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Statistical Mechanics of Liquids and Solutions

Intermolecular Forces, Structure and Surface Interactions Volume I

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## Book Description

The statistical mechanical theory of liquids and solutions is a fundamental area of physical sciences with important implications in other fields of science and for many industrial applications. This book introduces equilibrium statistical mechanics in general, and statistical mechanics of liquids and solutions in particular. A major theme is the intimate relationship between forces in a fluid and the fluid structure – a relationship that is paramount for the understanding of the subject of interactions in dense fluids. Using this microscopic, molecular approach, the text emphasizes clarity of physical explanations for phenomena and mechanisms relevant to fluids, addressing the structure and behavior of liquids and solutions under various conditions. A notable feature is the author’s treatment of forces between particles that include nanoparticles, macroparticles, and surfaces. The book provides an expanded, in-depth treatment of simple liquids and electrolytes in the bulk and in confinement.

- Provides an introduction to statistical mechanics of liquids and solutions with special attention to structure and interactions.
- Offers an extensive presentation starting with the basics of statistical mechanics to modern aspects of the theory of liquids and solutions, including intermolecular interactions in fluids.
- Treats both homogeneous bulk fluids and inhomogeneous fluids near surfaces and in confinement.
- Takes a microscopic, molecular approach that combines physical transparency, theoretical sharpness and a pedagogical and accessible style.
- Gives explicit and clear textual explanations and physical interpretations for any mathematical relationships and derivations.
- Goes deeper than the available texts on interactions in fluids, by taking the discussion beyond simple approximations and mean field approaches.

The book will be an invaluable resource for advanced undergraduate, graduate, and postgraduate students in physics, chemistry, soft matter science, surface and colloid science and related fields, as well as professionals and instructors in those areas of science.

