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. Theories of slow dynamics and the dynamical heterogeneities of the glassy state are presented, as well as nonequilibrium dynamics and thermodynamic phase transitions at deep supercooling. Mathematical treatments are given in full detail so that readers can learn the basic techniques. SHANKAR 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), 1959Statistical 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 Acknowledgements 1

page xiii xvi

Statistical physics of liquids 1.1 Basic statistical mechanics 1.1.1 Thermodynamic functions 1.1.2 The classical N -particle system 1.1.3 The BBGKY hierarchy equations 1.1.4 The Boltzmann equation 1.2 Equilibrium properties 1.2.1 The Gibbs H-theorem 1.2.2 The equilibrium ensembles 1.2.3 The static structure factor 1.2.4 Integral equations for g(r) 1.3 Time correlation functions 1.3.1 The density correlation function 1.3.2 The self-correlation function 1.3.3 The linear response function 1.4 Brownian motion 1.4.1 The Langevin equation 1.4.2 The Stokes–Einstein relation

1 3 3 6 7 9 11 12 14 19 24 36 38 41 47 50 50 53

Appendix to Chapter 1 A1.1 The Gibbs inequality A1.2 The force–force correlation A1.3 Brownian motion A1.3.1 The noise correlation A1.3.2 Evaluation of the integrals

55 55 55 56 56 57

2

58 58 60

The freezing transition 2.1 The density-functional approach 2.1.1 A thermodynamic extremum principle

vii

viii

Contents

2.2

2.3

2.4

2.1.2 An approximate free-energy functional 2.1.3 The Ramakrishnan–Yussouff model Weighted density functionals 2.2.1 The modified weighted-density approximation 2.2.2 Gaussian density profiles 2.2.3 The hard-sphere system Fundamental measure theory 2.3.1 Density-independent weight functions 2.3.2 The free-energy functional Applications to other systems 2.4.1 Long-range interaction potentials 2.4.2 The solid–liquid interface

64 68 72 75 76 78 85 86 88 90 91 99

Appendix to Chapter 2 A2.1 Correlation functions for the inhomogeneous solid A2.2 The Ramakrishnan–Yussouff model A2.3 The weighted-density-functional approximation A2.4 The modified weighted-density-functional approximation A2.5 The Gaussian density profiles and phonon model

105 105 106 109 113 115

3

117 117 118 121 129 131 133 134 137 137 141 145 150 152 155

Crystal nucleation 3.1 Classical nucleation theory 3.1.1 The free-energy barrier 3.1.2 The nucleation rate 3.1.3 Heterogeneous nucleation 3.2 A simple nonclassical model 3.2.1 The critical nucleus 3.2.2 The free-energy barrier 3.3 The density-functional approach 3.3.1 The square-gradient approximation 3.3.2 The critical nucleus 3.3.3 The weighted-density-functional approach 3.4 Computer-simulation studies 3.4.1 Comparisons with CNT predictions 3.4.2 The structure of the nucleus

Appendix to Chapter 3 A3.1 The schematic model for nucleation A3.1.1 Critical nucleus A3.1.2 The free-energy barrier A3.2 The excess free energy in the DFT model

160 160 160 161 162

4

164 164 165

The supercooled liquid 4.1 The liquid–glass transition 4.1.1 Characteristic temperatures of the glassy state

Contents

4.2

4.3

4.4

5

4.1.2 The free-volume model 4.1.3 Self-diffusion and the Stokes–Einstein relation Glass formation vs. crystallization 4.2.1 The minimum cooling rate 4.2.2 The kinetic spinodal and the Kauzmann paradox The landscape paradigm 4.3.1 The potential-energy landscape 4.3.2 The free-energy landscape Dynamical heterogeneities 4.4.1 Computer-simulation results 4.4.2 Dynamic length scales

Dynamics of collective modes 5.1 Conservation laws and dissipation 5.1.1 The microscopic balance equations 5.1.2 Euler equations of hydrodynamics 5.1.3 Dissipative equations of hydrodynamics 5.1.4 Tagged-particle dynamics 5.1.5 Two-component systems 5.2 Hydrodynamic correlation functions 5.2.1 Self-diffusion 5.2.2 Transport coefficients 5.3 Linear fluctuating hydrodynamics 5.3.1 The generalized Langevin equation 5.3.2 The liquid-state dynamics 5.4 Hydrodynamics of a solid

ix

170 171 175 176 177 181 182 188 192 193 198 204 205 205 207 209 211 212 215 217 218 225 225 236 246

Appendix to Chapter 5 A5.1 The microscopic-balance equations A5.1.1 The Euler equations A5.1.2 The entropy-production rate A5.2 The second fluctuation–dissipation relation

260 260 263 264 269

6

271 271 271 274 287 287 293 295 300 303 303

Nonlinear fluctuating hydrodynamics 6.1 Nonlinear Langevin equations 6.1.1 Coupling of collective modes 6.1.2 Nonlinear Langevin equations 6.2 The compressible liquid 6.2.1 The one-component fluid 6.2.2 The nonlinear diffusion equation 6.2.3 A two-component fluid 6.2.4 The solid state 6.3 Stochastic balance equations 6.3.1 Smoluchowski dynamics

x

Contents

6.3.2

Fokker–Planck dynamics

308

Appendix to Chapter 6 A6.1 The coarse-grained free energy

310 310

7

318 319 320 327 328 330 334 335 339 343

Renormalization of the dynamics 7.1 The Martin–Siggia–Rose theory 7.1.1 The MSR action functional 7.2 The compressible liquid 7.2.1 MSR theory for a compressible liquid 7.2.2 Correlation and response functions 7.3 Renormalization 7.3.1 Fluctuation–dissipation relations 7.3.2 Nonperturbative results 7.3.3 One-loop renormalization

Appendix to Chapter 7 A7.1 The Jacobian of MSR fields A7.2 The MSR field theory A7.3 Invariance of the MSR action A7.4 The memory-function approach A7.4.1 The projection-operator method A7.4.2 The mode-coupling approximation

348 350 351 355 357 358 361

8

363 363 365 369 376 383 388 391 400 400 406 407 409 411 412 414 417 419 420 426

The ergodic–nonergodic transition 8.1 Mode-coupling theory 8.1.1 The schematic model 8.1.2 Effects of structure on dynamics 8.1.3 Tagged-particle dynamics 8.1.4 Dynamical heterogeneities and MCT 8.1.5 Linking DFT with MCT 8.1.6 Dynamic density-functional theory 8.2 Evidence from experiments 8.2.1 Testing with schematic MCT 8.2.2 Glass transition in colloids 8.2.3 Molecular-dynamics simulations 8.2.4 Discussion 8.3 Ergodicity-restoring mechanisms 8.3.1 Ergodic behavior in the NFH model 8.3.2 The hydrodynamic limit 8.3.3 Numerical solution of NFH equations 8.4 Spin-glass models 8.4.1 The p-spin interaction model 8.4.2 MCT and mean-field theories

Contents

xi

Appendix to Chapter 8 A8.1 Calculation of the spring constant A8.2 Field-theoretic treatment of the DDFT model A8.3 The one-loop result for vˆ vˆ (0, 0)

430 430 432 442

9

443 443 443 444 449 451 451 456 459 460 465 469 471 471 473

The nonequilibrium dynamics 9.1 The nonequilibrium state 9.1.1 A generalized fluctuation–dissipation relation 9.1.2 Computer-simulation studies 9.2 The effective temperature 9.2.1 The phenomenological approach 9.2.2 A simple thermometer 9.3 A mean-field model 9.3.1 The mode-coupling approximation 9.3.2 The FDT regime 9.3.3 The aging regime 9.3.4 Quasi-ergodic behavior 9.4 Glassy aging dynamics 9.4.1 Thermalization 9.4.2 Aging dynamics: experiments

Appendix to Chapter 9 A9.1 The energy of the oscillator A9.2 Evaluation of integrals A9.2.1 Integral IR for the FDT solution A9.2.2 Integrals for the aging solution

479 479 481 481 482

10

486 486 487 488 490 491 494 498 498 501 505 507 509 514 519 522

The thermodynamic transition scenario 10.1 The entropy crisis 10.1.1 The Adam–Gibbs theory 10.1.2 Dynamics near TK 10.2 First-order transitions 10.2.1 Metastable aperiodic structures 10.2.2 Random first-order transition theory 10.3 Self-generated disorder 10.3.1 Effective potential and overlap functions 10.3.2 A model calculation 10.4 Spontaneous breaking of ergodicity 10.4.1 The replica method for self-generated disorder 10.4.2 Free energy of the Replicated liquid 10.4.3 An example: The φ 4 model 10.5 The amorphous solid 10.5.1 The Mézard–Parisi model

xii

Contents

Appendix to Chapter 10 A10.1 Matrix identity A10.2 Matrix identity II A10.3 Computation of the vibrational contribution Iv A10.4 Computation of Tr ln M∗ References Index

531 531 532 533 536 540 558

Preface

This book is aimed at teaching the important concepts of the various theories of statistical physics of dense liquids, freezing, and the liquid–glass transition. Both thermodynamic and time-dependent phenomena relating to transport properties are discussed. The standard tools of statistical physics of the dense liquid state and the associated technicalities needed to learn them are included in the presentation. Details of some of the calculations have been included, whenever needed, in the appendices at the ends of chapters. I hope this will make the book more accessible to beginners in this very active field of research. The book is expected to be useful for graduate students and researchers working in the area of soft-condensed-matter physics, chemical physics, and the material sciences as well as for chemical engineers. We now give a brief description of what is in the book. The first chapter reviews the basics of statistical mechanics necessary for studying the physics of the liquid state. Key concepts of equilibrium and nonequilibrium statistical mechanics are presented. The topics covered here have been chosen keeping in mind the theories and concepts covered in the subsequent chapters of the book. Following this introductory chapter, we focus on the physics of liquids near freezing. In Chapter 2, we demonstrate how the disordered liquid state as well as the crystalline state of matter with long-range order can be understood in a unified manner using thermodynamic extremum principles. Our primary focus in discussing the freezing transition here is the classical density-functional approach using the density as the order parameter. The model is constructed from a basic statistical-mechanical description of the equilibrium liquid close to the freezing point. It predicts the location (in terms of thermodynamic parameters such as temperature and density) of the transformation into the crystalline states coming from the liquid side. This is in contrast to the traditional lattice-instability theories of melting of the solid state. The approach is termed microscopic since it uses the two-body interaction between the particles in the classical Hamiltonian as the starting point. After introducing the model for the broken symmetric state in terms of the inhomogeneous density function, we analyze the process of transformation from one phase to another. As the disordered liquid is quenched to a lower temperature at which the stable equilibrium state would be a crystal, the metastable liquid transforms into the state with long-range order through a nucleation process. A model for this transformation using a xiii

xiv

Preface

purely thermodynamic approach is given in terms of the classical nucleation theory. The latter is an important idea that is also applied in a somewhat different context in understanding the deeply supercooled state. Next, extensions of the density-functional theories introduced in the previous chapter are applied to identify the critical nucleus formation in the melt. Comparisons of the theories of freezing and nucleation with corresponding computer-simulation results are discussed at every step. Simulating nucleation of the crystalline phase from the melt has always been difficult. In recent years special techniques have been developed to study the formation and structure of the critical nucleus. We discuss these developments in Chapter 3. In Chapter 4, we begin considering the supercooled state, i.e., liquid kept at metastable equilibrium at temperatures beyond the freezing point. Competition between supercooling into a metastable state and the process of crystallization as well as the standard phenomenology of the glassy state are discussed here. The computer models of the liquid developed to study various aspects of the supercooled state are also described in this chapter. Concepts of dynamical heterogeneities and growing length scales associated with the supercooled liquid are introduced at this point. Next we turn to the discussion of the microscopic theories of the slow dynamics which develops in the supercooled liquid as it is increasingly supercooled. We present the theoretical developments in this field starting from the basics. Hence three chapters have been devoted to introducing the reader to the rather involved formalism necessary for treating this topics. The first, Chapter 5, presents the formulation of the basic equations of hydrodynamics for a set of slow modes in the many-body system. The dissipative equations are constructed using phenomenological arguments, and linear transport coefficients are defined. We introduce here the idea of generalizing hydrodynamic equations to short length scales in the dense liquid. In Chapter 6, we discuss the formulation of the nonlinear fluctuating hydrodynamics for several model systems. These equations control the dynamics of a set of slow modes in a manner that includes collective effects from semi-microscopic length scales. In Chapter 7, we present methods for formulating a renormalized theory of the dynamics. The effects of the nonlinear coupling of the slow modes are obtained in a systematic manner by using diagrammatic methods of quantum field theory. This Martin–Siggia–Rose (MSR) field-theoretic model obtains the dynamic density correlation function in terms of renormalized transport coefficients, which are themselves expressed self-consistently in terms of dynamic correlation functions. A new approach to studying the complex behavior of the supercooled liquid started with the idea of a nonlinear feedback mechanism on its transport properties from the coupling of slowly decaying correlation functions. That the renormalized dynamics for a compressible liquid obtains this model in a natural way is demonstrated in Chapter 7. The formalism also facilitates the analysis of the full implications of the nonlinearities in the equations of motion on the asymptotic dynamics. Chapters 5, 6, and 7 are technical and may be skipped by those not interested in understanding the construction of the mathematical models for the dynamics in full rigor. Some simpler deductions of the basic mode-coupling model are also presented in an appendix.

Preface

xv

The above renormalized model gives rise to the idea of an ergodic–nonergodic (ENE) transition in which the long-time limit of the density correlation function freezes. This ENE transition, its implications for the dynamics, and supporting evidence from experiments and simulation are discussed in full detail in Chapter 8. The possible role of ergodicity-restoring mechanisms and removal of the sharp ENE transitions follows thereafter. Finally, we make a critical evaluation of the mode-coupling theory for structural liquids and its link with models of dynamics for mean-field systems. Up to this point the discussion of the dynamics is only in terms of equilibrium correlations. In Chapter 9 we deal with the nonequilibrium aspects of the glassy state. We present the modification of the standard results of equilibrium statistical mechanics, such as the fluctuation–dissipation theorem and the concept of effective temperatures for the glassy state. Related computer-simulation studies are presented. Theoretical models of nonequilibrium dynamics in terms of mean-field spin models are also worked out. The relaxation time for the supercooled liquid increases drastically as it is supercooled and eventually vitrification occurs when the liquid behaves like a frozen solid. Apart from having a characteristic large viscosity, the supercooled liquid shows a discontinuity in specific heat due to freezing of the translational degrees of freedom in the liquid. The difference of the entropy of the supercooled liquid from that of the solid having only vibrational motion around a frozen structure represents the entropy due to large-scale motion and is called the configurational entropy Sc of the supercooled liquid. Theoretical analysis of the rapid disappearance of Sc with supercooling (the so-called “entropy crisis”) is essential for our understanding of the physics of the glass transition. The connections between structure and dynamics and the possibilities of an underlying thermodynamic phase transition in the deeply supercooled liquid are discussed in the last chapter of the book.

Acknowledgements

I gratefully acknowledge Kyozi Kawasaki for many helpful discussions, support and exchange of views on important concepts. His work and ideas appear in several chapters of this book and have been an inspiration for the author. Discussions and associations with various scientists at different stages of my career have been useful experience and helped me in writing this book. In particular I would like to mention A. Angell, B. Bagchi, J.L. Barrat, J.K. Bhattacharjee, C. Dasgupta, D. Dhar, J.W. Dufty, J.P. Dyre, M.H. Ernst, T.R. Kirkpatrick, R. Kühn, H. Löwen, F. Mezei, U. Mohanty, S.R. Nagel, T. Odagaki, T.V. Ramakrishnan, S. Ramaswamy, S. Sarkar, R. Schilling, Y. Singh, A. Solokov, A.K. Sood, J. Yeo, and S. Yip. During the course of writing this book I have benefited from discussions on various related topics with G. Biroli, C. Chakravarty, T. Engami, S. Franz, H. Hayakawa, B. Kim, V. Kumaran, A. Loidl, P. Lunkenheimer, K. Miyazaki, D. Reichman, S. Sastry, F. Sciortino, H. Tanaka, G. Tarajus, R. Yamamoto, and A. Yoshimori. I acknowledge my colleagues R. Ramaswamy, A.K. Rastogi, and S. Sarakar for their support. I acknowledge all my students for their research which has been very much needed in writing this book. In particular I acknowledge help from my students L. Premkumar, M. Priya, B. Sen Gupta, and S. P. Singh in preparing some of the figures and the manuscript. I thank Mr. R. N. Saini for his help with the reproduction of the figures. Big thanks are due to little Bodhisattva Das for helping me to prepare the figure presented on the front cover of the book. I am grateful to the authors and publishers who allowed figures from their publications to be used in the book. I am very thankful to Dr. Simon Capelin and Dr. Graham Hart from Cambridge University Press for allowing me time well beyond the initially agreed limit to finish writing this book. Lindsay Barnes, Laura Clark, S. Holt, M. Waddington, and Emma Walker of Cambridge University Press are acknowledged for their constant help, reminders, and patience – for keeping the project alive even though I kept missing deadlines repeatedly. Sehar Tahir is acknowledged for providing very useful help with LATEX and CUP style files. The UGC Capacity-Build-up grant of JNU, the PURSE grant from DST, and the CSIR research grant 03(1036)/05/EMRII are acknowledged for financial support. I acknowledge the Institute for Molecular Sciences (IMS) in Okazaki, Japan for support during my visit xvi

Acknowledgements

xvii

there in 2009–10. I am deeply grateful to Professor Hirata and his group there, in particular Dr. Yoshida, and others for their kind hospitality and help. The Physics Department of the University of Florida, Gainesville is acknowledged for hospitality and support during my visit there in summer 2009. The Joint Theory Institute of the University of Chicago is acknowledged for hospitality and support during my visits there in the summers of 2008 and 2010. Finishing this long project would not have been possible without the constant support and help of my wife Soma and my son Rhivu. It gave me the strength to carry on doing this work in the midst of the many other responsibilities of daily life. Finally, the acknowledgment which precedes all others is that to Professor Gene Mazenko, who introduced me to the statistical physics of liquids. I have greatly benefited from my association with him, and his teaching is reflected in the pages of this book. I gratefully acknowledge him for his kindness and support over all these years. Shankar Prasad Das, School of Physical Sciences, Jawaharlal Nehru University, New Delhi 110067.

1 Statistical physics of liquids

A liquid in thermodynamic equilibrium is described at the macroscopic level in terms of a few characteristic variables such as temperature T , pressure P, volume V , the (average) number of particles N , mass density ρ0 , etc. (Fermi, 1956). The thermodynamic properties listed above are not all independent and are related through a relation termed the equation of state. The equilibrium liquid transforms into a crystalline solid when its temperature T falls below a characteristic value Tm . The latter is termed the freezing point of the liquid at the corresponding pressure. The liquid state involves random motion of the constituent particles, each of which represents a basic microscopic unit for the system. The normal liquid is isotropic in a time-averaged sense. At freezing this isotropic symmetry of the normal liquid state is spontaneously broken. The crystalline state has a characteristic long-range order of the individual microscopic units. The transformation of the liquid into a crystalline solid involves the absorption of latent heat. This transformation is somewhat distinct from the condensation of the gaseous state into the liquid state. In the later case both the states are disordered. In Fig. 1.1 we display schematically a phase diagram showing the different equilibrium states for the corresponding values of the thermodynamic properties T and P. For example, the horizontal arrow in Fig. 1.1 indicates that as we cross the line separating the phases a transition from the liquid to the crystalline state occurs. This is accompanied by a discontinuous change of the thermodynamic properties (such as the volume V ) and is called a first-order transition. The coexistence line of the liquid and gaseous states terminates at the critical point. Beyond this point the transformation from one state to another occurs through a continuous, or second-order, phase transition. For the solid and the liquid states no such termination of the phase line occurs. The corresponding diagram showing the existence of the phases in the P–T plane is presented in Fig. 1.2. The part of the P–T plane which is not physically accessible to a single phase is marked as the coexistence region. Transformation from one state to another occurs here. The phase diagrams describe the equilibrium states as a function of the thermodynamic parameters. However, almost all liquids can, with varying degrees of ease, be supercooled below Tm without developing long-range order of the crystalline state. The supercooled liquid develops very slow dynamics. This slowing down is far more drastic than the decrease in the average speed of the liquid particles for the cooling to lower temperatures. The time taken for a characteristic fluctuation to relax to equilibrium increases by orders of magnitude as 1

2

Statistical physics of liquids

Fig. 1.1 The equilibrium phase diagram (schematic) of a simple fluid in the pressure (P) and temperature (T ) plane.

Fig. 1.2 The equilibrium phase diagram (schematic) of a simple fluid in the density n (equivalent to n 0 in the text) and temperature (T ) plane. The regions marked for coexistence are unavailable for a single phase and represent transformation.

the liquid is increasingly supercooled. The viscosity of the liquid increases below Tm until a situation in which the liquid stops flowing and transforms into an amorphous solid is reached. We will study in this book models for understanding the behavior of the liquid at the freezing point Tm and beyond. Our treatment involves primarily a coarse-grained picture of the microscopic-level description of the liquid.

1.1 Basic statistical mechanics

3

1.1 Basic statistical mechanics The microscopic description of the liquid is formulated here by treating the individual microscopic units in terms of the laws of classical physics. The justification of the classical approximation depends on the value of the thermal de Broglie wavelength 0 = √ h/ 2π mkB T corresponding to the average momentum of the liquid particles (of mass m) at a temperature T . kB = 1.38 × 10−16 erg/K is the Boltzmann constant. If the mean −1/3 is such that the ratio ξTr = nearest-neighbor separation of the liquid particles l0  ρ0 0 /l0 1 then the classical description is justified. For most liquids (Hansen and McDonald, 1986; Barrat and Hansen, 2003) barring hydrogen and neon, this ratio is small near the triple point. For molecular liquids another characteristic quantity, namely the ratio ξR = Rot /T between the characteristic rotational temperature Rot = 2 /(2I kB ) (I is the molecular moment of inertia of the molecule) and temperature T should also be small. However, as the temperature T falls and density ρ0 increases, ξTr increases. In the classical approximation the contributions to the thermodynamic properties from the kinetic and interaction parts of the Hamiltonian can be obtained separately. If Re represents the absolute value of the ratio of the kinetic and potential parts of the total energy of the system of particles then Re ∼ 1 corresponds to the liquid state. On the other hand, for the gaseous and solid states we have Re  1 and Re 1, respectively. The liquid state thus represents an intermediate between two extremes. Alternatively, this characterization of the liquid state can be done in terms of the interaction potential. Let a pair of particles in the system interact via a potential of depth ∼ and range ∼σ . The liquid state will correspond to the particles being cohesive, which requires that the total volume V ∼ N σ 3 and the average kinetic energy kB T ∼ . Indeed, the intermediate nature of the liquid state makes it particularly difficult to develop a quantitatively accurate model for its thermodynamic or dynamic properties at high densities. Both for dilute gases and for low-density solids the corresponding idealized model, namely the perfect gas and the harmonic solid, can be treated exactly. Such a reference state is lacking for a high-density liquid. In this book we will describe how the methods of classical statistical mechanics have been used to develop models for strongly interacting liquids and further extended to gain an understanding of the properties of deeply supercooled states.

1.1.1 Thermodynamic functions The first law of thermodynamics provides us with a definition of the internal energy U as a property of the equilibrium thermodynamic state. The second law of thermodynamics defines another state function S representing the thermodynamic entropy of the system. The first two laws are expressed in terms of the basic equation treating the variations of U and S as exact differentials, T d S = dU + P d V − μ d N¯ ,

(1.1.1)

where V is the volume and N¯ is the number of particles N in the system. Note that the number of particles N¯ is used here as a macroscopic (extensive) thermodynamic property

4

Statistical physics of liquids

of the system (similar to the volume V ). In order to distinguish it from the number of particles N in the system in a microscopic picture (to be used later) we denote it with a bar. μ is the chemical potential. The internal energy U can be treated as a function of T and V , giving the differential,   ∂U d V, (1.1.2) dU = cv dT + ∂V T where cv = (∂U /∂ T )V is the specific heat at constant volume. Using eqn. (1.1.2) in eqn. (1.1.1) and the fact that d S is an exact differential, we obtain the thermodynamic relation 

∂U ∂V



 T

∂P =T ∂T

 − P.

(1.1.3)

V

From the second law of thermodynamics follows the entropy function S of the thermodynamic state. As a direct consequence of the second law it follows that for an isolated system the entropy S never decreases. Equivalently, for a change from a state A to another state B (Fermi, 1936)  B dQ S(B) − S(A) ≤ , (1.1.4) A T where the equality holds for a reversible change. Thus for an isolated system going through an infinitesimal change the heat supplied Q ≤ T S. Using the first law, this inequality leads to U + P V − T S ≤ 0.

(1.1.5)

For a constant-temperature and constant-volume process therefore we have (U −T S) ≤ 0. Hence for an isolated system the function F = U −T S always decreases and equilibrium is reached when the function F reaches a minimum. On the other hand, if the temperature T and pressure P remain constant, the inequality (1.1.5) tells us that (U − T S + P V ) ≤ 0. Therefore in constant-temperature, constant-pressure processes the thermodynamic function G = U − T S + P V ≤ 0 is minimized. F and G are called the Helmholtz and Gibbs free energies, respectively. Using the result (1.1.1), the differentials of F and G are obtained as d F = dU − T d S − S dT = −P d V + μ d N¯ − S dT

(1.1.6)

dG = dU − T d S − S dT + P d V + V d P = μ d N¯ + V d P − S dT.

(1.1.7)

From the above equations it follows that the thermodynamic potentials F and G can be treated as functions of the corresponding sets of thermodynamic variables, F ≡ F(T, V, N¯ )

1.1 Basic statistical mechanics

5

and G ≡ G(T, P, N¯ ). For the Gibbs free energy the situation is somewhat special since, apart from N¯ , the other two thermodynamic variables on which it depends are both intensive properties, i.e., independent of the size of the system. We can write the Gibbs free energy per particle as G(T, P, N¯ ) = N¯ g(T, P). We obtain, from eqn. (1.1.7), the chemical potential μ as   G ∂G (1.1.8) = g(T, P) = , μ= ¯ N¯ ∂ N P,T which is the Gibbs free energy per particle. Using the relation G = μ N¯ in eqn. (1.1.7), we obtain the very useful Gibbs–Duhem relation, N¯ dμ − V d P + S dT = 0.

(1.1.9)

The above relation implies that, at the transition for the coexisting phases at equal temperature and pressure (d P = dT = 0 ), the chemical potentials of the two phases are also equal, μ1 = μ2 .

(1.1.10)

The Gibbs free energy per particle is continuous through the phase transition. However, from eqn. (1.1.9) it follows that   ∂μ V − = = v, (1.1.11) ∂P T N¯   S ∂μ = s. (1.1.12) = ∂T P N¯ Thus the first derivative of the Gibbs free energy is discontinuous at the transition (hence the name first-order transition). The respective amounts of discontinuity are equal to the changes in specific volume and entropy for the two phases across the transition. We therefore obtain     ∂μ s ∂μ − = . (1.1.13) ∂T P ∂P T v Since the change in Gibbs free energy (per particle) μ is a function of the intensive variables P and T , we can use the chain rule to write the LHS of eqn. (1.1.13) as (∂ P/∂ T )μ . At the transition μ is zero and hence the corresponding derivative is obtained as   d P(T ) s ∂P = = . (1.1.14) dT ∂ T μ=0 v Using the fact that at the transition the temperature T remains constant, the difference between the specific entropies is s = L/T , where L is the latent heat of the transition. Hence, denoting the change of specific volume between the two phases as v = v2 − v1 ,

6

Statistical physics of liquids

the above equation for the rate of change of pressure P at coexistence with temperature T is obtained as d P(T ) L = . dT T (v2 − v1 )

(1.1.15)

The above equation is termed the Clapeyron equation.

1.1.2 The classical N-particle system We begin with a review of a few basic results concerning a fluid as a classical statistical mechanical system. We consider a fluid as a system of volume V containing N particles of mass m and following Newton’s laws of motion. The Hamiltonian for the system is given by the sum of a kinetic part K (p1 , . . . , p N ) and a potential part U (r1 , . . . , r N ), H= =

N  pα2 + U (r1 , . . . , r N ) 2m

α=1 N  α=1



pα2 1 u(rαβ ), + 2m 2

(1.1.16)

α,β

where rα and pα , respectively, denote the position and momentum of the αth particle and rαβ = |rα − rβ |. We assume that the particles of the fluid interact through a two-body potential u(rαβ ) dependent on the vector rαβ connecting the two interacting particles. In an isotropic system u is a function of only rαβ . We treat here the simplified case in which there is no external potential. The prime in the summation in the second term on the RHS of (1.1.16) indicates that the α = β term is excluded. The equations of motion of the particles are given by r˙ α ≡

drα ∂H , = dt ∂pα

p˙ α ≡

dpα ∂H , =− dt ∂rα

(1.1.17)

for α = 1, . . . , N . These equations with appropriate initial conditions determine the classical dynamics of the N -particle system. In the phase-space description the system is described in terms of a single phase point in the 6N -dimensional phase space involving the variables {rα , pα } for the N particles. The dynamics of the system is then represented by the trajectory of a single phase point in the phase space. The physically measured value of a dynamic variable is then given by an average on the phase-space trajectory. In equilibrium such an average is replaced, using the ergodic hypothesis (Dorfman, 1999), by an average over an appropriate ensemble representative of the system. Each member of the ensemble is an identical copy of the system evolving independently under the above equations of motion. The “fluid” constituted by these points in the phase space has a characteristic density termed the phase-space density f (N ) (r N , p N , t). This is the density of representative points corresponding to the members of the ensemble in any particular part of the phase space. Thus f (N ) dr1 dp1 · · · dr N dp N is the number of phase points in an

1.1 Basic statistical mechanics

7

elementary volume dr N dp N at time t. It also represents the probability that a particular member of the ensemble will be in a state in which the position and momentum coordinates of the ith particle (for i = 1, . . . , N ) lie in the ranges ri to ri + dri and pi to pi + dpi , respectively. Since the total number of ensemble members is fixed, we can maintain the normalization  dr N dp N f (N ) (r N , p N , t) = 1,

(1.1.18)

where we have used the notation dr1 dp1 · · · dr N dp N ≡ dr N dp N . 1.1.3 The BBGKY hierarchy equations f (N )

The time evolution of is given by the Liouville equation, which is obtained by applying the continuity equation for the “incompressible fluid” formed by the phase points of the ensemble. Therefore   N ∂ f (N ) ∂ f (N )  ∂ f (N ) · r˙ α + · p˙ α = 0. + (1.1.19) ∂t ∂rα ∂pα α=1

Using the equation of motion eqn. (1.1.17), we obtain from (1.1.19) ∂ f (N ) + {H, f (N ) } = 0, ∂t with the Poisson bracket between variables A and B being defined as

N  ∂A ∂B ∂A ∂B { A, B} = · − · . ∂rα ∂pα ∂pα ∂rα

(1.1.20)

(1.1.21)

α=1

The above equation for the time evolution of f (n) is often written as

∂ + iL f (N ) = 0 ∂t

(1.1.22)

in terms of the Liouville operator L: L ≡ i{H, }.

(1.1.23)

The formal solution of the above equation for the N -particle distribution function is obtained as f (N ) (t) = exp[−iLt] f (N ) (0).

(1.1.24)

It is useful to compare the above result for the time evolution of f (N ) (t) with the time evolution of an operator A(r N , p N ). Since A is not explicitly dependent on t its change with time is controlled by that of {r N , p N }. Therefore the equation of motion for A is obtained as

N ∂A  ∂A ∂A · r˙ α + · p˙ α ≡ iLA. = (1.1.25) ∂t ∂rα ∂pα α=1

8

Statistical physics of liquids

The solution of the above equation gives for the time dependence of A(t) the result A(t) = exp[+iLt] A(0).

(1.1.26)

The time dependence of f (N ) is therefore different from that of a dynamic variable. It represents the motion of the phase points in the phase space. The Liouville equation obtained for eqns. (1.1.17) makes this dynamics similar to that of an incompressible fluid. Let us consider the number of representative points present in a small elementary volume at t = 0. With the progress of time this elementary volume stretches out, changing its shape, but its total volume remains unchanged. In mathematical terms, in the Hamiltonian dynamics, the Jacobian of the corresponding transformation of the phase-space variables is unity, J [r N (0), p N (0) → r N (t), p N (t)] = 1.

(1.1.27)

This is termed the Liouville theorem. The phase-space trajectories of the different points never cross each other since the dynamics at this level is completely deterministic. Starting from an initial configuration the evolution is always unique. The Liouville equation describes the dynamics at the level of N particles. Using the definition (1.1.16) of the Hamiltonian for the N -particle system interacting through the two-body potential, the operator L in eqn. (1.1.23) is further simplified. The derivative of H with respect to the particle coordinates is given by 

 ∂u(rαβ ) ∂H = ≡ −Fαβ , ∂rα ∂rα

(1.1.28)

β

Fαβ being the force on the αth particle due to interaction with the βth particle. The prime in the sum indicates that the α = β term is absent. Note that in the definition (1.1.16) for H we have dropped for simplicity the presence of any possible external potential and hence there is no external force on the particles here. This operator L is expressed as a sum of one-body and two-body terms: L=

N  α=1

N  1  θα,β , 2

(1.1.29)

α,β=1

pα · ∇rα , m = Fαβ · (∇pα − ∇pβ ),

Sα ≡ θα,β

Sα +

(1.1.30) (1.1.31)

where the prime in the sum indicates that the terms for α = β are omitted. The above form of the Liouville operator is conveniently used to obtain a description of the dynamics at a reduced level. Starting from the level of N particles, the reduced distribution function giving the conditional probability of finding a reduced number of particles is defined. Thus the s-particle (s < N ) reduced distribution function f (s) is defined as  N! f (N ) (x N , t)dxs+1 . . . dx N , (1.1.32) f (s) (xs , t) = (N − s)!

1.1 Basic statistical mechanics

9

where the position rα and momentum pα for the αth particle are jointly denoted by the single variable xα . The notation xn ≡ {x1 , . . . , xs } is used as the argument in eqn. (1.1.32). Starting from the Liouville equation (1.1.22) and integrating out the positions and momenta for the (N − s) particles, the dynamical equation for the reduced function f (s) is obtained as ⎫ ⎧ s s ⎬ ⎨∂  pi 1  θi, j f (s) (xs , t) + · ∇ri + ⎭ ⎩ ∂t m 2 i=1

=−

s  

i, j=1

dxs+1 θi,s+1 f (s+1) (x(s+1) , t),

(1.1.33)

i=1

where the prime in the double sum indicates the absence of the i = j term. The distribution function f (N ) vanishes when the coordinate xα is at the limits of the integration, i.e., when rα is outside the volume of the system or pα → ∞. Hence the terms in which the integrand is a total derivative with respect to the integrated variable of position and momentum have been set equal to zero in deriving eqn. (1.1.33). With some further simplifications eqn. (1.1.33) for f (s) also reduces to the following form in terms of the force Fi j between the ith and jth particles: ⎧ ⎛ ⎡ ⎞ ⎤⎫ s s ⎬ ⎨∂   ⎣ pi · ∇ri + ⎝ Fi j ⎠ · ∇pi ⎦ f (s) (xs , t) + ⎭ ⎩ ∂t m i=1

=−

s  

j=1

dxs+1 Fi,s+1 · ∇pi f (s+1) (x(s+1) , t).

(1.1.34)

i=1

Equations (1.1.33) and (1.1.34) relate the distribution function f (s) to its counterpart at the next level, i.e., f (s+1) , and thus for s = 1, . . . , N eqn. (1.1.33) gives rise to a hierarchy of N equations for the reduced distribution functions. This is known as the BBGKY hierarchy of equations for classical dynamics. At the simplest level, an approximate solution for the one-particle distribution function f (1) (r, p, t) is obtained from the first equation of the BBGKY hierarchy:     p1 ∂ (1) (1.1.35) + · ∇r1 f (x1 , t) = − dr2 dp2 F12 · ∇p1 f (2) (x1 , x2 , t). ∂t m 1.1.4 The Boltzmann equation In order to reach a closed equation for the one-particle distribution function f (1) (x1 , t) from the above equation a relation linking the two-point function f (2) (x1 , x2 , t) to f (1) is needed. The simplest approximation in this regard is to ignore all correlation between the particles and use, in the mean-field approximation, f (2) (x1 , x2 , t) ≈ f (1) (x1 , t) f (1) (x2 , t).

(1.1.36)

10

Statistical physics of liquids

Using this closure relation in eqn. (1.1.35), we obtain what is termed the Vlasov equation,   p1 ∂ ¯ (1.1.37) + · ∇r1 + F1 (r1 , p1 , t) · ∇p1 f (1) (r1 , p1 , t) = 0, ∂t m where we have defined the average force exerted on particle 1 by other particles as   (1.1.38) F¯ 1 (r1 , p1 , t) = dr2 dp2 F¯ 12 f (1) (r1 , r2 , p1 , p2 , t). The term involving F12 in eqn. (1.1.38) is similar to the contribution which would appear in the presence of an external potential. Hence the Vlasov equation represents a collisionless approximation in the presence of an effective mean field. The above description of the dynamics is useful in plasma physics, in which long-range Coulomb forces are present. However, this is not a good approximation for studying a liquid with short-range interactions. In general, the RHS of eqn. (1.1.35) represents the contribution from the twobody interaction term to the dynamic evolution of f (1) (x1 , t) and is termed the collision contribution. The above equation is therefore written in the schematic form  

(1) ∂ ∂ f p1 . (1.1.39) + · ∇r1 f (1) (r1 , p1 , t) = ∂t m ∂t coll

The RHS of the equation involving interaction of two particles is approximated through different schemes with a proper closing relation for the hierarchy in terms of the one-particle function. The most studied among the various approximations is the case of the Boltzmann equation in which the RHS (∂ f (1) /∂t)coll is approximated in terms of binary-collision contributions only. At low densities only two particles are likely to collide at a given point. The evaluation of the two-point functions before collision is done with the approximation of molecular chaos. Thus the factorization of eqn. (1.1.36) is applied. This again remains valid at low densities when successive binary collisions are completely uncorrelated. The collision of two particles at a point r is governed by the laws of classical mechanics. momenta of the particles before and that after collisions are denoted by {p1 , p2 } and  The   p1 , p2 , respectively. The RHS of eqn. (1.1.39) is approximated as (Dorfman, 1999)      ∂ f (1) 1 dσ = dp2 d |p1 − p2 | ∂t m d coll      × f (1) r, p1 , t f (1) r, p2 , t − f (1) (r, p1 , t) f (1) (r, p2 , t) . (1.1.40) where dσ /d denotes the differential scattering cross section in solid angle d defined around the (p1 −p2 ) vector of relative motion between the two particles before the collision. The approximation (1.1.40) gives rise to the Boltzmann H theorem which implies that the many-particle system evolves irreversibly to an equilibrium state. The time evolution

1.2 Equilibrium properties

11

of the HB function is defined as a functional of the one-particle distribution function f (r, p, t) as  HB (t) = dp f (p, t) ln f (p, t). (1.1.41) Note that it is consistent to assume that the distribution function is independent of r since there is no external field present. On taking the time derivative of eqn. (1.1.41), we obtain  " ∂ f (p, t) ! d HB (t) = dp 1 + ln f (p, t) . (1.1.42) dt ∂t It is straightforward to establish (Huang, 1987) using the Boltzmann equation that d HB (t) ≤ 0, dt

(1.1.43)

showing that the H function always decreases. This is known as the Boltzmann H theorem. For f (p, t) equal to the Maxwell–Boltzmann distribution function  3/2

β β p2 exp − (1.1.44) φMB ( p) = 2πm 2m the H function HB (t) is independent of time and is equal to zero. The function φMB ( p) is normalized and the constant β is determined from the average kinetic energy as  p 2 /(2m) = 3/(2β). Hence β is identified with the temperature T of the equilibrium fluid β = 1/(kB T ). In general the nature of the interaction potential plays an important role in evaluating the collision integral on the RHS of eqn. (1.1.39). The irreversible kinetic equation obtained from closing the BBGKY hierarchy is used to compute the basic transport equations for fluid with dissipative dynamics. The most widely studied system in this regard is a system of hard spheres. We discuss this further in Section 5.2.2 below.

1.2 Equilibrium properties By definition for thermodynamic equilibrium, the distribution function f (N ) depends on the coordinates rα (t) and pα (t) but is independent of time t. The equilibrium state of a fluid corresponds to the stationary solutions of the distribution functions, ∂ (N ) N N f (r , p , t) = 0. ∂t

(1.2.1)

Therefore from eqn. (1.1.20) it follows # that the$ N -particle equilibrium distribution func(N ) (N ) (N ) tion f EQ must satisfy the condition f EQ , H = 0. Hence f EQ must be expressed as a functional of the Hamiltonian H or of function {Ai } (say), all of which have vanishing Poisson bracket with the Hamiltonian. These stationary distribution functions determine the Gibbsian ensembles. In this regard it is particularly useful to consider the following theorem for the equilibrium state.

12

Statistical physics of liquids

1.2.1 The Gibbs H-theorem The H function in terms of the N -particle distribution function f (N ) is defined as   (1.2.2) HG = dr N dp N f (N ) (r N , p N ) ln f (N ) (r N , p N ). In some cases the ensemble may include variation of the total number of particles N or the total volume of the system. We denote by Tr an averaging over the whole phase space. If the different members of the ensemble have only energy as the variable with both the total number of particles N and the volume V fixed (a canonical ensemble) then Tr stands for Tr ≡

N 1 % drα dpα , N! h 3N

(1.2.3)

α=1

where the factor of h 3N is included to take into account the quantum-mechanical counting of possible states and the factor of N ! to account for the permutation among the N identical particles. If the ensemble is one in which, for example, variation of the total number of particles as well as of the energy is allowed (a grand-canonical ensemble) then we define   ∞  1 N dr dp N . (1.2.4) Tr ≡ h 3N N ! N =0

We now define the H function HG as  HG = Tr f (N ) (r N , p N )ln f (N ) (r N , p N ) .

(1.2.5)

Let us consider the problem of minimizing the above H function subject to the following constraints in terms of a specific set of functions Ai (r N , p N ), i = 1, . . . , m, given by {A¯ i } such that  A¯ i = Tr Ai f (N ) (r N , p N ) . (1.2.6) In equilibrium the time-independent f (N ) (r N , p N ) describes the probability for the different members of the corresponding ensemble. The condition (1.2.6) implies that the ensemble averages {A¯ i } characterize the thermodynamic equilibrium state. The H theorem at the N -particle level in this case states that, subject to the constraints (1.2.6), the (N ) normalized distribution function f 0 corresponding to which the value of HG reaches a minimum is given by    (N ) N −1 N N N αi Ai (r , p ) . (1.2.7) f 0 (r , p ) = W0 exp − i

The set of parameters {αi } for i = 1, . . . , m defines the thermodynamic state characterized by optimum f (0) . W0 is the normalization constant     N N W0 = Tr exp − αi Ai (r , p ) (1.2.8) i

1.2 Equilibrium properties

13

which ensures the condition Tr f 0(N ) (r N , p N ) = 1.

(1.2.9)

In order to prove the above theorem we first consider an arbitrary choice of the distri(N ) bution function f (N )  = f 0 . Since both the distribution functions represent probabilities, both of them are real positive numbers and x = f (N ) / f 0(N ) is also a positive number. Since both the distributions are normalized, we must have (N )

Tr f (N ) (r N , p N ) = Tr f 0

(r N , p N ) = 1.

(1.2.10)

Now let us consider the difference of the function HG as evaluated with the arbitrary functional f (N ) from that corresponding to f 0(N ) :  HG [ f ] − H [ f 0 ] = Tr f 0 x ln{x f 0 (r N , p N , t)} − f 0 ln f 0 , (1.2.11) where we have dropped the superscript N on f to keep the notation simple. Using the normalization condition (1.2.10), the RHS of eqn. (1.2.11) reduces to " ! HG [ f ] − H [ f 0 ] = Tr f 0 {x ln x − x + 1} + ( f − f 0 )ln f 0 = I1 + I2 .

(1.2.12)

The first integral, I1 , on the RHS of eqn. (1.2.12) is always positive since, according to the Gibbs inequality (see Appendix A1.1), the quantity (x ln x − x + 1) is positive definite for positive x. The second integral, I2 , on the RHS of eqn. (1.2.12) vanishes. To demonstrate this we use the expression (1.2.7) for evaluating ln f 0 ,  & '  N N I2 = −Tr ( f − f 0 ) ln W0 + αi Ai (r , p ) i

= 0.

(1.2.13)

The above result for I2 follows easily from the normalization condition (1.2.10) and the constraints (1.2.6) for the ensemble averages of the set of observable {Ai }. Note that W0 and the parameters {αi } are not dependent on the phase-space variables. We have thus proved that HG for all arbitrary choices of f (N ) increases from its value corresponding to (N ) the function f 0 . The latter, as given by the RHS of eqn. (1.2.7), therefore minimizes the function HG . If we identify −kB HG with the thermodynamic entropy of the equilibrium state, the other parameters are also readily identified from the relation  S = − ln W0 − αi A¯ i . (1.2.14) − kB i

Since for the equilibrium state the entropy is a maximum, we obtain, on taking the derivative of eqn. (1.2.14) with respect to αi , the following result for the thermodynamic average: ∂ ln W0 . A¯ i = − ∂αi

(1.2.15)

14

Statistical physics of liquids

On taking a second derivative we obtain ∂2 ∂ A¯ i = ln W0 . ∂αi ∂αi2

(1.2.16)

Using the expression (1.2.8) for ln W0 , the RHS of eqn. (1.2.16) is evaluated as the average of the mean-square fluctuation of Ai ,  ∂2 2 ln W = Tr Ai2 − {Tr[Ai ]}2 = Tr[{Ai − A¯ i } ] = Ai2 , (1.2.17) 0 2 ∂αi giving the relative fluctuation of Ai as Ai2 = −

∂ A¯ i . ∂αi

(1.2.18)

The result (1.2.18) implies that the relative fluctuation of Ai is of the order of the square root of the size S of the system, # $1/2   Ai2 1 ∼O . (1.2.19) ¯ S Ai The RHS approaches zero in the thermodynamic limit. Different representative statistical ensembles for the equilibrium state of the system are obtained with corresponding specific choices of the set of extensive variables {Ai }. The ensemble-averaged quantities {A¯ i } remain fixed at their experimentally measured values in the corresponding thermodynamic state which is characterized by the corresponding set of intensive thermodynamic variables {αi }. The above result (1.2.19) therefore demonstrates the equivalence of the different ensembles since the fluctuation of a specific property Ai around its average value is negligible in the thermodynamic limit of a large system, e.g., lim N → ∞, lim V → ∞ but N /V remaining finite. 1.2.2 The equilibrium ensembles Let us consider some of the specific ensembles in the following. For a completely isolated system we have the micro-canonical ensemble in which the assumption of equal a-priori probability for each state is applied. The canonical ensemble For the canonical ensemble the system is connected to an external heat bath allowing exchange of energy. The average energy is fixed and is determined by the equilibrium temperature T of the thermodynamic state. In the present case we have A ≡ H and the corresponding parameter is αi ≡ β. The number of particles N is fixed in this case. Evaluation of the H function gives   HG = dr N dp N f 0 ln f 0 = −ln Z N − H¯ , (1.2.20)

1.2 Equilibrium properties

15

where Z N ≡ W0 . Upon identifying the average energy H¯ as the thermodynamic internal energy U , the above relation (1.2.14) reduces to −

S = −ln Z N − βU. kB

(1.2.21)

We identify β = 1/(kB T ) and the Helmholtz free energy of the system F as − kB T ln Z N = U − T S = F.

(1.2.22)

The equilibrium probability distribution is therefore given by (N )  N

f0

 r , pN =

1 exp[−β H ] , ZN h 3N N !

with the partition function Z(N , V, T ) defined as    1 dr N dp N exp −β H (r N , p N ) . Z(N , V, T ) ≡ Z N = 3N h N!

(1.2.23)

(1.2.24)

The key relation connecting the thermodynamic property (the Helmholtz free energy F) with the statistical-mechanical quantity (the canonical partition function Z N ) is given by F = −kB T ln Z N .

(1.2.25)

For a noninteracting system (ideal gas) the Hamiltonian H has only a kinetic part and it is straightforward to obtain the partition function Z N by evaluating the Gaussian integrals involving the pα , α = 1, . . . , N : ZN =

VN N !30

,

(1.2.26)

√ with the thermal de Broglie wavelength 0 = h/ 2π mkB T . The free energy for the noninteracting system is obtained as  ( ) Fid = N kB T ln n30 − 1 . (1.2.27) By using the relation P = −(∂ F/∂ V )T we obtain from eqn. (1.2.27) the ideal-gas equation of state P V = N kB T . The Gibbs free energy is given by ( ) G = F + P V = N kB T ln n30 , (1.2.28) and the chemical potential for the ideal gas is

( ) μid = kB T ln n30 .

(1.2.29)

The average energy, which is identified as the thermodynamic internal energy U , is obtained as * * ∂ ¯ (1.2.30) ln Z N ** . U=H=− ∂β V

16

Statistical physics of liquids

The mean-square fluctuation of the internal energy is obtained from the general formula (1.2.18) as   ∂U 2 H = − = N kB T 2 cv , (1.2.31) ∂β V,N where cv is the specific heat per particle √  at constant volume. The relative root-mean-square fluctuation per particle is of O 1/ N , making it negligible in the thermodynamic limit. The grand-canonical ensemble The grand-canonical ensemble describes the system connected to an external bath allowing exchange of both energy and particles. The ensemble average of the total energy and the total particle numbers of the system are fixed. In this case we have A ≡ {H, N } and the corresponding αi is denoted by {β, −βμ}. Equation (1.2.14) reduces to the form −

S = −ln  − H¯ + βμ N¯ , kB

(1.2.32)

where  ≡ W0 in this case. The average energy H¯ and N¯ , respectively, correspond to the thermodynamic internal energy U and the average number of particles at temperature T . If we identify β = 1/(kB T ) and μ as the chemical potential, eqn. (1.2.32) reduces to the thermodynamic relation − kB T ln  = U − T S − μ N¯ = F − G = −,

(1.2.33)

where G is the Gibbs free energy of the system and  = − P V is the thermodynamic potential. The equilibrium probability distribution in the grand-canonical ensemble is therefore given by (N )

f0

(r N , p N ) =

1 exp[−β(H − μN )] ,  h 3N N !

(1.2.34)

where the grand-canonical partition function (μ, V, T ) is obtained as (μ, V, T ) =

  ∞   1 dr N dp N exp −β(H (r N , p N ) − μN ) . (1.2.35) N!

N =0

The average energy and the average number of particles are respectively obtained as * * ∂ U = − ln ** , ∂β μ,V * * * * ∂ ∂ * = z ln ** . (1.2.36) ln * N¯ = ∂(βμ) ∂z T,V T,V where z = exp(βμ) is the fugacity. Note that the averaging in the grand-canonical ensemble can be interpreted as a weighted average with respect to the canonical-ensemble

1.2 Equilibrium properties

17

N -particle partition function Z N with all possible values of N . Thus we can express the grand-canonical partition function as (μ, V, T ) =

∞  zN ZN . N!

(1.2.37)

N =0

The average number of particles as given by eqn. (1.2.36) is expressed as N¯ =

∞ 

N P(N ),

(1.2.38)

N =0

where P(N ) is the probability for the N -particle state given by P(N ) =

z N ZN . N! 

(1.2.39)

The key relation in the case of the grand-canonical ensemble connecting the thermodynamic property (the thermodynamic potential  ≡ −P V ) with the statistical-mechanical quantity (the grand-canonical partition function ) is given by  = −kB T ln .

(1.2.40)

The mean-square fluctuation of the number of particles is obtained from the general formula (1.2.18) as *  ¯ ∂ N¯ ** ∂N 2 N = = kB T . (1.2.41) * ∂(βμ) T,V ∂μ T  √  The relative root-mean-square fluctuation per particle is of O 1/ N , making it negligible in the thermodynamic limit. The RHS of eqn. (1.2.41) is further simplified by using the Gibbs–Duhem relation (McQuarie, 2000) N¯ dμ = V d P − S dT, and hence



∂μ ∂ N¯

 = T

V N¯



∂P ∂ N¯

 = T

1 N¯



(1.2.42)

∂P ∂n

 ,

since n = N¯ /V . However, (∂ P/∂n)T can also be written as         ∂P V2 ∂P 1 ∂V ∂P = =− = , ¯ ∂n T ∂ V T ∂n T ∂ V nκ N T T where κT = −

1 V¯

 ¯ ∂V ∂P T

(1.2.43)

T

(1.2.44)

(1.2.45)

18

Statistical physics of liquids

is the isothermal compressibility of the system. Using the result (1.2.44) in eqn. (1.2.41), we obtain N 2 − N¯ 2 n (1.2.46) = κT . β N¯ √ The root-mean-square fluctuation N is therefore of O(1/ N ) and negligible in the thermodynamic limit – except near the critical point, where the compressibility is diverging and hence the role of fluctuations becomes important there. The isobaric–isothermal ensemble The isobaric ensemble describes the system connected to an external bath allowing both exchange of energy and variation of the volume V of the system. The number of particles remains constant in this case. The ensemble average of the total energy and the volume of the system are fixed. In this case we have Ai ≡ {H, V } and the corresponding αi is denoted by {β, βα}. Evaluation of the H function gives   N dp N f 0 ln f 0 = −ln  − βH  − βαV , (1.2.47) HG = dr where  ≡ W0 . Upon identifying the average energy H  as the thermodynamic internal energy U the above relation (1.2.14) reduces to −

S = −ln  − β(U + α V¯ ), kB

(1.2.48)

where V¯ is the volume in the thermodynamic equilibrium state at temperature T . We identify β = 1/(kB T ) and α as the thermodynamic pressure P so that eqn. (1.2.48) reduces to the thermodynamic relation − kB T ln  = U − T S + α V¯ = U − T S + P V = G.

(1.2.49)

The equilibrium probability distribution in the isobaric ensemble is therefore given by  (N )  (1.2.50) f 0 r N , p N = −1 exp[−β{H + P V }], where (N , P, T ) is the isobaric partition function. The latter is obtained as a Laplace transform (see eqn. (1.3.10) for its definition) at “frequency” β P with respect to the volume V ,  ∞   0 N dp N exp[−β{H + P V }], (1.2.51) d V dr (N , P, T ) = 3N h N! 0 where 0 is a constant with the dimension of volume introduced so as to make the partition function  dimensionless. The average volume and the mean square of the volume fluctuations are, respectively, obtained as * * ∂ , (1.2.52) ln ** V¯ = −kB T ∂P T,N

1.2 Equilibrium properties

19

Table 1.1 Different ensembles, the corresponding partition functions, and how they relate to the appropriate thermodynamic function. Ensemble type Canonical

Partition function

Thermodynamic function

Z N (N , V, T )

Helmholtz free energy, F F = −kB T ln Z N

Grand-canonical

(μ, V, T )

Thermodynamic potential,   = −P V = −kB T ln 

Isothermal–isobaric

(N , P, T )

Gibbs free energy, G G = −kB T ln 

* (V )2 κT 1 ∂ V¯ ** = . =− * ¯ ¯ ∂(β P) β V V T,N

(1.2.53)

The above results for the different thermodynamic properties corresponding to the canonical, grand-canonical, and isobaric ensembles are shown in Table 1.1. 1.2.3 The static structure factor In the present section we discuss the correlation functions of fluctuations of dynamic variables at different spatial points but at the same time. These represent the static or the structural properties of the liquid. For a high-density liquid the most basic quantity characteristic of its structural or thermodynamic properties is the pair correlation function. The pair correlation function g(r ) For an N -particle system the equilibrium solution to the Liouville equation (1.1.22) is given by the Gibbsian distributions discussed in the previous section. Let us consider s (rs , ps ) of s particles as defined in eqn. (1.1.32), the reduced probability distributions f eq specifically for the equilibrium fluid. We use the short-hand notation rs ≡ {r1 , r2 , . . . , rs }, s in this case of a etc. here. The momentum dependence of the distribution function f eq thermodynamic equilibrium state is Maxwellian, &N ' % (s) s s f eq (r ) = φMB (ps ) n (s) (1.2.54) N (r ), s=1

where φMB (ps ) is the normalized Maxwell distribution function as defined in eqn. (1.1.44). The conditional probability of there being s particles simultaneously at {r1 , . . . , rs } irre(s) by integrating out the spective of their momenta is obtained from the corresponding f eq momentum variables { ps },  s (s) s n (s) (r ) = dps f eq (r , ps ). (1.2.55) N The simplest example of the above distribution function is the s = 2 case, i.e., g N (r1 , r2 ), which is termed the radial distribution function. It represents the conditional probability

20

Statistical physics of liquids

of a particle being at r2 while another is at r1 . In general a distribution function at the (s) s-particle level is defined from n N (rs ), g (s) N (r1 , r2 , . . . , rs ) =

(s)

n N (r1 , r2 , . . . , rs ) , (n N )s

(1.2.56)

where we define n N = N /V , the number of particles per unit volume in the N -particle system. In the grand-canonical-ensemble description, we average the reduced distribution s function n (s) N (r ) over the total number of particles N in the system to obtain the averaged quantity ∞ + ,  (s) s (s) s P(N )n N (rs ). (1.2.57) n (r ) ≡ n N (r ) = N =s

The simplest and most important is the s = 2 case. The thermodynamic quantity g(r1 , r2 ) termed the pair correlation function is defined as n (2) (r1 , r2 ) = n 2 g(r1 , r2 ),

(1.2.58)

where the factor n 2 on the LHS is for normalization of g(r ); n is the average number of particles per unit volume of the liquid, n = N¯ /V = n N . Some useful properties of this function follow directly from simple physical considerations. Thus, in the equilibrium state translational invariance holds and for an isotropic system g(r1 , r2 ) is a function only of the distance between the two points and can be denoted as g(r12 ). For a small separation r between two points g(r ) vanishes due to the repulsive core of the interaction potential, whereas, for large r , g(r ) → 1 so that the probability of two particles being at the two points goes to the trivial limit n 2 . The radial distribution function g(r ) is the most fundamental quantity in describing the thermodynamic properties of the equilibrium fluid. In this respect it is important to note two very basic relations linking the thermodynamic properties of a liquid to the pair correlation function. The virial equation For a liquid in which the interaction between the particles involves only two-body potentials as introduced in the expression (1.1.16) for the Hamiltonian, the thermodynamic pressure is obtained in terms of the pair correlation function only. To demonstrate this, we begin from the thermodynamic relation P = −(∂ F/∂ V )T with the free energy being given by F = −kB T ln Z N . The canonical partition function Z N is obtained for the interacting system as  1 dr N exp[−βU (r1 , . . . , r1 )], (1.2.59) ZN = N !3N 0 where 0 is the thermal de Broglie wavelength as defined with eqn. (1.2.26). The pressure is obtained as * * 1 1 ∂ * P= (1.2.60) Z N* . * Z ∂ V N !3N N 0 T

1.2 Equilibrium properties

21

The calculation of the thermodynamic pressure P involves computing the derivative of Z N with respect to the volume. The volume dependence of the partition function has two components. Let us first consider the 3N -dimensional integral over the spatial coordinates. Let us assume that the volume of the system is a box of size 2L for each dimension and that the spatial integral is done over the linear range −L to +L. The volume of the system V ∼ L 3 . A change of the integration variables from x to x  = x/L reduces the configuration integral on the RHS of eqn. (1.2.59) to  VN

+1 −1

dr1 . . .



+L −L

drN exp[−βU (r1 , . . . , r N )].

Using this form in eqn. (1.2.60), we obtain for the pressure   +1  +L 1 P N −1  dr1 . . . drN e−βU (r1 ,...,r N ) = NV kB T N !Z N 3N −1 −L 0     −βU (r1 ,...,r N ) ∂U −β dr1 . . . dr N e ∂V    β ∂U dr N e−βU (r1 ,...,r N ) =n− . 3N ∂V N !Z N 0

(1.2.61)

(1.2.62)

For systems in which the potential part U of the Hamiltonian involves only two-body interactions the integral Ivirial (say) in the second term on the RHS is simplified. In this case the spatial dependence of U comes through the two-body interaction potential u(r), N  1  U= u(ri j ). 2

(1.2.63)

i, j=1

Since the spatial dependence of U comes through that of the two-body potential, we write the derivative on the RHS in the form Ivirial =

N   e−βU (r1 ,...,r N ) du(ri j ) dri j β  · dr1 . . . dr N . 2 dri j dV N !Z N 3N 0 i, j=1

(1.2.64)

On changing the integration variable from x to x  = x/L it follows that ri j → V 1/3ri j and hence the derivative dri j /d V = ri j /(3V ). For a system of identical particles all the different N (N − 1) choices for the pairs in the sum above give the same integral. For the spherically symmetric potential u(ri j ) we obtain

 −βU (r1 ,...,r N )  e 1 N! du(r12 ) · r12 dr3 . . . dr N dr1 dr2 6V (N − 2)! dr12 N !Z N 3N 0  du(r ) 2πβ 2 dr r 3 = n g(r ). (1.2.65) 3 dr

Ivirial =

22

Statistical physics of liquids

We obtain the pressure in terms of the pair correlation functions as P 2πβ = n − n2 kB T 3



∞ o

du(r ) g(r )r 3 dr. dr

(1.2.66)

In the above equation we have dropped the subscript N from g in the thermodynamic limit. This is termed the virial equation. The compressibility condition The isothermal compressibility κT of the liquid defined in eqn. (1.2.45) is also related to the pair correlation function g(r ). In order to demonstrate this, we integrate with respect to (N ) all the variables {rs , ps } the definition (1.1.32) of the f (s) and obtain, in terms of f eq (r N ),    N! (s) (N ) N dr N dp N f eq drs n N (rs ) = (r ). (1.2.67) (N − s)! On taking the s = 2 case, eqn. (1.2.67), and averaging over the number of particles N in terms of the probability P(N ) defined in eqn. (1.2.39), we obtain - .   (2) dr1 dr2 n N (r1 , r2 ) = d N P(N )(N 2 − N ) = N 2 − N¯ , (1.2.68) where the angular brackets imply averaging over the number of particles N in the system. Using the definition (1.2.58) for the pair correlation function, the above relation gives us the mean-square fluctuation of N for an isotropic fluid,   2 2 2 ¯ ¯ dr1 dr2 g(r1 , r2 ) − N¯ 2 N − N = N +n  = N¯ + n 2 V dr g(r ) − N¯ nV

 ¯ = N 1 + n dr{g(r ) − 1} . (1.2.69) Now, using the relation (1.2.46), we obtain the compressibility condition  n κT = 1 + n (g(r ) − 1)dr. β

(1.2.70)

Note that the above relation (1.2.70) is obtained from consideration of the grand-canonical ensemble and, unlike eqn. (1.2.66), is not limited only to pairwise-additive interaction potentials. The static correlation function Starting from the level of N particles, the reduced distribution function giving the conditional probability of finding a smaller number of particles has been defined in eqn. (1.1.32).

1.2 Equilibrium properties

23

Thus the s-particle (s < N ) reduced distribution function f (s) is defined as  N! f (N ) (r N , p N , t)drs+1 dps+1 . . . dr N dp N . f (s) (rs , ps , t) = (N − s)!

(1.2.71)

Furthermore, the conditional probability of there being s particles simultaneously at {r1 , . . . , rs } irrespective of their momenta is obtained from the corresponding f (s) by integrating out the momentum variables { ps },   N! (s) f (N ) (r N , p N , t)dp N drs+1 . . . dr N , n N (rs ) = dps f (s) (rs , ps ) = (N − s)! (1.2.72) where we have used the notation p N ≡ {p1 , . . . , p N }. Note that the above definition of n N can be rewritten in the form  N! (s) d r˜ N d p˜ N f (N ) (r N , p N , t){δ(r1 − r˜ 1 ) . . . δ(rs − r˜ s )} n N (rs , t) = (N − s)! / N 0  = δ(r1 − r˜ α1 ) . . . δ(rs − r˜ αs ) , (1.2.73) α1 ,...,αs =1

where the prime in the summation in (1.2.73) indicates that none of the αi are equal. The simplest examples of the above distribution function are the s = 1, 2, . . . cases, / N 0  (1) δ(r − rα ) , (1.2.74) n (r) = α

n

(2)



(r, r ) =

/N  

0 

δ(r − rα )δ(r − rβ ) ,

(1.2.75)

α,β

and so on, where we have dropped the subscript N on n (s) for simplicity and {r1 , . . . , r N } are now the space coordinates of the particles labeled as α = 1, . . . , N respectively. Note that the above definitions for the average distribution functions are not restricted to systems in equilibrium. Let us focus on the simplest of these spatial distribution functions, namely the oneˆ particle quantity n (1) (r), simply written as the ensemble average of n(r), n(r) ˆ =

N 

δ(r − rα ),

(1.2.76)

α

where we have put the hat on n to indicate that it depends on the phase-space variables. The integral of this one-particle density over the whole volume is N , the total number of particles. For an isotropic fluid in equilibrium 1in the2 absence of any external field the average density at every point is a constant so that n(r) ˆ = N /V = n 0 . The product of the density fluctuations δ nˆ = nˆ − n 0 at two different spatial points but at the same time, averaged over the equilibrium ensemble, is referred to as the equal-time correlation function. For the fluid

24

Statistical physics of liquids

in equilibrium translation invariance holds and the equal-time correlation function χnn is a function of (r − r ) only: / N N 0  1 2    ˆ n(r ˆ ) = δ(r − rα )δ(r − rβ ) − n 20 χnn (r − r ) = δ n(r)δ / 

= δ(r − r )

α=1 β=1 N 

0

δ(r − rα ) +

α=1

/

N  

0 

δ(r − rα )δ(r − rβ ) − n 20 ,

α,β=1

(1.2.77) where the prime indicates that the α = β term is dropped from the sum. However, using now eqns. (1.2.74) and (1.2.75), we obtain from the above equation the result ! " (1.2.78) χnn (r − r ) = n 0 δ(r − r ) + n 0 {g(r − r ) − 1} . Now, on taking the Fourier transform (see eqn. (1.3.9) for its definition) of the eqn. (1.2.78), we obtain the result

 χnn (k) = n 0 1 + n 0 dr eik · r {g(r) − 1} ≡ n 0 S(k),

(1.2.79)

where the Fourier transform of the radial distribution function g(r ) is expressed in terms of the Fourier-transformed quantity S(k) for the isotropic liquid,  (1.2.80) S(k) = 1 + n 0 eik · r h(r )dr, with h(r ) = g(r ) − 1. S(k) is termed the static structure factor for the liquid. This is a quantity directly measurable from the coherent part of elastic scattering intensities using a radiation source suitable for the length scale probed in the system. For experimental methodologies used in scattering experiments see Berne and Pecora (1976) and Hansen and McDonald (1986).

1.2.4 Integral equations for g(r) Both (1.2.66) and (1.2.70) give rise to the equation of state for the liquid, via two different routes, and in terms of the pair correlation function g(r ). Computation of the radial distribution function g(r ) for a liquid thus forms a major part of the study of its equilibrium thermodynamic properties. The determination of g(r ) is carried out by the solution of appropriate integral equations. We discuss below the broad outlines of some of the methods for computing g(r ) for a liquid. The Kirkwood approximation One of the earlier approaches to computing g(r ) comes from the stationary limit of the BBGKY hierarchy, from which one can obtain a set of equations for the static functions

1.2 Equilibrium properties

25

(s)

g N defined in eqn. (1.2.56) for the N -particle system. The hierarchy equation for the s-particle distribution function (in the absence of any external field) is obtained from eqn. (1.1.34) as ⎧ ⎫⎞ ⎤ ⎡⎛ s s ⎨ ⎬  pi ⎣⎝ (s) s ⎦ kB T ∇ri + Fi j ⎠ f eq (x ) · ⎩ ⎭ m i=1

=−

s  pi i=1

m

j=1





dxs+1 Fi,s+1 ·

·

(s+1) (s+1) f eq (x )

.

(1.2.81)

Since eqn. (1.2.81) is independent of s and the value of the momentum ps , the equality must hold term by term and the space-dependent parts within the square brackets should (s) be equal. We write the equation now in terms of the spatial distribution functions n N (rs ) as follows:   (s) s ˜ kB T ∇1 + F1 n N (r ) = − drs+1 F1,s+1 · n (s+1) (r(s+1) ), (1.2.82) N where we have used the notation F˜ 1 =

s  l=1

F1l =

s 

∇1 u(r1 , rl )

(1.2.83)

l=1

and ∇1 represents the derivative with respect to r1 . Now, using the definition (1.2.56) for (s) the s-particle distribution function g N (rs ), we obtain the hierarchy equations s −kB T ∇1 g (s) N (r ) =

s  l=1

+n

s ∇1 u(r1 , rl )g (s) N (r )



(rs+1 )drs+1 . ∇1 u(r1 , rs+1 )g (s+1) N

(1.2.84)

The set of equations described by (1.2.84) is known as the Yvon–Born–Green (YBG) hierarchy. To solve this, a suitable closure scheme is needed. For an isotropic fluid the first equation of this set, i.e., the s = 2 case, reduces to an integral equation for g N (r1 , r2 ) of the form (we drop the superscript s for simplicity)  kB T ∇1 g N (r1 , r2 ) + g N (r1 , r2 )∇1 u(r1 , r2 ) = −n ∇1 u(r1 , r3 )g N (r1 , r2 , r3 )dr3 . (1.2.85) (3)

The hierarchy is closed by writing the function g N , involving three particles, in the Kirkwood superposition approximation (Kirkwood, 1935) as (2) (2) (2) g (3) N (r1 , r2 , r3 ) = g N (r1 , r2 )g N (r2 , r3 )g N (r3 , r1 ).

(1.2.86)

26

Statistical physics of liquids

For the isotropic liquid the pair correlation function depends only on the distance between the two points, g N (r1 , r2 ) ≡ g N (r12 ). Using (1.2.86) in eqn. (1.2.85), we obtain the following equation:  ∇1 {kB T ln[g N (r12 )] + u(r12 )} = −n dr3 ∇1 u(r13 ){g N (r23 ) − 1}g N (r13 ). (1.2.87) The last term added on the RHS gives a vanishing contribution since for the isotropic liquid every point is equivalent on average, and hence  dr3 u(r1 , r3 )g N (r13 ) = 0. (1.2.88) The integral equation (1.2.87) known as the Yvon–Born–Green equation is good for computing the pair correlation functions in a low-density fluid. Once g(r ) is known, the other thermodynamic properties readily follow, as discussed above. For systems interacting with two-body potentials the integral-equation approach has been applied widely and the corresponding theoretical formulation of the problem is specific to the nature of the interaction. We discuss below the broad outlines of the calculation of the pair correlation function for two generic cases: (a) for a short-range repulsive potential like the hard-sphere interaction; and (b) for an interaction having an attractive part effective over a long distance. Hard-sphere interaction Subsequent to the Kirkwood approximation for closing the hierarchy, a new class of integral equations for g(r ) was obtained for fluids interacting through a pairwise additive potential by considering different methods of closing the hierarchy. This involved introducing a new type of correlation function, termed the direct correlation function c(r ), which was defined from the Ornstein–Zernike relation (Ornstein and Zernike, 1914),  h(r ) = c(r ) + n c(|r − r |)h(r )dr . (1.2.89) Note that from eqn. (1.2.79) it follows that eqn. (1.2.89) also leads to the relation S(k) = [1 − n 0 c(k)]−1 .

(1.2.90)

To solve for g(r ) and c(r ), for a given pair potential u(r ), eqn. (1.2.89) is supplemented with a closure relation. Two standard prescriptions for this, namely the Percus–Yevick (PY) (Percus and Yevick, 1958) solution and hypernetted chain closure (HNC) (van Leeuwen et al., 1959), respectively, give y(r ) = 1 + h(r ) − c(r ) y(r ) = exp[h(r ) − c(r )]

(PY), (HNC),

(1.2.91) (1.2.92)

1.2 Equilibrium properties

27

where the function y on the LHS is defined as y(r ) = eβu(r ) g(r ). The motivation for both of the proposed closures can be reached through different approaches, for which we refer the reader to Percus (1962, 1964) and Stell (1963). Of the two proposed closures, the PY solution is most suitable for describing the correlations between particles when the interaction potential is harshly repulsive or of short range. The hard-sphere potential is a typical example of this. The HNC, on the other hand, is well suited for long-range potentials such as Coulombic systems. However, for strongly coupled systems like high-density or lowtemperature liquids, both these closures lead to severe thermodynamic inconsistency since the virial and compressibility equations given by (1.2.66) and (1.2.70), respectively, lead to different equations of state. For investigating equilibrium theories of simple liquids, the hard-sphere system has been the most widely studied system, both through computer simulations and using theoretical methods. For the hard-sphere system the integral equation for g(r ) can be solved exactly. The hard-sphere potential is given by & ∞ for r ≤ σ, (1.2.93) u HS (r ) = 0 for r > σ. Since the barrier height is infinite, with the hard-sphere potential g(r ) = 0 for r ≤ 0 and g(r ) = y(r ) for r ≥ 0. The PY closure eqn. (1.2.91) in this case takes the form & −y(r ) for r ≤ σ, c(r ) = (1.2.94) 0 for r ≥σ. With the above closure relation the Ornstein–Zernike equation (1.2.89) is solved using Laplace-transform methods and the direct correlation function is obtained as (Wertheim, 1963, 1964; Thiele 1963) & −α0 − α1 (r/σ ) − α2 (r/σ )3 for r ≤ σ, c(x) = (1.2.95) 0 for r > σ . The constants αi appearing in eqn. (1.2.95) are obtained in terms of the packing fraction ϕ = π nσ 3 /6 as follows: α0 =

(1 + 2ϕ)2

, (1 − ϕ)4 (1 + ϕ/2)2 α1 = −6ϕ , (1 − ϕ)4 ϕα0 α2 = . 2

(1.2.96) (1.2.97) (1.2.98)

Often it is more convenient to express the Fourier transform of the above expression in such a way as to obtain the static structure factor S(k) using the relation (1.2.90) (see Fig. 1.3).

28

Statistical physics of liquids

Fig. 1.3 The static structure factor from MD simulations of Lennard-Jones system at T = 0.723 and ρ = 0.844 (dots). The solid line corresponds to the hard-sphere mdoel.

Thus the Fourier transform of (1.2.95) gives the result (Ashcroft and Lekner, 1966)

sin(skσ ) ds s {α0 + α1 s + α2 s 3 } nc(kσ ) = −4πnσ skσ 0 24ϕ  α1 = − 3 α0 (sin x − x cos x) + {2x sin x − (x 2 − 2)cosx − 2} x x α3 3 4 + 3 {(4x − 24x)sinx − (x − 12x 2 + 24)cosx + 24} (1.2.99) x 

3



1

2

in terms of the dimensionless variable x = kσ . The above result is useful in obtaining the compressibility of the hard-sphere fluid using the relation nc(0) = −ϕ

(4 − ϕ)(2 + ϕ 2 )

(1.2.100)

(1 − ϕ)4

and hence, from eqn. (1.2.90) for the corresponding static structure factor, S(0) is obtained as S −1 (0) = 1 − nc(0) =

(1 + 2ϕ)2 (1 − ϕ)4

.

(1.2.101)

Since the compressibility is expressed as κT−1 = n(∂ P/∂n), we obtain from the compressibility condition, i.e., eqn. (1.2.70), the relation 

∂P ∂n



= β −1

(1 + 2ϕ)2 (1 − ϕ)4

.

(1.2.102)

1.2 Equilibrium properties

29

By integrating the above expression we obtain the equation of state from the compressibility equation as P 1 + ϕ + ϕ2 . = nkB T (1 − ϕ)3

(1.2.103)

On the other hand, we obtain the equation of state directly from eqn. (1.2.66), giving that  ∞ du(r ) 2 2πβ βP = n − n g(r )r 3 dr 3 0 dr  ∞ 2 2π =n+n δ(r − σ ) g(r )r 3 dr 3 0 ! " = n 1 + 4ϕg(σ ) . (1.2.104) In reaching the above result we used the fact that the discontinuous hard-sphere potential u HS (r ) given by eqn. (1.2.93) decreases discontinuously from ∞ to 0 as r increases from σ − to σ + . Hence the derivative β du(r )/dr = −δ(r − σ ). The pair correlation function at contact g(σ ) directly follows from the PY closure. In the limit r → σ + , g(r )  = 0 while c(r ) = 0. On the other hand, for r → σ − , g(r ) = 0 while c(r )  = 0. Since the function y(r ) is continuous through r = σ , we obtain from eqn. (1.2.91) that g(σ + ) = −c(σ − ). It then follows from eqn. (1.2.95) that g(σ ) = α0 + α1 + α2 =

1 + ϕ/2 (1 − ϕ)2

(1.2.105)

and hence the corresponding equation of state is obtained from eqn. (1.2.104): P 1 + 2ϕ + 3ϕ 2 . = nkB T (1 − ϕ)2

(1.2.106)

The two different equations of state given by (1.2.103) and (1.2.106) match up to O(ϕ 2 ), beyond which the difference grows. An accurate equation of state for a hard-sphere system has been developed by studying the Mayer cluster expansion for such a system. The latter is generally defined as ∞

 P Bi (T )n i−1 =1+ nkB T

(1.2.107)

i=2

in terms of virial coefficients Bi . These coefficients can be computed order by order using cluster diagrams. In fact, for the hard-sphere system B2 , B3 , and B4 are known analytically. However, the complexity of the calculation increases rapidly with the order. The numbers of diagrams entering the computation of Bi for i = 2, . . . , 6 are, respectively, 1, 3, 10, 56, and 468. Explicit calculations are therefore limited to low orders. For the hard-sphere

30

Statistical physics of liquids

system this calculation has been extended to B8 (van Rensburg, 1993; Vlasov et al., 2002) to obtain P = 1 + 4ϕ + 10ϕ 2 + 18.365ϕ 3 + 28.325ϕ 4 + 39.74ϕ 5 + 53.5ϕ 6 + 70.8ϕ 7 + · · · nkB T ∞  Cm ϕ m . (1.2.108) ≡1+ m=1

The above series has been summed by making the useful observation that the first seven terms on the RHS are well reproduced if the coefficient Cm is approximated with the formula Cm = 3m + m 2 .

(1.2.109)

The coefficients evaluated with this formula are C1 = 4 (4), C2 = 10 (10), C3 = 18 (18.365), C4 = 28 (28.325), C5 = 40 (39.74), C6 = 54 (53.5), and C7 = 70 (70.8), where the numbers in parentheses are the actual values of the coefficient in the series presented in eqn. (1.2.108). The formula (1.2.109) for Cm is therefore taken as a good approximation for the virial coefficient to all orders. We obtain the series in the form ∞

 P (m 2 + 3m)ϕ m . =1+ nkB T

(1.2.110)

m=1

The sums can be easily evaluated using the results ∞ 

∞ d  m ϕ ϕ = , dϕ (1 − ϕ)2 m=1 m=1   ∞ ∞  d 2 m ϕ(1 + ϕ) 2 m m ϕ = ϕ ϕ = . dϕ (1 − ϕ)3

m=1

mϕ m = ϕ

(1.2.111)

(1.2.112)

m=1

On evaluating the series on the RHS of eqn. (1.2.110) with the results (1.2.111) and (1.2.112), we obtain the equation of state as P 1 + ϕ + ϕ2 − ϕ3 . = nkB T (1 − ϕ)3

(1.2.113)

which is termed the Carnahan–Starling equation of state (Carnahan and Starling, 1969; Mansoori et al., 1971). Though it was obtained in an ad-hoc manner in the above simplified analysis, eqn. (1.2.113) presents a very accurate equation of state valid for a hard-sphere liquid up to high packing fractions close to freezing, having discrepancies from moleculardynamics simulation results of at most around 1%. The predictions from this equation of state lie in between those from the other two equations of state given by (1.2.103) and (1.2.106), respectively, following from the compressibility condition (1.2.70) and the virial equation (1.2.66). This is shown in Fig. 1.4. Several other equations of state have been devised (Mulero et al., 1999), though the form (1.2.113) is the simplest and most widely

1.2 Equilibrium properties

31

Fig. 1.4 The equation of state giving P/(nkB T ) − 1 vs. the packing fraction η for hard spheres according to the Percus–Yevick theory using the compressibility (PY-C) (eqn. (1.2.103)) and virial (PY-V) (eqn. (1.2.106)) routes. The result from the Carnahan–Starling equation is shown as a dashed line.

used. The Helmholtz free energy of the liquid is obtained by integrating the thermodynamic relation P = −(∂ F/∂ V )T ,    ϕ   d ϕ¯ β P dn P = N kB T . (1.2.114) F = − P dV = N ϕ¯ n n2 0 The corresponding expression for the excess free energy of the hard-sphere liquid (in addition to the ideal-gas or noninteracting part) given in eqn. (1.2.114) is obtained by integrating the Carnahan–Starling equation of state,

 ϕ βF d ϕ¯ β P ϕ(4 − 3ϕ) . (1.2.115) = −1 = N ϕ¯ n (1 − ϕ)2 0 On the other hand, on using either the PY or the HNC closure we will obtain two equations of state for the virial and compressibility conditions. This problem was addressed by proposing bridge functions to interpolate between the two schemes given in (1.2.91) and (1.2.92). The bridge function is adjusted to achieve full consistency between the two equations of state (Rogers and Young, 1984). Long-range interaction For interaction potentials that have a long-range attractive part in addition to the shortrange repulsive part, the closure schemes described above are not very useful. In this case the two-body interaction potential u(r ) is divided into a harshly repulsive short-range u 0 (r )

32

Statistical physics of liquids

part and a weak attractive part u 1 (r ) effective over long distances. In the conventional mean-spherical approximation (MSA) (Lebowitz and Penrose, 1966) this situation was initially treated by taking u 0 as a hard-sphere potential so that the resulting integral equation reduces to the PY equation with u 1 = 0. This approach was subsequently generalized to schemes in which u 0 is continuous though strongly repulsive (Chihara, 1973; Madden and Rice, 1980). This is referred to as “soft-core” MSA (SMSA). We consider here the scheme developed by Weeks, Chandler, and Andersen for systems with long-range attractive potentials (Chandler and Weeks, 1970; Weeks et al., 1971; Andersen et al., 1971). In the WCA theory the two-body interaction potential u(r) is divided into a short-range repulsive part u 0 (r) and a long-range attractive part u a (r), u(r) = u 0 (r) + u a (r).

(1.2.116)

A typical example is the spherically symmetric Lennard-Jones (LJ) interaction potential, for which such a division (Weeks et al., 1971) is made as follows. The LJ interaction potential u(r ) ≡ u LJ (r ) for two particles at a distance r is defined as  12  6  σ σ − ≡ w(r ), (1.2.117) u LJ (r ) = 40 r r where 0 is the depth of the potential at its minimum and σ is the scale associated with the potential. u LJ is written as a sum of two parts, a hard-core repulsive part u LJ R and an , respectively defined as attractive part u LJ P & w(r ) + 0 , for r < σ0 , LJ u 0 ≡ u R (r ) = (1.2.118) 0, for r > σ0 , and

& ua ≡

u LJ P (r )

=

−0 ,

for r < σ0 ,

w(r ),

for r > σ0 ,

(1.2.119)

where σ0 = 21/6 σ . For the homogeneous liquid state, its thermodynamic properties are obtained by treating the weak attraction u LJ P as a perturbation of a reference system having . an interaction potential u LJ R The thermodynamic property of the system with the given interaction potential u(r) is obtained as a sum of two contributions: first, that of a reference system having the purely repulsive interaction u 0 (12); and second, the contribution due to the attractive part of the interaction u a (12), which is obtained by treating the latter as a weak perturbation. In the first part, the properties of the reference system are usually known from independent models. The most commonly used reference system is the one with hard-sphere potential interaction potential u H d (r ), with diameter d characterizing the hard-sphere potential (for its definition see eqn. (1.2.93)). This choice of the hard-sphere system is motivated by the fact that there exist several models that accurately describe its thermodynamic properties

1.2 Equilibrium properties

33

for densities ranging from low to very high values. Some of these have already been discussed above. The diameter of the reference hard-sphere liquid system is chosen such that the corresponding thermodynamic free energy is equal to that of the original system with interaction potential u 0 (12). The dependence of the free energy F on the interaction potential u 0 (r) is treated in terms of the function e0 (r ) = exp(−βu 0 (r )), which changes sharply at the effective range of the harshly repulsive potential u 0 (r). The corresponding quantity for the hard-sphere potential ed (r) = exp(−βu H d (r )), on the other hand, changes discontinuously from 0 to 1 at r = d. The difference between the “e” functions corresponding to the reference and the hard-sphere systems defines the function e0 (r) = e0 (r) − ed (r).

(1.2.120)

The “blip function” e0 (r ) is nonzero only in a small range of O(d) (say) around r = d which is the diameter of the equivalent hard-sphere system. This is shown schematically in Fig. 1.5. Since the blip function is identically zero if u 0 (r) = u H d (r), the free energy of the reference system is expressed in a functional Taylor expansion in powers of e0 (r ) around the free energy of the equivalent hard-sphere system (EHS): * δ F ** e0 (r) δe0 (r) *e0 =ed *   * 1 δ2 F * dr dr + e0 (r)e0 (r ) + · · ·.  2 δe0 (r)δe0 (r ) *e0 =ed 

F[u 0 ] = F[u d ] +

dr

(1.2.121)

Fig. 1.5 The Boltzmann factors e0 (r ) for the potential u 0 (r ); ed (r ) for the hard-sphere potential uH d (r ); and the blip function e0 (r ) vs. r .

34

Statistical physics of liquids

We note first that the functional derivative in the second term on the RHS above is evaluated, noting that F = −kB T ln Z N , as  δF δ F δu 0 (r ) = dr δe0 (r) δu 0 (r ) δe0 (r)

 δu 0 (r ) 1 δZ N  = kB T dr − . (1.2.122) Z N δu 0 (r ) δe0 (r) Since −βu 0 (r) = ln e0 (r) the functional derivative −β

δu 0 (r ) = eβu 0 (r) δ(r − r ). δe0 (r)

(1.2.123)

If the potential-energy part U of the Hamiltonian involves only two-body interactions as in the definition (1.2.63), the integral I˜ (say) in the square bracket on the RHS of eqn. (1.2.122) is further simplified. In this case the derivative of U with respect to the two-body potential u 0 (r) is simply a sum of delta functions,     β δU −βU (r1 ,...,r N ) dr I˜ = . . . dr e 1 N δu 0 (r ) N !Z N 3N 0 N  e−βU (r1 ,..., r N )  β  = δ(r − ri j )dr1 . . . dr N . (1.2.124) 2 N !Z N 3N 0 i, j=1 For a system of identical particles all the different N (N − 1) choices for the pairs in the sum above give the same integral. For the translationally invariant system the integral dr1 dr2 ≡ V dr12 , giving   −βU (r1 ,...,r N ) N! β e dr1 dr2 δ(r − r12 ) dr3 . . . dr N I˜ = 2 (N − 2)! N !Z N 3N 0  −βU (r12 =r ; r3 ,...,r N ) βV N! e = dr3 . . . dr N 2 (N − 2)! N !Z N 3N 0 V = β n 2 g(r ). (1.2.125) 2 Hence, using eqns. (1.2.123) and (1.2.125) in eqn. (1.2.122), we obtain the result 4 3 2  δF   n V  = −kB T dr g(r ) eβu 0 (r ) δ(r − r ) δe0 (r) 2 1 = − N nkB T y(r). (1.2.126) 2 Using this relation, the expansion (1.2.122) for the free energy is obtained as  n dr yd (r )e0 (r) + · · ·. F[u 0 ] = F[u d ] − N kB T 2

(1.2.127)

The free energy of the reference system is made equal to that of the equivalent hard-sphere system by setting the first-order term in the RHS of eqn. (1.2.121) equal to zero. Thus the

1.2 Equilibrium properties

35

diameter d is determined from the condition  dr yd (r )e0 (r) = 0.

(1.2.128)

The diameter d is obtained by numerically solving the above equation using the hardsphere pair correlation functions and the y function. Since the function e0 (r) is nonzero only in a small region around r = d, the above condition is evaluated to leading order by replacing yd with its value at this point. The integral on the LHS of eqn. (1.2.128) is then obtained (using the definition (1.2.120)) as   ∞  ∞ dr{e0 (r) − ed (r)} = dr e−βu 0 (r) − dr. (1.2.129) 0

d

We have used the fact that ed (r ) equals 0 for r < d and 1 for r > d. From eqn. (1.2.129) one obtains for d the result (Barker and Henderson, 1967)  ∞ ( ) dr 1 − e−βu 0 (r) . (1.2.130) d= 0

In fact, with the condition (1.2.128) satisfied, the free energy of the reference system is obtained in terms of that of the equivalent hard-sphere system (Chandler and Weeks, 1970; Weeks et al., 1971, Andersen et al., 1971; Barker and Henderson, 1971) F[u 0 ] = F[u d ] + O( 4 ),

(1.2.131)

where  is the range around r = d over which the blip function is nonzero. Note that the value of d obtained from the solution of eqn. (1.2.128) depends both on the density n and on the temperature T , whereas the d obtained from the Barker–Henderson relation (1.2.130) depends only on the temperature. From eqns. (1.2.126) and (1.2.127) we obtain that to leading orders y(r) = yd (r), and hence the pair correlation function for the reference system is obtained as g0 (r ) = e−βu 0 (r ) yd (r ) + O( 2 ).

(1.2.132)

The pair correlation function g(r ) for the Lennard-Jones system is obtained in terms of that of the equivalent hard-sphere system of diameter d from eqn. (1.2.132). This is shown in Fig. 1.6. The corresponding g(r ) obtained from molecular-dynamics simulations as well as numerical solution of the Percus–Yevick equations for the LennardJones potential is also shown for comparison. In Section 2.4.1 we will discuss the application of the WCA approach to obtain the thermodynamic properties of the inhomogeneous solid state for a Lennard-Jones system. The contribution to the free energy from the attractive part of the potential is added as a mean-field term (see eqn. (2.4.9) in Chapter 2) in this theory. Further improvements for the thermodynamic properties (of the liquid) at higher densities are obtained by using the bridge functions mentioned above to interpolate between the SMSA and the HNC equations (Zerah and Hansen, 1986; Due and Haymet, 1995; Due and Henderson, 1996). These methods lead to values of the static structure factor for the liquid that are in agreement with simulations at high densities.

36

Statistical physics of liquids

Fig. 1.6 The radial distribution function g(r ) for the Lennard-Jones liquid at ρ ∗ = 0.85 and T ∗ = 0.88. MD simulations by Verlet (1968) (open circles); the Percus–Yevick solution for the pair function for the Lennard-Jones potential by Mandel et al. (1970) (dashed line); and the theory c American Institute of Physics. of Weeks et al. (1971) described in the text (solid line). 

1.3 Time correlation functions The time correlation of the microscopic densities, averaged over the equilibrium ensemble, is an important ingredient in the study of the dynamics of a fluid. Essentially all experiments probing the dynamic properties of the fluid are also analyzed in terms of time correlation functions. Let ψˆ a be a dynamic variable dependent on the phase-space coordinates, denoted with the symbol {r N (t), p N (t)} ≡  N (t), defined as ψˆ a (t) = ψa [ N (t)].

(1.3.1)

For clarity we drop any possible spatial dependence of ψˆ a . The hat on the variable is used to denote that it is dependent on the phase-space variables. The time correlation function is defined as the correlation function of the fluctuation of ψa (t) and ψb (t  ) (t > t  ) around the corresponding equilibrium average, Cab (t, t  ) = δ ψˆ a (t)δ ψˆ b (t  ),

(1.3.2)

where the angular brackets denote the average in the equilibrium ensemble in the following way:   ˆ ˆ δ ψa (t)δ ψb (t ) = d N (t  )δ ψˆ a (t)δ ψˆ b (t  ) f eqN [ N (t  )]  = d N (t  )exp[iL(t − t  )]δ ψˆ a (t  )δ ψˆ b (t  ) f eqN [ N (t  )]. (1.3.3) The equilibrium distribution f eqN is independent of time and hence Cab is dependent only on the difference of the two times Cab (t, t  ) = Cab (t − t  ). Of particular interest for the

1.3 Time correlation functions

37

discussion of the dynamics of liquids is the equilibrium averaged autocorrelation function, which is defined as + , (1.3.4) Caa (t) = δ ψˆ a (t)δ ψˆ a∗ (0) . Since Caa (t) = Caa (−t) it follows from the above definition that Caa (t) is real for all times. Using Schwarz’s inequality (Dennery and Krzwicki, 1967) for the autocorrelation function, + , (1.3.5) |Caa (t)| ≤ ψˆ a ψˆ a∗ = Caa (0), i.e., the magnitude of the function is bound by its initial value. Indeed, in the long-time limit, t → ∞, the dynamic variables become uncorrelated and we have lim

(t−t  )→∞

|Caa (t − t  )| = δ ψˆ a (t)δ ψˆ a (t  ) = 0.

(1.3.6)

An important property of the autocorrelation function is that its time derivative at t = 0 vanishes. In order to prove this, we note that, since ψa (t + s)ψb (s) is independent of s, d ψa (t + s)ψb (s) = 0, ds ψ˙ a (t + s)ψb (s) + ψa (t + s)ψ˙ b (s) = 0, where the dot implies a derivative with respect to s. On setting s = 0 we obtain the result ψ˙ a (t)ψb (0) = −ψa (t)ψ˙ b (0). For the autocorrelation at t = 0 we have the result * * d = C˙ aa (0) = 0. ψa (t)ψa (0)** dt

(1.3.7)

(1.3.8)

t=0

For our discussions in this book it will be particularly useful to define the two-sided Fourier transform  +∞ Cab (t)eiωt dt (1.3.9) C˜ ab (ω) = −∞

and the one-sided Laplace transform  +∞ Cab (t)ei zt dt, Cab (z) = −i

Im(z) > 0.

0

The two transforms are related in the following way:  +∞ Cab (t)ei zt dt Cab (z) = −i 0

 +∞  +∞ d ω¯ i(z−ω)t ¯ dt e dt C˜ ab (ω) ¯ −i = −∞ 2π 0    +∞ d ω¯ C˜ ab (ω) ¯ = . 2π z − ω ¯ −∞

(1.3.10)

38

Statistical physics of liquids

The above relation is further analyzed for the autocorrelation function using the standard identity   1 1 =P ∓ iπ δ(ω − ω), ¯ (1.3.11) lim →0 ω ± i − ω ¯ ω − ω¯ where P indicates the principal value. In this case, i.e., a ≡ b and the Fourier transform C˜ aa (ω) is real, we obtain the following useful relations between the Fourier and Laplace transforms of the correlation function:  C˜ aa (ω) = −2Caa (ω + i), 

 (ω¯ + i) d ω¯ Caa  Caa (ω + i) = − , π ω − ω¯

(1.3.12) (1.3.13)

for  → 0. 1.3.1 The density correlation function For the fluid generally we define for a microscopic variable ψˆ a (r, t), corresponding to the microscopic property aα (t) for the αth particle at position rα (t), ψˆ a (r, t) =

N 

aα δ(r − rα (t)).

(1.3.14)

α=1

In Table 1.2 we list examples of a few types of experiments with the corresponding dynamic variable aα whose time correlation is measured using the respective method. Our discussion here will primarily involve the microscopic variables of mass density and momentum density, defined, respectively, as  mδ(r − rα (t)), (1.3.15) ρ(r, ˆ t) = α

gˆ (r, t) =



pα δ(r − rα (t)).

(1.3.16)

α

We put a hat on the slow variable to indicate the dependence on the phase-space coordinates of this variable. Note that the mass density ρ(r, ˆ t) is similar to the single-particle number density n(r, ˆ t). For the one-component system in fact we have ρˆ = m n. ˆ We show in Appendix A5.1 that the mass and momentum densities are related by the continuity equation ∂ ρˆ + ∇ · gˆ = 0. ∂t

(1.3.17)

The most important correlation function for our purpose is that of the density fluctuations, denoted by δ ρ(r, ˆ t) = ρ(r, ˆ t) − ρ0 ,

(1.3.18)

1.3 Time correlation functions

39

Table 1.2 Examples of correlation functions of a few dynamic variables and the corresponding experiments which measure them.  implies a sum over the particle index α = 1, . . . , N while   implies a sum over both particle indices α and β, with α  = β. Dynamic variable

Correlation function

v: velocity of a tagged particle

v(t) · v(0)

5 + ik · (r (t)−r (0)) , α α e 5 + ik · (r (t)−r (0)) , α β e (1/N )

ij

Polarizability tensor να ii i να

Rotational diffusion

(1/N )

rα (t): position of the αth particle at t

5

Translational diffusion

i (t) j (0)

: angular velocity about the center of mass n(r, t): density

Trace ναT =

Experiment

(1/N ) (1/N )

Neutron scattering

, 5+ T να (0)ναT (t)eik · (rα (t)−rα (0))

, 5 + x y xy να (0)να (t)eik · (rα (t)−rβ (0))

Polarized Brillouin scattering Depolarized Brillouin scattering

with ρ0 = ρ ˆ being the average density. The average densities ρ0 = mn 0 are constants in the isotropic liquid state. The density–density correlation function involving fluctuations at two different space and time points is defined as ˆ t)δ n(r ˆ  , 0). G nn (|r − r |, t) = V δ n(r,

(1.3.19)

It is often more convenient to describe the density–density correlation function of the Fourier component of the density variable, n(k, ˆ t) =

1 V



dr eik · r n(r, ˆ t) = V −1



eik · rα (t) .

(1.3.20)

α

The Fourier-transformed correlation function is then defined as F(k, t) =

1 δ n(k, ˆ t)δ n(−k, ˆ 0), N

(1.3.21)

where V is the volume. Using the above definition of the density fluctuations, we obtain that the correlation function F(k, t) can be written in two separate parts involving

40

Statistical physics of liquids

incoherent and coherent parts as follows: Fd (k, t) =

N  1  + ik · (rα (t)−rβ (0)) , e , N

(1.3.22)

α,β=1

N 1  ik · (rα (t)−rα (0)) e , Fs (k, t) = N

(1.3.23)

α=1

where the prime indicates that α  = β in the sum. The first part is often called the distinct part of density correlation functions while the second part is referred to as the self part of the correlation function or the van Hove correlation function. The spectral quantity for the density–density correlation function S(k, ω), S(k, ω) =

1 2π



+∞

−∞

F(k, t)eiωt dt,

(1.3.24)

is called the dynamic structure factor and is measurable in inelastic light-, X-ray-, and neutron-scattering experiments (Mezei, 1991). The correlation function F(k, t = 0) for density fluctuations at the same time refers to the structural or thermodynamic property of the liquid. The static correlation function F(k, t = 0) = n 0 S(k), where S(k) is the static structure factor defined in eqn. (1.2.79). In the theories considered in the subsequent chapters in this book, the implicit effect of the interaction potential for the fluid particles on the dynamic properties will always be considered in terms of the corresponding static structure factor and higher-order static correlations. Another important correlation function for our discussion is the current–current correlation function. The current gˆ (x, t) is a vector field and hence, for an isotropic system, the corresponding correlation function is divided into a longitudinal part and a transverse part. The spatial Fourier transform of the correlation function G i j (q, t) between gi (x, t) and gi (x , 0) for the isotropic fluid is expressed as G i j (k, t) = kˆi kˆ j G L (k, t) + (δi j − kˆi kˆ j )G T (k, t),

(1.3.25)

with subscripts L and T referring to the longitudinal and transverse parts, respectively. If we choose both indices i and j of coordinates to be along the direction of the vector k then kˆi = kˆ j = 1 and G i j = G L . If both i and j are in a direction transverse to k then G i j = G T . On the other hand, if i( j) is along k and j (i) transverse to k, then G i j = 0. From the continuity equation (1.3.17) and the definitions (1.3.19) and (1.3.25) it follows directly that ω2 G nn (k, ω) = k 2 G L (k, ω),

(1.3.26)

linking the two-sided Fourier transforms of the density and (longitudinal) current autocorrelation functions.

1.3 Time correlation functions

41

1.3.2 The self-correlation function The dynamics of a single particle moving in a surrounding fluid can be described in terms of the density of the single particle nˆ s (r, t) defined as the tagged particle (α) density, nˆ s (r, t) = δ(r − rα (t)).

(1.3.27)

The average nˆ s  of the tagged particle density is obtained as n s (r, t) =

1 V

(1.3.28)

and goes to zero in the thermodynamic limit V → ∞. The correlation of the taggedparticle density at two different times defines the van Hove correlation function, G s (|r − r |, t) = V δ nˆ s (r, t)δ nˆ s (r , 0) = V δ(r − rα (t))δ(r − rα (0)) − V −1 .

(1.3.29)

The tagged-particle density nˆ s (r, t) is δ(r − rα (t)), with rα (t) denoting the position of the tagged particle at time t. In an isotropic fluid G s (r, t) is a function of r only. The Fourier transform of G s is the self-part of the density correlation function defined above in eqn. (1.3.23). From eqn. (1.3.29) G s (r, t) can be interpreted as the probability per unit volume of finding a particle at a distance r at a given time t, given that it was at the origin r = 0 at t = 0. Note that, using eqn. (1.3.28), it follows from the definition of G s (r, t) that  G s (r, 0) = δ(r )

and

dr G s (r, t) = 1.

(1.3.30)

Equation (1.3.30) is a statement of the conservation of probability. In an isotropic fluid G s is a function of r only and 4πr 2 G s (r, t) represents the probability of finding a particle at r at time t given that it was at the origin at time t = 0. In Fig. 1.7 we show schematically how this probability distribution spreads out with the passage of time. The frequency transform Ss (k, ω) of G s is defined, with  Fs (k, t) = Ss (k, ω) =

+∞

−∞  +∞ −∞

dr eik · r G s (r, t),

(1.3.31)

dt eiωt Fs (k, t).

(1.3.32)

The tagged-particle correlation is obtained directly from the measurement of Ss (k, ω) in neutron-scattering experiments (Fig. 1.8). The coherent component of the scattered intensity is proportional to the self part of the correlation function and from it one directly obtains the information on the single-particle dynamics.

42

Statistical physics of liquids

Fig. 1.7 A schematic representation of how G s (r, t) spreads out with time.

¯ Fig. 1.8 Dependence of the correlation function S(Q, E) for coherent (solid) and incoherent (dashed) neutron-scattering analysis of argon at the frequency E (in meV) and the wave-number c American Physical Society. Q (in the Å−1 ) plane. From Sköld et al. (1972). 

The free-particle limit It is straightforward to demonstrate that the function G s (r, t) and hence its Fourier transform Fs (k, t) have a Gaussian distribution in the t → 0 limit. In the very-short-time regime, the particle moves as a free one, and G s (r, t) varies spatially as a narrow Gaussian distribution function. In the free-particle limit it is straightforward to use the above definition in

1.3 Time correlation functions

43

terms of probability to compute

  p dp φMB ( p)δ r − t m     1 1 r 2 = , 3/2 exp − 2 v t 0 2π v 2 t 2 

G s (r, t) =

(1.3.33)

0

where φMB is the Maxwellian momentum distribution. For the free particle motion the particle with momentum p moves a distance r = pt/m. The corresponding Fourier transform Fs (k, t) and the frequency transform Ss (k, ω) are obtained as the following Gaussian functions:

1 Fs (k, t) = exp − (kv0 t)2 , 2  √    2π 1 ω 2 Ss (k, ω) = exp − . (1.3.34) kv0 2 kv0 The behavior predicted by (1.3.34) is seen in Fig. 1.8. Therefore in the short-time or freeparticle limit the correlation functions remain Gaussian. The time correlation of the velocities of a tagged particle at two different times plays an important role in the description of the single-particle dynamics in a fluid. This is defined in three dimensions as 1 (1.3.35) ϕv (t) = v(t) · v(0). 3 For short times once again we can expand v(t) at t = 0 and compute the above correlation function. The first-order term goes to zero on using eqn. (1.3.7), and the leading-order contribution reads

t 2 20 2 + ··· , (1.3.36) ϕv (t) = v0 1 − 2 where v02 = kB T /m is the thermal velocity of a particle, 1 v(0) · v(0). 3 The quantity 0 has the dimension of frequency and is obtained as v02 =

20 =

1 1 {m v˙ } · {m v˙ } = F · F, 3mkB T 3mkB T

(1.3.37)

(1.3.38)

where F is the force exerted on the particle α by its neighbors. For a system in which the interactions between the particles are described by the pairwise additive potential v(12), we show in Appendix A1.2 by evaluating the force–force correlation function that 0 can be expressed in terms of the pair correlation function g(r ) by the following relation:  n0 20 = (1.3.39) ∇ 2 v(r )g(r )dr. 3m

44

Statistical physics of liquids

0 is termed the Einstein frequency, since for short times it represents the frequency of vibration of the tagged particle undergoing small oscillations in the potential well produced by its neighbors. Note that the above analysis holds only for continuous potentials. For hard-sphere systems such an analysis does not hold. In this case the linear-order term contributes, and it has been shown (Lebowitz et al., 1969) that the autocorrelation function has the exponentially decaying form ϕv (t) = v02 [1 − 0 t + · · ·] ≈ e−0 t ,

(1.3.40)

where the frequency 0 = 2/(3ta ), with ta being the Boltzmann collision time, given by 4πn 0 σ 2 . ta−1 = √ βmπ

(1.3.41)

For higher densities the structure of ta is replaced by the corresponding Enskog time tE = ta /g(σ ). The decay of the tagged-particle correlation for a dense hard-sphere system is therefore exponential and is given by

2t 2 . (1.3.42) ϕv (t) = v0 exp − 3tE The exponential decay of ϕv (t) is a result of the completely random nature of uncorrelated binary collisions of the particles. Cooperative effects coming from longer length scales cause nonexponential or power-law decays in the correlation functions. We discuss such contributions to correlation functions in subsequent chapters. Long-time dynamics From the general definition (1.3.23) of the tagged-particle correlation (since all the particles are identical) we obtain + , 1 2 (1.3.43) Fs (k, t) = eik · (rα (t)−rα (0)) = eikd(t) , where d(t) = kˆ · (rα (t) − rα (0)), with kˆ being the unit vector in the direction of the wave vector. The terms with odd powers of k vanish in the above expansion, meaning that the equilibrium correlation function Fs (k, t) for the isotropic fluid is a function of |k| only, Fs (k, t) = 1 −

k2 2 k4 d (t) + d 4 (t) + · · ·. 2! 4!

(1.3.44)

In order to organize the Gaussian and non-Gaussian parts of Fs (k, t), we write the above expansion in terms of time-dependent functions n (t), for n = 1, 2, . . . (Rahman et al., 1962),  (1.3.45) Fs (k, t) = exp −k 2 1 (t) + k 4 2 (t) − k 6 3 (t) + · · · .

1.3 Time correlation functions

45

On comparing coefficients of the different powers of k in eqns. (1.3.44) and (1.3.45), we obtain for the different n (t) 1 2 d (t), 2! 1  4 2 d (t) − 3d 2 (t) , 2 (t) = 4! 1  6 3 d (t) − 15d 4 (t)d 2 (t) + 30d 2 (t) . 3 (t) = 6! 1 (t) =

(1.3.46) (1.3.47) (1.3.48)

The Gaussian part of Fs (k, t) is determined by 1 (t), while the n for n ≥ 2 represent the non-Gaussian part. We demonstrate in the following that the function 1 (t) can be expressed in terms of the two-point current correlation function ϕv (t). In order to see the link between the tagged-particle correlations and the velocity correlations, we express the displacement of the tagged particle as an integral over its velocity. Thus, if we denote by u(t) the component of the velocity v of the tagged particle α in the direction of k, d(t) is obtained in terms of u as  t d(t) = kˆ · (rα (t) − rα (0)) = d t¯ u(t). (1.3.49) 0

Since Fs (k, t) = exp(ikd(t)), we obtain the exponential on the RHS as a formal solution of the equation ∂ ikd(t) = iku(t)eikd(t) e ∂t

(1.3.50)

in the following form by iteration:  t  dt  u(t  )eikd(t ) eikd(t) = 1 + ik 0  t  t  = 1 + ik dt1 u(t1 ) + (ik)2 dt1 u(t1 ) 0

t1

dt2 u(t2 ) + · · ·.

(1.3.51)

0

0

On taking an average of the above equation, since in the isotropic fluid in equilibrium on average odd powers of u vanish, we obtain the tagged-particle correlation in terms of the expansion Fs (k, t) =

∞  n=0

(−k 2 )

n





t

dt1 0

0

t

 dt2 . . .

t

dt2n u(t1 )u(t2 ) . . . u(t2n ). (1.3.52)

0

Now straightforward comparison of powers of k 2 with the definition (1.3.45) shows that  t1  t dt1 dt2 u(t1 )u(t2 ), (1.3.53) 1 (t) = 0 0  t1  t2  t 1 dt1 dt2 dt3 u(t1 )u(t2 )u(t3 )u(t4 ) − 12 (t), (1.3.54) 2 (t) = 2 0 0 0

46

Statistical physics of liquids

and so on. Using the property of time translational invariance of the equilibrium correlation functions of u, the integral expression for 1 (t) is expressed in terms of the two-point velocity correlation functions ϕv (t1 − t2 ) as follows:  t1  t1  t  t 1 (t) = dt1 dt2 u(t1 )u(t2 ) = dt1 ds ϕv (s) 0 0 0 0  t  t  t  t dt1 ds ϕv (s) − dt1 ds ϕv (s) = 0

0



t

=t 0

ds ϕv (s) −

 0

0

t

t1

sϕv (s)ds,

(1.3.55)

where in reaching the final result we integrated the second term on the RHS by parts. Therefore, if the non-Gaussian effects are ignored, the tagged-particle correlation is entirely determined by ϕv (t). For an isotropic liquid d 2 (t) = 13 r 2 (t) in three dimensions, giving the result  t r 2 (t) = 6t ϕv (s)ds, (1.3.56) 0

where we have ignored the contribution from the second term on the RHS of eqn. (1.3.55). The latter is a constant when the velocity correlation function decays in an exponential manner. However, it has been observed from the computer molecular-dynamic simulations (Alder and Wainwright, 1967, 1970) of fluids that the velocity autocorrelation ϕv (t) has a long-time tail decaying as t −3/2 in three dimensions. This long-time tail in ϕv (s) makes the second integral grow as t 1/2 . This makes the long-time (t → ∞), long-distance (k → 0) behavior of Fs (k, t) very different from the simple exponential prediction. However, the long-time decay of the tagged-particle correlation is still well behaved since we consider the limit k → 0 and t → ∞ such that k 2 t remains finite. The factor k 2 1 (t) on the RHS of eqn. (1.3.45) is finite in this limit. Therefore, for a system in which the dynamics is Gaussian, all the n (t) for n ≥ 2 in the exponent on the RHS of (1.3.45) vanish and the tagged-particle correlation is obtained in the form

 t  2 2 Fs (k, t) = exp −k t1 (t) = exp −k t ϕv (s)ds . (1.3.57) 0

Hence in the Gaussian limit Fs (k, t) is determined in terms of the velocity correlation function ϕv (t) only. Non-Gaussian effects The non-Gaussian effect in the single-particle dynamics is related to higher (than-two-point) correlation functions in the fluid and was first studied for moleculardynamics simulations (Rahman, 1964) of argon. The expansion (1.3.45) is reorganized in terms of parameters αn (t) for n = 1, 2, 3, . . . for the isotropic system as (Nijboer and Rahman, 1966)   3k (t) 2k (t) −k (t) (1.3.58) α2 (t) − {α3 (t) − 3α2 (t)} . . . , 1+ Fs (k, t) = e 2! 3!

1.3 Time correlation functions

47

Fig. 1.9 Non-Gaussian parameters αn (t) for n = 2, 3, and 4 vs. time t (in units of 10−12 s) obtained from results of molecular-dynamic simulations of argon. From Rahman (1964).

where we have defined k (t) = k 2 1 (t). The αn are obtained as r 2n (t) n − 1, cn r 2 (t) 1 · 3 · 5 · · · · · (2n + 1) cn = . 3n

αn (t) =

(1.3.59) (1.3.60)

In particular, the above definition gives α1 (t) = 1 and the first non-Gaussian parameter α2 (t) as 1 2 3 r 4 (t) α2 (t) = 1 (1.3.61) 2 − 1. 5 r 2 (t) 2 In Fig. 1.9 the so-called non-Gaussian parameters αn (t) for n = 2, 3, and 4 are shown, with all quantities peaking at an intermediate time. The non-Gaussian parts are computed from molecular-dynamic simulation of argon (Rahman, 1964).

1.3.3 The linear response function An important application of the equilibrium correlation functions lies in the formulation of a linear response to an external perturbation. Let us consider the case in which

48

Statistical physics of liquids

a time-dependent external field h a (r, t) couples to the dynamic variable a(r). ˆ The Hamiltonian of the system in the presence of the external field is obtained as a sum of two parts, H = H0 + H  (t), where H0 is the unperturbed system and H  is the perturbation,  H  (t) = − a(r)h ˆ a (r, t)dr.

(1.3.62)

(1.3.63)

ˆ The fluctuation in the average value of a dynamic variable b(x) is obtained by taking a (N ) phase-space average with respect to the distribution function f (t). The time evolution of the phase-space probability density f (N ) (t) for the perturbed density is determined from the Liouville equation (1.1.22). In order to keep the notation simple, we drop the dependence of f (N ) on the phase-space variables. We define the fluctuation from the equilibrium  f (N ) as (N ) f (N ) (t) = f eq +  f (N ) (t),

(1.3.64)

(N ) where f eq is the equilibrium distribution which satisfies (N ) # $ ∂ f eq (N ) (N ) = H0 , f eq = −iL0 f eq = 0. (1.3.65) ∂t The symbol { } here denotes the Poisson brackets between the corresponding dynamic variables and L0 is the Liouville operator with the unperturbed Hamiltonian H0 . From the Liouville equation the time evolution of  f (N ) is obtained as

∂ f (N ) = −iL f (N ) = {H, f (N ) } ∂t = {H0 , f (N ) } + {H  , f (N ) }  # $ (N ) (N ) = −iL0  f − dr a(r ˆ  ), f eq h a (r , t).

(1.3.66)

(N ) The solution of the above equation with the boundary condition f (N ) (−∞) = f eq is   t  # $  (N )  f (N ) (t) = − dt  dr e−i(t−t )L0 a(r (1.3.67) ˆ  ), f eq h a (r , t  ). −∞

ˆ The fluctuation in a dynamic variable b(r) at time t is obtained from the corresponding (N ) phase-space average with  f (, t) by integrating over all the 6N phase-space variables, collectively denoted as ,  ˆ t) = d b(r) ˆ b(r, f (N ) (, t)    t  # $  (N )   ˆ dt  dr d e−i(t−t )L0 a(r ˆ  ), f eq b(r)h =− a (r , t )  ≡

−∞ +∞

−∞

dt



dr Rba (r − r , t − t  )h a (r , t  ),

(1.3.68)

1.3 Time correlation functions

49

where Rba (r − r , t − t  ) is the linear response function in the liquid with time and spatial translational invariance. From the above expression the response function is obtained as  # $ (N ) ˆ Rba (r − r , t − t  ) = −(t − t  ) d a(r ˆ  , t  ), f eq b(r, t), (1.3.69) where (t) is the Heaviside step function, which is equal to 1 and 0 for x < 0 and x > 0, respectively. An important result of equilibrium statistical mechanics is a relation between the linearresponse function and correlation functions that is termed the fluctuation–dissipation theorem (FDT). It follows directly then, using the definition of Poisson brackets, that the response function can be expressed in a more symmetric form in terms of the equilibrium average of the Poisson bracket, ˆ t), a(r ˆ  , t  )}0 , Rba (r − r , t − t  ) = −(t − t  ){b(r,

(1.3.70)

where the subscript 0 on the angular bracket implies averaging with respect to the unper(N ) turbed Hamiltonian H0 . Using the canonical distribution function f eq ∼ exp(−β H0 ), it can be shown directly from (1.3.69) that $ # ˆ , t ) (N ) (N ) ∂ a(r . (1.3.71) = −β f eq a(r ˆ  , t  ), f eq ∂t  Using the above result in eqn. (1.3.69) gives the following relation between the correlation function and the corresponding response function: Rba (t − t  ) = (t − t  )

1 ∂Cba , kB T ∂t

(1.3.72)

where Cba is the correlation function averaged over the equilibrium distribution. In equilibrium we assume that time translational invariance holds for the correlation function, i.e., Cba (t, t  ) ≡ Cba (t − t  ). The relation (1.3.72) is the fluctuation–dissipation theorem (FDT). We have dropped the dependence on spatial coordinates in eqn. (1.3.72) for the sake of simplicity. The Laplace transform (Abramowitz and Stegun, 1965) of the R(z) response function is defined as  ∞ dt ei zt Rba (t), Im(z) > 0. (1.3.73) Rba (z) = −i 0

Since aˆ and bˆ are both real functions, it follows directly that for the real and imaginary parts of the response function R(z), denoted by R  and R  , respectively, one has the following symmetry properties:   (ω) = Rba (−ω), Rba

(1.3.74)

 (ω) Rba

(1.3.75)

=

 −Rba (−ω).

Equations (1.3.72)–(1.3.74) hold both for real space and for the corresponding Fouriertransformed quantities, and it is not explicitly mentioned which for simplicity.

50

Statistical physics of liquids

1.4 Brownian motion In the earlier sections our discussion of the dynamic correlation functions involved dealing with the phase-space variables in the 6N -dimensional space at the microscopic level. The kinetic theory of many-particle systems evolving under Liouville dynamics provides the basic tool for calculating the correlation functions in this case. Computer moleculardynamics simulations dealing with the microscopic coordinates of a small number of particles also constitute another useful tool for theoretically obtaining the correlation functions. An alternative approach to these methods is to study the dynamics in terms of a chosen set of dynamic variables. The basic criterion involved in this method is separating the dynamics into a slow part and a fast part. The choice of the set of variables is motivated in order to exploit the widely varying time scales in the dynamics. The correlations of the slow modes are then computed by treating the rest of the degrees of freedom as noise, this noise being completely random on the time scale of the slow modes. A classic example of the treatment of dynamics with stochastic noise is the Brownian motion which we will introduce in the following. The theory of Brownian motion discussed here was developed by Einstein (1905), Smoluchowski (1906), Planck (1917), and others (Chandrasekhar, 1943). It constitutes an approach to the dynamics of a many-particle system that is an alternative to the Liouvillian or Newtonian dynamics. In formulating the dynamics of a fluid we will largely follow a generalization of such a scheme in the subsequent chapters of this book.

1.4.1 The Langevin equation Small grains of pollen immersed in a fluid undergo a kind of perpetual irregular motion. This is termed Brownian motion after the name of the botanist Robert Brown, who first observed this phenomenon in 1827. The irregular motion is due to the incessant random collisions of the molecules of the surrounding fluid with the pollen grain or the so-called Brownian particle. The mass of this particle is much higher than that of a colliding particle of the surrounding fluid. The equation of motion of the Brownian particle is obtained in terms of a Langevin equation involving a stochastic part distinct from the completely deterministic Newtonian dynamics. This formulation of the problem is based on the assumption that the motion of the Brownian particle is controlled by forces varying on very different time scales and hence the equation of motion consists of a slow part and a fast part. The latter is termed noise, and calculation of the correlations of dynamic variables now implies treating different realizations of the noise. The equation of motion of a heavy Brownian particle of mass m moving in a sea of particles is written in the following form: dvi = −ζ vi (t) + f i (t), dt

(1.4.1)

1.4 Brownian motion

51

where we have chosen the mass m of the Brownian particle as unity to keep the algebra simple. Here i denotes the corresponding Cartesian component. The first term on the RHS represents a frictional drag term on the Brownian particle proportional to the velocity v. ζ is a controlling kinetic coefficient, which in this case is taken to be the same for all components i for simplicity. The second term f i (t), on the other hand, denotes the effect of all the random kicks the big particle gets from the smaller particles in the surrounding fluid. The above equation is the simplest form of the Langevin equation written in one dimension. To solve eqn. (1.4.1) we multiply it by the factor exp(ζ t), to write d [vi (t)eζ t ] = eζ t f i (t). dt It is straightforward to integrate from t0 to time t to obtain  t dτ eζ τ f i (τ ), vi (t)eζ t = vi0 eζ t0 +

(1.4.2)

(1.4.3)

t0

where vi (t0 ) = vi0 . The ith component of the velocity of the heavy particle is therefore obtained as  t 0 −ζ (t−t0 ) + dτ f i (τ )e−ζ (t−τ ) . (1.4.4) vi (t) = vi e t0

The first term on the RHS of eqn. (1.4.4) gives the contribution from the initial value of v0 decaying exponentially, while the second term gives the integrated effect from the force f i (τ ) acting over this time interval. To make further progress, we need to prescribe the nature of the random force f i (t). Since it keeps pushing the big particle at random from all directions, it is plausible to assume that, for all times t,  f i (t) = 0.

(1.4.5)

Using this result, we obtain that the average velocity at time t simply follows from the exponential decay from the initial value, since the average noise is zero: + , vi (t) = vi0 e−ζ (t−t0 ) .

(1.4.6)

In thermal equilibrium vi0  = 0 and hence vi (t) remains zero at all times. To compute the correlation of the velocity at different times, we need to know how the noise values f i (t) at different times are correlated with each other. If the system is in equilibrium, the correlation between the random forces at times t and t  depends only on the difference of the two times, i.e., time translational invariance holds. For the isotropic system we define this correlation in terms of the time-dependent function D¯ 0 (t − t  ),  f i (t) f j (t  ) = δi j D¯ 0 (t − t  ).

(1.4.7)

In Appendix A1.3 we consider some general properties of the function D¯ 0 (t), and show ¯ that it is symmetric around t = 0. It remains at all times bounded between limits + D(0) ¯ and − D(0), decaying to zero in the long-time limit. As pointed out above, in the Langevin

52

Statistical physics of liquids

description of the dynamics the velocity v of the massive Brownian particle is treated as a slow variable and the force f (t) changes randomly over time scales much shorter than the time scale of variation of v(t). As a simplification we will consider the extreme case in which the correlation of the noise is defined in terms of a delta function,  f i (t) f j (t  ) = δi j D¯ 0 δ(t − t  ),

(1.4.8)

giving what is termed white noise. Furthermore, since the physical process of driving the Brownian particle with the noise must follow causality, i.e., f i (t) has no influence on vi (t  ), where t > t  , we have  f i (t)v j (t  ) = 0.

(1.4.9)

The correlation of the velocities at two different times t and t  is obtained from the general solution given by (1.4.4), + ,  (1.4.10) Ci j (t, t  ) = vi (t)v j (t  ) = vi0 v 0j e−ζ (t+t −2t0 ) + Ii j , where the integral Ii j is obtained from the relation  t  t   dτ dτ  e−ζ (t−τ ) e−ζ (t −τ )  f i (τ ) f j (τ  ). Ii j =

(1.4.11)

t0

t0

On the RHS of eqn. (1.4.10) a cross term is set equal to zero using the causality relation (1.4.9). Now, with the correlation of the noise as defined above in the diagonal form eqn. (1.4.8), the velocity correlation function at two different times is obtained. For an isotropic system both terms on the RHS of eqn. (1.4.11) are diagonal, reducing to Ci j (t, t  ) ≡ δi j C(t, t  ) and Ii j = δi j IB . We obtain for C(t, t  ) the result 

C(t, t  ) = e−ζ (t+t −2t0 ) C(t0 , t0 ) + IB , where we define the integral IB as  t  ¯ IB = D 0 dτ t0

t





dτ  e−ζ (t−τ ) e−ζ (t −τ ) δ(τ − τ  ).

(1.4.12)

(1.4.13)

t0

The last integral is evaluated in Appendix A1.3 to obtain the following result for the velocity correlation function:

D¯ 0 −ζ |t−t  | D¯ 0  C(t, t  ) = + e−ζ (t+t −2t0 ) C(t0 , t0 ) − e . (1.4.14) 2ζ 2ζ The equal-time correlation C(t, t) is obtained by setting t = t  :

D¯ 0 D¯ 0 C(t, t) = + e−2ζ (t−t0 ) C(t0 , t0 ) − . 2ζ 2ζ

(1.4.15)

The above relation implies that if the system reaches equilibrium it continues to remain at equilibrium for all times t unless disturbed. The equal-time correlation C(t, t) should then

1.4 Brownian motion

53

remain fixed for all t as given by eqn. (1.4.15) at C(t0 , t0 ) =

D¯ 0 . 2ζ

(1.4.16)

If the Brownian particle is in equilibrium in the liquid at temperature T at time t0 , then the initial distribution for the velocities is Maxwell–Boltzmann, giving the equal-time correlation as C(t0 , t0 ) = kB T,

(1.4.17)

where we have taken the mass m for the particle to be unity as above. The correlation C(t, t) is then obtained from eqn. (1.4.14) as

D¯ 0 D¯ 0 −2ζ (t−t0 ) +e . (1.4.18) kB T − C(t, t) = 2ζ 2ζ For the noise correlation (1.4.7), the corresponding velocity correlation function C(t, t  ) for the Brownian particle follows from eqn. (1.4.14). In the equilibrium fluid at temperature T we obtain the correlation function C(t, t  ) in the time translational invariant form, 

C(t, t  ) ≡ C(t − t  ) = kB T e−ζ |t−t | .

(1.4.19)

Finally, according to the condition (1.4.17), we have, for the Brownian particle moving in the fluid in equilibrium at temperature T , D¯ 0 = 2ζ C(t0 , t0 ) = 2ζ kB T.

(1.4.20)

The noise correlation as defined in eqn. (1.4.7) is therefore given by  f i (t) f j (t  ) = 2kB T ζ δ(t − t  )δi j .

(1.4.21)

1.4.2 The Stokes–Einstein relation Using the result for the velocity correlation function for the Brownian particles, straightforward integration gives the displacement xi (t) for a particle after time t. Since vi (t) = d xi /dt, we can integrate the above result to compute the root-mean-square displacement of the particle. We rewrite eqn. (1.4.19) as . d d  (1.4.22) xi (t) xi (t  ) = kB T e−ζ |t−t | . dt dt Note that in the above expression and in what follows below the repeated index i is not summed over. The above equation is integrated to obtain the following results for the correlation between the displacements at times t and t  :   t kB T t   dτ dτ  e−ζ (τ −τ ) ≡ IB , (1.4.23) xi (t)xi (t  ) = ζ t0 t0

54

Statistical physics of liquids

where we have denoted the displacement at time t as xi (t) = xi (t) − xi (t0 ). We evaluate  the integral IB in Appendix A1.3 to obtain xi (t)xi (t  ) = kB T ζ −1 t + t  − |t − t  | − 2t0 $

#   + ζ −1 e−ζ (t−t0 ) + e−ζ (t −t0 ) − e−ζ |t −t| − 1 . (1.4.24) For the case t = t  the result for the mean-square displacement is  xi2 (t) = 2kB T ζ −1 t − t0 + ζ −1 {e−ζ (t−t0 ) − 1} .

(1.4.25)

For short times relative to the characteristic time ζ −1 , i.e., ζ (t − t0 ) 1, we can expand the exponential on the RHS and obtain that the root-mean-square displacement is directly proportional to the time (t − t0 ), #+ ,$1/2 = v0 (t − t0 ), (1.4.26) [xi (t) − xi (t0 )]2 where v02 = kB T is the root-mean-square average velocity of the Brownian particle (its mass is taken as unity). This is free-particle behavior. On the other hand, for long times relative to the characteristic time ζ −1 , i.e., ζ (t − t0 )  1, we obtain , + (1.4.27) [xi (t) − xi (t0 )]2 = 2kB T ζ −1 (t − t0 ) ≡ 2 D¯ 0 (t − t0 ), showing that the mean-square displacement grows linearly with time. This is termed the Einstein relation, and the constant of proportionality is defined as the diffusion coefficient D¯ 0 . From eqn. (1.4.27) we obtain a relation between the diffusion coefficient D¯ 0 and the friction coefficient ζ , namely D¯ 0 = kB T ζ −1 . In fact, the friction coefficient ζ is a property related to the resistance to the motion of the Brownian particle from the surrounding liquid. If the Brownian particle has a radius a and the surrounding fluid has shear viscosity η, then the frictional drag on the sphere is given by Stokes’ law, i.e., ζ = 0.6πaη. The diffusion coefficient D¯ 0 for the Brownian particle is then inversely proportional to the viscosity of the surrounding medium according to what is termed the Stokes–Einstein relation (Einstein, 1956): kB T . (1.4.28) D¯ 0 = 0.6πaη It is important to note at this point that the relation (1.4.28) is established here for the motion of a heavy Brownian particle in the surrounding fluid. However, it has also been found that even in the case of a liquid particle moving in a sea of identical particles the Stokes–Einstein relation of an inverse dependence of the diffusion coefficient on the viscosity seems to holds in liquids. We present a phenomenological argument for justifying the Stokes–Einstein relation with respect to the self-diffusion process in Chapter 4, Section 4.1.2, using the free-volume theory.

Appendix to Chapter 1

A1.1 The Gibbs inequality For any variable x we write the identity  g ln g − g + 1 = =

g

dx 1 g

d [x ln x − x + 1] dx

d x ln x.

(A1.1.1)

1

If g > 1 then the integrand ln x is positive in the integration range and hence the integral on the RHS is positive definite. If, on the other hand, g < 1, then with a change of variable y = 1/x the integrand reduces to ln y/y 2 , with y as the integration variable changing from y = 1 to y = 1/g > 1. Over this range the integrand is positive and hence the integral is again positive definite. For g = 1 the RHS is zero. Thus we have proved that the quantity g ln g − g + 1 is always positive as long as g is positive.

A1.2 The force–force correlation We obtain here the expression for the force–force correlation function for the αth particle for a fluid interacting through a pairwise-additive continuous interaction potential. This is used in computing the Einstein frequency for a liquid particle as presented in eqn. (1.3.38). In general, if the Hamiltonian is H and the total potential energy is V , the derivatives with respect to the position coordinates rα of the αth particle are equal since the kinetic part of H is only momentum-dependent. We obtain for the force–force correlation for the αth particle F · F = ∇α V · ∇α V  = ∇α V · ∇α H .

(A1.2.1)

In order to compute the above average, let us first obtain for the general dynamic variable A the following result: 55

56

Appendix to Chapter 1



∂H dp A · A · ∇α H  = dr f eq ∂rα

  ∂ −1 N N dr dp A · = −β f eq ∂rα

  ∂ = β −1 dr N dp N · A f eq . ∂rα 



N

N

(A1.2.2)

In reaching the last equality we ignored a surface term. In the present case, A ≡ ∇α V and we obtain from eqn. (A1.2.1)   F · F = kB T dr N dp N f eq ∇α2 V. (A1.2.3) For a fluid in which the particles are interacting through a two-body potential only the total potential energy V is expressed as a pairwise sum of pair interactions u(r1 , r2 ) ≡ u(12) between particles 1 and 2 and the RHS of eqn. (A1.2.3) is obtained as   N dp N f eq ∇α2 V F · F = kB T dr 3 4    exp[−β H ] 2 = kB T dr1 dr2 ∇1 u(12) (N − 1) dr3 . . . dr N ZN   kB T dr1 dr2 ∇12 u(12)n 20 g(12) = N  # $ (A1.2.4) = n 0 kB T dr ∇ 2 u(r ) g(r ).

A1.3 Brownian motion A1.3.1 The noise correlation In equilibrium the correlation between the noise values f (t) in the Langevin equation at two different times t and t  depends only on the difference of the two times, i.e., time translational invariance holds. We define for the isotropic system  f i (t) f j (t  ) = δi j D¯ 0 (t − t  ).

(A1.3.1)

Since the correlation matrix is diagonal, in this case D¯ 0 (t) =  f i (t) f i (0) =  f i (0) f i (−t) (A1.3.2) =  f i (0) f i (−t) = D¯ 0 (−t). 1 2 2 Thus D¯ 0 is a symmetric function. Also D¯ 0 (0) = f i (t) is by definition a positive definite quantity. In fact, D¯ 0 (0) presents a limit for the function D¯ 0 (t): + , + , { f i (t) ± f i (0)}2  = f i2 (t) + f i2 (0) ± 2 f i (t) f i (0) = 2{ D¯ 0 (0) ± D¯ 0 (t)}.

(A1.3.3)

A1.3 Brownian motion

57

Since the LHS of (A1.3.3) is always greater than or equal to zero, we have that D¯ 0 (t) cannot go beyond the limits ± D¯ 0 (0). Finally, for times t large relative to the correlation time for the noise, lim  f i (t) f i (0) =  f i (t) f i (0) = 0.

(A1.3.4)

t→∞

The correlation of the noise represented by D¯ 0 (t) therefore is a symmetric function bounded between limits ±D(0) and decaying to zero in the long-time limit.

A1.3.2 Evaluation of the integrals We present here the evaluation of some of the integrals involved in the discussion of the Langevin dynamics. First we consider the integral IB defined in eqn. (1.4.13),  t  t   ¯ dτ dτ  e−ζ (t−τ ) e−ζ (t −τ ) δ(τ − τ  ) IB = D0 t0 t 0   t  t    dτ  e−ζ (t+t −2τ ) + θ (t  − t) dτ e−ζ (t+t −2τ ) = D¯ 0 θ (t − t  ) t0

= D¯ 0

 e−ζ (t+t )

t0





θ (t − t  ){e2t − e2t0 } + θ (t  − t){e2t − e2t0 }

2ζ D¯ 0  −ζ |t−t  |  = − e−ζ (t+t −2t0 ) . e 2ζ

(A1.3.5)



Next we consider the integral IB defined in eqn. (1.4.23),    t  t τ    −ζ (τ −τ  )  −ζ (τ  −τ ) IB = θ (t − t) dτ dτ e + dτ e t0





t0

t



τ

τ

 −ζ (τ −τ  )



t

 −ζ (τ  −τ )



+ θ (t − t ) dτ dτ e + dτ e t0 t0 τ  ) (   = θ (t  − t) 2(t − t0 ) + ζ −1 e−ζ (t−t0 ) + e−ζ (t −t0 ) − e−ζ |t −t| − 1  ) (   + θ (t − t  ) 2(t  − t0 ) + ζ −1 e−ζ (t−t0 ) + e−ζ (t −t0 ) − e−ζ |t−t | − 1 = 2tθ (t  − t) + 2t  θ (t − t  ) − 2t0    + ζ −1 e−ζ (t−t0 ) + e−ζ (t −t0 ) − e−ζ |t −t| − 1    = t + t  − |t − t  | − 2t0 + ζ −1 e−ζ (t−t0 ) + e−ζ (t −t0 ) − e−ζ |t −t| − 1 .

(A1.3.6)

2 The freezing transition

In this chapter we discuss a theory for the freezing of an isotropic liquid into a crystalline solid state with long-range order. The transformation is a first-order phase transition with finite latent heat absorbed in the process. We focus on a first-principles orderparameter theory of freezing that originated from the pioneering work of Ramakrishnan and Yussouff (1979). The theory approaches the problem from the liquid side and views the crystal as a liquid with grossly inhomogeneous density characterized by a lower symmetry of the corresponding lattice. This is in contrast to description of the crystal in terms of phonons. The crucial quantity characterizing the physical state of the system in this non-phonon-based model is the average one-particle density function n(x). The thermodynamic description of either phase involves a corresponding extremum principle for a relevant potential. The latter, obtained as a functional of the one-particle density n(x) and the stable thermodynamic state of the system, is identified by the corresponding density required for invoking the extremum principle. This approach, which is generally referred to as the density-functional theory (DFT) of freezing (Haymet, 1987; Baus, 1987, 1990; Singh, 1991; Löwen, 1994; Ashcroft 1996), has been improved over the years and successfully applied for the study of liquid-to-crystal transitions in various simple liquids, the solid–liquid interface, two-dimensional systems, metastable glassy states, etc. For applications of density-functional methods in statistical mechanics there exist general reviews (Evans, 1979; Henderson, 1992). The theoretical formulation presented here also becomes the starting point of consideration of the dynamics of the dense liquid near freezing point and beyond that in the supercooled region. In the present chapter we focus on the treatment of the thermodynamic phase transition from the liquid to the crystalline state using DFT. 2.1 The density-functional approach In the liquid state of matter the constituent particles move randomly so that the timeaveraged density is the same at all points. As the temperature of the liquid is lowered (or equivalently the average density increased), the particles abandon the random spatial arrangement and form the crystalline state having a spatially varying density n(x) according to the symmetry of the corresponding lattice. At zero temperature, minimizing the 58

2.1 The density-functional approach

59

potential energy of the system corresponding to the optimum distribution of particles provides the choice of the relevant lattice structure. Understanding the transformation at finite temperature is more subtle. Here the phase transformation is a result of the competition between the energetic and entropic contributions. Predicting (in a quantitative manner) the thermodynamic condition under which such a transformation occurs in a many-particle system is the primary issue in the theory of freezing. This refers to the temperature of freezing as well as the volume change in the freezing process. Subsequent to some initial attempts (Lindemann, 1910; Born, 1939; Kirkwood and Monroe, 1941; Jancovici, 1965; Brout, 1963, 1966; Thouless and Kosterlitz, 1973) at understanding this very general phenomena of freezing, an order-parameter theory using the equilibrium density as the relevant variable was proposed by Ramakrishnan and Yussouff (1979). This theory adopts an approach that is intermediate between one of purely microscopic origin and a phenomenological description of the liquid near the transition. The thermodynamic properties of the inhomogeneous crystalline state are obtained in terms of the corresponding quantities for the homogeneous liquid state. The thermodynamics of the dense uniform liquid is well understood through integral-equation theories or simulations, which are used as an input in the theory described below. In the density-functional theory for the freezing transition the interaction potential between the liquid particles constitutes the microscopic-level description of the manyparticle system. The basic characteristics of the two-body potential for which a crystalline state appears (under appropriate conditions of density and temperature) include (a) a strongly repulsive part at short range and (b) an attractive part effective at long range. In such a condition the Hamiltonian can be written in a harmonic expansion around the equilibrium sites which correspond to the minimum-potential-energy configuration. The attractive part of the potential seemingly appears to play an important role in stabilizing the solid in a crystalline state in which each of the individual particles is localized around its mean position. However, even for the hard-sphere interaction, the liquid-tocrystal transition occurs, as was shown by computer simulations (Alder and Wainwright, 1962; Hoover and Ree, 1968). In this case the system must be stabilized externally by application of appropriate pressure. The hard-sphere liquid transforms into an f.c.c. solid at the reduced density n 0 σ 3 = 0.96 while the close-packed f.c.c. structure corresponds to a much higher value of n 0 σ 3 = 1.41. There are thus aspects of the freezing transition that go beyond the details of the interaction potential. Indeed, the geometric nature of the freezing transition is apparent in the structural similarity of classical liquids at high densities. Close to the freezing-transition point the static structure factor S(q) (see Chapter 1 for its definition) of many liquids near the first peak (wave vector q → qm ) has approximately the value S(qm ) = 2.85 (Verlet, 1974). This includes computer-simulation results for a Lennard-Jones liquid (Lennard-Jones and Devonshire, 1939) along the melting curve (Hansen and Verlet, 1969), a hard-sphere fluid (Verlet, 1968), and a one-component plasma (Hansen, 1973), and experimental results for materials such as Ar (Page et al., 1969), Na (Greenfield et al., 1971), and Pb (North et al., 1968). The value of S(qm ) lies in the range 2.8–3.1. This is the analogue of the corresponding result in the solid phase, namely the

60

The freezing transition

Lindemann (1910) criterion of melting, i.e., the mean-square displacement u 2  scaled by the square of the interatomic separation is 0.01 at melting (Ross and Alder, 1966). Let us briefly outline the classical density-functional scheme adopted in the following. In the order-parameter theory for the freezing transition presented here the free energy of an inhomogeneous system is uniquely determined from the properties of its homogeneous counterpart. The thermodynamic property is obtained as a functional of the one-particle density n(x). The latter is expressed in terms of a suitable set of parameters, treated as the order parameters of the transition. The liquid and the crystalline states correspond to orderparameter values from which one obtains the appropriate density functions representing the corresponding phase. The thermodynamic properties are computed assuming the system to be in a single phase (either liquid or crystal) and fluctuations of the order parameter are not taken into account. The density-functional approach is mean-field-like in this respect and it ignores the effects of fluctuations which make the crystalline state unstable in lower dimensions. In what follows, we first describe the formulation of an extremum principle for the appropriate thermodynamic potential. This is followed by the construction of the appropriate free-energy functional in terms of the inhomogeneous density function. Testing the optimization of the thermodynamic property then naturally leads to identification of the corresponding equilibrium phase. 2.1.1 A thermodynamic extremum principle The equilibrium state of the many-particle system is identified from the minimization of the appropriate thermodynamic free energy. For uniform systems the minimization is done with respect to the variation of the average density. For example, in the liquid-tovapor transformation both phases are characterized by their respective densities. In the grand-canonical-ensemble description of the uniform system the equilibrium state is characterized by the minimum value of the thermodynamic potential EQ = −P V , where P and V , respectively, denote the thermodynamic pressure and volume (McQuarie, 2000) of the fluid. In the case of the freezing transition, on the other hand, the crystalline state is characterized by a density function n(x) that is space-dependent and hence the corresponding free energy is treated as a functional of the density function. The appropriate free-energy functional required for the identification of the equilibrium state depends on the type of ensemble chosen in the statistical averaging. In the present case of solid–liquid transition it is appropriate to consider the grand-canonical ensemble so that the number of particles in a given volume can change abruptly. We present here, following Haymet and Oxtoby (1986) and Oxtoby (1991), the analysis for the proper extremum principle and the construction of the free-energy functional for the inhomogeneous state. Let us consider a system of N identical particles of mass m and the position and momentum coordinates {rα , pα }, for α = 1, . . . , N . The whole set of phase-space variables is to be denoted as {r N , p N }. Its Hamiltonian is given by HN =

N  pα2 + U + Uext ≡ H N0 + Uext , 2m

α=1

(2.1.1)

2.1 The density-functional approach

61

where H N0 is the intrinsic part of the Hamiltonian in which U (r1 , . . . , r N ) is the interaction energy of the N -particle system – as has already been introduced in Chapter 1 in eqn. (1.1.16). Uext is the external field contribution and is expressed in terms of the 5 interaction with a local field obtained from a one-body potential φ, Uext = α φ(rα ). Following the example of equilibrium statistical mechanics, we define the functional [ f ] of f (r N , p N ) as [ f ] = Tr f (H N − μN + β −1 ln f ),

(2.1.2)

where μ is the chemical potential and β = 1/(kB T ) denotes the inverse temperature characterizing the grand-canonical ensemble. The symbol Tr in (2.1.2) refers to the classical trace defined in terms of the integration of the phase-space variables (r N , p N ) (defined in (1.2.3)) as     ∞  1 dr dp . . . dr . . . dp N . (2.1.3) Tr ≡ 1 N 1 h 3N N ! N =0

If we choose f (r N , p N ) to be a trial distribution function representing the probability of the system in the phase-space region (r N , p N ), then, according to the above definition, Tr f (r N , p N ) = 1.

(2.1.4)

The external field contribution Uext , which is a sum of single-particle contributions, is expressed as an integral,  N  Uext = φ(rα ) = dx n(x)φ(x), ˆ (2.1.5) α=1

in terms of the microscopic one-particle density n. ˆ At the microscopic level, the oneparticle density n(x) ˆ is formally defined as a function of the phase-space variables (r N , p N ), n(x) ˆ =

N 

δ(x − rα ).

(2.1.6)

α=1

The average density function corresponding to the distribution function f is then obtained as n(x) = Tr n(x) ˆ f. The density-functional model uses the equilibrium averaged density as the order parameter. The statistical-mechanical formulation of the problem is often conveniently done in terms of the grand-canonical ensemble, implying that the volume of the system remains constant while the number of particles changes. The equilibrium (McQuarie, 2000) distribution function f 0 corresponding to this ensemble is obtained as f 0 = −1 exp[−β(H N − μN )],

(2.1.7)

where (β, μ) is the grand-canonical partition function. Using (2.1.7) for f 0 in the definitions (2.1.2) and (2.1.4) it follows that [ f 0 ] = −β −1 ln (β, μ) = EQ ,

(2.1.8)

62

The freezing transition

where EQ is the equilibrium grand potential for the system. EQ = −P V , where P and V , respectively, denote the pressure and volume of the system in equilibrium. The corresponding density n 0 (x) is obtained by averaging over the grand-canonical ensemble, n 0 (x) = Tr n(x) ˆ f0 =

δ ln  , δβu(x)

(2.1.9)

where u(x) = μ − φ(x). For any arbitrary distribution function f satisfying the normalization (2.1.4) the corresponding functional [ f ] is obtained as [ f ] = Tr f (H N − μN + β −1 ln f ) = [ f 0 ] + β −1 Tr[ f ln( f / f 0 )] > [ f 0 ] = EQ .

(2.1.10)

Two points are to be noted with respect to the above results. (a) The second equality in (2.1.10) is obtained using the definition of the grand-canonical ensemble being restricted to those distribution functions f for which the ensemble average of the Hamiltonian H N and the number of particles N remain fixed. (b) The last inequality in (2.1.10) follows from the fact that Tr[ f ln( f / f 0 )] is always positive. To see this, we consider  ∞  1 d f 0 ()[x ln(x)] N! N =0  ∞  1 d f 0 ()[x ln(x) − x + 1] , = N!

Tr[ f ln( f / f 0 )] =

(2.1.11)

N =0

where we define f ()/ f 0 () ≡ x (say) with d denoting the elementary volume in the 6N -dimensional phase space. The last equality in eqn. (2.1.11) follows since both f () and f 0 () are normalized (see eqn. (2.1.4)), canceling out the last two terms on the RHS. According to the Gibbs inequality (see Appendix A1.1) the quantity within the square brackets on the RHS of eqn. (2.1.11) is always positive and hence the integral is also positive definite. Therefore the functional [ f ] attains its minimum value for the choice f = f 0 . We now consider the following question: can two different external potentials φ and φ  correspond to the same equilibrium density n 0 (x)? Let the equilibrium distribution functions with two such external potentials be f 0 and f 0 , respectively. The Hamiltonian corresponding to the external potential φ  is denoted as  , H N = H N0 + Uext

5

(2.1.12)

 =  where Uext α φ (rα ). The functional [ f ] now has its minimum value for the corresponding equilibrium distribution f = f 0 . By comparing the values of the functional 

2.1 The density-functional approach

corresponding to f = f 0 and f 0 , respectively, we obtain the inequality ( ) EQ = Tr f 0 H N − μN + β −1 ln f 0 ( ) < Tr f 0 H N − μN + β −1 ln f 0  = EQ + dx n 0 (x)[φ  (x) − φ(x].

63

(2.1.13)

The inequality in the second line of (2.1.13) follows from the fact that the choice f 0 minimizes the functional [ f ] in this case. To simplify the analysis we assume the chemical potentials both in the primed and in the unprimed cases to be the same μ. The above argument also applies in the reverse order on interchanging the primed and unprimed quantities, i.e., we evaluate the functional [ f ] first for f = f 0 corresponding to the Hamiltonian H N with external potential φ to obtain the inequality  (2.1.14) EQ < EQ + dx n 0 (x)[φ(x) − φ  (x)]. On adding (2.1.13) and (2.1.14) we obtain the contradictory result EQ + EQ < EQ + EQ . Therefore the original assumption linking φ(x) and φ  (x) to the same n 0 (x) is not valid. We have thus proved the important result that the equilibrium density n 0 (x) is uniquely determined by the external potential φ(x). The inverse, i.e., a given n 0 (x) uniquely determining the corresponding field φ(x), also holds in the DFT description for the thermodynamic state. This is implied from the finite-temperature generalization (Mermin, 1965) of the ground-state theorem (Hohenberg and Kohn, 1964). On the other hand, from (2.1.8) it follows that for a given system characterized by H N0 , i.e., by the corresponding interaction U (as defined in (2.1.1), the equilibrium distribution function f 0 is entirely determined by the external potential φ(x). On combining these two results we obtain that the equilibrium distribution function f 0 is uniquely determined by the one-particle density n 0 (x). Using the unique correspondence between f 0 and n 0 , a functional φ of the one-particle density n(x) for the system in an external potential φ is obtained as φ [n] = [ f ] ≡ Tr f (H N − μN + β −1 ln f ).

(2.1.15)

In terms of φ [n] we further define a functional F[n(x)],   φ [n] = F[n] + dx φ(x)n(x) − μ dx n(x)  = F[n] − dx n(x)u(x),

(2.1.16)

where u(x) is as defined above with eqn. (2.1.9). It follows directly from the definition (2.1.2) for [ f ] that F[n] is obtained from the corresponding distribution function f as ) ( (2.1.17) F[n(x)] = Tr f HN0 + β −1 ln f .

64

The freezing transition

For the equilibrium distribution f = f 0 the corresponding one-particle density is n 0 (x), and we obtain for the value of φ [n 0 ] φ [n 0 ] ≡ [ f 0 ] = EQ .

(2.1.18)

For any other f  = f 0 since [ f ] is higher, i.e., [ f ] > [ f 0 ]. If n(x) is the one-particle density corresponding to f , we must have then φ [n] > φ [n 0 ]. Thus we reach the important result that the equilibrium density n 0 (x) minimizes the functional φ [n] and its value at the minimum is equal to the equilibrium grand potential. This result is conveniently expressed in terms of the functional variational principle,

δφ [n] = 0. (2.1.19) δn(x) n(x)=n 0 (x) Note that, if we were considering the canonical ensemble, then it would follow in an exactly similar manner that the corresponding functional (of density) to be minimized would be the one involving the first two terms on the RHS of (2.1.16).  Fφ [n] = F[n] + dx n(x)φ(x). (2.1.20) For the equilibrium density n 0 (x) the minimum value of this functional Fφ [n] is equal to the Helmholtz free energy of the system much in the same way as φ [n] is equal to the grand potential EQ for the grand-canonical ensemble considered above. The results (2.1.18) and (2.1.19) form the basis for calculation of the equilibrium densities in studying the freezing transition. This is described next.

2.1.2 An approximate free-energy functional Having identified the thermodynamic extremum principle we now construct the proper free-energy functional F[n] in terms of the density n(x). This functional is then used to determine the appropriate inhomogeneous density function n 0 (x) at equilibrium satisfying the extremum principle (2.1.21). Using the definition (2.1.16) and the extremum condition (2.1.19), we obtain the result

δ F[n] = μ − φ(x) ≡ u(x). (2.1.21) δn(x) n=n 0 The free-energy function F[n] has contributions from two different parts, F = Fid + Fex ,

(2.1.22)

to be defined as follows. (a) The ideal-gas part Fid is the free-energy functional for a noninteracting system of particles. (b) The excess part Fex is due to interaction between the particles.

2.1 The density-functional approach

65

The ideal-gas part Fid [n] as a functional of density n(x) is obtained by using the definition (2.1.20). This also involves obtaining the functional relation between the equilibrium density and the field u(x). For the ideal-gas case with the interaction potential U = 0 this can be done 5 exactly. In this case H N0 has only the kinetic-energy term K = α pα2 /(2m) and hence, on explicitly integrating out the 3N momentum variables, the grand-canonical partition function  is obtained as  & '   = Tr exp −β H0 − u(rα ) α



N  ∞  1 −3 βu(x) = dx e  N! 0 N =0

 −3 βu(x) dx e = exp 0 ,

(2.1.23)

√ where 0 = h/ 2πmkB T is the de Broglie or thermal de Broglie wavelength for the liquid particles. From eqns. (2.1.9) and (2.1.23) one obtains for the noninteracting case the equilibrium density for the inhomogeneous state in the presence of the field φ(x) as n 0 (x) =

exp[βu(x)] 30

,

(2.1.24)

" ! i.e., βu(x) = ln 30 n 0 (x) . We obtain the free-energy functional F[n 0 ] by evaluating the expression (2.1.16) at n = n 0 as   # $ (2.1.25) β Fid [n 0 ] = dx n 0 (x) ln 30 n 0 (x) − 1 , where we have used for the ideal-gas equation of state  −βEQ = β P V = N¯ ≡ dx n 0 (x)

(2.1.26)

for the average number of particles N¯ . We choose the ideal-gas part of the free energy Fid [n] as a functional of the density function n(x) such that for the equilibrium density n(x) = n 0 (x) it reduces to the expression (2.1.25) for Fid [n 0 ]. Fid [n] is thus obtained by generalizing the above result as   ( ) −1 dx n(x) ln 30 n(x) − 1 . (2.1.27) Fid = β The equilibrium density for the noninteracting system is obtained as n 0 (x) = z exp[−βφ(x)], where the normalization constant z = exp(βμ)/30 .

