Physics of Liquid Matter

Preface
Contents
Acronyms
1 An Introduction to the Liquid State of Matter
1.1 Liquid State of Matter
1.1.1 Examples of Phase Diagram of Pure Substances: CO2 and Water
1.1.2 Phase Diagram of Binary Mixtures
1.2 Structure and Dynamics of Liquids: Experiments and Correlation Functions
1.3 Microscopic Models for Liquids
1.3.1 Classical Approximation
1.3.2 Different Models
1.4 Potential Energy Landscape
1.5 Approximate Theories and Computer Simulation
1.6 Water and Hydrogen Bond
1.7 Metastable States and Disordered Solid Matter
1.8 Soft Matter
1.8.1 Colloids
1.8.2 Biomolecules
References
2 Thermodynamics and Statistical Mechanics of Fluid States
2.1 Extensive and Intensive Functions
2.2 Energy and Entropy
2.3 Gibbs-Duhem Relation
2.4 Equilibrium Conditions
2.5 Equilibrium Conditions and Intensive Quantities
2.6 Macroscopic Response Functions and Stability Conditions
2.7 Legendre Transforms and Thermodynamic Potentials
2.7.1 Helmholtz Free Energy
2.7.2 Gibbs Free Energy
2.7.3 Enthalpy
2.7.4 Grand Canonical Potential
2.7.5 Tabulated Thermodynamic Potentials
2.8 Stability Conditions for Thermodynamic Potentials
2.9 Coexistence and Phase Transitions
2.10 Phase Transitions and Their Classifications
2.11 Van der Waals Equation
2.12 General Form of the Van der Waals Equation and Corresponding States
2.13 Critical Behaviour of the Van der Waals Equation
2.14 Ensembles in Statistical Mechanics
2.14.1 Microcanonical Ensemble
2.14.2 Canonical Ensemble
2.14.3 Grand Canonical Ensemble
2.14.4 Isobaric-Isothermal Ensemble
2.15 Fluctuations and Thermodynamics
References
3 Microscopic Forces and Structure of Liquids
3.1 Force Field for Atoms in Liquids
3.2 Local Structure of a Liquid
3.3 Distribution Functions in the Canonical Ensemble
3.4 Relation of the RDF with Thermodynamics
3.4.1 Energy
3.4.2 Pressure from the Virial
3.5 Distribution Functions in the Grand Canonical Ensemble
3.6 Hierarchical Equations
3.7 Qualitative Behaviour of the Radial Distribution Function
3.8 Experimental Determination of the Structure of Liquids
3.9 Neutron Scattering on Liquids
3.10 Static Limit and the Structure of Liquid
3.11 The Static Structure Factor
3.12 The Structure Factor and the RDF of Liquid Argon
3.13 The Structure Factor Close to a Critical Point
3.14 Structure of Multicomponent Liquids
3.14.1 Partial Structure Factor of Multicomponent Liquids
3.14.2 Isotopic Substitution
3.14.3 An Example: Molten Salts
3.15 Structure of Molecular Liquids
3.15.1 Structure of Liquid Water
References
4 Theoretical Studies of the Structure of Liquids
4.1 Virial Expansion in the Canonical Ensemble
4.1.1 From Hard Spheres to the Van der Waals Equation
4.2 The Mean Force Potential
4.3 Kirkwood Approximation
4.4 Radial Distribution Function from the Excess Free Energy
4.5 Density Distributions from the Grand Partition Function
4.6 Grand Potential as Generating Functional
4.7 Classical Density Functional Theory
4.7.1 Equilibrium Conditions
4.7.2 The Ornstein-Zernike Equation
4.7.3 The Ornstein-Zernike Equation in k-Space
4.7.4 Free Energy Calculation
4.7.5 Expansion from the Homogeneous System
4.8 Closure Relations from the Density Functional Theory
4.9 An Exact Equation for the g(r)
4.10 HNC and Percus-Yevick Approximations
4.10.1 RPA and MSA
4.11 Properties of the Hard Sphere Fluid
4.12 Equation of State and Liquid-Solid Transition of Hard Spheres
4.13 Percus-Yevick for the Hard Sphere Fluid
4.14 Equation of State and Thermodynamic Inconsistency
4.15 Routes to Consistency: Modified HNC and Reference HNC
4.16 Perturbation Theories: Optimized RPA
4.17 Models for Colloids
References
5 Methods of Computer Simulation
5.1 Molecular Dynamics Methods
5.1.1 Molecular Dynamics and Statistical Mechanics
5.1.2 Algorithms for the Time Evolution
5.1.3 Predictor/Corrector
5.1.4 Verlet Algorithms
Velocity Verlet
Leapfrog
5.1.5 Calculation of the Forces
