Geometry and Phase Transitions in Colloids and Polymers

✍ Scribed by William Kung

• Publisher: World Scientific Publishing Company
• Year: 2009
• Tongue: English
• Leaves: 214
• Series: World Scientific Lecture Notes in Physics
• Category: Library

✦ Synopsis

This monograph represents an extension of the author's original PhD thesis and includes a more thorough discussion on the concepts and mathematics behind his research works on the foam model, as applied to studying issues of phase stability and elasticity for various non-closed packed structures found in fuzzy and colloidal crystals, as well as on a renormalization-group analysis regarding the critical behavior of loop polymers upon which topological constraints are imposed. The common thread behind these two research works is their demonstration of the importance and effectiveness of utilizing geometrical and topological concepts for modeling and understanding soft systems undergoing phase transitions.

✦ Table of Contents

Contents......Page 12
Preface......Page 8
List of Figures......Page 16
List of Tables......Page 23
The Big Picture......Page 24
1. Modern Physics at a Glance......Page 26
Geometry and Phase Transitions, in General......Page 38
2.1 Introduction......Page 40
2.1.1 Evolution of the Universe: Decoupling of the Four Fundamental Forces......Page 41
2.1.2 Three States of Water......Page 42
2.1.3 Spins and Magnetism......Page 44
2.2 Modern Classi.cation of Phase Transitions......Page 46
2.3 First-Order Phase Transitions: Solid-Liquid Transition......Page 47
2.4 Second-Order Phase Transitions: Scaling and Universality......Page 48
2.5.1 Kadano. Picture: Coarse-Graining of Spin Blocks......Page 49
2.5.2 General Formulation......Page 51
2.5.3 Critical Exponents......Page 54
2.5.4 Origin of Universality Class......Page 55
2.5.5 Wilsonian Picture: Momentum-Space Renormalization Group......Page 56
2.6.1 Semi-groups......Page 57
2.7 Conclusion......Page 58
References......Page 59
3.2 Electronic Density-Functional Theory......Page 61
3.3 ClassicalDensity-Functional Theory......Page 65
3.4 Conclusion......Page 69
References......Page 70
4.1 Introduction......Page 72
4.2 Lattice Symmetry Groups......Page 73
4.3 Two-Dimensional Space Groups......Page 76
4.3.1 Hermann-Mauguin Crystallographic Notation......Page 78
4.3.2 Orbifold notation......Page 80
4.3.3 Why Are There Exactly 17 Wallpaper Groups?......Page 101
4.3.4 Other Aspects of Topology in Physics......Page 107
4.4.1 Face-centered Cubic (FCC) Lattices......Page 108
4.4.2 Body-Centered Cubic (BCC) Lattices......Page 111
4.4.3 A15 Lattices......Page 112
4.5 Conceptual Framework of the Foam Model......Page 113
4.6 The Kelvin Problem and the Kepler Conjecture......Page 115
4.7 Conclusion......Page 120
References......Page 121
Geometry and Phase Transitions, in Colloidal Crystals......Page 124
5.1 Introduction......Page 126
5.2 Bulk Free Energy......Page 127
5.3.1 Charged Colloidal Crystals......Page 132
5.3.2 Fuzzy Colloidal Crystals......Page 134
5.4 Conclusion......Page 135
References......Page 136
6.1 Introduction......Page 138
6.2 Phase Transitions of Charged Colloids......Page 140
6.3 Foam Analogy and Charged Colloids......Page 142
References......Page 143
7.1 Introduction......Page 145
7.2 Foam Analogy and Cubic Elastic Constants......Page 147
7.3 Elasticity of Charged Colloidal Crystals......Page 152
7.4 Elasticity of Fuzzy Colloids......Page 160
References......Page 166
Geometry and Phase Transitions, in Topologically Constrained Polymers......Page 168
8.1 Introduction......Page 170
8.2 O(N)-Symmetric φ6-Theory......Page 171
8.3 Chern-Simons Theory and Writhe......Page 177
8.4 One-Loop Scaling of Closed Polymers......Page 182
8.5 Two-Loop Results......Page 186
8.6 Conclusion......Page 193
References......Page 194
Summary......Page 198
9. Final Thoughts......Page 200
Index......Page 202