(2.1.28)

66

The freezing transition

The ideal-gas part of the free energy is an entropic contribution. See also Appendix A1.1 for a deduction of Fid for a coarse-grained density function n(x) using an occupationnumber representation. It follows from the result (2.1.27) that an inhomogeneous density function implies a higher degree of mass localization and hence reduction of entropy. For a system of N particles the difference between the Fid values corresponding to a uniform state of density n 0 and one with an inhomogeneous density distribution n(x) is given by  ˜ n(x)], ˜ (2.1.29) β Fid = n 0 dx n(x)[ln where we assume that n(x) ˜ = n(x)/n 0 . Since by definition n(x) ˜ is always positive, from the Gibbs inequality (see Appendix A1.1) it follows that  ! " dx n(x)ln ˜ n(x) ˜ − n(x) ˜ +1 ≥0 (2.1.30) 6 for any density distribution n(x). For a fixed number of particles we have dx[n(x) ˜ − 1] = 0, we have from the relation (2.1.29) Fid ≥ 0. Entropy always drops upon localization of the particles in a state with inhomogeneous density profiles, as a result of the restriction of available phase space. The interaction part For the interacting system we include in F the contribution from the excess part of the free energy Fex . The functional-extremum principle (2.1.21) now reduces to the form  (2.1.31) ln 30 n 0 (x) − c(1) (x; n 0 ) = β{μ − φ(x)} = βu(x), where we have used the definition β

−1 (1)

c



δ Fex (x; n 0 (x)) = δn(x)

n(x)=n 0 (x)

.

(2.1.32)

Using the result (2.1.31), we obtain for the equilibrium density n 0 (x) n 0 (x) = z exp[−βφ(x) + c(1) (x; n 0 (x))],

(2.1.33)

showing that c(1) (x; n 0 (x)) acts as a one-body potential due to the interaction between the fluid particles. The higher-order direct correlation functions are defined in terms of functional derivatives of c(1) with respect to n 0 (x). We demonstrate in Appendix A2.1 that the two-point function c(2) (x1 , x2 ) is related to the pair correlation function g (2) (x1 , x2 ) in the fluid by a relation that reduces to the Ornstein–Zernike relation for the uniform liquid discussed in Chapter 1. The expression for the one-body potential in (2.1.33) is simplified by expanding it around its value cl for the uniform liquid state of density n l = N¯ /V . For the uniform liquid in the absence of any external field we have from eqn. (2.1.33) for the uniform density n l = z exp(cl ). The inhomogeneous density function n 0 (x) is then obtained in terms

2.1 The density-functional approach

67

of the corresponding one-particle direct correlation function c(1) (r ), # $ n 0 (x) = n l exp c(1) (x; n 0 (x)) − cl − βφ(x) .

(2.1.34)

A simple Taylor expansion for c(1) (x; n 0 (x)) around the uniform liquid state gives  (1) c [x1 ; n 0 (x)] = cl (n l ) + dx2 c(2) [x1 , x2 ; n l ]δn 0 (x2 )  1 dx2 dx3 c(3) [x1 , x2 , x3 ; n l ]δn 0 (x2 )δn 0 (x3 ) + · · ·, (2.1.35) + 2 where δn 0 (x) = n 0 (x) − n l is the fluctuation of the equilibrium density in the inhomogeneous state from the uniform liquid-state density n l . In eqn. (2.1.35) we use the following definitions for the direct correlation functions c(i) as the successive functional derivatives of Fex evaluated at the liquid-state density n l ,

δ i Fex (i) c (x1 , . . . , xi ; n l ) = . (2.1.36) δn(x1 ) . . . δn(xi ) n=n l For practical calculations one usually adopts the simplest approximation, keeping terms only up to the second-order term (i = 2) in the expansion (2.1.35) to obtain the following expression for the inhomogeneous density in equilibrium:  n 0 (x1 ) = n¯ l exp

dx2 c(2) (x1 , x2 ; n l )δn 0 (x2 ) ,

(2.1.37)

where we identify n¯ l = z exp[−βφ + cl ] ≡ n l exp[−βφ]. For eqn. (2.1.37) the trivial solution is then the uniform density n 0 (x1 ) = n l in the absence of any external field. The solution of eqn. (2.1.37) is the starting point for the subsequent analysis for testing the possibility of an inhomogeneous density state. The two-point kernel function c(2) (x1 , x2 ; n l ) which is defined in terms of the functional derivative of the one-body potential c(1) is required in order to completely specify eqn. (2.1.37) for the inhomogeneous density. The excess free energy in the solid state Fex is expressed as a functional Taylor expansion (Courant and John, 1965; Abramowitz and Stegun, 1970) in the density fluctuations δn 0 (x) = n 0 (x) − n l ,  Fex = F(n l ) − dx1 c(1) (x1 ; n l )δn 0 (x)   1 dx1 dx2 c(2) (x1 , x2 ; n l )δn 0 (x1 )δn 0 (x2 ) + · · ·. (2.1.38) − 2 The series involves the functions c(i) defined in (2.1.36) at the liquid-state density n 0 (x) = n l . In the present case we have only kept terms up to second order in the density difference.

68

The freezing transition

2.1.3 The Ramakrishnan–Yussouff model We now discuss the density-functional approach to the freezing transition of an isotropic liquid into a crystalline state, using an order-parameter description in terms of the density. We begin with the eqn. (2.1.37) for the inhomogeneous density function n(x) obtained from the functional extremum principle (2.1.19). This equation is solved using the kernel function c(x1 , x2 ) ≡ c(|x1 − x2 |) for the isotropic homogeneous liquid state with density n l . The stable solid state at freezing is identified from the solution of the implicit relation (2.1.37) obtained by using test density functions corresponding to a chosen lattice structure. The present theoretical approach therefore does not solve the problem of spontaneous breaking of the translational symmetry; rather it allows one to identify the crystal symmetry in the inhomogeneous state by picking up the appropriate density function n 0 (x) which satisfies the functional extremum principle (2.1.19). The density function defined with a suitable parametrization constitutes the order parameter of the transition. In this regard there are two main choices for the test density function that have been made in the literature and will be considered below. Before discussing specific details of the solution, we first outline the general scheme of locating the freezing-transition point. At low densities there is only one trivial solution, n(x) = n l , of eqn. (2.1.37) referring to the homogeneous liquid state. For densities n l above a certain value n ∗l (dependent on the temperature T and the chemical potential μ of the liquid), inhomogeneous solutions n(x) with spatially periodic densities corresponding to one or more crystalline states are possible. Such inhomogeneous solutions are obtained for a continuous range of densities above n ∗l above which for every n l there is a corresponding inhomogeneous state with density n 0 (x). The homogeneous state and the inhomogeneous state are at the same temperature and chemical potential. The particular pair which represents true phase coexistence is then chosen from the corresponding Maxwell construction (Huang, 1987) in the present case of a grand-canonical ensemble. In the context of liquid–vapor transition the total number of particles N remains fixed, and for the isobaric ensemble the corresponding Gibbs free energies of the two states are equal. In the grand-canonical ensemble for the density-functional theory considered here, the volume V is fixed and the corresponding thermodynamic potential which is minimized is the grand potential . The coexisting states with respective densities n l and n 0 (x) have the same temperature T and chemical potential μ. The location of the true transition point is inferred by equating the grand potentials in the two states. The difference between the grand potentials in the inhomogeneous crystalline state with density n 0 (x) and the homogeneous liquid state with density n l (in the absence of the external potential φ) is obtained as  ≡ [n 0 (x)] − [n l ] = Fid [n 0 (x)] + Fex [n 0 (x)] − μ

 dx[n 0 (x) − n l ].

(2.1.39)

The difference Fid = Fid [n 0 (x)] − Fid (n l ) in the ideal-gas part of the free energy is directly calculated from (2.1.27). The difference Fex = Fex [n 0 (x)] − Fex (n l ) between

2.1 The density-functional approach

69

the excess free energies of the liquid and solid states is expressed as a functional Taylor expansion in the density fluctuations δn 0 (x) = n 0 (x) − n l from (2.1.38) as  Fex = − dx1 c(1) (x1 ; n l )δn 0 (x)   1 (2.1.40) dx1 dx2 c(2) (x1 , x2 ; n l )δn 0 (x1 )δn 0 (x2 ) + · · ·. − 2 On substituting (2.1.40) into (2.1.39) and using the relation (2.1.31) for the case in which the external field φ = 0, we obtain the following result for the grand potential difference between the crystalline and liquid states:

  n 0 (x1 ) − dx1 (n 0 (x1 ) − n l )  = dx1 n 0 (x1 )ln nl   1 dx1 dx2 c(2) (x1 , x2 ; n l )δn 0 (x1 )δn 0 (x2 ) − · · ·. − (2.1.41) 2 Therefore, by equating  in (2.1.41) to zero the the coexisting density at the freezingtransition point is obtained. Since the volume V is kept constant, the equality of the grand potential  = −P V for the two states also implies the equality of pressure P. In summary, among all pairs of points with equal temperature and chemical potential, the one with equal pressure (up to the same order in perturbation theory) marks the phase-transition point. This identification is displayed schematically in Fig. 2.1 with a phase diagram. In the scheme described above for locating the freezing point of the liquid into a crystal, a crucial ingredient is the choice of the test density function which involves the order parameters of the associated thermodynamic phase transition. Two functional forms for

Fig. 2.1 The density-functional theory in the grand-canonical ensemble relates each of the liquid states to a corresponding solid state having the same chemical potential and temperature. (Schematic c American Institute of Physics. diagram from Haymet and Oxtoby (1986).) 

70

The freezing transition

the density have been used extensively in the existing literature. The formulation originally adopted by Ramakrishnan and Yussouff (1979) as well as by Haymet and Oxtoby (1981) and Oxtoby and Haymet (1982) uses the expansion in the reciprocal-lattice-vector (RLV) space. The first choice for the density function in terms of RLVs is described in some detail below to clarify the dependence of the order parameter on the lattice symmetry of the crystal structure. It is also useful for the discussion on the nucleation process for the growth of the crystal phase, which will be described in the next chapter. The second choice for the density function in terms of the Gaussian profiles located at the lattice points is discussed in the next section together with the weighted-density-functional theories. In the Ramakrishnan–Yussouff theory the solid-state density with underlying crystal symmetry is expressed exactly in terms of an order-parameter expansion,   ∞  Am eiKm · x . n 0 (x) = n l 1 + η + (2.1.42) m=1

The sum in (2.1.42) is over the RLVs {Km } of the corresponding crystal structure (Ashcroft and Mermin, 1976) being tested. The amplitudes Am represent the order parameters of the transition. If all of the Am and η are set to zero, the uniform liquid state is obtained. The average density of the solid state is defined as  1 (2.1.43) dx n 0 (x). ns = V η = (n s − n l )/n l denotes the fractional change in density at freezing. Using the expansion (2.1.42) for the density function, the integral in the exponent on the RHS in (2.1.37) is obtained as  ∞  dx c(x − x )δn 0 (x ) = c˜0 η + c˜m Am eiKm · x . (2.1.44) m=1

We have simplified the notation above by defining the dimensionless quantity c˜m ≡ n l c(|Km |) (for m = 0, 1, . . .), where c(K m ) is the Fourier transform of the direct correlation function c(r ) of the uniform liquid evaluated at the RLV Km . For the isotropic liquid c˜m is a function only of the magnitude of the wave vector. At the phase-transition point the grand potentials for the two phases are equal. In Appendix A2.2, the expression (2.1.41) for the difference  of the thermodynamic potential between the two phases is obtained in terms of the order parameters η and Am : ⎡ ⎤ 2  ξm ⎦ 1 1  = n l V ⎣(c˜0 − 1)η + c˜0 η2 + , (2.1.45) 2 2 c˜m m=0

where ξm denotes the rescaled amplitudes defined as ξm = c˜m Am . A nonzero set of solutions for the amplitude parameters in the inhomogeneous density function (2.1.42) from the above DFT equations marks a stable crystalline phase. The amplitudes η and Am corresponding to the crystalline state in which the liquid freezes are obtained with the following two inputs.

2.1 The density-functional approach

71

Table 2.1 Comparison of the freezing parameters for hard spheres as obtained in the Ramakrishnan–Yussouff theory, using the Percus–Yevick (PY) structure factor (with and without the Verlet–Weiss (VW) correction) for the uniform liquid state, and the simulation of Hoover and Ree (1968). From Haymet and Oxtoby (1986). Structure factor (PY with VW) Average liquid density n l Average solid density n s Fractional density change η Entropy change s/kB

0.9460 1.037 0.0907 −1.047

Structure factor (PY without VW) 0.9665 1.033 0.0690 −0.944

Simulation 0.939–0.948 1.036–1.045 0.0928–0.1129 −1.16

(a) A suitable set of RLVs Km representing the crystal lattice into which the isotropic liquid changes at the freezing point. Although the ansatz (2.1.42) for the inhomogeneous density can in principle involve an infinite number of wave vectors, for practical computations this series is terminated with a finite number of RLV values. (b) The structural information of the uniform liquid state in terms of the c j . These quantities are obtained as the Fourier transforms of the Ornstein–Zernike direct correlation function c(r ). In the present discussion we are restricting the computation to terms of up to second order in density fluctuations in the expression for the free energy and hence only two-point direct correlation functions are involved. The required input information of uniform liquid structure for obtaining the order parameters of the freezing transition is in terms of the direct correlation function c(r ). In principle any simple liquid with available structural information (from experiment, computer simulations or theoretical models) can therefore be studied with this model. In practice, however, the most widely investigated system is the hard-sphere fluid. In their seminal paper Ramakrishnan and Yussouff (1979) studied the hard-sphere system, restricting the set of RLVs to two shells only. Subsequently Haymet and Oxtoby (1986) included up to 20 sets of RLVs. We discuss this treatment in more detail in Appendix A2.2 and the results are listed in Table A2.4. The scaled direct correlation functions c j of the hard-sphere system are approximated with the Percus–Yevick (PY) (Percus and Yevick, 1958; Percus, 1962, 1964) structure factor with Verlet–Weiss (VW) corrections (described in Chapter 1). Following the scheme described above, this calculation tells us that the freezing transition occurs at n l σ 3 = 0.946 (σ is the hard-sphere diameter), to an f.c.c. solid with average density n s σ 3 = 1.03 and a fractional density change η = 0.09. These numbers are in good agreement with the corresponding computer-simulation results as indicated in Table 2.1. The entropy change (per particle) on freezing is obtained from the Clapeyron equation (McQuarie, 2000),   s 1 dP −η = . (2.1.46) kB n l 1 + η dT

72

The freezing transition

The results with two different structural inputs of c j (with and without Verlet–Weiss corrections to the structure factor) are listed in Table 2.1. The derivative d P/dT at the transition point is directly computed from the corresponding equation of state, i.e., the empirical Carnahan–Starling and the PY equation of state, respectively (described in Chapter 1). The Ramakrishnan–Yussouff theory outlined above is a basic step towards our understanding of freezing phenomena. It provides a formulation of the first-order phase transition through identification of the proper order parameters of the transformation process. The density-functional formalism involved provides the thermodynamic properties of the inhomogeneous crystalline state in terms of those of the liquid state. However, in spite of the usefulness of this model mentioned above, some severe approximations are made. The most important is the truncation of the functional expansion of the excess free energy Fex in terms of the density difference δn 0 (x) = n 0 (x) − n l . The particles in the crystal are highly localized about the respective lattice sites and hence the average solid-state density is strongly inhomogeneous, implying that the density difference δn is not small. Attempts to include higher-order corrections in this series (Haymet, 1983) only make the situation worse and result in poor predictions for the crystal–liquid coexistence conditions for the hard-sphere system (Curtain, 1989). Furthermore, the density distributions in the crystal as obtained from the calculation outlined above are much sharper than those found in simulation studies. The improvement of the theory of the freezing transition comes from the weighted-density-functional theory for studying inhomogeneous fluids. This is discussed in the next section. 2.2 Weighted density functionals The DFT for freezing involves studying the stability of the inhomogeneous solid by comparing its thermodynamic properties with those of the homogeneous liquid. In the previous section we have discussed the calculation of the excess contribution to the free energy of the crystal (over the so-called ideal-gas contribution for a noninteracting system) by retaining low-order terms in a perturbation series. This excess part due to the interaction between the particles is computed using the basic interaction potential between the particles as an input. Improvement of the DFT has been achieved by making nonperturbative approximations for the excess free energy in terms of that of an equivalent system. The characteristics of the coarse-grained system are determined from a weighted average of the inhomogeneous density of the original system. The corresponding weighting function(s) therefore constitute a basic ingredient of this approach to studying inhomogeneous solids. It is either self-consistently dependent on the density itself or chosen to be density-independent, being determined from geometric considerations of the constituent particles. The densityfunctional formulation with weighting is termed the weighted-density-functional theory. The equivalent system is generally determined in terms of a local density, but schemes with a position-independent density have also been formulated. The theory also requires as an input a standard framework for computing the thermodynamic properties of the equivalent low-density system. This makes a hard-sphere fluid particularly suitable for

2.2 Weighted density functionals

73

developing such theories. In our treatment of the weighted-density-functional theories we follow closely the developments due to Ashcroft and coworkers (Curtain and Ashcroft, 1985, 1986; Curtain, 1987; Denton and Ashcroft 1989) and also the fundamental-measure theory (Rosenfeld, 1988, 1989; Rosenfeld et al., 1990) applicable to hard-sphere systems. In the following we also discuss applications for systems whose interaction potentials have a long-range attractive part in addition to the short-range repulsion. In this approach the thermodynamic properties of the crystal are approximated with a free-energy functional constructed using the following mapping. The free-energy density of the inhomogeneous system at a given point x1 is the same as that of a homogeneous system, but evaluated at an auxiliary density n¯ 0 (x). The excess free energy Fex is obtained as  Fex =

dx n 0 (x) f WDA [n¯ 0 (x)].

(2.2.1)

The weighted density function n¯ 0 (x) is constructed by weighting the physical density n 0 (x ) over a relevant region around the point x with a suitable weighting function w. The dependence of n¯ on n should reflect the nature of the inhomogeneity present in a solid phase and yet be general enough to be applicable to the different phases of the system. For example, if the structured phase is a crystal, then the high degree of correlation in it should be reflected in the corresponding coarse-grained system. On the other hand, for a homogeneous liquid of constant density, the coarse-grained density should be the same as the original density. This is achieved by making the functional dependence of n¯ 0 (x) self-consistent and nonlocal:  n¯ 0 (x) =

dx w(x − x , n¯ 0 (x))n 0 (x ).

(2.2.2)

To satisfy the constraint that for a uniform system with density n 0 the weighted density should also be n 0 , we have the normalization condition 

dx w(x − x , n 0 ) = 1.

(2.2.3)

Equations (2.2.1) and (2.2.2), which form the basis of the weighted-density approximation (WDA), can in fact be justified from general considerations of the free-energy functionals. We demonstrate this in Appendix A2.3. The simplest functional dependence between the coarse-grained density n¯ 0 (x) and the physical n 0 (x) is that provided by Tarazona (1984, 1985). Here w is assumed to be independent of density. For a hard-sphere system, if the sphere of influence affecting the density at a point is taken to be the hard-sphere diameter σ , then w has the form w(r ) = 3/(4π σ 3 ) for r ≤ σ and is zero otherwise. Other forms have also been studied in the literature (Nordholm et al., 1980; Baus and Colot, 1985). As we will explain below, the weighting

74

The freezing transition

function can be computed in a self-consistent manner taking into account the correlations present in the inhomogeneous solid. The basic idea behind the WDA is to take into account how the short-ranged nonlocal effects in the interacting, inhomogeneous system (the crystal) influence its thermodynamic properties. In the WDA the excess free energy of the solid is interpreted as that of an effective liquid of much lower density than the average solid density. It follows then in a natural way that the low-density liquid is what is required to account for the correlations in the solid state. By choosing the weighting function in an optimal way, an approximate re-summation of functional expansion (in terms of density) of the exact excess free energy can be reached in the WDA. In general the weighting function serves to smooth out the sharp density variations over a spatial range by virtue of its being nonlocal in nature. Since the available information for the inhomogeneous liquid is minimal, the weighting function is fixed by using the known properties of the uniform liquid state. w is chosen such that the known relation between the two-point direct correlation function c(2) (x1 , x2 ; n 0 ) and the second functional derivative in density of the excess free energy Fex is strictly obeyed at the homogeneous limit of n 0 (x) → n 0 : 2 WDA

δ Fex [n] (2) . (2.2.4) c (x1 , x2 ; n 0 ) = −β δn(x1 )δn(x2 ) n(x)=n 0 By definition of the weighted density function n(x), ¯ the excess free energy is defined as  WDA [n 0 (x)] = dx n 0 (x) f 0 (n¯ 0 (x)), (2.2.5) Fex where f ex (n 0 ) is the free-energy density for the homogeneous liquid of density n 0 . The knowledge of a suitable function for f ex (n 0 ) is a required input in formulating the weighteddensity-functional theory of the inhomogeneous liquid. Using the WDA form of the excess free energy and evaluating the RHS of (2.2.4), one obtains (for the proof see Appendix A2.3) the following relation between the direct correlation function c(2) (x, n 0 ) of the homogeneous liquid and the WDA weighting function w,  −β −1 c(2) (x − x ; n 0 ) = 2 f ex (n 0 )w(x − x )   (n 0 ) dx1 w(x − x1 )w(x1 − x ) + n 0 f ex    + n 0 f ex (n 0 ) dx1 w(x − x1 )w  (x1 − x )

 + w (x − x1 )w(x1 − x ) .

(2.2.6)

Since the direct correlation function c(2) (x − x ; n 0 ) of the homogeneous liquid is known, the weighting function w is obtained from the solution of (2.2.6). Alternatively, the above equation is Fourier-transformed to obtain   −β −1 c(2) (k; n 0 ) = 2 f ex (n 0 )w(k; n 0 ) + n 0 f ex (n 0 )w 2 (k; n 0 )  (n 0 )w(k; n 0 )w  (k; n 0 ). + 2n 0 f ex

(2.2.7)

2.2 Weighted density functionals

75

In the zero-wave-number limit, since the normalization condition (2.2.3) implies w(0; n 0 ) → 1, the above relation reduces to the compressibility rule (Hansen and McDonald, 1986)   β −1 c(2) (0; n 0 ) = −2 f ex (n 0 ) − n 0 f ex (n 0 ).

(2.2.8)

Using c(2) (k; n 0 ) as input, eqn. (2.2.7) is solved for w(k), which is then inverse Fouriertransformed to obtain w(r ). Hence the weighted density n¯ 0 (x) is also obtained from (2.2.2). Let us consider the main features of the weighted-density-functional approach for the thermodynamic description of the inhomogeneous crystalline state outlined above. With the weighting function w determined from the solution of (2.2.6), a functional expansion of WDA about a uniform reference state is guaranteed to be identical to the corresponding −Fex expansion of −Fex to second order in density differences. The higher-order terms in this functional expression can also be obtained within the WDA by taking further functional WDA , since the relation derivatives of −Fex c(m) (k, 0, . . . ; n 0 ) =

∂ m−2 ∂n n−2 0

c(2) (k; n 0 )

(2.2.9)

between c(2) (k; n 0 ) and c(n) (k, k  , . . . ; n 0 ) is preserved for all m > 2. The WDA approximates the excess part of the free energy in a manner that is exact to lowest order in density variations with arbitrarily large spatial fluctuations typical of the crystalline state. It also includes terms to all higher orders in a density expansion, since it provides the approximate expressions for three-point and all higher-point direct correlation functions using the two-point functions of the homogeneous liquid.

2.2.1 The modified weighted-density approximation The weighted-density-functional approach for computing the free energy of the inhomogeneous liquid is simplified to a large extent by further refining the theory into a form that has been termed the modified weighted-density-functional approximation (MWDA) by Denton and Ashcroft (1989). In the MWDA the effective liquid in terms of which the correlations in the solid are expressed has a constant density nˆ 0 rather than the n¯ 0 (x) of the WDA. The excess free energy of the solid is now obtained in the form Fex = N f ex [nˆ 0 ]. The density nˆ 0 of the equivalent liquid is obtained from the global average   1 dx dx w(x ˜ − x , nˆ 0 ) nˆ 0 = N

(2.2.10)

(2.2.11)

in terms of a weighting function w˜ that is analogous to the corresponding quantity w of the WDA. Similarly to in the case of the WDA, the basic MWDA equations (2.2.10) and (2.2.11) can also be justified at a formal level. This is shown in Appendix A2.4. We also

76

The freezing transition

demonstrate in Appendix A2.4 that the weighting function w(x) ˜ of the MWDA is obtained in the form

1 nˆ 0   −1 (2)  w(x, ˜ x , nˆ 0 ) = −  β c (x, x ; n s ) + (2.2.12) f (nˆ 0 ) , 2 f ex V ex where f  represents the second derivative of the free-energy density f (n 0 ) with respect to the density. Now, on substituting this equation into the defining relation for the basic MWDA equation (2.2.11), we obtain the following self-consistent expression for the density nˆ 0 :

   1  nˆ 0 = −  nˆ 0 n s f ex (nˆ 0 ) + β −1 dx n 0 (x) dx n 0 (x ) c(2) (x, x ; nˆ 0 ) . (2.2.13) 2 f ex The above self-consistent relation is solved for a chosen density function n 0 (x) to obtain nˆ 0 . The only input required is the two-point direct correlation function c(2) for the uniform liquid state. The free energy of the inhomogeneous solid is obtained in terms of the relation (2.2.10). The necessary steps required to compute the free energy of the inhomogeneous solid in the WDA and in the MWDA are similar. The following inputs are necessary in both cases. (a) A suitable prescription for the inhomogeneous density function n 0 (x) of the solid state. (b) The direct correlation function c(2) (r ) of the homogeneous liquid. (c) The free-energy function f ex (n 0 ) for the homogeneous liquid of density n 0 . Parametrization of the inhomogeneous density function constitutes the basic step in the DFT model. We consider in the next section a standard choice of the inhomogeneous density function n 0 (x) for the crystalline state. 2.2.2 Gaussian density profiles In the stable crystalline phase for standard materials the root-mean-square displacements of the atoms are only a small fraction of the nearest-neighbor separations. The Hamiltonian for the crystal reduces to a harmonic expansion in terms of displacement from the lattice sites. The corresponding excitations of the dynamic crystalline state are the phonons. The inhomogeneous density function is suitably approximated as a sum of Gaussians,  3/2  α 2 e−α(x−R) , (2.2.14) n 0 (x) = π R

where the parameter α is the inverse width of the Gaussian. The α = 0 limit corresponds to the uniform liquid with infinitely wide Gaussian profiles and increasing values of α correspond to more strongly localized density profiles. The Lindemann parameter L0 associated with the melting of a solid is defined as the ratio of the root-mean-square displacement of an atom of the solid from its lattice site to the nearest-neighbor distance in the crystal at coexistence. The width parameter α is inversely proportional to L0 = 3/(αa 2 ),

2.2 Weighted density functionals

77

Fig. 2.2 (a) The radial density function 4πr 2 n 0 (r)σ for the Lennard-Jones particles at the freezing point measured from a lattice site with r in the (100) and (111) directions (solid line) and in the (110) direction (dashed line). The deviations of n 0 (r) from the Gaussian form n G is shown in (b) by the c American Institute of Physics. corresponding curves. Adapted from Laird et al. (1987). 

where a = (4/n s )1/3 is the lattice constant of the f.c.c. lattice. For the hard-sphere system the single-particle density function is also described quite accurately in terms of the above expression of the sum of Gaussian profiles. The suitability of the Gaussian density approximation is further clarified in Fig. 2.2, where the radial density function 4πr 2 n 0 (r)σ for a system of particles interacting through the Lennard-Jones (L-J) potential is shown (see eqn. (1.2.117) for the definition of the L-J potential). Figure 2.2 clearly shows that the density profile in the crystalline phase is well approximated in the Gaussian form. The single-parameter formulation (2.2.14) of the density function is simply related to the earlier expansion (2.1.42) in terms of RLVs. To see this, let us consider the Ramakrishnan– Yussouff parametrization of the inhomogeneous density function n 0 (x),1 1 Note that this expression is slightly different from that of (2.1.42).

78

The freezing transition

 n 0 (x) = n s 1 +

∞ 

 A˜ m eiKm · x .

(2.2.15)

m=1

By Fourier transforming the density function as given by both of the above two expressions, (2.2.14) and (2.2.15), we obtain for n 0 (Km ) Ns A˜ m = n 0 (Km )  3/2   α 2 = dx e−α(x−R) −Km · x . π

(2.2.16)

R

On completing the square in the exponential and evaluating the Gaussian integral we obtain the result  (2.2.17) A˜ m = exp −K m2 /(4α) , which links the order parameters A˜ m in terms of the width parameter α. The parametrization (2.2.14) for the density function is closely associated with the Debye model (McQuarie, 2000) of the crystal and is further discussed in Appendix A2.5. The parametrization of the density in terms of the Gaussian profiles has been particularly useful for describing the inhomogeneous solid, as we will demonstrate in a number of applications in this book.

2.2.3 The hard-sphere system In the weighted-density-functional theories (both WDA and MWDA) the free energy of the solid is obtained in terms of an effective liquid of much lower density. Accurate knowledge of the thermodynamic properties (equilibrium correlation functions) of the homogeneous liquid is therefore essential for computing those properties for the solid state. The hardsphere fluid satisfies the criteria very well since the various integral-equation theories provide a very good account of the correlations in a hard-sphere system at low densities. The self-consistent equations (2.2.7) and (2.2.13) of the WDA (Curtain and Ashcroft, 1985, 1986) and the MWDA (Denton and Ashcroft, 1989), respectively, are solved, using the direct correlation function c(r ) of the Percus–Yevick (PY) solution (see Chapter 1) for the hard-sphere fluid. The the excess-free-energy function f ex is obtained by computing the pressure P of the system from the corresponding equation of state which relates P to the density n 0 of the liquid. For the PY solution this is given by βP 1 + ϕ + ϕ2 = . n0 (1 − ϕ)3

(2.2.18)

ϕ = πn 0 σ 3 /6 is the packing fraction of the fluid and σ is the hard-sphere diameter. The excess free energy f ex is computed by integrating the equation

2.2 Weighted density functionals

β P/n 0 = 1 + ϕ

∂β f ex , ∂ϕ

79

(2.2.19)

leading to the result f ex

3 1 = − 1 − ln(1 − ϕ). 2 (1 − ϕ)2

(2.2.20)

The calculation of the thermodynamic properties of the inhomogeneous solid using the weighted-density-functional theory therefore involves the following basic steps. Calculation of the free energy of the solid For a chosen density function, determined by the width parameter α, the WDA (MWDA) free energy is calculated by obtaining the corresponding weight function w (w). ˜ Hence we obtain the density for the corresponding equivalent liquid n¯ 0 (nˆ 0 ) of the WDA (MWDA). The free energy thus computed is then minimized with respect to the parameter α of the test density function to obtain the density distribution for the thermodynamic state. By comparing the free energies of the liquid and solid states, the thermodynamic parameters for the freezing point are obtained. To determine the coexisting densities of the two phases, the respective grand potentials () are equated for two states having the same chemical potential (μ). The thermodynamically stable inhomogeneous structure for the crystalline state is identified from the minimum of the total free energy F = Fid + Fex . At a given density (below the freezing point) this free energy corresponds to a certain value of the width parameter α lying between the homogeneous-liquid limit of α → 0 and the strong-localization case of α → ∞. The key aspect of the weighted-density-functional approach for studying inhomogeneous solids can be understood as follows. For the very localized density profiles characteristic of the crystal, the expression (2.1.25) for the ideal-gas part Fid of the free energy reduces to the form ( )

5 α 3/2 β Fid (α)/N = − + ln 30 (2.2.21) 2 π corresponding to the large-α limit. It is monotonically increasing with α. On the other hand, the excess or interaction part of the free energy evaluated with the MWDA or the WDA approximation monotonically decreases, favoring strong localization of the particles around the lattice sites. Physically this implies that with increasing localization each particle feels the effects of its neighbor to a lesser extent. This is very different from the case of a high-density liquid in which the particles are strongly interacting. It is therefore possible to map the interaction contribution to the free energy of the solid to that of a uniform liquid of much lower density. The solid-state correlations are modeled by an effective liquid of much lower density.

80

The freezing transition

Let us now consider the explicit computation of the excess free energy of the hard-sphere crystal in the WDA and MWDA schemes. A. In the WDA the equation (2.2.7) is solved to obtain w(k), which is then Fouriertransformed to obtain w(r ). The density n¯ 0 (x) of the effective liquid is obtained for a given choice of the density function n 0 (x) (which is fixed by a corresponding value of α). The excess free energy of the liquid is obtained by computing locally the function f ex (n¯ 0 (x)). B. In the MWDA the density nˆ 0 for the equivalent liquid is obtained by solving selfconsistently the single equation (2.2.13) for a chosen value of the width parameter α. The excess free energy of the solid is then calculated as f ex (nˆ 0 ). Clearly the WDA calculation is computationally more cumbersome since calculation of the weight function w requires solving a differential equation. This is much simpler in the case of the MWDA, for which the density nˆ 0 is obtained at much less computational effort. Furthermore, unlike in the MWDA, in the WDA the excess free energy f ex has to be computed locally and then integrated over the system. In order to illustrate the weighted-density-functional approximation it will be useful to discuss the details of the calculation in the simplest case, i.e., the MWDA model for the freezing transition in a hard-sphere system. For the f.c.c. solid into which the liquid freezes, the lattice constant is given by a = (4/n s )1/3 . Self-consistent solution of (2.2.13) gives the equivalent liquid density nˆ 0 for a given value of α. The free-energy contributions corresponding to the ideal and excess parts are computed as functions of α, and the minimum of the total free energy at the optimum parameter value α ≈ 127 corresponds to the equilibrium density distribution. The corresponding weight function w˜ of the MWDA is obtained by using the self-consistent solution nˆ 0 on the RHS of (2.2.12). The curves for the liquidand solid-state free energies (see Fig. 2.3) cross at an intermediate density that marks the phase transition of the system from a homogeneous liquid state to a more localized density distribution with f.c.c. crystal structure. The free energy of the liquid state shown in Fig. 2.3 is computed from the Carnahan–Starling equation of state (see Chapter 1), which provides an accurate description of the thermodynamics near freezing point. A higher degree of mass localization (higher α values) occurs with increasing average density n s . Determination of the densities of the coexisting phases The densities of the coexisting phases in the MWDA are determined by a common-tangent construction to ensure that the appropriate thermodynamic conditions are satisfied. The chemical potential and the thermodynamic potential of the solid phase are computed using the relations ∂ f (n s ) , μ(n s ) = ∂n s (n s ) = F(n s ) − μ(n s )Ns . (2.2.22) The corresponding quantities for the uniform liquid state are obtained using the Carnahan– Starling equation of state. The coexisting solid and liquid densities satisfy simultaneously

2.2 Weighted density functionals

81

Table 2.2 Freezing parameters for the hard-sphere liquid–f.c.c.-solid transition: the average solid density n s , liquid density n l , change in density n 0 , latent heat per particle s, and Lindemann parameter L0 for the f.c.c. solid. RY, Ramakrishnan–Yussouf. Reproduced from Denton and Ashcroft (1989).

Simulation MWDA WDA RY

nsσ 3

nlσ 3

n 0 σ 3

s/kB

L0

1.04 1.036 1.045 1.147

0.94 0.910 0.916 0.967

0.10 0.13 0.13 0.18

1.16 1.35 1.41 2.24

0.126 0.097 0.093 0.055

7

f

6

5

4

3

0.9

1 n0

1.1

Fig. 2.3 Total free energies of the liquid (dashed curve) and f.c.c. solid (solid curve) phases of the hard-sphere system. The crossing point of the two curves indicates the freezing density above which c American the f.c.c. solid is the equilibrium state. Reproduced from Denton and Ashcroft (1989).  Physical Society.

the conditions (see Fig. 2.1) μ(n s ) = μ(n l )

and

(n s ) = (n l )

(2.2.23)

giving the coexisting pairs of liquid and solid densities. Thermodynamic properties such as the Lindemann parameter for the f.c.c. solid and the change of entropy at the freezing point as obtained from the different density-functional models as well as the results from the computer simulation of the hard-sphere system are compared in Table 2.2. The corresponding results for the free energy of the hard-sphere fluid and the thermodynamic pressures are compared in Table 2.3. It is worth noting that the WDA and MWDA give almost comparable results, though the computational effort for determining the WDA density n¯ 0 (x) is considerably more than that for nˆ 0 in the MWDA.

82

The freezing transition

Table 2.3 A comparison of results from different density-functional models for the free energy per particle F/N (in units of kB T) and the ratio of pressure to solid density n s , P/(n s kB T ) for the hard-sphere f.c.c. solid over a range of solid densities. Reproduced from Denton and Ashcroft (1989). F/(N kB T )

P/(n s kB T )

nsσ 3

MWDA

WDA

Simulation

MWDA

WDA

Simulation

1.000 1.025 1.050 1.075 1.100

4.412 4.629 4.853 5.090 5.347

4.449 4.674 4.908 5.156 5.423

4.661 4.868 5.099 5.354 5.663

8.51 8.96 9.61 10.58 11.90

8.83 9.40 10.09 11.06

10.25 10.81 11.49 12.30 13.26

The one-particle direct correlation function The one-particle direct correlation function c(1) defined in eqn. (2.1.32) represents the potential seen by a single particle in the inhomogeneous state and is an important thermodynamic property worth considering. In the case of the hard-sphere solid this is obtained using the MWDA. On substituting the expression (2.2.15) for the inhomogeneous density function n 0 (x) into the basic equation of the MWDA, i.e., eqn. (2.2.11) for the density nˆ 0 , we obtain 1  ˜2 nˆ 0 = n s + Am w˜ m (nˆ 0 ). (2.2.24) ns Km =0

As already noted above, the amplitude A˜ m is related to the width parameter α of the Gaussian approximation in (2.2.17). In order to obtain a suitable expression for c(1) it is convenient to treat the weight function w(r ˜ ) in terms of its Fourier transform w˜ m (nˆ 0 ). The latter is obtained from eqn. (2.2.12) as  1 (2)  w˜ m = −  + nˆ 0 δ K ,0 f ex (nˆ 0 ) , (2.2.25) β −1 cm 2 f ex (n) ˆ (2) is the Fourier transform of the two-point direct correlation function for the where cm homogeneous liquid of density nˆ 0 and δ K ,0 denotes the Kronecker delta function. Starting from the basic definition of the one-particle direct correlation function, we obtain the result (1)

δ f ex (nˆ 0 ) δ Ns f ex (nˆ 0 ) = f ex (nˆ 0 ) + Ns δn 0 (x) δn 0 (x) δ n ˆ 0  (nˆ 0 ) , = f ex (nˆ 0 ) + Ns f ex δn 0 (x)

−β −1 cMWDA (x, [n 0 ]) =

(2.2.26)

where we have used the relation δ Ns /δn 0 (x) = 1. The functional derivative [δ nˆ 0 /δn 0 (x)] in the second term on the RHS is computed by taking a derivative with respect to n 0 (x) of

2.2 Weighted density functionals

83

the basic equation (2.2.11) of the MWDA. Thus we obtain δ nˆ 0 1 = δn 0 (x) Ns



 6 2 dx1 w(x ˜ − x1 ; nˆ 0 )n 0 (x1 ) − nˆ 0 . 6 6 1 − Ns−1 dx1 dx2 w˜  (x1 − x2 ; nˆ 0 )n 0 (x1 )n 0 (x2 )

(2.2.27)

Using the Fourier transform of the two integrals appearing respectively in the numerator and the denominator of the RHS of the above expression, we obtain the following expression for c(1) in the MWDA: (1)  (nˆ 0 ) −β −1 cMWDA (x, [n]) = f ex (nˆ 0 ) + n s f ex  5 2 n s + m A˜ m w˜ m (nˆ 0 )eiKm · x − nˆ 0 × . 5  (nˆ ) n s − m A˜ 2m w˜ m 0

(2.2.28)

5 We have simplified the notation in the above equation by denoting m as the sum over RLVs Km with the prime implying the exclusion of the Km = 0 case. The RHS of the above formula is easily computed once the density nˆ and the corresponding weight function w˜ are also known from the self-consistent solution of the basic MWDA equations. The result for the one-particle direct correlation function c(1) is displayed in Fig. 2.4 for the hard-sphere f.c.c. solid. The one-body potential represented by c(1) is lowest at the lattice sites (where the density is highest) and highest in the interstitial regions (where the density is lowest). This is also plausible according to relation (2.1.33), since the location of the minimum potential represents the equilibrium position for a particle.

(1)

Fig. 2.4 The one-particle direct correlation function cMWDA (r ) in the MWDA approximation of a hard-sphere solid vs. the separation between two points along the three symmetry directions of the c American Physical Society. cubic unit cell. Reproduced from Denton and Ashcroft (1989). 

84

The freezing transition

An interesting application of the density-functional model for the hard-sphere solid is first-principles computation of the elastic constants starting from the basic interaction potential between the particles. The elastic properties of the equilibrium crystalline state are obtained from the second functional derivative of the free-energy functional with respect to the density function. When the free energy of the solid state was evaluated using perturbative expansions like the Ramakrishnan–Yussouff model described above, unphysical results of negative elastic constants (Jari´c and Mohanty, 1987; Jones, 1987) were obtained. Subsequently this was improved by performing nonperturbative calculations using weighted-density-functional approaches (Velasco and Tarazona, 1987; Xu and Baus, 1988). These results are in good agreement with simulation results (Frenkel and Ladd, 1987; Runge and Chester, 1987). The weighted-density-functional models WDA and MWDA that we employed above to describe the inhomogeneous state below freezing represent the primary development in this approach. They constitute the ingredients of the theoretical approach for dealing with the nonuniform state in terms of an equivalent uniform liquid. Another approach, which is somewhat similar to the MWDA, is the generalized effective-liquid approximation (GELA) (Lutsko and Baus, 1990). This model uses a different prescription for the positionindependent weighted density for the effective liquid in terms of which the crystalline state is described. The purely repulsive hard-core systems considered in this section have some specific characteristics peculiar to them. These can be attributed to the success of the weighteddensity approximation (keeping up to second order in density fluctuations) at describing accurately hard-core systems in terms of an equivalent low-density fluid. For the Hamiltonian with a purely hard-sphere interaction no expansion in terms of displacements from equilibrium sites exists. The hard-sphere crystal is therefore somewhat anomalous when viewed from the perspective of usual descriptions of lattice dynamics. The system is entirely controlled by collisions and the motion of the particles between collisions loses coherence very rapidly. This particular aspect of the hard-sphere crystal is reflected in its thermodynamic properties being successfully computed in terms of an equivalent liquid of much lower density. The ballistic motion of the freely moving hard spheres in the crystal between collisions is quite analogous to the corresponding motion of the particles in the low-density fluid. However, the above-described similarity in the case of the purely anharmonic hardsphere crystal is peculiar to itself. For the 1/r n -type potential (n → ∞ is the hard-sphere potential), as n approaches values more typical of short-range interactions in real systems, the coherence in the motion of the particles increases. In such cases, unlike for the hardsphere solid, the similarity between the correlations in the liquid and those in the solid is much less. As a result the success of the weighted-density-functional theories at providing understanding of freezing in systems with softer interactions has been limited. For the hard-sphere crystal the average domain of motion of a particular sphere is constrained in space over a scale determined largely by the range of the direct correlation function at the corresponding density. The range of the direct correlation function c(2) increases considerably with that of the interaction potential (n becoming smaller)

2.3 Fundamental measure theory

85

and hence the coarse-graining length scale for the corresponding weight function (over which the inhomogeneous density n 0 (x) should be averaged to obtain the weighted density n¯ 0 (x)) increases. This implies that it is appropriate include higher-order correlations like c(3) in the calculation of the weight function for the systems with softer potentials. For example, in the MWDA formulation for plasma (n = 1) inclusion of the three-particle correlation is required to obtain the freezing (Likos and Ashcroft, 1992). For hard-sphere systems (n → ∞), on the other hand, the third-order extension merely leads to additional improvement of already satisfactory results. 2.3 Fundamental measure theory A generalization of the scheme of describing the inhomogeneous state using the idea of weighted density has been formulated for the hard-sphere fluid and is termed the fundamental measure theory (FMT) (Rosenfeld 1988, 1989; Rosenfeld et al., 1990). In the fundamental measure theory the free energy of the inhomogeneous state is obtained in terms of not just one density but several weighted densities. The corresponding weight functions are independent of density, and are constructed by making use of the geometric properties of the constituent particles. Let us consider a one-component system of hard spheres of radius R (diameter σ = 2R). In the FMT the excess free energy Fex is expressed as  (2.3.1) β Fex [n 0 (x)] = dx ex [{n¯ i (x)}]. The (excess) free-energy density ex is assumed to be a functional of several weighted densities n¯ i (for i = 1, 2, 3, . . .) respectively defined in terms of density-independent weight functions wi (x),  n¯ i (x) = dx wi (x − x )n 0 (x). (2.3.2) The corresponding functional derivative of n¯ i with respect to the density n(x) is obtained as δ n¯ i (x ) (2.3.3) = wi (x − x ). δn 0 (x) Using (2.3.3) in the definition (2.1.36) for the two-point direct correlation function as the second functional derivative of the excess free energy, we obtain  δ 2 ex δ 2 Fex c(x1 , x2 ) = −β dx wi (x1 − x)w j (x2 − x). (2.3.4) =− δn 0 (x1 )δn 0 (x2 ) δ n¯ i δ n¯ j i, j

In the uniform-fluid limit, the functional derivative reduces to the partial derivatives and we obtain for the direct correlation function  ∂ 2 ex

c(x1 , x2 ) = (2.3.5) wi ⊗ w j , ∂ n¯ i ∂ n¯ j i, j

where ⊗ represents the convolution of the two weighting functions wi (x1 ) and w j (x2 ).

86

The freezing transition

A key observation in the formulation of the FMT is that the Percus–Yevick (PY) expression for the direct correlation function of the homogeneous hard-sphere fluid can be written in the form of the RHS of (2.3.5). For this a proper choice has to be made for the set of weight functions, which are defined in terms of the characteristic geometry of two overlapping hard spheres.

2.3.1 Density-independent weight functions The characteristic functions which relate to the geometry of a single hard sphere of radius R are identified as 4 R(3) = π R 3 , R(2) = 4π R 2 , 3 R(1) = R,

R(0) = 1.

(2.3.6)

Around the center of a sphere there is a sphere of radius σ = 2R that the center of another hard sphere cannot enter without producing an overlap. The overlap volume V and the overlap surface area S of two identical spheres with their centers at a distance r < 2R are obtained in terms of x = r/(2R) as  V(r ) = (r  − R)(|r − r | − R)dr = R3 (1 − 3x + 4x 3 )(r  − 2R),  S(r ) = 2 (r  − R)δ(|r − r | − R)dr = R2 (1 − r/(2R))(r − 2R),

(2.3.7)

(2.3.8)

where (x) is the step function, which is equal to 1 and 0 for x < 0 and x > 0, respectively. The expression (1.2.95) for c(r ) presented in Chapter 1 is given in terms of the overlap volume V and overlap surface S as −c(r ) = χ (3) V(r ) + χ (2) S(r ) + χ (1) R(r ) + (r − 2R)χ0

(2.3.9)

by using the following two definitions. 1. The density-dependent coefficients χ (i) are expressed as χ (1) = 2 χ02 ,

χ (2) = 1 χ02 +

χ (3) = 0 χ02 + 21 2 χ03 +

23 4 χ , 4π 0

22 3 χ , 4π 0 (2.3.10)

where χ0 and i for i = 0, . . . , 3 are defined in terms of the fundamental measures R(i) of a sphere defined in (2.3.6) as χ0 = 1/(1 − 3 ), (i)

i = n 0 R .

(2.3.11) (2.3.12)

2.3 Fundamental measure theory

87

Note that 3 and 0 are equal to the packing fraction ϕ and the density n 0 of the uniform hard-sphere fluid, respectively. 2. The so-called overlap radius R(r ) of the two spheres with their centers at a distance r is defined as R = R − r/4 = R(1 − x/2). R is expressed in terms of S as   r S(r ) R R(r ) = R − (r − 2R) = + (r − 2R). (2.3.13) 4 8π R 2 The expression (2.3.9) for the direct correlation function reduces to the form (2.3.5) with a proper choice for the wi . In this respect we define the functions w(3) and w (2) which are related to the volume and surface area, respectively, of a sphere of radius R, w (3) (r) = (|r| − R),

w(2) = δ(|r| − R).

(2.3.14)

The characteristic functions V and S for two overlapping spheres defined in eqns. (2.3.7) and (2.3.8) are obtained as simple convolutions of w(2) and w(3) as  (3) (3) (2.3.15) V(r ) = w ⊗ w ≡ dr w (3) (r − r )w (3) (r ),  (2.3.16) S(r ) = 2w (3) ⊗ w(2) ≡ 2 dr w (3) (r − r )w (2) (r ). In order to obtain the function R defined in (2.3.13) in terms of the characteristic functions of a sphere, we need to have a representation of the step function (r − 2R) using basic functions similar to those defined in (2.3.14). This requires extending the set of w(i) to include two more scalar functions defined as w (0) (r) , 4π R 2 and two more vector functions denoted by w (0) (r) =

w (1) (r) =

w (0) (r) 4π R

(2.3.17)

r w(2) (r) = ∇w (3) (r) ≡ δ(|r| − R), r w(2) (r) . (2.3.18) w(1) (r) = 4π R The step function  is represented in terms of the basis functions which include the scalar functions w(i) , i = 0, 1, 2, 3, and the vector functions w(1) and w(2) ,   = 2 w(3) ⊗ w(0) + w (2) ⊗ w(1) + w(2) ⊗ w(1) . (2.3.19) The convolution involving w(1) and w(2) appearing in the last term on the RHS implies that of the scalar product of the two vectors. Using the relations (2.3.19) and (2.3.15), the expression (2.3.9) for the direct correlation function for the homogeneous hard-sphere fluid reduces to the form (2.3.5) with the weight functions wi and w j both running over the whole set {w (0) , w (1) , w (2) , w (3) , w(1) , w(2) } constructed from the characteristic volume and surface function of the hard sphere. The partial-derivative matrix [∂ 2 ex /∂ n¯ i ∂ n¯ j ] for the homogeneous fluid is obtained as a function of the density n 0 and the radius R of the hard sphere.

88

The freezing transition

2.3.2 The free-energy functional The free-energy density ex is constructed in terms of the linear combinations of the weighted densities and their products. The general form for ex in the spirit of a virial expansion is obtained as   ex [{n¯ i (x)}] = 1 [n¯ i (x)] + 2 [n¯ i (x)n¯ j (x)] i

+



i, j

3 [n¯ i (x)n¯ j (x)n¯ k (x)] + · · ·.

(2.3.20)

i, j,k

The weighted densities n i for the inhomogeneous system are defined in terms of the corresponding (density-independent) weight functions as  i = 0 to 3, (2.3.21) n¯ i (x) = dx wi (x − x )n 0 (x),  n¯ i (x) = dx wi (x − x )n 0 (x), i = 1, 2. (2.3.22) The dimension of the scalar density n¯ i (x) is [L]i−3 (for i = 0 to 3) while the dimension of n¯ i is the same as that of n¯ i (for i = 1, 2). Since ex on the LHS of (2.3.20) has the dimension of the density L −3 (see the definition (2.3.1) for ex ), the free-energy density is expressed (from dimensional considerations) as a linear combination of n¯ 0 , n¯ 1 n¯ 2 , n¯ 32 , n¯ 1 · n¯ 2 , and n¯ 2 (n¯ 2 · n¯ 2 ), keeping terms of up to third order in the expansion (2.3.20), ex [{n¯ i (x)}] = γ (0) n¯ 0 (x) + γ (1) n¯ 1 (x)n¯ 2 (x) + γ (2) n¯ 32 (x) + γ (3) [n¯ 1 (x) · n¯ 2 (x)] + γ (4) n¯ 2 (x)[n¯ 2 (x) · n¯ 2 (x)],

(2.3.23)

where the coefficients γ (i) (i = 1, . . . , 5) in the expansion are functionals of the dimensionless density n¯ 3 (x). In the uniform-fluid limit, the expression (2.3.23) reduces to the form ex [{i }] = γ (0) 0 + γ (1) 1 2 + γ (2) 23 ,

(2.3.24)

since in this case the scalar density n¯ i (for i = 0 to 3) reduces to i and n¯ i = 0 (for i = 1, 2). Determining the coefficients γ (i) (i = 1, . . . , 5) in the expansion (2.3.23) requires the introduction of further approximations. Here once again the analogy with the uniformfluid state is used. By considering a single solute particle in a uniform hard-sphere fluid, the bulk pressure P is obtained. The bulk pressure is identified with the partial derivative of ex with respect to the packing fraction 3 , using the formulation of the scaled-particle theory (Reiss et al., 1959), ∂ex = β P ≡ 0 + β Pex , ∂3

(2.3.25)

2.3 Fundamental measure theory

89

where Pex denotes the contribution to the pressure from the excess part of the free energy. This relation is now assumed to be valid for the inhomogeneous fluid, in order to obtain the following differential equation: δex (x) = n¯ 0 (x) + β Pex [n 0 (x)]. δ n¯ 3 (x)

(2.3.26)

The functional Pex [n 0 (x)] is obtained from the definition of the thermodynamic potential functional 0 [n(x)] as   δ Fex [n 0 (x)] ≡ − Pex [n 0 (x)]dx = Fex − n 0 (x) dx. (2.3.27) δn 0 (x) On computing the functional derivative of Fex with density in terms of the corresponding derivatives with respect to the weighted densities n¯ i (x) and using (2.3.3), the relation (2.3.27) gives the following result for Pex [n(x)]: β Pex [n 0 (x)] = −ex [n¯ i (x)] +



n¯ i (x)

i

δex (x) , δ n¯ i (x)

(2.3.28)

with i running over the whole set of weighted densities denoted in (2.3.21) and (2.3.22). On combining eqns. (2.3.26) and (2.3.28) we obtain the following equation for ex :  ∂ex ∂ex = n¯ 0 − ex [n¯ i ] + n¯ i , ∂ n¯ 3 ∂ n¯ i

(2.3.29)

i

where we have replaced the functional derivatives appearing in the local relation by corresponding partial derivatives with respect to the weighted densities. Now, substituting the expression (2.3.23) for the excess free energy ex , and taking the terms involving the weighted densities and their products to be independent, we obtain the following five differential equations for the coefficients γ (i) for i = 0, . . . , 5: ∂γ (0) 1 = , ∂ n¯ 3 1 − n¯ 3 2γ (i) ∂γ (i) = ∂ n¯ 3 1 − n¯ 3

∂γ (i) γ (i) = ∂ n¯ 3 1 − n¯ 3

for i = 1, 3,

for i = 2, 4.

(2.3.30)

γ (0) = − ln(1 − n¯ 3 ) + c0 , c1 c2 , γ (2) = , γ (1) = 1 − n¯ 3 (1 − n¯ 3 )2 c3 c4 γ (3) = , γ (4) = . 1 − n¯ 3 (1 − n¯ 3 )2

(2.3.31)

The solutions of these equations are

(2.3.32)

The constants ci (for i = 1, . . . , 5) are determined by ensuring that for the uniform hardsphere fluid the two-point function c(r ) obtained from eqns. (2.3.5) and (2.3.24) matches with the corresponding Percus–Yevick expression, i.e., the RHS of eqn. (2.2.20). From this

90

The freezing transition

one obtains the constants as c0 = 0, c1 = 1, c2 = 1/(24π ), c3 = −1, and c4 = −1/(8π ). Using these results, the excess-free-energy functional is obtained as ex [{n¯ i (x)}] = −n¯ 0 ln(1 − n¯ 3 ) +

n¯ 3 − 3n¯ 2 (n¯ 2 · n¯ 2 ) n¯ 1 n¯ 2 − n¯ 1 · n¯ 2 + 2 . 1 − n¯ 3 24π (1 − n¯ 3 )2

(2.3.33)

The free-energy expression (2.3.24) can be changed in such a manner that in the uniformfluid limit the Carnahan–Starling (instead of the Percus–Yevick) expression of the free energy is obtained (Roth et al., 2002). The theory is easily generalized (Rosenfeld et al., 1990; Rosenfeld et al., 1997) to mixtures of hard spheres by defining a set of weight functions and hence weighted densities for each species. In formulating the FMT optimal use of the geometric considerations is made in obtaining the correlations in the uniform hard-sphere fluid in terms of the weight functions wi . The FMT by construction ensures that in the uniform-fluid limit the Percus–Yevick (PY) result for the direct correlation function is reproduced. An alternative approach by Kierlik and Rosinberg (1990) avoids use of the vector weight functions discussed above and obtains similar results with only four scalar weight functions and hence four weighted densities. Here the description of the inhomogeneous state in terms of density-independent weight functions replaces that with a density-dependent single weight function in the WDA (Curtain and Ashcroft, 1985). It is not clear a priori whether this is necessarily a better description of the inhomogeneous state. When applied to the freezing problem, the FMT indicates a very sharp decrease of the free energy, with the width of the Gaussian density profiles becoming unrealistically large. For large values of the width parameter α, i.e., for extremely localized density profiles, the local packing fraction given by the weighted density ϕ¯ 0 = n¯ 3 (x) is almost unity, the underlying reason being the short range (radius σ/2 of the hard sphere) of w3 in this case. In the WDA this problem is avoided since the corresponding weight function has a range σ and the local packing fraction π n¯ 0 σ 3 /6 is less than 0.5, implying that the inhomogeneous solid is mapped into a low-density uniform liquid. Thus application of the FMT is not very suitable for the freezing problem. In fact, here the free energy of the solid is never less than that of the liquid (which corresponds to the case α = 0). The more useful application of the FMT is to the calculation of the higher-order correlations in the uniform-liquid state (Kierlik and Rosinberg, 1990), for which good agreement with Monte Carlo results is obtained. The FMT has recently been applied in the study of systems beyond simple hard-core repulsion by Lutsko (2008).

2.4 Applications to other systems The basic idea of the density-functional theory outlined above has been applied for studying different systems including mixtures, liquid crystals, flux-lattice melting in superconductors, quasicrystals, glasses, fluids in confined geometries, wetting, adsorption and the nature of interfaces. It has also been applied to the study of liquid–solid phase transitions in systems having realistic nonsingular interactions. The inhomogeneous density function

2.4 Applications to other systems

91

used to describe the crystalline state has been modified to include the presence of vacancy defects in a crystal and obtains an estimation of the equilibrium density of vacancies. Here the basic interaction potential between the particles is the only input used in the theory. A general description of these extensions can be found in Henderson (1992). In the present section we discuss two important applications linked to the basic theme of this chapter, namely the study of systems with long-range attractive forces and the nature of liquid–solid interfaces.

2.4.1 Long-range interaction potentials So far we have discussed the density-functional theories of the freezing of a hard-sphere fluid. Geometric packing considerations play an important role in determining the thermodynamic properties of the condensed state. However, inclusion of only the hard-core form of the interaction potential in the Hamiltonian is inadequate for understanding the symmetry-breaking transition in real systems and hence consideration of more physical interactions is required. The crystalline phase is generally characterized by strong coherence, which is absent in the hard-sphere case. In the present section we discuss recent developments of models applicable to liquids having more realistic interaction potentials going beyond the simple hard-sphere type. For the Ramakrishnan and Yussouff model discussed in Section 2.1.3, of course, there is no specific limitation to the hard-sphere interaction potential. Only the structural properties of the homogeneous liquid state in terms of the direct correlation functions are required as input in solving the model equations and hence the theory can, in principle, be applied to systems with any interaction potential. However, the weighted-density-functional models described above require accurate knowledge of the equation of state of the homogeneous liquid in order to obtain the correlations corresponding to the inhomogeneous crystalline state. For the hard-sphere fluid such information is readily available from various standard integral-equation approaches. Extending the density-functional model to systems with attractive potentials, e.g., Lennard-Jones interaction, requires special treatment of the long-range attractive forces between the particles. The perturbation theory We now discuss formulation of the density-functional theory for a system with a realistic interaction potential that has a long-range attractive part. The treatment is analogous to the earlier perturbative treatment of Weeks et al. (1971) (referred to above as the WCA theory) for computing the thermodynamic properties of such systems in the homogeneous liquid state. Let us consider a system interacting through a two-body potential that is characterized in terms of a reference potential and a perturbation, involving a small parameter κ, as u(x12 ; κ) = u R (x12 ) + κu P (x12 ),

(2.4.1)

representing the reference and the perturbative components of the two-body potential, respectively. To formulate the density-functional theory of the inhomogeneous state of the system with a realistic interaction potential u(x12 ) as described above, we first construct

92

The freezing transition

the appropriate free-energy functional. A formal expression for the free energy is obtained by starting from the exact definition of the two-point distribution function f 2 (x1 , x2 ) as f 2 (x1 , x2 ) = n(x ˆ 1 )n(x ˆ 2 ),

(2.4.2)

where nˆ represents the microscopic one-particle density and the angular brackets refer to the average over the equilibrium distribution. The two-point distribution function represents the probability of particles 1 and 2, respectively, being simultaneously at x1 and x2 . The Boltzmann factor for obtaining the statistical average on the RHS of (2.4.2) is determined by the potential-energy part V of the Hamiltonian H. The dependence of the total potential energy V on spatial coordinates is expressed in terms of the two-body potential u(x12 ; κ) as follows:   1 dx1 dx2 n(x ˆ 1 )n(x ˆ 2 )u(x12 ; κ). (2.4.3) V(2) = 2 From eqn. (2.4.3) it follows that the functional derivative of the partition function  Z= exp[−β H ] (2.4.4) with respect to the two-body potential u(x12 ) involves the product of two one-point density functions appearing between the angular brackets on the RHS of (2.4.2). Since the Helmholtz free energy is given by F = −kB T ln Z ,

(2.4.5)

the corresponding two-point function denoted by f 2 (x1 , x2 ; κ) is obtained formally in terms of a functional derivative with respect to the two-body potential, f 2 (x1 , x2 ; κ) =

δ F[n 0 (x)] . δu(x12 ; κ)

We integrate the relation (2.4.6) formally along κ = 0 to 1 to obtain the result   1 1 dκ dx1 dx2 f 2 (x1 , x2 ; κ)u P (x12 ) ≡ FP , F[n 0 (x)] − FR [n 0 (x)] = 2 0

(2.4.6)

(2.4.7)

where FR [n(x)] is the Helmholtz free energy of the reference system having the interaction potential u R . To lowest order in κ we obtain FP as  1 (2.4.8) dx1 dx2 f 2R (x1 , x2 )u P (x12 ), FP = 2 where f 2R (x1 , x2 ) denotes the two-particle distribution function for the system with interaction potential u R . Thus the free energy of the solid with interaction potential u(x12 ) is obtained as  1 (2.4.9) dx1 dx2 f 2R (x1 , x2 )u P (x12 ). F[n 0 (x)] = FR [n 0 (x)] + 2

2.4 Applications to other systems

93

In general an angularly averaged quantity 7 f 2R (x12 ) for the two-point function f 2R (x1 , x2 ) is defined as   d dx1 R 2 7R n 0 f 2 (x) = (2.4.10) f (x1 , x1 + x), 4π V 2 d being the differential solid angle around x. Using the definition (2.4.10) for the angularly averaged correlation function 7 f we obtain from (2.4.9) the following expression for the free energy per particle of the inhomogeneous state for the system with interaction potential u:  f [n 0 (x)] = f R [n 0 (x)] + 2π n 0 dr r 2 7 (2.4.11) f 2R (r )u P (r ). Thus both the free energy f R and the angularly averaged two-point distribution function 7 f 2R (r ) of the reference system are required in order to construct the free-energy functional for the system with interaction potential u(x 12 ). Since the thermodynamic properties of the hard-sphere system are well known, it becomes a natural choice for the reference system. The interaction potential for the liquid is divided into a harshly repulsive part (representing a hard core) and a long-range attractive part to be treated as a weak perturbation. The next step in this calculation is therefore to choose a suitable description for the reference system with interaction potential u R in terms of an equivalent hard-sphere system. The diameter of the hard-sphere system is determined by setting suitable conditions, to be described below. The reference system For a short-range repulsive u R (x12 ), the free-energy functional FR [n(x)] of the reference system is obtained in terms of an equivalent hard-sphere system. The diameter dE of this equivalent hard-sphere system is treated as an adjustable parameter in the theory and is determined by applying a suitable criterion obtained from thermodynamic considerations. To identify dE , the free energy of the reference system is written in terms of an expansion around that for the hard-sphere system in powers of a blip function e(x1 , x2 ),   δ FHS [n 0 (x)] 1 FR = FHS + (2.4.12) dx1 dx2 e(x1 , x2 ) + · · ·, 2 δeHS (x1 , x2 ) where e(x1 , x2 ) = exp(−βu(x12 ))

(2.4.13)

is the Boltzmann factor for the interaction potential u(x12 ). The blip function is defined as e(x1 , x2 ) = eR (x1 , x2 ) − eHS (x1 , x2 ),

(2.4.14)

namely as the difference between the Boltzmann factors for the reference potential u R and a hard-sphere potential u HS . If the range of the reference potential is comparable to the hardsphere diameter then the function e is nonzero only over a range of ξ dHS with ξ < 1, whence the name blip function. Since the potential u is related to the Boltzmann factor e

94

The freezing transition

as −βu = ln e, it follows from (2.4.6) that the functional derivative of the Helmholtz free energy with respect to eHS (x1 , x2 ) is related to the two-point distribution function, −β

δ F[n 0 (x)] = f 2 (x1 , x2 )eβu HS (x12 ) ≡ yHS (x1 , x2 ). δe(x1 , x2 )

(2.4.15)

On using the relation (2.4.15) and evaluating the angular part of the integral on the RHS of (2.4.12) we obtain the result for the free energy as  (2.4.16) FR [n 0 (r)] = FHS [n 0 (r)] + 2π V dr r 2 y˜HS (r )e(r ) + O((e)2 ), where y˜HS (r ) denotes the angular average of the hard-sphere function yHS (x1 , x2 ) and is obtained by averaging the latter over all orientations. The diameter dE of the equivalent hard-sphere system with potential u HS is adjusted so as to ensure that the first-order term in the expansion of eqn. (2.4.16) for the free energy vanishes. This is done by solving the relation  ∞  ∞ dr r 2 y˜HS (r ) = dr r 2 y˜HS (r )e−βu R , (2.4.17) dE

0

where dE is the diameter of the equivalent hard-sphere system. The discussion so far is applicable to an equivalent hard-sphere system irrespective of whether the system is homogeneous or not. For the isotropic liquid state translational invariance holds and the angularly averaged quantity y˜HS (r ) → gHS (r ), the uniform liquid-state radial distribution function for the reference system. From the solution of eqn. (2.4.17) using the pair correlation function gHS we obtain the corresponding value of the diameter dE of the equivf 2R (r ) the alent (uniform) hard-sphere system. For the above two key quantities f R and 7 respective functional dependences on density are assumed to be the same as those of the corresponding quantities for an equivalent hard-sphere system of radius dE . The free energy The free energy of the system with interaction potential u in the inhomogeneous solid state is computed by making use of the above perturbation theory. We will discuss below two approaches along these lines. Model A The free energy f [n 0 (r)] per particle of the inhomogeneous system with interaction potential u is obtained as an expansion around the corresponding uniform liquid state (of density n 0 ) constructed in terms of direct correlation functions of the liquid,   ∞  1 f [n 0 (r)] = f [n 0 ] − dr . . . drn c(n) (r1 , . . . , rn )n 0 (r1 ) . . . n 0 (rn ), β N n! n=2

(2.4.18) where c(n) is the n-particle direct correlation function of the liquid of density n 0 . For the free energy f (n 0 ) of the uniform state we substitute the corresponding limit of the RHS of eqn. (2.4.11),

2.4 Applications to other systems



f [n 0 (r)] = f R [n 0 ] + 2πn 0

dr r 2 7 f 2R (r )u P (r ).

95

(2.4.19)

We now choose the uniform reference system of density n 0 , in terms of an equivalent hard-sphere system. The expansion (2.4.18) reduces in terms of the equivalent hard-sphere system (as 7 f 2R (r ) → gHS (r )) to  f [n 0 (r)] = f HS [n 0 ] + 2πn 0 −

∞  n=2

1 β N n!

dr r 2 gHS (r ; n 0 )u P (r )





drn c(n) (r1 , . . . , rn )n 0 (r1 ) . . . n 0 (rn ), (2.4.20)

dr . . .

Next we write the direct correlation function of the system in terms of that of the hard(n) sphere system as c(n) = c(n) + cHS . Substituting this form of the direct correlation function into the above equation gives f [n(r )] = f HS [n 0 ] −  + 2πn 0 −

∞  n=2

∞ 

1 β N n!

n=2



 dr . . .

(n) drn cHS (r1 , . . . , rn )n 0 (r1 ) . . . n 0 (rn )

dr r 2 gHS (r )u P (r )

1 β N n!



 dr . . .

drn c(n) (r1 , . . . , rn )n 0 (r1 ) . . . n 0 (rn ). (2.4.21)

The sum of the first two terms on the RHS of eqn. (2.4.21) is equal to the inhomogeneousstate free energy of the hard-sphere system, giving the result  f [n(r)] = f HS [n 0 ] + 2πn 0 −

∞  n=2

1 β N n!

dr r 2 gHS (r )u P (r )



 dr . . .

drn c(n) (r1 , . . . , rn )n 0 (r1 ) . . . n 0 (rn ). (2.4.22)

Let us now consider the example of the Lennard-Jones (LJ) interaction potential u(r ) ≡ u LJ (r ) which is defined for two particles at a separation r in eqn. (1.2.117). u LJ is divided LJ into two parts, namely a hard-core repulsive part, u LJ R , and an attractive potential, u P , respectively defined as & u LJ R

=

w(r ) + 0 ,

for r < σ0 ,

0,

for r > σ0 ,

(2.4.23)

96

The freezing transition

and

& u LJ P

=

−0 ,

for r < σ0 ,

w(r ), for r > σ0 ,

(2.4.24)

where σ0 = 21/6 σ and w(r ) is the RHS of eqn. (1.2.117). For the homogeneous liquid state, its thermodynamic properties are obtained by treating the weak attraction u LJ P as a . perturbation over a reference system having an interaction potential u LJ R The free energy of the solid is determined by evaluating the RHS of eqn. (2.4.22). As a typical case the first term on the RHS for the free energy of a hard-sphere system is computed using the MWDA method. The second term is a mean-field term (de Kuijper et al., 1990) that depends on the pair correlation function gHS (r ) of the uniform hard-sphere fluid of diameter dE . The third term depends on the density function of the inhomogeneous system and on the form of the c(n) . For simplicity we neglect the higher-order terms in c(n) (n > 2). Estimations of c(2) are made for large and small distances with simple approximations. For short distances both systems have a hard core and hence c(2) → 0. This ensures that the self-interaction does not influence the results. On the other hand, for large r , e.g., beyond the potential-well minimum at σ0 , the direct correlation functions are approximately equal to the potentials and hence the difference is simply w(r ). Over intermediate distances of the order of the diameter of the equivalent hard sphere, c(2) (r ) is different. The following approximation for c(2) has been used (Curtain and Ashcroft, 1986): ⎧ ⎪ 0 < r < rnn , ⎪ ⎨ 0, (2) (2.4.25) c = −β0 , rnn ≤ r < 21/6 , ⎪ ⎪ ⎩−βw(r ), 21/6 ≤ r, where rnn is a parameter related to the inter-particle spacing in the crystal and 0 is the depth of the potential well. The free energy of the stable solid is obtained by minimizing it with respect to the variations of the parameters of the density function n(r). In Fig. 2.5 the temperature–density phase diagram of the Lennard-Jones system as calculated by application of the present theory is shown. The Lindemann parameter L0 remains nearly constant at about 0.12–0.13 along the coexistence line. Model B A somewhat different scheme for computing the free energy of the inhomogeneous state for systems with interaction potentials having a long-range attractive part has been developed by Rascón et al. (1996). In this case the WCA approach is applied directly to the inhomogeneous solid state and the free energy is computed from eqn. (2.4.11) in terms of the reference state which itself is inhomogeneous. By using the hard-sphere 7 f 2 (r ) functional of the inhomogeneous solid state described above, a better estimation for the mean-field contribution to the free energy is obtained. The calculation of the corresponding two-particle distribution function f 2 requires some special steps. An approximate evaluation of the two-point distribution function f 2 (x1 , x2 ) for the hard-sphere system as a functional of the density n(x) for a Lennard-Jones interaction potential is done as follows.

2.4 Applications to other systems

97

Fig. 2.5 The phase diagram of a Lennard-Jones system obtained from model A is shown as a solid line. The corresponding computer-simulation results are shown as circles. Reproduced from Curtain c American Physical Society. and Ashcroft (1986). 

The two-point distribution function f 2 . Since the individual particles in a crystal are sharply localized to their corresponding lattice sites, f 2 (x1 , x2 ) is approximated at the simplest level as the product of the individual average single-particle densities, f 2 (x1 , x2 ) = n 0 (x1 )n 0 (x2 ).

(2.4.26)

The above approximation for f 2 , which is defined in (2.4.2), amounts to neglecting the twopoint correlation in the crystal and is better applicable when the separation |x1 −x2 | is large. To proceed further we need the inhomogeneous density function for the solid state. The inhomogeneous density n 0 (x) for the hard-sphere system is expressed as a sum of Gaussian profiles as defined in (2.2.14) parametrized in terms of the WP αH (say). On substituting this ansatz for n 0 (x) into the expression (2.4.26), and taking an angular average as defined in (2.4.10), we obtain an approximate expression 7 f 20 (r ) for the two-point function:  7 f 2i0 (r ), (2.4.27) f 20 (r ) = i

where the functions f 2i0 (r ) =

n0 4π

f 2i0 (r ) 9

represent Gaussian peaks,  

αH wi αH 2 (r − Ri ) , exp − 2π r Ri 2

i = 2, 3, . . .

(2.4.28)

Here Ri is the radius of the ith shell around a lattice site chosen as the origin and wi is the number of sites in the ith shell in the lattice. In deriving (2.4.28) we neglected contributions 2 involving exponential terms like e−αH (r +Ri ) /2 , since for large αH values (corresponding to the highly localized structures in the crystal) such factors are negligible at finite r

98

The freezing transition

values. Note that in (2.4.28) the angularly averaged two-point function is a function of the hard-sphere packing fraction ϕE for the equivalent hard-sphere system. The parameter αH in the expression for f 2 represents its functional dependence on the density function n 0 (x). In definition (2.4.28) above we have not included the case i = 1 corresponding to the peak at shortest r in 7 f 20 (r ). The approximation used in obtaining (2.4.27) is valid at large r but proves inadequate for including the correlations in the system at short length scales. For short distance 7 f 2 (r ) is parametrized using the known thermodynamic properties of the corresponding hard-sphere solid as follows: & " ! 2 , (A if r ≥ σc , /r ) exp −(α /2)(r − r ) 0 0 0 7 (2.4.29) f 21 (r ) = 0, otherwise. The three parameters A0 , α0 , and r0 are determined from the numerical solutions of three coupled nonlinear equations obtained by imposing the following three constraints on 7 f 21 (r ). 6 (a) The normalization of the angular-average two-point distribution function dr 7 f 2 (r ) over the first shell is equal to the coordination number of that shell in the corresponding crystal structure. (b) The virial equation relating the pressure P of the system to the 7 f 2 (σHS ) of the corresponding hard-sphere crystal (σHS is the hard-sphere diameter) at contact applies. f 210 . (c) The average r  calculated with f˜(21) (r ) should be the same as that computed with 7 In the scheme for obtaining 7 f 2 (r ) for small r , outlined above, the pressure P at packing fraction ϕE of the equivalent hard-sphere system is required as an input. The pressure is obtained by computing the free energy of the solid with any of the standard densityfunctional methods discussed above for hard-core systems. For an optimum value of the width parameter αH = α¯ in the hard-sphere density function, the free-energy functional FHS [n] of the equivalent hard-sphere system is a minimum. This minimum value corresponds to the equilibrium free energy of the hard-sphere system of packing fraction ϕ. The corresponding pressure P is determined from the numerical derivative with respect to the density n 0 of the free energy FHS at the minimum. Using this value of the pressure P as well as the other constraints described above, the three constants in (2.4.29) are computed. The two formulas, given by (2.4.28) and (2.4.29), respectively, lead finally to the angularly averaged two-point distribution function,  7 7 f 21 (r ) + (2.4.30) f 2i0 (r ). f 2 (r ) = 7 i=2

In Fig. 2.6 the angularly averaged correlation function 7 f 2 (r ) of the hard-sphere f.c.c. solid at packing fraction ϕ = 0.537, is plotted against the distance r scaled with the corresponding hard-sphere radius. For the results presented here we have used the modified weighted-density-functional approximation (MWDA) for computing the hard-sphere free energy. It is important to note in this respect that the value of the equilibrium free energy FHS for the hard-sphere system considered above is different from FR of the reference system. The latter is just a part of the total free energy of the LJ system. The DFT treatment of the hard-sphere system (at packing fraction ϕE for the equivalent system) with

2.4 Applications to other systems

99

6 5

2

~f

4 3 2 1 0

2

r

4

6

Fig. 2.6 The angularly averaged correlation function 7 f 2 (r ) defined in eqn. (2.4.10) vs. the radial distance r/σ for a hard-sphere system at packing fraction 0.52. σ is the hard-sphere diameter.

density function characterized by Gaussian width αH as developed above is solely aimed at determining the functional dependence of the free energy FR and the two-point distribution function 7 f 2 (r ) of the reference system on the inhomogeneous density function n 0 (x). The total free energy for the inhomogeneous crystalline state is now obtained from (2.4.11) for a chosen density n 0 and temperature T of the LJ system. This involves comf 2R in terms of the equivalent hard-sphere functionals obtained puting the two parts f R and 7 above. The density function n 0 (x) is characterized by a width parameter denoted by α. The equilibrium state corresponds to the minimum of the free energy f per particle obtained for an optimum value of α. The hard-sphere system’s free energy for computing f R is obtained here using the MWDA. The free energy of the Lennard-Jones crystal obtained theoretically with the DFT approach described above is in very good agreement with corresponding computer-simulation results (van der Hoef, 2000).

2.4.2 The solid–liquid interface The density-functional theory of freezing outlined above has also been applied for modeling a number of related physical systems. Among those of particular interest to our present discussion is the study of the interface between coexisting solid and liquid phases (Evans, 1990, 1992). The structure of the interface between the crystal and the liquid is characterized by the sharp density peaks of the solid phase becoming gradually broader with a corresponding decrease in height as the homogeneous-liquid side is approached. The density peaks become more strongly overlapping, with the interface appearing like a perturbed liquid and gradually smoothing into the homogeneous-liquid state of uniform density. The nature of the density profiles determines the overall width of the interface. Following the general approach of the DFT of inhomogeneous solids, the density distribution for the interface is determined from the minimization of the corresponding excess grand potential ,

100

The freezing transition



 = F[n(x)] − μ

dx n(x) + P V,

(2.4.31)

with the density n(x) approaching asymptotically the uniform-liquid density n l and the solid-state density n 0 (x) on the two sides of the interface. Both phases are at the coexistence values of the set of thermodynamic parameters (μ, P, T ). The first step in the formulation of a density-functional model for a theoretical description of the liquid–solid interface involves parametrization of the density n 0 (x) in the interface region. Following the prescription (2.1.42) for the inhomogeneous density in the bulk phase, we write down the density n 0 (x) in the interface region with the function  A˜ m Bm (z)eiKm · x , (2.4.32) n 0 (x) = n l + (n s − n l )B0 (z) + m

with the z axis being chosen perpendicular to the interface. In (2.4.32) the functions B0 and Bm are introduced to take into account the variation of the density n 0 (x) across the interface. The criteria followed in constructing them are as follows. (i) Each of the Bm (for m = 0, 1, 2, . . .) is constructed so as to have values 0 and 1 corresponding to the bulk liquid and bulk solid phases, respectively. (ii) The Bm associated with the corresponding reciprocal wave vector Km must decay over a shorter distance as the magnitude of K m increases. To maintain positive density, it is necessary also that the larger K m is, the closer to the crystal the corresponding function Bm vanishes. We consider here the ansatz for the Bm as follows: ⎫ ⎧ 1, z 0 > |z|, ⎪ ⎪ ⎪ ⎪ ⎪   ⎪ ⎪ ⎪ ⎬ ⎨ 1 z − z0 1 + cos π Bm (z) = , m ⎪ ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎪ ⎭ ⎩ 0, z m < |z|.

z 0 < |z| < z m ,

(2.4.33)

The quantity m = |z m − z 0 | which represents the width of the density-variation range corresponding to wave vector K m is parametrized in the form   K1 ν m = |z m − z 0 | = 0 , 0 ≤ ν ≤ 1, K m ≥ K 1 , (2.4.34) Km with K 1 denoting the smallest nonzero RLV. 0 is the width of the interface. Equations (2.4.33) and (2.4.34) represent an interface slab geometry with the bulk solid phase in |z| < z 0 , the bulk liquid phase in |z| > z 0 + 0 , and the interfacial region completely confined within the region z 0 < |z| < z m . This is shown schematically in Fig. 2.7. The quantities n l and {n s , A˜ m } for a suitably chosen set of RLVs Km correspond to the coexisting bulk liquid and solid phases, respectively. These are inputs for the test density function n 0 (x) to be used in the free-energy calculation of the interface. In the parametrization of

2.4 Applications to other systems

101

Fig. 2.7 A schematic diagram showing the spatial variations of Bm (z) order parameters describing the interfacial density profile given in eqn. (2.4.33) for an interface of width z. Adapted from c American Physical Society. Curtain (1989). 

the interfacial density function n 0 (x) depicted above the order parameters all begin their decay from the point z = z 0 . The set of parameters for the optimum density distribution in the interface region is therefore reduced to 0 , and ν. In principle z 0 can also be treated as a free parameter in characterizing the density profile. The function B0 is another variational parameter representing the change of average density in the interfacial region on going from n s to n l . In the present context B0 is chosen to be the same as the order parameter corresponding to the shortest RLV K 1 , i.e., B0 ≡ B1 . With the above parametrization of the density function n 0 (x), the weight function w(x − x ) of the WDA is determined by finding the self-consistent solution of the basic equation of the WDA (2.2.6) or equivalently of (2.2.7) in the Fourier space. The weighted density n(x) ¯ is obtained using the relation (2.2.2) and the excess free energy Fex is computed from (2.2.5). Note that, instead of using the WDA approach described above, the excess free energy can be computed with a direct functional expansion (Oxtoby and Haymet, 1982). However, this requires one to adopt the square-gradient approximation. While this assumption of slow density variation allows an analytic solution of the problem, in the present context this can be avoided in the WDA formulation through the direct minimization of free energy as outlined above. The square-gradient approximation, which involves a different parametric function n 0 (x) for the interfacial density, will be discussed in the next chapter while we treat the nucleation of a spherical droplet of crystal in the liquid melt. The DFT discussed here is strictly at the mean-field level. Hence the role of fluctuations, the existence of long-wavelength capillary waves, and roughening transitions are ignored in the present treatment. The grand potential for the interface is minimized with respect to the parameters {0 , ν} of the density function. In Fig. 2.8 the results obtained for a hard-sphere system in terms

102

The freezing transition

Fig. 2.8 The equilibrium plane-averaged interfacial density ρ(z) defined in eqn. (2.4.35) for the (100) f.c.c.–liquid interface. Solid line, the theoretical result for the hard spheres showing four nonbulk layers; dashed line, simulation results for soft spheres showing six to seven non-bulk layers. The centers of the theoretical and simulated profiles have been approximately aligned. From Curtain c American Physical Society. (1989). 

of the plane averaged density profiles at the interface are shown. The average density is defined as   1 d x dy n 0 (x), (2.4.35) n 0 (z) = A where A is the area of the interface. The interface considered here is a (100) plane of the crystal. The value of z 0 appearing in the definition of n 0 (x) is set by choosing it to be at the middle of the two layers of the crystal interfaces. The Percus–Yevick approximation is used throughout for the structure factor and equation of state. This also allows use of the coexistence conditions stated in Table 2.2. The surface energy is given by γsrf = /A, where A is the area of the interface. The numerical computation indicates that the situation of an interface with four atomic layers each of width a/2 (where a is the cubic-lattice spacing) and the exponent ν = 0.25 corresponds to the state of lowest free energy (Curtain, 1989). The surface free energy is

2.4 Applications to other systems

103

Fig. 2.9 Layer-by-layer contributions to the surface free energy βγ σ 2 of the equilibrium hard-sphere (111) f.c.c.–liquid interface. The interface spreads between layer 16 and layer 19. From Curtain c American Physical Society. (1989). 

computed to be γsrf = 0.66 ± 0.02 in units of βσ 2 , where σ is the hard-sphere diameter. The (111)-plane-averaged interface is also found to have the lowest surface free energy corresponding to the same conditions, namely a width of four layers, each having width √ a/ 3, and an exponent ν = 0.25. The surface free energy γsrf = 0.63±0.02 is nearly equal to that of the (100) plane. An analysis of the layer-by-layer contribution to the surface free energy of the (111) interface is presented in Fig. 2.9. The largest contribution to the free energy occurs in the middle of the interface. There is also an asymmetry in the free-energy contributions, with the near-crystal region costing more free energy than that near the liquid side. The results of the density-functional model for the interfaces show very little anisotropy, with the surface free energies differing by less than 5%. However, it is useful to note that the two-parameter form of the density variation depicted above does not include anisotropy. The above results on the structure of the interfaces are comparable to those obtained from computer simulations of similar systems. For soft-sphere systems with an r −12 interaction potential the (100) (Cape and Woodcock, 1980) and (111) (Tallon, 1986) planes and for the Lennard-Jones system the (100) and (111) planes all obtain interfaces that consist of six or seven atomic layers of density variation. The surface free energy γsrf for the hard-sphere system computed above can be shown to be in reasonable agreement with simulation results for the soft-sphere systems by expressing the simulation results in terms of an equivalent hard-sphere system (Curtain, 1989). The above method of studying the interface is extended to the Lennard-Jones systems by computing the contribution due to the attractive part of the interaction in the mean-field approximation. Broughton and Gilmer (1986) obtained the surface tension from simulation of the Lennard-Jones system, which showed that the (111), (100), and (110) faces have the same surface free energies. The work mentioned is discussed in Chapter 3, in which we treat nucleation.

104

The freezing transition

The basic step in the weighted-density-functional theories involves a mapping of the inhomogeneous system (crystal) into a coarse-grained system that has inhomogeneity over a different scale (WDA) or has constant density (MWDA). This procedure represents a reduction in number of the associated degrees of freedom at the shortest length scales. The basic interaction in the original and coarse-grained system, however, remains the same here. The present treatment of the first-order freezing transition is in some way analogous to the renormalization-group (RG) approach (Wilson and Kogut, 1974; Wilson, 1975) which is applied to the theory of critical phenomena of second-order phase transitions. For first-order transitions application of the RG has not been common (Nienhuis and Nauenberg, 1975).

Appendix to Chapter 2

A2.1 Correlation functions for the inhomogeneous solid We demonstrate below that the two-point function c(2) (x1 , x2 ) for the inhomogeneous state is related to the pair correlation in the fluid. The one-body potential c(1) (x1 , n 0 (x)) defines the inhomogeneous density n(x1 ) through the relation (2.1.33) or equivalently through (2.1.31). By taking a functional derivative of (2.1.31) with respect to n 0 (x2 ) we obtain the following result for the two-point function c(2) (x1 , x2 ; n 0 (x)): c(2) (x1 , x2 ; n 0 (x)) ≡

δ δu(x1 ) δ(x1 − x2 ) c(1) (x1 ; n 0 (x)) = β − . δn 0 (x2 ) δn 0 (x2 ) n 0 (x1 )

(A2.1.1)

However, using the definition (2.1.9) of the equilibrium density n 0 (x1 ) and taking one more functional derivative of it with respect to the field u(x2 ), we obtain the following result for the average of the product of the microscopic densities nˆ at two points, β −1

δn 0 (x1 ) ˆ 2 )0 − n 0 (x1 )n 0 (x2 ), = n(x ˆ 1 )n(x δu(x2 )

(A2.1.2)

where the microscopic density n(x) ˆ is defined as in (2.1.6) and the subscript 0 on the angular bracket refers to the equilibrium average with the grand-canonical ensemble. The first term on the RHS of (A2.1.2) can be expressed in terms of one- and two-point contributions as follows: /



ˆ 2 )0 = n(x ˆ 1 )n(x / =

0 δ(x1 − xα )δ(x1 − xβ )

α,β



00 δ(x1 − xα )δ(x1 − xβ )

α=β

+ δ(x1 − x2 )n(x ˆ 1 )0 0

= n 0 (x1 )n 0 (x2 )g(x1 , x2 ) + n 0 (x1 )δ(x1 − x2 ), 105

(A2.1.3)

106

Appendix to Chapter 2

where we have used the definition (1.2.58) for the pair correlation function introduced in Chapter 1. Using (A2.1.3), the above relation (A2.1.2) reduces to the form β −1

δn 0 (x1 ) = h(x1 , x2 )n 0 (x1 )n 0 (x2 ) + n 0 (x1 )δ(x1 − x2 ). δu(x2 )

(A2.1.4)

We now use the two functional derivatives in eqns. (A2.1.1) and (A2.1.4) in the functional identity (u(x) and n 0 (x) being uniquely related to each other)

 δu(x1 ) δu(x1 ) δn 0 (x3 ) dx3 (A2.1.5) = = δ(x1 − x2 ), δn 0 (x3 ) δu(x2 ) δu(x2 ) to obtain the following integral relation of the two-point kernel c(2) (x1 , x2 ) with h(x1 , x2 ) = g(x1 , x2 ) − 1:  (2) h(x1 , x2 ) = c (x1 , x2 ) + dx3 {c(2) (x1 , x3 )}h(x3 , x2 )n 0 (x3 ). (A2.1.6) For the isotropic homogeneous liquid with constant density the relation (A2.1.6) reduces to the standard Ornstein–Zernike relation (1.2.89) presented in Chapter 1. The kernel function c(2) appearing in eqn. (2.1.37) is now evaluated in terms of the pair correlation function g(r ) or the static structure factor S(k) for the isotropic liquid. These structure functions are known from various integral-equation theories of the liquid state (discussed in Chapter 1), from experiments or from computer simulations, and form the starting point of the DFT for the freezing transition. Similarly, any one of the higher-order generalizations of the direct correlation function (as defined in eqn. (2.1.36)) (Barrat et al., 1987, 1988) is also related to the corresponding multi-particle correlations in the fluid.

A2.2 The Ramakrishnan–Yussouff model We consider here the deduction and solution scheme of the density-functional equations in the Ramakrishnan–Yussouff model. We begin with the form of the inhomogeneous density function given by   ∞  iKm · x n 0 (x) = n l 1 + η + Am e (A2.2.1) m=1

defining the crystalline state in terms of the order parameters η and {Am }. By taking a Fourier transform of (2.1.37) the following set of equations for the order parameters η and Am is obtained: ∞    dx c˜0 η iKm · x 1+η =e c˜m Am e exp , (A2.2.2) V m=1 ∞    dx −iK j · x c˜0 η iKm · x exp c˜m Am e e . (A2.2.3) Aj = e V m=1

A2.2 The Ramakrishnan–Yussouff model

107

To simplify the discussion of the above equations, we define the integrals κm in terms of the rescaled amplitudes ξm = c˜m Am as ⎡ ⎤   dx −iK j · x (A2.2.4) exp ⎣ ξs eiKs · x ⎦ . e κj = V s=0

Equations (A2.2.2) and (A2.2.3) are now obtained in the simplified forms 1 + η = ec˜0 η κ0 , κj ξj = [1 + η] . c˜ j κ0

(A2.2.5) (A2.2.6)

Equations (A2.2.5) and (A2.2.6) together with the definition (A2.2.4) form a set of coupled equations for the order parameters η and Am of the freezing transition. At the phase-transition point the grand potentials for the two phases are equal. By substituting the ansatz (A2.2.2) into the RHS of (2.1.41), the difference  of the thermodynamic potential between the two phases is obtained in terms of the order parameters η and Am . Using for the integral the result ⎡ ⎤    ξ2 ⎦ dx1 dx2 c(2) (x1 , x2 ; n l )δn 0 (x1 )δn 0 (x2 ) = n l V ⎣c˜0 η2 + (A2.2.7) c˜m m=0

and some trivial algebra, we obtain from (2.1.41) the following expression for : ⎡ ⎤ 2  ξm ⎦ 1 1 . (A2.2.8)  = n l V ⎣(c˜0 − 1)η + c˜0 η2 + 2 2 c˜m m=0

It is useful to note here that the RLV |K j | is expressed in units of the inverse of the lattice constant a and the direct correlation function is expressed as a function of |K j |σ , hence the ratio σ/a enters the numerical computation. Since

1/3 σ (1 + η)n l σ 3 (A2.2.9) = a 3 the input structural entity c j on the RHS of (A2.2.8) is dependent on η. The next step in the calculation is to use a set of RLVs for the crystalline structure. The choice of a proper set of RLVs is related to the symmetry of the chosen crystal lattice whose stability is being tested. For an f.c.c. lattice, for example, the first set of eight nearestneighbor RLVs is given by K1 =

2π {±1, ±1, ±1}, a

(A2.2.10)

√ which is a set of eight vectors, each of length |K1 | = (2π/a) 3. In general the RLV Km is expressed in terms of its components, Km =

2π {±m x , ±m y , ±m z }, a

(A2.2.11)

108

Appendix to Chapter 2

where m x , m y , and m z are integers. The length of this vector is |Km | = (2π/a) : m 2x + m 2y + m 2z . We adopt a notation in which the RLV Km denotes a set of vectors Kmα , α = 1, 2, . . . , 2 p ( p is an integer), forming a shell of radius K m . For all the RLVs belonging to a particular shell s (say) the common amplitude is denoted by As . Hence all the ξi belonging to this shell are assumed to be the same and denoted by ξs . Using this symmetry of the different RLVs belonging to a particular shell s, we simplify the sum 5 iKs · x (appearing on the RHS of eqn. (A2.2.4)) and obtain the following result for s=0 ξs e κ0 corresponding to the case K 0 = 0:     dx ωs (x)ξs , exp (A2.2.12) κ0 = V s where ωm (x) =



exp[iKmα · x] ≡

α



exp[−iKmα · x].

(A2.2.13)

α

Thus, for example, with {K1α } being the first set of eight lattice vectors denoted by eqn. (A2.2.10), we obtain that ω1 (x) = 8 cos(π x)cos(π y)cos(π z) and so on. We obtain κ0 using the result (A2.2.13), giving that    1  1  1  dx dy dz exp ωs (x)ξs , κ0 = 0

0

0

(A2.2.14)

(A2.2.15)

s

where the volume integral is performed over one primitive cell of the lattice and the index s in the sum denotes the different concentric shells in the RLV space. For computing κ j for j  = 0, we make use of the relation κj =

∂ κ0 , ∂ξ j

(A2.2.16)

which follows from using the symmetry of the RHS of (A2.2.4) for all the RLVs {K j } belonging to a particular shell. With a suitably chosen set of RLVs extending to a certain number of shells, the corresponding amplitudes κ j and η are obtained from numerical solution of eqns. (A2.2.5) and (A2.2.6) together with the defining relations (A2.2.4) and (A2.2.16). The average solid-state density n s and hence the lattice constant (assuming the solid to be free of defects) is obtained in terms of the fractional density change, n s = n l (1 + η) =

4 a3

(A2.2.17)

(for the f.c.c. structure). The lattice constant thus follows self-consistently from the minimization of the free-energy functional. The results are listed in Table A2.4.

A2.3 The weighted-density-functional approximation

109

Table A2.4 Freezing parameters for hard spheres in the Ramakrishnan–Yussouff theory, using the Percus–Yevick structure factor (with Verlet–Weiss correction) for the uniform liquid state. From Haymet and Oxtoby (1986). Number of RLVs Nn 1 8 6 12 24 8 6 24 24 24 8 24 12 48 24 6 24 24 24 8

Lattice vectors (±i, ± j, ±k)

Order parameter As

Lindemann ratio L0

0 (1, 1, 1) (2, 0, 0) (2, 2, 0) (3, 1, 1) (2, 2, 2) (4, 0, 0) (3, 3, 1) (4, 2, 0) (4, 2, 2) (3, 3, 3) (5, 1, 1) (4, 4, 0) (5, 3, 1) (4, 4, 2) (6, 0, 0) (6, 2, 0) (5, 3, 3) (6, 2, 2) (4, 4, 4)

0.0907 1.0436 1.0299 0.9780 0.9427 0.9309 0.8857 0.8540 0.8444 0.8053 0.7767 0.7775 0.7324 0.7081 0.6995 0.7018 0.6701 0.6464 0.6398 0.6103

0.0668 0.0660 0.0644 0.0635 0.0633 0.0629 0.0626 0.0624 0.0620 0.0618 0.0617 0.0615 0.0612 0.0612 0.0610 0.0608 0.0608 0.0607 0.0606

A2.3 The weighted-density-functional approximation Equations (2.2.1) and (2.2.2), which form the basis of the WDA, can in fact be justified from general considerations (Ashcroft, 1996). In order to demonstrate this we start from the following definition of functional derivatives of the excess free energy:  δ Fex [n] Fex [n λ+λ (x)] − Fex [n λ (x)] = dx (A2.3.1) n λ (x), δn λ (x) where we have used for the deviation n λ (x) ≡ n λ+λ (x) − n λ (x). The parameter λ associated with the density function n λ (x) specifies the nature or the “path” along which the density is being varied. It follows from the definition (2.1.36) that the functional derivative on the RHS of (A2.3.1) is the one-particle distribution function c(1) [x; n λ (x)]. A suitable expression for the excess free energy Fex is obtained by formally integrating the density along the “path” n λ (x) = λn 0 (x) from λ = 0 to 1. Along this “path” n λ (x) =

∂n λ dλ = n 0 (x)dλ. ∂λ

(A2.3.2)

110

Appendix to Chapter 2

The expression (A2.3.1) for the excess free energy reduces to the form  Fex = −



1

dx

dλ c(1) [x; λn 0 (x)]n 0 (x).

(A2.3.3)

0

In obtaining the result (A2.3.3) we also used the fact that the excess free energy Fex is zero in the limit of zero density. In an exactly similar way, by applying the definition (2.1.36) for the case n = 2 at the appropriate density, the following formal expression for the one-particle function c(1) [x, n 0 (x)] is obtained: c(1) [x; n 0 (x)] = c(1) [x; n¯ 0 (x)] +



1

dλ



0

dx c(2) [x1 , x2 ; n λ (x)]n λ (x),

(A2.3.4)

where the functional integration in (A2.3.4) is performed along the “path” n λ (x) = n¯ 0 (x) + λ {n 0 (x) − n¯ 0 (x)}

(A2.3.5)

from λ = 0 to 1, n¯ 0 (x) being some suitable density function to be further specified below. Here c(1) [x; n¯ 0 (x)] is the one-particle direct correlation function of a homogeneous liquid of density n¯ 0 (x). On substituting the exact expression (A2.3.4) for the one-particle function c(1) into (A2.3.3) we obtain for the excess free energy  Fex =

dx n 0 (x) f 0 [n¯ 0 (x)]     −1  dx dx n 0 (x) dλ dλ c(2) [x1 , x2 ; λn λ (x)]n λ (x), −β

(A2.3.6)

where the function f 0 (n¯ 0 ) in the first term represents the excess free energy of the homogeneous liquid at density n¯ 0 . Using the definition for n λ along this chosen path (A2.3.5), Fex is obtained as  Fex = dx n 0 (x) f 0 [n¯ 0 (x)]   ! " (A2.3.7) − β −1 dx dx n 0 (x) n 0 (x ) − n¯ 0 (x) W[x, x ; n 0 (x), n¯ 0 (x)], where we have adopted the following definition for the function W: W[x, x ; n 0 (x), n¯ 0 (x)] ≡





1

dλ{λ} 0

0

1

dλ c(2) (x, x ; λn λ ).

(A2.3.8)

The density in the argument of c(2) is λn λ (x) = λ{n¯ 0 (x) + λ [n 0 (x) − n¯ 0 (x)]}. If we demand that the excess free energy is given by only the first term on the RHS of (A2.3.7) in a form similar to (2.2.1), then the second term must go to zero. Using this condition,

A2.3 The weighted-density-functional approximation

111

a defining relation for the density n¯ 0 (x) reduces to the form (2.2.2), with the corresponding weight function w(x1 − x2 ) obtained as w(x − x ; n¯ 0 (x)) = 6

W[x, x ; n 0 (x), n¯ 0 (x)] . dx W[x, x ; n 0 (x), n¯ 0 (x)]

(A2.3.9)

The weighting function w obtained above automatically satisfies the normalization condition (2.2.3). Equation (A2.3.9) also demonstrates the dependence of the weighting function on the direct correlation function of the liquid.

The homogeneous liquid Next, we obtain the relation (2.2.6) for the weighting function in the uniform-density limit. By virtue of the definition of the weighted density function n(x), ¯ the excess free energy is defined as  WDA [n 0 (x)] = dx n 0 (x) f ex (n¯ 0 (x)), (A2.3.10) Fex where f ex (n 0 ) is the free-energy density for the homogeneous liquid of density n 0 . On taking a functional derivative of (A2.3.10) with respect to n 0 (x), we obtain  WDA δ n¯ 0 (x1 ) δ Fex  [n¯ 0 (x1 )]n 0 (x1 ) = f 0 [n¯ 0 (x)] + dx1 f ex , (A2.3.11) δn 0 (x) δn 0 (x) where the prime on f ex implies a partial derivative with respect to the corresponding density n 0 . On taking a second functional derivative we obtain WDA δ n¯ 0 (x) δ n¯ 0 (x ) δ 2 Fex   [n¯ 0 (x)] [n¯ 0 (x )] = f ex + f ex   δn 0 (x)δn 0 (x ) δn 0 (x ) δn 0 (x)  δ n¯ 0 (x1 )  δ n¯ 0 (x1 ) + dx1 n 0 (x1 ) f ex [n¯ 0 (x1 )] δn 0 (x) δn 0 (x )  δ 2 n¯ 0 (x1 )  [n¯ 0 (x1 )]n 0 (x1 ) . + dx1 f ex δn 0 (x)δn 0 (x )

(A2.3.12)

Now, in order to simplify further the above equation, we need to evaluate the first and second functional derivatives of the weighted density n¯ 0 (x) with respect to the actual density n 0 (x). Starting from the defining relation  (A2.3.13) n¯ 0 (x) = dx w(x − x , n¯ 0 (x))n 0 (x ), we obtain the following results for the functional derivatives: δ n¯ 0 (x ) = w(x − x; n¯ 0 (x)) δn 0 (x)

 δ n¯ 0 (x ) + dx1 n 0 (x1 )w  (x − x1 ; n¯ 0 (x )) . δn 0 (x)

(A2.3.14)

112

Appendix to Chapter 2

On taking a second derivative we obtain

δ 2 n¯ 0 (x1 ) δ n¯ 0 (x)  = w (x1 − x; n¯ 0 (x)) δn 0 (x )δn 0 (x) δn 0 (x )

δ n¯ 0 (x1 )  + w (x1 − x; n¯ 0 (x1 )) δn 0 (x )

 δ n¯ 0 (x1 ) δ n¯ 0 (x1 )  + dx2 n 0 (x2 )w (x1 − x2 ; n¯ 0 (x1 )) δn 0 (x) δn 0 (x )

 δ 2 n¯ 0 (x1 ) . (A2.3.15) + dx2 n(x2 )w  (x1 − x2 ; n¯ 0 (x1 )) δn 0 (x)δn 0 (x ) We now evaluate the above functional derivatives in the homogeneous-liquid limit n 0 (x) → n 0 and correspondingly the weighted density n¯ 0 (x) → n 0 (obtained using the normalization (2.2.3)). For the first derivative we obtain, from eqn. (A2.3.10), * δ n¯ 0 (x ) ** = w(x − x; n 0 ), (A2.3.16) δn 0 (x) *n 0 with the second term on the RHS going to zero, since * *   * * δ    *   * dx1 w (x − x1 ; n¯ 0 (x ))* ≡ dx1 w(x − x1 ; n¯ 0 (x ))* = 0, ) δ n ¯ (x 0 n0 n0

(A2.3.17)

as a result of the normalization condition (2.2.3). Similarly, the second functional derivative is evaluated from (A2.3.15), giving the following result in the homogeneous-liquid limit: * δ 2 n¯ 0 (x1 ) ** = w  (x1 − x ; n 0 )w(x1 − x; n 0 ) + w  (x1 − x; n 0 )w(x1 − x ; n 0 ), δn 0 (x )δn 0 (x ) *n 0 (A2.3.18) since the last two terms on the RHS are equal to zero, using similar arguments regarding normalization of the weight function w to those used in (A2.3.17). We now evaluWDA in the homogeneous-liquid ate (A2.3.12) for the second functional derivative of Fex limit by substituting results (A2.3.17) and (A2.3.18) for the first and second derivatives, respectively, of the weighted density function and obtain * WDA * δ 2 Fex * = 2 f  (n 0 )w(x − x ; n 0 ) ex  δn 0 (x)δn 0 (x ) *n 0   + n 0 f ex dx1 w(x − x1 ; n 0 )w(x1 − x ; n 0 )    dx1 w(x − x1 ; n 0 )w  (x1 − x ; n 0 ) + n 0 f ex  + w  (x − x1 ; n 0 )w(x1 − x ; n 0 ) . (A2.3.19)

A2.4 The modified weighted-density-functional approximation

113

In writing eqn. (A2.3.19) we assume that the weighting function w(x; n 0 ) in the homogeneous liquid depends only on |x|. Upon identifying the functional derivative on the RHS as a two-point direct correlation function we obtain for the homogeneous liquid the relation  (n 0 )w(x − x ) −β −1 c(2) (x − x ; n 0 ) = 2 f ex   + n 0 f ex (n 0 ) dx1 w(x − x1 )w(x1 − x )    (n 0 ) dx1 w(x − x1 )w  (x1 − x ) + n 0 f ex

 + w  (x − x1 )w(x1 − x ) ,

(A2.3.20)

where w(x) is the weighting function in the uniform liquid of density n 0 .

A2.4 The modified weighted-density-functional approximation The formulation of the MWDA approach in terms of eqns. (2.2.10) and (2.2.11) can be be justified at a formal level. The analysis is similar to what has been presented above for the basic equations of the WDA in Appendix A2.3. We take the reference density to be a constant, n¯ 0 (x) = nˆ 0 by setting the second term in eqn. (A2.3.7) equal to zero. This leads to the following expression for the effective liquid density nˆ 0 in the MWDA,   nˆ 0 = dx dx n 0 (x)n 0 (x )w[x, ˜ x ; n 0 (x), nˆ 0 ], where the weighting function w˜ is now expressed in a form equivalent to (A2.3.9) for the WDA case, w(x ˜ − x ) = 6

˜ W[x, x ; n 0 (x), n¯ 0 (x)] , 6 ˜ x ; n 0 (x), n¯ 0 (x)] dx n 0 (x) dx n 0 (x )W[x,

(A2.4.1)

˜ being the corresponding function W ˜ W[x, x ; n 0 (x), nˆ 0 ] ≡

 0

1



1

dλ λ

! " dλ c(2) x, x ; λ{nˆ 0 + λ [n 0 (x) − nˆ 0 ]} .

(A2.4.2)

0

Note that the weighting function w˜ in the present case of the MWDA is different from the weighting function w (see eqn. (A2.3.9)) for the WDA. However, it also satisfies a similar normalization condition to (2.2.3), namely  dx w(x ˜ − x , nˆ 0 ) = 1 (A2.4.3) at all densities n 0 .

114

Appendix to Chapter 2

The homogeneous liquid As in the case of the WDA, the weighting function w˜ for the MWDA is determined by considering the limiting case of the homogeneous liquid. Starting from the basic eqn. (2.2.10) for the MWDA, we take the functional derivative with respect to n 0 (x) to obtain MWDA δ nˆ 0 δ Fex  (nˆ 0 ) = f 0 (nˆ 0 ) + N f ex . δn 0 (x) δn 0 (x)

(A2.4.4)

On taking a second functional derivative, we obtain

MWDA δ nˆ 0 δ nˆ 0 δ nˆ 0 δ nˆ 0 δ 2 Fex   ( n ˆ ) [nˆ 0 (x )] = f + + N f ex ex 0 δn 0 (x)δn 0 (x ) δn 0 (x) δn 0 (x ) δn 0 (x) δn 0 (x )  [nˆ 0 (x )] + N f ex

δ 2 nˆ 0 . δn 0 (x)δn 0 (x )

(A2.4.5)

The first and second functional derivatives of the weighted density nˆ 0 with respect to the actual density n 0 (x) are evaluated starting from the defining relation (2.2.11) as    1 2 δ nˆ 0   ˜ − x ; nˆ 0 ), (A2.4.6) = dx w(x − x ; nˆ 0 ) − 2 dx dx w(x δn 0 (x) N N  2 2 δ 2 nˆ 0 (A2.4.7) = w(x − x ; nˆ 0 ) − 2 dx w(x − x ; nˆ 0 ).  δn 0 (x)δn 0 (x ) N N We now evaluate the above functional derivatives in the homogeneous-liquid limit n 0 (x) → n s ( the solid density) and correspondingly the weighted density nˆ 0 → n s (obtained using the normalization (A2.4.3)). For the first derivative we obtain from eqn. (A2.4.6) * 2n s ns ns δ nˆ 0 ** = 2 − = . (A2.4.8) * δn 0 (x) n s N N N Similarly, the second functional derivative is evaluated from (A2.4.5), giving the following result in the homogeneous-liquid limit: * * 2 δ 2 nˆ 0 * = 2 w(x ˜ − x ; n s ) − . (A2.4.9) δn 0 (x )δn 0 (x ) *n s N NV MWDA in the We now evaluate (A2.4.5) for the second functional derivative of Fex homogeneous-liquid limit by substituting results (A2.4.8) and (A2.4.9) for the first and second derivatives, respectively, of the weighted density function and obtain ns  −β −1 c(2) (x, x ; n s ) = w(x, ˜ x ; n s ), (A2.4.10) + 2 f ex V giving the following result for the weighting function: 1  n s  w(x, ˜ x ; n s ) = −  β −1 c(2) (x, x ; n s ) + (A2.4.11) f (nˆ 0 ) . 2 f ex V ex

The self-consistent solution (for the appropriate nˆ 0 ) of (A2.4.11) provides the weighting function for the MWDA.

A2.5 The Gaussian density profiles and phonon model

115

A2.5 The Gaussian density profiles and phonon model Note that the Gaussian approximation (2.2.14) for the density of the inhomogeneous solid used in the liquid-based DFT here can be linked to a phonon description of the solid in a self-consistent Debye model, a one-parameter (TD ) theory for the dynamic crystal. In the Debye model (Huang, 1987) the phonon frequency ω = ck for wave number k < kD or, correspondingly, can be defined for the Debye temperature TD using kB TD = hωD = hckD . The speed of sound in the crystal, both for longitudinal and for transverse waves, is assumed here to be c. The density of states g(ω) in the Debye solid (the number of modes between ω and ω + dω is g(ω)dω) is proportional to ω2 and, in the normalized form, is expressed as 3 3 /ω2 , 9N ωD for ω ≤ ωD , (A2.5.1) g(ω) = 0, for ω > ωD , where ωD is the Debye frequency. For high temperatures, TD /T 1, so that quantum effects are negligible. The energy of the solid in terms of the 3N normal ) modes is obtained 5 ( 1 in terms of the occupation numbers {n i } as E{n i } = i n i + 2 ωi , ignoring any constant contribution to the energy. The partition function for the 3N phonon modes is obtained as   % e−β ωi /2 Q= , (A2.5.2) 1 − e−β ωi i where the frequencies range from 0 up to a maximum of ωD . On taking the corresponding logarithm of the partition function, we obtain

 βωi −β ωi }+ ln Q = − ln{1 − e 2 i

 ωD βωi −β ωi }+ g(ω)dω. (A2.5.3) ln{1 − e =− 2 0 Using the standard thermodynamic identities, we obtain the entropy of the phonon gas in terms of the dimensionless parameter xD = TD /T as   ! " ∂F −T S = −β = Nβ −1 3 ln(1 − e xD ) − 4I(xD ) , (A2.5.4) ∂β V where I is the Debye integral, I(x) =

3 x3

 o

x

t3 dt. et − 1

(A2.5.5)

The energy of the phonons is obtained as U =−



3 ∂ ln Q = 3Nβ −1 I(xD ) + xD . ∂β 8

(A2.5.6)

116

Appendix to Chapter 2

In the high-temperature limit (xD 1), the integral I has the asymptotic behavior 3 I(xD ) ≈ 1 − xD + · · ·. (A2.5.7) 8 In the high-temperature limit the energy goes to the value 3N kB T . For the free energy of the noninteracting phonons we take only the kinetic-energy contribution E = 3N kB T /2. Using the result (A2.5.4) for the entropy in this high-temperature limit, we obtain for the scaled free energy per particle the result ( ) E −TS 5 β = β f = 3 ln xD − + O xD2 . (A2.5.8) N 2 This form reduces to the ideal-gas free-energy expression (2.2.21) if we identify 9 α 0 xD = π with the Gaussian model of eqn. (2.2.14), 0 being the thermal wavelength. At higher temperatures (hence smaller xD ), 0 ∼ T −1/2 becomes smaller and thus refers to smaller values of α, indicating a lower degree of mass localization in the crystalline state.

3 Crystal nucleation

If the liquid is cooled beyond the corresponding freezing point Tm at which the liquid and crystalline phases coexist in equilibrium, a thermodynamic driving force builds up towards forming the crystal. In this chapter we will discuss how the liquid transforms into a crystal, focusing on how the changes in the liquid are initiated and on the nature of the crystalline region that is formed. This process is referred to as nucleation. The thermodynamic force favoring the formation of the crystal seed in the supercooled liquid competes with the process of forming an interface between the solid and the liquid. The cost of the interfacial free energy therefore presents a barrier to the formation of the new phase. Only when the driving force is made large enough by moving deep into the supercooled state does crystallization occur on laboratory time scales. Thus pure water can be cooled to −20 ◦ C or below without freezing. Our focus here will be mainly on the process of crystallization of solid from the melt. The condensation of vapor into liquid is a very thoroughly studied process that has been discussed in various reviews (Stanley, 1971; Evans, 1979; ten Wolde et al., 1998). For condensation from a low-density gas or crystallization from dilute solution, it is easier to identify the nucleating bubbles since they differ widely in composition from the surrounding phase. In a crystal forming from a melt the nucleating droplet is far less distinct. It is somewhat problematic to define the cluster in the bulk of the melt in terms of a certain set of particles. One possible route is to identify the coordination numbers, which differ between the liquid and the crystalline phase. We will simplify our discussion here by limiting the discussion to one-component systems and also focus on what is termed homogeneous nucleation (Oxtoby, 1992a, 1992b; Gunton, 1999), which occurs in the bulk of the pure liquid phase. We also briefly explore here the issue of heterogeneous nucleation (Turnbull, 1969) started by impurities or occurring on the surfaces. 3.1 Classical nucleation theory In the classical nucleation theory we associate a number of particles with the nucleus of the new phase having a sharp interface with the bulk liquid. In this theory of the nucleation process the nucleus of the crystal, no matter how small, is treated using macroscopic thermodynamic principles. Let us first consider the formation of Ni clusters, each consisting of i monomers, and N single monomers (N  Ni ). Assuming that clusters mix ideally 117

118

Crystal nucleation

with the monomers, the number of equivalent states is (N + Ni )!/(N !Ni !). Hence the total change in free energy on forming the clusters, at temperature T , is    

 Ni Ni Ni G i + kB T Ni ln + N ln , (3.1.1) G = Ni + N Ni + N i

where G i is the free energy of formation of a single cluster of i molecules from the bulk. On minimizing the free-energy cost G, we find that the equilibrium number of clusters of size i is given by ! " Ni = N exp −G i /(kB T ) . (3.1.2)

3.1.1 The free-energy barrier Next we compute the free-energy cost G i for a cluster of size i from the bulk liquid. We assume here that the clusters are spherical in shape and consist of i particles. The pressure difference between the two phases at the interface with principal radii of curvature R1 and R2 is given by   1 1 + , (3.1.3) P = γs R1 R2 where γs is the surface tension. For a spherical interface R1 = R2 = R, eqn. (3.1.3) becomes PrS − P L =

2γs , R

(3.1.4)

where PrS and P L , respectively, denote the pressures in the crystalline phase at the spherical interface of radius r and in the bulk liquid phase. By applying the Gibbs–Duhem relation (1.1.9) discussed in Chapter 1, dμS = vd P S for the crystalline phase, at a constant temperature, we obtain for the difference between the chemical potential on the interface at r and that of the bulk  PrS ( ) S S v dP S = vm PrS − P S , (3.1.5) μr − μ = PS

assuming that the volume per particle vm for the crystal remains constant in the incompressible limit. In obtaining (3.1.5) we approximate the properties of the nucleus at the center as being same as those of the bulk solid and the surface tension of the small droplet with curved surface is taken to be the same as that of a plane interface. This is termed the capillary approximation. This applies better for large droplets and is a weak approximation in the case of small droplets. On combining (3.1.5) and (3.1.4) we obtain for the solid-phase chemical potential μrS at the interface   2γs μrS = μS + vm + PL − PS . (3.1.6) r

3.1 Classical nucleation theory

Since 2γs /r  P L − P S , the above relation approximates to   2γs . μrS = μS + vm r

119

(3.1.7)

For a spherically shaped nucleating bubble of volume V and radius R containing a total number of molecules i, we have ivm ≡ V = (4π/3)R 3 . The total Gibbs free energy of the nucleating bubble is then given by     2γs S S dN G r = μ d N + vm r¯    2γs . (3.1.8) = μS i + 4π r¯ 2 d r¯ r¯ Here we take the cluster to be stationary and ignore the rotational and vibrational contributions to the free energy. The difference of the Gibbs free energy from that of the surrounding bulk liquid is obtained as   4 S L 3 (3.1.9) π R G v + 4π R 2 γs , G i ≡ G r − G = 3 where G v is the difference of the bulk values for the Gibbs free energy per unit volume in the two phases, ( ) (3.1.10) G v = μS − μL /vm = μ/vm . Since the crystalline phase is more stable (i.e., μS < μL ), G v is negative. In terms of the number of particles i in the nucleus of volume V , we have vm i = (4π/3)R 3 . Using this relation, the surface contribution to the free energy represented by the second term on the RHS of (3.1.9) is expressed in terms of i, giving for the total free-energy difference of the nucleus of i molecules in the two phases ; 3

G i = vm G v i + aγs i 2/3 ,

(3.1.11)

2 i 2/3 is the surface area of the nucleus containing i monomers. where ai 2/3 = 36π vm If we do not assume a completely spherical shape of the interface, the free energy of the cluster can be argued to have the form (Turnbull and Fisher, 1949)  (aϕ γϕs )i 2/3 , (3.1.12) G i = vm G v i + ϕ

where the summation in the second term on the RHS is carried out over all facets of the cluster with corresponding interfacial energy γϕs and aϕ is a geometric factor (which will reduce to a for a spherical nucleus). The expression (3.1.12) for the free energy is schematically written in the form G i = Ai 2/3 − Bi kB T

(3.1.13)

120

Crystal nucleation

Fig. 3.1 Schematic representations of the surface and bulk contributions to the free energy of the incompressible embryo in the classical nucleation theory (CNT). The sum of the two terms on the RHS of (3.1.11) reaches a maximum corresponding to the unstable critical nucleus.

5 using A = ϕ (aϕ γϕs )/(kB T ) and B = vm |G v |/(kB T ). Keeping the shape of the nucleus fixed, the free energy G i reaches a maximum for i = i ∗ , i.e., dG i /di = 0. We show this behavior of the free-energy difference G i (denoted as wmin ) with the number of particles i in the cluster schematically in Fig. 3.1. Using (3.1.13) we obtain 3 5

2 (aϕ γϕs ) 3 2A ∗ = , (3.1.14) i = 3B 3vm |G v | with the corresponding maximum value of the free energy as 5 4 (aϕ γϕs )3 . G ∗ = 27vm |G v |2

(3.1.15)

For the special case of the spherical nucleus we have, for all the facets, γϕs ≡ γs and 5 aϕ ≡ a. The result (3.1.12) for the barrier in terms of the radius R of the nucleus then reduces to the form given on the RHS of eqn. (3.1.9). For the critical nucleus the number of monomers and the barrier heights are respectively obtained as i∗ =

32π 3vm



γs |G v |

3 ,

(3.1.16)

3.1 Classical nucleation theory

G i ∗ =

16π γs3 3|G v |2

.

121

(3.1.17)

By equating the volume of i ∗ vm of the critical nucleus to the spherical volume of (4/3)π R ∗ 3 , we obtain for the radius of the critical nucleus R∗ =

2γs . |G v |

In this case the constants A and B in eqn. (3.1.13) are given by aγs A= , kB T vm |G v | . B= kB T

(3.1.18)

(3.1.19) (3.1.20)

The maximum at n = n ∗ in the free-energy barrier and the corresponding radius R ∗ of the cluster represent a limiting situation. Note that the above expression (3.1.17) for the free energy computed from thermodynamic consideration of reversible processes represents the minimum work needed to form the cluster of i monomers. Furthermore, it is also a maximum with respect to the variation of the number of monomers i. The critical nucleus is therefore in a state of unstable equilibrium. A cluster of radius less than R ∗ (having fewer than n ∗ monomers) is subcritical and shrinks spontaneously. On the other hand, clusters with radius greater than R ∗ (having more than n ∗ monomers) are supercritical and increasingly likely to grow. The classical nucleation theory (CNT) described above involves making approximations that are questionable. First, it assumes that even a cluster of molecular size has bulk-like properties at the center. Second, the clusters of monomers forming the nucleus are defined with a sharp interface between the two phases. More specifically, the CNT uses a surface free energy for the nuclei that is the same as that of the infinite planar interface, whereas the actual interface for the small nucleus is in fact sharply curved. Thus, while the basic ideas used in constructing the CNT capture the essential physics, there is a need for improvement.

3.1.2 The nucleation rate Since the supercooled liquid can be maintained in the metastable liquid state well below the freezing point, considerations of the dynamics become important in understanding the process of crystal formation from the melt. This dynamic process generally involves clusters of various sizes of the new phase gaining or losing more molecules from the bulk liquid. This process is idealized in terms of rate equations that are local in time, controlling the loss or gain of particles from the clusters of different sizes. At the simplest level, namely the description of the rate process of growth (of small clusters), the role of memory is ignored. This essentially amounts to ignoring the correlation in the events which

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Crystal nucleation

cause the growth or decay of the clusters. The temperature is assumed to remain constant in the steady state as the cluster grows or decays. This implies that the clusters are in thermal equilibrium while the numbers of particles belonging to them change. The latent heat evolved in the process of phase change does not affect this condition. We also assume that the clusters grow or decay by attaching or losing single units or monomers. In the following we consider two cases of nucleation of the crystal: (a) transition from a fluid to a solid, i.e., crystallization from a low-concentration solution; and (b) crystallization from a high-density melt, whereby the growth of the nucleus takes place through an activated process. Nucleation from low monomer concentration Let the number of n-particle clusters of the crystalline phase present in the parent phase at n and  n denote the rates of monomer addition and loss time t be denoted by Nn (t). If + − from the n-particle cluster, respectively, the net number of clusters changing per unit time from size n to size n + 1 at time t is given by n+1 n Nn (t) − − Nn+1 (t). Jn (t) = +

(3.1.21)

Equilibrium is reached when Nn (for all n) no longer changes with time. The equilibrium distribution is given by Jn (t) = 0, i.e., n+1 e n e N n = − Nn+1 . +

(3.1.22)

On the other hand, a steady state is reached when the rates of change of the numbers of differently sized clusters become constant, i.e., independent of time and size. This is characterized by Nn (t) ≡ Nns , giving for the steady-state nucleation rate n+1 s n s N n − − Nn+1 . J s ≡ Jn (t) = +

(3.1.23)

In the steady state the rate of change of the number of clusters of a particular size Nn is independent of time but not necessarily equal to zero. The latter is a particular case corresponding to the equilibrium state. Using the relation (3.1.22), the steady-state nucleation rate J s defined above satisfies the relation    Nie s i s s J = + Ni − Ni+1 . (3.1.24) e Ni+1 Since the equilibrium density Nie of the clusters of size i is given by eqn. (3.1.2), we obtain from (3.1.24) the nucleation rate 3 4

G i+1 − G i i s exp . J s = + Nis − Ni+1 kB T

(3.1.25)

3.1 Classical nucleation theory

123

Using the Taylor expansions d Nis , di d = G i + i G i , di

s Ni+1 = Nis + i

G i+1

(3.1.26) (3.1.27)

and keeping terms up to lowest order, with i = 1, we obtain from (3.1.25) the following differential equation for Ni :

d d Ni i + + β Ni G i = −J, (3.1.28) di di where we drop from now on the superscript s for the steady state for simplification and i = use the notation β = 1/(kB T ). Now in general the nucleation rate is defined as + 2/3 ai λ+ , where λ+ is the dynamic factor denoting the number of monomers arriving at the nucleus per unit time per unit area and ai 2/3 is the surface area of the nucleus containing i i , we obtain monomers. Using this result for + d Ni d + β Ni G i = − J¯i −2/3 , di di

(3.1.29)

where J¯ = J/(aλ+ ). The formal solution of this equation is obtained by multiplying (3.1.29) by the factor exp(β G i ) and then integrating from an initial point i 0 up to i to obtain

 i βG i 0 −βG i βG w −2/3 ¯ dw e w + Ni 0 e −J , (3.1.30) Ni = e i0

where we define the number of clusters Ni = Ni0 for i = i 0 . Now we can further simplify the above expression for Ni by choosing i 0 to be small enough relative to the critical nucleus (i = i ∗ ) that the number of critical nuclei can be expressed in terms of the number of monomers N per unit volume as given in eqn. (3.1.2), i.e., Ni0 = N exp(β G i0 ). Hence

 i −βG i βG w −2/3 ¯ dw e w +N . (3.1.31) −J Ni = e i0

Now, for the number of clusters Ni with a very large number of particles we should have limi→∞ Ni = 0. On the other hand, since G i reaches a maximum at an intermediate i = i ∗ and falls to a negative value for large values of i, we must also have limi→∞ exp(−β G i ) = ∞. Therefore, in the large-i limit of eqn. (3.1.30), we obtain the result

 i dw eβG w w −2/3 + N = 0. (3.1.32) lim − J¯ i→∞

i0

The nucleation rate J = J¯aλ+ is therefore given by J=

N aλ+ , I∗

(3.1.33)

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Crystal nucleation

where the integral I∗ is obtained as  I∗ =

∞

exp [β G n ] n −2/3 dn.

(3.1.34)

i0

The free energy G n is maximum at n = n ∗ and the dominant contribution of the integral I∗ in the denominator of the RHS of eqn. (3.1.34) comes from values of n close to n ∗ . We evaluate this integral by replacing G n with a Taylor expansion up to quadratic terms, G n = G n ∗ −

* 1 ** G n ∗ * (n − n ∗ )2 , 2

(3.1.35)

and take the limit over n from n = n ∗ to ∞ since the contribution from n = i 0 to n = i ∗ is negligible. This gives  1/2

G n ∗ ∗ −2/3 *2π kB T* I∗ = n exp . (3.1.36) *G ∗ * kB T n The steady-state nucleation rate J is obtained as * *

*G ∗ * 1/2 G n ∗ n n∗ J = N + exp − , 2πkB T kB T

(3.1.37)

∗

n = (an ∗ 2/3 )λ represents the number of monomers that hit the nucleus of surwhere + + face area A0 = an ∗ 2/3 per unit time. The nucleation rate is thus expressed in the form of the general expression ! " J = J0 exp −G n ∗ /(kB T ) . (3.1.38)

The prefactor J0 in the above expression for the nucleation rate is proportional to the number of monomers N per unit volume of the mother phase from which the nucleating cluster is formed. J0 also has the following two components. ∗

n represents the number of monomers attaching to the nucleus (i) The dynamic factor + per unit time. It is given by A0 λ+ , where A0 is the area of the cluster and λ+ denotes the rate of arrival of monomers on the cluster per unit area. (ii) The statistical prefactor Z , called the Zel’dovich factor (Zel’dovich, 1943), * * *G ∗ * 1/2 n Z= , (3.1.39) 2π kB T

is determined by the shape of the free-energy surface. Using the expression (3.1.13) * * for G n ∗ and taking the second derivative gives *G n ∗ * = |μ|/(3n ∗ ). Hence the Zel’dovich factor is expressed in terms of the chemical potential difference as

1/2 μ . (3.1.40) Z= 6πkB T n ∗

3.1 Classical nucleation theory

125

The expression for the nucleation rate J is now obtained in the form ∗

n J = N +



μ 6πkB T n ∗

1/2



G n ∗ exp − . kB T

(3.1.41)

Note that, in the above expression for the nucleation rate per unit volume per unit time, N denotes the number of monomers in unit volume. In order to further simplify the above expression, we need to estimate the dynamic prefactor present in the expression for the nucleation rate. In the case of transition from a dilute fluid to a solid, the dynamic prefactor is determined by taking the ideal-gas approximation for the monomers hitting the surface of the nucleus. The number of particles λ+ falling on the surface per unit time is obtained from the simple kinetic theory of ideal-gas particles in terms of the equilibrium pressure Pe as  ∞  π/2 dv φ(v)(mv)(2π ) d(cos θ )cosθ Pe = λ+ 0 0 ; (3.1.42) = λ+ (2πmkB T ), where φ(v) denotes Maxwell’s velocity distribution for the ideal gas. Using the ideal-gas equation of state Pe = Ne kB T and the monomer concentration Ne at pressure Pe is reduced from its value N at the pressure P of nucleation by a factor S, where S = P/Pe . Thus we n ∗ for the nucleus of area A obtain for + 0   kB T 1/2 N n∗ + = (3.1.43) A0 . 2πm S For a spherically shaped critical nucleus the area is A0 = 4π R ∗ 2 with the radius R ∗ given by (3.1.18). Using (3.1.15), (3.1.16), and (3.1.10), we obtain for the nucleation rate J the result 9  

2 γ3 2γs N 2 vm 16π vm s J= (3.1.44) exp − , πm S 3(kB T )3 (ln S)2 in which the chemical potential difference μ between the two phases is substituted in terms of the supersaturation as μ = ln S/(kB T ). Crystallization from the melt For nucleation from the condensed phase, the dynamic prefactor in the nucleation rate is approximated using the reaction-rate approach (Turnbull and Fisher, 1949). In this model the process of addition of the monomer to the nucleus is assumed to take place through transition into an intermediate activated state that represents an unstable maximum in the free energy as shown schematically in Fig. 3.2. This maximum at A corresponds to the nucleus in the activated state and is intermediate between the two minima corresponding to the nuclei with i and i + 1 particles, respectively. These two minima correspond to

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Crystal nucleation

Fig. 3.2 A schematic representation of the free energy of activation corresponding to the process of adding a monomer to a nucleus.

the states of the nucleus before and after the monomer attachment. We assume that the free energies of formation of the nuclei corresponding to i, A, and i +1 are respectively denoted by G i , G A , and G i+1 . The free-energy difference between the states corresponding to A and i is given by 1  f i∗ ≡ G A − G i =  f ∗ + (G i+1 − G i ) 2 i dG i ∗ , = f + 2 di

(3.1.45)

where we use the definition  f ∗ = G A − (G i+1 + G i )/2. Similarly, the free-energy difference between the nuclei corresponding to A and i + 1 is 1 ∗ ≡ G A − G i+1 =  f ∗ − (G i+1 − G i )  f i+1 2 i dG i ∗ . = f − 2 di

(3.1.46)

The steady-state rate of the nucleation J in this case is obtained in a manner similar to that used in formulating eqn. (3.1.23), as a sum of two opposite processes. These respectively ∗ and  f ∗ . The correspond to rate processes (at temperature T ) of crossing barriers  f i+1 i nucleation rate is obtained as  !    " ∗ Ni+1 , (3.1.47) J = ai 2/3 ν0 exp −β  f i∗ Ni − exp −β  f i+1 where ν0 = kB T / h is the primary frequency corresponding to the thermal energy kB T and ai 2/3 is the surface area of the nucleus with i monomers. Using the Taylor expansions as

3.1 Classical nucleation theory

127

presented in eqn. (3.1.26) and keeping terms of up to lowest order in i, we obtain the following differential equation for the variation of Ni :

d d Ni (3.1.48) + β Ni G i = − J¯i −2/3 , di di where J¯ = J/{aν0 exp[− f ∗ /(kB T )] exp[− f ∗ /(kB T )]}. Note that (3.1.48) is identical to eqn. (3.1.29) in the earlier section except that now the rate of monomer attachment per unit area λ+ is replaced by ν0 exp[− f ∗ /(kB T )]. Hence the nucleation rate J in this case is obtained from (3.1.41) in the form

 1/2 G n ∗ +  f ∗ μ J = N ν0 n ∗ 2/3 exp − , (3.1.49) 6πkB T n ∗ kB T where N denotes the number of monomers per unit volume. In the above expression for the nucleation rate the exponent term involving the free-energy barrier G n ∗ is further simplified. Using (3.1.17), G n ∗ is expressed in terms of μ ≡ vm G v , where the chemical potential difference μ between the two phases is related to the molar enthalpy h and the molar entropy s through the thermodynamic relation μ = h − T s = s(Tm − T ).

(3.1.50)

We have used here the result that close to the melting temperature Tm , h = Tm s. Using (3.1.50) in (3.1.17) and the definition μ = vm G v , we obtain for the free-energy barrier G i ∗ =

2 γ3 16π vm s

3 s 2 (Tm − T )2

.

(3.1.51)

  2 1/3 The exponential factor in (3.1.49) simplifies further with the definition a = 36π vm and the expression (3.1.16) for i ∗ . We obtain the following result for the nucleation rate:  

2 (kB T γs )1/2 f ∗ −16π γs3 vm J = 2N vm exp − exp h kB T 3kB T s 2 (Tm − T )2

(3.1.52)

for the spherical nucleus. From eqn. (3.1.51) it follows that the free-energy barrier G v to be surmounted for the critical nucleus formation falls sharply as the undercooling of the liquid is increased, i.e., G v ∝ TR−2 , where the extent of undercooling is defined as Tm − T T =1− ≡ 1 − TR ≡ TR Tm Tm

(3.1.53)

in terms of the scaled temperature TR = T /Tm . At T = Tm the barrier G v is infinite, making the nucleation rate zero. To initiate the formation of the crystal seed the temperature of the liquid has to be well below the freezing point Tm , i.e., TR of the liquid has to be large. We now consider this dependence of the nucleation process on the extent of undercooling.

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Crystal nucleation

Dependence on undercooling The temperature dependences of the two exponential factors in the above expression for the nucleation rate J result in opposite trends. The first factor, exp[− f ∗ /(kB T )] decreases with lowering of T . On the other hand, the second exponential factor increases as T decreases, since the difference T = Tm − T grows with lowering of the temperature. In fact, the latter factor in the exponent produces the dominant effect and for an increase of T by a few degrees the exponential factor in the nucleation rate J changes by orders of magnitude. To see this, we rewrite eqn. (3.1.52) in the form



G i ∗ a0 α 3 ζ ≡ J0 exp − (3.1.54) J = J0 exp − kB T TR (TR )2 using the definitions 2/3

α=

vm γs , h

ζ =

h . kB Tm

(3.1.55)

The constant a0 = 16π/3 for a spherical droplet. In order to obtain this value of J0 the energy barrier  f ∗ has to be estimated. J0 is inversely related to the time τJ taken by a monomer to cross this barrier at the interface and become attached to the nucleus. Let us write J0 in the form CN /τJ , where CN is a constant, and hence the nucleation rate J is

CN G i ∗ J= exp − . (3.1.56) τJ kB T τJ is inversely proportional to the diffusion constant Ds and hence directly proportional to the viscosity if we assume that the Stokes–Einstein law holds. A lower value of the viscosity implies a shorter passage time and hence a lower barrier. We ignore the temperature dependence of the viscosity of the undercooled liquid at first to keep J0 within its upper bound. The prefactor J0 is estimated as 1032 (Turnbull and Fisher, 1949), taking the viscosity of the liquid as 10−2 P. The effect of undercooling on the crystallization is demonstrated through a plot of the nucleation rate with respect to the scaled temperature TR . This is shown in Fig. 3.3 (Turnbull, 1969). The curves are fixed by choosing constant values of the parameter αζ 1/3 (≡ 0 , say). Close to Tm , i.e., at low undercooling (TR ≈ 1), the curve for J corresponding to a fixed 0 rises steeply. The curve reaches a maximum at TR = 1/3 and falls off to zero as we approach T = 0. In order to be able to observe crystallization experimentally, the rate J must exceed some minimum practical value, say one nucleus per cubic meter per second (10−6 cm−3 s−1 ). This threshold value for J is shown by a horizontal line in Fig. 3.3. We notice in this plot of Fig. 3.3 that at low undercooling or close to Tm , corresponding to most values of the parameter 0 , the nucleation rate J falls much below the horizontal line and cannot be observed experimentally. In this case for 0 > 0.9 the crystallization practically never occurs through a homogeneous nucleation process. On the other hand, crystallization is almost impossible to suppress for 0 < 0.25. For intermediate values of 0 crystallization can remain suppressed only up to moderate undercooling.

3.1 Classical nucleation theory

129

  Fig. 3.3 The logarithm of the rate J in cm−3 s−1 of homogeneous nucleation in the undercooled liquid given by (3.1.54) with reduced temperature TR for various values of 0 ≡ αζ 1/3 . The temperature dependence of the viscosity η present in the prefactor J0 in (3.1.54) is ignored here. From c Taylor & Francis Group, http://www.informaworld.com Turnbull (1969). 

The above analysis is based on two assumptions: first, that the Stokes–Einstein relation holds; and second, that the temperature dependence of the viscosity can be ignored. Both are violated in the liquid deeply supercooled below Tm . A more realistic scenario for the crystallization process in the supercooled liquid emerges when we take into account (a) the sharp temperature dependence of the viscosity of the metastable liquid below the freezing point and (b) the violation of the Stokes–Einstein law. This is considered in the next chapter in our discussion of the glass transition. 3.1.3 Heterogeneous nucleation In our discussion of the crystallization process we have so far considered the phenomena to be taking place in the bulk of the liquid phase. This is generally termed homogeneous nucleation. In reality, however, the process occurs in a heterogeneous manner around foreign particles or impurities distributed in the sample. The surface of the liquid droplet is also often the source of heterogeneous nucleation. Turnbull (1950) extended the CNT for homogeneous nucleation to the heterogeneous process by considering the liquid and crystalline phases in contact with a solid substrate. The outcome of this is a modification of

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Crystal nucleation

the free-energy barrier to the nucleation as obtained in the CNT in eqn. (3.1.17) in the following form: G i ∗ =

16π γs3 3|G v |2

f HN (θ ).

(3.1.57)

The factor f HN (θ ) is the heterogeneous nucleation factor given by f HN (θ ) =

1 (2 + cos θ )(1 − cos θ )2 , 4

(3.1.58)

where θ is the contact angle between the substrate and the crystal droplet nucleating from the liquid phase. This contact angle θ is determined in terms of the surface free energies of the different phases, cos θ =

γLS − γCS , γs

(3.1.59)

where γs denotes the surface free energy of the liquid–crystal interface while γLS and γCS , respectively, represent the surface free energies for the liquid–substrate interface and the crystal–substrate interface. If the substrate on which the nucleation is occurring is identical to the liquid phase we have (γLS − γCS ) → −γs and cos θ → 1, i.e., θ → π. In this limit we have homogeneous nucleation occurring in the liquid and f HN (θ ) = 1. The main implication of the heterogeneous nucleation is therefore represented in terms of a modification of the nucleation barrier. There is also a corresponding modification in the pre-exponential term J0 for the nucleation rate as given in (3.4.7) later. In a bulk heterogeneous process expectedly the factor modifying the nucleation rate signifies the fraction of the impurity sites which has been used up. Gránásy and Iglól (1997) assumed a modified nucleation rate in the following phenomenological form:

G ∗ J = {ϑ J0 } exp − f HN , (3.1.60) kB T where ϑ < 1 is the fraction of molecules active in nucleation. f HN < 1 is the factor representing the reduction of the potential barrier due to the presence of heterogeneities, as has ∗ already been discussed above. From (3.1.17) the exponential factor f HN [G /(kB T )]  eqn. 3 3 in (3.1.60) is proportional to γT / T G v , where γT is the surface free energy at temperature T . In the CNT the quantity γT is usually taken to be independent of temperature and this is rectified in terms of the quantity χ (T ) = γT /γs , which is the ratio of the the two temperature-dependent surface tension γs to the bulk surface tension γs between  phases. Therefore a plot of log[J/J0 ] against the quantity X ≡ χ (T )3 / T G 2v is a straight line with an intercept on the y axis equal to log ϑ. Using estimates of X from the different types of theories described above, Gránásy and Iglól (1997) computed the corresponding prediction for ϑ in each case. Since ϑ should be less than unity by definition, its values obtained for the various theoretical models are useful for judging the appropriateness of the corresponding theory.

3.2 A simple nonclassical model

131

3.2 A simple nonclassical model We now discuss the process of formation of the critical nucleus in terms of a simple schematic model (Bagdassarian and Oxtoby, 1994) in which the solid–liquid interface is described in terms of a single order parameter. The present model is treated analytically and also captures the important aspects of a more realistic density-functional approach to be described in subsequent sections. The theory involves choosing simple approximate forms for the free-energy profile of the undercooled liquid having minima corresponding to the stable and metastable bulk phases. The single order parameter ψ(r, t) used in the present model is assumed to have ψ(r, t) = 0 for the metastable bulk liquid phase and some characteristic constant value, say ψs , for the stable bulk solid phase. The relative depth of the deeper minima denotes the extent of supercooling of the system. The time evolution of this order parameter is written in the standard time-dependent Ginzburg–Landau (TDGL) (Ma, 1976) form δ[ψ] ∂ψ(r, t) = −0 , (3.2.1) ∂t δψ where [ψ] is the grand-canonical free energy expressed as a functional of the order parameter ψ(r, t) and 0 is the bare kinetic coefficient. The term δ/δψ on the RHS of (3.2.1) acts like a driving force (de Groot and Mazur, 1984) controlling the time variation of the order parameter. Hence the equation δ[ψ] =0 δψ

(3.2.2)

corresponds to a stationary nucleus. Now we write the grand potential functional of the inhomogeneous solid liquid in the square-gradient approximation (Cahn and Hilliard, 1958) of the order-parameter fluctuation in terms of the simple expression  # $ 1 dr ω[ψ] + |∇ψ(r, t)|2 . [ψ] = (3.2.3) 2 Using this form of free energy in eqn. (3.2.1), the time evolution of ψ(r, t) becomes − 0−1

∂ψ(r, t) δω[ψ] = − ∇ 2 ψ(r, t). ∂t δψ

For the free-energy density ω(ψ) we choose the suggestive form 3 4 1 1 ω(ψ) = min λ0 ψ 2 , λs (ψ − ψs )2 + T 2 2

(3.2.4)

(3.2.5)

which represents two intersecting parabolas as shown in Fig. 3.4, with their respective minima at ψ = 0 and ψ = ψs . The depths of the two minima differ by an amount T , which is a measure of the undercooling of the system. T = 0 corresponds to the situation in which the two minima in ω are equal, signifying the coexistence of the liquid and crystalline phases. The curvatures of the two parabolas corresponding to the liquid

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Crystal nucleation

Fig. 3.4 The schematic form of the potential energy taken as a combination of two parabolas as given by eqn. (3.2.5). The height T of the free-energy barrier between the two minima of the free energy represents the extent of supercooling of the system.

and crystalline states are denoted by λ0 and λs , respectively. The parabolas intersect at the order parameter value ψ = ψc , which can be related to the undercooling T by T =

1 1 λ0 ψc2 − λs (ψs − ψc )2 . 2 2

(3.2.6)

ψ(r, t) is the order parameter through the solid–liquid interface. The critical nucleus is represented by eqn. (3.2.2), which represents a saddle point for the free energy [ψ] (Bender and Orszag, 1978). Note that

  δ[ψ] 2 δ[ψ] ∂ψ ∂ , (3.2.7) = dr = −0 dr ∂t δψ ∂t δψ so that  always decreases while the optimum order-parameter profile is being determined by the condition (3.2.2). In solving this equation for ψ, we note that for the part ψ > ψc we need to use the solid part of the free-energy profile (the parabola with its minimum at ψc ), whereas for ψ < ψc the liquid part (the parabola with its minimum at 0) is to be used. In the above choice of the free energy in the double-parabola form there is automatically the limit in which the liquid loses its metastability with respect to the solid, i.e., ψc = 0. This limiting situation is characterized by the undercooling T obtained from (3.2.6) as sp ≡ T |ψc =0 = −

λs ψs2 2

(3.2.8)

and represents a spinodal in the present model. The basic equation describing the critical nucleus therefore follows from (3.2.4) in the form of a second-order differential equation for the order-parameter field in the steady state, δω[ψ] . (3.2.9) ∇ 2ψ = δψ

3.2 A simple nonclassical model

133

For representing the spherical nucleus we consider a spherically symmetric order-parameter profile. Using the corresponding spherically symmetric form of the ∇ 2 operator, we obtain the following differential equations for the order parameter from eqn. (3.2.2): d 2 ψ(r ) 2 dψ(r ) + − [ψ(r ) − ψs ] = 0, r dr dr 2 d 2 ψ(r ) 2 dψ(r ) + − ν¯ 2 ψ(r ) = 0, r dr dr 2

for ψ < ψc ,

(3.2.10)

for ψ > ψc .

(3.2.11) √ In writing the above two differential equations we express length r in units of 1 λs (i.e., √ √ λsr → r ) and define the parameter ν¯ = λ0 /λs . The two differential equations stated above are obtained by using in the equation (3.2.2) the respective parabolic form for the grand potential function  as stated in eqn. (3.2.5). The parameter ν¯ denotes the relative curvatures of these two parabolas representing the order-parameter dependence of the free energy . The crossover value of the order parameter ψ = ψc marks the proximity to the stable liquid or the crystalline minimum. The crystalline nucleus is identified with the order-parameter value ψ > ψc , since in this case the system is closer to the free-energy minimum at ψ = ψs . The order-parameter values ψ < ψc , on the other hand, correspond to the liquid state, which is characterized by the minimum at ψ = 0. 3.2.1 The critical nucleus Equations (3.2.10) and (3.2.11) are solved to obtain the following relation among ψc , ψs , and rc for the critical nucleus: ψc = ψs

1 − rc−1 tanh rc . 1 + ν¯ tanh rc

(3.2.12)

The solution of eqns. (3.2.10) and (3.2.11) under appropriate boundary conditions to reach the above result (3.2.12) is given in Appendix A3.1. The boundary r = rc therefore is a marker for the crystalline phase in this case and is a natural identification of the radius of the critical nucleus corresponding to the condition (3.2.2). As we see in eqn. (A3.1.4) of Appendix A3.1.1, ψin (0) → ψs , i.e., the order parameter at the center of the nucleus approaches the bulk-solid value. The relation (3.2.12) between ψc and ψs is useful in determining how the critical radius changes with the undercooling T . Two limiting cases are of interest here. Near coexistence In this limit the undercooling T → 0 and it then follows from (3.2.6) that ψs → ψc (1+ ν¯ ). Identification of the same relation from (3.2.12) requires tanh rc → 1 and rc−1 → 0. This corresponds to rc → ∞. In order to find the leading-order relation between rc and T , we simplify the relation (3.2.6) to leading order in T as

T ψs − ψc = ν¯ ψc 1 − . (3.2.13) λ0 ψc2

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Crystal nucleation

Upon inverting the above relation expressing ψc in terms of ψs with the approximation ψs ≈ (1 + ν¯ )ψc , we obtain ( ) ψs T + O 2T . + (3.2.14) ψc = 1 + ν¯ ν¯ λs ψs For large rc , to leading order (3.2.12) reduces to

r −1 1 − c . ψc = ψs 1 + ν¯ 1 + ν¯ On comparing (3.2.14) and (3.2.15) we obtain the relation

ν¯ rc T = − λs ψs2 . 1 + ν¯

(3.2.15)

(3.2.16)

In the above limit of coexistence T → 0 (rc → ∞) we have rc =

2 ˜T 



ν¯ , 1 + ν¯

(3.2.17)

˜ T = T /sp is the undercooling of the system relative to that at the spinodal point. where  The spinodal limit This is characterized by the situation ψc → 0, with the corresponding undercooling T being given by (3.2.8). In this case it follows from (3.2.12) that tanh rc → rc , i.e., rc → 0, implying that the liquid state is always unstable. In the limit rc → 0 the relation (3.2.12) reduces to the form ψc ≈ ψsrc2 /3. On using this in the definition (3.2.8) for the undercooling T we obtain



λs ψs2 2ψc 2 2 λs ψs2 1− 1 − rc . =− (3.2.18) T = − 2 ψs 2 3 The radius of the critical nucleus rc approaches zero with the supercooling sp corresponding to the spinodal point as,  1/2 ˜T . (3.2.19) rc ∼ 1 − 

3.2.2 The free-energy barrier The free energy for the formation of the critical nucleus relative to the liquid is computed in the present schematic model as

 ∞ |∇ψ(r, t)|2 dr r 2 ω[ψ(r, t)] + , (3.2.20) NC = 4π 2 0 where the subscript NC refers to a nonclassical quantity to distinguish it from the corresponding quantity in classical nucleation theory. In Appendix A3.1.1 we demonstrate that

3.2 A simple nonclassical model

135

evaluating the RHS of eqn. (3.2.20) corresponding to the critical nucleus profile leads to

) 4πrc3 1 − rc−1 tanh rc ( 2 T − 4πrc sp (3.2.21) ν¯ + rc−1 . NC = 3 1 + ν¯ tanh rc The last equality above is reached by using the relation (3.2.12) for ψc /ψs . Using the leading-order expansion of tanh rc ≈ 1 − 2 exp(−2rc ) for large rc the second term on the RHS can be further simplified to obtain NC =



4πrc3 ν¯ T − 4πrc2 + rc−1 sp . 3 1 + ν¯

(3.2.22)

Upon identifying the RHS of (3.2.22) with the surface term in the classical expression (3.1.9) for the free energy of the critical nucleus, we obtain the following expression for the surface free energy: γs = −sp

ν¯ 1 + ν¯

≡

λs ψs2 2



ν¯ . 1 + ν¯

(3.2.23)

The free energy NC and radius rc of the optimum cluster in the present schematic model can be compared with the corresponding results of the CNT. To make such a comparison we need to identify the relevant quantities in the present model with their corresponding counterparts in the CNT. This is done primarily on the basis of the similarities between the expression (3.2.22) for the free energy in the schematic model and the classical expression for the same quantity as given by (3.1.17). Thus rc is taken as the radius of the critical nucleus and T as the free-energy difference G v in the CNT. The surface tension γs is chosen from (3.2.23). Using the expression for the (3.1.17) for G ∗ we obtain the free energy (in units of sp ) as 16π G˜ ∗ = ˜2 3 T



ν¯ 1 + ν¯

3 .

