Phases of Matter and their Transitions

Gijsbertus de With, PhD, is Professor Emeritus of Materials Science at Eindhoven University of Technology in the Netherlands. His research is focused on the structure and interfacial phenomena related to the chemical and thermomechanical behavior of multi-phase materials.

• Preface xviList of Frequently Used Symbols and Abbreviations xxiSI Units, Physical Constants, and Conversion Factors xxviiSummary of Notation xxxi1 Introduction 11.1 Constituents of Matter 11.2 Matter and Energy: Interaction and Change 31.3 Mass and Charge 41.4 Macroscopic and Microscopic Approaches 61.5 Gases, Liquids, and Solids 71.6 What to Expect? 111.7 Units and Notation 12References 13Further Reading 142 Classical Mechanics 152.1 Frames, Particles, and Coordinates 152.2 From Newton to Hamilton 172.3 Hamilton’s Principle and Lagrange’s Equations 192.4 Conservation Laws 212.5 Hamilton’s Equations 242.6 Hamilton’s Principle for Continuous Systems 262.7 The Virial Theorem 272.8 Final Remarks 28References 28Further Reading 293 Quantum Mechanics 303.1 Quantum Concepts 303.1.1 Fundamental Quantum Kinematics 303.1.2 Operators and their Representation 333.1.3 Fundamental Quantum Kinetics 353.2 Interpretation and Some Exact Solutions 373.2.1 The Particle in a Box 393.2.2 The Harmonic Oscillator 403.2.3 The Rigid Rotator 413.2.4 Many Particles 423.3 Approximate Quantum Mechanics Solutions 433.3.1 The Born–Oppenheimer Approximation 433.3.2 The Variation Principle 443.3.3 The Hartree–Fock Method 473.3.4 Perturbation Theory 513.3.5 The Density Operator 533.4 Final Remarks 55References 55Further Reading 564 Intermolecular Interactions 574.1 The Semi-classical Approach 574.1.1 Electrostatic Interaction 594.1.2 Induction Interaction 624.1.3 Dispersion Interaction 634.1.4 The Total Interaction 644.2 The Quantum Approach 664.3 Model Interactions 694.4 Refinements 724.4.1 Hydrogen Bonding 724.4.2 Three-Body Interactions 744.4.3 Accurate Empirical Potentials 744.5 Final Remarks 75References 76Further Reading 775 Continuum Mechanics 785.1 The Nature of the Continuum 785.2 Kinematics 795.2.1 Material and Spatial Coordinates 795.2.2 General Deformations 805.2.3 The Small Displacement Gradient Approximation 815.3 Balance Equations 835.4 Kinetics 855.4.1 The Principle of Virtual Power 865.4.2 Linear Momentum 865.4.3 Angular Momentum 885.4.4 Cauchy’s Equations of Motion 885.5 The Stress Tensor 895.6 Mechanical Energy 905.7 Final Remarks 91References 92Further Reading 926 Macroscopic Thermodynamics 936.1 Classical Thermodynamics 936.1.1 The Four Laws 936.1.2 Quasi-Conservative and Dissipative Forces 996.1.3 Equations of State 1006.1.4 Mechanical and Thermal Equilibrium 1016.1.5 Auxiliary Functions 1016.1.6 Some Derivatives and their Relationships 1036.1.7 Chemical Content 1036.1.8 Chemical Equilibrium 1066.2 The Local State and Internal Variables 1106.2.1 The Behavior of Internal Variables 1116.2.2 The Local State 1136.3 Field Formulation 1156.3.1 The First Law 1156.3.2 The Second Law 1166.4 The Linear Approximation in Non-equilibrium Thermodynamics 1186.5 Final Remarks 122References 122Further Reading 1237 Microscopic Thermodynamics 1257.1 Basics of Statistical Thermodynamics 1257.1.1 Preliminaries 1257.1.2 Entropy and Partition Functions 1287.1.3 Fluctuations 1327.2 Noninteracting Particles 1347.2.1 Single Particle 1347.2.2 Many Particles 1347.2.3 Pressure and Energy 1357.3 The Semi-classical Approximation 1367.4 Interacting Particles 1417.5 Internal Contributions 1427.5.1 Vibrations 1427.5.2 Rotations 1457.5.3 Electronic Transitions 1477.6 Some General Aspects 1487.6.1 Mode or Average? 