A physics-based tessellation algorithm for particle assemblies on arbitrary surfaces

Abstract

Interacting particle assemblies embedded on a surface are often used to model biophysical systems and to study new colloidal materials. The configurations resulting from the particles interacting with each other and with their substrate affect the system's physical properties, which depend on local symmetries and defects. It is therefore important to identify the nature and location of defects in the particle assembly. This task is often achieved using either the Voronoi tessellation algorithm or order parameters. Although very useful, they present limitations, especially when the particle assemblies are embedded on irregular or highly deformed 3D surfaces. In this work, we present a novel algorithm to generate the tessellation of particle assemblies on 3D surfaces of arbitrary geometry. The algorithm is based on the particles' physical interactions and does not require a priori information on the surface geometry. The resulting cells in the tessellation represent each particle's interactions with its first ring of neighboring particles. The algorithm is tested using 2D and 3D surfaces, with or without periodic boundary conditions, with holes or fully covered by particles, of regular geometry or highly deformed shape, and in the presence of positive, zero, and negative Gaussian curvature. In all cases, the presented algorithm is capable of generating a tessellation representing the particle interactions and highlighting the location and nature of the assembly defects. Finally, the proposed algorithm is compared with both Voronoi and hexatic order parameters in several representative cases to highlight similarities with existing methods as well as the advantages of the newly proposed algorithm.