The radius of the critical nucleus in the CNT is obtained from (3.1.18) as

2γs 2 ν¯ ∗ R =− = , ˜ T 1 + ν¯ T 

(3.2.24)

(3.2.25)

which agrees with the corresponding result (3.2.17) in the schematic model close to coexistence. For a chosen free-energy landscape (characterized by ν¯ ) and a given undercooling ˜ T ), the CNT expressions for the free energy and the radius of the critical (determined by  nucleus are given by (3.2.24) and (3.2.25), respectively. In the nonclassical schematic model, on the other hand, estimates of the free energy and radius of the critical nucleus are obtained by directly solving eqns. (3.2.12) and (3.2.6). These two equations can be conveniently written as

136

Crystal nucleation

ϒ≡

ψc 1 − rc−1 tanh rc = , ψs 1 + ν¯ tanh rc

where ϒ is obtained from the solution of the equation ( ) ( ) ˜ T = 0. ϒ 2 1 − ν¯ 2 − 2ϒ + 1 − 

(3.2.26)

(3.2.27)

For a given value of ν¯ the critical radius rc is obtained as a function of the relative under˜ T . The free-energy barrier ∗ (expressed in units of sp ) to formation of the cooling  NC critical nucleus is obtained from (3.2.21) as ∗NC =

( ) 4πrc3 ˜ T − 4πrc2 ϒ ν¯ + rc−1 .  3

(3.2.28)

˜ T ) and ∗ (ν¯ ,  ˜ T ) are determined simultaneously. Therefore the critical radius rc (ν¯ ,  NC The radius rc of the critical nucleus in the present model can be either higher or lower than its counterpart R ∗ in the CNT depending on the value of the parameter ν¯ and the extent ˜ T . Results for the radius of the critical nucleus from these two theories of undercooling  agree in the limit of zero supercooling, i.e., at coexistence, at which rc → ∞. The same behavior is observed with respect to the height of the free-energy barrier to critical-nucleus formation computed in the above two theories. The dependence of the critical radius rc ˜ T is shown in Fig. 3.5. We choose for the parameter on the degree of undercooling  ν¯ = 0.5 here. In Fig. 3.5, we see that the predictions of the nonclassical theory can be higher and lower than the corresponding CNT predictions, depending on the undercooling. Both trends are also predicted within the different types of DFT models discussed in the next section. We have focused here primarily on the formation of the critical nucleus. The dynamics of its evolution is also obtained (Bagdassarian and Oxtoby, 1994) by solving the corresponding TDGL equations (3.2.1) for specific choices of the free-energy landscape defined in terms of ν¯ .

−1/2 ˜ T (see the text) Fig. 3.5 The critical radius rc in units of λs (see the text) vs. the undercooling  for two values of the parameter ν¯ (see text): (a) 1.0 and (b) 0.6. The corresponding values of the critical radius obtained from the CNT in the respective cases are also shown for comparison. From c American Institute of Physics. Bagdassarian and Oxtoby (1994). 

3.3 The density-functional approach

137

3.3 The density-functional approach The density-functional theory outlined in the previous chapter has been used for understanding the nucleation phenomena starting from basic principles of statistical mechanics. Compared with the model described in the previous section, here a more realistic description of the crystal and the liquid state is chosen. In this formulation the interaction potential between the constituent particles in the system is the primary input. As we have seen in Chapter 2 regarding studying the bulk crystalline phase or the interfaces, the grandcanonical potential is minimized with respect to the inhomogeneous density function in order to identify the critical nucleus. In the DFT formulation, unlike in the CNT described above, the effect of curvature of the interface in the computation of the free energy of the clusters is taken into account. The difference  between the grand-canonical potentials of the inhomogeneous state (consisting of the nucleating bubble) and of the homogeneous liquid state is optimized with respect to the inhomogeneous density function. This automatically identifies the critical nucleus, i.e., corresponding to the optimum density function n 0 (x) representing the crystal nucleus in the melt, we have [n 0 (x)] = l + G ∗ ,

(3.3.1)

where G ∗ denotes the free energy required for the formation of the critical nucleus (see eqn. (3.1.15) in the discussion of the CNT above). In formulating the appropriate DFT for the present case it is important to notice that the critical nucleus is in fact dynamically unstable. However, this instability is with respect to the variation of the number of monomers in the critical nucleus, which can either grow or shrink as a result of fluctuations in the number of monomers. This is therefore relevant when the grand-canonical ensemble is used to describe the critical nucleus (with a fixed number of particles, on the other hand, the critical nucleus is the state of minimum free energy and hence corresponds to a thermodynamically stable state). The critical nucleus represents a saddle point in the function space of density, having one unstable direction with respect to the particle number. Hence the optimum density distribution of the critical nucleus is determined from the solution of the Euler–Lagrange equation

δ = 0. (3.3.2) δn(r) n(r)=n 0 (r) The identification of the critical nucleus and its size as depicted through the optimum density profile follows in a natural way from the optimization of the thermodynamic function in the DFT approach to this problem. However, it is still a mean-field approach and the treatment of the smallest clusters containing a few molecules can only be approximate.

3.3.1 The square-gradient approximation A crucial ingredient of the DFT is the choice of the test density function with respect to which the appropriate thermodynamic functional is minimized. We have discussed in

138

Crystal nucleation

the previous chapter (see, for example, eqn. (2.4.32)) how the density-functional model is extended to study the plane interface of the liquid and the crystal. The parametrization of the density function for the critical nucleus is done in a manner similar to that done in the DFT studies for the plane interface of the liquid and crystal. The spatial dependences of the parameters in the inhomogeneous density function characterizing the nucleus are chosen in such a manner that, away from the surface of the nucleus, near the center, the structure is crystalline, while at large radial distances the uniform liquid state of the melt is obtained. A typical example follows from the ansatz (2.1.42) for the density function generalized in the following manner for the spherical nucleus (Harrowell and Oxtoby, 1984):    iKm · r Am (r )e , (3.3.3) n 0 (r) = n l 1 + η(r ) + m

in terms of a set of local order parameters, γ (r) ≡ {η(r ), Ai (r )}. The corresponding boundary conditions are η(r → ∞) = 0 for the liquid state and η(r → 0) = (n s − n l )/n l , which is the fractional change of density on solidification, n s being the average density of the solid state. In the discussion below we generalize this approach. We assume that the inhomogeneous density function n 0 (r; γ ) is parametrized in terms of locally varying amplitudes denoted by γ ≡ {γi (r)} for i = 1, . . . , m. The γi form a set of local order parameters. The grand potential function  for the inhomogeneous state is computed within the square-gradient approximation. The latter is applied to evaluate the nonlocal contributions to the thermodynamic potential involving the densities at two different spatial points. To illustrate this approximation, we consider the local and nonlocal parts of [n(r)],  [n] = Fid + Fex − μ n(r)dr. (3.3.4) The subscripts of the first and second terms on the RHS denote the ideal and the excess parts of the Helmholtz free energy, respectively. The ideal part Fid is represented in terms of a local functional of the density and the interaction or the excess part Fex involves nonlocal contributions. For the uniform state with constant order parameters these are given in Chapter 2 in eqns. (2.1.27) and (2.1.38), respectively. The grand potential  of the inhomogeneous state corresponding to the density function n 0 (x, {γi }) is obtained in terms of a functional Taylor-series expansion around the uniform liquid state.

  n 0 (x1 ; γ ) − dx1 (n 0 (x1 ; γ ) − n l ) β  = dx1 n 0 (x1 ; γ )ln nl  1 + (3.3.5) dx1 dx2 c(x1 , x2 )δn 0 (x2 ; γ (x2 ))δn 0 (x1 ; γ (x1 )) + · · ·, 2 where the direct correlation function c(r ) (Ohnesorge et al., 1991) is defined as

δ 2 Fex [n 0 ] c(x1 , x2 ) = −β . δn 0 (x1 ; γ )δn 0 (x2 ; γ ) n l

(3.3.6)

The above expression is similar to the expansion around the uniform liquid state shown in eqn. (2.1.41) in Chapter 2. Note that the RHS of (3.3.5) involves the inhomogeneous

3.3 The density-functional approach

139

densities and hence the corresponding order parameters γi at two different spatial points x1 and x2 in the interaction part. This part is simplified by adding to the local contribution corrections evaluated within the square-gradient approximation. On expanding the amplitudes {γ } up to second order in spatial derivatives, we obtain for the ith member of the set 1 (3.3.7) γi (x2 ) = γi (x1 ) + {x21 · ∇}γi (x1 ) + {x21 · ∇}2 γi (x1 ), 2 where x2 − x1 = x21 and ∇ indicates the derivatives with respect to the spatial variation of the local order parameters. The density δn 0 (x2 ; γ (x2 )) is now expressed within the above square-gradient approximation as

 ∂n 0 (x2 ) 1 {x21 · ∇}γi + {x21 · ∇}2 γi n 0 (x2 ; γ (x2 )) = n 0 (x2 ; γ (x1 )) + ∂γi 2 i

+

1  ∂ 2 n 0 (x2 ) 2

∂γi ∂γ j

ij

{x21 · ∇}2 γi γ j ,

(3.3.8)

where ∂n 0 (x2 )/∂γi denotes the partial derivative of the density n 0 (x2 ; γ (x1 )) with respect to its dependence on the order parameter γi . We will assume here a linear dependence of the field γi on the density function (similar to the relation (2.4.32) stated above for the interface problem) and hence ignore contributions from the last term on the RHS of (3.3.8). Substituting (3.3.8) for the density fluctuation δn 0 (x2 ; γ (x2 )) reduces (3.3.5) to the form

  n 0 (x1 ; γ ) − dx1 (n 0 (x1 ; γ ) − n l ) β  = dx1 n 0 (x1 ; γ )ln nl   1 dx1 dx2 c(x1 , x2 )δn 0 (x2 ; γ (x1 ))δn 0 (x1 ; γ (x1 )) + 2    ∂n 0 (x2 ) 1 c(x1 , x2 )δn 0 (x1 ; γ (x1 )){x21 · ∇}2 γi . (3.3.9) dx1 dx2 + 4 ∂γi i

In obtaining the above result we have set the contribution from the linear term in the expansion (3.3.7) to zero using the fact that the spatial Fourier transform c(k) of the direct correlation function c(r ) at the RLV is c (K n ) = 0 since the Kn correspond to peaks of the structure factor. Let us consider the different terms on the RHS of (3.3.9). (a) The first three terms on the RHS of (3.3.9) constitute to leading orders a contribution that is expressed in terms of a grand potential density D(x) as 3

 n 0 (x1 ; γ (x1 )) − δn 0 (x1 ; γ (x1 )) β u [γ ] = dx1 n 0 (x1 ; γ (x1 )) ln nl 4  1 dx2 c(x1 , x2 )δn 0 (x2 ; γ (x1 ))δn 0 (x1 ; γ (x1 )) + 2  ≡ dx D[γi (x)]. (3.3.10)

140

Crystal nucleation

D involves the order parameter γi (x) only locally. This part of the grand potential is denoted as u with the subscript u signifying the fact that the functional dependence of u is the same as that for the uniform system with constant order parameters. In (3.3.9) u is being evaluated with the Ramakrishnan–Yussouff functional Taylor-series expansion. Alternatively, this local contribution can be computed using the MWDA in the same manner as formulated for the bulk phases, for a more accurate evaluation of the grand potential. (b) The last term on the RHS of (3.3.9) is further simplified with a partial integration and using the substitution ∇α n 0 (x1 ; γ (x1 )) =

 ∂n 0 (x1 ) j

∂γ j

∇α γ j (x1 ),

(3.3.11)

where ∇α denotes the spatial derivative with respect to the corresponding spatial component xα . The coefficient of the square-gradient term, i.e., of ∇α γi ∇β γ j , is expressed αβ as a kernel #i j . The latter is evaluated to leading order in terms of the integral per6 formed with V −1 dx1 and is expressed in terms of the local order parameters γ (x), αβ #i j [γ (x)]

kB T =− 4V



β

α x21 x21

∂n 0 (x2 ) ∂n 0 (x1 ) c(x1 , x2 ; γ (x))dx1 dx2 . ∂γi ∂γ j

(3.3.12)

On combining the above terms we therefore obtain the following result for the difference between the grand potential of the inhomogeneous state and that of the uniform liquid:  = kB T



⎧ ⎨

dx D[γi (x)] + ⎩

αβ  i, j

αβ

⎫ ⎬

#i j [γ (x)]∇α γi (x)∇β γ j (x) . ⎭

(3.3.13)

Thus, in summary, the first term on the RHS of (3.3.13) represents the thermodynamic potential  obtained as a local functional of the order parameter γ (x) while the second term represents the contribution (evaluated within the square-gradient approximation) due to the variation of the local amplitudes γi through the interface region. The critical nucleus is identified by optimizing the thermodynamic grand-canonical potential , i.e., solving the corresponding Euler–Lagrange equation (3.3.2). In the present context these equations simplify to ordinary differential equations,

δ = 0. δγi (r)

(3.3.14)

3.3 The density-functional approach

141

The corresponding choice for the set of order parameters represents a saddle point for [n] in the function space of density. The ordinary differential equations obtained for the order parameters are given by αβ  ∂ ∂ αβ D[γ ] + # [γ ]{∇α γi (x)∇β γ j (x)} ∂γi ∂γ j i j j # $ αβ αβ − #i j [γ ]∇α ∇β γ j (x) − ∇α #i j [γ ] ∇β γ j (x) = 0

(3.3.15)

for the index i = 1, . . . , m running over the set. The above equations represent the spatial variation of the local amplitudes γi across the crystal–liquid interface in the critical nucleus. From what we have described above, the first step in the construction of the densityfunctional model of the nucleus in the melt therefore involves constructing (a) the density function parametrized in terms of a set of local order parameters that are chosen so as to represent a spherical nucleus in the melt and (b) the proper grand-canonical potential  as a functional of the inhomogeneous density function. In the simplest approximation the first term in (3.3.13) is computed in terms of a low-order functional Taylor-series expansion of the Ramakrishnan and Yussouff (1979) model described in Chapter 2. However, this would imply an ad-hoc truncation of the density expansion for the crystalline state in which the density fluctuations are not small. Alternatively, u is computed using the more sophisticated weighted-density-functional methods described in Chapter 2. We consider both cases here.

3.3.2 The critical nucleus In the simplest approximation u is computed (Harrowell and Oxtoby, 1984) in terms of the expansion (3.3.10) around the uniform liquid state and keeping terms of up to second order in the density fluctuations around the uniform liquid state. The functional  is to be optimized with respect to a suitably chosen density function, which represents the crystal nucleus. We consider for the density n 0 (r) representing the spherical nucleus the functional form (3.3.3) with the local amplitudes γ (r) ≡ {η(r), Ai (r)} being functions solely of the radial distance r . This implies spherical symmetry for the order parameters but not for the density function itself. Using the parametrization (3.3.3), we obtain for the density fluctuation δn 0 (x; γ ) = n 0 (x; γ ) − n l of the inhomogeneous state around the uniform liquid state density n l  δn 0 (x2 ; γ (x1 )) = n l η(x1 ) +

 i

 Ai (x1 )e

iKi · x2

.

(3.3.16)

142

Crystal nucleation

On substituting (3.3.17) into (3.3.10), the grand potential function u is obtained as ⎤ & ' ⎡    ˜ + Ai (x)eiKi · x ln ⎣η(x) A j (x)eiK j · x ⎦ u = n l dx η(x) ˜ +  − nl

i

&

dx η(x) +



' Ai (x)eiKi · x −

i

& × η(x) +



Ai (x)eiKi · x

'⎧ ⎨

i

⎩

η(x) +

n 2l 2





j



dx

dx c(x, x ; γ )

A j (x)eiK j · x

j

⎫ ⎬ 

,

⎭

(3.3.17)

where η˜ = 1 + η. Next, with the present choice of the density function, we obtain the following values for the partial derivatives: ∂n 0 (r) = nl, ∂η

and

∂n 0 (r) = n l eiKi · r . ∂Ai

(3.3.18)

αβ

Using the results (3.3.18) in the expression (3.3.12) the kernel #i j is obtained in a purely diagonal form, ⎧  ⎨ c0 δαβ , for γi , γ j ≡ η, 4 αβ (3.3.19) #i j =  ⎩ cn δ δ , for γ ≡ A , γ ≡ A . i i j j 4 i j αβ We represent here c(k) as the spatial Fourier transform of the direct correlation c(r ), which is assumed to be translationally invariant for the uniform liquid state. We have used here the notation cn = n l c(K n ) for the liquid density n l and the double-primed quantity, i.e., cn , is the corresponding second derivative of cn with respect to the wave vector K n . It is useful to note the following relations in this respect: 2

 d c(K ) n l dr c(r )rr = −n l I = −c0 I, (3.3.20) d K 2 K =0 2

 ˆ nK ˆ nK ˆ n d c(K ) ˆ n cn , n l dr c(r )eiKn · r rr = −n l K = −K (3.3.21) d K 2 Kn ˆ n is the unit vector along Kn . The subscript 0 in c0 denotes the same quantity where K for zero wave vector. Note that with the present choice (3.3.3) for the density function the αβ kernel #i j is not only diagonal but also determined solely in terms of the static correlation of the uniform liquid state. The total difference of the grand potentials between the inhomogeneous and homogeneous states is obtained by evaluating the RHS of (3.3.13) in this case as     nl 2  2  ˆ β  = β u + ci {Ki · ∇Ai (r)} . dr c0 |∇η(r)| + (3.3.22) 4 i

3.3 The density-functional approach

143

Using the above expression for  in (3.3.15), a set of ordinary differential equations for the order parameters {η, Ai } is obtained. Differentiating (3.3.22) with respect to η and Ai , respectively, gives     −1 iKm · x1 dx1 ln 1 + η + Am e (3.3.23) V = c0 η − βU0 , V

−1



 dx1 e

iKi · x1

ln η +

m



 Am e

iKm · x1

= ci Ai − βUi .

(3.3.24)

m

Note that the LHS of each of the above equations follows from the corresponding results for the uniform system with local order parameters {η, Ai }. We have used for simplification in the RHS of (3.3.23) the following definitions for U0 and Ui : c0 2 c ˆ i · ∇)2 Ai (x). βUi (x) = i (K (3.3.25) ∇ η(x), 2 2 U0 and Ui are the Fourier components of an external field U (x) that maintains the density profile in a uniform system with the space-dependent order-parameter fields {η(x), Ai },  βUi eiKi · x (3.3.26) βU (x) = βU0 + βU0 (x) =

i

 c c ˆ i · ∇)2 Ai (x). eiKi · x i (K = 0 ∇ 2 η(x) + 2 2

(3.3.27)

i

For the special case of the spherical nucleus considered here the order parameters η and Ai are merely radial functions and the defining relations for U0 and Ui in (3.3.25) are expressed with the following differential equations:

c0 d 2 2 d βU0 (r ) = + η(r ), (3.3.28) 2 dr 2 r dr

ci d 2 2 d + (3.3.29) Ai (r ). βUi (r ) = 6 dr 2 r dr In (3.3.28) we have averaged over the angular dependence present in (3.3.26) due to ˆ i · ∇)2 . These two equations are coupled to the two equations the directional operator (K (3.3.23) and (3.3.24) obtained above. In order to solve for the optimum density distribution in the critical nucleus corresponding to the saddle point in the space of the density function, the last two equations are expressed as a single relation, ' &   Am eiKm · x1 = c0 η − βU0 + eiKi · x1 {ci Ai − βUi }. (3.3.30) ln 1 + η + m

i

Equivalently, we have, 1+η+

 m

Am e



iKm · x1

=e

c0 η−βU0

  iKi · x1 exp {ci Ai − βUi }e . i

(3.3.31)

144

Crystal nucleation

Hence the order parameters η and Ai are obtained in terms of the self-consistent equations     ec0 η−βU0 dx1 exp {ci Ai − βUi }eiKi · x1 1+η = (3.3.32) V i

and ec0 η−βU0 Am = V



 dx1 e

iKm · x1

exp



 {ci Ai − βUi }e

iKi · x1

.

(3.3.33)

i

The coupled set of equations (3.3.26), (3.3.27), (3.3.32), and (3.3.33) constitutes the density distribution for the critical nucleus. To solve it the required input is the structure factor for the uniform liquid. The excess free energy per particle of the critical nucleus is then obtained from (3.3.1) by evaluating  for the optimum density distribution. We show in Appendix A3.2 that  is obtained as       ∞  c0 dη 2  ci dAi 2  2 r D0 [η, Ai ] − − (3.3.34) = 4π dr. n l kB T 4 dr 12 dr 0 i

D0 [η(r ), Ai (r )] represents the local contribution to the excess free energy. The latter is obtained from the corresponding functional for a uniform system characterized by {η, Ai }. The last two terms on the RHS of (3.3.34) represent the nonlocal contribution due to the variation of the order-parameter field. In the first application of the density-functional method to the nucleation problem Harrowell and Oxtoby (1984) solved the model equations for a simplified case by setting c0 = 0 and considered only a single RLV, K m , corresponding to the peak of the liquid structure factor. It follows from eqns. (3.3.28) and (3.3.29), respectively, that βU0 = 0 and

 d 2 cm 2 d βU1 (r ) = + (3.3.35) A1 (r ). 6 dr 2 r dr η is now an algebraic function of A1 as follows from (3.3.32). D0 exhibits two minima as a function of the order parameter A1 , corresponding to the crystalline and uniform liquid states, respectively. The liquid minimum exists down to the lowest supercooling temperature, indicating the absence of a spinodal point in the present model. From the r dependences on the order parameters η and A1 for the critical nucleus corresponding to various temperatures below the freezing point it follows that the crystal–liquid interface is a few atomic diameters thick and that this is essentially independent of supercooling. With lowering of the temperature the order parameter A1 remains essentially constant near r = 0, being equal to the corresponding value for the solid state. This indicates that the center of the nucleus has properties characteristic of the bulk solid. On the other hand, the order parameter η in the same region steadily decreases with supercooling. Thus for the critical nucleus the structure and average density, characterized by A1 and η, respectively, do not change at the same point in the interface. From the liquid side the crystalline

3.3 The density-functional approach

145

Fig. 3.6 The free-energy barrier A∗ (in units of kB T ) to nucleation vs. the extent of undercooling T , from the DFT (solid) and CNT (dashed) with the capillary approximation. From Harrowell and c American Institute of Physics. Oxtoby (1984). 

structure appears first in the interface region and the density changes more inside the nucleus. It is energetically less favorable to compress a disordered liquid and once the order appears it is easier to compress. The free energy of the critical nucleus is computed from the formula (3.3.34) in the present calculation. The result is shown in Fig. 3.6. The corresponding result according to the CNT is estimated from the formula (3.1.17). Here G v (noted as A in Fig. 3.6) was computed using the DFT model. For the surface free energy γs the DFT results for the planar interface were used. Figure 3.6 shows that the dependence of the free energy on the supercooling as predicted in the CNT is qualitatively the same as that in the present DFT model, though the former somewhat underestimates the free energy of the critical nucleus. In the CNT the radius of the corresponding critical nucleus is estimated from (3.1.18). The DFT approach by construction does not assume any sharp interface. However, a radius for the critical nucleus can be defined (Harrowell and Oxtoby, 1984) as the distance at which A1 falls to half of its value at the center of the nucleus. Such an estimate indicates that the CNT somewhat underestimates the radius of the critical nucleus in comparison with that predicted from the DFT.

3.3.3 The weighted-density-functional approach The DFT model for the critical nucleus using a simple Ramakrishnan–Yussouff-type perturbation expansion for the free energy was subsequently improved using more advanced techniques like the modified weighted-density-functional approximation (MWDA) by Shen and Oxtoby (1996a, 1996b). We have already discussed the MWDA calculation of the bulk free energy in the previous chapter. Here we will consider its specific application for

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identifying the critical nucleus. This involves the identification of a set of order-parameter fields {γi (x)} characterizing the density function for the nucleus forming in the melt. Let us first identify a suitable set of parameter fields in defining the density function. The inhomogeneous density is expressed in terms of the usual RLV expansion in the following form:  n 0 (x) = n¯ 0 + n s μm eiKm · x , (3.3.36) m

where n¯ 0 denotes the average density, n s is the average density of the solid and the μm denote the Fourier coefficients corresponding to the RLV Km . Note that the amplitude μm is dependent only on the magnitude of Km . The average density n¯ 0 changes from the density of the liquid (n l ) to that of the solid (n s ). The above expansion (3.3.36) in the RLVs can also be identified with a simple parametrization of the inhomogeneous density as a sum of the Gaussian density profiles centered at the lattice sites Ri , # $ ( α )3/2  2 n 0 (x) = 2a03 n¯ 0 e−α(x−Ri ) , π

(3.3.37)

{Ri }

where the inverse of α gives the width of the density profile. a0 is the lattice constant with the conventional unit cell being in the shape of a cube of side 2a0 and having four particles in each cell for the f.c.c. lattice structure. 2a03 is the volume of the primitive unit cell such that the factor 2a03 n¯ 0 on the RHS of (3.3.37) represents the dependence of the density function on the average density n¯ 0 . The two parametrizations (3.3.36) and (3.3.37) are equivalent to each other (see the discussion accompanying eqns. (2.2.15) and ! " 2 (2.2.16) in Chapter 2) with the choice μm = exp −K m /(4α) for the Fourier amplitudes. Therefore defining the density function in terms of Gaussian density profiles is equivalent to parametrizing all the RLV amplitudes μm (m > 1) in (3.3.36) in terms of the amplitude μ1 corresponding to the smallest nonzero K 1 , μm = (μ1 )(K m /K 1 ) . 2

(3.3.38)

Therefore the two order parameters n¯ 0 and μ1 completely define the inhomogeneous density function. The grand potential D for the bulk system is a functional of the inhomogeneous density function characterized by a pair of values for the parameters {n¯ 0 , μ1 }. For a Lennard-Jones system free energy is obtained as a sum of two different contributions. The Lennard-Jones interaction potential (see eqn. (1.2.117) for its definition) is divided into a short-range repulsive part and a long-range attractive part. The contribution to the free energy from the repulsive part of the interaction is computed in terms of an equivalent hard-sphere system using the MWDA as described in Chapter 2. (See eqns. (2.2.13) and (2.2.20).) The contribution to the free energy from the attractive part of the interaction is obtained in a mean-field approximation involving the pair correlation function for the uniform liquid state. For example, Shen and Oxtoby (1996a) obtained for a LennardJones system at T = 0.6 the stable bulk solid phase for n¯ 0 = 1.04 with the parameter

3.3 The density-functional approach

147

μ1 = 0.85 and the metastable bulk-liquid minimum for n¯ 0 = 0.89 and μ1 = 0. All densities and temperatures for the Lennard-Jones system were expressed in dimensionless units (see Chapter 1). Now let us consider the technique being applied to the critical nucleus. The spherically shaped critical nucleus nucleating from the melt is described in terms of the two order parameters {n¯ 0 , μ1 }, which are treated as functions of the radial distance r from the center. This form is similar to (2.4.32) used in the previous chapter for studying interfaces using the weighted-density-functional model. The spatial dependence of the order parameters fields {n¯ 0 , μ1 } is controlled by the corresponding Euler–Lagrange equations (3.3.14) obtained from the minimization of the grand potential. The optimum order-parameter functions therefore follow from the solution of the simple differential equations given by (3.3.15) as discussed above. The boundary conditions in this case are chosen in accord with the requirement that at the center of the nucleus (r = 0) and at large distance (r → ∞) the order parameters are equal to their characteristic values for the stable (crystalline) and metastable (liquid) phases, respectively. The order-parameter profiles for the critical nucleus obtained in the present case (Shen and Oxtoby, 1996a) are qualitatively similar to those obtained in the model described above, with the Ramakrishnan–Yussouff (Ramakrishnan and Yussouff, 1979) functional for the free energy. The results for the optimum density profile for the critical nucleus obtained with the MWDA are shown in Fig. 3.7. The density profile (n¯ 0 ) decays more rapidly across the interface than the structural order parameter (μ1 ). However, the results obtained from the Ramakrishnan–Yussouf (RY) free-energy functional and the MWDA approach follow opposite trends when compared

Fig. 3.7 The density (ρ) and structural order parameter (m), denoted by n s and μ1 , respectively, in the text, for the crystal nucleus at temperatures T = 75 K (solid), T = 70 K (long-dashed line), T = 60 K (short-dashed line), and T = 50 K (dotted line). Reproduced from Shen and Oxtoby c American Institute of Physics. (1996b). 

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Crystal nucleation

Fig. 3.8 An illustration of Bain’s distortion. Two f.c.c. cells (each of size a) outlined with thin black lines are shown. v, t, and u form an orthogonal triad of vectors. Along the t axis, the atoms are shown with open circles on the t = 0 plane, dot-filled circles on the t = a/2 plane, and solid circles on the t = a plane. With Bain’s distortion along the v axis, the cube outlined with thick dark lines becomes c American Physical Society. b.c.c. From Shen and Oxtoby (1996a). 

with the corresponding CNT predictions. The free energy of the critical nucleus and its size obtained with the CNT are less than the corresponding results from the DFT formulated with the RY expansion. On the other hand, the CNT results are greater than the corresponding quantities obtained in the MWDA model. The nature of the critical nucleus is further analyzed by using the MWDA model (Shen and Oxtoby, 1996b) to study the free-energy landscape for the system corresponding to different possible crystalline structures. In this extension of the theory the parameter space for the inhomogeneous density function includes a new order parameter, χ , which refers to the symmetry of the crystal lattice. χ changes in a continuous manner monitoring Bain’s distortion (Bain, 1924) (see Fig. 3.8) in the lattice structure, which is defined in terms of the following set of real-space vectors: a0 a1 = 2



 χ κˆ 1 + √ κˆ 2 , 2

a0 a2 = 2



 χ κˆ 3 + √ κˆ 2 , 2

a3 =

a0 (κˆ 1 + κˆ 3 ), (3.3.39) 2

where κˆ 1 , κˆ 2 , and κˆ 3 represent an orthogonal triad of basic vectors (indicated in Fig. 3.8 as v, t, and u, respectively). The lattice symmetry corresponds to b.c.c. and f.c.c. for χ √ taking the values 1 and 2, respectively. The RLVs corresponding to (3.3.39) are obtained as linear combinations of the following set of basis vectors: 2π b1 = a0 2π b3 = a0



 √ 2 κˆ 2 , −κˆ 3 + κˆ 1 + χ



 2 κˆ 2 . κˆ 3 + κˆ 1 − χ

2π b2 = a0



 √ 2 κˆ 2 , κˆ 3 − κˆ 1 + χ

√

(3.3.40)

3.3 The density-functional approach

149

Fig. 3.9 The landscape of the local grand-canonical free-energy density (ω) at coexistence temperature T = 83.1 K and n¯ 0 = 0.89. The symbol m denotes the parameter μ1 used in the text. The three minima shown correspond to (a) the stable liquid with m = 0, (b) the stable f.c.c. solid with χ = 1.414 and m = 0.818, and (c) the metastable b.c.c. solid with χ = 1.0 and m = 0.750. The metastable b.c.c. c American Physical solid creates a saddle point near χ = 1.0. From Shen and Oxtoby (1996a).  Society.

At fixed values of the density n¯ 0 and temperature T , the landscape for the local grandcanonical free-energy density F in the parameter space of √ μ1 exhibits (Shen and  χ and Oxtoby, 1996b) a minimum for a stable f.c.c. structure χ = 2 as well as one for a metastable b.c.c. structure (χ = 1). This landscape is displayed in Fig. 3.9 for n¯ 0 = 0.89 and coexistence temperature T = 83.1 K. In this extended description for studying the critical nucleus, the order parameters which characterize the nucleus are given by the set {n¯ 0 , χ, μ1 }. As the liquid is undercooled below the coexistence temperature, study of the MWDA model obtains the profile for χ (r ) corresponding to the critical nucleus. The spatial dependence of χ represents a substantial b.c.c. √  (χ = 1) of the nucleus  character near the interface changing over to f.c.c. structure χ = 2 towards the center. With increased supercooling the b.c.c. character spreads further inside, until at temperature 50 K the nucleus is essentially all b.c.c. We note at this point that the computer studies of nucleation of the crystal from a melt (ten Wolde et al., 1996) discussed next also suggest that in the critical nucleus the interfacial region displays strong signatures of the b.c.c. symmetry while near the center there is f.c.c. symmetry. This changes the surface free energy of the nucleus. For a small-sized nucleus even the nature of the crystal symmetry near the center is changed and is more b.c.c. like. Such a metastable b.c.c. phase has also been seen (Löser et al., 1994) in experiments on rapidly cooled liquid metals whose stable crystalline phase is f.c.c. An earlier theoretical study by Klein and Leyvraz (1986) using a field-theoretic model of the phase transition also suggested the possibility of a metastable b.c.c. phase in the first-order phase transition.

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Crystal nucleation

3.4 Computer-simulation studies The phenomenon of nucleation of a crystal from the melt has been studied using real experiments as well as computer simulations. There are some obvious problems in making an accurate study of the evolution of the nucleation process experimentally. It is difficult to study in an experiment the formation of the critical nucleus since the crystallite spends only a microscopic time in this stage of its evolution as the top of the potential barrier is approached. Computer simulation, which is a useful tool for studying the microscopic description of the fluid, has also been used for studying the nucleation process. In the present section we discuss some of the recent developments in the study of the nucleation process using such numerical methods. The straightforward approach of simulation for this problem is to supercool the liquid below the freezing point and watch the formation of the crystal nucleus. The free-energy barrier to the nucleation process falls as the inverse square of the supercooling (see eqn. (3.1.17) for the nucleation barrier in the CNT) so that at large supercooling the barrier is low. Mandell et al. (1976) simulated a small LennardJones system of 128 particles to study the nucleation process by monitoring the structure factor. Honeycutt and Andersen (1984) investigated the effect of system size on nucleation. Assuming the nucleation rate to be proportional to the volume, the observed nucleation rate in the simulation is expected to grow with the number of particles. The observed nucleation rate, however, decreased with the number of particles N , possibly indicating that the phenomenon being observed was not homogeneous nucleation. The simulations were strongly influenced by the periodic boundary conditions used in standard molecular-dynamic methods. Subsequent work by Swope and Andersen (1990) considered nucleation with one million Lennard-Jones particles. The study shows that, although b.c.c. and f.c.c. crystalline phases are formed in the early stage of nucleation, only the f.c.c. nuclei grow beyond the post-critical stage. In all these cases the nucleation is studied at deep supercooling (more than 40% below melting). The free-energy barrier to critical-nucleus formation is not small enough unless the system is deeply supercooled and as a consequence the rate of nucleation is too low to be observed. For example, at 20% supercooling in a typical simulation of 106 or so particles one has to wait up to 1030 simulation steps to observe one nucleation event (ten Wolde et al., 1996). Thus without deep supercooling the “brute-force” method for studying crystallization with computer simulations fails. At least 50% supercooling is needed in order to observe crystal formation over the time scale of a simulation. Studying the nucleation of the crystal from a melt using computer simulations at low undercooling (close to the freezing point) when the free-energy barrier is large requires the development of special techniques. In this regard it is useful to identify suitable reaction coordinates characterizing the crystallinity of the system. The free energy for the nucleus is obtained from the probability of occurrence of the corresponding value of the reaction coordinate, and this probability is computed through efficient sampling of the configuration space. By locating the maximum in the free-energy barrier against the reaction coordinate the critical nucleus is identified.

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151

For characterization of the solid-like phase being formed in the liquid an unambiguous numerical criterion needs to be specified. For analyzing the nature of a crystal structure various methods have been developed (Mandell et al., 1976; Honeycutt and Andersen, 1984; Yang et al., 1990). A widely used approach is the analysis of the Voronoi polyhedron (VP) (Ashcroft and Mermin, 1976) associated with a given particle in the crystal. This polyhedron for a particle is the set of all points of space that are closer to it than to any other particle. It is customary to represent the VP for a given crystalline structure in terms of a set of numbers {n l } ≡ (n 3 , n 4 , n 5 , . . .) called signatures. nl represents the number of faces of the polyhedron with l sides. For a perfect crystal the VP is the corresponding Wigner–Seitz cell (Wigner and Seitz, 1933). Thus a perfect b.c.c. structure is represented by signatures (0, 6, 0, 8, 0, . . .) representing six squares and eight hexagons. At finite temperature this representation will be modified by thermal vibrations. As a consequence a given crystal structure is described by a distribution of signatures of the corresponding VPs. A similar description can be constructed in terms of a suitable order parameter for the problem and will be considered in the present context. Liquid and solid-like clusters We describe below the scheme developed by Frenkel and coworkers (ten Wolde et al., 1996) for studying nucleation phenomena with computer simulations. This method uses the definition of a liquid-state order parameter by Steinhard et al. (1983). The main advantage of this scheme lies in the fact that it identifies the solid-like clusters from the disordered liquid state and is rather insensitive to the type of the specific crystalline order. In the metastable liquid undercooled below its freezing point the embryo of the crystal forming in the melt consists of particles (solid-like) that are in an environment qualitatively different from that of the particles in the melt (liquid-like). In order to classify the different particles of the system evolving during the simulation process as being solid-like or liquid-like, a local criterion is developed in terms of the local orientational order parameter q¯lm (i). The order parameter is defined as follows. We consider a set of neighbors Nc (i) of the particle i consisting of all the particles which lie within a radius of, say, rc around it. The unit vector rˆ i j along ri j depends on the polar and azimuthal angles, denoted by θi j and φi j , respectively. The local orientational order parameter is defined in terms of spherical harmonics as qlm (i) =

Nc (i) 1  Ylm (ˆri j ). Nc (i)

(3.4.1)

j=1

Using the above definition of the local order parameters, we average over all particles to obtain the global order parameter Q lm as Q lm =

N 1  Nc (i)qlm (i), N i=1

(3.4.2)

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Crystal nucleation

5 with the normalization factor N being equal to i Nc (i) summed over all the particles i in the system. From Q lm a rotationally invariant form of the order parameter is obtained as  Ql =

l 4π  |Q lm |2 2l + 1

1/2 .

(3.4.3)

m=−l

Owing to the presence of the sharp cutoff rc the spatial derivative of the order parameter Q l is not smooth and hence one must introduce suitable smoothing functions (ten Wolde et al., 1996). For studying the nucleation phenomena it is convenient to define for the many-particle system a suitable quantity that measures the degree of long-range order as it changes from the disordered liquid to the ordered crystalline state. The quantity Q 6 (Q l for l = 6) defined as above has the desirable property that it vanishes in the bulk liquid and acquires values O(1) for simple crystal lattices. Additionally, its value is insensitive to any specific type of crystal symmetry. Another suitable reaction coordinate for the nucleus is the number of particles n in a given cluster. The free-energy barrier for the cluster is obtained as a function of the reaction coordinate. In order to analyze the structure of the nucleus a local order parameter ql (i) (similar to the global quantity Q l ) is defined at the individual particle level,  ql (i) =

l 4π  |qlm (i)|2 2l + 1

1/2 ,

(3.4.4)

m=−l

representing the extent of local order around i. Focusing on the l = 6 case, corresponding ˆ is defined with properly to the particle i a normalized ((2 × 6) + 1)-dimensional vector O ˆ ˆ normalized components such that O(i) · O(i) = 1. Using this definition of the local orderparameter vectors, two particles i and j are treated as being “connected” if the dot product ˆ vectors exceeds a certain threshold value, say 0.5 (ten Wolde et al., of the corresponding O 1996). In order to classify a solid-like particle, it is checked whether the number of its neighbors to which it is “connected” exceeds a certain threshold value. In Fig. 3.10 we show in a Lennard-Jones system at coexistence (P = 5.68 and T = 1.15) the distribution of the number of connections per particle corresponding to the thermally equilibrated liquid, f.c.c., and b.c.c. structures. From Fig. 3.10 the usefulness of the above criteria in identifying solid-like particles is clear. If we take a threshold value of seven connected neighbors to characterize a particle as solid-like, then 99% of the particles in the f.c.c. structure and 97% of the particles in the b.c.c. structure are solid-like. On the other hand, for the liquid state less than 1% of the particles are solid-like.

3.4.1 Comparisons with CNT predictions The formation of the critical nucleus is studied with a biased Monte Carlo method (van Duijneveldt and Frenkel, 1992; ten Wolde et al., 1996) with the cluster size or the number

3.4 Computer-simulation studies

153

Fig. 3.10 In a Lennard-Jones system at coexistence (P = 5.68 and T = 1.15) the distribution of the numbers of connections per particle (see the text) corresponding to the thermally equilibrated liquid, f.c.c., and b.c.c. structures are shown. The distributions are based on averages over 50 independent c American Institute of Physics. atomic configurations. From ten Wolde et al. (1996). 

of particles n in it as a reaction coordinate for the nucleation process. The probability P(n) for its formation is computed to obtain the Gibbs free energy using the relation G(n) = G 0 − kB T ln[P(n)],

(3.4.5)

where G 0 is a constant. In order to obtain reliable results for all n values, an umbrella sampling technique (Torrie and Valleau, 1974) was adopted for sampling the configuration space. This involved using the biased Monte Carlo technique to sample a part of the configuration that would have been inaccessible in direct simulations. The simulations were carried out at constant temperature and pressure. In Fig. 3.11 the values of the Gibbs free energy of a hard-sphere system compressed to P = 15, 16, and 17 at volume fractions ϕ ≡ ϕliq = 0.5207, 0.5277, and 0.5343, respectively, are shown. These packing fractions correspond to a relatively low range of super-saturations and the pressures used in the simulation are all higher than the coexistence value Pcoex = 11.67. The maximum in the plot of the Gibbs free energy against the number of particles in the cluster represents the critical nucleus and gives the corresponding free-energy barrier. Compared with the experimental data from crystallization of liquids (colloidal systems) the simulation results are low by approximately a factor of 3 (Schätzel and Ackerson, 1993; Harland and Van Megen, 1997). The corresponding quantities as predicted from the CNT in eqn. (3.1.50) are obtained using phenomenological estimates of the specific volume vm and chemical potential difference μ for pressure P of the hard-sphere system (Hall, 1970). Using the surface tension γs as a free parameter the simulation data fit to the CNT predictions as shown in Fig. 3.11 and the best-fit estimate for γs differs from its value at the coexistence pressure obtained through independent simulation studies. However, if the above best-fit

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Crystal nucleation

Fig. 3.11 The free-energy barrier G(n) (in units of kB T ) to homogeneous crystal nucleation in a hard-sphere model of colloids vs. the number of particles in the largest solid-like cluster in the system. Results for the barrier computed using (3.4.5) are shown for three volume fractions ϕliq denoted as φ = 0.5207, 0.5277, and 0.5343, corresponding to pressures P = 15, 16, and 17, respectively. See also the schematic drawing of Fig. 3.1. The curves drawn are fits to the CNT expression (3.1.9) using the capillary approximation. The fits obtained for the surface tension γ give the values γs (P = 15) = 0.699, γs (P = 16) = 0.738, and γs (P = 17) = 0.748. From Auer and Frenkel (2001). c Nature Publishing Group. 

results obtained for γs from the simulations at various pressures are linearly extrapolated to the coexistence pressure, the agreement is good. This conforms to the usual criticism of the CNT, namely that it underestimates the free-energy barrier since it uses the value of the surface tension at the coexistence pressure. For the Lennard-Jones system the free-energy barrier to the nucleation is computed by using similar techniques to those mentioned above. In what follows we represent the temperature and pressure of the Lennard-Jones system in reduced units (see Chapter 1). A convenient way of identifying the critical nucleus is to plot the dependence of the Gibbs free energy on the reaction coordinate Q 6 to locate the maximum barrier height. For small values of Q 6 , corresponding to the liquid state, the solid-like clusters have few particles. At this stage of the simulation several small clusters are formed, signifying that it is entropically favorable to distribute a given amount of crystalline phase in several units. As the top of the barrier is approached, the surface energy dominates and the small solid-like clusters merge, forming a bigger cluster, which is the critical nucleus. The simulations are performed (ten Wolde et al., 1996) at two reduced pressures, P = 0.67 and P = 5.68, which correspond to the transition points of T = 0.75 and T = 1.15, respectively. The temperatures are kept at 20% undercooling from the respective transition points, i.e., at T = 0.60 and T = 0.92. The barrier height studied as a function of the reaction coordinate Q 6 displays a maximum for the corresponding critical nucleus. According to

3.4 Computer-simulation studies

155

the CNT the free-energy barrier height corresponding to the critical nucleus is given by (3.1.50). Using results from independent simulation data on Lennard-Jones systems near freezing for the enthalpy change (Hansen and Verlet, 1969) and the average surface free energy γs (Broughton and Gilmer, 1986), the CNT prediction for the barrier height is computed. This procedure leads to G/(kB T ) = 17.4 and 8.2 for pressures P = 0.67 and 5.68, respectively. The results from simulations done at those two pressure values are given by G/(kB T ) ≈ 19.4 and G/kB T ≈ 25.1, respectively. Clearly the agreement between the CNT prediction and the simulation is much better at P = 0.67 than at P = 5.68. This is somewhat expected since the simulation value of γs (Broughton and Gilmer, 1986) used in evaluating the formula of the CNT predictions was obtained at a pressure closer to P = 0.67. The discrepancy between the corresponding CNT result and that of the simulation at the higher pressure P = 5.68 is often ascribed to the fact that the surface free energy γs should be higher in this case. In fact, an increase of approximately 45% in γs will make the two results agree. More recently it has also been argued (Trudu et al., 2006) that the CNT predictions can be brought closer to the simulation data under such conditions by taking into account the fact that the critical nucleus is not exactly spherical. In a recent simulation of the Lennard-Jones system with techniques outlined above Trudu et al. (2006) studied the crystallization process from the stage of a small embryo to the critical stage. Here the following criteria were adopted: two particles are considered as neighbors if their separation is less than rmin , a distance equal to the position of the first minimum of the radial distribution function. For a solid-like particle (s) in the nucleus Css is the number of its neighbors (s ) that are also solid-like particles. In the liquid phase on average Css is close to zero until the formation of the embryo with a small number of particles occurs. This is signalled by an abrupt jump in Css , which takes place at a stage much earlier than that at which the critical nucleus is reached. This is shown in Fig. 3.12 with the dashed line. On the other hand, the average number of particles in the nucleus keeps steadily increasing and would not register this stage in which the coordination number jumps. Thus at moderate supercooling the crystallization is a two-step process, involving a precritical embryo followed by a slower growth to the critical nucleus. 3.4.2 The structure of the nucleus Once the solid-like particles in the undercooled melt have been identified using the criteria described above, the crystallites formed by them are determined with the standard cluster analysis. A typical scheme is to adopt the notion that two solid-like particles that are neighbors belong to the same cluster. It is useful to note that, according to the above definitions, every crystal structure formed in the melt is characterized by its own unique distribution of the value of the local bond-orientational order parameter q6 (i) which was introduced above in eqn. (3.4.4). This distribution is determined by constructing a histogram of the distribution functions of the bond-order parameter values of the particles corresponding to a given nucleus. From this histogram we construct a vector vˆ that has as many components as the number of bins in the histogram. For example, for a perfect crystal the local

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Crystal nucleation

Fig. 3.12 The formation of an embryo in the melt, showing the time evolution of the average coordination number Css between solid-like particles at T /Tm = 0.8 (solid line) and the number c American of solid-like particles in the largest cluster (dashed line). From Trudu et al. (2006).  Physical Society.

bond-order parameter q6 (i) is the same for all particles and the histogram has contributions in a single bin. In general the vector vˆ for a given cluster obtained in the simulation is then identified as a linear combination of the corresponding “basic” vectors which represent the equilibrated liquid, b.c.c., and f.c.c. structures in the same description. The appropriate values for these coefficients are obtained by minimizing ! "2 (3.4.6) 2st = vˆ − { f liq vˆ liq + f bcc vˆ bcc + f fcc vˆ fcc } , where vˆ bcc is the vector associated with the histogram of the b.c.c. structure and so on. The coefficients f liq , f bcc , and f fcc of the basic vectors in this expansion then respectively represent the relative abundance of the corresponding lattice symmetry in the given structure. For the Lennard-Jones system at 20% undercooling at T = 0.92 and pressure P = 5.68 the structural composition of the largest cluster has been identified in terms of f liq , f bcc , and f fcc . The amplitudes for the different structures are plotted in Fig. 3.13 with respect to the corresponding value of the reaction coordinate Q 6 . For small values of Q 6 , corresponding to precritical nuclei, the structure is predominantly “liquid-like” and has b.c.c. character. These nuclei are clearly more b.c.c. ordered than f.c.c. This result of the simulation is indicative of the Ostwald (1897) “step rule,” which states that crystallization from the melt takes place by transforming to the phase which is closest in free energy to the liquid state. Alexander and McTague (1978) extended the Landau theory, arguing from general symmetry considerations that in three dimensions for simple liquids freezing into the b.c.c. crystal phase is favored. For Q 6 values corresponding to the top of the free-energy barrier Fig. 3.13 shows that the nucleus has predominantly f.c.c. character, which is also in agreement with the simulations of Swope and Andersen (1990).

3.4 Computer-simulation studies

157

1.0 fliq 0.8

fbcc ffcc Δ2

f, Δ2

0.6

0.4

0.2

0.0 0.005

0.015

0.025

0.035

0.045

Q6

Fig. 3.13 The structural composition of the largest cluster in terms of f liq , f bcc , f fcc , and 2st displayed as a function of Q 6 at 20% undercooling, pressure P = 5.68, and temperature T = 0.92. c American This is based on 50 independent atomic configurations. From ten Wolde et al. (1996).  Physical Society.

Assuming the shape of the nucleus to be spherical, the above analysis is extended to study the nature of its crystallinity as a function of the distance from the center of the nucleus. The amplitudes f liq , f bcc , and f fcc are computed for a spherical shell of radius r and shown in Fig. 3.14. The core of the nucleus has mostly f.c.c. structure while in the liquid f fcc and f liq smoothly go to 0 and 1, respectively. The crystal–melt interface is rather broad, approximately 4σ , which is similar to what follows from the density-functional models discussed above and in contrast with the sharp-interface model of the CNT. Also the plot of the density in the different concentric shells as well as the coordination number Css defined above against distance indicates a broad interface as shown in Fig. 3.15. The density falls more sharply over the interface than does the structural order parameter – which is also in agreement with the predictions of the density-functional models (see Figs. 3.5 and 3.6). The amplitude f bcc exhibits a sharp increase near the interface, indicating its predominantly b.c.c. symmetry. Thus the simulation results indicate that the f.c.c. core of the critical nucleus is wetted by a shell having a more b.c.c. character. Indeed, such a behavior is also obtained from the density-functional model, as we have already discussed above (see Fig. 3.9). A similar analysis of the structure of the nuclei obtained in simulations of the hardsphere system at pressures P = 15, 16, and 17 above the coexistence value Pcoex = 11.67 has also been done by Auer and Frenkel (2001). The results obtained in this case are qualitatively different from those for the Lennard-Jones system described above. Analysis

158

Crystal nucleation 1.0 fliq fbcc

0.8

ffcc Δ2

f, Δ2

0.6

0.4

0.2

0.0 1.0

3.0

5.0 r

7.0

9.0

Fig. 3.14 The structure of the critical nucleus, indicated by f liq , f bcc , f fcc , and 2st , as a function of the distance r to the center of mass, at 20% undercooling, pressure P = 5.68, and temperature T = 0.92. This is based on 50 independent atomic configurations. From ten Wolde et al. (1996). c American Physical Society. 

density

11.0

1.04 9.0

1.02 7.0

1.00

0.98 1.0

NCP

density

NCP

5.0

3.0 3.0 RCNT

5.0 r

7.0

9.0

Fig. 3.15 The density and number of connections per particle (NCP) given by Css (see the text) as a function of the distance r to the center of mass, at 20% undercooling, pressure P = 5.68, and temperature T = 0.92. The vertical coordinate axes for density (left) and NCP (right) are chosen over a range that covers the liquid up to the bulk-solid value in each case. From ten Wolde et al. (1995). c American Physical Society. 

3.4 Computer-simulation studies

159

of the largest cluster formed in the simulation of the hard-sphere system shows that at the precritical stage the nucleus with a small number of particles has some liquid-like or b.c.c. structure while the random hexagonal close-packed (r.h.c.p.) structure (Sadoc and Mosseri, 1999; Nelson, 2002) is dominant for larger clusters. Thus, unlike for the Lennard-Jones liquid, the simple b.c.c. structure is not favored in the hard-sphere system during the first stage of crystallization. It nucleates into the metastable r.h.c.p. structure, which at a later stage transforms into a stable f.c.c. structure. While the simulation results are in quite good agreement with theoretical predictions for equilibrium structures, the situation is far less convincing in the case of the nucleation rate. Using the result (3.1.52), the nucleation rate is given by J = J0 e−G

∗ /(k

BT )

,

(3.4.7)

where the prefactor J0 = Z A(T ) is a product of two terms, namely (a) the Zel’dovich factor Z given in! (3.1.39) and (b) " a kinetic prefactor A(T ), which is determined by the frequency ν0 exp − f ∗ /(kB T ) of attachment of a monomer to the nucleus. The latter is approximated as the inverse of the time taken by a particle to traverse the average interparticle distance l0 (say). Assuming the Einstein relation of root-mean-square displacement in diffusive motion, this inverse time is given by 6Ds /l0 , where Ds is the self-diffusion coefficient. Ds is obtained directly through simulation. On approximating l0 ∼ N −1/3 , the prefactor A is obtained while the Zel’dovich factor Z is computed from the formula (3.1.40). The nucleation rate is directly " simulation (ten Wolde et al., 1996) ! estimated from and, by dividing it by the factor exp − f ∗ /(kB T ) as obtained in the CNT, the prefactor J0 can be estimated. The value of J0 obtained in this way exhibits a discrepancy of two orders of magnitude from the corresponding simulation results.

Appendix to Chapter 3

A3.1 The schematic model for nucleation We present below the calculation of the order-parameter profiles and the free-energy barrier for the critical nucleus in the schematic nonclassical model presented in Section 3.2.

A3.1.1 Critical nucleus Equations (3.2.10) and (3.2.11) are solved in terms of the Yukawa function Y(r ) = exp(±αr )/r , which satisfies the equation 1 d  2 r Y ≡ α 2 Y. r 2 dr

(A3.1.1)

The boundary conditions of ψ are obtained using the constraints that, for r → ∞, ψ → 0 representing the liquid minimum, and dψ/dr → 0 as r → 0, signifying that the nucleus at the center has ψ resembling the bulk-crystalline value. Thus eqns. (3.2.10) and (3.2.11) have respective solutions of the following forms which are valid in two different regimes controlled by the two free-energy minima at ψ = 0 and ψ = ψs : " A0 ! r for small r, e − e−r , r B0 −¯ν r for large r. e , ψ(r ) = r

ψ(r ) − ψs =

(A3.1.2) (A3.1.3)

Since the crossover of the free energy is marked by ψ = ψc , it is plausible that the above two solutions (A3.1.2) and (A3.1.3) should match at this value ψc . Let this crossover happen at r = rc , which is used to evaluate the constants A0 and B0 , leading to the following solutions in the two respective regimes:

ψc − ψs er − e−r (A3.1.4) ψin (r ) − ψs = , for r < rc , r/rc erc − e−rc ψout (r ) =

ψc −¯ν (r −rc ) e , r/rc 160

for r > rc .

(A3.1.5)

A3.1 The schematic model for nucleation

161

Both of the above two solutions for ψin and ψout are equal to ψc at r = rc . In addition, for continuity of the solution we require that the derivative dψ/dr is also continuous at r = rc , i.e., * * dψin ** dψout ** = . (A3.1.6) dr * dr * r =rc

r =rc

Upon applying this condition on the two solutions, we obtain a key relation linking ψc to ψs , ψc = ψs

1 − rc−1 tanh rc . 1 + ν¯ tanh rc

(A3.1.7)

The boundary r = rc therefore is a marker for the crystalline phase in this case and is a natural identification of the radius of the critical nucleus corresponding to the condition (3.2.2). From (A3.1.4) it follows that ψin (0) → ψs , i.e., the order parameter at the center of the nucleus approaches the bulk-solid value.

A3.1.2 The free-energy barrier The free energy for the formation of the critical nucleus relative to the liquid is computed in the present schematic model as

 ∞ |∇ψ(r, t)|2 2 NC = 4π dr r ω[ψ(r, t)] + , (A3.1.8) 2 0 where the subscript “NC” refers to a nonclassical quantity to distinguish it from the corresponding quantity in the CNT. On integrating the second term on the RHS of (A3.1.8) by parts we obtain

 ∞ ψ(r ) 2 2 r dr ω[ψ(r )] − (A3.1.9) NC = 4π ∇ ψ . 2 0 Using eqn. (3.2.9) for the second derivative of the order-parameter field ψ, we obtain

 ∞ ψ(r ) dω 2 NC = 4π dr r ω[ψ(r )] − . (A3.1.10) 2 dψ 0 Integration using the expressions for ω in the inner (r < rc ) and outer (r > rc ) regions as given by (3.2.5) gives the result

 rc λs 2 λs 2 NC = 4π dr r (ψ − ψs ) + T − (ψ − ψs )ψ 2 2 0  rc 3 4πrc = dr r 2 [ψ − ψs ]. T − 2π λs ψs (A3.1.11) 3 0 In writing the last equality we have used the fact that for the r > rc part of the integral in the second term on the RHS of (A3.1.11) the expression (3.2.5) for ω(ψ) produces a

162

Appendix to Chapter 3

vanishing contribution. Now, using the result (A3.1.4) in the second term on the RHS of (A3.1.11), the expression for the free energy of the critical nucleus reduces to the form

  4πrc3 1 − rc−1 tanh rc ψc −1 T − 2π λs ψs2rc2 3 ψs tanh rc

( ) 3 −1 4πrc 1 − rc tanh rc = T − 4πrc2 sp ν¯ + rc−1 . 3 1 + ν¯ tanh rc

NC =

(A3.1.12)

The last equality above is reached by using the relation (A3.1.7) for ψc /ψs . Using the leading-order expansion of tanh rc ≈ 1 − 2 exp(−2rc ) for large rc , the second term on the RHS can be further simplified to obtain NC



4πrc3 ν¯ 2 −1 = T − 4πrc − rc sp . 3 1 + ν¯

(A3.1.13)

A3.2 The excess free energy in the DFT model The excess free energy per particle of the critical nucleus is then obtained from (3.3.1) by evaluating  for the optimum density distribution. The latter is determined from the Euler–Lagrange equation, i.e., the density function characterized by the order-parameter fields {η, Ai } satisfying eqn. (3.3.30). Using this relation, the RHS of the expression (3.3.17) for u reduces to ⎤ & '⎡    n l dx 1 + η + Ai eiKi · x ⎣c0 η − βU0 + eiK j · x 1 {ci A j − βU j }⎦ &

 − nl

dx η +

& × η+

i



' Ai eiKi · x

i

 i

Ai eiKi · x

'⎧ ⎨

η+

⎩

− 

n 2l 2



A j eiK j

j

 dx · x

j

dx c(x, x ; γ )

⎫ ⎬ . ⎭

(A3.2.1)

The above expression easily reduces to the following convenient form for u as a functional of the order-parameter functions {η, Ai }:     u Ai (x)Ui , = dx D0 (x) − {1 + η(x)}U0 − (A3.2.2) n l kB T i

where D0 is given by D0 = η(c0 − 1) +

c0 2  ci 2 η + A . 2 2 i i

(A3.2.3)

A3.2 The excess free energy in the DFT model

163

To further clarify this expression for the excess free energy of formation of the nucleus, we rewrite the last two terms on the RHS of (3.3.22) as

 1 ˆ n l   2 2 ˆ dr {Ki · ∇Ai (r)} − {Ki · ∇Ai (r)} ci 2 2 i

 n l  1 2 2 + c0 dr |∇η(r)| − |∇η(r)| . (A3.2.4) 2 2 On carrying out a partial integration of the first and third terms and using the fact that the derivative of the order parameters vanishes at the boundaries of the integral, (A3.2.4) reduces to

 1 ˆ n l   2 2 ˆ dr Ai (r){Ki · ∇} Ai (r) + {Ki · ∇Ai (r)} ci − 2 2 i

 n l  1 2 2 − c0 dr {1 + η(r)}∇ η(r) + {∇η(r)} 2 2     = n l dr {1 + η(r)}U0 + Ai (r)Ui − nl

i

c0 4

{∇η(r)}2 +

ci 4

ˆ i · ∇Ai (r)}2 , {K

(A3.2.5)

where we have used (3.3.25) for substituting Ui in the first term. The third term on the RHS of (3.3.22) can be treated in an exactly similar manner to a positive definite form. On ˆ i · ∇}2 we obtain for the surface free energy the result taking an angular average for {K 2    2   ∞    c c  dη dA i i r 2 D0 [η, Ai ] − 0 − = 4π dr. (A3.2.6) n l kB T 4 dr 12 dr 0 i

D0 [η(r ), Ai (r )] gives the local contribution in terms of the corresponding functional for a uniform system characterized by {η, Ai }. The last two terms on the RHS of (A3.2.6) represent the nonlocal contribution due to the variation of the order-parameter field.

4 The supercooled liquid

In the previous chapters we have discussed the transition of the liquid from a disordered fluid state to an ordered crystalline state through a first-order phase transition at the melting or freezing point Tm . In the present chapter we consider the behavior of the liquid supercooled below Tm and the associated phenomenon of the liquid–glass transition. 4.1 The liquid–glass transition Almost all liquids can, under suitable conditions, be supercooled below the freezing point Tm while avoiding crystallization. The undercooled liquid continues to remain in the disordered state and is characterized by very rapidly increasing viscosity with decreasing temperature. The characteristic relaxation time τ of the liquid grows with increasing supercooling. Eventually, at low enough temperature, the supercooled liquid becomes so viscous that it can hold shear stress and behaves like a solid. At this stage the supercooled liquid is said to have transformed into a glass. The latter is an amorphous solid without long-range order. It is in fact in a nonequilibrium state on the time scale of the experiment. The relaxation time τ required for the supercooled liquid to equilibrate is longer than the typical time scale τexp of an experiment. Apart from the viscosity, other dynamic quantities such as the diffusion coefficient, dielectric response function, and conductivity change strongly with increasing supercooling. In contrast, thermodynamic properties such as the specific heat, enthalpy, compressibility, and static structure factor do not show any strong change with supercooling. The specific heat drops as the liquid transforms into an amorphous solid due to the freezing of the translational degrees of freedom. In the present chapter and the rest of this book we will discuss theoretical methods for understanding the physics of the formation of the amorphous solid state as one approaches it from the ergodic liquid side. Complementary treatments of the physics of the glassy state have been carried out. These refer to, for example, the models based on treating the glass as an elastic solid and termed shear-tranformation-zone theory of amorphous plasticity (Falk and Langer, 1998, 2010). Glasses have also been studied with models based on geometric approaches (Nelson, 2002) treating the frustration or incompatibility of long-range tiling with locally preferred order in the liquid. For a review of the geometric approaches to glassy behavior see Tarjus et al. (2005). 164

4.1 The liquid–glass transition

165

4.1.1 Characteristic temperatures of the glassy state There are several characteristic temperatures associated with the physics of supercooled liquids and glassy behavior. The calorimetric glass transition at Tg At this temperature the relaxation time becomes of the order of the experimental timescale τexp . The calorimetric glass transition Tg is fixed by setting τexp = 102 –103 s. Using the relation τR = ηG ∞ ,

(4.1.1)

where G ∞ is the high-frequency shear modulus, and taking the typical value G ∞ ∼ 1010 –1011 dynes/cm2 the shear viscosity at the glass transition is obtained as η(Tg ) ∼ 1013 poise.

(4.1.2)

Therefore the temperature at which the viscosity of the supercooled liquid reaches this generic value of 1013 P has been identified as the calorimetric glass-transition temperature Tg (Ediger et al., 1999). We refer to the out-of-equilibrium liquid below Tg as a structural glass. The ease of glass formation and avoidance of crystallization in a particular glass-forming material varies widely. From the above discussion on defining Tg , it follows that different choices of the experimental time scale or the cooling rates will correspond to different values of Tg . Therefore the accepted norm is to define Tg in the limit of zero cooling rate. Let us consider the cooling-rate dependence of the glass-transition point further. Indeed, a too sensitive dependence of Tg on the choice of the experimental time scale will make its location rather arbitrary. However, since the growth of the relaxation time is exponential in the vicinity of Tg , the variation in the cooling rate does not affect Tg in any drastic manner. To demonstrate this, we note that for an experimental time scale of τexp giving rise to the glass-transition temperature Tg we have (assuming an Arrhenius temperature dependence)   EA τexp = τ0 exp , (4.1.3) kB Tg where E A is an activation energy. Differentiation of this relation gives   kB Tg dτexp dTg =− . Tg EA τexp

(4.1.4)

Since τexp is large, a correspondingly large variation will be needed to make any appreciable shift in Tg . Diverging relaxation time at T0 The relaxation time sharply increases in the supercooled liquid. Many different expressions have been used to fit the experimentally observed temperature dependence of the viscosity

166

The supercooled liquid

of a supercooled liquid approaching Tg . A number of different choices of fitting functions and their relative merits have been listed by Angell et al. (1994). Besides the standard Arrhenius form for the enhancement of the relaxation time (described above in eqn. (4.1.3)) the other most studied form of standard fitting function is the Vogel–Fulcher (Vogel, 1921; Fulcher, 1925) form:

E VF , (4.1.5) τ ∼ τ0 exp T − T0 where where E VF is some characteristic energy scale of the material. For an alternative formula see Mauro et al. (2009). The temperature T0 , though purely phenomenological in nature, is an important characteristic property of glass-forming liquids. It represents the point at which the extrapolated relaxation time would diverge. It is lower than Tg by definition since the glass transition intervenes before the relaxation time of the liquid diverges. In the case of fragile systems T0 is close Tg , whereas for the strong glassy systems it is far below Tg and is close to absolute zero. The Vogel–Fulcher form has been used in fitting of the data over a wide range, including well above Tg . Note, however, that it is a very nonlinear fit of the temperature dependence of the relaxation time. This is different from similar fits for the thermodynamic properties like entropy. Hence extrapolation of the data for predicting a divergence at T0 is less reliable. Power-law relaxation Tc Over the initial range of growth of the relaxation time the power-law fit of the form τ ∼ (T − Tc )−a has often been used. a is a material-dependent exponent. The temperature Tc obtained in such power-law fits to relaxation times or diffusion coefficients has almost exclusively, irrespective of the situation, been called the transition temperature of the mode-coupling theory (MCT). The mode-coupling model for the slow dynamics will be discussed in detail in the subsequent chapters of this book. In its simplest form the model involves an ideal ergodic–nonergodic transition at a temperature Tc (say) that lies between the freezing point Tm and the glass-transition temperature Tg . This theory predicts a power-law growth of the relaxation time around this so-called transition temperature. The Kauzmann temperature TK From the definition of the glass-transition temperature Tg it is clear that if the waiting time required for the supercooled liquid to equilibrate in an experiment could be arbitrarily increased then thermodynamic measurements below Tg would be possible. Kauzmann (1948) tested such a scenario by extrapolating various properties of the supercooled liquid to temperatures below Tg . The metastable nature of the supercooled liquid with respect to the crystal was investigated by extrapolating various thermodynamic properties such as the entropy, enthalpy, free energy, and specific volume to low temperatures. Let us consider the change of entropy with temperature. The difference of entropies of the liquid state and the crystalline state is defined as S(T ) and the quantity S(T )/S(Tm ) was plotted

4.1 The liquid–glass transition

167

against T /Tm for low temperatures. Kauzmann observed that for some liquids the extrapolated entropy difference goes to zero at a finite temperature. This temperature, usually denoted TK , represents a limiting point. For T < TK the entropy of the supercooled liquid would be lower than that of the crystal. This is a paradoxical situation, since common intuition implies that entropy grows with disorder, and is termed the Kauzmann paradox. In this regard it is important to note that, unlike entropy, the free-energy difference between the supercooled liquid and the crystalline state does not decrease on being extrapolated to lower temperatures. This rules out the possibility that the temperature TK represents a point at which the metastable supercooled liquid undergoes a continuous phase transition into the crystalline state. The above issue regarding the behavior of the supercooled liquid near TK cannot be settled by experiment. Equilibrating the liquid near TK is impossible, since the glass transition intervenes at a higher temperature Tg > TK . In his original paper Kauzmann proposed the simplest resolution of this by asserting the existence of a spinodal point at a temperature Tsp > TK . Thus for T < Tsp the liquid always equilibrates into the crystalline state. We will discuss such a possibility in detail below. However, for cases in which (at least for certain liquids) the spinodal does not exist, the possibility of a transition to an ideal glassy state at some T ≥ TK remains and constitutes a subject of much current research interest. In Fig. 4.1 a schematic drawing of the change of a typical thermodynamic property such as the enthalpy H or the volume V of the supercooled liquid with temperature is shown.

Fig. 4.1 A schematic drawing of the behavior of a typical thermodynamic property such as the enthalpy H or the volume V of the supercooled liquid with temperature. The various characteristic temperatures of the glass physics at different stages of supercooling are marked on the temperature axis.

168

The supercooled liquid

The various characteristic temperatures of the glass physics are marked on the temperature axis. The structural relaxation time τα changes by a few orders of magnitudes at each stage on the temperature axis, ranging over 16 orders of magnitudes from freezing point Tm to the calorimetric glass-transition point Tg . Finally τα approaches divergence at the extrapolated temperature T0 ≈ TK , the Kauzmann temperature at which the configurational entropy vanishes. Strong and fragile liquids An instructive plot of the data of glassy relaxation was made by Angell (1984) of the relaxation time or equivalently the viscosity η vs. the inverse temperature Tg /T scaled with Tg (see Fig. 4.2). The increase of viscosity occurs in different ways in different materials. One extreme is a slow growth of η with lowering of temperature T over the temperature range T > Tg followed by a very sharp increase within a small temperature range close to Tg .

Fig. 4.2 Viscosity plots of various glass-forming liquids vs. Tg /T . Tg is defined as the temperature at which the viscosity reaches 1014 poise (1014 P). The inset shows the drop in the specific heat at Tg . From Angell (1984), reproduced by kind permission of the author.

4.1 The liquid–glass transition

169

In a number of systems termed fragile liquids a crossover in the temperature dependence of viscosity η was observed. A more uniform increase is seen over the whole temperature range for strong liquids such as B2 O3 and SiO2 . This behavior has been quantified by defining a fragility parameter m as the slope of the viscosity–temperature curve (Böhmer et al., 1993): * d log10 η ** m= . (4.1.6) d(Tg /T ) *T =Tg Thus, for example, m = 81 (for o-terphenyl) and m = 20 (for SiO2 ), respectively, denote two extreme cases of fragile and strong systems. While the strong liquids display the phenomenon of the undercooled liquid uniformly becoming more and more viscous with decreasing temperature, at the extreme fragile end the change of viscosity is extremely dramatic, with it growing by many orders of magnitude within a very narrow temperature range. The phenomenon of the glass transition in supercooled liquids as described above is based entirely on an experimental criterion, namely the point at which the supercooled liquid falls out of equilibrium on an experimental time scale. Understanding the true nature of the transformation will naturally require knowledge of in what ways the supercooled liquid close to the glass transition is different from the glassy state below Tg . At the macroscopic level both the crystal and the glass are solid states, i.e., have elastic properties. In a crystal the constituent particles are vibrating about their mean positions, which form a lattice with long-range order. A simple picture of the glassy state will portray individual constituent atoms vibrating around their mean positions on an amorphous or random lattice structure without any long-range order. Such a picture of the glass holds only over time scales smaller than the structural-relaxation time. Similarly to the case in a crystal, ergodicity is broken in the glass over this time scale. From a theoretical point of view the system is confined to a single equilibrium state having a minimum energy in the phase space. Beyond Tg the system can no longer explore the whole phase space and ergodicity is broken. Translational motion of the particles is practically absent and the system is confined to one local minimum of the energy. At Tg the specific heat c p at constant pressure drops, becoming close to that of the crystal (see the inset of Fig. 4.2). This is simply understood from the fact that in both cases, of the crystal and the glass, only vibrational motion of the particles around their mean positions is contributing. This view can be supported by invoking the fact that quantities such as volume and energy remain almost constant. However, what makes the two systems very different is that the crystal is a system in equilibrium, whereas a glass is strongly out of equilibrium. A dynamic correlation function of two times, e.g., the density–density correlation function, is a function of two times. The system is not invariant under time translation, which is a typical feature of nonequilibrium systems. This is termed aging. Over short time scales the system appears as if it were in equilibrium with a characteristic amorphous structure, whereas over longer times nonequilibrium effects show up.

170

The supercooled liquid

Fig. 4.3 A schematic two-dimensional picture of the free volume available for a particle in the fluid, assuming the neighbors to be fixed. The dark area represents the free area available to the center of the disk.

4.1.2 The free-volume model The idea of free volume in a liquid is a useful concept in providing phenomenological explanations of some of its observed behavior. In this approach, mass transport in the dense liquid or the glassy state is understood in terms of the movement of free volume in the system. The definition of free volume at high density, however, is not unique. As a typical example, let us consider a fluid in which the volume per particle is v and let v0 be the volume per particle which is excluded from all other particles. This is shown schematically for a single particle in Fig. 4.3. For example, in a low-density fluid of hard spheres of diameter σ in which only binary collisions occur v0 = 2π σ 3 /3 and the excess volume (v −v0 ) per particle is not shared with any other particle. In a dense liquid or an amorphous solid the excess volume remains largely fixed in the jammed state due to severe packing constraints. If the solid is heated, then at the initial stage the excess volume increases without any redistribution. Thermal expansion primarily increases the anharmonicity in the vibrational motion of the fluid particles around corresponding fixed positions, similarly to that which occurs in a crystal. A note of caution with this description, however, is that (as has already been pointed out above) such a picture in the glassy state can at best hold over certain time scales. As the fluid is heated, eventually a stage is reached in which a part of the excess volume is available for redistribution and this allows mass movement in the system with randomization of the particle motions. This part of the excess volume available for redistribution without any energy cost is referred to as the free volume vf in the system, v − v0 = vc + vf , where vc denotes the part which cannot be randomly distributed.

(4.1.7)

4.1 The liquid–glass transition

171

A useful quantity in this description is the probability of having an amount of free volume vf per particle on average. This can easily be calculated assuming the distribution of this volume to be totally random. Let vi be the free volume associated with Ni fluid particles, then for a total of N particles we have the average free volume vf given by  Ni = N , (4.1.8) i

γ



Ni vi = N vf ,

(4.1.9)

i

where γ is a geometric factor that corrects for the overlap of the free volumes. Note that the choice of γ as a single constant factor is a gross simplification of the situation of complex dynamics of a strongly interacting many-particle system. The number of different ways in which the free volumes can be distributed is given by the simple combinatorial factor N! = < . i Ni

(4.1.10)

By maximizing  for the equilibrium distribution subject to the above two constraints (4.1.8) and (4.1.9), we obtain the probability of having free volume v and v + dv (treated as a continuous variable) is obtained as   γv γ dv, (4.1.11) p(v)dv = exp − vf vf where the constants on the RHS are chosen so that the probability p(v) is normalized and the average of the free volume is vf . The above form for the probability (4.1.11) is a consequence of the random nature of the free-volume distribution in the dense liquid.

4.1.3 Self-diffusion and the Stokes–Einstein relation The above-described free-volume picture of the supercooled liquid is useful, albeit in a merely phenomenological way, for modeling the transport process in a dense liquid (Spaepen, 1977). At the microscopic level, a flow occurs in a dense fluid as a result of a number of individual atomic jumps. This process can be described using the following simple picture. Around a fluid particle a cage of volume v c is formed by the nearest neighbors. When a large enough hole occurs, allowing the particle to pass on, a jump occurs. In modeling this jump process we assume that the positions of the particle before and after the jump are positions of relative stability and represented by local free-energy minima for the particle. Let the activation energy or the height of the barrier required to be crossed be G c . This is shown by the diagram of a simple two-level system in Fig. 4.4. In the absence of an external force this energy is obtained from thermal fluctuations and hence equally many jumps occur in the two directions. In this case a tagged particle is undergoing diffusive motion. If, on the other hand, an external force like a shear is applied then the jumps in the direction of the shear (we call it the forward direction) are favored and

172

The supercooled liquid

Fig. 4.4 The two-level system, showing transport in a dense liquid. The nature of the free-energy barrier for a fluid particle (shown by the shaded disk) for transport to the position shown by the dashed circle in two situations, namely by diffusion (upper panel) and in flow in the presence of a shear (lower panel). Reproduced with kind permission from Spaepen (1977). Reprinted from Acta Metallurgica with permission from Elsevier.

the two-level free-energy curve for flow is biased in the way shown in Fig. 4.4. As a result there is a net forward flux of atoms, resulting in a flow. For a quantitative estimate of this process we need to take into account two factors. First, if the free volume available is greater than v c , the probability of a jump from a site occurring is obtained, by integrating eqn. (4.1.11), as      ∞  ∞ γ γv γ vc p(v)dv = exp − dv = exp − , (4.1.12) Ss = vf vf vc v c vf assuming the flow to be homogeneous, i.e., every site is taking part in the flow uniformly. Second, in the presence of an external shear κ (say) the net number of forward jumps is obtained using rate theory. The force on an atom is κα0 , where α0 is the projected area for

4.1 The liquid–glass transition

173

the particle on the shear plane. If the atom makes a jump of length l the work done is κα0l. Since l is roughly of the same magnitude as the atomic diameter we can take α0 l ∼ 0 , the volume of the atom. Therefore the work done is κ0 . If in the absence of the shear the free energies of the two states, before and after the jump, were the same, then in the presence of the shear they differ by an amount κ0 as shown in Fig. 4.4. Therefore the net number of jumps in the forward direction over the barrier of height G c − κ0 /2 is obtained as   

 G c + κ0 /2 G c − κ0 /2 ν = ν0 exp − − exp − , (4.1.13) kB T kB T where ν0 is the atomic vibration frequency. Now, if all the particles in the specimen make a forward jump of length one atomic diameter, a uniform shear κ = 1 occurs in the system. If just some of the atoms jump, then the shear rate is equal to the net number of forward jumps per second. From the above estimation, therefore, the shear rate ϑ˙ is obtained as   

 G c + κ0 /2 G c − κ0 /2 − exp − ϑ˙ = Ss ν0 exp − kB T kB T       γ vc G c κ0 = 2ν0 exp − exp − . (4.1.14) sinh vf 2kB T kB T For low shear rate κ we assume κ0 2kB T , such that sinh[κ0 /(2kB T )] ≈ κ0 / (2kB T ). The shear viscosity η is therefore obtained as the ratio of the applied stress κ and shear rate ϑ˙ as  c   κ γv G c kB T η= = exp . (4.1.15) exp ν0 0 vf kB T ϑ˙ In the case of diffusion of the particle in a fluid without any shear, the particle is undergoing a random motion. Hence the states before and after the jump have the same free energy as shown in Fig. 4.4 and the number of jumps per second (in either direction) from the above analysis is obtained as     γ vc G c νD = ν0 exp − . (4.1.16) exp − vf kB T The diffusion coefficient Ds is related to the mean-square displacement d 2 (t), per unit time by d 2 (t) = 6Ds . Assuming that each of the jumps has an average length l and that jumps occur with a frequency νD per second, this relation reduces to νDl 2 = 6Ds .

(4.1.17)

Using the expression (4.1.16) for νD , we obtain the diffusion coefficient for the tagged particle in a sea of identical particles as     ν0 l 2 γ vc G c exp − . (4.1.18) exp − Ds = 6 vf kB T

174

The supercooled liquid

From eqns. (4.1.15) and (4.1.18) we obtain the relation Ds η =

kB T , 60 /l 2

(4.1.19)

where 0 /l 2 is an atomic length. Equation (4.1.19) is the Stokes–Einstein relation for a liquid of identical particles. Note that we have obtained the relation given in eqn. (1.4.28) for the massive Brownian particle in a surrounding liquid of smaller particles. The above analysis demonstrates in a phenomenological manner that the same result holds for a liquid of identical particles. As we discuss in Section 5.2.2, with a model of the liquid that takes into account only short-time collisions in the transport behavior (the Enskog theory) no such relation holds. The above analysis applies only for a dense liquid. In the deeply supercooled state the above relation often breaks down due to decoupling of the rotational and translational motion of the fluid particle. A microscopic-level explanation of Stokes–Einstein-law violation in the dense liquid attributes it to the existence of spatial domains with different rates of atomic mobilities in this state. The translational diffusion is mostly confined to the fast domains, while the structural relaxation and hence viscosity is controlled by the reduced mobility in the slow domains (Stillinger and Hodgdon, 1994). The idea of free volume described above was used by Cohen and Turnbull (1959) and Turnbull and Cohen (1961, 1970) to model the liquid–glass transition as a thermodynamic phase transition. Subsequently Cohen and Grest (1979) and Grest and Cohen (1981) extended the free-volume picture to formulate the thermodynamics of the supercooled liquid. A phenomenological form of the free energy of the supercooled liquid (Debenedetti, 1997) was proposed using the ideal of a cell constituting a molecule and its surrounding cage. In the Cohen–Grest phenomenological picture cells with volume v > vc are termed liquid-like whereas those with v < vc are called solid-like. The excess amount vf = v − vc is the free volume associated with a liquid-like cell. Free exchange of free volume can occur from one liquid-like cell to another when the number of liquid-like cells neighboring the former exceeds a threshold value z ∗ . Such a combination of liquid-like cells is termed a liquid-like cluster. In the Cohen–Grest theory the glass transition is viewed as a dynamic percolation transition in this system. This occurs when the fraction of liquid-like cells drops below a critical value, causing the clusters to be isolated from one another. The phase above the percolation transition corresponds to the liquid state with an infinite liquid-like cluster. The theory describes the liquid–glass transition as an underlying firstorder transition that is masked by the system falling out of equilibrium at an earlier stage due to growing relaxation times. It provides detailed predictions for such as the thermodynamic quantities discontinuity of the specific heat and thermal expansion. For example, the temperature dependence of the viscosity η is expressed as log10 η = C0 +

Ds 1/2

T − T0 + {(T − T0 )2 + 4va ζ0 T }

.

(4.1.20)

The formula is arrived at with the use of a phenomenological expression (involving adjustable parameters) for the free energy of the supercooled liquid in terms of free

4.2 Glass formation vs. crystallization

175

volumes associated with the liquid-like and solid-like cells (Grest and Cohen, 1981). The above formula with a large number of adjustable parameters (C0 , Ds , va , ζ0 , and T0 ) has served as a useful tool for fitting experimental data on viscosity over 14 orders of magnitude. Molecular-dynamics simulations with simple liquids have also been used to study the free-volume model (Hiwatari, 1982). 4.2 Glass formation vs. crystallization The ease of glass formation in a given material depends on the likelihood of avoiding the onset of crystallization in the supercooled liquid. We now consider the conditions which control this process in the supercooled liquid approaching vitrification. Crystal formation from the melt takes place through the nucleation process in which a nucleus of the crystalline phase grows at the expense of adjacent liquid. We consider only homogeneous nucleation here. In Section 3.1.2, we discussed the dependence of the nucleation rate on the extent of undercooling. The prefactor J0 of the nucleation rate was linked there to the viscosity η of the undercooled liquid using the Stokes–Einstein relation and the role of the temperature dependence of the shear viscosity η was ignored. The sharp enhancement of the viscosity of the supercooled liquid with lowering of the temperature, however, has important implications relating to the prefactor J0 . The exponential factor exp[− f ∗ /(kB T )] in (3.1.52) for J0 becomes important near Tg . On approaching the transition the energy barrier  f ∗ grows substantially, making the time τJ taken by a monomer to cross the interface and become attached to a nucleus very long. This crossover time is inversely proportional to the self-diffusion coefficient Ds of the monomer. Assuming the Stokes–Einstein relation to hold, τJ grows linearly with η. Hence the prefactor J0 in eqn. (3.1.54) for the nucleation rate is inversely proportional to η. Near Tm the time τJ is finite but the barrier to nucleation G i ∗ is very high and hence, from (3.1.56), it follows that the nucleation rate J is very low. For low temperatures distant from Tm the barrier is low but τJ grows very fast due to a sharp fall in the self-diffusion coefficient and makes J very low. Thus, at an intermediate temperature J develops a maximum. This maximum sets the value of the minimum cooling rate since only if the sample is cooled at a faster rate does crystallization not intervene. To see this in more detail, let us assume here that the temperature dependence of η follows the Vogel–Fulcher temperature dependence (4.1.5). Let the temperatures be expressed relative to the freezing temperature Tm , i.e., TR = T /Tm and TR0 = T0 /Tm . In Fig. 4.5 the dependence of the nucleation rate J on the scaled temperature TR = T /Tm is displayed using the the constants η0 = 10−3.3 P and E VF = 3.34 in eqn. (4.1.5). All other parameters are kept the same as in Fig. 3.3. All of the curves displayed in Fig. 4.5 correspond to 0 = 0.5 (see Fig. 3.3 for the definition of 0 ). The variation of the nucleation rate over different stages of the undercooling is therefore strongly dependent on the value of TR0 characteristic of the different glass-forming materials (Binder and Kob, 2005). For higher TR0 the nucleation rate J changes more sharply and does so over a temperature range closer to Tm . The value of the nucleation rate J also decreases with increasing TR0 .

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The supercooled liquid

Fig. 4.5 The logarithm of the rate J (in cm−3 s−1 ) of homogeneous nucleation in the undercooled liquid vs. the reduced temperature TR = T /Tm for 0 = 0.5 (see Fig. 3.3). The temperature dependence of the viscosity η present in the prefactor J0 in (3.1.52) is taken here in the Vogel–Fulcher c Taylor & form (see the text) with the parameter TRg ≈ TR0 = T0 /Tm . From Turnbull (1969).  Francis Group, http://www.informaworld.com

For example, at TR0 ≈ 23 the nucleation rate is above the threshold value (see Fig. 4.5) only within a narrow temperature window in the supercooled region, 0.75 < TR < 0.80. On the other hand, for TR0 ≈ 0.5 the crystallization rate is much higher, opening up a wider temperature window in the supercooled region within which crystallization is likely to occur. For materials in which the glass-transition temperature Tg is closer to Tm (TRg ≈ TR0 ∼ 1), the temperature window within which nucleation is likely to start in the undercooled liquid becomes narrower. Hence the tendency toward glass formation grows as Tg approaches Tm . However, once crystallization has started, the liquid is warmed by the latent heat released in the crystallization process of the rapidly growing crystal and freezing continues up to a temperature slightly less than Tm . The resistance of a liquid to the formation of the critical cluster for nucleation is in fact much greater than its resistance to the growth of the cluster.

4.2.1 The minimum cooling rate Since the nucleation process occurs at the interface between the crystal and the surrounding liquid, at a given instant only a very small part of the system is involved in the crystallization process. For example, it is straightforward to see that within the volume of the liquid a surface area of 1 cm2 involves only 10−7 cm3 or less volume. From the definition of the steady-state nucleation rate J it follows that the number of crystalline nuclei δn appearing

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177

in a volume V of the liquid in time δt is given by J V δt. Hence the number of such nuclei formed in time tc is obtained as  tc J dt, (4.2.1) nC = V 0

where we have ignored the temperature dependence of V . Thus the possibility of forming a nucleus increases as we increase the volume V , or J , or the time tc . Let us assume that in order to have n C = 1 we need to go up to time tc = t ∗ . This implies that, in order to bypass the process of crystallization in this volume (so that not even a single nucleus is formed in V ) and succeed in forming the glass, one must undercool the liquid to the glass-transition temperature Tg in time less than t ∗ . This sets a minimum cooling rate for bypassing the crystallization process. The liquid must be cooled faster than this rate in order to suppress the nucleation of the crystalline phase from the liquid. With a steady nucleation rate, the time t ∗ is inversely proportional to the volume V in which the nucleation is taking place. For a nucleation rate J in the steady state for the molar volume vm , the corresponding time t ∗ ≡ τN is defined to be the nucleation time,

1 G n ∗ 0 τN = ≡ τN exp , (4.2.2) J vm kB T where we have used (3.1.38) in writing the last equality and τN0 = 1/(J0 vm ).1 From eqn. (4.2.1) it follows that in order to have n C < 1, so as to avoid crystallization, V and t ∗ should be correspondingly small. For example, Fig. 4.5 indicates that as TR0 increases the corresponding nucleation rate falls, making t ∗ lower. Therefore, for a material with higher Tg /Tm ≈ T0 /Tm , a comparatively higher cooling rate is necessary in order to avoid crystallization.

4.2.2 The kinetic spinodal and the Kauzmann paradox As indicated above, a simple resolution of the Kauzmann paradox comes from assuming the existence of a spinodal at a temperature Tsp > TK . The classical nucleation theory presented in the previous chapter does not consider the possibility of a spinodal (Trudu et al., 2006; Wang et al., 2007). The spinodal refers to a situation in which a metastable phase becomes locally unstable against fluctuations. Whereas such conditions generally appear in superheated liquids or overcompressed gases, for a supercooled liquid the situation is not clear. From a computer-simulation study of freezing in a Lennard-Jones system (Trudu et al., 2006) it appears that crystallization proceeds in a continuous and collective fashion with increased supercooling. At deep supercooling the liquid is unstable and crystallization is inevitable. 1 For volumes even lower than v the volume dependence of the nucleation rate is rather ambiguous, since the corresponding m

number of monomers involved is comparable to the critical size of the nucleus.

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The supercooled liquid

Violation of the Stokes–Einstein relation The possibility of the existence of a spinodal is controlled by two competing time scales, namely the structural relaxation τα of the undercooled liquid and the nucleation time τN . Generally, in order to make the supercooled liquid survive in the metastable equilibrium state, the characteristic relaxation time τα should be less than the corresponding nucleation time τN given in eqn. (4.2.2). This will ensure that fluctuations in the metastable state do not survive for long enough to start crystallization in the sample through nucleation. At the freezing point T = Tm the barrier to nucleation diverges (see eqn. (3.1.17)) and hence the corresponding nucleation time τN defined in eqn. (4.2.2) becomes infinite. The relaxation time τα , on the other hand, is finite at Tm . Hence, near Tm , τN  τα and crystallization is easily bypassed. As the liquid is further supercooled below Tm , if there exists a temperature Tsp (Tsp < Tm ) at which the relaxation time τα of the liquid exceeds the corresponding nucleation time τN , this will represent a limit of metastability. Below Tsp , the supercooled liquid is unable to equilibrate and crystallization cannot be bypassed. Here the crystalline state is the only equilibrium phase below Tsp . Such a temperature, if found for a given liquid, represents a kinetic spinodal point. The barrier to nucleation is nonzero at this temperature and the loss of stability is not of any thermodynamic origin. However, it is not even clear a priori whether such a point exists for all liquids. To test the above possibility, let us examine the prefactor τN0 on the RHS of eqn. (4.2.2) for τN . As indicated in eqn. (3.1.56), τN0 depends inversely on the average jump time τJ of the monomers in the interfacial region. The movement of the monomer across the interface is essential for the growth of the nucleus. The jump time τJ is inversely proportional to the self-diffusion coefficient, i.e., it is directly proportional to the viscosity η, assuming that the Stokes–Einstein relation (Einstein, 1956) holds. In this case τN0 (∼τα ) will tend to diverge at T0 introduced in the Vogel–Fulchure expression (4.1.5). However, it is clear from the relation (4.2.2) that τN will always be greater than τα and hence the supercooled liquid will seem to be able to stay along the metastable line without crystallization stepping in. Hence there will be no spinodal at an intermediate Tsp . This possibility is shown by dashed lines in Fig. 4.6. The crucial assumption involved in the above comparison of τN and τα is the validity of the Stokes–Einstein relation. If, however, there is a decoupling of the translational diffusion from the structural relaxation, the Stokes–Einstein relation breaks down and there is scope for an intermediate Tsp . We can estimate the nucleation time in an independent manner. Let u g denote the growth velocity of the crystallization nucleus front formed in an undercooled melt. It is expected that u g will be inversely proportional to the jump time τJ of the monomers in the interfacial region. As a simple approximation we assume the relation ug =

1 Fg (TR ), τJ

(4.2.3)

where Fg (TR ) is a function of the relative undercooling TR from the freezing point. For temperatures close to Tm , Fg should be directly proportional to TR , similarly to chemical potential difference δμ given by (3.1.50). The number of nuclei per unit volume formed

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179

Fig. 4.6 The role of violation of the Stokes–Einstein relation in resolving the Kauzmann paradox. Temperature dependences of three characteristic times of a glass-forming liquid: (a) the structural relaxation time τα follows the Vogel–Fulcher law diverging at a low temperature T0 ; (b) the translational diffusion time τD which decouples from τα at TB ; and (c) the nucleation time τN , which is controlled by τD below Tm (see the text). τN crosses τα at Tsp , below which crystallization always c occurs before the liquid can relax to metastable equilibrium. Reproduced from Tanaka (2003).  American Physical Society.

between τ and τ + dτ is J dτ . Therefore, after a time τN each of these J dτ nuclei has grown for time τN − τ at speed u g . Hence each of the nuclei in the unit volume grows to radius u g (τN − τ ). The total volume fraction crystallized in time τN is obtained as  τN 4π 3 π φc = dτ J (4.2.4) u g (τN − τ )3 = J u 3g τN4 . 3 3 0 Hence the time required for the crystallization of a small volume (fraction φc of the liquid) is obtained as  1/4  1/4 3φc 3φc exp [β G i ∗ ] τN = = τJ ≡ τJ f g (φc , T ), (4.2.5) πCN Fg (T )3 π J u 3g where we have used the result (3.1.56) in Section 3.1.2 for the nucleation rate J . τN has contributions from τJ and f g (φc , T ) with opposite trends. Near Tm the function Fg is large while τJ is finite. On the other hand, at low temperatures below Tm , Fg is finite but the time τJ grows sharply due to the fall of the self-diffusion in the deeply supercooled state. Thus the nucleation time τN has a minimum (nose) slightly below Tm . With increased supercooling τN , however, grows much less rapidly than the structural relaxation time τα . This is because τJ is related to the inverse of the self-diffusion constant Ds . The latter now falls less rapidly than τα since the Stokes–Einstein relation no longer holds. This

180

The supercooled liquid

is indicated in Fig. 4.6, showing that τN eventually meets the τα curve at a temperature Tsp > TK , making crystallization dominant below this temperature. Thus the existence of a kinetic spinodal above TK (and below Tg ) is feasible, leading to a possible resolution of the Kauzmann paradox. As an alternative to the above considerations, the existence of a kinetic spinodal has been argued even in situations where the Stokes–Einstein relation continues to hold. This involves taking into account the role of viscoelasticity of the melt in the formulation of the classical nucleation theory. Viscoelasticity of the melt In the classical nucleation theory (CNT) described in the previous chapter, the consequences of time-dependent stress relaxation (Turnbull, 1969) in the high-density melt due to the formation of the nucleus are ignored. The initial response to an applied stress is elastic over time scales short compared with that of structural relaxation. At high temperatures, close to the melting point Tm , stress relaxation is fast while the nucleation time is very large. At this stage the liquid quickly relaxes and viscoelastic effects are small. However, at low temperatures τα is large and the liquid develops solid-like behavior over shorter time scales. Including the viscoelastic effect in the formulation of CNT at the simplest level will involve taking into account the free-energy cost for the formation of the nucleus the elastic contribution E elastic . An extension of the CNT has been proposed by Cavagna et al. (2005). This involves modifying the expression (3.1.17) for the free-energy difference G v to include an elastic contribution to the change in energy (G v − E elastic ). The elastic energy E elastic is approximated with a simple viscoelastic model in the form E elastic = ε f (t),

(4.2.6)

where ε denotes the scale of the elastic contribution to the free energy of the nucleus and f (t) ≡ f (t/τα ) is a relaxation function satisfying the limiting values f (0) = 1 and f (∞) = 0. These limiting values of f (t) imply that for t τα , i.e., for time scales short compared with the structural relaxation time τα , the liquid exhibits solid-like behavior, whereas for t  τα the liquid is able to relax the stress, making the elastic contribution to the free energy negligible. The barrier to the nucleation is now time-dependent and is obtained as G i ∗ =

16π γs3 3{|G v | −ε f (t/τα )}2

.

(4.2.7)

The nucleation time defined in (4.2.2) is now determined by imposing the self-consistent relation

16π γs3 τN = τN0 exp (4.2.8) . 3{|G v | − ε f (τN /τα )}2

4.3 The landscape paradigm

181

Fig. 4.7 The nucleation time τN (solid line) and relaxation time τα (dashed line) in a schematic model. If the two do not cross (left panel) the liquid can equilibrate without nucleation setting in. If τα > τN at temperature Tsp nucleation starts before the liquid equilibrates. Tsp is the kinetic c American Physical Society. spinodal. Reproduced from Cavagna et al. (2005). 

According to eqn. (4.2.8) the nucleation time τN increases since the denominator of the exponent on the RHS is reduced as a result of including the elastic contribution to the free energy. The above relation has schematically been solved (Cavagna et al., 2005) assuming that the prefactor τN0 is inversely proportional to the shear viscosity and that the Stokes– Einstein relation continues to hold τN0 ≈ τα , the α-relaxation time. In the schematic model, the likelihood of the curve of the relaxation time τα crossing that of the nucleation time τN and hence the possibility of the spinodal point at an intermediate temperature Tsp has been demonstrated. In Fig. 4.7 this scenario of the existence of a kinetic spinodal due to the viscoelastic contribution to the free energy of the nucleus is shown.

4.3 The landscape paradigm A useful approach to the study of the complex behavior of a many-particle system has been studying the multi-dimensional potential-energy landscape or sometimes the so-called freeenergy landscape of a small number of particles. This has been particularly effective in recent years with the availability of fast computers. The application of energy landscapes has been a useful method for various disciplines of chemical physics (Wales, 2003) concerning clusters, glassy systems, and proteins. In the present section we mainly focus on some of the recent results obtained using the landscape paradigm for the structural glasses.

182

The supercooled liquid

4.3.1 The potential-energy landscape In 1969, in a seminal paper Goldstein presented a picture of the dynamics of a supercooled liquid that was based on the evolution of the system in the multi-dimensional phase space of all the configurational degrees of freedom. In a simple monatomic system of N particles in three dimensions this refers to a set of 3N coordinates in total. The total potential energy U (r N ) of the system defines a hypersurface that is often termed the potential-energy landscape (PEL). The PEL is characterized by different minima of the potential energy. In the consideration of the metastable supercooled liquid the lowest minimum, corresponding to the crystalline state, is ignored. The local minima represent only the amorphous structure here. The system is represented by a single point moving over the PEL. At a finite temperature there are fluctuations of the energy, allowing the system to move from one minimum to another by crossing barriers. For the equilibrium liquid, which is ergodic, these jumps occur of course between minima that on average have similar energy values. The average potential energy thus remains constant in time. The PEL is related to the mechanical properties of the N -particle system and is not dependent on the temperature T which refers to the thermodynamic state of the system. However, it is important to realize in this respect that, even though the PEL is not dependent on T , the part of the landscape explored by the system is strongly linked to the temperature of the system. The key aspect of the Goldstein picture is that at low temperature the the point representing the system remains confined to a local minimum until it moves out by making activated jumps over potential barriers. Thus the atoms are held in positions corresponding to small (vibrational) motions around a local minimum of the potential energy until a large enough fluctuation causes the system to jump over the potential barrier to move to another minimum. There are thus two different time scales characterizing the behavior of the system in this situation: first, vibrations for the system around a local minimum; and second, the system eventually crossing over from one minimum to another. At high temperature the thermal energy of the system is very comparable to the height of the barriers between different minima and the activated hopping does not control the dynamics. In this situation relaxation is fast and the normal fluid dynamics prevails. The above scenario indicates that with the fall of temperature there is a crossover in the dynamics of the liquid from continuous fluid-type motion to activated barrier hopping over potential barriers. From qualitative considerations Goldstein estimated that this occurs around a relaxation time scale of 10−9 s. This is fast compared with the relaxation times of the order of 102 s near Tg , but slow compared with the usual relaxation time of the order of 10−13 s near Tm . Since the growth of relaxation times is very fast near Tg , the above crossover occurs at a temperature intermediate between Tm and Tg . The activated barrier hopping in the PEL for the supercooled liquid was interpreted by Goldstein as a rearrangement of particles in a small region of the system in real space. Such rearrangements can occur independently in different parts of the system at low temperature. The crucial point here is that this activated barrier hopping of the system implies rearrangement of a number n of the liquid particles which is sub-extensive, i.e., n/N → 0

4.3 The landscape paradigm

183

in the thermodynamic limit. The height of the potential barrier which is to be surmounted in order to hop from one minimum to another in such a rearrangement is O(n) and is subextensive. Understanding the growth of the barrier height or n in this PEL picture with increasing supercooling is therefore a key aspect of the theory of vitrification. The configurational entropy Sc (eIS ) We discuss below a formal way of partitioning the configurational space as a sum of contributions from distinct basins of the potential-energy landscape. The partition function Z N for a system of N particles is  1 N e−βV (r ) dr N , (4.3.1) ZN = N !3N V 0 where 0 is the thermal de Broglie wavelength and β = 1/(kB T ). Assuming that the sum in Z N can be partitioned into basins, the integral on the RHS over the configurational space is expressed as a sum over individual basins,    i N ˜ N e−βV (r ) dr N = e−βeIS e−β V (r ) dr N , (4.3.2) V

i

∈i

where “∈i” implies that the integral is computed for the ith basin. V (r N ) = eIH + V˜ (r N ), so V˜ represents the vibrational part of the potential energy over the minimum at eHS . To sum the contributions from the different basins we denote the partition function for the system confined in the basin with an energy eIS as Z N (eIS , T, V ). This is obtained by doing a constrained sum in which only the basins with energy eIS are counted,   i 1 ˜ N −βeIS δei ,eIS e e−β V (r ) dr N , (4.3.3) Z N (eIS , T, V ) = 3N IS 0 (eIS ) i ∈i 5 where (eIS ) = i δei ,eIS counts the total number of basins of depth eIS . The above defIS inition allows us to formally write the full partition function defined in eqn. (4.3.1) as the sum over different basin energies eIS :  (eIS )Z N (eIS , T, V ). (4.3.4) ZN = eIS

The combinatorial factor of N ! in the denominator of the RHS of eqn. (4.3.1) disappears since the sum over all distinct basins and permutations of the particles does not change the basin. We define a basin free energy f basin (eIS , T, V ) as f basin (eIS , T, V ) = −kB T ln Z N (eIS , T, V ) to obtain the result ZN =

 eIS

(eIS )e−β fbasin (eIS ,T,V ) .

(4.3.5)

(4.3.6)

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The supercooled liquid

The number of local minima in the PEL (eIS ) corresponding to a given eIS defines the configurational entropy Sc of the supercooled liquid as Sc (eIS ) = kB ln (eIS ).

(4.3.7)

The partition function is written in terms of the basin free energy and the configurational entropy as  ZN = (eIS )e−β[ fbasin (eIS ,T,V )−T Sc (eIS )] . (4.3.8) eIS

The terms in the exponent on the RHS are extensive quantities, i.e., proportional to the number of particles. Hence, in the thermodynamic limit we evaluate the integral in terms of the corresponding saddle point at eIS = e¯IS (T, V ). The latter is obtained from the solution of the equation at given V and T , T

∂ Sc (eIS ) ∂ f basin (eIS ) = . ∂eIS ∂eIS

(4.3.9)

The free energy as a function of temperature and volume is obtained in terms of e¯IS as F = f basin (e¯IS ) − T Sc (e¯IS ).

(4.3.10)

The RHS is the free energy of the liquid constrained to remain in one of its characteristic basins plus an entropic term −T Sc (eIS ) that counts the number of basins explored at temperature T . Computer models for supercooled liquids Computer-simulation methods (Allen and Tildesley, 1987) have generally been extremely useful for studying the dynamics of supercooled liquids, including nonequilibrium phenomena such as aging. In traditional molecular-dynamics (MD) simulations, the classical equations of motion for a small number of particles (a few hundred) are solved in the computer. The maximum time scale over which the particles are simulated in a typical MD simulation is not comparable to the time scales of glassy relaxation seen in real experiments. However, MD simulations easily obtain a variety of correlation functions not accessible to experimental techniques. Real-space correlation functions are as easily computed as their Fourier-transformed counterparts. The motion of a single particle in the fluid and its heterogeneous dynamics over different time scales are also easily obtained in simulations. Monte Carlo (MC) methods in which a physical property of the fluid is obtained by generating a possible set of configurations used for ensemble averaging are also used. The particles interacting through a known potential are assigned an arbitrary set of coordinates, which evolves through successive random displacements. Here the particle momenta do not enter into the computation and no time scale is involved (unlike in MD). A problem often faced in computer simulation of liquids in the supercooled state is that crystallization intervenes. Traditionally nucleation is avoided more easily by studying binary mixtures of spherical particles of different diameters (Bernu et al., 1985; 1987). As an alternative approach, the dynamics of the monatomic system has also been studied (Angelani

4.3 The landscape paradigm

185

et al., 1998) with the master equation for a model system (consisting of a small number of particles) evolving in the PEL. Crystallization is avoided here by simply ignoring the corresponding minima from the landscape. We list below a few examples of simulated systems consisting of a small number of particles interacting through a simple interaction potential. 1. A binary mixture of Lennard-Jones (BMLJ) liquids: an 80 : 20 mixture of 8000 particles of two species A and B. The spherically symmetric interaction u αβ between two types of particles α, β ∈ {A, B} at a distance r is defined as (

σαβ )12 ( σαβ )6 u αβ (r ) = 4αβ − , (4.3.11) r r with the following choices for αβ and σαβ : AA = 1.0,

AB = 1.5,

BB = 0.5,

σAA = 1.0,

σAB = 0.8,

σBB = 0.88.

(4.3.12)

The potential u αβ (r ) is cut off at r > 2.5σαβ or minor variations of it in different cases. The AB interaction is stronger (deeper well) than both AA and BB interactions. The results of the simulation are generally expressed in reduced units: length is scaled with 1/2  2 /AA , respect to σAA , temperature in units of AA /kB , and time in units of mσAA where m is the mass of particles of either species A or species B. This system is usually termed the Kob–Andersen mixture or BMLJ type I. 2. A 50 : 50 mixture of BMLJ systems with different potential parameters (Sastry et al., 1998) has been used by various authors. Here all attractive interactions have the same strength AA = AB = BB and σAB = (σAA + σBB )/2. We refer to this as a BMLJ type-II mixture. 3. A 50 : 50 mixture of soft-sphere particles of species A and B with u αβ (r ) = 0 (σαβ /r )12 , where α, β ∈ {A, B}. Barrat and Latz (1990) simulated this mixture of soft spheres for studying liquid dynamics in the supercooled state. Yamamoto and Onuki (1998a, 1998b) simulated a soft-sphere mixture with σAB = (σAA + σBB )/2 and the interaction is truncated at r = 4.5σAA in two dimensions and r = 3σAA in three dimensions. The ratio of the sigmas is σAA /σBB = 1.4 (1.2) in two (three) dimensions. The mass ratio of the two species is chosen as m 2 /m 1 = 2. For the mixture of soft spheres an effective diameter is computed as σeff =

2  α,β=1

d xα xβ σαβ ,

(4.3.13)

where xs = Ns /(N1 + N2 ) for s = 1, 2. The state of the binary mixture is sometimes characterized by a single parameter eff defined as   0 d/12 d σeff , (4.3.14) eff = n 0 kB T where n 0 = (N1 + N2 )/V is the total number of particles per unit volume.

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The supercooled liquid

With the availability of modern computers it is possible to calculate the potential-energy function U (r N ) for a system with a large number (though much smaller than that for the thermodynamic limit) of particles. Stillinger and Weber (Stillinger and Weber, 1982, 1984; Stillinger 1995) developed a technique for studying the PEL in a systematic manner. The nature of this hypersurface is analyzed in a statistical way in terms of basic quantities such as the number of local minima and the density of such minima as a function of the energy. The shape of the surface near the local minimum and the hypervolume associated with a minimum are also important ingredients in the PEL approach to the physics of supercooled liquids. Starting from each point in the configuration space, a steepestdescent minimization of the potential-energy function is performed until a local minimum is found. The position of the local minimum in the configuration space defines an inherent structure denoted “IS.” The set of all points connected to a given local minimum in the above procedure defines a basin for the corresponding IS. Except for a set of points of zero measure accounting for the saddles and ridges between the different basins, all points in the configuration space are associated with a corresponding local minimum. Crossover in the dynamics Simulation of a Kob–Andersen BMLJ system of 256 particles has been done in order to analyze (Sastry et al., 1998) the dynamics within the IS structure approach. Three characteristic temperatures were observed by analyzing the nature of the dynamics of the representative point of the Kob–Andersen BMLJ system of 256 particles on the PEL. First, for T > Ta ≈ 1.0 the average energy e¯IS of the IS for the equilibrium liquid is insensitive to the temperature T . The corresponding self-intermediate function decays exponentially with time and the relaxation time grows with temperature in an Arrhenius manner. For T < Ta , the energy corresponding to the IS, e¯IS , falls with temperature. The relaxation is nonexponential and the temperature dependence of the relaxation time is non-Arrhenius in this region. Around TcIS ≈ 0.42 there is another crossover in the dynamics. The potential energy barriers between IS basins increase and the system remains largely in the harmonic regions of the basins. The dynamics below TcIS is controlled largely by activated hopping processes. The temperature range between Ta and TcIS has been called the landscapeinfluenced regime. At a much lower temperature T0IS the system gets trapped in a single low-energy basin. For the slowest cooling rate of the system T0IS ≈ 0.25. The temperature range from TcIS to T0IS has been called the landscape-dominated regime. In the landscapeinfluenced regime, as the temperature of the system approaches TcIS from above the interbasin hopping becomes increasingly rare. This temperature marks a clear separation of time scales between intra-basin dynamics and activated hopping between barriers. In a related study Schröder et al. (2000) studied the successive configurations produced in an MD simulation of a BMLJ type-II mixture containing 251 A particles and 249 B particles. The successive configurations of the particles were mapped into a corresponding time series of local minima (IS). The self-intermediate scattering functions both for the actual structure (representing the true dynamics) and for the corresponding IS (representing inherent dynamics) were compared. Beyond a crossover temperature Tx (say), it is found

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that the inherent dynamics is the same as the true dynamics from which the effects of vibrations has been removed. Beyond Tx the dynamics of the system in the configuration space consists of its fast vibrational motion (intra-basin) in a local minimum and slow diffusive motion among different basins. The landscape scenario has been used to study the thermodynamics of supercooled liquids in a number of works in recent years (Doliwa and Heuer, 2003; Sciortino, 2005; Heuer, 2008). Our understanding of the dynamics of the representative point of the system on the PEL grew clearer as the landscape was studied in terms of local minima as well as the saddles (Angelani et al., 2000; Broderix et al., 2000). The type-I BMLJ system of particles at fixed density 1.2 was equilibrated at a given temperature T , using standard MD techniques, and the saddle of the PEL U (x) was located by looking for the absolute minima of W (x) = ∇U · ∇U . All the extrema were classified in terms of their potential energy u and the number of unstable directions K(u) (number of negative eigenvalues of the Hessian matrix) called the index density, corresponding to the energy u. In Fig. 4.8 a plot of the index density K(u) vs. the corresponding energy density is shown. The data were obtained

Fig. 4.8 The index density vs. the potential energy. An average over all data was obtained by sampling at T ∗ ∈ [0.3, 2.0]. The solid line is a linear fit meeting the energy axis at u th . The inset shows the average density of the local minima u min (triangles) as a function of the temperature of the initial MD trajectory. Circles represent δT , the average potential-energy density U/N (T ) minus the vibrational energy 3T /2 in the harmonic approximation. The threshold energy u th = −4.55 is c American Physical Society. reached at T ∗ = 0.44 ≈ Tc . Reproduced from Broderix et al. (2000). 

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by (a) averaging the energy for all the extrema (irrespective of the temperature corresponding to the initial configuration) with a given index density (geometric average) and (b), at a given temperature T , averaging all the energies and the corresponding index densities (parametric average). The K(u) curve meets the energy axis at a threshold value u th that marks the border between the saddle-dominated part of PEL and the minima-dominated part. u th is above the lowest-lying minimum, u 0 , which is found in the PEL, indicating the existence of a finite energy-density interval within which there are overwhelmingly more minima than saddles. With increasing temperature, u 0 increases and eventually, at a critical temperature Tth , crosses u th . Thus, for T > Tth the representative point of the system is in the part of the PEL which is dominated by saddles. This temperature Tth therefore indicates a crossover from nonactivated dynamics (above Tth ) to activated dynamics (below Tth ). The evidence from the landscape studies on model systems strongly suggests that Tth is close to the Tc at which the extrapolated self-diffusion coefficient of a particle in the liquid goes to zero. For the BMLJ system at ρ = 1.2 one obtains Tth = 0.435 (Angelani et al., 2000), or 0.44 (Broderix et al., 2000). Computer simulation (Kob and Andersen, 1994, 1995a, 1995b) of the same system shows that the self-diffusion coefficient goes to zero with a power law at Tc = 0.435 (see Section 8.2.3 in Chapter 8 for more details). The results from another typical fragile liquid, i.e., soft-sphere, mixture (simulated using MC methods) indicate a similar relation: Tth = 0.242 ± 0.012 (Grigera et al., 2002) and the corresponding Tc = 0.226 (Roux et al., 1989). We will see in Chapter 8 that this temperature Tc can be identified with the dynamic transition point of mode-coupling theory (MCT) in the sense that power-law growth of the relaxation time occurs. 4.3.2 The free-energy landscape The thermodynamic free energy is a function of temperature and has a characteristic value corresponding to an equilibrium state. For the study of the glassy state this concept is often generalized to a coarse-grained Hamiltonian in terms of a set of suitable order parameters. The most obvious choice of the parameters for the coarse-grained Hamiltonian of a system with aperiodic density profiles is a set of spatial coordinates {Ri } that corresponds to the centers of the inhomogeneous density profiles. In other words the {Ri } denote the average positions of the constituent particles in a localized state. Such a parametrization defines a multi-dimensional landscape. With the entropic contribution included this is generally referred to as the free-energy landscape. Such models for supercooled liquids have been studied by evaluating the free-energy functional for a small number of particles. The size of the system considered is much smaller than the thermodynamic limit. The spirit of such models is similar to that of MD simulations with a finite-sized system. The optimization of the Ramakrishnan–Yussouff (RY) free-energy functional with respect to aperiodic density profiles was done numerically (Dasgupta, 1992) in the supercooled regime. This involved computing the free energy for a small volume L 3 having L = 4σ to 6σ , where σ is the hard-sphere diameter with periodic and free boundary conditions. The volume was divided into a cubic grid with lattice constant a0 = 0.2σ ,

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using the hard-sphere Percus–Yevick (PY) structure factor in the free-energy functional. The system underwent the freezing transition to an f.c.c. lattice at a somewhat lower packing fraction value of 0.43 (the corresponding simulation value is 0.49). The glassy minima were found to have free energies above that of the f.c.c. crystal and lower than that of the liquid for packing fractions below the crystallization transition point. The calculation of the local bond-orientational order parameters (see eqn. (3.4.1) for their definition) gave values indicative of local icosahedral order in the metastable glassy state. A lattice-gas model The dynamics of the dense fluid in terms of its evolution in the free-energy landscape was studied by Fuchizaki and Kawasaki (1998) by mapping the problem to a kinetic latticegas-type model. Here density is considered as the only relevant variable and a discretized version of the Fokker–Planck equation is considered in the form of a mesoscopic kinetic equation (Kawasaki, 1966a). In the lattice description the system is divided into an assembly of primitive cells with lattice constant h. The dynamics is governed by the following master equation for the probability distribution P0 [n, t] of having density profiles denoted as n at time t: ! " w0 (n|n  )P0 (n  , t) − w0 (n  |n)P0 (n, t) . (4.3.15) ∂t P0 [n, t] = n

The profile n ≡ {n i } denotes a set of occupation numbers with n i = 0 or 1 depending on whether the ith lattice site in the grid is vacant or occupied, respectively. The transition probability from {n  } to {n} is denoted as w0 (n|n  ), which satisfies the detailed-balance condition. The energy of the configuration {n} is obtained in terms of the Hamiltonian kB T  H0 [{n i }] = − c(|ri − r j |)n i n j , (4.3.16) 2 i= j

where ri denotes the position vector of the ith site and c(r ) is the two-point direct correlation function. Equations (4.3.15) and (4.3.16) for the kinetic Ising model are solved by the Monte Carlo method to obtain a time sequence of n(t) (Fuchizaki and Kawasaki, 1998). A system of size L˜ = 15 consisting of L˜ 3 lattice points with lattice constant h is chosen. c(r ) is taken from the solution of the PY equation with the hard-sphere diameter as σ = 3.3h. The incommensurability of σ/ h prevents crystallization setting in. The occupation number n i (t) in the lattice-gas formulation is related to the local particle den5 sity ρ(r, t) = α δ(r − rα ) in the continuum (for convenience we take the particle mass m = 1) by the relation n i τ = ρ(ri , t)τ ,

(4.3.17)

with the time average . . .τ taken over the typical relaxation time τ . The occupation numbers n i in the present context actually refer to entities that are different from a real fluid particle. On coarse-graining the system over a length 0 (say) larger than h, but smaller than σ , the identity of an individual particle is lost. The master equation (4.3.15) on coarse

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graining reduces (Fuchizaki and Kawasaki, 1999) to the Fokker–Planck equation (8.1.126) for a density distribution {ρα }, where the subscript α corresponds to the αth cell in the coarse-grained description. Since the PY direct correlation function c(r ) is generally negative, the equations for the dynamics are the same as those corresponding to a spin-one-half Ising Hamiltonian with anti-ferromagnetic interaction in an external magnetic field. The solution of the master equation (4.3.15) at the initial stage of the dynamics (Fuchizaki and Kawasaki, 1998) correctly captures the role of the slowly decaying density fluctuations. For low densities corresponding to packing fraction ϕ ≤ 0.490 the density correlation function follows the stretched exponential form exp[−(t/τ1 (q))βq ]. With increasing packing fraction the stretched exponential relaxation first appears at larger wave numbers. Let us focus on the results obtained for the late stage of the dynamics dominated by thermally activated transitions between different free-energy minima. For this a reference state is defined in terms of the state vector L˜  3

α

|n (t) =

n i (t)|ei ,

(4.3.18)

i=1

where n i (t) is the occupation variable, which is equal to 0 or 1 depending on whether i

the ith lattice site is empty or occupied at time t and |ei  = |0 . . . 0.1.0 . . . 0 is the ith orthonormal basis (of size L˜ 3 ) in the Fock space. The superscript α refers to a particular initial state. Now an average state vector is defined as *  0 * 1 t˜0 +τ * α2 1 1 * α  α  *% = |n¯  ≡ α * dt n (t ) , (4.3.19) ref Nα N * τ t˜0 where the origin t˜0 of the observation is chosen to be sufficiently large to ensure equilibration. The duration τ is much longer than the microscopic time scale (phonon characteristic time) but shorter than the time scale of thermally activated jumps. N α is the norm of the ˜ 3/2 , N being the number of particles in the fluid. state |n¯ α , so that N 1/2 ≥ N α ≥ N /( L) The first equality holds when the system is completely frozen during the time interval over which the average is taken, while the second one holds when the occupied sites * 2 are uniformly distributed. The state of the system at some later time t is denoted by *φtα with the averaging process done as defined in (4.3.19). In order to study how the system explores the free-energy landscape, an overlap function q(t) is considered, 2 1 α . (4.3.20) q(t) = φtα |%ref The time scale over which correlation is studied extends from t to t + τ1 , τ1 being the time scale of decay of density fluctuations. The result is shown in Fig. 4.9 for three packing fractions, ϕ = 0.312,*0.440, * α2 2 and 0.491. For ϕ = 0.312, q(t) is almost time-independent . This (close to unity) since *φtα is independent of t and is essentially the same as *%ref indicates that the system stays around a single global liquid-state minimum. As the system becomes more dense, ϕ = 0.440, beyond the equilibrium crystallization packing fraction of ϕ = 0.430, the amplitude of fluctuation of q(t) increases but the system is still trapped

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191

Fig. 4.9 The time evolution of the overlap function q(t) defined in (4.3.20) vs. time t for packing fractions η ≡ ϕ = 0.312 (solid line), 0.440 (short-dashed line), and 0.491 (long-dashed line). Time c is expressed in Monte Carlo steps per site. Reproduced from Fuchizaki and Kawasaki (1998).  Physical Society of Japan.

in the single minimum within the time scale of the computation. For values of the packing fraction beyond ϕ = 0.49 with the initial time scale τ1 ∼ 1000 Monte Carlo steps (MCSs) per site, the function q(t) relaxes at two different levels: first, at t ∼ 20 000 MCSs and then at t ∼ 60 000 MCSs per site. This is interpreted as the system being initially trapped in a local free-energy minimum and eventually relaxing towards the states with lower free energies through thermally activated hopping. Earlier than these works Dasgupta and Valls (1994) studied the dynamics of free-energy landscapes by locating the free-energy minimum of the RY functional mentioned above nearest to an initial metastable configuration of the density field ρ. These initial configurations {ρ(x)} were the ones reached by simulating the nonlinear Langevin equations (to be discussed in the next chapter) on a grid. Their results are also consistent with a crossover from a dynamics around the uniform liquid-state minimum at high temperature to an activated-hopping scenario at low temperatures. The above study of the dynamics suggests the following scenario for the dynamics for a hard-sphere system. At low densities the system essentially fluctuates around the liquid-state minimum. The relaxation time grows with increasing density. At this initial stage of supercooling the liquid-state dynamics is still controlled by continuous-fluid-type motion of the particles. As the packing fraction increases beyond a crossover value ϕA the dynamics is controlled by thermally activated hopping of the system in the free-energy landscape to configurations having inhomogeneous density distributions. In the high-density state the system is caught in a local minimum of the free energy with only vibrational motion in the free-energy basin. In real space this implies that the particles vibrate around their mean positions lying on a disordered lattice structure. As

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The supercooled liquid

the temperature of the liquid is supercooled to the glass-transition temperature Tg the liquid essentially freezes into a solid and attains a finite shear modulus.

4.4 Dynamical heterogeneities A characteristic behavior seen in a deeply supercooled liquid suggests that the dynamics in some regions of the system can be orders of magnitude faster than that in the neighboring regions. Indeed, with time the nature of the heterogeneity seen in one region of the space changes and is redistributed (Stillinger and Hodgdon, 1994; Cicerone and Ediger, 1996). This spatially heterogeneous nature of the dynamics of the undercooled liquid has been an area of much current research interest. The study of the dynamical heterogeneities generally refers to the length scales over which they occur and most naturally to the corresponding time scales over which they exist. Dynamical heterogeneities have important consequences for the transport properties of supercooled liquids. The relaxation of a typical correlation function in a supercooled liquid follows the stretched exponential form, " ! (4.4.1) C(t) ∼ exp −(t/τ )β , where τ is the relaxation time. The stretched-exponential exponent β decreases from its high-temperature value of unity (simple exponential relaxation or what is termed Debye relaxation) to smaller values near the glass-transition temperature Tg . Generally stretchedexponential relaxation is interpreted as a consequence of having several characteristic relaxation times, i.e., expressed in terms of the distribution of the relaxation time τ by g(τ ) as  ∞ C(t) ∼ dτ g(τ )e−(t/τ ) . (4.4.2) 0

The stretching of the relaxation in the supercooled liquid can be interpreted as a consequence of the distribution of relaxation modes (each being exponential) corresponding to the spatially heterogeneous dynamics from different regions of the system. The observation of such heterogeneities in deeply supercooled liquids has been made through dynamic hole-burning experiments (Schmidt-Rohr and Spiess, 1991; Cicerone and Ediger, 1995; Schiener et al., 1997). Fujara et al. (1992) showed by means of NMR experiments that a strong slowing down of translational diffusion occurs with supercooling. In a very standard glass former, o-terphenyl (OTP), translational diffusion has a weaker temperature dependence than the viscosity and rotational correlation time. The size of regions over which the dynamics becomes heterogeneous grows strongly with temperature on approaching Tg . The length scale over which the dynamics of a single particle becomes simple diffusive grows much longer as the temperature approaches Tg . NMR experiments have been used (Tracht et al., 1999) to measure directly the size of regions with heterogeneous dynamics. In our discussion in the present book we focus in particular on the aspects of dynamic heterogeneities observed in computer MD simulations. These studies

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are generally limited to much simpler molecules and much shorter time scales than those in real experiments. However, a number of interesting characteristics of the dynamics can be identified since the dynamics is probed here at a very microscopic level.

4.4.1 Computer-simulation results The simplest quantity representing the structure of the disordered liquid is the pair correlation function g(r ) or the static structure factor S(k) discussed in the earlier chapters. It has been widely found in simulations (Kob, 2003) that the liquid does not undergo any drastic change in its equilibrium structure with an increase in supercooling. On the other hand, the liquid dynamics becomes increasingly slow in the metastable supercooled state. The slowing down with fall of temperature in√the present context is far more drastic than the fall in the average speed of the particles (∝ T ). The characteristic relaxation time corresponding to the time correlation functions (introduced in Section 1.3) grows by orders of magnitude when the liquid is supercooled. This is generally manifested in a two-step relaxation process. At short times, i.e., times less than the average collision time between two particles, the dynamics is of free-particle type (see eqn. (1.3.33) for example). This is followed by the dissipative dynamics in which the correlation function decays to zero. At high temperature this final relaxation process is exponential and the corresponding relaxation time grows at lower temperatures. However, with increasing supercooling or an increase of density, the nature of the decay of the correlation function shows a qualitative change. The correlation function after the initial ballistic regime remains on a plateau for some time (on a logarithmic time axis) and finally decays to zero. This two-step process is a very typical feature of the dynamics of the dense liquid state. This behavior is most easily demonstrated by considering the tagged-particle correlation Fs (q, t) (see eqn. (1.3.23) for its definition) in a dense fluid. In Fig. 4.10(a) the decay of the tagged-particle correlation in a binary mixture is displayed for a wave vector near the structure-factor peak. For high temperature the correlation decays exponentially, and it develops a plateau as the temperature is lowered. The dynamics of the supercooled liquid becomes increasingly heterogeneous in different regions. To illustrate this further, let us consider the plot of the mean-square displacement r 2 (t) with respect to time. This is shown in Fig. 4.10(b). For short times the mean-square displacement grows as t 2 , conforming to the free-particle behavior. At low temperatures the mean-square displacement remains on a plateau over an intermediate time scale. This behavior signifies rattling of the tagged particle in the cage formed by surrounding particles. With increasing density the cages persist for longer time and hence the length of the plateau increases. However, the height of the plateau remains the same, indicating that the size of the cage does not drastically change with the fall of temperature. Over longer times the mean-square displacement grows linearly in time, indicating the self-diffusion process. Over this final time range the slope of the r 2 (t) vs. t curve gives the diffusion coefficient Ds through the Einstein relation (1.4.27). Ds falls by a few orders of magnitude in these systems. If the glass transition is considered as a jamming transition in which the

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The supercooled liquid

Fig. 4.10 Single-particle dynamics for the A particles in a BMLJ system. (a) The tagged-particle density correlation Fs (q, t) at q = qm , the first peak of the static structure factor. (b) The meansquare displacement r 2 (t) vs. time t. The curves from right to left in both (a) and (b) correspond to temperatures T = 5.0, 4.0, 3.0, 2.0, 1.0, 0.8, 0.6, 0.55, 0.50, 0.475, and 0.466. Reproduced from c American Physical Society. Kob and Andersen (1994). Both parts 

particles are trapped for long times then this will imply that with supercooling the breaking of the cage, or, for a tagged particle, escaping from its cage, becomes a very slow process. In the supercooled state the relaxation process becomes spatially heterogeneous. The detailed nature of this heterogeneity has been investigated in a number of studies using MD simulations (Perera and Harrowell, 1998; Yamamoto and Onuki, 1998a). A key aspect of this dynamical heterogeneity is violation of the Stokes–Einstein (SE) relation Ds η/T = constant which relates the translational diffusion Ds to the shear viscosity (see eqn. (1.4.28) in Chapter 1). The SE relation generally holds in normal liquids but breaks down with increased supercooling (Fujara et al., 1992; Chang et al., 1994; Cicerone and Ediger, 1995, Cicerone et al., 1995; Blackburn et al., 1996). The translational diffusion constant Ds is found to be larger than that expected from the SE relation, although the rotational diffusion continues to scale with the shear viscosity η. The temperature dependences of 1/Ds and η are different in the supercooled region. A phenomenological explanation of this behavior has been provided in terms of the existence of a spatially heterogeneous and transient distribution of local relaxation times (Perera and Harrowell, 1998). The mean-square displacement and hence Ds are controlled by the presence of a set of particles that are more mobile than the rest. The shear viscosity η, on the other hand, is dominated by the relaxation of the slower regions. Molecular-dynamics simulations of soft disks in two dimensions by Perera and Harrowell (1998) provided evidence for such a heterogeneous distribution. Simulations of a similar soft-sphere mixture with slightly different parameters (Yamamoto and Onuki, 1998a) demonstrated the heterogeneous nature of the diffusion of a tagged particle of a three-dimensional fluid. The van Hove self-correlation function G s (r, t) (see eqn. (1.3.29) for its definition) in the deeply supercooled state has a large-r tail that scales as r/t 1/2 for t ≤ 3τα (Yamamoto and Onuki,

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Fig. 4.11 The breakdown of the Stokes–Einstein relation seen in simulations of a soft-sphere binary mixture, in a plot of the self-diffusion coefficient Ds times the α-relaxation time τα vs. τα . The solid line represents the corresponding Stokes–Einstein value of Ds τα . Reproduced from Yamamoto and c American Physical Society. Onuki (1998a). 

1998a). This represents a departure from a simple diffusion process over time scales less than that for relaxation of the heterogeneous structures (τα ). For times long compared with τα the diffusion is homogeneous and the corresponding diffusion coefficient is determined from the mean-square displacement. In the high-temperature regime the stress-relaxation time τα and the self-diffusion coefficient Ds scale according to the SE law. Deviation from this is observed, with an enhanced diffusion coefficient, as the liquid is increasingly supercooled. This is displayed in Fig. 4.11. The van Hove self-correlation function G s (r, t) obtained from the computer simulation of a BMLJ type-I system in the supercooled region has been very useful for a careful analysis of the heterogeneous dynamics. In the discussion of G s (r, t) in Chapter 1 (Section 1.3.2) we have seen that for the isotropic fluid a strictly Gaussian form of this function is entirely determined in terms of the mean-square displacement of the tagged particle r 2 (t). The departure from Gaussian behavior is expressed in terms of the non-Gaussian parameter α2 (t) defined in eqn. (1.3.61). The latter involves the quantity r 4 (t) and in fact relates to four-point correlation functions in the liquid. Indeed, it follows from eqns. (1.3.53) and (1.3.54), respectively, that the self-diffusion coefficient Ds is determined in terms of a twopoint velocity correlation function, while the non-Gaussian parameter α2 (t) is linked to a four-point velocity correlation function. The non-Gaussian parameter α2 (t), obtained from the computer simulation (Kob et al., 1997) of the BMLJ type-I system, has a peak at times intermediate between the short-time ballistic regime and the structural-relaxation time τα . This is shown in Fig 4.12(a). The height of the peak as well as the time tp at which the peak appears increases with the fall of temperature.

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The supercooled liquid

Fig. 4.12 The non-Gaussian parameter α2 (t) vs. time t for T = 0.550, 0.480, and 0.451. The arrows indicate the peak tp of the corresponding α2 (t). (b) (r, t) defined in eqn. (4.4.3) vs. r at the time t = tp obtained in (a). The temperatures are the same as for the corresponding curves in (a). The arrow marks the location of rM for T = 0.451. The inset shows the same quantity on a log scale. c American Physical Society. Reproduced from Kob et al. (1997). Both parts 

The tagged-particle correlation G s (r, t) has been used by Kob et al. to establish a criterion for identifying a set of particles that are more mobile than the rest of the particles. The relative difference is defined as (r, t) =

G s (r, t) − G 0s (r, t) G 0s (r, t)

(4.4.3)

of the van Hove function from the corresponding Gaussian quantity G 0s (r, t). The latter is obtained as

3/2

3 3r 2 0 exp − 2 G s (r, t) = (4.4.4) 2πr 2 (t) 2r (t) in terms of the mean-square displacement r 2 (t) of the tagged particle of species A. From a plot of (r, tp ) vs. r (see Fig. 4.12(b)), one can identify a distance r = rM corresponding

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Fig. 4.13 (a) A demonstration of the string-like cooperative dynamics seen in simulation of a supercooled BMLJ system. Mobile particles (see the text for their definition) in the subregion of the simulation box are shown at two different times, t = 0 (light spheres) and the same particle at a later time (shaded spheres), with the bold dark lines connecting the corresponding pairs for identification. Arrows mark the paths followed by some of the particles. Reproduced from Donati et al. (1998). (b) From the BMLJ simulations of Fig. 4.12. The inset shows the radial distribution functions of particles of type A, gAA for the bulk particles and gAmAm for the corresponding mobile particles at temperature T = 0.451 in reduced units. The main figure of part (b) shows the ratio between the two types of correlation gAA and gAmAm for T = 0.550, T = 0.480, and T = 0.451. Reproduced c American Physical Society. from Kob et al. (1997). Both parts 

to the maximum r at which (r, tp ) vanishes. Particles that have traveled a distance r > rM within a time tp are then defined as “mobile” particles. Snapshots of these mobile particles show that they tend to form a cluster. The regions involved in these clusters are not compact but string-like. A superposition of two successive snapshots of a set of mobile particles is shown in Fig. 4.13(a). Explicit tracking of the mobile particles shows that they undergo a string-like motion. The average string length increases with decreasing temperature. The correlated nature of the dynamics of the mobile particles is manifested in the corresponding pair correlation function gAmAm of the mobile particles. The ratio RM of the pair correlation function gAmAm to that of the bulk particles gAA of species A is large (RM > 1) at small r and decays to 1 for large r . RM grows with lowering of the temperature, displaying enhanced correlation in the deeply supercooled state. The behavior of the ratio of pair correlation functions is shown in Fig. 4.13(b). The time correlation G M s (r, t) for the mobile particles decays faster than the corresponding bulk G s (r ). Finally, the probability P(t) that a particle that is mobile at t = 0 continues to remain so at t = t falls with time t. The time scale of the decay for P(t) is the same as that of the corresponding G M s for the mobile particles. Synchronous movements of a set of fluid particles have been used to search for a suitable mechanism for a fluid particle to escape from the surrounding cage (Langer, 2006a, 2006b) in the supercooled liquid. The formation of the string-like motion

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The supercooled liquid

as described above is a signature of the dynamical heterogeneities in the dense system over intermediate time scales. However, its relevance in producing the extremely slow dynamics near glass formation is unclear at this stage.

4.4.2 Dynamic length scales In the discussion of dynamical heterogeneities in the supercooled liquid in the previous section we have seen that in the supercooled liquid a set of particles is more mobile than the rest and they tend to form a cluster or string-like structure. The time scale of this stringlike cooperative dynamics defines the time over which dynamical heterogeneities persist. These observations naturally give rise to speculation about whether a growing correlation length can be associated with the supercooled liquid state. Various efforts (Dasgupta et al., 1991; Ernst et al., 1991) to identify such a length scale associated with the supercooled state have been made. In the following we will discuss some of the more recent results seen in simulations of liquids in the metastable supercooled state. In order to focus on the cooperative nature of the dynamics it is useful to consider the possible correlation in the respective displacements over time t between two particles situated at two spatial points separated by a distance r at an initial time “0.” The displacement μi (0, t) of the ith particle from time 0 to t is defined as μi (t, 0) = ri (t) − ri (0).

(4.4.5)

We have dropped the vector notation on μi and ri to avoid clutter. The average of μi (t, 0) is the same for all particles in the liquid at equilibrium. Hence the displacement fluctuation is obtained as μi (0, t) = μi (t, 0) − μ(t, 0).

(4.4.6)

The correlation of this displacement over time t for two different particles i and j is defined in terms of the correlation function Sμ (r, t) as +5 Sμ (r, t) =

i, j

, μi (0, t)μ j (0, t)δ(r − ri j (t)) , +5 , i, j δ(r − ri j (t))

(4.4.7)

where ri j (t) = ri (t) − r j (t). The denominator on the RHS is in fact the pair correlation function g(r ) in the liquid. The function Sμ (r, t) in fact involves the information from four different points, (0, 0), (0, t), (r, 0), and (r, t). If the fluctuations of μi and μ j are completely uncorrelated (for two points that are widely separated), we have Sμ (r, t) = μ2 → 0. At a given time t the correlation function Sμ (r, t) integrated over all distances r is a time-dependent function involving the correlation of four points and hence can be denoted as

4.4 Dynamical heterogeneities

199

Fig. 4.14 The mobility correlation function χμ (t) vs. time t at various temperatures from a c American Physical simulation of a BMLJ system. Reproduced from Donati et al. (1999).  Society.

 χμ (t) =

dr Sμ (r, t).

(4.4.8)

At a certain time t, if Sμ (r, t) is not falling off fast enough (long-ranged) with r then χμ (t) will develop a peak at that fixed time. From MD simulation of a BMLJ type-I system Donati et al. (1999) obtained the function χμ (t) at several temperatures in the supercooled region. We show in Fig. 4.14 that the correlation χμ (t) exhibits a distinct peak at t = t ∗ . Both t ∗ and the peak height increase as the liquid is increasingly supercooled. The height of the peak also increases with fall of temperature. Indeed, as noted above, the peak in χμ (t) at t = t ∗ indicates a slower spatial decay of the function Sμ (r, t ∗ ). For simulations of polymer melts similar results were obtained (Bennemann et al., 1999). The χμ at different temperatures scaled with respect to their corresponding values at the peak t ∗ collapse into a single curve. If the spatial decay of this function is associated with a length scale, then the increase in the peak of χμ at lower temperature implies a corresponding growth in the correlation length ξd (T ). Note that this enhanced correlation occurs only near t ∗ and therefore the associated correlation length ξd (T ) is purely dynamical. Multi-particle correlation functions have been investigated widely in order to reach a growing correlation length (Berthier et al., 2005, 2007a, 2007b). The above treatment in terms of the fourpoint functions can be generalized to identify an order parameter for the supercooled liquid state. Density fluctuations play a very crucial role in describing the liquid state, and the time correlation function of density fluctuations is therefore a key quantity of interest. We will see in subsequent chapters that a discontinuous change in the long-time limit of the density correlation function is used to define a dynamic transition in the supercooled liquid. Let us denote ψ(x, t) ≡ δρ(x, t)δρ(x, 0). For the ergodic liquid we have for the density correlation function lim δρ(x, t)δρ(x, 0) = lim ψ(x, t) = 0.

t→∞

t→∞

(4.4.9)

200

The supercooled liquid

On the other hand, the RHS of the definition (4.4.9) getting frozen at a nonzero value defines the nonergodic state. The discontinuous change in the time correlation function δρ(x, t)δρ(x, 0) represents the nonergodicity parameter.2 Let us now generalize this beyond the two-point level. Specifically, we consider the four-point correlation function, which has been of much current interest for understanding the physics of supercooled liquids. Starting from this, we define the four-point function G4 (x, t) in terms of the spatial correlation of ψ(x, t), G4 (x, t) = δρ(0, t)δρ(0, 0)δρ(x, t)δρ(x, 0) − δρ(0, t)δρ(0, 0)δρ(x, t)δρ(x, 0) = ψ(0, t)ψ(r, t) − ψ(0, t)ψ(r, t). If we define the bulk property φ(t) corresponding to ψ(x, t),  1 dx ψ(x, t), φ(t) = V then it follows that one can define a four-point susceptibility function χ4 (t) as  χ4 (t) = dx G4 (x, t)   ! " 1 dx dy ψ(x, t)ψ(y, t) − ψ(x, t)ψ(y, t) = V + , = V φ 2 (t) − φ(t)2 .

(4.4.10)

(4.4.11)

(4.4.12)

Using the above approach, the four-point correlation function of a BMLJ mixture of type I was been investigated by Berthier et al. (2007a, 2007b). In this case the correlations are defined using the tagged-particle density ρs (x, t). In terms of the Fourier-transformed quantities we define the two-point function ϕs (k, t) as ϕs (k, t) =

Nα 1  ρsi (k, t)ρsi (−k, 0), Nα

(4.4.13)

i=1

where ρsi (k, t) = exp(ik · ri (t)) and Nα is the total number of particles of the species A. The intermediate scattering function is obtained as Fs (k, t) = ϕs (k, t). Using the definition for the four-point function given in eqn. (4.4.12), the corresponding result for the tagged particle in the present case is obtained as ,  + (4.4.14) ϕs2 (k, t) − Fs2 (k, t) . χ4s (t) = Nα k

The sum over the wave vector on the RHS of the expression for χ4s (t) has been approximated by keeping only the dominant contribution at k = k0 corresponding to the peak 2 In general the characteristic order parameter of the ergodic–nonergodic (ENE) transition is expected to be defined in terms of

the difference between some property of the system being calculated using a complete ensemble average and the same property evaluated by taking an infinite-time average. We pursue here a different approach using a restricted phase-space average and define the transition in terms of a finite jump in the long-time limit of the crucial density–density correlation function. The average ψ(x, t) is therefore the order parameter for the ENE transition.

4.4 Dynamical heterogeneities

201

of the structure factor and is also angularly averaged for the isotropic system. A plot of the simulation data of χ4s (t) for the species A of the BMLJ type-I system displays a clear peak (Berthier et al., 2007a, 2007b) at t = t ∗ (say). This represents a time scale over which the correlation is strongest and is found to be less than the structural-relaxation time in the supercooled liquid. A correlation length ξ(T ) is also identified from the above analysis of the four-point correlation functions. For this we define the spatial fluctuation of the instantaneous value of the self-intermediate scattering function   ˜ i (t) − ri (0)))Fs (k, t) . δϕs (q, k, t) = eiq · ri (0) cos(k(r (4.4.15) i

Let us consider δϕs (q, k, t) for a fixed k = k0 near the structure-factor peak, angular averaged and denote this as δϕs (q, t). The four-point dynamic structure factor is obtained as S4 (q, t) =

1 δϕs (q, t)δϕs (−q, t). Nα

(4.4.16)

The above correlation has been angularly averaged over different directions of q to obtain the corresponding result for the isotropic liquid as a function of only the magnitude of q. The averaged function denoted simply as S4 (q, t) is directly obtained from the simulation data. S4 (q, t = t ∗ ) at the time t = t ∗ at which χ4s develops a peak near q → 0 and is fitted to the following Ornstein–Zernike form: S4 (q, t ∗ ) =

S4 (q = 0, t ∗ ) . 1 + (qξ )α

(4.4.17)

In the above fitting α is a fitting parameter. The four-point function is determined for different values of q lying within the range allowed by the size of the simulation box. The simulation data at different temperatures are compatible with the value α ≈ 2.4 in the above fitting form and give a dynamic correlation length ξ(T ). This is displayed in Fig. 4.15. Although the growth of the correlation length ξ with the fall of temperature is not as drastic as it is near the critical point, it increases with increasing supercooling. Besides the four-point functions, three-point correlation functions or response-type functions have also been linked to a corresponding correlation length that is similar to the above form. Berthier et al. (2007a, 2007b) considered the three-point correlation ST (q, t) =

1 δϕs (q, t)δe(−q, t) Nα

(4.4.18)

linking the tagged-particle density to the fluctuation of the energy density at time t according to  e(q, t) = eiq · ri (t) [ei (t) − e], (4.4.19) pi2 /(2m)

5

i

where ei (t) = + j u(ri j (t)) is the energy of the ith particle at time t and e is the average energy per particle. The fitting of the angularly averaged ST (q, t) to the Ornstein–Zernike form (4.4.17) corresponds to the parameter value α = 3.5 and the same

202

The supercooled liquid

Fig. 4.15 The length scale ξ obtained from analysis of the four-point correlation function S4 (q, t) as well as the three-point functions ST (q, t) (see the text) vs. the inverse temperature 1/T . The results correspond to MD as well as Monte Carlo simulations of a BMLJ system. The y axis is a linear scale. c American Institute of Physics. Reprinted from Berthier et al. (2007a). 

correlation length ξ(T ) as that displayed in Fig. 4.15. Fitting of the four-point functions calculated from MD simulations to finite-size scaling has also been used to extract a growing correlation length in the supercooled state (Karmakar et al., 2009). In all these cases the correlation length calculated from the computer simulations of a small number of particles in a box grows with supercooling and approaches the system size. Independently from the above analysis, a dynamic length scale has also been identified from considerations pertaining to the propagating shear waves in the supercooled liquid. In general, shear waves of very short wavelengths can exist even in a normal liquid, giving a characteristic solid-like response to a very-high-frequency perturbation. Thus, at a given temperature the liquid can sustain shear waves of a maximum wavelength. This maximum wavelength l0 (say) or corresponding minimum wave number qmin = 2π/l0 grows with the fall of temperature. This was observed in the MD simulation (Mountain, 1995) of a 50 : 50 soft-sphere mixture of size ratio 1.2. The length l0 grows with increasing supercooling. In general, for a soft-sphere potential of u(r ) = (σ/r )n , the thermodynamic properties at density n 0 and temperature T (T = (kB β)−1 ) are expressed in terms of a single dimensionless parameter eff = (β)3/n n 0 σ 3 , which increases with supercooling. A binary mixture of soft spheres with an equimolar mixture of particles of masses m 1 and 2m 1 and diameters σ11 and σ22 (σ12 = (σ11 + σ22 )/2), respectively, interacting with u i j (r ) = (σi j /r )12 , with the parameter eff = (β)1/4 n 0 σ 3 , is considered. The simulation of this mixture showed that, with a change of eff from 0.9 to 1.4, the length scale l0 is seen to grow by an order of magnitude. The increase of l0 is essentially linked to that of the shear viscosity, which sharply increases with supercooling. A common feature among the different types of dynamic length scales identified from computer simulations of the supercooled liquids is the following: all of them grow during the initial stages of supercooling and fit well to a power-law divergence around some lower temperature Tc . However, before Tc is reached the power-law divergence is cut off and the correlation length ξd is finite at Tc . We will see in Chapter 8 that at the theoretical level this

4.4 Dynamical heterogeneities

203

behavior of the dynamic correlation length is linked to a crossover in the dynamics of the liquid around a characteristic temperature Tc that lies between the freezing point Tm and the laboratory glass-transition point at Tg . This has been achieved by extending the statisticalmechanical model for the liquid-state dynamics to the supercooled region. In particular, this involves understanding the strongly correlated motion of the fluid particles as well as the effects of liquid structure on the dynamics at high density. We have discussed above the evidence for such correlations in the dynamics as seen in computer simulations. To develop the theoretical models for the dynamics, we will first introduce the proper formalism and derive an appropriate model for the time dependence of the correlation. This is discussed in the next three chapters.

5 Dynamics of collective modes

Theoretical developments on the dynamics of a dense liquid using a statistical-mechanical approach primarily involve a small set of slow collective densities termed hydrodynamic modes. The time scales of relaxation of these modes are much longer than those for the microscopic modes of the system. The basic approach adopted here is the analysis of the time correlation functions (introduced earlier in Chapter 1) of the slow modes. In the present chapter and the next two chapters we discuss microscopic methods for calculating the correlation functions involving the fluctuation or hydrodynamic approach. We focus primarily on the simplest type of correlation functions involving fluctuations at two different spatial and time coordinates. Owing to time translation invariance, equilibrium two-point correlation functions of hydrodynamic modes at the same time over different spatial points are time-independent and provide us with information on the thermodynamic behavior of the system. On the other hand, the dynamic behavior of the system is linked to the correlation of physically observable quantities at two different times. The time correlation function of density fluctuations is particularly important for our discussion of the slow dynamics in a liquid. In the simplest of the theoretical models, the decay of the correlation with time is exponential. We discuss here how such exponential relaxation behavior can be understood using linear dynamics of the fluctuations. The formalism developed in the later parts of this chapter allows in a natural way the extension of the macroscopic hydrodynamics to intermediate length and time scales, and is referred to as generalized hydrodynamics. At low densities or high temperature we obtain the standard forms of the correlation function for the normal liquid state. For liquids supercooled below the freezing point Tm , this approach finally leads to a set of microscopic models for the slow dynamics observed in those systems. This development is discussed in the next two chapters. The primary result predicted from these models is a crossover in the dynamic behavior of the liquid at a temperature below Tm , i.e., in the moderately supercooled state. The characteristic temperature around which this happens has been termed the mode-coupling-theory (MCT) transition temperature Tc . The present chapter is aimed at introducing some basic concepts for the liquid-state dynamics. We first obtain the basic conservation laws for the liquid and demonstrate how they give rise to the dissipative equations of hydrodynamics. The hydrodynamic correlation functions and the expressions for transport coefficients 204

5.1 Conservation laws and dissipation

205

follow next. We then introduce the linear equations of the fluctuating hydrodynamics, which form the basis of the discussion in the subsequent chapters. 5.1 Conservation laws and dissipation In a fluid with a large number of particles moving and colliding at random the average time between successive collisions of the fluid particles is extremely short, ∼10−15 s, even at moderate liquid densities, and it decreases with increasing density. The corresponding mean free path also shortens as the liquid becomes more dense. Within the background of this apparently chaotic dynamics there are some collective modes in the many-particle system relaxing over time scales that are many orders of magnitudes longer than those for the fast modes. The dynamic behavior of a fluid is described primarily in terms of the relaxation of a few such slow modes. As we will see in subsequent chapters, there are several reasons for the occurrence of the slow modes, such as microscopic conservation laws or breaking of some continuous symmetry of the microscopic Hamiltonian, or even some associated slow parameter such as the heavy mass of a Brownian particle. The most basic dynamic variables giving rise to the equations of macroscopic hydrodynamics for the fluid relate to conserved properties of the fluid. In an isotropic system there are five conserved densities corresponding to the total mass, momentum, and energy. 5.1.1 The microscopic balance equations In terms of the actual phase-space coordinates of the constituent particles, the microscopic density at a given point r at time t is defined as  mδ(r − rα (t)), (5.1.1) ρ(r, ˆ t) = α

gˆ (r, t) =



pα δ(r − rα (t)),

(5.1.2)

eα δ(r − rα (t)),

(5.1.3)

α

e(r, ˆ t) =

 α

5 where we have used the definition eα = pα2 /(2m) + (1/2) β u(rαβ ), with the prime in the sum implying that the α = β term is absent from the sum. We put a hat on the slow variable to indicate the dependence on the phase-space coordinates of this variable. Note that the mass density ρ(r, ˆ t) is similar to the single-particle density n(r, t) defined in eqn. (1.2.76). For the one-component system in fact we have ρˆ = m n. ˆ The significance of the above definitions for the microscopic variables is apparent in the coarse-grained picture. Thus, for example, with the above definition for the density ρ(r), ˆ its integral over all volume is equal to the total mass of the system m N , which is a conserved quantity. The respective conserved densities a(r) ˆ ≡ {ρ(r), ˆ gˆ (r), e(r)} ˆ satisfy the following microscopic-balance equations (for their deduction starting from the microscopic definitions (5.1.1)–(5.1.3) see Appendix A5.1):

206

Dynamics of collective modes

∂ ρˆ + ∇ · gˆ = 0, ∂t ∂ gˆi  ∇ j σˆ i j = 0, + ∂t

(5.1.4) (5.1.5)

j

∂ eˆ + ∇ · ˆje = 0, ∂t

(5.1.6)

with the corresponding currents being given by gˆ (r) =



pα δ(r − rα ),

(5.1.7)

α

σˆ i j (r) =

  pi pαj 1  ij α αβ αβ , δ(r − rα ) + m 2 α

(5.1.8)

α,β

ˆjie (r)

=

 α



pi 1  ij eα α δ(r − rα ) + αβ m 4 α,β



j

j pβ pα + m m

 αβ ,

(5.1.9)

where we have used the definitions of the following quantities in terms of the force Fαβ acting on the particle at rα due to the particle at rβ :  αβ =

1 0

ds δ(r − rα + srαβ ),

ij

j

i αβ = (rαβ · Fαβ )ˆrαβ rˆαβ .

