Towards a molecular theory of freezing. II. Study of bifurcation as a function of density

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Abstract
We study, as a function of density, crystalline solutions for the single particle probability density which bifurcate from the fluid solution of a hard sphere system. The BBGKY equation is used to describe the fluid phase, with its closure taken from computer simulations. As the density increase from zero, crystalline solutions bifurcate from the fluid with a periodicity determined by the density. Bifurcation is found to be characteristic of metastable states, and in general it does not occur at the equilibrium coexistence of two phases. When compared to the known hard disk and hard sphere isotherms, the bifurcation points are seen to be remarkably consistent with the density at which the metastable extension of the crystalline branch meets the equilibrium fluid branch. Metastable crystalline states are also predicted in one dimension. The controversial Kirkwood instability criterion is interpreted in terms of our theory. We show that our analysis applies to a large class of potential energy functions. The possibility that the bifurcation near the jamming densities observed in the amorphous packing of incompressible spheres and disks is also discussed.

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