Das Sh.P. Statistical Physics of Liquids at Freezing and Beyond
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STATISTICAL PHYSICS OF LIQUIDS AT FREEZING
AND BEYOND
Exploring important theories for understanding freezing and the liquid–glass transition,
this book is useful for graduate students and researchers in soft-condensed-matter physics,
chemical physics, and materials science. It details recent ideas and key developments,
providing an up-to-date view of current understanding.
The standard tools of statistical physics for the dense liquid state are covered. The
freezing transition is described from the classical density-functional approach. Classical
nucleation theory as well as applications of density-functional methods for nucleation of
crystals from the melt are discussed, and compared with results from computer simulation
of simple systems. Discussions of supercooled liquids form a major part of the book. The-
ories of slow dynamics and the dynamical heterogeneities of the glassy state are presented,
as well as nonequilibrium dynamics and thermodynamic phase transitions at deep super-
cooling. Mathematical treatments are given in full detail so that readers can learn the basic
techniques.
S
HANKAR PRASAD DAS is Professor of Physics at Jawaharlal Nehru University, New
Delhi. During the course of his career, he has made significant contributions to the field of
slow dynamics in supercooled liquids and the glass transition.
STATISTICAL PHYSICS OF LIQUIDS
AT FREEZING AND BEYOND
Shankar P. Das
Jawaharlal Nehru University
New Delhi, India
CAMBRIDGE UNIVERSITY PRESS
Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore,
São Paulo, Delhi, Dubai, Tokyo, Mexico City
Cambridge University Press
The Edinburgh Building, Cambridge CB2 8RU, UK
Published in the United States of America by Cambridge University Press, New York
www.cambridge.org
Information on this title: www.cambridge.org/9780521858397
c
S. Das 2011
This publication is in copyright. Subject to statutory exception
and to the provisions of relevant collective licensing agreements,
no reproduction of any part may take place without
the written permission of Cambridge University Press.
First published 2011
Printed in the United Kingdom at the University Press, Cambridge
A catalog record for this publication is available from the British Library
Library of Congress Cataloging-in-Publication Data
Das, Shankar P. (Shankar Prasad), 1959-
Statistical physics of liquids at freezing and beyond / Shankar P. Das.
p. cm.
ISBN 978-0-521-85839-7 (hardback)
1. Liquids–Thermal properties. 2. Crystallization–Mathematical models. 3. Nonequilibrium
thermodynamics. 4. Statistical thermodynamics. I. Title.
QC145.4.T5D37 2011
530.4
24–dc22
2010046159
ISBN 978-0-521-85839-7 Hardback
Cambridge University Press has no responsibility for the persistence or
accuracy of URLs for external or third-party internet websites referred to
in this publication, and does not guarantee that any content on such
websites is, or will remain, accurate or appropriate.
I dedicate this book to my parents,
Sudhir Kumar Das and Gita Rani Das
Contents
Preface page xiii
Acknowledgements xvi
1 Statistical physics of liquids 1
1.1 Basic statistical mechanics 3
1.1.1 Thermodynamic functions 3
1.1.2 The classical N -particle system 6
1.1.3 The BBGKY hierarchy equations 7
1.1.4 The Boltzmann equation 9
1.2 Equilibrium properties 11
1.2.1 The Gibbs H-theorem 12
1.2.2 The equilibrium ensembles 14
1.2.3 The static structure factor 19
1.2.4 Integral equations for g(r) 24
1.3 Time correlation functions 36
1.3.1 The density correlation function 38
1.3.2 The self-correlation function 41
1.3.3 The linear response function 47
1.4 Brownian motion 50
1.4.1 The Langevin equation 50
1.4.2 The Stokes–Einstein relation 53
Appendix to Chapter 1 55
A1.1 The Gibbs inequality 55
A1.2 The force–force correlation 55
A1.3 Brownian motion 56
A1.3.1 The noise correlation 56
A1.3.2 Evaluation of the integrals 57
2 The freezing transition 58
2.1 The density-functional approach 58
2.1.1 A thermodynamic extremum principle 60
vii
viii Contents
2.1.2 An approximate free-energy functional 64
2.1.3 The Ramakrishnan–Yussouff model 68
2.2 Weighted density functionals 72
2.2.1 The modified weighted-density approximation 75
2.2.2 Gaussian density profiles 76
2.2.3 The hard-sphere system 78
2.3 Fundamental measure theory 85
2.3.1 Density-independent weight functions 86
2.3.2 The free-energy functional 88
2.4 Applications to other systems 90
2.4.1 Long-range interaction potentials 91
2.4.2 The solid–liquid interface 99
Appendix to Chapter 2 105
A2.1 Correlation functions for the inhomogeneous solid 105
A2.2 The Ramakrishnan–Yussouff model 106
A2.3 The weighted-density-functional approximation 109
A2.4 The modified weighted-density-functional approximation 113
A2.5 The Gaussian density profiles and phonon model 115
3 Crystal nucleation 117
3.1 Classical nucleation theory 117
3.1.1 The free-energy barrier 118
3.1.2 The nucleation rate 121
3.1.3 Heterogeneous nucleation 129
3.2 A simple nonclassical model 131
3.2.1 The critical nucleus 133
3.2.2 The free-energy barrier 134
3.3 The density-functional approach 137
3.3.1 The square-gradient approximation 137
3.3.2 The critical nucleus 141
3.3.3 The weighted-density-functional approach 145
3.4 Computer-simulation studies 150
3.4.1 Comparisons with CNT predictions 152
3.4.2 The structure of the nucleus 155
Appendix to Chapter 3 160
A3.1 The schematic model for nucleation 160
A3.1.1 Critical nucleus 160
A3.1.2 The free-energy barrier 161
A3.2 The excess free energy in the DFT model 162
4 The supercooled liquid 164
4.1 The liquid–glass transition 164
4.1.1 Characteristic temperatures of the glassy state 165