(5.1.10) (5.1.11)

i is the ith component of the unit vector r ˆ αβ . Note that αβ = βα , and the quantity rˆαβ ij

αβ is also symmetric under exchange of {α, β} as well as {i, j}. The microscopic stress tensor is symmetric, σˆ¯ i j = σˆ¯ ji . The microscopic-balance equations obtained above with the formal manipulations are exact and symmetric under time reversal. However, they are not local in nature. In the case of short-range potentials the force Fαβ between the two particles is appreciable only if Rαβ is small. In this limit we can ignore the last part in the argument of the delta function in eqn. (5.1.10) such that αβ = δ(r − rα ) and obtain the currents in the local form as follows: ⎡ ⎤    pi pαj 1 i j ⎣ α αβ ⎦ δ(r − rα ), (5.1.12) + σˆ i j (r) = m 2 α ˆjie (r) =

 α

⎡

α,β 

i  ij ⎣eα pα + 1 αβ m 4 α,β



⎤ j  j pβ pα ⎦ δ(r − rα ). + m m

(5.1.13)

5.1 Conservation laws and dissipation

207

5.1.2 Euler equations of hydrodynamics The above microscopic equations for the slow conserved densities are reversible in time. These are exact balance equations. In order to obtain the hydrodynamic equations for the smoothly varying local densities of mass, momentum, and energy respectively denoted by {ρ(r, t), g(r, t), e(r, t)}, we need to average the above balance equations. The averaging in principle needs to be done over the nonequilibrium ensemble. Indeed, if the system is in equilibrium then by averaging the densities one obtains the corresponding time-independent equilibrium value. The nonequilibrium average is obtained by extending the notion of the Gibbsian ensemble to include systems that are out of equilibrium. In this regard the nonequilibrium system which we describe here is assumed to be in the time regime where it has reached a state of local equilibrium. This is termed the stage at which the set of local densities {a(r)} ˆ is sufficient to describe the state of the system. This is a plausible hypothesis, particularly at high densities, since the stage of local equilibrium is reached rapidly through frequent inter-particle collisions. The probability function for the local equilibrium state is obtained in analogy with that of the equilibrium state. For the equilibrium state with a set of extensive conserved quantities A = {N , H, P, . . .}, we have the canonical distribution (1.2.7), i.e., f eq (x N ) ∼ exp[−β(H − μN + P · v)].

(5.1.14)

The respective intensive thermodynamic variables given by the set b = {β, μ, v} refer to the temperature, chemical potential, and velocity, respectively. Analogously, the local equilibrium state is characterized by the nonuniform densities aˆ (r) ≡ {e, ˆ gˆ , n} ˆ with the distribution function 3  −1 ˆ − v(r, t) · g(r) ˆ f le ( N , t) = Q l exp − dr β(r, t) e(r) 

4 1 2 ˆ , − μ(r, t) − mv (r, t) n(r) 2 

(5.1.15)

where we denote the phase point  N by the set of 6N variables  N ≡ {r1 , p1 , . . . , r N , p N }. is the necessary normalization The number density nˆ is defined as m n(r) ˆ = ρ(r). ˆ Q −1 l constant, which therefore satisfies the relation ⎧ ⎫⎤ ⎡ ⎨  ⎬  ⎦. Q l = Tr ⎣exp − dr αa (r, t)a(r) ˆ (5.1.16) ⎩ ⎭ {a}

The conserved property A (extensive) defines the local density a(r), ˆ  A = dr a(r). ˆ

(5.1.17)

αa is the corresponding local thermodynamic property (intensive) in terms of which the local equilibrium is defined. The local ensemble average of aˆ is obtained from the relation

208

Dynamics of collective modes

a(r) ˆ le = −

δ ln Q l (t) . δαa (r, t)

(5.1.18)

In the form considered in eqn. (5.1.15) we therefore have the following αa corresponding to the set aˆ ≡ {e, ˆ gˆ , n}: ˆ

1 αn = β(r, t) μ(r, t) − mv 2 (r, t) , 2 αg = β(r, t)v(r, t),

αe = β(r, t).

(5.1.19)

In analogy with the equilibrium case, the set of local thermodynamic variables is given by {β(r, t), μ(r, t), v(r, t)}, referring to the local temperature, local chemical potential, and local velocity, respectively. These nonuniform parameters characterizing the local equilibrium are determined by imposing the self-consistency condition that the nonequilibrium state average of a local density aˆ can be approximated by taking the average over the local equilibrium distribution, a(r, t) = a(r) ˆ ˆ ne = a(r) le .

(5.1.20)

The averaged local densities a(r, t) then are treated as smooth functions of the space and time dependence, termed hydrodynamic fields. The time evolution of these nonuniform densities in a state perturbed from the thermodynamic equilibrium is given by the average of the balance equation for the corresponding microscopic densities {a}, ˆ i.e., ∂a(r, t) + ∇ · ja = 0. ∂t

(5.1.21)

The local equilibrium average ja of the microscopic currents ˆja (r, t) for the one-component fluid presented in eqn. (5.1.7) is obtained in Appendix A5.1.1. We have g ≡ ˆgle = ρv, σi j ≡ σˆ i j le = Pδi j + ρvi v j ,   v2 je ≡ ˆje le =  + ρ + P v, 2

(5.1.22) (5.1.23) (5.1.24)

where v(xt) is the local velocity so that the fluid appears to be at rest in the vicinity of this point x from a frame moving locally with velocity v(xt). On substituting these equations we obtain the Euler equations for the hydrodynamics for an ideal fluid without any dissipative effects, ∂ρ + ∇ · g = 0, ∂t   gi g j ∂gi  ∇j + + ∇i P = 0, ∂t ρ

(5.1.25) (5.1.26)

j



g ∂e + ∇ · (e + P) = 0. ∂t ρ

(5.1.27)

5.1 Conservation laws and dissipation

209

The above set of equations for the slow modes {ρ, g, e} is invariant under time reversal and represents reversible dynamics. In order to construct the equations for the irreversible dynamics we need to include the dissipative effects. The forms of the dissipative terms in the equations of motion are obtained in a phenomenological manner. To do this we consider the rate at which the entropy of the fluid is changing. The entropy of the fluid in the local equilibrium state is obtained by extending to this case the Boltzmann prescription relating the thermodynamic entropy to the mechanical probability of the state, S(t) = −ln f le ( N , t).

(5.1.28)

We define the entropy density s(r, t) whose spatial integral gives the total entropy S(t),  S(t) =

dr s(r, t).

(5.1.29)

The dynamics corresponding to the local-equilibrium description is described by the Euler equations. It has been shown (see Appendix A5.1.2) that the entropy density s(r, t) satisfies a continuity equation with a current sv: ∂s + ∇ · (sv) = 0. ∂t

(5.1.30)

There is no net production of entropy in the isolated system with reversible dynamics. The positive entropy production is a signature of the irreversible or dissipative effects in the fluid.

5.1.3 Dissipative equations of hydrodynamics The dissipative effects due to irreversible transport in the fluid are included in the dynamic equations for the local densities in a phenomenological manner. To make inferences about the form of the dissipative terms, we extend the currents in the equations for the dynamics in terms of a dissipative part, σi j ≡ σiRj + σiDj = Pδi j + ρvi v j + σiDj ,   v2 R D j e ≡ j e + je =  + ρ + P v + j D e. 2

(5.1.31) (5.1.32)

The corresponding equations of motion for the coarse-grained or averaged densities a(r, ˆ t) are given by

210

Dynamics of collective modes

∂ρ + ∇ · g = 0, ∂t ∂gi  ∇ j σi j = 0, + ∂t

(5.1.33) (5.1.34)

j

∂e + ∇ · je = 0. ∂t

(5.1.35)

There is no dissipative part in the mass current g, which is itself a conserved quantity. The continuity equation remains unaltered at the coarse-grained level. The other dissipative currents are expressed in terms of the gradients of the corresponding thermodynamic properties such as the local temperature T (r, t) = β −1 (r, t) and velocity v(r, t), σiDj = −ηi jkl ∇k vl , ! D" J i = −λi j ∇ j T,

(5.1.36) (5.1.37)

where the summation convention of repeated indices has been used. The tensor ηi jkl and λi j refer to the viscosity and thermal conductivity of the liquid, respectively. In Appendix A5.1.2 we demonstrate that the above forms for the dissipative currents (5.1.36) can be argued from general thermodynamic considerations such as that of positive entropy production in the irreversible process. For an isotropic fluid the transport matrices are further simplified,   2η0 (5.1.38) δi j δkl , ηi jkl = η0 (δik δ jl + δil δ jk ) + ζ0 − 3 λi j = λδi j ,

(5.1.39)

where η0 and ζ0 correspond to the shear and bulk viscosities, respectively, while λ is the thermal conductivity. The constant coefficients on the RHS of (5.1.38) are chosen such that the transverse modes couple to the shear viscosity η. The second term in the expression for σiRj in (5.1.31) keeps the equations of hydrodynamics invariant under a Gallelian transformation. A set of hydrodynamic equations for the averaged local densities {ρ(r, t), g(r, t), (r, t)} is then obtained in the following form: ∂ρ + ∇ · g = 0, ∂t

(5.1.40)

gi g j ∂gi + ∇j + ∇i P − L i0j v j = 0, ∂t ρ

(5.1.41)

∂ + v · ∇ + h ∇ · v − λ ∇ 2 T = 0, ∂t

(5.1.42)

where terms of up to linear order in the gradients of T (r, t) and v(r, t) have been kept in the dissipative parts of the currents. To close the set of equations we express the fluctuation

5.1 Conservation laws and dissipation

211

in the pressure P in terms of the density fluctuations (ignoring any coupling to the energy fluctuations for simplicity):   ∂P δP = δρ = c02 δρ, (5.1.43) ∂ρ T where c0 turns out to be the speed of sound in the fluid. Using the relations (1.2.52) and (1.2.70), the thermodynamic derivative on the RHS of (5.1.43) can be expressed in terms of the static structure factor S(0) as c02 = 1/(βm S(0)). The dissipation matrix L i0j relates to the viscosities, ( η0 ) L i0j = ζ0 + (5.1.44) ∇i ∇ j + η0 δi j ∇ 2 , 3 and h = e + P is the enthalpy density with e and P the energy density and pressure at equilibrium, respectively. Conventional hydrodynamics generally refers to the region of low frequency ω and small wave number k so that ωτc 1 and kl 1, where τc is the mean time between collisions and l is the mean free path. In eqns. (5.1.41) and (5.1.42) the dissipative terms representing transport in the nonuniform state are only up to linear order in gradients of the local thermodynamic fields and therefore only slow processes occurring over long length and time scales are considered. The transport coefficients appearing in the hydrodynamic equations are constants and are treated as known inputs. Equations (5.1.40) and (5.1.41) will be the primary focus in the subsequent chapters on our discussion of the dynamics of a one-component fluid.

5.1.4 Tagged-particle dynamics The dynamics of a single particle moving in a surrounding fluid can be described in terms of the density of the single particle nˆ s (r, t), which is defined by eqn. (1.3.27) in Chapter 1. The average nˆ s  of the tagged-particle density is obtained in eqn. (1.3.28) and is of O(1/V ). It goes to zero in the thermodynamic limit V → ∞. The velocity of this particle is vα . In the very-long-time limit the behavior of the tagged-particle correlation is expected to be diffusive. In order to demonstrate this we note that tagged-particle density n s (x, t) is a conserved density and its integral over the whole volume is fixed at unity. By taking a time derivative of eqn. (1.3.27) we obtain the following balance equation: ∂ nˆ s + ∇ · vˆ s = 0, ∂t

(5.1.45)

where the current vˆ s = vα δ(x − rα ). The above equation is closed by assuming for long times the constitutive relation between the averaged macroscopic particle current v and the density n s . This phenomenological relation linking the macroscopic particle current with the gradient of the tagged-particle density is given by Fick’s law, vs (x, t) = −Ds ∇n s (x, t),

(5.1.46)

212

Dynamics of collective modes

where the constant of proportionality Ds is a kinetic coefficient. Substituting the closing relation (5.1.46) into the coarse-grained version of eqn. (5.1.45), now leads to a closed equation for the average tagged-particle density n s (r, t): ∂n s − Ds ∇ 2 n s = 0. ∂t

(5.1.47)

The above diffusive equation represents the hydrodynamic behavior of the single-particle motion in terms of the diffusive process with the phenomenological constant Ds .

5.1.5 Two-component systems The equation of the hydrodynamics for the one-component system is easily extended to a two-component mixture. In this case the densities of the individual species are separately conserved quantities. The set of slow variables therefore consists of the two partial densities ρs (x), s = 1, 2, and the total momentum density g(x). The mass densities are defined microscopically as ρˆs (x) = m s

Ns 

δ(x − rαs (t)),

(5.1.48)

αs =1

where m s and rαs (t) denote the mass and the position of the αth particle of the sth species, respectively. The momentum densities gs for the two species are represented similarly as gˆ s (x) =

Ns 

pαs δ(x − rαs (t)),

s = 1, 2,

(5.1.49)

αs =1

where pαs is the momentum of the αth particle of species s (s = 1 or 2) and Ns is the number of particles of the sth species in the mixture. N is the total number of particles in the system, N = N1 + N2 . The relative abundances of the two species are given by xs = Ns /N for s = 1, 2, such that x1 + x2 = 1. The number of particles of species s per unit volume V is defined as n s0 = Ns /V with n 0 = N /V = n 10 + n 20 . The average mass density ρ0 is obtained as ρ0 = m 1 n 10 + m 2 n 20 . The total momentum density g = g1 + g2 in a binary mixture is a conserved property and hence a hydrodynamic variable. The total density ρ(x, ˆ t) = ρˆ1 (x, t) + ρˆ2 (x, t)

(5.1.50)

satisfies the continuity equation as in the one-component system. Since the individual densities ρ1 and ρ2 are separately conserved in the binary system, in addition to the total density there is an extra conserved quantity. This is the concentration variable cˆ defined as

5.1 Conservation laws and dissipation

c(x, ˆ t) = x2 ρˆ1 (x, t) − x1 ρˆ2 (x, t).

213

(5.1.51)

For simplicity we avoid inclusion of the energy variable in the set here. The set of microscopic equations for the dynamics now includes an additional equation for c. ˆ The equation for the momentum density is now modified from that corresponding to the one-component system, ∂ ρˆ + ∇ · gˆ = 0, ∂t ∂ gˆ i  ∇ j σˆ i j = 0, + ∂t

(5.1.52) (5.1.53)

j

with the corresponding currents being given by gˆ (r) =

Ns 2  

pαs δ(r − rαs ),

s=1 αs =1

σˆ i j (r) =

2  s=1

⎡ ⎣

j Ns  pαi s pαs αs =1

ms

(5.1.54)

⎤ Ns  Ns   1 ij δ(r − rαs ) + αs β  αs βs  ⎦, s 2

(5.1.55)

αs =1 βs  =1

where the sum over αs and βs  excludes the case αs = βs  . We have used the definitions of the following quantities in terms of the force Fαs βs  acting on the particle at rαs due to the particle at rβs  :  1 dw δ(r − rαs + wrαs βs  ), (5.1.56) αs βs  = 0

ij

j

αβ = (rαs βs  · Fαs βs  )ˆrαi s βs  rˆαs β  . s

(5.1.57)

The vector rαs βs  = rαs − rβs  and rˆαi s β  is the ith component of the unit vector rˆ αs βs  . Note s

ij

that αs βs  = βs  αs and the quantity αs βs  is also symmetric under exchange of {αs , βs  } as well as {i, j}. The microscopic stress tensor is symmetric, σˆ i j = σˆ ji , as in the case of the one-component liquid. The microscopic balance equation satisfied by the concentration variable c(r, t) is obtained in the form ∂ cˆ + ∇ · ˆjc = 0, ∂t

(5.1.58)

jc = x2 gˆ 1 − x1 gˆ 2 .

(5.1.59)

where the current ˆjc is obtained as

Considering as above the currents from the frame moving with the local velocity v(x, t), we obtain that the average current is given by + , ˆjc (x, t) = c(x, t)v(x, t) (5.1.60)

214

Dynamics of collective modes

and the corresponding reversible equation for the variable c(r, t) is obtained in the form ∂c(r, t) + ∇ · [v(r, t)c(r, t)] = 0. (5.1.61) ∂t The dissipative part of the concentration equation is computed by a similar procedure to that above. The local equilibrium distribution (5.1.15) now involves an extra coupling to the concentration variable, f le ( N , t)

3  = Q −1 exp − dr β(r, t) e(r) ˆ − v(r, t) · g(r) ˆ l 

4  x2 1 x1 μ1 (r, t) + μ2 (r, t) − v 2 (r, t) ρ(r) ˆ − μ(r, t)c(r) ˆ , − m1 m2 2 (5.1.62)

where μ = μ1 /m 1 − μ2 /m 2 is the difference between the chemical potentials of the two species. In the one-component limit the above expression for the distribution function reduces to the form (5.1.15). The corresponding part of the dissipative current is obtained for a positive definite rate of entropy production as + , ˆjD (5.1.63) c (x, t) = −γc ∇μ, where γc is a phenomenological kinetic coefficient introduced to include the dissipative effects. The equations for the total density ρ(x, t) and the total momentum density g(x, t) are respectively similar to eqns. (5.1.40) and (5.1.41) for the one-component case. By exploiting the thermodynamic relation of the differential for the pressure P and that for the chemical potential μ,     ∂P ∂P δρ + δc, (5.1.64) δP = ∂ρ c ∂c ρ     ∂μ ∂μ δμ = δρ + δc, (5.1.65) ∂ρ c ∂c ρ in terms of the concentration fluctuation δc and density fluctuation δρ a closed set of dissipative equations of motion for the slow variables in the binary mixture is obtained. The equation of motion for c(r, t) is obtained in the form ∂c(r, t) + ∇ · [v(r, t)c(r, t)] − Dc ∇ 2 c(r, t) − Dc ∇ 2 ρ(r, t) = 0, ∂t where

 D c = γc Dc

 = γc

are related to the inter-diffusion constant.

∂μ ∂c ∂μ ∂ρ

(5.1.66)

 ρ

,

(5.1.67)



(5.1.68) c

5.2 Hydrodynamic correlation functions

215

5.2 Hydrodynamic correlation functions In the previous section we obtained the deterministic equations for the time evolution of the coarse-grained densities ψa (r, t) ≡ {ρ, g, e}. These dissipative equations involve transport coefficients that act as 2 inputs in the theory. The correspond1 material-dependent ing time correlation functions ψa (r, t)ψa (r , t  ) of the coarse-grained densities (e.g., the autocorrelation of the density variable ρ(k, t)) can be computed with the understanding that the angular brackets now imply the average over the initial conditions. This averaging is generally done in terms of the probability of the initial distribution of the local densities and is determined by the local equilibrium distribution. The resulting correlation functions correspond to long distances and times (or, equivalently, small wave numbers and frequencies) and are termed the hydrodynamic correlation functions. They involve the transport coefficients representing the dissipation. Turning the argument the other way around, the correlation functions can be used to define the transport coefficients. Furthermore, if we work with the plausible hypothesis that these hydrodynamic correlation functions are identical to the microscopic correlation functions in the proper limit, then this leads us to a microscopic definition of the transport coefficients in terms of equilibrium correlation functions. This is also in agreement with the fact that the linear response of the system to an external perturbation is related to the equilibrium correlation functions. In the following we will obtain such hydrodynamic correlation functions for the dissipative equations for the one-component system discussed in the previous section. This will finally lead us to microscopic expressions for the transport coefficients in terms of equilibrium correlation functions. We begin with eqns. (5.1.40) and (5.1.41) for the density ρ and momentum density g, respectively. For simplicity and keeping in mind our primary focus in this book, we ignore coupling to the energy fluctuations. We also restrict our consideration to the linear form of the equations of motion for ρ and gi . On multiplying both eqn. (5.1.40) and eqn. (5.1.41) by ρ(x , 0) and taking the thermal average, we obtain, respectively,  ∂ ik j G g j ρ (k, t) = 0, G ρρ (k, t) + ∂t

(5.2.1)

 L0 ∂ il G gl ρ (k, t) = 0, G gi ρ (k, t) + ic02 ki G ρρ (k, t) − ∂t ρ0

(5.2.2)

j

k

where the spatial dependence has been Fourier-transformed with wave vector k. On multiplying eqn. (5.2.2) by ki and summing over i, we obtain, using eqns. (5.2.1), (1.3.25), and (5.1.44), the following equation for the density correlation function:

2 ∂ 2 ∂ 2 2 + 0 k (5.2.3) + c0 k G ρρ (k, t) = 0. ∂t ∂t 2 The above equation represents the damped sound waves of speed c0 in the fluid. The damping or sound attenuation is given by the kinetic viscosity 0 = DL /ρ0 , where

216

Dynamics of collective modes

DL = ζ0 + 4η0 /3 is the longitudinal viscosity of the fluid. By taking a Laplace transform, as defined in eqn. (1.3.73), of eqn. (5.2.3) and using the general result (1.3.8) for the intial time derivative of the density–density correlation function, we obtain G ρρ (q, z) in the form of a continued fraction,  −1 c02 k 2 G ρρ (k, z) = −G ρρ (k, t = 0) z − , (5.2.4) z + i0 k 2 where G ρρ (k, t = 0) = β −1 χ is the equal-time density–density correlation function. Next, on multiplying eqn. (5.1.41) by g j (x , 0) and taking the thermal average, we obtain for the time evolution of the current correlation function  L0 ∂ il G gl g j (k, t) = 0. G gi g j (k, t) + c02 ki G ρg j (k, t) − ∂t ρ0

(5.2.5)

l

Now we choose both of the components i and j in the direction transverse to the vector k such that kˆi = kˆ j = kˆT = 0. Using eqns. (1.3.25) and (5.1.44), we obtain the following equation for the transverse-current correlation function G T (q, t): ∂ G T (k, t) + ν0 k 2 G T (k, t) = 0, ∂t

(5.2.6)

with the kinetic viscosity ν0 = η0 /ρ0 . The Laplace transform G T (k, z) of the transversecurrent correlation function is obtained from the above equation as G T (k, z) =

G T (t = 0) , z + ik 2 ν0

(5.2.7)

where the equal-time correlation G T (t = 0) = ρ0 β −1 . The above relation leads to a useful expression for the shear viscosity, η0 =

ω2 β lim lim 2 G T (k, ω). 2 ω→0 k→0 k

(5.2.8)

As a consequence of the continuity equation (5.1.33), the longitudinal-current correlation function G L (k, ω) is related to the density correlation function through the relation ω2 G ρρ (k, ω) = k 2 G L (k, ω).

(5.2.9)

Using this and the expression (5.2.4) for the density correlation function, it is straightforward to reach the following limiting expression for the longitudinal viscosity, which is similar to eqn. (5.2.8) for the shear viscosity: 0 =

ω2 4 β η0 + ζ0 = lim lim 2 G L (k, ω), 3 2 ω→0 k→0 k

(5.2.10)

where we have used the relation χρρ (k) = mn 0 S(k) and in the small-k limit 1/(βm S(0)) = c02 .

5.2 Hydrodynamic correlation functions

217

5.2.1 Self-diffusion In an identical manner, by averaging over the initial distribution, we obtain from the coarse-grained equation (5.1.47) for the tagged-particle density n s (r, t) the equation for the corresponding hydrodynamic correlation function G s (r, t). For long times and long distances it satisfies the diffusion equation ∂G s − Ds ∇ 2 G s = 0. ∂t The solution for G s (r, t) is easily obtained in the form

1 r2 exp − , G s (r, t) = 2Ds t (4π Ds t)3/2

(5.2.11)

(5.2.12)

where the normalization constant has been chosen to satisfy the condition (1.3.30). The corresponding solution for the Fourier transform of G s (r, t) is obtained as  (5.2.13) Fs (k, t) = exp −Ds k 2 t . The frequency wave-vector transform of G s (r, t) is denoted as Ss (k, ω) and has the Lorentzian form Ss (k, ω) = Ss (k, ω) follows the sum rule



+∞

−∞

2Ds k 2 ω2 + (Ds k 2 )

2

.

dω Ss (k, ω) = 1 2π

(5.2.14)

(5.2.15)

to ensure that Fs (k, t = 0) = 1. Finally, the diffusion coefficient Ds for the decay of the tagged-particle correlation is obtained from the relation Ds =

ω2 1 lim lim 2 Ss (k, ω). 2 ω→0 k→0 k

(5.2.16)

In a similar manner we also reach an equation for the autocorrelation of the concentration variable c(r, t), which is an extra conserved density for a two-component system. The linearized equation of motion in this case is obtained by an averaging of eqn. (5.1.66),

∂ (5.2.17) + Dc k 2 c(k, t) G cc (k, t) + Dc k 2 G ρc (k, t) = 0. ∂t If the coupling with the density fluctuations is ignored then the Laplace transform of the concentration correlation is obtained in the simple diffusive-pole form as in eqn. (5.2.7). The corresponding inter-diffusion constant Dc is given by an expression similar to (5.2.16), Dc χcc =

ω2 1 lim lim 2 G cc (k, ω), 2 ω→0 k→0 k

where χcc is the equal-time correlation of the concentration variable c.

(5.2.18)

218

Dynamics of collective modes

5.2.2 Transport coefficients In the above discussion of the dynamics of the fluid in terms of the dissipative equations we have considered the transport coefficients such as the viscosity and thermal conductivity. The latter were introduced as constants that defined the dissipative parts of the corresponding currents representing a microscopic conservation law for the fluid. While the basic form of the hydrodynamic equations is the same for different fluids, the actual values of the transport coefficients appearing in these equations depend on the material properties of the corresponding fluid. A basic goal for the theory, therefore, is to obtain the transport coefficients in terms of the characteristic microscopic parameters for the system. As we have seen above, the linear response of a nonequilibrium system to an external perturbation is related to the corresponding equilibrium correlation of spontaneous fluctuations. Indeed, the linear transport coefficients appearing in the hydrodynamic equations can be obtained in terms of the equilibrium time correlation functions (see eqn. (1.3.3)). In eqns. (5.2.8), (5.2.10), and (5.2.16), respectively, we have expressed the shear, longitudinal, and self-diffusion coefficients of a one-component fluid. We demonstrate below that these relations reduce to integrals over time correlation functions of projected currents, which are termed the Green–Kubo relations for the transport coefficients. Green–Kubo relations We start from the expression (5.2.8) for the shear viscosity to obtain β lim lim η0 = 2 ω→0 k→0 1 = lim lim 2 ω→0 k→0

 

ω2 G T (k, ω) k2 +∞

−∞



d(t − t  )eiω(t−t )

1 ∂2 G T (k, t − t  ). k 2 ∂t ∂t 

(5.2.19)

The above steps follow directly from partially integrating twice. Now, using the definition for the Fourier transform of the tagged-particle correlation function, G T (k, t − t  ) =

1 gT (k, t)gT (−k, t  ), V

(5.2.20)

in eqn. (5.2.19), we obtain β lim lim η0 = 2V ω→0 k→0



+∞ −∞



d(t − t )e

iω(t−t  )

. 1 ∂ gT (k, t) ∂ gT (−k, t  ) . ∂t ∂t  k2

The spatial Fourier transform of the balance equation for the momentum density g in terms of the stress tensor is ∂ gi (k, t) = k j σi j (k, t). ∂t

(5.2.21)

5.2 Hydrodynamic correlation functions

219

Now, choosing the direction of k in the z direction, we obtain the transverse component along the x direction from the RHS of (5.2.21) as  +∞ β  d(t − t  )eiω(t−t ) σx z (k, t)σx z (−k, t  ). (5.2.22) lim lim η0 = 2V ω→0 k→0 −∞ Since the equilibrium correlation function of the stress on the RHS above is dependent only on the difference of the two times it is an even function of time. Hence we obtain β η0 = lim k→0 V



∞

dtσx z (k, t)σx z (−k, 0).

(5.2.23)

0

In an identical way, starting from the relation (5.2.10), we obtain for the longitudinal viscosity the corresponding relation  β ∞ dtσzz (k, t)σzz (−k, 0). 0 = lim k→0 V 0 However, the diagonal element of σi j (k, t) is nonzero in the long-time limit. As already defined in eqn. (A5.1.25), the nonzero average value of the diagonal element is the hydrodynamic pressure P in the fluid, lim σzz (k, t) = P V.

k→0

(5.2.24)

This makes the integral on the RHS of eqn. (5.2.24) diverging. This is rectified by subtracting out the constant part from the diagonal element of the stress, thereby obtaining finite transport coefficients in terms of the correlation of stress fluctuations: 0 = lim

k→0

β V



∞

dt{σzz (k, t) − P V }{σzz (−k, 0) − P V }.

(5.2.25)

0

The above examples for the Green–Kubo relation are easily extended to the self-diffusion coefficient Ds . We start from the expression (5.2.16) to obtain, in an identical manner to that above,  2 ω 1 Ss (k, ω) Ds = lim lim 2 ω→0 k→0 k2 .  +∞ 1 ∂n s (k, t) ∂n s (−k, t  )  1 d(t − t  )eiω(t−t ) 2 = lim lim 2 ω→0 k→0 −∞ ∂t ∂t  k  +∞ ki k j 1 d(t − t  ) 2 vi (k, t)v j (−k, t  ). (5.2.26) = lim 2 k→0 −∞ k In the small-k limit the equilibrium correlation of the velocity of the αth particle is diagonal and is proportional to δi j . Using this in eqn. (5.2.26),

220

Dynamics of collective modes

Ds =

1 3

 0

+∞

d(t)ϕv (t).

ϕv (t) = v(t) · v(0) represents the correlation of the single-particle velocities at two different times. The self-diffusion coefficient is expressed as an integral over the equilibrium time correlation function of tagged-particle velocities at two different times. Note that the hydrodynamic form (5.2.13) of the tagged-particle correlation is then identical to what is obtained for the microscopic quantity in eqn. (1.3.57), provided that we identify Ds from the Green–Kubo relation (5.2.27). For a purely Gaussian system the tagged-particle correlation Fs (k, t) is entirely determined by Ds or the current correlation function ϕv (t). Using the relation (1.3.56), it also follows that the mean-square displacement of the tagged particle is given by r 2 (t) = 6Ds t,

(5.2.27)

which is termed the Einstein relation. In a dense liquid the tagged particle is caught in a cage formed by its neighbors and the rattling motion of the particle in the cage causes a negative tail for the velocity correlation function. Eventually, over structural-relaxation times, the cage breaks and the correlation ϕv (t) decays to zero. The relation (5.2.27) shows that the negative tail results in a fall of the self-diffusion coefficient Ds . In Fig. 5.1 we show

Fig. 5.1 The tagged-particle velocity autocorrelation function ϕv (t) vs. time t for a Lennard-Jones fluid at various values of the temperature T ∗ and density n ∗ . The dimensionless units are as given in c American Physical Society. (5.2.28). From Levesque and Verlet (1970). 

5.2 Hydrodynamic correlation functions

221

the tagged-particle velocity correlation obtained from the simulations of a Lennard-Jones fluid characterized by the appropriate interaction potential (see eqn. (1.2.117)). For this system the time t, temperature T , and density ρ are expressed as dimensionless quantities using the following definitions: t t = τ0 ∗

T∗ =

 with τ0 =

kB T 

and

mσ 2 

1/2 ,

n ∗ = nσ 3 .

(5.2.28) (5.2.29)

Finally, for the inter-diffusion constant Dc of a two-component system we can obtain a similar Green–Kubo relation starting from eqn. (5.2.17). In this case, using the continuity equation (5.1.58) for the concentration variable, we obtain the Green–Kubo relation 1 Dc = 3χcc



+∞

dt ϕc (t),

(5.2.30)

0

where ϕc is the correlation of the current jc defined microscopically in eqn. (5.1.59). The two-component mixture reduces to a one-component system if the two species of the mixture are identical. In this case the above result for the inter-diffusion constant links to tagged-particle diffusion on taking N1 = 1 and N2 = N − 1. In the thermodynamic limit x2 → 1 and x2 → 1. In this case, χcc → 1 and the Green–Kubo relation (5.2.31) for Dc becomes identical to the relation (5.2.27) for the self-diffusion constant Ds of a one-component system. In the above treatment we have expressed the various transport coefficients as time integrals of currents in the hydrodynamic correlation functions. Now, using the hypothesis that these correlation functions are identical to their microscopic counterparts, the different currents involved in the Green–Kubo relations are obtained from the corresponding microscopic expressions. In this way the components of the stress correlation functions in eqns. (5.2.23) and (5.2.25) are obtained from the Fourier transform of the expression in eqn. (5.1.12). Similarly, the correlation of the current jc in the expression (5.2.31) for the inter-diffusion constant is obtained from the microscopic expression (5.1.59). This results in expressions for the transport coefficients in terms of the microscopic interaction parameters of the system and provides a very useful tool for computing theoretically the transport coefficients in computer MD simulation. An immediate implication of the Green–Kubo relations is the validity of linear constitutive relations and the nature of the corresponding transport coefficients in certain extreme cases. For exponentially decaying current–current correlation functions the transport coefficient is simply proportional to the corresponding relaxation time and is well behaved. However, it has been observed from computer MD simulations (Alder and Wainwright, 1967, 1970) that the velocity correlation function decays in a power-law form. In general let us consider the velocity correlation function ϕv (t) ∼ A0 t −d/2 in a d-dimensional

222

Dynamics of collective modes

system. This has been termed the long-time tail in the literature. That it has important implications regarding the transport coefficients of the fluid can be seen directly from the Green–Kubo relations. The transport coefficient λT is obtained as  t2 A0  1−d/2 t2 λT = ϕv (t¯)d t¯ = . (5.2.31) t t1 1 − d/2 t1 For d ≤ 2 the contribution from the upper limit of the integration becomes diverging. This implies that conventional hydrodynamics will be invalid in this situation. The origin of such effects, namely the power-law decay of correlation, comes from the collective effects in the fluid over semi-hydrodynamic length scales. This will be discussed in Chapter 6. Bare transport coefficients In the classical statistical-mechanical approach, a microscopic description of the dynamics is obtained from eqn. (1.1.39) for the time evolution of the one-particle distribution function f (x, v, t). The effects of the interaction between the particles on the dynamics are accounted for here through the collision term in this equation. In order to obtain the transport coefficient for a given fluid in terms of its microscopic properties, i.e., the interaction potential between the particles, the Boltzmann equation is a suitable starting point. These so-called bare transport coefficients correspond to the short-time dynamics due to uncorrelated binary collisions. Much progress on this has been achieved for the hard-sphere interaction potential given by ⎧ ⎨∞ for r ≤ σ, (5.2.32) u HS (r ) = ⎩0 for r > σ, where σ is the hard-sphere diameter. In a hard-sphere fluid the local conserved densities of mass, momentum, and energy {n(r, t), g(r, t), e(r, t)} are obtained from the moments of the one-particle distribution function f (v, x, t):  n(r, t) = dp f (1) (r, p, t),  g(r, t) =

dp p f (1) (r, p, t),

 e(r, t) =

dp

p2 (1) f (r, p, t). 2m

(5.2.33)

A general strategy for computation of the transport coefficients is to obtain the hydrodynamic balance equations for the dynamics of the conserved densities. This involves taking moments of eqn. (1.1.39). Alternatively, the Green–Kubo expression for the transport coefficient is obtained using the microscopic expressions for the stress tensor. The kinetic theory of the N -particle system (controlled by the Liouville operator L) is approximated here in terms of the two-body dynamics. For the hard-sphere system this involves approximating L in terms of the two-body hard-sphere collision operator (Résibois and de

5.2 Hydrodynamic correlation functions

223

Leener, 1977). The Boltzmann transport coefficients for the hard-sphere system, namely the shear viscosity ηB , the thermal conductivity λB , and the self-diffusion coefficient DsB of a tagged particle are obtained as  ηB = 

5 √ 16 π



m , σ t0

 75 kB , √ 64 π σ t0   1 3 B Ds = , √ 8 π n 0 σ t0 λB =

(5.2.34) (5.2.35) (5.2.36)

√ where t0 = σ/v0 is the time taken by the particle with thermal velocity v0 = 1/ βm to cover the length of the hard-sphere diameter σ and n 0 is the number of particles per unit volume. In the case of hard spheres the collisions are instantaneous and hence even at high densities multi-particle collisions are unlikely. Transport coefficients at high densities are therefore computed by improving the collision term in the Boltzmann equation with the Enskog approximation (Résibois and de Leener, 1977). The latter involves (a) accounting for the higher rate of collision in the dense system in terms of the pair correlation function g(r ) and (b) computing the transfer of flux in the transport through collision between the particles. In a dense system the collisional contribution to the transport process is comparable to and even dominant over that from actual movement of the fluid particles. Using the Enskog model, the corresponding Boltzmann transport coefficients are corrected (Hirschfelder et al., 1954) to 

 1 + 0.8 + 0.761Y , Y   1 λE = 4ϕ + 1.20 + 0.755Y , λB Y ηE = 4ϕ ηB

4ϕ DE = , DB Y

(5.2.37) (5.2.38) (5.2.39)

where the quantities with subscript E are the Enskog values, ϕ = π nσ 3 /6 is the packing fraction, and Y = 4ϕg(σ ), where g(σ ) is the hard-sphere pair correlation function at contact. The different terms in the above expressions for the transport coefficients can be understood from the corresponding Green–Kubo expressions. Thus, for example, the shear viscosity is obtained from the time correlation of the off-diagonal stress tensor σx z , which has a kinetic and a potential contribution (see eqn. (5.1.12)). Hence the correlation function involving products of two σ s has three parts: a purely kinetic contribution, corresponding to the transfer of momentum from free streaming of the particles; a potential

224

Dynamics of collective modes

term from the interaction of the particles, giving the purely collisional contribution to the transport; and a cross term. At liquid densities the collisional contribution is most dominant (the third term on the RHS of eqn. (5.2.37)). The bulk viscosity ζ0 , on the other hand, makes only a collisional contribution since the kinetic part is diagonal and is subtracted out in the expression (5.2.25). The Enskog approximation to the transport coefficients works well for liquids up to moderate densities n 0 σ 3 ≤ 0.3. The Enskog expression for the transport coefficients does not conform to the Stokes–Einstein relation. On approaching the freezing point the discrepancy with computer-simulation results grows. The Enskog value of shear viscosity becomes less than that from simulations almost by a factor of 2 at the freezing transition point. The self-diffusion coefficient, on the other hand, shows an opposite trend. The product Ds η calculated from simulations is almost constant for moderate densities n 0 σ 3 ≥ 0.3 and the value of this constant is close to that predicted from the Stokes–Einstein (SE) relation (within 10%). Thus, while at low densities the SE relation is violated, it holds for moderate liquid densities. In the supercooled liquid the discrepancies with Enskog theory grow much more rapidly, and the SE relation is again violated. At high density the disagreement of the shear viscosity with simulations stems from the fact that the latter has large contributions from slowly decaying long-time tails of the stress correlation function. This applies to the viscosities. The agreement between theory and simulation is much better in the case of the thermal conductivity. The origin of the long-time tails can be understood in terms of correlated dynamics in the fluid, which can be understood in terms of mode-coupling theories to be discussed later. For a general interaction potential multi-particle collisions are likely to contribute substantially to the transport process at high densities. Calculation of the transport coefficients in such cases is more involved and has been done along phenomenological lines with ad-hoc extension of the Enskog model in terms of the so-called modified Enskog theory or using numerical methods for computing the collision integrals (Fitts, 1966; Curtiss, 1967). Corrections to the short-time transport coefficients at higher densities have also been obtained by improving the Boltzmann equation in the corresponding situations. The transport coefficient is obtained in terms of a density expansion similar to the virial expansion for the equation of state of the fluid. The calculation involves treating the pair distribution function in the collision integral as a functional of the one-particle distribution function. The result of this rather complicated technical development is that we obtain the secondorder term ρ 2 ln ρ for the transport coefficient λ(ρ) as nonanalytic (Ernst et al., 1969; Curtiss, 1967),  λ(ρ0 ) = λB 1 + a1 ρ0 + a2 ρ02 ln ρ0 + · · · ,

(5.2.40)

where ρ0 is the equilibrium density and the ai are functions of temperature only. The origin of such nonanalytic behavior is a consequence of collective effects coming from semi-microscopic length and time scales. This will be discussed later, in Chapters 6–8.

5.3 Linear fluctuating hydrodynamics

225

5.3 Linear fluctuating hydrodynamics The mechanical description of the liquid consisting of N particles involves many degrees of freedom in the thermodynamic limit. The general strategy adopted in developing statisticalmechanical models for the dynamics is to focus on a small set of dynamic variables with time scales of variation much longer than those of the vast number of other degrees of freedom for the system. These slow modes are the hydrodynamic modes. The rest of the degrees of freedom are treated as noise. The simplest example of such a description of the dynamics is the case of Brownian motion discussed in Chapter 1. In that case the relevant variable, namely the velocity of the heavy particle, is slow due to its large inertia relative to the surrounding fluid particles. The origin of the slow mode can be different in different physical systems. We discuss in the following a general scheme for obtaining the dynamics for these slow modes. The equation of motion for a set of variables {φˆ i } is usually obtained through the standard projection-operator scheme of Zwanzig (1961) and Mori (1965a, 1965b). This involves a projection operator P, which is defined so as to “project” any dynamic variable along the chosen set in terms of a suitably defined scalar-product operation. The effects of the rest of the dynamic variables are treated as noise f i and spanned by the operator Q = 1 − P. The deduction of the equations of motion for the variables φˆ i with the help of P is described in the appendix of this chapter. We present below a somewhat different technique (Ma and Mazenko, 1975; Mazenko, 2006a) for obtaining the equation of motion. This formalism is referred to as the technique of fluctuating hydrodynamics at the level of the linear dynamics of slow modes. This is particularly useful in finally obtaining the nonlinear Langevin equations to be discussed in the next chapter. These nonlinear equations form the basis for constructing relevant theoretical models for the slow dynamics in the dense fluid near freezing and in the supercooled state.

5.3.1 The generalized Langevin equation Let us consider a classical system of N particles. Let {φˆ i } for i = 1, . . . , n denote the set of n collective densities that are functions of the phase-space variables {r1 , r2 , . . . , r N ; p1 , p2 , . . . , p N }. By suitably defining the φˆ i we maintain the condition that φˆ i (t) = 0. The time rates of variation of the collective variables are given by   ∂ φˆ i j ∂ φˆ i (t)  ∂ φˆ i j r˙ + p˙ , = j α j α ∂t ∂ rα ∂ pα

(5.3.1)

(5.3.2)

j,α

j

j

where pα and rα , respectively, denote the jth component of the momentum and the position coordinates of the αth particle in the system. Using Hamilton’s equations of motion for the particles in terms of the derivatives of the Hamiltonian H ,

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Dynamics of collective modes

r˙αi (t) =

∂H , ∂ pαi

p˙ αi (t) = −

∂H , ∂rαi

(5.3.3) (5.3.4)

the equation of motion of φˆ i is expressed in terms of Poisson’s bracket (Dzyloshinski and Volvick, 1980) as ∂ φˆ i (t) = {φˆ i , H }. ∂t

(5.3.5)

The Poisson bracket { A, B} between two dynamic variables A and B is defined as

 ∂A ∂B ∂A ∂B − i . (5.3.6) {A, B} = ∂rαi ∂ pαi ∂ pα ∂rαi i,α

ˆ Thus the equation of motion for the collective variable φ(t) is written in terms of the Liouville operator, ∂ φˆ i (t) = −{H, φˆ i } ≡ iLφˆ i (t), ∂t

(5.3.7)

which can be formally integrated to obtain φˆ i (t) = exp(iLt)φˆ i , where φˆ i = φˆ i (t = 0). The analysis which will follow is facilitated in terms of the Laplace transform defined (Abramowitz and Stegun, 1965) as  ∞ ˆ dt ei zt φˆ i (t) (5.3.8) φi (z) = −i 0

with Im(z) > 0. On taking the Laplace transform of eqn. (5.3.7) we obtain the relation φˆ i (z) = R(z)φˆ i ,

(5.3.9)

where the resolvent operator R(z) is given by R(z) = [z + L]−1 .

(5.3.10)

z φˆ i (z) = φˆ i − Lφˆ i (z).

(5.3.11)

Equivalently, we obtain

A key step in formulating the dynamics of the collective modes involves determining the effect of the operator L on the Laplace-transformed φˆ i (z). In general the net result of this operation can give rise to a complicated dynamics. The crucial approximation in this analysis is approximating the quantity Lφˆ i (z) as a sum of two parts. (a) The first part is linearly proportional to the fields φˆ i . (b) The rest of the time evolution of ϕi is treated as a noise f i (z).

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227

−Lφˆ i (z) ≈ Ki j (z)φˆ j (z) + i f i (z),

(5.3.12)

We write schematically

where Ki j (z) is a kernel function that represents the projection of Lφˆ i (z) on the subspace of the collective modes φˆ i (z). Here and in the rest of this section we adopt the Einstein summation convention that repeated indices are summed over. The leftover contribution or the noise denoted by f i (z) in the second term on the RHS of eqn. (5.3.12) behaves like a dynamic variable similar to φˆ i (z). The description of the dynamics of the many-particle system in the above scheme involves two sets of variables, {φi } and { f i }, which are independent at t = 0. At later times they mix through the nonlinearities in the dynamical equations. Thus φi (z) has components along both φi and f i (z). On taking the equilibrium average of (5.3.12) we obtain that the noise part f i (z) is zero on average,  f i (z) = 0.

(5.3.13)

The above result follows from two observations. (a) From (5.3.1) it follows that φˆ i (z) = 0. (b) We have, from the definition of the Liouville operator, the condition  ∞ d ˆ Lφi (z) = −i dt ei zt (−i) φˆ i (t) = 0. dt 0

(5.3.14)

Equation (5.3.14) is a consequence of the time translational invariance of the equilibrium average. A second requirement on the noise is that f i (z) has no projection on the mode φi (t) at t = 0, φˆ i f i (z) = 0.

(5.3.15)

The equation of motion for the collective modes maintains the two properties (5.3.13) and (5.3.15). Using the ansatz (5.3.12) in (5.3.11), we obtain ! " zδi j − Ki j (z) φ j (z) = φi + i f i (z). (5.3.16) Note that we have dropped the hat on the φ to imply that this approximate equation (obtained with the use of eqn. (5.3.12)) is satisfied for a coarse-grained form of the colˆ The inverse Laplace transform of (5.3.16) gives the equation of motion lective variable φ. for φi (t):  t ∂φi d t¯ Ki j (t − t¯ )φ j (t¯ ) = f i (t). (5.3.17) + ∂t 0 The dynamics of the many-particle system is now described in terms of (a) a set of slow modes denoted by {φ(t)}, which are coarse-grained functions with smooth space– time variations and not dependent on phase-space variables; and (b) the noise f i (t), which varies over time scales much shorter than that of φi (t) and is treated as random with average equal to zero. The dissipative parts in the generalized Langevin equation of motion for the

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Dynamics of collective modes

φi (t) are related to the correlation of the noise at two different times. This is similar to the Langevin equation introduced in Chapter 1 in reference to the discussion of the Brownian motion of a heavy particle moving in a liquid. The approximation of separating the time scales in the dynamics applies in that case because of the very large mass ratio between the Brownian particles and the surrounding liquid particles. The nature of the dependence of the dynamics on the φi is obtained by focusing on the kernel function Ki j . In general the kernel function Ki j can be divided into two parts, static and dynamic. For the Laplace-transformed quantity we write (s)

(d)

Ki j (z) = Ki j + Ki j (z).

(5.3.18)

Ki(s) j is independent of z and is taken as a static contribution. The second part, with a frequency dependence, is classified as the dynamic contribution K(d) . Using the above separation, the Langevin equation reduces to  t ∂φi ¯ ¯ ¯ (t − t )φ + d t¯ Ki(d) (5.3.19) + iKi(s) j j j (t − t )φ j (t ) = f i (t). ∂t 0 We now analyze the kernel matrix Ki j further by expressing it in terms of the correlation functions of the collective modes. The kernel function In evaluating the kernel function or the memory function Ki j we adopt the strategy of expressing it in terms of the correlation function of the slow variables. The correlation of φˆ i and φˆ j (t) is defined as Ci j (t) = φˆ i φˆ j (t). The Laplace transform of the correlation functions  ∞ Ci j (z) = −i dt ei zt Ci j (t)

(5.3.20)

(5.3.21)

0

is related to the resolvent R(z) defined above in eqn. (5.3.10) through the relation Ci j (z) = φˆ j φˆ i (z) = φˆ j R(z)φˆ i .

(5.3.22)

We now express the memory function Ki j in terms of the correlation functions Ci j . This involves multiplying (5.3.16) by φˆ j and taking the equilibrium average: zCi j (z) = Si j + Kik Ck j (z),

(5.3.23)

where Si j = φˆ i φˆ j  is the static or the equal-time correlation function among the slow variables. Note that in reaching (5.3.23) we have used the ansatz (5.3.15) of zero projection of the noise on the slow modes. On the other hand, on multiplying eqn. (5.3.11) by φˆ j and taking the equilibrium average we obtain zCi j (z) = Si j − φˆ j R(z)Lφˆ i .

(5.3.24)

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229

In the second term on the RHS of (5.3.24) we have used the identity LR(z) = L(z + L)−1 = R(z)L. On comparing (5.3.23) and (5.3.24) we obtain + , Kik Ck j (z) = − φˆ j R(z)Lφˆ i .

(5.3.25)

(5.3.26)

On multiplying (5.3.26) by z and substituting (5.3.23) on the LHS for zCk j (z) we obtain +  , + , Kik Sk j − φˆ j R(z)Lφˆ k = − φˆ j z R(z)Lφˆ i , + , + (5.3.27) = − φˆ j Lφˆ i + φˆ j LR(z)Lφˆ i , where we have used the identity z R(z) = 1 − LR(z). Since the Liouville operator L as defined in (5.3.3) is a derivative operator, the following property holds for the equilibrium average for two variables A and B: ALB = −{LA}B.

(5.3.28)

The above relation is also a consequence of the time translational invariance of the equilibrium distribution. The relation (5.3.27) now reduces to the following expression for the memory function K in terms of the correlation function C: + , , + + , Kik Sk j = − φˆ j Lφˆ i − {Lφˆ j }R(z){Lφˆ i } − Kik {Lφˆ j }R(z)φˆ k . (5.3.29) The RHS of (5.3.29) still contains the memory function. This is eliminated formally by inverting the relation (5.3.26) in terms of the inverse C −1 of the correlation matrix C, giving the relation + , , + Kik Sk j = − φˆ j Lφˆ i − {Lφˆ j }R(z){Lφˆ i } + , , + −1 (z) {Lφˆ j }R(z)φˆ k . (5.3.30) + φˆl R(z)Lφˆ i Clk The first term on the RHS of (5.3.30) is independent of z and is taken as a static contribution K(s) to the memory function K as defined in eqn. (5.3.18). The second and third terms with frequency dependences are classified as the dynamic contribution K(d) in eqn. (5.3.18). We therefore obtain the following definitions: (s) Kik Sk j = −φˆ j Lφˆ i ,

(5.3.31)

(d) Kik Sk j = −{Lφˆ j }R(z){Lφˆ i } −1 + {Lφˆ j }R(z)φˆ k Clk (z)φˆl R(z){Lφˆ i }.

(5.3.32)

We treat the static and the dynamic part of the kernel function K separately in the following.

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Dynamics of collective modes

The static part Since the static part transform

(s) Ki j

of Ki j (z) is a constant, we obtain by taking the inverse Laplace (s)

Ki j (t) = 2ii j δ(t),

(5.3.33)

where we denote i j ≡ φˆ k Lφˆ i Sk−1 j . This part of the kernel can be expressed in terms of static correlation functions. To compute this we express, using (5.3.7), the Liouville operator L in terms of the Poisson bracket to obtain − φˆ j Lφˆ i  = −iφˆ j {H, φˆ i },

(5.3.34)

i (5.3.35) Tr e−β H φˆ j {H, φˆ i }, Z where the equilibrium average is being considered in the canonical ensemble with partition function Z and the operation “Tr” denotes the classical trace (McQuarie, 2000), defined as     ∞  1 Tr ≡ (5.3.36) dr1 . . . dr N dp1 . . . dp N . N! =−

N =0

Since the Poisson brackets involve derivative operators, we have the relation e−β H {H, φˆ i } = −β −1 {e−β H , φˆ i }.

(5.3.37)

Using this result, we obtain from eqn. (5.3.34) that iβ −1 Tr φˆ j {e−β H , φˆ i } Z      ˆ ˆ iβ −1 ∂ ∂ φ ∂ φ ∂ i i = e−β H − e−β H Tr φˆ j . Z ∂rαs ∂ pαs ∂ pαs ∂rαs

−φˆ j Lφˆ i  = −

(5.3.38)

Now, partially integrating with the derivative of the Boltzmann factor e−β H simplifies the above expression to      ˆ ˆ ∂ φ ∂ φ iβ −1 ∂ ∂ i i Tr e−β H − s φˆ j − φˆ j Lφˆ i  = − + s φˆ j s Z ∂rα ∂ pαs ∂ pα ∂rα = iβ −1 {φˆ i , φˆ j }.

(5.3.39)

On denoting the Poisson bracket {φˆ i , φˆ j } between the slow variables as a function of the φˆ i , Q i j ≡ {φˆ i , φˆ j },

(5.3.40)

we obtain from the static part of the memory function the contribution 0 −1 Sk j , i j = iβ −1 Q ik

where Q i0j = Q i j  is the equilibrium average of the Poisson bracket Q i j .

(5.3.41)

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231

The dynamic part The dynamic part represented in eqn. (5.3.32) is a complicated expression and further analysis of this involves dealing with the many-body dynamics. First, we analyze the above (d) formal expression to prove that the contribution to Ki j comes only from the nonlinear part of Lφˆ i . To demonstrate this, we separate the linear and nonlinear parts of the latter as follows: Lφˆ i = ail φˆl + Nˆ i ,

(5.3.42)

with Nˆ i denoting the contributions involving (a) nonlinear products of the φˆ i and (b) those which cannot be expressed in terms of the φˆ i . On substituting (5.3.42) into the expression (5.3.32) for the part Lφˆ i we obtain (d) Sk j = −{Lφˆ j }R(z){ail φˆl + Nˆ i } Kik −1 + φˆ s R(z){ail φˆl + Nˆ i }Csk (z){Lφˆ j }R(z)φˆ k .

On collecting the coefficients of ail on the RHS we obtain the term  −1 (z){Lφˆ j }R(z)φˆ k  . ail {Lφˆ j }R(z)φˆl  − φˆ s R(z)φˆl Csk

(5.3.43)

(5.3.44)

The coefficient of ail given above vanishes. To prove this, we note that by definition we have −1 −1 (z) ≡ Cls (z)Csk (z) = δlk . φˆ s R(z)φˆl Csk

(5.3.45)

Using this relation, the two terms within the square brackets in (5.3.44) cancel out. Similarly, on expressing Lφˆ j in the rest of the RHS of (5.3.43) with the expansion (5.3.42) it follows in an exactly similar manner that the coefficients of a jl vanish. The dynamic part of the kernel is therefore entirely controlled by the Nˆ i : (d) −1 (z) Nˆ j R(z)φˆ k . Kik Sk j = − Nˆ i R(z) Nˆ j  + φˆ s R(z) Nˆ i Csk

(5.3.46)

The above analysis shows that the linear part of Lφˆ i does not contribute to Ki(d) j and the latter is controlled by the nonlinearities in the dynamics as well as fluctuations not included in the φˆ i . This contribution to the kernel Ki j is termed the one-particle irreducible part. The linear part of the dynamics, on the other hand, is included in the static contribution to Ki j . Using this scheme of separating the kernel function into a static and dynamic part, the Langevin equation (5.3.17) is now written as ∂φi + ii j φ j (t) + ∂t

 0

t

(d)

d t¯ Ki j (t − t¯)φ j (t¯) = f i (t).

(5.3.47)

From eqn. (5.3.47) an important relation between the noise and the dynamic part of the memory function follows: (d)

 f i (t) f j (t  ) = Kik (t − t  )Sk j .

(5.3.48)

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Dynamics of collective modes

The above relation is termed the second fluctuation–dissipation theorem (FDT) and follows from the assumption (5.3.15) that the noise has no projection onto the initial value of φˆ i (t). The proof is given in Appendix A5.2. In the linear fluctuating hydrodynamic approach discussed in this chapter the relation (5.3.48) between the noise correlation and the dynamic part of the memory function forms the basis for dealing with the latter. We assume that the dynamic part of the memory function is determined by the noise and represents the dynamics over a time scale that is assumed to be much shorter than that of slow modes. Ki(d) j is thus used to define short-time kinetic coefficients for the system. The bare transport coefficients determine the strength of the noise correlation in the system through the so-called second FDT and are used as an input in the theory. In the above discussion we have actually bypassed evaluating the dynamic part of the memory function and identified the latter with the physical (linear) transport matrix in the dissipative equations for the slow modes. Direct evaluation of the dynamic part using what is termed as the mode-coupling approximation, has been done by Résibois and de Leener (1966), de Leener and Résibois (1966), and Kawasaki (1970) (see also Andersen (2002, 2003a, 2003b) and Forster and Martin (1970) for further discussion on the dynamic part). The linear equation for the slow modes gives rise to a linear equation for the correlation function with a frequency-dependent memory function that satisfies the Green–Kubo relation (discussed in Chapter 1, see eqn. (5.2.23)) and is related to the noise through the second fluctuation–dissipation relation. In Appendix A7.4 we present, using the standard Mori–Zwanzig projection operator scheme, an alternative scheme for evaluating this memory function, in particular, for the dynamics of density correlation functions. This approach has traditionally been adopted in the literature for obtaining the simple mode-coupling model (Bengtzelius et al., 1984; Götze, 2009) for glassy dynamics. In the present book we will obtain the mode-coupling model starting from a set of nonlinear stochastic equations. In the next chapter we demonstrate how to obtain a set of nonlinear equations of motion for a chosen set of slow modes. The renormalization due to the nonlinearities in the equations of motion is obtained using field-theoretic techniques (Amit, 1999). The corresponding renormalized theory due to the nonlinear dynamics of the collective modes gives rise in a natural way to the self-consistent mode-coupling model. As we will see, the field-theoretic approach proves useful in understanding the full implications of the nonlinearities in the many-body dynamics. The Markov approximation In dealing with the physical problem of the dynamics of the liquids we assume a complete separation of time scale between the collective modes φˆ i and the f i . The fast modes or the noise f i are assumed to be correlated over much shorter time scales than the φˆ i . This implies that  f i (t) f j  decays to zero much faster than does φˆ i (t)φˆ j . Thus  f i (t) f j  is very sharply peaked near t = 0. The Markov approximation refers to the Markov process (Markov, 1907) in which the dynamic evolution of the system in every step is determined by the previous step only. In the present context, applying this approximation to the memory function amounts to assuming the absence of memory in the long-time limit. Thus in

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233

the Markov approximation the noise is in fact assumed to be delta-function correlated and defined in terms of a matrix L i0j which consists of the bare transport coefficients,  f i (t) f j (t  ) = 2β −1 L i0j δ(t − t  ).

(5.3.49)

f i is the white noise and its correlation is proportional to the temperature. From (5.3.48) (d) the dynamic part Ki j of the memory function is identified as (d)

0 −1 Ki j (t − t  ) = 2β −1 L ik Sk j δ(t − t  ).

(5.3.50)

Using (5.3.41) and (5.3.50) in (5.3.47) we obtain the linear Langevin equation for the slow mode φi including the white noise f i as # $ ∂φi 0 0 + L ik + β −1 −Q ik Sk−1 j φ j (t) = f i (t). ∂t

(5.3.51)

The correlation of the white noise f i is given in (5.3.49) by the matrix L i0j . We can identify the first and second terms on the RHS of (5.3.51) with the time-reversal behavior of the dynamics. (i) From the definition of the Poisson bracket it follows that Q i j (−t) = −i  j Q i j (t), where i denotes the time-reversal property of the slow mode φˆ i , i.e., φˆ i (−t) = i φˆ i (t). Therefore the second term in (5.3.51) remains invariant under time reversal and represents the reversible part of the dynamics. In the linearized Langevin approach the average of Q i j gives the static part of the memory function. (ii) Assuming that the matrix L i0j satisfies under time reversal the property L i0j (−t) = i  j L i0j (t), the third term on the LHS of (5.3.51) breaks the time-reversal symmetry of the equation of motion and hence represents dissipation. This dissipative term can be expressed in the form of a divergence ∇ · Ji , with the current Ji being proportional to the gradient of the slow mode φˆ j through a linear constitutive relation. The corresponding constants of proportionality are the linear transport coefficients for the system and are linked to the matrix L i0j . In the linearized Langevin equations the L i0j represent the physical transport coefficients. We clarify this further below in discussing the Langevin dynamics for the hydrodynamic modes of the liquid. The matrix Si j represents the equal-time correlation of the fields φˆ i φˆ j . We compute this static correlation function in terms of the correlation of the coarse-grained variables {φ} averaged over the function space with a coarse-grained free-energy functional F[φ] (Mazenko 2002a, 2002b). The equilibrium average is obtained in terms of the probability distribution determined with F, 6 Dφ φi φ j exp{−β F[φ]} . (5.3.52) Si j ≡ φi φ j  = 6 Dφ exp{−β F[φ]}

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Dynamics of collective modes

Using the functional identity  Dφ

δ  −β F φi e =0 δφ j

(5.3.53)

we obtain that φi

. δF = δi j . δφ j

(5.3.54)

Assuming F to be a quadratic functional of the fields φi , the above equation leads to the following expression:  1 ˆ = dx φi Si−1 F[φ] (5.3.55) j φj. 2 In terms of F the linear Langevin equations (5.3.51) for the fields φi reduce to the form # $ δF ∂φi 0 0 + L ik = f i (t). + β −1 −Q ik ∂t δφk

(5.3.56)

The above equation gives the linear Langevin equations for the dynamics of a chosen set of variables φi with a part being controlled by the noise denoted as f i . The functional F[φ] is the free-energy functional which determines the static correlation functions in equilibrium. Indeed, as we will see in the next chapter, the stationary distribution corresponding to the Langevin dynamics described by eqn. (5.3.56) is given by exp(−F[φ]). For constructing the Langevin equations one needs this free-energy functional expressed in terms of the coarse-grained densities φ. The construction of F is done mainly by maintaining consistency with the static description of the system. We now examine the relevant Langevin equations for the set of collective modes {φi } for the two typical cases of the isotropic fluid and the crystalline solid. Identification of slow modes Our discussion of the dynamics of fluids so far has been based on the identification of a set of modes {φˆ i } in the many-particle system. The characteristic time scales for these modes are much longer than those for the rest of the many other dynamic variables for the system. The latter are identified as noise { f i } and assumed orthogonal to the set of slow modes {φˆ i }, which is ensured with the orthogonality condition, see eqn. (5.3.15). We will now examine the physical situations in which such slow modes occur. The existence of slow modes in the fluid can be inferred from the nature of the eigenmodes of decay. Equivalently, the corresponding dispersion relations obtained from the pole structure of the dynamic correlation functions in the frequency space are useful for understanding the nature of the asymptotic decay. To explain this, let us consider here a single collective mode ˆ The Laplace transform of the normalized equilibrium correlation function denoted by φ.

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235

C(q, t) of φˆ at two different times corresponding to wave vector q is obtained from eqn. (5.3.23) as C(q, z) =

1 z + M(q, z)Sq−1

,

(5.3.57)

where the memory function M(q, z) is obtained in the following form using eqn. (5.3.30): ˆ φ ˆ + {Lφ}R(z){L ˆ ˆ M(q, z) = φL φ} ˆ ˆ ˆ −1 (z)φˆ R(z){Lφ}. − {Lφ}R(z) φC ˆˆ φφ

(5.3.58)

ˆ The corresponding pole of C(q, z) Sq is the equal time correlation function of the field φ. is given by z = −M(q, z)Sq−1

(5.3.59)

and represents a slow mode if the RHS → 0 in the small-q limit. This ensures that the correlation of φˆ over long distances decays at long times. Such a situation can result in one of the following two ways. (a) Conservation laws. Let φˆ be a microscopically conserved density evolving with Liouville operator L. It must satisfy the microscopic-balance equation ∂ φˆ ∂ φˆ + Lφˆ = + ∇ · jφˆ = 0, ∂t ∂t

(5.3.60)

ˆ Therefore the Fourier transform Lφˆ is of where jφˆ is a current corresponding to φ. O(q). From eqn. (5.3.58) it follows that the first term on the RHS is of O(q) while the second and third terms are of O(q 2 ). Thus M(q, z) → 0 as q → 0 while Sq remains finite in the same limit. Thus the dispersion relation in the spectrum for the correlation function is z → 0 as q → 0, implying that the fluctuations over long distances decay in the long-time limit. The density of the microscopically conserved property is a slow mode. For example, in an isotropic fluid the conservation laws of mass, momentum, and energy give rise to a set of hydrodynamic modes involving the corresponding microscopic densities. Similarly, spin diffusion in a paramagnet with conserved magnetization is a slow mode, and so on. (b) Long-range order. φˆ is a slow variable if it corresponds to a thermodynamic state with long-range order, i.e., its static correlation Sq diverges as q −2 for small q. This applies to a dynamic variable irrespective of whether or not it represents a conserved property. In this case, since the continuity equation of (5.3.60) no longer holds in general, we have lim M(q, z)  = 0.

q→0

(5.3.61)

However, since Sq−1 ∼ q 2 , in the hydrodynamic limit we still can satisfy similar dispersion relations as in case (a) above, i.e., z → 0 as q → 0. Hence φˆ represents

236

Dynamics of collective modes

a slow mode. Therefore diverging static correlation in the small-wave-number limit results in new slow modes in addition to the usual conservation laws. This behavior of the static correlation functions represents long-range correlations in the system and is associated with the breaking of a continuous symmetry. A classic example of this is the elastic-displacement degrees of freedom in a solid, which constitute the transverse sound modes. These are absent in the isotropic liquid, which has only longitudinal sound modes. This result is referred to as the Goldstone theorem (Goldstone, 1961; Nambu, 1960a, 1960b) and will be discussed in the next section. To summarize, in an isotropic liquid consisting of similar particles, the slow modes are the conserved densities of mass, momentum, and energy. In the present section we will consider these hydrodynamic modes for the liquid state and their plausible extensions to short wavelengths. In the next section we consider the case of the solid, which in addition to the conserved modes also has the Nambu–Goldstone modes due to long-range order in the crystalline or even in the glassy state.

5.3.2 The liquid-state dynamics The isotropic liquid provides the simplest set of slow modes in terms of the conserved densities of mass, momentum, and density {ρ, g, }. In the present analysis we make the further simplification of dealing with only the momentum and mass density fluctuations. The dynamical equations for the fluctuating variables for the isotropic fluid are obtained with the standard recipe outlined in Section 5.3.1. The description of the linearized dynamics in terms of a set of Langevin equations (Kim and Mazenko, 1990) is given by eqn. (5.3.56). The construction of these linearized equations requires the kernels Q i0j and L i0j , respectively, for the reversible and dissipative parts of the equations of motion for the momentum density. The collective densities are defined in terms of the phase-space variables {rα , pα } as  ρ(x, ˆ t) = mδ(x − rα ), (5.3.62) α

gˆ i (x, t) =

 α

pαi δ(x − rα ).

(5.3.63)

The Poisson brackets Q i j are computed from the corresponding definitions (5.3.62) of the respective slow modes in terms of the phase-space variables. Using the canonical Poissonbracket relation (Landau and Lifshitz, 1975a), # $ j rαi , pβ = δi j δαβ , (5.3.64) we obtain the following results for the Poisson brackets between hydrodynamic variables: Q ρρ ≡ {ρ(x), ˆ ρ(x ˆ  )} = 0, 

Q gi g j ≡ {gˆi (x), gˆj (x )}

(5.3.65)

5.3 Linear fluctuating hydrodynamics

= −∇x [δ(x − x )gˆi (x)] + ∇xi  [δ(x − x )gˆj (x )], j

Q ρg j ≡ {ρ(x), ˆ gˆi (x )} = −∇xi [δ(x − x )ρ(x)]. ˆ

237

(5.3.66) (5.3.67)

Using the above values of the Poisson brackets, we obtain for their equilibrium averages Q 0ρρ = 0,

(5.3.68)

Q 0gi g j = 0,

(5.3.69)

Q 0ρg j = −∇xi [δ(x − x )]ρ0 .

(5.3.70)

where the average values of ρ and gi are taken as ρ0 and 0, respectively, for the isotropic liquid. The dissipative coefficients L i0j are chosen so as to match the dissipative parts of the equations of motion for the momentum densities gi to those in the corresponding equations of macroscopic hydrodynamics discussed in Chapter 1 (see eqn. (5.1.36)). Upon identifying in the linear case the velocity field as vi = gi /ρ0 , we obtain   1 2 L 0gi g j ≡ L i0j = −η0 ∇ + δ ∇ (5.3.71) ∇i j − ζ0 ∇i ∇ j , ij 3 where ζ0 is the bare bulk viscosity and η0 is the bare shear viscosity. The longitudinal viscosity is defined as 0 = ζ0 + 4η0 /3. The other elements of the dissipative matrix L0 are chosen as L 0ρgi = L 0ρρ = 0,

(5.3.72)

a choice that is necessary in order to keep the form of the continuity equation unchanged. The other necessary ingredient for the construction of the equations of motion is the driving free-energy functional F expressed as a functional of the slow variables {ρ, g}. Let FK be the part dependent on the momentum density g and let the potential part FU be treated as a functional of the density ρ. We write the free energy in the following schematic form: 4 3 2  β −1 g (x) + f [ρ(x)] . (5.3.73) dx F = FK + FU = 2 ρ0 The kinetic or the momentum-density-dependent part FK in the first term on the RHS above gives the equilibrium correlation functions for the momentum density as gi (q)g j (q) = ρ0 kB T δi j ,

(5.3.74)

so that the average kinetic energy per particle is 12 kB T in accordance with the equipartition law (Huang, 1987). The density-dependent potential part FU is written here in terms of the

238

Dynamics of collective modes

functional f [ρ], which is the free-energy density. In the Gaussian approximation to the free energy, the simplest choice is χ −1 2 (5.3.75) δρ (x) 2 so as to correspond to a wave-vector-independent static structure factor or density correlation function χ . We adopt this approximation in order to keep the treatment simple. This is easily extended to the wave-vector-dependent structure factors by expressing the free energy in terms of direct correlation functions (Hansen and McDonald, 1986). f [ρ] ≡

Equations of linearized dynamics Using the values of the kernels Q i0j and L i0j discussed above as well as the Gaussian free-energy functional F[ρ, g], we obtain the linear equations of the fluctuating hydrodynamics. The equation of motion for the density remains the continuity equation even for the fluctuating variables, ∂ρ + ∇ · g = 0. (5.3.76) ∂t The linear equations for the components of the momentum density g are obtained in a generalized form of the Navier–Stokes equation:  gj ∂gi ∂f L i0j + θi . (5.3.77) = −ρ0 ∇i − ∂t ∂ρ ρ0 j

The first term on the RHS of eqn. (5.3.77) takes the form ∇i P[ρ], which is similar to the macro-hydrodynamic equation in terms of the pressure functional P with the identification   ∂f P = ρ0 . (5.3.78) ∂ρ For the choice (5.3.75) for the Gaussian free energy the pressure reduces to the form P = ρ0 χ −1 δρ. Hence the speed of sound c0 in this case is obtained as c02 = (∂ P/∂ρ) = ρ0 χ −1 . In the next chapter we will demonstrate that with the formulation of the nonlinear fluctuating hydrodynamics the relation (5.3.78) reduces to the usual thermodynamic relation   ∂F . (5.3.79) P=− ∂V T The stochastic θi term in the generalized Langevin equation (5.3.77) is assumed to represent Gaussian noise. The correlation of the noise is related to the bare damping matrix L i0j through the following fluctuation–dissipation relation: θi (r, t)θ j (r , t  ) = 2kB T L i0j δ(r − r )δ(t − t  ).

(5.3.80)

Equations (5.3.76) and (5.3.77) form the basic set of linear fluctuating hydrodynamic equations for the set {ρ, g}.

5.3 Linear fluctuating hydrodynamics

239

Correlation functions We construct the equations of motion for the correlation functions for the slow variables Ci j for the set φˆ i ≡ {ρ, g} using eqn. (5.3.23). The matrix Mi j is obtained in (5.3.18) as a (s) (d) sum of the static and dynamic parts, denoted by Ki j and Ki j , respectively. These kernels are obtained in terms of the average Poisson brackets Q i0j and the bare transport coefficients L i0j . The matrix of the equal-time or static correlation functions is diagonal with Sρρ = β −1 χ , Sgi g j = β

−1

ρ0 δi j .

(5.3.81) (5.3.82)

The static part is given by eqns. (5.3.33) and (5.3.41) and in the present case the only nonzero components are (s) (q, z) = −q j , Kρg j

2 K(s) gi ρ (q) = −qi c0 ,

(5.3.83)

where the speed of sound is given by c02 = ρ0 χ −1 . The dynamic part Ki(d) j is obtained by comparing (5.3.48) with (5.3.80) and the nonzero components are given by 0 K(d) gi g j (q, z) = −i L i j .

(5.3.84)

Using the above expressions for Ki j , the correlation-function matrix Ci j is solved from eqn. (5.3.23). In equilibrium time translational invariance holds and hence the correlation Ci j (t, t  ) of fluctuations at two different times t and t  depends only on the difference (t − t  ) of the two times. Similarly, using the spatial translational invariance in equilibrium, the Fourier transform of the correlation functions depends on a single wave vector q. The analysis of the correlation functions is particularly facilitated by using the isotropic symmetry of the normal liquid state. For the isotropic system the Fourier transform of the correlation function Ci j (q, t) is a function only of the magnitude of the wave vector q and the vector field g is split into components (a) gl parallel (longitudinal) and (b) gT perpendicular to the wave vector q. Functions involving the vector indices for Cartesian components are expressed in terms of longitudinal and transverse parts defined as Cρgi (q, ω) = qˆi Cρgl (q, ω), L T C gi g j (q, ω) = qˆi qˆ j C gg (q, ω) + (δi j − qˆi qˆ j )C gg (q, ω).

(5.3.85) (5.3.86)

Thus the density field couples to the longitudinal component gl . The matrix of the memory function Ki j (q) splits into a longitudinal and a transverse part decoupled from each other. The computation is straightforward using the relations (5.3.41) and (5.3.50), respectively, for the static and dynamic parts of Ki j . Equation (5.3.23) reduces to the following explicit form: 



G ρρ G Lρg 0 z q −1 χ . (5.3.87) = −β 0 ρ0 qc02 (q) z + i0 q 2 G Lgρ G Lgg

240

Dynamics of collective modes

We obtain the Laplace transform of the density correlation function as   z + i0 q 2 −1 G ρρ (q, z) = −β χ z 2 − c02 q 2 + i z0 q 2

(5.3.88)

with 0 = DL /ρ0 , where DL = ζ0 + 4η0 /3 is the longitudinal viscosity. The frequency (z)-dependent part within square brackets represents the decay behavior of the density fluctuations. This expression has two poles (up to leading order in q) at i z = ±qc0 + 0 q 2 , 2

(5.3.89)

representing the two decay modes of the density fluctuation. These correspond to the propagating sound modes in the isotropic fluid at speed c0 in directions ±q. 0 q 2 represents the attenuation of the corresponding sound waves. For later reference we will write the Laplace transform (for its definition, see eqn. (5.3.21)) of the density correlation function ψ(q, z) normalized with respect to its equal-time value in the form of a continued fraction,  ψ(q, z) = z −

q2

−1

z + iq 2 0

,

(5.3.90)

with q2 = c02 q 2 being the microscopic frequency of the liquid state. The transverse part of the current (gi ) correlation functions, on the other hand, is obtained as G Tgg (q, z) =

β −1 ρ0 , z + iq 2 ν0

(5.3.91)

where ν0 = η/ρ0 is the kinetic shear viscosity. The corresponding pole z = −iq 2 ν0

(5.3.92)

represents a diffusion process for the transverse current fluctuations. In three dimensions there are two diffusive shear modes for the two directions perpendicular to q. The transversecurrent correlation φT (q, z) normalized with respect to its equal-time value is expressed in the form of a simple diffusive pole, φ T (q, z) =

1 . z + iq 2 ν0

(5.3.93)

These expressions (5.3.88) and (5.3.91) are respectively identical to (5.2.4) and (5.2.7) for the density and current correlations obtained in Chapter 1, and both correspond to exponential decay of correlations. ˜ The Fourier transform C(q, ω) of the density correlation function is proportional to the intensity measured in scattering experiments. In Fig. 5.2 results from light-scattering and neutron-scattering experiments on normal liquids are shown in a schematic form. Figure 5.2(a) displays schematically the nature of this dependence at a fixed wave number q.

5.3 Linear fluctuating hydrodynamics

241

Fig. 5.2 (a) A schematic diagram showing the Fourier transform S(k, ω) of the density–density correlation function in a liquid. The two symmetrically placed “Brillouin peaks” at ω = ±cq are of Lorentzian form with half-width ∝ k 2 and are caused by propagating sound waves in the isotropic liquid. The central “Rayleigh peak” is of width ∝ k 2 and represents heat diffusion. Note that this heat mode has been ignored in the formulation of expression (5.3.87). Reproduced from Fleury and Boon (1969, 1973). (b) Brillouin peaks in liquid neon. Left panel: neutron-scattering data for T = 70 K, ρ = 0.48 g cm−3 shown by open circles. The dashed line is a fit with a hydrodynamic model similar to what is described in the text but including the heat mode. Right panel: light scattering data at T = 25 K and normal liquid density from Fleury and Boon (1969). The intensities of the scattering are shown in arbitrary units. The κ denotes corresponding wave-vector values in inverse Ångström c American Physical Society. units (Å−1 ). From Bell et al. (1975). Both parts 

The correspondence to experimental results is displayed in Fig. 5.2(b). The two Lorentzian peaks at ω = ±cq with half-width 0 q 2 correspond to the sound poles and are termed Brillouin peaks. The shear diffusion mode appears as a central diffusive peak of width ν0 q 2 . Note that we have considered only four of the hydrodynamic modes, out of the total of five modes present in the isotropic fluid, and ignored the energy conservation for simplicity. The latter gives rise to a diffusive heat mode (Forster, 1975) in the liquid and would have added a central diffusive peak, termed the Raleigh peak (as shown in Fig. 5.2(a)), of width proportional to the thermal conductivity of the liquid. The result for the maximum frequency ωm (k) of the longitudinal-current correlation function seen in MD simulations for various wave numbers k is shown in Fig. 5.3. The fluctuating-hydrodynamics description is

242

Dynamics of collective modes

Fig. 5.3 A plot of the maximum frequency ωm (k) of the longitudinal-current correlation function seen in MD simulations by Rahman (1968) of hard spheres vs. the wave vector in inverse Ångström units (Å−1 ). The solid line is a fit with an empirical model with adjustable fitting parameters. c American Physical Society. Reproduced from Akcasu and Daniels (1970). 

easily generalized to more complex systems, such as multi-component fluids (Cohen et al., 1971; March and Tosi, 1976) and molecular liquids (Gray and Gubbins, 1984). The generalized hydrodynamics In the discussion of hydrodynamic modes above we have ignored in the theoretical treatment for simplification any wave-vector dependence of the bare-transport-coefficient matrix L i0j as well as the structure factor χ of the liquid. In reality, however, the structure factor has a strong wave-vector dependence with a sharp peak at the wave number corresponding to the hard core of the particle interactions (see Chapter 1). The bare transport coefficients are also dependent on the corresponding length scales over which transport is being considered. The concept of hydrodynamic modes can be extended to short distances by considering the hydrodynamic equations at wave vectors that are large – comparable to that of the first peak in the static structure factor. Although the assumption of slow variations used in obtaining the hydrodynamic equations is not valid at short length and time scales, it is plausibly assumed that the deviations occur in a subtle and gradual manner. Thus, while retaining the basic structure of the equations, the hydrodynamic description can be extended by replacing the relevant thermodynamic properties or transport coefficients by functions that can vary in space or in both space and time (Chung and Yip, 1969; Akcasu and Daniels, 1970; Ailawadi et al., 1971). The resulting theory, which describes the dynamic behavior of the fluid over a wider range of spatial and temporal variations, is termed generalized hydrodynamics (Kadanoff and Martin, 1963; Boon and Yip, 1991). In applying this continuum hydrodynamic description to liquids, it is useful to note that

5.3 Linear fluctuating hydrodynamics

243

Fig. 5.4 A plot of kσ vs. the density nσ 3 for the hard-sphere system corresponding to kl0 = 0.5, where l0 is the corresponding mean free path. Hydrodynamics applies below this line. The positions of the structure-factor peaks are also shown with open circles. Reproduced from Lutsko et al. (1989). c American Physical Society. 

the length scale of the fluctuations does become comparable to the coarse-graining length involved in such a picture. However, with increasing density of the liquid the mean free path becomes smaller than the particle size and the validity of the continuum hydrodynamic treatment improves. In Fig. 5.4 we display for a hard-sphere system a comparison at various liquid densities between the structure-factor peak km and the k at which hydrodynamics is valid (i.e., kl0 < 1, where l0 is the mean free path at the corresponding density). For a dense liquid the two wave numbers are comparable. We discuss below the generalization of the hydrodynamics along these lines specifically for the case of a hard-sphere fluid. The nature of the dynamics of the fluctuations of the conserved densities at short wavelengths is obtained by making an extension of the conventional hydrodynamics to short length scales. A hard-sphere system provides the simplest example. We consider here the complete set of variables consisting of {ρ, g, T }, respectively denoting the density, momentum, and kinetic energy. There is no potential energy in this case of hard spheres. From the matrix eqn. (5.3.23) correlation functions are computed in the crucial short-time approximation. In this limit the contribution from the dynamic part K(d) of the memory function K is set equal to zero. For the hard-sphere system with instantaneous collisions this is still a reasonable approximation for studying the dynamics up to even long times. The treatment is considered at the level of uncorrelated binary collisions, which is a reasonable approximation at densities close to freezing. The static part of the memory function (d) Ki j is computed from (5.3.31) involving the Liouville operator L. For the special case

244

Dynamics of collective modes

of discontinuous hard-sphere potentials, the time dependence of the dynamic variable A(t) = ei L± t A is generated by operators L+ and L− for t > 0 and t < 0, respectively. L± respectively involve the corresponding hard-sphere collision operators T± (Résibois and de Leener, 1977). The operators L± are identical for fluids with continuous interaction potentials. The matrix elements K(s) are computed using the explicit forms of the collision operator. The equal-time correlation-function matrix Si j in this case is diagonal with the nonzero elements Sρρ = m 2 nS(x), 3n See = , 2β 2 mn , Sgl gl = Sgt gt = β

(5.3.94) (5.3.95) (5.3.96)

where we denote the wave number q by a dimensionless variable x = qσ involving the hard-sphere diameter σ . S(x) is the static structure factor for the hard-sphere liquid. A straightforward calculation gives for the static part, K(s) , ⎤ ⎡ 0 x 0 (5.3.97) K(s) = ⎣ xc02 (x) i0 (x) xξ(x) ⎦ , 0 −xξ(x) i DH (x) where c02 = 1/(βm S(x)) is now the wave-vector-dependent speed of sound. The various wave-vector-dependent functions are given by j1 (x) , x

(5.3.98)

DH (x) = ν0 [1 − j0 (x)],

(5.3.99)

ξ(x) = 1 + 3y

0 (x) = ν0 [1 − j0 (x) + 2 j2 (x)],

(5.3.100)

with jl denoting the spherical Bessel function of order l. DH (x) and 0 (x) are, respectively, the generalized heat-diffusion constant and sound attenuation. ν0 represents a microscopic frequency ν0 = 2/(3tE ), tE being the Enskog time between collisions (Résibois and de Leener, 1977) of the hard spheres, √ π σ , (5.3.101) tE = 6Y v0 where v0 = 1/(βm) is the thermal velocity of the fluid particles and Y = 4π ϕg(σ ) expressed in terms of the hard-sphere packing fraction ϕ = π nσ 3 /6. This is the same as the time defined in eqn. (1.3.41). The density–density correlation function F(q, t) (normalized with respect to its equal-time value) at large q is obtained in terms of the three exponentially relaxing modes, i.e., F(q, t) =

3  μ=1

Aμ (x)e−z μ (x)t ,

(5.3.102)

5.3 Linear fluctuating hydrodynamics

245

Fig. 5.5 The real part of the eigenvalues −z μ (k) (μ = 1, . . . , 5) vs. k for the hard-sphere fluid at packing fraction ϕ = π nσ 3 /6 = 0.471. The curves correspond to the pair of complex-conjugate sound modes (dotted), the heat mode (solid), and two degenerate shear modes (dashed). Reproduced c American Physical Society. from Das and Dufty (1992). 

where z μ (x) are the three eigenvalues of the matrix (5.3.97). These z μ (μ = 1, 2) in the small-q limit reduce to the propagating modes described above in eqn. (5.3.89). The wavenumber-dependent functions DH (x), 0 (x), and η0 (x) play the role of generalized transport coefficients representing dissipative effects over short length scales. These generalized transport coefficients in the small-wave-number limit reduce to the corresponding results obtained from the Enskog theory (Résibois and de Leener, 1977) for a hard-sphere system in which only the collision contribution has been kept (Kirkpatrick, 1985b). Similarly, the eigenvalues for the shear modes are z = −η0 (q), where ! " (5.3.103) η0 (q) = ν0 1 − j0 (x) − j2 (x) is the generalized shear viscosity. Figure 5.5 shows the real part of the eigenvalues −z μ (q) (μ = 1, . . . , 5) extended to large q, for the hard-sphere system at packing fraction ϕ = 0.471. The Percus–Yevick solution (Ashcroft and Lekner, 1966) with Verlet–Weiss correction (Verlet and Weiss, 1972; Henderson and Grundke, 1975) for the structure factor S(q) has been used in computing the matrix elements of (5.3.97). The propagating pair of sound modes (shown by the dotted line) crosses over to a pair of real modes for an intermediate wave-number range. The most interesting feature is the behavior of the heat mode (solid line), which close to the freezing point becomes soft near the peak of the structure factor of the liquid. This is a consequence of S −1 (q) being small there. Analysis of the amplitude factors of the eigenmodes of the matrix K(s) at this wave number indicates that the soft mode essentially represents density fluctuations. At high density the softening is pronounced and it corresponds to very slow dynamics near the diffraction peak (de Gennes, 1959) and is generally termed de Gennes narrowing in liquid-state physics. The two shear

246

Dynamics of collective modes

modes (dashed lines) are identical in the isotropic liquid. From the continuity equation it follows directly that the longitudinal-current correlation function is simply related to the density correlation function. The result for the maximum frequency ωm (k) of the longitudinal-current correlation function seen in MD simulations for various wave numbers k is shown in Fig. 5.3. For small wave numbers simple propagating sound waves are obtained and near the structure-factor peak there is strong slowing down as predicted from the theory. The nature of the fluctuations of the conserved densities at short length scales as obtained from the generalized hydrodynamic approach described above closely resembles that of a microscopic kinetic-theory approach (de Schepper and Cohen; 1980, 1982). 5.4 Hydrodynamics of a solid The fluctuating-hydrodynamics description of the crystal is obtained with the equations of motion for an extended set of slow modes consisting of the usual hydrodynamic modes due to the microscopic conservation laws (similar to those for an isotropic liquid) as well as the new modes which occur due to long-range order in the system. The latter are referred to as the Nambu–Goldstone modes. In the context of the freezing of a liquid into a crystal these are the transverse sound modes characteristic of the solid state and occur as a consequence of the breaking of the isotropic symmetry of the liquid state. Breaking of a microscopic symmetry and having long-range order makes the static correlation soft (i.e., diverging as q −2 ) and gives rise to the slow modes as described by the Goldstone theorem. We discuss this theorem below, starting from general considerations. This also provides a suitable recipe for defining the additional slow modes which occur due to symmetry breaking. We apply it to the case of the crystal in order to define the Goldstone modes for the crystal. The Goldstone theorem The Goldstone theorem in relativistic quantum field theory refers to the existence of a massless boson particle corresponding to the breaking of a continuous symmetry of the Lagrangian. We discuss this theorem here in the context of breaking of the continuous symmetry of a many-particle system. The operator Qˆ which generates the continuous symmetry is conserved, i.e., it commutes with the Hamiltonian operator Hˆ . In the classical description this will imply that the Poisson bracket of the conserved quantity Q with the Hamiltonian H vanishes, {Q, H } = 0. The density a(x) corresponding to A is defined as  Q = dx a(x),

(5.4.1)

(5.4.2)

and satisfies the continuity equation ∂a + ∇ · ja = 0, ∂t

(5.4.3)

5.4 Hydrodynamics of a solid

247

with the corresponding current being ja . In the Fourier-transformed form eqn. (5.4.3) takes the form a(q, ˙ t) + q jaL (q, t) = 0,

(5.4.4)

where jaL is the longitudinal component of the current ja (q) in the direction of q. Now, let us assume that there exist two fields ψ(x) and ϕ(x) ˜ such that the Poisson bracket of the former with Q is proportional to the latter, {ϕ(x), ˜ Q} = ξ0 ψ(x), where ξ0 is a constant. The equilibrium average of ψ(x) is given by  ψ(x) = d ψ(x)eq (),

(5.4.5)

(5.4.6)

where eq () is the equilibrium probability distribution for the phase-space element d. It is straightforward to prove that   ˜ Q}eq () d ψ(x)eq () = ξ0−1 d{ϕ(x), =

ξ0−1

 d ϕ(x){ ˜ eq (), Q},

(5.4.7)

where we have used eqn. (5.4.5) in writing the first equality. For the second equality we have used the general properties of the phase-space average, similarly to what was done in the mathematical manipulations involved in reaching eqns. (5.3.34)–(5.3.39) in Section 5.3.1. If the continuous symmetry generated by Q is maintained thermodynamically, then from eqn. (5.4.1) it follows that {eq (), Q} = 0

(5.4.8)

and hence, using the relation (5.4.7), we obtain that ψ(x) = 0.

(5.4.9)

On the other hand, when the continuous symmetry generated by Q is broken, the Poisson bracket on the LHS of eqn. (5.4.8) is nonzero and hence ψ(x) = ψ0  = 0.

(5.4.10)

Therefore, with the triad of fields {ϕ, ˜ ψ, a} satisfying eqn. (5.4.5), we can identify an order parameter ψ(x) to describe the broken symmetric state with long-range order. With the breaking of the continuous symmetry generated by Q the system passes into a phase that is characterized by the order parameter ψ0 . We now prove the important result that in the broken-symmetric state the Fourier transform of the static susceptibility corresponds to ϕ(x). ˜ Let ϕq be the spatial Fourier transform of ϕ(x), ˜ so that Sϕ (q) ≡ |ϕq |2 

(5.4.11)

248

Dynamics of collective modes

diverges as q −2 for small q. In real space this signifies long-range order in ϕ(x) ˜ correlation, i.e., the equilibrium averaged time correlation of ϕ(x) ˜ over long distances decays over long times. In order to establish the diverging of the correlation Sϕ (q) in the small-q limit, we make use of the Schwarz inequality (Dennery and Krzwicki, 1967) S A A (q)S B B (q) ≥ |S AB (q)|2 ,

(5.4.12)

where S A A , S AB , and S B B are the static correlations between two variables A(q) and B(q), with S A A (q) = A(q)B(q) etc. In particular, we make the choice A(q) = ϕq and B(q) = a(q). ˙ The diagonal elements of the static correlation matrix are now obtained as S A A = Sϕ (q),

(5.4.13)

S B B = q 2 SL (q).

(5.4.14) +* * , 2 The susceptibility SL (q) = * jaL * , which follows from the continuity eqn. (5.4.4). Since B(q) = a(q, ˙ t) = −{H, a(q, t)}

(5.4.15)

the susceptibility S AB is obtained as S AB = −ϕq a(−q) ˙ = ϕq {H, a(−q, t)}.

(5.4.16)

We use the following result involving the Poisson bracket (the proof is given in the appendix to this chapter): S AB = ϕq {H, a(−q)} = β −1 {ϕq , a(−q)}.

(5.4.17)

Since we are interested in the small-q limit, the equilibrium average on the RHS of (5.4.17) is obtained as  −1 dx dy{ϕ(x), ˜ a(y)} lim {ϕq , a(−q)} = V q→0

= {ϕ(x), ˜ Q},

(5.4.18)

assuming that the equilibrium average {ϕ(x), ˜ Q} is independent of x. Using the condition (5.4.5), the average of the Poisson bracket is expressed in terms of the longitudinal part of the order parameter, {ϕ(x), ˜ Q} = ξ0 ψ(x) ≡ ξ0 ψ0

(5.4.19)

(say). Now, using the Schwarz inequality (5.4.12), we obtain lim Sϕ (q) ≥

q→0

κ , q2

(5.4.20)

where κ = (kB T ξ0 ψ0 )2 /SL (0). In the absence of any long-range force in the system, SL (0) remains bounded.

5.4 Hydrodynamics of a solid

249

Goldstone modes in a crystal The order-parameter field for the crystalline state with inhomogeneous density ρ(x) is constructed to be proportional to δρ(x) = ρ(x) − ρ0 . The order-parameter field corresponding to the RLV K is defined as ψK (x) = [ρ(x) − ρ0 ] cos(K · x),

(5.4.21)

where ρ0 is the equilibrium density in the homogeneous liquid state. It is obvious that the above definition ensures that the crystal symmetry is maintained in the order parameter ψK (x) = ψK (x+R), where R is the lattice vector for the crystal. The continuous symmetry which is broken when the liquid freezes into a crystal is the translational invariance. The generator of the symmetry in this case is the total momentum Q ≡ P and the corresponding density a(x) ≡ g(x),  P(x) = dx g(x). (5.4.22) Now, we need to find the third field ϕ˜K (x), which forms a triad with ψK (x) and g(x) satisfying the commutation relation (5.4.5). We define ϕ˜K (x) ≡ u K (x) = ρ(x)

sin(K · x) K

(5.4.23)

and consider the Poisson bracket sin(K · x) (5.4.24) {ρ(x), Pi }. K We now adopt the definitions (5.1.1) and (5.1.2) for the density and the momentum density. Using the fundamental Poisson bracket (Landau and Lifshitz, 1975a) it is straightforward to obtain  (5.4.25) {ρ(x), Pi } = dx {ρ(x), gi (x)} = −∇xi ρ(x). {u K (x), Pi } =

Using eqn. (5.4.25) in the result (5.4.24) gives {u K (x), Pi } = −

sin(K · x) i ∇x ρ(x) K

i sin(K · x) ˆ = K i cos(K · x)[ρ(x) − ρ0 ] − ∇x δρ(x) K

i sin(K · x) ˆ δρ(x) . = K i ψK (x) − ∇x K

(5.4.26)

Taking the Fourier transform of eqn. (5.4.26), we write to leading order in the wave vector q the Poisson-bracket relation similar to the form (5.4.5), {u K (q), Pi } = Kˆ i ψK (q).

(5.4.27)

We can therefore interpret ψK (q) as the order-parameter field for the crystal and u K (q) as the new slow variable representing the Nambu–Goldstone modes. We attach the αth

250

Dynamics of collective modes

(α = 1, . . . , N ) particle to a parent lattice denoted by R0α displaced by a distance uα (t) from it such that rα (t) = R0α + uα (t).

(5.4.28)

Then, using eqn. (5.4.28) in the expression (5.4.23), we obtain     sin K · R0α + uα δ(x − rα (t)) u K (x) = K α

)  sin(K · uα ) ( δ x − R0α uα (t) K α ( )  uα (t)δ x − R0α (t) + O(u 3 ), = Kˆ · =

(5.4.29)

α

  where we have used sin K · R0α = 0 in reaching the last equality.  0 If the Rα represent aregular  crystalline lattice and K the corresponding RLVs, this 0 holds by definition. If the Rα represent a random structure for an amorphous solid, this holds only locally for large K. Assuming that the displacements u α (t) are small, the above definition for u K (x) reduces to the longitudinal component of the so-called displacement vector u(x, t), which is defined in terms of the displacement of the particles around their respective parent sites as follows: ( )  n 0 u(x, t) = uα (t)δ x − R0α , (5.4.30) α

so that u K (x) = n 0 Kˆ · u (x). Here ρ0 = mn 0 is the average density with n 0 = N /V being the average number of particles per unit volume. The extra slow mode for the crystal is therefore the displacement field u(x, t). If we ignore the presence of vacancies in the crystal, we can attach every particle (α) to its corresponding parent lattice site at R0α and its location is then described in terms of its displacement uα from the parent site. With this assumption of zero vacancy density, the number of lattice sites is equal to that of the particles. We write the inhomogeneous density function ρ(x, t) in the form ρ(x, t) =

N 

mδ(x − rα (t)) =

α=1

=

N  α=1

N  α=1

( ) mδ x − R0α − uα (t)

( )  # ( )$ mδ x − R0α + muα (t) · ∇R δ x − R0α

= ρ0 (x) − ρ0 ∇ · u(x).

α

(5.4.31)

Therefore the density fluctuation at a given point in the solid is simply the divergence of the displacement field, i.e., δρ ≡ −ρ0 ∇ · u(x). For a crystal in which defects are present, this relation no longer holds. The difference of these two quantities has been often treated

5.4 Hydrodynamics of a solid

251

(Cohen et al., 1976; Fleming and Cohen, 1976) as a measure for the defect density ρD (x, t) in the solid, ρD (x, t) = δρ + ρ0 ∇ · u(x).

(5.4.32)

In the next section we present the deduction of a set of nonlinear fluctuating hydrodynamic equations for the extended set of slow modes {ρ, g, u}. We drop the energy density from the set to keep the discussion simple. Linear dynamics A unified unified hydrodynamic approach to many-particle systems was developed (Martin et al., 1972; Cohen et al., 1976; Fleming and Cohen, 1976) to describe the linear transport in a crystal as well as in an amorphous solid. For a crystal the number of slow modes is extended to take into account the corresponding Nambu–Goldstone modes due to the breaking of the isotropic symmetry. As was pointed out earlier, the extra slow modes in the solid are the displacement variables u around a corresponding set of points forming the rigid lattice structure. The latter in the case of the crystal is a structure with longrange order representing various space groups. Thus, in a cubic crystal the total number of slow modes goes up to eight, which is the combination of the five conservation laws of an isotropic fluid and the three displacement variables u. The resulting formulation of the hydrodynamics of the cubic crystal gives rise to transverse and longitudinal sound modes (three each) as well as the energy diffusion and the very slow diffusion of vacancies in the crystal. For an amorphous solid the lattice structure mentioned above is a random set of points without any long-range order. An approach similar to that for the crystal is also adopted here, albeit with some obvious limitations. This description of the amorphous solid holds over time scales shorter than that for structural relaxation. The amorphous solid is treated as being similar to the crystal over length scales related to local structures, although the frozen state is isotropic over long distances. This approximation is further supported by the fact that in an experiment transverse sound modes similar to those in a crystal are observed in the glass. The hydrodynamic description of the glassy state is formulated in terms of a displacement field u(x) about the local metastable positions of the atoms, which remain unaltered for a long time in the glassy state. The dynamical equations for the fluctuating variables are obtained with the standard recipe outlined in Section 5.3.1. We discuss the equations of linearized dynamics here in terms of a set of Langevin equations obtained in the form (5.3.56) for a given set of collective modes {φˆ i } ≡ {ρ, g, u}. The construction of the dynamical equations has three main ingredients. I. The Gaussian free-energy functional In order to compute the nonlinear fluctuating hydrodynamic equations we need the explicit form of the effective Hamiltonian Fs for the solid. In addition to the usual terms that appear in the free-energy functional for an isotropic liquid, we include

252

Dynamics of collective modes

the energy cost due to distortion in the elastic solid. The free-energy functional is expressed as a sum of two contributions, Fs = F + Fel ,

(5.4.33)

where F is the free energy of the isotropic liquid. The elastic part Fel of the free energy of the solid state (Mazenko, 2002a, 2002b) is written as  1 Fel = (5.4.34) dx Ci jkl u i j (x)u kl (x), 2 where Ci jkl is the fourth-rank elastic tensor and u i j is the strain tensor expressed in terms of the symmetrized derivative of the displacement field u, " 1! ui j = (5.4.35) ∇i u j + ∇ j u i . 2 Fel depends only on the derivative of the u field. A simple translation of the system does not cost any extra free energy. The free-energy functional is kept Gaussian (quadratic) in the hydrodynamic fields {ρ, g, u}. Coupling of the density fluctuations to the displacement field u is being ignored here and will be considered later in the context of nonlinear dynamics. In general, the rank-four matrix Ci jkl in three dimensions has 81 components. However, the number of independent components of the elastic tensor is determined by symmetry considerations. Since u i j is symmetric by construction, it follows from the expression (5.4.34) for the free energy that the elastic tensor Ci jkl remains unchanged under the interchange of i ⇔ j as well as k ⇔ l, i.e., Ci jkl = C jikl = Ci jlk . In the Voigt (Weiner, 1983) notation the pair of indices i j is treated as a single index α (say). In three dimensions α is equated to any of six different combinations, namely α ≡ {11, 22, 33, 12, 23, 31}. Furthermore, the elastic tensor remains unchanged under the transformation {i j} ⇔ {kl}, i.e., Ci jkl = Ckli j = Clki j .

(5.4.36)

Denoting Ci jkl ≡ Cαβ , the above symmetries are equivalent to representing the elastic tensor with a 6 × 6 symmetric matrix Cαβ = Cβα . Thus Ci jkl has in general 21 components. For a cubic crystal the number of independent elastic constants reduces to three, ! " (5.4.37) Ci jkl = C1 δik δ jl + δil δ jk − δi j δkl + C2 δi j δkl + C3 δi4jkl , where δ 4 is the cubic invariant tensor which is equal to 1 if i = j = k = l and zero otherwise. C1 , C2 , and C3 are related to the corresponding elastic constants defined in the Voigt notation (Landau and Lifshitz, 1975b; Weiner, 1983) as C x x x x ≡ C11 = C1 + C2 + C3 ,

(5.4.38)

C x x x y ≡ C12 = C2 − C1 ,

(5.4.39)

C x yx y ≡ C44 = C1 .

(5.4.40)

5.4 Hydrodynamics of a solid

253

For an isotropic system this reduces to two independent elastic constants. We discuss this case separately below. II. The reversible part The reversible terms in the linear Langevin equation (5.3.56) in the generalized hydrodynamic model involve the Poisson brackets Q i j = {φi , φ j } between the different slow variables of the set {φi }. To evaluate these Poisson brackets one requires the microscopic definitions for the variables φˆi (x) corresponding to fields φi (x). In the present context we are dealing with a system of N classical particles, each of mass m, and the position and momentum of the αth particle at time t are denoted by rα (t) and pα (t), respectively. For the density ρˆ and the momentum density gˆ the standard prescriptions are stated in (5.3.34). The expression for the fluctuating variable u(x, t) is obtained in terms of the displacements of the individual particles uα (t) (α = 1, . . . , N ) from their respective mean positions denoted by R0α (see eqn. (5.4.28)), ρ(x, ˆ t)uˆ i (x, t) = m

N  α=1

u iα (t)δ(x − rα (t)).

(5.4.41)

The above definition of the Poisson bracket follows from a generalization of the relation (5.4.30) (we take m = 1) so as to include the presence of vacancies. Using the canonical Poisson-bracket relation # $ j (5.4.42) rαi , pβ = δi j δαβ , we obtain the following results for the various elements of the matrix Q i j involving the field u:  Q gi u j ≡ {gˆi (x), uˆj (x )} = −δ(x − x ) δi j − ∇xi  u j (x ) , Q u i u j ≡ {uˆ i (x), uˆ j (x )} = 0, Q u i ρ ≡ {uˆ i (x), ρ(x ˆ  )} = 0.

(5.4.43)

The other elements of the Q i j matrix are obtained from eqn. (5.3.65). The equilibrium averages of the above expressions for the various Poisson brackets are the constants and are useful inputs Q i j  in eqn. (5.3.56). III. The dissipative part The dissipative part of the dynamics of the slow variable is given by the third term on the LHS of the Langevin equation (5.3.56). It involves the bare-transport-coefficient matrix L i0j . The equations of motion for the Nambu–Goldstone modes now include new elements in the bare-transport-coefficient matrix in addition to the bare viscosities for the isotropic liquid. The dissipative term in the equation for the u field is assumed to have the simple time-dependent Ginzburg–Landau (TDGL) form, L 0u i u j ≡ ϒi j = ϒ0 δi j ,

(5.4.44)

254

Dynamics of collective modes

and we require that all other L i0j = 0. The transport coefficient ϒ0 relates to the very slow vacancy-diffusion process in the solid. For the momentum density equation, the irreversible part has the standard form with the dissipative coefficient L 0gi g j ≡ L i0j . It is straightforward to link the expression L i0j δ F/δg j for the irreversible term of the Langevin equation to the generalized viscosity tensor ηi jkl . The dissipative part of the stress tensor σiDj is expressed in terms of the constitutive relation σiDj = ηi jkl ∇k vl

(5.4.45)

involving the gradient of the velocity field v thermodynamically conjugate to the momentum density g. The dissipative term of the Langevin equation (5.3.56) is expressed in terms of the viscosity tensor ηi jkl as follows: ∇ j σiDj ≡ L i0j

δF . δg j

(5.4.46)

As we have already discussed in Section 5.1.3, the fourth-rank viscosity tensor ηi jkl has the same symmetries as the elasticity tensors and hence they have the same number of independent coefficients. The g-dependent part FK of the free energy F is obtained as in eqn. (5.3.73) for the case of an isotropic liquid. Assuming a purely quadratic form of the free-energy functional, ρ in the denominator of the expression for FK is replaced by ρ0 at the lowest order. Therefore, using the relation vi = δ F/δgi = gi /ρ0 and (5.4.46), the bare matrix L i0j is obtained in terms of the viscosity tensor as L i0j = ηikl j ∇k ∇l .

(5.4.47)

The number of independent elements of the fourth-rank viscosity tensor ηi jkl for the crystal is greater than that for an isotropic liquid. The actual number depends on the symmetry of the crystal. For example, in a uniaxial crystal the number of independent viscosity coefficients is five, with the following form for ηi jkl : ηi jkl = η1 δi z δ j z δkz δlz + η2 (δik δ jl + δil δ jk ) + η3 {δi z (δ jk δlz + δ jl δkz ) + δ j z (δil δkz + δik δlz )} + η4 δi j δkl + η5 (δi j δkz δlz + δkl δi z δ j z ).

(5.4.48)

The isotropic solid has two independent coefficients and we discuss this case separately below.

5.4 Hydrodynamics of a solid

255

For the general case of a system with broken symmetry we obtain the following equations for the linearized dynamics of the slow modes: ∂ρ + ∇ · g = 0, ∂t ∂gi  ∇ j σi j = θi , + ∂t

(5.4.49) (5.4.50)

j

∂u i δF gi + ϒ0 = ζi . − ∂t ρ0 δu i

(5.4.51)

The random parts in these equations are Gaussian noises and are related to the bare transport coefficients by θi (x, t)θ j (x , t  ) = 2ρ0 kB T L i0j δ(x − x )δ(t − t  ),

(5.4.52)

ζi (x, t)ζ j (x , t  ) = 2kB T δi j δ(x − x )δ(t − t  )ϒ0 2kBT ϒ0 δi j ,

(5.4.53)

ζi (x, t)θ j (x , t  ) = 0.

(5.4.54)

The different matrices of bare transport coefficients L i0j , ϒi j etc. for the inhomogeneous crystalline state can be obtained by formulating a theory for the solid similar to the Enskog theory for liquids. For the hard-sphere system such a calculation has been done (Kirkpatrick et al., 1990). The stress-energy tensor σi j in eqn. (5.4.50) has reversible and dissipative (irreversible) parts, denoted by σiRj and σiDj , respectively, such that σi j = σiRj + σiDj .

(5.4.55)

Using the expression (5.4.33) for the free-energy functional, we obtain its functional derivative with respect to u i as, δ Fs = Ci jkl ∇ j ∇k u l . δu i The reversible part of the stress-energy tensor is now given by  σiRj = ρ0 χ −1 δi j + Ci jkl u kl ,

(5.4.56)

(5.4.57)

where the strain tensor u i j is defined in eqn. (5.4.35). The dissipative part is obtained in terms of the strain tensor as σiDj = −ηi jkl u kl .

(5.4.58)

Using the above result (5.4.56) for the functional derivative of the free energy, eqn. (5.4.51) for the displacement field reduces to the form ∂u i gi − ϒ0 Ci jkl ∇ j ∇k u l + ζi . = ∂t ρ0

(5.4.59)

The set of fluctuating equations obtained above describes the dynamics of the slow modes for the elastic solid. It is generally applicable to a crystal in which long-range

256

Dynamics of collective modes

translational symmetry is broken. The corresponding decay of hydrodynamic fluctuations conforms to the existence of transverse sound modes in addition to the longitudinal sound modes in a solid. These equations have also been extended to study the dynamics of amorphous solids. In this case we treat the solid as a system in which freezing has occurred at the scale of local structure but overall translational invariance is maintained over longer distances. The isotropic solid For the isotropic solid the rank-four elasticity tensor Ci jkl is determined in terms of only two independent elastic constants, Ci jkl = λ0 δi j δkl + μ0 (δik δ jl + δil δ jk ),

(5.4.60)

where λ0 and μ0 are generally termed Lamé constants. They are, respectively, the bulk and shear modulus of the amorphous solid. The longitudinal modulus is given by ϑ0 = λ0 + μ0 . The expression (5.4.34) for the free energy now reduces to the form   1 dx λ0 (∇ · u)2 + 2μ0 (∇i u j )2 . Fel = 2

(5.4.61)

(5.4.62)

Similarly, for the isotropic solid the rank-four viscosity tensor ηi jkl is determined in terms of only two independent bare viscosities, ζ0 (bulk viscosity) and η0 (shear viscosity):

2 ηi jkl = −ζ0 δi j δkl − η0 δik δ jl + δil δ jk − δi j δkl . (5.4.63) 3 The second term on the RHS represents the traceless component. For the quantity in square brackets in this term, with any fixed i and j (k and l), if we set k = l (i = j) and sum over the common index k (i), the trace is zero. The longitudinal viscosity is defined as DL = ζ0 +

4η0 . 3

(5.4.64)

Using (5.4.63) in the relation (5.4.47), the dissipative matrix L i0j reduces to the form (5.3.71) discussed earlier for the isotropic case. In the fluctuating-hydrodynamics description, for the isotropic amorphous solid the bare transport coefficients ζ0 , η0 , and DL act as external input parameters in the theory. The equation of motion for the momentum density is given by (5.4.50), in which the stress-energy tensor σi j = σiRj + σiDj

(5.4.65)

is a sum of a reversible part and an irreversible part. The reversible part is a sum of a diagonal term and an off-diagonal term, σiRj = Pδi j + μ0 ∇i u j ,

(5.4.66)

5.4 Hydrodynamics of a solid

257

where the pressure P is expressed as a functional of density and the displacement field as P = c02 δρ + ϑ0 ∇ · u,

(5.4.67)

with c02 = ρ0 χ −1 being the speed of sound in the isotropic fluid. The dissipative part of the stress tensor in this case is obtained in the same form as for the isotropic fluid. In terms of the gradients of the momentum density field we write the σiDj in terms of the corresponding kinetic viscosities as ( ν0 ) ν0 σiDj = δi j ς0 + ∇ · g + ∇ j gi . 3 3

(5.4.68)

We have defined the kinetic bulk and shear viscosities of the solid as ς0 = ζ0 /ρ0 and ν0 = η0 /ρ0 , respectively. The corresponding sound attenuation is defined as 0 = ς0 + 4ν0 /3. Finally, the equation of motion for the displacement field u i now takes the form ( μ0 ) gi ∂u i − ϒ0 λ0 + = ∇i ∇ · u + μ0 ∇ 2 u i + ζi . ∂t ρ0 3

(5.4.69)

For the isotropic solid the vacancy density ρD is defined above in eqn. (5.4.32) and the dynamical equation for ρD (x, t) is obtained using eqns. (5.4.49)–(5.4.51) for the slow modes,

∂ 2 ¯ − ϒ0 ∇ ρD + ϒ¯ 0 ∇ 2 ρ = f D , ∂t

(5.4.70)

where we define ϒ¯ 0 = ϑ0 ϒ0 and the noise f D = ρ0 ∇ · f. The variable ρD (x, t) is related to the longitudinal part of u through the quantity ∇ · u. In the context of amorphous solids ρD has been interpreted as a density of free volumes in the amorphous solid (Cohen et al., 1976). The transverse part of u is related to the transverse sound modes in the amorphous solid. In order to demonstrate the hydrodynamic modes in the isotropic solid, we analyze the pole structure of the hydrodynamic correlation functions as was done in the earlier section for the isotropic fluid. Using the isotropic symmetry (in the present case of an isotropic solid), we split the correlation functions with the vector fields into longitudinal and transverse parts. Elements of the memory-function matrix Ki j for the extended set {ρ, g, u} are computed using the relations (5.3.41) and (5.3.50) for the static and dynamic parts of Ki j , respectively. The static correlation function matrix Si j is diagonal, with its diagonal elements the same as before. Of course, in this case we have an extra element Su i u j . To compute Si j we note the identity -

. δF u i (x) = δi j δ(x − x  ). δu j (x  )

(5.4.71)

258

Dynamics of collective modes

Using the definition (5.4.62) for the free energy of the isotropic solid, the longitudinal and transverse components of the correlation of the u field are obtained as S L (q) =

1 , ϑ0 q 2

(5.4.72)

S T (q) =

1 . μ0 q 2

(5.4.73)

The density field couples to the longitudinal components gl and u l , reducing (5.3.23) to the following explicit form: ⎤ ⎡ ⎤ ⎤⎡ L ⎡ χ 0 0 G ρρ G Lρg G Lρu z q 0 ⎥ ⎢ ⎢ ⎣qc2 (q) z + i0 q 2 0 ⎥ iϑ0 q 2 ⎦ ⎣G Lgρ G Lgg G Lgu⎦ = −β −1 ⎣0 ρ0 ⎦. 0 −1 0 −i/ρ0 z + i ϒ¯ 0 q 2 G Luρ G Lug G Luu 0 0 (ϑ0 q 2 ) (5.4.74) To analyze the hydrodynamic poles we consider the determinant of the matrix on the LHS of (5.4.74),  L D+ = (z + i ϒ¯ 0 q 2 ) z 2 − q 2 cL2 + i z0 q 2 + ics2 ϒ¯ 0 q 4 , (5.4.75) L gives rise to two poles where cs2 = ϑ0 /ρ0 and cL2 = c02 + ϑ0 /ρ0 ≡ c02 + cs2 . The zero of D+ (up to leading order in q), at z = ±qcL − (iq2 /2)˜ 0 , representing the two decay modes of the density fluctuation. These correspond to the propagating sound modes in the isotropic fluid at speed cL in directions ±q. The attenuation of the longitudinal sound wave (in either direction) is given by



 2

˜ 0 = 0 + αs ϒ¯ 0 ,

(5.4.76)

where αs = cs2 / c02 + cs . In addition to the longitudinal sound modes, there is an extra diffusive hydrodynamic mode given by the pole at z = −i ϒ¯ 0 q 2 . This is often interpreted as the diffusion of vacancies represented in eqn. (5.4.70) above. This is a characteristic mode present in the solid state. The notion of voids, however, is less ambiguous for the crystalline state with long-range order than for the amorphous glassy state. The transverse part of the matrix of the correlation functions is obtained as   

 T ρ 0 G gg G Tgu z + iν0 q 2 iμq 2 0 = −β −1 . (5.4.77) −1 −i/ρ0 z + iϒ0 G Tug G Tuu 0 (μ0 q 2 ) The corresponding pole in the transverse correlation functions is therefore obtained from the determinant of the matrix on the LHS of eqn. (5.4.77),  T = (z + i ϒ¯ 0 q 2 ) z + ν0 q 2 − cT2 q 2 , (5.4.78) D+ where cT2 = μ/ρ0 . By setting the determinant equal to zero we obtain two poles, z = ±qcT − (iq2 /2)˜ν0 , with ν˜ 0 = ν0 + ϒ¯ 0 . These poles correspond to the propagating sound

5.4 Hydrodynamics of a solid

259

modes in the solid state. In the solid state the two diffusive transverse shear modes change into transverse sound modes with speed and attenuation given by cT and ν˜ 0 , respectively.1 The nature of the vibrational modes in the amorphous glassy state and that of those in the crystalline state are different, which shows up in a number of anomalous properties of disordered solids at low temperature. The specific heat c p below 1 K does not follow the Debye T 3 law, but instead increases linearly with the temperature T . This has been explained phenomenologically in terms of two-level states (Anderson et al., 1972; Phillips, 1972, 1981). In the somewhat higher temperature range, there is a peak in the T dependence of c p /T 3 . A universal feature of disordered glasses is the so-called boson peak observed in neutron- and Raman-scattering experiments (Frick and Richter, 1995; Angell, 1995). These properties are generally ascribed to an excess density of vibrational states (Alba-siminoesco and Krauzman, 1995; Schirmachar et al., 1998, 2007; Grigera et al., 2003) in the amorphous solid over the predictions of the Debye model usually applied to the crystalline state. The density of vibrational modes scaled by the square of the frequency, i.e., g(ω)ω2 , goes through a peak. This anomalous excitation has been observed in disordered systems at terahertz (THz) frequencies using various techniques, e.g., neutron scattering (Buchenau et al., 1984), Raman scattering (Tao et al., 1991) dielectric-relaxation experiments (Lunkenheimer et al., 2000), X-ray techniques (Ruffle et al., 2003), and light scattering (Surovtsev et al., 2004). This THz peak in the density of states is termed the boson peak (Kanaya et al., 1994; Sokolov et al., 1994; Angell et al., 2003).

1 We have ignored in the above discussion the energy-conservation equation, which adds another diffusive hydrodynamic mode

(Martin et al., 1972).

Appendix to Chapter 5

A5.1 The microscopic-balance equations We obtain the equations of motion for the collective densities of mass, momentum, and energy as defined in eqns. (5.1.1) and (5.1.3). The time dependence in the conserved densities enters implicitly through the position and momentum coordinates of the phase-space variables and thus in the following we omit them unless they are specifically required. For the five conserved densities {ρ(r), ˆ gˆ (r), e(r)}, ˆ the corresponding currents are the mass current density gˆ (r), the momentum current density or stress tensor σˆ¯ (r), and the energy current ˆje (r). Microscopic expressions for the currents Jˆa in terms of the phase-space density are obtained using the equations of Hamiltonian dynamics. We begin with the equation for the mass density, for which we obtain  ∂ ρˆ m{∇rα δ(r − rα )} · r˙ α = ∂t α  = −∇r · δ(r − rα )pα = −∇ · gˆ .

(A5.1.1)

α

We have used the property ∇rα δ(r − rα ) = −∇r δ(r − rα ). Hence the current Jˆ ρ for mass density ρˆ is obtained as  pα δ(r − rα ) = gˆ (r). (A5.1.2) Jˆ ρ (r) = α

Similarly, for the momentum density gˆi we obtain  ∂ gˆi = {∇rα δ(r − rα )} · r˙ α pαi + δ(r − rα ) p˙ αi ∂t α = −∇r ·

 α

=−

 j

∇j



δ(r − rα )

 pα i i δ(r − rα )Fαβ pα + m αβ

  pi pαj 1 α i {δ(r − rβ ) − δ(r − rα )}Fαβ , δ(r − rα ) − m 2 α αβ

260

(A5.1.3)

A5.1 The microscopic-balance equations

261

where the prime in the sum implies that the α = β term is excluded. Fαβ is the force on the αth particle due to interaction with the βth particle. We can simplify the second term with the formal manipulation i {δ(r − rβ ) − δ(r − rα )}Fαβ & ' 1 ∂ i = ds δ(r − rα + srαβ ) Fαβ ∂s 0 ⎧ ⎫ ⎨ 1  ∂ ⎬ j i = ds δ(r − r + sr ) rαβ Fαβ α αβ j ⎩ 0 ⎭ ∂s r αβ

j

=



&

1

∇j 0

j

'

j i ds δ(r − rα + srαβ )rαβ Fαβ

.

(A5.1.4)

For an interaction potential that produces a central force field, i.e., Fαβ is in the direction of the vector rαβ , the second term on the RHS of (A5.1.3) reduces to the form & '  1 j ij i ∇j ds δ(r − rα + srαβ )rαβ Fαβ = −∇ j αβ αβ , (A5.1.5) 0

ij

where we have defined αβ and αβ as  αβ = ij

1 0

ds δ(r − rα + srαβ ),

(A5.1.6)

j

i αβ = (rαβ · Fαβ )ˆrαβ rˆαβ .

(A5.1.7) ij

It is straightforward to show that αβ = βα . The quantity αβ is also symmetric under exchange of {α, β} as well as {i, j}. The balance eqn. (A5.1.3) for the momentum density is obtained as ⎫ ⎧  ⎬    ⎨ pi pαj ∂ gˆ i 1 j α i ∇j αβ (ˆrαβ · Fαβ )ˆrαβ rˆαβ =− δ(r − rα ) + ⎭ ⎩ α ∂t m 2 αβ

j

=−



∇ j σˆ i j ,

(A5.1.8)

j

where the stress-energy tensor σˆ i j is σˆ i j (r, t) =

  pi pαj 1  ij α αβ αβ . δ(r − rα ) + m 2 α α,β

(A5.1.9)

262

Appendix to Chapter 5

The microscopic stress tensor σˆ i j satisfies the symmetry σˆ i j = σˆ ji , signifying conservation of angular momentum in the system. In a similar way we obtain the equation of motion for the energy density e(x): " ∂ eˆ  ! {∇rα δ(r − rα )} · r˙α eα + δ(r − rα )e˙α . = ∂t α

(A5.1.10)

To simplify the RHS of the above equation we first evaluate e˙α as   pα 1  ∂u(rαβ ) + · r˙ αβ e˙α = p˙ α · m 2 ∂rαβ β

=

 

Fαβ ·

β

 (p pβ ) pα 1 1 α Fαβ · − Fαβ · r˙ αβ = + . m 2 2 m m

(A5.1.11)

β

We have used the relations r˙ αβ = (pα − pβ )/m and the force on the particle at rα due to the particle at rβ is Fαβ = −∇rαβ u(rαβ ).

(A5.1.12)

Using eqn. (A5.1.11) in eqn. (A5.1.10), we obtain " ∂ eˆ  ! {∇rα δ(r − rα )} · r˙α eα + δ(r − rα )e˙α = ∂t α = −∇r ·

 α

= −∇r ·

 α

+

1 4

 (  α,β

pβ ) pα 1  ( pα eα + + · Fαβ δ(r − rα ) m 2 m m 

δ(r − rα )

α,β

δ(r − rα )

pα eα m

" pβ ) ! pα + · Fαβ {δ(r − rα ) − δ(r − rβ )} . m m

(A5.1.13)

Now, using the same manipulations as in eqns. (A5.1.4)–(A5.1.6), the part Fαβ {δ(r − rα ) − δ(r − rβ )} in the second term on the RHS can be formally expressed as a total derivative. This leads to the result ⎧   j ⎫ j    ⎬   ⎨ pi p 1 p ∂ eˆ β α i j α ∇i =− eα δ(r − rα ) + + αβ αβ . ⎩ α m ⎭ ∂t 4 m m i

α,β

j

(A5.1.14) We therefore obtain the energy-balance equation as ∂ eˆ = −∇ · ˆje , ∂t

(A5.1.15)

A5.1 The microscopic-balance equations

263

with the energy current ˆje being given by ˆjie (r) =



1   ij αβ δ(r − rα ) + m 4

 eα pi

α

α

α,β

j



j

j pβ pα + m m

 αβ .

(A5.1.16)

A5.1.1 The Euler equations In order to determine the local equilibrium averages of the microscopic currents presented in eqn. (5.1.7), we consider the dynamics of the fluid in a frame moving with the local velocity v(x, t) so that the fluid appears to be at rest in the vicinity of this point. We consider the conserved densities {ρˆ  (r), gˆ  (r), eˆ (r)} and the corresponding currents # $ gˆ  (r), σˆ  (r), ˆje (r) in the locally moving frame in the fluid. The phase-space coordinates of the particles in this frame are defined through the canonical transformation pα = pα + mv(r α )

and

rα = rα .

(A5.1.17)

Now, using the above transformation rules in the definitions (5.1.1)–(5.1.3), we obtain the relations between the conserved densities in the two frames: ρ(r) ˆ = ρˆ  (r),

(A5.1.18)

gˆ (r) = gˆ  (r) + ρˆ  (r)v(r),

(A5.1.19)

1 e(r) ˆ = eˆ (r) + v(r) · gˆ  (r) + ρˆ  (r)v 2 (r). 2

(A5.1.20)

Similarly, the transformation rules for the stress tensor σˆ i j and the energy current ˆje are obtained as σˆ i j (r) = σˆ ij (r) + gˆ j (r)vi (r) + gˆ j (r)vi (r) + ρˆ  (r)vi (r)v j (r), 3 4 ˆje (r) = ˆje (r) + eˆ (r) + v(r) · gˆ  (r) + 1 ρˆ  (r)v 2 (r) v(r) 2 1 + v 2 (r)ˆg (r) + σˆ · v(r). 2

(A5.1.21)

(A5.1.22)

It is important to note that the transformation rule (A5.1.22) for the heat current ˆje holds only for the local form of the heat current as described in eqn. (5.1.13), i.e., for shortrange potentials. Considering the transformation rules (A5.1.18)–(A5.1.20), the distribu      tion function f le (t) in terms of the primed variables  N ≡ r1 , p1 , . . . , rN , pN is obtained as      ! " (A5.1.23) f le  N ; t = Q −1 exp − dr β(r, t) eˆ (r) − μ(r, t)nˆ  (r) .

264

Appendix to Chapter 5

  On averaging with respect to the distribution f le  N ; t in the local rest frame, it follows directly from the pα → − pα antisymmetry (see the definitions (5.1.7) for the currents) that + , 1 2 ˆje = 0. (A5.1.24) g le = 0, le

σij ,

The average of the stress tensor on the other hand, is nonzero in the local frame and is linked to a time-dependent generalization of the corresponding thermodynamic pressure, + , σˆ ij = P(r, t)δi j . (A5.1.25) le

Similarly, the average of the energy density is linked to the internal energy density  in the fluid, 1 2 (A5.1.26) eˆ le = (r, t). On substituting these relations into eqns. (A5.1.19), (A5.1.21), and (A5.1.22), respectively, the averages of the microscopic currents are obtained as 1 2 gˆ le = ρv, (A5.1.27) 1 2 σˆ i j le = Pδi j + ρvi v j , (A5.1.28)   + , 2 ˆje =  + ρ v + P v, (A5.1.29) le 2 for the pressure P(r, t) and the internal energy density (r, t), respectively. On averaging eqn. (A5.1.20) we obtain the relation 1 2 1 e(r, t) = e(r) ˆ = (r, t) + ρ(r, t)v 2 (r, t). 2

(A5.1.30)

A5.1.2 The entropy-production rate The entropy of the fluid in the local equilibrium state is defined as S(t) = −ln f le ( N , t).

(A5.1.31)

Using the expression (5.1.15) for f le , we obtain  ln f le ( N , t) = −ln Q l − dr β(r, t)  

1 × e(r) ˆ − v(r, t) · g(r) ˆ − μ(r, t) − mv 2 (r, t) n(r) ˆ 2   = −ln Q l − dr αa (r, t)a(r), ˆ (A5.1.32) {a}

A5.1 The microscopic-balance equations

265

where the set of local densities is given by aˆ ≡ {e, ˆ gˆ , n}. ˆ On substituting the expression (5.1.15) on the RHS we obtain  dr αa (r, t)a(r, t). S(t) = ln Q l (t) + (A5.1.33) {a}

We define an entropy density s(r, t) whose spatial integral gives the total entropy S(t) of the system as given by eqn. (A5.1.33):  αa (r, t)a(r, t). (A5.1.34) s(r, t) = V −1 ln Q l (t) + {a}

The time rate of change of the entropy is obtained as  ∂ ∂ S(t) ∂ dr [αa (r, t)a(r, t)] . = ln Q l (t) + ∂t ∂t ∂t

(A5.1.35)

{a}

Since the time dependence of Q l (t) comes through that of the αl after all the phase-space variables have been integrated out, we write the first term on the RHS of eqn. (A5.1.35) in terms of time derivatives of the αa . Hence we obtain

   δ ln Q l  ∂αa (r, t) ∂ S(t) ∂ {αa (r, t)a(r, t)} = dr + ∂t δαa ∂t ∂t {a}

 =

dr

∂ ∂αa {αa a(r, t)} + −a(r, t) ∂t ∂t

{a}

 =



dr



αa (r, t)

{a}



∂ a(r, t). ∂t

(A5.1.36)

The LHS of the above equation is evaluated using the equation of motion for the slow variables. Let us first consider the Euler equations for reversible dynamics obtained from the local equilibrium average of the microscopic balance equations as given in eqn. (5.1.21),  + ,  ∂ S(t) αa (r, t)∇ · ˆja = − dr ∂t {a}

 =

dr

!

" −∇ · {αa (r, t)ja } + ja · ∇αa (r, t) ,

(A5.1.37)

{a}

where ˆja le ≡ ja are the local currents obtained in eqns. (5.1.22)–(5.1.24). If we define an entropy density s(r, t) whose spatial integral gives the total entropy S(t) of the system, then the above equation transforms into the form ⎧ ⎫⎤ ⎡   ⎨ ⎬  ∂s ⎣ ⎦ dr αa ja {ja · ∇αa (r, t)}. (A5.1.38) +∇· = dr ⎩ ⎭ ∂t {a}

{a}

266

Appendix to Chapter 5

For ˆ gˆ , n}, ˆ the corresponding αa are given by # the set of( conserved)$densities aˆ ≡ {e, β, −βv, −β μ − 12 mv 2 and the currents ja are listed in eqns. (5.1.22)–(5.1.24). Using these results, we obtain    1 2 αa ja = β(e + P)vi − βv j (Pδi j + ρvi v j ) − β μ − mv nvi 2 {a}

3   4 1 = βe − βv j g j − β μ − mv 2 n vi 2 ⎧ ⎫ ⎨ ⎬ = αa (r, t)a(r, t) vi (r, t) ⎩ ⎭ {a}

= {s(r, t) − V −1 ln Q l (t)}vi (r, t),

(A5.1.39)

where we have used eqn. (A5.1.34) in reaching the last equality. On substituting the above result into eqn. (A5.1.38), we obtain

    ∂s {ja · ∇αa (r, t)} + dr ∇ · {V −1 ln Q l v}. dr + ∇ · (sv) = dr ∂t {a}

(A5.1.40) Finally, we evaluate the first term on the RHS of eqn. (A5.1.40) explicitly:   {ja · ∇αa (r, t)} = (e + P)vi ∇i β {a}

i

−

 i, j

3  4

1 2 (Pδi j + ρvi v j )∇i (−βv j ) − nvi ∇i β μ − mv 2

= v · {( + P − nμ)∇β + (∇ P − n ∇μ)β} − ∇ ·(β Pv) = v · {T s ∇β + (∇ P − n ∇μ)β} − ∇ · (β Pv) = βv · {∇ P − s ∇T − n ∇μ} − ∇ · (β Pv).

(A5.1.41)

In the hydrodynamic regime all the local functionals T (r, t), μ(r, t), P(r, t), and v(r, t) are the same functionals of the conserved densities of mass, momentum, and energy as in the respective cases in equilibrium. Therefore, using the thermodynamic identity, i.e., the Gibbs–Duhem relation VdP − sdT − μdN = 0

(A5.1.42)

in this case, the quantity within the curly brackets on the RHS of eqn. (A5.1.41) is equal to zero. We define the entropy-production rate in the fluid as RE ,  d s(r, t)dr, (A5.1.43) RE = T dt

A5.1 The microscopic-balance equations

267

where the integration extends over the volume of the system. Using the results from (A5.1.41), we obtain

 ∂s + ∇ · (sv) RE = T dr ∂t    = T dr ∇ · vEQ V −1 (A5.1.44) where EQ = ln Q l −β P V . The RHS, which is expressed as a volume integral of the divergence, can be reduced to a surface integral, which is assumed to vanish when the system is not externally perturbed. Note that, for the equilibrium limit, f l ( N ) becomes the grandcanonical distribution and the corresponding partition function satisfies β P V = ln Q l (see eqn. (1.2.40)), making EQ = 0 identically zero. Therefore, we have for reversible dynamics that the entropy density satisfies the continuity equation ∂s + ∇ · (sv) = 0 ∂t

(A5.1.45)

with current sv. The net production of entropy of the isolated system with reversible dynamics is equal to zero. In the present situation we have completely reversible dynamics within the local-equilibrium approximation starting from the microscopic-level description in terms of Newton’s equations. Dissipative effects The dissipative effects due to the irreversible transport in the fluid are included in the dynamical equations in a phenomenological manner. To infer about the form of the dissipative terms, we extend the currents in the equations for the dynamics. Let us first define the total current as a sum of reversible and irreversible parts, denoted by superscripts R and D, respectively: σi j = Pδi j + ρvi v j + σiDj ≡ σiRj + σiDj ,   v2 R D je =  + ρ + P v + j D e ≡ j e + je . 2

(A5.1.46) (A5.1.47)

There is no dissipative part in the mass current g, which is itself a conserved quantity. The continuity equation remains unaltered at the coarse-grained level. The corresponding equations of motion for the coarse-grained or averaged densities a(r, ˆ t) are given by ∂ρ + ∇ · g = 0, ∂t ∂gi  ∇ j σi j = 0, + ∂t

(A5.1.48) (A5.1.49)

j

∂e + ∇ · je = 0. ∂t

(A5.1.50)

268

Appendix to Chapter 5

Now, using these equations of motion in eqn. (A5.1.36), we obtain ∂ S(t) = ∂t

 dr



αa (r, t)

{a}

 =−

⎡

∂ a(r, t) ∂t

⎤   1 ⎦ (βvi )∇ j σiRj + β ∇ · jR dr ⎣−β μ − mv 2 ∇ · g − e 2 

i, j

 −

⎤  ⎦. (βvi )∇ j σiDj + β ∇ · jD dr ⎣− e ⎡

(A5.1.51)

i, j

Now the first integral on the RHS has already been computed in the steps between eqns. (A5.1.36) and (A5.1.45). Using this, we obtain d dt





∂s s(r, t)dr ≡ dr + ∇ · (sv) ∂t ⎡ ⎤   ⎦. = dr ⎣ (βvi )∇ j σiDj − β ∇ · jD e 

(A5.1.52)

i, j

The entropy-production rate RE defined in eqn. (A5.1.43) in the presence of the dissipative effects is given by  RE = −

⎡

⎤  ∇T ⎦ dr ⎣ (∇ j vi )σiDj + · jD e , T

(A5.1.53)

i, j

ignoring terms O(v 3 ) and terms of higher order in gradients of T or v. Now, in order that RE ≥ 0 in the irreversible process, the dissipative currents must have the following forms to ensure positive definiteness: σiDj = −ηi jkl ∇k vl , ! D" je i = −λi j ∇ j T,

(A5.1.54) (A5.1.55)

where the positive constants ηi jkl and λi j represent the viscosity and the thermalconductivity tensors, respectively. Note that no cross term is present in the relation above linking the dissipative part of the stress tensor to ∇T or the heat current to the ∇i v j . This is linked to the fundamental property of the equations of motion, since the dissipative part breaks the time-reversal symmetry of the equations of motion.

A5.2 The second fluctuation–dissipation relation

269

A5.2 The second fluctuation–dissipation relation ˆ Let us consider the linear Langevin equation for the variable φ,  t ∂ (d) (d) d t¯ Ki j φˆ j (t¯) = f i (t), φˆ i (t) + iKi j φˆ j + ∂t 0

(A5.2.1)

where, as discussed in Section 5.3, the full memory function Ki j has been divided into a static and a dynamic part, denoted with superscripts (s) and (d), respectively, (d) Ki j (z) = Ki(s) j + Ki j (z).

(A5.2.2) (d)

Here we obtain a relation between the dynamic part of the memory kernel Ki j and the correlation of the noise f i (Mazenko, 2006). To obtain this so-called second fluctuation– dissipation relation we make use of the following identity for the solution φˆ i of the Langevin equation:  t −1 ˆ −1 ds Cik (t − s)Skl fl (s). (A5.2.3) φl + φˆi (t) = Cik (t)Skl 0

We establish the result (A5.2.3) by starting from the Laplace transform of the Langevin equation (A5.2.1) in the form (5.3.16). On the other hand, from eqn. (5.3.23) it follows directly that zδl j − Kl j = Slm Cm−1j (z).

(A5.2.4)

Using the above relation to eliminate the memory function Ki j from (5.3.16), we obtain Slm Cm−1j (z)φˆ j (z) = [zδl j − Kl j ]φˆ j (z) = φˆl + i fl (z).

(A5.2.5)

−1 from the left and summing over the repeated On multiplying the above equation by Cik Skl indices l and k, we obtain the result −1 ˆ {φl + i fl (s)}. φˆ i (z) = Cik (z)Skl

(A5.2.6)

On taking the inverse Laplace transform of (A5.2.6) the result (A5.2.3) follows. Now, in order to establish the second FDT (5.3.48), we rewrite the Langevin equation (A5.2.1) in the form  t (d) d t¯ Ki j (t − t¯)φˆ j (t¯) = f i (t), (A5.2.7) Hi (t) + 0

(s)

where we define Hi (t) in terms of the static part Ki j of the memory function, Hi (t) =

∂ φˆ i (s) + iKi j φˆ j . ∂t

(A5.2.8)

We multiply this equation by φˆ j and take the equilibrium average to obtain the relation  t (d) Hi (t)φˆ j  + ds Kik (t − s)Ck j (s) = 0, (A5.2.9) 0

270

Appendix to Chapter 5

since φˆ j is orthogonal to the noise f i (t). Next, using time translational invariance of equilibrium averages, we obtain that Hi (t)φˆ j  ≡ Hi φˆ j (−t),

(A5.2.10)

where Hi denotes the value of Hi (t) at t = 0. From the Langevin equation (A5.2.7) it follows that Hi = f i . Now we compute the RHS of (A5.2.10) by using (A5.2.3) for φˆ j (−t) and obtain - 3 4.  −t −1 ˆ −1 ˆ Hi (t)φ j  = f i C jk (−t)Skl φl + ds C jk (−t − s)Skl fl (s) . (A5.2.11) 0

The contribution from the first term in the curly brackets on the RHS gives zero since  f i φˆl  = 0. With a change of variables s → −s in the second term on the RHS, the above relation reduces to  t −1 ˆ Hi (t)φ j  = − ds f i fl (−s)C jk (s − t)Skl 0

 =−

t

−1 dy f i (t − y) fl C jk (−y)Skl .

0

(A5.2.12)

In obtaining the last equality we have made a change of variable s → t − y and used the time translational invariance of the equilibrium correlation functions  f i fl (−s) =  f i (s) fl .

(A5.2.13)

Similarly, on applying the same time translational invariance for the equilibrium correlation of the slow modes we have C jk (−y) = Ck j (y).

(A5.2.14)

The equal-time correlation matrix S satisfies −1 −1 Skl = Slk .

Using these two results, (A5.2.12) reduces to the form  t −1 Hi (t)φˆ j  = − dy f i (t − y) fl Slk Ck j (y).

(A5.2.15)

(A5.2.16)

0

On comparing the relations (A5.2.9) and (A5.2.16) we obtain the so-called second fluctuation–dissipation relation between the noise correlation and the dynamic part of the memory function, (d)

 f i (t) f j (t  ) = Kik (t − t  )Sk j .

(A5.2.17)

6 Nonlinear fluctuating hydrodynamics

The fluctuating-hydrodynamics approach discussed earlier takes into account only the transport properties at the level of completely uncorrelated motion of the fluid particles. The corresponding dissipative processes are expressed in terms of bare transport coefficients of the fluid. The strongly correlated motion of the fluid particles which occurs at high density is not take into consideration here. This is reflected through the Markov approximation of the transport coefficients and the short correlation of the corresponding noise representing the fast degrees of freedom in the system. The Markovian equations for the collective modes involving frequency-independent transport coefficients constitute a model for the dynamics of fluids with exponential relaxation of the fluctuations. The corresponding equations of motion for the collective modes are linear. However, exceptions occur in certain situations in which the description of the dynamics cannot be reduced to a set of linearly coupled fluctuating equations with frequency-independent transport coefficients. In this chapter we will consider the nonlinear dynamics of the hydrodynamic modes for studying the strongly correlated motion of the particles in a dense fluid. 6.1 Nonlinear Langevin equations We present in this section the formulation of a set of nonlinear stochastic equations for the dynamics of the many-particle system. We first discuss the physical motivation for extension of the fluctuating-hydrodynamics approach to include nonlinear coupling of the slow modes. It is important to note that the fluctuating nonlinear equations for the slow modes which we obtain below are treated as plausible generalizations of the macroscopic hydrodynamic laws and describe the actual motion of the hydrodynamic variables themselves, rather than just that of their mean values. 6.1.1 Coupling of collective modes The strong correlations that build up in a dense liquid show up in several ways. The striking anomalies seen in the transport properties near critical points are an example. Inclusion in the dynamical description of nonlinear coupling of the collective modes (Kawasaki, 1966a, 1966b) is required in order to explain the observed phenomena. Another example is the 271

272

Nonlinear fluctuating hydrodynamics

observation of long-time tails, which are related to power-law behavior in the asymptotic decay of a correlation function in a fluid, e.g., the 1/t decay of the velocity correlation of a tagged particle seen in computer simulations of hard disks (Alder and Wainwright, 1967, 1970). An implication of the power law in long-time tail decay is the breakdown of conventional hydrodynamics in low-dimensional systems. This is directly seen from the Green–Kubo-type relation in Section 5.1.2 in Chapter 5. If the integrand on the RHS of this relation decays as t −d/2 , the long-time diffusion coefficient diverges for d ≤ 2 (see eqn. 5.2.32). The origin of such observations lies in collective effects over semihydrodynamic length scales. Indeed, by measuring the hydrodynamic flow field around a tagged particle Alder and Wainwright (1970) showed that the initial momentum of this particle is partially transferred to the surrounding liquid, thereby setting up vortices around it. The flow field in turn transfers some of its momentum to the tagged particle by producing a kick from the back, which results in a positive tail in the correlation of v(t) · v(0) extending up to hydrodynamic time scales. A simple estimation of the above cooperative process can be carried out as follows. The initial velocity of the tagged particle is shared with the surrounding particles producing vortices in a volume t of radius Rt . After a short time the tagged particle is moving at velocity v(t) ∼ v(0)/(nt ), n being the number of particles in unit volume. This length Rt grows as t 1/2 , through diffusion of vortices, so that t ∼ t d/2 , producing the t −d/2 power-law tail. The theoretical models for the liquid state must take into account the collective effects present in the strongly interacting fluid. These local collective effects in a dense fluid are included in the theoretical description in several ways. (1) In the kinetic-theory approach the effects of correlated collision of the particles in the dense system are taken into account in terms of repeated binary collisions. This involves computing effects of the so-called ring- and repeated-ring-type collisions between the particles (Dorfman and Cohen, 1972, 1975). A general formalism for including the effects of correlated collisions was proposed through the fully renormalized kinetic theory (Mazenko, 1973a, 1973b, 1974; Mazenko and Yip, 1977; Sjögren and Sjölander, 1978). The dynamics of dense fluids is separated into a binary collision part and a contribution from ring collisions. These methods provided reasonable explanations of the features of the dense fluid dynamics seen in computer simulations and scattering experiments. It was demonstrated that the self-diffusion coefficient of a tagged fluid particle in a sea of similar particles decreases as the density increases. For very high densities, however, the traditional kinetic-theory models have not proved to be very effective, though they acted as an important precursor to reaching more innovative models for the dynamics in the strongly correlating liquid state. (2) In the so-called memory-function approach the correlated dynamics in a fluid is accounted for by introducing nonlinear coupling of the hydrodynamic modes in the dynamic description (Kawasaki, 1970b, 1971; Ernst et al., 1971, 1976; Ernst and Dorfman, 1976). These theories predicted a decay of t −d/2 in d dimensions for a variety of correlation functions connected with viscosity, diffusion, and thermal conductivity,

6.1 Nonlinear Langevin equations

273

through the Green–Kubo relations. A practical and feasible approach to the study of the liquid-state dynamics at high density involves focusing on a set of modes {φˆ i } with characteristic time scales different from that of the rest of the degrees of freedom for the many-particle system. The dynamics is controlled by generalized equations of motion for the chosen set of variables {φˆ i }. The deduction is outlined in Appendix A7.4 using the Mori–Zwanzig projection-operator formalism. To state the result briefly: ∂ φˆ i (t) − ii j φˆ i (t) + ∂t



t

Mi j (t − s)φˆ i (s)ds = Ri (t).

(6.1.1)

0

i j is the n × n frequency matrix obtained as , + i j = φˆ i∗ iLφˆ k S −1 , kj

(6.1.2)

k

where L is the Liouville + , operator for the microscopic dynamics of the N -particle sys∗ ˆ ˆ tem and Si j = φi φ j is the static or the equal-time correlation function of the slow variables. Ri (i = 1, . . . , n) represents the noise corresponding to the remaining large number of degrees of freedom other than {φˆ i }. The Ri (t) remain orthogonal to the space of φˆ i at all times, , + (6.1.3) φˆ i∗ R j (t) = 0. The kernel matrix M in (6.1.1) links the dynamics of φˆ i to its values at earlier times and is termed the memory-function matrix. The memory function Mi j in eqn. (6.1.1) is obtained as an autocorrelation of the noise Ri , Mi j = Ri (t)Rk Sk−1 j .

(6.1.4)

This is a generalized fluctuation–dissipation relation whose validity does not require the system to be close to the equilibrium state but is dependent only on the orthogonality condition (6.1.3). The linearized dynamical equations (6.1.1) for the slow variables are non-Markovian, i.e., have frequency-dependent transport coefficients Mi j . The generalized transport coefficients are then evaluated by computing the nonzero elements of the memory-function matrix M. These frequency- and wave-number-dependent functions are obtained in terms of the corresponding microscopic time correlation functions. The resulting model is known as the self-consistent mode-coupling theory (MCT) (Götze, 1991). It has proved to be an extremely useful step in the study of slow dynamics in the supercooled liquid state approaching the glass transition. The deduction of the MCT using this approach is outlined in the appendix following the discussion for the linear Langevin equations with nonlocal transport coefficients. We formulate the dynamics of the supercooled liquid taking an approach that is intermediate between the above two routes. The choice of a basic set {φˆ i } of slow variables for the system is dictated by various physical considerations as indicated in Section 5.3.1. The many remaining degrees of freedom are treated as noise fluctuating on much shorter

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Nonlinear fluctuating hydrodynamics

time scales. The nonlinear dynamics involves in the equations of motion the products of the slow modes, which are also treated as slow. The inclusion of the nonlinear interaction of the slow modes in the stochastic equations brings in collective effects. To demonstrate this, consider the following simple analysis. Take a slow variable ai , which, being a local density of a globally conserved property, has the property lim

q→0

∂ ai (q, t) = 0. ∂t

(6.1.5)

With this criterion, can the products of the slow variables also be treated as slow variables? The corresponding integral for a product of the slow variables, ai2 (r, t), is   ∂ dk dx eiq · x ai2 (x, t) = 2 lim i[k · Jai (k, t)]ai (−k, t), (6.1.6) q→0 ∂t (2π )3 where in reaching the second equality we have used the balance equation ∂t ai + ∇ · Jai = 0

(6.1.7)

for the slow variable ai and the corresponding current Jai . For long times t, the value of the integral in (6.1.6) is dominated by the small-k regime (since ai (k, t) is a hydrodynamic variable) and hence becomes small due to the factor of k in the integrand. Thus, by virtue of the presence of the nonlinear couplings of the slow variables in the equations for the dynamics, we essentially include collective effects in the dense systems over long length scales. 6.1.2 Nonlinear Langevin equations In the following, a set of nonlinear Langevin equations for the chosen set of slow modes ˆ describing the dynamics of dense fluids is obtained: {φ}  ∂ φˆ i (t) ∂F ˆ − L i0j + θi (t), = Vi [φ] ∂t ∂ φˆ j j

(6.1.8)

where (a) F is the equivalent Hamiltonian for the system expressed as a functional of the slow modes {φˆ i } such that the equilibrium distribution function at temperature T is P[φ] = Z −1 exp[−F/(kB T )], (b) the streaming current Vi represents the reversible part of the dynamics expressed in terms of the slow variables, and (c) the correlation of the noise θi (t) is given by θi (t)θ j (t  ) = 2kB T L i0j δ(t − t  )

(6.1.9)

in terms of the matrix of bare transport coefficients L i0j . We follow the treatment developed by Ma and Mazenko (1975) and Mazenko (2006b). The collective variable gφ (t) ˆ t)} for the system is dictated by The choice of a basic set of collective variables {φ(x, various physical considerations (which are discussed in Section 5.3.1). In the case of a

6.1 Nonlinear Langevin equations

275

classical fluid these collective variables are the conserved densities of the mass, momentum, and energy of the fluid. For a classical N -particle fluid, these collective variables are functions of the 6N phase-space variables. The many remaining degrees of freedom of the fluid are treated as noise fluctuating on a much shorter time scale. The equations for the time evolution of the variables {φˆ i } with nonlinear interactions are obtained by introducing the new variable gφ (t) in terms of the fields φi (x, t) as % ˆ gφ (t) = δ[φi − φˆ i (t)] ≡ δ[φ − φ(t)], (6.1.10) i

where the variables φi (x, t) are defined as field variables on a set of lattice points denoted by x. We suppress the spatial dependence of the fields in writing the above equations in order to keep the notation simple. In the present notation the hatted entities {φˆ i } are dependent on the phase-space coordinates {rα , pα } (α = 1, . . . , N ) while the {φi } are the field variables labeled by the space-time coordinates. The function gφ (t) satisfies the following condition by definition:  % D(φ)gφ (t) = dφi δ[φi − φˆ i (t)] = 1, (6.1.11) i

where we have used the notation D(φ) for the multiple integral over the set {φi } at all space points on a grid of size ,  % dφi , D(φ) ≡ lim (6.1.12) →0

i

termed the functional integral in the space of φ. On taking the thermal average of (6.1.11) we obtain the result   D(φ)P[φ] ≡ D(φ)gφ (t) = 1, (6.1.13) where the thermal average of gφ (t) over the phase-space variables for the φˆ i is defined as gφ (t) = P[φ]. From the definition (6.1.10) of gφ it is clear that its average represents the probability that the set {φˆ i } will have the corresponding values {φi } and hence P[φ] is a positive number. We define an equivalent Hamiltonian F[φ] for the system as a functional of the field variables φi such that e−β F[φ] , Z

(6.1.14)

D(φ)exp(−β F[φ])

(6.1.15)

P[φ] = where

 Z≡

is a normalization constant. The equal-time correlation of the field variable φ is defined by the relation Sφφ  = gφ gφ   = δ(φ − φ  )P[φ],

(6.1.16)

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Nonlinear fluctuating hydrodynamics

where in reaching the last equality we used the definition (6.1.10) of gφ in terms of delta functions. The function gφ is useful for determining all of the equal-time couplings of the collective variables φˆ i . For example,  D(φ)gφ (t)[φi φ j ] = φˆ i (t)φˆ j (t). (6.1.17) We obtain below a stochastic equation for the time evolution of gφ (t) following methods developed by Ma and Mazenko (1975) in the study of dynamic–critical phenomena. The equation of motion for gφ (t) is then used to construct a set of nonlinear Langevin equations for the collective modes {φˆ i (t)}. The corresponding recipe for obtaining the nonlinear Langevin equation is then applied in the subsequent sections for constructing the equations of motion for the relevant slow modes in the compressible liquid. These form the basis for the most widely studied model for slow dynamics in the high-density liquid and will be the focus of our discussion later in this chapter. We present this general scheme for obtaining the nonlinear Langevin equation in detail first. Dynamics of gφ (t) We obtain a linear Langevin equation for the generalized set of variables denoted as {gφ (t)} following the scheme described in Section 5.3.1 of Chapter 5. We work with the Laplace transform defined as  ∞ dt ei zt gφ (t) (6.1.18) gφ (z) = −i 0

with Im(z) > 0. The equation of motion for gφ is obtained as ∂gφ (t) = −{H, gφ } ≡ iLgφ (t), ∂t

(6.1.19)

which can be formally integrated to obtain gφ (t) = exp(iLt)gφ , where gφ = gφ (t = 0). On taking the Laplace transform of eqn. (6.1.19) we obtain the relation gφ (z) = R(z)gφ ,

(6.1.20)

where the resolvent operator R(z) is given by R(z) = [z + L]−1 .

(6.1.21)

Equivalently, we can express the above relation in the form zgφ (z) = gφ − Lgφ (z).

