Liquid Glass Transition: A Unified Theory From the Two Band Model

By Toyoyuki Kitamura

A glass is disordered material like a viscous liquid and behaves mechanically like a solid. A glass is normally formed by supercooling the viscous liquid fast enough to avoid crystallization, and the liquid-glass transition occurs in diverse manners depending on the materials, their history, and the supercooling processes, among other factors. The glass transition in colloids, molecular systems, and polymers is studied worldwide. This book presents a unified theory of the liquid-glass transition on the basis of the two band model from statistical quantum field theory associated with the temperature Green’s function method. It is firmly original in its approach and will be of interest to researchers and students specializing in the glass transition across the physical sciences.

• Examines key theoretical problems of the liquid-glass transition and related phenomena
• Clarifies the mechanism and the framework of the liquid-glass transition

• Language: English
• Publisher: Elsevier Science
• Release date: Dec 31, 2012
• ISBN: 9780124071704

Book preview

Liquid glass transition

A Unified Theory from the Two Band Model

First Edition

Toyoyuki Kitamura

Nagasaki Institute of Applied Science, Nagasaki, Japan

Table of Contents

Cover image

Title page

Copyright

Preface

Chapter 1: Introduction

1.1 The Structure of the Condensed States and the Quantum Regime

1.2 The Two Band Model for the Liquid-Glass Transition

1.3 Perspective of This Book

Chapter 2: Sound and Elastic Waves in the Classical Theory

2.1 Sound in the Classical Fluid Mechanics

2.2 Elastic Waves in the Classical Elastic Theory

2.3 Sound and Phonons in the Classical Microscopic Theory

2.4 The Kauzmann Entropy, the Vogel– Tamman– Fulcher Law and Specific Heat

Chapter 3: Fundamentals of Quantum Field Theory

3.1 The Number Representation and the Fock Space

3.2 An Example of Unitarily Inequivalent Representations; The Bogoliubov Transformation of Boson Operators

3.3 The Physical Particle Representation and the Dynamical Map

3.4 Free Physical Fields for Physical Particles

3.5 The Physical Particle Representation and Perturbation Theory

3.6 The Spectral Representations of Two-Particle Green’s Functions

3.7 Invariance, the Noether Current and the Ward-Takahashi Relations

Chapter 4: Temperature Green’s Functions

4.1 Definition of the Temperature Green’s Functions

4.2 Perturbation Theory and the Wick’s Theorem at Finite Temperature

4.3 Feynman Diagrams

4.4 Dyson’s Equation

Chapter 5: Real Time Green’s Functions and Temperature Green’s Functions

5.1 Various Kinds of Green’s Functions

5.2 Linear Response and Density Correlation Function

5.3 A Linear Response Theory at Finite Temperature

Chapter 6: The Structure of Glasses Associated with Phonons

6.1 The WT relations at finite temperature

6.2 The two band model and Green’s functions

6.3 The Nambu-Goldstone theorem and phonons

6.4 The structure of phonons

I Phonon dispersion curves

II The width of phonons

Chapter 7: The Liquid-Glass Transition

7.1 Random Scattering Processes and the Bethe–Salpeter Equation

7.2 Intra-Band Density Fluctuations: Sound and Diffusion

7.3 Inter-Band Density Fluctuations: Phonons and Viscosity

7.4 The Kauzmann Entropy Crisis and the VTF Law; Specific Heat, Relaxation Times, and Transport Coefficients

7.5 The Intermediate Scattering Function

7.6 A Generalized Navier-Stokes Equation

Chapter 8: Phonon Operators in Nonlinear Interaction Potentials

8.1 The Dynamical Equation for Phonon Operators in Nonlinear Interaction Potentials

8.2 Solitons and Bound States of the Self-Consistent Potential by the Boson Transformation Method

8.3 Localized Modes for a Quartic Potential in the One Loop Approximation

Chapter 9: Phonon and Sound Fluctuation Modes and Thermal Conductivities

9.1 The Effective Interaction Hamiltonian for Phonon Fields and the Elementary Scattering Processes of Phonons

9.2 Phonon Density Fluctuations: Phonon Entropy Fluctuation Modes and Thermal Conductivities

9.3 The Effective Interaction for Sound Fields

9.4 Sound Density Fluctuations; Sound Entropy Mode and Sound Thermal Conductivity

9.5 The Anomaly of Thermal Conductivity and Specific Heat in Low Temperature Glasses

Chapter 10: The Liquid-Glass Transition in Multi-Component Liquids

10.1 The Model Hamiltonian and the Random Scattering Hamiltonian

10.2 Sound and Diffusivity

10.3 Phonons, Boson Peaks, and Viscosities in Multi-Component Liquids

10.4 Phonons, Boson Peaks, and Viscosities in Two-Component Liquids

10.5 The Kauzmann Entropy Crisis and the VTF Law; Specific Heat, Relaxation Times, and Transport Coefficients

10.6 Concluding Remarks

Chapter 11: Extension of the Two Band Model

11.1 Excitations in a Bose-Condensed Liquid

11.2 A Model on the Origin of RNA

11.3 A Model on the Financial Panic

Copyright

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First edition 2013

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Preface

