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The Physics of Superionic Conductors and Electrode Materials NATO Advanced Science Institutes Series A series of edited volumes comprising multifaceted studies of contemporary scientific issues by some of the best scientific minds in the world, assembled in cooperation with NA TO Scientific Affairs Division. This series is published by an international board of publishers in conjunction with NATO Scientific Affairs Division A Life Sciences Plenum Publishing Corporation B Physics New York and London C Mathematical and D. Reidel Publishing Company Physical Sciences Dordrecht, Boston, and London 0 Behavioral and Martinus Nijhoff Publishers Social Sciences The Hague, Boston, and London E Applied Sciences F Computer and Springer Verlag Systems Sciences Heidelberg, Berlin, and New York G Ecological Sciences Recent Volumes in Series B: Physics Volume 87 -Relativistic Effects in Atoms, Molecules, and Solids edited by G. L. Malli Volume 88 -Collective Excitations in Solids edited by Baldassare Di Bartolo Volume 89a-Electrical Breakdown and Discharges in Gases: Fundamental Processes and Breakdown edited by Erich E. Kunhardt and Lawrence H. Luesen Volume 89b-Electrical Breakdown and Discharges in Gases: Macroscopic Processes and Discharges edited by Erich E. Kunhardt and Lawrence H. Luessen Volume 90 -Molecular Ions: Geometric and Electronic Structures edited by Joseph Berkowitz and Karl-Ontjes Groeneveld Volume 91 -Integrated Optics: Physics and Applications edited by S. Martellucci and A. N. Chester Volume 92 -The Physics of Superionic Conductors and Electrode Materials edited by John W. Perram The Physics of Superionic Conductors and Electrode Materials Edited by John W. Perram Odense University Odense, Denmark Plenum Press New York and London Published in cooperation with NATO Scientific Affairs Division Proceedings of a NATO Advanced Study Institute on The Physics of Superionic Conductors and Electrode Materials, held August 4-22, 1980, at Odense University, Odense, Denmark Library of Congress Cataloging in Publication Data NATO Advanced Study Institute on the Physics of Superionic Conductors and Electrode Materials (1980: Odense universiteit) The physics of superionic conductors and electrode materials. (NATO advanced science institutes series. Series B, Physics; v. 92) "Published in cooperation with NATO Scientific Affairs Division." Bibliography: p. Includes index. 1. Superionic .conductors-Congresses. 2. Electrodes-Congresses. I. Perram, John W. II. North Atlantic Treaty Organization. Scientific Affairs Division. III. Title. IV. Series. QC717.N37 1980 537.6/2 83-2332 ISBN-13: 978-1-4684-4492-6 e-ISBN-13: 978-1-4684-4490-2 DOl: 10.1007/978-1-4684-4490-2 © 1983 Plenum Press, New York Softcover reprint of the hardcover 15t edition 1 983 A Division of Plenum Publishing Corporation 233 Spring Street, New York, N.Y. 10013 All rights reserved. No part of this book may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording, or otherwise, without written permission from the Publisher PREFACE The following chapters present most of the lectures delivered at the NATO Advanced Studies Institute on "The Physics of Super ionic Conductors and Electrode Materials", held at Odense Univer sity's Mathematics Department between the 4th and 22nd of August, 1980. The aim of the organizing committee was to present in a rather detailed fashion the most recent advances in the computa tional mathematics and physics of condensed matter physics and to see how these advances could be applied to the study of ionically conducting solids. The first half of the meeting was mainly taken up with lectures. In the second week, working groups on the various aspects were set up, the students joining these groups being helped in the implementation of the lecture material. The leaders of these groups deserve special mention for the tremendous effort they put into this aspect of the meeting, particularly: Dr. Aneesur Rahman (Molecular Dynamics group) Dr. Fred Horne (Ion Transport group) Drs. Nick Quirke and David Adams (Monte Carlo methods) Dr. Heinz Schulz (Diffraction group) Dr. John Harding (Defect Calculations group) The Molecular Dynamics group achieved a certain amount of notoriety within the University by appearing to live in the terminal room. The computing aspect of the meeting would have been much less suc cessful without the assistance of Jan Kennett, of the Odense Data center, who gave a talk to participants on the computer's operating system and was always available to assist with programming and system problems. The Datacenter also