 750
200 Strength (MPa)
Annealed 0 538 ⬚C
Yield strength (MPa)
600
20% cold-worked 400
Yield strength (YS)
650 600 550
200
450
427 ⬚C
600
20% cold-worked Annealed
200 0
1
2
3
4
5
6
7
8
0
10
20 30 Dose (dpa)
40
50
Figure 4 Strength properties of 20% cold-worked type 316 stainless steel irradiated in EBR-II. Reproduced from Allen, T. R.; Tsai, H.; Cole, J. I.; Ohta, J.; Dohi, K.; Kusanagi, H. Effects of Radiation on Materials; ASTM: Philadelphia, PA, 2004; ASTM STP 1447, p 3. Reprinted, with permission, from Effects of Radiation on Structural Materials, copyright ASTM International, West Conshohocken, PA.
800
400
20% CW 316 stainless steel irradiation temp 375–385 ⬚C Test temperature 370 ⬚C
500
Annealed
0
0
700
9
10
Neutron fluence (1022 n cm−2 ) Figure 3 Yield strength of type 316 stainless steel irradiated in the EBR-II. Reproduced from Garner, F. A.; Hamilton, M. L.; Panayotou, N. F.; Johnson, G. D. J. Nucl. Mater. 1981, 103 & 104, 803.
of irradiation temperature on strength.14 Indeed, uncertainties in irradiation temperature are an inherent difficulty in neutron irradiation experiments.
shows yield strength resulting from the recovery of a cold-worked dislocation structure and the generation of a radiation-induced microstructure, resulting in a saturation strength independent of the initial condition of the alloy.10 Again, it is seen that the approach to saturation is faster with increasing temperature, with saturation achieved between 5 and 10 dpa at 538 and 650 C, but 15–20 dpa is necessary to achieve saturation at 427 C. Saturation is observed in yield strength curves for fluences as high as 9 1022 n cm2 in a fast reactor (45 dpa), but more recent data show a hint of softening above 50 dpa,11,12 and other fast reactor data have shown a reduction in strength even for displacements below 50 dpa, as shown in Figure 4.13 This could result from coarsening of the microstructure or depletion of interstitial elements from the matrix due to precipitation. This effect is also observed in martensitic steels irradiated to high dpa levels in the FFTF, but in this class of alloys, recovery of the martensitic lath structure is also a factor.12 However, even in austenitic steels, it is difficult to attribute such softening with certainty to an irradiation effect because of the strong influence
1.04.4 Effects of Neutron Spectrum This discussion has used neutron irradiations for illustration purposes. Reactors provide an effective instrument for achieving high neutron exposures under conditions relevant for most nuclear applications. However, reactor irradiations suffer from many difficult-to-control and, sometimes, uncontrolled variables. The neutron energy spectrum is responsible for large differences in irradiation effects between different reactors. The mechanism of atomic displacement is well understood.15 With a known neutron energy spectrum, neutron atomic displacements can be calculated as a function of fluence for a given reactor. Transmutation of elements in the material under study, which is a strong function of neutron spectrum, results in wide variation in some mechanical properties. This is of particular importance in applying fission reactor results to fusion. In a fusion device, helium and hydrogen will be generated through (n,a) and (n,p) reactions in nearly all common structural materials. Hydrogen has a very high diffusivity in metals so that an equilibrium concentration will be
Effect of Radiation on Strength and Ductility of Metals and Alloys
established at a level that is believed to be benign.16 By contrast, helium is insoluble in metals, segregating at grain boundaries and other internal surfaces and discontinuities. Although helium is produced in all nuclear reactors, the thermal spectrum is responsible for the highest concentrations. The largest contributors to helium in a thermal reactor are boron and nickel by the following reactions: 10
58
Bðn; aÞ7 Li
Niðn; gÞ59 Ni
59
Niðn; aÞ56 Fe
Boron is present as a trace element in most alloying elements but only at ppm levels. Nickel is a major constituent of many alloys and a minor constituent of still others. The two nickel reactions constitute a two-step generation process for helium, which starts slowly and accelerates as 59 Ni builds up in the alloy, limited only by the supply of 58Ni, which for practical purposes is often unlimited. In austenitic alloys, the high flux isotope reactor (HFIR) has generated over 4000 appm He in austenitic stainless steels. The generation rate is so high that multistep absorber experiments have been conducted to reduce the helium generation rate to that characteristic of fusion reactors, 12 appm He per dpa in austenitic stainless steels.17 (see Chapter 1.06, The Effects of Helium in Irradiated Structural Alloys). Other transmutation products may also complicate reactor irradiation studies. Examples are the transmutation of manganese to iron by the following reaction: 55Mn (n,g) 56Mn ! 56Fe and the transmutation of chromium to vanadium by 50Cr (n,g) 51 Cr ! 51V. The first reaction leads to loss of an alloy constituent, and the second leads to doping with an extraneous element. However, neither of these reactions has been shown to significantly affect mechanical properties of steels.18 Helium remains the most studied transmutation product, and it can have profound effects on tensile properties, especially at high temperatures. Experiments have been conducted in various reactors throughout the world to assess the effects of helium on mechanical properties of alloys.19 An interesting result is that helium has little effect on strength. This is illustrated in Figure 5 where a comparison has been made between austenitic steels irradiated in Rapsodie, a fast spectrum reactor, and steels irradiated in HFIR, a mixed-spectrum reactor with
103
a very high thermal flux. The saturation yield strength of all alloys remains within a single scatter band.20,21 The tramp impurity elements sulfur and phosphorus have significantly high (n,a) cross sections at high energies, as shown in Figure 6. Although the cross section for phosphorus is large only at energies characteristic of fusion, a boiling water reactor produces 500 appm He from sulfur and 40 appm He from phosphorus in eight years of operation. An Liquid Metal Fast Breeder Reactor (LMFBR) can produce 100 times these concentrations. All these elements are expected to enhance embrittlement when segregated to grain boundaries, but it remains to be determined which is more detrimental, helium, sulfur, or phosphorus.
1.04.5 Tensile Ductility Tensile ductility is a more vulnerable parameter than strength to radiation effects since it tends to be very high in unirradiated austenitic stainless steels and is often reduced to quite low levels by irradiation. It is also of more concern since strengthening, although not reliable due to its slow initiation, is usually a beneficial change. In contrast, embrittlement is always detrimental. Like strength, ductility exhibits saturation with increasing fluence, although the behavior is significantly more complex than that of strength. The general trends in type 316 stainless steel are shown in Figure 7 for material irradiated in the EBR-II. These data are for the same specimens for which the yield strength was shown in Figure 2.9 Fast reactor data are used here to avoid the complication of helium effects. Once stabilization of the dislocation microstructure is achieved, a smooth curve approaching an apparent saturation is observed. More information can be gleaned from ductility data if they are viewed in terms of irradiation and test temperature. Figure 822 shows total tensile elongation for a series of irradiated austenitic alloys at a displacement level of 30 dpa in both annealed and cold-worked conditions. The room temperature ductility exceeds 10%, but it decreases rapidly with increasing temperature up to approximately 300 C and then exhibits the expected increase with temperature observed for unirradiated alloys. Beyond 500 C, ductility again decreases with an onset of intergranular embrittlement resulting from helium introduced through transmutations in the thermal flux of the HFIR.
104
Effect of Radiation on Strength and Ductility of Metals and Alloys
900 1.4988 sa 1.4970 sa + cw + a AISI 316 cw AISI 304 sa
800
Saturation yield strength (MPa)
700
600
500
400
US 316 20% cw HFIR JPCA SA HFIR JPCA 15% cw HFIR
300
316 20% cw EBR-II US PCA SA + 800 ⬚C, 8 h HFIR
200 Typical yield strength values of unirradiated solution annealed austenitic stainless steel 100 350
400
450
500
550
600
650
Irradiation and test temperature (⬚C) Figure 5 Saturation yield strength as a function of temperature for austenitic alloys irradiated in Rapsodie, EBR-II, and high flux isotope reactor showing similar saturation strength. Reproduced from Grossbeck, M. L.; Ehrlich, K.; Wassilew, C. J. Nucl. Mater. 1990, 174, 264.
1.00E + 03
(n, a) cross-section (barns)
1.00E + 02
LMFBR flux (arb. units) LWR flux (arb. units) Sulfur (n, a) Phosphorus (n, a)
1.00E + 01
1.00E + 00
1.00E - 01
1.00E - 02
1.00E - 03 1.E - 11
1.E - 08
1.E - 05 1.E - 02 Energy (MeV)
Figure 6 Cross-section for (n,a) reactions as a function of neutron energy.
1.E + 01
1.E + 04
Effect of Radiation on Strength and Ductility of Metals and Alloys
105
17
Irrad. temp. » Test temp. Strain rate ~ 4 ´ 10-5 S−1
16 15 14
Test Temp. (⬚C) 371 427 483 538 593 649 704 760 816
Symbol
13 Total elongation (%)
12 11 10 9 8
593 ⬚C
7
538 ⬚C
6 5 4 3
816 ⬚C
2 1 0
760 ⬚C
0
1
2
649 ⬚C 371 ⬚C
704 ⬚C
3 4 5 6 7 8 Neutron fluence (n cm−2) (E > 0.1 MeV)
10 ´ 1022
9
Figure 7 Total elongation of 20% cold-worked type 316 stainless steel irradiated in EBR-II. Reproduced from Fish, R. L.; Cannon, N. S.; Wire, G. L. In Effects of Radiation on Structural Materials; Sprague, J. A., Dramer, K., Eds.; ASTM: Philadelphia, PA, 1979; ASTM STP 683, p 450. Reprinted, with permission, from Effects of Radiation on Structural Materials, copyright ASTM International, West Conshohocken, PA.
ORNL-DWG 89-13395 20 30 DPA
+ + J
+ Total elongation (%)
15
*
0 C
US PCA 25% CW HFIR JPCA ANN HFIR JPCA 15% CW HFIR US 316 20% CW HFIR US PCA B3 HFIR 316 20% CW EBR-II
J J
10
+ J J* 0 0
+
5
J J 0 C. 0
0
100
200
J C.
300 400 Temperature (⬚C)
500
C.J C. 00 600
700
Figure 8 Total elongation as a function of irradiation and test temperature for fast (EBR-II) and mixed-spectrum (high flux isotope reactor) reactor irradiation.
Uniform elongation, the elongation at the onset of plastic instability, or necking, appears to be most sensitive to the effects of irradiation and, in general, is less dependent on specimen geometry than other parameters such as total tensile elongation. The low values of uniform elongation are often cause for great concern, which is usually justified. However, it should be borne in mind that if stresses remain below the yield
stress of a metal, elongation becomes a secondary concern. As long as limited plastic deformation relieves the stress that produced it, a structure remains intact. The high level of irradiation strengthening observed at temperatures below 300 C, which is due to black dot defect clusters and small loops, also results in low ductility throughout this temperature range. Small helium bubbles and helium-defect
106
Effect of Radiation on Strength and Ductility of Metals and Alloys
20
Uniform elongation (%)
10 dpa 250 ⬚C Sym. (%)
15
0.20 0.20 0.33 0.29 0.31 0.28 0.23 0.18 0.30 0.20 0.10
10
5
316 SS
PCA
0 0
100
200
300 400 Temperature (⬚C) Symbols
500
600
316 20% CW EBR-II
316 ANN EBR-II
US 316 20% CW ORR
US PCA 25% CW HFIR
US PCA 25% CW ORR
US 316 20% CW HFIR
US PCA 25% CW HFR
US 316 20% CW HFR
316 20% CW DO HFIR
US PCA 25% CW BR2
US 316 20% CW BR2
EC 316 ANN HFIR
JPCA ANN HFIR
EC 316 ANN BR2
JPCA ANN HFR
EC 316 ANN HFR
JPCA 15% CW HFIR
US PCA B3 HFIR
700
Figure 9 Uniform elongation as a function of irradiation and test temperature at a displacement level of 10 dpa. The trend curves are for type 316 stainless steel and PCA. Reproduced from Grossbeck, M. L.; Ehrlich, K.; Wassilew, C. J. Nucl. Mater. 1990, 174, 264.
clusters also contribute to hardening and reduction in ductility, but this form of helium embrittlement is not related to the severe intergranular embrittlement that is observed above 500 C. Both these effects are apparent in Figure 9 where uniform elongation for an extensive set of austenitic alloys irradiated in thermal and fast spectrum reactors is shown.11 The specimens irradiated in the fast spectrum (<5 appm He) exhibit consistently higher ductility than the mixed-spectrum reactor specimens (500–1000 appm He) even at this low displacement level, especially above 600 C, where helium embrittlement is certain to control. A similar pattern is exhibited at 30 dpa where a very limited uniform elongation characteristic of lower temperatures is apparent. After a restoration of ductility above 400 C, ductility again decreases above 500 C due to the onset of intergranular helium embrittlement. Differences in alloy behavior, especially in the case of titanium-modified alloys somewhat clouds the understanding of helium
embrittlement observed in Figure 10.11 However, at 50 dpa, where helium levels exceed 4000 appm, the trend becomes clear with the fast reactor specimens showing uniform elongations several times larger than those observed in mixed-spectrum reactors (Figure 11).11 What is less expected is the recovery of ductility at 50 C at 50 dpa compared to the results at 30 dpa. This irradiation annealing effect has also been observed at 230 C by Ehrlich, where strength of the alloy 1.4988 decreased continuously from 10 to 30 dpa.20 Results from an experiment in the Oak Ridge Research Reactor (ORR), where the spectrum was tailored to produce a ratio of He per dpa characteristic of a fusion reactor, show similar low levels of uniform elongation for cold-worked alloys at low temperatures, but high uniform elongations were observed in annealed type 316 stainless steel at 60 C. This high ductility was drastically reduced between 200 and 330 C before the microstructure characteristic of higher temperatures became effective.21
Effect of Radiation on Strength and Ductility of Metals and Alloys
107
Uniform elongation (%)
20 316 20% CW EBR-II US 316 20% CW ORR US PCA 25% CW ORR US PCA 25% CW HFR 316 20% CW DO HFIR US 316 20% CW BR2 JPCA ANN HFIR JPCA ANN HFR JPCA 15% CW HFIR
15
10
316 ANN EBR-II US PCA 25% CW HFIR US 316 20% CW HFIR US 316 20% CW HFR US PCA 25% CW BR2 EC 316 ANN HFIR EC 316 ANN BR2 EC 316 ANN HFR US PCA B3 HFIR
30 dpa
5
0
0
100
200
300 400 Temperature (⬚C)
500
600
700
Figure 10 Uniform elongation of austenitic stainless steels irradiated in fast and thermal reactors to a displacement level of 30 dpa. Severe helium embrittlement is shown at 600 C. Reproduced from Grossbeck, M. L.; Ehrlich, K.; Wassilew, C. J. Nucl. Mater. 1990, 174, 264.
ORNL-DWG 90-14298
20
Uniform elongation (%)
50 dpa US PCA 25% CW HFIR JPCA SA HFIR JPCA 15% CW HFIR 316 20% CW EBR-II US PCA SA + 800 ⬚C, 8 h HFIR J 316 20% CW HFIR J 316 SA HFIR
15
10 64 dpa
5
0
78 dpa
0
100
200
300 400 Temperature (⬚C)
500
600
700
Figure 11 Uniform elongation of austenitic stainless steels irradiated to 50 dpa in high flux isotope reactor (HFIR) and 78 dpa in EBR-II showing embrittlement from helium generated in the mixed-spectrum reactor, HFIR. Reproduced from Grossbeck, M. L.; Ehrlich, K.; Wassilew, C. J. Nucl. Mater. 1990, 174, 264.
1.04.6 Effect of Test Temperature An interesting phenomenon is observed when irradiated alloys are tested at temperatures different from the irradiation temperature. Figure 12 shows total elongation data from cold-worked type 316 stainless steel irradiated to displacement levels of 48–63 dpa in the FFTF, where elongation is plotted
against the increment of the test temperature above the irradiation temperature.23 Although there is significant scatter in the data, the elongations below 1% obtained by test temperatures about 100 C above the irradiation temperature are cause for concern. This phenomenon has also been observed in higher nickel alloys. The cause of this phenomenon remains elusive, pending further testing with
108
Effect of Radiation on Strength and Ductility of Metals and Alloys
32 Fluence = 9.13 ´ 1022 n cm–2 (E > 0.1 MeV)
Total elongation (%)
28 24 20 16 12 8
f t = 12 ´ 1022 n cm–2
4 0 - 700 - 600 - 500 - 400 - 300 - 200 - 100
0
100
200
300
400 500
Test Irradiation (⬚C) – temperature temperature Figure 12 Total elongation of 20% cold-worked type 316 stainless steel irradiated in FFTF to displacement levels of 48–63 dpa. Hamilton, M. L.; Cannon, N. S.; Johnson, G. D. In Effects of Radiation on Materials; Brager, H. R., Perrin, J. S., Eds.; ASTM: Philadelphia, PA, 1982; ASTM STP 782, p 636. Reprinted, with permission, from Effects of Radiation on Structural Materials, copyright ASTM International, West Conshohocken, PA.
different holding times at various temperatures before tensile testing. Migration of interstitial solutes to moving dislocations is a candidate mechanism for this phenomenon.
1.04.7 Ferritic–Martensitic Alloys 1.04.7.1
Introduction
The class of ferritic–martensitic alloys with chromium concentrations in the range of 9–12% has attracted interest in the fast reactor programs because of its radiation resistance, in particular, very low swelling and low irradiation creep. Alloys such as Sandvik HT-9 (12Cr1Mo.6Mn.1Si.5W.3V)) and other alloys of this class were irradiated in the EBR-II,24 in research reactors25 and with heavy ions.26 The quantitative results from the ion irradiations in this class of alloys and the low neutron absorption cross section led to inclusion of ferritic alloys into the fast reactor alloy development programs, in particular in the United States in the mid-1970s. The radiation resistance has been confirmed to displacement levels of 70 dpa.12,14 Further interest in this class of alloys was initiated by the fusion reactor programs in Europe and the United States when the necessity for low neutron activation structural materials was realized. Further
research on martensitic alloys by fusion programs in Europe, the United States, and Japan led to the development of low-activation alloys by replacing elements that result in long-term activation products. Molybdenum and niobium, both of which result in longlived activation products, were replaced by tungsten and tantalum. This research led to radiation-resistant alloys with a fracture toughness superior to that of the commercial alloys even in the unirradiated condition.27 The compositions of representative members of this class of alloys referred to in this chapter are presented in Table 1. An excellent review of irradiation behavior of this class of alloys has been published by Klueh and Harries.27 Details of the metallurgy of martensitic alloys appears in Chapter 4.03, Ferritic Steels and Advanced Ferritic–Martensitic Steels. 1.04.7.2
Tensile Behavior
Unlike the tensile behavior of fcc metals, where there is a smooth increase in strength as plastic deformation proceeds and work hardening progresses, bcc metals typically exhibit a load drop almost immediately following the onset of plastic deformation. Interstitial solutes such as carbon in steels effectively lock dislocations leading to a longer period of elastic deformation after which generation of new dislocations results in a load drop, or yield point, until
Effect of Radiation on Strength and Ductility of Metals and Alloys
Table 1
109
Nominal or typical compositions of ferritic–martensitic alloys cited
Steel type
12Cr–MoVW 8Cr–2WVTa 9Cr–1WVTa 9Cr–1GeV 12Cr–1MoVNiNb 10Cr–MoVNiNb 9Cr–2WVTa 9Cr–2WVTa 9Cr–2WVTa 7Cr–2WVTa 2.25Cr–2WVTa 12Cr–2WVTa 12Cr–2WVTa 9Cr–1MoVNb 9Cr–2Mo–1Ni 9Cr–2W 9Cr–2W
Designation
HT-9 F82H OPTIFER Ia OPTIFER II MANET I MANET II 9Cr–2WVTa JLF-1 JLF-2 JLF-3 JLF-4 JLF-5 JLF-6 T91 JFMS NFL-0 NFL-1
Composition (wt%) C
Si
Mn
Cr
Ni
Mo
V
0.20 0.1 0.1 0.1 0.14 0.1 0.1 0.1 0.1 0.09 0.1 0.09 0.10 0.1 0.05 0.10 0.10
0.38
0.60
11.95 8.0 9.0
0.60
1.0
10.8 10.0 8.7 9.0 9.16 7.03 2.23 11.99 12.00 9.0 9.6 8.65 9.01
0.9 0.7
0.75 0.6
0.94
1.0 2.3
0.30 0.2 0.25 0.3 0.2 0.2 0.23 0.19 0.20 0.20 0.20 0.19 0.19 0.2 0.12 0.25 0.26
0.2
0.3 0.67 0.056 0.042
0.4 0.46 0.45 0.45 0.50 0.48 0.46 0.4 0.58 0.050 0.53
v
Nb
W
B
0.52 1.0 0.16 0.15
0.08 0.06
2.2 1.97 1.93 1.97 1.97 1.98 1.94 1.92 2.06
0.009 0.007
Other 0.04 Ta 0.07 Ta 1.1Ge 0.06 Zr 0.03 Zr 0.06 Ta 0.07 Ta 0.07 Ta 0.07 Ta 0.07 Ta 0.07 Ta 0.07 Ta
0.0032
Source: Maloy, S. A.; Toloczko, M. B.; McClellan, K. J.; et al. J. Nucl. Mater. 2006, 356, 62; Klueh, R. L.; Harries, D. R. High-Chromium Ferritic and Martensitic Steels for Nuclear Applications; ASTM: Philadelphia, PA, 2001; Kohno, Y.; Kohyama, A.; Hirose, T.; Hamilton, M. L.; Narui, M. J. Nucl. Mater. 1999, 271 & 272, 145; Kurishita, H.; Kayano, H.; Narui, M.; Kimura, A.; Hamilton, M. L.; Gelles, D. S. J. Nucl. Mater. 1994, 212–215, 730.
(´ 103) 120 1 ´ 1020 nvt
100
5 ´ 1018 nvt
Stress (psi)
80
Unirradiated 1.7 ´ 1019 nvt
60 40 20 0
0
0.05
0.10
0.15 0.20 0.25 Strain (in. in.–1)
0.30
0.35
0.40
Figure 13 Stress–strain curves for low-carbon steel weld material irradiated at 80 C (nvt is fluence in neutrons/cm2). Reproduced from Wilson, J. C. Effects of irradiation on the structural materials in nuclear power reactors. In Proceedings of the Second United Nations International Conference on the Peaceful Uses of Atomic Energy, United Nations, 1958; Vol. 5, p 431.
terminated by the work hardening mechanism of dislocation interaction.28 Upon irradiation, the load drop is frequently masked by an early termination of work hardening, leading to very low values of uniform elongation. This behavior is evident even at displacement levels
below 0.01 dpa and is illustrated in Figure 13 from research presented at the Second Atoms for Peace Conference in 1958.29 Extreme irradiation hardening and severe plastic instability are clearly illustrated by this early research. More recent alloys with more careful control of impurities and controlled processing
110
Effect of Radiation on Strength and Ductility of Metals and Alloys
1400
JFMS specimens tested at 25 ⬚C after irradiation in FFTF
Engineering stress (MPa)
1200
35.3 dpa, Tirr = 390 ⬚C 9.8 dpa, Tirr = 373 ⬚C 22.2 dpa, Tirr = 390 ⬚C 44 dpa, Tirr = 427 ⬚C
1000 800
0 dpa
600 400 200 0 0
2
4
12 14 16 8 10 Engineering strain (%)
6
18
20
22
Figure 14 Stress–strain curves for JFMS alloy irradiated in FFTF to 44 dpa at temperatures of 373–427 C and tested at 25 C. Reproduced from Maloy, S. A.; Toloczko, M. B.; McClellan, K. J.; et al. J. Nucl. Mater. 2006, 356, 62.
800 MATRON/tensile(S) 700
656–700 K
Yield stress (MPa)
600 500 400 300 200 100 0
0
10
20
30
40
50
60
70
Displacement damage (dpa)
Figure 15 Yield stress as a function of displacement level for martensitic alloys irradiated in FFTF or EBR-II. Reproduced from Kohno, Y.; Kohyama, A.; Hirose, T.; Hamilton, M. L.; Narui, M. J. Nucl. Mater. 1999, 271 & 272, 145.
have led to ferritic alloys with appreciable work hardening even at high displacement levels. Figure 14 shows tensile curves for the 9Cr–2Mo–1Ni steel, JFMS, neutron irradiated and tested at room temperature. Uniform elongations of several percent are evident, a reasonable value for irradiated steels.12 A plot of yield stress as a function of displacement damage level is shown in Figure 15 for low-activation
ferritic alloys irradiated in fast reactors,30 and plots of yield stress and total elongation are shown in Figure 16 for fast and mixed-spectrum reactors.31 Unlike the austenitic alloys, the martensitic alloys rapidly reach a peak in strength then soften with further irradiation followed by near saturation in strength beginning at about 30 dpa. Total elongation follows a corresponding pattern, demonstrating
111
Effect of Radiation on Strength and Ductility of Metals and Alloys
800
JLF-5 (12Cr–2WVTa)
Total elongation (%)
JLF-5 (12Cr–2WVTa)
600
Yield stress (MPa)
30
JLF-4 (2.25Cr–2WVTa)
700
500 F82H 10B
400 300
F82H STD
JLF-1 (9Cr–2WVTa)
F82H (8Cr–2WVTaB)
JLF-1 (9Cr–2WVTa)
20
F82H STD F82H (8Cr–2WVTaB)
10
JLF-3 (7Cr–2WVTa)
200
F82H 10B
: Irradiated in HFIR (spec. SS3) Others in FFTF (spec. TS(s))
100
JLF-4 (2.25Cr–2WVTa)
0 0
10
20
30
40
50
60
Displacement damage (dpa)
0
0
10
20
30
40
JLF-3 (7Cr–2WVTa)
50
60
Displacement damage (dpa)
Figure 16 Yield strength of martensitic alloys following irradiation at 400 C in FFTF or high flux isotope reactor. Reproduced from Kohyama, A.; Hishinuma, A.; Gelles, D. S.; Klueh, R. L.; Dietz, W.; Ehrlich, K. J. Nucl. Mater. 1996, 233–237, 138.
inverse behavior. This appears at first to be a phenomenon totally different from that which occurs in austenitic alloys. However, the operable mechanisms are really the same, just occurring at lower fluences. Hardening mechanisms are similar, occurring by point defect clusters at low irradiation temperatures and transitioning to loops and network dislocations and precipitation as temperature is increased. The martensitic alloys are more complex in that the initial microstructure is determined by the heat treatment. The alloys are used in the normalized and tempered condition produced by austenitizing the alloy and quenching below the A1 temperature to produce martensite. The martensite is then tempered below the A1 temperature. Precipitates, primarily carbides such as M23C6, form on prior austenite grain boundaries and on the martensite laths. In the case of the bcc alloys, more rapid radiation-enhanced diffusion results in irradiation-induced recovery and precipitate growth at lower fluence than in the fcc alloys. Recall that in the austenitic alloys, saturation was reached and sustained for a long period until microstructural coarsening resulted in a slight decrease in strength at high displacement levels. The pronounced peak in strength results from rapid hardening due to irradiation-produced defects, but the effect of irradiation hardening is offset by irradiation-enhanced recovery, resulting in a decrease in strength and hence a peak in strength.32 The martensitic alloys demonstrate the same phenomena but at a lower fluence.
Figure 17 shows tensile test results from alloys irradiated in FFTF to approximately 30 dpa. Tests at the irradiation temperatures show high and nearly constant values of total tensile elongation at temperatures above 425 C.33 Somewhat similar behavior of elongation was also exhibited by 9Cr–1MoVNb steel irradiated to 12 dpa in EBR-II.34 This alloy also exhibits an increase in total elongation at 550 C (Figure 18). The apparent saturation in strength above about 450 C is also in agreement with hardness measurements. In a study of irradiated HT-9 and 9Cr–1MoVNb, Hu and Gelles observed that hardness retained its unirradiated value upon irradiation to 26 dpa when irradiated at temperatures above about 450 C in EBR-II.35 From this behavior, it would normally be concluded that fracture properties would remain unchanged or improve with increasing temperature. Although fracture toughness testing is used extensively in the study and certification of irradiated alloys, the Charpy impact test is more commonly used in alloy development and fundamental research because the test requires smaller specimens and can be conducted more easily. Ductile to brittle transition temperature (DBTT) is a useful tool for comparison of alloys and assessment of radiation damage. Charpy impact testing was conducted by Hu and Gelles who observed that in the case of the 9Cr–1MoVNb, the DBTT did in fact retain essentially its unirradiated value for irradiation
112
Effect of Radiation on Strength and Ductility of Metals and Alloys
12 Total elongation (%)
Yield stress, sy (MPa)
800
600
400
Unirr.
200
8
4
0
(b)
600
700
800
900
1000
Uniform elongation (%)
Ultimate tensile stress, su (MPa)
(a)
0
800 600 Unirr. 400
Total elongation, er (%)
8
9Cr–1 MoVNb steel Unirradiated Aged Irradiated Test temperature @ Irradiation temperature @ Aging temperature
4
200 0 0
600
700
800
900
400
500 Test temperature (⬚C)
600
Figure 18 Elongation of 9Cr–1MoVNb irradiated in EBR-II to 0.9 dpa. Reproduced from Klueh, R. L.; Vitek, J. M. J. Nucl. Mater. 1985, 132, 27.
30
20 Unirr. 10
0 (c)
12
600 700 800 Irradiation temperature (K)
900
NFL-0 NFL-1 Figure 17 Tensile properties of two Fe–9Cr–2W steels with and without small additions of boron, irradiated in FFTF to approximately 30 dpa and tested at room temperature. Reproduced from Kurishita, H.; Kayano, H.; Narui, M.; Kimura, A.; Hamilton, M. L.; Gelles, D. S. J. Nucl. Mater. 1994, 212–215, 730.
temperatures above 450 C at 26 dpa but not for lower irradiation temperatures, as shown in Figure 19.35 However, the DBTT of HT-9 failed to retain its unirradiated value despite the absence of an increase in strength and hardness. In both cases the upper shelf energy was reduced by the irradiation, indicating some changes in the irradiation microstructure.
The shift in the DBTT for both alloys at 13 and 26 dpa is shown as a function of irradiation temperature in Figure 20. The retention of the increase in the DBTT at high temperatures illustrates the caution that must be used in assessing ferritic alloys. In impact testing, fracture is generally initiated at carbide particles. Even though coarsening of the carbides results in less impediment to dislocation motion, and thus less hardening, the stress intensity factor increases for a nucleating crack at a hard particle so that the effective crack length is the crack nucleus plus the diameter of the carbide particle.36 The result is that fracture toughness can increase with irradiation temperature. Despite the caution that must be taken in making generalizations based upon tensile behavior, development of low-activation martensitic alloys has led to alloys with very favorable fracture properties. The DBTT is shown in Figure 21 for two low-activation alloys, 9Cr–2WVTa and 9Cr–2WVTa–2Ni, with the conventional Ht-9 and 9Cr–1MoVNb for comparison. The irradiation was done in EBR-II at irradiation temperatures from 376 to 405 and to displacement levels of 23–33 dpa. After irradiation, the two
Effect of Radiation on Strength and Ductility of Metals and Alloys
113
9Cr–1Mo base metal (TV series), 26 dpa Ti = 390 ⬚C Ti = 500 ⬚C
Normalized fracture energy (J cm−2)
600
Control
480 TV11 360
240 TV22
120
0 -150
-100
-50 0 50 Test temperature (⬚C)
100
150
HT-9 base metal (TT series), 26 dpa
Normalized fracture energy (J cm−2)
400
320
Ti = 390 ⬚C Ti = 450 ⬚C Ti = 500 ⬚C Ti = 550 ⬚C
Control
240
160
TT23
TT15 TT20
80
0 -100
-50
0
TT28
50 100 150 Test temperature (⬚C)
200
250
Figure 19 Charpy impact test results for 9Cr–1Mo and HT-9 irradiated in EBR-II to 26 dpa. Reproduced from Hu, W. L.; Gelles, D. S. Influence of Radiation on Material Properties; ASTM: Philadelphia, PA, 1987; ASTM STP 956, p 83. Reprinted, with permission, from Influence of Radiation on Material Properties, copyright ASTM International, West Conshohocken, PA.
conventional alloys had DBTT values above room temperature, whereas the values for the two reduced-activation steels were below 75 C.32 The figure clearly shows that HT-9 is not the alloy of choice for nuclear applications. Despite the Ni content of 9Cr–2WVTa–2Ni, this class of alloys can prove useful in nuclear applications such as fast reactors where the thermal flux is small and the very high-energy neutrons characteristic of a fusion reaction are absent.
1.04.7.3
Helium Effects
Helium effects are important for systems that generate high-energy neutrons such as fusion reactors and spallation targets that encounter high-energy protons. The 14 MeV neutrons produced by D–T fusion will produce (n,a) reactions in nearly all common structural elements such that ferritic steels are not exempt from helium generation. However, in the absence of nickel, helium generation rates are lower.
114
Effect of Radiation on Strength and Ductility of Metals and Alloys
140 Shift in transition temperature (⬚C)
13 dpa HT9
120
26 dpa HT9
100
26 dpa Mod 9Cr–1Mo
13 dpa Mod 9Cr–1Mo
80 60 40 20 0 500 450 Irradiation temperature (⬚C)
400
550
Figure 20 Shift in ductile to brittle transition temperature as a function of irradiation temperature for 13 and 26 dpa following EBR-II irradiation. Reproduced from Hu, W. L.; Gelles, D. S. Influence of Radiation on Material Properties; ASTM: Philadelphia, PA, 1987; ASTM STP 956, p 83. Reprinted, with permission, from Influence of Radiation on Material Properties, copyright ASTM International, West Conshohocken, PA.
75
Transition temperature (⬚C)
50
Unirradiated Irradiated
25 0 -25
9Cr-1MoVNb 12Cr-1MoVW
-50 -75 -100 -125
9Cr-2WVTa 9Cr-2WVTa-2Ni
Figure 21 Ductile to brittle transition temperature before and after irradiation in EBR-II to 23–33 dpa at 376–405 C comparing conventional alloys with irradiation-resistant martensitic alloys. Reproduced from Klueh, R. L.; Sokolov, M. A.; Hashimoto, N. J. Nucl. Mater. 2008, 374, 220. Reprinted, with permission, from Influence of Radiation on Material Properties, copyright ASTM International, 100 Barr Harbor Drive, West Conshohocken, PA 19428.
In austenitic stainless steels, the high nickel content is used to introduce helium to simulate the fusion environment. However, the nickel content is so high that unrealistically high concentrations of helium are produced in a thermal or mixed-spectrum reactor. As a result, spectral tailoring is necessary to achieve the correct He per dpa ratio.37
In the case of martensitic alloys, several techniques have been used to generate helium in fission reactors. Doping with natural nickel has been used in 9Cr and 12Cr alloys27,38,39 and isotopically separated nickel has been used to discriminate against the effect of nickel as opposed to the effect of He. The isotope 59Ni has been used as it will generate He but 60Ni, used as a control, will not.40 Doping with 10B has also been used to introduce He into this class of alloys, leading to concentrations of several hundred parts per million.41–43 Nickel doping experiments with 12Cr–1MoVW were conducted in the HFIR using 1 and 2% Ni to generate up to about 300 appm He. This experiment demonstrated what appeared to be a strengthening effect of helium.44 However, it was determined that these results were clouded by the fact that doping with Ni lowers the A1 temperature of the alloy, leading to untempered martensite upon cooling from the tempering temperature. Adjustments were made in the tempering temperature, but the additional variables cast doubt on the results.44 The isotope separation experiments and the boron doping experiments found no clear indication of a helium effect.40,41 Several mechanisms for hardening by helium have been identified,45 but Klueh and Harries27 have, after a careful review, come to the conclusion that the effect of helium on strength and ductility below 500 C is inconclusive and that if there is an effect, it is probably small and of minor significance for the conditions examined. As with austenitic stainless steels, the effect of hardening, if it exists for ferritic alloys, is of minor importance compared with high-temperature intergranular embrittlement. High He concentrations and higher temperatures have been investigated in martensitic alloys by means of helium implantation with accelerators. Hasegawa implanted He at 600 C to levels of 500 appm and performed tensile tests at this temperature.46 However, all fractures were transgranular. Bae et al.47 implanted similar levels of helium, 500 appm, in the 12Cr-steel, MANET, at temperatures as high as 500 C. They observed hardening, shown in Figure 22, but again they were only transgranular fractures. Bae et al.47 also observed no synergistic effect of 500 appm H and 500 appm He and no effect of hydrogen alone at this level. Jung et al.48 found that He implanted in the 9Cr alloy, EUROFER97, to concentrations as high as 1250 appm produced both hardening and reduction in ductility when implanted at 250 C and tested at
Effect of Radiation on Strength and Ductility of Metals and Alloys
115
20
900 15 Total elongation (%)
MANET
Yield stress (MPa)
700
500
MANET
10
5 300
DSA 100
0 0
600 200 400 Irradiation and test temperature (⬚C)
800
0
200 400 600 Irradiation and test temperature (⬚C)
800
0.32 dpa, 500 appm H, 500 appm He 0.30 dpa, 500 appm He 0.02 dpa, 500 appm H Unirradiated
Figure 22 Yield strength and total elongation for the 12Cr steel, MANET 1 cyclotron implanted with He and H. Reproduced from Bae, K. K.; Ehrlich, K.; Mosalang, A. J. Nucl. Mater. 1992, 191–194, 905.
1200
0.125
EUROFER97 0.25
Timp = 250 ⬚C
s (MPa)
1000 0.25 0.06 0.125 0.06
800 600
cHe(at.%) 0 Ttest = 25 ⬚C
400 0 Ttest = 250 ⬚C
200 0
0
5 e (%)
10
Figure 23 Stress–strain curves for EUROFER97 for cyclotron implantation of helium. Jung, P.; Henry, J.; Chen, J. J. Nucl. Mater. 2005, 343, 275.
either 25 or 250 C. Their data demonstrate both hardening and loss of work hardening with helium, as clearly shown in Figure 23. Severe intergranular embrittlement was, in fact, demonstrated by Jung et al. in the 9Cr steels T91 and EM10 when levels of 5000 appm were attained by cyclotron implantation.49 In cases where He was implanted at 250 C and tested at 25 C and at 250 C, clear intergranular fracture was experienced. For implantation temperatures of 550 C, where
intergranular embrittlement would be expected in austenitic stainless steels, the fracture surfaces indicated a return of some ductility with little or no intergranular fracture, particularly when tested at 550 C, Figure 24. The latter case can perhaps be explained by the loss of strength at 550 C where plastic deformation blunts any nucleating crack before the local stress at a grain boundary is sufficiently high for grain separation. In the case of implantation at 550 C and testing at 25 C, the fracture mechanism is less clear. Here, the temperature is sufficiently high for diffusion of He to the grain boundaries, but perhaps capture by other sinks prevents high levels of He at the boundaries. More investigation of the high-temperature He embrittlement is necessary to determine the underlying mechanism of fracture. It is clear, however, that helium embrittlement is more effective in austenitic stainless steels than in ferritic–martensitic steels. Since nickel is the largest source of helium in a fast neutron environment, ferritic alloys clearly have a lower He generation rate than austenitic steels.45 The combination of higher resistance to helium embrittlement and the lower generation rate of He in ferritic alloys makes this class of alloys more favorable with respect to He embrittlement.
116
Effect of Radiation on Strength and Ductility of Metals and Alloys
250/25
550/25
250/250
550/550
Figure 24 Fracture surfaces of T91 following helium implantation at 250 and 550 C and tested at the implantation temperatures indicated. Reproduced from Jung, P.; Henry, J.; Chen, J.; Brachet, J. C. J. Nucl. Mater. 2003, 318, 241.
1.04.8 Refractory Metals 1.04.8.1
Tensile Behavior
The refractory metals are the metals in groups V and VI of the periodic table: vanadium, niobium, and tantalum in group V and chromium, molybdenum, and tungsten in group VI. All have the characteristic of a high melting point, hence the term refractory. The group VI metals are typically brittle, even without irradiation. For example, chromium is almost never used pure or as a major alloy element, although it is invaluable as a minor alloying element. Molybdenum and tungsten are both brittle in nature but can be made into useful structural alloys by controlling interstitial impurities and by the addition of minor elements. In contrast to the brittle behavior of the group VI metals, the group V metals are inherently ductile. Structural alloys based upon this group have been developed, primarily for very high temperature and space applications. The primary disadvantage of the refractory metals is their formation of volatile oxides as opposed to protective oxide layers. Vanadium and molybdenum oxides have melting points below metal working temperatures so that the metals become wet and can have liquid oxide drip off them. Unlike the tensile behavior of fcc metals, where there is a smooth increase in strength as plastic deformation proceeds and work hardening progresses, bcc
metals typically exhibit a load drop, or yield point, almost immediately following the onset of plastic deformation, as discussed in Section 1.04.7.2. In the case of refractory metals, mechanical properties are largely determined by interstitial solutes. High purity refractory metals do not exhibit a yield point but behave more like fcc metals. Niobium alloys irradiated in Li at 1200 C for over three months in EBR-II had total elongations of about 60%. Despite any irradiation hardening, the near absence of oxygen resulted in a very soft material at these high temperatures.50 However, since interstitials are nearly always present, tensile behavior is more typically characteristic of bcc metals. Irradiation-produced defects interact with interstitial elements, resulting, in some cases, in severe embrittlement. The tantalum alloy, Westinghouse T111 (Ta–8W–2Hf) is used in Figure 25 to illustrate a commonly observed phenomenon of plastic instability.51 Plastic deformation becomes local, with high levels of slip on closely spaced planes where dislocations sweep out the irradiation-generated defects giving rise to local channels of very high deformation. This phenomenon, called channel deformation, is very common in irradiated metals. The result is a sudden and severe load drop with the fracture surface showing what appears to be a completely ductile chisel point fracture.52 Addition of 405 wt ppm oxygen to T-111 results in a cleavage fracture with no measurable plastic deformation, as shown in Figure 26.51 In both Figures 25 and 26, corresponding unirradiated alloys are shown demonstrating ductile behavior. In the unirradiated condition, the addition of 400 wt ppm oxygen has only minor effects on strength and ductility, as can be concluded by a comparison of Figures 25 and 26. However, irradiation hardening superimposed upon the oxygen interstitial hardening appears to raise the yield stress above the cleavage stress for the alloy. It is suggested that interstitial solutes such as oxygen diffuse to irradiation-produced defect clusters, enhancing their hardening effect.53,54 All three behaviors are observed in irradiated refractory metals: ductile with hardening, plastic instability, and cleavage fracture in the elastic range.55 The synergism between interstitial hardening and irradiation hardening does not necessarily lead to immediate catastrophic embrittlement. This behavior is shown in Figure 27 for vanadium containing a very high level of oxygen, 2100 wt ppm. Irradiation to a fluence level of 1.5 1019 (E > 1 MeV) leads to the familiar plastic instability but with several per cent plastic strain.53
Effect of Radiation on Strength and Ductility of Metals and Alloys
117
1600 111T
1400
Unirradiated EBR-II irradiated ft (E > 0.1 MeV) = 1.6 × 1026 n m–2 Tirrad = Ttest = 873 K
Stress (MPa)
1200 1000 800
*
600 400 200 0
* 0
2
4
6
8
10
12 14 16 Strain (%)
18
20
22
24
26
28
Figure 25 Stress–strain curves for the Ta alloy, T-111 showing characteristic tensile behavior following irradiation in EBR-II to 8 dpa at 600 C. Reproduced from Grossbeck, M. L.; Wiffen, F. W. In Space Nuclear Power Systems; El-Genk, M. S., Hoover, M. D., Eds.; Orbit Book Co.: Malabar, FL, 1986; Vol. III, p 85.
1400 111 T + 405 wt. ppm
oxygen Unirradiated EBR-II irradiated ft (E > 0.1 MeV) = 1.4 × 1026 n m–2 Tirrad = Ttest = 853 K
*
1200
Stress (MPa)
1000 800 600 400
*
200 0
0
2
4
6
8
10
12 14 16 Strain (%)
18
20
22
24
26
28
Figure 26 Stress–strain curves for oxygen-doped T-111 T unirradiated and irradiated in EBR-II to 7 dpa at 580 C. Reproduced from Grossbeck, M. L.; Wiffen, F. W. In Space Nuclear Power Systems; El-Genk, M. S., Hoover, M. D., Eds.; Orbit Book Co.: Malabar, FL, 1986; Vol. III, p 85.
Interstitial solutes, especially oxygen, may be controlled by the addition of gettering elements. In the vanadium system, titanium has been successful. Alloys in the V–Cr–Ti system have been studied for application to fusion reactors. In refractory metal
alloys, it is the oxygen in solution that is detrimental, so that the oxygen must be combined with the titanium.56 This usually requires a heat treatment of sufficiently long times and high temperatures to precipitate the oxygen. In the vanadium–titanium system,
118
Effect of Radiation on Strength and Ductility of Metals and Alloys
90 Vanadium–2100 wt ppm oxygen Tensile tests at 300 ⬚K and 1.67 ´ 10-4 s-1
80
Irradiated, 1.5 ´ 1019 n cm−2 (E > 1 MeV)
70
Stress (Kpsi)
60 Unirradiated
50 40 30 20 10 0
Uniform elongation 0
2
4
6
8
12 14 10 Strain (%)
16
18
20
22
24
Figure 27 Stress–strain curves for oxygen-doped vanadium irradiated at 85 C at the Ames Laboratory Research Reactor. Reproduced from Wechsler, M. S.; Alexander, D. G.; Bajaj, R.; Carlson, O. N. In Defects and Defect Clusters in B.C. C. Metals and Their Alloys, Nuclear Metallurgy; Arsenault, R. J., Ed.; National Bureau of Standards: Gaithersburg, MD, 1973; Vol. 18, p 127.
a heat treatment of two hours at 950 C has been found sufficient to achieve ductility, whether or not it is to be irradiated.57,58 Confusion over oxygen present in solution and oxygen combined in precipitates is believed to be one reason for the disparity in tensile data for this class of alloys and perhaps accounts for the relatively high level of ductility observed in Figure 27. Upon irradiation of alloys in the range of V–3– 5Cr–3–5Ti in the HFIR, no cleavage fracture without plastic deformation was observed.59,60 However, plastic instability was commonly observed at irradiation and test temperatures below 400 C. Irradiations in the range of 4–6 dpa in the HFIR produced uniform elongations from 0.2 to 0.6% and total elongations below 4%. Corresponding irradiations at 500 C did not reveal plastic instability and produced uniform elongations in the range of 2–5%.59,60 Irradiations to 3–5 dpa in the advanced test reactor (ATR) demonstrated plastic instability for irradiation and test temperatures of about 200 C, with uniform elongations below 0.5%.61 Irradiations conducted in the high flux beam reactor (HFBR) at exposures of only 0.1 and 0.5 dpa corroborated these results and demonstrated a transition in the fracture mechanism between 300 and 400 C, resulting in a significant increase in ductility at temperatures above 400 C, Figure 28.62
1.04.8.2
Helium Effects
Helium is conveniently introduced into nickelbearing alloys through thermal neutron irradiation. Although helium is usually detrimental, especially if the material is to be subsequently welded,63 it offers a method to simulate the production of helium expected in the very hard spectrum of a fusion reactor. In the case of vanadium and other refractory metal alloys, the effect of helium has been studied using two primary methods of introduction of helium. One method is implantation of a-particles with an accelerator; the other is the use of the decay of tritium. Tritium rapidly diffuses into group V refractory metals at elevated temperatures. The elevated temperature serves more to dissolve the protective oxide layer than to accelerate the kinetics of dissolution. The tritium thus introduced is permitted to decay, by b-decay with a residual nucleus of 3 He. Helium-doped specimens have subsequently been neutron irradiated to study the synergistic effects of helium and atomic displacement damage. A limited number of experiments have used techniques to simultaneously implant He and produce atomic displacements through an irradiation environment of Li. The concept of introducing tritium into an irradiation capsule with the specimens in contact with lithium has been investigated to study vanadium
119
Effect of Radiation on Strength and Ductility of Metals and Alloys
12
140 0.0002
6 0.5 dpa (V1–V3)
4 2 0
Ttest = Tirr -2
0
100
200
300
Molybdenum
120
0.1 dpa V4
8
Ultimate tensile stress (1000 psi)
Uniform elongation (%)
10
400
500
600
100
700
40
0 60 Elongation (%)
alloys with the He/dpa ratio characteristic of a fusion environment. The tritium charge, the production of tritium from lithium, and the production of tritium from 3He are some of the important considerations in the design of the experiment.64 Although conceptually valid, the desired results have not yet been obtained with experiments of this type. Cyclotron-implanted helium has been used, also to study the effects of the fusion irradiation environment. Tanaka showed severe embrittlement with the introduction of 90 and 200 appm He at 700 C in V–20Ti.65 Grossbeck and Horak showed that a level of 80 at. ppm He implanted as part of the same experiment had no significant effect on elongation in V–15Cr–5Ti at 700 C.52 Braski also observed no significant effect on ductility in V–15Cr–5Ti at 600 C with similar levels of He introduced from decay of tritium.66 The alloys, Vanstar-7 and V–3Ti–1Si, were also investigated, in some cases with an improvement in ductility upon introduction of helium.66 Following irradiation, severe embrittlement was observed in V–15Cr–5Ti at 600 C in tritium trick samples by Braski66 and at 625 C in cyclotron-implanted samples by Grossbeck and Horak.52 Irradiation experiments with refractory metals, unless using a Li environment, frequently subject the specimens to contamination by interstitial impurities, also leading to embrittlement.52,67 Unalloyed molybdenum, Mo–0.5Ti, and Mo–50Re were irradiated in EBR-II by Wiffen at exposures of
0.0002
60
20
Test temperature (⬚C)
Figure 28 Uniform elongation of V–4Cr–4Ti irradiated in the high flux beam reactor. Reproduced from Snead, L. L.; Zinkle, S. J.; Alexander, D. J.; Rowcliffe, A. F.; Robertson, J. P.; Eatherly, W. S. Fusion Reactor Materials Semiannual Progress Report for Period Ending, Dec 31, 1997; DOE/ER-0313/23, p 81.
80
Controls 3.5 or 4.0 ´ 1022 n cm−2 at 455 ⬚C 6.1 ´ 1022 n cm−2 at 857–1136 ⬚C 4.0 ´ 1022 n cm−2 at 455 ⬚C + 1050 ⬚C anneal
40 20 0.0002 0 -200
0
200 400 Test temperature (⬚C)
0.0002 600
Figure 29 Ultimate tensile strength and total elongation for molybdenum in the unirradiated condition and irradiated in EBR-II to 20–30 dpa. Dashed lines connect results where irradiation conditions or strain rates were not constant. Reproduced from Wiffen, F. W. In Defects and Defect Clusters in B.C.C. Metals and Their Alloys, Nuclear Metallurgy; Arsenault, R. J., Ed.; National Bureau of Standards: Gaithersburg, MD, 1973; Vol. 18, p 176.
3.5–6.1 n cm2 (E > 0.1 MeV) (18–32 dpa).68 Although Mo alloys are known to exhibit increased ductility with increasing temperature in the unirradiated condition, at temperatures above 400–550 C, all three materials suffered plastic instability with uniform elongations below about 0.5%. This effect is shown in Mo in Figure 2968 where irradiation temperature is shown to be the critical parameter and where specimens irradiated at 455 and 1136 C were embrittled even in room temperature tests. This class of alloys is discussed further in Chapter 4.06, Radiation Effects in Refractory Metals and Alloys.
1.04.9 Amorphous Metals Stable metallic glasses may be produced, commonly in intermetallic compounds. Interest in the
Irradiation dose required for amorphization (dpa)
120
Effect of Radiation on Strength and Ductility of Metals and Alloys
16.0 14.0 12.0 40Ar irradiation
10.0 8.0 6.0 4.0 2.0 0
0
100
200 300 400 500 600 Irradiation temperature (K)
700
Figure 30 Irradiation displacement level as a function of temperature for 0.9 MeV electron and 0.5–1.5 MeV Ar ion irradiation. The family of curves is for several dpa rates of 1.04–1.83 mdpa s1. Reproduced from Howe, L. M.; Phillips, D.; Motta, A. T.; Okamoto, P. R. Surface Coatings Tech. 1994, 66, 411.
irradiation properties of this class of materials resulted from preliminary tests that showed that these materials actually became more ductile upon irradiation.69 Other intermetallic compounds have been shown to become amorphous upon irradiation. Although semiconductors such as Si and Ge are susceptible to amorphization under irradiation, the phenomenon is almost exclusively restricted to intermetallic compounds.70 To mention only a few, Zr3Al, Mo3Si, Nb3Ge, and Fe2Mo are compounds that have been studied in the amorphous state. Results and a detailed review of mechanisms and theories of amorphization have been published by Motta.70 In simple terms, the lattice disruption and defect generation from irradiation disrupts long-range order in the system. Thermal annealing competes with the disordering so that there is a critical temperature above which amorphization is not possible. Figure 30 shows a plot of the irradiation exposure necessary for amorphization as a function of temperature for Zr3Fe.71 The critical temperatures and the necessary exposures are both functions of the material as well as the impinging particle. Once formed, the amorphous phases are stable under irradiation, but the critical temperatures are typically lower than would be experienced for structural materials in nuclear systems. They are of interest, however, because some intermetallic phases, such as Fe2Mo and Fe3B found in commercial alloys, become amorphous under
irradiation.70,72 In the example of Zr3Fe, the critical temperature under argon ion irradiation is approximately 250 C, a temperature too low for most, but not all, reactors. The intermetallic alloys that can be produced in the amorphous state before irradiation are of more interest as potential structural materials, although they remain in the category of research interest at the present time. In addition to the increase in ductility upon irradiation, the absence of a crystalline structure with interacting dislocations was further incentive to investigate the irradiation properties of this class of materials. Metallic glasses containing boron, such as Fe40Ni40B20 and (Mo.6Ru.4)82B18 are a few examples, with the former receiving the most attention in terms of mechanical properties.69,73–75 Amorphous alloys are complex systems where changes in free volume and segregation into clusters of differing composition result in changes in behavior as irradiation proceeds. Investigation of the Fe–Ni–B alloy has shown that ductility first decreases and then increases with increasing fluence due to the competing effects of free volume and formation of regions of boron-depleted and boron-rich clusters.73 For sufficiently high fluences, the result is severe embrittlement. In the case of alloys based on the intermetallic Zr3Al, very severe embrittlement upon irradiation is attributed to the formation of new amorphous phases.76 Even though a crystal structure is absent, the atoms may be dislodged from their locations, creating additional free volume. Without the bonds present from a crystal lattice, the low binding energy results in high displacement levels for fluence levels that what would be considered low in crystalline alloys. Fluence levels in the range of 1016–1021 n cm2 have been investigated resulting in displacement levels exceeding 100 dpa. However, simply having similar displacement levels does not permit a true comparison with crystalline materials. Much research is necessary before this class of materials becomes of commercial importance.
1.04.10 Conclusions The relationship between the irradiation-induced microstructure and tensile properties has been briefly presented using representative classes of alloys. The austenitic stainless steels are an important class of alloys, and they are less complex than the martensitic steels. In the unirradiated condition, the austenitic
Effect of Radiation on Strength and Ductility of Metals and Alloys
alloys are primarily hardened by dislocation reactions leading to conventional work hardening, and the martensitic steels are hardened by phase transformations requiring careful heat treatments. The primary irradiation effects are similar, but they influence microstructure and, therefore, behavior in different ways. Both types of alloys have important applications in the nuclear field. Helium embrittlement might be the most important, considering the use of alloys in a neutron environment at high temperatures. For the proper conditions, helium can nearly always cause catastrophic failure. Repair welding of alloys with as little as 1–10 appm helium can lead to severe intergranular cracking. The refractory metals are useful for space reactor application because of their liquid metal compatibility and their high-temperature strength. Space reactors can lose heat only by thermal radiation, necessitating high temperatures. However, this class of alloys is most susceptible to embrittlement by interstitial impurities, and synergism of impurities with irradiation-induced defects is an area that must be addressed further. Amorphous alloys are a research curiosity in that they have interesting properties with respect to irradiation but little application at the present time. Any new class of alloys must be understood before it can be engineered, so research is the essential beginning.
References 1. 2. 3. 4. 5. 6. 7. 8. 9.
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Simons, R. L.; Hulbert, K. A. In Effects of Radiation on Materials; Garner, F. A., Perrin, J. S., Eds.; ASTM: Philadelphia, PA, 1985; Vol. II, p 820. Bement, A. L. In Proceedings on the Strength of Metals and Alloys, ASM: Petals Park, 1970; p 693. Koppenaal, T. J.; Kuhlmann-Wilsdorf, D. Appl. Phys. Lett. 1964, 4, 59. Taylor, G. I. Proc. Royal Soc. 1934, 145, 362. Seeger, A. Dislocations and Mechanical Properties of Crystals; Wiley: New York, 1957; p 243. Olander, D. R. Fundamental Aspects of Nuclear Reactor Fuel Elements; Technical Information Center, ERDA: Oak Ridge, TN, 1976; p 441. Grossbeck, M. L.; Maziasz, P. J.; Rowcliffe, A. F. J. Nucl. Mater. 1992, 191–194, 808. Kelly, P. M. Scripta Met. 1972, 6, 647. Fish, R. L.; Cannon, N. S.; Wire, G. L. In Effects of Radiation on Structural Materials; Sprague, J. A., Dramer, K., Eds.; ASTM: Philadelphia, PA, 1979; ASTM STP 683, p 450. Garner, F. A.; Hamilton, M. L.; Panayotou, N. F.; Johnson, G. D. J. Nucl. Mater. 1981, 103 & 104, 803. Grossbeck, M. L. J. Nucl. Mater. 1991, 179–181, 568. Maloy, S. A.; Toloczko, M. B.; McClellan, K. J. et al. J. Nucl. Mater. 2006, 356, 62.
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38. Klueh, R. L.; Vitek, J. M.; Grossbeck, M. L. J. Nucl. Mater. 1981, 103 & 104, 887. 39. Kasada, R.; Kimura, A.; Matsui, H.; Narui, M. J. Nucl. Mater. 1998, 258–263, 1199. 40. Gelles, D. S.; Hankin, G. L.; Hamilton, M. L. J. Nucl. Mater. 1997, 251, 188. 41. Shiba, K.; Suzuki, M.; Hishinuma, A.; Pawel, J. E. Effects of Radiation on Materials; ASTM: Philadelphia, PA, 1996; ASTM STP 1270, p 753. 42. Shiba, K.; Hishinuma, A. J. Nucl. Mater. 2000, 283–287, 474. 43. Kiumura, A.; Morimura, T.; Narui, M.; Matsui, H. J. Nucl. Mater. 1996, 233–237, 319. 44. Klueh, R. L.; Vitek, J. M.; Grossbeck, M. L. Effects of Radiation on Materials; ASTM: Philadelphia, PA, 1982; ASTM STP 782, p 648. 45. Mansur, L. K.; Grossbeck, M. L. J. Nucl. Mater. 1988, 155–157, 130. 46. Hasegawa, A.; Shiraishi, H.; Matsui, H.; Abe, K. J. Nucl. Mater. 1994, 212–215, 720. 47. Bae, K. K.; Ehrlich, K.; Mosalang, A. J. Nucl. Mater. 1992, 191–194, 905. 48. Jung, P.; Henry, J.; Chen, J. J. Nucl. Mater. 2005, 343, 275. 49. Jung, P.; Henry, J.; Chen, J.; Brachet, J. C. J. Nucl. Mater. 2003, 318, 241. 50. Grossbeck, M. L.; Heestand, R. L. Effect of irradiation on the tensile of niobium-base alloys. In Proceedings of Fourth Symposium on Space Nuclear Power Systems, Albuquerque, NM, Jan 15, 1987; CONF-870118, p 151. 51. Grossbeck, M. L.; Wiffen, F. W. In Space Nuclear Power Systems; El-Genk, M. S., Hoover, M. D., Eds.; Orbit Book Co.: Malabar, FL, 1986; Vol. III, p 85. 52. Grossbeck, M. L.; Horak, J. A. In Influence of Radiation on Material Properties; Garner, F. A., Henager, C. H., Igata, N., Eds.; ASTM: Philadelphia, PA, 1987; ASTM STP 956, p 291. 53. Wechsler, M. S.; Alexander, D. G.; Bajaj, R.; Carlson, O. N. In Defects and Defect Clusters in B.C.C. Metals and Their Alloys, Nuclear Metallurgy; Arsenault, R. J., Ed.; National Bureau of Standards: Gaithersburg, MD, 1973; Vol. 18, p 127. 54. Ohr, S. M.; Tucker, R. P.; Wechsler, M. S. Phys. Stat. Sol. A 1970, 2, 559. 55. Wiffen, F. W. Effects of irradiation on properties of refractory alloys with emphasis on space power reactor applications. In Proceedings of Refractory Alloy Technology for Space Nuclear Power Applications, USDOE, 1984; CONF-8308130, p 252. 56. DiStefano, J. R.; Hendricks, J. W. Nucl. Tech. 1995, 110, 145. 57. DiStefano, J. R.; DeVan, J. H. J. Nucl. Mater. 1997, 249, 150. 58. Grossbeck, M. L.; King, J. F.; Hoelzer, D. T. J. Nucl. Mater. 2000, 283–287, 1356.
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1.05 Radiation-Induced Effects on Material Properties of Ceramics (Mechanical and Dimensional) K. E. Sickafus University of Tennessee, Knoxville, TN, USA
ß 2012 Elsevier Ltd. All rights reserved.
1.05.1. 1.05.2. 1.05.2.1 1.05.2.2 1.05.2.3 1.05.2.3.1 1.05.2.3.2 1.05.2.3.3 1.05.2.3.4 1.05.2.3.5 1.05.2.3.6 1.05.2.3.7 1.05.2.3.8 1.05.2.3.9 1.05.2.3.10 1.05.2.3.11 1.05.2.4 1.05.3. 1.05.3.1 1.05.3.2 1.05.3.3 1.05.3.4 1.05.4. References
Introduction Radiation Effects in Ceramics: A Case Study – a-Alumina Versus Spinel Introduction to Radiation Damage in Alumina and Spinel Point Defect Evolution and Vacancy Supersaturation Dislocation Loop Formation in Spinel and Alumina Introduction to atomic layer stacking Charge on interstitial dislocations Lattice registry and stacking faults I: (0001) Al2O3 Lattice registry and stacking faults II: {111} MgAl2O4 Lattice registry and stacking faults III: {1010} Al2O3 Lattice registry and stacking faults IV: {110} MgAl2O4 Unfaulting of faulted Frank loops I: (0001) Al2O3 Unfaulting of faulted Frank loops II: {111} MgAl2O4 Unfaulting of faulted Frank loops III: {1010} Al2O3 Unfaulting of faulted Frank loops IV: {110} MgAl2O4 Unfaulting of faulted Frank loops V: experimental observations Amorphization in Spinel and Alumina Radiation Effects in Other Ceramics for Nuclear Applications Radiation Effects in Uranium Dioxide Radiation Effects in Silicon Carbide Radiation Effects in Graphite Radiation Effects in Other Ceramics Summary
Abbreviations dpa BF TEM i v ccp hcp SHI PKA CVD
Displacements per atom Bright-field Transmission electron microscopy Interstitial Vacancy Cubic close-packed Hexagonal close-packed Swift heavy ion Primary knock-on atom Chemical vapor deposition
1.05.1. Introduction Ceramic materials are generally characterized by high melting temperatures and high hardness values. Ceramics are typically much less malleable than
123 124 124 125 127 127 127 129 129 130 130 131 132 132 133 133 134 136 136 136 137 138 138 139
metals and not as electrically or thermally conductive. Nevertheless, ceramics are important materials in fission reactors, namely, as constituents in nuclear fuels, and are widely regarded as candidate materials for fusion reactor applications, particularly as electrical insulators in plasma diagnostic systems. These applications call for highly robust ceramics, materials that can withstand high radiation doses, often under very high-temperature conditions. Not many ceramics satisfy these requirements. One of the purposes of this chapter is to examine the fundamental mechanisms that lead to the relative radiation tolerance of a select few ceramic compounds, versus the susceptibility to radiation damage exhibited by most other ceramics. Ceramics are, by definition, crystalline solids. The atomic structures of ceramics are often highly complex compared with those of metals. As a consequence, we lack a detailed understanding of atomic 123
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processes in ceramics exposed to radiation. Nevertheless, progress has been made in recent decades in understanding some of the differences between radiation damage evolution in certain ceramic compounds. In this chapter, we examine the radiation damage response of a select few ceramic compounds that have potential for engineering applications in nuclear reactors. We begin by comparing and contrasting the radiation damage response of two particular (model) ceramics: a-alumina (Al2O3, also known as corundum in polycrystalline form, or ruby or sapphire in single crystal form) and magnesioaluminate spinel (MgAl2O4). Under neutron irradiation, alumina is highly susceptible to deleterious microstructural evolution, which ultimately leads to catastrophic swelling of the material. On the other hand, spinel is very resistant to the microscopic phenomena (particularly nucleation and growth of voids) that lead to swelling under neutron irradiation. We consider the atomic and microstructural mechanisms identified that help to explain the marked difference in the radiation damage response of these two important ceramic materials. The fundamental properties of point defects and radiation-induced defects are discussed in Chapter 1.02, Fundamental Point Defect Properties in Ceramics, and the effects of radiation on the electrical properties of ceramics are presented in Chapter 4.22, Radiation Effects on the Physical Properties of Dielectric Insulators for Fusion Reactors. It is important to be cognizant of the irradiation conditions used to produce a particular radiation damage response. Microstructural evolution can vary dramatically in a single compound, depending on the following irradiation parameters: (1) irradiation source–irradiation species and energies – these give rise to the so-called ‘spectrum effects,’ (2) irradiation temperature, (3) irradiation particle flux, and (4) irradiation elapsed time and particle fluence. Throughout this chapter, we pay particular attention to the variations in radiation damage effects due to differences in irradiation parameters. A single ceramic material can exhibit radiation tolerance under one set of irradiation conditions, while alternatively exhibiting damage susceptibility under another set of conditions. A good example of this is MgAl2O4 spinel. Spinel is highly radiation tolerant in a neutron irradiation environment but very susceptible to radiation-induced swelling when exposed to swift heavy ion (SHI) irradiation. Finally, it is important to note that radiation tolerance refers to two distinctly different criteria:
(1) resistance to a crystal-to-amorphous phase transformation; and (2) resistance to dislocation and void nucleation and growth. Both of these phenomena lead (usually) to macroscopic swelling of the material, but the causes of the swelling are completely different. The irradiation damage conditions that produce these two materials’ responses are also typically very different. We examine these two radiation tolerance criteria through the course of this chapter.
1.05.2. Radiation Effects in Ceramics: A Case Study – a-Alumina Versus Spinel 1.05.2.1 Introduction to Radiation Damage in Alumina and Spinel a-Al2O3 and MgAl2O4 are two of the most important engineering ceramics. They are both highly refractory oxides and are used as dielectrics in electrical applications (capacitors, etc.). Both a-Al2O3 and MgAl2O4 have been proposed as potential insulating and optical ceramics for application in fusion reactors.1–3 In a magnetically confined fusion device, these applications include (1) insulators for lightly shielded magnetic coils; (2) windows for radiofrequency heating systems; (3) ceramics for structural applications; (4) insulators for neutral beam injectors; (5) current breaks; and (6) direct converter insulators.3 Such devices in a fusion reactor environment will experience extreme environmental conditions, including intense radiation fields, high heat fluxes and heat gradients, and high mechanical and electrical stresses. A special concern is that under these extreme environments, ceramics such as a-Al2O3 and MgAl2O4 must be mechanically stable and resistant to swelling and concomitant microcracking. Over the last 30 years, many radiation damage experiments have been performed on a-Al2O3 and MgAl2O4 under high-temperature conditions by a number of different research teams. Figure 1 shows the results of one such study, where the high temperature, neutron irradiation damage responses of a-Al2O3 and MgAl2O4 are compared. The plot in Figure 1 was adapted from Figure 1 in an article by Kinoshita and Zinkle,4 based on experimental data obtained by Clinard et al. and Garner et al.5–8 The neutron (n) fluence on the lower abscissa in Figure 1 refers to fission or fast neutrons, that is, neutrons with energies greater than 0.1 MeV. Figure 1 also shows the equivalent displacement damage dose on the upper abscissa, in units of
Radiation-Induced Effects on Material Properties of Ceramics
5 4 Swelling (%)
Displacements per atom (dpa) 10 100
1
3
1000
a-Al2O3 1100 k 1025 k 925 k
2
MgAl2O4 658–1100 K
1 0 -1 1025
1026 1027 Neutron fluence (n m-2)
1028
Figure 1 Volume swelling versus neutron fluence in a-Al2O3 alumina and MgAl2O4 spinel.
displacements per atom (dpa). These dpa estimates are based on the approximate equivalence (for ceramics) of 1 dpa per 1025 n m2 (En > 0.1 MeV).9 Figure 1 shows a stark contrast between the radiation damage behavior, particularly the volume swelling behavior, between a-Al2O3 and MgAl2O4. Specifically, MgAl2O4 spinel exhibits no swelling in the temperature range 658–1100 K, for neutron fluences ranging from 3 1026 to 2.5 1027 n m2 (3–200 dpa). On the contrary, a-Al2O3 irradiated at temperatures between 925–1100 K exhibits significant volume swelling, ranging from 1 to 5% over a fluence range of 1 1025 to 3 1026 n m2 (1–30 dpa). The purpose of the following discussion is to reveal the reasons for the tremendous disparity in radiation-induced volume swelling between alumina and spinel. Figure 2 shows a bright-field (BF) transmission electron microscopy (TEM) image that reveals the microstructure of a-Al2O3 following fast neutron irradiation at T ¼ 1050 K to a fluence of 3 1025 n m2
c
100 nm Figure 2 Bright-field transmission electron microscopy image of voids formed in a-Al2O3 irradiated at 1050 K to a fluence of 3 1025 n m2 (3 dpa) (micrograph courtesy of Frank Clinard, Los Alamos National Laboratory).
125
(3 dpa). The micrograph reveals a high density of small voids (2–10 nm diameter), arranged in rows along the c-axis of the hexagonal unit cell for the a-Al2O3. When voids are arrayed in special crystallographic arrangements, as in Figure 2, the overall structure is referred to as a void lattice. Figure 2 shows the underlying explanation for the pronounced volume swelling of a-Al2O3 shown in Figure 1, namely the formation of a void lattice with increasing neutron radiation dose. This phenomenon is well known in many irradiated materials, both metals and ceramics, and is referred to as void swelling. Susceptibility to void swelling is a very undesirable material trait and basically disqualifies such a material from use in extreme environments (in this case, high temperature and high neutron radiation fields). It should be noted that TEM micrographs (not shown here) obtained from MgAl2O4 spinel irradiated under similar conditions to those in Figure 2 show no evidence of voids of any size. 1.05.2.2 Point Defect Evolution and Vacancy Supersaturation Voids are a consequence of a supersaturation of vacancies in the lattice and the tendency of excess vacancies to condense into higher-order defect complexes (either vacancy loops or voids). However, the root cause of void formation is actually not the vacancies, but the interstitials. Each atomic displacement event during irradiation produces a pair of defects known as a Frenkel pair. The constituents of a Frenkel pair are an interstitial (i) and a vacancy (v). Interstitials are more mobile than vacancies at most temperatures (at low-to-moderate temperatures, say less than half the melting point (0.5Tm), vacancies are essentially immobile in most materials), such that i-defects freely migrate around the lattice, while v-defects either remain stationary or move much smaller distances than i-defects. Because i-defects are highly mobile, they are able to diffuse to other lattice imperfections, such as dislocations, grain boundaries, and free surfaces, where they often are readily absorbed. This situation leads to a supersaturation of vacancies, that is, a condition in which the bulk vacancy concentration exceeds the complementary bulk interstitial concentration. This is a highly undesirable circumstance for a material exposed to displacive radiation damage conditions, because the v-defect concentration will continue to grow (unchecked) at approximately the Frenkel defect production rate, while the i-defect concentration
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will reach a steady-state concentration, determined by interstitial mobility and by the concentration of extended defects (extended defects presumably serve as sinks for interstitial absorption). The v-defect concentration will inevitably reach a critical stage at which the lattice can no longer support the excessive concentration of vacancies, at which point the v-defects will migrate locally and condense to form voids (or vacancy loops or clusters). This entire process, initiated by the supersaturation of vacancies, causes the material to undergo macroscopic swelling, and the material becomes susceptible to microcracking or failure by other mechanical mechanisms. This, indeed, is the fate suffered by a-Al2O3 when exposed to a neutron (displacive) radiation environment. It is interesting that a supersaturation of vacancies can even be established in a material devoid of extended defects, such as a high-quality single crystal or a very large-grained polycrystalline material. Single crystal a-Al2O3 (sapphire) is an example of just such a material.6 When freely migrating i-defects are unable to readily ‘find’ lattice imperfections such as grain boundaries and dislocations, they instead ‘find’ one another. Interstitials can bind to form diinterstitials or higher-order aggregates. Eventually, a new extended defect, produced by the condensation of i-defects, becomes distinguishable as an interstitial dislocation loop (also known as an interstitial Frank loop). Once formed, such a lattice defect acts as a sink for the absorption of additional freely migrating i-defects. With this, the conditions for a supersaturation of vacancies and macroscopic swelling are established. The defect situation just described can be conveniently summarized using chemical rate equations as described in detail in Chapter 1.13, Radiation Damage Theory. In eqn [1], we employ a simplified pair of rate equations to show the time-dependent fate of interstitials and vacancies produced under irradiation for an imaginary single crystal of A atoms: dCi ðt Þ dt
¼ Pi ðAA ! Ai þ VA Þ Riv ðAi þ VA ! AA Þ ½1a N ðnucleation rate for interstitial loopsÞ Gðgrowth rate for interstitial loopsÞ dCv ðt Þ dt
¼ Pv Riv
ðAA ! Ai þ VA Þ ðAi þ VA ! AA Þ
½1b
where Ci(t) and Cv(t) are the time-dependent concentrations of interstitials and vacancies, respectively; Pi and Pv are the production rates of interstitials and vacancies, respectively (equal to the Frenkel pair production rate); Ri–v is the recombination rate of
interstitials and vacancies (i.e., the annihilation rate of i and v point defects when they encounter one another in the matrix); N and G are the nucleation and growth rates, respectively, of interstitial loops; AA is an A atom on an A lattice site; Ai is an interstitial A atom; and VA is a vacant A lattice site (an A vacancy). (This equation for vacancies assumes low or moderate temperatures, such that vacancies are effectively immobile. Under high-temperature irradiation conditions, we would need to add nucleation and growth terms for voids, vacancy loops, or vacancy clusters. Reactions with preexisting defects are also ignored in eqn [1].) Note in eqn [1] that i–v recombination, Ri–v, is a harmless point defect annihilation mechanism (it restores, locally, the perfect crystal lattice). On the contrary, nucleation and growth (N and G) of interstitial loops are harmful point defect annihilation mechanisms, in the sense that these mechanisms leave behind unpaired vacancies in the lattice, thus establishing a supersaturation of vacancies, which is a necessary condition for swelling. It is interesting to compare and contrast the neutron radiation damage behavior shown in Figure 1 of alumina (a-Al2O3) and spinel (MgAl2O4) single crystals, in terms of the defect evolution described in eqn [1]. Alumina must be described as a highly radiation-susceptible material, due to its tendency to succumb to radiation-induced swelling. Spinel, on the other hand, is to be considered a radiationtolerant material, in view of its ability to resist radiation-induced swelling. According to eqn [1], we can speculate that mechanistically, nucleation and growth of interstitial dislocation loops are much more pronounced in alumina than in spinel. Also, eqn [1] suggests that harmless i–v recombination must be the most pronounced point defect annihilation mechanism in spinel so that a supersaturation of vacancies and concomitant swelling is avoided. Indeed, it turns out that nucleation and growth of dislocation loops are far more pronounced in alumina than in spinel, as discussed in detail next. The dislocation loop story described below is rich with the complexities of dislocation crystallography and dynamics. The unraveling of the mysteries of dislocation loop evolution in alumina versus spinel should be considered one of the greatest achievements ever in the field of radiation effects in ceramics, even though this was accomplished some 30 years ago! This story also illustrates the tremendous complexity of radiation damage behavior in ceramic materials, wherein point defects are created on both anion and cation sublattices, and where the defects generated often assume significant Coulombic charge states in
Radiation-Induced Effects on Material Properties of Ceramics
highly insulating ceramics (alumina and spinel are large band gap insulators). The earliest stages of the nucleation and growth of interstitial dislocation loops are currently impossible to interrogate experimentally. TEM has been used as a very effective technique for examining the structural evolution of dislocation in irradiated solids but only after the defect clusters have grown to diameters of about 5 nm. Interestingly, important changes in dislocation character probably occur in the early stages of dislocation loop growth, when loop diameters are only between 5 and 50 nm.10 Therefore, we must speculate about the nature of nascent dislocation loops produced under irradiation damage conditions. 1.05.2.3 Dislocation Loop Formation in Spinel and Alumina 1.05.2.3.1 Introduction to atomic layer stacking
Results of numerous neutron and electron irradiation damage studies suggest that two types of interstitial dislocation loops nucleate in a-Al2O3: (1) 1/3 [0001] (0001); and (2) 1=3h1011if1010g (see, e.g., the review by Kinoshita and Zinkle4). The first of these involves precipitation on basal planes in the hexagonal a-Al2O3 structure, while the second is due to precipitation on m-type prism planes. In MgAl2O4, similar studies indicate that primitive interstitial dislocation loops also have two characters: (1) 1/6 h111i {111} and (2) 1/4 h110i {110}.4 Though the crystal structure of spinel is cubic, compared with that of alumina, which is hexagonal, the nature of the dislocation loops formed in spinel is similar to those in alumina: {111} spinel loops are analogous to (0001) alumina loops; likewise, {110} spinel loops are analogous to f1010g alumina loops. We will first compare and contrast {111} spinel versus (0001) alumina loops and later discuss {110} spinel versus f1010g alumina loops. Both spinel h111i {111} and alumina [0001] (0001) interstitial dislocation loops involve insertion of extra atomic layers perpendicular to the h111i and [0001] directions, respectively. These layers are either pure cation or pure anion layers. In both spinel and alumina, anion layers along h111i and [0001] directions, respectively (i.e., along the 3 direction in both structures), are close packed (specifically, they are fully dense, triangular atom nets), while the cation layers contain ‘vacancies,’ which are necessary to accommodate the cation deficiency (compared with anion concentration) in both compounds (these ‘vacancies’ actually are interstices; they are ‘holes’ in the otherwise fully dense triangular atom nets
127
that make up each cation layer). Table 1 shows the arrangement of cation and anion layers in spinel and alumina, along h111i and [0001] directions, respectively.11 Both structures can be described by a 24-layer stacking sequence along these directions. Both spinel and alumina can be thought of as consisting of pseudo-close-packed anion sublattices, with cation layers interleaved between the anion layers. The anion sublattice in spinel is cubic close-packed (ccp) with an ABCABC. . . layer stacking arrangement, while alumina’s anion sublattice is hexagonal closepacked (hcp) with BCBCBC. . . layer stacking. In both structures, between each pair of anion layers there are three layers of interstices where cations may reside: a tetrahedral (t) interstice layer, followed by an octahedral (o) interstice layer, followed by another t layer. In spinel, Mg cations reside on t layers, while Al cations occupy the o layers. In alumina, all t layers are empty and Al occupies 2/3 of the o layer interstices. In spinel, cation interlayers alternate between a pure Al kagome´ layer and a mixed MgAlMg, threelayer thick slab. In alumina, each interlayer is pure Al in a honeycomb arrangement. 1.05.2.3.2 Charge on interstitial dislocations
In addition to spinel and alumina layer stacking sequences, Table 1 also shows the layer ‘blocks’ that have been found to comprise {111} and (0001) interstitial dislocation loops in spinel and alumina, respectively. An interstitial loop in spinel is composed of four layers such that the magnitude of the Burgers vector, b, along h111i is 1/6 h111i. The composition of each of these blocks has stoichiometry M3O4, where M represents a cation (either Mg or Al) and O is an oxygen anion. The upper 1/6 h111i block in Table 1 has an actual composition of Al3O4, while the lower 1/6 h111i block has a composition of Mg2AlO4. If Mg and Al cations assume their formal valences (2þ and 3þ, respectively), and O anions are 2, then the blocks described here are charged: (Al3O4)1þand (Mg2AlO4)1–. This may result in an untenable situation of excess Coulombic energy, as each molecular unit in the block possesses an electrostatic charge of 1 esu. It has been proposed that this charge imbalance is overcome by partial inversion of the cation layers in the 1/6 h111i blocks.12 (Inversion in spinel refers to exchanging Mg and Al lattice positions such that some Mg cations reside on o sites, while a similar number of Al cations move to t sites.) If a random cation distribution is inserted into either the upper or lower 1/6 h111i block shown in Table 1, then the block becomes charge neutral, that is, (MgAl2O4)x.
128
Layer #
24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0
Layer stacking of {111} planes along h111i in cubic spinel and (0001) planes along [0001] in hexagonal alumina Layer height
23/24 22/24 (11/12) 21/24 (7/8) 20/24 (5/6) 19/24 18/24 (3/4) 17/24 16/24 (2/3) 15/24 (5/8) 14/24 (7/12) 13/24 12/24 (1/2) 11/24 10/24 (5/12) 9/24 8/24 (1/3) 7/24 6/24 (1/4) 5/24 4/24 (1/6) 3/24 (1/8) 2/24 (1/12) 1/24 0/24 –1/24
Spinel (MgAl2O4) {111}-layer stacking along h111i direction
Alumina (a-Al2O3) (0001)-layer stacking along [0001] direction
O ¼ oxygen t ¼ tetrahedral interstices o ¼ octahedral interstices
Layer registry (ABCABC-type O-stacking)
Layer composition
O ¼ oxygen t ¼ tetrahedral interstices o ¼ octahedral interstices
t O t o t O t o t O t o t O t o t O t o t O t o t
C B A C B A C B A C B A C B A C B A C B A C B A C
– O4 Mg1 Al1 Mg1 O4 – Al3 – O4 Mg1 Al1 Mg1 O4 – Al3 – O4 Mg1 Al1 Mg1 O4 – Al3 –
Frank loop Burgers vectors
1/6<111> ¼ 4 layers (Al3O4)1þ
1/6<111> ¼ 4 layers (Mg2AlO4)1
t O t o t O t o t O t o t O t o t O t o t O t o t
Layer registry (BCBC-type O-stacking)
C a3 B a2 C a1 B a3 C a2 B a1
Layer composition
– O3 – Al2 – O3 – Al2 – O3 – Al2 – O3 – Al2 – O3 – Al2 – O3 – Al2 –
Frank loop Burgers vectors
1/3 [0001] ¼ 4 layers (excluding empty tetrahedral layers) (Al2O3)x
Radiation-Induced Effects on Material Properties of Ceramics
Table 1
Radiation-Induced Effects on Material Properties of Ceramics
129
Table 1 indicates that an (0001) interstitial dislocation loop in alumina consists of a four-layer block (excluding the empty t layers) such that the magnitude of the Burgers vector, b, along [0001] is 1/3 [0001]. The composition of each of these blocks is Al2O3, which is charge neutral, that is, (Al2O3)x. Thus, there are no Coulombic charge issues associated with interstitial dislocation loops along 3 in alumina. These dislocation loops consist simply of a pair of Al layers interleaved with two O layers.
specifically at the position of the red vertical line in the last sequence. Kronberg13 refers to this as an unsymmetrical electrostatic fault. This fault is seen to be intrinsic and only in the cation sublattice; the anion sublattice is undisturbed. In summary, the dislocation loop formed by 1/3 [0001] block insertion in alumina is an intrinsic, cation-faulted, interstitial Frank loop. This is also a sessile loop.
1.05.2.3.3 Lattice registry and stacking faults I: (0001) Al2O3
Now, we consider the formation of an interstitial dislocation loop along 3 in spinel. In spinel, O anion layers are fully dense triangular atom nets stacked in a ccp, ABCABC. . . geometry (A, B, and C are all distinct layer registries). Between adjacent O layers, 3/4 dense Al and MgAlMg layers are inserted, with registries labeled a, b, and c in Table 1 (a cations have the same registry as A anions; likewise, b same as B, c same as C). For stacking fault layer stacking assessments in spinel, it is conventional to simplify the layer notation for the cations (see, e.g., Clinard et al.6). The successive kagome´ Al layers are labeled a, b, g, while the MgAlMg mixed atom slabs are each projected onto one layer and labeled a0 , b0 , g0 . With these definitions, the registry of cation/anion stacking in spinel follows the sequence: a C b0 A g B a0 C b A g0 B. As with alumina, when extra pairs of cation and anion layers are inserted into the spinel 3 stacking sequence, a C b0 A g B a0 C b A g0 B, a fault in the stacking sequence is introduced. One can demonstrate how this works by inserting a 1/6 h111i b A block into the stacking sequence described above (this is equivalent to the upper Burgers vector for spinel shown in Table 1, which uses a kagome´ Al cation layer). We obtain:
Next, we must consider the lattice registry of the layer blocks inserted into alumina and spinel to form interstitial dislocation loops along 3. Registry refers to the relative translational displacements between successive layers in a stack. In alumina, O anion layers are fully dense triangular atom nets, stacked in an hcp, BCBCBC. . . geometry. B and C represent two distinct layer registries (displaced laterally with respect to one another). All the Al cation layers occur within the same registry, labeled a in Table 1 (a is displaced laterally relative to B and C). These Al layers are 2/3 dense, relative to the fully dense O layers, forming honeycomb atomic patterns. The successive Al layers are differentiated by where the cation ‘vacancies’ occur within each a layer. There are three possibilities that occur sequentially, hence the subscripted labels in Table 1 (a1, a2, a3).11 Thus, the registry of cation/anion stacking in alumina follows the sequence: a1 B a2 C a3 B a1 C a2 B a3 C. When extra pairs of Al and O layers are inserted into the stacking sequence, a1 B a2 C a3 B a1 C a2 B a3 C, a mistake in the stacking sequence is introduced. In other words, the dislocation loop formed by the block insertion is faulted (contains a stacking fault). Let us see how this works by inserting a 1/3 [0001] four-layer block, Al2O3Al2O3, into the stacking sequence described above. We obtain: ðbeforeÞ
a C b0 A g B a0 C b A g0 B b A a C b0 A g B
b0 A g B a0 C b A g0 B ðafter; showing stacking fault positionsÞ
a1 B a2 C a3 B a1 C a2 B a3 C a1 B a2 C a1 B a2 C a3 B a1 C a2 B a3 C ðafterÞ a1 a2 a3 a1 a2 a3 a1 a2 j a1 a2 a3 a1 a2 a3 ðafter; showing only cations and showing stacking fault positionÞ
a C b 0 A g B a0 C b A g 0 B a C b 0 A g B a0 C b A g0 B ðbeforeÞ a0 C b A g0 B ðafterÞ a C b 0 A g B a0 C b A g 0 B j b A j a C
a1 B a2 C a3 B a1 C a2 B a3 C a1 B a2 C a3 B a1 C a2 B a3 C
1.05.2.3.4 Lattice registry and stacking faults II: {111} MgAl2O4
½2
Notice in eqn [2] that after block insertion, the anion sublattice is not faulted (BCBC. . . layer stacking is preserved), whereas the cation sublattice is faulted,
½3
Notice in eqn [3] that after block insertion, both the anion sublattice (CABCAB. . . stacking is not preserved) and the cation sublattice are faulted. Also, notice that the cation and anion stacking sequences are faulted on both sides of the inserted b A block (the layer sequences are broken approaching the block from both the left and the right). Thus,
130
Radiation-Induced Effects on Material Properties of Ceramics
the b A block actually contains two stacking faults, on either side of the block. The positions of these stacking faults are denoted by vertical red lines in eqn [3]. The dislocation loop formed by 1/6 h111i block insertion in spinel is an extrinsic, cationþanion faulted, sessile interstitial Frank loop. We can also consider inserting a 1/6 h111i b0 A block into the spinel stacking sequence (i.e., the lower spinel Burgers vector shown in Table 1, which uses a mixed MgAlMg cation slab). We obtain: a C b0 A g B a0 C b A g0 B a C b0 A g B a0 C
ða1 BÞ ða2 CÞ ða3 BÞ ða1 CÞ ða2 BÞ ða3 CÞ ða1 BÞ ða2 CÞ ða3 BÞ ða1 CÞ ða2 BÞ ða3 CÞ ðbeforeÞ ða1 BÞ ða2 CÞ ða3 BÞ ða1 CÞ ða2 BÞ ða3 CÞ ða1 BÞ ða2 CÞ
b A g0 B ðbeforeÞ a C b0 A g B a0 C b A g0 B b0 A a C b0 A g B
ða1 BÞ ða2 CÞ ða3 BÞ ða1 CÞ ða2 BÞ ða3 CÞ ðafterÞ ða1 BÞ ða2 CÞ ða3 BÞ ða1 CÞ ða2 BÞ ða3 CÞ ða1 BÞ ða2 CÞ j
a0 C b A g0 B ðafterÞ a C b 0 A g B a 0 C b A g 0 B j b0 A j a C b0 A g B a0 C b A g0 B ðafter; showing stacking fault positionsÞ
anion and cations together, we can write the {1010} stacking sequence in alumina as (a1B) (a2C) (a3B) (a1C) (a2B) (a3C). Now, as with the basal plane story described earlier, when an extra 1=3h1010i two-layer block, (Al2O3)x(Al2O3)x, is inserted into the stacking sequence, (a1B) (a2C) (a3B) (a1C) (a2B) (a3C), a stacking fault occurs as follows:
ða1 BÞ ða2 CÞ ða3 BÞ ða1 CÞ ða2 BÞ ða3 CÞ ðafter; showing stacking fault positionÞ ½4
Once again, both the anion and cation sublattices are faulted, and we obtain an extrinsic, cationþanion faulted, sessile interstitial Frank loop. 1.05.2.3.5 Lattice registry and stacking faults III: {1010} Al2O3
So far we have considered Coulombic charge and faulting for 1/3 [0001] (0001) loops in alumina and 1/6 h111i {111} loops in spinel. Now, we must repeat these considerations for 1=3h1011if1010g prismatic loops in alumina and 1/4 h110i {110} loops in spinel. We begin with alumina prismatic loops. Alumina {1010} prism planes contain both Al and O in the ratio 2:3, that is, identical to the Al2O3 compound stoichiometry. Along the h1010i direction normal to the traces of the {1010} planes, the registry of the {1010} planes varies between adjacent planes, analogous to the registry shifts that occur between adjacent (0001) basal planes in alumina (discussed earlier). However, the patterns of Al atoms in all {1010} planes are identical. Similarly, the O atom patterns are identical in all {1010} planes. The registry of the O atom patterns between adjacent {1010} planes alternates every other layer, analogous to the BCBC. . . stacking of oxygen basal planes (Table 1, eqns [2–4]). On the other hand, the registry of the Al cation patterns is distinct from the O pattern registries (B and C), and the registry of the Al patterns only repeats every fourth layer. In other words, the stacking sequence of {1010} plane Al atom patterns can be described using the same nomenclature as in Table 1 and eqn [2] for (0001) alumina planes, that is, a1 a2 a3 a1 a2 a3. . .. Putting the
½5
Notice in eqn [5] that after block insertion, the anion sublattice is not faulted (BCBC. .. layer stacking is preserved), whereas the cation sublattice is faulted, specifically at the position of the red vertical line in the last sequence. Similar to the case of basal plane interstitial loop formation in alumina (discussed earlier), the dislocation loop formed by 1=3h1010i block insertion in alumina is an intrinsic, cation-faulted, sessile interstitial Frank loop. 1.05.2.3.6 Lattice registry and stacking faults IV: {110} MgAl2O4
Next, we consider 1/4 h110i {110} loops in spinel. Spinel {110} planes alternate in composition, (AlO2)(MgAlO2)þ. . ., such that each layer is a mixed cation/anion layer. To insert a charge-neutral interstitial slab along h110i in spinel requires that we insert a {110} double-layer block, (AlO2)(MgAlO2)þ, that is, a stoichiometric MgAl2O4 unit. The thickness of this slab is a/4 h110i, where a is the spinel cubic lattice parameter. Along the h110i direction normal to the traces of the {110} planes, the registry of the {110} planes varies between adjacent planes, analogous to the registry shifts that occur between adjacent {111} planes in spinel (discussed earlier). The O atom patterns are identical in all {110} planes, but the registry of the O atom patterns between adjacent {110} planes alternates every other layer, analogous to the BCBC. . . stacking described earlier. The Mg atom patterns are identical in each (MgAlO2)þ layer, while the registry of the Mg atom patterns alternates every other (MgAlO2)þ layer. We denote the Mg stacking sequence by a1 a2 a1 a2 . . .. There are two Al atom patterns along h110i: (1) the first
Radiation-Induced Effects on Material Properties of Ceramics
occurs in each (AlO2) layer with no change in registry between layers (we denote this Al pattern by b0 ); and (2) the second occurs in each (MgAlO2)þ layer, and the registry of these Al atom patterns alternates every other (MgAlO2)þ layer (we denote this Al stacking sequence by b1 b2 b1 b2 . . .). Combining all these considerations, we can write the {110} planar stacking sequence in spinel as follows: (b0 B) (a1b1C) (b0 B) (a2b2C). Now, as with the spinel {111} case described earlier, when an extra 1/4 h110i two-layer block, (AlO2)(MgAlO2)þ, is inserted into the spinel {110} stacking sequence, (b0 B) (a1b1C) (b0 B) (a2b2C), a stacking fault occurs as follows:
c
c 500 nm
ðb0 BÞ ða1 b1 CÞ ðb0 BÞ ða2 b2 CÞ ðb0 BÞ ða1 b1 CÞ ðb0 BÞ ða2 b2 CÞ 0
ðbeforeÞ 0
0
ðb BÞ ða1 b1 CÞ ðb BÞ ða2 b2 CÞ ðb BÞ ða1 b1 CÞ ðb0 BÞ ða1 b1 CÞ ðb0 BÞ ða2 b2 CÞ 0
0
ðafterÞ 0
ðb BÞ ða1 b1 CÞ ðb BÞ ða2 b2 CÞ ðb BÞ ða1 b1 CÞ j ðb0 BÞ j ða1 b1 CÞ ðb0 BÞ ða2 b2 CÞ ðafter; showing stacking fault positionsÞ
131
½6
Notice in eqn [6] that after block insertion, the anion sublattice is not faulted (BCBC. . . layer stacking is preserved), whereas the cation sublattice is faulted, specifically at the positions of the red vertical lines in the last sequence (the left-hand red line corresponds to the cation fault position for cation planar registries moving from right to left; likewise, the right-hand red line corresponds to the cation fault position for cation planar registries moving from left to right). Thus, the dislocation loop formed by 1/4 h110i twolayer block insertion in spinel is an extrinsic, cationfaulted, sessile interstitial Frank loop. Figure 3 shows an example of 1/4 h110i interstitial dislocation loops in spinel, produced by neutron irradiation.12 The alternating black-white fringe contrast within the loops is an indication of the presence of a stacking fault within the perimeter of each loop. The character of the {110} loops was determined by Hobbs and Clinard using the TEM imaging methods of Groves and Kelly,14,15 with attention to the precautions outlined by Maher and Eyre.16 These loops were determined to be extrinsic, faulted 1/4 h110i {110} interstitial dislocation loops. It is evident in Figure 3 that the extrinsic fault associated with these loops is not removed by internal shear, even when the loops grow to significant sizes (>1 mm diameter). This is the subject of our next topic of discussion, namely, the unfaulting of faulted Frank loops.
Figure 3 Bright-field transmission electron microscopy (TEM) image of {110} faulted interstitial loops in MgAl2O4 single crystal irradiated at 1100 K to a fluence of 1.9 1026 n m2 (20 dpa). Reproduced from Hobbs, L. W.; Clinard, F. W., Jr. J. Phys. 1980, 41(7), C6–232–236. The surface normal to the TEM foil is along h111i. The dislocation loops intersect the top and bottom surfaces of the TEM foil, which gives them their ‘trapezoidal’ shapes. The areas marked ‘C’ in the micrograph are regions where a ‘double-layer’ loop has formed, that is, a second Frank loop has condensed on planes adjacent to the preexisting faulted loop.
1.05.2.3.7 Unfaulting of faulted Frank loops I: (0001) Al2O3
In principle, faulted interstitial Frank loops can unfault by dislocation shear reactions. This should occur at a critical stage in interstitial loop growth, when the energy of the faulted dislocation loop, with a relatively small Burgers vector, becomes equal to an equivalently sized, unfaulted dislocation loop, with a larger Burgers vector. (In the absence of a stacking fault, the energy of a dislocation scales as b2, where b is the magnitude of the Burgers vector.) From this critical point on, the energy cost to incrementally grow the size of a dislocation loop favors the unfaulted loop, since there is no cost in energy due to a stacking fault within the loop perimeter. We examine first the unfaulting of 1/3 [0001] (0001) loops in alumina. To unfault a 1/3 [0001] (0001) dislocation loop in alumina, we must propagate a 1=3½1010 partial shear dislocation across the loop plane.6 This is described by the following dislocation reaction: 1 3½0001
þ 13½1010 !
1 3½1011
faulted loop
partial shear
unfaulted loop
ðbasalÞ
½7
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Radiation-Induced Effects on Material Properties of Ceramics
After propagating the partial shear through the loop, we are left with an unfaulted layer stacking sequence. The Burgers vector of the resultant dislocation loop, 1=3½1011, is a perfect lattice vector; therefore, the newly formed dislocation is a perfect dislocation. The resultant 1=3½1011 (0001) dislocation is a mixed dislocation, in the sense that the Burgers vector is canted (not normal) relative to the plane of the loop. 1.05.2.3.8 Unfaulting of faulted Frank loops II: {111} MgAl2O4 C
1/3 [1011]
1/3 [0001]
1/3 [1010] Figure 4 Alumina cation ‘honeycomb’ atom nets along the c-axis in Al2O3 corundum (anion sublattice not shown here). The black circles represent Al atoms. The gray squares represent cation ‘vacancies.’ The diagram shows the Burgers vectors involved in the partial shear unfaulting reactions for interstitial dislocation loops in alumina. Adapted from Howitt, D. G.; Mitchell, T. E. Philos. Mag. A 1981, 44(1), 229–238.
This reaction is shown graphically in Figure 4. Note that the magnitude of 1=3½1010 is approximately the Al–Al (and O–O) first nearest-neighbor spacing in Al2O3. When we pass a 1=3½1010 shear through a 1/3 [0001] (0001) dislocation loop, the cation planes beneath the loop assume new registries such that in eqn [2], a1, a2, and a3 commute as follows: a1 ! a3 ! a2 ! a1 . The anion layers beneath the loop are left unchanged (B ! B, C ! C). Taking the faulted (0001) stacking sequence in eqn [2] and assuming that the planes to the right are above the ones on the left, we perform the 1=3½1010 partial shear operation as follows: a1 B a2 C a3 B a1 C a2 B a3 C a1 B a2 |C a1 B a2 C a3 B a1 C a2 B a3 C
(faulted)
C a3 B a1 C a2 B a3 C a1 B a2 C a1 B a2 C a3 B a1 C a2 B a3 C a1 B a2 C a3 B a1 C a2 B a3 C a1 B a2 C
(unfaulted)
[8]
For this particular dislocation loop, it is thought that rather than unfaulting, 1/6 h111i {111} dislocations simply dissolve back into the lattice, in favor of the more stable 1/4 h110i {110} loops.12 As discussed earlier, the 1/6 h111i {111} dislocation can be presumed to be relatively unstable because it possesses both anion and cation faults, and in addition, it cannot preserve stoichiometry or charge balance in either normal or inverse spinel.12 Counter to this argument is the idea that if a 1/6 h111i {111} dislocation loop incorporates a partial inversion of its cation content, then this loop could be made both stoichiometric and charge neutral. Such a dislocation would arguably be more stable. However, {111} loops are never observed to grow very large (<100 nm) and are altogether absent in spinel samples irradiated at 1100 K.17 Therefore, it is likely that ‘disordered’ {111} interstitial loops are not an important aspect of radiation damage evolution in spinel. 1.05.2.3.9 Unfaulting of faulted Frank loops III: {1010} Al2O3
To unfault a 1=3½1010ð1010Þ dislocation loop in alumina, we must propagate a 1/3 [0001] partial shear dislocation across the loop plane.17 This is described by the following dislocation reaction: 1 3½1010
þ 13½0001 !
1 3½1011
faulted loop
partial shear
unfaulted loop
ðprismaticÞ
½9
This reaction is symmetric with that shown in eqn [8] for unfaulting a (0001) basal loop in alumina. The reaction in eqn [9] is shown graphically in Figure 4. When we pass a 1/3 [0001] shear through a 1=3½1010ð1010Þ dislocation loop, the cation planes beneath the loop assume new registries such that in eqn [6], a1b1 and a2b2 commute as follows: a1 b1 ! a2 b2 ! a1 b1 . The anion layers beneath the loop are left unchanged (B ! B, C ! C). In addition, the Al-only cation layers are unchanged (b0 ! b0 ). Taking the faulted (0001) stacking sequence in eqn [2] and assuming that the planes to the right
Radiation-Induced Effects on Material Properties of Ceramics
133
are above the ones on the left, we perform the 1/3 [0001] partial shear operation as follows:
(b⬘B) ( a1b1C) ( b⬘B) ( a2b2C) (b⬘B) ( a1b1C) |(b⬘B)| (a1b1C ) (b⬘B) ( a2b2C) (faulted) b⬘B a2b2C b⬘B a1b1C
(β⬘B) ( α1β1C) ( β⬘B) ( α2β2C) (β⬘B) ( α1β1C) |(β⬘B)| (α1β1C) ( β⬘B) ( α2β2C) (faulted)
(b⬘B) ( a1b1C) ( b⬘B) ( a2b2C) (b⬘B) ( a1b1C) (b⬘B) (a2b2C ) ( b⬘B) (a1b1C )
(α2β2C) ( β⬘B) ( α1β1C) (β⬘B) ( α1β1C) ( β⬘B) ( α2β2C) (β⬘B) ( α1β1C) (β⬘B) (α2β2C) ( β⬘B) ( α1β1C )
After propagating the partial shear through the loop, we are left with an unfaulted layer stacking sequence. The resultant dislocation loop, 1=3½1011, is a perfect dislocation. The resultant 1=3½1011 (0001) dislocation is a mixed dislocation, in the sense that the Burgers vector is canted (not normal) relative to the plane of the loop. 1.05.2.3.10 Unfaulting of faulted Frank loops IV: {110} MgAl2O4
To unfault a 1/4 [110] (110) dislocation loop in spinel, we must propagate a 1=4½112 partial shear dislocation across the loop plane.12 This is described by the following dislocation reaction: 1 4½110 faulted loop
þ
1 4½112
!
partial shear
1 2½101
½11
unfaulted loop
This reaction is shown graphically in Figure 5. When we pass a 1=4½112 shear through a 1/4 [110] (110) dislocation loop, the atomic planes beneath the loop assume new registries, such that in eqn [6], a1b1 and a2b2 commute as follows: a1 b1 ! a2 b2 ! a1 b1 . The anion layers beneath the loop are left unchanged (B ! B, C ! C). Also the Al b0 layers are left unchanged (b0 ! b0 ). Taking the faulted (110) stacking sequence in eqn [6] and assuming that the planes to the right are above the ones on the left, we perform the 1=4½112 partial shear operation as follows:
[0 0 1]
z
[1 0 0]
1/4 [112]
x 1/2[101] a [0 1 0]
y
[12]
(unfaulted)
[10]
1/4 [110]
Figure 5 Spinel unit cell showing the Burgers vectors involved in the partial shear unfaulting reaction for interstitial dislocation loops in spinel. The blue circles represent Mg atoms (Al and O are not shown here).
(unfaulted)
After propagating the partial shear through the loop, we are left with an unfaulted layer stacking sequence. The Burgers vector of the resultant dislocation loop, 1=2½101, is a perfect lattice vector; therefore, the newly formed dislocation is a perfect dislocation (equal to the Mg–Mg first nearestneighbor spacing). The resultant 1=2½101ð110Þ dislocation is a mixed dislocation, in the sense that the Burgers vector is canted (not normal) relative to the plane of the loop. 1.05.2.3.11 Unfaulting of faulted Frank loops V: experimental observations
Having examined the crystallography of unfaulting reactions in alumina and spinel, it is interesting now to compare and contrast what is experimentally observed so far as dislocation loop evolution in irradiated Al2O3 versus MgAl2O4 is concerned. First, it is observed that the 1/3 [0001] (0001) basal loops and 1=3h1010if1010g prismatic loops readily unfault under irradiation, by the reactions shown in eqns [7,8] and [9,10], respectively.6 These reactions occur when the loops reach 50 nm diameter,10 and each reaction produces an unfaulted loop with a 1=3h1011i perfect Burgers vector. Once formed, these unfaulted loops grow without bound until they intersect other growing dislocation loops, ultimately forming a dislocation network. Such a dislocation network in neutron irradiated Al2O3 is shown in Figure 6. Once the dislocation network in irradiated alumina is formed, it has been demonstrated that the product dislocations within the network are free to climb.6 The continuous climb of network dislocations in Al2O3 provides unsaturable sinks for Al and O interstitials arriving in stoichiometric proportions. All the conditions for a substantial supersaturation of vacancies are now in place. Al and O interstitials are readily absorbed at network dislocations, leaving behind numerous unpaired Al and O vacancies in the lattice. These unpaired vacancies inevitably condense to form voids. Under these conditions, void swelling must be the anticipated radiation response of the material. Contrast the evolution described above for alumina to the observed microstructural evolution in spinel. The predominant 1/4 h110i {110}
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Radiation-Induced Effects on Material Properties of Ceramics
100 10 0 nm
Figure 6 Weak-beam dark-field transmission electron micrograph showing the dislocation network formed in Al2O3 following neutron irradiation at 1015 K to a fluence of 3 1025 n m2 (3 dpa). Reproduced from Clinard, F. W., Jr.; Hurley, G. F.; et al. J. Nucl. Mater. 1982, 108/109, 655–670.
dislocations in spinel do not unfault under most experimental conditions tested to date.6 (Kinoshita et al.51 observed unfaulted 1/2 h110i {110} perfect loops in MgAl2O4 single crystals following neutron irradiations in the JOYO fast breeder test reactor (fast neutron fluences up to 6.5 1025 n m2 (equivalent to 6.5 dpa), and temperatures between 673 and 873 K). Kinoshita et al.51 also proposed a growth process of loops in spinel as follows: 1/6 [111] (111) 1/4 [110] (111) 1/4 [110] (101) 1/4 [110] (110) 1/2 [110] (110). Notice that this sequence ends in an unfaulted, perfect interstitial loop. This final loop configuration should be a good sink for interstitials, thus promoting a supersaturation of vacancies in the lattice. However, in neutron irradiations of MgAl2O4 single crystals in the fast flux test facility (FFTF), no evidence for 1/2 [110] (110) perfect dislocations was found, for neutron fluences ranging from 2.2 1026 to 2.17 1027 n m2 (equivalent to 22–217 dpa) in the temperature range 658–1023 K.51 Therefore, the proposed progression of spinel interstitial loop characteristics described above has, to date, been confirmed only under the JOYO irradiation conditions reported by Kinoshita et al.51) According to Clinard et al.:6 Persistence of the 1/4 h110i {110} stacking fault amounts to a failure of a 1=4h112i partial dislocation to nucleate, sweep across the loop plane, and so remove the fault.
The reason for this failure is paradoxical. Apparently, stacking fault energy cannot be the reason. The stacking fault energy estimate for 1/4 h110i {110} stacking faults in spinel, 180 mJ m2,18 is similar to the energy estimates for 1/3 [0001] (0001) and 1=3h1010if1010g stacking faults in alumina (320 and 750 mJ m2, respectively).10 Therefore, there seems to be a reasonable ‘driving force’ available to favor unfaulting of 1/4 h110i {110} stacking faults in spinel. Perhaps the explanation is simply that the magnitude of the partial shear vector required to unfault the faulted loops is prohibitively large. In spinel, the magnitude of the unfaulting 1=4h112i vector is 5 A˚, compared with the 1/3 [0001] (4.32 A˚) and 1=3h1010i (2.74 A˚) unfaulting vectors in alumina. Whatever the reason, spinel 1/4 h110i {110} stacking faults do not unfault, and this leads to void swelling resistance and impressive inherent radiation tolerance in spinel compared alumina. Hobbs and Clinard summarize the situation as follows: The absence of void swelling h in spinel i can be attributed to the failure of the loops to unfault and develop into dislocation networks; they therefore remain less than perfect interstitial sinks since the energy per added interstitial never drops below the fault energy. Vacancy-interstitial recombination thus remains the dominant mode of defect accommodation, and saturating defect kinetics inevitably ensue.
Therefore, in conclusion, the significant swelling of Al2O3 alumina at high temperatures is attributable to the unfaulting of interstitial dislocation loops and the subsequent formation of dislocation networks, which serve as efficient sinks for the absorption of interstitial atoms. This leaves behind a supersaturation of lattice vacancies, that is, an excess of unpaired vacancies in the bulk of the Al2O3. In irradiated MgAl2O4, only high-energy faulted loops are available as sinks for interstitials. Therefore, in this case, interstitial–vacancy (i–v) recombination is the dominant mechanism for defect accommodation, and negligible swelling results. 1.05.2.4 Amorphization in Spinel and Alumina Another response of materials to irradiation, not discussed up to now, is radiation-induced amorphization. Amorphization is a structural phase transformation from a crystalline solid to a solid that lacks any long-range order. Typically, the material still maintains a certain degree of short-range order, but as far
Radiation-Induced Effects on Material Properties of Ceramics
as diffraction techniques can discern, any long-range, crystalline order is destroyed following an amorphization transformation. Amorphization is a metastable process in which material is forced into a glass-like structure, which under thermodynamic equilibrium would be a prohibited structure. Amorphization transformations are most prevalent under ambient or low temperature irradiation conditions, such that kinetic recovery mechanisms are not effective at annihilating atomic displacements produced by irradiation. Typically, above a critical temperature, amorphization can be avoided in an otherwise amorphizable material, due to thermal recovery processes. Amorphization transformations can occur under both ballistic (displacive) and electronic (SHI) radiation damage conditions. Under ballistic conditions and depending on the material, amorphization can either occur within a single primary knock-on (PKA) ion track (or other irradiating particle track), or proceed through the accumulation of defects due to overlapping of damage tracks. Amorphization tends to be detrimental to materials employed in radiation environments, because the crystal-to-amorphous transformation is usually accompanied by significant volume swelling, mechanical softening, and microcracking, to name but a few deleterious effects. In ceramic materials, tendencies to radiationinduced amorphization are strongly dependent on crystal structure and chemistry, with the vast majority of ceramics exhibiting significant susceptibility to amorphization. One of the key properties that has been correlated quite well to amorphization resistance is ionicity: highly ionic compounds tend to resist amorphization; highly covalent compounds tend to readily succumb to amorphization at relatively low doses.19 Both spinel and alumina are relatively ionic compounds, but interestingly both can be amorphized by both ballistic and electronic damage mechanisms. Single crystal MgAl2O4 spinel was found to amorphize under ballistic ion irradiation conditions at a peak displacement damage level of 25 dpa (100 K irradiation temperature, 400 keV Xe2þions).20 A similar result was obtained under in situ ion irradiation conditions (30 K irradiation temperature, 1.5 MeV Xeþ ions).21 The critical temperature, Tc, for amorphization of spinel, has yet to be determined, but it is likely to be well below room temperature. (Only below Tc can the material be fully amorphized. Above Tc, kinetic recovery dominates and the material is partially to fully crystalline.) Single crystal a-Al2O3 (sapphire) has been observed to amorphize by a ballistic damage
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dose of about 3.8 dpa (20 K irradiation temperature, 1.5 MeV Xeþ ions, in situ).22 This is a significantly smaller amorphization dose than that for spinel irradiated under similar conditions. The critical temperature, Tc, for amorphization of a-Al2O3 was estimated to be about 170 K. In both alumina and spinel, the radiation-induced amorphization transformation does not occur by direct, ‘in-cascade’ amorphization but by damage accumulation by overlapping cascades (damage tracks). Presumably, neither a-Al2O3 nor MgAl2O4 can be amorphized at ambient temperature or above using displacive radiation damage conditions. However, there is a report of amorphization of a-Al2O3 at a ballistic damage dose of 3–7 keV per atom.23 Under SHI irradiation conditions, where electronic stopping predominates over nuclear stopping, both alumina and spinel undergo amorphization transformations, with significant concomitant volume swelling. In both materials, the transformation does not initiate until ion tracks are overlapped. In polycrystalline a-Al2O3, the threshold for amorphization was found to be at an accumulated electronic energy deposition of about 1.5 GGy (85 MeV I7þ ions at ambient temperature; amorphization was found to a depth of 4.5 mm, corresponding to energy deposition cross-sections ranging from 5 to 20 keV nm1 per ion.24 In single crystal sapphire irradiated under similar conditions (90.3 MeV 129Xe at room temperature), amorphization was found to initiate at the sample surface at an accumulated electronic energy deposition of about 0.3 GGy.25 These authors also observed a correlation between swelling (as measured by surface ‘pop-out’) and amorphization. However, the swelling values obtained from their measurements are too large to be realistic (more than 50% volume swelling). Nevertheless, the swelling associated with SHI radiation-induced amorphization in alumina is substantial. Matzke26 observed 30% free swelling in Al2O3 irradiated at 420 K with 72 MeV Iþ ions to fluences ranging from 1019 to 1021 ions m2 (5–500 GGy at the sample surface). In MgAl2O4, amorphization and significant surface pop-out were observed in SHI irradiations at 370 K using 72 MeV Iþ ions.27 The ion fluences where pop-out was observed were 1 1019 and 5 1019 ions m2 (5.3 and 27 GGy, respectively, at the sample surface). The volumetric swelling associated with this crystal-to-amorphous phase transformation was estimated to be 35%.28 In summary, huge volume changes appear to be associated with SHI amorphization transformations in model ceramics such as spinel and alumina.
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Radiation-Induced Effects on Material Properties of Ceramics
This concludes the comparison and contrast of radiation damage effects in two model ceramic oxides, namely, a-Al2O3 alumina and MgAl2O4 spinel. To make this chapter on radiation effects in nuclear reactor relevant materials as comprehensive as possible, we offer in the following section some notes on additional ceramic materials that are either important currently in nuclear reactor applications or have potential as advanced nuclear reactor materials with respect to future applications. In particular, we consider three representative ceramic materials, namely, urania, silicon carbide, and graphite. Subsequent chapters treat these materials in more detail: uranium dioxide (Chapter 2.02, Thermodynamic and Thermophysical Properties of the Actinide Oxides; Chapter 2.17, Thermal Properties of Irradiated UO2 and MOX; and Chapter 2.18, Radiation Effects in UO2), SiC (Chapter 2.12, Properties and Characteristics of SiC and SiC/SiC Composites and Chapter 4.07, Radiation Effects in SiC and SiC-SiC), and graphite (Chapter 2.10, Graphite: Properties and Characteristics; Chapter 4.10, Radiation Effects in Graphite; Chapter 4.11, Graphite in Gas-Cooled Reactors; and Chapter 4.18, Carbon as a Fusion Plasma-Facing Material).
1.05.3. Radiation Effects in Other Ceramics for Nuclear Applications In this section, we discuss briefly some important radiation effects in a few ceramics that are used in nuclear reactor applications. We will consider three representative ceramic materials: (1) urania (UO2); (2) silicon carbide (SiC); and (3) graphite (C). Unfortunately, we cannot provide a thorough review of the radiation damage studies that have been performed on many hundreds of other nonmetallic solids. 1.05.3.1 Dioxide
Radiation Effects in Uranium
UO2 is an important nuclear material because it is the fuel form of choice for conventional light water reactors. Unfortunately, our knowledge of pure radiation effects in UO2 is somewhat limited. This is because 235 U undergoes fission in a thermal neutron radiation environment, and consequently, the radiation response of UO2 is dictated by chemical evolutionary effects rather than by conventional, point defect condensation effects. Swelling in UO2 during service as a nuclear fuel can be significant; several percent for some
percent burnup of heavy metal.29 This swelling is primarily due to the accumulation of gaseous fission products as well as to some degree, solid fission products. It is not believed that UO2 is susceptible to void swelling (described in previous sections). Under ballistic radiation damage conditions, UO2 exhibits polygonization, that is, a grain subdivision process in which UO2 grains initially 10 mm diameter subdivide into 104 to 105 new small grains of 0.2 mm size.30 These authors demonstrated that polygonization is initiated at a critical ballistic damage dose, apparently independent of temperature. In particular, irradiation of single crystal UO2 with 300 keV Xe ions at 77 K, 300 K, and 773 K, to a fluence of 4 1020 Xe m2 or higher, produces the polygonization transformation.30 However, these authors concluded that this transformation cannot be due to radiation damage alone but is probably also related to the implanted impurity atoms (Xe), which reach a concentration of 5–7% at the critical fluence described above. Despite the polygonization transformation in UO2, no amorphization transformation, induced by ballistic damage conditions, has ever been observed. Under SHI (electronic stopping) irradiation damage conditions, once again amorphization was not observed (even with overlapped ion tracks), and the SHI-induced swelling is negligible.31 These experiments included numerous ion species (Zn, Mo, Cd, Sn, Xe, I, Pb, Au, and U) and energies ranging from 72 MeV to 2.7 GeV. Latent tracks were visible by TEM for electronic stopping powers greater than 29 keV nm1, but all tracks were crystalline. Lattice parameter expansion and polygonization were also observed. 1.05.3.2 Radiation Effects in Silicon Carbide SiC is an important engineering ceramic because of its high-temperature stability, high thermal conductivity, and special electronic properties. It has been proposed for use in nuclear applications including structural components in fusion reactors, cladding material for gas-cooled fission reactors, and as an inert matrix for the transmutation of plutonium and other transuranics.32 In high-temperature gas-cooled reactors, SiC is the primary barrier material for TRISO coated fuel particles.33 Also, SiC fiber, SiC matrix (SiC/SiC) composites are attractive candidate materials for first wall and blanket components in fusion reactors.34
Radiation-Induced Effects on Material Properties of Ceramics
Only limited studies of elevated-temperature microstructural evolution (dislocation loops, voids, etc.), based on neutron or ion irradiations, have been performed on SiC. In pyrolytic b-SiC (cubic, 3C), Price35 found small (2–5 nm diameter) {111} Frank loops following neutron irradiation at 900 C to 2.4 1021 n cm2 (E > 0.18 MeV) (2.4 dpa). Yano and Iseki36 found the same loops in b-SiC irradiated at 640 C to 1.0 1023 n cm2 (E > 0.10 MeV) (100 dpa) and, using high-resolution TEM, determined these to be 1/3 h111i {111} interstitial Frank loops. These loops are constructed by inserting a single extra Si-C layer into the CABCAB Si-C stacking sequence. This produces the sequence CAjC 0 B 0 jCAB, where the prime denotes a p rotation of the tetrahedral unit (note that an adjacent Si-C layer is modified by the insertion of the extra Si-C layer). In 6H-type hexagonal a-SiC, Yano and Iseki36 found ‘black spot’ defects lying on (0001) planes following neutron irradiation at 840 C to 1.7 1021 n cm2 (E > 0.10 MeV) (1.7 dpa). They coarsened these defects using high-temperature annealing and determined the defects to be interstitial Frank loops. The stacking sequence along (0001) in 6H a-SiC is ABCA 0 B 0 C 0 . Yano and Iseki proposed that the Frank loops are formed by a mechanism similar to b-SiC (described above), wherein insertion of an extra Si-C layer modifies an adjacent Si-C layer to produce a sequence such as ABCjB 0A 0 jC 0 B 0 . Such a defect is described as a 1/6 [0001] (0001) interstitial Frank loop. For low temperatures (150–800 C), small amounts of swelling (0–2%) are observed in monolithic SiC samples produced by chemical vapor deposition (CVD).33 It should be noted that CVD-SiC is cubic and highly faulted.37 This swelling saturates at low damage levels (a few dpa) and the saturated swelling is lower, the higher the temperature. Much of this swelling is due to strain caused by surviving interstitials formed during ballistic damage cascades. As the irradiation temperature approaches 1000 C, the surviving defect fraction diminishes because interstitial mobility increases with temperature and i–v recombination is enhanced. Newsome et al.33 found swelling values of 1.9, 1.1, and 0.7% for neutron irradiations at 300, 500, and 800 C, respectively. Above 1000 C, neutron irradiation-induced void formation in b-SiC was first observed by Price35 at 1250 C (4.3–7.4 dpa) and 1500 C (5.2–8.8 dpa). Interestingly, no dislocation loops were observable by TEM in these samples. Price35 postulated that
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the interstitials may have been annihilated at stacking faults. Alternatively, he suggested that interstitial defects were present following irradiation, but they were too small and the contrast too weak to detect them. Nevertheless, at present it is not clear whether void formation in SiC is due to vacancy supersaturation produced by a dislocation bias. In any case, swelling of a few percent was observed for irradiations at temperatures greater than 1000 C, and Price38 speculated that this swelling probably does not saturate with dose. At low temperatures (60 C), Snead and Hay37 observed both a- and b-SiC to amorphize following a total fast neutron fluence of 2.6 1021 n cm2 (2.6 dpa). This amorphization transformation was accompanied by a large reduction in density (10.8%), that is, volumetric swelling of nearly 11%. Snead and Hay37 estimated that the critical temperature for amorphization (the temperature above which amorphization is not possible) is 125 C (a lower limit for the threshold amorphization temperature). The critical temperature is dose rate dependent. In the study above, the dose rate was 8 10–7 dpa s1. In other electron and ion irradiation experiments with dose rates of 1 10–3 dpa s1, researchers found critical temperatures ranging from 20 to 70 C for 2 MeV electron irradiations,39–41 150 C for energetic Si ions,42 and 220 C for 1.5 MeV Xe ions.43,44 1.05.3.3
Radiation Effects in Graphite
Graphite (C) is a very important material for nuclear energy applications. Graphite is a moderator used to thermalize neutrons in thermal gas and water-cooled reactors in the United Kingdom and the Soviet Union, respectively.45 Pyrolitic graphite is one of the barrier coating materials used in TRISO coated fuel particles.33 Graphite and carbon composites are also used as plasma-facing materials in fusion reactors.46 Numerous radiation effects studies have been performed on graphite. Nevertheless, the behavior of graphite in a radiation damage environment remains poorly understood. This is due primarily to the fact that graphite comes in so many forms and is produced in so many different ways, that in fact, the structure and chemistry of graphite used in nuclear applications is not a well-defined constant. Nevertheless, there are some aspects of the crystal structure of graphite and the changes in this structure induced by irradiation that are somewhat analogous to the discussion of Al2O3 versus MgAl2O4, presented earlier in this chapter.
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Radiation-Induced Effects on Material Properties of Ceramics
Graphite is a hexagonal crystalline material, with an ABAB. . . layer stacking arrangement of carbon sheets. These carbon layers have obvious hexagonal atom patterns in them. However, they are not fully dense triangular atom nets, as would be the case in a close-packed structure. They are so-called graphene sheets, in which the atom pattern is a honeycomb pattern, identical to the cation layer patterns in Al2O3 (see Section 1.05.2.3.1). Each C atom is surrounded by three nearest-neighbor C atoms, and the bonding linking each C atom with its neighbors is characterized by sp2 hybridization. The bonding that links adjacent graphene layers is weak, Van der Waals-type bonding. The interstitial dislocation loops that form in irradiated graphite, by the condensation of freely migrating interstitial point defects, form (not surprisingly) on (0001) basal planes, between adjacent graphene layers. In some of the earliest work on radiation effects in graphite, this was described as follows47: When subjected to bombardment with fission neutrons, primary collisions displace carbon atoms from their normal sites in the layers, driving them to sites between planes (interstitial or interlamellar positions).
This loop nucleation is analogous to the (0001) basal interstitial loops that form in Al2O3 during the initial stages of irradiation (Section 1.05.2.3.1). However, the basal loops in graphite do not grow to any significant size. Instead, the graphene layers adjacent to interlamellar loop nuclei buckle, which causes a net increase in the c-dimension of the hexagonal material and a concomitant decrease in the a-dimension.48 This buckling is believed to be due to sp3 bond formation between C interstitials and C atoms in the graphene planes.49,50 The overall macroscopic effect of c-axis expansion and a-axis shrinkage is dimensional changes of crystallites within the graphite. Macroscopic radiation damage effects in graphite are discussed in detail in Chapter 4.10, Radiation Effects in Graphite. 1.05.3.4 Radiation Effects in Other Ceramics Numerous additional ceramics have been either used or proposed for nuclear reactor materials applications. These include graphite (discussed in other chapters in this volume) as well as carbides and nitrides, such as ZrC and ZrN, which have higher thermal conductivities than their sister oxide compound, ZrO2. Research into the radiation damage
properties of these materials is in its infancy, and therefore, these compounds are not described in further detail here.
1.05.4. Summary The response of ceramic materials to radiation is especially complex because ceramics (with the exception of graphite) are made up of anions and cations (sometimes several different cations) such that the atomic defects that initiate radiation damage are different in their size, chemistry, charge, mobility, and so on. Thus, it is difficult to predict how the microstructure of a ceramic will evolve under irradiation and, in turn, how properties such as structural stability will change in response to the radiation-induced microstructural alterations. Nevertheless, we present a case study (described below) wherein researchers have succeeded in explaining the extraordinary differences between the radiation responses of two important engineering ceramics. We devoted much of this chapter to comparing and contrasting the high-temperature radiation damage response of two quite similar refractory, dielectric ceramics: a-alumina (Al2O3) and magnesio-aluminate spinel (MgAl2O4). Al2O3 is highly susceptible to radiation-induced swelling, whereas MgAl2O4 is not. The swelling of Al2O3 is due to excessive void formation in the crystal lattice. We considered in detail in this chapter the atomic and microstructural mechanisms that help to explain why voids nucleate and grow in Al2O3 to a very significant degree, whereas in MgAl2O4, this problem is much less pronounced. We showed that the reasons for the great differences between the radiation damage behavior of Al2O3 and MgAl2O4 have mainly to do with differences in the way interstitial loops nucleate and grow in these two oxides. The hope is that by understanding these differences, we will by analogy be able to understand the radiation damage behavior of other ceramic materials. In this chapter, we also examined two different phenomena that lead to degradation in the mechanical properties of ceramics: (1) nucleation and growth of interstitial dislocation loops and voids and (2) crystal-to-amorphous phase transformations. Both these phenomena cause macroscopic swelling of materials. This ultimately leads to the failure of materials because of unacceptable dimensional changes, microcracking, excessive increases in hardness (or alternatively, softening in the case of amorphization), and so on.
Radiation-Induced Effects on Material Properties of Ceramics
We concluded this chapter with brief discussions of a few ceramics additionally important for nuclear energy applications, namely silicon carbide (SiC), uranium dioxide (UO2), and graphite (C).
25. 26. 27. 28.
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30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51.
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Kabir, A.; Meftah, A.; et al. Nucl. Instrum. Meth. Phys. Res. B 2008, 266, 2976–2980. Matzke, H. Nucl. Instrum. Meth. Phys. Res. B 1996, 116, 121–125. Sickafus, K. E.; Matzke, H.; et al. J. Nucl. Mater. 1999, 274, 66–77. Zinkle, S. J.; Skuratov, V. A.; et al. Nucl. Instrum. Meth. Phys. Res. B 2002, 191, 758–766. Olander, D. R. Fundamental Aspects of Nuclear Reactor Fuel Elements; Technical Information Center, Office of Public Affairs, Energy Research and Development Administration: Springfield, Virginia, 1976. Matzke, H.; Turos, A.; et al. Nucl. Instrum. Meth. Phys. Res. B 1994, 91(1–4), 294–300. Matzke, H.; Lucuta, P. G.; et al. Nucl. Instrum. Meth. Phys. Res. B 2000, 166–167, 920–926. Jiang, W.; Weber, W. J. Phys. Rev. B 2001, 64, 125206. Newsome, G.; Snead, L. L.; et al. J. Nucl. Mater. 2007, 371, 76–89. Taguchi, T.; Igawa, N.; et al. J. Nucl. Mater. 2007, 367–370, 698–702. Price, R. J. J. Nucl. Mater. 1973, 48, 47–57. Yano, T.; Iseki, T. Philos. Mag. A 1990, 62(4), 421–430. Snead, L. L.; Hay, J. C. J. Nucl. Mater. 1999, 273, 213–220. Price, R. Nucl. Technol. 1977, 35, 320. Inui, H.; Mori, H.; et al. Acta Metall. 1989, 37(5), 1337–1342. Inui, H.; Mori, H.; et al. Philos. Mag. B 1990, 61(1), 107–124. Inui, H.; Mori, H.; et al. Philos. Mag. B 1992, 66(6), 737–748. Zinkle, S. J.; Snead, L. L. Nucl. Instrum. Meth. Phys. Res. B Beam Interact. Mater. Atoms 1996, 116(1–4), 92–101. Weber, W. J.; Wang, L. M. Nucl. Instrum. Meth. Phys. Res. B 1995, 106, 298–302. Weber, W. J.; Wang, L. M.; et al. Nucl. Instrum. Meth. Phys. Res. B 1996, 116, 322–326. Burchell, T. D. In Carbon Materials for Advanced Technologies; Burchell, T. D., Ed.; Elsevier Science: Oxford, 1999. Snead, L. L. In Carbon Materials for Advanced Technologies; Burchell, T. D., Ed.; Elsevier Science: Oxford, 1999; pp 389–427. Billington, D. S.; Crawford, J. H. Radiation Damage in Solids; Princeton University Press: Princeton, NY, 1961; p 396. Koike, J.; Pedraza, D. F. J. Mater. Res. 1994, 9, 1899–1907. Wallace, P. R. Solid State Commun. 1966, 4, 521–524. Jenkins 1973. Kinoshita, C.; Fukumoto, K.-I.; et al. J. Nucl. Mater. 1995, 219, 143–151.
1.06
The Effects of Helium in Irradiated Structural Alloys
Y. Dai Paul Scherrer Institut, Villegen PSI, Switzerland
G. R. Odette and T. Yamamoto University of California, Santa Barbara, CA, USA
ß 2012 Elsevier Ltd. All rights reserved.
1.06.1 1.06.2 1.06.2.1 1.06.2.2 1.06.2.3 1.06.2.4 1.06.2.5 1.06.2.6 1.06.3 1.06.3.1 1.06.3.2 1.06.3.3 1.06.3.4 1.06.3.5 1.06.3.6 1.06.3.7 1.06.4 1.06.4.1 1.06.4.2 1.06.4.2.1 1.06.4.2.2 1.06.4.3 1.06.4.4 1.06.5 1.06.5.1 1.06.5.2 1.06.5.3 1.06.5.4 1.06.5.5 1.06.6 1.06.6.1 1.06.6.2 1.06.7 References
Introduction and Overview Experimental Approaches to Studying He Effects in Structural Alloys Single, Dual, and Triple-Beam CPI Neutron Irradiations with B or Ni Doping In Situ He Implantation Spallation Proton–Neutron Irradiations, SPNI Proposed Future Neutron-Irradiation Facilities Characterization of He and He Bubbles A Review of Helium Effects Models and Experimental Observations Background Historical Motivation for He Effects Research Void Swelling and Microstructural Evolution: Mechanisms The CBM of Void Nucleation and RT Models of Swelling Summary: Implications of the CBM to Understanding He Effects on Swelling and Microstructural Evolution HTHE Critical Bubble Creep Rupture Models Experimental Observations on HTHE Recent Observations on Helium Effects in SPNI Microstructural Changes Mechanical Properties of FMS After SPNI Helium effects on tensile properties and He-induced hardening effects Helium effects on fracture properties and He-induced embrittlement effects Mechanical Properties of AuSS After SPNI Summary of Effects of Irradiation on Tensile and Fracture Properties Atomistic Models of He Behavior in Fe He Energetics and He–Defect Complex Interactions He Interactions with Other Defects Helium Migration Master Models of He Transport, Fate, and Consequences Dislocation–Cavity Interactions Radiation Damage Tolerance, He Management, Integration of Helium Transport and Fate Modeling with Experiment ISHI Studies and Thermal Stability of Nanofeatures in NFA MA957 Master Models of He Transport Fate and Consequences: Integration of Models and Experiment Summary and Some Outstanding Issues
Abbreviations ANL appm APT
Argon National Laboratory Atomic parts per million Accelerator Production of Tritium
AT AuSS bcc BF
142 146 146 147 148 149 149 150 151 151 152 155 156 162 163 166 168 168 172 172 174 177 178 178 179 180 180 182 182 183 183 185 186 189
As tempered Austenitic stainless steels Body-centered cubic Bright field
141
142 CBM CD CPI CT CVN CW DBTT
The Effects of Helium in Irradiated Structural Alloys
Critical bubble model Cluster dynamics Charged-particle irradiations Compact-tension Charpy V-notch Cold-worked Ductile-to-brittle transition temperature DDBTT Shifts in DBTT DFT Density functional theory dpa Displacement per atom EAM Embedded atom method EELS Electron energy-loss spectroscopy fcc Face-centered cubic FFTF Fast Flux Test Facility FMS Ferritic–martensitic steels FNSF Fusion Nuclear Science Facility FZJ Forschungscentrum Ju¨lich GB Grain boundary HFIR High Flux Isotope Reactor HFR High Flux Reactor (Petten) HTHE High temperature helium embrittlement IFMIF International Fusion Material Irradiation Facility IG Intergranular ISHI In situ He implantation KMC/KLMC Kinetic Monte Carlo–lattice Monte Carlo LANSCE Los Alamos Neutron Science Center LBE Liquid lead-bismuth eutectic LTHE Low temperature hardening-helium embrittlement MD Molecular dynamics MS Molecular statics NF Nanofeatures NFA Nanostructured ferritic alloys ODE Ordinary differential equation OEMS (Positron) orbital electron momentum spectra ORNL Oak Ridge National Laboratory PAS Positron annihilation spectroscopy PIE Postirradiation examination RED Radiation-enhanced diffusion REP Radiation enhanced precipitation RIP Radiation-induced precipitation RIS Radiation-induced segregation RT Rate theory SANS Small-angle neutron scattering SIA Self-interstitial atom SINQ Swiss Spallation Neutron Source
SP SPN SPNI STIP TDS TEM TMS V
Small punch Spallation proton–neutron Spallation proton–neutron irradiations SINQ Target Irradiation Program Thermal desorption spectroscopy Transmission electron microscopy Tempered martensitic steels Vacancy
1.06.1 Introduction and Overview This chapter reviews the profound effects of He on the bulk microstructures and mechanical properties of alloys used in nuclear fission and fusion energy systems. Helium is produced in these service environments by transmutation reactions in amounts ranging from less than one to thousands of atomic parts per million (appm), depending on the neutron spectrum, fluence, and alloy composition. Even higher amounts of H are produced by corresponding n,p reactions. In the case of direct transmutations, the amount of He and H are simply given by the content weighted sum of the total neutron spectrum averaged energy dependent n,a and n,p cross-sections for all the alloy isotopes (hsn,ai) times the total fluence (ft). The spectral averaged cross-sections for a specified neutron spectrum can be obtained from nuclear database compilations such as SPECTER,1 LAHET,2 and MCNPX3 codes. He and H are also produced in copious amounts by very high-energy protons and neutrons in spallation targets of accelerator-based nuclear systems (hereafter referred to as spallation proton–neutron (SPN) irradiations, SPNI).4,5 The D–T fusion first wall spectrum includes 14 MeV neutrons (20%), along with a lower energy spectrum (80%). The 14 MeV neutron energy is far above the threshold for n,a (5 MeV) and n, p (1 MeV) reactions in Fe.6 Note that some important transmutations also take place by multistep nuclear reactions. For example, thermal neutrons (nth) generate large amounts of He in Ni-bearing alloys by a 58Ni(nth,g)59Ni(nth,a) reaction sequence. These various irradiation environments also produce a range of solid transmutation products. High-energy neutrons also produce radiationinduced displacement damage in the form of vacancy and self-interstitial atom (SIA) defects. Vacancies and SIA are the result of a neutron reaction and scatteringinduced spectrum of energetic primary recoiling
The Effects of Helium in Irradiated Structural Alloys
Table 1 Typical dpa, He, and H production in nuclear fission, fusion, and spallation facilities Irradiation facility
Fission reactor
Fusion reactor first wall
Spallation targets
dpa range (in Fe) He per dpa (in Fe) H per dpa (in Fe) Temperature ( C)
<200–400 <1 <1 270–950
50–200 10 40 300–800
<35 <100 <500 50–600
Source: Dietz, W.; Friedrich, B. C. In Proceedings of the OECD NEA NSC Workshop on Structural Materials for Innovative Nuclear Systems, 2007, p 217; Mansur, L. K.; Gabriel, T. A.; Haines, J. R.; Lousteau, D. C. J. Nucl. Mater. 2001, 296, 1; Vladimirov, P.; Moeslang, A. J. Nucl. Mater. 2006, 356, 287–299.
Dimensional instability irradiation creep and swelling
Life time
atoms with energies ranging from less than 1 keV, in neutron irradiations, up to several MeV in SPN irradiations.7 The high-energy primary recoils create cascades of secondary displacements of atoms from their crystal lattice positions, measured in a calculated dose unit of displacements per atom (dpa). As in the case of n,a transmutations, dpa production can also be evaluated using spectral averaged displacement cross-sections8 that are calculated using the codes and nuclear database compilations cited above. Typical operating conditions of various fission, fusion, and spallation facilities are summarized in Table 1. Notably, He (and H) generation in fast fission (He/dpa << 1), fusion (He/dpa 10), and spallation proton–neutron (He/dpa up to 100) environments differs greatly and this is likely to have significant effects on the corresponding microstructural and mechanical property evolutions. The primary characteristic of He, which makes it significant to a wide range of irradiation damage phenomena, is that it is essentially insoluble in solids. Hence, in the temperature range where it is mobile, He diffuses in the matrix and precipitates to initially form bubbles, typically at various microstructural trapping sites. The bubbles can serve as nucleation sites of growing voids in the matrix and creep cavities on grain boundaries (GBs), driven by displacement damage and stress, respectively. While He effects are primarily manifested as variations in the cavities, all microstructural processes taking place under irradiation are intrinsically coupled; hence, difference in the He generation rate can also affect precipitate, dislocation loop, and network dislocation evolutions as well (see Section 1.06.3). Figure 1, adopted from Molvik et al.,9 schematically illustrates the effects of high He as a function of lifetime-temperature limits in a fusion first wall structure for various irradiation-induced degradation phenomena. At high temperatures, lifetimes (green curve) are primarily dictated by chemical compatibility, fatigue, thermal creep, creep rupture, and creep–fatigue limits. In this regime, He can further degrade the tensile ductility and the other hightemperature properties, primarily by enhancing grain boundary cavitation, in some cases severely. In austenitic stainless steels (AuSS), high-temperature He embrittlement (HTHE) has been observed at concentrations as low as 1 appm.10,11 In contrast, 9Cr ferritic–martensitic steels (FMS), which are currently the prime candidate alloy for fusion structures, are much more resistant to HTHE.12,13
143
He embrittlement, thermal creep, corrosion Hardening, fracture
Window
Temperature Figure 1 Illustration of the materials design window for the fusion energy environment, as a function of temperature. Reproduced from Molvik, A.; Ivanov, A.; Kulcinski, G. L.; et al. Fusion Sci. Technol. 2010, 57, 369–394.
At intermediate temperatures (blue curve), growing voids form on He bubbles, and He accumulation largely controls the incubation time prior to the onset of rapid swelling (see Section 1.06.3). FMS are also much more resistant to swelling than standard austenitic alloys,14,15 although the microstructures of the latter can be tailored to be more resistant to void formation by He management schemes.16 High He concentrations can also extend irradiation hardening and fast fracture embrittlement to intermediate temperatures.17 At lower temperatures (red curve), where irradiation hardening and loss of tensile uniform ductility are severe, high He concentrations enhance large positive shifts in the ductile-to-brittle transition temperature (DBTT) in bcc (body-centered cubic) alloys.18–20 This low-temperature fast fracture embrittlement phenomenon is believed to be primarily the result of
144
The Effects of Helium in Irradiated Structural Alloys
He-induced grain boundary weakening, manifested by a very brittle intergranular (IG) fracture path, interacting synergistically with irradiation hardening.20,21 High concentrations also increase the irradiation hardening at dpa levels that would experience saturation in the absence of significant amounts of He.17 A significant concern for fusion is that the dpatemperature window may narrow, or even close, for a practical fusion reactor operating regime. What is sketched above is only a very broad-brush, qualitative description of some of the important He effects. The quantitative effects of He, displacement damage, temperature and stress, and their interactions, which control the actual positions of the schematic curves shown in Figure 1, depend on the combination of all the irradiation variables, as well as details of the alloy type, composition, and starting microstructure (material variables). The effects of a large number of interacting variables, the complex interactions of a plethora of physical mechanisms, and the implications to the wide range of properties of concern are not well understood; and even if they were, such
complexity would beg easy description. Therefore, a first priority is to develop a good understanding of and models for the transport and fate of He at the point when it is effectively immobilized in bubbles and voids, often at various microstructural sites. Such insight provides a basis for developing microstructures that can manage He and thus mitigate its deleterious effects. To this end we next briefly outline key radiation damage processes, including the role of He. Figure 2 schematically illustrates the combined effects of He and displacement damage on irradiationinduced microstructural evolutions.22 Figure 2(a) shows a molecular dynamics simulation of primary displacement damage produced in displacement cascades. Most of the initially displaced atoms return to a lattice site (self-heal). Residual cascade defects include single and small clusters of vacancies and SIA. In the temperature range of interest, vacancies (red circles) and SIA (green dumbbells) are mobile. SIA clusters, in the form of dislocation loops, are also believed to be mobile in some cases, undergoing one-dimensional diffusion on their glide prisms.
Recombination
Precipitate – void (b) Growing SIA loop/ climbing dislocation
Cascade (a)
Vacancy
SIA (c)
Stably growing matrix bubble
He GB h +
Unstably growing matrix void
m = m*
drc /dt
rc
0 rb −
sn
(e)
r c* (m*)
(f)
Jv
Stress-driven grain boundary cavity growth
r*v m < m*
(d) Figure 2 Illustration of the combined effects of He and displacement damage on irradiation-induced microstructural evolutions.
The Effects of Helium in Irradiated Structural Alloys
However, the cascade loops may also be trapped by interactions with solutes. Small cascade vacancy clusters may coarsen in the cascade region by Ostwald ripening and diffusion coalescence mechanisms. Both isolated and clustered defects interact with alloy solutes forming cascade complexes. The cascade vacancy clusters dissolve over a time associated with cascade aging, which depends strongly on temperature. The concentration of cascade vacancy clusters, which act as sinks (or recombination centers) for migrating vacancies and SIA, scales directly with the damage rate. Thus, the overall defect production microstructures can be viewed as being composed of steady-state concentrations of diffusing defects, small loops, and cascade vacancy clusters; the latter are important if the irradiation time is much less than the cluster annealing time. Vacancy–SIA recombination at clusters, in the matrix and at vacancy trapping sites, can give rise to important damage rate, or flux, effects. Figure 2(b) shows that SIA can recombine with diffusing and trapped vacancies, in this case one trapped on a precipitate interface. Figure 2(b) also shows that both bubbles (blue part circle) and voids (orange part circle) often form on precipitates. Figure 2(c) shows that dislocation loops (green hexagon) nucleate and grow due to preferential absorption of SIA (bias). Preferential accumulation of SIA also takes place at network dislocation segments (inverted green T), causing climb. Loop growth and dislocation climb can lead to creation (loops and Herring–Nabarro sources) and annihilation (of oppositely signed network segments) of dislocations, ultimately leading to quasi-steady-state densities, as is observed in the case of AuSS.
Figure 2(d) shows that He precipitates to form bubbles (larger blue circles) at various sites, in this case in the matrix. Small bubbles are stable since they absorb and emit vacancies in net numbers that exactly equal the number of SIA that they absorb; thus bubbles grow only by the addition of diffusing He atoms (small blue circle). However, Figure 2(e) shows that when bubbles reach a critical size they convert to unstably growing underpressurized voids (large orange circle containing blue He atoms) due to an excess flux of vacancies over SIA arising from the dislocation bias for the latter defect. Figure 2(e) shows the corresponding growing creep cavities transformed from critical He bubbles on stressed GBs. Designs of microstructures that mitigate, or even fully suppress, these various coupled evolutions are described in Section 1.06.6 and discussed in references.22,23 Therefore, a master overarching framework for measuring, modeling, and managing He effects must be based on developing and understanding the dominant mechanisms controlling its generation, transport, fate, and consequences, as mediated by the irradiation conditions and the detailed alloy microstructure. Figure 3 illustrates such a framework for He generation, transport, and fate. In this framework, experiments and models can be integrated to establish how He is transported to various microstructural trapping (-detrapping) features and how He locally clusters to form bubbles at these sites, as well as in the matrix. The master models must incorporate parameters that describe He diffusion coefficients under irradiation, binding energies for trapping at the various sites and He–vacancy cluster and other interaction energies.
Multiscale modeling-experiment framework Generate mobile He by transmutation and emission from traps Matrix transport of He by various mechanisms and partitioning to subregion sinks controlled by vacancy and SIA defects, matrix properties, and trap-sink microstructures – Nucleation and growth of matrix cavities
Grain boundaries
Internal subregion structure
Fine-scale precipitates
145
Dislocation substructures
Other precipitates
Transport of He within and between interconnected subregions Emission of He from subregions Formation of subregion cavities
Figure 3 Illustration of a multiscale master modeling–experiment framework for He generation, transport, and fate.
146
The Effects of Helium in Irradiated Structural Alloys
Given the length and comprehensive character of this chapter, it is useful to provide the reader a guide to what follows. Notably, we have tried to develop useful semi-standalone sections. Section 1.06.2 describes the various experimental approaches to studying He effects in structural alloys including both neutrons and various types of charged-particle irradiations (CPI). Section 1.06.3 reviews the historical knowledge base on He effects, which has been developed over the past 40 years, with emphasis on bubble evolution, void swelling, and HTHE processes. While less of current interest, the examples included here primarily pertain to standard AuSS, discussions of experiment and modeling are closely integrated to emphasize the insight that can be derived from such coupling. Particular attention is paid to the critical bubble mechanism for the formation of growing voids and grain boundary cavities and the corresponding consequences to swelling and creep rupture. The implications of the coupled models and experimental observations to designing irradiation-tolerant alloys that can manage He are discussed in some detail. Section 1.06.4 focuses on a much more recent body of observations on He effects in SPNI. The emphasis here is on descriptions of defect and cavity microstructures in both FMS and AuSS irradiated at low to intermediate temperatures and the corresponding effects on their strength, ductility, and fast fracture resistance. Similarities and differences between the SPNI effects and those observed for fission irradiations are drawn where possible. Section 1.06.5 summarizes some key examples of atomistic modeling of He behavior, which has been the focus of most recent modeling efforts. Insight into mechanisms and critical parameters provided by these models will form the underpinning of the comprehensive master models of He transport, fate, and consequences. Section 1.06.6 builds on the discussion in Section 1.06.3 regarding managing He by trapping it in a population of small stable bubbles. A specific example comparing FMS to a new class of hightemperature, irradiation-tolerant nanostructured ferritic alloys (NFA) irradiated in a High-Flux Isotope Reactor (HFIR) at 500 C to 9 dpa and 380 appm He is described. The results of this study offer proof in principle of the enormous potential for developing irradiation-tolerant NFA that could turn He from a liability to an asset. Section 1.06.6 again couples these experimental observations with a master multiscale model of the transport and fate of He in both
FMS and NFA. The predictions of the master model, that is both microstructurally informed and parameterized by atomistic submodels, are favorably compared to the HFIR data. Section 1.06.7 briefly summarizes the status of understanding of He effects in structural alloys and concludes with some outstanding issues. Reading this summary first may be helpful to general readers who then can access the more detailed information at their own discretion.
1.06.2 Experimental Approaches to Studying He Effects in Structural Alloys 1.06.2.1
Single, Dual, and Triple-Beam CPI
Single (He), dual (typically heavy ions to produce dpa and He), and triple (typically heavy ions, He and H) beam CPI have been extensively used to study He effects for a wide variety of materials and conditions. The number of facilities worldwide, both current and historically, and the large resulting literature cannot be fully cited and summarized in this chapter, but some examples are given in Section 1.06.3. A more complete overview of these facilities can be found in a recent Livermore National Laboratory Report.24 Extensive high-energy He implantation studies of creep properties were carried out at Forschungszentrum Ju¨lich using a 28 MeV He cyclotron.25 Major dual- and triple-beam studies were previously carried out at Oak Ridge National Laboratory (180 keV H, 360 keV He, 3.5 MeV Fe)26 and many other facilities around the world.24 The new JANNUS facility at Saclay couples a 3 MV Pelletron with a multicharged ion source and a 2.5 MV single Van de Graaff and a 2.25 MeV tandem accelerator.27 Another multibeam facility at Orsay couples a 2 MV couple, a tandem accelerator, and a 190 kV ion implanter to a 200 kV transmission electron microscope (TEM) to allow simultaneous co-irradiation and observation.27 The advantages of He implantation and multibeam ion irradiations include the following: (a) conditions can be well controlled and in many cases selectively and widely varied; (b) high dpa, He, and H levels can be achieved in short times; (c) the specimens are often not, or only minimally, activated; and (d) in situ TEM observations are possible in some cases. The disadvantages include the following: (a) highly accelerated damage rates compared with neutron irradiations; and in the case of
The Effects of Helium in Irradiated Structural Alloys
(nth,a) (bred from 58Ni with a n,a cross-section of 10 barns) cited in Section 1.06.1; (b) or by the 10 B þ nth ! 7Li þ a reaction (20% of elemental B with a cross-section of 4010 barns) (1 barn ¼ 10–24 cm2). Significant quantities of He can also be generated by epithermal–fast spectrum neutron reactions with B as well as prebred 59Ni.29 Figure 4(a) shows calculated and measured He production in natural Ni in the HFIR target capsule position.30 Figure 4(b) shows the corresponding He/dpa ratio for a Fe alloy doped with 2% natural Ni. Two Ni doping characteristics are evident: (a) there is a transient phase in He production regime prior to a He/dpa peak at about 20 dpa in HFIR; (b) if the alloy contains more than a few percent Ni, like in AuSS, the He/dpa is much higher than that for fast fission and higher than that for fusion spectra but is comparable to, or slightly less than, the He/dpa for SPNI. Modifying the amounts of 58Ni and 60Ni (isotope tailoring) can control and target He/dpa ratios (e.g., to fusion).29,31,32 An approximately constant He generation rate can be obtained by using irradiated Ni pre-enriched in 59Ni.29,31 Various amounts of 58Ni, 59Ni, and 60Ni can also be used to control the He/dpa ratio in fast spectrum reactors, like the Fast Flux Test Facility (FFTF), as well as in mixed spectrum reactors, like HFIR.29,31,32 Boron is not normally added to steels used for nuclear applications, but it has been used in a number of doping studies.33,34 A major advantage of B doping is that significant amounts of He are produced by the 10 B, but not the 11B, isotope. Thus, the effect of doping with 10B versus 11B can be used to isolate this effect of
multibeam ion irradiations, (b) shallow damage depths and the proximity of free surfaces; (c) nonuniform damage production and the deposition of foreign ions; and (d) inability to measure bulk properties. High-energy He implantation can be used on bulk specimens tested, either in situ or postimplantation, to measure tensile, creep, and creep rupture properties. The corresponding disadvantages are that He implantation results in high He/dpa ratios (6000 appm He/dpa).28 The differences between CPI and neutron irradiation can significantly affect microstructural evolution. Thus, it must be emphasized that He implantation and multibeam CPI do not simulate neutron irradiations. Although it has been argued that CPI reveal general trends and that corrections, like temperature adjustments, allow extrapolations to neutron-irradiation conditions, both assertions are problematic. The proper role of He implantation and multibeam CPI is to help inform and calibrate models and to identify and quantify key processes based on carefully designed mechanism experiments. 1.06.2.2 Neutron Irradiations with B or Ni Doping The effects of high He levels on microstructure and mechanical properties have been extensively studied in mixed fast–thermal spectrum fission reactor irradiations of alloys naturally containing, or doped with, Ni and B. In these cases, high He levels are produced by thermal neutron nth,a reactions, either by (a) the two-step reaction with 58Ni(nth,g) (68% of elemental Ni with a nth,g cross-section of 0.7 barns) and 59Ni
(a)
14 12
59 dpa
Helium (appm/dpa)
Helium production (appm)
40 000
44
30 000 34 20 000
10 000
10 8 6 4 Total Incremental
2 0
147
0
5 10 Thermal neutron fluence (1022 n cm–2)
0
15 (b)
0
20
40
60
80
100
dpa
Figure 4 (a) Measured and calculated He production from Ni irradiated in HFIR. The solid line is calculated using the evaluated 58Ni and 59Ni cross-sections. (b) The He/dpa ratio in Fe-based 2% Ni alloy for accumulated total (solid red line) and incremental (dashed blue line) He. Reproduced from Greenwood L. R.; ASTM STP 1490 and the data provided by Greenwood L. R.
148
The Effects of Helium in Irradiated Structural Alloys
He, in a B-containing alloy. However, the issues associated with B doping are even more problematic than those for Ni. In mixed spectrum reactors, all the 10 B is quickly converted to He and Li by the thermal neutrons. In this case, the He is initially introduced at much too high a rate per dpa but then saturates at the 10B content. The other major limitations are that B is virtually insoluble in steels and primarily resides in Fe and alloy boride phases.35 Boron also segregates to GBs. Thus, He from B reactions is not homogeneously distributed. Recently, nitrogen additions to FMS steels to form fine-scale BN phases have been used to increase the homogeneity of B and He distributions.36 Varying the He/dpa ratio in Ni- and B-containing alloys can also be achieved by attenuating thermal neutron fluxes (spectral tailoring) in mixed spectrum reactors as well as selecting appropriate fast reactor irradiation positions.31,37,38 Spectral tailoring, either by attenuating thermal neutrons or irradiating in epithermal–fast reactor spectra, is especially helpful in B doping.33,39,40 However, doping alloys that do not normally contain Ni or B can affect both their properties and microstructures, including their response to He and displacement damage. For example, transformation kinetics during heat treatments (hardenability) and the baseline properties of FMS are strongly affected by both Ni and B. Ni also has a strong effect on refining irradiation-induced microstructures and enhancing irradiation hardening.20,41–44 As noted previously, to some extent these confounding factors can be evaluated by comparing the effects of various amounts of 10 B/11B45 and 58Ni/60Ni. However, doped alloys are inherently ‘different’ from those of direct interest. Note that excess dpa due to n,a reaction recoils must be accounted for,46 and in the case of B doping the Li reaction product may play some role as well.
the isotope decay technique produces few dpa at a very high He/dpa. The first proposal ISHI in a mixed fast (dpa)–thermal (He) spectrum proposed using 235U triple fission reactions to inject 16 MeV a-particles uniformly in steel specimens up to 50 mm thick; the 50 mm thickness permits tensile and creep testing as well as microstructural characterization and mechanism studies at fusion relevant dpa rates and He/dpa ratios.31 The triple fission technique was applied to implanting ferritic steel tensile specimen, albeit without complete success.48 A much more practical approach is to use thin Ni-bearing implanter foils to uniformly deposit He up to a depth of 8 mm in Fe in a thick specimen at controlled He/dpa ratios.49 As illustrated in Figure 5(a)–5(c), there are at least three basic approaches to implanter design. Here we will refer to thin and thick, specifically meaning a specimen (ts) or implanter layer (ti) thickness that is less than or greater than the corresponding a-particle range, respectively. Ignoring easily treated difference in the a-particle range (Ra) and atom densities in the injector and specimens for simplicity, thick implanter layers on one side of a thick specimen produce linearly decreasing He concentration (XHe) profiles, with the maximum concentration at the specimen surface that is one half the concentration in the bulk injector material, XHeo ¼ XHei/2 (Figure 5(a)). If a thin specimen is implanted from both sides by thick layers, the He concentration
XHe
XHe
Xmax =
2
In Situ He Implantation
In situ He implantation (ISHI) in mixed spectrum fission reactors is a very attractive approach to assess the effects of He–dpa synergisms in almost any material that avoids most of the confounding effects of doping. The basic idea is to use an implanter layer, containing Ni, Li, B, or a fissionable isotope, to inject high-energy a-particles into an adjacent sample simultaneously undergoing neutron-induced displacement damage. Early work proposed implanting He using the decay of a thin layer of a-emitting isotope adjacent to the target specimen.47 However,
XHei Ri 2 Ra
Ra
x 0
1.06.2.3
XHe =
XHei Ri
(a)
x
0
Rα
Rα
(b) XHe
XHe = tiXHei/2Ra
ti
x 0
Xu Ra
(c) Figure 5 Illustration of three basic approaches to the ISHI design.