Near the liquid-glass transition, a particle locates in a randomly distributed well-localized potential in a quantum state. A particle makes up and down transitions between two levels for a while and hops to a neighbouring vacancy. This picture yields the two band model in the random system. In this book, we develop the unified theory of the liquid-glass transition on the basis of the two band model by statistical quantum field theory associated with the temperature Green’s function method. In order to develop in a self-contained manner, we explain all the tools we use in this book in Chapters 3–5. The concept of free physical fields plays a key role in statistical quantum field theory. The Ward-Takahashi relations associated with the spontaneous breakdown of the symmetry also play an important role in establishing the Hamiltonian, the Nambu-Goldstone theorem and the approximation procedures among the Green’s functions, the self-energy parts and the vertex parts.

The dynamical and dissipative processes in random systems are treated at the two stages: at the first stage, we consider the configurationally averaged Hamiltonian, by which we treat the dynamical processes. At the second stage, we take into account the randomness by introducing the scattering processes of particles by the random potentials due to random frequencies and random hopping amplitudes. The configuration average of the simultaneously scattering processes of two particles by the same random potentials constrains the dynamical processes, which yields the relaxation and dissipative processes.

In the two band systems, bubbles of intra-band and inter-band particle-hole pairs are created. The sum of an infinite series of interacting bubbles of intra-band and inter-band particle-hole pairs yield sound and phonons, respectively. The extra bubbles of intra-band and inter-band particle-hole pairs not constituting the sound and phonons, make the intra-band and inter-band fluctuation entropies, which compensate with diffusion and viscosity, respectively, and then the system reaches the thermal equilibrium. The liquid and glassy states have a kind of the long range order; the pair distribution function looks the same to each observer at his own particle in a sense of the configuration average. In the glassy state, the pair distribution function holds the same every time so that it breaks the symmetry of translation, which leads to the Nambu-Goldstone bosons, phonons; the up and down transitions of particles between the two levels yield phonons. On the other hand, in the liquid state, the pair distribution function changes with time so that the hopping of particles is essential, which yields the density fluctuation mode, sound. Sound is not the Nambu-Goldstone boson.

Conventional theories have not discriminated between sound and phonons clearly. Sometimes, sound is considered to be a longitudinal phonon in the long wave length regime. This is not correct. Despite the limitations of the macroscopic and microscopic classical theories for sound and elastic waves (phonons), macroscopic classical theories in the long wavelength regime have discriminated between sound and elastic waves; the classical fluid mechanics has dealt with sound and the classical elastic theory with elastic waves. The microscopic classical theory can also discriminate between them, where elastic waves are valid over the long and short wavelength regimes, while sound is limited in the long wavelength regime. The macroscopic and microscopic classical theories for sound and elastic waves and their limitations are presented in Chapter 2.

Near the liquid-glass transition, since the energy bands of particles are so narrow that there appear different situations from the classical regime: first, the sound keeps alive over the long and short wavelength regimes. The sound disappears at a disappearing temperature and a reciprocal particle distance. This fact relates with the Kauzmann entropy crisis. Second, in a supercooling process in a liquid, enormous bubbles of intra-band particle-hole pairs not constituting sound are created so that the extra bubbles of intra-band particle-hole pairs not constituting the sound form the intra-band fluctuation entropy, which mixes with the sound entropy. This mixing entropy with the wave vectors near the reciprocal particle distance yields just the Kauzmann entropy, which has the form of the Curie law with the sound disappearing temperature as the critical temperature and a negative sign. The hopping probability of a particle is proportional to the exponent of the Kauzmann entropy per perticle from the Einstein relation on probability and entropy. Thus the hopping amplitude (the exponent of the Kauzmann entropy per particle) manifests the Vogel-Tamman-Fulture (VTF) law. The Kauzmann entropy crisis seems to occur as the temperature decreases towards the sound disappearing temperature, but the inter-band fluctuation entropy increases as the temperature decreases. Eventually, the inter-band fluctuation entropy crosses over the Kauzmann entropy, where the glass transition occurs above the sound disappearing temperature so that it prevents the entropy crisis.