assisted the meeting with a generous allocation of computer time. The meeting also owes a debt of gratitude to the University's Rektor, Dr. Aage Trommer, for ensuring that the University's facil ities were available to us, and to the Mayor of Odense, Mr. Werner Dalskov and the City for their hospitality during the meeting. Finally, the Director would like to express his gratitude to the participants, whose appetite for work and tolerance of red cabbage ensured the success of the meeting. v CONTENTS Introduction • • • • • • • • • • • • • • 1 1. Relations Between Crystal Structures and Ionic Conductivity • • • • • • • • • 5 Heinz Schulz 2. Thermodynamic and Transport Properties of Super ionic Conductors and Electrode Materials • 27 J. H. Harding 3. Evaluation and Meaning of Ionic and Dipolar Lattice Sums . • • • • • • • • • • • • 41 E.R. Smith and John W. Perram 4. Evaluation of Lattice Sums in Disordered Ionic Systems. • • • • . • • . • • • • • 57 E.R. Smith and John W. Perram 5. On the Crystalline, Liquid, Glassy, Gaseous, and Superfluid States of Simple Substances 79 R.M.J. Cotterill 6. Molecular Dynamics Studies of Superionic Conductors 93 A. Rahman and P. Vashishta 7. Same Applications of Conditionally Convergent Lattice Sums • • • • • • . . • •• •.•••••••••• 143 E.R. Smith and John W. Perram 8. Boundary Conditions in the Simulation of Ionic Systems. • • • • • • • • • • • • • • 163 S. W. de Leeuw 9. Introduction to Monte Carlo Simulation Techniques. • • • • 177 David Adams vii viii CONTENTS 10. The calculation of Free Energies Using Computer Simulation • • • • • • 197 N. Quirke 11. Monte carlo Study of Solid Electrolytes. • • • • • • • • • 211 Y. Hiwatari and A. Ueda 12. Molecular Dynamics with Constraints. • ••••••• 221 H.J.C. Berendsen and W.F. van Gunsteren 13. Stochastic Dynamics of Polymers •••••••••••••• 241 W.F. van Gunsteren and H.J.C. Berendsen 14. Application of Irreversible Thermodynamics to Mass Transport in Ionic Conductors. • • • • . • • 257 Frederick H. Horne 15. Ion Transport Boundary Conditions. •• • ••.•••• 273 Frederick H. Horne, John H. Leckey, and John W. Perram INDEX • • • • • • • • • • • • . • • • . • • • • • • • • • . • 279 INTRODUCTION John W. Perram Matematisk Institut Odeijse Universitet DK 5230, Odense, Denmark The price of oil and the political instabilities associated with its sources of supply have led technologists to look at other forms of portable energy, the most obvious candidate being electro chemical energy stored in batteries. This has generated, in turn, interest in the scientific investigation of electrolytes other than aqueous ionic solutions as candidates for inclusion in the conduct ing phase of advanced batteries. It has been known for a long time that certain crystals become ionic conductors above a transition temperature. Most of the well known cases, such as silver iodide, the alkaline earth fluorites, and lithium nitride have little potential technical application themselves. However, in studying them, we are seeking to understand how they work, with a view to applying our knowledge of mechanism to devising new materials with technological applications. There are far too many crystals for a project of exhaustive conductivity testing to have much chance of success. It is to the elucidation of mechanism that this workshop and this volume are directed. The chapters can be regarded as a source book in methodology. There are any number of excellent reviews and books on the state of the art in the field of ionic conductors. However, the scientist who wishes to understand how these materials work will find it difficult to get access to the necessary theor etical background. It is thus hoped that the budding theoretician, in this and other areas, will find these lectures useful. Some of the topics require a fair amount of mathematical expertise for their understanding, although they contain enough detail for the reader to be able to follow the argument. For this, the editor makes no apology: the theoretical problems in this area are not easy. 2 INTRODUCTION In this task, no amount of experimental work can tell us any thing in the absence of a proper theoretical framework. However, theories must be grounded in reality, and the early lectures of the meeting were directed towards phenomenology. Of these, the first, by Professor Schulz, gives a clear and elegant elucidation of what can be learnt by diffraction studies, and how the rationale for conducting experiment is to discover mechanism. In Chapter 2, Dr. Harding discusses the conduction mechanism from a quasi-thermodynamic point of view, describing the so-called "hopping