Structural Analysis Algorithms for Nanomaterials

Abstract

Denne afhandling præsenterer en omformulering af eksisterende problemer inden for materialevidenskab med brug af velkendte metoder fra anvendt matematik: grafteori, numerisk geometri, og heltalsprogrammering.Centrosymmetriparameteren omformuleres som et grafparringsproblem, hvormed uoverensstemmelserne i de eksisterende beregningsmetoder løses.Ved at omformulere en afstandsfunktion mellem forskellige krystalgittre (root mean square distance, RMSD) som et parringsproblem i todelte grafer, bevises det, at RMSD i krystalgitre kan beregnes i polynomisk tid, hvilket udgør en forbedring ift. den eksisterende forventning om fakultets-tidsforbrug. Denne metode er derefter udvidet til todimensionelle monolag.En metode præsenteres til genkendelse af ordnede krystallinske faser i molekuledynamiske simuleringer. En robust klassikation opnås ved brug af template matching, som også formuleres som et parringsproblem i todelte geometriske grafer. Matrix dekomponeringer bruges til at udvikle en geometrisk gitterparringsalgoritme, der kan identicere alle grænseflader med lav tøjning. De resulterende stabile og lavenergiske grænseflader som algoritmen finder kan bruges til design og konstruktion af topologiske superledere, som har mange anvendelsesmuligheder inden for kvantecomputere.Cluster expansion modeller bruges til at finde grundtilstande i guld-sølv nanopartikler, som bruges i en bred vifte af katalytiske processer. Udover denne konkrete anvendelse, udvikles teoretiske metoder optimal konstruktion af cluster expansion modeller, præcis afgørelse af grundtilstanden i større modeller, samt udtømmende bestemmelse af alle mulige grundtilstande i mindre modeller.Til sidst præsenteres en metode til en næsten-optimal sampling af orienteringer.Blandt de mange anvendelsesmuligheder inden for natur- og ingeniørvidenskab,beskrives her specifikt indeksering af diffraktionsmønstre til brug for eksperimentel materialekarakterisering. Ved at anvende metoder fra numerisk geometri opnås en markant forbedret sampling.This thesis presents a reformulation of existing problems in materials science in terms of well-known methods from applied mathematics: graph theory, computational geometry, and mixed integer programming.The centrosymmetry parameter is reformulated as a graph matching problem, and resolves the inconsistencies in the existing calculation methods as a consequence. By formulating the distance function of lattices as a bipartite graph matching problem, it is shown that the similarity between crystal lattices (root mean square distance, RMSD) can be calculated in polynomial time, which improves upon the existing factorial-time bound. This method is subsequently extended to two-dimensional monolayers.A method is presented for the identication of ordered crystalline phases in molecular dynamics simulations. A robust classication is obtained by the use of template matching, also formulated as a bipartite matching problem on geometric graphs. This method is adapted for two-dimensional materials, in order that e.g. defect structures in polycrystalline graphene can be studied.Matrix decompositions are used to develop a geometric lattice matching algorithm, which can exhaustively identify all low-strain interfaces. The stable, low-energy interfaces which are found as a result are intended for use in the design and construction of topological superconductors, which have important applications in quantum computing.Cluster expansion models are used to nd ground-state structures in gold-silver nanoparticles, which are used in a variety of catalysis processes. In addition to this concrete application, theoretical methods are developed for the optimal construction of cluster expansion models, the exact determination of ground states in a large model, and the exhaustive determination of all possible ground states in a small model.Lastly, a method for nearly-optimal sampling of orientations is presented. Whilst this has many applications in science and engineering, the use-case described here is the indexing of diffraction patterns for experimental materials characterization. Signicantly improved sampling is achieved by applying methods from computational geometry