The Effects of Helium in Irradiated Structural Alloys
is uniform and equal to one half that in the bulk injector material (Figure 5(b)). In contrast, a thin layer implants a uniform concentration of He to a depth of the Ra ti. In this case, the He concentration in both the implantation layer and specimen is equal and lower than in the bulk (XHei) as XHes ¼ tiXHei/ 2Ra (Figure 5(c)). Thus, the He/dpa ratio can be controlled by varying the concentration of the isotope that undergoes n,a reactions with thermal neutrons, ti, and the thermal to fast flux ratio. ISHI experiments were, and continue to be, carried out in HFIR using thin (0.8–4 mm) NiAl coating layers on TEM disks for a large matrix of Fe-based alloys for a wide range of dpa, He/dpa (<1–40 appm He/dpa), and irradiation temperatures. In this case, 4.8 MeV a-particles produce uniform He concentration to a depth of 5–8 mm (Figure 5(c)). Further details are given elsewhere.50 The first results of in situ implantation experiments in HFIR have been reported and are discussed in Section 1.06.6.23,51–53 The technique has also been used to implant SiC fibers irradiated in HFIR.50 More recently, the two-sided thick Ni implanter method was used to produce He/dpa ratios 25 appm/dpa in thin areas of wedge-shaped specimen alloys irradiated in the advanced test reactor to 7 dpa over a range of high temperatures.54 1.06.2.4 Spallation Proton–Neutron Irradiations, SPNI High fluxes of neutrons can be generated by highenergy and current (power) proton beams via spallation reactions that fragment the atomic nuclei heavily in a heavy metal target (like W, Pb, and Hg). At 500 MeV, these reactions produce 10 neutrons per proton. Applications of spallation sources include neutron scattering, nuclear waste transmutation, and driving subcritical fission reactors. A key challenge to developing advanced high-power, longlived spallation source targets is the ability of structural alloys to withstand severe radiation damage, corrosive fluids, and mechanical loading. Most notably, radiation damage in spallation source irradiations, produced by the neutrons and protons, results in both high dpa and concentrations of transmutation products, including He and H (see Table 1). As a consequence, there have been and continue to be international programs on radiation effects in SPNI environments, beginning with a large program in Los Alamos Neutron Science Center (LANSCE) in 1996 and 1997,55 followed by a continuing SINQ (Swiss Spallation Neutron Source) Target
149
Irradiation Program – STIP, started in 1998, that continues to this day, at the Paul Sherrer Institute, in Switzerland, involving an international collaboration of ten institutions in China, Europe, Japan, and the United States.56,57 Because of the accelerator production of tritium target application, the irradiation temperature LANSCE experiment was up to 164 C. The highest damage levels, mostly produced by protons, were 12 dpa and 180 appm He/dpa.4 About 20 materials were irradiated in a variety of specimen configurations in this study. The maximum damage levels in the STIP-I to -IV irradiations56,57 were 25 dpa and 2000 appm He. The corresponding temperatures ranged from 80 to 800 C, but most specimens were nominally irradiated between 100 and 500 C. The temperatures directly depend on the high nuclear heating rates in the target, and both varied by 15% during the 2-year irradiation; and, in the case of STIP-I and-IV, some capsules experienced a significant overtemperature transient. The high heating rates also result in fairly large uncertainties in the temperatures of individual specimens. Note that the temperature control in the most recent STIP-V experiment was significantly better than that in previous studies. Over 60 elemental metals and alloys, ceramics, and composites have been irradiated in the STIP-I to -V, in the form of miniaturized specimens for both microstructural studies and mechanical testing, including tensile, fatigue, fracture toughness, and Charpy V-notch (CVN) measures of the DBTT. Some specimens were irradiated in contact with stagnant liquid Hg, PbBi eutectic, and Pb. The STIP database is discussed in Section 1.06.4 1.06.2.5 Proposed Future Neutron-Irradiation Facilities The proposed International-Fusion-MaterialIrradiation-Facility (IFMIF) is an accelerator-driven neutron source that is based on the proton-stripping reaction.58,59 Neutrons are generated by a beam of 40 MeV deuterons that undergo a proton-stripping reaction when they interact with a flowing liquid lithium jet target. The resulting neutron beam has a spectrum with a high-energy tail above a peak around 14.60 As in the case of D, T, and spallation reactions, these neutrons are well above the threshold energy for n,a reactions; thus, IFMIF produces fusion like He/dpa ratios at high dpa rates. The nuclear reaction kinematics and limited neutron
150
The Effects of Helium in Irradiated Structural Alloys
target source dimensions result in an IFMIF irradiation volume with large gradients over a high-flux region just behind the target. Two 125 mA beams on the Li target produce an 500 cm3 region with dpa rates of 20–50 fpy1 (full power year) at He/ dpa 12 appm dpa1. The medium flux region, from 1.0 to 20 dpa fpy1, is much larger with a volume of 6000 cm3. The Materials Test Station is a new spallation neutron source, proposed by Los Alamos National Laboratory, that is primarily intended to irradiate fast reactor materials and fuels.61 The LANSCE linear accelerator will produce a 1-MW proton beam to drive the spallation neutron source with a fast reactor like spectrum and a high-energy tail up to 800 MeV. The high-energy tail neutrons produce a He/dpa 6–13 appm dpa1 close to that for a fusion first wall. The dpa rates are 7.5–15 fpy1 in a 200 cm3 irradiation volume and 2.5–12.5 fpy1 in an additional volume of 450 cm3. An accelerator upgrade to 3.6 MW would increase these dpa rates to 20–40 fpy1 and 5–16 fpy1, respectively. In both cases, the limited volume for high-flux accelerated irradiations presents a great challenge to developing small specimen mechanical test methods62,63 and experimental matrices64 that can produce the database needed for materials qualification. The database will require irradiations over a range of temperatures for tensile, fracture toughness, fatigue, and creep property characterization. Indeed, it is clear that qualifying materials for fusion applications will require a new paradigm of linking comprehensive microstructural characterization and physically based predictive modeling tools to multiscale models and experiments of structure-sensitive properties as input into engineering models of materials performance. A variety of proposals have been made to develop volumetric D–T fusion devices such as the Fusion Nuclear Science Facility (FNSF), which would provide a basis to test components and materials.65 In some cases, these devices would address a much broader array of issues, such as tritium breeding and extraction. In most cases, the fusion source would be driven by external energetic D beams. Discussing the details of such proposed devices is far beyond the scope of this chapter. However, we note that from a materials development perspective, such devices would be useful to the extent that they are steady state, operate with very high-duty factors, and produce sufficient wall loading to deliver high He and dpa exposures.
1.06.2.6 Characterization of He and He Bubbles The primary techniques used to characterize the behavior of He and He bubbles in materials include TEM, small-angle neutron scattering (SANS), positron annihilation spectroscopy (PAS), and thermal desorption spectroscopy (TDS). All of these techniques, and their numerous variants, have individual limitations. Complete and accurate characterization of He transport and fate requires a combination of these methods; however, such complementary tools are seldom employed in practice. Note that there are also a variety of other methods of studying helium in solids that cannot be discussed due to space limitations. TEM, with a practical resolution limit of about 1 nm, is the primary method for characterizing He bubbles. Bubbles and voids are most frequently observed by bright field (BF) ‘through-focus’ imaging in thin regions of a foil. The Fresnel fringe contrast changes from white (under) to black (over) as a function of the focusing condition. The bubble size is often taken as the mid diameter of the dark under focus fringe. Two critical issues in such studies are artifacts introduced by sample preparation, which produce similar images and determine the actual size, especially below 2 nm.66–68 Electron energyloss spectroscopy (EELS) can be used to estimate the He pressure in bubbles.69,70 SANS provides bulk measures of He bubble microstructures. In ferromagnetic steels, both nuclear and magnetic scattering cross-sections can be measured by applying a saturating magnetic field (2T) perpendicular to the neutron beam. The coherent scattering cross-section variations with the scattering vector are fit to derive the bubble size distribution, with a potential subnanometer resolution limit.71 The magnitude of the scattering crosssection is proportional to the square of the scattering length density contrast factor between the matrix and the bubble times the total bubble volume fraction. Since the magnetic scattering factor contrast is known (He is not magnetic), the bubble volume fraction, and corresponding number densities, can be directly determined by SANS. The nuclear scattering crosssection provides a measure of the He density in the bubbles. Thus, the variation in the ratio of the nuclear (He dependent) to magnetic (He independent) scattering cross-sections with the scattering vector can be used to estimate the He pressure (density) as a function of the bubble size.71,72 Some studies have shown
The Effects of Helium in Irradiated Structural Alloys
that SANS bubble size distributions are in good agreement with TEM observations,73,74 while others show considerable differences for small (<2 nm) bubbles.72 Limitations of SANS include distinguishing the bubble scattering from the contributions of other features; note that, in many cases, these features may be associated with the bubbles. Other practical issues include measurements over a sufficient range of scattering vectors and handling of radioactive specimens. Note that small-angle X-ray scattering studies can also be used to characterize He bubbles, and this technique is highly complementary to SANS measurements. PAS is a powerful method for detecting cavities that are smaller than the resolution limits of TEM and SANS. Indeed, positrons are very sensitive to vacancy type defects, and even single vacancies can be readily measured in PAS studies.75,76 PAS can also be used to estimate the He density, or He/vacancy ratio, in bubbles.77 In the case of He-free cavities, the positron lifetime increases with increasing the nanovoid size, saturating at several tens of vacancies. However, in the case of bubbles, the lifetime decreases with increasing He density. In principle, positron orbital electron momentum spectra (OEMS) can also provide element-specific information about the annihilation site.78 Thus, for example, OEMS might detect the association of a bubble with another microstructural feature. Limitations of positron methods include that they generally do not provide quantitative and unique information about the cavity parameters. The application of PAS to studying He in steels has been very limited to date. TDS measures He release from a sample as a function of temperature during heating or as a function of time during isothermal annealing. The time–temperature kinetics of release provides indirect information about He transport and trapping/ detrapping processes. For example, isothermal annealing experiments on low-dose (<2 appm) a-implanted thin Fe and V foils showed that substitutional helium atoms migrate by a dissociative mechanism, with dissociation energies of about 1.4 eV, and that dihelium clusters are stable up to 637 K in Fe and up to 773 K in V.79 At higher concentrations in irradiated alloys, He can be deeply trapped in cavities (bubbles and voids); in this case, He is significantly released only close to melting temperatures.80,81 Given the complexity and multitude of processes encountered in many studies, it is important to closely couple TDS with detailed physical models.82,83 Techniques that can quantify He concentrations at small levels
151
used in TDS can also be used to measure the total He contents in samples that are melted.81 In summary, a variety of complementary techniques can be used to characterize He and He bubbles in structural materials. A good general reference for these techniques and He behavior in solids can be found in Donnelly and Evans.84 TEM and SANS can measure the number densities, size distributions, and volume fractions of bubbles, subject to resolution limits and complicating factors. The corresponding density of He in bubbles can be estimated by TEM–EELS, SANS, and PAS. TDS can provide insight into the He diffusion and trapping/detrapping processes. Unfortunately, there have been very limited applications in which various methods have been applied in a systematic and complementary manner. Major challenges include characterizing subnanometer bubbles in complex structural alloys, including their association with various microstructural features.
1.06.3 A Review of Helium Effects Models and Experimental Observations 1.06.3.1
Background
Clearly, it is not possible to cite, let alone describe in detail, the extensive literature on He effects in irradiated alloys. This literature encompasses both mechanical properties, especially HTHE, and the effects of He on microstructural evolutions, particularly void swelling. There is also a more limited literature on fundamental processes and properties related to He in solids, like desorption measurements and He solution, binding, and diffusion activation energies. Much of previous work pertains to fcc (face-centered cubic) AuSS, which is one of interest for fast reactor cladding applications. However, standard AuSS, like AISI 316 (Fe–0.17Cr–0.12Ni–bal Mo, Si, Mn, . . .) are highly prone to both HTHE and void swelling. Thus, advanced AuSS and bcc FMS have supplanted conventional AuSS as the leading candidates for nuclear applications. Nevertheless, conventional AuSS alloys nicely illustrate the damaging effects of He (see Section 1.06.3.2 and following), which are both subtle and significantly mitigated in advanced steels. Swelling and HTHE resistance are largely due to microstructural designs that manage He. Particular emphasis in this section is placed on the critical bubble model (CBM) concept of the
The Effects of Helium in Irradiated Structural Alloys
transition of stable He bubbles to unstably growing voids, both under irradiation-driven displacement damage, and stress-driven growth of grain boundary creep cavities. We believe this focus is appropriate, since it seems that many current modeling efforts have lost connection with the basic thermodynamic–kinetic foundation for understanding He effects provided by the CBM concept and the large body of earlier related research. The organization of this section is as follows. Section 1.06.3.2 outlines the historical motivation for concern about He effects in structural alloys, including examples of HTHE and void swelling. Section 1.06.3.3 describes the mechanisms of swelling and its relation to He and He bubbles, especially in AuSS. Section 1.06.3.4 presents a quantitative CBM for void nucleation and a simple rate theory (RT) model of swelling. Section 1.06.3.5 summarizes the implications of the experimental observations and models, and the development of irradiation-resistant alloys. Sections 1.06.3.6 and 1.06.3.7 discuss the application of the CBM to HTHE and corresponding experimental observations, respectively. 1.06.3.2 Historical Motivation for He Effects Research The primary motivation for the earliest research was the observation that even a small concentration of bulk He, in some cases in the range of one appm or less, generated in fission reactor irradiations of AuSS, could lead to HTHE, manifested as significant reductions in tensile and creep ductility and creep rupture times. The degradation of these properties coincided with an increasing transition from transgranular to intergranular rupture.10,85–89 HTHE is attributed to stress-driven nucleation, growth, and coalescence of grain boundary cavities formed on the He bubbles. The early studies included mixed spectrum neutron irradiations that produce large amounts of He in alloys containing Ni and B. Figure 6 shows one extreme example of the dramatic effect of HTHE on creep rupture times for a 20% cold-worked (CW) 316 stainless steel tested at 550 C and 310 MPa following irradiations between 535 and 605 C in the mixed spectrum HFIR that produced up to 3190 appm He and 85 dpa.88 At the highest He concentration, the creep rupture time is reduced by over four orders of magnitude, from several thousand to less than 0.1 h. A comprehensive review of the large early body of research on He effects on mechanical properties of AuSS can be found in Mansur and Grossbeck.11
104 1000 100 tr (h)
152
10 1 0.1 0.01
0
20 48 He (appm)
3190
Figure 6 Creep rupture time for CW 316 AuSS for various He contents following HFIR irradiation. Reproduced from Bloom, E. E.; Wiffen, F. W. J. Nucl. Mater. 1975, 58, 171.
The early fission reactor irradiations research on HTHE was later complemented by extensive accelerator-based He ion implantation experiments, primarily carried out in the 1980s (see Schroeder and Batfalsky90 and Schroeder, Kesternich and Ullmaier91 as examples) but that have continued to recent times.92 HTHE models were developed during this period, primarily in conjunction with the He ion implantation experiments.93–100 The He implantation studies and models are discussed further in Sections 1.06.3.6 and 1.06.3.7. A more general review of He effects, again primarily in AuSS, can be found in Ullmaier99 and a comprehensive model-based description of the behavior of He in metals in Trinkaus.96 Research on He effects was also greatly stimulated by the discovery of large growing voids in irradiated AuSS.101 As an example, Figure 7(a) shows swelling curves for a variety of alloys used in reactor applications.102–104 Figure 7(b) illustrates macroscopic consequences of this phenomenon in an AuSS.105 Figure 8 shows a classical micrograph of a solution annealed (SA) AuSS with dislocation loops and line segments, precipitates, precipitate-associated and matrix voids, and possibly He bubbles (the small cavities). RT-based modeling studies of void swelling began in the early 1970s,106,107 peaking in the 1980s, and continuing up to recent times.108 Most of the earliest models emphasized the complex effects of He on void swelling.109,110 As discussed in more detail below, these and later models rationalized many observed swelling trends and also suggested approaches to developing more swelling-resistant AuSS, largely based on trapping
The Effects of Helium in Irradiated Structural Alloys
153
16 HT-9 9Cr–1Mo 2 1/4Cr–1Mo 316SS PCA
Void swelling (%)
14 12
Unirradiated fuel cladding tube
1 cm
10 8 6 4 2 0
(a)
0
50
100 150 200 Displacement dose (dpa)
250
(b)
Figure 7 (a) Typical swelling versus dpa curves for standard 316 AuSS (316SS), a swelling-resistant AuSS (PCA), and various ferritic–martensitic steels (HT9, 9C–1Mo, and 21/4C–1Mo). Reproduced from Gelles, D. S. J. Nucl. Mater. 1996, 233, 293; Garner, F. A.; Toloczko, M. B.; Sencer, B. H. J. Nucl. Mater. 2000, 276, 123; Klueh, R. L.; Harries, D.R. High-Chromium Ferritic and Martensitic Steels for Nuclear Applications; American Society for Testing and Materials: Philadelphia, 2001. (b) Illustration of macroscopic swelling. Reproduced from Straalsund, J. L.; Powell, R.W.; Chin, B. A. J. Nucl. Mater. 1982, 108–109, 299.
Figure 8 Typical microstructures observed in irradiated solution annealed (SA) AuSS composed of dislocation loops, network dislocations, precipitates, and voids, including both those in the matrix and associated with precipitates (by courtesy of J. Stiegler).
He in small bubbles at the interfaces of fine-scale precipitates. Reviews summarizing mechanisms and modeling of swelling carried out during this period, including the role of He, can be found in Odette,111 Odette, Maziasz and Spitznagel,112 Mansur,113 Mansur and Coghlan,114 Freeman,115 and Mansur.116
Reviews of experimental studies of void swelling can be found in later studies by Maziasz16 and Zinkle, Maziasz and Stoller.117 Further motivation for understanding He effects was stimulated by a growing interest in the effects of the very high transmutation levels produced in fusion reactor spectra (see Section 1.06.1).89,99,111,112,118 Experimental studies comparing microstructural evolutions in AuSS irradiated in fast (lower He) and mixed spectrum (high He) reactors provided key insight into the effects of He.16,119,120 Helium effects were also systematically studied using dual-beam He–heavy ion CPI.26,121–129 Beginning in the mid-1970s, a series of studies specifically addressed the critical question of how to use fission reactor data to predict irradiation effects in fusion reactors,15,109–112,118,130–133 and this topic remains one of intense interest to this day. An indication of the complexity of He effects is illustrated in Figure 9, showing microstructures in a dual-beam He–heavy ion irradiation of a SA AuSS to 70 dpa and 625 C at different He/dpa.123 In this case, voids do not form in the single heavy ion irradiation without He. At intermediate levels, of 0.2 appm/dpa, large voids are observed, resulting in a net swelling of 3.5%. At even higher levels of 20 appm/dpa, the voids are more numerous, but smaller, resulting in less net swelling of 1.8%. These observations show that some He promotes the formation of voids, but that higher amounts can reduce swelling. Figure 10 shows the effect of various conditions for
154
The Effects of Helium in Irradiated Structural Alloys
100 nm (a)
(b)
(c)
Figure 9 The effects of the He/dpa ratio on void swelling in a dual ion-irradiated AuSS at 70 dpa and 625 C. The void volume is largest at the intermediate He/dpa ratio of 0.2 appm dpa1, which falls between the limits of 0 and 20 appm dpa1. Reproduced from Kenik, E. A.; Lee, E. H. In Irradiation Effects on Phase Stability; Holland, J. R., Mansur, L. K., Potter, D. I., Eds.; TMS-AIME, Pittsburgh PA, 1981; p 493.
(a)
100 nm
S = 18%
(b)
S = 11%
(c)
(d)
S = 4%
S = 1%
Figure 10 The effects of He and He implantation conditions on the cavity microstructure in ion-irradiated pure Fe–Ni–Cr SA AuSS (note that this is a different alloy from the one in Figure 9) at 70 dpa and 625 C, where the swelling is given in : (a) no He (18%); (b) 1400 appm He co-implanted (11%); (c) 1400 appm He hot preinjected at 900 K (4%); and (d) 1400 appm He cold preimplanted at 20 C (1%). Reproduced from Packan, N. H.; Farrell, K. J. Nuc. Mat. 1979, 85–86, 677–681.
introducing 1400 appm He coupled with a 4 MeV Ni ion irradiation of a swelling-prone model SA AuSS to 70 dpa at 625 C.125 In this case, the swelling is largest (18% due to voids) with no implanted He and smallest (1%) with He preimplanted at ambient temperature due to the very high density of bubbles. These results also show that voids can form
at sufficiently high CPI damage rates without He, probably assisted by the presence of impurities like oxygen and hydrogen. Most notably, however, the swelling decreases with increasing bubble number densities. The emphasis of more recent experimental work has been on SPNI that generate large amounts
The Effects of Helium in Irradiated Structural Alloys
of He, compared with fission reactors, as well as displacement damage (see Section 1.06.4). The SPNI studies have focused on mechanical properties and microstructures, primarily at lower irradiation temperatures, nominally below the HTHE regime. In addition, as discussed in Section 1.06.2, a previously proposed in situ He injection technique31,49 has recently been developed and implemented to study He–displacement damage interactions in mixed spectrum reactor irradiations (e.g., HFIR) at reactorrelevant dpa rates.23,51–53 As discussed in Section 1.06.5, recent modeling studies have emphasized electronic and atomistic evaluations of the energy parameters that describe the behavior of He in solids, including interactions with point and extended defects134–136 (and see Section 1.06.5). The refined parameters are being used in improved RTand Monte Carlo models of He diffusion and clustering to form bubbles on dislocations, precipitates, and GBs, as well as in the matrix, as discussed in Section 1.06.6. It is again important to emphasize that the broad framework for predicting He effects is an understanding and modeling of its generation, transport, and fate, as well as the multifaceted consequences of this fate. We begin with a discussion of the role of He in void swelling and other microstructural evolution processes. We then return to the issue of HTHE. 1.06.3.3 Void Swelling and Microstructural Evolution: Mechanisms The previous section included examples of void swelling. Voids result from the clustering of vacancies produced by displacement damage, as characterized by the number of dpa. Atomic displacements produce equal numbers of vacancy and SIA defects. As noted previously, descriptions of swelling mechanisms, including the role of He, can be found in excellent reviews.113–116 Early RT models showed that swelling is due to an excess flux of vacancies to voids, which is a consequence of a corresponding excess flux of SIA to biased dislocation sinks.106,107 Typical displacement rates (Gdpa) in high-flux reactors (HFR) are 10–6–10–7 dpa s1. Hence, an irradiation time of 108 s (3 years) produces up to 100 dpa. Only about 30% of the primary defects survive shorttime cascade recombination.137 The residual defects undergo long-range migration and almost all either recombine with each other or annihilate at sinks. However, a small fraction of SIA and vacancies cluster to form dislocation loops and cavities,
155
respectively. Ultimate survival of only 0.1% of the dpa in the form of clustered vacancies leads to 10% swelling at 100 dpa. Classical models138,139 demonstrated that for the low Gdpa in neutron irradiations, homogeneous void nucleation rates are very low at temperatures in the peak swelling regime for AuSS between about 500 and 600 C. However, heterogeneous void nucleation on He bubbles is much more rapid than homogeneous nucleation.109 Indeed, nucleation is not required when the He bubbles reach a critical size (r*) and He content (m*). The CBM concept has provided a great deal of insight into the effects of He on swelling.15,109– 112,114,118,130–133,140–151 In particular, the CBM rationalized the extended incubation dpa in fast reactor irradiations prior to the onset of rapid swelling. As previously shown in Figure 2(d) and 2(e), here we clearly distinguish between bubbles, which shrink or grow only by the addition of He, from larger voids, which grow unstably by the continuous accumulation of vacancies. In the case of bubbles, the gas pressure (p) plus a chemical stress due to irradiation (see Section 1.06.3.4) just balances the negative capillary stress 2g/ rb, where g is the surface energy and rb the bubble radius. By definition drb/dt ¼ 0 for bubbles, while the growth rate is positive and negative for cavities that are slightly smaller and larger than rb, respectively. In the case of voids (v), drv/dt is positive at all rv greater than the critical radius. Voids are typically underpressurized with p << 2g/rv. More generally, cavities include both bubbles and voids and can contain an arbitrary number of vacancies (n) and He atoms (m). The evolution of the number of discrete vacancy (n)–He (m) cavities, N(n,m), in a two-dimensional nm space can be numerically modeled using cluster dynamics (CD) master equations. In the simplest case of growth or shrinkage by the absorption or emission of the monomer diffusing species (He, vacancies, and SIA), an ordinary differential equation (ODE) for each n,m cluster, dN(n,m)/dt, tracks the transitions from and to all adjacent cluster classes (n 1 and m 1), as characterized by He, vacancy, and SIA rates of being absorbed (bHe,v,i) and the vacancy emission (av) rate, as dN ðn; mÞ=dt ¼ bHe ðn; m 1ÞN ðn; m 1Þ þ bv ðn 1; mÞN ðn 1; mÞ av ðn þ 1; mÞN ðn þ 1; mÞ þ bi ðn þ 1; mÞN ðn þ 1; mÞ ½bHe ðn; mÞ bv ðm; nÞ bi ðn; mÞN ðn; mÞ
½1
Note that thermal SIA and He emission rates are low and need not be included in eqn [1]. However,
156
The Effects of Helium in Irradiated Structural Alloys
He may be dynamically resolutioned by displacement cascades.152,153 There are a total of nmax mmax such coupled ODEs. The rate coefficients, a and b, are typically computed from solutions to the diffusion equation, to obtain cavity sink strengths,107,113–116 along with the concentrations of the various species in the matrix and vacancies in local thermodynamic equilibrium with the cavity surface. The local vacancy concentrations are controlled by the surface energy of the void, g, via the Gibbs Thomson effect, and the He gas pressure.109,139,141 Conservation equations are used to track the matrix concentrations of the mobile He, vacancies, and SIA based on their rates of generation, clustering, loss to all the sinks present, and, for the point defects, vacancy–SIA recombination.144 Similar RT CD methods can also be used to simultaneously model SIA clustering to form dislocation loops, as well as climb driven by the excess flux of SIA to network dislocations.111,144 In AuSS, loop unfaulting produces network dislocations, and network climb results in both production and annihilation of the network segments with opposite signs. Thus, dislocation structures evolve along with the cavities. However, the a and b rate coefficients depend on a number of defect and material parameters that were not well known during the period of intense research on swelling in the 1970s and 1980s, and integrating a very large number of nmax mmax coupled ODEs was computationally prohibitive at the time these models were first proposed. One simplified approach, based on analytically calculating the rate of void nucleation on an evolving distribution of He bubbles, coupled to a void growth model provided considerable insight into the role of He in void swelling.109,111 These early models, which also included parametric treatments of void and bubble densities,110–112 led to the correct, albeit seemingly counterintuitive, predictions that higher He may decrease, or even totally suppress, swelling in some cases, while in other cases swelling is enhanced, or remains unaffected. These early models also predicted the formation of bimodal cavity size distributions, as confirmed by subsequent modeling studies and many experimental observations.111,112,114,118,131,133,134,148,151 Most aspects of void formation and swelling incubation can be approximately modeled based on the CBM concept. A critical bubble is one that has grown to a radius (r*) and He content (m*), such that, upon the addition of a single He atom or vacancy, it immediately transforms into an
unstably growing void (see Figure 2(d) and 2(e)) without the need for statistical nucleation. Note that while a range of n and m clusters are energetically highly favorable compared with equal numbers of He atoms and vacancies in solution, bubbles represent the lowest free energy configuration in the vacancy-rich environments, characteristic of materials experiencing displacement damage. That is, in systems that can swell due to the presence of sink bias mechanisms that segregate excess fractions SIA and vacancies to different sinks and at low reactor relevant damage rates, cavities primarily evolve along a bubble path that can ultimately end in a conversion to voids.
1.06.3.4 The CBM of Void Nucleation and RT Models of Swelling For purposes of discussion and simplicity, the effects of cascade defect clustering and recombination are ignored, and we consider only single mobile vacancies and SIA defects in the simplest form of RT to illustrate the CBM. At steady state, isolated vacancies and SIA are created in equal numbers and annihilate at sinks at the same rate. Dislocation–SIA interactions due to the long-range strain field result in an excess flow of SIA to the ‘biased’ dislocation sinks and, thus, leave a corresponding excess flow of vacancies to other neutral (or less biased) sinks, (DvXv DiXi). Here, D is the defect diffusion coefficient and X the corresponding atomic fraction. Assuming that the defect sinks are restricted to bubbles (b), voids (v), and dislocations (d), the DX terms are controlled by the corresponding sink strengths (Z): Zb (4prbNb), Zv (4prvNv) for both vacancies and SIA; Zd (r) for vacancies and Zdi (r [1 þ B]) for SIA. Here, r and N are the size and number densities of bubbles and voids, r is the dislocation density, and B is a bias factor. At steady state, Dv Xv Di Xi ¼ ½Gdpa Zdi =fðZb þ Zv þ Zd Þ ðZb þ Zv þ Zd ½1 þ BÞg þ Dv Xve ½2 Here, DvXve represents thermal vacancies that exist in the absence of irradiation and (1/3) is the ratio of net vacancy to dpa production. In the absence of vacancy emission, the excess flow of vacancies results in an increase in the cavity radius (r) at a rate given by dr =dtþ ¼ ðDv Xv Di Xi Þ=r
½3
The Effects of Helium in Irradiated Structural Alloys
However, cavities also emit vacancies, resulting in shrinkage at a rate given by the capillary approximation as
m/m*= 0.60 0.75
½4
The Xve exp[(2g/r p)O/kT] term is the concentration of vacancies in local equilibrium at the cavity surface, and O is the atomic volume. Thus, the net cavity growth rate is dr =dt ¼fDv Xv Di Xi Dv Xve exp½ð2g=rc pÞO=kT g=r
10
½5
5
dr/dt (10-5 nm s-1)
dr =dt ¼ Dv Xve exp½ð2g=r pÞO=kT =r
157
1.0 1.25
0
-5
Growth stability and instability conditions occur at the dr/dt ¼ 0 roots of eqn [5], when Dv Xv Di Xi Dv Xve exp½ð2g=r pÞO=kT ¼ 0 ½6a Note that DvXve is approximately the self-diffusion coefficient, Dsd. The He pressure is given by p ¼ 3kmkT =4pr 3
½6b
Here, k is the real gas compressibility factor. Equation [6a] can be expressed in terms of the effective vacancy supersaturation, L ¼ ðDv Xv Di Xi Þ=Dsd
½6c
The bubble and critical radius occur at L exp½ð2g=r pÞO=kT ¼ 0
½6d
In the absence of irradiation (or sink bias), L ¼ 1 and all cavities are bubbles in thermal equilibrium, at p ¼ 2g/rb. Assuming an ideal gas, k ¼ 1, eqn [6d] can be written as 2g=r ð3mkT Þ=ð4pr 3 Þ kT lnðLÞ=O ¼ 0
½7a
Note that kT ln(L)/O is equivalent to a chemical hydrostatic tensile stress acting on the cavity. Rearranging eqn [7a] leads to a cubic equation with the form, rc3 þ c1 r 2 þ c2 ¼ 0
½7b
c1 ¼ ½2gO=½kT lnðLÞ
½7c
c2 ¼ ½3mO=½4p lnðLÞ
½7d
As shown in Figure 2(d) and 2(e), eqn [7b] has up to two positive real roots. The smaller root is the radius of a stable (nongrowing) bubble containing m He atoms, rb, and the larger root, rv , is the corresponding critical radius of a (m*,n*) cavity that transforms to a growing void. Voids can, and do, also form by classical heterogeneous nucleation
-10 0.8 0.9 1
2
3
rc (nm) Figure 11 The CBM predictions of radial growth rate of cavities as a function of their He content, m, normalized by the critical He content for conversion of bubbles to growing voids, m*. The effective supersaturation is (L ¼ 4.57), temperature is (T ¼ 500 C), and surface energy is (g ¼ 1.6 J m2) . The two roots in the case of m < m* are for bubbles and voids, respectively. Cavities can transition from bubbles to voids by classical nucleation or reach a m* by He additions. The effect of He on the growth of voids is minimal at sizes larger than about 2.5 nm in this case.
on bubbles between rb and rv .109,132,141 However, as shown in Figure 2(d) and 2(e), as m increases, rb increases and rv decreases, until rb ¼ rv ¼ r* at the critical m*. An example of the dr/dt curves assuming ideal gas behavior taken from Stoller133 is shown in Figure 11 for parameters typical of an irradiated AuSS at 500 C with L ¼ 4.57. The corresponding r* and m* are 1.50 nm and 931, respectively. The critical bubble parameters can be evaluated for a realistic He equation of state using master correction curves, y1(ln L) for m* and y2(ln L) for r*, based on high-order polynomial fits to numerical solutions for the roots of eqn [7b].143 A simpler analytical method to account for real gas behavior based on a Van der Waals equation of state can also be used.151 The results of the two models are very similar.143 Voids often form on critical bubbles located at precipitate interfaces at a smaller m* than in the matrix.142 This is a result of the surface–interface tension balances that determine the wetting angle between the bubble and precipitate interface (see Figure 20(b)). Formation of voids on precipitates can be accounted for by a factor Fv 4p/3, reflecting the smaller volume of a precipitate-associated critical
158
The Effects of Helium in Irradiated Structural Alloys
bubble at r*, compared with a spherical bubble in the matrix, with Fv ¼ 4p/3. Note that the critical matrix and precipitate-associated bubble have the same r*. The m* and r* are given by
3
2
m ¼ ½32Fv y1 g O =½27ðkT Þ ðlnðLÞ Þ 3
2
L f½Gdpa BZd =½ðZb þ Zd Þ ðZd ð1 þ BÞ þ Zb Þ=Dsd g þ 1
Figure 13 shows the corresponding m* and r* as a function of the concentration of 1 nm bubbles, Nb, at 500 and 600 C again using the AuSS parameters given in Stoller.133 Clearly, high Nb can lead to large critical bubble sizes requiring high He contents for void formation. Thus, to a good approximation, the primary mechanism for void formation in neutron irradiations is the gradual and stable, gas-driven growth of bubbles by the addition of He up to near the critical m*.
½8a
r ¼ ½4y2 gO=½3kT lnðLÞ
½9
½8b
Figure 12 shows m*, r* as a function of temperature for typical parameters for SA AuSS steels taken from Stoller.133 More generally, L can simply be related to Dsd, , Gdpa, B, and the sink’s various strengths. Assuming Zv 0 during the incubation period, 107
103
106 102 rc* (nm)
m*He
105 104
101
103 100
Ideal gas 102
HSEOS
101 300
400
500
HSEOS Ideal gas
600
700
10-1 300
400
500
600
700
Temperature (⬚C)
Temperature (⬚C)
Figure 12 Critical bubble model predictions of m* and r* as a function of temperature for parameters typical of an AuSS modified from Stoller.133
10−4
1013
1011
773 K 873 K
10−5 10−6
rc* (m)
m*
109
107
10−7
105
10−8
103
10−9
101 20 10
773 K 873 K
1021
1022
Nb (m−3)
1023
1024
10−10 20 10
1021
1022
1023
1024
Nb (m−3)
Figure 13 Critical bubble predictions of m* and r* as a function of the bubble density (Nb) at 773 and 873 K for parameters typical of a solution annealed AuSS taken from Stoller.133 At low Nb the bubble sink strength is lower than that for dislocations, hence bubbles have little effect on m* and r*. However, at higher bubble densities the bubbles become the dominant sink resulting in rapid increases in m* and r*.
The Effects of Helium in Irradiated Structural Alloys
20
1013 1011
773 K 873 K
109 Voids per interval (%)
15
dpa*
107 105 103
159
400 dpa
101
10
5
10−1 10−3 1020
1021
1022
1023
1024
Nb (m−3) Figure 14 Predicted incubation dpa* for the onset of void swelling as a function of the density (Nb) of 1 nm at 773 and 873 K for parameters typical of a solution annealed AuSS taken from Stoller.133 The dpa* increases linearly with Nb at lower bubble densities, simply because the He partitions to more sites. However, in the bubble sink dominated regime, dpa* scales with N5b . The horizontal dashed line shows a dose of 400 dpa.
Although nucleation is rapid on bubbles with m close to m*, modeling void formation in terms of evaluating the conditions leading to the direct conversion of bubbles to voids is a good approximation.132 The corresponding incubation dpa (dpa*) needed for Nb bubbles to reach m* is given by dpa ¼ ½m Nb =½He=dpa
½10
Figure 14 shows dpa* for He/dpa ¼ 10 appm dpa1 and the same AuSS parameters used in Figure 13. Clearly high Nb increases the dpa*, both by increasing the neutral sink strength, thus decreasing L, and partitioning He to more numerous bubble sites. Indeed, in the bubble-dominated limit, Zb >> Zd and Zv, the dpa* scales with N5b! The CBM also predicts bimodal cavity size distributions, composed of growing voids and stable bubbles. Once voids have formed, they are sinks for both He and defects, and thus slow and eventually stop the growth of the bubbles to the critical size and further void formation. Figure 15 shows a bimodal cavity versus size distribution histogram plot for a Ni–He dual ion irradiation of a pure stainless steel,114 and many other examples can be found in
0
0
20
40 60 Void diameter (nm)
80
Figure 15 A typical example of a bimodal cavity size distribution composed of small bubbles and large voids in a Ni–He dual ion-irradiated AuSS at 670 C, 10 dpa, and a 20 appm He/dpa. Reproduced from Mansur, L. K.; Coghlan, W. A. J. Nucl. Mater. 1983, 119, 1.
the literature111,112,114,133,153 Figure 16(a) shows low He favors the formation of large voids in a CW stainless steel irradiated in experimental breeder reactor-II (EBR-II) to 40 dpa at 500 C and 43 appm He, resulting in 12% swelling, while Figure 16(b) shows that the same alloy irradiated in HFIR at 515– 540 C to 61 dpa and 3660 appm He has a much higher density of smaller cavities, resulting in only 2% swelling.16 Thus, while He is generally necessary for void formation, very high bubble densities can actually suppress swelling for the same irradiation conditions as also shown previously in Figures 9 and 10. This can lead to a nonmonotonic dependence of swelling on the He/dpa ratio. One example of a model prediction of nonmonotonic swelling is shown in Figure 17.154 Note that unambiguous interpretations of neutron-irradiation data are often confounded by uncertainties in irradiation temperatures and complex temperature histories.155,156 However, the suppression of swelling by high Nb is clear even in these cases. Bubble sinks can also play a significant role in the post-incubation swelling rates. Neglecting vacancy emission from large voids, and using the same assumptions described above, leads to a simple expression _ the rate of for the overall normalized swelling rate S,
160
The Effects of Helium in Irradiated Structural Alloys
0.25 mm
0.10 mm
(b)
(a)
Figure 16 Comparison of swelling in a 20% CW AuSS irradiated in fast spectrum Experimental Breeder Reactor-II (EBR-II, low He) and the mixed spectrum HFIR (high He) reactors: (a) the EBR-II irradiation at 510 C to 69 dpa and 43 appm He produced 12% swelling; (b) the HFIR irradiation at 515–540 C to 61 dpa an appm He/dpa ratio 60 produced 2% swelling. Reproduced from Maziasz, P. J. J. Nucl. Mater. 1993, 205, 118.
50
5 PCA, 500–520 ⬚C, 11–13 dpa
Swelling (DV/VO, %)
3
SA CW FFTF ORR HFIR Model predictions 550 ⬚C, 75 dpa CW 316
40
30
2
20
1
10
0 0
20 40 60 He/dpa ratio (appm He/dpa)
Swelling (DV/V, %)
Fusion range
4
0 80
Figure 17 Data for irradiations at 500–520 C of CW and SA AuSS suggesting that swelling peaks at an intermediate He/dpa ratio, reasonably consistent with the trend of model predictions (lines) at higher temperature and dpa. Reproduced from Stoller, R. E. J. Nucl. Mater. 1990, 174(2–3), 289.
increase in total void volume per unit volume divided by the displacement rate as S_ ¼ ½BZd Zv =½ðZb þ Zv þ Zd ÞðZd ð1 þ B Þ þ Zv þ Zb Þ ½11 Figure 18 shows S_ for B ¼ 0.15 and ¼ 0.3 as a function of Zv/Zd, with a peak at Zd ¼ Zv and
Zb 1, representing the case when nearly all the bubbles have converted to voids and balanced void and dislocation sink strengths. The S_ decreases at higher and lower Zv/Zd. Figure 18 also shows S_ as a function of Zv/Zd for a range of Zb/Zv. Increasing Zb with the other sink strengths fixed reduces the S_ in the limit scaling with 1=Zb2 . These results again show that significant swelling rates require some
The Effects of Helium in Irradiated Structural Alloys
10-1 10-2 10-3
· S
10-4 10-5 10-6
Zb/Zv = 0.1 Zb/Zv = 1
10-7
Zb/Zv = 10 Zb/Zv = 100
10-8 10-3
10-2
10-1
100
101
102
103
Zv / Zd _ for various bubble Figure 18 Predicted swelling rate (S) to void sink strength ratios (Zb/Zv) as a function of the void to dislocation sink ratio (Zv/Zd). The highest S_ is for a low Zb/Zv at a balanced void and dislocation sink strengths Zv Zb. S_ decreases with increasing Zb/Zv and the corresponding peak rate shifts to lower Zb/Zv.
bubbles to form voids with a sink strength of Zv that is not too small (or large) compared with Zd. However, a large population of unconverted bubbles, with a high sink strength Zb, can greatly reduce swelling rates. A significant advantage of the CBM is that it requires a relatively modest number of parameters, and parameter combinations, that are generally reasonably well known, including for defect production, recombination, dislocation bias, sink strengths, interface energy, and Dsd. Potential future improvements in modeling bubble and void evolution include better overall parameterization using electronic–atomistic models, a refined equation of state at small bubble sizes, and precipitate specific estimates of Fv based on improved models and direct measurements. Further, it is important to note that the CBM parameters can be estimated experimentally as the pinch-off size between the small bubbles and larger voids.114,124,157 Application of CBM to void swelling requires treatment of the bubble evolution at various sites, including in the matrix, on dislocations, at precipitate interfaces, and in GBs. Increasing the He generation rate (GHe) generally leads to higher bubble concenp trations, scaling as Nb / GHe .111,112,131–133,140,144,172 The exponent p varies between limits of 0, for totally heterogeneous bubble nucleation on a fixed number of deep trapping sites, to >1 when the
161
dominant He fate is governed by trap binding energies, large He bubble nucleus cluster sizes (most often assumed to be only two atoms), and loss of He to other sinks. Assuming the dominant fate of He is to form matrix bubbles, p has a natural value of 1=2 for the condition that the probability of diffusing He to nucleate a new matrix bubble as a di-He cluster is equal to the probability of the He being absorbed in a previously formed bubble.158 Bubble formation is also sensitive to temperature and depends on the diffusion coefficient and mechanism, as well as He binding energies at various trapping sites. Substitutional He (Hes) diffuses by vacancy exchange with an activation energy of Ehs 2.4 eV.159 For bimolecular nucleation of matrix bubbles, Nb scales as exp(Ehs/2kT ). Helium can also diffuse as small n 2 and m 1 vacancy–He complexes, but bubbles are essentially immobile at much larger sizes. Helium is most likely initially created as interstitial He (Hei), which diffuses so rapidly that it can be considered to simply partition to various trapping sites, including vacancy traps, where Hei þ V ! Hes. Note that, for interstitial diffusion, the matrix concentrations of Hei are so low that migrating Hei–Hei reactions would not be expected to form He bubbles. Thermal detrapping of Hes from vacancies to form Hei is unlikely because of the high thermal binding energy160 and see Section 1.06.5 for other references) but can occur by a Hes þ SIA ! Hei reaction, as well as by direct displacement events.152,161 If Hei and Hes maintain their identities at trapping sites, they can detrap in the same configuration. Clustering reactions between Hes, Hei, and vacancies form bubbles at the trapping sites. Thus, He binding energies at traps are also critical to the fate of He and the effects of temperature and GHe. Traps include both the microstructural sites noted above as well as deeper local traps within these general sites, such as dislocation jogs and grain boundary junctions136 (and see Section 1.06.5 for other references). If the trapping energies are low, or temperatures are high, He can recycle between various traps and the matrix a number of times before it forms or joins a bubble. However, once formed bubbles are very deep traps, and at a significant sink density, they play a dominant role in the transport and fate of He. In principle, the binding energies of He clusters are also important to bubble nucleation. Recent ab initio simulations have shown that even small clusters of Hei in Fe are bound, although not as strongly as Hes–V complexes. Indeed, the binding energies of
162
The Effects of Helium in Irradiated Structural Alloys
small HemVn complexes with n m are large (2.8– 3.8 eV),134,135 suggesting that the bi- or trimolecular bubble nucleation mechanism is a good approximation over a wide range of irradiation conditions. Further, for neutron-irradiation conditions with low GHe and Gdpa that create a vacancy-rich environment, it is also reasonable to assume that He clusters initially evolve along a bubble-dominated path. As discussed previously, the effects of higher bubble densities on overall microstructural evolutions are p complex. The observation that Nb scales as GHe relation has been used in many parametric studies of the effects of varying bubble and void microstructures. Bubble nucleation and growth and void swelling are suppressed at very low GHe. However, as noted above, swelling can sometimes decrease beyond a critical GHe due to higher Nb. Indeed, void formation and swelling can be completely suppressed by a very high concentration of bubbles. High bubble concentrations can also suppress the formation of dislocation loops and irradiation-enhanced, induced, and modified precipitation associated with solute segregation, by keeping excess concentrations of vacancies and SIA very low.16,26,111,112,162 1.06.3.5 Summary: Implications of the CBM to Understanding He Effects on Swelling and Microstructural Evolution Void swelling is only one component of microstructural and microchemical evolutions that take place in alloys under irradiation. In addition to loops and network dislocations, other coevolutions include solute segregation and irradiation–enhanced–induced– altered precipitation. In the mid-1980s, CBM and RT models of dislocation loop and network evolution were self-consistently integrated in the computer code MicroEv, which also included a parametric treatment of precipitate bubble–void nucleation sites.133,144 Later work in the 1990s further developed and refined this code.163 A major objective of much of this research was to develop models to make quantitative predictions of the effect of the He/dpa ratios on void swelling for fusion reactor conditions. CBMs have been used to parametrically evaluate the effects of many irradiation variables and material parameters15,114,118,128,129,140,149,150 as well as to model swelling as a function of temperature, dpa and dpa rates, and the He/dpa ratio (see both Stoller and Odette references). The CBMs have also been both informed by and compared with data from experiments in both fast and mixed thermal–fast
spectrum test reactors, including EBR-II (fast), FFTF (fast), and HFIR (mixed),16,119 complemented by extensive dual ion CPI results.26,124,125,128,129,157,164a,164–171 The semiempirical CBM models and concepts rationalize a wide range of seemingly complex and sometimes disparate observations, including the following:
Void nucleation on bubbles
The general trends in the temperature, dpa, and He/dpa dependence of the number densities of bubbles and voids
Incubation dpa and postincubation swelling rates, including the effects of temperature and stress
The occurrence of bimodal cavity size distributions of small He bubbles and larger voids
Bubble nucleation on dislocations and precipitate interfaces
Swelling that is increased, decreased, or unaltered by increasing GHe, depending on the combination of other irradiation and material variables
Suppression of void swelling by a very high number of densities of bubbles
Highly coupled concurrent evolutions of all the microstructural features, resulting in weaker trend toward refinement of precipitate and loop structures at higher GHe and, in the limit of very high Nb, suppression of loops and precipitation
Strong effects of the schedule and temperature history of He implantation in CPI
Effects of alloying elements on swelling incubation associated with corresponding influence on precipitation, solute segregation, and the self-diffusion coefficient
Swelling resistance of AuSS that have stable finescale precipitates that trap He in small interface bubbles
The much higher swelling resistance of bcc FMS compared with fcc AuSS The concept of trapping He in a high number density of bubbles to enhance the swelling and HTHE resistance (and creep properties in general) was implemented in the development of AuSS containing fine-scale carbide and phosphide phases. Figure 19 shows the compared cavity microstructures resulting in 6% void swelling in a conventional AuSS (Figure 19(a)) to an alloy modified with Ti and heat treated to produce a high density of fine-scale TiC (Figure 19(b)) phases with less than 0.2% bubble swelling following irradiation to 45 dpa and 2500 appm He at 600 C.172 There are many other examples of swelling-resistant AuSS that were successful in delaying the onset of swelling to much
The Effects of Helium in Irradiated Structural Alloys
(a)
(b)
163
0.25 mm
Figure 19 Comparison of a conventional AuSS (a) to a swelling-resistant (b) Ti-modified alloy for HFIR irradiations at 600 C to 45 dpa and 2500 appm He. Reproduced from Maziasz, P. J.; J. Nucl. Mater. 1984, 122(1–3), 472.
higher dpa than in conventional AuSS. However, as illustrated in Figure 7, these steels also eventually swell. This has largely been attributed to thermalirradiation instability and coarsening of the fine-scale precipitates that provide the swelling resistance.172 FMS are much more resistant to swelling than advanced AuSS.15,102,104,116,128,129,162,169,174,175 The swelling resistance of FMS, compared with AuSS, can be attributed to a combination of their (a) lower dislocation bias; (b) higher sink densities for partitioning He into a finer distribution of bubbles, thus increasing m*; (c) low void to dislocation sink ratios; (d) a higher self-diffusion coefficient that increases m*; and (e) lower He/dpa ratios.15,176 However, void swelling does occur in FMS, as well as in unalloyed Fe,177 and is clearly promoted by higher He/dpa ratios. Higher He can decrease incubation times for void formation and increase Zv/Zd ratios closer to 1, resulting in higher swelling rates.52,157,168–171 Recent models predict significant swelling in FMS,178 and the potential for high postincubation swelling rates in these alloys remains to be assessed. Swelling in FMS clearly poses a significant life-limiting challenge in fusion first wall environments in the temperature range between 400 and 600 C. NFA, which are dispersion strengthened by a high density of nanometer-scale Y–Ti–O-enriched features, are even more resistant to swelling and other manifestations of radiation damage than FMS.22,23,51,179,180 Irradiation-tolerant alloys will be discussed in Section 1.06.6. 1.06.3.6 HTHE Critical Bubble Creep Rupture Models The CBM concept can also be applied to the effects of grain boundary He on creep rupture properties. Stress-induced dislocation climb also results in generation excess vacancies that can accumulate at
growing voids. In particular, tensile stresses normal to GBs (s) generate a flux of vacancies to boundary cavity sinks, if present, and an equal, but opposite, flux of atoms that plate out along the boundary as illustrated in Figure 20(a). The simple capillary condition for the growth of empty cavities is the s > 2g/ r. In this case of cavities containing He, the growth rate is given by dr =dt ¼ ½ðDgb dÞ=ð4pr 2 Þ f1 exp½ð2g=r P sÞO=kT g
½12
Here Dgb and d are the grain boundary diffusion coefficient and thickness, respectively. The corresponding dr/dt ¼ 0 conditions also lead to a stable bubble (rb) and unstably growing creep cavity (r*) roots. As noted previously, a vacancy supersaturation, L, produces a chemical stress that is equivalent to a mechanical stress s ¼ kT ln(L)/O. Thus, replacing ln(L) in eqn [8a] and [8b] with sO/kT directly leads to expressions for m* and r* for creep cavities m ¼ ½32Fv pg3 =½27kT s2
½13a
r ¼ 0:75g=s
½13b
This simple treatment can also be easily modified to account for a real gas equation of state. Note that it is usually assumed that GBs are perfect sinks for both vacancies and SIA. Thus, it is generally assumed that displacement damage does not contribute to the formation of growing creep cavities. Understanding HTHE requires a corresponding understanding of the basic mechanisms of creep rupture in the absence of He. At high stresses and short rupture times, the normal mode of fracture in AuSS is transgranular rupture, generally associated with power law creep growth of matrix cavities.181,182 However, at lower stresses IG rupture occurs in a
164
The Effects of Helium in Irradiated Structural Alloys
sa sb Plating atoms Jv
Ja
Creeping grain cage
Matrix
sn
Cavities
GB
sn sb
GB particle
sa
(a)
(b)
Figure 20 (a) A schematic illustration of cavity growth by vacancy diffusion and atom plating when subject to an applied stress, sa. The surrounding cage of uncavitated grains must creep to accommodate the displacements caused by the cavitated grain boundary. This results in a back stress, sb, that reduces the net stress, sn, on the grain boundary (sn ¼ sa sb) so that the deformation processes come to a steady-state balance, where the creep rate controls the cavity growth rate. (b) A schematic illustration of the differences in the volume of cavities with the same radius of curvature that are located in the matrix, on grain boundaries, and on grain boundary particles. Smaller volumes reduce the critical m* for conversion of bubbles to creep cavities due to the applied stress. The same mechanism occurs for bubble to void conversions associated with chemical stresses due to irradiation-induced vacancy supersaturation.
wide range of austentic and ferritic alloys. Although space does not permit proper citation and review, it is noted that a large body of literature on IG creep rupture emerged in the late 1970s and early 1980s. Briefly, this research showed that under creep conditions a low to moderate density of grain boundary cavities forms (1010–1012 m2), usually in association with second-phase particles and triple-point junctions.183–184a Grain boundary sliding results in transient stress concentrations at these sites, and interface energy effects at precipitates also reduce the critical cavity volume (Fv 4p/3) relative to matrix voids, as illustrated in Figure 20(b). Once formed, however, creep cavities can rapidly grow and coalesce if unhindered vacancy diffusion and atom plating take place along clean GBs. Such rapid cavity growth rates lead to short rupture times in low creep strength, single-phase alloys. Thus, useful high-temperature multiphase structural alloys must be designed to constrain creep cavity nucleation and growth rates by a variety of mechanisms. For example, grain boundary phases can inhibit dislocation climb and atom plating.185 As illustrated in Figure 20(a), growth cavities, which are typically not uniformly distributed on all grain boundary facets, can be greatly inhibited by the constraint imposed by creep in the surrounding cage of grains, which is necessary to accommodate the cavity swelling and grain boundary displacements.186 Creep
stresses in the grains impose back stresses on the GBs that result in compatible deformation rates. Thus, it is the accommodating matrix creep rate that actually controls the rate of cavity growth, rather than grain boundary diffusion itself. Creep-accommodated, constrained cavity growth rationalizes the Monkman–Grant relation187 between the creep rate (e0 ), the creep rupture time (tr), and a creep rupture strain (ductility) parameter (er) as e0 tr ¼ er
½14a
Thus, in high-strength alloys, low dislocation creep rates (e0 ) lead to long tr. The typical form of e0 e0 ¼ Asr expðQcr =kT Þ
½14b
The effective stress power r for dislocation creep is typically much greater than 5 for creep-resistant alloys, and the activation energy for matrix creep of Q cr 250–350 kJ mol1 is on the order of the bulk self-diffusion energy.181 These values are much higher than those for unconstrained grain boundary cavity growth, with r 1–3 and Q gb 200 kJ mol1. A number of creep rupture and grain boundary cavity growth models were proposed based on these concepts.186,188,189 Note that there are also conditions, when grain boundary vacancy diffusion and atom plating are highly restricted and cavities are well separated, where matrix creep enhances, rather than constrains, cavity growth. As noted above, power law creep controls matrix cavity growth at high stress,
The Effects of Helium in Irradiated Structural Alloys
leading to transgranular fracture.181,182 Models of the individual, competing, and coupled creep and cavity growth processes have been used to construct creep and creep rupture maps that delineate the boundaries between various dominant mechanism regimes. However, further discussion of this topic is beyond the scope of this chapter. Accumulation of significant quantities of grain boundary He has a radical effect on creep rupture, at least in extreme cases. First, at high He levels, the number density of grain boundary bubbles (Ngb) and creep cavities (Nc) is usually much larger than the corresponding number of creep cavities in the absence of He; the latter is of the order 1010–1012 m2.181,190 Figure 21 shows the evolution of He bubbles and grain boundary cavities under stress.191 Indeed, Ngb of more than 1015 m2 have been observed in high-dose He implantation studies.100,192 Although Ngb is not well known for neutron-irradiated AuSS, it has been estimated to be of the order 1013 m2 or more.193,194 At high He levels, a significant fraction of the grain boundary bubbles convert to growing creep cavities, resulting in high Nc. Of course, both Ngb and Nc depend on stress as well as many material parameters and irradiation variables, especially those that control the amount of He that reaches and clusters on GBs. As less growth is required for a higher density of cavities to coalesce, creep rupture strains,
YE-11560
YE-11611
(a)
(b) 0.1 mm
0.1 mm
Figure 21 The growth of grain boundary bubbles and their conversion to creep cavities in an AuSS: (a) bubbles on grain boundaries of a specimen injected with 160 appm and annealed at 1023 K for 6.84 104 s; (b) the corresponding cavity distribution for an implanted specimen annealed at 1023 K for 6.84 104 s under a stress of 19.6 MPa. Reproduced from Braski, D. N.; Schroeder, H.; Ullmaier, H. J. Nucl. Mater. 1979, 83(2), 265.
165
er, roughly scale with Nc1=2 . Bubble-nucleated creep cavities are also generally more uniformly distributed on various grain boundary facets. More uniform distributions and lower er decrease accommodation constraint, thus, further reducing rupture times associated with cavity growth. Equation [13a] suggests that m* scales with 1/s2. If the GB bubbles nucleate quickly and once formed the creep cavities rapidly grow and coalesce, then creep rupture is primarily controlled by gas-driven bubble growth to r* and m*.93–95,97 In the simplest case, assuming a fixed number of grain boundary bubbles Ngb and flux of He to the grain boundary, JHe, the creep rupture time, tr, is approximately given by tr ¼ f½Fv 32pg3 =½27kT s2 g½Ngb =JHe Ngb =½GHe s2 ½15
Note that this simple model, predicting tr / 1/s2 scaling, is a limiting case primarily applicable at (a) low stress; (b) when creep rupture is dominated by He bubble conversion to creep cavities by gas-driven bubble growth to r*; and (c) when diffusion (or irradiation) creep-enhanced stress relaxations are sufficient to produce compatible deformations without the need for thermal dislocation creep in the grains. More generally, scaling of tr / 1/sr, r 2 is expected for bubbles containing a distribution of m He atoms. For example, if Ngb scales as mq, then Nc would scale as s2q.194,195 Further, at higher s, hence lower tr , there is less time for He to collect on GBs. Thus in this regime, intragranular dislocation creep, with a larger stress power, r, may return as the rate-limiting mechanism controlling the tr – s relations. Equation [15] also provides important insight into the effect of both the grain boundary and matrix microstructures. Helium reaches the GBs (JHe) only if it is not trapped in the matrix. Matrix bubbles are, by far, the most effective trap for He.95,97 If it is assumed that the number of matrix bubbles, Nb, is proportional to √GHe while the grain boundary bubble number density (Ngb) is fixed, a scaling relation for tr can be approximated as pffiffiffiffi tr / Ngb =½ G He s2 ½16 if the number of grain boundary bubbles also scales 1=4 with √JHe, tr scales with GHe . At the other extreme, if Nb and Ngb are both independent of the He con1 (eqn [15]). Thus, microcentration, tr scales with GHe structures with high Nb (and Zb) that are resistant to void swelling are also likely to be resistant to HTHE. HTHE models for AuSS were developed based on these concepts and various elaborations93–96,182,190,194,196,197 as well as methane
166
The Effects of Helium in Irradiated Structural Alloys
gas-driven constrained growth of grain boundary cavities.198 The HTHE models developed by Trinkaus and coworkers were closely integrated with the extensive He implantation and creep rupture studies discussed further below. It should be emphasized that the HTHE models cited above are only qualitative and primarily represent simple scaling concepts that must be validated and calibrated using microstructural, creep rate, and creep rupture data. For example, more quantitative models require detailed treatment of He accumulation and redistribution at the GBs into a stably growing population of bubbles, with a time-dependent fraction that ultimately converts to growing creep cavities. 1.06.3.7 Experimental Observations on HTHE The results of experimental studies on He embrittlement of AuSS are broadly consistent with the concepts described here. However, the literature for neutron irradiations is much more limited than in the case of microstructural evolution and matrix swelling, especially for the most pertinent data from reliable inreactor creep rupture tests. Indeed, there is little quantitative characterization of grain boundary cavity and other microstructures for neutron-irradiated alloys. The most consistent trend for neutron irradiations is that high-temperature postirradiation tensile tests show significant to severe reductions in tensile ductility and creep rupture times and IG rupture along GBs.11
As noted above, there is a much more significant body of work for well-characterized high-energy He ion implantation studies. Helium can be preimplanted at various temperatures and further subjected to various postimplantation annealing treatments, prior to tensile or creep testing, or simultaneously with creep testing. The different modes of He implantation result in very different creep rupture behavior.90 Helium implantation during high temperature in-beam creep is perhaps the most relevant, controlled, and systematic approach to studying HTHE. A series of implantation studies carried out at the Research Center Ju¨lich in Germany, coupled with the models described above, are the most comprehensive and insightful examples of this research.90,91,99,100,192,199–201 Figure 22(a) shows the mean trend lines for tr versus applied stress for SA 316SS at 1023 K for in-beam creep, at an implantation rate of 100 appm He/h, compared with unimplanted controls.90 Clearly, HTHE leads to a very large reduction in the tr especially at lower stress. The stress power is r 4 for the in-beam creep condition, compared with 9 for the unimplanted control. Figure 22(b) shows a corresponding plot for a Ti-modified AuSS (DIN 1.4970) in-beam creep tested at 1073 K.90 HTHE is observed, but the magnitude of the reduction in tr is less in this case. The stress power in-beam creep condition is r 2.85 compared with 5.7 for the control. As expected, HTHE also reduces er; in the case of Ti-modified steel, er decreases from 10% to 1%90 and for the annealed 316SS from more than 30% to 1% or less.99 Similar
104
1000
Unimplanted control In-beam Implanted at 1073 K
No He beam 100 appm He/h 1000
100
tr (h)
tr (h)
100 10
10 1
1
0.1 10
(a)
100
sa (MPa)
0.1 80
1000
(b)
100
120
140
160
180
200
sa (MPa)
Figure 22 (a) The creep rupture time versus stress for a 316 AuSS tested at 1023 K under He implantation at a rate of 100 appm h1 and the corresponding unimplanted control showing severe HTHE. (b) The creep rupture time versus stress for a Ti-modified AuSS tested at 1073 K also under He implantation at a rate of 100 appm h1 and the corresponding unimplanted control. Data for He preimplantation to 100 appm at 1073 K is also shown.90 Note that the Ti-modified AuSS is much stronger than the 316 alloy in spite of the higher test temperature and that the effect of HTHE is mitigated at lower stresses in this alloy.
The Effects of Helium in Irradiated Structural Alloys
q
XHer / tr / GHe
½17
The in-beam creep tests suggest that q 1/3. Thus, for example, if tr ¼ 10 h at XHe ¼ 1000 appm (see above), for in-beam creep tests, this scaling predicts tr 2700 h at XHe 60 appm for He generated at fusion reactor rates of 200 appm year1. Note that this is not a reliable absolute estimate of the creep rupture time since the in-beam irradiation experiments involved very thin specimens (0.1 mm). Figure 23 shows tr versus stress for in-reactor creep tests at 993 K under neutron irradiation. The corresponding ‘control’ curve at 993 K is based on a logarithmic interpolation between curves for
104
1000
tr (h)
comparisons for a test of these two alloys at 873 K also show severe reductions in tr and er in 316SS, whereas there is a much smaller effect in the Ti-modified AuSS.192 This work also showed that the Ti-modified alloy is much stronger in CW condition and suffers only moderate HTHE. Helium implantation of unstressed specimens at 1023 K to levels between 10 and 1000 appm was used to evaluate the critical bubble size resulting in rapid rupture in subsequent creep tests at the same temperature.91 A rapid drop-off in tr and er occurred between 300 and 1000 appm at a stress of 90 MPa. This He concentration correlated with an average grain boundary bubble size of 13–17 nm, which provides a reasonable estimate of the corresponding critical bubble size for these conditions. However, it must be emphasized that in-beam creep tests do not ‘simulate’ neutron-irradiation conditions since the He implantation rates are highly accelerated and yield very high He/dpa ratios. Further, the duration of these tests is limited by schedules of the ion accelerators. The in-beam creep data show that the tr and er decrease with 1=3 decreasing implantation rates scaling as GHe 91 (Schroeder et al. and see previous discussion). This may be due to the effect of GHe on the number density of matrix bubbles that help shield GBs from He accumulation, as suggested by in eqn [16]. Indeed, matrix bubble densities, Nb, in 316SS increase with the concentration of He preimplanted at 1023 K, 1=2 scaling as XHe at greater than 100 appm He. The corresponding number of grain boundary bubbles, Ngb, is insensitive to XHe.199 Of course, the scaling applies only to high implantation rates that result in significant He concentrations at all GHe. A more relevant scaling law would be based on the He required for creep rupture (XHer), which also scales with tr , as
167
100
10 In-pile 720 ⬚C Control
1 10
100
1000
sa (MPa) Figure 23 Creep rupture time (tr) versus stress (s) trends at 993 K for a Ti-modified AuSS under neutron irradiation (in pile) that generates from 12 to 95 appm He showing severe HTHE at low stresses. Reproduced from Schroeder, H.; Batfalsky, P. J. Nucl. Mater. 1983, 117, 287.
specimens preimplanted with up to 80 appm He at ambient temperature and tested at 973 and 1073 K. Note that data at 1073 K suggest that at such lowtemperature preimplantation has little effect on tr . Severe HTHE is again observed in this case at stresses below about 200 MPa for in-reactor creep conditions. The key results of these studies can be summarized as follows:
High implantation rates of the order 100 appm h1 result in the formation of a high number density of bubbles in the matrix and on GBs up to and in excess of 1015 m2.
Matrix and grain boundary bubbles form both as isolated cavities as well as in association with other features such as second-phase particles, dislocations and grain edges, and triple points.
Essentially all the implanted He precipitates in bubbles.
Matrix bubbles influence the amount of He flowing to GBs.
The grain boundary bubbles grow stably with the addition of He, until some reach the critical size where they convert to stress-driven growing creep cavities.
Postimplantation tests at 90 MPa and 750 C for a wide range of He contents suggest a critical size of 13–17 nm.
168
The Effects of Helium in Irradiated Structural Alloys
Creep cavity growth and coalescence kinetics are rapid, and tr is dominated by the time needed to establish a population of bubbles and grow (gas driven) them to the critical size.
The creep rupture time is generally lower for implantation coincident with creep, compared to preimplantation followed by creep testing.
The stress power r relating rupture time and the applied stress (tr / sr) is decreased for in-beam and in-reactor creep tests, with r 2–4 when compared with postimplantation and unimplanted values of r 6–9.
The tr and XHer decrease with decreasing He gen1=3 eration rates with a scaling law GHe .
However, significant He is required to cause HTHE in all cases.
Alloys with fine-scale matrix and grain boundary precipitates (e.g., TiC) that trap He in a larger number of smaller bubbles mitigate HTHE.
The CBM also rationalizes the degradation of high-temperature fatigue properties at high He levels.91,99,204 The implantation studies show that HTHE is most severe in conventional AuSS, like 316SS, that contain only coarse carbides. Fine-scale TiC (and phosphide) phases that trap He in a high density of finescale matrix bubbles N90,192,203–205 provide greatly enhanced HTHE resistance. Key issues are optimizing the ability of the precipitates to trap He in small bubbles and ensuring their thermal and irradiation stability needed for long-term service. Note that a fine-scale matrix also increases the strength of AuSS alloys, thus enhancing creep constraint reductions of cavity growth rates; and fine grain boundary phases can also impart further resistance to grain boundary cavitation. Indeed, Maziasz and coworkers extended these concepts to developing a new series of AuSS alloys with remarkable unirradiated creep strength (in terms of both creep rates and rupture times) based on precise control of microalloyed fine-scale matrix and grain boundary phases.206 A number of studies have also shown that the FMS are very resistant to HTHE as well as void swelling.12,13,192,201,207 The most obvious explanation is that the sink densities and numerous trapping sites for He keep bubbles finely distributed and protect prior austenite GBs from He accumulation.176 Lath boundaries may be especially effective if, due to their special nature, they are effective in trapping He in small bubbles, but at the same time, resistant to cavity growth by vacancy accumulation. However, it has
been suggested that the bubble microstructures are not that dissimilar in AuSS and FMS and that at least part of the difference in the HTHE sensitivity is that the FMS are inherently weaker at the same temperatures and that the corresponding lower grain boundary stresses increase m*.12,199
1.06.4 Recent Observations on Helium Effects in SPNI The most recent source of significant information on He effects on bulk microstructures and properties is from SPN irradiations of both AuSS and FMS to maximum doses of 20 dpa and 1800 appm He at temperatures up to 450 C. 1.06.4.1
Microstructural Changes
The microstructures of FMS alloys irradiated in STIP at temperatures below about 400 C are dominated by defect clusters and bubbles, characterized by their respective number densities (Nd/b) and diameters (dd/b). Table 2 summarizes some recent TEM observations on FMS and AuSS after neutron and SPN irradiations. Figure 24 shows the development of defect structures in FMS F82H and T91 irradiated in STIP-I209 at irradiation temperatures up to 360 C. Note that the temperatures, dpa, and He levels (dose) are correlated with one another due to their mutual dependence on the proton flux. The microstructures in both FMS alloys are similar and are composed of small (1–3 nm) defects (likely small SIA cluster–dislocation loops, Figures 24(a)–24(d)), along with a lower number of larger dislocation loops (>3 nm). Figure 25(a) plots the loop data for F82H as a function of irradiation temperature for the STIP-I and -II irradiations, along with some neutron data, including from an ISHI experiment. The STIP-I irradiation conditions were 10–12 dpa with He levels that increase with temperature from 560 to 1115 appm. The STIP-II data were at 10–19 dpa with between 750 and 1790 appm He. Note that the STIP-I irradiation initially ran at a much lower irradiation temperature while the STIP-II irradiation was more isothermal. Ignoring these confounding factors, the overall trends show that the dd increases with temperature while Nd decreases, after apparently peaking at 250 C. The neutron data point at 10 dpa and 310 C, with a low He content, following irradiation in the Peten High
The Effects of Helium in Irradiated Structural Alloys
169
Table 2 Some recent TEM data of FMS and AuSS after neutron (N) and spallation proton–neutron (SPN) irradiations at temperatures between 250 and 425 C Materials
F82H F82H T91 F82H T91 F82H F82H 316 316 316LN 316LN 316 JPCA 316LN 316LN
Irradiation
N SPN SPN SPN SPN SPN SPN N N SPN SPN N N SPN SPN
Tirr ( C)
Dose (dpa)
310 295 295 255 250 350 400 275 275 285 245 400 400 355 425
10 9.7 10.1 10.1 8.3 18.6 20.3 5.7 13.3 9.3 11.7 17.3 17.3 11 19.4
He (appm)
670 725 735 540 1640 1790 705 1010 200 200 1025 1800
Defect clusters/loops
Cavities
Mean size d (nm)
Density (1022 m3)
Mean size d (nm)
Density (1022 m3)
6.9 6.5 5.4 5.5 4.5 9.5 8.5 9.1 9.4 3.7 2.45 20.2 15.7 3.9 2.53
2.8 2.7 3.3 3.8 3.6 1.4 2.8 19 19 58 33.5 0.95 1.0 24.6 36.1
1.4 1.2 1.2 1.1 3.3 5.0
40 53 42 54 33 25
3.6 3.0 1.6 1.9
0.62 1.1 53 51
Flux Reactor (HFR), is similar to the STIP-II data at higher He levels. The neutron data 400 C from an ISHI study with 90 appm He at 3.8 dpa215 appears to be more consistent with the overall data trends than the corresponding STIP-II data point; however, this may be because of the lower dose in this latter case (Figure 25(b)). As shown in Figure 26, the He bubble populations in the two FMS irradiated in STIP-I to 10 dpa at 295 C are also similar. Due to the resolution limits, conventional TEM cannot image bubbles smaller than about 1 nm. Thus, bubbles are visible only at He concentrations and temperatures above 500 appm and 170 C, respectively, in the STIP database.66 Figure 25(b) shows that db increases and Nb decreases with increasing temperature in the STIPI and -II irradiations. The neutron data at 500 C, from an ISHI study with 380 appm He at 9 dpa, is consistent with the STIP data trends.215 The smallest bubbles found in the SPNI studies, with an average diameter of 0.7 nm, were observed in F82H from the STIP-I irradiation to 9.9 dpa/560 appm He at 175 C. Bubbles were not observed at lower temperatures and He levels.17,66,209,216,217 In this case, the He is presumably located in a very high concentration of subvisible He–V clusters, which may be overpressurized. At higher temperatures of 350 and 400 C in the STIP-II irradiation, the apparent sizes of the bubbles are much larger than those for STIP-I. Figure 27 shows the cavity structure in two F82H samples irradiated in STIP-II (left) and STIP-III
References
210 211 211 211 211 212 212 213 213 214 215 216 216 214 215
(right) to similar dose and He concentration at nominal temperatures of 400 50 and 440 50 C. Some of the cavities in the 400 C STIP-II irradiation have transitioned from bubbles to larger voids, forming a bimodal size distribution, while there is a larger Nb of smaller bubbles in the 440 C STIP-III case, with a monotonic size distribution. These differences are believed to be due to the fact that the STIP-III irradiation also ran at lower temperatures of about 200 C during the initial phase of the irradiation up to 0.3 dpa and 30 appm. It is believed that the high Nb (5.1 1023 m3) nucleated during this transient and then grew at higher temperatures and He levels. Thus, due to the large Nb, voids did not form in this case. In contrast, voids formed in the STIP-II irradiation, since it was more isothermal, resulting in a lower Nb (2.4 1023 m3) and closer to the peak swelling temperature for FMS of 400 C.15 Large voids that are associated with precipitates are shown in Figure 28(a) for a STIP-II irradiation at 400 50 C. The corresponding microstructure for irradiations at lower irradiation temperatures of 350 50 C is again composed of a high density of small bubbles. Thus, the increase in the average sizes of the cavities in Figure 25 at 350 and 400 C in STIP-II is due to the bimodal mix of bubbles and voids. The strings of bubbles seen in Figure 28(b) also indicate a strong association between bubbles and dislocations. Further, neither bubble denuded zones near GBs nor a significant number of grain boundary
170
The Effects of Helium in Irradiated Structural Alloys
(a)
(b) 3.8 dpa/225 appm He/110 ⬚C
3.8 dpa/225 appm He/110 ⬚C
(c)
(d) 5.7 dpa/375 appm He/200 ⬚C
5.8 dpa/380 appm He/205 ⬚C
(e)
(f) 9.7 dpa/670 appm He/295 ⬚C
10.1 dpa/725 appm He/295 ⬚C
(g)
(h) 11.8 dpa/1115 appm He/360 ⬚C
F82H
11.8 dpa/1115 appm He/360 ⬚C
30 nm
T91
Figure 24 Defect cluster and loop structures of irradiated F82H (left column) and T91 (right column) samples. The irradiation conditions are indicated below the micrographs. The scale for all the micrographs is the same as indicated at the bottom. Reproduced from Jia, X.; Dai, Y. J. Nucl. Mater. 2003, 318, 207.
bubbles have been found in STIP samples investigated to date. This suggests that grain boundary bubbles may be too small to image. SPN irradiations also produce defect clusters, faulted Frank loops, and bubbles in AuSS (316L
and SS304L).212,218,219 Below 285 C, only defect clusters and Frank loops (no bubbles) are observed. At temperatures between 245 and 425 C, the STIP irradiations produce a high number density, between 2 1023 and 6 1023 m3, of small loops with
171
The Effects of Helium in Irradiated Structural Alloys
15
8
8
4 5
50
Bubbles and voids 30
4
Bubbles 20
2
2
0 0 150 200 250 300 350 400 450 500 550 Tirr (⬚C) (a)
ISHI N STIPII N STIPI N
40 6 d (nm)
6
60 ISHI d STIPII d STIPI d
N (1022 m-3)
Loop d (nm)
10
Loop N (1022 m−3)
10
10 ISHI N STIPII N STIPI N HFR N ISHI d STIPII d STIPI d HFR d
10
0 0 150 200 250 300 350 400 450 500 550 (b)
Tirr (⬚C)
Figure 25 Irradiation temperature dependences of (a) loop size and density and (b) bubble size and density of F82H specimens irradiated in STIP with SPN, in HFR with fission neutrons, and in HIFR with in situ He implantation (ISHI).
20 nm (a)
(b)
Figure 26 He bubble structures of (a) F82H irradiated to 9.7 dpa/670 appm He and (b) T91 irradiated to 10.1 dpa/725 appm He at 295 C. The scale shown in (b) is the same for (a). Reproduced from Jia, X.; Dai, Y. J. Nucl. Mater. 2003, 318, 207.
(a)
30 nm
(b)
Figure 27 He bubble structures of irradiated F82H samples irradiated in STIP-II to 20.3 dpa/1800 appm He at 400 50 C (left) (reproduced from Jia, X.; Dai, Y. J. Nucl. Mater. 2006, 356, 105–111) and in STIP-III to 20.3 dpa/1725 appm He at 440 50 C but at 200 C for the first 0.3 dpa (right). The scale shown in (a) is the same for (b). Reproduced from Tong, Z.; Dai, Y. J. Nucl. Mater. 2010, 398, 43.
172
The Effects of Helium in Irradiated Structural Alloys
(a)
35 nm
(b)
20 nm
Figure 28 Helium bubble structure in the samples irradiated in STIP-II: (a) 20.3 dpa/1800 appm He at 400(50) C and (b) 18.6 dpa/1570 appm He at 350(30) C. Reproduced from Jia, X.; Dai, Y. J. Nucl. Mater. 2006, 356, 105.
(a)
25 nm
(b)
Figure 29 He bubble structure of SS 316LN samples irradiated to 9.3 dpa/705 appm He at 285 C (left) and 19.4 dpa/1800 appm He at 425 C (right). The scale shown in (a) is the same for (b). Reproduced from Dai, Y.; Wagner, W. J. Nucl. Mater. 2009, 389, 288.
average sizes between 2.5 and 3.9 nm. In contrast, neutron irradiations of AuSS produce a lower density of larger loops. For example, mixed spectrum reactor irradiations at 400 C, which produce smaller but significant amounts of He compared with the SPNI case, result in 30 times fewer and 6 times larger loops. The differences are even larger for fast reactor irradiations with much lower He levels. Thus, it appears that high He results in significant refinement of the loop structures in AuSS. Figure 29 shows small 1–2 nm bubbles in 316LN AuSS for both 285 C irradiations to 9.3 dpa and 705 appm He at 285 C and 19.4 dpa and 1800 appm He at 425 C. Unlike the case of FMS irradiated at 400 C, no large voids were observed in this case. The refinement of the bubble structures in SPNI with high He levels is even more profound. The mixed spectrum reactor irradiations spectrally tailored to produce 11 appm He/dpa produced 60 times fewer cavities with a bimodal distribution of bubbles and voids compared with the SPNI case with 7 times more He. The average cavity diameter is about 2.4 times larger in the mixed spectrum
neutron case. These observations are highly consistent with the concepts described in Section 1.06.3 with regard to higher He and Nb suppressing void formation. Again, however, the likely effects of the temperature history confound quantitative interpretations of both the loop and cavity microstructures in the AuSS as well as FMS. 1.06.4.2 Mechanical Properties of FMS After SPNI The SPNI have produced a very large database on changes in strength and toughness properties up to 20 dpa, 1800 appm He, and 450 C. The focus here is on two topics: (1) helium-induced hardening and (2) helium embrittlement. 1.06.4.2.1 Helium effects on tensile properties and He-induced hardening effects
The tensile properties of various FMS and AuSS after SPNI have been extensively investigated in the last decade.17,216,217,220–231 The tests were conducted at either ambient or the irradiation temperature.
The Effects of Helium in Irradiated Structural Alloys
The SPNI data are first compared with those for neutron irradiations of FMS, which show significant irradiation-induced hardening at 325 C, decreasing hardening between 325 and 400 C, and little or no hardening at greater than 400 C.20 In the neutron case, the hardening measured as changes in the yield stress (Dsy), initially increases with the square root of dpa but approaches a saturation level at higher doses. The saturation hardening depends on the irradiation temperature, and is 480 MPa at 10 dpa and 200–300 C. The uniform elongation (eu) following neutron irradiation decreases to less than 1% within several dpa. The postnecking strains are less affected, and the corresponding total elongation (et) decreases to between 3 and 10%. Figure 30 compares the SPNI yield strength increase data (Dsy) with neutron data trends. Up to 10–13 dpa, the SPNI Dsy are generally similar to, or slightly lower than, the neutron data trends. However, the SPNI Dsy do not saturate up to the maximum dose of 20 dpa, where the hardening reaches remarkable levels in excess of 700 MPa. The higher increment of SPNI hardening is even more pronounced at 350 C and extends to well above 400 C.17,231 The additional hardening above 10 dpa is primarily attributed to helium bubbles and, perhaps, with an additional contribution from the
800 700
N-irradiation
Tirr: 80–350 ⬚C Ttest = RT
200 ⬚C 250 ⬚C 300 ⬚C 350 ⬚C 400 ⬚C 450 ⬚C 500 ⬚C
SPN irrad.
600
DYS (MPa)
500 400 300 200 100 0 −100 0
1
2
3
4
5
6
7
8
9
10
(dpa)1/2 Figure 30 A comparison between the hardening induced by SPN irradiations and neutron irradiations. For neutron irradiations, the trend line ‘200 C’ is for data irradiated and tested at 200 C. Model curves are reproduced from Yamamoto, T.; Odette, G. R.; Kishimoto, H.; Rensman, J.-W.; Miao, P. J. Nucl. Mater. 2006, 356, 27. Reproduced from Dai, Y.; et al. J. Nucl. Mater. (2011), doi:10.1016/ j.jnucmat.2011.04.029.
173
higher loop density. Again note that, in some cases, the refined microstructures may be due to variable temperature history effects noted previously. Figure 31 shows that the corresponding et of FMS after SPNI (symbols)221,222 is generally similar to, or only slightly less than, for neutron irradiations at 325 C (solid lines)232 up to about 10–12 dpa for irradiations between 80 and 350 C and room temperature tests (note that the modest differences in total elongation may be at least partly due to differences in the size and geometry of the tensile specimens). However, at higher doses between 10 and 18 dpa and 750–1300 appm He, the et of FMS following SPNI approaches 0 and, in some cases, the tensile specimens break during elastic loading at fracture stresses less than the yield stress, sy. The fracture surfaces of the high-dose SPN-irradiated samples show a mixed brittle IG and transgranular cleavage fracture,227 similar to that observed in T91 and EM10 FMS after implantation at 250 C with 2500 appm He producing 0.4 dpa.28 In contrast to the 20 dpa and 1800 appm He data reported here, high levels of hardening that can be attributed to He have generally not been observed previously, either in high-energy implantation studies, at less than 500 appm He,233,234 or in low-temperature SPNI.225 Indeed, excess hardening was not observed in the LANSCE irradiations at <100 C at He levels up to 2000 appm. Helium implantation at 200 C235,236 and 250 C28 indicated that significant hardening due to He occurred only at high He concentration levels above 5000 appm. All these results suggest that at less than 400 C and 600 appm He, irradiation hardening is dominated by defect clusters and loops. Coupled with the small size of He–vacancy clusters (<1 nm), a partial explanation may be found in recent molecular dynamics (MD) simulations237,238 showing that He bubbles can cause significant hardening but that their contributions are reduced if they are overpressurized. The data in Figure 32 more directly show the hardening contributions of He in STIP irradiated alloys that were annealed at 600 C for 1 h to remove the defect clusters and loops, while leaving the more stable bubbles unaffected.239 The residual bubble hardening is significant and increases with the square root of dpa and He. Note that the latter provides a crude measure of the volume fraction of bubbles. The data in Figure 32 were combined with TEM measurements of db and Nb and were used to evaluate the dislocation obstacle strength (a) of the 1 nm bubbles, based on the relation Dsy 3DDPH
174
The Effects of Helium in Irradiated Structural Alloys
25
STIP data
Ttest = RT
Ti: 80–350 ⬚C F82H Optimax-A F82H EBW T91 EM10 Optifer-V HT9 EP823 EM10 T91
Total elongation (%)
20 T91, EM10 Ti = 325 ⬚C Eurofer, JLF-1 9Cr2WVTa Ti = 325 ⬚C
15
10
5
0 0
5
10 15 Irradiation dose (dpa)
20
25
Figure 31 Dose dependence of total elongation of FMS irradiated in STIP at 350 C and tested at RT. Trend lines for fission reactor irradiations are shown for comparison. Reproduced from Dai, Y.; et al. J. Nucl. Mater. (2011), 415, 306.
120
200 F82H Optimax-A
ΔHV0.05 (kg mm−2)
ΔHV0.05 (kg mm−2)
F82H Optimax-A
600 ⬚C-annealed
100 80 60 40
As-irradiated 600 ⬚C-annealed
150
100
50
20 0 0
0
5
10
15 20 25 (CHe, appm)1/2
30
35
40
0
1
2
3
4
(dpa)1/2
Figure 32 The dose and CHe dependences of the hardening of the F82H and Optimax-A irradiated in STIP-I in the as-irradiated and annealed at 600 C/2 h conditions. Reproduced from Peng, L.; Dai, Y. J. Nucl. Mater. (2011), doi:10.1016/ j.jnucmat.2010.12.208.
sy 3DDPH 3aGb√Nbdb, yielding an estimated a 0.1. Note that this value of a is much lower than estimates based on MD simulations discussed in Section 1.06.5. This difference may be because strength superposition effects for combining bubble obstacles with preexisting strengthening features in the FMS were not accounted for in this evaluation. Strength superposition effects may also help rationalize the smaller hardening from bubbles below about 500 appm He. Combining estimates of a 0.1 with the TEM data
discussed in the previous section suggests that significant hardening by bubbles (and voids) will extend to temperatures up to 500 C at high He levels. 1.06.4.2.2 Helium effects on fracture properties and He-induced embrittlement effects
The effects of He on fast fracture, typically characterized by shifts in the DBTT measured in CVN impact tests (DT ), has long been a subject of
The Effects of Helium in Irradiated Structural Alloys
significant controversy. This controversy has been fueled by studies that were interpreted to suggest that even small to moderate amounts of He result in increases in DBTT.14,34,240–242 However, it has been shown that at temperatures below about 400 C embrittlement is primarily due to irradiation hardening (Dsy), resulting from fine-scale irradiationinduced dislocation obstacles.20,21 The simplest relation is hardening–shift relation, which is given by
The first data that clearly indicated a nonhardening role of He were generated in the early STIP experiments, showing a transition from ductile and cleavage fracture modes to extremely brittle IG fracture20,220 and somewhat larger than expected DT. Analyses of a large database on irradiation hardening and embrittlement, including the STIP data,20 showed that He does not produce significant nonhardening embrittlement at less than about 500 appm. However, above this rough threshold the hardening-shift coefficient C (¼DT/Dsy) increases due to weakening of the GBs associated with He accumulation, to the point where they became the preferred fracture path. The database was used to derive a simple semiempirical model for CVN DT for 300 C irradiations as
½18
DT ¼ CDsy
Here C depends on a number of variables but for irradiated FMS has an average value of 0.4 C MPa1 for subsized CVN tests. Thus, it is obvious that He would contribute to embrittlement of FMS to the extent that it contributes to hardening. However, as noted previously, He effects on hardening are minimal up to levels of about 500 appm. Further, most of the data on He effects on embrittlement are confounded by the experimental techniques, like Ni and B doping, or use of atypical fracture test methods. Irradiation embrittlement can also be induced by nonhardening mechanisms associated with changes in the local fracture properties that are controlled by coarse-scale microstructural features, like brittle trigger particles for cleavage, and segregation of elements that weaken GBs.20,21
CHe (appm)
175
C ¼ 0:4 þ 7 104 ðXHe 500Þð C MPa1 Þ ½19 As shown in Figure 33 the model prediction (dashed curve)242a is remarkably consistent with SPNI and neutron data including more recent results. The STIP data are based on subsized CVN tests (KLST and 1/3 CVN) on different FMS irradiated in STIPI–III up to about 17 dpa at temperatures below 300 C.19 The solid symbols are small punch test data converted to CVN DT. The neutron data were taken from the literature,14,240–244 and these results
400
800
1300 300
Ti < 380 ⬚C
CVN
SP
700
F82H
250
600
T91 Optimax-A/C
200
Optifer-V/-IX Eurofer97
400
150
300
DDBTTSP (⬚C)
DDBTTCVN (⬚C)
500
100 200 100
27–70 dpa T91, F82H, E-97
Neutron irradiations 0 0
2
4
6
8
10
12
14
16
18
50
0 20
Displacement (dpa) Figure 33 DBTT shift as a function of irradiation dose for different FMS irradiated in STIP. Neutron-irradiation data are included for comparison. Reproduced from Dai, Y.; Wagner, W. J. Nucl. Mater. 2009, 389, 288. The dashed curve is drawn according to the model prediction (eqn [19]).
176
The Effects of Helium in Irradiated Structural Alloys
sc*
* sig,He
Msy,s*
Msy,s*
Msy,i
Msy,u
DTi Tu
Ti
T
Msy,u
DTi + He Tu
T
Ti+He
Figure 34 A sketch showing the mechanisms for irradiation-induced hardening (increase of yield stress, Dsy) and helium-induced grain boundary weakening effects (decrease in the intergranular fracture stress, s ig ) that elevate the brittle to ductile transition temperature.
are also consistent with the analysis of a larger database.20 As schematically illustrated in Figure 34, the synergistic low-temperature hardening–helium embrittlement (LTHE) He threshold can be rationalized as follows. Cleavage fracture occurs when the stress concentrated at the tip of a blunting crack, Msy, exceeds a critical local stress, s c , over a critical volume needed to activate a brittle trigger particle.21 Here, M is a stress concentration factor. Likewise, brittle IG fracture occurs when the crack tip stress exceeds the critical local stress s ig over a sufficient volume needed to crack GBs. The s ig is initially higher than s c ; thus, fracture occurs by transgranular cleavage (Figure 34(a)). However, s ig decreases with increasing He GB accumulation, and beyond a bulk threshold level, ca. 500 appm, s ig falls below s c (Figure 34(b)). Thus, the grain boundary becomes the favored crack path. The s ig continues to decrease with increasing He accumulation, resulting in an increasing increment of DT, even in the absence of additional hardening. The transition to IG crack paths is marked by a larger fraction of grain boundary facets on the fracture surface. Note that the continued increase in Dsys with higher He was not recognized at the time that this simple model was developed, thus the new insight and expanded database will be used to refine the model. Helium that is not clustered into bubbles is likely the most damaging condition, with a monolayer coverage producing essentially complete grain boundary decohesion. The actual amount and distribution of helium on GBs has not been established and is a function of the temperature and microstructure as well as bulk XHe. However, even at 400 C boundary bubbles are less than 1 nm in diameter. Assuming that grain boundary helium derives from regions in the
100 ⬚C
200 KJq (MPam1/2)
sc*
Msy,i
HFIR data HT9, F82H 90–250 ⬚C
250
sig*
150
T91, STIP-I T91, STIP-I F82H, STIP-I Optimax, STIP-I T91, STIP-III F82H, STIP-III
170 ⬚C 250 ⬚C
100
250 ⬚C
LANSCE T91 data 50–160 ⬚C
50
25 ⬚C
400 ⬚C
0 0
2
4
6
8 10 12 Dose (dpa)
14
16
18
20
Figure 35 Fracture toughness as a function of irradiation dose for different FMS irradiated in STIP. The temperature values indicated are for testing temperatures, which are equal to or close to irradiation temperatures. Data bands from LANSCE SPN irradiation223 and HFIR neutron irradiation are shown for comparison.247
adjoining matrix and is located in spherical bubbles with equal numbers of mHe atoms and vacancies, the fractional grain boundary coverage can be estimated as fHe ¼ tHeXHe/[10–4 mHe]; here, tHe is the thickness (mm) of the layers that feed helium to the grain boundary. Thus for example, fHe 0.25, assuming tHe ¼ 0.25 mm and mHe ¼ 5 and XHe ¼ 500 appm. Note that this tHe may be too large considering that denuded zones are not evidently observed at GBs in STIP samples. However, the data are not sufficient to reach firm conclusions, and a combination of models and mechanism experiments is needed to determine the partitioning of He to GBs for various microstructures and irradiation conditions. Other studies245,246 reached similar conclusions regarding the effect of He on grain boundary strength. Indeed, simple and direct evidence is provided by the brittle fracture stresses measured in the tensile tests cited previously, which decreased from 1850 to 1640 MPa with increasing He levels from 1250 to 2500 appm. These helium-degraded s ig are well below the cleavage s c 2000 MPa. Embrittlement and DT are most properly evaluated by fracture toughness tests that are expected to show hardening–He synergisms that are similar to those measured in CVN tests. Figure 35 shows the estimated fracture toughness (KJq) of various FMS after SPNI based on three-point bend tests on small precracked bars at test temperatures approximately equal to the irradiation temperatures.210,222 Note that at high dose, KJq decreases to less than 40 MPa √m, close to lower shelf fracture toughness of FMS, even at the maximum irradiation temperature of 400 C.
The Effects of Helium in Irradiated Structural Alloys
Figure 35 also shows that the KJq of the T91 steel irradiated at LANSCE are degraded at lower doses (up to about 4.3 dpa)223 than in the STIP irradiations. This may be the result of the combination of the lower irradiation temperatures and higher helium generation rates in this case. Note that at 25 C irradiation of T91 in STIP-I to 4.3 dpa also resulted in low KJq. Figure 36 shows the predicted shifts in the master curve reference temperature (DT0) at 100 MPa√m for FMS F82H (similar to that for Eurofer97) neutron irradiated at temperatures from 200 to 400 C as a function of the square root of dpa. The corresponding
1000
ΔTo (⬚C)
800
n (200 ⬚C) n (300 ⬚C) n (400 ⬚C) 100 ⬚C 170 ⬚C 250 ⬚C 400 ⬚C
1600 appm
600 700 appm 400 1200 appm
500 appm 200
350 appm
0
0
2
4
6
8
10
dpa1/2 Figure 36 Comparison of the predicted shift in the Master Curve reference temperature, DT0, with the data shown in Figure 35 showing the drastic embrittling effect of high concentrations of He.
STIP SPNI DT0 data shown in Figure 35 are estimated by adjusting the measured KJc to 100 MPa√m based on the master curve shape and further taking the unirradiated T0 as 100 C.248 These approximate, but semiquantitatively correct comparisons show that the synergistic hardening–He mechanism also results in much larger fracture toughness DT0 when compared with neutron irradiation with low He. Most notably, the estimated DT0 for the 400 C irradiation is of the order 700 C. 1.06.4.3 Mechanical Properties of AuSS After SPNI Figure 37 shows tensile data of various AuSS (316L, 304L, 316F, JPCA, and 316L-EBW) after SPN irradiations at 100–350 C223,228,229,249–253 as well as neutron irradiations at around 300 C.254–257 The increase in yield and tensile stress for tests at the irradiation temperature appears to be somewhat lower following SPNI compared to mixed spectrum neutron irradiation, which also produces high levels of He. However, since the alloys used in the different studies are not the same, no clear conclusion can be drawn from this observation. The strength increases plateau above about 10 dpa. In all cases, the uniform elongation falls to less than 1% above about 5 dpa.252 Similar trends are also generally observed in fast reactor irradiations of AuSS.257a,258 Figure 38 shows the effect of LANSCE irradiations below 160 C on the fracture toughness of SS316L tested between 50 and 80 C.223 The fracture toughness for tests at the indicated temperatures for EC316LN irradiated in STIP below 340 C and SS316L and JPCA irradiated at HFIR at 300 C is
1000
50
800
40
Elongation (%)
Strength (MPa)
100 £ Tt » Ti £ 350 ⬚C
600
400
200 100 £ Tt » Ti £ 350 ⬚C
0
0
5
YS, SPN UTS, SPN YS, neutron UTS, neutron
20 10 15 Irradiation dose (dpa)
25
177
STN, SPN TE, SPN STN, neutron TE, neutron
30
20
10
0
0
5
10 15 20 Irradiation dose (dpa)
25
Figure 37 Irradiation dose dependence of yield stress (YS), ultimate tensile stress (UTS), strain to necking (STN), and total elongation (TE) of AuSS irradiated with SPN and fission neutrons and tested at temperatures between 100 and 350 C.
178
The Effects of Helium in Irradiated Structural Alloys
500 LANSCE, 50–164 ⬚C STIP-III, 150–400 ⬚C HFIR, 100/250 ⬚C
450 400 100 ⬚C
KJq (MPam1/2)
350
150 ⬚C
300
0.11 dpa
250 300 ⬚C
250 ⬚C
200 150 100 50
400 ⬚C
200 ⬚C
0 0
2
4
6
8 10 12 Dose (dpa)
14
16
18
20
Figure 38 Fracture toughness as a function of irradiation dose for different AuSS after irradiation with SPN (LANSCE227 and STIP259) and neutrons (HFIR257).
also shown. The decreases KJq with dpa have similar trends, although the STIP data show a less regular pattern; however, at higher dpa the toughness falls at 50 MPa√m or less. A similar trend has been previously reported for both fast reactor and mixed spectrum reactor irradiations and is considered to be due to reductions in the uniform strain and strain hardening rates that are not strongly influenced by He.258 1.06.4.4 Summary of Effects of Irradiation on Tensile and Fracture Properties Comparison of the STIP data (high helium) with neutron-irradiation (lower He) data for FMS leads to the following conclusions:
Up to about 500 appm, excess irradiation hardening and ductility loss in FMS due to He are modest.
At higher damage levels, due to contributions from bubbles, hardening continues to increase with dpa and He in SPNI but saturates in neutron irradiations.
At high He levels, bubbles contribute to hardening even more significantly at higher irradiation temperatures.
Reductions in the uniform and total elongation strains are also similar in neutron and SPN irradiations up to 500 appm, but elastic fracture and increasing brittle IG fracture occur even in tensile tests at higher He levels.
The effect of He on fast fracture and DBTT shifts is also modest below 500 appm, but at higher
levels it increases rapidly due to both extra hardening and reduction in the critical stress for IG fracture that fall below that for cleavage.
Synergistic hardening and nonhardening embrittlement lead to enormous DBTT shifts and an increase in the maximum temperature for significant DBBT shifts up to irradiation temperatures that may be well in excess of 400 C. The effects of He on tensile and fracture properties are less apparent in AuSS. Unpublished results from SPNI also show that He is trapped in small bubbles in NFA in a way that appears to protect GBs from IG embrittlement for the same irradiation conditions that result in severe synergistic He-hardening DBTT shifts.
1.06.5 Atomistic Models of He Behavior in Fe Multiscale modeling hierarchically links various computational techniques over widely differing length and time scales.159 Multiscale modeling of He transport, fate, and consequences requires linking ab initio electronic structure, MD, kinetic Monte Carlo/Lattice Monte Carlo (KMC/KLMC), and mean field RT simulations to predict microstructural evolutions as an ultimate basis for modeling mechanical behavior. (see Chapter 1.08, Ab Initio Electronic Structure Calculations for Nuclear Materials; Chapter 1.09, Molecular Dynamics; Chapter 1.14, Kinetic Monte Carlo Simulations of Irradiation Effects; Chapter 1.15, Phase Field Methods and Chapter 1.16,
The Effects of Helium in Irradiated Structural Alloys
Dislocation Dynamics) In this section, we emphasize electronic–atomistic models of He behavior in Fe lattices.
increasing temperature and bubble size as expected. However, the MD studies suggest that the pressure in small bubbles predicted by MD is a factor of 2 times lower than that predicted by the simple capillary approximation for Van der Waals and hard sphere equations of state.263 The reasons for this discrepancy are not yet understood. Although quantitative details differ, the most important results of the atomistic models are that (a) both substitutional and interstitial He have very high formation energies; (b) there are large positive binding energies for He and vacancies in HemVn clusters even at m/n > 1; (c) at high m/n, the clusters relax and increase n by emitting SIA, or punching SIA loops; clusters with m/n 1 are very stable. Ab initio calculations in the framework of density functional theory (DFT) have been used to obtain formation and binding energies (Eb) of small clusters typically containing up to combinations of four He atoms and vacancies.82,134 The results are summarized in Table 3. Here, the Eb of the He in the cluster is referenced to tetrahedral Hei, with an energy that is slightly lower than that for other interstitial configurations. The ab initio calculations show that Hei clusters are bound to even absent vacancies. However, the Eb for He–V clusters is much larger. The Eb for He monotonically decreases, while the vacancy Eb monotonically increases, with larger helium to vacancy ratios, m/n, except for the case HemV, where the He Eb do not change much for m 4. These trends reflect a high level of overpressurization in the small clusters relative to a relaxed bubble with n > m. The He Eb also increases with cluster size, from 2.3 eV at m ¼ n ¼ 1 to 3.05 eV at m ¼ n ¼ 4. The most important implications of
1.06.5.1 He Energetics and He–Defect Complex Interactions The results of atomistic modeling of He–defect interaction energies are presented in this section for both ab initio and MD and molecular statics (MS) methods. The ab initio results provide interaction energies 0 K that can be used in Monte Carlo simulations that are generally restricted to very small, nonequilibrium, high-pressure (solid) He–vacancy (V) clusters. This contrasts with the models described in Section 1.06.3 that treat He as a continuum phase within the capillary approximation. In this latter case at sufficiently large bubble sizes and high temperatures, and in the absence of irradiation, the pressure in a relaxed strain-free bubble is given by the capillary approximation as P ¼ 2g/rb, where the gas pressure is balanced by the average surface tension. Note that this approximation ignores the faceted shape of small bubbles.260 This is a lower energy cluster configuration than those at higher He pressures that are balanced by a contribution from an elastically strained matrix, P ¼ (2g þ mb)/rb, where m is the shear modulus and b is the Burgers vector. MD can simulate gas bubbles at finite temperatures.260–263 A recent MD study at temperatures up to 700 K shows that the He pressure is very high in 2 nm clusters but saturates at 25 GPa for a He/vacancy ratio of 3, due to spontaneous SIA emission at the theoretical strength of the Fe lattice.260 The MD simulations show that the equilibrium bubble He/vacancy ratio (m/n) is <1 and decreases with Table 3
179
He and V binding energies to small HemVn clusters (0 m, n 4)
HemVn ® HemVn−1 + V n
0
m
1
0.3
0 HemVn ® Hem−1Vn + He
1 2 3 4
3
2
1.61
3.71 0.43
1.84
0.95
1.83
0.98
1.91
0.83 3.3
2.85 2.75
1.04 1.8
2.36
1.57 3.16
2.03
2.3
5.52
1.32 3.12
2.91
2.07
1.16 3.84
2.96 1.85
4.59
0.62
0.37
0.78
2.3
4
2.57
1.97 3.05
Source: Fu, C. C.; Willaime, F. Phys. Rev. B 2005, 72, 064117; Ortiz, C. J.; Caturla, M. J.; Fu, C. C.; Willaime, F. Phys. Rev. B 2007, 75(10), 100102.
180
The Effects of Helium in Irradiated Structural Alloys
these results are that, except at very high temperatures, m 2 and n 2 clusters are likely the stable nucleation sites for He–vacancy cluster formation and subsequent bubble evolution, as assumed in the models described in Section 1.06.3. Once formed and equilibrated, even small He–vacancy clusters are extremely stable up to very high temperatures. MD simulations of larger HemVn (m 20, n 20) clusters predict vacancy Eb trends that are very similar to those found in the ab initio calculations for small clusters.265 However, the Eb for He derived from these MD simulations are consistently 1 eV higher than those from ab initio calculations. The MD simulations used an embedded atom method (EAM) potential for Fe–Fe266 and pair potentials for Fe–He267 and He–He.268 While this set of potentials has been widely used, they predict significantly larger differences between Hes and Hei formation energies than the ab initio results.269 Improved EAM potentials have been developed and a three-body potential for Fe–He–Fe264,269 predicts cluster formation and binding energies that are generally closer to the ab initio results. Note that, while we are not aware of specific simulation results, overpressurized bubbles would be expected to be biased sinks for vacancies and against SIA. Taken together, these results further reinforce the assumption that, in vacancy-rich environments produced by displacement damage, stable He–V clusters nucleate at very small sizes (m n ¼ 2) and grow along a near-equilibrium bubble path. Thus, a critical issue is how He is transported and interacts with other microstructural features. 1.06.5.2
He Interactions with Other Defects
Interactions with defects, like dislocations and interfaces, play a dominant role in the fate of He in complex multiconstituent, multiphase structural alloys. MD, MS, and ab initio calculations have been carried out to characterize the interactions of He with Fe SIA and their clusters.159,270,271 These simulations revealed various reactions–interactions involving (a) a spontaneous recombination–replacement reaction, where a single SIA replaces a Hes atom leaving a Hei; and, (b) SIA cluster trapping–detrapping reactions with single Hes and Hei atoms, as well as with small He clusters. The simulations also showed that small SIA clusters are strongly bound by a single Hei and Hei clusters with high Eb from 1.3 to 4.4 eV. Such trapping significantly retards the primarily
one dimensional motion of SIA clusters, which otherwise are highly mobile with a migration energy of less than 0.1 eV in pure Fe. Interactions of He with microstructural features such as dislocations, GBs, and nanometer-scale precipitates have also been modeled.136 The Dimer method272 was used to efficiently identify saddle point activation energies that were then used in MD simulations to observe interaction mechanisms and reaction paths. Energy landscapes for He around dislocations were modeled for (a/2)[111] and [1–12] edge dislocations as well as (a/2)[111] screw dislocations.273,274 Similar calculations have also been carried out for four symmetric tilt boundaries with a common h101i axis (S3{112}, S11{323}, S9{114}, and S3 {111}) using a two-part rectangular cell.275 The interactions of He with coherent nanometer-scale Cu precipitates have also been modeled.136 Figure 39(a) shows the Hes Eb and excess volume per unit area (Vex) for edge and screw dislocations as a function of the distance from the core. The maximum Eb is much larger for edge (0.5 eV) compared with screw dislocations (0.15 eV). The Eb closely correlate with the Vex. As shown in Figure 39(b), a similar relationship is found for different GBs. The Eb of substitutional and interstitial He varies from 0.2 to 0.8 eV and 0.6 to 2.7 for screw and edge dislocations, respectively, increasing linearly with increasing Vex. Figure 39(c) shows the Eb for both a single vacancy and a Hes atom as a function of distance from a 2 nm coherent Cu precipitate.136 Both the Hes and a single vacancy have very similar energy– distance relations, with the maximum Eb 0.6 eV at the precipitate surface. Table 4 summarizes the results for these extended defect models. The qualitative implications of these results are that Hei is strongly trapped at various common microstructural features, while Hes is more weakly trapped. Thus, it is likely that detrapping of Hei involves a Hei þ V ! Hes reaction. 1.06.5.3
Helium Migration
Migration of He in Fe has been studied using ab initio134 and EAM MD/MS methods.159,160 Interstitial Hei diffusion is almost athermal, with very low migration energy, Em 0.06 eV134 to 0.08 eV.159,160 Hes diffuses by either (a) a classical vacancy exchange (Hes–V) mechanism; or (b) by dissociation mechanism that involves a Hes ! Hei þ V reaction, followed by the diffusion of Hei. The activation energy (Ea) for the dissociation mechanism is estimated
The Effects of Helium in Irradiated Structural Alloys
0.75 Eb (edge)
Vex (screw)
Eb (screw)
1
0.25
0
-0.5 -20 (a)
0
-0.25 20
10 -10 0 Distance normal to glide plane (A)
S3 {111}
int He
0.5
0.5
3.5
He(s) binding energy, Eb (eV)
Vex (edge)
Maximum He binding energy (eV)
Excess volume, Vex (A3)
1.5
181
3
S9 {114}
Sub He
2.5 S11 {323}
2 1.5 S3 {112}
1 0.5 0 0
0.01
(b)
0.02 0.03 Vgb/A (nm)
0.04
0.05
1.0 V1 (100) V1 (101) He1V1 (100) He1V1 (101)
Binding energy (eV)
0.8
0.6
0.4
0.2
0.0 0
2
(c)
8 10 4 6 Distance from cluster (Å)
12
Figure 39 (a) Substitutional He atom binding energies to edge and screw dislocations along with the excess volume as a function of the distance from the core. (b) Maximum binding energy of interstitial and substitution He atoms as a function of the excess volume per unit GB area. (c) Binding energies of vacancies and substitutional He to a 2-nm coherent Cu precipitate as a function of the distance. Table 4 Binding energies of He atoms to various microstructural features Feature
Edge dislocation Screw dislocation h101i axis tilt GB
Coherent NF
Maximum binding energy (eV)
(a/2)[111][112] (a/2)[111] S3{112} S11{323} S9{114} S3{111} 2 nm
Hei
Hes
2.29 1.05 0.55 1.40 2.29 2.66 –
0.5 0.25 0.17 0.42 0.70 0.78 0.58
to be 2.4 to 3.70 eV based on and EAM MD/MS methods, respectively.134,159 The vacancy exchange mechanism is similar to that for any substitutional solute, except that there is an unusually high binding energy for HesV complex. In bcc crystals Hes diffuses by a sequence of an initial Hes–V exchange, followed by a jump of the vacancy from the first nearest neighbor (NN) to a second NN position. Thus, Hes diffusion then requires that a (the same or another) vacancy jump back to a different NN position than the one involved in the initial exchange. The rates of exchange, including jumps to a third NN position, are needed to model the Hes diffusion coefficient,
182
The Effects of Helium in Irradiated Structural Alloys
DHes. Indeed, a minimum of five jump frequencies must be considered in modeling any substitutional solute diffusion coefficient in a bcc lattice, in this case Hes (DHes).276 The analytical five-frequency model provides both the correlation coefficient and net migration energy (Em) for Hes diffusion expressed in terms of the individual vacancy exchange activation energies. The activation energies for the various exchanges have been evaluated by both MD159,160 simulations and ab initio calculations.134 These activation energies have been used in the five-frequency model to estimate DHe as well as in direct KLMC simulations. The KLMC model yields: DHes ¼ 2:8 104 expð2:35=kTÞðm2 s1 Þ
Table 5 The Em for He, V, and Hes–V in model S3{112} and S11{323} GB and a-Fe matrix Em (eV)
Hei
V-Ackland
V-Mendelev
HesV
S3{112} S11{323} Edge dislocations Screw dislocations a-Fe matrix
0.46 0.47 0.4–0.5
0.48 0.74 –
0.38 0.61 –
0.9 0.92 –
0.4
0.43
–
1.1
0.08
0.78
0.64
1.13
Source: Gao, F.; Heinisch, H. L.; Kurtz, R. J. J. Nucl. Mater. 2007, 367–370, 446–450; Heinisch, H. L.; Gao, F.; Kurtz, R. J.; Le, E. A. J. Nucl. Mater. 2006, 351, 141–148; Heinisch, H. L.; Gao, F.; Kurtz, R. J.; Phil. Mag. 2010, 90, 885.
½20
He atoms also diffuse quasi one dimensionally along a dislocation core. The detailed mechanisms and activation energies have been studied by MD and Dimer method.136 For example, interstitial Hei trapped on an a/2h111i{110}edge dislocation in a-Fe is in a h111i crowdion configuration. Thus, He atoms can migrate along the dislocation line by jumping as a crowdion to an adjacent close-packed row with the migration energy of 0.4–0.5 eV.273 Hei also migrates along a a/2h111i screw dislocation within or near the core with a similar migration energy of 0.4 eV, in this case via exchanges between octahedral interstitial sites. Hes migrates near the core of the screw dislocations by vacancy mechanism. The migration energy of 1.1 eV is associated with vacancy jumps from NN to second NN positions.274 Thus, the Em for Hei is higher on dislocations than in the matrix and lower for Hes. Diffusion of He atoms on two symmetric tilt GBs, S3{112}, S11{323}, was also studied using MD and Dimer methods.277–279 Hei diffusion was found to be one to three dimensional depending on the boundary characteristics. Hei diffuses one dimensionally along <1–13> direction in the S11{323} GBs at temperatures from 600 to 1200 K. In the S3{112} GBs, Hei diffuses two or three dimensionally at lower and higher temperatures, respectively.278 The mean square displacement in a long-term MD simulation indicated in Em ¼ 0.28 (eV) for interstitial He migration on S3{112} boundary and Em ¼ 0.34 (eV) for migration on S11{323} boundary. Both of these GBs Em are higher than the value 0.087 eV in a-Fe lattice. The preexponential coefficient was found to be 4.35 10–8 (m2 s1) in both cases. Dimer saddle point searches of possible migration paths of the Hei yield somewhat different Em but rationalize the slightly different results for the two
GBs. Possible migration paths of vacancies and Hes– vacancy complexes have also been studied using Dimer method. An important observation is that both tend to migrate one dimensionally, especially at low temperatures.279 These results depend on the EAM potentials and cannot be considered to be quantitative. However, the trends provide considerable insight, and it is notable that the estimated grain boundary vacancy and Hes–vacancy Em are lower in the GBs than bulk Fe matrix, while the Hei Em is higher. These Em are summarized in Table 5. 1.06.5.4 Master Models of He Transport, Fate, and Consequences The overall approach to modeling He behavior in complex structural materials, such as FMS and NFA, that accounts for real microstructures was illustrated in Figure 3 and is briefly described in Section 1.06.6.2 below. The atomistic simulation results detailed above were used to inform the higher spatial- and temporal-scale microstructure evolution models of iron-based structural alloys. 1.06.5.5
Dislocation–Cavity Interactions
MD techniques have also been used to study dislocation–cavity interactions.238 The results of this work, where the cavities range from voids, to underpressurized, equilibrium, and overpressurized He bubbles, can be described in terms of an obstacle strengthening parameter (a) defined as a ¼ tc ðL d Þ=Gb
½21
Here, tc is the MD critical resolved shear stress for cavities with a diameter d that is spaced L apart,
The Effects of Helium in Irradiated Structural Alloys
G is the shear modulus, and b is the Burgers vector. In summary, a
Depends on the helium to vacancy ratio, m/n, and is highest for m/n for near-equilibrium bubbles and is lower for overpressurized bubbles.
Decreases with increasing temperature.
Increases with cavity size and at 300 K the peak a increases from about 0.2 to 0.4 in the diameter range of 1–4 nm.
1.06.6 Radiation Damage Tolerance, He Management, Integration of Helium Transport and Fate Modeling with Experiment 1.06.6.1 ISHI Studies and Thermal Stability of Nanofeatures in NFA MA957 In this section, we describe the status of developing a potentially transformational new class of materials, we call nanostructured ferritic alloys (NFA), with emphasis on He management for radiation tolerance as discussed in Section 1.06.3.22,23 NFA manifest high tensile, creep and fatigue strengths, unique thermal stability, and remarkable irradiation tolerance. The outstanding characteristics of NFA result from the presence of an ultrahigh density of Y–Ti–O rich nanofeatures (NF). The multifunctional NF, which are remarkably thermally stable, impede dislocation climb and glide, enhance SIA–vacancy recombination, and, perhaps most importantly, trap He in small, high-pressure gas bubbles. The bubbles reduce the amount of He reaching GBs, thus mitigating toughness loss at lower temperatures and potential degradation of creep rupture properties at higher temperature. He trapped in a high number density of small bubbles also mitigates many other manifestations of irradiation effects, including void swelling. As discussed in Section 1.06.2, ISHI in mixed spectrum fission reactor irradiations provides an attractive approach to assessing the effects of He–dpa synergisms. To reiterate, the basic idea is to use Ni (or B or Li)-bearing implanter layers to inject high-energy a-particles into an adjacent material that is simultaneously undergoing fast neutron-induced displacement damage. The a-particles can be produced by two-step 58Ni(nth,g)59Ni(nth,a) thermal neutron (nth) reactions. A series of ISHI irradiation experiments have been carried out in HFIR. Micronscale NiAl injector coatings were used to uniformly implant a-particles to a depth of 5–8 mm in TEM
183
disks for a large matrix of alloys irradiated over a wide range of temperatures and dpa at controlled He/dpa ratios ranging from 1 to 40 appm dpa1. Here, we compare the cavity structures in a 14Cr NFA, MA957, to those in an 8Cr FMS, F82H, following HFIR irradiation at 500 C to 9 dpa and 380 appm He. The experimental details are given elsewhere.23,50,51 Through-focus sequence TEM images were used to characterize the bubbles and voids, with care taken to avoid surface artifacts. Bubble-like features were generally not found in the unimplanted regions of either MA957 or F82H. As illustrated in a typical underfocused image in Figure 40(a), a high number density (Nb 4.3 1023 m3) of very small (average db 1.2 nm) bubbles are observed in the NFA. The inserts in Figure 40(a) show examples of the decoration of larger features with cavities. Image overlap analysis suggests that most bubbles are associated with a similar number density (6.5 1023 m3) of NFs.51,280 However, the degree of bubble–NF association has not yet been fully demonstrated and quantified. The boundary in MA957 in Figure 40(a) appears to be relatively cavity free, and there does not seem to be a large nearby NF-cavity denuded zone. Assuming equal partitioning of all the 380 appm He to 4.3 1023 m3 bubbles (80 He atoms/bubble), g ¼ 2 J m2 is consistent with rb 2g/kmkT 0.6 nm at 500 C, where k is the real gas compressibility factor, which is in remarkable agreement with the measured average cavity size. Thus, we conclude that the He is primarily stored in near-equilibrium bubbles at a capillary pressure of 2g/rb 6500 MPa in this case. A higher He content of 2000 appm partitioned to the same number of bubbles (400 He atoms/bubble) increases rb to 1.1 nm, still far below the critical size for void formation, which is estimated to be well over 10 nm. The 9 dpa irradiation at 500 C has no observable effect on the NFs. As shown in Figure 40(b) and 40(c), a lower number density of (Nb 5.3 1022 m3) of somewhat larger (db 2.1 nm) bubbles (the smaller population of cavities in this case) are observed in F82H, along with much larger faceted cavities; the larger cavities are likely voids. The smaller matrix bubbles in F82H are clearly formed on dislocations, as highlighted by the black and white contrast insert in Figure 40(c). Figure 40(d) shows that the cavity size distribution is much narrower in the MA957, with a maximum diameter of less than 2.5 nm. In contrast, the largest diameters exceed 10 nm in F82H and it appears that a bimodal bubble–void
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The Effects of Helium in Irradiated Structural Alloys
(a)
50 nm (b)
50 nm
(c)
50 nm 100
100
davg = 1.2 ± 0.2 nm
davg = 2.1 nm
23 m−3
N = 5.3 ⫻ 1022 m−3
10
MA957 1
0.1 0
(d)
Frequency (%)
Frequency (%)
N = 4.3 ⫻ 10
5
10
15
dbubble (nm)
10
F82H 1
0.1 0
5
10
15
dbubble (nm)
Figure 40 (a) Underfocused TEM image of in situ He injected MA957; (b) and (c) underfocused TEM image of in situ He injected F82H; (d) cavity size distributions in MA957 versus F82H.
cavity size distribution is developing.51 Model-based extrapolation of these results suggests that significant swelling may develop at higher He and dpa. The ISHI results suggest that NF are effective in trapping He in fine-scale bubbles at least up to 500 C. The sink strength of 4.3 1023 m3, 1.2 nm bubbles is Zb 3.2 1015 m2, which is significantly higher than the typical total sink strength in FMS alloys (<1015 m2). Presumably, Zb could be increased by an additional factor of at least two to three in alloys with larger numbers of NFs and bigger associated bubbles at higher He levels.
It is important to emphasize that both He bubbles and NFs are key to highly irradiation-tolerant alloys. The primary role of the NFs is to provide preferred and thermally stable sites for forming bubbles. In principle high densities of stable bubbles can sequester He up to high levels and serve as sinks that, in principle, mitigate all manifestations of displacement damage. Helium management schemes based on these principles are critical to developing fusion energy and may also play a role in fission applications intended to reach very high dpa levels. The effect of He bubbles on defect damage accumulation is being
The Effects of Helium in Irradiated Structural Alloys
investigated using the in situ implantation technique, including at higher He and dpa, for a wide range of irradiation temperatures and large alloy matrix. 1.06.6.2 Master Models of He Transport Fate and Consequences: Integration of Models and Experiment Complex structural materials, such as FMS and NFA, are composed of a wide range of types, morphologies, size scales, and associations of various microstuctural features. A general master model approach to treating He transport, fate, and consequences in such alloys is shown in Figure 3. He transport and clustering reactions within the matrix and in various microstructural regions are treated. In this framework, RT models, or KMC simulations, are used to transport and partition He to the various microstructural regions, such as dislocations, grain and subgrain boundaries, and heterophase interfaces. He is further transported within a region to and from internal subregions. For example, dislocations are regions, and precipitates and jogs on dislocations constitute subregions. He can recycle within or between regions. The accumulation of He in the various sites results in the formation of He bubbles (and in some cases voids). Preliminary development of a master model code to implement this framework is underway.178 In its current RT formulation, He is generated by transmutations as both Hei and Hes at relative fractions that can be based on a physically motivated parameter that needs to be established. These interstitial and substitutional forms can switch by interaction with vacancies and SIA, respectively. The Hei rapidly diffuses to the various regions where it is trapped or captured by a matrix vacancy to form Hes. Hes diffuses more slowly to the various regions or is displaced by a SIA to form Hei. The differences in the diffusion rate are reflected in the steady-state matrix concentrations of Hei (very low) and Hes (higher). A CD model (see Section 1.06.3) is used to track the fate of He and bubble evolution by reactions He þ Hem ! Hem þ 1 in the various regions and subregions. Although the framework of the CD models allows the disassociation of clusters by He emission, this is not implemented in the results described below. That is, a He2 cluster is taken as a stable nucleus for the formation and evolution of larger bubbles. A further assumption is that the HemVn clusters grow with He addition as equilibrium bubbles (m/n < 1), along the lowest free-energy path, with a real gas equation of state p ¼ 2g/rb, as described in Section 1.06.3. This
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approximation is valid at low damage rates and vacancy-rich environments associated with neutron irradiations. The current implementation focuses on bubble evolution, and since the dislocation bias is set to B ¼ 0, the model does not directly treat void formation and swelling. However, this capability will be implemented in future versions of the code. Helium atoms trapped at precipitates, dislocations, and GBs may either be emitted back to the matrix, at a rate determined by their binding energies, or diffuse to be captured by deeper subregion traps. The He þ He ! He2 reactions form a bubble nucleus in all regions and subregions, either heterogeneously with other trapped He or homogeneously with reactions between two freely diffusing He atoms. Nucleation of bubbles on dislocations is a very important process. Dislocations are modeled as a size distribution of segments bounded by deeper traps than the dislocation itself, such as junctions, jogs, and attached precipitates. The initial distribution of dislocation segments is resegmented (split) as bubbles homogeneously nucleate on them. The master model contains many parameters. Where possible, microstructural observations were used to provide microstructural parameters for grain sizes, dislocation densities, and precipitates, as summarized in Table 6. In general, the binding and activation energies were obtained from the models described in Section 1.06.5. Details are presented elsewhere.178 Figure 41 shows an example of the master model predictions of bubble radii (a,c) and number densities (b,d) compared with the ISHI data described previously in this section for 40 appm He/dpa at 500 C up to 10 dpa for the FMS F82H (a,b) and NFA MA957 (c,d) microstructure variables shown in Table 6. The data are shown for F82H in both as tempered (AT) and 20% CW conditions. The overall agreement is quite good. The model predicts that almost all of the bubbles form on dislocations in F82H and on dislocations and dislocation associated NF in MA957, broadly consistent with observations. The model predicts a smaller number of larger Table 6 Typical microstructural parameters for FMS and NFA models Region
Parameter
Nanoprecipitates Radius (rp) Density (Np) Dislocations Density (r) Grain size
Diameter (dg)
FMS
NFA
n/a 1.5 (nm) n/a 7 1023 (m3) 1 1015 1 1015 (m2) (m2) 20 (mm) 2 (mm)
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The Effects of Helium in Irradiated Structural Alloys
10-8
1024
r = 1015 m-2, dg = 20 mm (no NF)
Dislocation
40 appm He/dpa @ 500 ⬚C
1022
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Rbubble (m)
F82H mod.3 AT CW20%
10-9
Matrix
Matrix 1020
F82H mod.3 AT CW20%
1018
GB
Dislocation 1016
GB
r = 1E15 m-2, dg = 20 mm (no NF) 40 appm He/dpa @ 500 1/4C
10-10 0.0001
0.001
0.01
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1
10
dpa
(a)
1014 0.0001
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r = 1 ´ 1015 (m-2)
Disl. + DNF*
40 appm He/dpa @ 500 ⬚C
dg = 2 (mm)
1023
NNF = 7 ´ 1023 (m-3) rNF = 1.5 (nm)
1022
NF
Matrix Nbubble (m-3)
Rbubble (m)
Disl. + DNF* R(average) MA957 12YWT
0.01
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1
GB
1019
1017
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NF
1020
40 appm He/dpa @ 500 ⬚C
1018
Matrix
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Total MA957 12YWT
1021
GB
(c)
0.001
1016 0.0001
10
dpa
(d)
r = 1 ´ 1015 (m-2)
NNF = 7 ´ 1023 (m-3)
dg = 2 (mm)
rNF = 1.5 (nm)
0.001
0.01
0.1
1
10
dpa
Figure 41 Model predictions corresponding in situ He-implanter data for the (a) average bubble radius; (b) number density in F82H; (c) average bubble radius; and (d) number density in MA957also show the observations in experiments.
bubbles than observed in the F82H in the AT condition. The agreement is better for MA957, and the model predictions are consistent with the observation that a higher number density of smaller bubbles form in this case. The MA957 model predicts that there is a lower number of smaller bubbles in the matrix and especially on GBs. Note that the models do not yet contain lath boundaries that are observed to contain a high concentration of bubbles in F82H. The predicted size distribution of bubbles is shown in Figure 42. The agreement with the experimental results is again quite good and reflects the significant differences that are observed in the two alloys.
1.06.7 Summary and Some Outstanding Issues Developing fusion as a large-scale energy source and high-energy proton accelerator-based technologies are the primary motivations for studying He effects in structural alloys. These environments produce copious quantities of He (and H) by transmutation reactions. High levels of He, coupled with displacement (dpa) radiation damage, lead to a wide variety of property degradation phenomena over a wide range of irradiation temperatures, including both severe embrittlement and dimensional instabilities of various types.
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The Effects of Helium in Irradiated Structural Alloys
100 AT
12YWT
CW20%
MA957
Model 10 dpa
Model@10 dpa
9 dpa 360 appmHe 500 ⬚C
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Frequency (%)
Frequency (%)
100
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10
1 9 dpa 360 appmHe 500 ⬚C
0.1 0
5
10
15
dbubble (nm)
0.1
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5
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Figure 42 Predicted bubble size distributions in F82H and MA957 models along with the observations in the in situ implantation experiments.
Indeed, there is growing evidence that He–dpa synergisms will severely limit the operating window for leading candidate FMS in fusion first wall structures. Ultimate resolution of He effect issues in fusion applications will require a dedicated high-energy neutron source to develop an information base to construct and verify rigorous physically based predictive models of the effects of the fusion environment on performance-sustaining properties. These models must account for the interactions of a large number of variables, characterizing both the irradiation service environment and alloy of interest. However, in the interim, there are a number of irradiation techniques that can be used to simultaneously introduce high levels of He and dpa into materials. The most notable irradiation techniques include multibeam charged particle irradiations (CPI), ISHI in mixed spectrum fission reactors, and spallation neutron sources. Each of these methods has limitations, but significant progress in understanding He effects has been, and will continue to be, achieved by closely integrating all of these irradiation tools with advanced physical models. A primary objective of such modeling, and associated experiments, is to understand and predict the transport, fate, and consequences of He, and its interactions with displacement damage. There is a very large historical literature on irradiation effects in AuSS, including the key role played by He. Helium is critical to void swelling and high-temperature embrittlement (HTHE), where the latter is manifested as severe reductions in creep rupture times and strains. Standard AuSS provide an excellent basis
for developing an understanding of these phenomena, since they are (a) sensitive to many manifestations of irradiation damage and especially He effects; and (b) generate high levels of He from two-step Ni thermal neutron reactions in mixed spectrum fission reactors (typically of the order 50 appm He/dpa) and much lower, but significant, amounts of He (<1 appm He/dpa) in fast reactors. Both void formation and HTHE are characterized by a significant incubation period prior to forming growing cavities followed by rapid swelling or creep rupture. Helium bubbles are typically the formation sites for both voids and grain boundary creep cavities. RT-based thermodynamic–kinetic models rationalize many important trends in these phenomena. In particular, the critical bubble concept relates the combination irradiation variables, of temperature and dpa rates, along with a number of material-defect variables and parameters, to the critical size and helium content of bubbles that convert to voids, due to dislocation bias for SIA or grain boundary creep cavities due to stress. The critical bubble concept rationalizes and provides a basis to quantitatively model the incubation periods for both of these phenomena. RT can also be used to model post-incubation swelling rates and creep rupture times and strains. These models predict a number of important observations like bimodal bubble–void cavity size distributions and precipitate-associated bubbles and voids. Critical bubble–cavity growth models highlight the critical roles played by the overall He bubble microstructures on irradiation effects. Bubbles are necessary for the formation of voids and large
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The Effects of Helium in Irradiated Structural Alloys
numbers of creep cavities. However, high concentrations of bubbles, as generally associated with high He generation rates, can prolong incubation times by increasing the number of He partitioning–trapping sites in small highly subcritical bubbles; and, in the case of swelling and other manifestations of matrix radiation displacement damage. High bubble densities reduce the excess vacancy supersaturations and excess defect fluxes by acting as dominant defect sinks. These insights provide important guidance to developing irradiation-tolerant alloys. Enhanced damage tolerance can be achieved by creating finescale and stable microstructural features that can form small, harmless bubbles that sequester high levels of He suppressing swelling and protecting GBs from HTHE and grain boundary decohesion, leading to enormous DBTT shifts. Swelling-resistant advanced stainless steels, with extended incubation times, have used alloy carbide and phosphide phases to manage He in this manner. FMS, with high dislocation densities and fine lath structures, are intrinsically more damage resistant than standard austenitic alloys for a variety of reasons. These reasons include low He generation rates in fission reactors. However, the rates of He generation are much higher in D–T fusion spectra, and the irradiation damage tolerance of FMS may be significantly degraded in this case. SPNI have been the primary source of recent insight into the effects of He in structural alloys, especially mechanical property effects. Perhaps the most significant result of the SPNI studies is that high He-hardening synergisms can lead to enormous shifts in the DBTT shifts and IG fracture in FMS. The mechanisms responsible for such synergistic embrittlement, which can lead to transition temperature elevations of 600 C or more, are He-induced weakening of GBs as well as enhanced hardening at higher temperatures and higher dpa levels. Thus, there is concern that severe embrittlement and void swelling may close the window for useful application of these alloys in fusion environments. A parallel set of activities is needed to develop predictive models of He effects. Integrated master models, based on a multiscale–multiphysics paradigm, are being developed to predict He transport, fate, and consequences in realistic alloy microstructures. These master models contain a number of parameters and must be both mechanistically and microstructurally informed. First-principles electronic structure theory and atomistic simulations can provide required model parameters and mechanistic insights. These tools include DFT, embedded atom-based
MD and MS, various Monte Carlo methods, and RT. For example, recent first principles and atomistic research has provided important information on He–vacancy cluster energetics, He interactions with a range of microstructural features, He diffusion mechanisms, and rates in the matrix, along dislocations and in GBs. These models show high He–vacancy binding energies even at the smallest cluster sizes, strong interactions between He and dislocations and dislocation jogs, and both homo- and heterophase interfaces. MD methods have also been used to characterize phenomena such as resolutioning of He from bubbles, showing this to be a minor mechanism; and determining the dislocation interaction strength of cavities, ranging from under to overpressurized conditions, showing that near-equilibrium bubbles are the strongest obstacles. Recent MD studies have also suggested that capillary models may overpredict He gas pressures in equilibrium bubbles. As noted above, the concepts and models described above can be used to guide the development of irradiation-tolerant alloys. The most promising class of such materials have been dubbed NFA, which contain an ultrahigh density of remarkably stable, nanometer-scale Ti–Y–O NF. The NF are now believed to be primarily a complex oxide Y2Ti2O7 pyrochlore phase. The NF provide remarkable hightemperature creep strength, so that NFA can operate above the displacement damage regime, where recovery processes are much faster than defect accumulation rates. More significantly the NF trap helium in a very high density of very fine-scale bubbles. In principle, the bubbles suppress or mitigate essentially all manifestation of radiation damage and property degradation. Limited proof in principle validation of the ability of NFA to manage He has been provided by ISHI irradiations that were carried out in the HFIR. For example, an irradiation of NFA MA957 to 9 dpa and 380 appm He produced a very high density of nanometer-scale bubbles on the NFs and dislocations. The same irradiation of the FMS F82H produced roughly an order of magnitude lower density of cavities, composed of a bimodal distribution of bubbles and larger voids. Bubbles primarily form on dislocations in this case. Further, limited data showed relatively bubble-free interfaces in MA957, in contrast to interfaces in F82H that are highly decorated with small bubbles. The results of the ISHI experiments compare very favorably with a He transport and fate master model that is under development. The master model is both microstructurally informed and incorporates
The Effects of Helium in Irradiated Structural Alloys
parameters derived from the atomistic/electronic models cited above. Preliminary RT models have also been used to extrapolate the ISHI data to predict void swelling at higher dpa and He levels. These models suggest that FMS may experience significant swelling at damage levels greater than 50 dpa, while the NFA will remain void free. Indeed the combination of the experimental and modeling results suggest that He can be transformed from a liability to an asset in NFA. Although these conclusions represent an optimistic view of the status of research on measuring, modeling, and managing He–dpa effects in structural alloys, especially for fusion applications, there are a number of outstanding issues that require additional science-based research to resolve. Space does not permit a complete listing, but these issues and research needs can generally be classified into broad categories related to He effects per se and those related to developing, optimizing, and qualifying alloys that can manage He in a way that may provide near immunity to radiation damage. He-related issues and research needs include the following:
Extension of experimental observations of He transport, fate, and consequences to more alloys for a wide range of microstructures and to higher damage (He and dpa) levels and higher temperatures over a range of He/dpa, using all the irradiation techniques: CPI, ISHI, and SPNI. Characterization of the management of He in NFA at temperatures up to 750 C, or more, is particularly critical.
Intercomparisons of accelerated high-rate CPI and lower reactor relevant rate ISHI and SPNI data and development of experimental–modeling approaches that will permit reliable predictions of He–dpa effects at very high damage levels, up to several hundred dpa, that cannot be accessed practically in neutron ISHI and SPNI.
Very detailed characterization of He and He bubble distributions, along with the balance of irradiation microstructures, with particular emphasis on quantitative evaluations, like the distribution of He and bubbles on interfaces, using a suite of advanced characterization methods tailored to this application.
Focused mechanism studies that can provide both parameters and insights into key mechanisms that govern the transport, fate, and consequences of He in various complex alloy and model systems.
Continued development and refinement of multiscale master models of He transport fate and consequences, along with the narrower first principles
189
and atomistic studies (models and experiments) needed to better parameterize and mechanistically inform the master models. One of many examples of modeling needs is the resolution of differences between continuum capillary versus atomistic MD-based models of helium–vacancy cluster and bubble properties.
Developing models and basic experiments that relate the microstructural consequences of the fate of He to the degradation of performancesustaining mechanical properties. One of many examples is the detailed model of the behavior of He on GBs and the corresponding reduction in the grain boundary fracture strength. However, it is clear that He must be managed as well as understood. Thus, issues and research needs also include development of practical NFA or alloys with similar attributes. These include the following items:
Identify alloy composition-synthesis designs and thermal mechanical processing paths that optimize the NF, and the balance of NFA microstructures, so as to provide a balanced suite of outstanding and isotropic properties.
Resolve issues of low fracture toughness and anisotropic properties in extruded NFA product forms.
Demonstrate and understand the thermal and irradiation stability of far-from-equilibrium NF and NFA microstructures.
Develop practical fabrication and joining methods, which preserve optimal NFA microstructures and yield defect-free components.
Reduce costs, improve alloy homogeneity and reproducibility, and establish industrial-scale supply sources.
Qualify new alloys for nuclear service for extended lifetimes. We believe that there is a science-based framework to resolve these and other critical issues related to the effects and management of He in practical alloy systems. (see Chapter 1.03, Radiation-Induced Effects on Microstructure and Chapter 1.09, Molecular Dynamics)
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Jitsukawa, S.; Maziasz, P. J.; Ishiyama, T.; Gibson, L. T.; Hishinuma, A. J. Nucl. Mater. 1992, 191–194, 771. Jitsukawa, S.; Grossbeck, M. L.; Hishinuma, A. J. Nucl. Mater. 1992, 191–194, 790. Pawel, J. E.; Rowcliffe, A. F.; Alexander, D. J.; Grossbeck, M. L.; Shiba, K. J. Nucl. Mater. 1996, 233–237, 202–206. Odette, G. R.; Frey, D. J. Nucl. Mater. 1979, 85–86, 817. Odette, G. R.; Lucas, G. E. J. Nucl. Mater. 1991, 179–181, 572. Dai, Y.; et al. to be published. Hafez Haghighat, S. M.; Lucas, G.; Schaublin, R. EPL 2009, 85, 60008. Gao, N.; Victoria, M.; Chen, J.; Van Swygenhoven, H. Journal of Physics: Condensed Matter. 2011, 23, 245403. Stewart, D. M.; Osetsky, Yu. N.; Stoller, R. E.; Golubov, S. I.; Seletskaia, T.; Kamenski, P. J. Phil. Mag. 2010, 90, 935–944. Stewart, D. M.; Osetskiy, Yu. N.; Stoller, R. E. J. Nucl. Mater. 2011, doi:10.1016/j.jnucmat.2010.12.217. Stoller, R. E.; Golubov, S. I.; Kamenski, P. J.; Seletskaia, T.; Osetsky, Yu. N. Phil. Mag. 2010, 90, 923–934. Morishita, K.; Sugano, R.; Wirth, B. D.; de la Rubia, T. D. Nucl. Instrum. Meth. Phys. Res. B 2003, 202, 76–81. Ackland, G. J.; Bacon, D. J.; Calder, A. F.; Harry, T. Phil. Mag. A 1997, 75, 713. Wilson, W. D.; Johnson, R. D. In Interatomic Potentials and Simulation of Lattice Defects; Gehlen, P. C., Beeler, J. R., Jr., Jaffee, R. I., Eds.; Plenum: New York, 1972; p 375. Beck, D. E. Mol. Phys. 1968, 14, 311. Seletskaia, T.; Osetskiy, Yu. N.; Stoller, R. E.; Stocks, G. M. J. Nucl. Mater. 2007, 367–370, 355. Ventelon, L.; Wirth, B.; Domain, C. J. Nucl. Mater. 2006, 351, 119–132. Shim, J. H.; Kwon, S. C.; Kim, W. W.; Wirth, B. D. J. Nucl. Mater. 2007, 367–370, 292–297. Henkelman, G.; Jonsson, H. J. Chem. Pbys. 1999, 111, 7010. Heinisch, H. L.; Gao, F.; Kurtz, R. J.; Le, E. A. J. Nucl. Mater. 2006, 351, 141–148. Heinisch, H. L.; Gao, F.; Kurtz, R. J. Phil. Mag. 2010, 90, 885. Kurtz, R. J.; Heinisch, H. L. J. Nucl. Mater. 2004, 329–333, 1199–1203. Le Claire, A. D. J. Nucl. Mater. 1978, 69&70, 70. Gao, F.; Heinisch, H. L.; Kurtz, R. J. J. Nucl. Mater. 2006, 351, 133–140. Gao, F.; Heinisch, H. L.; Kurtz, R. J. J. Nucl. Mater. 2007, 367–370, 446–450. Gao, F.; Heinisch, H. L.; Kurtz, R. J. J. Nucl. Mater. 2009, 386–388, 390–394. Miao, P.; Odette, G. R.; Yamamoto, T.; Alinger, M.; Klingensmith, D. J. Nucl. Mater. 2008, 377, 59–64.
1.07
Radiation Damage Using Ion Beams
G. S. Was University of Michigan, Ann Arbor, MI, USA
R. S. Averback University of Illinois at Urbana-Champagne, Urbana, IL, USA
ß 2012 Elsevier Ltd. All rights reserved.
1.07.1 1.07.2 1.07.3 1.07.3.1 1.07.3.2 1.07.3.3 1.07.3.4 1.07.4 1.07.4.1 1.07.4.1.1 1.07.4.1.2 1.07.4.2 1.07.4.2.1 1.07.4.2.2 1.07.4.2.3 1.07.4.2.4 1.07.4.2.5 1.07.4.2.6 1.07.4.3 1.07.5 1.07.5.1 1.07.5.2 1.07.5.3 1.07.6 References
Introduction Motivation for Using Ion Beams to Study Radiation Damage Review of Aspects of Radiation Damage Relevant to Ion Irradiation Defect Production Primary and Weighted Recoil Spectra Damage Morphology Damage Rate Effects Contributions of Ion Irradiation to an Understanding of Radiation Effects Electron Irradiations Displacement threshold surfaces Point defect properties Ion Irradiations The damage function Freely migrating defects Alloy stability under ion irradiation Mechanical properties Multiple ion beams Swift ions Comparison with Neutrons Advantages and Disadvantages of Irradiations using Various Particle Types Electrons Heavy Ions Light Ions Practical Considerations for Radiation Damage Using Ion Beams
Abbreviations AES APT bcc BWR dpa fcc FMD FP IASCC IGSCC LWR MD NRT
Auger electron spectroscopy Atom probe tomography Body-centered cubic Boiling water reactor Displacements per atom Face-centered cubic Freely migrating defect Frenkel pair Irradiation assisted stress corrosion cracking Intergranular stress corrosion cracking Light water reactor Molecular dynamics Norgett–Robinson–Torrens
NWC PKA RCS RIS SCC STEM/EDS
TEM
195 196 197 197 199 200 202 204 204 204 205 206 206 207 207 209 209 209 211 215 216 217 219 219 221
Normal water chemistry Primary knock-on atom Recoil collision sequence Radiation induced segregation Stress corrosion cracking Scanning transmission electron microscopy/energy dispersive spectrometry Transmission electron microscopy
1.07.1 Introduction Radiation effects research has been conducted using a variety of energetic particles: neutrons, electrons, protons, He ions, and heavy ions. Energetic ions 195
196
Radiation Damage Using Ion Beams
can be used to understand the effects of neutron irradiation on reactor components, and interest in this application of ion irradiation has grown in recent years for several reasons including the avoidance of high residual radioactivity and the decline of neutron sources for materials irradiation. The damage state and microstructure resulting from ion irradiation, and thus the degree to which ion irradiation emulates neutron irradiation, depend upon the particle type and the damage rate. This chapter will begin with a summary of the motivation for using ion irradiation for radiation damage studies, followed by a brief review of radiation damage relevant to charged particles. The contribution of ion irradiation to our understanding of radiation damage will be presented next, followed by an account of the advantages and disadvantages of the various ion types for conducting radiation damage studies, and wrapping up with a consideration of practical issues in ion irradiation experiments.
1.07.2 Motivation for Using Ion Beams to Study Radiation Damage In the 1960s and 1970s, heavy ion irradiation was developed for the study of radiation damage processes in materials. As ion irradiation can be conducted at a well-defined energy, dose rate, and temperature, it results in very well-controlled experiments that are difficult to match in reactors. As such, interest grew in the use of ion irradiation for the purpose of simulating neutron damage in support of the breeder reactor program.1–3 Ion irradiation and simultaneous He injection were also used to simulate the effects of 14 MeV neutron damage in conjunction with the fusion reactor engineering program. The application of ion irradiation (defined here as irradiation by any charged particle, including electrons) to the study of neutron irradiation damage caught the interest of the light water reactor community to address issues such as swelling, creep, and irradiation assisted stress corrosion cracking of core structural materials.4–6 Ion irradiation was also being used to understand the irradiated microstructure of reactor pressure vessel steels, Zircaloy fuel cladding, and materials for advanced reactor concepts. There is significant incentive to use ion irradiation to study neutron damage as this technique has the potential for yielding answers on basic processes in addition to the potential for enormous savings in time and money. Neutron irradiation experiments are not amenable to studies involving a wide range
of conditions, which is precisely what is required for investigations of the basic damage processes. Simulation by ions allows easy variation of the irradiation parameters such as dose, dose rate, and temperature over a wide range of values. One of the prime attractions of ion irradiation is the rapid accumulation of end of life doses in short periods of time. Typical neutron irradiation experiments in thermal test reactors may accumulate damage at a rate of 3–5 dpa year1. In fast reactors, the rates can be higher, on the order of 20 dpa year1. For low dose components such as structural components in boiling water reactor (BWR) cores that typically have an end-of-life damage of 10 dpa, these rates are acceptable. However, even the higher dose rate of a fast reactor would require 4–5 years to reach the peak dose of 80 dpa in the core baffle in a pressurized water reactor (PWR). For advanced, fast reactor concepts in which core components are expected to receive 200 dpa, the time for irradiation in a test reactor becomes impractical. In addition to the time spent ‘in-core,’ there is an investment in capsule design and preparation as well as disassembly and allowing for radioactive decay, adding additional years to an irradiation program. Analysis of microchemical and microstructural changes by atom probe tomography (APT), Auger electron spectroscopy (AES) or microstructural changes by energy dispersive spectroscopy via scanning transmission electron microscopy (STEM-EDS) and mechanical property or stress corrosion cracking (SCC) evaluation can take several additional years because of the precautions, special facilities, and instrumentation required for handling radioactive samples. The result is that a single cycle from irradiation through microanalysis and mechanical property/SCC testing may require over a decade. Such a long cycle length does not permit for iteration of irradiation or material conditions that is critical in any experimental research program. The long cycle time required for design and irradiation also reduces flexibility in altering irradiation programs as new data become available. The requirement of special facilities, special sample handling, and long irradiation time make the cost for neutron irradiation experiments very high. In contrast to neutron irradiation, ion (heavy, light, or electrons) irradiation enjoys considerable advantages in both cycle length and cost. Ion irradiations of any type rarely require more than several tens of hours to reach damage levels in the 1–100 dpa range. Ion irradiation produces little or no residual radioactivity, allowing handling of samples without
Radiation Damage Using Ion Beams
the need for special precautions. These features translate into significantly reduced cycle length and cost. The challenge then is to verify the equivalency between neutron and ion irradiation in terms of the changes to the microstructure and properties of the material. The key question that needs to be answered is how do results from neutron and charged particle irradiation experiments compare? How, for example, is one to compare the results of a component irradiated in-core at 288 C to a fluence of 1 1021 n cm2 (E > 1 MeV) over a period of one year, with an ion irradiation experiment using 3 MeV protons at 400 C to 1 dpa (displacements per atom) at a dose rate of 105 dpa s1 (1 day), or 5 MeV Ni2þ at 500 C to 10 dpa at a dose rate of 5 103 dpa s1 (1 h)? The first question to resolve is the measure of radiation effect. In the Irradiation assisted stress corrosion cracking (IASCC) problem in LWRs, concern has centered on two effects of irradiation: radiation-induced segregation of major alloying elements or impurities to grain boundaries, which may cause embrittlement or enhance the intergranular stress corrosion cracking (IGSCC) process, and hardening of the matrix that results in localized deformation and embrittlement. The appropriate measure of the radiation effect in the former case would then be the alloy concentration at the grain boundary or the amount of impurity segregated to the grain boundary. This quantity is measurable by analytical techniques such as AES, APT, or STEM-EDS. For the latter case, the measure of the radiation effect would be the nature, size, density, and distribution of dislocation loops, black dots, and the total dislocation network, and how they impact the deformation of the alloy. Hence, specific and measurable effects of irradiation can be determined for both neutron and ion irradiation experiments. The next concern is determining how ion irradiation translates into the environment describing neutron irradiation. That is, what are the irradiation conditions required for ion irradiation to yield the same measure of radiation effect as that for neutron irradiation? This is the key question, for in a postirradiation test program, it is only the final state of the material that determines equivalence, not the path taken. Therefore, if ion irradiation experiments could be devised that yielded the same measures of irradiation effects as observed in neutron irradiation experiments, the data obtained in postirradiation experiments will be equivalent. In such a case, ion irradiation experiments can provide a direct substitute for neutron irradiation. While neutron irradiation will always be required to qualify materials for reactor
197
application, ion irradiation provides a low-cost and rapid means of elucidating mechanisms and screening materials for the most important variables. A final challenge is the volume of material that can be irradiated with each type of radiation. Neutrons have mean free paths on the order of centimeters in structural materials. One MeV electrons penetrate about 500 mm, 1 MeV protons penetrate about 10 mm, and 1 MeV Ni ions have a range of less than 1 mm. Thus, the volume of material that can be irradiated with ions from standard laboratory-sized sources (TEMs, accelerators), is limited.
1.07.3 Review of Aspects of Radiation Damage Relevant to Ion Irradiation 1.07.3.1
Defect Production
The parameter commonly used to correlate the damage produced by different irradiation environments is the total number of displacements per atom (dpa). Kinchin and Pease7 were the first to attempt to determine the number of displacements occurring during irradiation and a modified version of their model known as the Norgett–Robinson–Torrens (NRT) model8 is generally accepted as the international standard for quantifying the number of atomic displacements in irradiated materials.9 According to the NRT model, the number of Frenkel pairs (FPs), nNRT(T ), generated by a primary knock-on atom (PKA) of energy T is given by nNRT ðT Þ ¼
kED ðT Þ 2Ed
½1
where ED(T ) is the damage energy (energy of the PKA less the energy lost to electron excitation), Ed is the displacement energy, that is, the energy needed to displace the struck atom from its lattice position, and k is a factor less than 1 (usually taken as 0.8). Integration of the NRT damage function over recoil spectrum and time gives the atom concentration of displacements known as the NRT displacements per atom (dpa): ðð ½2 dpa ¼ fðEÞvNRT ðT ÞsðE; T ÞdT dE where f(E) is the neutron flux and s(E,T ) is the probability that a particle of energy E will impart a recoil energy T to a struck atom. The displacement damage is accepted as a measure of the amount of change to the solid due to irradiation and is a much better measure of an irradiation effect than is the particle fluence. As shown in Figure 1, seemingly different effects of
198
Radiation Damage Using Ion Beams
300 LASREF, 40 ⬚C RTNS-II, 90 ⬚C OWR, 90 ⬚C
250
Yield stress change (MPa)
Yield stress change (MPa)
300
200 150 100 50 0
1017
1018
1019
LASREF, 40 ⬚C RTNS-II, 90 ⬚C OWR, 90 ⬚C
250 200 150 100 50
0
1020
10−3
Neutron fluence, E > 0.1 MeV
10−2
DPA
Figure 1 Comparison of yield stress change in 316 stainless steel irradiated in three facilities with very different neutron energy flux spectra. While there is little correlation in terms of neutron fluence, the yield stress changes correlate well against displacements per atom (dpa). Reprinted, with permission, from ASTM, copyright ASTM International, 100 Barr Harbor Drive, West Conshohocken, PA 19428.
7.5 MeV tantalum 10−15
1012
5 MeV nickel 1010 Protons ITER be first wall HFIR target FFTF mid-core PWR 1/4-T RPV
108 106 104 −9 10
10−7
10−5 10−1 10−3 Particle energy (MeV)
10
Figure 2 Energy spectrum for neutrons from a variety of reactor types and a monoenergetic proton beam. Reproduced from Stoller, R. E.; Greenwood, L. R. J. Nucl. Mater. 1999, 271–272, 57–62.
Calculated dpa/(incident particle) (cm2)
Neutron flux/lethergy (n cm−2 s−1) proton flux (ions cm−2 s−1)
10−14 1014
10−16
20 MeV carbon
10−17 1.3 MeV hydrogen
10−18
10−19 14 MeV neutrons −20
1 MeV neutrons
10
irradiation on low temperature yield strength for the same fluence level (Figure 1(a)) and disappear when dpa is used as the measure of damage (Figure 1(b)). A fundamental difference between ion and neutron irradiation effects is the particle energy spectrum that arises because of the difference in the way the particles are produced. Ions are produced in accelerators and emerge in monoenergetic beams with very narrow energy widths. However, the neutron energy spectrum in a reactor extends over several orders of magnitude in energy, thus presenting a much more complicated source term for radiation damage. Figure 2 shows the considerable difference in neutron and ion energy spectra and also between neutron spectra in different reactors and at different locations within the reactor vessel.
10−21
0
2
4 6 8 Distance into solid (m)
10
12
Figure 3 Displacement–damage effectiveness for various energetic particles in nickel. Reproduced from Kulcinski, G. L.; Brimhall, J. L.; Kissinger, H. E. In Proceedings of Radiation-Induced Voids in Metals; Corbett, J. W., Ianiello, L. C., Eds.; USAEC Technical Information Center: Oak Ridge, TN, 1972; p 453, CONF-710601.
