Theory of Liquids: From Excitations to Thermodynamics [1 ed.] 1009355473, 9781009355476, 9781009355483

Table of contents :
1. Introduction
2. Experimental heat capacity
3. Excitations in solids and gases
4. Energy and heat capacity of solids
5. First-principles description of liquids: exponential complexity
6. Liquid energy and heat capacity from interactions and correlation functions
7. Frenkel theory and Frenkel book
8. Collective excitations in liquids post-Frenkel
9. Molecular dynamics simulations
10. Liquid energy and heat capacity in the liquid theory based on collective excitations
11. Quantum liquids: excitations and thermodynamic properties
12. Sui Generis
13. Connection between phonons and liquid thermodynamics: a historical survey
14. The supercritical state
15. Minimal quantum viscosity from fundamental physical constants
16. Viscous liquids
17. A short note on theory
18. Excitations and thermodynamics in other disordered media: spin waves and spin glass systems
19. Evolution of excitations in strongly-coupled systems
References
Index.

Citation preview

THEORY OF LIQUIDS

Of the three basic states of matter, liquid is perhaps the most complex. While its flow properties are described by fluid mechanics, its thermodynamic properties are often neglected, and for many years it was widely believed that a general theory of liquid thermodynamics was unattainable. In recent decades that view has been challenged, as new advances have finally enabled us to understand and describe the thermodynamic properties of liquids. This book explains the recent developments in theory, experiment and modelling that have enabled us to understand the behaviour of excitations in liquids and the impact of this behaviour on heat capacity and other basic properties. Presented in plain language with a focus on real liquids and their experimental properties, this book is a useful reference text for researchers and graduate students in condensed matter physics and chemistry, as well as for advanced courses covering the theory of liquids. K O S T YA T R A C H E N K O is a professor at Queen Mary University of London working in condensed matter physics. He received his PhD from the University of Cambridge and held a research fellowship at Darwin College, Cambridge, and an Engineering and Physical Sciences Research Council (EPSRC) advanced fellowship. His work was awarded the CCP5 Prize and was one of the top 10 Physics World Breakthroughs in 2020.

T H E O RY O F LIQUIDS From Excitations to Thermodynamics KOSTYA TRACHENKO Queen Mary University of London

Shaftesbury Road, Cambridge CB2 8EA, United Kingdom One Liberty Plaza, 20th Floor, New York, NY 10006, USA 477 Williamstown Road, Port Melbourne, VIC 3207, Australia 314–321, 3rd Floor, Plot 3, Splendor Forum, Jasola District Centre, New Delhi – 110025, India 103 Penang Road, #05–06/07, Visioncrest Commercial, Singapore 238467 Cambridge University Press is part of Cambridge University Press & Assessment, a department of the University of Cambridge. We share the University’s mission to contribute to society through the pursuit of education, learning and research at the highest international levels of excellence. www.cambridge.org Information on this title: www.cambridge.org/9781009355476 DOI: 10.1017/9781009355483 c Kostya Trachenko 2023

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 & Assessment. First published 2023 A catalogue record for this publication is available from the British Library. ISBN 978-1-009-35547-6 Hardback Cambridge University Press & Assessment 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.

Frenkel discusses his theory of liquids with Mott. Based on Mott’s recollection in Section 7.2. Illustrated by Ivan Shkoropad.

Contents

Preface

page xi

1

Introduction

1

2

Experimental Heat Capacity

9

3

Excitations in Solids and Gases 3.1 Harmonic Solids and Quadratic Forms 3.2 Continuum Approximation: Elasticity and Hydrodynamics 3.3 Excitations in Quantum Fluids 3.4 Intermediate Summary

21 26 27

4

Energy and Heat Capacity of Solids 4.1 Harmonic Solids 4.2 Weak Anharmonicity

29 29 32

5

First-Principles Description of Liquids: Exponential Complexity

35

Liquid Energy and Heat Capacity from Interactions and Correlation Functions 6.1 The Method 6.2 The Problem 6.3 A Better Alternative? 6.4 Structure and Thermodynamics

39 39 43 45 48

Frenkel Theory and Frenkel’s Book 7.1 Liquid Relaxation Time 7.2 Collective Excitations in the Frenkel Theory

50 50 57

6

7

16 16

vii

Contents

viii

7.3

8

Continuity of Liquid and Solid States and the Frenkel– Landau Argument: First Ignition of the Glass Transition Debate

Collective Excitations in Liquids Post-Frenkel 8.1 The Telegraph Equation and Gapped Excitations in Liquids 8.2 Another Solution of the Telegraph Equation 8.3 The First Glimpse into Liquid Thermodynamics and Collective Modes 8.4 Symmetry of Liquids: Viscoelasticity and Elastoviscosity 8.5 Collective Modes in Generalised Hydrodynamics 8.6 Experimental Evidence of Collective Excitations in Liquids 8.7 Decay of Excitations in Liquids and Solids

62 64 64 69 71 72 74 79 86

9

Molecular Dynamics Simulations

10

Liquid Energy and Heat Capacity in the Liquid Theory Based on Collective Excitations 10.1 Liquid Energy and Heat Capacity 10.2 Comparison to the Integral over Interaction and Correlation Functions 10.3 Anharmonic Effects 10.4 Quantum Effects 10.5 Verification of the Liquid Theory Based on Collective Excitations 10.6 Molecular Liquids and Quantum Excitations

109 112

Quantum Liquids: Excitations and Thermodynamic Properties 11.1 Liquid 4 He 11.2 Liquid 3 He 11.3 Quantum Indistinguishability

114 114 118 121

12

Sui Generis

125

13

Connection between Phonons and Liquid Thermodynamics: A Historical Survey 13.1 Early Period

127 127

11

90

97 97 103 105 107

Contents

13.2 14

15

Late Period

The Supercritical State 14.1 What Happens at High Temperature? 14.2 The Supercritical State and Its Transition Lines 14.3 Applications of Supercritical Fluids 14.4 C Transition 14.5 Excitations and Thermodynamic Properties 14.6 Liquid Elasticity Length and Particle Mean-Free Path 14.7 Heat Capacity of Matter: From Solids to Liquids to Gases Minimal Quantum Viscosity from Fundamental Physical Constants 15.1 Viscosity Minima and Dynamical Crossover 15.2 Minimal Quantum Viscosity and Fundamental Constants 15.3 Elementary Viscosity, Diffusion Constant and Uncertainty Principle 15.4 Viscosity Minima in Quantum Liquids 15.5 Lower Bound of Thermal Diffusivity 15.6 Practical Implications 15.7 The Purcell Question: Why Do All Viscosities Stop at the Same Place? 15.8 Fundamental Constants, Quantumness and Life 15.9 Another Layer to the Anthropic Principle

ix

130 133 133 134 140 141 145 149 150

152 152 155 159 161 163 165 165 166 168

16

Viscous Liquids 16.1 Collective Excitations, Energy and Heat Capacity 16.2 Entropy 16.3 Heat Capacity at the Liquid–Glass Transition 16.4 Glass Transition Revisited

171 171 175 177 181

17

A Short Note on Theory

185

18

Excitations and Thermodynamics in Other Disordered Media: Spin Waves and Spin Glass Systems

188

Evolution of Excitations in Strongly Coupled Systems

194

References Index

197 207

19

Preface

In 1946, Born and Green [1] wrote: It has been said that there exists no general theory of liquids.

About 60 years later, Granato [2] observed that nothing is said about liquid specific heat in standard introductory textbooks, and there is little or nothing in advanced texts, adding that there is little general awareness of what the basic experimental facts to be explained are. This was reflected in his teaching: Granato recalls how he enjoyed teaching theories of the specific heat of gases and solids but lived in constant fear that a student might ask “How about liquids?”. That question was never asked in lectures over many years. Granato attributes this to an issue with our teaching practice, that is, that unsolved problems in physics are not sufficiently covered. The problems involved in liquid theory are fundamental. These problems are well illustrated by two related assertions. These assertions are important to quote in full. The first quote is from the Statistical Physics textbook by Landau and Lifshitz [3]. Landau and Lifshitz discuss the general thermodynamic properties of liquids and explain why they cannot be calculated, contrary to solids and gases. Section 66 opens with a forceful statement of the problem: Unlike gases and solids, liquids do not allow a calculation in a general form of the thermodynamic quantities or even of their dependence on temperature. The reason lies in the existence of a strong interaction between the molecules of the liquid while at the same time we do not have the smallness of the vibrations which makes the thermal motion in solids especially simple. The strength of the interaction between molecules makes it necessary to know the precise law of interaction in order to calculate the thermodynamic quantities, and this law is different for different liquids.

Landau and Lifshitz emphasise this point later in the opening of Section 74 when discussing the van der Waals equation. xi

xii

Preface

The second quote is from Pitaevskii [4]: Differently from a gas, the interaction terms in the equation of state for liquids are not small. Therefore liquid properties depend strongly on the specific character of interaction between molecules. Liquid theory generally lacks a small parameter which could be used to simplify a theory. It is impossible to obtain any analytical formulas for thermodynamic properties of liquids.

These problems seemingly seal the fate of liquid theory and preclude its development at a level nowhere near that of the well-advanced theories of the two other states of matter, solids and gases, which we have been enjoying for a long time. Appreciating the problems in the preceding quotes, I wondered whether these problems were as arresting as they sounded. Are liquids really so complex that we cannot hope for a theory explaining their most basic properties and cannot teach liquids to our students? Could it be that we became too attached to ideas used in theories of solids and gases and did not manage to move on with liquids using a different approach? Would this approach require a modification of solid state and gas theories or do we need to come up with something entirely different in order to understand liquids? It is important to ask at this point why the problems outlined by Landau, Lifshitz and Pitaevskii do not emerge in the solid state theory, in which interactions are as strong and system-specific as in liquids? An important premise of statistical physics is that an interacting system is governed by its excitations [3, 5]. In solids, these are collective excitations, phonons. Understanding phonons and relating them to solid properties is at the heart of the remarkable success of the solid state theory developed over 100 years ago. Why can’t we use the same approach to liquids based on phonons? One quick answer could be that liquids flow and hence do not have fixed reference points, which we can use to perform the harmonic expansion and derive these phonons. This is not quite the case: liquid particles oscillate around quasiequilibrium points at times shorter than the liquid relaxation time before jumping to new positions. Discussing the theory of this motion and its implications for phonons in liquids and liquid thermodynamic properties constitutes a large part of this book. The second important observation is related to the predominant approach adopted in liquid theories in the last century. The first attempt at liquid theory using phonons was undertaken by Sommerfeld and Brillouin over 100 years ago. This was around the same time that the papers by Einstein and Debye were published and laid the foundations of the modern solid state theory based on phonons. Developing this line of inquiry in liquids largely came to a halt soon after, and theories of liquids and solids diverged in their fundamental approaches. Whereas the solid state theory continued to be developed on the basis of phonons, theories

Preface

xiii

of liquids started to use an approach based on inter-atomic interactions and correlation functions. As discussed in this book in detail, this approach faced several inherent limitations and fundamental problems. This was largely because the problems stated by Landau, Lifshitz and Pitaevskii were inevitable in this approach. From the perspective of this book, the important problem is that the approach based on inter-atomic interactions and correlation functions has not resulted in a physical understanding of the most basic experimental properties of real liquids such as specific heat. A set of new results has emerged in the last few decades related to collective excitations in liquids. It has taken a combination of experimentation, theory and modelling to understand phonons in liquids well enough to connect them to liquid thermodynamic properties. Recall that this connection between phonons and thermodynamic properties was the basis of the Einstein and Debye approaches to solids, which laid the foundations of the modern solid state theory. The upshot is that the liquid theory can still be constructed on the basis of phonons, but the key point is that, differently from solids, the phase space available to phonons is not fixed but is variable instead. In particular, this phase space reduces with temperature. This reduction quantitatively explains the experimental liquid data and in particular the reduction of liquid specific heat from the solid-like value of cv = 3kB to the ideal gas value of cv = 32 kB with temperature. It has taken another decade to obtain an independent verification of this theory. The small parameter in the liquid theory is therefore the same as in the solid state theory: small phonon displacements. However, an important difference from solids is that this small parameter operates in a variable phase space. This addresses the problems stated by Landau, Lifshitz and Pitaevskii. We can see why earlier liquid theories diverged from the solid state theory in their fundamental approaches about 100 years ago: the nature of collective excitations in liquids was not known at the time when this divergence took place. With the liquid theory operating in terms of collective excitations as in the solid state theory, we are now in a strong position to understand liquids. This book reviews this research, starting with the early work by Sommerfeld and Brillouin and ending with recent independent verifications of the liquid theory. I will follow the variation of the phase space in liquids in a wide range of parameters on the phase diagram, from low-temperature liquids to high-temperature supercritical fluids. I will then come back to low-temperature viscous liquids approaching the liquid–glass transition. The liquid theory and its independent verifications discussed in this book focus on real liquids and their experimental properties rather than on model systems. This importantly differentiates this book from others.

xiv

Preface

I will also show how developments in liquid theory resulted in new unexpected insights. For example, the variation of the phase space available to phonons is related to liquid viscosity, which quantifies the ability to flow. Viscosity has a minimum related to the crossover of particles dynamics from liquid-like to gaslike. It turns out that this minimum is governed by fundamental physical constants including ~. This, in turn, shows that water and water-based life are well attuned to fundamental physical constants including the degree of quantumness of the physical world set by ~. Discussing the implications of this adds another layer to the anthropic principle. Tabor calls liquids the “neglected step-child of physical scientists” and “Cinderella of modern physics” compared with solids and gases [6]. Although this observation was made nearly 30 years ago, Tabor would have reached the same conclusion regarding liquid thermodynamics on the basis of more recent textbooks (as discussed in Chapter 1, liquid textbooks mostly focus on liquid structure and dynamics and do not discuss most of the basic thermodynamic properties such as liquid energy and heat capacity). An important aim of this book is to make liquids a full family member on a par with the other two states of matter, if not more sophisticated due to the ability of liquids to sustain a variable phase space. The selection of topics in this book was helpfully aided and narrowed down by adopting a well-established approach in physics where by an interacting system is fundamentally understood on the basis of its excitations [3, 5]. Consequently, a large part of this book discusses collective excitations in liquids and their relation to basic liquid properties throughout the history of liquid research. This shows how earlier and more recent ideas physically link to each other and in ways not previously considered. I expect that there will be several groups of readers who will benefit from better understanding liquids. In his recent review, Chen [7] observes that our current understanding of thermophysical properties of liquids is very unsatisfactory, adding, on a more optimistic note, that progress will be helped by what has been done earlier. This book therefore reaches out to scientists at any stage of their career who are interested in the states of matter and the history of the long-standing problem of understanding liquids theoretically. The second group targeted by this book includes researchers and graduate students working in the area of liquids and related areas, such as soft condensed matter physics and systems with strong dynamical disorder. The third group are lecturers looking to include liquids in their undergraduate and graduate courses, such as those on statistical or condensed matter physics, as well as students who can use this book as a reference. As observed by Granato [2],

Preface

xv

there is nothing said about [liquid specific heat] in the standard introductory textbooks, and little or nothing in advanced texts as well. In fact, there is little general awareness even of what the basic experimental facts to be explained are.

This observation has been recently shared and highlighted, emphasising the lack of undergraduate instruction in fluids [8]. This book aims to fix this issue for the benefit of both lecturers and their students. I am grateful to M. P. Allen, V. Brazhkin, B. Carr, G. Chen, L. Noirez, J. C. Phillips and U. Winderberg for discussion and comments, to I. Shkoropad for illustrations and to my family for support.

1 Introduction

I had a memorable library day trying to find an answer to a question that is simple to formulate: What is a theoretical value of energy and heat capacity of a classical liquid? I looked through all of the textbooks dedicated to liquids, as well as statistical physics and condensed matter textbooks, in the Rayleigh Library at the Cavendish Laboratory in Cambridge, UK. To my surprise, they had either very little or nothing to say about the matter. I then took a short walk to Cambridge University Library and did a thorough search there. This returned the same result. My surprise quickly grew closer to astonishment, for two reasons. First, heat capacity is one of the central properties in physics. Constant-volume heat capacity is the temperature derivative of the system energy, the foremost property in physics including statistical physics. Heat capacity informs us about the system’s degrees of freedom and regimes in which the system is in, classical or quantum. It is also a common indicator of phase transitions, their types and so on. Understanding the energy and heat capacity of solids and gases is a central and fundamental part of theories of these two phases. Thermodynamic properties such as energy and heat capacity are also related to important kinetic and transport properties, such as thermal conductivity. Not having this understanding in liquids, the third basic state of matter, is a glaring gap in our theories. This is especially so in view of the enormous progress made in condensed matter research in the last century. The second reason for my surprise was that the textbooks did not mention the absence of a discussion of liquid heat capacity as an issue. It is harder to solve a problem if we don’t know it exists. Around the same time, I met Professor Granato from the University of Illinois who shared my observations. He notes [2] that nothing is said about liquid specific heat in standard introductory textbooks, and there is little or nothing in advanced texts, adding that there is little general awareness of what the basic experimental facts to be explained are. This was reflected in his teaching: Granato recalls how he enjoyed teaching theories of the specific heat of gases and solids but lived in 1

2

Introduction

constant fear that a student might ask “How about liquids?”. That question was never asked over many years of teaching involving about 10,000 students. Granato attributes this to an issue with our teaching practice, that is, that unsolved problems in physics are not sufficiently covered. The available textbooks are mostly concerned with liquid structure and dynamics. They do not discuss most basic thermodynamic properties, such as liquid energy and heat capacity, or explain whether the absence of this discussion is related to a fundamental theoretical problem. Textbooks in which we might expect to find this discussion but don’t include those dedicated to liquids [6, 9–22] and related systems [23–28], advanced condensed matter texts [29–33] and statistical physics textbooks [3, 5, 34–41].1 This list has a notable outlier: the Statistical Physics textbook by Landau and Lifshitz [3]. Landau and Lifshitz discuss the general thermodynamic properties of liquids and explain why they cannot be calculated, contrary to solids and gases. This explanation is quoted in the present book’s preface, together with a related quote from Pitaevskii that discusses the absence of a small parameter in liquids. The absence of a small parameter implies that we can’t use well-known tools such as quadratic forms based on the harmonic expansion in solids and perturbation theory in gases in order to understand liquids theoretically. Other authors have made similar observations, which we will encounter throughout this book. For example, Tabor says that “the liquid state raises a number of very difficult theoretical problems” [6]. As set out by Landau, Lifshitz and Pitaevskii, the liquid problem is formidable. It could explain why liquid energy and heat capacity are not discussed in textbooks, although in my experience, an experience shared by Granato, the lack of these discussions may be related to the low awareness of which experimental facts need to be explained, let alone of the fundamental problem of liquid theory. The problem of liquids was well recognised in the area of molecular modelling and molecular dynamics simulations. It is interesting that a classic textbook on molecular dynamics simulations is entitled Computer Simulation of Liquids [42]. This method can simulate solids, gases or any other system in which an inter-atomic interaction is known, yet the textbook title focuses on liquids. As the book’s authors explain, an important motivation for such computer simulations was the need to understand liquids, because they are not amenable to a theoretical understanding using approximate perturbative approaches or virial expansions. The authors note that, for some liquid properties, it may not even be clear how to begin constructing an approximate theory in a reasonable way. Computer modelling was seen as a way to help fix this problem. 1 I will discuss two books by D. Wallace and J. Proctor separately.

Introduction

3

In view of the enormous progress made in condensed matter research, it is perhaps striking to realise that we do not have a basic understanding of liquids as the third state of matter and certainly not on a par with solids and gases. This was one of the reasons I decided to look into this problem. As its title suggests, this book follows the path from excitations in liquids to their thermodynamic properties. This general path is well established in statistical physics: an interacting system has collective excitations, or collective modes. Thermodynamic properties are then calculated on the basis of these collective excitations. The solid state theory is a celebrated case in which the solid energy is calculated on the basis of phonons in the Einstein or Debye model, with all of the important consequences that follow. In liquids, this path was largely unexplored in the past (with an exception of liquid helium, for which the quantum nature of the system, perhaps surprisingly, simplifies the theory). Until fairly recently, collective excitations in liquids were not understood well enough, and it was unclear if and how they could be related to liquid thermodynamics on the basis of a quantitative theory. As is often the case, a difficult problem needs to be addressed from different perspectives, and the first aim of this book is to synthesise the results coming from three different lines of enquiry: experimentation, theory and molecular modelling. Some of these results are fairly new, while others date back nearly two centuries. I will show that liquids have a long and extraordinary history of research, but this history has been fragmented and sometimes undervalued. When considered from a longer term perspective, the history of collective excitations in liquids reveals a fascinating story that involves physics luminaries and includes milestone contributions from Maxwell in 1867, followed by Frenkel and Landau. A separate and largely unknown line of enquiry aiming to connect phonons in liquids and liquid thermodynamics involved the work of Sommerfeld, published in 1913, one year after the Debye’s paper on the heat capacity of solids. This line of enquiry was later developed by Brillouin and by Wannier and Piroué. When we get to the equation written (but not solved) by Frenkel to describe collective excitations in liquids, we will see that this equation was introduced by Kirchhoff in 1857 and discussed by Heaviside and Poincaré. The second and most technical aim of this book is to connect excitations in liquids to their thermodynamic properties. In the process, a distinct liquid story will emerge, and this will achieve the third aim of this book: to put real liquids and an understanding of their experimental properties in to the spotlight. Tabor calls liquids the “neglected step-child of physical scientists” and “Cinderella of modern physics” compared with solids and gases [6]. Although this observation was made nearly 30 years ago, Tabor would have reached the same conclusion today with regard to liquid thermodynamics on the basis of more recent textbooks (listed

4

Introduction

earlier). The third aim of this book is therefore to make liquids a full family member on a par with the other two states of matter, if not more sophisticated and refined due to the ability of liquids to combine oscillatory solid-like and diffusive gas-like components of particle motion. We will see how these two components endow the phase space for collective excitations in liquids with an important property: its ability to change with temperature. The focus of this book is on understanding real liquids and their experimental properties, rather than on model systems. This importantly differentiates this book from others. The experimental properties include specific heat as an important indicator of the degrees of freedom in the system and a quantity that can be directly measured experimentally and compared with theory. I start, in Chapter 2, with a brief account of the experimental data that we will be mostly concerned with. It will quickly become apparent that the common models used to understand liquids, such as the van der Waals and hard-sphere models, are irrelevant for understanding the specific heat of real liquids, bringing about the realisation that something quite different is needed. I will then recall our current understanding of collective modes in solids and gases and how these modes underly the theory of their thermodynamic properties in Chapters 3 and 4. An important observation here is that quadratic forms greatly simplify theoretical description. We particularly appreciate this simplification when we deal with liquids where we don’t have them. The small parameter problem discussed by Landau, Lifshitz and Pitaevskii does not necessarily mean that liquids cannot be understood in some other way. However, in Chapter 5, we will see that a head-on first-principles description of liquids, which we use in the solid state theory, is exponentially complex and therefore intractable. An alternative approach to liquids is based on an integral involving interactions and correlation functions, and I review this approach in Chapter 6. This will show that earlier liquid theories and the solid state theory diverged in their fundamental approaches. Early theories considered that the goal of the statistical theory of liquids was to provide a relation between liquid thermodynamic properties and the liquid structure and intermolecular interactions. Working towards this goal involved ascertaining the analytical models for structure and interactions in liquids. Developing these models then became the essence of earlier liquid theories. I will review the general problems involved in this approach, including the inevitable problems set out by Landau, Lifshitz and Pitaevskii in the this book’s preface. I will then highlight that these problems are absent in the solid state theory because this theory does not operate in terms of correlation functions and interactions and is based on phonons instead. This makes the solid state theory tractable, physically transparent and predictive. One of the main aims of this book is to show that liquids can be understood using a similar theory based on phonons.

Introduction

5

In Chapter 7, I will show how Frenkel’s ideas broke new ground for understanding liquids. This notably involves the concept of liquid relaxation time and its implications for liquid dynamics and collective excitations. This resulted in several important predictions, including the propagation of solidlike transverse waves in liquids at high frequency. The main difference between liquids and solid glasses was thought to be resistance to shear stress; however, Frenkel proposed that this was not the case: liquids too can support shear stress, albeit at a high frequency. Frenkel also proposed that the liquid–glass transition is a continuous, rather than a discontinuous, process. As we will see in Chapter 16, this continues to be the topic of current glass transition research. In Chapter 8, I will discuss the equation governing liquid transverse modes that Frenkel wrote but, surprisingly, did not solve. Once simplified, this equation becomes the “telegraph equation” discussed by Kirchhoff, Heaviside and Poincaré. Surprisingly, in view of its wide use, the telegraph equation was not fully explored in terms of the different dispersion relations it supports. I will show how the solution gives rise to an important property of collective transverse modes in liquids: these modes exist above the threshold wavevector only and correspond to gapped momentum states. In terms of propagating waves, the gap also exists in the frequency domain. The gap increases with temperature, and this brings about the key property of liquids: the volume of the phase space available to collective excitations reduces with temperature. This is in striking difference from solids, for which this phase space is fixed. I will discuss the evidence for this reduction of the phase space on the basis of experimental and modelling data. I will highlight that the hydrodynamic description was historically adopted as a starting point of discussing liquids and their viscoelastic properties. In this approach, hydrodynamic equations are modified to account for a solid-like response. I will show that this is related to our subjective perception rather than liquid physics and that both hydrodynamic and solid-like elastic theories are equally legitimate starting points of liquid description, both resulting in the telegraph equation. In this sense, liquids have a symmetry of their description where by the “elastoviscosity” term is as justified as the commonly used term “viscoelasticity” to discuss liquid properties. As mentioned earlier in this chapter, widely used molecular dynamics simulations were stimulated by the need to understand liquids. This was because liquids were not thought to be amenable to theory. In Chapter 9, I will review the molecular dynamics simulations method and focus on two sets of molecular dynamics data related to the main topic of this book: thermodynamic properties and constant-volume specific heat (cv ) in particular, as well as collective excitations. Armed with preceding results, I will discuss the calculation of liquid energy and specific heat on the basis of propagating phonons in Chapter 10. I will demonstrate the advantage of this approach over that based on inter-atomic potentials

6

Introduction

and correlation functions. Theoretical results will be compared to a wide range of experimental data, universally showing the decrease of the liquid specific heat with temperature. This decrease is related to the reduction of the volume of the phase space available for collective excitations due to the progressively smaller number of transverse phonons at high temperature. I will also discuss independent verifications of this theory. In Chapter 11, I will examine the relation between collective excitations in quantum liquid 4 He and its specific heat as discussed by Landau and Migdal. I will also discuss 3 He, for which, differently from other systems, low-temperature thermodynamic properties are governed by localised quasi-particles rather than collective modes. In Chapter 12, I will recall Tabor’s discussion of what constitutes the sui generis approach to liquids. In the light of preceding results and our current understanding, the key to sui generis in liquids is to observe that these are systems in a mixed dynamical state involving both oscillations and the diffusion of particles, in contrast with pure dynamical states in gases and solids where by particle motion is purely diffusive and oscillatory. The balance between oscillatory and diffusive motion shifts with temperature (pressure), representing the sophistication of liquids compared with solids and gases. This shift is related to the variation of the volume of phase space available to collective excitations mentioned earlier. This, in turn, provides the key to understanding liquid thermodynamics and the universal decrease of liquid specific heat with temperature, in particular. In Chapter 13, I will review the history of research aiming to link liquid thermodynamic properties to liquid collective excitations. Involving physics luminaries, this history dates back over a century. Quite remarkably, the first attempt at the theory of liquid thermodynamics based on phonons was done by Sommerfeld only one year after the Debye theory of solids and six years after Einstein’s theory were published. Whereas the Debye theory has become part of every textbook in which solids are mentioned, we had to wait for about a century until we started understanding phonons in liquids well enough to be able to connect them to liquid thermodynamic properties. Sommerfeld’s line of enquiry was taken up by Brillouin, who reconciled his theory of liquids and experimental data by making a fascinating proposal that liquids may consist of small crystallites. Wannier and Piroué later expanded on this work, as did other authors. I will follow the link between excitations in liquids and their thermodynamics in a wide range of pressures and temperatures on the phase diagram, from hightemperature supercritical fluids through subcritical liquids to low-temperature viscous systems approaching the glass transition. In Chapter 14, I will examine the reduction of the phase space available to collective excitations above the critical point. Above the Frenkel line corresponding to the disappearance of the

Introduction

7

oscillatory component of particle motion and two transverse modes, this reduction is due to the progressive disappearance of the remaining longitudinal mode. In view of the increasing deployment of supercritical fluids in environmental, cleaning and extracting applications, this discussion is of environmental and industrial relevance, as Chapter 14 will show. The phase space available to collective excitations depends on the Frenkel hopping frequency, which, in turn, depends on viscosity. The temperature dependence of viscosity shows universal minima, and the minima themselves are related to the dynamical crossover of particle dynamics discussed in Chapter 14. This leads us to Chapter 15, where I will show that the viscosity minimum is a universal quantity for each liquid and is fixed by fundamental physical constants, an interesting result considering that viscosity is strongly system and temperature dependent. I will show how this provides an answer to the question asked by Purcell and considered by Weisskopf in the 1970s, namely “Why does viscosity never drop below a certain value comparable to that of water?”. The minimal viscosity turns out to depend on ~ and hence be a quantum property. This has immediate consequences for water viscosity and essential processes in living organisms and cells. Water and life appear to be well attuned to the degree of quantumness of the physical world and other fundamental physical constants. This adds another layer to the anthropic principle. Considering another extreme of large viscosity at low temperature brings us to the realm of viscous liquids, as discussed in Chapter 16. Here, the link between excitations and system thermodynamics becomes fairly simple because the reduction of the phase space available to collective excitations can be safely ignored. As a result, liquid energy and specific heat to a very good approximation are given by 3N phonons as in solids. Together with the dynamical disappearance of the viscous response at the glass transition temperature, this explains the observed jump of liquid specific heat at the glass transition temperature, which logarithmically increases with the quench rate. In Chapter 17, I will discuss Mott’s recollection of the Frenkel work. Mott says that Frenkel was interested in real systems and what is really happening in those systems. This prompts a discussion of what we aim to achieve through a physical theory and what this theory should do. The final two chapters, Chapters 18 and 19, explore how the insights we gained from the liquid theory can be used in other areas. Chapter 18 discusses how spin glasses can be understood on the basis of spin waves as is done in the theory of ferromagnets and anti-ferromagnets and similarly to how structural glasses are understood on the basis of phonons. In Chapter 19, I discuss the problem of strongly coupled field theories and make an analogy with strongly interacting liquids.

8

Introduction

I have not discussed all of the sources I have consulted in this book. The selection of topics was helpfully aided and narrowed down by the guiding principle I used in writing this book. This principle is based on a well-established approach in physics where by an interacting system is fundamentally understood on the basis of its excitations [3, 5]. I have also taken a cautious attitude towards proposals that have not yet shown themselves to be well founded or constructive (I briefly touch upon this issue in Chapter 17). I may have missed subtle or more important points, and for this reason I recommend that an inquisitive reader consult references for details and look elsewhere. This is particularly so given that this book includes the results of my own group and collaborators. However, where I have really made a difference is in writing this overview as more than a collection of ideas, concepts and models related to liquids. By focusing on collective excitations in liquids and basic liquid properties, I have shown that earlier and more recent research and ideas physically link to each other and in ways not previously considered. I end several chapters with questions and proposals for future research to either complement the already existing results or pursue a new line of enquiry.

2 Experimental Heat Capacity

Our understanding of the basic states of matter is illustrated by the phase diagram shown in Figure 2.1. Solids are located in the low-temperature and high-pressure part of the phase diagram. Gases are in the opposite, high-temperature and lowpressure, part. Liquids are sandwiched between solids and gases: the liquid state starts above the triple point and is bound by melting and boiling lines. As discussed in the preface and Chapter 1, there is a large and striking gap between the theoretical understanding of the three states of matter. Theories of solids and gases are well developed. In stark contrast, there is no theoretical understanding of the most basic liquid properties, such as energy and specific heat. This book seeks to reduce this gap and is concerned with the fundamental understanding of liquids, with an emphasis on real liquids and experimental data rather than on model systems. We will adopt a generic approach of statistical physics where by the properties of an interacting system are fundamentally related to excitations. The solid state theory is one notable example of this relation being used in many important ways. Our broad goal in this book is to ascertain the nature of collective excitations in liquids and then relate them to liquid thermodynamic properties. Returning briefly to the phase diagram in Figure 2.1, we see that the boiling line ends at the critical point, which conditionally marks the beginning of the supercritical state. A large part of this book deals with subcritical liquids. Towards the end of this book, I will also discuss how insights from the theory of subcritical liquids can be used to understand the supercritical state and its dynamical and thermodynamic properties. One of my talks in Cambridge was interrupted by a keen theoretician, who asked me to show the experimental data early on in my talk. I later thought that this was a good idea because it can quickly tell us what we are dealing with. Figure 2.2 shows the experimental specific heats of liquid mercury and argon. These two liquids

9

10

Experimental Heat Capacity

Figure 2.1 Phase diagram of matter with triple and critical points shown.

represent different liquid types: metallic and noble. Here and throughout this book, we adopt the notations of Landau and Lifshitz [3] and set kB = 1 everywhere. The cv of liquid mercury, with the electronic contribution subtracted, is shown in Figure 2.2 in the temperature range where mercury is subcritical. We also show the cv of liquid argon. Argon is well described by a simple Lennard-Jones potential. For this reason, liquid argon has become a common case study in numerous theoretical approaches and molecular dynamics simulations. We show the cv of argon at three different pressures. The lowest pressure exceeds the critical pressure Pc ≈ 4.9 MPa by about 10 times in order to expand the temperature range where argon exists as a fluid (at room pressure this range is quite narrow) and to avoid critical and near-critical anomalies affecting cv . This book considers the generic behaviour of cv unaffected by critical and near-critical anomalies. We will discuss the effects related to the critical point in Section 14.2 and the supercritical state in general in Chapter 14. We observe that the liquid cv of liquid mercury is close to 3 at low temperature and tends to about 2 at high temperature. A similar trend is seen in liquid argon: cv starts just below 3 at low temperature and decreases to 2 and below at high temperature. We also observe that at a fixed temperature, cv increases with pressure. In argon, viscosity increases with pressure and, therefore, higher viscosity is related to a larger cv . It is useful to keep this in mind for now. We will revisit the pressure dependence of cv in Section 10.5. We will later see that the decrease of cv from 3 to 2 and below to its limiting ideal-gas value of 23 is a universal behaviour seen in other metallic, noble, molecular and hydrogen-bonded liquids. A large part of this book is about understanding this behaviour theoretically.

Experimental Heat Capacity

11

Figure 2.2 Top: experimental specific heat cv of liquid mercury (Hg) with the electronic contribution subtracted as a function of reduced temperature, where Tm is the melting temperature [43, 44]. Bottom: experimental cv of argon (Ar) at different pressures as a function of temperature (Kelvin) [45].

There are three further quick observations from Figure 2.2. First, the decrease of cv is different from what is seen in solids, where cv either increases with temperature due to phonon excitations or stays constant in the classical harmonic regime. Second, cv at low temperature is close to the Dulong–Petit value of 3. This result, revisited later in this chapter on the basis of the Wallace plot, immediately suggests that the physics of liquids close to melting is governed by 3N phonons (N is the number of particles) as is the case in solids. Another possibility is that some other

12

Experimental Heat Capacity

mechanism operates in liquids that nevertheless gives the Dulong–Petit result cv = 3, or that this other mechanism and phonons work together to give cv = 3. This is unlikely on general physical grounds, since it would involve fine tuning to give the same universal low-temperature value of cv = 3 in all liquids. We will later see that no mechanism other than phonons is needed to account for the experimental value of specific heat. We will find that this assertion requires several important sets of results from theory, experiment and modelling. The third and for now final observation from Figure 2.2 is that common models widely used to discuss liquids are irrelevant for understanding the energy and heat capacity of real liquids. These models are notable workhorses of liquid physics and include the widely discussed van der Waals (vdW) model [19] and the hard-sphere model (see, for example, [28, 46, 47]). This may perhaps come as surprising to the reader, in view of the wide discussion of these models in the literature and textbooks. For this reason, it is worth expanding upon this point. Our first example is the vdW model, which is widely used to discuss liquidrelated effects, including the liquid–gas transitions, critical points, compressibility, liquid’s ability to withstand negative pressure and as a starting point of liquid theories involving perturbation methods [19]. The vdW equation of state reads:   N 2a P + 2 (V − Nb) = NT (2.1) V where P, V , N and T are pressure, volume, number of atoms and temperature, respectively, and a and b are constants related to the pair interaction potential and atomic size, respectively. Eq. (2.1) corresponds to the free energy [3]:   Nb N 2a F = Fid − NT ln 1 − − (2.2) V V where Fid is the free energy of the ideal gas. Eq. (2.2) follows from (a) assuming a dilute system with small density and corresponding to the first two terms in the virial expansion in powers NV , (b) calculating the temperature-dependent term of the second virial coefficient in the high-temperature approximation UT0  1, where U0 is the cohesion energy and (c) interpolating between the liquid and gas state and imposing a finite compressibility in the liquid where the liquid volume V should always be larger than Nb in (2.2) (or else the argument of
the logarithm becomes negative) [3]. The entropy S = − ∂F depends on temperature through the first ideal gas ∂T V term only:   Nb S = Sid + N ln 1 − (2.3) V

Experimental Heat Capacity

13

The energy E = F + TS is: E = Eid −

N 2a V

(2.4)

resulting in the specific heat of the ideal gas [3]: cv =

3 2

(2.5)

in stark contrast with the experimental data in Figure 2.2. We see that the vdW model describes an ideal gas from the point of view of the system’s specific heat. This model is a good example of how the virial expansion and high-temperature approximations, while accounting for some liquid properties and introducing interactions in the system, are nevertheless unable to alter cv and the associated number of degrees of freedom in the ideal gas reference system. We might think that cv = 32 of the vdW model is related to the system being dilute. However, the vdW equation, Eq. (2.1), is often derived and discussed by substituting V by V − Nb without assuming that Nb is small compared with V [48]. Hence a system can have a large solid-like concentration comparable to the inverse volume per atom and still have cv = 23 . This brings us to the second workhorse of liquid physics, the hard-sphere model. The hard-sphere model consists of spheres of fixed volume set by an infinite interaction potential at distances shorter than the sphere radius and zero potential outside. This model has been widely used to account for the structure of liquids, giving rise to the approaches where short-range repulsion is dominant in governing liquid physics. We will review these theories in Chapter 6 in the context of perturbation approaches to liquids. The hard-sphere system has also been used to discuss liquid–solid transitions, equation of state, melting, communal entropy and so on (see, for example, [13, 19, 23, 28, 46, 47]). This included the work of pioneers of computer simulations such as Alder, Wainwright and Hoover [49–52], laying the foundation for molecular dynamics simulations. We discuss these simulations in Chapter 9. The specific heat of the hard-sphere system is that of the ideal gas: cv = 32 as is the case of the vdW system in Eq. (2.5). One easy way to understand this is to consider the hard-sphere system as the large n limit of the interaction U(r) ∝ r1n . Molecular dynamics simulations show that at large n, the ratio between the potential and kinetic energy becomes small, corresponding to the onset of the ideal gas behaviour [53]. This happens because harsh repulsion results in a particle spending an increasingly large proportion of its time moving outside the potential range, resulting in the average potential energy becoming small. With only kinetic energy remaining, the specific heat becomes equal to that of the ideal gas.

14

Experimental Heat Capacity

Figure 2.3 The Wallace plot showing the experimental cv of 18 liquids at the melting point against the cv of crystals at the melting point. For metals, the electronic contribution is subtracted from the experimental data. The dashed line represents equality between liquid and crystal values. The ideal gas result cv = 23 for the hard-sphere model is shown. The plot is adapted from [44, 54] with permission from the American Physical Society. The cv for the vdW and the ideal-gas systems are added to the Wallace plot.

The irrelevance of the hard-sphere models for understanding liquid energy and cv is further illustrated in the Wallace plot. Wallace [44, 54] constructs a striking plot illustrating the similarity of the experimental cv of liquids and their parent crystals for 18 systems shown in Figure 2.3. We have already noted this similarity earlier in this chapter. We observe that similarly to the vdW model, the hard-sphere model is unable to change the number of degrees of freedom from the ideal-gas value and to account for the experimental data. Figure 2.3 importantly shows that the cv of real liquids is about twice as large as the cv of the hard-sphere and the vdW models, where cv = 23 as in the ideal gas. This large discrepancy also applies to liquid theories that go beyond the hardsphere and vdW models while using these models as a starting point of a perturbation theory. A perturbation theory is tractable and meaningful in the sense of convergence only if terms in the series are small. Hence, adopting cv = 32 in the vdW or hard-sphere system as a starting point of the perturbation theory will not account for the experimental data in Figures 2.2 and 2.3, where cv = 3 at low temperature: to be consistent with experiments, the perturbation terms should account for another 32 , which is not small compared with the zeroth starting term. For this reason, perturbation methods are unable to describe the experimental cv shown in

Experimental Heat Capacity

15

Figures 2.2 and 2.3 or explain the physical origin of behaviour seen in these figures. We will revisit this point in Chapter 6 where we will discuss liquid theories based on the hard-sphere and vdW models. Our discussion of common liquid models such as the vdW and hard-sphere models sets us up for the realisation that something quite different is needed to understand the experimental behaviour of real liquids. This realisation is constructive and invites us think about what the starting point of liquid theory should be.

3 Excitations in Solids and Gases

In this chapter, we describe how collective excitations emerge in two other states of matter, solids and gases, and focus on different methods of their derivation. We will discuss harmonic phonon excitations in solids, plane waves in crystals and excitations in the continuum approximation including in the theory of elasticity and hydrodynamics. We will then discuss how these excitations are related to thermodynamic properties including energy and heat capacity. It is useful to mention a piece of terminology related to phonons and excitations. The term “phonon” was conceived by Tamm in 1930 [55] and first appeared in print in Frenkel’s book on wave mechanics four years later [56]. Originally, the term “phonon” was used to refer to sound quanta of harmonic oscillations, which can be considered as quasi-particles propagating in the lattice with energy ~ω [3]. A related concept is that of an “excitation”, which refers to a weakly excited state of an interacting macroscopic quantum system, with the emphasis that this state corresponds to the collective motion of atoms and cannot be associated with individual atoms [3]. The subsequent use of the terms “phonons” and “collective excitations” extended outside quantum physics and into classical regime. These terms are used interchangeably with “modes” and “waves” depending on context. It is also not uncommon to find the term “anharmonic phonons” being used to refer to collective motions, which is different from harmonic plane waves. In this book, we consider “collective excitations” in the general sense of collective states in an interacting system, with the important associated benefit that the system energy becomes the sum of energies of these excitations. In most cases, the excitations discussed are harmonic, and we will make separate references to cases where they are not.

3.1 Harmonic Solids and Quadratic Forms We recall the Landau and Lifshitz [3] quote in the Preface: “The strength of the interaction between molecules makes it necessary to know the precise law 16

3.1 Harmonic Solids and Quadratic Forms

17

of interaction in order to calculate the thermodynamic quantities, and this law is different for different liquids.” Interactions in solids are also strong and can be of very different natures: manybody interactions in metals, long-range Coulomb interactions in ionic solids such as NaCl or MgO, short-range covalent interactions in, for example, Si or C, mixed ionic–covalent bonding in the remaining majority of solids such as SiO2 or TiO2 , weak dispersive interactions in molecular crystals, hydrogen bonding and so on. Why don’t strong interactions and system specificity represent a problem in a solid state theory? The reason is that in solids we use a harmonic (quadratic) approximation for the interaction energy, regardless of the nature of inter-particle interactions. As a result, we are able to derive universally applicable results for energy, heat capacity and other thermodynamic properties, as well as their temperature dependence. The quadratic form for the interaction energy and resulting phonons frees us from interactions and makes the solid state theory tractable, general and predictive. We will discuss this in Chapter 6 in more detail. More generally, the quadratic forms and ensuing harmonic motions are benefiting us in many areas of physics, from mechanical vibrations to gravitational waves, particle physics and the quantisation of fields in the quantum field theory. We are so used to them that they might not strike us as something too profound. However, we really start appreciating the quadratic forms in physical models when we no longer have them, such as for liquids. Having come across their inapplicability in several important cases, I am reminded of lectures by Professor Sir Sam Edwards that I attended, where he emphasised how fortunate we are if we can reduce a physical problem to a quadratic form, and how quickly we become stuck if we can’t. In the following we briefly recall how we use quadratic forms in the solid state theory. The important starting point for describing dynamics in solids is to consider small oscillations around stable equilibrium points. These two conditions, the smallness of oscillations and the stability of an equilibrium, do not seemingly apply to liquids. Indeed, liquid particles must undergo large diffusive-like displacements in order to flow. Conversely, we might think that the existence of equilibrium points would disable the flow. In Chapter 7, we will see that this traditional picture is importantly incomplete and deprives us of understanding essential liquid properties. We will see that liquid particles both diffuse and oscillate around quasiequilibrium (rather than equilibrium) positions, and in later chapters we will find that this combined motion is the key to liquid properties. Coming back to solids, the two conditions above enable us to (a) expand the potential energy U(q) around the stable equilibrium and (b) retain only first nonvanishing terms. If q is a generalised coordinate and q0 is its equilibrium value, the

Excitations in Solids and Gases

18

, is zero at the equilibrium point q = q0 because the force is first linear term, − dU dq zero at equilibrium. The first non-zero term is the second quadratic term, giving: 1 U(q) − U(q0 ) = k(q − q0 )2 2

(3.1)

where k = U 00 (q0 ). Counting the energy from the value U(q0 ) and introducing the displacement x = q − q0 , the potential energy is: 1 U(x) = kx2 2 The total energy is therefore quadratic: 1 1 E(x) = m˙x2 + kx2 2 2 with the following equation of motion:

(3.2)

(3.3)

m¨x + kx = 0

(3.4)

x = Re [A exp(iωt)]

(3.5)

and the solution:

where A is a complex constant and ω is frequency. Using the quadratic form for the interaction energy between particles, we arrive at a remarkably simple and physically transparent model of a solid, the harmonic solid. This is a system of s coupled harmonic oscillators (s is the number of degrees of freedom) with the potential energy given by a positive definite quadratic form: U=

1X kik xi xk 2 i,k

(3.6)

The total energy is also a quadratic form: E=

1X (mik x˙i x˙k + kik xi xk ) 2 i,k

(3.7)

with symmetrical coefficients mik = mki and kik = kki . Eq. (3.7) gives rise to the equations of motion, which form a set of s linear homogeneous differential equations: X mik x¨ k + kik xk = 0 (3.8) k

3.1 Harmonic Solids and Quadratic Forms

19

Seeing the solutions as xk = Ak exp(iωt) gives the following system of algebraic equations: X
(3.9) kik − ω2 mik Ak = 0 k

which for non-zero Ak should have a vanishing determinant: |kik − ω2 mik | = 0

(3.10)

Eq. (3.10) is a characteristic equation with s real positive roots ωα2 , which give the eigenfrequencies of the system. In terms of these eigenfrequencies, the solution for xk is: X xk = 1kα 2α (3.11) α

where 1kα are the minors of the determinant (3.10) and 2α = Re [Cα exp(iωα t)]

(3.12)

and Cα are arbitrary complex constants. 2α are new normal coordinates for s independent, decoupled, oscillators satisfying the following equation: ¨ α + ωα2 2α = 0 2

(3.13)

The system energy can now be written as the sum of energies of independent normal oscillations (normal modes):
1 X ˙2 E= Qα + ωα2 Q2α (3.14) 2 α √ where Qα = mα 2α . The great benefit of Eq. (3.14) is that it gives a way in to calculate the energy of a solid in a harmonic approximation. As far as solid’s thermodynamic properties are concerned, this turns out to be a very good approximation. Importantly, the reason we can write Eq. (3.14) is that the starting point (3.7) is a quadratic form. We now add a point that will be come important in later parts of this book, the point that is not emphasised in textbooks. The preceding discussion and Eqs. (3.13) and (3.14), in particular, do not assume that the arrangement of atoms is periodic and crystalline. The only assumption we have made so far is the harmonic expansion around the equilibrium points in Eq. (3.7), but the points themselves do not need to be organised in an ordered structure and can correspond to the topologically disordered system, for example glass or spin glass. This has several important implications as we will see in later parts of this book. Two of these are worth keeping in mind for now. First, we obtain 3N normal frequencies in our

20

Excitations in Solids and Gases

disordered harmonic system and can therefore rigorously introduce and define the vibrational density of states g(ω) normalised to 3N. Second, the system energy becomes the sum of normal mode energies in Eq. (3.14). If we now consider a crystal, we can introduce another parameter characterising each normal mode, the wavevector k. Then, we seek the solutions of Eq. (3.8) (suitably modified to account for elementary cells [3]) that are narrowed down as plane waves: xtn = xn exp [i (krt − ωt)]

(3.15)

where rt is the radius vector running over elementary cells, n enumerates atoms in the elementary cell and t enumerates the cells. As a result, we find the dispersion relations in the system, which include three acoustic branches, two transverse branches and one longitudinal branch, as well as optic branches. These relations depend on the crystal type. The simplest onedimensional problem with N atoms corresponds to wave propagation along certain symmetry directions in the cubic lattice [57]. We can set up the first-principles treatment of the problem in the harmonic approximation based on the quadratic form (3.3), where the force acting on an atom in plane s is Fs = C(us+1 − us ) + C(us−1 −us ), where us is the displacement of the atom from its equilibrium position in plane s and C is the force constant. Then, the first-principles equations of motion are: d 2 us m 2 = C(us+1 + us−1 − 2us ) (3.16) dt where m is the atom mass. Seeking the solution of Eq. (3.16) as plane waves gives the dispersion relation as: 1 ω = 2ω0 | sin ka| 2 (3.17)   12 C ω0 = m n where k = 2π is the wavevector and n is an integer. a N In this description, the eigenstates in crystals are plane waves with wavelengths extending roughly from the system size all the way to the inter-atomic separation a and a frequency extending from the fundamental frequency Lc (c is the speed of sound) to the highest frequency set by ω0 in Eq. (3.17). For small ka, Eq. (3.17) becomes the linear dispersion relation:

ω = ck where the speed of sound is c = aω0 .

