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.
Table of Contents
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