5.1.6 Initial Configuration
5.1.7 Temperature in the Microcanonical Ensemble
5.1.8 Equilibration Procedure
5.1.9 Thermodynamic and Structure
5.1.10 Long-Range Corrections
5.1.11 Ewald Method
5.2 Monte Carlo Simulation
5.2.1 Monte Carlo Integration and Importance Sampling
5.2.2 Integrals in Statistical Mechanics
5.2.3 Importance Sampling in Statistical Mechanics
5.2.4 Markov Processes
5.2.5 Ergodicity and Detailed Balance
5.2.6 Metropolis Method
5.2.7 Averaging on Monte Carlo Steps
5.2.8 MC Sampling in Other Ensembles
Isobaric-Isothermal MC
Grand Canonical MC
5.2.9 MC in the Gibbs Ensemble
5.3 MD in Different Ensembles
5.3.1 Controlling the Temperature: MD in the Canonical Ensemble
The Nosé Method
Hoover Equations
5.3.2 Pressure Control
5.4 Molecular Liquids
5.5 Microscopic Models for Water
5.6 Some More Hints
5.7 Direct Calculations of the Equation of State
5.8 Free Energy Calculation from Thermodynamic Integration
5.9 An Example: Liquid-Solid Transition
5.10 Calculation of the Chemical Potential: The Widom Method
5.11 Sampling in a Complex Energy Landscape
5.12 Umbrella Sampling
5.13 Histogram Methods
5.14 Free Energy Along a Reaction Coordinate
5.14.1 Umbrella Sampling for Reaction Coordinates
5.14.2 Metadynamics
5.15 Simulation of Critical Phenomena
References
6 Dynamical Correlation Functions and Linear Response Theory for Fluids
6.1 Dynamical Observables
6.2 Correlation Functions
6.2.1 Further Properties of the Correlation Functions
6.3 Linear Response Theory
6.4 Dynamical Response Functions
6.5 Fluctuation-Dissipation Theorem
6.6 Response Functions and Dissipation
6.7 Density Correlation Functions and Van Hove Functions
6.8 Neutron Scattering to Determine the Liquid Dynamics
6.9 Dynamic Structure Factor
6.9.1 Static Limit
6.9.2 Incoherent Scattering
6.10 Density Fluctuations and Dissipation
6.10.1 Detailed Balance
6.11 Static Limit of the Density Fluctuations
6.12 Static Response Function and the Verlet Criterion
References
7 Dynamics of Liquids
7.1 Thermal Motion in Liquids
7.2 Brownian Motion and Langevin Equation
7.3 Diffusion and Self Van Hove Function
7.4 Limit of the Dilute Gas
7.5 Short Time Expansion of the Self-Intermediate Scattering Function
7.6 Correlation Functions of the Currents
7.7 The Hydrodynamic Limit
7.8 Diffusion in the Hydrodynamic Limit
7.9 Velocity Correlation Function
7.10 Liquid Dynamics in the Hydrodynamic Limit
7.10.1 Transverse Current
7.10.2 Equations Under Isotherm Conditions, Longitudinal Current and Sound Waves
7.10.3 Longitudinal Current in Presence of Thermal Diffusion and Brillouin Scattering
7.11 Different Regimes for the Liquid Dynamics: The De Gennes Narrowing
7.12 Introduction of Memory Effects
7.12.1 The Langevin Equation and Memory Effects
7.12.2 Viscoelasticity: The Maxwell Model
7.12.3 Generalized Viscosity and Memory Effects
7.13 Definition of Memory Functions
7.14 Memory Function for the Velocity Correlation Function
References
8 Supercooled Liquids: Glass Transition and Mode Coupling Theory
8.1 Phase Transitions and Metastability of Liquids
8.2 Liquids Upon Supercooling: From the Liquid to the Glass
8.3 Angell Plot
8.4 Kauzmann Temperature
8.5 Adam and Gibbs Theory
8.5.1 Cooperative Rearranging Regions
8.5.2 Calculation of the Configurational Entropy
8.6 Energy Landscape and Configurational Entropy
8.7 Dynamics of the Supercooled Liquid and Mode Coupling Theory
8.7.1 Dynamics Upon Supercooling
8.7.2 Mode Coupling Theory and Cage Effect
8.7.3 Formulation of the Theory
8.7.4 Glass Transition as Ergodic to Non-ergodic Crossover
8.7.5 The β-Relaxation
8.7.6 α-Relaxation
References
9 Supercooled Water
9.1 Supercooled and Glassy Water
9.2 The Hypothesis of a Liquid-Liquid Critical Point
9.3 The Widom Line at the Liquid-Liquid Transition
9.4 Water as a Two-Component Liquid
9.5 Dynamical Properties of Water Upon Supercooling
9.6 Widom Line and the Fragile to Strong Crossover
References
Index