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SPRINGER BRIEFS IN THE MATHEMATICS OF MATERIALS 2 Akihiko Hirata Kaname Matsue Mingwei Chen Structural Analysis of Metallic Glasses with Computational Homology SpringerBriefs in the Mathematics of Materials Volume 2 Editor-in-chief Motoko Kotani, Sendai, Japan Series editors Yasumasa Nishiura, Sendai, Japan Masaru Tsukada, Sendai, Japan Samuel M. Allen, Cambridge, USA Willi Jaeger, Heidelberg, Germany Stephan Luckhaus, Leipzig, Germany More information about this series at http://www.springer.com/series/13533 Akihiko Hirata Kaname Matsue (cid:129) Mingwei Chen Structural Analysis of Metallic Glasses with Computational Homology 123 AkihikoHirata MingweiChen AdvancedInstitute for Materials Research AdvancedInstitute for Materials Research Tohoku University Tohoku University Sendai, Miyagi Sendai, Miyagi Japan Japan Kaname Matsue TheInstitute of Statistical Mathematics Tachikawa, Tokyo Japan ISSN 2365-6336 ISSN 2365-6344 (electronic) SpringerBriefs inthe Mathematics of Materials ISBN978-4-431-56054-8 ISBN978-4-431-56056-2 (eBook) DOI 10.1007/978-4-431-56056-2 LibraryofCongressControlNumber:2016935963 ©TheAuthor(s)2016 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpart of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission orinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar methodologynowknownorhereafterdeveloped. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfrom therelevantprotectivelawsandregulationsandthereforefreeforgeneraluse. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authorsortheeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinor foranyerrorsoromissionsthatmayhavebeenmade. Printedonacid-freepaper ThisSpringerimprintispublishedbySpringerNature TheregisteredcompanyisSpringerJapanKK Preface In this short book, we introduce the application of computational homology for structure analysis of metallic glasses. Metallic glasses, relatively new materials in thefieldofmetals,arethenext-generationstructuralandfunctionalmaterialsowing to their excellent properties. Understanding the properties and developing novel metallic glass materials will require uncovering their atomic structures, which, unlike crystals, have no periodicity. Although numerous experimental and simu- lationstudieshavebeenperformedtorevealthesestructures,itisextremelydifficult toperceivearelationshipbetweenstructuresandpropertieswithouttheappropriate viewpoint. Our purpose is to show how our new approach using computational homology provides useful insight into the interpretation of the atomic structures. It is noted that computational homology is now widely applied to various data analyses. We start with a basic, brief survey of metallic glasses and computational homology and then move on to the detailed procedures and interpretation of analyses based on computational homology for metallic glasses. The authors are grateful to T. Fujita and P.F. Guan for providing structural models of metallic glasses. Sendai, Japan Akihiko Hirata 2015 v Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Crystal and Amorphous Structures. . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Benefits of Topological Analysis. . . . . . . . . . . . . . . . . . . . . . . . 4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2 Metallic Glasses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2 Structure Models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.3 Experimental Difficulty in Determining Structures . . . . . . . . . . . . 12 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3 Overview of Cubical Homology . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.2 The Fundamental Idea of Homology . . . . . . . . . . . . . . . . . . . . . 17 3.3 Cubical Sets and Chains. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 3.4 Boundary and Cubical Chain Complexes . . . . . . . . . . . . . . . . . . 24 3.5 Cubical Homology and Betti Numbers. . . . . . . . . . . . . . . . . . . . 29 3.6 Remarks and Guide to the Literature . . . . . . . . . . . . . . . . . . . . . 36 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 4 Application of Computational Homology to Metallic Glass Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 4.2 Calculation Procedures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 4.2.1 Preparation of Cubical Dataset from Atomic Configurations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 4.2.2 Effect of Atomic Size. . . . . . . . . . . . . . . . . . . . . . . . . . . 