## Table of Contents

Contents

Preface, xi

Overview of Contents, xv

Author, xix

PART I **Basis of Equilibrium Statistical Mechanics**

CHAPTER 1 Introduction 3

1.1 The Microscopic Definitions of Entropy and Temperature 3

1.1.1 A Simple Illustrative Example 5

1.1.2 Microscopic Definition of Entropy and Temperature for

Isolated Systems 12

1.2 Quantum vs Classical Mechanical Formulations of Statistical

Mechanics: An Example 17

1.2.1 The Monatomic Ideal Gas: Quantum Treatment 18

1.2.2 The Monatomic Ideal Gas: Classical Treatment 27

Appendix 1A: Alternative Expressions for the Entropy of an Isolated

System 31

CHAPTER 2 Statistical Mechanics from a Quantum Perspective 33

2.1 Postulates and Some Basic Definitions 33

2.2 Isolated Systems: The Microcanonical Ensemble 38

2.3 Thermal Equilibria and the Canonical Ensemble 52

2.3.1 The Canonical Ensemble and Boltzmann’s Distribution Law 52

2.3.2 Calculations of Thermodynamical Quantities; the Connection

with Partition Functions 56

*2.3.2.1 The Helmholtz Free Energy *56

*2.3.2.2 Thermodynamical Quantities as Averages *59

*2.3.2.3 Entropy in the Canonical Ensemble *63

2.4 Constant Pressure: The Isobaric-Isothermal Ensemble 66

2.4.1 Probabilities and the Isobaric-Isothermal Partition Function 66

2.4.2 Thermodynamical Quantities in the Isobaric-Isothermal

Ensemble 71

*2.4.2.1 The Gibbs Free Energy *71

*2.4.2.2 Probabilities and Thermodynamical Quantities *73

*2.4.2.3 The Entropy in the Isobaric-Isothermal Ensemble *76

2.5 Open Systems: Chemical Potential and the Grand Canonical

Ensemble 79

2.5.1 Probabilities and the Grand Canonical Partition Function 79

2.5.2 Thermodynamical Quantities in the Grand Canonical

Ensemble 83

2.6 Fluctuations in Thermodynamical Variables 88

2.6.1 Fluctuations in Energy in the Canonical Ensemble 88

2.6.2 Fluctuations in Number of Particles in the Grand Canonical

Ensemble 89

2.6.3 Fluctuations in the Isobaric-Isothermal Ensemble 90

2.7 Independent Subsystems 91

2.7.1 The Ideal Gas and Single-Particle Partition Functions 91

2.7.2 Translational Single-Particle Partition Function 95

Appendix 2A: The Volume Dependence of *S *and Quasistatic Work 99

Appendix 2B: Stricter Derivations of Probability Expressions 103

CHAPTER 3 Classical Statistical Mechanics 109

3.1 Systems with *N *Spherical Particles 110

3.2 The Canonical Ensemble 112

3.3 The Grand Canonical Ensemble 122

3.4 Real Gases 125

CHAPTER 4 Illustrative Examples from Some Classical Theories of

Fluids 131

4.1 The Ising Model 131

4.2 The Ising Model Applied to Lattice Gases and Binary Liquid

Mixtures 134

4.2.1 Ideal Lattice Gas 135

4.2.2 Ideal Liquid Mixture 136

4.2.3 The Bragg-William Approximation 138

*4.2.3.1 Regular Solution Theory *138

*4.2.3.2 Some Applications of Regular Solution Theory *142

*4.2.3.3 Flory-Huggins Theory for Polymer Solutions *151

PART II **Fluid Structure and Interparticle Interactions**

CHAPTER 5 Interaction Potentials and Distribution Functions 165

5.1 Bulk Fluids of Spherical Particles. The Radial Distribution Function 166

5.2 Number Density Distributions: Density Profiles 172

5.3 Force Balance and the Boltzmann Distribution for Density:

Potential of Mean Force 175

5.4 The Relationship to Free Energy and Chemical Potential 181

5.5 Distribution Functions of Various Orders for Spherical Particles 184

5.5.1 Singlet Distribution Function 184

5.5.2 Pair Distribution Function 185

5.5.3 Distribution Functions in the Canonical Ensemble 188

5.6 The structure factor for homogeneous and inhomogeneous fluids 192

5.7 Thermodynamical Quantities from Distribution Functions 199

5.8 Microscopic density distributions and density-density correlations 212

5.9 Distribution Function Hierarchies and Closures, Preliminaries 215

5.10 Distribution Functions in the Grand Canonical Ensemble 218

5.11 The Born-Green-Yvon Equations 222

5.12 Mean Field Approximations for Bulk Systems 225

5.13 Computer Simulations and Distribution Functions 227

5.13.1 General Background 227

*5.13.1.1 Basics of Molecular Dynamics Simulations *228

*5.13.1.2 Basics of Monte Carlo Simulations *231

5.13.2 Bulk Fluids 235

*5.13.2.1 Boundary Conditions *235

*5.13.2.2 Distribution Functions *237

*5.13.2.3 Thermodynamical Quantities *242

5.13.3 Inhomogeneous Fluids 250

5.13.3.1 Density Profiles Outside Macroparticles or Near

*Planar Surfaces *250

*5.13.3.2 Pair Distribution Functions *252

Appendix 5A: The Dirac Delta Function 255

CHAPTER 6 Interactions and Correlations in Simple Bulk

Electrolytes 257

6.1 The Poisson-Boltzmann (PB) Approximation 258

6.1.1 Bulk Electrolytes, Basic Treatment 258

6.1.2 Decay of Electrostatic Potential and Effective Charges of

Particles 275

*6.1.2.1 The Concept of Effective Charge *275

*6.1.2.2 Electrostatic Potential from Nonspherical Particles *278

6.1.2.3 The Decay of Electrostatic Potential from Spherical

*and Nonspherical Particles *283