1487.6.2 Fluctuations and Other Ensembles 1497.6.3 Equipartition of Energy 1507.6.4 The Gibbs–Bogoliubov Inequality 151References 152Further Reading 1548 Gases 1558.1 Basic Kinetic Theory of Gases 1558.2 The Virial Expansion 1598.2.1 Some Further Remarks 1628.3 Equations of State 1648.4 The Principle of Corresponding States 1688.4.1 The Extended Principle 1718.5 Transition State Theory 1748.5.1 Chemical Kinetics Basics 1748.5.2 The Equilibrium Constant 1758.5.3 Potential Energy Surfaces 1768.5.4 The Activated Complex 1778.5.5 The Link to Experiment 1798.6 Dielectric Behavior 1808.6.1 Basic Aspects 1808.6.2 The Debye–Langevin Equation 1828.6.3 Frequency Dependence 1858.6.4 Estimating μ and α 190References 193Further Reading 1969 Liquids 1979.1 Approaches to Liquids 1979.2 Distribution Functions, Structure, and Energetics 1989.2.1 Structure 2009.2.2 Energetics 2039.3 The Integral Equation Approach 2069.3.1 The Ornstein–Zernike Equation 2069.3.2 The Yvon–Born–Green Equation 2099.3.3 Other Integral Equations 2109.3.4 The Potential of Mean Force 2129.4 Comparison: Hard-Sphere and Lennard-Jones Results 2149.5 Scaled-Particle Theory 2179.6 Structural Models 2189.6.1 Cell Models 2209.6.2 Hole Models 2269.6.3 Some Other Implementations of Hole Theory 2319.7 The Generalized van der Waals Model 2379.8 Phonon Theory of Liquids 2409.9 The Quantum Cluster Equilibrium Model 2449.10 Some Continuum Aspects 2459.11 Dielectric Behavior 249References 255Further Reading 25910 Solids 26010.1 Inorganics and Metals 26010.2 Polymers 26310.3 Lattice Concepts 26510.4 Crystalline Structures 26710.5 Bonding: The Quantum-mechanical Approach 27010.5.1 The Nearly Free Electron Approximation 27010.5.2 The Tight Binding Approximation 27510.5.3 Density Functional Theory 27810.6 Bonding: The Empirical Approach 28210.6.1 Atoms, Ions, and Electronegativity 28210.6.2 Covalent and Molecular Crystals 28610.6.3 Ionic Crystals: The Classical Approach 28710.6.4 Ionic Crystals: Electronegativity Approaches 29010.6.5 Metallic Crystals 29410.7 Lattice Dynamics 29610.8 Two Simple Models 29910.9 Properties 30010.9.1 Heat Capacity 30010.9.2 Thermal Expansivity 30210.9.3 Bulk Modulus 30310.10 Defects 30410.10.1 Zero-dimensional Defects 30510.10.2 One-dimensional Defects 30810.10.3 Other Defects 31010.11 Thermo-elasticity 31210.11.1 Elastic Behavior 31210.11.2 Stress States and the Associated Elastic Constants 31310.11.3 Elastic Energy 31410.11.4 A Matter of Notation 31510.11.5 Anisotropic Materials 31610.11.6 The Effect of Temperature 31910.12 Final Remarks 320References 320Further Reading 32511 Interfaces 32611.1 Thermodynamics of Interfaces 32611.2 One-Component Surfaces: Semiempirical Considerations 33111.3 One-Component Surfaces: Theoretical Considerations 33611.3.1 Density Functional Theory 33611.3.2 Capillary Wave Theory 34111.4 Solid Surface Structure 34311.4.1 Surface Roughening 34511.5 Adsorption at Interfaces 34911.5.1 Solutions 34911.5.2 Thermodynamics of Adsorption 35511.5.3 Statistics of Adsorption 35711.5.4 Adsorption Isotherms 36011.6 Final Remarks 366References 366Further Reading 37012 Phase Transitions: General Aspects 37112.1 Some General Considerations 37112.2 The Clapeyron and Clapeyron–Clausius Equation 37512.3 The Mosselman Solution for the Clapeyron Equation 37612.4 The Ehrenfest–Prigogine–Defay Equations 37812.5 Landau and Landau-like Theory 380References 383Further Reading 38413 Discontinuous Phase Transitions: Liquids ↔ Gases 38513.1 Thermodynamics of Evaporation 38513.1.1 Evaporation in the Presence of an Inert Gas 38713.2 Kinetics of Evaporation 38813.2.1 Classical Kinetic Theory 38813.2.2 Secondary Effects 39313.2.3 Other Approaches 39413.3 The Reverse Transition: Condensation 39513.3.1 Drops and Bubbles 39513.3.2 Classical Nucleation Theory 39713.3.3 Nucleation Kinetics 39913.3.4 Modifications 40113.3.5 Molecular Aspects 404References 408Further Reading 41014 Discontinuous Phase Transitions: Solids ↔ Liquids 41114.1 Melting or Fusion 41114.2 Mechanical or Bulk Melting 41414.2.1 Vibrational Instability 41414.2.2 Lattice Instability 41814.2.3 Vacancies 41814.2.4 Interstitials 41914.2.5 Dislocations 42214.2.6 Interstitialcies 42414.2.7 Simulations 42714.3 Thermodynamic or Surface-Mediated Melting 42814.3.1 Melting of Nanoparticles 42814.3.2 Vacancies Revisited 43014.3.3 Dislocations Revisited 43214.4 Polymer Melting 43414.5 The