model". In later lectures, Dr. Catlow developed these ideas and described the methodology of defect calculations in ionic crystals, discussing in a general way the techniques and assumptions used in developing computer codes of the HADES and PLUTO families. In the next two chapters, Dr. Smith gives a clear, detailed exposition of the calculation of lattice sums in both perfect and defective crystals. These techniques turn out to be crucial for the carrying out and interpretation of simulations of all types. In Chapter 5, Professor Cotterill gives an overview of the application of computer simulation techniques to the study of the transition from ordered solid to disordered fluid. This lecture is included because it sheds light on how solids melt, and whether sub-lattice melting could be responsible for ionic conductivity. In Chapter 6, Drs. Rahman and Vashishta describe in admirable detail how one goes about performing a molecular dynamics simula tion of ionic systems in general and superionic conductors in particular. This article reviews all the main molecular dynamics work and shows how the method may be extended to operate at con stant pressure and how structural phase transitions may be studied. It will undoubtedly become a standard reference work in this area. In Chapter 7, Dr. Smith addresses the difficult problem of including the effects of electronic polarizability in the calcula tions. It should be stressed that all molecular dynamics calcula tions treat the ions as non-polarizable. Here one obtains a closer approximation to reality at the cost of acquiring a truly many-body interaction. In Chapter 8, Dr. de Leeuw describes the role of boundary conditions in ionic simulations. He shows how these act in a very subtle way to determine the calculated conductivity. In so doing, he answers one of the long-standing puzzles as to why a computer simulation of a molten salt gives a D.C. conductivity at all. In Chapter 9, Dr. Adams introduces the second method of com puter simulation, the so-called Monte Carlo method. He discusses how one obtains structural and thermodynamic information by sampling INTRODUCTION 3 the phase space rather than evaluating time averages. Again the exposition is clear and detailed and all significant variants of the method are described. In Chapter 10, Dr. Quirke shows how the Monte Carlo method may be used to calculate free energies, the one quantity inaccessible by the molecular dynamics technique. He describes recent work on the umbrella sampling method and discusses how these methods may be applied to systems of interest here. In Chapter 11, Drs. Hiwatari and Ueda present details of a Monte Carlo study of calcium fluoride and silver iodide. Their work supplements the molecular dynamics studies of the same systems in Chapter 6. In Chapter 12, Drs. Berendsen and van Gunsteren take up a prob lem which has recently attracted a lot of interest in the field, namely the question of polymeric electrolytes. It seems that sub stances such as polyethylene oxide which have a relatively high dielectric constant can act as solvents for small inorganic ions. Such plastic substances have more desirable materials properties than either solids or liquids. The authors describe how the molec ular dynamics method can be applied to such systems, using the method of constraint dynamics pioneered by them. In Chapter 13, the same authors describe the method of stochas tic dynamics, a type of hybrid technique. They show how the forces from the most rapid motions in the system may be treated in a random fashion while focussing on the slower dynamic processes and thus saving a lot of computer time. Complementary to understanding the microscopic dynamics of of conductors is the problem of predicting fluxes and fields in macroscopic samples of them under various conditions. In Chapter 14, Dr. Horne gives a survey of the application of irreversible thermodynamics to mass transport in ionic conductors. He partic ularly stresses the important role of the Onsager coefficients and reviews recent exciting work on the application of finite element numerical techniques to the solution of electrodiffusion equations. A particular topic of discussion in the ion transport group was the question of proper boundary conditions for the migration of ions across phase boundaries. Chapter 15 describes briefly some progress made on this problem during and after the symposium. This work would not be complete without the editor expressing his enormous debt of gratitude to Lisbeth Larsen, who retyped sev eral of the manuscripts.
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