(3.18)

3.2 Continuum Approximation: Elasticity and Hydrodynamics

21

In the three-dimensional monatomic solid, we need to consider three different polarisations of the displacement vector  and seek the solution to the equations of motion as plane waves polarised in three spatial directions as u(r, t) =  exp(kr − ωt). As a result, we find [58] three acoustic phonon branches with the dispersion relation similar to Eq. (3.17), including one longitudinal branches and two transverse branches. The longitudinal and transverse branches have different values of c in Eq. (3.18). This is understandable because the longitudinal wave is related to density fluctuations, whereas the transverse wave does not, to first order, involve density fluctuations so that the deformation associated with the transverse wave can be considered as pure shear. This analysis is in full accord with the experimental data. In Figure 3.1 we show representative dispersion curves in Al, K, Na and Ar measured in neutron scattering experiments. We observe the longitudinal and transverse phonons along different directions. The transverse modes are not degenerate along some directions ([110] direction for Na, Al and Ar), which is seen as two branches with different values of c. The transverse modes are degenerate along other directions, resulting in their coincidence in Figure 3.1. We observe the essential features of the dispersion curves predicted by Eqs. (3.17) and (3.18): the phonon dispersion relation starts as a linear dependence at short k and curves over close to the zone boundary. In Section 8.6, we will see that dispersion curves in liquids share these features and are remarkably similar to those in solids, except that the transverse modes become gapped at high temperature. This will enable us to relate collective excitations in liquids to their thermodynamic properties. We note that, in the most general case considered in Eqs. (3.7) to (3.14), including a structurally disordered solid, the eigenstates are not plane waves. In a disordered solid (glass), these eigenstates can substantially differ from plane waves at distances approaching the atomistic scale and large k. On the other hand, the eigenstates of a disordered solid are close to plane waves for small k and ω and describe two transverse and one longitudinal sound waves. This result also follows from the continuum elastic approximation discussed in the next section.

3.2 Continuum Approximation: Elasticity and Hydrodynamics Equations such as Eq. (3.16) give the first-principles description of dynamics at the atomistic level. It provides the most detailed information about atomic dynamics, with k extending to their largest value km = πa in the first Brillouin zone and corresponding to the shortest wavelength available in the system due to the “granular” structure of condensed matter at distances comparable to a. Similarly, ω at km in Eq. (3.17) achieves its largest value possible, ω = ω0 , again due to the finite value

22

Excitations in Solids and Gases

Figure 3.1 Dispersion curves measured by neutron scattering experiments in solid Na [59], K [60], Al [58] and Ar [61] along different directions ν is the frequency and. ξ is the reduced wave vector component. Circles and diamonds correspond to longitudinal and transverse phonons, respectively.

3.2 Continuum Approximation: Elasticity and Hydrodynamics

23

of inter-atomic separation a. In field theories, a is known as a “UV cutoff ” setting the largest energy scale in the system. The continuum approximation detaches from the atomistic structure of condensed matter and instead considers it at larger length scales x  a. We will consider two common continuum approximations, elasticity and hydrodynamics. We will see that these approximations give solutions that can too be written as plane waves, similarly to what we found in the first-principles atomistic approach in the previous section. The elasticity theory considers the mechanics of a solid body as continuous media where the constitutive relation, the linear Hooke law, is related to the quadratic form of the free energy F [62]:  2 1 1 F = G uik − δik ull + Ku2ll (3.19) 3 2 where K and G are bulk and shear moduli and uik is the strain tensor.   ∂F as: The constitutive relation is found by writing the stress tensor σik = ∂u ik T

  1 σik = Kull δik + 2G uik − δik ull 3 The equations of motion, ρ u¨ = uy,z :

∂σik , ∂xk

(3.20)

result in two different equations for ux and

∂ 2 ux ∂ 2 ux = ∂x2 ∂t2 2 ∂ 2 uy,z ∂ uy,z = c2t ∂x2 ∂t2 4 K + 3G c2l = ρ G c2t = ρ c2l

(3.21)

where the first equation corresponds to the longitudinal wave with the displacement ux in the direction of propagation and the second equation corresponds to two transverse waves with the displacement uy and uz in the direction perpendicular to the direction of propagation. The general solution of a wave equation such as in Eqs. (3.21) is u = f1 (x − ct) + f2 (x + ct), where f1 and f2 are arbitrary functions. The widely used form of the solution is the monochromatic plane wave: u ∝ exp (i(ωt − kx))

(3.22)

Excitations in Solids and Gases

24

The second commonly considered continuum (in the above sense) approximation is the most general one. It does not make use of the constitutive relations such as Eq. (3.20) and instead considers the propagating sound wave as the oscillating small changes of pressure p and density ρ. These changes can be related to each other via the continuity and the Euler equations, resulting in a sound wave as follows [63]. Introducing small deviations of pressure p0 and ρ 0 from their equilibrium values p0 and ρ0 , we write: p = p0 + p0

(3.23)

ρ = ρ0 + ρ 0

We note that the smallness of p0 and ρ 0 in Eq. (3.23) is analogous to the smallness of atomic displacements in the microscopic approach considering atomic displacements in Eq. (3.8) and in the elasticity theory assuming small strains in Eq. (3.20). Both of these assumptions are related to their corresponding quadratic forms, Eqs. (3.7) and (3.19). Using the second relation in Eq. (3.23) in the continuity equation ∂ρ + ∂t div(ρv) = 0, where v is velocity, and assuming v to be of the same first order of smallness as p0 and ρ 0 , gives: ∂ρ 0 + ρ0 div(v) = 0 ∂t In the same approximation, the Euler equation, ∂v ∇p0 + =0 ∂t ρ0

∂v ∂t

(3.24) + (v∇) v = − ∇p , gives: ρ (3.25)

0 0 0 0 Eqs.  (3.24)  and (3.25) have three variables: v, p and ρ ·p and ρ can be related as ∂p p0 = ∂ρ ρ 0 , assuming that the motion in the ideal lossless medium is adiabatic. 0 s Using it in Eq. (3.24) gives:   ∂p ∂ρ 0 + ρ0 div v = 0 (3.26) ∂t ∂ρ0 s

Eqs. (3.25) and (3.26) contain two variables, v and p0 , and define the sound wave. It is convenient to define the potential of velocity as v = gradφ. This gives p0 = −ρ ∂φ from Eq. (3.25) (the lower indexes in p0 and ρ0 are dropped for brevity ∂t from now on). Using it in Eq. (3.26) gives the wave equation for the sound wave in this approximation: ∂ 2φ (3.27) c2 4φ = 2 ∂t   ∂p . where c2 = ∂ρ s

3.2 Continuum Approximation: Elasticity and Hydrodynamics

25

Applied to the plane wave where density, pressure and velocity depend on one coordinate x only, Eq. (3.27) gives: ∂ 2φ ∂ 2φ = ∂x2  ∂t2 ∂p c2 = ∂ρ s

c2

(3.28)

Eq. (3.28) predicts the propagation of the compressional sound wave, similarly to the first equation in the elasticity theory in Eq. (3.21). This can be viewed as a reduction of Eq. (3.21) if the system loses its rigidity so that G = 0 and transverse waves are removed from the spectrum. This can be made more specific by considering a gas described by the equation of state PV = NT. In this case, the bulk ∂p ∂P modulus K = −V ∂V = P and, according to Eq. (3.28), c2 = ∂ρ = Pρ = Kρ , where and m is atomic mass. We now see that the elastic result in Eq. (3.21) ρ = mN V reduces to the hydrodynamic one (3.28) if G = 0. This gives ct = 0 in Eq. (3.21), leaving the first equation for ux identical to Eq. (3.28) with c2l = Kρ . We have not yet discussed the decay of collective excitations and considered the systems above as ideal in a sense that excitations do not experience losses. Decay and dissipation will be discussed in detail in Section 8.7. Here, it is important to note one implication of wave decay in the purely hydrodynamic approach. Once we account for the deviation of the fluid from being ideal and introduce viscosity, we find that shear waves are over-damped and non-propagating – in contrast with the longitudinal sound wave related to pressure and density oscillations, which, while decaying due to viscosity, remains propagating. This is seen as follows. The constitutive relation between the viscous shear stress σv and shear strain in ∂v a viscous liquid is σv = η ∂xy , where η is viscosity and vy is the velocity in the direction perpendicular to x. This can be written as: σv = η

∂s ∂t

(3.29)

where s = ∂y is the shear strain. ∂x Differently from the Hooke law, where stresses and strains are linearly related in the elasticity theory (see Eq. (3.20)) or to the force linearly dependent on displacement (see Eq. (3.8) or Eq. (3.16)), the viscous stress in Eq. (3.29) is proportional ∂v to the time derivative of the strain. The constitutive relation σv = η ∂xy gives the equation of motion in the form of the Navier–Stokes equation in the absence of an external pressure gradient: ∂v ∂ 2v =ν 2 (3.30) ∂t ∂x where ν = ρη is the kinematic viscosity and we dropped the lower index in v.

Excitations in Solids and Gases

26

Comparing this with the wave equations in Eq. (3.21) or Eq. (3.28), we see that Eq. (3.30) misses one time derivative. If we now seek the solution of Eq. (3.30) as v = v0 exp (i(kx − ωt)), we find iω = νk 2 . Assuming real ω andpcomplex k ω corresponds to wave decay in space, and in this case we have k = 2ν (1 + i). Then: x

x

v = v0 e− d ei( d −ωt) where d =

q

2ν ω

(3.31)

is the dissipation length. x

In Eq. (3.31), one spatial modulation in the factor ei( d −ωt) corresponds to x = 2πd. At this distance, the wave amplitude decays by a factor of e2π ≈ 535, resulting in an over-damped non-propagating wave. 3.3 Excitations in Quantum Fluids We have so far discussed excitations in classical systems. In quantum fluids, excitations are affected by quantum statistics. This is a large area of study, and here we consider only one example of excitations in dilute fluids with Bose–Einstein statistics. We will use the insight from this calculation to discuss the thermodynamics and heat capacity of liquid 4He in Chapter 11. We will see in that chapter that the energy and specific heat of 4He are directly related to phonon excitations, as is the case for the classical liquids mostly discussed in this book. The approach is due to Bogoliubov and starts with the Hamiltonian describing weakly perturbed states of the Bose gas: H=

X p2 1 X p01 p02 + + a+ a + Up1 p2 ap0 ap0 ap2 ap1 p 1 2 2m p 2 p

(3.32)

where the first and second terms represent kinetic and potential energy, a+ p and ap p0 p0

are creation and annihilation operators, and Up11p22 is the matrix element of the pair interaction potential U(r). Without the second term, the ground state of the system is related to the kinetic energy of the gas with Bose–Einstein statistics. For weak interactions, the energy levels of the system can be calculated in the perturbation theory. As a result, the diagonalised Hamiltonian reads [3]: X H = E0 + (p)b+ (3.33) p bp where s (p) =

 c2 p2

+

p2 2m

2

gives the dispersion relation for excitations in this interacting system.

(3.34)

3.4 Intermediate Summary

27

c in Eq. (3.34) is the speed of sound: r

4π~2 na (3.35) m2 m where n is concentration, a = 4π~ 2 U0 is the scattering length and U0 is the volume integral of the pair interaction potential between particles. p2 We observe that for large momenta, p  mc, the energy becomes  = 2m like for free particles. At small momenta, p  mc, Eq. (3.34) gives the dispersion relation for the sound wave: c=

 = pc

(3.36)

We have already encountered sound waves and their linear dispersion in the previous sections of this chapter discussing classical systems in Eq. (3.18). We therefore find that the low-lying excitations in the quantum Bose fluid are sound waves. This is an important result for understanding thermodynamic properties of liquid helium 4 He, which we will discuss in Chapter 11. 3.4 Intermediate Summary It is now good time to take stock and summarise excitations in the microscopic, elastic and hydrodynamic approaches. This is done in Table 3.1. In the microscopic first-principles approach to crystals, collective excitations have the largest degree of “structure” to them: they are plane waves with wavelengths extending from the system size all the way to the inter-atomic separations. As we move down the table, this structure gradually disappears. In harmonic disordered solids, collective excitations are harmonic but are not plane waves like in crystals. These excitations can, however, be approximated by plane waves at small k and ω. In the continuum elastic approximation, collective excitations become the longitudinal and transverse sound waves with wavelengths λ  a. In the hydrodynamic approximation, we are only left with the long-wavelength longitudinal sound wave. Our discussion so far leaves out an important question of what is the nature of collective excitations in real liquids, like those shown in Figure 2.2? Where do these excitations fit in the general picture of collective modes discussed earlier in this chapter? We will dedicate several of the next chapters to this question. For now, the result is summarised in the final two entries in Table 3.1. Briefly, these entries can be understood as follows. An elastic approach that predicts both longitudinal and transverse modes (entry 3 in Table 3.1) is seemingly inapplicable to liquids because liquids flow. We might think that this leaves us with the next, hydrodynamic, result giving one longitudinal mode only and no transverse modes at all. However, we will see that this

Excitations in Solids and Gases

28

Table 3.1 Types of waves propagating in different states of matter and theories used to describe this propagation. Medium and theory

Type of propagating waves

1. Harmonic crystals: first-principles theory

3N plane waves (one longitudinal and two transverse) with wavelengths extending to the inter-atomic separation a

2. Harmonic disordered solids: first-principles theory

3N normal modes including plane waves at small k and ω

3. Solids: continuum elasticity

One longitudinal and two transverse waves including plane waves with λ  a

4. Fluids and gases: continuum hydrodynamics

One longitudinal sound wave including plane waves with λ  a and no transverse waves

5. Subcritical liquids and low-temperature supercritical fluids: viscoelastic theory

Two gapped transverse waves and one longitudinal wave

6. High-temperature supercritical fluids

One gapped longitudinal wave and no transverse waves

approximation is too crude and inappropriate for real liquids. Instead, we will see that liquids, their excitations and their thermodynamics are described by the combination of elastic and hydrodynamic properties. This combination is provided by the Maxwell–Frenkel viscoelastic theory, and we will see that transverse modes do in fact exist in this theory (as they do in the elastic approach), albeit these modes become gapped. This is shown in the penultimate entry in Table 3.1. The gap increases with temperature, resulting in the progressive disappearance of transverse waves. As the gap widens beyond the zone boundary, these excitations disappear, and we are left with one longitudinal phonon only (the final entry in Table 3.1). On further temperature increase in the supercritical gas-like state, the wavelength of this longitudinal phonon begins disappearing, starting from the shortest wavelength. In this sense, the longitudinal phonon becomes gapped too, similarly to the transverse phonon. This leaves us with the long-wavelength longitudinal sound wave. At this point, we find ourselves in the gas-like hydrodynamic regime where excitations have the least “structure” among all of the excitations we discussed.

4 Energy and Heat Capacity of Solids

4.1 Harmonic Solids The harmonic approximation assumes small displacements from equilibrium. This is a reasonable assumption in most cases and is a greatly simplifying one. As noted by Ashcroft and Mermin, this assumption “is forced on us by computational necessity” [58]. These authors add that, in going beyond a harmonic approximation, we have to resort to very complex approximate schemes whose validity is far from clear. Solid state theory takes great advantage of harmonic approximation and quadratic forms. A spectacular result is a quick calculation of solid thermodynamic properties, producing universally applicable results widely consistent with experiments. This calculation has become one of the cornerstones of the solid state theory and has a long history: the main idea is Einstein’s, who proposed the use of quantised collective excitations to calculate thermodynamic properties of solids in 1907 [64]. In addition to appreciating the ease of this calculation, this section serves two further purposes. First, we will compare the ease of first-principles calculations in the solid state theory with the great difficulty arising in liquids due to the absence of quadratic forms and strong non-linearity governing the essential liquid physics. This will be discussed in Chapter 5. Second, we will later see that, despite this difficulty, there is a way to calculate liquid thermodynamic properties using the Maxwell–Frenkel theory. This calculation uses important concepts from the solid state theory discussed in this section. The solid state theory makes use of two properties of the harmonic approximation. First, the excitations in the system are 3N independent phonons, enabling us to write the partition function Z as the sum over these phonon states as: X En,α Z= e− T (4.1) n,α

29

Energy and Heat Capacity of Solids

30

where the sum is over the energy states of each phonon (n) and over different phonons (α). The second important property of the harmonic approximation
is that the energy states of a quantum oscillator are equidistant: En = ~ω n + 12 . Then, the sum over n becomes a geometric progression that can be summed. As a result, the free energy F = −T ln Z due to phonons becomes:  X  ~ωα ln 1 − e− T F = E0 + T (4.2) α

where E0 is the zero-point energy: E0 =

X ~ωα

(4.3)

2

α

The energy E = F − T ∂F is: ∂T E = E0 +

~ωα

X α

e

~ωα T

(4.4)

−1

The high-temperature limit corresponds to T exceeding the energy of the highest frequency in the system. This frequency is ω0 = ac (see Eq. (3.18)), hence the highP . In this limit, Eq. (4.4) with E0 in (4.3) gives E = T temperature limit is T  ~c a α

or: E = 3NT

(4.5)

since the sum is over 3N phonons. This gives the Dulong–Petit result for specific ∂E heat cv = N1 ∂T in the classical case: cv = 3

(4.6)

Dating back to 1812, the Dulong–Petit value in Eq. (4.6) is probably the earliest result discussed in this book. In the low-temperature limit, only low-frequency terms corresponding to sound waves contribute to the energy in Eq. (4.4). The sum in Eq. (4.4) can be calculated as the integral over ω with the density of states g(ω). It is possible to calculate g(ω) in the general case of an isotropic solid such as glass because, as we have seen in Section 3.2, this system supports one longitudinal and two transverse waves with velocities cl and ct , respectively. This becomes more complicated and system-dependent in an anisotropic solid, the point to which we will return in Chapter 10 where we discuss the applicability of g(ω) to liquids. 2 dk , where V is The number of sound waves in the wavevector interval dk is V 4πk (2π )3 ω ω volume. Assuming k = cl for one longitudinal wave and k = cl for two transverse wave gives g(ω) as:   1 2 V + 3 ω2 (4.7) g(ω) = 2π 2 c3l cl

4.1 Harmonic Solids

31

It is important that g(ω) is split into the longitudinal and transverse part, which, in turn, means that the energy (4.4) similarly splits into the longitudinal and transverse terms. We will make use of this separation in Chapter 10 where we will calculate the energy of transverse waves in liquids where, differently from solids, these waves become gapped. In the solid state theory, we can carry this separation into the final result; however, it is more convenient to introduce the average speed of sound c¯ as c¯33 = c23 + c13 , in which case Eq. (4.7) becomes: t

l

3V 2 ω (4.8) 2π 2 c¯ 3 The energy (4.4) can now be calculated as an integral over ω with g(ω) in Eq. (4.8) and with the upper limit extended to infinity because the integral becomes fast converging at low temperature. This gives the Debye result [3]: g(ω) =

E = E0 +

π 2 VT 4 10 (~¯c)3

(4.9)

and cv =

2π 2 T3 5(~¯c)3 n

(4.10)

where n = N/V is concentration. Differently from high- and low-temperature regimes, a calculation of thermodynamic properties in the intermediate temperature range requires the knowledge of the entire frequency spectrum, and this spectrum is system specific. The Debye interpolation formula gives qualitatively correct results in the low- and high-temperature range, by making the assumption that g(ω) in Eq. (4.8) operates in the entire frequency range and up to Debye frequency ωD , provided this frequency corresponds to 3N waves in the system. Using this condition in Eq. (4.8) gives the density of states: ω2 (4.11) g(ω) = 9N 3 ωD where 6π 2 N ωD = c¯ V 

 13 (4.12)

Thermodynamic properties can now be calculated as an integral with g(ω) in Eq. (4.11) up to the upper frequency ωD . For example, the energy (4.4) becomes:   ~ωD E = E0 + 3NTD (4.13) T Rx 3 dz where D(x) = x33 ezz −1 is the Debye function. 0

Energy and Heat Capacity of Solids

32

At high temperature T  ~ωD , D(x) is close to 1, giving the energy as E0 + 3NT and specific heat cv = 3 like in Eq. (4.6). At low temperature T  ~ωD , D(x) =  3 T π4 12π 4 , giving c = and coinciding with the low-temperature cv in Eq. v 5 ~ωD 5x3 (4.10) in view of Eq. (4.12).

4.2 Weak Anharmonicity In the harmonic approximation, excitations do not decay. In Section 8.7, we will discuss different reasons for the decay of excitations in solids and liquids. One important reason is the anharmonicity, that is, the deviation of the interaction potential from quadratic harmonic form (see Eqs. (3.3) and (3.7)). In solids, this deviation is assumed to be small in the sense that the energy associated with the anharmonic terms is small compared with the harmonic energy. In this case, the anharmonic effects can be treated by a perturbation theory. In liquids, the deviation from anharmonicity is strong and cannot be accounted for by a perturbation theory: in the next chapter, we will see that the overall inter-atomic potential qualitatively changes from single- to multi-well. Despite the assumed smallness of anharmonic terms, their consideration is important and qualitatively changes the results of the harmonic theory. This includes the deviation of cv from the Dulong and Petit value, thermal expansion, phonon scattering and the associated finite value of thermal conductivity, elastic properties, phase transitions and defect mobility. Equally important are thermalisation and ergodicity, the two properties related to anharmonic interactions. The amount of research into anharmonic effects is large [65–67]. Here, we focus on the effect of anharmonicity on energy and heat capacity, two central properties discussed in this book. The perturbation theory starts with expanding the potential energy U in Taylor series over atomic displacements. Retaining the third and fourth orders gives: U=

1X 1X 0 0 φ(r0ll ) + φx (ll0 )(qlx − qlx ) 2 0 2 0 ll ll x 1X 0 0 0 + φxy (ll )(qlx − qlx )(qly − qly ) 4 0 ll xy

+

1 X 0 0 0 φxyz (ll0 )(qlx − qlx )(qly − qly )(qlz − qlz ) 12 0 ll xyz

1 X 0 0 0 0 + φxyzω (ll0 )(qlx − qlx )(qly − qly )(qlz − qlz )(qlω − qlω ) 48 0 ll xyzω

(4.14)

4.2 Weak Anharmonicity

33

where the anharmonic coefficients φx... are given by the derivatives at equilibrium separations in a usual way. Assuming a face-centred cubic lattice with pair and short-range (nearestneighbour) interactions, a tour de force calculation [68] obtains corrections of cv to the Dulong–Petit value (4.6) in the high-temperature limit as: 0000 000   1 φ (r0 ) 172.3 (φ (r0 ))2 1 ~2 1 00 −3 cv = 3 1 − T + T − φ (r ) + O(T ) (4.15) 0 8 (φ 00 (r0 ))2 4608 (φ 00 (r0 ))3 3 M T 2

which can be either positive or negative depending on φ. Obtaining the anharmonic correction in a closed form such as Eq. (4.15) is certainly important. At the same time, it is important to discuss the predictive power of the perturbation approach. As Cowley notes [65], φx... are very complicated to evaluate even if the potential functions are known. Complications related to evaluating φx... necessitated approximations, which, as Cowley further notes, are quite inadequate for real systems and are useful in order-of-magnitude calculations only. The problem of the anharmonic theory relying on the knowledge of interatomic interaction models and φ has been noted earlier [65, 68]. Cowley continues, “undoubtedly the unsatisfactory nature of these models is the limiting factor in our understanding of many anharmonic properties”, and that if anharmonic calculations are to have any quantitative significance, the realistic models are necessary [65]. Consequently, theoretical work on inter-atomic potentials was stated as an essential future effort at the time. We note in passing that, despite the progress in materials modelling since that time, the problem remains. Indeed, apart from a relatively small number of materials (a negligible fraction of all of the known ones), it has not been possible to develop a general recipe for successfully mapping all of the inter-atomic interactions onto the sets of empirical functions to be used in expansion such as Eq. (4.14). The problem is particularly acute with modern materials, which often have complicated interactions in the form of hydrogen-bonded, polymeric, long-range and many-body interactions, magnetic correlations, non-trivial band gap changes with temperature, anisotropy, layered structures, large numbers of distinct atoms in organic and biological systems, and other factors that severely limit the development of high-quality interaction models. Importantly, because the anharmonic effects are believed to be small, a small departure of a potential model from a high-quality one renders the partitioning into harmonic and anharmonic parts in Eq. 4.14 and subsequent interpretation of anharmonicity meaningless. Experimental data such as phonon lifetimes and frequency shifts can provide quantitative estimates for anharmonicity effects and anharmonic expansion coefficients in particular. This involves complications and limits the predictive power of the theory [65].

34

Energy and Heat Capacity of Solids

The quantitative evaluation of anharmonicity effects has remained a challenge, with the frequent result that the accuracy of leading-order anharmonic perturbation theory is unknown and the magnitude of anharmonic terms is challenging to justify [65, 68–71]. As was noted from the early studies onwards [68, 69], the main problem with the approach based on expansions such as Eq. (4.14) and the subsequent understanding of anharmonic effects is that good-quality models for inter-atomic forces are not generally available. We stress this point of the need to have explicit knowledge of the inter-atomic interactions and structure here because, in Chapter 6 and Section 10.2, we will see that the same problem of inter-atomic interactions exists in the approach to liquids based on the inter-atomic forces and correlation functions. Why doesn’t the problem of inter-atomic interactions exist in the harmonic solid state theory? The reason is that considering collective excitations in the harmonic approximation has freed us from the need to have explicit knowledge of systemspecific interactions. As a result, the system energy (and its derivatives such as heat capacity) in the classical regime is universal for all solids, like in Eq. (4.5). Accounting for quantum effects introduces only one system-specific parameter ωD in Eq. (4.13) while maintaining the universally applicable temperature dependence of thermodynamic properties, one of the main tasks of statistical physics [3]. The main aim of this book is to develop the same approach to liquids and similarly express liquid thermodynamic properties in terms of collective excitations. Before concluding this chapter, we note that an alternative approach to treat anharmonicity is not based on the perturbation theory but on the Grüneisen approximation, where the
softening of phonon frequencies ωi is quantified by i parameters γi = − Vω ∂ω [66]. We will use this approximation in Section 10.3 ∂V T to account for anharmonic effects in liquids in a way that does not require the knowledge of interactions and correlation functions.

5 First-Principles Description of Liquids: Exponential Complexity

We recall the discussion in the Preface stating that the main problem precluding a liquid theory was seen as the absence of a simplifying small parameter. Later we will see that liquids in fact do have a small parameter in the Maxwell–Frenkel theory, albeit at short times. For now, let us not worry about it and instead ask just how hard a first-principles theory of liquids is without a simplification provided by the small parameter? What happens if we try to treat microscopic dynamics head-on as we do in solids? We have seen in Chapters 3 and 4 that, in solids, the main feature simplifying the theory is the smallness of atomic oscillations around their equilibrium positions and ensuing harmonic approximations involving quadratic forms. Small deviations from the harmonic case can be treated in a perturbation theory, similar to how small anharmonicity was treated in Section 4.2. This does not apply to liquids, where the anharmonicity is strong in the following sense. In the anharmonic solid theory, Eq. (4.14) and the accompanying discussion assume that atoms continue to vibrate at their fixed lattice points. On the other hand, atoms in high-temperature liquids move diffusively between quasi-equilibrium points. This means that the inter-atomic potential experienced by liquid atoms has the multi-well structure shown in Figure 5.1. This potential cannot be derived by adding small terms to a harmonic well. This constitutes strong anharmonicity and strong non-linearity of interactions, as discussed in this chapter. Strong non-linearity brings us to the new realm of non-linear physics, which, among other problems, deals with coupled non-linear oscillators [72, 73]. Perhaps the simplest model involves two coupled Duffing oscillators with the energy [72]:  2  X 1 2 α 2 β 4 2 q˙ i + qi − qi + (q1 − q2 )2 (5.1) E= 2 2 4 2 i=1

35

First-Principles Description of Liquids: Exponential Complexity

36

Figure 5.1 Multi-well potential related to the particle motion in liquids and involving jumps between different quasi-equilibrium positions. Schematic illustration.

and the equations of motion: q¨1 + αq1 +  2 (q1 − q2 ) − βq31 = 0 q¨2 + αq2 +  2 (q2 − q1 ) − βq32 = 0

(5.2)

where  is the coupling strength. This model is not integrable, and cannot be solved analytically but using approximations only.  However,a simpler model can be written in terms of variables 1 ψn = √2ω ω0 qn + i dqdtn , where ω0 is the frequency of the uncoupled oscillator: 0

dψ1 = ω0 ψ1 + dt dψ2 = ω0 ψ2 + i dt i

 (ψ1 − ψ2 ) − α |ψ1 |2 ψ1 2  (ψ2 − ψ1 ) − α |ψ2 |2 ψ2 2

(5.3)

where the last terms represent the non-linearity and  = ω 0 . Eq. (5.3) is known as the discrete self-trapping (DST) model, and is one of the rare examples in non-linear physics that is exactly solvable analytically. The important results can be summarised as follows. At low energy, the stationary points on the map (q, q˙ ) (or on the map of two other independent dynamical variables) do not change, and the motion remains oscillatory and qualitatively similar to the linear case. On the other hand, the character of motion qualitatively changes at a certain energy that depends on α : the old stationary point becomes an unstable saddle point, and a new pair of stable stationary points emerge, separated by the energy barrier [72]. This corresponds to the bifurcation point, the emergence of new solutions as a result of changes of parameters in the dynamical system such as energy. This is illustrated in Figure 5.2. 2

First-Principles Description of Liquids: Exponential Complexity

37

Figure 5.2 The phase portrait of the DST (Eq. (5.3)) at two different energies. Independent variables u and v are functions of ψ1 and ψ2 in Eq. (5.3) and describe the trajectories of two coupled non-linear oscillators. At low energy (top), two stationary points, “a” and “b”, remain unchanged as in the case of two coupled linear oscillators, corresponding to weak non-linearity. At high energy (bottom), the bifurcation takes place: point “a” becomes an unstable saddle point and two new stationary points “c” appear. Schematic illustration, adapted from [74] with permission from the Institute of Physics Publishing.

An accompanying interesting insight is that, in contrast with the linear harmonic case, the energy is not equally partitioned between the oscillating particles but can localise at one of them. This reflects the more general insight that the superposition principle no longer works in non-linear systems in general. In terms of liquid dynamics, the bifurcation corresponds to the jump of liquid particles from one quasi-equilibrium position to the next. The importance of the bifurcation emerging in the non-linear theory is that it shows how the firstprinciples treatment of the non-linear equations of motion gives rise, via the bifurcation at high energy, to a new qualitatively different solution: instead of oscillating around a fixed position at low energy as in a solid, a particle starts to move between two stable stationary points at high energy, corresponding to the liquidlike motion of particles between two neighbouring minima in Figure 5.1. It proves that, in the most simple non-linear system, the liquid-like motion emerges as a bifurcation of the solid-like motion. The bifurcation in the original model (Eq. (5.2)) emerges at energies E ∝  2 or amplitudes x ∝ . This is to be expected: the energy of coupling needs to be surmounted in order to break away from the low-energy solid-like solution. The DST model (Eq. (5.3)) is not identical to the original system of coupled Duffing oscillators (Eq. (5.2)). The difference with the DST model is that the

38

First-Principles Description of Liquids: Exponential Complexity

islands of chaotic dynamics appear on the phase map and grow with the system energy. This is related to the non-integrability of Eq. (5.2). The excitations in the original model (Eq. (5.2)) can be found using approximate techniques only. However, the DST model is close to (Eq. (5.2)) provided that oscillation amplitudes and couplings  are small. This proximity between the two models is used to assert the same result, the emergence of the bifurcations of solutions. The real problem arises when the number of non-linear oscillators, N, increases. The analysis of the case N = 3 is complicated from the outset by the fact that the corresponding DST model is non-integrable to begin with. The approximations involved in the increasingly complicated analysis of stationary states, new bifurcations emerging from these states and corresponding collective motions become hard to control. Further analysis heavily relies on computer modelling. This modelling indicates the emergence of many unanticipated modes and chaotic behaviour at higher energy. The problem significantly increases for N = 4 in terms of finding new stationary points, related collective modes, analysing non-trivial branching of next-generation bifurcations and so on. For larger N, only approximate qualitative observations can be made regarding the energy spectrum, energy localisation, bifurcations and emerging collective modes. This is done using approximations and insights from N = 2 − 4 [72]. Notably, the number of stationary states and bifurcations exponentially increases with N. Therefore, the problem of finding stationary states, bifurcations and collective modes related to the new stationary states, as well as to their evolution with the system’s energy, becomes exponentially complex and hence intractable for large N [72]. Therefore, the failure of the head-on first-principles treatment of liquids is related to the intractability of the exponentially complex problem of calculating bifurcations, stationary points and collective modes in a large system of coupled non-linear equations. The exponential complexity of non-linear systems such as liquids invites us to appreciate quadratic forms. The exponential and other complexities were entirely avoided in the harmonic approximation based on quadratic forms discussed in Chapter 3.

6 Liquid Energy and Heat Capacity from Interactions and Correlation Functions

6.1 The Method In this chapter, we discuss the approach to liquid theory based on interactions and correlation functions. We will see that this approach diverged from the solid state theory at the fundamental level. It may be helpful to revisit this chapter when we review the history of liquid theories based on collective excitations in Chapter 13. It is useful at this point to recall the quotes from Landau, Lifshitz and Pitaevskii from the Preface. These quotes contain several important references. The first reference is to the absence of a small parameter. The second reference relevant to this chapter is to the strong dependence of liquid properties on the type of interaction, implying that no universally applicable formulas can be derived for liquid thermodynamic properties, in contrast with solids. We have also come across the dependence of energy on the type of interactions in Section 4.2, which discusses how weak anharmonicity in solids can be treated using the perturbation theory. In liquids, the problem gets worse because the interactions are strong, resulting in stronger dependence of energy on the liquid type as noted by Landau, Lifshitz and Pitaevskii. In Section 4.2, we noted that the reason we are not concerned with this dependence in the solid state theory is that the approach based on phonons gives universal results for solid thermodynamic properties and their temperature dependence. This approach makes the solid state theory tractable, transparent and universally applicable. If we don’t know the nature of phonons in liquids, the remaining option is to consider structure and interactions in liquids and see how far we can get in understanding liquids in this approach. In this chapter, we discuss several representative examples illustrating this approach. We don’t aim to be comprehensive but rather we intend to give a list of representative examples illustrating the general nature of this approach.

39

40

Liquid Energy and Heat Capacity from Interactions and Correlation Functions

Figure 6.1 Pair distribution function g(r) of argon simulated with the LennardJones potential at density ρ = 1.9 g/cm3 and temperatures T=250 K, 500 K, 1,000 K, 2,000 K, 4,000 K and 8,000 K. Peaks of g(r) decrease with temperature. Reproduced from [77] with permission from the American Physical Society.

In a system with pair interactions, the average number of atoms in the thin sphere of thickness dr is 4πnr2 g(r)dr, where n is concentration and g(r) is the pair distribution function. Then, the interaction energy can be calculated as an integral over 4πng(r)u(r)r2 dr, where u(r) is the pair interaction energy. Counting each pair once, the total energy reads: Z n 3 g(r)u(r)dV (6.1) E = NT + 2 2 which can also be derived using a more rigorous statistical physics argument [16]. An expanded theory of Eq. (6.1) relating distribution functions and inter-atomic potentials to liquid energy and equation of state and involving higher order correlation functions and integro-differential equations was discussed by, for example, Born and Green [1], Green [75], Kirkwood [11] and Yvon [10, 46]. The cv corresponding to Eq. (6.1) is: cv =

3 n + 2 2

Z 

∂g(r) ∂T

 u(r)dV

(6.2)

v

The calculation of cv in this approach requires the knowledge of temperature dependence of g(r) and ∂g(r) . This dependence can be calculated from molecular ∂T dynamics simulations, as discussed in Chapter 9. An example of g(r) calculated at different temperatures is shown in Figure 6.1. Calculating cv analytically using ∂g(r) involves higher order distribution func∂T tions [16]. March [13] quotes the result of Schofield [76] who uses the fluctuation equation for cv :

6.1 The Method

  hEnihnEi 1 cv = hEEi − nT 2 hnni

41

(6.3)

where n and E are concentration and energy, respectively, as before and hi denote averages of fluctuating properties, and obtains cv as:  Z Z 3 1 n 2 2 cv = + 2 g(r)u (r)dr + n g3 (r, s)u(r)u(s)drds (6.4) 2 T 2 Z n3 + g4 (r, s, t − g(r)g(|t − s|)u(r)u(|t − s|)drdsdt 4 R
2 ! 2 R n g(r)u(r)dr + n2 (g3 (r, s) − g(r))u(r)drds R − 1 + n (g(r) − 1)dr where g3 and g4 are three- and four-body correlation functions, respectively. Born and Green start their paper with observing [1]: It has been said that there exists no general theory of liquids.

We met this quote in the Preface, where we discussed general problems involved in liquid theory. Similarly to Landau, Lifshitz and Pitaevskii, Born and Green say that no general theory of liquids exists because it is impossible to use simplifying conditions either of the kinetic theory of gases where the density is small or of the solid state theory with a high degree of spatial order. Born and Green add that a mathematical formulation of the problem should still be possible without making these simplifying assumptions, although a solution “may be extremely difficult” (recall our discussion of the complexity of the head-on approach to liquids in Chapter 5). These authors develop a system of integro-differential dynamical equations for the coordinate and velocity distribution functions involving higher order correlations. In the binary collision approximation, this theory reduces to the Boltzmann equation in the kinetic gas theory. If the interactions are taken as pair interactions, the liquid energy in this theory reduces to Eq. (6.1). Born and Green subsequently used this approach to calculate transport properties such as viscosity using approximations in a simple model [78]. The approach to liquids based on interactions and structure involving correlation functions was subsequently used in several different lines of enquiry (for a review, see, for example, the review by Barker and Henderson [46]). For example, Weeks, Chandler and Anderson start with the van der Waals model of liquids and accordingly focus on strong short-range repulsive interactions setting the liquid structure [79, 80]. Weeks, Chandler and Anderson separate the intermolecular interaction u(r) into the repulsive u(r0 ) and attractive parts w(r): u(r) = u0 (r) + w(r)

(6.5)

42

Liquid Energy and Heat Capacity from Interactions and Correlation Functions

The first result of the Weeks, Chandler and Anderson theory is structural: the liquid structure at high density is predictably unaffected by the attractive part. The second result is related to thermodynamic properties: assuming that u0 (r) plays the main part at high density, Weeks, Chandler and Anderson consider the difference of the liquid free energy and the ideal gas free energy 1F as: Z n 1F = 1F0 + g0 (r)u(r)dr (6.6) 2 where 1F0 is the excess free energy in the reference fluid and g0 (r) is the distribution function in the system with purely repulsive interactions. Corrections to Eq. (6.6) are related to the difference g(r) − g(r0 ) so that Eq. (6.6) is exact when the interactions are purely repulsive and u(r) = u0 (r). Calculating the second term in Eq. (6.6) energy numerically with model g0 (r), such as the hard-sphere model (see Chapter 2 for the discussion of the hard-sphere model), and using the Lennard-Jones potential to make the separation (Eq. (6.5)), Weeks, Chandler and Anderson find a good agreement with Eq. (6.6). In a similar vein, Rosenfeld and Tarazona separate the interaction potential into the reference and perturbation parts as in Eq. (6.5), corresponding to the reference and perturbation free energy terms similar to Eq. (6.6). Using the hard-sphere system as the reference, they apply the perturbation theory to asymptotically expand the free energy [81]. This gives the first temperature-dependent correction to the 3 energy as ∝ T 5 originating from the singularity of the hard sphere free energy 2 functional and to cv varying as ∝ T − 5 . Other similar ideas involving representing the repulsive part as a reference point and the attractive part as a perturbation and subsequently calculating different terms using the interaction potential and correlation functions were discussed prior to Weeks, Chandler and Anderson [46]. For example, Zwanzig uses the hard-sphere system and the Lennard-Jones potential to separate the interactions into the reference repulsive and perturbation parts and employs a perturbation theory in powers 1 [82]. He finds the correction to the equation of state as: T PV = T



PV T



 2 a1 1 + +O T T 0

(6.7)

where the first term on the right-hand side is the equation of the reference system and a1 depends on the integral over the perturbation potential and correlations functions similar to the second term in Eq. (6.1). This predictably gives a good agreement with experimental data at high temperature but not at low. This point is discussed by Barker and Henderson in detail [46].

6.2 The Problem

43

6.2 The Problem The approaches based on interaction potentials and correlation functions discussed in the previous section are interesting in the sense of making links between different properties and model ingredients. They are not very useful in understanding the behaviour of real liquids, such as the universal behaviour illustrated in Figure 2.2, for several reasons. The first and main reason from the perspective of this book is that this approach has not provided a physical understanding of the universal behaviour of liquid specific heat such as that shown in Figure 2.2, qualitatively or quantitatively. Other issues facing this approach are discussed in this section. We now recall one of the problems stated by Landau, Lifshitz and Pitaevskii in the Preface regarding liquid physics: no generally applicable theory of liquids is possible because interactions in liquids are both strong and system-specific. These interactions are explicitly present in Eqs. (6.1), (6.2) and (6.3), as well as in the approaches where these interactions are split into system-dependent repulsive and attractive parts such as Eqs. (6.5) and (6.6). Strong system dependence also applies to any perturbative expansion: even if this expansion could apply to realistic dense liquids, the coefficients in front of perturbative terms are still system-specific, and so are the results. In fact, Eqs. (6.1)–(6.6) serve as mathematical expressions of the problems set out by Landau, Lifshitz and Pitaevskii. Indeed, the energy and heat capacity explicitly depend on interactions and correlation functions in these equations. Strong and system-specific interactions imply that the liquid energy and other properties are strongly system-dependent and hence cannot be part of a general theory. The problems stated by Landau, Lifshitz and Pitaevskii therefore appear inevitable in the approach to liquids based on inter-atomic interactions and correlation functions. It was probably for this reason that the short four-paragraph Preface to the Statistical Physics textbook by Landau and Lifshitz [3] includes the following separate paragraph: “We have not included in this book the various theories of ordinary liquids and of strong solutions, which to us appear neither convincing nor useful.”

This assessment might come across as too short, in view of useful insights discussed in the previous section. Nevertheless, the issues raised by Landau, Lifshitz and Pitaevskii, combined with other problems listed in this section, are probably insurmountable in the approach to liquids based on interactions and correlation functions. It is fair to say that this approach has not demonstrated how these problems can be resolved despite being around for a long time. In Section 4.2 on the weak anharmonicity in solids, we met another problem discussed by Cowley: apart from several simple cases such as Lennard-Jones systems,

44

Liquid Energy and Heat Capacity from Interactions and Correlation Functions

inter-atomic interactions are generally unknown. Interactions in liquids are generally complex, including many-body interactions like in liquid metals, long-range electrostatic forces, and molecular and hydrogen bonding, or involve large molecule sizes in organic and biological fluids and so on. These interactions can in principle be calculated using quantum-mechanical simulations; however, the energy integral requires an explicit knowledge of analytical functions to be used in equations such as Eqs. (6.1)–(6.6). In some cases, the quantum-mechanical energy surface can be mapped onto a system of empirical functions, and this work has been and continues to be at the forefront of physical chemistry. However, from the point of view the physical understanding of liquids, the need to calculate system-specific interactions reduces the predictive power of a theory. In cases where we can map inter-atomic interactions onto a set of analytical functions, using them in the approach involving Eqs. (6.1)–(6.6) or their modifications can be prohibitive if these functions are not simple. This includes, but is not limited to, interactions and correlations involving many-body interactions in liquid metals, long-range forces in ionic or partially ionic liquids, hydrogen bonding and so on. We will discuss these different types of liquids and their specific heat in a different approach in Chapter 10. The same system-specificity problem applies to correlation functions: Eqs. (6.1)–(6.6) depend on system-specific correlation functions that enter the energy integral. These correlation functions are available for a small class of models or simple systems such as the Lennard-Jones system, but not for the vast majority of liquids where correlations can be higher order and complex. These correlation functions can be simulated or obtained from experiments. This is a hard task, which, if achievable, again reduces the predictive power of a theory. Importantly, even if we could analytically calculate the energy based on interactions and correlation functions where they are available for model systems or from molecular dynamics simulations (see Figure 6.1), this does not mean that we would have a physical understanding. We would still need to develop a physical model explaining the universal behaviour of liquid specific heat such as that shown in Figure 2.2. The approach based on interactions and correlation functions has not helped us in developing such a model, and the reason for this brings us back again to the problems outlined by Landau, Lifshitz and Pitaevskii in the Preface. A physical model and associated physical understanding are aided by an analytical equation expressing, for example, the energy as a function of temperature, as is the case in the solid state theory (see, for example, Eqs. (4.5) and (4.9)). Having such an equation enables us to discuss essential physics. However, such an equation cannot be derived for liquids in general form in the approach based on interactions and correlation functions because these are both strong and system-specific as stated by Landau, Lifshitz and Pitaevskii.

6.3 A Better Alternative?

45

Assuming that we know interactions and correlation functions in a simple case, we can try to calculate the liquid energy and other thermodynamic functions numerically on a computer using the equations in the preceding section of this chapter. Let’s assume that this calculation gives us cv = 3 in liquids close to melting as found experimentally. The question of where this value comes from and how it is related to g(r) and u(r) in Eq. (6.1) would still remain (in solids, of course, we know that cv = 3 is due to 3N phonons). We would still need a physical model. The same applies to calculating cv in molecular dynamics simulations: we can routinely calculate cv = 3 in liquids close to melting as well as its decrease with temperature (see Chapter 9). However, this does not mean that we understand the physical origin of this cv or its temperature dependence. Volker Heine, my PhD advisor, used to say “if the only result of your simulation is an agreement with experiment then you wasted computer time”. 6.3 A Better Alternative? It is important to realise that Eq. (6.1) and the general approach based on structure and interactions is as applicable to solids as it is to liquids. Solid glasses are a particularly good example here because their structure is nearly identical to that in liquids [83]. This raises two related questions. First, why are we not using this approach in the solid state theory? Second, why are we not concerned with the issues listed in the previous section related to structure and interactions in the solid state theory? We could choose to use Eq. (6.1) or its extension to calculate the energy and cv of a solid, with the proviso that inter-atomic interactions and correlation functions are available. However, if this is all we had in terms of describing solids theoretically, this would clearly be a poor-man’s theory of solids due to all of the issues listed in the previous section. In solids, both crystalline and amorphous, we have a theory based on collective excitations, phonons (see Section 3.1), which is general, predictive and physically transparent. We have had this theory to understand solids for such a long time now (the Debye paper was published in 1912 [84] and the Einstein paper was published five years earlier [64]) that it is hard to imagine physics without it. In Chapter 4, we have seen that a description in terms of collective excitations in the system is the way to understand solid properties in universal terms. Quite remarkably, there is no need to have knowledge of structure and interactions to understand the most basic thermodynamic properties of solids. As discussed in Chapter 4 and summarised in this section, most important results such as the temperature dependence of energy and heat capacity readily come out in the approach based on phonons and without the need to consider structure and

46

Liquid Energy and Heat Capacity from Interactions and Correlation Functions

interactions in a particular system. Wouldn’t it be great if we could do the same in liquids? It clearly would,1 but we need to understand the nature of phonons in liquids. As discussed in Chapters 3 and 4, quadratic forms and associated great simplifications giving rise to phonons in solids were not previously thought to apply to liquids. There is nevertheless a way to understand phonons in liquids and relate them to liquid thermodynamic properties. A large part of this book discusses a general programme of how to do that. This programme largely resolves the small parameter problem outlined by Landau, Lifshitz and Pitaevskii in the Preface. This is because the small parameter does exist in liquids but, differently from solids, it operates in a reduced phase space available to phonons. We will discuss this in Chapters 7, 8 and 10. The approach to liquids based on phonons also resolves another problem stated by Landau, Lifshitz and Pitaevskii, namely that interactions in liquids are strong and system-specific, making liquid properties strongly system-specific. The approach to liquids based on phonons frees us from system-specific interactions and correlation functions and all associated problems listed in the previous section, just as it frees us from these problems in the solid state theory. We note that interactions and structure are implicitly present in the phonon approach as well, but their explicit knowledge is not needed to understand solid thermodynamic properties and their temperature dependence, one of the main tasks of statistical physics [3]. As discussed in Chapter 3, the interactions in the phonon approach are approximated by a quadratic form with coefficients kki , which depend on both interactions and structure because these coefficients are set by second derivatives of the interaction energy evaluated at equilibrium separation in Eq. (3.7). Once we derive phonons in the harmonic approximation, phonon frequencies is all we need to know in the solid state theory based on the free energy in Eq. (4.2) (we can bypass interactions and structure altogether if we obtain the set of phonon frequencies ωα in Eq. (4.2) independently involving, for example, a neutron scattering experiment). It then turns out that operating in terms of phonons, we can obtain the most important results even without knowing the set ωα . We have seen that these frequencies drop out from the system energy (Eq. (4.5)) and specific heat (Eq. (4.6)) in the classical case at high temperature, resulting in universally applicable temperature dependence (independence) of these properties. At low temperature, we obtain the energy (Eq. (4.9)) and heat capacity (Eq. (4.10)) with similarly universal temperature dependence. The actual values of energy and heat capacity in this regime depend on only one system-specific parameter, c¯ (or 1 A recent paper reviewing the state of the field [7] compares different liquid theories and concludes that

theories based on the knowledge of liquid structure and interactions will not be fruitful, adding that the approach based on excitations is a more constructive way forward.

