Stacking entropy of hard-sphere crystals

Classical hard spheres crystallize at equilibrium at high enough density. Crystals made up of stackings of two-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 falloff 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 $0.00115\pm 0.00004k_B$ per sphere, which is roughly $30\%$ higher than the same quantity measured near the melting transition.