Mixed Triangular-Square Lattices on a Spherical Surface

Abstract

Defects play a crucial role in determining the structures and properties of materials. When putting lattices onto a curved surface that has non-trivial Euler characteristics, defects must appear due to the geometric frustrations. Extensive studies have shown the singular lattices on curved surfaces presenting point- and scar- disclinations with high symmetry. Of recent interest is the mixed ordered lattices on flat surfaces. The phase separation of multi-ordered lattices produces a complicated, maze-like structure, where defect also plays an important role. It makes us curious about the defect properties when arranging mixed lattices on a curved geometry. In the thesis, we propose and discuss a computational and theoretical approach to study the properties of the mixed two-dimensional triangular-square lattices on a spherical surface.

First, we introduce Hertzian interactions in molecular dynamics simulations to stabilize the coexistence of triangular and square domains and enable soft particles to self-assemble on a sphere. The simulations reveal novel defect morphologies beyond conventional point and scar defects—domains of one lattice type acting as defects within the bulk of the other, arranged with unexpected symmetry.

To analyze these assemblies further, we develop tiling methods to arrange mixed triangular-square lattices onto spherical geometries, where defects type and locations are determined. we generalize the Caspar–Klug construction for triangulations of the sphere and introduce two operations---Face‐Rotation (FR) and Cut‐and-Rotate (CR)---to generate mixed tilings with minimal defects and high symmetries.

This tiling methods enables the comparison of the defective energies of these structures with a coarse-grained model developed by Bowick \textit{et al.}. A state diagram is plotted to show the energetically favored structure at a given area fraction of square lattices. Finally, we compare these mixed configurations to singular lattices via a limiting approximation to their energy landscapes.

Our work presents a new angle to understand the tripartite tug-of-war among crystalline orders, defects, and topology—an interplay that occurs across scales, from biomolecular assemblies to architectural frameworks.