6.3 A Better Alternative?

47

ωD in view of Eq. (4.12)). This also applies to the Debye model describing the case intermediate between high and low temperature. Once the harmonic approximation is made and phonons are derived, they emerge as a basic universal concept in many areas of physics, including solid state physics. Other effects such as anharmonicity can be added on top to harmonic phonons to describe, for example, phonon scattering, finite phonon lifetimes or particle scattering in the quantum field theory. Anharmonicity can be introduced by either using the perturbation theory discussed in Chapter 4, where corrections to the phonon energy are calculated, or by using the Grüneisen approximation describing softening of phonon frequencies with temperature or volume [66]. In this and many other areas of physics and different physical scenarios, we use phonons as a basic selfsufficient concept to describe collective excitations and add other effects on top as needed. Imagine for a moment that we did not discover the quadratic forms and harmonic theory of solids and instead used the approach based on correlation functions and inter-atomic interactions, with all the issues mentioned earlier. Our mathematical picture of the physical world would be quite different. We have come to appreciate the quadratic forms discussed in Section 3.1 and all of the benefits they have given us. Now imagine what would happen if we had a similar theory for liquids, the theory based on phonons? We would no longer be content with considering correlation functions and inter-atomic interactions. We would not even need to consider them at all. We would be as happy with this liquid theory as we have been with the solid state theory. It is therefore interesting to observe how earlier liquid theories and the solid state theory diverged at the point of a fundamental approach. Early theories [1, 11, 46, 82] considered that the goal of the statistical theory of liquids was to provide a relation between liquid thermodynamic properties and the liquid structure and intermolecular interactions. Working towards this goal involved ascertaining the analytical models for liquid structure and interactions. Developing these models has then become the essence of earlier liquid theories [6, 11–13, 15, 16, 18–20]. On the other hand, the solid state theory does not aim to predict the solid structure and its characteristics such as g(r). For a given chemical composition, crystal structure can be predicted in quantum-mechanical calculations by finding the energy minima [85, 86]. However, crystal structures cannot be analytically predicted. Instead, the structure is an input to the theory (such as assuming the cubic structure for calculating phonon dispersion curves or anharmonic contributions in Eq. (4.15)). Similarly, the solid state theory does not aim to predict inter-atomic interactions. Some simple models of these interactions play a useful role in the solid state theory; however, it is recognised that the variety of interactions (ionic, covalent and their combinations, metallic, dispersion, hydrogen-bond interactions

48

Liquid Energy and Heat Capacity from Interactions and Correlation Functions

and so on) belongs to the realm of computational chemistry and is not something that can be addressed by a purely theoretical approach. At the same time, this variety and other system-specific properties do not present an issue in the solid state theory, which is based on collective excitations, phonons. Although the approach to the liquid theory diverged from the solid state theory in its fundamental perspective, there were notable exceptions. As discussed later in Chapter 13, Sommerfeld and Brillouin considered that the liquid energy and thermodynamic properties are fundamentally related to phonons like in solids. The first Sommerfeld paper discussing this was published only one year after the Debye theory of solids and six years after Einstein’s “Planck’s theory of radiation and the theory of the specific heat” in 1907 [64]. This line of enquiry largely came to a halt in the years that followed, and liquid theories based on structure and interactions were pursued instead. Whereas the Debye and Einstein theories have become part of nearly every textbook in which solids and phonons are mentioned, a theory of liquid thermodynamics has remained unworkable for the century that followed. One potential reason for this is that, differently from solids, the nature of collective excitations in liquids remained unclear for a long time, lasting about a century.

6.4 Structure and Thermodynamics Although there are issues involved in the approach based on correlation functions and interactions, it is interesting to explore this approach further. What makes this exploration interesting is not an attempt at a theory of liquids, but an attempt to understand the general link between structure and thermodynamics. In a simple case where interactions and correlation functions are pairwise, Eqs. (6.1) and (6.2) are applicable to both liquids and solids including crystals and glasses. If ∂g(r) ∂T is available from detailed experiments or molecular modelling and is used in the second term in Eq. (6.2), this term should become 32 in solids in order to give the Dulong–Petit result in a solid cv = 3 (a small deviation from cv = 3 is expected due to anharmonicity). should Once the solid melts and becomes a liquid at higher temperature, ∂g(r) ∂T give a decrease of cv , as shown in Figure 2.2. Molecular dynamics simulations show the reduction of g(r) peaks with temperature as seen in Figure 6.1 and indicate that ∂g(r) decreases too [77]. This could account for the decrease of cv in Eq. (6.2); how∂T ever, we note that (a) the structures of solid glasses and liquids is nearly identical [83] and (b) the reduction of g(r) peaks with temperature in glasses should not result in the decrease of cv in Eq. (6.2): the calculated cv in glasses should either be cv = 3 or slightly deviate from it due to anharmonicity as discussed earlier. Therefore, the temperature behaviour of g(r) in glasses and liquids must be different. It is currently unclear how this difference is manifested in g(r) and how it gives rise

6.4 Structure and Thermodynamics

49

Figure 6.2 Relationships between three general properties: thermodynamics, structure and dynamics.

to a very different temperature behaviour of cv in glasses and liquids on the basis of Eq. (6.2). This general relationship between structure and thermodynamics is interesting and has not been explored across both solid and liquid states. Indeed, let us consider pair relationships between three general properties of condensed matter: structure, dynamics and thermodynamics, as illustrated in Figure 6.2. The relationship between dynamics and thermodynamics has been well understood in solids. This is mainly related to our discussion in Section 3.1. The relationship between structure and dynamics is a subject of ongoing research [87]. The remaining relationship, that of structure and thermodynamics, can be explored too, in both solids and liquids, where cv should have a different temperature behaviour according to Eq. (6.2). Linking it to the two other pairs, dynamics–thermodynamics and structure– dynamics, in both solids and liquids is an interesting programme of future research, particularly to those researchers with an interest in the structure of condensed matter phases.

7 Frenkel Theory and Frenkel’s Book

7.1 Liquid Relaxation Time In this chapter, we discuss a set of new ideas related to liquids that were put forward by Frenkel. These ideas represent a distinct approach of Frenkel, which is characterised by Mott [88]: Frenkel was a theoretical physicist. By this I am stressing that he was primarily and most of all interested in what is happening in real systems, and the mathematics he used served his physics and not otherwise as is sometimes the case for the modern generation of scientists . . . He asks: what is really happening and how can this be explained?

Condensed matter physics as a term originated from adding liquids to the thenexisting field of solid state physics. Proposals to do so precede what is often thought, and date back to the 1930s when J. Frenkel proposed to develop liquid theory as a generalisation of solid state theory and unify the two states under the term “condensed bodies” [9]. How are solids different from liquids? If a solid is a crystal, then the solid–liquid transition is accompanied by symmetry changes and a phase transition. However, the structure of the solid glass is very similar to that of a liquid [83]. A commonly discussed difference between a solid and a liquid is that the former can support shear stress but the latter cannot. Frenkel proposed that this is not the case and that liquids too can support shear stress and associated solid-like transverse waves, but at a frequency above a certain frequency ωF . In doing that, Frenkel’s physical intuition enabled him to surpass the exponential complexity of a first-principles description of liquids discussed in Chapter 5 and make an important prediction about liquid collective modes. Frenkel’s work is not unknown in liquid physics; however, it is not uncommon to see his ideas being forgotten or reinvented in different disguises but with essentially the same physical meaning. This was also observed by Mott [88]. However, 50

7.1 Liquid Relaxation Time

51

Figure 7.1 Illustration of a particle jump between two quasi-equilibrium positions in a liquid. These jumps take place with a period of τ on average.

this previous interpretation (or lack thereof) of Frenkel’s work is only one of the reasons to recall those ideas. Frenkel’s book [9] is a set of groundbreaking proposals, of which we recall only those relevant to understanding collective excitations in liquids. It is fitting to discuss terms such as “collective modes”, “phonons” and other quasi-particles in relation to J. Frenkel’s work because he was involved in coining and disseminating these terms. For example, the term “phonon”, attributed to Tamm [55], first appeared in print in Frenkel’s book on wave mechanics [56]. Frenkel’s book [9] is not always an easy read. At times I felt that a commentary book on “how to read Frenkel” would be useful, similarly to how our contemporaries interpret the work of other pioneers of scientific thought [89]. Frenkel’s work on liquid dynamics was naturally related to his research related to defect migration in solids, and so he viewed the two processes as sharing important qualitative properties. The migration rate of defects in a solid (crystalline or amorphous) is governed by the potential energy barrier U (sometimes referred to as the “activation energy”) set by the surrounding atoms. At a fixed volume of the “cage” formed by the nearest neighbours, U needs to be very large for the diffusion event to occur in any reasonable time. However, the cage thermally oscillates and periodically opens up fast local diffusion pathways as illustrated in Figure 7.1. If τ0 is the elementary “rattling” time and τ is the time between consecutive diffusion events, the two times are related as: U

τ = τ0 e T

(7.1)

where U can be temperature dependent as discussed later in Section 16.3. If 1r is the increase of the cage radius required for the atom to jump from its cage (see Figure 7.1), U is equal to the work needed to expand the radius of the cage from r to r + 1r, where r is the cage radius. This gives U as [9]:

52

Frenkel Theory and Frenkel’s Book

U = 8πGr1r2

(7.2)

where G is shear modulus. Note that when a sphere expands in a static elastic medium, no compression takes place at any point. Instead, the system expands by the amount equal to the increase of the sphere volume [9], resulting in a pure shear deformation. The strain components u from an expanding sphere (noting that u →0 as r → ∞) are urr = −2b/r3 , uθθ = uφφ = b/r3 [62], giving pure shear uii = 0. As a result, the energy to statically expand the sphere depends on shear modulus G only. Frenkel considered the this picture of particle jumps to be applicable to liquids as well as solids, and introduced liquid relaxation time τ as the average time between particle jumps at one point in space in a liquid [9, 90]. In real liquids, there may be a distribution of relaxation times as established in experiments such as dielectric spectroscopy (see, for example, [91]). In this case, τ implies average relaxation time. The operation of particle jumps in liquids is often referred to as “relaxation process” or “structural relaxation” because structural correlations and externally induced perturbations in liquids decay (relax) on the time scale of τ . The range of τ is bound by two important values. If crystallisation is avoided, τ increases at low temperature until it reaches the value at which the liquid stops flowing at the experimental timescale, corresponding to τ = 102 − 103 s and the liquid–glass transition [83] (τ of course keeps growing at low temperature, albeit becomes experimentally unobservable). At this point, the liquid becomes glass as discussed in Chapter 16 in more detail. At high temperature, τ approaches its minimal value given by the inverse of the maximal frequency. This frequency, ω0 in Eq. (3.17), is approximately equal to Debye frequency ωD in Eq. (4.12). To simplify the discussion and avoid the need to operate in terms of both ω0 and ωD , we will refer to the maximal frequency in the system as ωD . This will not affect assertions we will be making. The picture of molecular motion envisaged by Frenkel and shown in Figure 7.1 is a combination of solid-like oscillatory motion around quasi-equilibrium positions and flow-enabling diffusive jumps between those positions. Conceptually simple, this has been an important insight because it underpins liquid “viscoelasticity”, the combination of viscous and elastic properties at the level of molecular motion. Throughout this book, we will see how the development of this microscopic picture explains many important dynamical and thermodynamic properties of liquids. Molecular dynamics simulations have been useful in ascertaining the combined oscillatory and diffusive motion in liquids, starting from the early days of computer modelling [92]. We will discuss molecular dynamics simulations and their

7.1 Liquid Relaxation Time

53

Figure 7.2 y-coordinate of three atoms involved in a large-scale diffusive jump event as a function of the time-step number. The y-axis is in the units of the parameter σ featuring in the Lennard-Jones potential for argon. σ is approximately equal to the nearest-neighbour distance of about 3.4 Å. The dashed horizontal lines correspond to the average positions around which atoms oscillate like in solids. The vertical dotted line corresponds to large-scale flow-enabling diffusive jumps of these atoms. The data are generated by molecular dynamics simulations. The data are from [93].

role in ascertaining different properties of liquids in Chapter 9. One of the main utilities of molecular dynamics simulations is to provide a visual representation of real molecular motion. Here, we briefly show one result from molecular dynamics simulations that illustrates the Frenkel picture in a striking way [93]. Figure 7.2 shows the molecular dynamics data and in particular the y-coordinate of three different atoms involved in a large-scale diffusive jump event shown schematically in Figure 7.1. We clearly see an oscillatory motion of these atoms, followed by large sudden rearrangements to new quasi-equilibrium positions and subsequent oscillations around those positions. With a remarkable physical insight, Frenkel proposed the following simple picture of vibrational states in the liquid. At times significantly shorter than τ , no particle rearrangements take place. Hence, the system is a solid glass describable by Eqs. (3.19)–(3.21) and supports one longitudinal mode and two transverse modes. At times longer than τ , the system is a flowing liquid, and hence does not support shear stress or shear modes but one longitudinal mode only as any elastic

54

Frenkel Theory and Frenkel’s Book

medium (in a dense liquid, the wavelength of this mode extends to the shortest wavelength comparable to inter-atomic separations as discussed in Section 8.6). This is equivalent to asserting that the only difference between a liquid and a solid glass is that the liquid does not support all transverse modes as the solid, but only those above the Frenkel hopping frequency ωF : ω > ωF =

1 τ

(7.3)

where we omit the factor of π in ω = πτ for brevity (τ corresponds to the halfperiod of a conditional periodic motion in Figure 7.1) and because in liquids the range of τ spans 16 orders of magnitude (see Chapter 16), making a small constant factor unimportant. Eq. (7.3) implies that liquids have the solid-like ability to support shear waves and shear stress, with the only difference that this ability exists not at zero frequency like in solids but at a frequency larger than the hopping frequency ωF (in this chapter and later in the book, we often use the term “solid-like” to denote this property in Eq. (7.3)). This was a breakthrough in understanding liquids, especially in the absence of the experimental and modelling support we have now. As discussed in Chapter 8, it took many decades to collect the evidence proving Frenkel right. In Chapter 4 discussing the solid state theory, we came across the Debye frequency ωD , which is close to the intra-cage oscillation frequency, sometimes referred to as the “rattling frequency”. This frequency importantly features in several solid properties such as energy and heat capacity. In liquids, we see that there is an additional frequency emerging: the inter-cage hopping frequency ωF that quantifies the frequency of particle motion between different cages. In Chapter 10, we will see that both frequencies, ωD and ωF , govern the liquid thermodynamic properties, including energy and heat capacity. In the classical case, the liquid energy depends on the ratio ωωDF only. The longitudinal excitation continues to be propagating in Frenkel’s picture: we can recall the discussion of continuum approximation in Section 3.2, where we saw that density fluctuations exist in any interacting system. In Section 8.6, we will see that experiments have ascertained the propagation of longitudinal phonons with frequency extending to the largest Debye frequency as in solids. However, the presence of the relaxation process and τ differently affects the propagation of the longitudinal excitations in different regimes ω > τ1 and ω < τ1 , as discussed in the next section. In Frenkel’s mind, liquids are inherently capable of both viscous and elastic response, but their rigidity is “masked” by their viscosity. This is apparent in the preceding general picture of molecular flow. To make this idea more explicit,

7.1 Liquid Relaxation Time

55

Frenkel recalls the Maxwell paper on the dynamical theory of gases [94]. Maxwell wrote an elastic response for stress F, strain s and a coefficient of elasticity E as: dF ds =E (7.4) dt dt and proposed that a finite viscosity will result in F decaying with time at a certain rate. If this rate is proportional to F, Eq. (7.4) modifies to: dF ds F =E − dt dt T

(7.5)

t

with the solution F ∝ e− T for constant strain s. is constant, Maxwell gives the solution of Eq. (7.5) as: If the rate ds dt ds t + Ce− τ (7.6) dt Maxwell subsequently calls the product ET the coefficient of viscosity and T the time of relaxation of the elastic force. Frenkel develops the idea that liquids are capable of combined viscous and elastic response as follows. Let’s consider a liquid under shear flow (we will see that the same idea applies to a bulk compression separated into elastic and viscous components in Section 7.2). The y-gradient of horizontal velocity vx due to viscous deformation is: Pxy ∂vx = (7.7) ∂y η F = ET

where Pxy is shear stress and η is viscosity. The gradient of velocity due to elastic deformation is: ∂s 1 dP ∂vx = = ∂y ∂t G dt

(7.8)

where s is shear strain and G is shear modulus and where we dropped the lower index of P for brevity. We now recall the microscopic picture of molecular dynamics in liquids where each molecule combines the oscillatory and diffusive components of motion. These components are related to elastic and viscous response, respectively. When both of these are present, the velocity of a layer vx is the sum of the two velocities, giving: 1 dP P ∂vx = + ∂y G dt η

(7.9)

Frenkel calls this combined response a “viscoelastic effect”, the term commonly used at present. We will later see that Eq. (7.9) is more than just an “interpolation” between two constitutive relations in Eqs. (7.7) and (7.8) describing hydrodynamic and elastic

56

Frenkel Theory and Frenkel’s Book

regimes. We will find that, from the point of view of understanding liquids and explaining the experimental facts, Eq. (7.9) can be viewed as a constitutive relation for the liquid state of matter. When external perturbation stops and vx = 0, Eq. (7.9) gives:   t P = P0 exp − τM (7.10) η τM = G where τM is the time that Frenkel calls “Maxwell relaxation time”. In the next important step, Frenkel identifies τM with the microscopic relaxation time he introduced earlier, the time between particle rearrangements τ in Eq. (7.3): τM ≈ τ . We slightly rephrase his discussion for better clarity while keeping the main idea intact. x Let’s rewrite ∂v in Eq. (7.9) as ∂s , where s is shear strain, and consider a response ∂y ∂t of Eq. (7.9) to a periodic force P = Aeiωt : ∂vx ∂s 1 = = (1 + iωτ M ) P ∂y ∂t η

(7.11)

Let’s assume that τM is the same, or about the same, as the microscopic τ featuring in Eq. (7.3) and consider the motion of a molecule during the time interval . 2  τ (ωτ  1) means that the local environment of equal to the period 2 = 2π ω a molecule remains unchanged during 2. This corresponds to the elastic equilibrium and to the solid-like (non-hydrodynamic) regime of liquid dynamics. When ωτ  1, Eq. (7.11) gives ∂s = η1 (iωτ M ) P = G1 ∂P , or purely elastic response. ∂t ∂t In fact, this regime ωτ  1 is just what Frenkel expressed in Eq. (7.3) on the basis of microscopic dynamics. The opposite regime, 2  τ (ωτ  1), implies that a molecule undergoes many diffusive jumps during time 2, corresponding to the hydrodynamic equilibrium and hydrodynamic response. When ωτ  1, Eq. x (7.11) gives ∂v = Pη . This is related to the Navier–Stokes equation describing a ∂y purely hydrodynamic response. Therefore, concludes Frenkel, the microscopic origin of τM is related to τ and both relaxation times must be close. He backs up this assertion by evaluating the actual values of η and G in Eq. (7.10) using his theory. Applied to liquids, G featuring in η = GτM ≈ Gτ has the meaning of the shear modulus at high frequency, which can be taken as the maximal frequency of shear waves present in the liquid, comparable to the Debye frequency [95]. Highfrequency G is usually denoted as G∞ to imply an “instantaneous” value. G∞ in liquids is of the same order as elastic moduli in corresponding solids. For example, Wallace estimates G∞ to be 1 GPa and 5 GPa in liquid water and mercury, respectively [96]. We will return to different potential and kinetic terms contributing to

7.2 Collective Excitations in the Frenkel Theory

57

G∞ in Chapter 14 where we will discuss thermodynamic properties of supercritical fluids. The identification of τM with τ has become one of the central points of research in viscoelasticity because it connected macroscopic properties such as η and G∞ to the microscopic dynamics in liquids. This identification was confirmed on the basis of numerous experiments [83] and molecular dynamics simulations. For example, if τ is calculated as the time during which a molecule changes its local bonding environment as implied by Frenkel, τ = τM follows with good precision [97]. τ can also be calculated using a cut-off distance deciding how far a molecule moves before this motion counts as Frenkel’s jump. If this cut-off is on the order of the inter-atomic separation, the calculated τ is close to the time decay of the correlation function setting the structural relaxation time, which, among other properties, governs the relaxation of elastic stresses in Eq. (7.10) [98]. In what follows, we will drop the lower index in τM for brevity and in view of the connection between τM and τ . In the equations that follow, τ is assumed to be τM because it is based on Eq. (7.11). Its microscopic interpretation as the time between molecular jumps will be apparent from context. At this point it is interesting to recall the discussion of exponential complexity of the first-principles of liquid description discussed in Chapter 5 and put it in the context of results in this section. The Frenkel approach overcomes this problem and singles out the main physical property of liquids (τ ), which, as we will see later, defines the relative contributions of oscillatory and diffusive motion in the liquid and ultimately governs collective excitations. This approach makes a physically reasonable assumption that quasi-equilibrium states and the local particle surroundings of jumping atoms in a homogeneous liquid are equivalent. In the language of non-linear theory, this implies that emerging new bifurcations and stationary states at all generations produce physically equivalent states on average. This also implies that the conditions governing local particle jumps can be considered approximately the same everywhere in the system. In the next section, we will see how far Frenkel advanced in understanding collective excitations in liquids using Eq. (7.9).

7.2 Collective Excitations in the Frenkel Theory Considering that liquids are capable of both viscous and elastic responses, Frenkel used two approaches to discuss excitations in liquids: 1. Modifying the equations of elasticity to account for the hydrodynamic response of liquids.

Frenkel Theory and Frenkel’s Book

58

2. Modifying the equations of hydrodynamics to account for the elastic response of liquids. The second approach constitutes the main topic of a separate and large field of liquid research called generalised hydrodynamics. We will discuss different and characteristic ways in which generalised hydrodynamics has developed in Section 8.5. Approach 1 is explored a lot less by comparison and has not advanced too much beyond Frenkel’s analysis. It is important to note that Eq. (7.9) contains the sum of two terms, elastic and viscous, each of them being equally legitimate from a physical point of view. Yet it is the second approach that has been overwhelmingly preponderant, and it is interesting to ask why. The reason for this was probably related to our subjective experience and familiarity with flow properties of low-viscous liquids, whereas liquid elasticity is “masked” by the flow as noted by Frenkel. We will return to this point in Section 8.4, where we will see that both approaches 1 and 2 give equivalent results. ∂vx ∂s Recalling that = , we write Eq. (7.9) as: ∂y ∂t ∂s 1 dP P = + ∂t G dt η

(7.12)

and introduce the following operator: d dt

(7.13)

ds 1 = AP dt η

(7.14)

A=1+τ where τ =

η , G

to rewrite Eq. (7.9) as:

. d = A−1 from Eq. (7.13) If A−1 is the reciprocal operator to A, P = ηA−1 ds dt dt τ −1 and hence P = G(1 − A )s. Comparing this with the elastic constitutive relation P = Gs, we find that the presence of the hydrodynamic viscous term is equivalent to the substitution of G by the operator M = G(1 − A−1 ): G → M = G(1 − A−1 )

(7.15)

Let us now consider the propagation of the wave of P and s with time dependence exp(iωt). Differentiation gives multiplication by iω. Then, A = 1 + iωτ , and M is: G M= (7.16) 1 1 + iωτ

7.2 Collective Excitations in the Frenkel Theory

If M = R exp(iφ), the inverse complex velocity is

1 v

=

q

59 ρ M

=

q

ρ (cos φ2 R

−

φ ), 2

i sin where ρ is density. P and s depend on time and position x as f = exp(iω(t − x/v)). Using the expression for 1v : f = exp(iωt) exp(−ikx) exp(−βx)

(7.17)

where k=ω

r

ρ φ cos R 2

(7.18)

ρ φ sin R 2

(7.19)

and the absorbtion coefficient is: β=ω

r

Combining Eqs. (7.18) and (7.19) and introducing the propagation length d = 1/β so that f ∝ exp(−x/d) in Eq. (7.17), we write: d=

λ 2π tan φ2

(7.20)

where λ = 2π is the wavelength. k 1 . For high-frequency waves ωτ  1, tan φ ≈ φ = From Eq. (7.16), tan φ = ωτ 1 λωτ , giving d = π . Therefore, this theory gives propagating transverse waves in ωτ the solid-like elastic regime ωτ  1, with the propagation length: ωτ π (ωτ  1) d=λ

(7.21)

We note that this result is derived for plane waves, and it approximately holds in disordered systems for wavelengths that are large enough. At smaller wavelengths comparable to structural inhomogeneities, d is reduced due to the dissipation of plane waves in the disordered medium. We will return to this point in Section 8.7. In the hydrodynamic regime ωτ  1, we find φ = π2 and: λ 2π (ωτ  1) d=

(7.22)

Differently from the solid-like elastic regime ωτ  1, this means that lowfrequency transverse waves are overdamped because they are dissipated over the in Eq. distance smaller than the wavelength by a factor 2π . Using β = d1 = 2π λ (7.17), we find that the wave decays by a factor e2π ≈ 535 over one spatial period λ.

60

Frenkel Theory and Frenkel’s Book

We have already encountered this very result in the purely hydrodynamic case discussed in Eq. (3.31) in Section 3.2. Here, this happens because k in Eq. (7.18) and β in Eq. (7.19) become equal when φ = π2 in the regime ωτ  1. Then, Eq. (7.17) becomes equivalent to the wave propagation in the purely hydrodynamic case represented by Eq. (3.31). The existence of propagating shear waves in liquids in the regime ωτ  1 is equivalent to the prediction based on the microscopic picture of molecular motion in Eq. (7.3). Propagating waves with frequency ω above τ1 means that these wave have an energy gap. We will revisit this important effect in Chapter 8. The discussion so far was related to the shear response of liquids and the propagation of shear waves. Frenkel extended the viscoelastic picture to longitudinal waves where the generalised bulk modulus is: K → K1 +

K2 1 1 + iωτ

(7.23)

where K1 is the static frequency-independent term, τ is the bulk viscosity relaxation time of the same magnitude as the shear relaxation time in Eq. (7.16) and the second term is similar to Eq. (7.16). Repeating the same steps as in the shear case above, we find the propagation length in the solid-like elastic regime ωτ  1 to be the same as in Eq. (7.21). In the hydrodynamic regime ωτ  1, the propagation length becomes: λ ωτ (ωτ  1)

d≈

(7.24)

The absorption coefficient β = d1 ∝ ω2 , in agreement with the result from hydrodynamics [63]. In an important difference from transverse waves, the longitudinal waves remain propagating at low frequency, consistent with the result in the continuum approximation discussed in Section 3.2. However, ωτ < 1 and ωτ > 1 give different dissipation regimes. Indeed, d decreases with τ and viscosity in the regime ωτ  1 in Eq. (7.24). This is consistent with the result from hydrodynamics [63]. On the other hand, d increases with τ and viscosity in the solid-like elastic regime ωτ  1 as follows from Eq. (7.21). In this regime, large τ corresponds to the solid glass and propagating waves. We therefore find that the longitudinal mode remains propagating in the liquid: for a given τ , its frequency range is split into two regimes, with a crossover point ωτ ≈ 1. We will use this result when we calculate the liquid energy in Chapter 10. We now come to approach 2 listed at the beginning of this section. Eqs. (7.14)– (7.16) modify (generalise) elasticity equations by including the hydrodynamic

7.2 Collective Excitations in the Frenkel Theory

61

response. Equally, argued Frenkel, one can use Eq. (7.12) to generalise hydrodynamic equations by endowing the system with the solid-like property to sustain shear stress. We write the Navier–Stokes equation as:   1 dv 2 ∇ v= ρ + ∇p (7.25) η dt where v is velocity, p is pressure, η is shear viscosity, ρ is density and the full derivative is dtd = ∂t∂ + v∇. From Eqs. (7.12)–(7.14), we see that generalising viscosity to include the elastic response amounts to making the following substitution:   1 1 d → 1+τ (7.26) η η dt Using Eq. (7.26) in Eq. (7.25) gives:    d dv 2 η∇ v = 1 + τ ρ + ∇p dt dt

(7.27)

Having proposed Eq. (7.27), Frenkel did not seek to solve it. This is surprising because the solution gives an important property of liquids: the gapped momentum state. I will show the solution and discuss its implications for liquid excitations in Chapter 8. Why didn’t Frenkel attempt to solve Eq. (7.27), similar to how he solved the generalisation of elasticity? We don’t know; however, a potentially useful insight comes from Tamm [99]: He [Frenkel] wrote the book under the difficult wartime and evacuation conditions, when his “office” at the Physico-technical Institute was a corner fenced off by several file cabinets in the hall of the Ethnographic Museum of the Kazan’ University . . . Incidentally, some of the exhibits of that museum were used at that time for their original purpose; in particular, one of the staff members of the Institute used a primitive grain mill of an Indian tribe to mill some rye which he managed to acquire. Naturally, Frenkel had likewise no office at home, using his landlady’s laundry room as a working place. This laundry room, of course, had no table, which was replaced by a plywood board held on the knee. Under these conditions Frenkel wrote in the summer and fall of 1943 his “Kinetic Theory of Liquids,” one of his best books (by his own estimate). This book summarized twenty years of work on the theory of liquids and real crystals. There are few examples in the history of science where a physicist developed such an extensive branch of science with so much use of his own ideas and work.

We will solve Eq. (7.27) in Chapter 8 and will find that the solution gives the gapped momentum state of phonons in liquids. The discussion in this section completes our review of the Frenkel theory. In a fairly small amount of space, we have learned quite a lot about particle dynamics in

62

Frenkel Theory and Frenkel’s Book

liquids and ways to understand how these dynamics affects phonons. This forms an important part of Frenkel’s book. Frenkel’s excitement about his theory is apparent in Mott’s [88] recollection: Professor Frenkel invited me for lunch which, according to him, should start very soon, at 4 pm. In the UK, lunch is at 1 pm so I was pretty hungry. When we came to Prof Frenkel’s flat, he said with a beaming smile: “I was mistaken, lunch will be at 6.30, but this is great because now we have whole two hours to discuss my recent work on the theory of liquids.” And indeed that was great, and I hope I learned from Prof Frenkel that physics, and not food, should be preferred.

This recollection is illustrated in the first picture of this book. Having proposed the theory of high-frequency propagating transverse waves in liquids, Frenkel observes that the liquid transverse spectrum is in the frequency range between the inverse of τ and the highest frequency in the system and adds that the frequency distribution can be well approximated by the Debye model because this model is designed for isotropic systems such as liquids and glasses rather than crystals. We will return to this point when we review the experimental phonon dispersion curves in liquids in Chapter 8. Frenkel is not fully free from doubt though. He says “it seems improbable that in ordinary liquids at room temperature transverse vibrations with a frequency of the order of 1010 s−1 should exist”. It is fascinating to see this combination of ground breaking ideas and self-doubt in the absence of experimental or any other support. Being meticulous and conscientious in his book, Frenkel is open to a possibility that he might be wrong. In Chapter 8, we will discuss experimental and modelling evidence proving that propagating high-frequency collective excitations exist in liquids up to the largest wavevectors and frequency, both transverse and longitudinal. The experimental evidence for this came more than 70 years after Frenkel’s prediction.

7.3 Continuity of Liquid and Solid States and the Frenkel–Landau Argument: First Ignition of the Glass Transition Debate To finish the discussion of Frenkel’s book, we recall an important prediction he made regarding the liquid–glass transition in his microscopic theory of liquid motion based on τ . If a liquid does not crystallise on cooling, τ increases beyond the observation time, at which point the system is a solid glass from the experimental point of view. In view of (a) the debate and controversy that followed and (b) reiterating the basic tenet of Frenkel’s view, it is better to quote the following paragraph directly (the final paragraphs of Section 4.1 of Frenkel’s book [9]) rather than restating it:

7.3 Continuity of Liquid and Solid States and the Frenkel–Landau Argument

63

The possibility of such a continuous transition from the liquid state into the solid amorphous one, which is actually realized in the case of a number of complex substances on rapid cooling, must be considered as a direct proof of the fact that the properties of fluidity and rigidity can coexist in the same body, the classification of condensed bodies into solids and liquids having thus but a relative meaning convenient for practical purposes but devoid of scientific value.

In this view, the difference between a liquid and a solid glass is quantitative rather than qualitative. In Chapter 16, we will discuss how liquid thermodynamic properties and heat capacity in particular change at the liquid–glass transition temperature Tg . We will see that this change is related to the viscous response dynamically disappearing at the experimental timescale. This is consistent with the view that the liquid–glass transition involves quantitative rather than qualitative changes. In the preceding quote, Frenkel clearly refers to amorphous solids. In 1935, he published a paper in Nature entitled “Continuity of the solid and the liquid states” [100]. Similarly to the quote, he talks about amorphous solids as supercooled liquids with a very large relaxation time τ , implying the continuity between liquid and solid glass states. About midway through the paper, he also brings crystals into the discussion and, referring to mobile defects appearing in the crystal close to melting and to a structural similarity between a microcrystalline solid and a liquid close to melting, adds that crystal–liquid melting may not be sharp either. This elicited an objection from Landau [101, 102] who observed that the crystal–liquid transition involves symmetry changes and hence cannot be continuous because the symmetry elements are either present or absent, without an intermediate state. Interestingly, the current debate of the glass transition problem [83] is largely related to the questions discussed by Frenkel and Landau: is the liquid–glass transition a continuous dynamical effect or is it related to a discontinuous thermodynamic phase transition, hidden or avoided? Is there a glass phase distinctly different from the liquid phase and, if there is, what is the associated change of symmetry? What is not commonly known is that, to a large extent, this debate has now being going on for nearly 100 years. We will return to the problem of the liquid–glass transition in Section 16.3.

8 Collective Excitations in Liquids Post-Frenkel

The career of a young theoretical physicist consists of treating the harmonic oscillator in ever-increasing levels of abstraction. Sidney Coleman [103]

8.1 The Telegraph Equation and Gapped Excitations in Liquids As mentioned in the previous chapter, Frenkel wrote Eq. (7.27) but did not solve it. The solution appeared much later [74] and turned out to be very informative for understanding phonons in liquids. In Eq. (7.27), we consider no external forces, p = 0 and the slowly flowing fluid in x-direction so that dtd = ∂t∂ . Then, Eq. (7.27) reads: η

∂ 2v ∂ 2v ∂v = ρτ +ρ 2 2 ∂x ∂t ∂t

(8.1)

where v can be y or z velocity components perpendicular to x. In contrast with the Navier–Stokes equation, the generalised hydrodynamic equation Eq. (8.1) contains the second time derivative of v and hence allows for propagating waves. Indeed, Eq. (8.1) without the final term q the wave q reduces to η equation for propagating shear waves with velocity cs = τρ , or cs = Gρ in view of Eq. (7.10). The final term represents dissipation. Using η = Gτ = ρc2 τ , where G and c correspond to their solid value or approximately to their high-frequency values in the liquid, we rewrite Eq. (8.1) as: c2

64

∂ 2v ∂ 2 v 1 ∂v = + ∂x2 ∂t2 τ ∂t

(8.2)

8.1 The Telegraph Equation and Gapped Excitations in Liquids

65

Setting τ → ∞ in Eq. (8.2) corresponds to a solid and gives the wave equation with propagating non-decayed waves. Eq. (8.2) has a form of the “telegraph” (or “telegraphist”) equation. According to Koshlyakov [104], this term was coined by Poincaré. I was able to find the Poincaré paper [105] where he writes the equation: ∂ 2φ ∂ 2φ ∂φ = B +C (8.3) 2 2 ∂x ∂t ∂t where φ is the electric potential and A, B and C are constants related to capacitance, induction and resistance, respectively. Poincaré says that Eq. (8.3) is known as the “telegraphist equation”, implying that the term was already used before his paper. Eq. (8.3) was derived by Heaviside [106] in 1876 and earlier by Kirchhoff [107] in 1857 in the context of electric currents. The Kirchhoff paper precedes the Maxwell paper [94] by 10 years, and is probably the earliest publication that contains an equation relevant to understanding liquids. An interested reader may consult a review by Mawhin [108] who discusses the early history of the telegraph equation including the works of Poincaré, Heaviside and Kirchhoff. The telegraph equation is widely used to discuss a number of effects, including heat waves, thermal conductivity, diffusion and electricity transmission losses. It is a surprising fact that, despite the long history of the telegraph equation, its different dispersion relations and their properties are not commonly discussed (see, for example, [109–112]). We found the solution of the telegraph equation giving the gapped momentum state fairly recently [74, 113]. This solution is discussed in this section. Seeking the solution of Eq. (8.2) as v ∝ exp (i(kx − ωt)) gives: A

i − c2 k 2 = 0 (8.4) τ The full solution of Eqs. (8.2) and (8.4) depends on initial and boundary conditions. To solve Eq. (8.4), we need to consider a physical problem in question. This involves the choice of setting ω and k to be real or complex. We will discuss this in the next section in more detail. Considering real k and complex ω corresponds to decay in time. A model example corresponding to these boundary conditions is propagation of the wave, which is set up in an elastic rod and subsequently immersed in a viscous liquid. The displacements and its derivatives are periodic with distance but decay with time until the wave oscillations cease. Apart from this model example, this case also corresponds to phonons in liquids, which we discuss in this book. In particular, this gives phonon decay in liquids and describes a typical inelastic neutron or ω2 + ω

66

Collective Excitations in Liquids Post-Frenkel

X-ray scattering experiment where the scattering intensity at a given real k gives the position of the phonon peak frequency (real part of ω) and the peak width gives the phonon lifetime or decay (imaginary part of ω). For real k and complex ω, solving Eq. (8.4) gives: r 1 i ω = − ± c2 k 2 − 2 (8.5) 2τ 4τ giving the time decay of transverse waves as:   t v ∝ exp − exp (i(kx − ωr t)) (8.6) 2τ where r

1 (8.7) 4τ 2 We observe that ωr in Eq. (8.7) is real if k is larger than the threshold value kg given by: 1 kg = (8.8) 2cτ No real ω exist for k < kg . Therefore, Eqs. (8.7) and (8.8) describe the gapped momentum state (GMS). The GMS can be compared to the gapped energy state discussed in the previous chapter (see Eq. (7.3)), and we will discuss the relationship between the two gaps in this chapter. The dispersion relation (Eq. (8.7)) showing the GMS is shown in Figure 8.1. For comparison, we also show the gapless phonon or photon dispersion relation and the relation corresponding to p the gapped energy state such as the dispersion relation of a massive particle E = p2 + m2 , where m is particle mass. We recognise that the GMS in Figure 8.1 is not commonly discussed in physics, including in the areas of mathematical physics and partial differential equations, where the telegraph equation is used [104, 109–112], as compared with the other two dispersion relations. One area where Eq. (8.7) is encountered is related to a hypothetical tachyon with an imaginary mass [114] or an instability not present in equilibrium systems. A recent review [113] compiles several areas in physics and physical scenarios where the GMS is encountered. In field-theoretical terms, the second viscous term in the telegraph equation (Eq. (8.2)) does not readily lend itself to a Lagrangian formulation because it involves dissipation. The field-theoretical description of the GMS requires two scalar fields [115] and results in Eq. (8.2). How can we understand the microscopic origin of the gapped momentum state in liquids? Microscopically, the gap in k-space can be related to a finite propagation ωr =

c2 k 2 −

8.1 The Telegraph Equation and Gapped Excitations in Liquids

67

Figure 8.1 Three different dispersion relations: dependencies of energy E or frequency ω on momentum p or wavevector k. The middle straight line shows the gapless dispersion relation for a massless particle (photon) or a phonon in solids. The bottom curve shows the dispersion relation of the gapped momentum state (Eq. (8.7)). The top curve shows the dispersion relation of the gapped energy state. Schematic illustration.

length of waves in a liquid: if τ is the time during which stress (e.g. shear stress) relaxes in Eq. (7.10), cτ gives the shear wave propagation length, or liquid elasticity length del : del = cτ

(8.9)

1 approximately corresponds to transverse Therefore, the condition k > kg = 2cτ waves with wavelengths shorter than the propagation length. We note that del in Eq. (8.9) is approximately equal to d in Eq. (7.21). Indeed, using ω = 2πc in Eq. (7.21) gives d = π cτ . Therefore, our analysis is consistent λ with the previous Frenkel result. This is to be expected, since the starting point for Eq. (7.21) is Eq. (7.9), the same equation that leads to our telegraph equation (Eq. (8.2)). We now recall the discussion in the previous chapter where the transverse waves in liquids propagate above the hopping frequency ωF = τ1 but not below. We first encountered this in Frenkel’s picture of liquid dynamics in Eq. (7.3) and then later in Eqs. (7.16)–(7.22) showing that the transverse waves propagate in the regime ωτ  1 but not when ωτ  1. This result implies the frequency (energy) gap for transverse excitations, whereas Eq. (8.7) indicates that liquids have the gap in the wavevector (momentum) space. How can this be reconciled? The answer is that Eqs. (7.16)–(7.22) explicitly use the condition ωτ  1 for propagating waves and demonstrate that these waves are propagating in this regime

68

Collective Excitations in Liquids Post-Frenkel

but not when ωτ  1. On the other hand, considering Eq. (8.7) only does not tell the full story. In terms of an inelastic scattering experiment, Eq. (8.7) tells us the position of the peak frequency in scattering intensity only; however, this equation does not inform us about the peak width or decay. The full picture is given by Eq. (8.6), which involves both the frequency of the plane waves and its decay. If decay time is much shorter than then the wave period, the waves become over-damped and non-propagating. We came across an extreme example of this effect at the end of Section 3.2: a purely hydrodynamic description gives the result that the amplitude of transverse waves decays by a factor e2π ≈ 535 over one spatial period. The same can happen in Eqs. (8.6) and (8.7) when k becomes close to kg : in this case ωr in Eq. (8.7) becomes arbitrarily close to 0, meaning that the decay time 2τ in Eq. (8.6) gets much shorter than the wave period. According to Eq. (8.6), the decay time (lifetime) and decay rate are 2τ and 0 = 1 . If, as is often assumed, the crossover between propagating and non-propagating 2τ modes is conditionally given by ω = 0 (here we drop the lower index in ωr in Eq. (8.7) for brevity and assume that ω is real), Eq. (8.7) gives for the propagating regime ω > 0: k>

√ 1 √ = kg 2 cτ 2

(8.10)

Using Eq. (8.10) in Eq. (8.7) gives the waves propagating above the frequency: ω>

1 2τ

(8.11)

which is the same as Frenkel’s result for the frequency gap in Eq. (7.3) for propagating transverse waves bar the factor of 2. In view of this, Frenkel’s original assumption for propagating transverse waves based on microscopic dynamics, Eq. (7.3), can be interpreted as ω > 0, where 0 ≈ τ1 . This interpretation is enabled by our solving the telegraph equation (Eq. (8.2)) and discussing the solution (Eqs. (8.6) and (8.7)). Another insight from comparing the frequency and decay rate is related to the density of states. The dispersion relation (8.7) corresponds to the quadratic Debye density of states in the propagating regime (8.11) and to the linear density of states in the opposite non-propagating over-damped regime (the crossover between the two regimes is seen in the numerical instantaneous mode analysis [164]). The liquid energy related to propagating phonons is therefore related to the quadratic Debye density of states which we will use in Chapter 10. We will also discuss the quadratic density of states used by Migdal and Landau to calculate the energy and heat capacity of liquid helium in Chapter 11.

8.2 Another Solution of the Telegraph Equation

69

The interplay between ω and 0 helps understand another interesting property of the GMS. The group velocity corresponding to Eq. (8.7) is: dω c2 k =q dk c2 k 2 −

(8.12) 1 4τ 2

When k → kg , dω increases without bound and becomes superluminal, consistdk ent with the idea that a tachyon, with the same dispersion relation as Eq. (8.7), moves faster than light [114]. However, this does not represent a problem for liquid physics: the ratio of ω in Eq. (8.7) and the decay rate 0 = 2τ1 in Eq. (8.6) is √ 4c2 τ 2 k 2 − 1. This goes to 0 as k → kg , implying non-propagating waves. So far, we have considered the GMS emerging in the transverse sector of liquid excitations. In Chapter 3, we saw that the sound wave with large wavelengths and small k exists in all media we considered (see Table 3.1). This implies no gap at small k, in contrast with transverse waves where the k-gap emerges at small k. This can be understood in the Maxwell–Frenkel viscoelastic theory on the basis of Eq. (7.23). According to this equation, the full bulk modulus is K = K1 + 1+K21 and iωτ includes the static part K1 . Using this K in the Fourier-transformed wave equation ω2 = c2 k 2 = Kρ k 2 gives: ! K 2 k2 (8.13) ρω2 = K1 + 1 1 + iωτ This gives a cubic equation for ω, in an interesting difference from the quadratic equation (Eq. (8.4)) which sets the gap for transverse waves. The difference is due to the static term K1 in the longitudinal case. We recover the same quadratic equation (Eq. (8.4)) if we set K1 = 0 in Eq. (8.13). Writing the solution of Eq. (8.13) for the case K1 6= 0 is cumbersome; however, we can consider two different regimes of this equation instead. In the gas-like hydrodynamic regime ωτ  1, K is given by its static term only: K = K1 . This gives the sound wave and explains why no gap emerges in the longitudinal wave at small k: the sound at small k remains a propagating mode. In the solid-like elastic regime ωτ  1, K = K1 + K2 . Therefore, the speed of sound increases at the viscoelastic crossover when K2 kicks in at high frequency. This is analogous to the fast sound or positive dispersion effect discussed in Section 8.6.

8.2 Another Solution of the Telegraph Equation We recall our earlier observation that the dispersion relations of the telegraph equation (Eq. (8.2)) were not fully explored. This is surprising considering that the

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Collective Excitations in Liquids Post-Frenkel

telegraph equation is commonly included in textbooks on mathematical physics and partial differential equations (see, for example, [104, 109, 110]). The novelty involved in our solutions in Eqs. (8.5)–(8.8) is that we considered complex ω and real k, which gave the new solution of the telegraph equation not known previously: the GMS. This consideration was necessitated by physics relevant to phonons in liquids as mentioned earlier. Textbooks on mathematical physics and partial differential equations consider the opposite case: complex k and real ω. This corresponds to decay in space (see, for example, [109, 116]). A model example of this is different from the first case we considered: it corresponds to the boundary condition where a wave is induced from one end of an elastic rod immersed in a viscous liquid. Using complex k and real ω in Eq. (8.4) and accompanying decay in space also corresponds to the propagation of electromagnetic waves in a conductor and gives the skin effect where the incoming wave loses energy and dissipates due to induced currents. This is commonly discussed in textbooks related to electrodynamics (see, for example, [116]). In this case, the wave persists at all times but decays with distance. It was this scenario that was commonly considered previously [104, 109, 110, 116]. This discussion in textbooks has surprisingly missed the GMS solution of the telegraph equation shown in the previous section. The solution of Eq. (8.4) for complex k = k1 + ik2 and real ω is: s  12  2 1 ω  k1 = √ 1+ + 1 ωτ c 2

(8.14)

and s

ω k2 = √  1 + c 2



1 ωτ

2

 12 − 1

(8.15)

resulting in the space-decaying field as: v ∝ e−k2 x ei(k1 x−ωt)

(8.16)

The dispersion relation in this case is different from Eq. (8.7): the dispersion relation (Eq. (8.14)) is gapless, which is easier to see if it is written as: 2c2 k12 ω= q 4c2 k12 +

(8.17) 1 τ2

This may come as surprising, Eq. (8.17) and Eq. (8.7) come from the same equation, namely Eq. (8.2) or (8.4), yet it was noted that different dispersion relations

8.3 The First Glimpse into Liquid Thermodynamics and Collective Modes

71

can originate from the same equation depending on boundary conditions [117] and experimental set-up [118]. Even though the dispersion relation (Eq. (8.14)) is formally different from Eq. (8.7), the GMS still follows: in the propagating regime ωτ  1, k1 = ωc and 1 k2 = 2cτ , giving the space-decaying field as: x

v ∝ e− 2cτ ei(k1 x−ωt)

(8.18)

For travelling distance x smaller than cτ , the wave propagation in Eq. (8.18) proceeds like in the absence of dissipation. However, the k-gap emerges implicitly (rather than explicitly as in Eq. (8.7)) because Eq. (8.18) implies the decay range 1 2cτ and hence no propagating plane waves with k < 2cτ , or below kg in Eq. (8.8).