Influence of Pressure 43614.6 Other Aspects 44014.7 Melting in Perspective 44214.8 The Reverse Transition: Freezing or Solidification 44414.8.1 Nucleation and Growth 44414.8.2 Some Further Remarks 44614.8.3 Polymers and Metals 44814.8.4 Water 451References 452Further Reading 45715 Continuous Phase Transitions: Liquids ↔ Gases 45815.1 Limiting Behavior 45815.2 Mean-Field Theory: Landau Theory 46115.2.1 Landau-Like Theory: Fluid Transitions 46315.3 Scaling 46515.3.1 Homogeneous Functions 46515.3.2 Scaling Potentials 46615.3.3 Scaling Lattices 46715.4 Renormalization 46915.5 Final Remarks 475References 476Further Reading 47616 The Liquid Crystal Transformation 47816.1 Nature and Types 47816.2 The Nematic–Isotropic Transformation 48016.2.1 The Orientation as Internal Variable 48016.2.2 The Discontinuous Transformation 48116.3 Alternative Approaches 48216.3.1 Maier–Saupe Theory 48316.3.2 The Coil–Helix Transformation 48516.3.3 Onsager Theory 48616.4 Some Extensions 48916.5 Elastic Energy and Defects 49116.6 The Fréedericksz Transformation 494References 496Further Reading 49717 Dielectric Behavior and the Ferroelectric Transformation 49817.1 Preliminaries and Dielectric Materials 49817.1.1 General Remarks 49817.1.2 Dielectric Materials 50017.2 Electronic Polarization 50117.3 Vibrational Polarization 50317.3.1 Three Models 50717.4 Orientational Polarization 51017.5 Space–Charge Polarization 51117.6 Ferroelectric Materials 51217.7 Ferroelectric Behavior 51617.7.1 The Thermodynamic Approach 51617.7.2 The Microscopic Approach 518References 521Further Reading 52318 The Glass Transition 52518.1 What Is a Glass? 52518.1.1 Glassy Materials 52818.1.2 Property Changes at Tg 52918.2 The Thermodynamic Approach 53018.3 The Structural Approach 53518.3.1 Free Volume Theory 53618.3.2 Continuous Transition Theory 53918.4 The Lattice Gas Approach 54118.5 Phonon Theory for Glasses 54318.6 Mode-Coupling Theory 54618.7 Final Remarks 549References 550Further Reading 55319 Irreversibility and the Return to Equilibrium 55519.1 Some Considerations 55519.2 The Boltzmann Approach 55719.2.1 Time Invariance 55819.2.2 Recurrence 56019.3 The Gibbs Approach 56119.4 The Formal Approach 56319.5 The Physical Approach 56719.6 The Information Theory Approach 57119.6.1 A Brief Review 57119.6.2 High and Low Probability Manifolds 57219.7 Closure 578References 580Further Reading 583Appendix A Guide to Mathematics Used 584A. 1 Symbols and Conventions 584A. 2 Derivatives, Differentials, and Variations 584A. 3 Composite, Implicit, Homogeneous, Complex, and Analytic Functions 586A. 4 Extremes and Lagrange Multipliers 588A. 5 Legendre Transforms 588A. 6 Coordinate Axes Rotations 589A. 7 Change of Variables 590A. 8 Calculus of Variations 591A. 9 Matrices and Determinants 592A. 10 The Eigenvalue Problem 594A. 11 Matrix Decompositions 597A. 12 Scalars, Vectors, and Tensors 598A. 13 Tensor Analysis 601A. 14 Gamma, Dirac, and Heaviside Functions 603A. 15 Laplace and Fourier Transforms 604A. 16 Some Useful Expressions 606Further Reading 607Appendix B Elements of Special Relativity Theory 608B.1 Lorentz Transformations 608B.2 Velocities, Contraction, Dilatation, and Proper Quantities 610B.3 Relativistic Lagrange and Hamilton Functions 611References 612Further Reading 612Appendix C The Lattice Gas Model 613C. 1 The Lattice Gas Model 613C. 2 The Zeroth or Mean-Field Approximation 613C. 3 The First or Quasi-Chemical Approximation 615C. 4 Athermal Entropy for Chain-Like Molecules 619References 621Further Reading 621Appendix D Elements of Electrostatics 622D.1 Coulomb, Gauss, Poisson, and Laplace 622D.2 A Dielectric Sphere in a Dielectric Matrix 624D.3 A Dipole in a Spherical Cavity 626Further Reading 627Appendix E Elements of Probability and Statistics 629E.1 Probability 629E.2 Single Variable 631E.3 Multiple Variables 632E.4 The Normal Distribution and the Central-Limit Theorem 633References 635Further Reading 635Appendix F Selected Data 636References 650Appendix G Answers to Selected Problems 652Index 659