A colloidal viewpoint on the finite sphere packing problem: the sausage catastrophe

TL;DR

This study explores finite sphere packing using colloids in lipid vesicles, revealing various conformations and identifying specific cluster sizes that pack more efficiently than linear arrangements, challenging traditional assumptions.

Contribution

It provides experimental and simulation evidence of diverse sphere packings and identifies specific cluster sizes with superior packing efficiency, advancing understanding of finite sphere packing.

Findings

Identified bistable states due to membrane fluctuations.

Mapped out a state diagram of sphere conformations.

Found clusters of 56-70 spheres with higher packing efficiency.

Abstract

It is commonly believed that the most efficient way to pack a finite number of equal-sized spheres is by arranging them tightly in a cluster. However, mathematicians have conjectured that a linear arrangement may actually result in the densest packing. Here, our combined experimental and simulation study provides a realization of the finite sphere packing problem by studying non-close-packed arrangements of colloids in a flaccid lipid vesicle. We map out a state diagram displaying linear, planar and cluster conformations of spheres, as well as bistable states which alternate between cluster-plate and plate-linear conformations due to membrane fluctuations. Finally, by systematically analyzing truncated polyhedral packings, we identify clusters of $56\leq N \leq 70$ spheres, excluding $N=57$ and 63, that pack more efficiently than linear arrangements.