8.3 The First Glimpse into Liquid Thermodynamics and Collective Modes Looking at Eq. (8.8) or Eq. (8.11), we get our first glimpse into the origin of the universal behaviour of liquid specific heat shown in Figure 2.2. The hopping frequency ωF and kg in Eq. (8.8) or (8.11) increase with temperature because τ becomes shorter. Therefore, the phase space available to collective excitations in liquids reduces with temperature. Using this result, we will calculate liquid energy and heat capacity in Chapter 10. Here, we can already attribute the experimental decrease of cv in Figure 2.2 to the reduction of the phonon phase space in liquids. The important difference with the solid state theory discussed in Chapter 4 is that the phase space for collective excitations in solids is fixed: the number of collective excitations, phonons, is always 3N. This fixes the volume of the phase space d0 as [3]: Y 1 d0 = dpα dqα (8.19) (2π ~)3N α where pα and qα are momenta and coordinates of 3N normal modes and gives the energy of the solid in Eq. (3.14). On the other hand, the number of phonons and the phonon phase space are not fixed in liquids but vary with temperature or pressure (since pressure affects τ too). We will quantify this variability in Chapter 10. How wide can the k-gap get? According to Eq. (8.8), kg increases with temperature because τ decreases. In condensed matter systems and liquids in particular, there is an upper limit to kg set by the inter-atomic distance a playing the role of the ultraviolet cut-off. Recall that τ approximately corresponds to the average time between diffusive jumps in a liquid. Hence its shortest value cannot exceed

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an elementary, Debye, vibration period, τD . Using τ ≈ τD in Eq. (8.8) and noting that cτD ≈ a gives kg ≈ 1a . This is close to the largest, Debye, k set by the zone boundary in the spherical isotropic approximation, kD : kg ≈ kD

(8.20)

Eq. (8.20) sets the largest width of the k-gap and marks the point where two transverse modes disappear from the liquid spectrum. This disappearance of transverse modes is supported by the results of molecular dynamics simulations where transverse modes are directly calculated from the transverse current correlation functions [119, 120]. kg ≈ kD corresponds to an important crossover in the behaviour of both subcritical and supercritical fluids and to the Frenkel line discussed in Chapter 14.

8.4 Symmetry of Liquids: Viscoelasticity and Elastoviscosity We have found a number of important insights from solving the telegraph equation (Eq. (8.1)). We now recall that this equation is a simplified form of Eq. (7.27), which Frenkel obtained by generalising the viscosity in the Navier–Stokes equation (Eq. (7.25)) using Eq. (7.26). In turn, Eq. (7.26) follows from Eq. (7.12) where the hydrodynamic and solid-like elastic response terms are added. We observe that this addition makes each of these two terms equally legitimate from the point of view of choosing the starting point of liquid description. Yet Frenkel used the hydrodynamic Navier–Stokes equation as a starting point (Eq. (7.25)), which is also the approach used in generalised hydrodynamics as discussed in the next section. In view of the physical equality of hydrodynamic and elastic terms in Eq. (7.12), can we obtain the telegraph equation if we instead choose an elastic constitutive equation as a starting point? This is indeed possible to do. We consider the solid state as a starting point of liquid description and the wave equation describing a non-decayed propagation of transverse waves in the solid: G

∂ 2v ∂ 2v = ρ ∂x2 ∂t2

(8.21)

We now recall that Eq. (7.12) provides a way to generalise not only viscosity η in Eq. (7.26) but also the shear modulus G. We have already generalised G in Eq. (7.15). Using Eq. (7.15) in Eq. (8.21) gives: G(1 − A−1 )

∂ 2v ∂ 2v = ρ ∂x2 ∂t2

(8.22)

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73

We now act on both sides of Eq. (8.22) with operator A in Eq. (7.13): G (A − 1)

∂ 2v ∂ 2v = ρA ∂x2 ∂t2

(8.23)

According to Eq. (7.13), A − 1 in Eq. (8.23) is τ ∂d . Using G = ρc2 as earlier ∂t and rearranging the right side of Eq. (8.23) gives:   ∂ ∂v ∂ 2v ∂ ∂ 2v = + τ (8.24) c2 τ ∂t ∂x2 ∂t ∂t ∂t2 Integrating over time and setting the integration constant to 0 gives: ∂ 2v ∂ 2 v 1 ∂v = + (8.25) ∂x2 ∂t2 τ ∂t which is identical to the telegraph equation (Eq. (8.2)). We therefore see that the same telegraph equation can be obtained by adopting the solid state as a starting point of liquid description. This implies the symmetry of liquids as far as the starting point of their description is concerned. This is more than just a mathematical exercise. It importantly shows that adopting the hydrodynamic approach to liquids as a starting point is no more fundamental than adopting the solid elastic one. This has important implications from the point of view of the general theoretical approach to liquids and the type of theories we want to use to describe them. We will see that from the point of view of describing experimental liquid properties, Eq. (7.12) can be viewed as a constitutive equation for liquids, rather than a Navier–Stokes equation modified by an elastic term. In this view, we are not bound by the need to start with the hydrodynamic description as a starting point of theories explaining liquid properties. Historically, it was the hydrodynamic approach to liquids that was adopted as a starting point of liquid description, giving rise to a separate wide area of research, “generalised hydrodynamics”, which is reviewed in the next section. One can ask, why? One possible reason is that we observe familiar liquids flow on our subjective human timescales and hence are compelled to use the hydrodynamics and Navier– Stokes equation to describe this flow. Our common “observation time” is on the order of say seconds, whereas the elastic properties of familiar low-viscous liquids such as water are manifest at much shorter timescales close to picoseconds. In other words, our subjective experience often corresponds to the hydrodynamic regime t  τ , or ωτ  1. I invite the reader to do a simple experiment that illustrates this point and changes our perception that the correct starting point of liquid description is hydrodynamics. Honey at room temperature readily flows; however, it takes up to an hour for frozen honey to flow. Frozen honey appears as a solid: we can poke frozen honey with a spoon and the honey will move with its container as a whole, like a c2

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solid. The result of my own experiment is shown in Figure 8.2. I froze honey in my freezer at −18◦ C, turned the glass upside down and put it on a slightly inclined surface so that the flow was more easily visible on one side. Figure 8.2 shows the pictures I took 10, 20 and 30 minutes later. (I repeated the experiment with more viscous manuka honey; however, no flow was observed after one week. Suspecting crystallisation, I stopped the experiment.) Frozen honey does not flow and shows seemingly solid-like elastic properties at timescales of seconds and up to a few minutes. On this basis, we may be compelled to write the solid state equations to describe this system, as opposed to using the hydrodynamic Navier–Stokes equation describing flow. However, the only difference between room-temperature and frozen honey is quantitative and is due to different τ . This quantitative difference does not warrant making a definitive choice between two qualitatively different equations, the hydrodynamic Navier– Stokes equation and the elastic stress–strain equation, as the physically justified starting point of liquid description. Both hydrodynamic and elastic properties are equally legitimate mechanisms, albeit they can appear differently to us depending on τ and our observation period. One might object to this view by saying that, from a practical perspective, a hydrodynamic Navier–Stokes equation describes the flow properties of lowviscous liquids such as liquid mercury well. The problem is that this approach ignores other key properties, namely the short-time elastic properties and the ability to support transverse solid-like phonons. The main aim of generalised hydrodynamics, discussed in the next section, is to seek to fix this problem. We therefore see that from the physical point of view, “viscoelasticity” as a term is no more constructive or informative than “elastoviscosity”, and liquids are symmetrical with regard to the starting point of their description. We can obtain the same equations governing excitations in liquids regardless of whether the starting point is hydrodynamic or solid-like elastic.

8.5 Collective Modes in Generalised Hydrodynamics “Generalized hydrodynamics” as a term refers to a number of proposals to modify hydrodynamic equations by including solid-like elastic effects and solve the resulting equations. In this general sense, we have already met the earliest version of generalised hydrodynamics in Section 7.2, where we discussed Frenkel’s book. For example, Eq. (7.27) in Section 7.2 is a generalisation of the Navier-Stokes equation to include the solid-like elastic response. In generalised hydrodynamics, one often starts with hydrodynamic equations initially applicable to low ω and k and introduces a way to extend them by including the range of large ω and k. Generalised hydrodynamics is a large field (see, for

8.5 Collective Modes in Generalised Hydrodynamics

75

Figure 8.2 Flow of frozen honey. Once frozen, honey was kept in an upside-down glass on a slightly inclined surface in the freezer. The first image was taken 10 minutes after the glass was placed upside down. No visible flow was observed at this point. The second and third images were taken 10 and 20 minutes later, respectively.

example, [13, 14, 18, 19]), which we discuss fairly briefly, emphasising key starting equations and schemes of their modification to include higher energy effects, with the main aim to offer readers a feel for the methods used and physics discussed. The hydrodynamic description starts with viewing the liquid in continuum approximation as discussed in our earlier discussion in Section 3.2 and constraining

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76

it with continuity equation and conservation laws such as energy and momentum conservation. Accounting for thermal conductivity and viscous dissipation in the Navier–Stokes equation, the system of equations can be linearised and solved. This gives several dissipative modes. Using them to calculate the density–density correlation function gives the structure factor S(k, ω) in the Landau–Placzek form which includes several Lorentzians: S(k, ω) ∝

γ −1 2χ k 2
γ ω2 + χk 2 2 1 + γ

0k 2 (ω + ck)2 + 0k 2


2 +

!

0k 2 (ω − ck)2 + 0k 2

(8.26)


2

C

where χ is thermal diffusivity, γ = Cpv , Cp and Cv are constant-pressure and constant-volume heat capacities, and dissipation 0 depends on χ, γ , viscosity and density. The first term in Eq. (8.26) corresponds to the central Rayleigh peak and thermal diffusivity mode. The second two terms correspond to the Brillouin–Mandelstam peaks and describe acoustic modes with the adiabatic speed of sound c. The ratio of the intensity of the Rayleigh peak, IR , and the Brillouin–Mandelstam peak, R IBM , is the Landau–Placzek ratio: I IBM = γ − 1. Applied originally to lightscattering experiments, Eq. (8.26) is viewed as a convenient fit to high-energy experiments probing non-hydrodynamic processes as well. The fit may include several Lorentzians or their modifications. Generalising hydrodynamic equations and extending them to large k and ω is often performed in terms of correlation functions. The hydrodynamic Navier– Stokes equation for the transverse current correlation function Jt (k, t), ∂t∂ Jt (k, t) = −νk 2 Jt (k, t), where ν is kinematic viscosity, can be solved. The Fourier transform of the solution, Jt (k, ω), has a Lorentzian form similar to Eq. (8.26): Jt (k, ω) = 2v20

νk 2 ω2 + νk 2


2

(8.27)

where ν is kinematic viscosity and v20 = Jt (k, t = 0). The generalisation of hydrodynamics can now be done in terms of the memory function Kt (k, t) defined in the equation for Jt (k, ω) as: ∂ Jt (k, ω) = −k 2 ∂t

Zt

Kt (k, t − t0 )Jt (k, t0 )dt0

(8.28)

0

where Kt (k, t−t0 ) is the shear viscosity function or the memory function for Jt (k, ω), which describes its time dependence or “memory”.

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77

Introducing J˜t (k, s) as the Laplace transform Jt (k, ω) = 2Re[J˜t (k, s)]s=iω and taking the Laplace transform of Eq. (8.28) gives: J˜t (k, s) = v20

1 s+

k 2 K˜ t (k, s)

(8.29)

The generalisation of the current correlation function introduces the dependence on k and ω by writing K˜ t (k, s) as the sum of real and imaginary parts [K˜ t (k, s)]s=iω = Kt0 (k, ω) + iKt00 (k, ω). Then, Jt (k, ω) = 2v20

k 2 Kt0 (k, ω)
2
2 ω + k 2 Kt00 (k, ω) + k 2 Kt0 (k, ω)

(8.30)

resulting in the generalised hydrodynamic description of the transverse current correlation function with a resonance spectrum. The next important step is related to making the choice of time dependence of Kt (k, t). This is often postulated to have the form:   t Kt (k, t) = Kt (k, 0) exp − (8.31) τ (k) Eq. (8.31) decays with relaxation time τ . We recognise that this is essentially the same behaviour described by Eq. (7.10), except that the postulated form also assumes k-dependence of τ . In generalised hydrodynamics, Eq. (8.31) is used not only for Kt but also for several other types of correlation and memory functions. These may include modifications such as adding more exponentials with different decay times in order to improve the fit to experimental or simulation data. Mode-coupling schemes consider correlation functions for density and current density and factorise higher order correlation functions by expressing them as the product of two time correlation functions with coupling coefficients in the form of static correlation functions. This gives a better agreement for the relaxation function than the single exponential decay model. Neglecting the k-dependence of τ for the moment and taking the Laplace transforms of Eq. (8.31), we find Kt0 (k, ω) and Kt00 (k, ω). Using them in Eq. (8.30) gives Jt (k, ω) as [14]: Jt (k, ω) ∝

1 ω2 − k 2 Kt (k, 0) −

1 2τ 2



2

(8.32) + f (τ , Kt (k, 0))

where f is a non-essential function of τ and Kt (k, 0). The resonance frequency in Eq. (8.32) corresponds to the propagation of shear modes provided k 2 Kt (k, 0) > 2τ1 2 . We recognise this condition to be equivalent to

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the emergence of the gapped momentum state in Eqs. (8.7) and (8.8), which we derived as a solution of the telegraph equation. We observe that the route to this result from the telegraph equation (Eq. (8.2)) was considerably quicker and more physically transparent than the derivation summarised in Eqs. (8.27)–(8.32) in this section. Similar expressions can be derived for the longitudinal current correlation function, which also includes a static time-independent term that does not decay. This term corresponds to a non-zero bulk modulus, which gives propagating longitudinal waves in the hydrodynamic regime and is physically equivalent to Eq. (7.23) written by Frenkel. An alternative approach used in generalised hydrodynamics involves making a phenomenological assumption that a dynamical variable in the liquid is described by the generalised Langevin equation: ∂a(t) + ia(t) + ∂t

Zt

a(t0 )K(t − t0 )dt0 = f (t)

(8.33)

0

where the first two terms reflect the possibility of propagating modes, the third term plays the role of friction with the memory function K, and f is the random force. This approach proceeds by treating a(t) not as a single variable but as a collection of variables of choice. In this case, a(t) becomes a vector including, in its simplest forms, conserved density, current density and energy variables. These variables are further generalised to include their dependence on k. This gives a set of coupled equations that can be solved in the matrix form. The set of dynamical variables can be expanded to include, for example, the stress tensor and heat currents. In this case, the generalised viscosity is found to have the same exponential decay as in Eq. (8.31) once the stress tensor is explicitly introduced as a dynamical variable, an assumption is made regarding the stress correlation function and a number of approximations are made. Then, similar viscoelastic effects are found as in the approach based on current correlation functions [14]. The propagation of shear and longitudinal modes is also discussed in the modecoupling theories mentioned earlier. This approach aims to take a more general approach in the following sense. Considering that correlation functions are related to density and current density correlators, the theory represents K˜ t (k, s) in Eq. (8.29) by the second-order memory functions Mt (k, t) and Ml (k, t) for transverse and longitudinal currents, so that the transverse function J˜t (k, s) and longitudinal function J˜l (k, s) acquire the forms of damped oscillators. J˜l (k, s) differs from J˜t (k, s) by the presence of a non-zero static term, giving a finite static restoring force for the longitudinal mode. As in the earlier discussion (see Eq. (7.23) in Section 7.2), this gives propagating longitudinal modes in the hydrodynamic regime. Rather than

8.6 Experimental Evidence of Collective Excitations in Liquids

79

postulating the relaxation functions Mt (k, t) and Ml (k, t) as in Eq. (8.31), the modecoupling theory introduces higher order correlation functions and approximates them by the products of two-time correlation functions. Memory functions can then be calculated using the input from molecular dynamics simulations producing static correlation functions and other required parameters. The onset of shear wave propagation can be related to certain shoulder-like features in the calculated memory function for simple systems [14]. The amount of ongoing research in generalised hydrodynamics has markedly decreased compared with several decades ago [14]. Interestingly, the strong steer towards going beyond this approach came from the experimental, and not theoretical, community. This happened after the solid-like properties of liquids were experimentally ascertained and problems with the theory became apparent, including its phenomenological and empirical nature [121, 122]. Some generalised hydrodynamics results continue to be used for fitting the experimental spectra, including the form of the dynamic structure factor from inelastic scattering experiments.

8.6 Experimental Evidence of Collective Excitations in Liquids In this section, we will review the experimental evidence showing the propagation of collective modes in liquids, including longitudinal and transverse waves. We will also discuss more recent evidence showing the development of the GMS in the transverse sector. We discussed this state at some length on the basis of the telegraph equation in Section 8.1. The experimental evidence supporting the propagation of phonons in liquids at high ω and k includes inelastic X-ray, neutron and Brillouin scattering experiments. Most of this evidence is fairly recent and has benefited from the deployment of powerful synchrotron sources of X-rays. Early experiments detected the presence of high-frequency propagating modes and mapped dispersion curves, which bore a striking resemblance to those in solids [123]. This and similar results were generated at temperatures around melting. The measurements were later extended to high temperatures considerably above the melting point, confirming the same result. It is now well established that liquids sustain propagating phonons extending to wavelengths comparable to inter-atomic separations in solids [121–131]. This is a remarkable fact asserting the similarity of vibrational states in liquids and solids. In this Chapter, we have seen that volume-conserving transverse phonons in liquids differ from those in solids because they become gapped. For this reason, we emphasise transverse phonons in our discussion of experimental results. Initially detected in viscous liquids (see, for example, [132–134]), transverse modes

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were later found in low-viscous liquids on the basis of positive sound dispersion [121, 124, 125, 127, 135]. This effect, sometimes referred to as the “fast sound”, represents the increase of the longitudinal speed of sound over its hydrodynamic value. Predicted by Frenkel [9], this effect corresponds to the increase of the sound velocity from its hydrodynamic value: s B c= (8.34) ρ to the solid-like elastic value: s c=

B + 34 G ρ

(8.35)

where B and G are bulk and shear moduli, respectively. We have previously encountered Eq. (8.35) in Section 3.2, where we discussed the wave equations in elasticity theory. The same idea is also present in Eq. (7.23) for the frequency-dependent bulk modulus in Section 7B: an additional term for the bulk modulus, K2 , is absent when ωτ  1 but kicks in at high frequency in the regime ωτ  1. In later work, transverse waves in low-viscous liquids were directly measured in the form of distinct dispersion branches [122, 128–131]. This work includes inelastic scattering experiments where the scattered X-ray or neutron intensity is related to the dynamic structure factor S(k, ω) measured as a function of the wavevector and frequency. For a fixed k, the scattered intensity has a peak as a function of ω. This provides a point (k, ω) on the dispersion curve. In liquids, the scattered intensity shows a well-defined longitudinal peak as a function of ω for a given k. The transverse signal is relatively weak and is often seen as a shoulder in the scattered intensity. Both longitudinal and transverse peaks are fitted to find a peak frequency at a given k. This gives longitudinal and transverse dispersion curves ω(k). In Figure 8.3, we show the dispersion curves measured in inelastic X-ray scattering experiments in liquid Na, Ga and Sn, together with SiO2 glass for comparison. In Figure 8.4, we show the dispersion curves measured in liquid Fe, Cu and Zn using the experimental set-up to study liquids with high melting points. We observe a striking similarity between liquids and solids (both crystals and glasses) in terms of longitudinal and transverse dispersion curves. This similarity is evident by comparing the liquid dispersion curves in Figures 8.3 and 8.4 and the dispersion curves in solids in Figure 3.1 in Section 3.1. Like in solids (see the discussion in Section 3.1), the phonon dispersion relations in liquids start as a linear dependence at small k and curve over close to the zone boundary. If we dropped the

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81

Figure 8.3 Experimental dispersion curves. (a) Longitudinal (bullets) and transverse (diamonds) dispersion curves in SiO2 glass [136]. (b) Longitudinal (filled bullets) and transverse (filled diamonds) phonons in liquid Na at 393 K. Open bullets and diamonds correspond to longitudinal and transverse phonons in polycrystalline Na, and dashed lines correspond to longitudinal and transverse branches along the [111] direction in Na single crystal [122]. (c) Longitudinal (bullets) and transverse (diamonds) phonons in liquid Ga at 295 K and 0.8 GPa. The bullets and diamonds are bracketed by the highest and lowest frequency branches measured in bulk crystalline β-Ga along high symmetry directions [130]. (d) Longitudinal dispersion curves from different experiments (bullets) and transverse dispersion curves (diamonds) in liquid Sn [131] at 533 K. Triangles are results from ab initio simulations. Dispersion curves in Na and Ga are reported in reduced zone units.

labels and captions in Figure 3.1 in and in Figures 8.3 and 8.4, a reader wouldn’t be able to tell in most cases which figure corresponded to solids and which to liquids. This point is further evidenced by the closeness of the dispersion curves in liquid Na and Ga to their parent solids, both bulk crystals and polycrystals, in Figure 8.3.

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Collective Excitations in Liquids Post-Frenkel

Figure 8.4 Experimental longitudinal (bullets) and transverse (diamonds) dispersion curves in (a) liquid Fe at 1823 K, (b) liquid Cu at 1423 K and (c) liquid Zn at 713 K [129]. Dashed lines indicate the slope corresponding to the hydrodynamic sound in the limit of low k.

The data exemplified by Figures 8.3 and 8.4 constitute important experimental evidence regarding collective excitations in liquids: despite topological and dynamical disorder, longitudinal and transverse phonons exist in liquids in a wide range of k and up to the largest k corresponding to inter-atomic separations, as is the case in solids. Finally, we observe a good agreement between the experimental dispersion curves and those calculated in molecular dynamics simulations in Fig. 8.3. In the next section, we will discuss the results from molecular dynamics simulations in more detail. We now recall the central result from Section 8.1, the emergence of the GMS in the transverse sector of the liquid spectrum given (see Eqs. (8.7) and (8.8)). The experimental evidence for this is fairly scarce. The first experimental confirmation of the GMS was reported not in liquids but in dusty plasma [137] where particles were imaged by a camera. This was possible because, in contrast with liquids, this system has long inter-particle distances and low frequencies: typical inter-particle distance and characteristic frequencies are of the order of 1 mm and

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83

Figure 8.5 Experimental transverse dispersion relations in dusty plasma in the solid crystalline state (bullets) and the liquid state (diamonds). a = 0.325 mm. The emergence of the gap in k-space is seen in the liquid state. The data are from [137].

10 s−1 , respectively. Dispersion relations were then derived from calculated transverse current correlation functions using the imaged trajectories of dusty plasma particles. Figure 8.5 shows a gapless transverse dispersion relation in the solid crystalline state of dusty plasma and the emergence of the gap in the liquid state. Theoretically, the GMS in plasma is rationalised using generalised hydrodynamics, which is very similar to what we discussed in Section 8.5 for liquids (see, for example, [138–140]). Diaw and Murillo [139] trace the generalised hydrodynamics approach used to discuss plasma back to Frenkel’s viscoelastic theory of liquids. Around the time of writing this book, the first experimental evidence for the GMS was reported in liquid Sb [141]. Figure 8.6 shows longitudinal and transverse branches measured in liquid Sb by inelastic X-ray scattering experiments at 973 K. Although the transverse points do not extend all the way to zero frequency at small k, a reasonable extrapolation of low-lying points strongly indicates the presence of the GMS. A similar effect is seen in liquid Bi at 573 K, although the number of low-k points is smaller than liquid Sb. There are also several indirect pieces of experimental evidence supporting the GMS in liquids. Recall the positive sound dispersion effect where the speed of q B sound increases from its hydrodynamic value c = ρ to the solid-like elastic value r c=

B+ 34 G ρ

and where the ability to sustain transverse excitations gives non-zero

G. According to the discussion in Section 8.1, these transverse excitations start kicking in for k > kg , implying positive sound dispersion at these k-points. This further implies that positive sound dispersion should disappear with temperature

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Collective Excitations in Liquids Post-Frenkel

Figure 8.6 Experimental dispersion relations in liquid Sb (top) and Bi (bottom). Bullets and diamonds correspond to longitudinal and transverse excitations, respectively. The data are from [141].

starting from small k because the k-gap increases with temperature. This is confirmed experimentally [124]: inelastic X-ray experiments in liquid Na show that positive sound dispersion is present in a wide range of k at low temperature. As temperature increases, positive sound dispersion disappears starting from small k, in agreement with the picture of the GMS. At high temperature, positive sound dispersion is present at large k only. Another indirect piece of experimental evidence for GMS comes from lowfrequency shear elasticity of liquids at small scale. According to Eq. (8.7), the frequency at which a liquid supports shear stress can be small, provided k is close to kg . This implies that small systems are able to support shear stress at low frequency. This has been ascertained experimentally [142, 143]. Before concluding this section, we make two remarks related to the similarity between the vibrational properties of liquids and solids and the universality of

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these properties. First, we have observed that, apart from the GMS developing in liquids in the transverse spectrum, the dispersion relations in liquids and their solid counterparts, both amorphous and crystalline, are very similar. Interestingly, the similarity (and the lack thereof) between vibrations in disordered glasses and their parent crystals have also been widely discussed. The widely discussed “Boson” peak in the low-frequency range has long been thought to be present in glasses only but not in crystals and to originate from disorder. This has led to theories attributing the Boson peak to non-crystalline effects (see [144] for a review). However, after many years of research, it transpired that similar vibrational features are present in crystals as well, provided that glasses and crystals have a similar density [145, 146]. This seems to imply that these properties are generic and are unrelated to disorder [147]. Second, we observe the “roton” minimum in the dispersion curves in Figure 8.6 as well as in Fig. 8.3 for liquid Na. The minimum follows the decrease of the dispersion curve and its levelling off around 20–5 nm−1 . The roton minimum was originally introduced to discuss the anomalous properties of liquid He [3] and was viewed as being intrinsic to liquid He and related to quantum effects [148]. On the basis of recent inelastic scattering data, we see that the roton minimum is universally seen in classical liquids and is probably related to the washed-out higher Brillouin zones in their parent crystals [149]. Balibar [149, 150] provides insightful reviews of the history of helium research. We will discuss liquid helium and its thermodynamic properties in Chapter 11. By way of summary, experimental evidence implies the propagation of longitudinal and transverse waves in liquids with wavelengths extending to inter-atomic separation like in solids. The GMS in the transverse sector is evidenced to a smaller degree: the experimental data such as that in Figure 8.6 are recent and not plentiful. In Chapter 9, we will show how the GMSs are seen in liquids in molecular dynamics simulations at a fine level of detail. Experimentally, it would be interesting and important to observe similar details of emergence and evolution of GMS with temperature. It would also be important to see the GMS effect in several different liquids. Our final point in this section is related to theoretical approaches to liquids. The experimental evidence for solid-like dispersion curves and propagating phonons in liquids such as that shown in Figures 8.3, 8.4 and 8.6 appeared much later than the phonon dispersion curves in solids shown in Figure 3.1. The examples of phonon dispersion curves in solids in Figure 3.1 and liquids in Figures 8.3, 8.4 and 8.6 are from papers in the early 1960s and 2010s, respectively, with a gap of at least 50 years and probably longer. During this gap, a large proportion of liquid theories were developed that were unrelated to phonons. We discussed these theories in Chapter 6. In Chapter 13, we will see that the phonon approach to liquids

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was largely dormant during this period, even though this approach was first discussed by Sommerfeld and Brillouin around the same time when the Einstein and Debye theories of solids were published. It is likely that the liquid theory may have taken a different route had the experimental phonon dispersion curves in liquids come about earlier and closer to the time when the same data in solids became available.

8.7 Decay of Excitations in Liquids and Solids We have encountered several results of wave decay in liquids so far, and it is now a good time to take stock and discuss more details and mechanisms of wave decay in liquids and solids. The cases we have encountered are as follows: 1. Over-damped transverse waves in the purely hydrodynamic description, with the decay of the wave amplitude by a factor of e2π ≈ 535 over one spatial period (Section 3.2). 2. Propagating transverse waves in the regime ωτ  1, with the propagation range d ≈ λ · ωτ , and over-damped transverse waves in the regime ωτ  1, with the λ propagation range d = 2π and the same result as in point (1) (Section 7.2). 3. Propagating longitudinal waves in the regimes ωτ  1 and ωτ  1, with the λ propagation ranges of about d = λ · ωτ and d = ωτ , respectively (Section 7.2). 4. The solutions of the telegraph equation giving propagating transverse waves for √ ω > 0 = 2τ1 and k > kg 2 and over-damped transverse waves for k close to kg where ω  0 (Section 8.1). These results, points (1–4), are due to a finite viscosity and relaxation time τ in liquids. However, there are other mechanisms of wave decay, which we will discuss in the following. The first observation we make is related to collective excitations in solids, both crystals and glasses. Quite generally, let’s consider a system of particles where a state of thermal equilibrium has been achieved so that the system temperature can be introduced. Let us now isolate this system from its environment and consider the system at given thermodynamic parameters such as temperature and pressure. The energy of such a closed system is conserved. Because the energy is conserved, there are no channels for energy dissipation. Consequently, the eigenstates of this system, its collective excitations, do not decay. This does not rely on the existence of crystallinity or plane waves: for example, we can consider a topologically disordered system, glass, and assume that interactions are well described by the harmonic quadratic interaction. In this chapter, we have seen that this system has 3N normal modes with frequencies ωi , the linear superposition of which gives the

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collective excitations in the disordered system. Being linear combinations of eigenstates, these collective excitations clearly do not decay. This result does not rely on the existence of order or disorder in the system. We can consider non-linear interactions that are not quadratic. We have seen an example of this in Chapter 5. In this case, the eigenstates cannot be represented by simple harmonic functions, and may acquire a complicated mathematical description. The resulting collective excitations can no longer be represented by a superposition of non-interacting normal modes. However, even in this case, these eigenstates do not decay in our conservative system. If we now consider propagating waves, and in particular the plane waves in the form given by Eq. (3.15), periodicity becomes important. In a crystal, collective excitations can be represented by plane waves as discussed in Chapter 3, with k corresponding to the entire possible range of the wavelengths λ, from the size of the crystal to the shortest inter-atomic separation. Inter-atomic interactions in real crystals are not entirely harmonic, and we have discussed the effects of anharmonicity in Section 4.2. As a result, the collective excitations are no longer represented by a set of harmonic plane waves. To describe the effects of anharmonicity, multiphonon processes are introduced, where, for example, one phonon decays into two, and where plane waves have a finite lifetime. Apart from anharmonicity, the reasons for finite phonon lifetimes can also include imperfections of the crystal lattice: defects, local strains and so on. The decay of phonons in real crystals has consequences for transport properties including, for example, finite thermal conductivity. The decay and lifetimes of phonons, magnons and other collective excitations are routinely measured experimentally using inelastic scattering. The absence of periodicity in disordered media (glasses, liquids, spin glasses and so on) precludes the representation of collective excitations in terms of harmonic plane waves with a set of k. If we continue to use plane waves as an approximation, we encounter decay and finite lifetimes. These turn out to be small for large wavelengths where the continuum approximation applies. As mentioned in Section 3.2, plane waves are the eigenstate solutions in the continuum approximation. For smaller wavelengths, decay of phonons increases, and is commonly seen as widening of the intensity of the dynamic structure factor. The difference between this model and the ordered system is the absence of a well-defined relationship between ω and k inasmuch as k is not defined well due to the lack of periodicity, especially at small wavelengths. Even though the system is disordered, we can still rigorously introduce the vibrational density of states g(ω) because there is a well-defined set of normal frequencies ωi in the harmonic approximation. We therefore encounter different cases and types of dissipation. Three examples are:

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1. Non-plane wave eigenstates in disordered harmonic systems (e.g. harmonic glass): non-decayed. 2. Plane wave eigenstates in ordered harmonic systems: non-decayed. 3. Plane waves in disordered systems: decayed. Let’s now inject a set of plane waves with well-defined energy and k into our disordered system and follow its propagation. This set will transform into excitations with different frequencies ωi travelling in different directions. This gives the decay of plane waves and sets the decay time, the time during which the eigenstates can be represented by plane harmonic waves. In disordered systems at large wavelengths where the continuum approximation works well, this time can be quite long and, consequently, decay and damping are small. At wavelengths smaller than the medium-range order where the details of structure become important (say about 10–20 inter-atomic distances), this time is short and damping is large. It is not possible to calculate the propagation range analytically so we must rely on experimental or modelling results. Both indicate that the high-frequency plane waves at 1 THz and above become fairly localised in the disordered glass structure so that the wavelength becomes comparable to the propagation range (see, for example, [151, 152]). In crystals, high-frequency plane waves can become localised as well, due to anharmonicity, defects and so on, as mentioned earlier. For example, the lifetimes of 9 THz phonons in Si and Al are in the very small range: 0.3–0.4 ps in Al and 1 ps in Si at room temperature already [153]. Similar lifetimes are seen for 15 THz phonons in Si and GaN on the basis of time-resolved Raman scattering experiments [154, 155]. In hybrid perovksites, the lifetime of high-frequency phonons is similarly in the picosecond range [156]. Whereas the localisation of high-frequency plane waves in solids (both crystals and glasses) is important for transport and other properties as mentioned earlier, it bears no significance for their thermodynamic properties. We know that the hightemperature specific heat of classical solids, both crystals or glasses, is universally close to the Dulong–Petit result c = 3 (recall that crystallinity is not needed to obtain 3N modes in disordered harmonic systems and the accompanying result c = 3, as discussed in Section 3.1). Here, all 3N modes contribute to the system energy, and in the Debye model with the density of states g(ω) ∝ ω2 , it is exactly these high-frequency modes that make the largest contribution to energy. In liquids, we might expect the phonon lifetimes and propagation ranges to decrease further than their parent solids due to at least two factors. First, the wave propagation range generally decreases at higher temperature where liquids exist. Second, finite viscosity and τ limit the propagation range as we have seen in points (1–4) at the beginning of this section. Similarly to solids, we can’t analytically predict the lifetimes and propagation ranges in liquids and have to turn to experiments.

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A fairly universal result emerges from the experiments discussed in Section 8.6. The lifetime of the highest frequency transverse waves in liquid Sn in Fig. 8.3 is about 0.7 ps, corresponding to ω > 0 and the propagation range of about 10 Å [131], comparable to the cage size shown in Figure 7.1. In liquid Fe, Cu and Zn shown in Fig. 8.4, the lifetimes of high-frequency modes are in a similar range, 0.3–0.6 ps, for both transverse and longitudinal waves, corresponding to about 9 Å of propagation range [129]. The condition ω > 0 also applies to the lower frequency phonons, both transverse and longitudinal. The same is seen in liquid Ga, both at room and high pressure [130]. Very similar numbers can be deduced from frequencies and lifetimes calculated in molecular dynamics simulations [119], as discussed in the next chapter. We therefore see that the experimental lifetimes and propagation ranges of phonons in liquids are similar to those in solids, including glasses and crystals. In solids, each of these phonons contributes T to the classical energy to give the high-temperature value of the solid’s specific heat c = 3. Therefore, the experimentally measured phonons in liquids contribute to the liquid energy to the same extent as in solids. More generally, let’s consider an excitation characterised by a pair of numbers (ω, 0) that are close in different systems. This excitation makes the same contribution to the system energy, regardless of whether the system is a crystal, glass, liquid or gas. In Chapter 10, we will calculate the liquid energy on the basis of phonons.

9 Molecular Dynamics Simulations

In Chapters 5–8, we discussed theory and experiments related to collective excitations and thermodynamic properties of liquids. Since the development of computers and algorithms, computer modelling and molecular dynamics (MD) simulations in particular have become a third distinct line of enquiry, alongside experiment and theory. MD simulations are now widely used to test theories, interpret experiments, predict new effects or even substitute experiments in conditions that are not feasible. What is interesting is that MD simulations were stimulated by the problem of liquids. The classic MD book [42] is called Computer Simulation of Liquids for a reason. This reason is in fact a collection of related reasons discussed in the Preface, Introduction and Chapters 5 and 6. As discussed by Allen and Tildesley in their textbook [42], an important motivation for computer simulations as such was the need to understand liquid’s because they are not amenable to traditional theoretical approaches such as perturbation theories, virial expansions and so on. Indeed, pioneering MD work was aimed specifically at liquids [49–52, 157–159]. We have already encountered our first example of how MD simulations provide insights into liquid dynamics in Chapter 7. A simple yet powerful plot such as that in Figure 7.2 confirms and illustrates the Frenkel picture of molecular motion in liquids, where a particle undergoes many oscillations around quasi-equilibrium positions, followed by flow-enabling jumps to new positions. We will see that taking full advantage of such microscopic dynamics provides the key to understanding thermodynamics and excitations in liquids. We will not delve into the details of the MD method and instead we refer the reader to comprehensive texts [42, 160]. At the heart of the method is the iterative solving of the equations of motion for a system of interacting particles. The interaction potential is specified at the beginning and is often fitted to experimental properties such as structure, energy and its derivatives. Alternatively, the potential can be calculated by solving quantum-mechanical equations. The most 90

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important outcome of an MD simulation is the set of particle coordinates, velocities and forces at each time step, providing the full trajectory of the system in phase space. MD simulations of liquids could be the subject of more than one book. This is especially so considering that simulations of liquids started at about the same time that the MD method became available. Having started with modelling the hard-sphere systems mentioned in Chapter 2, these simulations soon involved more realistic liquids [157–159]. Here, we will focus on the set of MD data related to the main topic of this book: thermodynamic properties and cv in particular, as well as collective excitations. Using the total energy E calculated in MD simulations, cv can be calculated as dE cv = N1 dT , where N is the number of particles. cv , as well as other quantities, can be also calculated from fluctuations of other properties. For example, calculating the fluctuations of kinetic energy K in the constant-energy ensemble gives cv from the following equation [42]:   3 2 2 3kB 2 2 hK i − hKi = NkB T 1 − (9.1) 2 2cv Both methods of calculating cv give similar results, as is seen in Figure 9.1. Similarly to the experimental behaviour discussed in Chapter 2, we observe a decrease of cv from about 3 to 2 and below. While this gives us confidence in the MD method, it does not help us to understand the physical origin of the calculated values of cv or its temperature dependence. To do so, we need a physical model. We have already met this problem in Chapter 6, where we talked about the problems involved in calculating liquid energy and cv from interactions and correlation functions. We now come to the second set of MD results relevant to the scope of this book: collective excitations. We have discussed inelastic scattering experiments in liquids in Section 8.6, which generate the data for dispersion curves. The dynamic structure factor and dispersion curves can be also calculated in MD simulations. These simulations serve to interpret the experimental data and add to them in several ways. Whereas the longitudinal signal and associated dispersion curves are usually well resolved, the transverse signal in the experimental scattered intensity is fairly weak. The MD simulations are particularly useful for understanding these excitations in two ways. First, the transverse modes can be calculated directly and without the need to decompose the scattered intensity into the longitudinal and transverse parts. Second, there is little experimental evidence for the gapped momentum state (GMS) predicted by the telegraph equation in Section 8.1, and MD simulations can be used to address this effect in detail. As mentioned in Section 8.6, the transverse signal often needs to be fitted to find the peak frequency of transverse phonons at a given wavevector. In MD

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Figure 9.1 Specific heat of argon from MD data at density 1.9 g/cm3 . The solid line is the temperature derivative of energy. Diamonds correspond to cv calculated using the fluctuation equation (Eq. (9.1)). The data are from [77].

simulations, the transverse waves can instead be directly calculated from the transverse current correlation function C(k, t): k2 hJx (−k, t)Jx (k, 0)i N N X
J (k, t) = k × vj (t) exp −ikrj (t) C(k, t) =

(9.2)

j=1

where J (k, t) is the transverse current, v is particle velocity and k is the wavevector along the z-axis. Using positions and velocities generated by MD simulations, the current correlation function is readily calculated. The Fourier transform of C(k, t) gives the intensity map C(k, ω), which can be directly compared to the dynamic structure factor measured in inelastic scattering experiments. The calculated intensity maps for simulated Ar and CO2 are shown in Figure 9.2 at different temperatures. The dispersion curves corresponding to the maxima of C(k, ω) are shown in Figure 9.3. Figures 9.2 and 9.3 include both subcritical liquids and supercritical fluids. We will discuss the supercritical state and its collective excitations in Chapter 14. Here,

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Figure 9.2 Transverse intensity maps of C(k, ω) for supercritical Ar at 250 K (top), 350 K (middle) and 450 K (bottom) and supercritical CO2 at 300 K (top), 400 K (middle) and 500 K (bottom). The maximal intensity corresponds to the middle part of each graph and reduces away from it towards the darker background on either side. The data are adapted from [119] with permission from the American Physical Society.

the only relevant difference between the supercritical and subcritical states is that in the former case we can explore a considerably wider temperature range without the boiling phase transition intervening.

Figure 9.3 Transverse dispersion curves of supercritical Ar at 200–500 K and 550 K (top), subcritical liquid Ar at 85–120 K (middle) and supercritical CO2 at 300– 600 K (bottom). The temperature increment is 30 K, 5 K and 30 K for supercritical Ar, subcritical liquid Ar and supercritical CO2 , respectively. The deviation from the linearity (curving over) of dispersion curves at large k is related to probing the effects comparable to inter-atomic separations (in the solid this corresponds to the curving over of ω ∝ sin ck at large k). This effect is not accounted for in the theory leading to the gap in Section 8.1 and describing the k-gap in the linear part of the dispersion relation. We show k in the range slightly extending the first pseudo-zone boundary (FPZB) at low temperature. As volume and inter-atomic separation increase with temperature, the FPZB shrinks, resulting in the decrease of ω ∝ sin ck at large k beyond the FPZB. This effect is unrelated to the k-gap. The data are reproduced from [119] with permission from the American Physical Society.

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Figures 9.2 and 9.3 make two points in support of the theoretical and experimental discussion in Sections 8.1 and 8.6. The first observation is the clear presence of the transverse dispersion relation at low temperature, which looks very similar to that seen in solids. The second observation is related to the emergence of the GMS at higher temperature. This effect was noted in early MD studies: the calculated peaks of transverse current correlation functions were seen at large k but not at low [161]. The emergence of the gap was subsequently confirmed in several other MD simulations. In Figures 9.2 and 9.3, we see that the gap in k-space develops and increases with temperature. This confirms the theoretical prediction in Eq. (8.8): higher temperature decreases τ and results in the larger gap in momentum space. Plotted as a function of τ1 , kg shows an approximately linear behaviour as predicted by Eq. (8.8) [119]. Later MD simulations analysed the emergence and evolution of the GMS and found essentially the same effect [162]. The emergence of kg has also been ascertained in hard disk models used to understand the rigidity transition in glasses and the liquid–glass transition [163]. Here, the k-gap increases at low packing fractions. Because rigidity is related to propagating shear modes, the value of kg was interestingly proposed as an order parameter quantifying the rigidity transition. The MD results make an important point: the phase space for collective excitations reduces with temperature, importantly confirming the theoretical prediction of Eqs. (8.7) and (8.8). At the Frenkel line discussed in Chapter 14, the transverse waves disappear. This disappearance is supported by the results of MD simulations where transverse modes are calculated from the transverse current correlation functions [120]. This insight will be used to develop the theory of liquid thermodynamics in both liquid-like and gas-like regimes in Chapters 10 and 14. The reduction of the phase space with temperature is also seen in a different kind of MD simulation. Eq. (3.10) in Section 3.1 can be diagonalised, once the coefficients kik are known. These coefficients can be found from the second derivatives of energy of the liquid structure generated in MD simulations. Then, the normal frequencies give the spectrum of excitations in the system. In solids, these frequencies are all real. In liquids, some frequencies become imaginary, corresponding to unstable configurations related to the diffusive motion of liquid particles. In this analysis, the number of real and imaginary frequencies decrease and increases with temperature, respectively (see, for example, [164]). This is another way of illustrating how the phase space for phonon excitations in liquids reduces with temperature.

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We have so far discussed MD simulations related to complementing, understanding and interpreting the experimental data related to cv and collective excitations in liquids. We will return to MD simulations in Chapter 14, where we will find a different use for these simulations: generating data at very high temperature and pressure where no experimental data currently exist.

10 Liquid Energy and Heat Capacity in the Liquid Theory Based on Collective Excitations

10.1 Liquid Energy and Heat Capacity We now possess all of the ingredients required to calculate the liquid energy on the basis of collective excitations. In Section 7.2, we saw that longitudinal waves continue propagating in liquids. Flow-enabling particle jumps in liquids only result in different decay properties in two regimes: ωτ > 1 and ωτ < 1. On the other hand, particle jumps have a more dramatic effect on transverse waves: these waves become gapped. In the Frenkel theory discussed in Section 7.2, transverse waves propagate above the frequency gap ωF = τ1 . In Chapter 8, we saw that transverse waves exist above the threshold value in momentum space k√ g (in terms of the existence of the peak frequency) and are propagating above kg 2, approximately corresponding to the same frequency gap ωF . In either case, transverse excitations in liquids have a gap that increases with temperature because τ decreases. As a result, the volume of the phase space available to collective excitations reduces with temperature. We recall our discussion in Section 8.3: the phase space is fixed in solids because the number of phonons is 3N at any temperature. In an important difference from solids, the phonon phase space in liquids reduces with temperature. This reduction is the key to calculating the liquid energy. To make this explicit, we write the solid energy Es as: Es = Kl + Pl + Kt + Pt

(10.1)

where K and P are kinetic and potential parts of the phonon energy and subscripts l and t refer to longitudinal and transverse excitations. Eq. (10.1) can be rewritten as: Es = Kl + Pl + Kt (ω < ωF ) + Pt (ω < ωF ) + Kt (ω > ωF ) + Pt (ω > ωF )

(10.2) 97

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where ω < ωF and ω > ωF refers to parts below and above the inter-cage hopping frequency ωF in each term, respectively. In liquids, diffusive particle jumps have two implications for the system energy. First, particle jumps result in the reduction of the phase space and the disappearance of transverse excitations with frequency ω < ωF as discussed earlier. This implies no restoring forces for excitations with frequency ω < ωF and, as a result, Pt (ω < ωF ) becomes 0. Second, particle jumps contribute the energy of diffusing particles Ed , which can be written as the sum of kinetic and potential parts Kd + Pd , where Pd is related to the energy of interaction of diffusing particles with other parts of the system, which is not already contained in the potential energy of the phonon terms, Pl and Pt (ω > ω F ). The sum of Kd and the rest of the kinetic terms in Eq. (10.2) gives the total kinetic energy of the system, K = 32 NT in the classical case, because this energy depends only on temperature and not on the way in which the motion partitions into the oscillatory phonon motion and diffusive jump motion. Then, E = K + Pl + Pt (ω > ωF ) + Pd

(10.3)

Pd is small compared with the other terms in Eq. (10.3). Let’s consider a gradual transition from solid to liquid dynamics during, for example, the softening of glass. Instead of pure oscillations, atoms in the system start jumping between quasi-equilibrium positions with frequency ωF as illustrated in Figure 10.1. This large-amplitude diffusive motion can be viewed as a “slipping” motion due to the smallness of restoring forces at frequency ω < ωF . This implies that the potential energy at frequency ω < ωF is small compared with the potential energy at frequency ω > ωF , where the restoring forces are strong and enable wave propagation instead of slipping: Pt (ω < ωF )  Pt (ω > ωF ). The nature of inter-atomic interactions involved in the diffusive slipping motion is similar to that in the solid oscillatory case, and the only difference with the oscillatory case is the larger amplitude of the slipping motion. Hence Pd ≈ Pt (ω < ωF ). Combining this with Pt (ω < ωF )  Pt (ω > ωF ), Pd  Pt (ω > ωF ) follows. Rephrasing this, were Pd large and comparable to Pt (ω > ωF ), strong restoring forces at low frequency would result, and would lead to the existence of low-frequency vibrations instead of diffusion. We also note that because Pl ≈ Pt , Pd  Pt (ω > ωF ) gives Pd  Pl , further implying that Pd can be omitted in Eq. (10.1). Then, E = K + Pl + Pt (ω > ωF )

(10.4)

Neglecting small Pd in Eq. (10.3) is the only approximation in the theory; all subsequent transformations serve to make the calculations convenient only. We note that in the viscous regime τ  τD , the justification for the smallness of Pd in the two previous paragraphs becomes unnecessary. Indeed, a rigorous statistical

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Figure 10.1 Oscillatory particle motion in solids (top) becomes diffusive motion in liquids (bottom) at low frequency. This process can be thought of as a “slipping” of oscillatory phonon motion due to the smallness of restoring forces at low frequency.

mechanical argument shows that the total energy of diffusing atoms (the sum of their kinetic and potential energy) can be ignored to a very good approximation if τ  τD . This will be discussed Chapter 16. Eq. (10.4) can be rewritten using the virial theorem Pl = E2l and Pt (ω > ω F ) = Et (ω>ωF ) (here, P and E refer to their average values) and by additionally noting 2 that the total kinetic energy K is equal to the value of the kinetic energy of a solid and can therefore be written, using the virial theorem, as the sum of kinetic terms related to longitudinal and transverse waves: K = E2l + E2t . This gives: E = El +

Et (ω > ωF ) Et + 2 2

(10.5)

where El and Et are the energies of longitudinal and transverse modes in the solid. In the classical case, El = NT and Et = 2NT. We will consider the quantum case in Section 10.4. Et (ω > ωF ) can be calculated as the usual phonon energy: Z Eph = E(ω, T)g(ω)dω (10.6) where g(ω) is the phonon density of states. In Chapter 8, we saw that the experimental phonon dispersion curves in liquids and solids are very similar. This readily implies close similarity of the phonon density of states in liquids and solids. In Section 8.1, we also saw that the density of states of propagating phonons in liquids is quadratic as in the Debye model

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(the non-propagating over-damped modes have instead the linear density of states). We therefore use the quadratic density of states in the Debye model, as is done in solids. Considering that the Debye model works particularly well for isotropic systems, this is a good approximation. The theory developed in this section therefore involves the same level of approximation as the Debye theory of solids. The density of states of transverse modes is gt (ω) = ω6N3 ω2 , where ωmt is the Debye frequency mt of transverse modes, and we have taken into account that the number of transverse modes in the solid-like density of states is 2N. ωmt can be somewhat different from the longitudinal Debye frequency; for simplicity, we assume ωmt ≈ ωD . We first consider a classical liquid where the phonon energy is E(ω, T) = T. Then, !  Zω D ωF 3 (10.7) Et (ω > ωF ) = Et (ω, t)gt (ω)dω = 2NT 1 − ωD ωF

We note that the same result for Et in Eq. (10.7) follows in the model where the transverse modes are gapped in k-space due to a finite phonon propagation range del = cτ as discussed in the previous chapter. Considering the reciprocal k-space where the phonons operate and g(k) ∝ k 2 gives:  Et

1 k> cτ

ZkD



E(ω, T)

= kg = cτ1

6N 2 k dk k 3D

(10.8)

Evaluating the integral gives a result identical to that of Eq. (10.7) once we use ω D = ckD . This is easy to understand: the lower limit in Eq. (10.8) kg = cτ1 corresponds in the Debye model ω = ck to ωF = τ1 , the lower limit in Eq. (10.7). Using Eq. (10.7) in Eq. (10.5), together with El = NT and Et = 2NT, gives the liquid energy as [165]:  3 ! ωF E = NT 3 − (10.9) ωD We now discuss the implications of Eq. (10.9) for the liquid energy and specific heat. Eq. (10.9) is a fairly simple equation. An advantage of having this simplicity in the theory is that it makes it fairly easy to understand and interpret the experimental data, as discussed in this chapter. In addition to the intra-cage rattling, or Debye, frequency ωD , Eq. (10.9) contains the inter-cage frequency ωF . This frequency quantifies the rate of particle jumps between neighbouring quasi-equilibrium positions in the liquid. The presence of ω F in the liquid theory is understandable because liquids flow, and ωF quantifies this flow.

