Phases of Matter and their Transitions: Concepts and Principles for Chemists, Physicists, Engineers, and Materials Scientists

An all-in-one, comprehensive take on matter and its phase properties. In Phases of Matter and their Transitions, accomplished materials scientist Dr. Gijsbertus de With delivers an accessible textbook for advanced students in the molecular sciences. It offers a balanced and self-contained treatment of the thermodynamic and structural aspects of phases and the transitions between them, covering solids, liquids, gases, and their interfaces. The book lays the groundwork to describe particles and their interactions from the perspective of classical and quantum mechanics and compares phenomenological and statistical thermodynamics. It also examines materials with special properties, like glasses, liquid crystals, and ferroelectrics. The author has included an extensive appendix with a guide to the mathematics and theoretical models employed in this resource. Readers will also find: Thorough introductions to classical and quantum mechanics, intermolecular interactions, and continuum mechanics. Comprehensive explorations of thermodynamics, gases, liquids, and solids. Practical discussions of surfaces, including their general aspects for solids and liquids. Fulsome treatments of discontinuous and continuous transitions, including discussions of irreversibility and the return to equilibrium. Perfect for advanced students in chemistry and physics, Phases of Matter and their Transitions will also earn a place in the libraries of students of materials science. Cover Half Title Phases of Matter and their Transitions: Concepts and Principles for Chemists, Physicists, Engineers, and Materials Scientists Copyright Dedication Contents Preface List of Frequently Used Symbols and Abbreviations SI Units, Physical Constants, and Conversion Factors Summary of Notation 1. Introduction 1.1 Constituents of Matter 1.2 Matter and Energy: Interaction and Change 1.3 Mass and Charge 1.4 Macroscopic and Microscopic Approaches 1.5 Gases, Liquids, and Solids 1.6 What to Expect? 1.7 Units and Notation References Further Reading 2. Classical Mechanics 2.1 Frames, Particles, and Coordinates 2.2 From Newton to Hamilton 2.3 Hamilton's Principle and Lagrange's Equations 2.4 Conservation Laws 2.5 Hamilton's Equations 2.6 Hamilton's Principle for Continuous Systems 2.7 The Virial Theorem 2.8 Final Remarks References Further Reading 3. Quantum Mechanics 3.1 Quantum Concepts 3.1.1 Fundamental Quantum Kinematics 3.1.2 Operators and their Representation 3.1.3 Fundamental Quantum Kinetics 3.2 Interpretation and Some Exact Solutions 3.2.1 The Particle in a Box 3.2.2 The Harmonic Oscillator 3.2.3 The Rigid Rotator 3.2.4 Many Particles 3.3 Approximate Quantum Mechanics Solutions 3.3.1 The Born–Oppenheimer Approximation 3.3.2 The Variation Principle 3.3.3 The Hartree–Fock Method 3.3.4 Perturbation Theory 3.3.5 The Density Operator 3.4 Final Remarks References Further Reading 4. Intermolecular Interactions 4.1 The Semi‐classical Approach 4.1.1 Electrostatic Interaction 4.1.2 Induction Interaction 4.1.3 Dispersion Interaction 4.1.4 The Total Interaction 4.2 The Quantum Approach 4.3 Model Interactions 4.4 Refinements 4.4.1 Hydrogen Bonding 4.4.2 Three‐Body Interactions 4.4.3 Accurate Empirical Potentials 4.5 Final Remarks References Further Reading 5. Continuum Mechanics 5.1 The Nature of the Continuum 5.2 Kinematics 5.2.1 Material and Spatial Coordinates 5.2.2 General Deformations 5.2.3 The Small Displacement Gradient Approximation 5.3 Balance Equations 5.4 Kinetics 5.4.1 The Principle of Virtual Power 5.4.2 Linear Momentum 5.4.3 Angular Momentum 5.4.4 Cauchy's Equations of Motion 5.5 The Stress Tensor 5.6 Mechanical Energy 5.7 Final Remarks References Further Reading 6. Macroscopic Thermodynamics 6.1 Classical Thermodynamics 6.1.1 The Four Laws 6.1.2 Quasi‐Conservative and Dissipative Forces 6.1.3 Equations of State 6.1.4 Mechanical and Thermal Equilibrium 6.1.5 Auxiliary Functions 6.1.6 Some Derivatives and their Relationships 6.1.7 Chemical Content 6.1.8 Chemical Equilibrium 6.2 The Local State and