(6.1.22)

Following steps analogous to the earlier case, from (6.1.22) we obtain a Langevin equation for gφ by approximating the quantity Lgφ (z) as a sum of two parts: (a) the first part is linearly proportional to the gφ and (b) the rest is treated as a noise f φ (z). Thus we write, schematically, − Lgφ (z) = Kφ φ¯ (z)gφ¯ (z) + i f φ (z),

(6.1.23)

6.1 Nonlinear Langevin equations

277

where Kφ φ¯ (z) is a kernel function that represents the projection of Lgφ (z) on gφ (z). In eqn. (6.1.23) and in the rest of this section we adopt the Einstein summation convention that repeated (barred) functions are integrated over in the function space, i.e.,  ¯ φ φ¯ (z)gφ¯ (z). (6.1.24) Kφ φ¯ (z)gφ¯ (z) = D(φ)K The leftover contribution in (6.1.23) denoted by f φ (z) is the noise, which behaves like a dynamical variable similar to gφ (z). The noise f φ (z) has no projection on the slow mode gφ  (t) at t = 0, gφ  f φ (z) = 0,

(6.1.25)

or, equivalently, we obtain in terms of time correlations

Integration of (6.1.26) with noise f φ ,

6

gφ  f φ (t) = 0 for t ≥ 0.

(6.1.26)

Dφ  gives the equivalent condition for the average of the  f φ (t) = 0.

(6.1.27)

Using (6.1.23) in eqn. (6.1.22), we obtain the equation of motion for the Laplace transform of gφ (t) in the frequency space, zgφ (z) − Kφ φ¯ (z)gφ¯ (z) = gφ + i f φ (z).

(6.1.28)

Equivalently, eqn. (6.1.28) reduces to a linear Langevin equation in the time space, ∂ gφ (t) + iKφ φ¯ gφ¯ (t) = f φ (t). ∂t Let us focus now on the memory function Kφ φ¯ .

(6.1.29)

The memory function The Laplace transform of the correlation function of the gφ (t) at two different times is defined as G φφ  (z) = gφ  gφ (z) = gφ  R(z)gφ .

(6.1.30)

The correlation function G φφ  (z) satisfies the memory-function equation zG φφ  (z) − Kφ φ¯ G φφ ¯  (z) = Sφφ  ,

(6.1.31)

where Sφφ  is the static or the equal-time correlation function of the slow variables as defined above in eqn. (6.1.16). The memory function Kφφ  is divided into two parts, (s)

(d)

Kφφ  = Kφφ  + Kφφ  , (s)

(6.1.32)

(d)

where Kφφ  is the static part and Kφφ  is the dynamic part. Using the ansatz (6.1.32) in (6.1.28), we obtain the following version of the Langevin equation (6.1.29) for gφ (t): ∂ (s) (d) gφ (t) + iKφ φ¯ gφ¯ (t) + iKφ φ¯ gφ¯ (t) = f φ (t). ∂t

(6.1.33)

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Nonlinear fluctuating hydrodynamics

Following steps similar to those used in obtaining the static and dynamic parts of the memory function in the case of linearized dynamics (respectively given by eqns. (5.3.31) and (5.3.32) in Chapter 6), we obtain (s) Kφ φ¯ Sφφ ¯  = −gφ  Lgφ ,

(6.1.34)

Kφ(d)φ¯ (z)Sφφ ¯  = −{Lgφ }R(z){Lgφ } (z)gφ¯  R(z){Lgφ }. + {Lgφ  }R(z)gφ¯ G −1 φ¯ φ¯ 

(6.1.35)

The complicated expressions for the memory functions presented above finally lead to the Langevin equation for the variable gφ . To analyze the memory function further, we use a special property of the Liouville operator L due to its derivative form,

 ∂H ∂ ∂H ∂ · − · . (6.1.36) L≡i ∂pα ∂rα ∂rα ∂pα α Since delta functions are involved in the definition (6.1.10) of gφ , the derivative of the latter with respect to φˆ i is the negative of that with respect to φi . Therefore, on replacing the derivative of gφ with respect to the phase-space variables rα or pα in terms of a derivative with respect to the phase-space variable φˆ i , i.e.,    ∂gφ ∂ φˆ i ∂ H  ∂ ∂ H ∂gφ ∂ φˆ i ∂ H · → · =− gφ · , (6.1.37) ∂pα ∂rα ∂φi ∂rα ∂pα ∂ φˆ i ∂rα ∂pα i

i

we obtain the result Lgφ = −

 ∂ {gφ Lφˆ i }, ∂φi

(6.1.38)

i

since Lφˆ i in the curly bracket is a function of phase-space variables only and does not involve φi . Equation (6.1.38) is a key result, which we use below in our analysis of the static and dynamic parts of the memory function. The static part Let us first focus on the static part of the memory function, which is frequency-independent and hence considerably simpler. Since the equal-time correlation Sφφ  defined in (6.1.16) is diagonal in φ, we obtain using (6.1.38) on the RHS of the expression (6.1.34) for the static part of the memory function  (s) ¯ (s) δ(φ¯ − φ  )P[φ  ] Kφ φ¯ Sφφ D(φ)K ¯  ≡ φ φ¯ (s)

= Kφφ  P[φ  ] =

 ∂ gφ  gφ Lφˆ i . ∂φi i

(6.1.39)

6.1 Nonlinear Langevin equations

279

Now, making use of the definition (6.1.10) of gφ (t), the RHS of the above expression for the static part of the memory function simplifies to (s)

Kφφ  P[φ  ] =

 ∂ ! " δ(φ − φ  )gφ Lφˆ i  . ∂φi

(6.1.40)

i

The equilibrium average gφ Lφˆ i  on the RHS of eqn. (6.1.40) is expressed in terms of the Poisson bracket of the corresponding quantities. This is similar to the steps that led to the relation (5.3.39) obtained earlier in the case of linearized dynamics, gφ Lφˆ i  = −iβ −1 {φˆ i , gφ }.

(6.1.41)

Using the properties of the linear operator L as discussed in the deduction of (6.1.38), we obtain  ∂ gφ {φˆ i , φˆ j }. (6.1.42) {φˆ i , gφ } = − ∂φ j j

The RHS of (6.1.41) is further simplified to obtain gφ Lφˆ i  = iβ −1 = iβ −1

 ∂ ∂  ˆ = iβ −1 Q i j (φ)gφ  gφ Q i j (φ) ∂φ j ∂φ j  ∂  Q i j P[φ] , ∂φ j

(6.1.43)

where Q i j is as defined in eqn. (5.3.40) in the previous chapter to denote the Poisson bracket as a function of the collective modes. Thus eqn. (6.1.39) reduces to the form

 ∂ ∂ (s) Q i j P[φ] δ(φ − φ  ) Kφφ  P[φ  ] = iβ −1 ∂φi ∂φ j i, j

= iβ

−1

 

 ∂ ∂ Qi j ∂ P[φ]  Q i j + P[φ] δ(φ − φ ) ∂φi ∂φ j ∂φ j i, j

 

 ∂ ∂ F[φ]  −1 ∂ Q i j +β δ(φ − φ ) −Q i j P[φ] , =i ∂φi ∂φ j ∂φ j

(6.1.44)

i, j

where we have used the definition (6.1.14) for P[φ] to make the substitution ∂ F[φ] ∂ P[φ] = −β P[φ] . ∂φ j ∂φ j

(6.1.45)

The above relation implies that the equilibrium probability distribution P[φ] satisfies the ( j) following condition defined in terms of an operator D˜ φ :

∂ ∂ F[φ] ( j) +β (6.1.46) P[φ] ≡ D˜ φ P[φ] = 0. ∂φ j ∂φ j

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Nonlinear fluctuating hydrodynamics

On the RHS of (6.1.44) the factor P[φ] can be replaced with P[φ  ] due to the presence of δ(φ − φ  ) and this cancels out with the same term on the LHS, leading to the following expression for the static part of the memory function: (s) Kφφ  = −i

 ∂ ! " δ(φ − φ  )Vi [φ] . ∂φi

(6.1.47)

i

The quantity Vi [φ] within the square brackets is termed the streaming velocity term in the space of the functions {φi } and is given by

 ∂ Qi j ∂ F[φ] − β −1 Qi j . (6.1.48) Vi [φ] = ∂φ j ∂φ j j

Vi [φ] satisfies an important divergence property,  ∂ {Vi (φ)P[φ]} = 0, ∂φi

(6.1.49)

i

which follows directly on substituting the expression (6.1.48) for Vi [φ] into (6.1.49): ⎤ ⎡    ∂ ∂ Q ∂ F[φ] ij ⎣ − β −1 Qi j P[φ]⎦ ∂φi ∂φ j ∂φ j i

j

= −β

−1

 ∂ ∂ P[φ] ∂ Q i j + P[φ] Qi j ∂φi ∂φ j ∂φ j ij

 ∂ ∂   = −β −1 Q i j P[φ] = 0. ∂φi ∂φ j

(6.1.50)

ij

We have used in reaching the first equality above eqn. (6.1.45) for (∂ F[φ]/∂φ j ), while the last equality follows from the antisymmetric property of the Poisson bracket, Q i j [φ] = −Q ji [φ]. The condition (6.1.49) ensures that the probability current vector, which is equal to the product of the probability P[φ] and the streaming velocity Vi , is divergence-free in the function space of {φi }. We make use of this divergence condition further below. Using the results (6.1.47) for the static part of the memory function in the second terms on the LHS of the linear Langevin equation (6.1.33) for gφ (t), the latter reduces to the form (s)

iKφ φ¯ gφ¯ (t) =

 ∂  ¯ i [φ]gφ¯ (t) δ(φ − φ)V ∂φi i

 ∂ = Vi [φ]gφ (t), ∂φi

(6.1.51)

i

where the streaming current Vi [φ] is given by eqn. (6.1.48) in terms of the Poisson brackets of the slow variables {φi }.

6.1 Nonlinear Langevin equations

281

The dynamic part (d)

Let us now consider the dynamic part Kφ φ¯ (z) of the memory function given by (6.1.35). This part is in general frequency-dependent and is considerably more complicated. From our discussion in the previous section on the memory function it is known that the linear part of the dynamics of the collective variables {φˆ i } does not contribute to this part. In the present chapter we treat this part within the framework of the Markov approximation. We will work with the assumption that the dynamic part of the memory function depends only on the fast variables in the system. At the formal level the expression on the RHS of (6.1.35) simplifies to the following form on using the result (6.1.38) for Lgφ :  ∂ ∂  ij (d)  Kφφ (6.1.52) φφ  (z) ,  (z)P[φ ] ≡ ∂φi ∂φ j ij

is a functional of φ and φ  . In reaching the expression on the LHS of eqn. where (6.1.52) the diagonal property of the static correlation Sφφ ¯  is used in a manner similar to what has been done in the case of the static part. We finally obtain  ¯ (d) (z)δ(φ¯ − φ  )P[φ  ] D(φ)K Kφ(d)φ¯ (z)Sφφ ¯  ≡ φ φ¯ ij φφ  (z)

(d)

= Kφφ  (z)P[φ  ].

(6.1.53)

ij

A detailed expression for φφ  (z) may be obtained from the defining relation (6.1.35) (d)

for the dynamic part Kφφ  of the memory function. However, in the present discussion of the dynamics of the many-particle system using the nonlinear fluctuating-hydrodynamics approach we treat this dynamic part in an indirect manner. We approximate the dynamic component of the memory function only in terms of the short-time dynamics of the system. The Markov approximation ij

In the simplest approximation the matrix elements φφ  (z) are assumed to be frequencyindependent and are obtained in terms of the short-time transport coefficients of the system. In Section 5.3.1 we have shown in eqn. (5.3.48) that for the linear Langevin equation the correlation of the noise, which is assumed orthogonal to the slow modes, is obtained in terms of the dynamic part of the memory function. On applying this in the present context for the Langevin equation (6.1.33), the noise correlation satisfies (d)

 f φ (t) f φ  (t  ) = Kφφ  (t − t  )P[φ  ] = 2i

 ∂ ∂  ij φφ  δ(t − t  ). ∂φi ∂φ j

(6.1.54)

ij

To obtain the RHS we have taken the inverse Laplace transform of (6.1.52) to obtain (d) Kφφ  (t − t  ). In the fluctuating-hydrodynamics approach these short-time transport coefficients denoted by L i0j act as phenomenological inputs in the theory and may be determined from a detailed microscopic kinetic-theory approach to the system’s dynamics. The bare

282

Nonlinear fluctuating hydrodynamics

kinetic coefficients L i0j (φ) in general can depend on the field φ. We assume as a defining relation φφ  = −iβ −1 L i0j (φ)δ(φ − φ  )P[φ  ], ij

(6.1.55)

which is designed so that the contribution from the dynamic part of the memory function represents dissipation. L i0j (φ) is the matrix of generalized transport coefficients for the system. The frequency-independent form of the dynamic part influences the construction of the linear Langevin equation for gφ (t) in two important ways. (1) The correlation of the noise f φ of the Langevin equation (6.1.33) is now given by (d)

 f φ (t) f φ  (t  ) = Kφφ  (t − t  )P[φ  ] $  ∂ ∂ # 0   (φ)δ(φ − φ )P[φ ] δ(t − t  ). (6.1.56) L = 2β −1 ∂φi ∂φ j i j ij

(2) The ansatz (6.1.55) for the dynamic part of the memory function is useful in expressing the corresponding contribution in the Langevin equation (6.1.33) (given by the third term on the LHS of this equation) in terms of the bare transport coefficients L i0j . On (d)

substituting this ansatz into (6.1.55) we obtain for Kφ φ¯ (z)

 ∂ ∂  L i0j (φ)δ(φ − φ  )P[φ  ] ∂φi ∂φ j ij & '  ]  ∂ ∂ ∂ P[φ = −iβ −1 L 0 (φ) δ(φ − φ  ) P[φ  ] + δ(φ − φ  ) ∂φi i j ∂φ j ∂φ j ij

 ∂ ∂ ∂ F[φ] −1 0   = iβ L (φ) δ(φ − φ ) + β δ(φ − φ ) P[φ  ]. ∂φi i j ∂φ j ∂φ j ij (6.1.57)

(d)  −1 Kφφ  (z)P[φ ] = −iβ

On canceling out the common factor of P[φ  ] from both sides of (6.1.57) we obtain the result

 ∂ ∂ F[φ] ∂ (d) −1 0 ¯ L (φ) β + δ(φ − φ) Kφφ  (z) = iβ ∂φi i j ∂φ j ∂φ j ij $  ∂ # ( j) ¯ . = iβ −1 (6.1.58) L i0j (φ)D˜ φ δ(φ − φ) ∂φi ij

Using the result (6.1.58) in the dynamic part of the memory function, the third term on the LHS of the linear Langevin equation (6.1.33) for gφ (t) reduces to $  ∂ # ( j) (d) ¯ φ¯ (t) L i0j (φ)D˜ φ δ(φ − φ)g iKφ φ¯ gφ¯ (t) = −β −1 ∂φi ij

 ∂ ( j) L 0 (φ)D˜ φ gφ (t), = −β −1 ∂φi i j ij

(6.1.59)

6.1 Nonlinear Langevin equations

283

with the L i0j (φ) being bare kinetic coefficients defining the correlation of the noise f φ (t) through eqn. (6.1.56). The Fokker–Planck equation On adding the static and dynamic parts given by (6.1.51) and (6.1.59), respectively, we obtain for the contribution of the memory function to the Langevin equation  iKφ φ¯ gφ¯ (t) = i Kφ(s)φ¯ + Kφ(d)φ¯ gφ¯ (t) ⎡ ⎤   ∂ ( j) ⎣Vi [φ] − β −1 = L i0j (φ)D˜ φ ⎦ gφ (t) ≡ −Dφ gφ (t), (6.1.60) ∂φi i

ij

where the operator Dφ is defined as ⎤ ⎡     ∂ ∂ ∂ F[φ] ⎦ ⎣Vi [φ] − β −1 L i0j (φ) +β Dφ = − ∂φi ∂φ j ∂φ j i

j

⎡ ⎤   ∂ ( j) ⎣Vi [φ] − β −1 L i0j (φ)D˜ φ ⎦ . = ∂φi i

(6.1.61)

j

On substituting the result (6.1.60) into the generalized Langevin equation (6.1.33) for gφ (t), the latter reduces to the form ∂ gφ (t) = Dφ gφ (t) + f φ (t). ∂t

(6.1.62)

The above equation is a generalized form of the Fokker–Planck equation (Fokker, 1914; Planck, 1917; Risken, 1996) with noise f φ whose correlation is given by (6.1.56) in terms of the bare kinetic coefficients. This is the key result of the present section and is essential in obtaining the nonlinear Langevin equation for the collective variables. The average quantity gφ (t) represents the probability that the collective modes {φˆ i } have the value {φi }. Taking the average of (6.1.62) and using (6.1.27) gives a Fokker– Planck equation: ∂ gφ (t) = Dφ gφ (t) ∂t

(6.1.63)

with Dφ as the corresponding driving operator. For the equilibrium state, gφ (t) ≡ P[φ] do not have any explicit time dependence, so Dφ gφ (t) = 0. Indeed, this equation is satisfied by P[φ], as can be seen from direct substitution. Using (6.1.61) we have Dφ P[φ] = −

 ∂ ∂  0 ( j) {Vi [φ]P[φ]} + β −1 L i j (φ)D˜ φ P[φ], ∂φi ∂φi i

j

(6.1.64)

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Nonlinear fluctuating hydrodynamics

where the operator Dφ is as defined in (6.1.61). Using (6.1.46) and (6.1.49), the RHS of (6.1.61) is equal to zero. The nonlinear Langevin equation To summarize the discussion so far, we have reached a linear Langevin equation (6.1.62) for the collective variable gφ (t) defined through the relation (6.1.10). The dynamics is assumed to have a clear difference in the time scale for the collective modes and that for the noise f φ . The correlation of the noise is given in eqn. (6.1.56) by the short-time transport coefficients L i0j . A set of nonlinear Langevin equations for the slow modes {φˆ i (t)} is obtained by integrating (6.1.62) with respect to the function {φi }. Considering the equation for the collective variable φˆ i , we note that  (6.1.65) φˆ i (t) = D(φ)φi gφ (t). On taking the first moment of eqn. (6.1.62) we obtain the nonlinear Langevin equation,  ∂ φˆ i (t) (6.1.66) = D(φ)φi Dφ gφ (t) + θi (t). ∂t The noise θi (t) is obtained from the first moment of f φ as  θi (t) = D(φ)φi f φ .

(6.1.67)

The first term on the RHS of (6.1.66) is given by  D(φ)φi Dφ gφ (t)  =−

 ∂ D(φ)φi ∂φ j

 V j [φ] −



j

 L 0jk (φ)

β

−1

k

∂ ∂ F[φ] + ∂φk ∂φk

 gφ (t). (6.1.68)

Upon integrating by parts with the derivative ∂/∂φ j acting on φi on the RHS, eqn. (6.1.68) reduces to the form ⎡ ⎤      ∂ ∂ F[φ] ⎦ gφ (t) D(φ)φi Dφ gφ (t) = D(φ) ⎣Vi [φ] − L i0j (φ) β −1 + ∂φ j ∂φ j j

ˆ + = Vi [φ]

 j

 β

−1

ˆ ∂ L i0j (φ) ∂ φˆ j

−

L i0j

∂F ∂ φˆ j

 .

The corresponding Langevin equation is given by   ˆ  ∂ L i0j (φ) ∂ φˆ i (t) −1 0 ∂F ˆ − Li j = Vi [φ] + β + θi (t). ∂t ∂ φˆ j ∂ φˆ j j

(6.1.69)

(6.1.70)

6.1 Nonlinear Langevin equations

285

The correlation of the noise θi (t) is computed from the definition (6.1.67) as 

θi (t)θ j (t  ) =

 D(φ)

D(φ  )φi φ j  f φ (t) f φ  (t  ).

(6.1.71)

Using on the RHS of eqn. (6.1.72) the ansatz (6.1.56) which defines the correlation of the noise f φ in terms of the bare transport coefficients L i0j , it reduces to the following relation θi (t)θ j (t  )  $  ∂ ∂ # = 2β −1 D(φ)D(φ  )φi φ j L 0kl (φ)δ(φ − φ  )P[φ  ] δ(t − t  ). (6.1.72)  ∂φk ∂φl kl

On partially integrating (6.1.72) we obtain, θi (t)θ j (t  ) = 2β −1 = 2β −1



# $ D(φ)D(φ  ) L i0j (φ)δ(φ − φ  )P[φ  ] δ(t − t  )



D(φ)L i0j (φ)P[φ]δ(t − t  ) 1 2 = 2β −1 L i0j (φ) δ(t − t  ),

(6.1.73)

1 2 where L i0j (φ) denotes the equilibrium averaged form of the matrix of the bare transport coefficients. For the somewhat simplified case in which the dependence of the bare transport coefficients on the slow modes φˆ i is ignored, we obtain from (6.1.71) the nonlinear Langevin equation for the dynamics of the slow variables {φˆ i },  ∂ φˆ i (t) ∂F ˆ − L i0j + θi (t), = Vi [φ] ∂t ∂ φˆ j j

(6.1.74)

where (a) F is the equivalent Hamiltonian for the system expressed as a functional of the slow modes {φˆ i } such that the equilibrium distribution function at temperature T is P[φ] = Z −1 exp[−F/(kB T )], (b) the streaming current Vi is given by (6.1.48), and (c) the correlation of the noise θi (t) is given by θi (t)θ j (t  ) = 2kB T L i0j δ(t − t  )

(6.1.75)

in terms of the matrix of bare transport coefficients L i0j . Similarly to the case of the linearized Langevin equation, the first and second terms on the RHS of (6.1.75) have opposite signatures with respect to the time reversal. The above Langevin equation is further simplified to the form

286

Nonlinear fluctuating hydrodynamics

∂ φˆ i (t)  δF , = Q i j − L i0j ∂t δψ j

(6.1.76)

where the second term on the RHS of eqn. (6.1.48) has been ignored. (i) From the definition of the Poisson bracket it follows that Q i j (−t) = −i  j Q i j (t), where i denotes the time-reversal property of the slow mode φˆ i , i.e., φˆ i (−t) = i φˆ i (t). It follows from the expression (6.1.48) for Vi that this part of the equation of motion for φˆ i remains invariant under time reversal. Hence Vi represents the reversible part of the dynamics. (ii) Assuming that the matrix L i0j satisfies under time reversal the property L i0j (−t) = i  j L i0j (t),

(6.1.77)

the second term on the LHS of (5.3.51) breaks the reversal symmetry of the equation of motion and hence represents dissipation. The Gaussian noise The noise correlation given by (6.1.76) for the case of the bare-transport-coefficient matrix L i0j is independent of the slow variables φˆ and in fact corresponds to the noise θi having a Gaussian distribution function. Thus the average of a dynamic variable A[θ ] with respect to the noise is defined as  A[θ] =

D(θ )A[θ ]P[θ ],

where the Gaussian distribution function P[θ] for the noise is given by ⎡ ⎤   +∞ β 1 −1 exp⎣− d t¯ θi (t¯)[L 0 ]i j θ j (t¯)⎦ . P[θ ] = Zθ 4 −∞

(6.1.78)

(6.1.79)

ij

6 Z θ is a normalization constant so as to ensure that D(θ )P[θ ] = 1. In order to compute the noise correlation with respect to P[θ ], we start from the functional identity  δ {P[θ] A[θ]} = 0. D(θ ) (6.1.80) δθi (t) With the form (6.1.80) for P[θ ], its derivative with respect to the noise is given by δ P[θ] β −1 = − [L 0 ]ik θk (t)P[θ ]. δθi (t) 2

(6.1.81)

The identity (6.1.81) then implies

. δ A[θ ] β 0 −1 [L ]ik θk (t)A[θ ] = . 2 δθi (t)

(6.1.82)

On multiplying eqn. (6.1.83) by the matrix L 0 from the left we obtain the equivalent relation . −1 0 δ A[θ ] θi (t)A[θ ] = 2β L ik . (6.1.83) δθk (t)

6.2 The compressible liquid

Now, on choosing A[θ] ≡ θ j (t  ), we obtain  . 0 δθ j (t ) = 2β −1 L i0j δ(t − t  ), θi (t)θ j (t  ) = 2β −1 L ik δθk (t)

287

(6.1.84)

which is the same as (6.1.76) obtained above. In all subsequent treatment of the fluctuating hydrodynamics in this book we confine the discussion to the case of Gaussian white noise.

6.2 The compressible liquid We consider an appropriate set of slow variables {φˆ i } for the compressible liquid and construct the corresponding equations for the dynamics of these collective modes. These equations, which are generalizations of the corresponding macroscopic hydrodynamic laws, are applied here to study phenomena at microscopic frequencies and wavelength. The collective modes are deviations from the equilibrium state whose properties are assumed to be known. For the chosen set of variables there is a vast difference in the time scales of variation in comparison with that of the large number of microscopic variables of the fluid. The origin of these slow variables for a specific system can vary, e.g., microscopic conservation laws as in the present case, the heavy mass of a Brownian particle or breaking of a continuous symmetry (Forster, 1975) of a many-particle system. In the present chapter we discuss the formulation of these nonlinear fluctuating equations for a few generic cases. The fluctuating nonlinear Langevin equations for the liquid are Markovian in form and involve bare transport coefficients related to the noise through a standard fluctuation– dissipation relation,   ∂ φˆ i (r, t) ∂F ˆ + dr L i0j (r, r ) (6.2.1) = θi (r, t). + Vi [φ] ∂t ∂ φˆ j (r , t) j The matrix of bare transport coefficients L i0j of the liquid defines the correlation of the noise θi . The transport properties of the liquid for short times are determined by L i0j . For long-time dynamics the bare coefficients are renormalized due to the nonlinearities in the equations of motion leading to generalized transport coefficients. In the theory we discuss here the equations of nonlinear fluctuating hydrodynamics (NFH) are conveniently used for computing the corrections to the linear model. Before we turn to the development of the formalism for renormalization of the nonlinear theory in the next chapter, we obtain the appropriate set of nonlinear fluctuating hydrodynamic equations for the compressible liquid in the present chapter.

6.2.1 The one-component fluid We consider the model for the compressible liquid with a set of slow variables consisting of (a) the mass density ρ(r, t) and (b) the momentum density g(r, t) (Das and Mazenko,

288

Nonlinear fluctuating hydrodynamics

1986; Das, 1987, 1990). These collective densities are defined in terms of the phase-space variables {rα , pα } as  mδ(x − rα ), (6.2.2) ρ(x, ˆ t) = α

gˆ i (x, t) =

 α

pαi δ(x − rα ).

(6.2.3)

The energy fluctuations are ignored in the present analysis for simplicity. The nonlinear Langevin equations for these slow variables are constructed using the standard prescriptions given above in eqn. (6.2.1). The two main ingredients in reaching the nonlinear equations of motion are the Poisson brackets and the coarse-grained free-energy functional. The Poisson brackets The reversible part of the generalized Langevin equation involves the classical Poisson brackets Q i j . These are computed from the corresponding microscopic definitions (6.2.2) of the respective slow modes in terms of the phase-space variables. Using the canonical Poisson-bracket relation, # $ j rαi , pβ = δi j δαβ , (6.2.4) we obtain the following results for the Poisson brackets between hydrodynamic variables in the one-component fluid: {ρ(x), ˆ ρ(x ˆ  )} = 0, {gˆi (x), gˆj (x )} = −∇x [δ(x − x )gˆi (x)] + ∇xi  [δ(x − x )gˆj (x )], j

(6.2.5)

{ρ(x), ˆ gˆi (x )} = −∇xi [δ(x − x )ρ(x)]. ˆ The Poisson brackets are by definition antisymmetric, Q i j = −Q ji . The coarse-grained free-energy functional In order to compute the reversible streaming-velocity terms as well as the dissipative terms, we need the driving free-energy functional F[a] = F[ρ, g] as a functional of the coarsegrained variables. In Appendix A6.1 we discuss the computation of F for the compressible liquid as a functional of the coarse-grained densities. The functional dependence of F on the hydrodynamic fields, i.e., ρ and g, is divided into two parts, F[ρ, g] = FK [g, ρ] + FU [ρ]. The momentum-density-dependent kinetic part FK is obtained as  g 2 (x) . FK [g, ρ] = dx 2ρ(x)

(6.2.6)

(6.2.7)

The derivation of the kinetic part FK as given in eqn. (6.2.7) can also be obtained from a microscopic approach (Langer and Turski, 1973) starting from a calculation of the

6.2 The compressible liquid

289

equilibrium partition function. This is discussed in Appendix A6.1. The potential part FU of F is treated as a functional of the density ρ only. In Appendix A6.1 we have discussed the simplest approximation in which F[ρ] is obtained from the bare particle interactions as U [ρ]. In the following we discuss a scheme in which the FU [ρ] in terms of the coarse-grained density ρ(x) is constructed by maintaining the analogy with the static or thermodynamic description of the liquid. We use the mass density ρ(x) here instead of the number density n(x). In order to be consistent with the discussions in the earlier chapters and to avoid clutter, we take the two to be the same, i.e., the mass of the fluid particles m = 1. For the one-component system ρ(x) = mn(x). A simple way of constructing the free-energy functional is to choose a form that preserves the static or the equal-time correlation function in the equilibrium state. Consider a formal expansion for FU that is quadratic in density fluctuations,   1 (6.2.8) dx dx δρ(x)M(x − x )δρ(x ). β FU = 2 The equal-time correlation of the density fluctuations at two different points is obtained by taking an average over the function space with respect to the equilibrium distribution exp[−F],  e−FU [ρ] Dn δρ(x)δρ(x ) , (6.2.9) χρρ (x − x ) = δρ(x)δρ(x ) = ZN where ZN is the partition function defined as  ZN = Dρ e−FU [ρ] .

(6.2.10)

In order to compute the static correlation function χρρ , we use the functional identity

 δ e−β FU Dn ρ(x) = 0, (6.2.11) δρ(x ) ZN to obtain the result

. δ FU = δ(x − x ). δρ(x) δρ(x )

-

(6.2.12)

δρ(x) = ρ(x) − ρ0 denotes the density fluctuation. Using eqn. (6.2.12) and definition (6.2.9), −1 . The Gaussian expression (6.2.8) for the free-energy functional F in we obtain M ≡ χρρ U terms of the fluctuations of the coarse-grained density is therefore obtained in terms of the static correlation function χρρ . For liquids supercooled below the freezing point it is well known that there is no drastic change in the equal-time correlation function. Theories of the liquid state can therefore be extended to metastable densities to compute the static correlation function and the latter can be used as an input in the expression for FU . The simplest choice for the correlation χρρ (x − x ) would be to take a completely uncorrelated structure, i.e., ∼δ(x − x ), corresponding to a wave-vector-independent structure factor.

290

Nonlinear fluctuating hydrodynamics

However, at high density the structure is strongly correlated. Another possible choice for the coarse-grained free-energy functional is to reproduce the first peak at wave vector q = q0 of the static structure factor in a Lorentzian form. This involves including terms with the derivatives of density in FU , β FU =

1 2





# $2 dx A0 δρ 2 (x) + κ0 δρ(x) ∇ 2 + q02 δρ(x) .

(6.2.13)

The constants A0 , κ0 , and q0 are temperature- and density-dependent fitting parameters to match the static correlation functions (Das, 1987). A more appropriate form of the coarse-grained free-energy functional is obtained by making use of the expressions for the static correlations in terms of the Ornstein–Zernike direct correlation function c(r ). In the Fourier-transformed form, χnn (k), or equivalently the static structure factor S(k) defined in eqn. (1.2.79), relates to c(k) through the relation (1.2.90). In general FU is split into an ideal-gas or entropic contribution Fid and an interaction term Fin (see Appendix A6.1). On expanding this entropic contribution in (A6.1.30) to quadratic order in density fluctuations and using the relation (1.2.90) between S(k) and c(k), we obtain the following expression for the interaction part of the free-energy functional up to quadratic order of the coarse-grained density: 1 β Fin [ρ] = − 2 2m



dx dx c(2) (x − x  )δρ(x)δρ(x  ).

(6.2.14)

The coarse-grained free-energy functional FU [n] therefore takes the form referred to as the Ramakrishnan–Yussouff form in Chapter 2. The total free energy is obtained as F[ρ, g] = FK [ρ, g] + FU [ρ]       ρ(x)30 ρ(x) g 2 (x) + ln −1 = dx 2ρ(x) βm m  1 dx dx c(2) (x − x  )δρ(x)δρ(x  ). − 2βm 2

(6.2.15)

The mode-coupling models for glassy dynamics in this book will be largely based on the above Gaussian form of FU [ρ]. Langevin equations Let us first consider the equation for mass density ρ(x, t). The momentum dependence of F[ρ, g] is such that the continuity equation ∂ρ + ∇ ·g = 0 ∂t

(6.2.16)

6.2 The compressible liquid

291

is maintained even for the fluctuating variables. The reversible term for the equation for ρ is obtained as  δF dx{ρ(x), g j (x )} Vρ = δg j (x ) j

=−

 j

δF ∇ j ρ(x) . δg j (x)

(6.2.17)

In order to obtain the continuity eqn. (6.2.16) we require that g j (x) δF = , δg j (x) ρ(x)

(6.2.18)

which was obtained from a detailed microscopic consideration in Appendix A6.1. Using the Poisson-bracket relations (6.2.5), the streaming velocity for the momentum density is given by  gi g j δ FU ∇j Vgi (x) = −ρ(x)∇i − . (6.2.19) δρ(x) ρ j

The remaining steps in constructing the equations of motion for the hydrodynamic modes concern obtaining dissipative terms in the momentum equations. This requires that the element of the dissipative matrix L corresponding to one of the elements being ρ is L 0ρa = 0. The matrix elements L 0gi g j are chosen to be the same as those presented in Section 5.1.3 for the isotropic fluid (see eqn. (5.1.44)). The nonlinear equations for the components of the momentum density g are therefore obtained in a generalized form of the Navier–Stokes equation (Zuberav et al., 1997),    gi g j gj δ FU  ∂gi ∇j L i0j (6.2.20) = −ρ∇i − + + fi . ∂t δρ ρ ρ j

j

The second term on the RHS of eqn. (6.2.20) refers to the well-known Navier–Stokes nonlinearity and is a result of coupling of convective currents. The stochastic term in the generalized Langevin equation (6.2.20) is denoted by the noise f i , which is assumed to be Gaussian. The correlation of the noise is related to to the bare damping matrix L i0j through the following fluctuation–dissipation relation:  f i (r, t) f j (r , t  ) = 2kB T L i0j (r, r )δ(t − t  ).

(6.2.21)

The above set of equations (6.2.16) and (6.2.20) forms the basic set of nonlinear fluctuating hydrodynamic equations for the set {ρ, g}. The presence of 1/ρ in various terms on the RHS of (6.2.20) presents an odd kind of nonlinearity that creates an infinite set of nonlinear terms. For compressible liquids both the convective term (the second term) and the dissipative term (the third term) of the RHS of this equation contain the ρ −1 nonlinearity.

292

Nonlinear fluctuating hydrodynamics

Typically, dealing with the 1/ρ nonlinearity is avoided by working with the velocity field v(x, t) defined through the relation g(x, t) = ρ(x, t)v(x, t).

(6.2.22)

For an incompressible liquid this is a trivial relation, since ρ(x, t) = ρ0 is a constant. However, for a compressible liquid this imposes a nonlinear constraint on the fluctuating fields. The density nonlinearity present in the reversible part of the equation of motion (6.2.20) has important consequences for the dynamics of compressible liquids. This is included in the first term representing the pressure functional. Assuming a local functional form of FU with respect to the density ρ,  (6.2.23) FU = dx f [ρ(x)], the first term in eqn. (6.2.20) is expressed as ∇i P[ρ], with the pressure functional P[ρ] satisfying the standard thermodynamic relation   ∂f P=ρ − f. (6.2.24) ∂ρ For a set of three variables {ρ, g, v}, the equations of motion are eqn. (6.2.16), eqn. (6.2.20) expressed as  ∂gi  ∇ j (gi v j ) + ∇i P − L i0j v j = f i , (6.2.25) + ∂t j

j

and the nonlinear constraint (6.2.22). By expressing the free-energy functional FU [ρ] as a polynomial in terms of the density fluctuations, the corresponding nonlinear equations for the dynamics of the momentum densities gi are obtained. As indicated above, in the simplest case we choose for the freeenergy functional FU , which corresponds to a wave-vector-independent structure factor χ . In this case the free-energy density is given by f [ρ] ≡

χ −1 (δρ)2 . 2

(6.2.26)

The corresponding expression for the pressure functional P is obtained from (6.2.24) as P = c02 δρ + χ −1

(δρ)2 , 2

(6.2.27)

where c02 = ρ0 χ −1 gives the speed of sound in the liquid. Thus, even with a purely Gaussian free energy, we have a nonlinearity in the dynamic equation for the momentum density and hence the momentum density is termed a nonlinearity of purely dynamic origin. For a more realistic description of the dynamics a standard choice for the interaction

6.2 The compressible liquid

293

part Fin of the free-energy functional FU [ρ] is as stated above in eqn. (6.2.14) in terms of the direct correlation functions. Using the free-energy functional given in eqn. (6.2.15), the equations of motion for the gi are obtained as   ∂ ∇ j [ρvi (x, t)v j (x, t)] + ∇i dx U (x, x  )δρ(x  , t) gi (x, t) + ∂t j

 dx1 dx2 V i (x, x1 , x2 )δρ(x1 , t)δρ(x2 , t)

+ −



d x  L i0j (x − x  )v j (x  , t) = f i (x, t).

(6.2.28)

j

The kernels U and V i in the above equation are obtained as U (x, x  ) = V i (x, x1 , x2 ) =

1 [δ(x − x  ) − n 0 c(x − x  )], βm

(6.2.29)

1 [δ(x − x2 )∇i c(x − x1 ) + δ(x − x1 )∇i c(x − x2 )], βm 2 (6.2.30)

where we denote ∇i ≡ ∂/∂ xi . The above equation for the momentum density gi obtained with a quadratic form of Fin is therefore consistent with a thermodynamic description of the liquid used in the density-functional theory of the freezing transition (see also Appendix A6.1). The nonlinear terms in the dynamic equations result from the special Poisson-bracket structure of the conserved densities of mass and momentum. The presence of these nonlinearities modifies the dynamic behavior of the fluid obtained from the linear theory. In the next chapter we present the scheme for the construction of such a renormalized perturbation theory of the dense liquids.

6.2.2 The nonlinear diffusion equation We have considered a set of nonlinear Langevin equations for the liquid involving the slow modes of mass and momentum density, denoted by ρ and g, respectively. We consider in the present section a formulation in which the dynamics is described in terms of the equation of motion of the single conserved variable density ρ(x, t). We demonstrate here how, under some suitable approximations, the momentum density g(x, t) can be integrated out of the problem, with the equation of motion ρ(x, t) being obtained as a nonlinear Langevin equation similar to a diffusion equation. Such models for the dynamics of fluids have found many applications in the case of colloids. The microscopic-level dynamics of the colloid particles in a surrounding medium is considered to have a random component and is intrinsically dissipative.

294

Nonlinear fluctuating hydrodynamics

The nonlinear Langevin equations for a one-component fluid obtained in the previous section for the set of collective modes {ρ, g} are ∂ρ + ∇ ·g = 0 ∂t

(6.2.31)

δ FU  0 g j ∂gi Li j = −ρ∇i − + fi , ∂t δρ ρ

(6.2.32)

and

j

where we have ignored the convective nonlinearity involving coupling of momentum densities g, since the primary aim here is to focus on the dynamics of the density fluctuations. From eqns. (6.2.31) and (6.2.32) the momentum density g can be eliminated (Munakata, 1994, 1996). For this we assume that the momentum density relaxes fast compared with the density. This is termed the adiabatic approximation. A crucial step in this elimination involves assuming that the bare transport coefficient L i0j is dependent on the fluctuating variable ρ(x, t). With the above ad-hoc assumption of making the noise multiplicative, the equation for the momentum density reduces to the form δ FU ∂gi = −ρ∇i + 0 gi + ρ 1/2 ζi . ∂t δρ

(6.2.33)

The noise correlation given by eqn. (6.2.21) is now replaced with the FDT relation ζi (r, t)ζ j (r , t  ) = 2kB T i0j δ(r − r )δ(t − t  ).

(6.2.34)

The noise in the equations is now therefore multiplicative. The correlation matrix is assumed to be diagonal for simplicity: i0j = 0 δi j , where 0 has dimensions of inverse time, representing a microscopic time scale of the problem considered here. Note that, from the structure of the above equation of motion for the momentum density g, it is apparent that the latter is no longer being treated like a conserved mode. Now, in the adiabatic approximation we put the LHS of eqn. (6.2.33) equal to zero, thereby obtaining

δ FU −1 gi = 0 −ρ∇i + fi . (6.2.35) δρ By substituting this into the continuity equation (6.2.31) we obtain the equation of motion for the density variable as

δ FU ∂ρ √ −1 (6.2.36) = 0 ∇ · ρ ∇ + ρθ. ∂t δρ The noise θ (x, t) has the correlation θ (r, t)θ (r , t  ) = 2Ds ∇ · ∇  δ(r − r )δ(t − t  ),

(6.2.37)

6.2 The compressible liquid

295

where we have the bare diffusion coefficient Ds in terms of the kinetic coefficient 0 as Ds =

kB T . 0

(6.2.38)

Using 0−1 as the basic unit of time, we obtain the basic equation for the density fluctuation,

∂ρ δ FU √ =∇· ρ∇ + ρθ. ∂t δρ

(6.2.39)

Note that the deduction outlined above involves taking the ad-hoc step of making the noise multiplicative. Indeed, this removes the 1/ρ nonlinearity present in the dynamic equation (6.2.32), reducing it to the form (6.2.33). In Section 8.1.6 we will further discuss these interchanges between the 1/ρ nonlinearity and the multiplicative noise. Kawasaki and Miyazima (1997) demonstrated from a proper MSR field-theoretic analysis that, on integrating out the momentum density g from the problem in the adiabatic approximation, the multiplicative noise follows in a natural way.

6.2.3 A two-component fluid Let us now consider the equations of nonlinear fluctuating hydrodynamics for a binary mixture. The slow variables in this system are the two partial densities ρs (x), s = 1, 2, and the total momentum density g(x). The two densities representing conservation of the two species are defined microscopically as ρs (x) = m s

Ns 

  δ x − rαs (t) ,

(6.2.40)

α=1

where m s and rαs (t) denote the mass and the position of the αth particle of the sth species, respectively. The momentum densities gs for the two species are represented similarly as gs (x) =

Ns 

  pαs δ x − rαs (t) ,

s = 1, 2,

(6.2.41)

α=1

where psα is the momentum of the αth particle of species s (s = 1 or 2) and Ns is the number of particles of the sth species in the mixture. The total number of particles is given by N=

2 

Ns .

(6.2.42)

s=1

The relative abundance of each of the two species is given by xs = Ns /N for s = 1, 2, such that x1 + x2 = 1. The number of particles of species s per unit volume V is defined as n s0 = Ns /V and the total number density as

296

Nonlinear fluctuating hydrodynamics

n0 =

N = n 10 + n 20 . V

(6.2.43)

Note that the average mass density ρ0 is obtained as ρ0 = m 1 n 10 + m 2 n 20 .

(6.2.44)

g = g1 + g2 ,

(6.2.45)

The total momentum density,

in a binary mixture is a conserved property and hence a hydrodynamic variable. The equations for the dynamics of the slow variables consisting of the set {ρ1 , ρ2 , g} are obtained using the standard procedure outlined above. We present the equations of nonlinear fluctuating hydrodynamics here with an alternative choice of slow modes. The theory is formulated in terms of the total density ρ(x, t) = ρ1 (x, t) + ρ2 (x, t)

(6.2.46)

and the concentration variable defined as c(x, t) = x2 ρ1 (x, t) − x1 ρ2 (x, t).

(6.2.47)

The average of c(x, t) is obtained as c0 = n 0 (m 1 − m 2 )x1 x2 .

(6.2.48)

The set of slow modes in terms of which the dynamics is formulated is ψ ≡ {δρ, δc, g}, where δρ = ρ − ρ0 and δc = c − c0 . The fluctuating equations for this set of slow variables are obtained using the standard procedure outlined in Section 6.1.2. The generalized Langevin equations for the slow mode ψ(x, t) are obtained in the standard form δF ∂ψi (t)  = θi (t), (6.2.49) + Q i j + L i0j ∂t δψ j where Q i j = {ψi , ψ j } denotes the Poisson bracket between the slow variables ψi and ψ j . The noise θi is assumed to be white and Gaussian, with its correlation defining the bare transport matrix L0 as θi (x, t)θ j (x , t  ) = 2kB T L i0j δ(x − x )δ(t − t  ).

(6.2.50)

The deduction of the above nonlinear equations for the binary mixture therefore requires as inputs (a) the Poisson bracket Q i j between the slow modes {ψ}; (b) the free-energy functional F, which specifies the equilibrium state of the binary liquid; and (c) the dissipative matrix L0 defined by the corresponding noise correlations.

6.2 The compressible liquid

297

We consider these quantities in the following. (A) The Poisson brackets. The reversible, or Euler, part of the equations of motion (6.2.49) is obtained in terms of the Poisson brackets Q i j between the slow variables. Using the basic relation # $ j rsi , ps  = δss  δαβ , (6.2.51) we obtain the following Poisson brackets between the slow variables: {ρ(x), gi (x )} = −∇i [δ(x − x )ρ(x)],

(6.2.52)

{c(x), gi (x )} = −∇i [δ(x − x )c(x)],

(6.2.53)

{c(x), ρ(x )} = 0.

(6.2.54)

The corresponding Poisson-bracket relations between the partial densities {ρs } are obtained as {ρs (x), gi (x )} = −∇i [δ(x − x )ρs (x)],

(6.2.55)

{gi (x), g j (x )} = −∇ j [δ(x − x )gi (x)] + ∇i [δ(x − x )g j (x )].

(6.2.56)

(B) The free-energy functional. The equilibrium behavior of a system with Langevin dynamics is determined by the functional F. Similarly to the one-component case, it has two parts, the kinetic and the “potential,” denoted by FK and FU , respectively, F = FK + FU .

(6.2.57)

The kinetic part FK of the free-energy functional is expressed in terms of the slow variables g and ρ following the same standard procedure (Langer and Turski, 1973) as that described for the one-component system in Appendix A6.1. The details of this deduction, beginning from a microscopic Hamiltonian for the binary fluid, are available in Harbola and Das (2003). We obtain 2

 g (x) 1 dx . (6.2.58) FK = 2 ρ(x) Note that the total density ρ = ρ1 + ρ2 appears in the denominator of (6.2.58). This term is essential for generating the Gallelian invariant form of the generalized Navier–Stokes equation obtained for the momentum density g discussed below. For the potential part FU of the free energy, there is an ideal-gas contribution together with the interaction term, FU = Fid [ρ, c] + Fin [ρ, c].

(6.2.59)

The ideal-gas part Fid is obtained as a generalization of the corresponding onecomponent result in the form  

 1  ρs (x) 3 dx ρs (x) ln Fid [ρ1 , ρ2 ] = s − 1 , (6.2.60) ms ms s

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Nonlinear fluctuating hydrodynamics

√

where s = h β/(2πm s ) denotes the corresponding thermal de Broglie wavelength. In terms of variables {ρ, c}ρ1 and ρ2 are expressed as ρs (x, t) = xs ρ(x, t) + as c(x, t),

(6.2.61)

where the constant as is defined as as = ±1 for s = 1 and s = 2, respectively. We express Fid as Fid [ρ, c] =

2   s=1



x s ρ + as c dx ms

  

x s ρ + as c 3 s − 1 . ln ms

(6.2.62)

The interaction part Fin is obtained in the standard form in terms of the Ornstein– Zernike direct correlation functions for a two-component system. The latter are denoted by css  between species s and s  . Fin for the two-component system is written as a generalization of the corresponding result (6.2.14) for the one-component case,  1 dx dx css  (x − x )δρs (x)δρs  (x ). (6.2.63) Fin (ρ) = − 2m s m s  Using the transformation (6.2.61), the free energy FU is expressed as an expansion in terms of the corresponding fluctuations δρ and δc:  ! 1 dx dx cρρ (x − x )δρ(x)δc(x) FU (ρ) = Fid [ρ, c] − 2 " + ccc (x − x )δc(x)δc(x) + 2ccρ (x − x )δc(x)δρ(x) . (6.2.64) The coefficients cρρ , ccc , and cρc in the expansion (6.2.64) are obtained in terms of the Ornstein–Zernike direct correlation functions css  for a binary mixture defined in eqn. (6.2.63) as follows: cρρ = x12 c˜11 + x22 c˜22 + 2x1 x2 c˜12 ,

(6.2.65)

cρc = x1 c˜11 − x2 c˜22 + (x2 − x1 )c˜12 ,

(6.2.66)

ccc = c˜11 + c˜22 − 2c˜12 ,

(6.2.67)

where we have defined c˜ss  =

1 css  , m s m s

for s, s  = 1, 2.

(6.2.68)

The nonlinearities in the fluctuating equations relevant for producing slow dynamics (as we will see in the next chapter) are present even for a purely quadratic or Gaussian free-energy functional FU of δρ and δc. This is similar to the corresponding result for the one-component system. The origin of the nonlinearities is therefore purely

6.2 The compressible liquid

299

Gaussian in nature. If we consider the binary mixture for which m 1 = m 2 and the two species are identical, we have css  ≡ c, where c is the direct correlation function of the one-component system. It follows from (6.2.65)–(6.2.67) that cρρ = c and ccc = cρc = 0, and the expression (6.2.64) for FU reduces to the one-component result (6.2.14). (C) The bare transport coefficients. The bare-transport-coefficient matrix L0 is constructed using simple physical considerations. Since the current of the total density, i.e., the momentum density itself, is a conserved quantity, there is no dissipative part in the equation for the total density ρ and the form of the continuity equation is maintained. Thus we have L 0ρα = L 0αρ = 0 for α ≡ {ρ, c, gi }. The matrix elements L 0gi g j ≡ L i0j corresponding to the momentum current density involve the viscosity tensor given by   1 0 2 (6.2.69) ∇i ∇ j + δi j ∇ − ζ0 ∇i ∇ j , L i j (x) = −η0 3 where η0 and ζ0 are the bare shear and bulk viscosities, respectively. The longitudinal viscosity is given by 0 = (4η0 /3 + ζ0 ). On the other hand, L 0cc = γ0 ∇ 2 , with γ0 representing the inter-diffusion constant. For the matrix elements L 0cgi , we note from the relation (6.1.77) that, using the time-reversal property of the fields c and gi , L 0cgi (t) = −L 0cgi (−t).

(6.2.70)

Since we have white noise, in order to satisfy the above condition we must have L 0cgi = L 0gi c = 0. With the above choice of the free-energy functional, and the Q i j and L i j matrices, we obtain the equations of motion for {ρ, c, gi }. The equation corresponding to ρ is just the continuity equation (6.2.16) with g as the current, ∂ρ + ∇ · g = 0. ∂t

(6.2.71)

The Langevin equation for c is a diffusive equation with a coupling to the total density ρ,

δ FU gi ∂c + ∇i c + γ0 ∇ 2 = f. (6.2.72) ∂t ρ δc The corresponding equation for the momentum density gi is obtained as

gi g j gj δ FU δ FU ∂gi + ∇j + ρ∇i + c∇i + L i0j = θi . ∂t ρ δρ δc ρ

(6.2.73)

The noise terms in the Langevin equations (6.2.73) and (6.2.72) for gi and c are denoted by θi and f , respectively. These are assumed to be white Gaussian and are related to the corresponding bare transport coefficients by θi (x, t)θ j (x , t  ) = 2kB T L i0j δ(x − x )δ(t − t  ),

(6.2.74)

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Nonlinear fluctuating hydrodynamics

 f (x, t) f (x , t  ) = 2kB T γ0 ∇ 2 δ(x − x )δ(t − t  ),

(6.2.75)

θi (x, t) f (x , t  ) = 0.

(6.2.76)

The functional derivatives of the free energy FU with respect to ρ or c needed in the construction of the equations of motion are obtained from the expression (6.2.64) for this quantity as an expansion in terms of δρ and δc. 6.2.4 The solid state The extended set of slow variables of the solid includes the conserved modes as well as the Nambu–Goldstone modes discussed in the previous chapter (see Section 5.4) in the context of linear dynamics. The nonlinear Langevin equations for these collective modes are obtained using the standard techniques outlined above. The reversible part of the dynamics is expressed in terms of the Poisson brackets Q i j = {ψi , ψ j } between the different slow modes of the set {ψi }. For evaluating these Poisson brackets mentioned above we need the microscopic definitions for the variables ψˆi (x) corresponding to fields ψi (x). We consider a system of N classical particles each of mass m. rα (t) and pα (t), respectively, are the position and the momentum of the αth particle at time t. For the density ρˆ and the momentum density gˆ we use the standard prescription (5.3.34). The extra slow modes or the Nambu–Goldstone modes to be added to the set of conserved densities take into account the broken symmetry of the solid state with elastic properties. We define the fluctuating variable u(x, t) by considering the displacements of the individual particles uα (t) (α = 1, . . . , N ) from their respective mean positions denoted by r0α , ρ(x, ˆ t)uˆi (x, t) = m

N  α=1

u iα (t)δ(x − rα (t)),

(6.2.77)

# $ j such that rα (t) = r0α + uα (t). Using the canonical Poisson-bracket relation rαi , pβ = δi j δαβ we obtain the following relations for the hydrodynamic variables:  {gˆi (x), uˆj (x )} = −δ(x − x ) δi j − ∇xi  u j (x ) , {uˆ i (x), uˆ j (x )} = 0,

(6.2.78)

{uˆ i (x), ρ(x ˆ  )} = 0. Using eqns. (6.2.78) and (6.2.58), we obtain the following forms for the streaming velocity V [ψ] in the nonlinear Langevin eqn. (6.2.1): Vρ = −∇ · g,

) δF gi g j δ FU ( U i − ρ∇xi − δi j − ∇xi u j (x) , Vg = −∇ j ρ δρ δu j (x) Vui =

gi g − · [∇x u i ]. ρ ρ

(6.2.79)

6.2 The compressible liquid

301

The dissipative parts of the dynamical equations are expressed in terms of the bare transport coefficients and are assumed to be the same as those in the case of the linear dynamics. For the momentum-density equation, the irreversible part has the standard form with the dissipative coefficient in eqn. (6.2.1) given by  L gi g j ≡

L i0j

= −η0

 1 2 ∇i ∇ j + δi j ∇ − ζ0 ∇i ∇ j , 3

(6.2.80)

where ζ0 is the bare bulk viscosity and η0 is the bare shear viscosity. We define the longitudinal viscosity as 0 = ζ0 + 4η0 /3. The dissipative term in the equation for the u field is assumed to have the simple time-dependent Ginzburg–Landau (TDGL) form, L u i u j ≡ i j = 0 δi j .

(6.2.81)

We require that all other L i j = 0. In the fluctuating-hydrodynamics description the bare transport coefficients ζ0 , η0 , and 0 act as external parameters. For the bare viscosities Green–Kubo-type relations can be evaluated in a kinetic-theory approach with suitable models. The time scale corresponding to the quantity 0 which relates to the vacancy diffusion is, however, much longer. The construction of the nonlinear Langevin equation (6.2.1) involves the free energy. The effective Hamiltonian F contains, in addition to the usual terms for an isotropic liquid, the energy cost due to distortion in the elastic solid. The elastic part is constructed in terms of the gradient of the displacement field u using the nonlinear strain field si j =

1 1 (∇i u j + ∇ j u i ) − ∇i u m ∇ j u m . 2 2

(6.2.82)

λ0 and μ0 are related to the bulk and shear modulus of the amorphous solid. The longitudinal modulus is given by ϑ0 = λ0 + μ0 . We also consider a term representing the coupling of the density fluctuations to the elastic strain field si j . In the simplest form it is a coupling between the density and the trace S of si j . At the linear level in u, S is simply ∇ · u, which in our discussion of the linear dynamics we have seen to be related to the defect density (ρD ) in the solid state. The free energy F is obtained as the sum of two parts as in the case of the liquid in eqn. (6.2.6). The kinetic part FK is given by (6.2.7) as before. The potentialenergy part FU [ρ, u] is constructed preserving the rotational and translational invariance of the system, FU =

1 2



 dx χ −1 (δρ)2 + 2ς0 S δρ + λ0 S 2 + 2μ0 (si j s ji ) .

(6.2.83)

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Nonlinear fluctuating hydrodynamics

The couplings χ and ς0 in the effective Hamiltonian relate to the static structure factor for the system. Using eqn. (6.2.83), the following set of fluctuating nonlinear equations is obtained: ∂ρ + ∇ · g = 0, ∂t ∂gi  ∇ j σi j = θi , + ∂t j

∂u i δF + fi , + v · ∇u i = vi − i j ∂t δu j

(6.2.84)

together with the nonlinear constraint g = ρv.

(6.2.85)

The stress-energy tensor σi j has reversible and dissipative (irreversible) parts, denoted by σiRj and σiDj , respectively, such that σi j = σiRj + σiDj , where

σiRj

= ρVi V j +

c02 δρ

− [λ0 S + ς0 δρ]

(6.2.86)

χ −1 λ0 2 μ0 2 + (δρ) + ρ0 ς0 S − S − (slm sml ) δi j 2 2 2 ! " ∂S ∂S − μ0 si j − 2sim s jm . ∂∇i u m ∂∇ j u m

The dissipative part is related to gradients of the velocity field,

2 σiDj = −η0 ∇i V j + ∇ j Vi − δi j (∇ · v) − ζ0 δi j (∇ · v). 3

(6.2.87)

(6.2.88)

The stress tensor satisfies σi j = σ ji , which guarantees conservation of angular momentum in the system. The random parts in the above fluctuating equations are Gaussian noise terms and are related to the bare transport coefficients by θi (x, t)θ j (x , t  ) = 2ρ0 kB T L i0j δ(x − x )δ(t − t  ),  f i (x, t)θ j (x , t  ) = 0,

(6.2.89)

 f i (x, t) f j (x , t  ) = 2kB T i j δ(x − x )δ(t − t  ). The set of nonlinear Langevin equations obtained above gives the dynamics of the slow modes for a solid with elastic properties. It is generally applicable to a crystal in which long-range translational symmetry is broken. In this case the free energy defined in eqn. (6.2.83) involves the rank-four elastic tensor Ci jkl . Similarly, the viscosity tensor ηi jkl appears in the dissipative stress tensor σiDj . These will respectively involve more elastic

6.3 Stochastic balance equations

303

and transport coefficients as required by the symmetry of the corresponding crystal, making the formulation extremely complicated. The simplified description given above for the isotropic system is more relevant to the case of amorphous solids in which freezing has occurred at the scale of local structure but overall translational invariance is maintained over longer distances. However, this inherently brings into the above description an underlying time scale over which the definition of the displacement field is applicable. The particles vibrate around their mean positions, forming a random lattice structure that corresponds to a metastable minimum of the potential energy of the many-particle system. In the deeply supercooled glassy state the solid-like behavior persists for very long times over which the above solid-like description applies.

6.3 Stochastic balance equations In the previous sections of this chapter we have considered the dynamics for a system in which the individual particles follow time-reversible equations of Hamiltonian mechanics. We discussed the formulation of linear as well as nonlinear Langevin equations in terms of collective densities like the mass density ρ(x, t) and g(x, t). These coarse-grained densities or averaged quantities whose space and time variations are smooth have dynamics described in terms of partial differential equations, which have in general reversible as well as irreversible parts. On the other hand, we have non-coarse-grained densities corresponding (see definitions (5.1.1)–(5.1.3) in Chapter 5) to exact balance equations (see eqns. (5.1.4)–(5.1.6) with the respective currents). These equations are reversible since the microscopic dynamics is reversible. Irreversibility in the Langevin equations for the coarse-grained densities in the case of Newtonian dynamics is phenomenologically introduced. In the present section we will discuss the situation in which the dynamics at the microscopic level is chosen to be irreversible. This can represent the dynamics of the colloid particles in a solvent. We describe the dynamics of the individual particle approximately with an equation of motion that is a Langevin equation with a dissipative as well as fluctuating term representing noise. This approximates the effect of the solvent molecules of much smaller inertia constantly colliding with the bigger colloidal particles. With this dissipative microscopic dynamics we obtain here a corresponding set of exact balance equations, similarly to the case of Newtonian dynamics discussed in the earlier sections.

6.3.1 Smoluchowski dynamics In the present section we consider an N -particle system in which the microscopic dynamics of the constituent particles is described with the Smoluchowski equations (Smoluchowski, 1915) involving only the particle coordinates. The momentum dependence of the particles is ignored in the over-damped limit. A colloidal system with heavy particles in a solution is a typical example of such a system. Let each of the N particles be of mass m, with their

304

Nonlinear fluctuating hydrodynamics

position coordinates given by {xα } for α = 1, . . . , N . The density at position x at time t is formally defined as1 ρ(x, ˆ t) =

N 

δ(x − xα (t)) ≡

α=1

N 

ρˆα (x, t),

(6.3.1)

α=1

where we put the hat on ρ to indicate that it is a microscopic function dependent on the phase-space densities {xα }. Since the number N of particles is constant, ρ(x, ˆ t) is a microscopically conserved quantity. The microscopic dynamics of the N -particle system is assumed to follow Brownian dynamics. The momentum of the individual particles does not enter the description of the dynamics in the so-called over-damped limit. The equation of motion for each particle is given by the stochastic equation representing the irreversible dynamics,  dxα (t) ∇α U (xα (t) − xβ (t)) + fα (t), =− dt N

ζB

(6.3.2)

β=1

where we have used the following notation to represent the derivative operator ∇α = ∂/∂xα . We adopt the notation that the β = α contribution of the summation in the second term on the RHS is zero. The stochastic part or the noise f is white and Gaussian, with its correlation given by + , j f αi (t) f β (t  ) = 2kB T δαβ δi j δ(t − t  ). (6.3.3) In the dimensionality-correct form of the equation of motion (6.3.2), we note that the factor ζB on the LHS denotes a friction coefficient associated with the microscopic dynamics. It has the dimension of inverse time. To keep the notation simple, we take ζB = 1 in the following. To obtain an equation for the density ρ(x, ˆ t) defined above, we make use of the following chain rule of stochastic differential equations in Itô calculus (Oksendal, 1992). Let us consider a set of stochastic variables xi (t) (i = 1, . . . , m) satisfying the stochas2 x˙i = h i + gi j ξ j , where the correlation of the white noise ξi is defined as 1tic equation ξl (t)ξm (t  ) = δlm δ(t − t  ). The Itô chain rule gives the stochastic differential equation for the variable y({xi }) in the form y˙ =

 ∂y  1 ∂2 y x˙i + gik gk j . ∂ xi 2 ∂ xi ∂ x j i

(6.3.4)

i, j,k

This above result follows directly on taking the deviation of the function y(xi ) in terms of the corresponding variation in the xi . We apply the above chain rule for the stochastic differential equations for the variables xα (t) for α = 1, . . . , N with the respective differential equations being given by (6.3.2). Here we adopt the notation that the Greek indices 1 In dealing with the nonlinear dynamics of the density field we take the mass m of the particles as unity in order to keep the

notation simple. This makes the number density n(x) and mass density ρ(x) the same.

6.3 Stochastic balance equations

305

correspond to the particle labels while the Roman indices are for the Cartesian components. In this case we define the function y as ˆ t) = y({xα }) ≡ ρ(x,

N 

δ(x − xα ).

(6.3.5)

α=1

Note that the coordinate x acts as a label on ρ. ˆ In this case the matrix gi j = δi j is diagonal. The corresponding stochastic differential equation for the density variable ρ(x, ˆ t) is then obtained as ⎡ ⎤   ∂ {∇α ρˆα (x, t)} · ⎣− ∇α U (xα − xβ ) + fα (t)⎦ ρ(x, ˆ t) = ∂t α β

+ kB T



∇α · ∇α ρˆα (x, t)

α

= −∇ ·



⎡ ρˆα (x, t) ⎣fα (t) − ∇α



d x  U (xα − x )

α



⎤ δ(x − xβ )⎦

β

+ kB T ∇ 2 ρ(x, ˆ t) = −∇ ·





 ρˆα (x, t) fα (t) − ∇α d x  U (xα − x )ρ(x ˆ  , t)

α

+ kB T ∇ 2 ρ(x, ˆ t)

3

ˆ t) + ∇ · ρ(x, ˆ t)∇ = kB T ∇ 2 ρ(x, & −∇·



4

d x  U (x − x )ρ(x ˆ  , t)

' ρˆα (x, t)fα (t) .

(6.3.6)

α

In reaching the above equation, we have generally used the trick for the derivative of the function that ∇α ρˆα ≡ ∇α δ(x − xα ) = −∇δ(x − xα ).

(6.3.7)

Hence we obtain an equation for the time evolution of the fluctuating density ρ(x, ˆ t): 3 4  ∂ ρ(x, ˆ t) ˆ t) + ∇ · ρ(x, ˆ t)∇ d x  U (x − x )ρ(x ˆ  , t) = kB T ∇ 2 ρ(x, ∂t − η (x, t).

(6.3.8)

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Nonlinear fluctuating hydrodynamics

In eqn. (6.3.8) we have defined the random force denoted by η (x, t) as η (x, t) = −

N 

∇ · {ρˆα (x, t)fα (t)}.

(6.3.9)

α=1

The correlation of the noise η is obtained as 2 1  ˆ t)ρ(x ˆ , t ) η (x, t)η (x , t  ) = 2kB T δ(t − t  )∇x · ∇x  ρ(x, = 2kB T δ(t − t  )∇x · ∇x  [δ(x − x )ρ(x, ˆ t)],

(6.3.10)

where the angular brackets on the LHS indicate the average over the noise f α in the microscopic equations of motion. We have used in obtaining eqn. (6.3.10) the following property of the delta function: ˆ t)ρ(x ˆ  , t)} = ∇x · ∇x  δ(x − x )ρ(x, ˆ t). ∇x · ∇x  {ρ(x,

(6.3.11)

We now redefine the noise η(x, t) as η(x, t) = ∇i {ρˆ 1/2 (x, t)θi (x, t)}, where θi (x, t) is a global noise with the correlation 2 1 θi (x, t)θ j (x , t  ) = 2kB T δ(t − t  )δi j δ(x − x ).

(6.3.12)

(6.3.13)

It then easily follows that the two noises η and η are statistically identical. The equation of motion (6.3.8) for the density field then takes the form    ∂ ρ(x, ˆ t) δ FˆBD = DB ∇i ρ(x, + ∇i {ρˆ 1/2 (x, t)θi (x, t)}, ˆ t)∇i ∂t δ ρ(x, ˆ t)

(6.3.14)

where we have written ζB (previously set equal to 1) explicitly in terms of the constant DB , kB T DB = . (6.3.15) ζB √ DB has the dimension of a diffusion constant. The noise ∇ · { ρθ } in the Langevin equation is multiplicative. The quantity FˆBD in the first term on the RHS of eqn. (6.3.14) is a functional of the density ρ(x, ˆ t),   1 id FˆBD = FˆBD + ˆ U˜ (x − x )ρ(x ˆ  ), (6.3.16) dx dx ρ(x) 2 id is the corresponding functional where U˜ = βU is the interaction scaled with kB T and FˆBD for the noninteracting system (ideal gas),  id ˆ ρ(x)) ˆ − 1]. (6.3.17) FˆBD = β −1 dx ρ(x)[ln(

6.3 Stochastic balance equations

307

id on ρˆ is identical to that of the free energy obtained The functional dependence of FˆBD as a functional of the coarse-grained density ρ in eqn. (2.1.27) in Chapter 2. The equilibrium distribution for the system follows from the stationary solution of the Fokker–Planck equation corresponding to the Langevin equation (6.3.14) and is given by

PEQ =

exp[−FBD ] , Z

(6.3.18)

where Z is a normalizing factor. Equation (6.3.14) for the microscopically conserved quantity density follows the form of a continuity equation, ∂ ρ(x, ˆ t) + ∇ · jρˆ = 0. ∂t The current

 jρˆ = −DB ρˆ ∇

δ FBD δ ρˆ

 −

; ρθ ˆ

(6.3.19)

(6.3.20)

has a regular part as well as a random part. Even with a Gaussian or quadratic interaction part for the free-energy functional FˆBD , eqn. (6.3.14) is nonlinear. However, the ideal gas ˆ produces a linear term in the part of FBD , which involves the non-Gaussian term ln ρ, equation of motion. The functional FˆBD [ρ] ˆ in eqn. (6.3.16) is similar to the free energy FU [ρ] expressed in eqn. (6.2.15) as a functional of the corresponding coarse-grained density ρ in a Newtonian fluid. This similarity holds provided that we identify the bare interaction U˜ (r ) with the direct correlation functions −c(r ) at the lowest order. With this identification, eqn. (6.3.14) for the density ρ(x, ˆ t) (Dean, 1996) is identical to eqn. (6.2.39) for the coarse-grained density for ρ(x, t) (Kawasaki, 1998). Microscopic derivations (Yoshimori, 2005) of similar equations using projection-operator methods have also been done. Owing to the apparent similarity in structure between the nonlinear diffusion equation (6.2.39) for ρ(x, t) and eqn. (6.3.14) for ρ(x, ˆ t), they have often been conflated as the Dean–Kawasaki equation. However, it is important to note that Kawasaki’s eqn. (6.2.39) is for the coarse-grained density and involves making the adiabatic approximation regarding the dynamics of the momentum fluctuations. On the other hand, the microscopic dynamics itself is chosen as dissipative in the case of Dean’s eqn. (6.3.14). The latter is an exact balance equation for the microscopic density function ρˆ similar to the Euler equations (5.1.4)– (5.1.6) obtained in the Newtonian case. The idea of mixing these two different equations should therefore be treated with caution. We refer to the formulation of the dynamics in terms of the stochastic eqn. (6.2.39) as dynamic density-functional theory (DDFT). In Section 8.1.6 we discuss a possible ergodic–nonergodic transition in this so-called dynamic density-functional theory using density as the single variable. The tagged-particle density ρs (x, t) is also a conserved quantity, and a similar balance equation for it follows as well. The density ρs is defined as ρˆs (x, t) ≡ ρˆα = δ(x − xα (t)),

(6.3.21)

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Nonlinear fluctuating hydrodynamics

where the αth particle is tagged. Starting from (6.3.6), we obtain 3 4

 ∂ ρˆs (x, t) = DB ∇ 2 ρˆs (x, t) + ∇ · ρˆs (x, t)∇ d x  U˜ (x − x )ρ(x , t) + ηs (x, t). ∂t (6.3.22) The correlation of the noise ηs is obtained as 2 1 ηs (x, t)ηs (x , t  ) = 2DB δ(t − t  )∇x ∇x  {δ(x − x )ρs (x, t)}.

(6.3.23)

Equation (6.3.22) obtained above is similar to the equations for the tagged-particle dynamics in the Newtonian-dynamics case to be discussed later, in Section 8.1 of Chapter 8. However, for the Newtonian-dynamics case approximations have to be made in dealing with the momentum density of the tagged particle, which is not a conserved quantity.

6.3.2 Fokker–Planck dynamics The exact balance equations or Dean equations discussed above are not peculiar to the case of microscopic Smoluchowski dynamics only. Other types of microscopic dynamics, such as Fokker–Planck dynamics in terms of the position xα and momentum xα of the N -particle system have also been considered, dxα (t) = pα , dt

(6.3.24)

 dpα (t) ∇α U (xα (t) − xβ (t)) − γ¯0 pα + ξα (t), =− dt N

(6.3.25)

β=1

where γ¯0 is a dissipative coefficient with the dimension of inverse time and U (xα − xβ ) is the interaction potential between the particles α and β. The particle mass m is chosen as unity. The correlation of the noise ξα is related to the dissipative coefficient γ¯0 by + , j ξαi (t)ξβ (t  ) = 2kB T γ¯0 δαβ δi j δ(t − t  ). (6.3.26) The microscopic density ρ(x, ˆ t) and the momentum density g(x, ˆ t) are defined in this case as in eqns. (5.1.1) and (5.1.2), respectively. We take the mass m of the fluid particle to be unity in order to keep the notation simple. Using the above definitions, the following balance equations for ρˆ and gˆ have been obtained by Nakamura and Yoshimori (2009): ∂ ρ(x, ˆ t) + ∇ · gˆ (x, t) = 0, ∂t ∂ ˆ i j (x, t) + ρ(x, ˆ t)∇i gˆi (x, t) + ∇ j # ∂t − γ¯0 gˆi (x, t) = ξi (x, t),

(6.3.27) 

d x  U (xα − x )ρ(x ˆ  , t) (6.3.28)

6.3 Stochastic balance equations

5

309

ˆ i j (x, t) = α pαi pα δ(x−xα (t)). The noise in eqn. (6.3.28) is defined as ξi (x, t) = where # ; 5 i ˆ t)θi (x, t) and is multiplicative in nature. θi is the Gaussian α δ(x − xα (t))ξα (t) = ρ(x, white noise with correlation, 1 2 θi (x, t)θ j (x , t  ) = 2kB T γ¯0 δ(x − x )δ(t − t  ). (6.3.29) j

The first term on the RHS of eqn. (6.3.28) can be adjusted, allowing us to write 5 5 i j − xα (t)) ν δ(x − xν (t)) α pα pα δ(x ˆ 5 #i j (x, t) = ν δ(x − xν (t)) 5 = =

α

5 j pαi δ(x − xα (t)) ν pν δ(x − xν (t)) 5 ν δ(x − xν (t))

gˆi (x, t)gˆ j (x, t) , ρ(x, ˆ t)

(6.3.30)

where we have used the presence of delta functions in the numerator to make the α and ν particles the same when they are both at the same position x. Thus the balance equation (6.3.28) reduces to

gˆi gˆ j ∂ gˆ i δFU √ (6.3.31) + ∇j + ρ∇ ˆ i − γ¯0 gˆi = ρθi . ∂t ρˆ δ ρˆ FU [ρ] is related to the bare interaction potential through the relation   1 dx dx ρ(x)U FU = ˆ (x − x )ρ(x ˆ  ). 2

(6.3.32)

The exact balance equations discussed above are similar in form to the corresponding fluctuating-nonlinear-hydrodynamics (FNH) equations (see eqns. (6.2.16)–(6.2.20)) for reversible microscopic dynamics, but with some subtle differences. First, the functional FU in the microscopic balance equation (6.3.31) does not contain an entropic (ideal-gas) ˆ i j term, term, unlike eqn. (6.3.16) in the case of microscopic Brownian dynamics. The # averaged over the pα with a Maxwell distribution, gives rise to such a term in FU . Second, the stochastic part in eqn. (6.3.31) involves multiplicative noise. It is related to the noise in the microscopic equations and is different from the phenomenological frictional term in (6.2.20) involving bare transport coefficients in FNH (with reversible microscopic dynamics).

Appendix to Chapter 6

A6.1 The coarse-grained free energy We present here the calculation of the kinetic part, Fk [ρ, g], of the coarse-grained freeenergy functional of the one-component system. The macroscopic system is divided into cells of volume cl much larger than the average volume per particle. The linear dimensions of a cell (cl ) are much smaller than any correlation length in the system. The characteristic function ψa (r) for the cell labeled a is defined as follows: ⎧ ⎨1, if r ∈ a, (A6.1.1) ψa (r) = ⎩0, if r  a. The mass density ρa and the total momentum ga corresponding to the ath cell are given in terms of the characteristic function ψa as ρa =

N 

mψa (rα ),

ga =

α=1

N 

pα ψa (rα ).

(A6.1.2)

α=1

In the limit cl → 0 and a → ∞, the cell values of the densities of mass and momentum can be treated as continuous fields, ρa → ρ(x)cl ,

(A6.1.3)

ga → g(x)cl .

(A6.1.4)

On the other hand, the microscopic density ρˆ of the particles at every given space-time point is defined with eqn. (1.3.15), considering the particles to be point-like. The process of coarse graining therefore can be considered as a mapping T [ρ|ρ] ˆ from the microscopic density ρˆ to the smooth field ρ at every point. The coarse-grained free-energy functional F in terms of the ρa and ga is defined using the statistical-mechanical partition function as     N N % %   k −β F −β H = Tr e δ ρ − mψ (r ) δ g − p ψ (r ) . (A6.1.5) e a

a

a

α

α=1

310

α

a

a

α=1

a

α

A6.1 The coarse-grained free energy

311

We have used above the notation “Tr” for the differential N 1 % drα dpα , Tr ≡ δr δp = N! h3

(A6.1.6)

α=1

where h is Planck’s constant. The Hamiltonian H for the fluid is written as H=

N  pα2 + U (r1 , r2 , . . . , r N ), 2m

(A6.1.7)

α=1

where U is the interaction term and is a function of the spatial coordinates and rα is the position of the αth particle (α = 1, . . . , N ) and will be simply denoted as U (r ). The RHS of eqn. (A6.1.5) is further simplified by using integral representations for the Kronecker delta δ k and the Dirac δ given, respectively, by  ?      N N   dz a mψa (rα ) = exp i ρa − mψa (rα ) ln z a (A6.1.8) δ k ρa − 2πi z a α=1

and



δ ga −

N 

α=1

 pα ψa (rα ) =

α=1



∞ −∞

  N  dya exp iya · ga − ya · pα ψa (rα ) , (A6.1.9) (2π )3 α=1

@

where the integral in (A6.1.8) is with ln z a and denotes the integral along a closed contour around the origin (Carrier et al., 1966). On substituting the representations (A6.1.8) and (A6.1.9) into eqn. (A6.1.5) we obtain % ? dz a  ∞ dya ! " −β F = exp i(ya · ga + ρa ln z a ) + φ(z, y) . (A6.1.10) e 3 2πi z (2π ) a −∞ a The function φ(z, y) is obtained as eφ(z,y) =





δr δp e−β H exp −i

N 

 (m ln z(rα ) + pα · y(rα )) .

(A6.1.11)

α=1

We have introduced above the functions ln z(r) and y(r),  ln z(r) = {ln z a }ψa (r), a

y(r) =



ya ψa (r).

(A6.1.12)

a

The momentum integral in eqn. (A6.1.11) is computed by completing the perfect square and then performing the Gaussian integral. This gives the result    N  1 φ(ζ ) 3 dr1 . . . dr N exp −βU (r ) − N ln 0 − im = ln ζ (rα ) , (A6.1.13) e N! α=1

312

Appendix to Chapter 6

where



y 2 (r) ζ (r) = z(r) exp −i 2β

(A6.1.14)

√ and 0 = h/ 2πmkB T is the thermal de Broglie wavelength (Huang, 1987). We identify the variables ζa for the cells, which are similar to z a and ya in eqn. (A6.1.12), in terms of the characteristic functions ψa in the following manner:  {ln ζa }ψa (r). (A6.1.15) ln ζ (r) = a

From the property (A6.1.1) it follows easily that ln z a = ln ζa + i

ya2 . 2β

(A6.1.16)

Equation (A6.1.13) now reduces to the form  1 3 % −i ln ζ {m 5 ψ (r )} φ(ζ ) a α a α . dr1 . . . dr N e−βU (rα )−N ln 0 e = e N! a

(A6.1.17)

On using eqn. (A6.1.16) in (A6.1.10) and changing the integration variable from z a to ζa , we obtain 

 % ? dζa  ∞ dya ρa 2 exp i y · g − ) , (A6.1.18) y e−β F = + θ (ζ a a a 2πiζa −∞ (2π )3 2β a a where e

θ(ζa )

1 = N!

 &

 dr1 . . . dr N e

−βU(rα )−N ln 30

exp i ρa − m



'



ψa (rα ) ln ζa .

α

(A6.1.19) The variable y in eqn. (A6.1.18) is now integrated out by completing the perfect square and performing the Gaussian integral, obtaining e

−β F

 ? %  β 3/2 dζa θ(ζa ) ga2 = exp −β e . 2πρa 2ρa 2πiζa a

(A6.1.20)

The coarse-grained free energy F ≡ F[ρ(x), g(x)] is obtained by taking the continuum limit, i.e., cl → 0 and a → ∞. In this limit the cell values of the densities of mass and momentum can be treated as continuous fields, ρa → ρ(x)cl ,

(A6.1.21)

ga → g(x)cl .

(A6.1.22)

A6.1 The coarse-grained free energy

313

The momentum-density-dependent part of the free energy in the exponent on the RHS of eqn. (A6.1.20) is now obtained in the form of a volume integral involving the field variables,   2

 g2 % ga a lim exp = exp cl →0 2ρa 2ρa a a    g 2 (x) cl ≡ exp 2ρ(x) a 

= exp

g 2 (x) dx . 2ρ(x)

(A6.1.23)

The density-dependent part of the coarse-grained free energy is separated into two parts, F = FK [ρ, g] + FU [ρ].

(A6.1.24)

The part FU [ρ] is dependent only on the density ρ, while the kinetic part FK is dependent on the momentum density as well. The latter is given in units of β −1 = kB T as  g 2 (x) . (A6.1.25) FK [g, ρ] = dx 2ρ(x) On substituting for eθ(ζ ) from eqn. (A6.1.19), the integral in eqn. (A6.1.18) reduces to the form    N %  1 k β FU [ρ] −βU (rα )−N ln 30 dr1 . . . dr N e = δ ρa − mψa (rα ) . e N! a a=1

(A6.1.26) An approximate form of the coarse-grained form of the free-energy functional is obtained by evaluating the integrals over the spatial variables {rα } in terms of occupation numbers {n a } of the cells, where n a = ρa /m. The volume element dr1 . . . dr N /N ! with the delta-function constraints on the RHS of eqn. (A6.1.26) is evaluated in terms of a localized distribution of the particles in the cells, in which n a identical particles are in a given cell of volume cl . The volume element is obtained as a product over the cells given by < na a cl /n a ! . The effect of interaction between the particles is included through Fin [{ρa }] involving the densities ρa in the cells. The coarse-grained function FU [ρ] is written in terms of cell contributions in the form   % n a     3 cl (A6.1.27) e−N ln 0 exp −β Fin [{ρa }] . exp −β FU [ρ] = na ! a Thus FU [ρ] is obtained as a sum of an ideal-gas part Fid and an interaction contribution Fin , F = Fid + Fin .

(A6.1.28)

314

Appendix to Chapter 6

The ideal-gas contribution corresponding to a noninteracting system (U = 0) Fid is obtained from eqn. (A6.1.27) using the Starling approximation,  

 na β Fid = (A6.1.29) n a ln − n a + N ln 30 .  cl a In the continuum limit the RHS of (A6.1.29) (using the constraint the form    β Fid = n(x) ln n(x)30 − n(x) cl a

1 → m



 

ρ(x)30 dx ρ(x) ln m



5 a

n a = N ) reduces to

 −1 ,

(A6.1.30)

where 0 is the thermal de Broglie wavelength. The above result is identical to the idealgas contribution to the free energy obtained in Chapter 2. The interaction part Fin is a consequence of the interaction U . For example, the two-body interaction term defined in the Hamiltonian (1.1.16) is expressed as   1 dx dx ρ(x ˆ  )u(x − x )ρ(x ˆ  ) ≡ U [ρ(x)] ˆ (A6.1.31) U [rα ] = 2m 2 in terms of the microscopic density functions ρ(x). ˆ If the coarse graining were an exact mapping T [ρ|ρ] ˆ at every point, we would be able to write the interaction part defined in eqn. (A6.1.27) above as Fin = U [ρ(x)].

(A6.1.32)

In reality this is the simplest approximation since the coarse graining maps the densities over the coarse-graining volume into a smooth average function. The approximate coarsegrained free-energy functional is therefore obtained as F[ρ, g] = FK [ρ, g] + FU [ρ]        ρ(x)30 ρ(x) g 2 (x) + kB T ln − 1 + U [ρ(x)] = dx 2ρ(x) m m

(A6.1.33)

in terms of the bare-interaction-potential function. The free-energy functional F[ρ, g, v] We have seen that in an appropriate formulation of the dynamics of the compressible liquid to take into account all the relevant nonlinearities a new field v enters the fluctuatinghydrodynamics description. It is defined in terms of the nonlinear constraint g = ρv.

(A6.1.34)

A6.1 The coarse-grained free energy

315

Let us consider the implications of extending the space of coarse-grained variables to {ρ, g, v}. The corresponding driving free-energy functional F[ρ, g, v] is connected to its counterpart F[ρ, g] through the relation     ˜ · g, v], exp −β F[ρ, g, v] = exp −β F[ρ, g] [ρ (A6.1.35) where the free-energy functional in terms of variables {ρ, g} is obtained as F[ρ, g] = ˜ · g, v] on the RHS of eqn. (A6.1.35) ensures that the FK [ρ, g] + FU [ρ]. The function [ρ nonlinear constraint (A6.1.34) is maintained, i.e.,  ˜ · g, v] = 1. Dg [ρ (A6.1.36) ˜ Now we define the density-dependent function F[ρ] by integrating out the variables g and v from the fluctuating-hydrodynamics description as     ˜ exp −β FU [ρ] = Dg Dv exp {−β F[ρ, g, v]} . (A6.1.37) The multiple integral on the RHS of eqn. (A6.1.37) can be evaluated in different orders yielding an identical result. Let us demonstrate this for consistency. First, we integrate g. Using the explicit form of the momentum-dependent part FK [ρ, g] (see eqn. (A6.1.25) above) and the definition (A6.1.36), the integral on the RHS in eqn. (A6.1.37) reduces to the form   exp −β F˜U [ρ] = Iv [ρ]e−FU [ρ] , (A6.1.38) where Iv [ρ] is the Gaussian integral on the RHS with respect to g, 3  4  v 2 (x) Dv exp − dx ρ(x) . Iv [ρ] = 2

(A6.1.39)

On the other hand, we obtain, on integrating out v first,   exp −β F˜U [ρ] = I0 [ρ]Ig [ρ]e−FU [ρ] ,

(A6.1.40)

where the integrals I0 [ρ] and Ig [ρ] are defined as  ˜ · g, v], Dv [ρ I0 [ρ] =

(A6.1.41)

 Ig [ρ] =

3  4 g 2 (x) Dg exp − dx . 2ρ(x)

(A6.1.42)

For the integral I0 we use the representation of the delta function in terms of a normalized Gaussian function, ˜ [ρ, g, v] ∼ δ(g − ρv) = lim

σ →0

1 (2π σ 2 )

3/2

 (g − ρv)2 exp − 2σ 2

(A6.1.43)

316

Appendix to Chapter 6

We now make a change of variable v→v + g/ρ. With another change of variable v/σ →v˜ (before taking the delta-function limit of width σ going to zero), the integral I0 reduces to a Gaussian form     v˜ 2 (x) I0 = D v˜ exp − dxρ 2 (x) . (A6.1.44) 2 All three integrals I0 , Ig , and Iv have the same form,     ϑ 2 (x) Aϑ [ f ] 2 Iϑ [ f ] ≡ e = Dϑ exp − dx f (x) . 2

(A6.1.45)

with f (x) ∼ ρ(x), ρ −1/2 (x) and ρ 1/2 (x), respectively. ϑ(x) is a vector field of the same < dimension as g or v. Dϑ ≡ i Dϑi . The integral (A6.1.45) is evaluated using the identity   

 δ f 2ϑ 2 Dϑ ϑi (x) exp − dx , (A6.1.46) δϑ j (x ) 2 which leads to the result     f 2ϑ 2 2  = δi j δ(x − x )Iϑ . f (x) Dϑ ϑi (x)ϑ j (x ) exp − dx 2

(A6.1.47)

In the limit x → x , on setting i = j and summing over i, we obtain for d dimensions     f 2ϑ 2 f 2 (x) Dϑ ϑ 2 (x) exp − dx (A6.1.48) = dδ(0)Iϑ , 2 6 where δ(0) = 3 /(2π )d for a short-wavelength cutoff of 2π/ in the theory. On the other hand, on taking a derivative of (A6.1.45) with respect to f and using eqn. (A6.1.48), one obtains  6 ϑ 2 (x) δ Iϑ δ Aϑ 2 ≡ Iϑ [ f ] = − f (x) Dϑ ϑ 2 (x)e− dx f (x) 2 δ f (x) δ f (x) = −dδ(0)

Iϑ [ f ] . f (x)

On solving eqn. (A6.1.49), one obtains Aϑ [ f ] as Aϑ [ f (x)] = −dδ(0) ln



 f (x) , f0

(A6.1.49)

(A6.1.50)

where f 0 is the value of f for which Aϑ [ f ] = 0. I0 , Ig , and Iv , respectively, correspond to f (x) being ρ(x), ρ −1/2 (x), and ρ 1/2 (x), and satisfy  

d ρ(x) I0 Ig = Iv = exp − δ(0) ln , (A6.1.51) 2 ρ0 rendering the RHS of eqn. (A6.1.38) and that of eqn. (A6.1.40) identical. We obtain the result ) (   − d δ(0) ln ρ(x) ρ0 , (A6.1.52) Dg Dv e−β F[ρ,g,v] = e−β FU [ρ] e 2

A6.1 The coarse-grained free energy

317

which is consistent with the fluctuating-hydrodynamics description in terms of the fluctuating fields {ρ, g, v}. We consider the density-dependent part as FU [ρ] and ignore the second factor on the RHS of eqn. (A6.1.45). The latter is absorbed as a renormalization of the static behavior in terms of an extra field ϑ(x),  ϑ 2 (x) β F[ρ, g, v, ϑ] = β F[ρ, g, v, ϑ] + dx (A6.1.53) 2ρ(x) so as to write



 Dg

 Dv

Dϑ e−β F[ρ,g,v,ϑ] = e−β FU [ρ] .

(A6.1.54)

By suitably choosing the dynamics of the field ϑ, the equations of motion for {ρ, g, v} are kept unchanged and decoupled from that for ϑ. Thus, the dynamics at the level of ρ, g, and v remains the same even though the equilibrium distribution is renormalized. This choice, however, cannot be considered definitive. The lack of uniqueness in the problem 6 is also seen from the fact that a transformation F[ρ, g, v, ϑ] → F[ρ, g, v, ϑ] − μ dx ρ(x), where μ is a constant, leaves the equations for the dynamics invariant. The final equilibrium probability distribution of the coarse-grained fields is undetermined up to a factor of 6 exp{−μ dx ρ(x)}. The dynamical equations do not uniquely determine ρ0 .

7 Renormalization of the dynamics

We have discussed the construction of the nonlinear Langevin equations for the slow modes in a number of different systems in the previous chapter. Next, we analyze how the nonlinear coupling of the hydrodynamic modes in these equations of motion affects the liquid dynamics. In particular, we focus here on the case of a compressible liquid in the supercooled region. In this book we will primarily follow an approach in which the effects of the nonlinearities are systematically obtained using graphical methods of quantum field theory. Such diagrammatic methods have conveniently been used for studying the slow dynamics near the critical point (Kawasaki, 1970; Kadanoff and Swift, 1968) or turbulence (Kraichnan 1959a, 1961a; Edwards, 1964). The present approach, which is now standard, was first described by Martin, Siggia, and Rose (1973). The Martin–Siggia–Rose (MSR) field theory, as this technique is named in the literature, is in fact a general scheme applied to compute the statistical dynamics of classical systems. The field-theoretic method presented here is an alternative to the so-called memoryfunction approach. The latter in fact involves studying the dynamics in terms of nonMarkovian linearized Langevin equations (see, for example, eqn. (6.1.1) which are obtained in a formally exact manner with the use of so called Mori–Zwanzig projection operators. This projection-operator scheme is described in Appendix A7.4). The generalized transport coefficients or the so-called memory functions in this case are frequency-dependent and can be expressed in terms of Green–Kubo forms of integrals of time correlation functions. These memory functions account for the long-time effects, i.e., the role of correlated dynamics of the particles in the dense fluid. The renormalized perturbation expression for the memory function finally leads us to the model for the slow dynamics characteristic of the dense liquid. The latter is termed the mode-coupling theory (MCT) for liquids and has proved to be an essential tool for understanding the slow dynamics in a metastable liquid below the freezing point Tm . The evaluation of the memory function in terms of correlation functions using the projection-operator method is also presented in the appendix. This approach, though somewhat ad hoc, is often taken as an alternative way of obtaining the self-consistent mode-coupling model. The MSR field-theory approach discussed below is particularly aimed towards analyzing the full implications of the nonlinearities in the equations of generalized hydrodynamics introduced in Section 6.2.1. These fluctuating nonlinear Langevin equations are 318

7.1 The Martin–Siggia–Rose theory

319

Markovian. The random forces or the noises in the nonlinear stochastic equations of motion are related to the bare transport coefficients of the system through the usual fluctuation– dissipation relations.1 The corrections to the bare transport coefficients due to the nonlinearities provide what is termed the mode-coupling contributions. With the field-theoretic methods described below, these corrections can be computed in a systematic manner in successive orders in a perturbation theory. Finally, apart from the above two approaches, the simplest form of the mode-coupling model is also obtained directly from the equations of nonlinear fluctuating hydrodynamics without getting into an involved formalism. We present this deduction (Kawasaki, 1995) in Appendix A7. 7.1 The Martin–Siggia–Rose theory The MSR theory is constructed in terms of a generating functional from which both the time correlation and the response functions are obtained. The original theory (Martin et al., 1973; Dekker and Haake, 1975; Phythian, 1975, 1976; Andersen, 2000) was developed ˆ with an operator formalism involving the field ψ and an additional adjoint operator ψ. The ψ operator does not commute with its hatted counterpart. The correlation between ψ and ψˆ corresponds to the response functions, which are computed together with the usual correlation functions between the ψs. The MSR field theory applied to the case of classical continuous fields satisfying a linear or nonlinear Langevin equation has subsequently been formulated using the functional integral approach (Janssen, 1976; Bausch et al., 1976; Phythian, 1977; De Dominicis and Pelti, 1978). By construction this formulation also involves, in addition to the field variable, an associated hatted field in a manner that is very similar to the operator approach of the original MSR theory. In the following we will adhere to the functional integral formulation of the MSR. It has been applied extensively in a variety of problems: to mention a few examples, for the study of diffusion in disordered media (Bouchaud and Georges, 1990); polymer solution dynamics (Fredrickson and Helfand, 1990); domain-growth problems in dynamics of phase transition (Mazenko, 1990); diffusion in random media (Deem and Chandler, 1994); driven diffusive systems (Garrido et al., 1990); diffusion of a Brownian particle in a fluid with random timeindependent velocity fields (Deem, 1995); critical dynamics of driven interfaces in random media (Narayan and Fisher, 1993); interface growth using Kadar–Parisi–Zhang (Kadar et al., 1986; Medina et al., 1989) equations in field-theoretic models (Sun and Plischeke, 1994; Frey and Tauber, 1994); dynamics of liquid crystals (Milner and Martin, 1986); study of turbulence (Eyink, 1994; Mou and Weichman, 1995); and roughening surface transitions in the presence of a disordered pinning potential (Scheidl, 1995). Our primary focus in the present discussion is on the formulation of the MSR model for studying the dynamics of supercooled liquids. The field-theoretic model for the dense liquid has been studied by various authors (Das and Mazenko, 1986, 2009; Kim and Mazenko, 1991, Mazenko and Yeo, 1994; Yeo and Mazenko, 1995, Das and Schilling, 1993, Kawasaki and Miyazima, 1997; 1 These bare transport coefficients L 0 s introduced here are treated as input parameters to be determined from kinetic theories ij

dealing with the short-time dynamics of the fluid.

320

Renormalization of the dynamics

Kawasaki and Kim, 2001; Harbola and Das, 2002; Andreanov et al., 2006; Miyazaki and Reichman, 2005; Kim and Kawasaki, 2008; Mazenko, 2010; Yeo, 2010). The formulation of the MSR theory presented here gives rise in a very natural way to a model for the renormalized (due to the nonlinearities) liquid-state dynamics in a self-consistent form. As we will see in the next chapter, this gives rise to the feedback mechanism of mode-coupling theory (MCT) for slow dynamics. Let us first outline the construction of the field-theoretic model for renormalization of the dynamics of the compressible liquid (Das and Mazenko, 1986). The associated MSR field theory involves two sets of fields {ρ, g, v} and their corresponding hatted counterparts {ρ, ˆ gˆ , vˆ }. The noise-averaged correlation functions of the field variables are expressed in terms of generalized transport coefficients with wave-vector and frequency dependences. Such dependences result in the MSR theory from the renormalization of the bare transport coefficients due to the nonlinearities in the equations of motion. The corrections to the transport coefficients are obtained as a perturbation series expansion involving graphs that are evaluated in terms of correlation and response functions. For a dense liquid approaching the glass transition, the evaluation of the perturbation series for the transport coefficient is done assuming that the effects of coupling of strong density fluctuations are dominant.

7.1.1 The MSR action functional Let us begin with a field ψ whose dynamics is given by the stochastic equation of motion, ∂ψ(1) ¯ ¯ + U3 [12¯ 3]ψ( ¯ ¯ ¯ + θ (1) = −{U1 [1] + U2 [12]ψ( 2) 2)ψ( 3)} ∂t1 ≡ −W [ψ] + θ (1).

(7.1.1)

¯ . . . etc. label space, time, and any other index In the above equation the numbers 1, 2, associated with the fields {ψ} representing the slow variables. In eqn. (7.1.1) it is implied that the repeated barred indices are integrated or summed over. The noise θ is assumed to be simply additive. The correlation of the noise is defined in terms of the matrix L 0 of bare or short-time kinetic coefficients for the system, θ (1)θ (2) = 2kB T L 0 (1, 2).

(7.1.2)

The above description of the dynamics can be generalized to the case in which the noise is multiplicative (Jensen, 1981). The functional f [ψ] of the field ψ which satisfies the equation of motion (7.1.1) is formally written in the form  f [ψ] =

Dψ  δ(ψ − ψ  ) f [ψ  ],

(7.1.3)

7.1 The Martin–Siggia–Rose theory

321

where the functional integral with Dψ  is according to the definition of (6.1.12). The functional delta function is defined as % δ(ψ − ψ  ) = lim δ(ψ(i) − ψ  (i)), (7.1.4) →0

i

where i ∈ d+1 belongs to a (d +1)-dimensional lattice (including d, the number of spatial dimensions, and time). Since ψ is a solution of the equation of motion (7.1.1) we replace the delta function on the RHS of eqn. (7.1.3) with a change of coordinates to    ∂ψ(1) f [ψ] = Dψ J [ψ, θ] f [ψ]δ + W [ψ] − θ (1) , (7.1.5) ∂t1 where

* * * δθ * * * J [ψ, θ] = det * δψ *

(7.1.6)

is the Jacobian of the transformation due to the change of the argument of the delta function. Using a causal connection in the time discreatization, the Jacobian2 of this transformation is treated as a constant C0 (say). Finally, on replacing the delta function on the RHS of (7.1.6) by its functional Fourier transform in terms of a conjugate field, we obtain  4

3   ∂ψ(1) ˆ + W [ψ] . (7.1.7) f [ψ] = C0 Dψ D ψˆ f [ψ]exp −i ψ(1) ∂t1 We can now take an average over the randomness and obtain the corresponding average over the noise as    D ψˆ f [ψ]  f [ψ] = C0 Dψ    . ∂ψ(1) ˆ × exp −i d1 ψ(1) + W [ψ] − θ (1) . (7.1.8) ∂t1 Further simplification of the RHS will require details of the nature of the randomness described by θ . This should lead to an action functional A for developing an appropriate field-theoretic model that takes into account the role of the nonlinearities in the equation of motion. The renormalized theory is then constructed order by order in a standard diagrammatic approach. An example is the case of supercooled liquids. Let us now assume that the noise is white and Gaussian. The corresponding average is then taken over the distribution P[θ] = I0 exp[−A0 (θ )], where A0 is a quadratic function of θ , 4 3 β −1 d1 d2 θ (1)[L 0 ] (1, 2)θ (2) , A0 (θ ) = (7.1.9) 4 and the constant I0 is defined such that 1 = 1. Equation (7.1.9) gives the proper fluctuation–dissipation relation of the average correlation of the noise in terms of the bare damping matrix L 0 . 2 The evaluation of the Jacobian has some ambiguity. We discuss this in Appendix A7.1.

322

Renormalization of the dynamics

The result (7.1.2) follows directly from the identity  δ [θ (2)e−A0 (θ) ] = 0 D(θ ) δθ(1)

(7.1.10)

−1

and taking [L 0 ] to be a symmetric matrix. Using eqn. (7.1.1), the exponential term within angular brackets on the RHS of eqn. (7.1.8) for the average of f (ψ) is written as 3

4  ∂ψ(1) (7.1.11) + W [ψ] − θ − A0 (θ ) . exp −i d1 ψˆ i (1) ∂t The exponent in (7.1.11) is reduced to a perfect square by adding an extra term quadratic in ψˆ and the corresponding Gaussian integral in θ is easily performed. Denoting this integral as IG , eqn. (7.1.8) is expressed in the form    ˆ , (7.1.12)  f [ψ] = I0 C0 IG Dψ D ψˆ f [ψ]exp −A[ψ, ψ] where the action functional A is obtained as 3 4    ∂ψ(1) −1 0 ˆ ˆ ˆ ˆ A[ψ, ψ] = d1 d2 ψ(1)β L (12)ψ(2) + i d1 ψ(1) + W [ψ] . ∂t1 (7.1.13) The quantity I0 C0 IG on the RHS of eqn. (7.1.12) gives a normalization constant, which is fixed to ensure that 1 = 1, such that  6 6 ˆ Dψ D ψˆ f [ψ]exp −A[ψ, ψ]  . (7.1.14)  f [ψ] = 6 6 ˆ Dψ D ψˆ exp −A[ψ, ψ] The expression (7.1.14) for the average of the functional f (ψ) is used to write the averages of fields and higher-order correlation functions in terms of a generating functional Z ξ . Assuming f (ψ) ≡ ψ, we write ψ(1) = with

 Zξ =

 Dψ

δ ln Z ξ , δξ(1)

 ˆ . D ψˆ exp −Aξ [ψ, ψ]

(7.1.15)

(7.1.16)

We have defined the generating functional Z ξ by including a linear current term in the corresponding action functional Aξ ,  ˆ = A[ψ, ψ] ˆ − d1 ξ(1)ψ(1), (7.1.17) Aξ [ψ, ψ] where A is given by (7.1.13).

7.1 The Martin–Siggia–Rose theory

323

The multi-point correlation functions of the variables ψ are obtained from the generating functional, * * 1 δ δ ψ(1) . . . ψ(m) = . (7.1.18) ... Z ξ ** Z δξ(1) δξ(m) ξ

ξ =0

Note that, from the expression for the MSR action (7.1.13) and the equation of motion (7.1.11), it follows that with the linear part of the dynamics, i.e., terms in W [ψ] linear in the fields, ψ produces an MSR action functional that is quadratic (Gaussian) in the fields. The MSR response function In the RHS of the equation of motion (7.1.1) the constant term U1 does not involve the field variable ψ. Let us consider the change in the field ψ corresponding to an increment in U1 → U1 + h˜ ψ with an external field term in the MSR action:   6 1 ˜ ˆ Dψ D ψˆ e−A+i d2ψ(2)h ψ (2) − e−A ψ(1) δψ(1) = Z  ( ) ˆ (7.1.19) = i ψ(1)ψ(2) h˜ ψ (2)d2 + O h˜ 2ψ . Since according to causality the field ψ(1) at time t1 is dependent only on h˜ ψ and noise at ˆ The correlation an earlier time, we must have t2 < t1 for nonzero values of ψ(1)ψ(2). between the hatted and unhatted fields is therefore like a response function and hence time-ordered. It is important to note that the above-mentioned response function is generally not the response function associated with a physical field h ψ which couples to the field ψ in the equivalent Hamiltonian, i.e.,  Fext ∼ dx ψ(x)h ψ (x, t). (7.1.20) If h˜ ψ is simply proportional to h ψ then the MSR response function becomes the same as the physical linear response function. In the equilibrium statistical mechanics the correlation and linear response functions are related through fluctuation–dissipation relations. (See eqn. (1.3.70) in Section 1.3.3.) Hence in such a situation the elements G ψψ and G ψ ψˆ of the matrix G are related in MSR theory. In order to check whether this is the case, we note that h˜ ψ is present in the driving term in the equation of motion for ψ. In the fluctuatinghydrodynamics description, the equation of motion for the field ψ as given in eqn. (6.1.76) has driving terms like  δF Q i j − L i0j . (7.1.21) δψ j According to eqn. (7.1.20), the functional derivative (δ F/δψ j ) is proportional to h ψ . Hence h˜ ψ is proportional to h ψ only if the Poisson bracket Q i j and the bare transport coefficients L i0j are constants, i.e., independent of the fields ψ. We have seen in Chapter 6 that (in general) both these conditions can hold if the dynamics is linear. In this case the

324

Renormalization of the dynamics

correlation G ψ ψˆ between a hatted and an unhatted field is related through a fluctuation– dissipation relation to the correlation G ψψ between two corresponding unhatted fields. In the case of nonlinear dynamics, however, such linear FDTs are not always satisfied. In particular, neither of the two models we consider in the present book, namely the FNH model in Section 6.2 and the DDFT model in Section 8.1.6, belongs to this category. In the former the Poisson bracket Q i j depends on the fluctuating fields, whereas in the latter the transport coefficient is dependent on the density. However, certain relations between G ψψ and G ψ ψˆ still hold as generalized fluctuation–dissipation relations. We discuss this in Section 7.3.1.

The Schwinger–Dyson equation To facilitate the discussion of the renormalization due to the nonlinearities in the dynamics, the action functional Aξ is written in a polynomial form. We consider here the form Aξ [%] =

1 1  %(1)G −1 (12)%(2) + V (123)%(1)%(2)%(3) 0 2 3 1,2 1,2,3  %(1)ξ(1), −

(7.1.22)

1

with the corresponding partition function given by the compact form  Zξ =

D% e−Aξ [%] .

(7.1.23)

We have adopted a compact notation above in which the spatial coordinate x1 and time t1 both for the hatted and for the unhatted fields are all incorporated into a single index 1 of the vector-field variable %(1). The vertex functions V (123) are defined in such a way that they are symmetric under the exchange of the indices. For simplicity we have included in the above expression for the MSR action (7.1.22) terms of up to cubic order vertex in the action. The more generalized treatment with quartic nonlinearity in the action is available elsewhere (Das and Mazenko, 1986). The Gaussian terms in the action consist of two types of terms: first, with two ψˆ fields as a result of averaging over the noise; and second, a product of ψˆ and ψ coming from the linear part of the equations of motions. If we make √ √ ˆ β and ψ → ψ β, the Gaussian terms become of O(1) while a transformation ψˆ → ψ/ nonlinear terms of nth order in the ψs in the equation of motion are O[(kB T )(n−1)/2 ] in the action. The renormalized theory which takes into account the role of the nonlinearities (beyond Gaussian terms) is obtained using standard Feynman-graph (Amit, 1999) methods of field theory. In the functional-integral formulation (Bausch et al., 1976; De Dominicis and Pelti, 1978; De Dominicis and Pelti, 1978; Janssen, 1979; Jensen, 1981) of the MSR theory, the renormalized expressions for the transport coefficients appear in a fully selfconsistent form.

7.1 The Martin–Siggia–Rose theory

325

The one-point function G(1) = %(1) is obtained from the generating function Z x i in terms of the derivative, %(1) =

" δ ! ln Z ξ . δξ(1)

(7.1.24)

We include the density variable in the set of slow variables % as δρ(1) = ρ(1) − ρ(1). G(1) vanishes as ξ → 0 both for ρ and for g. The two-point function G(12) is given by G(12) =

δ G(1) = δ%(1)δ%(2), δξ(2)

(7.1.25)

where δ%(1) = %(1) − %(1) ≡ %(1). The inverse of the two-point correlation matrix G(12) is defined through the relation  G −1 (13)G(32) = δ(12). (7.1.26) 3

For a Gaussian action without any cubic or higher-order terms in the action, G(12) is equal to G 0 (12). The role of the nonlinearities is expressed in terms of the so-called “self-energy” matrix  through the Schwinger–Dyson equation, G −1 (12) = G −1 0 (12) − (12).

(7.1.27)

In Appendix A7.2 we show that, for the action functional (7.1.22) involving cubic vertices V (123), the “self-energy” matrix  is self-consistently expressed in terms of the correlation functions,  (12) = V (134)G(35)G(46)R(526). (7.1.28) 3,4,5,6

The renormalized three-point vertex function denoted by R(123) in (7.1.28) is obtained from the solution of a self-consistent integral equation,  T (1453)G(46)G(57)(726), (7.1.29) R(123) = (123) + 4,5,6,7

where (123) = 2V (123) +



U (1453)G(46)G(57)(726).

(7.1.30)

4,5,6,7

The four-point kernel T (1234) is to be determined from the integral equation  U (1564)G(57)G(68)T (7238). T (1234) = U (1234) +

(7.1.31)

5,6,7,8

The four-point vertex function U (1234) is defined as U (1234) =

δ(12) . δG(34)

(7.1.32)

326

Renormalization of the dynamics

Equations (7.1.28) and (7.1.29) are conveniently expressed in terms of the Feynman graphs shown in Fig. A7.1 in Appendix A7.2. The zeroth-order matrix G 0 in the MSR action (7.1.22) corresponds to a Gaussian form. The effects of nonlinearities in the equations of motion are calculated by evaluating the different self-energies expressed in terms of Feynman graphs. A graphical expansion for  in terms of the bare vertices V and the full correlation functions is straightforward to obtain. In Fig. A7.2 the diagrammatic expansion up to second order in kB T is shown. In practice the renormalization has generally been considered only up to one-loop order. To lowest order we obtain the following one-loop expression for the self-energy ,  2V (134)G(35)G(46)V (526). (7.1.33) (12) = 3,4,5,6

The bare vertex function V (123) is determined from the nonlinearities in the equations of motion for the collective modes. In the present case the approach outlined above provides the renormalization of the bare transport coefficients in a self-consistent form in terms of full correlation functions. We will apply the above-described general framework of the MSR field theory to compute systematically the corrections to the linear theory due to the nonlinear coupling of the slow modes. In particular, our focus will be on the dynamics of a compressible liquid for which the construction of the nonlinear Langevin equations was discussed in the previous chapter. The MSR field theory outlined above involves a matrix G of correlation functions between the different fields. The elements of this matrix broadly belong to two categories: the correlation functions G ψψ and the “response” functions G ψ ψˆ . As has already been pointed out (see eqns. (7.1.19)–(7.1.21)), the correlation between a hatted and an unhatted field is like a linear response function corresponding to an equivalent field and is time-retarded in order to maintain causality. The response functions satisfy the relation G α βˆ (q, ω) = −G ∗βα ˆ (q, ω).

(7.1.34)

From the Schwinger–Dyson equation (7.1.27) it then follows that the self-energy matrix elements satisfy ∗ α βˆ (q, ω) = −βα ˆ (q, ω).

(7.1.35)

In the MSR theory the element of the correlation function matrix G corresponding to hatted fields is always zero. This can be seen in the following way. From the construction of the MSR action the Gaussian part [G −1 0 ]αβ is zero. Also, as a result of causality (Mazenko and Yeo, 1994), the αβ element of the self-energy matrix is also zero. Therefore elements G −1 αβ between two unhatted fields are zero and hence the inverse G αˆ βˆ elements between two hatted fields are zero. The renormalization scheme can be summarized as follows. The matrix G of full correlation and response functions is obtained from the Schwinger–Dyson equation (7.1.27). The zeroth-order matrix G 0 corresponds to the Gaussian part of the MSR action functional

7.2 The compressible liquid

327

and nonlinearities in the equations of motion give rise to nonlinear vertices. The renormalized theory is obtained by evaluating the different self-energies expressed in terms of a corresponding set of Feynman graphs determined by the vertex structure of the action functional or equivalently by the nonlinearities in the equations of motion for the slow variables. A graphical expansion for  in terms of the bare vertices V and the full correlation functions is straightforward to obtain. Equation (7.1.33) is the one-loop expression for the self-energy in terms of the bare vertices V (123) in the MSR action functional. Specific self-energy contributions provide the renormalization of the corresponding bare transport coefficients in a self-consistent form, i.e., in terms of full correlation functions G. This renormalized theory involving self-consistent expressions for the transport coefficients in terms of correlation functions constitutes an important feedback mechanism. The latter forms the basis of the self-consistent mode-coupling theory for the cooperative dynamics of a dense liquid and is discussed in the next chapter.

7.2 The compressible liquid Let us now focus on the case of compressible liquids and consider the structure of the renormalized theory when the dynamics is described by the nonlinear hydrodynamic equations presented in Section 6.2.1. Since the self-consistent equations for the time correlation functions are essential in the discussion of the mode-coupling models, we describe this analysis in some detail. Before we turn to the full analysis of the equations for the compressible liquid with the nonlinear constraint, let us briefly take note of the case of the incompressible fluid, i.e., the case of constant density. The incompressible case In this case, since the density ρ is constant in time, ∇ · g = 0, i.e., k · g(k) = 0. So the momentum field is transverse to the direction of the wave vector k and is denoted by gT . The equation for the momentum conservation reduces to the form

∂giT T gk ∇k g j + η0 ∇ 2 giT + θiT , (7.2.1) = −Pi j ∂t ρo where θiT is the transverse component of the noise. The nonlinear term in (7.2.1) is due to the convective nonlinearity, which in this case will be the transverse part of (g · ∇)g. The transverse projection is done by the operator PiTj (x), whose Fourier transform is given by ˆ = δi j − kˆi kˆ j . PiTj (k)

(7.2.2)

The effect of this nonlinearity on the bare viscosity η0 was studied using standard theoretical techniques (Forster et al., 1977). The shear viscosity η develops a long-time tail of power-law decay, η(t) = Aη t −d/2 .

(7.2.3)

328

Renormalization of the dynamics

The exponent of the power-law behavior, i.e., d/2, is the same as that obtained in computer simulations as well as in kinetic-theory models. Moreover, the exponent Aη was computed to be kB T  −3/2 −3/2 Aη = + D0 7ν0 , (7.2.4) 120π 3/2 which is the same as the result obtained from a detailed microscopic kinetic-theory calculation (Pomeau and Résibois, 1975).

7.2.1 MSR theory for a compressible liquid For the compressible case let us first consider the explicit form the action (7.1.13) takes. In this case, while g satisfies a Langevin equation with a noise term, the equation for the density ρ is the continuity equation (6.2.16). Therefore the appearance of ρˆ in the action is

 ∂ρ(1) + ∇1 · g(1) , (7.2.5) i d1 ρ(1) ˆ ∂t1 so that the functional integral over ρˆ reduces to a delta functional enforcing the continuity equation. Similarly, the nonlinear relation g = ρv is also taken in the form of a deltafunction constraint, 3  4 % ˆ δ[g − ρv]. (7.2.6) exp i d1 ψ(1)W [ψ] D(v) x

This delta function is then represented in terms of another conjugate field vˆ as  ! " i d1 vˆi (1) gi (1) − ρ(1)vi (1) .

(7.2.7)

Thus, with the set of fields consisting of the slow modes {ρ, g, v} and the corresponding hatted fields {ρ, ˆ gˆ , vˆ }, the MSR action is given by ⎡      ∂ρ −1 0 ⎣ A = dt dx gˆi β L i j gˆ j + i ρˆ + ∇ ·g ∂t i, j

+i

 i

+i

 i

⎞ # $  ∂g δ F i U gˆi ⎝ + ρ∇i + ∇ j (ρvi v j ) + L i0j v j ⎠ ∂t δρ ⎛

j

 vˆi (gi − ρvi ) ,

(7.2.8)

7.2 The compressible liquid

329

where we have set the linear (in the fields) term involving the currents ξ explicitly to zero. The nonlinearities in the generalized Langevin equation for g are determined from the functional FU . If FU is chosen to be a local quadratic functional of density with a wave-vector-independent structure factor, the function f in (6.2.23) is given by f (x) =

χ0−1 2 δρ (x). 2

(7.2.9)

The corresponding equal-time density correlation function is then obtained as & δρ(q)δρ(−q) =

ρ0 /c02 ,

for q ≤ ,

0,

for q > ,

(7.2.10)

where  is a large-wave-number cutoff corresponding to the shortest length scale for the model. With the above choice (7.2.9) of the free energy FU , we obtain that the pressure functional is given by P[δρ] = ρ0 χ0−1 δρ +

χ0−1 2 δρ . 2

(7.2.11)

Hence for the choice (7.2.9) of the free energy the Gaussian part of the MSR action is now obtained in the form ⎡      ∂ρ −1 0 ⎣ gˆi β L gˆ j + i ρˆ + ∇ ·g A0 = dt dx ∂t i, j  4  3 ∂gi  2 0 +i gˆi vˆi (gi − ρ0 vi ) , + c0 ∇i ρ + L i j v j + i ∂t i

i

(7.2.12) where c02 = ρ0 χ0−1 is the hydrodynamic speed of sound. Note that the Gaussian free energy FU gives rise to a nonlinear dynamics as a result of the Poisson-bracket structure for the slow modes in the liquid. In this case we have a cubic term in the non-Gaussian part AI (which contributes beyond the Gaussian order) of the action involving the coupling of the density fluctuations,  AI = i

 dt

dx

 i

gˆi (x, t)

χ0−1 ∇i δρ 2 (x, t). 2

(7.2.13)

This cubic nonlinearity in the action is a consequence of the structure of the Poisson brackets (6.2.5) between the slow variables. The nonlinearity in eqn. (7.2.13) is related purely to the reversible part of the dynamics.

330

Renormalization of the dynamics

7.2.2 Correlation and response functions The renormalized theory for the dynamics of the compressible liquids is developed in terms of the correlation functions, G αβ (12) = ψβ (2)ψα (1),

(7.2.14)

G α βˆ (12) = ψˆ β (2)ψα (1).

(7.2.15)

and the response functions,

The averages denoted here by the angular brackets are functional integrals over all the fields weighted by e−A . The nonlinearities in the equations of motion (6.2.20) and (6.2.22) give rise to non-Gaussian terms in the action (7.2.8) involving products of three or more field variables. Linear dynamics The inverse of the matrix G 0 in the Gaussian part of the MSR action (7.2.12) is obtained from the part which is of quadratic order in the fields. This part corresponds to the linearized equations for the dynamics. The corresponding inverse of the matrix G 0 for the model equations of the compressible liquid is shown in Table 7.1. For an isotropic liquid we can treat the transverse and longitudinal parts of the vector fields, i.e., g, v, and their hatted counterparts, separately. The transverse components correspond to q · gT = 0, and they do not couple into the density or its hatted conjugate ρ. ˆ The longitudinal and transverse components of the correlation functions in the linearized dynamics, denoted with suffix L and suffix T, respectively, are obtained by inverting the corresponding components of G −1 0 shown in Table 7.1. The longitudinal part of the Gaussian matrix G 0 is obtained in the block-matrix form involving 3 × 3 matrices C and R. The matrix of correlation functions described below involves the longitudinal and transverse parts of the dissipation matrix L i0j . These are denoted by 0 and η0 , respectively (see eqn. (5.3.71) for their definition),   C R L G0 = , (7.2.16) R†  where  denotes the 3 × 3 null matrix with all the elements equal to zero. C and R denote the matrices of the correlation and response functions, respectively, and R† is the Hermitian conjugate of R. The correlation-function matrix C is expressed up to a common factor in the following form: ⎡ ⎤ vL ρ gL ⎢ ⎥ ⎢ ⎥ 2 ⎢ q 2 ρ 2 qωρ 2 qωρ0 ρ ⎥  q 0 0 −1 0 ⎢ ⎥, C = 2β (7.2.17) ∗ ⎢ D0 D0 ⎢ qωρ 2 ω2 ρ 2 ω2 ρ0 gL ⎥ ⎥ 0 0 ⎣ ⎦ 2 2 qωρ0 ω ρ0 ω vL

7.2 The compressible liquid

331

Table 7.1 The matrix G −1 0 corresponding to the MSR action (7.2.12).

ρ gi vi ρˆ gˆi vˆi

ρ

g

v

ρˆ

gˆ

vˆ

0 0 0 ω −qi c02 0

0 0 0 −q j ωδi j iδi j

0 0 0 0 i Li j −iρ0 δi j

−ω qi 0 0 0 0

q j c02 −ωδi j i Li j 0 2β −1 L i j 0

0 iδi j −iρ0 δi j 0 0 0

where D0 is obtained as

( ) D0 = ρ0 ω2 − q 2 c02 + iωq 2 0 .

(7.2.18)

Similarly, for the response part of the matrix G involving correlations between hatted and unhatted fields we obtain ⎡ 1 R= D0

ρˆ

⎢ ⎢ ρ ω + i q 2 ⎢ 0 0 ⎢ ⎢ 2 ⎢ qc0 ρ0 ⎣ qc02

gˆ L

vˆL

qρ0

0 q 3

ωρ0

ω0 q 2

ω

  i ω2 − q 2 c02

⎤ ⎥ ρ ⎥ ⎥ ⎥. ⎥ gL ⎥ ⎦ vL

(7.2.19)

Let us focus on the expression for the density–density correlation function at zeroth order: G 0ρρ (q, ω) = 2β −1

ρ02 0 q 4 D0 D0∗

(7.2.20)

The poles in the correlation functions represent the corresponding hydrodynamic modes. The zeros of D0 thus indicate the existence of two propagating sound modes. In the hydrodynamic limit this is given by z = ±c0 q − q 2 0

(7.2.21)

with c0 and 0 denoting the speed and attenuation, respectively, of sound waves due to the density fluctuations. The longitudinal part of the current correlations is related to the density correlations through the relation ω2 G ρρ (q, ω) = q 2 G Lgg (q, ω),

(7.2.22)

which follows from the continuity equation (6.2.16). Since in the linear case the momentum density g and current v are trivially connected, g = v/ρ0 , the correlation and response

332

Renormalization of the dynamics

functions are trivially connected. Thus, from the inversion of the matrix G −1 0 , it directly follows that G vv (q, ω) =

1 1 G gg (q, ω) = G gv (q, ω). 2 ρ0 ρ0

(7.2.23)

In a similar way the transverse part of the correlation functions is obtained in block-matrix form by inverting the matrix G −1 0 shown in Table 7.1:   C RT T (7.2.24) G0 = RT† T involving 2×2 matrices C T and RT . T denotes the 2×2 null matrix with all the elements equal to zero. C T and RT denote the matrices of the transverse correlation and response functions, respectively, and RT† is the Hermitian conjugate of RT . From straightforward T inversion of the transverse part of the matrix G −1 0 we obtain the following results for C T and R : ⎡ ⎤ gT vT ⎥ η0 q 2 ⎢ ⎢ 2 ⎥ C T = 2β −1 (7.2.25) ρ ρ g ⎢ ⎥ 0 T ⎦ W0 W0∗ ⎣ 0 ρ0 1 vT and

⎡ RT =

1 W0

gT

⎢ ⎢ ⎢ ρ0 ⎣ 1

⎤

vT

⎥ ⎥ gT ⎥ , ⎦ vT

η0 q 2 iω

(7.2.26)

where the denominator W0 is given by W0 = ρ02 ω2 + q 4 η02 .

(7.2.27)

From the above result, let us focus on the typical quantity of transverse-current correlations, !

G T0

"

(q, ω) = gg

2β −1 ρ02 η0 q 2 ω2 + η02 q 4

.

(7.2.28)

At the level of linear dynamics this represents the diffusive decay mode of transverse shear in the isotropic liquid. This is expressed in the poles on the RHS of eqn. (7.2.28). From the explicit form of the zeroth-order (Gaussian) correlation-function matrix we note that the response function G 0αβ satisfies the relation  G 0α βˆ = − G 0ˆ βα

∗

.

(7.2.29)

7.2 The compressible liquid

333

The correlation and response functions in the linear case are related through the fluctuationdissipation relations:  (7.2.30) G 0αβ (q, ω) = −2β −1 Im G 0αβ ˆ (q, ω) . This follows directly from the Green-function matrix G 0 presented above. We shall come back to the issue of fluctuation–dissipation relations in the nonlinear case later on in this chapter. The expressions for longitudinal and transverse correlations given by eqns. (7.2.17) and (7.2.25), respectively, are the same as the corresponding expressions for the correlation functions in Section 5.3.2 (see eqns. (5.3.88) and (5.3.93), respectively). Nonlinear dynamics The nonlinear coupling of the hydrodynamic modes in the equations of motion gives rise to the non-Gaussian part of the action (7.2.8), and their effects on the correlation and response functions are obtained in terms of the self-energy matrix  defined through the Schwinger– Dyson equation (7.1.27). Let us first consider the cases in which both external indices in the matrix of eqn. (7.1.27) correspond to the unhatted fields. In this case we take note of the following properties of the general structure of the full Green-function matrix G−1 . (a) [G0 −1 ]αβ = 0, which follows from the action (7.2.8) obtained in the MSR field theory. (b) αβ = 0, which follows from the causal nature of the response functions in the MSR field theory. Since each of the vertices in the MSR action functional consists of at least one hatted field, the diagrams corresponding to the self-energy matrix elements for which both indices are unhatted will involve a closed loop of response functions and hence will vanish by causality. We therefore obtain that the elements of the G−1 matrix corresponding to the unhatted fields, [G−1 ]αβ = 0, are similar to those in the Gaussian case. The structure of the full G −1 is similar to the form (7.2.16) in the linear case. Upon inverting the matrix G−1 , we obtain for the correlation functions of the physical, unhatted field variables  G αβ = − G α μˆ Cμˆ (7.2.31) ˆ ν G νˆ β , μν

where Greek-letter subscripts take the values ρ, g, and v, and the self-energy matrix Cμˆ ˆν is given by −1 Cμˆ ˆ ν = 2β 0 δμˆ νˆ δμˆ gˆ − μˆ ˆν.

(7.2.32)

The double-hatted self-energies μˆ ˆ ν vanish if either index corresponds to the density. Equation (7.2.31) is a key relation expressing the correlation functions in terms of the response functions. From the Schwinger–Dyson equation (7.1.27) we obtain that the response functions G ψψ ˆ satisfy ) ( ¯ − αμ ¯ ¯ (13) (7.2.33) G −1 ˆ (13) G μβˆ (32) = δ(12)δαˆ βˆ 0 αμ ˆ

where we use the convention that the barred index 3, is summed over.

334

Renormalization of the dynamics

Table 7.2 The matrix of the coefficients Nα βˆ in the numerator on the RHS of eqn. (7.2.34) for the response functions.

ρ g v

ρˆ

gˆ

vˆ

ωρL + i L q(ρL c2 + Lγ ) q(c2 + iωγ )

ρL q ρL ω ω + iq 2 γ

Lq Lω i(ω2 − q 2 c2 )

The self-energies αμ ˆ are expressed in perturbation theory in terms of the two-point correlation and response functions. Using the explicit polynomial form of the action (7.2.8), the response functions are expressed in the general form G α μˆ (q, ω) =

Nα μˆ (q, ω) , D(q, ω)

(7.2.34)

where the matrix N is given in Table 7.2 and the determinant D in the denominator is given by D(q, ω) = ρL (ω2 − q 2 c2 ) + i L(q, ω)(ω + iq 2 γ ).

(7.2.35)

The various quantities are defined such that ρL , c2 , and L are identified as the corresponding renormalized quantities for the bare density ρ0 , speed of sound squared c02 , and longitudinal viscosity 0 , respectively. We have, in terms of single-hatted or response self-energies, ρL (q, ω) = ρ0 − ivv ˆ (q, ω),

(7.2.36)

L(q, ω) = 0 + igv ˆ (q, ω),

(7.2.37)

qc (q, ω) = 2

qc02

+ gρ ˆ (q, ω),

(7.2.38)

and γ is defined in terms of the self-energy element vρ ˆ ≡ qγ . 7.3 Renormalization In statistical mechanics the linear response to an external disturbance on a system in equilibrium and the corresponding correlation functions of fluctuations are related. These are generally termed fluctuation–dissipation theorems (FDTs). We have discussed in Section 7.3.1 that in the correlation matrix G of the MSR field theory the elements G ψψ and G ψ ψˆ are related through certain fluctuation–dissipation relations. In the case of linear dynamics such relations follow directly. However, in the case of nonlinear dynamics the FDT relations are more restricted. Deker and Haake (1975) demonstrated that such FDT relations exist in a number of specific situations. Our interest here is focused on the FDT relations which are satisfied for the case of compressible liquids (Das and Mazenko, 1986; Miyazaki and Reichman, 2005; Andreanov et al., 2006). The formulation of the FDT in the

7.3 Renormalization

335

MSR field theory is closely linked to the time-reversal symmetry of the action functional corresponding to the nonlinear Langevin equation for the dynamics.

7.3.1 Fluctuation–dissipation relations Let us consider how the MSR action changes under time reversal. This is closely linked to how the Langevin equation changes under time reversal. Let us define the time-reversal operation on the field ψi as T ψi (t) = i ψi (−t),

(7.3.1)

where i = ±1. For simplicity we suppress the spatial dependence of the fields and focus on the time dependence only. The Langevin equation for the time evolution of the slow mode ψi is obtained as δ Ft ∂ψi − Vi [ψ] + L i0j = ζi , ∂t δψ j (t)

(7.3.2)

where ζi is the Gaussian random noise correlated in time by a delta function. We adopt here the convention that the repeated indices are summed over. The reversible part of the dynamics is obtained in terms of the Poisson bracket Q i j between the slow modes, Vi [ψ] = Q i j [ψ] ∂ψi  − Q i j [ψ] − L i0j ∂t

δ Ft , δψ j (t) δ Ft = ζi . δψ j (t)

(7.3.3) (7.3.4)

We also assume here that L i0j is not dependent on the fields ψi and can be treated as a constant input in the dynamic description. Under time reversal the irreversible part involving the dissipative coefficient L i0j changes sign, ∂ψi (−t)  δ F−t − Q i j [ψ(−t)] + L i0j = ζi , ∂(−t) δψ j (−t)

(7.3.5)

where we have used the following time-reversal property for Q i j which follows from the definition of the Poisson bracket in eqn. (5.3.40): Q i j [ψ(−t)] = −i  j Q i j [ψ(t)].

(7.3.6)

We have also assumed that the bare transport coefficients satisfy the following time-reversal symmetry: L i0j (−t) = i L i0j  j .

(7.3.7)

Therefore the term involving the Poisson bracket Q i j remains invariant under time reversal while the term involving the dissipation coefficient L i0j changes sign and represents the

336

Renormalization of the dynamics

irreversible part of the dynamics. We now find a transformation T that will maintain the ˆ invariance of the MSR action A[ψ, ψ]:  t2 ˆ = A[ψ, ψ] dt ψˆ i (t)β −1 L i0j ψˆ j (t) t1

+ i ψˆ i (t)



$ δF ∂ψi # t − Q i j [ψ] − L i0j ∂t δψ j (t)



.

(7.3.8)

ˆ which In Appendix A7.3 we show that the transformation rules for the fields {ψ, ψ} maintain this property are given by ψi (x, −t) → i ψi (x, t), ψˆ i (x, −t) → −i ψˆ i (x, t) − iβ

δF . δψi (x, t)

(7.3.9) (7.3.10)

We can now use this to obtain the fluctuation–dissipation relation between the response function (involving one hatted field) and a corresponding correlation function. Let us consider the response function ψˆ i (x, t)ψ j (x , t  ). This response function is nonzero for t > t  . On the other hand, for t → −t and t  → −t  the corresponding response function ψˆ i (x, −t)ψ j (x, −t  ) vanishes since −t < −t  . Now, using the time-reversal properties of ψˆ i and ψ j given by (7.3.10) and (7.3.9), respectively, one obtains + , 1 2 ψˆ i (x, t)ψ j (x , t  ) − iβ θi (x, t)ψ j (x , t  ) = 0, (7.3.11) where the field θ is defined as θi (x, t) =

δF . δψi (x, t)

(7.3.12)

The response function is time-ordered and is nonzero for t > t  . The FDT which follows from eqn. (7.3.11) is expressed as G ψˆ i ψ j (t, t  ) = iβ(t − t  )G θi ψ j (t, t  ),

(7.3.13)

where (t) is the step function. In the stationary state we assume time translational invariance and obtain the one-sided 2 (see eqn. (1.3.10) for its definition) Cθi ψ j of the correlation function 1Laplace transform θi (x, t)ψ j (x, t  ) as Cθi ψ j (q, ω) = β −1 G ψˆ i ψ j (q, ω).

(7.3.14)

In the above equation we assumed spatial translational invariance and have taken a Fourier transform at wave vector q with respect to the spatial coordinates. Equation (7.3.14) leads to the following useful relation between the Fourier transforms of the correlation and response functions of the MSR field theory involving the nonlinear Langevin equation: G θi ψ j (q, ω) = −2β −1 Im G ψˆ i ψ j (q, ω).

(7.3.15)

7.3 Renormalization

337

The form of the FDT (7.3.13) provides a relation between the response functions (involving a hatted and an unhatted field) and the correlation functions (involving two unhatted fields). However (unlike in the case of linear dynamics), this is not a relation involving only two-point functions. It reduces to that form only in certain special situations. Deker and Haake (1975) described this in terms of processes of classes A and B, respectively. Class A. In this case the reversible part of the dynamics is zero and we have a purely dissipative dynamics. So the equation for the dynamics is obtained from eqn. (7.3.2) as ∂ψi δF + L i0j = ζi . ∂t δψ j (t)

(7.3.16)

In this case the dissipative part is not necessarily linear. Using the equation of motion (7.3.16), the FDT relation (7.3.13) reduces to the form -

. −1 ∂ψk (t)   0  G ψˆ i ψ j (t − t ) = −iβ(t − t ) L ik − ζi (t) ψ j (t ) ∂t

 −1 ∂ 1 2 1 2  0   = −iβ(t − t ) L ik ψk (t)ψ j (t ) − ζi (t)ψ j (t ) , (7.3.17) ∂t assuming that the bare-transport-coefficient matrix L i0j is independent of the field variables {ψi }. The second term on the RHS of eqn. (7.3.17) is zero. This is justified with the following argument: (a) for t < t  1it is equal to 2 zero because of the step function (t −t  ) and (b) for t > t  on average ζi (t)ψ j (t  ) is zero. The delta-functioncorrelated noise ζ (t) can have an influence on the field ψ j (t  ) only if t  is greater than t. Therefore the second term is always zero. We obtain the simplified FDT relation  −1 ∂ 1 2 0 G ψˆ i ψ j (t − t  ) = −iβ(t − t  ) L ik ψk (t)ψ j (t  ) ∂t  −1 ∂  0 (7.3.18) G ψk ψ j (t − t  ) ≡ −i(t − t )β L ik ∂t linking the response function to the time derivative of the correlation functions. The above FDT relation therefore holds in purely dissipative systems having (a) constant bare transport coefficients and (b) a driving free functional that is non-Gaussian. This FDT relation therefore applies only to cases in which the noise is additive in nature. Class B. In this case the dissipative term in the equation of motion is linear in the fields. This corresponds to the bare transport coefficient L i0j being independent of fields and the free-energy functional being quadratic in the fields. The Poisson bracket Q i j involved depends on the fields and gives rise to the nonlinear dynamics. Dynamical equations of this type have been considered by Kawasaki (1970b) for studying critical dynamics. The form of the dissipative term in the Langevin equation (7.3.2) implies that, for constant (independent of the fields ψi ) bare transport coefficients L i0j , the functional derivative [δ F/δψi ] ≡ θi can be expressed as a linear combination of the fields: θi =

δF −1 = β −1 Sik ψk . δψi

(7.3.19)

338

Renormalization of the dynamics

The above relation follows directly from the definition of the equal-time correlation matrix Si j . The FDT (7.3.15) is expressed in the linear form  −1 Sik G ψk ψ j (t − t  ). (7.3.20) G ψˆ i ψ j (t − t  ) = i(t − t  ) k

This analysis was further extended by Miyazaki and Reichman (2005) to the case in which both Q i j and L i0j , controlling the reversible and irreversible parts of the dynamics, respectively, have a linear dependence on the fields: Q i j [ψ] = Q i0j + Q i1jk ψk , L i0j [ψ]

=

L˜ i0j

+

(1) L˜ i jk ψk .

(7.3.21) (7.3.22)

It was shown from a one-loop analysis that the linear FDT given by eqn. (7.3.20) still holds in the extension of case B. For recent work on the fluctuation dissipation relation in Nonlinear Langevin equation with field dependent transport coefficients see Kumaran (2011). The compressible liquid We now come back to the discussion of the model for compressible liquids. The set of equations (6.2.16) and (6.2.20) constitutes a dynamics that does not match class A or class B described above. The bare transport coefficients are taken to be independent of the fields but the free energy is not Gaussian due to the presence of the 1/ρ nonlinearity in the kinetic term (see eqn. (6.2.7)). Therefore, this model does not have a complete set of FDRs linearly relating correlation and response functions. However, using the timetranslational-invariance properties of the action (7.2.8), some useful fluctuation–dissipation relations between correlation and response functions involving the field g are obtained. This basically involves observation of the invariance of the MSR action under the following transformations: ρ(x, −t) → ρ(x, t),

(7.3.23)

gi (x, −t) → −gi (x, t),

(7.3.24)

δF , δρ(x, t) = −ρ(x, ˆ t) + iβπ(x, t), δF , gˆi (x, −t) → −gˆi (x, t) + iβ δgi (x, t)

(7.3.25)

ρ(x, ˆ −t) → −ρ(x, ˆ t) + iβ

= −gˆi (x, t) + iβvi (x, t),

(7.3.26)

δF = vi (x, t) δgi (x, t)

(7.3.27)

where we have used the result

following the dependence of the free-energy functional on the momentum density gi as given by eqn. (6.2.7). The field π(x, t) is defined as

7.3 Renormalization

339

δF = π(x, t). δρ(x, t)

(7.3.28)

Using the transformation (7.3.26), we obtain from the relation (7.3.15) the useful FDT relation G vi α (q, ω) = −2β −1 Im G gˆi α (q, ω),

(7.3.29)

where α indicates any of the fields {ρ, g, v}. The second FDT relation, which follows in a similar manner from the transformation (7.3.25), is G π α (q, ω) = −2β −1 Im G ρα ˆ (q, ω),

(7.3.30)

where α indicates any of the fields {ρ, g, v}. While eqns. (7.3.29) and (7.3.30) constitute the two basic FDT relations in the fluctuating-hydrodynamics model, they differ in an important way. The field δ F/δgi = vi which is involved in the first FDT relation also appears in the equation of motion (6.2.20) of the momentum density in the fluctuating-hydrodynamics description. Indeed, it is this FDT which proves to be most useful in the analysis presented in the next section. We show that this FDT is crucial in establishing the renormalizability of the dynamics in the hydrodynamic limit. The second FDT, on the other hand, involves the introduction of a new field, π = (δ F/δρ), which is absent from the equations of motion of the slow variables, i.e., eqns. (6.2.16) and (6.2.20). The nonlinear part of this field π , which is the functional derivative of the free energy F with respect to density, comes from (δ FK /δρ), where FK is the kinetic part involving the 1/ρ nonlinearity. However, this term finally leads to the wellknown Navier–Stokes nonlinearity ∇ j (gi v j ) in the equations of motion and π drops out from the dynamics. As a result the MSR theory does not have a linear FDT-type relation between G ρρ and G ρ ρˆ that is valid for all wave vectors. Such a relation can be obtained only in the hydrodynamic limit (see eqn. (8.3.10) in Section 8.3.2). 7.3.2 Nonperturbative results The field theory outlined above involves a large number of correlation functions between the different fields. We divide them broadly into two categories, namely the correlation functions denoted by G ψψ and the “response” functions G ψ ψˆ with the symbols ψ and ψˆ referring to any one of the field variables or the hatted ones, respectively. Several simple relations are obtained (Das and Mazenko, 1986) in this respect. The first important result is G α βˆ (q, ω) = −G ∗βα ˆ (q, ω).

(7.3.31)

From eqn. (7.3.31) and the Schwinger–Dyson equation (7.1.27) the following nonperturbative result for the self-energy matrix elements then follows directly: ∗ α βˆ (q, ω) = −βα ˆ (q, ω).

(7.3.32)

As a consequence of the continuity equation we obtain the following relation between the correlation functions involving the density field ρ and those involving the current g: ωG ρβ (q, ω) − q · G gβ (q, ω) = δβ ρˆ .

(7.3.33)

340

Renormalization of the dynamics

Using the FDT relation (7.3.29), we obtain a number of relations between the correlation functions and the response functions in a nonperturbative manner. For the isotropic fluid the description of the renormalized theory is facilitated by treating the longitudinal and transverse parts separately in the same manner as we did for the case of linearized dynamics. The self-energy matrix can also be split into a longitudinal and a transverse part in the following way: L T (q, ω) + (δi j − qˆi qˆ j )αβ (q, ω), αi β j (q, ω) = qˆi qˆ j αβ

(7.3.34)

where the superscripts L and T denote the longitudinal and transverse parts, respectively. The transverse case We will first discuss the relatively simpler transverse case in which the transverse components do not couple to the density field. Since the cubic and quartic vertices in the MSR action from eqns. (6.2.16) and (6.2.25) do not contain the g field, the self-energy matrix elements with a gi vanish, i.e., gi β = βgi = 0. The transverse correlations are obtained from the inversion of the transverse part of the correlation matrix presented on the RHS of eqn. (7.1.27). It follows directly from the fluctuation–dissipation theorem (7.3.29) that the correlation functions G vv and G vg are related to the response functions G g gˆ and G v gˆ , respectively. In the renormalized theory these correlation functions are obtained as T (q, ω) = Cvv

(βρT )−1 , ω + iηR (q, ω)

T Cvg (q, ω) =

β −1 , ω + iηR (q, ω)

(7.3.35)

where

where the renormalized function ηR (q, ω) is defined as

i T ηR (q, ω) = η0 + 2 gv (q, ω) ρT−1 (q, ω), ˆ q

(7.3.36)

T ρT (q, ω) = ρ0 − ivv ˆ (q, ω).

It is useful to check for the static correlation functions from the limiting values of the renormalized correlation functions. For example, the equal-time velocity correlation function is obtained from the following high-frequency limit: T T (q) = lim ωCvv (q, ω) = (βρ0 )−1 , χvv ω→∞

(7.3.37)

T (q, ω) vanishes as 1/ω for large ω. Simiwhere we have assumed that the self-energy vv ˆ T = β −1 . larly, we also obtain χvg

7.3 Renormalization

341

The renormalized theory takes a simpler form in the hydrodynamic limit of small wave number and frequency. In the transverse case in which the hydrodynamic modes are diffusive the self-energies  T (q, ω) are analyzed for ω ∼ q 2 in the limit of small q. In doing the analysis in this limit, we first identify the explicit factors of q and ω in the different self-energies necessitated by symmetry and conservation laws. We have, for example, that T 2 T gv ˆ = −iq γgv ˆ (q, ω),

(7.3.38)

gTˆ gˆ gTˆ vˆ

(7.3.39)

= =

−q 2 γgˆTgˆ (q, ω), −iq 2 γgˆTvˆ (q, ω).

(7.3.40)

As a consequence of conservation of momentum, every external gˆi vertex contribution to T supplies a factor of q . Since the system is isotropic, all of the  T then must be of gv i ˆ gβ ˆ

O(q 2 ). Using the fluctuation–dissipation theorem the following nonperturbative relations between the self-energies are obtained (Das and Mazenko, 1986) in the hydrodynamic limit: T γgˆTgˆ (0, 0) = 2β −1 γgv ˆ (0, 0),

(7.3.41)

−1 −1 T vT ˆ vˆ (0, 0) = 2β ω vv ˆ (0, ω).

(7.3.42)

T (0, 0) = 0 since  T (0, 0) is nonzero The last equation presented above tells us that vv ˆ vˆ vˆ and finite. Therefore, in the hydrodynamic limit we can write down the transverse correlation functions in the same form as the corresponding zeroth-order quantity obtained for those in the Gaussian action, with the renormalized viscosity being given in terms of    −1  (7.3.43) ηR = η0 + γgv ˆ (0, 0) ρT

or, in terms of the gˆ gˆ self-energy, as

 −1 β , ηR = η0 + γgˆ gˆ (0, 0) ρT 2

(7.3.44)

where T ρT = ρ0 + vv ˆ (0, 0).

(7.3.45)

The results (7.3.43) and (7.3.44) indicate that, in the hydrodynamic limit, the transverse correlation and response functions corresponding to the dynamics with the full nonlinearities are obtained in terms of the renormalized (generalized) shear viscosity. The renormalization of the latter is obtained from either the correlation self-energy (involving both hatted field indices) gTˆ gˆ or equivalently from the response self-energy (involving a single

T . This result is a direct consequence of the fluctuation–dissipation hatted field index) gv ˆ result (7.3.29).

342

Renormalization of the dynamics

The longitudinal case The longitudinal case is somewhat more complicated than the transverse one since here we have to deal with three fields {ρ, g, v} both in the hatted and in the unhatted sets. The correlation functions are obtained by inverting the longitudinal part of the G −1 matrix. The response functions are presented in the form (7.2.34). The fluctuation–dissipation theorem (7.3.29) once again is used to link the response function to the correlation function similarly to what we have already discussed in the transverse case. The crucial relation used in this regard is given by (7.2.31). In the longitudinal case there are traveling modes and hence the self-energies  L are analyzed in the limit ω ∼ q as q → 0. We first define the self-energy elements in terms of the explicit factors of q and ω necessitated by conservation laws and symmetry. We have, for example, that gLˆ gˆ = −q 2 γgˆ gˆ ,

(7.3.46)

ρLvˆ = qγρ vˆ ,

(7.3.47)

L gv ˆ

2

(7.3.48)

ρ gˆ = qγρ gˆ .

(7.3.49)

= −iq γgv ˆ ,

Relations between the different elements of the self-energy matrix follow directly from the analysis of the fluctuation–dissipation theorem (Das and Mazenko, 1986). We will discuss here two results that are particularly useful for our subsequent analysis of the self-consistent mode-coupling theory. First, we consider the relation between the matrix elements which are involved in the renormalization of the viscosity,   (7.3.50) γgˆ gˆ (0, 0) = 2β −1 γgv ˆ (0, 0) + γ˜ρ gˆ , where we have defined the limiting quantity γ˜ρ gˆ = lim ω−1 γρgˆ (0, ω). ω→0

(7.3.51)

The renormalization of the longitudinal viscosity 0 is computed in terms of the response self-energy gv ˆ as  −1  L(q, ω) = 0 + γgv (7.3.52) ˆ (0, 0) + γ˜ρ gˆ ρL or in term of the correlation self-energy gˆ gˆ as

β L(q, ω) = 0 + γgˆ gˆ (0, 0) ρL−1 , 2

(7.3.53)

with ρL given by  ρL = ρ0 + vv ˆ (0, 0).

(7.3.54)

Thus for both the shear and the longitudinal viscosities the renormalization can be computed by analyzing either gˆ gˆ or the response self-energies gv ˆ and gρ ˆ . We have therefore established nonperturbatively the renormalizability of the theory in the hydrodynamic

7.3 Renormalization

343

limit. This is obtained using only the fluctuation–dissipation theorem (7.3.29). In this limit of small ω and q the renormalization of the full correlation functions involves the replace (0, 0) and ments c0 → c and 0 → L(q, ω) in the zeroth-order expressions. Since both vv ˆ ρgˆ (0, 0) vanish in the hydrodynamic limit, the renormalized quantities ρL and c are real. Finally, we consider a useful relation for the self-energy matrix element vρ ˆ . Note that the latter involves the nonlinear vertex concerning the conjugate field v. ˆ This vertex is a consequence of the nonlinear constraint (6.2.22) introduced in the theory to deal with the 1/ρ nonlinearities in the dynamics, vˆ vˆ (0, 0) =

2ρL  γ (0, 0), βc2 ρ vˆ

(7.3.55)

where the renormalized speed of sound is given by c2 = c02 − γρ gˆ (0, 0). This result also follows from the analysis of the fluctuation–dissipation theorem (7.3.29) in the hydrodynamic limit.

7.3.3 One-loop renormalization We now consider the renormalization of the transport coefficients at one-loop order. This calculation in a natural way gives rise to the self-consistent mode-coupling model which forms the basis of our discussion in the subsequent chapters on equilibrium and nonequilibrium dynamics in the glassy state. The MSR action A for the specific case of the compressible liquid can now be written from (7.2.8) in the symmetrized form     −1 1 ˆ = d1 d2 ψα (1) G 0αβ (12)ψβ (2) A[ψ, ψ] 2 α,β     1 + Vαβγ (123)ψα (1)ψβ (2)ψγ (3) d1 d2 d3 3 αβγ      1 d1 d2 d3 d4 Vαβγ μ (1234)ψα (1)ψβ (2)ψγ (3)ψμ (4), + 4 αβγ μ

(7.3.56) where the repeated indices α, β, γ , and μ in all three terms are summed over. We adopt a notation in which we denote the different fields as ψα (1) with the index 1 representing the space-time point (x1 , t1 ), where α represents the corresponding slow variable and runs over ρ, ρ, ˆ gi , gˆi , vi , and vˆi . Let us focus on the non-Gaussian part of the action. The symmetrized cubic vertices in the action (7.3.56) are obtained as Vαβγ (123) =

1˜ Vαβγ (123) + V˜βαγ (213) + V˜αβγ (321) 2 + V˜αγβ (132) + V˜βγ α (231) + V˜γ αβ (312) ,

(7.3.57)

344

Renormalization of the dynamics

where V˜αβγ (123) =

3 

(i) (123). V˜αβγ

(7.3.58)

i=1

The last line on the RHS of eqn. (7.3.56) represents a four-point vertex function. This arises from the convective nonlinearity ∇ j [gi g j /ρ] ≡ ∇ j (ρvi v j ) in the equation of motion for the momentum density. From the polynomial form of the action discussed in the previous section the vertex functions are obtained. Specifically, for the flat-structure-factor case, (1) V˜αβγ (123) = i

 i

(2) V˜αβγ (123) = iρ0

δα,gˆi ∇1i



χ −1 δ(1, 2)δ(1, 3)δβ,ρ δγ ,ρ , 2

(7.3.59)

δα,gˆi ∇1i δ(1, 2)δ(1, 3)δβ,vi δγ ,v j ,

(7.3.60)

δα,vˆi δβ,ρ δγ ,vi .

(7.3.61)

i, j

(3) V˜αβγ (123) = −i

 i

The vertices presented above arise from the pressure term, the convective term in the generalized Navier–Stokes equation (6.2.20), and the nonlinear constraint (6.2.22), respectively. Similarly the quartic vertex is a sixth of the sum of all permutations of the variables (α, 1), (β, 2), (γ , 3), and (μ, 4) labeling the unsymmetrized vertex V˜αβγ μ (1234), where  δα,gˆi δβ,ρ δγ ,vi δμ,v j ∇1i [δ(1, 2)δ(1, 3)δ(1, 4)]. (7.3.62) V˜αβγ μ (1234) = i i, j

The role of these four-point vertices with respect to the long-time tails in viscosities has already been indicated in Section 6.1.1. In three-dimensional systems the contributions from such terms can be taken into account through a redefinition of the bare transport coefficient. In the following we ignore this nonlinearity. Using the above expressions for the vertex functions, the different elements of the self-energy matrix  are obtained in a self-consistent form in terms of the full correlation function from the formula (7.1.28). This has so far been achieved only at one-loop order. We saw in the earlier section that the renormalizations of the viscosities 0 (longitudinal) and η0 (shear) are obtained from the corresponding components of the self-energy gˆi gˆ j . A detailed diagrammatic analysis of the corresponding self-energy diagrams (shown in Fig. 7.1) gives the renormalized longitudinal viscosity at one-loop order as  L(q, z) = 0 + dt ei zt MC (q, t)   ∞ dk L dt ei zt VMC (q, k)G ρρ (k, t)G ρρ (q − k, t), (7.3.63) = 0 + (2π )3 0

7.3 Renormalization

345

Fig. 7.1 The one-loop diagrams at O(kB T ) for the self-energy gˆi gˆ j . The lines joining both ends with the vertices represent fully renormalized correlation functions as labeled. The vertices with three legs attached to them are as indicated in (7.3.59)–(7.3.61). L (q, k) denotes vertex functions determined in terms of thermodynamic propwhere VMC erties of the liquid. For example, in the wave-vector-independent model with the form (7.3.59) we obtain L VMC ≡

βχ −2 . 2ρ0

(7.3.64)

It is important to note that the above one-loop correction to the longitudinal viscosity is obtained with some simplifications and approximations. In the supercooled state the slow decay of the density fluctuations is assumed to produce the dominant contribution to the transport properties. The contributions to gˆ gˆ from diagrams involving the convective vertices (resulting from nonlinear coupling of currents in the equations of motion) are relatively small in the supercooled state and are absorbed by redefining the corresponding bare transport coefficient. This is a key approximation made in the study of slow dynamics using the mode-coupling models.3 The renormalized density correlation function A key quantity of interest in all subsequent discussions will be the density–density correlation function. We define the normalized correlation function with respect to the equal-time value as G ρρ (q, t) ψ(q, t) = . (7.3.65) G ρρ (q, t = 0) The renormalization is easily extended to the case in which a realistic structure factor with a proper wave-vector dependence is taken into account. With the wave-vector-dependent 3 This approximation is further justified with the dynamic density-functional model. This is formulated in terms of the density

only and the momentum density is integrated out of the model using the adiabatic approximation. The renormalized theory in this case has a very similar wave-vector dependence to that obtained above ignoring the role of the convective nonlinearity. This is discussed later in Section 8.1.6.