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In the classical case, these two frequencies enter as the ratio R: R=

ωF ωD

(10.10)

which shows how close the inter-cage frequency of diffusive jumps, ωF , is to its maximal value ωD . At low temperature where ωF  ωD (τ  τD ), Eq. (10.9) gives the liquid energy as: E = 3NT

(10.11)

dE as in the harmonic solid and cv = N1 dT close to 3. This is consistent with the experimental result in Figure 2.2, which is also shown in Figure 10.2. The closeness of liquid cv to the Dulong–Petit value at low temperature can be interpreted as follows. At low temperature, the phase space for collective excitations is only slightly different from that in the solid case, due to small ωF (large τ ). As a result, liquid cv is close to the Dulong–Petit value. ωF increases with temperature because the frequency of molecular hopping becomes higher. As a result, the number of transverse excitations with frequency above ω F decreases. Consequently, the energy (Eq. (10.9)) decreases its slope and cv is predicted to decrease with temperature. As the number of transverse modes decreases with temperature, cv decreases from about 3 to 2, in agreement with Figure 2.2. Concomitantly, Et in Eq. (10.7) decreases and becomes 0 when ωF = ωD . This disappearance of transverse modes is supported by the results of molecular dynamics simulations where transverse modes are directly calculated from the transverse current correlation functions [120]. A quantitative agreement in the entire temperature range can be studied by using ω F = Gη∞ in Eq. (10.9), where η is taken from an independent experiment and dE calculating cv = N1 dT = 3 [165]. The result is shown in Figure 10.2, where we see a fairly good agreement that is within the experimental uncertainty of cv of about 0.1–0.2 [43]. The agreement improves once we account for the effect of anharmonicity on the mode energy (recall that Eq. (10.9) assumes that each phonon contributes the energy T to the system energy in the harmonic case). We will discuss anharmonicity in Section 10.3. We recall that Eq. (10.9) has no free fitting parameters. Using ω F = Gη∞ introduces a parameter Gω∞ which is fixed by the system properties. In Figure 10.2, this D ratio is 0.57 × 10−3 Pa·s [165]. Taking G∞ = 5 GPa for mercury [96] gives ωD of about 10 THz. This is a typical value of the Debye frequency. This is an important point: the agreement with experiments is achieved using sensible values of physical parameters fixed by system properties. We will return to this point in Sections 10.3 and 10.5.

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Liquid Energy and Heat Capacity in the Liquid Theory

Figure 10.2 Solid line: experimental specific heat of liquid mercury with electronic contribution subtracted [43, 44]. Dashed line: the prediction of Eq. (10.9). Adapted from [165] with permission from the American Physical Society.

In Section 10.4, we will discuss experimental data of cv for several types of liquids, including metallic, noble and molecular liquids, and will see that their cv similarly decreases with temperature, as Eq. (10.9) predicts. The same trend, the decrease of cv with temperature, has been experimentally found in complex liquids, including systems such as toluene, propane, ether, chloroform, benzene, methyl cyclohexane and cyclopentane, hexane, heptane, octane and so on [166]. This gives us confidence that the physical origin of the experimental behaviour of cv is the universal reduction of the phase space of transverse excitations as discussed earlier. We therefore see that the decrease of cv is related to the reduction of the volume of the phase space available to phonons in liquids. This is an important point. In solids, the phase space of phonons remains unchanged at all state points: the number of phonons is always 3N, where N is the number of particles. On the other hand, the phase space available to collective modes in liquids is temperature dependent. The phase space is also pressure dependent inasmuch as pressure changes ω F . We have seen that the reduction of the phase space operates somewhat differently depending on the model. In the original Frenkel model discussed in Chapter 7, transverse excitations disappear above the frequency ωF . The solution of the telegraph equation predicts the disappearance of these modes above the threshold wavevector kg in Eq. (8.8). Earlier in this section, we saw that both models give the same result for the energy of transverse modes in the classical case. In the quantum case discussed in Section 10.4, the results of the two models may differ; however, this does not affect the general effect: a temperature increase results in the reduction of the volume of the phase space, and it is this reduction that is fundamentally related to the universal decrease of liquid cv .

10.2 Comparison to the Integral over Interaction and Correlation Functions

103

When ωF reaches its maximal value of ωD in the system at high temperature, Eq. (10.9) predicts: cv = 2

(10.12)

where we noted that ωF flattens off at high temperature and ωdTF becomes small. We will encounter the value cv = 2 several times in this book and will see that it corresponds to a special state of liquid dynamics and thermodynamics. Here, we add several further remarks regarding Eq. (10.9). First, Eq. (10.9) contains no free fitting parameters: R is fixed by the system properties. Second, Eq. (10.9) applies to liquids with different inter-atomic interactions and correlation functions. As long as ωF is the same for these different liquids, their energy and specific heat is predicted to be the same. This provides a wide range of applicability and universality of the theory based on excitations. This is to be expected on the basis of our earlier discussion in Chapter 4: we have seen how the solid state theory based on excitations gives universal results for the solid energy and heat capacity. We note that Eq. (10.9) requires the knowledge of ωF . On the one hand, this is reasonable on general physical grounds because the flow properties and ωF are expected to affect dynamics and collective excitations. On the other hand, this makes the liquid theory different from the solid state theory where the classical solid energy in Eq. (4.5) is self-sufficient and truly universal. This brings about the question of what we aim to achieve by a physical theory. Landau and Peierls observed [167]: “The essence of any physical theory is to use the results of an experiment to make assertions about the results of a future experiment.” This expresses the view that an important role of the theory is to provide a relation between different properties and observables (a similar view was held by Bohr; see [168] for a review). In this sense, Eq. (10.9) provides a relation between liquid thermodynamic properties such as energy and cv and the properties of liquid excitations governed by the dynamical property ωF . In deriving Eq. (10.9), we considered the simplest harmonic and classical case where each mode contributes T to the liquid energy. This condition can be relaxed, and we will consider how anharmonicity affects the energy of each excitation in Section 10.3. Before we do that, we compare Eq. (10.9) and its underlying theory to the approach based on interactions and correlation functions in the next section. 10.2 Comparison to the Integral over Interaction and Correlation Functions In Chapter 6, we discussed understanding liquid thermodynamics on the basis of intermolecular interactions and correlation functions. In the simple picture, this amounts to evaluating the integral (Eq. (6.1)):

104

Liquid Energy and Heat Capacity in the Liquid Theory

3 n E = NT + 2 2

Z g(r)u(r)dV

(10.13)

We have noted a few issues related to this approach. First, interactions and correlation functions are generally complex and unknown apart from simple cases such as Lennard-Jones and related systems. Even if they were available, it would still require developing a physical model to explain the origin of the experimental behaviour of the specific heat of real liquids. Perhaps for these reasons, the approach based on interactions and correlation functions has not provided a physical understanding of the universal behaviour of liquid specific heat such as that shown in Figure 2.2, qualitatively or quantitatively. In notable contrast with this approach, Eq. (10.9) does not require the knowledge of interactions and correlation functions. It depends on one parameter only, R = ωωDF in Eq. (10.10). This also means that the liquid description based on excitations is more universal: lets consider liquids with very different interactions and correlation functions: Ar, Hg, CH4 , O2 . Eq. (10.9) predicts that as long as R is the same for these liquids at a certain pressure and temperature, their energy is the same. Similarly, if ωF has similar temperature dependence for all these liquids in a certain range, Eq. (10.9) predicts the same cv . While this issue is somewhat practical in character, the second issue is more foundational and is related to the fundamental understanding of the liquid state. This issue is related to the observation by Landau, Lifshitz and Pitaevskii discussed in the Preface and in Chapter 6: interactions in liquids in Eq. (10.13) are strong and system-specific; therefore, liquid energy and other thermodynamic properties are also system-specific, precluding the calculation of liquid thermodynamic properties in general form, in contrast with solids and gases. This issue is also related to the “no small parameter” problem quoted in the Preface and discussed in Chapter 6. How is this problem circumvented by Eq. (10.9)? We have seen that liquids too have the small parameter: these are small solid-like oscillations. However, this small parameter does not apply to all 3N phonons like in solids. Instead, it applies to a reduced phase space available to phonons that become gapped in liquids. Indeed, we have seen in Chapter 8 that the excitation phase space reduces with temperature, and the small parameter operates in this reduced phase space. Therefore, the approach based on excitations solves the small parameter problem. Does it solve system-specificity, the first problem stated by Landau, Lifshitz and Pitaevskii in the Preface? It does not solve it fully in the sense that we don’t have a system-independent expression for the liquid energy on par with that in solids where E = 3NT in Eq. (4.5). This is probably impossible to achieve in liquids on general grounds. Indeed, the flow properties of liquids are quantified by

10.3 Anharmonic Effects

105

viscosity, which strongly depends on temperature. Therefore, the property of this flow, ωF , must enter thermodynamic properties as it does Eq. (10.9). Nevertheless, the approach to liquids based on excitations goes a long way towards describing liquids in much more general terms. This is achieved by getting rid of systemspecific interactions and correlation functions featuring in Eq. (10.13) and instead operating in terms of one single parameter ωF . This parameter describes the liquid flow and, importantly, governs the phase space of collective excitations available in the liquid. In Chapter 6, we observed that Eq. (10.13) is as applicable to solids as it is to liquids. We can use Eq. (10.13) to calculate the energy and cv of solids, with the proviso that we know the inter-atomic interactions and correlation functions. However, this appears to us as quite a poor theory of solids. This is because we have a theory of solids based on collective excitations, phonons. This theory is general, predictive and physically transparent. We have had this theory for such a long time now (recall that the Debye paper was published in 1912 [84]) that we can’t imagine our life without it. Imagine for a moment that we did not discover quadratic forms and the harmonic theory of solids and instead used the approach based on correlation functions and inter-atomic interactions. Our lives as physicists would be a lot harder because of all of the issues mentioned earlier. With the new understanding of collective excitations in liquids discussed in Chapter 8, we can relate these excitations to the liquid energy and other thermodynamic properties as was done in solids a long time ago. As a result, this theory has the same attributes as the solid state theory. In particular, it is physically transparent and general: the liquid energy and heat capacity are related to collective excitations as is the case in solids, with the important difference that the phase space for these excitations is not fixed but reduces with temperature. This theory is also predictive, and we will discuss its predictions and their detailed tests in Section 10.5.

10.3 Anharmonic Effects We have so far assumed that each phonon contributes T to the liquid energy as is the case for harmonic solids. Anharmonicity becomes important at high temperature, and we discussed its effects in solids in Section 4.2. Anharmonicity is even more important in liquids, since liquids exist at higher temperature. In the harmonic case, phonon frequencies are independent of temperature. The effect of anharmonicity generally results in phonon frequencies reducing with temperature. This takes place at both constant pressure and constant volume. At constant volume, a reduction of frequencies is related to inherent anharmonicity and increased vibration amplitudes. This can be accounted for as follows. Let’s

Liquid Energy and Heat Capacity in the Liquid Theory

106

consider the partition function Z2 corresponding to, for example, Et + Et (ω > ωF ) in Eq. (10.5): ! Z 2N 1 X 2 −N 0 2 2 Z2 = (2π ~) exp − (p + ωti qi ) dpdq 2T i=1 i (10.14) ! Z 2N 1 X 2 (p + ωti2 q2i ) dpdq × exp − 2T ω >ω i ti

F

where ωti are frequencies of transverse excitations, N is the number of atoms and N 0 is the number of phonon states that include 2N waves and N1 transverse waves with frequency ω > ωF . Eq. (10.14) illustrates the point we mentioned several times earlier, including in Chapter 8 and Section 10.1: the volume of the phase space available to collective excitations reduces with temperature. This is due to the condition ωti > ω F = τ1 , which reduces the phase space over which the integral in the partition function is carried out. In solids, this phase space is fixed and is due to 3N phonons. Integrating Eq. (10.14), we find: !−1 !−1 2N 2N Y Y 2N N1 ~ωti T ~ωti (10.15) Z2 = T i=1

ωti >ω0

In the harmonic approximation, frequencies ωti are temperature-independent. d Then, Eq. (10.15) gives the energy E = T 2 dT ln Z = 2NT + N1 T. The final term gives the same result as Eq. (10.7) in Section 10.1. Indeed, N1 can be calculated as ω RD
3
N1 = gt (ω)dω = 2N 1 − ωωDF . This gives the N1 T equal to Et in Eq. (10.7). ωF

In the anharmonic case, frequencies become temperature-dependent, and we write: N1 2N X X 1 dωti 1 dωti 2 + N1 T − T Et + Et (ω > ω F ) = 2NT − T ωti dT ωti dT i=1 i=1 2

(10.16)

where the derivatives are at constant volume. Using Grüneisen approximation [66], it is possible to derive a useful approxi
α i = − , where α is the coefficient of thermal expansion mate relation: ω1 dω dT v 2 [169, 170]. Using this in Eq. (10.16) gives:   αT Et + Et (ω > ωF ) = (2N + N1 )T 1 + (10.17) 2 The first term
in Eq. (10.5), El , can be calculated in the same way, giving El = NT 1 + αT . Using this and Eq. (10.17) with N1 from Eq. (10.5) gives the 2 anharmonic liquid energy as:

10.4 Quantum Effects

αT E = NT 1 + 2 



 3−

107

ωF ωD

3 ! (10.18)

which reduces to Eq. (10.9) when α = 0.
The presence of the anharmonic term in Eq. (10.18), 1 + αT , explains why the 2 experimental cv of liquids may differ from the Dulong–Petit value cv = 3 close to the melting point [43, 44]. Eq. (10.18) has been found to quantitatively describe the cv of five commonly studied liquid metals (Hg, In, Cs, Rb and Sn) in a wide temperature range where cv decreases from about 3 around the melting point to 2 at high temperature [171]. Similarly to the harmonic case, the agreement is seen for physically reasonable values of parameters Gω∞ and for α close to the experimental α in these liquids. D When ωF  ωD (τ  τD ) at low temperature, Eq. (10.18) gives:   αT (10.19) E = 3NT 1 + 2 and the cv is: cv = 3 (1 + αT)

(10.20)

Eq. (10.20) is equally applicable to solids and viscous liquids where ωF  τ D , and has been found to be consistent with several simulated crystalline and amorphous systems [169]. We note that Eqs. (10.19) and (10.20) don’t need to be derived from Eq. (10.18), and also follow from considering the solid as a starting point where all 3N excitations are present.

10.4 Quantum Effects Eqs. (10.9) and (10.18) are readily generalised to the case where quantum phonon excitations become important at low temperature. In this case, each term in Eq. (10.5) can be calculated using the free energy of quasi-harmonic excitations as:   X  ~ωi Fph = E0 + T ln 1 − exp − (10.21) T i where E0 is the energy of zero-point vibrations. In calculating the energy, Eph = dF i Fph − T dTph , we assume dω 6= 0 like in the previous section, giving for the phonon dT energy: Eph = E0 + ~ where E0 is the zero-point energy.

X ωi − T dωi
dT ~ωi exp −1 T i

(10.22)

Liquid Energy and Heat Capacity in the Liquid Theory

108

Using

dωi dT

i = − αω as before gives: 2   αT X ~ωi Eph = E0 + 1 +
i 2 exp ~ω −1 T i

(10.23)

In this form, Eq. (10.23) can be used to calculate each of the three terms in Eq. (10.5). The energy of one longitudinal mode, the first term in Eq. (10.5), can P be calculated by substituting the sum in Eq. (10.23), , with the Debye vibrational density of states for longitudinal phonons, g(ω) = ω3N3 ω2 , where ωD is D
P the Debye frequency. Integrating from 0 to ωD gives = NTD ~ωT D , where Rx z3 dz D(x) = x33 exp(z)−1 is the Debye function. The energy of two transverse waves in 0

the second term of Eq. (10.5) can similarly be calculated as the integral with
the P = 2NTD ~ωT D . The density of states for transverse phonons g(ω) = ω6N3 ω2 to be D energy of two transverse modes with ω > ωF in the third term of Eq. (10.5) can be calculated by integrating the expression in the sum in Eq. (10.23) from ωF to ωD with the same density of states. This integral is the difference between the integral from 0 to ωD and the integral from 0 to ωF . This gives the harmonic Et (ω > ωF ) (harmonic in the a sense that the anharmonicity is not included) as: X ~ωi Et (ω > ωF ) =
~ωi −1 ω>ω F exp T (10.24)       ωF 3 ~ωD ~ωF − 2NT = 2NTD D T ωD T Putting all three terms together in Eq. (10.5) gives the liquid energy that accounts for both anharmonicity and quantum effects as:     3  !  ~ω D ωF ~ω F αT 3D (10.25) − D E = NT 1 + 2 T ωD T where we dropped E0 . We note that E0 is temperature-dependent because it depends on ωF and therefore T. However, this becomes important at temperatures of several K only, whereas in most liquids, including those discussed in the next section, we deal with significantly higher temperatures where E0 and the associated contribution to cv are small compared with the terms in Eq. (10.25). In the high-temperature classical limit where ~ωT D  1 and, therefore, ~ωT F  1 (this is because ωF < ωD ), the Debye functions become 1, and Eq. (10.25) reduces to the energy of the classical liquid, Eq. (10.18). If anharmonic effects are not accounted for and α = 0, which is a good approximation at low temperature where α is small, Eq. (10.25) becomes:

10.5 Verification of the Liquid Theory Based on Collective Excitations



~ωD E = NT 3D T



 −

ωF ωD

3



~ωF D T

109

! (10.26)

which reduces to Eq. (10.9) in the high-temperature limit where the Debye functions are 1. For some of the liquids discussed in the next section, the high-temperature classical approximation ~ωT D  1 does not hold in the low part of the temperature range considered. In this case, quantum effects at those temperatures become significant, and Eq. (10.25) should be used to calculate liquid cv . Throughout this book, we have mostly discussed liquid energy and specific heat and paid less attention to the entropy. This is primarily because we would like the theory to be directly compared with the primary experimental data of cv . The entropy can be written using the partition functions in Section 10.3 though. In Chapter 16, we will discuss the entropy and its change in the viscous liquid regime.

10.5 Verification of the Liquid Theory Based on Collective Excitations There have been several approaches to verify and scrutinise the liquid theory based on phonons. The first approach was to compare theory predictions to experimental data in many liquids in a wide range of temperatures and pressures. cv was evaluated from Eq. (10.25) where ωF was calculated from independent viscosity experiments as in the case of Hg in Figure 10.2 and compared with the experimental cv of 21 liquids including metallic (Cs, Ga, Hg, In, K, Na, Pb, Rb and Sn), noble (Ar, Kr, Ne and Xe), molecular (CH4 , CO, F2 , H2 S, N2 and O2 ) and network (H2 O and D2 O) liquids [172]. A good agreement between theory and experiment was found. This is illustrated in Figure 10.3 where a characteristic subset of these liquids is shown, including three liquids from each class: metallic, noble and molecular. Since the aim is to compare theoretical predictions with experiments in the widest temperature range possible, the data are used at pressures exceeding the critical pressures. As a result, many of the liquids studied are supercritical. We will discuss the supercritical state in more detail in Chapter 14. As discussed in the next section, the Debye model is not a good approximation in molecular and hydrogen-bonded systems where the frequency of intra-molecular vibrations considerably exceeds the rest of the frequencies in the system (e.g. 3,572 K in CO and 2,260 K in O2 ). However, the intra-molecular modes are not excited in the temperature range of experimental cv (see Figure 10.3). Therefore, the contribution of intra-molecular motion to cv is purely rotational, crot . The rotational motion is excited in the considered temperature range, and is classical, giving crot = R for linear molecules such as CO and O2 and crot = 3R for molecules with three 2

110

Liquid Energy and Heat Capacity in the Liquid Theory

Figure 10.3 Experimental cv (circles) in metallic, noble and molecular liquids. Experimental cv is measured on isobars. Theoretical cv (dashed line) was calculated using Eq. (10.25). The data for molecular and noble liquids are taken at high pressure to increase the temperature range where these systems exist in the liquid form [172]. The data are shown in two graphs to avoid overlapping and are adapted from [74] with permission from the Institute of Physics Publishing.

rotation axes such as CH4 . Consequently, cv for liquid CO shown in Figure 10.3 corresponds to the heat capacity per molecule, with crot subtracted from the experimental data. In this case, N in Eqs. (10.9), (10.18) and (10.25) refers to the number of molecules. Figure 10.3 shows good agreement between experiments and theoretical predictions in a wide temperature range of about 50–1,300 K. The agreement supports the interpretation of the universal decrease of liquid cv with temperature discussed earlier: this is due to the reduction of the phase space available for transverse excitations. As in the earlier example in Section 10.1, the agreement between theoretical and experimental cv involves physically sensible values of parameters ωD on the order of THz, G∞ in the GPa and sub-GPa range characteristic of liquids, and α = (10−4 − 10−3 ) K−1 in the range typical for liquids [172]. In addition to the temperature variation of liquid cv , we can also see how this cv is affected by pressure. In Chapter 2, we noted that the cv of liquid Ar increases with pressure in Figure 3.2. This can be understood as follows. Higher pressure reduces inter-atomic distances, making it harder for a particle to escape its cage, as shown in Figure 7.1. As a result, the activation barrier U in Eq. (7.1) grows, increasing τ

10.5 Verification of the Liquid Theory Based on Collective Excitations

111

and η. Larger η implies lower hopping frequency ωF = Gη∞ . As a result, the number of transverse modes propagating above ωF increases or, in other words, the phase space for excitations becomes larger. According to this theory, this means larger cv . This also follows directly from Eq. (10.9): lower ωF implies a larger energy slope and larger cv . We note that viscosity increases under pressure as long as the liquid structure and bonding remain unaltered. In some cases, viscosity can also decrease with pressure. This decrease can be large and can involve a drop of η of several orders of magnitude along the melting curve. This takes place when high pressure results in the transformation from the viscous covalent network to the low-viscous metallic state as happens, for example, in As-S liquids [173], or in the appearance of higher local coordinations and associated reduction of network covalency in liquid B2 O3 [174]. The second approach to verify and scrutinise the liquid theory involves an independent analysis. Proctor has independently undertaken “detailed and rigorous” verification of the liquid theory based on phonons in [175] as well as in his recent textbook [176]. As mentioned in the Introduction, textbooks dedicated to liquids either do not mention liquid specific heat or mention it very briefly. Proctor’s textbook is an exception in that it discusses liquid cv in both subcritical and supercritical states. We will come to this book and the supercritical state in Chapter 14. Proctor asks several questions: 1. Comparing theoretical and experimental cv involves parameters (depending on the level approximation, these are Gω∞ in the classical harmonic case, Gω∞ and D D α in the classical anharmonic case, and (G∞ ,ωD ,α) in the quantum anharmonic case). What if the best fit to the data is provided by parameters that are not physically reasonable? 2. The comparison between theoretical and experimental data such as that shown in Figure 10.3 is done on isobars. How does this comparison hold for other different (P,T) paths on the phase diagram? What if the best fit along these paths does not give the expected trends of parameters used? 3. Theoretical evaluation of cv involves the derivative of ωF and viscosity fitting. This fitting might affect cv , and a clear and unambiguous way to prepare the viscosity data to calculate ωF is needed. How is the agreement with experimental cv affected as a result? 4. For a physical model to have a value, it must be falsifiable. Is it falsifiable and can it be shown to fail? To answer these questions, Proctor performs a detailed analysis of experimental data and theoretical predictions based on the phonon theory of liquid

112

Liquid Energy and Heat Capacity in the Liquid Theory

thermodynamics and Eq. (10.25) in particular. He finds that “the phonon theory of liquid thermodynamics could have failed in a multitude of ways were it not correct” [175]. Proctor observes that theory could have required the parameters in the preceding list to have physically unrealistic values or to follow the wrong trend as a function density along a certain path on the phase diagram in order to fit the data. He concludes that this did not happen and, on the contrary, that the theory constrains the parameters to lie in the narrow range of physically sensible values and, moreover, to exhibit the correct trend with changing density.

10.6 Molecular Liquids and Quantum Excitations The decrease of liquid cv with temperature discussed in the previous section is a generic effect related to the reduction of the phase space available to collective excitations. However, system-specific effects may alter this behaviour. For example, the intra-molecular vibrations in molecular liquids become progressively excited at high temperature due to quantum effects. As a result, the cv of molecular liquids can first decrease like in Figure 10.3, followed by the increase due to quantum excitations of intra-molecular vibrations at high temperature. Depending on the result of the competition between these two effects in a specific molecular liquid, cv can be non-monotonic and, for example, have a minimum, as is the case for O2 and N2 [45]. In other and less frequent cases with weak bonding (e.g. H2 or He), quantum excitations of inter-particle degrees of freedom and the associated increase of cv can already be strong at low temperature and can become a dominant effect. We will discuss these effects in quantum liquids in the next chapter. Quantum excitations of intra-molecular motion in molecular liquids are a trivial effect similar to the effect operating in their parent molecular solids and are therefore unrelated to liquids and liquid physics. For this reason, we do not discuss this effect in detail and observe that accounting for inter-molecular vibrations in molecular liquids is readily done. To account for quantum excitations of the intra-molecular motion, we note that the Debye model assuming a continuous distribution of frequencies is not a good approximation. Instead of using the Debye model for the entire frequency spectrum like we did in the previous section, we should consider the actual phonon spectrum where the high-frequency intramolecular part is well separated from the rest and perform the sum over this spectrum. This applies to both liquids and solids and can be done by extending the sum in Eq. (10.23) to the upper frequency representing the Debye frequency for the inter-molecular part of the spectrum only, separately calculating the sum of energies Ein of high-frequency intra-molecular motion and adding the two terms.

10.6 Molecular Liquids and Quantum Excitations

113

For a linear molecule such as N2 or O2 with one intra-molecular frequency ωin , Ein in the harmonic case is simply: Ein = E0,in +

exp

~ωin
~ωin T

−1

(10.27)

Eq. (10.27) can be readily generalised to the case of non-linear molecules with several intra-molecular frequencies such as CH4 . The liquid energy is then the sum of the energy of these intra-molecular motions and the inter-molecular energy discussed in previous sections.

11 Quantum Liquids: Excitations and Thermodynamic Properties

11.1 Liquid 4 He Quantum liquids are liquids where the effects of quantum statistics, Fermi or Bose, become operative at low temperature on the order of ∼ 1 K. Quantum liquids is a large area of research, and here we only recall the results related to the focus of this book: the relationship between collective excitations and thermodynamic properties. This relationship turns out to be particularly simple for Bose liquids, where the energy is the sum of energy of excitations. In Fermi liquids, this is not generally the case: the quasi-particle energy is not an independent quantity but is a functional derivative of the system energy with respect to the distribution function [3]. The calculation of thermodynamic properties of Bose liquids such as 4 He proceeds as follows. In quantum mechanics, any weakly perturbed quantum state is a set of elementary quantum excitations. In Bose liquids, excitations can appear and disappear singly, in contrast with Fermi liquids where excitations appear and disappear in pairs. The elementary excitations with small momenta p are the sound waves, the phonons, with the linear dispersion relation  = pc. We have seen this result emerging in the Bogoliubov theory in Section 3.3. We therefore find that, at temperatures close to zero, the elementary excitations are phonons, and the system energy is then the sum of these excitations. This greatly simplifies the calculation of energy and other thermodynamic properties. In Chapter 4, we saw how the solid state theory benefits from this approach. Before calculating the energy and specific heat of 4 He, let’s look at the dispersion relation in this system. Figure 11.1 shows the dispersion curve from neutron scattering experiments in liquid 4 He below the lambda transition. The dashed line in this figure corresponds to the linear part of the dispersion relation ω = ck. Later experiments have ascertained the same shape of the dispersion curve in a wider range of k and with more experimental points (see, for example, [178]).

114

11.1 Liquid 4 He

115

Figure 11.1 The phonon dispersion curve in 4 He from neutron scattering experiments at 1.1 K. The dashed line corresponds to the speed of sound of 237 m/s. The data are from [177].

Figure 11.1 shows the “roton minimum”, which played an important role in the early development of the theory of liquid 4 He [3]. Later developments related this minimum to intrinsic properties of 4 He and quantum statistics [148]. Having seen the dispersion curves in classical liquids in Section 8.6, we observe that the same roton minimum is observed there too (see, for example, Figures 8.3 and 8.6). It is therefore a generic feature of the liquid spectrum, which is probably related to the washed-out higher Brillouin zones in their parent crystals rather than to lowtemperature quantum-statistical effects. We note that the two-fluid model set forth by Tisza and Landau gives rise to another type of waves: this is the “second sound” mode related to the entropy wave and the relative motion of the normal and superfluid components [179] (see [148– 150, 179, 180] for the original papers related to the two-fluid model, reviews of this model and the history of helium research). The next step in constructing a thermodynamic theory of liquid 4 He is to assume that this liquid does not support two transverse phonon branches but only one longitudinal branch (we will discuss this assumption in this section in more detail). Then, the liquid energy is given by Eq. (4.9) derived using the quadratic Debye density of states and where the second temperature-dependent term is divided by 3: E = E0 +

π 2 VT 4 30 (~¯c)3

(11.1)

116

Quantum Liquids: Excitations and Thermodynamic Properties

Figure 11.2 Experimental cv of liquid 4 He. The data are from [181].

and cv =

2π 2 T3 15(~¯c)3 n

(11.2)

Landau attributes this theory to Migdal, who performed the calculation in 1940 [179]. This is probably the earliest calculation relating collective excitations and thermodynamics in a liquid, classical or quantum, which withstood the test of time and experiment. It is interesting that this relationship was made in a quantum, rather than a classical, liquid considering that a quantum liquid is a lot more complex. That the energy is quantised in quantum systems turns out to be the simplifying circumstance. These quantum energy levels are populated by the lowest-energy phonon excitations, and we don’t need to consider anything else and so are able to write Eq. (11.2) for liquid cv straight away. Figure 11.2 shows the experimental cv of 4 He at vapour pressure, corresponding to the nearly ∝ T 3 temperature dependence [181], in agreement with Eq. (11.2). The crossover at high temperature is attributed to rotons considered to dominate at higher temperature. An interesting question concerns the assumption made in deriving Eqs. (11.1) and (11.2), namely that only the longitudinal mode is included in the calculation, but not the two transverse modes. In view of the experimental and

11.1 Liquid 4 He

117

modelling evidence for transverse modes in many other liquids, as discussed in Chapters 8 and 9, this becomes an important question. Experimentally, neutron scattering experiments have not revealed these modes in liquid He. However, more informative X-ray scattering experiments that showed transverse modes in classical liquids (see Section 8.6) are not suitable for liquid helium. On the other hand, path integral simulations of liquid helium show minima of the velocity autocorrelation function at low temperature [182]. These indicate the presence of the oscillatory component of particle motion, which, in turn, is related to the propagation of transverse modes as discussed in Section 7.2 and Chapter 8. Whether or not transverse phonons propagate in liquid He is therefore unclear. On the basis of our earlier discussion in Chapter 8, we can nevertheless conclude that transverse excitations, even if they exist in the temperature range shown in Figure 11.2, do not contribute to the energy and cv because these excitations are gapped from below. Indeed, transverse phonons that are excited at low temperature must be in the low-frequency range, but these phonons are non-propagating as discussed in Chapter 8. More explicitly, we recall Eq. (10.24): 

~ωD Et = 2NTD T



 − 2NT

ωF ωD

3



~ωF D T

 (11.3)

The Debye energy ~ωD of solid He is about 26 K, and ~ωF is not far from this value, given that the normal component of 4 He has very low viscosity. Then, ~ωD ~ωF , T  1 for the temperature range in Figure 11.2. The Debye function T π4 D(x) = 5x 3 for large x. As a result, the two terms in Eq. (11.3) cancel each other out, giving Et = 0. Et = 0 in the low-temperature quantum liquid is due to gapped excitations not contributing to the system energy at low temperature. We therefore find that not accounting for transverse excitations is correct in the low-temperature limit, even though at the time of deriving Eqs. (11.1) and (11.2) the reason for this was not understood. On the other hand, not accounting for transverse excitations in the high-temperature classical liquid is not justified. We have seen this in Chapter 8. As mentioned at the beginning of this chapter, research into quantum liquids has a long history. Despite this, some central problems remain not understood. Pines and Nozieres [148] observe that “microscopic theory does not at present provide a quantitative description of liquid He II” (“II” here refers to helium below the superfluid transition temperature). A microscopic theory exists only for models of dilute gases or models with weak interactions where perturbation theory applies. We have considered this model in Section 3.3 and Eqs. (3.32)–(3.34). The absence of the microscopic theory of superfluidity is in contrast with superconductivity, where superconducting properties emerge from a microscopic Hamiltonian [183].

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Griffin broadly agrees with the assessment of Pines and Nozieres and says that we can’t make quantitative predictions of superfluid 4 He on the basis of existing theories and depend on experimental data for guidance [184]. Interestingly, Griffin attributes the theoretical problems of understanding the superfluid helium to the “difficulties of dealing with a liquid, whether Bose-condensed or not”. In other words, Griffin recognises that the general problems of liquid theory discussed in the Preface, the Introduction and subsequent chapters have a long reach and extend to the low-temperature superfluid system. Compared with several decades ago, research into liquid 4 He and its superfluidity has been slowing down. We have a set of important results and we know that several fundamental problems remain unresolved, but we don’t know what to do next or where the next important insight is likely to come from. One insight we have gained from classical liquids is that considering microscopic details of their dynamics and the combined oscillatory and diffusive components of particle motion in particular is the key to understanding liquids. It may well be that these details will similarly need to be incorporated in the future microscopic theory of liquid helium. We will return to this point in Chapter 15, where we will discuss viscosity minima in quantum liquids and their relation to microscopic dynamics. 11.2 Liquid 3 He In liquid 3 He, where atoms are fermions rather than bosons as in 4 He, the relationship between thermodynamic properties and collective excitations is different. Similarly to Bose liquids, Fermi liquids is a large area of research, and here we only recall the results related to thermodynamic properties and collective excitations, that is, the focus of this book. We discuss the main results in this section and refer the reader to a more detailed discussion in, for example, [5], [148] and [185]. There are two types of excitations in a Fermi liquid, localised quasi-particles and collective modes [148]. Introduced by Landau in the Fermi liquid theory [186], quasi-particles assume the role of gas particles when interactions in the gas are turned on. The starting point of the theory is the assumption that the classification of energy levels in the Fermi system is not affected by turning on the interactions between the particles in the gas. The energy of the interacting Fermi system, E, is the primary concept. E is not the sum of quasi-particle energies; however, the quasi-particle energy can be defined as a functional derivative of E with respect to the quasi-particle distribution. The quasi-particle picture and the condition for a quasi-particle to have a welldefined momentum apply when the excited states of the liquid are described by the distribution function differing from the step function in a small region near

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the Fermi surface. If τq is the relaxation time related to the mean free path of quasi-particles due to their collisions, the quantum uncertainty of the quasi-particle energy is τ~q . Then, the quasi-particle picture applies if: ~ T τq

(11.4)

since the width of the transitional zone where quasi-particles can scatter is on the order of T (i.e. for sufficiently long-lived quasi-particles) [5]. τq depends on temperature as: τq ∝

1 T2

(11.5)

because the collision rate above the Fermi surface at low temperature is ∝ T 2 . At low enough temperature, the condition in Eq. (11.4) is satisfied. At low temperature, where the Fermi distribution is close to the step function, the quasi-particle distribution function can be approximated by the Fermi distribution, like in the ideal non-interacting Fermi gas. As a result, thermodynamic properties of the Fermi liquid at low temperature become those of the ideal Fermi gas, except that the particle mass is substituted by the quasi-particle effective mass. In particular, this theory predicts the linear specific heat: cv ∝ T

(11.6)

at low temperature, where the quasi-particle concept applies. This linear dependence of cv on temperature similarly applies to the specific heat of electrons in metals. The prediction of the linear behaviour cv ∝ T is in agreement with the experimental data showing that the specific heat of 3 He approximately approaches the linear temperature dependence below about 0.1 K. This is shown in Figure 11.3 and is viewed as a confirmation of the Fermi liquid theory discussed earlier [187, 188]. We therefore see that, in an interesting difference from 4 He and other liquids discussed earlier, the thermodynamic properties of 3 He are not related to collective modes but to localised quasi-particles instead. This is due to the governing role of the Fermi statistics and exclusion principle, which force the properties of the Fermi liquid to be close to those of the Fermi gas at low temperature. In this sense, the Fermi liquid is a unique system where thermodynamic liquid properties are close to those of the gas. We have seen that this is markedly not the case for 4 He or the classical liquids considered in this book. As discussed in the previous section, the Migdal and Landau result cv ∝ T 3 in 4 He is due to collective excitations, phonons, like in solids. In Chapter 10, we saw how specific heat in classical liquids is similarly governed by phonons.

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Figure 11.3 Experimental cv of liquid 3 He [188]. The points include four different experimental data sets.

The collective excitations are nevertheless operative in the Fermi liquid and are related to the dynamics of quasi-particles. Considering these dynamics, we interestingly find two regimes of wave propagation that are analogous to those we encountered in classical liquids in Section 7.2, where we discussed the solid-like elastic regime ωτ  1 and the hydrodynamic regime ωτ  1. In 3 He, ωτq  1 corresponds to the hydrodynamic regime of wave propagation where the hydrodynamic equilibrium is established. This is the longitudinal sound wave discussed in Chapter 3, sometimes referred to as the “first sound” [148]. In the opposite non-equilibrium case ωτq  1, wave propagation is unaffected by quasi-particle collisions. This wave was called “zero sound” by Landau [189]. There are two similarities between the zero and first sounds in the Fermi liquid, on the one hand, and the wave propagation in ordinary liquids in the hydrodynamic and solid-like elastic regimes, on the other hand. First, the velocity of the zero sound in the regime ωτq  1 is larger than that of the first sound (by a factor of √ 3 in the weakly non-ideal Fermi gas). This is similar to the “fast sound” (positive sound dispersion effect) in the ordinary classical liquids discussed in Section 8.6, where the wave velocity increases above its hydrodynamic value in the regime ωτ  1. Second, wave dissipation in the Fermi liquid increases with τq and viscosity, as well as frequency, in the first sound regime ωτq  1: the absorbtion coefficient β is β ∝ τq ω2 . This is the same as in the case of the longitudinal sound in the hydrodynamic regime of ordinary liquids as is seen in Eq. (7.24) in Section 7.2, where we found β = d1 ∝ ω2 . On the other hand, β ∝ T 2 ∝ τ1q does not depend on frequency and decreases with τq and viscosity in the zero sound regime ωτq  1 (this assumes that collisions are classical at high temperature;

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the opposite quantum regime yields temperature-independent β) [189]. This is the same effect as the increase of the wave propagation length with τ and viscosity in the solid-like elastic regime ωτ  1 in ordinary liquids, as discussed in Section 7.2. The existence of these collective modes in the Fermi liquid and the zero sound in particular imply that these excitations can occupy their quantum states in any numbers and therefore obey Bose statistics, corresponding to the Bose branch of the Fermi liquid spectrum. However, this branch contributes higher powers of temperature to thermodynamic properties (e.g. cv ∝ T 3 as in the case of 4 He discussed earlier), and therefore should not be considered in the Landau Fermi liquid theory [5]. Similarly to 3 He, the two different regimes of wave propagation apply to 4 He (discussed in the previous section). ωτq  1 corresponds to the hydrodynamic regime and the “first sound”, whereas the regime ωτq  1 is analogous to the zero sound in the Fermi liquid [148].

11.3 Quantum Indistinguishability This section is largely explorative and is related to one particular unresearched aspect of quantum liquids. It is not directly related to the link between liquid thermodynamics and collective excitations, the main topic of this book. It is nevertheless related to the connection between thermodynamics and dynamics of individual particles. We have seen an example of this connection in the previous section. Here, we discuss this connection from a different point of view. In Chapters 7 and 8, we saw how particle dynamics govern collective modes in liquids, which, in turn, govern liquid thermodynamic properties. In quantum liquids, particle dynamics additionally and directly affect thermodynamic properties via quantum statistics. In the previous two sections, we have seen that quantum statistics play a governing role in thermodynamic properties of 4 He and 3 He liquids. We have seen the implication of different quantum statistics in these two liquids in the form of very different temperature dependence of cv in Figures 11.2 and 11.3. Quantum statistics stem from the indistinguishability of identical particles in quantum mechanics. When particle wave packets overlap, or when the de Broglie wavelength λdB is larger than the inter-particle separation a, particle distinguishability is lost. To reflect this, the wave function needs to be symmetrised with respect to particles exchanging places and becomes either symmetric for Bose or antisymmetric for Fermi particles [190]. For example, the latter case corresponds to the following wave function:

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1 ψ = √ (f (x1 − xa )g(x2 − xb ) − f (x2 − xa )g(x1 − xb )) 2

(11.7)

where f (x1 − xa ) and g(x1 − xb ) are the wave packets of the first particle and second particles in the regions of space near x = xa and x = xb , respectively. Physically, symmetrisation (Eq. (11.7)) means that no new quantum state is produced as a result of particle exchange where x1 ↔ x2 . The corresponding probability function P is: P=

1 {|f (x1 − xa )g(x2 − xb )|2 + |f (x2 − xa )g(x1 − xb )|2 2 + [f ∗ (x1 − xa )g(x1 − xb )g∗ (x2 − xb )f (x2 − xa )

(11.8)

+ complex conjugate]} The first two terms represent “classical probability” terms, whereas the remaining terms are quantum-mechanical interference effects. Significant interference implies that particles cannot be assigned a definite identity [190]. The consequences of indistinguishability are far-reaching and include different statistical distributions (Bose–Einstein or Fermi–Dirac), Pauli principles, exchange interactions and so on. The textbook discussions of quantum indistinguishability tacitly assume that particles are always able to physically exchange places. A large part of this discussion concerns electrons. We now observe that the condition λdB > a holds in both liquid and solid helium. However, thermodynamic properties of both 4 He and 3 He solids are largely governed by lattice effects, like in conventional quantum solids, where the Debye result cv ∝ T 3 due to phonons is taken as the main mechanism, with various corrections operating on top [191–193], whereas the properties of 4 He and 3 He liquids are markedly different. Hence the ability of particles to flow and exchange places is important for quantum indistinguishability and statistics to manifest. The difference in the properties of 4 He and 3 He liquids due to different statistics and the similarity of 4 He and 3 He solids is also noted by Leggett, who notes that, in order to observe the effects of indistinguishability and quantum statistics, particles need to physically exchange places and “find out” that they are indistinguishable [185, 194]. Leggett uses another example, the difference in the vibration and rotation spectra of heteronuclear and homonuclear molecules, related to a difference between rotation where by identical particles physically change places and vibration where by they do not. Discussing supersolidity, he points to a related open question of the dynamics of atomic exchanges: How is the overlap of atomic wavefunctions in the symmetrised many-body wavefunction in solid 4 He related to atomic exchanges, and what are the consequences [185]?

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As discussed in Chapter 7, the flow of particles in liquids is a dynamical process characterised by liquid relaxation time τ . τ separates the short-time solid-like elastic regime t  τ and the long-time diffusive hydrodynamic regime t  τ . In Chapter 8 and Section 8.4, we saw that both regimes are important from the physical point of view and that neglecting one of them misses essential physics and observed effects. Since the diffusive particle motion enabling physical exchanges takes place on the timescale t  τ only, an interesting question is: What happens to quantum indistinguishability and its consequences at short time t  τ when particle dynamics are purely oscillatory like in solids and no exchanges are possible? If τ is one year, measuring an observable during one second would return the same result as in the solid because the two systems are the same during the time of measurement. Let’s assume that the quantum system is prepared at time t = t0 , the wave function starts evolving in response to changing external parameters such as temperature or pressure and an observable is measured in the time window between t0 and t0 + 1t. The preceding discussion suggests that experimental consequences of quantum indistinguishability, statistics and related effects should not be observed in the short-time regime of liquid dynamics 1t  τ , similarly to how they are not in solids [195]. It is perhaps surprising that this area has remained unexplored. One possible reason for this is that the observables in quantum liquids and gases are often measured during times well in excess of the relaxation time τ so that particle exchanges are always operative. From the fundamental point of view, it is nevertheless important to know what happens to quantum indistinguishability, statistics and their experimental consequences in the short-time regime and how these properties start to develop at later times exceeding the relaxation time τ . This leads to an interesting question: What kind of experiment could reveal the dynamical onset of indistinguishability and its implications? Compared with liquids, such an experiment may be easier to perform in cold gases where relaxation times are relatively long [196]. Condition t  τ describes the equilibrium state of the system and the commonly discussed hydrodynamic regime (see Section 8.4) where particle exchanges are operative. In the opposite regime, t  τ , the liquid is in the non-equilibrium solid-like elastic regime. As discussed in Chapters 7 and 8, the solid-like regime has remained relatively unexplored compared with the hydrodynamic regime. In Chapters 8 and 10, we saw how the exploration of this regime results in an important new understanding of classical liquids. In quantum liquids, exploring this regime might similarly bring new and interesting insights. In the equilibrium state, the dynamics of particle exchanges and τ are related to the degree of overlap of particle wave functions and associated energy terms such as the exchange energy. For example, the transition rate 0t in a system of two particles between the state |ii = φ1 (r1 )φ2 (r2 ) and the state with particles swapped,

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|f i = φ1 (r2 )φ2 (r1 ), is 0t ∝ |hf |U|ii|2 according to the Fermi golden rule, where U is the interaction operator and hf |U|ii is its matrix element: Z hf |U|ii = Uφ1 (r1 )φ1∗ (r2 )φ2 (r2 )φ2∗ (r1 )dV1 dV2 (11.9) This is the exchange energy J setting the energy levels of two particles and depending on the overlap of the wave functions φ1 (r1 ) and φ2 (r2 ). In this simple model, 0t ∝ J 2 . J depends on U, implying that τ and J are related as well because (a) τ depends on the activation energy barrier U˜ and (b) U and U˜ are closely related. Unless tunnelling is involved, τ can additionally depend on other parameters such as temperature. Exploring the implications of dynamical quantum indistinguishability generally connects to the currently topical research in anyons, particles with statistics that are neither Fermi nor Bose but are fractional. At the heart of the emergence of anyons and their different statistics is the discussion of the actual process of particle exchanges and in particular consideration of different exchange pathways and topologically different paths in two dimensions [197]. This is generally similar to what we discussed in this section, in that we also look into details of the actual exchange process. Our discussion is related to the question of what happens to the system (in terms of experimental consequences of indistinguishability and statistics) at short times and before particles have a chance to exchange places, and how the onset of indistinguishability develops once exchanges become operative. This applies to three dimensions as well as two.

