STATISTICAL MECHANICS OF PHASES, INTERFACES, AND THIN FILMS

H. Ted Davis

VCH

CONTENTS

1 / Kinetic Theory of Dilute Gases in Equilibrium 1 1.1 Introduction 1 1.2 Equations of State for Pressure and Energy

of an Ideal Monatomic Gas 3 1.3 Replacement of Time Averages by Ensemble Averages 6 1.4 Some Definitions from Probability Theory 8 1.5 Maxwell Velocity Distribution 12 1.6 Distribution and Mean Values Derived from

Maxwell Distribution 17 1.7 Mean Free Path and Mean Collision Frequency for

Rigid Sphere Molecules 23 1.8 Effusion 29 1.9 Evaporation and Chemical Reaction Rates 34 1.10 Experimental Tests of Maxwell's Distribution Law 40 1.11 The Boltzmann Factor and Barometric Formula 43 1.12 Waterson's Contribution to Kinetic Theory 47

Supplementary Reading 49 Exercises 49 References 52

2 I The Elements of Ensemble Theory 55 2.1 Introduction 55 2.2 Intermolecular Forces 55 2.3 Quantum Mechanics of Simple Systems 59

2.3.1 Particle in a Box 60 2.3.2 Noninteracting Particles in a Box 62 2.3.3 Rigid Rotator 62 2.3.4 One-Dimensional Harmonic Oscillator 64 2.3.5 A Model Diatomic Molecule in a Box 65 2.3.6 N Noninteracting Diatomic Molecules of the Type

Considered in Case 5 69 2.3.7 Energy of Hydrogen Atom in a Box 70 2.3.8 N Noninteracting Hydrogen Atoms in a Box 71 2.3.9 A Polyatomic Model That Includes Electronic States

2.4 Postulates of Ensemble Theory 73 2.5 Canonical Ensemble 74 2.6 Grand Canonical Ensemble 83 2.7 Microcanonical Ensemble 87

2.8 Isobaric Ensemble 88 2.9 Classical Mechanical Limit 89

Supplementary Reading 93 Exercises 93 References 95

-—- 3 / Statistical Mechanics of Some Simple Systems 97 3.1 Boltzmann Statistics 97 3.2 Monatomic Gases 102 3.3 Polyatomic Gases 106 3.4 Ideal Solids 113 3.5 Paramagnetism 122 3.6 Langmuir Adsorption 126 3.7 One-Dimensional Ising Model 128 3.8 Two-Dimensional Ising Model 131 3.9 The Tonks-Takahashi Fluid 134 3.10 Ideal Photon Gas (Black Body Radiation) 138 3.11 Ideal Gases Obeying Fermi-Dirac and Bose-Einstein

Statistics 142 Supplementary Reading 145 Exercises 145 References 152

—— 4 / Statistical Thermodynamics of Simple Classical Fluids 153 4.1 Distribution Functions and Thermodynamic Quantities 153 4.2 One-Dimensional Fluid of Rigid Rods 163 4.3 Van der Waals Model 168 4.4 Phase Behavior of Pure Fluids 174 4.5 Semiempirical and Empirical Equations of State 181 4.6 Computation of Liquid-Vapor Phase Diagrams from

Equations of State 194 4.7 Microscopic Derivation of Law of Corresponding States 797

Supplementary Reading 198 Exercises 799 References 207

— 5 I Imperfect Gases 203 5.1 Virial Expansion of Thermodynamic Functions 203 5.2 Theory of Virial Coefficients 272 5.3 Potential Models and Virial Coefficients 277

5.3.1 Hard Sphere Fluid 277 5.3.2 Square-Well Fluid 279 5.3.3 LJ Fluid 227 5.3.4 Kihara Fluid 223 5.3.5 Stockmayer Fluid 227 5.3.6 Ionic Fluid 231 5.3.7 Empirical Inert Gas Potential 233

xii / CONTENTS

5.4 Density Expansion of Correlation Functions: One-Component, Simple Fluid 237 Supplementary Reading 243 Exercises 243 References 245

6 / Phase Equilibria 247 6.1 Thermodynamics of Phase Equilibria 247 6.2 Some Observed Patterns of Phase Behavior 266

6.2.1 One-Component Phase Equilibria 266 6.2.2 Two-Component Phase Equilibria 270 6.2.3 Three-Component Phase Equilibria 279

6.3 Peng-Robinson Equation of State 286 6.3.1 Two-Component or Binary Phase Equilibria 289 6.3.2 Three-Component or Ternary Phase Behavior 296 6.3.3 Four-Component or Quaternary Phase Behavior 301

6.4 Lattice Theory of Solutions: Regular Solutions 302 6.4.1 Binary Phase Equilibria 304 6.4.2 Ternary Phase Behavior 308

6.5 Lattice Theory of Solutions: Quasichemical Approximation 314 6.6 Lattice Theory of Solutions: Directed Bond and

Decorated Lattice Models 319 6.1 Lattice Theory of Solutions: Flory-Huggins Theory

of Polymer Solutions 325 6.8 LatticeTheory of Solutions: Association Colloids 329

Supplementary Reading 330 Exercises 330 References 331

7 / Capillarity and Interfacial Thermodynamics 333 7.1 Manifestations of Surface or Interfacial Tension 333 7.2 The Young-Laplace and Kelvin Equations 339 7.3 Capillary Hydrostatics 344 7.4 Interfacial Thermodynamics 352 7.5 Thermodynamics of Wetting 362 7.6 Thin Films 370