Internal Variables 6.2.1 The Behavior of Internal Variables 6.2.2 The Local State 6.3 Field Formulation 6.3.1 The First Law 6.3.2 The Second Law 6.4 The Linear Approximation in Non-equilibrium Thermodynamics 6.5 Final Remarks References Further Reading 7. Microscopic Thermodynamics 7.1 Basics of Statistical Thermodynamics 7.1.1 Preliminaries 7.1.2 Entropy and Partition Functions 7.1.3 Fluctuations 7.2 Noninteracting Particles 7.2.1 Single Particle 7.2.2 Many Particles 7.2.3 Pressure and Energy 7.3 The Semi‐classical Approximation 7.4 Interacting Particles 7.5 Internal Contributions 7.5.1 Vibrations 7.5.2 Rotations 7.5.3 Electronic Transitions 7.6 Some General Aspects 7.6.1 Mode or Average? 7.6.2 Fluctuations and Other Ensembles 7.6.3 Equipartition of Energy 7.6.4 The Gibbs–Bogoliubov Inequality References Further Reading 8. Gases 8.1 Basic Kinetic Theory of Gases 8.2 The Virial Expansion 8.2.1 Some Further Remarks 8.3 Equations of State 8.4 The Principle of Corresponding States 8.4.1 The Extended Principle 8.5 Transition State Theory 8.5.1 Chemical Kinetics Basics 8.5.2 The Equilibrium Constant 8.5.3 Potential Energy Surfaces 8.5.4 The Activated Complex 8.5.5 The Link to Experiment 8.6 Dielectric Behavior 8.6.1 Basic Aspects 8.6.2 The Debye–Langevin Equation 8.6.3 Frequency Dependence 8.6.4 Estimating μ and α References Further Reading 9. Liquids 9.1 Approaches to Liquids 9.2 Distribution Functions, Structure, and Energetics 9.2.1 Structure 9.2.2 Energetics 9.3 The Integral Equation Approach 9.3.1 The Ornstein–Zernike Equation 9.3.2 The Yvon–Born–Green Equation 9.3.3 Other Integral Equations 9.3.4 The Potential of Mean Force 9.4 Comparison: Hard‐Sphere and Lennard‐Jones Results 9.5 Scaled‐Particle Theory 9.6 Structural Models 9.6.1 Cell Models 9.6.2 Hole Models 9.6.3 Some Other Implementations of Hole Theory 9.7 The Generalized van der Waals Model 9.8 Phonon Theory of Liquids 9.9 The Quantum Cluster Equilibrium Model 9.10 Some Continuum Aspects 9.11 Dielectric Behavior References Further Reading 10. Solids 10.1 Inorganics and Metals 10.2 Polymers 10.3 Lattice Concepts 10.4 Crystalline Structures 10.5 Bonding: The Quantum‐mechanical Approach 10.5.1 The Nearly Free Electron Approximation 10.5.2 The Tight Binding Approximation 10.5.3 Density Functional Theory 10.6 Bonding: The Empirical Approach 10.6.1 Atoms, Ions, and Electronegativity 10.6.2 Covalent and Molecular Crystals 10.6.3 Ionic Crystals: The Classical Approach 10.6.4 Ionic Crystals: Electronegativity Approaches 10.6.5 Metallic Crystals 10.7 Lattice Dynamics 10.8 Two Simple Models 10.9 Properties 10.9.1 Heat Capacity 10.9.2 Thermal Expansivity 10.9.3 Bulk Modulus 10.10 Defects 10.10.1 Zero‐dimensional Defects 10.10.2 One‐dimensional Defects 10.10.3 Other Defects 10.11 Thermo‐elasticity 10.11.1 Elastic Behavior 10.11.2 Stress States and the Associated Elastic Constants 10.11.3 Elastic Energy 10.11.4 A Matter of Notation 10.11.5 Anisotropic Materials 10.11.6 The Effect of Temperature 10.12 Final Remarks References Further Reading 11. Interfaces 11.1 Thermodynamics of Interfaces 11.2 One‐Component Surfaces: Semiempirical Considerations 11.3 One‐Component Surfaces: Theoretical Considerations 11.3.1 Density Functional Theory 11.3.2 Capillary Wave Theory 11.4 Solid Surface Structure 11.4.1 Surface Roughening 11.5 Adsorption at Interfaces 11.5.1 Solutions 11.5.2 Thermodynamics of Adsorption 11.5.3 Statistics of Adsorption 11.5.4 Adsorption Isotherms 11.6 Final Remarks References Further Reading 12. Phase Transitions: General Aspects 12.1 Some General Considerations 12.2 The Clapeyron and Clapeyron–Clausius Equation 12.3 The Mosselman Solution for the Clapeyron Equation 12.4 The Ehrenfest–Prigogine–Defay Equations 12.5 Landau and Landau‐like Theory References Further Reading 13. Discontinuous Phase Transitions: Liquids ↔ Gases 13.1 Thermodynamics of Evaporation 13.1.1 Evaporation in the Presence of an Inert Gas 13.2 Kinetics of Evaporation 13.2.1 Classical Kinetic Theory 13.2.2 Secondary Effects 13.2.3 Other Approaches 13.3 The Reverse Transition: Condensation 13.3.1 Drops and Bubbles 13.3.2 Classical Nucleation Theory 13.3.3 