Self-Assembly and Real-Space Modeling in Binary Hard-Particle and Quasicrystalline Systems

Abstract

In this thesis, I will show a fundamentally new way to understand and construct quasicrystal tilings based on a new formulation of the quasi-unit cell called layering. There are many methods for generating quasitilings including Ammann/pentagrid methods, cut-and-project higher-dimensional projections, deflation/inflation substitutions, and various approaches using matching rules and non-local forced moves. The only other method which has a real-space quasi-unit cell comes from covering theory. However, covering approaches, much like matching-rule approaches, do not provide a recipe for error-free construction of the quasitiling and lack a description of the phason flips—a new type of local particle movement only observed in quasicrystals. In this thesis, I will showcase that layering does provide a way to construct perfect quasitiling from a quasi-unit cell and naturally gives rise to a real-space description of the phason mode of particle movement. This new method was applied to the three Penrose pentagonal tilings, the Ammann-Beenker octagonal tiling, the Tübingen triangle decagonal tiling, the Niizeki-Gähler dodecagonal tiling, and finally the Ammann-Kramer-Neri icosahedral tiling. In addition, simulations were performed using a patchy hard-particle set of Penrose rhombuses as a demonstration of the power of the analysis method. The last chapter of this thesis reports the formation of a binary crystal of hard polyhedra due solely to entropic forces. Although the alternating arrangement of octahedra and tetrahedra is a known space-tessellation from Maurolyctus in 1529 (Lagarias, 2015), it had not previously been observed in self-assembly simulations. Both known one-component phases—the dodecagonal quasicrystal of tetrahedra and the densest-packing of octahedra in the Minkowski lattice—are found to coexist with the binary phase. Apart from an alternative, monoclinic packing of octahedra, no additional crystalline phases were observed.