The Gregory-Newton Problem for Kissing Sticky Spheres

TL;DR

This paper explores all possible arrangements of 12 spheres around a central sphere in the sticky-hard-sphere model, analyzing their geometric and energetic properties, and how these arrangements relate to Lennard-Jones potentials and symmetry breaking.

Contribution

It systematically characterizes all non-isomorphic kissing sphere arrangements in the SHS model and examines their relation to Lennard-Jones potential minima and symmetry breaking phenomena.

Findings

All 737 SHS contact graphs are subgraphs of the icosahedral graph.

The LJ(6,12) potential has only one minimum: the icosahedral structure.

Symmetry breaking occurs only with highly repulsive short-range potentials.

Abstract

All possible non-isomorphic arrangements of 12 spheres kissing a central sphere (the Gregory-Newton problem) are obtained for the sticky-hard-sphere (SHS) model, and subsequently projected by geometry optimization onto a set of structures derived from an attractive Lennard-Jones (LJ) type of potential. It is shown that all 737 derived SHS contact graphs corresponding to the 12 outer spheres are (edge-induced) subgraphs of the icosahedral graph. The most widely used LJ(6,12) potential has only one minimum structure corresponding to the ideal icosahedron where the 12 outer spheres do not touch each other. The point of symmetry breaking away from the icosahedral symmetry towards the SHS limit is obtained for general LJ($a,b$) potentials with exponents $a,b\in \mathbb{R}_+$. Only if the potential becomes very repulsive in the short-range, determined by the LJ hard-sphere radius $\sigma$, symmetry broken solutions are observed.