Calculating the symmetry number of flexible sphere clusters

TL;DR

This paper introduces a theoretical and computational framework to determine the symmetry number of flexible sphere clusters in three dimensions, which is crucial for understanding their equilibrium properties and self-assembly behavior.

Contribution

It defines the sticky symmetry group for flexible sphere clusters, develops a numerical algorithm to compute it, and extends the method to partially distinguishable spheres, enabling efficient analysis of complex clusters.

Findings

The algorithm accurately computes the symmetry group for test cases.

The method efficiently extends to spheres with non-identical interactions.

Partition functions for all 6-sphere clusters were successfully calculated.

Abstract

We present a theoretical and computational framework to compute the symmetry number of a flexible sphere cluster in $\mathbb{R}^3$, using a definition of symmetry that arises naturally when calculating the equilibrium probability of a cluster of spheres in the sticky-sphere limit. We define the sticky symmetry group of the cluster as the set of permutations and inversions of the spheres which preserve adjacency and can be realized by continuous deformations of the cluster that do not change the set of contacts or cause particles to overlap. The symmetry number is the size of the sticky symmetry group. We introduce a numerical algorithm to compute the sticky symmetry group and symmetry number, and show it works well on several test cases. Furthermore we show that once the sticky symmetry group has been calculated for indistinguishable spheres, the symmetry group for partially distinguishable spheres (those with non-identical interactions) can be efficiently obtained without repeating the laborious parts of the computations. We use our algorithm to calculate the partition functions of every possible connected cluster of 6 identical sticky spheres, generating data that may be used to design interactions between spheres so they self-assemble into a desired structure.