Another major difference in the characteristics of ions and neutrons is their depth of penetration. As shown in Figure 3, ions lose energy quickly because of high electronic energy loss, giving rise to a spatially nonuniform energy deposition profile caused
199
Radiation Damage Using Ion Beams
where Rd is the number if displacements per unit volume per unit time, N is the atom number density, and f is the particle flux (neutron or ion). In the case of neutron–nuclear interaction described by the hardsphere model, eqn [3] becomes Rd gE ss ¼ ½4 Nf 4Ed where g ¼ 4mM/(m þ M)2, M is the target atom mass, m is the neutron mass, E is the neutron energy, and ss is the elastic scattering cross-section. For the case of ion– atom interaction described by Rutherford scattering, eqn [3] becomes Rd pZ2 Z2 e4 M1 gE ¼ 1 2 ln ; ½5 NI 4EEd M2 Ed where e is the unit charge, M1 is the mass of the ion, and M2 is the mass of the target atom. As shown in Figure 3, for comparable energies, 1.3 MeV protons cause over 100 times more damage per unit of fluence at the sample surface than 1 MeV neutrons, and the factor for 20 MeV C ions is over 1000. Of course, the damage depth is orders of magnitude smaller than that for neutron irradiation. 1.07.3.2 Primary and Weighted Recoil Spectra A description of irradiation damage must also consider the distribution of recoils in energy and space. The primary recoil spectrum describes the relative number of collisions in which the amount of energy between T and T þ dT is transferred from the primary
recoil atom to other target atoms. The fraction of recoils between the displacement energy Ed, and T is ð 1 T sðE; T 0 ÞdT 0 ½6 PðE; T Þ ¼ N Ed where N is the total number of primary recoils and s(E,T ) is the differential cross-section for a particle of energy E to create a recoil of energy T. The recoil fraction is shown in Figure 4, which reveals only a small difference between ions of very different masses. Figure 5 shows the difference in the types of damage that are produced by different types of 1.0
0.8 Fraction of recoils
by the varying importance of electronic and nuclear energy loss during the slowing down process. Their penetration distances range between 0.1 and 100 mm for ion energies that can practically be achieved by laboratory-scale accelerators or implanters. By virtue of their electrical neutrality, neutrons can penetrate very large distances and produce spatially flat damage profiles over many millimeters of material. Further, the cross-section for ion–atom reaction is much greater than for neutron–nuclear reaction giving rise to a higher damage rate per unit of particle fluence. The damage rate in dpa per unit of fluence is proportional to the integral of the energy transfer cross-section and the number of displacements per PKA, nNRT(T): ð gE Rd ¼ sðE; T ÞnNRT ðT ÞdT ½3 Nf Ed
He H Kr Ar
0.6 Ne 0.4
Fraction of recoils with energy above Ed and below T
0.2 1 MeV ions ® Cu 0 101
102
103
104
T (eV) Figure 4 Integral primary recoil spectra for 1 MeV particles in copper. Curves plotted are the integral fractions of primary recoils between the threshold energy and recoil energy, T from eqn [6]. Reproduced from Averback, R. S. J. Nucl. Mater. 1994, 216, 49.
1 MeV electrons T = 60 eV e = 50−100%
106
E
105 104
1 MeV protons T = 200 eV e = 25%
Ti
103 102
1 MeV heavy ions T = 5 keV e = 4%
Tn
Ed
Tp Te
101 E
1 MeV neutrons T = 35 keV e = 2% Figure 5 Difference in damage morphology, displacement efficiency, and average recoil energy for 1 MeV particles of different types incident on nickel. Reproduced from Was, G. S.; Allen, T. R. Mater. Char. 1994, 32, 239.
Radiation Damage Using Ion Beams
particles. Light ions such as electrons and protons will produce damage as isolated FPs or in small clusters while heavy ions and neutrons produce damage in large clusters. For 1 MeV particle irradiation of copper, half the recoils for protons are produced with energies less than 60 eV while the same number for Kr occurs at about 150 eV. Recoils are weighted toward lower energies because of the screened Coulomb potential that controls the interactions of charged particles. For an unscreened Coulomb interaction, the probability of creating a recoil of energy T varies as 1/T2. However, neutrons interact as hard spheres and the probability of creating a recoil of energy T is independent of recoil energy. In fact, a more important parameter describing the distribution of damage over the energy range is a combination of the fraction of defects of a particular energy and the damage energy. This is the weighted average recoil spectrum, W(E,T ), which weights the primary recoil spectrum by the number of defects or the damage energy produced in each recoil: ðT 1 sðE; T 0 ÞED ðT 0 ÞdT 0 ½7 W ðE; T Þ ¼ ED ðEÞ Ed ED ðEÞ ¼
ð T^ Ed
sðE; T 0 ÞED ðT 0 ÞdT 0
½8
^ is the maximum recoil energy given by where T ^ T ¼ gEi ¼ 4EiM1M2/(M1 þ M2)2. Ignoring electron excitations and allowing ED(T ) ¼ T, then the weighted average recoil spectra for Coulomb and hard sphere collisions are WCoul ðE; T Þ ¼
lnT lnEd ^ lnEd lnT
½9
T 2 Ed2 Ed2
½10
WHS ðE; T Þ ¼
Equations [9] and [10] are graphed in Figure 6 for 1 MeV particle irradiations of copper. The characteristic energy, T1/2 is that recoil energy below which half of the recoils are produced. The Coulomb forces extend to infinity and slowly increase as the particle approaches the target; hence the slow increase with energy. In a hard sphere interaction, the particles and target do not interact until their separation reaches the hard sphere radius at which point the repulsive force goes to infinity. A screened Coulomb is most appropriate for heavy ion irradiation. Note the large difference in W(E,T ) between the various types of irradiations at E ¼ 1 MeV.
1.0 Copper 0.8
Protons
Ne
0.6
Kr
W (T)
200
0.4 Neutrons 0.2
0 101
102
103
104 T (eV)
105
106
107
Figure 6 Weighted recoil spectra for 1 MeV particles in copper. Curves representing protons and neutrons are calculated using eqns [9] and [10], respectively. W(T ) for other particles were calculated using Lindhard cross-sections and include electronic excitation. Reproduced from Averback, R. S. J. Nucl. Mater. 1994, 216, 49.
While heavy ions come closer to reproducing the energy distribution of recoils of neutrons than do light ions, neither is accurate in the tails of the distribution. This does not mean that ions are poor simulations of radiation damage, but it does mean that damage is produced differently and this difference will need to be considered when designing an irradiation program that is intended to produce microchemical and microstructural changes that match those from neutron irradiation. There is, of course, more to the description of radiation damage than just the number of dpa. There is the issue of the spatial distribution of damage production, which can influence the microchemistry and microstructure, particularly at temperatures where diffusion processes are important for microstructural development. In fact, the ‘ballistically’ determined value of dpa calculated using such a displacement model is not the appropriate unit to be used for dose comparisons between particle types. The reason is the difference in the primary damage state among different particle types. 1.07.3.3
Damage Morphology
The actual number of defects that survive the displacement cascade and their spatial distribution in solids will determine the effect on the irradiated microstructure. Figure 7 summarizes the effect of
Radiation Damage Using Ion Beams
201
Total dpa Particle type and energy Loss to displacement cascades
Freely migrating defects
Mutual recombination outside of cascade
Loss to sinks in matrix
Loss at grain boundaries
Void swelling loop structure
Defect diffusion matrix chemistry
Boundary structure and micro chemistry Radiation-induced segregation
Figure 7 History of point defects after creation in the displacement cascade.
damage morphology from the viewpoint of the grain boundary and how the defect flow affects radiationinduced grain boundary segregation. Of the total defects produced by the energetic particle, a fraction appears as isolated, or freely migrating defects, and the balance is part of the cascade. The fraction of the ‘ballistically’ produced FPs that survive the cascade quench and are available for long-range migration is an extremely important quantity and is called the migration efficiency, e. These ‘freely migrating’ or ‘available migrating’ defects10 are the only defects that will affect the amount of grain boundary segregation, which is one measure of radiation effects. The migration efficiency can be very small, approaching a few percent at high temperatures. The migration efficiency, e, comprises three components: gi,v: the isolated point defect fraction, di,v: clustered fraction including mobile defect clusters such as di-interstitials, and z: fraction initially in isolated or clustered form after the cascade quench that is annihilated during subsequent short-term (>1011 s) intracascade thermal diffusion. They are related as follows: e ¼ di þ g i þ z i ¼ d v þ g v þ z v
½11
Figure 8 shows the history of defects born as vacancies and interstitials as described by the NRT model.
Displacement cascade efficiency (x) Intracascade thermal recombination (z ) Surviving defect fraction (QDF) (x – z ) Isolated point defect fraction (IDF) (g i,v)
Clustered point defect fraction (CDF) (d i,v)
Mobile clusters
Immobile clusters
Evaporating defects Available defects (li,v)
Figure 8 Interdependence of isolated point defects, mobile defect clusters, and thermally evaporating defect clusters that contribute to the fraction of surviving defects that are ‘available’ for radiation effects. Reproduced from Zinkle, S. J.; Singh, B. N. J. Nucl. Mater. 1993, 199, 173.
Due to significant recombination in the cascade, only a fraction (30%) is free to migrate from the displacement zone. These defects can recombine outside of the cascade region, be absorbed at sinks in the
202
Radiation Damage Using Ion Beams
matrix (voids, loops), or be absorbed at the grain boundaries, providing for the possibility of radiationinduced segregation. The fraction of defects that will be annihilated after the cascade quench by recombination events among defect clusters and point defects within the same cascade (intracascade recombination), z, is about 0.07, for a migration efficiency of 0.3 (see below for additional detail).10 The clustered fraction, d includes large, sessile clusters and small defect clusters that may be mobile at a given irradiation temperature and will be different for vacancies and interstitials. For a 5 keV cascade, di is about 0.06 and dv is closer to 0.18.10 Some of these defects may be able to ‘evaporate’ or escape the cluster and become ‘available’ defects (Figure 8). This leaves g, the isolated point defect fraction that are available to migrate to sinks, to form clusters, to interact with existing clusters, and to participate in the defect flow to grain boundaries that gives rise to radiation-induced segregation. Owing to their potential to so strongly influence the irradiated microstructure, defects in this category, along with defects freed from clusters, make up the freely migrating defect (FMD) fraction. Recall that electrons and light ions produce a large fraction of their defects as isolated FPs, thus increasing the likelihood of their remaining as isolated rather than clustered defects. Despite the equivalence in energy among the four particle types described in Figure 5, the average energy transferred and the defect production efficiencies vary by more than an order of magnitude. This is explained by the differences in the cascade morphology among the different particle types. Neutrons and heavy ions produce dense cascades that result in substantial recombination during the cooling or quenching phase. However, electrons are just capable of producing a few widely spaced FPs that have a low probability of recombination. Protons produce small widely spaced cascades and many isolated FPs due to the Coulomb interaction and therefore, fall between the extremes in displacement efficiency defined by electrons and neutrons. The value of g has been estimated to range from 0.01 to 0.10 depending on PKA energy and irradiation temperature, with higher temperatures resulting in the lower values. Naundorf12 estimated the freely migrating defect fraction using an analytical treatment based on two factors: (1) energy transfer to atoms is only sufficient to create a single FP, and (2) the FP lies outside a recombination (interaction)
Table 1 Efficiency for producing freely migrating defects, g, in nickel by different kinds of irradiations (Ed ¼ 40 eV, riv ¼ 0.7 nm) using Lindhard’s analytical differential collision cross-section Irradiation
(%)
1 MeV Hþ 2 MeV Hþ 2 MeV Liþ 1.8 MeV Neþ 300 keV Niþ 3 MeV Niþ 3.5 MeV Krþ 2 keV Oþ
24.0 19.2 16.9 8.7 2.3 3.8 3.0 9.8
Source: Naundorf, V. J. Nucl. Mater. 1991, 182, 254.
radius so that the nearby FPs neither recombine nor cluster. The model follows each generation of the collision and calculates the fraction of all defects produced that remain free. Results of calculation using the Naundorf model are shown in Table 1 for several ions of varying mass and energy. Values of Z range between 24% for proton irradiation to 3% for heavy ion (krypton) irradiation. Recent results,13 however, have shown that the low values of FMD efficiency for heavy ion or neutron irradiation cannot be explained by defect annihilation within the parent cascade (intracascade annihilation). In fact, cascade damage generates vacancy and interstitial clusters that act as annihilation sites for FMD, reducing the efficiency of FMD production. Thus, the cascade remnants result in an increase in the sink strength for point defects and along with recombination in the original cascade, account for the low FMD efficiency measured by experiment. 1.07.3.4
Damage Rate Effects
As differences in dose rates can confound direct comparison between neutron and ion irradiations, it is important to assess their impact. A simple method for examining the tradeoff between dose and temperature in comparing irradiation effects from different particle types is found in the invariance requirements. For a given change in dose rate, we would like to know what change in dose (at the same temperature) is required to cause the same number of defects to be absorbed at sinks. Alternatively, for a given change in dose rate, we would like to know what change in temperature (at the same dose) is required to cause the same number of defects to be absorbed at sinks. The number of defects per unit volume, NR, that have recombined up to time t, is given by Mansur14
Radiation Damage Using Ion Beams
ðt NR ¼ Riv Ci Cv dt
½12
0
where Riv is the vacancy–interstitial recombination coefficient and Ci and Cv are interstitial and vacancy concentrations, respectively. Similarly, the number of defects per unit volume that are lost to sinks of type j, NSj, up to time t, is ðt NSj ¼ kSj Cj dt
½13
0
where kSj is the strength of sink j and Cj is the sink concentration. The ratio of vacancy loss to interstitial loss is RS ¼
NSv NSi
½14
where j ¼ v or i. The quantity NS is important in describing the microstructural development involving total point defect flux to sinks (e.g., RIS), while RS is the relevant quantity for the growth of defect aggregates such as voids that require partitioning of point defects to allow growth. In the steady-state recombination dominant regime, for NS to be invariant at a fixed dose, the following relationship between ‘dose rate (Ki) and temperature (Ti)’ must hold: 2 kT1 K2 Evm ln K1 T2 T1 ¼ ½15 K2 1 ln 1 kT Evm K1 where Evm is the vacancy migration energy. In the steady-state recombination dominant regime, for RS to be
invariant at a fixed dose, the following relationship between ‘dose rate and temperature’ must hold: kT12 K2 Evm þ2Evf ln K1 ½16 T2 T1 ¼ K2 1 ln 1 EvmkT þ2Evf K1 where Evf is the vacancy formation energy. In the steadystate recombination dominant regime, for NS to be invariant at a fixed temperature, the following relationship between ‘dose (F) and dose rate’ must hold: 1=2 F2 K2 ¼ ½17 F1 K1 Finally, in the steady-state recombination dominant regime, for NS to be invariant at a fixed dose rate, the following relationship between ‘dose and temperature’ must hold: 2kT12 2 ln F Evm F1 ½18 T2 T1 ¼ F2 1 ln 1 kT Evm F1 Figure 9 shows plots of the relationship between the ratio of dose rates and the temperature difference required to maintain the same point defect absorption at sinks (a), and the swelling invariance (b). The invariance requirements can be used to prescribe an ion irradiation temperature–dose rate combination that simulates neutron radiation. We take the example of irradiation of stainless steel under typical BWR core irradiation conditions of 4.5 108 dpa s1 at 288 C. If we were to conduct a proton irradiation with a characteristic dose rate of 7.0 106 dpa s1, then using eqn [15] with a vacancy formation energy of 1.9 eV and a vacancy migration
50
700
40
500
Em ν = 0.5
400 300 200
1.0 1.5
100
(a)
DTemperature (⬚C)
DTemperature (⬚C)
600
0
203
1
10 100 Ratio of dose rates
Eνm = 0.5 30 1.0 1.5 20
10
0
1000
1
(b)
10 100 Ratio of dose rates
1000
Figure 9 Temperature shift from the reference 200 C required at constant dose in order to maintain (a) the same point defect absorption at sinks, and (b) swelling invariance, as a function of dose rate, normalized to initial dose rate. Results are shown for three different vacancy migration energies and a vacancy formation energy of 1.5 eV. Adapted from Mansur, L. K. J. Nucl. Mater. 1993, 206, 306–323; Was, G. S. Radiation Materials Science: Metals and Alloys; Springer: Berlin, 2007.
204
Radiation Damage Using Ion Beams
energy of 1.3 eV, the experiment will be invariant in NS with the BWR core irradiation (e.g., RIS) at a proton irradiation temperature of 400 C. Similarly, using eqn [16], a proton irradiation temperature of 300 C will result in an invariant RS (e.g., swelling or loop growth). For a Ni2þ ion irradiation at a dose rate of 103 dpa s1, the respective temperatures are 675 C (NS invariant) and 340 C (RS invariant). In other words, the temperature ‘shift’ due to the higher dose rate is dependent on the microstructure feature of interest. Also, with increasing difference in dose rate, the DT between neutron and ion irradiation increases substantially. The nominal irradiation temperatures selected for proton irradiation, 360 C and for Ni2þ irradiation, 500 C represent compromises between the extremes for invariant NS and RS.
1.07.4 Contributions of Ion Irradiation to an Understanding of Radiation Effects Ion irradiations have been critical to the development of both our fundamental and applied understanding of radiation effects. As discussed in Sections 1.07.2 and 1.07.3, it is the flexibility of such irradiations and our firm understanding of atomic collisions in solids that afford them their utility. Principally, ion irradiations have enabled focused studies on the isolated effects of primary recoil spectrum, defect displacement rate, and temperature. In addition, they have provided access to the fundamental properties of point defects, defect creation, and defect reactions. In this section, we highlight a few key experiments that illustrate the broad range of problems that can be addressed using ion irradiations. We concentrate our discussion on past ion irradiations studies that have provided key information required by modelers in their attempts to predict materials behavior in existing and future nuclear reactor environments, and particularly information that is not readily available from neutron irradiations. In addition, we include a few comparative studies between ion and neutron irradiations to illustrate, on one hand, Table 2
the good agreement that is possible, while on the other, the extreme caution that is necessary in extrapolating results of ion irradiations to long-term predictions of materials evolution in a nuclear environment. 1.07.4.1
Electron Irradiations
The unique feature of electron irradiations in comparison to ions and neutrons is that they create defects in very low-energy recoil events. As a consequence, nearly all FPs are produced in isolation. This has been of foremost importance in developing our understanding of radiation damage, as it made studies of defect creation mechanisms as well as the fundamental properties of FPs possible. Recall that the properties of vacancies andvacancy clusters, for example, formation and migration energies, stacking fault energies, etc., could be determined from quenching studies. It is not possible, however, to quench in interstitials in metals. Very little was therefore known about this intrinsic defect prior to about 1955 when irradiation experiments became widely employed. In this section, we highlight some of the key findings derived from these past studies. 1.07.4.1.1 Displacement threshold surfaces
The creation of a stable FP requires that a lattice atom receives an energy greater than Tm, which is the minimum displacement energy. This value has been determined experimentally in many materials by measuring the change in some physical property, such as electrical resistivity or length change, as a function of maximum recoil energy of a target atom. Such experiments are practical only for electron irradiations for which recoil energies can be kept low, but with the irradiation particles still penetrating deeply into, or through, the specimen. Typical values are shown in Table 2. As a crystal is not homogeneous, the threshold energy depends on the crystallographic direction in which the knock-on atom recoils. The anisotropy of the threshold energy surface has been mapped out in various crystals by measuring the production rate of defects as a function of both the electron energy, near threshold, and the orientation of single crystalline
Minimum displacement energies in pure metals, semiconductors, and stainless steel (SS)
Materials
Al
Cgraph
Cu
Fe
Ge
Mo
Ni
W
Si
SS
Tm (eV)
16
25
19
17
15
33
23
41
13
18
Source: Lucasson, P. In Fundamental Aspects of Radiation Damage in Metals; Robibnson, M. T., Young, F. W., Jr., Eds.; ERDA Report CONF-751006; 1975; p 42; Andersen, H. H. Appl. Phys. 1979, 18, 131.
Radiation Damage Using Ion Beams
specimens with respect to the electron beam direction.15,16 The total cross-section for FP production rate is given by the expression 2ðp p=2 ð
sd ðY1 ; F1 ; E1 Þ ¼ 0
0
dsðy2 ; E1 Þ df2 2p dy2
½19
nðY2 ; F2 ; T Þdy2 where the subscripts 1 and 2 refer to incoming electron and recoiling ion, respectively, and Y1, F1, Y2, F2 are the polar and azimuthal angles of the electron beam relative to the crystal axis; y2, f2, are these same angles relative to the beam direction; n is the anisotropic damage function. Near threshold, n ¼ 1 for T > Tm, and 0 for T < Tm. By measuring the production rate for many sample orientations and energies, the damage function can be obtained using eqn [19], although various approximations are required in the deconvolution. The results are illustrated in Figure 10 for Cu.17 It is noteworthy that the minimum threshold energy is located in the vicinity of close-packed directions. This is also true for bcc metals. The anisotropy reflects the basic mechanism of defect production, viz., replacement collision sequences (RCSs), which had been identified by molecular dynamics simulations as early as 1960.18 [111] 56
55 406
18 2
43 683
a b 22 6
23 4 18 2
23 2
47 81 43
43 208
150 61
215
253
31
103 28 9
60
11
45 69
25 3
24 4 20 3
26 0 22 4
27 11
30 10
26 3
24 4
23 6
20 3
20 4
23 4
25 5
25 12
29 9
23 4
21 4
20
23 2
25 4
23 3
21 3
29 10
[100]
20 2
3 22 3 [110]
(a) Figure 10 Displacement energy threshold surface for Cu. The general anisotropy is typical of all fcc metals, although specific values vary. bcc metals show similar behavior of minima along close-packed directions. Reproduced from King, W. E.; Merkle, K. L.; Meshii, M. Phys. Rev. B 1981, 23, 6319.
205
The primary knock-on atom in an RCS recoils in the direction of its nearest neighbor, h110i in fcc crystals, and replaces it, with the neighbor recoiling also in the h110i and replacing its neighbor. A vacancy is left at the primary recoil site, and an interstitial is created at the end of the sequence. Replacement sequences are the most efficient way to separate the interstitial far enough from its vacancy, 2–3 interatomic spacings, for the FP to be stable. While the lengths of these sequences are still debated, it is clear that the mechanism results in both defect production and atomic mixing. For neutron irradiations, higher energy recoils are numerous, and the average displacement energy, Ed, becomes more relevant for calculations of defect production (see eqn [1]). This value, which can be obtained by averaging over the threshold displacement energy surface, is usually difficult to determine experimentally. A rough estimate, however, can be obtained from, Td 1.4Tm in fcc metals and 1.6Tm in bcc metals.19 1.07.4.1.2 Point defect properties
As FPs are produced in isolation during electron irradiation, the properties of single point defects and their interactions with impurities and sinks can be systematically investigated. An example is shown in Figure 11(a), where the results of lowtemperature isochronal annealing of Cu are shown following 1.4 MeV electron irradiation at 6 K.20 Recovery is observed to occur in ‘stages.’ These studies have revealed that interstitial atoms become mobile at very low temperatures, always below 100 K, in so-called Stage I, while vacancies become mobile at higher temperatures, Stage III. The various substages IA–IE seen in Figure 11(a) arise from the interaction between interstitial–vacancy pairs, which are produced in close proximity. Stage IE refers to the free migration of interstitials in the lattice, away from its own vacancy, and annihilation at distant vacancies; these interstitials are freely migrating as discussed earlier. For comparison, Stage I annealing of Cu following neutron irradiation is shown in Figure 11(b).21 Notice that the close pair substages are suppressed during neutron irradiation, illustrating the dramatic difference in the defect production process for these types of irradiation. Similarly, annealing studies on electron-irradiated Al doped with Mg or Ga impurities are shown in Figure 12.22 For these, it is observed that Stage I recovery is suppressed as interstitials trap at impurities and do not recombine. The recovery at higher temperature, in Stage II, reveals distinct subannealing stages.
206
Radiation Damage Using Ion Beams
ID
IB
ID
IC dDr (% K–1) dT
7 dDr (% K–1) dT
40
6 5
IE IA
3
30
20
1 − Dr °
1 − Dr °
4
Cu
IC
10
2
IB
IA
1 0
10
20
30
(a)
40 50 T (K)
60
70
0
80
(b)
10
20
40
30
50
60
70
T (K)
Figure 11 Low-temperature isochronal annealing of Cu following (a) electron (reproduced from Corbett, J. W.; Smith, R. B.; Walker, R. M. Phys. Rev. 1959, 114, 1452) or (b) fast neutron irradiation (reproduced from Burger, G.; Isebeck, K.; Volkl, J.; Schilling, W.; Wenzl, H. Z. Angew. Phys. 1967, 22, 452).
100 Dr [nW.cm] ° 2.5 2.7 Al-(99.995%pure) 3.3
90
Al-0.06° at.%Mg Al-0.085 at.%Ga
Dr/Dr0 (%)
80 70 60 50 40 30
1.07.4.2.1 The damage function
20 10 0 10
30
100
300
T (K)
Figure 12 Recovery of electrical resistivity in Al, Al–0.06 at.% Ga, and Al–0.085 at.% Ga following 1 MeV electron irradiation. Reproduced from Garr, K. R.; Sosin, A. Phys. Rev. 1969, 162, 669.
These annealing stages are generally attributed to either the interstitial dissociating from the impurity, or the interstitial–impurity complex migrating to a vacancy or a defect sink. Migrating interstitial–solute complexes lead to segregation. A compilation of the properties of point defects for many metals, and their interactions with impurities can be found in Ehrhart.23 This information has played a crucial role in developing an understanding of radiation damage in more complex engineering alloys and under more complex irradiation conditions. 1.07.4.2
threshold energies using low energy protons, to tens of keV using MeV self-ions. In addition, defect production rates can be varied over many orders of magnitude, reaching values over 0.1 dpa s1. Moreover, by using more than one ion beam, the primary recoil spectrum can be tailored to closely match that produced by an arbitrary fission neutron spectrum.
Ion Irradiations
Ion irradiations are the most flexible method for irradiating materials. As discussed in Section 1.07.2, the primary recoil spectrum can be shifted from near
Calculations of defect production, eqn [2], require knowledge of the damage function, n(T ). While it is not possible to measure this function directly, as no irradiation creates monoenergetic recoils except near the surface, it can be obtained by measuring defect production for a wide range of ion irradiations and subsequently deconvoluting eqn [3]. Low-energy light ions, for example, weight the recoil spectrum near the threshold energy, 25–100 eV, while more energetic heavy ions weight it at high energies. Results are shown for Cu in Figure 13. Here, electrical resistivity measurements are employed to monitor the absolute number of FPs produced per unit dose of irradiation. Included in this figure are the damage efficiency function, x(T1/2), deduced from the experiments and x(T ) calculated using molecular dynamics computer simulation. The damage efficiency function is defined as nðT Þ ¼ xðT ÞnNRT ðT Þ;
½20
where nNRT(T ) is the NRT damage function defined by eqn [1]. The good agreement between experiment and simulations illustrates that the damage function in Cu is now well understood. This is now true for many other pure metals as well.24 In alloys and ceramic
Radiation Damage Using Ion Beams
207
Cu H
Experiment Calculation
0.8
He
x
Li 0.6
CN O
0.4
Ne
Cu
Kr Ar Fe Ag Bi
1.0 Relative efficiency
1
1 MeV H
0.8 0.6 2 MeV He 0.4
2 MeV Li
FF 0.2 0
10
2
0.2
FN
MD simulation 3
10
10
4
5
10
T, T1/2 (eV)
Figure 13 Damage function efficiency factor of Cu (see eqn [20]) showing the decrease in efficiency versus cascade energy. The experimental data (solid squares) represent efficiencies for different ion irradiations plotted versus the characteristic cascade energy for the irradiation, T1/2 (see text). The open triangles represent the efficiency versus cascade energy, T, obtained by molecular dynamics (MD) simulation. The open circles represent the calculated efficiencies for the different irradiations using the MD efficiency function and eqn [2]. Reproduced from Averback, R. S.; de la Rubia, T. D. In Solid State Physics; Ehrenreich, H., Spaepen, F., Eds.; Academic Press: New York, 1998; pp 281–402.
materials, however, the damage function remains poorly known. 1.07.4.2.2 Freely migrating defects
The damage function refers to the number of FPs created within the first several picoseconds of the primary recoil event. At longer times, defects migrate from their nascent sites and interact with other defects and microstructural features. As noted earlier, many radiation effects, such as radiation-enhanced diffusion, segregation, and void swelling, depend more strongly on the number of defects that escape their nascent cascades and migrate freely in the lattice before annihilating, trapping, or forming defect clusters. The same general approach used to determine the damage function has been employed to determine the relative fraction of freely migrating defects, that is, e/nNRT, as illustrated by Figure 14. Here, the relative number of Si atoms segregating to the surface during irradiation, per dpa, is plotted versus a characteristic energy of the recoil spectrum, T1/2. It is seen that the fraction decreases rapidly with increasing recoil energy. Similar experiments were performed using radiationenhanced diffusion, as described in Section 1.07.2. While ion irradiation has proved extremely useful in illustrating the spectral effects on freely migrating
0 102
3 MeV Ni 3.25 MeV Kr 103
104 105 T1/2 (eV)
106
107
Figure 14 Relative efficiencies for producing freely migrating defects plotted as a function of the characteristic recoil energy, T1/2. Reproduced from Rehn, L. E.; Okamoto, P. R.; Averback, R. S. Phys. Rev. 1984, B30, 3073.
defects, extracting quantitative information about freely migrating defects from such experiments is difficult. These measurements, unlike the damage function, require very high doses, and several dpa; the buildup of the sink structure must be adequately taken into account. It is also difficult to estimate, for example, how many interstitials are required to transport one Si atom to the surface. We mention in passing that experiments performed using ordering kinetics in order–disorder alloys have provided a more direct measure of the number of freely migrating defects (vacancies in this case), as these experiments require doses less than 107 dpa so that no damage build-up can occur.25 These experiments show similar effects of primary recoil spectrum on the fraction of freely migrating defects, although the fractions of such defects were found to be somewhat higher in these experiments, 5–10%. These fractions are in good agreement with radiation-enhanced diffusion experiments using self-ions on Ni, when the effect of sink strength is taken into account.26 1.07.4.2.3 Alloy stability under ion irradiation
Irradiation of materials with energetic particles drives them from equilibrium, and in alloys, this becomes manifest in a number of ways. One of them concerns nonequilibrium segregation. The creation of large supersaturations of point defects leads to persistent defect fluxes to sinks. In many cases, these point defect fluxes couple with solutes, resulting in either the enrichment or depletion of solutes at these sinks. This effect was first discovered by using in situ electron
208
Radiation Damage Using Ion Beams
irradiations in a high voltage electron microscope,27 and it has been systematically investigated subsequently using ion irradiations,28 as the surface sink provides a convenient location to measure composition changes. Unlike neutron irradiation, moreover, the damage created by ions is generally inhomogeneous, reaching a peak level at some depth in the sample. As a consequence, point defect fluxes emanate from these regions. An example of this effect is shown in Figure 15 where a Ni–12.7 at.% Si alloy was irradiated with protons. As the alloy is supersaturated with Si prior to irradiation, Ni3Si precipitates
Ni plating
Peak damage region
Ni3Si surface film Bombarded surface
Figure 15 Behavior of silicon in a Ni–12.7 Si alloy following irradiation with protons. Note the region depleted of Ni3Si precipitates at the peak damage location and just below the surface. Courtesy of P. R. Okamoto.
form in the sample. At the location of peak damage, the concentration of interstitials is the highest, and hence these defects flow outward from this region. These interstitials form interstitial–solute complexes with Si, resulting in a Si flux out of this area as well, depleting the region of Si. As a consequence, a region depleted of Ni3Si precipitates is observed at the peak damage depth. Note too that the surface sink for interstitials leads to enrichment of Si, resulting in a surface layer of Ni3Si. The region just below the surface accordingly becomes depleted of Si, leaving a zone depleted of Ni3Si precipitates. While irradiation induced segregation can lead to nonequilibrium segregation and precipitation in single phase alloys, irradiation can also lead to dissolution of precipitates in nominally two-phase alloys. An interesting example of this behavior concerns Ni–12 at.% Al alloys irradiated with 300 keV Ni ions.29 These alloys were first annealed at high temperatures to develop a two-phase structure of Ni3Al (g0 ) and Ni–10.5 at.% Al (g). The initial precipitate size, depending on the annealing time was 2.5 or 4.6 nm. As shown in Figure 16, the precipitates disorder during irradiation at room temperature, owing to atomic mixing in cascades. The rate of disordering depends on the size of the precipitates, being slowest for homogeneous Ni3Al sample and fastest in the alloy with the smallest precipitates. The authors
Ni3AI NiAI (r = 4.6 nm) NiAI (r = 2.5 nm)
0.8 0.6 0.4 0.2
Degree of LRO S/S0
Degree of LRO S/S0
1.0 1.0
Ni3AI NiAI (r = 4.6 nm)
0.5
0.0 0.0 0.0 0.00 0.05 0.10 0.15 0.20 0.25 0.30 (b) (a) Irradiation dose F (dpa)
550 ⬚C 450 ⬚C 2.0 4.0 Irradiation dose F (dpa)
6.0
0.8
f(r )
0 dpa 0.4
0.0 0.0 (c)
5 dpa
2 dpa
4.0
8.0
0.0
4.0 8.0 Radius (nm)
0.0
4.0
8.0
Figure 16 (a) Disordering rate Ni3Al precipitates in two-phase Ni–12 at.% Al alloys and homogeneous Ni3Al during 300 keV Ni bombardment at room temperature; (b) same as (a) but irradiation at 550 C; (c) size distribution of Ni3Al precipitates after irradiation to two doses. After 5 dpa, a steady state size is obtained. Reproduced from Schmitz, G.; Ewert, J. C.; Harbsmeier, F.; Uhrmacher, M.; Haider, F. Phys. Rev. B 2001, 63, 224113.
Radiation Damage Using Ion Beams
suggest that the reason for this dependence on precipitate size is that atomic mixing reduces the concentration of Al in the precipitates, which thereby accelerates the disordering. When the same irradiation is performed at higher temperatures, and radiation-enhanced diffusion takes place, the system does not completely disorder, but rather remains partially ordered, owing to a competition between disordering in the displacement cascades and reordering by radiation-enhanced diffusion. Noteworthy, however, is the size of the precipitate, as shown in Figure 16(c), where it is observed that the precipitates initially shrink in size, but then reach a steady state radius. Therefore, unlike in thermal aging, precipitates in irradiated alloys can reach a stable steady state size that is a function of irradiation intensity and temperature. Similar behavior has been observed in two-phase immiscible alloys in which case a steady state size of precipitates is formed.30 This so-called ‘patterning’ phenomenon has been explained on the basis of a competition between disordering by atomic mixing in energetic collision events and reordering during thermally activated diffusion. For patterning, however, it is required that the atomic relocation distances during collisional mixing are significantly larger than the nearest neighbor distance. An interesting consequence of this requirement in regard to the present discussion of using ion irradiation to simulate neutron damage is that electron and proton irradiations, which do not produce energetic cascades or long relocation distances, should not induce compositional patterning, but heavy ions and fast neutron irradiation, which do produce cascades, will cause patterning. Further details can be found in Enrique31 and Enrique et al.32 1.07.4.2.4 Mechanical properties
Measurements of mechanical properties on irradiated materials usually require bulk samples and therefore neutron irradiation. Ion beams, however, can be employed for some measurements, such as plastic deformation. Typically, these experiments employ high energy protons, E > 2 MeV, or He ions, E > 7 MeV, as these particles can penetrate through thin foils, such as Fe or steel, that are greater than 15 mm in thickness. Moreover, displacement rates 105 dpa s1 are obtainable without excessive beam heating.33 Deformation experiments have also been performed using GeV heavy ions, as these penetrate targets several microns in thickness. The displacement rates, however, are low as most of the beam energy is lost through electronic excitations. Heavy
209
ions with lower energies, E 1–4 MeV, have also been used in deformation studies; for these, however, specimen must be very thin, 200 nm, and effects of the surface must be taken into account.34,35 1.07.4.2.5 Multiple ion beams
One of the difficulties in using ion beams to simulate neutron irradiation damage is the potential for missing certain synergistic behaviors in the damage evolution. For example, neutron irradiation leads to transmutation products and the generation of He and fission gases in addition to displacement damage. Generation of gas is particularly relevant to 14 MeV neutron irradiation for which large amounts of He and H are produced. Ion beams, however, offer the opportunity of using two or even three beams simultaneously and thus to tailor test irradiations to meet expected reactor conditions; see, for example, Serruys et al.36 This is often not possible in existing test reactor facilities, and the building of new test facilities for fusion machines has been formidably expensive. The application of multiple ion beams is illustrated in Figure 17 in a study of void swelling in vanadium. Here, the synergistic effects of simultaneously implanting 350 keV H and 1 MeV He, while irradiating with 12 MeV Ni ions are shown. Without the He beam, swelling is negligible, even with the implantation of H, but with it, the H greatly enhances the swelling. H implantation, on the other hand, is seen to reduce the density of cavities. 1.07.4.2.6 Swift ions
An important contribution to the damage in nuclear fuels derives from fission fragments. There are two groups of fission products: one group with atomic number near 42 (Mo) and energy 100 MeV and the other with atomic number near 56 (Ba) and energy 70 MeV. The maximum electronic stopping powers of these energetic particles, 18 keV nm1 for the heavier and 22 keV nm1 for the lighter, are far greater than their respective nuclear stopping powers. Similar to ion irradiation studies described above, where the primary recoil spectrum can be systematically varied, the masses and energies of ions can be varied to examine effects of electronic stopping power. An example is shown in Figure 18 where the electronic stopping power is plotted as a function of energy (per nucleon) for different ion irradiations of UO2. The two boxes in the figure indicate stopping powers associated with the fission fragments and the heavy particle recoils of a emitters. One of the questions addressed by such studies
Radiation Damage Using Ion Beams
Swelling (%)
20
10
20
0 20
10
Cavity density (1020 m−3)
210
20 15 10
20
5 15 0 10
pa )d pm (ap He
pa )d pm (ap He
20
10
5
10
0
0
0
0 0
–1
–1
0
(a)
20 10 a–1 p d ) m p p H(a
(b)
20 10 pa–1 d ) m p p (a H
dE/dx (keV nm–1)
70
60
20
1.4 1.2 Heavy FP
Light FP
1.0 0.8
10
0.6
238U
208Pb
0.4
50 U Fission
0 40
197Au
0.2
235
dE/dx (keV nm–1)
Energy (MeVamu–1)
Figure 17 Cavity volume fraction (a) and cavity density (b) in pure vanadium irradiated with 12 MeV Ni3þ ions to 30 dpa at 873 K with and without simultaneous irradiation of He and H. Reproduced from Sekimura, N.; Iwai, T.; Arai, Y.; et al. J. Nucl. Mater. 2000, 283–287, 224–228.
−8 −6 −4 −2 0 2 4 FPs range (μm)
0.0 8
6
129Xe
116Sn
30 GANIL HMI
GSI TASCC
106Cd 100Mo 127I
dE/dxFP
20
70Zn
Recoils 10
Zn70Zn Efission 0 10−5
10−4
10−3
10−2
10−1
1
10
102
Energy (MeVamu–1) Figure 18 Plot of dE/dx as a function of the energy for a series of ions. The circle indicates the conditions for 72 MeV ions of 127I. The two large squares show dE/dx representative of fission products and for the heavy recoil atoms of a-decaying actinides. The inset shows the energy loss and the remaining energy of typical light and heavy fission products along their range of 7 mm length. Reproduced from Matzke, Hj.; Lucuta, P. G.; Wiss, T. Nucl. Instrum. Meth. B 2000, 166–167, 920.
Radiation Damage Using Ion Beams
1.07.4.3
Comparison with Neutrons
Proton irradiation has undergone considerable refinement as a radiation damage tool. Numerous experiments have been conducted and compared to equivalent neutron irradiation experiments in order to determine whether proton irradiations capture the effects of neutron irradiation on microstructure, microchemistry, and hardening. In some cases, benchmarking exercises were conducted on the same native alloy heat as neutron irradiation in order to eliminate heat-to-heat variations that may obscure comparison of the effects of the two types of irradiating particles. The following examples cover a number of irradiation effects on several alloys in an effort to demonstrate the capability of proton irradiation to capture the critical effects of neutron irradiation. Figures 19–23 show direct comparisons of the same irradiation feature on the same alloy heats (commercial purity (CP) 304 and 316 stainless steels) following either neutron irradiation at 275 C or
8
24 CP-316 SS Protons at 360 ⬚C to 1.0 dpa Neutrons at 275 ⬚C to 1.1⫻1021 n cm–2 (~1.5 dpa)
20
7 6 5
16
Cr 4 Ni
3
12
Measured Si (wt%)
Measured Cr or Ni (wt%)
has been the formation of fission fragment tracks. Tracks have not yet been observed in the bulk of UO2 due to fission; however, by using ion irradiation, the stopping powers could be increased. The dashed line at 29 keV nm1 in Figure 18 represents the threshold stopping power for track formation.37 This value is 30% greater than the maximum for fission fragments, thus helping to explain why fission fragment tracks are not seen in the bulk. Such tracks are observed, however, close to the surface. They are explained by fission products passing near or parallel to the surface and creating shock waves which interact with the surface.38 These studies have also been useful in gaining important data for understanding fission gas evolution in nuclear fuels. For example, 72 MeV iodine ions (see Figure 18), approximate very closely the stopping power of fission fragments. Such studies have shown that 72 MeV I irradiations cause Kr atoms preimplanted into UO2 to nucleate into bubbles, and preformed bubbles to undergo resolution. A radiation-enhanced diffusion coefficient for the Kr was estimated from these studies to be D 1.2 1030 cm5 F_ , where F_ is the fission rate per cubic centimeter, and found independent of temperature below 500 C (see Matzke et al.37 for details). The importance of such studies as these is that the basic processes in complex nuclear fuels can be elucidated by studies that carefully control singly the irradiation conditions and materials parameters in the fuel, such as fission gas concentration, damage, etc.
211
2 Si 1
8 −12
0 4 8 −8 −4 Distance from grain boundary (nm)
0 12
Figure 19 Comparison of grain boundary segregation of Cr, Ni, and Si in commercial purity 16 stainless steel following irradiation with either protons or neutrons to similar doses. From Was, G. S.; Busby, J. T.; Allen, T.; et al. J. Nucl. Mater. 2002, 300, 198–216.
3 MeV proton irradiation at 360 C to similar doses. Figure 19 compares the RIS behavior of Cr, Ni, and Si in a 316 stainless steel alloy following irradiation to approximately 1 dpa. Neutron irradiation results are in open symbols and proton irradiation results are in solid symbols. This dose range was chosen as an extreme test of proton irradiation to capture the ‘W’-shaped chromium depletion profile caused by irradiation of a microstructure, which contained grain boundaries that were enriched with chromium prior to irradiation. Note that the two profiles track each other extremely well, both in magnitude and spatial extent. Good agreement is obtained for all three elements. Figure 20 shows a comparison of the dislocation microstructure as measured by the dislocation loop size distribution (Figure 20(a)) and the size and number density of dislocation loops (Figure 20(b)) for 304 SS and 316 SS. The main features of the loop size distributions are similar for the two irradiations, viz. a sharply peaked distribution in the case of 304 SS and a flatter distribution with a tail in the case of 316 SS. The agreement in loop size is good for the 304 SS alloy, while loops are smaller for the protonirradiated 316 alloy. The loop density is about a factor of 3 less for the proton-irradiated case than the neutron-irradiated case, which is expected as the proton irradiation temperature was optimized to track RIS (higher temperature) rather than the
212
30
Fraction of loop population (%)
Fraction of loop population (%)
Radiation Damage Using Ion Beams
Protons at 360 ⬚C (1.0 dpa) Neutrons at 275 ⬚C (0.7 dpa)
20
10
CP 304 SS 0 0
10 20 15 Loop diameter (nm)
5
(a)
25
30
12
50 Protons at 360 ⬚C (1.0 dpa)
40
Neutrons at 275 ⬚C (1.1 dpa)
30 20
CP 316 SS
10 0 0
5
10 15 20 Loop diameter (nm)
25
30
1024
Loop density (m−3)
Loop diameter (nm)
10 8 6 4 Protons at 360 ⬚C
2
304
1023
1022
316
Protons at 360 ⬚C
Neutrons at 275 ⬚C
0
0
1
2
(b)
304
316
Neutrons at 275 ⬚C
3 4 Dose (dpa)
5
1021
6
0
1
2
3 4 Dose (dpa)
5
6
Figure 20 Comparison of (a) loop size distributions and (b) loop diameter and loop number density for commercial purity 304 and 316 stainless steels irradiated with neutrons or protons to similar doses. From Was, G. S.; Busby, J. T.; Allen, T.; et al. J. Nucl. Mater. 2002, 300, 198–216. 1500
1500
1000
500
0
(a)
CP 316 SS
Yield strength (MPa)
Yield strength (MPa)
CP 304 SS
0
1
2
3 4 Dose (dpa)
5
1000
500
0
6
0
1
(b)
2
3 4 Dose (dpa)
5
6
Protons at 360 ⬚C (hardness) Neutrons at 275 ⬚C (hardness) Neutrons at 275 ⬚C (shear punch)
Figure 21 Comparison of hardening in commercial purity 304 (a) and 316 (b) stainless steel irradiated with neutrons or protons to similar doses. From Was, G. S.; Busby, J. T.; Allen, T.; et al. J. Nucl. Mater. 2002, 300, 198–216.
dislocation loop microstructure. That the loop sizes and densities are even close is somewhat remarkable considering that loop density is driven by in-cascade clustering, and cascades from proton irradiation are
much smaller than those from neutron irradiation. The surviving fraction of interstitial loops, however, is greater for proton irradiation, partially compensating the greater loop formation rate under neutron
Radiation Damage Using Ion Beams
0
1.4
Fast neutron fluence (E > 1 MeV) ⫻1025 n m−2 0.5 1 1.5 2 2.5 3 3.5 4
Neutrons With He Without He
1.2
100
CP 304 SS
Protons at 360 ⬚C Neutrons at 275 ⬚C
NWC
213
1 0.8 s/s0
Measured IG percentage
80
60
0.6 0.4
40
0.2
20 0
0 0
1
2
3 Dose (dpa)
4
5
6
Figure 22 Comparison of the extent of intergranular stress corrosion cracking in commercial purity 304 stainless steel following similar stress corrosion cracking tests of either neutron- or proton-irradiated samples from the same heat. From Was, G. S.; Busby, J. T.; Allen, T.; et al. J. Nucl. Mater. 2002, 300, 198–216.
0.14 Ni+ irradiation 675 ⬚C 140 dpa
70 60
0.13 0.12
Proton irradiation 400 ⬚C 3.0 dpa
50 40 30
0.11 0.10 0.09
Neutron irradiation 510 ⬚C 2.6 ⫻ 1026 n m−2 E > 0.1 MeV
20
0.08
10 0
Swelling (%) protons
Swelling (%) neutron and Ni ion
80
0.07 0
20
40
60
80
0.06 100
Bulk nickel concentration (at.%)
Figure 23 Effect of bulk nickel concentration on swelling resulting from irradiation with different particles: neutrons, nickel ions, and protons. Reproduced from Allen, T. R.; Cole, J. I.; Gan, J.; Was, G. S.; Dropek, R.; Al Kenik, E. J. Nucl. Mater. 2005, 341, 90–100.
irradiation and resulting in loop densities that are within a factor of 3.39 Figure 21 shows a comparison of irradiation hardening between the two types of irradiation. The results
0
0.5
1 1.5 Dose (dpa)
2
2.5
Figure 24 Comparison of relaxation in residual stresses between neutron- and proton-irradiated stainless steel after removing the effect of thermally-induced relaxation. From Sencer, B. H.; Was, G. S.; Yuya, H.; Isobe, Y.; Sagasaka, M.; Garner, F. A. J. Nucl. Mater. 2005, 336, 314–322.
are again similar, with proton irradiation resulting in slightly lower hardness. Figure 22 shows the IASCC susceptibility of CP 304 SS as measured by the %IG on the fracture surface following constant load testing (neutron-irradiated samples) and constant extension rate testing (proton-irradiated samples) in BWR normal water chemistry (NWC). Despite the significantly different testing mode, the results are in excellent agreement in that both proton and neutron irradiation result in the onset of IGSCC, at about 1 dpa.40 Figure 23 shows the swelling behavior in austenitic stainless steels as a function of nickel content for proton, Ni ion, and neutron irradiation. While these experiments were conducted on different sets of alloys, and under highly disparate irradiation conditions, they all show the same dependence of nickel on swelling. In the two commercial purity alloys, no voids were formed in either neutron or protonirradiated samples. As a last example of stainless steel alloys, Figure 24 shows the relaxation of residual stress by neutron and proton irradiation. Here again, results are from different alloys and different types of tests, but both show the same dependence of stress relaxation on dose. The next examples are from reactor pressure vessel steel and Zircaloy. Figure 25 shows an experiment on model reactor pressure vessel alloys in which the
214
Radiation Damage Using Ion Beams
500
300
Tin = 300 ⬚C (all)
−7
Proton: 3–7 ⫻ 10 dpa s–1 −10 Neutron: 3 ⫻ 10 dpa s–1 −9 Electron: 7 ⫻ 10 dpa s–1
280
Vickers hardness (Hv)
Change in yield strength (MPa)
400
300 VA, Fe, neutron VA, Fe, proton VA, Fe, electron VD, Fe–0.9Cu–1.0 Mn, neutron VD, Fe–0.9Cu–1.0 Mn, proton VD, Fe–0.9Cu–1.0 Mn, electron VH, Fe–0.9Cu, neutron VH, Fe–0.9Cu, proton
200
100
0
−100 10−5
260 240 220 200
p - Zircaloy 4, 350 ⬚C n - Zircaloy 2, 350–400 ⬚C p - Zircaloy 4, 310 ⬚C
180
10−4
10−3 Dose (dpa)
10−2
10−1
160
0
1
2
3
4 5 Dose (dpa)
6
7
8
Figure 25 Irradiation hardening in model reactor pressure vessel steels following neutron, proton, and electron irradiation at about 300 C. From Was, G. S.; Hash, M.; Odette, G. R. Philos. Mag. 2005, 85(4–7), 703–722.
Figure 26 Hardening of Zircaloy-4 irradiated with 3 MeV protons at 310 and 350 C and comparison to neutronirradiated Zircaloy-2. From Zu, X. T.; Sun, K.; Atzmon, M.; et al. Philos. Mag. 2005, 85(4–7), 649–659.
same model alloy heats were irradiated with neutrons, electrons, or protons at 300 C to doses spanning two orders of magnitude. The alloys include a high-purity Fe heat (VA) that hardens very little under irradiation, an Fe–0.9Cu (VH) heat that hardens rapidly initially, followed by a slower hardening rate above 0.1 mpda, and a Fe–0.9Ce–1.0Mn alloy (VD) in which the hardening rate is greatest over the dose range studied. Despite the very different compositions and hardening rates, the results of the three types of irradiation agree well. Figure 26 shows hardening for Zircaloy-2 and Zircaloy-4 irradiated with either neutrons or protons. Although the irradiations were not conducted on the same heats of material, or using similar irradiation parameters, there is good agreement in the magnitude and dose dependence of hardening. Proton irradiation also induced amorphization of a Zr(Fe,Cr)2 precipitate after irradiation to 5 dpa at 310 C, similar to that observed in reactor. These examples represent a comprehensive collection of comparison data between proton and neutron irradiation and taken together serve as a good example for the capability of charged particles to emulate the effect of neutron irradiation on the alloy microstructure. As a final example, to emphasize the care that must be exercised in extrapolating the results of one type of irradiation to make predictions for another, we discuss a comparison of void swelling in Cu due to
2.5 MeV electrons, 3.0 MeV protons, and fission neutrons.41 An attempt was made to keep all irradiation variables constant during the experiments, sample purity, defect production rate, and temperature; only the primary recoil spectrum was varied. The results for nucleation rates of voids and void swelling are shown in Figure 27(a) and 27(b), respectively. Clearly observed is that void swelling and void nucleation are significantly enhanced for neutron irradiation in comparison to proton or electron irradiation. This result is notably in strong contrast to the efficiencies obtained for defect production and radiation-induced segregation (or FMDs) for these three types of irradiation. The reduced efficiency of the production of FMDs was attributed to defect annihilation within the cascade core; these results for void swelling, however, indicate that the defect clustering process is also critical to microstructural evolution in irradiated alloys. Singh and coworkers41,42 argue that the clustering of interstitials in cascades, and their collapse into dislocation loops, result in interstitial migration by one-dimensional glide of loops, the so-called production bias model.43 As a consequence, interstitials and vacancies become efficiently separated. Swelling therefore is more severe for irradiations that produce energetic cascade, for example, neutrons, than for those that do not, electrons. Proton irradiation is intermediate; that is, small cascades are produced.
215
Radiation Damage Using Ion Beams
100
Copper 523 K 10−1 1022
523 K Void density (m−3)
Swelling (%)
Copper 10−2
10−3
10−4
10−5 −4 10
10−2 10−1 Dose (NRT dpa)
Fission neutrons 10
3 MeV protons
20
1019
Fission neutrons 3 MeV protons 2.5 MeV electrons 10−3
1021
2.5 MeV electrons
100
1018 −4 10
10−3 10−2 10−1 Dose (NRT dpa)
100
Figure 27 Void swelling as a function of dose in oxygen-free high conductivity (OFHC)-copper during irradiations with electrons, protons, and fission neutron. Reproduced from Singh, B. H.; Eldrup, M.; Horsewell, A.; Ehrhart, P.; Dworschak, F. Philos. Mag. A 2000, 80, 2629.
1.07.5 Advantages and Disadvantages of Irradiations using Various Particle Types
ton = 50 ms 50
toff = 2000 ms
40 K0/K0,avg
Each particle type has its advantages and disadvantages for use in the study of radiation effects or for emulating neutron irradiation damage. Common disadvantages of charged particle beams are the lack of transmutation reactions and the need to use a rasterscanned beam. With the exception of some minor transmutation reactions that can occur with light ion irradiation, charged particles do not reproduce the types of transmutation reactions that occur in reactor core materials due to the interaction with neutrons. The most important of these is the production of He by transmutation, particularly in alloys that contain elements such as Ni or B. But a second consideration is that of a raster-scanned beam in which any volume element of the target is exposed to the beam for only a fraction of the raster-scan cycle. For a typical beam scanner and beam parameters, the fraction of time that any particular volume element in the solid is being bombarded is 0.025. Thus, the instantaneous dose rate during the beam-on portion of the cycle is 40 times that of the average, Figure 28. The result is that the defect production rate is very high and defects can anneal out in the remaining 0.975 portion of the cycle before the beam again passes through the volume element. As such, the effective defect production
60
30
20
10 K0,avg 0
0
1
2
3 4 Time (ms)
5
6
7
Figure 28 The effect of a raster-scanned beam on the instantaneous production rate of point defects with the same time averaged rate as a continuous source. From Was, G. S.; Allen, T. R. In Radiation Effects in Solids, NATO Science Series II: Mathematics, Physics and Chemistry; Sickafus, K. E., Kotomin, E. A., Uberuaga, B. P., Eds.; Springer: Berlin, 2007; Vol. 235, pp 65–98.
rate in raster-scanned systems will be less, and must be accounted for. One objective of ion irradiation is to emulate the effect of neutrons, and a second is to understand basic physical radiation damage processes, for which
216
Radiation Damage Using Ion Beams
neutron irradiation is often less well suited. While ion irradiation can be conducted with great control over temperature, dose rate, and total dose, such control is a challenge to reactor irradiations. For example, instrumented tubes with active temperature control are expensive to design, build, and operate. Even so, frequent power changes can be difficult to handle as the flux–temperature relationship will change and this can result in artifacts in the irradiated microstructure.44 On the other hand, temperatures in cheaper irradiation vehicles that use passive gas gaps and gamma heating (such as ‘rabbit’ tubes) are known with even less certainty. While neutron dosimetry is used in some experiments, doses and dose rates are often determined by neutronic models of the core locations and are not verifiable. As such, ion irradiations enjoy the advantage of better control and verification of irradiation conditions as compared to neutron irradiation. Table 3 provides a list for each of three particle types: electrons, heavy ions, and light ions (protons), and they are discussed in detail in the following sections. 1.07.5.1
Electrons
Electron irradiation is easily conducted in a highvoltage transmission electron microscope using either
Table 3
a hot filament or a field emission gun as an electron source. An advantage is that the same instrument used for irradiation damage can be used to image the damage. Another advantage is that the high dose rate requires very short irradiation time, but will also require a large temperature shift as explained in the Section 1.07.3. There are several disadvantages to electron irradiation using a TEM. First, energies are generally limited to 1 MeV. This energy is sufficient to produce an isolated FP in transition metals, but not cascades. The high dose rate requires high temperatures that must be closely monitored and controlled, which is difficult to do precisely in a typical TEM sample stage. Another drawback is that as irradiations are often conducted on thin foils, defects are created in close proximity to the surface and their behavior may be affected by the presence of the surface. Perhaps the most serious drawback is the Gaussian shape to the electron beam that can give rise to strong dose rate gradients across the irradiated region. Figure 29 shows the composition profile of copper around a grain boundary in Ni–39%Cu following electron irradiation. Note that while there is local depletion at the grain boundary (as expected), the region adjacent to the minimum is strongly enriched in copper because of the strong defect flux out of the irradiated
Advantages and disadvantages of irradiations with various particle types
Advantages Electrons Relatively ‘simple’ source – TEM Uses standard TEM sample High dose rate – short irradiation times
Heavy ions High dose rate – short irradiation times High Tavg Cascade production
Light ions Accelerated dose rate – moderate irradiation times Modest DT required Good depth of penetration Flat damage profile over tens of microns
Disadvantages Energy limited to 1 MeV No cascades Very high beam current (high dpa rate) leading to large temperature shifts relative to neutrons Poor control of sample temperature Strong ‘Gaussian’ shape (nonuniform intensity profile) to beam No transmutation Very limited depth of penetration Strongly peaked damage profile Very high beam current (high dpa rate) leading to large temperature shifts relative to neutrons Potential for composition changes at high dose via implanted ion No transmutation Minor sample activation Smaller, widely separated cascade No transmutation
Source: Was, G. S.; Allen, T. R. In Radiation Effects in Solids, NATO Science Series II: Mathematics, Physics and Chemistry; Sickafus, K. E., Kotomin, E. A., Uberuaga, B. P., Eds.; Springer: Berlin, 2007; Vol. 235, pp 65–98.
Radiation Damage Using Ion Beams
217
8
9 7
8 Si concentration (at.%)
6
494 ⬚C
Cu concentration (at.%)
5 e-beam diameter
4 6
400 ⬚C
D+
7 6 8
e−
7
5
6 4
5 6
−4
400 ⬚C 5 4 −6
0 2 4 −4 −2 Distance from grain boundary (μm)
6
Figure 29 Enrichment of copper surrounding a local depletion at the grain boundary. The enrichment is caused by the high defect flux away from the irradiated region defined by the horizontal line. From Ezawa, T.; Wakai, E. Ultramicroscopy 1991, 39, 187.
Solute concentration (wt%)
60
50 Iron Chromium Nickel
40
30
−3
0 1 2 3 −2 −1 Distance from grain boundary (μm)
4
Figure 31 Comparison of (a) deuteron and (b) electron irradiation showing the greater amount of segregation and the narrower profile for the deuteron irradiation. From Wakai, E. Trans. J. Nucl. Mater. 1992, 33(10), 884.
zone defined by the horizontal line below the spectrum. This outward-directed defect flux causes a reversal in the direction of segregation from that caused by a defect flux to the sink. Another often observed artifact in electron irradiation is very broad grain boundary enrichment and depletion profiles. Figure 30 shows that the enrichment profile for Ni and the depletion profiles for Fe and Cr in stainless steel have widths on the order of 75–100 nm, which is much greater than the 5–10 nm widths observed following neutron irradiation under similar conditions and model simulations of radiationinduced segregation. A similar effect was observed by Wakai45 using electron and Dþ irradiation of the same alloy in which the segregation profile was much higher and narrower around the grain boundary in the deuteron-irradiated sample as compared to the electron irradiation (Figure 31).
20
1.07.5.2 10
0
100
200 300 400 Distance (nm)
500
600
Figure 30 Broad grain boundary enrichment and depletion profiles in Fe–20Cr–25Ni–0.75Nb–0.5Si following irradiation with electrons at 420 C to 7.2 dpa. From Ashworth, J. A.; Norris, D. I. R.; Jones, I. P. J. Nucl. Mater. 1992, 189, 289.
Heavy Ions
Heavy ions enjoy the benefit of high dose rates resulting in the accumulation of high doses in short times. Also, because they are typically produced in the energy range of a few MeV, they are very efficient at producing dense cascades, similar to those produced by neutrons. The disadvantage is that as with electrons, the high dose rates require large
218
Radiation Damage Using Ion Beams
dpa versus depth for various ions incident on nickel
Ni
5
9 Al
4
6
3
8.1 MeV aluminum ions (dpa per 1016 ions cm−2)
1.2 1.0 0.8 0.6
2
0.4
1
0.2
0
0
3
0
0
0.5
1
1.5
2
2.5
3
5 MeV carbon ions (dpa per 1016 ions cm−2)
C
12
Al Ni C
14 MeV nickel ions (dpa per 1016 ions cm−2)
15
Depth (μm) Figure 32 Damage profiles for C, Al, and Ni irradiation of a nickel target at energies selected to result in the same penetration depth. From Whitley, J. B. Ph.D. Thesis, University of Wisconsin-Madison, Madison, WI, 1978.
1.5
250
3.2
1.5
200 Swelling (%)
2.4
Observed
2
150
1.6 dpa
1.2 0.8
100
50
0.4 0 0 (a)
0.2 0.4 0.6 0.8 1 Depth (μm)
Displacement rate (10−3 dpa s–1)
5 MeV 2.8
1
0.5
0 0
0 1.2 1.4 1.6 (b)
Ni2+
on Ni
1
0.5
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 Depth (μm)
þ
Figure 33 (a) Subsurface swelling resulting from 5 MeV Ni ion irradiation of Fe–15Cr–35Ni at 625 C and (b) displacement rate and ion deposition rate calculated for 5 MeV Ni2þ on nickel. Adapted from Garner, F. A. J. Nucl. Mater. 1983, 117, 177–197; Lee, E. H.; Mansur, L. K.; Yoo, M. H. J. Nucl. Mater. 1979, 85&86, 577–581.
temperature shifts so that irradiations must be conducted at temperatures of 500 C in order to create similar effects as neutron irradiation at 300 C. Clearly, there is not much margin for studying neutron irradiations at higher reactor temperature as higher ion irradiation temperatures will cause annealing. Another drawback is the short penetration depth and the continuously varying dose rate over the penetration depth. Figure 32 shows the damage profile for several heavy ions incident on nickel. Note that the damage rate varies continuously and
peaks sharply at only 2 mm below the surface. As a result, regions at a very well-defined depth from the surface must be isolated and sampled in order to avoid dose or dose rate variation effects from sample to sample. Small errors (500 nm) made in locating the volume to be characterized can result in a dose that varies by a factor of 2 from the target value. A problem that is rather unique to nickel ion irradiation of stainless steel or nickel-base alloys is that in addition to the damage they create, each bombarding Ni ion constitutes an interstitial. Figure 33(a) shows
Radiation Damage Using Ion Beams
that 5 MeV Ni2þ irradiation of a Fe–15Cr–35Ni alloy resulted in high swelling in the immediate subsurface region compared to that near the damage peak. As shown in Figure 33(b), the Ni2þ ions come to rest at a position just beyond the peak damage range. So even though the peak damage rate is about 3 that at the surface, swelling at that location is suppressed by about a factor of 5 compared to that at the surface.46 The reason is that the bombarding Ni2þ ions constitute interstitials and the surplus of interstitials near the damage peak results in a reduction of the void growth rate.47,48 In the dose rate–temperature regime where recombination is the dominant point defect loss mechanism, interstitials injected by Ni2þ ion bombardment may never recombine as there is no corresponding vacancy production. 1.07.5.3
Light Ions
In many ways, proton irradiation overcomes the drawbacks of electron and neutron irradiation. The penetration depth of protons at a few MeV can exceed 40 mm and the damage profile is relatively flat such that the dose rate varies by less than a factor of 2 over several tens of micrometers. Further, the depth of penetration is sufficient to assess such properties as irradiation hardening through microhardness measurements, and stress corrosion cracking through crack initiation tests such as the slow strain rate test.
219
Figure 34 shows schematics of 3.2 MeV proton and 5 MeV Ni2þ damage profiles in stainless steel. Superimposed on the depth scale is a grain structure with a grain size of 10 mm. Note that with this grain size, there are numerous grain boundaries and a significant irradiated volume over which the proton damage rate is flat. The dose rate for proton irradiations is 2–3 orders of magnitude lower than that for electrons or ions, thus requiring only a modest temperature shift, but as it is still 102–103 times higher than neutron irradiation, modest doses can be achieved in reasonably short irradiation time. The disadvantages are that because of the small mass of the proton compared to heavy ions, the recoil energy is smaller and the resulting damage morphology is characterized by smaller, more widely spaced cascades than with ions or neutrons. Also, as only a few MeV are required to surmount the Coulomb barrier for light ions, there is also a minor amount of sample activation that increases with proton energy.
1.07.6 Practical Considerations for Radiation Damage Using Ion Beams In the process of setting up an ion irradiation experiment, a number of parameters that involve beam
105 Ion 10−15 5 MeV Ni
10−17
Calculated range (μm)
dpa/(ion cm−2)
10−16
1000
2+
3.2 MeV protons
10−18 10−19 10−20
10
0.1 1 MeV neutrons
10−21 10−22 0
H He Ni
10
20 30 Depth (μm)
40
Figure 34 Damage profiles for 1 MeV neutrons, 3.2 MeV protons, and 5 MeV Ni2þ ions in stainless steel. From Was, G. S.; Allen, T. R. In Radiation Effects in Solids, NATO Science Series II: Mathematics, Physics and Chemistry; Sickafus, K. E., Kotomin, E. A., Uberuaga, B. P., Eds.; Springer: Berlin, 2007; Vol. 235, pp 65–98.
Calculated by SRIM 2000 Stainless steel (Fe–20Cr–10Ni)
0.001 0.01
0.1
1
10
100
Energy (MeV) Figure 35 Range of hydrogen, helium, and nickel ions in stainless steel as a function of ion energy. From Was, G. S.; Allen, T. R. In Radiation Effects in Solids, NATO Science Series II: Mathematics, Physics and Chemistry; Sickafus, K. E., Kotomin, E. A., Uberuaga, B. P., Eds.; Springer: Berlin, 2007; Vol. 235, pp 65–98.
Radiation Damage Using Ion Beams
Time to reach 1 dpa (h)
(a)
Energy deposited (W)
(b)
Beam current (μA)
(c)
10−4 10−5 10−6 10−7 600 500 400 300 200 100 0 400
500
300
400 Residual activity
200 100 0 100 Maximum current at 360 ⬚C Maximum current at 400 ⬚C
80 60 40 20 0
(d)
At 360 ⬚C At 400 ⬚C
behavior during proton irradiation vary with energy, dose rate, the time to reach 1 dpa, deposited energy, and the maximum permissible beam current (which will determine the dose rate and total dose), given a temperature limitation of 360 C. With increasing energy, the dose rate at the surface decreases because of the drop in the elastic scattering cross-section (Figure 36(a)). Consequently, the time to reach a target dose level, and hence the length of an irradiation, increases rapidly (Figure 36(b)). Energy deposition scales linearly with the beam energy, raising the burden of removing the added heat in order to control the temperature of the irradiated region (Figure 36(c)). The need to remove the heat due to higher energies will limit the beam current at a specific target temperature (Figure 36(d)), and a limit on the beam current (or dose rate) will result in a longer irradiation to achieve the specified dose. Figure 37 summarizes how competing features of an irradiation vary with beam energy, creating tradeoffs in the beam parameters. For example, while greater depth is generally favored in order to increase the volume of irradiated material, the higher energy required leads to lower dose rates near the surface and higher residual radioactivity. For proton irradiation, the optimum energy range, achieved by balancing these factors, lies between 2 and 5 MeV as shown by the shaded region.
Range (μm) Residual activity (arbitrary units)
Dose rate (dpa s−1)
characteristics (energy, current/dose) and beamtarget interaction must be considered. ASTM E 521 provides standard practice for neutron radiation damage simulation by charged-particle irradiation49 and ASTM E 693 provides standard practice for characterizing neutron exposures in iron and low alloy steels in units of dpa.9 One of the most important considerations is the depth of penetration. Figure 35 shows the range versus particle energy for protons, helium ions, and nickel ions in stainless steel as calculated by SRIM.50 The difference in penetration depth between light and heavy ions is over an order of magnitude in this energy range. Figure 36 shows how several other parameters describing the target
5
10 Energy (MeV)
15
400
300
Energy minimum due to depth penetration
350
Energy maximum due to residual activity
300 250 Range
200 200
150 Time to 1 dpa
100
50
20
Figure 36 Behavior of beam-target parameters as a function of beam energy proton irradiation at 360 C; (a) dose rate, (b) time to reach 1 dpa, (c) energy deposition, and (d) beam current limit to maintain a sample temperature of 360 C. From Was, G. S.; Allen, T. R. In Radiation Effects in Solids, NATO Science Series II: Mathematics, Physics and Chemistry; Sickafus, K. E., Kotomin, E. A., Uberuaga, B. P., Eds.; Springer: Berlin, 2007; Vol. 235, pp 65–98.
100
0
0 1
2
5 10 Energy (MeV)
20
Figure 37 Variation of ion range, residual activity, and time to reach 1 dpa as a function of proton energy. Reproduced from Was, G. S.; Allen, T. R. In Radiation Effects in Solids, NATO Science Series II: Mathematics, Physics and Chemistry; Sickafus, K. E., Kotomin, E. A., Uberuaga, B. P., Eds.; Springer: Berlin, 2007; Vol. 235, pp 65–98.
Time to reach 1 dpa (h)
220
Radiation Damage Using Ion Beams
References 1. 2. 3.
4. 5.
6. 7. 8. 9.
10. 11.
12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24.
Garner, F. A. J. Nucl. Mater. 1983, 117, 177. Mazey, D. J. J. Nucl. Mater. 1990, 174, 196. Standard Practice for Neutron Irradiation Damage Simulation by Charged Particle Irradiation, Designation E521-89, American Standards for Testing and Materials, Philadelphia, 1989; p D–9. Was, G. S.; Andresen, P. L. JOM 1992, 44(4), 8. Andresen, P. L.; Ford, F. P.; Murphy, S. M.; Perks, J. M. In Proceedings of the Fourth International Symposium on Environmental Degradation of Materials in Nuclear Power Systems – Water Reactors; National Association of Corrosion Engineers: Houston, TX, 1990; pp 1–83. Andresen, P. L. In Stress Corrosion Cracking, Materials Performance and Evaluation; Jones, R. H., Ed.; ASM International: Meals Park, OH, 1992; p 181. Kinchin, G. H.; Pease, R. S. Prog. Phys. 1955, 18, 1. Norgett, M. J.; Robinson, M. T.; Torrens, I. M. Nucl. Eng. Des. 1974, 33, 50. ASTM E693-01. Standard Practice for Characterizing Neutron Exposures in Iron and Low Alloy Steels in Terms of Displacements Per Atom (DPA), E 706(ID); American Society for Testing and Materials: West Conshohocken, PA, 2007. Zinkle, S. J.; Singh, B. N. JNM 1993, 199, 173. Kulcinski, G. L.; Brimhall, J. L.; Kissinger, H. E. In Proceedings of Radiation-Induced Voids in Metals; Corbett, J. W., Ianiello, L. C., Eds.; USAEC Technical Information Center: Oak Ridge, TN, 1972; p 453, CONF-710601. Naundorf, V. J. Nucl. Mater. 1991, 182, 254. Iwase, A.; Rehn, L. E.; Baldo, P. M.; Funk, L. J. Nucl. Mater. 1996, 238, 224–236. Mansur, L. K. J. Nucl. Mater. 1994, 216, 97. Jung, P.; Chaplin, R. L.; Fenzl, H. J.; Reichelt, K.; Wombacher, P. Phys. Rev. B 1973, 8, 553. Vajda Rev, P. Mod. Phys. 1977, 49, 481. King, W. E.; Merkle, K. L.; Meshii, M. Phys. Rev. B 1981, 23, 6319. Gibson, J. B.; Goland, A. N.; Milgram, M.; Vineyard, G. H. Phys. Rev. 1960, 120, 1229. Lucasson, P. In Fundamental Aspects of Radiation Damage in Metals; Robibnson, M. T., Young, F. W., Jr., Eds.; ERDA Report CONF-751006; 1975, p 42. Corbett, J. W.; Smith, R. B.; Walker, R. M. Phys. Rev. 1959, 114, 1452. Burger, G.; Isebeck, K.; Volkl, J.; Schilling, W.; Wenzl, H. Zeitschrift Angew. Phys. 1967, 22, 452. Garr, K. R.; Sosin, A. Phys. Rev. 1969, 162, 669. Ehrhart, P. In Landolt –Bornstein New Series, Group III; Ullmaier, H., Ed.; Springer: Berlin, 1991; Vol. 25, p 115. See e.g., Bacon, D. In Computer Simulations in Materials; Kirchner, H. O., et al. Eds.; Kluwer: The Netherlands, 1996; p 189.
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.
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Wei, L. C.; Lang, E.; Flynn, C. P.; Averback, R. S. Appl. Phys. Lett. 1999, 75, 805. Fielitz, P.; Macht, M. P.; Naundorf, V.; Wollenberger, H. J. Nucl. Mater. 1997, 251, 123. Okamoto, P. R.; Harkness, S. D.; Laidler, J. J. ANS Trans. 1973, 16, 70. Okamoto, P. R.; Wiedersich, H. J. Nucl. Mater. 1974, 53, 336. Schmitz, G.; Ewert, J. C.; Harbsmeier, F.; Uhrmacher, M.; Haider, F. Phys. Rev. B 2001, 63, 224113. Krasnochtchekov, P.; Averback, R. S.; Bellon, P. Phys. Rev. B 2005, 72(17), 174102. Enrique, R. A.; Bellon, P. Phys. Rev. Lett. 2000, 84, 2885. Enrique, R. A.; Nordlund, K.; Averback, R. S.; Bellon, P. J. Appl. Phys. 2003, 93, 2917. See e.g., Jung, P.; Schwarz, A.; Sahu, H. K. Nucl. Instrum. Meth. A 1985, 234, 331. Mayr, S. G.; Averback, R. S. Phys. Rev. B 2003, 68, 214105. Brongersma, M. L.; Snoeks, E.; van Dillen, T.; Dillen, A. J. Appl. Phys. 2000, 88, 59. Serruys, Y.; Trocellier, P.; Miro, S.; et al. J. Nucl. Mater. 2009, 386–388, 967. Matzke, Hj.; Lucuta, P. G.; Wiss, T. Nucl. Instrum. Meth. B 2000, 166–167, 920. Ronchi, C. J. Appl. Phys. 1973, 44, 3573. Gan, J.; Was, G. S.; Stoller, R. E. J. Nucl. Mater. 2001, 299, 53–67. Onchi, T.; Dohi, K.; Soneda, N.; Navas, M.; Castano, M. L. In Proceedings of the 11th International Conference on Environmental Degradation of Materials in Nuclear Power Systems – Water Reactors; American Nuclear Society: La Grange Park, IL, 2003; p 1111. Singh, B. H.; Eldrup, M.; Horsewell, A.; Ehrhart, P.; Dworschak, F. Philos. Mag. A 2000, 80, 2629. Golubov, S. I.; Singh, B. N.; Trinkaus, H. Philos. Mag. A 2001, 81, 2533. Singh, B. N.; Foreman, A. J. E. Philos. Mag. A 1992, 66, 975. Garner, F. A.; Sekimura, N.; Grossbeck, M. L.; et al. J. Nucl. Mater. 1993, 205, 206–218. Wakai, E. Trans. J. Nucl. Mater. 1992, 33(10), 884. Garner, F. A. J. Nucl. Mater. 1983, 117, 177–197. Lee, E. H.; Mansur, L. K.; Yoo, M. H. J. Nucl. Mater. 1979, 85&86, 577–581. Brailsford, A. D.; Mansur, L. K. J. Nucl. Mater. 1977, 71, 110–116. ASTM E521-96. Standard Practice for Neutron Radiation Damage Simulation by Charged-Particle Irradiation; American Society for Testing and Materials: West Conshohocken, PA, 2009. Ziegler, J. F.; Biersack, J. P.; Littmark, U. The Stopping and Range of Ions in Matter; Pergamon: New York, 1996.
1.08 Ab Initio Electronic Structure Calculations for Nuclear Materials J.-P. Crocombette and F. Willaime Commissariat a` l’Energie Atomique, DEN, Service de Recherches de Me´tallurgie Physique, Gif-sur-Yvette, France
ß 2012 Elsevier Ltd. All rights reserved.
1.08.1 1.08.2 1.08.2.1 1.08.2.2 1.08.2.2.1 1.08.2.2.2 1.08.2.3 1.08.2.3.1 1.08.2.3.2 1.08.2.3.3 1.08.3 1.08.3.1 1.08.3.1.1 1.08.3.1.2 1.08.3.2 1.08.3.2.1 1.08.3.2.2 1.08.3.2.3 1.08.3.2.4 1.08.3.2.5 1.08.3.3 1.08.3.3.1 1.08.3.3.2 1.08.3.4 1.08.4 1.08.4.1 1.08.4.1.1 1.08.4.1.2 1.08.4.1.3 1.08.4.2 1.08.4.2.1 1.08.4.2.2 1.08.4.2.3 1.08.4.2.4 1.08.4.2.5 1.08.4.3 1.08.5 1.08.5.1 1.08.5.1.1 1.08.5.1.2 1.08.5.1.3 1.08.5.1.4 1.08.5.1.5 1.08.5.2
Introduction Methodologies and Tools Theoretical Background Codes Basis sets Pseudoization schemes Ab Initio Calculations in Practice Output Cell sizes and corresponding CPU times Choices to make Fields of Application Perfect Crystal Bulk properties Input for thermodynamic models Defects Self-defects Hetero-defects Point defect assemblies Kinetic models Extended defects Ab Initio for Irradiation Threshold displacement energies Electronic stopping power Ab Initio and Empirical Potentials Metals and Alloys Pure Iron and Other bcc Metals Self-interstitials and self-interstitial clusters in Fe and other bcc metals Vacancy and vacancy clusters in Fe and other bcc metals Finite temperature effects on defect energetics Beyond Pure Iron helium–vacancy clusters in iron and other bcc metals From pure iron to steels: the role of carbon Interaction of point defects with alloying elements or impurities in iron From dilute to concentrated alloys: the case of Fe–Cr Point defects in hcp-Zr Dislocations Insulators Silicon Carbide Point defects Defect kinetics Defect complexes Impurities Extended defects Uranium Oxide
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1.08.5.2.1 1.08.5.2.2 1.08.5.2.3 1.08.5.2.4 1.08.6 References
Bulk electronic structure Point defects Oxygen clusters Impurities Conclusion
Abbreviations bcc CTL DFT DLTS EPR fcc FLAPW FP GGA LDA LSD LVM PAW PL RPV SIA SQS TD-DFT
Body-centered cubic Charge transition levels Density functional theory Deep level transient spectroscopy Electron paramagnetic resonance Face-centered cubic Full potential linearized augmented plane waves Fission products Generalized gradient approximation Local density approximation Local spin density approximation Local vibrational modes Projector augmented waves Photo-luminescence Reactor pressure vessel Self-interstitial atom Special quasi-random structures Time dependent density functional theory
1.08.1 Introduction Electronic structure calculations did not start with the so-called ab initio calculations or in recent years. The underlying basics date back to the 1930s with an understanding of the quantum nature of bonding in solids, the Hartree and Fock approximations, and the Bloch theorem. A lot was understood of the electronic structure and bonding in nuclear materials using semiempirical electronic structure calculations, for example, tight binding calculations.1 The importance of these somewhat historical calculations should not be overlooked. However, in the following sections, we focus on ‘ab initio’ calculations, that is, density functional theory (DFT) calculations. One must acknowledge that ‘ab initio calculations’ is a rather vague expression that may have different meanings depending on the community. In the present chapter we use it, as most people in the materials science community do, as a synonym for DFT calculations.
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The popularity of these methods stems from the fact that, as we shall see, they provide quantitative results on many properties of solids without any adjustable parameters, though conceptual and technical difficulties subsist that should be kept in mind. The presentation is divided as follows. Methodologies and tools are briefly presented in the first section. The next two sections focus on some examples of ab initio results on metals and alloys on one hand and insulating materials on the other.
1.08.2 Methodologies and Tools 1.08.2.1
Theoretical Background
In the following a very basic summary of the DFT is given. The reader is referred to specialized textbooks2–4 for further reading and mathematical details. Electronic structure calculations aim primarily at finding the ground state of an assembly of interacting nuclei and electrons, the former being treated classically and the latter needing a quantum treatment. The theoretical foundations of DFT were set in the 1960s by the works of Hohenberg and Kohn. They proved that the determination of the ground-state wave function of the electrons in a system (a function of 3N variables if the system contains N electrons) can be replaced by the determination of the ground-state electronic density (a function of only three variables). Kohn and Sham then introduced a trick in which the density is expressed as the sum of squared single particle wave functions, these single particles being fictitious noninteracting electrons. In the process, an assembly of interacting electrons has been replaced by an assembly of fictitious noninteracting particles, thus greatly easing the calculations. The electronic interactions are gathered in a one-electron term called ‘the exchange and correlation potential,’ which derives from an exchange and correlation functional of the total electronic density. One finally obtains a set of one-electron Schro¨dinger equations, whose terms depend on the electronic density, thus introducing a self-consistency loop. No exact formulation exists for this exchange and correlation functional, so one has to resort to
Ab Initio Electronic Structure Calculations for Nuclear Materials
approximations. The simplest one is the local density approximation (LDA). In this approximation, the density of exchange and correlation energy at a given point depends only on the value of the electronic density at this point. Different expressions exist for this dependence, so there are various LDA functionals. Another class of functionals pertains to the generalized gradient approximation (GGA), which introduces in the exchange and correlation energy an additional term depending on the local gradient of the electronic density. These two classes of functionals can be referred to as the standard ones. Most of the ab initio calculations in materials science are performed with such functionals. Recently effort has been put into the development of a new kind of functional, the so-called hybrid functionals, which include some part of exact exchange in their expression. Such functionals, which have been used for years in chemistry, have begun to be used in the nuclear materials context, though they usually involve much more time-consuming calculations. One of their interests is that they give a better description of the properties of insulating materials. We finish this very brief theoretical introduction by mentioning the concepts of k-point sampling and pseudoization. In the community of nuclear materials, most calculations are done for periodic systems, that is, one considers a cell periodically repeated in space. Bloch theorem then ensures that the electronic wave functions should be determined only in the irreducible Brillouin zone, which is in practice sampled with a limited number of so-called k points. A fine sampling is especially important for metallic systems. Most ab initio calculations use pseudopotentials. Pseudoization is based on the assumption that it is possible to separate the electronic levels in valence orbitals and core orbitals. Core electrons are supposed to be tightly bound to their nucleus with their states unaffected by the chemical environment. In contrast, valence electrons fully participate in the bonding. One then first considers in the calculation that only the valence electrons are modified while the core electrons are frozen. Second, the true interaction between the valence electrons and the ion made of the nucleus and core electrons is replaced by a softer pseudopotential of interaction, which greatly decreases the calculation burden. Various pseudoization schemes exist (see Section 1.08.2.2.2). Beyond ground-state properties, other theoretical developments allow the ab initio calculations of additional features. Detailing these developments is
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beyond the scope of this text; let us just mention among others time-dependent DFT for electron dynamics, GW calculations for the calculation of electronic excitation spectra, density functional perturbation theory for phonon calculations, and other second derivatives of the energy. 1.08.2.2
Codes
Ab initio calculations rely on the use of dedicated codes. Such codes are rather large (a few hundred thousand lines), and their development is a heavy task that usually involves several developers. An easy, though oversimplified, way to categorize codes is to classify them in terms of speed on one hand and accuracy on the other. The optimum speed for the desired accuracy is of course one of the goals of the code developers (together with the addition of new features). Codes can primarily be distinguished by their pseudoization scheme and the type of their basis set. We will not describe many other numerical or programming differences, even though they can influence the accuracy and speed of the codes. The possible choices in terms of basis sets and pseudoization are discussed in the following paragraphs. Pseudoization scheme and basis set are intricate as some bases do not need pseudoization and some pseudoizations presently exist only for specific basis sets. These methodological choices intrinsically lead to accurate but heavy, or conversely fast but approximate, calculations. We also mention some codes, though we have no claim to completeness on that matter. Furthermore, we do not comment on the accuracy and speed of the codes themselves as the developing teams are making continuous efforts to improve their codes, which make such comments inappropriate and rapidly outdated. 1.08.2.2.1 Basis sets
For what concerns the basis sets we briefly present plane wave codes, codes with atomic-like localized basis sets, and all-electron codes. All-electron codes involve no pseudoization scheme as all electrons are treated explicitly, though not always on the same footing. In these codes, a spatial distinction between spheres close to the nuclei and interstitial regions is introduced. Wave functions are expressed in a rather complex basis set made of different functions for the spheres and the interstitial regions. In the spheres, spherical harmonics associated with some kind of radial functions (usually Bessel functions) are used, while in the interstitial
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regions wave functions are decomposed in plane waves. All electron codes are very computationally demanding but provide very accurate results. As an example one can mention the Wien2k5 code, which implements the FLAPW (full potential linearized augmented plane wave) formalism.6 At the other end of the spectrum are the codes using localized basis sets. The wave functions are then expressed as combinations of atomic-like orbitals. This choice of basis allows the calculations to be quite fast since the basis set size is quite small (typically, 10–20 functions per atom). The exact determination of the correct basis set, however, is a rather complicated task. Indeed, for each occupied valence orbital one should choose the number of associated radial z basis functions with possibly an empty polarization orbital. The shape of each of these basis functions should be determined for each atomic type present in the calculations. Such codes usually involve a norm-conserving scheme for pseudoization (see the next section) though nothing forbids the use of more advanced schemes. Among this family of codes, SIESTA7,8 is often used in nuclear material studies. Finally, many important codes use plane waves as their basis set.9 This choice is based on the ease of performing fast Fourier transform between direct and reciprocal space, which allows rather fast calculations. However, dealing with plane waves means using pseudopotentials of some kind as plane waves are inappropriate for describing the fast oscillation of the wave functions close to the nuclei. Thanks to pseudopotentials, the number of plane waves is typically reduced to 100 per atom. Finally, we should mention that other basis sets exist, for instance Gaussians as in the eponymous chemistry code10 and wavelets in the BigDft project,11 but their use is at present rather limited in the nuclear materials community. 1.08.2.2.2 Pseudoization schemes
As explained above, pseudoization schemes are especially relevant for plane wave codes. All pseudoization schemes are obtained by calculations on isolated atoms or ions. The real potential experienced by the valence electrons is replaced by a pseudopotential coming from mathematical manipulations. A good pseudopotential should have two apparently contradictory qualities. First, it should be soft, meaning that the wave function oscillations should be smoothened as much as possible. For a plane wave basis set, this means that the number of plane waves needed to
represent the wave functions is kept minimal. Second, it should be transferable, which means that it should correctly represent the real interactions of valence electrons with the core in any kind of chemical environment, that is, in any kind of bonding (metallic, covalent, ionic), with all possible ionic charges or covalent configurations conceivable for the element under consideration. The generation of pseudopotentials is a rather complicated task, but nowadays libraries of pseudopotentials exist and pseudopotentials are freely available for almost any element, though not with all the pseudoization schemes. One can basically distinguish norm-conserving pseudopotentials, ultrasoft pseudopotentials, and PAW formalism. Norm-conserving pseudopotentials were the first ones designed for ab initio calculations.12 They involve the replacement of the real valence wave function by a smooth wave function of equal norm, hence their name. Such pseudopotentials are rather easy to generate, and several libraries exist with all elements of the periodic table. They are reasonably accurate although they are still rather hard, and so they are less and less used in plane wave codes but are still used with atomic-like basis sets. Ultrasoft pseudopotentials13 remove the constraint of norm equality between the real and pseudowave functions. They are thus much softer though less easy to generate than norm-conserving ones. The Projector Augmented Wave14 formalism is a complex pseudoization scheme close in spirit to the ultrasoft scheme but it allows the reconstruction of the real electronic density and the real wave functions with all their oscillations, and for this reason this method can be considered an all-electron method. When correctly generated, PAW atomic data are very soft and quite transferable. Libraries of ultrasoft pseudopotentials or PAWatomic data exist, but they are generally either incomplete or not freely available. Plane wave codes in use in the nuclear materials community include VASP15 with ultrasoft pseudopotentials and PAW formalism, Quantum-Espresso16 with norm-conserving and ultrasoft pseudopotentials and PAW formalism, and ABINIT17 with normconserving pseudopotentials and PAW formalism. Note that for a specific pseudoization scheme many different pseudopotentials can exist for a given element. Even if they were built using the same valence orbitals, pseudopotentials can differ by many numerical choices (e.g., the various matching radii) that enter the pseudoization process. We present in the following a series of practical choices to be made when one wants to perform
Ab Initio Electronic Structure Calculations for Nuclear Materials
ab initio calculations. But the first and certainly most important of these choices is that of the ab initio code itself as different codes have different speeds, accuracies, numerical methods, features, input files, and so on, and so it proves quite difficult to change codes in the middle of a study. Furthermore, one observes that most people are reluctant to change their usual code as the investment required to fully master the use of a code is far from negligible (not to mention the one to master what is in the code). 1.08.2.3
Ab Initio Calculations in Practice
In this paragraph, we try to give some indication of what can be done with an ab initio code and how it is done in practice. The calculation starts with the positioning of atoms of given types in a calculation cell of a certain shape. That would be all if the calculations were truly ab initio. Unfortunately, a few more pieces of information should be passed to the code; the most important ones are described in the final section. The first section introduces the basic outputs of the code, and the second one deals with the possible cell sizes and the associated CPU times. 1.08.2.3.1 Output
We describe in this section the output of ab initio calculations in general terms. The possible applications in the nuclear materials field are given below. The basic output of a standard ab initio calculation is the complete description of the electronic ground state for the considered atomic configuration. From this, one can extract electronic as well as energetic information. On the electronic side, one has access to the electronic density of states, which will indicate whether the material is metallic, semiconducting, or insulating (or at least what the code predicts it to be), its possible magnetic structure, and so on. Additional calculations are able to provide additional information on the electronic excitation spectra: optical absorption, X-ray spectra, and so on. On the energetic side, the main output is the total energy of the system for the given atomic configuration. Most codes are also able to calculate the forces acting on the ions as well as the stress tensor acting on the cell. Knowing these forces and stress, it is possible to chain ground-state calculations to perform various calculations: Atomic relaxations to the local minimum for the atomic positions.
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From the relaxed positions (where forces are zero), one can calculate second derivatives of the energy to deduce, among other things, the phonon spectrum. This can be done either directly, by the socalled frozen phonon approach, or by first-order perturbation theory (if such feature is implemented in the code). In this last case, the third-order derivative of the energy (Raman spectrum, phonon lifetimes) can also be computed. Starting from two relaxed configurations close in space, one can calculate the energetic path in space joining these two configurations, thus allowing the calculation of saddle points. The integration of the forces in a Molecular Dynamics scheme leads to so-called ab initio molecular dynamics (see Chapter 1.09, Molecular Dynamics). Car–Parrinello molecular dynamics18 calculations, which pertain to this class of calculations, introduce fictitious dynamics on the electrons to solve the minimization problem on the electrons simultaneously with the real ion dynamics.
1.08.2.3.2 Cell sizes and corresponding CPU times
The calculation time of ab initio calculations varies – to first order – as the cube of the number of atoms or equivalently of electrons (the famous N3 dependence) in the cell. If a fine k-point sampling is needed, this dependence is reduced to N2 as the number of k points decreases in inverse proportion with the size of the cell. On the other hand, the number of selfconsistent cycles needed to reach convergence tends to increase with N. Anyway, the variation of calculation time with the size of the cell is huge and thus strongly limits the number of atoms and also the cell size that can be considered. On one hand, calculations on the unit cell of simple crystalline materials (with a small number of atoms per unit cell) are fast and can easily be performed on a common laptop. On the other hand, when larger simulation cells are needed, the calculations quickly become more demanding. The present upper limit in the number of atoms that can be considered is of the order of a few hundreds. The exact limit of course depends on the code and also on the number of electrons per atoms and other technicalities (number of basis functions, k points, available computer power, etc.), so it is not possible to state it precisely. Considering such large cells leads anyway to very heavy calculations in which the use of parallel versions of the codes is
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almost mandatory. Various parallelization schemes are possible: on k points, fast Fourier transform, bands, spins; the parallelization schemes actually available depend on the code. The situation gets even worse when one notes that a relaxation roughly involves at least ten ground-state calculations, a saddle point calculation needs about ten complete relaxations, and that each molecular dynamics simulation time step (of about 1 fs) needs a complete ground-state calculation. Overall, one can understand that the CPU time needed to complete an ab initio study (which most of the time involves various starting geometry) may amount up to hundreds of thousands or millions of CPU hours. 1.08.2.3.3 Choices to make
Whatever the system considered and the code used, one needs to provide more inputs than just the atomic positions and types. Most codes suggest some values for these inputs. However, their tuning may still be necessary as default values may very well be suited for some supposedly standard situations and irrelevant for others. Blind use of ab initio codes may thus lead to disappointing errors. Indeed, not all these choices are trivial, so mistakes can be hard to notice for the beginner. Choices are usually made out of experience, after considering some test cases needing small calculation time. One can distinguish between choices that should be made only once at the beginning of a study and calculation parameters that can be tuned calculation by calculation. The main unchangeable choices are the exchange and correlation functional and the pseudopotentials or PAW atomic data for the various atomic types in the calculation. First, one has to choose the flavor of the exchange and correlation functional that will be used to describe the electronic interactions. Most of the time one chooses either an LDA or a GGA functional. Trends are known about the behavior of these functionals: LDA calculations tend to overestimate the bonding and underestimate the bond length in bulk materials, the opposite for GGA. However, things can become tricky when one deals with defects as energy differences (between defect-containing and defect-free cells) are involved. For insulating materials or materials with correlated electrons, the choice of the exchange and correlation functional is even more difficult (see Section 1.08.5). The second and more definitive choice is the one of the pseudopotential. We do not mean here the choice of the pseudoization scheme but the choice
of the pseudopotential itself. Indeed, calculated energies vary greatly with the chosen pseudopotential, so energy differences that are thermodynamically or kinetically relevant are meaningless if the various calculations are performed with different pseudopotentials. The determination of the shape of the atomic basis set in the case of localized bases is also of importance, and it is close in spirit to the choice of the pseudopotential except that much less basis sets than pseudopotentials are available. More technical inputs include the k-point sampling. The larger the number of k points to sample the Brillouin zone, the more accurate the results but the heavier the calculations will be. This is especially true for metallic systems that need fine sampling of the Brillouin zone, but convergence with respect to the number of k points can be accelerated by the introduction of a smearing of the occupations of electronic levels close to the Fermi energy. The shape and width of this smearing function is then an additional parameter.19 the number of plane waves (obviously for plane wave codes but also for some other codes that also use FFT). Once again the larger the number of plane waves, the more accurate and heavier the calculation. the convergence criteria. The two major convergence criteria are the one for the self-consistent loop of the calculation of the ground-state electronic wave functions and the one to signal the convergence of a relaxation calculation (with some threshold depending on the forces acting on the atoms).
1.08.3 Fields of Application Ab initio calculations can be applied to almost any solid once the limitations in cell sizes and number of atoms are taken into account. Among the materials of nuclear interest that have been studied one can cite the following: metals, particularly iron, tungsten, zirconium, and plutonium; alloys, especially iron alloys (FeCr, FeC to tackle steel, etc.); models of fuel materials, UO2, U–PuO2, and uranium carbides; structural carbides (SiC, TiC, B4C, etc.); waste materials (zircon, pyrochlores, apatites, etc.). In this section, we rapidly expose the types of studies that can be done with ab initio calculations. The last two sections on metallic alloys and insulating materials will allow us to go into detail for some specific cases.
Ab Initio Electronic Structure Calculations for Nuclear Materials
1.08.3.1
Perfect Crystal
1.08.3.1.1 Bulk properties
Dealing with perfect crystals, ab initio calculations provide information about the crystallographic and electronic structure of the perfect material. The properties of usual materials, such as standard metals, band insulators, or semi-conductors, are basically well reproduced, though some problems remain, especially for nonconductors (see Section 1.08.5.1 on SiC). However, difficulties arise when one wishes to tackle the properties of highly correlated materials such as uranium oxide (Section 1.08.5.2). For instance, no ab initio code, whatever the complexity and refinements, is able to correctly predict the fact that plutonium is nonmagnetic. In such situations, the nature of the chemical bonding is still poorly understood, so the correct physical ingredients are probably not present in today’s codes. These especially difficult cases should not mask the very impressive precision of the results obtained for the crystal structure, cohesive energy, atomic vibrations, and so on of less difficult materials. 1.08.3.1.2 Input for thermodynamic models
The information on bulk materials can be gathered in thermodynamical models. Most ab initio calculations are performed at zero temperature. Even with this restriction, they can be used for thermodynamical studies. First, ab initio calculations enable one to consider phases that are not accessible to experiments. It is thus possible to compare the relative stability of various (real or fictitious) structures for a given composition and pressure. Considering alloys, it is possible to calculate the cohesive energy of various crystallographic arrangements. Solid solutions can also be modeled by so-called special quasi-random structures (SQS).20 Beyond a simple comparison of the energies of the various structures, when a common underlying crystalline network exists for all the considered phases, the information about the cohesive energies can be used to parameterize rigid lattice inter-atomic interaction models (i.e., pair, triplet, etc., interactions) that can be used to perform computational thermodynamics (see Chapter 1.17, Computational Thermodynamics: Application to Nuclear Materials). These interactions can then be used in mean field or Monte-Carlo simulations to predict phase stabilities at nonzero temperature.21 As examples of this kind of studies one can cite the determination of solubility limits (e.g., Zr and Sc in aluminum22) and the exploration of details of the
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phase diagrams (e.g., the inversion of stability in the iron-rich side of the Fe–Cr diagram23). Directly considering nonzero temperature in ab initio simulation is also possible, though more difficult. First, one can calculate for a given composition and structure the electronic and vibrational entropy (through the phonon spectrum), which leads to the variation in heat capacity with temperature. Nontrivial thermodynamic integrations can then be used to calculate the relative stability of various structures at nonzero temperature. Second, one can perform ab initio molecular dynamics simulations to model finite temperature properties (e.g., thermal expansion). 1.08.3.2
Defects
Point defects are of course very important in a nuclear complex as they are created either by irradiation or by accommodation of impurities (e.g., fission products (FP)) (see Chapter 1.02, Fundamental Point Defect Properties in Ceramics and Chapter 1.03, Radiation-Induced Effects on Microstructure). More generally, they have a tremendous role in the kinetic properties of the materials. It is therefore not surprising if countless ab initio studies exist on point defects in nuclear materials. Most of them are based on a supercell approach in which the unit cell of the perfect crystal is periodically repeated up to the largest possible simulation box. A point defect is then introduced, and the structure is allowed to relax. By difference with the defect-free structure, one can calculate the formation energy of the defect that drives its equilibrium concentration. Some care must be taken in writing this difference as the number and types of atoms should be preserved in the process. Point defects are also the perfect object for the saddle point calculations that give the energy that drives their kinetic properties. Ab initio permits accurate calculation of these energies and also consideration of (for insulating materials) the various possible charge states of the defects. They have shown that the properties of defects can vary greatly with their charge states. Many different kinds of defects can be considered. A list of possible defects follows with the characteristic associated thermodynamical and kinetic energies. 1.08.3.2.1 Self-defects
Vacancies and interstitials, with the associated formation energy driving their concentration and migration energy driving their displacement in the solid; the sum of these two energies is the activation energy for diffusion at equilibrium. For such simple defects,
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Ab Initio Electronic Structure Calculations for Nuclear Materials
In the nuclear context, such defects can be fission products in a fuel material, actinide atoms in a waste material, helium gases in structural materials, and so on; ab initio gives access to the solution energy of these impurities, which allows one to determine their most favored positions in the crystal: interstitial position, substitution for host atoms, and so on. The kinetic energies of migration of interstitial impurities are accessible as well as the kinetic barrier for the extraction of an impurity from a vacancy site.
ab initio molecular dynamics. We are aware of studies in GaN27 and silicon carbides.28,29 The procedure is the same as that with empirical potentials: one initiates a series of cascades of low but increasing energy and follows the displacement of the accelerated atom. The threshold energy is reached as soon as the atom does not return to its initial position at the end of the cascade. Such calculations are very promising as empirical potentials are usually imprecise for the orders of energies and interatomic distances at stake in threshold energies. However, they should be done with care as most pseudopotentials and basis sets are designed to work for moderate interatomic distances, and bringing two atoms too close to each other may lead to spurious results unless the pseudopotentials are specifically designed.
1.08.3.2.3 Point defect assemblies
1.08.3.3.2 Electronic stopping power
it is possible to go beyond the 0 K energies and to access the free energies of formation and migrations by calculating the vibrational spectra in the presence of the defect in the stable position and at the saddle point (see Section 1.08.4.2.3). 1.08.3.2.2 Hetero-defects
In this class, one can include the calculation of interstitial assemblies as well as the complexes built with impurities and vacancies. One then has access to the binding of monoatomic defects to the complexes,24 possibly with the associated kinetic energy barriers. 1.08.3.2.4 Kinetic models
As for perfect crystals, the information obtained by ab initio calculations can be gathered and integrated in larger scale modeling, especially, kinetic models. Many kinetic Monte-Carlo models were thus parameterized with ab initio calculations (see e.g., the works on pure iron25 or FeCu26and Chapter 1.14, Kinetic Monte Carlo Simulations of Irradiation Effects). 1.08.3.2.5 Extended defects
Even if the cell sizes accessible by ab initio calculations are small, it is possible to deal with some extended defects. Calculations then often need some tricks to accommodate the extended defect in the small cells. Some examples are given in the next section on studies on dislocations. 1.08.3.3
Ab Initio for Irradiation
Irradiation damage, especially cascade modeling, is usually preferentially dealt by larger scale methods such as molecular dynamics with empirical potentials rather than ab initio calculations. However, recently ab initio studies that directly tackle irradiation processes have appeared. 1.08.3.3.1 Threshold displacement energies
First, the increase in computer power has allowed the calculations of threshold displacement energies by
Second, recent studies have been published in the ab initio calculations of the electronic stopping power for high-velocity atoms or ions. The framework best suited to address this issue is time-dependent DFT (TD-DFT). Two kinds of TD-DFT have been applied to stopping power studies so far. The first approach relies on the linear response of the system to the charged particle. The key quantity here is the density–density response function that measures how the electronic density of the solid reacts to a change in the external charge density. This observable is usually represented in reciprocal space and frequency, so it can be confronted directly with energy loss measurements. The density– density response function describes the possible excitations of the solid that channel an energy transfer from the irradiating particle to the solid. Most noticeably the (imaginary part of the) function vanishes for an energy lower than the band gap and shows a peak around the plasma frequency. Integrating this function over momentum and energy transfers, one obtains the electronic stopping power. Campillo, Pitarke, Eguiluz, and Garcia have implemented this approach and applied to some simple solids, such as aluminum or silicon.30–32 They showed that there is little difference between the usual approximations of TD-DFT: the random phase approximation, which means basically no exchange correlation included, or adiabatic LDA, which means that the exchange correlation is local in space and instantaneous in time. The influence of the band structure of the solid accounts for noticeable deviations from the homogeneous electron gas model.
Ab Initio Electronic Structure Calculations for Nuclear Materials
The second approach is more straightforward conceptually but more cumbersome technically. It proposes to simply monitor the slowing down of the charged irradiated particle in a large box in real space and real time. The response of the solid is hence not limited to the linear response: all orders are automatically included. However, the drawback is the size of the simulation box, which should be large enough to prevent interaction between the periodic images. Following this approach, Pruneda and coworkers33 calculated the stopping power in a large band gap insulator, lithium fluoride, for small velocities of the impinging particle. In the small velocity regime, the nonlinear terms in the response are shown to be important. Unfortunately, whatever the implementation of TD-DFT in use, the calculations always rely on very crude approximations for the exchange-correlation effects. The true exchange-correlation kernel (the second derivative of the exchange-correlation energy with respect to the density) is in principle nonlocal (it is indeed long ranged) and has memory. The use of novel approximations of the kernel was recently introduced by Barriga-Carrasco but for homogeneous electron gas only.34,35 1.08.3.4
Ab Initio and Empirical Potentials
Ab initio calculations are often compared to and sometimes confused with empirical potential calculations. We will now try to clarify the differences between these two approaches and highlight their point of contacts. The main difference is of course that ab initio calculations deal with atomic and electronic degrees of freedom. Empirical potentials depend only on the relative positions of the considered atoms and ions. They do not explicitly consider electrons. Thus, roughly speaking, ab initio calculations deal with electronic structure and give access to good energetics, whereas empirical potentials are not concerned with electrons and give approximate energetics but allow much larger scale calculations (in space and time). Going into some details, we have shown that ab initio gives access to very diverse phenomena. Some can be modeled with empirical potentials, at least partly; others are completely outside the scope of such potentials. In the latter category, one will find the phenomena that are really related to the electronic structure itself. For instance, the calculations of electronic excitations (e.g., optical or X-ray spectra) are conceptually impossible with empirical potentials. In the
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same way, for insulating materials, the calculation of the relative stability of various charge states of a given defect is impossible with empirical potentials. Other phenomena that are intrinsically electronic in nature can be very crudely accounted for in empirical potentials. The electronic stopping power of an accelerated particle is an example. As indicated above, it can be calculated ab initio. Conversely, from the empirical potential perspective one can add an ad hoc slowing term to the dynamics of fast moving particles in solids whose intensity has to be established by fitting experimental (or ab initio) data. In a related way, some forms of empirical potentials rely on electronic information; for instance, the Finnis–Sinclair36 or Rosato et al.37 forms. In the same spirit, a recent empirical potential has been designed to reproduce the local ferromagnetic order of iron.38 However, this potential assumes a tendency for ferromagnetic order, while ab initio calculation can (in principle) predict what the magnetic order will be. Therefore, ab initio is very often used as a way to get accurate energies for a given atomic arrangement. This is the case for the formation and migration energies of defects, the vibration spectra, and so on. These phenomena are conceptually within reach of empirical potentials (except the ones that reincorporate electronic degrees of freedom such as charged defects). Ab initio is then just a way to get proper and quantitative energetics. Their results are often used as reference for fitting empirical potentials. However, the fit of a correct empirical remains a tremendous task especially with the complex forms of potentials nowadays and when one wants to correctly predict subtle, out of equilibrium, properties. Finally, one should always keep in mind that cohesion in solids is quantum in nature, so classical interatomic potentials dealing only with atoms or ions can never fully reproduce all the aspects of bonding in a material.
1.08.4 Metals and Alloys The vast majority of DFT calculations on radiation defects in metallic materials have been performed in body-centered cubic (bcc) iron-based materials, for obvious application reasons of ferritic steels but also because of the more severe shortcoming of predictions based only on empirical potentials. A number of accurate estimates of energies of formation and migration of self-interstitial and vacancy defects as well as small defect clusters and solute-vacancy or solute-interstitial complexes have been obtained.
Ab Initio Electronic Structure Calculations for Nuclear Materials
DFT calculations have been intensively used to predict atomistic defect configurations and also transition pathways. An overview of these results is presented below, complete with examples in other bcc transition metals, in particular tungsten, as well as hcp-Zr. These examples illustrate how DFT data have changed the more or less admitted energy landscape of these defects and also how they are used to improve empirical potentials. In the final part of this chapter, a brief overview of typical works on dislocations (in iron) is presented. 1.08.4.1
55 Fe
50 45 P-bcc (PW)
40 35 E (mRy)
232
30
P-bcc (LSD)
25
Pure Iron and Other bcc Metals
Ferritic steels are an important class of nuclear materials, which include reactor pressure vessel (RPV) steels and high chromium steels for elevated temperature structural and cladding materials in fast reactors and fusion reactors, see Chapter 4.03, Ferritic Steels and Advanced Ferritic–Martensitic Steels. From a basic science point of view, the modeling of these materials starts with that of pure iron, in the ferromagnetic bcc structure. Iron presents several difficulties for DFT calculations. First, being a three dimensional (3D) metal, it requires rather large basis sets in plane wave calculations. Second, the calculations need to be spin polarized, to account for magnetism, and this at least doubles the calculation time. But most of all, it is a case where the choice of the exchangecorrelation functional has a dramatic effect on bulk properties. The standard LDA incorrectly predicts the paramagnetic face-centered cubic (fcc) structure to be more stable than the ferromagnetic bcc structure. The correct ground state is recovered using gradient corrected functionals,39 as illustrated in Figure 1. Finally, it was pointed out that pseudopotentials tend to overestimate the magnetic energy in iron,40 and therefore, some pseudopotentials suffer from a lack of transferability for some properties. In practice, however, in the large set of the results obtained over the last decade for defect calculations in iron, a quite remarkable agreement is obtained between the various computational approaches. With a few exceptions, they are indeed quite independent on the form of the GGA functional, the basis set (plane wave or localized), and the pseudopotential or the use of PAWapproaches. 1.08.4.1.1 Self-interstitials and selfinterstitial clusters in Fe and other bcc metals
The structure and migration mechanism of selfinterstitials in iron is a very good illustrative example of the impact of DFT calculations on radiation defect
P-fcc (PW)
20
F-bcc (LSD)
15 10 5
P-fcc (LSD)
F-bcc (PW)
0 2.40 2.45 2.50 2.55 2.60 2.65 2.70 2.75 s arbitrary units (a.u.) Figure 1 Calculated total energy of paramagnetic (P) bcc and fcc and ferromagnetic (F) bcc iron as a function of Wigner–Seitz radius (s). The dotted curve corresponds to the local spin density (LSD) approximation, and the solid curve corresponds to the GGA functional proposed by Perdew and Wang in 1986 (PW). The curves are displaced in energy so that the minima for F bcc coincide. Energies are in Ry (1 Ry ¼ 13.6057 eV) and distances in bohr (1 bohr ¼ 0.5292 A˚). Reproduced from Derlet, P. M.; Dudarev, S. L. Prog. Mater. Sci. 2007, 52, 299–318.
studies. Progress in methods, codes, and computer performance made this archetype of radiation defects accessible to DFT calculations in the early 2000s, since total energy differences between simulation cells of 128þ1 atoms could then be obtained with a sufficient accuracy. In 2001, Domain and Becquart reported that, in agreement with the experiment, the h110i dumbbell was the most stable structure.41 Quite unexpectedly, the h111i dumbbell was predicted to be 0.7 eV higher in energy, at variance with empirical potential results that predicted a much smaller energy difference. DFT calculations performed in other bcc metals revealed that this is a peculiarity of Fe,42 as illustrated in Figure 2, and magnetism was proposed to be the origin of the energy increase in the h111i dumbbell in Fe. The important consequence of this result in Fe, which has been confirmed repeatedly since
Ab Initio Electronic Structure Calculations for Nuclear Materials
Defect energy relative to the <111> (eV)
2.5 2 1.5
2.5 V Nb Ta Fe
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<110>
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-1 Octa. <111>
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<110>
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<100>
Octa.
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Figure 2 Formation energies of several basic SIA configurations calculated for bcc transition metals of group 5B (left) and group 6B (right), taken from Nguyen-Manh et al.42 Data for bcc Fe are taken from Fu et al.43 Reproduced from Nguyen-Manh, D.; Horsfiels, A. P.; Dudarev, S. L. Phys. Rev. B 2006, 73, 020101.
then, is that it excludes the SIA migration to occur by long 1D glides of the h111i dumbbell followed by on-site rotations of the h110i dumbbell, as predicted previously from empirical potential MD simulations. Moreover, DFT investigation of the migration mechanism yielded a quantitative agreement with the experiment for the energy of the Johnson translation–rotation mechanism (see Figure 3), namely 0.3 eV.43 These DFT calculations were followed by a very successful example of synergy between DFT and empirical potentials. The DFT values of interstitial formation energies in various configurations and interatomic forces in a liquid model have indeed been included in the database for a fit of EAM type potentials by Mendelev et al.45 This approach has resulted in a new generation of improved empirical potentials, albeit still with some limitations. When considering SIA clusters made of parallel dumbbells, the Mendelev potential agrees with DFT for predicting a crossover as a function of cluster size from the h110i to the h111i orientation between 4 and 6 SIA
clusters.44 However, discrepancies are found when considering nonparallel configurations.46 More precisely, new configurations of small SIA clusters were observed in MD simulations performed at high temperature with the Mendelev potential. The energy of the new di-interstitial cluster, made of a triangle of atoms sharing one site (see Figure 4), is even lower than that of the parallel configuration within DFT but higher by 0.3 eV with the Mendelev potential (see also Section 1.08.4.3 on dislocations). The new triand quadri-interstitial clusters, with a ring structure (see Figure 4), are one of the few examples in which a significant discrepancy is found between various DFT approaches. Calculations with the most accurate description of the ionic cores predict that the new tri-interstitial configuration is slightly more stable than the parallel configuration, whereas more approximate ones predict that it is 0.7 eV higher. The first category includes calculations in the PAW approach, performed using either the VASP code or the PWSCF code and also ultrasoft pseudopotential calculations. The second one includes calculations
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Ab Initio Electronic Structure Calculations for Nuclear Materials 0.8 DFT-GGA Mendelev Ackland
Energy barrier (eV)
0.6
0.4
0.2
0.0 [011]
[110]
[111] Crowd.
[111]
Figure 3 Left: Johnson translation–rotation mechanism of the h110i dumbbell; white and black spheres indicate the initial and final positions of the atoms, respectively. Reproduced from Fu, C. C.; Willaime, F.; Ordejon, P. Phys. Rev. Lett. 2004, 92, 175503. Right: Comparison between the DFT-GGA result and two EAM potentials for the energy barriers of the Johnson mechanism and the h110i to h111i transformation. Reproduced from Willaime, F.; Fu, C. C.; Marinica, M. C.; Torre, J. D.; Nucl. Instrum. Meth. Phys. Res. B 2005, 228, 92.
<011> <101> <101>
<011>
<110>
<110>
Figure 4 New low-energy configurations of SIA clusters in Fe, which revealed discrepancies between DFT and empirical potentials and between various approximations within DFT. Reproduced from Terentyev, D. A.; Klaver, T. P. C.; Olsson, P.; Marinica, M. C.; Willaime, F.; Domain, C.; Malerba, L. Phys. Rev. Lett. 2008, 100, 145503.
with less transferable ultrasoft pseudopotentials with VASP and norm-conserving pseudopotentials with SIESTA.46 Such a discrepancy is not common in defect calculations in metals. Further investigations are required to understand more precisely its origin, in particular the possible role of magnetism. The structures of the most stable SIA clusters in Fe, and more generally of their energy landscape, remain an open question. One would ideally need to combine DFT calculations with methods for exploring the energy surface, such as the Dimer47 or ART48 methods. Such a combination is possible in principle, and it has indeed been used for defects in semiconductors,49 but due to computer limitations this is not the case yet in Fe. The alternative is to develop new empirical potentials in better agreement with
DFT energies in particular for these new structures, to perform the Dimer or ART calculations with these potentials, and to validate the main features of the energy landscape thus obtained by DFT calculations. To summarize, the energy landscape of interstitial type defects has been revisited in the last decade driven by DFT calculations, in synergy with empirical potential calculations. 1.08.4.1.2 Vacancy and vacancy clusters in Fe and other bcc metals
DFT has some limitations in predicting accurate vacancy formation energies in transition metals. The exceptional agreement with the experiment obtained initially within DFT-LDA50 was later shown to result from a cancellation between two effects. First, the
Ab Initio Electronic Structure Calculations for Nuclear Materials
Ackland et al.94 Mendelev et al.45 DFT-GGA
1.2 Migration energy (eV)
structural relaxation, which was neglected by Korhonen et al.50 is now known to significantly reduce the vacancy formation energy, in particular in bcc metals.51 Second, due to limitations of exchange-correlation functionals at surfaces, DFT-LDA tends to underestimate the vacancy formation energy. This discrepancy is even larger within DFT-GGA, and it increases with the number of valence electrons. It is therefore rather small for early transition metals (Ti, Zr, Hf,), but it is estimated to be as large as 0.2 eV in LDA and 0.5 eV in GGA-PW1 for late transition metals (Ni, Pd, Pt).52 However, the effect is much weaker for migration energies.52 A new functional, AM05, has been proposed to cope with this limitation.53 Less spectacular effects are expected in vacancytype defects than in interstitial-type defects when going from empirical potentials to DFT calculations. The discussion on vacancy-type defects in Fe will be restricted to the results obtained within DFT-GGA, due to the superiority of this functional for bulk properties. For pure Fe, DFT-GGA vacancy formation and migration energies are in the range of 1.93–2.23 eVand 0.59–0.71 eV.41,43,54 These values are in agreement with experimental estimates at low temperatures in ultrapure iron, namely 2.0 0.2 eV and 0. 55 eV, respectively. These values can be reproduced by empirical potentials when included in the fit, but one discrepancy remains with DFT concerning the shape of the migration barrier. It is indeed clearly a single hump in DFT25 and usually a double hump with empirical potentials. Concerning vacancy clusters, the structures predicted by empirical potentials, namely compact structures, were confirmed by DFT calculations, but there are discrepancies in the migration energies. In both cases, the most stable divacancy is the nextnearest-neighbor configuration, with a binding energy of 0.2–0.3 eV.25,55,56 The migration can occur by two different two-step processes, with an intermediate configuration that is either nearest neighbor or fourth nearest neighbor.56 A quite unexpected result of DFT calculations was the prediction of rather low migration energies for the tri- and quadrivacancies, namely 0.35 and 0.48 eV.25 Depending on the potential, this phenomenon is either not reproduced or only partly reproduced (see Figure 5).57 Stronger deviations from empirical potential predictions for divacancies are observed in DFT calculations performed in other bcc metals. The most dramatic case is that of tungsten, where the nextnearest-neighbor interaction is strongly repulsive (0.5 eV) and the nearest-neighbor interaction is
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1.0 0.8 0.6 0.4 0.2 0.0
V
V2
V3
V4
V5
Figure 5 Migration energies of vacancy clusters in Fe, as a function of cluster size. Reproduced from Fu, C. C.; Willaime, F. (2004) Unpublished.
vanishing.58 This result does not explain why voids are formed in tungsten under irradiation. 1.08.4.1.3 Finite temperature effects on defect energetics
The properties of radiation defects at high temperature may change due to three possible contributions to the free energy: electronic, magnetic, and vibrational. These three effects can be well modeled in bulk bcc iron,59 but they are more challenging for defects. The electronic contribution, which exists only in metals, arises due to changes in the density of states close to the Fermi level. The electronic entropy difference between, for example, two configurations is, to first order, proportional to the temperature, T, and the change in density of states at the Fermi level. This electronic effect is straightforward to take into account in DFT calculations. It was shown in tungsten to decrease the activation free energy for self-diffusion by up to 0.4 eV close to the melting temperature. Thus, although this effect is relatively small in general, it cannot be neglected at high temperature. The magnetic contribution is important in iron. Spin fluctuations were shown to be the origin of the strong softening of the C0 elastic constant observed as the temperature increases up to the ag transition temperature,60 and it drives, for instance, the temperature dependence of relative abundance of <100> and <111> interstitial loops formed under irradiation.61 It is also known to have a small effect on vacancy properties, but to the authors’ knowledge there is presently no tractable method to predict this effect for point defects quantitatively from DFT calculations. This is probably one of the important challenges in the field.
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Ab Initio Electronic Structure Calculations for Nuclear Materials
Finally, vibrational entropy effects can in principle be obtained either in the quasi-harmonic approximation from phonon frequency calculations or directly from first-principles molecular dynamics. There are very few examples of such calculations in the literature. The vibrational modes of vacancies and self-interstitials in iron have been investigated by DFT calculations, and their formation entropies have been estimated.62 As illustrated recently in Mo, it is also possible to calculate the temperature dependence of the vacancy formation enthalpy, from DFT molecular dynamics simulations, including anharmonic effects, as well as the defect jump frequency, going beyond the transition state approximation.63 1.08.4.2
Beyond Pure Iron
1.08.4.2.1 helium–vacancy clusters in iron and other bcc metals
Irradiation of metals by neutrons produces, besides point defects, rare gases by transmutation reactions. Helium is a major concern since it has a very low solubility in metals, see Chapter 1.06, The Effects of Helium in Irradiated Structural Alloys. It is deeply trapped by vacancies, and helium–vacancy clustering can ultimately lead to bubble formation and void swelling. At variance with empirical potential predictions, DFT calculations showed that interstitial helium is unambiguously located on tetrahedral sites, not only in iron, but also in all other bcc metals.64–67 An improved Fe–He pair potential was then obtained by fitting to the DFT results.68 DFT (b)
calculations account for the very fast migration of interstitial helium as well as for its deep trapping to vacancies, although not as deep as predicted by empirical potentials. Note that an unexpected effect was observed in Vanadium, where the helium atom in a vacancy is found to be off centered.66 The energy landscape of the helium–divacancy complex also revealed unexpected configurations, in particular, for the lowest energy configuration where the helium atom is located halfway between two nearestneighbor vacancies (see Figure 6). More generally, a systematic study of the energetics of all small HenVm clusters in iron was performed,64 giving very useful data for the kinetic modeling of helium–vacancy clustering and dissociation.69 Quite interestingly, interstitial He atoms are found to attract one another, even in the absence of vacancies.24,64 This clustering of helium atoms may then yield the emission of selfinterstitials. Finally, the interaction of helium with self-interstitials is, as expected, much weaker but also attractive.24 Similar studies have been performed on small helium–vacancy clusters in tungsten58,70 and also on the behavior of hydrogen in iron and tungsten. It should be noted that at low temperature, quantum effects must be taken into account in the migration properties in particular for hydrogen.71 1.08.4.2.2 From pure iron to steels: the role of carbon
In steels, the presence of carbon, even though its concentration is very low, considerably affects defect properties because of the strong carbon-defect (d)
(a)
(e)
Energy (eV)
1.2
(f)
(c)
0.8
0.4
0.0 3 nn
1 nn Reaction coordinate
2 nn
2 nn
Figure 6 Schematic representation of the energy landscape of the HeV2 complex. Reproduced from Fu, C. C.; Willaime, F. Phys. Rev. B 2005, 72, 064117.
Ab Initio Electronic Structure Calculations for Nuclear Materials
interaction. DFT calculations reproduce the wellknown fact that carbon is located in octahedral sites, and they also confirm the strong attraction between interstitial carbon and a monovacancy, with a binding energy of about 0.5 eV.72–74 This strong attraction is the origin of the confusing discrepancy between the vacancy migration energy in ultrapure iron, 0.6 eV, and the effective vacancy migration energy in iron with carbon or in steels, that is, 1.1 eV, which corresponds to first order to the sum of the vacancy migration energy and the carbon-vacancy binding energy.74 More interestingly, DFT calculations predict that the complex formed by a vacancy and two carbon atoms, VC2, is extremely stable, due to the formation of a strong covalent bond between the carbon atoms. The VC–C binding energy is indeed close to 1 eV,72–74 and VC2 complexes are expected to play a very important role. The interaction between carbon and selfinterstitials is also attractive but weaker. In agreement with experiments,75 DFT calculations confirmed a binding energy of 0.2 eV76 and predict, at variance with initial empirical potential results, that the nearest-neighbor configurations are repulsive and that the most attractive configuration is that shown in Figure 7. This shortcoming of empirical potentials was overcome recently with an improved potential derived taking into account information from the electronic structure.77 The strong interaction of carbon with vacancies also affects the energetics of helium–vacancy clusters, and it is important to take this into account to reproduce, for example, thermal helium desorption experiments performed in iron.78 Similar calculations have been performed with nitrogen.72 1.08.4.2.3 Interaction of point defects with alloying elements or impurities in iron
The diffusion of point defects produced by irradiation may induce fluxes of solutes, for example, toward
237
or away from defect sinks, depending on the defect– solute interactions. DFT is again a very powerful tool to predict such interactions, which can then be used in kinetic models. This approach is also useful in the absence of irradiation, and a very interesting example has been obtained in the simulation of the first stages of the coherent precipitation of copper in bcc–Fe. DFT calculations predicted that the vacancyformation energy in metastable bcc–Cu (which is not known experimentally since bulk Cu is fcc) is 0.9 eV, that is, much smaller than that in bcc iron, namely 2.1 eV. This leads to strong trapping of vacancies by the Cu precipitates. As a result, precipitates containing up to several tens of copper atoms are quite surprisingly predicted to be much more mobile than individual copper atoms in the iron matrix.26 Another very illustrative example is given by the study of atomic transport via interstitials in dilute Fe–P alloys. DFT results indeed predict that Fe–P mixed dumbbells are highly mobile but that they can be deeply trapped by a substitutional P atom.79 A systematic study of the interaction of monovacancies and self-interstitials with all transition-metal solutes has been reported recently (see Figure 8).80 1.08.4.2.4 From dilute to concentrated alloys: the case of Fe–Cr
In the approach described earlier, which considers low concentrations of solutes and defects, the number of independent configurations is rather small, and they can be easily taken into account in kinetics model. The situation is much more complex when considering Fe–Cr with Cr concentration in the range 10–20%. Nevertheless, first results have been obtained by considering the interaction of defects with one or two Cr atoms in the Fe matrix.81 These data could ideally be used to fit an improved empirical potential, but the Fe–Cr system is rather difficult to model because of the strong interplay between magnetic and chemical interactions. This is also clearly one of the challenges in the field.
Figure 7 Structure of carbon–vacancy and carbon–self-interstitial complexes in iron, predicted from DFT calculations. Reproduced from Fu, C. C.; Meslin, E.; Barbu, A.; Willaime, F.; Oison, V. In Theory, Modeling and Numerical Simulation of Multi-Physics Materials Behavior, 2008; Vol. 139, pp 157–164, 168.
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Blochl14 Zunger et al.20 and Becquart and Domain55 Djurabekova et al.56 Fu and Willaime57 and Becquart and Domain58
ce
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st an
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Figure 8 DFT-GGA solute-vacancy binding energies in iron for 3D, 4D, and 5D elements for 1–5 nn relative positions. Reprinted with permission from Olsson, P.; Klaver, T. P. C.; Domain, C. Phys. Rev. B 2010, 81, 054102. Copyright (2010) by the American Physical Society.
1.08.4.2.5 Point defects in hcp-Zr
Point defects in hcp-Zr have also been studied via DFT calculations. It was found in particular that the vacancy migration energy is lower by 0.15 eV within the basal plane than out of the basal plane.82 The situation for the self-interstitial is quite complex, since among the known configurations, at least three configurations are found to have almost the same formation energy (within 0.1 eV): the octahedral (O), split dumbbell (S), and basal octahedral (BO) configurations.83,84 1.08.4.3
Dislocations
The collective behavior of dislocations can be described thanks to dislocation dynamics codes. In order to reinforce the physical foundation, input data such as mobility laws can be obtained from atomistic calculations of individual dislocations. These defects can now be investigated using more accurate ab initio electronics structure methods. We exemplify these studies by focusing in the following section on the properties of dislocations in bcc metals and especially
iron. In these materials, dislocation properties are known to be closely related to their core structure. When dealing with dislocations, special care should be taken in the positioning of the dislocations and in the boundary conditions of the calculations. For instance, considering h111i screw dislocations, the two cell geometries proposed in the literature – the cluster approach85 and the periodic array of dislocation dipoles86 – have been thoroughly compared.87 The calculations of dislocations are extremely demanding as they can include up to 800 atoms, so studies usually use fast codes such as SIESTA.8 The construction of simulation cells appropriate for such extended defects should be optimized for cell sizes accessible to DFT calculations, and the cell-size dependence of the energetics evidenced in both the cluster approach and the dipole approach for various cell and dipole vectors should be rationalized. The quadrupolar arrangement of dislocation dipoles is most widely used for such calculations87 although the cluster approach with flexible boundary conditions can be considered a reference method when no energies are necessary (i.e., only structures).
Ab Initio Electronic Structure Calculations for Nuclear Materials
DFT calculations in bcc metals such as Mo, Ta, Fe, and W85,87–91 predict a nondegenerate structure for the core, as illustrated in Figure 9 using differential displacement maps as proposed by Vitek.92 The edge component reveals the existence of a significant core dilatation effect in addition to the Volterra field, which can be successfully accounted for by an anisotropic elasticity model.93 Thanks to good control of energy, it is also possible to obtain quantitative results on the Peierls potential; namely, the 2D energy landscape seen by a straight screw dislocation as it moves perpendicular to the Burgers vector. This is exemplified in the following Figure 10(a), where a high symmetry direction of the Peierls potential is sampled: the line going between two easy core positions along the glide direction, that is, the Peierls barrier. These calculations
239
were performed by simultaneously displacing the two dislocations constituting the dislocation dipole in the same direction and by using a constrained relaxation method. In the same work, the behavior of the Ackland–Mendelev potential for iron,45 which gives the correct nondegenerate core structure unlike most other potentials, has been tested against the obtained DFT results. It appears that it compares well with the DFT results for the g-surfaces, but discrepancies exist on the deviation from anisotropic elasticity of both edge and screw components and on the Peierls potential. Indeed, the empirical potential results do not predict any dilatation elastic field exerted by the core. Besides, the Peierls barrier displayed by the Ackland–Mendelev potential yields a camel hump shape, as illustrated in Figure 10(a), and at the halfway position, the core spreads between
[110]
(a)
[111]
[112]
(b)
(c)
Figure 9 (a) Differential displacement map of the nondegenerate core structure of a <111> screw dislocation in Fe, as obtained from SIESTA GGA. (b) Same as (a) after subtraction of the Volterra anisotropic elastic field and magnified by a factor of 20. (c) Same as (b) for the displacement in the (111) plane (or edge component) and a magnification by a factor of 50. Reproduced from Ventelon, L.; Willaime, F. J. Comput. Aided Mater. Des. 2007, 14, 85–94. 200
100
40 30 20
SIESTA GGA Ackland Ackland–Mendelev Dudarev–Derlet
0
-100
10 0 0.0
(a)
SIESTA GGA SIESTA LDA Ackland–Mendelev
Energy (meV/b)
Energy barrier (meV/b)
50
0.2
0.4 0.6 Reaction coordinate
0.8
-200 0.0
1.0 (b)
0.2
0.4 0.6 Polarity
0.8
1.0
Figure 10 (a) Peierls barrier in Fe calculated with the Ackland–Mendelev potential45 and with SIESTA using the two exchange-correlation functionals, LDA and GGA. Reproduced from Ventelon, L.; Willaime, F. J. Comput. Aided Mater. Des. 2007, 14, 85–94. (b) Dependence of the dislocation core energy with the modulus of its polarization calculated using SIESTA and the three empirical potentials, namely, the Ackland,94 Ackland–Mendelev,45 and Dudarev–Derlet38 potentials. Reproduced from Ventelon, L.; Willaime, F. Philos. Mag. 2010, 90, 1063–1074.
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Ab Initio Electronic Structure Calculations for Nuclear Materials
two easy core positions, whereas it exhibits a single hump barrier within DFT and a nearly hard-core structure at halfway position. The effect of the exchange-correlation functional within DFT appears to be significant.87 More insight into the stability of the core structure can be gained by looking at the response of the polarization of the core, as represented in Figure 10(b). In the Ackland–Mendelev and DFT cases, these calculations confirm that the stable core is completely unpolarized, and they prove that there is no metastable polarized core.95 Finally, the methodology exists for calculating the structure and formation and migration energies of single kinks, but using it with DFT96 remains challenging because cells with about 1000 atoms are needed, together with a high accuracy.
1.08.5 Insulators From the atomistic and electronic structure point of view, it is legitimate to distinguish between electrically conducting materials on one hand and insulating or semiconducting materials on the other. Indeed, insulating materials exhibit specific behaviors, especially for the point defects. Due to the existence of a gap in the electronic density of states, the point defects may be charged. There is recent evidence that the properties of the point defects, especially their kinetic properties, such as the migration energy, depend a lot on their charge state. The charge of a given point defect depends on the position of the Fermi level within the band gap: a low lying Fermi level (close to the valence band) favors positively charged defects, whereas a Fermi level close to the conduction band favors negatively charged defects. The positions of the Fermi level corresponding to transitions between charge states are called charge transition levels (CTL). The correct determination of these CTL allows the correct prediction of the charge states of the defects, as piloted by Fermi level position, that is, the doping conditions for the material. Standard DFT methods fail to reproduce accurately these CTL, and the research of more accurate methods is presently a very active field in the electronic structure community, with major implications for microelectronic research as well as for nuclear materials, especially in view of the aforementioned variation of point defect kinetic properties with their charge state. All these charge aspects of point defects are completely out of range for empirical potentials.
In the last two sections, we exemplify the research on insulating materials by summarizing the available results for two important insulating nuclear materials: silicon carbide and uranium dioxide. Silicon carbide is an important candidate material for fusion and fission applications. Even if it arguably a less crucial material than UO2, we start with this material as its electronic structure is simpler. UO2 is obviously the basic model material for the nuclear fuel of usual reactors. 1.08.5.1
Silicon Carbide
This brief survey exemplifies the kind of calculations that can be performed on common insulating materials (as opposed to correlated ones such as UO2) in a nuclear context. Specificities of insulating materials when compared with metallic systems will clearly appear, especially for what concerns the possible charge states of the defects and the difficulties standard DFT calculations have in satisfactorily reproducing the quantities that govern them. SiC exists in many different structures. Nuclear applications are interested with the so-called b structure (3C–SiC), a zinc blende crystal cubic form. We shall therefore focus on this structure, although many additional calculations have been performed on other structures of the hexagonal type, which are more of interest for microelectronics applications. Silicon carbide is a band insulator whose bulk structural properties are well reproduced by usual DFT calculations. The electronic structure of the bulk material is also well reproduced except for the usual underestimation of the band gap by DFT calculations. Indeed, the measured gap is 2.39 eV,97 whereas standard DFT-LDA calculations give 1.30 eV.97 1.08.5.1.1 Point defects
The first DFT calculations of point defects in silicon carbide,98 dating back to 1988, were burdened by strong limitations in computing time. For this reason, they were performed with relatively small supercells (16 and 32 atoms), largely insufficient basis sets (plane waves with energy up to 28 Ry), and further approximations, namely for the relaxation of atomic positions. Moreover, they were limited to high symmetry configurations. The results were only qualitative; however, it was already clear that vacancies and antisites could be relatively abundant, at equilibrium, with respect to interstitial defects. The authors dared to approach some defect complexes and could predict that antisite pairs and divacancies were bound.
Ab Initio Electronic Structure Calculations for Nuclear Materials
Vacancies were thoroughly studied at the turn of the century.97,99–101 The most prominent result may be the metastability of the silicon vacancy. Indeed, following a suggestion coming from a self-consistent DFT-based tight-binding calculation by Rauls and coworkers,102 the electron paramagnetic resonance (EPR) spectra of annealed samples of irradiated SiC were measured103 and compared with calculated hyperfine parameters. This showed that silicon vacancies are metastable with regard to a carbon vacancy–carbon antisite complex (VC–CSi); a fact that has since been consistently confirmed by the other calculations. Interstitials were less studied than vacancies. One should however mention a study104 devoted to carbon and silicon in interstitials in silicon carbide. Beyond these studies dedicated to one type of defect, very complete and comprehensive work on both vacancies and interstitials was also published. One should cite Bernardini et al.105 devoted to the formation energies of defects, while Bockstedte et al.106 goes further as it also covers migration energetics of basic intrinsic defects (vacancies, interstitials, antisites). It is worth noting that in such covalent compounds there are many possible atomic structures for defects as simple as a monointerstitial and that all these structures must be considered in the calculation (see Figure 11). As examples, the results of these various studies on what concerns formation energies and CTL of vacancies are summarized in the following tables. One can see a general agreement in the formation energies of the neutral defects, especially in the recent references. The small differences are related to k-point sampling or cell size in the calculations. Larger discrepancies appear between the various predicted CTL. They relate to the inaccuracy of standard DFT calculations in treating empty or defect states.
Si
CTC
Front
A simple example relates directly to the underestimation of the band gap: the silicon interstitial (in the ISi TC configuration) in the neutral state shows up as metallic in standard calculations, the defect states lying inside the conduction band. This fact, on one hand, calls for a better description of the exchangecorrelation potential for these configurations; on the other, it makes the convergence with k points and cell size very slow, as has recently been pointed out.107 This drawback of standard DFT-LDA/GGA supercell calculations is common to other defects in SiC. Even when calculated defect states fall within the band gap, their position inside it can be grossly miscalculated with standard DFT calculations. The errors produced by standard DFT calculations for the CTL are well known nowadays. The determination of an accurate method to calculate these CTL is an active field of research with works on advanced methods such as GW (e.g., the results on SiO2108) or hybrid functionals.109 For what concerns nuclear materials, and especially SiC, GW corrections and excitonic effects will allow further comparisons with experiments (Table 1).110 1.08.5.1.2 Defect kinetics
Before the aforementioned work by Bockstedte and coworkers106 almost no work was devoted to migration properties of point defects in SiC. We should, however, cite previous preliminary works by the same group,111,112 a work on the mechanisms of formation of antisite pairs,113 and a work on vacancy migration published in 2003.114 The comprehensive study of migration barriers in Bockstedte et al.106 showed, first of all, that vacancies have much higher migration energies than those of interstitials: higher than 3 eV for the former in the neutral state, around 1 eV for the latter (0.5 for IC, 1.4 for ISi). Another
C
CTSi
CHex
241
Csp<110>
Csp<100>
CspSi<100>
Side
Figure 11 Possible geometries for a carbon interstitial in cubic SiC. Reprinted with permission from Bockstedte, M.; Mattausch, A.; Pankratov, O. Phys. Rev. B 2003, 68, 205201. Copyright (2003) by the American Physical Society.
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Ab Initio Electronic Structure Calculations for Nuclear Materials
Table 1
Formation energies for vacancies and their charge transition levels according to various authors VC
VSi
References
0
þ/0
þ/þþ
0
þ/0
0/
/
98 191 99 97 192 106 105
5.6 – 4.01 – 3.74 3.78 3.84
1.7 – 1.41 – 1.18 – –
1.9 – 1.72 – 1.22 1.29 –
7.6 – 8.74 7.7 8.37 8.34 8.78
– 0.54 0.43 0.50 – 0.18 0.41
– 1.06 1.11 0.56 0.57 0.61 0.88
– 1.96 1.94 1.22 1.60 1.76 1.40
The values are for the 3C-polytype in silicon-rich conditions. Values are expressed in eV.
remarkable finding is the strong variation of the migration energy with the charge state; indeed, the migration energy for the carbon vacancy is raised by almost 2 eV going from the neutral to the 2þ charged state, whereas the silicon vacancy finds its migration barrier reduced by 1 eV when its charge goes from neutral to 2. Interstitials are reported to have their lowest migration barriers in the neutral state, except for the ISi TC configuration, which is expected to have an almost zero energy barrier of migration in the 2þ and 3þ charge states. Such large changes in the migration energies of defects with their charge should induce tremendous variations in their kinetic behavior under different charge states. The energy barriers of recombinations of close interstitial vacancy pairs have also been tackled.115–117 It appears that the energetic landscape for the recombination of Frenkel pairs is extremely complex. One should distinguish the regular recombination of an homo interstitial-vacancy pairs from those of hetero interstitial-vacancy pairs, which leads to the formation of an antisite. Recent works tend to suggest that the latter may, in certain conditions, have a lower energy than the recombination of a regular Frenkel pair. A kinetic bias for the formation of antisites, preliminary to decomposition, may thus be active in SiC under irradiation.118 Calculations of threshold displacement energies from first-principles molecular dynamics29 have also been reported. Their results show that this quantity is strongly anisotropic, and they found average values (38 eV for Si and 19 eV for C) that are in agreement with currently accepted values (coming from experimental evaluations that are, however, largely dispersed). These calculations prove that available CPU power is now large enough to calculate TDE from ab initio molecular dynamics. This is good news as empirical potentials are basically not reliable in the prediction of TDE.
1.08.5.1.3 Defect complexes
Several defect complexes have been studied by first-principles calculations in silicon carbide. The identification of EPR signals, deep level transient spectroscopy (DLTS), or photoluminescence (PL) experiments based on calculated properties have been attempted for some of them. Crucial to these identifications is the reliability of the predictions of charge transition levels (for the position of DLTS peaks) and of annealing temperatures, through more or less complicated mechanisms. One of the first, and simplest, defect complex identified through comparison of theory and experiment was the VC–CSi coming from the annealing of silicon vacancies in 6H-SiC, as previously mentioned. More complex antisite defects or antisite complexes119,120 as well as divacancy complexes121–123 were called upon for the attribution of PL or EPR peaks. Various kinds of carbon clusters were studied in detail theoretically.124–127 The cited works deal with the stability, electrical properties, and local vibrational modes (LVM) of several structures. It was shown that the aggregation of carbon interstitials with carbon antisites can lead to various bound configurations. In particular, two, three, or even four carbon atoms can substitute one silicon atom forming very stable structures. The binding energy of these structures is high: from 3.9 to 5 eV, according to the charge state, for the (C2)Si, and further energy is gained when adding further carbon atoms. Silicon clusters did not raise as much interest as carbon ones; however, a recent work107 deals with the stability and dynamics of such silicon clusters (see Figure 12). 1.08.5.1.4 Impurities
The interest in SiC as a large band gap semiconductor for electronic applications has promoted works on typical dopants. Most of the calculations focus on hexagonal SiC, but one can reasonably assume that
Ab Initio Electronic Structure Calculations for Nuclear Materials
Ie2
Energy barrier (eV)
ITC + ITC
Ia2
Ie2
Ia2
Ib2
Ia2
Ic2
Ia2
Id2
ISisp<110> + I
243
Id2
3.0
3.0
2.5
2.5 2.0
2.0 Emigra(I) 1.5 Ebind(I2e)
1.5
Etrans(I2e-a)
1.0 0.5
Emigra(I2a)
Ebind(I2d)
Etrans(I2c-a) in-plane
Erotation(I2a)
Eopen(I2a-c)
out-plane
Erotation (I2a)
1.0 0.5 0.0
0.0
Reaction coordinate Figure 12 Energetic landscape of silicon mono- and di-interstitial in cubic SiC. Reproduced from Liao, T. (2009) Unpublished.
the results would not be very different in cubic SiC. One can find calculations dealing with boron129,130 as an acceptor and nitrogen131,132 or phosphorus133 as a donor. Other impurities were studied: transition metals,134–136 oxygen,137 important for the behavior of the SiO2/SiC interface, hydrogen,138–140 rare gases,141 and palladium.142 A systematic study of substitutional impurities has recently appeared,143 which focuses on the trends of carbon vs. silicon substitution according to the position of species in the periodic table. 1.08.5.1.5 Extended defects
Another major subject, which has attracted much interest for the hexagonal types of SiC, is related to the electronic properties of extended defects, surfaces/interfaces, stacking faults, and dislocations. The reason why extended defects have been mainly studied in the hexagonal types of silicon carbide lies in the fact that electronic properties of dislocations and stacking faults are particularly important for understanding the degradation of hexagonal SiC devices144 and the remarkable enhancement of dislocation velocity under illumination in the hexagonal phase.145 Nevertheless, some studies have been done for cubic SiC on the electronic structure of stacking faults146–151 and various types of dislocations.152–155 Obviously, a lot of work remains to be done on the extended defects in b SiC. 1.08.5.2
Uranium Oxide
1.08.5.2.1 Bulk electronic structure
Due to its technological importance and the complexity of its electronic structure, uranium oxide has become one of the test cases for beyond
LDA methods. Indeed, UO2 comes out as a metal when its electronic structure is calculated with LDA or GGA. This result has been found by many authors using many different codes or numerical schemes (the primary calculation being the work of Arko and coworkers156). The physical difficulty lies in the fact that UO2 is a Mott insulator. f electrons are indeed localized on uranium atoms and are not spread over the material as usual valence electrons are. The first correction that has been applied is the LDAþU correction in which a Hubbard U term acting between f electrons is added ‘by hand’ to the Hamiltonian.157,158 This method allows the opening of an f–f gap.157 However, it suffers from the existence of multiple minima in the calculations, so the search for the real ground state is rather tricky as the calculation is easily trapped in metastable states.159 Hybrid functionals are another type of advanced methods that are very often used nowadays in the quantum chemistry community. Their principle is to mix a part of Hartree–Fock exact exchange with a DFT calculation; it has been applied to UO2 has been made by Kudin et al.160 These methods are very promising for solid-state nuclear materials. However, the same problem of metastability as in LDAþU exists for such hybrid functionals,161 and the computational load is much heavier than that in common or LDAþU calculations. Recently, an alternative to LDAþU has been proposed: the so-called local hybrid functional for correlated electrons162 in which the hybrid functional is applied only to the problematic f electrons. An application on UO2 is available.163
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Ab Initio Electronic Structure Calculations for Nuclear Materials
1.08.5.2.2 Point defects
While UO2 comes out as a metal with LDA or GGA DFT calculations, its structural properties are quite well reproduced by these standard methods. Based on this observation, some studies, using this standard framework, have been published on point defects.164–166 The values obtained for the formation energies for the composite defects (oxygen and uranium Frenkel pairs and Schottky defect) compare well with experimental estimates. However, as UO2 is predicted to be a metal with such methods, it is impossible to consider the charge state of the defects. More recent studies using the þU correction have been published. Most of them still focus on neutral defects.159,167–169 The discrepancies between the results obtained in these various studies are larger than the spread usually observed in ab initio calculations; for instance, the formation energy of the oxygen Frenkel pair is found anywhere between 2.6170 and 6.5 eV.159 This suggests some hidden problem in the calculations, probably related to the possible occurrence of metastable minima in the calculations. We are aware of only one study of the charge state of point defects. This work,171 done within LDAþU, predicts the following charge states: 4 charge for uranium vacancy, 2 for oxygen interstitial, and from þ2 to 0 for oxygen vacancy depending on the position of the Fermi level. However, in this last work, the formation energies of the composite defects (Frenkel and Schottky) built from charged defects are in not as good agreement with experiments as the ones obtained with neutral defects. Together with the large spread of values mentioned here, this underestimation shows that the correct reproduction of point defects in UO2 with ab initio seems not yet at hand. Beyond the formation energy of isolated defects, some studies focus on their migration. Gupta and
coworkers172 calculated the migration energy of the oxygen vacancy (1.0 eV) and interstitial (1.1 eV). In this last case, they found that the stable position for a monointerstitial is a dumbbell configuration. This point is refuted by others173 who calculated the migration energy of oxygen mono- (0.81 eV) and di-interstials (0.47 eV) and implemented this information in a Kinetic Monte-Carlo model, showing that the di-interstitial configuration, though less abundant than the single interstitial, may play a dominant role in oxygen diffusion in hyperstoichiometric oxide. 1.08.5.2.3 Oxygen clusters
Another point of interest beyond point defects is the clustering of oxygen interstitials. Indeed, oxygen interstitial clustering has been deduced from diffraction experiments174 many years ago. However, a debate remains on the exact shape of such clusters. Two configurations are contemplated: the so-called Willis clusters174,175 or cubo-octahedral clusters that have been observed by neutron diffraction in U4O9176 and U3O7.177 These clusters are made of 12 oxygen and 8 uranium atoms and amount for 4 oxygen interstitials. An additional oxygen interstitial may reside in the center of the cluster, forming a so-called filled cube-octahedral cluster (with five interstitials). Recent calculations have proved that Willis clusters are in fact unstable and transform upon relaxation into assemblies of three or four interstitials surrounding a central vacancy cluster (Figure 13).178 The three interstitial–1 vacancy cluster has been found independently by other authors173,180 who refer to it as split di-interstitials. These clusters prove in fact to have a formation energy higher than the cube-octahedral cluster (Figure 14), especially the filled one.178,179
O
O
OV O⬘ U
U
O⬘⬘
OV
OV
O⬘⬘
(a)
(b)
Figure 13 Relaxation process of a Willis cluster of oxygen interstitials in UO2. Reproduced from Geng, H. Y.; Chen, Y.; Kaneta, Y.; Kinoshita, M. Appl. Phys. Lett. 2008, 93.
Ab Initio Electronic Structure Calculations for Nuclear Materials
Figure 14 Cubo-octahedral cluster of oxygen interstitials in UO2. Reprinted with permission from Geng, H. Y.; Chen, Y.; Kaneta, Y.; Kinoshita, M. Phys. Rev. B 2008, 77, 180101(R).180 Copyright (2008) by the American Physical Society.
1.08.5.2.4 Impurities
Lattice sites and solution energies of FP are of major importance in fundamental studies of nuclear fuels, see Chapter 2.20, Fission Product Chemistry in Oxide Fuels. They pilot the dependence of the behavior of FP on fuel stoichiometry and temperature as well as their possible release from the fuel in the context of a direct storage of spent fuel. As experimental studies in this field are very difficult, ab initio results are of great value. In such studies, one considers the insertion of a fission atom in interstitial or vacant sites of UO2. A difficulty arises for the latter case.181 Indeed, one then has to distinguish between the incorporation energy, defined as the energy to incorporate the FP in a preexisting vacancy site, and the solution energy, which is the one relevant for full thermodynamical equilibrium, in which the amount of available vacant site is taken into account. One then adds to the incorporation energy the so-called apparent formation energy, which is defined as the logarithm of the vacancy concentration multiplied by the temperature. Such apparent formation energies depend on the stoichiometry of UO2þx. A positive (respectively, negative) solution energy then means that the FP is insoluble (respectively, soluble) in UO2þx. The first DFT study of the incorporation of a FP in UO2 is the one by Petit et al.182 on krypton in the late 1990s. It was performed within the LMTO-ASA formalism, which could give only qualitative results.
245
Crocombette181 used more modern plane wave formalism to calculate the insertion of some FP (krypton, iodine, cesium, strontium, and helium) but neglected atomic relaxation, which limits the accuracy of the results. Freyss et al.183 considered He and Xe. All these calculations were performed with standard LDA. More recent works always included a þU correction. While the first papers dealt only with interstitial and monovacancy sites, more recent works may also consider divacancy or tri-vacancy sites that often appear to be the most stable sites for FPs. Many FPs have been recently considered. At the time of writing one could find in the literature, beyond the works already mentioned, calculations on helium,184 iodine,185 xenon,186 strontium,186 cesium,186–188 molybdenum,189 and zirconium.189 Yun et al.190 dealt with helium and went beyond the solution energies as they also considered migration and clustering energies.
1.08.6 Conclusion It is hoped that the examples discussed above have shown the tremendous interest of ab initio calculations for nuclear materials. Indeed, they allow the qualitative and most of the time quantitative calculations of the basic energetic and kinetic properties that have a major influence on the behavior of the materials at the atomic scale. For metallic materials, the common theoretical framework works quite well. One can thus nowadays tackle objects of increasing complexity, for example, assemblies of defects or dislocations. The main limit for these materials is the severe restriction in possible cell sizes. Silicon carbide exemplifies the successes of ab initio methods in modeling the properties of a band insulator of interest for the nuclear industry. However, some difficulties remain, especially for what concerns the correct prediction of CTL in these materials. In actinide materials, the case of uranium oxide, by far the most studied of the actinide compounds of interest as a nuclear material, shows that a lot of information can be obtained, for example, for the solution energies of FP or the structure of oxygen interstitial clusters. However, this information remains only qualitative, due to the very complex electronic structure of such actinide compounds with localized f electrons. The solution for these difficulties with insulating materials should come from the current developments of hybrid functional or GW calculations, with the drawback that these advanced methods
246
Ab Initio Electronic Structure Calculations for Nuclear Materials
are at least one order of magnitude heavier than the standard ones. Ab initio methods have thus brought a lot of information for nuclear materials and will certainly continue to do so. Conversely, nuclear materials are a very challenging field for the use of these ab initio methods in many aspects: physical principles, numerical schemes, practical implementation, and so on.
Acknowledgments
31. 32. 33. 34. 35. 36. 37. 38.
The authors thank Drs Fabien Bruneval, Chu Chun Fu, Guido Roma, and Lisa Ventelon for their valuable input.
39.
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1.09
Molecular Dynamics
W. Cai Stanford University, Stanford, CA, USA
J. Li University of Pennsylvania, Philadelphia, PA, USA
S. Yip Massachusetts Institute of Technology, Cambridge, MA, USA
ß 2012 Elsevier Ltd. All rights reserved.
1.09.1 1.09.2 1.09.3 1.09.3.1 1.09.4 1.09.5 1.09.5.1 1.09.5.2 1.09.6 1.09.6.1 1.09.6.1.1 1.09.6.1.2 1.09.6.2 1.09.6.2.1 1.09.6.2.2 1.09.7 References
Introduction Defining Classical MD Simulation Method The Interatomic Potential An Empirical Pair Potential Model Book-keeping Matters MD Properties Property Calculations Properties That Make MD Unique MD Case Studies Perfect Crystal Zero-temperature properties Finite-temperature properties Dislocation Peierls stress at zero temperature Mobility at finite temperature Perspective
Abbreviations bcc CSD EAM FS MD NMR nn NPT
Body-centered cubic structure Central symmetry deviation Embedded Atom Method potential Finnis–Sinclair potential Molecular dynamics simulation Nuclear Magnetic Resonance experiment Nearest-neighbor distance Ensemble in which number of atoms, pressure and temperature are constant NVE Ensemble in which number of atoms, volume and total energy are constant NVT Ensemble in which number of atoms, volume and temperature are constant PBC Periodic boundary condition
1.09.1 Introduction A concept that is fundamental to the foundations of Comprehensive Nuclear Materials is that of microstructural evolution in extreme environments. Given
249 250 252 252 253 255 255 255 256 256 257 258 259 259 261 262 264
the current interest in nuclear energy, an emphasis on how defects in materials evolve under conditions of high temperature, stress, chemical reactivity, and radiation field presents tremendous scientific and technological challenges, as well as opportunities, across the many relevant disciplines in this important undertaking of our society. In the emerging field of computational science, which may simply be defined as the use of advanced computational capabilities to solve complex problems, the collective contents of Comprehensive Nuclear Materials constitute a set of compelling and specific materials problems that can benefit from science-based solutions, a situation that is becoming increasingly recognized.1–4 In discussions among communities that share fundamental scientific capabilities and bottlenecks, multiscale modeling and simulation is receiving attention for its ability to elucidate the underlying mechanisms governing the materials phenomena that are critical to nuclear fission and fusion applications. As illustrated in Figure 1, molecular dynamics (MD) is an atomistic simulation method that can provide details of atomistic processes in microstructural evolution. 249
250
Molecular Dynamics
As the method is applicable to a certain range of length and time scales, it needs to be integrated with other computational methods to span the length and time scales of interest to nuclear materials.9 The aim of this chapter is to discuss in elementary terms the key attributes of MD as a principal method of studying the evolution of an assembly of atoms under well-controlled conditions. The introductory section is intended to be helpful to students and nonspecialists. We begin with a definition of MD, followed by a description of the ingredients that go into the simulation, the properties that one can calculate with this approach, and the reasons why the method is unique in computational materials research. We next examine results of case studies obtained using an open-source code to illustrate how one can study the structure and elastic properties of a perfect crystal in equilibrium and the mobility of an edge dislocation. We then return to Figure 1 to provide a perspective on the potential as well as the limitations of MD in multiscale materials modeling and simulation.