41 4.2.3 Effect of Voxel Size . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 4.3 Homological Analysis for Metallic Glass Structures. . . . . . . . . . . 42 4.4 Homological Analysis for Crystal Structures. . . . . . . . . . . . . . . . 45 4.5 Structural Features of Metallic Glasses Viewed by Homology. . . . 46 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 vii viii Contents 5 Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 Appendix A: Several Topics About Homology. . . . . . . . . . . . . . . . . . . 51 List of Figures Figure 1.1 Crystal structures of (a) a metallic solid (gold, Au), (b) an ionic solid (rock salt, NaCl), and (c) a covalent solid (silica, SiO ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2 Figure 1.2 Atomic configurations of (a) crystal silica and (b) amorphous silica. Red and blue spheres denote silicon and oxygen atoms, respectively. . . . . . . . . . . . . . . . . 3 Figure 1.3 Electron diffraction patterns from amorphous (left) and crystal (right) structures. . . . . . . . . . . . . . . . . . . . . . . . 3 Figure 1.4 Construction of a Voronoi polyhedron for a two-dimensional structure (upper left), a Voronoi polyhedron indexed by h0 0 12 0i (lower left), an atomic configuration of a metallic glass (upper right), and a Voronoi analysis for a metallic glass (lower right). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Figure 1.5 Typical atomic clusters with several coordination numbers (CN) in metallic glasses. Voronoi indices are also shown . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Figure 1.6 Distribution of atoms for (a and a′) imperfect (amorphous) structures and (b) perfect (crystal) structure characterized by homological analysis. . . . . . . . . . . 5 ix x ListofFigures Figure 1.7 Two different groupings for four atomic configurations including a regular triangle, a regular tetragon, a distorted triangle, and a distorted tetragon. a We classify atomic configurations according to atomic numbers. In this case, each category contains both symmetric and distorted configurations. b We classify atomic configurations according to connectivity. In this case, categories are characterized based on whether or not atoms are connected to each other. This classifica- tion is independent of the number of atoms. Moreover, in the category of distorted configurations (below), the characterization is robust under small perturbations of atomic positions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Figure 2.1 Bulkmetallicglass(left),ribbonmetallicglass(middle), and window glass (right) . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Figure 2.2 Atomic coordinates in a face-centered cubic, b hexagonal close packed, and c body-centered cubic structures (ball-and-stick style) . . . . . . . . . . . . . . . . . . . . . . 11 Figure 2.3 Two-dimensional picture of the dense random packing model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Figure 2.4 Typical example of reduced interference function Qi (Q) (upper) and pair distribution function g(r) (lower) obtained from a metallic glass. . . . . . . . . . . . . . . . . . . . . . . 13 Figure 2.5 Schematic of Angstrom beam electron diffraction method (left) and electron diffraction patterns obtained from a metallic glass with different beam sizes [13]. . . . . . . . 14 Figure 3.1 Can you estimate and explain the complexities of (a) and (b)? The complexity of these objects is outside analytic and geometric structures such as singularities or curvatures. . . . . . . . . . . . . . . . . . . . . . . . . . 16 Figure 3.2 Homological identification and distinguishment. aArubberband(left)andaJapanese5-yencoin(right). b A coffee cup (left) and a (filled) doughnut (right). c A measurement rule (left) and a pair of chopsticks (right) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 Figure 3.3 Representation of a ball-like set by a finite number of elementary cubes (a cubical set) . . . . . . . . . . . . . . . . . . . 19 Figure 3.4 A cubical set X in R2. X ¼ð½1;2(cid:2)(cid:3)½1;2(cid:2)Þ[ð½1;2(cid:2)(cid:3) ½2;3(cid:2)Þ[ð½2;3(cid:2)(cid:3)½2(cid:2)Þ[ð½2;3(cid:2)(cid:3)½3(cid:2)Þ[ð½3(cid:2)(cid:3) ½2;3(cid:2)Þ[ð½3(cid:2)(cid:3)½1(cid:2)Þ. For sets A and B, A[B denotes the union of A and B. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21