Correlations in randomly stacked solids

TL;DR

This paper analyzes layer correlations in randomly stacked sphere structures, revealing exponential decay in correlations for both Barlow and Torquato-Stillinger stackings, with implications for material ordering and synthesis.

Contribution

It introduces a novel mapping of stacking correlations to Ising and Potts models, providing analytical insights into their decay behavior under randomness and bias.

Findings

Layer correlations decay exponentially in random Barlow stacking.

Bias favoring certain stackings does not induce long-range order.

Correlations in Torquato-Stillinger stacking behave similarly to Barlow stacking.

Abstract

Packing of spheres is a problem with a long history dating back to Kepler's conjecture in 1611. The highest density is realized in face-centred-cubic (FCC) and hexagonal-close-packed (HCP) arrangements. These are only limiting examples of an infinite family of maximal-density structures called Barlow stackings. They are constructed by stacking triangular layers, with each layer shifted with respect to the one below. At the other extreme, Torquato-Stillinger stackings are believed to yield the lowest possible density while preserving mechanical stability. They form an infinite family of structures composed of stacked honeycomb layers. In this article, we characterize layer-correlations in both families when the stacking is random. To do so, we take advantage of the H\"agg code -- a mapping between a Barlow stacking and a one-dimensional Ising magnet. The layer-correlation is related to a moment-generating function of the Ising model. We first determine the layer-correlation for random Barlow stacking, finding exponential decay. We next introduce a bias favouring one of two stacking-chiralities -- equivalent to a magnetic field in the Ising model. Although this bias favours FCC ordering, there is no long-ranged order as correlations still decay exponentially. Finally, we consider Torquato-Stillinger stackings, which map to a combination of an Ising magnet and a three-state Potts model. With random stacking, the correlations decay exponentially with a form that is similar to the Barlow problem. We discuss relevance to ordering in clusters of stacked solids and for layer-deposition-based synthesis methods.