Mesoscopic Thermodynamics for Scientists and Engineers

Mikhail A. Anisimov is a Distinguished University Professor Emeritus and Research Professor in the Department of Chemical and Biomolecular Engineering and the Institute for Physical Science and Technology at the University of Maryland, College Park. Dr. Anisimov is an internationally recognized scientist who has been investigating phase transitions and critical phenomena in soft condensed matter for more than fifty years. He is a Fellow of the American Physical Society, American Institute of Chemical Engineers, and American Association for the Advancement of Science. Thomas J. Longo is a Research Engineer at Barron Associates, Inc., focusing on machine learning applications to science and engineering. Dr. Longo completed a PhD in Chemical Physics from the University of Maryland, College Park in 2023, where he still serves as an Adjunct Research Associate. His research interests include theoretical and computational studies of thermodynamics and dynamics of phase transitions affected by chemical reactions, liquid polyamorphism, and dissipative mesoscopic strictures.

Preface xiii

Notations, Acronyms, and Units xv

Part I Thermodynamic Approach to Meso-Heterogeneous Systems 1

1 Macro, Meso, Micro 3

1.1 Thinner than a Hair 3

1.2 Where Does the Size Matter? 4

1.3 What Is Meso-Thermodynamics? 5

1.4 Probing Mesoscales 7

References 8

Exercises 11

2 Basics of Molecular Thermodynamics 13

2.1 Laws of Thermodynamics 13

2.2 Thermodynamic Potentials and Legendre Transformations 15

2.3 Maxwell Relations 20

2.4 “Fields” and “Coordinates,” Equilibrium Conditions, Gibbs Phase Rule 22

2.5 States of Matter, Crystal Polymorphism, and Fluid Polyamorphism 25

2.6 Phase Equilibria and Stability Criteria in Single-Component Substances 28

2.6.1 The Principle of Corresponding States 29

2.6.2 Maxwell’s Equal Area Rule 30

2.6.3 Metastability 32

2.7 Equations of State 33

2.7.1 Van der Waals Equation and Virial Expansion 33

2.7.2 Lattice Gas Model 35

2.8 Thermodynamics of Mixtures 38

2.8.1 Chemical Potentials and Osmotic Pressure 38

2.8.2 Thermodynamic Stability of Mixtures 40

2.8.3 Regular Solution Model 46

2.8.4 Phase Equilibria in Ternary Systems 49

2.9 Chemical-Reaction Equilibria 51

2.10 Connection to Partition Functions in Statistical Mechanics 53

2.11 Intermolecular Interactions 56

2.12 Ideal Gas and Condensed Matter; Quasiparticles 58

2.13 Elements of Nonequilibrium Thermodynamics 61

2.14 Absolute Values of Energy and Entropy 63

Appendix 2.A Calculating Phase Equilibria in Binary Mixtures 64

References 65

Exercises 68

3 Fluctuations and Brownian Motion 71

3.1 Probability and Work of Fluctuations 71

3.2 Fluctuations of Density 75

3.3 Fluctuations of Entropy 77

3.4 Fluctuations of Concentration in a Binary Mixture 78

3.5 Fluctuations of a Generalized Density 79

3.6 Fluctuations of a Generalized Field 81

3.7 Relaxation of Fluctuations 81

3.8 Fluctuations of Anisotropy 84

3.9 Brownian Motion 86

3.10 Brownian Motion Under Gravity 87

3.11 Probing Fluctuations and Brownian Motion 89

3.11.1 Rayleigh Molecular Scattering 89

3.11.2 Spectrum of Molecular Scattering 91

3.11.3 Dynamic Light Scattering 94

3.11.4 Acoustic Spectroscopy 97

References 99

Exercises 100

4 Landau Theory of Phase Transitions 103

4.1 First-Order and Second-Order Phase Transitions 103

4.2 The Concept of the Order Parameter 104

4.3 Truncated Landau Expansion 110

4.4 Extended Landau Expansion 116

4.4.1 Asymmetry of the Order Parameter in Fluids 117

4.4.2 First-Order Phase Transitions 118

4.4.3 Weakly First-Order Phase Transitions 119

4.4.3.1 Weakly First-Order Boiling-Condensation Transition 120

4.4.3.2 Weakly First-Order Transition Emerging from the Cubic Term 121

4.4.3.3 Weakly First-Order Transition Emerging from the Quartic Term 126

4.4.4 Tricritical Phase Transition 129

4.5 Meanfield Approximation 133

4.6 Spatial Correlations of Order-Parameter Fluctuations 134

4.7 Mesoscopic Susceptibility 136

4.8 Landau–Ginzburg Functional 137

4.9 Validity of Landau Theory: Ginzburg Criterion 139

4.10 Effects of the Dimensions of Space 142

4.11 Meanfield Dynamics of Phase Transitions 143

