Phases of Matter and their Transitions

Preface xvi List of Frequently Used Symbols and Abbreviations xxi SI Units, Physical Constants, and Conversion Factors xxvii Summary of Notation xxxi 1 Introduction 1 1.1 Constituents of Matter 1 1.2 Matter and Energy: Interaction and Change 3 1.3 Mass and Charge 4 1.4 Macroscopic and Microscopic Approaches 6 1.5 Gases, Liquids, and Solids 7 1.6 What to Expect? 11 1.7 Units and Notation 12 References 13 Further Reading 14 2 Classical Mechanics 15 2.1 Frames, Particles, and Coordinates 15 2.2 From Newton to Hamilton 17 2.3 Hamilton’s Principle and Lagrange’s Equations 19 2.4 Conservation Laws 21 2.5 Hamilton’s Equations 24 2.6 Hamilton’s Principle for Continuous Systems 26 2.7 The Virial Theorem 27 2.8 Final Remarks 28 References 28 Further Reading 29 3 Quantum Mechanics 30 3.1 Quantum Concepts 30 3.1.1 Fundamental Quantum Kinematics 30 3.1.2 Operators and their Representation 33 3.1.3 Fundamental Quantum Kinetics 35 3.2 Interpretation and Some Exact Solutions 37 3.2.1 The Particle in a Box 39 3.2.2 The Harmonic Oscillator 40 3.2.3 The Rigid Rotator 41 3.2.4 Many Particles 42 3.3 Approximate Quantum Mechanics Solutions 43 3.3.1 The Born–Oppenheimer Approximation 43 3.3.2 The Variation Principle 44 3.3.3 The Hartree–Fock Method 47 3.3.4 Perturbation Theory 51 3.3.5 The Density Operator 53 3.4 Final Remarks 55 References 55 Further Reading 56 4 Intermolecular Interactions 57 4.1 The Semi-classical Approach 57 4.1.1 Electrostatic Interaction 59 4.1.2 Induction Interaction 62 4.1.3 Dispersion Interaction 63 4.1.4 The Total Interaction 64 4.2 The Quantum Approach 66 4.3 Model Interactions 69 4.4 Refinements 72 4.4.1 Hydrogen Bonding 72 4.4.2 Three-Body Interactions 74 4.4.3 Accurate Empirical Potentials 74 4.5 Final Remarks 75 References 76 Further Reading 77 5 Continuum Mechanics 78 5.1 The Nature of the Continuum 78 5.2 Kinematics 79 5.2.1 Material and Spatial Coordinates 79 5.2.2 General Deformations 80 5.2.3 The Small Displacement Gradient Approximation 81 5.3 Balance Equations 83 5.4 Kinetics 85 5.4.1 The Principle of Virtual Power 86 5.4.2 Linear Momentum 86 5.4.3 Angular Momentum 88 5.4.4 Cauchy’s Equations of Motion 88 5.5 The Stress Tensor 89 5.6 Mechanical Energy 90 5.7 Final Remarks 91 References 92 Further Reading 92 6 Macroscopic Thermodynamics 93 6.1 Classical Thermodynamics 93 6.1.1 The Four Laws 93 6.1.2 Quasi-Conservative and Dissipative Forces 99 6.1.3 Equations of State 100 6.1.4 Mechanical and Thermal Equilibrium 101 6.1.5 Auxiliary Functions 101 6.1.6 Some Derivatives and their Relationships 103 6.1.7 Chemical Content 103 6.1.8 Chemical Equilibrium 106 6.2 The Local State and Internal Variables 110 6.2.1 The Behavior of Internal Variables 111 6.2.2 The Local State 113 6.3 Field Formulation 115 6.3.1 The First Law 115 6.3.2 The Second Law 116 6.4 The Linear Approximation in Non-equilibrium Thermodynamics 118 6.5 Final Remarks 122 References 122 Further Reading 123 7 Microscopic Thermodynamics 125 7.1 Basics of Statistical Thermodynamics 125 7.1.1 Preliminaries 125 7.1.2 Entropy and Partition Functions 128 7.1.3 Fluctuations 132 7.2 Noninteracting Particles 134 7.2.1 Single Particle 134 7.2.2 Many Particles 134 7.2.3 Pressure and Energy 135 7.3 The Semi-classical Approximation 136 7.4 Interacting Particles 141 7.5 Internal Contributions 142 7.5.1 Vibrations 142 7.5.2 Rotations 145 7.5.3 Electronic Transitions 147 7.6 Some General Aspects 148 7.6.1 Mode or Average? 