1.09.2 Defining Classical MD Simulation Method In the simplest physical terms, MD may be characterized as a method of ‘particle tracking.’ Operationally, it is a method for generating the trajectories of a system of N particles by direct numerical integration
of Newton’s equations of motion, with appropriate specification of an interatomic potential and suitable initial and boundary conditions. MD is an atomistic modeling and simulation method when the particles in question are the atoms that constitute the material of interest. The underlying assumption is that one can treat the ions and electrons as a single, classical entity. When this is no longer a reasonable approximation, one needs to consider both ion and electron motions. One can then distinguish two versions of MD, classical and ab initio, the former for treating atoms as classical entities (position and momentum) and the latter for treating separately the electronic and ionic degrees of freedom, where a wave function description is used for the electrons. In this chapter, we are concerned only with classical MD. The use of ab initio methods in nuclear materials research is addressed elsewhere (Chapter 1.08, Ab Initio Electronic Structure Calculations for Nuclear Materials). Figure 2 illustrates the MD simulation system as a collection of N particles contained in a volume O. At any instant of time t, the particle coordinates are labeled as a 3N-dimensional vector, r3N ðt Þ fr1 ðt Þ; r2 ðt Þ; . . . ; rN ðt Þg, where ri represents the three coordinates of atom i. The simulation proceeds with the system in a prescribed initial configuration, r3N ðt0 Þ, and velocity, r_ 3N ðt0 Þ, at time t ¼ t0 . As the simulation proceeds, the particles evolve through a sequence of time steps, r3N ðt0 Þ ! r3N ðt1 Þ ! where tk ¼ t0 þ kDt, r3N ðt2 Þ ! ! r3N ðtL Þ, k ¼ 1,2, . . ., L, and Dt is the time step of MD simulation. The simulation runs for L number of steps and covers a time interval of LDt . Typical values of L can range from 104 to 108 and Dt 1015 s. Thus, nominal MD simulations follow the system evolution over time intervals not more than 1–10 ns.
N y
Vj (t) rj (t)
z Figure 1 MD in the multiscale modeling framework of dislocation microstructure evolution. The experimental micrograph shows dislocation cell structures in Molybdenum.5 The other images are snapshots from computer models of dislocations.6–8
x
Figure 2 MD simulation cell is a system of N particles with specified initial and boundary conditions. The output of the simulation consists of the set of atomic coordinates r3N ðtÞ and corresponding velocities (time derivatives). All properties of the MD simulation are then derived from the trajectories, {r3N (t),r_ 3N(t)}.
Molecular Dynamics
The simulation system has a certain energy E, the sum of the kinetic and potential energies of the particles, E ¼ K þ U, where K is the sum of individual kinetic energies N 1 X v j vj K ¼ m 2 j ¼1
½1
and U ¼ U ðr Þ is a prescribed interatomic interaction potential. Here, for simplicity, we assume that all particles have the same mass m. In principle, the potential U is a function of all the particle coordinates in the system if we allow each particle to interact with all the others without restriction. Thus, the dependence of U on the particle coordinates can be as complicated as the system under study demands. However, for the present discussion we introduce an approximation, the assumption of a two-body or pair-wise additive interaction, which is sufficient to illustrate the essence of MD simulation. To find the atomic trajectories in the classical version of MD, one solves the equations governing the particle coordinates, Newton’s equations of motion in mechanics. For our N-particle system with potential energy U, the equations are 3N
m
d2 rj ¼ rrj U ðr3N Þ; j ¼ 1; . . . ; N dt 2
½2
where m is the particle mass. Equation [2] may look deceptively simple; actually, it is as complicated as the famous N-body problem that one generally cannot solve exactly when N is >2. As a system of coupled second-order, nonlinear ordinary differential equations, eqn [2] can be solved numerically, which is what is carried out in MD simulation. Equation [2] describes how the system (particle coordinates) evolves over a time period from a given initial state. Suppose we divide the time period of interest into many small segments, each being a time step of size Dt . Given the system conditions at some initial time t0, r3N ðt0 Þ, and r_ 3N ðt0 Þ, integration means we advance the system successively by increments of Dt , r3N ðt0 Þ ! r3N ðt1 Þ ! r3N ðt2 Þ ! ! r3N ðtL Þ
½3
where L is the number of time steps making up the interval of integration. How do we numerically integrate eqn [3] for a given U ? A simple way is to write a Taylor series expansion, rj ðt0 þ Dt Þ ¼ rj ðt0 Þ þ vj ðt0 ÞDt þ 1=2aj ðt0 ÞðDt Þ2 þ
½4
251
and a similar expansion for rj ðt0 Dt Þ. Adding the two expansions gives rj ðt0 þ Dt Þ ¼ rj ðt0 Dt Þ þ 2rj ðt0 Þ þ aj ðt0 ÞðDt Þ2 þ
½5
Notice that the left-hand side of eqn [5] is what we want, namely, the position of particle j at the next time step t0 þ Dt. We already know the positions at t0 and the time step before, so to use eqn [5] we need the acceleration of particle j at time t0. For this we substitute Fj ðr3N ðt0 ÞÞ=m in place of acceleration aj ðt0 Þ, where Fj is just the right-hand side of eqn [2]. Thus, the integration of Newton’s equations of motion is accomplished in successive time increments by applying eqn [5]. In this sense, MD can be regarded as a method of particle tracking where one follows the system evolution in discrete time steps. Although there are more elaborate, and therefore more accurate, integration procedures, it is important to note that MD results are as rigorous as classical mechanics based on the prescribed interatomic potential. The particular procedure just described is called the Verlet (leapfrog)10 method. It is a symplectic integrator that respects the symplectic symmetry of the Hamiltonian dynamics; that is, in the absence of floating-point round-off errors, the discrete mapping rigorously preserves the phase space volume.11,12 Symplectic integrators have the advantage of longterm stability and usually allow the use of larger time steps than nonsymplectic integrators. However, this advantage may disappear when the dynamics is not strictly Hamiltonian, such as when some thermostating procedure is applied. A popular time integrator used in many early MD codes is the Gear predictor– corrector method13 (nonsymplectic) of order 5. Higher accuracy of integration allows one to take a larger value of Dt so as to cover a longer time interval for the same number of time steps. On the other hand, the trade-off is that one needs more computer memory relative to the simpler method. A typical flowchart for an MD code11 would look something like Figure 3. Among these steps, the part that is the most computationally demanding is the force calculation. The efficiency of an MD simulation therefore depends on performing the force calculation as simply as possible without compromising the physical description (simulation fidelity). Since the force is calculated by taking the gradient of the potential U, the specification of U essentially determines the compromise between physical fidelity and computational efficiency.
252
Molecular Dynamics
Set particle positions
Assign particle velocities
For the nuclear motions, we consider an expansion of U in terms of one-body, two-body, . . . N-body interactions: U ðr3N Þ ¼
Save particle positions and velocities and other properties to file
þ
Reach preset time steps? Yes Save/analyze data and print results
Figure 3 Flow chart of MD simulation.
1.09.3 The Interatomic Potential This is a large and open-ended topic with an extensive literature.14 It is clear from eqn [2] that the interaction potential is the most critical quantity in MD modeling and simulation; it essentially controls the numerical and algorithmic simplicity (or complexity) of MD simulation and, therefore, the physical fidelity of the simulation results. Since Chapter 1.10, Interatomic Potential Development is devoted to interatomic potential development, we limit our discussion only to simple classical approximations to U ðr1 ; r2 ; . . . ; rN Þ. Practically, all atomistic simulations are based on the Born–Oppenheimer adiabatic approximation, which separates the electronic and nuclear motions.15 Since electrons move much more quickly because of their smaller mass, during their motion one can treat the nuclei as fixed in instantaneous positions, or equivalently the electron wave functions follow the nuclear motion adiabatically. As a result, the electrons are treated as always in their ground state as the nuclei move.
N X
N X
V2 ðri ; rj Þ
i<j
½6
V3 ðri ; rj ; rk Þ þ
i<j
The first term, the sum of one-body interactions, is usually absent unless an external field is present to couple with each atom individually. The second sum is the contribution of pure two-body interactions (pairwise additive). For some problems, this term alone is sufficient to be an approximation to U. The third sum represents pure three-body interactions, and so on. 1.09.3.1
No
V1 ðrj Þ þ
j ¼1
Calculate force on each particle
Update particle positions and velocities to next time step
N X
An Empirical Pair Potential Model
A widely adopted model used in many early MD simulations in statistical mechanics is the LennardJones (6-12) potential, which is considered a reasonable description of van der Waals interactions between closed-shell atoms (noble gas elements, Ne, Ar, Kr, and Xe). This model has two parameters that are fixed by fitting to selected experimental data. One should recognize that there is no one single physical property that can determine the entire potential function. Thus, using different data to fix the model parameters of the same potential form can lead to different simulations, making quantitative comparisons ambiguous. To validate a model, it is best to calculate an observable property not used in the fitting and compare with experiment. This would provide a test of the transferability of the potential, a measure of robustness of the model. In fitting model parameters, one should use different kinds of properties, for example, an equilibrium or thermodynamic property and a vibrational property to capture the low- and high-frequency responses (the hope is that this would allow a reasonable interpolation over all frequencies). Since there is considerable ambiguity in what is the correct method of fitting potential models, one often has to rely on agreement with experiment as a measure of the goodness of potential. However, this could be misleading unless the relevant physics is built into the model. For a qualitative understanding of MD essentials, it is sufficient to assume that the interatomic
Molecular Dynamics
potential U can be represented as the sum of twobody interactions X V ðrij Þ ½7 U ðr1 ; . . . ; rN Þ ffi i<j
where rij jri rj j is the separation distance between particles i and j. V is the pairwise additive interaction, a central force potential that is a function of only the scalar separation distance between the two particles, rij . A two-body interaction energy commonly used in atomistic simulations is the Lennard-Jones potential V ðr Þ ¼ 4e½ðs=r Þ12 ðs=r Þ6
½8
where e and s are the potential parameters that set the scales for energy and separation distance, respectively. Figure 4 shows the interaction energy rising sharply when the particles are close to each other, showing a minimum at intermediate separation and decaying to zero at large distances. The interatomic force F ðr Þ
dV ðr Þ dr
½9
is also sketched in Figure 4. The particles repel each other when they are too close, whereas at large separations they attract. The repulsion can be understood as arising from overlap of the electron clouds, whereas the attraction is due to the interaction between the induced dipole in each atom. The value of 12 for the first exponent in V(r) has no special significance, as the repulsive term could just as well be replaced by an exponential. The value of 6 for the second exponent comes from quantum mechanical calculations (the so-called London dispersion force) and therefore
V(r) F(r) rc
o
s
e
nn
ro
r
2nn
Figure 4 The Lennard–Jones interatomic potential V(r). The potential vanishes at r ¼ s and has a depth equal to e. Also shown is the corresponding force F(r) between the two particles (dashed curve), which vanishes at r0 ¼ 21=6 s. At separations less or greater than r0, the force is repulsive or attractive, respectively. Arrows at nn and 2nn indicate typical separation distances of nearest and second nearest neighbors in a solid.
253
is not arbitrary. Regardless of whether one uses eqn [8] or some other interaction potential, a short-range repulsion is necessary to give the system a certain size or volume (density), without which the particles will collapse onto each other. A long-range attraction is also necessary for cohesion of the system, without which the particles will not stay together as they must in all condensed states of matter. Both are necessary for describing the physical properties of the solids and liquids that we know from everyday experience. Pair potentials are simple models that capture the repulsive and attractive interactions between atoms. Unfortunately, relatively few materials, among them the noble gases (He, Ne, Ar, etc.) and ionic crystals (e.g., NaCl), can be well described by pair potentials with reasonable accuracy. For most solid engineering materials, pair potentials do a poor job. For example, all pair potentials predict that the two elastic constants for cubic crystals, C12 and C44, must be equal to each other, which is certainly not true for most cubic crystals. Therefore, most potential models for engineering materials include many-body terms for an improved description of the interatomic interaction. For example, the Stillinger–Weber potential16 for silicon includes a three-body term to stabilize the tetrahedral bond angle in the diamond-cubic structure. A widely used typical potential for metals is the embedded-atom method17 (EAM), in which the many-body effect is introduced in a so-called embedding function.
1.09.4 Book-keeping Matters Our simulation system is typically a parallelepiped supercell in which particles are placed either in a very regular manner, as in modeling a crystal lattice, or in some random manner, as in modeling a gas or liquid. For the simulation of perfect crystals, the number of particles in the simulation cell can be quite small, and only certain discrete values, such as 256, 500, and 864, should be specified. These numbers pertain to a facecentered-cubic crystal that has four atoms in each unit cell. If our simulation cell has l unit cells along each side, then the number of particles in the cube will be 4l3. The above numbers then correspond to cubes with 4, 5, and 6 cells along each side, respectively. Once we have chosen the number of particles we want to simulate, the next step is to choose the system density we want to study. Choosing the density is equivalent to choosing the system volume since density r ¼ N =O, where N is the number of particles
254
Molecular Dynamics
and O is the supercell volume. An advantage of the Lennard-Jones potential is that one can work in dimensionless reduced units. The reduced density rs3 has typical values of about 0.9–1.2 for solids and 0.6–0.85 for liquids. For reduced temperature kB T =e, the values are 0.4 – 0.8 for solids and 0.8–1.3 for liquids. Notice that assigning particle velocities according to the Maxwellian velocity distribution probability ¼ ðm=2pkB T Þ3=2 exp½mðvx2 þ vy2 þ vz2 Þ=2kB T dvx dvy dvz is tantamount to setting the system temperature T. For simulation of bulk properties (system with no free surfaces), it is conventional to use the periodic boundary condition (PBC). This means that the cubical simulation cell is surrounded by 26 identical image cells. For every particle in the simulation cell, there is a corresponding image particle in each image cell. The 26 image particles move in exactly the same manner as the actual particle, so if the actual particle should happen to move out of the simulation cell, the image particle in the image cell opposite to the exit side will move in and become the actual particle in the simulation cell. The net effect is that particles cannot be lost or created. It follows then that the particle number is conserved, and if the simulation cell volume is not allowed to change, the system density remains constant. Since in the pair potential approximation, the particles interact two at a time, a procedure is needed to decide which pair to consider among the pairs between actual particles and between actual and image particles. The minimum image convention is a procedure in which one takes the nearest neighbor to an actual particle as the interaction partner, regardless of whether this neighbor is an actual particle or an image particle. Another approximation that is useful in keeping the computations to a manageable level is the introduction of a force cutoff distance beyond which particle pairs simply do not see each other (indicated as rc in Figure 4). In order to avoid a particle interacting with its own image, it is necessary to set the cutoff distance to be less than half of the simulation cell dimension. Another book-keeping device often used in MD simulation is a neighbor list to keep track of who are the nearest, second nearest, . . . neighbors of each particle. This is to save time from checking every particle in the system every time a force calculation is made. The list can be used for several time steps before updating. In low-temperature solids where the particles do not move very much, it is possible to do an entire simulation without, or with only a few, updating, whereas in simulation of liquids, updation every 5 or 10 steps is common.
If one uses a naı¨ve approach in updating the neighbor list (an indiscriminate double loop over all particles), then it will get expensive for more than a few thousand particles because it involves N N operations for an N-particle system. For short-range interactions, where the interatomic potential can be safely taken to be zero outside of a cutoff rc, accelerated approaches exist that can reduce the number of operations from order-N2 to order-N. For example, in the so-called ‘cell lists’ approach,18 one partitions the supercell into many smaller cells, and each cell maintains a registry of the atoms inside (order-N operation). The cell dimension is chosen to be greater than rc, so an atom cannot possibly interact with more than one neighbor atom. This will reduce the number of operations in updating the neighbor list to order-N. With the so-called Parrinello–Rahman method,19 the supercell size and shape can change dynamically during a MD simulation to equilibrate the internal stress with the externally applied constant stress. In these simulations, the supercell is generally nonorthogonal, and it becomes much easier to use the so-called scaled coordinates sj to represent particle positions. The scaled coordinates sj are related to the real coordinates rj through the relation, rj ¼ H sj , when both rj and sj are written as column vectors. H is a 3 3 matrix whose columns are the three repeat vectors of the simulation cell. Regardless of the shape of the simulation cell, the scaled coordinates of atoms can always be mapped into a unit cube, ½0; 1Þ ½0; 1Þ ½0; 1Þ. The shape change of the simulation cell with time can be accounted for by including the matrix H into the equation of motion. A ‘cell lists’ algorithm can still be worked out for a dynamically changing H, which minimizes the number of updates.13 For modeling ionic crystals, the long-range electrostatic interactions must be treated differently from short-ranged interactions (covalent, metallic, van der Waals, etc.). This is because a brute-force evaluation of the electrostatic interaction energies involves computation between all ionic pairs, which is of the order N2, and becomes very timeconsuming for large N. The so-called Ewald summation20,21 decomposes the electrostatic interaction into a short-ranged component, plus a long-ranged component, which, however, can be efficiently summed in the reciprocal space. It reduces the computational time to order N3/2. The particle mesh Ewald22–24 method further reduces the computational time to order N log N.
Molecular Dynamics N L0 1X 1X vi ðtk Þ vi ðtk þ t Þ hvð0Þ vðt Þi ¼ N i¼1 L0 k¼1
1.09.5 MD Properties 1.09.5.1
Property Calculations
Let hAi denote a time average over the trajectory generated by MD, where A is a dynamical variable, A(t). Two kinds of calculations are of common interest, equilibrium single-point properties and timecorrelation functions. The first is a running time average over the MD trajectories ðt 1 dt 0 Aðt 0 Þ hA i ¼ lim t !1 t
½10
o
with t taken to be as long as possible. In terms of discrete time steps, eqn [10] becomes hA i ¼
L 1X Aðtk Þ L k¼1
½11
where L is the number of time steps in the trajectory. The second is a time-dependent quantity of the form L 1X Aðtk ÞBðtk þ t Þ L0 k¼1 0
hAð0ÞBðt Þi ¼
½12
where B is in general another dynamical variable, and L0 is the number of time origins. Equation [12] is called a correlation function of two-dynamical variables; since it is manifestly time dependent, it is able to represent dynamical information of the system. We give examples of both types of averages by considering the properties commonly calculated in MD simulation. * + N X V ðrij Þ potential energy ½13 U¼ i<j
* + N X 1 mi v i vi T¼ 3NkB i¼1
temperature
½14
0 1+ * N X @V ðrij Þ 1 X @mi vi vi P¼ r A pressure ½15 @rij ij 3O i¼1 j >i
* + N X X 1 gðr Þ ¼ dðr jri rj jÞ r4pr 2 N i¼1 j 6¼i
½16
radial distribution function
MSDðt Þ ¼
N 1X jri ðt Þ ri ð0Þj2 N i¼1
mean squared displacement
½17
255
½18
velocity autocorrelation function sab ¼
X va i
O
siab ; siab
* + X @V ðrij Þrij a rij b 1 ¼ mvia vib þ @rij rij va j >i
½19
Virial stress tensor In eqn [19], va is the average volume of one atom, via is the a-component of vector vi, and rij a is the a-component of vector ri rj . The interest in writing the stress tensor in the present form is to suggest that the macroscopic tensor can be decomposed into individual atomic contributions, and thus siab is known as the atomic level stress25 at atom i. Although this interpretation is quite appealing, one should be aware that such a decomposition makes sense only in a nearly homogeneous system where every atom ‘owns’ almost the same volume as every other atom. In an inhomogeneous system, such as in the vicinity of a surface, it is not appropriate to consider such decomposition. Both eqns [15] and [19] are written for pair potential models only. A slightly different expression is required for potentials that contain many-body terms.26 1.09.5.2
Properties That Make MD Unique
A great deal can be said about why MD is a useful simulation technique. Perhaps the most important statement is that, in this method, one follows the atomic motions according to the principles of classical mechanics as formulated by Newton and Hamilton. Because of this, the results are physically as meaningful as the potential U that is used. One does not have to apologize for any approximation in treating the N-body problem. Whatever mechanical, thermodynamic, and statistical mechanical properties that a system of N particles should have, they are all present in the simulation data. Of course, how one extracts these properties from the simulation output – the atomic trajectories – determines how useful the simulation is. We can regard MD simulation as an ‘atomic video’ of the particle motion (which can be displayed as a movie), and how to extract the information in a scientifically meaningful way is up to the viewer. It is to be expected that an experienced viewer can get much more useful information than an inexperienced one.
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Molecular Dynamics
The above comments aside, we present here the general reasons why MD simulation is useful (or unique). These are meant to guide the thinking of the nonexperts and encourage them to discover and appreciate the many significant aspects of this simulation technique. (a) Unified study of all physical properties. Using MD, one can obtain the thermodynamic, structural, mechanical, dynamic, and transport properties of a system of particles that can be studied in a solid, liquid, or gas. One can even study chemical properties and reactions that are more difficult and will require using quantum MD, or an empirical potential that explicitly models charge transfer.27 (b) Several hundred particles are sufficient to simulate bulk matter. Although this is not always true, it is rather surprising that one can get quite accurate thermodynamic properties such as equation of state in this way. This is an example that the law of large numbers takes over quickly when one can average over several hundred degrees of freedom. (c) Direct link between potential model and physical properties. This is useful from the standpoint of fundamental understanding of physical matter. It is also very relevant to the structure–property correlation paradigm in material science. This attribute has been noted in various general discussions of the usefulness of atomistic simulations in material research.28–30 (d) Complete control over input, initial and boundary conditions. This is what provides physical insight into the behavior of complex systems. This is also what makes simulation useful when combined with experiment and theory. (e) Detailed atomic trajectories. This is what one obtains from MD, or other atomistic simulation techniques, that experiment often cannot provide. For example, it is possible to directly compute and observe diffusion mechanisms that otherwise may be only inferred indirectly from experiments. This point alone makes it compelling for the experimentalist to have access to simulation. We should not leave this discussion without reminding ourselves that there are significant limitations to MD as well. The two most important ones are as follows: (a) Need for sufficiently realistic interatomic potential functions U. This is a matter of what we really know fundamentally about the chemical binding of the system we want to study. Progress is being
made in quantum and solid-state chemistry and condensed-matter physics; these advances will make MD more and more useful in understanding and predicting the properties and behavior of physical systems. (a) Computational-capability constraints. No computers will ever be big enough and fast enough. On the other hand, things will keep on improving as far as we can tell. Current limits on how big and how long are a billion atoms and about a microsecond in brute force simulation. A billion-atom MD simulation is already at the micrometer length scale, in which direct experimental observations (such as transmission electron microscopy) are available. Hence, the major challenge in MD simulations is in the time scale, because most of the processes of interest and experimental observations are at or longer than the time scale of a millisecond.
1.09.6 MD Case Studies In the following section, we present a set of case studies that illustrate the fundamental concepts discussed earlier. The examples are chosen to reflect the application of MD to mechanical properties of crystalline solids and the behavior of defects in them. More detailed discussions of these topics, especially in irradiated materials, can be found in Chapter 1.11, Primary Radiation Damage Formation and Chapter 1.12, Atomic-Level Level Dislocation Dynamics in Irradiated Metals. 1.09.6.1
Perfect Crystal
Perhaps the most widely used test case for an atomistic simulation program, or for a newly implemented potential model, is the calculation of equilibrium lattice constant a0, cohesive energy Ecoh, and bulk modulus B. Because this calculation can be performed using a very small number of atoms, it is also a widely used test case for first-principle simulations (see Chapter 1.08, Ab Initio Electronic Structure Calculations for Nuclear Materials). Once the equilibrium lattice constants have been determined, we can obtain other elastic constants of the crystal in addition to the bulk modulus. Even though these calculations are not MD per se, they are important benchmarks that practitioners usually perform, before embarking on MD simulations of solids. This case study is discussed in Section 1.09.6.1.1.
Molecular Dynamics
Following the test case at zero temperature, MD simulations can be used to compute the mechanical properties of crystals at finite temperature. Before computing other properties, the equilibrium lattice constant at finite temperature usually needs to be determined first, to account for the thermal expansion effect. This case study is discussed in Section 1.09.6.1.2. 1.09.6.1.1 Zero-temperature properties
In this test case, let us consider a body-centered cubic (bcc) crystal of Tantalum (Ta), described by the Finnis–Sinclair (FS) potential.31 The calculations are performed using the MDþþ program. The source code and the input files for this and subsequent test cases in this chapter can be downloaded from http://micro.stanford.edu/wiki/Comprehensive_ Nuclear_Materials_MD_Case_Studies. The cut-off radius of the FS potential for Ta is 4.20 A˚. To avoid interaction between an atom with its own periodic images, we consider a cubic simulation cell whose size is much larger than the cut-off radius. The cell dimensions are 5[100], 5[010], and 5[001] along x, y, and z directions, and the cell contains N ¼ 250 atoms (because each unit cell of a bcc crystal contains two atoms). PBC are applied in all three directions. The experimental value of the equilibrium lattice constant of Ta is 3.3058 A˚. Therefore, to compute the equilibrium lattice constant of this potential model, we vary the lattice constant a from 3.296 to 3.316 A˚, in steps of 0.001 A˚. The potential energy per atom E as a function of a is plotted in Figure 5. The data can be fitted to a parabola. The
−8.0990 −8.0992
E (eV)
−8.0994 −8.0996 −8.0998 −8.1 3.295
3.300
3.305
3.310
3.315
a0 (Å) Figure 5 Potential energy per atom as a function of lattice constant of Ta. Circles are data computed from the FS potential, and the line is a parabola fitted to the data.
257
location of the minimum is the equilibrium lattice constant, a0 ¼ 3.3058 A˚. This exactly matches the experimental data because a0 is one of the fitted parameters of the potential. The energy per atom at a0 is the cohesive energy, Ecoh ¼ 8.100 eV, which is another fitted parameter. The curvature of parabolic curve at a0 gives an estimate of the bulk modulus, B ¼ 197.2 GPa. However, this is not a very accurate estimate of the bulk modulus because the range of a is still too large. For a more accurate determination of the bulk modulus, we need to compute the E(a) curve again in the range of ja a0 j<104 A˚. The curvature of the E(a) curve at a0 evaluated in the second calculation gives B ¼ 196.1 GPa, which is the fitted bulk modulus value of this potential model.31 When the crystal has several competing phases (such as bcc, face-centered cubic, and hexagonalclosed-packed), plotting the energy versus volume (per atom) curves for all the phases on the same graph allows us to determine the most stable phase at zero temperature and zero pressure. It also allows us to predict whether the crystal will undergo a phase transition under pressure.32 Other elastic constants besides B can be computed using similar approaches, that is, by imposing a strain on the crystal and monitoring the changes in potential energy. In practice, it is more convenient to extract the elastic constant information from the stress–strain relationship. For cubic crystals, such as Ta considered here, there are only three independent elastic constants, C11, C12, and C44. C11 and C12 can be obtained by elongating the simulation cell in the x-direction, that is, by changing the cell length into L ¼ ð1 þ exx Þ L0 , where L0 ¼ 5a0 in this test case. This leads to nonzero stress components sxx, syy, szz, as computed from the Virial stress formula [19], as shown in Figure 6 (the atomic velocities are zero because this calculation is quasistatic). The slope of these curves gives two of the elastic constants C11 ¼ 266.0 GPa and C12 ¼ 161.2 GPa. These results can be compared with the bulk modulus obtained from potential energy, due to the relation B ¼ ðC11 þ 2C12 Þ=3 ¼ 196:1GPa. C44 can be obtained by computing the shear stress sxy caused by a shear strain exy. Shear strain exy can be applied by adding an off-diagonal element in matrix H that relates scaled and real coordinates of atoms. 2 3 L0 2exy L0 0 ½20 0 5 L0 H¼4 0 0 0 L0
258
Molecular Dynamics
30
sxx
Stress (MPa)
20
syy
10
sxy
0 −10 −20 −30
−1
0 Strain
1 ⫻ 10−4
Figure 6 Stress–strain relation for FS Ta: sxx and syy as functions of exx and sxy as a function of exy .
The slope of the shear stress–strain curve gives the elastic constant C44 ¼ 82.4 GPa. In this test case, all atoms are displaced according to a uniform strain, that is, the scaled coordinates of all atoms remain unchanged. This is correct for simple crystal structures where the basis contains only one atom. For complex crystal structures with more than one basis atom (such as the diamond-cubic structure of silicon), the relative positions of atoms in the basis set will undergo additional adjustments when the crystal is subjected to a macroscopically uniform strain. This effect can be captured by performing energy minimization at each value of the strain before recording the potential energy or the Virial stress values. The resulting ‘relaxed’ elastic constants correspond well with the experimentally measured values, whereas the ‘unrelaxed’ elastic constants usually overestimate the experimental values. 1.09.6.1.2 Finite-temperature properties
Starting from the perfect crystal at equilibrium lattice constant a0, we can assign initial velocities to the atoms and perform MD simulations. In the simplest simulation, no thermostat is introduced to regulate the temperature, and no barostat is introduced to regulate the stress. The simulation then corresponds to the NVE ensemble, where the number of particles N, the cell volume V (as well as shape), and total energy E are conserved. This simulation is usually performed as a benchmark to ensure that the numerical integrator is implemented correctly and that the time step is small enough. The instantaneous temperature T inst is defined in terms of the instantaneous kinetic energy K
through the relation K ð3N =2ÞkB T inst , where kB is Boltzmann’s constant. Therefore, the velocity can be initialized by assigning random numbers to each component of every atom and scaling them so that T inst matches the desired temperature. In practice, T inst is usually set to twice the desired temperature for MD simulations of solids, because approximately half of the kinetic energy flows to the potential energy as the solids reach thermal equilibrium. We also need to subtract appropriate constants from the x, y, z components of the initial velocities to make sure the center-of-mass linear momentum of the entire cell is zero. When the solid contains surfaces and is free to rotate (e.g., a nanoparticle or a nanowire), care must be taken to ensure that the center-of-mass angular momentum is also zero. Figure 7(a) plots the instantaneous temperature as a function of time, for an MD simulation starting with a perfect crystal and T inst ¼ 600 K, using the Velocity Verlet integrator13 with a time step of Dt ¼ 1 fs. After 1 ps, the temperature of the simulation cell is equilibrated around 300 K. Due to the finite time step Dt, the total energy E, which should be a conserved quantity in Hamiltonian dynamics, fluctuates during the MD simulation. In this simulation, the total energy fluctuation is <2 10–4 eV per atom, after equilibrium has been reached (t > 1 ps). There is also zero long-term drift of the total energy. This is an advantage of symplectic integrators11,12 and also indicates that the time step is small enough. The stress of the simulation cell can be computed by averaging the Virial stress for time between 1 and 10 ps. A hydrostatic pressure P ðsxx þ syy þ szz Þ=3 ¼ 1:33 0:01GPa is obtained. The compressive stress develops because the crystal is constrained at the zero-temperature lattice constant. A convenient way to find the equilibrium lattice constant at finite temperature is to introduce a barostat to adjust the volume of the simulation cell. It is also convenient to introduce a thermostat to regulate the temperature of the simulation cell. When both the barostat and thermostat are applied, the simulation corresponds to the NPT ensemble. The Nose–Hoover thermostat11,33,34 is widely used for MD simulations in NVT and NPT ensembles. However, care must be taken when applying it to perfect crystals at medium-to-low temperatures, in which the interaction between solid atoms is close to harmonic. In this case, the Nose–Hoover thermostat has difficulty in correctly sampling the equilibrium distribution in phase space, as indicated by periodic oscillation of the instantaneous temperature.
259
Molecular Dynamics
2
2
350
0
p (GPa)
Tinst (K)
1
400
p (GPa)
Tinst (K)
600
300 200
250 0
0
0.5
1
(a)
1.5 t (ps)
2
2.5
0 3
0 (b)
20
40
60
80
−2 100
t (ps)
Figure 7 (a) Instantaneous temperature Tinst and Virial pressure p as functions of time in an NVE simulation with initial temperature at 600 K. (b) Tinst and P in a series of NVT at T ¼ 300 K, where the simulation cell length L is adjusted according to the averaged value of P.
The Nose–Hoover chain35 method has been developed to address this problem. The Parrinello–Rahman19 method is a widely used barostat for MD simulations. However, periodic oscillations in box size are usually observed during equilibration of solids. This oscillation can take a very long time to die out, requiring an unreasonably long time to reach equilibrium (after which meaningful data can be collected). A viscous damping term is usually added to the box degree of freedom to accelerate the speed of equilibration. Here, we avoid the problem by performing a series of NVT simulations, each one lasting for 1 ps using the Nose–Hoover chain method with Velocity Verlet integrator and Dt ¼ 1 fs. Before starting each new simulation, the simulation box is subjected to an additional hydrostatic elastic strain of e ¼ hPi=B 0 , where hPi is the average Virial pressure of the previous simulation, where B 0 ¼ 2000GPa is an empirical parameter. The instantaneous temperature and Virial pressure during 100 of these NVT simulations are plotted in Figure 7(b). The instantaneous temperature fluctuates near the desired temperature (300 K) nearly from the beginning of the simulation. The Virial pressure is well relaxed to zero at t ¼ 20 ps. The average box size from 50 to 100 ps is L ¼ 16.5625 A˚, which is larger than the initial value of 16.5290 A˚. This means that the normal strain caused by thermal expansion at 300 K is exx ¼ 0.00203. Hence, the coefficient of thermal expansion is estimated to be a ¼ exx =T ¼ 6:8 106 K1 :35 1.09.6.2
Dislocation
Dislocations are line defects in crystals, and their motion is the carrier for plastic deformation of
crystals under most conditions (T < Tm/2).36,37 The defects produced by irradiation (such as vacancy and interstitial complexes) interact with dislocations, and this interaction is responsible for the change in the mechanical properties by irradiation (such as embrittlement).38 MD simulations of dislocation interaction with other defects are discussed in detail in Chapter 1.12, Atomic-Level Level Dislocation Dynamics in Irradiated Metals. Here, we describe a more basic case study on the mobility of an edge dislocation in Ta. In Section 1.09.6.2.1, we describe the method of computing its Peierls stress, which is the critical stress to move the dislocation at zero temperature. In Section 1.09.6.2.2, we describe how to compute the mobility of this dislocation at finite temperature by MD. 1.09.6.2.1 Peierls stress at zero temperature
Dislocations in the dominant slip system in bcc metals have h111i=2 Burgers vectors and {110} slip planes. Here, we consider an edge dislocation with Burgers vector b ¼ 1=2½111 (along x-axis), slip plane normal ½110 (along y-axis), and line direction ½1 12 (along z-axis). To prepare the atomic configuration, we first create a perfect crystal with dimensions 30½111, 40½110, 2½ 1 12 along the x-, y-, z-axes. We then remove one-fourth of the atomic layers normal to the y-axis to create two free surfaces, as shown in Figure 8(a). We introduce an edge dislocation dipole into the simulation cell by displacing the positions of all atoms according to the linear elasticity solution of the displacement field of a dislocation dipole. To satisfy PBC, the displacement field is the sum of the contributions from not only the dislocation dipole
260
Molecular Dynamics
Dislocation
Cut plane
b
y
(a)
x
(b)
Figure 8 (a) Schematics showing the edge dislocation dipole in the simulation cell. b is the dislocation Burgers vector of the upper dislocation. Atoms in shaded regions are removed. (b) Core structure of edge dislocation (at the center) and surface atoms in FS Ta after relaxation visualized by Atomeye.41 Atoms are colored according to their central-symmetry deviation parameter. Adapted from Li, J. In Handbook of Materials Modeling; Yip, S., Ed.; Springer: Dordrecht, 2005; pp 1051–1068; Mistake-free version at http://alum.mit.edu/www/liju99/Papers/05/Li05-2.31.pdf; Kelchner, C. L.; Plimpton, S. J.; Hamilton, J. C. Phys. Rev. B 1998, 58, 11085.
inside the cell, but also its periodic images. Care must be taken to remove the spurious term caused by the conditional convergence of the sum.26,40–42 Because the Burgers vector b is perpendicular to the cutplane connecting the two dislocations in the dipole, atoms separated from the cut-plane by <jbj=2 in the x-direction need to be removed. The resulting structure contains 21 414 atoms. The structure is subsequently relaxed to a local energy minimum with zero average stress. Because one of the two dislocations in the dipole is intentionally introduced into the vacuum region, only one dislocation remains after the relaxation, as shown in Figure 8(b). The dislocation core is identified by central symmetry analysis,13 which characterizes the degree of inversion-symmetry breaking. In Figure 8(b), only atoms with a central symmetry deviation (CSD) parameter larger than 1.5 A˚2 are plotted. Atoms with CSD parameter between 0.6 and 6 A˚2 appear at the center of the cell and are identified with the dislocation core. Atoms with a CSD parameter between 10 and 20 A˚2 appear at the top and bottom of the cell and are identified with the free surfaces. The edge dislocation thus created will move along the x-direction when the shear stress sxy exceeds a critical value. To compute the Peierls stress, we apply shear stress sxy by adding external forces on surface atoms. The total force on the top surface
atoms points in the x-direction and has magnitude of Fx ¼ sxy Lx Lz . The total force on the bottom surface atoms has the same magnitude but points in the opposite direction. These forces are equally distributed on the top (and bottom) surface atoms. Because we have removed some atoms when creating the edge dislocation, the bottom surface layer has fewer atoms than the top surface layer. As a result, the external force on each atom on the top surface is slightly lower than that on each atom on the bottom surface. We apply shear stress sxy in increments of 1 MPa and relax the structure using the conjugate gradient algorithm at each stress. The dislocation (as identified by the core atoms) does not move for sxy < 27 MPa but moves in the x-direction during the relaxation at sxy ¼ 28 MPa. Therefore, this simulation predicts that the Peierls stress of edge dislocation in Ta (FS potential) is 28 1 MPa. The Peierls stress computed in this way can depend on the simulation cell size. Therefore, we will need to repeat this calculation for several cell sizes to obtain a more reliable prediction of the Peierls stress. There are other boundary conditions that can be applied to simulate dislocations and compute the Peierls stress, such as PBCs in both x- and y-directions,42 and the Green’s function boundary condition.44 Different boundary conditions have different size dependence on the numerical error of the Peierls stress. The simulation cell in this study contains two free surfaces and one dislocation. This is designed to minimize the effect of image forces from the boundary conditions on the computed Peierls stress. If the surfaces were not created, the simulation cell would have to contain at least two dislocations so that the total Burgers vector content was zero. On application of the stress, the two dislocations in the dipole would move in opposite directions, and the total energy would vary as a function of their relative position. This would create forces on the dislocations, in addition to the Peach–Koehler force from the applied stress, and would lead to either overestimation or underestimation of the Peierls stress. On the contrary, the simulation cell described above has only one dislocation, and as it moves to an equivalent lattice site in the x-direction, the energy does not change due to the translational symmetry of the lattice. This means that, by symmetry, the image force on the dislocation from the boundary conditions is identically zero, which leads to more accurate Peierls stress predictions. However, when the simulation cell is too small, the free surfaces in the y-direction
Molecular Dynamics
and the periodic images in the x-direction can still introduce (second-order) effects on the critical stress for dislocation motion, even though they do not produce any net force on the dislocation. 1.09.6.2.2 Mobility at finite temperature
The relaxed atomic structure from Section 1.09.6.2.1 at zero stress can be used to construct initial conditions for MD simulations for computing dislocation mobility at finite temperature. The dislocation in Section 1.09.6.2.1 is periodic along its length (z-axis) with a relatively short repeat distance ð2½1 12Þ. In a real crystal, the fluctuation of the dislocation line can be important for its mobility. Therefore, we extend the simulation box length by five times along z-axis by replicating the atomic structure before starting the MD simulation. Thus, the MD simulation cell has dimensions 30[111], 40½110, 10½1 12 along the x, y, z axes, respectively, and contains 10 7070 atoms. In the following section, we compute the dislocation velocity at several shear stresses at T ¼ 300 K. For simplicity, the simulation in which the shear stress is applied is performed under the NVT ensemble. However, the volume of the simulation cell needs to be adjusted from the zero-temperature value to accommodate the thermal expansion effect. The cell dimensions are adjusted by a series of NVT simulations using an approach similar to that used in Section 1.09.6.1.2, except that exx , eyy, ezz are allowed to adjust independently. As we have found in Section 1.09.6.1.2 that for a perfect crystal, the thermal strain at 300 K is e ¼ 0.00191, exx , eyy , ezz are initialized to this value at the beginning of the equilibration.
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After the equilibration for 10 ps, we perform MD simulation under different shear stresses sxy up to 100 MPa. The simulations are performed under the NVT chain method using the Velocity Verlet algorithm with Dt ¼ 1 fs. The shear stress is applied by adding external forces on surface atoms, in the same way as in Section 1.09.6.2.1. The atomic configurations are saved periodically every 1 ps. For each saved configuration, the CSD parameter45 of each atom is computed. Due to thermal fluctuation, certain atoms in the bulk can also have CSD values exceeding 0.6 A˚2. Therefore, only the atoms whose CSD value is between 4.5 and 10.0 A˚2 are classified as dislocation core atoms. Figure 9(a) plots the average position hxi of dislocation core atoms as a function of time at different applied stresses. Due to PBC in x-direction, it is possible to have certain core atoms at the left edge of the cell with other core atoms at the right edge of the cell, when the dislocation core moves to the cell border. In this case, we need to ensure that all atoms are within the nearest image of one another, when computing their average position in x-direction. When the configurations are saved frequently enough, it is impossible for the dislocation to move by more than the box length in the x-direction since the last time the configuration was saved. Therefore, the average dislocation position hxi at a given snapshot is taken to be the nearest image of the average dislocation position at the previous snapshot so that the hxiðt Þ plots in Figure 9(a) appear as smooth curves. Figure 9(a) shows that all the hxiðt Þ curves at t ¼ 0 have zero slope and nonzero curvature,
100 MPa
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Figure 9 (a) Average position of dislocation core atoms as a function of time at different shear stresses. (b) Dislocation velocity as a function of sxy at T ¼ 300 K.
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indicating that the dislocation is accelerating. Eventually, hxi becomes a linear function of t, indicating that the dislocation has settled down into steady-state motion. The dislocation velocity is computed from the slope of the hxiðt Þ in the second half of the time period. Figure 9(b) plots the dislocation velocity obtained in this way as a function of the applied shear stress. The dislocation velocity appears to be a linear function of stress in the low stress limit, with mobility M ¼ v=ðsxy bÞ ¼ 2:6 104 Pa1 s1. Dislocation mobility is one of the important material input parameters to dislocation dynamics (DD) simulations.46–48 For accurate predictions of the dislocation velocity and mobility, MD simulations must be performed for a long enough time to ensure that steady-state dislocation motion is observed. The simulation cell size also needs to be varied to ensure that the results have converged to the large cell limit. For large simulation cells, parallel computing is usually necessary to speed up the simulation. The LAMMPS program49 (http://lammps.sandia.gov) developed at Sandia National Labs is a parallel simulation program that has been widely used for MD simulations of solids.
1.09.7 Perspective The previous sections give a hands-on introduction to the basic techniques of MD simulation. More involved discussions of the technical aspects may be found in the literature.30 Here, we offer comments on several enduring attributes of MD from the standpoint of benefits and drawbacks, along with an outlook on future development. MD has an unrivalled ability for describing material geometry, that is, structure. The Greek philosopher Democritus (ca. 460 BCE–370 BCE) recognized early on that the richness of our world arose from an assembly of atoms. Even without very sophisticated interatomic potentials, a short MD simulation run will place atoms in quite ‘reasonable’ locations with respect to each other so that their cores do not overlap. This does not mean that the atomic positions are correct, as there could be multiple metastable configurations, but it provides reasonable guesses. Unlike some other simulation approaches, MD is capable of offering real geometric surprises, that is to say, providing new structures that the modeler would never have expected before the simulation run. For this reason, visualization of atomistic
structure at different levels of abstraction is very important, and there are several pieces of free software for this purpose.13,50,51 As the ball-and-stick model of DNA by Watson and Crick52 was nothing but an educated guess based on atomic radii and bond angles, MD simulations can be regarded as ‘computational Watson and Crick’ that are potentially powerful for structural discovery. This remarkable power is both a blessing and a curse for modelers, depending on how it is harnessed. Remember that Watson and Crick had X-ray diffraction data against which to check their structural model. Therefore, it is very important to check the MD-obtained structures against experiments (diffraction, high-resolution transmission electron microscopy, NMR, etc.) and ab initio calculations whenever one can. Another notable allure of MD simulations is that it creates a ‘perfect world’ that is internally consistent, and all the information about this world is accessible. If MD simulation is regarded as a numerical experiment, it is quite different from real experiments, which all practitioners know are ‘messy’ and involve extrinsic factors. Many of these extrinsic factors may not be well controlled, or even properly identified, for instance, moisture in the carrier gas, initial condition of the sample, the effects of vibration, thermal drift, and so on. The MD ‘world’ is much smaller, with perfectly controlled initial conditions and boundary conditions. In addition, real experiments can only probe a certain aspect, a small subset of the properties, while MD simulation gives the complete information. When the experimental result does not work out as expected, there could be extraneous factors, such as a vacuum leak, impurity in the reagents, and so on that could be very difficult to trace back. In contrast, when a simulation gives a result that is unexpected, there is always a way to understand it, because one has complete control of the initial conditions, boundary conditions, and all the intermediate configurations. One also has access to the code itself. A simulation, even if a wrong one (with bugs in the program), is always repeatable. Not so with actual experiments. It is certainly true that any interatomic potential used in an MD simulation has limitations, which means the simulation is always an approximation of the real material. It also can happen that the limitations are not as serious as one might think, such as in establishing a conceptual framework for fundamental mechanistic studies. This is because the value of MD is much greater than simply calculating material
Molecular Dynamics
parameters. MD results can contribute a great deal towards constructing a conceptual framework and some kind of analytical model. Once the conceptual framework and analytical model are established, the parameters for a specific material may be obtained by more accurate ab initio calculations or more readily by experiments. It would be bad practice to regard MD simulation primarily as a black box that can provide a specific value for some property, without a deeper analysis of the trajectories and interpretation in light of an appropriate framework. Such a framework, external to the MD simulation, is often broadly applicable to a variety of materials; for example, the theory and expressions of solute strengthening in alloys based on segregation in the dislocation core. If solute strengthening occurs in a wide variety of materials, then it should also occur in ‘computer materials.’ Indeed, the ability to parametrically tune the interatomic potential, to see which energetic aspect is more important for a specific behavior or property, is a unique strength of MD simulations compared with experiments. One might indeed argue that the value of science is to reduce the complex world to simpler, easier-to-process models. If one wants only all the unadulterated complexity, one can just look at the world without doing anything. Thus, the main value of simulation should not be in the final result but also in the process, and the role of simulations should be to help simplify and clarify, not just to reproduce, the complexity. According to this view, the problem with a specific interatomic potential is not that it does not work, but that it is not known which properties the potential can describe and which it cannot, and why. There are also fundamental limitations in the MD simulation method that deserve comment. The algorithm is entirely classical, that is, it is Newtonian mechanics. As such, it misses relativistic and quantum effects. Below the Debye temperature,53 quantum effects become important. The equipartition theorem from classical statistical mechanics, stating that every degree of freedom possesses kBT/2 kinetic energy, breaks down for the high-frequency modes at low temperatures. In addition to thermal uncertainties in a particle’s position and momentum, there are also superimposed quantum uncertainties (fluctuations), reflected by the zero-point motion. These effects are particularly severe for light-mass elements such as hydrogen.54 There exist rigorous treatments for mapping the equilibrium thermodynamics of a quantum system to a classical dynamics system. For instance, path-integral molecular dynamics
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(PIMD)55,56 can be used to map each quantum particle to a ring of identical classical particles connected by Planck’s constant-dependent springs to represent quantum fluctuations (the ‘multi-instance’ classical MD approach). There are also approaches that correct for the quantum heat capacity effect with single-instance MD.53,57 For quantum dynamical properties outside of thermal equilibrium, or even for evaluating equilibrium time-correlation functions, the treatment based on an MD-like algorithm becomes even more complex.58–60 It is well recognized in computational material research that MD has a time-scale limitation. Unlike viscous relaxation approaches that are first order in time, MD is governed by Newtonian dynamics that is second order in time. As such, inertia and vibration are essential features of MD simulation. The necessity to resolve atomic-level vibrations requires the MD time step to be of the order picosecond/100, where a picosecond is the characteristic time period for the highest-frequency oscillation mode in typical materials, and about 100 steps are needed to resolve a full oscillation period with sufficient accuracy. This means that the typical timescale of MD simulation is at the nanosecond level, although with massively parallel computer and linear-scaling parallel programs such as LAMMPS,49 one may push the simulations to microsecond to millisecond level nowadays. A nanosecond-level MD simulation is often enough for the convergence of physical properties such as elastic constants, thermal expansion, free energy, thermal conductivity, and so on. However, chemical reaction processes, diffusion, and mechanical behavior often depend on events that are ‘rare’ (seen at the level of atomic vibrations) but important, for instance, the emission of a dislocation from grain boundary or surface.61 There is no need to track atomic vibrations, important as they are, for time periods much longer than a nanosecond for any particular atomic configuration. Important conceptual and algorithmic advances were made in the so-called Accelerated Molecular Dynamics approaches,62–66 which filter out repetitive vibrations and are expected to become more widely used in the coming years. . . . Above all, it seems to me that the human mind sees only what it expects.
These are the words of Emilio Segre´ (Noble Prize in Physics, 1959, for the discovery of the antiproton) in a historical account of the discovery of nuclear fission by O. Hahn and F. Strassmann,67 which led to a Nobel Prize in Chemistry, 1944, for Hahn. Prior to the
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discovery, many well-known scientists had worked on the problem of bombarding uranium with neutrons, including Fermi in Rome, Curie in Paris, and Hahn and Meitner in Berlin. All were looking for the production of transuranic elements (elements heavier than uranium), and none were open minded enough to recognize the fission reaction. As atomistic simulation can be regarded as an ‘atomic camera,’ it would be wise for anyone who wishes to study nature through modeling and simulation to keep an open mind when interpreting simulation results.
14. 15. 16. 17. 18. 19. 20. 21. 22.
Acknowledgments W. Cai appreciates the assistance from Keonwook Kang and Seunghwa Ryu in constructing the case studies and acknowledges support by NSF grant CMS-0547681, AFOSR grant FA9550-07-1-0464, and the Army High Performance Computing Research Center at Stanford. J. Li acknowledges support by NSF grant CMMI-0728069 and DMR1008104, MRSEC grant DMR-0520020, and AFOSR grant FA9550-08-1-0325.
23. 24. 25. 26. 27. 28. 29. 30. 31.
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Gue´rin, Y.; Was, G. S.; Zinkle, S. J. MRS Bull. 2009, 34(1), 10–19. Basic Research Needs for Advanced Nuclear Energy Systems: Report of the Basic Energy Sciences Workshop on Basic Research Needs for Advanced Nuclear Energy Systems; U.S. Department of Energy Office of Basic Energy Sciences, 2006. Simulation Based Engineering Science – Revolutionizing Engineering Science Through Simulation; National Science Foundation, 2006. Science Based Nuclear Energy Systems Enabled by Advanced Modeling and Simulation at the Extreme Scale; U.S. Department of Energy’s Offices of Science and Nuclear Energy, 2009. Kopetskii, C. V.; Pashkovskii, A. I. Phys. Stat. Solidif. A 1974, 21. Marian, J.; Cai, W.; Bulatov, V. V. Nat. Mater. 2004, 3, 158. Bulatov, V. V.; Cai, W. Phys. Rev. Lett. 2002, 89, 115501. Bulatov, V. V.; Hsiung, L. L.; Tang, M.; et al. Nature 2006, 440, 1174. Integrated Computational Materials Engineering: A Transformational Discipline for Improved Competitiveness and National Security; National Research Council, 2008. Verlet, L. Phys. Rev. 1967, 159(1), 98–103. Frenkel, D.; Smit, B. Understanding Molecular Simulation: From Algorithms to Applications, Academic Press: New York, 2002. Yoshida, H. Phys. Lett. A 1990, 150(5–7), 262–268. Li, J. In Handbook of Materials Modeling; Yip, S., Ed.; Springer, 2005; pp 565–588; Mistake free version at http://alum.mit.edu/www/liju99/Papers/05/Li05-2.8.pdf.
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Finnis, M. Interatomic Forces in Condensed Matter; Oxford University Press: Oxford, 2003. Born, M.; Oppenheimer, R. Ann. Phys. 1927, 84(20), 457–484. Stillinger, F. H.; Weber, T. A. Phys. Rev. B 1985, 31, 5262–5271. Daw, M. S.; Baskes, M. I. Phys. Rev. B 1984, 29, 6443–6453. Allen, M. P.; Tildesley, D. J., Computer Simulation of Liquids; Clarendon Press: New York, 1987. Parrinello, M.; Rahman, A. J. Appl. Phys. 1981, 52(12), 7182–7190. Ewald, P. P. Ann. Phys. 1921, 64(3), 253–287. de Leeuw, S. W.; Perram, J. W.; Smith, E. R. Proc. Roy. Soc. Lond. A 1980, 373, 27–56. Darden, T.; York, D.; Pedersen, L. J. Chem. Phys. 1993, 98, 10089–10093. Essmann, U.; Perera, L.; Berkowitz, M. L.; Darden, T.; Lee, H.; Pedersen, L. G. J. Chem. Phys. 1995, 103(19), 8577–8593. Deserno, M.; Holm, C. J. Chem. Phys. 1998, 109(18), 7678–7701. Srolovitz, D.; Vitek, V.; Egami, T. Acta Metall. 1983, 31(2), 335–352. Bulatov, V. V.; Cai, W. Computer Simulations of Dislocations; Oxford University Press: Oxford, 2006. Streitz, F. H.; Mintmire, J. W. Phys. Rev. B 1994, 50(16), 11996–12003. Yip, S. In Molecular-Dynamics Simulation of StatisticalMechanical Systems; Ciccotti, G.; Hoover, W. G., Eds.; North-Holland: Amsterdam, 1986; pp 523–561. Yip, S. Nat. Mater. 2003, 2(1), 3–5. Yip, S. Handbook of Materials Modeling; Springer: Dordrecht, 2005. Finnis, M. W.; Sinclair, J. E. Philos. Mag. A Phys.Condens. Matter Struct. Defects Mech. Prop. 1984, 50(1), 45–55. Muller, M.; Erhart, P.; Albe, K. J. Phys. Condens. Matter 2007, 19(32), 326220–326243. Nose, S. Mol. Phys. 1984, 52(2), 255–268. Hoover, W. G. Phys. Rev. A 1985, 31(3), 1695–1697. Martyna, G. J.; Klein, M. L.; Tuckerman, M. J. Chem. Phys. 1992, 97(4), 2635–2643. Ryu, S.; Cai, W. Model. Simul. Mater. Sci. Eng. 2008, 16(8), 085005–085017. Frost, H. J.; Ashby, M. F. Deformation-Mechanism Maps; Pergamon Press: Oxford, 1982. Hirth, J. P.; Lothe, J. Theory of Dislocations; Wiley: New York, 1982; 2nd edn. de la Rubia, T. D.; Zbib, H. M.; Khraishi, T. A.; Wirth, B. D.; Victoria, M.; Caturla, M. J. Nature 2000, 406(6798), 871–874. Li, J. Model. Simul. Mater. Sci. Eng. 2003, 11(2), 173–177. Cai, W.; Bulatov, V. V.; Chang, J. P.; Li, J.; Yip, S. Phys. Rev. Lett. 2001, 86(25), 5727–5730. Cai, W.; Bulatov, V. V.; Chang, J. P.; Li, J.; Yip, S. Philos. Mag. 2003, 83(5), 539–567. Li, J.; Wang, C. Z.; Chang, J. P.; Cai, W.; Bulatov, V. V.; Ho, K. M.; Yip, S. Phys. Rev. B 2004, 70(10), 104113–104121. Woodward, C.; Rao, S. Philos. Mag. A 2001, 81, 1305–1316. Kelchner, C. L.; Plimpton, S. J.; Hamilton, J. C. Phys. Rev. B 1998, 58, 11085–11088. van der Giessen, E.; Needleman, A. Model. Simul. Mater. Sci. Eng. 1995, 3, 689–735. Tang, M.; Kubin, L. P.; Canova, G. R. Acta Mater. 1998, 46, 3221–3235. Cai, W.; Bulatov, V. V.; Chang, J.; Li, J.; Yip, S., In Dislocations in Solids; Nabarro, F. R. N.; Hirth, J. P., Eds.; Elsevier: Amsterdam, 2004; Vol. 12, pp 1–80.
Molecular Dynamics 49. Plimpton, S. J. Comput. Phys. 1995, 117(1), 1–19. 50. Bhattarai, D.; Karki, B. B. J. Mol. Graph. 2009, 27(8), 951–968. 51. Stukowski, A. Model. Simul. Mater. Sci. Eng. 2010, 18(1), 015012. 52. Watson, J. D.; Crick, F. H. C. Nature 1953, 171, 737. 53. Li, J.; Porter, L.; Yip, S. J. Nucl. Mater. 1998, 255(2–3), 139–152. 54. Mills, G.; Jonsson, H.; Schenter, G. K. Surf. Sci. 1995, 324(2–3), 305–337. 55. Chandler, D.; Wolynes, P. G. J. Chem. Phys. 1981, 74(7), 4078–4095. 56. Sprik, M. In Computer Simulation in Materials Science: Interatomic Potentials, Simulation Techniques and Applications; Meyer, M., Pontikis, V., Eds.; Kluwer: Dordrecht, 1991; pp 305–320. 57. Dammak, H.; Chalopin, Y.; Laroche, M.; Hayoun, M.; Greffet, J. J. Phys. Rev. Lett. 2009, 103(19), 190601.
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Jang, S.; Voth, G. A. J. Chem. Phys. 1999, 111(6), 2357–2370. Miller, W. H. J. Phys. Chem. A 2001, 105(13), 2942–2955. Poulsen, J. A.; Nyman, G.; Rossky, P. J. Proc. Natl. Acad. Sci. USA 2005, 102(19), 6709–6714. Zhu, T.; Li, J.; Samanta, A.; Leach, A.; Gall, K. Phys. Rev. Lett. 2008, 100(2), 025502–025506. Voter, A. F. Phys. Rev. Lett. 1997, 78(20), 3908–3911. Voter, A. F.; Montalenti, F.; Germann, T. C. Annu. Rev. Mater. Res. 2002, 32, 321–346. Laio, A.; Parrinello, M. Proc. Natl. Acad. Sci. USA 2002, 99(20), 12562–12566. Miron, R. A.; Fichthorn, K. A. J. Chem. Phys. 2003, 119(12), 6210–6216. Hara, S.; Li, J. Phys. Rev. B 2010, 82(18), 184114–184121. Segre, E. G. Phys. Today 1989, 42(7), 38–43.
1.10
Interatomic Potential Development
G. J. Ackland University of Edinburgh, Edinburgh, UK
ß 2012 Elsevier Ltd. All rights reserved.
1.10.1
Introduction
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1.10.2 1.10.3 1.10.4 1.10.4.1 1.10.4.2 1.10.4.2.1 1.10.4.2.2 1.10.4.2.3 1.10.4.2.4 1.10.4.2.5 1.10.4.2.6 1.10.5 1.10.5.1 1.10.5.2 1.10.5.3 1.10.6 1.10.6.1 1.10.6.2 1.10.6.3 1.10.7 1.10.7.1 1.10.7.2 1.10.7.3 1.10.7.4 1.10.7.5 1.10.8 1.10.8.1 1.10.8.2 1.10.8.3 1.10.8.4 1.10.9 1.10.10 1.10.10.1 1.10.10.2 1.10.10.3 1.10.11 1.10.12 1.10.12.1 1.10.12.2 1.10.12.2.1 1.10.12.2.2 1.10.12.3
Basics Hard Spheres and Binary Collision Approximation Pair Potentials LJ Phase Diagram Necessary Results with Pair Potentials Outward surface relaxation Melting points Vacancy formation energy Cauchy pressure High-pressure phases Short ranged From Quantum Theory to Potentials Free Electron Theory Nearly Free Electron Theory Embedded Atom Methods and Density Functional Theory Many-Body Potentials and Tight-Binding Theory Energy of a Part-Filled Band The Moments Theorem Key Points Properties of Glue Models Crystal Structure Surface Relaxation Cauchy Relations Vacancy Formation Alloys Two-Band Potentials Fitting the s–d Band Model Magnetic Potentials Nonlocal Magnetism Three-Body Interactions Modified Embedded Atom Method Potentials for Nonmetals Covalent Potentials Molecular Force Fields Ionic Potentials Short-Range Interactions Parameterization Effective Pair Potentials and EAM Gauge Transformation Example: Parameterization for Steel FeCr FeC Austenitic Steel
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1.10.13 1.10.13.1 1.10.14 References
Analyzing a Million Coordinates Useful Concepts Without True Physical Meaning Summary
Abbreviations BCA bcc DFT DOS EAM fcc FS GGA hcp LDA LJ MD MEAM NFE
Binary collision approximation Body-centered cubic Density functional theory Density of electronic states Embedded atom method Face-centered cubic Finnis–Sinclair Generalized gradient approximation for exchange and correlation Hexagonal closed packed Local density approximation for exchange and correlation Lennard Jones Molecular dynamics Modified embedded atom method Nearly free electron
Symbols Ec Cohesive energy KB Boltzmann constant KF Fermi wavevector
1.10.1 Introduction Nuclear materials are subject to irradiation, and their behavior is therefore not that of thermodynamic equilibrium. To describe the behavior that leads to radiation damage at a fundamental level, one must follow the trajectories of the atoms. Since millions of atoms may be involved in a single event, this must be done by numerical simulation, either molecular dynamics (MD) or kinetic Monte Carlo. For either of these, a description of the energy is needed: this is the interatomic potential. Now that accurate quantum mechanical force calculations are available, one might ask whether there is still a role for atomistic potentials. In practice, ab initio calculations are currently limited to a few hundred atoms and a few picoseconds, or a few thousands at T ¼ 0 K. With MD and interatomic potentials, one can run calculations of millions of atoms for nanoseconds. With
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kinetic Monte Carlo calculation, timescales extend for years. These methods still provide the only method of atomistic simulation in these regimes. The issue is not whether they are still necessary; rather it is which of their predictions are correct. Before embarking on any simulation, it is essential to consider whether the potential contains enough of the right physics to describe the problem at hand. For example, in studying copper, one might ask about the following: Color (total electronic structure) Conductivity (electronic structure around Fermi level) Crystal structure (ground state electronic structure) Freezing mechanism (bond formation and crystal structure) Dislocation dynamics (stacking fault energy) Surface structures (bond breaking energies) Interaction with primary radiation (short-range atom–atom interactions) Phonon spectrum (curvature of potential) Corrosion (chemical reaction) In each case, a totally different aspect of physics is required. An essential issue for empirical potentials is transferability: can potentials fitted to one set of properties describe another? In general they cannot, so one has to be careful to use a potential in the type of application that it was intended for. For many physical problems such as dislocation dynamics, swelling, fracture, segregation, and phase transitions, much of the physics is dominated by geometry. Here, finite size effects are more important than accurate energetics and empirical potentials find their home.
1.10.2 Basics Materials relevant to reactors are held together by electrons. An interatomic potential expresses energy in terms of atomic positions: the electronic and magnetic degrees of freedom are integrated out. Most empirical potentials are derived on the basis of some approximation to quantum mechanical energies. If they are subsequently used in MD of solids, then what
Interatomic Potential Development
is actually used are forces: the derivative of the energy. For near-harmonic solids, it is actually the second derivative of the energy that governs the behavior. A dilemma: Does the primary term covering energetics also dominate the second derivative of the energy? To take an extreme example, an equation which calculates the energy of a solid exactly for all configurations to 0.1% is: E ¼ mc 2 : most of the energy is in the rest mass of the atoms. But this is patently useless for calculating condensed matter properties. We encounter the same problem in a less extreme form in metals: should we concentrate the energy gained in delocalizing the electrons to form the metal, or is treating perturbations around the metallic state more useful? In general, the issue is ‘What is the reference state.’ Most potentials implicitly assume that the free atom is the reference state.
1.10.3 Hard Spheres and Binary Collision Approximation The simplest atomistic model is the binary collision approximation (BCA), which simulates cascades as a series of atomic collisions. The ‘potential’ is then basically a hard sphere, with collisions either elastic or inelastic and the possibility of adding friction to describe an ion’s progress through an electron field. There is no binding energy, so the condensed phase is stabilized due to confinement by the boundary conditions. This is a poor approximation, since the structures and energies of the equilibrium crystal and defects do not correspond to real material. However, it allows for quick calculation and the scaling of defect production with energy of the primary knock-on atom. The best-known code for this type of calculation is probably Oak Ridge’s MARLOWE.1
1.10.4 Pair Potentials Pairwise potentials are the next level in complexity beyond BCA. They allow soft interactions between particles and simultaneous interaction between many atoms. Pair potentials always have some parameters that relate to a particular material, requiring fitting to experimental data. This immediately introduces the question of whether the reference state should be free atoms (e.g., in argon), free ions (e.g., NaCl), or ions embedded in an electron gas (e.g., metals). The two classic pair potentials used for modeling are the Lennard Jones (LJ) and Morse potential. Each
269
consists of a short-range repulsion and a long-range attraction and has two adjustable parameters. s 12 s6 VLJ ¼ e r r VMorse ¼ e½2e2ar ear s6 cohesion is on the basis of van der Waals The r interactions, while the ear is motivated by a screenedCoulomb potential. The repulsive terms were invented ad hoc. Because they have only two parameters, all simulations using just LJ or Morse are equivalent, the values of e, s, or a , which simply rescale energy and length. Hence, these potentials cannot be fitted to other properties of particular materials. Both LJ and Morse potentials stabilize closepacked crystal structures, and both have unphysically low basal stacking fault energies. Equivalently, the energy difference between fcc and hcp is smaller than for any real material. For materials modeling, this introduces a problem that the (110) dislocation structure is split into two essentially independent partials. For radiation damage, this means that configurations such as stacking fault tetrahedra are overstabilized, and unreasonably large numbers of stacking faults can be generated in cascades, fracture, or deformation. If the energy scale is set by the cohesive energy of a transition metal, then the vacancy and interstitial formation energy tend to be far too large; if the vacancy energy is fitted, then the cohesive energy is too small. For binary systems, these potentials can stabilize a huge range of crystal structures, even without explicit temperature effects; some progress has been made to delineate these, but it is far from complete.2 Once one moves to N-species systems, there are e and s parameters for each combination of particles, that is, N(N þ 1) parameters. Now it is possible to fit to properties of different materials, and the rapid increase in parameters illustrates the combinatorial problem in defining potentials for multicomponent systems. 1.10.4.1
LJ Phase Diagram
In practice, pair potentials are cut off at a certain range, which can have a surprising effect on stability as shown in Figure 1 While the LJ fluid is very well studied, the finite temperature crystal structure has only recently been resolved. The problem is that the fcc and hcp structures are extremely close in energy (see Figure 1(b)), so the entropy must be calculated extremely accurately.
270
Interatomic Potential Development
400 500 600 700 800 900 1000
300
200
0.5
100
0
ps 3/e
fcc 0.4
kT/e
0.3 hcp Harmonic N = 6 Harmonic N = 123 Harmonic+fit N = 123 LS-NPT N = 63 LS-NVT N = 63 LS-NPT N = 123 Salsburg and Huckaby 3
0.2
2
3
rc/s
4
5
6
Figure 1 Energy difference between hcp and fcc for the Lennard-Jones potential at 0 K, as a function of cutoff (rc ) with either simple truncation or with the potential shifted to remove the energy discontinuity at the cutoff. Without truncation the difference is 0.0008695 e, with hcp more stable.
This has been done by Jackson using ‘latticeswitch Monte Carlo3 (see Figure 2). The equivalent phase diagram for the Morse potential remains unsolved. The LJ potential has been used extensively for fcc materials, and it still comes as a surprise to many researchers that fcc is not the ground state. 1.10.4.2 Necessary Results with Pair Potentials Apart from the specific difficulties with Morse and LJ potentials, there are other general difficulties that are common to all pair potentials, which make them unsuitable for radiation damage studies. Expanding the energy as a sum of pairwise interactions introduces some constraints on what data can be fitted, even in principle. It is important to distinguish this problem from a situation where a particular parameterization does not reproduce a feature of a material. There are many features of real materials that cannot be reproduced by pair potential whatever the functional form or parameterization used. 1.10.4.2.1 Outward surface relaxation
For a single-minimum pair potential, the nearest neighbors repel one another, while longer ranged neighbors attract. When a surface is formed, more long-range bonds are cut than short-range bonds, so there is an overall additional repulsion. Hence, the surface layer is pushed outward. But in almost all metals, the surface atoms relax toward the bulk, because the bonds at the surface are strengthened. Similarly, pair potentials
0.1
0
1
1.5
2 rs 3
Figure 2 Pressure versus temperature phase diagram for the crystalline region of the Lennard-Jones system in reduced units where p is pressure and r is density. The equilibrium density is at rs3 ¼ 1:0915. Filled squares are the harmonic free energy integrated to the thermodynamic limit from Salsburg, Z. W.; Huckaby, D. A. J. Comput. Phys. 1971, 7, 489–502. All other points are from lattice-switch Monte Carlo simulations with N atoms, lines showing the phase boundary deduced from the Clausius–Clapeyron equation, from Jackson, A. N. Ph. D. Thesis, University of Edinburgh, 2001; Jackson, A. N.; Bruce, A. D.; Ackland, G. J. Phys. Rev. E 2002, 65, 036710.
give too large a ratio of surface to cohesive energy, again consistent with the failure to describe the strengthening of the surface bonds. 1.10.4.2.2 Melting points
With LJ, the relation between cohesive energy and melting is Ec =kB Tm 13, other pair potentials being similar. Real metals are relatively easier to melt, with values around 30. One can fit the numerical value of the e parameter to the melting point, and accept the discrepancy as a poor description of the free atom. 1.10.4.2.3 Vacancy formation energy
For a pair-potential, removing an atom from the lattice involves breaking bonds. The cohesive energy of a lattice comes from adding the energies of those bonds. Hence, the cohesive energy is equal to the vacancy formation energy, aside from a small difference from relaxation of the atoms around the vacancy.
Interatomic Potential Development
In real metals, the vacancy formation energy is typically one-third of the cohesive energy, the discrepancy coming yet again from the strengthening of bonds to undercoordinated atoms. 1.10.4.2.4 Cauchy pressure
Pairwise potentials constrain possible values of the elastic constants. Most notably, it is the ‘Cauchy’ relation which relates C12 –C66 . In a pairwise potential, these are given by the second derivative of the energy with respect to strain, which are most easily treated by regarding the potential as a function of r 2 rather than r ; whence for a pair potential V ðr 2 Þ, it follows: 2 X 00 2 2 2 V ðrij Þxij yij C12 ¼ C66 ¼ O ij where i; j run over all atoms and O is the volume of the system. In metals, this relation is strongly violated (e.g., in gold, C12 ¼ 157GPa; C44 ¼ C66 ¼ 42GPa).
and simplify. Quantum mechanics can be expressed in any basis set, so there are several possible starting points for such a theory. Thus, a picture based on atomic orbitals (i.e., tight binding) or plane waves (i.e., free electrons) can be equally valid: for potential development, the important aspect is whether these methods allow for intuitive simplification. When a potential form is deduced from quantum theory, approximations are made along the way. An aspect often overlooked is that the effects of terms neglected by those approximations are not absent in the final fitted potential. Rather they are incorporated in an averaged (and usually wrong) way, as a distortion of the remaining terms. Thus, it is not sensible to add the missing physics back in without reparameterizing the whole potential. 1.10.5.1
Free Electron Theory
For a free electron gas with Fermi wavevector kF, the energy U of volume O is5 U¼
1.10.4.2.5 High-pressure phases
Many materials change their coordination on pressurization (e.g., iron from bcc (8) to hcp (12)) and some on heating (e.g., tin, from fourfold to sixfold). This suggests that the energy is relatively insensitive to coordination – for pair potentials, it is proportional. These problems suggest that a potential has to address the fact that electrons in solids are not uniquely associated with one particular atom, whether the bonding be covalent or metallic. Ultimately, bonding comes from lowering the energy of the electrons, and the number of electrons per atom does not change even if the coordination does.
271
h2 kF5 O 10p2 me
This contribution to the energy of the condensed phase generates no interatomic force since U is independent of the atomic positions. However, its contribution is significant: metallic cohesive energy and bulk moduli are correct to within an order of magnitude. Consideration of this term gives some justification for ignoring the cohesive energy and bulk modulus in fitting a potential, and fitting shear moduli, vacancy, or surface energies instead. The discrepancy is absorbed by a putative free electron contribution which does not contribute to the interatomic atomic forces in a constant volume ensemble calculation.
1.10.4.2.6 Short ranged
It is worth noting that some properties that are claimed to be deficiencies of pair potentials are actually associated with short range. So, for example, the diamond structure cannot be stabilized by nearneighbor potentials, but a longer ranged interaction can stabilize this, and the other complex crystal structures observed in sp-bonded elements.4
1.10.5 From Quantum Theory to Potentials To understand how best to write the functional form for an interatomic potential, we need to go back to quantum mechanics, extract the dominant features,
1.10.5.2
Nearly Free Electron Theory
In nearly free electron (NFE) theory, the effects of the atoms are included via a weak ‘pseudopotential.’ The interatomic forces arise from the response of the electron gas to this perturbation. To examine the appropriate form for an interatomic potential, we consider a simple weak, local pseudopotential V0 ðr Þ. The total potential actually seen at ri due to atoms at rj will be as follows: X V0 ðrij Þ þ W ðr Þ V ðri Þ ¼ j
where W ðr Þ describes how the electrons interact with one another. Given the dielectric constant,
272
Interatomic Potential Development
we can estimate W in reciprocal space using linear response theory: W ðqÞ ¼ V0 ðqÞ=eðqÞ where eðqÞ is the dielectric function. In Thomas Fermi theory, eðqÞ ¼ 1 þ ð4kF =pa0 q 2 Þ where a0 is the Bohr radius. A more accurate approach due to Lindhard: 4kF 1 1 x2 1 þ x ln j þ j eðqÞ ¼ 1 þ pa0 q 2 4x 2 1x where x ¼ q=2kF accounts for the reduced screening at high q, and r0 is the mean electron density. From this screened interaction, it is possible to obtain volume-dependent real space potentials.6 The contributions to the total energy are as follows: The free electron gas (including exchange and correlation) The perturbation to the free electron band structure Electrostatic energy (ion–ion, electron–electron, ion–electron) Core corrections (from treating the atoms as pseudopotentials)
Soft phonon instabilities are an extreme case of the Kohn anomaly. They arise when the lowering of energy is so large that the phonon excitation has negative energy. In this case, the phonon ‘freezes in,’ and the material undergoes a phase transformation to a lower symmetry phase. Quasicrystals are an example where the atoms arrange themselves to fit the Friedel oscillation. This gives well-defined Bragg Peaks for scattering in reciprocal space, and includes those at 2kF but no periodic repetition in real space. Charge density waves refer to the buildup of charge at the periodicity of the Friedel oscillation. ‘Brillouin Zone–Fermi surface interaction’ is yet another name for essentially the same phenomenon, a tendency for free materials from structures which respect the preferred 2kF periodicity for the ions – which puts 2kF at the surface. ‘Fermi surface nesting’ is yet another example of the phenomenon. It occurs for complicated crystal structures and/or many electron metals. Here, structures that have two planes of Fermi surface separated by 2kF are favored, and the wavevector q is said to be ‘nested’ between the two. Hume-Rothery phases are alloys that have ideal composition to allow atoms to exploit the Friedel oscillation.
This arises from the singularity in the Lindhard function at q ¼ 2kF : physically, periodic lattice perturbations at twice the Fermi vector have the largest perturbative effect on the energy. The effect of Friedel oscillations is to favor structures where the atoms are arranged with this preferred wavelength. It gives rise to numerous effects.
NFE pseudopotentials enabled the successful prediction of the crystal structures of the sp3 elements. It is tempting to use this model for ‘empirical’ potential simulation, using the effective pseudopotential core radius and the electron density as fitting parameters; indeed such linear-response pair potentials do an excellent job of describing the crystal structures of sp elements. For MD, however, there are difficulties: the electron density cannot be assumed constant across a free surface and the elastic constants (which depend on the bulk term) do not correspond to long-wavelength phonons (which do not depend on the bulk term). Since most MD calculations of interest in radiation damage involve defects (voids, surfaces), phonons, and long-range elastic strains, NFE pseudopotentials have not seen much use in this area. They may be appropriate for future work on liquid metals (sodium, potassium, NaK alloys). The key results from NFE theory are the following:
Kohn anomalies in the phonon spectrum are particular phonons with anomalously low frequency. The wavevector of these phonons is such as to match the Friedel oscillation.
The cohesive energy of a NFE system comes primarily from a volume-dependent free electron gas and depends only mildly on the interatomic pair potential.
In this model, interatomic pair potential terms arise only from the band structure and the electrostatic energy (the difference between the Ewald sum and a jellium model) and give a minor contribution to the total cohesive energy. However, these terms are totally responsible for the crystal structure. A key concept emerging from representing the Lindhard screening in real space is the idea of a ‘Friedel oscillation’ in the long-range potential: V ðr Þ /
cos 2kF r ð2kF r Þ3
Interatomic Potential Development
The pair potential is density dependent: structures at the same density must be compared to determine the minimum energy structure. The pair potential has a long-ranged, oscillatory tail. These potentials work well for understanding crystal structure stability, but not for simulating defects where there is a big change in electron density. The reference state is a free electron gas: description of free atoms is totally inadequate.
1.10.5.3 Embedded Atom Methods and Density Functional Theory In the density functional theory (DFT), the electronic energy of a system can be written as a functional of its electron density: U ¼ F ½rðrÞ 7
The embedded atom model (EAM) postulates that in a metal, where electrostatic screening is good, one might approximate this nonlocal functional by a local function. And furthermore, that the change in energy due to adding a proton to the system could be treated by perturbation theory (i.e., no change in r). Hence, the energy associated with the hydrogen atom would depend only on the electron density that would exist at that point r in the absence of the hydrogen. UHðrÞ ¼ FH ðrðrÞÞ The idea can be extended further, where one considers the energy of any atom ‘embedded’ in the effective medium of all the others.8 Now, the energy of each (ith) atom in the system is written in the same form, Ui ¼ Fi ðrðri ÞÞ To this is added the interionic potential energy, which in the presence of screening, they took as a short-ranged pairwise interaction. This gives an expression for the total energy of a metallic system: X X Fi ðrðri ÞÞ þ V ðrij Þ Utot ¼ i
ij
To make the model practicable, it is assumed that r can be evaluated P as a sum of atomic densities fðr Þ, that is, rðri Þ ¼ j fðrij Þ and that F and V are unknown functions which could be fitted to empirical data. The ‘modified’ EAM incorporates screening of f and additional contributions to r from many-body terms.
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1.10.6 Many-Body Potentials and Tight-Binding Theory 1.10.6.1
Energy of a Part-Filled Band
An alternate starting point to defining potentials is tight-binding theory. As this already has localized orbitals, it gives a more intuitive path from quantum mechanics to potentials. Consider a band with a density of electronic states (DOS) n(E) from which the cohesive energy becomes ð Ef U ¼ ðE E0 ÞnðEÞ dE where E0 is the energy of the free atom, which to a first approximation lies at the center of the band. For example, a rectangular d-band describing both spin states and containing N electrons, width W has nðEÞ ¼ 10=W , and EF ¼ W ðN 5Þ=10 þ E0 whence (Figure 3), N ð10 N Þ W ½1 U ¼ 10 This gives parabolic behavior for a range of energyrelated properties across the transition metal group, such as melting point, bulk modulus, and Wigner– Seitz radius. For a single material, the cohesion is proportional to the bandwidth. Even for more complex band shapes, the width is the key factor in determining the energy. The width of the band can be related to its second moment9 here: ð m2 ¼ ðE E0 Þ2 nðEÞ dE ¼
ð E0 þW =2 E0 W =2
10ðE E0 Þ2 =W dE
¼ 5W 2 =6
½2
To build a band in tight-binding theory, we set up a matrix of onsite and hopping integrals (Figure 4). For a simple s-band ignoring overlap, n(E)
W
E0
EF E
Figure 3 Density of states for a simple rectangular band model.
274
Interatomic Potential Development
Hopping integral h(r) h
h
s
s
Onsite term s
s
h
h
0
0
h
s
h
0
0
h
h
s
h
h
0
0
h
s
h
0
0
h
h
s
1. hhhhh 2. hhshh
Figure 4 Matrix of onsite and hopping integrals for a planar five-atom cluster – in tight binding this gives five eigenstates, each of which contributes one level to the ‘density of states’: five delta functions. In an infinite solid, the matrix and number of eigenstates become infinite, so the density of states becomes continuous. Of course, tricks then have to be employed to avoid diagonalizing the matrix directly. Figure 5 Dashed and dotted lines show two of the chains of five hops which contribute to the fifth moment of the tight-binding density of states.
S ¼ hi jVi ji i hðrij Þ ¼ hi jVi jj i The electron eigenenergies come from diagonalizing this matrix (there are, of course, cleverer ways to do this than brute force). Typically, we can use them to create a density of states, n(E), which can be used to determine cohesive energy (as above). The width of this band depends on the off-diagonal terms (in the limit of h ¼ 0, the band is a delta function). One can proceed by fitting S and h, or move to a further level of abstraction. 1.10.6.2
The Moments Theorem
A remarkable result by Ducastelle and CyrotLackmann10 relates the tight-binding local density of states to the local topology. If we describe the density of states in terms of its moments where the pth moment is defined by ð1 E p nðEÞ dE mp ¼ 1
and recall that by definition X nðEÞ ¼ dðE Ei Þ i
i
i
where H is the Hamiltonian matrix written on the basis of the eigenvectors. But, the trace of a matrix is invariant with respect to a unitary transformation, that is, change of basis vectors to atomic orbitals i. Therefore,
X X ½H p ii mðiÞ p i
i
ðiÞ
A sum of local moments of the density of states mp . These diagonal terms of H p are given by the sum of all chains of length p of the form Hij Hjk Hkl . . . Hni . These in turn can be calculated from the local topology: a prerequisite for an empirical potential. They consist of all chains of hops along bonds between atoms which start and finish at i (e.g., see Figure 5). By counting the number of such chains, we can build up the local density of states. Unfortunately, algorithms for rebuilding DOS and deducing the energy using higher moments tend to converge rather slowly, the best being the recursion method.11 The zeroth moment simply tells us how many states there are. The first moment tells us where the band center is. Taking the band center as the zero of energy, the second moment is as follows: X X ðiÞ ½Hij Hji ¼ hðrij Þ2 ½3 m2 ¼ ½H 2 ii ¼ j
where i labels the eigenvalues, we get ð1 X X p Ep dðE Ei Þ dE ¼ Ei ¼ Tr½H p mp ¼ 1
mp ¼ Tr½H p ¼
j
where h is a two-center hopping integral, which can therefore be written as a pairwise potential. This result, that the second moment of the tightbinding density of states can be written as a sum of pair potentials, provides the theoretical underpinning for the Finnis–Sinclair (FS) potentials. Referring back to the rectangular band model, we can take the ðiÞ second moment of the local density of states m2 as a measure of the bandwidth.
Interatomic Potential Development
This gives the relationship between cohesive energy, bandwidth, and number of neighbors ðzi Þ. In qffiffiffiffiffiffiffi ðiÞ the simplest form Wi / m2 pffiffiffi N ð10 N Þ Wi / z ½4 20 that is, the band energy is proportional to the square root of the number of neighbors. Note that this is only a part of the total energy due to valence bonding. There is also an electrostatic interaction between the ions and an exclusionprinciple repulsion due to nonorthogonality of the atomic orbitals – it turns out that both of these can be written as a pairwise potential V ðr Þ. The moments principle was laid out in the late 1960s.12 To make a potential, the squared hopping integral is replaced by an empirical pair potential fðrij Þ, which also accounts for the prefactor in eqn [4] and the exact relation between bandwidth and second moment. Once the pairwise potential V ðrij Þ is added, these potentials have come to be known as FS potentials.13 X X X sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Ecoh ¼ V ðrij Þ fðrij Þ ½5 Ui ¼
i
ij
j
where V and f are fitting functions.
275
Further work14 showed that the square root law held for bands of any shape provided that there was no charge transfer between local DOS and that the Fermi energy in the system was fixed. For bcc, atoms in the second neighbor shell are fairly close, and are normally assumed to have a nonzero hopping integral. Notice that the first three moments only contain information about the distances to the shells of atoms within the range of the hopping integral. Therefore, a third-moment model with near neighbor hopping could not differentiate between hcp and fcc (in fact, only the fifth moment differentiates these in a nearneighbor hopping model!). This led Pettifor to consider a bond energy rather than a band energy, and relate it to Coulson’s definition of chemical bond orders in molecules.15 Generalizing this concept leads to a systematic way of going beyond second moments and generating bond order potentials. One can investigate the second-moment hypothesis by looking at the density of states of a typical transition metal, niobium, calculated by ab initio pseudopotential plane wave method, Figure 6, and comparing it with the density of states at extremely high pressure. The similarity is striking: as the material is compressed, the band broadens but the structure with five peaks remains unchanged. The s-band is displaced slightly to higher energies at high pressure, but still provides a
Density of states (electrons eV–1)
2
1.5
1
0.5
0 -5
0
5
10
15
Energy (eV) Figure 6 Density of states for bcc Nb. Dotted blue line is at ambient pressure, and solid red line is for 32% reduction in volume. The Fermi energy is set to zero in each case.
276
Interatomic Potential Development
low, flat background, which extends from slightly below the d-band to several electron volts above.
Table 1 Neighbor distances in fcc, hcp (c=a ¼ and bcc, in units of the nearest neighbor separation
pffiffiffiffiffiffiffiffi 8=3),
Structure
1.10.6.3
Key Points
fcc
In a second-moment approximation, the cohesive (bond) energy is proportional to the square root of the coordination. Other contributions to the energy can be written as pairwise potentials.
1(12)
bcc
1(8)
hcp
1(12)
pffiffiffi 2ð6Þ qffiffi 4 3 (6) pffiffiffi 2 (6)
pffiffiffi 3 (24) pffiffiffi 2 (12) qffiffi 8 3 (2)
2(12) qffiffiffiffi 11 3 (24) pffiffiffi 3 (18)
pffiffiffi 5ð24Þ pffiffiffi 3 (8) qffiffiffiffi 11 3 (6)
2(6)
The number of neighbors pffiffiffiffi at that distance is given in brackets. For fcc the shells fall at N for all integers up to 30 except one. As fcc and hcp structures have identical numbers and ranges of first and second neighbors, glue or pair potentials can only distinguish them via long-range interactions.
1.10.7 Properties of Glue Models The embedded atom and FS potentials fall into a general class of potentials of the form: " # X X X Ui ¼ Fi fðrij Þ þ V ðrij Þ i
j
j
with a many-body cohesive part and a two-body repulsion. Both fðrij Þ and V ðrij Þ are short ranged, so MD with these potentials is at worst only as costly as a simple pair potential (computer time is proportional to number of particles). These models are sometimes referred to as glue potentials,16 the many-body F term being thought of as describing how strongly an atom is held by the electron ‘glue’ provided by its environment. The pragmatic approach to fitting in all glue schemes is to regard the pair potential as repulsive at short-range with long-range Friedel oscillations. Compared with most pair potential approaches, this is unusual in that the repulsive term is longer ranged than the cohesive one. 1.10.7.1
Crystal Structure
According to the tight-binding theory on which the FS potentials are based, the relative stability of bcc and fcc is determined by moments above the second, which in turn relate to three center and higher hops. These third and higher moments effects are explicitly absent in second-moment models, and so by implication, the correct physics of phase stability is not contained in them. There is no such clear result in the derivation of the EAM; however, since the forms are so similar, the same problem is implicit. In glue models, energy is lowered by atoms having as many neighbors as possible; thus, fcc, hcp, and bcc crystal structures (and their alloy analogs) are normally stable (see Table 1); bcc is normally stable in potentials when the attractive region is broad enough
to include 14 neighbors, fcc/hcp are stable for narrower attractive regions in which only the eight nearest bcc-neighbors contribute significantly to the bonding. Indeed, without second neighbor interactions, bcc is mechanically unstable to Bain-type shear distortion. The fcc–hcp energy difference is related to the stacking fault energy: it is common to see MD simulations with too small an hcp–fcc energy difference producing unphysically many stacking faults and over widely separated partial dislocations. Phase transitions are observed in some potentials. As free energy calculations are complicated and time consuming,17 it is impractical to use them directly in fitting – one would require the differential of the free energy with respect to the potential parameters, and this could only be obtained numerically. Consequently, most potentials are only fitted to reproduce the zero temperature crystal structure, and high-temperature phase stability is unknown for the majority of published potentials. One counterexample is in metals such as Ti and Zr, where the bcc structure is mechanically unstable with respect to hcp, but becomes dynamically stabilized at high temperatures. Here, the transition temperature is directly related to a single analytic quantity: the energy difference between the phases. Although about half of this difference comes from electronic entropy,18 which suggests a temperature-dependent potential, phase transition calculations have been explicitly included in some recent fits.19 The case of iron is also anomalous, as the phase transition is related to changes in the magnetic structure. 1.10.7.2
Surface Relaxation
Glue models atoms seek to have as many neighbors as possible; therefore, when a material is cleaved, the surface atoms tend to relax inward toward the bulk to increase cohesion. This effect also arises because of
277
Interatomic Potential Development
the longer range of the repulsive part of the potential: at a surface, the further-away atoms are absent. This is in contrast to pair potentials and in agreement with real materials. 1.10.7.3
Cauchy Relations
The functional form of the glue model places fewer restrictions on the elastic constants of materials than pair potentials do; for example, the Cauchy pressure for a cubic metal is as follows20: " #2 d2 F ðxÞ X 0 2 2 C12 C66 ¼ f ðr Þxj dx 2 j If the ‘embedding function’ F (minus square root in FS case) has positive curvature, the Cauchy pressure must be positive, as it is for most metals. A minority of metals have negative Cauchy pressure. It is debatable whether this indicates negative curvature of the embedding function, or a breakdown of the glue model. There are also some Cauchy-style constraints on the third-order elastic constants. But in general, ‘glue’ type models can fit the full anisotropic linear elasticity of a crystal structure. 1.10.7.4
Vacancy Formation
In a near-neighbor second-moment model for fcc, breaking one of twelve bonds reduces the cohesive energy of each atom ffi adjacent to the vacancy by a pffiffiffiffiffiffiffiffiffiffiffi factor of ð1 11=12Þ ¼ 4:25%. Other glue models give a similar result. Meanwhile, the pairwise (repulsive) energy is reduced by a full 1=12 ¼ 8:3%. Thus, energy cost to form a vacancy is lower in glue-type models than in pairwise ones. For actual parameterizations, it tends to be less than half the cohesive energy. 1.10.7.5
the atom i In the EAM, the function Fi depends on P being embedded, while the charge density j fj ðrij Þ into which it is embedded depends on the species and position of neighboring atoms. By contrast for FS potentials, the function F is a given (square root), while fðrij Þ is the squared hopping integral, which depends on both atoms. There is no obvious way to relate this heteroatomic hopping integral to the homoatomic ones, but a practical approach is to take a geometric mean21: one might expect this form from considering overlap of exponential tails of wavefunctions.
1.10.8 Two-Band Potentials In the second-moment approximation to tight binding, the cohesive energy is proportional to the square root of the bandwidth, which can be approximated as a sum of pairwise potentials representing squared hopping integrals. Assuming atomic charge neutrality, this argument can be extended to all band occupancies and shapes22 (Figure 7). The computational simplicity of FS and EAM follows from the formal division of the energy into a sum of energies per atom, which can in turn be evaluated locally. Within tight binding, we should consider a local density of states projected onto each atom. The preceding discussion of FS potentials concentrates solely on the d-electron binding, which dominates transition metals. However, good potentials are difficult to make for elements early in d-series (e.g., Sc, Ti) where the s-band plays a bigger role. An extension to the second-moment model, which keeps the idea of
DOS
Alloys
To make alloy potentials in the glue formalism, one needs to consider both repulsive and cohesive terms. Thinking of the repulsive part as the NFE pair potential, it becomes clear that the long-range behavior depends on the Fermi energy. This is composition dependent – the number of valence electrons is critical, so it cannot be directly related to the individual elements. The short-ranged part should reflect the core radii and can be taken from the elements. Despite this obvious flaw, in practice, the pairwise part is usually concentration-independent and is refitted for the ‘cross’ heterospecies interaction.
Ef E
DOS
Ef E Figure 7 In second-moment tight binding, the band shape is assumed constant at all atoms, the effect of changing environments being a broadening of the band.
278
Interatomic Potential Development
locality and pairwise functions, is to consider two separate bands, for example, s and d. This was first considered for the alkali and alkaline earth metals, where s-electrons dominate. These appear at first glance to be close-packed metals, forming fcc, hcp, or bcc structures at ambient pressures. However, compared with transition metals, they are easily compressible, and at high pressures adopt more complex ‘open’ structures (with smaller interatomic distances). The simple picture of the physics here is of a transfer of electrons from an s- to a d-band, the d-band being more compact but higher in energy. Hence, at the price of increasing their energy (U ), atoms can reduce their volumes (V ). Since the stable structure at 0 K is determined by minimum enthalpy, H ¼ U þ PV at high pressure, this sd transfer becomes energetically favorable. The net result is a metal–metal phase transformation characterized by a large reduction in volume and often also in conductivity, since the s-band is free electron like while the d-band is more localized. Two-bands potentials capture this transition, which is driven by electronic effects, even though the crystal structure itself is not the primary order parameter. Materials such as cerium have isostructural transitions. It was thought for many years that Cs also had such a transition, but this has recently been shown to be incorrect,23 and the two-band model was originally designed with this misapprehension in mind.24 For systems in which electrons change, from an s-type orbital to a d-type orbital as the sample is pressurized, one considers two rectangular bands of widths W1 and W2 as shown in Figure 8 with widths evaluated using
eqn [3]. The bond energy of an atom may be written as the sum of the bond energies of the two bands on that atom as in eqn [4], and a third term giving the energy of promotion from band 1 to band 2 (see eqn [8]): X Wi1 ni1 ðni1 N1 Þ Ubond ¼ 2N1 i Wi2 þ ni2 ðni2 N2 Þ þ Eprom ½6 2N2 where N1 and N2 are the capacities of the bands (2 and 10 for s and d respectively) and ni1 and ni2 are the occupation of each band localized on the ith atom. For an ion with total charge T, assuming charge neutrality, ni1 þ ni2 ¼ T
½7
The difference between the energies of the band centers a1 and a2 is assumed to be fixed. The values of a correspond to the appropriate energy levels in the isolated atom. Thus, a2 a1 is the excitation energy from one level to another. For alkali and alkaline earth metals, the free atom occupies only s-orbitals; the promotion energy term is therefore simply Eprom ¼ n2 ða2 a1 Þ ¼ n2 E0
½8
where E0 ¼ a2 a1 . Thus, the band energy can be written as a function of ni1 , ni2 , and the bandwidths (evaluated at each atom as a sum of pair potentials, within the secondmoment approximation). Defining, i ¼ ni1 ni2
½9
and using eqn [7], we can write as follows:
D(E) N/W1 + N/W2
d-band
N/W1 s-band a1 − W1/2
a1
Ef
a2 − W2/2
a1 + W1/2
E
a2 a + W /2 2 2
Figure 8 Schematic picture of density of electronic states in rectangular two-band model. Shaded region shows those energy states actually occupied.
Interatomic Potential Development
X T i ðWi1 Wi2 Þ ðWi1 þ Wi2 Þ 4 4 i 2 2 þ T Wi1 Wi2 þ þ i 8 N1 N2 T Wi1 Wi2 þ i N1 N2 4 T i E0 ½10 þ 2 Although this expression looks unwieldy, it is computationally efficient, requiring only two sums of pair potentials for Wi and a minimization at each site independently with respect to i , which can be done analytically. In addition to the bonding term, a pairwise repulsion between the ions, which is primarily due to the screened ionic charge and orthogonalization of the valence electrons, is added. In general, this pair potential should be a function of i and j . But to maintain locality, one has to write this pairwise contribution to the energy in the intuitive form, as the sum of two terms, one from each ‘band,’ proportional to the number of electrons in that band: Ubond ¼
V ðrij Þ ¼ ðni1 þ nj 1 ÞV1 ðrij Þ þ ðni2 þ nj 2 ÞV2 ðrij Þ
½11
We rearrange this to give the energy as a sum over atoms: " # X X X Upair ¼ ni1 V1 ðrij Þ þ ni2 V2 ðrij Þ ½12 i
j 6¼i
j 6¼i
The total energy is now simply Utot ¼ Upair þ Ubond
½13
This depends on i , which takes whatever values to minimize the energy, s @Utot ¼0 @ i
½14
explicitly for i0 independently at each atom, with the constraint that ji j cannot be greater than the total number of electrons T per atom. The fixed capacities of the bands (N1 and N2 ) can also prohibit the realization of i0 . It is therefore necessary to limit the values which i may have to those where i0 does not imply negative band occupation. The expressions for i involves only constants and sums of pair potentials, and can be evaluated independently at each atom at a similar computational cost to a standard many-body type potential.
279
The variational property expressed in eqn [14] can be exploited to derive the force on the ith atom: dUtot fi ¼ dri @Utot @Utot @ ¼ j @ri @ dri @Utot ¼ j @ri
½15
Hence, the force is simply the derivative of the energy at fixed . Basically, this is the Hellmann– Feynman theorem25 which arises here because is essentially a single parameter representation of the electronic structure. This result means that, like the energy, the force can be evaluated by summing pairwise potentials. Hence, the two-band second-moment model is well suited for large-scale MD. The force derivation itself is somewhat tedious, and the reader is referred to the original papers. There is no Hellman–Feynman type simplification for the second derivative, so analytic expressions for the elastic constants in two-band models are long ranged and complicated. Consequently, elastic constants are best evaluated numerically. 1.10.8.1
Fitting the s–d Band Model
To make a usable potential, the functional forms of f and V must be chosen. Although this is somewhat arbitrary, the physical picture of hopping integral and screened ion–ion potential suggests that both should be short ranged, continuous, and reasonably smooth. Popular choices are cubic splines, power series, and Slater orbitals. The promotion energy E0 is simply that required to promote an electron from the s level into the d level of an isolated atom. The band capacities are Ns ¼ 2, Nd ¼ 10 and the total number of electrons per atom depends on the element, for example, in Cs T ¼ 1. In the first application, parameters were fitted to the energy–volume relations for bcc and fcc cesium and the transition pressure between phases Cs-II and Cs-III. Figure 9 shows the energy–volume curves for the fcc and bcc structures calculated using the model. At ambient conditions for Cs, there are no d-electrons, so the fitting process is just like a normal FS potential. This determines the s-band parameters, and the d-band parameters are then fitted to the high-pressure phase data, where both s- and d-electrons contribute. Although an isostructural phase transition is likely to be accompanied by instability of the bulk modulus, there may also be a precursor shear instability. Thus,
Interatomic Potential Development
1.0 0.5 0.0 -0.5
9.0 fcc lattice parameter (Å)/ cohesive energy (eV per atom)
h
280
-1.0
Energy per atom (eV)
1.5 fcc bcc 4.3 GPa
1.0 0.5 0.0 -0.5
8.0 7.0 6.0 5.0 4.0 3.0 2.0 1.0 0.0
-1.0 -1.5
0
50
100 150 Volume per atom (Å3)
200
Figure 9 Top: Variation of i with compression, showing s!d transfer in model cesium (T ¼ 1). Bottom: An energy– volume curve for the two-band potential. The minimum lies at –1.3163 eV and 115.2 A˚3 per atom. The gradient of the straight dash–dotted line is the experimental Cs-II–Cs-III transition pressure. In reality, cesium also has a bcc–fcc phase transition at 2.3 GPa; however, first principles calculations show that these two structures are almost degenerate in energy at 0 K.
the mechanism may involve shearing rather than isostructural collapse, particularly if a continuous interface between the two phases exists, as in a shockwave. With the two-band model, the transition is first order, the volume collapse occurring before the bulk modulus becomes negative in the unstable region. Although the shear and tetragonal shear decrease in the unstable region, neither actually goes negative. The two-band model is applicable to transition metals, but since the d-band is occupied at all pressures, electron transfer is continuous and there is no phase transition. This makes the empirical division of the energy into s and d components challenging. However, once appropriately scaled for ionic charge and number of electrons T, in principle, the method could be extended to alloys with noninteger T. The results of such an extrapolation are extraordinarily good (Figure 10), considering that there is no fitting to any material other than Cs. The extrapolation breaks down at high Z where the amount of sp hybridization is not fully captured in the parameterization. As with FS, no information about band shape is included, and so the sequence of crystal structures cannot be reproduced. While the extrapolated potentials do not represent the optimal parameterization for specific transition
1
2
3
4 5 6 7 8 9 Number of sd electrons
10 11
Figure 10 Extrapolation of two-band model fitted to Cs to various {6s5d}-band materials, with the lengths being scaled according to the Fermi vector (Z 1=3 ) and the energies by (Z 1=2 ). Lattice parameters are shown by the dashed line (calculated) and squares (experiment), and the cohesive energies by the solid line (calculated) and circles (fcc at experimental density).
metals, the recovery of the trends across the group lends weight to the idea that the two-band model correctly reproduces the physics of this series. The s–d two-band approach has also been applied with considerable success by considering the s-band as an alloying band.26 This has been applied to the FeCr system, which we discuss in more detail later. 1.10.8.2
Magnetic Potentials
The two-band approach can be applied to magnetic materials, where the bands spin up and spin down bands have the same capacity (N ¼ Nd" ¼ Nd# ¼ 5). If in addition we assume that the bands have the same width and shape (see, e.g., Figure 11), there is a remarkable collapse of the model onto the singleband EAM form, with a modified embedding function. The formalism here extends to the two-band model, but the physics is analogous to other magnetic potentials.27 For simplicity, consider a rectangular d-band of full width W centered on E0 . The bond energy for a single spin-up band relative to the free atom is given by " ð Ef ¼ Z=N 1 W " " Z 2 1 W =2 NE=W dE ¼ Z U ¼ N W =2 ½16 where Z" is the occupation of the band and uparrows denote ‘spin up.’
Interatomic Potential Development
281
2.5
Density of states
2
1.5
1
0.5
0 -8
-6
-4
-2 Energy (eV)
0
2
Figure 11 bcc iron density of states for majority and minority spin bands from ab initio spin-dependent GGA pseudopotential calculations with 4913 k ¼ points, adjusted so that the Fermi energy lies at the zero of energy. The two-band model assumes that the bands have the same shape and width, but are displaced in energy relative to each other: the figure shows this to be reasonable.
To describe the ferromagnetic case, it is assumed that there are two independent d-bands corresponding to opposite spins, and that these can be projected onto an atom to form a local density of states. For a free atom atomic case, Hund’s rules determine a high-spin case (e.g., S ¼ 2 for iron), and there is an energy U x associated with transferring an electron to a lower spin state. In the solid, the simplest method is to set U x to be proportional to the spin with the coefficient of proportionality being an adjustable parameter, E0 . U x ¼ E0 jZ" Z# j
½17
Defining the spin, S ¼ jZ" Z# j and assuming charge neutrality (T ¼ ðZ" þ Z# Þ), the two-d-band binding energy on a site i is then as follows: Ui ¼ Ui" þ Ui# þ Uix ¼ þ
Wi 2 ðT þ Si2 Þ TWi =2 E0 Si 4N ½18
Differentiating this equation about Si gives us the optimal value for the magnetization of a given atom of Si ¼ 2NE0 =Wi , and the many-body energy of an atom with T ¼ 6, N ¼ 5 (suppressing the i label) as U ¼ 6W =5 5E02 =W E0 =Wi 0:4 ¼ 2W =5 4E0 E0 =Wi 0:4
½19
where neither band is allowed to have occupancy more than 5 or less than 0. For a material with T d-electrons (where T > 5), transfer of electrons between the spin bands becomes advantageous for W > 10E0 =ð10 T Þ. For smaller W, the spin " band is full and the energy is simply proportional to the bandwidth of the # band as in the FS model. Similar cases apply to the T < 5 case when the minority band may be empty. There has been some controversy about the expression for U x . In the two-band model, this is a promotion energy from the minority spin band to the majority. In the atomic case, it is the energy to violate Hund’s rule, and the implicit reference state is the high-spin atom. Electron transfer is bound by the number of electron, so the function has discontinuous slope at Si ¼ 0:4. By contrast, the approach of Dudarev and coworkers27 uses a Stoner model for the spin energy, which introduces quadratic and quartic terms in U x ðSi Þ. In that case, the implicit reference state is the nonmagnetic solid, and any value of Si is acceptable. Within the second-moment model, the bandwidth W is given by the square root of r, the sum of the squares of the hopping integral. Applying this, and the usual pairwise repulsion V ðr Þ, gives an expression for the two-band energy
282
Interatomic Potential Development
Ui ¼
X j
B=
V ðrij Þ
pffiffiffiffiffi rj
pffiffiffiffiffi rj H ð2W 5E0 Þ
4E0 H ð5E0 2W Þ
½20
where H is the Heaviside step function, B is a constant, and the zero of energy corresponds to the nonmagnetic atom. Note that this form does not explicitly include S, and that it has the pffiffiffiEAM form with an embedding function FI ðxÞ ¼ x ð1 B=xÞ. Although this model incorporates magnetism and provides a way to calculate the magnetic moment at each site, it is possible to use it without actually calculating S. The additional many-body repulsive term is similar to, for example, the many-body potential method of Mendelev et al. It is also interesting that in the original FS paper, it was not possible to fit the properties of the magnetic elements Fe and Cr; an extra term was added ad hoc. Later parameterizations of FS potentials for iron with a pure square root for F have not exactly reproduced the elastic constants.28 The implication of this work is that for secondmoment type models, there should be a one-toone relation between the local density r and the magnetic moment. Figure 12 shows this relation for two parameterizations, r from Dudarev–Derlet and Mendelev et al. and the magnetic moment calculated with spin-dependent DFT projected onto atoms. It shows that there are two cases. For atoms associated with local defects, the density varies quite sharply with r, while for the crystal under pressure the variation is slower. It is noteworthy that the same broad features are present in both potentials, even though the Mendelev et al. potential was fitted without consideration of magnetic properties, albeit with a FS-type embedding function. This suggests that the magnetic effects were unwittingly captured in the fitting process. 1.10.8.3
Nonlocal Magnetism
The two-band model projects the magnetism onto each atom. It does not properly describe magnetic interactions, so it cannot distinguish between ferro, para, and antiferromagnetism. In order to do so, we need to include an interionic exchange term. Pauli repulsion arises from electron eigenstates being orthogonal. While its nature on a single atom is complex, its interatomic effects can be modeled as
a pairwise effect of repulsion between electrons of similar spins. The secondary effect of magnetization is that there are more electrons in one band than in another, and more same-spin electron pairs to repel one another, and so the repulsion between those bands is enhanced. Conceptually, this can be captured in two pairwise effects, the standard nonmagnetic screened-Coulomb repulsion of the ions plus the core–core repulsion, and an additional Si dependent term arising from Pauli repulsion between like-spin electrons. V ðrij Þ ¼ V0 ðrij Þ þ ðSi" Sj" þ Si# Sj# ÞVm ðrij Þ
½21
Note that an antiferromagnetic state with Si þ Sj ¼ 0 would have a lower repulsive energy. In the tightbinding picture, this would be compensated by a much reduced hopping integral, and hence, lower W. If we insist on Si > 0, then we suppress these solutions and can model ferromagnetic or diamagnetic iron. Also, as with DFT-GGA/LDA, the spin is Ising-like. At the time of writing, no good parameterization of this type of potential exists. The difficulty is that determining the spins Si is a nonlocal process: the optimal value of the spin on site i depends on the spin at site j. The only practical way to proceed appears to be to treat the spins as dynamical variables, in which case it is probably better to treat them as noncollinear Heisenberg moments. 1.10.8.4
Three-Body Interactions
It is worth noting that the ‘glue’ type potentials cannot be expanded in a sum of two-body, threebody, four-body, etc. terms. Three-body terms enter into the free electron picture through nonlinear response of the electron gas, and into the tightbinding picture in the fourth moment description and beyond. At constant second moment, increasing the fourth and higher even moments of the DOS tends to lead to a bimodal distribution. Bimodal distributions will be favored by materials with half-filled bands. Thus, three-body terms are likely to be important in structures with few small-membered rings of atoms, and hence, small low moments. The bcc structure is a borderline example of this, but the classic is the diamond structure. Diamond has no rings of less than six atoms, resulting in a strongly bimodal DOS. This bimodal structure in the tight-binding representation is also interpreted as bonding and antibonding states in a covalent picture.
Interatomic Potential Development
283
Magnetic moments in iron versus r Dudarev/Derlet potential case2 3 bcc Fe perfect crystal Vacancy 100 sia 110 sia 111 sia Octa Tetra
m /mB
2
1
0
0
0.5
1
2
1.5 r
(a)
2.5
3
Magnetic moments in iron versus r Ackland/Mendelev potential 3 bcc Fe perfect crystal Vacancy 100 sia 110 sia 111 sia Octa Tetra
m /mB
2
1
0
20
30
40
(b)
50
60
70
r
Figure 12 Relationship between ab initio calculated magnetic moment per atom, and r from (a) Dudarev–Derlet magnetic potential and (b) Mendelev et al.
Interatomic potentials for carbon and silicon fall into this category. However, once a band gap is opened, the Fermi energy and perfect screening are lost, and the rigid band approximation is less appropriate.
1.10.9 Modified Embedded Atom Method The modified embedded atom method (MEAM) is an empirical extension of EAM by Baskes, which
284
Interatomic Potential Development
includes angular forces. As in the EAM, there are pairwise repulsions and an embedding function. In the EAM, the ri is interpreted as a linear supposition of species-dependent spherically averaged atomic electron densities (here designated by fðr Þ); in MEAM ri is augmented by angular terms. The spherically symmetric partial electron density rð0Þ is the same as the electron density in the EAM: ð0Þ
ri
¼
X
fð0Þ ðrij Þ
j
where the sum is over all atoms j, not including the atom at the specific site of interest i. The angular contributions to the density are similar to spherical harmonics: they are given by similar formulas weighted by the x; y; and z components of the distances between atoms (labeled by a; b; g): 2 32 X X a ij 4 ðrð1Þ Þ2 ¼ fð1Þ ðrij Þ 5 rij a j 2 32 2 32 X X X a b 1 ij ij 4 ðrð2Þ Þ2 ¼ fð2Þ ðrij Þ 2 5 4 fð2Þ ðrij Þ5 3 r ij j j a;b 2 32 2 32 X X X aij bij gij 1 ð3Þ 2 ð3Þ ð2Þ 4 5 4 ðr Þ ¼ f ðrij Þ f ðrij Þ5 3 j rij3 j a;b;g
The fðlÞ are so-called ‘atomic electron densities,’ which decrease with distance from the site of interest, and the a; b; and g summations are each over the three coordinate directions (x, y, z). The functional forms for the partial electron densities were chosen to be translationally and rotationally invariant and are equal to zero for crystals with inversion symmetry about all atomic sites. Although the terms are related to powers of the cosine of the angle between groups of three atoms, there is no explicit evaluation of angles, and all the information required to evaluate the MEAM is available in standard MD codes. Typically, atomic electron densities are assumed to decrease exponentially, that is, fðlÞ ðRÞ ¼ exp b ðlÞ ðR=re 1Þ where the decay lengths (re and b ðlÞ ) are constants. While there is no derivation of the MEAM from electronic structure, it also introduced the physically reasonable idea of many-body screening, which is missing in pair-functional forms such as EAM. Thus, fðRij Þ is reduced by a screening factor determined by the other atoms k forming three-body triplets with i and j : primarily those lying between i and j. This eliminates the need for an explicit cut-off in the ranges of V ðr Þ and fðlÞ ðr Þ.
For close-packed materials, the improvement of MEAM over standard EAM is marginal; the angular terms come out to be small. For sp-bonded materials, a large three-body term can stabilize tetrahedrally coordinated structures, but since the physics arises from preferred 109 angles rather than preferred fourfold coordination, it suffers problems similar to Stillinger– Weber type potentials (discussed below). Very high angular components enable one to fit the complex phases of lanthanides and actinides. It is tempting to attribute this to the correct capture of the f-electron physics, although the additional functional freedom may play a role in enabling fits to low symmetry structures.
1.10.10 Potentials for Nonmetals While much of the work on structural materials has concentrated on metals, there are important issues involving nonmetallics for coatings, corrosion, and fuel. In this section, we review other types of potentials. 1.10.10.1
Covalent Potentials
Empirical potentials for covalent materials have been much less successful than for metals. As with the NFE pair potentials, the bulk of the energy is contained in the covalent bond, and potentials which well describe distortions from fourfold coordination tend to fail when applied to other bond situations, such as surfaces, high pressure phases, or liquids. A commonly used example, the Stillinger–Weber potential,29 is written as 2 X XX 1 V ðrij Þ þ F ðrij ÞF ðrik Þ cos yijk Ui ¼ 3 j j k where V ðrij Þ and F ðrij Þ are short-ranged pair potentials. The form of the three-body term, with its minimum at 109 , ensures the stability of a tetrahedrally bonded network. The stacking fault energy (equivalently, the difference between cubic and hexagonal diamond) is zero, so the ground state crystal structure is not unique; however, this is not far from correct, and hexagonal diamond can be found in carbon and silicon. Moreover, it has little effect in many simulations since the large kinetic barrier against cubic–hexagonal phase transitions prevents them occurring in simulations. Solidification and recovery from cascade damage are counterexamples. Stillinger–Weber works well for fourfoldcoordinated amorphous networks, and vibrational properties of the diamond structures. It gives too low
Interatomic Potential Development
density (and coordination) for the liquid and highpressure phases, because it fails to reproduce the rebonding of atoms at the surface to remove dangling bonds. An alternate approach to stabilizing diamond via the 109 angle is to do so through its tetravalent nature. The simplest type is the restricted bond pair potential30: X X Ui ¼ Aðrij Þ Bðrij Þ j
j ¼1;4
where the attractive part of the potential is summed over at most four neighbors (one per electron). This formalism describes well the collapse of the network under pressure or melting, but lacks shear rigidity (only the repulsion of second neighbors provides shear rigidity). There is also some ambiguity over which four neighbors should be chosen, which makes implementation difficult. An embellishment on this is the bond-charge model, in which the electrons in the bonds repel one another. This adds a three-center term of the form X Cðrjk Þ i
where j and k are bonded neighbors of i. This approach avoids explicitly introducing the tetrahedral angle into the potential. Note that although this term is associated with atom i (and is often interpreted as a bondbending term at i ), in the simplest form forces derived from this term are independent of i. The problem of defining ‘bonded neighbors’ can be circumvented, in the spirit of the embedded atom method, by having an embedding function that effectively cuts off after the bonding reaches four neighbors worth31 as in the Tersoff approach: X X Aðrij Þ Bðrij Þ Ui ¼ j
where Bðrij Þ ¼ fðrij Þ
j
X
Gðrik ; rjk ; rij Þ
k
The bond ij is weakened by the presence of other bonds ik and jk involving atoms i and j. The protetrahedral angular dependence is still necessary to stabilize the structure, and further embellishment by Brenner32 corrects for overbinding of radicals. These potentials give a good description of the liquid and amorphous state, and have become widely used in many applications, in addition to elements such as Si and C, as well as covalently bonded compounds such as silicon carbide33 and tungsten carbide.34
1.10.10.2
285
Molecular Force Fields
Potentials based on bond stretching, bond bending, and long-ranged Coulomb interactions are widely used in molecular and organic systems. Chemists call these potentials ‘force fields.’ They cannot describe making and breaking chemical bonds, but by capturing molecular shapes, they describe the structural and dynamical properties of molecules well. There are many commercial packages based on these force fields, for example, CHARMM35 and AMBER.36 They are primarily useful for simulating molecular liquids and solvation, but have seen little application in nuclear materials, on account of the long-range Coulomb forces, which are costly to evaluate in large simulations. 1.10.10.3
Ionic Potentials
With no delocalized electrons, ionic materials should be suitable for modeling with pair potentials. The difficulty is that the Coulomb potential is long ranged. This can be tackled by Ewald or fast multipole methods, but still scales badly with the number of atoms. The simplest model is the rigid ion potential, where charged (q) ions interact via long-range Coulomb forces and short-ranged pairwise repulsions V ðr Þ. X qi qj V ðrij Þ þ U¼ 4pe0 rij ij For example, a common form of the pair potential in oxides consists of the combination of a (6-exp) Buckingham form and the Coulomb potential: U¼
X ij
A expðarij Þ b=rij6 þ
qi qj 4pe0 rij
where a and b are parameters and rij is the distance between atoms i and j. As with other potential, various adjustments are needed in order to obtain reasonable forces at very short distance; see for example, recent reviews of UO2.37 For nuclear applications, the most commonly studied material in the open literature is UO2, which is widely used as a reactor fuel. It adopts a simple fluorite structure with a large bandgap, which makes potentialfitting to get the correct crystal structure reasonably straightforward. Early work fitted the potentials to lattice parameter and compressibility, and later to elastic constants and the dispersion relation. The elastic constants are c11 ¼ 395 GPa, c12 ¼ 121 GPa, and c44 ¼ 64 GPa.38 As previously described, the Cauchy relation generally applies to a pairwise potential,
286
Interatomic Potential Development
C12 ¼ C44 , which is seldom true experimentally for oxides. However, the Cauchy relation is on the basis of the assumption that all atoms are strained equally, which is not the case for a crystal such as UO2 where some atoms do not lie at centers of symmetry. Thus, the violation of the Cauchy relation in UO2 can be fitted by attributing it to internal motions of the atoms away from their crystallographic positions. (The violation of the Cauchy relation is similar in oxides with and without this effect, so it is debatable whether this is the correct physical effect.) The earlier potentials were based on the Coulomb charge plus Buckingham described above; more recent parameterizations include a Morse potential. While this gives more degrees of freedom for fitting, having two exponential short-range repulsions with different exponents appears to be capturing the same physics twice. Comparison of the parameters39 shows that the prefactor for the U–U Buckingham repulsion varies by ten orders of magnitude when fitted. Moreover, the original Catlow parameterization sets this term to zero. This difference tells us that the small U atoms seldom approach one another close enough for this force to be significant. Even the ionic charges vary between potentials by almost a factor of two, with more recent potentials taking lower values. Despite the huge disparity in parameters, the size of cascades is similar and the recombination rate is high. Polarizability is not incorporated in rigid ion potentials; they will always predict a high-frequency dielectric constant of 1, which is much smaller than typical experimental values. The main consequence of this for MD appears in the longitudinal optic phonon modes. The solution is that ions themselves change in response to environment. A standard model for this is the shell model in which the valence electrons are represented by a negatively charged shell, connected to a positively charged nucleus by a spring. (Typically this represents both atomic nucleus and tightly bound electrons.) In a noncentrosymmetric environment (e.g., finite temperature), the shell center lies away from the nuclear center, and the ion has a net dipole moment – it is polarized. X X qi qj V ðrij Þ þ þ ki ðri rishell Þ2 U¼ 4pe r 0 ij i ij In this case, rij may refer to the separation between nuclei i and j or the centers of the shells associated with i and j. In MD, the shells have extremely low mass, and are assumed to always relax to their equilibrium position: this is a manifestation of the Born–Oppenheimer approximation used in DFT calculation.
Shell model potentials,40 which capture the dipole polarizability of the oxygen molecules, were developed by Grimes and coworkers, and have been through many extensions and reparameterizations since then. Again, there have been many successful parameterizations with wildly differing values for the parameters; even the sign of the charge on the U core and shell changes.41 A particular issue with ionic potentials is that of charge conservation. A defect involving a missing ion will lead to a finite charge. If the simulation is carried out in a supercell with periodic boundary conditions, this will introduce a formally infinite contribution to the energy. The simple way to deal with this is to ignore the long wavelength (k ¼ 0) term in the Ewald sum. This, under the guise of a ‘neutralizing homogeneous background charge’ is routinely done in first principles calculation. Alternately, a variable charge approach can be used42 in which the extra charge is added to adjacent atoms. The original approach then involved minimizing the total energy with respect to these additional charges, which is computationally demanding. A promising new development is to limit the range of the charge redistribution.43 While this screening approximation is difficult to justify fully in an insulator, it is very computationally efficient or a system involving dilute charged impurities, and appears to reproduce most known features of AlO.
1.10.11 Short-Range Interactions In radiation damage simulations, atoms can come much closer together than in any other application. Interatomic force models are often parameterized on data that ignore very short-ranged interactions, and the physics of core wavefunction overlap is seldom well described by extarpolation. In radiation damage, we are normally dealing with a case where two atoms come very close together, but the density of the material does not change. By analogy with a free electron gas, we see that the energy cost of compressing the valence wavefunctions is important for high-pressure isotropic compression but absent in the collision scenario. In the first case, the ‘short range’ repulsion is a many-body effect, while in the second case, it is primarily pairwise. Thus, we should not expect that fitting to the high-pressure equation of state will give a good representation of the forces in the initial stages of a cascade, or even for interstitial defects. Indeed, when there was no accurate measurement of interstitial
Interatomic Potential Development
formation energies, older potentials gave a huge range of values based on extrapolation of near-equilibrium data and uncertain partition of energy between pairwise and many-body terms. Accurate values for these formation energies are now available from first principles calculations and are incorporated in the fitting; therefore, a symptom of the problem is resolved. At very short range, the ionic repulsive interaction can be regarded as a screened-Coulombic interaction, and described by multiplying the Coulombic repulsion between nuclei with a screening function wðr =aÞ: V ðr Þ ¼
Z1 Z2 e 2 wðr =aÞ 4pe0 r
where wðr Þ ! 1 when r ! 0 and Z1 and Z2 are the nuclear charges, and a is the screening length. The most popular parameterization of w is the Biersack– Ziegler potential, which was constructed by fitting a universal screening function to repulsions calculated for many different atom pairs (Ziegler 1985). The Biersack–Ziegler potential has the form wðxÞ ¼ 0:1818e3:2x þ 0:5099e0:9423x þ 0:2802e0:4029x þ 0:02817e0:2016x where 0:8854a0 Z10:23 þ Z20:23 and x ¼ r =a and a0 ¼ 0:529 A˚ is the Bohr radius. This potential must then be joined to the longer ranged fitted potential. There are many ways to do this, with no guiding physical principle except that the potential should be as smooth as possible. Typical implementations ensure that the potential and its first few derivatives are continuous. The short-range interactions arising from high pressure come mainly from isotropic compression, and should be fitted to the many-body part. To achieve this division in glue models, the function F ðrÞ should become repulsive at large r, but fðr Þ should not become very large at small Rij . a¼
1.10.12 Parameterization Having deduced the functional form of the potential from first principles, it remains to choose the fitting functions and fit their parameters to empirical data. Most papers simply state that ‘the potential was parameterized by fitting to . . . .’ The reality is different.
287
Firstly, one must decide what functions to use for the various terms. Here, one may be guided by the physics (atomic charge density tails in EAM, square root embedding in FS, Friedel oscillations), by the anticipated usage (short-range potentials will speed up MD, and discontinuities in derivatives may cause spurious behavior), or simply by practicality (Can the potential energy be differentiated to give forces?). Secondly, one must decide what empirical data to fit. Cohesive energies, elastic moduli, equilibrium lattice parameters, and defect energies are common choices. Accurate ab initio calculations can provide further ‘empirical’ data, notably about relative structural stability, but now increasingly about point defect properties. (It is, of course, possible to calculate all the fitting data from ab initio means. Potentials fitted in this way are sometimes referred to as ab initio. While this is pedantically true, the implication that these potentials are ‘better’ than those fitted to experimental data is irritating.) Ab initio MD can also give energy and forces for many different configurations at high temperature. Force matching44 to ab initio data is one of best ways to produce huge amounts of fitting data. There have been many attempts to fully automate this process, but to date, none have produced reliably good potentials. This is in part because of the fact that although MD only uses forces (differential of the potential), many essential physical features (barrier heights, structural stability, etc.) do depend on energy, which in MD ultimately comes from integrating the forces. Small systematic errors in the forces, which lead to larger errors in the energies, can then cause major errors in MD predictions. Furthermore, if the potential is being used for kinetic Monte Carlo, the forces are irrelevant. By using least squares fitting, all the data may be incorporated in the fit, or some data may be fitted exactly and others approximately. However, since the main aim of a potential is transferability to different cases, the stability of the fitting process should be checked. The best way to proceed is to divide the empirical data arbitrarily into groups for fitting and control, to fit using only a part of the data, and then to check the model against the control data. This process can be done many times with different divisions of fitting and control. Any parameter whose value is highly sensitive to this division should be treated with suspicion. Structural stability is the most difficult thing to check, since one simply has to check as many structures as possible. In addition to testing the ‘usual suspects,’ fcc, bcc, hcp, A15, o-Ti, MD, or lattice
288
Interatomic Potential Development
dynamics can help to check for mechanical instability of trial structures. 1.10.12.1 Effective Pair Potentials and EAM Gauge Transformation Although the various glue-type potentials attribute different aspect of the physics to the N-body and pairwise terms in the potential, if one has complete freedom in choosing the functions for V ðr Þ, F ðrÞ, and f, then it is possible to move energy between the two terms. Johnson and Oh noted that the EAM potential Ui ¼
X
Vij ðrij Þ þ Fi
hX
fj ðrij Þ
i
is invariant under a transformation hX i hX i X fj ðrij Þ ! Fi fj ðrij Þ þ A fj ðrij Þ Fi For an alloy,45 Vab ðr Þ ¼
1 fb ðr Þ f ðr Þ Va ðr Þ þ a Vb ðr Þ 2 fa ðr Þ fb ðr Þ
Thus, it is possible to choose a ‘gauge’ for the potential, for example, by setting F 0 ðr0 Þ ¼ 0 for some reference density r. The advantage of the gauge transformation is that it simplifies fitting the potential. It eliminates terms in F 0 ðr0 Þ for pressure and elastic moduli at the equilibrium volume: these terms are nonlinear in the fitting parameters. Thus, the fitting process can be done by linear algebra. The downside of the gauge transformation is that it destroys the physical intuition behind the form of the many-body term. Moreover, the gauge is determined by a particular reference configuration, a simple concept for elements, but one which does not transfer readily to alloys. The FS potentials do not have this freedom, because the function F is predefined as a square root. However, they introduced the ‘effective pair potential’ pffiffiffiffiffi Veff ðrij Þ ¼ V ðr Þ fðr Þ= r0 where r0 is a reference configuration (typically the equilibrium crystal structure). Many of the equilibrium properties which they used for fitting depend only on this quantity. In addition to gauge transformation, MD depends only on the derivative of the total energy. Energy can be partitioned between atoms in any way one likes, without changing the physical results. However, on-atom properties, such as the magnetic moment in magnetic
potentials, typically do depend on the partition of energy between atoms. Such quantities do not have the gauge-invariance property. 1.10.12.2 Steel
Example: Parameterization for
1.10.12.2.1 FeCr
Steel is of particular importance to radiation damage. Stainless steel is based on FeCr alloys, which have been observed by first principles calculation to exhibit unusual energy of solution. For small Cr concentrations, the energy of solution is negative; however, once the concentration exceeds about 10%, it changes sign. Thus, the FeCr system has a miscibility gap, but even at 0 K, there is a finite Cr concentration in the Fe-rich region. The underlying physics of this is that it is favorable for a Cr atom to dissolve in ferromagnetic Fe, provided the Cr spin is opposite to the Fe. Two adjacent Cr cannot be antiparallel to each other and to the Fe matrix. Thus, nearby Cr atoms suffer magnetic frustration, which leads to repulsion between Cr atoms in FeCr not seen in pure Fe or pure Cr. Reproducing this effect in a potential is a challenging problem. In early work, EAM was regarded as being inappropriate for bcc metals (this turned out to be due to the use of rapidly decaying functions). The original FS functional form stabilized bcc elements, but they were unable to obtain a good fit for the elastic constants in Fe and Cr without introducing further parameters. The two-band model can be applied to the FeCr system46 by assuming that the material can be treated as ferromagnetic, and using s and d as the two bands. They adopted the functional form of the interactions from the iron potential by Ackland and Mendelev, scaling the Cr electron density by the ratio of the atomic numbers 24/26. The CrCr potential was refitted to elastic and point defect properties. As the previous Fe parameterization incorporated with effect of s-electrons in a single embedding function, the socalled s-band density of this model in fact depends only on the FeCr cross potential. It described the excess energy of alloying by a many-body rather than pairwise additive effect. By choosing values which favor Fe atoms with a single Cr neighbor, this potential gives the skew solubility. This is an ingenious solution: magnetic frustration is essentially a 2 þ N-body effect. Cr atoms repel when in an Fe rich ferromagnetic environment; this is neatly captured by the longranged Slater orbital used for the s-electron. It is
Interatomic Potential Development
debatable whether this term is really capturing physics associated with the s-band. A related approach47 created a potential in which the embedding function depends directly on the local Cr concentration. The skew embedding function readily reproduced the phase diagram, which was the intention of the work. However, the short-range ordering and the Cr–Cr repulsion which appears to underlie the physics of radiation damage are less well reproduced. 1.10.12.2.2 FeC
Carbon dissolves readily in iron, producing a strengthening effect that underlies all steel. The physics of this is rather complex: the solution energy is very high (6 eV), and carbon adopts an interstitial position in bcc Fe with a barrier of 0.9 eV to migration. It is attracted to tensile regions of the crystal and to vacancies. It is repelled from compressive regions, including interstitial atoms, although the asymmetry of the interstitial means there are some tensile sites at larger distances which are favorable. First principles calculation also shows that the carbon forms covalently bonded pairs in a vacancy site, and the energy gained from the bond more than compensated for the reduced space available to the second carbon atom. These criteria prove rather demanding for parameterizing FeC potentials, even though they only cover compositions with vey low carbon concentrations. An early pair potential by Johnson48 proved extremely successful, and it was only once first principles calculation revealed the repulsion between C and interstitials that a major problem was revealed. Although interstitial atoms are specific to radiation damage applications, there is a strong implication that the binding to other overcoordinated regions such as dislocation cores may be wrong. It appears to be very difficult to obtain the correct bonding of carbon in all the cases above with smooth EAM-type functions. Even in recent potentials,49 like those by Johnson, carbon binds chemically to the interstitial. There is a qualitative explanation for this. Electronic structure calculation50 shows that the electrons pile up between the two nearest neighbors in the octahedral configuration, essentially forming two FeC bonds. However, all the simple potentials described above obtain similar bonding from all six neighbors, stabilizing the octahedral site because the tetrahedral site has only four neighbors. This approach favors carbon bonding to highly coordinated defects, and underlies the bonding to interstitials. An EAM
289
potential with a Tersoff–Brenner style saturation in the C cohesion has addressed this problem.51 This is tuned to saturate at two near neighbors, and so favors the octahedral site but not overcoordination. As a consequence, it does not bind carbon to the interstitial. 1.10.12.3
Austenitic Steel
Few potentials exist for fcc iron. Calculating hightemperature phase transitions is a subtle process involving careful calculation of free energy differences, which makes it difficult to incorporate in the fitting process. Although the bcc–fcc transition has been reported for one EAM iron potential,52 it is probably fortuitous and has been disputed.53 In any case, it is at far higher temperature than experimentally observed. Worse, it is likely that magnetic entropy plays a significant role,54 and the magnetic degrees of freedom are seldom included in potentials. Some very recent progress has been made; an analytic bond order potential53 shows bcc–fcc–bcc transitions for iron and an MEAM parameterization by Baskes successfully reproduces the bcc–fcc–bcc phase transitions in iron on heating by using temperature-dependent parameters. It seems certain that the challenge of austenitic steel will be receiving more attention in the next few years.
1.10.13 Analyzing a Million Coordinates 1.10.13.1 Useful Concepts Without True Physical Meaning For very large simulations, imaging is a problem, since showing all atoms in a massive simulation is likely to obscure the important regions. There are a number of heuristic quantities arising from the interatomic potential which can be used to pick out the atoms associated with atypical configurations of interest. Most empirical potentials define the energy per atom. The atomic level stress55 1 X a b sab f r mi via vib i ¼ 2Oi j ij ij where f is the force on atom i due to atom j which determines the glass transition.56 The concept of ‘local crystal structure’ can be used to locate twin boundaries, phase transitions, etc. This may be done by common neighbor analysis, or
290
Interatomic Potential Development
bespoke investigation of pair and angular distribution functions to search for particular configurations.57 The balance between pair and many-body energies in glue-type models, or more significantly the various angular density functions of MEAM. The magnetization in ‘magnetic’ potentials.
15.
Although uniquely defined for a given potential, many of these are not well-defined concepts in electronic structure. Yet, they can be extremely useful in identifying the atoms in far-from-equilibrium environments.
19.
1.10.14 Summary
16. 17. 18.
20.
21. 22. 23.
Interatomic potential development is a continuing challenge for materials modeling. They represent the only way to perform MD, which in turn is crucial for the nonequilibrium and off-lattice processes, which dominate radiation damage. Despite best efforts, few potentials can be reliably employed to predict quantitative energies beyond where they are fitted. Their most useful role is to reveal processes and topologies that might be of importance in real materials.
References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.
13. 14.
Robinson, M. T. Phys. Rev. B 1989, 40, 10717. Elliott, R. S.; Shaw, J. A.; Triantafyllidis, N. Int. J. Solids Struct. 2002, 39(13–14), 3845–3856. (a) Bruce, A. D.; Wilding, N. B.; Ackland, G. J. Phys. Rev. Lett. 1997, 79, 3002; (b) Jackson, A. N.; Bruce, A. D.; Ackland, G. J. Phys. Rev. E 2002, 65(3), 036710. Heine, V. In Solid State Physics; Ehrenreich, H., Seitz, F., Turnball, D., Eds.; Academic Press: New York, 1980; Vol. 35. See e.g., Ashcroft, N. W.; Mermin, N. D. Solid State Physics; Thomson Learning: Stamford, CT. See e.g., (a) Cohen, M. H.; Heine, V. Phys. Rev. 1961, 122, 1821; (b) Heine, V.; Weaire, D. Solid State Phys. 1971, 24, 247. Daw, M. S.; Baskes, M. I. Phys. Rev. Lett. 1983, 50, 1285. (a) Daw, M. S.; Baskes, M. I. Phys. Rev. B 1984, 29, 6443; (b) Jacobsen, K. W.; Norskov, J. K.; Puska, M. J. Phys. Rev. B 1987, 35, 7423–7442. (a) Ducastelle, F. J. Phys. 1970, 31, 1055; (b) Ducastelle, F.; Cyrot-Lackmann, F. J. Phys. Chem. Solids 1971, 32, 285. Ducastelle, F.; Cyrot-Lackmann, F. Adv. Phys. 1967, 16, 393; J. Phys. Chem. Solids 1970, 31, 1295. Haydock, R. Solid State Phys. 1980, 35, 216. (a) Cyrot-Lackmann, F. Surf. Sci. 1968, 15, 535; (b) Ducastelle, F. J. Phys. 1970, 31, 1055; (c) Ducastelle, F.; Cyrot-Lackmann, F. J. Phys. Chem. Solids 1971, 32, 285; see e.g., ‘The Physics of Metals’ by J. Friedel. Finnis, M. W.; Sinclair, J. E. Philos. Mag. A 1984, 50, 45. Ackland, G. J.; Vitek, V.; Finnis, M. W. J. Phys. F 1988, 18, L153.
24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40.
41.
(a) Coulson, C. A. Proc. R. Soc. A 1939, 169, 413–428; (b) Pettifor, D. G. Phys. Rev. Lett. 1989, 63, 2480–2483; (c) Finnis, M. W. Prog. Mater. Sci. 2007, 52(2–3), 133–153. (a) Ercolessi, F.; Parrinello, M.; Tosati, E. Philos. Mag. A 1988, 58, 213; (b) Robertson, I. J.; Heine, V.; Payne, M. C. Phys. Rev. Lett. 1993, 70, 1944–1947. Ackland, G. J. J. Phys. CM 2002, 14, 2975. Willaime, F.; Massobrio, C. Phys. Rev. Lett. 1989, 63, 2244; Phys. Rev. B 1991, 43, 11653–11665. Mendelev, M. I.; Ackland, G. J. Philos. Mag. Lett. 2007, 87, 349–359. Here, as in molecular dynamics codes, it is convenient to write the potential as a function of r2, which avoids the need for square roots, see Ackland et al. Philos. Mag. A 1987, 56, 735. Ackland, G. J.; Vitek, V. Phys. Rev. B 1990, 41, 10324. Ackland, G. J.; Finnis, M. W.; Vitek, V. J. Phys. F 1988, 18, L153. McMahon, M. I.; Nelmes, R. J.; Rekhi, S. Phys. Rev. Lett. 2001, 87, 255502. Ackland, G. J.; Reed, S. K. Phys. Rev. B 2003, 67, 174108. (a) Hellmann, H. Einfuhrung in die Quantenchemie; Franz Deuticke: Leipzig, 1937; p 285; (b) Feynman, R. P. Phys. Rev. 1939, 56(4), 340. Olsson, P.; Wallenius, J.; Domain, C.; Nordlund, K.; Malerba, L. Phys. Rev. B 2005, 72, 214119. Dudarev, S. L.; Derlet, P. M. J. Phys. Condens. Matter 2005, 17, 7097. Ackland, G. J.; Bacon, D. J.; Calder, A. F.; Harry, T. Philos. Mag. A 1997, 75, 713–732. Stillinger, F. H.; Weber, T. A. Phys. Rev. B 1985, 31, 5262. Ackland, G. J. Phys. Rev. B 1989, 40, 10351. Tersoff, J. Phys. Rev. B 1988, 38, 9802. (a) Tersoff, J. Phys. Rev. B 1988, 37, 6991; (b) Brenner, D. W. Phys. Rev. B 1990, 42(15), 9458. (a) Tersoff, J. Phys. Rev. B 1994, 49, 16349 (b) Gao, F.; Weber, W. J. Nucl. Instr. Meth. Phys. Res. B 2002, 191, 504–508. Juslin, N.; et al. J. Appl. Phys. 2005, 98, 123520. Brooks, B. R.; et al. J. Comp. Chem. 2009, 30, 1545–1615. Cornell, W. D.; et al. J. Am. Chem. Soc. 1995, 117, 5179. Govers, K.; Lemehov, S.; Hou, M.; Verwerft, M. J. Nucl. Mater. 2007, 366, 161. Wachtman, J. B.; Wheat, M. L.; Anderson, H. J.; Bates, J. L. J. Nucl. Mater. 1965, 16(1), 39–41. Devanathan, R.; Yu, J.; Weber, W. J. J. Chem. Phys. 2009, 130, 174502. (a) Lewis, G. V.; Catlow, C. R. A. J. Phys. C 1985, 18, 1149; (b) Jackson, R. A.; et al. Philos. Mag. A 1986, 53, 27; (c) Morelon, N. D.; Ghaleb, D.; Delaye, J. M.; Van Brutzel, L. Philos. Mag. 2003, 83, 1533; (d) Basak, C. B.; Sengupta, A. K.; Kamath, H. S. J. Alloys Compd. 2003, 360, 210; (e) Arima, T.; Yamasaki, S.; Inagaki, Y.; Idemitsu, K. J. Alloys Compd. 2005, 400, 43; (f) Yakub, E.; Ronchi, C.; Staicu, D. J. Chem. Phys. 2007, 127, 094508. (a) Catlow, C. R. A.; Grimes, R. W. J. Nucl. Mater. 1989, 165, 313; (b) Grimes, R. W.; et al. J. Am. Ceram. Soc. 1989, 72; (c) Grimes, R. W.; Miller, R. H.; Catlow, C. R. A. J. Nucl. Mater. 1990, 172, 123; (d) Ball, R. G. J.; Grimes, R. W. J. Chem. Soc. Faraday Trans. 1990, 86, 1257; (e) Grimes, R. W.; Catlow, C. R. A. Philos. Trans. R. Soc. Lond. A 1991, 335, 609; (f) Grimes, R. W.; Ball, R. G. J.; Catlow, C. R. A. J. Phys. Chem. Solids 1992, 53, 475; (g) Grimes, R. W. Mat. Res. Soc. Symp. Proc. 1992, 257, 361; (h) Busker, G.; Grimes, R. W.; Bradford, M. R. J. Nucl. Mater. 2000, 279, 46; (i) Busker, G.; Grimes, R. W.; Bradford, M. R. J. Nucl.
Interatomic Potential Development
42. 43. 44. 45. 46. 47. 48. 49.
Mater. 2003, 312, 156; (j) Meis, C.; Chartier, A. J. Nucl. Mater. 2005, 341, 25. Streitz, F. H.; Mintmire, J. W. Phys. Rev. B 1994, 50, 11996. Elsener, A.; et al. Model. Simul. Mater. Sci. Eng. 2008, 16, 025006. Ercolessi, F.; Adams, J. B. Europhys. Lett. 1994, 26, 583. Johnson, R. A. Phys. Rev. B 1989, 39, 12554; Phys. Rev. B 1988, 37, 3924; Phys. Rev. B 1990, 41, 9717. Olsson, P.; Wallenius, J.; Domain, C.; Nordlund, K.; Malerba, L. Phys. Rev. B 2005, 72, 214119. Caro, A.; Crowson, D. A.; Caro, M. Phys. Rev. Lett. 2005, 95, 075702. Johnson, R. A. Phys. Rev. 1964, 134, 1329. Lau, T. T.; Forst, C. J.; Lin, X.; Gale, J. D.; Yip, S.; Van Vliet, K. J. Phys. Rev. Lett. 2007, 98, 215501; (b) Becquart, C. S.; et al. Comp. Mater. Sci. 2007, 40, 119.
50.
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Domain, C.; Becquart, C. S.; Foct, J. Phys. Rev. 2004, B69, 144112. 51. Hepburn, D. J.; Ackland, G. J. Phys. Rev. B 2008, 78, 165115. 52. Lopasso, E. M.; Caro, M.; Caro, A.; Turchi, P. E. A. Phys. Rev. B 2003, 68, 214205. 53. Muller, M.; Erhart, P.; Albe, K. J. Phys. CM. 2007, 19, 326220. 54. Muller et al. JPCM 2007, 19, 326220. 55. Maeda, K.; Egami, T.; Vitek, V. Philos. Mag. A 1980, 41, 883. 56. Kulp, D. T.; et al. Model. Simul. Mater. Sci. Eng. 1992, 1, 315. 57. (a) Honeycutt, J. D.; Andersen, A. C. J. Phys. Chem. 1987, 91, 4950; (b) Pinsook, U. Phys. Rev. B 1998, 58, 11252; (c) Ackland, G. J.; Jones, A. P. Phys. Rev. B 2006, 73, 054104.
1.11
Primary Radiation Damage Formation
R. E. Stoller Oak Ridge National Laboratory, Oak Ridge, TN, USA
ß 2012 Elsevier Ltd. All rights reserved.
1.11.1 1.11.2 1.11.3 1.11.4 1.11.4.1 1.11.4.2 1.11.4.3 1.11.4.3.1 1.11.4.3.2 1.11.4.4 1.11.4.4.1 1.11.4.4.2 1.11.4.4.3 1.11.5 1.11.5.1 1.11.5.2 1.11.5.3 1.11.5.4 1.11.6 References
Introduction Description of Displacement Cascades Computational Approach to Simulating Displacement Cascades Results of MD Cascade Simulations in Iron Cascade Evolution and Structure Stable Defect Formation In-cascade Clustering of Point Defects Interstitial clustering Vacancy clustering Secondary Factors Influencing Cascade Damage Formation Influence of preexisting defects Influence of free surfaces Influence of grain boundaries Comparison of Cascade Damage in Other Metals Defect Production in Pure Metals Defect Production in Fe–C Defect Production in Fe–Cu Defect Production in Fe–Cr Summary and Needs for Further Work
Abbreviations BCA COM D MC MD NN NRT PKA RCS SIA T TEM
Binary collision approximation Center of mass Deuterium Monte Carlo Molecular dynamics Nearest neighbor Norgett, Robinson, and Torrens Primary knock-on atom Replacement collision sequences Self-interstitial atom Tritium Transmission electron microscope
1.11.1 Introduction Many of the components used in nuclear energy systems are exposed to high-energy neutrons, which are a by-product of the energy-producing nuclear reactions. In the case of current fission reactors, these neutrons are the result of uranium fission,
293 294 297 300 303 305 308 308 312 315 316 318 319 323 324 325 328 328 328 329
whereas in future fusion reactors employing deuterium (D) and tritium (T) as fuel, the neutrons are the result of DT fusion. Spallation neutron sources, which are used for a variety of material research purposes, generate neutrons as a result of spallation reactions between a high-energy proton beam and a heavy metal target. Neutron exposure can lead to substantial changes in the microstructure of the materials, which are ultimately manifested as observable changes in component dimensions and changes in the material’s physical and mechanical properties as well. For example, radiation-induced void swelling can lead to density changes greater than 50% in some grades of austenitic stainless steels1 and changes in the ductile-to-brittle transition temperature greater than 200 C have been observed in the low-alloy steels used in the fabrication of reactor pressure vessels.2,3 These phenomena, along with irradiation creep and radiation-induced solute segregation are discussed extensively in the literature4 and in more detail elsewhere in this comprehensive volume (e.g., see Chapter 1.03, Radiation-Induced Effects on Microstructure; Chapter 1.04, Effect of Radiation
293
294
Primary Radiation Damage Formation
on Strength and Ductility of Metals and Alloys; and Chapter 1.05, Radiation-Induced Effects on Material Properties of Ceramics (Mechanical and Dimensional)). The objective of this chapter is to describe the process of primary damage production that gives rise to macroscopic changes. This primary radiation damage event, which is referred to as an atomic displacement cascade, was first proposed by Brinkman in 1954.5,6 Many aspects of the cascade damage production discussed below were anticipated in Brinkman’s conceptual description. In contrast to the time scale required for radiationinduced mechanical property changes, which is in the range of hours to years, the primary damage event that initiates these changes lasts only about 1011 s. Similarly, the size scale of displacement cascades, each one being on the order of a few cubic nanometers, is many orders of magnitude smaller than the large structural components that they affect. Although interest in displacement cascades was initially limited to the nuclear industry, cascade damage production has become important in the solid state processing practices of the electronics industry also.7 The cascades of interest to the electronics industry arise from the use of ion beams to fabricate, modify, or analyze materials for electronic devices. Another related application is the modification of surface layers by ion beam implantation to improve wear or corrosion resistance of materials.8 The energy and mass of the particle that initiates the cascade provide the principal differences between the nuclear and ion beam applications. Neutrons from nuclear fission and DT fusion have energies up to about 20 MeV and 14.1 MeV, respectively, while the peak neutron energy in spallation neutron sources reaches as high as the energy of the incident proton beam, 1 GeV in modern sources.9 The neutron mass of one atomic mass unit (1 a.m.u.1.66 10–27 kg) is much less than that of the mid-atomic weight metals that comprise most structural alloys. In contrast, many ion beam applications involve relatively low-energy ions, a few tens of kiloelectronvolts, and the mass of both the incident particle and the target is typically a few tens of atomic mass unit. The use of somewhat higher energy ion beams as a tool for investigating neutron irradiation effects is discussed in Chapter 1.07, Radiation Damage Using Ion Beams. This chapter will focus on the cascade energies of relevance to nuclear energy systems and on iron, which is the primary component in most of the alloys employed in these systems. However, the description of the basic physical mechanisms of displacement
cascade formation and evolution given below is generally valid for any crystalline metal and for all of the applications mentioned above. Although additional physical processes may come into play to alter the final defect state in ionic or covalent materials due to atomic charge states,10 the ballistic processes observed in metals due to displacement cascades are quite similar in these materials. This has been demonstrated in molecular dynamics (MD) simulations in a range of ceramic materials.11–15 Finally, synergistic effects due to nuclear transmutation reactions will not be addressed; the most notable of these, helium production by (n,a) reactions, is the topic of Chapter 1.06, The Effects of Helium in Irradiated Structural Alloys.
1.11.2 Description of Displacement Cascades In a crystalline material, a displacement cascade can be visualized as a series of elastic collisions that is initiated when a given atom is struck by a high-energy neutron (or incident ion in the case of ion irradiation). The initial atom, which is called the primary knock-on atom (PKA), will recoil with a given amount of kinetic energy that it dissipates in a sequence of collisions with other atoms. The first of these are termed secondary knock-on atoms and they will in turn lose energy to a third and subsequently higher ordered knock-ons until all of the energy initially imparted to the PKA has been dissipated. Although the physics is slightly different, a similar event has been observed on billiard tables for many years. Perhaps the most important difference between billiards and atomic displacement cascades is that an atom in a crystalline solid experiences the binding forces that arise from the presence of the other atoms. This binding leads to the formation of the crystalline lattice and the requirement that a certain minimum kinetic energy must be transferred to an atom before it can be displaced from its lattice site. This minimum energy is called the displacement threshold energy (Ed) and is typically 20 to 40 eV for most metals and alloys used in structural applications.16 If an atom receives kinetic energy in excess of Ed, it can be transported from its original lattice site and come to rest within the interstices of the lattice. Such an atom constitutes a point defect in the lattice and is called an interstitial or interstitial atom. In the case of an alloy, the interstitial atom may be referred to as a self-interstitial atom (SIA) if the atom is the primary
Primary Radiation Damage Formation
Em ¼ 4Eo A1 A2 =ðA1 þ A2 Þ2
½1
where A1 and A2 are the atomic masses of the two particles. Two limiting cases are of interest. If particle 1 is a neutron and particle 2 is a relatively heavy element such as iron, Em 4E0/A. Alternately, if A1 ¼ A2, any energy up to E0 can be transferred. The former case corresponds to the initial collision between a neutron and the PKA, while the latter corresponds to the collisions between lattice atoms of the same mass. Beginning with the work of Brinkman mentioned above, various models were proposed to compute the total number of atoms displaced by a given PKA as a function of energy. The most widely cited model was that of Kinchin and Pease.17 Their model assumed that between a specified threshold energy and an upper energy cut-off, there was a linear relationship between the number of Frenkel pair produced and the PKA energy. Below the threshold, no new displacements would be produced. Above the high-energy cut-off, it was assumed that the additional energy was dissipated in electronic excitation and ionization. Later, Lindhard and coworkers developed a detailed theory for energy partitioning that could be used to compute the fraction of the PKA energy that was dissipated in the nuclear system in elastic collisions and in electronic losses.18 This work was used by Norgett, Robinson, and Torrens (NRT) to develop a secondary displacement model that is still used as a standard in the nuclear industry and elsewhere to compute atomic displacement rates.19
The NRT model gives the total number of displaced atoms produced by a PKA with kinetic energy EPKA as nNRT ¼ 0:8Td ðEPKA Þ=2Ed0
½2
where Ed is an average displacement threshold energy.16 The determination of an appropriate average displacement threshold energy is somewhat ambiguous because the displacement threshold is strongly dependent on crystallographic direction, and details of the threshold surface vary from one potential to another. An example of the angular dependence is shown in Figure 1,20 for MD simulations in iron obtained using the Finnis–Sinclair potential.21 Moreover, it is not obvious how to obtain a unique definition for the angular average. Nordlund and coworkers22 provide a comparison of threshold behavior obtained with 11 different iron potentials and discusses several different possible definitions of the displacement threshold energy. The factor Td in eqn [2] is called the damage energy and is a function of EPKA. The damage energy is the amount of the initial PKA energy available to cause atomic displacements, with the fraction of the PKA’s initial kinetic energy lost to electronic excitation being responsible for the difference between EPKA and Td. The ratio of Td to EPKA for iron is shown in Figure 2 as a function of PKA energy, where the analytical fit to Lindhard’s theory described by Norgett and coworkers19 has been used to obtain Td. Note that a significant fraction of the PKA energy is dissipated in electronic processes even for energies 80 ?
Stable Unstable Knock-on energy (eV)
alloy component (e.g., iron in steel) to distinguish it from impurity or solute interstitials. The SIA nomenclature is also used for pure metals, although it is somewhat redundant in that case. The complementary point defect is formed if the original lattice site remains vacant; such a site is called a vacancy (see Chapter 1.01, Fundamental Properties of Defects in Metals for a discussion of these defects and their properties). Vacancies and interstitials are created in equal numbers by this process and the name Frenkel pair is used to describe a single, stable interstitial and its related vacancy. Small clusters of both point defect types can also be formed within a displacement cascade. The kinematics of the displacement cascade can be described as follows, where for simplicity we consider the case of nonrelativistic particle energies with one particle initially in motion with kinetic energy E0 and the other at rest. In an elastic collision between two such particles, the maximum energy transfer (Em) from particle (1) to particle (2) is given by
295
?
?
60
40
20
0 [100]
[110] [210]
[111] [221]
[100] [211]
Knock-on direction
Figure 1 Angular dependence of displacement threshold energy for iron at 0 K. Reproduced from Bacon, D. J.; Calder, A. F.; Harder, J. M.; Wooding, S. J. J. Nucl. Mater. 1993, 205, 52–58.
296
Primary Radiation Damage Formation
as low as a few kiloelectronvolts. The factor of 0.8 in eqn [2] accounts for the effects of realistic (i.e., other than hard sphere) atomic scattering; the value was obtained from an extensive cascade study using the binary collision approximation (BCA).23,24 The number of stable displacements (Frenkel pair) predicted by both the original Kinchin–Pease model and the NRT model is shown in Figure 3 as a function of the PKA energy. The third curve in the figure will be discussed below in Section 1.11.3. The MD results presented in Section 1.11.4.2 indicate that nNRT overestimates the total number of
Ratio: damage energy to PKA energy
0.9 0.85 0.8 0.75 0.7 0.65 0.6
0
20
40
60
80
100
PKA energy (keV)
Figure 2 Ratio of damage energy (Td) to PKA energy (EPKA) as a function of PKA energy.
Frenkel pair that remain after the excess kinetic energy in a displacement cascade has been dissipated at about 10 ps. Many more defects than this are formed during the collisional phase of the cascade; however, most of these disappear as vacancies and interstitials annihilate one another in spontaneous recombination reactions. One valuable aspect of the NRT model is that it enabled the use of atomic displacements per atom (dpa) as an exposure parameter, which provides a common basis of comparison for data obtained in different types of irradiation sources, for example, different neutron energy spectra, ion irradiation, or electron irradiation. The neutron energy spectrum can vary significantly from one reactor to another depending on the reactor coolant and/or moderator (water, heavy water, sodium, graphite), which leads to differences in the PKA energy spectrum as will be discussed below. This can confound attempts to correlate irradiation effects data on the basis of parameters such as total neutron fluence or the fluence above some threshold energy, commonly 0.1 or 1.0 MeV. More importantly, it is impossible to correlate any given neutron fluence with a charged particle fluence. However, in any of these cases, the PKA energy spectrum and corresponding damage energies can be calculated and the total number of displacements obtained using eqn [2] in an integral calculation. Thus, dpa provides an environmentindependent radiation exposure parameter that in
14 Kinchin–Pease model NRT model with PKA energy NRT model with NRT damage energy
10 8 1200 Number of Frenkel pair
Number of Frenkel pair
12
6 4
1000 800 600 400 200
2
0 0
20
40
60
80
100
120
140
PKA energy (keV)
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
PKA energy (keV) Figure 3 Predicted Frenkel pair production as a function of PKA energy for alternate displacement models (see text for explanation of models).
Primary Radiation Damage Formation
many cases can be successfully used as a radiation damage correlation parameter.25 As discussed below, aspects of primary damage production other than simply the total number of displacements must be considered in some cases.