Appendix 4.A Emergence of Tricriticality 147

References 148

Exercises 151

Part II Applications of Mesoscopic Thermodynamics 155

5 Polymer Solutions: A Meanfield Description 157

5.1 Polymer Solutions 157

5.2 Random Walk of a Polymer Chain 158

5.3 Flory–Huggins Theory of Polymer Solutions 160

5.3.1 Thermodynamics of Phase Separation 160

5.3.2 Theta Point 161

5.4 Liquid–Liquid Critical Point vs. Theta Point 163

5.4.1 Landau Expansion of the Flory–Huggins Free Energy 163

5.4.2 Osmotic-Pressure Virial Expansion 165

5.5 Widom’s Crossover Between Meanfield Criticality and Theta-Point Tricriticality 166

5.6 Theta Point as a Tricritical Point 167

5.7 Self-Avoiding Walk of a Polymer Chain 170

5.8 Phase Separation in Polymer Blends 173

5.9 Diffusion of Polymer Chains in Solution 174

5.10 Polymer Chain Near the Critical Point of a Binary Solvent 175

Appendix 5.A Derivation of Landau Coefficients in the Flory-Huggins Model 176

References 179

Exercises 180

6 Surfaces and Interfaces 183

6.1 Surface and Interfacial Tension 183

6.2 Van der Waals–Landau Theory of Smooth Interfaces 185

6.3 Curved Interfaces and Tolman’s Length 189

6.4 Wetting Transitions 191

6.5 Nucleation and Spinodal Decomposition 194

6.5.1 Gibbs Theory of Homogeneous Nucleation 195

6.5.2 Cahn–Hilliard Theory of Spinodal Decomposition 196

6.5.3 Spinodal Decomposition in Mixtures with Interconverting Species 198

Appendix 6.A Derivation of the Growth Rate in Spinodal Decomposition 201

References 202

Exercises 203

7 Self-Assembly and Modulated Phases 205

7.1 Supramolecular Self-Assembly 205

7.2 Equilibrium Polymerization 207

7.3 Micellization 209

7.3.1 Thermodynamics of Micellization 209

7.3.2 Building a Nanoparticle Through Micellization 213

7.3.3 Fluctuations of Aggregation Number 214

7.4 Gelation 215

7.5 Landau Theory of Modulated Phase Formation 218

7.6 Dissipative Modulated Structures 222

References 225

Exercises 226

8 Critical Phenomena 229

8.1 Challenges in Quantifying Critical Anomalies 230

8.2 Critical Singularities 233

8.2.1 Order Parameter 233

8.2.2 Heat Capacity 234

8.2.3 Susceptibility 236

8.2.4 Interfacial Tension 237

8.2.5 Thermal Conductivity and Shear Viscosity 237

8.3 Nature of Critical Singularities 239

8.3.1 Divergence of the Correlation Length 239

8.3.2 Fluctuation-Cluster Model 240

8.3.3 Universal Scaling Relations 242

8.3.4 Classes of Critical-Point Universality 243

8.4 Phenomenology of Critical Behavior 246

8.4.1 Isomorphism of Critical Phenomena 246

8.4.2 Scaling Free Energies and Susceptibilities 248

8.4.3 Linear Model of the Scaled Equation of State 250

8.4.4 Corrections to Asymptotic Scaling Behavior 251

8.4.4.1 Symmetric Corrections 252

8.4.4.2 Asymmetric Corrections for Fluids: Complete Scaling 253

8.4.4.3 Consequences of Asymmetry 254

8.4.5 Crossover Between Meanfield and Scaling Critical Behavior 256

8.5 Critical Phenomena in Fluid Mixtures 258

8.5.1 Complete Scaling for Binary Fluids 260

8.5.2 Weakly Compressible Liquid Mixtures 263

8.5.2.1 Singular Diameter of Phase Coexistence in Liquid Mixtures 264

8.5.2.2 Closed-Loop Phase Diagrams 264

8.5.3 Application to Liquid-Gas Critical Loci 266

8.5.3.1 Critical Line Condition 266

8.5.3.2 Peculiar Points on the Liquid–Gas Critical Loci 266

8.5.4 Renormalization of Critical Exponents 269

8.6 Effects of Fluctuations on Near-Critical Interfaces 271

8.7 Divergence of Tolman’s Length 273

8.8 Critical Phenomena in Polymer Solutions 274

8.8.1 Ising Criticality and de Gennes–Sanchez Relations 274

8.8.2 Crossover Between Ising Criticality and Theta-Point Tricriticality 276

8.9 Finite-Size Scaling 279

8.10 Critical Dynamics 283

8.10.1 Critical Slowing Down and Classes of Universality in Dynamics 283

8.10.2 Isomorphism of Transport Properties in Fluids and Fluid Mixtures 284

8.10.3 Finite-Time (Frequency) Effects 286

8.10.4 Coupling of Critical Dynamic Modes: Avoided Crossing 289

8.10.4.1 Coupling of Diffusive Modes in Binary Fluids 290

8.10.4.2 Dynamic Coupling in Polymer Solutions 291

Appendix 8.A Crossover Between Critical Singularities in Pure Fluids and Binary Mixtures 293

References 295

Exercises 301

Index 305