346

Renormalization of the dynamics

model in which the dynamics is given by eqns. (6.2.28) and (6.2.29), the renormalized transport coefficient L(q, z) is obtained from the relevant set of self-energies. The latter are expressed in terms of the correlation functions and response functions. A renormalized perturbation expansion for the transport coefficients at arbitrary order in the nonlinearities can thus be constructed in principle. The renormalized longitudinal viscosity with the mode-coupling contribution at one-loop order is given by   ∞ dk i zt L L(q, z) = 0 + dt e Vq,k ψ(k, t)ψ(q − k, t), (7.3.66) (2π )3 0 L is determined in terms of the where 0 is the bare viscosity. The vertex function Vq,k thermodynamic properties of the liquid as  "2 n0 dk ! L (qˆ · k)c(k) + qˆ · (q − k)c(|q − k|) S(k)S(|q − k|), (7.3.67) = Vq,k 3 2mβ (2π )

where S(q) denotes the static structure factor for the liquid. Using the renormalizability of the nonlinear theory (in the hydrodynamic limit) which was proved in the previous section, the density correlation function is obtained using the renormalized transport coefficient L(q, z) in the expression given in eqn. (7.2.20),  −1 q2 , (7.3.68) ψ(q, z) = z − z + iq 2 L(q, z) where q represents a microscopic frequency characteristic of the liquid state. Similarly, the normalized transverse current correlation function φT (q, z) takes the form φT (q, z) =

1 , z + iq 2 ηR (q, z)

with the renormalized shear viscosity ηR (q, z) obtained as  ηR (q, z) = η0 + dt ηMC (q, t)ei zt .

(7.3.69)

(7.3.70)

For the wave-vector-dependent model, the mode-coupling contribution ηMC obtained from the transverse part of the self-energy gˆ gˆ at one-loop order is  dk 1 T ηMC (q, t) = Vq,k G ρρ (k1 , t)G ρρ (k, t). (7.3.71) 2βρ0 (2π )3 T is obtained as The vertex function Vq,k T ˆ 2 }. Vq,k = [c(k) − c(k1 )]2 {1 − (qˆ · k)

(7.3.72)

Another simple and systematic approach to computing the memory function was taken by (Zaccarelli et al., 2001) by considering the equation for ρq (Fourier transform of density ρ(x, t)). This is obtained in the form of a linear equation  t ρ¨q + q2 ρq (t) + L q (t − s)ρ˙q (s)ds = f q (t). (7.3.73) o

7.3 Renormalization

347

A self-consistent expression for the memory function L q is obtained in terms of a nonlinear function of ρq s exploiting the proper fluctuation–dissipation relation. The final form (8.1.33) is reached here by assuming that the noise f q is an additive Gaussian process. A similar approach was used by Wu and Cao (2003) for obtaining the memory kernel for a linear molecular liquid. Note that in the wave-vector-dependent models the length scales over which the fluctuations are considered are very small. Near the static structure-factor peak the length scale is of the order of the diameter of a constituent particle. This involves extending the theory to much shorter length and time scales beyond the hydrodynamic limit. This is the regime of generalized hydrodynamics. It is, however, important to note that the validity of the renormalized perturbation theory in terms of correlation functions has so far been established only in the hydrodynamic limit, and the extension of the equations of fluctuating nonlinear hydrodynamics to large wave vector is merely a plausible assumption at this point. At high densities the mean free path of the liquid particles is small and hence the validity of hydrodynamics is pushed to short length scales. Both in the field-theoretic model and in the so-called memory-function approach (discussed in Appendix A7.4) the wave-vector dependence of the vertex functions at large k involves invoking this limit. This indeed goes back to the very basic problem of extending the dense-liquid-state theory to the finite-wave-number and -frequency limit. Recently a fundamental theory for the kinetics of systems of classical particles has been presented (Mazenko, 2010). This involves a unification of kinetic theory, Brownian motion, and the MSR field theory. Here, instead of following the standard method of constructing the field theory with the conserved densities, one works with the microscopic equations of motion. It is the dynamic generalization of the functional theory of fluids in equilibrium. In this model the conjugate set of (hatted) MSR fields is introduced at the microscopic level and the perturbation theory is organized self-consistently in terms of the interaction potential. At the second order, the renormalized equations for the density correlation functions constitute a dynamic feedback mechanism and give rise to the ergodic–nonergodic (ENE) transition. This is very similar to ENE transition in the standard mode-coupling theory to be discussed in the next chapter. While the model with microscopic Brownian dynamics applies naturally to colloidal systems, the approach in general allows for compatible approximations for higher-order correlation functions and is in fact applicable to a large set of dynamical systems. These include reversible and dissipative systems with Newtonian and Fokker–Planck dynamics. To summarize, we have obtained here, using the field-theoretic approach, the expressions for the density–density correlation in terms of generalized transport coefficients, including the effects from the coupling of the slow modes. The MSR field-theoretic model presents a suitable technique by which to obtain the renormalized perturbation theory in a self-consistent form. In the next chapters we discuss how this self-consistent model is used for understanding the slow dynamics characteristic of the dense liquid approaching vitrification.

Appendix to Chapter 7

In this Appendix we present a simple deduction (Kawasaki, 1995) for the basic model of mode-coupling theory without going into many technical complications. In the following, we simplify the notation by writing the vectors q ≡ q etc. The two basic equations of nonlinear fluctuating hydrodynamics (6.2.16) and (6.2.28) are written in the simplified form ∂ρ + ∇.g = 0 ∂t ∂gi + ∇i ∂t



(A7.1)







d x U (x, x )δρ(x , t) −



d x  L i0j (x − x  )

j

g j (x  , t) ρ0

 +

dx1 dx2 V i (x, x1 , x2 )δρ(x1 , t)δρ(x2 , t) = f i (x, t).

(A7.2)

First, we ignore the convective nonlinearity involving products of the momentum field gi from the momentum equation (6.2.28) and focus on the role of density couplings relevant for the supercooled liquid. Second, we simplify the dissipative term in the momentum equation by taking the bare transport coefficient matrix as diagonal, i.e., L i0j (x − x  ) ∼ ρ0  B δi j ∇ 2 δ(x − x  ), and ignore the 1/ρ nonlinearity produced by the corresponding equilibrium quantity 1/ρ0 . The noise f i (x, t) is related to the bare transport matrix through the FDT relation  f i (r, t) f j (r , t  ) = 2k B T L i0j (r, r )δ(t − t  ).

(A7.3)

On taking the divergence of eqn. (A7.2) and combining with the continuity equation (A7.1), we obtain a second-order equation for the density fluctuation as  ∂ρ(x  , t) 1 2 ∂ 2ρ dx S −1 (x − x  )δρ(x  , t) +  B − ∇ 2 βm ∂t ∂t

 + ∇. dx1 dx2 V i (x, x1 , x2 )δρ(x1 , t)δρ(x2 , t) = θ (x, t), 348

(A7.4)

Appendix to Chapter 7

349

where the random noise is defined in terms of the divergence θ (x, t) = ∇. f i (x, t). The first term on the RHS of eqn. (A7.4) involves the inverse of the two-point function S(r ), whose Fourier transform is the static structure factor S(k). Using eqn. (6.2.29), it follows that the latter is related to the Fourier transform of the two-point direct correlation function c(k) through the Ornstein–Zernike relation S(k) = (1 − n 0 c(k))−1 . On taking a Fourier transform of eqn. (A7.4), we obtain

2 ∂ 2 ∂ 2 +  q (A7.5) +  B q ρq (t) + Rq (t) = θ (q, t), ∂t ∂t 2 where ρq (t) and θ (q, t), respectively, denote the Fourier transforms of the density fluctu√ ation δρ(r) and the noise θ (x, t). q = 1/ βm S(k) represents a microscopic frequency of the liquid state. The linear terms in the above equation represent damping of density fluctuations controlled by the bare transport coefficient  B . The nonlinear term Rq in the above equation is obtained using the expression (6.2.30) for V i as

 dk 1 (A7.6) q.kc(k) + q.(q − k)c(q − k) ρk ρq−k . Rq (t) = βm (2π )3 In order to obtain the renormalization of the dynamics due to Rq we interpret the latter as a sum of two terms,  t ∂ρq (t  ) dt   R (q, t − t  ) + θ˜ (q, t), (A7.7) Rq (t) = − ∂t  0 where the first term on the RHS is a regular contribution representing memory effects in ˜ Note that, terms of a kernel function  R (t) and the second is the random component θ. for a memory function with the property  R (t) =  R (−t), the first term remains invariant under time reversal. Indeed, the source of the nonlinear term Rq is from the reversible part of the dynamics. The random part θ˜ defined in eqn. (A7.7) is related to the corresponding dissipative part involving  R through the fluctuation–dissipation relation −1  R (q, t) = θ˜ (q, t)θ˜ (−q, 0)|ρq |2  .

(A7.8)

Using the definitions (A7.6) and (A7.7) in the relation (A7.8), it is straightforward to show that

2  1 dk  R (q, t) = q.kc(k) + q.(q − k)c(q − k) S(k)S(q − k) 2ρ0 β (2π )3 × φk (t)φq−k (t), (A7.9) where φk (t) is the normalized density correlation function defined as φk (t) = (ρq )(t) −1

ρ−q (0)|ρq |2  . In reaching eqn. (A7.9) one needs to make the crucial approximation of equating four-point correlations of density fluctuation to products of two-point density correlation functions (see Appendix A7.4.2). This closes the mode-coupling equation at the lowest level. Using eqn. (A7.8) in eqn. (A7.5), we obtain a second-order differential equation for ρq (t). On multiplying this equation by ρ−k (0) and averaging with respect

350

Appendix to Chapter 7

to the noise θ we obtain the following differential equation for the normalized density correlation function:  t 2 ¨ ˙ dt   R (q, t − t  )φ˙q (t  ) = 0, (A7.10) φq (t) + q φq (t) +  B φq (t) + 0

where  R (t) is given by eqn. (A7.9). The above equation for φq (t) constitutes the basic equation for the mode-coupling model discussed in Chapter 8 for the ergodicity– nonergodicity transition.

A7.1 The Jacobian of MSR fields In obtaining the action in terms of the physical field ψ and its hatted conjugate ψˆ in the MSR field theory we treat as constant the functional Jacobian which appears in the calculation of the generating function for computing statistical averages. More specifically, the Jacobian is associated with the functional delta function which ensures that the field variable ψ is a solution of the equation of motion. Thus the Jacobian links the field variable ψ with the noise θ through the equation of motion for the field variable. It therefore naturally depends on the manner in which the equation of motion is discretized in time. To explain this, let us consider a one-dimensional case (Jensen, 1981) of a single variable ψ:   ∂ψ(1) ¯ ¯ + W3 [12¯ 3]ψ( ¯ ¯ ¯ + θ (1), 2) 2)ψ( 3) = − U1 [1] + U2 [12]ψ( ∂t

(A7.1.1)

where we adopt the convention that the repeated (barred) indices are summed or integrated over. On a discrete-time grid the above equation of motion is written in the form (for example) " 1! ψ(ti ) − ψ(ti−1 ) = a1 U1 [ti ] + b1 U1 [ti−1 ] + · · ·  + an Un [ti , t2¯ , . . ., tn¯ ]ψ(t2¯ ). . .ψ(tn¯ ) ˜ i−1 ), (A7.1.2) ˜ i + bθ + bn Un [ti−1 , t2¯ ]ψ(t2¯ ). . .ψ(tn¯ ) + · · · + (aθ with the constraint that the numbers ai and bi for all i satisfy ai + bi = 1 and a˜ + b˜ = 1. The discretization scheme is arbitrary with these constraints and in the limit  → 0 we expect that all of them should become identical. In the one-dimensional case, for example, if we choose, for all i, ai = 0 and bi = 1, then the Jacobian is a constant, % 1 (A7.1.3) ≡ C0 . J=  d+1 i∈

Although in the  → 0 limit C0 is diverging, it is canceled out by a similarly large quantity from the denominator in the final expression for a statistically averaged quantity. In general the arbitrariness in the discretization of the equation of motion can be used to set the Jacobian J equal to a constant. What is important is to note that the Jacobian links the field ψ at time ti to its values as well as that of the noise for an earlier time ti  < ti . The role of

A7.2 The MSR field theory

351

the Jacobian is therefore essential in maintaining causality. The latter also ensures that the response functions are time-ordered. In our discussion of the models described in this book, we treat the Jacobian as a constant. In some cases (see the discussion with a p-spin model in Section 8.4, eqn. (8.4.10)) the Jacobian can also be simply absorbed as a correction term in the action functional (De Dominics, 1978; De Dominicis and Pelti, 1978) in the thermodynamic limit.

A7.2 The MSR field theory Here we establish the crucial relations (7.1.28)–(7.1.31) which express self-consistently the self-energy  of the MSR field theory in terms of the full correlation functions G. We start from the standard functional identity,  δ (A7.2.1) D(%) [e−A x i(%) ] = 0, δ%(1) corresponding to the action functional (7.1.22) having cubic vertices. The above identity is equivalent to . δ Aξ = 0. (A7.2.2) δ%(1) Using the expression (7.1.22) for the MSR functional, we obtain ¯ ¯ ¯¯ ¯ ¯ G −1 0 (12)G(2) + V (123)%(2)%(3) = ξ(1).

(A7.2.3)

We have adopted in the above equation the summation convention, i.e., the repeated indices (with a bar) are summed over. We also use the fact that the three-point vertex function V (123) is symmetric under exchange of the indices. This is an important property of the vertex function V and we use it repeatedly in this chapter:  δ e−Aξ δ G(1) = D% %(1) δξ(2) δξ(2) Z = %(1)%(2) − %(1)%(2) = δ%(1)δ%(2) ≡ G(12).

(A7.2.4)

The above equation reduces to the standard result %(1)%(2) = %(1)%(2) + G(12).

(A7.2.5)

On reorganizing the terms in (A7.2.3), we obtain ! " ¯ ¯ ¯¯ ¯ ¯ ¯¯ G −1 0 (12)G(2) + V (123) G(2)G(3) + G(23) = ξ(1).

(A7.2.6)

352

Appendix to Chapter 7

On taking one more derivative of eqn. (A7.2.6) w.r.t. the current ξ(2), we obtain the equation

¯ ¯ ¯ + V (12¯ 3) ¯ G(22)G( ¯ ¯ + G(2)G( ¯ ¯ + δ G(32) = δ(12). (1 2)G( 22) 3) 32) G −1 0 ¯ δξ(2) (A7.2.7) Starting from the definition of the inverse of a matrix, ¯ ¯ = δ(12), 21) G −1 (12)G(

(A7.2.8)

and differentiating with respect to ξ(3), we obtain ¯ ¯ δG −1 (12) ¯ δG(22) = 0, ¯ + G −1 (12) G(22) δξ(3) δξ(3)

(A7.2.9)

or, equivalently, −1 ¯ ¯ δG(23) ¯ δG (34) G(43). ¯ = −G(23) δξ(4) δξ(4)

(A7.2.10)

By substituting (A7.2.9) into (A7.2.7) we obtain ¯ ¯ ¯¯ ¯ ¯ G −1 0 (12)G(22) + 2V (132)G(3)G(22)   −1 (4 ¯ 5) ¯ δG ¯ ¯ ¯ = δ(12). − V (12¯ 3)G( 2¯ 4) G(52) ¯ δξ(3)

(A7.2.11)

On formally defining the self-energy matrix  in terms of the inverse of the full Green function, we obtain  ¯ ¯ ˜ ¯ (A7.2.12) G −1 0 (12) − (12) G(22) = δ(12), where the inverse of the full Green-function matrix in the form of the Schwinger–Dyson equation is given by: ¯ ¯ G −1 = G −1 0 (12) − (12). The self-energy matrix  is obtained as

 −1 (42) ¯ δG ¯ ¯ + V (12¯ 3)G( ¯ ¯ , (12) = −2V (123)G( 3) 2¯ 4) ¯ δξ(3)

(A7.2.13)



(A7.2.14)

with the renormalized three-point vertex function R(123) defined as −

δG −1 (12) δ(12) ¯ ¯ ≡ = R(123)G( 33). δξ(3) δξ(3)

(A7.2.15)

The following self-consistent expression is obtained for the self-energy  matrix in terms of correlation functions: ¯ ¯ + V (12¯ 3)G( ¯ ¯ ¯ 5)G( ¯ ¯ (12) = −2V (123)G( 3) 2¯ 4)R( 42 5¯ 3).

(A7.2.16)

A7.2 The MSR field theory

353

Next we obtain a self-consistent equation for computing the vertex function R as a perturbation series in the bare vertex V . From eqn. (A7.2.6) we see that ξ(1) can be expressed in terms of G(1) and G(12), order by order in perturbation theory. We therefore make a change of variables from ξ(1) to {G(1), G(12)} and use the chain rule to write ¯ ¯ δG −1 (12) δG(4) δG −1 (12) δG(3¯ 4) δG −1 (12) = + . ¯ ¯ ¯ δξ(3) δG(4) δξ(3) δG(34) δξ(3)

(A7.2.17)

Now, on substituting the useful identity (A7.2.10), the above relation reduces to −1 ¯ ¯ δG −1 (12) δG −1 (12) ¯ − δ(12) G(3¯ 6) ¯ δG (65) G(5¯ 4). ¯ G(43) = ¯ ¯ δξ(3) δξ(3) δG(4) δG(3¯ 4)

(A7.2.18)

We use the Schwinger–Dyson equation (A7.2.13) to replace the functional derivatives δ(12) δG −1 (12) −→ − δξ(3) δξ(3)

(A7.2.19)

and obtain eqn. (A7.2.18) in the form ¯¯ δ(12) δ(12) ¯ + δ(12) G(3¯ 6) ¯ δ(65) G(5¯ 4). ¯ G(33) = ¯ ¯ δξ(3) δξ(3) δG(3) δG(3¯ 4)

(A7.2.20)

The above relation is written as the following self-consistent equation for the three-point vertex function R(123): ¯ ¯ ¯ ¯ R(123) = (123) + U (123¯ 4)G( 3¯ 5)G( 4¯ 6)R( 6¯ 53),

(A7.2.21)

with the three- and four-point vertex functions (123) and U (1234) defined as δ(12) , δG(3) δ(12) U (1234) ≡ . δG(34) (123) ≡

(A7.2.22) (A7.2.23)

Equation (A7.2.21) for the renormalized vertex function can easily be rearranged with the following formal rearrangements (to avoid clutter we drop the arguments of the various functions, with the understanding that all the indices and coordinates except the external three points (123) are summed over): R =  + U GGR =  + U GG + U GGU GGR =  + U GG + U GGU GG + U GGU GGU GGR =  + [U + U GGU + U GGU GGU + · · · ] GG =  + T GG,

(A7.2.24)

where the new four-point vertex T is now obtained in terms of the integral equation T = U + U GGU + U GGU GGU + · · · ≡ U + U GGT .

(A7.2.25)

354

Appendix to Chapter 7

Hence we obtain the following self-consistent integral equations for the vertex functions: ¯ ¯ ¯ R(123) = (123) + T (14¯ 53)G( 4¯ 6)G( 5¯ 7¯ )(7¯ 62).

(A7.2.26)

The four-point kernel T (1234) is to be determined from the integral equation ¯ ¯ ¯ (5¯ 423). ¯ T (1234) = U (1234) + U (12¯ 34)G( 2¯ 4)G( 3¯ 5)T

(A7.2.27)

The three-point vertex function (123) is obtained by solution of the following selfconsistent equation: ¯ δ(12) δ(12) δG(4¯ 5) δG −1 (12) =− − ¯ ¯ δG(3) δG(3) δG(45) δG(3)

δG −1 (6¯ 7¯ ) δ(12) ¯ G(4¯ 6)G( 5¯ 7¯ ) . = 2V (123) + ¯ δG(3) δG(4¯ 5)

(123) =

(A7.2.28)

In obtaining the last equality we express the derivative in the second term on the RHS as −1 ¯ ¯ δG(12) ¯ δG (23) G(32). ¯ = −G(12) δG(3) δG(3)

(A7.2.29)

The above result follows from the identity (A7.2.10). Equation (A7.2.28) is now obtained in the form of a self-consistent equation for the three-point vertex function (123), ¯ ¯ ¯ 7¯ ). (123) = 2V (123) + U (124¯ 5)G( 4¯ 6)G( 5¯ 7¯ )(63

(A7.2.30)

Equations (A7.2.16), (A7.2.26), (A7.2.27), and (A7.2.30) provide a formal structure of the renormalization of the linear theory in terms of the self-energy matrix  expressed in terms of correlation functions. The lowest-order contribution (in the perturbation theory in terms of the bare vertex function V ) to (12) is obtained by replacing both R and  with the bare vertex 2V , ¯ (42 ¯ 5)G( ¯ ¯ ¯ 5¯ 3), (12) = 2V (12¯ 3)G( 2¯ 4)V

(A7.2.31)

where we assume that the symmetric state corresponds to current ξ = 0 and that the onepoint function G(1) = 0, and hence the first term in (A7.2.16), disappears. Using this form for  in the definition (A7.2.22), the lowest-order contribution to the four-point vertex function is obtained as ¯ ¯ (344). ¯ U (1234) = 4V (123)G( 3¯ 4)V

(A7.2.32)

The above result for U , when substituted into the renormalization for the vertex function  (and hence R), gives the two-loop contribution to the self-energy. The bare vertex function V (123) is determined from the nonlinearities in the equations of motion for the slow modes. The standard set of graphs representing the coupled equations (A7.2.16) and (A7.2.26) is shown in Fig. A7.1. A graphical expansion for  in terms of the bare vertices V and the full correlation functions G can be obtained to any arbitrary order in principle. In Fig. A7.2 the diagrammatic expansion up to second order in kB T is shown. In practice the renormalization has been considered only up to one-loop order.

A7.3 Invariance of the MSR action

355

Fig. A7.1 The diagrammatic expansion for the self-energy  as given by the self-consistent expressions in eqns. (A7.2.16)–(A7.2.27). Open and closed  each with three attached legs represent V (123) and R(123), respectively. Closed  with three attached legs represent (123). Open and closed squares with four attached legs represent U (1234) and T (1234), respectively. The lines joining both ends with the vertices represent fully renormalized correlation or response functions.

Fig. A7.2 The expansion for the self-energy  as given in Fig. A7.1 up to second order in kB T .

A7.3 Invariance of the MSR action Now let us consider the MSR action (7.1.13) for this Langevin dynamics:   t2 $ δ F 

∂ψi # ˆ = dt ψˆ i (t)β −1 L i0j ψˆ j (t) + i ψˆ i (t) − Q i j [ψ] − L i0j . A[ψ, ψ] ∂t δψ j (t) t1 (A7.3.1) ˆ depends on how the fields ψ and The time-reversal property of the MSR action A[ψ, ψ] ˆ ψ change under T . While the change for ψ is given by eqn. (A7.3.1), we need to specify ˆ For this we note that the action functional A involves, apart from ψi and the change for ψ. ψˆ i , the functional derivative δ F/δψi . Therefore we use the following prescription for ψˆ i :  

δF , (A7.3.2) T ψˆ i (t) = ψˆ i (−t) = −κi ψˆ i (t) − γ δψi

356

Appendix to Chapter 7

where κ and γ are open parameters that can be chosen so as to control the behavior of A under T . We now consider using (A7.3.1) and (A7.3.2), the effect of time reversal on the MSR action:   



 t2 δF δF dt ψˆ i (−t) − γ T A = β −1 κ 2 L i0j ψˆ j (−t) − γ δψi −t δψ j −t t1  

 t2 δF + iκi dt ψˆ i (−t) − γ δψi −t t1   # $  δF  ∂ψi (−t) 0 × i − i  j Q i j [ψ(−t)] + L i j  j ∂(−t) δψ j −t = β −1 κ 2 I1 + iκI2 .

(A7.3.3)

The two integrals I1 and I2 are evaluated in the following forms: 



t2

I1 =

dt t1

L i0j

ψˆ i ψˆ j − 2γ ψˆ i



δF δψ j



 +γ

2

δF δψi



δF δψ j



(A7.3.4) −t

and

) δ F 4

∂ψi ( 0 I2 = − Q i j [ψ] + L i j ∂t δψ j −t t1

  t2   t2 $ δF

δ F ∂ψi δF # −γ dt +γ dt Q i j [ψ] + L i0j δψi ∂t −t δψi δψ j −t t1 t1 

t2

dt ψˆ i

3

= I2 + J1 + J2 ,

(A7.3.5)

where we have denoted on the RHS of both equations by use of the subscript “−t” that the fields ψ and ψˆ within the square brackets are evaluated at time −t. We first evaluate the integrals J1 , and J2 ,  J1 = −γ



t2

dt 

t1 −t1

δ F ∂ψi δψi ∂t

−t

" ! ∂ Ft = −γ dt = −γ F−t1 − F−t2 , ∂t −t2

 t2 δF 0 δF J2 = γ dt L [ψ] . δψi i j δψ j −t t1

(A7.3.6) (A7.3.7)

In evaluating J2 we have used the fact that 

t2

t1



δF δF dt Q i j [ψ] δψi δψ j

−t

= 0,

(A7.3.8)

A7.4 The memory-function approach

357

since Q i j is odd under the exchange of i and j. Using eqns. (A7.3.6) and (A7.3.8) in eqn. (A7.3.5), we obtain for the integral I2 the result  t2 3 ) δ F 4

∂ψi ( dt ψˆ i I2 = − Q i j [ψ] + L i0j ∂t δψ j −t t1

 t2 " ! δF 0 δF − γ F−t1 − F−t2 + γ dt L [ψ] . (A7.3.9) δψi i j δψ j −t t1 Using the results of (A7.3.4) and (A7.3.9), in eqn. (A7.3.3), we obtain  t2  " ! ˆ = κ2 T A[ψ, ψ] dt ψˆ i β −1 ψˆ j − γ F−t1 − F−t2 −t t 1  

δF δF ∂ψi + iκ ψˆ i + i{2iβ −1 κγ − 1}L i0j − Q i j [ψ] ∂t δψ j δψ j −t

 t2 δF 0 δF + (iκγ + β −1 γ 2 κ 2 ) dt L i j [ψ] . (A7.3.10) δψi δψ j −t t1 We now adjust the parameters so that the MSR action remains invariant under the transformation T . We therefore require that κ 2 = 1,

2iβ −1 κγ − 1 = 1,

iκγ + β −1 γ 2 κ 2 = 0.

(A7.3.11)

More specifically, we choose κ = −1 and β −1 γ = i so that   −t2 $ δ F 

∂ψi # −1 ˆ 0 ˆ ˆ ˆ dt ψi β ψ j + i ψi − Q i j [ψ] − L i j T A[ψ, ψ] = ∂t δψ j −t1 " ! " ! ˆ (A7.3.12) − F−t1 − F−t2 = A[ψ, ψ] + F−t2 − F−t1 , since finally we take the limit t1 = −t2 → ∞. This makes the transformation T which leaves the MSR action invariant (up to a constant) as follows: ψi (x, −t) → i ψi (x, t), ψˆ i (x, −t) → −i ψˆ i (x, t) − iβ



δF . δψi (x, t)

(A7.3.13) (A7.3.14)

If F is a non-Gaussian functional of the fields {ψi } then the above transformations which leave the MSR action invariant are nonlinear.

A7.4 The memory-function approach We present below an alternative deduction of the mode-coupling model using the projection-operator method.

358

Appendix to Chapter 7

A7.4.1 The projection-operator method The projection-operator scheme provides us with a way of describing the dynamics of a many-particle system in terms of a reduced set of variables, given a proper definition of their inner product. In the present context of the dynamics of fluids, we use the set of local densities of the conserved properties of mass, momentum, and energy to constitute the chosen set of slow modes. The average w.r.t. equilibrium ensemble is taken as the inner product. In general, for the deterministic equations of hydrodynamics the conserved densities refer to the corresponding fluctuating quantity averaged over some suitable statistical ensemble. The projection-operator technique, on the other hand, constructs the equations of motion which apply to the time evolution of the corresponding fluctuating property in any single member of the ensemble. Let us consider a set of dynamic variables { Ai (r, t)} whose time evolution follows from the Liouville equation, ∂ Ai (A7.4.1) + LAi = 0, ∂t involving the Liouville operator L (Hansen and McDonald, 1986). The formal solution of the above equation gives Ai (t) = exp(iLt) Ai . We define Ai such that its equilibrium average is equal to zero and use the notation Ai (t = 0) ≡ Ai . The correlation function Ci j of the variables Ai is defined as 1 2 Ci j (t) = Ai∗ (t)A j (0) , (A7.4.2) where the angular brackets denote an average over the equilibrium distribution. The projection operator P is defined through its action on a dynamic variable B(t), ,+ ,−1 + P B(t) = Ak . (A7.4.3) B(t)A∗j A∗j Ak j,k

The operator Q=1−P

(A7.4.4)

denotes the projection in the orthogonal subspace. The projection operator satisfies the properties P 2 = P,

Q2 = Q,

PQ = QP = 0.

Next we make use of the following operator identity:  t it L it Q L e =e + ds ei(t−s)L PLeis QL .

(A7.4.5)

(A7.4.6)

0

The above identity is proved easily by taking the Laplace transform of the RHS and then applying the convolution theorem. Defining the Laplace transform of a function f (t) as  ∞ f (z) = (−i) dt ei zt f (t), Im(z) > 0, (A7.4.7) 0

A7.4 The memory-function approach

we obtain 1 1 1 1 1 1 {L − QL} − PL = − z + QL z + L z + QL z + QL z + L z + QL

z + L − L + QL 1 1 = . = z+L z + QL z+L

359

(A7.4.8)

The final expression reached is the Laplace transform of the LHS of (A7.4.6). Next we operate with both sides of this operator identity on QLAi . The LHS gives, using Q = 1 − P, eit L QLAi = eit L LAi − eit L PLAi + ,+ ,−1 ∂ = eit L A − eit L (LAi ) A∗j A∗j Ak Ak ∂t ,+ ,−1 ∂ Ai (t) + = Ak (t). (A7.4.9) − (LAi ) A∗j A∗j Ak ∂t In eqn. (A7.4.9) we have used the fact that the averaged quantities are independent of phase-space variables and the operator L does not act on them. Using the identity (A7.4.6) for eit L on the LHS,  t + ,+ ,−1 ds ei(t−s)L (LRi (s))A∗j A∗j Ak Ak Ri (t) + 0  t + , = Ri (t) + ds (LRi (s))A∗j A j Ak −1 Ak (t − s), (A7.4.10) 0

where we have defined Ri (t) = eit QL QLAi = eit QLQ QLAi .

(A7.4.11)

We have used the property (A7.4.5) of the operator Q in reaching the last equality. Therefore, on equating the two sides of eqn. (A7.4.9), an equation for the time evolution of the dynamic variables Ai (t) is obtained in the following matrix form:  t ˙ M(t − s) · A(s)ds = R(t). (A7.4.12) A(t) − i · A(t) + 0

A refers to a column vector with the Ai , i = 1, . . . , n (say) as elements.  is the n × n frequency matrix. Using the anti-Hermitian nature of the operator L, the matrix elements i j are obtained as 1 21 2−1 i j = Ai∗ iLAk A∗k A j . (A7.4.13) k

R is a column matrix with elements Ri (i = 1, . . . , n) given by eqn. (A7.4.11). Equation (A7.4.12) is in fact a simple consequence of the Liouville equation arranged in a different form. The construction of this mathematical identity involves the projection operator P, which is based on a suitably chosen inner product. Its physical meaning is clear only when we interpret this equation as a Langevin equation for a set of “slow” variables { Ai }. The many remaining degrees of freedom other than {Ai } for the system are spanned

360

Appendix to Chapter 7

by the operator Q. Their role in the dynamics is given by Ri (t) in (A7.4.12), and they are treated as “noise.” The identification of this equation with the Langevin equation (A7.4.12) is therefore closely linked to Ri (t) having the characteristics of the Langevin noise with respect to the slow variables { Ai }. The average of the noise over the initial nonequilibrium ensemble must be zero. Using the maximum-entropy principle, similarly to what we did in the case of the equilibrium ensemble, we write the nonequilibrium distribution ne as ! " ne (, 0) = eq () 1 + μi Ai∗ () + · · · , (A7.4.14) where the {μi } represent the Lagrange multipliers to be determined in terms of the corresponding averages Ai  in the nonequilibrium ensemble. Using (A7.4.14), we obtain for the average of the noise over the initial nonequilibrium ensemble ⎡ ⎤   (A7.4.15) Ri (t) = d Ri (t)eq () ⎣1 + μ j A∗j () + O(μ2 )⎦ . j

The first term on the RHS gives a vanishing contribution, since for the equilibrium state, for any time, A(t) = Aeq = 0,

(A7.4.16)

and hence, from the equation of motion, we obtain Ri (t)eq = 0. For the second term on the RHS to vanish we must have  Ri (t) = d Ri (t)eq ()A∗j () = 0.

(A7.4.17)

(A7.4.18)

At this point the equilibrium average is taken as the inner product in the definition of the projection operator. This identification ensures that the noise average is zero to linear order in deviation from the equilibrium, with Ri (t) remaining orthogonal to the space of A at all times, 2 1 ∗ (A7.4.19) Ai R j (t) = 0. The kernel matrix M in (A7.4.12) links the dynamics of A to its values at earlier times and is termed the memory-function matrix. M is expressed in terms of the correlation of the force R(t) in the orthogonal subspace of Q. The correlation function in the second term on the RHS of (A7.4.10) is simplified using the anti-Hermitian nature of L, , + (LRi (s))A∗j = −Ri (s)(LA j ) = −Ri (s)(1 − P)(LA j ).

(A7.4.20)

In obtaining the last equality, we have made use of the property Ri (s)P(LA j ) = 0,

(A7.4.21)

A7.4 The memory-function approach

361

which follows from the orthogonality condition (A7.4.19). Using the initial value Rk ≡ Rk (0) = QLAk ,

(A7.4.22)

the memory function Mi j in eqn. (A7.4.12) is obtained as an autocorrelation of the noise Ri , 2−1 1 Mi j = Ri (t)Rk  A∗k Ai .

(A7.4.23)

This is a generalized fluctuation–dissipation relation whose validity does not require the system to be close to the equilibrium state but is dependent on the orthogonality condition (A7.4.19). The dynamical equation for the time evolution of the correlation functions Ci j (t) defined in (A7.4.2)) is reached in a matrix form by multiplying eqn. (A7.4.12) from the right by A∗ · A∗ A−1 and averaging to obtain  t ∂C M(t − s) · C(s)ds = 0, (A7.4.24) − iC(t) + ∂t 0 where we have used the fact that A R(t) = 0. The one-sided Laplace transform of the memory-function equation (A7.4.24) reduces to the form [zI +  + M]C(z) = χ .

(A7.4.25)

The equation χ ≡ C(t = 0) represents the initial conditions for (A7.4.24). Note that the equation is linear in correlation functions with Green–Kubo transport coefficients.

A7.4.2 The mode-coupling approximation The dynamical equations (A7.4.12) linear in the fluctuations Ai involve the matrix M(t) of non-Markovian transport coefficients, which are also expressed in the Green–Kubo form in (A7.4.23). The collective effects that influence the long-time dynamic behavior are now included in the memory-function matrix M. In the mode-coupling theories the frequencyand wave-number-dependent functions in M are obtained in terms of the products of time correlation functions (Götze and Lücke, 1975; Munakata and Igarashi, 1977, 1978). The memory effects are thus computed in terms of correlation functions themselves rather than being approximated through phenomenological time-dependent functions (Boon and Yip, 1991). The analysis of the memory function in the projected space of an extended set of slow modes (involving coupling of slow modes) gives the frequency-dependent part in terms of hydrodynamic correlation functions. We analyze here the expression (A7.4.23) for the non-Markovian transport coefficient in the Mori equations (A7.4.12) corresponding to a specific set of collective modes. The generalized force corresponding to density, K i = QLgi ,

(A7.4.26)

362

Appendix to Chapter 7

is expressed (Götze, 1991) in terms of its projection on the coupled hydrodynamic modes C(kp . . .) (say) following definition (A7.4.3) as  

k p  K i |C(k p )χ −1 (kp, k p )C(kp), Ki =

(A7.4.27)

kp

where

χ −1

is the inverse of the correlation matrix C(kp)C(k p ).

(A7.4.28)

On substituting (A7.4.27) into (A7.4.23), the memory function is obtained in terms of time correlation functions like C(kp) exp(−iQLQ)C(k p )

and

C(kp)Lgi .

(A7.4.29)

These are further simplified with some crucial approximations described below. (a) Near the glass transition, the density fluctuations are assumed to be most dominant, leading to the simple choice C(kp) = δρ(k)δρ(p). The corresponding equal-time correlation matrix is obtained as χ −1 (k p ) = δkk δpp /[β N 2 S(k)S( p)].

(A7.4.30)

(b) The next important approximation involves equating four-point correlations of density fluctuations evolving in time with generator QLQ to corresponding two-point correlations evolving in time with L: δρ(k)δρ(p)e−i QLQ δρ(k )δρ(p )  δρ(k, t)∗ δρ(k )δρ(p, t)∗ δρ(p ). (A7.4.31) This reduction has been termed the Gaussian approximation or Kawasaki approximation in the MCT literature. The memory function is now obtained in the form  |(qˆ · K)C(kk1 )|2 S(k)S(k1 )φ(k, t)φ(k1 , t), (A7.4.32) m L (q, t)  (1/n 2 N ) k

where k1 = q − k. The equilibrium averages K i C(kk1 ) ≡ ρ(k)ρ(k1 )Lgi 

(A7.4.33)

are evaluated using classical Poisson-bracket relations between ρ and g to obtain the mode-coupling kernel (7.3.66) and (7.3.67).

8 The ergodic–nonergodic transition

We now discuss a model for the slow dynamics of supercooled liquids that follows from the analysis of the renormalized theory of liquid dynamics discussed in the previous chapter. This theoretical development has been termed in the literature the mode-coupling theory (MCT) of supercooled liquids.

8.1 Mode-coupling theory Let us first describe two basic steps in the formulation of the MCT starting from the model for the dynamics of the liquid developed in the last chapter. I. The liquid dynamics is formulated in terms of the nonlinear equations of the generalized hydrodynamics for a set of slow modes. Renormalized expressions for the correlation functions of the slow modes involving generalized transport coefficients that are wave-number- and frequency-dependent generalizations of the hydrodynamic transport coefficients are obtained. The Laplace transform of the density–density correlation function ψ(q, z) is obtained in terms of the renormalized longitudinal viscosity L(q, z) as  ψ(q, z) = z −

q2 z + iq 2 L(q, z)

−1 ,

(8.1.1)

where q represents a microscopic frequency that is characteristic of the liquid state. II. As a consequence of the nonlinear coupling of the slow modes the bare transport coefficient 0 (say) that appears in the equations of linearized dynamics is renormalized to L(q, z). The MSR field theory outlined in the previous chapter provides a systematic scheme for computing these corrections from a set of self-energies, which are expressed in terms of the correlation functions and response functions. A renormalized perturbation expansion for the transport coefficients at arbitrary order in the nonlinearities can thus be constructed in principle. For example, the renormalized longitudinal viscosity L(q, z) at one-loop order is given by eqns. (7.3.66) and (7.3.67). In general, the 363

364

The ergodic–nonergodic transition

renormalized transport coefficient L(q, z) is expressed as a self-consistent functional Mq of the density correlation function of the {ψ}: L(q, z) ≡ Mq [{ψ}].

(8.1.2)

Therefore, with the coupled set of equations (8.1.1) and (8.1.2) obtained in steps I and II above, the time correlation function of the density fluctuations is expressed in terms of renormalized transport coefficients, which themselves are expressed in terms of correlation functions. This gives rise to a self-consistent feedback mechanism for the density correlation function with characteristic slow dynamics. It is useful to note at this point that originally non-self-consistent mode-coupling models were used in the study of relaxation in systems near second-order phase-transition points (Fixman, 1962; Kadanoff and Swift, 1968; Kawasaki, 1970a, 1970b). In this approach the renormalizing contributions to the transport coefficients are approximated as a functional Mq [{C 0 }] of the correlation functions C 0 between the different hydrodynamic modes in the linear theory which involve only the bare transport coefficients. In the case of critical phenomena, the static correlations grow indefinitely near the transition point and, through the mode-coupling terms, the cooperative effects over different length and time scales are probed. For the normal liquid state, mode-coupling contributions to transport coefficients (Alley et al., 1983) and diffusivity (Curkier and Mehaffey, 1978) were computed at lowest order to account for the substantial difference between computer-simulation data and corresponding results obtained from the Enskog theory (Résibois and de Leener, 1977). The latter takes into account only short-time uncorrelated dynamics of the particles and is applicable for low-density fluids. More exotic results from inclusion of mode-coupling effects indicated breakdown of conventional hydrodynamics, e.g., the long-time tails in the dynamic correlation functions (Forster et al., 1977), or divergence of components of the dynamic viscosity tensor in smectic-A liquid crystals (Mazenko et al., 1983; Milner and Martin, 1986). The relevance of the MCT to the theory of the glass transition came with the idea of expressing the mode-coupling terms, i.e., the functional Mq [{C}] for the generalized transport coefficient, solely in terms of the full density–density correlation function ψ itself. The functional Mq [{C}] and the defining relation for ψ in terms of the generalized transport coefficient gave a closed equation for ψ (Geszti, 1983). From expression (7.3.66) we see that, if the density correlation freezes in the long-time limit, L(q, z) has a 1/z pole. Such a pole in the viscosity, when substituted into (7.3.68), appears to be self-consistent with the 1/z pole for the correlation function. The mode-coupling model that we discuss in this chapter is based on this crucial dynamic feedback mechanism. We note here in this regard that the model equations follow from a generalized-hydrodynamics approach extended to the finite-wave-vector or corresponding short-length-scale regime. The glass transition is not intrinsically related to the long distance and long time scales which concern hydrodynamics in the traditional sense. The transition to an amorphous solid-like state is characterized by the freezing process occurring first over small or intermediate

8.1 Mode-coupling theory

365

length scales. However, the effects of the jamming process should finally propagate to hydrodynamic regimes. We present an outline of the present chapter. As a result of the nonlinear feedback mechanism outlined above, a dynamic transition to a nonergodic state is predicted in the simple MCT. The long-time limit of the density correlation ψ freezes at a nonzero value beyond the transition point, while the thermodynamic properties of the liquid vary smoothly through the transition. The viscosity follows a power-law divergence approaching the dynamic transition point, accompanied by a finite shear modulus in the glassy state. Subsequent works (Das and Mazenko, 1986) demonstrated that the sharp dynamic transition to an ideal glassy state predicted in the preliminary works by Leutheusser (1984), Bengtzelius et al. (1984), and Das et al. (1985a) is finally cut off and ergodicity is maintained at all densities. However, the above-described feedback mechanism from the coupling of density fluctuations causes a substantial enhancement of the viscosity. Thus, although the sharp transition is cut off, there are strong remnants of it, causing a qualitative change in the supercooled liquid dynamics around a temperature Tc that is higher than the usual glass-transition temperature Tg . Identification of the new temperature Tc (Tc > Tg ) for liquids, generally of the fragile type, has been one of the main outcomes from selfconsistent MCT. Subsequent theoretical works as well as experimental works have focused on finding the signatures of the mode-coupling transition at Tc .

8.1.1 The schematic model The complex relaxation scenario that results from the nonlinear feedback mechanism of density fluctuations is conveniently demonstrated in terms of a schematic model in which all wave-vector dependences are dropped. The q-independent density correlation function, normalized with respect to its equal-time value, is obtained as ψ(q, t) =

G ρρ (q, t) ≡ ψ(t). S(q)

(8.1.3)

The Laplace transform ψ(z) of the the density correaltion function ψ(t) satisfies the equation

1 ψ(z) = z − z + im(z)

−1

,

(8.1.4)

where time has been rescaled to t0 , with 0 ≡ cq treated as a characteristic microscopic frequency in the model, dropping all wave-vector dependence. The renormalized viscosity L(q, z) has been written here in a rescaled form in terms of m(z). In the schematic models the one-loop contribution to the viscosity is obtained as  ∞ m(z) = m 0 + dt ei zt c2 ψ 2 (t), (8.1.5) 0

366

The ergodic–nonergodic transition

where m 0 = 0 γ0 /c2 is the scaled bare transport coefficient. From the field-theoretic model presented in the previous chapter, the constant c2 is determined in terms of the vertex functions of cubic and higher order in the action functional (see Section 7.3.3, eqns. (7.3.63), (7.3.64), (7.3.66), and (7.3.67)). The actual dependence of c2 on the thermodynamic properties of the fluid can be determined by taking into account the proper wave-vector dependence in the model, to be considered in the next section. In order to keep the analysis at a more general level we will in fact consider renormalization of the transport coefficient beyond one-loop order and generalize the integral in the second term on the RHS of eqn. (8.1.5) to  ∞ dt ei zt H [ψ(t)], (8.1.6) m(z) = m 0 + 0

where H [ψ(t)] is a local functional of the density correlation function ψ(t) (Das et al., 1985a; Götze, 1991; Kim and Mazenko, 1992), H [ψ(t)] =

M 

cn ψ n (t),

(8.1.7)

n=1

with the cn being determined by the thermodynamic state of the fluid. Thus the one-loop result considered above corresponds to H [φ] = c2 φ 2 . Equations (8.1.4) and (8.1.5) form a coupled set of equations for the density autocorrelation function. In the time space they can be simply expressed as one closed nonlinear integral equation for ψ(t),  t ˙ ˙ ¨ ψ(t) + ν0 ψ(t) + ψ(t) + ds H [ψ(t − s)]ψ(s) = 0, (8.1.8) 0

where ν0 is the bare transport coefficient. Initially, eqn. (8.1.8) was derived (Leutheusser, 1984) from a physical picture for the supercooled liquid dynamics as follows. The deeply supercooled state is characterized by strong density fluctuations. The basic quantity describing the dynamic properties of density fluctuations is the dynamic structure factor, whose decay is controlled by the longitudinal viscosity. The dynamic transport coefficients at high density are strongly influenced by correlated motion of the fluid particles and are expressed mainly through the mode-coupling terms involving products of strong density fluctuations. Therefore, through this feedback process, the relaxation time of density fluctuations is enhanced by the slow decay of the density fluctuations themselves. A similar enhancement was argued also for the shear viscosity (Geszti, 1983). Bengtzelius et al. (1984) discussed a model in which the feedback mechanism for the liquid was considered in greater detail, and in their model the proper structure factor of a real liquid was included. The basic assumption in the analysis of eqn. (8.1.4) is that, depending on the the kernel H [ψ], i.e., the coefficients cn , there is a time range over which ψ(t) is approximately time-independent and has the form ψ(t) = f + (1 − f )ψν (t),

(8.1.9)

8.1 Mode-coupling theory

367

where f is the value of ψ(t) in some metastable state that is yet to be specified. We consider time scales on which the inequality |zψν (z)| 1 is valid for the !Laplace " transform L z [ψν (t)]. Using eqn. (8.1.9) in eqn. (8.1.4) and keeping terms up to O ψν2 (z) , we obtain  (1 − f )2  0 + 1 ψν (z) − (1 − f )H  ( f )zψν 2 (z) + H ( f )L z ψν 2 (t) = 0, (8.1.10) z 2 with 0 = H ( f ) − 1 =

f , 1− f

" ∂ ! (1 − f )H ( f ) − f . ∂f

(8.1.11)

In eqn. (8.1.10) the primes indicate derivatives with respect to f . An ideal metastable state is obtained when both 0 and 1 are zero, giving a solution for the decaying function ψν (t). We can determine f by setting 1 = 0. This is obtained for H( f ) + 1 =

C0 , 1− f

(8.1.12)

where C0 is a constant not depending on f . Under this condition 0 is given by 0 =

C0 − 1 . 1− f

(8.1.13)

  An ideal nonergodic state is obtained when, for a given set of cn ≡ cn∗ , we have 0 cn∗ = 0. Thus the ideal transition takes place for C0 = 1, giving 0 = 0. The decay of the density–density correlation ψ(t) becomes extremely slow in a certain part of the parameter space (involving the cn ) and eventually it freezes at a nonzero value f , indicating a dynamic transition of the fluid to a nonergodic state. For the Leutheusser model we obtain the nonergodicity parameter f = 12 when the coupling c2 has a critical value of c2∗ = 4. For the model that includes both c1 and c2 (Götze, 1991) having a linear term in the kernel, i.e., H [ψ] = c1 ψ + c2 ψ 2 ,

(8.1.14)

the critical values are parametrized in terms of a single quantity λG , 2λG − 1 , λ2G 1 c2∗ = 2 , λG c1∗ =

and correspond to a line in the c1 − c2 space given by √ ∗ c1∗ = 2 c2 − c2∗

(8.1.15)

(8.1.16)

368

The ergodic–nonergodic transition

representing the ergodic–nonergodic transition. For higher-order models there are critical surfaces. Note that, with 1 = 0, eqn. (8.1.10) is given by  0 (1 − f )2  C − zψν 2 (z) + H ( f )L z ψν 2 (t) = 0. z 1− f 2

(8.1.17)

For the high-frequency region where the term proportional to 0 can be neglected, i.e., (|0 |)1/2 |zψν (z)|, the above equation reduces to the simple form  zψν 2 (z) − λs L ψν 2 (t) = 0,

(8.1.18)

(8.1.19)

with λs at the transition point determined from the relation λs =

(1 − f )3  1 H  ( f ) . H (f) ≡ 2 2 [H  ( f )]3/2

(8.1.20)

This equation is satisfied by the function ψν (t) = A(t0 )−a ,

(8.1.21)

where the exponent a is given in terms of the parameter λs by the equation  2 (1 − a) = λs . (1 − 2a)

(8.1.22)

The inequality (8.1.18) and the solution (8.1.21) obviously hold in the time region t0 t τa ≡ t0 |0 |−1/(2a) ,

(8.1.23)

where t0 = 0 −1 represents a microscopic time scale and τa diverges at the ideal transition point, i.e., 0 = 0. Therefore ψ(t) decays algebraically towards the metastable value f . For 0  = 0, i.e., points away from the transition, the dynamics is quite different for the 1/2 cases 0 > 0 and 0 < 0. For 0 > 0, eqn. (8.1.17) has the solution ψν (z) ∼ 0 /z as z → 0 and f = f 0 + c(0 )1/2 + O(0 ),

(8.1.24)

where f 0 is the value of f at the ideal transition point and c is a constant. For 0 < 0 we have from eqn. (8.1.17) a solution for ψν (z) that is more singular than 1/z as z → 0 and has the following form in the time regime: ψ(t) = f − B(t/τα )b ,

(8.1.25)

where B is a positive constant. Similarly to a in eqn. (8.1.21), b satisfies the transcendental equation  2 (1 + b) = λs . (1 + 2b)

(8.1.26)

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369

Equation (8.1.25) is referred to as the von Schweidler relaxation law. However, this relaxation eventually violates the inequality |zψν (z)| 1 and this decay mechanism is valid in the region τa t τα ,

(8.1.27)

where τα = |0 |−[1/(2a)+1/(2b)] represents the time scale for the power law or the β-process. For t ≥ τα , the system enters into the α-relaxation regime. In the α-relaxation regime no fully analytic solution of the mode-coupling equations is known. Numerical solution in this regime is well fit by a stretched exponential. Assuming the form ψ(t) = f e−(t/τα )

β

(8.1.28)

to be the solution of eqn. (8.1.8), we obtain an approximate expression for the stretching exponent β in terms of the parameters cn : M 

cn f n −1/β = 1.

(8.1.29)

n=1

For the Leutheusser model at the ideal transition c2∗ = 4 and f = 12 we get βL = ln 2/ ln(c2 f ) = 1, i.e., relaxation is exponential. For the model with c1 and c2 , referred to as the φ12 model (Sjögren, 1986; Götze, 1991), βG =

ln 2 ln 2 =− . ln [c2 f /(1 − c1 )] ln (1 − λG )

(8.1.30)

Therefore βG varies along the transition line given by (8.1.16). Let us summarize here the complex relaxation scenario that characterizes the dynamics driven by the feedback mechanism of the schematic model discussed above. For 0 < 0, first there is power-law decay, followed by the von Schweidler relaxation and finally the stretched exponential relaxation. This is the ergodic behavior. At the transition point, 0 = 0, the power-law decay extends to the longest time scale. In the nonergodic state, 0 > 0, the density correlation function freezes in a state of structural arrest, after the initial power-law decay. The sequence of relaxations in the ergodic state is shown schematically in Fig. 8.1.

8.1.2 Effects of structure on dynamics The discussion of the schematic model outlined above demonstrates that the feedback mechanism central to MCT is insensitive to the wave-vector dependence. The transition predicted is essentially dynamic in the sense that the divergence of the viscosity at the critical point is not accompanied by any sharp change in the thermodynamic properties of the liquid. Certain mean-field models with intrinsic (quenched) disorder, namely equations similar to that of the schematic model, also provide a useful description of the dynamics (see Section 8.4). However, the MCT for the glass transition is not just the self-consistent

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The ergodic–nonergodic transition

Fig. 8.1 The relaxation behaviors over different time scales as predicted for the liquid in the ergodic state below the ideal dynamic transition point of the schematic mode-coupling model.

treatment of an integral equation with one degree of freedom. The actual impact of the feedback mechanism becomes clear when we consider the wave-vector-dependent model (Kirkpatrick, 1985a). Using the static structure factor as an input, the predictions of the theory are directly linked to thermodynamic properties such as temperature and density. In the wave-vector-dependent model, even with the simplest form of quadratic coupling of density fluctuations, many different time scales (corresponding to different wave numbers) are coupled. This as a result produces stretched exponential relaxation behavior. In the present section we consider the effects of including structure on the dynamics predicted in MCT. In the wave-vector-dependent model the Laplace transform of the density correlation function φ(q, z) is now expressed with a generalization of the memory function m L (q, z), in the form φ(q, z) = z + im L (q, z), zφ(q, z) − 1

(8.1.31)

where we have rescaled time to a dimensionless form with a characteristic microscopic frequency 0 ≡ cq. The memory function is expressed as a sum of two parts,  ∞ L L dt ei zt m˜ L (q, t), (8.1.32) m (q, z) = m 0 (q) + 0

m L (q, t)

is the mode-coupling contribution to the viscosity and m L0 corresponds where to the bare or short-time contribution to the viscosity. For practical calculations of the self-consistent model the renormalizations are computed from the relevant self-energies at one-loop order (see eqns. (7.3.66) and (7.3.67) discussed earlier),  n dk [VL (q, k)]2 S(k)S(k1 )φ(k1 , t)φ(k, t), (8.1.33) m˜ L (q, t) = 2mβ (2π )3

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371

with the notation k1 = q − k. The vertex function VL is defined as VL (q, k) = (qˆ · k)c(k) + (qˆ · k 1 )c(k1 ).

(8.1.34)

The renormalization of the transport coefficients in terms of density correlation functions is expressed in a self-consistent form here. In an intuitive picture this self-consistent treatment is justified by appealing to the nature of the single-particle motion in the strongly correlated liquid: the tagged particle rattles around in the cage formed by its surrounding particles, which are also trapped. The tagged particle thus needs to be treated in the same manner as the ones forming the cage around it and hence the renormalized transport coefficient is taken to be a self-consistent functional of the density correlation function. The feedback process is a manifestation of the fact that the motion of a particle in a dense medium influences the surroundings, which will in turn react and influence its subsequent motion. The nonergodic phase is defined in terms of the φ(q, t) having a nondecaying part f (q) in the long-time limit, φ(q, t) = f (q) + (1 − f (q))φν (q, t).

(8.1.35)

φν (q, t) goes to zero for large t. The f (q), which are termed nonergodicity parameters (NEPs), are determined from a set of coupled nonlinear integral equations, fq 1 = 2 m˜ L (q, t → ∞) ≡ Hq [ f k ]. 1 − fq q

(8.1.36)

In writing the set of eqns. (8.1.36) we have treated the integral over the wave vector k in the mode-coupling term as a sum over a discrete set of k values uniformly distributed over a grid of size, say, M, and extending up to a suitable upper cutoff value. In the discrete form we denote, for a q = 1, . . . , M grid, f (q) → f q , φ(q, t) → φq (t). The functional Hq on the RHS of (8.1.36) is obtained from the long-time limit of the expression for m˜ L (q, t) in eqn. (8.1.33) and is a functional of all the f q . Equation (8.1.36) now represents a set of M coupled integral equations, which are numerically solved using iterative methods. The necessary input for this is the static structure factor S(q) for the liquid. For low densities, only the trivial solution set with all of the f q equal to zero (corresponding to the ergodic liquid state) is obtained. The critical density at which all of the f q simultaneously converge to a nonzero set of values marks a dynamic transition of the fluid to a nonergodic state. For example, let us consider a system of hard spheres of diameter σ with the packing fraction ϕ = π nσ 3 /6. The Percus–Yevick solution with the Verlet–Weiss correction (Verlet and Weiss, 1972; Barker and Henderson, 1976) for S(k) of a hard-sphere system is then used as an input in eqn. (8.1.36). The fluid undergoes an ergodic-to-nonergodic transition at the critical value of the packing fraction ϕ = ϕ c = 0.525. The contributions to the mode-coupling integral (8.1.36) from k values above the upper cutoff value 50σ −1 do not affect the transition point much.

372

The ergodic–nonergodic transition

1. The cusp behavior of the NEP. In the close vicinity of this dynamic transition point, the order parameter of this transition, i.e., the f q , exhibits a cusp behavior similar to eqn. (8.1.24) in the schematic model. Understanding this behavior in the wave-vectordependent model involves an asymptotic analysis of the integral equation (8.1.36) near the ideal transition point (Götze, 1985). The starting point of such an exercise is the stability matrix Cqk , which is defined as Cqk =

δHq (1 − f k )2 , δ fk

(8.1.37)

all elements of which are positive numbers by definition. Let ekc and eˆkc , respectively, c at the transition point corredenote the right and left eigenvectors of the matrix Cqk sponding to the eigenvalue 1:  c c Cqk ek = eqc , (8.1.38) k



c eˆqc Cqk = eˆkc .

(8.1.39)

q

The nonergodicity parameters f q are related to the corresponding f qc at the dynamic transition point as 1/2

f q = f qc + gλ h q 0

+ O(),

(8.1.40)

where the amplitude is given by gλ = (1 − λ)−1/2 , ( ) h q = 1 − f qc 2 eqc .

(8.1.41)

(8.1.42) 5 The separation parameter 0 is defined in terms of , namely 0 = C, where C = q eˆqc Cqc and  is the relative distance from the dynamic transition point in terms of a suitable control parameter. For a hard-sphere system the packing fraction is the controlling thermodynamic variable,  = (ϕ − ϕ c )/ϕ c . The parameter λ in (8.1.40) is obtained as  c eˆqc Cq,kp ekc ecp , (8.1.43) λ= qkp

with



Cqc c Cq,kp

∂ Hq = n , ∂n n=n c   2 1 2 2 δ Hq = (1 − f k ) (1 − f p ) 2 δ fk δ f p

(8.1.44) .

(8.1.45)

n=n c

It should be noted that the square-root cusp behavior of the NEP is an asymptotic result. Results extracted from experimental data for the Debye–Waller factor are used to locate the existence of a possible mode-coupling transition point. Since the sharp dynamic

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373

transition of the supercooled liquid gets smeared in a real system, the square-root cusp in reality is an idealization and its existence can be concluded only indirectly. Furthermore, even if we confine the discussion to the ideal model and ignore the cutoff mechanism, the cusp behavior still gets modified due to leading-order corrections to the asymptotic result (8.1.40):  1/2 1/2 f q = f qc + gλ h q 0 1 + 0 K˜ q . (8.1.46) The amplitude functions K¯ q have been computed by Franosch et al. (1997) for the simple form of the memory function (8.1.33). 2. Scaling in power-law relaxations. The decay of the density correlation function φq (t) has distinct relaxation regimes similar to those occurring in the case of the schematic model described above. First, φq (t) decays to a plateau value f q algebraically, φq (t) = f qc + h q (τβ /t)a .

(8.1.47)

As in the schematic case, (8.1.47) is valid in the time window t0 t τβ . The time scale τβ diverges algebraically with the separation parameter 0 such that τβ = t0 |0 |−1/(2a) . Above the critical density, in the nonergodic glassy phase φq (t) decays by a power law t −a to the corresponding value of the NEP f q . For densities less than the critical value, the initial power-law relaxation crosses over to the von Schweidler relaxation, φq (t) = f qc − h q (t/τα )b .

(8.1.48)

This occurs in the time window τβ t τα (which is the same as in the schematic case in (8.1.25)). The exponents a and b of the power-law relaxations follow from the same transcendental equation as in the corresponding schematic case. The parameter λs in the schematic case now corresponds to the exponent parameter λ and is computed using the formula presented in eqn. (8.1.43). For hard-sphere systems with one-loop vertex contributions, this is computed as λ = 0.74 (Fuchs et al., 1992). The power-law decays in different time windows as described above can be expressed in the form of a single scaling function,  ; h q−1 φq (t) − f qc = 0 g± (tˆ), (8.1.49) where g± (tˆ) refer to the master functions in terms of the reduced time tˆ = t/τβ , above and below the dynamic transition point, respectively. The LHS of eqn. (8.1.49) scaled √ with a factor of 0 will follow one master curve independently of the wave number q. The power-law relaxations then correspond to the asymptotic behavior of the scaling functions. For short times, when t τβ , to leading orders g± (tˆ 1) = tˆ−a .

(8.1.50)

374

The ergodic–nonergodic transition

In the other limit of t  τβ , the scaling functions are different above and below the transition point. In the nonergodic state g+ → gλ = (1 − λ)−1/2

for tˆ  1.

(8.1.51)

In this time regime eqn. (8.1.49) reduces to the cusp behavior of the NEP described by eqn. (8.1.40). A crossover time τˆ+ is sometimes defined when the monotonically decreasing function reaches within, say, 0.1% of its asymptotic value, i.e., g+ (τˆ+ ) = 1.001gλ . For the liquid side g− for large tˆ approaches the von Schweidler law g− (tˆ  1) = −B tˆb ,

(8.1.52)

where B is a constant. In this case a corresponding crossover time τˆ− can be defined where g− (τˆ− ) = 0. 3. The factorization property. Note that both of the exponents a and b in the time regimes corresponding to the power-law relaxations are independent of temperature and density. From (8.1.49) it is clear that the space and time dependence of the quantity φ(q, t) − f (q) factorizes. This rather remarkable prediction of the MCT has been termed the factorization property. This is an asymptotic result that holds in the close vicinity of the dynamic transition point (Das, 1993). The higher-order corrections can be presented in the form of a series with factorized terms (Franosch et al., 1997) in powers of (τβ /t)b , ( τ )a ( τ )a  β β φq (t) − f qc = h q 1 + K q(1) (a) , (8.1.53) t t and a similar expansion in powers of (t/τα )b for the von Schweidler relaxation,  b   b  t t c (1) φq (t) − f q = −h q 1 − K q (−b) . (8.1.54) τα τα (1)

The q-dependent amplitudes K q (x) are the so-called “corrections to scaling.” One will need to keep adding higher-order terms as the distance from the transition increases. The wave-vector independence implied in the factorization property is also affected due to other reasons. For example, the microscopic time scale t0 in the q-dependent model now assumes special significance (Das, 1993). The time t0 quantifies the matching of the short-time decays for all the q values to the corresponding transients which are obviously dependent on the wave number. The microscopic frequency for wave number q is given by q2 = q 2 [βm S(q)]−1 . The time window between the crossover from the transient to the power-law behavior given by eqn. (8.1.53) is now strongly dependent on wave number. Furthermore, within the range of q values over which the amplitude of the correction term becomes small, the factorization property is naturally valid over longer times. Thus the range of validity of the so-called factorization property is strongly dependent on q.

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375

With the next-to-asymptotic-order terms in eqn. (8.1.49) taken into account, the new scaling functions consist of corresponding correction terms, $  ; # (1) ;  h q−1 φq (t) − f qc = 0 g± (tˆ) + 0 g± (q, tˆ) , (8.1.55) (1)

where tˆ = t/τβ is the rescaled time. The leading-order correction terms g± for the master functions (Franosch et al., 1997) are dependent on the wave number q. As we move away from the dynamic transition point, the q-independence of the RHS of eqn. (8.1.55) becomes invalid. In short, closer to the dynamic transition point, the time window over which the correlation function φ(q, t) remains near the plateau f qc expands, and hence the range of validity of the factorization property grows. Finally, we note that the corrections to the scaling described here are for the simple MCT model in which all aspects of the ergodicity-restoring mechanism have been ignored. In the presence of such physically relevant processes (to be described next), which remove the sharp dynamic transition, it remains unclear whether the system will ever get close enough to the transition point for it to clearly display the above behaviors. 4. α-Relaxation scaling. In the region asymptotically close to the dynamic transition at ϕ c with the time scale τα tending to ∞, the α-relaxation in the MCT is described in terms ˆ q (t/τα ). All the relaxation of a temperature-independent master function φq (t) ≡  curves at different temperatures will merge into a single curve with a temperaturedependent relaxation time τα . This property is termed the time–temperature superposition principle. The α-relaxation is asymptotically defined in the limit of the control parameter ϕ c approaching the critical value ϕ → ϕ c , with z → 0, τα → ∞, so that zˆ = zτα remains finite. The mode-coupling prediction for the α-relaxation is obtained through numerical solution of the model equations. The equation for the master function  is obtained from (8.1.31)–(8.1.33) in the scaling limit z → 0, τα → ∞, so that ˆ q (ˆz ) = τα−1 φ(q, z) is obtained as zˆ = zτα . The equation for the Laplace transform  ˆ q (ˆz ) = 

mˆ q (ˆz ) , 1 + zˆ mˆ q (ˆz )

(8.1.56)

where we define mˆ q (ˆz ) = − im L (q, z)/(τα q2 ). The mode-coupling integrals in mˆ are evaluated at the transition point ϕ = ϕ c . The microscopic and power-law relaxations are ˆ is equal to f q . Therefore eliminated here, so the tˆ = 0 value of the scaling function  eqn. (8.1.36) for locating the dynamic transition now corresponds to the tˆ → 0 limit of (8.1.56). The scaling function q (tˆ) is obtained very accurately by numerically solving (Fuchs et al., 1992) eqn. (8.1.56) in the time space as  tˆ d ˆ q (s) ˆ ds mˆ q (tˆ − s) (8.1.57) q (tˆ) = mˆ q (tˆ) − d tˆ 0 and using the the Percus–Yevick solution for the hard-sphere-system input static structure factor. An asymmetrically stretched shape of the spectra in the frequency space is obtained due to the overlap of the von Schweidler relaxation. The stretched-exponential

376

The ergodic–nonergodic transition

form is a good fit for the master function at all wave numbers, with better quality of agreement at large q. The stretching exponent β is wave-number-dependent and βq shows a minimum at the first peak of the static structure factor. The corresponding relaxation times τα (q) also show a peak at this wave vector. Similarly to the scaling function for the initial power-law regimes, the relaxation behavior described by the master function  is an asymptotic result that is valid in the close vicinity of the transition point. The time–temperature superposition is a prediction of the simple MCT only near the dynamic transition point. As we move away from the transition, the temperature-independent master function requires correction terms very much in the same way as for the power-law regime, ( ) ( )2 ˆ q (tˆ) + 0 1 − f qc  ˆ q(1) (tˆ) + O 20 . (8.1.58) φq (t) =  The temperature independence of the α-relaxation master function is lost with the inclusion of the correction term. In contrast to the case of power-law relaxations, the leadingorder correction to the α master function in eqn. (8.1.58) is of O( 0 ). The correspond1/2  ing correction to the power-law scaling function is of O 0 . Indeed, the numerical solutions for φq (t) from the simple mode-coupling model show a marked departure from time–temperature superposition except in the close vicinity of the dynamic transition point. The solutions of the mode-coupling equations away from the transition follow power-law or stretched-exponential forms with effective exponents that are dependent both on temperature and on the wave number.

8.1.3 Tagged-particle dynamics The single-particle density ρs (x, t) in a fluid is a microscopically conserved mode very much like the total density ρ(x, t) or the total momentum density g(x, t). This conservation law gives rise to a new slow mode in the fluid. Over hydrodynamic length and time scales (large compared with microscopic length and time scales, respectively) this mode corresponds to the tagged-particle diffusion or self-diffusion process in the fluid. In this section we discuss the implications of slowly decaying density fluctuations in a dense fluid on the tagged-particle motion. The equation of motion First the tagged-particle density is defined microscopically through a relation similar to eqn. (5.3.62) for the total density, ρˆs (x, t) = mδ(x − rα ). The corresponding density correlation or the van Hove correlation function, 1 2 G s (|r − r |, t − t  ) = ρˆs (r, t)ρˆs (r , t  )

(8.1.59)

(8.1.60)

8.1 Mode-coupling theory

377

represents the time-dependent spatial distribution of the tagged particle. The constraint  dr G s (r, t) = 1 (8.1.61) ensures the particle conservation. In the thermodynamic limit the average ρs  is zero and hence δρs ≡ ρs . It is rather straightforward to see, by taking a time derivative of the RHS of eqn. (8.1.59), that the conserved tagged-particle density ρs satisfies the microscopic continuity equation, ∂ ρˆs + ∇ · gˆ s = 0. ∂t

(8.1.62)

The corresponding current density gˆ s is obtained as gˆ is (x, t) = pαi δ(x − rα ).

(8.1.63)

The coarse-grained quantities ρs and gs are related through a phenomenological constitutive relation: gis (x, t) = −DB ∇i ρs (x, t),

(8.1.64)

linking the tagged-particle density and this current at large length scales. On combining eqns. (8.1.59) and (8.1.64) we obtain the diffusion equation for the tagged-particle density ρs ,

∂ 2 (8.1.65) − DB ∇ ρs = 0, ∂t where DB is the tagged-particle diffusion constant. However, unlike the total momentum density gˆ , the momentum density gˆ s of the tagged particle is not a conserved property. It is also not a slow variable in the sense a heavy Brownian particle is, since a tagged particle is in the sea of identical particles in the fluid. Obtaining a fluctuating equation for the tagged-particle current density gs requires a phenomenological approach. The equation of motion for the current gs is obtained by following the standard procedure described above with respect to the dynamics of {ρ, g}. The reversible part of the equation for gsi is obtained from the usual Poisson brackets. From the above microscopic definitions the Poisson brackets are obtained as $ # (8.1.66) gˆ si (x), ρˆs (x ) = ∇i [δ(x − x )ρˆs (x )], $   # j j (8.1.67) gˆ si (x), gˆ s (x ) = −∇ j δ(x − x )gˆ si (x) + ∇i δ(x − x )gˆ s (x ) . The other ingredient required for construction of the equations of fluctuating nonlinear hydrodynamics for the tagged particle is the driving free-energy functional in terms of the coarse-grained densities. For this we consider a functional similar to the one used for the one-component system, F s = FUs [ρs ] + FKs [ρs , gs ],

(8.1.68)

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The ergodic–nonergodic transition

where FUs is the so-called potential or density (ρs )-dependent part of the free-energy functional F s for the single-particle dynamics. FKs is the kinetic or the momentum density gs -dependent part of the free-energy functional and is assumed to take the form 2

 gs (x, t) s . (8.1.69) FK [ρs , gs ] = dx 2ρs (x, t) This is very similar to the corresponding kinetic-energy part FK of the driving free energy (see eqn. (6.2.7)) in our earlier discussion of the fluctuating-hydrodynamics description with respect to the total variables {ρ, g}. This form (8.1.69) gives for the equation of motion for ρs a continuity equation similar to (8.1.62), namely ∂ρs (8.1.70) + ∇ · gs = 0. ∂t Now, using the standard prescription outlined in the earlier chapter (see Section 6.3), we obtain the generalized Langevin equation for the tagged-particle momentum density,   j  δ FUs δFs gsi gs ∂gsi − ∇j + ζi . (8.1.71) = −ρs ∇i + isj j ∂t δρs ρs δgs j

Here we have taken the nonzero elements in the bare transport matrix to correspond to gi g j ≡ isj . The bare matrix determines the correlation of the Gaussian noise ζi through s s the fluctuation–dissipation relation, 1 2 ζi (x, t)ζ j (x , t  ) = 2kB T isj δ(x − x )δ(t − t  ). (8.1.72) Since gs is not a conserved density, s does not involve any spatial gradient operator. The corresponding Fourier transform is nonzero in the small-wave-number limit. The first term on the RHS represents the reversible part of the dynamics, Vis = −ρs ∇i

δ FUs . δρs

(8.1.73)

With the above choice for FKs , the dissipative part of the single-particle dynamics given by the third term on the RHS of eqn. (8.1.71) is obtained as isj

δFs j

δgs

j

= γ0 gs ,

(8.1.74)

where we have defined the dissipative coefficients for the tagged-particle dynamics to be proportional to the tagged-particle density, isj ≈ ρs γ0 δi j

(8.1.75)

as the simplest approximation (Kirkpatrick and Nieuwoudt, 1986a). Equation (8.1.71) for the tagged-particle momentum density is now obtained in the form δ FUs ∂gsi δFs + s i = ζi (x, t). + ρs ∇i ∂t δρs δgs

(8.1.76)

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379

The above equation of motion is similar to the balance equation (6.3.31) which we obtained in Section 6.3.2 of Chapter 6 for the microscopic current gˆ s (x, t) with Fokker–Planck dynamics. This similarity also requires identifying the bare interaction potential βU with the Ornstein–Zernike direct correlation function −c(r ). The primary difference between these two equations is that, with dissipative Fokker–Planck dynamics, the equations deal with the microscopic densities ρˆs and gˆ s , whereas in the Newtonian case presented above we are dealing with the coarse-grained densities. With the above choice of (8.1.75) of the dissipative coefficients the noise becomes multiplicative in this case as in the case of eqn. (6.3.31). To complete the construction  of the equation of motion for gs we need to evaluate the  functional derivative δ FUs /δρs which appears on the RHS of (8.1.73). Similarly to in the case of the total density, we obtain FUs as FUs [ρs ] = Fids [ρs ] + Fins [ρs ].

(8.1.77)

The first term Fids on the RHS comes from the ideal-gas part while the second term Fins represents the contribution from the interaction. The ideal part Fids is approximated from the one-component limit1 of the result (6.2.60) for the binary mixture,  

 1 ρs (x)3 dx ρs (x) ln −1 , (8.1.78) β Fids [ρs ] = m m where the subscript signifies the self-part here. Using the above expression for the ideal gas part Fids , we obtain the functional derivative as δFs δ FUs kB T [ln ρs (x, t)] + in . = δρs m δρs

(8.1.79)

The first term actually produces a linear term for the dynamics. Linear dynamics Let us first consider the linear equation of motion for gs . The ideal-gas contribution (the first term on the RHS of eqn. (8.1.79)) contributes a linear term in the reversible current in the equation for gs , ∂ gsi (8.1.80) + s gsi + v02 ∇i ρs = ζsi . ∂t We ignore the convective nonlinearity given by the second term on the RHS of eqn. (8.1.71) in the treatment below. It should be noted here that, in our earlier discussion of the tagged– particle momentum-density equation in Section 6.3.2 with Fokker–Planck dynamics, we saw that in the adiabatic approximation a similar nonlinear convective term gives rise to a term that is linear in the density fluctuation. This is identical to the ideal-gas contribution or the excluded-volume term in the dynamics involving ∇i ρs , i.e., the third term on the LHS of eqn. (8.1.80). The single-particle dynamics at the linear level is now completely decoupled 1 This is equivalent to making the two species in the mixture identical and setting N and N equal to N − 1 and 1, respectively. 1 2

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The ergodic–nonergodic transition

from the collective dynamics involving {ρ, g}. Equations (8.1.70) and (8.1.80) are similar to the corresponding equations (6.2.16) and (6.2.20), respectively, for ρ and g. To keep the treatment simple, we take s as a constant (independent of frequency) and replace it by its value in the small-q limit. The time correlation functions of the tagged-particle density ρs for these two linear equations of motion are easily computed using the MSR technique outlined above, or using directly a memory-function approach (Ma and Mazenko, 1975). The Fourier–Laplace transform Fs (q, z) of the van Hove correlation function G s (r, t) is obtained in the form of a two-step continued fraction 

q 2 v02 Fs (q, z) = z − z + i0s

−1 .

(8.1.81)

This result for Fs (q, z) is similar to the expression (7.3.68) for the total density correlation. Here v0 is the average speed of the fluid particle. In the long-time limit (small z) Fs develops a diffusive pole Fs (q, z) =

1 , z + iq 2 DB

(8.1.82)

with the bare self-diffusion coefficient DB = v02 / 0s . For the hard-sphere system 0s computed from the Enskog model gives the Enskog result, 0s = 2/(3tE ). Mode-coupling effects Since the average value of ρs  ∼ V −1 , where V is the volume of the system, it is negligible in the thermodynamic limit. The second term on the RHS of eqn. (8.1.79) involving the interaction part of FUs therefore makes a contribution nonlinear in ρs . The crucial input here is the functional derivative of FUs with respect to ρs . Using for the interaction part Fins an expansion in terms of the density fluctuations and involving the Ornstein–Zernike direct correlation functions of the one-component fluid, we obtain δ Fins 1 =− δρs βm



c(x − x )δρ(x , t)dx ,

(8.1.83)

keeping only the lowest-order contributions involving the two-point direct correlation functions. See the Ramakrishnan–Yussouff free-energy functional given by eqns. (2.1.38) and also (6.2.14). This form was considered earlier with respect to both equilibrium densityfunctional calculations and the mode-coupling dynamics of the collective variables in Chapters 2 and 6, respectively. The above form (8.1.83) of the dynamic equation includes in the single-particle dynamics the feedback effects resulting from coupling of the taggedparticle density ρs with the total density fluctuations, i.e., δρ. Following standard procedures (see Section 7.1.1 for how to compute the perturbative corrections to the transport properties) we obtain the mode-coupling contribution to the kernel s . Upon including

8.1 Mode-coupling theory

381

the nonlinear coupling of δρs and δρ in the equation of motion for the tagged-particle momentum, we obtain ∂ gsi + s gsi + v02 ∇i ρs + Vsi [q − k, k]ρs (k)δρ(|q − k|) = ζsi . ∂t

(8.1.84)

The vertex function Vs in the third term on the LHS is given by Vsi (q − k, k) =

ki c(k). βm 2

(8.1.85)

The RHS of the expression (8.1.81) for the tagged-particle correlation is obtained as 

q 2 v02 Fs (q, z) = z − z + iRs (q, z)

−1 .

(8.1.86)

The kernel 0s in (8.1.81) is now renormalized due to the nonlinear coupling of the taggedparticle density ρs with ρ in the equation of motion (8.1.84) for the tagged-particle density,  s (q, t)dt. (8.1.87) Rs (q, z) = s + ei zt ˜ mc At one-loop order the mode-coupling contribution is obtained as  n0 dk s ˜ mc (q, t) = [(qˆ · k)c(k)]2 S(k)Fs (q − k, t)φ(k, t) ≡ m˜ Ls (q, t), βm (2π )3

(8.1.88)

where qˆ denotes the unit vector. The correlation of the tagged-particle momentum also follows in a straightforward manner from the above analysis. The renormalized expression for the longitudinal tagged-particle current–current correlation is obtained as Fs (q, z) =

1 . z + iRs (q, z)

(8.1.89)

The transverse current correlation has a similar form. It is useful to note here that the renormalized model given by (8.1.86)–(8.1.88) for the tagged-particle correlation is also obtained in the so-called microscopic approach of the projection-operator formalism by following a procedure similar to that described in Appendix A7.4 (Götze, 1991). This involves taking projections on the space of coupled slow modes Cs (kp . . .) = δρ(k)δρs (p) . . . The same model for tagged-particle correlation was studied in order to argue that the self-diffusion coefficient goes to zero (Bengtzelius et al., 1984) at the dynamic transition point of MCT. Let us consider the implication of having a mode-coupling kernel of the form (8.1.88) described above. The time evolution of the tagged-particle correlation function is now completely enslaved to that of the density correlation function. The self-correlation Fs (q) in (8.1.81), driven by the memory function (8.1.88), freezes to f qs at the dynamic transition

382

The ergodic–nonergodic transition

point of the MCT described by (8.1.31)–(8.1.33). The corresponding equation for the nonergodicity parameter f qs (the long-time limit of the tagged-particle correlation Fs (q, t)) is f qs 1−

f qs

=

1 L m˜ (q, t → ∞) ≡ Hqs [ f k ], 2sq s

(8.1.90)

with 2sq = q 2 v02 . This is similar to the corresponding equation (8.1.36) for the total density correlation function. In the close vicinity of the dynamic transition, on the liquid side Fs (q, t) follows a sequence of relaxation similar to that of the density correlation function as discussed earlier in Section 8.1.1. Away from the transition, a correction similar to what was already discussed for the density correlation function applies (Fuchs et al., 1998) in this model for the tagged-particle motion. The hydrodynamic behavior of the tagged-particle motion is obtained from the q → 0, z → 0 limit of the correlation function Fs (q, z) given by eqn. (8.1.86). There is diffusive behavior, as follows from the result Fs (q, z) =

1 , z + iq 2 D R (0, 0)

(8.1.91)

where the renormalized diffusion coefficient is obtained as D R (0, 0) =

1

. βmRs (0, 0)

(8.1.92)

At the dynamic transition point, the mode-coupling contribution given by eqn. (8.1.88) develops a 1/z pole in Rs . This is a consequence of the freezing of the density correlation function at the transition. Hence the self-diffusion coefficient D R → 0 in the long1 2 2 time limit. The corresponding mean-square displacement r (t) freezes in time, signifying complete localization of the particle. The above model for the tagged-particle correlation has been very widely used in the literature for studying slow single-particle dynamics in a dense liquid. In this theory the tagged-particle correlation completely freezes and the self-diffusion coefficient goes to zero at the mode-coupling or the ENE transition point. This is a consequence of the similarity of the renormalized model for the self-correlation function given by (8.1.86)–(8.1.88) to its counterpart for the total density correlation. In its present form the memory kernel for the tagged-particle dynamics is completely enslaved to the collective density correlations and causes the self-diffusion coefficient to vanish at the MCT transition point. This coupling of the autocorrelation of the total density fluctuations and that of the tagged-particle density in the memory function is rooted in construction of the model for single-particle dynamics as described above. The two-step continued-fraction form for the self-correlation follows from the equations for the tagged-particle density ρs and its current gs . These equations are similar to the corresponding equations for the total mass density ρ and momentum density g, both of which are microscopically conserved quantities. In the case of single-particle dynamics, only the tagged-particle density ρs (and not its current gs ) is a microscopically conserved property.

8.1 Mode-coupling theory

383

The microscopic variables ρˆs and gˆ s are connected by the simple continuity equation (8.1.62), but it is not possible to justify the relation (8.1.70) connecting the coarse-grained quantities ρs and gs . Since gs is not a slow variable, the dynamic equation for ρs should in fact involve dissipation and noise. There is also no basis for assuming a separation of time scales between the “noise” and “regular” parts in the equation for gs . The theory of the single-particle dynamics should in fact follow in a natural way from the one-particle limit of the binary mixture. Of related interest is the Smoluchowski level description for single-particle dynamics in a colloid, which is formulated (Schweizer and Saltzman, 2003, 2004) by treating the displacement r (t) of the particle as a slow variable. 8.1.4 Dynamical heterogeneities and MCT The MCT takes into account the effects of correlated dynamics of the particles in a highdensity liquid. The different aspects of dynamical heterogeneities seen in the computer simulations should follow in a natural way from these mode-coupling models. Thus, for example, the non-Gaussian parameter α2 (t) (see eqn. (1.3.61) for its definition) is obtained by evaluating the tagged-particle correlation function Fs (q, t) for very small q truncated at O(q 4 ) (Kaur and Das, 2002; Fuchs et al., 1998). From the expansion given in eqn. (1.3.58), it follows that for small q the curve (1/q 2 )(1 − Fs (q, t)) vs. q 2 is a straight line with intercept r 2 (t)/6 and slope −r 4 (t)/120. The term α2 (t) can then be evaluated using these two quantities from eqn. (1.3.61). In a simple hard-sphere system, for example, the bare contribution to s (q, t) is given by s0 = 2/(3tE ), tE being the Enskog collision time defined in eqn. (5.3.101). The direct correlation function c(k) and the static structure factor S(k) are obtained as the Percus–Yevick solution with Verlet–Weiss correction. To evaluate the Fs (q, t) in the small-q range, the memory function ˜ smc (q, t) is expressed as an expansion in q given by smc (q, t) = ˜ s(0) (t) + q 2 ˜ s(2) (t) + q 4 ˜ s(4) (t) + · · ·.

(8.1.93)

 t) q − k|, The successive ˜ s(n) are obtained by using the Taylor-series expansions of Fs (| and Vs (q − k, k) in eqn. (8.1.88). The α2 (t) obtained from the mode-coupling equation develops a peak over a time scale longer than the microscopic times and is similar to the one observed in the computer-simulation studies of Kob et al. (1997) discussed in Section 4.4.1. In the simple MCT model with the ideal dynamic transition at ηc , the time at which this peak of α2 (t) occurs keeps growing on approaching ηc . Above ηc this peak never occurs; instead, α2 (t) attains a constant value for long times for the packing fraction of 0.540 for the hard-sphere system. In reality, however (as we will see later in this chapter), the dynamic transition at ηc is finally removed due to ergodicity-restoring mechanisms operating in the compressible liquid. The peak in α2 (t) therefore survives at high density in an extended version of the MCT. The behavior of α2 (t) with increasing density is shown in Fig. 8.2(a). The result for the tagged-particle correlation G s (r, t) vs. r obtained from the MCT is used to set a criterion for identifying a set of particles that are more mobile than the rest

384

The ergodic–nonergodic transition

Fig. 8.2 (a) The non-Gaussian parameter α2 vs. t ∗ for the hard-sphere system at packing fractions η = 0.550 (dotted), η = 0.565 (solid), and η = 0.580 (dot–dashed) as obtained from solution of extended MCT equations. In the inset, the variation of tp2 vs. η is shown for this model. (b) The fraction of mobile particles obtained using the MCT: G s (r, tp2 ) vs. r ∗ = r/σ for the hard-sphere system at packing fraction η = 0.565. The dashed curve is the corresponding Gaussian distribution function G 0s (r, tp2 ) (see the text). The inset shows the variation of the fraction of mobile particles f M c American Physical Society. vs. η. Reproduced from Kaur and Das (2002). Both parts 

of the particles. This was already described in Section 4.4.1 with eqns. (4.4.3) and (4.4.4) for the BMLJ systems (Kob et al., 1997). The Gaussian function G 0s (r, t) is defined in terms of r 2 (t) and compared with the full G s (r, t). The distance rM is identified from the largest value of r at which the two curves G s and G 0s intersect (see Fig. 8.2(b)). The time tp2 at which the tagged-particle correlation is considered is chosen (similarly to what

8.1 Mode-coupling theory

385

is done in simulations) to be the one at which the corresponding non-Gaussian parameter α2 (t) shows a large peak. The fraction f M of particles that have traveled a distance larger than rM is obtained from the area under the G s (r, tp2 ) vs. r curve beyond this distance rM and is displayed in the inset. The mode-coupling model has also been extended (Biroli and Bouchaud, 2004) to study the four-point correlation function which is associated with the dynamic correlation length ξd discussed in Section 4.4.2. The density being the dominant variable, one can focus on the four-point correlation function defined in terms of the collective density G4 (r, t; r0 , t0 ) = ρ(0, 0)ρ(r0 , t0 )ρ(r, t)ρ(r + r0 , t + t0 ) − ρ(0, 0)ρ(r0 , t0 ) ρ(r, t)ρ(r + r0 , t + t0 ) .

(8.1.94)

G4 (r, t; r0 , t0 ) is a measure of the cooperativity of the dynamics. The corresponding Fourier–Laplace transform χ4 of the four-point function in the small-wave-vector limit is obtained using diagrammatic methods. The four-point function has been dealt with by making simplifications such as that there is only one type of cubic vertex involving the density ρ and its conjugate MSR field ρ. ˆ The four-point correlation function is expressed as a sum of ladder diagrams (shown in Fig. 8.3). The diagrammatic expansion to all orders then takes the form of an infinite geometric series, ˜ ˜ 2 [ψ(t)] + · · · χ4 ≡ 1 + M[ψ(t)] +M 1 = , ˜ 1 − M[ψ(t)]

(8.1.95)

˜ where M[ψ(t)] is the corresponding one-loop contribution in the ladder diagrams in terms of the correlation function C. This also requires assuming the validity of the fluctuation– dissipation theorem for the relation between the correlation and response functions. The ideal transition point corresponds to the solution of the one-loop equation for the non˜ = 1. For T < Tc the ergodicity parameter { f c } at the transition point Tc as M[ψ(t)] density correlation ψ(t) = f + ψν (t) (see eqn. (8.1.9)). By expanding the denominator of the RHS of eqn. (8.1.95) around f = f c and using the result that in the nonergodic phase √ f − f c ∼  (see eqn. (8.1.24)), we obtain that the four-point correlation χ4 for T < Tc √ √ −1 diverges like 1/ . The wave-vector-dependent propagator behaves like (  k 2 + ) . For T > Tc , on the other hand, a simple Taylor-function expansion around the criti−1 cal f c shows that the propagator behaves like (k 2 + ) . Using the analogy with the typical Ornstein–Zernike form for the two-point functions, the above result for the propagators indicates that the corresponding dynamic correlation length ξd diverges like  −ν with ν = 14 and ν = 12 for T < Tc and T > Tc , respectively. Later, three-point correlation functions similar to eqn. (4.4.18) have been considered in the so-called inhomogeneous mode-coupling theory (Biroli et al., 2006), and it was predicted that the exponent ν = 12 ; above is in fact ν = 14 , predicting a slower growth of ξd on approaching the transition from above. In a related work Szamel (2008) did an analysis of the four-point correlation function for interacting Brownian particles, claiming that the former can be expressed in terms of three-point functions closely related to the three-point susceptibility introduced

386

The ergodic–nonergodic transition

Fig. 8.3 (a) The crucial cubic vertex in the MSR action functional constructed with the DDFT equation for density. (b) The typical ladder diagram for the four-point correlation function constructed c Europhysics Letters. with the cubic vertex of (a). From Biroli and Bouchaud (2004). Both parts 

by Biroli et al. (2006) and the standard two-point correlation function. Since the order parameter in MCT is a two-point function relating fluctuations at two different times, the corresponding susceptibility is a four-point function. The divergence of the latter is used as an indicator of diverging correlation length. However, as outlined above, obtaining this result involves using rather simple arguments and approximate treatment of the diagrammatic expansion for the four-point correlation function with a class of ladder diagrams only. Finally, since the transition at Tc is cut off, the divergence predicted here does not imply a sharp increase of ξd , in agreement with findings in simulations. The normalized four-point correlation function χ (t) = |δρ(km,t )δρ(−km,0 )|2 /|δρ(km )δρ(−km |2 evaluated at the peak of the structure factor (km ) displays a peak that grows with density. This quantity has recently been computed from a direct solution of the nonlinear Langevin equations (see Section 8.3.3) for a one-component Lennard-Jones system. The result for this four-point function showing a growing peak is shown in Fig. 8.4. Other cases of growing correlation length have also appeared in mode-coupling models. The first of such cases refers to the random diffusion model (RDM) (Mazenko, 2008; 2006), which is somewhat similar to the so-called DDFT model with a single density variable. Here, unlike in the above example of higher-order correlations, the growing correlation length has been identified from the dynamic structure factor. We discuss this

8.1 Mode-coupling theory

387

Fig. 8.4 Results from numerical solution of the equations of nonlinear fluctuating hydrodynamics. The four-point function χ4 (qm , t) at the peak of the structure factor qm (see eqn. (8.3.17) for its definition) is shown vs. time t in reduced Lennard-Jones units. The reduced density is fixed at 1.10 and the corresponding reduced temperatures are shown in the figure. The lines are best fits of the points to a Lorentzian form (Sen Gupta and Das, 2010, unpublished work).

below at the end of Section 8.1.6. Another example of a growing length scale refers to the viscoelastic behavior of the supercooled liquid. The MCT, which provides a microscopic explanation for visco-elasticity, has been used (Alhuwalia and Das, 1998; Das 1999) to identify a growing length scale l0 . The latter is associated with the maximum wavelength of the shear waves. Identification of this length scale from computer simulations of an equimolar mixture of soft-sphere liquids was discussed in Section 4.3.1. The crossover in the behavior of the diffusive shear mode to propagating shear waves is obtained from the solution of the self-consistent set of MCT equations given by (7.3.69)–(7.3.72). The relaxation of the shear mode is obtained here using the static structure factor of the liquid as the sole input. The solution of the MCT equations identifies the minimum wave number qmin above which the shear mode is propagating. This crossover in the behavior of the shear modes is intimately connected to the shear viscosity of the supercooled liquid. If the shear viscosity diverges, implying that the corresponding Laplace transform develops a pole, η(z) ∼ 1/z, then from eqn. (7.3.69) it follows that the shear modes are propagating for all wave numbers, i.e., qmin = 0. Hence, as the shear viscosity diverges at the ideal transition point, the corresponding l0 also diverges. Therefore, in the ideal nonergodic phase, for all temperatures below the dynamic transition point l0 → ∞. This is unlike a correlation length, which shows divergence only at the transition point. In reality, however, since the

388

The ergodic–nonergodic transition

ideal transition is cut off, the divergence of l0 is avoided, but l0 grows with increasing shear viscosity. Recently evidence of such length scale was reported (Torchinsky et al., 2011) from study of shear acoustic phonons in supercooled triphenylphosphite (TPP).

8.1.5 Linking DFT with MCT In this section we demonstrate that some of the results from the mode-coupling models for the dynamics of the frozen supercooled liquid are linked to a static density-functional description of the amorphous solid state. This link is established here, focusing on the tagged-particle dynamics as predicted from the two different descriptions of the frozen solid. For the mode-coupling model we have seen that, within the approximations made in the naive mode-coupling theory, the self-diffusion coefficient D R of a tagged particle is zero below the dynamic transition point. In the thermodynamic picture, on the other hand, the glassy state is characterized as one in which the individual particles are oscillating around a set of lattice sites {Ri } situated on a random lattice. In the case of harmonic motion of these particles, the system is termed a harmonic solid. The present discussion, which is based on Kirkpatrick and Wolynes (1987), bridges the two basic theories we are dealing with in this book, i.e., the density-functional theory (DFT) and the mode-coupling theory (MCT). The key quantity for analyzing the single-particle dynamics is the tagged-particle density correlation function given by (8.1.86). The Laplace transform of the related quantity tagged-particle current correlation is obtained in the form (8.1.89). The mode-coupling s (0, t) given by eqn. (8.1.88). In effects are included here in terms of the self-energy ˜ mc the naive mode-coupling model, the transition of the supercooled liquid to the nonergodic glassy state occurs at a critical density. This is driven by a key feedback mechanism as a result of which the density correlation function freezes in the long-time limit. As was pointed out above, due to the form (8.1.88) of the mode-coupling contribution the freezing of the density correlation function implies that the Laplace transform of the memory s (0, t) develops a 1/z pole. In the small-z limit we write function ˜ mc Rs (z) =

˜ ∞ + RT, z

(8.1.96)

s (0, t), where ˜ ∞ is the long-time limit of ˜ mc s (t → ∞), ˜ ∞ = ˜ mc

(8.1.97)

and RT represents the regular terms. The self-diffusion coefficient of the tagged particle goes to zero, implying complete localization. The development of the 1/z pole or nonzero value for ∞ is therefore crucial for the localized motion of the single particle. Assuming the latter to be harmonic in nature, a simple relation is reached between the corresponding spring constant and ˜ ∞ . Using the form (8.1.96) for the Laplace transform of the selfs (t), it easily follows from the denominator of the RHS of eqn. (8.1.89) that energy ˜ mc the current correlations represent damped harmonic waves signifying vibration around the

8.1 Mode-coupling theory

389

lattice sites. The corresponding frequency of oscillation ωs and hence the spring constant κs of the vibration can be obtained from the pole as mκs = ωs2 = ˜ ∞ .

(8.1.98)

The above relation follows from a purely dynamic route (MCT) for the localized singleparticle dynamics. In the DFT described earlier in Chapters 2 and 3 the equilibrium free energy of the solid state is obtained by a minimization of the suitable functional with respect to the inhomogeneous density function. The solid-state density is characterized by particles oscillating around their mean positions forming a lattice. In the case of a crystal this lattice has longrange order, whereas for an amorphous solid this is a random structure representing a metastable state. The test density function which has been most effectively used in the DFT approximates the density profiles in terms of normalized Gaussian functions centered around a set of lattice points {Ri }, n(x) =

( α )3/2  π

e−α(x−Ri ) ≡ 2

N 

φα (|x − Ri |),

(8.1.99)

i=1

Ri

where φα (r ) = (α/π)3/2 exp(−αr 2 ). In the harmonic solid the spring constant of oscillation of a single particle around the lattice site is simply related to the width parameter α for the Gaussian profiles. To see this, we first note that for the harmonic solid the average kinetic and potential energies are the same, each being equal to kB T /2, i.e., kB T κs  p2  = x 2  = . (8.1.100) 2m 2 2 The average mean-square displacement for the single particle in this case is given by x 2  = 1/(2α). On combining this with eqn. (8.1.100) we obtain the relation κs = 2(kB T )α

(8.1.101)

linking the spring constant with the width parameter. Now, using the relation (8.1.98), the width parameter α is related to the self-energy ˜ ∞ by βm (8.1.102) ˜ ∞ . 2 The above relation then serves as the key link between the DFT description of the harmonic solid and the mode-coupling kernel. In the ergodic state ˜ ∞ → 0 and hence α vanishes, corresponding to the liquid state in the DFT description. The long-time limit ˜ ∞ for the memory kernel ˜ s (t) follows from the result (8.1.88):  n0 dk 2 2 k c (k)S(k)Fs (−k, t)φ(k, t). (8.1.103) ˜ s (0, t) = 3βm (2π )3 α=

Equation (8.1.102) for α is now obtained in the form  dk 2 2 n0 α= k c (k)S(k)Fs (−k, t → ∞)φ(k, t→∞). 6 (2π )3

(8.1.104)

390

The ergodic–nonergodic transition

A self-consistent equation for the width parameter α is thus obtained by applying the Vineyard approximation φ(k, t) = Fs (k, t). For the localized Gaussian density profiles the long-time limit of the tagged-particle density correlation is expressed in terms of the width parameter as 2

k . (8.1.105) Fs (k, t → ∞) = exp − 4α Using this approximation, the RHS of the expression (8.1.104) for the width parameter α reduces to the form 2

 n0 k dk 2 α= k c(k)h(k)exp − (8.1.106) . 6 2α (2π )3 We define h(k) = c(k)S(k). The above result is an expression for the width parameter of the localized Gaussian density profiles obtained from dynamic considerations using the simple mode-coupling model for tagged-particle dynamics. In the DFT approach, for the harmonic solid we can estimate the spring constant κs also using equilibrium considerations. Here we start from the corresponding expression for the free-energy functional treated as a function of the inhomogeneous density of the harmonic solid. In Appendix A8.1 we compute the variation of the free energy of the solid corresponding to a displacement δR1 for a single particle tagged as 1. The free energy as a quadratic functional of the displacement δR1 is obtained as   1 dx dx c(|x − x |) β Fin = 2   N  φα (|x − Ri |)∇x ∇x φα (|x − R1 |) · δR1 δR1 + · · · × i=2

1 ≡ κisj δ R1i δ R1 j . 2

(8.1.107)

In the last equality we treat the quadratic expression for the free energy to obtain κisj as the tensor of spring constants for the harmonic motion of the tagged particle 1 around R1 . For an isotropic system we denote this as κ s δi j , which is a scalar. We obtain from (8.1.107), with some simple algebra, the following result for α: α=

n0 6



2

k 2 k c(k)h (k)exp − , 0 2α 2π 3 dk

(8.1.108)

where c(k) is the Fourier transform for the direct correlation function c(x) for the isotropic liquid. The function h 0 (k) on the RHS of (8.1.108) is obtained as h 0 (k) =

N 1  ik ·(Ri −R j ) e , N i= j=1

(8.1.109)

8.1 Mode-coupling theory

391

where the bar denotes an average over the aperiodic distribution of the lattice points {Ri }. The similarity of the two expressions (8.1.106) and (8.1.108) for the width parameter α obtained in the two different pictures is a signature of the link between the corresponding static and dynamic descriptions. The structure functions h(q) and h 0 (q) corresponding to the dynamic (MCT) and the static (DFT) description are associated with the uniform liquid state and the frozen solid state, respectively. Indeed, the frozen lattice structure in the DFT description of localized density profiles is applicable only over the time scale of the glassy state during which the underlying lattice {Ri } remains time-independent. For a deeper understanding of the two approaches discussed above, a self-consistent treatment of the problem is necessary.

8.1.6 Dynamic density-functional theory Our discussion so far involving the mode-coupling models for the dynamics has been formulated in terms of the set of conserved variables consisting of the density ρ and the momentum density g. A somewhat simpler strategy is to work with a single variable, namely the fluctuating density ρ. In the supercooled state strong density fluctuations play the most dominant role in producing the slow dynamics characteristic of these systems. This is particularly significant since, when the supercooled liquid transforms into a frozen amorphous state, the freezing starts at short length scales. At short wavelength the energy and momentum fluctuations in the many-particle system are quickly transferred among the particles while the individual density fluctuations still decay much more slowly. It is therefore plausible that the dynamics for such a system at a somewhat simplified level can be considered in terms of the density fluctuations only. This has been referred to in the literature as the dynamic extension of the density-functional theory (DDFT) of freezing which is used for the study of the freezing transition and was discussed in detail in the earlier chapters of this book (see Section 6.2.2). The same model equation for the dynamics of density fluctuations has also been used in studying the kinetics of the freezing transition into an ordered phase (Munakata, 1978, 1989, 1990; Bagchi, 1987). In the present section we present the analysis of a hydrodynamic model for which the equation of motion for the density is given by a nonlinear Langevin equation with a diffusive kernel and multiplicative noise (Risken, 1996). As we demonstrate below, this DDFT model also captures some of the essential properties of the theory formulated in terms of both the mass and the momentum densities. In particular, this DDFT model also undergoes the ENE transition discussed above in the context of the fluctuating-hydrodynamics model. The nonlinear equation of motion In Section 6.2.2 we obtained the nonlinear Langevin equation for the coarse-grained density field ρ(x, t) for systems of particles following Newtonian dynamics at microscopic level. This involved starting from the nonlinear fluctuating-hydrodynamic (NFH) description of classical liquids in terms of {ρ, g} and integrating out the momentum density g from

392

The ergodic–nonergodic transition

the NFH equations with the adiabatic approximation. The final outcome was a single equation of motion for ρ with multiplicative noise. However, in our earlier discussion of the nonlinear diffusion equation the multiplicative noise was introduced in an ad-hoc manner. Subsequently it was demonstrated by Kawasaki and Miyazima (1997) that the same equation follows in a natural way on using the field-theoretic model for the dynamics in terms of {ρ, g} and integrating out the momentum variable with the adiabatic approximation. We now briefly discuss this deduction. A convenient starting point for this deduction is the Martin–Siggia–Rose action described in eqn. (7.2.8), including both of the fluctuating variables, density ρ and momentum g. By integrating out the momentum density field g, the dynamics is formulated in terms of the density field ρ and its hatted counterpart ρ. ˆ Let us first consider the equation of motion for g as given by eqn. (6.2.20) in the form #−1 g + fρ = θ,

(8.1.110)

where θ is the Gaussian white noise and we have defined #−1 = ∂t + 0 ρ −1 ≡ ∂t + τ −1 .

(8.1.111)

τ −1 = 0 /ρ represents a relaxation time scale for the momentum fluctuations. On comparing this with eqn. (6.2.20) the force fρ is obtained as   δ FU [ρ] fρ = ρ ∇ . (8.1.112) δρ(r) The convective nonlinearity term ∇ j [gi g j /ρ] in (6.2.20) has been ignored in (8.1.110). FU denotes the free energy of the system expressed as a functional of density ρ and determines the equilibrium state of the liquid, as we discuss further below. FU is written as a sum of an ideal-gas part and an interaction part in the standard Ramakrishnan–Yussouff form given in eqn. (2.1.38) in terms of the direct correlation form  FRY ≡ β FU = dx ρ(x)[ln ρ(x) − 1]   1 (8.1.113) dx1 dx2 c(x1 , x2 ; n l )δρ(x1 )δρ(x2 ) + · · · − 2 measured with respect to the free energy of the uniform liquid state at density ρ0 . The momentum field g is formally integrated out of the problem by performing a trivial Gaussian integral and the action A[ρ, ρ] ˆ is written only in terms of the density field ρ and its hatted conjugate ρ. ˆ Let us write the expression (7.2.8) for the action A in the fluctuating-hydrodynamics formulation involving {ρ, ρ, ˆ g, gˆ } in the form

 β −1 ˆ ρ˙ + ∇ · g) . (8.1.114) (# g + fρ )0−1 (#−1 g + fρ ) − i ρ( A[ρ, ρ, ˆ g, gˆ ] = − d1 4 The quadratic and linear terms in the momentum field g in the first term on the RHS of (8.1.114) are combined as a perfect square to form a Gaussian integral. The momentum

8.1 Mode-coupling theory

393

field g is then formally integrated out of the problem giving the action in terms of ρ and ρ. ˆ Treating the Jacobian as a constant, we obtain

 1 ˜ · J } − βfρ · L −1 · f ρ + i ρˆ · ρ˙ , (8.1.115) {J · #0 · # A[ρ, ρ] ˆ = d1 0 β ˜ is the adjoint of # and the current J is now expressed as where # ˜ −1 L −1 fρ − i ∇ ρ. J = β# ˆ 0

(8.1.116)

The complicated result of (8.1.115) is simplified by making some rather strong assumptions. In the so-called adiabatic approximation the momentum density is assumed to relax very fast so that the operator #−1 can be replaced by 0 ρ −1 ≡ τ −1 . The action (8.1.115) is simplified to the form

 τ ˆ ρ˙ + τ ∇ · fρ } . (8.1.117) ρ(∇ ρ) ˆ 2 + i ρ{− A[ρ, ρ] ˆ = − d1 β The action functional given by (8.1.117) reduces to a convenient form with the identification of the bare transport coefficient D0 = τ/β (with the particle mass taken to be unity). ˆ we obtain for the MSR action Using the transformation ρˆ → D0−1 ρ,   # $ A[ρ, ρ] ˆ = − d1 D0−1 ρ(∇ ρ) ˆ 2 + i ρˆ −D0−1 ρ˙ + β∇ · fρ (8.1.118) as a functional of ρ and ρ. ˆ On the RHS of eqn. (8.1.118) we denote a space-time integral over variables r and t simply as d1 and the arguments (r, t) for different functions are dropped for the sake of brevity. This form of the action (8.1.118) conforms to the following stochastic Langevin equation for the density field:

δ FU ∂ρ √ (8.1.119) = D0 ∇ · ρ ∇ + ρθ, ∂t δρ where we have multiplicative noise. The correlation of the noise θ is 1 2 θ (x, t)θ (x , t  ) = 2D0 δ(t − t  )∇x ∇x  {δ(x − x )}.

(8.1.120)

The FNH description in terms of the additive noise is thus reduced to the DDFT model equation with multiplicative noise and without the 1/ρ nonlinearities in the equations of motion. FU is a functional of the density field ρ that controls the equilibrium state of the fluid (see below) and is often identified with the Ramakrishnan and Yussouff (1979) free-energy functional introduced in the equilibrium density-functional theories discussed in Chapter 2. The functional2 F is expressed as a sum of an ideal-gas part Fid and an interaction contribution in terms of the direct correlation functions c(n) : F = Fid [ρ] + Fin [ρ],

(8.1.121)

2 From now on we drop the subscript in F etc. and simply write it as F ≡ F[ρ], understood to be a functional of the density ρ . U

394

The ergodic–nonergodic transition

where



β Fid [ρ] =

# $ dx ρ(x)[ln 30 ρ(x) − 1],

β Fin [ρ] = −

 ∞  1 dx1 . . . dxn c(n) (x1 , . . . , xn )δρ(x1 ) . . . δρ(xn ), n!

(8.1.122) (8.1.123)

n=1

where 0 is the thermal de Broglie wavelength and δρ denotes the fluctuation of density around the density of a uniform liquid state. Using this general expression for the freeenergy functional in eqn. (8.1.119), we obtain the corresponding equation of motion for the density field,  ∂ρ (8.1.124) = D0 ∇ 2 ρ − D0 ∇ρ(x, t) ∇ dx veff (x − x )ρ(x , t), ∂t veff (x − x ) =

∞ 

1 (n − 1)!

n=2 (n)

×c

 dx3 . . . dxn

(x, x , x3 , . . . , xn )δρ(x3 ) . . . δρ(xn ).

(8.1.125)

This is generally termed the Vlasov–Smoluchowski equation (Smoluchowski, 1915) and has been used for studying the dynamics of density fluctuations in different systems (Calef and Wolynes, 1983; Munakata, 1978, 1989; Bagchi, 1987). If the free-energy expansion is formally truncated at the two-point level, we obtain veff (x − x ) = c(2) (x − x ). Equation (8.1.119) for the coarse-grained density ρ(x, t) is identical in form to the exact ˜ t) for a system of particles with balance eqn. (6.3.14) for the microscopic density3 ρ(x, Brownian dynamics (BD). For this, see the deduction of eqn. (6.3.14) in Section 6.3.1. This similarity also requires identifying the Ornstein–Zernike two-point direct correlation functions c(r ) appearing in the Kawasaki equation of motion (8.1.119) for the coarse-grained density ρ with the bare interaction potential βU (r ) in the corresponding exact balance equation of Dean for the microscopic density ρ. ˜ Even though it is often presented as the Dean–Kawasaki model, the corresponding densities involved are different. The balance equation is for ρ(x, ˜ t), which is not the coarse-grained density ρ(x, t). The Fokker–Planck description The DDFT model for the fluid in terms of the density ρ(x) is alternatively described with the corresponding probability density functional P[ρ, t]. The time evolution of P is given by the corresponding Fokker–Planck equation. We obtained earlier eqn. (6.1.63) for the time evolution of the probability distribution of a set of fields {φi } with the corresponding Fokker–Planck operator Dφ being given by eqn. (6.1.61). In the present case the bare transport operator is of the form L i0j ≡ ∇i ρ∇i and the reversible current Vi [φ] is 3 Here we call the microscopic density ρ˜ , instead of ρˆ as was done in Section 6.3.1, to avoid confusion with the MSR conjugate

field to ρ(x, t).

8.1 Mode-coupling theory

395

zero. Hence the Fokker–Planck equation for the time evolution of the probability P[ρ] is obtained as ∂ P[ρ, t] = −Dρ P[ρ, t], ∂t where the operator Dρ is given by

 δ δ δ F[ρ] Dρ = D0 dr ∇ · ρ(r)∇ kB T + . δρ(r) δρ(r) δρ(r)

(8.1.126)

(8.1.127)

The positive constant D0 is treated as a bare transport coefficient. The stationary solution of (8.1.127) is given by exp[−β F] and therefore F is termed the free-energy functional in terms of the density function. F has the same form as is used in the density-functional models discussed in Chapter 3. The above Fokker–Planck equation for the dynamics of the density distribution is obtained (Kawasaki, 1994) from the Smoluchowski (1915) equation for interacting colloidal particles without hydrodynamic interactions through a suitable coarse-graining procedure. The H -theorem It is possible to reach a suitable H -theorem in the present problem (Munakata, 1994) of density fluctuations. First, we consider the deterministic form of the dynamic equation, ignoring the noise in eqn. (8.1.119). The rate of change of the free energy is obtained as  δ F ∂ρ(x, t) dF = dx dt δρ(x, t) ∂t

 δ F[ρ] 2 = −β D0 dx ρ(x, t) ≤ 0, (8.1.128) δρ(x, t) since the density ρ(x, t) is always positive. F reaches a minimum when the functional derivative of F[ρ] with respect to density is a constant, i.e., δF = μ, δρ(x, t)

(8.1.129)

where μ is a chemical potential. Next we consider the equation of motion (8.1.119) including the random part. The corresponding H function is obtained as  Dρ{F[ρ]P[ρ, t] + kB T P[ρ, t]lnP[ρ, t]}. (8.1.130) Hρ [P] = The time rate of variation of Hρ is given by  dHρ [P] ∂ P[ρ, t] = Dρ[F[ρ] + kB T ln P[ρ, t]] dt ∂t  d + kB T Dρ P[ρ, t]. dt

(8.1.131)

396

The ergodic–nonergodic transition

The last term vanishes due to conservation of probability P[ρ, t]. Using the Fokker–Planck eqns. (8.1.126) and (8.1.127), we obtain the result,    dHρ = −D0 Dρ dx F[ρ] + kB T ln P[ρ, t] dt 3 3 4 4

δ δP δF × ∇ · ρ(x)∇ P[ρ, t] + kB T ∇ · ρ(x)∇ δρ(x) δρ(x) δρ(x)

  δF kB T δ P = D0 Dρ dx + δρ(x) P δρ(x) 3 3 4 4

δF δP × ∇ · ρ(x)∇ P + kB T ∇i ρ(x)∇i . (8.1.132) δρ δρ The RHS is further simplified by doing a partial integration w.r.t. x to obtain   3 4   dHρ δF δP δF = −D0 Dρ dx ∇i P + kB T ρ∇i dt δρ δρ δρ  3 

4 δP kB T δ P δF + ∇i + kB T ρ∇i δρ(x) P δρ(x) δρ 3

42   δF δ ln P P[ρ, t]ρ(x) + kB T = −D0 Dρ dx ∇ δρ(x) δρ(x) ≤ 0,

(8.1.133)

since both ρ and P are always positive. Therefore the generalized H function always decreases. The RHS of eqn. (8.1.132) is zero for the stationary distribution 3 4

 Peq ∼ exp −β F[ρ] − μ dx ρ(x) , (8.1.134) where μ is the chemical potential for the equilibrium state. The random part of the equation of motion (8.1.119) therefore drives the system to its lowest free-energy minimum. Next we focus on the relevance of the DDFT model described by the nonlinear diffusion equation (8.1.119) for the coarse-grained density ρ with respect to the ENE transition discussed above. Renormalization Let us now consider the nature of the dynamics resulting from the nonlinear stochastic equation for density obtained in the previous section. If we ignore the interaction between the particles and replace the free-energy functional by the ideal part Fid given by eqn. (6.3.17), then eqn. (6.3.14) represents simple diffusion: ∂ρ(x, t) 1/2 (8.1.135) = D0 ∇ 2 ρ(x, t) + ρ0 θ (x, t). ∂t The correlation of the noise θ is given by (6.3.13), with the density ρ being replaced with ρ0 at the linear level. For the linear dynamics the correlation functions are straightforward to compute in the MSR formalism in terms of ρ and the corresponding hatted field ρ. ˆ For the

8.1 Mode-coupling theory

397

density ρ variable we define the correlation functions in terms of the density fluctuations, i.e., ρ = ρ0 + δρ. The frequency and wave-number transform of the zeroth-order matrix G −1 0 is obtained from the action given in (8.1.118), ⎤ ⎡ ρ ρˆ ⎥ ⎢ ⎢ −1 2 ρ ⎥ (8.1.136) G −1 (q, ω) = 0 ωD − iq ⎥, ⎢ 0 0 ⎦ ⎣ ωD0−1 + iq 2 α0 D0−1 q 2 ρˆ where we denote α0 = 2β −1 ρ0 . Let us consider the implications of the nonlinearities in the equations of motion involving coupling of density fluctuations. For this, we obtain the full matrix G −1 of two-point correlation functions. This matrix is a generalization of the Gaussian matrix G −1 0 and is computed in terms of the corresponding self-energy matrix  using the Schwinger–Dyson equation (7.1.27). The renormalizability of the theory here will imply that the correlation function G −1 in the nonlinear theory maintains the form of the matrix (8.1.136) in which the transport coefficient D0 is replaced with the renormalized quantity DR . The bare transport coefficient D0−1 is corrected either from the self-energy ρˆ ρˆ or from the response −1 matrix, respectively. A self-energy ρρ ˆ , in the correlation and response parts of the G consistent form of the renormalized theory is obtained if we assume that the two types of self-energies are related through a fluctuation–dissipation theorem ω ∗ 2β −1 ρ0 ρρ (8.1.137) ˆ (ω) = 2 ρˆ ρˆ (ω) = −ρ ρˆ (ω) . q Upon inverting the full correlation function matrix G −1 (q, ω), we obtain for the normalized density correlation function ψ(q, z) =

1 z

+ iq2 /DR−1 (q, z)

.

(8.1.138)

By evaluating the self-energies up to one-loop order we obtain the generalized diffusion coefficient DR for the renormalization of D0−1 as (Kawasaki and Miyazima, 1997)  dk ˜ −1 −1 Vq,k G ρρ (k, t)G ρρ (q − k, t). (8.1.139) DR (q, ω) = D0 + (2π )3 To understand the above one-loop correction term, we note that, in the notation of (7.3.56), the crucial vertex function Vρρρ ˆ [q, q − k] represents the coupling of the density fluctuations. It gives rise to a cubic nonlinearity in the non-Gaussian part of the action and is in fact identical to the vertex function VL (q, k) defined in (8.1.34) in the fluctuatinghydrodynamics model. The vertex function V˜q,k is obtained as "2 S(q) ! ˆ ˆ , ( q · k)c(k) + q ·(q − k)c(|q − k|) V˜q,k = 2ρ0 q 2

(8.1.140)

in terms of the two-point Ornstein–Zernike direct correlation function c(r ) which defines the interaction part of F in eqn. (8.1.113).

398

The ergodic–nonergodic transition

The dynamic transition is characterized by the point at which the density correlation function develops a 1/z pole as DR−1 diverges for small z. The asymptotic behavior of the density correlation function as predicted by the eqns. (8.1.138) and (8.1.139) is identical to what we obtained with the mode-coupling model involving {ρ, g}. To demonstrate this, we consider the nonergodicity parameter f q as the long-time limit of the normalized density correlation function, lim

t→∞

G ρρ (q, t) ≡ fq . Sρρ (q)

(8.1.141)

On taking the long-time limit of eqns. (8.1.138) and (8.1.139), we obtain that the f q satisfy the integral equation fq = m(q, ˜ t → ∞), 1 − fq where the kernel m(q, ˜ t) at one-loop order is given by  dk ˜ 1 Vq,k S(k)S(k1 ) f k1 f k . m(q, ˜ t) = 2ρ0 β (2π )3

(8.1.142)

(8.1.143)

The integral equations (8.1.142) for the nonergodicity parameters in the DK model are the same as the corresponding eqns. (8.1.36) in the FNH model. The detailed wave-vector dependences of the vertex functions are identical to one-loop level in the two models as described in eqns. (7.3.67) and (8.1.140). This implies that the DDFT model undergoes an identical ergodic–nonergodic transition. However, in the case of DDFT the effect of the mode coupling now appears in the form of a lifetime renormalization. The model described above is primarily based on the assumption that the full correlation function G (with all nonlinearities taken into account) is expressed in a form (involving renormalized transport coefficients) similar to that of G 0 , i.e., the assumption is that the theory is renormalizable in terms of a generalized diffusion coefficient D R (q, z). The assumed FDT relation (8.1.137) is ad hoc. It is easy to see that this relation holds if we consider the one-loop diagram for the respective self energies (similar to Fig. 7.1) and evaluate the graphs with the zeroth-order Green function G 0 . In other words, the FDT holds at O(kB T ) in a non-selfconsistent calculation. The construction of the self-consistent renormalized model (for the ergodic–nonergodic transition) with all the associated nonlinearities is intricate but keeps the basic result unchanged at the lowest order (Kim and Kawasaki, 2008). We present this in Appendix A8.2. At a somewhat more general level the equation of motion (8.1.119) for the density function ρ(x, t) has often been considered in a form with a density-dependent transport coefficient D(ρ): 

 ∂ρ(x, t) δF = ∇i D(ρ)∇i + η(x, t). (8.1.144) ∂t δρ(x, t)

8.1 Mode-coupling theory

399

The noise η in the Langevin equation obtained above is multiplicative and its correlation is given by 1 2 η(x, t)η(x , t  ) = 2kB T δ(t − t  )∇x ∇x  {D(ρ)δ(x − x )}.

(8.1.145)

The free-energy functional F in eqn. (8.1.144) is simply written as a functional of density ρ(x). The DDFT model is a particular case with D(ρ) ≡ ρ so that the bare diffusion coefficient is linear in density. Another variant of this model is the hindered-diffusion or random-diffusion model (RDM), in which the driving free energy is chosen to be purely quadratic (Gaussian) (Mazenko, 2008),   1 (8.1.146) dx1 dx2 δρ(x1 )χ −1 (x1 − x2 ) δρ(x2 ), F[ρ] = 2 making the static behavior trivial. However, a high-density constraint is imposed on the bare diffusion coefficient with the following choice for D(ρ): D(ρ) = D0

ρc − ρ θ (ρc − ρ), ρc

(8.1.147)

where D0 is the diffusion constant in the low-density limit and ρc is a positive parameter in the theory. This is a departure from the usual approach of nonlinear fluctuating hydrodynamics, in which bare transport coefficients are treated as constants. This model is analogous to the facilitated-spin models (Fredrickson and Andersen, 1984; Garrahan and Chandler, 2002, 2003; Whitelam et al., 2005), in which kinetic coefficients depend on the local environment in a lattice. In the continuum limit with the conserved density the corresponding diffusion coefficient is density-dependent in the RDM. The self-consistent renormalized perturbation theory has been worked out to two-loop order in the RDM. This treatment goes beyond that of the above models, which are generally treated to one-loop order. There are two ways of analyzing this perturbation-theory result for the RDM, giving rise to an interesting situation. Using the Kawasaki rearrangement described in appendix A8.2, at one-loop order, the mode-coupling theory with an ergodic–nonergodic transition is obtained. However, at two-loop order it has been shown that the ergodic–nonergodic transition is not supported. This is possibly indicating that ENE transition of the DDFT model might not be viable at higher orders. It has further been demonstrated in the RDM that the corresponding mode-coupling theory gives rise to a growing length scale (Mazenko, 2008; McCowan and Mazenko, 2010). The dynamic density correlation function computed in this theory at one-loop order leads to a slowing down as the relevant nonlinear coupling increases. Eventually one hits a critical coupling at which the time decay becomes algebraic. Near this critical coupling a weak peak develops at a wave number well above the zero-wave-number peak associated with the conservation law. The width of this pre-peak in Fourier space decreases with time and can be identified with a characteristic kinetic length that grows with a power law in time. The new peaked state is maintained at two-loop order.

400

The ergodic–nonergodic transition

8.2 Evidence from experiments For the supercooled liquids, MCT now plays the role which van der Waals theory did for the theories of phase transition at the mean-field level. The theory offers a microscopic model for the dynamics of supercooled liquids in the initial stages of viscous slow-down. With the development of better experimental tools and fast computers, it has now become possible to probe the liquid-state dynamics over a wide range of time scales, ranging up to 18 decades. This has revealed a rich set of hitherto unknown features of the dynamics of the supercooled state. It is here that a microscopic theory like MCT has provided the clue to understanding the experimental (laboratory or computer) observations. The main achievement of MCT has been that, starting from a statistical-mechanical approach, the existence of a crossover temperature Tc that is greater than the calorimetric glass-transition temperature Tg has been identified in the supercooled liquid. Tc signifies a dynamic transition in the liquid, and, in its close vicinity, interesting scaling behavior is predicted over varying length and time scales. In the present section we briefly review the predictions of the MCT that have been put to the test through detailed analysis of data from experiments. The experimental techniques used in such studies include neutron- and light-scattering methods, dielectric spectroscopy, and measurements of the mechanical response of the supercooled liquid to external perturbations. In making such comparisons, it is generally assumed that correlations of other variables such as, for example, the dielectric moment, couple to the density, both obeying the same type of relaxation law. These comparisons with experimental data, barring a few exceptions, are mainly done w.r.t. the predictions of the above-described idealized model which involves a sharp transition of the supercooled liquid to a nonergodic glassy state.

8.2.1 Testing with schematic MCT For comparing the scattering data from experiments, often the schematic theoretical model without wave-vector dependence (described in the schematic model of the previous section) is chosen. In most cases the data analysis is done by treating the relevant quantities (such as the NEP f , and the exponent parameter λ) appearing in the MCT as fitting parameters. Also it is generally assumed that the role of the ergodicity-restoring mechanisms is minimal, at least over the time scale of the specific effect being studied. The nonergodicity parameter The cusp behavior of the nonergodicity parameters (NEPs) indicated in eqn. (8.1.46) is an asymptotic result and is valid only in the close vicinity of the transition point. The temperature dependences of the NEPs f q and f qs , corresponding to the long-time limit of the density and tagged-particle correlations, respectively, have been studied by a number of workers (Frick et al., 1990; Li et al., 1993) in order to locate the mode-coupling transition point Tc indirectly. Theoretically the NEP is the weight of the contribution by the

8.2 Evidence from experiments

401

Table 8.1 A list of Tc and the corresponding Tg in degrees Kelvin for some commonly studied glass-forming materials. Material

Tm

Tc

Tg

Tc /Tg

CKN PC OTP Salol

483 218 329 315

378 190 290 263

333 160 243 218

1.1 1.2 1.2 1.2

Fig. 8.5 The effective Debye–Waller factor for CKN vs. temperature T at q = 0. The q = 0 value is obtained from the corresponding results at different q values measured with impulsive lightscattering spectroscopy. The curves are fits to the square-root cusp. Reproduced from Yang and c American Institute of Physics. Nelson (1996). 

delta function π f q δ(ω) to the dynamic structure factor at densities beyond the dynamic transition point. In reality, however, the NEP is determined from the area under the α-peak having finite width. If the correlation function is obtained with respect to time, the NEP is approximated by the plateau value of the density correlation function or the taggedparticle correlation function, for f q and f qs , respectively. By projecting the temperature dependence of the NEP into a cusp-like behavior, the mode-coupling transition temperature Tc is identified. The elastic modulus measured in the impulsive stimulated scattering of light from different wave vectors q (Yang and Nelson, 1996) is converted to obtain NEPs at q = 0 relating to density fluctuations for macroscopic wavelengths. In Fig. 8.5, a plot of

402

The ergodic–nonergodic transition

the effective Debye–Waller factor vs. temperature T is shown for CKN (0.4Ca(NO3 )2 – 0.6KNOB3 ) at q = 0. The critical temperature estimated is Tc = 378 ± 2 K. In Table 8.1, the mode-coupling temperature Tc and the corresponding Tg for a few commonly studied materials are listed. The cusp behavior was similarly analyzed to identify Tc in other materials listed in Table 8.1: o-terphenyl (OTP) (Tölle et al., 1997), CKN (Kartini et al., 1996; Pimenov et al., 1996), salol (Toulouse et al., 1993; Yang and Nelson, 1995), and propylene carbonate (PC) (Yang et al., 1996). The relative change of the NEP in the asymptotic range where the cusp behavior dominates is not very large and hence the identification of Tc tracing a cusp-like behavior (see the inset of Fig. 8.5) in the NEP is somewhat arbitrary. Power-law relaxations 1. The critical decay: The decay (t/τβ )−a given by eqn. (8.1.47) has been observed in several systems beyond microscopic time scales. The dielectric-loss data for the molten salt CRN (0.4Ca(NO3 )2 –0.6RbNO3 ) show such a decay with an exponent a = 0.20 over three decades in time (Lunkenheimer et al., 1997b). The NEP f , the amplitude of the power law h, and the time τβ are treated here as fitting parameters. This critical power-law type relaxation is expected only over a limited time or frequency window where the correction terms indicated in eqn. (8.1.53) can be neglected. However, direct numerical solution of the MCT equations for a one-component Lennard-Jones system (Smolej and Hahn, 1993) indicates that the critical decay is observed for a relative distance from the dynamic transition (in terms of density) of  ≈ 4.2 × 10−4 only. 2. Von Schweidler (VS) relaxation. Like the critical decay described above, identification of this second power-law regime is also problematic due to several factors. For times t that are not long enough, the correlation function is above the critical plateau value, merging the VS relaxation (8.1.48) with the critical decay (8.1.47). On the other hand, higher-order corrections dominate at long times. Even the choice of the plateau is also, in many cases, a bit arbitrary. The critical decay is observed only in the close vicinity of the transition, while the range of validity of the VS relaxation is not known a priori, thus making their observation rather subtle. Owing to problems with the direct fit of the time correlation-function data, it is sometimes convenient to do the data analysis for identifying the power-law behaviors in terms of the frequency-dependent (dynamic) susceptibility function. The susceptibility function is defined in terms of the Laplace-transformed quantity χ (z) = φ(t = 0) + zφ(z).

(8.2.1)

The corresponding Fourier-transformed function is given by the imaginary part χ  . The two power-law decays of t −a and t b in the time correlation function give rise to corresponding frequency dependences in χ  of the form ωa and ω−b , respectively. As a result of these two opposite trends, χ  reaches a minimum at an intermediate frequency ω = ωmin . In the region of this minimum the susceptibility function is interpolated with the schematic form

8.2 Evidence from experiments

   b  a  χ ω ω min χ  (ω) = min b +a . a+b ωmin ω

403

(8.2.2)

The frequency corresponding to the susceptibility minimum ωmin is inversely proportional 1/(2a) . Therefore the plot of (ωmin )2a vs. the corto the β-process time scale τβ , i.e., to 0 responding temperature T of the liquid is expected to follow a straight line extrapolating to meet the temperature axis at the dynamic transition point Tc . The susceptibilities for CKN at temperatures in the range T = 468–383 K fit well (Li et al., 1992) with the scaling function for λ = 0.81. This choice of λ corresponds to power-law exponents a = 0.27 and b = 0.46, respectively. In comparison, the susceptibility curves from the dielectric-loss spectra (Lunkenheimer et al., 1997b) of CKN exhibit the minimum for T ≤ 417 K with the master function corresponding to a somewhat lower value of λ = 0.76. In general, the range of validity of the scaling behavior over the time or the frequency scale increases as the temperature T of the system approaches Tc from above. Similarly to the behavior ωmin ∼ τβ−1 ∝ |0 |1/(2a) for the minimum of the susceptibility function, the position of the α-relaxation peak of the susceptibility function at ωα also corresponds to the time scale τα and ωα ∼ τα−1 ∝ |0 |γ , where γ = 1/(2a) + 1/(2b). √ 2a , ω1/γ or  is proportional to Furthermore, the intensity χmin 0 . Therefore plots of ωβ α   2 χmin with temperature T should follow a straight line and meet the T axis on extrapolation at T = Tc (0 = 0) (Götze and Sjögren, 1992). This analysis is suitable for estimating Tc from the high-temperature side. This was applied to dielectric-loss spectra for CKN (Lunkenheimer et al., 1997b). The exponents a and γ were used as fitting parameters. Figure 8.6 shows this estimation of Tc for CKN and CRN, which was done by plotting the temperature dependence of, respectively, from the top panel down, the height, the position of the dielectric-loss minimum, and the α-peak position for CKN (closed symbols) and CRN (open symbols). The best-fit results a = 0.30 and γ = 2.6 were obtained for CKN (corresponding to exponent parameter λ = 0.76). The transition temperature Tc for CKN obtained by extrapolating the dielectric-loss data is close to 375 K. This is comparable to the corresponding result Tc = 378 ± 2 K obtained by studying the location of the square-root cusp as given by eqn. (8.1.24) or eqn. (8.1.40) for the NEP from the lowtemperature side. Similar fits with ωmin extracted from light-scattering data (Li et al., 1992) for CKN give a = 0.28 and γ = 2.9, corresponding to λ = 0.81, roughly in agreement with the above dielectric-spectroscopy results. For CRN these exponent values are respectively a = 0.20 and γ = 4.3 (corresponding to λ = 0.91) and the estimate for Tc lies between 360 K and 375 K. Consider the case of another typical glass-forming material, propylene carbonate (PC). Schneider et al. (1999) studied this material both in the normal and in the supercooled state using dielectric spectroscopy over more than 18 decades of frequency. Analysis done with the minimum of the dielectric-susceptibility data as well as the α-relaxation time scales gives Tc = 187 K. The minimum in the dielectric-susceptibility function is described with the scaling form (8.2.2) corresponding to a = 0.29 and b = 0.50 and is consistent with the

404

The ergodic–nonergodic transition

Fig. 8.6 Identification of the mode-coupling temperature Tc in two supercooled liquids, CKN (solid symbols) and CRN (open symbols), from plots of the temperature (T ) dependences of three different quantities: top panel, the height of the dielectric-loss minimum; central panel, the position of the dielectric-loss minimum; and bottom panel, the α-peak position. Representations along the vertical axes have been chosen so as to produce straight lines according to the predictions of MCT. The solid lines are consistent with a critical temperature of Tc = 375 K for CKN, and the dashed lines c American Physical with Tc = 365 K for CRN. Reproduced from Lunkenheimer et al. (1997b).  Society.

exponent parameter value λ = 0.78. The value Tc = 187 K is also in agreement with previous results from light-scattering studies by Du et al. (1994) and solvation-dynamics experiments (at 176 K) by Ma et al. (1996) on the same material. The neutron-scattering studies by Börjesson et al. (1990), however, gave a much higher value for Tc (180–200 K). Subsequent reexamination of PC by Wuttke et al. (2000) with neutron-scattering measurements improved this result. Those authors obtained Tc ≈ 182 K (corresponding to λ = 0.72). The α-relaxation regime The relaxation of the correlation function beyond the corresponding plateau f c of the so-called NEP and subsequent to the von Schweidler decay is generally referred to as αrelaxation. In the close vicinity of the dynamic-transition point the α-relaxation is

8.2 Evidence from experiments

405

described to leading order in terms of a temperature-independent master function. This also implies time–temperature superposition. However, as the distance from the transition increases, correction terms of O(0 ) to this master function become significant (see eqn. (8.1.58)). Furthermore, the effects of ergodicity-restoring mechanisms of the extended MCT will also influence the predictions for final relaxation in the theoretical model. A few characteristic properties of α-relaxation can be listed as follows. 1. The temperature dependence of the α-peak. The time scale of α-relaxation τα increases as |0 |−γ with γ = 1/(2a) + 1/(2b). The temperature dependence of the peak frequency is therefore of the form ωα ∼ (T − Tc )γ . This is shown in the low1/γ est panel of Fig. 8.6, in which ωα for CKN is plotted against temperature. The values of a = 0.28 and γ = 2.9 used are kept the same as those obtained in fitting ωβ and χmin as described above. Similar fits with τβ and τα extracted from light-scattering data (Li et al., 1992) for CKN give a = 0.28 and γ = 2.9 with the corresponding λ = 0.81. Schönhals et al. (1993) carried out an extensive study of the α-relaxation in PC. The result for the transition temperature Tc = 186.6 K is comparable to the corresponding transition temperature Tc = 180 K from neutron-scattering results (Börjesson et al., 1990; Elmroth et al., 1992). The temperature dependence of the α-peak frequency νp (∼ωα ) for PC fits to a power law proposed by MCT, νp ∼ (T − Tc )γ with exponent γ = 2.45, as shown in Fig. 8.7 (Schönhals et al., 1993). The data also fit an Arrhenius behavior. Beyond a temperature TA , the power-law behavior crosses over to a Vogel– Fulcher form. The MCT Tc (186.6 K) lies in between the temperatures T0 and TA , which are equal to 130 K and 216.6 K, respectively, for PC. 2. The scaling in the α-regime. Time–temperature superposition for the α-relaxation is a leading-order asymptotic behavior in the simple MCT. Theoretically, even within the simple MCT, away from the transition temperature-dependent corrections add to the master function. The role of ergodicity-restoring processes can further affect the simple result of time–temperature superposition. Experimentally, there have been claims and counter-claims regarding the validity of this. For depolarized light-scattering spectroscopy of o-terphenyl (OTP) (Cummins et al., 1997), the α-peak can be fit with a Kohlrausch–William–Watts (KWW) form. A temperature-independent stretching exponent β = 0.78 was found to fit the data, provided that the higher-frequency wing was interpreted as an overlap from the β-process or the von Schweidler-type decay. The corresponding VS exponent b = 0.6 is obtained from the analysis of the β-process (Cummins et al., 1997). Similarly, the temperature-independent β for T > Tc was obtained (a) from light-scattering-spectroscopy data for CKN (Li et al., 1992), glycerol (Wuttke et al., 1994), and PC (Du et al., 1994); and (b) from the dielectric-loss data for CKN (Pimenov et al., 1996), glycerol (Lunkenheimer et al., 1996), and PC (Lunkenheimer et al., 1997a). Arguments in favor of the temperature independence of β are based on the reasoning that the observed temperature dependence of β, if any, is a result of ignoring the β-process. It has also been argued that the α-relaxation data conform to a temperature-dependent stretching exponent. From dielectric-susceptibility

406

The ergodic–nonergodic transition 0.30

TA

(f∞/fP)

0.25

0.30

1/log10 (f∞/fP)

0.20

log10

0.20

0.10 150

0.15

200 250 T [K] MCT behavior

Arrhenius behavior 0.10

TC

0.05

VFT behavior T0

MCT behavior

0.00 0

50

100

150

200

250

300

T [K]

Fig. 8.7 A plot of 1/ log10 (ν∞ /νp ) vs. temperature T for propylene carbonate (PC) with log10 ν∞ = 13.11. The Vogel–Fulcher–Tamann (VFT) temperature T0 , the MCT Tc , and the crossover temperature TA are indicated by arrows. The crossover behavior is demonstrated more c American Physical Society. clearly in the inset. Reproduced from Schönhals et al. (1993). 

data for low-relative-molecular-mass glass-forming liquid PC over 15 decades, Schönhals et al. (1993) obtained a temperature-dependent stretching exponent β – a result that contradicted the earlier light-scattering results (Elmroth et al., 1992) of a temperatureindependent β.

8.2.2 Glass transition in colloids Among the various experimental systems studied for testing signatures of the ergodic– nonergodic transition predicted in the MCT, the nature of the slow dynamics of colloidal suspensions shows the most prominent agreement. A colloid consists of suspended particles in a solvent liquid treated as an inert continuous background. Although colloids are very different from an atomic system, certain characteristic properties are similar. The effective interactions between the suspended particles in the colloid are made steeply repulsive in the suspension using suitable techniques and hence its equilibrium phase behavior mimics that of a hard-sphere system. For example, the melting volume fraction ϕ = ϕm = 0.542 ± 0.003 in this system is in agreement with that obtained from simulation

8.2 Evidence from experiments

407

for hard-sphere crystals (Hoover and Ree, 1968). The microscopic dynamics in the colloid particles involves frequent collisions of solvent molecules, but the positions of these suspended particles change on a much slower time scale. This vast separation of time scales allows use of the stochastic Langevin equations for the dynamics of the colloid particles, in contrast to the deterministic Newtonian equations of motion in an atomic system. However, over long time scales the dynamics of the two types of systems are similar, showing essentially diffusive behavior. The diffusive dynamics becomes extremely slow at high density, leading to effective freezing in an amorphous configuration. The predictions of the fully wave-vector-dependent MCT model were compared extensively with the data from dynamic-light-scattering studies on colloidal systems (van Megen and Underwood, 1993a, 1993b, 1994). The normalized density correlation function φ(q, t) computed in the light-scattering experiment on the colloids almost freezes to a constant value for a change of the packing fraction within 1% around ϕ c = 0.574. This structural arrest (within the range 0.574 < ϕ c < 0.581) in colloids has been identified with the dynamic transition of the simple MCT. From the relaxation data the β-scaling of MCT is demonstrated in Fig. 8.8. This plot also tests the q independence of the LHS of (8.1.55) over a suitable range of time. The theoretical curve shown in Fig. 8.8 consists of the contribution from the βprocess (given by (8.1.49)) as well as the α-relaxation master function . The experimental data fit to a β-scaling function obtained with the exponent parameter λ = 0.758. Using the hard-sphere structure factor in the formula (8.1.43), the corresponding value of the exponent parameter is obtained as λ = 0.77. The agreement between theory and experiments is thus remarkable in the regime of the β-process. However, it is worth noting that many parameters are involved in the fitting procedure (see the discussion below). Also, the comparisons described above are all based on the simple MCT predicting a dynamic transition. It has generally been argued that, since the momentum exchange with the colloid particles occurs on much shorter time scales, the cutoff mechanism is weak in these systems. The data, even at the highest density, do not rule out the existence of an ergodicity-restoring process. 8.2.3 Molecular-dynamics simulations Computer simulation of the dynamics of a small number of particles has been used as an effective tool for testing the predictions of the MCT. Here the theory can be tested without any adjustable parameter and using the structural data directly as input. The predictions of MCT were tested more extensively with MD simulation of a binary soft-sphere system by Roux et al. (1989, 1990). These works confirmed the factorization property and predicted deviations from the Stokes–Einstein relation in the vicinity of the transition. The extrapolated self-diffusion coefficient vanishes for both components of the mixture with a power-law exponent roughly equal to 2 (Bernu et al., 1987). Kob and Andersen (1994, 1995a, 1995b) later investigated a binary Lennard-Jones mixture (BMLJ) extensively against various predictions of MCT. The mixture consisted of 800 type-A and 200

408

The ergodic–nonergodic transition

Fig. 8.8 The density correlation function φq (t) ≡ f (q, τ ) in the scaled form [φq (t) − f qc ]/ h q vs. time t = τ . R is the hard-sphere radius; σ is the separation parameter, and φ is the packing fraction. The solid line corresponds to the MCT master function σ 1/2 [g− (tˆ)] (see eqns. (8.1.49)–(8.1.52)) corresponding to the exponent parameter λ = 0.758; dashed lines are for λ = 0.70 and 0.80. The vertical arrow on the time axis corresponds to the β-relaxation time scale τβ . Reproduced from van c American Physical Society. Megen and Underwood (1994). 

type-B particles of the same mass m, interacting through the Lennard-Jones potential (see eqn. (4.3.11) for details). Length is rescaled in units of σAA , temperature in units ! 2 "1/2 /(48AA ) . The temperature range studied was of AA /kB , and time in units of mσAA from T = 5.0 to a lowest value of T = 0.466. A characteristic transition temperature Tc has been identified for the Kob–Andersen mixture by extrapolating to zero the power-law fit for the self-diffusion coefficient obtained from simulation. Analysis of the simulation data of a BMLJ gave Tc = 0.435. On the other hand, Nauroth and Kob (1997) solved the mode-coupling equations of Bosse and Thakur (1987) using the corresponding required structure factor directly from the simulation results and obtained a transition temperature Tc = 0.922. This was subsequently improved by application of a different version of the mode-coupling model (Harbola and Das, 2002) obtained from an approach that takes into account proper momentum conservation in the system. Solution of the equations of the latter model with the same input

8.2 Evidence from experiments

409

structure factors shows that down to T = 0.435 there is no nonzero solution for the NEPs. This model has so far not been extended to compute the mode-coupling effects on the self-diffusion coefficient in a binary system. However, the diffusion coefficient Ds has been computed using a model for the dynamics of the tagged particle as described in Section 8.1.3. In this case the memory function for the tagged particle is simply enslaved to the decay of the density correlation functions Css  (see eqn. (8.1.88)). Using results for Css  obtained from the model of Harbola and Das (2002), Ds has been calculated (Harbola and Das, 2003) and found to extrapolate to zero at Tc ≈ 0.44. The MD simulation of the Kob–Andersen mixture has also revealed signatures of the two-step relaxation process predicted from MCT. A generic feature of the tagged-particle correlation, signifying cage formation in the dense fluid, is the plateau it reaches over intermediate times prior to final relaxation. In the MD simulations the observed plateau ranges over almost three decades at T = 0.466. The height of the plateau has been identified to be the value of the NEP and the corresponding time interval treated as the β-process regime (Kob and Andersen, 1994). Over the time range 3 ≥ t ≥ 1868 corresponding to the so-called β-relaxation (process) regime and close to the transition point, the simulation data agree with the factorization property of MCT (see Eq. (8.1.53)). The critical power-law relaxation (t −a ) is not seen in the simulation data. This is rationalized by assuming a strong influence of the microscopic dynamics on the early part of the power-law regime. In the later part of the β-regime, the von Schweidler relaxation with a positive exponent is visible. Beyond the power-law behaviors, on the scale of α-relaxation the correlation-function data fit to a stretched-exponential form with stretching exponent β, which is different from the VS exponent discussed above. At low temperature the relaxation time in the stretchedexponential form τα (T ) follows a power-law increase, appearing to diverge (Kob and Andersen, 1995b) at the same temperature Tc as that at which the diffusion coefficient extrapolates to zero. The exponent of the divergence, γ = 2.7, approximately follows the MCT prediction. However, the exponents of divergence of τ and Ds−1 are not the same. For the self-diffusion coefficients, both the temperature Tc and the exponent γ  (say) are found to be the same for both types of species in the mixture. However, the exponent γ  does not match the corresponding exponent γ found from fitting the diverging behavior of the α-relaxation time scale around Tc . A similar behavior was reported by Schröder et al. (2000). This presumably indicates that the mechanism of the slowing down for the density fluctuations is not the same as that for the self-diffusion. 8.2.4 Discussion In general, the above-cited items of evidence from experiments or simulations in support of the MCT involve a large number of fitting parameters. In most cases, especially for systems like CKN and OTP, the crucial inputs of the mode-coupling model are treated as free fitting parameters. The typical adjustable quantities used are the time scale of transients t0 , the NEP f , the amplitude h, the transition point Tc , the prefactor of the α-relaxation time scale C, and the exponent parameter λ – there are up to as many as six such adjustable

410

The ergodic–nonergodic transition

parameters. In addition, most of the tested relaxation behaviors are predictions from the theory at leading order, in the close vicinity of the transition. Often this requires careful adjustment of the time range or temperature range in order to be able to observe the predicted effect. Thus, for example, the predicted factorization property is valid only close to the transition and over the time window of power-law relaxation. It is not valid in the longer-time part of the α-relaxation. Similarly, the predicted temperature independence of the stretching exponent β (and hence time–temperature superposition) is expected to hold only when the liquid is neither too close to Tc (so that the ergodicity-restoring mechanisms do not intervene) nor too far from the transition (in which case the leading-order behavior would not hold). Additional correction terms to the leading-order results of the β-scaling, as well as the α-scaling, become important (Hinze et al., 2000) as the distance from the transition increases. It is important to note that in the above analysis of experimental data the existence of a dynamic transition point is inferred only indirectly. The critical exponents related to the divergences in the MCT are completely system-dependent and cannot be classified into universality classes. In fact, the experiment really never showed a direct signature of Tc . Assuming that there is a substantial blurring or rounding of the transition, which masks it, fitting the experimental data with scaling assumptions is usually possible, within some error around the rounded “singularity.” This fitting procedure which seeks consistency between the two steps of relaxation also identifies, indirectly, the location of the dynamic-transition point. On the other hand, using the static structure factor of the liquid as an input, the corresponding MCT equations can be solved to obtain the location of the dynamic-transition point. This is a first-principles calculation without any adjustable parameter. However, the transition density thus obtained from the MCT for a simple hard-sphere system does not seem to agree well with the corresponding control-parameter value at which a possible structural arrest is interpreted in the simulations or experiments. This is presumably linked with the low-order perturbative nature of MCT, in which the one-loop model is used to obtain the renormalized transport coefficient. It should be noted here that the two-step relaxation of MCT is different from Goldstein’s picture of two time scales, which was discussed in Section 4.3. The former occurs above Tc , where barrier hopping is not dominating the dynamics, and is related to the initial stages of supercooling. The vibrational and configurational parts of the dynamics depicted in Goldstein’s picture of an energy-landscape description, on the other hand, occur when barrier hopping is strongly influencing the dynamics. The MCT, at least in its present form, is not suitable for understanding the behavior of the deeply supercooled liquid in which the viscosity rises by 14 orders of magnitude (equivalent to time scales of the order of 102 s or so). More recently, phenomenological extensions of MCT have been proposed in order to extend the theory further in this direction (Bhattacharyya et al., 2005). Another related and important issue here is the intervention of the crystallization process, which becomes more likely with increasing supercooling. In the MCT approach depicted above, the liquid dynamics is considered to be in a stationary state not too far below its freezing point and it is

8.3 Ergodicity-restoring mechanisms

411

simply assumed that crystallization does not occur. Note that at Tm the barrier to nucleation is infinite. However, in theoretical analysis for the deeply supercooled state an appropriate model for the dynamics should include the competition with the crystallization process and a logical resolution of the Kauzmann paradox.

8.3 Ergodicity-restoring mechanisms So far we have considered the simple form of the mode-coupling model in which the ergodic–nonergodic (ENE) transition occurs at a critical density or temperature. The static or thermodynamic properties of the system do not undergo any drastic change on passing through the transition. These structural properties merely determine the strength of the mode-coupling effects and hence the critical values of the control parameters for the transition point. The ENE transition is therefore essentially a dynamic transition. However, as discussed above, evidence of such an ENE transition in experiments on glass-forming liquids has been obtained only through indirect means. At best one can only conclude that there is a crossover in the dynamic behavior of the liquid around the mode-coupling Tc . A very pertinent question to ask then is that of whether the sharp ENE transition described in different models described above finally survives in a more careful analysis of the theory. From a theoretical point of view, the occurrence of this sharp dynamic transition in a real liquid having a finite number of participating degrees of freedom is unexpected. This criticism of the ENE transition also holds even for the wave-vector-dependent model. On the other hand, the ENE transition in the self-consistent model is not an artifact of some approximate solution of the dynamics, although this may be the case in certain specific models, e.g., in the case of a nonlinear oscillator (Siggia, 1985). Here the nonlinear equations for the dynamics are linked to an effective potential energy, which is non-Gaussian, and the corresponding ENE transition predicted from the perturbative solutions of the dynamic model is destroyed due to the nature of the equilibrium states. An important point to note here is that the ENE-transition model discussed in the earlier section is obtained with a completely Gaussian interaction free energy for which the possibility of such disordering agents is absent. This is clear from the special form of the nonlinearities in the reversible part of the dynamics of the generalized Navier–Stokes equation (Das et al., 1985a). The crucial issue which needs to be addressed here is that of whether the MCT model really preserves this simple form (with the prediction of an ideal transition) under proper consideration of the equations governing the dynamics of the dense fluid. In the following, we address the question: do the equations of fluctuating hydrodynamics applied to a compressible liquid allow the ENE transition to occur? A careful analysis (Das and Mazenko, 1986) of the nonlinearities in the equations governing the dynamics of compressible liquids shows that ergodicity holds at all densities. This is the case even with the liquid dynamics being controlled by an effective Gaussian potential energy with a single minimum and without any thermally activated hopping.

412

The ergodic–nonergodic transition

8.3.1 Ergodic behavior in the NFH model In order to facilitate the discussion, we note the equivalence of the following asymptotic behaviors of the density correlation function and its Laplace transform: lim G ρρ (t)  = 0,

t→∞

G ρρ (z) ∼ 1/z, G ρρ (ω) ∼ δ(ω).