Nucleation Kinetics 13.3.4 Modifications 13.3.5 Molecular Aspects References Further Reading 14. Discontinuous Phase Transitions: Solids ↔ Liquids 14.1 Melting or Fusion 14.2 Mechanical or Bulk Melting 14.2.1 Vibrational Instability 14.2.2 Lattice Instability 14.2.3 Vacancies 14.2.4 Interstitials 14.2.5 Dislocations 14.2.6 Interstitialcies 14.2.7 Simulations 14.3 Thermodynamic or Surface‐Mediated Melting 14.3.1 Melting of Nanoparticles 14.3.2 Vacancies Revisited 14.3.3 Dislocations Revisited 14.4 Polymer Melting 14.5 The Influence of Pressure 14.6 Other Aspects 14.7 Melting in Perspective 14.8 The Reverse Transition: Freezing or Solidification 14.8.1 Nucleation and Growth 14.8.2 Some Further Remarks 14.8.3 Polymers and Metals 14.8.4 Water References Further Reading 15. Continuous Phase Transitions: Liquids ↔ Gases 15.1 Limiting Behavior 15.2 Mean‐Field Theory: Landau Theory 15.2.1 Landau‐Like Theory: Fluid Transitions 15.3 Scaling 15.3.1 Homogeneous Functions 15.3.2 Scaling Potentials 15.3.3 Scaling Lattices 15.4 Renormalization 15.5 Final Remarks References Further Reading 16. The Liquid Crystal Transformation 16.1 Nature and Types 16.2 The Nematic–Isotropic Transformation 16.2.1 The Orientation as Internal Variable 16.2.2 The Discontinuous Transformation 16.3 Alternative Approaches 16.3.1 Maier–Saupe Theory 16.3.2 The Coil–Helix Transformation 16.3.3 Onsager Theory 16.4 Some Extensions 16.5 Elastic Energy and Defects 16.6 The Fréedericksz Transformation References Further Reading 17. Dielectric Behavior and the Ferroelectric Transformation 17.1 Preliminaries and Dielectric Materials 17.1.1 General Remarks 17.1.2 Dielectric Materials 17.2 Electronic Polarization 17.3 Vibrational Polarization 17.3.1 Three Models 17.4 Orientational Polarization 17.5 Space–Charge Polarization 17.6 Ferroelectric Materials 17.7 Ferroelectric Behavior 17.7.1 The Thermodynamic Approach 17.7.2 The Microscopic Approach References Further Reading 18. The Glass Transition 18.1 What Is a Glass? 18.1.1 Glassy Materials 18.1.2 Property Changes at Tg 18.2 The Thermodynamic Approach 18.3 The Structural Approach 18.3.1 Free Volume Theory 18.3.2 Continuous Transition Theory 18.4 The Lattice Gas Approach 18.5 Phonon Theory for Glasses 18.6 Mode‐Coupling Theory 18.7 Final Remarks References Further Reading 19. Irreversibility and the Return to Equilibrium 19.1 Some Considerations 19.2 The Boltzmann Approach 19.2.1 Time Invariance 19.2.2 Recurrence 19.3 The Gibbs Approach 19.4 The Formal Approach 19.5 The Physical Approach 19.6 The Information Theory Approach 19.6.1 A Brief Review 19.6.2 High and Low Probability Manifolds 19.7 Closure References Further Reading Appendix A. Guide to Mathematics Used A.1 Symbols and Conventions A.2 Derivatives, Differentials, and Variations A.3 Composite, Implicit, Homogeneous, Complex, and Analytic Functions A.4 Extremes and Lagrange Multipliers A.5 Legendre Transforms A.6 Coordinate Axes Rotations A.7 Change of Variables A.8 Calculus of Variations A.9 Matrices and Determinants A.10 The Eigenvalue Problem A.11 Matrix Decompositions A.12 Scalars, Vectors, and Tensors A.13 Tensor Analysis A.14 Gamma, Dirac, and Heaviside Functions A.15 Laplace and Fourier Transforms A.16 Some Useful Expressions Further Reading Appendix B. Elements of Special Relativity Theory B.1 Lorentz Transformations B.2 Velocities, Contraction, Dilatation, and Proper Quantities B.3 Relativistic Lagrange and Hamilton Functions References Further Reading Appendix C. The Lattice Gas Model C.1 The Lattice Gas Model C.2 The Zeroth or Mean‐Field Approximation C.3 The First or Quasi‐Chemical Approximation C.4 Athermal Entropy for Chain‐Like Molecules References Further Reading Appendix D. Elements of Electrostatics D.1 Coulomb, Gauss, Poisson, and Laplace D.2 A Dielectric Sphere in a Dielectric Matrix D.3 A Dipole in a Spherical Cavity Further Reading Appendix E. Elements of Probability and Statistics E.1 Probability E.2 Single Variable E.3 Multiple Variables E.4 The Normal Distribution and the Central‐Limit Theorem References Further Reading Appendix F. Selected Data References Appendix G. Answers to Selected Problems Index