6.1.3 Interaction between two Particles Treated on an Equal Basis 290

*6.1.3.1 Background *290

6.1.3.2 The Decay of Interaction between Two Nonspherical

*Macroions *291

6.1.4 The Interaction between Two Macroions for all Separations 294

*6.1.4.1 Poisson-Boltzmann Treatment *294

6.1.4.2 Electrostatic Part of Pair Potential of Mean Force,

*General Treatment *297

6.1.5 One Step beyond PB:What HappensWhen *all *Ions are

Treated on an Equal Basis? 299

6.2 Electrostatic Screening in Simple Bulk Electrolytes, General Case 304

6.2.1 Electrostatic Interaction Potentials 306

*6.2.1.1 Polarization Response and Nonlocal Electrostatics *307

*6.2.1.2 The Potential of Mean Force and Dressed Particles *311

*6.2.1.3 Screened Electrostatic Interactions *313

6.2.2 The Decay Behavior and the Screening Decay Length 318

6.2.2.1 Oscillatory and Monotonic Exponential Decays:

*Explicit Examples *318

6.2.2.2 Roles of Effective Charges, Effective Dielectric

*Permittivities and the Decay Parameter κ *320

6.2.2.3 The Significance of the Asymptotic Decays: Concrete

*Examples *330

6.2.3 Density-Density, Charge-Density and Charge-Charge

Correlations 339

Appendix 6A: The Orientational Variable *ω *345

Appendix 6B: Variations in Density Distribution When the External

Potential is Varied; the First Yvon Equation 346

Appendix 6C: Definitions of the HNN, HQN and HQQ Correlation

Functions 349

CHAPTER 7 Inhomogeneous and Confined Simple Fluids 353

7.1 Electric Double-Layer Systems 354

7.1.1 The Poisson-Boltzmann (Gouy-Chapman) Theory 355

7.1.1.1 The Poisson-Boltzmann Equation for Planar Double

*Layers *355

*7.1.1.2 The Case of Symmetric Electrolytes *358

7.1.1.3 Effective Surface Charge Densities and the Decay of

*the Electrostatic Potential *362

7.1.2 Electrostatic Screening in Electric Double-Layers, General

Case 366

*7.1.2.1 Decay of the Electrostatic Potential Outside aWall *367

7.1.2.2 Decay of Double-Layer Interactions: Macroion-Wall

*andWall-Wall *369

7.1.3 Ion-Ion Correlation Effects in Electric Double-Layers: Explicit

Examples 372

7.1.4 Electric Double-Layers with Surface Polarizations (Image

Charge Interactions) 379

7.1.5 Electric Double-Layers with Dispersion Interactions 385

7.2 Structure of Inhomogeneous Fluids on the Pair Distribution Level 392

7.2.1 Inhomogeneous Simple Fluids 393

*7.2.1.1 Lennard-Jones Fluids *393

*7.2.1.2 Hard Sphere Fluids *396

7.2.2 Primitive Model Electrolytes 406

*7.2.2.1 Pair Distributions in the Electric Double-Layer *406

7.2.2.2 Ion-Ion Correlations Forces: Influences on Density

*Profiles *410

Appendix 7A: Solution of PB Equation for a Surface in Contact with a

Symmetric Electrolyte 415

Appendix 7B: Electric Double-Layers with Ion-Wall Dispersion

Interactions in Linearized PB Approximation 416

CHAPTER 8 Surface Forces 421

8.1 General Considerations 421

8.1.1 The Disjoining Pressure and the Free Energy of Interaction 422

8.1.2 Electric Double-Layer Interactions, Some General Matters 424

8.2 Poisson-Boltzmann Treatment of Electric Double-Layer

Interactions 427

8.2.1 Equally Charged Surfaces 428

8.2.2 Arbitrarily Charged Surfaces 435

8.2.3 Electrostatic part of double-layer interactions, general

treatment 439

8.3 Surface Forces and Pair Correlations, General Considerations 440

8.4 Structural Surface Forces 447

8.5 Electric Double-Layer Interactions with Ion-Ion Correlations 451

8.5.1 Counterions between Charged Surfaces 451

8.5.2 Equilibrium with Bulk Electrolyte 459

8.6 Van der Waals Forces and Image Interactions in Electric

Double-Layer Systems 460

8.6.1 Van derWaals Interactions and Mean Field Electrostatics: The

DLVO Theory 461

8.6.2 The Effects of Ion-Ion Correlations on Van derWaals

Interactions. Ionic Image Charge Interactions 463

8.6.3 The Inclusion of Dispersion Interactions for the Ions 467

Appendix 8A: Solution of PB Equation for Counterions between Two

Surfaces 472

Appendix 8B: Proofs of Two Expressions for *P*slit

⊥ 474

List of Symbols, 479

Index, 487

## Author(s)

### Biography

Roland Kjellander acquired a master’s degree in chemical engineering, a Ph.D. in physical chemistry, and the title of docent in physical chemistry from the Royal Institute of Technology, Stockholm, Sweden. He is currently a professor emeritus of physical chemistry in the Department of Chemistry and Molecular Biology at the University of Gothenburg, Sweden. His previous appointments include roles in various academic and research capacities at the University of Gothenburg, Sweden; Australian National University, Canberra; Royal Institute of Technology, Stockholm, Sweden; Massachusetts Institute of Technology, Cambridge, USA; and Harvard Medical School, Boston, USA. He was awarded the 2004 Pedagogical Prize from the University of Gothenburg, Sweden, and the 2007 Norblad-Ekstrand Medal from the Swedish Chemical Society. Professor Kjellander’s field of research is statistical mechanics, in particular liquid state theory.