148 7.6.2 Fluctuations and Other Ensembles 149 7.6.3 Equipartition of Energy 150 7.6.4 The Gibbs–Bogoliubov Inequality 151 References 152 Further Reading 154 8 Gases 155 8.1 Basic Kinetic Theory of Gases 155 8.2 The Virial Expansion 159 8.2.1 Some Further Remarks 162 8.3 Equations of State 164 8.4 The Principle of Corresponding States 168 8.4.1 The Extended Principle 171 8.5 Transition State Theory 174 8.5.1 Chemical Kinetics Basics 174 8.5.2 The Equilibrium Constant 175 8.5.3 Potential Energy Surfaces 176 8.5.4 The Activated Complex 177 8.5.5 The Link to Experiment 179 8.6 Dielectric Behavior 180 8.6.1 Basic Aspects 180 8.6.2 The Debye–Langevin Equation 182 8.6.3 Frequency Dependence 185 8.6.4 Estimating μ and α 190 References 193 Further Reading 196 9 Liquids 197 9.1 Approaches to Liquids 197 9.2 Distribution Functions, Structure, and Energetics 198 9.2.1 Structure 200 9.2.2 Energetics 203 9.3 The Integral Equation Approach 206 9.3.1 The Ornstein–Zernike Equation 206 9.3.2 The Yvon–Born–Green Equation 209 9.3.3 Other Integral Equations 210 9.3.4 The Potential of Mean Force 212 9.4 Comparison: Hard-Sphere and Lennard-Jones Results 214 9.5 Scaled-Particle Theory 217 9.6 Structural Models 218 9.6.1 Cell Models 220 9.6.2 Hole Models 226 9.6.3 Some Other Implementations of Hole Theory 231 9.7 The Generalized van der Waals Model 237 9.8 Phonon Theory of Liquids 240 9.9 The Quantum Cluster Equilibrium Model 244 9.10 Some Continuum Aspects 245 9.11 Dielectric Behavior 249 References 255 Further Reading 259 10 Solids 260 10.1 Inorganics and Metals 260 10.2 Polymers 263 10.3 Lattice Concepts 265 10.4 Crystalline Structures 267 10.5 Bonding: The Quantum-mechanical Approach 270 10.5.1 The Nearly Free Electron Approximation 270 10.5.2 The Tight Binding Approximation 275 10.5.3 Density Functional Theory 278 10.6 Bonding: The Empirical Approach 282 10.6.1 Atoms, Ions, and Electronegativity 282 10.6.2 Covalent and Molecular Crystals 286 10.6.3 Ionic Crystals: The Classical Approach 287 10.6.4 Ionic Crystals: Electronegativity Approaches 290 10.6.5 Metallic Crystals 294 10.7 Lattice Dynamics 296 10.8 Two Simple Models 299 10.9 Properties 300 10.9.1 Heat Capacity 300 10.9.2 Thermal Expansivity 302 10.9.3 Bulk Modulus 303 10.10 Defects 304 10.10.1 Zero-dimensional Defects 305 10.10.2 One-dimensional Defects 308 10.10.3 Other Defects 310 10.11 Thermo-elasticity 312 10.11.1 Elastic Behavior 312 10.11.2 Stress States and the Associated Elastic Constants 313 10.11.3 Elastic Energy 314 10.11.4 A Matter of Notation 315 10.11.5 Anisotropic Materials 316 10.11.6 The Effect