Computation of the free energy for alternative crystal structures of hard spheres

Abstract

A single-occupancy cell (SO-cell) method has been applied to calculate the free energies of different crystal structures of hard spheres via molecular dynamics (MD). The objectives are (i) to examine the nature of the phase transition in the SO-cell model, (ii) to determine the thermodynamic stability of the bcc crystal phase relative to the fcc and the free fluid, (iii) to establish the relative stability of the fcc and hcp crystal structures, and (iv) to investigate hybrid structures of the unit stacking type ‘–ABCAB–'. MD computations are reported for the pressures of the SO-cell models of all these structures. The SO-cell phase transition for fcc and hcp is first-order, with ordered and disordered phases coexisting at the same p, V and T. The transition is much weaker for bcc. The metastable fluid–bcc phase transition parameters are determined; the bcc phase is everywhere unstable compared with fcc. The bcc solid melts to the metastable fluid at a pressure of 14.5 k_BT/σ³, and has a melting volume of 0.95 Nσ³, i.e., very close to that of the fcc crystal. A more precise numerical estimate for the fcc–hcp entropy difference is reported. At close packing the fcc phase is the more stable by 0.0026(1) Nk_BT; the Gibbs and Helmoltz energy differences are the same at close packing. For expanded volumes close to melting, the hcp crystal has a slightly higher pressure than the fcc; the enthalpy difference at melting is 0.0030(5) Nk_BT. Consequently the Gibbs energy difference approaching melting becomes less than the uncertainty in the computations, i.e. <0.001 Nk_BT.