1.11.3 Computational Approach to Simulating Displacement Cascades Given the short time scale and small volume associated with atomic displacement cascades, it is not currently possible to directly observe their behavior by any available experimental method. Some of their characteristics have been inferred by experimental techniques that can examine the fine microstructural features that form after low doses of irradiation. The experimental work that provides the best estimate of stable Frenkel pair production involves cryogenic irradiation and subsequent annealing while measuring a parameter such as electrical resistivity.26,27 Less direct experimental measurements include small angle neutron scattering,28 X-ray scattering,29 positron annihilation spectroscopy,30 and field ion microscopy.31 More broadly, transmission electron microscopy (TEM) has been used to characterize the small point defect clusters such as microvoids, dislocation loops, and stacking fault tetrahedra that are formed as the cascade collapses.32–36 The primary tool for investigating radiation damage formation in displacement cascades has been computer simulation using MD, which is a computationally intensive method for modeling atomic systems on the time and length scales appropriate to displacement cascades. The method was pioneered by Vineyard and coworkers at Brookhaven National Laboratory,37 and much of the early work on atomistic simulations is collected in a review by Beeler.38 Other methods, such as those based on the BCA,20,21 have also been used to study displacement cascades. The binary collision models are well suited for very highenergy events, which require that the interatomic potential accurately simulate only close encounters between pairs of atoms. This method requires substantially less computer time than MD but provides less detailed information about lower energy collisions where many-body effects become important. In addition, in-cascade recombination and clustering can only be treated parametrically in the BCA. When the necessary parameters have been calibrated using the results of an appropriate database of MD cascade results, the BCA codes have been shown to reproduce the results of MD simulations reasonably well.39,40
297
A detailed description of the MD method is given in Chapter 1.09, Molecular Dynamics, and will not be repeated here. Briefly, the method relies on obtaining a sufficiently accurate analytical interatomic potential function that describes the energy of the atomic system and the forces on each atom as a function of its position relative to the other atoms in the system. This function must account for both attractive and repulsive forces to obtain the appropriate stable lattice configuration. Specific values for the adjustable coefficients in the function are obtained by ensuring that the interatomic potential leads to reasonable agreement with measured material parameters such as the lattice parameter, lattice cohesive energy, single crystal elastic constants, melting temperature, and point defect formation energies. The process of developing and fitting interatomic potentials is the subject of Chapter 1.10, Interatomic Potential Development. One unique aspect arises when using MD and an empirical potential to investigate radiation damage, viz. the distance of closest approach for highly energetic atoms is much smaller than that obtained in any equilibrium condition. Most potentials are developed to describe equilibrium conditions and must be modified or ‘stiffened’ to account for these short-range interactions. Chapter 1.10, Interatomic Potential Development, discusses a common approach in which a screened Coulomb potential is joined to the equilibrium potential for this purpose. However, as Malerba points out,41 critical aspects of cascade behavior can be sensitive to the details of this joining process. When this interatomic potential has been derived, the total energy of the system of atoms being simulated can be calculated by summing over all the atoms. The forces on the atoms are obtained from the gradient of the interatomic potential. These forces can be used to calculate the atom’s accelerations according to Newton’s second law, the familiar F ¼ ma (force ¼ mass acceleration), and the equations of motion for the atoms can be solved by numerical integration using a suitably small time step. At the end of the time step, the forces are recalculated for the new atomic positions and this process is repeated as long as necessary to reach the time or state of interest. For energetic PKA, the initial time step may range from 1 to 10 1018 s, with the maximum time step limited to 1–10 1015 s to maintain acceptable numerical accuracy in the integration. As a result, MD cascade simulations are typically not run for times longer than 10–100 ps. With periodic boundary conditions, the size of the simulation cell needs to be
298
Primary Radiation Damage Formation
large enough to prevent the cascade from interacting with periodic images of itself. Higher energy events therefore require a larger number of atoms in the cell. Typical MD cascade energies and the approximate number of atoms required in the simulation are listed in Table 1. With periodic boundaries, it is important that the cell size be large enough to avoid cascade self-interaction. For a given energy, this size depends on the material and, for a given material, on the interatomic potential used. Different interatomic potentials may predict significantly different cascade volumes, even though little variation is eventually found in the number of stable Frenkel pair.42 Using a modest number of processors on a modern parallel computer, the clock time required to complete a high-energy simulation with several million atoms is generally less than 48 h. Longer-term evolution of the cascade-produced defect structure can be carried out using Monte Carlo (MC) methods as discussed in Chapter 1.14, Kinetic Monte Carlo Simulations of Irradiation Effects. The process of conducting a cascade simulation requires two steps. First, a block of atoms of the desired size is thermally equilibrated. This permits the lattice thermal vibrations (phonon waves) to be established for the simulated temperature and typically requires a simulation time of approximately 10 ps. This equilibrated atom block can be saved and used as the starting point for several subsequent cascade simulations. Subsequently, the cascade simulations are initiated by giving one of the atoms a defined amount of kinetic energy, EMD, in a specified direction. Statistical variability can be introduced by either
Table 1
further equilibration of the starting block, choosing a different PKA or PKA direction, or some combination of these. The number of simulations required at any one condition to obtain a good statistical description of defect production is not large. Typically, only about 8–10 simulations are required to obtain a small standard error about the mean number of defects produced; the scatter in defect clustering parameters is larger. This topic will be discussed further below when the results are presented. Most of the cascade simulations discussed below were generated using a [135] PKA direction to minimize directional effects such as channeling and directions with particularly low or high displacement thresholds. The objective has been to determine mean behavior, and investigations of the effect of PKA direction generally indicate that mean values obtained from [135] cascades are representative of the average defect production expected in cascades greater than about 1 keV.43 A stronger influence of PKA direction can be observed at lower energies as discussed in Stoller and coworkers.44,45 In the course of the simulation, some procedure must be applied to determine which of the atoms should be characterized as being in a defect state for the purpose of visualization and analysis. One approach is to search the volume of a Wigner–Seitz cell, which is centered on one of the original, perfect lattice sites. An empty cell indicates the presence of a vacancy and a cell containing more than one atom indicates an interstitial-type defect. A more simple geometric criterion has been used to identify defects in most of the results presented below. A sphere with a radius equal to 30% of the iron lattice parameter is
Typical iron atomic displacement cascade parameters
Neutron energy (MeV)
Average PKA energy (keV)a
Corresponding Td (keV)b EMD
NRT displacements
Ratio: Td/EPKA
0.00335 0.00682 0.0175 0.0358 0.0734 0.191 0.397 0.832 2.28 5.09 12.3 14.1c
0.116 0.236 0.605 1.24 2.54 6.6 13.7 28.8 78.7 175.8 425.5 487.3
0.1 0.2 0.5 1.0 2.0 5.0 10.0 20.0 50.0 100.0 200.0 220.4
1 2 5 10 20 50 100 200 500 1000 2000 2204
0.8634 0.8487 0.8269 0.8085 0.7881 0.7570 0.7292 0.6954 0.6354 0.5690 0.4700 0.4523
a
This is the average iron recoil energy from an elastic collision with a neutron of the specified energy. Damage energy calculated using Robinson’s approximation to LSS theory.19 Relevant to D–T fusion energy production.
b c
Typical simulation cell size (atoms) 3 456 6 750 54 000 128 000 250 000 500 k 2.5 M 5–10 M 10–20 M
Primary Radiation Damage Formation
centered on the perfect lattice sites, and a search similar to that just described for the Wigner–Seitz cell is carried out. Any atom that is not within such a sphere is identified as part of an interstitial defect and each empty sphere identifies the location of a vacancy. The diameter of the effective sphere is slightly less than the spacing of the two atoms in a dumbbell interstitial (see below). A comparison of the effective sphere and Wigner–Seitz cell approaches found no significant difference in the number of stable point defects identified at the end of cascade simulation, and the effective sphere method is faster computationally. The drawback to this approach is that the number of defects identified by the algorithm must be corrected to account for the nature of the interstitial defect that is formed. In order to minimize the lattice strain energy, most interstitials are found in the dumbbell configuration; the energy is reduced by distributing the distortion over multiple lattice sites. In this case, the single interstitial appears to be composed of two interstitials separated by a vacancy. In other cases, the interstitial configuration is extended further, as in the case of the crowdion in which an interstitial may be visualized as three displaced atoms and two empty lattice sites. These interstitial configurations are illustrated in Figure 4, which uses the convention adopted throughout this chapter, that is, vacancies are displayed as red spheres and interstitials as green spheres. A simple postprocessing code was used to determine the true number of point defects, which are reported below. Most MD codes describe only the elastic collisions between atoms; they do not account for energy
[111] z
[111]
[110]
Figure 4 Typical configurations for interstitials created in displacement cascades: [110] and [111] dumbbells and [111] crowdion.
299
loss mechanisms such as electronic excitation and ionization. Thus, the initial kinetic energy, EMD, given to the simulated PKA in MD simulations is more analogous to Td in eqn [2] than it is to the PKA energy, which is the total kinetic energy of the recoil in an actual collision. Using the values of EMD in Table 1 as a basis, the corresponding EPKA and nNRT for iron, and the ratio of the damage energy to the PKA energy, have been calculated using the procedure described in Norgett and coworkers.19 and the recommended 40 eV displacement threshold.16 These values are also listed in Table 1, along with the neutron energy that would yield EPKA as the average recoil energy in iron. This is one-half of the maximum energy given by eqn [1]. As mentioned above, the difference between the MD cascade energy, or damage energy, and the PKA energy increases as the PKA energy increases. Discussions of cascade energy in the literature on MD cascade simulations are not consistent with respect to the use of the term PKA energy. The third curve in Figure 3 shows the calculated number of Frenkel pair predicted by the NRT model if the PKA energy is used in eqn [2] rather than the damage energy. The difference between the two sets of NRT values is substantial and is a measure of the ambiguity associated with being vague in the use of terminology. It is recommended that the MD cascade energy should not be referred to as the PKA energy. For the purpose of comparing MD results to the NRT model, the MD cascade energy should be considered as approximately equal to the damage energy (Td in eqn [2]). In reality, energetic atoms lose energy continuously by a combination of electronic and nuclear reactions, and the typical MD simulation effectively deletes the electronic component at time zero. The effects of continuous energy loss on defect production have been investigated in the past using a damping term to slowly remove kinetic energy.46 The related issues of how this extracted energy heats the electron system and the effects of electron–phonon coupling on local temperature have also been examined.47–50 More recently, computational and algorithmic advances have enabled these phenomena to be investigated with higher fidelity.51 Some of the work just referenced has shown that accounting for the electronic system has a modest quantitative effect on defect formation in displacement cascades. For example, Gao and coworkers found a systematic increase in defect formation as they increased the effective electron–phonon coupling in 2, 5, and 10 keV cascade simulations in iron,50 and a similar effect was reported
300
Primary Radiation Damage Formation
by Finnis and coworkers.47 However, the primary physical mechanisms of defect formation that are the focus of this chapter can be understood in the absence of these effects.
1.11.4 Results of MD Cascade Simulations in Iron MD simulations have been employed to investigate displacement cascade evolution in a wide range of materials. The literature is sufficiently broad that any list of references will be necessarily incomplete; Malerba,41 Stoller,43 and others52–70 provide only a representative sample. Additional references will be given below as specific topics are discussed. The recent review by Malerba41 provides a good summary of the research that has been done on iron. These MD investigations of displacement cascades have established several consistent trends in primary damage formation in a number of materials. These trends include (1) the total number of stable point defects produced follows a power-law dependence on the cascade energy over a broad energy range, (2) the ratio of MD stable displacements divided by the number obtained from the NRT model decreases with energy until subcascade formation becomes prominent, (3) the in-cascade clustering fraction of the surviving defects increases with cascade energy, and (4) the effect of lattice temperature on the MD results is rather weak. Two additional observations have been made regarding in-cascade clustering in iron, although the fidelity of these statements depends on the interatomic potential employed. First, the interstitial clusters have a complex, three-dimensional (3D) morphology, with both sessile and glissile configurations. Mobile interstitial clusters appear to glide with a low activation energy similar to that of the monointerstitial (0.1–0.2 eV).71 Second, the fraction of the vacancies contained in clusters is much lower than the interstitial clustering fraction. Each of these points will be discussed further below. The influence of the interatomic potential on cascade damage production has been investigated by several researchers.42,72–74 Such comparisons generally show only minor quantitative differences between results obtained with interatomic potentials of the same general type, although the differences in clustering behavior are more significant with some potentials. Variants of embedded atom or Finnis–Sinclair type potential functions (see Chapter 1.10, Interatomic Potential Development) have most often been used.
However, more substantial differences are sometimes observed that are difficult to correlate with any known aspect of the potentials. The analysis recently reported by Malerba41 is one example. In this case, it appears that the formation of replacement collision sequences (RCS) (discussed in Section 1.11.4.1) was very sensitive to the range over which the equilibrium part of the potential was joined to the more repulsive pair potential that controls short-range interactions. This changed the effective cascade energy density and thereby the number of stable defects produced. Therefore, in order to provide a self-consistent database for illustrating cascade damage production over a range of temperatures and energies and to provide examples of secondary variables that can influence this production, the results presented in this chapter will focus on MD simulations in iron using a single interatomic potential.43,53,54,64–68 This potential was originally developed by Finnis and Sinclair21 and later modified for cascade simulations by Calder and Bacon.58 The calculations were carried out using a modified version of the MOLDY code written by Finnis.75 The computing time with this code is almost linearly proportional to the number of atoms in the simulation. Simulations were carried out using periodic, Parrinello–Rahman boundary conditions at constant pressure.76 As no thermostat was applied to the boundaries, the average temperature of the simulation cell was increased as the kinetic energy of the PKA was dissipated. The impact of this heating appears to be modest based on the observed effects of irradiation temperature discussed below, and on the results observed in the work of Gao and coworkers.77 A brief comparison of the iron cascade results with those obtained in other metals will be presented in Section 1.11.5. The primary variables studied in these cascade simulations is the cascade energy, EMD, and the irradiation temperature. The database of iron cascades includes cascade energies from near the displacement threshold (100 eV) to a 200 keV, and temperatures in the range of 100–900 K. In all cases, the evolution of the cascade has been followed to completion and the final defect state determined. Typically this is reached after a few picoseconds for the low-energy cascades and up to 15 ps for the highest energy cascades. Because of the variability in final defect production for similar initial conditions, several simulations were conducted at each energy to produce statistically meaningful average values. The parameters of most interest from these studies are the number of surviving point defects, the fraction of
Primary Radiation Damage Formation
these defects that are found in clusters, and the size distribution of the point defect clusters. The total number of point defects is a direct measure of the residual radiation damage and the potential for longrange mass transport and microstructural evolution. In-cascade defect clustering is important because it can promote microstructural evolution by eliminating the cluster nucleation phase. The parameters used in the following discussion to describe results of MD cascade simulations are the total number of surviving point defects and the fraction of the surviving defects contained in clusters. The number of surviving defects will be expressed as a fraction of the NRT displacements listed in Table 1, whereas the number of defects in clusters will be expressed as either a fraction of the NRT displacements or a fraction of the total surviving MD defects. Alternate criteria were used to define a point defect cluster in this study. In the case of interstitial clusters, it was usually determined by direct visualization of the defect structures. The coordinated movement of interstitials in a given cluster can be clearly observed. Interstitials Table 2
301
bound in a given cluster were typically within a second nearest-neighbor (NN) distance, although some were bound at third NN. The situation for vacancy clusters will be discussed further below, but vacancy clustering was assessed using first, second, third, and fourth NN distances as the criteria. The vacancy clusters observed in iron tend to not exhibit a compact structure according to these definitions. In order to analyze the statistical variation in the primary damage parameters, the mean value (M), the standard deviation about the mean (s), and the standard error of the mean (e) have been calculated for each set of cascades conducted at a given energy and temperature. The standard error of the mean is calculated as e ¼ s/n0.5, where n is the number of cascade simulations completed.78 The standard error of the mean provides a measure of how well the sample mean represents the actual mean. For example, a 90% confidence limit on the mean is obtained from 1.86e for a sample size of nine.79 These statistical quantities are summarized in Table 2 for a representative subset of the iron cascade database.
Statistical analysis of primary damage parameters derived from MD cascade simulations
Energy (keV)
Temperature (K)
Number of cascades
Surviving MD displacements (mean / standard deviation / standard error)
Clustered interstitials (mean / standard deviation / standard error)
Number
Number
per NRT
per NRT
per MD surviving defects
0.5
100
16
3.94 0.680 0.170
0.790 0.136 0.0340
1.25 1.39 0.348
0.250 0.278 0.0695
0.310 0.329 0.0822
1
100
12
6.08 1.38 0.398
0.608 0.138 0.0398
2.25 1.66 0.479
0.225 0.166 0.0479
0.341 0.248 0.0715
1
600
12
5.25 2.01 0.579
0.525 0.201 .0579
1.92 2.02 0.583
0.192 0.202 0.0583
0.307 0.327 0.0944
1
900
12
4.33 1.07 0.310
0.433 0.107 0.031
1.00 1.28 0.369
0.100 0.128 0.0369
0.221 0.287 0.0829
2
100
10
10.1 2.64 0.836
0.505 0.132 0.0418
4.60 2.80 0.884
0.230 0.140 0.0442
0.432 0.0214 0.00678
5
100
9
22.0 2.12 0.707
0.440 0.0424 0.0141
11.4 2.40 0.801
0.229 0.0481 0.0160
0.523 0.113 0.0375 Continued
302
Table 2
Primary Radiation Damage Formation
Continued
Energy (keV)
Temperature (K)
Number of cascades
Surviving MD displacements (mean / standard deviation / standard error)
Clustered interstitials (mean / standard deviation / standard error)
Number
Number
per NRT
per NRT
per MD surviving defects
5
600
13
19.1 3.88 1.08
0.382 0.0777 0.0215
9.77 4.09 1.13
0.195 0.0817 0.0227
0.504 0.187 0.0520
5
900
8
17.1 2.59 0.915
0.343 0.0518 0.0183
8.38 1.85 0.653
0.168 0.0369 0.0131
0.488 0.0739 0.0261
10
100
15
33.6 5.29 1.37
0.336 0.0529 0.0137
17.0 4.02 1.04
0.170 0.0402 0.0104
0.506 0.101 0.0261
10
600
8
30.5 10.35 3.66
0.305 0.104 0.0366
18.1 8.46 2.99
0.181 0.0846 0.0299
0.579 0.115 0.0406
10
900
7
27.3 5.65 2.14
0.273 0.0565 0.0214
18.6 6.05 2.29
0.186 0.0605 0.0229
0.679 0.0160 0.00606
20
100
10
60.2 8.73 2.76
0.301 0.0437 0.0138
36.7 6.50 2.06
0.184 0.0325 0.0103
0.610 0.0630 0.0199
20
600
8
55.8 5.90 2.09
0.281 0.0290 0.0103
41.6 5.85 2.07
0.211 0.0285 0.0101
0.746 0.0796 0.0281
20
900
10
51.7 9.76 3.09
0.259 0.0488 0.0154
35.4 8.94 2.83
0.177 0.0447 0.0141
0.682 0.0944 0.0299
30
100
16
94.9 13.2 3.29
0.316 0.0440 0.0110
57.2 11.5 2.88
0.191 0.0385 0.00963
0.602 0.0837 0.0209
40
100
8
131.0 12.6 4.45
0.328 0.0315 0.0111
74.5 15.0 5.30
0.186 0.0375 0.0133
0.570 0.102 0.0361
50
100
9
168.3 12.1 4.04
0.337 0.0242 0.00807
93.6 6.95 2.32
0.187 0.0139 0.00463
0.557 0.0432 0.0144
100
100
10
329.7 28.2 8.93
0.330 0.0283 0.0089
184.8 20.5 6.47
0.185 0.0205 0.00650
0.561 0.0386 0.0122
100
600
20
282.4 26.6 5.95
0.282 0.0266 0.00595
185.5 26.9 6.01
0.186 0.0269 0.00601
0.656 0.0556 0.0124
100
900
18
261.0 17.5 4.13
0.261 0.0175 0.00413
168.7 17.3 4.08
0.169 0.0173 0.00408
0.646 0.0498 0.0117
200
100
9
676.7 37.9 12.6
0.338 0.0190 0.00632
370.3 29.5 9.83
0.185 0.0147 0.00491
0.548 0.0464 0.0155
Primary Radiation Damage Formation
1.11.4.1
Cascade Evolution and Structure
The evolution of displacement cascades is similar at all energies, with the development of a highly energetic, disordered core region during the initial, collisional phase of the cascade. Vacancies and interstitials are created in equal numbers, and the number of point defects increases sharply until a peak value is reached. Depending on the cascade energy, this occurs at a time in the range of 0.1–1 ps. This evolution is illustrated in Figure 5 for a range of cascade energies, where the number of vacancies is shown as a function of the cascade time. Many vacancy– interstitial pairs are in quite close proximity at the time of peak disorder. An essentially athermal process of in-cascade recombination of these close pairs takes place as they lose their kinetic energy. This leads to a reduction in the number of defects until a quasisteady-state value is reached after about 5–10 ps. As interstitials in iron are mobile even at 100 K, further short-term recombination occurs between some vacancy-interstitial pairs that were initially separated by only a few atomic jump distances. Finally, a stage is reached where the remaining point defects are sufficiently well separated that further recombination is unlikely on the time scale (a few hundred picoseconds) accessible by MD. Note that the number of stable Frenkel pair is actually somewhat lower than the value shown in Figure 5 because the values obtained using the effective sphere identification
procedure were not corrected to account for the interstitial structure discussed above. A mechanism known as RCS may help explain some aspects of cascade structure.24,41 An RCS can be visualized as an extended defect along a closepacked row of atoms. When the first atom is pushed off its site, it dissipates some energy and pushes a second atom into a third, and so on. When the last atom in this chain is unable to displace another, it is left in an interstitial site with the original vacancy several atomic jumps away. Thus, RCSs provide a mechanism of mass transport that can efficiently separate vacancies from interstitials. The explanation is consistent with the observed tendency for the final cascade state to be characterized by a vacancy-rich central region that is surrounded by a region rich in interstitial-type defects. However, although RCSs are observed, particularly in low-energy cascades, they do not appear to be prominent enough to explain the defect separation observed in higher energy cascades.58 Visualization of cascade dynamics indicates that the separation occurs by a more collective motion of multiple atoms, and recent work by Calder and coworkers has identified a shockwave-induced mechanism that leads to the formation of large interstitial clusters at the cascade periphery.80 This mechanism will be discussed further in Section 1.11.4.3.1. Coherent displacement events involving many atoms have also been reported by Nordlund and coworkers.81
100 000 100 K MD simulations in iron
Number of Frenkel pair
10 000
300 eV 1 keV 5 keV 10 keV 20 keV 100 keV
1000
100
10
1 0.001
0.01
303
0.1 MD simulation time (ps)
Figure 5 Time evolution of defects formed during displacement cascades.
1.0
10
304
Primary Radiation Damage Formation
1 keV cascade in Figure 6(a). However, at higher energies, some channeling82,83 of recoil atoms may occur. This is a result of the atom being scattered into a relatively open lattice direction, which may permit it to travel some distance while losing relatively little energy in low-angle scattering events. The channeling is typically terminated in a highangle collision in which a significant fraction of the recoil atom’s energy is transmitted to the next generation knock-on atom. When significant subcascade formation occurs, the region between high-angle collisions can be relatively defect-free as the cascade develops. This evolution is clearly shown in Figure 7 for a 40 keV cascade, where the branching due to high-angle collisions is observed on a time scale of a few hundreds of femto seconds. One practical implication of subcascade formation is that very high-energy cascades break up into what looks like a group of lower energy cascades. An example of subcascade formation in a 100 keV cascade is shown in Figure 8 where the results of 5 and 10 keV cascades have been superimposed into the same block of atoms for comparison. The impact of subcascade formation on stable defect production will be discussed in the next section.
Defect production tends to be dominated by a series of simple binary collisions at low PKA energies, while the more collective, cascade-like behavior dominates at higher energies. The structure of typical 1 and 20 keV cascades is shown in Figure 6, where parts (a) and (b) show the peak damage state and (c) and (d) show the final defect configurations. The MD cells contained 54 000 and 432 000 atoms for the 1 and 20 keV simulations, respectively. Only the vacant lattice sites and interstitial atoms identified by the effective sphere approach described above are shown. The separation of vacancies from interstitials can be seen in the final defect configurations; it is more obvious in the 1 keV cascade because there are fewer defects present. In addition to isolated point defects, small interstitial clusters are also clearly observed in the 20 keV cascade debris in Figure 6(d). In-cascade clustering is discussed further in Section 1.11.4.3. The morphology of the 20 keV cascade in Figure 6(b) exhibits several lobes which are evidence of a phenomenon known as subcascade formation.82 At low energies, the PKA energy tends to be dissipated in a small volume and the cascades appear as compact, sphere-like entities as illustrated by the
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(c)
(d)
z Vacancy
Interstitial
Figure 6 Structure of typical 1 keV (a,c) and 20 keV (b,d) cascades. Peak damage state is shown in (a and b) and the final stable defect configuration is shown in (c and d).
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50 fs
100 fs
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540 fs
Figure 7 Evolution of a 40 keV cascade in iron at 100 K, illustrating subcascade formation.
MD cascade simulations in iron at 100 K: peak damage
Y
10 keV 100 keV
5 keV X Z
5 keV – 0.26 ps 10 keV – 0.63 ps 100 keV – 0.70 ps Figure 8 Energy dependence of subcascade formation.
1.11.4.2
Stable Defect Formation
Initial work of Bacon and coworkers indicated that the number of stable displacements remaining at the end of a cascade simulation, ND, exhibited a powerlaw dependence on cascade energy.84 For example, their analysis of iron cascade simulations between 0.5 and 10 keV at 100 K showed that the total number of surviving point defects could be expressed as 0:779 ND ¼ 5:67E MD
½3
where EMD is given in kiloelectronvolts. This relationship is not followed below about 0.5 keV because true cascade-like behavior does not occur at these
low energies. Subsequent work by Stoller64–67 indicated that ND also begins to deviate from this energy dependence above 20 keV when extensive subcascade formation occurs. This is illustrated in Figure 9(a) where the values of ND obtained in cascade simulations at 100 K is plotted as a function of cascade energy. At each energy, the data point is an average of between 7 and 26 cascades, and the error bars indicate the standard error of the mean. It appears that three well-defined regions with different energy dependencies exist. A power-law fit to the points in each energy region is also shown in Figure 9(a). The best-fit exponent in the absence of true cascade conditions below 0.5 keV is 0.485. From 0.5 to
306
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1000 Surviving MD displacements
Average and standard error Iron, 100 K MD simulations 1.03
EMD 100
0.75
10
EMD 0.5
EMD 1
0.1
(a)
1 10 MD cascade energy (keV)
100
Surviving displacements per NRT
1.6 Iron, 100 K simulations 1.4 1.2 1 0.8 0.6 0.4 0.2 0 (b)
0.1
1 10 Cascade energy (keV)
100
Figure 9 Cascade energy dependence of stable point defect formation in iron MD cascade simulations at 100 K: (a) total number of interstitials or vacancies and (b) ratio of MD defects to NRT displacements. Data points indicate mean values at each energy, and error bars are standard error of the mean.
20 keV, the exponent is 0.75. This is marginally lower than the value in eqn [2], possibly because the 20 keV data were used in the current fitting. An exponent of 1.03 was found in the range above 20 keV, which is dominated by subcascade formation. Only in the highest energy range do the MD results approach the linear energy dependence predicted by the NRT model. The range of plus or minus one standard error is barely detectable around the data points, indicating that the change in slope is statistically significant. The data from Figure 9(a) are replotted in Figure 9(b) where the number of surviving displacements is divided by the NRT displacements at each energy. The rapid decrease in this MD defect survival
ratio at low energies was first measured in 1978 and is well known.57,85 The error bars again reflect the standard error and the dashed line through the points is only a guide to the eye. The MD/NRT ratio is greater than 1.0 at the lowest values of EMD, indicating that the NRT formulation underestimates defect production in this energy range. This is consistent with the low-energy (near threshold) simulations preferentially producing displacements in the ‘easy’ directions.26 The actual displacement threshold varies with crystallographic direction and is as low as 19 eV in the [100] direction.20,84 Thus, using the recommended average value of 40 eV Ed in eqn [2] predicts fewer defects at low energies. The average value is more appropriate for the higher energy events where true cascade-like behavior occurs. In the cascade-dominated regime, the defect density within the cascade increases with energy. Although many more defects are produced, their close proximity leads to a higher probability of in-cascade recombination and a lower defect survival fraction. The surviving defect fraction shows a slight increase as the cascade energy increases above 20, and the indicated standard errors make it arguable that the increase is statistically significant. If significant, the increase appears to be associated with subcascade formation, which becomes prominent above 10–20 keV. In the channeling regions between the high-angle collisions that produce the subcascades shown in Figures 7 and 8, the moving atom loses energy in many low-angle scattering events that produce low-energy recoils. These are essentially like low-energy cascades, which have higher-than-average defect survival fractions (Figure 9). These events could contribute to the incremental increase in defect survival at the highest energies. The average defect survival fraction of 0.3 NRT shown for cascade energies greater than about 10 keV is consistent with values of Frenkel pair formation obtained from resistivity change measurements following low-temperature neutron irradiation and ion irradiation.26,27,57,85 The effect of irradiation temperature is shown in Figure 10, which compares the defect survival fractions obtained from simulations at 100, 600, and 900 K. Although it is difficult to discern a consistent effect of temperature between the 600 and 900 K data points, the defect survival fraction at 100 K is always somewhat greater than at either of the two higher temperatures. A similar result for iron was reported in Bacon and coworkers.84 In addition to an interest in radiation temperature itself, the effect of temperature is relevant to the
Primary Radiation Damage Formation
simulations presented here because no thermostat was applied to the simulation cell to control temperature. As mentioned above, the energy introduced by the PKA will lead to some heating if the simulation cell temperature is not controlled by a thermostat. For example, in a 1 keV cascade simulation with 54 000 atoms, the average temperature rise will be about 140 K when all the kinetic energy of the PKA is distributed in the system. This change in temperature should be more significant at 100 K than at higher temperatures. The fact that defect survival at 600 and 900 K is lower than at 100 K suggests that the 100 K results may be
Surviving displacements per NRT
1.6
Iron cascade simulations Mean value and standard error
1.4
100 K 600 K 900 K
1.2 1 0.8 0.6 0.4 0.2 0
0.1
1 10 Cascade energy (keV)
100
Figure 10 Temperature dependence of stable defect formation in MD simulations: ratio of MD defects to NRT displacements.
0.7
MD parameter ratio
0.6 0.5
somewhat biased toward lower survival values by the PKA-induced heating. This is in agreement with the effect of temperature reported by Gao and coworkers77 in their study of 2 and 5 keV cascades with a hybrid MD model that extracted heat from the simulation cell. On the other hand, the difference between the 100 and 600 K results is not large, so the effect of 200 K of cascade-induced heating may be modest. A simple assessment of this cascade-induced heating was carried out using 10 keV cascades at 100 K. Two independent sets of simulations were carried out, seven simulations in a cell of 128 k atoms and eight simulations in a cell of 250 k atoms. A 10 keV cascade will raise the average temperature by 604 and 309 K, respectively, for these two cell sizes. The results of these simulations are summarized in Figure 11, where the parameters plotted are the surviving defect fraction (per NRT), the fraction of interstitials in clusters (per NRT), and the fraction of interstitials in clusters (per surviving MD defect). In each case, the range of values for the two populations are shown, along with their respective mean values with the standard error indicated. The mean and standard error for the combined data sets is also shown. Although the heating differed by a factor of two, it is clear that the defect survival fraction is essentially identical for both populations. There is a slight trend in the interstitial clustering results, which indicates that a higher temperature (due to a smaller number of atoms) promotes interstitial clustering. This is consistent with the results that will be discussed below.
7–10 keV in 128 k atom 8–10 keV in 250 k atoms 10 keV, all 15
Iron at 100 K Averages and standard error for indicated data points
0.4 0.3 0.2 0.1 0
MD defects/NRT
307
Clustered interstitials per NRT
Clustered interstitials per MD defect
Figure 11 Effect of cascade heating on defect formation in 10 keV cascades at 100 K.
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1.11.4.3 In-cascade Clustering of Point Defects Among the features visible in the two cascades shown in Figure 6 are a number of small interstitial clusters. For example, the cascade debris from the 1 keV cascade in Figure 6(c) contains only seven stable interstitials, but five of them (71%) are in clusters: one di-interstitial and one tri-interstitial. This tendency for point defects to cluster is characteristic of energetic displacement cascades, and it differentiates neutron and ion irradiation from typical 0.5 to 1 MeV electron irradiation, which primarily produces only isolated Frenkel pair defects. The differences between in-cascade vacancy and interstitial clustering discussed below, and the fact that their migration behavior is also quite different, have a profound influence on radiation-induced microstructural evolution at longer times. This impact of point defect clusters on microstructural evolution is discussed in detail in Chapter 1.13, Radiation Damage Theory. 1.11.4.3.1 Interstitial clustering
The dependence of in-cascade interstitial clustering on cascade energy is shown in Figure 12 for simulation temperatures of 100, 600, and 900 K, where the average number of interstitials in clusters of size two or larger at each energy has been divided by the total number of surviving interstitials in part (a), and by the number of displaced atoms predicted by the NRT model for that energy in part (b). The data points and error bars in Figure 12 indicate the mean and standard error at each energy. The error bars can be used to make two significant comments. First, the relative scatter is much higher at lower energies, which is similar to the case of defect survival shown in Figure 10. Second, comparing again with Figure 10, the standard errors about the mean for interstitial clustering are greater at each energy than they are for defect survival. The fact that the interstitial clustering fraction exhibits greater variability between cascades at a given energy than does defect survival is essentially related to the variety of defect configurations that are possible. A given amount of kinetic energy tends to produce a given number of stable point defects; this simple observation is embedded in the NRT model, that is, the number of predicted defects is linear in the ratio of the energy available to the energy per defect. However, any specific number of point defects can be arranged in many different ways.
At the lowest energies, where relatively few defects are created, some cascades produce no interstitial clusters and this is primarily responsible for the larger error bars at these energies. The average fraction of interstitials in clusters is about 20% of the NRT displacements above 5 keV, which corresponds to about 60% of the total surviving interstitials. Although it is not possible to discern a systematic effect of temperature below 10 keV, there is a trend toward greater clustering with increasing temperature at higher energies. This can be more clearly seen in Figure 12(a) where the ratio of clustered interstitials to surviving interstitials is shown, and in the high-energy values in Table 2. This effect of temperature on interstitial clustering in these adiabatic simulations is consistent with the observations of Gao and coworkers77 mentioned above, that is, they found that the interstitial clustering fraction increases with temperature. The interstitial cluster size distributions exhibit a consistent dependence on cascade energy and temperature as shown in Figure 13 (where a size of 1 denotes the single interstitial). The cascade energy dependence at 100 K is shown in Figure 13(a), where the size distributions from 10 and 50 keV are included. The influence of cascade temperature is shown for 10 keV cascades in Figure 13(b), and for 20 keV cascades in Figure 13(c). All interstitial clusters larger than size 10 are combined into a single class in the histograms in Figure 13. The interstitial cluster size distribution shifts to larger sizes as either the cascade energy or temperature increases. An increase in the clustering fraction at the higher temperatures is most clearly seen as a decrease in the number of mono-interstitials. Comparing Figures 13(b) and 13(c) demonstrates that the temperature dependence increases as the cascade energy increases. The largest interstitial cluster observed in these simulations was contained in a 20 keV cascade at 600 K as shown in Figure 14. This large cluster was composed of 33 interstitials (<111> crowdions), and exhibited considerable mobility via what appeared to be a 1D glide in a <111> direction.64,66 Although the number of point defects produced and the fraction of interstitials in clusters was shown to be relatively independent of neutron energy spectrum,82 the increase in the number of large clusters at higher energies suggested that the in-cascade cluster size distributions may exhibit more sensitivity to neutron energy spectrum than did these other parameters. At 100 K, there are no interstitial clusters larger than 8 for cascade energies of 10 keV or
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309
Clustered interstitials per surviving MD defects
0.9 0.8 0.7
Iron cascade simulations Mean and standard error: 100 K 600 K 900 K
0.6 0.5 0.4 0.3 0.2 0.1 0 0.1
1
(a)
10 Cascade energy (keV)
100
0.45 Iron cascade simulations Mean and standard error: 100 K 600 K 900 K
Clustered interstitials per NRT
0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 0.1 (b)
1
10 Cascade energy (keV)
100
Figure 12 Fraction of surviving interstitials contained in clusters at 100 K; the fraction in (a) is relative to the total number of MD defects created and in (b) is relative to the NRT displacements.
less. Therefore, the fraction of interstitials in clusters of 10 or more was chosen as an initial parameter for evaluation of the size distributions. This partial interstitial clustering fraction is shown in Figure 15. As the large clusters are relatively uncommon, the fraction of interstitials contained in them is correspondingly small. This leads to the relatively large standard errors shown in the figure. However, it is clear that the energy dependence of the formation of these large clusters is much stronger than simply the
total fraction of interstitials in clusters. Infrequent large clusters such as the 33-interstitial cluster shown in Figure 14 play a significant role in the sharp increase in this clustering fraction observed between 100 and 600 K for the 20 keV cascades. One unusual observation reported by Wooding and coworkers60 and Gao and coworkers86 was that some of the interstitial clusters exhibited a complex 3D morphology rather than collapsing into planar dislocation loops which are expected to have lower
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Average interstitial cluster distributions: 10 keV cascades at 100 and 900 K
0.5
100 K, 10 keV 50 keV
0.4 0.3 0.2 0.1
0
(a)
Fraction of interstitials in cluster size
Fraction of interstitials in cluster size
Average interstitial cluster distributions: cascades at 100 K
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Average interstitial cluster distributions: 20 keV cascades at 100 and 600 K 20 keV, 100 K 600 K
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(c)
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Number of interstitials in cluster
Figure 13 Fractional size distributions of interstitial clusters formed directly within the cascade, comparison of (a) 10 and 50 keV cascades at 100 K, (b) 10 keV cascades at 100 and 900 K, and (c) 20 keV cascades at 100 and 600 K.
0.18 Interstitials in clusters ³ 10 (per NRT)
Iron cascade simulations
Y
Vacancy Interstitial
Z
X
0.16
Mean and standard error: 100 K 600 K 900 K
0.14 0.12 0.1 0.08 0.06 0.04 0.02 0
0
50
100 150 Cascade energy (keV)
200
Figure 14 Residual defects at 30 ps from a 20 keV cascade at 600 K containing a 33-interstitial cluster.
Figure 15 Cascade energy dependence of interstitials contained in clusters of 10 or more: clustered interstitials divided by NRT displacements.
energy. Similar clusters have been seen in materials such as copper, although they appear to be less frequent in copper.54 The existence of such clusters has been confirmed with interatomic potentials that were developed more recently and with ab initio
calculations.87 Representative examples of these clusters are shown in Figure 16, where a ring-like four-interstitial cluster is shown in (a) and a fiveinterstitial cluster is shown in (b). Unlike the mobile clusters that are composed of [111] crowdions such
Primary Radiation Damage Formation
y
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x z
z
5 SIA: 20 keV, 600 K
4 SIA: 20 keV, 600 K
(a)
(b)
Figure 16 Two examples of sessile interstitial configurations formed in 20 keV, 600 K displacement cascades: clusters in (a) and (b) consist of 4 and 5 SIA, respectively (cf. the glissile cluster configuration in Figure 14).
(a)
(b)
(c)
Atom on lattice site Vacant lattice site Atom in interstitial position (d) Figure 17 Three-dimensional sessile interstitial cluster in 10 keV, 100 K cascade: (a) [010] projection normal to five adjacent {110} planes, (b–d) projections through three of the {110} planes.
as the one shown in Figure 14, the SIA clusters in Figure 16 are not mobile. As such, they have the potential for long lifetimes in the microstructure and may act as nucleation sites for larger interstitial-type defects. Figure 17 shows a somewhat larger sessile cluster containing eight SIA. This particular cluster was examined in detail by searching a large number of low-order crystallographic projections in an attempt to find a projection in which it would appear as a loop. Such a projection could not be found. Rather, the cluster was clearly 3D with a single di-, tri-, and di-interstitial on adjacent, close-packed (110) planes as shown in the figure. The eighth interstitial is a [110] dumbbell that lies perpendicular to the others and on the left side in Figure 17(a). Figure 16(b–d) are [101] projections through the three center (101) planes in Figure 17(a).
It is possible that the typical 10–15 ps MD simulation was not sufficient for the cluster to reorient and collapse. To examine this possibility, the simulation time of a 10 keV cascade at 100 K that contained a similar eight SIA cluster was continued up to 100 ps. Very little cluster restructuring was seen over the time from 10 to 100 ps. In fact, the cluster had coalesced into nearly its final configuration by 10 ps. Gao and coworkers86 carried out a more systematic investigation of sessile cluster configurations with extended simulations at 300 and 500 K. They found that many sessile clusters had converted to glissile within a few hundred picoseconds, but at least one eight SIA cluster remained sessile for 500 ps even after aging at temperatures up to 1500 K. Given the impact that stable sessile clusters would have on the longer timescale microstructural evolution as
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discussed in Chapter 1.13, Radiation Damage Theory, further research is needed to characterize the long-term evolution of cascade-created point defect clusters. It is significant to point out that the conversion of glissile SIA clusters into sessile clusters has also been observed. For example, in a 20 keV cascade at 100 K, a glissile eight SIA cluster was trapped and converted into a sessile nine SIA cluster when it reacted with a single [110] dumbbell. The simulation was continued for more than 200 ps and the cluster remained sessile. The mechanism responsible for interstitial clustering has not been fully understood. For example, it has not been possible to determine whether the motion and agglomeration of individual interstitials and small interstitial clusters during the cascade event contributes to the formation of the larger clusters that are observed at the end of the event. Alternate clustering mechanisms in the literature include the suggestion by Diaz de la Rubia and Guinan88 that large clusters could be produced by a loop punching mechanism. Nordlund and coworkers62 proposed a ‘liquid isolation’ model in which solidification of a melt zone isolates a region with excess atoms. However, a new mechanism has recently been elucidated by Calder and coworkers,80 which seems to explain how both vacancy and interstitial clusters are formed, particularly the less frequent large clusters. Their analysis of cluster formation followed an investigation of the effects of PKA mass and energy, which demonstrated that the probability of producing large vacancy and SIA clusters increases as these parameters increase.89 The conditions of this study produced a unique dataset that motivated the effort to unravel how the clusters were produced. They developed a detailed visualization technique that enabled them to connect the individual displacements of atoms that resulted in defect formation by comparing the start and end positions of atoms in the simulation cell. This defined a continuous series of links between each vacancy and interstitial that were ultimately produced by a chain of displacements. These chains could be displayed in what are called lines of ‘spaghetti.’80 Regions of tangled spaghetti define a volume in which atoms are highly agitated and a certain fraction of which are displaced. Stable interstitials and interstitial clusters are observed on the surface in this volume. From their analysis of cascade development and the final damage state, Calder and coworkers were able to demonstrate a correlation between the production of large SIA clusters and a process taking place very
early in the development of a cascade. Specifically, they established a direct connection between such clusters and the formation of a hypersonic recoil atom that passed through the supersonic pressure wave created by the initiation of the cascade. This highly energetic recoil may create a subcascade and a secondary supersonic shockwave at an appropriate distance from the primary shockwave. In this case, SIA clusters tend to be formed at the point where the primary and secondary shockwaves interfere with one another. This process is illustrated in Figure 18.80 Atoms may be transferred from the primary shockwave volume into the secondary shockwave volume, creating an interstitial supersaturation in the latter and a vacancy supersaturation in the former. In this case, the mechanism of creating large SIA clusters early in the cascade process correspondingly leads to the formation of large vacancy clusters by the end of the thermal spike phase, that is, after several picoseconds. It is notable that the location of the SIA cluster is determined well before the onset of the thermal spike phase, by about 0.1 ps. Calder’s spaghetti analysis provides the opportunity for improved definition of parameters such as cascade volume and energy density; the interested reader is directed to Calder and coworkers80 for more details. 1.11.4.3.2 Vacancy clustering
As discussed elsewhere,59,63,65 in-cascade vacancy clustering in iron is quite low (10% of NRT) when a NN criterion for clustering is applied. This was identified as one of the differences between iron and copper in the comparison of these two materials reported by Phythian and coworkers.59 However, when the coordinates of the surviving vacancies in 10, 20, and 40 keV cascades were analyzed, clear spatial correlations were observed. Peaks in the distributions of vacancy-vacancy separation distances were obtained for the second and fourth NN locations.64 These radial distributions are shown in Figure 19. Similar results were obtained from the analysis of the vacancy distributions in higher energy cascades at 100 and 600 K. The peak observed for vacancies in second NN locations is consistent with the di-vacancy binding energy being greater for second NN (0.22 eV) than for first NN (0.09 eV).90 The reason for the peak at fourth NN is presumably related to this also since two vacancies that are second NN to a given vacancy would be fourth NN. In addition, work discussed by Djurabekova and coworkers91 indicates that there is a small binding energy between two vacancies at the fourth NN distance.
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313
Primary supersonic shock wave (destructive)
PKA
Hypersonic recoil(s)
(a)
Secondary supersonic shock waves (destructive)
(b) High density
Primary shock wave injection of atoms Sonic shock wave (nondestructive) separation
Low density
(c)
Transonic shock- = Spaghetti wave limit zone
(d)
Injected atoms trapped during recovery Lattice recovery
Large interstitial clusters
Spaghetti zone
(e)
(f)
Figure 18 Schematic representation of cascade development leading to the formation of interstitial and vacancy clusters formation. Reproduced from Calder, A. F.; Bacon, D. J.; Barashev, A. V.; Osetsky, Yu. N. Phil. Mag. 2010, 90, 863–884.
Number of nth neighbor pairs
100
Iron cascade simulations at 100 K 50 keV 40 keV 20 keV 10 keV
80
60
40
20
0
1-NN
2-NN 3-NN 4-NN 5-NN 6-NN >6NN Vacancy separation by nearest-neighbor distance
Figure 19 Spatial correlation of all vacancies observed in 10, 20, 40, and 50 keV cascades at 100 K.
An example of a locally vacancy-rich region in a 50 keV, 100 K cascade is shown in Figure 20, where the region around a collection of 14 vacancies has
been extracted from the larger simulation cell. This appears to be a nascent or uncollapsed vacancy cluster. Each of the vacancies has at least one other
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vacancy within the fourth NN spacing of 1.66a0, where a0 is the iron lattice parameter. The ‘cluster’ is shown in two views: a 3D perspective view and an orthographic projection (x) in Figure 20. Such an arrangement of vacancies is similar to some of the vacancy clusters observed by Sato and coworkers in field ion microscope images of irradiated tungsten.92 Since the time period of the MD simulations is too short to allow vacancies to jump (<100 ps), the possibility that these closely correlated vacancies might collapse into clusters over somewhat longer times has
Y
X
Arrows indicate 4-NN spacing = 1.66ao
Z
Figure 20 Typical uncollapsed or nascent vacancy cluster from 50 keV cascade at 100 K; 14 vacancies are contained, each of which is within the fourth nearest-neighbor distance (1.66a0).
been investigated using MC simulations. The vacancy coordinates at the end of the MD simulations were extracted and used as the starting configuration in MC cascade annealing simulations. The expectation of vacancy clustering was confirmed in the MC simulations, where many of the isolated vacancies had clustered within 70 ms.90,93 The energy and temperature dependence of in-cascade vacancy clustering as a fraction of the NRT displacements is shown in Figure 21 for cascade energies of 10–50 keV. Results are shown for clustering criteria of first, second, third, and fourth NN. A comparison of Figure 21 and Figure 12 demonstrates that in-cascade vacancy clustering in iron remains lower than that of interstitials even when the fourth NN criterion is used. This is consistent with the experimentally observed difficulty of forming visible vacancy clusters in iron as discussed by Phythian and coworkers,59 and the fact that only relatively small vacancy clusters are found in positron annihilation studies of irradiated ferritic alloys.94 However, it should be pointed out that work with more recently developed iron potentials finds less difference between vacancy and interstitial clustering.74 The cascade energy dependence of vacancy clustering is similar to that of interstitials; there is essentially zero clustering at the lowest energies but it rapidly increases with cascade energy and is relatively independent of energy above 10 keV. However, vacancy clustering decreases as the temperature increases,
0.25
Vacancy clustering fraction (per NRT)
Iron cascade simulations 100 K 600 K 900 K
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Based on 1st-NN 2nd-NN 4th-NN
0.15
0.1
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40
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Figure 21 Cascade energy dependence of vacancy clustering: clustered vacancies divided by NRT displacements. Data points indicate mean values, and error bars are standard error of the mean.
Primary Radiation Damage Formation
which is consistent with vacancy clusters being thermally unstable. Fractional vacancy cluster size distributions are shown in Figure 22, for which the fourth NN clustering criterion has been used. Figure 22(a) illustrates that the vacancy cluster size distribution shifts to larger sizes as the cascade energy increases from 10 to 50 keV. This is similar to the change shown for interstitial clusters in Figure 13(a). There is a corresponding reduction in the fraction of single vacancies. However, as mentioned above, the effect of cascade temperature shown in Figure 22(b) and 22(c) is the opposite of that observed for interstitials. The magnitude of the temperature effect on the vacancy cluster size distributions also appears to be weaker than in the case of interstitial clusters. The fraction of single vacancies increases and the size distribution shifts to smaller sizes as the temperature increases from 100 to 900 K for the 10 keV cascades, and from 100 to 600 K for 20 keV cascades. Similar to the case of interstitial clusters, the effect of temperature seems to be greater at 20 keV than at 10 keV.
1.11.4.4 Secondary Factors Influencing Cascade Damage Formation The results of simulations such as those presented above should not be viewed as being quantitatively accurate. As already mentioned, subtle changes in the fitting of the interatomic potential can alter the cascade simulations both qualitatively and quantitatively. Even if a sufficiently accurate potential can be identified, the results represent a certain limiting case of what may be observed experimentally. This is because all the simulations mentioned so far were carried out in perfect material – computer-pure material. Nowhere in nature can such perfect metal be found, particularly for iron, which is easily contaminated with minor interstitial impurities such as carbon. In this section, a few examples will be discussed to illustrate how reality may influence cascade damage production relative to the perfect material case. The examples include the influence of preexisting defects, free surfaces, and grain boundaries.
(a)
4th-nearest neighbors, 100 K, 10 keV 50 keV
0.6 0.5 0.4 0.3 0.2 0.1 0
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4 5 6 7 8 9 Number of vacancies in cluster
Average vacancy cluster distribution: 100 versus 900 K 10 keV cascades
Fraction of vacancies in cluster size
Fraction of vacancies in cluster size
Average vacancy cluster distribution: 10 versus 50 keV cascades at 100 K
0.7
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Fraction of vacancies in cluster size
Average vacancy cluster distribution: 100 versus 600 K 20 keV cascades
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(c)
4th-nearest neighbors, 20 keV, 100 K 600 K
0.6
1
2
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4 5 6 7 8 9 Number of vacancies in cluster
10 ³11
Figure 22 Fractional size distributions of loosely coupled vacancy clusters (all within fourth NN) formed directly within the cascade, comparison of (a) 10 and 50 keV cascades at 100 K, (b) 10 keV cascades at 100 and 900 K, and (c) 20 keV cascades at 100 and 600 K.
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1.11.4.4.1 Influence of preexisting defects
Even if a well-annealed, nearly defect-free, single crystal material is selected for irradiation, radiationinduced defects will rapidly change the state of the material. A simple calculation employing typical elastic scattering cross-sections for fast neutrons and the cascade volumes observed in MD simulations will demonstrate that by the time a dose of 0.01 dpa is reached, essentially the complete volume will have experienced at least one cascade. There have been relatively few studies on how cascade damage production may be different in material with defects.95–97 The results of cascade simulations reported in Stoller and Guiriec97 were carried out at 10 keV and 100 K to expand the range of previous work carried out using 1 keV simulations in copper95 and 0.40, 2.0, and 5.0 keV simulations in iron.96 A 10 keV cascade energy is high enough to initiate in-cascade clustering, is near the plateau region of the defect survival curve, and involves a limited degree of subcascade formation. For these conditions, the database discussed above (see Figure 11) includes two independent sets of cascades, seven in a 128 k atom cell and eight in a 250 k atom cell that can be used to provide a basis of comparison. A cell size of 250 k atoms was used for the cascade simulations with preexisting damage. The study in Stoller and Guiriec97 involved three simple configurations of preexisting damage that were all derived from cascade debris. This is perhaps the simplest possible damage structure, a collection of point defects and point defect clusters. The first configuration was simply the as-quenched debris from a 10 keV cascade in perfect crystal. A total of 30 vacancies and interstitials were present, including one di- and one 7-interstitial cluster. The second case was similar, but the point defects were reconfigured so that the 30 vacancies included a 6-vacancy void and a 9-vacancy loop, and the interstitial clusters included four di-, one tri-, and one 8-interstitial cluster. The third configuration contained only a single 30-vacancy void. These configurations are shown in Figure 23. Eight simulations were carried out with different initial PKAs and the same <135> PKA direction. The selected PKA were 15–20 lattice parameters from the center of the cascade debris and located such that the <135> direction pointed them toward the center of the debris field. The same set of PKAs was used for all three defect configurations. As expected, substantial variation was observed between the different simulations for any given preexisting defect configuration; in some cases the cascade produced more defects than in perfect crystal,
y
x z
(a)
y
x z
(b)
y
x z
(c) Figure 23 Initial defect distributions for investigation of effects of preexisting damage on defect formation in 10 keV cascades: (a) 10 keV cascade debris with 30 SIA and 30 vacancies, (b) same number of defects as in (a) but clustering artificially increased, (c) 30-vacancy void.
while in others fewer were produced. The most dramatic visible effects were observed for the 30-vacancy void. In one case, the void was completely intact after the second cascade, while in the others it was destroyed to varying degrees. The impact of preexisting damage on stable defect formation in the 10 keV cascades is shown in Figure 24, where results from the three different defect configurations are compared with those obtained in perfect crystal. The variation between two sets of perfect crystal simulations is provided for comparison purposes. The statistical information from analysis of defect survival
Primary Radiation Damage Formation
317
0.5
10 keV cascade simulations in iron at 100 K Surviving MD defects per NRT
0.45 0.4
Perfect crystal
0.35 0.3 0.25
Material with defects
0.2 0.15 0.1 7:128 k 8:250 k atoms atoms
All 15
30 i,v cascade debris
30 i,v with small loops and void
30 vacancy void only
Figure 24 Comparison of defect survival values for cascades in perfect crystal and material with preexisting defects (i,v denotes interstitial and vacancy).
Table 3 vacancy)
Summary of defect production results from cascades with preexisting damage (i,v denotes interstitial and
Perfect crystal (all 15 128 k and 250 k atoms) Defective crystal 30 i,v: cascade debris, with one di- and one 7-interstitial cluster 30 i,v: cascade debris with four di-, one tri-, and one 8-interstitial cluster; 6-vacancy void, 9-vacancy loop 30-vacancy void only
and interstitial clustering is summarized in Table 3. On average, a significant reduction in defect formation was observed for the two configurations most typical of random cascade debris. A slight increase (that may not be statistically significant) in defect production was observed when the cell contained only a small void. Only the second defect configuration led to a significant change in interstitial clustering. Although the approach in Stoller’s investigation of preexisting damage was slightly different, the results are consistent with previous studies by Foreman and coworkers95 and Gao and coworkers.96 They observed substantial reductions in defect production when a cascade was initiated in material containing defects. The reductions in defect production observed by Stoller (Figure 24 and Table 3) are somewhat
Survival fraction (per NRT)
Standard error
Interstitial cluster fraction (per NRT)
Standard error
0.336
0.0137
0.170
0.0155
0.260
0.0214
0.179
0.0119
0.279
0.0258
0.110
0.0191
0.370
0.0288
0.190
0.0188
smaller. This difference may partially be due to the higher cascade energy employed here (10 keV vs. 0.4– 5 keV), but the statistical nature of cascade evolution is also a factor. Gao and coworkers analyzed the results of several simulations as a function of distance between the center of mass (COM) of the new cascade and that of the preexisting damage. A good correlation was found between this spacing and the number of defects produced. In the work of Stoller and Guiriec,97 the distance between PKA location and the preexisting damage was nearly constant. As the morphology of each cascade is quite different, the COM spacings varied. This is certainly part of the reason for the variety of behaviors mentioned above for the case of the small void. The average behavior for a fixed initial separation cannot be
318
Primary Radiation Damage Formation
directly compared to any of Gao’s results for the average at a fixed distance. Many more simulations need to be carried out at different energies to develop a more complete picture of cascade damage formation in material with typical defect densities, particularly to assess the clustering behavior. Overall, the reduced defect survival observed in material containing defects suggests that it may be appropriate to employ defect formation values that are somewhat lower than the perfect crystal results in the kinetic models used to simulate microstructural evolution over long times. 1.11.4.4.2 Influence of free surfaces
The rationale for investigating the impact of free surfaces on cascade evolution is the existence of an influential body of experimental data provided by experiments in which thin foils are irradiated by high-energy electrons and/or heavy ions.98–106 In most cases, the experimental observations are carried out in situ by TEM and the results of MD simulations are in general agreement with the data from these experiments. For example, some material-to-material differences observed in the MD simulations, such as differences in in-cascade clustering between bcc iron and fcc copper, also appear in the experimental data.59,107,108 However, the yield of large point defect clusters in the simulations is lower than would be expected from the thin foil irradiations, particularly for vacancy clusters. It is desirable to investigate the source of this difference because of the influence this data has on our understanding of cascade damage formation. Both simulations81,97,109,110 and experimental work105,106 indicate that the presence of a nearby free surface can influence primary damage formation. For example, interesting effects of foil thickness
have been observed in some experiments.105 Unlike cascades in bulk material, which produce vacancies and interstitials in equal numbers, the number of surviving vacancies in surface-influenced cascades can exceed the number of interstitials because of interstitial transport to the surface. This could lead to the formation of larger vacancy clusters and account for the differences in visible defect yield observed between the results of MD cascade simulations conducted in bulk material and the thin-film, in situ experiments. Initial modeling work reported by Nordlund and coworkers81 and Ghaly and Averback109 demonstrated the nature of effects that could occur, and Stoller and coworkers97,100 subsequently conducted a study involving a larger number of simulations at 10 and 20 keV to determine the magnitude of the effects. To carry out the simulations,97,100 a free surface was created on one face of a cubic simulation cell containing 250 000 atom sites. Atoms with sufficient kinetic energy to be ejected from the free surface (sputtered) were frozen in place just above the surface. Periodic boundary conditions are otherwise imposed. Two sets of nine 100 K simulations at 10 keV were carried out to evaluate the effect of the free surface on cascade evolution. In one case, all the PKAs selected were surface atoms and, in the other, PKA were chosen from the atom layer 10a0 below the free surface. The PKA in eight 20 keV, 100 K simulations were all surface atoms. Several PKA directions were used, with each of these directions slightly more than 10 off the [001] surface normal. Figure 25 provides a representative example of a cascade initiated at the free surface. The peak damage state at 1.1 ps is shown in (a), with the final damage state at 15 ps shown in (b). The large number of apparent vacancies and interstitials in
y
y
x
x
1.142E-12
(a)
z
1.588E-11
(b)
z
Figure 25 Defect evolution in typical 10 keV cascade initiated by a surface atom: (a) peak damage state at 1.1 ps, and (b) final damage state at 15 ps.
Primary Radiation Damage Formation
Figure 25(a) is due to the pressure wave from the cascade reaching the free surface. With the constraining force of the missing atoms removed, this pressure wave is able to displace the near-surface atoms by more than 0.3a0, which is the criterion used to choose atom locations to be displayed. As mentioned above, a similar pressure wave occurs in bulk cascades, making the maximum number of displaced atoms much greater than the final number of displacements. Most of these displacements are short-lived, as shown in Figure 26, in which the time dependence of the defect population is shown for three typical bulk cascades, one surface-initiated cascade, and one cascade initiated 10a0 below the surface. The effect of the pressure wave persists longer in surface-influenced cascades, and may contribute to stable defect formation. The number of surviving point defects (normalized to NRT displacements) is shown in Figure 27 for both bulk and surface cascades, with error bars indicating the standard error of the mean. The results are similar at 10 and 20 keV. Stable interstitial production in surface cascades is not significantly different than in bulk cascades; the mean value is slightly lower for the 10 keV surface cascades and slightly higher for the 20 keV case. However, there is a substantial increase in the number of stable vacancies produced, and the change is clearly significant. It is particularly worth noting that the number of surviving interstitials and vacancies is no longer equal for cascades initiated at the surface because interstitials can be lost by sputtering or the diffusion of interstitials and small glissile
interstitial clusters to the surface. Reducing the number of interstitials leads to a greater number of surviving vacancies, as less recombination can occur. In-cascade clustering of interstitials is also relatively unchanged in the surface cascades (e.g., see Figures 4 and 5 in Stoller110). The effect on incascade vacancy clustering was more substantial. The vacancy clustering fraction per NRT (based on the fourth NN criterion discussed above) increased from 0.15 to 0.18 at 10 keV and from 0.15 to 0.25 at 20 keV. Moreover, the vacancy cluster size distributions changed dramatically, with larger clusters produced in the surface cascades. The free surface effect on the vacancy cluster size distributions obtained at 20 keV bulk is shown in Figure 28. The largest vacancy cluster observed in the bulk cascades contained only six vacancies, while the surface cascades had clusters as large as 21 vacancies. This latter size is near the limit of visibility in TEM, with a diameter of almost 1.5 nm. Overall, these results imply that cascade defect production in bulk material is different from that observed in situ using TEM. More research such as that by Calder and coworkers111 is required to fully assess these phenomena, particularly for higher cascade energies, in order to improve the ability to make quantitative comparisons between simulations and experiments. 1.11.4.4.3 Influence of grain boundaries
Depending on the complexity of the microstructure, internal interfaces such as grain boundaries, twins,
Number of displaced atoms
10 000
1000
100
10 Bulk cascades Surface cascade 10a0 from surface 1
10-14
319
10-13
10-12
10-11
Time (s) Figure 26 Time dependence of displaced atoms in 10 keV cascades, three typical cascades initiated near the center of the cell are compared with a cascade initiated by an atom on a free surface and one initiated by an atom 10a0 below the free surface.
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Primary Radiation Damage Formation
0.55 Iron, 10 keV, 100 K
Surviving MD defects per NRT
0.5 Vacancies 0.45
0.4
7 in 128 k atoms
0.35 All 15 0.3 8 in 250 k atoms 0.25
In bulk
(a)
Interstitials 10*a0 from surface
At surface
0.5
Surviving MD defects per NRT
Iron, 20 keV, 100 K 0.45
0.4 Surface cascades 0.35 Bulk cascades 0.3
0.25 (b)
Frenkel pair
Interstitials
Vacancies
Figure 27 Average stable defect production in 10 and 20 keV cascades. 10 keV data compares two populations of bulk cascades, cascades initiated 10a0 below the free surface, and cascades initiated at the free surface, 20 keV cascades. 20 keV results compare bulk and free surface cascades.
and lath and packet boundaries (in ferritic/martensitic steels) can provide a significant sink in the material for point defects. As such, they may play a significant role in radiation-induced microstructural evolution. For example, the effect of grain size on austenitic stainless steels was observed as early as 1972.112–114 The swelling effect was more closely associated with damage accumulation than damage production, but current understanding of the role of mobile interstitial clusters has provided a link to damage production as well (Singh and coworkers115 and Chapter
1.13, Radiation Damage Theory). More recently, there has been considerable interest in the properties of nanograined materials because the high sink strength could lead to very efficient point defect recombination and improved radiation resistance. It is reasonable to expect that primary damage production could be influenced in nanograined material because the grain sizes can be of comparable size to high-energy displacement cascades. Moreover, investigation of grain size effects by MD would be computationally limited to nanograin sizes in any case.
Primary Radiation Damage Formation
321
10
Average number of clusters
Bulk PKA Surface PKA
1
0.1
2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 Vacancy cluster size (4-NN)
Figure 28 Comparison of in-cascade vacancy cluster size distribution in 20 keV, 100 K cascades initiated by a PKA near the center (bulk) and at the free surface of the simulation cell.
To date, there have been a limited number of studies carried out to investigate whether and how primary damage formation would be altered in nanograined metals,116–121 and quite strong effects have been observed.116 The work from Stoller and coworkers122 will be used here to illustrate the phenomenon because the results of that study can be directly compared with the existing single crystal database that has been discussed above. A sufficient number of simulations were carried out at cascade energies of 10 and 20 keV and temperatures of 100 and 600 K to obtain a statistically significant comparison. The results demonstrate that the creation of primary radiation damage can be substantially different in nanograined material due to the influence of nearby grain boundaries. To create the nanocrystalline structure, grain nucleation sites were chosen, and the grains were filled using a Voronoi technique.123 A 2 2 2 lattice parameter face-centered cubic (fcc) unit cell system was used to obtain the grain nucleation sites, resulting in 32 grains in the final sample. Each Voronoi polyhedron was filled with atoms placed on a regular bcc iron crystalline lattice, with the lattice orientation randomly selected. Grain boundaries occur naturally when the atomic plains in adjacent polyhedra impinge on one another, and overlapping atoms at the grain boundaries were removed. The final system was periodic and had an average grain size of 10 nm, system box length of 28.3 nm, and contained roughly 1.87 million atoms. More details on the procedure can
5 keV
20 keV
Figure 29 MD simulation cell, 32 10 nm grains. Shaded red circle and green ellipse indicate approximate size of 5 and 10 keV cascades, respectively.
be found in Stoller and coworkers.122 The system was equilibrated for over 200 ps including a heat treatment up to 600 K. Figure 29 illustrates a typical grain structure with each grain shown in a different color. The approximate sizes of 5 and 20 keV cascades have been projected on to the face of the simulation cell. MD cascade simulations were carried out in the same manner discussed above, although the analysis was somewhat more difficult due to the need to differentiate cascade-produced defects from the defect structure associated with the grain boundaries. It was common for the cascade volume to cross from one grain to another.122 The number of vacancies and
322
Primary Radiation Damage Formation
interstitials surviving in the nanograin simulations is compared to the single crystal results in Figure 30. A wider range of cascade energies is included in Figure 30(a) to show the trend in the single crystal data, while Figure 30(b) highlights the differences at the temperature and energy of the nanograin simulations. Mean values are indicated by the symbols in Figure 30(a) and the height of the bars in Figure 30(b), and the error bars indicate the standard error in both cases. Similar to the case for surfaceinfluenced cascades, the number of surviving vacancies and interstitials is not the same for cascades in nanograined material. The number of vacancies surviving in the nanograined material is similar to the single crystal data for 10 keV cascades, but higher at
20 keV. Much lower interstitial survival is observed in nanograined material under all conditions. Consistent with the overall reduction in interstitial survival shown in Figure 30(a), the number of interstitials in clusters is dramatically reduced in nanograined material for all the conditions examined. As the number of surviving point defects, particularly interstitials, is so strongly reduced in the nanograin material, it is helpful to compare the fraction of defects in clusters in addition to the absolute number. Such a comparison is shown in Figure 31 where the fractions of surviving interstitials and vacancies contained in clusters in both nanograined and single crystal iron are compared for all the conditions simulated. The relative change in the clustering fraction is somewhat
Number of defects
180 160
Single crystal Fe database, 100 K Nanograin Fe
140
Vacancies: 10 keV, 100 K Interstitials Vacancies: 20 keV, 100 K Interstitials
120
Vacancies: 20 keV, 600 K Interstitials
100 80 60 40 20 0
0
10
(a)
20 30 Cascade energy (keV)
40
50
90 80
Single crystal: Vacs = Ints Nanograin: vacancies Nanograin: interstitials
Surviving point defects
70 60 50 40 30 20 10 0 (b)
10 keV, 100 K
20 keV, 100 K
20 keV, 600 K
Figure 30 Number of stable interstitials and vacancies created by displacement cascades in single crystal and nanograined iron.
Primary Radiation Damage Formation
323
0.8 Single crystal Nanograin Surviving fraction in clusters
0.7 0.6 0.5 0.4 0.3 0.2 0.1 Interstitial Vacancy 10 keV, 100 K
Interstitial Vacancy 20 keV, 100 K
Interstitial Vacancy 20 keV, 600 K
Figure 31 Fraction of surviving interstitials and vacancies contained in clusters in single crystal and nanograined iron.
less than the change in the total number of defects in clusters, but are still substantial for interstitial defects. Notably, the temperature dependence of clustering between 100 and 600 K observed in the single crystal 20 keV cascades is reversed in nanograined material. Between 100 and 600 K, the fraction of interstitials in clusters increases for single crystal iron but decreases for nanograined iron. Conversely, the vacancy cluster fraction decreases for single crystal iron and increases for nanograined iron. Although the range of this study was limited in temperature and cascade energy, the results have demonstrated a strong influence of microstructural length scale (grain size) on primary radiation damage production in iron. Both the effects and the mechanisms appear to be consistent with previous work in nickel,116,120 in which very efficient transport of interstitial defects to the grain boundaries was observed. In both iron and nickel, this leads to an asymmetry in point defect survival. Many more vacancies than interstitials survive at the end of the cascade event in nanograined material while equal numbers of these two types of point defects survive in single grain material. Similar to single crystal iron,59,64 few of the vacancies have collapsed into compact clusters on the MD timescale. The vacancy clusters in both single and nanograined iron tend to be loose 3D aggregates of vacancies bound at the first and second NN distances as shown above in Figure 20. The size distribution of such vacancy clusters was not significantly different between the single and nanograin material. In contrast, the
interstitial cluster size distribution was altered in the nanograined iron, with the number of large clusters substantially reduced. There appears to be both a reduction in the number of large interstitial clusters formed directly in the cascade and less coalescence of small mobile interstitial clusters as the latter are being transported to the grain boundaries. The changes in defect survival observed in these simulations are qualitatively consistent with the limited available experimental observations.117–119 For example, Rose and coworkers117 carried out roomtemperature ion irradiation experiments of Pd and ZrO2 with grain sizes in the range of 10–300 nm, and observed a systematic reduction in the number of visible defects produced. Chimi and coworkers118 measured the resistivity of ion irradiated gold specimens following ion irradiation and found that resistivity changes were lower in nanograined material after room-temperature irradiation. However, they observed an increased change in nanograined material following irradiation at 15 K. The low-temperature results could be related to the accumulation of excess vacancy defects as they would be immobile at 15 K.
1.11.5 Comparison of Cascade Damage in Other Metals Differences in cascade damage formation between different metals was among the topics discussed at a workshop in 1998 entitled ‘Basic Aspects of Differences in Irradiation Effects Between fcc, bcc, and hcp
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Primary Radiation Damage Formation
Metals and Alloys.’124 The papers collected in that volume of the Journal of Nuclear Materials can be consulted to obtain the details on both damage production and damage evolution. A brief summary of the observed differences and similarities will be presented in this section. Although the development of alloy potentials is relatively recent, there have been a sufficient number of investigations to provide a comparison of displacement cascade evolution in pure iron with that in three binary alloys, Fe–C, Fe–Cu, and Fe–Cr.125–138 The motivation for each of these binary systems is clear. Carbon must be added to iron to make steel, and as a small interstitial solute it could interact with and influence interstitial-type defects. Copper is of interest largely because it is a primary contributor to reactor pressure vessel embrittlement when it is present as an impurity in concentrations greater than about 0.05 atom% (Chapter 4.05, Radiation Damage of Reactor Pressure Vessel Steels). Steels containing 7–12 atom% chromium are the basis of a number of modern ferritic and ferritic– martensitic steels that are of interest to nuclear energy systems (see Klueh and Harries133). 1.11.5.1
point defects produced in many materials follows a simple power-law dependence over a broad range of cascade energies (see eqn [3]). This behavior is shown in Figure 32 for several pure metals and Ni3Al.107 This figure also includes a line labeled NRT that is obtained from eqn [2] if the displacement threshold is taken as 40 eV, which is the recommended average value for iron.16 The difference between the NRT and Fe lines reflect the ratio plotted in Figure 9. As the displacement threshold is different for different metals (e.g., 30 eV is recommended for Cu16), the other lines should not be compared directly with the NRT values. When normalized using the appropriate NRT displacements, the difference in the survival ratio between Fe and Cu can be seen in Figure 33.61 Although the stable defect production in the other metals may be either somewhat lower or higher than in iron, the behavior is clearly similar across this group of bcc, fcc, and hcp materials. As the energies involved in displacement cascades are so much greater than the energy per atom in a perfect lattice or the vacancy and interstitial formation energies, it is not surprising that ballistic defect production would be similar. In-cascade clustering behavior shows a stronger variation between metals than does total defect survival. The fraction of surviving interstitials contained in clusters is shown in Figure 34 for some of these
Defect Production in Pure Metals
As mentioned above in Section 1.11.4.2, Bacon and coworkers84,107 have shown that the number of stable
100
NF = A(Ep)m
Fe
NRT Ti Al
NF
Ni
10 Zr Ni3Al Cu
Metal
A
m
Ti Fe
6.01 5.57
0.80 0.83
Ni3Al
5.47
0.71
Cu Zr Ni
5.13
0.75 0.74
Al
8.07
4.55 4.37
0.74 0.83
1 1
10 Ep (keV)
Figure 32 Stable defect formation as a function of cascade energy for several pure metals and Ni3Al 100 K, and for Al and Ni at 10 K. The inset table shows the values of m and A (with Ep in keV) yielding the best power-law fit to the data. Reproduced from Bacon, D. J.; Gao, F.; Osetsky, Yu. N. J. Nucl. Mater. 2000, 276, 1–12.
Primary Radiation Damage Formation
0.8 a-Fe Cu
0.7 0.6
NF/NNRT
0.5 0.4 0.3 0.2 0.1 0.0 1
100
10 Ep (keV)
Figure 33 Total surviving Frenkel pair divided by the corresponding number of NRT displacements for Fe and Cu.61 Displacement thresholds of 40 eV and 30 eV were used for Fe and Cu, respectively.16
1.0 100 K 0.8 Cu
Zr
Ti
0.6
Fe
f icl
Ni3Al 0.4
0.2
0.0 0
10
30
20
40
50
Ep (keV)
Figure 34 The fraction of surviving interstitials in clusters of two or more as a function of cascade energy at 100 K for Cu, a-Fe, a-Ti, a-Zr, and Ni3Al at 100 K. Reproduced from Bacon, D. J.; Gao, F.; Osetsky, Yu. N. J. Nucl. Mater. 2000, 276, 1–12.
same metals as in Figure 32.107 Although defect formation does not seem to correlate with crystal structure in Figure 32, there is some indication that this may not be the case with interstitial clustering. The lowest clustering fraction is seen in bcc Fe, while the close-packed Cu (fcc) and hcp (Ti, Zr) materials yield higher values. Ti and Zr exhibit nearly the same value. Ni3Al, which is nominally close-packed is more similar to iron. This may be a result of the ordered structure and some impact of antisite defects on interstitial clustering. However, there is insufficient data available to make any definitive conclusions.
325
A further comparison of vacancy and interstitial clustering in Fe and Cu is provided by Figure 35,61 which provides histograms of the cluster size distributions for two different cascade energies at 100 K. The interstitial cluster size distributions are shown in (a) and (c) for Fe and Cu, respectively, and the corresponding vacancy cluster size distributions are shown in (b) and (d). Note the scale difference on the abscissa between Figure 35(a and b) and Figure 35(c and d). In addition to having a higher fraction of surviving defects in clusters, copper clearly produces much larger clusters of both types. It is clear that some of these differences are related to either the crystal structure and/or such basic parameters as the stacking fault energy. Many of the large vacancy clusters in fcc copper, which has a low stacking fault energy, are large stacking fault tetrahedra (generally imperfect). Similarly, large faulted SIA loops are observed in Cu. Figure 36 illustrates the difference observed between iron and copper in typical 20 keV cascades at 100 K. The final damage state is shown for Fe in (a) and Cu in (b). The simulation cells have an edge length of 50 lattice parameters in both cases. The copper cascade is clearly more compact and exhibits more point defect clustering. While comparisons of iron and copper have been thoroughly explored in the literature,59,61,107,139 there have also been studies on materials such as zirconium, which is relevant to nuclear fuel cladding.70,107,140,141 Figure 37 provides an example of the differences in point defect clustering between Fe and hcp Zr. The average number of SIAs and vacancies in clusters per cascade as a function of cascade energy at 100 K is shown for (a) zirconium and (b) iron.107 Note the difference in scale on both the number in clusters and the cluster size, and that the highest cascade energy is 20 keV in (a) and 49 keV in (b). In both metals the probability of clustering increases with cascade energy, and the size of the largest cluster similarly increases. As indicated by the fact that there are more single vacancies than single interstitials, a greater fraction of SIAs are in clusters. Similar to the Fe–Cu comparison, there is significantly more clustering in close-packed Zr than in bcc Fe. 1.11.5.2
Defect Production in Fe–C
Calder and coworkers examined the effect of carbon on defect production in the Fe–C system with the carbon concentration between 0 and 1.0 atom%.125 The Fe potential was developed by Ackland and coworkers.134 The form of this potential is similar to
326
Primary Radiation Damage Formation
0.6
Fraction of vacancies in clusters
Fraction of SIAs in clusters
0.5 a-Fe, 100 K: 10 keV 50 keV
0.4
0.3
0.2
0.1
a-Fe, 100 K: 10 keV 20 keV
0.5 0.4 0.3 0.2 0.1 0.0
0.0 2
4
(a)
6
8
10
2
12
(b)
Number of SIAs in cluster
4
6
8
10
Number of vacancies in cluster 0.20
0.25
Cu 100 K 10 keV Cu 100 K 10 keV
0.20
0.15
0.15
Fraction of SIAs in clusters
0.10
0.05
0.00 Cu 100 K 25 keV 0.20
0.15
Fraction of vacancies in clusters
0.10
0.05
0.00 Cu 100 K 25 keV
0.15
0.10 S = 0.13 NV = 25-40
S = 0.06
0.10
NSIA > 41
S = 0.06 NV > 41
0.05 0.05
0.00
0.00 10
(c)
20
30
40
Number of SIAs in cluster
10
(d)
20
30
40
Number of vacancies in cluster
Figure 35 Comparison of in-cascade interstitial (a,c) and vacancy (b,d) cluster size distributions at 100 K for Fe (a,b) and Cu (c,d). Reproduced from Bacon, D. J.; Osetsky, Yu. N.; Stoller, R. E.; Voskoboinikov, R. E. J. Nucl. Mater. 2003, 323, 152–162.
the Finnis–Sinclair potential discussed throughout this chapter, but the absolute level of defect production is somewhat lower. Simulations were carried out at temperatures of 100 and 600 K for cascade energies of 5, 10, and 20 keV. Thirty simulations were carried out at each condition to ensure a good statistical sampling. No systematic effect of carbon was observed on either stable defect formation or the clustering of
vacancies and interstitials. Analysis of the octahedral sites around vacancies and interstitials revealed a statistically significant association of carbon atoms with both vacancies and SIAs. This indicates an effective trapping, which is consistent with the solute–defect binding energies. Although primary damage formation was not affected by carbon, the trapping mechanism could have an effect on damage accumulation.
Primary Radiation Damage Formation
327
Cu
Fe y
y
x
x
z
z
20 keV, 100 K iron
(a)
20 keV, 100 K copper
(b)
Figure 36 Comparison of stable defect production from a 20 keV cascade at 100 K in Fe (a) and Cu (b). Note larger SIA clusters in (b).
T = 100 K
Zr
Number of defects per cascade
25 20 15 10 5 0 1 2 3 4 5 6 7 8 9 10 11
Clus
ter s
(a)
ize
12 13
14
25
2
30
2
5
5
10
10
20
40
eV)
y (k
nerg
e PKA
40
20
Fe
Number of defects per cascade
100 90 80 70 60 50 40 30 20 10 0 1 2 3 4 5 6
7 8 9 10 11 12
Clus
ter s
ize
13 14
18
(b) Vacancy
24
2
2
5
5
PKA
10
10
rgy ene
20
20
40
40
)
(keV
Interstitial
Figure 37 The number of SIAs and vacancies in clusters per cascade as a function of cascade energy in (a) a-zirconium and (b) a-iron at 100 K. The values were obtained by averaging over all cascades at each energy. Reproduced from Bacon, D. J.; Gao, F.; Osetsky, Yu. N. J. Nucl. Mater. 2000, 276, 1–12.
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Primary Radiation Damage Formation
1.11.5.3
Defect Production in Fe–Cu
Copper concentrations as high as 0.4 atom% were found in early reactor pressure steels, largely due to both steel recycling and the use of copper as a corrosion-resistant coating on steel welding rods. Research that began in the 1970s demonstrated that this minor impurity was responsible for a significant fraction of the observed vessel embrittlement due to its segregation into a high density of very small (a few nanometer diameter) copper-rich solute clusters (Becquart and coworkers,126 Chapter 4.05, Radiation Damage of Reactor Pressure Vessel Steels). Becquart and coworkers employed MD cascade simulations to determine whether displacement cascades could play a role in the Cu-segregation process, for example, by coalescing with vacancies in the cascade core during the cooling phase. The set of interatomic potentials used is described in Becquart and coworkers.126 Cascade energies of 5, 10, and 20 keV were employed in simulations at 600 K, with copper concentrations of 0, 0.2, and 2.0 atom%. Similar to the case for Fe–C, no effect of copper was found on either stable defect formation or point defect clustering. The tendency for copper to be found bound with either a vacancy or an interstitial in solute–defect complex was observed. The copper–vacancy complexes may play a role in the formation of copperrich clusters over longer times, but no evidence for copper clustering was observed in the cascade debris. Similar results were found in an earlier study by Calder and Bacon.127 Overall, the results of the Fe–Cu studies completed to date are consistent with the fact that Fe and Cu have similar masses and do not strongly interact. 1.11.5.4
Defect Production in Fe–Cr
Interest in ferritic and ferritic–martensitic steels has stimulated the development of Fe–Cr potentials such as those discussed by Malerba and coworkers.129 These potentials have been applied to investigate the influence of Cr on displacement cascades130,131 and on point defect diffusion.132 The MD cascade study by Malerba and coworkers130 involved cascade energies from 0.5 to 15 keV at 300 K. In contrast to the Fe–C and Fe–Cu results discussed above, a slight increase in stable defect formation was observed in Fe–10%Cr relative to pure Fe. The asymptotic value of the defect survival ratio (relative to the NRT) at the highest energies was 0.28 for Fe and 0.31 for Fe–10%Cr. In a later study by the same
authors, which involved a larger number of simulations and energies up to 40 keV, they also concluded that the presence of 10%Cr did not lead to a change in the collisional phase of the cascade but rather reduced the amount of recombination during the cooling phase.131 Additional detailed studies performed with more recent Fe–Cr potentials essentially confirmed the absence of any significant effect of Cr on primary damage in Fe–Cr alloys as compared to pure Fe.136–138 The lack of a Cr effect on the collisional or ballistic phase of the cascade may be expected because, like Cu, the mass of Cr is similar to Fe. The reduced recombination appears to be related to the formation of highly stable mixed Fe–Cr dumbbell interstitials. About 60% of interstitial dumbbells contain a Cr atom, which is substantially higher than the overall Cr concentration of 10%. In spite of the strong mixed dumbbell formation, the fraction of point defects in clusters did not seem to be significantly different than in pure Fe. However, if the stability and mobility of the mixed dumbbells and clusters containing them proves to be appreciably different than pure iron dumbbells,132 there could be an influence on damage accumulation at longer times. Experimental results that are consistent with this hypothesis135 are mentioned in Terentyev and coworkers.138
1.11.6 Summary and Needs for Further Work The use of MD to simulate primary damage formation has become widespread and relatively mature. In addition to the research involving metals discussed above, the approach has also been applied to common structural ceramics11–14,142 and ceramics of interest to the nuclear fuel cycle.15,143,144 However, there are a number of areas that require further research. Some of these have to do with the most basic aspect of MD simulations, that is, the interatomic potentials that are used. In addition to the Finnis–Sinclair potential for iron that was used as a reference case in this chapter, results from several other iron potentials were mentioned. The choice of potential is never an obvious one, and there have been few studies to systematically compare them. In one of the studies mentioned in Section 1.11.3, the details of how one joins the equilibrium part of the potential to a screened Coulomb potential to account for shortrange interactions were shown to significantly
Primary Radiation Damage Formation
influence cascade evolution and defect formation.41 Although a clear difference has been demonstrated, there is no clear path to determining what constitutes the ‘correct’ way to join these potentials. In the case of iron and other magnetic elements such as chromium, research to address the issue of how magnetism may influence defect formation and behavior has only recently begun.145–147 The effect may be modest in the ballistic phase of the cascade when energies are high, but magnetism must certainly influence the configuration and properties of stable defects. Magnetic effects may also determine critical properties of interstitial clusters such as their migration energy and primary diffusion mechanism, which will strongly influence the nature of radiation damage accumulation. As the standard density functional theory fails to fully account for magnetic effects, further developments in electronic structure theory are required in order to provide data for fitting new and more accurate potentials. The interaction between the atomic and electronic systems has largely been neglected in most of the work discussed above. This may impact the results of MD simulations in at least two ways. First, energetic atoms lose energy in a continuous slowing down process that involves both the elastic collisions MD currently models and electronic excitation and ionization between these elastic collisions with lattice atoms. Because of the energy dependence of elastic scattering cross-sections, neglecting the energy loss between atomic collisions could lead to more diffuse cascades and higher predicted defect survival. The second effect is related to inaccuracies in temperature when energy transfer between the electronic and atomic systems (electron–phonon coupling) is neglected. To first order, the atoms remain hotter when energy loss to the electron system is not accounted for. Given the temperature dependence of defect survival and defect clustering discussed above, this clearly has the potential to be significant in any one material. In addition, as electron–phonon coupling varies from one material to another, its neglect may obscure real differences in defect formation between materials. Finally, the issue of rare events requires more investigation. The need to carry out sufficient simulations at a given condition to obtain an accurate estimate for mean behavior was emphasized in the chapter. However, it may be that rare events are also important for the prediction of radiation damage accumulation at longer times or higher doses. If nucleation of extended defects is difficult, which is
329
typically the case at higher temperatures and lower point defect supersaturations, rare events that seed the microstructure with large clusters may largely control the process. One example of a potentially significant rare event is provided by the work of Soneda and coworkers.148 They carried out one hundred 50-keV simulations at 600 K to obtain a good statistical description of defect formation at this condition. In one of these simulations, 223 stable point defects were created, which was much greater than the average of 130 defects. In addition, a <100> vacancy loop containing 153 vacancies was created. The diameter of the loop was about 2.9 nm, which is large enough to be visible by TEM. The impact of the one-in-a-hundred type events should not be underestimated without further study.
Acknowledgments The author would like to acknowledge the fruitful collaboration and discussions on cascade damage for many years with Drs. David Bacon and Andrew Calder (University of Liverpool, UK), Lorenzo Malerba (SCK/CEN, Mol, Belgium), and Yuri Osetskiy (ORNL). He was first introduced to MD cascade simulations by Drs. Alan Foreman (deceased) and William Phythian during a short-term assignment at the AEA Technology Harwell Laboratory, UK, in the summer of 1994. Various aspects of the author’s research discussed in this chapter were supported by the Division of Materials Sciences and Engineering, and the Office of Fusion Energy Sciences, US Department of Energy and the Office of Nuclear Regulatory Research of the US Nuclear Regulatory Commission at the Oak Ridge National Laboratory under contract DE-AC05–00OR22725 with UT-Battelle, LLC.
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Vo¨rtler, K.; Bjo¨rkas, C.; Terentyev, D.; Malerba, L.; Nordlund, K. J. Nucl. Mater. 2008, 382, 24. 137. Bjo¨rkas, C.; Nordlund, K.; Malerba, L.; Terentyev, D.; Olsson, P. J. Nucl. Mater. 2008, 372, 312. 138. Terentyev, D.; Malerba, L.; Chakarova, R.; et al. J. Nucl. Mater. 2006, 349, 119. 139. Voskoboinikov, R. E.; Osetsky, Yu. N.; Bacon, D. J. J. Nucl. Mater. 2008, 377, 385–395. 140. Voskoboinikov, R. E.; Osetsky, Yu. N.; Bacon, D. J. Nucl. Instr. Meth. Phys. Res. B 2006, 242, 68–70. 141. Voskoboinikov, R. E.; Osetsky, Yu. N.; Bacon, D. J. In Effects of Radiation on Materials; Allen, T. R., Lott, R. G., Busby, J. T., Kumar, A. S., Eds.; American Society for Testing and Materials International: West Conshohocken, PA, 2006; pp. 299–313, STP 1475. 142. Bacorisen, D.; Smith, R.; Uberuaga, B. P.; Sickafus, K. E.; Ball, J. A.; Grimes, R. W. Nucl. Instr. Meth. B 2006, 250, 28–35. 143. Bishop, C. L.; Grimes, R. W.; Parfitt, D. C. Nucl. Instr. Meth. B 2010, 268, 2915–2917. 144. Aidhy, D. S.; Millett, P. C.; Desai, T.; Wolf, D.; Phillpot, S. R. Phys. Rev. B 2009, 80, 104107. 145. Dudarev, S. L.; Derlet, P. M. J. Phys. Cond. Mat. 2005, 17, 7097–7118. 146. Ackland, G. J. J. Nucl. Mater. 2006, 351, 20–27. 147. Malerba, L.; Ackland, G. J.; Becquart, C. S.; et al. J. Nucl. Mater. 2010, 406, 7–18. 148. Soneda, N.; Ishino, I.; Diaz de la Rubia, T. Phil. Mag. Lett. 2001, 81, 649–659. 136.
1.12 Atomic-Level Dislocation Dynamics in Irradiated Metals Y. N. Osetsky Oak Ridge National Laboratory, Oak Ridge, TN, USA
D. J. Bacon The University of Liverpool, Liverpool, UK
Published by Elsevier Ltd.
1.12.1 1.12.2 1.12.2.1 1.12.2.2 1.12.3 1.12.3.1 1.12.3.2 1.12.3.3 1.12.3.4 1.12.4 1.12.4.1 1.12.4.1.1 1.12.4.1.2 1.12.4.2 1.12.4.2.1 1.12.4.2.2 1.12.4.3 1.12.5 References
Introduction Radiation Effects on Mechanical Properties Radiation-Induced Obstacles to Dislocation Glide Effects on Mechanical Properties Method Why Atomic-Scale Modeling? Atomic-Level Models for Dislocations Input Parameters Output Information Results on Dislocation–Obstacles Interaction Inclusion-Like Obstacles Temperature T ¼ 0 K Temperature T > 0 K Dislocation-Type Obstacles Stacking fault tetrahedra Dislocation loops Microstructure Modifications due to Plastic Deformation Concluding Remarks
Abbreviations
Symbols
bcc DD DL EAM ESM fcc hcp IAP MBP MC MD MMM MS PAD PBC SFT SIA TEM
b bL D G L t T vD « «˙ g w n rD t tc tP
Body-centered cubic Dislocation dynamics Dislocation loop Embedded atom model Equivalent sphere method Face-centered cubic Hexagonal close-packed Interatomic potential Many-body potential Monte Carlo Molecular dynamics Multiscale materials modeling Molecular statics Periodic array of dislocations Periodic boundary condition Stacking fault tetrahedron Self-interstitial atom Transmission electron microscope
334 334 334 335 336 336 336 338 338 339 339 339 342 345 345 348 352 353 355
Dislocation Burgers vector Dislocation loop Burgers vector Obstacle diameter Shear modulus Dislocation length Simulation time Ambient temperature Dislocation velocity Shear strain Shear strain rate Stacking fault energy Angle between dislocation segments Poisson’s ratio Dislocation density Shear stress Critical resolved shear stress Peierls stress
333
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Atomic-Level Dislocation Dynamics in Irradiated Metals
1.12.1 Introduction Structural materials in nuclear power plants suffer a significant degradation of their properties under the intensive flux of energetic atomic particles (see Chapter 1.03, Radiation-Induced Effects on Microstructure). This is due to the evolution of microstructures associated with the extremely high concentration of radiation-induced defects. The high supersaturation of lattice defects leads to microstructures that are unique to irradiation conditions. Irradiation with high energy neutrons or ions creates initial damage in the form of displacement cascades that produce high local supersaturations of point defects and their clusters (see Chapter 1.11, Primary Radiation Damage Formation). Evolution of the primary damage under the high operating temperature (600 K to >1000 K) leads to a microstructure containing a high concentration of defect clusters, such as voids, dislocation loops (DLs), stacking fault tetrahedra (SFTs), gas-filled bubbles, and precipitates, and an increase in the total dislocation network density (see Chapter 1.13, Radiation Damage Theory; Chapter 1.14, Kinetic Monte Carlo Simulations of Irradiation Effects and Chapter 1.15, Phase Field Methods). These changes affect material properties, including mechanical ones, which are the subject of this chapter. A general theory of radiation effects has not yet been developed, and currently the most promising way to predict materials behavior is based on multiscale materials modeling (MMM). In this framework, phenomena are considered at the appropriate length and times scales using specific theoretical and/or modeling approaches, and the different scales are linked by parameters/mechanisms/rules to provide integrated information from a lower to a higher level. Research on the mechanical properties of irradiated materials, a topic of crucial importance for engineering solutions, provides a good example of this. The lowest level treats individual atoms by first principles, ab initio methods, by solving Schro¨dinger’s equation for moving electrons and ions. Calculations based on electron density functional theory (DFT) (see Chapter 1.08, Ab Initio Electronic Structure Calculations for Nuclear Materials) and its approximations, such as bond order potentials (BOPs), can consider a few hundred atoms over a very short time of femtoseconds to picoseconds. Delivery of the resulting information to higher level models can be achieved through effective interatomic potentials (IAPs) (see Chapter 1.10, Interatomic Potential Development), in which the adjustable
parameters are fitted to the basic chemical and structural properties obtained ab initio. IAPs are required for atomic-scale modeling methods such as molecular statics (MS) and molecular dynamics (MD), which are used to simulate millions of atoms. Time spanning nanoseconds to microseconds can be simulated by MD if the number of atoms is not large (see Section 1.12.3.3). This level can provide properties of point and extended defects and interactions between them (see Chapter 1.09, Molecular Dynamics). For mechanical properties, important interactions are between moving dislocations, which are responsible for plasticity, and defects created by irradiation. Mechanisms and parameters determined at this level can then inform dislocation dynamics (DD) models based on elasticity theory of the continuum (see Chapter 1.16, Dislocation Dynamics). DD models can simulate processes at the micrometer scale and mesh with the mechanical properties of larger volumes of material used in finite elements (FEs) methods, that is, realistic models for the design of core components. In this chapter, we consider direct interactions at the atomic scale between moving dislocations and obstacles to their motion. The structure of the chapter is as follows. First, we summarize the main features of the irradiation microstructure of concern. Then we provide a short description of atomic-scale methods applied to dislocation modeling, bearing in mind the details presented in Chapter 1.09, Molecular Dynamics. This is followed by a review of important results from simulations of the interaction between dislocations and obstacles. We then describe how dislocations modify microstructure in irradiated metals. Finally, we indicate some issues that will hopefully be resolved by atomic-scale modeling in the near future. Our main aim is to give the reader a general picture of the phenomena involved and encourage further research in this area. The following sources1–4 provide a more general and deeper understanding of dislocations and modeling of plasticity issues.
1.12.2 Radiation Effects on Mechanical Properties 1.12.2.1 Radiation-Induced Obstacles to Dislocation Glide Primary damage of structural materials is initiated by the interaction of high-energy atomic particles with material atoms to cause the energetic recoil and displacement of primary knock-on atoms (PKAs). PKA energy can vary from a few tens to tens of
Atomic-Level Dislocation Dynamics in Irradiated Metals
thousands of electron volt and the PKA spectrum can be calculated for a particular position in a particular installation.5 A PKA with energy >1 keV gives rise to a displacement cascade that produces a localized distribution of point defects (vacancies and self-interstitial atoms, SIAs) and their clusters (see Chapter 1.11, Primary Radiation Damage Formation). Further evolution of these defects produces specific microstructures that depend on the irradiation type, ambient temperature, and the material and its initial structure (see Chapter 1.13, Radiation Damage Theory). This radiation-induced microstructure consists typically of voids, gas-filled bubbles, DLs (that can evolve into a dislocation network), secondary-phase precipitates, and other extended defects specific to the material, for example, SFTs in face-centered cubic (fcc) metals. These features are generally obstacles to the dislocation motion. Their size is typically6 in the range of nanometers to tens of nanometers and their number density may reach 1024 m3. At this density, the mean distance between obstacles can be as short as 10 nm, and such a high density of small defects, particularly those with a dislocation character, makes the mechanisms of radiation effects on mechanical properties very different from those due to other treatments. 1.12.2.2
Effects on Mechanical Properties
Radiation-induced defects, being obstacles to dislocation glide, increase yield and flow stress and reduce ductility (see Chapter 1.04, Effect of Radiation on Strength and Ductility of Metals and Alloys for experimental results). Furthermore, if the obstacle density is sufficiently high to block dislocation motion, preexisting Frank-Reed dislocation sources are unable to operate and plastic deformation requires operation of sources that are not active in the unirradiated state. These new sources operate at much higher stress and give rise to new mechanisms such as yield drop, plastic instability, and formation of localized channels with high dislocation activity and high local plastic deformation. Understanding these phenomena is necessary for predicting material behavior under irradiation and the design and selection of materials for new generations of nuclear devices. Obstacles induced by irradiation affect moving dislocations in a variety of ways, but can be best categorized as one of two types, namely inclusionlike obstacles and those with dislocation properties. The first type includes voids, bubbles, and precipitates, for example. They usually have relatively
335
short-range strain fields and their properties may not be changed significantly by interaction with dislocations. (Copper precipitates in iron are an exception to this – see Sections 1.12.4.1.1–1.12.4.1.2.) Those that are not impenetrable are usually sheared by the dislocation and steps defined by the Burgers vector b of the interacting dislocation are created on the obstacle– matrix interface. Unstable precipitates, such as Cu in Fe, may also suffer structural transformation during the interaction, which can change their properties. These obstacles do not usually modify dislocations significantly, although they may cause climb of edge dislocations (see Sections 1.12.4.1.1–1.12.4.1.2). Their main effect is to create resistance to dislocation glide. Obstacles such as voids and bubbles are among the strongest, and as a result of their high density, they contribute significantly to radiation-induced hardening.7 Materials designed to exploit oxide dispersion strengthening (ODS) are produced with a high concentration of rigid, impenetrable oxide particles, which introduce extremely high resistance to dislocation motion.8 These obstacles are also considered here as their scale, typically a few nanometers, is similar to that of obstacles formed under irradiation. The second obstacle type consists of those with a dislocation character, for example, DLs and SFTs, and so dislocation reactions occur when they are encountered by moving dislocations. Loops have relatively long-range strain fields and hence interact with dislocations over distances much greater than their size. SFTs are three-dimensional (3D) structures and have short-range strain fields. Loops with perfect Burgers vectors are glissile, in principle, whereas SFTs and faulted loops, for example, Frank loops in fcc metals, are sessile. In addition to causing hardening, the reaction of these defects with a gliding dislocation can modify both their own structure and that of the interacting dislocation. As will be demonstrated in Section 1.12.4.2, their effect depends very much on the geometry of the interaction, that is, their position and orientation relative to the moving dislocation, and the nature of the mutual dislocation segment that may form in the first stage of interaction. The contribution of these obstacles to strengthening can be significant, for their density can be high. Modification of irradiation-induced microstructure due to plastic deformation is an additional possibly important effect. If mechanical loading occurs during irradiation, it can contribute significantly to the overall microstructure evolution and therefore to change in material properties. Accumulation of internal stress during irradiation is unavoidable in real structural
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Atomic-Level Dislocation Dynamics in Irradiated Metals
materials and so this effect should not be ignored. The effects of concurrent deformation and irradiation on microstructure are far from clear, for only a few experimental studies of in-reactor deformation have been performed.9 This area, therefore, provides a good example of how atomic-scale modeling can help in understanding a little-studied phenomenon.
dislocation is effectively infinite in length. If the model contains one obstacle, the length, L, of the model in the periodic direction represents the center-to-center obstacle spacing along an infinite row of obstacles. It is the treatment of the boundaries in the other two directions that distinguishes one method from another. A versatile atomic-scale model should allow for the following.10
1.12.3 Method
1. Reproduction of the correct atomic configuration of the dislocation core and its movement under the action of stress. 2. Application of external effects such as applied stress or strain, and calculation of the resultant response such as strain (elastic and plastic) or stress and crystal energy. 3. Possibility of moving the dislocation over a long distance under applied stress or strain without hindrance from the model boundaries. 4. Simulation of either zero or non-zero temperatures. 5. Possibility of simulating a realistic dislocation density and spacing between obstacles. 6. Sufficiently fast computing speed to allow simulation of crystallites in the sizes range where size effects are insignificant.
1.12.3.1
Why Atomic-Scale Modeling?
First principle ab initio methods for self-consistent calculation of electron-density distribution around moving ions provide the most accurate modeling techniques to date. They take into account local chemical and magnetic effects and provide significant potential for predicting material properties. They are used with success in applications where the properties are limited to the nanoscale, for example, microelectronics, catalysis, nanoclusters, and so on (Chapter 1.08, Ab Initio Electronic Structure Calculations for Nuclear Materials). A typical scale for this is of the order of a few nm. However, this leaves a significant gap between ab initio methods and those required to model properties of bulk materials arising from radiation damage. These involve phenomena acting over much longer scales, such as interactions between mobile and sessile defects, their thermally activated transport, their response to internal and external stress fields and gradients of chemical potential. Models for bulk properties are based on continuum treatments by elasticity, thermal conductivity, and rate theories where global defect properties such as formation, annihilation, transport, and interactions are already parameterized at continuum level. The only technique that currently bridges the gap in the scales between ab initio and the continuum is computer simulation of a large system of atoms, up to 106–108. Atoms move as in classical Newtonian dynamics due to effective forces between them calculated from empirical interatomic potentials and respond to internal and external fields due to temperature, stress, and local imperfections. Atomic-scale modeling has provided the results presented in this chapter. In the following section, we present a short description of typical models for simulation of dislocations and their interactions with defects formed by radiation. 1.12.3.2 Atomic-Level Models for Dislocations All models use periodic boundary conditions in the direction of the dislocation line so that the
A comprehensive review of models developed so far is to be found in Bacon et al.,4 and so here we merely present a short summary of the pros and cons of some models used most commonly. Historically, the earliest models consisted of a small region of mobile atoms surrounded in the directions perpendicular to the dislocation direction by a shell of atoms fixed in the positions obtained by either isotropic or anisotropic elasticity for displacements around the dislocation of interest.11 This model was used successfully to investigate dislocation core structure and, being simple and computationally efficient, can use a mobile region large enough to simulate interaction between static dislocations and defects and small defect clusters. Its main deficiencies are its inability to model dislocation motion beyond a few atomic spacings because of the rigid boundaries (condition 3) and its restriction to temperature T ¼ 0 K (condition 4). The desirability of allowing for elastic response of the boundary atoms due to atomic relaxation in the inner region, for example, when a dislocation moves, has led to the development of several quasicontinuum models. The elastic response can be accounted for by using either a surrounding FE mesh or an elastic Green’s function to calculate the response of boundary atoms to forces generated by the inner region. Such models are accurate but computationally
337
Atomic-Level Dislocation Dynamics in Irradiated Metals
inefficient and have not found wide application so far.4 Furthermore, their use for simulation of T > 0 K (condition 4) is still under development.12 Nevertheless, quasicontinuum models, especially those based on Green’s function solutions, can be employed in applications where calculation of forces on atoms is computationally expensive and a significant reduction in the number of mobile atoms is desirable.13 The models now most widely applied to simulate dislocation behavior in metals are based on the periodic array of dislocations (PAD) scheme first introduced for simulating edge dislocations.14,15 In this, periodic boundary conditions are applied in the direction of dislocation glide as well as along the dislocation line, that is, the glide plane is periodic. This means that the dislocation is one of a periodic, 2D array of identical dislocations. The success of PAD models is because of their simplicity and good computational efficiency when applied with modern empirical IAPs, for example, embedded atom model (EAM) type. They can be used to simulate screw, edge, and mixed dislocations.4,10,16 With a PAD model containing 106–107 mobile atoms, essentially all conditions 1–6 can be satisfied. Their ability to simulate interactions with strong obstacles of size up to at least 10 nm makes PAD models efficient for investigating dislocation–obstacle interactions relevant to a radiation damage environment. Practically all important radiation-induced obstacles can be simulated on modern computers using parallelized codes and most can even be treated by sequential codes. Details of model construction for different dislocations can be found elsewhere.4,10,16 Here we just present an example of system setup for screw or edge dislocations in bcc and fcc metals interacting with dislocations loops and SFTs, as presented in Figure 1.
There are two types of DL in an fcc metal: glissile perfect loops with bL ¼ 1/2h110i and sessile Frank loops with bL ¼ 1/3h111i. There are two types of glissile loop with Burgers vectors 1/2h111i and h100i in a body-centered cubic (bcc) metal. Visualizing interaction mechanisms is a strong feature of atomic scale modeling. The main idea is to extract atoms involved in an interaction and visualize them to understand the mechanism. Usually these atoms are characterized by high energy, local stresses, and lattice deformation. The techniques used are based on analysis of nearest neighbors,17 central symmetry parameter,18 energy,19 stress,20 displacements,10 and Voronoi polyhedra.21 A relatively simple and fast technique, for example, was suggested for an fcc lattice.16 It is based on comparison of position of atoms in the first coordination of an atom with that of a perfect fcc lattice. If all 12 neighbors of the analyzed atom are close to that position, it is assigned to be fcc. If only nine neighbors correspond to perfect fcc coordination, the atom is taken to be on a stacking fault. Other numbers of neighbors can be attributed to different dislocations. Modifications of this method have been successfully applied in hexagonal close-packed (hcp)22 and bcc23 crystals. Another improvement of this method for MD simulation at T > 0 K was introduced24 in which the above analysis was applied periodically (every 10–50 time_ and over a certain steps depending on strain rate e) time period (100–1000 steps). A probability of an atom to be in different environment was estimated and the final state was assigned to the maximum over the analyzed period. Such a probability analysis can be applied to other characteristics such as energy or stress excess over the perfect state and it provides a clear picture when the majority of thermal fluctuations are omitted.
L
L
t
t Screw
L S
S t
⬍111⬎
Obstacle: loop or SFT
Ad
b
L
b = 1/3
dC
dC
Edge
b = ⬍100⬎
t
b = 1/2 ⬍111⬎
D b = 1/2 ⬍110⬎ A C B
fcc bcc Figure 1 Examples of periodic array of dislocation model setup for screw and edge dislocations in body-centered cubic and face-centered cubic crystals. Examples of dislocation loops, a stacking fault tetrahedron, and sense of applied resolved shear stress, t, are indicated.
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Atomic-Level Dislocation Dynamics in Irradiated Metals
1.12.3.3
Input Parameters
The IAP is a crucial property of a model for it determines all the physical properties of the simulated system. Discussion on modern IAPs is presented in Chapter 1.10, Interatomic Potential Development and so we do not elaborate on this subject here. Another important property is the spatial scale of the simulated system. The periodic spacing, Lg, in the direction of the dislocation glide has to be large enough to avoid unwanted effects due to interaction between the dislocation and its periodic neighbors in the PAD; 100–200b is usually sufficient.10 Furthermore, the model should be large enough to include all direct interactions between the dislocation and obstacle and the major part of elastic energy that may affect the mechanism under study. MD simulations have demonstrated that a system with a few million atoms is usually sufficient to satisfy conditions for simulating interaction between a dislocation and an obstacle of a few nanometers in size. The biggest obstacles considered to date are 8 nm voids,23 10 nm DLs,25 and 12 nm SFTs26 in crystals containing 6–8 million mobile atoms. It should be noted that static simulation (T ¼ 0 K) usually requires the largest system because most obstacles are stronger at low T and the dislocation may have to bend strongly and elongate before breaking free.23 Simulation of a dynamic system, that is, T > 0 K, introduces another important and limiting factor for atomic-scale study of dislocation behavior, namely the simulation time, t, which can be achieved with the computing resource available. Under the action of increasing strain applied to the model, the time to reach a given total strain determines the minimum applied strain rate, e,_ that can be considered. This parameter defines in turn the dislocation velocity. Consider a typical simulation of dislocation–obstacle interaction in an Fe crystal, for which b ¼ 0.248 nm. For L ¼ 41 nm, a model containing 2 106 mobile atoms would have a cross-section area of 5.73 1016 m2; that is, a dislocation density rD ¼ 1.75 1015 m2. For Lg ¼ 120b ¼ 29.8 nm, the model height perpendicular to the glide plane would be 19 nm. At e_ ¼ 5 106 s1 , the steady state velocity, vD, of a single dislocation estimated from the Orowan relation _ D b is 11.6 m s1. The time for the dislocation vD ¼ e=r to travel a distance Lg at this velocity would be 2.6 ns. Thus, even if the dislocation breaks away from the void without traversing the whole of the glide plane, the total simulated time would be 1 ns.
The lowest strain rate for dislocation-obstacle interaction reported so far27 is 105 s1 and it resulted in vD ¼ 48 cm s1. This strain rate is about six to ten orders of magnitude higher than that usually applied in laboratory tensile experiments and more than ten orders higher than that for the creep regime. This presents an unresolvable problem for atomic-scale modeling and even massive parallelization gains only three or four orders in e_ or vD. We conclude that the possibilities of modern atomic-scale modeling are limited to dislocation velocity of at least 0.1 cm s1. Nevertheless, atomic-scale modeling, particularly using MD (T > 0 K), is a powerful, and sometimes the only, tool for investigating processes associated with lattice defect interactions and dynamics. The main advantage of MD is that, if applied properly to a large enough system, it includes all classical phenomena such as evolution of the phonon system and therefore free energy, rates of thermally activated defect motion, and elastic interactions. It is, therefore, one of the most accurate techniques for investigating the behavior of large atomic ensembles under different conditions. We reemphasize that the realism of atomic-scale modeling is limited mainly by the validity of the IAP and restricted simulation time. 1.12.3.4
Output Information
Atomic-scale methods and particularly MD can provide a wide range of valuable information on the processes simulated. The most important are 1. Information on the physical state of the system. This includes temperature and stress and their distribution; displacement of atoms and their transport; interaction energy and therefore force between defects; and evolution of internal, elastic, and free energies. Extraction of this information is well understood and procedures can be found in Chapter 1.04, Effect of Radiation on Strength and Ductility of Metals and Alloys; Chapter 2.13, Properties and Characteristics of ZrC; Chapter 5.01, Corrosion and Compatibility and Chapter 1.09, Molecular Dynamics. 2. Detail of atomic mechanisms. This includes analysis of the position and environment of individual atoms based on calculation of their energy, site stress, or local atomic configuration. Atoms can then be identified with particular features such as constituents of defect clusters, stacking faults, dislocation cores, and so on. Having this information at particular times provides unique knowledge of
Atomic-Level Dislocation Dynamics in Irradiated Metals
defect structure, transformation.
motion,
interactions,
and
The information summarized in 1 and 2 can be used to determine how the mechanisms involved depend on parameters such as obstacle type and size and dislocation type, material temperature, and applied stress or strain.
1.12.4 Results on Dislocation– Obstacles Interaction 1.12.4.1
Inclusion-Like Obstacles
1.12.4.1.1 Temperature T ¼ 0 K
Voids in bcc and fcc metals at T ¼ 0 K and >0 K are probably the most widely simulated obstacles of this type. Most simulations were made with edge dislocations.10,25–34 A recent and detailed comparison of strengthening by voids in Fe and Cu is to be found in Osetsky and Bacon.34 Examples of stress–strain curves (t vs. e) when an edge dislocation encounters and overcomes voids in Fe and Cu at 0 K are presented in Figures 2 and 3, respectively. The four distinct stages in t versus e for the process are described in Osetsky and Bacon10 and Bacon and Osetsky.23 The difference in behavior between the two metals is due to the difference in their dislocation core structure, that is, dissociation into Shockley partials in Cu but no splitting in Fe (for details see Osetsky and Bacon34).
339
Under static conditions, T ¼ 0 K, voids are strong obstacles and at maximum stress, an edge dislocation in Fe bows out strongly between the obstacles, creating parallel screw segments in the form of a dipole pinned at the void surface. A consequence of this is that the screw arms cross-slip in the final stage when the dislocation is released from the void surface and this results in dislocation climb (see Figure 4), thereby reducing the number of vacancies in the void and therefore its size. In contrast to this, a Shockley partial cannot cross-slip. Partials of the dissociated dislocation in Cu interact individually with small voids whose diameter, D, is less than the partial spacing (2 nm), thereby reducing the obstacle strength. Stress drops are seen in the stress–strain curve in Figure 3. The first occurs when the leading partial breaks from the void; the step formed by this on the exit surface is a partial step 1/6h112i and the stress required is small. Breakaway of the trailing partial controls the critical stress tc. For voids with D larger than the partial spacing, the two partials leave the void together at the same stress. However, extended screw segments do not form and the dislocation does not climb in this process. Consequently, large voids in Cu are stronger obstacles than those of the same size in Fe, as can be seen in Figure 6 and the number of vacancies in the sheared void in Cu is unchanged. Cu-precipitates in Fe have been studied extensively23,27–29 due to their importance in raising the yield stress of irradiated pressure vessels steels35 and
200
Shear stress (MPa)
150
100
50
0 -50
D (nm): 0.0
0.9 1.0 1.5 2.0 3.0 4.0 5.0 0.5
1.0
6.0 1.5
Strain (%) Figure 2 Stress–strain dependence for dislocation–void interaction in Fe at 0 K with L ¼ 41.4 nm. Values of D are indicated below the individual plots. From Osetsky, Yu. N.; Bacon, D. J. Philos. Mag. 2010, 90, 945. With permission from Taylor and Francis Ltd. (http://www.informaworld.com).
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Atomic-Level Dislocation Dynamics in Irradiated Metals
350 300
Shear stress (MPa)
250 200 150 100 50 0 -50
D (nm):
1 1.5 2
3
4 5 6
7
8
-100 0.0
0.2
0.4
0.6 0.8 Strain (%)
1.0
1.2
Figure 3 Stress–strain dependence for dislocation–void interaction in Cu at 0 K with L ¼ 35.5 nm. Values of D are indicated below the individual plots. From Osetsky, Yu. N.; Bacon, D. J. Philos. Mag. 2010, 90, 945. With permission from Taylor and Francis Ltd. (http://www.informaworld.com).
20 5.0 nm
16
4.0 nm
[110], a
12
3.0 nm 8 2.0 nm 4
1.0 nm 0.9 nm
0 -40
-30
-20
-10
0 [112], a
10
20
30
40
Figure 4 [111] projection of atoms in the dislocation core showing climb of a1/2h111i{110} edge dislocation after breakaway from voids of different diameter in Fe at 0 K. Climb-up indicates absorption of vacancies. The dislocation slip plane intersects the voids along their equator. From Osetsky, Yu. N.; Bacon, D. J. J. Nucl. Mater. 2003, 323, 268. Copyright (2003) with permission from Elsevier.
the availability of suitable IAPs for the Fe–Cu system.36 These precipitates are coherent with the surrounding Fe when small, that is, they have the bcc structure rather than the equilibrium fcc structure of Cu. Thus, the mechanism of edge dislocation interaction with small Cu precipitates is similar to that of voids in Fe. The elastic shear modulus, G, of bcc Cu is lower than that of the Fe matrix and the dislocation is attracted into the precipitate by a reduction in its strain energy. Stress is required to overcome the
attraction and to form a 1/2h111i step on the Fe–Cu interface. This is lower than tc for a void, however, for which G is zero and the void surface energy relatively high. Thus, small precipitates (3 nm) are relatively weak obstacles and, though sheared, remain coherent with the bcc Fe matrix after dislocation breakaway. tc is insufficient to draw out screw segments and the dislocation is released without climb. The Cu in larger precipitates is unstable, however, and their structure is partially transformed toward
8
8
6
6
4
4
2
2
0
0
-2
-2
-4
-4
-6
-6
-8
-8 -50
-45
-40 <111>, ao
-35
4
341
_ [112], ao
_ <112>, ao
Atomic-Level Dislocation Dynamics in Irradiated Metals
2 _ 0 -2 -4 <110>, ao
Figure 5 Position of Cu atoms in four consecutive ð110Þ planes through the center of a 4 nm precipitate in Fe after dislocation breakaway at 0 K. The figure on the right shows the dislocation line in [111] projection after breakaway; climb to the left/right indicates absorption of vacancies/atoms by the dislocation. From Bacon, D. J.; Osetsky, Yu. N. Philos. Mag. 2009, 89, 3333. With permission from Taylor and Francis Ltd. (http://www.informaworld.com).
1.2 - Voids in Fe 1.0
- Cu-precipitates in Fe - Voids in Cu
tC Gb/L
0.8
0.6
Dvoid = 1.52
0.4
DOrowan = 0.77
0.2
0.0 5
10
15
20
25
30
(D-1 + L-1)-1, b Figure 6 Critical stress tc (in units Gb/L) versus the harmonic mean of D and L (unit b) for voids and Cu-precipitates in Fe and voids in Cu at 0 K.
the more stable fcc structure when penetrated by a dislocation at T ¼ 0 K. This is demonstrated in Figure 5 by the projection of atom positions in four {110} atomic planes parallel to the slip plane near the equator of a 4 nm precipitate after dislocation breakaway. In the bcc structure, the {110} planes have a twofold stacking sequence, as can be seen by the upright and inverted triangle symbols near the outside of the precipitate, but atoms represented by
circles are in a different sequence. Atoms away from the Fe–Cu interface are seen to have adopted a threefold sequence characteristic of the {111} planes in the fcc structure. This transformation of Cu structure, first found in MS simulation of a screw dislocation penetrating a precipitate,37,38 increases the obstacle strength and results in a critical line shape that is close to those for voids of the same size.34 Under these conditions, a screw dipole is created
342
Atomic-Level Dislocation Dynamics in Irradiated Metals
and effects associated with this, such as climb of the edge dislocation on breakaway described above for voids in Fe, are observed.23,27 The results above were obtained at T ¼ 0 K by MS, in which the potential energy of the system is minimized to find the equilibrium arrangement of the atoms. The advantage of this modeling is that the results can be compared directly with continuum modeling of dislocations in which the minimum elastic energy gives the equilibrium dislocation arrangement. An early and relevant example of this is provided by the linear elastic continuum modeling of edge and screw dislocations interacting with impenetrable Orowan particles39 and voids.40 By computing the equilibrium shape of a dislocation moving under increasing stress through the periodic row of obstacles, as in the equivalent MS atomistic modeling, it was shown that the maximum stress fits the relationship tc ¼
Gb ½lnðD1 þ L1 Þ þ D 2pAL
½1
where G is the elastic shear modulus and D is an empirical constant; A equals 1 if the initial dislocation is pure edge and (1 n) if pure screw, where n is Poisson’s ratio. Equation [1] holds for anisotropic elasticity if G and n are chosen appropriately for the slip system in question, that is, if Gb2/4p and Gb2/4pA are set equal to the prelogarithmic energy factor of screw and edge dislocations, respectively.39,40 The value of G obtained in this way is 64 GPa for h111i{110} slip in Fe and 43 GPa for h110i{111} slip in Cu.41 The explanation for the D- and L-dependence of tc is that voids and impenetrable particles are ‘strong’ obstacles in that the dislocation segments at the obstacle surface are pulled into parallel, dipole alignment at tc by self-interaction.39,40 (Note that this shape would not be achieved at this stress in the line-tension approximation where self-stress effects are ignored.) For every obstacle, the forward force, tcbL, on the dislocation has to match the dipole tension, that is, energy per unit length, which is proportional to ln(D) when D L and ln(L) when L D.39 Thus, tcbL correlates with Gb2ln(D1 þ L1)1. The correlation between tc obtained by the atomic-scale simulations above and the harmonic mean of D and L, as in eqn [1], is presented in Figure 6. A fairly good agreement can be seen across the size range down to about D < 2 nm for voids in Fe and 3–4 nm for the other obstacles. The explanation for this lies in the fact that in the atomic simulation, as in the earlier continuum modeling, obstacles with D > 2–3 nm are strong at T ¼ 0 K and result in a dipole alignment at tc.
Smaller obstacles in Fe, for example, voids with D < 2 nm and Cu precipitates with D < 3 nm, are too weak to be treated by eqn [1]. Thus, the descriptions above and the data in Figure 6 demonstrate that the atomicscale mechanisms that operate for small and large obstacles depend on their nature and are not predicted by simple continuum treatments, such as the line-tension and modulus-difference approximations that form the basis of the Russell–Brown model of Cuprecipitate strengthening of Fe,42 often used in predictions and treatment of experimental observations. The importance of atomic-scale effects in interactions between an edge dislocation and voids and Cu-precipitates in Fe was recently stressed in a series of simulations with a variable geometry.43 In this study, obstacles were placed with their center at different distances from the dislocation slip plane. An example of the results for the case of 2 nm void at T ¼ 0 K is presented in Figure 7. The surprising result is that a void with its center below the dislocation slip plane is still a strong obstacle and may increase its size after the dislocation breaks away. This can be seen in Figure 7, where a dislocation line climbs down absorbing atoms from the void surface. More details on larger voids, precipitates, and finite temperature effects can be found in Grammatikopoulos et al.43 1.12.4.1.2 Temperature T > 0 K
In contrast to the T ¼ 0 K simulations above, modeling by MD provides the ability to investigate temperature effects in dislocation–obstacle interaction. (The limit on simulation time discussed in
Glide plane Dz =
R
R/2
0
-R/2
-R
Climb-up - atoms left inside void
Climb-down - atoms taken off void Figure 7 Schematic representation of the configurations studied for voids of radius R in Grammatikopoulos et al.43 and the corresponding shape of the edge dislocation line seen in [111] projection after breakaway. Dz is the distance of the center of the void from the dislocation glide plane. From Grammatikopoulos, P.; Bacon, D. J.; Osetsky, Yu. N. Model. Simulat. Mater. Sci. Eng. 2011, 19, 015004. With permission from IOP Publishing Ltd.
Atomic-Level Dislocation Dynamics in Irradiated Metals
Section 1.12.3.3 prevents study of the creep regime controlled by dislocation climb.) Results on the temperature dependence of tc from simulation of interaction between an edge dislocation and 2 and 6 nm voids in Fe,29,30,34 Cu-precipitates in Fe,27,29 and voids in Cu30,34 are presented in Figure 8. In general, the strength of all the obstacles becomes weaker with increasing temperature, although the mechanisms involved are not the same for the different obstacles. The temperature-dependence of void strengthening in Fe has been analyzed by Monnet et al.44 using a mesoscale thermodynamic treatment of MD data in the point obstacle approximation to estimate activation energy and its temperature dependence. In this way, the obstacle strength found by atomic-scale modeling can be converted into a mesoscale parameter to be used in higher level modeling in the multiscale framework. More investigations are required to define mesoscale parameters for more complicated cases such as voids in Cu and Cu-precipitates in Fe. Void strengthening in Cu exhibits specific behavior in which the temperature-dependence is strong at low T < 100 K but rather weak at higher T (for more details see Figure 8 in Osetsky and Bacon34). The reason for this is as yet unclear. Interestingly, MD simulation has been able to shed light on thermal effects in strengthening due to Cu-precipitates in Fe, as in Figure 8 (for more details see Figure 5 in Bacon and Osetsky23). Small precipitates, D < 3 nm, are stabilized in the bcc coherent state by the Fe matrix, as noted above 1.0
2 and 6 nm obstacles: . ε = 5 × 106 s-1 Voids in Cu
0.8
Voids in Fe
tc Gb/L
Cu-prpt in Fe 0.6
6 nm
0.4
2 nm
0.2
0.0 0
100
200
300
400
500
600
700
T (K)
Figure 8 Plot of tc versus T for voids and Cu-precipitates in Fe and voids in Cu. D is as indicated, L ¼ 41.4 nm, and e_ ¼ 5 106 s1 .
343
for T ¼ 0 K, and are weak, shearable obstacles. The resulting temperature-dependence of tc is small. Larger precipitates were seen to be unstable at T ¼ 0 K with respect to a dislocation-induced transformation toward the fcc structure. This transformation is driven by the difference in potential energy of bcc and fcc Cu. The free energy difference between these two phases of Cu decreases with increasing T until a temperature is reached at which the transformation does not occur. Thus, large precipitates are strong obstacles at low T and weak ones at high T. This is reflected in the strong dependence of tc on T shown in Figure 8. More explanation of this effect can be found in Bacon and Osetsky.23 These simulation results showing the different behavior of small and large Cu-precipitates suggest that the yield stress of underaged or neutron-irradiated Fe–Cu alloys, which contain small, coherent Cu-precipitates, should have a weak T-dependence, whereas that in an overaged or electron-irradiated alloy, in which the population of coherent precipitates has a larger size, should be stronger. Some experimental observations support this.45 One is a weak change in the temperature dependence of radiation-induced precipitate hardening in ferritic alloys observed after neutron irradiation when only small (<2 nm) precipitates are formed. The other is the experimentally-observed temperature and size dependence of deformation-induced transformation of Cu-precipitates in Fe.46 Other obstacles with inclusion properties, such as gas-filled bubbles and other types of precipitates, have been studied less intensively, and we present just a few examples here. The effect of chromium precipitates on edge dislocation motion in matrices of either pure Fe or Fe–10 at.% Cr solid solution was studied by Terentyev et al.47 Cr and Cr-rich precipitates have the bcc structure and are coherent with the matrix. Unlike Cu-precipitates in Fe, G of Cr is higher than that of both matrices and so the dislocation is repelled by Cr precipitates. Under increasing strain, the dislocation moves until it reaches the precipitate– matrix interface where it stops until the stress reaches the maximum, tc, just before the dislocation enters the precipitate (see Figure 2 in Terentyev et al.47). The t versus e behavior is similar to that for voids in Cu, but without stress drops associated with partial dislocations, and no softening effects similar to voids and Cuprecipitates in Fe were observed. At tc, the dislocation shears the obstacle without acquiring a double jog. Only 2.8 and 3.5 nm precipitates in the size range D ¼ 0.6–3.5 nm had tc comparable with values given
344
Atomic-Level Dislocation Dynamics in Irradiated Metals
by eqn [2] (see Figure 4 in Terentyev et al.47); the others were much weaker. Separate contributions from the chemical strengthening (CS) and shear modulus difference (SMD) between Fe and Cr were estimated and their sum was found to be close to the tc found in simulation. It was also found that tc for the alloy with Cr precipitates in an Fe–Cr solid solution is the sum of tc for the same precipitate in a pure Fe matrix and the maximum stress for glide of the dislocation motion in the Fe–Cr solid solution alone. Helium-filled bubbles created by vacancies and helium formed in transmutation reactions are common features of the irradiated microstructure of structural materials (see Chapter 1.13, Radiation Damage Theory). However, there is a lack of information on the properties of He bubbles and their contribution to changes in mechanical properties. The main problem is the uncertainty regarding the equilibrium state of bubbles of different sizes and at different temperatures, that is, their He-to-vacancy ratio (He/Vac). A small (0.5 M-atom) model was used48–50 to simulate interaction between an edge dislocation and a row of 2 nm cavities with He/Vac ratios of up to 5 in Fe at T between 10 and 700 K. It was found that tc has a nonmonotonic dependence on the He/Vac ratio, dislocation climb increases with this ratio, and interstitial defects are formed in the vicinity of the bubble. Recent work to clarify the equation of state of bubbles using a new Fe–He three-body interaction potential51 has shown that the equilibrium concentration of He is much lower than expected; for example the He/Vac ratio is 0.5 for a 2 nm bubble at 300 K in Fe.52 Simulation of an edge dislocation interacting with 2 nm bubbles using the new potential for He/Vac ratio in the range 0.2–2 and T between 100 and 600 K has now been performed53 and preliminary conclusions drawn. The dislocation interaction with underpressurized bubbles (He/Vac < 0.5) is similar to that with voids described above, that is, the dislocation climbs up by absorbing some vacancies on breakaway and tc increases with increasing values of He/Vac ratio up to 0.5. The interaction with overpressurized bubbles (He/Vac > 0.5) is different. The dislocation climbs down and tc decreases with increasing value of He/Vac ratio. At the highest ratio, the dislocation stress field induces the bubble to emit interstitial Fe atoms from its surface into the matrix toward the dislocation before it makes contact. The bubble pressure is reduced in this way and interstitials are absorbed by the dislocation as a double superjog. Equilibrium bubbles are therefore the strongest. Some of
these conclusions, such as formation of interstitial clusters around bubbles with high He/Vac ratios, are similar to those observed earlier,48–50 others are not. More modeling is necessary to clarify these issues. As noted in Section 1.12.2.2, impenetrable obstacles such as oxide particles and incoherent precipitates represent another class of inclusion-like features. Although these obstacles are usually preexisting and not produced by irradiation, they are considered to be of potential importance for the design of nuclear energy structural materials and should be considered here. Atomic-level information on their effect on dislocations is still poor, however, and we can only refer to some recent work on this. The interaction between an edge dislocation and a rigid, impenetrable particle in Cu was simulated by Hatano54 using the Cu–Cu IAP as for the Fe–Cu system36 and a constant strain rate of 7 106 s1 at T ¼ 300 K. The particle was created by defining a spherical region in which the atoms were held immobile relative to the surrounding crystal. The Hirsch mechanism2 was found to operate. In the sequence shown in Figures 1 and 2 of Hatano,54 several stages can be observed such as (1) the dislocation under stress approaching the obstacle from the left first bows round the obstacle to form a screw dipole; (2) the screw segments cross-slip on inclined {111} planes at tc; (3) they annihilate by double cross-slip, allowing the dislocation, now with a double superjog, to bypass the obstacle; (4) a prismatic loop with the same b is left behind and (5) the dragged superjogs pinch-off as the dislocation glides away, creating a loop of opposite sign to the first on the right of the obstacle. tc varies with D and L as predicted by the continuum modeling that led to eqn [1], but is over 3 times larger in magnitude. Hatano argues that this could arise from either higher stiffness of a dissociated dislocation or a dependence of tc on the initial position of the dislocation. It is also possible that the requirement for the dislocation to constrict and the absence of a component of applied stress on the crossslip plane results in a high value of tc. Simulation of 2 nm impenetrable precipitates in Fe has been carried out by Osetsky (2009, unpublished). The method is different from that used by Hatano54 in that the precipitate, constructed from Fe atoms held immobile relative to each other, was treated as a superparticle moving according to the total force on precipitate atoms from matrix atoms. The interaction mechanism observed is quite different from those reported earlier for Cu,54 for the Hirsch mechanism and formation of interstitial clusters does
Atomic-Level Dislocation Dynamics in Irradiated Metals
not occur. Instead, the mechanism observed was close to the Orowan process, with formation of an Orowan loop that either shrinks quickly if the obstacle is small and spherical, or remains around it if D is large (4nm) or becomes elongated in the direction perpendicular to the slip plane. It is interesting to note that if the model of a completely immobile precipitate in Hatano54 is applied to Fe, the same Hirsh mechanism is observed as that in the earlier study. Comparison of strengthening due to pinning of a 1/2h111i{110} edge dislocation by 2 nm spherical obstacles of different nature simulated at 300 K is presented in Figure 9. One can see that the coherent Cu precipitate is the weakest whereas a rigid impenetrable precipitate when Orowan loop is stabilized by its shape is the strongest. A surprising result is that an equilibrium He-bubble is a stronger obstacle than the equivalent void. The reason for this is not clear yet. Little is known on the interactions between screw dislocations and inclusion-like obstacles. In fact, we are aware of only one published study of screw dislocation–void interaction in fcc Cu.55 It was found that voids are quite strong obstacles and their strength and interaction mechanisms are strongly temperature dependent. Thus at low temperature, the 1/2h110i {111} screw dislocation keeps its original slip plane 250 MD modeling in Fe: Edge dislocation 1/2⬍111⬎{110} Crystal 2 074 000 atoms Temperature 300 K Obstacle size 2 nm Obstacle spacing 42 nm
225
tc (MPa)
200 175 150 125 100 75 50
Cu-prpt
Void
He-bubble He/Vac = 0.5
Rigid particle
Orowan mechanism
Figure 9 Critical stress for an edge dislocation penetrating through different 2 nm obstacles in Fe at T ¼ 300 K. L ¼ 41.4 nm and e_ ¼ 5 106 s1 . Table 1
345
when it breaks away from the void. However, at high T > 300 K, cross-slip is activated and plays an important role in dislocation–void interaction. Several depinning mechanisms involving dislocation crossslip on the void surface were simulated and formation of DLs was observed in some cases. Interestingly, the void strength increases with increasing temperature and the authors explain this by changing interaction mechanisms. Intensive cross-slip was observed54 that propagated through the periodic boundaries along the dislocation line direction, with the result that the void was interacting with its images. Similar effects have been observed in the interaction of a screw dislocation and SIA loops and SFTs, and the possible significance of this for understanding the simulation results has been discussed elsewhere.4 1.12.4.2
Dislocation-Type Obstacles
Extensive simulations of interactions between moving dislocations and dislocation-like obstacles such as DLs and SFTs has demonstrated that the reactions involved follow the general rules of dislocation– dislocation reaction, for example, Frank’s rule for Burgers vectors,1,2 even though the reacting segments are of the nanometer scale in length. Results of these interactions are in the range from no effect on both dislocation and obstacle to complete disappearance of the obstacle and significant modification of the dislocation. A detailed analysis of reactions was made for SFTs an fcc metal56 and later for SIA loops in Fe.57 In general, five types of reaction were identified, as summarized in Table 1. The outcomes in Table 1 were observed for different obstacles under different reaction conditions such as interaction geometry, strain rate, ambient temperature, and so on. We give some examples in the following section. 1.12.4.2.1 Stacking fault tetrahedra
Reactions of type R1 have been observed for both screw and edge dislocations and all the defects with dislocation character. Interestingly, the strength effect of this reaction varies from minimum to
Description of main reactions between dislocations and obstacles with dislocation character4
Reaction
Dislocation type
Overall result
R1 R2 R3 R4 R5
Edge or screw Edge or screw Edge Screw Edge and screw
Dislocation and obstacle remain unchanged Obstacle changed but dislocation unchanged Partial or full absorption of obstacle by edge dislocation (superjog formation) Temporary absorption of obstacle by screw dislocation (helix formation) Dislocation drags glissile defects
346
Atomic-Level Dislocation Dynamics in Irradiated Metals
maximum. For example, it is insignificant in the case of a 1/2h110i{111} edge dislocation interacting with an SFT58 and maximum for a screw dislocation interacting with a DL when the loop is fully absorbed into a helical turn on the dislocation.57 The mechanism for the way both the obstacle and dislocation remain unchanged is different for each case. An edge dislocation interacting with an SFT close to its tip creates a pair of ledges on its surface that are not stable and annihilate athermally.56,58 An example of this reaction is presented in Figure 10. If the dislocation slip plane is far enough from the SFT tip in the compressive region of the dislocation (for details of geometrical definitions see Bacon et al.4), the ledges can be stabilized.56,58,59 This can be seen in Figure 11 (1). If the dislocation passes through the SFT several times in the same slip plane, it can detach the portion of the SFT above the slip plane, as shown in Figure 11 (2–4). Both the above mechanisms are common for small SFTs, low T, fast dislocations, and the position of the SFT tip above the slip plane of an edge dislocation. If, however, the SFT tip is below the dislocation slip plane, and T is high enough and the dislocation speed low enough, D
reaction R3 can be activated. The stages of this reaction are presented in Figure 12. An example of effects of SFT orientation and temperature for the interaction of an edge dislocation with an SFT is presented in Figure 13. In this study, the dislocation slip plane intersected a 4.2 nm SFT through its geometrical center at the applied e_ ¼ 5 106 s1 in a wide temperature range from 0 to 450 K.59 Reaction R1 was observed (see Figure 10) at all temperatures when the SFT was oriented with its tip up relative to the dislocation slip plane (orange triangles up in Figure 13) and at the two lowest temperatures when it was oriented in the opposite sense. At T ¼ 300 K and orientation with tip down, a couple of ledges were formed on the SFT surface (see Figure 11). It may be noted that the R2 mechanism requires higher applied stress even though the temperature is increased. At higher T ¼ 450 K, the interaction mechanism is changed and the whole portion of the SFT above the slip plane is absorbed by the dislocation (Figure 12), that is, reaction R3 occurs, creating a pair of superjogs on the dislocation line. Some vacancies were also found to form to accommodate the glissile configuration of the superjogs.
dC Ad C
A
(a)
(b)
(c)
(d)
Figure 10 An example of reaction R1 for an edge dislocation passing through 4.2 nm SFT (136 vacancies) oriented with apex above the slip plane in Cu at 300 K. From Osetsky, Yu. N.; Rodney, D.; Bacon, D. J. Philos. Mag. 2006, 86, 2295. With permission from Taylor and Francis Ltd. (http://www.informaworld.com).
Atomic-Level Dislocation Dynamics in Irradiated Metals
0
1
2
3
347
4
Figure 11 Shear of a 2.4 nm SFT (45 vacancies) by an edge dislocation in Cu at 300 K. 0: Initial SFT; (1): creation of two ledges (reaction R1); (2–4): evolution of the configuration due to additional passes of the dislocation. From Osetsky, Yu. N.; Stoller, R. E.; Matsukawa, Y. J. Nucl. Mater. 2004, 329–333, 1228. Copyright (2004) with permission from Elsevier.
B
A C
ad
dA
ab
dA
Cd Cd
(a)
(b) Ca
Cd Cb
dA
ad
bd
Cd
(c)
(d)
Figure 12 An example of reaction R3 for an edge dislocation and 4.2 nm SFT (136 vacancies) with its tip below the slip plane in Cu at 450 K. From Osetsky, Yu. N.; Rodney, D.; Bacon, D. J. Philos. Mag. 2006, 86, 2295. With permission from Taylor and Francis Ltd. (http://www.informaworld.com).
This is discussed later in Section 1.12.4.3. More details on interactions between screw and edge dislocations and SFT can be found elsewhere.60–63 In general, it can be concluded that the SFTs created under irradiation, that is, <4 nm in size,6 are very stable objects and unlikely to be eliminated by a simple interaction with either edge or screw dislocations. Numerous attempts have been made to find a mechanism responsible for formation of clear, defect-free channels in irradiated fcc materials.64 One of the most used models considers absorption of an SFT by screw and mixed 60 dislocations.65 The absorption by conversion of an SFT into a helical turn on a screw dislocation has been observed by in situ transmission electron microscope (TEM) deformation experiments,64,66–68 but only partial absorption has been found in MD modeling. As observed and discussed elsewhere,4,59 SFT separation into parts due to temporary absorption of part of an SFT as a helical turn and its expansion along a screw dislocation line can occur, but complete annihilation of an SFT has not been reproduced by atomic-scale
modeling of bulk material. Possible reasons, including inability of MD to reproduce the whole set of experimental conditions such as stress state, scale, strain rate, and so on, were discussed in Matsukawa et al.66–68 An alternative interpretation was suggested as a result of MD simulation of interaction between a screw dislocation and an SFT in a thin film,26 that is, the conditions realized experimentally for in situ TEM deformation. A thin film of fcc Cu was simulated and a 1/2h110i screw dislocation with an end on each surface was moved toward the SFT (size 12 and 18 nm) placed in the film center. A number of steps occurred that resulted in elimination of the vast portion of the SFT26: 1. The dislocation glided toward the SFT and partially absorbed it as a helical turn. 2. Edge segments of the turn glided toward the free surfaces and were annihilated there. 3. Glide of edge segments provided mass transport; in this particular case, transport of vacancies to the free surfaces.
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Atomic-Level Dislocation Dynamics in Irradiated Metals
175 4.2 nm SFT-edge disl.: - Tip down
150
- Tip up
CRSS (MPa)
125
R2 (ledges formation)
100
75
R3 (partial absorption)
50 R1 (no SFT damage) 25 0
100
200 300 Temperature (K)
400
500
Figure 13 Temperature dependence of the critical stress for an edge dislocation penetrating through a 4.2-nm SFT (136 vacancies) in one of two different orientations in Cu. The type of reaction is indicated for all cases.
4. The screw character of the dislocation was restored but in another slip plane. The dislocation then glided away leaving behind a small portion of the original SFT. Although this mechanism provides a mechanistic understanding of in situ TEM observations, it can operate only between surfaces or interfaces where the ends of the screw dislocation can cross-slip and, therefore, cannot be applied to bulk material and is unlikely to be responsible for the clear channel formation observed in bulk samples. We will return to this question later. 1.12.4.2.2 Dislocation loops
DLs are common objects formed in metals under irradiation. Depending on metal properties and irradiation conditions, loops of different types can be formed. They are mainly interstitial in nature although vacancy loops can appear in some metals under specific conditions. Here we consider only interstitial DLs. The shortest Burgers vector in bcc metals is bL ¼ 1/2h111i and loops with this are the most common. In neutron and heavy-ion irradiated Fe, loops can have bL ¼ h100i,69 particularly at T above about 300 C. Note that both have a perfect Burgers vector and are glissile. The most common loops in fcc metals have bL ¼ 1/2h110i and are also perfect and glissile. In metals and alloys with a low stacking fault energy, Frank loops with bL ¼ 1/3h111i can form. They are faulted and sessile. In the following section, we consider examples of reactions R1–R4 in fcc and bcc metals.
1.12.4.2.2.1
Face-centered cubic metals
Interaction between a 1/2h110i{111} edge dislocation and a 1/2h110i SIA loop in Ni was first studied by Rodney and Martin,70 and then followed a series of more detailed investigations involving glissile and sessile SIA loops and screw and edge dislocations.16,70–74 Reaction R3 was observed with small loops having bL ¼ 1/2h110i intersecting the dislocation slip plane.70,75 Glissile clusters were attracted by the dislocation core and absorbed there athermally, creating a pair of superjogs. Superjog segments have different structure, as depicted in Figure 14, and to accommodate this a few vacancies are formed, as in the case of the R3 reaction with an SFT (see Figure 12). Each superjog has different mobility, for example, the Lomer-Cottrell segment on the left of Figure 14(b) has high Peierls stress. Therefore, although the jogged dislocation continues gliding under applied stress, it has a significantly lower velocity because it now experiences higher effective phonon drag.71 An interesting example of an R2 reaction was observed during interaction between a 1/2h110i screw dislocation and a 1/3h111i Frank loop.16 If the loop is not too large and the dislocation is not too fast, the dislocation can absorb the whole loop into a helical turn (see stages in Figure 15). The turn can expand only along the dislocation line and, therefore, if applied stress is maintained, the helix constricts; finally a perfect DL, with the same b as the dislocation, is released as the screw breaks away.
Atomic-Level Dislocation Dynamics in Irradiated Metals
349
z y x
AC
(a)
bC
bC
db dC
Ab
db
bC Ad Ab
Ad
db/AC
AC
Ab
dC
(b) Figure 14 (a) Structure of an edge dislocation after absorption of a 37-interstitial loop in Ni at 300 K, resulting in reaction R2 and (b) the corresponding Burgers vector geometry of the superjogs labeled using the Thompson tetrahedron notation. Reprinted with permission from Rodney, D.; Martin, G. Phys. Rev. B 2000, 61, 8714. Copyright (2000) by the American Physical Society.
Note that the same mechanism occurs when a screw dislocation intersects a glissile 1/2h110i loop with Burgers vector different from that of the dislocation, that is, absorption into a helical turn and release of a loop with the same Burgers vector as the dislocation.76 In the case when the Frank loop is large or the dislocation (either screw or edge) is fast, the dislocation simply shears the loop and creates a step on its surface (see Figure 8 in Rodney16). The probability of this reaction is higher for an edge dislocation, whereas transformation of the loop into a perfect loop is more probable for the screw dislocation.73 The strengthening effect due to dislocation–SIA loop interactions can be significant, especially when a helical turn is formed on a screw dislocation. The total contribution of dislocation–SIA loop interaction to the flow stress under irradiation has not been considered so far because of the large number of possible reactions and the sensitivity of their outcomes in terms of mechanism and strengthening effect to parameters such as interaction geometry, loop size and Burgers vector, strain rate, and temperature. An estimate of the flow stress contribution for the case of reaction R2 in Ni was made in Rodney and Martin.75
1.12.4.2.2.2
Body-centered cubic metals
Many atomic-scale simulations of interaction between a 1/2h111i{110} dislocation and an SIA loop in bcc metals have been made, particularly using EAM potentials for Fe. They have shown that when the defects come into contact, a new dislocation segment is formed with one of two possible Burgers vectors 1/2h111i and h100i, and further evolution depends on features such as loop size, b, position, dislocation character, temperature, and strain rate. Thus, many different outcomes have been observed by atomic scale modeling, but here we can present only a few common examples. Reaction R3 is most common for small SIA loops, that is, loop absorption on the dislocation line resulting in a pair of superjogs. It occurs when the loop is initially below the dislocation slip plane (tension region of the dislocation strain field) and bL ¼ 1/2h111i is inclined to the slip plane. An important feature of the interaction is the ability of such loops to glide quickly toward the core of the approaching dislocation. SIA loops up to 5 nm in diameter have been simulated.77–80 When small loops (a few tens of SIAs) reach the core, they are fully absorbed athermally creating a double superjog of the size equivalent to the number of SIAs in the
350
Atomic-Level Dislocation Dynamics in Irradiated Metals
(a)
(b)
(c)
(d)
(e)
(f)
Figure 15 Unfaulting of a 1/3h111i Frank loop by interaction with a screw dislocation in Ni at 300 K and transformation into a helical turn on the dislocation line: (a) initial configuration, (b) first cross-slip, (c) and (d) successive cross-slip events, (e) configuration at the end of unfaulting, (f) configuration after relaxation and elongation of the helical turn. From Rodney, D. Nucl. Instrum. Meth. Phys. Res. B 2005, 228, 100. Copyright (2005) with permission from Elsevier.
loop. This process does not pin the dislocation and these loops are very weak obstacles to dislocation glide.77 Large loops (more than 100 SIAs) also glide to make contact with the dislocation line but are not absorbed athermally. Instead, a new dislocation segment is formed due to the following energetically favorable Burgers vector reaction between the dislocation and loop: 1 1 ½111 ½111 ¼ ½010 2 2
½2
Five stages of the interaction are presented in Figure 16 for a 5-nm loop containing 331 SIAs. The Burgers vectors are indicated and Figure 16(b) corresponds to the occurrence of the reaction of eqn [2]. The new segment with b ¼ [010] cannot glide in the dislocation slip plane ð110Þ and therefore acts as a strong obstacle to further glide of the dislocation. Under increasing applied strain, the dislocation bows out until two long segments of 1/2[111] screw dislocations are formed as shown in Figure 16(c). Figure 16(d) shows the same configuration in [111]
Atomic-Level Dislocation Dynamics in Irradiated Metals
(010)
(111)
t = 50 ps
(d)
[111]
_ (111)
_ (111)
(a)
(111)
351
(b)
t = 655 ps
t = 55 ps
(e)
(c)
t = 655 ps
t = 765 ps
Figure 16 Visualization of stages in the interaction between a 1=2½111 edge dislocation and a 5 nm (331 SIAs) 1=2½111 loop in Fe at 300 K. From Bacon, D. J.; Osetsky, Yu. N.; Rong, Z. Philos. Mag. 2006, 86, 3921. With permission from Taylor and Francis Ltd. (http://www.informaworld.com).
projection. High stress at junctions connecting the dislocation line and loop, the remainder of the original loop, and the new segment induces the latter to slip down on the ð101Þ plane, and glide of this [010] segment over the loop surface results in the following reaction with the remaining loop: 1 1 ½111 þ ½010 ¼ ½111 ½3 2 2 This concludes with the formation of a pair of superjogs on the original dislocation and results in complete absorption of the 5 nm loop. Large loops are strong obstacles in this reaction, stronger than voids with the same number of vacancies.4,80 It should be noted, however, that the interaction just described depends rather sensitively on temperature because of the low mobility of screw segments in the bcc metals. Cross-slip of the screw segments is required to allow the [010] segment to glide down. Simulations at T ¼ 100 K showed that although the stages in Figure 16(a)–16(c) still occurred, the [010] segment did not glide under the strain rate imposed in MD and the screw dipole created by the bowing dislocation was annihilated without the loop being transformed according to eqn [3]. The resulting reaction was of type R1, for both dislocation and loop were unchanged after dislocation breakaway.78 More details and examples can be found.4,77–82 Competition between reactions R1, R2, and R3 for a 1/2h111i{110} edge dislocation and 1/2h111i and h100i SIA loops has been considered in detail.83,84 We cannot describe all the reactions here but some pertinent features are underlined; note that the favorable Burgers vector reaction between a 1/2h111i dislocation and a h100i loop results in a 1/2h111i segment, for example,
1 1 ½111 þ ½100 ¼ ½111 2 2
½4
Thus, a perfect loop with bL ¼ [100] can be converted into a sessile complex of 1=2½111 and 1=2½111 loop segments joined bya [100] dislocation segment.83 Similar conjoined loop complexes were observed in simulations of interactions between two glissile 1/2h111i loops85,86 in bcc Fe. Competition between reaction R3 on one side and R1 and R2 on the other was discussed in Bacon and Osetsky.82 The earlier conclusion that small loops (1 nm) can be absorbed easily and not present strong obstacles to dislocation glide was confirmed. The strength and reaction mechanism for larger loops depends on their size and the loading conditions. At low T 100 K, both 1/2h111i and h100i loops are strong obstacles that are not absorbed by 1/2h111i dislocations. As with obstacles that result in the true Orowan mechanism, the dislocation unpins by recombination of the screw dipole, and the critical stress is determined by the loop size, similar to eqn [1]. At higher Tand/or low strain rate (<107 s1), the mechanism changes from Orowanlike bypassing to complete loop absorption, irrespective of bL. The absorption mechanism for R3 involves propagation of the reaction segment over the loop surface. This requires cross-slip of the arms of the screw dipole drawn out on the pinned dislocation and involves dislocation reactions. Thermally activated glide and/or decomposition of the pinning segment, in turn, depends on loop size, temperature, and bL. Therefore, the obstacle strength of a 1/2h111i loop is, in general, higher than that of a h100i loop because a h100i dislocation segment associated with the former (see eqn [3], Figure 16 and related text) is much less mobile than a 1/2h111i segment involved in reactions with the latter (eqn [4]).
352
Atomic-Level Dislocation Dynamics in Irradiated Metals
Much less is known about interaction mechanisms involving screw dislocations in bcc metals. We are aware of only two studies reported to date that considered 1/2h111i and h100i SIA loops.87,88 The main feature in these cases is the ability of a 1/2h111i screw dislocation to absorb a complete or part loop into a temporary helical turn before closing the turn by bowing forward and breaking away. As in the case of fcc metals described above, this provides a powerful route to reorienting the Burger vectors of different loops to that of the dislocation. An example of such a reaction is visualized in Figure 17.88 Other reactions observed in these studies87,88 include a reforming of the original glissile loop into a sessile complex of two segments having different b, as described above (reaction R2), and complete restoration of the initial loop (reaction R1). As is clear from the examples discussed above, the obstacle strength associated with different mechanisms can depend strongly on parameters such as loop size and Burgers vector, interaction geometry, e_ and T. Unlike the situation revealed for inclusionlike obstacles discussed in Section 1.12.4.1, there does not appear to be a simple correlation between size and strength for loops. Some data related to particular sets of conditions can be found in Terentyev et al.,83,84,88 as well as a comparison of the obstacle strength of voids and DLs.84 There are cases when loop strengthening is compatible to or even exceeds that of voids containing the same number of point defects, making DLs an important component of radiation-induced hardening. 1.12.4.3 Microstructure Modifications due to Plastic Deformation In this section we consider some other cases, such as those involved in reaction R5 ignored above, and refocus some conclusions already made on other
reactions. We note here that so far the need to investigate DD in irradiated metals was led mainly by the need to create multiscale modeling tools for predicting changes in mechanical properties due to irradiation. This is desirable for practical estimations in engineering support of real nuclear devices. However, there is another consequence of dislocation activity that is related to microstructure changes that occur. While this may not be important in postmortem experiments on irradiated materials, it can be important in real devices operating under irradiation. It is obvious that internal and external stresses can accumulate during irradiation of complicated devices due to high temperature, radiation growth, swelling, and transition periods of operation, for example, shut down and restart. Creep is usually taken into account but microstructure changes due to dislocation activity during irradiation are not. This activity affects the whole process of microstructure evolution and should therefore be taken into account in predicting the effects of irradiation. The validity of this statement is demonstrated by recent in-reactor straining experiments on some bcc and fcc metals and alloys.9 It is not possible at the moment to formulate unambiguous conclusions and more experimental work will be necessary for this. Nevertheless, it is clear that dislocation activity during irradiation directly affects the radiation damage process. In the following section, we describe some mechanisms that can contribute to this at the atomic-scale level. First, consider reaction R5, omitted in Section 1.12.4.2, for dislocation obstacles. Drag of glissile interstitial loops by moving edge dislocations was first observed in the MD modeling of bcc and fcc metals.89 It is well known that SIA clusters in the form of small perfect DLs exhibit thermally activated glide in the direction of their Burgers vector.90,91 It is characterized by very low activation energy 0.01– 0.10 eV. An edge dislocation, having a long-range
C B C D z
x
A
B A
y
(a)
(b)
(c)
(d)
Figure 17 Interaction between a 1/2[111] screw dislocation, gliding in the y direction shown by the double arrow in (a), and a [010] SIA loop. The loop is absorbed (b,c) and then reformed in (d) with b ¼ 1/2[111]. From Terentyev, D. A.; Bacon, D. J.; Osetsky, Yu. N. Philos. Mag. 2010, 90, 1019. With permission from Taylor and Francis Ltd. (http://www.informaworld.com).
