Density of Binary Disc Packings: Playing with Stoichiometry

TL;DR

This paper rigorously determines the densest packings of binary disc mixtures with specific size ratios, revealing optimal stoichiometries and phase behaviors through computer-assisted proofs.

Contribution

It provides the first rigorous classification of densest packings for binary disc mixtures with a specific size ratio, including phase separation and mixing phenomena.

Findings

Maximum density at 1:1 stoichiometry with square grid and nested small discs

Chaotic mixing of discs in excess of large discs

Phase separation with hexagonal phases when small discs are in excess

Abstract

We consider hard-disc mixtures with disc sizes within ratio $\sqrt{2}-1$, that is, the small disc exactly fits in the hole between four large discs. For each prescribed stoichiometry of large and small discs, the densest packings are rigorously determined via a computer-assisted proof. The density is maximal for the 1:1 stoichiometry: the large discs then form a square grid in each interstitial site of which a small disc nests. When there is an excess of large discs, the densest packings are made of a single phase which mixes the two types of discs in a chaotic way (it can be described by square-triangle tilings). When there is an excess of small discs, on the contrary, a phenomenon of phase separation appears: the large discs are involved in the densest 1:1 stoichiometry phases while the excess of small discs form compact hexagonal phases.