12 Sui Generis

Tabor interestingly lists three ways to approach and understand liquids [6]. The first two approaches involve modifying equations of gases and solids. This is related to our discussion in Chapter 8, where we discussed different ways to modify the hydrodynamic and elasticity equations. The third approach mentioned by Tabor is sui generis (“of its own kind”), where liquids are discussed for what they are. Tabor lists two examples of this: earlier structural models of liquids and molecular dynamics simulations. We can now be much more specific as to what sui generis means for liquids. We have seen that the key to understanding liquids is to appreciate from the outset that they are systems with a “mixed” character of particle motion [74]. In solids and gases, the particle motion is “pure” in the sense that it is purely oscillatory in solids and purely diffusive in gases. In liquids, particle motion combines the oscillation around quasi-equilibrium positions and diffusive motion between these positions. This combined, mixed, oscillatory-diffusive particle motion has presented a challenge for liquid theory because well-developed theories existed for two states of matter where particle dynamics are “pure”. These two states are solids, where particle dynamics are purely oscillatory, and gases, where particle dynamics are purely diffusive. We have seen that accounting for the mixed character of particle motion is not unique: it can be done by either modifying hydrodynamic equations to account for the solid-like elastic response or modifying elastic equations to account for the hydrodynamic liquid flow (see Chapter 8). Appreciating this, we can solve the resulting equations and find that the phase space for liquid collective excitations reduces with temperature. A convenient parameter quantifying this reduction is the ratio of the inter-cage hopping frequency ωF and the intra-cage maximal, or rattling, frequency ωD in Eq. (10.10): ωF R= (12.1) ωD 125

126

Sui Generis

which enters Eqs. (10.9), (10.18) or (10.25) for the liquid energy in different approximations. This R-parameter quantifies the mixed state of liquid dynamics. R close to 0 corresponds to ωF  ωD and to mostly oscillatory motion like in solids. This is characteristic of viscous liquids and liquids approaching the glass transition. We will discuss this in Chapter 16. R close to 1 corresponds to ωF ≈ ωD and to the case where the oscillatory component of particle motion mostly disappears at the Frenkel line, corresponding to the gas-like dynamics. We will discuss this regime in Chapter 14. We will see that these two cases, R = 0 and R = 1, are particularly simple from the point of view of understanding liquid thermodynamics because they correspond to the “pure” rather than mixed regimes of particle dynamics. On the other hand, the variation of R between 0 and 1 corresponds to the mixed regime of liquid dynamics, and this has been the most challenging regime from the point of view of understanding liquids. In Chapter 8, we saw that liquid thermodynamic properties undergo the largest change in this regime: cv changes from 3 (small R) to 2 (R close to 1). After R = 1 is reached, particle dynamics change and becomes gas-like and purely diffusive. This regime is easier to understand because particle dynamics are no longer mixed. In this regime, cv continues to decrease from 2 to its smallest value possible: the ideal-gas value of 23 . We will discuss this in Chapter 14. We therefore see that it is the mixed character of liquid dynamics (where R varies between 0 and 1) that makes liquids qualitatively different from solids and gases and sui generis (to be of their own kind). This mixed character is probably the most informative single property at the microscopic level that defines what sui generis means for liquids from the physical point of view. As discussed in Chapter 8, this mixed character of particle dynamics is responsible for the variability of the phase space of phonons in liquids. As discussed in this book, finding out what sui generis for liquids actually is and appreciating its implications for liquid excitations and thermodynamics has taken a long time. It required the combination of theory, experiment and modelling, with some central contributions from over 150 years ago [94].

13 Connection between Phonons and Liquid Thermodynamics: A Historical Survey

13.1 Early Period As discussed in Chapter 8, the key observation about liquid physics was related to the gradual reduction of the phase space available for collective excitations with temperature. Using a combination of insights from theory, experiments and modelling, we were able to make this reduction specific and quantitative. This, in turn, enabled us to predict and explain another well-defined effect in Chapter 8: the universal decrease of liquid cv with temperature. Implicit in this discussion is that thermodynamics and collective excitations in liquids are related, as they are in solids. In a more general sense, the connection between liquid thermodynamics and collective excitations is not entirely new and was discussed in a number of approaches. These approaches were different from each other and from the discussion in Chapters 8 and 10. This is perhaps not surprising considering that key results from experiments, modelling and theory emerged only recently (see, for example, the final paragraph in Section 8.6). It is nevertheless important to review this research, not least because it is largely unknown and not commonly discussed. Involving physics luminaries, some early proposals to connect liquid thermodynamics to phonons date back more than a century and reveal a fascinating story. Loosely speaking, this period can be described as “early”, corresponding to the ideas of Sommerfeld and Brillouin put forward around the same time when the solid theory was developed by Einstein and Debye linking phonons to solid thermodynamics. We first recall that the relationship between phonons and liquid thermodynamics in quantum liquids is simplified (see the discussion by Landau and Migdal in Section 6). The simplifying factor is the quantum nature of the system where the low-lying energy levels are related to phonons. In classical liquids, matters turned out to be more complicated.

127

128 Connection between Phonons and Liquid Thermodynamics: A Historical Survey

My first entry into this history started with an unlikely source, the Brillouin textbook Tensors in Mechanics and Elasticity [198]. It is unlikely because we don’t expect to find a discussion of liquids in a book with such a title. Brillouin discusses liquids in the elasticity textbook because he considers solids (rather than gases) as an appropriate starting point for developing a theory of liquids. He starts with the observation that extending the kinetic theory of gases to the liquid state gives useful results close to the critical point only, whereas understanding low-temperature dense liquids requires solid-state concepts. He mentions a related model by Bernal and Bethe involving a short-range order in liquids close to that in crystals, which washes out at longer distance. Brillouin then asks how the solid state theory needs to be modified in order to explain liquid properties. We recall that this is generally similar to what Frenkel did when he modified the solid state equation to enable the liquid flow on the basis of Maxwell ideas. We saw this in Section 7.2 discussing how Frenkel modifies the shear and bulk moduli to enable liquid flow and, on this basis, obtains propagating waves in liquids in different regimes. Frenkel, however, did not relate these phonons to liquid thermodynamic properties. Brillouin assumes that the shear rigidity vanishes at melting and that the transverse motions become “free” as a result. Then, the liquid energy can be calculated as follows. In solids, the energy is:   NT NT Es = 3NT = NT + 2 + (13.1) 2 2 where NT is the energy of the longitudinal phonon and each term in brackets corresponds to the kinetic and potential part of the transverse phonon. In liquids, the potential part is zero because no transverse phonons propagate, giving the liquid energy as:   NT = 2NT (13.2) E = NT + 2 2 and cv as: cv = 2 (13.3) which we recognise to be the high-temperature limit in Eq. (10.12). Brillouin notes that his result cv = 2 contradicts cv = 3 of real liquids close to melting. As a way out of this contradiction, Brillouin proposes that at small length scale, liquids possess an order close to crystalline where transverse modes propagate (thus explaining cv = 3) and certain easy cleavage planes (explaining liquid flow). Even though this crystallite picture of liquids is not what scientists currently believe, it is notable that it is very difficult to exclude the presence of crystallites, or cybotactic groups, even in solid glasses on the basis of experimental evidence

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[199]. It took a combination of several state-of-the art experiments and modelling to ascertain the structure of the paracrystalline diamond where very small crystallites are embedded in a disordered network. This is a very recent result [200]. I am not saying this to suggest that high-temperature liquids might have small crystallites present, but rather to observe that the Brillouin proposal was (a) truly ingenious and represented the kind of disruptive thinking that in other instances results in breakthroughs and (b) not something that could be easily dismissed on the basis of experimental evidence, even today. Brillouin subsequently compares this idea to the earlier work aimed at relating the liquid energy to phonons. This work similarly started with the solid-like approach to liquids and considered removing the transverse phonons from the liquid spectrum to account for the seeming inability of liquids to support shear stress. According to Brillouin, this was discussed in his PhD thesis in 1920 and in his papers in 1922–32 [201–203], as well as in Sommerfeld’s preceding pioneering publications in 1913–14 [204]. In those papers, the extension of the Debye theory of solids to liquids was similarly based on the hypothesis of transverse velocity becoming 0; however, the total number of degrees of freedom, 3N, was kept intact. This kept the liquid cv = 3 like in solids, but resulted in the Debye wavelength in the liquid becoming shorter than that in the solids. Noting that it is “embarrassing” to have to assume that disorder in liquids results in shorter propagating waves than in solids, Brillouin favours the new result cv = 2 in Eq. (13.3) over the previous model. The work of Brillouin, Sommerfeld and Debye related to liquids and solids is understandably interrelated. Debye was a student and colleague of Sommerfeld and published his paper on heat capacity of solids in 1912 [84] where he introduced the widely known “Debye model”. Sommerfeld’s publication [204] followed the year after, where he aimed to extend this line of enquiry to liquids. Brillouin too was a student of Sommerfeld and was probably aware of his ideas related to liquids, which he sought to develop in the decades that followed. It is quite remarkable that the first attempt at the theory of liquid thermodynamics based on phonons was published by Sommerfeld only one year after the Debye theory of solids and six years after Einstein’s paper “Planck’s theory of radiation and the theory of the specific heat” came out in 1907 [64]. The Debye and Einstein theories have become part of nearly every textbook in which solids and phonons are mentioned [3, 38, 58]. A theory of liquid thermodynamics, on the other hand, remained unworkable for the century that followed. Coming back to the pioneering work of Sommerfeld and Brillouin, we observe a notable difference between their discussion and Frenkel’s theory discussed in Chapter 7. In Frenkel’s theory, transverse modes in the liquid do not suddenly disappear but remain propagating above the frequency ωF and hence disappear

130 Connection between Phonons and Liquid Thermodynamics: A Historical Survey

gradually below ωF as ωF increases with temperature. On the other hand, all transverse waves disappear abruptly at melting in Brillouin’s theory, resulting in cv = 2 in Eq. (13.3). As discussed in Chapter 10, the Brillouin result cv = 2 in Eq. (13.3) plays an important role; however, this value is reached at high temperature only when the hopping frequency ωF reaches its maximal value comparable to ωD in Eq. (10.9). Another way to state the disappearance of transverse waves is to set G = 0, where G is shear modulus. Brillouin used this condition to discuss melting and the liquid state: melting corresponds to a vanishing rigidity of the crystal [205]. The decrease of G with temperature and its eventual disappearance at melting was subsequently explored by Born [206]. Frenkel was aware of this work and disagreed [9] with the idea that the shear modulus becomes zero in liquids on melting. Interestingly, the first-principles theory of melting has turned out to be a tough problem and is yet to be developed. The problem of liquid thermodynamics and in particular its relationship to phonons extended over a substantial period of Brillouin’s research. About 20 years after his first published idea on this topic, he had a short paper [205] where he discussed the decrease of shear modulus with temperature and noted that, whereas melting corresponds to zero macroscopic rigidity, the microscopic rigidity may remain finite so that the liquid is able to sustain hypersonic transverse waves. This can then explain both the liquid flow and the experimental result of cv = 3 in liquids close to melting. This is of course much closer to Frenkel’s picture discussed in Chapter 7. Notably, Sommerfeld and Brillouin consider that the liquid energy and thermodynamic properties are fundamentally related to phonons like in solids. Just how this works in liquids was not clear at the time. This turned out to be a formidable problem. Indeed, we have seen in Chapters 7 and 8 that it took about a century until we started understanding phonons in liquids well enough to be able to connect them to liquid thermodynamic properties. This understanding was assisted by the early theoretical work of Frenkel, although Frenkel himself did not connect phonons to liquid thermodynamics. A long pause, lasting about 50 years, followed, after which important results from experiments, modelling and theory started to emerge and helped to place the link between phonons and liquid thermodynamics on a solid quantitative basis. 13.2 Late Period Apart from isolated cases, Brillouin’s ideas were not developed. One example of such a case is the work by Wannier and Piroué who took up the Brillouin idea that transverse modes disappear in the liquid. Wannier and Piroué sought to describe this disappearance in a gradual, rather than abrupt, manner [207]. They recall the

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131

Maxwell proposal that liquids are capable of both elastic and viscous response. On this basis, they consider an oscillator that loses energy through the viscous relaxation process and propose, without derivation, that the potential energy P of this oscillator is: 1 P=K (13.4) 1 + ω21τ 2 where K = T2 is the kinetic energy of this oscillator. Wannier and Piroué then consider this relaxing oscillator as a phonon in the liquid and calculate the liquid energy by integrating the energy of phonons with potential terms (Eq. (13.4)) and Debye density of states, with τ calculated from id , where ηid is the ideal gas viscosity. Wannier and Piroué viscosity as τ = η−η G observe a good agreement with the experimental cv of liquid mercury, adding that the agreement could be misleading due to the approximations made. We can see what Wannier and Piroué were aiming to achieve. The fraction in Eq. (13.4) is present in the real part of the complex shear modulus M in Eq. (7.16) in Section 7.2. That M was derived by Frenkel as a modification of elasticity equations to allow for the liquid flow. The fraction in Eq. (13.4) can be viewed as a viscoelastic function with two different trends in the regimes ωτ  1 and ωτ  1. The solid-like elastic regime ωτ  1 gives the solid result P = K in Eq. (13.4). On the other hand, ωτ  1 gives the hydrodynamic result P = 0 corresponding to no propagating waves. There are important differences between the approach of Wannier and Piroué and that of Frenkel. Apart from the seemingly ad hoc nature of relation (Eq. (13.4)), this equation results in the potential part of the phonon energy becoming 0 for all phonons in the regime ωτ  1, including transverse and longitudinal. As discussed in Chapter 7, the Frenkel theory predicts that only transverse phonons become non-propagating in the regime ωτ  1. The longitudinal low-frequency phonons remain propagating as expected for the usual sound waves. Second, Eq. (13.4) implies that the classical phonon energy explicitly depends on frequency as well as on τ . It is unclear what is the physical basis for this dependence. In the Frenkel theory, τ only sets the range of propagation of transverse phonons, but the phonon itself is the same as in the solid where it is independent of τ and frequency. It appears that Eq. (13.4) plays the role of a convenient viscoelastic function interpolating between solid-like elastic and liquid-like hydrodynamic regimes. Using this function achieves a result that is not generally dissimilar to the mechanism discussed in Chapter 8: smaller τ at high temperature gradually reduces P for each phonon in Eq. (13.4), this effect being larger for low-frequency phonons. This gives a smaller contribution of low-frequency phonons to the liquid energy, which could be viewed as an effective reduction of the phase space available for collective excitations discussed in Chapter 10. However, this effect is implemented

132 Connection between Phonons and Liquid Thermodynamics: A Historical Survey

quite differently in the Wannier and Piroué theory, where the energy of each phonon acquires the explicit and unphysical dependence on ω and τ in Eq. (13.4). In the Introduction, we mentioned that textbooks dedicated to liquids either do not mention liquid specific heat or mention it very briefly. Faber’s textbook [12] has one condensed and informative paragraph talking about liquid cv . Faber first observes that liquid cv is close to the Dulong–Petit value cv = 3. Noting that this may be surprising, since a good part of cv = 3 is attributed to transverse modes that ideal fluids are unable to support, he adds that real liquids are viscous, which is sufficient for transverse modes with wavelengths comparable to atomic dimensions to propagate with little attenuation. Faber observes that, on heating, viscosity decreases and transverse modes become hindered, adding that “It is no doubt for this reason, as Wannier and Piroue (1956) have pointed out, that cv diminishes on heating in the way that it does.” Faber further notes say that cv tends to cv = 2 at high temperature close to the critical point, consistent with the idea that by this stage transverse excitations are no longer contributing. There are two exceptions to the list of liquid textbooks (or any textbooks, for that matter) in which liquid specific heat is discussed in some detail. In both of these cases, the specific heat is linked to phonons, albeit in a different way. The first textbook is by Wallace [54]. As mentioned in Chapter 2, Wallace recognises the closeness of liquid cv close to melting to cv in parent crystals and attributes this closeness to 3N propagating phonons in liquids. Wallace notes the decrease of liquid cv with temperature. The Wallace theory is based on the system vibrating in each minimum of the multi-valley energy landscape and transits between these minima. In this theory, the liquid free energy has the term related to all 3N phonons like in the solid, giving a contribution to the liquid energy as 3NT and cv = 3 at all temperatures. Calculating the corresponding entropy and adding to it a fixed term equal to the entropy of melting, Wallace finds a good agreement with the experimental data [54]. Another important term is the boundary term resulting from the truncation of the potential valleys at places where two valleys intersect. This boundary term depends on the amplitude of the phonon motion. Adding the two terms, Wallace finds cv that decreases with temperature due to the increasing role of the boundary term [44]. The second more recent textbook is by Proctor [176]. Similarly to Wallace, Proctor links liquid thermodynamic properties to phonons. This link is direct and is based on the theory in Chapter 10. Proctor additionally undertakes a “detailed and rigorous” verification of this theory, exploring different ways in which this theory could be falsified, including the values of physical parameters used, such as the Debye frequency, shear modulus, relaxation time and so on [175, 176]. Proctor concludes that the theory passes these tests in a wide range of pressures and temperatures, including both subcritical liquids and supercritical fluids.

14 The Supercritical State

14.1 What Happens at High Temperature? In Chapter 10, we discussed the decrease of liquid cv due to the reduction of the phase space available to phonon excitations. As the hopping inter-cage frequency ωF increases with temperature, the number of transverse excitations reduces. This gives the liquid energy in Eq. (10.9) as:  E = NT 3 −

ωF ωD

3 ! (14.1)

As discussed in Section 10.1, the smallest value of cv that this equation gives is F cv = 2 (neglecting the term with dω that is small at high temperature). cv = 2 dT is reached when the hopping frequency ωF becomes comparable to the highest frequency ωD : ωF ≈ ωD . What happens if we increase temperature even higher? On general grounds, cv should tend to its ideal-gas value of 23 : at high temperature the potential energy becomes small compared with the kinetic energy and can be ignored. Two things can happen: if the system is below the critical point, the liquid will boil and become a gas, and its specific heat abruptly changes to that of the gas. A more interesting case is when the system is above the critical point because here we can follow the gradual evolution of cv from 2 to 32 and therefore have a new physical picture of thermodynamics in the supercritical state. This necessitates the discussion of how particle dynamics change in high-temperature liquids. Let’s suppose we are in the supercritical state at temperature where ωF becomes comparable to ωD , corresponding to cv = 2 according to Eq. (14.1). The hopping frequency ωF cannot exceed the rattling frequency ωD ; ωD is the largest frequency in the system by definition. What happens to Eq. (14.1) and cv on further temperature increase? 133

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The Supercritical State

The answer is that Eq. (14.1) stops being applicable altogether because the character of particle motion qualitatively changes. Recall that Eq. (14.1) was derived in Chapter 10 in the regime where the Frenkel picture applies: each particle performs solid-like oscillations around quasi-equilibrium positions and diffusive jumps between those positions. These combined oscillatory and diffusive dynamics are quantified by the hopping frequency ωF (inverse of liquid relaxation time τ ). ω F ≈ ωD means that a particle spends about the same time oscillating as diffusing. In other words, the oscillatory component of particle motion disappears, and the particle motion becomes purely diffusive and gas-like. Further temperature increase results in the particle mean free path increasing beyond the inter-atomic separation, and this has consequences for system thermodynamics as discussed in Section 14.5. The disappearance of the oscillatory component of particle motion in the supercritical state corresponds to the Frenkel line (FL) and the crossover of dynamical, thermodynamic and structural properties of the system. Proposed about a decade ago, the FL has subsequently become an experimental reality [208]. The FL importantly affects collective excitations and thermodynamics, the main topic of this book, and for this reason we summarise the main effects related to the FL in the next section.

14.2 The Supercritical State and Its Transition Lines Until fairly recently, there was no reason to survey matter well above the critical point in any detail or understand it from a physical point of view. That part of the phase diagram was widely considered to be unremarkable and physically homogeneous, with no discernible differences between liquid-like and gas-like states [3, 19, 20, 37, 209]. This statement was just about all that was ever mentioned about the supercritical state in textbooks from a theoretical point of view. The first textbook explaining that the supercritical state of matter is in fact rich with exciting physics was published only recently, in 2021 [176]. Several proposals discussed the physical basis for separating liquid-like and gaslike states not far above the critical point. For example, this included the formation of “ridges” corresponding to the maxima of density fluctuations [210–213]. A later proposal introduced the Widom line, the line of constant-pressure heat capacity maxima persisting above the critical point on isobars [214]. This was based on the observation that the lambda-anomaly of heat capacity at the critical point does not disappear abruptly away from the point but instead becomes a maximum that progressively smears away from the critical point. This is seen in Figure 14.1 showing the experimental constant-pressure specific heat cp of Ar just below the critical pressure (Pc = 4.86 MPa) and at several pressures above it.

14.2 The Supercritical State and Its Transition Lines

Figure 14.1 Constant-pressure specific heat cp (a) and constant-volume specific heat cv (b) at different pressures. The graphs show the experimental data for Ar [45].

135

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The Supercritical State

We observe that the divergence and maxima in Figure 14.1 are considerably stronger in cp than in the constant-volume specific heat cv . We also see that the maxima and anomalies of cv disappear faster and at lower pressure than those of cv , implying that the (P, T) line of maxima of cv is different from that of cp and the Widom line. The difference in behaviour of constant-pressure and constant-volume heat capacities Cp and Cv can be attributed to the equation: Cp = Cv + VTα 2 B

(14.2)

where α is thermal expansion and B is bulk modulus, which shows that the anomalies of α and B contribute to the anomaly of cp . Interestingly, the overall decrease of cv from about cv = 3 to its limiting ideal-gas value at high temperature cv = 32 is seen across the entire pressure range in Figure 14.1(b), except for a fairly narrow range of critical anomalies at low pressure. This is similar to what we have seen in Figure 2.2 at higher pressures, and it shows that cv is a more informative and generic property of a system’s physical behaviour than cp . Indeed, differently from cp , cv is unaffected by system-specific α and B (see Eq. (14.2)). Our final observation from Figure 14.1 is that both cp and cv tend to their limiting ideal-gas values at high temperature: cv = 32 and cp = 52 . We will discuss the high-temperature supercritical behaviour of specific heat in Section 14.5. The line of persisting maxima is not unique: in addition to cp , one can similarly define lines of maxima of other properties such as cv , compressibility, thermal expansion and density fluctuations. These lines, as well as the Widom line related to cp , are close to each other in the vicinity of the critical point, but they diverge away from the critical point [176, 215]. There are two further important properties of these lines. First, the lines of maxima depend on the path taken on the phase diagram and are different on isobars, isotherms, isochores and so on. Second, the maxima of the heat capacity, compressibility, thermal expansion and so on disappear around (2.5-3)Tc , where Tc is the critical temperature, and so do the corresponding lines, including the Widom line [176, 215]. More recently, the FL was introduced [53, 216, 217] as a dynamical crossover line in the supercritical state. Recall the Frenkel picture of particle dynamics in Chapter 7: a particle oscillates around a quasi-equilibrium point and performs jumps between two nearest points with a timescale set by the liquid relaxation time τ . Frenkel was thinking about usual subcritical liquids; however, the same picture applies above the critical point as well. Let’s fix pressure above the critical point and start in the low-temperature part of the phase diagram in Figure 14.2. τ decreases with temperature, and the proportion of time that a particle spends oscillating decreases too. At some point, a particle stops oscillating, and its motion becomes purely diffusive. This corresponds to the dynamical crossover

14.2 The Supercritical State and Its Transition Lines

137

Figure 14.2 The FL on the phase diagram showing the combined oscillatory and diffusive particle motion below the line and purely diffusive motion above the line. Schematic illustration.

that separates the combined oscillatory and diffusive motion below the FL from purely diffusive above the line. The FL is illustrated in Figure 14.2. Introduced about 10 years ago, the FL has been experimentally confirmed in several supercritical fluids using different experimental techniques [208]. We will return to this point later. As discussed in the previous section, the disappearance of the oscillatory component of motion corresponds to ωF ≈ ωD or τ ≈ τD . This is a qualitative criterion of the FL, which quickly illustrates what happens to particle dynamics, and is approximate. There are two quantitative criteria of the FL, which give the same line on the phase diagram [217]. The first, dynamical, criterion is based on the form of the velocity autocorrelation function (VAF). We note that calculating and analysing VAF in liquids dates back to an early era of computer modelling [157]. Oscillatory behaviour and monotonic decay of VAF was previously linked to liquid-like and gas-like behaviour, respectively [218, 219]. In a similar spirit, the FL was defined as the line where the solid-like oscillatory behaviour of the VAF and its minima disappeared. The second quantitative criterion of the FL is thermodynamic and defines the line where cv = 2 in the classical case. This value is our old friend: we first encountered it in Eq. 10.12 in Section 10.1. As discussed earlier, cv = 2 is the limiting high-temperature result from Eq. (14.1). Physically, this is because τ ≈ τD at the dynamical crossover corresponds to the complete disappearance of transverse modes from the liquid spectrum. This also corresponds to the Brillouin result, which he derived assuming that all transverse modes disappear in the liquid on melting (see Chapter 13). The reason the dynamical crossover corresponds to the disappearance of transverse waves can be understood as follows. Setting τ = τD in the equation for k-gap (Eq. (8.8)) gives the requirement for transverse modes

138

The Supercritical State

to have k larger than kg = 2cτ1 D ≈ 1a , where a is the inter-atomic separation. This results in the disappearance of all transverse waves from the system spectrum because the wavelength cannot become shorter than the inter-atomic separation. The same of course follows from the Frenkel theory stating that propagating transverse excitations exist above the frequency ωF = τ1 in Eq. (7.3). Setting τ ≈ τD corresponds to the highest frequency in the system, ωD , in which case we again find the disappearance of transverse waves from the spectrum. This disappearance of transverse modes is supported by the results of molecular dynamics simulations where transverse modes are directly calculated from the transverse current correlation functions [120]. We note that the thermodynamic criterion applies to harmonic or quasi-harmonic excitations where the energy of each mode is T as assumed in Eq. (14.1), but not to idealised and exotic systems such as soft-sphere systems with a large repulsion exponent or the hard-sphere system as its limit [53, 217]. As discussed in Chapter 2, these systems are unrelated to real liquids in terms of their thermodynamic properties, since their cv = 32 like in the ideal gas. That the system loses shear resistance at all frequencies in its spectrum is an important transition point. As discussed in Chapter 7, the ability of liquids to flow is often associated with their zero rigidity that markedly distinguishes liquids from solid glasses (recall our discussion of the earlier work of Brillouin and Born in Chapter 13 who used the absence of rigidity as the physical criterion of melting). However, this implies zero rigidity at small frequency or wavevectors only. At higher frequency ω and wavevector k, a liquid supports shear stress as discussed in Chapters 7 and 8. On the other hand, the FL corresponds to the loss of shear resistance at all frequencies in its spectrum and to the transition between the “rigid” liquid-like state (in a sense of its ability to support shear modes) and nonrigid gas-like fluid. We note that the disappearance of shear modes does not imply that the high-frequency shear modulus G∞ becomes zero. G∞ is always non-zero, including in the non-interacting ideal gas state where its given by the purely kinetic term G∞ = nT [220]. The FL idea stimulated a series of experiments, which consistently found the transitions at the FL in supercritical Ne [221], N2 [222, 223], CH4 [224], C2 H6 [225] and CO2 [226] using X-ray, neutron and Raman scattering experiments and their combinations. The transitions were found at the state points previously mapped using the dynamical VAF criterion [227] and in the deeply supercritical state (10–1,000 times the critical pressure) where the near-critical anomalies and the Widom line do not exist. In addition to these systems listed, the transition at the FL was also found to operate in supercritical H2 O on the basis of earlier experiments [208].

14.2 The Supercritical State and Its Transition Lines

139

We do not review this experimental evidence for the FL here because this evidence is structural and outside the main topic of this book; instead, we refer the reader to the recent review [208]. Here, we make two points about the experimental evidence related to the FL. First, the reason the earlier experiments focused on supercritical structure is that structure is easier to measure and analyse than dynamics or thermodynamics in extreme conditions. Second, structure and dynamics are related on general grounds. Below the FL, particles oscillate around quasiequilibrium positions and occasionally jump between them below the FL. The average time between jumps is given by the liquid relaxation time, τ . Therefore, a static structure exists during time τ for a large number of particles below the FL, giving rise to the well-defined short- and medium-range order comparable to that existing in disordered solids. On the other hand, particles lose the oscillatory component of motion and start to move in a purely diffusive manner like in gases above the FL. This implies that the features of the pair distribution function g(r) and its Fourier transform, the structure factor S(q), are expected to be liquid-like below the FL and gas-like above the line. As a result, the features of g(r) and S(q) have a different temperature dependence below and above the FL. In addition, structure, dynamics and thermodynamics are all inter-related as discussed in Chapter 6. Indeed, the disappearance of transverse modes implies that cv crosses over from the temperature dependence set by Eq. (14.1) to a different dependence governed by the remaining longitudinal mode (see the next section). Such a crossover of cv is indeed seen in molecular dynamics simulations [228]; however, a more dramatic manifestation of the thermodynamic crossover will be shown in Section 14.3. We now recall Eq. (6.2) and see that the crossover of cv implies the crossover of g(r) and the supercritical structure. We make three further remarks related to the FL. First, the crossover of particle dynamics affects other physical quantities of the supercritical system. Viscosity, speed of sound and thermal conductivity all decrease with temperature in the liquid-like regime below the FL but increase with temperature above the line like in gases (i.e. have minima). The diffusion constant crosses over from exponential temperature dependence in the liquid-like regime to power-law dependence like in gases. Depending on the path chosen on the phase diagram, the crossovers and minima of these properties may deviate from the FL but often remain close to it [229, 230]. In Chapter 15, we will elaborate on viscosity minima in more detail and will discuss their important property: the viscosity minima turn out to be universal and quantum, expressible in terms of fundamental physical constants including ~. Second, differently from the Widom line, the FL extends to arbitrarily high pressure and temperature (as long as the system remains chemically unaltered; a

140

The Supercritical State

chemical and/or structural transformation or ionisation may result in a new and different FL). Another important difference from the Widom line is that the FL is physically unrelated to the critical point and exists in systems such as soft-sphere systems without a critical point or a boiling line [217]. Third, in systems where the boiling line and critical point exist, the FL continues below the critical point and touches the boiling line slightly below the critical temperature [217]. Crossing the FL below the critical point was experimentally confirmed in [225]. Below the critical point, crossing the FL by increasing the temperature at a fixed pressure corresponds to the transition between gas-like and liquid-like dynamics. On further temperature increase, crossing the boiling line corresponds to a phase transition between two gas-like states with different densities, where the lower temperature phase is regarded as a dense gas because a cohesive liquid state ceases to exist close to the critical point [231]. Close to the critical point itself, particle dynamics is gas-like in the sense discussed earlier because the FL touches the boiling line below the critical temperature as just mentioned. However, this is inconsequential for system properties near the critical point because these properties are largely governed by large critical fluctuations and anomalies, which set the thermodynamic, dynamical and elastic behaviour of the system [37, 232, 233]. In this regard, it is interesting to recall the Fisher–Widom line (FWL), which, similarly to the FL, touches the boiling line slightly below the critical point [176]. The FWL separates the fluid state where the pair distribution function has an oscillatory space decay from the state where the pair distribution function has one single decaying peak only. Experimentally, the FWL has not been detected [176]. Originally deduced for one-dimensional systems [234], the FWL was later studied in a three-dimensional system using modelling where the truncation of inter-atomic potential at short distance was found to be essential to obtain the transition between oscillatory and decaying pair distribution function behaviour [235, 236]. It is unclear how far the FWL extends above the critical point [176]. 14.3 Applications of Supercritical Fluids Building on the fundamental understanding of the phase diagram and new physical effects in its supercritical part, recent developments have come at an important time from the application point of view. Supercritical fluids are increasingly employed in a wide variety of applications in industry [176, 209, 237–239]. For example, supercritical CO2 and H2 O are used for chemical extraction (e.g. producing decaffeinated coffee), biomass decomposition, dry cleaning, fluid chromatography, chemical reactions, impregnation and dyeing, supercritical drying, pharmacology, microparticle formation, power generation, biodiesel production, carbon capture and storage, and refrigeration and as antimicrobial agents. The environment is one

14.4 C Transition

141

area where the application of supercritical fluids is particularly important. There is growing pressure from the increasing amount of organic and toxic waste entering the environment, and the need for remediation and effective green solutions for waste processing is pressing. The range of waste is wide and includes toxic chemicals such as biphenyls, dioxins, pharmaceutical waste, highly contaminated sludge from the paper industry, bacteria and stockpiles of warfare agents. Current technologies such as landfilling and incineration have problems related to environmental pollution that are becoming increasingly difficult and costly to handle due to higher treatment standards and limitations introduced by state regulations. It is here where supercritical fluids are useful: they have been deployed as effective and environmentally friendly cleaning and dissolving systems. Destroying hazardous waste using supercritical water has been shown to be a viable industrial method and superior to other existing methods [209, 237, 238]. The high extracting and dissolving efficiency of supercritical fluids is due to a combination of two factors. First, the density of supercritical fluids at typical industrial pressures and temperatures is much higher than that of gases, and is close to that of subcritical liquids. High density ensures efficient interaction of the molecules of the supercritical fluid and the solute. Second, the diffusion coefficients in a supercritical fluid at typical industrial conditions are 10–100 times higher than in subcritical liquids, promoting solubility and dissolving. Therefore, supercritical fluids have the best of both worlds: they possess the high density of liquids and the large diffusion constants of gases. Despite widening industrial use of supercritical fluids, it has been noted that their theoretical understanding has been lacking [209, 237, 238]. Instead, the use of supercritical fluids has been informed by empirical observations and test results. It has been widely noted that improving theoretical knowledge of the supercritical state is important for scaling up, widening and increasing the reliability of these applications (see, for example, [209, 240–244]). In this sense, an earlier observation and explanation that the solubility maxima (“ridges”) correspond to the FL [227] is industrially relevant. Astronomy and planetary science is another area where supercritical fluids are important [176, 208, 245–247]. For example, gas giants such as Jupiter and Saturn contain supercritical hydrogen, which governs the properties of these and other planets. A theoretical understanding of the supercritical state is therefore useful for understanding astronomical objects and their behaviour. 14.4 C Transition The separation of the supercritical state into two different states using the FL discussed in Section 14.2 involves a physical model that uses the VAF dynamical criterion or thermodynamic criterion cv = 2. It is interesting to ask whether

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The Supercritical State

this separation can also be done in a way that is model-free? A related question is whether the separation involves universality across all supercritical systems in terms of suitably identified physical parameters? It turns out that the results in Chapter 8 enable us to construct a plot that gives an unambiguous, model-free and universal separation of the supercritical diagram into the liquid-like and gas-like states. The central thermodynamic property we consider here, as we do throughout this book, is cv . The second parameter is the dynamical elasticity length: del = cτ

(14.3)

which plays an important role in liquid dynamics and thermodynamics as we saw in Eq. (8.9) in Section 8.1. Molecular dynamics simulations show that cv versus η and cv versus τ have very different dependencies along the isochores, isobars and isotherms [248, 249]. However, cv versus del = cτ turns out to be a nearly universal plot where all paths approximately collapse on the same c-shaped graph as shown at the top of Figure 14.3. The bottom of Figure 14.3 shows cv (del ) in several different types of fluids: noble Ar, molecular N2 and CO2 , and metallic Pb. We recall Chapter 9 on molecular dynamics simulations, where we discussed different ways in which these simulations can be used to understand, interpret and complement the experimental data, provided we have a physical model. Molecular dynamics simulations are particularly useful to study the supercritical state at extreme conditions because no experimental data enabling us to construct the plot such as that in Figure 14.3 currently exist. This range of temperature and pressure is too high for current experiments, and we can compare molecular dynamics and experimental data of viscosity at fairly low pressure and temperature only [248]. Moreover, the values of high-frequency G needed to calculate τ are scarce even at room temperature and pressure [96], let alone at extreme supercritical conditions. The key feature of the C-transition in Figure 14.3 is the inversion point corresponding to the change of the sign of the derivative of cv with respect to del . The origin of the inversion point is as follows. The dynamical length del (Eq. (14.3)) always has a minimum as a function of temperature when crossing from liquidlike to gas-like regimes of particle dynamics. Recall that this crossover is related to the dynamical crossover at the FL discussed in Section 14.2 earlier. Indeed, τ and del = cτ in Eq. (14.3) decrease with temperature below the line. In the gaslike regime above the FL, del becomes the particle mean free path, which increases with temperature (see the end of the next section). The inversion point is therefore related to the transition between liquid-like and gas-like particle dynamics. The coordinates of the inversion point are physically significant. cv ≈ 2 at the inversion point is our old friend. It is the high-temperature limit of the liquid-like

14.4 C Transition

Figure 14.3 Top: specific heat cv as a function of del = cτ across nine different paths (isotherms, isochors and isobars) spanning the supercritical state up to 330 Tc and 8,000 Pc in argon. The data are reproduced from [248] with permission from the American Physical Society. Bottom: the same but in different supercritical systems: Ar, N2 , CO2 and Pb. In molecular systems, cv is heat capacity per molecule and has an additional rotational term contributing 1 to cv subtracted from the calculated cv for comparison. Different paths for different systems have a fixed path- and system-independent inversion point close to cv = 2 and del = cτ = 1 Å. From [249]: Cockrell and Trachenko, some rights reserved; exclusive licensee AAAS. Distributed under a Creative Commons Attribution NonCommercial License 4.0 (CC BY-NC).

143

144

The Supercritical State

regime where transverse waves propagate, according to Eq. (14.1). It is also the Brillouin result discussed in Chapter 13. del = cτ ≈ 1 Å at the inversion point is important too: it is close to the interatomic separation (“ultraviolet cut-off ” in condensed matter) marking the shortest wavelength in the system. Figure 14.3 shows the role played by the dynamical length (Eq. (14.3)) in both liquid-like and gas-like regimes of the supercritical state. del decreases with temperature, reducing the phase space available for transverse phonons as discussed in Chapter 8. When τ approaches its shortest value comparable to the Debye vibration period τD , kg = 2d1el (see Eq. (8.8)) approaches the Brillouin zone boundary because cτD = a, where a is the inter-atomic separation on the order of angstrom. At this point, all transverse modes disappear, corresponding to cv = 2 as Eq. (14.1) predicts. On further temperature increase, the system crosses over to the gas-like regime where the phonon phase space continues to reduce, albeit now for longitudinal phonons as discussed in the next section. The associated potential energy of the longitudinal phonons reduces, eventually resulting in cv = 23 in Figure 14.3. We will discuss this in the next section in more detail. Due to its path-independence, the inversion point in the c-transition serves as an unambiguous and model-free way to separate the liquid-like state at the top part of the c-plot from the gas-like state at the bottom part. The line of inversion points on the phase diagram lies close to the FL determined from the dynamical VAF criterion as expected [248]. The supercritical plot in terms of parameters (del , cv ) has an interesting double universality, as seen at the bottom of Figure 14.3. The first universality is that, for each system, the c-transition plot has an inversion point that is fixed and corresponds to about λd = 1 Å and cv = 2 for all paths on the phase diagram, including isobars, isochores and isotherms, spanning orders of magnitude of temperature and pressure. The second universality is that this behaviour is generic on the supercritical phase diagram and is the same for all fluids. Figure 14.3 shows a relationship between a thermodynamic quantity, cv , and a dynamical quantity, del . In this sense, the c-transition can be considered a dynamical–thermodynamic transition with double universality, as mentioned earlier. The fixed point of this dynamical–thermodynamic transition approximately corresponds to del = 1 Åand cv = 2. Figure 14.3 importantly demonstrates one of the main points discussed in this book: the relationship between collective excitations and thermodynamic properties of liquids. Above the inversion point, the transverse phonon states below the FL are governed by del = cτ as shown in Chapter 8. As discussed in Chapter 10, these phonons contribute to the liquid energy and cv . Therefore, del serves as a relevant physical parameter governing cv .

14.5 Excitations and Thermodynamic Properties

145

An interesting question is why cv is also governed by del below the inversion point in Figure 14.3, where the transverse modes disappear and the only propagating wave is longitudinal? The answer is that del below the inversion point becomes the particle mean free path, which governs the wavelength of the remaining longitudinal phonon. This is discussed in Section 14.6.

14.5 Excitations and Thermodynamic Properties From the point of view of the main topic of this book, namely liquid thermodynamic properties and collective excitations, the most important aim of the preceding sections in this chapter was to ascertain collective excitations in the supercritical state. We now possess all of the required ingredients and data to discuss thermodynamic properties of the supercritical state and relate those properties to collective excitations. In the supercritical state below the FL, particle dynamics combine the oscillatory and diffusive components of motion. This is the Frenkel picture of particle dynamics in subcritical liquids as discussed in Chapter 7. This has the same consequences for excitations in the system as in the case of subcritical liquids: two transverse modes operate above kg and become propagating above the frequency ωF . Therefore, the same assertions about liquid thermodynamic properties including its energy can be readily made with no new calculations. In particular, we can use the results in Chapter 10 and Eq. (14.1) in particular to calculate the cv of supercritical fluids below the FL and compare this cv to experiments. We have already done this comparison in Figure 10.3 where the temperature range includes the supercritical state below the FL. At the FL, two transverse modes disappear as discussed in Section 14.2. What happens to excitations in the purely diffusive regime of particle motion above the FL? The remaining excitation is the longitudinal mode. Similarly to transverse modes in subcritical liquids and supercritical fluids, the longitudinal mode becomes gapped; however, the gap is not on the low k or low ω side. Instead, it is on the side of large k. Indeed, the purely diffusive character of molecular motion above the FL means that we can now operate in simple terms of the kinetic gas theory where the key parameter is the particle mean free path L. L increases with temperature because particle energy increases. In the hard-sphere approximation, L = nσ1 , where n is concentration and σ is the cross section. L increases on the isobar and is mostly related to the decrease of n. On the isochore, L decreases slower and due to the effective decrease of σ at higher energy. An important consequence of L for the longitudinal wave is that its wavelength cannot be shorter than the particle mean

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The Supercritical State

free path L because the motion at distances shorter than L is not oscillatory by definition. However, the oscillatory motion exists at distances larger than L, and the condition λ  L serves as a condition to apply the continuum approximation discussed in Section 3.2. We therefore arrive at the condition of propagating longitudinal modes: λ>L

(14.4)

Eq. (14.4) is the longitudinal analogue of the condition for transverse waves to emerge, k > kg in Eq. (8.8), or Frenkel’s condition ω > ωF . Eq. (14.4) is supported by extensive molecular dynamics simulations showing the crossover of longitudinal waves at λ = L as Eq. (14.4) predicts [250]. At large λ, we get the expected result that the system is able to support long-wavelength longitudinal waves, sound. We now recall the results in Chapters 7 and 8 where we saw that the transverse modes in low-temperature subcritical liquids are gapped, with the gap developing starting from low ω and k. Eq. (14.4) shows that the longitudinal mode in high-temperature supercritical fluids above the FL is also gapped. In an interesting difference from the transverse modes in low-temperature liquids below the FL, the gap in the longitudinal mode starts developing from high ω and k. We will develop Eq. (14.4) later in this chapter. Here, we note that this equation represents the same general effect we observed in Chapter 8: the reduction of the phase space available to phonons with temperature. Before discussing the cv of gas-like supercritical fluids, it is useful to summarise collective excitations in both subcritical and supercritical liquids. Figure 14.4 illustrates that, at low temperature, liquids have the same set of collective modes as in solids: one longitudinal phonon and two transverse phonons. In the viscous regime at low enough temperature where τ  τD or ωF  ωD , the liquid energy is almost entirely given by the vibrational energy due to these phonons, as discussed in Chapter 16. On temperature increase, the number of transverse modes propagating above the frequency ωF decreases, reducing the phonon phase space. At the FL where particles lose the oscillatory component of motion and start moving diffusively as in a gas, the two transverse modes disappear. Further temperature increase reduces the phonon phase space further, namely the phase space available for longitudinal phonons according to Eq. (14.4). In the limit of large L, we are left with the long-wavelength longitudinal sound wave. We summarised this process in Table 3.1. We are now ready to calculate the thermodynamic properties and energy of the supercritical system above the FL. The absence of the longitudinal mode with λ < L implies that the potential energy associated with that mode is 0. Then the energy is given by the system kinetic energy and the potential energy of the

14.5 Excitations and Thermodynamic Properties

147

Figure 14.4 Evolution of transverse and longitudinal waves in liquids and supercritical fluids. The variation of colour from darker at the bottom to lighter at the top corresponds to temperature increase. The figure shows that the transverse waves (left) start disappearing with temperature increases, starting with longwavelength modes and completely disappearing at the FL. The longitudinal waves (right) exist up to the FL but start disappearing above the line, starting with the shortest wavelength, with only long-wavelength longitudinal modes propagating at high temperature. Reproduced from [74] with permission from the Institute of Physics Publishing.

longitudinal mode with λ > L, Pl (λ > L). For comparison, we first recall the energy for subcritical liquids, Eq. (10.4): E = K + Pl + Pt (ω > ωF )

(14.5)

In our current case of liquids with only one longitudinal mode, Pt is zero at all frequencies, and the energy is: E = K + Pl (λ > L)

(14.6)

148

The Supercritical State

or

El (λ > L) (14.7) 2 where El (λ > L) is the energy of the longitudinal mode with wavelength λ > L. F) Similarly to how we calculated Pt (ω > ωF ) = Et (ω>ω in Chapter 10, we calcu2 late El (λ > L) as the integral of the phonon energy E(ω, T) with the Debye density of states gl (ω) = ω3N3 ω2 , where ωD is the Debye frequency and we have taken into D account that the number of longitudinal modes is N. This gives: E=K+

3N El = 3 ωD

Zω0

E(ω, T)ω2 dω

(14.8)

0

is the upper frequency cut-off due to Eq. (14.4). where ω0 = k 2π L Taking the classical case E(ω, T) = T is well justified at high supercritical tem 3
3 = NT La , where a is perature above the FL. Integrating gives El = NT ωωD0 the inter-atomic distance on the order of angstrom set by the cohesive interaction as before. Using this in Eq. (14.6) gives the energy of the supercritical fluid in the gas-like regime above the FL as: 3T T  a 3 E= + (14.9) 2 2 L For L  a corresponding to a dilute gas, Eq. (14.9) gives E = 3T and the 2 3 ideal-gas value cv = 2 as expected. The kinetic theory of gases (see, for example, [251]) considers dilute systems where L  a is a good approximation. For this reason, the cv of monatomic dilute gases is close the ideal-gas value (this gets corrected by internal degrees of freedom in molecular gases). On the other hand, L  a does not apply to dense supercritical fluids. In particular, L becomes comparable to a close to the FL where the particle motion changes from combined solid-like oscillatory and diffusive to purely diffusive. When L = a, Eq. (14.9) gives E = 2NT and cv ≈ 2, matching the high-temperature result of the liquid energy below the FL (see Eq. (14.1)). In deriving Eq. (14.9), we have assumed that the phonon energy is T in the harmonic case. Similarly to what we did in Section 10.3, we can consider the effect of anharmonicity on the phonon energy and relate it to the coefficient of thermal expansion α. Repeating the same steps as in Section 10.3, we find the energy of the supercritical fluids above the FL corrected for anharmonicity as:   3T T αT  a 3 E= + 1+ (14.10) 2 2 2 L which reduces to Eq. (14.9) if α = 0.

14.6 Liquid Elasticity Length and Particle Mean-Free Path

149

Eqs. (14.9) and (14.10) have no free fitting parameters. Indeed, in the purely diffusive regime above the FL, L is related to the mean-free path of the gas-like viscosity as: 1 η = ρvL (14.11) 3 where ρ is density and v is the average speed of particles. We can therefore calculate L from experimental viscosity using Eq. (14.11), using this L in Eq. (14.10) and calculating cv . This is the approach we used to calculate the liquid cv of subcritical liquids in Chapter 10 where ωF was calculated from the experimental viscosity. The calculation shows good agreement between the theoretical and experimental cv of supercritical Ne, Ar, Kr, Xe, N2 and CO. As in the case for subcritical fluids, the agreement is seen for the physically justified and typical values of parameters such as α ≈ 10−3 K−1 in fluids and so on [252].

14.6 Liquid Elasticity Length and Particle Mean-Free Path We add several remarks regarding dynamics and thermodynamics in the hightemperature gas-like state. These remarks involve the relationship between the liquid elasticity length del in Eq. (14.3) and the particle mean-free path L in the gas-like regime above the FL and below the inversion point in Figure 14.3. First, Eq. (14.7) and its consequence, Eq. (14.9), are based on Eq. (14.4) asserting that longitudinal phonons in the high-temperature gas-like state propagate with wavelengths larger than L. This gives a gap in the momentum state. We now recall the gapped momentum state in the transverse sector involving kg in Eq. (8.8). We saw in Chapter 8 that kg is related to the elasticity length del = cτ in Eq. (8.9) because it sets the wave propagation range. This brings the question of whether del can be related to the longitudinal gap set by Eq. (14.4) as well? This relationship can indeed be made. In the absence of transverse waves, the low-frequency longitudinal speed of sound is set by the static bulk modulus K1 in Eq. (7.23), where τ is related to bulk relaxation time τB . The corresponding dynamical length del is del = cτB = c KηB0 , where ηB is the bulk viscosity in the gaslike regime, ηB ≈ η = 31 ρvL as in Eq. (14.11). Recalling that the speed of sound c in the high-temperature gas-like regime becomes the thermal speed of particles v and that K1 in the gas-like state is K1 = P = nT (see Section 3.2), del becomes: del =

1 ρv2 L 3 nT

(14.12)

or, in view of 12 ρv2 = 32 nT, del = L

(14.13)

150

The Supercritical State

Therefore, the liquid elasticity length del , which defines the k-gap in the transverse sector at low temperature below the FL, becomes the particle mean-free path in the high-temperature gas-like regime above the FL. In view of Eq. (14.4), this implies that del = L governs the range of wavevectors of the longitudinal waves in the gas-like state above the FL. Our second remark brings us back to the C transition in Section 14.4. We can now understand the reason for the near collapse of cv as a function of del = cτ in Figure 14.3. Recall that the reason for choosing the dynamical length del = cτ as the relevant control parameter was that it governs the propagation of transverse phonons in the liquid-like regime below the FL. Why would the collapse work below the inversion point and above the FL where these transverse modes have long disappeared? This can be understood from Eq. (14.13): del becomes the particle mean-free path in the high-temperature gas-like regime and therefore governs the longitudinal phonon states below the inversion point in Figure 14.3 according to Eq. (14.4). The only caveat here is that τ in Figure 14.3 was calculated using the shear, rather than the bulk, modulus and viscosity. However, the shear modulus in a high-temperature gas-like fluid where the interaction energy is small compared with kinetic energy, G∞ = nT [220], and is the same as K1 = nT in the gas-like state used in the derivation of Eq. (14.13). Therefore, Eq. (14.13) applies to calculations in the hightemperature gas-like state involving both bulk and shear moduli.

