Structure of finite sphere packings via exact enumeration: Implications for colloidal crystal nucleation

TL;DR

This paper systematically enumerates and analyzes all possible sphere packings to understand their structures, stability, and implications for colloidal crystal nucleation, revealing limitations of classical nucleation theory for small clusters.

Contribution

It provides a complete enumeration of sphere packings, analyzing their structures and stability, and highlights the importance of considering unconstrained nuclei in classical nucleation theory.

Findings

Number of nonisomorphic isostatic packings grows exponentially with N

Maximally contacting packings are not always densest or most symmetric

Classical nucleation theory fails for small N due to diverse structures

Abstract

We analyze the geometric structure and mechanical stability of a complete set of isostatic and hyperstatic sphere packings obtained via exact enumeration. The number of nonisomorphic isostatic packings grows exponentially with the number of spheres $N$, and their diversity of structure and symmetry increases with increasing $N$ and decreases with increasing hyperstaticity $H \equiv N_c - N_{ISO}$, where $N_c$ is the number of pair contacts and $N_{ISO} = 3N-6$. Maximally contacting packings are in general neither the densest nor the most symmetric. Analyses of local structure show that the fraction $f$ of nuclei with order compatible with the bulk (RHCP) crystal decreases sharply with increasing $N$ due to a high propensity for stacking faults, 5- and near-5-fold symmetric structures, and other motifs that preclude RHCP order. While $f$ increases with increasing $H$, a significant fraction of hyperstatic nuclei for $N$ as small as 11 retain non-RHCP structure. Classical theories of nucleation that consider only spherical nuclei, or only nuclei with the same ordering as the bulk crystal, cannot capture such effects. Our results provide an explanation for the failure of classical nucleation theory for hard-sphere systems of $N\lesssim 10$ particles; we argue that in this size regime, it is essential to consider nuclei of unconstrained geometry. Our results are also applicable to understanding kinetic arrest and jamming in systems that interact via hard-core-like repulsive and short-ranged attractive interactions.