of Temperature 319 10.12 Final Remarks 320 References 320 Further Reading 325 11 Interfaces 326 11.1 Thermodynamics of Interfaces 326 11.2 One-Component Surfaces: Semiempirical Considerations 331 11.3 One-Component Surfaces: Theoretical Considerations 336 11.3.1 Density Functional Theory 336 11.3.2 Capillary Wave Theory 341 11.4 Solid Surface Structure 343 11.4.1 Surface Roughening 345 11.5 Adsorption at Interfaces 349 11.5.1 Solutions 349 11.5.2 Thermodynamics of Adsorption 355 11.5.3 Statistics of Adsorption 357 11.5.4 Adsorption Isotherms 360 11.6 Final Remarks 366 References 366 Further Reading 370 12 Phase Transitions: General Aspects 371 12.1 Some General Considerations 371 12.2 The Clapeyron and Clapeyron–Clausius Equation 375 12.3 The Mosselman Solution for the Clapeyron Equation 376 12.4 The Ehrenfest–Prigogine–Defay Equations 378 12.5 Landau and Landau-like Theory 380 References 383 Further Reading 384 13 Discontinuous Phase Transitions: Liquids ↔ Gases 385 13.1 Thermodynamics of Evaporation 385 13.1.1 Evaporation in the Presence of an Inert Gas 387 13.2 Kinetics of Evaporation 388 13.2.1 Classical Kinetic Theory 388 13.2.2 Secondary Effects 393 13.2.3 Other Approaches 394 13.3 The Reverse Transition: Condensation 395 13.3.1 Drops and Bubbles 395 13.3.2 Classical Nucleation Theory 397 13.3.3 Nucleation Kinetics 399 13.3.4 Modifications 401 13.3.5 Molecular Aspects 404 References 408 Further Reading 410 14 Discontinuous Phase Transitions: Solids ↔ Liquids 411 14.1 Melting or Fusion 411 14.2 Mechanical or Bulk Melting 414 14.2.1 Vibrational Instability 414 14.2.2 Lattice Instability 418 14.2.3 Vacancies 418 14.2.4 Interstitials 419 14.2.5 Dislocations 422 14.2.6 Interstitialcies 424 14.2.7 Simulations 427 14.3 Thermodynamic or Surface-Mediated Melting 428 14.3.1 Melting of Nanoparticles 428 14.3.2 Vacancies Revisited 430 14.3.3 Dislocations Revisited 432 14.4 Polymer Melting 434 14.5 The Influence of Pressure 436 14.6 Other Aspects 440 14.7 Melting in Perspective 442 14.8 The Reverse Transition: Freezing or Solidification 444 14.8.1 Nucleation and Growth 444 14.8.2 Some Further Remarks 446 14.8.3 Polymers and Metals 448 14.8.4 Water 451 References 452 Further Reading 457 15 Continuous Phase Transitions: Liquids ↔ Gases 458 15.1 Limiting Behavior 458 15.2 Mean-Field Theory: Landau Theory 461 15.2.1 Landau-Like Theory: Fluid Transitions 463 15.3 Scaling 465 15.3.1 Homogeneous Functions 465 15.3.2 Scaling Potentials 466 15.3.3 Scaling Lattices 467 15.4 Renormalization 469 15.5 Final Remarks 475 References 476 Further Reading 476 16 The Liquid Crystal Transformation 478 16.1 Nature and Types 478 16.2 The Nematic–Isotropic Transformation 480 16.2.1 The Orientation as Internal Variable 480 16.2.2 The Discontinuous Transformation 481 16.3 Alternative Approaches 482 16.3.1 Maier–Saupe Theory 483 16.3.2 The Coil–Helix Transformation 485 16.3.3 Onsager Theory 486 16.4 Some Extensions 489 16.5 Elastic Energy and Defects 491 16.6 The Fréedericksz Transformation 494 References 496 Further