(8.3.1)

Let us now consider the model equations for the compressible liquid introduced in Section 3.3.1 to study the nature of the asymptotic dynamics. In the nonlinear fluctuatinghydrodynamics formulation discussed above, renormalization of the viscosity is obtained from the self-energy gˆ gˆ and the one-loop contribution to this self-energy involves the product of the density correlation functions. As we have seen in earlier chapters, the integral relation between L(ω) and G ρρ (t) (see eqn. (7.3.63)) gives rise to the nonlinear feedback mechanism which forms the very basis of the self-consistent mode-coupling approach to the glass physics. Therefore an ENE transition is characterized by a persistent time dependence of the density correlation function and this implies a diverging viscosity. This is equivalent to saying that the self-energy gˆ gˆ blows up at small frequencies: gˆ gˆ = −Aδ(ω) + RT,

(8.3.2)

where RT represents terms that are regular in the ω → 0 limit and A is the amplitude of the singular term. In writing the above expression we are not ignoring the wave-vector dependence but suppressing it to keep the notation simple. Let us now enquire whether the above assumption (8.3.2) is compatible with the set of Schwinger–Dyson equations corresponding to the MSR action (7.2.8). On putting eqn. (8.3.2) back into eqn. (7.2.31) we obtain a δ(ω) peak in G ρρ as long as the response function G ρ gˆ is not zero in the ω → 0 limit. This result follows simply on setting both α and β equal to ρ in eqn. (7.2.31). It is straightforward to obtain that the singular contribution of G ρρ comes from gˆ gˆ in the form G ρρ = G ρ gˆ gˆ gˆ G gρ ˆ + RT.

(8.3.3)

For an ENE transition to occur, it is necessary that the response function G ρ gˆ not vanish as ω → 0. The response functions G ψ ψˆ are calculated from the relation (7.2.34), where the Nψ ψˆ are as given in Table 7.2. The response function G ρ gˆ therefore has the form G ρ gˆ =

Nρ gˆ ρL q = , D D

(8.3.4)

where D is given by eqn. (7.2.35). This requires that ρL goes to a nonzero value in the zero-frequency limit and that the determinant D does not blow up as ω → 0. We assume,

8.3 Ergodicity-restoring mechanisms

413

with no reason to expect otherwise, that the ω → 0 limits of ρL , γ , c2 , and L are nonzero. As a result D(ω → 0) is not infinite4 and hence G ρ gˆ  = 0 in the low-frequency limit. From the relation (7.2.31) it also follows that the correlation functions G ρv and G vv have a delta-function contribution coming from gˆ gˆ , provided that G gv ˆ is nonzero in the small-frequency limit. This is the case if the self-energy contribution γ (ω = 0)  = 0. To demonstrate this, we consider the case in which α and β in eqn. (7.2.31) are both equal to the current v, to obtain G Lvv = G v gˆ gˆ gˆ G gv ˆ + RT.

(8.3.5)

Similarly, G ρv is obtained by setting α and β equal to ρ and v, respectively, G Lρv = G ρ gˆ gˆ gˆ G gv ˆ + RT.

(8.3.6)

Using the expressions for Nα βˆ provided in Table 7.2, it follows from eqns. (8.3.5) and (8.3.6), respectively, that the correlation functions G vv and G ρv both have a diverging contribution in the ω → 0 limit if the quantity γ (or equivalently vρ ˆ ) is nonvanishing in the same limit. To summarize, from eqn. (7.2.31) it follows that all three correlation functions G ρρ , G ρv , and G vv exhibit a δ(ω) component, provided that γ is nonzero. On the other hand, the correlation functions involving a momentum index g do not have a delta-function peak at zero frequency. To demonstrate this, we note that if either index α or β on the LHS of (7.2.31) is the momentum density g then the singular contribution due to gˆ gˆ is coupled to the response function G g gˆ . However, from Table 7.2 it follows that G g gˆ =

ρL ω D(ω)

(8.3.7)

vanishes as ω → 0 as long as D(ω = 0)  = 0. Therefore, the correlation functions involving a momentum index g do not have a delta-function peak at zero frequency. Next we consider the implications of the FDT (7.3.29) on the above results. Since G vρ and G vv blow up, it follows that the imaginary parts of the response functions G gρ ˆ and G gv ˆ , respectively, blow up. Considering the explicit form of these response functions from Table 7.2, we obtain Im(ρL D ∗ ) , DD∗ Im{ρL (ω + iq 2 γ )D∗ } = −2β −1 . DD∗

G vρ = −2β −1

(8.3.8)

G vv

(8.3.9)

This implies that we require simultaneously that D∗ D is bounded and that the imaginary parts of both ρL qD∗ and (ω + iq 2 γ )D∗ diverge. But, since both D and D , denoting the real and imaginary parts of D, respectively, are bounded, ρL and γ must diverge in order to make this possible. However, in that case it follows from (7.2.35) that D must also blow up and we have a contradiction. This leads to the obvious conclusion that the 4 It is useful to note that for the z → 0 limit the quantity L(z + iγ q 2 ) in D does not diverge even when L ∼ 1/z becomes large, since Lγ q 2 remains finite in the nonhydrodynamic regime ω ∼ q 2 .

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The ergodic–nonergodic transition

original assumption of a nonergodic phase is not supported in the model. The key selfenergy contribution is therefore vρ ˆ , or equivalently γ . If for some reason this quantity vanishes at zero frequency, then G ρv and G vv vanish as ω goes to zero. Then G ρv and G vv do not have a δ(ω) component and one does not have the constraints on ρL , γ , and D. In this case one may have an ENE transition. The presence of the nonzero self-energy ρ vˆ is therefore crucial and ensures the absence of the ENE transition. This self-energy element is a direct consequence of the presence of 1/ρ nonlinearities in the NFH equations. The nonlinear constraint g = ρv takes care of this nonlinearity and allows construction of the MSR action functional to all orders in perturbation theory. This avoids any truncation of a nonpolynomial action in the perturbation action (Yeo, 2010). It is important to note that the cutoff mechanism results from the nonperturbative treatment of the density nonlinearities and does not violate any time-reversal symmetry. The ergodicity-restoring mechanism which removes the sharp ENE transition has often been referred to in the literature as one signifying hopping processes. Such a description can (somewhat vaguely) be justified as follows. The sharp ENE transition, which is a result of the nonlinear interaction of the density fluctuations, signifies the cage effect for the single-particle motion in the dense system. At higher densities the persistence of the cages grows in time until at a critical density a dynamic transition of the liquid to a nonergodic glassy phase occurs. This is similar to other models for the glass transition, such as the freevolume model (Grest and Cohen, 1981) and the trapping-diffusion model (Odagaki, 1995; Odagaki and Hiwatari, 1990). The process of trapping of a particle in the cage formed by its neighbors can be interpreted as being produced by a static random potential. The ergodicity-restoring mechanism is then viewed as a relaxation via jumps over the almost static potential barriers due to cage formation. However, calling this a “hopping process” is clearly a matter of interpretation and could even be misleading.

8.3.2 The hydrodynamic limit For small q and ω we can further analyze the ergodicity-restoring mechanism described above. The density correlation function, which is the central quantity in the MCT, is computed from the corresponding response function in this case,5 G ρρ (q, ω) = 2β −1 χρρ (q)Im G ρρ ˆ (q, ω),

(8.3.10)

where χρρ (q) is the Fourier transform of the equal-time correlation function. The Laplace transform of the density–density correlation function normalized w.r.t. to its equal-time value is obtained by inverting the G −1 matrix as G ρ ρˆ (q, z) =

zρL + i L . ρL (ω2 − q 2 c2 ) + i L(z + iq 2 γ )

(8.3.11)

5 Note that this relation holds only in the hydrodynamic limit and hence the one-loop expression for the cutoff function

presented in the following is valid only for small q and ω.

8.3 Ergodicity-restoring mechanisms

415

The renormalization of the longitudinal viscosity L(q, z) is obtained from the longitudinal L of the corresponding self-energy matrix  part gv gˆi v j of the isotropic liquid, ˆ L(q, z) = L 0 +

β L  ˆ (q, z). 2 gv

(8.3.12)

As already discussed in the previous section, the absence of the ENE transition is primarily a consequence of the nonzero self-energy vρ ˆ . If we ignore the self-energy vρ ˆ , the expression (8.3.11) is identical to the conventional expression (8.1.31) for the density correlation function. This crucial self-energy element vρ ˆ appears in the renormalized theory as a consequence of the nonlinear constraint (6.2.22) which is introduced in order to deal with the 1/ρ nonlinearity in the hydrodynamic equations. For computing the effect of the ergodicity-restoring processes on the final decay of the density correlation function, we start from (8.3.11) in the following form:  φ(q, z) = z + iδ(q, z) − where we have used the definition



δ(q, z) = γ (q, z)

−1

q2 z + i L(q, z)

i L(q, z) . z + i L(q, z)

,

(8.3.13)

(8.3.14)

In the asymptotic limit, for long enough times or correspondingly for small enough z, when the longitudinal viscosity is large due to the feedback effect from the coupling of density fluctuations we have z L(q, z). The density–density correlation function φ(q, z) given by eqn. (8.3.13) reduces to the form φ(q, z) =

z+i

1

(

c02 L˜ −1

+ γ˜ q 2

).

(8.3.15)

We have used above the leading-order result in q, L(q, z) ∼ q 2 L˜ and γ (q, z) ∼ q 2 γ˜ . If the term involving γ˜ in the denominator of the RHS of eqn. (8.3.15) is ignored, the longest time scale of relaxation for the density correlation function will be given by L˜ −1 . A dynamic transition occurs in this model, with φ(q, z) ∼ 1/z corresponding to the situation in which L˜ tends to diverge beyond a critical density. The enhancement of L˜ occurs due to the feedback effects from slowly decaying density fluctuations. For example, we have seen in Section 8.1.2 that such a situation occurs at a packing fraction ϕ c = 0.525 if the model equations are considered with a hard-sphere structure factor. Owing to the presence of the term γ the 1/z pole in φ is avoided even if L˜ becomes arbitrarily large due to feedback effects. In this case, γ˜ q 2 determines the pole of the density correlation function. However, this is not a new hydrodynamic mode in the system and there is no extra conservation law involved. Since the viscosity remains finite, to leading order the final relaxation of the density correlation is controlled self-consistently by L˜ −1 (Latz and Schmitz, 1996).

416

The ergodic–nonergodic transition

Similar ergodicity-restoring mechanisms removing the sharp ENE transition of the MCT have been discussed by other authors (Götze and Sjögren, 1987; Schmitz et al., 1993; Das, 1996; Chong, 2008). So far our discussion of the cutoff function is qualitative and treats the self-energy vρ ˆ in a nonperturbative manner. For estimating the extent of slowing down due to the density feedback mechanism from the theoretical model, a quantitative approach is needed. This is done by evaluating the relevant self-energies at one-loop order. The evaluation of the contribution to γ˜ at one-loop level is outlined in Appendix A8.3. In doing this we make use of the relation (7.3.55), which follows from the FDT (7.3.29) in the hydrodynamic limit. Thus γ is computed from the self-energy vˆ vˆ . The one-loop expression that we use for the cutoff function is valid only in the hydrodynamic limit in which the relation (7.3.55) holds. The asymptotic behavior of the density correlation functions φ(q, t) is obtained from the solution of eqn. (8.3.11) retaining the cutoff function γ (q, z). Both these equations are combined in time and space in the form of a coupled set of integrodifferential equations (Das, 1987, 1990). With this extended model a qualitative change in the dynamics is observed around the ideal transition point in calculations. A similar behavior in experimental data obtained from the study of a large number of liquids classified as fragile liquids was reported (Taborek et al., 1986). Finally, the above discussion on the ergodicity-restoring mechanism applies only to models involving both mass and momentum densities. What about the DDFT models, which are formulated in terms of density fluctuations only? The analysis presented in Section 8.1.6 demonstrates that the DDFT model at one-loop level gives rise to the same dynamic transition as that of the fluctuating-hydrodynamics model. It is important to stress here that the Kawasaki rearrangement (see Section A8.2) applied for this analysis is ad hoc and can give rise to contradictory results if one extends it to two-loop order (Mazenko, 2008). In the NFH description the inclusion of the 1/ρ nonlinearities or the current gives rise to a theory that is generally inconsistent with an ENE transition. The 1/ρ nonlinearity is absent in the DDFT model. As we have seen above, by integrating out the momentum variable from the NFH equations and applying what is termed the adiabatic approximation, the fluctuating equation for the single density variable can be obtained. The 1/ρ nonlinearities play a very nontrivial role in reaching the final form of the stochastic equation for ρ in DDFT. As the momentum variable is integrated out, the noise, which is of additive nature in the NFH case, becomes multiplicative in the stochastic nonlinear equation for the density ρ. The full nature of the dynamics which follows from the DDFT equation with multiplicative noise and, in particular, the issue of restoring the ergodic behavior in this case still remain unclear. Even the very feasibility of such a scheme is not evident, since in the DDFT model the dynamics is formulated in terms of density fluctuations only and the momentum density has been integrated out of the problem at an early stage. It would also be useful to note in this regard that, in computer simulations with Newtonian dynamics or Brownian dynamics of the particles, an ENE transition to the nonergodic phase has never been observed. It is therefore natural to expect that ergodic behavior will be restored in the DDFT model finally.

8.3 Ergodicity-restoring mechanisms

417

Let us summarize the discussion above. The standard MCT with a sharp dynamic transition is obtained when the cutoff function is ignored. In the full theory or the so-called extended MCT the ENE transition is removed. We have shown here that this result is nonperturbative and is reached without using the hydrodynamic limit. However, the explicit one-loop expression for the cutoff function (of the ergodicityrestoring mechanism) in terms of correlation functions has been reached only in the hydrodynamic limit. The origin of this ergodic behavior has been shown to be the 1/ρ nonlinearities appearing in the equations of motion. An important point in this respect is the validity of the basic equations for describing compressible liquids. We have discussed in the previous chapters how these equations, which are taken as plausible generalizations of the hydrodynamics, are obtained. This involves the extension of the NFH description to large wave vector or short wavelengths and these equations constitute the primary tools available for studying the dynamics of a many-particle system. It is important to note in this respect that, be it through the so-called memory-function formalism or kinetic-theory approaches, a renormalized theory for self-consistent MCT for all wave vectors remains elusive. Phenomenological extensions of the mode-coupling theory have been made by extending the set of slow modes by including transverse modes and defect densities (Kim, 1992; Das and Schilling, 1994; Yeo and Mazenko, 1995) or order parameters for structural relaxation (Liu and Oppenheim, 1997). A viable mechanism for understanding the boson peak has also been constructed (Das and Schilling, 1999; Götze and Mayr, 2000; Cang et al., 2005) using mode-coupling models involving transverse sound modes in the presence of defects in a disordered solid (see also Kojima and Novikov (1996)). Evidence for the role of such vibrational modes has indeed been found in recent computer experiments on glass-forming liquids (Shintani and Tanaka, 2008).

8.3.3 Numerical solution of NFH equations The equations of fluctuating nonlinear hydrodynamics we have discussed in detail in the earlier chapters give rise to the mode-coupling model. However, the standard MCT model is a result obtained by considering the renormalized theory for the nonlinear dynamics only up to one-loop order. Thus one is considering only to first order in perturbation theory an effect that is becoming large as the dynamic transition is approached. Hence, while the conclusions reached can be qualitatively correct, quantitatively definite results are unlikely to be obtained. The MCT for the structural relaxation has not been extended to consider renormalization to higher order in a controlled manner. Indeed, this goes back to the usual problem of dense-liquid-state theory where no suitable small parameter exists for constructing a quantitatively accurate theory. The NFH equations have also been solved numerically (Lust et al., 1993; Sen Gupta et al., 2008) to obtain results from a nonperturbative approach. Stable numerical solutions for the density field on a lattice of finite size have been obtained with a local coarse graining of the fluctuating density field. The

418

The ergodic–nonergodic transition

normalized dynamic correlation function of density fluctuations δρ at times tw and t + tw is defined as C(t + tw , tw ) =

δn(t + tw )δn(tw ) . δn(tw )δn(tw )

(8.3.16)

For large waiting time tw , the system equilibrates and time translational invariance holds. In this case C(t + tw , tw ) ≡ C(t). The decay of C(qm , t) is in agreement with the corresponding MD-simulation results of Ullo and Yip (1985) for a one-component fluid interacting through the purely repulsive cut Lennard-Jones potential. The input c(r ) used in the numerical solution of the fluctuating-hydrodynamics equations is taken from the standard integral-equation solutions for the structure of a uniform liquid interacting through the same cut Lennard-Jones potential. The bare transport coefficients which determine the correlations of Gaussian noise in the NFH equations are such that the dynamics agrees with computer-simulation results at short times. The agreement between the correlation functions obtained from the MD simulations (Ullo and Yip, 1985) and those from numerical solution of the NFH equations improves considerably at higher density. This reflects the fact that at higher densities the mean free path of the fluid particles gets smaller and approaches the atomic length scale. As a result, the validity of generalized hydrodynamic equations at short length scales (corresponding to wave vector q ∼ qm ) improves with increasing density. The four-point correlation function χ4 (km , t) at the structure-factor peak is defined as χ4 (km , t) =

ˆ ˆ m , t) F(−k  F(k m , t) ˆ m , 0)|2 | F(k

,

(8.3.17)

ˆ m , t) = δρ(km , t)δρ(km , 0) shows a peak near the α-relaxation time scale (see where F(k Fig. 8.4). The peak height and position grow with supercooling roughly in same the manner as that in which the α-relaxation grows. This four-point function in fact acts as a signature of the departure from Gaussian nature of collective density fluctuations. Note that a similar behavior is seen at the single-particle level from the study of the so-called non-Gaussian parameter (see Figs. 4.12(a) and 8.4). Therefore the NFH equations (which give rise to the MCT) do indeed conform to the behavior reflecting the dynamical heterogeneities. The numerical solution of the NFH equations for a one-component Lennard-Jones system does not display many features of the two-step relaxation process. The various relaxation regimes predicted in the simple MCT model are distinct only when the different characteristic time scales of the dynamics are separated and do not overlap. Indeed, as is clear from our discussion above, the asymptotic behavior of the density correlation function strongly depends on how small the cutoff function γ is. The smaller γ is, the longer is the time scale over which the feedback mechanism is effective in enhancing the transport coefficient. For small enough γ , the time scale of final decay is widely separated from the two-step relaxation of simple MCT described in Sections 8.1.1 and 8.1.2. On the other hand, if γ does not become small enough self-consistently, a lot of the relaxation scenarios

8.4 Spin-glass models

419

that are predicted in the simple MCT will be screened out (Liu and Oppenheim, 1997), since the system does not get close enough to the dynamic transition point. 8.4 Spin-glass models We introduce in this section models of supercooled liquids formulated in terms of models developed for disordered systems referred to as spin glasses (Binder and Young, 1986; Mézard et al., 1987; Fischer and Hertz, 1991; Binder and Kob, 2005). In particular, we focus our discussion on the aspects of the spin-glass dynamics which are related to the feedback mechanism of the mode-coupling theory and the associated ENE transition discussed earlier in this chapter. It is useful to note at the outset a few points about the two types of disordered system, namely the structural and spin glasses, between which the analogy is being made here. Indeed, both represent systems without any long-range order. In each case the difference between the glassy state and the corresponding high-temperature disordered state, i.e., the normal liquid and the paramagnet, respectively, is in terms of the time-dependent behavior. However, there are obvious differences between these systems as well. In the structural glass the most stable state having the lowest free energy is the crystalline state with long-range order, whereas the supercooled liquid is metastable with respect to the crystal. No such state exists corresponding to the disordered spin-glass phase. Furthermore, in the spin glass the disorder is “quenched,” depicting the situation in which the exchange-interaction coupling constants of the spins are random variables and are time-independent on all experimental time scales. On the other hand, in the structural glass the randomness is self-generated. It is important to note in this regard that spinglass-like behavior has been seen also in systems without any intrinsic disorder (Bouchaud and Mézard, 1994; Marinari et al., 1994a, 1994b; Chandra et al., 1995). The presence of frustration in such systems is crucial in order to produce the spin-glass state rather than quenched randomness (Bouchaud et al., 1997; Cugliandolo, 2003). Indeed, systems with and without quenched disorder have similar behavior as long as a large number of uncorrelated metastable states are associated. Another point of difference between the two types of disordered systems lies in the nature of the transition from the high-temperature state to the corresponding glassy state in the two cases. In the case of a spin glass a second-order phase transition (in zero magnetic field) characterized by a diverging relaxation time and nonlinear magnetic susceptibility has been seen in experiments (Mydosh, 1993; Binder and Young, 1986). The existence of an ideal thermodynamic structural glass transition, on the other hand, remains speculative. This issue of a thermodynamic phase transition will be discussed further in Chapter 10. We first discuss the scenario of the dynamic feedback mechanism of the self-consistent MCT in the mean-field spin models. Here we will consider the dynamics around the equilibrium state similarly to the MCT for the supercooled liquids, which is almost exclusively done for fluctuations around the equilibrium state. Time translation invariance (TTI) holds in such a situation, and the two-point correlation function is dependent only on the time difference between the points at which the fluctuations are considered. In recent years

420

The ergodic–nonergodic transition

good progress has been made in our understanding of the nonequilibrium problem through spin models that reduce to a similar form (at least in terms of the mathematical structure) to the corresponding MCT for supercooled liquids. These nonequilibrium aspects will be considered in Chapter 9.

8.4.1 The p-spin interaction model The connection between structural and spin glasses is proposed (Kirkpatrick and Thirumalai, 1987a, 1987b) through the p-spin ( p > 2) interaction Hamiltonian. These theories show that the dynamical equations for the spin fluctuations in the soft version of the mean-field p-spin interaction model are similar to the self-consistent mode-coupling equations for the liquids. Unlike the Ising-type spin glasses, p-spin models lack reflection symmetry, and this is somewhat reminiscent of the case of real liquids. The spin– spin interaction Hamiltonian with quenched disorder and involving interaction of p spins ( p > 2) is of the form  HJ = − Ji1 ···i p σi1 · · · σi p . (8.4.1) i 1
The above form of the Hamiltonian needs to be regularized with a constraint to include all possible values of the random variable Ji1 ···i p . The regularization used in the soft-spin version (8.4.2) is done by including the function H˜ (σ ) with a σ 4 term. The Hamiltonian of the mean-field p-spin spin-glass model takes the following soft-spin version: H = HJ +

N  

H˜ (σi ) − h i σi ,

(8.4.2)

i=1

where the part H˜ of the Hamiltonian is given by  r¯0 σ 2 + uσi4 . H˜ (σ ) = 2 i

(8.4.3)

i

The couplings {Ji1 ···i p } are independent random interactions whose probability distribution function is given by   p−1 1/2 −Ji21 ···i p N p−1 N exp . (8.4.4) P(Ji1 ···i p ) = π p! p! Ji1 ···i p therefore has zero mean and variance (Ji1 ···i p )2 = p!/(2N p−1 ). The dependence of the distribution on N in (8.4.4) is chosen such that the relative fluctuation in energy in the limit N → ∞ is vanishing, making the thermodynamic limit well behaved. The length of the soft spin σi is allowed to vary continuously from −∞ to +∞. This is a mean-field model since all spins interact with all other spins. The strength of the couplings does not decay with distance and hence there is no role of an underlying lattice. These are termed fully connected or infinite-dimensional mean-field models. Such models with

8.4 Spin-glass models

421

random bonds are frustrated since it is impossible to satisfy all bonds at the same time. The dynamics for σi (t) is given by the Langevin equation, νk−1

∂ δH σi (t) = − + ςi (t), ∂t δσi (t)

(8.4.5)

where νk is a bare kinetic coefficient representing the microscopic time scales for the dynamics. It is related to the variance of the Gaussian random noise by ςi (t)ς j (t  ) = 2νk−1 δi j δ(t − t  ).

(8.4.6)

For studying the dynamics, the physical quantity of interest is the two-spin correlation function Ci j (t − t  ) = σi (t)σ j (t  ).

(8.4.7)

The linear response function Ri j to an external field h j , on the other hand, is defined as Ri j (t − t  ) =

∂ σi (t) ∂h j (t  )

for t > t  .

(8.4.8)

From the equation of motion (8.4.5) it is clear that the physical field h i which couples to the spin σi in the Hamiltonian H also appears as a driving field in the equation of motion. The latter couples to the conjugate MSR field σˆ i in the action. Hence the physical response function is identical to the MSR function R defined above.6 Following the MSR approach outlined above, the value of the functional A[σ ] of the field σ which satisfies the field equation (8.4.5) is obtained as  A[σ ] = Dσ  A[σ  ]δ(σ − σ  )    δβ H −1 ∂σi (t) ≡ Dσ A[σ ]J [σ, ς ]δ νk + − ςi (t) ∂t δσi (t)  3 4    δH −1 ∂σi = Dσ D σˆ A[σ ]J [σ, ς ]exp − dt i σˆ i νk − ςi + . ∂t δσi i

(8.4.9) The functional Jacobian J , which is defined as * * * δςi (t) * *, * J = det * δσ (t) *

(8.4.10)

j

can be evaluated in the thermodynamic limit as *

* * δ δ H ** −1 ∂ * + ν J = det * δσ j (t) k ∂t  δσi * *  * * * 2H ∂ δ *  * δ(t − t ) = det *δi j νk−1  δ(t − t  ) + *. * * ∂t δσi2 6 See the discussion in Section 7.3.1 with respect to the fluctuation–dissipation theorem in MSR theory.

(8.4.11)

422

The ergodic–nonergodic transition

The Jacobian is added as an extra term in the action A J ,   δ2 H A J = −lnJ = G 0 (0) dt νk , 2 (t) δσ i i

(8.4.12)

where G 0 is the Green function satisfying ∂  δ(t − t  ) = G −1 0 (t − t ). ∂t 

(8.4.13)

We first note from (8.4.13) δ(t − t  ) = −



t −∞

¯ d t¯ G −1 0 (t − t )

  ¯ ¯ = − d t¯ G −1 0 (t − t )θ (t − t )   ¯ ¯ ≡ d t¯ G −1 0 (t − t )G 0 (t − t ),

(8.4.14)

where θ represents the step function. Thus G 0 (t) = −θ (−t) satisfies the equation ∂ G 0 (t − t  ) = δ(t − t  ). ∂t The RHS of eqn. (8.4.11) is then obtained as, *  * * δ 2 H ** * −1 −1 J = det *δi j νk G 0 1 − G 0 νk *. * δσi2 *

(8.4.15)

(8.4.16)

Using the result for (8.4.16), A J follows easily (up to a constant term in the action):   δ2 H 1 νk dt , (8.4.17) A J = −lnJ = 2 δσi2 (t) i where we have taken the value θ (0) = 12 for the step function (Zinn-Justin, 2002). Note that the contribution from the counter-term in A J is independent of the random interaction Ji1 ···i p . By averaging expression (8.4.9) over the noise ς we obtain the result    ˆ − AJ , (8.4.18) A[σ ] = Dσ D σˆ A[σ ]exp −A Ji1 ···i p [u, u] ˆ being given by with the action A Ji1 ···i p [u, u] 3 4

  ∂σi δH ˆ = dt + σˆ i (t)νk−1 σˆ i (t) + i σˆ i (t) νk−1 A Ji1 ···i p [u, u] ∂t δσi i   + dt [σi u i + i σˆ i uˆ i ]. (8.4.19) i

8.4 Spin-glass models

423

The average in eqn. (8.4.19) is therefore computed in terms of a generating functional   # $ ˆ = Dσ D σˆ exp −A Ji1 ···i p [u, u] ˆ − AJ . (8.4.20) Z Ji1 ···i p [u, u] For all values of the random variable Ji1 ···i p we have the corresponding generating function satisfying Z Ji1 ···i p [u i = 0, uˆ i = 0] = I0 ,

(8.4.21)

where I0 is a constant obtained after doing the trivial Gaussian integrals. The time correlation of the spins averaged over the quenched random interactions is calculated using the MSR field theory outlined earlier in Section 7.1. From the condition (8.4.21) it follows that the disorder-averaged correlation functions can be obtained by averaging Z Ji1 ···i p itself over the random distribution (8.4.4). Since the random variable Ji1 ...i p is linear in the equation of motion (see eqn. (8.4.19)), the corresponding MSR action functional is averaged out (De Dominicis, 1978; Ma and Rudnick, 1978; Kirkpatrick and Thirumalai, 1987a, 1987b) with respect to the Gaussian probability distribution for the Ji1 ...i p . The quenched averaging is done directly on Z , giving  % ¯ d Ji1 ...i p P[Ji1 ,i2 ,...,i p ]Z Ji1 ...i p [u, u] ˆ Z [u, u] ˆ =  ≡

Ji 1





D σˆ exp −A[σ, σˆ ] −

Dσ



  [σi u i + σˆ i uˆ i ] . dt i

Using the standard field-theoretic techniques of Section 7.1 and assuming that Ji1 ...i p is symmetric under exchange of the site indices, the action of the MSR theory is obtained in terms of the spin σ and the corresponding conjugate field σˆ as 3 4

  −1 −1 ∂σi 3 A[σ, σˆ ] = dt + r¯0 σi + 4uσi − h i σˆ i νk σˆ i + i σˆ i νk ∂t i

+ A J + V [σ, σˆ ],

(8.4.22)

where the part V involving nonlinearities of cubic and higher order is obtained as    μ V (σ, σˆ ) = dt dt  i σˆ i (t)σ j (t) 2N p−1 i, j,i 3 ,...,i p   × i σˆ i (t  )σ j (t  ) + ( p − 1)i σˆ j (t  )σi (t  ) σi3 (t)σi3 (t  ) . . . σi p (t)σi3 (t  ), (8.4.23) with μ = pβ 2 /2. The integration over the random interactions generates the 2 p-spin couplings (nonlocal in time) in the action functional.

424

The ergodic–nonergodic transition

Using a generalization (Kirkpatrick and Thirumalai, 1987a, 1987b) of the saddle-point method (Gross and Mézard, 1984; Gardner, 1985) yields the following equation of motion for the spin σi (t) averaged over the quenched random interactions:  t ∂ (t, t¯)σi (t¯)d t¯ = f i (t) + h i (t). (8.4.24) νk−1 σi (t) + r¯0 σi + 4uσi3 (t) + ∂t o The correlation of the renormalized noise f i in the new Langevin equation is obtained in terms of a self-consistent expression involving the correlation function C,  f i (t) f j (t  ) = δi j (t − t  ) = 2νk−1 δ(t − t  ) + μC p−1 (t − t  ).

(8.4.25)

The last term on the LHS of the renormalized equation (8.4.24) for the dynamics represents a memory term involving the kernel . The latter is expressed self-consistently in terms of the correlation and response functions as (t − t  ) = μ( p − 1)R(t − t  )C p−2 (t − t  ).

(8.4.26)

In eqns. (8.4.25) and (8.4.26) the linear response function R and the correlation function C are defined as C(t, t  ) =

N 1  σi (t)σi (t  ), N i=1

R(t, t  ) =

. N N 1  δσi (t) 1  σi (t)i σˆ i (t  ), = N δςi (t  ) N i=1

(8.4.27)

i=1

with the overbar indicating the average over the quenched randomness and the angular brackets in the expression above stand for the average over the noise ς. Note that the renormalizing contributions both in eqn. (8.4.25) and in eqn. (8.4.26) result from the non-Gaussian part of the action V (σ, σˆ ). The first term on the RHS of eqn. (8.4.23) renormalizes the bare kinetic coefficient νk which determines the correlation of bare noise ς, while the second term contributes to the memory term . With p = 2 this model would also correspond to the equations obtained by Sompolinsky and Zippelius (1982). The above expressions for the kernel functions  and  are related in equilibrium. For fluctuations around equilibrium TTI holds and the correlation and response functions are related through the fluctuation–dissipation theorem (FDT) ∂ (8.4.28) C(t − t  ). ∂t From the above equations, it follows quite straightforwardly that the correlation function C(t) normalized with respect to its equal-time value, i.e., φ(t) = C(t)/C(t = 0), satisfies the dynamical equation  t ˙ + φ(t) + λ p ˙ d0 φ(t) ds φ p−1 φ(s) = 0, (8.4.29) R(t − t  ) = −θ (t − t  )

0

8.4 Spin-glass models

425

where d0 = (¯r0 νk )−1 and λ p = μ¯r0 with the equal-time correlation being given by C(t = 0) = r¯0−1 . For p = 3, eqn. (8.4.29) is identical to the mode-coupling eqn. (8.1.8) (apart from an inertial term involving second derivatives with respect to time) of the structural-glass problem with the memory function H = c2 φ 2 . Keeping both p = 2 and p = 3 gives rise to a model similar to the φ12 model discussed in Section 8.1.1. We have demonstrated earlier in this chapter that these models undergo a dynamic transition at a low enough temperature T = Tc . The corresponding behavior of the normalized density correlation φ(t) is given by 2− p

φ(t) =

p−2 + At −a . p−1

(8.4.30)

In the long-time limit φ(t) goes to a nonzero value. The power-law exponent a satisfies (for p = 2) the equation 1  2 (1 − a) = (1 − 2a) 2

(8.4.31)

in terms of the  function. This is the same as the equation satisfied by the power-law exponent in the Leutheusser model with solution a = 0.395. The renormalized kinetic coefficient is  ∞ eiωt φ p−1 (t). (8.4.32) dR (ω) = d0 + λ p o

In the zero-frequency limit this satisfies dR (T → Tc ) ∼ (T − Tc )−γ

(8.4.33)

with γ = 1.765. The similarity of these results to those of MCT has led to the speculation that this model is in the same universality class as the structural glass transition along the lines of ideas introduced in connection with the second-order phase-transition point (Ma, 1976). It is worth noting here that the dynamic transition predicted for the p-spin models is removed in finite dimensions due to nucleation processes or activation over finite freeenergy barriers (Kirkpatrick and Thirumalai, 1989; Parisi et al., 1999; Drossel et al., 2000). This is similar to what happens in MCT of structural glass, where finally the ergodicity is maintained and the dynamic transition is absent. Kirkpatrick and Thirumalai (1987a) have argued from consideration of the statics of the p-spin models that at a temperature Ts < Td there is a thermodynamic transition. The static transition was identified with the one-step replica-symmetry-breaking transition. Another standard choice for regularization of the model preventing σi from exploding in an unstable direction of the random coupling Ji1 ...i p of the Hamiltonian would be to impose a constraint N 1  2 σi (t) = 1, N i=1

(8.4.34)

426

The ergodic–nonergodic transition

which also makes C(t, t) = 1. For the p-spin Hamiltonian this constraint gives what is called the spherical p-spin model (Crisanti and Sommers, 1992; Crisanti et al., 1993).

8.4.2 MCT and mean-field theories In spite of the similar mathematical structure of the MCT for structural glass and the meanfield p-spin glass models, there are some obvious differences. A subtle point to note here is that the natures of the nonlinearities in the corresponding dynamical equations giving rise to the mode-coupling models in the respective cases of structural and spin glasses are quite different. For the structural glasses, we have seen in the earlier sections of the present chapter that the driving nonlinearities are in the reversible part of the dynamics. They are generated from a Gaussian Hamiltonian and are essentially of dynamic origin. This is due to the special Poisson-bracket structure of the relevant hydrodynamic variables contributing to the reversible part of the equations of motion (Das et al., 1985b). In the infinite-range p-spin models, on the other hand, the relevant nonlinearities are dissipative and come from the non-Gaussian part of the effective Hamiltonian. The analogy of the structural glasses with the infinite p-spin models was extended by Kirkpatrick and Thirumalai (1989) to propose for the structural case a density-functional form of the free-energy functional in units of kB T ,   1 FST = dx1 dx2 δρ(x1 )χ0−1 (x1 − x2 )δρ(x1 ) 2   1 1 + g3 dx1 [δρ(x1 )]3 + g4 dx1 [δρ(x1 )]4 3 4  − μ dx1 ρ(x1 ), (8.4.35) where μ is the chemical potential. The constants g3 and g4 are coupling constants in the non-Gaussian free energy. These terms are related to three- and four-point correlations in the fluid. The equation of motion for the time dependence of the density fluctuations δρ(x, t) is chosen in the form of a relaxational dynamics for the conserved quantity (density), 0−1

∂ δFST [δρ] δρ(x, t) = −∇ 2 + ξ(x, t), ∂t δρ(x, t)

(8.4.36)

which is similar to the form of eqn. (8.4.5) in the p-spin model. ξ is the Gaussian white noise. The mode-coupling model constructed for the above dynamics shows a transition similar to the ENE transition described in the case of the p-spin model. On the other hand, multiple solutions of the density-functional theory appear and each metastable solution is given a canonical weight P[δρ] for the corresponding density configuration. A static order parameter is defined in terms of the average of the equal-time density correlation Q(x1 , x2 ) = δρ(x1 )δρ(x2 ),

(8.4.37)

8.4 Spin-glass models

427

where the angular brackets denote the average with weight P[δρ]. It was shown that Q(k) satisfies the nonlinear integral equation

I † (k) , (8.4.38) Q(k) = χ0 (k) 1 + I † (k) where

 I † (k) = 2g3

dk  Q(k − k  )Q(k  ).

From the mode-coupling equations corresponding to the nonlinear Langevin equations given by (8.4.35) and (8.4.36) it was shown (Kirkpatrick and Thirumalai, 1989) that the nonergodicity parameter qEA (k), i.e., the long-time limit of the density–density correlation function δρ(k, t)δρ(−k, 0), satisfies the same integral equation, (8.4.38), as Q(k) does. Both these order parameters are proportional to the lowest-order non-Gaussian coupling, i.e., g3 . The equation of motion for the density fluctuations here (eqn. (8.4.36)) is very different from those considered in the previous sections and is rather similar to what has been considered for the mean-field spin models. The possibility of a sharp ENE transition in the various mode-coupling models must be viewed carefully. The multi-spin model involves an infinite-dimensional mean-field approach since every spin is interacting equally with every other spin. The existence of the dynamic transition in this case is generally associated with the appearance of many metastable minima (Thirumalai et al., 1989). Ergodicity breaks when the system gets trapped in one minimum for infinitely long times. Since the barrier between the metastable states is infinite, reaching the true equilibrium state is forbidden. In analogy with the p-spin model, the mode-coupling model for a real liquid (with short-range forces) is often termed a mean-field theory. If we consider only the schematic form of the MCT for the structuralglass case, with the one-dimensional nonlinear integral equation then such an analogy is possibly justified. However, in the structural case the uncorrelated binary-collision contributions to the transport coefficients are already included in the bare or short-time part and the mode-coupling contribution in fact represents a correction taking into account correlated motions in the many-particle system. Although these are accounted for only at the level of the Kawasaki approximation or one-loop approximation in the simple MCT, it is still a correction beyond the uncorrelated dynamics. Another important aspect of this analogy between mean-field spin models and MCT of structural glasses is the often-cited reference to activated hopping processes. Indeed, the dynamic transition predicted for the p-spin models is removed in finite dimensions due to nucleation processes or activation over finite free-energy barriers (Kirkpatrick and Thirumalai, 1989; Parisi et al., 1999; Drossel et al., 2000). In analogy with p-spin models, if we term MCT a mean-field theory where the system is caught in a single free-energy minimum, then restoration of the ergodic behavior will come from activated jumps over the barriers. Hence the absence of the sharp ENE transition in a supercooled liquid (which is also a fact supported by results from experiments and simulations) is often ascribed to “hopping processes” or activated hopping over energy barriers. However, as we have

428

The ergodic–nonergodic transition

already seen in Section 8.3, the origin of the restoration of ergodicity in the various models of liquid-state dynamics (Das and Mazenko, 1986; Schmitz et al., 1993) arises from a simple Hamiltonian, without inclusion of hopping processes. Finally, let us consider a toy model (Kawasaki and Kim, 2001) involving an N -component fluid consisting of N density variables ρi , i = 1, 2, . . . ,N and M momentum variables gα , α = 1, 2, . . . , M with (M < N ). This model is characterized by a built-in quenched disorder. This has been done in the spirit of an earlier description of fluids due to Kraichnan (1959). The equations of motion for the density and momentum variables are written in the form ω (8.4.39) ρ˙i = K iα gα + √ Ji jα ρ j gα , N ω (8.4.40) g˙ α = −ω2 ρ j K jα − √ Ji jα {ω2 ρi ρ j − kB T δi j } − γ0 bα + f α , N with the noise correlation given by  f α (t) f β (t  ) = 2kB T γ0 δ(t − t  ).

(8.4.41)

ω, γ0 , and kB T are constants for the model. The above equations follow from the formalism developed in Chapter 7 for obtaining the nonlinear Langevin equations. To construct the equations we define the nonzero Poisson brackets between the density and momentum variables as

ω (8.4.42) {ρi , gα } = − K iα + √ Ji jα ρ j , N where Ji jα denote the quenched variables assumed to follow a Gaussian distribution. The driving free-energy functional for the stationary solution of the Fokker–Planck equation corresponding to the above Langevin equation is of the form exp[−F/(kB T )] with the free-energy functional given by M  N 1  2  2 2 gα + ω ρi . (8.4.43) F [{ρi }, {gα }] = 2 α=1

i=1

The above nonlinear Langevin equations (8.4.39) and (8.4.40) follow directly from eqns. (6.1.48) and (6.1.71). The nonlinear terms (quadratic in the fields) in the dynamical equations are proportional (by construction) to the quenched variables J . In the present model an important difference from the dissipative dynamics of the earlier examples of p-spin models (Section 8.4.1) and the DFT model (Section 8.4.2) is that the nonlinearities in eqns. (8.4.39) and (8.4.40) are all in the reversible part of the dynamics, as can easily be seen by doing a time reversal in the above equations. From the analysis of this model it was shown (Kawasaki and Kim, 2001) that for the case of N = M no sharp transition is seen in this toy model, even in the limit N → ∞. The ideal glass transition is recovered in this model only when the number N of density variables is chosen to be different from the number of momentum components (M) (with N > M). In this case δ = M/N < 1, which restricts the available phase space and presumably facilitates the sharp dynamic transition.

8.4 Spin-glass models

429

The ergodicity-restoring process in the present case also does not involve any hopping processes. One of the basic ingredients which are common to these models (in which the sharp dynamic transition is cut off) is that they all have momentum fluctuations taken into account in the fluctuating-hydrodynamics description. It is instructive in this respect to consider the explicit form of the Hamiltonian in the two cases. In the p-spin case every spin interacts with every other spin, as a result of which the resulting model is of infinite-dimensional mean-field type. In comparison, the liquid or the structural glass is a finite-dimensional system, since the interaction part of its Hamiltonian 5 is usually a sum of short-ranged two-body interactions, Hliq = iN= j = 1 U (|Ri − R j |), where Ri represents the position of the ith particle. At low temperatures the individual particles merely vibrate around their mean positions on a random lattice structure, which corresponds to a local minimum of the Hamiltonian. The disorder is self-generated in the amorphous solid-like state since each particle experiences a different environment. By defining vibrations around an amorphous lattice Hliq can be mapped to a multi-spin interaction-type Hamiltonian (Kühn and Horstmann, 1997; Singh and Das, 2009) involving soft spins. Imposing the frozen structure automatically implies the system being caught in a single minimum and that it is of mean-field nature. Those subtle differences notwithstanding, the analogy of the p-spin interaction spin models with the MCT for structural glasses has led to the assumption that there is a dynamic similarity between the two types of systems with and without intrinsic disorder. The landscape studies of prototype fragile liquids have further strengthened the analogy between structural and p-spin models. Similar observations have been made in different studies. For example, a finite energy-density interval where local minima dominate overwhelmingly in number over saddles occurs in mean-field models (Cavagna et al., 1998), which is similar to the case of simple atomic systems. The analogy is also extended to the nonequilibrium dynamics. The off-equilibrium solutions of the spherical p-spin model (discussed in the next chapter) agree well with the MD-simulation data on the nonequilibrium state of the structural glass. As we will see from this point onwards in the remaining chapters of this book, the analogy of the mean-field spin-glass models with the molecular systems has been often appealed to in order to aid understanding of the behavior of the supercooled liquids. This ranges from nonequilibrium dynamics (discussed in Chapter 9) to the scenario of an underlying phase transition at low temperatures (Chapter 10).

Appendix to Chapter 8

A8.1 Calculation of the spring constant Let us consider the density function in the form of a sum of Gaussian profiles around a set of lattice points {Rα }, n(x) =

 3/2  N  α 2 e−α(x−Ri ) ≡ φα (|x − Ri |), π

(A8.1.1)

i=1

Ri

where φi (r ) = (α/π)3/2 exp(−αr 2 ). We want to compute here the response due to a displacement δR1 of a single lattice site R1 (say). We write the density as n(x) = φα (|x − R1 − δR1 |) +

N 

φα (|x − Ri |)

i=2

= φ˜ α (|x − R1 ) +

N 

φα (|x − Ri |),

(A8.1.2)

i=2

where we have introduced the notation φ˜ α (|x − R1 |) ≡ φα (|x − R1 − δR1 |). The ideal-gas part of the free energy remains the same under this displacement. However, the interacting part is changed. We compute this as   1  dx dx n(x)c(|x − x |)n(x ) β Fin = − 2     N  1  ˜ φα (|x − Ri |) dx dx φα (|x − R1 |) + =− 2 i=2   N     φα (|x − Ri |) × c(|x − x |) φ˜ α (|x − R1 |) + 1 =− 2 



 dx

i=2 



dx c(|x − x |)

× φ˜ α (|x − R1 |)φ˜ α (|x − R1 |) + φ˜ α (|x − R1 |)

N  i=2

430

φα (|x − Ri |)

A8.1 Calculation of the spring constant

+ φ˜ α (|x − R1 |)

N 

φα (|x − Ri |) +

i=2

≡

4 

N N  

431

⎤ φα (|x − Ri |)φα (|x − R j |)⎦

i=2 j=2

Ii .

(A8.1.3)

i−1

With an interchange of the integrated variables x and x  , it follows that the second and third integrals, denoted by I2 and I3 , respectively, are the same. Using (A8.1.2), the sum of these two integrals is written as  I2 + I3 = −

 dx

! " dx c(|x − x |)φ˜ α (|x − R1 |) n(x ) − φα (|x − R1 |) ,

(A8.1.4)

and   1 dx dx [n(x) − φα (|x − R1 |)]c(|x − x |) 2 !  " × n(x ) − φα (|x − R1 |) ,   = − dx dx c(|x − x |)

I4 = −

! × n(x)n(x ) − φα (|x − R1 |)n(x )

"

− φα (|x − R1 |)n(x) + φα (|x − R1 |)φα (|x − R1 |)   = Fin + dx dx c(|x − x |)

1  × φα (|x − R1 |)n(x ) − φα (|x − R1 |)φα (|x − R1 |) . 2



(A8.1.5)

On adding the four integrals, we obtain the result  β Fin =

 dx

dx c(|x − x |)

 # × n(x ) φα (|x − R1 |) − φ˜ α (|x − R1 |) +  =

1 φ˜ α (|x − R1 |)φα (|x − R1 |) + φ˜ α (|x − R1 |)φα (|x − R1 |) 2 

dx

− φ˜ α (|x − R1 |)φ˜ α (|x − R1 |) − φα (|x − R1 |)φα (|x − R1 |) dx c(|x − x |)

 # × n(x ) φα (|x − R1 |) − φ˜ α (|x − R1 |)

$

432

Appendix to Chapter 8

1

φ˜ α (|x − R1 |) − φα (|x − R1 |) 2 $  × φ˜ α (|x − R1 |) − φα (|x − R1 |) . +

(A8.1.6)

We now expand the difference of the Gaussian functions as a Taylor-series expansion in δR1 in the form φ˜ α (|x − R1 |) − φα (|x − R1 |) = ∇x φα (|x − R1 |) · δR1 1 + ∇x ∇x φα (|x − R1 |) · δR1 δR1 + · · · . 2

(A8.1.7)

On substituting this expansion into eqn. (A8.1.6) we obtain for the variation in the free energy   β Fin = dx dx c(|x − x |) 3 × n(x )∇x φα (|x − R1 |) · δR1 1 + n(x )∇x ∇x φα (|x − R1 |) · δR1 δR1 2 4 1  + ∇x φα (|x − R1 |)∇x φα (|x − R1 |) · δR1 δR1 + · · · . 2

(A8.1.8)

The first term on the RHS goes to zero since {Ri } are the equilibrium (metastable) positions and represent a minimum of the thermodynamic function. The i = 1 term in the summation expression for n(x ) appearing in the second term on the RHS of eqn. (A8.1.8) actually cancels out with the third term. The second and third terms combine to give the following result for the free energy as a quadratic functional of the displacement δR1 :   1 β Fin = dx dx c(|x − x |) 2   N   φα (|x − Ri |)∇x ∇x φα (|x − R1 |) · δR1 δR1 + · · · × i=2

1 ≡ κisj δ R1i δ R1 j , 2

(A8.1.9)

where κisj is the tensor of spring constants. For an isotropic system we denote this as κ s , which is a scalar.

A8.2 Field-theoretic treatment of the DDFT model The analysis presented in Section 8.1 with the dynamic density-functional model provides the basic nature of the slow dynamics in the nonlinear case; the FDT relation (8.1.137)

A8.2 Field-theoretic treatment of the DDFT model

433

(Kawasaki and Miyazima, 1997) used is ad hoc. Let us consider the associated renormalization of the field-theoretic model established in subsequent works (Andreanov et al., 2006; Kim and Kawasaki, 2008). The time-reversal symmetry We first construct, following the standard methods introduced in Chapter 7, the action functional of the MSR field theory for the equation of motion (8.1.144). Using the steps followed in Section 7.1, the average of a dynamic variable f [ρ] is defined as   f [ρ] = Dρ  f [ρ  ]δ(ρ − ρ  ) -   

.  δF ∂ρ   . (A8.2.1) − ∇i D(ρ )∇i  − η = Dρ f [ρ ]J [ρ] δ ∂t δρ η In the above equation the Jacobian of the transformation is treated as a constant by adopting the Itô (Oksendal, 1992) interpretation of the multiplicative noise. The average of the functional f [ρ] is then given by  4  3     δF ∂ρ  f [ρ] = − ∇i D(ρ)∇i Dρ D ρˆ f [ρ]exp i dx dt ρˆ ∂t δρ   .  × exp −i dr dt ρ(x, ˆ t)η(x, t)  ≡

 Dρ

  D ρˆ f [ρ] exp −A[ρ, ρ] ˆ .

η

(A8.2.2)

By averaging with respect to the Gaussian noise η, the MSR action A is obtained as   4 3   ∂ρ δF ˆ 2 − i ρˆ − ∇ · D(ρ)∇ A[ρ, ρ] ˆ = dx dt β −1 ρ(∇ ρ) ∂t δρ 3   4   ∂ρ δF = dx dt −β −1 ρˆ ∇ · (D(ρ)∇ ρ) ˆ − i ρˆ − ∇ · D(ρ)∇ ∂t δρ 3   4   ∂ρ δ F = dx dt −i ρˆ − β −1 ρˆ ∇ · D(ρ)∇ ρˆ − iβ . (A8.2.3) ∂t δρ The expression (A8.2.3) for the action A is written as a sum of two parts:   ∂ρ(x, t) A[ρ, ρ] ˆ = −i dx dt ρ(x, ˆ t) ∂t   ! " + β −1 dx dt ρ(x, ˆ t)∇ · D(ρ)∇ ρ(x, ˆ −t) .

(A8.2.4)

The above MSR action functional A[ρ, ρ] ˆ is invariant under the transformation ρ(x, −t) = ρ(x, t), ρ(x, ˆ −t) = −ρ(x, ˆ t) + iβ

δF . δρ(x, t)

(A8.2.5)

434

Appendix to Chapter 8

The second integral on the RHS of (A8.2.4) clearly remains invariant under the transformation (A8.2.5), as can easily be seen by partially integrating twice on the spatial coordinates. The first term under the same transformation changes to     ∂ρ(x, t) ∂ρ(x, t) → i dx dt ρ(x, ˆ −t) − i dx dt ρ(x, ˆ t) ∂t ∂t

  ∂ρ(x, t) δF = −i dx dt ρ(x, ˆ t) − iβ δρ(x, t) ∂t

  ∂ρ(x, t) δ F ∂ρ(x, t) = −i dx dt ρ(x, ˆ t) +β ∂t δρ(x, t) ∂t    ∂F ∂ρ(x, t) + β dt = −i dx dt ρ(x, ˆ t) ∂t ∂t   ∂ρ(x, t) . (A8.2.6) = −i dx dt ρ(x, ˆ t) ∂t The MSR action (A8.2.3) therefore remains invariant under the transformation (A8.2.5) up to a constant. Note that we have ignored the last term on the RHS of (A8.2.6) since it can be expressed with boundary terms and hence is treated as a constant. This is similar to the case of the time-reversal symmetry transformation in the fluctuating-hydrodynamics case discussed in Section A7.3. The linear response function Let us consider the response to an external field that couples to density in the free-energy functional,   (A8.2.7) Fext = − dx dt δρ(x, t)h e (x, t). The corresponding term in the equation of motion (8.1.144) has the form ∇i D(ρ)∇i h e (x, t) and hence produces a contribution to the MSR action   A = i dx dt ρ(x, ˆ t)∇{D(ρ)∇h e (x, t)}. (A8.2.8) Upon integrating partially twice with respect to spatial coordinates, A is obtained in a form proportional to the external field,   A = i dx dt ∇ · {D(ρ)∇ ρ(x, ˆ t)}h e (x, t). (A8.2.9) The corresponding induced change linear in the external field is given by    ρ(x, t) = i dx dt  ρ(x, t)∇  · {D(ρ)∇  ρ(x ˆ  , t)}h e (x , t)}   (A8.2.10) ≡ dx dt  R(x, t; x , t  )h e (x , t  ).

A8.2 Field-theoretic treatment of the DDFT model

435

The last equality follows from the definition of the linear response function R(x, t; x , t  ). ∇  represents the derivative operator with respect to the components of x . Thus we obtain R(x, t; x , t  ) = iρ(x, t)∇  · {D(ρ)∇  ρ(x ˆ  , t  )}.

(A8.2.11)

Note that R is the linear response function to an external field h e directly coupling to the density ρ. As discussed earlier (see Section 7.3.1) since the dissipative coefficient D(ρ) is dependent on density here (multiplicative noise), the physical response function is not a simple two-point correlation function between a hatted and an unhatted field of the MSR theory. The fluctuation–dissipation theorem It is straightforward to show that R is related to a correlation function by the standard FDT of equilibrium statistical mechanics discussed in Chapter 1, eqn. (1.3.72). We begin with the identity . δA ρ(x, t) = 0. (A8.2.12) δ ρ(x ˆ , t ) Using the expression given in eqn. (A8.2.3) for the MSR action, we obtain from (A8.2.12) 0 = ρ(x, t) 2β −1 ∇ · {D(ρ)∇ ρ(x ˆ  , t  )}   . ∂ρ(x , t  ) δF   +i − ∇ · D(ρ)∇ ∂t  δρ(x , t  ) 2 ∂ 1 ˆ  , t  )} = i  ρ(x, t)ρ(x , t  ) + β −1 ρ(x, t)∇ · {D(ρ)∇ ρ(x ∂t 4. 3 δF −1     ρ(x, t)∇ · D(ρ)∇ ρ(x ˆ , t ) − iβ +β δρ(x , t  ) ∂ ˆ  , t  )} = i  G ρρ (x, t; x , t  ) + β −1 ρ(x, t)∇ · {D(ρ)∇ ρ(x ∂t − β −1 ρ(x, t)∇  · {D(ρ)∇  ρ(x ˆ  , −t  )}. (A8.2.13) In reaching the last equality we have used the transformation (A8.2.5). From the definition (A8.2.11) of the response function R we obtain β

∂ G ρρ (x, t; x , t  ) = R(x, t; x , t  ) − R(x , t  ; x, t). ∂t 

(A8.2.14)

For t > t  , causality requires that the response function R(x , t  ; x, t) = 0, and hence we obtain the standard FDT of equilibrium statistical mechanics relating the physical response functions and the correlation function, R(x, t; x , t  ) = (t − t  )

1 ∂ G ρρ (x, t; x , t  ), kB T ∂t 

where (t) represents the Heaviside step function.

(A8.2.15)

436

Appendix to Chapter 8

In Section 7.3.1, in the MSR field theory, we showed that there exists a certain set of MSR-FDT relations linking the correlation functions between (a) two unhatted fields and (b) a hatted field and an unhatted field. The latter are termed the MSR response functions and in a nonlinear theory these are different from the physical response functions. From the time-reversal symmetry (A8.2.5) we obtain the following MSR-FDT relations: . δF −1    β G ρρ (A8.2.16) . ˆ (x, t; x , t ) = i(t − t ) ρ(x, t) δρ(x , t  ) The time-reversal invariance transformations given by (A8.2.5) are nonlinear since the functional derivative δ F/δρ is nonlinear in ρ. The ergodic–nonergodic transition Let us now consider the structure of the renormalized theory due to the nonlinear coupling of the density fluctuations in the MSR action (A8.2.3) and how it affects the ergodic– nonergodic (ENE) transition. Linear relations between correlation and response functions are obtained for the nonlinear model by introducing into the fluctuating-hydrodynamics description the nonlinear constraints involving the larger set of fields. For the nonlinear fluctuating-hydrodynamics (NFH) model described in Chapter 6 with the set {ρ, g}, development of the appropriate field-theoretic model involves introducing the new field δ F/δgi = vi . The field v appears in the equations of motion for g. In general, introducing the variable δF θi = (A8.2.17) δψi gives rise to a nonlinear constraint in the theory if F is non-Gaussian. For example, in the NFH model the free energy is expressed as a sum of kinetic (FK ) and potential (FU ) parts. The dependence on the momentum density (g) is in the kinetic part FK . This term is non-Gaussian, as shown in expression (6.2.7) for FK . Including v thereby imposes the nonlinear constraint g = ρv in this model. The appropriate field theory in the case of the DDFT equation (6.3.14) requires ψi ≡ ρ. The contribution from the non-Gaussian part of the free-energy functional F[ρ] in δ F[ρ]/δρ is now treated as a new variable θ (Kim and Kawasaki, 2008). We express the free energy as a sum of two parts, F = Fid [ρ] + Fin [ρ], where

(A8.2.18)



 # $ dx ρ(x) ln 30 ρ(x) − 1 ,   1 dx dx U˜ (x − x )δρ(x, t)δρ(x , t), β Fin [ρ] = 2

β Fid [ρ] =

(A8.2.19) (A8.2.20)

where 0 is the thermal de Broglie wavelength and δρ denotes the fluctuation of the density around the density of a uniform liquid state. U˜ (x) represents an interaction term at the Gaussian level.

A8.2 Field-theoretic treatment of the DDFT model

437

Then β

δ F[ρ] δ Fid [ρ] δ Fin [ρ] =β +β δρ(x, t) δρ(x, t) δρ(x, t)    ρ(x, t) = ln + dx U˜ (x − x )δρ(x , t) ρ0

 δ(x − x )  ˜  = dx U (x − x ) + δρ(x , t) + f [δρ(x, t)] ρ0  = dx B(x − x )δρ(x , t) + θ (x, t), (A8.2.21)

where we have defined the kernel B as δ(x) . B(x) = U˜ (x) + ρ0

(A8.2.22)

The field θ introduced in eqn. (A8.2.21) is defined in terms of the function f as   ∞  1 δρ(x, t) n θ (x, t) = f (δρ) = − , (A8.2.23) − n ρ0 n=2

or, equivalently,



δρ(x, t) δρ(x, t) δ Fid [ρ] + θ (x, t) = ln . ≡β ρ0 ρ0 δρ(x, t)

(A8.2.24)

The field θ introduced above comprises a bigger set of fluctuating variables {ρ, θ } and the corresponding hatted variables then lead to a set of linear FDTs. Note that the equilibrium average of the field θ is equal to zero: . . . δ F[ρ] δ Fin [ρ] δ Fid [ρ] =β −β θ (x) = β δρ(x) δρ(x) δρ(x)  δ (A8.2.25) = − Dρ e−β F = 0. δρ(x) Since FU is taken to be a quadratic functional of δρ, the functional derivative (δ Fin /δρ) is linear in δρ and is equal to zero. For the static correlation . . δ F[ρ] δ Fid [ρ]  = β δρ(x) Sρθ (x − x ) ≡ δρ(x) δρ(x, t) δρ(x ) . δ F[ρ] = β δρ(x) (A8.2.26) − δρ(x)B ⊗ δρ(x ), δρ(x ) where B ⊗ δρ denotes the convolution on the RHS of eqn. (A8.2.21). On taking the Fourier transform of the above equation and using the result (A8.2.22) we obtain the result Bq Sρρ (q) + Sρθ (q) = 1,

(A8.2.27)

where Bq = U˜ (q) + ρ0−1 is the Fourier transform of B(x). In the noninteracting case Sρθ = 0.

438

Appendix to Chapter 8

The MSR action for the new set of variables {ρ, ρ, ˆ θ, θˆ } with the nonlinear constraint (A8.2.23) is obtained as 3   ˆ 2 − i θˆ (θ − f (δρ)) A[ρ, ρ, ˆ θ, θˆ ] = dx dt β −1 ρ(∇ ρ)

4 ∂ρ − i ρˆ − ∇ · (ρ ∇{B ⊗ δρ + θ }) ∂t ˆ = Ag [ρ, ρ, ˆ θ, θ ] + Ang [ρ, ρ, ˆ θ, θˆ ]. (A8.2.28) The Gaussian and non-Gaussian parts, denoted by Ag and Ang , respectively, are obtained as 3

4   ∂ρ 2 −1 2 ˆ ˆ − i θ θ − i ρˆ − ρ0 ∇ {B ⊗ δρ + θ} , (A8.2.29) Ag = dx dt β ρ0 (∇ ρ) ∂t   # $ Ang = dx dt β −1 δρ(∇ ρ) ˆ 2 + i θˆ f (δρ) + i ρˆ ∇ · δρ ∇ [B ⊗ δρ + θ ] . (A8.2.30) With the extended set of field variables, the set of linear transformations which keeps the MSR (A8.2.30) action invariant is obtained as ρ(x, −t) = ρ(x, t),

(A8.2.31) 

ρ(x, ˆ −t) = −ρ(x, ˆ t) + iB ⊗ δρ(x , t) + iθ (x, t),

(A8.2.32)

θ (x, −t) = θ (x, t),

(A8.2.33)

ˆ −t) = θ(x, ˆ t) + i θ(x,

∂ ρ(x, t). ∂t

(A8.2.34)

Both the Gaussian and the non-Gaussian parts of the action remain separately invariant under the above set of linear transformations, and the following set of FDTs holds in the nonlinear theory just as the FDTs (7.3.29) and (7.3.30) hold in the NFH case. We write them in terms of the spatial Fourier transform as ∂ G ρρ (q, t), ∂t ! " G ρ ρˆ (q, t) = i(t) Bq G ρρ (q, t) + G ρθ (q, t) , ! " G θ ρˆ (q, t) = i(t) Bq G θρ (q, t) + G θθ (q, t) , ∂ G θ θˆ (q, t) = i(t) G θρ (q, t). ∂t G ρ θˆ (q, t) = i(t)

(A8.2.35) (A8.2.36) (A8.2.37) (A8.2.38)

The time-reversal transformations (A8.2.31)–(A8.2.34) are linear with an extended set of slow modes. However, this inevitably brings in the nonlinear constraint (A8.2.23) linking θ and δρ. The dynamics of the various correlation and response functions in the associated field theory is conveniently obtained by the Schwinger–Dyson equation,  ¯ ¯ ¯ ¯ (A8.2.39) G −1 0 (12)G(22) − (12)G 22) = δ(12). 2¯

A8.2 Field-theoretic treatment of the DDFT model

439

From the large number of equations for the different elements of the correlation function matrix G, Kim and Kawasaki (2008) showed that the density correlation function G ρρ in the nonlinear theory satisfies the equation

 t ∂ ˜ ρˆ ρˆ (q, t − t  )G ρρ (q, t  ) dt   G ρρ (q, t) = −q2 G ρρ (q, t) − ∂t 0  t ∂  − dt ρˆ θˆ (q, t − t  )  G ρρ (q, t  ), (A8.2.40) ∂t 0 where q2 = q 2 /(βm S(q)) is a microscopic frequency of the liquid state. S(q) is the static structure factor for the liquid, which is related to the interaction term U˜ (q) through the Ornstein–Zernike (Ornstein and Zernike, 1914) relation ρ0 , (A8.2.41) Sρρ (q) ≡ ρ0 S(q) = 1 + ρ0 U˜ (q) ρ0 being the equilibrium number density of particles (the mass m is unity in the present ˜ ρˆ ρˆ in eqn. (A8.2.40) in terms of the self-energy matrix notation). We define the quantity  element ρˆ ρˆ : ˜ ρˆ ρˆ (q, t) = 

1 ρˆ ρˆ (q, t). ρ0 q2 S(q)

(A8.2.42)

In developing the diagrammatic perturbation series for the renormalized theory, a useful identity for the fields ρ and θ follows from (A8.2.24):   

 ρ δρ +θ ρ∇i ln = (ρ0 + δρ)∇i ρ0 ρ0 δρ = ∇i δρ + ∇i δρ + ρ0 ∇i θ + δρ ∇i θ. (A8.2.43) ρ0 Since the LHS is identically equal to the first term on the RHS, we obtain the identity that ρ0 ∇i θ +

δρ ∇i δρ + δρ ∇i θ = 0. ρ0

(A8.2.44)

Note that the first term on the LHS of (A8.2.44) is linear while the other two terms are nonlinear in the field variables. In the field-theoretic model involving the four fields {ρ, ρ, ˆ θ, θˆ }, as a consequence of (A8.2.44) cancelations occur between the contributions coming from the Gaussian and non-Gaussian parts of the action (A8.2.28). Kawasaki rearrangement An important consequence of the above equation of motion for the density correlation function is the following: The positive sign of the second term on the RHS of eqn. (A8.2.40) renders the dynamics unstable if its contribution is large. This equation therefore requires further tuning in order for it to be physically relevant. This is done through a reorganization of the kernels, a scheme proposed by Kawasaki (1995, 1997), with the introduction of an irreducible memory function (Cichocki and Hess, 1987). We will refer to this as Kawasaki

440

Appendix to Chapter 8

rearrangement. Let us first ignore the last term on the RHS of eqn. (A8.2.40) involving ρˆ θˆ and consider the dynamics of a correlation function C with the memory-function equation

 t ∂C (t − s)C(s) , (A8.2.45) = −τ0−1 C(t) − τ0 ∂t 0 where τ0 > 0 is the relaxation time in the absence of any nonlinearity whose effect is represented by the kernel . We also ignore the field indices associated with the selfenergies and correlation functions. On taking the Laplace transform of eqn. (A8.2.45) we obtain C(0) ˜ = C(z) , (A8.2.46) −1 z + τ0 [1 − τ0 (z)] where we have defined the frequency transform of the kernel  as  (z) = dt ei zt (t).

(A8.2.47)

We now replace the memory kernel in the denominator on the RHS of (A8.2.46) with its irreducible counterpart denoted by I such that eqn. (A8.2.46) takes a more physically meaningful form,  −1 τ0−1 ˜ = C(0) z + C(z) . (A8.2.48) 1 + I (z) The corresponding relation between the kernel  and its irreducible counterpart I is given by 1 − τ0 (z) =

1 , 1 + I (z)

(A8.2.49)

or, equivalently, (z) = τ0−1

I (z) . 1 + I (z)

(A8.2.50)

˜ develops For  → τ0−1 , we obtain I → ∞. Hence from eqn. (A8.2.46) it follows that C(z) a 1/z pole. In the time space eqn. (A8.2.50) implies the relation  t dt  (t − t  )I (t  ). (A8.2.51) (t) = I − τ0 0

The equation of motion for the correlation function in terms of the irreducible memory function is obtained as  t ∂C(s) ∂C I (t − s) = −τ0−1 C(t) − τ0 , (A8.2.52) ∂t ∂s 0 which, on Laplace transforming, reduced to eqn. (A8.2.48). Note that eqn. (A8.2.52) follows if we replace C in the second term on the RHS of eqn. (A8.2.45) in the lowest-order approximation with −τ0 ∂C/∂t.

A8.2 Field-theoretic treatment of the DDFT model

441

We now go back to the full equation (A8.2.40) for the dynamics of the density correlation function. Following the above scheme, we introduce an irreducible memory function MI with the equation

 t ∂ 2   ∂ dt MI (q, t − t ) G ρρ (q, s) . G ρρ (q, t) = −q G ρρ (q, t) + ∂t ∂s 0 (A8.2.53) Direct comparison with eqn. (A8.2.40) shows that MI satisfies  t ∂G ρρ (q, s) dt  MI (q, t − t  ) ∂s 0  t  ∂ ˜ ρˆ ρˆ (q, t − t  ) +  ˆ (q, t − t  ) dt  q−2  G (q, t  ) = ρˆ θ  ρρ ∂t 0  t  t ∂  ˜  + dt ρˆ ρˆ (q, t − t ) ds MI (q, t  − s) G ρρ (q, s). ∂s 0 0

(A8.2.54)

Laplace transformation of eqn. (A8.2.54) gives the following self-consistent relation for MI : ˜ ρˆ ρˆ (q, z), MI (q, z) = R (q, z) + MI (q, z)

(A8.2.55)

where we have defined R (q, t) as

 ˜ ρˆ ρˆ (q, t) +  ˆ (q, t) . R (q, t) = q−2  ρˆ θ

In the time space this reduces to the relation for the irreducible part MI :  t MI (q, t) = R (q, t) + ds MI (q, t − s)ρˆ ρˆ (q, s).

(A8.2.56)

(A8.2.57)

0

To summarize, the dynamics of the density correlation function G ρρ is obtained from the equation of motion (A8.2.53) in terms of the irreducible kernel MI which satisfies eqn. (A8.2.57). To lowest order MI (q, t) = R (q, t).

(A8.2.58)

The equation satisfied by the density correlation function is therefore obtained as

 t ∂ 2   ∂ dt R (q, t − t ) G ρρ (q, s) = 0. G ρρ (q, t) = −q G ρρ (q, t) + ∂t ∂s 0 (A8.2.59) The above equation represents the feedback mechanism of MCT if we express R selfconsistently in terms of the density correlation function G ρρ . From eqn. (A8.2.59) it follows that slowing the dynamics of the density correlation function increases R , eventually causing the ENE transition, beyond which the density correlation function is frozen and R (q, 0) diverges. In other words, a 1/z pole in G ρρ (z) self-consistently gives rise to the

442

Appendix to Chapter 8

Fig. A8.1 The one-loop diagrams at O(kB T ) for the self-energy vˆi vˆ j . The lines joining at both ends with the vertices represent fully renormalized correlation or response functions as labeled. The corresponding three-leg vertices are both given by (7.3.61).

1/z pole in R (z). Kim and Kawasaki (2008) reported that at one-loop level even the wavevector dependence of R (in its self-consistent form) controlling the ENE transition point is identical to that predicted from the simplified treatment reported in eqns. (8.1.139) and (8.1.140). A8.3 The one-loop result for vˆ vˆ (0, 0) The cutoff function γ is also obtained self-consistently in terms of the hydrodynamic correlation functions. We state the results in terms of the kernel in time γ (q, t), where  ∞ dt ei zt γ (q, t). (A8.3.1) γ (q, z) = o

γ (q, z) is expressed as the sum of two parts, γ L and γ T , referring to the longitudinal and transverse parts in the isotropic fluid, respectively. Since the cutoff function γ is defined in the hydrodynamic limit as q 2 γ (q, z) ∼ qvρ ˆ , we evaluate the former in terms of the self-energy vˆ vˆ using the relation (7.3.55). From the one-loop diagrams for the self-energy vˆ vˆ shown in Fig. A8.1 we obtain the one-loop contributions as (Das and Mazenko, 1986)

 dk u u 1 u 1 S(k)S(k1 ) ˙ t), ˙ 1 , t)φ(k, γ L (q, t) = (A8.3.2) + φ(k 2n 0 k1 S(q) (2π )3 k k  dk 1 S(k1 ) γ T (q, t) = (1 − u 2 ) φ(k1 , t)φ T (k, t), 3 2n 0 S(q) (2π ) ˆ and u 1 = qˆ · kˆ 1 . where φ˙ is the time derivative of the function φ, k1 ≡ q − k, u = qˆ · k, The hatted quantity denotes the corresponding unit vector. This one-loop-order result for the cutoff function is valid only in the hydrodynamic limit.

9 The nonequilibrium dynamics

In the present chapter we discuss the nonequilibrium dynamics and aging (Struik, 1978) in structural glasses. The nonequilibrium state has been studied quite extensively through computer simulations and application of mean-field models. We discuss here the characterization of the nonequilibrium state through plausible extensions of the equilibriumthermodynamics description and consider possible applications of the new results obtained for the dynamics. In this regard the model based on the similarity between structural glasses and spin glasses has proved to be very useful for understanding the simulation results for the structural glass. Indeed, such findings further consolidate the underlying similarity between disordered systems with and without intrinsic disorder.

9.1 The nonequilibrium state A characteristic behavior of the nonequilibrium state is the violation of the fluctuation– dissipation theorem (FDT) discussed in Chapter 1. In the following we first consider the FDT violation seen in simulation studies and in certain mean-field theoretical models.

9.1.1 A generalized fluctuation–dissipation relation The fluctuating-hydrodynamics model discussed in the previous chapter refers to fluctuations around the equilibrium state for which the FDT equation R(t, t  ) =

1 ∂C(t, t  ) kB T ∂t 

linking the correlation function C(t, t  ) and the response function R(t, t  ) holds. (See eqn. (1.3.72) in Chapter 1 for this standard result of equilibrium statistical mechanics.) For simplicity we omit the subscripts indicating the dynamic variables associated with C and R. For glassy dynamics the FDT is generalized in a parametric form by defining a quantity X (t, t  ) (for t > t  ) as R(t, t  ) =

X (t, t  ) ∂C(t, t  ) . kB T ∂t  443

(9.1.1)

444

The nonequilibrium dynamics

The crucial assertion here is that, taking the similarity to the mean-field structural-glass (SG) models, in the limit of long times t  , t → ∞, X (t, t  ) is expressed as a function of the correlation function C(t, t  ) only, i.e., X (t, t  ) ≡ x[C(t, t  )] (say). Equation (9.1.1) is expressed in terms of the integrated response function M defined as  1  t 1 ds R(t, s) = − x(c)dc, (9.1.2) M(t, t  ) = kB T C t where we have used a normalization C(t, t) = 1. It is clear that a plot of −kB T M vs. C has slope −x(C). When the FDT is obeyed, kB T M(t − t  ) = C(t − t  ) − 1,

(9.1.3)

and hence the slope x = −1. A more general form of x(C) corresponds to violation of the FDT. In the parametric representation in terms of x(C) the violation of the FDT is presented in terms of correlation windows rather than time windows.

9.1.2 Computer-simulation studies The aging behavior in a supercooled liquid was studied by Kob and Barrat (1998, 2000) using molecular-dynamics simulations. These authors used a binary mixture of LennardJones particles (BMLJ) – referred to as the Kob–Anderson mixture introduced earlier in Section 4.3.1. Binary systems continue to remain in the liquid state without crystallization and are suitable for studying the dynamics below the freezing transition. We had earlier discussed (see Section 8.2.3) computer-simulation studies of the equilibrium dynamics of the BMLJ system and their use in testing the predictions of the mode-coupling theory. These studies are done mainly in the temperature range 0.45 < T < 0.8. The nonequilibrium dynamics was obtained with the simulation of 1000 particles at constant volume V = L 3 in cubic box length L = 9.4. The aging behavior was studied in terms of the two-point correlation functions C(τ + tw , tw ) for different waiting times tw . Following results from the mean field theories and experimental data of spin glasses, the behavior of the correlation function in the aging regime is tested for the form C(tw + τ ; tw ) ∼ C A [h(tw + τ )/ h(tw )], where h(x) is a monotonically increasing function of its argument x. The correlation function depends on two times in the absence of time-translational invariance (which holds for short times). Thus the τ and tw dependences of the correlation function in the aging regime enter only through the combination h(tw + τ )/ h(tw ). The form of the function h(t) can be used to distinguish between different theoretical models to describe the aging dynamics, e.g., the droplet model of Fisher and Huse (1986, 1988) for spin glasses predicts h(t) ∼ log(t). It was argued (Müssel and Rieger, 1998) that the Lennard-Jones model showed this type of aging dynamics. Aging in the BMLJ system was considered by first equilibrating the system in the high-temperature phase at a temperature Ti > Tc and then quenching at time t = 0 to a temperature T f < Tc . Following the quench, the system was allowed to evolve for a waiting time tw , after which the measurements of the quantities of

9.1 The nonequilibrium state

445

Fig. 9.1 The two time correlation function as a function of the variable [h(τ + tw )/ h(tw ) − 1] for different wave vectors k = 3.0, 7.23, and 12.5. In each case the waiting times tw = 0, 10, 100, 1000, 10 000, and 63 100 time units are considered. The function h(t) was chosen to make the curves for k = 7.23 collapse at long times and its t dependence is shown in the inset. Reproduced c Europhysics Journal B. from Kob and Barrat (2000). 

interest were started. Figure 9.1 shows the scaling of the correlation functions for different values of tw = 0, 10, 100, 1000, 10 000, and 63 100 time units at three different wave vectors, k = 3.0, 7.23, and 12.5. In the inset of the figure the h(t) function corresponding to the structure factor peak, i.e., k = 7.23, is shown. This h(t) is chosen to make the best collapse of the different tw data at this wave vector. To a first approximation h(t) is close to a logarithm, but significant deviations occur. For wave vectors different from the structurefactor peak, the curves for the different tw values do not collapse nicely at long times. This indicates that the scaling function h(t) depends on the corresponding observable – which in this case is density fluctuations at a particular wave length. Let us now consider the violation of the FDT as seen in the computer simulations of the supercooled liquids in the nonequilibrium state. For the structural glasses the correlation and response functions for the liquid have been studied extensively by Kob and Barrat (2000) using molecular-dynamics simulations of binary mixtures of the LennardJones (BMLJ) system. The linear response function is studied by adding a small fictitious “charge” κ = ±1 randomly to each particle and defining the observable,  κi cos(k · Ri (t)), (9.1.4) Ok (t) = i

where Ri (t) is the position of the ith particle at time t. An additional term of the form  κi cos(k · Ri ) (9.1.5) H ≡ h 0 i

is added to the Hamiltonian to represent coupling of O to the field h 0 . In the computer simulation the liquid is quenched at t = 0 from the initial high temperature Ti to a low temperature Tf , and the evolution of the system is followed by keeping the field h 0 off until time t = tw . At time t = tw + τ the average of the variable Ok (over different realizations

446

The nonequilibrium dynamics

Fig. 9.2 A parametric plot of the integrated response function M(tw + τ, tw ) and the correlation function C(tw +τ, tw ) for wave vector k = 7.25, at the quenched temperature (see the text) Tf = 0.3: circles, tw = 1000; and triangles, tw = 10 000. Time and wave vector are expressed in the LennardJones units (see the text). The straight lines have slopes −1.0 and −0.45. Reproduced from Kob and c Europhysics Letters. Barrat (1999). 

of the random distributions of κ) in the presence of the conjugate field h 0 (which couples to Ok ) is denoted in terms of the integrated response function M(τ + tw , tw ):  tw Ok (τ + tw , tw ) = h 0 R(τ + tw , s)ds = h 0 M(τ + tw , tw ). (9.1.6) τ +tw

The correlation and response functions obtained are presented in Fig. 9.2 in a parametric plot of −kB T M(t, t  ) vs. C(t, t  ) over different ranges of C (Kob and Barrat, 1997). This is shown here for Tf = 0.3 referring to the temperature to which the system is initially quenched. For short times (C close to 1) the data for different t  collapse onto a single curve of slope −1, showing that the FDT holds. For smaller values of C the slope of this curve is not unity, indicating FDT violation. The corresponding values of the parameter X ≡ m obtained by Kob and Barrat for three temperatures Tf = 0.4, 0.3, and 0.1 were independent of the wave vector and were compatible with a linear dependence of m on Tf . One-component Lennard-Jones systems have also been simulated for studying FDT violation. The problem of crystallization of a one-component system makes it difficult to study it in the supercooled state. Di Leonardo et al. (2000) studied the FDT in a system of 256 particles interacting with a Lennard-Jones potential modified by inclusion of a small manybody term in its potential energy to inhibit crystallization. The off-equilibrium dynamics of the system was studied after sudden isothermal (at temperature T ) compressions from well-equilibrated liquid configurations towards the glassy state. The response functions

9.1 The nonequilibrium state

447

Fig. 9.3 The temperature dependence of the FDT-violation factor in the nonequilibrium state for two densities of a one-component LJ system: nσ 3 = 1.14 (open circles) and nσ 3 = 1.24 (filled squares). Fits shown with dashed lines give the corresponding effective temperatures Teff (see the text) as 0.86 c American Physical Society. and 1.43, respectively. Reproduced from Di Leonardo et al. (2000). 

were computed by introducing dynamic variables similar to Ok discussed above for the BMLJ case. The generalized FDT was studied and the violation parameter m was obtained as a function of temperature T . The result for the FDT-violation parameter is shown in Fig. 9.3. At low temperature, m(T ) is proportional to T , crossing over to m = 1 at a characteristic temperature Tc . The linear temperature dependence of m below a characteristic temperature conforms to the predictions in the one-step replica-symmetry-breaking scenario proposed for the spin glasses (Mézard et al., 1987). An interesting question in this regard is that of how the temperature Tc for the liquid compares with other characteristic temperatures of the dynamics of structural glasses. The calorimetric glass-transition temperature Tg at which the time scale of equilibration of the supercooled liquid crosses the laboratory time scale indicates the liquid falling out of equilibrium. On the other hand, from consideration of the FDT, above Tc the system falls out of equilibrium and this is expected to correspond to the calorimetric glass-transition temperature. In the case of the MD simulation of a BMLJ system Kob and Barrat (2000) report a crossover temperature Tc ≈ 0.7 that is much higher than Tg . In the one-component Lennard-Jones system the agreement between these two temperatures is much better. The crossover temperature from liquid- to solid-like behavior has been obtained by analyzing the temperature dependence of the potential energy. In the glassy state the time scale of relaxation becomes so large that the liquid behaves like an amorphous solid. Equilibrium MD studies show that the potential energy for the liquid crosses over from T 5/3 behavior (liquid-like) to linear dependence (solid-like) at a temperature close to Tc . The FDT violation in supercooled liquids has also been studied using (see Section 4.3.1) an approach based on inherent structures in the potential-energy landscape (PEL). A new temperature Tint is identified for the nonequilibrium system. The free energy of the liquid is obtained here as a sum of configurational and vibrational parts, f (eIS ) = eIS − Tint sconf + f vib (eIS , T ),

(9.1.7)

448

The nonequilibrium dynamics

where eIS is the energy of the local minima corresponding to a basin of attraction in the PEL, sconf counts the number of minima with energy eIS , and f vib is the average free energy of a basin of depth eIS . T denotes the temperature of the bath (which is attained by the fast degrees of freedom) and Tint is the Lagrange multiplier used to extremize f . In equilibrium Tint = T , the temperature of the bath. From the landscape studies of the typical fragile BMLJ system it was shown (Sciortino and Tartaglia, 2001) that Tint is close to T /m, where m is denoted as the FDT violation parameter. A steady perturbation applied to the system at t = tw is given by the H  (t) = −V0 Bδ(t − tw ) due to the coupling of the dynamic variable B with a constant field. The Fourier transform of the√ density of the 5 Nα ik.Rα e i / N . Let B = particles of species α at wave vector k is obtained as ρkα = i=1 ρkα + ρkα∗ be identified with the real part of ρkα . Since the applied field is constant here, the linear response function will be proportional to the perturbed value of the coupled variable at time τ , i.e., ρkα (τ ). The corresponding correlation function is denoted by Skαα (τ ) = ρkα (τ )ρkα (0)0 . The correlation and response functions are obtained at different values of τ corresponding to two waiting times tw =1024 and 16 384. In Fig. 9.4 the behaviors of the correlation and response functions are displayed, demonstrating the violation of the FDT. In equilibrium m = 1, whereas for the glassy state m < 1. For strong liquids such as

Fig. 9.4 FDT violation using PEL study. Left: the time dependences of the response function (see the text for its definition) ρkα (τ ) (open symbols) and the correlation function Skαα (τ ) (filled symbols) for the two waiting times tw = 1024 (circles) and tw = 16 384 (squares). Right: a parametric plot of correlation and response functions similar to Fig. 9.2. The solid lines have slope proportional to 1/kB Tint with Tint (see the text for its definition) respectively corresponding to the two inherent structures characterized by e I S (tw = 1024) = −7.576 and e I S (tw = 16 384) = −7.602, both with the same bath temperature T = 0.25. The two dashed lines correspond to the equilibrated system with slope proportional to 1/kB T . The wave vector corresponds to k = 6.7. From Sciortino and Tartaglia c American Physical Society. (2001). 

9.2 The effective temperature

449

Fig. 9.5 The quantity [1 − Ck (t)]/T involving the correlation function (squares) and the integrated response function Mk (t) (circles) vs. time t. The temperature and shear rates are shown in the legend. The two curves overlap when the FDT is satisfied for small t and for larger values of the correlation functions. The inset shows a plot of C and M similar to Fig. 9.1. From Berthier and Barrat (2002). c American Institute of Physics. 

silica (Scala et al., 2003), however, the unexpected result m > 1 was found in the out-ofequilibrium state. The out-of-equilibrium state of the liquid can also be obtained by applying a steady-state perturbation to the system, e.g., applying a homogeneous steady shear flow (Berthier and Barrat, 2002). The steady shear flow is characterized by the corresponding shear rate γ0 , which creates a nonequilibrium steady state with time translational invariance. The inverse of the shear rate γ0−1 introduces a time scale and acts as a control parameter in studying the nonequilibrium state that is similar to the waiting time tw in relaxational dynamics. In Fig. 9.5 the violation of the FDT is shown by comparing the two sides of eqn. (9.1.3) for a fixed tw (i.e., γ −1 ) for the BMLJ system. (See the text near eqns. (4.3.11) and (4.3.12) for its definition.) The FDT is tested by computing the Fourier transform of the self-part of the density correlation function Ck (t) for wave vector k = km ez corresponding to the peak of the structure factor. The liquid is subjected to a steady shear γ = 10−3 expressed in units of τ0 for the BMLJ system. The response functions are computed using the random field defined in eqn. (9.1.4). For small t the FDT is satisfied, but it ceases to hold for large t as the system enters the aging regime. We discuss such behavior in the context of mean-field theory below.

9.2 The effective temperature For a liquid in thermodynamic equilibrium its macroscopic behavior is described in terms of a small number of thermodynamic variables such as temperature, pressure, volume,

450

The nonequilibrium dynamics

and chemical potential. The statistical-mechanical or microscopic description of a manyparticle system, on the other hand, is formulated in terms of a suitable ensemble of a large number of similar systems. Each member of this ensemble corresponds to a state of the system defined by the specific values of the coordinates of the large number of variables for the constituent particles. For a system in equilibrium the probability density for a particular member of the corresponding ensemble is characterized in terms of a set of conserved properties. For a set of conserved quantities A0 = {N , H, P, . . .}, respectively denoting the total energy, number of particles, momentum, etc., the corresponding probability density is given by f (N , H, P · · · ) ∼ exp [B0 · A0 ] ,

(9.2.1)

as has already been discussed in Section 5.1.2, eqn. (5.1.14). The quantities for B0 = {ν, −β, v, . . .}, in (9.2.1) respectively correspond to the chemical potential, temperature, and velocity fields etc. describing a particular thermodynamic state of the system. The equilibrium ensemble describing the possible microscopic states for the system is thus characterized by a small set of time-independent thermodynamic properties. For systems that are out of equilibrium, such ideas are extended to modify the definition of thermodynamic variables such as temperature. In the local equilibrium ensemble the definition (9.2.1) is extended in terms of time-independent local thermodynamic variables {ν(r), −β(r), v(r), . . .}. For nonequilibrium systems in which such time independence cannot be invoked, developing an equivalent description is not straightforward. We now require extensions of the basic ideas applied in formulating the thermodynamics of the equilibrium system. In the out-of-equilibrium state, the issue of time independence applies only for a class of “fast” variables that relax over the time scale of observation while the slow variables in the many-particle system are still evolving with time. For systems showing aging, e.g., for glassy dynamics, such an approach is very appropriate (Cugliandolo and Kurchan, 1994). Consider the case of a supercooled liquid that has been quenched from a high to a low temperature a long time ago and is currently in a weakly perturbed state. The glassy state has reached relaxation times beyond the laboratory time scales. The driven system reaches a stationary state after some time and an effective temperature is associated with it. It is useful to note here that “stationarity” in the present context implies a loss of memory of the initial conditions. While this property is associated with the equilibrium state of a system, the converse is not true, i.e., a stationary state does not have all of the other properties of the Gibbsian ensemble. The time scale associated with the description of such a stationary (weakly perturbed) state is related to the strength of the perturbation. For example, for a system exposed to a steady shear γ0 , the corresponding time scale is proportional to γ0−1 . The out-of-equilibrium system with extremely slow dynamics is linked to a “temperature” that is suitably defined for the nonequilibrium state. This has been attempted in two ways.

9.2 The effective temperature

451

9.2.1 The phenomenological approach The nonequilibrium state of the supercooled liquid in the glass-transition region (near the calorimetric glass-transition temperature Tg ) has time scales of relaxation that are much larger than the laboratory time scales. The nonequilibrium liquid is in a frozen state that is characterized in terms of a time-dependent fictive temperature Tfic . The nature of this dependence for the fictive temperature is defined from a phenomenological approach. In the limiting case of equilibrium liquid at high temperature the time dependence is absent and Tfic = T . Near the glass-transition range generally Tfic > T . The time dependence of Tfic is prescribed in terms of the nonlinear differential equation Tfic − T dTfic =− . dt τ (T, Tfic )

(9.2.2)

The time τ (T, Tfic ) depends self-consistently on the fictive temperature Tfic (t) itself, as well as on the temperature of the bath T . The resulting Tfic (t) is obtained by making some suitable phenomenological choices for τ (T, Tfic ) in eqn. (9.2.2). An example (Tool, 1946; Gardon and Narayanaswamy, 1970; Narayanaswamy, 1971; Moynihan et al., 1976; Jäckle, 1986) is

x A (1 − x)A + , (9.2.3) τ (T, Tfic ) = τ0 exp T Tfic where the parameter x ∈ [0, 1]. A and τ0 are constants related to the dynamics of the system. For very low temperatures Tfic → Tg . This time-dependent fictive temperature therefore follows from a purely phenomenological basis.

9.2.2 A simple thermometer Here some generic property of the nonequilibrium state, e.g., violation of the fluctuation– dissipation theorem (FDT), is used in extending the usual thermodynamic description of the equilibrium state to systems that are out of equilibrium. Let us consider the most typical thermodynamic property, namely the “temperature” of the nonequilibrium system. We note that in equilibrium the temperature is measured with some auxiliary device (thermometer) linked to the original system in a state of equilibrium. The measurement of the “temperature” of the system is then done in terms of the properties of the “contents” of the device. Similarly, for the nonequilibrium system the measuring device or “thermometer” couples to some observable O (say) of the system. A crucial step for describing the nonequilibrium state involves the extension of the equipartition law of equilibrium statistical mechanics to the out-of-equilibrium state of the thermometer. The average kinetic or potential energy per degree of freedom for the thermometer is equal to (kB Teff )/2, where Teff is the effective temperature of the system. The underlying assumption here in extending the thermodynamic concept of temperature, of course, is that the system has reached a state in which the flow of heat is very small. We demonstrate below that the effective temperature Teff of the nonequilibrium state defined in this manner is linked to the fluctuations and responses of

452

The nonequilibrium dynamics

the observable(s) O of the nonequilibrium system (Cugliandolo et al., 1997a; Cugliandolo, 2003). In equilibrium these are related through the FDT. Thus the effective temperature in the equilibrium limit reduces to the bath temperature. For the out-of-equilibrium case this is closely linked to the FDT-violation factor. The measurement of the effective temperature Teff requires a device (thermometer) whose characteristic time scale matches with that of Teff . The simplest example of such a device we consider is a harmonic oscillator with characteristic oscillation frequency ω0 . It measures the temperature of a nonequilibrium system S by coupling to an observable denoted by O, and the system is kept in contact with a heat bath at temperature T . If S is in equilibrium at temperature T , then the energy of the oscillator will satisfy E osc  = kB T (Cohen-Tannoudji et al., 1977). The Hamiltonian for the system plus the thermometer is obtained as Htot = H (S) + Hosc + Hint ,

(9.2.4)

where the energy of the oscillator is ω2 x 2 x˙ 2 + 0 . 2 2 Assuming linear coupling, we obtain for the interaction part Hosc =

Hint = −εO x.

(9.2.5)

(9.2.6)

The oscillator’s equation of motion is given in terms of the characteristic frequency, x¨ = −ω02 x + εO(t).

(9.2.7)

Assuming that εx(t) is small, the effect of the thermometer (oscillator) on the observable O is computed from the linear response theory. Let us assume that the fluctuating part of ˜ The observable O(t) is the observable O in the off-equilibrium system is denoted by O. therefore obtained as  t ˜ dt  R O (t, t  )x(t  ), (9.2.8) O(t) = O(t) +ε 0

where the response function R O for the observable O is defined as R O (t, t  ) =

δ O(t). δ{εx(t  )}

(9.2.9)

The angular brackets represent the average over the fluctuations. The correlation function for the observable O (in the absence of the coupling to the oscillator) is defined as ˜ O(t ˜  ). C O (t, t  ) =  O(t)

(9.2.10)

Using the relation (9.2.8) in eqn. (9.2.7), we obtain the following fluctuating equation for the oscillator variable x(t):  t ˜ dt  R O (t, t  )x(t  ) + ω02 x(t) = ε O(t). (9.2.11) x¨ − ε2 0

9.2 The effective temperature

453

The above equation of motion for the oscillator indicates that it takes up energy through fluctuations of O˜ and dissipates through the response of the system. We solve eqn. (9.2.11) in Appendix A9.1 to obtain the average mean-square displacement after a time tw that is much larger than the characteristic time scale ω0−1 of the oscillator. The average potential energy of the oscillator coupling to the system through the observable O, measured at time tw , is given by  (ω , t ) ω02 2 ω0 C O 0 w x tw = ,  2 2χ (ω0 , tw )

where have we defined  CO (ω, t) 

 = Re

t

(9.2.12)



dt  C O (t, t  )e−ω(t−t ) ,

(9.2.13)

0



χ (ω, t) = Im

t



dt  R(t, t  )e−ω(t−t ) .

(9.2.14)

0

The effective temperature defined above is analogous to the kinetic temperature of the system and is obtained by equating the average potential energy to kB /2 times the “temperature” TO (ω0 , tw ) of the out-of-equilibrium system. Thus we obtain the result TO (ω0 , tw ) =

 (ω , t ) ω0 C O 0 w χ  (ω0 , tw )

(9.2.15)

defining the temperature in units of kB . The above example of the thermometer being identified with a single oscillator can easily be generalized to a more general measuring device (Cugliandolo et al., 1997b). The effective temperature obtained above is thus dependent on (a) the waiting time tw at which the temperature is being measured,   (b) the characteristic frequency (ω0 ) or time ω0−1 of the thermometer, and (c) the observable O of the system to which the thermometer is coupled. In equilibrium the effective temperature is independent of all of the above three quantities and becomes equal to the thermodynamic temperature of the bath. The above expression for the effective temperature also shows that it is compatible with the Fourier-transformed expression for the FDT-violation parameter X (C), assuming that it does not vary too fast with ω0 . The latter condition is generally satisfied in systems with slow dynamics (Cugliandolo et al., 1997b). From the MD studies of FDT violation (Kob and Barrat, 1997) discussed above, the effective temperature has been obtained for the BMLJ mixture after an initial quench. The effective temperature was studied in a laboratory glass-forming system, glycerol, by Grigera and Israeloff (1999), who did an experiment similar to the oscillator model described above. These authors studied the FDT below the glass-transition temperature Tg by measuring the dielectric susceptibility and the noise correlations. Figure 9.6 displays the effective temperature Teff of glycerol following a quench to T = 179.8 K against the corresponding waiting time tw . Teff relaxes to T on a time scale that is comparable to that for the aging of the susceptibility as seen in the experiment.

454

The nonequilibrium dynamics

Fig. 9.6 The effective temperature of glycerol Teff (see the text) vs. the waiting time tw following a quench to T = 179.8 K. The points display the experimental data while the solid line presents the exponential decay to the equilibrium temperature. Reproduced from Grigera and Israeloff (1999). c American Physical Society. 

We have considered above the case for the nonequilibrium state with relaxational dynamics. As already indicated, the out-of-equilibrium situation can also be created by applying a steady-state perturbation to the system, e.g., applying a homogeneous steady shear flow (Berthier and Barrat, 2002). The steady shear flow is characterized by the corresponding shear rate γ0 , which creates a nonequilibrium steady state with time translational invariance. The inverse of the shear rate γ0−1 introduces a time scale and acts as a control parameter in studying the nonequilibrium state, similarly to the waiting time tw in relaxational dynamics. In this case a simple harmonic oscillator is used as a thermometer by coupling it to the system through the observable O. This analysis gives rise to the definition of an effective temperature TO (ω0 ) for the liquid in the steady state (with time translational invariance), TO (ω0 ) = where we have used the definitions  (ω) = Re CO 

 (ω ) ω0 C O 0 ,  χ (ω0 )

 

∞

(9.2.16)

dt C O (t)eωt ,

(9.2.17)

dt R(t)eωt .

(9.2.18)

0 ∞

χ (ω) = Im 0

To test the above idea of effective temperature being measured with an oscillator, Berthier and Barrat (2002) simulated an 80 : 20 mixture of N = 2916 Lennard-Jones particles of types A and B with the same interaction as that described in Section 4.3.1. Lees–Edwards

9.2 The effective temperature

455

boundary conditions are used in a cubic simulation box of size L = 13.4 in units of the length σAA defined by the Lennard-Jones interaction potential. The liquid is in a state of steady shear with γ0 = 10−3 and in contact with a heat bath at temperature T = 0.3. The units of time and temperature are as defined for the binary Lennard-Jones mixture in Chapter 1. The simulated mixture includes 10 massive tracer particles, whose masses are given by m tr ∈ [1, 107 ]. They are otherwise identical to the particles of type A. The equilibrium structure of the simulated system is the same as that of the system with normal “light” particles. The Einstein frequency ωE of the tracer particles is defined by

n0 ωE = 3m tr

1/2

 dr g(r )∇ U (r ) 2

,

(9.2.19)

where g(r ) is the pair correlation function and U (r ) represents the interaction potential between the particles. ωE characterizes the oscillation frequency for a fluid particle in the cage formed by its neighbors. From eqn. (9.2.19) it follows that the corresponding oscillation frequency falls by three orders of magnitude for the most massive tracer particle. Berthier and Barrat computed the effective temperature from the average kinetic energy or the average mean-square velocity 1in 2the direction transverse to the flow of the tracer particles. In Fig. 9.7 the plot of m tr vz2 ≡ Teff vs. the mass m tr shows how the effective temperature varies with the mass of the tracer particles. While for the light particles the effective temperature is the same as that of the bath (T = 0.3 in this simulation), it crosses over to Teff = 0.65 for the heaviest particles. The dynamics of the tracer particles

Fig. 9.7 The mean kinetic energy in the z direction vs. the mass m tr (m tr ∈ [1, 107 ]) of the tracer particle in a BMLJ system for Tf = 0.3 and shear rate γ = 10−3 . The horizontal lines are for c American Institute of T = 0.3 and Teff = 0.65. Reproduced from Berthier and Barrat (2002).  Physics.

456

The nonequilibrium dynamics

Fig. 9.8 The probability distribution function P(vz ) of the velocity vz for various massive tracers (m tr ∈ [1, 107 ] as shown on the figure) of type A in the BMLJ system. Plots of P(vz ) for different tracer particles follow Gaussian shapes (shown with full lines) with corresponding effective temperatures. For m tr = 102 , Teff = 0.3 (the equilibrium case); for m tr = 104 –105 , Teff = 0.44–0.54 (the crossover region); and for m tr = 106 –107 , Teff = 0.65 (the asymptotic effective temperature). c American Institute of Physics. Reproduced from Berthier and Barrat (2002). 

represents a low-frequency filter coupled to fluctuations in the host fluid. Hence they act as thermometers measuring the corresponding effective temperatures in the nonequilibrium steady state over the characteristic time scales (determined by the corresponding inverse frequencies) of the respective tracer particles. In Fig. 9.8 it is shown that the probability distribution of the velocity P(vz ) follows a Gaussian form,   9 m tr vz2 m tr P(vz ) = exp − , (9.2.20) 2π Teff 2Teff which implies a Maxwellian distribution form with Teff replacing the temperature.

9.3 A mean-field model In this section we consider a theoretical model for the slow dynamics in the nonequilibrium state. The mean-field theory considered here reduces in the equilibrium limit to a form very similar to the mode-coupling model discussed in the previous chapter. So far in our discussion of the mode-coupling theory we have assumed that time translational invariance (TTI) holds in the supercooled liquid state, i.e., the correlation of the fluctuations at times t and t  depends only on the difference t − t  of the two times. However, in the case of glassy relaxation, when the supercooled liquid is out of equilibrium, the TTI property does not hold. To be more specific, the dynamic correlation C(t, t  ) between two times depends both on t and on t  and is not a function of t − t  only. We consider here the mean-field

9.3 A mean-field model

457

dynamics of the spherical p-spin model in a situation in which TTI does not hold. The time evolution of the p-spin spherical model is given by the dissipative Langevin equation, 0−1

∂ δH σi (t) = z(t)σi (t) − + ςi (t), ∂t δσi (t)

(9.3.1)

where 0−1 is a bare kinetic coefficient. Here z(t) is the Lagrange multiplier which ensures the spherical constraint on the spins {σi }, N 1  2 σi = 1. N

(9.3.2)

i=1

To reach the dynamical equations for the correlation function averaged over the noise ςi , in a situation in which TTI does not hold, we start from the Schwinger–Dyson equation, A 2). G0−1 (1, 2) = G −1 (1, 2) − (1,

(9.3.3)

In the MSR formalism G has elements corresponding to a set of slow variables {ψα } (to ˆ The be denoted as α) as well as a conjugate set of hatted fields {ψˆ α } (to be denoted as α). matrix structure of this equation contains both the correlation and the response functions as elements of the matrix G. An important property of this matrix is that the elements of −1 , with both the indices as unhatted fields must be zero. This is required its inverse, i.e., Gψψ by the structure of the MSR field theory. See the text given above eqn. (7.2.31) for proof of this. Then [G −1 ]αα = .

(9.3.4)

Hence the inverse of the full matrix G −1 takes the form ⎡ ⎤  [R0−1 − ]α βˆ ⎢ ⎥ G −1 ≡ ⎣ ⎦. ˜ ˆ [R −1 − ] − 0

αβ ˆ

(9.3.5)

αˆ β

In writing the above expression for G −1 , we denote the elements of the self-energy matrix corresponding to the response functions, i.e., with the matrix indices being hatted–unhatted ˆ ˆ fields, by  and the elements of the G −1 matrix between hatted fields (i.e., ψˆ ψ) (i.e., ψ ψ) by . Considering the general form of the MSR action (see eqn. (8.4.22)) and taking into account the Gaussian random nature of the noise terms whose correlations are expressed 0 , we obtain for  ˜ on the RHS of (9.3.5) in terms of the bare transport coefficients αβ 0 ˜ ˆ (t, t  ) = 2αβ  δ(t − t  ) + αˆ βˆ (t, t  ). αˆ β

(9.3.6)

The explicit form of the operator R0−1 in the case of the p-spin model is obtained from the linear part of the equation of motion (9.3.1) for a single-spin variable σ as

∂ + z(t) . (9.3.7) R0−1 ≡ ∂t

458

The nonequilibrium dynamics

On taking the inverse of the above matrix to obtain G we have ⎤ ⎡ ˜ ˆ Rˆ Rα νˆ  R ˆ νˆ δ δβ αβ ⎥ ⎢ G≡⎣ ⎦, Rαβ  ˆ

(9.3.8)

˜ in terms of the single σˆ σˆ element  of the self-energy matrix as where we write  ˜ t  ) = 2δ(t − t  ) + (t, t  ), (t,

(9.3.9)

with time being rescaled with the bare transport coefficient 0 . Therefore the correlation of the hatted fields Gψˆ ψˆ remains equal to zero (Martin et al., 1973; Kirkpatrick and Thirumalai, 1987a) in order to maintain causality. For the correlation function elements we obtain the relation ˜ C = R R. (9.3.10) On applying the operator R to the Schwinger–Dyson equation for the response elements we obtain R0−1 R = δ(t − t  ) +  R.

(9.3.11)

For the correlation function matrix C we obtain from eqn. (9.3.10) ˜ R −1 C = R.

(9.3.12)

Now, on replacing R −1 = R0−1 −  using the Schwinger–Dyson equation, we obtain ˜ + C. R0−1 C = R

(9.3.13)

From eqns. (9.3.13) and (9.3.11), respectively, we obtain the following equations for the dynamics of the correlation and the response functions between times t and t  :  t ∂ dt  (t, t  )R(t  , t  ) C(t, t  ) = −z(t)C(t, t  ) + 2R(t  , t) + ∂t 0  t + dt  (t, t  )C(t  , t  ), (9.3.14) 0

∂ R(t, t  ) = −z(t)R(t, t  ) + δ(t − t  ) + ∂t

 t

t

dt  (t, t  )R(t  , t  ).

(9.3.15)

For the sake of definiteness we take t > t  . The fact that the time-ordered property of the response function, i.e., R(t, t  ), is zero for t < t  due to causality has been used in writing eqns. (9.3.14) and (9.3.15). The nonequilibrium dynamics for all times t and t  are thus given by the first-order integro-differential equations which admit a unique solution for finite times. At equal times the conditions maintained are C(t, t) = 1, ∂ C(t, t ± ) = ±1. ∂t

R(t, t − ) = 1,

(9.3.16) (9.3.17)

9.3 A mean-field model

459

The spherical constraint and the boundary conditions at t = t  require that the Lagrange multiplier z(t) is determined in terms of the kernel functions. We apply eqn. (9.3.14) for the case t  = t + and obtain, using the above boundary conditions (9.3.16),  t ! " ds (t, s)R(t + , s) + (t, s)C(t + , s) . (9.3.18) z(t) = 1 + 0

We have not specified yet the form of the self-energy matrices or the kernels  and  which represent the effect of nonlinearities in the equations of motion. Further discussion of the dynamics requires specification of these kernels.

9.3.1 The mode-coupling approximation In our discussion so far we have absorbed all the effects of the nonlinearities into the selfenergy matrix elements  and . For the p-spin Hamiltonian (8.4.2) discussed in Chapter 8, the kernels  and  are obtained as (t, t  ) = μ( p − 1)C p−2 (t, t  )R(t, t  ), (t, t  ) = μC p−1 (t, t  ),

(9.3.19)

pβ 2 /2

with β = 1/T the inverse temperature. For subsequent treatment of where μ = the nonequilibrium discussion we generalize the expressions (9.3.19) for the self-energy or the memory-kernel matrix elements (which are consequences of the nonlinear dynamics) to the following general forms involving polynomials expressed in terms of a series of coupling constants {a p }: (t, t  ) =

∞ 

a p ( p − 1)C p−2 (t, t  )R(t, t  ),

p=2

(t, t  ) =

∞ 

a p C p−1 (t, t  ).

(9.3.20)

p=2

The above polynomial form for the memory functions in the context of MCT of liquids was proposed by Das et al. (1985a). It has very often been used in the mode-coupling literature with respect to schematic models (Götze, 1991). Recently, such forms for memory kernels have also been argued (Singh and Das, 2009) to apply for amorphous solids, on treating individual particle displacements as soft spins. We use this general form for the memory function for discussion of the asymptotic dynamics here. This approach of writing the kernel functions  and  as functionals of the correlation and response functions constitutes the basic ingredient of the mode-coupling approximation. It has also been argued (Bouchaud et al., 1996) that, in systems where the force term in the Langevin equation of the slow modes is obtained from a potential and detailed balance holds, the following relation holds: 

(t, t  ) =  [C(t, t  )]R(t, t  ),

(9.3.21)

460

The nonequilibrium dynamics 

with  (x) = d(x)/d x. For systems in which the equality (9.3.21) does not hold, there is violation of the detailed balance. The mode-coupling dynamics with kernels satisfying a modified version of eqn. (9.3.21) in the simple form (with a parameter μ) ¯ 

(t, t  ) = μ ¯ [C(t, t  )]R(t, t  )

(9.3.22)

has been taken (Cugliandolo et al., 1997a) to describe driven systems. The asymptotic dynamics for both t and t  large is obtained from the solution of the differential equations (9.3.15) and (9.3.14). The solutions of these equations are obtained (Cugliandolo and Kurchan, 1993) in analytic form for the following two broad regimes. A crucial factor driving the nonequilibrium dynamics comes from z(t), which is explicitly time-dependent. A. Both t and t  are large such that (t − t  )/t → 0. Over this time scale the difference between the two time arguments in the two-point correlation functions is small and translational invariance is satisfied. This is the region in which the fluctuation– dissipation theorem (FDT) holds, and will be referred to as the FDT regime. B. Both t and t  are large such that (t − t  )/t ∼ O(1). Over this time scale the difference between the two time arguments in the two-point correlation functions is large and nonequilibrium dynamics does not obey time translational invariance. This will be referred to as the aging regime. We discuss the dynamics in these two regimes below.

9.3.2 The FDT regime In the FDT regime, the correlation function C(t, s) and the response function R(t, s) are only functions of the difference of the two times (t − s) as a result of time translational invariance (TTI). Let these be denoted by CF and RF , respectively, being related through the FDT expression RF (t − s) = (t − s)∂s CF (t − s),

(9.3.23)

where ∂s denotes the derivative with respect to s and we have absorbed the kB T factor into the definition of the correlation function. In this case, it follows directly from the expressions (9.3.20) for the memory functions that the corresponding TTI quantities are related: ∂ (9.3.24) F (t − t  ) =  F (t − t  ). ∂t In the following we will show that eqns. (9.3.14) and (9.3.15) for the correlation and response functions, respectively, reduce to a single equation for their corresponding TTI counterparts CF and RF . Let us consider the equation for the correlation function (9.3.14). The two integrals on the RHS are rearranged to write the equation in the form

9.3 A mean-field model





∂ + z(t) C(t, t  ) = ∂t



t

461

! " ds (t, s)R(t  , s) + (t, s)C(s, t  )

0



−

t t

ds (t, s)R(t  , s).

(9.3.25)

We consider first the second integral on the RHS. In the FDT limit with s lying between t and t  , TTI holds and R(t  , s) ≡ R(t  − s) is related to the correlation function C(t  − s) through the relation (9.3.23). Hence the second term on the RHS reduces to the form  t ∂ ds F (t − s) CF (t  − s), (9.3.26) − ∂s t where the subscript F denotes the FDT regime in which TTI holds. Next we consider the first integral on the RHS of eqn. (9.3.25). The limits of this definite integral are split into two parts,  t  t−δ  t ! " I (t, t  ) = ds (t, s)R(t  , s) + (t, s)C(s, t  ) ≡ + . (9.3.27) 0

0

t−δ

t )

The time scale δ is chosen such that for the FDT regime (of t and in the second integral on the RHS of (9.3.27) the relation (9.3.23) holds. Thus δ represents a time scale much smaller than t or t  . On the other hand, for the first integral on the RHS of (9.3.27) the correlation and response functions have time arguments that are sufficiently separated that TTI does not hold. This is the aging regime. The RHS of (9.3.27) is thus obtained as  t−δ ! " I (t, t  ) = ds A (t, s)RA (t  , s) + A (t, s)CA (s, t  ) 0



+

t

! " ds F (t, s)RF (t  , s) + F (t, s)CF (s, t  )

t−δ

≡ IA + IF ,

(9.3.28)

where the subscripts A and F, respectively, denote the corresponding quantities for the FDT and aging regimes. In the aging regime the correlation and response functions are written in terms of scaling functions as 



h(t ) h  (t  ) h(t )   , RA (t, t ) = R , (9.3.29) CA (t, t ) = qφ h(t) h(t) h(t) . where h(t) is a monotonically ascending function of time and the prime indicates a derivative with respect to its argument. q is the nonergodicity parameter and the scaling function has the property φ(λ → 1) = 1. This is in keeping with the form of the correlation and response functions seen in simulations and experiments on aging. In the simplest case, when h(t) ∼ t (Bouchaud, 1992; Bouchaud and Dean, 1995), the above relations reduce to CA (t, t  ) = qφ(t  /t),

RA (t, t  ) = t −1 R(t  /t).

(9.3.30)

462

The nonequilibrium dynamics

Using the ansatz (9.3.29), the integral IA in (9.3.28) reduces to the form  IA =

t−δ

dt 

0

∞  p=2

dλ a p q p−1 φ p−1 (λ )  R(λ ) dt + ( p − 1)q

 =

1−δ/t

dλ

0

∞ 

p−2

φ

p−2

dλ   (λ )  R(λ )qφ(λ /λ) dt 

 a p q p−1 R(λ ) φ p−1 (λ ) + ( p − 1)φ p−2 (λ )φ(λ /λ) ,

p=2

(9.3.31) where we have changed the integration variable t  to λ = h(t  )/ h(t) and defined λ = h(t  )/ h(t). In the limit t, t  → ∞ and with δ much smaller than both t and t  , this leads to λ → 1. The integral IA is now obtained as IA =

∞ 

a p pq p−1 Iq ,

(9.3.32)

dλ φ p−1 (λ )R(λ ).

(9.3.33)

p=2

where the integral Iq is defined as  Iq =

1

0

The integral IF in (9.3.28) is calculated by applying the relation (9.3.23). Using the explicit forms for  and , we obtain IF =

∞ 

 ap

p=2

=

∞  p=2

=

∞ 





∂ ∂ p−1 p−1 ds CF (t − s) CF (t  − s) + CF (t  − s) CF (t − s) ∂s ∂s t−δ t

t

ap

ds t−δ

∂  p−1 CF (t − s)CF (t  − s) ∂s

! " a p CF (t − t  ) − q p .

(9.3.34)

p=2

In writing the last equality we have used the constraint condition that C(t, t) = CF (0) = 1 and that the time scale δ by definition ensures that both CF (δ) and CF (δ − (t − t  )) → q. Now, by combining the results of (9.3.32) and (9.3.34), we obtain for the integral I (t, t  ) the result ∞   I (t, t  ) = a p {CF (t − t  ) − q p } + pq p−1 Iq . (9.3.35) p=2

9.3 A mean-field model

463

Evaluating the RHS of eqn. (9.3.25), using the results of (9.3.26) and (9.3.35), and making some rearrangement of variables, eqn. (9.3.14) finally reduces to the form  τ ∂ ∂ dτ1 F (τ − τ1 ) CF (τ1 ) CF (τ ) = −z ∞ CF (τ ) − ∂τ ∂τ1 0 +

 a p CF (τ ) − q p + pq p−1 Iq ,

∞ 

(9.3.36)

p=2

where z ∞ denotes the Lagrange multiplier z(t) in the limit t → ∞. We can rearrange the above equation further to obtain the following result for the TTI correlation function C(t) (we drop the subscript F from now on for simplicity):

 t ∂ ∂ dτ (t − τ ) C(τ ) + z˜ ∞ [C(t) − 1] + 1 C(t) + ∂t ∂τ 0 = 1 − z∞ +

∞ 

 a p 1 − q p + pq p−1 Iq ,

(9.3.37)

p=2

where z˜ ∞ is defined as z˜ ∞ = z ∞ − 1 −

∞ 

ap.

(9.3.38)

p=2

The RHS of eqn. (9.3.37) is computed from the expression for z(t) given by eqn. (9.3.18). From the definition (9.3.27) it follows that z(t) = 1 + I (t, t + ).

(9.3.39)

On taking the t → ∞ limit of the above equation, we obtain, by applying the constraint C(t, t) = 1 and using the result (9.3.35), z∞ = 1 +

∞ 

 a p 1 − q p + pq p−1 Iq ,

(9.3.40)

p=2

implying that the RHS of eqn. (9.3.37) is zero. The TTI equation for the correlation function in the FDT regime is therefore obtained as 

  t ∂C(τ ) ∂ dτ (t − τ ) + 1 C(t) + + z˜ ∞ [C(t) − 1] = 0. ∂t ∂τ 0

(9.3.41)

Equation (9.3.15) for the response function R also reduces to the same TTI equation (9.3.41) in the FDT limit. To see this, it is best to work with the integrated response function M(t, t  ) defined as  t M(t, t  ) = R(t, s)ds. (9.3.42) t

464

The nonequilibrium dynamics

In the FDT regime, on applying (9.3.23), the above definition reduces to  t ∂  C(t, s)ds = 1 − C(t, t  ), M(t, t ) = t  ∂s

(9.3.43)

since the spherical constraint ensures that C(t, t) = 1. Now, by integrating eqn. (9.3.15) for the response function R(t, s) from s = t  to t, we obtain the equation  t  t ∂ ds dt  (t, t  )R(t  , s). (9.3.44) M(t, t  ) = −z(t)M(t, t  ) + 1 + ∂t t s Let us focus on the integral in the last term on the RHS of (9.3.44). We show in Appendix A9.2 that this integral reduces to the form  τ ∞  ∂ dτ  (τ − τ  )  C(τ ) + [1 − C(τ )] ap, (9.3.45) ∂τ 0 p=2

where τ = t − t  . Thus eqn. (9.3.44) reduces in the FDT regime to the following form:  τ ∞  ∂ ∂ dτ  (τ − τ  )  C(τ ) + [1 − C(τ )] ap. M(τ ) = −z(t)M(τ ) + 1 + ∂τ ∂τ 0 p=2

(9.3.46) Using the relation (9.3.43) in the FDT limit as M(τ ) = 1 − C(τ ), for large t the above equation reduces to

 τ ∂ ∂ dτ  (τ − τ  )  C(τ  ) = 0, (9.3.47) + 1 C(τ ) + z˜ ∞ [C(τ ) − 1] + ∂τ ∂τ 0 where z˜ ∞ is as given by eqn. (9.3.38). Thus both the correlation-function and the responsefunction equations, (9.3.14) and (9.3.15), respectively, reduce in the FDT limit to the identical form given by either eqn. (9.3.47) or eqn. (9.3.41). This form is very similar to the basic equation (8.1.8) of the mode-coupling theory for structural glasses (discussed earlier in Section 8.1.1), apart from there being an inertial term in the latter. On the other hand, the term proportional to z˜ ∞ is peculiar to the case of the spherical spin model. The latter makes the dynamics somewhat different in this case below the dynamic-transition point. From the long-time limit of eqn. (9.3.47) we obtain for the nonergodicity parameter (NEP), i.e., the long-time limit of the correlation function q, the following equation: − z∞ +

∞  p=2

a p (1 − q p−1 ) = −

1 , 1−q

(9.3.48)

where z ∞ is given by the relation (9.3.40). The solution of these equations gives the value of the NEP as well as the dynamic-transition point (in terms of the critical couplings a p ). However, for obtaining z ∞ we need the solution for the correlation and response functions in the aging regime involved in the definition of Iq (see eqn. (9.3.33)). This is discussed next.

9.3 A mean-field model

465

9.3.3 The aging regime For t and t  both going to infinity in such a way that (t − t  )/t  ∼ O(1) analytic solutions for the correlation and response functions in the aging regime, denoted by CA (t, t  ) and RA (t, t  ), respectively, are obtained from (9.3.14) and (9.3.15) by separating the integrals appearing in each equation into contributions coming from the aging and FDT regimes. In the present case both t and t  are large, and also they are very widely separated. Two times δ and δ  enter the problem, respectively separating (as in the previous section) the corresponding time scales t and t  into FDT and aging regimes. Since t and t  are very widely separated, the time derivatives on the LHS of both (9.3.14) and (9.3.15) can be ignored. We analyze the two equations below. The equation for the correlation function From eqn. (9.3.14) for the correlation function C(t, t  ) we obtain, on setting the LHS of this equation to zero in the aging regime, 



0 ≈ −z ∞ C(t, t ) +  +

t−δ

t  −δ 





ds A (t, s)RA (t , s) +

0

ds A (t, s)C(s, t  ) +

0



t

t

t  −δ 

ds A (t, s)RF (t  , s)

ds F (t, s)CA (s, t  ).

(9.3.49)

t−δ

The subscripts A and F in the kernels  and  denote the corresponding quantities in the aging and FDT regimes, respectively. The integrals on the RHS of eqn. (9.3.49) are denoted as Ii (i = 1, . . . , 4). Equation (9.3.49) therefore reduces to 0 = −z ∞ qφ(λ) +

4 

Ii .

(9.3.50)

i=1

Note that in the third integral (I3 ) on the RHS of eqn. (9.3.49) we do not indicate a subscript on the correlation function C(s, t  ). As s varies from 0 to t − δ the correlation function C(s, t  ) passes through both the aging and the FDT regime. For 0 ≤ s ≤ t  − δ  and also for t  + δ  ≤ s ≤ t − δ, the correlation function has the aging behavior CA ; and for the range t  − δ  ≤ s ≤ t  + δ  it has FDT behavior CF . Therefore this integral I3 separates into three integrals as follows: 

t  −δ 

I3 =



ds A (t, s)CA (s, t ) +

0



+

t−δ t  +δ 



t  +δ 

t  −δ 

ds A (t, s)CF (s, t  )

ds A (t, s)CA (s, t  )

≡ I31 + I32 + I33 .

(9.3.51)

466

The nonequilibrium dynamics

We compute the integrals Ii (i = 1, . . . , 4) on the RHS of eqn. (9.3.50) in Appendix A9.2.2 to obtain the following result: ⎤ ⎡ ∞ ∞   a p (1 − q p−1 ) + (1 − q) a p q p−2 φ p−2 (λ)⎦ 0 = qφ(λ) ⎣−z ∞ + +

∞ 

&

p=2

p=2



1

a p q p−1 ( p − 1)



dλ R(λ )φ p−2 (λ )φ({λ /λ}sg(λ−λ ) )

0

p=2



1

+

'

dλ φ p−1 (λλ )R(λ ) .

(9.3.52)

0

We substitute for the first two terms within the square brackets on the RHS of (9.3.52) using the relation (9.3.48) and obtain ⎤ ⎡ ∞  1 a p q p−2 φ p−2 (λ)⎦ 0 = qφ(λ) ⎣− + (1 − q) 1−q p=2 &  1 ∞   a p q p−1 ( p − 1) dλ R(λ )φ p−2 (λ )φ({λ /λ}sg(λ−λ ) ) + 0

p=2



1

+



dλ φ

' p−1





(λλ )R(λ ) .

(9.3.53)

0

The equation for the response function From eqn. (9.3.15) for the response function R(t, t  ) we obtain, on setting the LHS of this equation to zero in the aging regime,  t  +δ ∂   0≈ ds A (t, s)RF (s, t  ) RA (t, t ) = −z ∞ RA (t, t ) + ∂t t  t−δ  t + ds A (t, s)RA (s, t  ) + ds F (t, s)RA (s, t  ). (9.3.54) t  +δ

t  −δ

In writing the expression for the response function in the aging regime, we express the RHS of the above equation in terms of integrals Ji (i = 1, . . . , 3) as 0 = −z ∞ t −1 R(λ) + J1 + J2 + J2 .

(9.3.55)

We evaluate the integrals J1 , J2 , and J3 in Appendix A9.2.2 and, using J1 , J2 , J3 , and z ∞ in eqn. (9.3.55), we obtain the relation ⎡ ⎤ ∞ $ #  0 = R(λ) ⎣−z ∞ − a p ( p − 1)q p−2 φ p−2 (λ)[q − 1] + [1 − q p−1 ] ⎦ p=2

+

∞  p=2

a p ( p − 1)q



1

p−2 λ

dλ p−2  φ (λ )R(λ )R(λ/λ ). λ

(9.3.56)

9.3 A mean-field model

467

Now, using the relation (9.3.48) in the RHS of (9.3.57), we simplify it further to the form ⎤ ⎡ ∞  1 a p ( p − 1)q p−2 φ p−2 (λ)⎦ + (1 − q) 0 = R(λ) ⎣− 1−q p=2

+

∞ 

& a p ( p − 1)q

1

p−2 λ

p=2

' dλ p−2    φ (λ )R(λ )R(λ/λ ) . λ

(9.3.57)

We have now obtained from the above analysis eqns. (9.3.53) and (9.3.57) treating eqns. (9.3.14) and (9.3.15), respectively, in the aging regime. These two relations between the correlation and response functions in the aging regime should also be consistent with necessary boundary conditions so as to match with the corresponding solutions in the FDT regime. Therefore in the limit t → t  or λ → 1 we have φ(λ = 1) ∼ 1 and R(λ = 1)  = 0. Using these conditions and eqns. (9.3.53) and (9.3.57) in the limit λ → 1, we obtain ⎡ ⎤ ∞ ∞   1 a p q p−2 ⎦ + a p pq p−1 Iq = 0, (9.3.58) + (1 − q) q ⎣− 1−q p=2

p=2

∞

−

 1 + a p ( p − 1)q p−2 = 0. 2 (1 − q)

(9.3.59)

p=2

The dynamic-transition point and the corresponding NEP q are computed from the solution of the equation (1 − q)−2 =

∞ 

a p ( p − 1)q p−2 .

(9.3.60)

p=2

By substituting eqn. (9.3.59) into eqn. (9.3.58) we obtain (1 − q)

∞ 

a p ( p − 2)q p−1 =

p=2

∞ 

a p pq p−1 Iq .

(9.3.61)

p=2

To evaluate the integral Iq on the RHS we need more information on the correlation and response functions. The correlation and response functions in the aging regime are no longer linked through the FDT. We consider here the modified relation (9.1.1) in terms of a constant FDT-violation parameter m as R A (t, t  ) = m

∂ C A (t, t  ). ∂t 

(9.3.62)

Using the solution (9.3.29) in the aging regime, the corresponding relation of the scaling functions is obtained as ∂ R(λ) = mq φ(λ). (9.3.63) ∂λ

468

The nonequilibrium dynamics

Now we evaluate the RHS of (9.3.61) involving the integral Iq , ∞ 

a p pq p−1 Iq = m

p=2

∞ 



∞ 



∞ 

1

a pq p

dλ

0

p=2

=m

dλ φ p−1 (λ )

0

p=2

=m

1

a p pq p

dφ(λ ) dλ

*1 * d p  * φ (λ ) * dλ 0

a pq p.

(9.3.64)

p=2

Therefore we obtain from eqn. (9.3.61) the following expression for the FDT-violation parameter m: 5∞ p−2 ( p − 2) p=2 a p q 5∞ m = (1 − q) . (9.3.65) p−1 p=2 a p q Let us now consider two special cases. (a) Case I Here we include only the quadratic term in the expression for the memory kernel containing only the p = 3 term, i.e., (t, t  ) = a3 C 2 (t, t  ). In this case we have the following two equations for q and m: (1 − q)−2 = 2a3 q,

(9.3.66)

m = q −1 − 1.

(9.3.67)

For the state in which the FDT is satisfied we have m = 1 and hence q = 12 and a3 = 4. The coupling-constant value a3 = 4 marks the transition to the nonergodic state. Note that for this case we have, from (9.3.48) and (9.3.38), that the constant z˜ ∞ = 0, rendering eqn. (9.3.47) in the FDT regime the same as that in the schematic model of a structural glass. (b) Case II We consider the situation in which the kernel includes the p = 2 and 3 terms in the expansion (t, t  ) = a2 C(t, t  ) + a3 C 2 (t, t  ).

(9.3.68)

The two equations involving a2 and a3 are obtained as (1 − q)−2 = a2 + 2a3 q,

m=

a3 q(1 − q) . a2 q + a3 q 2

(9.3.69)

For the ergodic state in which the FDT is satisfied, i.e., m = 1, the above two equations for a2 and a3 are solved to obtain a2 =

1 − 2q , (1 − q)2

a3 =

1 . (1 − q)2

9.3 A mean-field model

469

On eliminating q from the above relation we obtain the following line of dynamic transition: √ (9.3.70) a2 = 2 a3 − a3 . In the equilibrium limit this model is identical to what is termed the φ12 model in the literature and gives rise to the stretched-exponential relaxation. The relaxation behavior for the equilibrium correlation function for this model has been discussed in detail in Chapter 8. Numerical solution of eqns. (9.3.14) and (9.3.15) (Singh and Das, 2010, unpublished) gives the correlation and response functions, respectively, in the nonequilibrium states. The different stages of the dynamics as depicted in the above analysis are displayed in Fig. 9.9 in a form similar to the corresponding results from computer simulations shown in Fig. 9.5.

9.3.4 Quasi-ergodic behavior Let us consider the behavior of the correlation and response functions in the two regimes which emerges from the above analysis. First, within the FDT regime the behavior follows the typical mode-coupling equation given by (9.3.47). The solution for the correlation function as discussed in the previous chapter is of the form

1.0

-M

0.5

0.5

0.0

10–3

0.0 0.0

1.0

0.5

C

100

t

103

106

Fig. 9.9 Results from the theoretical model with the kernel given by eqn. (9.3.68), showing the quantity [1 − Ck (t)]/T involving the correlation function (solid line) and the integrated response function −Mk (t) (dashed line) vs. time t. The parameter values are a2 = 0.5 and a3 = 6.0. The two curves overlap when the FDT is satisfied for small t and for larger values of the correlation functions. This is qualitatively similar to the simulation results displayed in Fig. 9.4. The two curves become identical in equilibrium and the corresponding solution is shown with the dot–dashed line (Singh and Das, 2010, unpublished).

470

The nonequilibrium dynamics

CF (t, t  ) ≡ CF (t − t  ) = q + B1 τ −a + · · ·.

(9.3.71)

The correlation function decays over the time scale of the FDT regime to the plateau value given by the nonergodicity parameter q. On the other hand, within the aging regime the scaling form for the correlation function follows C(t, t  ) ∼ qφ(λ), where λ = h(t  )/ h(t). For τ = t −t  small compared with t, this result is expressed in terms of a Taylor expansion around λ = 1 as  C(t, t  ) ≡ qφ(λ) = q + Ai (1 − λ)i = q − A1 {τ κ(t)} + · · ·. (9.3.72) i

By expanding λ ≡

h(t  )/ h(t)

around λ = 1 (t = t  ) we obtain κ(t) as κ(t) =

d ln h(t) . dt

(9.3.73)

κ(t) ∼ t −1 corresponds to the case of simple aging, i.e., h(t) ∼ t γ with exponent γ . However, a more general form of the asymptotic dynamics is envisaged with the choice κ(t) = t −μ , which gives for the aging function h(t) the result 1−μ

t , (9.3.74) h(t) = exp 1−μ the corresponding integration constant being chosen to be unity. The case of μ = 0 corresponds to the case h(t) ∼ exp(t) and hence h(t + tw )/ h(tw ) ∼ exp(t), implying time-translational invariance of the correlation function. On the other hand, μ = 1 corresponds to h(t) ∼ t, which is termed simple aging (h(t + tw )/ h(tw ) ∼ f (t/tw ). Thus, the sub-aging behavior will give 0 < μ < 1. Numerical solutions of eqns. (9.3.14) and (9.3.15) were obtained by Kim and Latz (2003) for the p = 3 case in which the memory functions are given by eqns. (9.3.18) and (9.3.19) with μ = pβ 2 /2 (β = 1/kB T is the inverse temperature). For T = 0.61 the numerical solution conforms to the above empirical form of h(t) with μ = 0.82. For lower temperatures the exponent μ becomes larger to make the data for correlation and response functions collapse on a single curve. The behavior of the correlation function over the intermediate time scales is obtained by matching the solutions in the FDT regime and the aging regime. Let there be a crossover time scale t δ expressed in terms of the exponent δ (δ < 1). Over this time the nature of the correlation function changes from the FDT behavior (9.3.71) to the form given by (9.3.72). For large enough τ ∼ O(t) the decay crosses over to the aging behavior. The above two forms are similar to the power-law relaxation of equilibrium correlation functions discussed in the previous chapter. The von Schweidler-like exponent b is unity in this case. For the plateau region the solution is written in a general form in terms of the scaling functions gi as C(t, t  ) = C(t − t  ) = q + g1 (τ/t δ )t −α + · · ·.

(9.3.75)

The correlation function (9.3.75) reduces to the above two behaviors, respectively given by (9.3.71) and (9.3.72), in two limiting cases. In the FDT limit, matching (9.3.75) with

9.4 Glassy aging dynamics

471

the solution (9.3.71) requires that α = δa and gi (x) ∼ Bi x −ia for x 1, where the variable x is defined as τ = t − t  = xt δ . On the other hand, for the aging regime x  1, on comparing with (9.3.72) we obtain gi (x) ∼ x i . In this case the exponent δ for the crossover time scale is related to the exponent κ of the aging behavior (9.3.74) through the relation δ + α = μ (Andreanov and Lefèvre, 2006). The corresponding relation to the power-law exponent a of the FDT regime is δ(1 + a) = μ. In the final relaxation process over the time scale of τ ∼ O(t), the behavior of the correlation function is controlled by φ(λ), always decaying to zero since h(t) is a monotonically ascending function of time. Thus, for any time t  , if one waits long enough, there is always a t for which the correlation function decays to zero. This is termed quasi-ergodic behavior. The nonequilibrium dynamics of the mean-field p-spin model discussed so far in this section has not been extended to deal with the structural-glass problem involving multiple slow variables. However, in a somewhat simplified approach, these mean-field equations for the time evolutions of the correlation and response functions have been applied to study nonequilibrium dynamics in some cases, e.g., in the problem of nonlinear rheology (Berthier et al., 2000) and for the dynamics of hetero-polymers (Pitard and Shakhnovich, 2001). 9.4 Glassy aging dynamics The effective temperature Teff for the nonequilibrium state defined above refers to a particular mode O of the system. However, as the system equilibrates the effective temperatures linked to the different such observables should all approach the common bath temperature. The process of thermalization in the dynamics of the nonequilibrium state can be understood using the equations of motion of the mean-field model discussed above. 9.4.1 Thermalization To generalize the above formulation, let us consider here a set of slow modes labeled as a, b, c, . . . for which we write down the dynamical equations for the correlation C = (Cab ) and the response function R = (Rab ). This is done in the same manner as was done before for the single-spin model (see eqns. (9.3.14) and (9.3.15)),  ∂ z ac (t)Ccb (t, t  ) + 2Rab (t  , t) Cab (t, t  ) = − ∂t c   t dt  ac (t, t  )Rcb (t  , t  ) + +

c t

0

dt 



0

c

t

c

ac (t, t  )Ccb (t  , t  ),

 ∂ z ac (t)Rcb (t, t  ) + δ(t − t  )δab Rab (t, t  ) = − ∂t ct  + dt  ac (t, t  )Rcb (t  , t  ).

(9.4.1)

(9.4.2)

472

The nonequilibrium dynamics

For the sake of definiteness we take t > t  . The time-ordered property of the response function, i.e., Rab (t, t  ) = 0 for t < t  due to causality, has been used in writing eqns. (9.4.1) and (9.4.2). As before, we have two qualitatively different types of regimes characterizing the dynamics. (a) The quasi-equilibrium or FDT regime for t, t  → ∞ and t − t  finite. In this case TTI holds and we have the usual FDT relation 1 ∂ Cab (t − t  ), (9.4.3) Rab (t − t  ) = T ∂t  where we have used the units in which kB = 1. Rab (t, t  ) vanishes for t  > t due to causality. The memory functions of the corresponding TTI quantities are related, ab (t − t  ) =

1 ∂ ab (t − t  ). T ∂t 

(9.4.4)

In this regime eqns. (9.4.1) and (9.4.2) for the correlation functions and response functions reduce to a single equation of the equilibrium MCT. (b) The aging regime in which t, t  → ∞ such that (t − t  )/t ∼ O(1). The corresponding correlation and response functions obey the relations

h ab (t  )  ˜ ˜ , (9.4.5) Cab (t, t ) = Cab h ab (t) m ab ∂ ˜ Cab (t − t  ), R˜ ab (t − t  ) = T ∂t 

(9.4.6)

where the h ab (t) are monotonically ascending functions of time and the m ab are constants. The equations for the time evolution of the correlation and response functions reduce to a closed set with a suitable ansatz for the different effective temperatures. Cugliandolo et al. (1997b) proposed two types of scenarios concerning the effective temperatures for the nonequilibrium dynamics. Each scheme represents a corresponding characteristic thermalization process. I. In the quasi-equilibrium or FDT regime the effective temperatures corresponding to two different observables O1 and O2 are equal to each other and equal to the temperature of the bath. Here the FDT is satisfied and we have for the memory functions, to which the corresponding TTI quantities are related, ab (t − t  ) =

1 ∂ ab (t − t  ). T ∂t 

(9.4.7)

In the aging regime the effective temperatures corresponding to O1 and O2 are equal to each other but not necessarily equal to that of the heat bath, m 11 ∂ C11 (t − t  ), R˜ 11 (t − t  ) = T ∂t  m 22 ∂ C22 (t − t  ), R˜ 22 (t − t  ) = T ∂t 

(9.4.8) (9.4.9)

9.4 Glassy aging dynamics

473

with m 11 = m 22 . The behavior of effective temperature in the aging regime being independent of the observable is termed thermalized aging. In this situation O1 and O2 are strongly coupled, and for the off-diagonal elements of the correlation and response functions we have m¯ ∂ ˜ R˜ 12 (t − t  ) = C12 (t − t  ), (9.4.10) T ∂t  where m¯ > 0 and is of the same order as the similar quantities appearing in the corresponding relations for self-response functions R11 , R22 , etc. Thus m 11 = m 22 = ¯ The corresponding memory functions in the aging regime satisfy m 12 = m 21 ≡ m. ˜ ab (t − t  ) = 

m¯ ∂ ab (t − t  ), T ∂t 

∀a, b.

(9.4.11)

The scaling functions h ab (t) for the aging dynamics are the same for all a and b in this case, such that C˜ ab (t, t  ) = C˜ ab [h(t  )/ h(t)]. II. In the quasi-equilibrium or FDT regime the effective temperatures corresponding to two different observables O1 and O2 are equal to each other and equal to the temperature of the bath. This is the same as in scheme I. However, in the aging regime the effective temperatures are not equal to each other and neither is equal to that of the heat bath. Thus m 11  = m 22 . Therefore the effective temperature in the aging regime is different for different observables. In this situation O1 and O2 are effectively decoupled and hence for the off-diagonal elements of the correlation and response functions we have m 12 ∂ ˜ R˜ 12 (t − t  ) = C12 (t − t  ), (9.4.12) T ∂t  where m 12 , m 21 → 0. This is termed unthermalized aging behavior.

9.4.2 Aging dynamics: experiments In deeply supercooled liquids the relaxation is typically nonexponential, a behavior that has been linked in the recent literature to the heterogeneous nature of the dynamics of the liquid particles. Spatially extended and structurally heterogeneous regions appear in the nonequilibrium state and the dynamic properties of the undercooled liquid change with its aging. Lehny and Nagel (1998) studied the dielectric response of a typical glass-forming material, glycerol (fragility index m = 53), following a quench from a high temperature to one below the glass-transition temperature Tg . Those authors studied the dielectric response of the liquid for different values of the aging time tage , which is measured conveniently from the instant of quenching. The dielectric susceptibility (ν, tage ) =   (ν, tage ) + i  (ν, tage )

(9.4.13)

is a function of the frequency as well as the aging time. In the equilibrated system, for large enough tage the spectrum is time-independent and the imaginary part   (ν) displays a broad peak whose position νp decreases with temperature following a Vogel–Fulcher

474

The nonequilibrium dynamics

Fig. 9.10 Plots of   (ν, tage ) of glycerol vs. frequency ν following a quench from 206.2 K to 177.6 K corresponding to two different aging times: tage = 200 s (circles) and tage = 2 × 104 s (squares). The lines are equilibrium spectra at three temperatures: 177.6 K (solid), 180.1 K (dotted), and 182.5 K (dot–dashed). The spectra after 200 s and 2 × 104 s for the nonequilibrium liquid following this quench closely approximate the equilibrium spectra at 182.5 K and 180.1 K, respectively. c American Physical Society. Reproduced from Lehny and Nagel (1997). 

law. At T = Tg = 185 K the peak frequency for glycerol is νp ≈ 10−3 Hz. The aging experiment for the nonequilibrium state was done for frequencies above this value. For quenches to not far below Tg the   (ν, tage ) vs. ν curves for different values of tage mimic  (ν) vs. ν curve at a higher temperature. This behavior is disthe equilibrium response eq played in Fig. 9.10 for a quench from T = 206.2 K to T = 177.6 K. The lower the value of tage , the higher the temperature for the corresponding equilibrium curve. This important observation indicates that, at least for quenches close to the glass-transition temperature Tg , the spectral shape of the dielectric susceptibility changes similarly with time during the aging process. The identification of the aging system with the corresponding equilibrium one provides us with a way of associating a temperature with the out-of-equilibrium glassy state. This is similar to the phenomenological fictive temperature Tfic discussed above. For temperatures far below Tg such a mapping does not follow, however. Modified Kohlrausch–William–Watts (MKWW) relaxation Let us consider another important aspect of the aging behavior, i.e., the manner in which a typical response function for the nonequilibrium state (in this case that of the dielectric response) changes with the aging time tage . The dielectric-loss response function   for glycerol quenched to temperature 179 K (Tg = 185 K ) fits well to the Kohlrausch– Williams–Watts (KWW) form over the frequency range ν = 1Hz − 105 Hz. The fitted form is )   ( " !   .   tage = st − eq exp −(tage /τage )βage + eq

(9.4.14)

9.4 Glassy aging dynamics

475

The subscripts “st” and “eq” in the above formula respectively refer to the initial (tage → 0)  are obtained as correand long-time (tage → ∞) limiting values of   . Both st and eq sponding parameter values to obtain the best fit of the experimental data to the above form. The relaxation time τage and the stretching exponent βage are also treated as free fit parameters. Generally all these parameters for data fitting are dependent on the frequency ν for the corresponding response function. The τage or βage obtained in this procedure, however, do not agree with the corresponding equilibrium (extrapolated) parameter values. τα and βα .1 The stretching exponent βage is usually much smaller than βα . For example, in the case of glycerol βage ≈ 0.29 while the extrapolated βα = 0.55. An important observation in this regard was made by Lunkenheimer et al. (2005) who demonstrated that using a modified KWW (MKWW) fitting function the aging-time dependence of the above dielectric-response-function data for the different frequencies (ν) can be simultaneously fitted very well with frequency-independent τage and βage . The relaxation time τage in the stretched exponential function is itself dependent on the aging time tage (Zotev et al., 2003; Bissig et al., 2003). )     ( β     tage = st − eq . exp − tage /τage (tage ) age + eq

(9.4.15)

The aging-time dependence of τage (tage ) can be obtained in two different ways, implying two very different mechanisms for the aging process in the supercooled liquid. Let us consider both. I. Lunkenheimer et al. (2005) defined the dependence of the relaxation time τage on tage in the MKWW in terms of a corresponding “time-dependent” frequency νage introduced as 1 τ (tage ) = . (9.4.16) 2π νage (tage ) The aging-time dependence of νage is chosen in the parametric form       β νage tage = νst − νeq exp − 2π ν(tage )tage age + νeq .

(9.4.17)

In the above definition τage → 1/(2π νst ) and 1/(2π νeq ) as tage → 0 and ∞, respectively. An almost-perfect fit for the dielectric data over the whole frequency range is obtained with the above choice of the time dependence for the relaxation time. βage and βage are now the same for the response-function data at different frequencies ν = 1 − 105 Hz. This scaling of the response function is shown in Fig. 9.11. An important feature of the above fitting procedure is that the best-fit values for the stretching exponent βage for a number of glass-forming systems are found to be the same as those of the corresponding α-relaxation βα . Furthermore, the best-fit value obtained for νeq in (9.4.17) corresponds to a time τeq = 1/(2π νeq ), which agrees well with the corresponding α-relaxation time τα (for a given substance) obtained by extrapolation 1 At the sub T temperature, the equilibrium counterpart of τ , denoted τ , is obtained by extrapolating the results of the g age α corresponding equilibrium α -relaxation times at T > Tg .

476

The nonequilibrium dynamics

Fig. 9.11 Scaling of the dielectric-response function  of glycerol at 179 K for different frequencies. The fits are obtained with τage (tage ) in the modified KWW ansatz using two schemes (see the text): (i) eqns. (9.4.18) and (9.4.19) (the solid line) and (ii) eqns. (9.4.15)–(9.4.17) (the dashed line almost coincident with the solid line). Also shown are the curves obtained with the normal KWW form, in both of which (i) the stretching exponent is fixed at βα , and (ii) the relaxation time τage is respectively equal to τst (short-dashed line) and τeq (long-dashed line). From Lunkenheimer et al. (2005) and Sen c American Physical Society. Gupta and Das (2007). 

from higher temperatures (T > Tg ) to sub-Tg regions. In Fig. 9.12 the variation τα with inverse temperature 1/T is shown for a number of glass formers. The same figure also displays (with stars) the corresponding value of τeq = 1/(2π νeq ) obtained from the best-fit value in the above MKWW fitting scheme. It lies on the extrapolated curve of the α-relaxation times in the sub-Tg regime. The equilibrium α-relaxation thus seems to be closely related to the aging process. The stretching exponents being the same in both cases (βage = βα ) possibly implies that the stretching of the relaxation remains unaffected by aging and hence conforms to the validity of the time–temperature superposition during the aging process. Thus, if the aging behavior is indeed due to the structural rearrangements of heterogeneous regions, we expect that τage and βage are primarily controlled by the α-relaxation mechanism. II. As an alternative to the above scheme, an equally good fit to the aging data is obtained (Sen Gupta and Das, 2007) by choosing the tage dependence of the relaxation time τage in the MKWW function in a more natural manner τage (tage ) = (τst − τfn ) F(tage ) + τfn .

(9.4.18)

9.4 Glassy aging dynamics

477

Fig. 9.12 Plots of α-relaxation times τα from measurements of equilibrium dielectric properties for different glass formers (open symbols). The lines are fit with the Vogel–Fulcher–Tammann law. For CKN the structural-relaxation times from mechanical spectroscopy are also displayed (+ signs). The closed symbols show the best-fit values obtained from the fitting of the aging-time dependence of the c nonequilibrium data using eqns. (9.4.15)–(9.4.17). From Lunkenheimer et al. (2005). American Physical Society.

The function F (t) reduces to the value 1 or 0, for t → 0 and ∞, respectively, so that the relaxation time τage attains the asymptotic values τst and τfn respectively in the above two limits. A perfect fit is obtained for all the   (tage ) data at different frequencies using eq. (9.4.14) with τage (tage ) being determined with the ansatz (9.4.18). Among the different choices of the function f (t) in (9.4.18) the best-quality fit is obtained with (Sen Gupta and Das, 2007)

2 f (t) = 1 + exp{2t/τage (t)}

βage (9.4.19)

The fittings over the whole frequency range of the dielectric data are shown in Fig. 9.11, in which all the data scale into a single master curve. This fitting scheme suggests the following alternative scenario for explaining the dielectric-relaxation data as follows: The aging process involves two basic steps. In the first stage, the aging data fits to the MKWW form with a time-dependent relaxation time τage . The time dependence of τage (tage ) (given by eqn. (9.4.18)) is somewhat more natural than that of Lunkenheimer et al. (given by eqn. 9.4.17). As tage reaches a characteristic time scale τfn the data can now be described in this second stage with a simple KWW form having a constant relaxation time comparable to τfn . This holds simultaneously for relaxation data for the response functions at different frequencies (ν). The aging-time dependence of the

478

The nonequilibrium dynamics

Fig. 9.13 The MKWW relaxation time τage (tage ) vs. tage as obtained by fitting the dielectricrelaxation data using the scheme given by eqn. (9.4.18) for various glass-forming materials. From c American Physical Society. Sen Gupta and Das (2007). 

relaxation time τage is shown in Fig. 9.13 for various glass-forming materials. However, unlike the first fitting scheme, the limiting value of τfn obtained here is not the same as τα discussed above. The stretching exponent is also very different from that of the α-relaxation. This might equally well be indicative of a different mechanism controlling the nonequilibrium dynamics apart from the equilibrium α-relaxation.

Appendix to Chapter 9

A9.1 The energy of the oscillator We present here the calculation of the potential energy of the oscillator being driven by eqn. (9.2.11). The oscillator variable x(t) is coupled to the observable O. To solve the inhomogeneous partial differential equation we use the following definition in terms of a corresponding Green function G:  t ˜  ). dt  G(t, t  ) O(t (A9.1.1) x(t) = ε 0

The Laplace transform of the Green function G(t, t  ) is then obtained as G(ω, t) =

−ω2

+ ω02

1 . −  2 χ (ω, t)

In eqn. (A9.1.2) we have used the following definitions:  t  dt  G(t, t  )e−ω(t−t ) , G(ω, t) = 

(A9.1.2)

(A9.1.3)

0

χ (ω, t) =

t



dt  R O (t, t  )e−ω(t−t ) .

(A9.1.4)

0

In order to take the inverse transform of (A9.1.2) one must find the poles of the denominator on the RHS. Assuming that the coupling is weak enough (small ε), we consider the poles in the vicinity of ω = ±ω0 in the following form:

i , (A9.1.5) ω ≈ ±ω0 1 − ω0 τc (t) where the characteristic time τc (t) for the damped oscillator is given by τc (t) =

2ω0 . ε2 χ  (ω0 , t)

(A9.1.6)

Now, evaluating the contour integral in terms of the residue at the relevant pole, we obtain for the Green function in real time

t − t G(t, t  ) = exp − (A9.1.7) sin[ω0 (t − t  )]θ (t − t  ). τc (t) 479

480

Appendix to Chapter 9

Now let us consider the mean-square displacement x 2 (t) of the oscillator (t  ω0−1 ) measured in a short interval at time t. Using the definition (A9.1.1), we now obtain for the average of the square of the displacement  t  t , + ˜  ) O(t ˜  ). dt  dt  G(t, t  )G(t, t  ) O(t (A9.1.8) x 2 (t) = ε2 0

0

On making changes of variables τ = t − t  and τ  = t − t  , we obtain  t  t 2 2 dτ dτ  G(t, t − τ )G(t, t − τ  )C O (t − τ, t − τ  ). x (t) = ε 0

(A9.1.9)

0

Since causality requires that G(t, t  ) = 0 for t  > t, we can extend the lower limits of τ and τ  to −∞. Also, since G(t, t  ) decays exponentially with t − t  for large t, we can extend the upper limits for τ and τ  to +∞. With these approximations we obtain  +∞  +∞ 2 2 x (t) = ε dτ G(t, t − τ ) dτ  G(t, t − τ  )  ×

−∞

+∞

−∞

−∞

dω  C O (ω, t − τ )eiω(τ −τ ) . 2π

(A9.1.10)

The exponential decay of the Green function implies that the major contribution to the above integral comes from the small-τ part and hence C O (ω, t −τ ) in (A9.1.10) is replaced by C O (ω, t), to obtain  +∞ dω G(ω, t)G(−ω, t)C O (ω, t) x 2 (t) = ε2 −∞ 2π

 +∞ dω 1 C O (ω, t) 1 = − , (A9.1.11)   D+ −∞ 2π χ (ω, t) − χ (−ω, t) D− where we have used the definitions D± = ω2 − ω02 + ε2 χ  (±ω, t). The integral on the RHS of the above expression for x 2  is computed by the method of residues. The poles are obtained from the zeros of D± which appear in the denominators of the two terms within square brackets on the RHS of (A9.1.11). The only singularities which contribute to the integral are obtained by closing the latter on the upper half-plane. They lie in the vicinity of ω = ±ω0 + iε2

χ  (ω0 , t) . 2ω0

(A9.1.12)

Using the fact that the imaginary part χ  (ω) is an odd function of ω, we obtain from the above the average of the square of the displacement x(t) to leading order in ε as x 2 (t) =

 (ω , t) CO 0 , ω0 χ  (ω0 , t)

(A9.1.13)

A9.2 Evaluation of integrals

481

 (ω , t) is defined as where C O 0  (ω0 , t) = Re CO



t



dt  C O (t, t  )eiω(t−t ) =

0

1 [C O (ω) + C O (−ω)] . 2

(A9.1.14)

A9.2 Evaluation of integrals We present here the evaluation of the various integrals used in the analysis of the asymptotic behavior of correlation and response functions in the FDT and aging regimes.