Atomic-Level Dislocation Dynamics in Irradiated Metals
elastic field, can interact with such clusters and, if bL is parallel to the dislocation glide plane, can drag or push them as it moves under applied stress/strain. The dynamics of this process have been investigated in detail and correlations between cluster and dislocation mobility analysed.92–94 Additional friction due to cluster drag reduces dislocation velocity, an effect that is stronger in fcc than bcc metals because of features of cluster structure.89 The maximum speed at which a dislocation can drag an SIA cluster is achieved by a compromise between dislocation-cluster interaction force and cluster friction and varies at T ¼ 300 K from 180 m s1 in fcc Cu to >1000 m s1 in bcc Fe for loops containing a few tens of SIAs.89 An important consequence of this drag process is that a moving dislocation can sweep glissile clusters and transport them through the material. Other reactions that may affect microstructure evolution involve both inclusion- and dislocationlike obstacles. The relevant reaction for the latter is denoted R3 in Table 1. In this case, an edge dislocation climbs and the formation of superjogs by defect absorption changes its structure and total line length. This changes its mobility and its cross-section for interaction with other defects, such as point defects, their clusters and impurities, and this in turn affects microstructure evolution. Reactions of type R2 can also be important for they change properties of obstacles. In thermal aging without stress, obstacles such as voids, SIA clusters and SFTs evolve towards their equilibrium low-energy state, that is voids/precipitates into faceted near-spherical shapes, SFTs into regular tetrahedron shape, and so on, whereas shearing creates interface steps on voids/precipitates and creates ledges on SFT faces. These surface features change the properties of the defects by putting them into a higher energy state. Reaction R4 introduces another mechanism of mass transport, for the helical turn (representing the absorbed defects) can only extend or translate in the direction of its Burgers vector, that is, along the screw dislocation line. The case of dislocation-SFT interaction in a thin film considered in Section 1.12.4.2.1 has demonstrated that this may introduce completely new mechanisms. This effect can also play a role when a dislocation ends on an internal interface where it can cross-slip. Reaction R4 also orders the orientation of DLs left behind by a gliding screw dislocation for it changes bL of these loops to b of the dislocation, irrespective of their initial orientation. Finally, we describe a case when several of the above mechanisms may have a significant effect on
353
microstructure changes if operating at the same time on different defects. It is known from experimental studies6 that under neutron irradiation, Cu accumulates a high density of SFTs and this density saturates with dose at a high level (1024 m3), close to conditions under which displacement cascades overlap. Taking into account that SIA clusters are necessarily accumulated in the system, this high saturation density implies that annihilation reactions between the vacancy population in SFTs and SIA loops is suppressed. MD modeling in which an SIA cluster was placed between two SFTs about 10 nm apart and intersecting the loop glide cylinder has confirmed that annihilation reaction does not occur even after 50 ns at T < 900 K.95 The result is not surprising for each vacancy of an SFT is distributed over the four faces of the tetrahedron within the stacking faults. However, simulations show that an annihilation reaction can be promoted by the involvement of a gliding dislocation. Two cases have been considered. In one, an edge dislocation under applied stress dragged a 1/2h110i SIA loop toward an SFT placed 7 nm below the dislocation slip plane and intersecting the glide cylinder of the loop. The overlapping fractions of SFT and dragged SIA loop annihilated by recombination. Different obstacle sizes (SFT from 45 to 61 vacancies and SIA loops from 37 to 91 SIAs) and geometries with different levels of overlap were simulated and recombination occurred in all cases. In the other situation, the same SFT and SIA loop 10–20 nm apart were placed on the slip plane of a screw dislocation. On approaching the obstacles, the dislocation absorbed a portion of each to form two helical turns (reaction R4). The turns of vacancy and interstitial character had opposite sign and the smaller was annihilated by recombination with part of the larger. On moving farther ahead, the dislocation released the unrecombined portion of the remaining helix to leave a small defect. This was usually part of the original SIA loop because a loop can be completely absorbed as a helical turn, whereas only a part of an SFT can be absorbed in this way. Thus, the overall result of interstitial loop drag under applied stress was annihilation of a significant part of both clusters by a reaction between helices of opposite signs.
1.12.5 Concluding Remarks Atomic-scale simulation by computer has become a powerful tool for investigation of material properties
354
Atomic-Level Dislocation Dynamics in Irradiated Metals
and processes that cannot be achieved by experimental techniques or other theoretical methods. This is due to the increasing power of computers, the development of new efficient modeling codes, and the extensive usage of ab initio calculations for probing atomic mechanisms and generating data for design of new IAPs. Moreover, increasing length and time scales attainable by atomistic modeling provides overlap with experimental scales in some cases, thereby allowing direct verification of modeling results.26 In this chapter, we have described only a selection of the results obtained within the last decade by atomic-scale modeling of DD in irradiated bcc and fcc metals. The examples presented and references cited demonstrate how detailed insight into mechanisms can be gained by such modeling. For some obstacles to dislocation motion, for example, many inclusion-like obstacles, strengthening is controlled by the dislocation line shape at breakaway and can be parameterized using the existing elasticity theory models. In other cases, for example, dislocation-like obstacles, reactions and their results, including obstacle strength, depend very much on the material, and dislocation character and core structure; dislocation behavior is also sensitive to conditions such as interaction geometry, temperature, and strain rate. The variety of outcomes for dislocationlike obstacles is complicated and wide, and although features of these reactions can be understood within the framework of the continuum theory of dislocations, for example, Frank’s rule, a general formalization of the reactions in terms of obstacle strength and reaction product does not exist. Nevertheless, it has been seen that the insight gained by simulation has allowed the outcome of reactions to be classified in a meaningful way (Table 1). This will allow for validation of higher-level DD modeling of these reactions using the continuum approximation. An excellent example of the way in which this can be done has been provided by Martinez and coworkers,96,97 who used MD and DD to simulate the same dislocation– SFT interactions. The continuum modeling of these nanometer-scale obstacles was verified by the atomic simulation, and this enabled a large number of interaction geometries and conditions to be investigated successfully by DD. Unfortunately, successful overlaps in scale of atomistic modeling and experiment or/and continuum modeling are still rare. Efforts in all techniques are necessary to progress understanding of mechanisms and their parameterization for predictive modeling tools that can be applied to irradiated materials.
Investigations such as those just discussed bring assurance that atomic-scale modeling is correct at least qualitatively and is invaluable in cases where scale overlap of techniques is not yet achieved. Two main problems exist with regard to the quality of its quantitative outcomes. One is concerned with time scale. As already mentioned, the limit on time scale is the main disadvantage of current atomic-scale modeling. The maximum simulation time achieved so far is of the order of a few hundred nanoseconds. For dislocation studies, this allows dislocations with velocity as low as 0.5 m s1 to be modeled – and it may be that some processes are insensitive to velocity at this level27 – but the overall strain rate (105 s1) is fast compared with experiment, and the interaction time (100 ns) with an obstacle is too short for thermally-activated processes to be sampled. Development of new methods for keeping atomic-level accuracy over at least microsecond to second time scales is necessary to progress to the next step toward predictive modeling for engineering applications. An example of a new generation technique for simulating realistic strain rates whilst retaining atomic-scale detail was published recently.98 The new technique combines atomic-scale modeling for estimating vacancy migration barriers in the vicinity of an edge dislocation and kinetic Monte Carlo (MC) for simulating vacancy kinetics in a crystal with a specified dislocation density. The technique was successfully applied to simulate the process of power-law creep over a macroscopic time scale with microscopic fidelity. The other problem is concerned with accuracy in describing interatomic interactions. Much of the research described in this chapter, based as it is on empirical EAM IAPs, is more related to the behavior of model metals with bcc or fcc crystal structure in general than to the elements Fe or Cu in particular. This difficulty will become more acute with the demand for more sophisticated, radiationresistant alloys, and future investigations of chemical effects on plasticity will require IAPs that incorporate chemistry in a meaningful way. More on this is presented in Chapter 1.10, Interatomic Potential Development. We would like to conclude on an optimistic note. It is clear that significant progress has been achieved over the last decade in understanding the details of the atomic-scale mechanisms involved in dislocations dynamics in structural metals in a reactor environment. The small, nanoscale nature of the obstacles created by radiation damage is such that the techniques described here provide uniquely valuable
Atomic-Level Dislocation Dynamics in Irradiated Metals
information, despite the limitations they currently experience.
Acknowledgments
22. 23. 24. 25. 26.
Much of the work described in this chapter was carried out with support of the Division of Materials Sciences and Engineering, U.S. Department of Energy, under contract with UT-Battelle, LLC (YO, modeling and results analysis).
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1.13
Radiation Damage Theory
S. I. Golubov Oak Ridge National Laboratory, Oak Ridge, TN, USA
A. V. Barashev Oak Ridge National Laboratory, Oak Ridge, TN, USA; University of Tennessee, Knoxville, TN, USA
R. E. Stoller Oak Ridge National Laboratory, Oak Ridge, TN, USA
Published by Elsevier Ltd.
1.13.1 1.13.2 1.13.3 1.13.3.1 1.13.3.2 1.13.3.2.1 1.13.3.2.2 1.13.4 1.13.4.1 1.13.4.2 1.13.4.3 1.13.4.3.1 1.13.4.3.2 1.13.4.3.3 1.13.4.4 1.13.4.4.1 1.13.4.4.2 1.13.4.4.3 1.13.5 1.13.5.1 1.13.5.1.1 1.13.5.1.2 1.13.5.1.3 1.13.5.1.4 1.13.5.1.5 1.13.5.1.6 1.13.5.1.7 1.13.5.1.8 1.13.5.2 1.13.5.2.1 1.13.5.2.2 1.13.5.2.3 1.13.5.3 1.13.6 1.13.6.1 1.13.6.1.1 1.13.6.1.2 1.13.6.1.3 1.13.6.1.4 1.13.6.1.5 1.13.6.1.6
Introduction The Rate Theory and Mean Field Approximation Defect Production Characterization of Cascade-Produced Primary Damage Defect Properties Point defects Clusters of point defects Basic Equations for Damage Accumulation Concept of Sink Strength Equations for Mobile Defects Equations for Immobile Defects Size distribution function Master equation Nucleation of point defect clusters Methods of Solving the Master Equation Fokker–Plank equation Mean-size approximation Numerical integration of the kinetics equations Early Radiation Damage Theory Model Reaction Kinetics of Three-Dimensionally Migrating Defects Sink strength of voids Sink strength of dislocations Sink strengths of other defects Recombination constant Dissociation rate Void growth rate Dislocation loop growth rate The rates P(x) and Q(x) Damage Accumulation Void swelling Effect of recombination on swelling Effect of immobilization of vacancies by impurities Inherent Problems of the Frenkel Pair, 3-D Diffusion Model Production Bias Model Reaction Kinetics of One-Dimensionally Migrating Defects Lifetime of a cluster Reaction rate Partial reaction rates Reaction rate for SIAs changing their Burgers vector The rate P(x) for 1D diffusing self-interstitial atom clusters Swelling rate
358 360 361 361 362 362 363 363 363 364 364 364 365 366 367 368 368 368 370 371 371 372 372 373 373 374 374 375 375 375 377 378 378 379 379 379 380 380 381 381 382
357
358
Radiation Damage Theory
1.13.6.2 1.13.6.2.1 1.13.6.2.2 1.13.6.2.3 1.13.6.3 1.13.6.3.1 1.13.6.3.2 1.13.7 References
Main Predictions of Production Bias Model High swelling rate at low dislocation density Recoil-energy effect GB effects and void ordering Limitations of Production Bias Model Swelling saturation at random void arrangement Absence of void growth in void lattice Prospects for the Future
Abbreviations bcc BEK fcc F–P FP FP3DM GB hcp kMC MD ME MFA NRT PBM PD PKA RDT RIS RT SDF SFT SIA
Body-centered cubic Bullough, Eyre, and Krishan (model) Face-centered cubic Fokker–Plank equation Frenkel pair Frenkel pair three-dimensional diffusion model Grain boundary Hexagonal close-packed Kinetic Monte Carlo Molecular dynamics Master equation Mean-field approximation Norgett, Robinson, and Torrens (standard) Production bias model Point defect Primary knock-on atom Radiation damage theory Radiation-induced segregation Rate theory Size distribution function Stacking-fault tetrahedron Self-interstitial atom
Symbols Ca Da f(ri ) Ga N r R
rd S
L
Concentration of a-type defects Diffusion coefficient for a-type defects Size distribution function Production rate of a-type defects by irradiation Number density Mean void radius Reaction rate Dislocation capture radius for an SIA cluster Void swelling level Total trap density in one dimension
Lj rd t
383 383 384 385 387 387 387 387 389
Partial density of traps of kind j ( j ¼ c; d) Dislocation density Lifetime
1.13.1 Introduction The study of radiation effects on the structure and properties of materials started more than a century ago,1 but gained momentum from the development of fission reactors in the 1940s. In 1946, Wigner2 pointed out the possibility of a deleterious effect on material properties at high neutron fluxes, which was then confirmed experimentally.3 A decade later, Konobeevsky et al.4 discovered irradiation creep in fissile metallic uranium, which was then observed in stainless steel.5 The discovery of void swelling in neutron-irradiated stainless steels in 1966 by Cawthorne and Fulton6 demonstrated that radiation effects severely restrict the lifetime of reactor materials and that they had to be systematically studied. The 1950s and early 1960s were very productive in studying crystalline defects. It was recognized that atoms in solids migrate via vacancies under thermalequilibrium conditions and via vacancies and selfinterstitial atoms (SIAs) under irradiation; also that the bombardment with energetic particles generates high concentrations of defects compared to equilibrium values, giving rise to radiation-enhanced diffusion. Numerous studies revealed the properties of point defects (PDs) in various crystals. In particular, extensive studies of annealing of irradiated samples resulted in categorizing the so-called ‘recovery stages’ (e.g., Seeger7), which comprised a solid basis for understanding microstructure evolution under irradiation. Already by this time, which was well before the discovery of void swelling in 1966, the process of interaction of various energetic particles with solid
Radiation Damage Theory
targets had been understood rather well (e.g., Kinchin and Pease8 for a review). However, the primary damage produced was wrongly believed to consist of Frenkel pairs (FPs) only. In addition, it was commonly believed that this damage would not have serious long-term consequences in irradiated materials. The reasoning was correct to a certain extent; as they are mobile at temperatures of practical interest, the irradiation-produced vacancies and SIAs should move and recombine, thus restoring the original crystal structure. Experiments largely confirmed this scenario, most defects did recombine, while only about 1% or an even smaller fraction survived and formed vacancy and SIA-type loops and other defects. However small, this fraction had a dramatic impact on the microstructure of materials, as demonstrated by Cawthorne and Fulton.6 This discovery initiated extensive experimental and theoretical studies of radiation effects in reactor materials which are still in progress today. After the discovery of swelling in stainless steels, it was found to be a general phenomenon in both pure metals and alloys. It was also found that the damage accumulation takes place under irradiation with any particle, provided that the recoil energy is higher than some displacement threshold value, Ed, (30–40 eV in metallic crystals). In addition, the microstructure of different materials after irradiation was found to be quite similar, consisting of voids and dislocation loops. Most surprisingly, it was found that the microstructure developed under irradiation with 1 MeV electrons, which produces FPs only, is similar to that formed under irradiation with fast neutrons or heavy-ions, which produce more complicated primary damage (see Singh et al.1). All this created an illusion that three-dimensional migrating (3D) PDs are the main mobile defects under any type of irradiation, an assumption that is the foundation of the initial kinetic models based on reaction rate theory (RT). Such models are based on a mean-field approximation (MFA) of reaction kinetics with the production of only 3D migrating FPs. For convenience, we will refer to these models as FP production 3D diffusion model (FP3DM) and henceforth this abbreviation will be used. This model was developed in an attempt to explain the variety of phenomena observed: radiation-induced hardening, creep, swelling, radiation-induced segregation (RIS), and second phase precipitation. A good introduction to this theory can be found, for example, in the paper by Sizmann,9 while a comprehensive overview was produced by Mansur,10 when its development was
359
already completed. The theory is rather simple, but its general methodology can be useful in the further development of radiation damage theory (RDT). It is valid for 1 MeV electron irradiation and is also a good introduction to the modern RDT, see Section 1.13.5. Soon after the discovery of void swelling, a number of important observations were made, for example, the void super-lattice formation11–14 and the micrometer-scale regions of the enhanced swelling near grain boundaries (GBs).15 These demonstrated that under neutron or heavy-ion irradiation, the material microstructure evolves differently from that predicted by the FP3DM. First, the spatial arrangement of irradiation defects voids, dislocations, second phase particles, etc. is not random. Second, the existence of the micrometer-scale heterogeneities in the microstructure does not correlate with the length scales accounted for in the FP3DM, which are an order of magnitude smaller. Already, Cawthorne and Fulton6 in their first publication on the void swelling had reported a nonrandomness of spatial arrangement of voids that were associated with second phase precipitate particles. All this indicated that the mechanisms operating under cascade damage conditions (fast neutron and heavy-ion irradiations) are different from those assumed in the FP3DM. This evidence was ignored until the beginning of the 1990s, when the production bias model (PBM) was put forward by Woo and Singh.16,17 The initial model has been changed and developed significantly since then18–28 and explained successfully such phenomena as high swelling rates at low dislocation density (Section 1.13.6.2.2), grain boundary and grain-size effects in void swelling, and void lattice formation (Section 1.13.6.2.3). An essential advantage of the PBM over the FP3DM is the two features of the cascade damage: (1) the production of PD clusters, in addition to single PDs, directly in displacement cascades, and (2) the 1D diffusion of the SIA clusters, in addition to the 3D diffusion of PDs (Section 1.13.3). The PBM is, thus, a generalization of the FP3DM (and the idea of intracascade defect clustering introduced in the model by Bullough et al. (BEK29)). A short overview of the PBM was published about 10 years ago.1 Here, it will be described somewhat differently, as a result of better understanding of what is crucial and what is not, see Section 1.13.6. From a critical point of view, it should be noted that successful applications of the PBM have been limited to low irradiation doses (<1 dpa) and pure metals (e.g., copper). There are two problems that
360
Radiation Damage Theory
prevent it from being used at higher doses. First, the PBM in its present form1 predicts a saturation of void size (see, e.g., Trinkaus et al.19 and Barashev and Golubov30 and Section 1.13.6.3.1). This originates from the mixture of 1D and 3D diffusion–reaction kinetics under cascade damage conditions, hence from the assumption lying at the heart of the model. In contrast, experiments demonstrate unlimited void growth at high doses in the majority of materials and conditions (see, e.g., Singh et al.,31 Garner,32 Garner et al.,33 and Matsui et al.34). An attempt to resolve this contradiction was undertaken23,25,27 by including thermally activated rotations of the SIA-cluster Burgers vector; but it has been shown25 that this does not solve the problem. Thus, the PBM in its present form fails to account for the important and common observation: the indefinite void growth under cascade irradiation. The second problem of the PBM is that it fails to explain the swelling saturation observed in void lattices (see, e.g., Kulchinski et al.13). In contrast, it predicts even higher swelling rates in void lattices than in random void arrangements.25 This is because of free channels between voids along close-packed directions, which are formed during void ordering and provide escape routes for 1D migrating SIA clusters to dislocations and GBs, thus allowing 3D migrating vacancies to be stored in voids. Resolving these two problems would make PBM self-consistent and complete its development. A solution to the first problem has recently been proposed by Barashev and Golubov35,36 (see Section 1.13.7). It has been suggested that one of the basic assumptions of all current models, including the PBM, that a random arrangement of immobile defects exists in the material, is correct at low and incorrect at high doses. The analysis includes discussion of the role of RIS and provides a solution to the problem, making the PBM capable of describing swelling in both pure metals and alloys at high irradiation doses. The solution for the second problem of the PBM mentioned above is the main focus of a forthcoming publication by Golubov et al.37 Because of limitations of space, we only give a short guide to the main concepts of both old and more recent models and the framework within which radiation effects, such as void swelling, and hardening and creep, can be rationalized. For the same reason, the impact of radiation on reactor fuel materials is not considered here, despite a large body of relevant experimental data and theoretical results collected in this area.
1.13.2 The Rate Theory and Mean Field Approximation The RDT is frequently but inappropriately called ‘the rate theory.’ This is due to the misunderstanding of the role of the transition state theory (TST) or (chemical reaction) RT (see Laidler and King38 and Ha¨nggi et al.39 for reviews) in the RDT. The TST is a seminal scientific contribution of the twentieth century. It provides recipes for calculating reaction rates between individual species of the types which are ubiquitous in chemistry and physics. It made major contributions to the fields of chemical kinetics, diffusion in solids, homogeneous nucleation, and electrical transport, to name a few. TST provides a simple way of formulating reaction rates and gives a unique insight into how processes occur. It has survived considerable criticisms and after almost 75 years has not been replaced by any general treatment comparable in simplicity and accuracy. The RDT uses TST as a tool for describing reactions involving radiationproduced defects, but cannot be reduced to it. This is true for both the mean-field models discussed here, and the kinetic Monte Carlo (kMC) models that are also used to simulate radiation effects (see Chapter 1.14, Kinetic Monte Carlo Simulations of Irradiation Effects). The use of the name RT also created an incorrect identification of the RDT with the models that emerged in the very beginning, which assumed the production of only FPs and 3D migrating PDs to be the only mobile species, that is, FP3DM. It failed to appreciate the importance of numerous contradicting experimental data and, hence, to produce significant contribution to the understanding of neutron irradiation phenomena (see Barashev and Golubov35 and Section 1.13.6). A common perception that the RDT in general is identical to the FP3DM has developed over the years. So, the powerful method was rejected because of the name of the futile model. This caused serious damage to the development of RDT during the last 15 years or so. Many research proposals that included it as an essential part, were rejected, while simulations, for example, by the kMC etc. were aimed at substituting the RDT. The simulations can, of course, be useful in obtaining information on processes on relatively small time and length scales but cannot replace the RDT in the largescale predictions. The RDT and any of its future developments will necessarily use TST. An important approximation used in the theory is the MFA. The idea is to replace all interactions in a
Radiation Damage Theory
many-body system with an effective one, thereby reducing the problem of one-body in an effective field. The MFA is used in different areas of physics on all scales: from ab initio to continuum models. In the RDT, the main objective is to describe diffusion and interaction between defects in a self-consistent way. So, the primary damage is produced by irradiation in the form of mobile vacancies, SIAs, SIA clusters, and immobile defects. The latter together with preexisting dislocations and GBs, and those formed during irradiation, for example, voids and dislocation loops of different sizes represent crystal microstructure and change during irradiation. The complete problem of microstructure evolution is, thus, too complex; some approximations are necessary and the MFA is the most natural option. It should be emphasized that a particular realization of the MFA depends on the problem and it can be employed even in cases with spatial correlations between defects. For example, in this way Go¨sele40 demonstrated that the absorption rates of 3D migrating vacancies by randomly distributed and ordered voids are significantly different; and then it was shown in Barashev et al.25 that the effect is even stronger for 1D diffusing SIA clusters. In some specific cases, however, when the time and length scales of the problem permit, numerical approaches such as kMC can be a natural choice for studying spatial correlations.
361
The total production rate of displacements per atom (dpa), G NRT , can be calculated using this equation, by integrating the flux of projectile particles, ’ðEÞ, (E is the particle energy), as max E~ð
1 ð
G NRT ¼
dE’ðEÞ Ed
0
dsðE; E~Þ ~ ~ nðE ÞdE dE~
½2
where sðE; E~Þ is the cross-section of reactions, in which an incident particle transfers energy E~ to an max is the maximum transferable energy. atom and E~ For a head-on collision of a nonrelativistic projectile of mass m and a target atom of mass M max E~ ¼
4Mm E ðM þ mÞ2
while for relativistic electrons, 2me E max þ 2 E E~ ¼ M me c 2
½3a
½3b
where me is the electron mass and c is the speed of light. The NRT model is accepted as an international standard for quantifying the number of atomic displacements produced under cascade damage conditions. It is based on the theory of isolated binary collisions and, hence, cannot be used to characterize the defects formed during the collision phase and survive at the end of the cool-down phase of cascades. Description of the latter is considered below.
1.13.3 Defect Production Interaction of energetic particles with a solid target is a complex process. A detailed description is beyond the scope of the present paper (Robinson41). However, the primary damage produced in collision events is the main input to the RDT and is briefly introduced here. Energetic particles create primary knock-on (or recoil) atoms (PKAs) by scattering either incident radiation (electrons, neutrons, protons) or accelerated ions. Part of the kinetic energy, EPKA , transmitted to the PKA is lost to the electron excitation. The remaining energy, called the damage energy, Td , is dissipated in elastic collisions between atoms. If the Td exceeds a threshold displacement energy, Ed , for the target material, vacancy-interstitial (or Frenkel) pairs are produced. The total number of displaced atoms is proportional to the damage energy in a model proposed by Norgett et al.42 and known as the NRT standard E nðE~Þ ¼ 0:8
PKA
ðE~Þ
2Ed
½1
1.13.3.1 Characterization of Cascade-Produced Primary Damage The NRT displacement model is most correct for irradiation such as 1 MeV electrons, which produce only low-energy recoils and, therefore, the FPs. At higher recoil energies, the damage is generated in the form of displacement cascades, which change both the production rate and the nature of the defects produced. Over the last two decades, the cascade process has been investigated extensively by molecular dynamics (MD) and the relevant phenomenology is described in Chapter 1.11, Primary Radiation Damage Formation and recent publications.43,44 For the purpose of this chapter the most important findings are (see discussion in the Chapter 1.11, Primary Radiation Damage Formation): For energy above 0.5 keV, the displacements are produced in cascades, which consist of a collision and recovery or cooling-down stage.
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Radiation Damage Theory
A large fraction of defects generated during the collision stage of a cascade recombine during the cooling-down stage. The surviving fraction of defects decreases with increasing PKA energy up to 10 keV, when it saturates at a value of 30% of the NRT value, which is similar in several metals and depends only slightly on the temperature. By the end of the cooling-down stage, both SIA and vacancy clusters can be formed. The fraction of defects in clusters increases when the PKA energy is increased and is somewhat higher in face-centered cubic (fcc) copper than in bcc iron. The SIA clusters produced may be either glissile or sessile. The glissile clusters of large enough size (e.g., >4 SIAs in iron) migrate 1D along closepacked crystallographic directions with a very low activation energy, practically a thermally, similar to the single crowdion.45,46 The SIA clusters produced in iron are mostly glissile, while in copper they are both sessile and glissile. The vacancy clusters produced may be either mobile or immobile vacancy loops, stacking-fault tetrahedra (SFTs) in fcc metals, or loosely correlated 3D arrays in bcc materials such as iron.
cluster type, PKA energy and material, and is connected with the fractions e as 1 X xGa ðxÞ ¼ ea G NRT ð1 er Þ ½6
As compared to the FP production, the cascade damage has the following features.
Single vacancies and other vacancy-type defects, such as, SFTs and dislocation loops, have been considered quite extensively since the 1930s because it was recognized that they define many properties of solids under equilibrium conditions. Extensive information on defect properties was collected before material behavior in irradiation environments became a problem of practical importance. Qualitatively new crystal defects, SIAs and SIA clusters, were required to describe the phenomena in solids under irradiation conditions. This has been studied comprehensively during the last 40 years. The properties of these defects and their interaction with other defects are quite different compared to those of the vacancytype. Correspondingly, the crystal behavior under irradiation is also qualitatively different from that under equilibrium conditions. The basic properties of vacancy- and SIA-type defects are summarized below.
The generation rates of single vacancies and SIAs are not equal: Gv 6¼ Gi and both smaller than that given by the NRT standard, eqn. [2]: Gv ; Gi < G NRT . Mobile species consist of 3D migrating single vacancies and SIAs, and 1D migrating SIA and vacancy clusters. Sessile vacancy and SIA clusters, which can be sources/sinks for mobile defects, can be formed. The rates of PD production in cascades are given by Gv ¼ G NRT ð1 er Þð1 ev Þ
½4
Gi ¼ G NRT ð1 er Þð1 ei Þ
½5
where er is the fraction of defects recombined in cascades relative to the NRT standard value, and ev and ei are the fractions of clustered vacancies and SIAs, respectively. One also needs to introduce parameters describing mobile and immobile vacancy and SIA-type clusters of different size. The production rate of the clusters containing x defects, GðxÞ, depends on
x ¼2
where a ¼ v; i for the vacancy and SIA-type clusters, respectively. The total fractions ev and ei of defects in clusters are given by the sums of those for mobile and immobile clusters, ea ¼ esa þ ega
½7
where the superscripts ‘s’ and ‘g’ indicate sessile and glissile clusters, respectively. In the mean-size approximation Gaj ðxÞ ¼ Gaj d x hxaj i ½8 where j ¼ s; g; dðxÞ is the Kronecker delta and hxaj i is the mean cluster size and 1
Gaj ¼ hxaj i G NRT ð1 er Þeja
½9 46
Also note that although MD simulations show that small vacancy loops can be mobile, this has not been incorporated into the theory yet and we assume that they are sessile: egv ¼ 0 and esv ¼ ev . 1.13.3.2
Defect Properties
1.13.3.2.1 Point defects
The basic properties of PDs are as follows: 1. Both vacancies and SIAs are highly mobile at temperatures of practical interest, and the diffusion coefficient of SIAs, Di , is much higher than that of vacancies, Dv : Di Dv .
Radiation Damage Theory
2. The relaxation volume of an SIA is much larger than that of a vacancy, resulting in higher interaction energy with edge dislocations and other defects. 3. Vacancies and SIAs are defects of opposite type, and their interaction leads to mutual recombination. 4. SIAs, in contrast to vacancies, may exist in several different configurations providing different mechanisms of their migration. 5. PDs of both types are eliminated at fixed sinks, such as voids and dislocations. The first property leads to a specific temperature dependence of the damage accumulation: only limited number of defects can be accumulated at irradiation temperature below the recovery stage III, when vacancies are immobile. At higher temperature, when both PDs are mobile, the defect accumulation is practically unlimited. The second property is the origin of the so-called ‘dislocation bias’ (see Section 1.13.5.2) and, as proposed by Greenwood et al.,47 is the reason for void swelling. A similar mechanism, but induced by external stress, was proposed in the so-called ‘SIPA’ (stress-induced preferential absorption) model of irradiation creep.48–53 The third property provides a decrease of the number of defects accumulated in a crystal under irradiation. The last property, which is quite different compared to that of vacancies leads to a variety of specific phenomena and will be considered in the following sections. 1.13.3.2.2 Clusters of point defects
The configuration, thermal stability and mobility of vacancy, and SIA clusters are of importance for the kinetics of damage accumulation and are different in the fcc and bcc metals. In the fcc metals, vacancy clusters are in the form of either dislocation loops or SFTs, depending on the stacking-fault energy, and the fraction of clustered vacancies, ev , is close to that for the SIAs, ei . In the bcc metals, nascent vacancy clusters usually form loosely correlated 3D configurations, and ev is much smaller than ei . Generally, vacancy clusters are considered to be immobile and thermally unstable above the temperature corresponding to the recovery stage V. In contrast to vacancy clusters, the SIA clusters are mainly in the form of a 2D bundle of crowdions or small dislocation loops. They are thermally stable and highly mobile, migrating 1D in the close-packed crystallographic directions.45 The ability of SIA clusters to move 1D before being trapped or absorbed by a dislocation, void, etc. leads to entirely different reaction kinetics as compared with that for 3D migrating
363
defects, and hence may result in a qualitatively different damage accumulation than that in the framework of the FP3DM (see Section 1.13.6). It should be noted that MD simulations provide maximum evidence for the high mobility of small SIA clusters. Numerous experimental data, which also support this statement, are discussed in this chapter, however, indirectly. One such fact is that most of the loops formed during ion irradiations of a thin metallic foil have Burgers vectors lying in the plane of the foil.54 It should also be noted that recent in situ experiments55–58 provide interesting information on the behavior of interstitial loops (>1 nm diameter, that is, large enough to be observable by transmission electron microscope, TEM). The loops exhibit relatively low mobility, which is strongly influenced by the purity of materials. This is not in contradiction with the simulation data. The observed loops have a large cross-section for interaction with impurity atoms, other crystal imperfections and other loops: all such interactions would slow down or even immobilize interstitial loops. Small SIA clusters produced in cascades consist typically of approximately ten SIAs and have, thus, much smaller cross-sections and consequently a longer mean-free path (MFP). The influence of impurities may, however, be strong on both the mobility of SIA clusters and, consequently, void swelling is yet to be included in the theory.
1.13.4 Basic Equations for Damage Accumulation Crystal microstructure under irradiation consists of two qualitatively different defect types: mobile (single vacancies, SIAs, and SIA and vacancy clusters) and immobile (voids, SIA loops, dislocations, etc.). The concentration of mobile defects is very small (1010–106 per atom), whereas immobile defects may accumulate an unlimited number of PDs, gas atoms, etc. The mathematical description of these defects is, therefore, different. Equations for mobile defects describe their reactions with immobile defects and are often called the rate (or balance) equations. The description of immobile defects is more complicated because it must account for nucleation, growth, and coarsening processes. 1.13.4.1
Concept of Sink Strength
The mobile defects produced by irradiation are absorbed by immobile defects, such as voids, dislocations, dislocation loops, and GBs. Using a MFA, a crystal
364
Radiation Damage Theory
can be treated as an absorbing medium. The absorption rate of this medium depends on the type of mobile defect, its concentration and type, and the size and spatial distribution of immobile defects. A parameter called ‘sink strength’ is introduced to describe the reaction cross-section and commonly designated as kv2 , ki2 , 2 ðxÞ for vacancies, SIAs, and SIA clusters of size x and kicl (the number of SIAs in a cluster), respectively. The role of the power ‘2’ in these values is to avoid the use of square root for the MFPs of diffusing defects between production until absorption, which are correspond1 ðxÞ. There are a number of ingly kv1, ki1 , and kicl publications devoted to the derivation of sink strengths.40,59–61 Here we give a simple but sufficient introduction to this subject. 1.13.4.2
Equations for Mobile Defects
For simplicity, we use the following assumptions: The PDs, single vacancies, and SIAs, migrate 3D. SIA clusters are glissile and migrate 1D. All vacancy clusters, including divacancies, are immobile. The reactions between mobile PDs and clusters are negligible. Immobile defects are distributed randomly over the volume. Then, the balance equations for concentrations of mobile vacancies, Cv , SIAs, Ci , and SIA clusters, g Cicl ðxÞ, are as follows dCv ¼ G NRT ð1 er Þð1 ev Þ þ Gvth dt kv2 Dv Cv mR Di Ci Cv
½10
½11
g
dCicl ðxÞ g g 2 ¼ Gicl ðxÞ kicl ðxÞDicl Cicl ; dt x ¼ 2; 3; . . . xmax Gvth
g
dCicl g 1 g g 2 ¼ hxi i G NRT ð1 er Þei kicl Dicl Cicl dt
½13
where eqn [9] is used for the cluster generation rate. To solve eqns [10]–[13], one needs the sink strengths 2 , the rates of vacancy emission from kv2 , ki2 , and kicl various immobile defects to calculate Gvth , and the recombination constant, mR . The reaction kinetics of 3D diffusing PDs is presented in Section 1.13.5, while that of 1D diffusing SIA clusters in Section 1.13.6. In the following section, we consider equations governing the evolution of immobile defects, which together with the equations above describe damage accumulation in solids both under irradiation and during aging. 1.13.4.3
Equations for Immobile Defects
The immobile defects are those that preexist such as dislocations and GBs and those formed during irradiation: voids, vacancy- and SIA-type dislocation loops, SFTs, and second phase precipitates. Usually, the defects formed under irradiation nucleate, grow, and coarsen, so that their size changes during irradiation. Hence, the description of their evolution with time, t, should include equations for the size distribution function (SDF), f ðx; t Þ, where x is the cluster size. 1.13.4.3.1 Size distribution function
dCi ¼ G NRT ð1 er Þð1 ei Þ ki2 Di Ci dt m R D i Ci Cv
rather weak,45,46 the mean-size approximation for the SIA clusters may be used, where all clusters are g assumed to be of the size hxi i. In this case, the set of eqn [12] is reduced to the following single equation
½12
where is the rate of thermal emission of vacancies from all immobile defects (dislocations, GBs, voids, etc.); Dv , Di , and Dicl ðxÞ are the diffusion coefficients of vacancies, single SIAs, and SIA clusters, respectively; and mR is the recombination coefficient of PDs. Since the dependence of the cluster diffusivity, 2 ðxÞ, on size x is Dicl ðxÞ, and sink strengths, kicl
The measured SDF is usually represented as a function of defect size, for example, radius, x R : f ðR; t Þ. In calculations, it is more convenient to use x-space, x x, where x is the number of defects in a cluster: f ðx; t Þ. The radius of a defect, R, is connected with the number of PDs, x, it contains as: 4p 3 R ¼ xO 3 pR2 b ¼ xO
½14
for voids and loops, respectively, where O is the atomic volume and b is the loop Burgers vector. Correspondingly, the SDFs in R- and x-spaces are related to each other via a simple relationship. Indeed, if small dx and dR correspond to the same cluster group, the number density of this cluster group defined by two functions f ðxÞdx and f ðRÞdR must be equal, f ðxÞdx ¼ f ðRÞdR, which is just a differential form
Radiation Damage Theory
of the equality of corresponding integrals for the total number density: N¼
1 X
1 ð
f ðxÞ
x ¼2
1 ð
f ðxÞdx ¼ x ¼2
f ðRÞdR ½15 R ¼ Rmin
The relationship between the two functions is, thus, dx f ðRÞ ¼ f ðxÞ dR For voids and dislocation loops 1=3 36p 3 fc ðRÞ ¼ x 2=3 fc ðxÞ x ¼ 4pR O 3O 1=2 4pb 2 fL ðRÞ ¼ x 1=2 fL ðxÞ x ¼ pbR O O
½16
Note the difference in dimensionality: the units of f ðxÞ are atom1 (or m3), while f ðRÞ is in m1 atom1 (or m4), as can be seen from eqn [15]. Also note that these two functions have quite different shapes, see Figure 1, where the SDF of voids obtained by Stoller et al.62 by numerical integration of the master equation (ME) (see Sections 1.13.4.3.2 and 1.13.4.4.3) is plotted in both R- and x-spaces. 1.13.4.3.2 Master equation
The kinetic equation for the SDF (or the ME) in the case considered, when the cluster evolution is driven by the absorption of PDs, has the following form @f s ðx; tÞ ¼ G s ðxÞ þ J ðx 1; t Þ J ðx; t Þ; x 2 ½18 @t
where G s ðxÞ is the rate of generation of the clusters by an external source, for example, by displacement
1021
Diameter (nm) 1
0
2
1023
1022
1021 1019 1020
fvcl(x) Fvcl(r)
1018
1019
Number density (m–3, nm–1)
Void number density (m–3)
T = 373 K, FP 1020
E2V = 0.3 eV 1017 100
200
300
400
500
cascades, and J ðx; t Þ is the flux of the clusters in the size-space (indexes ‘i’ and ‘v’ in eqn [18] are omitted). The flux J ðx; tÞ is given by J ðx; t Þ ¼ Pðx; t Þf ðx; t Þ Q ðx þ 1; t Þf ðx þ 1; t Þ
1018 600
Number of vacancies
Figure 1 Size distribution function of voids calculated in x-space, fvcl(x) (x is the number of vacancies), and in d-space, fvcl(d) (d is the void diameter). From Stoller et al.62
½19
where Pðx; t Þ and Q ðx; tÞ are the rates of absorption and emission of PDs, respectively. The boundary conditions for eqn [18] are as follows f ð1Þ ¼ C f ðx ! 1Þ ¼ 0
½17
365
½20
where C is the concentration of mobile PDs. If any of the PD clusters are mobile, additional terms have to be added to the right-hand side of eqn [19] to account for their interaction with immobile defect which will involve an increment growth or shrinkage in the size-space by more that unity (see Section 1.13.6 and Singh et al.22 for details). The total rates of PD absorption (superscript !) and emission ( ) are given by ! ¼ Jtot
1 X
PðxÞf ðxÞ;
x¼2
Jtot ¼
1 X
Q ðxÞf ðxÞ ½21
x¼2
where the superscript arrows denote direction in the ! and Jtot are related to the sink strength size-space. Jtot of the clusters, thus providing a link between equations for mobile and immobile defects. For example, when voids with the SDF fc(x) and dislocations are only presented in the crystal and the primary damage is in the form of FPs, the balance equations are dCv ¼ G NRT ð1 er Þ dt mR Di Ci Cv þ Zvd rd Dv ðCv Cv0 Þ Pc ð1Þfc ð1; t Þ Q vc ð2Þfc ð2; t Þ xX ¼1 ðPc ðxÞfc ðx; t Þ
x¼1 Q vc ðx
þ 1Þfc ðx þ 1; t ÞÞ
½22
dCi ¼ G NRT ð1 er Þ dt mR Di Ci Cv þ Zid rd Di Ci
xX ¼1 x¼1
Q i c ðx þ 1Þfc ðx þ 1; t Þ
½23
d are the dislocation density and where rd and Zi;v its efficiencies for absorbing PDs, mR , is the recombination constant (see Section 1.13.5); the last two terms in eqn [22] describe the absorption and
366
Radiation Damage Theory
emission of vacancies by voids and the last term in eqn [23] describes the absorption of SIAs by voids. The balance equations for dislocation loops and secondary phase precipitations can be written in a similar manner. Expressions for the rates Pðx; t Þ; Q ðx; t Þ, d , and mR are the dislocation capture efficiencies, Zi;v derived in Section 1.13.5. 1.13.4.3.3 Nucleation of point defect clusters
Nucleation of small clusters in supersaturated solutions has been of significant interest to several generations of scientists. The kinetic model for cluster growth and the rate of formation of stable droplets in vapor and second phase precipitation in alloys during aging was studied extensively. The similarity to the condensation process in supersaturated solutions allows the results obtained to be used in RDT to describe the formation of defect clusters under irradiation. The initial motivation for work in this area was to derive the nucleation rate of liquid drops. Farkas63 was first to develop a quantitative theory for the so-called homogeneous cluster nucleation. Then, a great number of publications were devoted to the kinetic nucleation theory, of which the works by Becker and Do¨ring,64 Zeldovich,65 and Frenkel66 are most important. Although these publications by no means improved the result of Farkas, their treatment is mathematically more elegant and provided a proper background for subsequent works in formulating ME and revealing properties of the cluster evolution. A quite comprehensive description of the nucleation phenomenon was published by Goodrich.67,68 Detailed discussions of cluster nucleation can also be found in several comprehensive reviews.69,70 Generalizations of homogeneous cluster nucleation for the case of irradiation were developed by Katz and Wiedersich71 and Russell.72 Here we only give a short introduction to the theory. For small cluster sizes at high enough temperature, when the thermal stability of clusters is relatively low, the diffusion of clusters in the size-space governs the cluster evolution, which is nucleation of stable clusters. In cases where only FPs are produced by irradiation, the first term on the right-hand side of eqn [18] is equal to zero and cluster nucleation, for example, voids, proceeds via interaction between mobile vacancies to form divacancies, then between vacancies and divacancies to form trivacancies, and so on. By summing eqn [18] from x ¼ 2 to 1, one finds dNc ¼ J ðxÞjx¼1 Jcnucl dt
½24
where Nc ¼
1 P x¼2
f ðxÞ is the total number of clusters.
The nucleation rate in this case, Jcnucl , is equal to the rate of production of the smallest cluster (divacancies in the case considered); hence the flux J ðxÞjx¼1 is the main concern. When calculating Jcnucl , one can obtain two limiting SDFs that correspond to two different steadystate solutions of eqn [18]: (1) when the flux J ðx; t Þ ¼ 0, for which the corresponding SDF is n(x), and, (2) when it is a constant: J ðx; t Þ ¼ Jc , with the SDF denoted as g(x). Let us first find n(x). Using equation PðxÞnðxÞ Q ðx þ 1Þnðx þ 1; t Þ ¼ 0 and the condition n(1) ¼ C, one finds that nðxÞ ¼ C
x 1 Y
PðyÞ ;x 2 Q ðy þ 1Þ y¼1
½25
Using function nðxÞ, the flux J ðx; t Þ can be derived as follows f ðxÞ f ðx þ 1Þ ½26 J ðx; t Þ ¼ PðxÞnðxÞ nðxÞ nðx þ 1Þ The SDF g(x) corresponding to the constant flux, J ðx; t Þ ¼ Jc , can be found from eqn [26]: gðxÞ ¼ Jc nðxÞ
1 X y¼x
1 PðyÞnðyÞ
½27
Using the boundary conditions gð1Þ ¼ nð1Þ ¼ C one finds that Jcnucl is fully defined by n(x): Jcnucl ¼ P 1 x¼1
1 ½PðxÞnðxÞ1
½28
Generally, nðxÞ has a pronounced minimum at some critical size, x ¼ xcr , and the main contribution to the denominator of eqn [28] comes from the clusters with size around xcr . Expanding nðxÞ in the vicinity of xcr up to the second derivative and replacing the summation by the integration, one finds an equation for Jcnucl , which is equivalent to that for nucleation of second phase precipitate particles.64,65 Note that eqn [28] describes the cluster nucleation rate quite accurately even in cases where the nucleation stage coexists with the growth which leads to a decrease of the concentration of mobile defects, C. This can be seen from Figure 2, in which the results of numerical integration of ME for void nucleation are compared with that given by eqn [28].73 In the case of low temperature irradiation, when all vacancy clusters are thermally stable (C ¼ Cv in the case) and only FPs are produced by irradiation,
Radiation Damage Theory
10–3 10–6 dpa s–1
Fe 200 ⬚C
10–4
rd = 1014 m–2
Nucleation rate (dpa–1)
10–5 250 ⬚C 10–6 275 ⬚C
10–7 10–8
300 ⬚C
10–9 Numerical
10–10
Equation [30] 10–11 10–6
10–5
10–4
10–3
10–2
10–1
100
101
Irradiation dose (dpa)
Figure 2 Comparison of the dependences of the void nucleation rate as a function of irradiation dose calculated using master equation, eqns [18] and [28]. From Golubov and Ovcharenko.73
the void nucleation rate, eqn [21], can be calculated analytically. Indeed, in the case where the binding energy of a vacancy with voids of all sizes is infinite, Evb ðxÞ ¼ 1 (see eqn [75]), it follows from eqn [25] that the function n(x) is equal to Cv Dv Cv x1 ½29 nðxÞ ¼ 1=3 D i Ci x Substituting eqn [29] in eqn [28], one can easily find that the nucleation rate, Jcnucl , takes the form 1 Jcnucl ¼ wCv Dv Cv P
x1 1 Di Ci x¼1
½30
Dv Cv
where w ¼ ð48p2 =O2 Þ1=3 is a geometrical factor of the order of 1020 m2 (see Section 1.13.5). The sum in the dominant eqn [30] is a simple geometrical progression and therefore it is equal to 1 X Di Ci x1 x¼1
Dv Cv
¼
1 Dv Cv 1 Di Ci =Dv Cv Dv Cv Di Ci
½31
Substituting eqn [31] to eqn [30], one can finally obtain the following equation Jcnucl ¼ wCv ðDv Cv Di Ci Þ
½32
Note that the function g(x) in this case takes a very simple form, g(x) ¼ Cc/x1/3, and hence decreases with increasing cluster size. In contrast, in R-space,
367
g(R) (see eqn [16]) increases with increasing cluster size: gðRÞ ¼ ð36p=OÞ1=3 Cc R (see also eqns [43] and [44] in Feder et al.69). The real time-dependent SDF builds up around the function g(x) with the steadily increasing size range (see, e.g., Figure 2 in Feder et al.69). Also note that homogeneous nucleation is the only case where an analytic equation for the nucleation rate exists. In more realistic scenarios, the nucleation is affected by the presence of impurities and other crystal imperfections, and numerical calculations are the only means of investigation. Such calculations are not trivial because for practical purposes it is necessary to consider clusters containing very large numbers of defects and, hence, a large number of equations. This can make the direct numerical solution of ME impractical. As a result several methods have been developed to obtain an approximate numerical solution of ME (see Section 1.13.4.4 for details). The equations formulated in this section govern the evolution of mobile and immobile defects in solids under irradiation or aging and provide a framework, which has been used for about 50 years. Application of this framework to the models developed to date is presented in Sections 1.13.5 and 1.13.6. 1.13.4.4 Methods of Solving the Master Equation The ME [18] is a continuity equation (with the source term) for the SDF of defect clusters in a discreet space of their size. This equation provides the most accurate description of cluster evolution in the framework of the mean-field approach describing all possible stages, that is, nucleation, growth, and coarsening of the clusters due to reactions with mobile defects (or solutes) and thermal emission of these same species. The ME is a set of coupled differential equations describing evolution of the clusters of each particular size. It can be used in several ways. For short times, that is, a small number of cluster sizes, the set of equations can be solved numerically.74 For longer times the relevant physical processes require accounting for clusters containing a very large number of PDs or atoms (106 in the case of one-component clusters like voids or dislocation loops and 1012 in the case of two-component particles like gas bubbles). Numerical integration of such a system is feasible on modern computers, but such calculations are overly time consuming. Two types of procedures have been developed to deal with this situation: grouping techniques (see, e.g., Feder et al.,69
368
Radiation Damage Theory
Wagner and Kampmann,70 and Kiritani75) and differential equation approximations in continuous space of sizes (see, e.g., Goodrich67,68, Bondarenko and Konobeev,76 Ghoniem and Sharafat,77 Stoller and Odette,78 Hardouin Duparc et al.,79 Wehner and Wolfer,80 Ghoniem,81 and Surh et al.82). The correspondence between discrete microscopic equations and their continuous limits has been the subject of an enormous amount of theoretical work. The equations of thermodynamics, hydrodynamics, and transport equations, such as the diffusion equation, are all examples of statistically averaged or continuous limits of discrete equations for a large number of particles. The extent to which the two descriptions give equivalent mathematical and physical results has been considered by Clement and Wood.83 In the following two sections, we briefly discuss these methods. 1.13.4.4.1 Fokker–Plank equation
In the case where the rates Pðx; t Þ; Q ðx; t Þ are sufficiently smooth, it is reasonable to approximate them by continuous functions P~ðx; t Þ; Q~ ðx; t Þ and to replace the right-hand sides of eqns [18] and [19] by continuous functions of two variables, J ðx; t Þ and f ðx; t Þ. The Fokker–Plank equation can be obtained from the ME by expanding the right-hand side of eqn [18] in Tailor series, omitting derivatives higher than the second order @f S ðx; t Þ @ ¼ G s ðxÞ ½V ðx; t Þf ðx; t Þ @t @x þ
@2 ½Dðx; t Þf ðx; t Þ @x 2
½33
where V ðx; t Þ ¼ P~ðx; t Þ Q~ ðx; t Þ 1 DðxÞ ¼ P~ðx; t Þ þ Q~ ðx; t Þ ½34 2 The first term in eqn [33] describes the hydrodynamiclike flow of clusters, whereas the second term accounts for their diffusion in the size-space. Note that for clusters of large enough sizes, when the cluster evolution is mainly driven by the hydrodynamic term, the functions P~ðx; t Þ; Q~ ðx; t Þ are smooth; hence the ME and F–P equations are equally accurate. For sufficiently small cluster sizes, when the diffusion term plays a leading role, eqn [33]) provides only poor description.67,68,83 As the cluster nucleation normally takes place at the beginning of irradiation, that is, when the clusters are small, the results obtained using F–P equation are expected to be less accurate compared to that of ME.
1.13.4.4.2 Mean-size approximation
In eqn [24], the term with V ðx; t Þ is responsible for an increase of the mean cluster size, while the term with DðxÞ is responsible for cluster nucleation and broadening of the SDF. For large mean cluster size, most of the clusters are stable and the diffusion term is negligible. This is the case when the nucleation stage is over, and the cluster density does not change significantly with time. A reasonably accurate description of the cluster evolution is then given in the mean-size approximation, when fc ðx; t Þ ¼ Nc dðx hxðt ÞiÞ where dðxÞ is the Kronecker delta and Nc is the cluster density. The rate of change of the mean size in this case can be calculated by omitting the last term in the right-hand side of eqn [24], multiplying both sides by x, integrating over x from 0 to infinity, and taking into account that f ðx ¼ 1; t Þ ¼ 0 and f ðx ¼ 0; t Þ ¼ 0 dhxi ¼ V ðhxi; t Þ dt
½35
1.13.4.4.3 Numerical integration of the kinetics equations
The main idea of the grouping methods for numerical evaluation of the ME is to replace a group of equations described by the ME with an ‘averaged’ equation. Such a procedure was proposed by Kiritani75 for describing the evolution of vacancy loops during aging of quenched metals. Koiwa84 was the first to examine the Kiritani method by comparing numerical results with the results of an analytical solution for a simple problem. Serious disagreement was found between the numerical and analytical results, raising strong doubts regarding the applicability of the method. The main objection to the method75 in Koiwa84 is the assumption used by Kiritani75 that the SDF within a group does not depend on the size of clusters. However, Koiwa did not provide an explanation of where the inaccuracy comes from. The Validity of the Kiritani method was examined thoroughly by Golubov et al.85 The general conclusion of the analysis is that the grouping method proposed by Kiritani is not accurate. The origin of the error is the approximation that the SDF within a group is constant as was predicted by Koiwa.84 Thus, the disagreement found in Koiwa84 is fundamental and cannot be circumvented. Because it is important for understanding the accuracy of the other methods suggested for numerical calculations of cluster evolution,
369
Radiation Damage Theory
@N ¼ J ð1; t Þ @t 1 X @S J ðx; t Þ ¼ J ð1; t Þ þ @t x¼1
½36 ½37
where the generation term in eqn [18] is dropped for simplicity. Equations [36] and [37] are the conservation laws which can be satisfied when one uses a numerical evaluation of the ME. When a group method is used, the conservation laws can be satisfied for reactions taking place within each group.69 However, this is not possible within the approximation used by Kiritani75 because a single constant can be used to satisfy only one of the eqns [36] and [37]. To resolve the issue, Kiritani75 used an ad hoc modification of the flux J ðxi Þ; therefore, the final set of equations for the density of clusters within a group, Fi , are as follows dFi 1 ¼ ½Ji1 Ji dt Dxi
Ji ¼
2Dxi 2Dxiþ1 Pi Fi Qiþ1 Fiþ1 Dxi þ Dxiþ1 Dxi þ Dxiþ1
0.5 Steady-state SDF K-method 0.4 Void density (1021 m–3 nm–1)
the analysis performed in Golubov et al.85 is briefly highlighted below. It follows from P eqn [18] that the total number of clusters, N ðt Þ ¼ 1 x¼2 f ðx; t Þ and P total number of defects in the clusters, Sðt Þ ¼ 1 x¼2 xf ðx; t Þ, are described by the following equations:
New method
0.3
0.2
0.1dpa 0.05 dpa
0.1 0.01 dpa 0.0 0
1
2
4 3 Void diameter (nm)
5
6
7
Figure 3 Size distribution function of voids calculated in copper irradiated at 523 K with the damage rate of 107 dpa s1 for doses of 102–101 dpa. The dashed and solid lines correspond to the Kiritani method and the new grouping method, respectively. The thick line corresponds to the steady-state function, gðxÞ. Reproduced from Golubov, S. I.; Ovcharenko, A. M.; Barashev, A. V.; Singh, B. N. Philos. Mag. A 2001, 81, 643–658.
½38
½39
where Dxi is the width of the ‘i ’ group. Equations [38] and [39] indeed satisfy both the conservation laws. However, they do not provide a correct description of cluster evolution described by the ME because the flux Ji in eqn [39] depends on the widths of groups and these widths have no physical meaning. An example of a comparison of the calculation results obtained using the Kiritani method with the analytical and numerical calculations based on a more precise grouping method is presented in Figure 3. Note that in the limiting case where the widths of group are equal, Dxi ¼ Dxi þ 1 , the flux Ji is equal to the original one, J ðx; t Þ. In this limiting case, eqns [38] and [39] correspond to those that can be obtained by a summation of the ME within a group and therefore they provide conservation of the total number of clusters, N ðt Þ, only. This limiting case is probably the simplest way to demonstrate the inaccuracy of the Kiritani method.
It is worth noting that this comparison also sheds light on the relative accuracy of other numerical solutions of the F–P equation such as in Bondarenko and Konobeev,76 Ghoniem and Sharafat,77 Stoller and Odette,78 and Hardouin Duparc et al.79 Equations [36] and [37] provide a way of getting a simple but still reasonably correct grouping method for numerical integration of the ME. Indeed, the two conservation laws, eqns [36] and [37], require two parameters within a group at least. The simplest approximation of the SDF within a group of clusters (sizes from xi1 to xi ¼ xi1 þ Dxi 1) can be achieved using a linear function fi ðxÞ ¼ Li0 ðx hxii Þ þ Li1
½40
where hxii ¼ xi 1=2ðDxi 1Þ is the mean size of the group. Equations for Li0 ; Li1 are as follows69 dLi0 1 ¼ ½J ðxi1 Þ Jx ðxi Þ dt Dxi dLi1 ¼ dt Dxi 1 2s2i Dxi
!
½41
1 Jx ðxi1 Þ þ J ðxi Þ 2J hxi i ½42 2
370
Radiation Damage Theory
where s2i ¼
1 Dxi
"
xi X k¼xi1 þ1
k2
1 Dxi
xi X
!2 # k
½43
k¼xi1 þ1
is the dispersion of the group. Equations [41] and [42] describe the evolution of the SDF within the group approximation. Note that the last term in the brackets on the right-hand side of eqn [42] follows from the corresponding term in eqn [38] in Golubov et al.85 when the rates Pðx; t Þ; Q ðx; t Þ are independent from x within the group. Note also that the factor ‘1=Dxi ’ is missing in eqn [38] in Golubov et al.85 As can be seen from eqns [41] and [42], in the case where Dxi ¼ 1, eqns [41] and [42] transform to eqn [18], that is f ðxi Þ ¼ Li0 and Li1 ¼ 0 in contrast with Kiritani’s method, where the equation describing the interface number density of clusters between ungrouped and grouped ones has a special form (see, e.g., eqn [21] in Koiwa84). It has to be emphasized that this grouping method is the only one that has demonstrated high accuracy in reproducing well-known analytical results such as those by Lifshitz–Slezov–Wagner86,87 (LSW) and Greenwood and Speight88 describing the asymptotic behavior of SDF in the case of secondary phase particle evolution89 and gas bubble evolution90 during aging. A different approach for calculating the evolution of the defect cluster SDF is based on the use of the F–P equation. Note that the use of eqn [33] as an approximate method for treating cluster evolution is not new, for the work initiated by Becker and Do¨ring64 has been brought into its modern form by Frenkel.66 An advantage of the F–P equation over the ME is based on the possibility of using the differential equation methods developed for the case of continuous space. Quite comprehensive applications of the analytical methods to solve the F–P have been done by Clement and Wood.83 It has been shown83 that convenient analytical solutions of the F–P equation cannot be obtained for the interesting practical cases. Thus, several methods have been suggested for an approximate numerical solution for it. The simplest method is based on discretization of the F–P equation76–79 that transforms it to a set of equations for the clusters of specific sizes similar to the ME; in both the cases the matrix of coefficients of the equation set is trigonal. This method is convenient for numerical calculations and allows calculating cluster evolution up to very large cluster sizes (e.g., Ghoniem81). However, this method is not accurate because it is identical to the approach used by Kiritani75 in
which SDF was approximated by a constant within a group. Thus, all the objections to Kiritani’s method discussed above are valid for this method as well. Also note that the method has a logic problem. Indeed a chain of mathematical transformations, namely ME to F–P and F–P to discretized F–P, results in a set of equations of the same type, which can be obtained by simple summation of ME within a group. Moreover, the last equation is more accurate compared to the discretized F–P because it is a reduced form of the ME. Another approach for numerical integration of the F–P equation was suggested by Wehner and Wolfer (see Wehner and Wolfer80). The method allows calculating cluster evolution on the basis of a numerical path-integral solution of the F–P equation which provides an exact solution in the limiting case where the time step of integration approaches zero. For a finite time step, the method provides an approximate solution with an accuracy that has not been verified. Moreover, there was an error in the calculation presented in Wehner and Wolfer80,91 and so the accuracy of the method remains unclear. A modification of this method according to which the evolution of large clusters is calculated by employing a Langevin Monte Carlo scheme instead of the path integral was suggested by Surh et al.82 The accuracy of this method has not been verified as an error was also made in obtaining the results presented in Surh et al.82,91 The momentum method for the solution of the F–P equation used by Ghoniem81 (see also Clement and Wood83) is quite complicated and may provide only an approximate solution. So far, none of the methods suggested for numerical evaluation of the F–P equation has been developed and verified to a sufficient degree to allow effective and accurate calculations of defect cluster evolution during irradiation in the practical range of doses and temperatures.
1.13.5 Early Radiation Damage Theory Model The chemical reaction RT was used very early to model the damage accumulation under irradiation (Brailsford and Bullough92 and Wiedersich93). The main assumptions were as follows: (1) the incident irradiation produces isolated FPs, that is, single SIAs and vacancies in equal numbers, (2) both SIAs and vacancies migrate 3D, and (3) the efficiencies of the SIAs and vacancy absorption by different sinks are different because of the differences in the strength of
Radiation Damage Theory
the corresponding PD-sink elastic interactions. Thus, the preferential absorption of SIAs by dislocations (i.e., the dislocation bias) is the only driving force for microstructural evolution in this model, which is a variant of the FP3DM. It should be emphasized that, in the framework of the FP3DM, no distinction is made between different types of irradiation: 1 MeV electrons, fission neutrons, and heavy-ions. It was believed that the initial damage is produced in the form of FPs in all these cases. Now we understand the mechanisms operating under different conditions much better and make clear distinction between electron and neutron/heavy-ion irradiations (see Singh et al.,1,22 Garner et al.,33 Barashev and Golubov,35 and references therein for some recent advances in the development of the so-called PBM). However, the FP3DM is the simplest model for damage production and it correctly describes 1 MeV electron irradiation. It is therefore useful to consider it first. The more comprehensive PBM includes the FP3DM as its limiting case. 1.13.5.1 Reaction Kinetics of Three-Dimensionally Migrating Defects
ka2 ¼
2 kaj
following section, we present examples of such a treatment based on the so-called lossy-medium approximation.61 1.13.5.1.1 Sink strength of voids
Consider 3D diffusion of mobile defects near a spherical cavity of radius R, which is embedded in a lossy-medium of the sink strength k2 : G k2 DðC C eq Þ rJ ¼ 0
½46
where C eq is the thermal-equilibrium concentration of mobile defects and the defect flux is C rU ½47 J ¼ D rC þ kB T Here, D is the diffusion coefficient, U is the interaction energy of the defect with the void, kB is the Boltzmann constant, and T the absolute temperature. The boundary conditions for the defect concentration, C, at the void surface and at infinity are CðRÞ ¼ C eq
½48
G ½49 k2 D Equation [49] follows from eqn [46] and the requirement that the gradients vanish at large distances. Here, all other sinks in the system, voids, dislocations, etc. are considered in the MFA and contribute to the total sink strength k2 . This procedure is selfconsistent. The interaction energy of a defect with the void in eqn [47] is small and usually neglected. The solution of eqn [46] for a void located at the origin of the coordinate system, r = 0, is then C 1 ¼ C eq þ
In the case considered, eqns [10]–[12] for mobile defects are reduced to the following form dCv ¼ G NRT þ Gvth kv2 Dv Cv mR Di Ci Cv dt dCi ¼ G NRT ki2 Di Ci mR Di Ci Cv ½44 dt In order to predict the evolution of mobile PDs and their impact on immobile defects, one needs to know the sink strength of different defects for vacancies and SIAs and the rate of their mutual recombination. The reaction kinetics of 3D migrating defects is considered to be of the second order because the rate equations contain terms with defect concentrations to the second power.40 An important property of such kinetics is that the leading term in the sink strength of any individual defect depends on the characteristics of this defect only. Thus, N X
371
½45
j ¼1
where a ¼ v; i and N is the total number of sinks per unit volume. For example, the total sink strength of an ensemble of voids of the same radius, R, is equal to ka2 ¼ Nka2 ðRÞ. The individual sink strength such as a void or a dislocation loop may be obtained from a solution to the PD diffusion equation. In the
R Cðr Þ ¼ C eq þ ðC 1 C eq Þ 1 exp½k ðr RÞ ½50 r
The total defect flux, I , through the void surface S ¼ 4pR2 is given by I ¼ SJ ðRÞ ¼ kC2 ðRÞDðC 1 C eq Þ
½51
where the void sink strength is kC2 ðRÞ ¼ 4pRð1 þ kRÞ
½52
The sink strength of all voids in the system is obtained by integrating over the SDF, f ðRÞ: ð hR2 i ½53 kC2 ¼ dRkC2 ðRÞf ðRÞ ¼ 4phRiNC 1 þ k hRi Ð where NC ¼ dRf ðRÞ is the void number density, hRi is the void mean radius and hR2 i is the mean radius squared. Typically, k2 1014 m2 , that is,
372
Radiation Damage Theory
k1 100 nm, while the void radii are much smaller, so that one can omit the term proportional to the radius squared: kC2 ¼ 4phRiNC
½54
Equation [52] is derived by neglecting the interaction between the void and mobile defect. There is a difference between the interaction of SIAs and vacancies with voids due to differences in the corresponding dilatation volumes. As a result, the void capture radius for an SIA is slightly larger than that for a vacancy (see, e.g., Golubov and Minashin94). However, this difference is usually negligible compared to that for an edge dislocation, which is described below. 1.13.5.1.2 Sink strength of dislocations
An equation for the dislocation sink strength can be derived the same way as for voids. In this case, eqn [46] is solved in a cylindrical coordinate system and the interaction between PDs and dislocation is significant and not omitted. For an elastically isotropic crystal and PDs in the form of spherical inclusions, the interaction energy has the form95 U ðr ; yÞ ¼
A sin y r
½55
where A¼
mb 1 þ n DO 3p 1 n
½56
m is the shear modulus, n the Poisson ratio and DO the dilatation volume of the PD under consideration. The solution of eqn [35] in this case was obtained by Ham95 but is not reproduced here because of its complexity. It has been shown that a reasonably accurate approximation is obtained by treating the dislocation as an absorbing cylinder with radius Rd ¼ Ae g =4kB T , where g ¼ 0:5772 is Euler’s constant.95 The solution is then given by
G K0 ðkr Þ ½57 Cðr Þ ¼ 2 1 Dk K0 ðkRd Þ where K0 ðxÞ is the modified Bessel function of zero order. Using eqns [47] and [57], one obtains the total flux of PDs to a dislocation and the dislocation sink strength as I ¼ 2pRd rd DJ ðRd Þ ¼ kd2 DðC 1 C eq Þ ¼ rd Z 2p Zd ¼ lnð1=kRd Þ kd2
½58
d
½59
where rd is the dislocation density and Zd the capture efficiency. The capture efficiencies for vacancies and SIAs, Zvd and Zid , are different because of the difference in their dilatation volumes (see eqn [56]) 2p lnð1=kRad Þ
½60
mb 1 þ n e g DOa 3p 1 n 4kB T
½61
Zad ¼ where a ¼ v; i and Rad ¼
The dilatation volume of SIAs is larger than that of vacancies, hence RiD > RvD and the absorption rate of dislocations is higher for SIAs: Zid > Zvd . This is the reason for void swelling, which is shown below in Section 1.13.5.2.1. A more detailed analysis of the sink strengths of dislocations and voids for 3D diffusing PDs can be found in a recent paper by Wolfer.96 1.13.5.1.3 Sink strengths of other defects
The sink strengths of other defects can be obtained in a similar way. For dislocation loops of a toroidal shape97 2 kLðv;iÞ ¼ 2pRL ZLv;i 2p ZLv;i ¼ v;i lnð8RL =rcore Þ
½62
v;i are the loop radius and the effecwhere RL and rcore tive core radii for absorption of vacancies and SIAs, respectively. Similar to dislocations, the capture efficiency for SIAs is larger than that of vacancies, ZLi > ZLv , for loops. For a spherical GB of radius RG (see, e.g., Singh et al.98) 2 ¼ kGB
1 3x2 ðxcothx 1Þ R2G x2 3ðxcothx 1Þ
½63
where x ¼ kRG . In the limiting case of x 1, that is, when the GB is the main sink in the system, 2 ¼ kGB
15 R2G
½64
For the surfaces of a thin foil of thickness L (see eqn [7] in Golubov99) 2 ¼ kfoil
k2 kL=2cothðkL=2Þ 1
½65
In the limiting case of kL 1, that is, when the foil surfaces are the main sinks, 2 ¼ kfoil
12 L2
½66
Radiation Damage Theory
1.13.5.1.4 Recombination constant
Equation [35] can be used to obtain the rate of recombination reactions between vacancies and SIAs. In a coordinate system where the vacancy is immobile, the SIAs migrate with the diffusion coefficient Di þ Dv and, hence, the total recombination rate is R ¼ 4preff ðDi þ Dv ÞCi nv mR Di Ci Cv
½67
where nv ¼ Cv =O and the fact that Di Dv at any temperature is used. In this equation, reff is the effective capture radius of a vacancy, defining an effective volume where recombination occurs spontaneously (athermally). The recombination constant, mR , in eqn [67] is, hence, equal to 4preff ½68 mR O MD calculations show that a region around a vacancy, where such a spontaneous recombination takes place, consists of 100 lattice sites.100,101 From 3 =3 ¼ 100O, one finds that reff is approximately 4preff two lattice parameters, hence mR 1021 m2 . Dissociation of vacancies from voids and other defects is an important process, which significantly affects their evolution under irradiation and during aging. Similar to the absorption rate eqn [54], it has been shown that the dissociation rate is proportional to the void radius. Such a result can readily be obtained by using the so-called detailed balance condition. However, as the evaporation takes place from the void surface, the frequency of emission events is proportional to the radius squared. In the following lines, we clarify why the dissociation rate is proportional to the void radius and elucidate how diffusion operates in this case. Consider a void of radius R, which emits ndiss ¼ t1 diss vacancies per second per surface site in a spherical coordinate system. Vacancies migrate 3D with the diffusion coefficient Dv ¼ a2 =6t, where a is the vacancy jump distance and t is the mean time delay before a jump. The diffusion equation for the vacancy concentration Cv is r2 Cv ¼ 0
½69
To calculate the number of vacancies emitted from the void and reach some distance R1 from the void surface, we use absorbing boundary conditions at this distance Cv ðR1 Þ ¼ 0
An additional boundary condition must specify the vacancy–void interaction. Assuming that vacancies are absorbed by the void, which is a realistic scenario, the vacancy concentration at one jump distance a from the surface can be written as Cv ðR þ aÞ Cv ðR þ 2aÞ ¼ ndiss þ ½71 t 2t The left-hand side of the equation describes the frequency with which vacancies leave the site. The first term on the right-hand side accounts for the production of vacancies due to evaporation from the void. The last term on the right-hand side accounts for vacancies coming to this site from sites further way from the void surface. After representing the latter term using a Taylor series, in the limit of R a, the boundary condition, eqn [71], assumes the following form Cv ðRÞ ¼ 2tndiss þ arCv ðRÞ
½70
½72
Using this condition and eqns [69] and [70], one finds the vacancy concentration, Cv ðr Þ, is equal to Cv ðr Þ ¼ 2tndiss
1.13.5.1.5 Dissociation rate
373
r 1 ðR1 Þ1 R1 ðR1 Þ1
½73
It can readily be estimated using the last two equations that the gradient of concentration in eqn [72] is smaller than the other terms by a factor of a=r0 and does not contribute to eqn [73]. This means that most vacancies emitted from the void return to it. As a result, the equilibrium condition for the concentration near the void surface is defined by the equality of the frequency of evaporation and the frequency of jumps back to the surface and is not affected by the flux of vacancies away from the surface. The vacancy equilibrium concentration at the void surface is readily obtained from eqn [73] as Cveq ðRÞ ¼ Cv ðRÞ ¼ 2tndiss . The total number of vacancies passing through a spherical surface of radius R and area S ¼ 4pR2 per unit time, that is, the rate of vacancy emission from the void, is equal to SDv rCv ðr Þjr ¼R O Dv Cveq Dv Cveq 4pR 4pR ¼ 1 O 1 R=R O
Jvem ¼
½74
There are three points to be made. First, eqn [73] becomes independent of the distance r from the surface, when r R. Thus, vacancies reaching this distance are effectively independent of their origin and can be counted as dissociated from the void. Second, despite the fact that the total vacancy
374
Radiation Damage Theory
emission frequency is proportional to the void surface area, the total vacancy flux far away from the surface is proportional to the void radius. This is a well-known result of the reaction–diffusion theory40 considering the void capture efficiency. Third, as can be seen from eqn [74], significant deviation from the proportionality to the void radius occurs at distances of the order of the void radius. As discussed above, most emitted vacancies return to the void. The fraction of vacancies which do not return is equal to the ratio of the frequency defined by eqn [63] and the total frequency of vacancy emission 4pR2 ndiss =a 2 . It is thus equal to a=R. The same result can be demonstrated considering another, although unrealistic, scenario in which vacancies are reflected by the voids.102 We also note that the first nonvanishing correction to the proportionality of the vacancy flux to the void radius is positive and proportional to the void radius squared, see eqn [74], where Rð1 R=R1 Þ1 R þ R2 =R1 . The same result was obtained previously by Go¨sele40 when considering void capture efficiency. Thus, with increasing volume fraction more and more vacancies become absorbed at other voids and the proportionality to the void radius squared would be restored. The first correction term just shows the right tendency. 1.13.5.1.6 Void growth rate
The concentration of vacancies in equilibrium with a void of radius R, Cveq ðRÞ, which enters eqn [74], can be obtained by considering the free energy of a crystal with a void and a solution of vacancies. Let x be the number of vacancies taken from a solution of vacancies to make a spherical void of a radius R ¼ ð3xO=4pÞ1=3 . The associated free energy change is given by 4pR3 mv þ 4pg~R2 ½75 DF ¼ 3O where mv ¼ kB T ln Cv =Cvth is the chemical potential of a vacancy (Cvth is the equilibrium concentration in a perfect crystal) and g~is the void surface energy. By differentiating this equation with respect to radius and equating the result to zero, one can find the equilibrium vacancy concentration, which is given by 2Og~ ½76 Cveq ðRÞ ¼ Cvth exp RkB T Absorption and emission of PDs change a void volume on the basis of the flux of PDs dDV =dt ¼ 4pR2 ðdR=dt Þ ¼ ðJv Ji Jvem Þ. With the aid of eqns [51], [52], [74] and keeping the leading term
proportional to R only and [76], the growth rate of a void due to absorption of vacancies and SIAs and vacancy emission can be written as
dR 1 2Og~ th ½77 ¼ Dv Cv Di Ci Dv Cv exp dt R RkB T Neglecting the entropy factor for simplicity, one can find that Cvth ¼ expðEvf =kB T Þ, where Evf is the vacancy formation energy. The last term in the square brackets on the right-hand side of eqn [66] can be then represented in the following form 2Og~ Eb th Dv exp ½78 Dv Cv exp RkB T RkB T where 2Og~ ½79 R is a well-known equation for the binding energy of a vacancy with a void that is valid for large enough radius. For voids of small sizes, the value Eb has to be calculated by using ab initio or MD methods. Equation [77] is used in calculations of void swelling. Note that the vacancy and SIA fluxes, the first and second terms, enter this equation symmetrically and this is because of the neglect of the difference in the interactions of SIAs and vacancies with voids. Also, when the sum of the second and third terms in the right-hand side of this equation is larger than the first term, the voids shrink. Such a shrinking takes place during annealing of preirradiated samples or, in some cases, during irradiation, if the irradiation conditions are changed. However, in the majority of cases, voids grow under irradiation because dislocations interact more strongly with SIAs than vacancies. Eb ¼ Evf
1.13.5.1.7 Dislocation loop growth rate
The concentration of vacancies, ðCveq Þvl;il , in equilibrium with the dislocation loop of radius R of vacancy (subscript ‘vl’) and SIA (subscript ‘il’) type can be obtained in the same way as in the previous subsection (e.g., Bullough et al.29) ðg þ Eel Þb 2 ½80 ðCveq Þvl;il ¼ Cvth exp sf kB T where gsf ; Eel ; and b are the stacking-fault energy, the interaction energy of PDs with dislocation and the dislocation Burgers vector, respectively. The ‘þ’ and ‘’ in the exponent correspond to the cases of vacancy and SIA loops, respectively. In the case
Radiation Damage Theory
when both PDs are considered as spherical dilation centers, the interaction energy Eel is given by mb 2 Rþb ln Eel ¼ ½81 4pð1 nÞðR þ bÞ b where m and n are the shear modulus and Poisson ratio, respectively. Hence, the growth rates of vacancy and SIA loops are
dRvl 1 v ¼ ZL Dv Cv ZLi Di Ci dt b ðgsf þ Eel Þb 2 v th ZL Dv Cv exp kB T
dRil 1 i ¼ ZL Di Ci ZLv Dv Cv dt b ðg þ Eel Þb 2 ½82 þZLv Dv Cvth exp sf kB T 1.13.5.1.8 The rates P(x) and Q(x)
Equations [54] and [62] for the sink strengths of voids and dislocation loops for mobile PDs permit the calculation of rates P(x) and Q(x), which determine the cluster evolution described by the ME (see Section 1.13.4.3.2). For example, the total rate of absorption of vacancies by voids is equal to kc2 Dv Cv (see eqns [10] and [45]). The same quantity is given 1 P Pc ðxÞfc ðxÞ. By equating these two rates one by x¼2
where
Dv Cv kc2 ¼
1 X
Pc ðxÞfc ðxÞ
½83
x¼2
Taking into account eqns [14] and [54], the following expression for the rate Pc ðxÞ can readily be obtained Pc ðxÞ ¼ wc x 1=3 Dv Cv
½84
where
2 1=3 48p ½85 wc ¼ O2 The rate Qc(x), which consists of two terms, the SIA absorption and vacancy emission rates, can be obtained the same way
b Ev ðxÞ ½86 Qc ðxÞ ¼ wc x 1=3 Di Ci þ Dv exp kB T For dislocation loops of SIA type, the rates Pil(x) and Qil(x) take the following form
b Eil ðxÞ Pil ðxÞ ¼ wl x 1=2 ZLi Di Ci þ ZLv Dv exp kB T Qil ðxÞ ¼ wl x 1=2 ZLv Dv Cv
½87
4p Ob
1=2
Eilb ðxÞ ¼ Evf þ ðgsf þ Eel ðxÞÞb 2
½88
For vacancy loops, the rates Pvl(x) and Qvl(x) are given by Pvl ðxÞ ¼ wl x 1=2 ZLi Dv Cv
b Evl ðxÞ Qvl ðxÞ ¼ wl x 1=2 ZLi Di Ci þ ZLv Dv exp ½89 kB T
where Eilb ðxÞ ¼ Evf ½gsf þ Eel ðRÞb 2
½90
The equations given above have been obtained by neglecting mutual recombination between vacancies and SIAs. Accounting for recombination makes the diffusion equations for the concentrations of PDs nonlinear, an approximate solution for which has been obtained using a linearization procedure.103 The correction is, however, insignificant for conditions of practical importance. 1.13.5.2
obtains
wl ¼
375
Damage Accumulation
Damage accumulation in pure metals during irradiation primarily takes place in the formation and evolution of vacancy and SIA-type defects. At temperatures higher than recovery stage III, which is the main interest for practical purposes, vacancy clusters normally take the form of voids that result in the change of a volume, that is, swelling. Owing to limitations of space, in the following section we focus only on a description of void evolution. 1.13.5.2.1 Void swelling
The solution obtained from eqns [44] depends on the irradiation temperature. Temperatures below recovery stage II will not be considered here. At temperatures smaller than that corresponding to the recovery stage III, when vacancies are immobile and the interstitials are mobile, the concentration of vacancies will build up. At some irradiation dose, the vacancy concentration will become high enough that mutual recombination of PDs may become the dominant mechanism of the defect loss, thus controlling defect accumulation. In this case,
376
Radiation Damage Theory
the dose dependence of PD concentrations can be calculated analytically104
G NRT Di Ci ðt Þ ¼ 2mR
2 31=2 1=2 ðt 2 4 k ðtÞdt5 0
t NRT 1=2 ð
G Dv Cv ðt Þ ¼ 2mR
2t 31=2 ð 2 4 k ðtÞ k ðt1 Þdt1 5 dt 2
0
½91
0
Because the sink strength, k2 ðtÞ, changes very slowly (the vacancy-type defects shrink and SIA-type defects grow because of the SIA absorption), it follows from eqn [91] that Di Ci ðt Þ / ðG NRT t Þ1=2 Dv Cv ðt Þ / ðG NRT t Þ1=2
Cv
4 ⫻ 107
3 ⫻ 107 Nv 2 ⫻ 107 C2v
10–11 Ci 10–14 10–10
10–8
10–6 10–4 10–2 Irradiation dose (dpa)
100
Vacancy supersaturation
5 ⫻ 107 300 ⬚C 10–6 dpa s–1 rd = 1014 m–2
10–8
G kc2 Di Ci Zid rd Di Ci ¼ 0
½93
The defect concentrations, Cv and Ci , are then G Dv ðkc2 þ Zvd rd Þ G Ci ¼ 2 Di ðkc þ Zvd rd Þ
Cv ¼
½94
Hence, taking into account that Zvd Zid , Dv Cv Di Ci ¼
G kc2 þ Zvd rd
½95
The swelling rate is equal to the net (excess) flux of vacancies to voids: dS ¼ kc2 ½Dv Cv Di Ci df ¼ Bd
kc2 Zvd rd kc2 Zvd rd B d ðkc2 þ Zvd rd Þðkc2 þ Zid rd Þ ðkc2 þ Zvd rd Þ2
½96
where S ¼ ð4p=3ÞNc hrc i3 and f ¼ Gt are the total volume of voids and the irradiation dose in dpa, respectively; and Bd is the dislocation bias factor
Fe
Sw ellin g
PD concentration/swelling (atom–1)
10–5
G kc2 Dv Cv Zvd rd Dv Cv ¼ 0
½92
At temperatures higher than that corresponding to recovery stage III, both vacancies and SIAs are mobile. Hence, after a certain time of irradiation, called the ‘transient period’, their concentrations reach a steady state. A comprehensive analysis of the time (irradiation dose) dependence of PD concentrations for different sink strength can be found in Sizmann.9 The dose dependence of PD concentrations and void swelling obtained by the numerical integration of ME73 is presented in Figure 4. As can be seen, the vacancy supersaturation, ðDv Cv Di Ci Þ=Dv Cveq , becomes positive when the PD concentrations reach steady state and this gives rise to void growth. Also, note that in the transient regime only divacancies are
10–2
formed. In the following discussion we concentrate on the irradiation doses beyond the transient period, which are of more practical interest. If only voids and edge dislocations are present in the system, and mutual recombination and thermal emission of vacancies from voids and dislocations are both negligible, the balance equations for the concentrations of vacancies and SIAs, Cv and Ci , are given by
1 ⫻ 107
0
Figure 4 Dose dependences of the concentrations of point defects, void swelling, vacancy supersaturation, and void number density calculated in the framework of FP3DM by numerical integration of the master equation, eqn [18]. From Golubov and Ovcharenko.73
Bd ¼
Zid Zvd Zvd
½97
The maximum value of the ratio in the right-hand side of eqn [96] is 1/4, when the sink strengths of voids and dislocation are equal to each other, kc2 ¼ Zvd rd . Thus the maximum swelling rate is dS Bd ½98 ¼ 4 df max It is easy to show that the swelling rate described by eqn [96] depends only weakly on the variation of the sink strength of voids and dislocations: a difference of an order of magnitude results in a decrease of the swelling rate by a factor of 3 only. To obtain the steady-state swelling rates of 1% per NRT dpa, which are observed in high-swelling
Radiation Damage Theory
fcc materials, one would need the bias factor to be about several percent. Data on swelling in electronirradiated metals resulted in Bd 2 4% for the fcc copper24,105,106 (data reported by Glowinski107 were used in Konobeev and Golubov106), 2% for pure Fe–Cr–Ni alloys,108 and orders of magnitude lower values for bcc metals (e.g., swelling data for molybdenum109). Because the electron irradiation produces FPs, it is reasonable to accept these values as estimates of the dislocation bias. Note that the first attempt to determine Bd by solving the diffusion equations with a drift term determined by the elasticity theory for PD–dislocation interaction as described in Section 1.13.5 showed that the bias is significantly larger than the empirical estimate above. Several works have been devoted to such calculations,96,110–113 which predicted much higher Bd values, for example, 15% for the bcc iron and 30% for the fcc copper. With these bias factors, the maximum swelling rates based on Bd =4 should be equal to about 4% and 8% per dpa but such values have never been observed. An attempt to resolve this discrepancy can be found in a recent publication.114 Surprisingly, the steady-state swelling rate of 1% per NRT dpa has been found in neutron- (and ion-) irradiated materials, for example, in various stainless steels, even though the primary damage in these cases is known to be very different and the void swelling should be described in the framework of the PBM, which gives a rather different description of the process. An explanation of this is proposed in Section 1.13.6. 1.13.5.2.2 Effect of recombination on swelling
Mutual annihilation of PDs happens either by direct interaction between single vacancies and SIAs in the matrix or within a certain type of neutral sink which we call ‘saturable.’ The fluxes of vacancies and SIAs to them are equal. An example of such sinks is vacancy loops, which were considered in the framework of the BEK model29 and PBM,22 that is, in the case where the vacancy clusters are generated in cascades. The BEK model is not discussed further in the present work because it does not correspond to any realistic situation in solids under irradiation; vacancy clustering in cascades is always accompanied with the SIA clustering, which is accounted for in the framework of the PBM but not in the BEK model. The balance equations in the case considered are as follows 2 D C k2 D C Zd r D C ¼ 0 G mR Di Ci Cv kN i i c v v v d v v 2 D C k2 D C Zd r D C ¼ 0 G mR Di Ci Cv kN i i c i i i d i i
½99
377
2 where kN is the strength of neutral sinks. Note that absorption rate of both vacancies and SIAs in 2 D i Ci , eqn [99] is described by the same quantity, kN which reflects neutrality of this sink with respect to vacancies and SIAs.115,116 The defect concentrations and swelling rate are
Dv Cv Di Ci ¼
G 1 1 kc2 þ Zvd rd 1 þ fR 1 þ fN
dS k2 Zd r 1 1 ¼ Bd c v d 2 2 d df 1 þ f 1 þ fN R kc þ Zv rd
½100
where 1 fR ¼ 2 fN ¼
"sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi # 4mR G 1þ 1 2 Þ2 D ðkc2 þ Zvd rd þ kN v
2 kN kc2 þ Zvd rd
½101
In the absence of an effect on the sink structure, mutual recombination reactions are important at low temperature, when the vacancy diffusion is slow, and for high defect production rates, when the vacancy concentration is sufficiently high to provide higher sink strength for SIAs than that of existing extended defects. This can be expressed mathematically by an inequality fR 1 or more explicitly as a temperature 2 2 boundary kB T < Evm =ln½2Dv0 ðkc2 þ Zvd rd þ kN Þ =mR G where Dv0 is the preexponential factor in the vacancy diffusion coefficient and Evm is the effective activation energy for the vacancy migration. In practice, this situation is unlikely to occur because the radiationinduced sink strength rapidly increases at low temperatures. In this case recombination at sinks is of greater importance. One of the important aspects that recombination reactions introduce to microstructural evolution is the appearance of a temperature dependence; at low temperatures, an increase of the swelling rate with increasing temperature is predicted, which is also observed experimentally in the fcc-type materials. The question of whether it was possible to explain the experimental reduction of swelling rate with decreasing temperature by recombination was addressed.29 It was found that the observed temperature effect on swelling rate was much stronger than predicted by recombination alone. The impact of neutral sinks on swelling rate is significant when they represent the dominant sink
378
Radiation Damage Theory
2 in the system: kN kc2 þ Zvd rd . The swelling rate in the case is given by
kc2 Zvd rd dS ¼ Bd 2 2Þ d ðkc þ Zv rd Þðkc2 þ Zvd rd þ kN df Bd
ðkc2
kc2 Zvd rd 2 þ Zvd rd ÞkN
Dveff ¼
½102
Such a situation may occur, for example, at low enough temperature, when the thermal stability of vacancy loops and SFTs becomes high enough, leading to their accumulation up to extremely high concentrations. Another possibility is when a high density (about 1024 m3) of second phase particles exists, as in the oxide dispersion strengthened (ODS) steels. 1.13.5.2.3 Effect of immobilization of vacancies by impurities
The diffusion coefficient of vacancies is an important parameter for microstructural evolution, for it determines the rate of mutual recombination of PDs. Migrating vacancies can also meet solute or impurity atoms and form immobile complexes, which can then dissociate. In quasi-equilibrium, when the rates of complex formation and dissociation events are equal to each other: znþ Cv0 Cs0 ¼ n Cvs
½103
Here, Cvs and Cs are the concentrations of complexes and solute atoms, respectively, Cs0 and Cv0 are the concentrations of free (unpaired) solute atoms and vacancies, respectively, nþ and n are the frequencies of complex formation and dissociation events, respectively, and z is a geometrical factor, which is of the order of the coordination number for complexes with a short-range (first-nearest neighbor) interaction and unity for long-range interacb , is tions. The binding energy of the complex, Evs þ b defined from n =n ¼ expðbEvs Þ. The solute concentration is generally much higher than that of vacancies, hence Cs0 Cs Cv0 ¼ Cv Cvs
½104
Substituting these into eqn [103], one obtains Cvs ¼
b aCv Cs expðbEvs Þ bÞ 1 þ aCs expðbEvs
The effective diffusion coefficient of vacancies may be defined as
½105
The total vacancy concentration is, therefore, b Cv ¼ Cv0 þ Cvs ¼ Cv0 1 þ aCs exp bEvs ½106
Dv bÞ 1 þ aCs expðbEvs Dv0 b exp bðEvm þ Evs Þ aCs
½107
While the vacancy concentration is approximately equal to b ½108 Cv Cv0 aCs exp bEvs The vacancy flux is, thus, equal to that in the absence of impurities, Dveff Cv ¼ Dv Cv0
½109
which is supported by the measurements of the self-diffusion energy, which is almost independent of the presence of impurities. The main conclusion is that the total vacancy flux does not depend on the presence of impurity atoms. However, impurity trapping may affect the recombination rate and hence Cv may be increased. 1.13.5.3 Inherent Problems of the Frenkel Pair, 3-D Diffusion Model Many observations contradict the FP3DM. These include the void lattice formation11–14 and higher swelling rates near GBs than in the grain interior in the following cases: high-purity copper and aluminum irradiated with fission neutrons or 600 MeV protons (see original references in reviews117,118); aluminum irradiated with 225 MeV electrons119 and neutron-irradiated nickel120 and stainless steel.121 Furthermore, the swelling rate at very low dislocation density in copper is higher,122–124 and the dependence of the swelling rate on the densities of voids and dislocations is different,125 than predicted by the FP3DM. It gradually became clear that something important was missing in the theory. There was evidence that this missing part could not be the effect of solute and impurity atoms or the crystal structure. Indeed, austenitic steels of significantly different compositions and swelling incubation periods exhibit similar steady-state swelling rates of 1% per NRT dpa.32,33 And, although generally the bcc materials show remarkable resistance to swelling,31,33 the alloy V–5% Fe showed the highest swelling rate of 2% per dpa: 90% at 30 dpa.34 As outlined in Section 1.13.3.1, the primary damage production under neutron and ion irradiations is more complicated; in addition to PDs, both vacancy
Radiation Damage Theory
and SIA clusters are produced in the displacement cascades. This is the reason the FP3DM predictions fail to explain microstructure evolution in solids under cascade damage conditions. In fact, it has been shown that it is the clustering of SIAs rather than vacancies that dominates the damage accumulation behavior under such conditions. The PBM proposed in the early 1990s and developed during the next 10 years (see Section 1.13.1) essentially resolved many of the issues; the phenomena mentioned have been properly understood and described. This model is described in the next section.
interval t1 < t < t2 is given by the integral over this interval. For particles undergoing random walk, this function is found to be equal to i 2 p2 D1D t ipx uðt ; x; xÞ ¼ 2p i exp sin 2 x x i¼1 1 X
1.13.6.1 Reaction Kinetics of One-Dimensionally Migrating Defects The 1D migration of the SIA clusters along their Burgers vector direction results in features that distinguish their reaction kinetics from 3D diffusing defects. These were first noticed in and theoretically analyzed for annealing experiments (Lomer and Cottrell,126 Frank et al.,127 Go¨sele and Frank,128 Go¨sele and Seeger,129 and Go¨sele40) and, then, under irradiation (Trinkaus et al.19,20 and Borodin130). In this section, we consider the reaction kinetics of 1D migrating clusters with immobile sinks and follow the procedure employed in Barashev et al.25 Detailed information about the diffusion process of a 1D migrating particle is given by the function uðt ; x; xÞ, which is known as Furth’s formula for first passages and has the following probabilistic significance.131 In a diffusion process starting at the point x > 0, the probability that the particle reaches the origin before reaching the point x > x in the time
½110
where D1D is the diffusion coefficient. Using this function, one can write the probability for a particle to survive until time t, that is, not to be absorbed by the barriers placed at the origin and at the point x, as ðt; x; xÞ ¼ ¼
1.13.6 Production Bias Model The continuous production of SIA clusters in displacement cascades is a key process, which makes microstructure evolution under cascade conditions qualitatively different from that during FP producing 1 MeV electron irradiation. In this case, eqns [10]–[12] should be used for the concentration of mobile defects. The equations for isolated PDs have been considered in detail in the previous section. In order to analyze damage accumulation under cascade irradiation, one needs to define the sink strengths of various defects for the SIA mobile clusters in eqn [12]. We give examples of such calculations for the case when cluster migrates 1D rather than 3D in the following section.
379
ð1 t
dt 0 uðt 0 ; x; xÞ þ uðt 0 ; x x; xÞ
1 exp ð2i 1Þ2 p2 D1D t =x 2 4X pxð2i 1Þ sin p i¼1 x 2i 1
½111
The expected duration of the particle motion until its absorption is given by: 1 ð
truin ðx; xÞ ¼
ðt ; x; xÞdt ¼ 0
xðx xÞ 2D1D
½112
Equation [112] is the classical result of the ‘gambler’s ruin’ problem considered by Feller.131 1.13.6.1.1 Lifetime of a cluster
In order to obtain the lifetime of 1D migrating clusters, one should average truin ðx; xÞ over all possible distances between sinks and initial positions of the clusters, that is, over x and x. For this purpose, the corresponding probability density distribution, ’ðx; xÞ, is required. Let us assume that all sinks are distributed randomly throughout the volume and introduce the 1D density of traps (sinks), L, that is, the number of traps per unit length. In this case, ’ðx; xÞ can be represented as a product of the probability density for a cluster to find itself between two sinks separated by a distance x, L2 x expðLxÞ, and the probability density to find a cluster at a distance x from one of these sinks, 1=x: ’ðx; xÞ ¼ L2 expðLxÞ; 0 < x < 1; 0 < x < x ½113
With this distribution, the cluster lifetime, t1D , and the mean-free path to sinks, l, are: t1D ¼ htruin ðx; xÞix;x ¼ 1=2D1D L2 l ¼ hxix;x ¼ 1=L
½114 ½115
where the brackets denote averaging: hix;x ¼ 1 Ð Ðx dx dx’ðx; xÞ 0
0
380
Radiation Damage Theory
1.13.6.1.2 Reaction rate
It follows from eqn [114] that the reaction rate between 1D migrating clusters and immobile sinks (e.g., Borodin130) is given by: R1D ¼ 2L2 D1D C ¼
2 D1D C l2
½116
This equation defines the total reaction rate as a function of L, determined by the concentration and geometry of sinks. If there are different sinks in the system, L is a sum of corresponding contributions Lj from traps of type j. In a crystal containing dislocations and voids only, L ¼ Ld þ Lc
½117
where subscripts ‘d’ and ‘c’ stand for dislocations and voids, respectively. These partial trap densities are found below. Consider voids of a particular radius ri randomly distributed over the volume. Without loss of generality, the capture radius of a void for a cluster is assumed here to be equal to its geometrical radius, that is, rci ¼ ri . A void of radius ri is available to react with mobile clusters that lie in a cylinder of this radius around the cluster path. Hence, the partial 1D density of voids of any particular radius, Lci , and the total 1D void density, Lc , are given by prci2 f
Lci ¼ ðri Þ X Lci ¼ prc2 Nc Lc ¼ i
where f ðri Þ is the SDF of voids (
P i
½118 ½119
f ðri Þ ¼ Nc is the
total void number density) and rc2 is the mean square of the void capture radius. For dislocations Ld ¼ prd r d r d
½120
is the dislocation density defined as the where mean number of dislocation lines intersecting a unit area (surface density) and rd is the corresponding capture radius. This can be shown in the following way. The mean number of dislocation lines intersecting the cylinder of unit length and radius rd around the cluster path equals the area of the cylinder surface, 2prd , times the dislocation density divided by 2. (The factor 2 arises because each dislocation intersects the cylinder twice.) It should be noted that the dislocation sink strength for 3D diffusing defects is usually expressed through the dislocation density, rd , defined as the total length of dislocation lines per unit volume of crystal (volume density). The relationship between r d and rd depends on the
distribution of the dislocation line directions. For a completely random arrangement, the volume density is twice the surface density, rd 2r d (see, e.g., Nabarro132). In this case, eqn [120] is the same as found by Trinkaus et al.19,20 Substituting eqns [117]–[120] into eqn [116], the total reaction rate of the clusters in a crystal containing random distribution of voids and dislocations is found to be130: pr r
2 d d þ prc2 Nc D1D C ½121 R1D ¼ 2 2 For the case, in which immobile vacancy and SIA clusters are also taken into account, the sink strength for 1D diffusing SIA clusters, kg2 , is equal to pr r
2 d d þ prc2 Nc þ svcl Nvcl þ sicl Nicl kg2 ¼ 2 ½122 2 where svcl and sicl are the interaction cross-sections and Nvcl and Nicl the number densities of the sessile vacancy and SIA clusters, respectively. svcl and sicl are proportional to the product of the loop circumference and the corresponding capture radius similar to rd for dislocations. 1.13.6.1.3 Partial reaction rates
A detailed description of the microstructure evolution requires the partial reaction rates, Rj, of the clusters with each particular sink, for example, dislocations or voids of various sizes.22 According to the definition of the parameters Lj and L, the ratio Lj =L is the probability for a trap to be of type j. Hence, the partial reaction rates are Lj R ½123 Rj ¼ L A similar relation between total and partial reaction rates was used in Go¨sele and Frank.128 Using eqn [116], one can write the partial reaction rate of clusters with sinks of type j Rj ¼ 2Lj LD1D C ¼
2 D1D C llj
½124
where lj ¼ 1=Lj is the mean distance between a cluster and a sink of type j in 1D, cf. eqn [116]. Thus, the partial reaction rate of a specific type of sink depends on the density of that sink and also on the density of all other sinks. This correlation between sinks is characteristic of pure 1D diffusion– reaction kinetics in contrast to 3D diffusion where the leading term of the sink strength of any defect is not correlated with others (see eqn [54]).
Radiation Damage Theory
1.13.6.1.4 Reaction rate for SIAs changing their Burgers vector
It has been suggested that deviations of the SIA cluster diffusion from pure 1D mode may significantly alter their interaction rate with stable sinks.23 These deviations could have different reasons, such as thermally activated changes of the Burgers vector of glissile SIA clusters, as observed in MD simulation studies for clusters of two and three SIAs. The reaction rate in the case has been calculated previously25,27; here we present the main result only. time delay before Burgers vector If tch is the mean pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi change and l ¼ 2D1D tch is the corresponding MFP, then the reaction rate can be approximated by the following function25: " 1=2 # l2 8l2 C R 2 1þ 1þ 2 ½125 l tch 2l which gives the correct value in the limiting case of pure 1D diffusion, when tch ! 1, and a correct description of increasing reaction rate with decreasing tch . The analysis is valid for values of l larger than the mean void and dislocation capture radii, and overestimates reaction rates in the limiting case of 3D diffusion, see paragraph 6 in Barashev et al.25 for details. Similar functional form of the reaction rate is obtained by employing an embedding procedure,27 which gives a correct description over the entire range of l in the case when voids are the dominant sinks in the system. 1.13.6.1.5 The rate P(x) for 1D diffusing self-interstitial atom clusters
In the case where 1D migrating SIA clusters are generated during irradiation in addition to PDs, the ME has to account for their interaction with the immobile defects. In the simplest case where the mean-size approximation is used for the clusters, g Gicl ðxÞ ¼ G g dðx xg Þ, the ME for the defects such as voids or vacancy and SIA loops takes a form22 @f s ðx; t Þ=@t ¼ G s ðxÞ þ J ðx 1; t Þ J ðx; tÞ P1D ðxÞf s ðxÞ P1D ðx xg Þf s ðx xg Þ; x 2
½126
where P1D ðxÞ is the rate of glissile loop absorption by the defects. The and in eqn [126] are used to distinguish between vacancy-type defects (voids and vacancy loops/SFT) and SIA type because capture of SIA glissile clusters leads to a decrease in the size in the former case and an increase in the latter one.
381
The rate P1D ðxÞ depends on the type of immobile defects. In the case of voids, their interaction with the SIA clusters is weak and therefore the cross-sections may be approximated by the corresponding geometrical factor equal to pR2v Nv . The rate P1D ðxÞ in this case is given by (see eqn [11c] in Singh et al.22) pffiffiffi2=3 LDg Cg 2=3 3 p P1D ðxÞ ¼ 2 x ½127 4 O1=3 qffiffiffiffiffiffiffiffiffi where L ¼ kg2 =2. Note that the factor 2 in eqn [127] was missing in Singh et al.22 In the case of dislocation loops, the situation is more complicated as the cross-section is defined by long-range elastic interaction. A fully quantitative evaluation is rather difficult because of the complicated spatial dependence of elastic interactions, in particular, for elastically anisotropic media. For loops of small size, the effective trapping radii turn out to be large compared with the geometrical radii of the loops and hence the ‘infinitesimal loop approximation’ may be applied. It is shown (see Trinkaus et al.20) that in this case the cross-section is proportional to ðxxg Þ1=3 thus the rate P1D ðxÞ is equal to 2:25p xg Tm 2=3 LDg Cg x 2=3 ½128 P1D ðxÞ ¼ 1=3 T O where T and Tm are temperature and melting temperature, the multiplier is a correction factor which is introduced because eqn [4] in Trinkaus et al.20 was obtained using some approximations of the elastically isotropic effective medium and, consequently, it can be considered as a qualitative estimate of the crosssection rather than a quantitative description. The factor is of order unity and was introduced as a fitting parameter. Since sessile SIA and vacancy clusters have different structures (loops in the case of the SIA clusters and frequently SFTs in the case of vacancy clusters), the multiplier and, consequently, the appropriate cross-sections may be slightly different. Also note that mO ¼ kB Tm has been used in Trinkaus et al.20 as an estimate on a homologous basis. In the case of large size dislocation loops, the cross-section of their interaction with the SIA glissile clusters can be calculated in a way similar to that of edge dislocations. Namely, it is proportional to the product of the length of dislocation line, that is, 2pRl , and the capture radius, bl . The rate P1D ðxÞ in that case is given by rffiffiffiffiffiffi p ½129 b1 LDg Cg x 1=2 P1D ðxÞ ¼ 4 Ob
382
Radiation Damage Theory
Note that in the more general case where different sizes of the SIA glissile clusters are taken into account, the last term on the right side of eqn [126] has to be replaced with the sum y¼xgmax P P1D ðx yÞf ðx yÞ. y¼xgmin
1.13.6.1.6 Swelling rate
By omitting the recombination term, eqns [10]–[12] for mobile defects can be rewritten as dCv 2 þ Zvcl k2 Þ ¼ Gv þ Gvc Dv Cv ðkc2 þ Zvd rd þ Zvicl kicl v vcl dt dCi 2 þ Zvcl k2 Þ ¼ Gi Di Ci ðkc2 þ Zivcl rd þ Ziicl kicl i vcl dt g 2 dC icl ¼ G g 2D C prd rd þ pr 2 N þ s N þ s N g g c vc c ic icl vcl i dt 2
½130
It has been shown that, under conditions in which swelling is observed, the vacancy and SIA clusters produced by cascades reach steady-state size distributions at relatively small doses.22 This is because vacancy clusters have far lower thermal stability than voids. The growth of sessile SIA clusters is restricted on account of the high vacancy supersaturation, which builds up due to rapid 1D diffusion of mobile SIA clusters to sinks. Consequently, at relatively low doses, the SDF of the sessile SIA clusters achieves steady state. After reaching steady state, both types of sessile clusters start to serve as recombination centers for PDs and glissile SIA clusters. Analytical expressions for the steady-state SDFs of vacancy and SIA clusters can be found (see eqns [23] and [24] in Singh et al.22) and the corresponding sink strengths of the clusters at the steady state are given by (eqn [25] in the same reference) Esv Gv 2 ¼ kvcl g Dv expðEvcl =kB T Þ kc2 þ Zdv rd Ei Gv 2 1 kc þ Zdv rd 1 s xvcl 2 kicl
¼
Esi
2 g k Ei c
!
þ Zdi rd
½131
! 1 1 s xicl
½132
where Evcl is an effective binding energy of vacancies s with the vacancy clusters and xvcl;icl are the mean size of the vacancy and SIA glissile clusters (see eqn [8]). It should be noted that the SDF of sessile interstitial clusters, the sink strength of which is described by eqn [132], is limited by the maximum size of the clusters produced in displacement cascades (see Figure 2 in Singh et al.22). This is because the clusters
produced are reduced in size due to the high vacancy supersaturation. Fluctuations in the defect arrival to the clusters produce a tail in the SDF extending beyond the maximum size formed in cascades. The tail is characterized by very small concentrations and cannot describe the observed nucleation and growth of SIA clusters and the consequent formation of the dislocation network (see, e.g., Garner32 and Garner et al.33). The most probable reason for this failure is that the cluster–cluster interaction leading to their coalescence is neglected in the current theoretical framework. Sessile interstitial clusters are produced in cascades at rates comparable to those of PDs. The evolution of concentrations of mobile species (PDs and glissile clusters) in this case may be described by nonstationary equations because of the very fast evolution of the sessile cluster population. High vacancy supersaturation will drive the evolution of the sessile SIA clusters toward quasisaturation state, beyond which the steady-state equations for the mobile species become valid. Similar steady state for vacancy clusters will be achieved because of the thermal instability of the clusters.22 2 Gv ¼ Dv Cv ðkc2 þ Zvd rd Þ þ Dv Cv Zvicl kicl 2 þ Di Ci Zvvcl kvcl þ Dg Cg Lxg sicl Nicl
½133
2 Gi ¼ Di Ci ðkc2 þ Zid rd Þ þ Dv Cv Zvicl kicl
½134
2 þ Di Ci Zvvcl kvcl 2Dg Cg Lxg sicl Nicl
g
Gi ¼ 2Dg Cg
prd rd þ prc2 Nc þ svc Nvcl þ sic Nicl 2
2 ½135
In the framework of PBM, the balance equations for PDs depend on the concentration of glissile clusters and, thus, are very different from those in the FP3DM. The vacancy supersaturation is obtained from the difference between DvCv and DiCi as given by eqns [133] and [134] Dv Cv Di Ci Zvd rd Dv Cv þ Zid rd g e G NRT ð1 er Þ svcl Nvcl þ sicl Nicl þ i 2 1 ½136 kc þ Zvd rd Lg qffiffiffiffiffiffiffiffiffi where Lg ¼ kg2 =2 ¼ prd rd =2 þ prc2 Nc þ svcl Nvcl þ ¼ Bd
kc2
sicl Nicl . The first and the second terms on the right-hand side of eqn [136] correspond to the actions
Radiation Damage Theory
of the dislocation bias and the production bias, respectively. As can be seen, the first term depends on the vacancy concentration, and hence on the total sink strength of all defects including PD clusters. The second term also depends on the sink strength of all defects but differently, and describes the distribution of excess of vacancies between voids and dislocations, and their recombination at PD clusters. In the PBM, the swelling rate is given by dS ¼ kc2 ðDv Cv Di Ci Þ 2Dg Cg xg Lg prc2 Nc ½137 dt and, with the aid of eqn [136], can be represented as follows
8 < kc2 Zvd rd dS ¼ ð1 er Þ Bd : ðk2 þ Zd rÞðk2 þ Zd r þ Zicl k2 þ Zvcl k2 Þ df v v d v icl v vcl c c " ! #9 kc2 prc2 Nc = svcl Nvcl þ sicl Nicl g þei 1 Lg Lg ; k2 þ Zd r c
v d
½138
where f ¼ G NRT t is the NRT irradiation dose. The first term in the brackets on the right-hand side of eqn [138] corresponds to the influence of the dislocation bias and the second one to the production bias. The factor ð1 er Þ describes intracascade recombination of defects, which is a function of the recoil energy and may reduce the rate of defect production by up to an order of magnitude that can be compared to the NRT value: ð1 er Þ ! 0:1 at high PKA energy (see Section 1.13.3). As indicated by this equation, the swelling rate is a complicated function of dislocation density, dislocation bias factor, and the densities and sizes of voids and PD clusters. It also demonstrates the dependence of the swelling g rate on the recoil energy, determined by ei , which increases with increasing PKA energy up to about 10–20 keV. The main predictions of the PBM are discussed below. 1.13.6.2 Main Predictions of Production Bias Model As can be seen from eqn [138], the action and consequences of the two biases, the dislocation and production ones, is quite different. As shown in Section 1.13.5, the dislocation bias depends only slightly on the microstructure and predicts indefinite void growth. In contrast, the production bias can be positive or negative, depending on the microstructure. The reason for this is in negative terms in eqn [138]. The first term decreases the action of the
383
production bias due to recombination of the SIA clusters at sessile vacancy and SIA clusters, while the second one arises from the capture of SIA clusters by voids. The latter term may become equal to zero or even negative, hence the combination of the two bias factors does not necessarily lead to a higher swelling rate, as shown in Barashev and Golubov.35 1.13.6.2.1 High swelling rate at low dislocation density
As shown in Section 1.13.5, in the framework of FP3DM, the swelling rate depends on the dislocation density and becomes small for a low dislocation density, dS=df Bd rd =kc2 ! 0 at rd ! 0 (see eqn [96]). Thus, it was a common belief that the swelling rate in well-annealed metals has to be low at small doses, that is, when the dislocation density increase can be neglected. Under neutron irradiation, the effect of dislocation bias on swelling is even smaller because of intracascade recombination: ðdS=dfÞFPP3D neutr ¼ FPP3D FPP3D ðdS=dfÞelectr ð1 er Þ ðdS=dfÞelectr . It has been found experimentally, however, that the void swelling rate in fully annealed pure copper irradiated with fission neutrons up to about 102 dpa (see Singh and Foreman18) is of 1% per dpa, which is similar to the maximum swelling rate found in materials at high enough irradiation doses. This observation was one of those that prompted the development of the PBM. The production bias term in eqn [138] allows the understanding of these observations. Indeed, at low doses of irradiation, the void size is small, and therefore, the void cross-section for the interaction with the SIA glissile clusters is small (prc2 Nc =Lg 1). As a result, the last term in the production bias term is negligible and thus the swelling rate is driven by the production bias: dS k2 g ð1 er Þei 2 c d ½139 kc þ Zv rd df max As in the case Zvd rd kc2 , the swelling rate is determined by the cascade parameters dS=df g g ð1 er Þei kc2 =ðkc2 þ Zvd rd Þ ð1 er Þei . It has been 22,24 shown that a good agreement with observations is achieved with the following parameters: 1 er ¼ 0:1 g and ei ¼ 0:2, which are in good agreement with the results of MD simulations of cascades. g It is worth emphasizing that the value ð1 er Þei determines the maximum swelling rate, which can be produced by the production bias. Indeed, assuming that for some reason (see Section 1.13.7) there is no interaction of the mobile SIA clusters with voids and
384
Radiation Damage Theory
sessile vacancy and SIA clusters, the swelling rate is g given by dS=df 1=2ð1 er Þei where the sink strength ratio, kc2 =ðkc2 þ Zvd rd Þ, is taken to be equal to 1/2, as frequently observed in experiments. Taking into account the magnitude of the cascade parag meters er and ei estimated in Golubov et al.24 and neglecting the dislocation bias term in eqn [138], one may conclude that the maximum swelling rate under fast neutron irradiation may reach about 1% per dpa. As pointed out in Section 1.13.5, in the case of FP production, that is, in the FP3DM, the maximum swelling rate is also 1% per dpa. This coincidence is one of the reasons why an illusion that the FP3DM model is capable of describing damage accumulation in structural and fuel materials in fission and future fusion reactors has survived despite the fact that nearly 20 years have passed since the PBM was introduced. Note that the production bias provides a way to understand another experimental observation, namely, that the swelling rate in some materials decreases with increasing irradiation dose (see, e.g., Figure 5 in Golubov et al.24). Such a decrease is predicted by eqn [138], as the negative term of the production bias, prc2 Nc =Lg , increases with an increase in the void size. As the first term in the
101
100
Swelling (%)
10–1
10–2
Tirr = 523 K
Copper Experiments Neutrons Protons Electrons Calculations (1) Neutron (2) Proton (3) Electron
(1)
(2)
10–3
production bias is proportional to the void radius and the second to the radius squared, the swelling rate may finally achieve saturation at a mean void radius equal to Rmax 2prd .19,30,35 Finally, the cascade production of the SIA clusters may strongly affect damage accumulation. As can be seen from eqn [132], the steady-state sink strength of the sessile SIA clusters is inversely proportional to the fraction of SIAs produced in cascades in the form g 2 ! 1 when ei ! 0. of mobile SIA clusters, thus kicl This limiting case was considered by Singh and Foreman18 to test the validity of the original framework of the PBM,16,17 where all the SIA clusters produced by cascades were assumed to be immobile g (hereafter this case of ei ¼ 0 is called the Singh– Foreman catastrophe). If for some reasons this case is realized, void swelling and the damage accumulation in general would be suppressed for the density of SIA clusters, hence, their sink strength would reach a very high value by a relatively low irradiation dose, f 1dpa, (see Singh and Foreman18). Thus, irradiation with high-energy particles, such as fast neutrons, provides a mechanism for suppressing damage accumulation, which may operate if the SIA clusters are immobilized. In alloys, the interaction with impurity atoms may provide such an immobilization. The so-called ‘incubation period’ of swelling observed in stainless steels under neutron irradiation for up to several tens of dpa (Garner32,33) might be due to the Singh–Foreman catastrophe. A possible scenario of this may be as follows: during the incubation period, the material is purified by RIS mainly on SIA clusters because of their high density. At high enough doses, that is, after the incubation period, the material becomes clean enough to provide the recovery of the mobility of small SIA clusters created in cascades that triggered on the production bias mechanism. As a result, the high number density of SIA clusters decreases via the absorption of the excess of vacancies, restoring conditions for damage accumulation.
(3)
10–4
1.13.6.2.2 Recoil-energy effect 10–5 10–4
10–3
10–2 Dose (dpa)
Figure 5 Experimentally measured133 and calculated24 levels of void swelling in pure copper after irradiation with 2.5 MeV electrons, 3 MeV protons, and fission neutrons. The calculations were performed in the framework of the FP3DM for the electron irradiation and using the production bias model as formulated in Singh et al.22 for irradiations with protons and fission neutrons. From Golubov et al.24
The recoil energy enters the PBM through the casg cade parameters er and ei (see eqn [138]). Direct experimental evaluation of the recoil energy effect on void swelling was made by Singh et al.,133 who compared the microstructure of annealed copper irradiated with 2.5 MeV electrons, 3 MeV protons, and fission neutrons at 520 K. For all irradiations, the damage rate was 108 dpa s1. The average recoil energies in those irradiations were estimated133
Radiation Damage Theory
to be about 0.05, 1, and 60 keV for electron, proton, and neutron irradiations, respectively, thus, producing the primary damage in the form of FPs (electrons), small cascades (protons), and well-developed cascades (neutrons). The cascade efficiency, 1 er , hence, the real damage rate, was highest for electron irradiation (no cascades, the efficiency is equal to unity) and minimal for neutron irradiation (0.1, see Section 1.13.3). If dislocation bias is the mechanism responsible for swelling, the swelling rate is proportional to the damage rate and therefore must be highest after electron and lowest after neutron irradiation. However, just the opposite was found; the swelling level after neutron irradiation was 50 times higher than after electron irradiation, with the value for proton irradiation falling in between (see Figure 5). These results represent direct experimental confirmation that damage accumulation under cascade damage conditions is governed by mechanisms that are entirely different from those under FP production. The results obtained in this study can be understood as follows. Under electron irradiation, only the first term on the right-hand side of eqn [138] operg ates, as ei ¼ 0. The swelling rate is low in this case because of the low dislocation density, as discussed in Section 1.13.6.2.1. Under cascade damage conditions, the damage rate is smaller because of the low g cascade efficiency. In this case ei 6¼ 0 and the second term on the right-hand side of eqn [138] plays the main role, which is evident from the theoretical treatment of the experiment carried out in the following section.24 1.13.6.2.3 GB effects and void ordering
As shown in the previous section, several striking observations of the damage accumulation observed in metals under cascade damage conditions can be rationalized in the framework of the PBM. This became possible because of the recognition of the importance of 1D diffusion of SIA clusters, which are continuously produced in cascades. The reaction kinetics in this case are a mixture of those for 1D and 3D migrating defects. Here, we emphasize that 1D transport is the origin of some phenomena, which are not observed in solids under FP irradiation. One such phenomenon is the enhanced swelling observed near GBs. It is well known that GBs may have significant effect on void swelling. For example, zones denuded of voids are commonly observed adjacent to GBs in electron-, ion-, and neutron-irradiated materials.134–137 Experimental observations on the
385
effect of grain size on void concentration and swelling in pure austenitic stainless steels irradiated with 1 MeV electrons were also reported.138,139 In these experiments both void concentration and swelling were found to decrease with decreasing grain size. Theoretical calculations are in good agreement with the grain-size dependence of void concentration and swelling measured experimentally in austenitic stainless steel irradiated with 1 MeV electrons.139,140 However, there is a qualitative difference between grain-size dependences of void swelling for electron irradiation and that for higher recoil energies. In particular, in the latter case, in the region adjacent to the void-denuded zone, void swelling is found to be substantially enhanced.134,136,141–147 Furthermore, in neutron-irradiation experiments on high-purity aluminum, the swelling in the grain interior increases strongly with decreasing grain size.144 This is opposite to the observations under 1 MeV electron irradiation139 and to the predictions of a model based on the dislocation bias.140 An important feature of the enhanced swelling near GBs under cascade irradiation is its large length scale. The width of this enhanced-swelling zone is of the order of several micrometers, whereas the mean distance between voids is of the order of 100 nm. Thus, the length scale is more than an order of magnitude longer than the mean distance between voids. The MFP of 3D diffusing vacancies and single SIAs is given by rffiffiffiffi 1=2 2 pffiffiffi d 3D ¼ 2 Z rd þ 4prc Nc ½140 L ¼ 2 k and is of the order of the mean distance between defects. Hence, 3D diffusing defects cannot explain the length scale observed. In contrast, the MFP of 1D diffusing SIA clusters is given by sffiffiffiffi pr r
1 2 d d 1D 2 þ pr N ½141 ¼ L ¼ c c 2 kg2 and is of the order of several micrometers, hence, exactly as required for explanation of the GB effect (see Figure 6). A possible explanation for the observations would be as follows. The SIA clusters produced in the vicinity of a GB, in the region of the size L1D , are absorbed by it, while 3D migrating vacancies give rise to swelling rates higher than that in the grain interior. The impact of the GB on the concentration of 1D diffusing SIA clusters can be understood by using local sink strength, that is, the sink strength that depends on the distance of a local
386
Radiation Damage Theory
0.70
102
0.60
1D
0.50 Local swelling (%)
Mean range/cavity spacing
Nv = 5 ⫻ 1018 m–3
101 2D
100 10–3
10–2
10–1 Swelling (%)
100
101
area to the GB, l. It has been shown22 that the local sink strength in a grain of radius RGB is given by prd rd 1 þ prc2 Nc þ 2 ðlð2RGB lÞÞ1=2
r = 2 ⫻ 1011 m–2
0.40 5 ⫻ 1011 m–2 0.30 2 ⫻ 1012 m–2
0.10 ⫻10
Figure 6 Ratio of the mean-free path of the self-interstitial atom clusters and the distance between voids as a function of void swelling level for 1D, 2D, and 3D migration of the clusters. From Trinkaus, H.; Singh, B. N.; Foreman, A. J. E. J. Nucl. Mater. 1993, 206, 200–211.
kg2 ðRGB ; lÞ ¼ 2
TEM SRT PBM
0.20
3D
Copper 623 K 0.3 dpa
!2
½142
As can be seen from eqn [142], the sink strength has a minimum at the center of the grain, that is, at l ¼ RGB, and increases to infinity near the GB, when l ! 0. The so-called grain-size effect, an increase of the swelling rate in the grain interior in grains of relatively small sizes (less than about 5 mm) with decreasing grain size, has the same origin as the GB effect discussed above. The swelling rate at the center of a grain may increase with decreasing grain size, when the grain size becomes comparable with the MFP of 1D diffusing SIA clusters and the zones of enhanced swelling of the opposite sides of GBs overlap. The swelling in the center of a grain as a function of grain size is presented in Figure 7.26 For comparison purposes, the values of the local void swelling (see Table 3 in Singh et al.26) determined in the grain interiors by TEM are also shown. The PBM predicts a decrease of swelling with increasing grain size for grain radii bigger than 5 mm, which is in accordance with the experimental results. Note that the swelling values calculated by the FP3DM (broken curve in Figure 7) are magnified by a factor of 10. Another striking phenomenon observed in metals under cascade damage conditions is the formation
0.00 0.1
1
10 Grain radius (mm)
100
Figure 7 Calculation results on the grain-size dependence of the void swelling in the grain interior in copper irradiated at 623 K to 0.3 dpa. The results are for both the production bias model (PBM) and the FP3DM. The FP3DM values are magnified by a factor of 10. Filled triangles are the measured values. Open circles show calculations using PBM for specific grain sizes and experimental values for void densities and a dislocation density of 12 1012 m2. From Singh, B. N.; Eldrup, M.; Zinkle, S. J.; Golubov, S. I. Philos. Mag. A 2002, 82, 1137–1158.
of void lattices. It was first reported in 1971 by Evans148 in molybdenum under nitrogen ion irradiation, by Kulchinski et al.149 in nickel under selenium ion bombardment, and by Wiffen150 in molybdenum, niobium, and tantalum under neutron irradiation. Since then it has been observed in bcc tungsten, fcc Al, hcp Mg, and some alloys.151–155 Ja¨ger and Trinkaus156 reviewed the characteristics of defect ordering and analyzed the theories proposed at that time, including those based on the elastic interaction between voids and phase instability theory. They concluded that in cubic metals, the void ordering is due to the 1D diffusion of SIA clusters along close-packed crystallographic directions (first proposed by Foreman157). Two features of void ordering support this conclusion. First, the symmetry and crystallographic orientation of a void lattice are always the same as those of the host lattice. Second, the void lattices are formed under neutron and heavy-ion but not electron irradiation. This conclusion is also supported by theoretical analysis performed in Ha¨hner and Frank158 and Barashev and Golubov.159
Radiation Damage Theory
1.13.6.3 Limitations of Production Bias Model Successful applications of the PBM have been limited to low irradiation doses (<1 dpa) and pure metals (e.g., copper). There are two apparent problems preventing the general application of the model at higher doses. 1.13.6.3.1 Swelling saturation at random void arrangement
The PBM predicts a saturation of void size.30 This originates from the mixture of 1D and 3D diffusion– reaction kinetics under cascade damage conditions, the assumption lying at the heart of the model. More specifically, it stems from the fact that the interaction cross-section with a void is proportional to the void radius, R, for 3D migrating vacancies and to R2 for 1D diffusing SIA clusters. As a result, above some critical radius, the latter becomes higher than the former and the net vacancy flux to such voids is negative. In contrast, experiments demonstrate indefinite void growth in the majority of materials and conditions.31– 34 An attempt to resolve this contradiction was undertaken by including thermally activated rotations of the SIA-cluster Burgers vector23,25,27; but it has been shown by Barashev et al.25 that this is not a solution. Thus, the PBM fails to account for this important and common observation, that is, the indefinite void growth under cascade irradiation. A way of resolving this issue is discussed in Section 1.13.7. 1.13.6.3.2 Absence of void growth in void lattice
Another problem of the PBM is that it fails to explain swelling saturation at rather low swelling levels (approximately several percent) observed in void lattices. In fact, it even predicts an increase in the swelling rate when a random void arrangement is changed to that of a lattice.25 This is because the free channels between voids along close-packed directions, which are formed during void ordering, provide escape routes for 1D migrating SIA clusters to dislocations and GBs, thereby allowing 3D migrating vacancies to be stored in voids. A possible explanation of the problem is discussed in a forthcoming paper by Golubov et al.37
387
materials when the initial damage is in the form of only FPs and under neutron irradiation, when thermally stable glissile SIA clusters are continuously produced in cascades. The successful applications of the PBM have been limited to low irradiation doses (<1 dpa) and pure metals (e.g., Cu). Furthermore, it predicts the saturation of void size with increasing irradiation dose. Thus, it fails to account for the most important observation under neutron or heavy-ion irradiation: continuous increase in void swelling. The observed continuous void growth may be explained by the development of spatial correlations between voids and other lattice defects. Such as, precipitates and dislocations, that shadow voids from the SIA clusters (see Figures 8–10). It has been argued that this must be the case and the very absence of a void lattice (i.e., a particular case of spatial correlation, which is between voids) must be an indication that spatial correlations with other defects prevail.35
2 Precipitate 1
3 Void
4 Figure 8 Schematic diagram illustrating screening of a void from self-interstitial atom clusters by a precipitate. The close-packed directions of the cluster Burgers vectors are indicated by arrows. From Barashev, A. V.; Golubov, S. I. Philos. Mag. 2009, 89, 2833–2860.
2
Compression 1
3 Void
1.13.7 Prospects for the Future Edge dislocation
As discussed in the previous section, the PBM changed the concept of RDT by recognizing that qualitatively different mechanisms operate in
Figure 9 Same as in Figure 8 but for a void in the compression side of edge dislocation. From Barashev, A. V.; Golubov, S. I. Philos. Mag. 2009, 89, 2833–2860.
388
Radiation Damage Theory
[143] with an appropriate choice of the dependence of c on the irradiation dose (see Figure 10). At high doses, the voids must be completely shielded from the SIA clusters: c ¼ 0, and the steady-state swelling rate of 1% per dpa observed in austenitic steels33 can be interpreted as being equal to about half of the production bias, that is, the fraction of SIAs produced as 1D mobile clusters:
h =0 V
20
Swelling (%)
15
10
dS 1 g esurv ei dfNRT 2
5 h = 0.5 V
h =1 V
0 0
5
10 15 Irradiation dose (NRT dpa)
20
25
Figure 10 Dependence of swelling on irradiation dose calculated using eqn [143] for Nc ¼ 1022 m3 , rm ¼ 5 nm, egi ¼ 0:2, esurv ¼ 0:1 and different values of the correlation-screening factor of voids, c . The curve with full squares has been calculated for correlations developing with irradiation dose, when c ¼ 1 in the beginning and c ¼ 0 by 10 dpa and higher dose. From Barashev, A. V.; Golubov, S. I. Philos. Mag. 2009, 89, 2833–2860.
To account for this effect, a new parameter, c , has been introduced, called the ‘correlation-screening factor,’ which is equal to unity in the absence of shadowing effects and zero when voids are screened completely from the SIA clusters. The swelling rate is then given by dS r g F ½143 ¼ ei 1 c df rm where F is a proportionality coefficient, which is a function of all parameters involved.35 Experimental evidence on the association of large voids with various precipitates (G, , Laves, etc.)120,160–162 and the compression side of edge dislocations163,164 has been available for a long time. More recent evidence can be attributed to Portnykh et al.165 who studied the microstructure of 20% cold-worked 16Cr–15Ni– 2Mo–2Mn austenitic steel irradiated up to 100 dpa in a BN-600 fast reactor in the temperature range from 410 to 600 C. TEM studies revealed voids of three types: a-type associated with dislocations, b-type associated with G-phase precipitates and c-type distributed homogeneously. The c-type voids were the smallest and made practically no contribution to swelling, while the a-type voids were the largest. Such spatial correlations must be a common feature under cascade irradiation. As discussed in Barashev and Golubov,35 the experimental data on void swelling can be fit by eqn
½144
where esurv ð1 er Þ 0:1 is the survival fraction of defects in displacement cascade. The weak dependence on steel composition observed is probably because the final defect structure is defined by early stages of cascades, when the energies involved are much higher than the binding energies of defects with solute atoms. The observed correlation of the incubation period prior to swelling with the formation of a dislocation network may be connected with an increase of the volume for the nucleation and growth of voids in which voids are screened from the SIA clusters. Higher dislocation density also corresponds to a smaller dislocation climb rate, which might be essential for preserving void-dislocation correlations. Another distinguishing feature of neutron irradiation is transmutation of atoms, which transform even pure metals into alloys with increasing irradiation dose. The atmospheres of solute (or transmuted) elements near voids may repel SIA clusters and, hence, assist or even solely explain the unlimited void growth. It was shown (see, e.g., Golubov,166 Golubov et al.,167 and references therein) that RIS can provide an additional mechanism of preferential absorption of mobile defects even in the framework of FP3DM, causing a ‘segregation’ bias, which must be different for immobile defects (e.g., voids) and mobile defects, such as dislocations. In the PBM, the interaction of the mobile SIA clusters with different defects may even be more important. Solute atoms may also decrease the mobility of SIA clusters, thereby increasing the recombination rate with migrating vacancies. In the case of very high binding energy of SIA clusters with impurity atoms, the ‘Singh–Foreman catastrophe’18 discussed in Section 1.13.6.2.1 may occur. Thus, two additional features beyond those already in the PBM distinguish the microstructure evolution under neutron compared to electron irradiation at high enough doses: transmutation of atoms and development of spatial correlations. A fully predictive theory must account for these effects.
Radiation Damage Theory
The development of a predictive theory requires revisiting all its essential elements: nucleation, growth, movement of voids, and other lattice defects in the presence of spatial correlations, etc. Carefully planned experiments spanning different temperatures, defect production rates, etc., must be a central part of these future studies. Development of the RIS theory for accounting for the SIA clusters is necessary for understanding the sensitivity of microstructure to material composition. Generally, the challenge is to create a theory, where the mean-field approach in its conventional form is abandoned, a task not attempted before.
Acknowledgments The authors would like to acknowledge the fruitful collaboration and discussions on the physics of radiation damage for many years with Professor Yu.V. Konobeev (IPPE, Russia), Drs. B.N. Singh (Risø National Laboratory, Denmark), H. Trinkaus (Forschungscentrum Ju¨lich, Germany), S.J. Zinkle and Yu.N. Osetsky (Oak Ridge National Laboratory, USA), and Professor D.J. Bacon (The University of Liverpool, UK). Various aspects of the author’s research discussed in this chapter were supported by the Division of Material Science and Engineering and the Office of Fusion Energy Sciences, Department of Energy and the Office of Nuclear Regulatory Research, US (SIG and RES) and the UK Engineering and Physical Science Research Council (AVB).
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Garner, F. A., Perrin, I. S., Eds.; American Society of Testing and Materials: Philadelphia, PA, 1985; ASTM STP 870, pp 248–261. 145. Foreman, A. J. E.; Singh, B. N.; Horsewell, A. Mater. Sci. Forum 1987, 15–18, 895–900. 146. Singh, B. N.; Horsewell, A. J. Nucl. Mater. 1994, 212–215, 410–415. 147. Horsewell, A.; Singh, B. N. Influence of Grain and Subgrain Boundaries on Void Formation and Growth in Aluminum Irradiated with Fast Neutrons; American Society of Testing and Materials: Philadelphia, PA, 1987; pp 220–229, ASTM Special Technical Publication 955. 148. Evans, J. H. Nature 1971, 229, 403–405. 149. Kulchinski, G. L.; Brimhall, J. L.; Kissinger, H. E. J. Nucl. Mater. 1971, 40, 166–174. 150. Wiffen, F. W. In Proceedings of the International Conference on Radiation-Induced Voids in Metals; Corbet, J. W., Iannielo, L. C., Eds.; U.S. Atomic Energy: Albany, NY, 1972; pp 386–396, USAEC-CONF-71061. 151. Mazey, D. J.; Francis, S.; Hudson, A. I. J. Nucl. Mater. 1973, 47, 137–142. 152. Risbet, A.; Levy, V. J. Nucl. Mater. 1974, 50, 116–118. 153. Loomis, B. A.; Gerber, S. B.; Taylor, A. J. Nucl. Mater. 1977, 68, 19–31. 154. Garner, A. F.; Stubbins, J. F. J. Nucl. Mater. 1994, 212, 1298–1302. 155. Horsewell, A.; Singh, B. N. Radiat. Eff. 1987, 102, 1–5. 156. Ja¨ger, W.; Trinkaus, H. J. Nucl. Mater. 1993, 205, 394–410. 157. Foreman, A. J. E. A Mechanism for the Formation of a Regular Void Array in an Irradiated Metal; Harwell Report AERE-R 7135; 1972. 158. Ha¨hner, P.; Frank, W. Solid State Phenom. B 1992, 23–24, 203–219. 159. Barashev, A. V.; Golubov, S. I. Philos. Mag. 2010, 90, 1787–1797. 160. Rowcliffe, A. F.; Lee, E. H. J. Nucl. Mater. 1982, 108–109, 306–318. 161. Pedraza, D. F.; Maziasz, P. J. Void-Precipitate Association During Neutron Irradiation of Austenitic Stainless Steel. In Radiation-Induced Changes in Microstructure: 13th International Symposium (Part I), ASTM STP 955; Garner, F. A., Packan, N. H., Kumar, A. S., Eds.; American Society for Testing and Materials: Philadelphia, 1987; pp 161–194. 162. Boothby, R. M.; Williams, T. M. J. Nucl. Mater. 1988, 152, 123–138. 163. Farrell, K.; Houston, J. T. J. Nucl. Mater. 1970, 35, 352–355. 164. Risbet, A.; Brebec, G.; Lanore, J. M.; Levy, V. J. Nucl. Mater. 1975, 56, 348–354. 165. Portnykh, I. A.; Kozlov, A. V.; Panchenko, V. L.; Chernov, V. M.; Garner, F. A. J. Nucl. Mater. 2007, 367–370, 925–929. 166. Golubov, S. I. Metallofizika 1989, 11, 10–18, [in Russian]. 167. Golubov, S. I.; Ginkin, V. P.; Strokova, A. M. Mater. Sci. Forum 1992, 97–99, 97–103.
1.14
Kinetic Monte Carlo Simulations of Irradiation Effects
C. S. Becquart Ecole Nationale Supe´rieure de Chimie de Lille, Villeneuve d’Ascq, France; Laboratoire commun EDF-CNRS Etude et Mode´lisation des Microstructures pour le Vieillissement des Mate´riaux (EM2VM), France
B. D. Wirth University of Tennessee, Knoxville, TN, USA
ß 2012 Elsevier Ltd. All rights reserved.
1.14.1 1.14.2 1.14.3 1.14.4 1.14.4.1 1.14.4.2 1.14.4.3 1.14.4.4 1.14.5 1.14.6 1.14.7 1.14.8 1.14.9 References
Introduction Modeling Challenges to Predict Irradiation Effects on Materials KMC Modeling KMC Modeling of Microstructure Evolution Under Radiation Conditions Irradiation Rate Transmutation Rate Diffusion Rate Emission/Dissociation Rate Atomistic KMC Simulations of Microstructure Evolution in Irradiated Fe–Cu Alloys OKMC Example: Ag Fission Product Diffusion and Release in TRISO Nuclear Fuel Some Limits of KMC Approaches Advanced KMC Methods Summary and a Look at the Future of Nuclear Materials Modeling
Abbreviations AKMC BKL EKMC FP HTR KMC MC MD NEB NRT OKMC PKA RPV RTA SIA TRISO
Atomistic kinetic Monte Carlo Boris, Kalos, and Lebowitz Event kinetic Monte Carlo Frenkel pairs High-Temperature Reactor Kinetic Monte Carlo Monte Carlo Molecular dynamics Nudged elastic band Norgett, Robinson, and Torrens Object kinetic Monte Carlo Primary knockon atom Reactor pressure vessel Residence time algorithm Self-interstitial atoms Tristructural isotropic fuel particle
1.14.1 Introduction Many technologically important materials share a common characteristic, namely that their dynamic behavior is controlled by multiscale processes. For
393 394 395 396 397 397 397 398 399 404 406 407 408 408
example, crystal growth, plasma processing of materials, ion-beam assisted growth and doping of electronic materials, precipitation in structural materials, grain boundary and dislocation evolution during mechanical deformation, and alloys driven by high-energy particle irradiation all experience cluster nucleation, growth, and coarsening that impact the evolution of the overall microstructure and, correspondingly, property changes. These phenomena involve a wide range of length and time scales. While the specific details vary with each material and application, kinetic processes at the atomic to nanometer scale (especially related to nucleation phenomena) are largely responsible for materials evolution, and typically involve a wide range of characteristic times. The large temporal diversity of controlling processes at the atomic to nanoscale level makes experimental identification of the governing mechanisms all but impossible and clearly defines the need for computational modeling. In such systems, the potential benefits of modeling are at a maximum and are related to reduction in time and expense of research and development and introduction of novel materials into the marketplace. Systems in which the materials microstructure can be represented by multiple particles experiencing 393
394
Kinetic Monte Carlo Simulations of Irradiation Effects
Timescale µs–s ns–µs ps–ns
1012 and 103 s. The goal of this chapter is to describe the state of the art in kinetic Monte Carlo (KMC) simulation, as well as to identify a number of priority research areas, moving toward the goal of accelerating the development of advanced computational approaches to simulate nucleation, growth, and coarsening of radiation-induced precipitates and defect clusters (cavities and/or dislocation loops). It is anticipated that the approaches will span from atomistic molecular dynamics (MD) simulations to provide key kinetic input on governing mechanisms to fully three-dimensional (3D) phase field and KMC models to larger scale, but spatially homogeneous cluster dynamics models.
1.14.2 Modeling Challenges to Predict Irradiation Effects on Materials The effect of irradiation on materials is a classic example of an inherently multiscale phenomenon, as schematically illustrated in Figure 1. Pertinent processes span over more than 10 orders of magnitude in length scale from the subatomic nuclear to
Nano/microstructure and local chemistry changes; nucleation and growth of extended defects and precipitates
Irradiation temperature, n/g energy spectrum, flux, fluence, thermal cycling, and initial material microstructure inputs:
s–year
Decades
Brownian motion and occasional collisions against one another and systems with other defects (dislocations, grain boundaries, surfaces, etc.) are in particular amenable to multiscale modeling. Within a multiscale approach, atomistic simulations (utilizing either electronic structure calculations or semiempirical potentials) investigate controlling mechanisms and occurrence rates of diffusional and reactive interactions between the various particles and defects of interest, and inform larger length scale kinetic (Monte Carlo, phase field, or chemical reaction rate theory) models, which subsequently lead to the development of constitutive models for predictive continuum scale models. Simulating long-time materials dynamics with reliable physical fidelity, thereby providing a predictive capability applicable outside limited experimental parameter regimes is the promise of such a computational multiscale approach. A critical need is the development of advanced and highly efficient algorithms to accurately model nucleation, growth, and coarsening in irradiated alloys that are kinetically controlled by elementary (diffusive) processes involving characteristic time scales between
50nm
Long-range defect transport and annihilation at sinks Gas diffusion and trapping
Radiation enhanced diffusion and induced segregation of solutes
Cascade aging and local solute redistribution
Defect recombination, He and H clustering, and generation migration Primary defect production and short-term annealing
Atomic–nm
Underlying microstructure (preexisting and evolving) impacts defect and solute fate
nm–µm
µm–mm
mm–m
Lengthscale Figure 1 Illustration of the length and time scales (and inherent feedback) involved in the multiscale processes responsible for microstructural changes in irradiated materials. Reproduced from Wirth, B. D.; Odette, G. R.; Marian, J.; Ventelon, L.; Young, J. A.; Zepeda-Ruiz, L. A. J. Nucl. Mater. 2004, 329–333, 103.
Kinetic Monte Carlo Simulations of Irradiation Effects
structural component level, and span 22 orders of magnitude in time from the subpicosecond of nuclear collisions to the decade-long component service lifetimes.1,2 Many different variables control the mix of nano/microstructural features formed and the corresponding degradation of physical and mechanical properties of nuclear fuels, cladding, and structural materials. The most important variables include the initial material composition and microstructure, the thermomechanical loads, and the irradiation history. While the initial material state and thermomechanical loading are of concern in all materials performancelimited engineering applications, the added complexity introduced by the effects of radiation is clearly the distinguishing and overarching concern for materials in advanced nuclear energy systems. At the smallest scales, radiation damage is continually initiated by the formation of energetic primary knock-on atoms (PKAs) primarily through elastic collisions with high-energy neutrons. Concurrently, high concentrations of fission products (in fuels) and transmutants (in cladding and structural materials) are generated and can cause pronounced effects in the overall chemistry of the material, especially at high burnup. The PKAs, as well as recoiling fission products and transmutant nuclei quickly lose kinetic energy through electronic excitations (that are not generally believed to produce atomic defects) and a chain of atomic collision displacements, generating a cascade of vacancy and self-interstitial defects. High-energy displacement cascades evolve over very short times, 100 ps or less, and small volumes, with characteristic length scales of 50 nm or less, and are directly amenable to MD simulations if accurate potentials are available. The physics of primary damage production in high-energy displacement cascades has been extensively studied with MD simulations.3–8 (see Chapter 1.11, Primary Radiation Damage Formation) The key conclusions from the MD studies of cascade evolution have been that (1) intracascade recombination of vacancies and self-interstitial atoms (SIAs) results in 30% of the defect production expected from displacement theory, (2) many-body collision effects produce a spatial correlation (separation) of the vacancy and SIA defects, (3) substantial clustering of the SIAs and to a lesser extent the vacancies occur within the cascade volume, and (4) high-energy displacement cascades tend to break up into lobes or subcascades, which may also enhance recombination.4–7 Nevertheless, it is the subsequent diffusional transport and evolution of the defects produced during displacement cascades, in addition to solutes and
395
transmutant impurities, that ultimately dictate radiation effects in materials and changes in material microstructure.1,2 Spatial correlations associated with the displacement cascades continue to play an important role in much larger scales as do processes including defect recombination, clustering, migration, and gas and solute diffusion and trapping. Evolution of the underlying materials structure is thus governed by the time and temperature kinetics of diffusive and reactive processes, albeit strongly influenced by spatial correlations associated with the microstructure and the continuous production of new radiation damage. The inherently wide range of time scales and the ‘rare-event’ nature of many of the controlling mechanisms make modeling radiation effects in materials extremely challenging and experimental characterization often unattainable. Indeed, accurate models of microstructure (point defects, dislocations, and grain boundaries) evolution during service are still lacking. To understand the irradiation effects and microstructure evolution to the extent required for a high fidelity nuclear materials performance model will require a combination of experimental, theoretical, and computational tools. Furthermore, the kinetic processes controlling defect cluster and microstructure evolution, as well as the materials degradation and failure modes may not entirely be known. Thus, a substantial challenge is to discover the controlling processes so that they can be included within the models to avoid the detrimental consequences of in-service surprises. High performance computing can enable such discovery of class simulations, but care must also be taken to assess the accuracy of the models in capturing critical physical phenomena. The remainder of this chapter will thus focus on a description of KMC modeling, along with a few select examples of the application of KMC models to predict irradiation effects on materials and to identify opportunities for additional research to achieve the goal of accelerating the development of advanced computational approaches to simulate nucleation, growth, and coarsening of microstructure in complex engineering materials.
1.14.3 KMC Modeling The Monte Carlo method was originally developed by von Neumann, Ulam, and Metropolis to study the diffusion of neutrons in fissionable material on the Manhattan Project9,10 and was first applied to simulate radiation damage of metals more than
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Kinetic Monte Carlo Simulations of Irradiation Effects
40 years ago by Besco,11 Doran,12 and later Heinisch and coworkers.13,14 Monte Carlo utilizes random numbers to select from probability distributions and generate atomic configurations in a stochastic process,15 rather than the deterministic manner of MD simulations. While different Monte Carlo applications are used in computational materials science, we shall focus our attention on KMC simulation as applied to the study of radiation damage. The KMC methods used in radiation damage studies represent a subset of Monte Carlo (MC) methods that can be classified as rejection-free, in contrast with the more classical MC methods based on the Metropolis algorithm.9,10 They provide a solution to the Master Equation which describes a physical system whose evolution is governed by a known set of transition rates between possible states.16 The solution proceeds by choosing randomly among the various possible transitions and accepting them on the basis of probabilities determined from the corresponding transition rates. These probabilities are calculated for physical transition mechanisms as Boltzmann factor frequencies, and the events take place according to their probabilities leading to an evolution of the microstructure. The main ingredients of such models are thus a set of objects (which can resolve to the atomic scale as atoms or point defects) and a set of reactions or (rules) that describe the manner in which these objects undergo diffusion, emission, and reaction, and their rates of occurrence. Many of the KMC techniques are based on the residence time algorithm (RTA) derived 50 years ago by Young and Elcock17 to model vacancy diffusion in ordered alloys. Its basic recipe involves the following: for a system in a given state, instead of making a number of unsuccessful attempts to perform a transition to reach another state, as in the case of the Metropolis algorithm,9,10 the average time during which the system remains in its state is calculated. A transition to a different state is then performed on the basis of the relative weights determined among all possible transitions, which also determine the time increment associated with the selected transition. According to standard transition state theory (see for instance Eyring18) the frequency Gx of a thermally activated event x, such as a vacancy jump in an alloy or the jump of a void can be expressed as: Ea ½1 GX ¼ nX exp kB T
where nX is the attempt frequency, kB is Boltzmann’s constant, T is the absolute temperature, and Ea is the activation energy of the jump. During the course of a KMC simulation, the probabilities of all possible transitions are calculated and one event is chosen at each time-step by extracting a random number and comparing it to the relative probability. The associated time-step length dt and average time-step length Dt are given by: lnr 1 and Dt ¼ P dt ¼ P GX GX n
½2
n
where r is a random number between 0 and 1. The RTA is also known as the BKL (Bortz, Kalos, Lebowitz) algorithm.19 Other techniques are possible, as described by Chatterjee and Vlachos.20 The basic steps in a KMC simulation can be summarized thus: 1. Calculate the probability (rate) for a given event to occur. 2. Sum the probabilities of all events to obtain a cumulative distribution function. 3. Generate a random number to select an event from all possible events. 4. Increase the simulation time on the basis of the inverse sum1 of the rates of all possible events 0 @Dt ¼ P w A, Ni Ri
where w is a random deviate that
i
assures a Poisson distribution in time-steps and N and R are the number and rate of each event i. 5. Perform the selected event and all spontaneous events as a result of the event performed. 6. Repeat Steps 1–4 until the desired simulation condition is reached.
1.14.4 KMC Modeling of Microstructure Evolution Under Radiation Conditions KMC models are now widely used for simulating radiation effects on materials.21–50 Advantages of KMC models include the ability to capture spatial correlations in a full 3D simulation with atomic resolution, while ignoring the atomic vibration time scales captured by MD models. In KMC, individual point defects, point defect clusters, solutes, and impurities are treated as objects, either on or off an underlying crystallographic lattice, and the evolution of these objects is modeled over time. Two general approaches have been used in KMC simulations, object KMC (OKMC) and event KMC (EKMC),35,36 which differ in the
Kinetic Monte Carlo Simulations of Irradiation Effects
397
treatment of time scales or step between individual events. Within the OKMC designation, it is also possible to further subdivide the techniques into those that explicitly treat atoms and atomic interactions, which are often denoted as atomic KMC (AKMC), or lattice KMC (LKMC), and which were recently reviewed by Becquart and Domain,45 and those that track the defects on a lattice, but without complete resolution of the atomic arrangement. This later technique is predominately referred to as object Monte Carlo and used in such codes as BIGMAC27 or LAKIMOCA.28 More recently, several algorithmic ideas have been identified that, in combination, promise to deliver breakthrough KMC simulations for materials computations by making their performance essentially independent of the particle density and the diffusion rate disparity, and these will be further discussed as outstanding areas for future research at the end of the chapter. KMC modeling of radiation damage involves tracking the location and fate of all defects, impurities, and solutes as a function of time to predict microstructural evolution. The starting point in these simulations is often the primary damage state, that is, the spatially correlated locations of vacancy, self-interstitials, and transmutants produced in displacement cascades resulting from irradiation and obtained from MD simulations, along with the displacement or damage rate which sets the time scale for defect introduction. The rates of all reaction–diffusion events then control the subsequent evolution or progression in time and are determined from appropriate activation energies for diffusion and dissociation; moreover, the reactions and rates of these reactions that occur between species are key inputs, which are assumed to be known. The defects execute random diffusion jumps (in one, two, or three dimensions depending on the nature of the defect) with a probability (rate) proportional to their diffusivity. Similarly, cluster dissociation rates are governed by a dissociation probability that is proportional to the binding energy of a particle to the cluster. The events to be performed and the associated time-step of each Monte Carlo sweep are chosen from the RTA.17,18 In these simulations, the events which are considered to take place are thus diffusion, emission, irradiation, and possibly transmutation, and their corresponding occurrence rates are described below.
per unit time and volume) of randomly distributed displacement cascades of different energies (5, 10, 20, . . . keV) as well as residual Frenkel pairs (FPs). New cascade debris are then injected randomly into the simulation box at the corresponding rate. The cascade debris can be obtained by MD simulations for different recoil energies T, or introduced on the basis of the number of FP expected from displacement damage theory. In the case of KMC simulation of electron irradiation, FPs are introduced randomly in the simulation box according to a certain dose rate, assuming most of the time that each electron is responsible for the formation of only one FP. This assumption is valid for electrons with energies close to 1 MeV (much lower energy electrons may not produce any FP, whereas higher energy ones may produce small displacement cascades with the formation of several vacancies and SIAs). The dose is updated by adding the incremental dose associated with the scattering event of recoil energy T, using the Norgett–Robinson–Torrens expression8 for the number of displaced atoms. In this model, the accumulated displacement per atom (dpa) is given by:
1.14.4.1
Usually the rates of diffusion can be obtained from the knowledge of the migration barriers which have to be known for all the diffusing ‘objects’; that is, for the point defects in AKMC, OKMC, and EKMC or the clusters in OKMC or EKMC. For isolated point
Irradiation Rate
The ‘irradiation’ rate, that is, the rate of impinging particles in the case of neutron and ion irradiation, is usually transformed into a production rate (number
Displacement per subcascade ¼
0:8T 2ED
½3
where T is the damage energy, that is, the fraction of the energy of the particle transmitted to the PKA as kinetic energy and ED is the displacement threshold energy (e.g., 40 eV for Fe and reactor pressure vessel (RPV) steels51). 1.14.4.2
Transmutation Rate
The rate of producing transmutations can also be included in KMC models, as deduced from the reaction rate density determined from the product of the neutron cross-section and neutron flux. Like the irradiation rate, the volumetric production rate is used to introduce an appropriate number of transmutants, such as helium that is produced by (n, a) reactions in the fusion neutron environment, where the species are introduced at random locations within the material. 1.14.4.3
Diffusion Rate
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Kinetic Monte Carlo Simulations of Irradiation Effects
defects, the migration barriers can be from experimental data, that is, from diffusion coefficients, or theoretically, using either ab initio calculations as described in Caturla et al.49 and Becquart and Domain50 or MD simulations as described in Soneda and Diaz de la Rubia.22 Since the migration energy depends on the local environment of the jumping species, it is generally not possible to calculate all of the possible activation barriers using ab initio or even MD simulations. Simpler schemes such as broken bond models, as described in Soisson et al.,52 Le Bouar and Soisson,53 and Schmauder and Binkele,54 are then used. Another kind of simpler model is based on the calculation of the system configurational energies before and after the defect jump. In this model, the activation energy is obtained from the final Ef and the initial Ei as follows: Ef Ei DE ¼ Ea0 þ ½4 2 2 where Ea0 is the energy of the moving species at the saddle point. The modification of the jump activation energy by DE represents an attempt to model the effect of the local environment on the jump frequencies. Indeed, detailed molecular statics calculations suggest that this represents an upper-bound influence of the effect,55 and although this is a very simplified model, the advantage is that this assumption maintains the detailed balance of jumps to neighboring positions. The system configurational energies Ei and Ef , as well as the energy of the moving species at the saddle point Ea0 can be determined using interatomic potentials as described in Becquart et al.,26 Bonny et al.,44 Wirth and Odette,55 and Djurabekova et al.56 when they exist. However, at present, this situation is only available for simple binary or ternary alloys. This approach allows one to implicitly take into account relaxation effects as the energy at the saddle point which is used in the KMC and is obtained after relaxation of all the atoms. The challenge in that case is the total number of barriers to be calculated, which is determined by the number of nearest neighbor sites included in the definition of the local atomic environment. Without considering symmetries, this number is sN, where s is the number of species in the system. In spite of using the fast techniques that were developed to find saddle points on the fly such as the dimer method,57 the nudged elastic band (NEB) method,58 or eigen-vector following methods,59 this number quickly becomes unmanageable. Ideally, the alternative should be to find patterns in the dependence of the energy barriers on the configuration. This is the Ea ¼ Ea0 þ
approach chosen by Djurabekova and coworkers,56 using artificial intelligence systems. For more complex alloys, for which no interatomic potentials exist, Ei and Ef can be estimated using neighbor pair interactions.60– 63 A recent example of the fitting procedure of a neighbor pair interactions model can be found in Ngayam Happy et al.63 A discussion of the two approaches applied to the Fe–Cu system has been published by Vincent et al.64 Also note that in the last 10 years, methods in which the possible transitions are found in some systematic way from the atomic forces rather than by simply assuming the transition mechanism a priori (e.g., activation–relaxation technique (ART) or dimer methods)65–68 have been devised. The accuracy of the simulations is thus improved as fewer assumptions are made within the model. However, interatomic potentials or a corresponding method to obtain the forces acting between atoms for all possible configurations is necessary and this limits the range of materials that can be modeled with these clever schemes. The attempt frequency (nX in eqn [1]) can be calculated on the basis of the Vineyard theory69 or can be adjusted so as to reproduce model experiments. 1.14.4.4
Emission/Dissociation Rate
The emission or dissociation rate is usually the sum of the binding energy of the emitted particle and its migration energy. As in the case of migration energy, the binding energies can be obtained using either experimental studies, ab initio calculations, or MD. As stated previously, three kinds of KMC techniques (AKMC, OKMC, and EKMC) have been used so far to model microstructural evolutions during radiation damage. In atomistic KMC, the evolution of a complex microstructure is modeled at the atomic scale, taking into account elementary atomic mechanisms. In the case of diffusion, the elementary mechanisms leading to possible state changes are the diffusive jumps of mobile point defect species, including point defect clusters. Typically, vacancies and SIAs can jump from one lattice site to another lattice site (in general first nearest neighbor sites). If foreign interstitial atoms such as C atoms or He atoms are included in the model as in Hin et al.,70,71 they lie on an interstitial sublattice and jump on this sublattice. In OKMC, the microstructure consists of objects which are the intrinsic defects (vacancies, SIAs, dislocations, grain boundaries) and their clusters (‘pure’ clusters, such as voids, SIA clusters, He or C clusters), as well as mixed clusters such as clusters containing both He atoms, solute/impurity atoms, and
Kinetic Monte Carlo Simulations of Irradiation Effects
interstitials, or vacancies. These objects are located at known (and traced) positions in a simulation volume on a lattice as in LAKIMOCA or a known spatial position as in BIGMAC and migrate according to their migration barriers. In the EKMC approach,72,73 the microstructure also consists of objects. The crystal lattice is ignored and objects’ coordinates can change continuously. The only events considered are those which lead to a change in the defect population, namely clustering of objects, emission of mobile species, elimination of objects on fixed sinks (surface, dislocation), or the recombination between vacancy and interstitial defect species. The migration of an object in its own right is considered an event only if it ends up with a reaction that changes the defect population. In this case, the migration step and the reaction are processed as a single event; otherwise, the migration is performed only once at the end of the EKMC time interval Dt. In contrast to the RTA, in which all rates are lumped into one total rate to obtain the time increment, in an EKMC scheme the time delays of all possible events are calculated separately and sorted by increasing order in a list. The event corresponding to the shortest delay, ts, is processed first, and the remaining list of delay times for other events is modified accordingly by eliminating the delay time associated with the particle that just disappeared, adding delay times for a new mobile object, etc. To illustrate the power of KMC for modeling radiation effects in structural materials and nuclear fuels, this chapter next considers two examples, namely the use of AKMC simulations to predict the coupled evolution of vacancy clusters and copper precipitates during low dose rate neutron irradiation of Fe–Cu alloys and the use of an OKMC model to predict the transport and diffusional release of fission product, silver, in tri-isotropic (TRISO) nuclear fuel. These two examples will provide more details about the possible implementations of AKMC and OKMC models.
