Entropy difference between crystal phases

In a recent Letter1, Woodcock reported the results of a molecular dynamics study in which he claims to have finally determined the free-energy difference between the hexagonal close-packed (h.c.p.) and face-centred cubic (f.c.c.) phases of a crystal of (classical) hard spheres. Woodcock reports a small positive difference in the reduced Gibbs free-energy, which is equivalent to a difference in the reduced Helmholtz free-energy of ΔF*≡(Fhcp −Ffcc)/RT=0.005(1) at the melting density (R is the gas constant, T is the absolute temperature, and the number in parentheses is the estimated error in the last digit). As Woodcock correctly points out, the calculation of the relative stability of the f.c.c. and h.c.p. phases of hard spheres is a long-standing problem in statistical physics. Attempts to resolve it date back to the work of Alder, Hoover and colleagues2,3,4,5, and most recently, a direct simulation by Frenkel and Ladd6, obtaining the bounds of Helmholtz free-energy of −0.001⩽ΔF*⩽0.002. Woodcock's estimate is incompatible with this latter result.