This scenario in the supercooling process naturally elucidates the universal features in the liquid-glass transition such as the Kauzmann entropy on the gap of the specific heat at the glass transition temperature and the VTF law on the relaxation and transport coefficients. This is why we call this scenario the unified theory of the liquid-glass transition. Conventionally, the Adam-Gibbs formula on the relaxation and dissipative processes has been used to arrange the experimental results. The Adam-Gibbs formula is a phenomenological one. The formula describes the hopping amplitude as the exponent of the inverse configuration entropy. Since the configuration entropy corresponding to the Kauzmann entropy is involved in the exponent of the hopping amplitude in the reverse order, the configuration entropy cannot explain the Kauzmann entropy crisis and the gap of the specific heat at the glass transition temperature. We develop the unified theory of the liquid-glass transition in Chapter 6–10.

The two level systems are ubiquitous. The two band model plays an essential role in the two level systems, where the constituent matter hops. If we identify the matter and the inter-matter interaction potentials in a two level system, with a particle and inter-particle interaction potentials in a liquid, we can extend the unified theory of the liquid-glass transition to the matter system. We call the Kauzmann entropy crisis in the system, a panic. Thus panics are also ubiquitous as the two level systems. Panics occur in financial market, psychology, … , and even in nature. In this context, the liquid-glass transition is regarded as one of panics in nature. We develop the financial panics as the extension of the two band model in Chapter 11.

I would like to dedicate this book in the memory of Dr. Shozo Takeno.

Chapter 1

Introduction

Abstract

A brief survey on (1) the structure of liquids and glasses in the condensed states and (2) the two band model for the liquid-glass transition: (1) The structure of the liquid and glassy states is represented by the pair distribution function. The liquid-glass transition is in the quantum regime. (2) The two band model is established. First, sound and phonons associated with boson peak are derived. Secondly, the dissipative (diffusivity and viscosities) and relaxation processes are investigated. Finally the liquid-glass transition in the supercooling process associated with the Kauzmann entropy crisis and the Vogel–Tamman–Fulture (VTF) law are investigated.

Keywords

Liquids; Glasses; Pair distribution function; Two band model; Sound; Phonons; Boson peak; Liquid-glass transition; Supercooling; Kauzmann entropy; Vogel-Tamman-Fulture (VTF) law

A glass is a disordered material like a viscous liquid and behaves mechanically as a solid [1–6]. A glass is normally formed by supercooling the viscous liquid fast enough to avoid crystallization. The liquid-glass transition occurs in diverse manner depending on materials, supercooling processes, history of materials, and so on. Among the diversity, there are universal features such as:

1.The Kauzmann paradox on the entropy crisis [7]:

As the temperature decreases toward the glass transition temperature, the entropy of the system seems to drop to a negative value.

2.The Vogel-Tamman-Fulcher (VTF) law [8–10] on the relaxation and the transport processes:

.

3.The jump of specific heat at the glass transition temperature:

There is a gap in the values of the specific heats between just below and just above the glass transition temperature.

4.The boson peak [11]:

There appears a peak of the density states at the harmonic frequency of phonons below the glass transition temperature.

, at which the specific volume decreases with the temperature keeping constant till the completion of crystallization, where the first order phase transition accompanied with the emission of the latent heat takes place, and then the specific heat decreases with the temperature decreasing as schematically illustrated in Figure 1.1.

Figure 1.1 ) in the supercooled liquid state and then E to G in the glassy state. The specific volume changes upward concavely around the glass transition temperature.

, where the liquid-glass transition takes place, and then the specific volume decreases at first upward concavely, then linearly with the temperature decreasing as schematically illustrated in depends on the supercooling processes.

In order to establish a unified theory to elucidate the universal features in such complex systems, at the first step we must start with a simple model in a simple liquid composed with a kind of isotropic particles and develop the theory in a most tractable way; we construct the two band model as a simple model and make the mean field approximation (the random phase approximation) as a tractable way. Despite the simple model and the crude approximation, once we have established a unified theory, it plays a guiding role in establishing more rigorous theories, rigorously simulating the dynamical equations including nonlinear interactions, extending to more complex systems, arranging experimental results, and so on.

