Self Assembled Clusters of Spheres Related to Spherical Codes

TL;DR

This paper explores the thermodynamic self-assembly of spheres into well-defined clusters related to spherical codes, revealing how temperature and size ratios influence cluster structure and dynamics, with implications for colloidal material design.

Contribution

It introduces a model linking sphere assembly to spherical codes and demonstrates how temperature and size ratios control cluster morphology and collective particle rearrangements.

Findings

Cluster structures relate to spherical codes.

Temperature and size ratios dictate cluster shape.

Unique collective rearrangements occur in specific clusters.

Abstract

We consider the thermodynamically driven self-assembly of spheres onto the surface of a central sphere. This assembly process forms self-limiting, or terminal, anisotropic clusters (N-clusters) with well defined structures. We use Brownian dynamics to model the assembly of N-clusters varying in size from two to twelve outer spheres, and free energy calculations to predict the expected cluster sizes and shapes as a function of temperature and inner particle diameter. We show that the arrangements of outer spheres at finite temperatures are related to spherical codes, an ideal mathematical sequence of points corresponding to densest possible sphere packings. We demonstrate that temperature and the ratio of the diameters of the inner and outer spheres dictate cluster morphology and dynamics. We find that some N-clusters exhibit collective particle rearrangements, and these collective modes are unique to a given cluster size N. We present a surprising result for the equilibrium structure of a 5-cluster, which prefers an asymmetric square pyramid arrangement over a more symmetric arrangement. Our results suggest a promising way to assemble anisotropic building blocks from constituent colloidal spheres.