Entropy difference between crystal phases

Woodcock replies —I reported the discovery a substantial area of pressure difference (ΔP) between the f.c.c. and h.c.p. single-occupancy-cell models, which arises from a difference in order-disorder transition pressures. The result was a free-energy difference in favour of f.c.c., corresponding to an entropy difference 0.005NkB, over the range V=1.00Nσ3 to 1.25Nσ3, with a generous uncertainty (±0.001), estimated by integrating the standard deviations of subaverages of ΔP for individual data points. Extension of the computations on either side of the phase transition have since revealed a tail in the pressure difference for V>1.25σ3 in favour of h.c.p. There is also a weak pressure difference for volumes below melting. I have now obtained more accurate data for these tails, including new data points on both sides of the single-occupancy-cell phase transition (Fig. 1). Figure 1 Latest molecular dynamic data for the pressure difference as a function of volume at constant temperature, ΔP(. V)T, between the h.c.p. and f.c.c. single-occupancy-cell crystal structures for hard spheres; V0 is the close-packed crystal volume and Vm is the volume at melting. The area under this curve is the Helmholtz free-energy difference between the two crystal structures at close packing in units of NkBT.