In order to construct the two band model, we must clarify the structure of condensed states. The structure of a system relates with the long range order of the system, which means that the same order holds over the whole system. The long range order spontaneously breaks the translational symmetry; the symmetry broken state is associated with the Nambu-Goldstone (NG) bosons (phonons) [12–15]. In the crystalline state, the long range order is represented by the periodic structure; the pair distribution function looks exactly the same to each observer at his own particle [16]. A particle locates in a periodically distributed localized potential making up and down transitions between exactly the same two levels for a long time. The modes of the propagation of the up and down transitions through the interactions are a longitudinal and two transverse phonons. In the liquid and glassy states, although particles are randomly distributed, there is a kind of the long range order; the pair distribution function looks the same to each observer at his own particle in a sense of the configuration average [17]. A particle locates in a randomly distributed localized potential making up and down transitions between the two levels and then hops to a neighboring vacancy. The hopping yields the two levels bandwidths. The modes of the propagation of the up and down transition through the interactions are also phonons. In the glassy state, since the hopping seldom occurs, the long range order holds; the pair distribution function holds the same every time so that phonons are essential, while in the liquid state, since the hopping always occurs, the long range order does not holds; the pair distribution function changes every time and the hopping causes the density fluctuations so that the sound mode is essential. The sound is not the NG boson.

The sound plays an essential role in the liquid-glass transition. In the two band model, the sum of an infinite series of interacting bubbles of intra-band particle-hole pairs constitutes sound mode, while the sum of an infinite series of interacting bubbles of inter-band and a reciprocal particle distance K K, where the Kauzmann entropy crosses over the fluctuation entropy due to the bubbles of inter-band particle-hole pairs. The probability corresponding to the Kauzmann entropy per particle from the Einstein relation manifests the VTF law; the probability is proportional to the exponent of the Kauzmann entropy per particle.

In Introduction, we briefly survey the structure of the condensed states, the two band model, the collective modes (sound and phonons), and the mechanism of the liquid-glass transition.

1.1 The Structure of the Condensed States and the Quantum Regime

A pair distribution function and an inter-particle interaction potential yield a well-localized potential of a particle, which can be approximated as a harmonic potential. The structure of the pair distribution function determines the structure of phonons. On the other hand, when a particle has a vacancy at the nearest neighbor, the potential of the particle has another minimum potential in the direction to the vacancy so that the particle can hop to the position of the vacancy.

The long range order spontaneously breaks down the translational symmetry, which yields the NG bosons, a longitudinal and two transverse phonons. In the glassy state, phonons are essential, while in the liquid state, sound is essential due to the hopping so that the sound is not NG bosons.

In this section, we also verify the liquid-glass transition occurs in the quantum regime.

1.1.1 The Structure of Condensed States, the Pair Distribution Function, and Hopping of Particles

called the radial distribution function has a peak at a mean inter-particle distance a and with the distance R is schematically illustrated in Figure 1.2.

Figure 1.2 -component.

The potential between particles has a hard core and a valley with a minimum near a mean inter-particle distance a illustrated in feels a well-localized potential made up by the surrounding particles. In order to see how the potential well is made up by the surrounding particles, we consider a one-dimensional system with the xand a, is well localized as illustrated as thick line. We can regard the well-localized potential as a harmonic potential. Figure 1.3c shows the case for the existence of a vacancy at a, has double minimums so that the particle at the 0-site can hop to the vacancy at the a-site.

Figure 1.3 The potential between particles and the potentials made up by the surrounding particles in a one-dimensional system with the x is schematically illustrated. It has a hard core and a valley with a minimum near a mean inter-atomic distance a and a , is a well localized as schematically illustrated by a thick line. The potential made by each particle is illustrated by its own dotted line. (c) For the case of the existence of a vacancy at a , has double minimums as schematically illustrated by a thick line.

as illustrated in Figure 1.4, since the mand the wall which should be made by the n-particle, the potential of the m. The mas illustrated in Figure 1.4.

Figure 1.4 are the hopping magnitude between the ground states and that between the excited states, respectively.

is essential in the glassy state.

1.1.2 The Nambu-Goldstone Bosons, Phonons, and Sound

In the condensed states, the liquid, glassy, and crystalline ones, the Hamiltonians are of the same form invariant under the spatial translation. However, the long range orders in the glassy and crystalline states spontaneously break down the symmetry of the spatially translational invariance. The spontaneous breakdown of the symmetries is associated with the Nambu-Goldstone bosons [12–15]. The Nambu-Goldstone bosons are a longitudinal and two transverse phonons. The Nambu-Goldstone bosons play a role in recovering the symmetries.

looks exactly the same to each observer at his own particle has the structure starting with a peak at an inter-particle distance and oscillatorily damping toward the mean particle density in distance R in the glass, the symmetries of the spatially translational invariance spontaneous break down. This spontaneously breakdown of the symmetry is associated with the Nambu-Goldstone bosons, which are a longitudinal and two transverse phonons.

. The vagueness is substantial in a liquid due to the hopping of particles.

The hopping of particles plays an essential role in the liquid state. The hopping of particles causes density fluctuations. The density fluctuation waves are just sound. Conventionally, phonons and sound have not been discriminated clearly. Sometimes sound has been confused with a longitudinal phonon