Stacking Entropy of Hard Sphere Crystals

TL;DR

This study uses advanced Monte Carlo algorithms to precisely measure entropy differences among various stacking sequences in hard sphere crystals, revealing that fcc stacking has the highest entropy, with differences varying near melting and close-packing densities.

Contribution

The paper introduces two multicanonical Monte Carlo methods to directly compare entropies of different stacking sequences in hard sphere crystals, providing detailed insights into interlayer entropic interactions.

Findings

fcc stacking has the highest entropy among all stackings

Entropic interactions involve 3-5 layers and decay with distance

Entropy difference between fcc and hcp is ~0.00115 k_B per sphere at maximum density

Abstract

Classical hard spheres crystallize at equilibrium at high enough density. Crystals made up of stackings of 2-dimensional hexagonal close-packed layers (e.g. fcc, hcp, etc.) differ in entropy by only about $10^{-3}k_B$ per sphere (all configurations are degenerate in energy). To readily resolve and study these small entropy differences, we have implemented two different multicanonical Monte Carlo algorithms that allow direct equilibration between crystals with different stacking sequences. Recent work had demonstrated that the fcc stacking has higher entropy than the hcp stacking. We have studied other stackings to demonstrate that the fcc stacking does indeed have the highest entropy of ALL possible stackings. The entropic interactions we could detect involve three, four and (although with less statistical certainty) five consecutive layers of spheres. These interlayer entropic interactions fall off in strength with increasing distance, as expected; this fall-off appears to be much slower near the melting density than at the maximum (close-packing) density. At maximum density the entropy difference between fcc and hcp stackings is $0.00115 +/- 0.00004 k_B$ per sphere, which is roughly 30% higher than the same quantity measured near the melting transition.