14.7 Heat Capacity of Matter: From Solids to Liquids to Gases Let us now take a wider view of the specific heat of matter. In Chapter 4, we discussed the Dulong–Petit result for the specific heat of solids, cv = 3. Not considering anharmonic effects, this is the largest value of classical specific heat among all three states of matter (solid, liquid, gas) due to the motion of ions. In Chapter 10, we discussed the decrease of cv from 3 to 2 in the liquid-like regime of liquid dynamics, which includes both subcritical and supercritical systems below the FL (see Eq. (14.1)). In the previous section, we saw that Eq. (14.9) gives the subsequent decrease of cv from 2 to its smallest value possible, the ideal-gas value 23 . We have therefore considered and rationalised all possible classical values of cv on the phase diagram of matter: from cv = 3 in solids to the ideal-gas value cv = 23 . As discussed in Chapters 3 and 4, understanding solids was greatly simplified by quadratic forms and collective modes, phonons, that readily emerge. The thermodynamic properties of solids and in particular their cv are then readily related to these phonons, resulting in cv = 3. To understand the decrease of cv from 3 to 32 involves the liquid state. We have seen that liquids are more sophisticated and refined than solids and gases

14.7 Heat Capacity of Matter: From Solids to Liquids to Gases

151

due to their ability to combine oscillatory solid-like and diffusive gas-like components of motion. We have also seen these two components endow the phase space for phonons in liquids with an important property: its ability to change with temperature. We have seen that the decrease of cv from 3 to 2 and then to 23 is due to the same general effect: the reduction of the phase space available to collective excitations in the system. However, this reduction proceeds differently in different dynamical regimes. The reduction of the volume of the phase space first starts with the disappearance of transverse excitations starting from the lowest k and ω. When these excitations disappear at the FL, the phase space continues to reduce due the remaining longitudinal mode disappearing starting from the shortest k as shown in Figure 14.4.

15 Minimal Quantum Viscosity from Fundamental Physical Constants

15.1 Viscosity Minima and Dynamical Crossover This chapter is not directly related to the main topic of this book: understanding liquid thermodynamic properties on the basis of collective excitations. However, it is related to the crossover of particle motion itself. We have seen that this motion ultimately determines collective excitations in two different regimes of liquid dynamics. In Chapters 7 and 8, we saw that the phonon states in low-temperature subcritical liquids are governed by liquid relaxation time (or hopping frequency ωF ). Later, in Chapter 14, we saw that the phonon states in high-temperature supercritical fluids are governed by the particle mean-free path L. ωF and L are related to liquid-like and gas-like viscosity, and the crossover between the liquid-like and gas-like states results in the emergence of an interesting property: the minimal quantum viscosity. As discussed in this chapter, this has several interesting and wide-reaching implications. A number of important physical properties seem to have lower or upper bounds in the sense that they never fall below or exceed certain values. Understanding the origin of these bounds has enthralled physicists, including those interested in collective dynamics [253–261]. More generally, finding and understanding the bounds often means that we understand or clarify the basic underlying physics of the effect or property in question. An interesting lower bound was found in holography (see, for example, [262] for a review), the approach that maps strongly interacting field theories with weakly interacting gravity duals. Holography seeks to overcome the problem of treating strongly coupled field theories where perturbation theories do not work (we will return to this problem in Chapter 19). In this sense, the problem of strongly interacting field theories is similar to the liquid theory problem discussed in the Preface, the Introduction and Chapter 6 where it was noted that the traditional theories fail in liquids because interactions are strong and system-specific.

152

15.1 Viscosity Minima and Dynamical Crossover

153

Holography seeks to overcome the problem of strong interactions by finding a weakly interacting gravity dual that can be solved. Treating a weakly interacting gravity model using string theory methods yields the lower bound for the ratio of viscosity η and the volume density of entropy s as [253]: η ~ ≥ s 4π

(15.1)

This has generated ample interest in the area of holography and string theory. The bound (Eq. (15.1)) is referred to as the “perfect fluidity” bound and is being explored in different systems, including strongly interacting Bose liquids, ultracold Fermi gases and quark–gluon plasma [263]. Interestingly, the bound (Eq. (15.1)) was compared with viscosity minima in real liquids such as water and nitrogen [253]. We now know where these minima come from: recall our discussion of the dynamical crossover in Section 14.2. There we noted that viscosity, speed of sound and thermal conductivity all decrease with temperature in the liquid-like regime but increase with temperature above the line like in gases (i.e. have minima). What we can now do is calculate the value of viscosity at the minima. We did not need to do this calculation in the context of collective excitations and understanding liquid thermodynamics. However, we will see that this calculation reveals several interesting properties, makes a connection to an earlier open question posed by Purcell and has implications for living matter. Let’s start with recalling the origin of viscosity minima. Both dynamic viscosity η and kinematic viscosity ν have minima as a function of temperature. These minima are convenient to study in the supercritical state where no boiling phase transition intervenes; however, the same results for the viscosity minima follow for subcritical liquids [264]. Figure 15.1 shows minima of kinematic viscosity of several liquids as an example. As discussed in Chapter 7, particle dynamics in low-temperature liquids consists of oscillations around quasi-equilibrium positions and jumps between those positions after time τ , the liquid relaxation time, has elapsed. τ decreases with temperature as τ = τ D exp UT , where τD is related to the inverse attempt frequency and U is the activation energy barrier (U can be temperature-dependent). The same applies to viscosity η (we recall that η = τ G∞ ): 

U η = η0 exp T

 (15.2)

Particle dynamics and temperature dependence of η qualitatively change in the high-temperature gas-like state of liquids. As discussed in Section 14.2, there is no

154

Minimal Quantum Viscosity from Fundamental Physical Constants

Figure 15.1 Experimental kinematic viscosity of noble, molecular and network fluids showing minima. ν for He, H2 , O2 , Ne, CO2 and H2 O are shown at 20 MPa, 50 MPa, 30 MPa, 50 MPa, 30 MPa and 100 MPa, respectively. The experimental data are from [45].

oscillatory component of motion in this state, and therefore no “attempt” frequency. Instead, the particle motion becomes purely diffusive, and viscosity is given by Eq. (14.11) in Section 14.5: 1 η = ρvL 3

(15.3)

This equation predicts that η in gases does not depend on density or pressure because L ∝ ρ1 . It also predicts that gas viscosity increases with temperature because v increases. The increase of η at high temperature and its decrease at low temperature imply that η has a minimum, and we expect this minimum to correspond to the crossover between the gas-like and liquid-like viscosity. In the next section, we calculate the value of viscosity at the minimum. If temperature is increased at pressure below the critical point, the system crosses the boiling line and undergoes the liquid–gas phase transition. Viscosity still has a minimum, but undergoes a sharp change at the transition, rather than a smooth minimum as in Figure 15.1. In order to avoid effects related to the phase transition itself, it is convenient to consider matter above the critical point, the supercritical state, which we discussed in Chapter 14 in some detail.

15.2 Minimal Quantum Viscosity and Fundamental Constants

155

15.2 Minimal Quantum Viscosity and Fundamental Constants There are two ways in which ηm , viscosity at the minimum, can be evaluated. We can either take the low-temperature limit of the gas-like viscosity (Eq. (15.3)) or the high-temperature limit of the liquid-like viscosity given by the Maxwell relation η = Gτ discussed in Chapter 7. We start with the first approach and consider how η = ρvL changes with a temperature decrease (we drop the factor 13 in Eq. (15.3), since our calculation evaluates the order of magnitude of viscosity minimum as discussed in this section). η and the mean-free path L decrease when the temperature is lowered. L is bound by an ultraviolet cut-off in condensed matter systems: inter-particle separation a. From this point on, L has no room to decrease further. Instead, the system enters the liquidlike regime where η starts increasing on further temperature decreases according to Eq. (15.2) because the diffusive motion of molecules crosses over to thermally activated, as discussed in Section 14.2. Therefore, the minimum of η approximately corresponds to L ≈ a. When L becomes comparable to a, v can be evaluated as v = τaD because the time for a molecule to move distance a in this regime is given 1 by the characteristic timescale set by τD . Setting L = a, v = τaD = 2π ωD a, where m ωD is the Debye frequency, and ρ ≈ a3 gives ηm as: 1 mωD (15.4) 2π a We note that Eq. (15.3) applies in the regime where L is larger than a, and in this sense our evaluation of viscosity minimum is an order-of-magnitude estimation, as are our other results in this section. In this regard, we note that theoretical models can only describe viscosity in a dilute gas limit where perturbation theory applies [251], but not in the regime where L is close to a and where the energy of inter-particle interaction is comparable to the kinetic energy. In view of these theoretical issues as well as many orders of magnitude by which η can vary, this approximation for viscosity minimum is meaningful. In addition to be informative, an order-of-magnitude evaluation is unavoidable if a complicated property such as viscosity is to be expressed in terms of fundamental constants only as we intend to do in this section. ηm in Eq. (15.4) approximately matches the result obtained by approaching the viscosity minimum from low temperature where η is given by Eq. (15.2) and considering the Maxwell relationship η = Gτ discussed in Chapter 7. η and τ decrease with temperature according to Eq. (15.2), but this decrease is bound from below because τ starts approaching the shortest timescale in the system set by the Debye vibration period, τD . From this point on, τ has no room to decrease further. Instead, the system enters the gas-like regime where η starts increasing with temperature as in Eq. (15.3) because the thermally activated motion of molecules crosses over ηm =

156

Minimal Quantum Viscosity from Fundamental Physical Constants

to diffusive. Therefore, the minimum of η can be evaluated from τ ≈ τD . G can be estimated as G = ρc2 , where c ≈ τaD is the transverse speed of sound. Then, 2

1 mωD as in Eq. (15.4), where we used ρ = am3 as before. ηm = GτD = ρ τaD = 2π a It is useful to consider the kinematic viscosity ν = ρη . ν governs liquid flow as follows from the Navier–Stokes equation (Eq. (3.30)). ν also describes momentum diffusivity, analogous to thermal diffusivity involved in heat transfer and gives the diffusion constant in the gas-like regime of molecular dynamics [9]. Another benefit of considering ν is that it makes the link to the high-energy result (Eq. (15.1)) where η is divided by the volume density of entropy. 1 Using ν = ρη = vL, v = 2π aωD and L = a as before gives the minimal value of ν, νm , as: 1 νm = ωD a2 (15.5) 2π

We now recall that the properties defining the ultraviolet cut-off in condensed matter can be expressed in terms of fundamental physical constants. Two quantities of interest are the Bohr radius, aB , setting the characteristic scale of inter-particle separation on the order of Angstrom: aB =

4π 0 ~2 me e2

(15.6)

2

and the Rydberg energy, ER = 8πe 0 aB [58], setting the characteristic scale for the cohesive energy in condensed matter phases on the order of several eV: ER =

me e4 32π 2 02 ~2

(15.7)

where e and me are electron charge and mass, respectively. We now recall the known ratio between the characteristic phonon energy ~ωD
1 and the cohesive energy E, ~ωED . Approximating ~ωD as ~ maE 2 2 , taking the ratio ~ωD and using a = aB from Eq. (15.6) and E = ER from Eq. (15.7) gives, up to a E factor close to 1: ~ωD  me  21 (15.8) = E m The same ratio (Eq. (15.8)) follows by combining two known relations in metal
1 lic systems: ~ωED ≈ vcF , where vF is the Fermi velocity, and vcF ≈ mme 2 , providing an order-of-magnitude estimation ~ωED in other systems too [58]. Combining Eqs. (15.5) and (15.8) gives: 1 Ea2  me  21 νm = (15.9) 2π ~ m

15.2 Minimal Quantum Viscosity and Fundamental Constants

157

a and E in Eq. (15.9) are set by their characteristic scales aB and ER as discussed earlier. Using a = aB from Eq. (15.6) and E = ER from Eq. (15.7) in Eq. (15.9) gives a remarkably simple νm as: νm =

1 ~ √ 4π me m

(15.10)

The same result for νm in Eq. (15.10) can be obtained without explicitly using aB and ER in Eq. (15.9). Indeed, the cohesive energy, or the characteristic energy of electromagnetic interaction, is: E=

~2 2me a2

(15.11)

Using Eq. (15.11) in Eq. (15.9) gives Eq. (15.10). Another way to derive Eq. (15.10) is to consider the “characteristic” viscosity η∗ [265]: 1

(Em) 2 (15.12) η = a2 η∗ is used to describe viscosity scaling on the phase diagram: the ratio between viscosity and η∗ is the same for systems described by the same interaction potential in equivalent points of the phase diagram. For systems described by the LennardJones potential, the experimental and calculated viscosity near the triple point and close to the melting line is about three times larger than η∗ [265, 266]. Near the critical point, η∗ is about four times larger than viscosity and is expected to give the right order of magnitude of viscosity at the minimum at moderate pressure. The kinematic viscosity corresponding to Eq. (15.12) is: ∗

1

η∗ E2a = 1 (15.13) ρ m2 Using a = aB from Eq. (15.6) and E = ER from Eq. (15.7) in Eq. (15.13) gives the same result as Eq. (15.10) up to a constant factor on the order of unity. As before, we can also use Eq. (15.11) in Eq. (15.13) to get the same result. Eq. (15.10) is consistent with the experimental data: the ratio between experimental and predicted νm is in the range of about 0.5–3 [264]. Setting m = mp , where mp is the proton mass, corresponds to atomic H, and in this case Eq. (15.10) gives a new property: the fundamental quantum viscosity: νf =

~ 1 m2 ≈ 10−7 √ 4π me mp s

(15.14)

The value of νf is in agreement with the viscosity minima observed in Figure 15.1.

158

Minimal Quantum Viscosity from Fundamental Physical Constants

Figure 15.2 Fundamental kinematic viscosity νf depends on three values only: ~, me and mp .The other fundamental constants shown as examples are the electron charge e, speed of light c and gravitational constant G.

νf depends on three parameters: ~, me and mp , as illustrated in Figure 15.2. ~ and me are fundamental constants. Although mp depends on other standard model parameters, the dimensionless number mmpe is attributed a fundamental importance [267], as discussed in Section 15.8 in more detail. νf interestingly does not depend on electron charge e, contrary to what one might expect considering that viscosity is set by the inter-particle forces, which are electromagnetic in origin. Although e enters Eqs. (15.6), (15.7) and (15.11) for the Bohr radius, Rydberg and cohesive energy, it cancels out in Eq. (15.10) for νm and in Eq. (15.14) for νf . We observe that the minimal viscosity (Eq. (15.10)) turns out to be quantum. This may seem surprising and at odds with our thinking of high-temperature liquids as classical. Eq. (15.10) reminds us that the nature of interactions in condensed matter is quantum-mechanical in origin, with ~ affecting the Bohr radius and Rydberg energy. We will return to this point in Section 15.8. We note that our derivation of the viscosity minimum (Eq. (15.10)) and fundamental viscosity (Eq. (15.14)) involves more than a dimensional analysis. First, the dimensionless analysis is not unique in the absence of a physical model. As mentioned earlier, it is not a priori clear that νm should involve ~ and be a quantum property, especially so given that most liquids are considered classical. Second, a purely dimensional analysis can give a quantity with the right dimensions
but the wrong value. Indeed, multiplying the right-hand side of Eq. (15.10) by f mme , where f is an arbitrary function, gives the result consistent with the dimensional analysis but produces any desired value of νm with a suitable choice of f . Third, we have

15.3 Elementary Viscosity, Diffusion Constant and Uncertainty Principle

159

used a specific physical model to derive νm . We started by attributing the viscosity minimum to the crossover of microscopic particle dynamics, from combined oscillatory and diffusive to purely diffusive. This consideration led us to a particular regime of particle dynamics where the particle speed is set by the inter-atomic separation and elementary vibration period. Dimensional analysis alone does not have anything to say about why this regime would correspond to the minimum of viscosity. We next evaluated the minimal value of ηm using two approaches involving the Maxwell relation and the gas kinetic theory. Each of these approaches is based on a specific physical mechanism. We then related νm to the length and energy
1 scales using the ratio ~ωED = mme 2 in Eq. (15.8). The dimensionality analysis does not predict this ratio and is consistent with ~ωED taking any dimensionless number. We finally expressed the length and energy scales in terms of fundamental constants. Most of these steps involved in the derivation of Eq. (15.10) are physically guided and incorporate a lot more information than would be available from purely dimensional considerations. Our final observation in this section is that Eq. (15.14) is surprisingly close to the kinematic viscosity of a very different system, the quark–gluon plasma [268]. This is remarkable, given that viscosity and density in ordinary liquids and the quark–gluon plasma differ by 16 orders of magnitude and that the two systems have disparate interactions and fundamental theories. A hint for this remarkable similarity comes from the universality of the dynamical crossover discussed earlier. At the crossover, particle dynamics are neither liquid-like with many oscillations and occasional jumps nor gas-like where L  a, but instead are at the border between the two regimes. At this border, kinematic viscosity turns out to be fixed by the fundamental constants only and independent of charge as mentioned earlier. This can help explain the similarity of νm of ordinary liquids and the quark–gluon plasma [268].

15.3 Elementary Viscosity, Diffusion Constant and Uncertainty Principle Corresponding to atomic H, Eq. (15.14) gives the maximal value of the minimal kinematic viscosity. It is interesting to find a viscosity-related quantity that has an absolute minimum. This can be done by introducing the “elementary” viscosity ι (“iota”) defined as the product of ηm and elementary volume a3 : ι = ηm a3 or, equivalently, ι = νm m. Using Eq. (15.10), ι is: ~ ι= 4π



m me

 21 (15.15)

160

Minimal Quantum Viscosity from Fundamental Physical Constants

Eq. 15.15 has the absolute lower bound, ιm , for m = mp in H: ~ ιm = 4π



mp me

 12 (15.16)

which is on the order of ~ and interestingly involves the proton-to-electron mass ratio, one of a few dimensionless numbers of general importance [267]. For two of the lightest liquids, H2 and He, the minimum of νm, νm m = ι, should be close to the lower bound (Eq. (15.16)). The minima of experimental νm are in the range (1.5–3.5)~ for He and H2 , consistent with ιm in Eq. (15.16) [264]. ι is a convenient property to discuss the uncertainty relation and its implication for the lower bound. As discussed in the previous section, the minimum of ν can be evaluated as νm = va, corresponding to ι = mva = pa, where p is particle momentum. According to the uncertainty principle applied to a particle localised in the region set by a, ι ≥ ~. This is consistent with the bound ιm in Eq. (15.16), although a more general equation (Eq. (15.15)) gives a stronger bound, which increases for heavier molecules. An important difference between the lower bound (Eqs. (15.10) or (15.15)) and bounds based on the uncertainty relation is that Eqs. (15.10) and (15.15) correspond to a true minimum as seen in Figure 15.1 (in the sense that the function has an extremum), whereas the uncertainty relation compares a product ( px or Et) to ~ but the product does not necessarily correspond to a minimum of a function and can apply to a monotonic function. The uncertainty relation can also be used to evaluate the diffusion constant D and its lower bound in the gas-like regime of particle dynamics. In the gas-like and liquid-like regimes of particle dynamics (see Section 15.1), D = ν and D ∝ ν1 , respectively [9]. This implies that, differently from η and ν, D does not have a minimum and monotonically increases with temperature, albeit with a crossover at the Frenkel line marking the transition from gas-like to liquid-like dynamics as discussed in Section 14.2. However, the lower bound in the gas-like regime can be found by using the same approach we used for viscosity minimum earlier and by equating the particle mean-free path to a: Dm = νm = va. Combining this with the uncertainty relation pa ≥ ~, we find the lower bound of D as: Dm ≥

~ m

(15.17)

Eq. (15.17) is found to be consistent with diffusion experiments in Fermi gases [269].

15.4 Viscosity Minima in Quantum Liquids

161

Figure 15.3 Kinematic viscosity of liquid 4 He above and below the λ-point at about 2.2 K. Dotted lines are from [45]. Solid circles are from [270]. Open diamonds correspond to the viscosity of He II at atmospheric pressure from [271].

15.4 Viscosity Minima in Quantum Liquids The viscosity minima in Figure 15.1 are seen in classical liquids. It is interesting to ask whether the same minima exist in quantum liquids. This is particularly so in view of our earlier discussion of excitations and thermodynamics of quantum liquids in Chapter 11. Figure 15.3 shows several sets of experimental data of kinematic viscosity of liquid 4 He. The data represented by finely spaced points and lines above the superfluid transition temperature Tc = 2.17 K (λ-point) are from the US National Institute of Standards and Technology (NIST) [45]. These lines include interpolation artefacts at low temperature. These artefacts are usually unimportant in a wider temperature range; however, here we are interested in liquid helium, which exists in a narrow temperature range only. For this reason, I also show experimental points on which the NIST curves are based [270] as solid circles in Figure 15.3. In addition, I show the viscosity of He II below Tc , attributed to the normal component with non-zero viscosity [271]. The kinematic viscosity is calculated using density from [270]. There are several observations from Figure 15.3. We first observe that viscosity has minima in both phases of He, He I and He II, similarly to the other classical liquids in Figure 15.1. Viscosity minima with similar values at the minimum are also seen in 3 He [272]. We next observe that in He I above the critical pressure (Pc = 0.23 MPa), the viscosity minima in Figure 15.3 are not far from those seen in Figure 15.1 in classical

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Minimal Quantum Viscosity from Fundamental Physical Constants

liquids. The minima are somewhat lower in helium because (a) the pressure in Figure 15.3 is lower and (b) the viscosity minima in helium are understandably lower than in other liquids because the inter-particle interactions in He are particularly weak. Similarly to Figure 15.1, the viscosity minima increase with pressure. Viscosity has a jump due to the liquid–gas transition in the subcritical state. The jump starts close to the viscosity minima in the supercritical regime at 1 and 3 MPa. The viscosity still has minima, albeit these are not smooth like in the supercritical state. Although the viscosity minimum of He II is lower than in liquid He I at atmospheric pressure by about a factor of two or three (this can be attributed to He II being a mixture of normal and superfluid components [148]), it is of the same order of magnitude. Earlier in this chapter, we saw that viscosity minima in classical liquids are set by fundamental physical constants. The similarity between the viscosity minima in He I and He II in Figure 15.3 suggests that the minimum in He II is similarly set by these constants. This, in turn, indicates that the mechanism setting the viscosity minimum in He II may be similar. Indeed, the calculation of the viscosity minima in terms of fundamental constants in Section 15.2 is based on a particular regime of particle dynamics corresponding to the crossover between liquid-like and gas-like regimes. The closeness of calculated and observed viscosity minima is therefore informative in the sense of microscopic dynamics: viscosity decreasing with temperature is related to the combined oscillatory and diffusive particle motion, whereas viscosity increasing with temperature is indicative of purely diffusive motion. This picture is consistent with path-integral simulations of He: the minima of velocity autocorrelation function, associated with the liquid-like combined oscillatory and diffusive motion [217], are seen at 1.2 K [182] where viscosity decreases in Figure 15.3. Clearly, more work is needed to ascertain the nature of microscopic motion in liquid He and its relation to observed properties including superfluidity. Here, we see how the discussion of viscosity minima, their origin and value in terms of fundamental constants has the potential to provide interesting insights into microscopic dynamics in quantum liquids. This is important for constructing a microscopic theory of He II. In Chapter 11, we quoted the observation of Pines and Nozieres regarding the absence of a microscopic theory of He II. Such a theory would have to incorporate the microscopic particle dynamics in liquid helium, and viscosity minima provide an insight into these dynamics. Previously, the behaviour of helium viscosity was discussed in terms unrelated to particle dynamics. Landau and Khalatnikov calculated viscosity due to scattering of phonons and rotons by each other, with the result that viscosity decreases with temperature [273]. This includes a provision that this result does not hold in

15.5 Lower Bound of Thermal Diffusivity

163

the range where viscosity increases with temperature because of the proximity of the λ-point. Tisza, on the other hand, considered the part of the temperature range where viscosity increases with temperature, and attributes it to the gas-like behaviour described by the gas kinetic theory [180]. This was done in one of Tisza’s pioneering papers introducing the two-fluid model of liquid helium (see insightful reviews by Balibar [149, 150] related to the development of the two-fluid model and history of helium research). Dash [274] considers the entire regime where viscosity first decreases, goes through the minimum and then increases, as in Figure 15.3, and explains this non-monotonic behaviour by combining the Landau and Khalatnikov model with the model where viscosity increases due to the increasing normal fluid fraction. 15.5 Lower Bound of Thermal Diffusivity It turns out that Eq. (15.10) also sets the minimum of an unrelated property: thermal diffusivity α = ρcκ p , where κ is thermal conductivity, ρ is density and cp is heat capacity per mass unit. Whereas ν sets momentum diffusivity, α plays the role of the diffusion constant quantifying the propagation of thermal energy. Considering different regimes of heat propagation by collective excitations, phonons, and using arguments similar to those in Section 15.1 including setting the phonon mean-free path to a, it is possible to show that the theoretical minimal value of α, αm , is: αm = νm =

1 ~ √ 4π me m

(15.18)

where νm is given by Eq. (15.10) [275]. This gives rise to the fundamental thermal diffusivity as in Eq. (15.14): αf =

2 1 ~ −7 m ≈ 10 √ 4π me mp s

(15.19)

Eq. (15.18) is consistent with the experimental data: the ratio between experimental and predicted αm is in the range of about 0.9–4 [275]. To illustrate the closeness of αm and νm , we plot α and ν for several noble and molecular liquids in Figure 15.4. This closeness is interesting, considering that the two properties are physically distinct and are measured very differently. There are interesting and important similarities and differences between the two properties, ν and α, with regard to their low- and high-temperature behaviour. I refer the reader to [275] for details of this discussion because thermal conductivity is outside the scope of this book. Here, it is interesting to ask to what extent αm and νm represent the true (global) minima on the phase diagram. It turns out that νm gives the global minimum on the entire phase diagram. αm gives the minimum on the phase diagram except in the vicinity of the critical point [275].

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Minimal Quantum Viscosity from Fundamental Physical Constants

Figure 15.4 Experimental thermal diffusivity α (solid lines) and kinematic viscosity ν (dashed lines) for He (20 MPa), N2 (10 MPa), Ar (20 MPa) and CO2 (30 MPa). The experimental data are from [45]. Reproduced from [275] with permission from the American Physical Society.

Our discussion of thermal conductivity so far applies to systems where the dominant contribution is related to the motion of atoms, ions or molecules. We can also consider systems where electron conductivity is important or dominant, such as metals. Similarly to thermal diffusivity in insulators, the minimum of electron thermal diffusivity, αme , corresponds to setting l = a in thermal diffusivity α = vl: αme = va, where v is the electron speed. This corresponds to the Ioffe–Regel crossover (see, for example,q[276] and the references therein). The electron velocity can be estimated as v = m2Ee , where E is given by the Rydberg energy (Eq. (15.7)). Estimating a as aB in Eq. (15.6) as before gives αme as: αme

 = aB

2ER me

 12 (15.20)

Using aB from Eq. (15.6) and ER from Eq. (15.7) in Eq. (15.20) gives: αme =

~ m2 ≈ 10−4 me s

(15.21)

Set by fundamental physical constants, the bound in Eq. (15.21) is universal and does not depend on the system, in contrast with Eq. (15.18). Compared with the fundamental thermal diffusivity due to the motion of ions in Eq. (15.19), we see that αme is about 103 times larger. This is due to the electron mass being smaller than the proton mass.

15.7 The Purcell Question: Why Do All Viscosities Stop at the Same Place?

165

Similarly to kinematic viscosity, as discussed in Section 15.3, αme is consistent with an uncertainty relation applied to an electron located within a distance a, me va ≥ ~. We finally note that m~ , where m is the particle mass, has been discussed as the lower diffusivity bound for spin transport [259, 277–280]. 15.6 Practical Implications Apart from ascertaining theoretical minima of the phase diagram, the fundamental limits of viscosity and thermal conductivity have practical implications. For example, designing a low-viscosity liquid in lubricating applications benefits from knowing that viscosity cannot be lower than the fundamental bound. On the other hand, if viscosity is substantially larger than the bound, there is room for improvement, which can be pursued. Similarly, low viscosity and associated high diffusion is relevant for using supercritical fluids in cleaning, extracting and dissolving processes, including environmental and green applications [176, 209, 237–239]. We discussed this in Section 14.3 where we also noted that a better understanding of the supercritical state is perceived to be important for scaling up, widening and increasing the reliability of these applications (see, for example, [209, 240–244]). The earlier results in this chapter show how runny a supercritical fluid can ever get and inform us about whether or not it’s worth changing pressure and temperature conditions to reduce the viscosity further in order to increase molecular mobility. The same applies to the minima of thermal conductivity and diffusivity. For example, knowing the lower limit of thermal conductivity is helpful for designing an insulating material: this limit tells us whether it’s worth trying to reduce the thermal conductivity (or diffusivity) further. Another example is the figure of merit measuring the efficiency of thermoelectricity. This figure is inversely proportional to thermal conductivity. Therefore, the minimal thermal conductivity corresponding to Eq. (15.18) gives the maximal possible figure of merit, keeping all other factors unchanged. 15.7 The Purcell Question: Why Do All Viscosities Stop at the Same Place? After the result for minimal quantum viscosity was worked out, I realised that this result gave the answer to an interesting question posed earlier. In 1977, Purcell noted that there was almost no liquid with viscosity much lower than that of water and observed (original italics preserved) [281]: “The viscosities have a big range but they stop at the same place. I don’t understand that.” In the first footnote of that paper, Purcell says that Weisskopf has explained this to him. I did not find published Weisskopf ’s explanation; however, the same

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Minimal Quantum Viscosity from Fundamental Physical Constants

year, Weisskopf published the paper “About liquids” [282]. That paper starts with a story often recited by conference speakers: imagine a group of isolated theoretical physicists trying to deduce the states of matter using quantum mechanics only. They are able to predict the existence of gases and solids, but not liquids. Weisskopf implies that liquids are hard to understand. We have discussed the reasons for this throughout this book, starting from the Preface. Our discussion in Sections 15.1 and 15.2 and Eq. (15.10) in particular help answer the Purcell question. The fundamental constants help keep νm in Eq. (15.10) from moving up or down too much. νm are not universal due to νm ∝ √1m mass dependence, although this does not change νm too much for most liquids. For different fluids, such as those in Figure 15.1, Eq. (15.10) predicts νm in the range 2 (0.3–1.5)·10−7 ms . This fairs well with the minima in Figure 15.1 and is somewhat lower, but not far, from ν in water at room temperature and pressure. Water at room temperature happens to be runny enough and close to the minimum. This is what Purcell noted: viscosities of most liquids do not go much lower than in water. Therefore, the answer to the Purcell question has two parts. First, viscosities “stop” because they have minima. Second, the minima are fixed by fundamental physical constants. An interesting implication of this discussion is related to our everyday experience of dealing with water and water-based products. We saw earlier that water viscosity is not far from what Eq. (15.14) predicts. This prompts an interesting thought: our daily experience is set by three fundamental constants in Eq. (15.14). We will develop this thinking in the next section. 15.8 Fundamental Constants, Quantumness and Life Our search for the source of consistency and predictability of the observed physical world has led us to physical laws and related fundamental theories. Parameters in these theories are fundamental physical constants such as ~ or me , which feature in Eqs. (15.10) and (15.14). Dimensionless combinations of these constants, pure numbers, play a particularly important role in physics and give the observed universe its distinctive character and differentiate it from others we might imagine [267, 283]. There is a truly fascinating history of earlier and ongoing efforts to understand the origin and rationalise the values of fundamental constants. Given that we don’t know anything more fundamental [284], this is one of the ultimate grand challenges in physics. Referring to fundamental constants as “barcodes of ultimate reality”, Barrow proposes that these constants will one day unlock the secrets of the universe [267]. Understanding fundamental constants is viewed as one of the greatest challenges in modern science.

15.8 Fundamental Constants, Quantumness and Life

167

Fundamental constants play a profound role in a number of processes in highenergy physics where they, for example, govern nuclear reactions and nuclear synthesis in stars, including the synthesis of carbon, oxygen and so on, which can then form molecular structures essential to life. An interesting realisation is that these processes require a finely tuned balance between the values of several fundamental constants. Here, dimensionless numbers play an important role. For example, the fine structure constant α ≈ 1/137 and the electron-to-proton mass ratio β = mmpe ≈ 1/1,836 (although mp is related to other Standard Model parameters, the ratio mmpe is a convenient parameter in this discussion) play a role in making the centres of stars hot enough to initiate nuclear reactions. Unless the values of α and β satisfy a certain relation, there would be no stars. Unless α lies in a fairly narrow range, protons are predicted to decay long before the stars can form. There are other examples of what would happen as a result of altering fundamental constants, all showing that there is a fairly narrow “habitable zone” in the parameter space (α,β). In this zone, matter can remain stable long enough for stars to evolve and produce essential biochemical elements including carbon, planets can form and life-supporting molecular structures can emerge. For this reason, Barrow calls the observed fundamental constants “biofriendly” [267] and Adams refers to our universe as “biophilic” [283]. Adams gives a detailed review of the fine-tuning of fundamental constants in our universe and possible other universes. Focusing primarily on astronomy, cosmology and particle physics, Adams’ review [283] discusses variations of fundamental constants that can support life. Our discussion in Section 15.2 and Eq. (15.10) in particular enable us to add another observation. The fundamental constants are friendly to life at a higher level too: biological processes, including those in cells, rely heavily on water. Let’s consider what would happen if fundamental constants were to take different values. According to Eq. (15.10), the minimum of ν and hence the minimum of η will change accordingly. Using ηm = νm ρ, ρ ∝ am3 , m ∝ mp and Eqs. (15.6) and (15.10), B we find: e6 q ηm ∝ 5 mp m5e (15.22) ~ Let’s consider what happens if we dial ~ and set it smaller than the current value. η in Eq. 15.22 is quite sensitive to ~ and increases if ~ is smaller. Raising the viscosity minimum implies that viscosity increases at all conditions of pressure and temperature. Larger viscosity means that water now flows slower, dramatically affecting life processes such as blood flow, vital flow processes in cells and intercellular processes. At the same time, diffusion strongly decreases, implying a slowing down of all diffusive processes of essential substances and molecular

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Minimal Quantum Viscosity from Fundamental Physical Constants

structures in and across cells. This affects, for example, protein mobility, active transport involving protein motors and cytoskeletal filaments, molecular transport, cytoplasmic mixing and the mobility of cytoplasmic constituents, and sets the limits at which molecular interactions and biological reactions can occur. Diffusion is also essential for cell proliferation. These processes have been of interest in life and biochemical sciences [286]. Physically, the origin of this slowing down due to smaller ~ is related to the decrease of the Bohr radius (Eq. (15.6)) as the classical regime with smaller ~ is approached. This results in the increase of energy in Eq. (15.7) via Eq. (15.11), making it harder to flow and diffuse. A large viscosity increase (think of the viscosity of tar or higher) may mean that life might not exist in its current form or might not exist at all. One might hope that cells could still survive in such a universe by finding a hotter place where the overly viscous and bio-unfriendly water is thinned. This would not help though: ηm sets the minimum below which viscosity cannot fall, regardless of temperature or pressure. This applies to any liquid, not just water, and therefore to all life forms using the liquid state to function. We therefore see that water and life are well attuned to the degree of quantumness of the physical world (in conjunction with other fundamental constants and parameters). The same applies to other fundamental constants in Eq. (15.22), such as e, me and, to a smaller degree, mp . 15.9 Another Layer to the Anthropic Principle The results in this chapter add another layer to the discussion of the anthropic principle, sometimes referred to as the anthropic argument or anthropic observation. Eliciting different views [267, 283, 285, 287–289, 293], this term is a collection of related ways to rationalise the observed values of fundamental constants by proposing that these constants serve to create conditions for an observer to emerge and hence are not unexpected. Developing this argument often involves an ensemble of disjoint universes and a physical mechanism to generate this ensemble. Then, a relatively small number of universes have the right values of fundamental constants, and we find ourselves in one of those universes and measure those constants. Alternatives include introducing the natural selection argument into cosmology, explaining the observed values of fundamental constants. In these discussions, there are several types of conditions that need to be met for life and observers to exist. These conditions involve the range of effects, starting from cosmological processes and ending with nuclear synthesis. One of remarkable effects is Hoyle’s prediction of the energy level of carbon nucleus of about 7.65 MeV. This resonance level is required in order to explain carbon abundance

15.9 Another Layer to the Anthropic Principle

169

and in particular the synthesis of carbon from fusing three alpha particles in stars. Following Hoyle’s prediction, the required energy level was experimentally confirmed. This carbon resonance-level coincidence is considered striking. A related important effect is a slightly lower resonance level in oxygen, which enables carbon to survive further resonant reactions. This finely balanced sequence of coincidences enables carbon-based life [267]. In this process, the production of carbon and oxygen importantly depends on their nuclear energy levels, which, in turn, depend on the fine structure constant and strong nuclear force constant. A small change of these constants (more than 0.4% and 4% for the nuclear and fine structure constant) results in almost no carbon or oxygen produced in stars [267]. Another example is the finely tuned balance between the masses of up and down quarks: larger up-quark mass gives the neutron world without protons and, hence, no atoms consisting of nuclei and electrons around them; larger down-quark mass gives the proton world without neutrons, where only light atoms can form, not heavy atoms. Our world with many heavy atoms with electronic orbitals, which endow complex chemistry, would disappear with only a few per cent fractional change in the mass difference of the two quarks [293]. Nuclear reactions are high-energy processes. Condensed matter physics involves much lower energies, and our earlier discussion showed how fundamental constants govern water viscosity. This adds a biological and biochemical aspect to the discussion of the anthropic principle. What change of water viscosity and diffusion constant from their current values is needed to disable cellular and inter-cellular biological processes essential to life. For example, this can happen if water became too viscous due to the lower viscosity bound getting larger, necessitating higher viscosity at all conditions. Once this is known, we can readily calculate the corresponding change of fundamental constants setting this lower bound using, for example, Eq. (15.10) or (15.14). One might think that the constraints on fundamental constants from star formation or nuclear synthesis are already tight enough to keep water viscosity from taking unwanted values not conducive to life. There are two points to consider here. First, it is possible to substantially change the lower bounds for kinematic and e2 and dynamic viscosity and at the same time keep the fine structure constant α = ~c me the electron-to-proton mass ratio β = mp intact, with no consequences for star formation and nuclear synthesis. There are several ways of doing this. For example, increasing me and mp proportionally increases ηm in Eq. (15.10) but leaves α and β intact. Alternatively, one can change ~ and e in Eq. (15.10) but keeping α and β the same, and so on. Second, different effects involved in the existing hierarchy of observed fundamental constants and operating at different levels [267, 288, 289] have different tolerances to life-disabling variations. While stars producing heavy elements and

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Minimal Quantum Viscosity from Fundamental Physical Constants

energy are quite robust, atomic nuclei require a narrower range of fundamental parameters to be stable and hence more stringent conditions on quark masses [283]. A small, compared with a large, range of allowed fundamental constants is interesting because it tells us how special our universe is and sets the weight of the anthropic argument [283]. We have seen that sustaining liquid-based life (including water-based life) imposes constraints on fundamental physical constants that are additional to and different from what has been discussed before in nuclear synthesis. These constraints come from condensed matter physics and involve biology and biochemistry, adding a higher level to the hierarchy of life-enabling effects [267, 288, 289]. It remains to be seen how tight these constraints are compared to constraints discussed in particle physics and cosmology. Exploring these and related issues further is important and invites an interdisciplinary research. This interdisciplinarity has previously included some chemical and physical aspects of life [292]; however, the overall focus was on particle physics, astronomy and cosmology and on production of heavy elements in stars [267, 283, 287, 292, 293]. On the other hand, fundamental insights from condensed matter and liquid physics were not explored, and this overview illustrates the benefits of this consideration. We have been led to the discussion of bounds and fundamental physical constants by the need to understand liquids better. We started from basic effects such as liquid energy and phonons and have seen that not only does this brings us closer to understanding liquids but it also suggests unexpected links to other areas and raises new questions. It is likely that, as we proceed, more unexpected links will be uncovered in the future.

16 Viscous Liquids

16.1 Collective Excitations, Energy and Heat Capacity Before proceeding to discuss viscous liquids, it is useful to recall the overall structure and logic followed in this book. We started discussing liquid energy and specific heat in Chapter 10 in the most challenging regime, where particle dynamics are “mixed” and combine oscillatory and diffusive motion. We then asked how these mixed dynamics change at high temperature, and saw that they become “pure” and diffusive in high-temperature supercritical fluids later Chapter 14. We have seen that, as temperature increases, the phase space for collective excitations reduces, resulting in the decrease of liquid cv from its solid-like value of 3 to the ideal-gas value of 32 . As discussed in Section 14.7, this has exhausted all possible values of classical cv . It is now a good time to reverse the temperature and ask what happens to liquid dynamics and thermodynamics in the low-temperature viscous regime. There is no strict definition of what a “viscous” liquid is. Loosely speaking, water and mercury at room temperature (η ≈ 10−3 Pa·s) are considered to have low viscosity. On the other hand, we talk about glycerine and honey (η ≈ 1 Pa·s) as viscous liquids. In physical terms, a distinction between viscous and low-viscous liquids can be made on the basis of closeness of the liquid relaxation time τ to its shortest value τD and, accordingly, viscosity η close to its “elementary” value η0 = G∞ τD . Water and mercury at room temperature and pressure have viscosities that are not far from their elementary values. Then, τ  τD

(16.1)

or η  η0 can be taken as conditions for a liquid to be “viscous”. τ and η increase at low temperature. This increase stops if a liquid crystallises. Quite often it doesn’t, due to a range of factors. For example, the quench rate may be too fast or the viscosity may be too high for crystallisation to proceed. 171

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Viscous Liquids

Some compounds such as AsS and B2 O3 or systems with large molecules including organics are virtually impossible to crystallise from the melt. For example, crystallisation of B2 O3 and AsS requires high pressure, which acts to reduce viscosity by changing the bonding from covalent to ionic or metallic [173, 174]. If the liquid doesn’t crystallise on cooling, τ and η continue to increase until the liquid forms a solid glass at the glass transition temperature Tg where τ ≈ 102 − 103 s. Recalling that η = G∞ τ and taking G∞ ≈ 10 GPa, this corresponds to η ≈ 1012 − 1013 Pa· s. Of course τ and η continue to increase below Tg as well, and τ can become very long indeed. Let’s take SiO2 glass with Tg ≈ 1,500 K and ask what τ is at room temperature T = 300 K. A straightforward calculation gives τ equal to about the fourth power of the age of the universe [170]. Large numbers such as this are common when we deal with the exponential dependence such as that of τ in Eq. (7.1). This large increase of τ below Tg is inconsequential as far as the system dynamics are concerned: particle dynamics remain purely oscillatory in the liquid below Tg like in the solid, as long as the observation time is shorter than τ . For silica glass at room temperature, it is safe to assume that this condition always applies to our experiment. When discussing liquid thermodynamic properties and their relation to collective excitations, we needed to address a difficult problem: how are we to approach and treat these excitations in liquids? In the preceding chapters, we have done this using a combination of theory, experiments and modelling. It turns out that we don’t need to worry about collective excitations in viscous liquids because these excitations are much simpler to understand than those in lowviscous liquids. Unlike for low-viscous liquids, the reduction of the phase space for excitations in viscous liquids is very small and can be safely ignored. As a result, we are dealing with the same excitations as in solid glass, and can readily calculate the energy and specific heat. This follows from a simple statistical physics argument as set out in the following. Let’s recall the Frenkel theory of molecular jumps in the liquid illustrated in Figure 7.1. This figure illustrates a particle oscillating in one “cage” and jumping to the neighbouring quasi-equilibrium position. The time between two consecutive jumps of a particle is τ . The jump probability is the ratio between the time spent diffusing and oscillating. The jump event lasts on the order of the Debye vibration period τD ≈ 0.1 ps. τ is the time that the particle spends oscillating because it is the time between two consecutive jumps. Therefore, the jump probability is ττD . In statistical equilibrium, this probability is equal to the ratio of diffusing atoms, Ndif , and the total number of atoms, N. Then, at any given moment of time: Ndif τD = N τ

(16.2)

16.1 Collective Excitations, Energy and Heat Capacity

173

If Edif is the energy associated with diffusing particles, Edif ∝ Ndif . Together with Etot ∝ N, Eq. (16.2) gives: Edif τD = (16.3) Etot τ Eq. (16.3) implies that under the condition of Eq. (16.1), the contribution of Edif to the total energy at any moment of time is negligible. Eq. (16.3) corresponds to the instantaneous value of Edif . From the physical point of view, the instantaneous state is set by the shortest timescale in the system, τD . During time τD , the system is not in equilibrium. The equilibrium state, by its definition, is reached when the observation time exceeds system relaxation time, τ . After time τ , all particles in the statistical system undergo jumps. Therefore, we need to calculate Edif that is averaged over time τ . We divide time τ into m time av periods of duration τD each, so that m = ττD . Edif , averaged over time τ , Edif , is then: m 2 E1 + Edif + · · · + Edif av Edif = dif (16.4) m i where Edif are the instantaneous values of Edif in Eq. (16.3).

av Edif Etot

is then:

av 2 E1 + Edif + · · · + Emdif Edif = dif Etot Etot · m Ei

in Eq. (16.5) is equal to According to Eq. (16.3), each of the terms Edif tot are m terms in the sum in Eq. (16.5). Therefore: av Edif τD = Etot τ

(16.5) τD . τ

There

(16.6)

Thus, under the condition of Eq. (16.1), the ratio of the average energy of diffusion motion to the total energy is negligibly small, like in the instantaneous case. Consequently, the energy of the liquid under the condition of Eq. (16.1) is, to a very good approximation, given by the remaining vibrational part. Similarly, the liquid l constant-volume specific heat, cv,l = N1 dE , is entirely vibrational in the regime of dT Eq. (16.1): El = Elvib cv,l = cvib v,l

(16.7)

The vibrational energy and specific heat of liquids in the regime of Eq. (16.1) is readily found. The liquid vibrational energy, Elvib , is given by 3N phonons: Elvib = 3NT in the harmonic case. This is of course consistent with the gapped states of transverse excitations in liquids expressed by Eq. (7.3) or Eq. (8.14): large τ means

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Viscous Liquids

1 in Eq. (8.8) is small and can be safely ignored. Therefore, in that the gap kg = 2cτ the regime of Eq. (16.1), Elvib = 3NT to a very good approximation, like in a solid. We now consider Eq. (16.7) in harmonic and anharmonic cases. In the harmonic case, Eq. (16.7) gives the energy and specific heat of a liquid as 3NT and 3, respectively, namely the same as in a harmonic solid:

Elh = Esh = 3NT chv,l = chv,s = 3

(16.8)

where s corresponds to the solid and h to the harmonic case. In the anharmonic case, Eq. (16.8) is modified by the anharmonicity related to softening of phonon frequencies as discussed in Section 10.3. This results in Eqs. (10.19) and (10.20):   αT (16.9) E = 3NT 1 + 2 and cv = 3 (1 + αT)

(16.10)

Several pieces of evidence support these results of cv in viscous liquids. First, we recall that the experimental specific heat of liquids is close to 3 close to melting where τ  τD , consistent with Eqs. 16.8 and 16.10. Second, the condition τ  τD becomes particularly pronounced in viscous liquids approaching the liquid–glass transition where ττD becomes as small as ττD ≈ 10−15 . Experiments have shown that in the highly viscous regime just above Tg , the heat capacity measured at high frequency and representing the vibrational part of heat capacity coincides with the total low-frequency heat capacity usually measured [290, 291], consistent with Eq. (16.7). In the glass transformation range close to Tg , the two heat capacities start to differ due to non-equilibrium effects and freezing of configurational entropy, and coincide again below Tg in the solid glass. Third, representing cv by its vibrational term in the highly viscous regime gives the experimentally observed change of heat capacity in viscous liquids at Tg . This is discussed in Section 16.3. We recall that the only condition used to derive these results for E and cv is Eq. (16.1). For practical purposes, this condition is satisfied for τ & 10τD already. Perhaps not widely recognised, the condition τ ≈ 10τD holds even for low-viscous liquids such as liquid metals (Hg, Na, Rb and so on) and noble liquids such as Ar near their melting points [45], let alone for more viscous liquids such as roomtemperature olive or motor oil, honey and so on. On lowering the temperature, τ increases from its shortest limiting value of τ = τD ≈ 0.1 ps to τ ≈ 103 s where, by definition, the liquid forms glass at the glass transition temperature Tg . Here, τ changes by 16 orders of magnitude.