Reading 497 17 Dielectric Behavior and the Ferroelectric Transformation 498 17.1 Preliminaries and Dielectric Materials 498 17.1.1 General Remarks 498 17.1.2 Dielectric Materials 500 17.2 Electronic Polarization 501 17.3 Vibrational Polarization 503 17.3.1 Three Models 507 17.4 Orientational Polarization 510 17.5 Space–Charge Polarization 511 17.6 Ferroelectric Materials 512 17.7 Ferroelectric Behavior 516 17.7.1 The Thermodynamic Approach 516 17.7.2 The Microscopic Approach 518 References 521 Further Reading 523 18 The Glass Transition 525 18.1 What Is a Glass? 525 18.1.1 Glassy Materials 528 18.1.2 Property Changes at Tg 529 18.2 The Thermodynamic Approach 530 18.3 The Structural Approach 535 18.3.1 Free Volume Theory 536 18.3.2 Continuous Transition Theory 539 18.4 The Lattice Gas Approach 541 18.5 Phonon Theory for Glasses 543 18.6 Mode-Coupling Theory 546 18.7 Final Remarks 549 References 550 Further Reading 553 19 Irreversibility and the Return to Equilibrium 555 19.1 Some Considerations 555 19.2 The Boltzmann Approach 557 19.2.1 Time Invariance 558 19.2.2 Recurrence 560 19.3 The Gibbs Approach 561 19.4 The Formal Approach 563 19.5 The Physical Approach 567 19.6 The Information Theory Approach 571 19.6.1 A Brief Review 571 19.6.2 High and Low Probability Manifolds 572 19.7 Closure 578 References 580 Further Reading 583 Appendix A Guide to Mathematics Used 584 A. 1 Symbols and Conventions 584 A. 2 Derivatives, Differentials, and Variations 584 A. 3 Composite, Implicit, Homogeneous, Complex, and Analytic Functions 586 A. 4 Extremes and Lagrange Multipliers 588 A. 5 Legendre Transforms 588 A. 6 Coordinate Axes Rotations 589 A. 7 Change of Variables 590 A. 8 Calculus of Variations 591 A. 9 Matrices and Determinants 592 A. 10 The Eigenvalue Problem 594 A. 11 Matrix Decompositions 597 A. 12 Scalars, Vectors, and Tensors 598 A. 13 Tensor Analysis 601 A. 14 Gamma, Dirac, and Heaviside Functions 603 A. 15 Laplace and Fourier Transforms 604 A. 16 Some Useful Expressions 606 Further Reading 607 Appendix B Elements of Special Relativity Theory 608 B.1 Lorentz Transformations 608 B.2 Velocities, Contraction, Dilatation, and Proper Quantities 610 B.3 Relativistic Lagrange and Hamilton Functions 611 References 612 Further Reading 612 Appendix C The Lattice Gas Model 613 C. 1 The Lattice Gas Model 613 C. 2 The Zeroth or Mean-Field Approximation 613 C. 3 The First or Quasi-Chemical Approximation 615 C. 4 Athermal Entropy for Chain-Like Molecules 619 References 621 Further Reading 621 Appendix D Elements of Electrostatics 622 D.1 Coulomb, Gauss, Poisson, and Laplace 622 D.2 A Dielectric Sphere in a Dielectric Matrix 624 D.3 A Dipole in a Spherical Cavity 626 Further Reading 627 Appendix E Elements of Probability and Statistics 629 E.1 Probability 629 E.2 Single Variable 631 E.3 Multiple Variables 632 E.4 The Normal Distribution and the Central-Limit Theorem 633 References 635 Further Reading 635 Appendix F Selected Data 636 References 650 Appendix G Answers to Selected Problems 652 Index 659