Theory of Disordered Solids: From Atomistic Dynamics to Mechanical, Vibrational, and Thermal Properties

Description

This book presents a consistent mathematical theory of the non-electronic physical properties of disordered and amorphous solids, starting from the atomic-level dynamics and leading to experimentally verifiable descriptions of macroscopic properties such as elastic and viscoelastic moduli, plasticity, phonons and vibrational spectra, and thermal properties. This theory begins with the assumption of the undeniable existence of an “amorphous lattice”, which allows one to relegate the theoretical uncertainties about the ultimate nature of the glass transition to a subsidiary role and thus take a more pragmatic approach towards the modelling of physical properties.  The book introduces the reader not only to the subtle physical concepts underlying the dynamics, mechanics, and statistical physics of glasses and amorphous solids, but also to the essential mathematical and numerical methods that cannot be readily gleaned from specialized literature since they are spread out among many often technically demanding papers. These methods are presented in this book in such a way as to be sufficiently general, allowing for the mathematical or numerical description of novel physical phenomena observed in many different types of amorphous solids (including soft and granular systems), regardless of the atomistic details and particular chemistry of the material.   This monograph is aimed at researchers and graduate-level students in physics, materials science, physical chemistry and engineering working in the areas of amorphous materials, soft matter and granular systems, statistical physics, continuum mechanics, plasticity, and solid mechanics. It is also particularly well suited to those working on molecular dynamics simulations, molecular coarse-grained simulations, as well as ab initio atomistic and DFT methods for solid-state and materials science. Preface Acknowledgments Contents Acronyms 1 A Bird's-Eye View of Amorphous Solids 1.1 Microscopic Bonding and Interactions in Disordered Solids 1.1.1 Central-Force (Non-covalent) Interactions 1.1.2 Covalent Bonding 1.1.3 Metallic Bonding 1.1.4 Anharmonicity 1.2 The Structure of Disordered Solids 1.2.1 Static Structure Factor and Radial Distribution Function 1.2.2 Structural Paradigms for Disordered Solids 1.2.2.1 The Random Network Model 1.2.2.2 The Random Close Packing Paradigm 1.2.2.3 Monodisperse Random Close Packing 1.2.2.4 Polydisperse Random Packings 1.2.3 The Fractal Model 1.3 Dynamic Correlation Functions 1.4 The Glass Transition 1.5 Structural Order Parameters 1.5.1 Bond-Orientational Order Parameter 1.5.2 Inversion Symmetry 1.6 Constraint Counting and Isostaticity 1.7 Anharmonic Potential of Mean Force in Glasses References 2 Elasticity 2.1 Introduction 2.2 The Concept of Affine and Nonaffine Deformations 2.3 Born-Huang Formulae for the Affine Elastic Moduli 2.4 Elastic Moduli of Amorphous Solids 2.4.1 General Theory 2.4.2 The Shear Modulus of Random JammedSphere Packings 2.4.3 Elasticity of Random Networks with Internal Stresses 2.5 The Shear Modulus of Glasses 2.5.1 The Shear Modulus of Polymer Glasses 2.5.2 The Shear Modulus of Lennard-Jones Glasses 2.6 The Shear Modulus of Colloidal Gels 2.6.1 Elasticity of Fractal Gels 2.6.2 Intermediate Dense Gels, and Cluster Glasses 2.7 The Bulk Modulus 2.8 Stress-Fluctuation Formalism for the Elastic Moduli of Thermal Systems 2.8.1 General Formalism 2.8.2 The Temperature-Dependent Shear Modulus of Perfect Crystals 2.8.3 The Case of Liquids 2.8.4 The Temperature-Dependent Shear Modulus of Glasses Revisited References 3 Viscoelasticity 3.1 Fundamentals of Linear Viscoelasticity 3.2 Microscopic Nonaffine Theory of Viscoelasticity 3.2.1 Nonequilibrium Dissipative Equation of Motion 3.2.2 Derivation of Microscopic Viscoelastic Moduli 3.3 Case Study: Polymer Glasses 3.4 Case Study: Metallic Glasses 3.4.1 Application to Cu50Zr50 Metallic Glass 3.4.2 Experiments 3.4.3 MD Simulations with EAM Potentials 3.4.4 Memory Kernel for the Friction 3.4.5 Dynamic Viscoelastic Young's Moduli of Cu50Zr50 3.5 Microscopic Theory of Relaxation Modulus and Creep 3.5.1 Theory of Power-Law Creep in Disordered Solids References 4 Wave Propagation and Damping 4.1 Sound Attenuation 4.2 Akhiezer Damping 4.3 Microscopic Theory of Sound Attenuation in Amorphous Solids 4.3.1 Linear Response Theory 4.3.2 Dynamic Response Function: From the Stress to the Displacement Correlator 4.3.3 From Nonaffine Motions to the Susceptibility 4.3.4 Acoustic Wave Propagation 4.3.5 Acoustic Wave Attenuation 4.4 Ioffe-Regel Crossover References 5 Phonons and Vibrational Spectrum 5.1 The Debye Model of Solids 5.1.1 The Debye Density of States 5.1.2 Debye Frequency, Momentum, and Temperature 5.1.3 Van Hove Singularities 5.2 The Boson Peak in the VDOS 5.3 Case Study: Simple Lattices with Randomness 5.4 Theoretical Models 5.4.1 The Random Matrix Theory (RMT) Model 5.4.2 Singular Behavior Near Marginal Stability 5.4.3 The Damped Phonon Model of the VDOS References 6 Thermal Properties 6.1 Specific Heat 6.2 Thermal Conductivity 6.3 The Tunneling Model References 7 Viscosity of Supercooled Liquids 7.1 The Viscosity of Liquids 7.2 The Vogel-Fulcher-Tammann Empirical Expression 7.3 Fragility 7.4 The Adam-Gibbs Model and Growing Length Scales 7.5 Microscopic Theory of Viscosity 7.5.1 Frenkel's Theory 7.5.2 Dyre's Shoving Model 7.5.3 Linking Viscosity to Bonding and Structure: The KSZ Model 7.6 First-Principles Frameworks 7.6.1 Green-Kubo Formalism 7.6.2 Nonaffine Response Theory References 8 Plastic Deformation 8.1 Plasticity of Crystals 8.1.1 Dislocations 8.1.2 Schmid's Law 8.1.3 Dislocations as Topological Defects 8.1.4 Volterra Construction 8.2 Theory of Plastic Deformation in Amorphous Solids Mediated by Dislocation-Type Defects 8.2.1 Dislocation-Type Defects in Amorphous Solids from Nonaffine Displacements 8.3 Plasticity Mediated by Dislocation-Type Defects in Amorphous Solids: Polymer Glasses 8.4 Shear Banding and Eshelby-Like Quadrupoles References 9 Confinement Effects 9.1 Elasticity and Waves Under Confinement 9.2 Comparison with Experimental and Simulations Data 9.2.1 Confined Liquids 9.2.2 Confined Amorphous Solids 9.3 Vibrational Density of States of Confined Solids References A A Brief Reminder of Elasticity Theory B Lattice Dynamics of Metallic Glasses with the EAM Potential C Generalized Langevin Equation Derived from Caldeira-Leggett Hamiltonians C.1 Caldeira-Leggett Hamiltonian C.2 Derivation of the Generalized Langevin Equation C.3 The Fluctuation-Dissipation Theorem C.4 The Memory Kernel References Glossary Index