1.14.5 Atomistic KMC Simulations of Microstructure Evolution in Irradiated Fe–Cu Alloys Cu is of primary importance in the embrittlement of the neutron-irradiated RPV steels. It has been observed to separate into copper-rich precipitates within the ferrite matrix under irradiation. As its role was discovered more than 40 years ago,74–76 Cu precipitation in a-Fe has been studied extensively
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under irradiation as well as under thermal aging using atom probe tomography, small angle neutron scattering, and high resolution transmission electron microscopy. Numerical simulation techniques such as rate theory or Monte Carlo methods have also been used to investigate this problem, and we next describe one possible approach to modeling microstructure evolution in these materials. The approach combines an MD database of primary damage production with two separate KMC simulation techniques that follow the isolated and clustered SIA diffusion away from a cascade, and the subsequent vacancy and solute atom evolution, as discussed in more detail in Odette and Wirth,21 Monasterio et al.,34 and Wirth et al.77 Separation of the vacancy and SIA cluster diffusional time scales naturally leads to the nearly independent evolution of these two populations, at least for the relatively low dose rates that characterize RPVembrittlement.1,78–81 The relatively short time (100 ns at 290 C) evolution of the cascade is modeled using OKMC with the BIGMAC code.77 This model uses the positions of vacancy and SIA defects produced in cascades obtained from an MD database provided by Stoller and coworkers82,83 and allows for additional SIA/ vacancy recombination within the cascade volume and the migration of SIA and SIA clusters away from the cascade to annihilate at system sinks. The duration of this OKMC is too short for significant vacancy migration and hence the SIA/SIA clusters are the only diffusing defects. These OKMC simulations, which are described in detail elsewhere,77 thus provide a database of initially ‘aged’ cascades for longer time AKMC cascade aging and damage accumulation simulations. The AKMC model simulates cascade aging and damage evolution in dilute Fe–Cu alloys by following vacancy – nearest neighbor atom exchanges on a bcc lattice, beginning from the spatial vacancy population produced in ‘aged’ high-energy displacement cascades and obtained from the OKMC to the ultimate annihilation of vacancies at the simulation cell boundary, and including the introduction of new cascade damage and fluxes of mobile point defects. The potential energy of the local vacancy, Cu–Fe, environment determines the relative vacancy jump probability to each of the eight possible nearest neighbors in the bcc lattice, following the approach described in eqn [4]. The unrelaxed Fe–Cu vacancy lattice energetics are described using Finnis–Sinclair N-body type potentials. The iron and copper potentials are from Finnis and Sinclair84 and Ackland et al.,85 respectively; and the iron–copper potential was developed
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by fitting the dilute heat of the solution of copper in iron, the copper vacancy binding energy, and the iron–copper [110] interface energy, as described elsewhere.55 Within a vacancy cluster, each vacancy maintains its identity as mentioned above, and while vacancy–vacancy exchanges are not allowed, the cluster can migrate through the collective motion of its constituent vacancies. The saddle point energy, which is Ea0 in eqn [4], is set to 0.9 eV, which is the activation energy for vacancy exchange in pure iron calculated with the Finnis–Sinclair Fe potential.84 The time (DtAKMC) of each AKMC sweep (or step) is determined by DtAKMC ¼ (nPmax)1, where Pmax is the highest total probability of the vacancy population and n is an effective attempt frequency. This is slightly different than the RTA, in which an event chosen at random sets the timescale as opposed to always using the largest probability as done here. In this work, n ¼ 1014 s1 to account for the intrinsic vibrational frequency and entropic effects associated with vacancy formation and migration, as used in the previous AKMC model by Odette and Wirth.21 As mentioned, the possible exchange of every vacancy (i) to a nearest neighbor is determined by a Metropolis random number test15 of the relative vacancy jump probability (Pi/Pmax) during each Monte Carlo sweep. Thus at least one, and often multiple, vacancy jumps occur during each Monte Carlo sweep, which is different from the RTA. Finally, as mentioned above, as the total probability associated with a vacancy jump depends on the local environment, the intrinsic timescale (DtAKMC) changes as a function of the number and spatial distribution of the vacancy population, as well as the spatial arrangement of the Cu atoms in relation to the vacancies. The AKMC boundary conditions remove (annihilate) a vacancy upon contact, but incorporate the ability to introduce point defect fluxes through the simulation volume that result from displacement cascades in neighboring regions as well as additional displacement cascades within the simulated volume. The algorithms employed in the AKMC model are described in detail in Monasterio et al.34 and the remainder of this section will provide highlights of select results. The AKMC simulations are performed in a randomly distributed Fe–0.3% Cu alloy at an irradiation temperature of 290 C and are started from the spatial distribution of vacancies from an 20 keV displacement cascade. The rate of introducing new cascade damage is 1.13 105 cascades per second, with a cascade vacancy escape probability of 0.60 and a
vacancy introduction rate of 1 104 vacancies per second, which corresponds to a damage rate of this simulation at 1011 dpa s1. Thus a new cascade (with recoil energy from 100 eV to 40 keV) occurs within the simulated volume (a cube of 86 nm edge length) every 8.8 104 s (1 day), while an individual vacancy diffuses into the simulation volume every 1 104 s (3 h). AKMC simulations have also been performed to study the effect of varying the cascade introduction rate from 1.13 103 to 1.13 107 cascades per second (dpa rates from 1 109 to 1 1013 dpa s1). The simulated conditions should be compared to those experienced by RPVs in light water reactors, namely from 8 1012 to 8 1011 dpa s1, and to model alloys irradiated in test reactors, which are in the range of 109–1010 dpa s1. Figures 2 and 3 show representative snapshots of the vacancy and Cu solute atom distributions as a function of time and dose at 290 C. Note, only the Cu atoms that are part of vacancy or Cu atom clusters are presented in the figure. The main simulation volume consists of 2 106 atoms (100a0 100a0 100a0) of which 6000 atoms are Cu (0.3 at.%). Figure 2 demonstrates the aging evolution of a single cascade (increasing time at fixed dose prior to introducing additional diffusing vacancies or new cascade), while Figure 3 demonstrates the overall evolution with increasing time and dose. The aging of the single cascade is representative of the average behavior observed, although the number and size distribution of vacancy-Cu clusters do vary considerably from cascade to cascade. Further, Figure 2 is representative of the results obtained with the previous AKMC models of Odette and Wirth21 and Becquart and coworkers,24,26 which demonstrated the formation and subsequent dissolution of vacancy-Cu clusters. Figure 3 represents a significant extension of that previous work21,24,26 and demonstrates the formation of much larger Cu atom precipitate clusters that results from the longer term evolution due to multiple cascade damage in addition to radiation enhanced diffusion. Figure 2(a) shows the initial vacancy configuration from an aged 20-keV cascade. Within 200 ms at 290 C, the vacancies begin to diffuse and cluster, although no vacancies have yet reached the cell boundary to annihilate. Eleven of the initial vacancies remain isolated, while thirteen small vacancy clusters rapidly form within the initial cascade volume. The vacancy clusters range in size from two to six vacancies. At this stage, only two of the vacancy clusters are associated with copper atoms, a divacancy
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Figure 2 Representative vacancy (red circles) and clustered Cu atom (blue circles) evolution in an Fe–0.3% Cu alloy during the aging of a single 20 keV displacement cascade, at (a) initial (200 ns), (b) 2 ms, (c) 48 ms, (d) 55 ms, (e) 83 ms, and (f) 24.5 h.
cluster with one Cu atom and a tetravacancy cluster with two Cu atoms. From 200 ms to 2 ms, the vacancy cluster population evolves by the diffusion of isolated vacancies through and away from the cascade region, and the emission and absorption of isolated vacancies in vacancy clusters, in addition to the diffusion of the small di-, tri-, and tetravacancy clusters. Figure 2(b) shows the configuration about 2 ms after the cascade. By this time, 14 of the original vacancies have diffused to the cell boundary and annihilated, while 38 vacancies remain. The vacancy distribution includes six isolated vacancies and seven vacancy clusters, ranging in size from two divacancy clusters to a ten vacancy cluster. The number of nonisolated copper atoms has increased from 223 in the initial random distribution to 286 following the initial 2 ms of cascade aging.
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Figure 3 Representative vacancy (red) and clustered Cu atom (blue) evolution in an Fe–0.3% Cu alloy with increasing dose at (a) 0.2 years (97 udpa), (b) 0.6 years (0.32 mdpa), (c) 2.1 years (1.1 mdpa), (d) 4.0 years (2.0 mdpa), (e) 10.7 years (4.4 mdpa), and (f) 13.7 years (5.3 mdpa).
The evolution from 2 to 48.8 ms involves the diffusion of isolated vacancies and di- and trivacancy clusters, along with the thermal emission of vacancies from the di- and trivacancy clusters. Over this time, 7 additional vacancies have diffused to the cell boundary and annihilated, and 20 additional Cu atoms have been incorporated into Cu or vacancy clusters. Figure 2(c) shows the vacancy and Cu cluster population at 48.8 ms, which now consists of three isolated vacancies and four vacancy clusters, including a 4V–1Cu cluster, a 6V–4Cu cluster, a 7V cluster, and an 11V–1Cu cluster. Figure 2(d) and 2(e) shows the vacancy and Cu cluster population at 54.8 and 82.8 ms, respectively. During this time, the total number of vacancies has been further reduced from 31 to 21 of the original 52 vacancies, the vacancy cluster
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population has been reduced to three vacancy clusters (a 4V–1Cu, 7V, and 9V–1Cu), and 30 additional Cu atoms have incorporated into clusters because of vacancy exchanges. Over times longer than 100 ms, the 4V–1Cu atom cluster migrates a short distance on the order of 1 nm before shrinking by emitting vacancies, while the 7V and 9V–1Cu cluster slowly evolve by local shape rearrangements which produces only limited local diffusion. Both the 7V and 9V–1Cu cluster are thermodynamically unstable in dilute Fe alloys at 290 C and ultimately will shrink over longer times. The vacancy and Cu atom evolution in the AKMC model is now governed by the relative rate of vacancy cluster dissolution, as determined from the ‘pulsing’ algorithm, and the rate of new displacement damage and the diffusing supersaturated vacancy flux under irradiation. Figure 2(f ) shows the configuration about 8.8 104 s (24 h) after the initial 20 keV cascade. Only 17 vacancies now exist in the cell, an isolated vacancy which entered the cell following escape from a 500 eV recoil introduced into a neighboring cell plus two vacancy clusters, consisting of 7V–1Cu and 9V–1Cu. Three hundred and forty-five Cu atoms (of the initial 6000) have been removed from the supersaturated solution following the initial 24 h of evolution, mostly in the form of di- and tri-Cu atom clusters. Figure 3(a) shows the configuration at about 0.1 mdpa (0.097 mdpa) and a time of 7.1 106 s (82 days). Ten vacancies exist in the simulation cell, consisting of eight isolated vacancies and one 2V
cluster, while 807 Cu atoms have been removed from solution in clusters, although the Cu cluster size distribution is clearly very fine. The majority of Cu clusters contain only two Cu atoms, while the largest cluster consists of only five Cu atoms. Figure 3(b) shows the configuration at a dose of 0.33 mdpa and time of 2.1 107 s (245 days). Only one vacancy exists in the simulation volume, while 1210 Cu atoms are now part of clusters, including 12 clusters containing 5 or more Cu atoms. Figure 3(c) shows the evolution at 1 mdpa and 7.2 107 s (2.3 years). Again, only one vacancy exists in the simulation cell, while 1767 Cu atoms have been removed from supersaturated solution. A handful of well-formed spherical Cu clusters are visible, with the largest containing 13 Cu atoms. With increasing dose, the free Cu concentration in solution continues to decrease as Cu atoms join clusters and the average Cu cluster size grows. Figure 3(d) and 3(e) shows the clustered Cu atom population at about 2 and 4.4 mdpa, respectively. The growth of the Cu clusters is clearly evident when Figure 3(d) and 3(e) is compared. At a dose of 4.4 mdpa, 48 clusters contain more than 10 Cu atoms, and the largest cluster has 28 Cu atoms. The accumulated dose of 5.34 mdpa is shown in Figure 3(f ). At this dose, more than one-third of the available Cu atoms have precipitated into clusters, the largest of which contains 42 Cu atoms, corresponding to a precipitate radius of 0.5 nm. Figure 4 shows the size distribution of Cu atom clusters at 5.34 mdpa, corresponding to the configuration shown in Figure 3(f ). The vast majority of the
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Cu clusters consist of di-, tri-, tetra-, and penta-Cu atom clusters. However, as shown in the inset of Figure 4 and as visible in Figure 3(f ), a significant number of the Cu atom clusters contain more than five Cu atoms. Indeed, 29 clusters contain 15 or more Cu atoms (a number density of 1.2 1024 m3), which corresponds to a cluster containing a single atom with all first and second nearest neighbor Cu atoms and a radius of 0.29 nm. An additional 45 clusters contain at least nine Cu atoms (atom þ all first nearest neighbors), while 9 clusters contain 23 or more atoms (number density of 3.8 1023 m3). This AKMC simulation is currently continuing to reach higher doses. However, the initial results are consistent with experimental observations and show the formation of a high number density of Cu atom clusters, along with the continual formation and dissolution of 3D vacancy-Cu clusters. Figure 5 shows a comparison of varying the dose rate from 109 to 1013 dpa s1. Each simulation was performed at a temperature of 290 C and introduced additional vacancies into the simulation volume at the rate of 104 s1. The effect of increasing dose
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(c) Figure 5 Comparison of the representative vacancy (red) and clustered Cu atom (blue) population at a dose of 1.9 mdpa and 290 C as a function of dose rate, at (a) 1011 dpa s1, (b) 109 dpa s1, (c) 1013 dpa s1.
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rate at an accumulated dose of 1.9 mdpa is especially pronounced when comparing Figure 5(c) (1013 dpa s1) with Figure 5(a) (1011 dpa s1) and Figure 5(b) (109 dpa s1). At the highest dose rate, a substantially higher number density of small 3D vacancy clusters is observed, which are often complexed with one or more Cu atoms. Vacancy cluster nucleation occurs during cascade aging (as described in Figure 2) and is largely independent of dose rate, but cluster growth is dictated by the cluster(s) thermal lifetime at 290 C versus the arrival rate of additional vacancies, which is a strong function of the damage rate and vacancy supersaturation under irradiation. Thus, the higher dose rates produce a larger number of vacancies arriving at the vacancy cluster sinks, resulting in the noticeably larger number of growing vacancy clusters. Also, there is a corresponding decrease in the amount of Cu removed from the solution by vacancy diffusion. In contrast, the effect of decreasing dose rate is greatly accelerated Cu precipitation. Already at 1.9 mdpa, a number of large Cu atom clusters exist at a dose rate of 1013 dpa s1, with the largest containing 35 Cu atoms, as shown in Figure 5(c). The increased Cu clustering caused by a decrease in dose rate results from a reduction in the number of cascade vacancy clusters, which serve as vacancy sinks. Thus, a higher number of free or isolated vacancies are available to enhance Cu diffusion required for the clustering and precipitation of copper. While these flux effects are anticipated and have been predicted in rate theory calculations performed by Odette and coworkers,78,79 the spatial dependences of cascade production and microstructural evolution, in addition to correlated diffusion and clustering processes involving multiple vacancies and atoms are more naturally modeled and visualized using the AKMC approach. While the results just presented in Figures 2–5 have shown the formation of subnanometer Cu-vacancy clusters and larger growing Cu precipitate clusters that result from AKMC simulations, which only consider vacancy-mediated diffusion, Becquart and coworkers have shown that Cu atoms in tensile positions can trap SIAs and therefore the Cu clustering behavior may also be influenced by interstitialmediated transport. Ngayam Happy and coworkers63,86 have developed another AKMC model to model the behavior of FeCu under irradiation. In this model, diffusion takes place via both vacancy and selfinterstitial atoms jumps on nearest neighbor sites. The migration energy of the moving species is also determined using eqn [4], where the reference activation
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energy Ea0 depends only on the type of the migrating species. Ea0 has been set equal to: the ab initio vacancy migration energy in pure Fe when a vacancy jumps towards an Fe atom (0.62 eV); the ab initio solute migration energy in pure Fe when a vacancy jumps towards a solute atom (0.54 eV for Cu); and the ab initio migration-60 rotation energy of the migrating atom in pure Fe when a dumbbell migrates (0.31 eV). Ei and Ef are determined using pair interactions, according to the following equation: X X eðiÞ ðSj Sk Þ þ Edumb ½5 E¼ i¼1;2
j
where i equals 1 or 2 and corresponds to first or second nearest neighbor interactions, respectively, and where j and k refer to the lattice sites and Sj (respectively Sk) is the species occupying site j (respectively k): Sj in {X, V} where X ¼ Fe or Cu. A more detailed description of the model can be found in Ngayam Happy et al.63 In this study, various Cu contents were simulated (0, 0.18, 0.8, and 1.34 at.%) at three different temperatures (300, 400, and 500 C). Without going into too much detail here, one can state that these AKMC simulation results are qualitatively similar to those presented in Figures 2–5, which showed the formation of small, vacancy-solute clusters and copper enriched cluster/precipitate formation at 300 C. Similarly, the effect of decreasing dose rate in high Cu content alloys was also found to accelerate Cu precipitation. This model does show that the formation of the Cu clusters/precipitates during neutron irradiation takes place via two different mechanisms depending upon the Cu concentration. In a highly Cu supersaturated matrix, precipitation is accelerated by irradiation, whereas in the case of low Cu contents, Cu precipitates form by induced segregation on vacancy clusters. The influence of temperature was investigated for an Fe–0.18 wt% Cu alloy irradiated at a flux of 2.3 105 dpa s1. At 400 and 500 C, neither Cu precipitates nor Cu-vacancy clusters were formed, in agreement with the results of Xu et al.87 At these temperatures, the model indicates that the vacancy clusters are not stable and induced segregation is thus hindered. Another interesting result obtained with this model is that the presence of Cu atoms in the matrix was found to decrease the point defect cluster sizes because of the strong interactions of Cu with both vacancies and SIAs.
1.14.6 OKMC Example: Ag Fission Product Diffusion and Release in TRISO Nuclear Fuel The second example demonstrates mesoscale KMC model simulations of the diffusion of silver (Ag) through the pyrolytic carbon and silicon carbide containment layers of a TRISO nuclear fuel particle. The model atomically resolves Ag, but provides a nonatomic, mesoscale medium of carbon and silicon carbide that includes a variety of defect features including grain boundaries, the carbon–silicon carbide interfaces, cracks, precipitates, and nanocavities. These defect features can serve as either fast diffusional pathways or traps for the migrating silver. The model consists of a 2D slab geometry incorporating the pyrolytic carbon and silicon carbide layers, with incident silver atoms placed at the innermost pyrolytic carbon layer, as described in more detail in Meric de Bellefon and Wirth.88 The key input parameters to the model (diffusion coefficients, trap binding energies, interface characteristics) are determined from available experimental data, or parametrically varied, until more precise values become available from lower length scale modeling. The predicted results, in terms of the time/temperature dependence of silver release during postirradiation annealing and the variability of silver release from particle to particle have been compared to available experimental data from the German High-Temperature Reactor (HTR) Fuel Program,89 as shown below in Figure 7, and studies performed by the Japan Atomic Energy Research Institute ( JAERI).90 Figure 6 presents KMC simulation results, which shows the effect of different grain geometries in SiC on silver release during postirradiation annealing. In this model, the grains are considered to have a rectangular geometry. The smaller dimension is parallel to the interfaces and has a fixed length of 1 mm. The longer dimension, parallel to the radial direction in a TRISO fuel particle, has a variable length that is uniformly distributed among grains over a range from 1 to 40 mm, as shown in the upper plot of Figure 6. Such a grain distribution mimics a highly columnar structure, as is often observed experimentally.91 The diffusion coefficient for silver transport within the grain boundaries has been assumed to be three orders of magnitude higher than in bulk SiC. As expected, within this model, the presence of grains provides fast diffusion paths for silver transport and accelerates the released fraction. Adding a
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Figure 6 Kinetic Monte Carlo simulations of the fractional release of Ag through the pyrolytic carbon and silicon carbide layers at 1700 C, which demonstrate the influence of the grain geometry in SiC on silver release. The rectangular to columnar grains have a height of 1 mm and width in the range of 1–40 mm (along the entire SiC layer).
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AVR experiments: AVR 82/9 (1600 ⬚C) AVR 74/11 (1700 ⬚C) AVR 76/18 (1800 ⬚C) Simulation: Particle 23 (1600 ⬚C) Particle 24 (1700 ⬚C) Particle 25 (1800 ⬚C)
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Heating time (h) Figure 7 Kinetic Monte Carlo simulations of the fractional release of silver, as compared with the measured fractional release of silver at 1600, 1700, and 1800 C. The microstructure of the 1600 and 1700 C simulations is identical and contains an isotropic grain geometry in silicon carbide that consists of 1-mm long square grains with a grain boundary diffusivity 100 higher than in bulk. The microstructure of the 1800 C simulation is similar except that the SiC grains characteristics have been modified to match the measured release at 1800 C. Faster transport through grain boundaries is needed, which is obtained by implementing a highly columnar structure with grains that are 0.5 mm wide and with (radial) lengths between 10 and 40 mm, and a high grain boundary diffusivity that is 2000 times higher than in bulk.
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columnar structure in those grains further increases the release of silver; the released fraction increases from 50 to 80% when the grain distribution shifts from an isotropic structure (grains are 1 mm squares) to a highly columnar structure (length of grains uniformly distributed from 1 to 40 mm) after 270 h of heating at 1700 C. Figure 7 presents KMC simulation results of the transport of silver through a given PyC/SiC/PyC microstructure during postirradiation thermal annealing at 1600, 1700, and 1800 C, as well as results from three German experimental release measurements performed at annealing temperatures of 1600, 1700, and 1800 C. The simulated microstructures include reflective interfaces, trapping cavities, and a grain boundary structure in the SiC layer. The microstructures for the 1600 and 1700 C simulations (particle 23 and 24) are identical and contain an isotropic grain geometry in SiC that consists of 1-mm long square grains with a grain boundary diffusivity 100 higher than in bulk. In the 1800 C simulation (particle 25), the SiC grain characteristics are varied to match the measured release at 1800 C. Faster transport through grain boundaries is required to match the experimental results, which is obtained by implementing a highly columnar structure in which the grains are 0.5 mm-wide and have a radial length between 10 and 40 mm into SiC, as well as a much higher grain boundary diffusivity that is 2000 times higher than in bulk.
1.14.7 Some Limits of KMC Approaches AKMC is a versatile method that can be used to simulate the evolution of materials with complex microstructure at the atomic scale by modeling the elementary atomic mechanisms. It has been used extensively to study phase transformations such as precipitation, phase separation, and/or ordering in many systems, as discussed in a recent review.92 Despite the fact that the algorithm is fairly simple, the method is most of the time nontrivial to implement in the case of realistic materials (as opposed to AB alloys for instance). Indeed, the determination of the total potential energy of the system, that is, the construction of the cohesive model when the chemistry of the system under study is complex and involves many species or a complex crystallographic structure, is difficult to obtain. Furthermore, the knowledge of all the possible events and the rates at which they
occur, that is, the possible migration paths as well as their energies is nontrivial. On rigid lattices, the migration paths are more obvious to determine and cluster expansion type methods may be extended to determine the saddle point energies as a function of the local chemical environment. This can, however, take a very large amount of calculation time when there is a drastic difference in the local environment. Furthermore, complicated correlated motions such as the adatom diffusion on the (100) surfaces of fcc metals which occurs by a two-atom concerted displacement, in which the adatom replaces a surface atom, which in turn becomes an adatom, cannot be modeled within the simple scheme usually followed in AKMC of jumps to 1nn neighbor sites. Another drawback is that to be efficient, it is tempting to use rigid lattices as a large number of KMC steps have to be performed. This can lead to an approximate (or even completely unrealistic) treatment of microstructure elements such as incoherent carbide precipitates, SIA clusters, or interstitial dislocation loops. Note, however, that it is possible to perform off-lattice AKMC, which will of course require more time consuming simulations, as proposed recently by Mason et al.93 to investigate phase transformation in Al–Cu–Mg alloys. The authors noticed that the use of flexible lattices instead of rigid ones affected the mobility of the vacancies as well as the driving force of the reaction and therefore the rate at which phase separation took place. Furthermore, note that off-lattice AKMC also requires an equilibrium continuous cohesive model, which is difficult to build for multicomponent alloys. At the moment, OKMC methods have been mostly used to investigate the annealing of the primary damage as in Heinisch and Singh14 or Domain et al.28 or the effect of temperature change in the damage accumulation,94 but its strongest contribution in the field seems to be the study of parameters or assumptions such as the motion, 3D versus 1D motion, mobility of the SIA clusters,95–98 or corroboration of theoretical assumptions such as the analytical description of the sink strength.99 They have been used also to model as well as reexamine simple experiments such as He desorption in W100 or in Fe101 as well as the influence of C in isochronal annealing experiments. It can also be used to determine the production rate or source term (i.e., the ‘irradiation flux’) in mean field rate theory (MFRT) models, as discussed in the chapter on MFRT. As no spatial correlation is explicitly considered in these techniques, the source term has to take into account intracascade agglomeration and recombination. The
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amount of agglomeration can be obtained by annealing the cascade debris using OKMC.102 In the OKMC, the evolution of individual objects is simulated on the basis of time scales that encompass individual atomic diffusive jumps, dominated by the very fast events. This method is not efficient at high temperatures and/or high doses. The difficulty is the inability to model sufficiently high doses necessary for macroscopic materials behavior due to the focus on fast dynamics. The time-step between events is much longer in EKMC models, which require that a reaction (e.g., clustering among like defects, annihilation among opposite defects, cluster dissolution, or new cascade introduction) occur within each Monte Carlo sweep. EKMC can therefore simulate much longer times and therefore simulate materials evolution over higher doses. It is most efficient when few objects are present in the simulation box. But questions relate to whether the time-steps are too large to reliably capture the underlying fast dynamics and whether the assumed binary interactions are sufficient to reliably calculate interaction probabilities. Further, EKMC models developed to date have not included all of the relevant microstructural evolution mechanisms, but they do represent an interesting approach in the limit of long time-step Monte Carlo simulations.
1.14.8 Advanced KMC Methods To overcome some of the limitations described above, two techniques have been recently derived: the first-passage Green’s function and synchronous parallel KMC. The first-passage Green’s function approach has been successfully used in various subareas of computational science but so far has escaped the widespread attention of materials scientists. Preliminary work indicates that constructive use of the first-passage Green’s function approach for modeling radiation microstructures is possible46,47 but will require considerable effort to develop a time-dependent formulation of the method. However, the potential payoff is well worth it: preliminary estimates show that it should be possible to boost the effective performance of the (exact) Monte Carlo simulations by several orders of magnitude. The speedup of the simulation with the first-passage Green’s function approach can be estimated from a rough argument based on the number of events required to process the diffusion of a vacancy from one void to another in Oswald coarsening. For example, voids in irradiated
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alloys are separated by 0.2 mm. But as the atomic jump distance is typically on the order of 0.25 nm, the ratio between the required diffusion length and the atomic jump distance is around 103. The ripening process consists mainly of vacancies detaching from one void and diffusing to a neighbor. If this is done with a direct hop simulation, then 106 random walk diffusion hop events would be required, from vacancy emission to absorption. Each such event requires the generation of one or more random numbers and changes in bookkeeping tables that store current positions for each defect. Using the first-passage Green’s function algorithm, the vacancy in most cases will reach the vicinity of a void in several (10–30) steps, each of which will require <10 times the number of calculations for a simple hop. Thus, it is possible to conclude that the simulation of the ripening of voids (which is similar to modeling radiation-induced precipitation processes) at this spacing would require about three orders of magnitude less computer time than the current KMC programs. At the moment, the first-passage Green’s function KMC appears to work very well for some specific cases such as the one mentioned above, but when one tries to model a more realistic case such as the continuous introduction of displacement cascades in which all the defects are very close to each other and diffuse with very different diffusivities, the possible ‘protective domains’ become very small and the technique is not very efficient. One additional difficulty with KMC simulations is the fact that the current state-of-the-art simulation codes utilize serial computing only. Thus, there exists a critical need to accelerate the maturation of multiscale modeling of fusion reactor materials, namely the development of advanced and highly efficient Monte Carlo algorithms for the simulation of materials evolution when controlling processes occur with characteristic time scales between 1012 and 100 s. There has recently been some activity associated with synchronous parallel KMC48; however, the problem of dealing with highly inhomogeneous regions and species diffusing at rates that are disparate by many of orders of magnitude tends to greatly reduce the parallel algorithm performance. Clearly, more effort is needed on the development of advanced algorithms for KMC simulations. Further, it is imperative that the algorithms developed be highly efficient in today’s massively parallel computing architectures. At the moment, no single KMC method can efficiently treat the complex microstructures and kinetic evolution associated with radiation effects in
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multi-component materials, nor efficiently balance the computational requirements to treat inhomogeneous domains consisting of very different defect densities. It is possible that a combination of different techniques in the course of a single simulation will be the most efficient pathway.
1.14.9 Summary and a Look at the Future of Nuclear Materials Modeling This chapter has attempted to illustrate the power of KMC for modeling radiation effects in structural materials and nuclear fuels, following an introduction and review of the Monte Carlo technique. Monte Carlo modeling, first developed by Metropolis and coworkers9,10 during the Manhattan project does provide a physically satisfying technique to simulate the stochastic evolution of defect evolution in materials science and in fact has been used to simulate irradiation effects on materials for four decades. There are three main types of KMC modeling used in irradiation effects, namely event Monte Carlo, object Monte Carlo, and atomistic Monte Carlo. This chapter has focused on describing the atomistic KMC and OKMC methods by providing two examples of successful KMC simulations to predict the coupled evolution of vacancy clusters and copper precipitates during low dose rate neutron irradiation of Fe–Cu alloys and the transport and diffusional release of the fission product, silver, in TRISO nuclear fuel. These examples clearly demonstrate the power and ability of KMC models to capture the spatial correlations that can be an important component of microstructural evolution in nuclear materials. Yet, the further widespread application of KMC models will require algorithmic developments that can more readily treat the wide range of time scales inherent in microstructural evolution and yet effectively incorporate the rare-event dynamics in integrating system performance to realistic time and irradiation dose exposures. Thus, the challenges that must be overcome in future nuclear materials modeling include: bridging the inherently multiscale time and length scales which control materials degradation in nuclear environments; dealing with the complexity of multicomponent materials systems, including those in which the chemical composition is continually evolving due to nuclear fission and transmutation; discovering the unknown to prevent technical surprises;
transcending ideal materials systems to engineer materials and components; and incorporating error assessments within each modeling scale and propagating the error through the scales to determine the appropriate confidence bounds on performance predictions. Successful overcoming of these challenges will result in nuclear materials performance models that can predict the properties, performance, and lifetime of nuclear fuels, cladding, and components in a variety of nuclear reactor types throughout the full life cycle, and provide the scientific basis for the computational design of advanced new materials. While the current chapter is focused on the KMC modeling methodology, it is important to note the challenges of predictive materials models of irradiation effects. High performance computing at the petascale, exascale, and beyond is a necessary and indeed critical tool in resolving these challenges, yet it is important to realize that exascale computing on its own will not be sufficient. This is best recognized from a simple example considering the computational degrees of freedom in a MD simulation. Assuming that reliable, multicomponent interatomic potentials existed for the nuclear fuel rod and cladding in a nuclear power plant and that a constant time-step of 2 1015 s could sufficiently capture the physics of high-energy atomic collisions to conserve energy; then to simulate 1 day of evolution of 1 cm tall, 1 cm diameter fuel pellet clad and zirconium clad would require 6 1022 atoms for 4 1019 timesteps. For comparison, the LAMPPS MD code using classical force fields has been benchmarked with 40 billion atoms (4 1010) and 100 time-steps on 10 000 processors of the RedStorm at Sandia National Laboratory with a wall clock time of 980 s and on 64 000 processors of the BlueGene Light at Lawrence Livermore National Laboratory with a wall clock time of 585 s.103 Thus, even assuming optimistic scaling and parallelization, brute force atomistic simulation of the first full power day of a nuclear fuel pellet in a reactor by MD will remain well beyond the reach of high performance computing capabilities for the next decade.
References 1. Odette, G. R.; Wirth, B. D.; Bacon, D. J.; Ghoneim, N. M. MRS Bull. 2001, 26, 176. 2. Wirth, B. D.; Odette, G. R.; Marian, J.; Ventelon, L.; Young, J. A.; Zepeda-Ruiz, L. A. J. Nucl. Mater. 2004, 329–333, 103.
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1.15
Phase Field Methods
P. Bellon University of Illinois at Urbana-Champaign, Urbana, IL, USA
ß 2012 Elsevier Ltd. All rights reserved.
1.15.1
Introduction
411
1.15.2 1.15.3 1.15.4 1.15.4.1 1.15.4.2 1.15.4.2.1 1.15.4.2.2 1.15.4.2.3 1.15.4.2.4 1.15.5 References
General Principles and Applications of PF Modeling Quantitative PF Modeling PF Modeling Applied to Materials Under Irradiation Challenges Specific to Alloys Under Irradiation Examples of PF Modeling Applied to Alloys Under Irradiation Effects of ballistic mixing on phase-separating alloy systems Coupled evolution of composition and chemical order under irradiation Irradiation-induced formation of void lattices Irradiation-induced segregation on defect clusters Conclusions and Perspectives
412 418 420 420 421 421 423 426 427 428 430
Abbreviations 1D CVM KMC ME PF PFM SIA
One-dimensional Cluster variation method Kinetic Monte Carlo Master equation Phase field Phase field model Self-interstitial atom
1.15.1 Introduction Electronic and atomistic processes often dictate the pathways of phase transformations and microstructural evolution in solid materials. For quantitative modeling of these transformations and evolution, it is thus effective, and sometime necessary, to rely on methods using some representation of atoms and of their dynamics, as for instance in molecular dynamics simulations (see Chapter 1.09, Molecular Dynamics) and atomistic Monte Carlo simulations (see Chapter 1.14, Kinetic Monte Carlo Simulations of Irradiation Effects). While these atomistic methods can now simulate quite accurately the evolution of specific alloy systems, these simulations are nevertheless limited to small length scales, from a few to 100 nm. Molecular dynamics is furthermore limited to small time scales, typically in the nanosecond range, although in some cases, new developments
have made it possible to obtain atomistic simulations at much longer times (see Chapter 1.14, Kinetic Monte Carlo Simulations of Irradiation Effects). An alternative modeling approach is to replace the many microscopic degrees of freedom of the system of interest by the few mesoscopic variables that are sufficient to provide a realistic description. This approach has been widely used in many disciplines, and well-known examples are the Fourier and Fick equations, which describe the diffusive transport of heat and chemical species, respectively. This approach is also commonly used in modeling the evolution of point defects, in particular, during irradiation (see Chapter 1.13, Radiation Damage Theory and Sizmann1). The work of Cahn and Hilliard2– 5 and Landau and Lifshitz (see for instance Tole´dano and Toledano6) provided a way to include the contributions of interfaces to chemical evolution, thus making it possible to model heterogeneous and multiphase materials. Kinetic models based on these descriptions are broadly referred to as phase field (PF) methods, since the microstructure of a material is fully characterized by a few mesoscopic field variables such as concentration, magnetization, chemical order, or temperature. One key assumption of this approach is that the variables chosen to describe the state of the system vary smoothly across any interface or, in other words, that interfaces are diffuse. This assumption finds a natural justification in the theory of critical phenomena, since the
411
412
Phase Field Methods
interface thickness diverges at the critical temperature.7 Diffuse interface models offer some advantages over sharp interface models,8 in particular, for the modeling of complex microstructures. Furthermore, the PF approach can be extended to include macroscopic variables other than the local composition, making it possible to describe chemical order–disorder transitions, solid–liquid reactions, displacive transformations, and more recently dislocation glide. PF methods and applications have been recently reviewed by Chen,9 Emmerich,10 and Singer-Loginova and Singer.11 This chapter focuses on solid–solid phase transformations, with a particular emphasis on transformations and microstructural evolution relevant to irradiated materials. While conventional PF modeling lacks atomic resolution, the main interest in this technique comes from the fact that it can provide the evolution of large systems, exceeding the micrometer scale, over very long time scales, from seconds to centuries. Recent developments have led to the introduction of PF models (PFMs) that possess atomic resolution,12–26 the so-called PF crystal models. This model, which can be seen as a density functional theory for atoms, appears very promising, although at this time it is not clear whether it can reproduce correctly the discrete nature of pointdefect jumps from one lattice site to a neighboring lattice site. The PF crystal model is not covered in this chapter, so the interested reader should consult the above references. This chapter is organized as follows. Section 1.15.2 introduces the key concepts and steps employed in conventional, that is, phenomenological PF modeling, and provides some illustrative examples. Section 1.15.3 focuses on important recent developments toward quantitative PF modeling, whereby evolution equations are rigorously derived by coarse-graining a microscopic model. This approach provides a full treatment of fluctuations and thus makes it possible to study fluctuation-controlled reactions, such as nucleation of a second phase. The capability of PFMs to reach large time and length scales makes them an attractive tool for simulating the evolution of materials relevant to nuclear applications, in particular, for alloys subjected to irradiation. Applying PF modeling to these nonequilibrium materials, however, raises new challenges, as is discussed in Section 1.15.4.1. Some selected results of PF modeling applied to irradiated materials are presented in Section 1.15.4.2. Finally, conclusions and perspectives are given in Section 1.15.5.
1.15.2 General Principles and Applications of PF Modeling The first step in PF modeling lies in choosing and defining the fields of interest. These continuous variables are functions of space and time, and they are in most cases scalar fields, such as temperature, or the concentration of some chemical species of interest. In systems with solid–liquid interfaces, a phenomenological field variable is introduced in such a way that it varies continuously from 0 to 1 as one goes from a fully solid to a fully liquid phase. Multidimensional fields can be used as well, for instance, to describe the local composition of a multicomponent alloy, the local degree of chemical order, or the local crystallographic orientation of grains. These multidimensional fields may transform like vectors under symmetry operations, thus leading to a vectorial representation of the system and tensorial expressions for mobilities (as will be discussed later), but there are cases for which the multidimensional fields cannot be reduced to vectors.27 In all cases, an averaging procedure is necessary to define continuous field variables for systems that are intrinsically discrete at the atomic scale. Various averages can be used, including (1) a spatial average over representative volume elements, which will correspond to the cells used for evolving the PF variables; (2) a spatial and temporal average; or (3) a spatial and ensemble average. The spatial averaging method is used most often, although in many cases the exact conditions of the averaging procedure are not defined. Section 1.15.3 will cover a model where this coarse-graining is performed explicitly and rigorously. The last two averaging procedures are rarely explicitly invoked, although one of their advantages is that a smaller volume can be used for the spatial average, thanks to the additional averaging performed either in time or in the configuration space of a system ensemble. Turning now to the kinetic equations used to describe the evolution of these field variables, an important distinction is whether the field variable is conserved or nonconserved. For the sake of simplicity, the following discussion focuses on alloy systems. Let us consider two simple examples, one where the field variable is the local composition, C(r,t), in a binary A–B alloy system, and a second example, also for a binary alloy system, but this time with a fixed composition and where chemical ordering takes place. The degree of chemical order is described by the field S(r,t). For the sake of convenience, one
Phase Field Methods
may normalize that field such that S(r,t) ¼ 0 corresponds to a fully disordered state and S(r,t) ¼ 1 to a fully ordered state. The first field variable C(r,t) is globally conserved – assuming here that the system of interest is not exchanging matter with its environment. This imposes the constraint that the time evolution of the field variable at r is balanced by the divergence of the flux of species exchanged between the representative volume centered on r and the remainder of the system: @Cðr;t Þ ¼ rJ ðr;t Þ ½1 @t One then makes use of linear response theory in the context of thermodynamics of irreversible processes28 to linearly relate the flux J(r,t) to the driving force responsible for this flux. Here, this driving force is the gradient of the chemical potential mðr ; t Þ ¼ dF =dCðr ; t Þ, where F is the free energy of the system for a compositional field given by C(r,t). The resulting evolution equation is thus @Cðr;t Þ dF ¼ r Mr ½2 @t dCðr;t Þ where M is a mobility coefficient. In contrast, for the nonconserved order parameter S(r,t), its evolution is directly related to the free energy change as S(r,t) varies, so that by making use of linear response theory again @Sðr;t Þ dF ¼ L @t dSðr;t Þ
½3
where L is the mobility coefficient for the nonconserved field S(r,t). Two important consequences of eqns [2] and [3] are worth noticing. First, although all extrema of the system free energy (i.e., minima, maxima, saddle points) are stationary states, often in practice only the minima can be obtained at steady state due to numerical errors. Second, the stationary state reached from some initial state may not correspond to the absolute minimum of the free energy. In order to overcome this problem, noise can be added to transform these deterministic equations into stochastic (Langevin) equations, as will be discussed in Section 1.15.3. Following the work of Cahn and coworkers2–4 and Landau and Ginzburg,6 the free energy F is decomposed into a homogeneous contribution and an heterogeneous contribution. Treating the inhomogeneity contribution as a perturbation of a homogeneous state, one finds that, in the limit of small amplitude and long wavelength for this perturbation, the lowest order correction to the homogeneous free
413
energy is proportional to the square of the gradient of the field variable. For instance, returning to the simple example of an alloy described by the concentration field C(r,t), the total free energy can be written as ð ½4 F fCðr ; t Þg ¼ dV ½f ðCÞ þ kðrCÞ2 V
where f (C) is the free-energy density of a homogeneous alloy for the composition C, and k the gradient energy coefficient, which is positive for an alloy system with a positive heat of mixing. A similar expression can be used in the case of a nonconserved order parameter, for example, S(r,t), or more generally, in the case of an alloy described by nC conserved order parameters and nS nonconserved order parameters ð F ¼ dV f ðC1 ; C2 . . . CnC ; S1 ; S2 . . . SnS Þ V
þ
nC X p¼1
kp ðrCp Þ2 þ
nS X q¼1
q
ij ri Sq rj Sq
½5
An implicit summation over the indices i and j is assumed in the last term of eqn [5]. The number of nonzero and independent gradient energy coefficients q ij for the nonconserved order parameters is dictated by the symmetry of the ordered phase. Specific examples, for instance for the L12 ordered structure, can be found in Braun et al.27 and Wang et al.29 The free energy can also be augmented to include other contributions, in particular those coming from elastic fields using the elasticity theory of multiphase coherent solids pioneered by Khachaturyan,30 in the homogeneous modulus case approximation. This makes it possible to take into account the effect of coherent strains imposed by phase transformations or by a second phase, for example, a substrate onto which a thin film is deposited.31 Two important interfacial quantities, the excess interface free energy and the interface width, can be derived from eqn [5] for a system at equilibrium. We follow here the derivation given by Cahn and Hilliard.2 Considering the case of a binary alloy where two phases may form, referred to as a and b, and with respective B atom concentrations Ca and Cb, the existence of an interface between these two phases results in an excess free energy s ð s ¼ dV ½f ðCÞ þ kðrCÞ2 CmeB ð1 CÞmeA ½6 V
where meA and meB are the chemical potential of A and B species when the two phases a and b coexist at
414
Phase Field Methods
equilibrium. At equilibrium, this excess free energy is minimum. A homogeneous free energy Df referenced to the equilibrium mixture of a- and b-phases is introduced as Df ðCÞ ¼ f ðCÞ ½CmeB þ ð1 CÞmeA ¼ C½mB ðCÞ meB þ ð1 CÞ½mA ðCÞ meA ½7 (Note that the ‘D’ symbol in Df in eqn [7] does not refer to a Laplacian.) The variational derivative of this excess energy with respect to the concentration field is given by ds @Df @k ½8 ¼ 2kDC ðrCÞ2 dC @C @C At equilibrium, the excess free energy s is minimum, and the concentration field must be such that ds=dC ¼ 0. Thus, @Df @k @ ¼ 2kDC þ ðrCÞ2 ¼ ðkðrCÞ2 Þ ½9 @C @C @C Equation [9] must hold locally for any value of the concentration field along the equilibrium profile joining the a- and b-phases, and this can only be satisfied if kðrCÞ2 ¼ Df
½10
for all values of C(r). It is interesting to note that eqn [10] means that the equilibrium concentration profile is such that, at any point on this profile, the homogeneous and inhomogeneous contributions to the total free energy are equal. The interfacial excess free energy is thus given by ð ð ½11 s ¼ 2 dV Df ¼ 2 dV kðrCÞ2 V
V
This last integral over the spatial coordinates can be rewritten as an integral over the concentration field. Assuming a one-dimensional (1D) system for simplicity, Cðb
Cðb
dC krC ¼ 2
s¼2 Ca
dC
pffiffiffiffiffiffiffiffiffi kDf
½12
Ca
In order to proceed further, it is necessary to assume a functional shape for the concentration profile or for the homogeneous free energy Df. Expanding the free energy near the critical point Tc yields a symmetric double-well potential for the homogeneous free energy,2 which we write here as 2C 1 2 2C 1 2 Df ¼ Dfmax 1 1þ ½13 Cab Cab
with Cab ¼ Cb Ca ¼ 1 2Ca ¼ 2Cb 1. Using eqn [10], the equilibrium concentration profile can now be obtained: rffiffiffiffiffiffiffiffiffiffiffi ! Cab 2 Dfmax 1 tanh x þ ½14 CðxÞ ¼ 2 k Cab 2 Integration along this equilibrium profile from eqn [11] yields the interfacial energy 4 pffiffiffiffiffiffiffiffiffiffiffiffiffi s ¼ Cab kDfmax ½15 3 Furthermore, the width of the equilibrium profile we, which is defined as the length scale entering the argument of the hyperbolic tangent function in eqn [14], is given by rffiffiffiffiffiffiffiffiffiffiffi Cab k ½16 we ¼ 2 Dfmax In this conventional approach to PFMs, Dfmax and k are phenomenological coefficients. Equations [15] and [16] play an important role in assigning values to these coefficients for a specific alloy system. The excess interfacial energy may be known experimentally or it may be calculated separately, for instance by ab initio calculations.32 If the interfacial width we is also known, one can obtain Dfmax and k from an inverse solution of eqns [15] and [16] (note that Cab is given by the equilibrium phase diagram). Even if we is not known, values for Dfmax and k can be chosen to yield a prescribed value for s. In all cases, it is important to recognize that any microstructural feature that develops during the simulations is expressed in units of we. At elevated temperatures, as T ! Tc , s vanishes2 while we goes to infinity, and therefore, at high enough temperatures, interfaces are diffuse, thus meeting this essential requirement underlying the PF method. The PF eqns [2] and [3] are usually solved numerically on a uniform mesh with an explicit time integration, using periodic boundary conditions, when surface effects are not of interest. When the free energy contains an elastic energy contribution, it is quite advantageous to use semi-implicit Fourierspectral algorithms (see Chen9 and Feng et al.33 for details). Variable meshing can also be employed, in particular to better resolve interfaces when they tend to be sharp, for instance at low temperatures. A few examples selected from the literature serve to illustrate the capacity of PFMs to successfully reproduce a wide range of phenomena. In particular, Khachaturyan30 and his collaborators34–38 proposed a microelasticity theory of multiphase coherent solids,
Phase Field Methods
which has been widely used to include a strain energy in the overall free energy. A method for systems with strong elastic heterogeneity has been proposed by Hu and Chen,39 which includes higher order terms that are usually neglected in Khachaturyan’s approach. Figure 1 illustrates the anisotropic morphology of Al2Cu precipitates growing in an Al-rich matrix.32 Bulk-free energies were calculated using a mixedspace cluster expansion technique, with input from first-principle calculations for about 40 different ordered structures with full atomic relaxations. Interfacial energies were calculated at T ¼ 0 K from first-principle calculations as well, using configurations where the Al-rich solid solution and the tetragonal y0 -Al2Cu coexist. For the elastic strain energy calculations, the elastic constants of y0 -Al2Cu were calculated ab initio. An important feature of this system is that both elastic and interfacial energies are strongly anisotropic, and the PF approach makes it possible to include these anisotropies. Furthermore, when the high-aspect-ratio y0 -phase forms, its growth kinetics will be anisotropic as well, which can be included in a phenomenological way by introducing a dependence of the mobility on the orientation of the precipitate–matrix interface. Figure 1 illustrates that these three anisotropies, interfacial, elastic, and kinetic, are required to reproduce the morphology of y0 precipitates.
Isotropic
Figure 2 illustrates another effect of coherency stress on microstructural evolution, this time for an A1–L10 order–disorder transition in a Co–Pt alloy.40 The tetragonal distortion accompanying the ordering reaction leads to the formation of self-organized tweed patterns of coexisting (cubic) A1- and (tetragonal) L10-phases. As seen from Figure 2, the agreement between experimental and simulated microstructures is remarkable. Ni and Khachaturyan proposed recently that, in order to minimize elastic energy during transformations involving symmetry changes and lattice strain, a pseudospinodal decomposition is likely to take place, leading to 3D chessboard patterns.41 PF modeling has also been used extensively to study martensitic transformations,34–38,42–44 phase transformations in ferroelectrics45–57 (see also the recent review by Chen58 on that topic), transformations in thin films,47,59–65 grain growth and recrystallization,66–81 and microstructural evolution in the presence of cracks or voids.82–84 A recent extension of PFMs has been the inclusion of dislocations in the models,85–87 by taking advantage of the equivalence between dislocation loops and coherent misfitting platelet inclusions.88 This approach has been applied, for instance, to study the interaction between moving dislocations and solute atoms,89 or to study the influence of dislocation arrays on spinodal decomposition in thin films.61 Rodney et al.87 have pointed out,
Interface only
(b)
(a) Int + strain
Strain only
(c)
Int + strain + kinetics
Experiment
50 nm
50 nm
(d)
415
(e)
(f)
Figure 1 Phase field simulations of y0 -phase precipitation in Al–Si–Cu alloys at 450 C, illustrating that strain, interfacial, and kinetic anisotropies are required to reproduce experimental morphologies. Reprinted with permission from Vaithyanathan, V.; Wolverton, C.; Chen, L. Q. Phys. Rev. Lett. 2002, 88(12), 1255031–1255034. Copyright by the American Physical Society.
416
Phase Field Methods
(a)
(b)
(c) Time
(d)
(e)
(f)
Figure 2 Comparison between transmission electron microscopy experimental observations (a–c) and phase field modeling (d–f) of formation of chessboard pattern in Co39.5Pt60.5 cooled from 1023 K to (a) 963 K, (b) 923 K, and (c) 873 K. The scale bar corresponds to 30 nm. Reproduced from Le Bouar, Y.; Loiseau, A.; Khachaturyan, A. G. Acta Mater. 1998, 46(8), 2777–2788.
however, that the artificially wide dislocation cores required by the above approach lead to weak shortrange interactions. These authors have introduced a different PFM for dislocations, which allows for narrow dislocation cores. As an illustration of that model, Figure 3 shows the development of dislocation loops and their interaction with hard precipitates in a 3D g/g0 single crystal. It is interesting to note that dislocation loop initially expands by gliding in the soft g channels, until the local stresses are large enough for the dislocation to shear the hard g0 -phase. The above presentation of the PF equations leaves certain questions open. First, the maximum homogeneous free energy difference Dfmax involves the free energy of the unstable state separating the two minima at Ca and Cb. It is thus been questioned90 whether this quantity can be rigorously defined from thermodynamic principles. If one employs mean field techniques such as the cluster variation method (CVM)91–95 to derive the homogeneous free energy of an alloy, Dfmax is in fact very sensitive to the approximation used, and generally decreases as the size of the largest cluster used in the CVM increases.96 Kikuchi97–99 has argued that, in order to resolve this paradox, Df should not be considered as the free energy of any homogeneous state, but that it should be understood as the local contribution to the free energy of the system along the equilibrium composition profile.
The second set of questions relates to the gradient energy coefficients k and in eqn [5]. In many applications of PF modeling, these coefficients are taken as phenomenological constants that can be adjusted at will, as long as the microstructures are scaled in units of k1/2 or 1/2. Such an approach, however, is problematic for many reasons. First, when one scalar field variable is employed, for instance C(r,t), a regular solution model,2,100 or equivalently a Bragg–Williams approximation,101 establishes that the gradient energy coefficient is not arbitrary but that it is directly proportional to the interaction energy between atoms, that is, to the heat of mixing of the alloy. Furthermore, in the most general case, k should in fact be composition and temperature dependent. Starting from an atomistic model, rigorous calculations of k are possible by monitoring the intensity of composition fluctuations as a function of their wave vector, and using the fluctuation–dissipation theorem.100 In the case of a simple Ising-like binary alloy, it is observed that k varies as Cð1 CÞ, where C is the local composition of the alloy.100 Furthermore, when more than one field variable is employed, care should be taken to consider all possible contributions of field heterogeneities to the free energy of the system, as the different fields may be coupled. Symmetry considerations are important to identify the nonvanishing terms, but it
Phase Field Methods
(a)
(b)
the interfacial anisotropy can be fitted to experiments or to atomistic simulations. Let us now return to the mobility coefficients M and L introduced in eqns [2] and [3]. For the sake of simplicity, many PF calculations are performed while assigning an arbitrary constant value to these coefficients. An improvement can be made by relating the mobility to a diffusion coefficient. In the case of M, for instance, in order to make eqn [1] consistent with Fick’s second law for an ideal binary alloy system, one should choose M¼
(c)
(d)
(e)
(f)
Figure 3 Phase field modeling of the evolution of a dislocation loop (red line) in a g (dark phase)/g0 (white phase) under applied stress. Reproduced from Rodney, D.; Le Bouar, Y.; Finel, A. Acta Mater. 2003, 51(1), 17–30.
may remain challenging to assign values to these nonvanishing terms that are consistent with the thermodynamics of the alloy considered. Another important point is that interfacial energies are in general anisotropic. In order to obtain realistic morphological evolution, it is often important, and sometimes even absolutely necessary, to include this anisotropy, for example, in the modeling of dendritic solidification. The symmetry of the mesh chosen for numerically solving the PF equations introduces interfacial anisotropy but in an unphysical and uncontrolled way. One possible approach to introduce interfacial anisotropy is to let k vary with the local orientation of the interface with respect to crystallographic directions.11,32 Another approach is to rely on symmetry constraints27,30 to determine the number of independent coefficients in a general expression of the inhomogeneity term, see eqn [5]. In both approaches, the different coefficients entering
417
Cð1 CÞ ~ D kB T
½17
where C is the average solute concentration and D~ the interdiffusion coefficient. In both cases, the simulated times are expressed in arbitrary units of M1 or L1, thus precluding a direct connection with experimental kinetics. This problem is also directly related to the lack of absolute physical length scales in these simulations. Moreover, using a 1D Bragg–Williams model composed of atomic planes, Martin101 showed that M is not a constant but is in fact a function of the local composition along the equilibrium profile. A complete connection between atomistic dynamics and M will be made in Section 1.15.3. Similar to the discussion on coupling between various fields for the gradient energy terms, kinetic coupling is also expected in general. The kinetic couplings between composition (a conserved order parameter) and chemical ordering (a nonconserved order parameter) are revealed by including sublattices into Martin’s 1D model and deriving the macroscopic evolution of the fields from the microscopic dynamics. In that case, atoms jump between adjacent planes.102,103 As a result, instead of the mere superposition of eqns [2] and [3], the kinetic evolution of coupled concentration and chemical order in a binary alloy is given by @Cðr;t Þ dF ¼r M1 r @t dCðr;t Þ dF þ r M2 r dSðr;t Þ @Sðr;t Þ @F dF ¼ L1 þ r L2 r @t @Sðr;t Þ dSðr;t Þ dF þ r L3 r dCðr;t Þ
½18
where L1 is a mobility coefficient, and L2, L3, M1, and M2 are second-rank mobility tensors, since they
418
Phase Field Methods
relate diffusional fluxes (vectors) to chemical potential gradients (vectors). In the case of cubic crystalline phases, second-rank tensors reduce to scalars, but in many ordering reactions, noncubic phases form, thus leading to anisotropic mobility. Vaks and coworkers104 have also derived PFMs for simultaneous ordering and decomposition starting from microscopic models. These works, however, illustrate the fact that it would be quite difficult, especially for multidimensional field variables, to assign correct values to the kinetic coefficients for a given alloy system by relying solely on a phenomenological approach.
1.15.3 Quantitative PF Modeling The PF equations introduced in Section 1.15.2, that is, eqns [2] and [3], are phenomenological, and one particular consequence is that they lack an absolute length scale. All scales observed in PF simulations are expressed in units of the interfacial width we of the appropriate field variable. As discussed in the previous section, for the case of one scalar conserved order parameter, this width we and the excess interfacial free energy s are directly related to the gradient energy coefficient k and the energy barrier between the two stable compositions Dfmax (see eqns [15] and [16]). Beyond the difficulty of parameterizing k and Dfmax to accurately reflect the properties of a given alloy system, the phenomenological nature of these coefficients creates additional problems. In particular, as the number of mesh points used in a simulation increases, the interfacial width, expressed in units of mesh point spacing, remains constant if no other parameter is changed. Increasing the number of mesh points thus increases the physical volume that is simulated but does not increase the spatial resolution of the simulations. If the intent is to increase the spatial resolution, one would have to increase k so that the equilibrium interface is spread over more mesh points. Equilibrium interfacial widths in alloy systems typically range from a few nanometers at high temperatures to a few angstroms at low temperatures. In the latter case, if the interface is spread over several mesh points, it implies that the volume assigned to each mesh point may not even contain one atom. This raises fundamental questions about the physical meaning of the continuous field variables, and practical questions about the merits of PF modeling over atomistic simulations.
Another important problem related to the lack of absolute length scale in conventional PF modeling concerns the treatment of fluctuations. Fluctuations arise owing to the discrete nature of the microscopic (atomistic) models underlying PFMs. Furthermore, fluctuations are necessary for a microstructure to escape a metastable state and evolve toward its global equilibrium state, such as during nucleation. Fluctuations, or numerical noise, will also determine the initial kinetic path of a system prepared in an unstable state. The standard approach for adding fluctuations to the PF kinetic equations is to transform them into Langevin equations, and then to use the fluctuation– dissipation theorem to determine the structure and amplitude of these fluctuations. For instance, in the case of one conserved order parameter, the Cahn–Hilliard diffusion equation, that is, eqn [2], is transformed into the Langevin equation: @Cðr;t Þ dF ¼ r Mr þ xðr;t Þ ½19 @t dCðr;t Þ where xðr;t Þ is a thermal noise term. The structure of the noise term can be derived using fluctuation– dissipation105,106: hxðr;t Þi ¼ 0 hxðr;t Þxðr0 ;t 0 Þi ¼ 2kB TMr2 dðr r0 Þdðt t 0 Þ
½20
where the brackets h i indicate statistical averaging over an ensemble of equivalent systems. However, eqn [20] does not include a dependence of the noise amplitude with the cell size, which is not physical. Even if this dependence is added a posteriori, it is observed practically that this noise amplitude gives rise to unphysical evolution, as reported by Dobretsov et al.107 While these authors have proposed an empirical solution to this problem by filtering out the short-length-scale noise in the calculation of the chemical potentials, a physically sound treatment of fluctuations requires a derivation of the PF equations starting from a discrete description. Recently, Bronchart et al.100 have clearly demonstrated how to rigorously derive the PF equations from a microscopic model through a series of controlled approximations. We outline here the main steps of this derivation. The interested reader is referred to Bronchart et al.100 for the full derivation. These authors consider the case of a binary alloy system in which atoms migrate by exchanging their position with atoms that are first nearest neighbors on a simple cubic lattice. A microscopic configuration is defined by the ensemble of occupation variables, or
Phase Field Methods
spin values, for all lattice sites, C ¼ fsi g, where si ¼ 1 when the site i is occupied by an A or a B atom, respectively. The evolution of the probability distribution of the microscopic states is given by the following microscopic Master Equation (ME): X @PðCÞ ¼ W ðC ! Cij ÞPðCÞ @t i; j
þ
X i; j
W ðCij ! CÞPðCij Þ
½21
where the * symbol in the summation indicates that it is restricted to microscopic states that are connected to C through one exchange of the i and j nearest neighbor atoms, resulting in the configuration Cij . The next step is to coarse-grain the atomic lattice into cells, each cell containing Nd lattice sites. It is then assumed that local equilibrium within the cells is achieved much faster than evolution across cells. The composition of the cell n, cn, is given by the average occupation of its lattice site by B atoms, and thus cn ¼ 0; 1=Nd ; . . . ; Nd =Nd . A mesoscopic configuration is fully defined on this coarse-grained system ~ by C ¼ fcn g. A chemical potential can be defined within each cell and, if this chemical potential varies smoothly from cell to cell, the microscopic ME, eqn [21], can be coarse-grained into a mesoscopic ME: 2 ~ X @PðCÞ a ba ~ ~ ~ ~ y lmn ðCÞexp ¼ ðmm ðCÞ mn ðCÞÞ PðCÞ @t d 2d n;m þ gain term
½22
where a is the lattice parameter and d the number of lattice planes per cell (i.e., Nd ¼ (d/a)3), y is the ~ attempt frequency of atom exchanges, lmn ðCÞ is a mobility function that is directly related to the microscopic jump frequency, b ¼ ðkB T Þ1 , and ~ mn ðCÞ is the chemical potential in cell n. The * symbol over the summation sign indicates that the summation over m is only performed over cells that are adjacent to the cell n; the first term on the righthand side of eqn [22] represents a loss term, and there is a similar gain term, which is not detailed. The mesoscopic ME eqn [22] can be expanded to the second order using 1/Nd as the small parameter for the expansion. The resulting Fokker–Planck equation is then transformed into a Langevin equation for the evolution of the composition in each cell n: ðnÞ
@cn a 2 y X ~ ~ ~ lnm ðC Þ½mm ðC Þ mm ðC Þ þ zn ðt Þ ½23 ¼ 2 @t d kB T m
419
where the noise term zn ðt Þ is a Gaussian noise with first and second moments given by hzn ðt Þi ¼ 0 hzn ðt Þzn ðt 0 Þi ¼
2 a2 X ~ lnp ðC Þdðt t 0 Þ Nd d 2 p
hzn ðt Þzm ðt 0 Þi ¼
2 a2 ~ lnm ðC Þdðt t 0 Þ Nd d 2
ðnÞ
½24
While the structure of eqns [23] and [24] is quite similar to that of the phenomenological eqns [19] and [20], there are several key differences in these two descriptions. First, thermodynamic quantities such as the homogeneous free-energy density and the gradient energy coefficient are now cell-size dependent. These quantities can be evaluated separately using standard Monte Carlo techniques.100 Second, the mobility coefficients, and thus the correlations in the Langevin noise, are functions of the local concentration, as well as of the cell size. Bronchart et al.100 applied their model to the study of nucleation and growth in a cubic A1cBc system for various cell sizes, d ¼ 6a, d ¼ 8a, and d ¼ 10a. The supersaturation is chosen to be small so that the critical nucleus size is large enough to be resolved by these cell sizes. As seen in Figure 4, for a given supersaturation, the evolution of the volume fraction of precipitates is independent of the cell size and in very good agreement with fully atomistic kinetic Monte Carlo (KMC) simulations (not shown in Figure 4). The above results are important because they show that it is possible to derive and use PF equations that retain an absolute length scale defined at the atomistic level. The point will be shown to be very important for alloys under irradiation. On the other hand, the work by Bronchart et al.100 clearly highlights the difficulty in using quantitative PF modeling when the physical length scales of the alloy under study are small, as for instance in the case of precipitation with large supersaturation, which results in a small critical nucleus size, or in the case of precipitation growth and coarsening at relatively moderate temperature, which results in a small interfacial width. In these cases, one would have to reduce the cell size down to a few atoms, thus degrading the validity of the microscopically based PF equations since they are derived by relying on an expansion with respect to the parameter 1/Nd.
Phase Field Methods
Volume fraction
420
0.04
C = 0.160, d = 8a C = 0.165, d = 8a C = 0.170, d = 8a
0.02
C = 0.170, d = 6a C = 0.170, d = 10a 0.00
0
2 ´ 106
4 ´ 106 −1
Time (unit: q ) Figure 4 Evolution of the volume fraction of precipitates with time for a three-dimensional binary alloy A1cBc using the microscopically derived phase field eqns [23] and [24]. Parameters a and d are the lattice parameter and the number of lattice planes in h100i directions. For a given concentration, C ¼ 0.17, the precipitation kinetics is equally well resolved with three different cell sizes. Reprinted with permission from Bronchart, Q.; Le Bouar, Y.; Finel, A. Phys. Rev. Lett. 2008, 100(1). Copyright by the American Physical Society.
1.15.4 PF Modeling Applied to Materials Under Irradiation 1.15.4.1 Challenges Specific to Alloys Under Irradiation The PFMs discussed so far are broadly applied to materials as they relax toward some equilibrium state. In particular, the kinetics of evolution is given by the product of a mobility by a linearized driving force, see for instance eqns [2] and [3]. In the context of the thermodynamics of irreversible processes,28 the mobility matrix is the matrix of Onsager coefficients. Irradiation can, however, drive and stabilize a material system into a nonequilibrium state,108 owing to ballistic mixing and permanent defect fluxes, and so it may appear questionable at first whether linearized relaxation kinetics is applicable. A sufficient condition, however, is that these different fields undergo linear relaxation locally, and this condition is often met even under irradiation. A complicating factor arises from the presence of ballistic mixing, which adds a second dynamics to the system on top of the thermally activated diffusion of atoms and point defects. A superposition of linearized relaxations for these two dynamics is valid as long as they are sufficiently decoupled in time and space, so that in any single location, the system will evolve according to one dynamic at a time. KMC simulations indicate that, for dilute alloys, this decoupling is valid except for a small range of kinetic parameters where events from different dynamics interfere with one another.109
A second issue is that PFMs, traditionally, do not include explicitly point defects. Vacancies and interstitials are, however, essential to the evolution of irradiated materials, and it is thus necessary to include them as additional field variables. The situation is more problematic with point-defect clusters, which often play a key role in the annihilation of free point defects. Since the size of these clusters cover a wide range of values, it would be quite difficult to add a new field variable for each size, for example, for vacancy clusters of size 2 (divacancies), size 3 (trivacancies), size 4, etc. Moreover, under irradiation conditions leading to the direct production of defect clusters by displacement cascades, additional length scales are required to describe the distribution of defect cluster sizes and of atomic relocation distances. These new length scales are not physically related to the width of a chemical interface at equilibrium, we, and therefore, they cannot be safely rescaled by we. This analysis clearly suggests that one needs to rely on a PFM where the atomic scale has been retained. This is, for instance, the case in the quantitative PFM reviewed in Section 1.15.3. Another possible approach is to use a mixed continuous–discrete description, as illustrated below in Section 1.15.4.2.4. We note that information on defect cluster sizes and relocation distances should be seen as part of the noise imposed by the external forcing, here the irradiation, on the evolution of the field variables. The difficulty is thus to develop a model that can correctly integrate this external noise. It is
Phase Field Methods
well documented that, for nonlinear dissipative systems, the external noise can play a determinant role and, for large enough noise amplitude, may trigger nonequilibrium phase transformations.110–113 One last and important challenge in the development of PFMs for alloys under irradiation is the fact that in nearly all traditional models the mobility matrix is oversimplified, for instance Mirr ¼ Cð1 CÞD~irr =kB T , which is a simple extension to eqn [17] where D~ has been replaced by D~irr to take into account radiation-enhanced diffusion. In the common case of multidimensional fields, for instance for multicomponent alloys, or for alloys with conserved and nonconserved field variables, the mobility matrix is generally taken as a diagonal matrix, thus eliminating any possible kinetic coupling between these different field variables. As discussed at the end of Section 1.15.2, this approximation raises concerns because it misses the fact that these kinetic coefficients are related since they originate from the same microscopic mechanisms. This is, in particular, the case for the coupled evolution of point defects and chemical species in multicomponent alloys. This coupling is of particular relevance to the case of irradiated alloys since irradiation can dramatically alter segregation and precipitation reactions owing to the influence of local chemical environments on point-defect jump frequencies. While new analytical models have been developed recently using mean field approximations to obtain expressions for correlation factors in concentrated alloys,114–117 work remains to be done to integrate these results into PFMs. 1.15.4.2 Examples of PF Modeling Applied to Alloys Under Irradiation 1.15.4.2.1 Effects of ballistic mixing on phase-separating alloy systems
Consider the simple case where the external forcing produces forced exchanges between atoms (such relocations are found in displacement cascades), and let us assume for now that these relocations are ballistic (i.e., random) and take place one at a time. For this case, one can use a 1D PFM to follow the evolution of the composition profile C(x) during irradiation.118 This evolution is the sum of a thermally activated term, for which the classical Cahn diffusion model can be used, and a ballistic term: ð dCðxÞ dF ¼ Mirr r2 Gb CðxÞ wR ðx x 0 ÞCðx 0 Þdx 0 ½25 dt dC
421
where Mirr is the thermal atomic mobility, here accelerated by the irradiation, F the free energy of the system, Gb the jump frequency of the atomic relocations forced by the nuclear collisions, and wR is the normalized distribution of relocation distances, characterized by a decay length R. Since most of these atomic relocations take place between nearest neighbor atoms, in a first approximation one may assume that R is small compared to the cell size. In this case, the second ballistic term in eqn [25] reduces to a diffusive term: dCðxÞ dF ¼ Mirr r2 Gb a 2 r2 C dt dC
½26
In this case, the model thus reduces to the one initially introduced by Martin,119 and the steady state reached under irradiation is the equilibrium state that the same alloy would have reached at an effective irr temperature Teff ¼ T ð1 þ Gb =Girr th Þ, where Gth is an average atomic jump frequency, enhanced by the point-defect supersaturation created by irradiation. In particular, in the case of an alloy with preexisting precipitates, depending upon the irradiation flux and the irradiation temperature, this criterion predicts that the precipitates should either dissolve or continuously coarsen with time. Some relocation distances, however, extend beyond the first nearest neighbor distances,120,121 and it is interesting to consider the case where the characteristic distance R exceeds the cell size. An analytical model by Enrique and Bellon118 revealed that, when R exceeds a critical value Rc, irradiation can lead to the dynamic stabilization of patterns. To illustrate this point, one performs a linear stability analysis of this model in Fourier space, assuming here that the ballistic jump distances are distributed exponentially. The amplification factor w(q) of the Fourier coefficient for the wave vector q is given by oðqÞ=M ¼ ð@ 2 f =@C 2 Þq 2 2kq4 gR2 q 2 =ð1 þ R2 q 2 Þ
½27
where f (C) is the free-energy density of a homogeneous alloy of composition C, k the gradient energy coefficient, and g ¼ Gb =M is a reduced ballistic jump frequency. The analysis is here restricted to compositions and temperatures such that, in the absence of irradiation, spinodal decomposition takes place, that is, @ 2f/@C 2 < 0. The various possible dispersion curves are plotted in Figure 5. Unlike in the case of short R, it is now possible to find irradiation intensities g such that the
422
Phase Field Methods
Thermal w(q)
Total
qmin
128 q
Irradiation Figure 5 Sketch of the dispersion curve given by the linear stability analysis eqn [27], in the case when the ballistic relocation distance R is large. The total dispersion curve is decomposed into its thermal and irradiation components. Wave vectors below qmin are stable against decomposition.
ballistic term in eqn [27] is greater than j@ 2 f =@C 2 j at small q, but smaller than that at large q. In such cases, the amplification factor is first negative for small q values, but it becomes positive when q exceeds some critical value qmin, while for larger q, the amplification factor is negative again. Therefore, decomposition is still expected to take place, but only for wave vectors larger than qmin, that is, for wavelengths smaller than 2p/qmin. It can thus be anticipated that coarsening will saturate, since at large length scales, the alloy remains stable with respect to decomposition. Enomoto and Sawa122 have investigated this model using a 2D PFM based on eqn [25]. The interest here is that the PFM, unlike the above linear stability analysis, includes both linear and nonlinear contributions to the evolution of composition inhomogeneity and also permits following the morphology of the decomposition. Using this model, Enomoto and Sawa have confirmed the existence of the patterning regime, see Figure 6, and showed that this patterning can take place in the whole composition range. The PFM approach allows for a direct determination of the patterning length scale as a function of the irradiation conditions, as illustrated in Figure 7. Similar results have also been obtained using a variational analysis of eqn [25], leading to the dynamical phase diagram displayed in Figure 8. As seen in this diagram, when the characteristic length for the forced relocation is smaller than the critical value Rc, the system never develops patterns at steady state. Above Rc, patterning takes place when the irradiation
64
0
Figure 6 Irradiation-induced compositional patterning in a binary alloy with (a) CB ¼ 50% and (b) CB ¼ 35%, using a two-dimensional phase field model based on eqn [25] with 1282 cells. Reproduced from Enomoto, Y.; Sawa, M. Surf. Sci. 2002, 514(1–3), 68–73.
conditions are chosen so as to result in an appropriate g value. Another result obtained from the KMC simulations is that the steady state reached by an alloy is independent of its initial state. Experimental tests performed on a series of dilute Cu–M alloys, with M ¼ Ag, Co, Fe, have confirmed some of the key predictions of the above simulations and analytical modeling. In particular, irradiation conditions that result in atomic relocation distances exceeding a few angstroms do lead to the dynamical stabilization of precipitates at intermediate irradiation temperature.123 These results provide also a compelling rationalization of the puzzling results reported by Nelson et al.124 on the refinement of g0 precipitates in Ni–Al alloys under 100-keV Ni irradiation at 550 C. The origin of the above irradiation-induced compositional patterning lies in the finite range of the atomic mixing forced by nuclear collisions.125 Enrique and Bellon126,127 have shown that the effect of this finite-range dynamics can be formally recast as effective finite-range repulsive interaction between like particles. It is interesting to note that PF
Phase Field Methods
5
Patterning
423
g2
g1
R/Ö(C/A)
I(k,t)/(S02L04)
t/t0 = 500
2.5
Solid solution (Rc,gc)
1 Macroscopic phase separation 0.1
0
0
(a)
5 kL0
10
101
L0
-1/3
1 g /(A2/C)
Figure 8 Analytical dynamical phase diagram yielding the most stable steady state in a phase-separating A50B50 alloy as a function of the forced relocation distance R and the relative ballistic jump frequency g. Insets are (111) sections of three-dimensional kinetic Monte Carlo (KMC) results; the lateral size of the KMC inset is 17 nm. Reprinted with permission from Enrique, R. A.; Bellon, P. Phys. Rev. Lett. 2000, 84(13). Copyright by the American Physical Society.
(a)
(b)
(c)
(d)
(e)
(f)
100
10-1 1 10 (b)
102 t/t0
103
Figure 7 (a) Spherically averaged and rescaled structure factor for a low ballistic mixing frequency g ¼ 0.0005 (○) and a higher ballistic mixing frequency g ¼ 0.05 (●). (b) Evolution of the first moment of this structure factor as a function of rescaled time, demonstrating that the alloy undergoes continuous coarsening with the low mixing rate, but stabilizes at the finite length scale (irradiation-induced patterning) for the higher mixing rate. Reproduced from Enomoto, Y.; Sawa, M. Surf. Sci. 2002, 514(1–3), 68–73.
simulations of alloys with Coulomb interactions also predict a patterning of the microstructure.128 The parallel with the treatment of finite-range mixing is in fact quite strong, since a screened Coulomb repulsion is described by a decaying exponential, as also assumed for the probability of finite-range ballistic exchanges in deriving eqn [27]. The contribution of the Coulomb repulsion to the linear stability analysis is thus proportional to 1/(q2 þ qD2), where qD is the
Figure 9 Phase field modeling of the evolution of ordered precipitates in the presence of electrostatic (repulsive) interactions, with increasing time from A to F. The average precipitate size reaches a finite value at equilibrium. Reprinted with permission from Chen, L. Q.; Khachaturyan, A. G. Phys. Rev. Lett. 1993, 70(10), 1477–1480. Copyright by the American Physical Society.
screening wavelength. For reasons similar to the ones discussed in the case of finite-range mixing, it is then anticipated that these interactions will suppress coarsening. This is confirmed by PF simulations, as illustrated in Figure 9. 1.15.4.2.2 Coupled evolution of composition and chemical order under irradiation
Many engineering alloys contain ordered phases or precipitates to optimize their properties, in particular mechanical properties. It is thus important to
424
Phase Field Methods
investigate how these optimized microstructures evolve under irradiation. It is anticipated that, under appropriate conditions, ballistic mixing can lead to the dissolution of precipitates, and to the disordering of chemically ordered phases.129 Matsumura et al.130 used a 1D PF approach on a model binary alloy system to specifically investigate what evolution irradiation may produce. In that model, the composition field is represented by the globally conserved order parameter X(r), while the degree of order is represented by the nonconserved order parameter S(r). X is chosen to vary from 1 to þ1 for pure A and pure B composition, respectively, and S takes a value ranging from 0 to 1 for fully disordered and fully ordered phases, respectively. The free energy functional of the system is written as F ½fX ðrÞ; SðrÞ; T g 9 8 H ðt Þ > 2> > > > ðrX Þ f ðX ; S; T Þ þ ð> = < 2 dr ¼ > > > > K ðt Þ 2 > > : ðrSÞ ; þ ½28 2 where f (X, S, T ) is the mean field free energy of a homogeneous alloy, and H and K are positive constants of the interfacial energy coefficients in the presence of varying field X and S, respectively. The homogeneous free-energy density is given by a Landau expansion
f ðX ; S; T Þ
2 3 2 2 2 2 Þ bðT Þfx ðT Þ X gS ðX x m 0 aðT Þ4 5 ¼ f0 þ 2 þ bðT Þ2 x ðT Þ2 S 4 1
½29
where f0 is the mean field free energy of the disordered phase with composition xm, and a, b, x12 are positive constants depending on temperature. The equilibrium phase diagram for this model system is given in Figure 10. Notice, in particular, that at low temperature and for compositions sufficiently far from the equiatomic composition, an ordered phase coexists with a disordered phase. The kinetic evolution of these two fields is governed by @X ¼D0mix fr2 X @t
þ LðT ; fÞr
2
dF ðfX ; S; T gÞ m dX
½30
and @S dF ðfX ; S; T gÞ ¼ efS MðT ; fÞ @t dS
½31
where f is the atomic displacement rate, m is the chemical potential, D0mix and e are positive coefficients characterizing the efficiency of mixing and disordering
1.2 Disorder
Temperature, T/Tc
1.0
0.8
x0(T) Order
0.6
x2(T)
x2(T) 0.4 Order + disorder
0.2
x1(T)
Order + disorder
x1(T)
x0(T) 0.0 -1.0
x0(T) -0.5
0.0 Composition, X
0.5
1.0
Figure 10 Equilibrium phase diagram for model alloy system given by eqns [28] and [29]. The dotted lines correspond to the metastable extrapolation of the order–disorder transition into the miscibility gap. Reprinted with permission from Matsumura, S.; Muller, S.; Abromeit, C. Phys. Rev. B 1996, 54(9), 6184–6193. Copyright by the American Physical Society.
Phase Field Methods
by irradiation, and L and M are the mobility coefficients for the conserved and nonconserved fields. Note that here there is no kinetic coupling between these two fields. There is, however, thermodynamic coupling through the expression chosen for the homogeneous free-energy density, eqn [29]. Although point defects are not explicitly used as PF variables, the dependency of the mobility coefficients M and L with temperature, irradiation flux, and sink density (cS) is obtained from a rate theory model for the vacancy concentration under
425
irradiation in a homogeneous alloy.1 The steady-state phase diagrams for two irradiation flux values are given in Figure 11. At the higher flux, the phase diagram is composed of homogeneous disordered and ordered phases only. At low enough temperature, the ballistic mixing and disordering dominate the evolution of the alloy, leading to the destabilization of the ordered phase at and near the stoichiometric composition X ¼ 0, and to the disappearance of the two-phase coexistence domains for off-stoichiometric compositions.
1.2 CS = 10−5
Disorder
f = 10−5f c
Temperature, T/Tc
1.0
0.8
x1irr
x1irr
Order
0.6
0.4 x2
x0irr (T,f )
O+D
0.2 x1 0.0 -1.0
-0.5
Disorder
O+D x1
0.0 Composition, X
(a)
0.5
1.0
1.2 CS = 10−5
Disorder
f = 10−4f c
Temperature, T/Tc
1.0
0.8
Order
x1irr
x1irr
0.6
0.4
x0irr (T,f ) x2
x2
0.2 x1 0.0 -1.0 (b)
-0.5
Disorder
0.0 Composition, X
x1 0.5
1.0
Figure 11 Steady-state phase diagrams under irradiation for (a) a low irradiation flux and (b) an irradiation flux 10 times larger. The two-phase field is barely present in (a), and is no longer stable in (b). Reprinted with permission from Matsumura, S.; Muller, S.; Abromeit, C. Phys. Rev. B 1996, 54(9), 6184–6193. Copyright by the American Physical Society.
426
Phase Field Methods
The model has also been used to study the dissolution of ordered precipitates under irradiation. In agreement with prior lattice-based mean field kinetic simulations,131 it is found that two different dissolution paths are possible, depending upon the composition and irradiation parameters. Ordered precipitates may either disorder first and then slowly dissolve or they may dissolve progressively while retaining a finite degree of chemical order until their complete dissolution. These two kinetic paths have indeed been observed experimentally in Nimonic PE16 alloys irradiated with 300-keV Ni ions.132,133 1.15.4.2.3 Irradiation-induced formation of void lattices
The formation of voids in irradiated solids results from the clustering of vacancies, which can be assisted by vacancy clusters produced directly in displacement cascades and by the presence of gas atoms. Vacancy supersaturation under irradiation may locally reach a level large enough to trigger clustering owing to the biased elimination of interstitials on sinks, especially since interstitial atoms and small interstitial clusters usually migrate much faster than vacancies. Evans134 discovered in 1971 that under irradiation voids may self-organize into a mesoscopic lattice. The symmetry of the void lattice is identical to that of the underlying crystal, but with a void lattice parameter about two orders of magnitude larger than the crystalline lattice parameter (see also the reviews by Jager and Trinkaus135 and Ghoniem et al.136 for irradiation-induced patterning reactions). It has been suggested that the formation of the void lattice results from the 1D migration of self-interstitial atoms (SIAs) and of clusters of SIAs, although elastic interactions between voids could also contribute to self-organization.137 This 1D migration of SIAs would stabilize the formation of voids along directions of the SIAs migration by a shadowing
effect.138–140 The model proposed by Woo141 indicates, in particular, that the mean free path of SIAs needs to exceed a critical value for a void lattice to be stable. Atomistic KMC simulations have been performed142 to evaluate the dynamics of void formation, shrinkage, and organization during irradiation. Due to the large difference in mobility of vacancies and interstitials, the slow evolution of the microstructure, and the large range of length scales, assumptions had to be used, in particular, regarding the void position and size. Recently, Hu and Henager143 have approached the problem of void lattice formation in a pure metal using a PFM. Their model relies on the traditional approach presented in Section 1.15.2 for the evolution of the vacancy field, but it makes use of continuumtime random-walk kinetics for modeling the fast transport of interstitials. 2D simulations indicate that irradiation can stabilize a void lattice if the ratio of SIA to vacancy diffusion coefficients is large enough (see Figure 12) and if the defect production rate is not too large (see Figure 13). It would be interesting to extend this first model to include interstitial clusters. The model lacks an absolute length scale, for the reasons discussed in Section 1.15.2, and thus nucleation of new voids is treated in a deterministic and phenomenological manner based on the local vacancy concentration. It would clearly be beneficial to use a quantitative PFM of the type presented in Section 1.15.3 to treat void nucleation. This would also then make it possible to directly compare the void size stabilized by irradiation with experimental observations. We note also that Rokkam et al.144 recently introduced a simple PFM for void nucleation and coarsening in a pure element subjected to irradiation-induced vacancy production. In addition to the local vacancy concentration, these authors introduced a nonconserved order parameter to model the matrix–void interface, similar to the nonconserved order parameter
t * = 16 000
t * = 16 000
t * = 16 000
(a)
(b)
(c)
t * = 16 000
(d)
Figure 12 Phase field simulations of void distributions for a low generation rate of vacancies and self-interstitial atoms (SIAs), g_ V ¼ g_ SIA ¼ 105 , for different diffusivity ratios between SIA and vacancies, DSIA =DV , (a) 10, (b) 102, (c) 103, and (d) 104. Reproduced from Hu, S.; Henager, C. H., Jr. J. Nucl. Mater. 2009, 394, 155–159.
Phase Field Methods
t * = 16 000
t * = 16 000
t * = 16 000
t * = 16 000
(a)
(b)
(c)
(d)
427
Figure 13 Phase field simulations of void distributions for a high diffusivity ratio between self-interstitial atoms (SIAs) and vacancies, DSIA =DV ¼ 104 , and various generation rates of vacancies and SIAs, g_ V ¼ g_ SIA , (a) 2 103, (b) 5 104, (c) 104, and (d) 105. Reproduced from Hu, S.; Henager, C. H., Jr. J. Nucl. Mater. 2009, 394, 155–159.
used for solid–liquid interfaces. It is shown144,145 that this model reproduces many known phenomena, such as nucleation, growth, coarsening of voids, as well as the formation of denuded zones near sinks such as free surfaces and grain boundaries. This phenomenological model is currently limited by the absence of interstitial atoms in the description. It may also suffer from the fact that the void–matrix interfaces are intrinsically treated as diffuse, whereas real void–matrix interfaces are essentially atomically sharp. This problem is further discussed in the following section. 1.15.4.2.4 Irradiation-induced segregation on defect clusters
In order to circumvent the problems raised in the previous paragraph and in Section 1.15.4.1 for the inclusion of defect clusters in a PFM, Badillo et al.146 have recently proposed a mixed approach that combines discrete and continuum treatments of the defect clusters, so that each cluster is treated as a separate entity. Point-defect cluster size is treated as a discrete quantity for cluster production, whereas the long-term fate of clusters is controlled by a continuum-based flux of free point defects. New field variables are thus introduced to describe the size of these clusters: p
p
p
Nc;A p Nc;V p Nc;B ¼ d ; Cc;A ¼ d ; Cc;B ¼ d ½32 N N N p where Nc;V is the number of vacancies in the vacancy cluster in the cell p, and Nd is the number of substitup p tional lattice sites per cell; Nc;A and Nc;B are the numbers of A and B interstitial atoms, respectively, forming the interstitial cluster in the cell p. Each cell contains at most one cluster. The production of point defects by irradiation takes place at a rate dictated by the irradiation flux p Cc;V
f in dpa s1 and by the simulated volume. In the case of irradiation conditions leading to the intracascade clustering of point defects, the total number of point defects created in a displacement cascade, the fraction of those defects that are clustered, and the size and spatial distribution of these clusters are used as input data. The production of Frenkel pairs is treated in the same way as defect clusters, except that the variables affected are the free vacancy and interstitial concentrations. This treatment of defect and defect cluster production makes it possible to compare irradiation conditions with identical total defect production rates, that is, identical dpa s1 values, but with varied fractions of intracascade defect clustering and varied spatial distribution of these clusters. Furthermore, it is also very well suited for system-specific modeling, since all the above information can be directly and accurately obtained from molecular dynamic simulations sampling the primary recoil spectrum.121 In particular, one can build a library of such displacement cascades, so that the PFM will inherit the stochastic character in space and time of the production of defect clusters by displacement cascades. The continuous flux of free point defects to the clusters results in the growth or shrinkage of these clusters, which translates into the continuous p p evolution of the cluster field variables Cc;V , Cc;A , and p Cc;B . When any cluster field variable drops below 1=N d , this cluster is assumed to have dissolved, and the remaining one point defect is transferred to the corresponding free point-defect variable of that cell. For the sake of simplicity, defect clusters are treated as immobile, but the approach can be extended to include mobility, in particular for small interstitial clusters. Further details are available in Badillo et al.146 The potential of the above approach is illustrated by considering a 2D A8B92 alloy with a zero heat of
428
Phase Field Methods
mixing, so that at equilibrium, it always forms a random solid solution. The production of interstitials is, however, biased so that only A interstitial atoms are created. This could, for instance, simulate an alloy where there is a rapid conversion of B interstitial atoms into A interstitial atoms via an interstitialcy mechanism. The preferential transport of A interstitial atoms to defect clusters should lead to an enrichment of A species around defect clusters, since these clusters act as defect sinks. The effect of the primary recoil spectrum on this irradiation-induced segregation is studied by comparing two cases: the first one where a small fraction of cascades, 1/Ncas ¼ 5 104, produces defect clusters, and the second one where that fraction is 100 times higher, 1/Ncas ¼ 5 102. In both cases, however, the displacement rate per atom per second is the same, here 107 dpa s1. Figure 14 shows instantaneous concentration maps of the A solute atoms for 1/Ncas ¼ 5 104. In this case, the PFM uses 64 64 cells, each containing 7 7 lattice sites. Segregation of A species is clearly observed at a few locations, typically 2–5. This number is close to the average number of defect clusters. The sharp peaks with high levels of segregation correspond to segregation of existing defect clusters, either interstitial or vacancy ones. This is confirmed by visualizing the defect and defect cluster fields, see Figure 15. The broader segregation profiles in Figure 14 are the remnants of sharp segregation profiles after their corresponding clusters shrank
CA
1.0 400
0.5
300 0.0 0
200 100 200
100 300 400
0
Figure 14 The concentration field of A atom (CA ) for an A8B92 alloy with zero heat of mixing where all interstitials are created as A atoms. The two-dimensional model system contains 448 448 lattice sites, decomposed into 64 64 cells for defining the phase field variables, each cell containing 7 7 lattice sites. Irradiation displacement rate is 107 dpa s1; the cascade frequency rate is 1/Ncas ¼ 5 104, and the irradiation dose is 6 dpa. Reproduced from Badillo, A.; Bellon, P.; Averback, R. S. to be submitted.
and disappeared. As a result, the nonequilibrium segregation that build up on those clusters is being washed out by vacancy diffusion, as expected for an A–B alloy system with zero heat of mixing. In the case where defect sinks have a finite lifetime, as in the present case, one should thus expect a dynamical formation and elimination of segregated regions. In the case of much higher defect cluster production rate, 1/Ncas ¼ 5 102, a very different microstructure is stabilized by irradiation, as illustrated in Figure 16. Now a high density of clusters is present, typically 40 interstitial clusters and 20 vacancy clusters, as seen in Figure 17(a) and 17(b), and the segregation measured on these clusters is reduced by about one order of magnitude compared to the previous case. These results are reminiscent of the experimental findings reported by Barbu and Ardell147 and Barbu and Martin,148 showing that, with irradiation conditions producing displacement cascades (e.g., 500-keV Ni ion irradiation), the domain of irradiation-induced segregation and precipitation in undersaturated Ni–Si solid solutions is significantly reduced compared to the case where irradiation produces only individual point defects (e.g., 1-MeV electron irradiation).
1.15.5 Conclusions and Perspectives Thanks to fundamental advances, coupled with the development of efficient algorithms and fast computers, the PF technique has become a very powerful and versatile tool for simulating phase transformations and microstructural evolution in materials, as illustrated in this chapter. This technique provides simulation tools that are complementary to atomistic models, such as molecular dynamics and lattice Monte Carlo simulations, and to larger scale approaches, such as finite element models. With some modifications, it can also be employed for materials subjected to irradiation. In the case of materials subjected to irradiation, specific issues need to be addressed to fully realize the potential of PF modeling. First, a proper description of point defects and atom transport requires mobility matrices (or tensors) that capture the kinetic coupling between these different species. In particular, the models reviewed in this chapter do not account for the correlated motion of point defects and atoms, thus leading to unphysical correlation factors in the mobility coefficients. These correlation effects, however, play an essential role in phenomena
429
0.0004 0.0003 0.0002 0.0001 0.0000 0
400
0.00020 0.00015 0.00010 0.00005
Ci
CV
Phase Field Methods
300 200
400 300
0
100
200
100
200
300
0.08 0.06 0.04 0.02 0.00 0
400 300 200 100
(c)
0
400
400 300 200 100
(d)
300
0
0.05 0.04 0.03 0.02 0.01 0.00 0
100 200
400
(b)
CClus Int
CClus Vac
0
400
(a)
100
200
100 300
200
100 300 400
0
~V, (b) free interstitials Cp þ Cp Figure 15 Defect concentration fields corresponding to Figure 14: (a) free vacancies C int A int B (A and B atoms), (c) clustered vacancies Cc;V , and (d) clustered interstitials Cc;A þ Cc;B (A and B atoms). Reproduced from Badillo, A.; Bellon, P.; Averback, R. S. to be submitted.
CA
1.0 0.5
400 300
0.0 0
200
100
100
200 300 400
0
Figure 16 Concentration field of A atom CA for the same A8B92 alloy as in Figure 14, except at a higher cascade frequency, 1/Ncas ¼ 5 102. Notice the significant reduction in segregation on defect clusters compared to Figure 14. Reproduced from Badillo, A.; Bellon, P.; Averback, R. S. to be submitted.
such as irradiation-induced segregation and precipitation and are thus required in PFMs aiming for system-specific predictive power. Second, it remains challenging to include in a PFM all the elements of the microstructure relevant to evolution under irradiation, namely point-defect clusters, dislocations, grain boundaries, and surfaces, although it has been shown here that models handling adequately a subset of these microstructural elements are now becoming available. Third, the numerical integration of the evolution equations is more challenging than for conventional PFMs in the sense that the continuous defect production, as well as the large difference in vacancy and interstitial mobility, usually prevents the use of long integration time steps, even in coarse microstructures. Finally, materials under irradiation constitute nonequilibrium systems that are quite sensitive to the amplitude and the structure of fluctuations, in particular the fluctuations resulting from
Phase Field Methods
0.08 0.06 0.04 0.02 0.00 0
400 300 200 100
CClus Int
CClus Vac
430
100
200
400 300 200 100 200
300 (a)
0.05 0.04 0.03 0.02 0.01 0.00 0
100 300
400
0
(b)
400
0
Figure 17 Defect cluster concentration fields corresponding to Figure 16: (a) clustered vacancies, Cc;V and (b) clustered interstitials Cc;A þ Cc;B (A and B atoms). Reproduced from Badillo, A.; Bellon, P.; Averback, R. S. to be submitted.
point defect and point-defect cluster production, and from ballistic mixing of species. A self-consistent and tractable PFM that would include both thermal and irradiation-induced fluctuations is still missing. Such a model would be very beneficial for the study of microstructural evolution under irradiation, especially that involving the nucleation of new phases, defect clusters, or gas bubbles see Chapter 1.13, Radiation Damage Theory; Chapter 1.14, Kinetic Monte Carlo Simulations of Irradiation Effects; and Chapter 1.09, Molecular Dynamics.
Acknowledgments The author gratefully acknowledges stimulating discussions with Robert Averback, Arnoldo Badillo, Yan Le Bouar, Alphonse Finel, and Maylise Nastar. The author also thanks Robert Averback for his critical reading of the manuscript. The support from the US DoE-BES under Grant DEFG02-05ER46217 is acknowledged.
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Chen, L. Q. J. Am. Ceram. Soc. 2008, 91(6), 1835–1844. Chen, L. Q.; Geng, C.; Kikuchi, R. In TMS Annual Meeting; Minerals, Metals & Materials Society (TMS): Orlando, FL, 1997. Seol, D. J.; Hu, S. Y.; Li, Y. L.; Shen, J.; Oh, K. H.; Chen, L. Q. Met. Mater. Int. 2003, 9(1), 61–66. Hu, S. Y.; Chen, L. Q. Acta Mater. 2004, 52(10), 3069–3074. Seol, D. J.; Hu, S. Y.; Oh, K. H.; Chen, L. Q. Met. Mater. Int. 2004, 10(5), 429–434. Seol, D. J.; Hu, S. Y.; Liu, Z. K.; Chen, L. Q.; Kim, S. G.; Oh, K. H. J. Appl. Phys. 2005, 98(4), 1–5. Bouville, M.; Hu, S.; Chen, L. Q.; Chi, D.; Srolovitz, D. J. Model. Simulat. Mater. Sci. Eng. 2006, 14(3), 433–443. Sheng, G.; et al. Appl. Phys. Lett. 2008, 93(23), 2329041–232904-3. Chen, L. Q.; Fan, D. J. Am. Ceram. Soc. 1996, 79(5), 1163–1168. Venkitachalam, M. K.; Chen, L. Q.; Khachaturyan, A. G.; Messing, G. L. Mater. Sci. Eng. A 1997, 238(1), 94–100. Fan, D.; et al. Comput. Mater. Sci. 1998, 9(3–4), 329–336. Moelans, N.; Blanpain, B.; Wollants, P. Acta Mater. 2005, 53(6), 1771–1781. Suwa, Y.; Saito, Y. Mater. Trans. 2005, 46(6), 1214–1220. Harun, A.; Holm, E. A.; Clode, M. P.; Miodownik, M. A. Acta Mater. 2006, 54(12), 3261–3273. Wang, Y. U. Acta Mater. 2006, 54(4), 953–961. Moelans, N.; Blanpain, B.; Wollants, P. Phys. Rev. Lett. 2008, 101(2), 025502-1–025502-4. Kim, S. G.; Park, Y. B. Acta Mater. 2008, 56(15), 3739–3753. Suwa, Y.; Saito, Y.; Onodera, H. Mater. Trans. 2008, 49(4), 704–709. Chang, K.; Feng, W.; Chen, L. Q. Acta Mater. 2009, 57(17), 5229–5236. Chen, Y.; Kang, X. H.; Xiao, N. M.; Zheng, C. W.; Li, D. Z. Wuli Xuebao/Acta Phys. Sin. 2009, 58(Spec. Iss.). Mallick, A.; Vedantam, S. Comput. Mater. Sci. 2009, 46(1), 21–25. McKenna, I. M.; Gururajan, M. P.; Voorhees, P. W. J. Mater. Sci. 2009, 44(9), 2206–2217. Takaki, T.; Hisakuni, Y.; Hirouchi, T.; Yamanaka, A.; Tomita, Y. Comput. Mater. Sci. 2009, 45(4), 881–888. Vedantam, S.; Mallick, A. Acta Mater. 2010, 58(1), 272–281. Jin, Y. M.; Wang, Y. U.; Khachaturyan, A. G. Appl. Phys. Lett. 2001, 79(19), 3071–3073. Wang, Y. U.; Jin, Y. M.; Khachaturyan, A. G. Phil. Mag. 2005, 85(2–3 Spec. Iss.), 261–277. Millett, P. C.; Wolf, D.; Desai, T.; Rokkam, S.; El-Azab, A. J. Appl. Phys. 2008, 104(3), 033512-1–033512-6. Wang, Y. U.; Jin, Y. M.; Cuitin˜o, A. M.; Khachaturyan, A. G. Acta Mater. 2001, 49(10), 1847–1857. Rodney, D.; Finel, A. Phase field methods and dislocations. In Materials Research Society Symposium – Proceedings, Boston, MA, 2001. Rodney, D.; Le Bouar, Y.; Finel, A. Acta Mater. 2003, 51(1), 17–30. Nabarro, F. R. N. Philos. Mag. Ser. 7 1941, 42(334), 1224–1231. Hu, S. Y.; Choi, J.; Li, Y. L.; Chen, L. Q. J. Appl. Phys. 2004, 96(1), 229–236. Widom, B. J. Chem. Phys. 1965, 43(11), 3892. Gratias, D.; Sanchez, J. M.; De Fontaine, D. Phys. Stat. Mech. Appl. 1982, 113(1–2), 315–337. Sanchez, J. M.; Ducastelle, F.; Gratias, D. Phys. Stat. Mech. Appl. 1984, 128(1–2), 334–350. Kikuchi, R.; Brush, S. G. J. Chem. Phys. 1967, 47(1), 195–203.
432 94. 95. 96. 97. 98. 99. 100. 101. 102. 103. 104. 105. 106. 107. 108. 109. 110. 111. 112. 113. 114. 115. 116. 117. 118. 119. 120. 121. 122. 123.
Phase Field Methods Kikuchi, R. J. Phys. (Paris) Colloq. 1977, 7(12), 38 Colloq C. Kikuchi, R. J. Chem. Phys. 1976, 65(11), 4545–4553. Cenedese, P.; Kikuchi, R. Phys. Stat. Mech. Appl. 1994, 205(4), 747–755. Kikuchi, R. J. Chem. Phys. 1972, 57(2), 783–787. Kikuchi, R. J. Chem. Phys. 1972, 57(2), 787–791. Kikuchi, R. J. Chem. Phys. 1972, 57(2), 792–802. Bronchart, Q.; Le Bouar, Y.; Finel, A. Phys. Rev. Lett. 2008, 100(1), 015702-1–015702-4. Martin, G. Phys. Rev. B 1990, 41(4), 2279–2283. Martin, G. Phys. Rev. B 1994, 50(17), 12362–12366. Bellon, P.; Martin, G. Phys. Rev. B Condens. Matter Mater. Phys. 2002, 66(18), 1842081–1842087. Khromov, K. Y.; Pankratov, I. R.; Vaks, V. G. Phys. Rev. B Condens. Matter Mater. Phys. 2005, 72(9), 1–22. Cook, H. E. Acta Metall. 1970, 18(3), 297. Wang, Y.; Wang, H.-Y.; Chen, L.-Q.; Khachaturyan, A. G. J. Am. Ceram. Soc. 1995, 78(3), 657–661. Dobretsov, V. Y.; Pankratov, I. R.; Vaks, V. G. JETP Lett. 2004, 80(9), 602–607. Martin, G.; Bellon, P. In Advances in Research and Applications, Solid State Physics, Vol. 50; Academic Press: New York, 1997; pp 189–331. Roussel, J. M.; Bellon, P. Phys. Rev. B Condens. Matter Mater. Phys. 2002, 65(14), 1441071–1441079. Landauer, R. J. Appl. Phys. 1962, 33(7), 2209. Landauer, R. Phys. Today 1978, 31(11), 23–29. Horsthemke, W.; Lefever, R. Phys. Lett. A 1977, 64(1), 19–21. Horsthemke, W.; Lefever, R. Phys. Lett. A 1984, 106(1–2), 10–12. Nastar, M.; Dobretsov, V. Y.; Martin, G. Phil. Mag. A 2000, 80(1), 155–184. Nastar, M. Phil. Mag. 2005, 85(32), 3767–3794. Barbe, V.; Nastar, M. Phil. Mag. 2006, 86(11), 1513–1538. Barbe, V.; Nastar, M. Phys. Rev. B 2007, 76(5), 0542061–054206-8. Enrique, R. A.; Bellon, P. Phys. Rev. Lett. 2000, 84(13), 2885-1–2885-4. Martin, G. Phys. Rev. B 1984, 30(3), 1424–1436. Averback, R. S.; de la Rubia, T. D. Solid State Phys. 1998, 51, 281–402. Enrique, R. A.; Nordlund, K.; Averback, R. S.; Bellon, P. J. Appl. Phys. 2003, 93(5), 2917–2923. Enomoto, Y.; Sawa, M. Surf. Sci. 2002, 514(1–3), 68–73. Krasnochtchekov, P.; Averback, R. S.; Bellon, P. Phys. Rev. B Condens. Matter Mater. Phys. 2005, 72(17), 1–12.
124.
Nelson, R. S.; Mazey, D. J.; Hudson, J. A. J. Nucl. Mater. 1972, 44(3), 318. 125. Bellon, P. Precipitate and microstructural stability in alloys subjected to sustained irradiation. In Materials Science with Ion Beams; Bernas, H.; Springer: Berlin, 2010; pp 29–52. 126. Enrique, R. A.; Bellon, P. Phys. Rev. B Condens. Matter Mater. Phys. 1999, 60(21), 14649–14659. 127. Enrique, R. A. Bellon, P. Phys. Rev. B Condens. Matter Mater. Phys. 2004, 70(22), 224106-1–224106-6. 128. Chen, L. Q.; Khachaturyan, A. G. Phys. Rev. Lett. 1993, 70(10), 1477–1480. 129. Schulson, E. M. J. Nucl. Mater. 1979, 83(2), 239–264. 130. Matsumura, S.; Muller, S.; Abromeit, C. Phys. Rev. B 1996, 54(9), 6184–6193. 131. Martin, G.; Soisson, F.; Bellon, P. J. Nucl. Mater. 1993, 205(C), 301–311. 132. Bourdeau, F.; Camus, E.; Abromeit, C.; Wollenberger, H. Phys. Rev. B 1994, 50(22), 16205–16211. 133. Camus, E.; Abromeit, C.; Bourdeau, F.; Wanderka, N.; Wollenberger, H. Phys. Rev. B 1996, 54(5), 3142–3150. 134. Evans, J. H. Nature 1971, 229(5284), 403–404. 135. Jager, W.; Trinkaus, H. J. Nucl. Mater. 1993, 205, 394–410. 136. Ghoniem, N. M.; Walgraef, D.; Zinkle, S. J. J. Comput. Aided Mater. Des. 2001, 8(1), 1–38. 137. Yu, H. C.; Lu, W. Acta Mater. 2005, 53(6), 1799–1807. 138. Foreman, A. J. E. Harwell Report, AERE-R 7135; 1972. 139. Woo, C. H.; Frank, W. J. Nucl. Mater. 1985, 137(1), 7–21. 140. Golubov, S. I.; Singh, B. N.; Trinkaus, H. J. Nucl. Mater. 2000, 276, 78–89. 141. Woo, C. H. J. Nucl. Mater. 2000, 276, 90–103. 142. Heinisch, H. L.; Singh, B. N. Phil. Mag. 2003, 83(31–34), 3661–3676. 143. Hu, S.; Henager, C. H., Jr. J. Nucl. Mater. 2009, 394, 155–159. 144. Rokkam, S.; El-Azab, A.; Millett, P.; Wolf, D. Model. Simulat. Mater. Sci. Eng. 2009, 17(6), 0640021–064002-18. 145. Millett, P. C.; Rokkam, S.; El-Azab, A.; Tonks, M.; Wolf, D. Model. Simulat. Mater. Sci. Eng. 2009, 17(6), 064003-1–064003-12. 146. Badillo, A.; Bellon, P.; Averback, R. S. to be submitted. 147. Barbu, A.; Ardell, A. J. Scripta Metall. 1975, 9(11), 1233–1237. 148. Barbu, A.; Martin, G. Scripta Metall. 1977, 11(9), 771–775.
1.16
Dislocation Dynamics
N. M. Ghoniem University of California, Los Angeles, CA, USA
ß 2012 Elsevier Ltd. All rights reserved.
1.16.1
Introduction
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1.16.2 1.16.2.1 1.16.2.2 1.16.2.3 1.16.3 1.16.3.1 1.16.3.2 1.16.3.3 1.16.3.4 1.16.3.4.1 1.16.3.4.2 1.16.3.4.3 1.16.3.4.4 1.16.4 References
Computational Defect Mechanics Isotropic Crystals Anisotropic Crystals Illustrative Example: Initial Bow-Out of a Pinned Dislocation Applications to Modeling Radiation Effects Dislocation Interaction with Radiation-Induced Defects Formation of Dislocation Channels in Irradiated Materials Radiation Hardening The Ductile-to-Brittle Transition in Pressure Vessel Steels Background Crack-dislocation dynamics model Calculation techniques Results and discussion Outlook
435 435 437 438 439 439 441 443 447 447 448 449 450 452 452
Abbreviations BDT CISH CRSS DBTT DD EOM fcc FEA FEM HRR MC OFHC PSB RKR SFT SIA
Brittle–ductile transition Cascade-induced source hardening Critical resolved shear stress Ductile-to-brittle transition temperature Dislocation dynamics Equation of motion Face-centered cubic Finite element analysis Finite element method Hutchinson, Rice, and Rosengren Master curve Oxygen free high conductivity Persistent slip band Ritchie, Knott, and Rice Stacking fault tetrahedral Self-interstitial atom
1.16.1 Introduction A fundamental description of plastic deformation has been recently pursued in many parts of the world as a result of dissatisfaction with the limitations of continuum plasticity theory. Although continuum
models of plastic deformation are extensively used in engineering practice, their range of application is limited by the underlying database. The reliability of continuum plasticity descriptions is dependent on the accuracy and range of available experimental data. Under complex loading situations, however, the database is often hard to establish. Moreover, the lack of a characteristic length scale in continuum plasticity makes it difficult to predict the occurrence of critical localized deformation zones. In small volumes, or in situations where submicrometer resolution of material deformation is required (e.g., in thin films, micropillars, near dislocations, grain boundaries, etc.), the use of continuum plasticity models would be questionable. Although homogenization methods have played a significant role in determining the elastic properties of new materials from their constituents (e.g., composite materials), the same methods have failed to describe plasticity. It is widely appreciated that plastic strain is fundamentally heterogenous, displaying high strains concentrated in small material volumes, with virtually undeformed regions in between. Experimental observations consistently show that plastic deformation is heterogeneous at all length scales. Depending on the deformation mode, heterogeneous dislocation structures appear with definitive wavelengths. A satisfactory 433
434
Dislocation Dynamics
description of realistic dislocation patterning and strain localization has been rather elusive. Attempts aimed at this question have been based on statistical mechanics, reaction–diffusion dynamics, or the theory of phase transitions. Much of the effort has aimed at clarifying the fundamental origins of inhomogeneous plastic deformation. On the other hand, engineering descriptions of plasticity have relied on experimentally verified constitutive equations. Planar dislocation arrays are formed under monotonic stress deformation conditions, and are composed of parallel sets of dislocation dipoles. While persistent slip bands (PSBs) are found to be aligned in planes with their normal parallel to the direction of the critical resolved shear stress (CRSS), planar arrays are aligned in the perpendicular direction. Dislocation cell structures, on the other hand, are honeycomb configurations in which the walls have high dislocation density, while the cell interiors have low dislocation density. Cells can be formed under both monotonic and cyclic deformation conditions. However, dislocation cells under cyclic deformation tend to appear after many cycles. Direct experimental observations of these structures have been reported for many materials. Two main approaches have been advanced to model the mechanical behavior in this meso length scale. The first is based on statistical mechanics methods. In this approach, evolution equations for statistical averages (and possibly for higher moments) are to be solved for a complete description of the deformation problem. The main challenge in this regard is that, unlike the situation encountered in the development of the kinetic theory of gases, the topology of interacting dislocations within the system must be included. The second approach, commonly known as dislocation dynamics (DD), was initially motivated by the need to understand the origins of heterogeneous plasticity and pattern formation. An early variant of this approach (the cellular automata) was first developed by Lepinoux and Kubin,1 and this was followed by the proposal of DD by Ghoniem and Amodeo.2–4 In these early efforts, dislocation ensembles were modeled as infinitely long and straight in an isotropic, infinite elastic medium. The method was further expanded by a number of researchers,5 with applications demonstrating simplified features of deformation microstructure. DD has now become an important computer simulation tool for the description of plastic deformation at the micro- and mesoscales (i.e., the size range of a fraction of a micrometer
to tens of micrometers). The method is based on a hierarchy of approximations that enable the solution of relevant problems with today’s computational resources. In its early versions, the collective behavior of dislocation ensembles was determined by direct numerical simulations of the interactions between infinitely long, straight dislocations.5 Recently, several research groups have extended the DD methodology to the more physical, yet considerably more complex case of three-dimensional (3D) simulation. The method can be traced back to the concepts of internal stress fields and configurational forces. The more recent development of 3D lattice DD by Kubin and coworkers6–8 has resulted in greater confidence in the ability of DD to simulate more complex deformation microstructures. More rigorous formulations of 3D DD have contributed to its rapid development and applications in many systems.9–15 Many experimental observations have shown that neutron irradiation of metals and alloys at temperatures below recovery stage V causes a substantial increase in the upper yield stress (radiation hardening) and, beyond a certain dose level, induces a yield drop and plastic instability (see Chapter 1.03, RadiationInduced Effects on Microstructure and Chapter 1.04, Effect of Radiation on Strength and Ductility of Metals and Alloys). Furthermore, the postdeformation microstructure of a specimen showing an upper yield point has demonstrated two significant features. First, the onset of plastic deformation is generally found to coincide with the formation of ‘cleared’ channels, where practically all plastic deformation takes place. The second feature refers to the fact that the material volume in between cleared channels remains almost undeformed (i.e., no new dislocations are generated during deformation). In other words, the initiation of plastic deformation in these irradiated materials occurs in a very localized fashion. This specific type of plastic flow localization is considered to be one of many possible plastic instabilities in both irradiated and unirradiated materials.16,17 A theory of radiation hardening was proposed by Seeger18 in terms of dislocation interaction with radiation-induced obstacles (referred to as depleted zones). Subsequently, Foreman19 performed computer simulations of loop hardening, in which the elastic interaction between the dislocation and loops was neglected. The model is based on Orowan’s mechanism, which assumes that the obstacles are indestructible. In this view of matrix hardening, the stress necessary to overcome localized interaction barriers leads to the increase in the yield strength, while long-range elastic
Dislocation Dynamics
interactions are completely ignored. These short-range dislocation-barrier interaction models lead to an estimate of the increase of the CRSS of the form: Dt ¼ amb/l, where a is a numerical constant representing obstacle strength, m the shear modulus, b the magnitude of the Burgers vector, and l is the average interobstacle distance. Kroupa,20 on the other hand, viewed hardening as a result of the long-range elastic interaction between slip dislocations and prismatic loops. In their model of friction hardening, the force necessary to move a rigid, straight dislocation on its glide plane past a prismatic loop was estimated. In these two classes of hardening models, all dislocation sources are assumed to be simultaneously activated at the yield point, and plastic deformation is assumed to be homogeneous throughout the material volume. Thus, they do not address the physics of plastic flow localization. Singh et al.21 introduced the concept of ‘stand-off distance’ for the decoration of dislocations with small self-interstitial atom (SIA) loops, and proposed the ‘cascade-induced source hardening’ (CISH) model, in analogy with Cottrell atmospheres.22 The central question of the formation of dislocation decoration was treated analytically by Trinkaus et al.23 Subsequent detailed elasticity calculations showed that glissile defect clusters that approach dislocation cores within a stand-off distance are absorbed, while clusters can accumulate just outside this distance. Trinkaus et al. concluded that dislocation decoration is a consequence of defect cluster mobility and trapping in the stress field of grown-in dislocations. The CISH model was used to calculate the stress necessary to free decorated dislocations from the atmosphere of loops around them (upper yield, followed by yield drop), so that these freed dislocations can act as dislocation sources. The CRSS increase in irradiated Cu was shown by Singh et al.21 to be given by: Dt 0.1m(b/l )(d/y)2, where m is the shear modulus, and b, l, d, and y are the Burgers vector, interdefect distance, defect diameter, and standoff distance, respectively. Assuming that y d, and l 35b, they estimated Dt/m 5 103. The phenomenon of yield drop was proposed to result from the unpinning of grown-in dislocations, decorated with small clusters or loops of SIAs. Neutron irradiation of pure copper at 320 K leads to an increase in the upper yield stress and causes a prominent yield drop and the initiation of plastic flow localization. In this chapter, we first present a description of the mathematical and computational foundations of DD. We then assess the physical mechanisms, that are responsible for the initiation of plastic instability in irradiated face-centered cubic (fcc) metals
435
by detailed numerical simulations of the interaction between dislocations and radiation-induced defect clusters. Two distinct problems that are believed to cause the onset of plastic instability are addressed in the present study. First, we aim to determine the mechanisms of dislocation unlocking from defect cluster atmospheres as a result of the long-range elastic interaction between dislocations and sessile prismatic interstitial clusters situated just outside the stand-off distance. The main new feature of this analysis is that dislocation deformation is explicitly considered during its interaction with SIA clusters. Second, we investigate the mechanisms of structural softening in flow channels as a consequence of dislocation interaction with stacking fault tetrahedra (SFTs). Based on these numerical simulations, a new mechanism of channel formation is proposed, and the magnitude of radiation hardening is also computed. As an important application of DD simulations, we then present a model that describes the shift in the ductile-to-brittle transition temperature (DBTT) as a result of neutron irradiation, and compare this physically based model to experimental data. Finally, we discuss the future outlook for DD simulations and the role they may play in understanding radiation effects on the mechanical properties of structural materials.
1.16.2 Computational Defect Mechanics In this section, we develop the basic ingredients of computational DD, following closely the parametric method. The key idea here is that dislocations are divided into segments of smooth parametric curves in three dimensions. Each segment has two nodes, one on each end of the segment. Forces are computed on the segments, and effectively calculated at the nodes. An equation of motion (EOM) is then derived for each node, and the whole system is assembled in a matrix form, very similar to the finite element method (FEM). The matrix system is solved with time increments by either explicit or implicit integration methods. We will now go into some of the computational details. 1.16.2.1
Isotropic Crystals
When the Green’s functions are known, the elastic field of a dislocation loop can be constructed by a surface integration. The starting point in this
436
Dislocation Dynamics
calculation is the displacement field in a crystal containing a dislocation loop, which can be expressed as24,25 ð @ ui ðxÞ ¼ Cjlmn bm 0 Gij ðx; x0 Þnn dSðx0 Þ ½1 @x S l where Cjlmn is the elastic constants tensor, Gij (x, x0 ) are the Green’s functions at x due to a point force applied at x0 , S is the surface capping the loop, nn is a unit normal to S, and bm is the Burgers vector. The elastic distortion tensor, ui,j can be obtained from eqn [1] by differentiation. The symmetric part of ui,j (elastic strain tensor) is then used in the stress– strain relationship to find the stress tensor at a field point x, caused by the dislocation loop. In an elastically isotropic and infinite medium, a more efficient form of eqn [1] can be expressed as a line integral performed over the closed dislocation loop as26 bi ui ¼ rC Ak dlk 4p 1 1 Ekmn bn R;mi dlk ½2 þ rC Eikl bl R;pp þ 8p 1n where m and n are the shear modulus and Poisson’s ratio, respectively, b is the Burgers vector with Cartesian components bi, and the vector potential Ak ðRÞ ¼ Eijk Xi sj =½RðR þ R sÞ satisfies the differential equation Epik Ak;p ðRÞ ¼ Xi R3 , where s is an arbitrary unit vector. The radius vector R connects a source point on the loop to a field point, as shown in Figure 1, with Cartesian components Ri, successive partial derivatives R,ijk. . ., and magnitude R. The line integrals are carried out along a closed contour C defining the dislocation loop of differential arc length dl of components dlk. g3 = b b P
w=0
g2 = t
R g1 = e
Q
r w=1 1z
1x
1y
Figure 1 Parametric representation of dislocation segments. Reproduced from Ghoniem, N. M.; Huang, J.; Wang, Z. Philos. Mag. Lett. 2001, 82(2), 55–63.
Consider now the virtual motion of the dislocation loop. The mechanical power during the virtual motion is composed of two parts: (1) change in the elastic energy stored in the medium upon loop motion under the influence of its own stress (i.e., the change in the loop self-energy); (2) the work done on moving the loop as a result of the action of external and internal stresses, excluding the stress contribution of the loop itself. These two components constitute the work of Peach–Koehler.27 The main idea of DD is to derive approximate equations of motion from the principle of virtual power dissipation of the second law of thermodynamics,15 by finding virtual Peach–Koehler forces that would result in the simultaneous displacement of all dislocation loops in the crystal. A major simplification is that this many-body problem is reduced to the single loop problem. In this simplification, instead of moving all the loops simultaneously, they are moved sequentially, with the motion of each one against the collective field of all other loops. The approach is reminiscent of the single-electron simplification of the many-electron problem in quantum mechanics. If the material is assumed to be elastically isotropic and infinite, a great reduction in the level of required computations ensues. First, surface integrals can be replaced by line integrals along the dislocation. Second, Green’s functions and their derivatives have analytical solutions. Thus, the starting point in most DD simulations so far is a description of the elastic field of dislocation loops of arbitrary shapes by line integrals of the form proposed in de Wit26 as m 1 r R;mpp ðEjmm dli þ Eimm dlj Þ sij ¼ 4p C 2 1 þ Ekmn ðR;ijm dij R;ppm Þdlk ½3 1n where m and n are the shear modulus and Poisson’s ratio, respectively. The line integral is discretized, and the stress field of dislocation ensembles is obtained by a summation process over line segments, as shown in Figure 1. Once the parametric curve for the dislocation segment is mapped onto the scalar interval {o 2 [0,1]}, the stress field everywhere is obtained as a fast numerical quadrature sum.13 The Peach–Koehler force exerted on any other dislocation segment can be obtained from the total stress field (external and internal) at the segment as27 FPK ¼ s b t
½4
Dislocation Dynamics
The total self-energy of the dislocation loop is determined by double line integrals. However, Gavazza and Barnett28 have shown that the first variation in the self-energy of the loop can be written as a single line integral, and that the majority of the contribution is governed by the local line curvature. Based on these methods for evaluation of the interaction and self-forces, the weak variational form of the governing EOM of a single dislocation loop was developed by Ghoniem et al.14 as ð ðFkt Bak Va Þdrk jds j ¼ 0 ½5 G
Here, Fkt are the components of the resultant force, consisting of the Peach–Koehler force FPK (generated by the sum of the external and internal stress fields), the self-force Fs, and the osmotic force Fo (in case climb is also considered14). The resistivity matrix (inverse mobility) is Bak, Va are the velocity vector components, and the line integral is carried out along the arc length of the dislocation ds. To simplify the problem, let us define the following dimensionless parameters: r r ¼ ; a
f ¼
F ; ma
t ¼
mt B
Here, a is lattice constant, m the shear modulus, and t is time. Hence eqn [5] can be rewritten in dimensionless matrix form as ð dr T ½6 dr f jds j ¼ 0 dt G Here, f ¼ ½f1 ; f2 ; f3 T and r ¼ ½r1 ; r2 ; r3 T , are all dependent on the dimensionless time t*. Following Ghoniem et al.,14 a closed dislocation loop can be divided into Ns segments. In each segment j, we can choose a set of generalized coordinates qm at the two ends, thus allowing parameterization of the form r ¼ CQ
½7
Here, C ¼ ½C1 ðoÞ; C2 ðoÞ; . . . ; Cm ðoÞ; Ci ðoÞ, (i ¼ 1, 2,. . .,m) are shape functions dependent on the parameter (0 o 1), and Q ¼ ½q1 ; q2 ; . . . ; qm T , with qi a set of generalized coordinates. Substituting eqn [7] into eqn [6], we obtain Ns ð X > > > dQ ½8 dQ C f C C jds j ¼ 0 dt j ¼1 Gj Let,
ð fj ¼
>
C f jdsj; Gj
ð kj ¼
C> Cjds j Gj
437
Following a similar procedure to the FEM, we assemble the EOM for all contiguous segments in global matrices and vectors, as F¼
Ns X
fj;
K¼
j ¼1
Ns X
kj
j ¼1
Then, from eqn [8] we get dQ K ¼F dt 1.16.2.2
½9
Anisotropic Crystals
For an anisotropic, linearly elastic crystal, Mura29 derived a line integral expression for the elastic distortion of a dislocation loop, as ð ui;j ðxÞ ¼ Ejnk Cpqmn bm Gip;q ðx x0 Þnk dlðx0 Þ ½10 L
where nk is the unit tangent vector of the dislocation loop line L, dl is the dislocation line element, Ejnh is the permutation tensor, Cijkl is the fourth-order elastic constants tensor, Gij ;l ðx x0 Þ ¼ @Gij ðx x0 Þ=@xl , and Gij(x x0 ) are the Green’s tensor functions, which correspond to displacement components along the xi direction at point x due to a unit point force in the xj direction applied at point x0 in an infinite medium. It can be seen that the elastic distortion eqn [10] involves derivatives of the Green’s functions, which need special consideration. For general anisotropic solids, analytical expressions for Gij,k are not available. However, these functions can be expressed in an integral form (see e.g., Barnett,30 Willis,31 Bacon et al.,32 and Mura33), as Gij ;k ðx x 0 Þ ð 1 ½r k Nij ðkÞD1 ðkÞ ¼ 8p2 jr j2 Ck þ kk Clpmq ðr p kq þ kp r q ÞNil ðkÞNjm ðkÞD2 ðkÞdf
½11
where r ¼ x x0 , r ¼ r=jrj, k is the unit vector on the plane normal to r, the integral is taken around the unit circle Ck on the plane normal to r, and Nij(k) and D(k) are the adjoint matrix and the determinant of the second-order tensor Cikjlkkkl, respectively. At a point P on a dislocation line L, the external force per unit length is obtained by the Peach– Koehler formula FA ¼ (bsA) t, where sA is the sum of the stress from an applied load and that arising from internal sources at P, b is the Burgers vector of the dislocation, and t is the unit tangent vector of the element dl. For the special, yet important case of a
438
Dislocation Dynamics
planar dislocation loop lying on a glide plane with unit normal n, the glide force acts along the in-plane normal to L at P: m ¼ n t. The glide component FgA of the external force can be obtained by resolving FA along m as FgA ¼ FA m ¼ bsA n
½12
The in-plane self-force at the point P on the loop is also obtained in a manner similar to the external Peach–Koehler force, with an additional contribution from stretching the dislocation line upon a virtual infinitesimal motion34: F S ¼ kEðt Þ b s~S n
½13
where E(t) is the prelogarithmic energy factor for an infinite, straight dislocation parallel to t: Eðt Þ ¼ 12bSðt Þn, with S(t) being the stress tensor of an infinite straight dislocation along the loop’s tangent at P. ss is the self-stress tensor due to the dislocation L, and s~ ¼ 12½sS ðP þ EmÞ þ sS ðP EmÞ is the average self-stress at P, k is the in-plane curvature at P, and E ¼ jb j=2. Barnett and coworkers28,34 analyzed the structure of the self-force as a sum: 8 F S ¼kEðt Þ k Eðt Þ þ E 00 ðt Þln Ek J ðL; PÞ þ Fcore
3-Nodes 4-Nodes 5-Nodes
½14
where the second and third terms are line tension contributions, which usually account for the main part of the self-force, J(L, P) is a nonlocal contribution from other parts of the loop, and Fcore is due to the contribution to the self-energy from the dislocation core. 1.16.2.3 Illustrative Example: Initial Bow-Out of a Pinned Dislocation To illustrate the computational procedure involved in the present method, we consider here a very simple example in which the equations of motion can be solved analytically for one time step. Our purpose here is to highlight the essential features of the present computational method. More complex examples, which require extensive computations, will be given in the next section. Assume that we are interested in determining the shape of a dislocation line, pinned at two ends and under the influence of pure shear loading on its glide plane. The glide mobility is assumed to be isotropic and constant, and the segments will be taken as linear for illustrative purposes only. The dislocation line is pinned at points L and R,
1.5tbδt/B 6/5
6/7
9/5
tb L/4
L/4
L/4
L/4
Figure 2 Nodal displacements for the first time step of an initially straight segment.
with only two linear and equal segments connected at point A, as shown in Figure 2. We will compute the shape of the line, advancing it from its initial straight configuration to a curved position. Under these simplifications, the variation in Gibbs free energy dG for any one of the two segments is given by ð1 ð1 dG ¼ B V dr jdsj ¼ f t dr jdsj 0
½15
0
Now, we expand the virtual displacement and velocity in only two shape functions: C1 ¼ u; C2 ¼ 1 u. Thus drk ¼ dqik Ci
½16
Vk ¼ qik;t Ci
½17
Since we allow the displacement to be only in a direction normal to the dislocation line (y direction), we drop the subscript k as well. For arbitrary variations of dqik, the following equation is applicable to any of the two segments. ð1 ð1 Dt ðfPK þ fS ÞCi jds j ¼ B Dqm Cm Ci jds j 0
½18
0
Equation [18] can be explicitly integrated over a short time interval Dt. The resistivity matrix elements are Ð1 defined by gim B Ci Cm jds j, and the force vector Ð1
0
elements by fi Ci ðfPK þ fS Þjds j. With these 0
definitions, we have the following (2 2) algebraic system for each of the two elements: Dqm gim ¼ fi Dt
½19
Dislocation Dynamics
For any one linear element, the line equation can be determined by x u 0 q1 ¼ ½20 y 0 ð1 uÞ q2 And the resistivity matrix can be simplified as Bl 2 1 ½gmn ¼ 6 1 2
½21
Furthermore, as a result of the shear stress t and the absence of self-forces during the first time step only, the distributed applied force vector reads tbl 1 ½22 ffm g ¼ 2 1 Since the dislocation line is divided into two equal segments, we can now assemble the force vector, stiffness matrix, and displacement vector in the global coordinates, and arrive at the following equation for the global nodal displacements DQi : 9 8 9 8 9 38 2 1 0 < DQ1 = < F1 = <1= Bl 4 tblDt 1 4 1 5 DQ2 ¼ 2 þ Dt 0 ½23 : ; : ; 12 4 : ; DQ3 F3 0 1 2 1 2
An important point to note here is that, at the two fixed ends, we know the boundary conditions but the reaction forces needed to satisfy overall equilibrium are unknown. These reactions act on the fixed obstacles at L and R, and are important in determining the overall stability of the configuration (e.g., if they exceed a critical value, the obstacle is destroyed, and the line is released). If DQ1 ¼ DQ3 ¼ 0 at both fixed ends, we can easily solve for the nodal displacement DQ2 ¼ 32 rbDt B and for the unknown reaction forces at the two ends: F1 ¼ F3 ¼ 118tbl. If we divide the dislocation line into more equal segments, the size of the matrix equation expands, but the nodal displacements and reaction forces can be calculated similarly. Results of analytical solutions for successively larger number of nodes on the dislocation segment are shown in Figure 2. In the following section, we present several applications of the computational method to irradiated materials, where modeling defect behavior has contributed to a greater understanding of radiation effects.
1.16.3 Applications to Modeling Radiation Effects Radiation interaction with materials results in the production of lattice defects. The motion and interaction between radiation-induced lattice defects among
439
themselves and the existing microstructure have direct consequences to the macroscopic response of the material to the radiation environment. We present in this section several applications of computational DD to modeling of radiation-induced defects and their effects on mechanical properties. 1.16.3.1 Dislocation Interaction with Radiation-Induced Defects Using an infinitesimal loop approximation, Kroupa found the stress tensor of a prismatic loop to be of the form sij ¼ kij m bR2/2r3, where kij is an orientation factor of order unity, R is the loop radius, and r the distance from the loop center. The total force and its moment on an SIA cluster can be expressed respectively as Fi ¼ n0j sjk;i bk0 dA0 Mi ¼ Eijk n0j bl0 sik dA0 where n0j , bk0 , dA0 refer to the Cartesian components of the normal vector, the Burgers vector, and the habit plane area of the cluster, respectively. As a mobile SIA dislocation loop moves closer to the core of the slip loop, the turning moment on its habit plane increases. When the mechanical work of rotation exceeds a critical value of 0.1 eV per crowdion, we assume that the cluster changes its Burgers vector and habit plane, and moves to be absorbed into the dislocation core. Thus, the mechanical work for cluster rotation is equated to a Ðy critical value (i.e., dW ¼ y12 Mi dyi ¼ DUerit ) and used as a criterion to establish the stand-off distance.35 The two main aspects of dislocation interaction with defect clusters that affect both hardening and ensuing plastic flow localization are (1) dislocation unlocking from defect cluster atmospheres; and (2) destruction of SFTs on nearby slip planes by gliding dislocations. The interaction between grown-in dislocations and trapped defect clusters has been shown to lead to unfaulting of vacancy clusters in the form of vacancy loops. It can also result in rotation of the habit plane of mobile SIA clusters. Once either of these possibilities is realized for a vacancy or SIA cluster, it is readily absorbed into the dislocation core. Ghoniem et al.35 used these conditions to determine an appropriate stand-off distance from the dislocation core, which is free of irradiation-induced defect clusters. It is estimated that clusters within a distance of 3–9 nm from the dislocation core in Cu will be absorbed, either by rotation of their Burgers vector or by unfaulting. We will use this estimate as a guide to calculations of long-range interactions of dislocations with sessile
440
Dislocation Dynamics
prismatic SIA clusters situated outside the stand-off distance. While the experimentally observed average SFT size is 2.5 nm for oxygen free high conductivity (OFHC) copper, the radius of a sessile interstitial cluster, which results from coalescence of smaller mobile clusters, is assumed to be in the range 4–20 nm. The local density of interstitial defect clusters at the standoff distance is taken to be in the range 0.6–4 1024 m3, giving an average intercluster spacing of 18–35a. In subsequent computer simulations, we use the following set of material data for Cu: lattice constant a ¼ 0.3615 nm, shear modulus m ¼ 45.5 GPa, Poisson’s ratio n ¼ 0.35, and F–R source length L ¼ 1500–2000a. Consider now the more complex interaction between an expanding Frank–Read (F–R) dislocation source, and the full field of multiple sessile SIA clusters (loops) present in a region of decoration. As the F–R source expands in the elastic field of SIA clusters, each point on the dislocation line will experience a resistive (or attractive) force, which must be overcome for the dislocation to move further. The dislocation line curvature and, hence, the local self-force also change dynamically. To determine the magnitude of collective cluster resistance, systematic calculations for the dynamics of interaction between
SIA clusters are presented (Figure 3). When the SIA clusters are all attractive, the dislocation line is immediately pulled into their atmosphere, but as the applied stress is increased, the dislocation remains trapped by the force field of SIA clusters. When the stress is increased to 200 MPa, the line develops an asymmetric configuration as a result of its Burgers vector orientation, and an unzipping instability eventually unlocks the F–R source from the collective cluster atmosphere. This asymmetric unlocking mode is characteristic of a high linear cluster density on smaller sections of the F–R source decoration, where the linear density of SIA clusters is 501 (cluster/lattice constant). Figure 3 shows the detailed dynamics of the collective cluster interaction with an expanding F–R source. A fluctuation in the line shape is amplified by the combined effects of the applied and self-forces on the middle section of the F–R source, and the dislocation succeeds in penetrating through the collective cluster field at a critical tensile stress of s11 ¼ 180 MPa (or equivalently at CRSS of t/m ¼ 0.0015). The critical shear stress (in units of the shear modulus) to unlock the F–R source is shown in Figure 4 as a function of the stand-off distance for a fixed intercluster distance of 50a. The results of current calculations are
-600
-700 -200
[-1 -1 2] (a)
-800
-900
-1000
bc
-1100 b -1200
-500
0 [-1 1 0] (a)
500
Figure 3 Dynamics of the symmetric unlocking mechanism, initiated by small fluctuations in the dislocation line as it passes near the cluster atmosphere. The Burgers vector is b ¼ 12½110.
Dislocation Dynamics
The current estimates for the unlocking stress are thus consistent with the experimental data, and indicate that the operation of F–R sources from decorated dislocations can be initiated by one (or both) of the following possibilities:
0.2 Trinkaus et al. Present results (L = 50a) Present results (L = 75a) 0.15
t/m (%)
441
1. Activated F–R sources are decorated with a statistically low linear defect cluster density; 2. Dislocation sources are initiated at stress singularities in regions of internal stress concentration.
0.1
0.05
1.16.3.2 Formation of Dislocation Channels in Irradiated Materials 0
0
0.0002
0.0004
0.0006
1/d*2 Figure 4 Scaling of the critical shear stress with the stand-off distance for a fixed intercluster spacing of 50a.
compared with the analytical estimates of Trinkaus et al.23 For larger stand-off distances, the current results show a larger critical stress as compared to the analytical estimates, while for stand-off distances smaller than 60a, a smaller critical stress is required to unlock the F–R source. When the stand-off distance is large, the applied stress must overcome the self-force, which results from the finite length of the F–R source, in addition to the collective cluster elastic field. At smaller stand-off distances, however, the dislocation easily unlocks by one of the two unzipping instability modes discussed earlier, and the predicted CRSS is smaller than analytical estimates. At intercluster distances smaller than 70a, the dislocation shape instability results in a CRSS value that is smaller than the corresponding analytical result. It is estimated that the required CRSS is 0.001m (50 MPa for copper), for an average intercluster distance of l 50a, and a standoff distance of 40a. It is experimentally difficult to determine the local value of l in the decoration region of dislocations, which is likely to vary considerably, depending on the character of the dislocation Burgers vector. However, l 50a is an upper bound, while l 20–30a is more likely. Since the CRSS is roughly inversely proportional to l, the most likely value of the CRSS to unlock dislocations and start the operation of F–R sources would be tCRSS 100–150 MPa. Depending on the local value of the Schmidt factor, the corresponding uniaxial applied stress is thus likely to be on the order of 200–300 MPa, under conditions of heavy decoration (i.e., at a displacement damage dose of 0.1 dpa).
Once dislocation sources are unlocked from their decoration atmospheres causing a yield drop, they additionally interact with the surrounding random field of defect clusters (e.g., vacancy loops or SFTs). We investigate here the interaction between emitted F–R dislocations and vacancy-type defect clusters as a possible mechanism of radiation softening immediately beyond the yield point. Numerical computer simulations are performed for the penetration of undissociated slip dislocation loops emitted from active F–R sources (i.e., unlocked from defect decorations) against a random field of SFTs or sessile Frank loops of the type: 13h111if111g. At the present level of analysis, there is no distinction between SFTs and vacancy loops as they are modeled as point obstacles to dislocation motion that can be destroyed once an assumed critical force on them is reached. The long-range elastic field of these small obstacles is ignored. The random distribution of SFTs is generated as follows: (1) The volumetric density of SFTs is used to determine the average 3D position of each generated SFT; (2) A Gaussian distribution function is used (with the standard deviation being 0.1–0.3 of the average spacing) to assign a final position for each generated SFT around the mean value. (3) The intersection points of SFTs with glide planes are computed by finding all SFTs that intersect the glide plane. For simplicity, we perform this procedure assuming that SFTs are spherical and uniform in size. Initially, one slip dislocation loop is introduced between two fixed ends and a search is performed for all neighboring SFTs on the glide plane. Subsequent nodal displacements (governed by the local velocity) are adjusted such that a released segment interacts with only one SFT at any given time. The interaction scheme is a dynamic modification to Friedel statistics,36 where the asymptotic maximum plane resistance is found by assuming steady-state propagation
Dislocation Dynamics
225 MPa 145 MPa 4000
0
b
200 0 X (a )
Z (a)
2000 0
1000
0
0
of quasistraight dislocation lines. While Friedel calculates the area swept as the average area per particle on the glide plane, we adjust the segment line shape dynamically over several time steps after it is released from an SFT. When a segment is within 5a from the center of any SFT, it is divided into two segments with an additional common node at the point of SFT intersection with the glide plane. The angle between the tangents to the two dislocation arms at the common node is then computed, and force balance is performed. When the angle between the two tangents reaches a critical value of Fc, the node is released, and the two open segments are merged into one. If the force balance indicates that the segment is near equilibrium, no further incremental displacements of the node are added, and the segment of the loop is temporarily stationary. However, if a net force acts on that segment of the loop, it is advanced and the angle recomputed. It is possible that the angle between tangents will reach the critical value even though the segment is out of equilibrium. Sun et al.37 have shown that the elastic interaction energy between a glissile dislocation and an SFT is not sufficient to transform the SFT into a glissile prismatic vacancy loop. They proposed an alternate mechanism for the destruction of SFTs by passage of jogged and/ or decorated dislocations close to the SFT. The energy released from recombination of a small fraction of vacancies in the SFT was estimated to result in its local rearrangement. In the present calculations, we assume that vacancies in the SFT are absorbed in the dislocation core of the small contacting dislocation segment, forcing it to climb and form atomic jogs. With this mechanism, the entire SFT is removed from the simulation space, and jogged dislocations continue to glide on separate planes, thus dragging atomic-size jogs with them. Successive removal of SFTs from nearby glide planes can easily lead to channel formation and flow localization in the channel, because the passage of consecutive dislocation loops emitted from the F–R source is facilitated with each dislocation loop emission. The matrix density of SFTs in irradiated copper at low temperature (0.22–0.27Tm) is taken from experimental data (Singh et al.38). Figure 5 shows the results of computer simulations for propagation of plastic slip emanating from a single Frank–Read source in copper irradiated and tested at 100 C (Singh et al.38). The density of SFTs is 4.5 1023 m3 and the average size is 2.5 nm. In this simulation, the crystal size is set at 1.62 mm, while the initial F–R source length is 1600a (576 nm). A uniaxial applied tensile stress
2000 Y (a)
3000
4000
400
442
Figure 5 Propagation of plastic slip emanating from a single Frank–Read source in copper irradiated and tested at 100 C. Displacement damage dose ¼ 0.1 dpa, stacking fault tetrahedron density ¼ 4.51023 m3, size ¼ 2.5 nm. Simulated crystal size ¼ 4500a (1.62 mm). Initial F–R source length ¼ 1600a (576 nm). Stress is applied along [100]. ‘Unzipping’ of curved dislocation segments is clear during the initial stages of deformation, where long segments can get ‘stuck’ till they are unzipped by increasing the applied stress.
along [100] (s11) is incrementally increased, and the dislocation line configuration is updated until equilibrium is reached at the applied stress. Once full equilibrium of the dislocation line is realized, the stress is increased again, and the computational cycle repeated. At a critical stress level (flow stress), the equilibrium dislocation shape is no longer sustainable, and the dislocation line propagates until it is stopped at the crystal boundary, which we assume to be impenetrable. All SFTs interacting with the dislocation line are destroyed, and plastic flow on the glide plane is only limited by dislocation–dislocation interaction through the pileup mechanism. During the initial stages of deformation, small curved dislocation segments unzip, forming longer segments, which are stuck until they are unzipped again by increasing the applied stress. It is noted that, particularly at higher stress levels, the F–R source dislocation elongates along the direction of the Burgers vector, as a result of the higher stiffness of screw dislocation segments as compared to edge components. The F–R source configuration is determined by (1) the character of its
Dislocation Dynamics
initial segment (e.g., screw or edge), (2) the distribution of SFTs intersecting the glide plane, and (3) any other dislocation–dislocation interactions. This aspect is illustrated in Figure 6, where two interacting F–R sources in copper are shown for a displacement dose of 0.01 dpa, an SFT density of 2.5 1023 m3, and an average size of 2.5 nm. All other conditions are the same as in Figure 5. The two F–R sources are separated by 20a (7.2 nm).
4000 3000 2000 1000 Z (a)
195 MPa b2
1.16.3.3 b1
0
00
100 MPa 20
00 Y ( 3000 a)
00 a) X(
20 40
00
00
40
Figure 6 Spread of plastic slip emanating from two interacting Frank–Read sources in copper irradiated and tested at 100 C. While the simulation conditions are the same in Figure 5, the two F–R sources are separated by 20a (7.2 nm). Significant deformation of the two loops is observed when corresponding segments meet, and a higher stress is required to overcome the additional forces generated by each F–R source on the other one.
240
Radiation Hardening
Penetration of activated F–R sources into a 3D field of destructible SFTs can be viewed as a percolation problem, first considered by Foreman19 on a single glide plane, and extended here to complex 3D climb/glide motion. The critical stress above which an equilibrium dislocation configuration is unsustainable corresponds to the percolation threshold, and is considered here to represent the flow stress of the radiation-hardened material. Since activated F–R sources may encounter nearby dislocations, such interactions should be considered in estimates of the flow stress. The effects of interplanar F–R source interactions on the flow stress in copper irradiated and tested at 100 C are shown in Figure 7. A plastic stress–strain curve is constructed from the computer
0 0
10
443
fc = 160⬚ 65
220
60
Two planes, d = 20a
Two planes, d = 500a
200
55 180
50 45
140
40
Single plane
35
120 fc
100
30
Δtapplied (MPa)
Δsapplied (MPa)
160
25
80
20 60 15 40
10
20 0
5 0
0.1
0.2
0.3
0.4 0.5 0.6 Local strain
0.7
0.8
0.9
1
0
Figure 7 The effects of interplanar F–R source interactions on the flow stress in copper irradiated and tested at 100 C, and a displacement damage dose of 0.1 dpa. Local strain is measured as the fractional area of swept glide planes.
444
Dislocation Dynamics
simulation data, where the local strain is measured in terms of the fractional area swept by expanding F–R sources on the glide plane. It is shown that, while the majority of the increase in applied stress of irradiated copper can be rationalized in terms of dislocation interaction with SFTs on a single glide plane, dislocation–dislocation dipole hardening can have an additional small component on the order of 15% for very close dislocation encounters on neighboring slip planes (e.g., separated by 20a). For larger separation (e.g., 500a) the additional effects of dipole hardening is negligible. Dislocation forest hardening does not seem to play a significant role in determination of the flow stress, as implicitly assumed in earlier treatments of radiation hardening (e.g., Seeger et al.40 and Foreman19). The influence of the irradiation dose on the local stress–strain behavior of copper irradiated and tested at 100 C is shown in Figure 8. In the present calculations, we do not consider strain hardening by dislocation– dislocation interactions, and make no attempt to reproduce the global stress–strain curve of irradiated copper. Computed values of the flow stress are in general agreement with the experimental measurements of Singh et al.38 as can be seen from Figure 9. A more precise correspondence with experimental data depends on the value of the critical interaction angle Fc, which is the only relevant adjustable parameter in the present calculations. In Figure 8, Fc ¼ 165 . Determination of the two adjustable parameters (Fc and B) requires atomistic computer simulations
beyond the scope of the present investigation. Additional calculations for the flow stress for OFHC copper, irradiated at 47 C and tested at 22 C, are given in Table 1 and compared with the experimental data of Singh et al.38 The flow stress value at a dose of 0.001 dpa has been extrapolated from the experimental results of Dai39 for single-crystal copper irradiated with 600-MeV protons. It is noted that, while the general agreement between the computer simulations and the experimental data of irradiated Cu is reasonable at both 22 and 100 C, the dose dependence of radiation hardening indicates that some other mechanisms may be absent from the current simulations. Investigations of dislocation interactions with full or truncated SFTs considered by Sun et al.37 indicated that local heating may be responsible for the dissolution of SFTs by interacting dislocations, and that their vacancy contents are likely to be absorbed by rapid pipe diffusion into the dislocation core. The consequence of this event is that the dislocation climbs out of its glide plane by the formation of atomic jogs, followed by subsequent glide motion of jogged dislocation segments on a neighboring plane. In Figure 10, we present results of computer simulations of this glide/climb mechanism of jogged F–R source dislocations. Figure 11 shows a side view of the glide/climb motion of a dislocation loop pileup, consisting of three successive loops, by projecting dislocation lines on the plane formed by the vectors [111] and ½112. SFTs have been removed for visualization clarity. It is noted that
200
300 0.1 dpa
250 0.01 dpa
30
100
0.001 dpa 20 165⬚
50
10
0
0.25
0.5
0.75
1
0
Local strain Figure 8 Dose dependence of the increase in flow stress in copper irradiated and tested at 100 C. The displacement damage dose varies in the range 0.001–0.1 dpa, and conditions correspond to the experimental data of Singh et al.38
Δs0.05 (MPa)
40
Δt applied (MPa)
Δs applied (MPa)
150
0
fc = 158⬚
50
Experiment
200 fc = 165⬚
150
fc = 170⬚
100 50 0
OFHC-Cu
0.1
0.2
0.3
Dose (dpa) Figure 9 Comparison between the dose dependence of computed increase in flow stress values and the experimental measurements of Singh et al.38
Dislocation Dynamics
445
Table 1 22 C
Experimental and calculated tensile hardening parameters for OFHC copper, irradiated at 47 C and tested at
Dap
SFT density (1023 m–3)
Inter-SFT distance (a)
Experimental Ds0.05 (MPa)
Calculated Ds (MPa) (Fc ¼ 160 )
Calculated Ds (MPa) (Fc ¼ 165 )
0.001 0.01 0.1 0.2
0.8 5.3 6.7 6.6
196 76 67.5 68.5
110 (Dai 1995) 134 238 249
145 240 250 245
110 205 210 205
Source: Singh, B.; Edwards, D.; Toft, P. J. Nucl. Mater. 1996, 238, 244.
Third loop
001
Second loop First loop
100
010
[112] [111]
7.5 mm
First loop Second loop Third loop
10 a
Figure 10 A front three-dimensional view of the glide/climb mechanism of the jogged dislocation pileup. Note the gradual destruction of stacking fault tetrahedra as the pileup develops.
Figure 11 h110i projection for glide/climb motion of a dislocation loop pileup consisting of three successive loops. stacking fault tetrahedra have been removed for visualization clarity.
for this simulated three-dislocation pileup, the first loop reaches the boundary and is held there, while the second and third loops expand on different slip planes. We assume that the simulation boundary is rigid, and no attempt is made to simulate slip transmission to neighboring grains. However, the force field of the first loop stops the motion of the second and third loop, even though the stress is sufficient to penetrate through the field of SFTs. This glide/ climb mechanism of jogged dislocations in a pileup can be used to explain two aspects of dislocation channel formation. As the group of emitted dislocations expands by glide, their climb motion is clearly determined by the size of an individual SFT. For the densities considered here, a climb step of nearly one atomic plane results from the destruction of a single SFT. The
climb distance is computed from the number of vacancies in an SFT and the length of contacting dislocation segments. The jog height is thus variable, but is generally of atomic dimensions for the conditions considered here. The width of the channel is a result of two length scales: (1) the average size of an SFT (2.5 nm); and (2) the F–R source-to-boundary distance (1–10 mm). Secondary channels, which are activated from a primary channel (i.e., source point), and which end up in a nearby primary channel (i.e., boundary) are thinner than primary channels. Further detailed experimental observations of the channel width dependencies are necessary before final conclusions can be drawn. The second aspect of experimental observations, and which can also be explained by the present mechanism, is that a small degree of hardening occurs once dislocation channels
446
Dislocation Dynamics
have been formed. Dislocation–dislocation interaction within the noncoplanar jogged pile up requires a higher level of applied stress to propagate the pileup into neighboring grains. While the initiation of a dislocation channel is simulated here, full evolution of the channel requires successive activation of F–R sources within the volume weakened by the first F–R source, as well as forest hardening within the channels themselves. The possibility of dislocation channel initiation on the basis of the climb/glide mechanism is further investigated by computer simulation of OFHC copper, irradiated to 0.01 dpa and tested at 100 C, is shown in Figure 12. Formation of clear channels is experimentally observed at this dose level (Singh et al.38). The figure is a 3D representation for the initial stages of multiple dislocation channel formation. For clarity of visualization, the apparent SFT density has been reduced by a factor of 100, since the total number of SFTs in the simulation volume is 3.125 107. The initial dislocation density is taken as r ¼ 1013 m2. To show the importance of spatial SFT density variations, a statistical spatial distribution within the simulation volume has
5 mm
Figure 12 Three-dimensional view for the formation of dislocation channels on glide planes which have low stacking fault tetrahedron (SFT) density, for irradiated copper at a dose of 0.01 dpa. For clarity of visualization, the apparent SFT density has been reduced by a factor of 100. Note that other dislocation segments are inactive as a result of the high density of surrounding clusters.
been introduced such that lower SFT densities are assigned near ten glide planes. All dislocation segments are inactive as a result of high density of surrounding SFTs, except for those on the specified glide planes. Search for nearby SFTs is performed only close to active channel volumes, which in this case totals 174 846. It is observed that within a 5-mm volume, the number of loops within a pileup does not exceed 5. It is expected that if the pileup continues across an entire grain (size 10 mm), a higher number of loops would be contained in a jogged dislocation pileup, and that the corresponding channel would be wider than in the present calculations. We have not attempted to initiate multiple F–R sources within the volume swept by the dislocation pileup to simulate the full evolution of channel formation. As a result, the channel shape created by a single active F–R source is of a wedge nature. In future simulations, we plan to investigate the full evolution of dislocation channels. The present investigations have shown that two possible mechanisms of dislocation unlocking have been identified: (1) an asymmetric unzipping-type instability caused by partial decoration of dislocations; (2) a fluctuation-induced morphological instability, when the dislocation line is extensively decorated by defect clusters. Estimated unlocking stress values are in general agreement with experimental observations, which show a yield drop behavior. It appears that unlocking of heavily decorated dislocations will be most prevalent in areas of stress concentration (e.g., precipitate, grain boundary, triple point junction, or surface irregularity). Computer simulations of the interaction between unlocked F–R sources and a 3D random field of SFTs have been used to estimate the magnitude of radiation hardening and to demonstrate a possible mechanism for the initiation of localized plastic flow deformation and cleared channels. Reasonable agreement with experimental hardening data has been obtained with the critical angle Fc in the limited range of 158–165 . Both the magnitude and dose dependence of the increase in flow stress by neutron irradiation at 50 and 100 C are reasonably well predicted. In spatial regions of internal high stress, or on glide planes of statistically low SFT densities, unlocked dislocation sources can expand and interact with SFTs. Dislocations drag atomic-size jogs and/or small glissile SIAs when an external stress is applied. High externally applied stress can trigger point-defect recombination within SFT volumes resulting in local high temperatures. A fraction of the vacancies contained in SFTs can
Dislocation Dynamics
therefore be absorbed into the core of a gliding dislocation segment, producing atomic-size jogs and segment climb. The climb height is a natural length scale dictated by the near-constant size of the SFT in irradiated copper. It is shown by the present computer simulations that the width of a dislocation channel is on the order of 200–500 atomic planes, as observed experimentally and is a result of a stresstriggered climb/glide mechanism. The atomic details of the proposed dislocation–SFT interaction and ensuing absorption of vacancies into dislocations need further investigation by atomistic simulations. Finally, it should be pointed out that at relatively high neutron doses, dense decorations of dislocations with SIA loops and a high density of defect clusters/ loops in the matrix are most likely to occur. As shown here, these conditions can lead to the phenomena of yield drop and flow localization. 1.16.3.4 The Ductile-to-Brittle Transition in Pressure Vessel Steels 1.16.3.4.1 Background
It is now well accepted that fracture of ferritic steels in the temperature range where it propagates by cleavage originates in microcracks (mostly in precipitates) ahead of macrocracks, which could be precracks in a test specimen or surface cracks in structures. To explain the extremely high cleavage fracture toughness (>20 MPa √m) compared with that ( 1 MPa √m) calculated from surface energy alone assuming pure cleavage of Fe matrix, it was originally postulated by Orowan41 that fracture in ferritic steels could occur due to cleavage originated in microcracks situated ahead of the main crack. Later, it was found by experiments that these microcracks originate in precipitates,42,43 and that the propagation of these microcracks into the matrix was assumed to be the controlling step in the fracture of ferritic steels. Another observation, though less well established, is that the cleavage stress at fracture on these microcracks is invariant with temperature.44,45 Ritchie, Knott, and Rice (RKR)46 used the HRR solutions (Hutchinson, Rice, and Rosengren) and finite element analysis (FEA) to simulate the plastic zone, and used a critical tensile stress achieved over a characteristic distance ahead of the crack as the failure criterion. This distance is essentially a fitting parameter, and RKR46 used a value equal to or twice the average the grain diameter. The model successfully predicts the lower-shelf fracture toughness, but fails to predict the upturn near the transition
447
temperature. Statistical models were introduced to predict the brittle–ductile transition (BDT) in steels starting with Curry and Knott,43 most notable among them were by Beremin47 and Wallin et al.48 In both of these models, FEA solutions of crack-tip plasticity were used to obtain the stress fields ahead of the crack. In the Beremin model,47 the maximum principal stress is calculated for each volume element in the plastic zone and a probability of failure is assigned. The total probability of failure is then obtained by summing over the entire plastic zone. Wallin et al.48 extended the modeling to the transition region by considering variation of the effective surface energy (gs þ gp) with temperature, where gs is the true surface energy and gp the plastic work done during propagation. This eventually led to the master curve (MC) hypothesis, which predicts that the BDT of all ferritic steels follows a universal curve.48,49 Even though the MC is used to check the reliability of structures under irradiation,50 a clear understanding of the physical basis of this methodology is still lacking.51 Odette and He52 explained the MC using a microscopic fracture stress varying with temperature. Most experimental findings53,54 indicate that the fracture stress is not sensitive to temperature, and more careful experiments and simulations may be required to resolve this issue. Discrete dislocation simulations of crack tips were successful in predicting the BDT of simple single crystalline materials.55,56 The advantage of this approach over the continuum methods is that fundamental material properties such as dislocation velocity and their mutual interactions can be treated dynamically. By these simulations, it has been found that the dislocation mobility plays a significant role in determining the transition temperature.56 However, the variation of dislocation mobility alone cannot explain the BDT behavior. An earlier attempt to model the BDT of complex materials like steels57 predicted the lower-shelf fracture toughness, substantiating Orowan’s postulate41 of high fracture toughness measured at low temperature. However, the model failed to predict the sharp increase of fracture toughness around the transition temperature region. Here, we present a discrete dislocation simulation in which crack-tip blunting is accounted for the first time. The effects of blunting are incorporated in the simulation using elastic stress fields of blunted cracks. As the crack tip is blunted due to dislocation emission, and the position of the ‘virtual sharp crack tip’ retreats from the blunted tip thereby reducing the field at the microcrack further in addition to the
448
Dislocation Dynamics
contribution from emitted dislocations. The critical particle is assumed to be at a fixed distance from the blunted macrocrack tip. 1.16.3.4.2 Crack-dislocation dynamics model
From numerous careful experimental studies conducted on the BDT behavior of steels, it is now established that precracks (macrocracks) blunt substantially before the fracture of the specimen occurs at the transition region. However, the examination of the fracture surface reveals that cracks propagate predominantly by cleavage.44 Several cracked brittle particles are found to be present in the broken samples,45,53 and the measured microscopic fracture stress (at the microcracks) is found to be a few orders of magnitude higher than that of the pure Griffith value.53,54 All these observations are considered in our model as follows: 1. We implemented the blunting of macrocracks by using the elastic crack-tip stress field for blunted cracks. As dislocations are emitted, the crack blunts and the radius of curvature increases. The notional crack tip, which is taken as reference for calculation of image stresses, retreats away from the blunted tip. 2. A microcrack is placed in the field of a macrocrack and the failure criterion used in the calculation of the cleavage crack propagation from this microcrack. 3. We consider the emission of dislocations and subsequent shielding from the microcrack tip (a detailed study of this and the observed constancy for microscopic fracture stress is reported in an earlier study19). The geometry of the model used for simulation is shown in Figure 13. A semi-infinite crack (macrocrack) with a finite microcrack situated ahead of it on its crack plane is loaded starting from K ¼ 0. Dislocation sources are assumed to exist at a distance x0 from the tip, and are situated on the slip planes passing through the crack tip. During loading, dislocations are emitted from source positions (x0) when the resolved shear stress reaches a value of 2ty. The resolved shear stresses are obtained using expressions based on derivations for a semi-infinite crack58 and a finite crack59 for the respective cases. The emitted dislocations move along the slip plane away from the crack tip, and the stress at the source increases until another dislocation is emitted. The emitted dislocations move with a velocity based on the following expression: jtxi j ty ðjtxi jÞm AeðEa =kT Þ ½24 vxi ¼ tx i
K
a
Macrocrack
a⬘
Microcrack t
Figure 13 The geometry of the crack and dislocations used in the macrocrack–microcrack simulation model. K is the applied load at the macrocrack, the slip planes angles are oriented at and to the crack planes of macrocrack and microcrack, respectively.