A9.2.1 Integral IR for the FDT solution We compute here integral I R (say) in the last term on the RHS of (9.3.44). The integration over t  extends from s to t, while that over s varies from t  to t (t > t  ). Therefore both the correlation and the response functions appearing in the integrand satisfy the FDT relation (9.3.23) and the FDT relation (9.3.24) between  and  applies. Furthermore, in I R we change the lower limit of the integral with respect to t  from s to t  . The extra contribution arising from this change is a definite integral with t  lying between t  and s. This is equal to zero since the integrand contains the response function R(t  , s), which vanishes for t  < s due to causality. Hence the integral I R now reduces to the form  IR =

ds

t

 =



t

t

t

t

t

dt  (t, t  )R(t  , s) =





dt (t, t )



t  t



t t

dt  (t, t  )



t t

ds R(t  , s)

ds R(t  , s).

(A9.2.1)

Once again we have changed the upper limit of s from t to t  in the last integral since the response function R(t  , s) is zero for s > t  . The expression for I R is further simplified with a partial integration as follows:  IR =

t

t

dt  (t, t  )M(t  , t  ) =

 t

t

dt 

∂ (t − t  )[1 − C(t  − t  )] ∂t 

 t *t ∂ * dt  (t − t  )  C(t  − t  ). (A9.2.2) = [1 − C(t  − t  )](t − t  )*  + t ∂t t After some trivial rearrangements I R is obtained as  IR = 0

where τ = t − t  .

τ

dτ  (τ − τ  )

∞

 ∂ C(τ ) + [1 − C(τ )] ap,  ∂τ p=2

(A9.2.3)

482

Appendix to Chapter 9

A9.2.2 Integrals for the aging solution Here we compute the following integrals appearing on the RHS of eqn. (9.3.49) for treating eqn. (9.3.14) in the aging limit: 

t  −δ 

I1 =  I2 = 

(A9.2.4)

0 t

ds A (t, s)RF (t  , s),

(A9.2.5)

ds A (t, s)C(s, t  ),

(A9.2.6)

ds F (t, s)CA (s, t  ).

(A9.2.7)

t  −δ  t−δ

I3 = 

ds A (t, s)RA (t  , s),

0 t

I4 =

t−δ

In the aging regime the times t and t  are such that the ratio λ = t  /t < 1. Both t and t  → ∞, with (t − t  )/t  of O(1). The time scales δ and δ  are chosen such that they are much smaller than t and t  , respectively. First we evaluate the integral I1 with the use of the scaling solution (9.3.29) in the aging regime, i.e.,

h(t  ) CA (t, t ) = qφ , h(t) 



h  (t  ) h(t ) RA (t, t ) = R , h(t) h(t) 

(A9.2.8)

. where h(t) is a monotonically ascending function of time and q is the nonergodicity parameter. I1 is obtained as 

t  −δ 

I1 =

dt 0

=



∞ 

a p q p−1 φ p−1 (λλ )

p=2

∞ 

 a pq

1

p−1

dλ R(λ ) dt 

dλ φ(λλ )R(λ ),

(A9.2.9)

0

p=2

where we have defined λ = h(t  )/ h(t  ) and λ = h(t  )/ h(t). Since t  is very large compared with δ  , the upper limit of the integral has been replaced as h(t  − δ  )/ h(t  ) → 1. The integral I2 is evaluated as  I2 = =

t t  −δ 

∞  p=2

ds A (t, s)RF (t  , s) ≈ A (t, t  ) 

a pq

p−1

φ

p−1

(λ)

t

t  −δ

ds



t

t  −δ 

∂CF (t  , s) . ∂s

ds RF (t  , s) (A9.2.10)

A9.2 Evaluation of integrals

483

For the range of integration of s from (t  − δ  ) to t  we have treated (t, s) as a constant (t, t  ), I2 =

∞ 

 a p q p−1 φ p−1 (λ)

p=2

=

∞ 

t t  −δ 

ds

∂ CF (t  − s), ∂s

a p q p−1 φ p−1 (λ)[CF (0) − CF (δ)]

p=2

=

∞ 

a p q p−1 φ p−1 (λ)(1 − q).

(A9.2.11)

p=2

Next we calculate the integral I3 by computing I31 , I32 , and I33 . The integral I31 is obtained from (9.3.51) as 

t  −δ 

I31 =

dt  A (t, t  )CA (t  , t  )

0

=

∞ 



=

=

p−2

 ( p − 1)a p q

h(t  −δ) h(t)

p−2

(t, t  )R(t, t  )CA (t  , t  )

dt  φ p−2 (λ )

0

p=2 ∞ 

dt  CA

0

p=2 ∞ 

t  −δ 

( p − 1)a p

 ( p − 1)a p q

λ

p−1

d(λ ) R(λ )qφ(λ /λ) dt 

dλ φ p−2 (λ )R(λ )φ(λ /λ),

(A9.2.12)

0

p=2

where we have defined λ = h(t  )/ h(t) and ignored terms of O(δ) in the limit δ t  , such that h(t  − δ)/ h(t) → λ. The integrals I31 and I33 are very similar, differing only in the limits. So the latter is evaluated in exactly the same way as the former, giving  I33 = =

t−δ 

t  +δ  ∞ 

dt  A (t, t  )CA (t  , t  ), 

( p − 1)a p q

1

p−1 λ

p=2

dλ φ p−2 (λ )R(λ )φ(λ/λ ),

(A9.2.13)

where we have used the fact that over the range of the integration t  > t  . On adding eqns. (A9.2.12) and (A9.2.13) we obtain I31 + I33 =

∞  p=2

 ( p − 1)a p q

p−1 0

1



dλ φ p−2 (λ )R(λ )φ((λ /λ)sg(λ−λ ) ). (A9.2.14)

484

Appendix to Chapter 9

Next we compute the integral I32 ,  I32 =

t  +δ 

t  −δ 

ds A (t, s)CF (s, t  ) 



= A (t, t )

t  +δ  t  −δ 

ds CF (s − t  ).

(A9.2.15)

Since t − t  ∼ O(t  ), and t and t  are very large, we take A (t, s) ≈ A (t, t  ) over the range of the integral. The contribution from integral I32 is therefore calculated as I32 =

∞ 

( p − 1)a p q p−1 φ p−1 (λ)R(λ)

p=2

dλ  (δ ) ≈ 0 dt 

(A9.2.16)

to leading orders. Now we consider the final integral I4 :  I4 =

t





ds F (t, s)CA (s, t ) ≈ CA (t, t )

t−δ

= qφ(λ) ∞ 

t

ds t−δ

∞ 

∂ F (t, s) ∂s

*t * a p C p−1 (t, s)*

t−δ

p=2

=



 a p q 1 − q p−1 φ(λ).

(A9.2.17)

p=2

Next we compute the integrals appearing on the RHS of eqn. (9.3.55), which follows from eqn. (9.3.15) in the aging regime. For J1 we obtain  J1 =

t  +δ  t

ds A (t, s)RF (s, t  ).

(A9.2.18)

Over the range t  to t  + δ  for variation of s, we approximate A (t, s) by A (t, t  ). For the response function in the FDT regime, since s > t  we use the corresponding relation RF (s, t  ) = −∂s CF (s − t  ) to write the integral J1 as 

J1 ≈ −A (t, t )



t  +δ  t

ds

∂ CF (s − t  ) ∂s

= −A (t, t  )[C(δ  )) − C(0)] = R(λ)

∞ dλ  a p ( p − 1)[1 − q]q p−2 φ p−2 (λ), dt  p=2

(A9.2.19)

A9.2 Evaluation of integrals

485

where λ = h(t  )/ h(t). For J2 we obtain  t−δ J2 = dt  A (t, t  )RA (t  , t  ) t  +δ 

=

∞ 

 a p ( p − 1)q

1

p−2

h(t  ) h(t)

p=2

dt  φ p−2 (λ )

dλ 1 dλ R(λ )R(λ/λ )   dt  λ dt

 1  ∞ dλ p−2  dλ  p−2 =  a p ( p − 1)q φ (λ )R(λ )R(λ/λ ).  dt λ λ

(A9.2.20)

p=2

For J3 we obtain



J3 =

t

ds F (t, s)RA (s, t  )

t−δ 

≈ RA (t, t )



t

ds t−δ

∞ 

p−2

a p ( p − 1)CF

(t − s)RF (t − s)

p=2

 t ∞ dλ  ∂ p−1 = R(λ)  ap ds CF (t − s) dt ∂s t−δ p=2

= R(λ)

∞ dλ  a p [1 − q p−1 ]. dt  p=2

(A9.2.21)

10 The thermodynamic transition scenario

In Chapter 4 we introduced the Kauzmann temperature TK as a possible limiting temperature for the existence of the supercooled liquid phase. The original hypothesis due to Kauzmann proposes eventual crystallization in the supercooled liquid at very low temperatures as a possible way out of the paradoxical situation in which the entropy of the disordered state becomes less than that of the crystal. Another possible explanation of the Kauzmann paradox could be that the simple extrapolation of the high-temperature result to very low temperature is not correct and the entropy difference between supercooled liquid and crystal remains finite down to very low temperature (Donev et al., 2006; Langer, 2006a, 2006b, 2007), finally going to zero only near T = 0. Either of these resolutions, however, leaves us with no understanding of the dramatic slowing down and associated phenomenology of the supercooled region above Tg . The difference of the entropy of the supercooled liquid from that of the solid having only vibrational motion around a frozen structure represents the entropy due to large-scale motion and is identified with the configurational entropy Sc of the liquid. The rapid disappearance of the configurational entropy of the disordered liquid or the so-called “entropy crisis” poses an important question that is essential for our understanding of the physics of the glass-transition phenomena and the divergence of the relaxation time at Tg . Apart from having a characteristic large viscosity, the supercooled liquid shows a discontinuity in specific heat c p at Tg due to freezing of the translational degrees of freedom in the liquid. The features described above are almost universally observed in all liquids. 10.1 The entropy crisis We will now explore some ideas proposed in recent times to understand the entropy-crisis characteristics of supercooled liquids. We discuss the physics of the deeply supercooled state on the basis of a plausible scenario for an underlying phase transition. Our focus is primarily on structural glasses interacting through short-range potentials. The ENE transition of the MCT at temperature Tc discussed in the earlier chapters marks a crossover in the dynamics of the supercooled liquid in a temperature range higher than Tg . Beyond Tc , with increasing supercooling the local motion of the molecules comes mainly to represent vibrations around frozen configurations and it is widely separated from large-scale 486

10.1 The entropy crisis

487

diffusive motion of the particles. In thermodynamic terminology the system remains trapped in a local minimum of the free energy. The entropy related to the large-scale motion is the configurational entropy (which is also called complexity in spin-glass terminology) Sc . It is related to the number of free-energy minima N of the N -particle system through the basic relation of statistical mechanics, N = exp[N Sc (T )].

(10.1.1)

10.1.1 The Adam–Gibbs theory In 1956 J. Gibbs proposed a resolution of the Kauzmann paradox with the hypothesis that the vanishing of the configurational entropy signifies (Gibbs, 1956) a second-order phase transition at T = TK . The configurational entropy comes down sharply to zero at TK and remains zero below TK . The entropy vs. temperature curve thus has a kink at this transition temperature. Let us consider the supercooled state in which crystallization is avoided until close to TK . Indeed, there are amorphous states in which crystallization is very unlikely. Even before reaching TK , it is clear that with increasing supercooling the entropy of the liquid rapidly approaches that of the crystal. The Gibbs–Di Marzio (Gibbs and Di Marzio, 1958) and subsequently Adam–Gibbs theory (Adam and Gibbs, 1965) is based on this hypothesis of a phase transition at TK . The model estimates the entropy of the given state using the relation (10.1.1). The Adam–Gibbs theory is based on the concept of cooperatively rearranging regions (CRRs). The term “cooperatively rearranging” here implies that by definition this is the smallest size of the system which cannot be further subdivided, and that each such region can be independently rearranged. The number of particles n involved in the CRRs changes with temperature. It is assumed that the number νc of independent configurations that a given CRR can assume is independent of its size and of the temperature. This simplifying assumption allows us to estimate the configurational entropy. Indeed, νc under general conditions should increase with n or as the temperature rises, but we ignore this dependence in the simplest form of the theory. With a total number of N particles, the number of CRRs each having n particles is equal to N /n. Hence the thermodynamic probability of a given state is obtained as W N = {νc } N /n .

(10.1.2)

Therefore the configurational entropy is estimated using the basic relation (10.1.1) connecting thermodynamics with statistical mechanics, Sc (T ) =

1 ln νc ln W N = . N n(T )

(10.1.3)

The configurational entropy and the number of particles involved in the CRRs are therefore inversely related. With the lowering of the temperature the number n of particles moving in a cooperative manner, i.e., the size of the CRRs, increases and hence Sc falls. From this purely thermodynamics-based argument it follows that the vanishing of Sc is directly related to a diverging size of the CRR.

488

The thermodynamic transition scenario

The configurational entropy Sc is related to the jump in the specific heat of the liquid at the transition. To demonstrate this, we need to extend the formalism of standard equilibrium thermodynamics to metastable states. Since Sc is defined as the difference between the entropies of the liquid and the crystalline states, denoted by SLQ and SCR , respectively, its derivative with respect to temperature is therefore

d Sc 1 d SLQ d SCR = T −T dT T dT dT c p = . (10.1.4) T c p is the difference of specific heats between the supercooled metastable liquid and the crystal. Upon integrating the above relation we readily obtain  T ¯ dT (10.1.5) c p . Sc (T ) − Sc (TK ) = TK T¯ Assuming c p to be weakly dependent on temperature and Sc (TK ) = 0 by definition, the above relation reduces to   T Sc (T ) = c p ln . (10.1.6) TK For low temperatures T ∼ TK the RHS of eqn. (10.1.6) reduces to the linear form Sc (T ) = c p

T − TK . TK

(10.1.7)

It is important to note here that the free energy of the supercooled liquid remains continuous through the transition at TK . The specific heat c p shows a jump but there is no latent heat absorbed. From the thermodynamic point of view, the transition envisaged at TK is thus a continuous second-order transition. However, in this transition we can also identify an order parameter that shows a finite jump at the transition. The inverse of the radius of the cage seen by each particle can be treated as an order parameter like this. In the liquid phase this is zero and jumps discontinuously to a finite value in the glassy phase, which is contradictory to the general nature of second-order transitions.

10.1.2 Dynamics near TK The theory of the deeply supercooled liquid state presented so far primarily deals with its thermodynamics properties, e.g., the calculation of the configurational entropy of the metastable state. An important aspect of the glassy behavior that has been left out of these discussions so far is the slowing down of the dynamics. With increasing supercooling the relaxation time for the liquid sharply increases. Relaxation in the present context implies that for a typical fluctuation around the disordered liquid state at a temperature T < Tm . The nature of the single-particle dynamics in the deeply supercooled liquid changes over from liquid-like continuous motions to activated hopping. In a structural glass this occurs

10.1 The entropy crisis

489

even without any external agent coming into play, i.e., the slow dynamics is self-generated. The growth of the relaxation in the Adam–Gibbs theory is considered in a simple way in the following. To understand the development of very long relaxation times in the deeply supercooled state, we need to introduce the idea that energy barriers build up to resist molecular rearrangement in the jammed state. Relaxation under such conditions occurs through thermally assisted hopping over the barrier (E B , say). The probability of such a jump will be controlled by the Boltzmann factor exp[−E B /(kB T )]. Thus estimation of the relaxation time is closely linked to that of the energy barrier E B which must be overcome so that a local fluctuation can relax. In a many-particle system with short-range forces, relaxation of a fluctuation takes place locally. To understand this, we consider the following simple picture. Every particle is affected by a set of n of its neighbors forming the CRR. Local relaxation of the fluctuations will involve rearranging n particles cooperatively. Therefore the barrier E B for this process is proportional to n. Using eqn. (10.1.3) we obtain a simple relation between E B and Sc , EB =

c , Sc (T )

(10.1.8)

where c is a constant. Note that the energy barrier E B is now temperature-dependent through that of the configurational entropy Sc . For temperature-independent barriers E B the relaxation time (e E B /(kB T ) ) will grow with temperature in accord with a simple Arrhenius law as is the case with the so-called strong liquids. However, the fragile glass-forming liquids show a much sharper change with temperature and the effective barrier height increases with lowering of the temperature. The present analysis is therefore more applicable to relaxation in fragile systems. The relaxation time τα (T ) corresponding to the barrier (10.1.8) is therefore obtained using the Arrhenius relation   βc . (10.1.9) τα = τ0 exp Sc (T ) Equation (10.1.9) represents the crucial link between the configurational entropy and the relaxation time in the deeply supercooled liquid state. For temperatures near the Kauzmann temperature the relaxation time τα (T ) is obtained as   E VF τα = τ0 exp , (10.1.10) T − TK where E VF = c /(kB c p ). This is the Vogel–Fulcher form for the relaxation time with the corresponding divergence temperature T0 ≡ TK . While the analysis above provides a natural resolution of the Kauzmann paradox and a basis for the empirical Vogel–Fulcher fitting form, the validity of the whole scenario is uncertain due to the rather simplified assumptions. The equality of the two temperatures T0 and TK has been tested in a wide number of glassy systems. As we have seen above, due to the intervention of the calorimetric glass transition at Tg , i.e., the liquid becoming too sluggish to equilibrate on laboratory time scales, it is impossible to locate precisely both

490

The thermodynamic transition scenario

of the temperatures T0 and TK . The temperature T0 is obtained by extrapolating a highly nonlinear fit of the data well above Tg to a different temperature regime below Tg . TK is also inferred through extrapolation of the entropy data below Tg . The above equality, however, suggests a link of deeper significance on considering the fact that the physics differs greatly for the two temperatures T0 and TK . T0 is the temperature at which the relaxation time for the supercooled liquid diverges and basically relates to the dynamics. On the other hand, the Kauzmann temperature TK is related to the vanishing of the thermodynamic property of configurational entropy of the metastable liquid. Hence the equality of T0 and TK , or, more appropriately, the linking of the sharp increase of relaxation time to the entropy crisis, actually represents effects of structure on the dynamics. It is plausible to assume that an increase of the number of particles n in a CRR implies growth of the size of the latter. This leads to the idea of a growing (static) correlation length in the supercooled liquid ξs , which is related to the number of particles n in the cluster: n ∼ ξsd for a d-dimensional system. Since the barrier E B for the relaxation process also grows with the number of particles n, we obtain that E B ∼ ξsd . From eqn. (10.1.9) we therefore obtain   B0 ξsd τα = τ0 exp , (10.1.11) kB T where B0 is a constant. It is important to compare this with the case of divergence of length and time scales in standard critical phenomena. It is well known that the relaxation time increases sharply on approaching the second-order phase-transition point and cooperativity over domains of the size of the correlation length requires that the relaxation time τR ∼ ξsz , where z is a dynamic exponent. Since the correlation length ξs in this case diverges with a power law on approaching the critical point, the relaxation time τR also follows a powerlaw divergence on approaching Tc . In a glass-forming liquid, on the other hand, using eqn. (10.1.11), it follows that growth of the static length scale by only an order of magnitude produces the required growth in the time scale near Tg . In the case of structural glasses, evidence of such a growing static length scale has been hard to find (see below for further discussion). For example, the magnitude of the structure factor of the liquid or the density does not acquire any special feature on supercooling (Dyre, 2006).

10.2 First-order transitions The freezing transition Tm of the disordered liquid to the crystalline state is a first-order phase transition with finite latent heat absorbed. The glass transition, however, is not associated with any latent-heat absorption. At Tg , as the system freezes into an amorphous solid-like structure, there is a drop in the specific heat due to the removal of the translational degrees of freedom. The free energy of the supercooled liquid, however, does not show any discontinuous change. At deep supercooling there is a large number of available metastable structures into which the liquid can freeze and hence a considerable entropic drive is present for the process. There have been theories for the vitrification process that

10.2 First-order transitions

491

have been built on possible scenarios involving first-order transitions in the special situation of there being a large number of available metastable states.

10.2.1 Metastable aperiodic structures In Chapter 2 we demonstrated the application of classical density-functional theory (DFT) to studying the first-order freezing transition of the liquid into an ordered crystalline state. DFT methods have also been applied to study the supercooled liquid state below the freezing point Tm for systems with aperiodic structures (Singh et al., 1985; Dasgupta, 1992; Kaur and Das, 2001a). The latter are modelled in terms of localized density profiles (over suitable time scales) around a disordered set of lattice points. The metastable states are identified as minima of the free-energy landscape intermediate between a crystal and a homogeneous liquid state. Computation of the thermodynamic free energy of such systems requires an explicit functional form for the order parameter, i.e., the inhomogeneous density function in terms of a suitable set of parameters. The free-energy functional is minimized with respect to these parameters. As we saw in Chapter 2, a very successful prescription of the density distribution for the crystalline case is obtained from the superposition of Gaussian density profiles centered on a lattice with long-range order,  φ(| r − Ri |), (10.2.1) ρ( r) = i

where { Ri } denotes the underlying lattice and the function φ is taken as the isotropic 2 Gaussian φ(r ) = (α/π)3/2 e−αr . In the case of an aperiodic structure {Ri } constitutes a random structure. α is the variational parameter such that its inverse characterizes the width of the peak and therefore signifies the degree of mass localization in the system. The homogeneous liquid state in this representation is characterized by Gaussian profiles of very large width corresponding to the limit α → 0. Each Gaussian profile provides the same contribution to the sum on the RHS of eqn. (10.2.1) corresponding to all spatial positions. The free energy of the liquid is obtained as a sum of two parts – the ideal-gas term and the interaction term, F[ρ] = Fid [ρ] + Fex [ρ].

(10.2.2)

The ideal-gas term of the free-energy functional (in units of β −1 ) is expressed as a functional of the inhomogeneous density function ρ(r) in the form  (  ) Fid [ρ] = d r ρ( r ) ln 30 ρ( r) − 1 , (10.2.3) 0 being the thermal de Broglie wavelength. For the highly localized structures which have been investigated, generally α was chosen to be large (greater than ≈ 50) and eqn. (10.2.3)

492

The thermodynamic transition scenario

was approximated by its asymptotic value for large α (see eqn. (2.2.21) discussed in Chapter 2), ( )

Fid 5 α 3/2 = f id [ρ] ≈ − + ln 30 . (10.2.4) N 2 π The above asymptotic formula for the case of large α is particularly applicable for a highly localized mass distribution of the crystalline state. For the aperiodic structures, however, the low-α range where overlapping Gaussian profiles from different sites contribute to Fid is also relevant. In this case Fid is evaluated numerically from the integral given in (10.2.3) as 3 

4  3 f id [ρ] = dr φ(r ) ln 0 dR φ(r − R)(δ(R) + ρ0 gB (R)) − 1 , (10.2.5) where the random structure on which the density profiles are centered is obtained on the average in terms of the site–site correlation function. A standard procedure generally followed here to obtain the latter is to use the gB (R) corresponding to Bernal’s random structure (Bernal, 1964), which is generated through Bennett’s algorithm (Bennett, 1972). The site–site correlation reduces to the pair correlation of the Bernal structure at the closepacked density. In this regard, it should be noted that for a hard-sphere system identification of the most closely packed random structure is somewhat anomalous (Torquato et al., 2000). In the present context, however, the Bernal structure is simply applied as a tool to evaluate the free energy for an inhomogeneous density profile centered at the random set of lattice points. Similar studies of the same free-energy functional when tested with random structures obtained from computer-simulation studies (Kim and Munakata, 2003) gave similar results. The ideal free-energy value evaluated with the expression given in eqn. (10.2.5) is shown in Fig. 10.1 (inset) for ρ0 = 1.12. The numerical result approaches the asymptotic (large-α) result (dashed line) for α > 20 within 5% as shown in the figure. The interaction part Fex of the free energy is evaluated using the standard formalism with the expression for the Ramakrishnan–Yussouff (RY) functional,   1 d r1 d r2 c(| r1 − r2 |; ρ0 )δρ( r1 )δρ( r2 ), (10.2.6) F ex = − 2 which gives the difference between the free energies of the solid and liquid phases of r ) is the density fluctuation from the average value ρ0 . The average density ρ0 . Here δρ( expansion (10.2.6) for the free energy of the liquid works as a better approximation for the free energy of the solid state in the low-α region of the density function than for the highly localized region for large α. The existence of metastable minima corresponding to low and high values of α has been seen using the above free-energy functional. The results we discuss here are for hard-sphere systems, and generally the solution of the Percus–Yevick equation with Verlet–Weiss correction for the direct correlation function, c(r ), of the reference liquid state is used in

10.2 First-order transitions

493

Fig. 10.1 The difference in free energy per particle denoted as  f /ρ between the solid and liquid states vs. α ∗ (ασ 2 ) at n ∗0 (n 0 σ 3 ) = 1.12. The inset shows the ideal-gas free-energy term vs. α ∗ for n ∗0 = 1.12. The solid line is obtained from the exact evaluation of formula (10.2.5) and the dashed line is the approximate formula (10.2.4) for large α. Reproduced from Kaur and Das (2001a). c American Institute of Physics. 

obtaining them. For the hard-sphere system, Singh et al. (1985) obtained metastable minima for the free energy corresponding to very large values of the width parameter α. This represents a highly localized inhomogeneous density distribution. A qualitatively different free-energy minimum with the optimum density function characterized by small α values has been obtained subsequently (Kaur and Das, 2001a). The low-α minimum appears at α ≈ 18 for ρ0 = 1.12, while the corresponding large-α minimum occurs for ασ 2 > 100. A typical result is displayed in Fig. 10.1. The free energy corresponding to the low-α minima is generally less than that for the highly localized (large-α) state throughout the density range above n 0 σ 3 = 1.15. The low-α minimum becomes more stable than the homogeneous liquid state (α → 0) beyond density n 0 σ 3 = 1.10. The corresponding “hard-sphere glass” state for high α values becomes more stable w.r.t. the homogeneous liquid state at an average density n 0 σ 3 = 1.14. The occurrence of the highly localized structures for large α is also sensitive to having special tails in the large-r behavior of the input c(r ) (Singh et al., 1985). The existence of the low-α minima, on the other hand, is found to be rather robust against different input direct correlation functions, and they occur even with the purely repulsive PY c(r ) without any tail. Treatments of the above free-energy functional using the MWDA described in Chapter 2 also obtain the metastable minima for the less localized metastable structure corresponding to small α values. Evaluation of the free energy of finite-size systems with computer models using fluctuating values of the width parameter α at different sites has also demonstrated (Chaudhary et al., 2005) the existence of similar metastable states.

494

The thermodynamic transition scenario

The two minima with the random structure for low and high values of α correspond to very different degrees of localization of the particles. The α corresponding to the minimum free-energy value is inversely proportional to the root-mean-square displacement of the particles from their sites and hence defines the Lindemann ratio for the state. The metastable supercooled state seen in computer simulations (Stillinger, 1995; LaViolette, 1985) shows better agreement with the low-α minimum. The Lindemann ratio of supercooled liquid seen in simulations is approximately three times that of the crystal at freezing (Stillinger, 1995). The height of the free-energy barrier between the metastable glassy structure and the uniform liquid state grows with increasing packing fraction. The extrapolated height of the barrier diverges with a power law at the packing fraction value ϕ c ≈ 0.62 with exponent 1.57 from the simple density-functional calculation (Kaur and Das, 2001a). In the next section our discussion of the mosaic theory implies a similar power-law divergence of the barrier (to coming out of a frozen amorphous structure) at the thermodynamic phase-transition point. In the case of the hard-sphere system this critical point is described in terms of the density. In comparison with the replicated-liquid-state theory for the glassy state (described in Section 10.5.1) we obtain the critical Kauzmann packing fraction ηK = 0.628 (Parisi and Zamponi, 2010), while an effective potential calculation (see Section 10.3.2) for the self-generated disorder gives ηK = 0.62 (Cardenas et al., 1999). 10.2.2 Random first-order transition theory The above studies of the density-functional models demonstrate that metastable free-energy minima exist in the supercooled region. Indeed, they correspond to a class of intermediate minima for different amorphous structures distributed on a lattice without any long-range order. Note that in the above analysis the width parameter α can also be varied (Odagaki and Yoshimori, 2000; Odagaki et al., 2002; Chaudhary et al., 2005). The metastable minimum occurs at a finite value of α, signifying a finite Lindemann ratio. Accordingly, the system is expected to undergo a first-order phase transition. However, such a scenario would imply a finite latent heat associated with the glass transition similar to that for the crystallization transition. For understanding the structural-glass transition without any latent heat, here we invoke an analogy to certain types of spin-glass models. These systems undergo a phase transition characterized by a jump in a locally defined order parameter without any absorption of latent heat. This has been termed a random first-order transition (RFOT). Such systems include Potts spin glasses (Gross et al., 1985), the p-spin glass (Mézard et al., 1987), and refined p-spin versions (Gardner, 1985) of the random-energy model (Derrida, 1981). Like the structural glasses, these spin-glass models also exhibit an entropy crisis, with the corresponding configurational entropy vanishing at the phasetransition point. In the case of the structural glass the RFOT is linked to the existence of a large number of aperiodic structures, in contrast to the unique crystalline state for the freezing transition. While drawing the parallel between structural and spin glasses it is worth remembering the obvious differences. The spin-glass models mentioned above are

10.2 First-order transitions

495

characterized by long-range interactions, whereas the structural glasses have short-range interactions (see the discussion in Section 8.4). The idea of the RFOT in supercooled liquids is based on the formation of an entropic droplet (Kirkpatrick et al., 1989) in the deeply supercooled liquid. A primary assumption in this theory is that the supercooled liquid is close to the metastable equilibrium. Hence it is applicable to more generic glass-forming systems that are unlike those formed by rapid quenching from a melt and are in a state far from equilibrium. In the mean-field description the liquid is generally assumed to be globally in a metastable state. However, in the finite-dimensional system such a description is valid only over a time scale beyond that on which the system comes out of the metastable state. The deeply supercooled liquid can be in any one of the extensive number of possible metastable states, each corresponding to an aperiodic structure. The study of the density-functional models described above has demonstrated the existence of such intermediate states. The possibility of an extensive number of aperiodic states (of the same free energy f ) gives rise to a finite configurational entropy Sc ( f ) for the supercooled liquid. The RFOT theory is based on the fact that for systems with finite-range interactions this entropy will lead to an instability for the global mean-field state, and the system will break up into a mosaic of structures for the supercooled liquid. As we have seen in the case of classical nucleation theory (see Chapter 3), there is a free-energy drive towards formation of the condensed phase, which is more stable than the parent phase. This is unlike the present case of a mosaic of different aperiodic states for which the free energies are the same. However, here the drive is entropic. There is a likelihood of formation of the new phase since there are so many of them available for the system to escape into, hence giving rise to the finite Sc ( f ). This entropy drive is −L d T sc where sc is the configurational entropy per unit volume in the supercooled domain of size L. However, there is a cost in forming the new phase that is proportional to σm L μ , which is like the surface-tension term in expression (3.1.9) for the free energy of formation of the critical nucleus in classical nucleation theory. The exponent μ here is kept at a more general level than the typical surface contribution and σm denotes the “generalized” surface tension between the two phases. We expect the value of the exponent μ to depend on the order parameters in terms of which the phases are described. For a simple Ising-like description μ = d − 1 and the energy cost is simply proportional to the surface area between the two phases. In general, μ ≤ d − 1. The total free-energy cost for formation of the bubble of the new phase is obtained as F = −T sc L d + σm L μ .

(10.2.7)

The above free-energy cost is optimized to obtain the size of the critical domain or the correlation length ξs as  ξs =

μσm T sc d

1/(d−μ) (10.2.8)

496

The thermodynamic transition scenario

and the corresponding value of the optimum free energy is obtained as E B ≡ F ∗ = σm (1 − κ)ξsμ ∼

σm 1/(1−κ) , (T sc )κ/(1−κ)

(10.2.9)

where κ = μ/d ≤ 1 − 1/d and is a constant less than 1 for d ≥ 2. E B represents the energy barrier to the formation of the new phase. The crucial aspect of this result is that the μ energy barrier to formation of this new phase grows with the correlation length ξs whereas in the Adam–Gibbs theory we have seen in eqn. (10.1.11) that the barrier E B grows as ξsd . The deeply supercooled liquid consists of glassy clusters separated by domain walls. As the temperature of the liquid is lowered their size (∼ξs ) grows. The correlation length ξs is inversely proportional to the configurational entropy Sc . At the Kauzmann temperature TK the correlation length ξs diverges as Sc → 0. The formation and growth of a nucleus in a first-order phase transition (described in Chapter 3) is qualitatively different from the above picture. In the earlier situation the nucleus simply consists of one phase, which grows in the parent phase. In the mosaic picture (Wolynes, 1989) even inside a glassy cluster another cluster can grow due to the entropic drive making the role of fluctuations very important. Using the concept of entropic droplets and assuming that there is an interface cost to putting different metastable states into contact in the mosaic picture, a glassy coherence length has been obtained (Biroli and Bouchaud, 2004). The dynamics of the liquid is controlled by activated processes in which there is constant formation of the amorphous clusters. At TK the barrier to such a transformation is infinite and hence the relaxation time diverges. The frozen liquid is now caught in a single aperiodic structure with finite shear modulus. In the original proposal of this theory (Kirkpatrick et al., 1989) scaling and renormalization-group ideas were used to claim that μ = d/2, i.e., κ = 12 . In this case the growth of the barrier E B ∼ 1/Sc and, assuming the relation (10.1.7), we obtain that the relaxation time τ (T ) ∼ 1/(T − TK ), which is the Vogel–Fulcher expression. However, using κ as a free parameter does not change the basic contents of the theory. The entropic-droplet theory was subsequently constructed for a realistic system of a random hetero-polymer model by Takada and Wolynes (1997). The density-functional model for the free energy with respect to the RFOT theory has been studied subsequently by Xia and Wolynes (2000, 2001) and by Biroli et al. (2008). The scenario of a thermodynamic phase transition at TK discussed above is linked to the existence of a static correlation length that diverges at the transition temperature. A qualitatively different kind of dynamic length scale related to the dynamics of the supercooled liquid has been identified, and we have already discussed this in Section 4.4.2. Indeed, the relaxation time in the supercooled liquid grows beyond any conceivable experimental time scale already near the calorimetric glass transition Tg , which is higher than TK . Evidence of a growing static correlation length in the measurable temperature range has been difficult to find, however. As we have already noted, the static structure factor of the liquid in terms of density correlation functions does not display any such building correlation. In fact, the characterization of a proper physical definition of the length scale is not clear in the present situation.

10.2 First-order transitions

497

In recent times some new insight into this search for possible correlation lengths has been gained by focusing on correlations of variables other than the simple density fluctuations. Yamamoto and Onuki (1998a,1998b) performed extensive MD simulations of a binary soft-sphere mixture of N1 = N2 = 5000 particles in two and three dimensions. The details of this interaction potential between the two species of the mixture are as described in Section 4.3.1. Bonds between two neighboring particles are defined in this system by using a length scale corresponding to the sharp first peak in the static correlation function. At a time t, two particles i and j of the liquid belonging to species α and β in a mixture are at ri (t) and r j (t), respectively. These two particles are considered bonded if the following constraint is satisfied: ri j (t) = |ri (t) − r j (t)| ≤ A1 σαβ ,

(10.2.10)

with A1 = 1.1 (in two dimensions) and 1.5 (in three dimensions). This bond is considered broken at a later time t +t  if the distance ri j (t +t  ) becomes larger than A2 σαβ . As a set of suitable parameters the following choice is made: A2 = 1.6 (in two dimensions) and 1.5 (in three dimensions). From the simulation data, the total number of surviving bonds starting from an initial configuration t is counted. Yamamoto and Onuki (1998b) observed that the breaking of the bonds in the supercooled liquid showed some intriguing correlations. In two dimensions a substantial increase in the bond-breakage time τb from 17 to 5 × 104 (expressed in the units indicated in Section 4.3.1) is seen for the liquid state on changing the thermodynamic parameter for the binary system from eff = 1 to eff = 1.4 (see eqn. (4.3.14) for the definition of eff ). In the supercooled state, the actual mapping of the broken bonds displays strong clustering. Although it was identified in terms of a dynamic quantity of bond breaking, a sort of static correlation function SB (q) of the broken bonds is defined with the following relation: /* *2 0 * 1 **  SB (q) = exp(iq · Ri j )** , (10.2.11) * NB i, j

where Ri j = (ri (t) + r j (t))/2 is the center position of a broken pair at a time t in the supercooled state and NB is the total number of broken bonds in the time interval [t, t +]. Yamamoto and Onuki (1998b) reported that the angle-averaged   form of this correlation function satisfies the Ornstein–Zernike form SB (0)/ 1 + ξs2 q 2 and identified a correlation length ξs that grows with the supercooling of the liquid. Hence, on considering a specific type of properties of the supercooled state (distinct from the standard order parameters such as density), signatures of a growing correlation are seen in the supercooled state. Recently, Tanaka et al. (2010) associated a growing static length scale with various supercooled liquids in simulations as well as from experiments. This correlation is again identified not in terms of the usual density correlation functions but in terms of a new structural order parameter related to bond-orientational order in the liquid. In some systems, such as poly-disperse colloids, one can identify static structural ordering via bondorientational order. In a two-dimensional binary soft-sphere mixture structural entropy has

498

The thermodynamic transition scenario

been used to identify the corresponding correlation function. However, such static order has not been seen in a three-dimensional binary soft-sphere mixture. The viability of a growing static correlation therefore remains somewhat questionable at this stage. 10.3 Self-generated disorder Our discussion of the previous section is primarily based on thermodynamic considerations of the supercooled liquid which is in a state of metastable equilibrium at a temperature T < TK . A full statistical-mechanical treatment of the problem (from a microscopic approach) for the analysis of the dynamic behavior and the development of the barrier to molecular rearrangement is still lacking. In the final sections of this book we discuss theories for the metastable state that have been formulated from a microscopic approach. This also results in the extension of the ideas of statistical mechanics to the ideal glassy state below TK . At moderate supercooling the dynamics is largely controlled by coupling of dominant density fluctuations, and is described in terms of the MCT discussed in the previous chapters. Beyond a crossover temperature, TA (say), the dynamics is controlled by thermally activated hopping of the system between different basins in the free-energy landscape. These basins correspond to configurations with inhomogeneous density distributions. As the temperature of the liquid is supercooled to the glass-transition temperature Tg , the liquid essentially freezes into a solid and attains a finite shear modulus. This is interpreted as that the system is caught in a local minimum of the free energy with only vibrational motion in the free-energy basin. In real space this implies that the particles vibrate around their mean positions, which form a disordered lattice structure. Note that this is a mean-field picture in which the system is caught in a metastable minimum corresponding to an amorphous structure. Using this picture, we now formulate a model for the amorphous solid in terms of the interaction potential between the constituent particles of this system. 10.3.1 Effective potential and overlap functions In a mechanical description the classical liquid of N particles is represented by a point in the 3N -dimensional configuration space (Huang, 1987). Let us denote by “y” the configuration (i.e., the collection of 3N coordinates of all of the particles) reached when the system is quenched to a temperature close to Tg . For temperatures below Tg the system will remain “close” (in the phase space) to y. Small-scale vibrations are allowed in order to equilibrate the system at the temperature of the heat bath, T (say). Thus we can view the deeply supercooled liquid as one in which the slow degrees of freedom are quenched and fast degrees of freedom thermalize at a temperature T . The parts of the phase space comprising the points at a fixed “distance” from y define a conditional statistical ensemble. In order to study the behavior of the supercooled system, we focus here on the free energy of the liquid corresponding to this ensemble. The ensemble is obtained in terms of the partition function Z y , which involves summing over the constrained Gibbs–Boltzmann

10.3 Self-generated disorder

499

measure. The existence of the glassy state in the present context is given the following interpretation: the free energy for the constrained state constitutes an “effective potential” for the system and the minima of this potential represent the “equilibrium” state of the glassy system. Evaluation of the effective potential or the constrained free energy, i.e., the average of ln Z y , requires using the replica trick developed for the spin-glass problem (Mézard et al., 1987). The replicated system reduces to a multi-component mixture in which analytic continuation on the number of components has to be done. But it is important to note at this point that the averaging in the spin glass is over the quenched impurities, which are absent in the case of a structural glass. The frozen reference configuration in fact plays the role of quenched variables in the structural case. We consider N identical point-like particles in a volume V and described by coordinates {xi } for i = 1, . . . , N . We denote this configuration simply as x. The interaction potential between the particles is v(xi − x j ). Let the configuration of this system at Tg be denoted y. Below Tg large-scale motions are frozen and only small-scale vibrations are allowed. We devise a criterion to define the part of the configuration space over which the distance from the point y is small. This can be quantified by introduction of the following normalized “distance” or “overlap” function: q(x, y) =

N 1  o(|xi − y j |), N

(10.3.1)

i, j=1

where the function o(r ) is defined as & o(r ) =

1,

for r ≤ σ 0 ,

0,

for r > σ 0 ,

(10.3.2)

with σ being the size of each particle and 0 a small number less than 1. Note that, between the two configurations, the ith particle in y can overlap with the jth particle in x. Thus the centers in the respective cases can lie within a distance less than σ 0 . For configurations x close to y the corresponding q(x, y) ≈ 1, whereas for widely separated configurations the value of q is small. Using the above definition of the overlap function q(x, y), a restricted Boltzmann–Gibbs distribution is defined as one consisting of points at a fixed “distance” q from y, P(x|y) =

e−β H (x) δ(q(x, y) − q), Z y (β)

(10.3.3)

where Z y (β) is the integral over x of the numerator on the RHS of eqn. (10.3.3), and is the partition function with respect to y as a frozen configuration, β = 1/(kB T ), where T is the temperature. In general, the above distribution will be strongly dependent on the parent location y. This makes the outcome of a cooling experiment completely arbitrary and impossible to associate with any measurement. Since by definition at the calorimetric

500

The thermodynamic transition scenario

glass-transition temperature Tg the liquid freezes for all practical time scales, we expect that the measure of y is obtained as μ(y) ≈

e−βg H (y) , Z(βg )

(10.3.4)

where βg = 1/(kB Tg ). The extensive quantities measured from the distribution function (10.3.3) will be independent of y in the thermodynamic limit. A suitable interpretation of this will be to take the frozen reference configuration y as being like a quenched variable that is effectively producing self-generated disorder in the structural glass. Let us now consider the free energy associated with the distribution P(x|y) averaged over the distribution of y, Gc (q, β) = −

kB T N

 dy

e−β H (y) ln Z(β)



d x e−β H (x) δ(q(x, y) − q) ,

(10.3.5)

where Z(β) is the integral of exp(−β H (y)) over y. Note that for simplicity we have assumed that the distribution of the frozen configuration y is controlled by the same temperature as the vibrational degrees of freedom. The delta-function constraint implies that for every y we restrict the measure of x to being only over the part of the phase space for which the global constraint q(x, y) is equal to some fixed value q. In order to devise a practical way to compute Gc (q, β) satisfying the global constraint, we now introduce a “field” γ coupling to the overlap function q(x, y) in the Hamiltonian and define the corresponding free energy as F(γ , β) = −

kB T N

 dy

e−β H (y) ln Z(β)



d x e−β{H (x)−γ q(x,y)} .

(10.3.6)

On taking a derivative of F with respect to γ we obtain ∂F (γ ) =− ∂γ

 dy

e−β H (y) Z(β)



d x q(x, y) 6

e−β{H (x)−γ q(x,y)} . d x e−β{H (x)−γ q(x,y)}

(10.3.7)

The global constraint of q(x, y) = q reduces the RHS of eqn. (10.3.7) to −q. Following the standard method of taking the Legendre transform in thermodynamics, we make a change of variable from γ to q by opting for the function Gc (q, β) defined as Gc (q, β) = min{F(γ ) + γ q}. γ

(10.3.8)

This is the free energy of the glassy state as a function of the “distance” q which characterizes the constrained Gibbs–Boltzmann measure. Gc is also referred to as the effective potential for the glassy state, with its minima characterizing the equilibrium (in the restricted sense) states of the supercooled liquid.

10.3 Self-generated disorder

501

Computation of the effective potential Gc (q, β) In order to compute the average (over the frozen configurations y) of the ln Z y we use the replica technique from spin-glass theory (Mézard et al., 1987). This is based on the identity Zn − 1 , n→0 n

ln Z = lim

(10.3.9)

where the overbar indicates the average of the corresponding quantity. On writing Z n as a product of n identical replicas we obtain  e−β H (x0 ) n Z = −kB T d x0 Z(β)   n    n   H (xα ) − γ q(xα , y0 ) . (10.3.10) × d x1 · · · d xn exp −β α=1

α=1

We have denoted above y ≡ x0 . Evaluation of the RHS of eqn. (10.3.10) involves computing thermodynamic properties of an (n + 1)-component mixture in the n → 0 limit. Note that the replica denoted by x0 is nonsymmetric with respect to the rest of the n replicas denoted by {xr } for r = 1, . . . , n. In the final limit, unlike in the typical spin-glass problem, in the present case the number of replicas tends to one, not zero. Similar situations occur in applying the replica trick to liquids (Stell, 1963; Goldbart et al., 1996) in which the disorder is external or quenched, rather than being self-generated disorder as in the present case. In the (n + 1)-component mixture which one has to deal with in order to evaluate the effective potential, the special component x0 interacts with all other replicas via the potential −N γ

n  α=1

q(x0 , xα ) = −γ

N n  

( ) o xi0 − x αj .

(10.3.11)

α=1 i, j=1

The effective potential is obtained using HNC closure for the (n + 1)-component mixture in terms of the pair correlation function gab between the different replicas a and b. The details of this calculation involving applying the HNC to the mixture can be found in the original references (Morita, 1960; Morita and Hiroike, 1961; Mézard and Parisi, 1996).

10.3.2 A model calculation We discuss here the results obtained by applying the above method to the case of a hardsphere potential (Cardenas et al., 1999) of diameter σ . With the choice r0 = 0.3 and o(r ) as a step function we obtain & 1, for r ≤ 0.3, o(r ) = (10.3.12) 0, for r > 0.3, where length has been scaled with respect to σ . At a fixed density of the hard-sphere system, for a given value of q (where 0 < q < 1), the corresponding value of γ is obtained from

502

The thermodynamic transition scenario

Fig. 10.2 (a) The plot of q vs. γ (see the text) like a pressure–volume diagram for a hard-sphere system using the hypernetted-chain-closure (HNC) structure. The different curves (from right to left) are for densities n 0 = 1.14, 1.17, 1.19, and 1.20. The lines connect the two branches corresponding to the same density. (b) The effective potential (see the text) calculated for the results shown in (a) for the hard-sphere system treated with HNC. The different curves from top to bottom correspond to the densities in the earlier figure in increasing order. For low densities a single minimum is present, whereas at high densities two minima exist, depicting the localized and delocalized states. c American Institute of Physics. Reproduced from Cardenas et al. (1999). Both parts 

the optimization of (F (γ ) + qγ ) for the pair of values of {q, γ }. The effective potential Gc (q) is then obtained by evaluating the RHS of (10.3.8) corresponding to this optimum value of γ . The plot of γ with q is shown in Fig. 10.2(a) for a few fixed densities. The expressed in units of σ 3 here so that the close-packed f.c.c. structure has a density n 0 is √ density n 0 = 2. Two qualitatively different types of minima are identified from this plot of q vs. γ . One is at a small q value, which signifies the delocalized liquid-like configurations with continuous motion of the fluid particles. At high densities a new minimum, corresponding to q ∼ 1, is observed. This corresponds to the glassy state in which the large-scale motions are frozen, behaving as self-generated disorder. Only vibrational modes in the frozen states occur, resulting in large overlap between the two replicas. Consider the specific results for the hard-sphere system (Cardenas et al., 1999) shown in Fig. 10.2(a). For n 0 ∼ 1.14, corresponding to γ > 0, there are two values of the overlap parameter q at which the minima are observed. Thus, at this density, in order to maintain the high-q state, the presence of the coupling γ between the replicas is required. At relatively low densities in the limit γ → 0, only the delocalized state corresponding to small q is present. This is clear from the curves corresponding to n 0 = 1.14 and 1.17 in Fig. 10.2(a). However, as the density is increased to n 0 ∼ 1.19, even in the γ → 0 limit, the high-q state appears. The latter represents the frozen state with a high value for the overlap function q(x, y).

10.3 Self-generated disorder

503

The corresponding plots of the effective potential are displayed in Fig. 10.2(b). For densities n 0 = 1.01, 1.14, and 1.17, only a single minimum in the effective potential Gc occurs. These correspond to low q values conforming to the continuous liquid-like dynamics with ergodic behavior. For the two densities n 0 = 1.19 and 1.20, on the other hand, a high-q minimum distinct from the one at low q appears. It is important to note that the relative height of the two minima at a given density of the effective potential in Fig. 10.2(b) is not an indicator of the relative stability of the two states. In fact, from our discussion above it is clear that both of them are manifestations of the same phase. At high densities the appearance of the secondary minimum at small q values signals the breaking of ergodicity. The phase space breaks into many regions, which are mutually inaccessible. The overlap between the configurations belonging to the different regions is small. The small-q minimum signifies this. On the other hand, the overlap between configurations belonging to the same region is high (q ≈ 1) and is manifested in the high-q minimum of the effective potential. The above calculation of the effective free-energy curve has been used to define the configurational entropy of the supercooled state. Let c denote the number of mutually inaccessible regions into which the phase space splits. By virtue of the definition of the configurational entropy Sc we can relate it to c as Sc = −kB ln c .

(10.3.13)

If we choose the configuration y randomly, the probability that x will be in the same region decreases as 1/c . From the perspective of the effective potential shown in Fig. 10.2(b), the occurrence of such an event will imply going from the small-q state to the large-q state. This implies activating over the barrier in the effective potential, i.e., Gc , the difference between the heights of the two minima. We therefore make the following link between the effective potential and the configurational entropy:

Sc 1 = exp[−Nβ Gc ], exp − = kB c

(10.3.14)

or T Sc /N = Gc . This result for Sc is shown for various densities in Fig. 10.3 for a hardsphere system and leads to the identification of the “Kauzmann density” ρK = 1.203 (corresponding to a packing-fraction value ϕK = 0.628) at which the configurational entropy becomes zero. This is the extrapolated point at which a thermodynamic transition to an ideal glassy phase is envisaged in the hard-sphere system. The above-described approach to the study of structural glasses is in close analogy with the methodology used for spin glasses. The idea of an effective potential for the liquids was originally developed to study metastable states in mean-field spin-glass models (Franz and Parisi, 1997, 1998). The p-spin interaction models are the typical mean-field systems displaying a Kauzmann-like entropy crisis. The configurational entropy is an increasing function of the temperature in the domain TK < T ≤ Tc and vanishes at TK . The overlap

504

The thermodynamic transition scenario

Fig. 10.3 The configurational entropy Sc as defined in eqn. (10.3.14) vs. density ρ0∗ for the hardsphere system using the effective potential. Sc extrapolates to zero, giving the Kauzmann density c American Physical Society. ρK = 1.203. Reproduced from Cardenas et al. (1999). 

function q(x, y) discussed above was also used (Franz and Parisi, 2000) for defining four-point functions χ4 (t) similar to those which have been introduced in Section 4.4.2 for the structural liquids. The definitions (10.3.1)–(10.3.2) for q(x, y) can be interpreted as a weighted product of densities ρ X (x) and ρY (y) corresponding to the configurations X and Y with weight function o(|xi − yi |). The four-point susceptibility is then a correlation of the q(X, Y ) at two different times 0 and t. χ4 (t) = β

+ , q 2 (X t , X 0 ) − q(X t , X 0 )q(X 0 , X 0 )

(10.3.15)

To compute this correlation of the q-functions, we consider a system in equilibrium at t = 0 with Hamiltonian H (X ) which evolves for positive times with a modified Hamiltonian Htot (X ) = H (X ) − εq(X, X 0 ).

(10.3.16)

Linear response theory then obtains the correlation function as χ4 (t) = β

+ , ∂ q(X t , X 0 ) q 2 (X t , X 0 ) − q(X t , X 0 )q(X 0 , X 0 ) = ∂ε

(10.3.17)

For the p-spin models the susceptibility diverges at the mean-field transition at Tc . The model liquids discussed here are generally characterized by short-range repulsive interactions. Since in the supercooled liquid slow dynamics takes place over enormous length and time scales, the implications of having long-range interaction producing the slow dynamics are important (they are discussed in the next section). Nucleation in disordered systems with large-but-finite-range KAC-type (Kac et al., 1963; van Kampen, 1964; Lebowitz and Penrose, 1966) interactions has been studied (Franz and Toninelli, 2004, 2005; Franz, 2005).

10.4 Spontaneous breaking of ergodicity

505

10.4 Spontaneous breaking of ergodicity We now discuss the statistical mechanics of the supercooled liquid in keeping with the scenario of an ideal glass transition that is characterized by the vanishing of the configurational entropy Sc at a point T = TK (say). This formulation, as we will see, also gives rise in a natural way to a first-principles approach to formulating the thermodynamics of the amorphous solid state or the glass. In a standard thermodynamic description, above the freezing point Tm , the disordered liquid state with a (time-averaged) constant density has the minimum free energy. The liquid cooled below its freezing point continues to remain in the disordered state, provided that crystallization is avoided. The free energy of the metastable liquid is obtained from a functional of a suitable order parameter ψ (say) for the state. Generally ψ is the inhomogeneous density ρ(x), which is written in a parametric form involving a set of coefficients. The latter defines the multidimensional-space freeenergy landscape (FEL) (see also Section 4.3.2 for discussion). The supercooled liquid is in a metastable state characterized by local minima of the free energy F[ρ], while the crystalline state is the most stable state for the deepest minimum in this landscape. Distinct basins exist in the FEL corresponding to different local minima of the free energy. This picture described here is somewhat analogous to the potential-energy landscape (PEL) of the N -particle system discussed earlier. However, the PEL is independent of temperature whereas the FEL changes with temperature. At T = 0 all the local minima coincide with the minimum of the potential energy as a function of the particle coordinates. In the vicinity of the freezing point the hight-temperature behavior of the liquid is described in terms of the single free-energy minimum for the uniform liquid state. In the deeply supercooled state, well below the freezing point, the corresponding FEL breaks up into an exponentially large number of basins with local minima. The metastable liquid is viewed as being caught in one of these many possible basins. This transformation in the supercooled liquid has been termed a spontaneous breakdown of ergodicity. In the case of structural glasses which is the subject of interest here, this occurs without the presence of any quenched disorder in the system. Let us consider evaluation of the partition function for such a system. The partition function is obtained by equating the summation of the Boltzmann factor over different possible states and has two main contributions. The first one comes from the evaluation of the Boltzmann weight with the system being confined to a given basin having a characteristic free-energy minimum. The number of such minima at a given temperature accounts for the configurational entropy of the supercooled liquid. The second contribution involves different states corresponding to a given basin, referred to as vibrations within the basin. This accounts for the vibrational contribution to the entropy of the supercooled liquid. Indeed, an idealized situation like this will require infinite barriers between different states such that the system confined within a given basin is undergoing only vibrations around the corresponding minimum. In mean-field p-spin models in which every spin interacts with every other spin (see Section 8.4 for more discussion) this is a more appropriate situation. We will apply this scheme for evaluating the partition function in the structural glasses in order to compute the thermodynamic properties of the glassy state.

506

The thermodynamic transition scenario

We begin with the first part, namely the configurational contribution. We adopt the notation in which a particular basin α has a free energy Fα and hence a free-energy density f α = Fα /N . With the above hypothesis of the FEL for the N -particle system being divided into separate basins, the partition function Z N for the N -particle system is obtained by summing up the contributions  ZN  d x e−β H (x) x∈α

α

=



e

−β N f α (T )

,

(10.4.1)

α

where H denotes the microscopic Hamiltonian as a function of phase-space variables denoted as x and β = 1/(kB T ) is the inverse temperature. At high temperature there is only one global minimum of the free energy F(β) contributing to Z N . Close to Tm there may appear an exponentially large number of metastable states with energies higher than F(β). However, the role of these metastable states is not “seen” in the evaluation of the Gibbs partition function as long as their number does not compensate for their small (compared with that of the global minimum) Boltzmann weight. At some temperature ∼T¯c (say) the number of metastable states becomes large enough to make up for the difference and contribute to the partition function. In a system with finite-range interactions the ergodicity breaking at T¯c is not complete. The dynamics is slow during this stage, but activated hopping processes can make the liquid reduce its free energy and finally relax to the liquid state.1 To describe the physics of the supercooled liquid below T¯c , we need to analyze the partition function evaluated as in eqn. (10.4.1). In the thermodynamic limit we will treat the free energy f α as a continuous variable spanning over the range f max (T ) > f α > f min (T ) at a given temperature T . The number of states corresponding to a given f is exponentially large and is given by

 N Sc ( f, T ) = δ( f − f α ), (10.4.2) νc ( f ) ≡ exp kB α where Sc ( f, T ) denotes the configurational entropy by definition.2 The partition function Z N is now obtained from the RHS of eqn. (10.4.1),  ZN =

d f νc ( f )e

−β N f

 =

f max

d f eβ N {T Sc ( f,T )− f } .

(10.4.3)

f min 1 This scenario has often led to the conjecture that T˜ is similar to the ideal transition temperature T of the MCT described in c c

the previous sections. Such an interpretation presumably is more appropriate for the mean-field p-spin model. For a structural glass, as we have seen, the dynamics is formulated for fluctuations around the uniform liquid state, and ergodic behavior is indeed recovered below Tc , even in the absence of any activated hopping processes. 2 It should be noted that this is only one particular way of defining the configurational entropy using the above description of the supercooled liquid in terms of the free-energy basins. Sc can also be defined, as we have seen in previous sections, in terms of local minima in the PEL or by dividing phase space at sufficiently low energy into many disconnected regions c as described in the previous section.

10.4 Spontaneous breaking of ergodicity

507

In the thermodynamic limit (N → ∞) the term which dominates the integral above corresponds to the free-energy density f = f ∗ for which the exponent {Sc ( f, T ) − β f } is a minimum. Hence the optimum value f ∗ is obtained from the solution of * 1 d Sc ( f ) ** = . (10.4.4) * df kB T f=f∗ For high enough temperatures (which includes temperatures below T < Tm but close to Tm ) the saddle-point energy f ∗ lies between f min and f max . In keeping with the scenario of a thermodynamic transition, the saddle point f ∗ shifts towards the corresponding f min (T ) as the liquid is further supercooled towards the Kauzmann temperature TK and finally reaches f ∗ = f min at TK . The temperature TK marks the point below which the saddle point sticks at f ∗ = f min (T ). In this scenario there is a phase transition in the liquid at T = TK . It is a second-order transition since the free energy is continuous through the transition and no latent heat is absorbed. The configurational entropy vanishes at TK and remains absent for all T < TK . Only a small (nonexponential) number of valleys will contribute to the partition function for T ≤ TK . These are the basins with free-energy minima at f min . 10.4.1 The replica method for self-generated disorder An important contribution to our understanding of the FEL of the glassy systems and spontaneously broken ergodicity in systems without any quenched disorder came from the work of Monasson (1995) in which an ingenious way of extending the replica trick to such systems was proposed. In this approach the configurational entropy Sc of the liquid is obtained by dealing with a composite system consisting of m replicas of the original system, with their being coupled through a weak interaction. As we will see, this is quite different from the replica approach which was applied to the spin-glass problem earlier (Mézard et al., 1987). To illustrate this, let us consider a free-energy functional Fφ in terms of a coarsegrained field φ(x). For example, F[n] denotes the free-energy functional which depends on the coarse-grained density function n(x) of the N -particle system treated as a field. For certain density distributions represented by the coarse-grained function n(x), the functional F reaches a local minimum representing a metastable state. The equilibrium Gibbs free energy for the system in terms of the field φ(x) is obtained as  −1 Fφ (β) = −β ln Dφ(x) exp{−β H [φ]}. (10.4.5) The function space of φ(x) covers the minima corresponding to the different basins of the FEL. Now, in order to compute the free-energy values at the bottom of a basin, we consider the presence of a pinning field ψ(x), which acts as an external quenched field and is coupled to φ(x) with a coupling g > 0. The corresponding free energy is obtained as a generalization of eqn. (10.4.5) to 3 4   g d x[ψ(x) − φ(x)]2 . (10.4.6) Fψ [g, β] = −β −1 ln Dφ(x) exp −β H [φ] − 2

508

The thermodynamic transition scenario

Note that the free energy Fψ [g, β] is minimum when we have the pinning field ψ(x) identical to the φ(x) corresponding to the minimum of the unperturbed free energy given by eqn. (10.4.5). For a state in which ergodicity is spontaneously broken, spanning the function space of ψ(x) should therefore be able to pick up the contribution from the minima of different basins in the FEL at temperature β −1 . Let us therefore use Fψ [g, β] as a “Hamiltonian” for the field ψ(x). The corresponding free energy F (in the limit in which the coupling g goes to zero) at some chosen temperature β˜ = mβ is obtained as  $

# −1 ˜ ˜ ˜ F[β] = −β lim ln Dψ(x) exp −βFψ [g, β] g→0+

˜ −1

= −β

˜ ln Z N (β).

(10.4.7)

The corresponding internal energy will be computed using the relation since U = −(∂/∂β ) ln Z ∂

˜ ln Z N (β) ∂ β˜ ∂ {mF[β, m]}. = ∂m

˜ =− U(β)

(10.4.8)

In writing the last equality we have chosen β as a constant. In general, it is difficult to compute the integral on the RHS of eqn. (10.4.7). However, if m is chosen to be a positive integer it is straightforward to compute the integral by introducing m identical replicas of the original system (Monasson, 1995). The field φ(x) for the composite system with different replicas is denoted as {φ} = {φ 1 }{φ 2 }{φ 3 } · · · {φ m }. It is straightforward to integrate out the Gaussian field ψ on the RHS of (10.4.7) to obtain the result ⎡  % m 1 Dφ μ (x) lim ⎣ln F[β, m] = − βm g→0+ μ=1 ⎧ ⎫⎤ m ⎨ ⎬   g [φ μ (x) − φ ν (x)]2 ⎦ × exp −β H[φ μ ] − ⎩ ⎭ 2m μ<ν μ=1

≡−

1 ln W, βm

(10.4.9)

where W is the partition function and H (together with the inter-replica coupling of strength g) is the Hamiltonian for the replicated system. In the next section we consider the calculation of the free energy of this replicated system, i.e., −β −1 ln W. The quantities F (β, m) and U(β, m) represent, at temperature β −1 , the (per-replica) free energies and the internal energies of the composite system, respectively. If we now consider the limit m = 1, the difference, i.e., β(U − F ), is equal to the “entropy” of the original system. The quotes around the word entropy are intended to indicate the peculiarity of the quantity obtained here with the special counting described above. This is achieved using the pinning field ψ(x), making it feasible to enumerate the

10.4 Spontaneous breaking of ergodicity

509

number of potential-energy basins in the FEL. This is the configurational entropy Sc of the supercooled liquid with spontaneously broken ergodicity. Note that it is different from the standard entropy β 2 (∂Fφ /∂β) associated with the field φ. From eqn. (10.4.8) we obtain the useful result

∂ . (10.4.10) F[β, m] Sc = β ∂m m=1 Below TK eqn. (10.4.10) cannot hold, since it predicts a negative configurational entropy for states having a lower free energy than Fφ . It follows from the definition (10.4.7) of ˜ Physically, even at T < TK , freezing into a small (nonextensive) number of states Z N (β). will still occur, and Sc remains zero in these states. However, the effective temperature β˜ = βm associated with the pinning field ψ remains pinned at βK for the physical system and different from the actual temperature β for φ. Understanding the physics for T < TK will require analytically continuing the theory to the region m < 1. We will discuss this case for the properties of the amorphous solid in the next section.

10.4.2 Free energy of the Replicated liquid The free-energy landscape (FEL) of a high-temperature liquid has a single minimum corresponding to a uniform liquid state. In the supercooled state the FEL transforms to one with an exponentially large number of metastable minima. This transformation is characterized by the liquid developing a finite configurational entropy Sc per particle. With the model described above Sc is obtained from eqns. (10.4.9) and (10.4.10). We express the free energy and hence the configurational entropy Sc in terms of the correlation matrix G in the replica space. In this respect it is useful to exploit the symmetry in the space of identical replicas to simplify the different elements of the matrix G of the correlation functions, Gab = φ a (x)φ a (y) = G(x − y) for a = b

(10.4.11)

= F(x − y) for a  = b.

(10.4.12)

We have assumed above that in the liquid state φ(x) = 0. Hence in this case the offdiagonal elements of the correlation matrix in the replica space are zero. With the spontaneous breakdown of the ergodicity there appear the distinct basins in the FEL, each of which with has a nonzero value of φ(x) for the liquid. In this case, for the replicated liquid, the off-diagonal elements of the correlation matrix φ a (x)φ b (y) (for a  = b) are also nonzero. If Pα is the weight factor for the α-th state or the basin in the FEL, then  Pα φ(x)φ(y)α , for a = b (10.4.13) Gab = α

=

 α

Pα φ(x)α φ(y)α , for a  = b.

(10.4.14)

510

The thermodynamic transition scenario

The small extensive interactions between the identical copies or replicas force all the m replicas to fall in the same state, i.e., simultaneously belong to the same free-energy basin. The breaking of ergodicity, however, implies a coupling between replicas and nonzero offdiagonal elements F(x − y) even in the g→0 limit.3 Assuming the symmetry of the replica indices, which is also present in the coupling (g) term on the RHS of (10.4.6), we obtain the matrix G of correlation functions in the convenient form G = (G − F)I + FE

(10.4.15)

where I is the identity matrix and E has all its elements as unity, both matrices being of size m × m. With purely Gaussian or quadratic Hamiltonian H, the m dependence of replicated quantity F [m] trivially drops out in the vanishing limit of the inter-replica coupling g. In that case we have Sc = 0. Hence consideration of non-Gaussian free energies is essential in investigating the ergodicity-breaking transition. With the special form (10.4.15) of the correlation matrix G, it follows that its inverse should also have a similar form (see below for a proof of this). The inverse of the correlation matrix G corresponding to a non-Gaussian theory is written in the form of the Schwinger–Dyson equation4 G −1 = G0−1 +  −

g E m

(10.4.16)

which defines the self-energy matrix . G0 is the correlation function in the absence both of the nonlinearities and of the g-coupling. Since the different replicas are completely decoupled in this case, we have for the zeroth-order matrix G0 = G 0 I. Therefore, from eqn. (10.4.16) we argue that the self-energy matrix  should also have the form (10.4.15)  = {G −  F }I +  F E.

(10.4.17)

Hence the inverse of the matrix G is obtained in the form G −1 = {G −1 0 + G −  F }I +  F E.

(10.4.18)

Note that the Gaussian part of the correlation matrix is assumed to keep the form G0−1 = G −1 0 I. We absorb the g-dependent part by redefining G and  F . We now use the fact that any two matrices of the form (10.4.15) will have a product that is also of the same form. Hence, with Ai = ai I + bi E for i = 1, 2, we obtain that the product has the form A1 A2 = a1 a2 I + (a1 b2 + b1 a2 + mb1 b2 )E.

(10.4.19)

3 In case the of spin glasses De Dominicis (1978a) has noted that the correlation in the replica space can be obtained in terms of

the long-time limit of dynamic correlations or the Edward–Anderson order parameters. Hence, using the MSR method for the dynamics, averages over the quenched randomness can be performed without relying on the corresponding replica trick in which the replica index is taken to zero at the end. 4 We adopt here an opposite sign compared with that in eqn. (7.1.27) in Chapter 7 for the MSR theory.

10.4 Spontaneous breaking of ergodicity

511

Hence we can express the inverse of the matrix A1 as A−1 1 =

1 b1 I− E. a1 a1 (a1 + mb1 )

(10.4.20)

By applying the above general result to the expression (10.4.18) for G −1 and then comparing with the form (10.4.15) we obtain the following equations for G and F: # $−1 G(q) = G −1 (q) +  (q) , G 0 F(q) = −

G 2 (q) F (q) ≡ G(q) − K (q), 1 − G(q) F (q)

(10.4.21) (10.4.22)

where K −1 (q) = G −1 (q) −  F (q). For the diagonal elements G(x − y) of the correlation matrix, contributions proportional to g can be neglected in the limit g→0. However, for the off-diagonal elements F(x − y) the g→0 limit is nontrivial. Corresponding to an infinitesimal coupling g if the self-energy matrix  has nonzero off-diagonal elements, G has off-diagonal components signaling nontrivial correlations in the replica space due to the spontaneous ergodicity breaking. From the general structure of the correlation matrix, G(q) as well as G is expected to be related to the static or thermodynamic properties and is not drastically affected at the ergodicity breaking and onset of glassy behavior. On the other hand, the quantity ζq = F(q)/G(q) behaves like a parameter that is nonzero beyond the spontaneous ergodicity breaking. From eqn. (10.4.22) we obtain the following equation for ζq : ζq = − F (q)G(q), 1 − ζq

(10.4.23)

implying that  F (q)  = 0 corresponds to a nonzero F(q). It is interesting to note that eqn. (10.4.23) is similar to eqn. (8.1.36) for the nonergodicity parameters f q of the ergodic– nonegodic transition for the MCT discussed in Chapter 8). It is straightforward to establish that at higher temperatures, above the ergodicity-breaking transition, the configurational entropy Sc vanishes. If the matrix G is diagonal in the replica space (i.e., F(q) = 0) then according to the Schwinger–Dyson equation  is a diagonal matrix. From eqn. (10.4.25) m will be a sum of (identical) diagonal terms of the m × m matrix and hence is proportional to m, making the RHS of eqn. (10.4.37) vanish. Therefore the configurational entropy Sc = 0 if the inter-replica correlation vanishes. The spontaneous ergodicity breaking is signalled by a nonzero F(q) obtained self-consistently from the solution of eqns. (10.4.21)–(10.4.22). Below this point Sc is nonzero and expected to approach zero at T = TK for nonzero values of F. In order do explicit calculations we need a closed set of equations for the F. A suitable closure for the self-energy  in terms of the  function is required at this point.

512

The thermodynamic transition scenario

The -derivable theory Before focusing on a specific form of the non-Gaussian free energy and the closure approximation for , we treat the model assuming that the self-energy  can be expressed in terms of the correlation function G with a functional [G]. In the context of quantum systems this has been termed the -derivable theory (Baym and Kadanoff, 1961; Baym, 1962): ab =

δ . δG ab

(10.4.24)

The above relation implies that , which is a functional of the correlation functions in G and the interaction V , is related to the self-energy by δ = Tr{δG}.

(10.4.25)

Note that the symbol "Tr" in the above equation (and also in the following) actually implies the trace of the matrix G(q) as well as the integral over all q values. In order to relate the functional  to the free energy of the system, we link the latter quantity to the correlationfunction matrix G by including a field Uab in the Gaussian part of the Hamiltonian H [{φa }] in eqn. (10.4.9), i.e., H0 [{φa }] =

1 1 −1 φa G0 ab φb + φa Uab φb . 2 2 a,b

(10.4.26)

a,b

It is then straightforward to see that the two-point function G is obtained by taking a derivative w.r.t. the current U , the logarithm of the partition function W defined with the above Hamiltonian, and finally setting the current equal to zero. For a symmetric current Uab = Uba , G ab = −

δ ln W. δUab

(10.4.27)

Hence the variation of the free-energy functional in terms of G and U is obtained as δ ln W = −Tr{GδU }.

(10.4.28)

Now  is related to the free-energy functional using the above relation. For this the RHS of eqn. (10.4.25) is simplified to δ = Tr{δG} = δTr{G} − Tr{[δ]G}. We now make use of the Schwinger–Dyson equation with the current  δ = δ G −1 − {G0−1 + U }

(10.4.29)

(10.4.30)

10.4 Spontaneous breaking of ergodicity

513

in eqn. (10.4.29) to obtain # $ δ = δ [Tr(G)] + Tr(GδU ) − Tr [δG −1 ]G  = δ [Tr(G)] − δ ln W − δ Tr ln G −1 .

(10.4.31)

In writing the last equality we have used the identity (A10.1.1) proved in Appendix A10.1. The free energy F defined in eqn. (10.4.9) is obtained up to an additive constant as F(m) =

1   + Tr ln G −1 − Tr(G) . βm

(10.4.32)

Using the above result for the free energy F , the configurational entropy Sc is obtained from eqn. (10.4.10). The derivative w.r.t. m in the expression for Sc in can be conveniently evaluated by making use of the special form of the matrix G given by eqn. (10.4.15). For matrices having this form we make use of two important identities, whose proofs are given in Appendix A10.2. For two m × m matrices B and A both of which have the special form (10.4.15), i.e., B = b1 I + b2 E and A = a1 I + a2 E, the following identities respectively hold:



∂ ∂ 1 = (10.4.33) TrB {b1 + b2 } ∂m m ∂m m=1 m=1 and, assuming that a1 and a2 are independent of m, ∂ ∂m



1 Tr ln A−1 m



=− m=1



a2 a2 − ln 1 − . a1 + a2 a1 + a2

(10.4.34)

The matrix G in the third term on the RHS of eqn. (10.4.32) has the form (10.4.15). Applying the identity (10.4.33), this term is obtained as  −

dq (2π )3

  F (q)G(q) =

dq (2π )3



ζq2 1 − ζq

 .

(10.4.35)

We have used in the above equation the key relation (10.4.22) which holds (for nonzero ζq = F(q)/G(q)) in the nonergodic state. The contribution from the second term on the RHS of eqn. (10.4.32) is evaluated using the identity (10.4.34) with A ≡ G. We obtain the following result for the configurational entropy Sc : Sc = 1 +

 q

  s ζq ,

(10.4.36)

514

The thermodynamic transition scenario

where we define s (x) = −x − ln(1 − x) − x 2 /(x − 1) and the function 1 is obtained in terms of the derivative of the  function, 1 =

∂ # (m) $** . * ∂m m m=1

(10.4.37)

It is straightforward to establish that above the ergodicity-breaking transition the configurational entropy Sc vanishes. If the matrix G is diagonal in the replica space (i.e., F(q) = 0) then, according to the Schwinger–Dyson equation,  is a diagonal matrix. Therefore, from eqn. (10.4.25) m will be a sum of (identical) diagonal terms of the m×m matrix and hence is proportional to m, making the RHS of eqn. (10.4.37) vanish. Therefore the configurational entropy Sc = 0 if the inter-replica correlation vanishes. The spontaneous ergodicity breaking is signalled by a nonzero F(q) obtained self-consistently from the solution of eqns. (10.4.21)–(10.4.22). In order to make it a closed set we need a suitable closure for the self-energy  in terms of the  function.

10.4.3 An example: the φ 4 model To make further progress we need to focus on specific models. We consider the φ 4 model with a suitable choice of  function to evaluate the free energy F of the replicated liquid and hence the configurational entropy Sc . The feasibility of the ergodicity-breaking transition in the quartic model in which the supercooled liquid develops a nonzero configurational entropy has been studied extensively (Monasson, 1999; Schmalian and Wolynes, 2000; Kawasaki, 2003; Loh et al., 2004). The model Hamiltonian for the system in terms of the vector field φ with n components (φ ≡ {φ 1 , . . . φ n }, and with the interaction of order u = u 0 /n (u 0 is a constant), is chosen in the form H[φ] =

$ u 1# [∇φ(x)]2 + r0 φ 2 (x) φ 4 (x) 2 2   + d x d x  φ(x)V (x − x  )φ(x  )

(10.4.38)

with V (x) being the interaction potential. In this the long-range nature of the interaction V (x) plays an important role in the occurrence of the transition. The closure of the selfenergy function  in terms of the correlation function G is done using the self-consistent screening approximation (SCSA) (Bray, 1974; Bray and Rickayzen, 1972, 1973). In this approximation the self-energy is expressed in terms of a screening interaction D through a self-consistent relation. We consider here the simplified case of the number of components of the field φ as n = 1 and write the i jth element of the self-energy matrix  in terms of that of D as  dk Di j (k)G ji (q − k). (10.4.39) i j (q) = 2 (2π )3

10.4 Spontaneous breaking of ergodicity

515

Di j (q) is approximated in the SCSA self-consistently in terms of the kernel function matrix # as follows:  −1 Di j (q) = v0−1 I + #(q) , (10.4.40) ij

where v0 = u 0 /β.5 # is obtained in terms of the correlation function to close the set of equations as  dk #i j (q) = G i j (k)G ji (q − k). (10.4.41) (2π )3 In writing the expression (10.4.39) for the self-energy we have absorbed a (diagonal) contribution, namely the so-called Hartree contribution 0 of the φ 4 field theory, into the Gaussian part G −1 0 , i.e., 2 ˜ G −1 0 (q) = q + r + V (q),

(10.4.42)

where the mass r0 in the Hamiltonian (10.4.38) is renormalized as r = r0 + 0 with  dk G(k) (10.4.43) 0 = u 0 (2π )3 and V˜ (q) is the Fourier transform of the potential V (x) in the Hamiltonian (10.4.38). The kernel # defined in eqn. (10.4.41) is represented in the form # = (πG − π F )I + π F E.

(10.4.44)

Using the definition (10.4.41) above and the form (10.4.15) for the matrix G it is straightforward to obtain the following expressions for πG and π F :  dp G( p)G(k − p) ≡ {G⊗G}q , (10.4.45) πG (k) = (2π )3  dp π F (k) = G( p)G(k − p)ζ p ζk− p ≡ {F⊗F}q , (10.4.46) (2π )3 where the symbol ⊗ denotes the convolution in the wave vector,  { f ⊗ g}q =

dk (2π )3

f (q − k)g(k).

(10.4.47)

The matrix [v0−1 I + #] in eqn. (10.4.40) and hence its inverse, i.e., D, also have the same form as (10.4.15). We write this as 5 In the evaluation of the partition function, temperature drops out of the problem if in the Hamiltonian given by eqn. (10.4.38) √

we apply the scaling φ →

βφ and u 0 → u 0 /β ≡ v0 .

516

The thermodynamic transition scenario

D = (DG − D F )I + D F E.

(10.4.48)

Using the inverse in the form given in eqn. (10.4.20) we obtain the following expressions for DG and D F : # $−1 DG (q) = v0−1 + πG (q) , D F (q) = −

(10.4.49)

2 (q) π F (q)DG . 1 − π F (q)DG (q)

(10.4.50)

Since the matrices D and G have the same matrix structure in terms of I and E, from the relation (10.4.39) it follows that the self-energy matrix also has the form  = {G −  F }I +  F E.

(10.4.51)

The diagonal and off-diagonal elements of the self-energy matrix are obtained as G (q) = 2[DG ⊗G]q

(10.4.52)

 F (q) = 2[D F ⊗F]q .

(10.4.53)

Starting from the above expression for  F and using the definition (10.4.46) for π F , it follows that for SCSA the following integral relation holds between the kernels:   dq dq F(q) (q) = 2 D F (q)π F (q). (10.4.54) F (2π )3 (2π )3 Using the SCSA (10.4.39) for , we can link the function  defined in eqn. (10.4.25) to the screening interaction D. For this let us consider the matrix identity (A10.1.1) given in Appendix A10.1 to obtain δ δG ab (q)

Tr{ln D−1 } =

 

=

δ Di−1 j (k) , D (k) ij 3 δG ab (q) (2π ) dk

dk (2π )3

Di j (k)

δ#i j (k) . δG ab (q)

(10.4.55)

On taking a functional derivative of eqn. (10.4.41) with respect to G ab , we obtain δ#i j (k) = δG ab (q)



 , δia δ jb δ(k  − q)G ji (k − k  ) (2π )3 dk

+ δib δ ja δ(k − k  − q)G i j (k  ) = δia δ jb G ji (k − q) + δib δ ja G i j (k − q).

(10.4.56)

10.4 Spontaneous breaking of ergodicity

517

By substituting the result (10.4.56) into eqn. (10.4.55) we obtain δ δG ab (q)

Tr{ln D

−1

 }=2

dk

(2π )3 = ab (q).

Dab (k)G ba (k − q) (10.4.57)

Following the definition (10.4.24) the  function for the φ 4 theory in the SCSA is obtained as $ # (10.4.58)  = Tr ln D−1 . Therefore the free-energy functional F for the replicated liquid is obtained as F=

1  Tr ln D−1 + Tr ln G −1 − Tr(G) . βm

(10.4.59)

The ergodicity-breaking transition Using the expression (10.4.52) for  F in the φ 4 -theory terms of elements of the correlation matrix G in the SCSA approximation we obtain from eqn. (10.4.23) the following relation below the ergodicity-breaking transition: ζq = 2G(q) 1 − ζq



dk (2π )3



 2 (k)π (k) DG F G(q − k)ζq−k . 1 − DG (k)π F (k)

(10.4.60)

The above set of self-consistent equations for the ζq gives nonzero solutions at the transition. Note that π F (q) and DG (q) are both related to static correlations G(q) in the liquid. These are taken as input in testing for nontrivial solutions for the ζq . It is interesting to note that eqn. (10.4.60) is similar to that for the non-ergodicity parameters corresponding to the dynamic transition of the simplified mode-coupling model. This ergodic–nonergodic transition is characterized by the long-time limit of the density-correlation function becoming nonzero. As we have discussed in Chapter 8, a complete analysis of the theory shows that this transition is finally smoothed off. However, strong signatures of it are seen in the first stage of viscous slowdown in a supercooled liquid. The nonlinearities driving the transition there came from both reversible (see Section 8.1.2 eqn. (8.1.36)) and purely dissipative types of dynamics (see Section 8.4.2 eqn. (8.4.38)). The origin of the nonlinearity in the present case of spontaneous ergodicity breaking is in the free-energy functional. For the special form (10.4.58) of the  function, the configurational entropy Sc is obtained from eqn. (10.4.36). Using the identity (10.4.34) for D ≡ A, the configurational entropy Sc is given by the formula

518

The thermodynamic transition scenario

 Sc =

! " dq  ! " S ζq − S ξq 3 (2π )

(10.4.61)

3x 2 . x −1

(10.4.62)

in terms of the function S(x) = −x − ln(1 − x) −

The argument ξq of S in eqn. (10.4.61) is defined in terms of ζ˜q = D F (q)/DG (q), ξq = − =

ζ˜q 1 − ζ˜q

= π F (q)DG (q)

π F (q) v0−1

(10.4.63) (10.4.64)

+ π F (q)

with the use of the respective expressions for DG and D F given in (10.4.49). Using eqns. (10.4.45) and (10.4.45), ξq is obtained in terms of the elements of the correlation matrix ξq =

{F ⊗ F}q v0−1

+ {G ⊗ G}q

.

(10.4.65)

Obtaining the expression (10.4.61) for the configurational entropy Sc requires using eqns. (10.4.23) and (10.4.63) in eqn. (10.4.54), leading to the useful relation       ζq2 ζ˜q2 dq dq =2 . (10.4.66) (2π )3 1 − ζq (2π )3 1 − ζ˜q The possibility of an ergodicity-breaking transition in the deeply supercooled liquid is therefore inferred by solving a set of self-consistent equations for nonzero ζq values. The nature of the interaction potential V (r ), which appears in the zeroth order in the expression (10.4.42) for the correlation G 0 , plays an important role here. For purely short-range interaction potentials, different choices of the mass term r0 in the effective Hamiltonian (10.4.38) do not produce a transition (i.e., a nonzero set ζ ) corresponding to any n ≥ 1 (Monasson, 1999). It was subsequently demonstrated (Schmalian et al., 2000) that the presence of a long-range component of the interaction potential is essential in this model. Competing interactions from the short-range forces and the long-range Coulomb interaction lead to finite configurational entropy Sc . This implies that, beyond a temperature at which a nonzero set of solutions for the ζq is found, there are exponentially many metastable states. The self-generated glassiness is manifested in the striped glassy phase (Westfahl et al., 2001). Subsequently Loh et al. considered the case of a KAC-type step interaction potential for V (r ) = 6α02 γ d ϑ(γ r ), (in d dimensions) with ϑ(y) = 1 for y ≤ 1 and zero otherwise. Here α02 ∝ d d r V (r ) controls the integral strength of the potential and γ −1 is the range of the potential. For large γ (short-range potential) the gradient term

10.5 The amorphous solid

519

Fig. 10.4 The configurational entropy Sc vs. temperature T with the ergodicity-breaking transition occurring at TA in the Kac-type step interaction potential. The parameter values are γ = 0.03 and r0 = 0.1494 (see the text). The units of energy and length are chosen such that α0 and the upper cutoff c American in wave-vector integration are both equal to unity. Reproduced from Loh et al. (2004).  Physical Society.

in the Hamiltonian (10.4.38) dominates and the role of V (r ) is negligible. As has already been stated, in this case no glassy state is observed. On the other hand, for small γ the transition occurs at a temperature TA . The configurational entropy Sc per unit volume is shown in Fig. 10.4. Near the transition Sc is proportional to γ 3 , implying that the longer the range of the interaction, the smaller the number of metastable states per unit volume and that it vanishes (i.e., becomes subexponential) in the van der Waals limit γ → 0 (Kawasaki, 2003).

10.5 The amorphous solid The theory described in the previous section provides us with a framework for understanding the entropy crisis in the supercooled liquid as being driven by an underlying thermodynamic phase transition at TK . The liquid transforms into an ideal glassy state with vanishing of the configurational entropy Sc . For temperatures T < TK in the ideal glassy state the configurational entropy of the supercooled liquid remains zero. In 1999 (Mézard and Parisi (1999) proposed a microscopic model of the glassy solid state below TK using this approach of a replicated liquid state. In the present section we demonstrate how the replica approach is applicable for computing thermodynamic properties of the

520

The thermodynamic transition scenario

glassy solid in terms of the basic interaction potential between the constituent particles. As has already been indicated above in the discussion of the ergodicity-breaking transition, the configurational entropy Sc remains zero below TK . This is ensured by keeping the effective temperature (βm)−1 (where m is the number of replicas) of the pinning field ψ (see eqn. (10.4.6) of earlier section) as TK . This indicates the possibility of understanding the glassy state by analytically continuing the above model to the region m < 1. For T < TK , the saddle-point free energy f ∗ corresponding to the optimum contribution in the partition function sticks at the extreme value f min of the range of variability of f . There are only subexponentially many such minima at f min . We can control the pinning of f ∗ at f min by working with a system consisting of m identical copies of the original N -particle system. On applying the above formulation of eqn. (10.4.3) for the partition function of the composite system of m replicas we obtain  Zm =

f max

eβ N {T Sc ( f,T )−m f } .

(10.5.67)

f min

The saddle point f ∗ for the cloned system is obtained from the solution of the equation m d Sc ( f ) ** = . * f=f∗ df kB T

(10.5.68)

For T > TK the corresponding value of the saddle point f ∗ lies in between the two limits at this temperature, i.e., f min (T ) < f ∗ < f max (T ). For m = 1, the saddle point occurs at f ∗ = f min , corresponding to temperature T = TK . Let us now consider the situation for T < TK . Equation (10.5.68) now can be used as the defining equation for f ∗ and by treating m as a tuning parameter we can make the saddle point coincide with the corresponding f min (T ). This requires making an analytic continuation to a region where the parameter m can be less than 1. Depending on the temperature T < TK , a critical value m = m ∗ (T ) < 1 is obtained. At this value of m for the composite system the saddle point reaches f ∗ = f min and configurational entropy Sc = 0. Alternatively, we can also describe the region m < 1 as follows. For fixed positive values of m less than unity, the corresponding temperature at which eqn. (10.5.68) is satisfied for f ∗ = f min is TmK which is less than TK . If the saddle point reaching the corresponding f min (T ) (and hence vanishing of Sc ) is considered synonymous with the transition point then the same occurs in an m-replicated system (m < 1) at a temperature TmK . Thus, for this chosen value of m, the replicated system has positive Sc until temperature TmK , below which Sc = 0. The m-replicated system considered above corresponds to a special liquid state of molecules of m atoms. In fact, for understanding the physics for T < TK , it is the cloned system with m values less than m ∗ (T ) that is of interest to us. This represents a liquid state with the various replicas forming a molecular bound state. Physically, at low temperatures the attraction between the different replicas forces all the m members to fall within the same free-energy minimum. However, the strength of the interaction between one such

10.5 The amorphous solid

521

molecule and another is effectively rescaled here by a factor m. For small enough m, the interaction between the particles is weak and the composite system is a liquid, which is rather peculiar in nature. The thermodynamic properties of the amorphous glassy state below TK are determined (Mézard and Parisi, 1999) in terms of those of this m-replicated liquid (m < 1). The free energy of the replicated system of m clones for T < TK is obtained from the saddle-point contribution in eqn. (10.5.67). We denote the free energy as mF(m, T ).6 Let f = f ∗ denote the saddle point in the integral on the RHS of eqn. (10.4.7) and be obtained from the solution of eqn. (10.5.68). Explicitly evaluating the saddle-point integral (10.5.67) at f = f ∗ gives mF (m, T ) = −

kB T ln Zm = m f ∗ − T Sc ( f ∗ , T ). N

(10.5.69)

Each of the m systems does not reach the lowest possible free energy, but the composite one does by balancing the free energy and the configurational entropy Sc as given in eqn. (10.5.69). The free energy f per replica and the configurational entropy Sc are evaluated from eqn. (10.5.69) and using the condition (10.5.68), f∗ =

∂ [mF (m, T )] ∂m

(10.5.70)

Sc =

m2 ∂ [F(m, T )] . T ∂m

(10.5.71)

The free energy of the cloned system F(m, T ) is a convex function of m having a maximum at m = m ∗ . This observation turns out to be consistent from the final result which follows. By virtue of eqn. (10.5.71), Sc = 0 at m = m ∗ . For m > m ∗ we run into an unphysical situation since the RHS of eqn. (10.5.71) becomes negative. How do we estimate the free energy in the glassy state at T < TK ? Studying the cloned liquid is useful in this respect since it helps us to locate the relevant minima in which the system is frozen in terms of m ∗ (T ). However, the free energy at T < TK cannot be obtained from the m→1 case of the cloned liquid described above. This is because the phase transition to the nonergodic state occurs at m = m ∗ (T ) < 1. However, we can determine the free energy below TK in terms of that of the cloned liquid state using the fact that in the glassy state the configurational entropy Sc is zero. It then follows from eqn. (10.5.71) that the free energy per replica F (m, T ) is independent of m. Hence the free energy F (m = 1, T ) of the original system is obtained by mapping from F (m = m ∗ (T ), T ). The latter is evaluated from the liquid-state free energy at the boundary m = m ∗ (T ) using the fact that the free energy is continuous through the transition at m ∗ . This is the crucial mapping of the free energy of the glass at T < TK into that of the replicated “liquid” consisting of molecules with m atoms. 6 We keep the notation of eqn. (10.4.7) introduced in the previous section for the liquid state above the thermodynamic

transition of vanishing entropy.

522

The thermodynamic transition scenario

To summarize the method, we are required to estimate the replicated free energy mF(m, T ) for the molecular liquid phase with each molecule consisting of m atoms. At T < TK , an analytic continuation in the unphysical range of m < 1 for F(m, T ) actually maps the replicated-liquid free energy into that of the glassy solid. In the present scenario, the supercooled liquid when cooled slowly enough undergoes a second-order phase transition at T = TK and the configurational entropy remains zero in the glassy phase following a kink at the transition point. The glassy state is caught in one of the subexponential number of free-energy minima for the system. In some sense this mapping is reminiscent of the mapping of the crystalline solid into a low-density liquid in the weighted-density-functional theories of the freezing transition discussed in Chapter 2. 10.5.1 The Mézard–Parisi model The idea sketched out above for obtaining the thermodynamic properties of the glass is simple and based on plausible hypotheses. The glassy state is mapped here into a replicatedliquid state with N molecules, each having m atoms. However, given the fact that one has to focus here on the unphysical domain m < 1, implementing the scheme appears somewhat strange. We are required to analytically continue the expressions for the free energy into the domain m < 1. In order to develop a statistical-mechanical formulation starting from a two-body interaction potential v(r ), we therefore need to start from a microscopic Hamiltonian. In the following we consider an explicit formulation (Mézard and Parisi, 1999a) of the scheme proposed above for summing the partition function in terms of contributions from different free-energy basins. The partition function for the N -particle system moving in a volume V in d dimensions is obtained as ⎡ ⎤  N N   β 1 % dri exp⎣− v(ri − r j )⎦. (10.5.72) ZN = N! 2 i=1

i, j=1

The thermodynamic limit in the present context involves taking N , V →∞, with the density n = N /V remaining fixed. We construct the replicated liquid by introducing the m identical replicas of the original system. We have m clones of each particle in the composite system forming a molecular liquid. We denote the position of the ith particle of replica a as ria with a ∈ 1, 2, . . . , m. To avoid clutter, we will write the position x dropping the vector notation here. The partition function Z m of this composite system is obtained as m  N % % ! "  dria Zm = (10.5.73) exp −β Hm ria . N! i=1 a=1

Hm is the Hamiltonian for the composite system consisting of the m clones of the original liquid, ⎧ ⎫ m N  N ( ( ) )⎬   ! a" 1 ⎨  Hm ri = v ria − r aj −  w ria − r bj . (10.5.74) ⎭ 2⎩ i, j=1

i, j=1 a,b=1

10.5 The amorphous solid

523

In the above, as well as in the following, we adopt the notation that the prime in the sum indicates the absence of the a = b term. w(r ) is the weak attractive potential of a short range, which is less than the typical inter-particle distance in the solid state. We now test whether a temperature TK can be identified from the above model such that for T < TK the free energy F (m, T ) of the replicated liquid reaches a maximum corresponding to m ∗ (T ) < 1. For this F (m, T ) has to be evaluated for real positive values of the parameter m. In the final calculation, we will take limw→0 after taking lim N →∞ . We begin by evaluating the partition function given in eqn. (10.5.74) with a simple-harmonic approximation for the Hamiltonian in terms of u. Several other methods have been developed (Mézard and Parisi, 1999b) for the problem, but we focus our discussion on the harmonic approximation. The location ria of the ith atom of replica “a” in the m-atom molecular liquid is expressed in terms of the center-of-mass coordinate ri , ria = ri + u ia ,

(10.5.75)

where ri is the center-of-mass coordinate for the ith molecule consisting of m identical atoms. By definition, ri is obtained as mri =



ria .

(10.5.76)

a

Hence, from eqn. (10.5.75), it follows that sum of individual displacements u ia for the ith molecule of the cloned system must satisfy 

u ia = 0.

(10.5.77)

a

In  expression   (10.5.73) for the partition function Zm , we make a change of variables  athe ri → ri , u ia for a = 1, 2, . . . , m, and i = 1, 2, . . . , N . It is useful to check some details of this coordinate transformation in order to keep track of the proper m dependence in the partition function Zm . There are in total m N d displacement variables u ia and N d center-of-mass coordinates ri . Taking into account the N d delta functions due to the constraints given by (10.5.77), the total number of independent variables is (m N − N + N ) d = m N d as expected. With this change of variables, taking into account the factor of m d due to the Jacobian, the partition function is obtained as 1 Zm = N!





 dr

N

du mN

N % i=1

 m δ d



 u ia

a

⎫ ⎧ m ( N   ( N  ⎨ β  ) )2 ⎬   v ria − r aj − u ia − u ib . × exp − ⎭ ⎩ 2 4 i, j=1

i

a,b=1

(10.5.78)

524

The thermodynamic transition scenario

In writing the above equation, we have simplified the notation by using on the RHS the following abbreviations for the differential elements: dr N ≡

N  %

dri ,

du mN ≡

i=1

m  N % %

du ia .

(10.5.79)

i=1 a=1

( ) The inter-particle potential v ria − r aj is expanded in terms of the displacements u ia around the respective centers of mass as ) ( ) ( v ria − r aj = v ri − r j + u ia − u aj ( )p ∞ u ia − u aj  = v(ri − r j ) + (10.5.80) v ( p) (ri − r j ), p! p=2

where v ( p) (r ) denotes the pth-order derivative of the two-body potential v(r ). Using the above expression for v in the partition function, we obtain     mNd N dr du mN δ N u mN Zm = N! ⎧ ⎤ ⎡   N  ∞ ua − ua p ⎨ β   i j ⎣v(ri − r j ) + v ( p) (ri − r j )⎦ × exp − ⎩ 2 p! i, j=1 p=2 ⎫ m N ⎬     a 2 − u i − u ib ⎭ 4 i a,b=1 ⎧ ⎫  N  ⎨ βm  ⎬ mNd v(ri − r j ) Iv , (10.5.81) dr N exp − = ⎩ 2 ⎭ N! i, j=1

  where we have denoted the product of N d delta-function constraints with δ N u mN as   N ( ) %  N N m δ um ≡ δ uN . (10.5.82) i=1

a

The integral Iv representing the contributions from the vibrations of the individual atoms about the corresponding center of mass is defined as  ( ) Iv = du mN δ N u mN exp[−Au ]. (10.5.83) The quantity Au in the exponent on the RHS of eqn. (10.5.83) represents the inter-replica interaction. For low temperatures we will make the crucial approximation that the u ia are small so that the expansion in the second term in the exponent on the RHS of (10.5.80) is considered only up to quadratic order ( p = 2). With this approximation the partition function reduces to an integral of the quantity Iv over the center-of-mass coordinates.

10.5 The amorphous solid

525

We express the full partition function Zm as an average over the center-of-mass coordinates, which we denote by putting a subscript ∗ on the angular brackets: ⎧ ⎫⎤ ⎡  N  ⎨ βm  ⎬ 1 Zm = ⎣ v(ri − r j ) ⎦ Iv dr N exp − ⎩ 2 ⎭ N! i, j=1

= Zcm (T ∗ )Iv ∗ ,

(10.5.84)

where Zcm denotes the partition function for a system involving only the motion of the center of mass of the m-atom molecules at a temperature T ∗ = T /m. The integral Iv represents the contributions from the vibration of the atoms around the corresponding center of mass and is obtained by performing the Gaussian m N d-dimensional integrals over the u ia variables with N d delta functions. We describe this calculation in Appendix A10.3. Here we just continue with the result,

m−1 Iv = Cm exp − Tr ln M , (10.5.85) 2 where the matrix M, which is dependent on the center-of-mass coordinates {ri } and the corresponding Hessian matrix for the interaction potential v(ri − r j ), is obtained as  v (μν) (ri − rk ) − v (μν) (ri − r j ), (10.5.86) Miμ, jν = δi j k

where we have used the abbreviation v μν ≡ ∂μ ∂ν v(ri − r j ). The constant Cm is obtained in Appendix A10.3 as Cm = m N d/2 (2π )(m−1)N d/2 .

(10.5.87)

On substituting this result and (10.5.85) into eqn. (10.5.84), we obtain the total partition function Zm evaluated in the harmonic approximation, - 3 4.

m−1 ∗ Zm = Zcm (T ) Cm exp − Tr ln M . (10.5.88) 2 ∗ The above expression therefore approximates the partition function of the molecular liquid with m replicas at temperature T as a sum of two contributions: first, the contribution Zcm from the center-of-mass motion of the ri , for i = 1, . . . , N molecules, each with m atoms at an effective temperature T ∗ = T /m; and second, the contribution in the square brackets on the RHS of eqn. (10.5.88) from the vibration modes giving rise to the Tr ln M term. Evaluating the second part described by M, which is related to the Hessian matrix of the glassy state, requires further approximations. This contribution, which accounts for the vibration of the individual atoms in the ith molecule around the center of mass at ri , is evaluated in a non-self-consistent approximation, i.e., we ignore the feedback effects from the vibration on the {ri } coordinates. In this approximation we write

.

1 1 exp − (m − 1)Tr ln M ≈ exp − (m − 1)Tr ln M∗ . (10.5.89) 2 2 ∗

526

The thermodynamic transition scenario

This is termed the quenched approximation (Mézard and Parisi, 1999b). Even after this approximation has been made, evaluation of the above term involving the “trace log” of the matrix M is difficult. An approximate evaluation of the RHS of eqn. (10.5.89) is given in Appendix A10.3. The definition of the free energy F(m, T ) per molecule of the replicated liquid is given in eqn. (10.5.69). The partition function Zm includes contributions separated into center-of-mass and vibrational parts as given by eqn. (10.5.88): 1 ln Zm mN

1 m−1 Tr ln M∗ . =− ln Zcm (T ∗ ) + ln Cm − mN 2

βF(m, T ) = −

(10.5.90)

The quantity f cm in the first term on the RHS of eqn. (10.5.92) denotes the free-energy density (per particle) of this center-of-mass system at a temperature T ∗ = T /m, f cm (T ∗ ) = −

kB T ln Zcm . mN

(10.5.91)

f cm (T ∗ ) is obtained from standard integral-equation theories of liquid-state physics. Using the results from eqns. (10.5.87) and (A10.4.16), we obtain the free energy F (m, T ) per replica as βF(m, T ) = β f cm (T ∗ ) +

" ! m−1 d ICM − ln (2π )m−1 m , 2m 2m

where ICM in the second term on the RHS is obtained as   dk  1 ICM = L3 [a˜ " (k)] + (d − 1)L3 [a˜ ⊥ (k)] d n0 (2π ) & '2 ⎤   v (μν) (r ) 1 ∗ ⎦ + d ln(βr0 ). dr gcm (r ) − 2 r¯0 μν

(10.5.92)

(10.5.93)

Using the above analytic expression in eqn. (10.5.92) requires application of the so-called quenched approximation, as well as treating multi-particle correlations with a superposition approximation. These are discussed in Appendices A10.3 and A10.4. It is important to note here that evaluating this approximate expression for the free energy requires only knowledge of the thermodynamic properties of the N -particle system in which the ith particle is located at the center of mass of the corresponding molecule in the replicated liquid. The interaction potential between the particles at i and j is v(ri − r j ). The integral ICM in the second term on the RHS is obtained in terms of the pair cor∗ and derivatives of the interaction potential v(r ). The function L is relation function gcm 3 defined as L3 (x) = ln(1 − x) + x + x 2 /2. The quantities a˜ " (k) and a˜ ⊥ (k) appearing in the ∗ (r )v (μν) (r ) with respect to arguments of L3 are obtained from the Fourier transform of gcm the wave vector k. In the isotropic system the transformed function is expressed in terms

10.5 The amorphous solid

527

of two components, which are referred to as “parallel” and “perpendicular” with respect to the wave vector, 2 

  ∂ v(r ) ik · r 1 ∗ ˆ ˆ ˆ ˆ dr gcm (r ) = kμ kν a" (k) + (10.5.94) δμν − kμ kν a⊥ (k). e ∂rμ ∂rν d For the arguments of the function L3 on the RHS of eqn. (10.5.92), we define a˜ " (k) = a" (k)/¯r0 and a˜ ⊥ (k) = a⊥ (k)/¯r0 . The m dependence of expression (10.5.92) is totally analytic and is therefore suitable for analytically continuing F (m, T ) to the region m < 1 of the parameter space. The above model was studied for different interaction potentials. For soft-sphere potentials v(r ) = 1/r 12 in three dimensions (Mézard and Parisi, 1999a). The Kauzmann temperature TK is obtained from eqn. (10.5.68) using the calculated value of the configurational entropy. By computing the free energy f cm (T /m) and the pair correlation function g ∗ (r ) with the hypernetted-chain approximation, the analytic expression for the replicated free energy is evaluated. By locating the maximum of the free energy, m ∗ (T ) is obtained. The Kauzmann temperature corresponds to m ∗ (TK ) = 1. In the present case this is obtained at TK = 0.194 and density n 0 = 1.0. This temperature and density together correspond to the dimensionless parameter eff = n 0 T −1/4  1.51. The corresponding result observed in computer simulations of the same system is  = 1.6 (Bernu et al., 1985; Roux et al., 1989). The various properties of the ideal glass phase follow from the model in a natural way. For the soft-sphere interaction potential the various parameters characterizing the glassy state are shown in Fig. 10.5(a). The effective temperature is given by T ∗ = T /m ∗ , where m ∗ is the value of m at which the free energy has a maximum. T ∗ varies very little in the glass phase, remaining close to TK . Below TK , in the glass phase the specific heat computed from the derivative of the internal energy remains almost constant at 32 . This is the result expected from the Dulong–Petit law for a classical solid, which is the model followed in the present analysis. The model was applied to other systems (which have more commonly been investigated in order to study the glass transition), namely binary mixture of soft spheres (Coluzzi et al., 1999). More recently, this model has been applied to the study of hard-sphere fluids (Parisi and Zamponi, 2010). The (Kauzmann) packing fraction ϕK at which the configurational entropy vanished is 0.62, which is comparable to the result of 0.628 obtained from Fig. 10.5(b) with application of the result (10.3.14). The nature of the proposed transition at TK is somewhat ambiguous. The configurational entropy Sc vanishes at TK and remains zero for T < TK . The thermodynamic transition at TK characterized by the configurational entropy Sc remains continuous on passing through the transition with a kink at TK . No latent heat is absorbed. In this sense the thermodynamic transition at TK is continuous. On the other hand, we define below an order parameter, which changes discontinuously on passing through the transition. The square of the size of the cage seen by each atom in the system is defined as A=

 1 1 2 2 x − xi 2 . 3 i

(10.5.95)

528

The thermodynamic transition scenario

Fig. 10.5 (a) The various quantities characteristic of the glass phase vs. temperature. From top to bottom: the inverse effective temperature 1/T ∗ = βm of the reference liquid, the internal energy, the specific heat, and the quantity 100 A/T , where A is the square of the cage radius defined in eqn. (10.5.95). Reproduced from Mézard and Parisi (1999a). (b) The configurational entropy Sc of the binary Lennard-Jones system. The full curve shows the theoretical prediction obtained from the Mézard–Parisi model using cloned liquid (Coluzzi et al., 1999). The dashed curve shows the results from the simulations of Sciortino et al. (2000) using an inherent-structure approach to the potentialenergy-landscape studies. The dotted curve shows the results from simulations by Coluzzi et al. c Institute of Physics. (2000). 

10.5 The amorphous solid

529

In the ergodic liquid state, when the particles are not localized, the inverse A−1 vanishes. This inverse cage size changes discontinuously at the transition, as is evident in the way the model is constructed. In the glass phase the inverse of A is finite and the partition function is evaluated by approximating the dynamics in terms of harmonic vibrations around the center-of-mass coordinates. Naturally, A ∝ T for the glass phase. The specific heat also jumps from its liquid-state value above the transition to one close to that of the crystalline state below the transition. The thermodynamic entropy is computed from the formula (10.5.71) obtained above using the analytic form (10.5.92) for the free energy, m2 ∂ [F(m, T )]m=1 T ∂m * m ∂ f cm (T ∗ ) ** kB = ∗ − [d ln(2π e) − ICM ] * T ∂m 2 m=1 = Sliq − Ssol ,

Sc =

(10.5.96)

where Sliq is the entropy of the liquid, ∂ f cm (T ∗ ) ∂T ∗ ∂ f cm (T ∗ ) = ∂m ∂m ∂ T ∗ T∗ (10.5.97) Sliq (T ∗ ), = m and T ∗ = T at m = 1. Ssol represents the entropy of the harmonic solid, which is given by kB {d ln(2π e) − ICM }. (10.5.98) 2 Indeed, as is clear from the above, we are computing the configurational entropy Sc of the supercooled liquid by subtracting from the liquid-state entropy the vibrational entropy in the supercooled state. The latter is obtained by performing a harmonic expansion around the center-of-mass coordinates. Sc is zero at T = TK . In Fig. 10.5(b) the theoretically calculated configurational entropy of the BMLJ system obtained from the formulation described above is compared with that of models studied with computers using potential-energy landscapes. The above calculation of the configurational entropy Sc for T > TK by going to the m = 1 limit of the center-of-mass contribution for the cloned molecular liquid is in fact similar to the analysis of the previous section. In the present context, however, the partition function is evaluated by performing the phase-space integrals in the harmonic approximation. The calculation presented here in terms of the short-range interaction potential is not motivated to locate the spontaneous breaking of ergodicity from the normal liquid state. Assuming the occurrence of such a transition, the thermodynamic properties of the amorphous solid (with Sc = 0) are obtained here using the microscopic interaction potential as an input. It is useful to note here that the replica index m ∗ (T ) depends linearly on the temperature T . This is similar to the case of one-step replica symmetry breaking, which was originally Ssol =

530

The thermodynamic transition scenario

introduced in the context of spin glasses (Mézard et al., 1987). Kirkpatrick and Thirumalai (1987a) observed the analogy between the phase transition of discontinuous spin glasses and the thermodynamic glass transition. We have discussed in Section 8.4 the close analogy between the (dynamic) ergodic–nonergodic transition in p-spin models and that predicted for structural glasses. Indeed, Kirkpatrick and Thirumalai in their 1987 work discussed the possibility of a thermodynamic transition as a one-step replica symmetry breaking at a temperature below the dynamic transition at Tc . A similar replica approach has been applied to the density-functional model for studying the phase diagrams of classical liquids in a quenched random pinning potential by Thalmann et al. (2000) as well as for computing correlations in a flux liquid in a strongly anisotropic, layered superconductor with random point pinning sites, as was proposed by Menon and Dasgupta (1994). The consequences of long-range interaction for glassy behavior have also been studied (Rostiashvili and Vilgis, 2000a, 2000b) using the old type of replica approach, in which the replica index going to the limit zero is to be invoked at the end. It is important to note here the difference between the uses of replicas in the cases of a spin glass and a structural glass. For spin glasses the replicas are used as a computational trick to calculate the average of the logarithm of the partition function, and in the end one needs to take the replica index to the limit zero. On the other hand, in the above discussion of the thermodynamics of structural glasses the idea of replicas has a more physical interpretation (Derrida, 1981; Gross and Mézard, 1984). In the structural glass there is no quenched disorder and hence there is no need to average over the logarithm of the partition function. Replicas are introduced here as a means to locate the spontaneous ergodicity-breaking transition or to study properties of the amorphous solid state within the framework of equilibrium statistical mechanics. We do not need to take the “zero-replica” limit. However, there is an analytic continuation in the number of replicas m becoming less than 1. An alternative way of describing the glassy state is to introduce a real coupling of strength γ (see Section 10.3.1) with another system, which has thermalized. In the latter case, however, the use of replicas to compute the logarithm of a partition function is in fact similar to that for the spin-glass problem.

Appendix to Chapter 10

A10.1 Matrix identity We prove here for the general square matrix G the following identity: # $  Tr G δG −1 = δ Tr ln G −1 .

(A10.1.1)

Let us first define a matrix A of the same size as G as A = ln G −1 . We denote the i jth element of G and that of A as G i j and Ai j , respectively, ∞ # $    1  n Tr G δG −1 = G i j δ[exp( A)] ji = Gi j δ A ji n! i, j

=



i, j

Gi j

i, j

×

∞  n=1

n 

1 n!

n=0

 j1 ,..., jn−1

A j j1 · · · {A jr−1 jr [δ A jr jr+1 ] A jr+1 jr+2 } · · · A jn−1 i .

(A10.1.2)

r =1

We now redefine the dummy variables i and j interchanging with jr and jr +1 , respectively, # $ Tr G δG −1 

=

{δ Ai j }

i, j, j1 ,..., jn−1

=

 {δ Ai j } i, j

=

∞  n=1

∞ n  1  A jr j1 . . . {A jr−1 i G jr jr+1 A j jr+2 } . . . A jn−1 jr+1 n! n=1

1 n!



n=1

n 

A jr j1 . . . {A jr−1 i G jr jr+1 A j jr+2 } . . . A jn−1 jr+1

j1 ,..., jn−1 r =1

∞ n   1  {δ Ai j } n! ij

r =1



A j jr+2 . . . A jn−1 jr+1 G jr+1 jr A jr j1 . . . A jr−1 i .

r =1 j1 ,..., jn−1

(A10.1.3) 531

532

Appendix to Chapter 10

The quantity within curly brackets represents the jith element of the product of n matrices and this is the same for all of the r = 1, . . . , n terms in the sum. Hence we obtain n ∞  # $   1  n−1−r {δ Ai j } G Ar A Tr G δG −1 = ji n! ij n=1 r =1 ∞   An−1  = {δ Ai j } G (n − 1)! ij n=1 ji   = {δ Ai j }[exp[ A]G] ji = {δ Ai j }δ ji ij

= δ{Tr A} = δ{Tr log G

i −1

}.

(A10.1.4)

A10.2 Matrix identity II We prove here identities (10.4.33) and (10.4.34) for matrices of the form A = a1 I + a2 E where I and E are both m × m matrices respectively represents the identity matrix and the matrix with all elements equal to unity. Let us first consider the i-th eigenvalues μi for the matrix with the corresponding eigenvector {x1i , . . . , xmi } (i = 1, . . . m). The eigenvalue equation is obtained as x1i + · · · + xmi = μi x1i = μi x2i = · · · = μi xmi

(A10.2.1)

The above equation can be satisfied either (a) For μi = 0, i.e., x1i + x2i + · · · + xmi = 0

(A10.2.2)

(b) For x1i = x2i = · · · = xmi , i.e, μi = m. Since the condition (A10.2.1) can be satisfied with (m − 1) independent choices of xαi for α = 1, . . . , m − 1, we have m − 1 of the eigenvalues of m × m matrix E equal to 0 and one eigenvalue equal to m. For these choices the corresponding eigenvalues of the matrix B will be λ = b1 , b1 , . . . b1 , {b1 + mb2 } B CD E

(A10.2.3)

respectively with the same eigenvectors described in cases (a) and (b) above. Therefore the trace of this matrix B is equal to m(b1 + b2 ) and hence the identity (10.4.33) follows. Next, for the matrix A defined as above, the eigenvalues for the inverse matrix A−1 are the reciprocals of those for the direct matrix. We denote them as {γi−1 }, 3 4 1 1 1 1 −1 . γ ≡ , ,... , a a a1 a1 + ma2 B 1 1CD E

A10.3 Computation of the vibrational contribution Iv

533

Hence eigenvalues for the matrix ln A−1 are

3 4 1 1 1 1 , ,... , . ln a1 ln a1 ln a1 ln(a1 + ma2 )

Therefore we obtain for the trace of this matrix ln A−1



1 1 −1 Tr(A ) = ln + (m − 1) ln a1 + ma2 a1

(A10.2.4)

(A10.2.5)

Hence, assuming a1 and a2 are not m dependent 3



* * 1 1 ∂ a2 −1 * = ln (A10.2.6) T r ln A − * ∂m m a a + ma2 1 1 m=1



4 1 1 − (m − 1) ln − ln a1 a1 + ma2 m=1



1 1 a2 = ln − ln − a1 a1 + a2 a1 + a2



a2 a2 =− − ln 1 − a1 + a2 a1 + a2 (A10.2.7)

A10.3 Computation of the vibrational contribution I v Here we compute the vibrational contribution Iv to the partition function Zm of the replicated liquid defined in eqn. (10.5.81). We evaluate the expression defined in eqn. (10.5.83) with the Gaussian approximation ⎧  N  m  d ⎨ β     a μ  a ν u i − u aj u i − u aj Iv = m N d du mN δ N u mN exp − ⎩ 4 i, j=1 a=1 μ,ν=1 ⎫ m N ⎬     a 2 u i − u ib × ∂μ ∂ν v(ri − r j ) − . ⎭ 4 i=1 a,b=1

(A10.3.1) By replacing the delta functions on the RHS by a corresponding integral representation, ⎡ ⎤    N N   %  N % N a −d a (A10.3.2) d Ji exp⎣i δ ui = (2π ) Ji · u i ⎦, δ um ≡ i=1

a

i=1

i,a

we obtain the vibrational contribution Iv as  ( m )N d  !  " du mN d J exp −Av u ia , Ji . Iv = 2π

(A10.3.3)

534

Appendix to Chapter 10

  We have used the notation dJ ≡ i=1 dJi above. The exponent Av u ia , Ji is a quadratic function of the u ia fields and a linear coupling to the current field Ji . Ignoring the crosscoupling terms in the  → 0 limit, we obtain the “action” Av in the exponent as
Av =

N d m  1   a a Miμ, jν u iμ u jν + i Ji · u ia . 2 a=1 i, j=1 μ,ν=1

(A10.3.4)

i,a

The matrix M has the form  & '   Miμ, jν = β δi j v (μν) (ri − rk ) − v (μν) (ri − r j ) + δi j δμν  ,

(A10.3.5)

k

where we have used the abbreviation v (μν) ≡ ∂μ ∂ν v(ri − r j ). From the “action” functional obtained in eqn. (A10.3.4) we identify from each of the m replica indices a a contribution Aav , such that Av =

m 

Aav .

(A10.3.6)

a=1

A particular replica component Aav is obtained in terms of the N d-dimensional field φia (i = 1, . . . , N d) and N d×N d symmetric matrix Mi j as Aav =

Nd Nd  1  a φi Mi j φ aj + i Ji φia 2 i, j=1

= =

1 2 1 2

i=1

#

Nd  i jkl=1 Nd  i, j=1

Nd $ # $ 1 Ji Mi−1 φia + Jl Mli−1 Mi j φ aj + Jk M−1 + kj j Jj 2

φ˜ ia Mi j φ˜ aj +

i j=1

1 2

Nd 

Ji Mi−1 j Jj.

(A10.3.7)

i, j=1

5 In the last equality we made a change of variables, φ˜ i = φia + j J j Mi−1 j . Using the results (A10.3.7) for each of the replica indices of Av in the expression (A10.3.3), we obtain the following result for Iv : ⎡ ⎤  N d   Nd m  Nd   1 m m ⎦ d φ˜ mN d d J exp⎣ Iv = Ji Mi−1 φ˜ ia Mi j φ˜ aj + j Jj 2π 2 2 a=1 i, j=1 i, j=1 ⎡ ⎤  √ N d   N d m Nd    1 m 1 ˜⎦ = d φ˜ mN d d J˜ exp⎣ φ˜ ia Mi j φ˜ aj + J˜i Mi−1 j Jj . 2π 2 2 a=1 i, j=1

i, j=1

(A10.3.8)

A10.3 Computation of the vibrational contribution Iv

535

In reaching the last equality we used for the differential of φ˜ ia for all the m N d variables d φ˜ mN d ≡

m Nd % %

d φ˜ ia

(A10.3.9)

i=1 a=1

√ and for all N d of the Ji variables a transformation m Ji = J˜i . The integrals over the φ˜ ia and Ji are all Gaussian and straightforward to do. To evaluate these integrals, we use the following general result for set n of variables {ψi }: ⎡ ⎤

n %  +∞  1 1 n/2 ⎣ ⎦ dψi exp − ψi Ai j ψ j = (2π ) exp − Tr ln A . (A10.3.10) 2 2 −∞ i

i, j=1

Let us consider the Gaussian integral for set n of variables {ψi }, ⎡ ⎤ n  %  +∞ 1 dψi exp⎣− ψi Ai j ψ j ⎦ . In = 2 −∞ i

(A10.3.11)

i, j=1

By means of a similarity transform A˜ = S −1 AS with the n × n matrix S constructed from the eigenvectors of the matrix A (assumed nonsingular), A is reduced to a diagonal form A˜ with the eigenvalues λi of A on the diagonal, i.e., with A˜ i j = δi j λi . With a corresponding linear transformation from {ψi } to {ψ¯ i } such that ψ¯ = Sψ, In reduces to (apart from a Jacobian) a diagonal form

 n/2 λi 2π d φ¯ i exp − φ¯ i2 = 2 λi i   1 = (2π )n/2 exp − ln λi . 2

In ∼

%

(A10.3.12)

i

In order to further analyze the above result, we define a matrix A , A = exp[A ],

(A10.3.13)

 or, equivalently, A = ln A. It is easy toshow  that the matrix S also diagonalizes A and  that the diagonal elements are λi = exp λi . Therefore we have   ln λi = λi = Tr A = Tr ln A. (A10.3.14) i

i

Hence we obtain from eqn. (A10.3.12) the useful result for the Gaussian integral in terms of the trace of the matrix ln A that

1 In = (2π )n/2 exp − Tr ln A . (A10.3.15) 2

536

Appendix to Chapter 10

Using this result, we obtain from eqn. (A10.3.8) the vibrational contribution as ⎡ ⎤  √ N d   Nd m  Nd   1 m 1 ˜⎦ d φ˜ mN d d J˜ exp⎣ Iv = φ˜ ia Mi j φ˜ aj + J˜i Mi−1 j Jj 2π 2 2 a=1 i j=1

i, j=1

 √ N d

√ (m+1)N d m m 1 −1 = exp − Tr ln M − Tr ln M 2π 2π 2 2

m − 1 = m N d/2 (2π )(m−1)N d/2 exp − Tr ln M 2

m−1 = Cm exp − Tr ln M . 2

(A10.3.16)

The factor Cm in front of the exponential is important for keeping track of the m dependence of the partition function Zm for the replicated liquid with m clones of the original system. The trace of the matrix M in the limit  → 0 is determined from & '  (μν) (μν) v (ri − rk ) − v (ri − r j ) . (A10.3.17) Miμ, jν = β δi j k

A10.4 Computation of Tr ln M∗ We present here an approximate calculation of Tr ln M∗ , which appears on the RHS of eqn. (10.5.89), after making the quenched approximation. In general the matrix M involves the Hessian matrix for the amorphous structure and constitutes a random matrix. Evaluation of the above term will involve computing the corresponding eigenvalues. These eigenvalues are also involved in analyzing the instantaneous normal modes (Keyes, 1997) in the disordered state. Indeed, the harmonic approximation made above makes sense only if M does not have any negative eigenvalue. The number of negative eigenvalues becomes very small for temperatures below Tc (Sciortino and Tartaglia, 1997). Hence the harmonic expansion is applicable for temperatures below Tc and close to Tg . We proceed here by adopting a simple resummation scheme of Mézard and Parisi, (1999a). This is based on the superposition approximation of multi-particle correlations in the amorphous solid. Let us first consider the diagonal element of the matrix M given by eqn. (10.5.86). We replace the first term in this expression for the matrix M by its average to obtain the simplified expression 0 /      1 (μν)   v (ri − rk ) ≈ δ(r − ri )δ(r − rk ) v (μν) (r − r  ) dr dr N k i,k   2 n ∗ (r − r  )∂μ ∂ν v(r − r  ) = 0 dr dr  gcm N  δμν n 0 r0 ∗ drgcm = n0 (r )∇ 2 v(r ) ≡ δμν , (A10.4.1) d β

A10.4 Computation of Tr ln M∗

where the length r0 is defined as r0 =

1 d



537

∗ drgcm (r )∇ 2 [βv(r )].

(A10.4.2)

To keep the notation simple we will now assume that v(r ) is scaled with the factor β and write βv(r ) as v(r ). In reaching eqn. (A10.4.1), we have used the isotropy of the system ∗ is the which is described in terms of the center-of-mass coordinates of the molecules. gcm pair correlation function at density n 0 and temperature T ∗ for this center-of-mass system. It should be noted that, with the present notation of writing the position vector as r , the differential dr actually denotes a d-dimensional elementary volume. With this approximation the diagonal elements of the matrix are of the form   1 Miμ, jν = n 0r0 δi j δμν − ∇μ ∇ν v(ri − r j ) . (A10.4.3) n 0 r0 For computing the Tr ln M term we expand the logarithms in powers of 1/(n 0 r0 ). We obtain the following expansion for Tr ln M: ∞ + ,( p) ,  Tr ln M , Tr ln M = N d ln(n 0r0 ) +

+

∗

p=2

∗

(A10.4.4)

where the pth-order term in the expansion is obtained in the form +

Tr ln M

,( p) ∗

d + N  ,  1 (ν1 ν2 ) (ν p ν1 ) =− (r − r ) . . . v (r − r ) v i i i i p 1 2 1 . (A10.4.5) p(n 0 r0 ) p {i}=1 {ν}=1

We have used above the notation {i} ≡ i 1 , i 2 , . . . i p and {ν} ≡ ν1 , ν2 , . . . ν p . The prime in the sum indicates that the terms with only different particles ri1 , ri2 , . . . , ri p are involved. On introducing the delta functions for each of the particle coordinates, we obtain   N , + + ,( p) 1 dr =− . . . dr δ(r − r ) . . . δ(r − r ) Tr ln M 1 p 1 i p i p 1 ∗ p(n 0 r0 ) p {i}=1

×

d 

v (ν1 ν2 ) (r1 − r2 )v (ν2 ν3 ) (r2 − r3 ) . . . v (ν p ν1 ) (r p − r1 )

{ν}=1

=− ×

1 p pr0



d 

 dr1 . . .

∗ dr p gcm (r1 . . . r p )

v (ν1 ν2 ) (r1 − r2 )v (ν2 ν3 ) (r2 − r3 ) . . . v (ν p ν1 ) (r p − r1 ). (A10.4.6)

{ν}=1

The lowest- p-value contribution on the RHS of eqn. (A10.4.4) comes from the p = 2 term in the expansion,  d + ,(2)  V 2 ∗ = − 2 drgcm (r ) {v (ν2 ν1 ) (r )} . (A10.4.7) Tr ln M ∗ 2r0 ν ν =1 1 2

538

Appendix to Chapter 10

For p > 2 terms we have to deal with higher-order correlation functions. We now make the ∗ (r . . . r ) for superposition approximation for the multi-particle correlation function gcm 1 p p ∗ ∗ (r − r ) p > 2, as n 0 times the products of pair correlation functions gcm (r1 − r2 ) . . . gcm p 1 to obtain7   + ,( p) 1 ∗ ∗ dr1 . . . dr p gcm =− p (r1 − r2 ) . . . gcm (r p − r1 ) Tr ln M ∗ pr0 ×

d 

v (ν1 ν2 ) (r1 − r2 )v (ν2 ν3 ) (r2 − r3 ) . . . v (ν p ν1 ) (r p − r1 )

{ν}=1

=−

1 p pr0

 dr1

d 

ξν1 ν2 (r1 − r2 ) ⊗ · · · ⊗ ξν p ν1 (r p − r1 ).

(A10.4.8)

{ν}=1

The ⊗ on the RHS of eqn. (A10.4.8) represent the convolution of two ξ functions with all the variables r2 , . . . , r p being integrated according to the following definition:  dr2 ξν1 ν2 (r1 − r2 )ξν2 ν3 (r2 − r3 ) ξν(2) (r − r ) = 3 1 ν3 1 ν2

≡



ξν1 ν2 (r1 − r2 ) ⊗ ξν2 ν3 (r2 − r3 ),

(A10.4.9)

ν2

where we have defined the quantity ∗ ξν1 ν2 (r1 − r2 ) = gcm (r1 − r2 )v (ν1 ν2 ) (r1 − r2 ).

(A10.4.10)

Since the Fourier transform of the convolution of two functions is equal to the product of the Fourier transforms of the two functions, we obtain (k) = ξν(2) 1 ν3

d 

ξν1 ν2 (k)ξν2 ν3 (k),

(A10.4.11)

ν2 =1

where the quantity ξμν (k) for the isotropic system is expressed as  ∗ ξμν (k) = drgcm (r )v (μν) (r )eik.r 1 ≡ kˆμ kˆν a" (k) + ( δμν − kˆμ kˆν )a⊥ (k). d

(A10.4.12)

kˆμ = kμ /k is the μth component of the unit vector in the direction of vector k. Note that it follows from the definition (A10.4.2) that r0 = a" (0) = a⊥ (0). We extend the definition (A10.4.11) to include convolutions of p (> 2) numbers of ξ functions:  ξν1 ν2 (k)ξν2 ν3 (k) . . . ξν p ν p+1 (k). (A10.4.13) ξν(1p)ν p+1 (k) = ν2 ...ν p

7 Note that this is an approximation which is usually good at low densities.

A10.4 Computation of Tr ln M∗

539

The above result of convoluted ξ functions is useful for evaluating the RHS of eqn. (A10.4.8). We note here that the latter is expressed as a chain of convolutions in the following manner: +

,( p)

Tr ln M

∗

d V  ( p) ξν ν (r1 − r1 )) p pr0 ν =1 1 1 1  Nd dk  p p =− a˜ (k) + (d − 1)a˜ ⊥ (k) , n0 p (2π )3 "

=−

(A10.4.14) where the functions a˜ " (k) and a˜ ⊥ (k) are obtained as a˜ " (k) =

a" (k) , r0

a˜ ⊥ (k) =

a⊥ (k) . r0

(A10.4.15)

Finally, on summing over the different values of p from eqns. (A10.4.7) and (A10.4.14) for all p values, we obtain the average of the vibrational contributions as  + ,  1 2 ∗ drgcm (r ) (v (μν) (r )) N −1 Tr ln M = d ln r0 − 2 ∗ 2n 0 r0 μν  $ d dk # + [ a ˜ (k)] + (d − 1)L [ a ˜ (k)] , (A10.4.16) L c " c ⊥ n0 (2π )3 where the function Lc is defined as Lc (x) = ln(1 − x) + x +

x2 2 .

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Index

-derivable theory, 512 φ 4 model, 514

activated process, 122, 182 activation energy, 165, 171, 425 Adam–Gibbs theory, 487 adiabatic approximation, 294, 345, 379, 416 adsorption, 90 aging, 169, 184, 449, 450, 460, 471, 473, 477, 481, 482 amorphous solid, 164, 170, 250, 251, 256, 257, 301, 303, 388, 429, 447, 459, 490, 498, 530, 536 amorphous structure, 169, 182, 494, 498, 536 angle azimuthal, 151 polar, 151 anharmonic crystal, 84 glass, 170 anti-ferromagnetic interaction, 190 aperiodic structure, 188, 491, 492, 494, 495 argon (Ar), 46 Arrhenius relaxation, 165 autocorrelation function concentration, 217 density, 39, 40, 169, 200, 215, 216, 244, 331, 345, 366, 414, 427 equilibrium averaged, 37 longitudinal current, 40 noise, 273, 361 tagged-particle velocity, 46 transverse current, 240 Bain’s distortion, 148 balance equations, 205 ballistic regime, 193 barrier effective potential, 503 free energy, 117, 118, 120, 121, 127, 128, 130, 134, 150, 152–154, 161, 411, 425, 494

hopping, 410, 489 potential energy, 27, 182, 186, 414, 489, 490, 496 barrier hopping, 182 basic vectors for cluster symmetry, 156 basin energy, 183 free energy, 184 free-energy landscape (FEL), 191, 448, 498, 505, 506, 510 hopping, 186 of attraction, 448 potential-energy landscape (PEL), 183, 186 vibrational motion, 187 BBGKY hierarchy, 24 Bennett’s algorithm, 492 binary collision, 10, 44, 170, 222, 243, 272, 427 mixture, 184, 193, 212, 214, 295, 296, 298, 299, 379, 383 soft-sphere mixture, 407, 497, 527 blip function, 33 Boltzmann collision term, 223 collision time ta , 44 constant kB , 3 equation, 10, 222, 224 factor, 92, 230 H -theorem, 10 transport coefficients shear self-diffusion coefficient, 223 shear viscosity, 223 thermal conductivity, 223 bond, 497 bond breaking, 497 bond-orientational order parameter, 155, 497 boson peak, 259 bridge function, 31 Brillouin peaks, 241 broken symmetry, 247 Brownian motion, 50, 56, 225, 304, 394, 416

558

Index cage, 171, 174, 193, 220, 371, 409, 414, 455, 529 capillary waves, 101 cell picture, 174 chemical potential at coexistence, 69 at interface, 118 binary mixture, 214 critical nucleus, 127 hard-sphere system, 153 ideal gas, 15 local equilibrium, 208 solid phase, 80 CKN, 403 Clapeyron equation, 6, 71 close-packed f.c.c. structure, 59 cluster, 181 coarse-graining length scale density function, 85 hydrodynamics, 243 collective density, 225 collision integral, 11 collisionless approximation, 10 colloids, 153, 293, 303, 395, 406 complexity, 487 compressibility equation, 22, 30, 75 isothermal, 18 compressible liquid 1/ρ nonlinearity, 292 concentration, 122 concentration equation, 214 concentration variable, 212, 221, 296 condensation, 117 conditional probability, 20 conductivity electrical, 164 thermal, 210, 224, 268 conjugate fields, 457 conservation energy, 207 mass, 207 momentum, 207, 341, 408 probability, 41, 396 tagged-particle density, 376 conservation laws, 205 conservation of angular momentum, 262 continued fraction, 216, 240, 380, 382 continuity equation, 7, 38, 209, 221, 235, 246, 267, 377 cooling rate, 165 cooperative effects, 44 cooperatively rearranging regions (CRRs), 487 correction to scaling in MCT, 375, 376, 382, 402, 405, 410 Coulomb force, 10

559

critical coupling, 464 decay, 402 density, 371 domain, 495 dynamic phenomena, 276 dynamics, 319, 337 exponent, 410 nucleus, 120, 131, 133–135, 138, 147, 495 phenomena, 104, 490 point, 201, 271 surface, 368 temperature, 402 CRN, 402 cusp in NEP, 373 damping, 215 damping matrix, 238, 321 de Broglie wavelength, 3, 15, 20, 65, 183, 298, 394, 436, 491 de Gennes narrowing, 245 Debye density of states, 115 frequency, 115 integral, 115 model, 78 relaxation, 192 temperature, 115 Debye T 3 law, 259 Debye model, 259 Debye–Waller factor, 372 defect density, 251 density expansion direct correlation function, 75 transport coefficients, 224 density-functional theory (DFT), 58, 59, 68, 70, 79, 81, 91, 106, 115 elastic constant, 84 detailed balance, 189, 459 determinant, 258 deterministic, 8, 50, 215, 358, 395 dielectric, 164, 400, 402, 453, 473 diffraction peak, 245 diffusion coefficient Brownian particle D¯ 0 , 54 collective density D0 , 399 tagged particle, 128, 173 diffusion in disordered media, 319 diffusion in random media, 319 diffusion of vortices, 272 diffusive heat mode, 241 direct correlation function from WDA, 75 higher-order, 66

560 one-particle, 67 Ornstein–Zernike, 71 two-particle, 26 dispersion relations, 234 displacement vector u, 250 domain growth problem, 319 double-well potential, 131 driven diffusive systems, 319 driven interfaces in random media, 319 dynamical heterogeneity, 192, 383 dynamics of liquid crystals, 319 effective Debye–Waller factor, 402 diameter of binary mixture, 185 interactions , 406 liquid in MWDA, 74 potential, 494, 500 temperature, 451, 453, 471, 472, 527 eigenvector, 372, 534 Einstein frequency, 44, 55, 455 relation for diffusion, 54, 159, 193, 220 summation convention, 227, 277 elastic constant, 84, 253 displacement, 236 energy, 180 fourth-rank tensor, 302 scattering, 24 solid, 252, 255 strain tensor, 252, 301 stress tensor, 252 embryo, 155 ensemble canonical, 20 equilibrium, 358 ergodic hypothesis, 6 grand-canonical, 60, 105 isobaric, 19 local equilibrium, 450 micro-canonical, 14 nonequilibrium, 207 thermodynamics, 19 trace, 12, 230 Enskog collision time tE , 44, 244, 383 tagged-particle dynamics, 380 transport coefficients self-diffusion coefficient, 223 shear viscosity, 223 thermal conductivity, 223 Enskog theory, 174, 223, 245 modified, 224

Index enthalpy, 127, 155, 167, 211 entropic droplet, 495 entropy, 13, 81, 166, 209 configurational, 183, 488, 503, 506, 521 crisis, 494, 503, 519 drive, 495 production rate, 264 supercooled liquid, 486 thermodynamic, 529 vibrational, 505, 529 equation of state Carnahan–Starling, 30, 72, 80 from compressibilty condition, 29 from virial condition, 29 ideal gas, 65, 125 Percus–Yevick, 72, 78, 102 equimolar mixture, 387 equipartition law, 237 Euler equations, 207, 263, 297 Euler–Lagrange equation, 137, 162 exchange interactions, 419 excluded volume, 379 facilitated spin models, 399 factorization property, 374 Feynman graphs, 326 Fick’s law, 211 fictive temperature, 451, 474 finite-size scaling, 202 fluctuating hydrodynamics, 205 binary mixture, 295 compressible liquid, 287 for crystal, 246 linear, 225, 238 nonlinear (NFH), 238, 271 fluctuation–dissipation relation bare transport matrix, 238, 287 MSR theory, 291, 321, 335 tagged-particle dynamics, 378 fluctuation–dissipation theorem p -spin model, 424 compressible liquid, 340 generalized, 443 Langevin equation, 232 linear-response theory, 49 MSR theory, 334, 435 violation, 443 flux-lattice melting, 90 Fokker–Planck equation, 189, 190, 283, 394, 395 Fokker–Planck operator, 394 four-point correlation functions, 195, 201, 362, 385, 426 four-point kernel, 325 four-point vertex functions, 344

Index Fourier transform, 37 fragile glasses, 166, 169 fragility index m , 169, 473 free-boundary conditions, 188 free energy Carnahan–Starling, 31, 90 excess, 31, 72 FMT functional, 85 Gibbs, 4, 15, 16, 68, 119, 153 Helmholtz, 4, 15, 31 ideal-gas contribution, 15 Ramakrishnan–Yussouff functional, 64, 68 reference system, 35 weighted density functional, 73 free particle, 42, 43, 193 free-volume theory, 54, 170, 257, 414 friction coefficient, 54, 304 frustration, 419 fugacity, 16 functional derivative, 34, 66, 85, 255, 300, 337, 436 functional Taylor expansion, 33, 138 fundamental measure theory (FMT), 73, 85 free-energy expansion, 88 Gallelian transformation, 210 Gaussian density profile, 70, 76, 115, 390, 491 distribution, 42, 286, 423 dynamics, 46 free energy, 238, 251, 298, 329, 411 integral, 15, 312 matrix, 397 MSR action functional, 325, 330, 341 noise, 286, 287, 296, 302, 378, 392, 421, 426 width parameter α , 99 generalized effective-liquid approximation (GELA), 84 generalized hydrodynamics, 204, 242, 246, 347, 418 geometric packing, 91 Gibbs H -theorem, 12 Gibbs inequality, 13, 55 Gibbs–Boltzmann measure, 499 Gibbs–Duhem relation, 5, 17, 118 Gibbsian ensemble, 11, 19, 207, 450 glassy relaxation α-process, 375, 405, 409, 475, 489 β-process, 369, 403, 405, 407, 409 global constraint, 500 global minimum, 190, 498 glycerol, 405, 453, 473, 475 Goldstone theorem, 246 grand potential, 64, 79, 131, 140 Green’s function, 333, 352, 422, 479 Green–Kubo relations, 218

inter-diffusion coefficient, 221 long-time tails, 222 self-diffusion coefficient, 219 viscosity, 219 ground-state theorem, 63 H -theorem in DDFT, 395 Hamilton’s equations, 225 Hamiltonian N -particle system, 61 p -spin interaction, 420 binary mixture, 297 cloned molecular liquid, 522 coarse-grained, 188 dynamics, 8, 260, 303 effective, 274 hard-sphere potential, 84 harmonic potential, 59 pairwise interaction, 6, 314 spin-one-half Ising, 190 with external potential, 48 hard sphere, 11, 26, 44, 78, 222, 371, 492 hard-sphere collision operators T± , 244 harmonic expansion, 76 oscillator, 452, 454 solid, 388, 389, 529 vibration, 529 waves, 388 Heaviside step function, 49, 86, 336, 422, 435 Hessian matrix, 187, 525 hole-burning experiments, 192 hopping process, 182, 186, 191, 414, 427, 428, 488 hydrodynamic interactions, 395 hydrodynamic limit, 235, 331, 339, 414, 442 hydrodynamic poles, 240 hypernetted chain closure (HNC), 26, 501

infinite geometric series, 385 inherent dynamics, 186 inherent structure, 186, 447 inhomogeneous mode coupling, 385 instantaneous normal mode, 536 integral equations, 24–26, 32, 59, 106 interface growth, 319 intermediate scattering function, 186, 200 irreducible memory function, 439 Ising-like description, 495 Ising model, 189, 190 Ising spin glass, 420 Itô calculus, 304, 433 Jacobian, 8, 321, 350, 393, 421, 433 for p-spin case, 351

561

562 Kac interaction potential, 504, 518 Kauzmann packing fraction, 494, 503, 527 paradox, 167, 177, 180, 486 temperature TK , 166, 168, 489 Kawasaki rearrangement, 439 kernel function dynamic part, 230 Marko approximation, 232 static part, 230 kinetic spinodal, 178 Kirkwood superposition approximation, 25, 26 Kohlrausch–William–Watts (KWW), 405, 474 ladder diagram, 385 Lagrange’s multiplier, 457 Landau theory, 156 landscape free energy, 135, 148, 149, 188, 191, 505 potential energy, 181, 183, 188, 410 landscape-dominated regime, 186 landscape-influenced regime, 186 Langevin equation, 50, 269 generalized, 227, 283 linear, 233, 253, 281 multiplicative noise, 391 nonlinear, 191, 274, 283, 284, 301, 303, 318 projected variable, 359 tagged-particle momentum, 378 Laplace transform, 37 latent heat, 490, 494 lattice-gas model, 189 lattice symmetry body-centered cubic (b.c.c.), 148 face-centered cubic (f.c.c.), 148 simple cubic (s.c.), 251 Lees–Edwards boundary condition, 455 Legendre transform, 500 length scale cooperatively rearranging region (CRR), 490 cooperativity, 272 critical phenomena, 490 of dynamic correlation, 198, 201, 385 of static correlation, 490, 496 propagating shear waves, 387 Stokes-Einstein relation, 174 validity of hydrodynamic description, 347, 418 Lennard-Jones (LJ) binary mixture (BMLJ), 185, 407, 444, 447, 453, 455, 529 coexistence, 152 crystal, 99 cut, purely repulsive, 418

Index interaction, 32, 59, 77, 91, 146, 177, 402, 446, 447 interface, 103 light-scattering experiments, 40, 240, 400, 403 Lindemann parameter, 60, 76, 81, 96, 494 Liouville dynamics, 50 equation, 7, 358 operator, 7, 8, 222 theorem, 8 liquid crystal, 90 local amplitudes, 141 icosahedral order, 189 minima free energy, 191 potential energy, 182 order parameter, 138, 140 orientational order parameter, 151 relaxation time, 194 temperature, 210 local coarse graining, 417 long-range interaction potential, 91 long-range order, 235 long-time tail, 46, 222, 272, 327, 364 Lorentzian, 217, 241, 290 many-body dynamics, 232 Markovian equations, 271, 319 Markovian transport coefficient, 361 Martin–Siggia–Rose (MSR) field theory, 318 massless boson, 246 master curve, 373, 477 equation, 185, 189, 190 function, 373, 375, 405 maximum-entropy principle, 360 Maxwell–Boltzmann distribution, 11, 53 Mayer cluster expansion, 29 mean spherical approximation (MSA), 32 mean-square displacement, 22, 76, 173, 193, 195, 220, 382, 453, 494 memory function, 228, 272, 281, 318, 357, 439 mesoscopic kinetic equation, 189 metastable b.c.c. structure, 149 microscopic densities, 205 minimum cooling rate, 177 mobile particles definition, 197 mean-square displacement, 194 pair correlation, 197 synchronous movements, 197 mode coupling 1/ρ nonlinearity, 416 α-relaxation, 375

Index mode coupling (cont.) φ12 model, 369, 425 applied to colloids, 406 binary mixture, 408 dynamic density-functional theory (DDFT), 391 dynamic transition at Tc , 365 dynamical heterogeneity, 383 ergodic–nonergodic (ENE) transition, 365 ergodicity-restoring mechanisms, 411 evidence from experiments, 400 factorization property, 407 four-point correlation, 385 glass transition, 364 in liquid crystals, 364 Kawasaki approximation, 362 link to density-functional theory (DFT), 388 longitudinal viscosity, 344, 366 MD simulations, 407 mean-field spin glass, 419 near second-order phase transition, 364 non-Gaussian parameter, 383 nonperturbative approach, 417 projection-operator technique, 357 shear viscosity, 346 spherical p-spin model, 456 stretched relaxation, 369 tagged-particle dynamics, 381, 390 toy model, 428 transition temperature Tc , 401 two-step relaxation, 410 wave-vector dependence, 371 mode-coupling theory (MCT), 166, 188, 204, 318 modified Kohlrausch–William–Watts (MKWW), 475 modified weighted-density approximation (MWDA), 75, 113, 145 modulus bulk, 256, 301 longitudinal, 256, 301 shear, 165, 192, 256, 301, 365, 496, 498 molecular dynamics (MD), 30, 46, 50, 184, 194, 241, 246, 407, 445 monomer concentration, 125 Monte Carlo (MC), 90, 152, 153, 189 MSR theory action functional, 322 compressible liquid, 328 fluctuation–dissipation theorem, 335 functional integral formalism, 319 Jacobian, 350 operator formalism, 319 quenched disorder, 423 renormalization, 320 renormalized vertex function, 353

response function, 323 time-reversal invariance, 338, 356 multiplicative noise, 294, 379, 416 Nambu–Goldstone modes, 236 Navier–Stokes equation, 238, 291, 339, 411 neutron-scattering experiments, 40, 240, 400 NMR experiments, 192 nonergodicity parameter (NEP), 371, 385, 402 nonpolynomial action in MSR, 414 non-Gaussian parameter, 47, 195, 383 nonlinear constraint, 292, 302, 314, 327 nonlinear oscillator, 411 normal mode, 115 nucleation capillary approximation, 118 classical theory (CNT), 117, 177, 180, 495 critical nucleus, 131 critical radius rc , 133 dependence on undercooling, 128 free-energy barrier, 118 from dilute solution, 125 from melt, 117, 125 heterogeneous, 129 homogeneous, 117 KAC interaction, 504 minimum rate, 128 rate process, 126 schematic model, 135 Stokes–Einstein relation, 129 nucleation rate J , 121 one-loop order, 326 order parameter, 59, 60, 68, 71, 131, 247, 372, 488, 527 order parameter in MCT, 386 Ornstein–Zernike relation, 26 O -terphenyl (OTP), 192, 405, 409 overlap function q(x, y), 499 overlap of free volumes, 171 overlap of hard spheres, 86 packing fraction, 27, 78, 90, 98, 153, 189, 190, 223, 245, 371, 372, 383, 415, 494 pair correlation, 66, 94, 106, 193, 223, 455 Bernal structure, 492 between replicas, 501, 526 pair correlation function, 19 pairwise additive, 55 paramagnet, 419 partition function canonical ensemble, 16 cloned molecular liquid, 520 free-energy basins, 522 grand-canonical ensemble, 17, 61

563

564 phonon modes, 115 volume dependence, 21 percolation, 174 Percus–Yevick (PY), 26, 71, 102, 189, 245, 371, 375, 492 periodic boundary conditions, 150, 188 perturbation theory, 69, 91, 293, 334, 347, 414 phase space, 6, 169, 225, 359, 428, 498 phonon, 58, 115 plasma physics, 10 Poisson bracket, 7, 48, 226, 233, 236, 246–249, 253, 279, 280, 286, 288, 296, 297, 426, 428 polymer solution dynamics, 319 Potts spin glass, 494 pressure functional, 257, 292 primitive cell, 108, 146, 189 projection operator, 225, 273, 357, 381 propylene carbonate (PC), 403 protein, 181 quasicrystals, 90 quasi-ergodic behavior, 471 quenched disorder, 369, 419, 530 Raleigh peak, 241 Ramakrishnan–Yussouff free energy, 392 Raman scattering, 259 random bonds, 421 close-packed structure, 492 cluster, 159 collisions, 50 diffusion model (RDM), 399 energy model, 494 field, 449 first-order transition, 494 force, 51, 319 heteropolymer, 496 hexagonal close packed (rhcp) structure, 159 interactions, 420 lattice, 388 matrix, 536 noise, 335, 421 point pinning sites, 530 potential, 414 quenched potential, 530 structure, 492 variables, 419 Rayleigh peak, 241 reaction coordinate, 150, 154 reciprocal lattice vector (RLV), 70, 77, 107, 249 reflection symmetry, 420 renormalization group, 104 replica cloned molecular liquid, 510

Index free energy, 521, 527 index, 529 interaction, 524 one-step replica symmetry breaking, 425, 447 overlap functions, 502 replica in spin glass, 499, 501 replica index, 533 replicated liquid-state theory, 494, 501, 522, 530 spin glass vs. structural glass, 530 replica method, 507 replicated liquid, 509 restricted phase space, 200 ring collision, 272 roughening surface transitions, 319 saddle, 132, 184, 424, 429, 520 scaled-particle theory, 88 Schwarz’s inequality, 37 Schwinger–Dyson equation, 324 second fluctuation–dissipation relation, 269 self-consistent mode coupling, 232, 327 Self-consistent screening approximation (SCSA), 514 self-diffusion coefficient Ds , 159, 175, 179, 193, 212, 220, 221, 295 self-energy matrix, 325 one-loop expression, 326 self-generated disorder, 500, 507 shear rate, 449 shear wave, 202, 387 SiO2 , 169 silica, 449 soft p -spin model, 420 disk, 194 heat mode, 245 sphere, 103, 185, 195, 202, 387, 407, 497, 527 solid–liquid interface, 90, 99 sound modes, 245 spatial heterogeneity, 192 specific heat at constant pressure c p , 169, 488 at constant volume cv , 16 spherical p-spin model, 426 spherical Bessel function, 244 spherical constraint, 425 spin diffusion, 235 spin-exchange dynamics spin glass p -spin interaction model, 420, 425, 426 spinodal, 132 spontaneous breaking of ergodicity, 505 square-gradient approximation, 101, 131, 138

Index static structure factor, 19, 106, 238, 290, 346, 370, 410, 439, 496 steepest-descent path, 186 Stokes–Einstein relation, 53, 128, 178, 407 Enskog theory, 224 Stokes–Einstein violation, 129, 178 streaming velocity, 233, 280, 288, 300 string-like structure, 198 strong glasses, 166, 169, 448, 489 structural arrest in colloids, 407 structural relaxation time τα , 178–180, 195 surface free energy, 102, 149 function for hard sphere, 87 hypersurface, 182, 186 impurity, 117 roughening transitions, 319 tension, 103, 118, 130, 135, 153, 495 term, 56 susceptibility α-peak, 403 p -spin, 504 aging behavior, 453 dielectric, 403 dynamic, 402 magnetic, 419 minimum, 403 nonlinear, 200 static, 247 tagged particle correlation, 41 density, 41, 211 diffusion, 171 velocity , 43 vibration frequency, 44 thermal energy, 126, 182 expansion, 170, 174 fluctuation, 171 velocity, 223, 244 vibration, 151 thermalization in aging, 472 thermally activated transitions, 190 thermodynamic limit, 14 thermodynamic potential, 17, 60, 107, 138 three-point correlation function, 201, 385, 426 three-point vertex function, 325, 353 threshold value energy, 188 liquid-like cell, 174 local order in nucleus, 152 nucleation rate, 128, 176 time correlation functions, 36

565

time-dependent Ginzburg–Landau (TDGL), 131, 253, 301 time translational invariance, 270 time–temperature superposition principle, 375 tracer particle, 455 transcendental equation, 368, 373 translational degrees of freedom, 164, 486 transverse current, 216, 240 modes, 210, 256, 381 trapping diffusion model, 414 turbulence, 319 two-level states, 259 two-level system, 171 two-point distribution function, 93 uniaxial crystal, 254 universality classes, 410, 425 vacancy diffusion mode, 251 valley, 507 van der Waals theory, 400 van Hove self-correlation function, 40, 194, 376 Verlet condition, 59 Verlet–Weiss correction, 71, 245, 371, 383, 492 vertex function cubic, 324 flat structure factor, 344 longitudinal, 341 self-correlation, 381 transverse correlation, 346 virial coefficient, 29, 30 equation, 20, 22, 27, 30 expansion, 224 free-energy expansion, 88 viscoelasticity, 180 viscosity Angell plot, 168 bare, 223, 301 bulk, 224, 237 effects of structure, 370 free-volume theory, 174 generalized, 245, 254, 341, 364 Green–Kubo relations, 218 incompressible liquid, 327 kinetic, 216, 240 liquid crystal, 364 long-time tails, 272 longitudinal, 216, 219, 237 nucleation rate, 175 power-law divergence, 365 shear, 54, 165, 194, 237 supercooled liquid, 164 tensor, 210, 254

566 Vlasov equation, 10 Vlasov–Smoluchowski equation, 394 Vogel–Fulcher equation, 166, 405, 473, 496 Voig notation, 252 von Schweidler relaxation, 409 Voronoi polyhedron, 151 waiting time, 418 Weeks–Chandlers–Andersen (WCA) theory, 32, 91 weighted-density-functional theory (WDA), 70, 72, 75, 78, 84, 109 weighting function coarse graining, 72

Index for the MWDA, 75 for uniform liquid, 113 for the WDA, 74 Fourier transform, 82 fundamental measure theory, 85 normalization condition, 73 wetting, 90 Wigner–Seitz cell, 151 X-ray scattering, 40 Yvon–Born–Green (YBG) hierarchy, 25 Zel’dovich factor, 124, 159