A New Direction in Mathematics for Materials Science (SpringerBriefs in the Mathematics of Materials Book 1)

This book is the first volume of the SpringerBriefs in the Mathematics of Materials and provides a comprehensive guide to the interaction of mathematics with materials science. The anterior part of the book describes a selected history of materials science as well as the interaction between mathematics and materials in history. The emergence of materials science was itself a result of an interdisciplinary movement in the 1950s and 1960s. Materials science was formed by the integration of metallurgy, polymer science, ceramics, solid state physics, and related disciplines. We believe that such historical background helps readers to understand the importance of interdisciplinary interaction such as mathematicsℓ́ℓmaterials science collaboration. The middle part of the book describes mathematical ideas and methods that can be applied to materials problems and introduces some examples of specific studiesℓ́ℓfor example, computational homology applied to structural analysis of glassy materials, stochastic models for the formation process of materials, new geometric measures for finite carbon nanotube molecules, mathematical technique predicting a molecular magnet, and network analysis of nanoporous materials. The details of these works will be shown in the subsequent volumes of this SpringerBriefs in the Mathematics of Materials series by the individual authors. The posterior section of the book presents how breakthroughs based on mathematicsℓ́ℓmaterials science collaborations can emerge. The authors' argument is supported by the experiences at the Advanced Institute for Materials Research (AIMR), where many researchers from various fields gathered and tackled interdisciplinary research Preface 6 Contents 9 1 A Historical View of Materials Science 11 1.1 Emergence of Materials Science as an Interdisciplinary Field 11 1.2 Classical Fields Within Materials Science 12 1.3 Peculiarity of Materials Science and Partnership with Mathematics 19 References 19 2 Influence of Mathematics on Materials Science Upto Date 20 2.1 Geometric Structures of Atomic Configurations 20 2.1.1 Atomism 20 2.1.2 The Miracle Year of 1669; The Emergence of Crystallography and Optocrystallography from Mineralogy 21 2.1.3 Quasicrystals 24 2.1.4 Aperiodic Tiling and Disordered System 25 2.1.5 Graph Modeling for Nano-Materials 27 2.1.6 Crystal Lattices and Their Standard Realizations 27 2.2 Quantum Materials 28 2.2.1 Electronic Characteristics of Periodic Materials System: Band Theory 29 2.2.2 Spin Current 31 2.2.3 Integer Quantum Hall Effect (IQHE) 32 2.2.4 Hofstadter's Butterfly 32 2.2.5 Central Limit Theorem for Magnetic Transition Operators 33 2.2.6 Topological Insulator 34 2.2.7 Non Commutative Bloch Theory 35 2.3 Pattern Formation 36 2.3.1 Patterns in Equilibrium: Soap Films, Soap Bubbles 37 2.3.2 Fundamentals of Crystal Growth 38 2.3.3 Reaction--Diffusion Equation 41 2.3.4 Mean Curvature Flow to Describe Crystal Growth 44 2.3.5 Level Set Method 45 2.3.6 Phase Field Method 45 2.4 Other Tools 47 2.4.1 Computed Tomography 47 2.4.2 Some Other Computational Tools 49 2.5 Global Trend to Encourage Mathematics--Materials Science Cooperation 52 References 55 3 Some Specific Examples of Mathematics--Materials Science Collaboration at AIMR 60 3.1 Elucidation of Metallic Glass Structure by Computational Homology 62 3.2 Application of a Stochastic Model 64 3.2.1 Stoichiometry Control Based on a Mathematical Model 64 3.2.2 Deformation Analysis of Bulk Metallic Glass Using a Stochastic Model 66 3.3 New Geometric Measures for Finite Carbon Nanotubes 68 3.4 Materials Having Network Structures 71 3.4.1 Mathematical Technique Predicts Molecular Magnet 71 3.4.2 Mixing Time of Molecules Inside of Nanoporous Gold 73 References 75 4 Breakthroughs Based on the Mathematics--Materials Science Collaboration 77 4.1 Real Interdisciplinary Integration 77 4.2 Organization Promoting Mathematics--Materials Science Collaboration 80 4.3 Specific Problems and Applications in the Future 82 5 Epilogue 84 References 85 Appendix A Supplements to “Quantum Materials” 86 This book is the first volume of the SpringerBriefs in the Mathematics of Materials and provides a comprehensive guide to the interaction of mathematics with materials science. The anterior part of the book describes a selected history of materials science as well as the interaction between mathematics and materials in history. The emergence of materials science was itself a result of an interdisciplinary movement in the 1950s and 1960s. Materials science was formed by the integration of metallurgy, polymer science, ceramics, solid state physics, and related disciplines. We believe that such historical background helps readers to understand the importance of interdisciplinary interaction such as mathematicsĺlmaterials science collaboration. The middle part of the book describes mathematical ideas and methods that can be applied to materials problems and introduces some examples of specific studiesĺlfor example, computational homology applied to structural analysis of glassy materials, stochastic models for the formation process of materials, new geometric measures for finite carbon nanotube molecules, mathematical technique predicting a molecular magnet, and network analysis of nanoporous materials. The details of these works will be shown in the subsequent volumes of this SpringerBriefs in the Mathematics of Materials series by the individual authors. The posterior section of the book presents how breakthroughs based on mathematicsĺlmaterials science collaborations can emerge. The authors' argument is supported by the experiences at the Advanced Institute for Materials Research (AIMR), where many researchers from various fields gathered and tackled interdisciplinary research Front Matter....Pages i-x A Historical View of Materials Science....Pages 1-9 Influence of Mathematics on Materials Science Upto Date....Pages 11-50 Some Specific Examples of Mathematics–Materials Science Collaboration at AIMR....Pages 51-67 Breakthroughs Based on the Mathematics–Materials Science Collaboration....Pages 69-75 Epilogue....Pages 77-78 Back Matter....Pages 79-86