16.2 Entropy

175

Consequently, the condition ττD  1, Eq. (16.1), or τ & 10τD applies in the range 103 –10−12 s, spanning about 15 orders of magnitude of τ . In this range, the Rparameter introduced in Eq. (10.10) and later discussed in Chapter 12 governing liquid excitations and thermodynamic properties is negligibly different from 0: R=0

(16.11)

We therefore see that the condition τ & 10τD corresponds to almost the entire range of τ where liquids exist as such. In this range, collective excitations govern the liquid energy to the same extent as they do in solids. In terms of the phase space available to these excitations, we can say that this space in viscous liquids is deformed very little compared with the phase space in solids. This is in contrast with the reduced phase space in low-viscous liquids discussed in Chapter 8.

16.2 Entropy As will have become apparent to the reader, the focus of this book is on the specific heat as an important indicator of the degrees of freedom in the system and a quantity that can be directly measured experimentally and compared to a theory. In this section, we also mention entropy, since it is often discussed in the context of the liquid–glass transition. Although Eq. (16.6), combined with Eq. (16.1), implies that the energy and cv of a liquid are entirely vibrational like in a solid, this does not apply to entropy: the diffusional component to entropy is substantial, and cannot be neglected. Indeed, if Zvib and Zdif are the contributions to the partition sum from vibrations and diffusion, respectively, the total partition sum in the regime τ  τD is Z = Zvib · Zdif . The product is justified because the smallness of the diffusive contribution to the partition function means that the two terms can be treated as independent and their d coupling can be ignored. Then, the liquid energy is E = T 2 dT (ln(Zvib · Zdif )) = 2 d 2 d T dT ln Zvib + T dT ln Zdif = Evib + Edif (here and in the remainder of this secEav

 1 from Eq. (16.6) tion, the derivatives are taken at constant volume). Next, Edif tot Edif also implies Evib  1, where, for brevity, we dropped the subscript referring to the dif average. Therefore, the smallness of the diffusional energy, EEvib  1, implies: d dT d dT

The liquid entropy, S = S=T

d dT

ln Zdif ln Zvib

1

(16.12)

(T ln(Zvib · Zdif )), is:

d d ln Zvib + ln Zvib + T ln Zdif + ln Zdif dT dT

(16.13)

Viscous Liquids

176

The condition of Eq. (16.12) implies that the third term in Eq. (16.13) is much smaller than the first one, and can be neglected. This gives: d ln Zvib + ln Zvib + ln Zdif (16.14) dT Eq. (16.14) implies that the smallness of Edif , expressed by Eq. (16.12), does not lead to the disappearance of all entropy terms that depend on diffusion because the term ln Zdif remains. This term is responsible for the excess entropy of liquids over that of solids. On the other hand, the smallness of Edif does lead to the disappeardS ance of terms depending on Zdif in the specific heat cv,l . Indeed, cv,l = T dT (here, S refers to entropy per atom or molecule), and from Eq. (16.14), we find:   d d d d cv,l = T T ln Zvib + T ln Zvib + T ln Zdif (16.15) dT dT dT dT S=T

Using Eq. (16.12) once again, we see that the third term in Eq. (16.15) is small compared with the second term and can be neglected. This gives:   d d d T ln Zvib + T ln Zvib (16.16) cv,l = T dT dT dT Therefore, cv does not depend on Zdif , and is given by the vibrational term that depends on Zvib only. As expected, Eq. (16.16) is consistent with cv in Eq. (16.8), which is purely vibrational. The inequality of liquid and solid entropies, Sl 6= Ss , originates because the entropy measures the total phase space available to the system, which is larger in the liquid due to the diffusional component present in Eq. (16.14). This can be quantified, for example, by the “communal” or “melting” entropy [23]. However, the diffusional component, ln Zdif , although large, is slowly varying with temperature according to Eq. (16.12), resulting in a small contribution to cv (see Eqs. (16.15) and (16.16)). On the other hand, the energy corresponds to the instantaneous state of the system (or averaged over τ ), and is not related to exploring the phase space. Consequently, El = Evib , resulting in Eq. (16.12) and the smallness of diffusional contribution to cv despite Sl 6= Ss . We note that the common thermodynamic description of entropy does not involve time: it is assumed that the observation time is long enough for the total phase space to be explored. In a viscous liquid with large τ , this exploration is due to particle jumps, and is complete at long times t  τ only, at which point the system becomes ergodic. If extrapolated below Tg , configurational entropy of viscous liquids reaches zero at a finite temperature, constituting an apparent problem known as the Kauzmann paradox. However, issues involved in separating configurational and vibrational entropy and interpreting experimental data became apparent later, affecting the way

16.3 Heat Capacity at the Liquid–Glass Transition

177

Figure 16.1 Heat capacity of poly(a-methyl styrene) measured in calorimetry experiments [294]. Glass transformation range operates in the interval of about 260–270 K. Reproduced from [170] with permission from the American Physical Society.

the Kauzmann paradox is viewed and the perceived extent of the problem (see, for example, [83], [294] and the references therein). 16.3 Heat Capacity at the Liquid–Glass Transition In this section, we continue to maintain our focus on liquid specific heat. In Section 16.1, we ascertained that the cv of viscous liquids is very close to its solid value and is given by 3N phonons like in solids. This discussion would be incomplete if we did not mention the behaviour of liquid heat capacity at the glass transition temperature Tg . Recall that Tg corresponds to temperature where τ becomes longer than the time during which an observable is measured. Heat capacity has an anomaly at Tg , and we need to understand the origin of this anomaly and its magnitude. Figure 16.1 shows a typical example of the change of the constant-pressure specific heat, cp , in the glass transformation range around Tg . Changes such as this are routinely measured in calorimetry experiments in different types of glasses (see, for example, [295] for a review). If clp and cgp correspond to the specific heat above and cl

below Tg on both sides of the glass transformation range, cpg = 1.1−1.8 for various p liquids [170, 296, 297]. The change of cp at Tg is viewed as the “thermodynamic” signature of the liquid–glass transition, and serves to define Tg in the calorimetry experiments. Tg measured as the temperature of the change of cp coincides with the temperature at which τ reaches 102 –103 s and exceeds the observation time.

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Viscous Liquids

As Dyre observes, the glass transition itself is not a big mystery [83]: when τ becomes longer than the time of experimental measurement, this measurement becomes related to the solid-like elastic properties of the system, and the system becomes glass. Having considered the wealth of experimentally observed effects related to the liquid–glass transition, Zanotto and Mauro propose [298] the following definition of what glass is: “Glass is a nonequilibrium, non-crystalline state of matter that appears solid on a short time scale but continuously relaxes towards the liquid state.” The emphasis here is on the dynamical origin of glass formation and nonequilibrium effects. We will return to this point in Chapter 18 where we will discuss the spin glass transition. For now, we observe that the view outlined earlier (expressed by Dyre, Mauro, Zanotto and others) of the dynamical origin of the non-equilibrium glass transition is just what Frenkel predicted (see the discussion in Section 7.3): the transition between the liquid and the glass is a continuous process related to the continuous increase of τ . This is supported by the wide range of data showing that the structure of the viscous liquid above Tg and the structure of glass are nearly identical. This is also in full agreement with the universally observed logarithmic change of Tg with the observation time or frequency discussed later in this section, which contrasts with what takes place at the phase transition. If the glass transition is a purely dynamical effect unrelated to a thermodynamic transition, we need to understand the “thermodynamic” signature related to the change of cp at Tg shown in Figure 16.1. We recall our discussion in Chapter 7: liquid response includes viscous response related to diffusive jumps and solid-like elastic response. When the viscous response stops at Tg during the experimental timescale (this is the definition of Tg ) and only the elastic response remains, the system’s bulk modulus B and thermal expansion coefficient α change. This results in different cp above and below Tg as we will now show. Let’s consider that pressure P is applied to a liquid. According to the Maxwell– Frenkel viscoelastic picture, the change of liquid volume, v, is v = vel + v r , where vel and vr are related to the solid-like elastic deformation and viscous relaxation processes, respectively. Let’s now define Tg as the temperature at which τ exceeds the observation time t. This implies that particle jumps are not operative at Tg during the time of observation. Therefore, v at Tg is given by purely the elastic response like in elastic solids. Then, we write: vel + vr Vl0 vg P = Bg 0 Vg

P = Bl

(16.17)

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179

where Vl0 and Vg0 are initial volumes of the liquid and the glass, vg is the elastic deformation of the glass and Bl and Bg are bulk moduli of the liquid and glass, respectively. Let 1T be a small temperature interval that separates the liquid from the glass  1. Then, Vl0 ≈ Vg0 . such that τ in the liquid, τl , is τl = τ (Tg + 1T) and 1T Tg Similarly, the difference between the elastic response of the liquid and the glass can be ignored for small 1T, giving vel ≈ vg . Combining the two expressions in Eq. (16.17) gives: Bg Bl = (16.18) 1 + 1 where 1 = vvelr is the ratio of the relaxational and elastic response to pressure. The coefficients of thermal expansion of the liquid and the glass, αl and α g , can be related in a similar way. Let’s consider liquid relaxation in response to the increase of temperature by 1T. We write: 1 vel + vr V0l 1T 1 vg αg = g V0 1T αl =

(16.19)

where vel and vr are volume changes due to the solid-like elastic and relaxational response, respectively, as in Eq. (16.17) but now in response to temperature variation and vg is the elastic response of the glass. Combining the two expressions for αl and αg and assuming Vl0 = Vg0 and v el = vg as before gives: αl = (2 + 1)αg

(16.20)

where 2 = vvelr is the ratio of relaxational and elastic response to temperature. Eqs. (16.18) and (16.20) provide the relationships between B and α in the liquid and the glass due to the presence of particle jumps in the liquid above Tg and their absence in the glass at and below Tg (insofar as Tg is the temperature at which t < τ ). Consistent with experimental observations, Eqs. (16.18) and (16.20) predict that liquids above Tg have larger α and smaller B compared with below Tg . We are now ready to calculate cp below and above Tg . In the previous section, we saw that, in the highly viscous regime, cv is given by the vibrational component of motion only (see Eq. (16.16)). Therefore, we use cv = 3(1 + αT) from Eq. (10.20), which accounts for phonon softening due to inherent anharmonicity. Writing constant-pressure specific heat cp as cp = cv + nTα 2 B, where n is the number density, we find cp above and below Tg as: clp = 3 (1 + αl T) + nTαl2 Bl , T > Tg
cgp = 3 1 + αg T + nTαg2 Bg , T < Tg

(16.21)

Viscous Liquids

180

Figure 16.2 Increase of Tg in Pd40 Ni40 P19 Si1 glass with the quench rate q. Circles are experimental points from [300].

According to Eq. (16.21), the temperature dependence of cp should follow that of α. This is in agreement with simultaneous measurements of cp and α showing that both quantities closely follow each other across Tg [299]. From Eq. (16.21),

clp g

cp

can be calculated using experimental B and α above and cl

below Tg . This gives good agreement with the experimentally observed cpg in the p range 1.1–1.8 [170]. We note that the anomaly of cp at Tg is a purely non-equilibrium effect related to the freezing of the viscous component of displacement at Tg . If we are able to wait longer, vr becomes operational both above and below Tg in Eqs. (16.17) and (16.19), resulting in Bl = Bg and αl = αg and hence clp = cgp in Eq. (16.21). An important effect related to the dynamical origin of the anomaly of cp at Tg is that Tg is not a fixed temperature. Tg universally decreases with the observation time, or increases with the quench rate q (see, for example, [91, 300]). This is a generic effect involved in many glass transition phenomena where the relaxation time exceeds the experimental timescale. In Figure 16.2, we show an example of how Tg , defined as the temperature of the jump of heat capacity, increases with the quench rate q. We observe that Tg varies in a large range depending on the quench rate (observation time). This variation can be explained as follows. Recall that the jump of cp at Tg takes place when the observation time t crosses liquid relaxation time τ . , τ at which the jump of heat capacity takes This implies that because q = 1T t place is τ (Tg ) = 1T , where 1T is the temperature interval of the glass transformq
ation range. Combining this with τ (Tg ) = τD exp U/Tg (here U is approximately

16.4 Glass Transition Revisited

181

constant because τ is nearly Arrhenius around Tg as discussed in Section 16.4) gives: U Tg = 1T (16.22) ln τ0 − ln q According to Eq. (16.22), Tg increases with the logarithm of the quench rate q. In particular, this increase is predicted to be faster than linear with ln q. This is consistent with experiments [91, 300] and the example shown in Figure 16.2. We note that Eq. (16.22) predicts no divergence of Tg because the maximal physically possible quench rate is set by the minimal internal time, the Debye vibration period in Eq. (16.22) is always larger than q. We will return to τD . This implies that 1T τD the logarithmic increase of Tg with the quench rate in Chapter 18 where we will discuss the spin glass transition. It is interesting at this point to recall the discussion in Section 14.2 where we saw that the Frenkel line separates the combined oscillatory and diffusive particle motion below the line from the purely diffusive particle motion above the line. A natural “symmetrical” line on the phase diagram could be the line separating the combined oscillatory and diffusive particle motion at high temperature and purely oscillatory motion at low temperature. Can the “glass transition line” on (P,T) serve to achieve this? As we have seen earlier in this section and in Eq. 16.22 in particular, Tg depends on the observation time (or frequency), and so no well-defined glass transition line exists on the phase diagram because the low-temperature glass state is a non-equilibrium liquid. The Frenkel line, on the other hand, separates two equilibrium states of matter. However, the liquid–crystal phase transition line is a fixed equilibrium line on the phase diagram and does correspond to the transition of the combined oscillatory and diffusive motion in the liquid and purely oscillatory motion in the crystal. 16.4 Glass Transition Revisited Having discussed the thermodynamic properties of viscous liquids and dynamical anomalies at Tg , it is interesting to recall Section 7.3, where we discussed the first ignition of the glass transition debate. Frenkel considered the liquid–glass transition as a continuous process involving only quantitative but no qualitative changes. In Frenkel’s picture, solid glass is just a very viscous liquid with relaxation time τ exceeding the observation time [100]. The jump of cp at Tg in Figure 16.1 is not an issue in this picture: this jump is a dynamical non-equilibrium effect as we discussed in the previous section. We recall Landau’s objection to Frenkel’s picture: melting involves symmetry changes and therefore cannot involve a continuous change [101, 102]. This is

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Viscous Liquids

indeed the case if there is a symmetry change; however, Frenkel was mostly discussing the liquid–glass rather than the liquid–crystal transition, and the structure of liquids and glasses is nearly identical. The question we now want to address is whether the liquid–glass transition has any properties, and in particular experimental properties, that can be attributed to a phase transition? A great amount of experimental data have been generated since Frenkel and Landau discussed the transition between liquids and solids. We can now look at these data and see what we have learned since the first ignition of the glass transition debate by Frenkel and Landau discussed in Section 7.3. The possibility of a phase transition, hidden or avoided due to very slow kinetics, was seemingly suggested by the Vogel–Fulcher–Tammann (VFT) equation used to fit the experimental data of relaxation time or viscosity [301, 302]:   A η, τ ∝ exp (16.23) T − T0 where A and T0 are constants. Systematic studies have found that the experimental data are fitted by the VFT equation no better than by functions involving no divergence at a finite temperature [302, 303]. However, even if the VFT function is preferred over others, there is now substantial experimental evidence that there is a crossover between the VFT dependence at high temperature and Arrhenius (or nearly Arrhenius) dependence well above Tg [304–308], implying that in fact two different functions are needed to explain the experimental η in the entire temperature range. This probably explains why the VFT equation shows a systematic error when it is used to fit the experimental law at low temperature [303]. The crossover between the VFT function and Arrhenius dependence takes place well above Tg . This means that we should not extrapolate Eq. (16.23) below Tg and consider a potential divergence at T0 , hidden or avoided due to slow kinetics. The function we need to be extrapolating is the Arrhenius one instead, in which case there is no divergence at a finite temperature. T0 is sometimes associated with the Kauzmann temperature at which the extrapolated liquid entropy approaches that of the crystal (see [83, 294, 302, 303] and citations therein). This was interpreted as an indication of a potential phase transition. However, this interpretation depends on several assumptions, including the equality of glass and liquid vibrational entropies, which was later found to be incorrect [83]. In the previous section, we saw that these two terms are different below and above Tg because α and B are different. Interestingly, the above crossover from the VFT equation to Arrhenius dependence can be rationalised by recalling our elasticity length del = cτ discussed in Chapter 8. Recall that this length served to quantify the propagation length of transverse phonons and the associated reduction of the phonon space in liquids due to

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183

the emergence of the gapped momentum state. This length becomes useful again in the context of the glass transition because del sets the length scale of cooperativity of molecular “relaxation” [309]. Indeed, a particle jump such as that shown in Figure 7.1 induces an elastic wave propagating in the liquid and affecting the jump in a different location. The length over which this wave propagates is set by del . Therefore, del sets the cooperativity of molecular relaxation and measures the range of elastic interaction between local jumps in the liquid. At high temperature where τ is close to its shortest value τD , del = cτD ≈ a, where a is the inter-atomic separation as before. This means that neighbouring local relaxation events do not elastically interact at high temperature. In this context, it is interesting to recall Section 14.4, where we saw that the double universality of the high-temperature plot cv (del ) is also related to del becoming equal to the fundamental length scale in the system, the inter-atomic separation a on the order of 1 Å (the ultraviolet cut-off). When del extends to the medium-range distance so that neighbouring relaxation events start to interact elastically, we expect to see an onset of cooperativity of interaction. τ at this crossover is about 1 ps, in agreement with experiments [309]. Both τ and del = cτ grow at low temperature, increasing the cooperativity of molecular relaxation. However, this growth is limited by the system size Ls . Therefore, del = Ls corresponds to all local jumps interacting in the liquid and maximal cooperativity, and should mark a crossover from the cooperative VFT equation to Arrhenius or nearly Arrhenius. τ at the crossover, τc , is: τ=

Ls c

(16.24)

This time corresponds to the fundamental frequency, or first harmonic, in the system ω = Lcs . For typical experimental values Ls = 0.1–1 mm and c = 103 m/s, τ = Lcs is 0.1–1 microseconds, in agreement with the nearly universal value of τ at the crossover seen experimentally [74, 304–308]. Considering that no experimental evidence points to a diverging behaviour or any other phase transition-related property, the authors of [302, 303, 308] justifiably conclude that theories of viscous liquids and the glass transition should not worry about the VFT-type or other potential divergences and a related phase transition. If this view is taken, we see that the research into the nature of the liquid–glass transition seems to have made a full circle over the last 90 years or so. In Section 7.3, we discussed Frenkel’s 1936 paper, where he introduced the concept of the liquid relaxation time τ and on its basis proposed that the liquid–glass transition is a continuous process. We now have a great deal more experimental and other data than was at Frenkel’s disposal at the time, and these data compel us to agree with his original view.

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Viscous Liquids

Focusing on liquids, this book does not discuss glasses below Tg . Glasses are interesting in their own right and have stimulated new models and ways of thinking mandated by their disordered structure [23]. For example, this interestingly includes thinking about glasses as networks with a balance between the degrees of freedom and the number of constraints [310–312]. This chapter dealing with viscous liquids finishes our discussion of the liquid state of matter. We have discussed the relationship between liquid collective excitations and liquid thermodynamic properties in a wide range of the phase diagram, including subcritical liquids, gas-like supercritical fluids and viscous liquids extending to glasses at low temperature. In Chapters 18 and 19, we will explore how we can use what we have learned from liquids in other areas in physics. Before doing that, I provide a short discussion of what I think a theoretical description should involve (Chapter 17).

17 A Short Note on Theory

This short chapter reflects a realisation that steadily grew as I was reviewing the history of research into liquids and related systems with structural and dynamical disorder. This observation is related to what we want to achieve through a physical theory. It is probably not new; however, my numerous discussions with colleagues have confirmed that this remains an interesting and relevant point to ponder about. In a physical theory, we aim for a mathematical description of a physical reality and have several criteria to judge a theory. Common criteria include agreement with experimental results. The wider the agreement in terms of the number of systems and experimental conditions (energy, temperature and so on), the wider the range of applicability of a theory. Another criterion is an ability to make a prediction, such as an outcome of a future experiment. Compared with explaining the existing set of data, predicting future outcomes is often more challenging, and this puts an additional pressure and constraints on a theory that must be falsifiable. If new experimental results are obtained as a result of these predictions, then this drives science forward and makes the theory useful, even if this theory is eventually falsified. We can add other criteria that strengthen a theory too: for example, we appreciate an internal consistency and economy. An important salient feature here is that a theory aims to provide a relationship between different experimental properties of a physical system. We have met this idea in Chapter 10 where we quoted Landau and Peierls [167]: “The essence of any physical theory is to use the results of an experiment to make assertions about the results of a future experiment.” An interesting realisation is that even this pragmatic view may not be sufficiently restrictive because there may be more than one way of achieving this, namely it is possible to predict an outcome of a future experiment on the basis of an earlier one using more than one theory.

185

186

A Short Note on Theory

Let’s recall the quote of Mott [88] about Frenkel in Chapter 7: “Frenkel was a theoretical physicist. By this I am stressing that he was primarily and most of all interested in what is happening in real systems, and the mathematics he used served his physics and not otherwise as is sometimes the case for the modern generation of scientists. . . He asks: what is really happening and how can this be explained?”

The key words here are real and really, and in the following I explain what I mean. Suppose we want to construct a theory of a measured effect. We propose that the measured effect involves several connected physical processes and provide a mathematical description of each process in our theory. To be “real” in the Mott sense, this theory should closely follow these actual processes. Rephrasing Zaanen [255], if “nature calculates” a process, our theory should follow the same calculation of this process. If this sounds opaque or, in contrast, too obvious, it is useful to consider cases where this does not happen. In many important cases, not only may we not know which existing mathematical model is realistically related to each physical step in nature’s “calculation” we may also not know what this model may look like at all. We may not even know what kind of tools we need to describe a particular physical effect. So we start experimenting and may invent a theory that we can solve. For example, we can consider a statistical theory of an object or effect in question in higher or infinite dimensions, solve it in those dimensions and see how the answer changes when we set the dimensionality back to its value in the system where the effect operates, 3 (4 in the relativistic case). This may be interesting theoretically and gives insights into the role of fluctuations; however, we don’t necessarily think that this is how real effects operate and how nature “calculates”, namely that an experiment measuring a property involves particles consulting higher or infinite dimensions before returning an observable. In other cases, our theories may involve a pathology or a non-physical behaviour, which we attempt to “cure” with various degrees of success. However, we don’t think that this is how a real physical process operates. A physical process proceeds without stumbling on pathologies and curing them every time. These issues reflect our mathematical constructs only, rather than a physical reality. Another example is simpler and is probably more familiar: these are theories involving assumptions and components that come at a cost that appears too high compared with everything else we know. Readers have probably encountered theories where they thought this was the case and where they wondered whether this was what was really happening in Mott’s sense. However, in other cases, we recognise upfront that a problem is too hard to deal with at the moment and map a physical process onto a completely different one that can be solved so that we can transfer the results back. This is the idea involved

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187

in the holographic correspondence: in certain cases, a strongly interacting field theory is equivalent to a weakly interacting gravity dual (see, for example, [262] for a review). This has been bringing important results and a realisation that our descriptions of different physical areas are mathematically related. Appreciating these connections is awe-inspiring and a very gratifying part of the physics enterprise. At the same time, we have not yet found a way to treat strongly interacting theories directly, and will be happy when we eventually will (whether or not solutions of this kind always need to emerge hinges upon a more philosophical question of whether the natural world is ultimately comprehensible in terms of our models). In these and other examples, we should eventually look for a physical model involving steps directly related to what is involved in a real measurable physical effect. We have learned this from the success of physics built from such models. Indeed, our most important theories forming the core of modern physics were born out of closely following details of actual processes involved in real systems and experimental data [313]. Reviewing the history of early and modern science, Weinberg observes that “the world acts on us like a teaching machine”, shaping our theories and reliable knowledge [313]. My point is that we should not remain content until we find out “what is really happening” in the sense that Mott discussed in relation to Frenkel’s work. Until that happens, it would be a mistake to think that we have a physical theory. We may leave the problem for some time while thinking about other ideas. In the meantime, reminding ourselves that ultimately we need to find out what is really happening in the Mott–Frenkel sense will help us focus and continue trying.

18 Excitations and Thermodynamics in Other Disordered Media: Spin Waves and Spin Glass Systems

This and the next chapter are not directly related to the physics of liquids. A reader interested in liquids only can skip the following discussion. The main intention here is to discuss how the insights from the liquid theory can be usefully applied to open problems in other areas. This discussion is therefore exploratory rather than definitive, with the main aim to draw analogies between different fields of research. In this chapter, we explore how the insights from disordered systems such as liquids and glasses can be used to understand spin glasses and in particular the relationship between collective excitations and thermodynamic properties. We recall our discussion in Section 16.1 noting that a liquid may be unable to crystallise due to either fast cooling or large viscosity. Instead, it forms glass, the structurally disordered solid. A similar state can emerge in the system of spins, magnetic moments, which form a frozen disordered spin configuration with zero magnetisation at low temperature, differently from ferromagnets or antiferromagnets. Understanding this state and the process of the spin glass transition has become a large, separate, area of research [314]. An interesting and important dichotomy has emerged between the areas of structural and spin glasses. This dichotomy is related to a fundamental approach we use to study interacting systems. It is not mentioned in textbooks, and for this reason it is worth discussing here. In Chapter 4, we saw how the thermodynamic properties of solids are governed by collective excitations, phonons. In the classical case, the central Dulong–Petit result cv = 3 is due to 3N phonons operating in the solid. We emphasised in Section 3.1 that this result and its underlying theory do not rely on the presence of crystalline order. In fact, the Debye model connecting low- and high-temperature properties of solids works best in homogeneous disordered systems rather than ordered crystals. As a result, thermodynamic properties of disordered glasses are governed by collective modes to the same extent as they are in crystals. There are

188

Excitations and Thermodynamics in Other Disordered Media

189

low-temperature anomalies in glasses [315]; however, these become unimportant in high-temperature glasses where cv = 3 is due to 3N phonons like in crystals. In other words, the structural disorder in glasses does not alter the relation between thermodynamic properties and collective excitations. In a system with strong interactions, ordered or disordered, thermodynamic properties are governed by collective excitations. This general premise is not limited to crystals or glasses, and forms a fundamental approach in statistical physics where system properties are consistently understood on the basis of excitations in a variety of systems [3, 5], including electronic, magnetic and so on. The state of affairs is markedly different when we consider ordered and disordered magnetic systems, spin glasses. In ordered magnetic systems, ferromagnets or anti-ferromagnets, spin waves are the well-understood collective excitations, magnons, which are analogous to phonons in solids. The spin waves govern major magnetic properties such as magnetisation, susceptibility and so on [38, 58]. On the other hand, the existing theories of spin glasses and their thermodynamic properties are divorced from spin waves. The reason for this is unclear. We will, however, elaborate on possible reasons in this chapter. The starting point of the spin glass theories is based on the Edwards–Anderson proposal [316] to consider that the disordered magnetic system undergoes a phase transition at the critical temperature Tc where the order parameter q becomes nonzero: q = hs1i s2i i 6= 0

(18.1)

The subscript i refers to a particular local spin. If s1i is the measured value of this spin and s2i is its value in infinitely remote time, q 6= 0 means a non-vanishing probability that the s2i will point in the same direction. Assuming random distribution of interactions between spins and representing this distribution as a Gaussian to average the free energy gives the result that q becomes non-zero below the critical temperature Tc , which depends on the distribution width. Thermodynamic properties such as heat capacity and susceptibility κ can then be expressed as a function of q below and above Tc . In particular, κ is found to be quadratic below Tc . Theories of spin glasses have been following the Edwards–Anderson approach [314] and discuss thermodynamic properties of spin glasses in terms of the lowtemperature phase and its order parameter(s). On the other hand, spin waves and their role in setting the low-temperature properties of spin glasses are barely discussed. One stated reason for this is that the evidence, and in particular experimental evidence, for spin glasses is fairly scarce [314]. Compared with emerging experimental and modelling evidence for collective excitations in liquids and structural glasses discussed in previous chapters, there is indeed less discussion of spin waves in spin glasses. Nevertheless, spin waves in spin glasses have been

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Excitations and Thermodynamics in Other Disordered Media

predicted theoretically, with the linear dispersion ω = ck, where c is the speed of magnons [317]. This has subsequently been confirmed in theoretical studies [318, 319], simulations [320] and neutron scattering and nuclear magnetic resonance experiments [321–324] including at temperatures above the spin glass transition temperature. Most of this evidence for spin waves is from the 1970s and 1980s and came around the same time that research into spin glasses was emerging. The exploration of spin waves in disordered magnetic systems did not seem to continue or expand, in notable difference from the large increase of research into collective excitations in liquids and structural glasses reviewed in earlier chapters. The reason for this difference is unclear, although we observe that a large part of the spin glass research, the spin glass theory, largely followed the general approach of Edwards and Anderson, where the focus has been on new, unconventional, phase transitions, order parameters and so on and where spin waves were not part of the theory [314, 325]. Another potential reason is that collective excitations in liquids were better understood on the basis of earlier Frenkel and the following approaches than spin glass systems. Let us recall our discussion in Chapter 3, where we saw that crystallinity is not required for the existence of 3N normal modes. In the harmonic approximation, these modes exist in both ordered and disordered systems. It is straightforward to calculate thermodynamic properties of spin glasses on the basis of spin waves. The free energy of the non-interacting gas of magnons is:   X  ~ωi (18.2) F = N0 + T ln 1 − exp − T i where ωi are magnon frequencies and N0 is the zero-point energy. As mentioned earlier, understanding ordered magnetic structures, their magnetic moment and susceptibility χ is based on spin waves. For example, using the dependence of magnon frequencies in a ferromagnet on external field H, ~ωk = 2JSk 2 a2 + gµB H, where J is the exchange parameter, S is spin and a is 3 lattice constant in Eq. (18.2), gives Bloch law: M ∝ −T 2 [326–328]. On the other hand, χ and M have not been calculated for spin glasses. From Eq. (18.2), χ is:   i X exp ~ω dωi 2 ~2 X 1 d2 ωi T χ = χ0 + − ~ (18.3) i T i exp ~ωi − 1
2 dH exp ~ω − 1 dH 2 T i T where χ0 is the susceptibility due to zero-point vibrations. field dependencies of ωi can be taken as ωl = ck and ωt = q For spin glasses,
2 ± µH , where ωl and ωt are longitudinal and transverse waves, (ck)2 + µH 2~ 2~

Excitations and Thermodynamics in Other Disordered Media

respectively, and µ is the spin magnetic moment [318]. This gives


dωi 2 dH

191

=


µ 2 2~

and ddHω2i = 4~µ2 ωi for small fields (H → 0) at which χ is measured in spin glasses [325]. Next, the linear ω ∝ k relationship for small H implies the quadratic density of states, g(ω), as for phonons. As before, we write g(ω) = 6N ω2 , where ω0 is ω03 the Debye frequency and the normalisation coefficient takes into account that 2N transverse waves contribute to the sums in Eq. (18.3) because field derivatives of the longitudinal frequency are zero and hence do not contribute to χ. Using these field derivatives, substituting the sums in Eq. (18.3) with integrals with g(ω) and noting that integration can be extended to infinity at low temperature due to the fast convergence of the integral gives [329]: 2

2

χ = χ0 +

π2 T2 Nµ2 3 4 TD

(18.4)

where TD = ~ω0 is the Debye temperature of spin waves. We find, therefore, that χ is quadratic with temperature and approaches a constant value χ0 at T → 0, consistent with experimental results (see, for example, [325, 330–333]). Interestingly, the same quadratic behaviour on the lower temperature side was found in the original Edwards–Anderson theory based on the description of the spin glass transition in terms of a phase transition and order parameters [316]. We pause for the moment here to make an important point regarding spin waves in spin glasses. We recall Chapter 6, where we discussed the problems involved in liquid theories based on inter-atomic interactions and correlation functions. There we saw how the approach to liquids based on collective excitations has freed us from these problems. The same applies to spin glasses: frustration-inducing interactions between spins involve randomness and can be of a complicated nature. Finding a way to treat these interactions and average over them was the main point of the original Edwards–Anderson paper and many others that have followed since. Adopting a different approach where thermodynamic properties of spin glasses are related to spin waves frees us from the problem to treat complicated spin interactions, as it does in liquids. At high temperature, χ is given by the Curie dependence: Nµ2 (18.5) T which crosses over to the spin wave χ at low temperature given by Eq. (18.4). The existing theories of spin glasses attribute the low-temperature behaviour to a phase transition, as in the original Edwards–Anderson paper. This was supported by early experiments showing sharp cusps of χ ; however, later experiments found that χ can also undergo a smooth crossover at the spin glass transition temperature χC =

192

Excitations and Thermodynamics in Other Disordered Media

Tg (see, for example, [331–340]). In either case, Tg is not fixed as expected for a phase transition but universally increases with the frequency (decreases with observation time) [325, 330, 333–335, 337–340]. This effect is exactly the same as in the case for the dynamical liquid–glass transition discussed in Section 16.3, where the effect is shown in Figure 16.2 and Eq. (16.22). This presents a problem for theories attributing the spin glass transition to a thermodynamic phase transition and order parameters, even though unconventional. Mydosh views profound dynamical effects in spin glasses as a problem for theories based on phase transitions [325]. Brazhkin observes that differently from spin glass theories, the real spin glass systems are metastable, implying that q in Eq. (18.1) always decays to 0 at a finite temperature once the equilibrium state is reached, as is the case in the Frenkel theory of liquids for t  τ , and raises the question of the extent to which the spin glass theories are related to real systems [341]. Sethna notes that basic conceptual questions in spin glasses remain unanswered and adds that its unclear what happens to the multitude of spin glass states on cooling [41]. The authors of [28] observe that it is currently unclear whether theories related to spin glasses apply to real systems, including structural glasses. We will not delve into the discussion of the nature of the spin glass transition, for our focus is on collective excitations in disordered media and their relationship to thermodynamic properties. What would certainly be interesting is to develop our experimental understanding of spin waves in spin glasses. Experiments have substantially improved since spin waves were first detected in spin glasses. It would be important and insightful to apply these experiments to an extended range of spin glass systems and measure spin waves and their properties. It would be equally interesting and important to study spin waves across the spin glass transition and above the glass transition temperature Tg . Above Tg , we expect to find the spin liquid or spin gas state, and it would be interesting to explore to what extent this state has viscoelastic properties in Frenkel’s sense as discussed in Chapter 7. It would also be interesting to explore how spin waves evolve across Tg . Recall our discussion in Chapter 8 where we saw that collective excitations become gapped in liquids, and that the phase space available to these excitations reduces with temperature. This leads to an important question: Does this happen in disordered spin systems as well? Another related interesting question is: What can we say about the Frenkel line (see Chapter 14) operating in spin liquids and separating the combined oscillatory and diffusive spin dynamics from purely diffusive dynamics? Considering that this area is underexplored, it is very likely that these experiments will bring about new and unanticipated results. These are important for developing a fundamental understanding of collective excitations in disordered media and their relation to system properties. We can then seek to compare and

Excitations and Thermodynamics in Other Disordered Media

193

potentially unify these insights with what we learned about collective modes in liquids as discussed in Chapter 8. Finding and exploring collective excitations in disordered spin systems above Tg interestingly connects to the currently topical research in spin liquids, both classical and quantum. In this area, ascertaining the nature of excitations and connecting them to spin liquid properties is a central and active line of enquiry [342, 343].

19 Evolution of Excitations in Strongly Coupled Systems

In this final chapter, we explore how we can use the lessons from liquid physics in a general setting of strongly interacting (strongly coupled) theories and make a connection to field theory. The subject of the quantum field theory is “rooted in a harmonic paradigm” [344] represented by the first two terms of the Lagrangian as:   X X X 1 L=  µ˙q2a − kab qa qb − gabc qa qb qc − . . .  2 a a,b a,b,c

(19.1)

where q are displacements corresponding to field variables in the field theory. Zee finds it notable that, since its inception, the quantum field theory has remained rooted in the harmonic paradigm operating in terms of basic notions such as oscillations and wave packets, and that this is taken on by the string theory, the heir to the quantum field theory [344]. These concepts are equally central in solid state and condensed matter physics, as illustrated by our earlier discussion, particularly in Section 3.1. The first two terms in Eq. (19.1) correspond to harmonic theory and plane wave solutions. These are related to our quadratic forms discussed in Section 3.1, which serve as the basis for the solid state theory. The remaining terms are due to anharmonic effects resulting in scattering, the interaction and the production of particles. These terms are analogous to the anharmonic terms in Eq. (4.14) discussed in Section 4.2, which give rise to phonon scattering, set the phonon lifetimes and so on. When the anharmonic terms are small, perturbation theory can be used. This gives small corrections to the harmonic result, such as the correction to harmonic cv in Eq. (4.15). In field theory, the anharmonic terms (often referred to as field coupling terms) are similarly treated in perturbation theory. This forms a large part 194

Evolution of Excitations in Strongly Coupled Systems

195

of the quantum field theory where the interaction terms are treated perturbatively to calculate the S-matrix, scattering, Feynman diagrams and so on. When the anharmonic terms are not small, perturbation theory does not apply. We recall that this problem, the absence of the “small parameter”, was central to the difficulty of theoretical liquid description. We have discussed this problem throughout this book, and in particular in the Preface, Introduction and Chapters 5 and 6. We have learned that a constructive way to deal with this problem in liquid physics is to consider how the phase space available to phonons evolves with the hopping frequency ω F corresponding to the inter-valley motion in Figure 5.1. This has brought two important insights. First, we saw that this phase space reduces with ωF . Second, from the point of view of liquid energy and other thermodynamic properties, we can treat the excitations in this contracted space as approximately unmodified compared with the harmonic solid case. These two results gave the key to calculating liquid thermodynamic properties and to understanding experimental results. In the quantum field theory, there are a number of important cases where the anharmonic terms in Eq. (19.1) are not small, where perturbation theory similarly does not apply and where the analytical solutions do not exist. This includes, for example, the Yang–Mills theory and other strongly coupled theories. Can we use the insights from the liquid theory to approach the strongly coupled problems in quantum field theory? Let’s assume that the anharmonic potential for the field in Eq. (19.1) is generally similar to the liquid-like multi-valley potential in Figure 5.1, as is the case for, for example, the axion potential [345]. We can then use the insights from liquid theory as follows. We have seen that this potential results in the field motion combining the oscillatory motion in each well (given by the first two terms in Eq. (19.1)) and inter-valley hopping motion with time τ or frequency ωF . Depending on the field theory and its setting, ωF can be related to tunnelling of the field between the minima or thermal hopping above the barrier. In either case, the dynamics of the field becomes viscoelastic and combine the oscillatory and diffusive motion as in the Frenkel theory discussed in Chapter 7. The equation of motion of the field becomes of a form similar to Eq. (8.2): c2

∂ 2v ∂ 2 v 1 ∂v = + ∂x2 ∂t2 τ ∂t

(19.2)

In the simple case of a scalar field, the oscillatory component of motion of the field is related to the Klein–Gordon equation, corresponding to the first term on the right-hand side in Eq. (19.2). The “viscous” component, related to the second term, does not readily lend itself to a Lagrangian formulation or a field-theoretical description because it involves dissipation. The field-theoretical

196

Evolution of Excitations in Strongly Coupled Systems

description requires two scalar fields [115] and results in Eq. (19.2) and the gapped momentum state discussed in Chapter 8. Eq. (19.2) gives a result with implications similar to those discussed in Chapter 8. Despite strong coupling, we still have the harmonic motion of the field and, therefore, the free theory, but this free theory operates in the reduced phase space. In particular, the transverse sector in this space is characterised by the gap in momentum space. The propagating excitations with frequency exceeding the decay rate have a frequency (energy) gap as discussed in Chapters 7 and 8, where the spectrum of propagating waves starts with the frequency: ω = ωF

(19.3)

and has no energy levels below ωF . This way of treating strongly coupled theories has an advantage because we can readily use the results from the free theory, including canonical quantisation, propagators and so on and subsequently apply them in the reduced space as we did in the liquid theory.

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Index

activation energy, 51 anharmonicity, 32, 34, 35, 48, 87, 88, 105, 148, 179 anthropic principle, 7, 168, 169 anyons, 124 argon, 9, 10, 142 bifurcation, 36–38, 57 Bloch law, 190 Bogoliubov approximation, 26, 114 Bohr, 103 Bohr radius, 156 Born, 130, 138 Born and Green, 40, 41 Bose liquid, 114, 153 Boson peak, 85, 158 Brillouin, 6, 48, 79, 127–130, 138, 144 bulk modulus, 25, 60, 69, 80, 178, 179 C transition, 142, 144 collective excitations, xiv, 3–9, 16, 21, 29, 39, 45, 48, 51, 57, 71, 79, 86, 87, 89–92, 95, 97, 114, 127, 145, 146, 172, 188 condensed matter, xiv, 2, 21, 49, 50, 71, 155, 156, 169 continuum approximation, 16, 23, 24, 27, 54, 60, 75, 87, 146 correlation function, 39, 43, 45, 47 critical point, 6, 9, 128, 133, 134, 136, 140, 154, 163 current correlation function, 72, 77, 78, 92, 95, 101 de Broglie wavelength, 121, 122 Debye frequency, 31, 54, 100, 101, 108, 148, 155, 191 Debye model, 3, 6, 31, 45, 48, 62, 88, 100, 105, 112, 127, 129, 188 Debye vibration period, 144, 149, 155, 172, 181 degrees of freedom, 1, 4, 13, 14, 112, 129, 184 density of states, 20, 30, 31, 68, 87, 88, 99, 108, 115, 148, 191 dimensional analysis, 158 dispersion relation, 20, 21, 26, 65, 66, 69, 70, 83, 85, 95, 114

Dulong–Petit, 11, 30, 32, 48, 88, 101, 107, 132, 150, 188 Einstein theory of solids, 3, 6, 29, 45, 48, 127, 129 elasticity, 16, 23, 24, 57, 60, 84 elasticity length, 67, 142, 144, 150, 182 elastoviscosity, 74 energy, xiv, 1, 2, 9, 12–14, 16, 18, 19, 30, 31, 40, 97, 100, 133, 148 energy gap, 60, 67 entropy, 109, 153, 156, 174–176 exponential complexity, 38, 50, 57 Faber, 132 fast sound, 80, 120 Fermi liquid, 118, 119 Fisher–Widom line, 140 free energy, 12, 23, 30, 42, 107, 132, 190 Frenkel, 3, 5, 16, 50–57, 61–64, 178, 181 Frenkel line, 6, 126, 134, 136, 138, 139, 141, 142, 144, 151, 181 frequency gap, 67, 68, 97 fundamental constants, 166, 167, 170 fundamental frequency, 183 fundamental viscosity, 157 gapped energy state, 66 gapped momentum state, 61, 67, 69, 70, 83, 95 gas giants, 141 generalised hydrodynamics, 58, 72, 74 glass transition, 5–7, 52, 62, 63, 172, 174, 175, 177, 178, 180, 181, 183, 188, 190–192 Grüneisen approximation, 34, 47, 106 hard-sphere model, 4, 12–15, 138 harmonic oscillator, 18, 64 harmonic solid, 18, 29, 34, 101, 174 heat capacity, xiv, 1–3, 12, 16, 32, 97 Heaviside, 3, 5, 65

207

208

Index

high-frequency response, 54, 56, 68 honey, 73, 74, 171, 174 hopping frequency, 7, 50, 54, 67, 71, 98, 125 hydrodynamics, 16, 23, 58, 60, 73 inversion point, 142, 144, 150 Kauzmann temperature, 182 Kirchhoff, 3, 5, 65 Landau, 3, 6, 63, 118, 127 Landau and Lifshitz, xi, 2, 4, 16, 41, 43, 46, 104 life, 167–169 lifetime, 33, 47, 66, 68, 87, 89 liquid 3 He, 118, 161 liquid 4 He, 114, 161, 162 liquid relaxation time, 52, 53, 56, 60, 63, 77, 86, 111, 123, 125, 130, 133, 134, 139, 152, 171–173, 195 longitudinal wave, 7, 20, 21, 23, 25, 27, 30, 53, 54, 60, 80, 97, 116, 145, 146, 148 Maxwell, 3, 28, 35, 55, 56, 65, 131, 155, 178 mean-free path, 119, 134, 142, 145, 152, 155 mercury, 9, 10, 56, 102, 171 metallic liquid, 10, 102, 109, 142 Migdal, 6, 116, 127 minimal viscosity, 7, 152, 153, 157 molecular dynamics simulations, 2, 13, 45, 52, 91, 101, 138, 142 molecular liquid, 102, 109, 112, 142, 163 momentum gap, 5, 28, 61, 65–68, 77, 79, 83, 84, 91, 95, 104 Mott, 50, 62, 186, 187 Navier–Stokes equation, 25, 56, 61, 64, 72–74, 76, 167 neutron scattering, 21, 79, 80, 114, 138, 190 noble liquid, 10, 102, 109, 142, 163 non-linearity, 29, 35, 37, 57, 87 normal modes, 19, 20, 71 nuclear magnetic resonance, 190 pair distribution function, 40, 139, 140 partition function, 29, 106 perturbation theory, 2, 12, 14, 32, 34, 42, 195 phase diagram, 6, 9, 134, 136, 137, 140, 144, 150, 163, 184 phase space, reduction, 5–7, 46, 71, 95, 97, 98, 101, 102, 104, 105, 110, 146, 151, 172, 192 phonon, 11, 16, 21, 29, 32, 46, 51, 71, 79, 102, 115, 129, 144, 148, 188 phonon dispersion, 21, 47, 62, 80 Pitaevskii, xii, 41, 43, 46, 104 plane waves, 16, 20, 21, 23, 27, 70, 88, 194 plasma, 82, 83 Poincaré, 3, 5, 65

Proctor, 2, 111, 132 propagation length, 59, 60, 67, 121, 182 Purcell, 7, 153, 165, 166 quadratic forms, 2, 4, 17, 24, 29, 35, 46, 47, 105, 150, 194 quantum field theory, 194 quantum liquids, 114, 117, 121, 127 quantum statistics, 26, 114, 121 quark–gluon plasma, 153, 159 Raman scattering, 88, 138 relaxation time, 52, 53, 56, 60, 63, 77, 86, 111, 119, 123, 125, 130, 133, 134, 139, 152, 171–173, 195 roton, 85, 115, 116 Rydberg energy, 156 shear modulus, 52, 55, 56, 72, 130, 138, 150 skin effect, 70 small parameter, xii, 4, 35, 39, 46, 104, 195 solid 3 He, 122 solid 4 He, 122 solid state theory, 3, 4, 9, 17, 29, 45, 48, 50, 128 solubility, 141 Sommerfeld, 3, 6, 48, 127, 129, 130 sound wave, 21, 24, 25, 27, 28, 30, 114 specific heat, xi, 1, 4, 6, 9, 11, 13, 30, 102, 143, 150, 173, 174, 177 speed of sound, 20, 27, 31, 80, 83, 139, 153, 156 spin dynamics, 192 spin glasses, 188, 190 spin waves, 7, 189, 190, 192 stationary state, 38 strong interactions, xi, 189, 195 structure factor, 76, 80, 87, 91, 92 subcritical liquid, 6, 9, 92, 132, 141, 145–147 supercritical fluid, 6, 92, 93, 109, 137, 140, 145, 148 supercritical state, 9, 10, 28, 133, 134, 136, 140, 141, 144–146, 153 superfluid, 115, 117, 118, 161, 162 susceptibility, 190 Tabor, xiv, 3, 6, 125 telegraph equation, 5, 65–68, 73, 78, 79, 102 thermal conductivity, 32, 65, 87, 139, 156, 163, 165 thermal diffusivity, 163, 165 thermal expansion, 32, 106, 136, 148, 157, 175, 178, 179 transverse wave, 5, 20, 27, 59, 67, 79, 80, 92, 97, 98, 101, 190 triple point, 9, 157 uncertainty relation, 160, 165 Van der Waals model, 4, 12, 14, 15, 41 velocity autocorrelation function, 117, 137

Index viscoelasticity, 5, 52, 55, 57, 60, 74, 178 viscosity, 7, 10, 25, 54, 55, 61, 88, 111, 165, 171 viscosity, fundamental, 158 viscosity, generalised, 61, 72, 78 viscosity, kinematic, 76, 154 viscosity, minimal, 7, 152, 153, 155, 157, 161 viscosity, water, 169 Vogel–Fulcher–Tammann equation, 182 Vogel–Fulcher–Tammann equation, crossover, 182

Wallace, 2, 11, 14, 132 Wallace plot, 14 Wannier and Piroué, 3, 6, 130, 131 water, 7, 56, 141, 153, 165–168, 171 Weisskopf, 7, 165 Widom line, 134, 136, 138, 139 X-ray scattering, 79, 80, 83, 138

209