Supplementary Reading 377 Exercises 378 References 379

8 / Structure and Stress in Simple Fluids and Their Mixtures: Applications to Interfaces 381 8.1 Density Distribution Functions: The Yvon-Born-Green

Hierarchy 381 8.2 Pressure Tensor and Fluid Mechanical Equilibrium:

Continuum Mechanics 388 8.3 Pressure Tensor and Fluid Mechanical Equilibrium:

Molecular Theory 390 8.4 Interfacial Tension of Planar Interfaces 397

CONTENTS / xiii

8.5 Gradient Theory of Inhomogeneous Fluids and Applications to Fluid-Fluid Interface 403

8.6 Fluids at Solid Surfaces or in Porous Media 416 Supplementary Reading 421 Exercises 422 References 423

= 9 / Density Functional Theory of Structure and Thermodynamics 425 9.1 Calculus of Variations and Functional Derivatives 425

9.1.1 Functional Differentiation 425 9.1.2 Functional Taylor's Series 431 9.1.3 Chain Rule and Inverse Functional Derivative 432 9.1.4 Implicit Functional Theorem 433 9.1.5 Functional Differential 433 9.1.6 Conditions for Extremum for a Functional 434 9.1.7 Functional Integration 434

9.2 Density Distributions and Correlation Functions 435 9.3 Homogeneous Fluids: Some Exact Results 442 9.4 Homogeneous Fluids: Approximate Theories 449

9.4.1 Mean Spherical Approximation 449 9.4.2 PY Approximation 454 9.4.3 Hypernetted Chain Approximation 454 9.4.4 Comparison of HNC, PY, and the BGYK

Approximations 455 9.4.5 Hard Sphere Mixtures 457 9.4.6 Perturbation Approximations 462

9.5 Inhomogeneous Fluids: Some Exact Results 466 Supplementary Reading 469 Exercises 469 References 472

= 10 / Confined One-Dimensional Fluids 473 10.1 Theormodynamic Properties of Hard-Rod Fluids

Between Rigid Walls 473 10.1.1 Evaluation of Partition Functions and

Thermodynamic Functions 473 10.1.2 Pore Occupancy and Disjoining Pressure of a Pore

Fluid 476 10.2 Density Distribution Functions for Hard-Rod Fluids in

Arbitrary External Fields 482 10.3 Computation of Density Distributions of Inhomogeneous

Hard-Rod Fluids 493 10.3.1 Reformulation of Integral Equations for Density

Distributions 493 10.3.2 Numerical Methods 496 10.3.3 Applications 499

10.4 Confined Tonks-Takahashi Fluids 507 10.4.1 Fluids Confined and in the Presence of an

Arbitrary External Field 507 10.4.2 One-Component Fluids: No External

Field (v(x) = 0) 511 Supplementary Reading 518

xiv / CONTENTS

Exercises 518 References 519

11 / Density Functional Theory of Fluid Interfaces 521 11.1 Local Density Functional Free Energy Model 521

11.1.1 The van der Waals Model 521 11.1.2 A Modified VDW Model 523 11.1.3 An Approximate Density Functional (ADF) Model 525 11.1.4 Density Gradient Theory 526

11.2 Local Density Functional Theory of Planar Fluid-Fluid Interfaces 528

11.3 Liquid-Vapor Interfaces: One-Component Fluids 532 11.4 Liquid-Vapor Interfaces: Multicomponent Fluids 536 11.5 Liquid-Liquid Interfaces 550

Supplementary Reading 553 Exercises 554 References 555

12 / Density Functional Theory of Confined Fluids 557 12.1 Nonlocal Density Functional Free Energy Models 557

12.1.1 The Generalized VDW Model 561 12.1.2 The Generalized Hard-Rod Model 562 12.1.3 The Tarazona Model 562 ПАЛ The Curtin-Ashcroft Model 564 12.1.5 The Meister-Kroll Model 565 12.1.6 Multicomponent Generalizations of the Models 567

12.2 Simple Fluids Confined to Slit Pores 570 12.3 Interactions Between Electrically Charged Confining

Surfaces 578 12.3.1 The Contact Theorem 578 12.3.2 Disjoining Pressure of Electrical Double Layer:

DLVO Theory 580 12.3.3 Disjoining Pressure of Electrical Double Layer: Density

Functional Theory 589 Supplementary Reading 595 Exercises 595 References 597

13 / Thin Films and Wetting Transitions 599 13.1 Introduction 599 13.2 Gradient Theory of Wetting Transitions 602 13.3 Nonlocal Density Functional Theory of Wetting Transitions 609 13.4 Local Density Functional Theory of Wetting Transitions 614 13.5 Experimental Studies of Wetting Transitions 621

Supplementary Reading 625 Exercises 625 References 626

CONTENTS / xv

14 / Ternary Amphiphilic Systems 627 Michael Schick

14.1 Introduction: The Systems and Their Behaviors 627 14.2 Lattice Gases and Lattice Fluids 632 14.3 Mean Field Theory 637 14.4 Structure and Correlation Functions of Middle Phase 649 14.5 Other Structures and Models 653

Supplementary Reading 655 Exercises 656 References 658

15 / Determination of Microstructure by Scattering Methods 659 15.1 The Theory of Scattering of Waves by Matter 659 15.2 Monatomic Fluids 664 15.3 Polyatomic Fluids 669 15.4 Crystalline Solids 678 15.5 Small Crystallites 688 15.6 Colloidal Dispersions and Small Angle Scattering 693

Supplementary Reading 700 Exercises 700 References 701

I Index 703