The values for the parameters were obtained by fitting the data of screw dislocation velocities in iron.59 The value of m has a linear dependence on temperature T: m ¼ 400/T þ 1.2, A ¼ 3.14 104, and Ea ¼ 0.316 eV. The first term restricts the motion of dislocations below the friction stress value (ty), making sure that v ¼ 0 for jtxi j< ty, and hence, most of the dislocations are in near-equilibrium positions at any given time. When the dislocations are in their equilibrium positions, the temperature and strain-rate dependence of measured fracture toughness (KF), plastic zone size (df), crack-tip opening displacement, etc. are determined only by the temperature and strain-rate dependence of the friction stress (ty). The friction stress used is chosen to be equal to the shear yield stress sy/2 when the Tresca yield criterion is assumed. Thus, the temperature dependence of fracture toughness is obtained by inputting the corresponding friction stress value for each temperature. Simulations were done for temperatures range from 180 to 60 C with corresponding yield stress values (sy) from 910 to 620 MPa. The arrays of emitted dislocations form the plastic zones of the crack. The crack may also get blunted due to dislocation emission. In our case, the effects of blunting will be negligible for microcracks since the number of dislocations emitted is only up to 102. However, the effects of blunting will be significant in the case of macrocracks, because the number of dislocations emitted is of the order 105; here the blunting effects are accounted for by using the elastic crack-tip field for a blunted crack.60 The plastic zone developed at the macrocrack modifies the field ahead
Dislocation Dynamics
of it so that it is the same as an elastic-plastic material with hardening.61 The microcrack placed in this field experiences a tensile stress and is assumed to propagate, leading to fracture when it reaches a critical value F (estimated on the basis of similar dislocation simulation of finite crack emitting dislocations along the slip planes). Computer simulations are performed in two stages. First, the microcrack is loaded to failure and the microscopic fracture stress is estimated for specific crystallographic orientations and crack sizes. The obtained microscopic fracture stress (sf) values are then used as the fracture criterion in the macrocrack simulation.
449
K 2r Xp
Microcrack
Macrocrack
r 2r Xp Notional crack tip
1.16.3.4.3 Calculation techniques
Figure 14 Schematic illustration of the crack-tip blunting. is the radius of the blunted crack tip, 2r the microcrack size, and Xp the distance of the microcrack from macrocrack tip. The distance of the microcrack from the notional sharp crack tip increases from Xp to Xp þ r.
A microcrack in the configuration shown on the right side of Figure 13 is loaded monotonically. To mimic the triaxial stress field surrounding the microcrack, we modify the expressions for the stress fields given in Wang and Lee59 by subtracting the applied normal stress (sAy) and adding the uniaxial yield stress (sy) for cases when sAy > sy. The value of the slip plane angle is chosen such that predominantly brittle crack configurations are simulated. The applied load is pffiffiffiffiffiffiffiffiffiffi incremented at a rate dK =dt ¼ 0:01MPa ms1 ; a variable time step is used to make sure dislocations move as an array, and numerical instabilities are avoided. Dislocations are emitted from source positions (here chosen as 4b, b ¼ 2.54 A˚) along the slip planes. Since source positions are equidistant from the respective crack tips, simultaneous emission occurs. The emitted dislocations move away from the crack tip along the slip planes with a velocity calculated using eqn [24]. The stress fields of these dislocations then shield the crack tip from external load. Once moved to their equilibrium position, the amount of shielding from each dislocation is calculated using expressions from Wang and Lee,59 and the total shielding at the crack tip is obtained by summation. We ignore antishielding dislocations in our simulations, since any dislocations nucleated in the present configuration around the crack tip will be absorbed by the crack. Thus, the number of dislocation absorbed is small, and any blunting effects due to them are neglected here. Because in our case dislocations are emitted from crack-tip sources, dislocations emitted from either end contribute to shield both crack tips.62 The applied load is increased monotonically and the crack-tip stress intensity is calculated at each time step. When the crack-tip stress intensity reaches a preassigned critical value, the crack is assumed to propagate and the corresponding
applied load is the microscopic fracture stress (sf). The following two predominantly brittle microcrack systems are considered here: (1) cleavage plane (001), crack front [110], slip system 12½111ð112Þ, hence a0 ¼ 35 160 , and (2) cleavage plane (110), crack front ½110; 12½111ð112Þ, hence a0 ¼ 54 440 . A schematic illustration of the macrocrack is shown on the left-hand side of Figure 14. The macrocrack is assumed to be semi-infinite, with dislocation sources close to the crack tip. Dislocations are emitted simultaneously along the two slip planes, symmetrically oriented to the crack plane. The plastic zone formed by emitted dislocations produces a field equivalent to an elastic-plastic crack with small scale yielding.61 The slip plane angle in this case is chosen to be 70.5 (the direction of maximum shear stress of the elastic crack-tip field), compared to the present slip plane angles, which match best with continuum estimates.61 The initial source position (x0) along the slip plane is chosen to be 4b, where b is the magnitude of Burgers vector. (For each positive dislocation emitted, a negative one is assumed to move into the crack.) The dislocation sources on either side of the crack plane are at equivalent positions x0 and operate simultaneously. During the simulation, the applied stress intensity is increased in small increments and the positions of dislocations are determined. It is found that the dislocations reach near-equilibrium positions. As the load is increased, more and more dislocations are emitted and the crack gets blunted. The radius of the blunted crack r is taken to be equal to Nb sin a, where N is the number of dislocations and a is the slip plane angle. The blunting of the macrocrack is illustrated in Figure 14. As blunting increases,
Dislocation Dynamics
1.16.3.4.4 Results and discussion
The (micro) fracture stress sF as a function of temperature was calculated for two different slip plane angles. The fracture criterion chosenpwas ffiffiffiffi a crack-tip stress intensity K ¼ KIC ¼ 1:0MPa m (value estimated for Fe with surface energy of 2 J m2). The source position (x0) is chosen to be 4b, and the microscopic fracture stress sF is estimated for different temperatures. The results indicate that sF is practically independent of temperature, consistent with many of the experiments (e.g., Kubin et al.6). The value of sF thus obtained for each temperature is used to calculate the macroscopic fracture toughness (KF) in the next stage of simulation. Figure 15 shows the typical behavior of the tensile stress at the microcrack (spyy ) as a function of the applied load (K). The fracture criterion in this case is spyy reaching the critical value sF calculated in the previous stage. For the case shown in this figure, the microcrack size isp1ffiffiffiffiffiffiffiffiffiffi mm, and the rate of loading dK =dt ¼ 0:01MPa ms1 . The distance of the
2000
-100
-180 -160 -140
-80
-60
1600 syy (MPa)p
1200 800 400 0
0
20
40
60
80
Applied K (MPa Öm) Figure 15 The tensile stress at the microcrack (spyy ) as a function of the (Kapp) or the simulation time for different yield stresses. The corresponding temperatures ( C) are shown in the plot.
30
20
Radius of curvature of blunted crack Plastic zone size at the macrocrack
40 30 20
10
d/102 mm
(1) the crack-tip fields are modified to be that of a blunted crack, (2) source position is chosen to be equal to the crack-tip radius, that is, x0 ¼ r, and (3) the notional crack tip from which the image stress of dislocations is calculated is moved back to the center of curvature of the blunted crack. At each time step, the stress ahead of the crack at a distance Xp along the crack plane (the microcrack is assumed to be at this position) is calculated using the expression from Trinkaus et al.23 The fracture criterion is the tensile stress ðspyy Þ at Xp reaching sF, calculated in the previous stage for the microcracks for the same temperature and the corresponding yield stress. Thus, when the microcrack tip stress intensity is K ¼ KIC, cleavage fracture of the matrix is assumed to occur. The applied load at the macrocrack then gives the fracture toughness (KF) at that given temperature/yield stress. Throughout the simulation, a microcrack of size 1 mm is used; this value is chosen as a typical (average) value in the range of sizes of microcracks found in experiments with which we compare our results.43 The detailed simulations reported here are obtained using Xp equal to 10 mm. However, the KF calculated using 20 mm is also shown in Figure 14. These values were chosen as typical of the order of grain sizes in this type of steel. It should be noted that changing these parameters will give different values for fracture toughness; however, the behavior of their variation with temperature will remain the same, as will be discussed next.
r (mm)
450
10 0
−180
−160 −140 −100 −80 Temperature (⬚C)
−60
0
Figure 16 The crack-tip radius (r) and the plastic zone size (df) calculated at KF for each temperature shown.
microcrack (particle) from the macrocrack (Xp) is 10 mm. As the temperature is increased and yield stress (friction stress) is decreased, the applied stress intensity K required for the tensile stress at Xp to reach the critical value sF increases exponentially. Two factors contributing to this exponential increase could be the decrease in the tensile stress at the microcrack due to crack-tip blunting and the increasing effects of stress field (predominantly compressive) from the emitted dislocations. This can be seen in Figure 16, where the plastic zone size (d ) and the radius of the blunted crack tip (r) for each temperature measured at fracture K-applied ¼ KF are shown. The plastic zone size is the distance measured along the slip plane to the farthest dislocation from the crack tip. The dislocation source distance (x0) is chosen as r for a crack-tip radius >4b ; else x0 ¼ 4b. Figure 17 shows the macroscopic fracture toughness
Dislocation Dynamics
90
90
60
Xp = 20 mm, 2r = 1 mm
With blunting
30
With out blunting
0 -200
-150
-100
-50
Temperature (⬚C)
10 000 Xp = 10 mm 2r = 1 mm
6000 4000
With blunting
2000 0 -200
-150 -100 Temperature (⬚C)
60
30
0 -200
Xp = 10 mm, 2r = 1 mm
-150
-100
-50
Temperature (⬚C)
Figure 17 The fracture toughness (KF) as a function of temperature for cases with and without blunting: microcrack size ¼ 1 mm.
8000
KF (MPa Öm)
KF (MPa Öm)
Xp = 10 mm, 2r = 1 mm
J2F (J m-2)
451
-50
Figure 18 The J2F integral calculated for the different temperatures for the blunting case shown in Figure 17.
KF as a function of temperature for cases with and without considering the effects of crack-tip blunting. For the case without blunting, the increase in the fracture toughness (KF) with temperature is small. However, a sharper increase in the fracture toughness is observed when blunting is accounted for in the simulation. This striking observation emphasizes the significant effect of blunting in the increase of fracture toughness with temperature. This exponential increase in the fracture toughness corresponds to the transition from brittle to ductile behavior. In Figure 18, J2F (J2 – integral value at fracture) calculated from the number of dislocations emitted at the corresponding load KF is shown. J2 is defined as the sum of the glide forces on all dislocations around the crack tip.26 In this case, the dislocations are in equilibrium against the friction stress (ty) and we can compute J2F as the product of the
Figure 19 The fracture toughness values from Figure 17 and for Xp ¼ 20 mm compared with experimentally determined values. Reproduced from Amodeo, R. J.; Ghoniem, N. M. Phys. Rev. 1990, 41, 6958.
total number of dislocations (N) and the friction stress (ty). Considering the fact that J2F is calculated from the number of dislocations emitted at an applied stress intensity factor KF, it is striking to note that the prefactor of the exponent of the J2F–temperature curve (0.0255) matches that of the KF–temperature curve (0.0123) to hold the known proportionality between K2 and J. In Figure 19, the calculated values of fracture toughness are compared with the fracture toughness measurements reported in Amodeo and Ghoniem3. The carbide found in these samples ranges in size from 0.44 to 1.32 mm, and in our calculations we have used microcracks of comparable size (1 mm). The results for the blunted case are shown in Figure 17, along with another set of values calculated for Xp ¼ 20 mm, shown here for comparison. We can see that the model predicts the rapid increase in fracture toughness at the transition temperature region, and reasonably fits the experimental data. Considering the simplicity of the present model, the agreement suggests that a good step has been taken in predicting the BDT behavior. The crack-tip behavior and the BDT predicted are in good agreement in the transition region where the fracture toughness increases rapidly with temperature. However, it should be noted that the model ceases to be valid at higher temperatures where ductile tearing effects will be significant. According to our model, the two factors that contribute to the sharp increase in the fracture toughness with temperature are (1) the increase in the mobility of the emitted dislocations and (2) the effect of macrocrack tip blunting. The mobility of emitted dislocations determines the equilibrium position of dislocations
452
Dislocation Dynamics
and thus determines the tensile stress at the microcrack. Also, the mobility of dislocations around the microcrack determines the crack-tip stress intensity at the microcrack and thus the microscopic fracture toughness (sf), which ultimately determines the fracture toughness of the material (KF). However, and as can be seen in Figure 17, this alone cannot explain the sharp upturn at the transition. The amount of crack-tip blunting is found to be a significant factor in capturing the rapid increase in the fracture toughness with temperature in the transition region.
1.16.4 Outlook While continuum approaches to modeling the mechanical properties of structural materials are limited to the underlying experimental database, DD methods offer new opportunities for modeling microstructure evolution from fundamental principles. The limitation to the method presented here is mainly computational, and much effort is needed to overcome several difficulties. First, the length and time scales represented by the present method are still short of many experimental observations, and methods of rigorous extensions are still needed. Second, the boundary conditions of real crystals are more complicated, especially when external and internal surfaces are to be accounted for. Thus, the present approach does not take into account large lattice rotations, and finite deformation of the underlying crystal, which may be important for explanation of plastic deformation at certain length scales. Finally, a much expanded effort is needed to bridge the gap between atomistic calculations of dislocation properties (as discussed by Osetsky and Bacon in Chapter 1.12, Atomic-Level Level Dislocation Dynamics in Irradiated Metals) on the one hand, and continuum mechanics formulations on the other. Nevertheless, with all of these limitations, the DD approach is worth pursuing, because it opens up new possibilities for linking the fundamental nature of the microstructure (especially of irradiated materials) with realistic deformation conditions. It can thus provide an additional tool to both theoretical and experimental investigations of plasticity and failure of materials.
through Grant # DE-FG02-03ER54708 with UCLA. The author would like to acknowledge the contributions of S. Noronha.
References 1. 2. 3. 4. 5. 6. 7.
8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20.
21. 22. 23. 24. 25.
Acknowledgments 26.
This work is supported by the US Department of Energy, Office of Nuclear Energy, through Grant # DE-FC07-06ID14748, and Office of Fusion Energy
27. 28.
Lepinoux, J.; Kubin, L. P. Scripta Metall. 1987, 21(6), 833. Ghoniem, N. M.; Amodeo, R. J. Solid State Phenom. 1988, 3&4, 377. Amodeo, R. J.; Ghoniem, N. M. Phys. Rev. 1990, 41, 6958. Amodeo, R. J.; Ghoniem, N. M. Phys. Rev. 1990, 41, 6968. Wang, H. Y.; LeSar, R. Philos. Mag. A 1995, 71(1), 149. Kubin, L. P.; Canova, G.; Condat, M.; Devincre, B.; Pontikis, V.; Brechet, Y. Diffusion Defect Data Solid State Data B Solid State Phenom. 1992, 23–24, 455. Canova, G.; Brechet, Y.; Kubin, L. P. In Modeling of Plastic Deformation and Its Engineering Applications; Anderson, S. I., Blide-Sorensen, J. B., Lorentzen, T., Pedersen, O. B., Sorensen, N. J., Eds.; RIS National Laboratory: Roskilde, Denmark, 1992; pp 27–33. Devincre, B.; Kubin, L. P. Model. Simulat. Mater. Sci. Eng. 1994, 2(3A), 559. Hirth, J. P.; Rhee, M.; Zbib, H. J. Comput. Aided Mater. Des. 1996, 3, 164. Schwarz, K. V.; Tersoff, J. Appl. Phys. Lett. 1996, 69(9), 1220. Zbib, R. M.; Rhee, M.; Hirth, J. P. Int. J. Mech. Sci. 1998, 40(2–3), 113. Rhee, M.; Zbib, H. M.; Hirth, J. P.; Huang, H.; de la Rubia, T. Model. Simulat. Mater. Sci. Eng. 1998, 6(4), 467. Ghoniem, N. M.; Sun, L. Z. Phys. Rev. B 1999, 60(1), 128–140. Ghoniem, N. M.; Tong, S. H.; Sun, L. Z. Phys. Rev. 2000, 61(2), 913–927. Ghoniem, N. M.; Huang, J.; Wang, Z. Philos. Mag. Lett. 2001, 82(2), 55–63. Kocks, U.; Argon, A.; Ashby, M. Prog. Mater. Sci. 1975, 19, 1. Luft, A. Prog. Mater. Sci. 1991, 38(1), 97. Seeger, A. In Proceedings of the 2nd United Nations International Conference on Peaceful Uses of Atomic Energy; IAEA: Geneva, 1958; Vol. 6, p 250. Foreman, A. Philos. Mag. 1968, 17, 353. Kroupa, F. In Theory of Crystal Defects, Proceedings of the Summer School Held in Hrazany in September 1964; Gruber, B., Ed.; Academia: Prague, 1966; pp 275–316. Singh, B. N.; Foreman, A. J. E.; Trinkaus, H. J. Nucl. Mater. 1997, 249(2–3), 103–115. Cottrell, A. Technical Report 30; Physical Society, London, University of Bristol, England, 1948. Trinkaus, H.; Singh, B. N.; Foreman, A. J. E. J. Nucl. Mater. 1997, 249(2–3), 91–102. Volterra, V. Ann. Sci. de l’E´cole Norm. Super. Paris 1907, 24, 401–517. Mura, T. The Continuum Theory of Dislocations, Advances in Materials Research; Interscience: New York, 1968; Vol. 3. de Wit, R. In Solid State PhysicsSeitz, F., Turnbull, D., Eds.; Academic Press: New York, 1960; Vol. 10, p 249. Peach, M. O.; Koehler, J. S. Phys. Rev. 1950, 80, 436. Gavazza, S.; Barnett, D. J. Mech. Phys. Solids 1976, 24, 171–185.
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Mura, T. Philos. Mag. 1963, 8, 843–857. Barnett, D. Phys. Status Solidi B 1972, 49, 741–748. Willis, J. J. Mech. Phys. Solids 1975, 23, 129–138. Bacon, D.; Barnett, D.; Scattergodd, R. In Progress in Materials Science; Chalmers, B. C. J. M. T., Ed.; Pergamon: Great Britain, 1980; Vol. 23, pp 51–262. Mura, T. Micromechanics of Defects in Solids, 2nd ed.; Martinus Nijhoff: Dordrecht, 1987. Barnett, D. Phys. Status Solidi A 1976, 38, 637–646. Ghoniem, N.; Singh, B.; Sun, L.; Diaz de la Rubia, T. J. Nucl. Mater. 2000, 276, 166–177. Friedel, J. Les Dislocations; Gauthier-Villars: Paris, 1956. Sun, L.; Ghoniem, N. M.; Tong, S.; Singh, B. J. Nucl. Mater. 2000, 283, 741. Singh, B.; Edwards, D.; Toft, P. J. Nucl. Mater. 1996, 238, 244. Dai, Y. Ph.D. Thesis, University of Lausanne, 1995. Seeger, A.; Diehl, J.; Mader, S.; Rebstock, H. Philos. Mag. 1957, 2(15), 323. Orowan, E. Trans. Inst. Eng. Shipbuilders Scotland 1945, 89, 165. McMahon, C. J., Jr.; Cohen, M. Acta Metall. 1965, 13, 591. Curry, D. A.; Knott, J. F. Met. Sci. 1979, 13, 341. Bowen, P.; Knott, J. F. Metall. Trans. A 1986, 17, 231. Veistinen, M. K.; Lindroos, V. K. Scripta Metall. 1984, 18, 185. Ritchie, R. O.; Knott, J. F.; Rice, J. R. J. Mech. Phys. Solids 1973, 21, 395. Beremin, F. Metall. Trans. 1983, A14, 2277. Wallin, K.; Sarrio, T.; Torronen, K. Met. Sci. 1984, 18, 13.
49. 50. 51. 52. 53. 54. 55. 56. 57. 58. 59. 60. 61. 62.
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McCabe, D. E.; Merkle, J. G.; Wallin, K. In Fatigue and Fracture Mechanics; Paris, P. C., Jerina, K. L., Eds.; ASTM: West Conshohocken, PA, 2000; Vol. 30, p 21. ASTM Standard Test Method E 1921-02. Annual Book of ASTM Standards; ASTM: West Conshohocken, PA, 2002; Vol. 03.01. Natishan, M. E.; Kirk, M. T. In Fatigue and Fracture Mechanics; Paris, P. C., Jerina, K. L., Eds.; ASTM: West Conshohocken, PA, 2000; Vol. 30, p 51. Odette, G. R.; He, M. Y. J. Nucl. Mater. 2000, 283–287, 120. Bowen, P.; Druce, S. G.; Knott, J. F. Acta Metall. 1987, 35, 1735. Ortner, S. R.; Hippsley, C. A. Mater. Sci. Technol. 1996, 12, 1035. Hirsch, P. B.; Roberts, S. G.; Samuels, J. Proc. R. Soc. Lond. 1989, A421, 25. Hirsch, P. B.; Roberts, S. G. Philos. Mag. A 1991, 64, 55. Hirsch, P. B.; Roberts, S. G. Philos. Trans. R. Soc. Lond. A 1997, 355, 335. Lakshmanan, V.; Li, J. C. M. Mater. Sci. Eng. 1988, A104, 95. Wang, S.; Lee, S. Mater. Sci. Eng. 1990, A130, 1. Creager, M.; Paris, P. C. Int. J. Fract. Mech. 1967, 3, 247. Noronha, S.; Ghoniem, N. Int. J. Mech. Mater. Des. 2007, Doi: 10.1007/s109999-007-9041-3. Zhang, T. Y.; Li, J. C. M. Acta Metall. Mater. 1991, 39, 2739.
1.17 Computational Thermodynamics: Application to Nuclear Materials T. M. Besmann Oak Ridge National Laboratory, Oak Ridge, TN, USA
Published by Elsevier Ltd.
1.17.1 1.17.2 1.17.3 1.17.4 1.17.4.1 1.17.4.2 1.17.4.3 1.17.4.4 1.17.4.5 1.17.4.6 1.17.5 1.17.6 1.17.7 References
Introduction Thermochemical Principles The CALPHAD Approach and Free Energy Minimization Treatment of Solutions Regular Solution Models Variable Stoichiometry/Associate Species Models Compound Energy Formalism Thermochemical Modeling of Defects Modified Associate Species Model for Liquids Ionic Sublattice/Modified Quasichemical Model for Liquids Thermochemical Data Sources Thermochemical Equilibrium Computational Codes Outlook
Abbreviations CALPHAD CEF DTA EMF NASA NIST MOX TRU
Calculation of phase diagrams Compound energy formalism Differential thermal analysis Electro-motive force National Aeronautics and Space Administration National Institute of Standards and Technology Mixed oxide fuel Transuranic
Symbols Cp E EBW Eij EQM G Gex Gid H Hmix
Heat capacity at constant pressure Energy of the system Bragg–Williams model energetic parameter Interaction energy between components i and j Quasichemical model energetic parameter Gibbs free energy Excess free energy Free energy contribution due to ideal entropy of mixing Heat or enthalpy Heat of mixing
L n P pO2*
R S s Sconfig T m V X y z Z
455 456 457 457 459 460 461 462 463 465 467 468 468 469
Interaction parameter, typically of the form aþbT Moles of a constituent Pressure A dimensionless quantity defined by the oxygen pressure divided by the standard state pressure Ideal gas law constant Entropy Index for a sublattice Configurational entropy Absolute temperature Chemical potential Volume Mole fraction The site fraction for species j Stoichiometric coefficient Nearest neighbor coordination number
1.17.1 Introduction Nuclear fuels and structural materials are complex systems that have been very difficult to understand and model despite decades of concerted effort. Even single actinide oxide or metallic alloy fuel forms have yet to be accurately, fully represented. The problem is 455
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Computational Thermodynamics: Application to Nuclear Materials
compounded in fuels with multiple actinides such as the transuranic (TRU) fuels envisioned for consuming long-lived isotopes in thermal or fast reactors. Moreover, a fuel that has experienced significant burnup becomes a very complex, multicomponent, multiphase system containing more than 60 elements. Thus, in an operating reactor the nuclear fuel is a high-temperature system that is continuously changing as fission products are created and actinides consumed and is also experiencing temperature and composition gradients while simultaneously subjected to a severe radiation field. Although structural materials for nuclear reactors are certainly complex systems that benefit from thermochemical insight, the emphasis and examples in this chapter focus on fuel materials for the reasons noted above. The higher temperatures of fuels quickly drive them to the thermochemical equilibrium state, at least locally, and their compositional complexity benefits from computational thermochemical analysis. Related information on thermodynamic models of alloys can be found in Chapter 2.01, The Actinides Elements: Properties and Characteristics; Chapter 2.07, Zirconium Alloys: Properties and Characteristics; Chapter 2.08, Nickel Alloys: Properties and Characteristics; Chapter 2.09, Properties of Austenitic Steels for Nuclear Reactor Applications; Chapter 1.18, Radiation-Induced Segregation; and Chapter 3.01, Metal Fuel. A major issue for nuclear fuels is that the original fuel material, whether the fluorite-structure phase for oxide fuels or the alloy for metallic fuels, has variable initial composition and also dissolves significant bred actinides and fission products. Thus, the fuel phase is a complex system even before irradiation and becomes significantly more complex as other elements are generated and dissolve in the crystal structure. Compounding the complexity is that, after significant burnup, sufficient concentrations of fission products are formed to produce secondary phases, for example, the five-metal white phase (molybdenum, rhodium, palladium, ruthenium, and technicium) and perovskite phases in oxide fuels as described in detail in Chapter 2.20, Fission Product Chemistry in Oxide Fuels. Thus, any chemical thermodynamic representation of the fuel must include models for the nonstoichiometry in the fuel phase, dissolution of other elements, and formation of secondary, equally complex phases. Dealing with the daunting problem of modeling nuclear fuels begins with developing a chemical thermodynamic (or thermochemical) understanding of the material system. Equilibrium thermodynamic states are inherently time independent, with the
equilibrium state being that of the lowest total energy. Therefore, issues such as kinetics and mass transport are not directly considered. Although the chemical kinetics of interactions are important, they are often less so in the fuel undergoing burnup (fissioning) because of the high temperatures involved and resulting rapid kinetics and can often be neglected on the time scales involved for the fuel in reactor. The time dependence of mass transport, however, does influence fuel behavior as evidenced by the significant compositional gradients found in high burnup fuel whether metal or oxide, and most notably by attack of the clad by fission products and oxidation by species released from oxide fuel. Although the equilibrium state provides no information on diffusion or vapor phase transport, it does provide source and sink terms for these phenomena. Thus, the calculation of local equilibrium within fuel volume elements can in principle provide activity/ vapor pressure values useful in codes for computing mass flux. Thermochemically derived properties of fuel phases also provide inherent thermal conductivity, source terms for grain growth, potential corrosion mechanisms, and gas species pressures, all important for fuel processing and in-reactor behavior. Thermochemical insights can therefore provide support for modeling species and thermal transport in fuels.
1.17.2 Thermochemical Principles Understanding the chemical thermodynamic behavior of reactor materials means describing multicomponent systems with regard to their relative free energies. For nuclear fuels that includes both stoichiometric phases as well as solid and liquid solutions containing multiple elements and the vapor species they generate. The total free energy determined from the thermochemical descriptions for all the potential phases is computed, and those phases/compositions that result in the lowest free energy state represent the equilibrium system. The expression of the free energy is in terms of the Gibbs free energy, G, at constant temperature and pressure, following the familiar relation G ¼ E þ PV TS
½1
where E is the energy of the system, P is pressure, V is volume, T is absolute temperature, and S is entropy. A convenient expression at equilibrium in a constant temperature and pressure system is G ¼ H TS
½2
Computational Thermodynamics: Application to Nuclear Materials
where H is the heat or enthalpy. The temperature dependence of the enthalpy is related to heat capacity, Cp , by ðT H ¼ H298 þ
Cp dT
½3
Cp =T dT
½4
298
and to entropy by ðT S ¼ S298 þ 298
The temperature dependence for Cp is expressed as a polynomial from which it is possible to generate what is termed the Gibbs free energy function, which is usually expressed as G ¼ A ¼ BT þ CT lnT þ DT 2 þ ET 3 þ F =T
½5
The Gibbs free energy function is a very convenient form to work with, particularly for free energy minimization software that computes an equilibrium state. That is defined as Gibbs free energy of a system that is at its minimum value, or @G ¼ 0. A very useful value to use when working with complex systems is the chemical potential, m, which is the partial derivative of the Gibbs free energy with respect to the moles or mole fraction of a constituent. Thus, at constant temperature and pressure mi ¼ ð@G=@ni ÞT ;P
½6
where n is the number of moles of the constituent. For constant temperature and pressure X mi dni ½7 @G ¼ i
A system’s equilibrium state is therefore computed by minimizing the total free energy expressed as the sum of the various Gibbs free energy functions constrained by the mass balance with a resulting assemblage of phases and their amounts.
1.17.3 The CALPHAD Approach and Free Energy Minimization The overall development of a consistent thermochemical representation for the phase equilibria and thermodynamics of a system utilizing all available information has been termed the CALPHAD (computer coupling of phase diagrams and thermochemistry) approach.1 Whether free energy and heat
457
capacity data are provided from first principles calculations or experimentally, for example, from differential scanning calorimetry, solution calorimetry, or thermogravimetric measurements, is irrelevant as long as the information is accurate and applicable. The situation is similar for phase equilibria, that is, what phases form under what conditions. The developed phase diagrams provide information that can be used to fit prospective thermochemical models. This data, together with current computational methods that facilitate development of accurate representations for systems reproducing observed behavior, define the CALPHAD methodology. The results ideally are databases for specific components that may also be used in the construction of systems with yet larger numbers of constituents. A schematic of the CALPHAD approach can be seen in Figure 1. The CALPHAD approach assumes that the systems being assessed are in equilibrium, that is, the lowest energy state under given conditions of temperature, pressure, and composition. The previous section describes the mathematical relationships that govern minimization of the total free energy. Traditionally, one determined the minimum free energy state by writing competing reactions related with equilibrium constants, with the phase assemblage from the reaction that yielded the most negative Gibbs free energy state being the most stable.3,4 A more generalized approach was developed in the 1950s by White et al.5 using Lagrangian multipliers. Zeleznik and Gordon6 investigated the major approaches to computing equilibrium states, which led to their development of a computer code for computing equilibrium at NASA. The techniques were further developed by van Zeggeren and Storey7,8 through the 1960s. Ultimately, Eriksson9–11 developed an approach that was generally applicable to a wide variety of systems and included solution phases that could be nonideal. This led to the widely used code SOLGASMIX,11 whose equilibrium calculational methodology remains central to many contemporary software packages. While SOLGASMIX appears to be the first, other codes for equilibrium calculations such as those noted in Section 1.17.6 had similar developmental histories.
1.17.4 Treatment of Solutions Whether it is the nonstoichiometry of fluoritestructure UO2x or variable composition orthorhombic or tetragonal U–Zr alloy fuel, the accurate thermochemical description of these phases has
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Computational Thermodynamics: Application to Nuclear Materials
Ab-initio calculations
Thermodynamic optimization
Theory quantum mechanics statistical thermodynamics
Estimates
Experiments DTA, calorimetry, EMF, vapor pressure metallography, X-ray diffraction, ...
Models with adjustable parameters
Adjusting the parameters
Thermodynamic functions G, H, S, Cp = ¦ (T, P, X, ...)
Storage databases, publications
Equilibrium calculations
Equilibria Graphical representation
Phase diagrams
Applications Figure 1 Diagram illustrating the computer coupling of phase diagrams and thermochemistry approach after Zinkevich.2
been through the use of solution models. Solid and liquid solution modeling from simple highly dilute systems to more complex interstitial and substitutional solutions with multiple lattices has been a rich field for some time, yet it is far from fully developed. To accurately describe the energetics of solutions will eventually require bridging atomistic models with the mesoscale treatments currently being used. Recent approaches have begun to deal with defect structures in phases, although only in a very constrained manner which limits clustering and other phenomena. Yet, when they are coupled with accurate data that allow fitting of the model parameters, the resulting representations have been highly predictive of phase relations and chemistry. A number of texts provide useful descriptions of solution models from the simple to the complex.12–15 While there are several relatively accurate but rather intricate approaches, such as the cluster variation method, the discussion in this chapter is confined to
simpler models that are easily used in thermochemical equilibrium computational software and thus applicable to large, multicomponent systems of interest in nuclear fuels. The simplest model is the ideal solution where the constituents are assumed to mix randomly with no structural constraints and no interactions (bonding or short- or long-range order). The standard Gibbs free energy and ideal mixing entropy contributions are X j ni Gi G ¼ X G id ¼ RT ni ln ni ½8
where G is the weighted sum of the standard Gibbs free energy for the constituents in the j phase solution, Gid is the contribution from the entropy increase due to randomly mixing the constituents, which is the configurational entropy, and R is the ideal gas law constant. For an ideal solution, the sum of the two represents the Gibbs free energy of the system.
Computational Thermodynamics: Application to Nuclear Materials
where L is the coefficient of the expansion in k, which can also have a temperature dependence typically of the form a þ bT. Thus, a regular solution is defined as k equals zero leaving a single energetic term. This approach is related to the Bragg–Williams description, with random mixing of constituents yet with enthalpic energetic terms such that
In cases where there are significant interactions (bonding or repulsive interaction energies) among mixing constituents, an energetic term or terms need to be added to the solution free energy. The inclusion of a simple compositionally weighted excess energy term, G ex, accounts for the additional solution energy for what is historically termed a regular solution XX Xi Xj Eij ½9 G ex ¼
G ex ¼ XA XB EBW
i¼1 j >1
1.17.4.1
½10
Regular Solution Models
A common formalism for excess energy expressions is the Redlich–Kister–Muggianu relation, which for a binary system can be written as X Lk;ij ðXi Xj Þk ½11 G ex ¼ Xi Xj k
2500
2500 Model of Gürler et al. Model of Jacob et al. 2236 Liquid 2000
2000 Temperature (K)
1827
1669 XRh = 0.33 1500
1500
Solid (fcc)
1210
XRh = 0.55
1183 Pd (fcc) + Rh (fcc) 1000
1000 0.0 Pd
0.2
½12
Here, XAXB represents a random mixture of A and B components and is thus the probability that A–B is a nearest-neighbor pair, and EBW is the Bragg– Williams model energetic parameter. In a relevant example, Kaye et al.16 have generated a solution model for the five-metal white phase noted above and more extensively discussed in Chapter 2.20, Fission Product Chemistry in Oxide Fuels. A binary constituent of the model is the fcc-structure Pd–Rh system, which at elevated temperatures forms a single solid solution across the entire compositional range. The phase diagram of Figure 2 also shows a lowtemperature miscibility gap, that is, two coexisting identically structured phases rich in either end member. The excess Gibbs free energy expression for
where X is the mole fraction and Eij the interaction energy between the components i and j. The system free energy is thus G ¼ G þ G id þ G ex
459
0.4
0.6
0.8
XRh
1.0 Rh
16
Figure 2 Computed Pd–Rh phase diagram with indicated data of Kaye et al. illustrating complete fcc solid-solution range. Reproduced from Kaye, M. H.; Lewis, B. J.; Thompson, W. T. J. Nucl. Mater. 2007, 366, 8–27 from High Temperature Materials Laboratory.
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Computational Thermodynamics: Application to Nuclear Materials -4
the fcc phase was determined from an optimization using tabulated thermochemical information together with the phase equilibria and yielded
-8
½13
1.17.4.2 Variable Stoichiometry/Associate Species Models As noted in Chapter 2.01, The Actinides Elements: Properties and Characteristics; Chapter 2.02, Thermodynamic and Thermophysical Properties of the Actinide Oxides; and Chapter 2.20, Fission Product Chemistry in Oxide Fuels, modeling of complex systems such as U–Pu–Zr and (U,Pu)O2x has been exceptionally difficult. For example, actinide oxide fuel is understood to be nonstoichiometric almost exclusively due to oxygen site vacancies and interstitials. As a result, the fluoritestructure phase has been treated as being composed of various metal-oxygen species with no vacancies on the metal lattice. An early and successful modeling approach has employed a largely empirical use of variable stoichiometry species that are mixed as subregular solutions to fit experimental information.17–20 This technique can be viewed as a variant of the associate species method.21 In the approach, thermochemical values were determined from the phase equilibria, that is, the phase boundaries, and data for the temperature– composition–oxygen potential [mO2 ¼ RT ln(pO2*)] where pO*2 is a dimensionless quantity defined by the oxygen pressure divided by the standard state pressure of 1 bar. The UO2x phase, for example, was treated as a solid solution of UO2 and UaOb where the values of a and b were determined by a fit to experimental data. Figure 3 illustrates the trial and error process using a limited data set to obtain the species stoichiometry which results in the best fit to the data. As can be seen, a variety of stoichiometries for the constituent species yield differing curves of ln(pO2*) versus f(x), with the most appropriate matching the slope of 2. Thus, for this example U10/3O23/3 provides for an optimum fit between U3O7 and U4O9, and its solution with UO2 best reproduces the observed oxygen potential behavior. Utilizing a much more extensive data set from a variety of sources resulted in a set of best fits to the data, yet they required three solid solutions to adequately represent the entire compositional range for UO2x. These are
-12 In (pO2* )
G ex ¼ XPd XRh ½21247 þ 2199XRh ð2:74 0:56XRh ÞT
-16
Raw data Hagemark and Broli, 1673 K Roberts and Walter, 1695 K [UO2] + [UO3] [UO2] + [U2O5] [UO2] + [U3O7]
-20
[UO2] + [U10/3 O23/3] [UO2] + [U4O9] Slope -2
-24 -8
-4
0 f (X)
4
8
Figure 3 The ln(pO2* ) dependence as a function of x for UO2þx and of f(x) for several solid-solution species’ stoichiometries for an illustrative oxygen pressure– temperature–composition data set. Coincidence with the theoretical slope of 2 indicates the proper solution model. Reproduced from Lindemer, T. B.; Besmann, T. M. J. Nucl. Mater. 1985, 130, 473–488.
UO2þx (high hyperstoichiometry, i.e., large values of x): UO2 þ U3O7, UO2þx (low hyperstoichiometry, i.e., smaller values of x): UO2 þ U2O4.5, and UO2x (hypostoichiometric): UO2 þ U1/3. The results of the models for UO2x are plotted in Figure 4 together with the entire data set used for optimizing the system. The above models for UO2x have been widely adopted, as has been a similar model of PuO2x .16 These have also been combined to construct a successful model for (U,Pu)O2x .16 Lewis et al.22 used an analogous technique for UO2x . Lindemer23 and Runevall et al.24 have generated successful models of CeO2x . Runevall et al.24 also used the method for NpO2x , AmO2x (with the work of Thiriet and Konings25), (U,Am)O2x, (Th,U)O2x, (U,Ce)O2x, (Pu,Am)O2x, and (U,Pu,Am)O2x. They noted that results for the (Th,U)O2þx were less successful perhaps because of the difficulty in the measurements
Computational Thermodynamics: Application to Nuclear Materials
)
UO
0
(
n
xi
x 2+
7
0.2
25
0.
0.2
461
U–O liquid region
0.15 0.1
0.03 0.01 0.006
-200
0.0 0.0 06 1
03 0.0
10-3
-400
0.0
10 -4
3
10 -3
Oxygen potential (kJ mol–1)
0.003
10 -
10 -5
10 -6
U–O liquid region
(UO
2-x )
X = 0.3
x in
(U
O
2)
ex
ac
t
6
0.2
10 -
-600
10 -4
0.1
5
-800 500
1000
1500
2000
2500
3000
Temperature (K) Figure 4 Oxygen potential plotted versus x for the models of UO2x of Lindemer and Besmann17 overlaid with the entire data set used for the optimization.
made near stoichiometry. Osaka et al.26–28 used the approach to successfully represent the (U,Am)O2x, (U,Pu,Am)O2x, and (Am,Th)O2x phases. 1.17.4.3
Compound Energy Formalism
Regular or subregular solution and variable stoichiometry representations, while relatively successful, lack a sense of reproducing physical processes. Specifically, they are constrained with regard to accurately dealing with entropy contributions because of the defect structures in nonstoichiometric phases and substitutional solutions. A practical advance has
been the sublattice approach, which has been further refined for crystalline systems in the compound energy formalism (CEF).29 As typical for cation– anion systems, the structure of a phase can be represented by a formula, for example, (A,B)k(D,E)l where A and B mix on one sublattice and D and E mix on a second sublattice. The constitution of the phase is made up of occupied site fractions, and allowing one of the constituents to be a vacancy permits treatment of nonstoichiometric systems. Even with a sublattice approach such as CEF, the relationship of eqn [10] is still applicable, but with an interpretation related to a sublattice model. The sum
462
Computational Thermodynamics: Application to Nuclear Materials
of the standard Gibbs free energies in this case is the sum of the values for the paired sublattice constituents, which for the example above might be AkDl. Each is a unique set with the Gibbs free energies for the constituents derived from the end-member standard Gibbs free energies, typically through simple geometric additions with any necessary additional configurational entropy contributions. The entropy contribution from mixing on the sublattice sites is defined as X zs y s lnðyjs Þ ½14 G id ¼ RT where z is the stoichiometric coefficient, s defines the lattice, and y is the site fraction for species j. Excess terms represent the interaction energetics between each set of sublattice constituents, for example, AkDl : BkDl. Again, a Redlich–Kister–Muggianu formulation that includes expansion terms for interactions between the constituents can be used: XXX yi1 yj2 yk1 Li; j :k G xs ¼ i
þ
j
k
XXX i
j
k
yk2 yi1 yj1 Lk:i; j
½15
where the sums are associated with components on each sublattice 1 and 2 and the L values are terms for the interaction energies between cations i and j on one sublattice when the other sublattice is occupied only by cation k, and vice versa for the second term. The PuO2x phase has been successfully represented by a CEF approach by Gueneau et al.30 The phase can be described by two sublattices with vacancies only on the anion sites (Pu4þ,Pu3þ)1(O2,Va)2. Including the end members, the constituent species are then
(Pu4þ)1(O2)2, (Pu4þ)1(Va)2, (Pu3þ)1(O2)2, and (Pu3þ)1(Va)2. A schematic of the relationship between the constituents is seen in Figure 5 where the corners represent each of the constituents listed above. The charged constituents must sum to neutrality, and the line designating neutrality is seen in Figure 5. Gibbs free energy expressions for each of the units can be determined from standard state values. Optimizations using all available thermochemical information, for example, oxygen potentials and phase equilibria, can thus yield the necessary corrections to the Gibbs free energies for the nonstandard constituents together with obtained interaction parameters (L values). The results are shown in Figure 6 where oxygen potential isotherms overlay the phase diagram and which shows mO2 results of models for other phases in the system. The CEF approach has recently begun to be more widely applied to nuclear fuels. Besides the PuO2x system noted above, Gueneau et al.31 also applied the model to accurately describe solid solution phases in the U–O system, as has Chevalier et al.32 who also addressed the U–O–Zr ternary system.33 Kinoshita et al.34 used a sublattice approach to model fluoritestructure oxides including ThO2x and NpO2x, although they did not include charged ionic cations and anions on the sublattices. Zinkevich et al.35 successfully modeled the CeO2x phase using the CEF approach in their comprehensive assessment of the Ce–O phase diagram. 1.17.4.4 Thermochemical Modeling of Defects Another way to view solid solutions and nonstoichiometry is as a function of defects in the ideal lattice.
(Pu3+)1(Va)2 (+3)
PuO1.5 = (Pu3+)1(Va1/4, O2-3/4)2 (0) (Pu3+)1(O2-)2 (-1)
(Pu4+)1(Va)2 (+4)
Neutral line
(Pu4+)1(O2-)2 = PuO2 (0)
Figure 5 Compound energy formalism sublattice model illustration of the components and their charge in a two-dimensional representation after Gueneau et al.31
Computational Thermodynamics: Application to Nuclear Materials
463
0 -4 -8
log10 (pO2) in bar
-12 -16 -20 -24 -28 -32 -36 1.5
1.6
1.7
1.8 O/Pu ratio
1.9
2.0
Figure 6 Oxygen potentials overlaying the phase equilibria for the Pu–O system as computed by Gueneau et al.31 showing the results of the fit to the compound energy formalism model and representative data for the PuO2x phase. Reproduced from Gueneau, C.; Chatillon, C.; Sundman, B. J. Nucl. Mater. 2008, 378, 257–272.
This has been of particular interest for oxide fuels as they are seen to govern dissolution of cations and nonstoichiometry in oxygen behavior and as a result, transport properties. Defect concentrations are inherent in the CEF, as vacancies and interstitials on the oxygen lattice for fluorite-structure actinide systems are treated as constituents linked to cations (see Section 1.17.4.3). A more explicit treatment of oxide systems with point defects has been applied to a wide range of materials such as high-temperature oxide superconductors, TiO2, and ionic conducting membranes, among others. For oxide fuels, point defects have been described thermochemically by a number of investigators starting as early as 1965 with more recent treatments in fuels by Nakamura and Fujino,36 Stan et al.,37 and Nerikar et al.38 Oxygen site defects, which dominate in the fluorite-structure fuels, are of course driven by the multiple possible valence states of the actinides, most notably uranium, which can exhibit Uþ3, Uþ4, Uþ5, and Uþ6. A simple example of the point defect treatment can be seen in Stan et al.37 They optimized defect concentrations from the defect reactions described in the Kroger–Vinck notation 00
OO ¼ Oi þ VO
00
‰O2 þ 2U U ¼ Oi þ 2UU A dilute defect concentration was assumed such that there were no interactions between defects and thus no excess energy terms. The results of the fit to literature data are seen in Figure 7(a), where the stoichiometry of the fluorite-structure hyperstoichiometric urania is plotted as a function of defect concentration xa. The relationships were also used to compute oxygen potentials as a function of stoichiometry and are plotted in Figure 7(b) illustrating relatively good agreement with values computed by Nakamura and Fujino.36 1.17.4.5 Modified Associate Species Model for Liquids The liquid phases in nuclear fuels are important to model so that the phase equilibria can be completely assessed through comparison of experimental and computed phase diagrams. The availability of solidus and liquidus information also provides necessary boundaries for modeling the solid-state behavior. Finally, safety analysis requirements with regard to the potential onset of melting will benefit from accurate representations of the complex liquids.
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Computational Thermodynamics: Application to Nuclear Materials
0.14 0.25
0.12
Oi 1373 K •• VO
0.15
U•
x in UO2+x
Xa
0.2
Oi
U
0.1
0.10
0.05
V• • O
0.15
0.1 x in UO2+x
0.05 (a)
1273 K
1373 K
0.06 Model
Model
0.02 0.00 -12
0
Nakamura and Fujino
0.08
0.04 1273 K
UO2+x
(b)
-10 -8 log10 pO2 (atm)
-6
Figure 7 (a) Concentrations of defect species in UO2þx relative to the concentration of oxygen sites in the perfect lattice, as a function of nonstoichiometry, calculated with a defect model. (b) UO2þx nonstoichiometry as a function of partial pressure of oxygen. (Dashed line is model-derived and solid line are results of Nakamura and Fujino36 and Stan et al.37)
Ideal, regular/subregular, or Bragg–Williams formulations are not very successful in representing metal and especially oxide liquids where there are strong interactions between constituents. The CEF model is designed for fixed lattice sites, and thus it too will not handle liquids. The issues for these complex liquids involve the short-range ordering that generally occurs and its effect on the form of the Gibbs free energy expressions. One approach to dealing with the issue of these strong interactions is the modified associate species method. The modified associate species technique for crystalline materials was discussed to an extent in Section 1.17.4.2. Its application to, for example, oxide melts has been more broadly covered recently by Besmann and Spear39 with much of the original development by Hastie and coworkers.40–43 The approach assumes that the liquid can be modeled by an ideal solution of end-member species together with intermediate species. The modified term refers to the fact that an ideal solution cannot represent a miscibility gap in the liquid as that requires repulsive (positive) interaction energy terms. Thus, when a miscibility gap needs to be included, interaction energies between appropriate associate species are added to the formulation. In the associate species approach, the system standard Gibbs free energy is simply the sum of the constituent end-member and associate free energies, for example, A, B, and A2B, where inclusion of the A2B associate is found to reproduce the behavior well,
G ¼ XA GA þ XB GB þ XA2 B GA2 B
½16
Consequently, ideal mixing among end members and associates generates the entropy contribution G id ¼ RT ðXA ln XA þ XB ln XB þ XA2 B ln XA2 B Þ ½17 Should a nonideal term providing positive interaction energies be needed to properly address a miscibility gap, it would be added into the total Gibbs free energy as in eqn [10]. For example, for an interaction between A and A2B in the Redlich–Kister–Muggianu formulation the excess term is expressed as X Lk;A:A2 B ðXA XA2 B Þk ½18 G ex ¼ XA XA2 B k
The associate species are typically selected from the stoichiometry of intermediate crystalline phases, but others as needed can be added to accurately reflect the phase equilibria even when no stable crystalline phases of that stoichiometry exist. Gibbs free energies for these species can be derived from fits to the phase equilibria and other data following the CALPHAD method with first estimates generated from crystalline phases of the same stoichiometry or weighted sums of existing phases when no stoichiometric phase exists. The application of the method for the liquid phase in the Na2O–Al2O3 is seen in the computed phase diagram in Figure 8. For this system, the associate species required to represent the liquid were only Na2O, NaAlO2, (1/3) Na2Al4O7, and Al2O3. In nuclear fuel systems, Chevalier et al.44 applied an associate species approach using the components O, U, and O2U, although it deviated from the associate species approach in using binary interaction parameters in a Redlich–Kister–Muggianu form. The computed
Computational Thermodynamics: Application to Nuclear Materials
465
2200
0.87 (0.89)
Liquid
2000
1869 (1867)
2054 (2054)
1885 (1976)
Liquid + NaAIO2
Liquid + b-alumina
1600
1443 (1443)
1200
NaAlO2
1400
0.01 1126
1000 0.0 Na2O
0.2
0.4 0.6 Mole fraction Al2O3
0.8
NaAl9O14 (b-alumina)
1584 0.68 (1585) (0.63) Na2Al12O18 (b⬘⬘-alumina)
Temperature (º C)
1800
1.0 Al2O3
Figure 8 Calculated phase diagram for the Na2O–Al2O3 system using the modified associate species approach for the liquid. Values in parentheses are the accepted phase equilibria temperatures or compositions shown for comparison with the results of the modeling. Reproduced from Chevalier, P. Y.; Fischer, E.; Cheynet, B. J. Nucl. Mater. 2002, 303, 1–28.
phase diagram showing agreement with the liquidus/ solidus data is seen in Figure 9. The use of the modified associate species model with ternary and higher order systems can require the use of ternary or possibly quaternary associates. Another issue with the modified associate species approach is that in the case of a highly ordered solution which requires an overwhelming content of an associate compared to an end-member, the relations do not follow what should be Raoult’s law for dilute solutions. At the other extreme, it is also apparent that in the case of essentially zero concentration of associates, the relationships do not default to an ideal solution as one would expect. 1.17.4.6 Ionic Sublattice/Modified Quasichemical Model for Liquids In contrast to using associates for liquid solutions is a sublattice approach in which cations and anions are mixed on respective lattice sites. With anions and cations assigned to specific sublattices, it is possible
to capture interactions and short-range order with species occupying the sites and additional energetic terms. The components can essentially be allowed to independently mix on each sublattice within the energetic constraints and the system free energy minimized.46 The approach has been successfully used by Gueneau et al.47 to model the liquid in the U–O and Pu–O systems where ionic metal species reside on one lattice and oxygen anions, neutral UO2 or PuO2, charged vacancies, and O species on the other. An improvement to the simple sublattice approach is the quasichemical method introduced by Fowler and Guggenheim48 and later further developed by Pelton and coworkers.49–52 It approaches short-range order in liquids through the formation of nearest-neighbor pairs on a quasilattice. It thus differs significantly from the modified associate species approach such that in the quasichemical method short-range order is accommodated by components pairing and the energetically described extent of like and unlike components pairing. The technique thus avoids the paradox where a high
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Computational Thermodynamics: Application to Nuclear Materials
T (K)
3600 L1
3200
L2 a
a
2800 fcc C
2400
UO2+5x
2000 O8U3 (s)
1600 1200
U1 (bcc A2) U1 (TET)
800
O9U4 (S) O3U1 (s)
400 0.0
U1 (ORT A20) &
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
x (mol)
Figure 9 The computed U–O phase diagram of Chevalier et al.44 for which the liquid was modeled using the associate species approach with selected experimental points indicated. Reproduced from Chevalier, P. Y.; Fischer, E.; Cheynet, B. J. Nucl. Mater. 2002, 303, 1–28.
degree of short-range order in the associate species approach causes minimal end-member species content and therefore fails in the limit to be a Raoult’s law solution.52 In the modified quasichemical approach for a simple binary A–B system, the components are treated as distributed on a quasilattice and that an energetic term governs exchange among the pairs. ðA AÞ þ ðB BÞ ¼ 2ðA BÞ
½19
A parameter, Z, represents the nearest-neighbor coordination number such that each component forms Z pairs. For one mole of solution, ZXA ¼ 2nAA þ 2nAB
½20
ZXB ¼ 2nBB þ 2nAB
½21
where the moles of each pair are nAA, nBB, and nAB. The relative proportions of each pair is Xij, where Xij ¼ nij =ðnAA þ nBB þ nAB Þ
½22
The configurational entropy contribution is captured from the random distribution of the pairs over the quasilattice positions. The result is a heat of mixing of Hmix ¼ ðXAB =2ÞEQM
½23
where EQM is the quasichemical model energetic parameter, and a configurational entropy
Sconfig ¼ RðXA ln XA þ XB ln XB Þ RðZ=2ÞðXAA lnðXAA =XA2 Þ þ XBB lnðXBB =XB2 Þ þ XAB lnðXAB =2XA XB ÞÞ
½24
Utilizing Gibbs free energy functions for the components and expanding EQM as a polynomial in a scheme for minimizing the system free energy provide a system for optimization of the liquid using known thermochemical values and phase equilibria. Issues such as the displacement of the composition of maximum short-range order from 50% composition are dealt with by assuming different coordination numbers for each component. Greater accuracy is obtained by inclusion of the Bragg–Williams model, thus incorporating lattice interactions beyond nearest neighbors. The modification to the quasichemical model yields Hmix ¼ ðEQM =2ÞXAB þ EBW XA XB
½25
The extension of the modified quasichemical model to ternary systems is directly possible using only binary model parameters. An issue for the modified quasichemical model is that it fails at high deviations from random ordering, although that is generally not a problem because immiscibility will occur before the deviations grow too large. The model can also predict a large amount
Computational Thermodynamics: Application to Nuclear Materials
of ordering that can result in a negative configurational entropy, a physical impossibility.53
1.17.5 Thermochemical Data Sources Tabulated thermochemical data have been available from a number of sources for several decades. For general substances, the most familiar have been the NIST-JANAF Thermochemical Tables54 and Thermochemical Data of Pure Substances.55 The data are generally given as 298.15 K values, and columns of values such as Gibbs free energy, heat, entropy, and heat capacity are listed incrementally with temperature. The NIST-JANAF Thermochemical Tables are also available online through the National Institute for Standards and Technology (NIST). One of the key issues in using thermochemical data is the consistency of the standard states. The current commonplace usage is that the standard state is defined as 298.15 K and 1 bar (100 kPa) pressure. Small, but potentially important, errors can arise if data with different standard states are combined, for example, values at standard state pressure of 1 atm and of 1 bar are used together. Much of the thermochemical data compilations are currently available as computer databases. In addition to the NIST-JANAF Thermochemical Tables54 is that of the Scientific Group Thermodata Europe (SGTE),56 which is well-established and has an ongoing program to assess data and add new species and phases. The same is true for the databases provided by THERMODATA57 in Grenoble, France, which has compound and solution values. Another source is MALT,46 supplied by Kagaku Gijutsu-Sha in Japan, which is more limited than SGTE,56 focusing on data that directly support industry issues. There have also been databases developed specifically for nuclear applications including THERMODATA,57 which has databases for both ex-vessel applications, NUCLEA, and for mixed oxide fuel (MOX). Kurata58 has developed a limited thermochemical database focused on metallic fuels. A database dedicated to zirconium alloys of interest for nuclear applications called ZIRCOBASE59,60 is available with fully developed representations of a number of zirconiumcontaining binary systems and some ternaries. The binaries and ternaries can be combined in generating higher order systems often with reasonably good accuracy. An SGTE56 nuclear materials database is also available containing most of the gaseous species and simple compounds of interest. An advanced nuclear
467
fuel-specific database initiated by the Commissariat a` l’Energie Atomique, FUELBASE,31 and which is expected to be moved under the auspices of the Nuclear Energy Agency with the Organization for Economic Cooperation and Development, is described in more detail in Chapter 2.02, Thermodynamic and Thermophysical Properties of the Actinide Oxides. Information on the most common compounds and, in recent years, solution phases for many important systems has become available in the literature and is included in databases such as those noted above. However, much important data and models are not available for nuclear systems, which have not received the same attention as, for example, commercial steels. With advances in first principles modeling, some stoichiometric compounds for which there is limited or no experimental information can have values computationally determined. This is more likely for gaseous species than for condensed phases because of the greater ease in modeling the vapor. Another approach to filling in needed data is to use simple estimation techniques. The heat capacity of a complex oxide can be fairly accurately represented by the linear summation of the values of the constituent oxides. A linear relationship with atomic number is often seen in the enthalpy of formation of analogous compounds. These and other methods are discussed extensively in Kubaschewski et al.14 Equilibrium computational software packages typically will automatically acquire the needed data from accompanying selected databases. The published and commercial databases are generally assessed, meaning that they are compatible with broadly accepted values for the systems and when used with other standard values in the database thus yield correct thermochemical and phase relations. However, caution is needed when using those data with additional values obtained from other sources such as published experimental or computed values so that fundamental relationships such as phase equilibria are preserved. Another very significant issue is the completeness of the information. A simple example is UO2 where calculations can be performed using database values for the phase, whereas in reality the phase varies in stoichiometry as UO2x and without including a representation for the nonstoichiometry any conclusions will be in doubt. Given the great complexity of the fuel and fission product phases described in Chapter 2.01, The Actinides Elements: Properties and Characteristics; Chapter 2.02, Thermodynamic and Thermophysical Properties of the Actinide Oxides; and Chapter 2.20, Fission
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Product Chemistry in Oxide Fuels, it is apparent that a thermochemical model of fuel undergoing burnup is far from complete. The metallic fuel composition U–Pu–Zr is reasonably well represented,61 largely from the constituent binaries, yet the fuel after significant burnup will also contain bred actinides and fission products. Similarly, the oxide fluorite fuel phase with uranium and plutonium has perhaps been completely represented (see Chapter 2.02, Thermodynamic and Thermophysical Properties of the Actinide Oxides), but it too has yet to be modeled containing other TRU elements and fission products. High burnup fuels will also generate other phases, as noted in Chapter 2.20, Fission Product Chemistry in Oxide Fuels, and these too are often complex solid solutions with numerous components. Thus, the critical question in thermochemical modeling is, does the database contain values and representations for all the species and phases of interest? Without inclusion of all important phases, the accuracy of any conclusions from calculations will be in question. As noted above, most databases are assessed, which implies that the included data have been evaluated with regard to the sources and methodologies used to obtain the data. It also implies that the data are consistent with information for other phases and species containing one or more of the same components/elements. Calculations of properties must return the appropriate relationships between phases and species (e.g., activities and phase equilibria). Thus, the use of data from multiple sources raises the specter of inconsistent values being used, leading to inaccurate representations. Assuring that the data are consistent between sources through checks of relationships such as known phase equilibria is important to maintaining confidence in the information providing accurate results.
1.17.6 Thermochemical Equilibrium Computational Codes There are a variety of software packages that will perform chemical equilibrium calculations for complex systems such as nuclear fuels. These have become quite versatile, able to compute the thermal difference in specific reactions as well as determining global equilibria at uniform temperature or in an adiabatic system. They also provide output through internal postprocessors or by exporting to text or spreadsheet applications. There are also a variety of output forms including activities/partial pressures,
compositions within solution phases, and amounts, which can include plotting of phase and predominance diagrams. The commercial products include FactSage62 and ThermoCalc63 which also contain optimization modules that allow use of activity and phase equilibria to obtain thermochemical values and fit to models for solutions. Other products include Thermosuite,64 MTDATA,65 PANDAT,66 HSC,67 and MALT.57
1.17.7 Outlook Computational thermodynamics as applied to nuclear materials has already substantially contributed to the development of nuclear materials ranging from oxide and metal fuel processing to assessing clad alloy behavior. Yet, in both development of data and models for complex fuel and fuel-fission product systems and in the application of equilibrium calculations to reactor modeling and simulation, there is much to accomplish. Databases containing accurate representations of both metallic and oxide fuels with minor actinides are lacking, and even less is known about more advanced fuel concepts such as carbide and nitride fuels. Representations for multielement fission products dissolved in fuel phases or as secondary phases generated after considerable burnup are also unavailable, although some simple binary and ternary systems have been determined. These are critically needed as they will help govern activities in metal and oxide fuels, influencing thermal conductivity and providing source terms for transport of important species such as those containing iodine. The other broad area that needs significant attention is the development of algorithms for computing chemical equilibria. Although there are robust and accurate codes for computing equilibria within the software packages discussed in Section 1.17.6, these suffer from relatively slow execution. That is not a problem for the codes noted above where only a few calculations are required at any time. However, incorporation of equilibrium state calculations in broad fuel modeling and simulation codes with millions of nodes to determine the spatial distribution of phases, solution compositions (e.g., local O/M in oxide fuel), and local activities poses a different problem. Current algorithms are far too slow for such use, and therefore, new techniques need to be developed to accomplish these calculations within the larger modeling and simulation codes.68
Computational Thermodynamics: Application to Nuclear Materials
Acknowledgments The author wishes to thank Steve Zinkle, Stewart Voit, and Roger Stoller for their valuable comments. Research supported by the U.S. Department of Energy, Office of Nuclear Energy, under the Fuel Cycle Research and Development and Nuclear Energy Advanced Modeling and Simulation Programs. This manuscript has been authored by UT-Battelle, LLC, under Contract No. DE-AC0500OR22725 with the U.S. Department of Energy. The U.S. Government retains and the publisher, by accepting the article for publication, acknowledges that the U.S. Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or reproduce the published form of this manuscript, or allow others to do so, for U.S. Government purposes.
22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37.
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Lewis, B. J.; Thompson, F. A.; Thompson, D. M.; Thurgood, C.; Higgs, J. J. Nucl. Mater. 2004, 328, 180–196. Lindemer, T. B. CALPHAD 1986, 10(2), 129–136. Runevall; et al. 2008 (unpublished work). Thiriet, C.; Konings, R. J. M. J. Nucl. Mater. 2003, 320, 292–298. Osaka, M.; Namekawa, T.; Kurosaki, K.; Yamanaka, S. J. Nucl. Mater. 2005, 344, 230–234. Osaka, M.; Kurosak, K.; Yamanaka, S. J. Alloys Compd. 2007, 428, 355–361. Osaka, M.; Kurosaki, K.; Yamanaka, S. J. Nucl. Mater. 2007, 362, 374–382. Hillert, M. J. Alloys Compd. 2001, 320, 161–176. Gueneau, C.; Chatillon, C.; Sundman, B. J. Nucl. Mater. 2008, 378, 257–272. Gueneau, C.; Baichi, M.; Labroche, D.; Chatillon, C.; Sundman, B. J. Nucl. Mater. 2002, 304, 161–175. Chevalier, P. Y.; Fischer, E.; Cheynet, B. J. Nucl. Mater. 2002, 303, 1–28. Chevalier, P. Y.; Fischer, E.; Cheynet, B. CALPHAD 2004, 28, 15–40. Kinoshita, H.; Setoyama, D.; Saito, Y.; et al. J. Chem. Thermodyn. 2003, 35, 719–731. Zinkevich, M.; Djurovic, D.; Aldinger, F. Solid State Ionics 2006, 177, 989–1001. Nakamura, A.; Fujino, T. J. Nucl. Mater. 1986, 140, 113–130. Stan, M.; Ramirez, J. C.; Cristea, P.; et al. J. Alloys Compd. 2007, 444–445, 415–423. Nerikar, P. V.; Liu, X. Y.; Uberuaga, B. P.; Stanek, C. R.; Phillpot, S. R.; Sinnott, S. B. J. Phys. Condens. Matter 2009, 21, 435602. Besmann, T. M.; Spear, K. E. J. Am. Ceram. Soc. 2002, 85(12), 2887–2894. Hastie, J. W.; Bonnell, D. W. High Temp. Sci. 1985, 19(3), 275–306. Hastie, J. W. Pure Appl. Chem. 1984, 56(11), 1583–1600. Hastie, J. W.; Plante, E. R.; Bonnell, D. W. Vaporization of Simulated Nuclear Waste Glass; NBSIR 83; NIST: Gaithersburg, MD, 1983; p 2731. Bonnell, D. W.; Hastie, J. W. High Temp. Sci. 1989, 26, 313–334. Chevalier, P. Y.; Fischer, E.; Cheynet, B. J. Nucl. Mater. 2002, 303, 1–28. Spear, K. S.; Allendorf, M. D. J. Electrochem. Soc. 2002, 149(12), B551–B559. Hillert, M.; Jansson, B.; Sundman, B.; Agren, J. Metall. Trans. 1985, 16A, 261–266. Gueneau, S.; Chatain, J. C.; Dumas, J.; et al. FUELBASE: A thermodynamic database for advanced nuclear fuels. In Proceedings HTR2006: 3rd International Topical Meeting on High Temperature Reactor Technology, Johannesburg, South Africa, 2006. Fowler, R. H.; Guggenheim, E. A. Statistical Thermodynamics; Cambridge University Press: Cambridge, 1939. Pelton, A. D.; Decterov, S. A.; Eriksson, G.; Robelin, C.; Dessureault, Y. Metall. Mater. Trans. B 2000, 31, 651–659. Blander, M.; Pelton, A. D. Geochim. Cosmochim. Acta 1987, 51, 85–95. Pelton, A. D.; Chartrand, P. Metall. Mater. Trans. A 2001, 32, 1355–1360. Pelton, A. D.; Kang, Y. B. Int. J. Mater. Res. 2007, 98(10), 907–917. Hillert, M.; Selleby, M.; Sundman, B. Acta Mater. 2009, 57, 5237–5244. Chase, M. W. J. Phys. Chem. Ref. Data. 1998, Monograph 9. Barin, I. Thermochemical Data for Pure Substances; VCH: Weinheim, 1989.
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56. Dinsdale, A. T. CALPHAD 1991, 15(4), 317–425. 57. Yokokawa, H.; Yamauchi, S.; Matsumoto, R. CALPHAD 2002, 26(2), 155–162. 58. Kurata, M. CALPHAD 1999, 23, 305–337. 59. Dupin, N.; Ansara, I.; Servant, C.; Toffolon, C.; Lemaignan, C.; Brachet, J. C. J. Nucl. Mater. 1999, 275(3), 287–295. 60. Toffolon-Masclet, C.; Brachet, J. C.; Servant, C.; et al. In Zirconium in the Nuclear Industry: 15th International Symposium; Kammenzind, B., Limback, M., Eds.; American Society for Testing and Materials Special Technical Publications: West Conshohocken, Pennsylvania, 2009; Vol. 1505, pp 754–775. 61. Bale, C. W.; Chartran, P.; Degterov, S. A.; et al. CALPHAD 2002, 26(2), 39.
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Sundman, B.; Jansson, B.; Anderson, J. CALPHAD 1985, 9(2), 37. Cheynet, B.; Chevalier, P. Y.; Fischer, P. CALPHAD 2002, 26(2), 167–174. Cheynet, B.; Chevalier, P.-Y.; Fischer, E. CALPHAD 2002, 26, 167–174. Davies, R. H.; Dinsdale, A. T.; Gisby, J. A.; Robinson, J. A. J.; Martin, S. M. CALPHAD 2002, 26(2), 229–271. Chen, S. L.; Zhang, F.; Daniel, S. J. Miner. Met. Mater. Soc. 2003, 55(12), 4. Wu, L.; Themelis, N. J. CIM Bull. 1988, 81(914), 97. Piro, M. H.; Welland, M. J.; Lewis, B. J.; Thompson, W. T.; Olander, D. R. Development of a self-standing numerical tool to compute chemical equilibria in nuclear materials. In Proceedings of Top Fuel, Paris, France 2009.
1.18
Radiation-Induced Segregation
M. Nastar and F. Soisson Commissariat a` l’Energie Atomique, DEN Service de Recherches de Me´tallurgie Physique, Gif-sur-Yvette, France
ß 2012 Elsevier Ltd. All rights reserved.
1.18.1
Introduction
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1.18.2 1.18.2.1 1.18.2.2 1.18.2.3 1.18.2.3.1 1.18.2.3.2 1.18.2.3.3 1.18.2.3.4 1.18.2.3.5 1.18.2.4 1.18.2.5 1.18.3 1.18.3.1 1.18.3.2 1.18.3.2.1 1.18.3.2.2 1.18.3.3 1.18.3.3.1 1.18.3.3.2 1.18.3.3.3 1.18.3.3.4 1.18.3.4 1.18.3.4.1 1.18.3.4.2 1.18.4 1.18.4.1 1.18.4.1.1 1.18.4.1.2 1.18.4.1.3 1.18.4.1.4 1.18.4.2 1.18.4.2.1 1.18.4.2.2 1.18.4.3 1.18.5 1.18.5.1 1.18.5.2 1.18.5.3 1.18.6 References
Experimental Observations Anthony’s Experiments First Observations of RIS General Trends Segregating elements Segregation profiles: Effect of the sink structure Temperature effects Effects of radiation particles, dose, and dose rates Impurity effects RIS and Precipitation RIS in Austenitic and Ferritic Steels Diffusion Equations: Nonequilibrium Thermodynamics Atomic Fluxes and Driving Forces Experimental Evaluation of the Driving Forces Local chemical potential Thermodynamic databases Experimental Evaluation of the Kinetic Coefficients Interdiffusion experiments Anthony’s experiment Diffusion during irradiation Available diffusion data Determination of the Fluxes from Atomic Models Jump frequencies Calculation of the phenomenological coefficients Continuous Models of RIS Diffusion Models for Irradiation: Beyond the TIP Manning approximation Interstitials Analytical solutions at steady state Concentration-dependent diffusion coefficients Comparison with Experiment Dilute alloy models Austenitic steels Challenges of the RIS Continuous Models Multiscale Modeling: From Atomic Jumps to RIS Creation and Elimination of Point Defects Mean-Field Simulations Monte Carlo Simulations Conclusion
473 473 474 475 475 475 476 476 476 476 477 479 480 480 480 481 481 482 482 482 483 483 483 484 486 486 487 487 487 488 488 488 489 490 490 490 491 491 494 495
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Radiation-Induced Segregation
Abbreviations AKMC bcc DFT dpa fcc IASCC
Atomic kinetic Monte Carlo Body-centered cubic Density functional theory Displacement per atom Face-centered cubic Irradiation-assisted stress corrosion cracking IK Inverse Kirkendall MIK Modified inverse Kirkendall nn Nearest neighbor NRT Norgett, Robinson, and Torrens PPM Path probability method RIP Radiation-induced precipitation RIS Radiation-induced segregation SCMF Self-consistent mean field TEM Transmission electron microscopy TIP Thermodynamics of irreversible processes
Symbols D Diffusion coefficient Phenomenological coefficient Lij, L or L-coefficient
1.18.1 Introduction Irradiation creates excess point defects in materials (vacancies and self-interstitial atoms), which can be eliminated by mutual recombination, clustering, or annihilation of preexisting defects in the microstructure, such as surfaces, grain boundaries, or dislocations. As a result, permanent irradiation sustains fluxes of point defects toward these point defect sinks and, in case of any preferential transport of one of the alloy components, leads to a local chemical redistribution. These radiation-induced segregation (RIS) phenomena are very common in alloys under irradiation and have important technological implications. Specifically in the case of austenitic steels, because Cr depletion at the grain boundary is suspected to be responsible for irradiation-assisted stress corrosion, a large number of experiments have been conducted on the RIS dependence on alloy composition, impurity additions, irradiation flux and time, irradiation particles (electrons, ions, or neutrons), annealing treatment before irradiation, and nature of grain boundaries.1–5
The first RIS models generally consisted of application of Fick’s laws to reproduce two specific effects of irradiation: diffusion enhancement due to the increase of point defect concentration, and the driving forces associated with point defect concentration gradients. According to these models, RIS is controlled by kinetic coefficients D or L (defined below) relating atomic fluxes to gradients of concentration or chemical potentials. It was shown that these coefficients are best defined in the framework of the thermodynamics of irreversible processes (TIPs) within the linear response theory. RIS models were then separated into two categories: models restricted to dilute alloys, and models developed for concentrated alloys. From the beginning until now, the dilute alloy models have benefited from progress made in the diffusion theory.6 The explicit relations between the phenomenological coefficients L and the atomic jump frequencies have been established, at least for alloys with first nearest neighbor (nn) interactions. In principle, such relations allow the immediate use of ab initio atomic jump frequencies and lead to predictive RIS models.7 While the progress of RIS models of dilute alloys is closely related to that of diffusion theory, most segregation models for concentrated alloys still use oversimplified diffusion models based on Manning’s relations.8 This is mainly because the jump sequences of the atoms are particularly complex in a multicomponent alloy on account of the multiple jump frequencies and correlation effects that are involved. Only very recently has an interstitial diffusion model been developed that could account for short-range order effects, including binding energies with point defects.9,10 Emphasis has so far been placed on comparisons with experimental observations. The continuous RIS models have been modified to include the effect of vacancy trapping by a large-sized impurity or the nature and displacement of a specific grain boundary. Most of the diffusivity coefficients of Fick’s laws are adjusted on the basis of tracer diffusion data. Paradoxically, the first RIS models were more rigorous11 than the present ones in which thermodynamic activities, particularly some of the cross-terms, are oversimplified. In this review, we go back to the first models starting from the linear response theory, albeit slightly modified, to be able to reproduce the main characteristics of an irradiated alloy. It is then possible to rely on the diffusion theories developed for concentrated alloys. Then again, lattice rate kinetic techniques12–14 and atomic kinetic Monte Carlo (AKMC) methods15–17
Radiation-Induced Segregation
have become efficient tools to simulate RIS. Thanks to a better knowledge of jump frequencies due to the recent developments of ab initio calculations, these simulations provide a fine description of the thermodynamics as well as the kinetics of a specific alloy. Moreover, information at the atomic scale is precious when RIS profiles exhibit oscillating behavior and spread over a few tens of nanometers. Discoveries and typical observations of RIS are illustrated in the first section. In the second section, the formalism of TIP is used to write the alloy flux couplings. It is explained that fluxes can be estimated only partially from diffusion experiments and thermodynamic data. An alternative approach is the calculation of fluxes from the atomic jump frequencies. The third section presents more specifically the continuous RIS models separated into the dilute and concentrated alloy approaches. The last section introduces the atomic-scale simulation techniques.
1.18.2 Experimental Observations 1.18.2.1
Anthony’s Experiments
RIS was predicted by Anthony,18 in 1969, a few years before the first experimental observations: a rare case in the field of radiation effects. The prediction stemmed from an analogy with nonequilibrium segregation observed in aluminum alloys quenched from high temperature. Between 1968 and 1970, in a pioneering work in binary aluminum alloys, Anthony and coworkers18–22 systematically studied the nonequilibrium segregation of various solute elements on the pyramidal cavities formed in aluminum after quenching from high temperature. They explained this segregation by a coupling between the flux of excess vacancies toward the cavities and the flux
473
of solute (Figure 1). Nonequilibrium segregation had been previously observed by Kuczynski et al.23 during the sintering of copper-based particles and by Aust et al.24 after the quenching of zone refined metals. Anthony suggested that similar coupling should produce nonequilibrium segregation in alloys under irradiation.18,19 He predicted that the segregation should be much stronger than after quenching because under irradiation, the excess vacancy concentration and the resulting flux can be sustained for very long times.19,25 As for the cavities formed by vacancy condensation in alloys under irradiation, which result in the swelling phenomenon (Chapter 1.03, Radiation-Induced Effects on Microstructure and Chapter 1.04, Effect of Radiation on Strength and Ductility of Metals and Alloys), he pointed out that with solute and solvent atoms of different sizes, segregation should generate strains around the voids.25 Finally, he predicted intergranular corrosion in austenitic steels and zirconium alloys, resulting from possible solute depletion near grain boundaries.25 Anthony also presented a detailed discussion on nonequilibrium segregation mechanisms, in the framework of the TIP,18–21 showing that the nonequilibrium tendencies are controlled by the phenomenological coefficients Lij of the Onsager matrix, which can be – in principle – computed from vacancy jump frequencies (see below Section 1.18.3). Clarifying previous discussions on nonequilibrium segregation mechanisms,23,24 he considered two limiting cases for the coupling between solute and vacancy fluxes in an A–B alloy (at the time, he did not apparently consider the coupling between solute and interstitial fluxes and its possible contribution to RIS). In both cases, the total flux of atoms must be equal and in the direction opposite to the vacancy flux:
Backscattered electrons
ZnKa
e–
Al2O3 film
Analyzed zone Z
Vacancy condensation cavity
Surface S JZn
JV
Aluminum matrix
Figure 1 One of Anthony’s experiments. After quenching of an Al–Zn alloy, vacancies condense in small pyramidal cavities (left), under an Al2O3 thin film covering the surface. Electronic probe measurements reveal enrichments in Zn around the cavities. Reproduced from Anthony, T. R. J. Appl. Phys. 1970, 41, 3969–3976.
JVA
(a)
JV JVB
(b)
JVA
Sink
JV JVB
Sink
Radiation-Induced Segregation
Sink
474
Ji JiB
JiA
(c)
Figure 2 Radiation-induced segregation mechanisms due to coupling between point defect and solute fluxes in a binary A–B alloy. (a) An enrichment of B occurs if dBV < dAV and a depletion if dBV > dAV . (b) When the vacancies drag the solute, an enrichment of B occurs. (c) An enrichment of B occurs when dBI > dAI .
1. If both A and B fluxes are in the direction opposite to the vacancy flux (Figure 2(a)), one can expect a depletion of B near the vacancy sinks if the vacancy diffusion coefficient of B is larger than that of A (dBV > dAV ); in the opposite case (dBV < dAV ), one can expect an enrichment of B (it is worth noting that this was essentially the explanation proposed by Kuczynski et al.23 in 1960). 2. But A and B fluxes are not necessarily in the same direction. If the B solute atoms are strongly bound to the vacancies and if a vacancy can drag a B atom without dissociation, the vacancy and solute fluxes can be in the same direction (Figure 2(b)): this was the explanation proposed by Aust et al.24 In such a case, an enrichment of B is expected, even if dBV > dAV . 1.18.2.2
First Observations of RIS
In 1972, Okamoto et al.26 observed strain contrast around voids in an austenitic stainless steel Fe–18Cr–8Ni–1Si during irradiation in a highvoltage electron microscope. They attributed this contrast to the segregation strains predicted by Anthony. This is the first reported experimental evidence of RIS. Soon after, a chemical segregation was directly measured by Auger spectroscopy measurements at the surface of a similar alloy irradiated by Ni ions.27 It was then realized that if the solute concentration near the point defect sinks reaches the solubility limit, a local precipitation would take place. In 1975, Barbu and Ardell28 observed such a radiationinduced precipitation (RIP) of an ordered Ni3Si phase in an undersaturated Ni–Si alloy. The analysis of strain contrast and concentration profiles measured by Auger spectroscopy suggested that undersized Ni and Si atoms (which can be more easily accommodated in interstitial sites) were diffusing toward point defect sinks, while oversized atoms (such as Cr) were diffusing away. Such a trend, later
confirmed in other austenitic steels and nickel-based alloys,29 led Okamoto and Wiedersich27 to conclude that RIS in austenitic steels was due to the migration of interstitial–solute complexes, and they proposed this new RIS mechanism, in addition to the ones involving vacancies (Figure 2(c)). Then again, Marwick30 explained the same experimental observations by a coupling between fluxes of vacancies and solute atoms, pointing out that thermal diffusion data showed Ni to be a slow diffuser and Cr to be a rapid diffuser in austenitic steels. We will see later that, in spite of many experimental and theoretical studies, the debate on the diffusion mechanisms responsible for RIS in austenitic steels is not over. Following these debates on RIS mechanisms, it became common to refer to the situation illustrated in Figure 2(a) as segregation by an inverse Kirkendall (IK) effect (the term was coined by Marwick30 in 1977) and to the one in Figure 2(b) as segregation by drag effects, or by migration of vacancy–solute complexes. In the classical Kirkendall effect,31 a gradient of chemical species produces a flux of defects. It occurs typically in interdiffusion experiments in A–B diffusion couples, when A and B do not diffuse at the same speed. A vacancy flux must compensate for the difference between the flux of A and B atoms, and this leads to a shift of the initial A/B interface (the Kirkendall plane). The IK effect is due to the same diffusion mechanisms but corresponds to the situation where the gradient of point defects is imposed and generates a flux of solute. The distinction between RIS by IK effect and RIS by migration of defect–solute complexes, initially proposed for the vacancy mechanisms, was soon generalized to interstitial fluxes by Okamoto and Rehn.32,33 RIS in dilute alloys, where solute–defect binding energies are clearly defined and often play a key role, is commonly explained by diffusion of solute–defect complexes, while the IK effect is often more useful to explain RIS in concentrated alloys. This distinction
Radiation-Induced Segregation
is reflected in the modeling of RIS (see Section 1.18.3). However, it is clear that RIS can occur in dilute alloys without migration of solute–defect fluxes. Moreover, such a terminology and sharp distinction can be somewhat misleading; the mechanisms are not mutually exclusive. In the case of undersized B atoms, for example, a strong binding between interstitial and B atoms can lead to a rapid diffusion of B by the interstitial (IK effect with DBi > DAi ) and to the migration of interstitial–solute complexes. More generally, one can always say that RIS results from an IK effect, in the sense that it occurs when a gradient of point defects produces a flux of solute. Nevertheless, because they are widely used, we will refer to these terms at times when they do not create confusion. 1.18.2.3
General Trends
Many experimental studies of RIS were carried out in the 1970s in model binary or ternary alloys, as well as in more complex and technological alloys (especially in stainless steels). It became apparent quite early on that RIS was a pervasive phenomenon, occurring in many alloys and with any kind of irradiating particle (ions, neutrons, or electrons). Extensive reviews can be found in Russell,1 Holland et al.,2 Nolfi,3 Ardell,4 and Was5: here, we present only the general conclusions that can be drawn from these studies. 1.18.2.3.1 Segregating elements
From the previous discussion, it is clear that it is difficult to predict the segregating element in a given alloy because of the competition between several mechanisms and the lack of precise diffusion data (especially concerning interstitial defects). As will be shown in Section 1.18.3, only the knowledge of the phenomenological coefficients Lij provides a reliable prediction of RIS. Nevertheless, on the basis of the body of RIS experimental studies, several general rules have been proposed. In dilute binary AB alloys, A and impurity thermal self-diffusion coefficients DA A diffusion coefficients DB are generally well known, at least at high temperatures. Tracer diffusion or intrinsic diffusion coefficients in some concentrated alloys are also available.34 RIS experiments do not reveal a systematic depletion of the fast-diffusing and enrichment of the slow-diffusing elements near the point defect sinks4,29: this suggests that the IK effect by vacancy diffusion is usually not the dominant mechanism. On the other hand, it seems that a clear correlation exists between RIS and the size effect33; undersized atoms usually segregate at point defect sinks, oversized
475
atoms usually do not. This suggests that interstitial diffusion could control the RIS, at least for atoms with a significant size effect. There are some exceptions: in Ni–Ge and Al–Ge alloys, the segregation of oversized solute atoms has been observed. Nevertheless, as pointed out by Rehn and Okamoto,33 no case of depletion of undersized solute atoms in dilute alloys has ever been reported. According to Ardell,4 this holds true even today. 1.18.2.3.2 Segregation profiles: Effect of the sink structure
Segregation concentration profiles induced by irradiation display some specific features. They can spread over large distances – a few tens of nanometers (see examples in Russell1 and Okamoto and Rehn29) – while equilibrium segregation is usually limited to a few angstroms. This is due to the fact that they result from a dynamic equilibrium between RIS fluxes and the back diffusion created by the concentration gradient at the sinks, while the scale of equilibrium segregation profiles is determined by the range of atomic interactions. Equilibrium profiles are usually monotonic, except for the oscillations, which can appear – with atomic wavelengths – in alloys with ordering tendencies.35 Segregation profiles observed in transient regimes are often nonmonotonic because of the complex interaction between concentration gradients of point defects and solutes. A typical example is shown in Section 1.18.5.3, where an enrichment of solute is observed near a point defect sink, followed by a smaller solute depletion between the vicinity of sink and the bulk. In this particular case, the depletion is due to a local increase in vacancy concentration, which results from the lower interstitial concentration and recombination rate. Other kinds of nonmonotonic profiles are sometimes observed, with typical ‘W-shapes.’ In some austenitic or ferritic steels, a local enrichment of Cr at grain boundaries survives during the Cr depletion induced by irradiation (see below). This could result from a competition between opposite equilibrium and RIS tendencies. However, the extent of the Cr enrichment often seems too wide to be simply due to an equilibrium property (around 5 nm, see, e.g., Sections 1.18.2.5 and 1.18.5.3). RIS profiles at grain boundaries are sometimes asymmetrical, which has been related to the migration of boundaries resulting from the fluxes of point defects under irradiation.37,38 The segregation is affected by the atomic structure and the nature of the sinks. It has been clearly shown that RIS in
476
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austenitic steels is much smaller at low angles and special grain boundaries than at large misorientation angles,39,40 the latter being much more efficient point defect sinks than the former.
RIS can occur only when significant fluxes of defects towards sinks are sustained, which typically happens only at temperatures between 0.3 and 0.6 times the melting point. At lower temperatures, vacancies are immobile and point defects annihilate, mainly by mutual recombination. At higher temperatures, the equilibrium vacancy concentration is too high; back diffusion and a lower vacancy supersaturation completely suppress the segregation. Temperature can also modify the direction of the RIS by changing the relative weight of the competing mechanisms, which do not have the same activation energy. In Ni–Ti alloys, for example, the enrichment of Ti at the surface below 400 C has been attributed to the migration of Ti–V complexes, and the depletion observed at higher temperatures should result from a vacancy IK effect.41 1.18.2.3.4 Effects of radiation particles, dose, and dose rates
RIS can be observed for very small irradiation doses; an enrichment of 10% of Si has been measured, for example, at the surface of an Ni–1%Si alloy, after a dose of 0.05 dpa at 525 C.32 Such doses are much lower than those required for radiation swelling5 or ballistic disordering effects.42 Increasing the radiation flux, or dose rate, directly results in higher point defect concentrations and fluxes towards sinks. The transition between RIS regimes is then shifted toward a higher temperature. But because point defect concentrations slowly evolve with the radiation flux (typically, proportional to its square root43 in the temperature range where RIS occurs), a high increase is needed to get a significant temperature shift. Radiation dose and dose rate are usually estimated in dpa and dpa s1, respectively, using the Norgett, Robinson, and Torrens model,44 especially when a comparison between different irradiation conditions is desired. It is then worth noting that the amount of RIS observed for a given dpa is usually larger during irradiation by light particles (electrons or light ions) than by heavy ones (neutrons or heavy ions). In the latter case, point defects are created by displacement cascades in a highly localized area, and a large fraction of vacancies and interstitials recombine or form
Back diffusion 0.6
T/Tm
1.18.2.3.3 Temperature effects
0.8
Radiation-induced segregation 0.4
0.2
0.0 10–6
Recombination
10–5
10–4
10–3
10–2
K0 (dpa s–1) Figure 3 Temperature and dose rate effect on the radiation-induced segregation.
point defect clusters. The fraction of the initially produced point defects that migrate over long distances and could contribute to RIS is decreased. On the contrary, during irradiation by light particles, Frenkel pairs are created more or less homogeneously in the material, and a larger fraction survive to migrate (Figure 3).45 1.18.2.3.5 Impurity effects
The addition of impurities has been considered as a possible way to control the RIS in alloys, for example, in austenitic steels. The most common method is the addition of an oversized impurity, such as Hf and Zr, in stainless steels,46 which should trap the vacancies (and, in some cases, the interstitials), thus increasing the recombination and decreasing the fluxes of defects towards the sinks. 1.18.2.4
RIS and Precipitation
As mentioned above, one of the most spectacular consequences of RIS is that it can completely modify the stability of precipitates and the precipitate microstructure.47 When the local solute concentration in the vicinity of a point defect sink reaches the solubility limit, RIP can occur in an overall undersaturated alloy. RIP of the g0 -Ni3Si phase is observed, for example, in Ni–Si alloys28 at concentrations well below the solubility limit (Ni3Si is an ordered L12 structure and can be easily observed in dark-field image in transmission electron microscopy (TEM)). In this case, it is believed that RIS is due to the preferential occupation of interstitials by undersized Si atoms.28 The g0 -phase
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(a)
(b)
477
(c)
Figure 4 Formation of Ni3Si precipitates in undersaturated solid solution under irradiation (a) in the bulk on preexisting dislocations and at interstitial dislocations (courtesy of A. Barbu), (b) at grain boundaries, and (c) at free surfaces. Reproduced from Holland, J. R.; Mansur, L. K.; Potter, D. I. Phase Stability During Radiation; TMS-AIME: Warrendale, PA, 1981.
can be observed on the preexisting dislocation network, at dislocation loops formed by self-interstitial clustering,28 at free surfaces45 or grain boundaries.48 The fact that the g0 -phase dissolves when irradiation is stopped clearly reveals the nonequilibrium nature of the precipitation. This is also shown by the toroidal contrast of dislocation loops (Figure 4(a)): the g0 -phase is observed only at the border of the loop on the dislocation line where self-interstitials are annihilated; when the loop grows, the ordered phase dissolves at the center of the loop, which is a perfect crystalline region where no flux of Si sustains the segregation. In supersaturated alloys, the irradiation can completely modify the precipitation microstructure. It can dissolve precipitates located in the vicinity of sinks when RIS produces a solute depletion. For example, in Ni–Al alloys,49 dissolution of g0 -precipitates is observed around the growing dislocation loops due to the Al depletion induced by irradiation (Figure 5), and in supersaturated Ni–Si alloys, Si segregation towards the interstitial sinks produces dissolution of the homogeneous precipitate microstructure in the bulk, to the benefit of the precipitate layers on the surfaces28 (Figure 6) and grain boundaries.50 In the previous examples, RIS was observed to produce a heterogeneous precipitation at point defect sinks. But homogeneous RIP of coherent precipitates has also been observed, for example, in Al–Zn alloys.51 Cauvin and Martin52 have proposed a mechanism that explains such a decomposition. A solid solution contains fluctuations of composition. In case of attractive vacancy–solute and interstitial– solute interactions, a solute-enriched fluctuation tends to trap both vacancies and interstitials, thereby favoring mutual recombination. The point defect concentrations then decrease, producing a flux of new defects toward the fluctuation. If the coupling with solute flux is positive, additional solute atoms
(110)
g001
Figure 5 Dissolution of g0 near dislocation loop precipitates in Ni–Al under irradiation. Reproduced from Holland, J. R.; Mansur, L. K.; Potter, D. I. Phase Stability During Radiation; TMS-AIME: Warrendale, PA, 1981.
arrive on the enriched fluctuations, and so it continues, till the solubility limit is reached. 1.18.2.5 Steels
RIS in Austenitic and Ferritic
We have seen that RIS was first observed in austenitic steels on the voids that are formed at large irradiation doses and lead to radiation swelling. The depletion of Cr at grain boundaries is suspected to play a role in irradiation-assisted stress corrosion cracking (IASCC); this is one of the many technological concerns related to RIS. The enrichment of Ni and the depletion of Cr can also stabilize the austenite near the sinks, and favor the austenite ! ferrite transition in the matrix.29 The segregation of minor elements can lead to the formation of g0 -precipitates (as in Ni–Si alloys), or various M23C6 carbides and other phases.1,29
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Radiation-Induced Segregation
(a)
(b)
Figure 6 Precipitate microstructure in Ni–Si alloys: the homogeneous distribution observed during thermal aging (a) is dissolved under electron irradiation and the surfaces of the transmission electron microscopy sample are covered by Ni3Si precipitates. (b) Ni–12%Si alloy under 1 MeV electron irradiation at 500 C, after a dose of 5 105 dpa. Courtesy of A. Barbu.
21 20 Cr concentration (wt%)
The segregation of major elements always involves an enrichment of Ni and a depletion of Cr at sinks over a length scale that depends on the alloy composition and irradiation conditions.5 The contribution of various RIS mechanisms is still debated. It is not clear whether it is the IK effect driven by vacancy fluxes, as suggested by the thermal diffusion coefficients DNi < DFe < DCr ,30 or the migration of interstitial– solute complexes, resulting in the segregation of undersized atoms,29 that is dominant. Some models of RIS take into account only the first mechanism,5 while others predict a significant contribution of interstitials.12 For the segregation of minor elements, the size effect seems dominant, with an enrichment of undersized atoms (e.g., Si27) and a depletion of oversized atoms (e.g., Mo53) (Figure 7). The effect of minor elements on the segregation behavior of major ones has been pointed out since the first experimental studies29; the effect of Si and Mo additions has been interpreted as a means of increasing the recombination rate by vacancy trapping. As previously mentioned, oversized impurity atoms, such as Hf and Zr, could decrease the RIS.46 RIS in ferritic steels has recently drawn much attention, because ferritic and ferrite martensitic steels are frequently considered as candidates for the future Generation IV and fusion reactors.54
Mill annealed Cr segregation
LWR-irradiated 316SS JEOL 2010F 0.7-nm probe
19 18 17 16 15 14 13 –20
Irradiated to ∼1.5 dpa W-shaped profile
Irradiated to ∼5 dpa Cr depletion
–15 –10 –5 0 5 10 15 Distance from grain boundary (nm)
20
Figure 7 Thermal and radiation-induced segregation profiles in 316 stainless steel. Reproduced from Bruemmer, S. M.; Simonen, E. P.; Scott, P. M.; Andresen, P. L.; Was, G. S.; Nelson, J. L. J. Nucl. Mater. 1999, 274, 299–314.
Experimental studies are more difficult in these steels than in austenitic steels, especially because of the complex microstructure of these alloys. Identification of the general trends of RIS behavior in these alloys
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13
479
465 ⬚C – irradiated
89
1.6 12
11
10
9
Concentration (wt%) (Fe)
Concentration (wt%) (Cr)
1.4 Iron
87
1.2 86 1.0 85
Chromium 0.8
84
0.6
83
0.4
Silicon 8
7
82
Concentration (wt%) (Ni, Si)
88
0.2
81
Nickel –100
–50
–25 –10 0 10 25
0 100
50
Distance from lath boundary (nm) Figure 8 Concentration profiles of Cr, Ni, Si, and Fe on either side of a lath boundary in 12% Cr martensitic steel after neutron irradiation to 46 dpa at 465 C. Reproduced from Little, E. Mater. Sci. Technol. 2006, 22, 491–518.
appears to be very difficult.55 Nevertheless, in some highly concentrated alloys, a depletion of Cr and an enrichment of Ni have been observed, reminding us of the general trends in austenitic steels54 (Figure 8). The RIS mechanisms are still poorly understood. The segregation of P at grain boundaries has been observed and, as in austenitic steels, the addition of Hf has been found to reduce the Cr segregation.55
1.18.3 Diffusion Equations: Nonequilibrium Thermodynamics In pure metals, the evolution of the average concentrations of vacancies CV and self-interstitials CI are given by: X dCV eq 2 ¼ K0 RCI CV kVs DV ðCV CV Þ dt s X dCI ¼ K0 RCI CV kIs2 DI CI dt s
½1
where K0 is the point defect production rate (in dpa s1) proportional to the radiation flux, R is the recombination rate, and DV and DI are the point defect diffusion coefficients. The third terms of the right hand side in eqn [1] correspond to point defect 2 annihilation at sinks of type s. The ‘sink strengths’ kVs 2 and kIs depend on the nature and the density of sinks
and have been calculated for all common sinks, such as dislocations, cavities, free surfaces, grain boundaries, etc.56,57 The evolution of point defect concentrations depending on the radiation fluxes and sink microstructure can be modeled by numerical integration of eqn [1], and steady-state solutions can be found analytically in simple cases.43 The evolution of concentration profiles of vacancies, interstitials, and chemical elements a in an alloy under irradiation are given by ]CV ¼ div JV þ K0 RCI CV ]t X eq 2 kVs DV ½CV C V s
X ]CI ¼ div JI þ K0 RCI CV kIs2 DI CI ]t s ]Ca ¼ div Ja ½2 ]t The basic problem of RIS is the solution of these equations in the vicinity of point defect sinks, which requires the knowledge of how the fluxes Ja are related to the concentrations. Such macroscopic equations of atomic transport rely on the theory of TIP. In this chapter, we start with a general description of the TIP applied to transport. Atomic fluxes are written in terms of the phenomenological coefficients of diffusion (denoted hereafter by Lij or, simply, L ) and the driving forces. The second part is
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Radiation-Induced Segregation
devoted to the description of a few experimental procedures to estimate both the driving forces and the L-coefficients. In the last part, we present an atomic-scale method to calculate the fluxes from the knowledge of the atomic jump frequencies. 1.18.3.1
Atomic Fluxes and Driving Forces
Within the TIP,58,59 a system is divided into grains, which are supposed to be small enough to be considered as homogeneous and large enough to be in local equilibrium. The number of particles in a grain varies if there is a transfer of particles to other grains. The transfer of particles a between two grains is described by a flux Ja , and the temporal variation of the local a concentration is given by the continuity equation ]Ca ¼ div Ja ]t
b
b
While thermodynamic data is usually available for the determination of driving forces, it is very difficult to determine the whole set of the L-coefficients from diffusion data. In the first part of the chapter, we define the driving forces as a function of concentration gradients. Then, we present the experimental diffusion coefficients in terms of the L-coefficients and thermodynamic driving forces. The last part of the section shows how to use first-principle calculations for building atomic jump frequency models to calculate macroscopic fluxes of specific alloys.
½3
The flux of species a between grains i and j is assumed to be a linear combination of the thermodynamic forces, Xb ¼ ðmjb mib Þ=kB T (i.e., of the gradient of chemical potentials rmb ) where mib is the chemical potential of species b on site i, T the temperature, and kB the Boltzmann constant. Variables Xb represent the deviation of the system from equilibrium, which tend to be decreased by the fluxes: X Lab Xb ½4 Ja ¼ b
The equilibrium constants are the phenomenological coefficients, and the Onsager matrix ðLab Þ is symmetric and positive. When diffusion is controlled by the vacancy mechanism, atomic fluxes are, by construction, related to the point defect flux: X JaV ½5 JV ¼ a
As gradients of chemical potential are independent, eqn [5] leads to some relations between the phenomenological coefficients and, if we choose to eliminate the LVVb coefficients, we obtain an expression for the atomic fluxes: X LVab ðXb XV Þ ½6 JaV ¼ b
Under irradiation, diffusion is controlled by both vacancies and interstitials. The flux of interstitials is also deduced from the atomic fluxes: X JbI ½7 JI ¼ b
Vacancy and interstitial contributions to the atomic fluxes are assumed to be additive: X X LVab ðXb XV Þ LIab ðXb þ XI Þ ½8 Ja ¼
1.18.3.2 Experimental Evaluation of the Driving Forces 1.18.3.2.1 Local chemical potential
The thermodynamic state equation defines a chemical potential of species i as the partial derivative of the Gibbs free energy G of the alloy, with respect to the number of atoms of species i, that is, Ni . The resulting chemical potential is a function of the temperature and molar fractions (also called concentrations) of the alloy components, Ci ¼ Ni =N , N being the total number of atoms. TIP postulates that local chemical potentials depend on local concentrations via the thermodynamic state equation. A chemical potential gradient rmi of species i is then equal to rmi 1 X1 qmi ¼ rCj Cj ½9 kB T kB T j Cj qCj where Ci is the local concentration of species i. In a binary alloy, concentration gradients of the two components are exactly opposite. The chemical potential gradient of component i is then proportional to the concentration gradient: rmi Fi ¼ rCi kB T Ci
½10
where Fi is called the thermodynamic factor. Furthermore, the Gibbs–Duhem relationship58 leads to interdependent chemical potential gradients: X Ck rmk ¼ 0 ½11 k ¼ 1;r
Radiation-Induced Segregation
where the sum runs over the number of species. Therefore, in a binary alloy there is one thermodynamic factor left: rmi F ¼ rCi kB T Ci
½12
where F ¼ FA ¼ FB . Note, that an alloy at finite temperature contains point defects. They are currently assumed to be at equilibrium with the local alloy composition, with the local chemical potential equal to zero. When calculating the thermodynamic factor, point defect concentration gradients are neglected. During irradiation, although point defects are not at equilibrium, one assumes that eqn [12] continues to be valid. Under irradiation, additional driving forces are involved. They correspond to the gradients of vacancy and interstitial chemical potentials, which are usually written in terms of their equilibrium eq eq concentrations CV and CI respectively: eq
eq
mV ¼ kB T lnðCV =CV Þ and mI ¼ kB T lnðCI =CI Þ
½13
leading to an expression of the associated driving force11: eq
rmV 1 x ]lnCV ¼ rCV VA rCA with xVA ¼ kB T CV CA ]lnCA
½14
The interstitial driving force has the same form, except that letter V is replaced by letter I. Note, that the equilibrium point defect concentrations may vary with the local alloy composition and stress. Although the variation of the equilibrium vacancy concentration is expected to be mainly chemical, the change of the elastic forces due to a solute redistribution at sinks should not be ignored for the interstitials.11 Due to the lack of experimental data, Wolfer11 introduced the equilibrium vacancy concentration as a contribution to a mean vacancy diffusion coefficient expressed in terms of the chemical tracer diffusion coefficients. Composition-dependent tracer diffusion coefficients could then account for the change of equilibrium vacancy concentration, with respect to the local composition. Within the framework of the TIP, a thermodynamic factor depends on the local value but not on the spatial derivatives of the concentration field. The use of this formalism for continuous RIS models deserves discussion. Indeed, a typical RIS profile covers a few tens of nanometers so that the cell size used to define the local driving forces does not exceed a few lattice parameters. Such a mesoscale
481
chemical potential is expected to depend not only on the local value, but also on the spatial derivatives of the concentration field. According to Cahn and Hilliard,60 the free-energy model of a nonuniform system can be written as a volume integral of an energy density made up of a homogeneous term plus interface contributions proportional to the squares of concentration gradients. Thus, all continuous RIS models that are derived from TIP retain only the homogeneous contribution to the energy density and cannot reproduce interface effects and diffuse-interface microstructures. In particular, an equilibrium segregation profile near a surface is predicted to be flat. 1.18.3.2.2 Thermodynamic databases
The thermodynamic factor in eqn [12] is proportional to the second derivative of the Gibbs free energy G of the alloy, with respect to the molar fraction of one of the components. It can be calculated on the basis of thermodynamic data. A database such as CALPHAD61 builds free-energy composition functions of the alloy phases from thermodynamic measurements (specific heats, activities, etc.). When available, the phase diagrams are used to refine and/or to assess the thermodynamic model. Although the CALPHAD free-energy functions are sophisticated functions of temperature and composition, it is interesting to study the simple case of a regular solution model. In the case of a binary alloy A1C BC with a clustering tendency, the Gibbs free energy is equal to G ¼ 2kB Tc Cð1 CÞ þ kB TC lnðCÞ þ kB T ð1 CÞ lnð1 CÞ
½15
where Tc is the critical temperature and C is the alloy composition. The regular solution approximation leads to a concentration-dependent thermodynamic factor equal to Tc ½16 T where concentration C now corresponds to a local concentration of B atoms, which varies in space and time. F ¼ 1 4Cð1 CÞ
1.18.3.3 Experimental Evaluation of the Kinetic Coefficients The L-coefficients characterize the kinetic response of an alloy to a gradient of chemical potential. In practice, what is imposed is a composition gradient.
482
Radiation-Induced Segregation
Chemical potential gradients, and therefore the fluxes, are assumed to be proportional to concentration gradients, (eqn [9]) leading to the generalized Fick’s laws X Dij rCj ½17 Ji ¼ j
A diffusion experiment consists of measurement of some of the terms of the diffusivity matrix Dij . These terms cannot be determined one by one because at least two concentration gradients are involved in a diffusion experiment. Note, that the L-coefficients can be traced back only if the whole diffusivity matrix and the thermodynamic factors are known. Furthermore, most of the diffusion experiments are performed in thermal conditions and do not involve the interstitial diffusion mechanism. In the following section, two examples of thermal diffusion experiments are introduced. Then, a few irradiation diffusion experiments are reviewed. The difficulty of measuring the whole diffusivity matrix is emphasized. 1.18.3.3.1 Interdiffusion experiments
In an interdiffusion experiment, a sample A (mostly composed of A atoms) is welded to a sample B (mostly composed of B atoms) and annealed at a temperature high enough to observe an evolution of the concentration profile. According to eqn [12], the flux of component i in the reference crystal lattice is proportional to its concentration gradient: Ji ¼ Di rCi
½18
where the so-called intrinsic diffusion coefficient Di is a function of the phenomenological coefficients and the thermodynamic factor: ! V LVii Lij F ½19 Di ¼ Ci Cj An interdiffusion experiment consists of measurement of the intrinsic diffusion coefficients as a function of local concentration. The resulting intrinsic diffusion coefficients are observed to be dependent on the local concentration. Within the TIP, while the driving forces are locally defined, the L-coefficients are considered as equilibrium constants. It is not easy to ensure that the experimental procedure satisfies these TIP hypotheses, especially when concentration gradients are large, and the system is far from equilibrium. When measuring diffusion coefficients, one implicitly assumes that a flux can be locally expanded
to first order in chemical potential gradients around an averaged solid solution defined by the local concentration. Starting from atomic jump frequencies and applying a coarse-grained procedure, a local expansion of the flux has been proved to be correct in the particular case of a direct exchange diffusion mechanism.62 An interdiffusion experiment is not sufficient to characterize all the diffusion properties. For example, in a binary alloy with vacancies, in addition to the two intrinsic diffusion coefficients, another diffusion coefficient is necessary to determine the three independent coefficients LAA , LAB , and LBB . 1.18.3.3.2 Anthony’s experiment
Anthony set up a thermal diffusion experiment involving vacancies as a driving force18–22,25,63 in aluminum alloys. The gradient of vacancy concentration was produced by a slow decrease of temperature. At the beginning of the experiment, the ratio between solute flux and vacancy flux is the following: JB LV þ LV ¼ V BB V AB V JV LAA þ LBB þ 2LAB
½20
The volume of the cavity and the amount of solute segregation nearby yield a value for the flux ratio. Note, that secondary fluxes induced by the formation of a segregation profile are neglected in the present analysis. This experiment, combined with an interdiffusion annealing, could be a way to estimate the complete Onsager matrix. Unfortunately, the same experiment does not seem to be feasible in most alloys, especially in steels. In general, vacancies do not form cavities, and solute segregation induced by quenched vacancies is not visible when the vacancy elimination is not concentrated on cavities. 1.18.3.3.3 Diffusion during irradiation
In the 1970s, some diffusion experiments were performed under irradiation.64 The main objective was to enhance diffusion by increasing point defect concentrations and thus facilitate diffusion experiments at lower temperatures. Another motive was to measure diffusion coefficients of the interstitials created by irradiation. In general, the point defects reach steady-state concentrations that can be several orders of magnitude higher than the thermal values. In pure metals, some experiments were reliable enough to provide diffusion coefficient values at temperatures that were not accessible in thermal conditions.64
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The analysis of the same kind of experiments in alloys happened to be very difficult. A few attempts were made in dilute alloys that led to unrealistic values of solute–interstitial binding energies.65 However, a direct simulation of those experiments using an RIS diffusion model could contribute to a better knowledge of the alloy diffusion properties. Another technique is to use irradiation to implant point defects at very low temperatures. A slow annealing of the irradiated samples combined with electrical resistivity recovery measurement highlights several regimes of diffusion; at low temperature, interstitials with low migration energies diffuse alone, while at higher temperatures, vacancies and point defect clusters also diffuse. Temperatures at which a change of slope is observed yield effective migration energies of interstitials, vacancies, and point defect clusters.66 In situ TEM observation of the growth kinetics of interstitial loops in a sample under electron irradiation is another method of determining the effective migration of interstitials.67 1.18.3.3.4 Available diffusion data
Interdiffusion experiments have been performed in austenitic and ferritic steels.34 The determination of the intrinsic diffusion coefficients requires the measurement of the interdiffusion coefficient and of the Kirkendall speed for each composition.68 In general, an interdiffusion experiment provides the Kirkendall speed for one composition only, leading to a pair of intrinsic diffusion coefficients in a binary alloy. Therefore, few values of intrinsic diffusion coefficients have been recorded at high temperatures and on a limited range of the alloy composition. Moreover, experiments such as those by Anthony happened to be feasible in some Al, Cu, and Ag dilute alloys. As a result, a complete characterization of the L-coefficients of a specific concentrated alloy (even limited to the vacancy mechanism) has, to our knowledge, never been achieved. In the case of the interstitial diffusion mechanism, the tracer diffusion measurements under irradiation were not very convincing and did not lead to interstitial diffusion data. The interstitial data, which could be used in RIS models,12 were the effective migration energies deduced from resistivity recovery experiments. 1.18.3.4 Determination of the Fluxes from Atomic Models First-principles methods are now able to provide us with accurate values of jump frequencies in alloys,
483
not only for the vacancy, but also for the interstitial in the split configuration (dumbbell). Therefore, an appropriate solution to estimate the L-coefficients is to start from an atomic jump frequency model for which the parameters are fitted to first-principles calculations. 1.18.3.4.1 Jump frequencies
In the framework of thermally activated rate theory, the exchange frequency between a vacancy V and a neighboring atom A is given by: ! mig DEAV ½21 GAV ¼ nAV exp kB T mig
if the activation energy (or migration barrier) DEAV is significantly greater than thermal fluctuations kB T (a similar expression holds for interstitial jumps). mig DEAV is the increase in the system energy when the A atom goes from its initial site on the crystal lattice to the saddle point between its initial and final positions. One of the key points in the kinetic studies is the description of these jump frequencies and of their dependence on the local atomic configuration, a description that encompasses all the information on the thermodynamic and kinetic properties of the system. 1.18.3.4.1.1
Ab initio calculations
In the last decade, especially since the development of the density functional theory (DFT), first-principle methods have dramatically improved our knowledge of point defect and diffusion properties in metals.69 They provide a reliable way to compute the formation and binding energies of defects, their equilibrium configuration and migration barriers, the influence of the local atomic configuration in alloys, etc. Migration energies are usually computed by the drag method or by the nudged elastic band methods. The DFT studies on self-interstitial properties – for which few experimental data are available – are of particular interest and have recently contributed to the resolution of the debate on self-interstitial migration mechanism in a-iron.70,71 However, the knowledge is still incomplete; calculations of point defect properties in alloys remain scarce (again, especially for self-interstitials), and, in general, very little is known about entropic contributions. Above all, DFT methods are still too time consuming to allow either the ‘on-the-fly’ calculations of the migration barriers, or their prior calculations, and tabulation for all the possible local configurations (whose
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number increases very rapidly with the range of interactions and the number of chemical elements). More approximate methods are still required, based on parameters which can be fitted to experimental data and/or ab initio calculations. 1.18.3.4.1.2 Interatomic potentials
Empirical or semiempirical interatomic potentials, currently developed for molecular dynamics simulations, can be used for the modeling of RIS, but two problems must be overcome: To get a reliable description of an alloy, the interatomic potential should be fitted to the properties that control the flux coupling of point defects and chemical elements. A complete fitting procedure would be very tedious and, to our knowledge, has never been achieved for a given system. The direct calculation of migration barriers with an interatomic potential, even though much simpler than DFT calculations, is still quite time consuming. Full calculations of vacancy migration barriers have indeed been implemented in Monte Carlo simulations,72 using massively parallel calculation methods, but they are still limited to relatively small systems and short times, for example, for the study of diffusion properties rather than microstructure evolution. It is possible to simplify the calculation of the jump frequency, for example, by not doing the full calculation of the attempt frequencies (their impact on the jump frequency must be less critical than that of the migration barriers involved in the exponential term) or by the relaxation of the saddle-point position.73 Malerba et al. have recently proposed another method where the point defect migration barriers of an interatomic potential are exactly computed for a small subset of local configurations, the others being extrapolated using artificial intelligence techniques. This has been successfully used for the diffusion of vacancies in iron–copper alloys.74,75 Such techniques have not yet been used to model RIS phenomena, but this could change in the future.
B atoms located on nth nn sites. Interactions between atoms and point defects can also be used to provide a better description of their formation energies and interactions with solute atoms, and other defects. Various approximations are used to compute the migration barriers: a common one77 is writing the saddle-point energy of the system as the mean energy between the initial energy Ei and final energy Ef , plus a constant contribution Q (which can depend on the jumping atom, A or B). The migration barrier for an A–V exchange is then: mig
DEAV ¼
Ef Ei þ QA 2
½22
where Ef Ei corresponds to the balance of bonds destroyed and created during the exchange. Another solution is to explicitly consider the SP of the jumping atom A with interaction energy eAV the system, when it is at the saddle point: X ðnÞ X ðnÞ mig SP eAi eVj ½23 DEAV ¼ eAV i;n
j ;n
SP eAV
itself can be written as a sum of interactions between A and the neighbors of the saddle point.12,78,79 Both approximations are easily extended to interstitial diffusion mechanisms, and their parameters can be fitted to experimental data and/or ab initio calculations. The first one has the drawback of imposing a linear dependence between the barrier and the difference between the initial and final energies, which is not justified and has been found to be unfulfilled in the very few cases where it has been checked72 (with empirical potentials). The second one should better take into account the effect of the local configuration and, according to the theory of activated processes, does not impose a dependence of the barrier on the final state. However, a model of pair interactions on a rigid lattice does not give a very precise description of the energetic landscape in a metallic solid solution, so, the choice of approximations [22] or [23] may not be crucial. Taking into account many-body interactions (fitted to ab initio calculations, using cluster expansion methods) could improve the description of migration barriers, but would significantly increase the simulation time.80,81
1.18.3.4.1.3 Broken-bond models
Because of these difficulties, simulations of diffusive phase transformation kinetics are commonly based on various broken-bond models, in the framework of rigid lattice approximations.5,76 The total energy of the system is considered to be a sum of constant pair ðnÞ interaction energies, for example, eAB between A and
1.18.3.4.2 Calculation of the phenomenological coefficients
Given an atomic jump frequency model, transitions of the alloy configurations are described by a master equation. With one point defect in the system, a Monte Carlo simulation produces a trajectory of the
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alloy in the configuration space. The L-coefficients can be obtained from a Monte Carlo simulation. Measurements are performed on an equilibrated system using the generalized Einstein formula of Allnatt.82 This numerical approach has proved its efficiency; however, the achievement of a predictive model using this method is limited to short ranges of composition and temperature. Simulations become rapidly unworkable when, for example, binding energies between interstitials and neighboring atoms are significant.9,83 The simulated trajectory can be trapped in some configurations due to the correlation effects of the diffusion mechanism; after a jump, an atom has a finite probability to exchange again with the same point defect and cancel its first jump. The escape probability of the point defect from an atom decreases with the binding energy between the two species. In the limited case of dilute alloys in which a few point defect jump frequencies are involved, it is possible to consider all the vacancy paths and deduce analytical expressions for the L-coefficients. On the other hand, diffusion models of concentrated alloys lead to approximate expressions of the transport coefficients. 1.18.3.4.2.1 Dilute alloys
The point defect jump frequencies to be considered are those that are far from the solute, those leaving the solute, those arriving at a nn site of the solute, and those jumping from nn to nn sites of the solute. Diffusivities are approached by a series in which the successive terms include longer and longer looping paths of the point defect from the solute. Using the pair-association method, the whole series has been obtained for the vacancy mechanism in bodycentered cubic (bcc) and face-centered cubic (fcc) binary alloys with nn interactions (cf. Allnatt and Lidiard6 for a review). However, there is still no accurate model for the effect of a solute atom on the self-diffusion coefficient.84 The L-coefficients as well as the tracer diffusivities depend on three jump frequency ratios in the fcc structure and two ratios in the bcc structure. For the irradiation studies, the same pair-association method has been applied to estimate the L-coefficients of the dumbbell diffusion mechanism in fcc85,86 and bcc83,87 alloys. Note, that the pair-association calculation in bcc alloys87 has recently been improved by using the self-consistent mean-field (SCMF) theory.83 The calculation includes the effect of the binding energy between a dumbbell of solvent atoms and a solute.
485
In the case of the vacancy diffusion mechanism, before the development of first-principles calculation methods, reliable jump frequency ratios could be estimated from a few experimental diffusion coefficients (three for fcc and two for bcc). Data were calculated from the solvent and solute tracer diffusivities, the linear enhancement of self-diffusion with solute concentration, or the electro-migration enhancement factors of tracer atoms in an electric field.88 Some examples of jump frequency ratios fitted to experimental data are displayed in tables.89 Currently, the most widely used approach is using first-principles calculations to calculate not only the vacancy, but also the interstitial jump frequencies. 1.18.3.4.2.2
Concentrated alloys
The first diffusion models devoted to concentrated alloys were based on a very basic description of the diffusion mechanism. The alloy is assimilated into a random lattice gas model where atoms do not interact and where vacancies jump at a frequency that depends only on the species they exchange with (two frequencies in a binary alloy). Using complex arguments, Manning8 could express the correlation factors as a function of the few jump frequencies. The approach was extended to the interstitial diffusion mechanism.90 At the time, no procedure was suggested to calculate these mean jump frequencies from an atomic jump frequency model. Such diffusion models, which consider a limited number of jump frequencies, already make spectacular correlation effects appear possible, such as a percolation limit when the host atoms are immobile.8,91,92 But they do not account for the effect of short-range order on the L-coefficients although one knows that RIS behavior is often explained by means of a competition between binding energies of point defects with atomic species, especially in dilute alloys. Some attempts were made to incorporate shortrange order in a Manning-type formulation of the phenomenological coefficients, but coherency with thermodynamics was not guaranteed.93,94 The current diffusion models, including shortrange order, are based on either the path probability method (PPM)95–97 or the SCMF theory.84,92,98–100 Both mean-field methods start from an atomic diffusion model and a microscopic master equation. While the PPM considers transition variables, which are deduced from a minimization of a pseudo freeenergy functional associated to the kinetic path, the SCMF theory introduces an effective Hamiltonian to represent the nonequilibrium correction to the
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distribution function probability and calculates the effective interactions by imposing a self-consistent constraint on the kinetic equations of the distribution function moments. Diffusion models built from the PPM were developed in bcc solid solution and ordered alloys, using nn pair transition variables, which is equivalent to considering nn effective interactions, and a statistical pair approximation for the equilibrium averages within the SCMF theory.95,96 The SCMF theory extended the approach to fcc alloys8,91,92 and provided a model of the composition effect on solute drag by vacancies.98 The interstitial diffusion mechanism was also tackled.10,83,101 For the first time, an interstitial diffusion model including short-range order was proposed in bcc concentrated alloys.10 It was shown that the usual value of 1 eV for the binding energy of an interstitial with a neighboring solute atom leads to very strong effects on the average interstitial jump frequency and, therefore, on the L-coefficients.
1.18.4 Continuous Models of RIS RIS is a phenomenon that couples the fluxes of defects created by irradiation and the alloy components. In RIS models, point defect diffusion mechanisms alone are considered, although it is accepted that displacement cascades produce mobile point defect clusters, which may contribute to the RIS. In the first section, we present the expressions used to simulate the RIS, the main results, and limits. Some examples of application to real alloys are discussed in the second part. The last section suggests some possible improvements in the RIS models. 1.18.4.1 Diffusion Models for Irradiation: Beyond the TIP RIS models have two main objectives: (1) to describe the reaction of a system submitted to unusual driving forces, such as point defect concentration gradients; and (2) to reproduce the atomic diffusion enhancement induced by an increase of the local point defect concentration. In comparison with the thermal aging situation, gradients of point defect chemical potential are nonnegligible. The L-coefficients are considered as variables that vary with nonequilibrium point defect concentrations. With L-coefficients varying in time, such models do not satisfy the TIP hypothesis. Instead, the authors of the first publications11,30,102 considered new quantities, the so-called
partial diffusion coefficients, as new constants associated with a temperature and a nominal alloy composition. The first authors to express the partial diffusion coefficients in terms of the L-coefficients were Howard and Lidiard103 for the vacancy in dilute alloys, Barbu86 for the interstitial in dilute alloys, and Wolfer11 in concentrated alloys. In the following, we use the formulation of Wolfer because the approximations made to calculate the L-coefficients and driving forces are clearly stated. In a binary alloy (AB), fluxes are separated into two contributions: the first one induced by the point defect concentration gradients, and the second one appearing after the formation of chemical concentration gradients near the point defect sink11: c C þ d c C ÞFrC þ C ðd rC d rC Þ JA ¼ ðdAV V V I A A AV AI AI I c c JV ¼ ðCA dAV þ CB dBV ÞrCV þ CV FðdAV rCA þ dBV rCB Þ c rC þ d c rC Þ ½24 JI ¼ ðCA dAI þ CB dBI ÞrCI CI FðdAI B A BI
with partial diffusion coefficients defined in terms of the L-coefficients and the equilibrium point defect concentration dAV ¼
LV þ LV AA AB CA CV
; dAI ¼
LIAA þ LIAB CA CI
LV LV 1 c dAV ¼ AA AB þ dAV zVA CA CV CB CV F c dAI ¼
LIAA LI 1 AB dAI zIA CA CI CB CI F
½25
where zVA is defined in terms of the local equilibrium vacancy concentration (see eqn [14]). Flux of B is deduced from the flux of A by exchanging the letters A and B. In a multicomponent alloy, equivalent kinetic equations are provided by Perks’s model.104 In this model, point defect concentrations are assumed to be independent of chemical concentrations: in other words, parameters zVA and zVB are set to zero. Most of the RIS models are derived from Perks’ model although they neglect the cross-coefficients.5 Flux of species i is assumed to be independent of the concentration gradients of the other species. In doing so, not only the kinetic couplings, but also some of the thermodynamic couplings are ignored. Indeed, as shown in eqn [9], a chemical potential gradient is a function of all the concentration gradients. An atomic flux results from a balance between the so-called IK effect, atomic fluxes induced by point defect concentration gradients (first term of the RHS of eqn [24]), and the so-called Kirkendall (K) effect
Radiation-Induced Segregation
reacting against the formation of chemical concentration gradients at sinks produced by the IK effect (last term of LHS of eqn [24]). Equation [24] can be used in both dilute and concentrated alloys. Differences between models arise when one evaluates specific partial diffusion coefficients. The first RIS model in dilute fcc alloys, designed by Johnson and Lam,105 introduced an explicit variable for solute–point defect complexes. The same kind of approach has been used by Faulkner et al.,106 although it has been shown to be incorrect in specific cases.86,107 A more rigorous treatment relies on the linear response theory, with a clear correspondence between the atomic jump frequencies and the L-coefficients. The first RIS model derived from a rigorous estimation of fluxes was devoted to fcc dilute alloys,108 and then to bcc dilute alloys.87 In concentrated alloys, due to the greater complexity and the lack of experimental data, further simplifications and more approximate diffusion models are used to simulate RIS. 1.18.4.1.1 Manning approximation
In Section 1.18.3.3, we mentioned the difficulty in measuring the L-coefficients of an alloy, especially those involved in fluxes induced by point defect concentration gradients. Diffusion data are even more difficult to obtain for the interstitials. This is probably why most of the RIS models emphasize the effect of vacancy fluxes, with the interstitial contribution assumed to be neutral. The first model proposed by Marwick30 introduced the main trick of the RIS models by taking out vacancy concentration as a separate factor of the diffusion coefficients. Expressions of the fluxes were obtained using a basic jump frequency model, which is equivalent to neglecting the cross-terms of the Onsager matrix. According to the random lattice gas diffusion model of Manning,8 correlation effects are added as corrections to the basic jump frequencies. The resulting fluxes30 are similar to eqn [24], except c c ; dAI Þ are that the c-partial diffusion coefficients ðdAV now equal to the partial diffusion coefficients ðdAV ; dAI Þ, which is equivalent to neglecting cross L-coefficients and the dependence of equilibrium point defect concentration on alloy composition. However, for the first time, the segregation of a major element, Ni, in concentrated austenitic steel was qualitatively understood in terms of a competition between fast- and slow-diffusing major alloy components. The partial diffusion coefficients associated with the vacancy mechanism are estimated using the
487
relations of Manning, deducing the L-coefficients from the tracer diffusion coefficients: 3 2 C D 1 f0 j j 7 6 P ½26 LVij ¼ Ci Di 4dij þ 5 f0 Ck Dk k
One observes that the Manning8 relations systematically predict positive partial diffusion coefficients: daV ¼ Da =CV
½27
Moreover, the three L-coefficients of a binary alloy are no longer independent. Both constraints are known to be catastrophic in dilute alloys, while they seem to be capable of satisfactorily describing RIS of major alloy components in austenitic steels.30 1.18.4.1.2 Interstitials
Wiedersich et al.102 added to Marwick’s model a contribution of the interstitials. The global interstitial flux is described by eqn [24], while preferential occupation of the dumbbell by a species or two is deduced from the alloy concentration and the effective binding energies. Such a local equilibrium assumption implies very large interstitial jump frequencies compared to atomic jump frequencies. This kind of model yields an analytical description of stationary RIS (see below eqn [28]). An explicit treatment of the interstitial diffusion mechanism was also investigated. From a microscopic description of the jump mechanism, one derives the kinetic equations associated with each dumbbell composition.109–112 Unlike previous models, there is no local equilibrium assumption, but correlation effects are neglected (except in Bocquet90). They could have used the interstitial diffusion model with the correlations of Bocquet.90 However, due to the lack of data for the interstitials, most of the recent concentrated alloy models neglect the interstitial contribution to RIS.104,113,114 1.18.4.1.3 Analytical solutions at steady state
An analytical solution of the coupled equations is obtained within steady-state conditions.102 At the boundary plane, variation of composition is controlled by a unique flux coming from the bulk. A steady-state condition implies that the latter flux is zero, and that, step by step, every flux is zero. In a binary alloy, eqn [24] leads to a relationship between concentration gradients102: CA CB dBI dAI dAV dAI rCV ½28 rCA ¼ ðCB dBI DA þ CA dAI DB Þ dBV dBI
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Radiation-Induced Segregation
where the intrinsic diffusion coefficient is equal to c c CV þ daI CI ÞF. The spatial extent of segreDa ¼ ðdaV gation coincides with the region of nonvanishing defect gradients. Note that, in the original paper,102 the partial and c-partial diffusion coefficients were taken to be equal. Such a simplification may change the amplitude of the RIS predictions. In a multicomponent alloy, not only the amplitude, but also the sign of RIS might be affected by this simplification. In dilute alloys, the whole kinetics can be approached by an analytical equation as long as the Kirkendall fluxes resulting from the formation of RIS are neglected.115
a new continuous model suggested a multifrequency formulation of the concentration-dependent partial diffusion coefficients. Instead of averaging the sums of interaction bonds in the exponential argument, every jump frequency corresponding to a given configuration is considered with a configuration probability weight.17 Predictions of the model are compared with direct RIS Monte Carlo simulations that rely on the same atomic jump frequency models. In the two presented examples, the agreement is satisfactory. However, the thermodynamic factor and correlation coefficients are yet to be defined clearly.17
1.18.4.1.4 Concentration-dependent diffusion coefficients
1.18.4.2
Most of the RIS models assume thermodynamic factors equal to 1, although in the first paper,11 a strong variation was observed between the thermodynamic factor and composition. Similarly, the quantities zVa and zIa of the point defect driving forces [14] are expected to depend on local concentration and stress field.11,116 For example, Wolfer11 demonstrated that RIS could affect the repartition between interstitial and vacancy fluxes and thereby, the swelling phenomena. The bias modification might be due to several factors: a Kirkendall flux induced by the RIS formation, or the dependence of the point defect chemical potentials on local composition, including the elastic and chemical effects. A local-concentration-dependent driving force is due to the local-concentration-dependent atomic jump frequencies. Following this idea, the modified inverse Kirkendall (MIK) model introduces partial diffusion coefficients of the form113 m EAV 0 ½29 dAV ¼ dAV exp kB T
1.18.4.2.1 Dilute alloy models
The migration energy is written as the difference between the saddle-point energy and the equilibrium energy. It depends on local composition through pair interactions calculated from thermodynamic quantities such as cohesive energies, vacancy formation energies, and ordering energies. In fact, the present partial diffusion coefficients correspond to a mesoscopic quantity deduced from a coarse-grained averaging of the microscopic jump frequencies. In principle, not only the effective migration energies, but also the mesoscopic correlation coefficients and thermodynamic factors should depend on local concentrations. Nevertheless, the thermodynamic factors, correlation coefficients, and the zVa factor are assumed to be those of the pure metal A. Recently,
Comparison with Experiment
RIS measurements in dilute alloys are less numerous than in concentrated alloys because the required grain boundary concentrations are usually smaller. However, some of the first RIS observations concerned the segregation of a dilute element, Si, in austenitic steels. In this specific case, observations were easy because the RIS of Si was accompanied by precipitation of Ni3Si. The first mechanism that was proposed to explain the observed solute segregation was the diffusion of solute–point defect complexes towards sinks.105 Since then, more rigorous models that rely on the linear response theory have been established and applied to the RIS description of Mn and P in nickel108 and P in ferritic steels.87,107 Although good precision of the microscopic parameters was still missing, the formulation of the kinetic equations was general enough to be used almost without modification.105 Recent ab initio calculations not only provided accurate atomic jump frequencies of P in Fe,7,70 but they also called into question the jump interstitial diffusion mechanism that had to be considered.7 Indeed, the octahedral and the (110) mixed dumbbell configurations have almost the same stability and migration enthalpies. The resulting effective diffusion energy estimated by the transport model was found to be smaller than the self-interstitial atom migration enthalpy, confirming the classical statement that a solute atom with a negative size effect tends to segregate at the grain boundary. However, as emphasized in Meslin et al.,7 the current interpretation of the interstitial contribution to RIS in terms of size effects is certainly oversimplified. A very large ab initio value of 1.05 eV for the binding energy between a mixed dumbbell and a substitutional P atom may lead to a large activation energy for P interstitials and a drastic reduction of P segregation predictions.7 To consider
Radiation-Induced Segregation
this new blocking configuration with two P atoms, a concentrated alloy diffusion model including shortrange-order effects is required. It is interesting to note that the same solute seems to have a positive coupling with the vacancy also (although the calculation was not as precise as for the interstitials as it was based on an empirical potential).117 In the same way, recent ab initio calculations showed that a Cu solute is also expected to be dragged by vacancy at low temperatures in Fe.118,119 1.18.4.2.2 Austenitic steels
Most of the RIS models for concentrated alloys were devoted to the ternary Fe–Cr–Ni system, which is a model alloy of austenitic steels. The diffusion ratios used in the fitting process are the ones extracted from the tracer diffusion coefficients measured by Rothman et al. (referenced in Perks and Murphy104 and Allen and Was113). In most of the studies, the input parameters are taken from Perks model.104 The more recent MIK model, which was initially based on the Perks model, used the CALPHAD database to fit its concentration-dependent migration energy model.113 A significant improvement in the predictive capabilities of RIS modeling was concluded after a systematic comparison with RIS, observed by Auger spectroscopy in Fe–Cr–Ni as a function of temperature, nominal composition, and irradiation dose.113 However, all the models were proved to be powerful enough to reproduce the correct tendencies of RIS in austenitic steels: a depletion of Cr and an enrichment of Ni near a grain boundary. When the binding energies of point defects with atoms are not so strong and the ratios between the tracer diffusion coefficients of the major elements are large enough (larger than 2–3), a rough estimation of the partial diffusion coefficients from tracer diffusion coefficients seems to be sufficient to reproduce the main tendencies. The interstitial contribution to RIS is usually neglected due to the lack of diffusion data. Stepanov et al.120 observed an electron-irradiated foil at a temperature low enough so that only interstitials contributed to the RIS. Segregation profiles were similar to the typical ones at higher temperature. Parameters of the interstitial diffusion model were estimated in such a way that the experimental RIS was reproduced. The migration energy of interstitials was assumed to be equal to 0.2 eV, which is quite low in comparison with the effective migration energy deduced from recovery resistivity measurements.121 The attempts of the MIK model to reproduce the characteristic ‘W-shaped’ Cr profiles observed at
489
intermediate doses were not conclusive36; transitory profiles disappeared after a dose of 0.001 dpa, while the experimental threshold value was around 1 dpa. A possible explanation may be the approximation used to calculate the chemical driving force. Indeed, a thermodynamic factor equal to 1 pushes the system to flatten the concentration profile in reaction to the formation of the RIS profile. A study based on a lattice rate theory pointed out that an oscillating profile was the signature of a local equilibrium established between the surface plane and the next plane.13 This kind of mean-field model predicts that the local enrichment of Cr at a surface persists at larger irradiation dose (0.1 dpa), though not as large as the experimental value. The role of impurities as point defect traps has been explored since the 1970s.122 In those models, impurities do not contribute to fluxes but to the sink population as immobile sinks with an attachment parameter depending on an impurity–point defect binding energy.123 Other models account for immobile vacancy traps by renormalizing the recombination coefficient with a vacancy–impurity binding energy.120 Whether by vacancy or by interstitial trapping, the result is a recombination enhancement and a decrease of point defect concentrations, leading to a reduction of RIS and swelling. Hackett et al.123 estimated some binding energies between a vacancy and impurities, such as Pt, Ti, Zr, and Hf in fcc Fe, using ab initio calculations. The energy calculations seem to have been performed without relaxing the structure, probably because fcc Fe is not stable at 0 K. Although the absolute values of the binding energies should be used with caution, one can expect the strong difference between the binding energies of a vacancy with Zr (1.08 eV) and Hf (0.71 eV) to persist after a relaxation of atomic positions. In a more rigorous way, the trapping of dumbbells could be modeled using the high migration energy associated with dumbbells bound to an impurity.12 Such a model would allow the impurity to migrate and change the sink density with dose. The same model could explain the recent experimental results observing that, after a few dpa, RIS of the major elements starts again.123 RIS in austenitic steels was observed to be strongly affected by the nature of the grain boundary, that is, by the misorientation angle and the S value.40 Differences between the observed RIS are explained by a so-called grain boundary efficiency, introduced in a modified rate equation model.40,109,114,124 Calculations of vacancy formation energies at different
490
Radiation-Induced Segregation
grain boundaries, for example, in nickel in which atomic interactions are described by an embedded atom method, have been used to model sink strength as a function of misorientation angle. The resulting RIS predictions around several tilt grain boundaries were found to be in good agreement with RIS data.124 Grain boundary displacement and its effect on RIS were considered too. Grain boundary displacement was explained by an atomic rearrangement process due to recombination of excess point defects at the interface. New kinetic equations including an atomic rearrangement process after recombination of point defects at the interface predict asymmetrical concentration profiles, in agreement with experiments.114 1.18.4.3 Models
Challenges of the RIS Continuous
Ab initio calculations have become a very powerful tool for RIS simulations. They have been shown to be able to provide not only the variation of the atomic jump frequencies with local concentration,125 but also new diffusion mechanisms.7 From a precise knowledge of the atomic jump frequencies and the recent development of diffusion models,98 a quantitative description of the flux coupling is expected to be feasible even in concentrated alloys. A unified description of flux coupling in dilute and concentrated alloys would allow the simultaneous prediction of two different mechanisms leading to RIS: solute drag by vacancies, and an IK effect involving the major elements. An RIS segregation profile spreads over nanometers. Cell sizes of RIS continuous models are then too small for the theory of TIP to be valid. A mesoscale approach that includes interface energy between cells, such as the Cahn–Hilliard method, would be more appropriate. A derivation of quantitative phase field equations with fluctuations has recently been published.62 The resulting kinetic equations are dependent on the local concentration and also cell-size dependent. However, the diffusion mechanism involved direct exchanges between atoms. The same method needs to be developed for a system with point defect diffusion mechanisms. Although it has been suggested since the first publications30 that the vacancy flux opposing the set up of RIS could slow down the void swelling, the change of microstructure and its coupling with RIS have almost never been modeled. Only very recently, phase field methods have tackled the kinetics of a concentration field and its interaction with a cavity population (see Chapter 1.15, Phase
Field Methods). The development of a simulation tool able to predict the mutual interaction between the point defect microstructure and the flux coupling is quite a challenge.
1.18.5 Multiscale Modeling: From Atomic Jumps to RIS The knowledge of the phenomenological coefficients L, including their dependence on the chemical composition, allows the prediction of RIS phenomena. Unfortunately, in practice, it is very difficult to get such information from experimental measurements, especially for concentrated and multicomponent alloys, and for the diffusion by interstitials. As we have seen, it is also quite difficult to establish the exact relationship between the phenomenological coefficients and the atomic jump frequencies because of the complicated way in which they depend on the local atomic configurations and because correlation effects are very difficult to be fully taken into account in diffusion theories. An alternative approach to analytical diffusion equations, then, is to integrate point defect jump mechanisms, with a realistic description of the frequencies in the complex energetic landscape of the alloy, in atomistic-scale simulations such as mean-field equations, or Monte Carlo simulations (molecular dynamics methods are much too slow – by several orders of magnitude – for microstructure evolution governed by thermally activated migration of point defects). Atomic-scale methods are appropriate techniques to simulate nanoscale phenomena like RIS. They are all based on an atomic jump frequency model. From this point of view, the difficulties are the same as for the modeling of other diffusive phase transformations (such as precipitation or ordering during thermal aging), complicated by the point defect formation and annihilation mechanisms and by the self-interstitial jump mechanisms, which are usually more complex than the vacancy ones.76 1.18.5.1 Creation and Elimination of Point Defects Because RIS is due to fluxes of excess point defects, modeling must take into account their creation and elimination mechanisms. It must, for example, reproduce the ratio between vacancy and interstitial concentrations that controls the respective weights of annihilation by recombination or elimination at sinks. The situation from this point of view is very different from phase transformations during thermal
Radiation-Induced Segregation
aging, where usually only vacancy diffusion occurs, and simulations can be performed with nonphysical point defect concentrations and a correction of the timescale (see, e.g., Le Bouar and Soisson78 and Soisson and Fu125). During electron or light ion irradiation, defects are homogeneously created in the material, with a frequency directly given by the radiation flux (in dpa s1), a condition that is easily modeled in continuous models, mean-field models,12,14 or Monte Carlo simulations.16,118 The formation of defects in displacement cascades during irradiation by heavy particles can also be introduced in kinetic models, using the point defect distributions computed by molecular dynamics.126 The annihilation mechanisms at sinks such as surfaces or grain boundaries are, for the time being, simulated using very simple approximations (perfects sinks, no formation/annihilation of kinks on dislocations, or steps on surfaces). This should not affect the basic coupling between diffusion fluxes, but the long-term evolution of the sink microstructure – which will eventually have an impact on the chemical distribution – cannot be taken into account. Finally, thermally activated point defect formation mechanisms that operate during thermal aging, are taken into account in some simulations.11–14 Simulations that do not include the thermal production are then valid only at sufficiently low temperatures, when defect concentrations under irradiation are much larger than equilibrium ones. 1.18.5.2
Mean-Field Simulations
The first mean-field lattice rate models included two thermally activated jump frequencies, one for the vacancy and the other for the interstitial. A direct interstitial diffusion mechanism14 and later a dumbbell diffusion mechanism12 have been modeled in detail. The vacancy jump frequency parameters are fitted to available thermodynamic and tracer diffusion data, and the interstitial parameters are fitted to effective migration energies derived from resistivity recovery measurements.121 The resulting localconcentration-dependent jump frequencies describe both the kinetics of thermal alloys toward equilibrium and irradiation-induced surface segregation in concentrated alloys. The surface and its vicinity are modeled by the stacking of N parallel atomic planes perpendicular to the diffusion axis, which is taken to be a [100] direction of an fcc alloy. A mirror boundary condition is used at one end, and a free surface, which can act as both a source and sink for point
491
defects at the other end. Above the surface, a buffer plane almost full of vacancies is added. Fluxes between the buffer plane and vacuum are forbidden. The resulting equilibrium segregation profiles are controlled by the nominal composition, temperature, and two interaction contributions, the first one expressed in terms of the surface tensions and the second in terms of the ordering energies. Note, that the predicted equilibrium vacancy concentration at the surface is much higher than in the bulk. Time dependence of mean occupations in an atomic plane of point defects and atoms results from a balance between averaged incoming and outgoing fluxes. Fluxes are written within a mean-field approximation, decoupling the statistical averages into a product of mean occupations and mean jump frequencies for which the occupation numbers in the exponential argument are replaced by the corresponding mean occupations. The resulting first order differential kinetic equations are integrated using a predictor corrector variable time step algorithm because of the high jump frequency disparities between vacancies and interstitials. It is observed that interstitial contribution to RIS is of the same order as that of the vacancy. The predicted formation of a ‘W-shaped’ profile as a transient state from the preirradiated enrichment to the strong depletion of Cr is shown to be governed by both thermodynamic properties and the relative values of the transport coefficients between Fe, Cr, and Ni (Figure 9). Thermodynamics not only plays a part in the transport coefficients but also arises in the establishment of a local equilibrium between the surface and the adjacent plane, explaining the oscillatory behavior of the Cr profile: an equilibrium tendency toward an enrichment of Cr at the grain boundary plane, which competes with a Cr depletion tendency under irradiation. However, the predicted profile is not as wide as the experimental one. What needs to be improved is the interstitial diffusion model. The lack of experimental and ab initio data leads to approximate interstitial jump frequencies. Coupling between fluxes is described partially as correlation effects are not accounted for. The recent mean-field developments98 should be integrated in this type of simulation. 1.18.5.3
Monte Carlo Simulations
AKMC simulations can be used to follow the atomic configuration of a finite-sized system, starting from a given initial condition, by performing successive
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0.24
0 dpa
0.24
1 dpa
0 dpa
0.01 dpa
0.2 Cr (at.%)
Cr (at.%)
0.2
0.16
13 dpa
0.12
0.16 0.12 0.08 1 dpa
0.04 0.08
–4 10–9
0 100
4 10–9
Distance from grain boundary (m)
−4 10–9
0 100
4 10–9
Distance from grain boundary (m)
Figure 9 Comparison of Cr segregation profiles as a function of irradiation dose in FeNi12Cr19 at T ¼ 635 K. Left cell represents typical experimental results of Busby et al.,36 right cell is mean-field predicted result starting from the experimental profile observed just before irradiation. Reproduced from Nastar, M. Philos. Mag. 2005, 85, 641–647.
point defect jumps.16,17,126,127 Then, the migration barriers are exactly computed (in the framework of the used diffusion model) for each atomic configuration using equations such as [22] or [23], without any mean-field averaging. The jump to be performed can be chosen with a residence time algorithm,128,129 which can also easily integrate creation and annihilation events.16 Correlation effects between successive point defect jumps, as well as thermal fluctuations, are naturally taken into account in AKMC simulations; this provides a good description of diffusion properties and of nucleation events (the latter being especially important for the modeling of RIP). The downside is that they are more time consuming than mean-field models, especially when correlation or trapping effects are significant. These advantages and drawbacks explain why AKMC is especially useful to model the first stages of segregation and precipitation kinetics. AKMC simulations were first used to study RIS and RIP in model systems,16,17 to highlight the control of segregation by point defect migration mechanism, and to test the results of classical diffusion equations. These studies show that it is possible – in favorable situations, where correlation and trapping effects are not too strong – to simulate microstructure evolution with realistic dose rates and point defect concentrations, up to doses of typically 1 dpa. In simple cases, AKMC simulations can validate the predictions of continuous models, on the basis of simple diffusion equations17,16: an example is given in Figure 10 for an ideal solid solution, that is, when RIS can be studied without any clustering or ordering tendency.16 In this simulation, the diffusion of
A atoms by vacancy jumps is more rapid than that of B atoms, and one observes an enrichment of B atoms at the sinks due to the IK effect (A and B atoms diffuse at the same speed by interstitial jumps; those jumps therefore do not contribute to the segregation). One can notice the nonmonotonous shape of the concentration profile, which corresponds to the prediction of Okamato and Rehn29 when the partial diffusion coefficients are dBV =dAV < dBI =dAI . In the more complicated case of a regular solution, Rottler et al.17 have shown that the RIS profiles of AKMC simulations can be reproduced by a continuous model using a self-consistent formulation, which gives the dependence of the partial diffusion coefficients with the local composition. In alloys with a clustering tendency, AKMC simulations have been used to study microstructure evolution when RIS and precipitation interact, either in under- or supersaturated alloys. The evolution of the precipitate distribution can be quite complicated as the kinetics of nucleation, growth, and coarsening depend on both the local solute concentrations and the point defect concentrations (which control solute diffusion), concentrations that evolve abruptly in the vicinity of the sinks.16 A case of homogeneous RIP, due to a mechanism similar to the one proposed by Cauvin and Martin52 (see Section 1.18.2.4), has been simulated with a simplified interstitial diffusion model.130,131 The application of AKMC to real alloys has just been introduced. Copper segregation and precipitation in a-iron has been especially studied, because of its role in the hardening of nuclear reactor pressure vessel steels.118,126,127,132,133 These studies are based on rigid
Radiation-Induced Segregation
1.00
493
(a) 0.002 dpa Point defect sink
CB
0.95 0.90
CB
(b) 0.078 dpa
10–4
0.95
10–5
0.90
10–6
1.00 (c) 0.500 dpa
10–8
Vacancies
10–7 Cd
CB
1.00
0.95
10–9 Interstitials
10–10 10–11
0.90
10–12 –40
–20
(a)
0 d (nm)
20
10–13
40 (b)
–40
–20
0 d (nm)
20
40
Cu concentration (at. fraction)
Figure 10 (a) Evolution of the B concentration profile in an A10B90 ideal solid solution under irradiation at 500 K and 103 dpa s1, when dAV > dBV and dAi ¼ dBi ; (b) Point defect concentration profiles in the steady state. Reproduced from Soisson, F. J. Nucl. Mater. 2006, 349, 235–250.
3.0 ⫻ 10–3 2.0 ⫻ 10–3 1.0 ⫻ 10–3 0.0
–10
–5
0 d (nm)
5
10
Figure 11 Concentration profile and formation of small copper clusters near a grain boundary, in a Fe–0.05%Cu alloy under irradiation at T ¼ 500 K and K0 ¼ 103 dpa s1.
lattice approximations, using parameters fitted to DFT calculations. The point defect formation energies are found to be much smaller in copper-rich coherent clusters than in the iron matrix,79,118 and there is a strong attraction between vacancies and copper atoms in iron, up to the second-nearest neighbor positions.70,125 AKMC simulations show that in dilute Fe–Cu alloys, the LCuV ¼ ½LCuCu þ LCuFe is positive at low temperatures, because of the diffusion of Cu–V pairs. At higher temperatures, Cu–V pairs dissociate and LCuV becomes negative.118,119 The resulting segregation behaviors have been simulated, with homogeneous formation of Frenkel pairs (i.e., conditions
corresponding to electron irradiations). Only vacancy fluxes are found to contribute to RIS; copper concentration increases near the sinks at low temperatures and decreases at high temperatures.118 In highly supersaturated alloys, RIS does not significantly modify the evolution of the precipitate distribution, except for the acceleration proportional to the point defect supersaturation. Figure 11 illustrates a simulation of RIS in a very dilute Fe–Cu alloy, performed with the parameters of Soisson and Fu118,125; the segregation of copper at low temperatures produces the preferential formation of small copper-rich clusters near the sinks, which could correspond to the beginning of a
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heterogeneous precipitation. However, simulations are limited to very short irradiation doses because of the trapping of defects as soon as the first clusters are formed. This makes it difficult to draw conclusions from these studies. Coprecipitation of copper clusters with other solutes (Mn, Ni, and Si) has been modeled by Vincent et al.126,127 and Wirth and Odette133 under irradiation at very high radiation fluxes and with formation of point defects in displacement cascades. AKMC simulations display the formation of vacancy clusters surrounded by copper atoms, which could result both from the Cu–V attraction (a purely thermodynamic factor) and from the dragging of Cu by vacancies (effect of kinetic coupling). The formation of Mn-rich clusters is favored by the positive coupling between fluxes of self-interstitials and Mn (DFT calculations show that the formation of mixed Fe–Mn dumbbells is energetically favored).
1.18.6 Conclusion We started this review with a summary of the experimental activity on RIS. Intensive experimental work has been devoted to austenitic steels and its model fcc alloys (Ni–Si, Ni–Cr, and Ni–Fe–Cr) and, more recently, to ferritic steels. A strong variation of RIS with irradiation flux and dose, temperature, composition, and the grain boundary type was observed. One study tried to take advantage of the sensitivity of RIS to composition to inhibit Cr depletion at grain boundaries. A small addition of large-sized impurities, such as Zr and Hf, was shown to inhibit RIS up to a few dpa in both austenitic and ferritic steels. On the other hand, the Cr depletion at grain boundaries was observed to be delayed when the grain boundary was enriched in Cr before irradiation. A ‘W-shaped’ transitory profile could be maintained until a few dpa before the grain boundary was depleted in Cr. The mechanisms involved in these recent experiments are still not well understood, although RIS model development was closely related to the experimental study. The main RIS mechanisms had been understood even before RIS was observed. From the first models, diffusion enhancement and point defect driving forces were accounted for. The kinetic equations are based on general Fick’s laws. While in dilute alloys one knows how to deduce such equations from atomic jump frequencies, in concentrated alloys more empirical methods are used. In particular, the definition of the partial diffusion coefficients of the Fick’s laws in
terms of the phenomenological L-coefficients of TIP has been lost over the years. This can be explained by the lack of diffusion data and diffusion theory to determine the L-coefficients from atomic jump frequencies. For years, the available diffusion data consisted mainly of tracer diffusion coefficients, and the RIS models employed empirical Manning relations, which approximated partial diffusion coefficients based on tracer diffusion coefficients. However, recent improvements of the mean-field diffusion theories, including short-range order effects for both vacancy and interstitial diffusion mechanism, are such that we can expect the development of more rigorous RIS models for concentrated alloys. It now seems possible to overcome the artificial dichotomy between dilute and concentrated RIS models and develop a unified RIS model with, for example, the prediction of the whole kinetic coupling induced by an impurity addition in a concentrated alloy. Meanwhile, first-principles methods relying on the DFT have improved so fast in the last decades that they are able to provide us with activation energies of both vacancy and interstitial jump frequencies as a function of local environment. Therefore, it now seems easier to calculate the Lcoefficients and their associated partial diffusion coefficients from first-principles calculations rather than estimating them from diffusion experiments. An alternative approach to continuous diffusion equations is the development of atomistic-scale simulations, such as mean-field equations or Monte Carlo simulations, which are quite appropriate to study the nanoscale RIS phenomenon. Although the mean-field approach did not reproduce the whole flux coupling due to the neglect of correlation effects, it predicted the main trends of RIS in austenitic steels, with respect to temperature and composition, and was useful to understand the interplay between thermodynamics and kinetics during the formation of an oscillating transitory profile. Monte Carlo simulations are now able to embrace the full complexity of RIS phenomena, including vacancy and split interstitial diffusion mechanisms, the whole flux coupling, the resulting segregation, and eventual nucleation at grain boundaries. But these simulations become heavy timeconsuming when correlation effects are important.
Acknowledgments Part of this work was performed in the framework of the FP7 Perform and GetMat projects. The authors want to thank M. Vankeerberghen for his useful remarks.
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