Contribution to the theory of freezing

Abstract

We report a theory of freezing based on analysis of the solutions of the equation describing the inhomogeneous density distribution at phase equilibrium. This analysis takes the form of a search for a bifurcation point, where the uniform density characteristic of the fluid phase becomes thermodynamically unstable relative to the periodic density distribution of the crystalline phase. The theory described is an extension of the work of Ryzhov and Tareeva, and it properly accounts for the jump discontinuity in the density at the liquid-solid transition. If carried through exactly, the bifurcation analysis will locate the true equilibrium transition and not display metastability in either the fluid or solid phases. However, as in other many body theories, the approximations needed to reduce the general equations to a tractable form lead to errors. Our approximations are: (a) truncation of the exact expansion for the density in an inhomogeneous system at the level of the direct correlation function for pairs of molecules; (b) the use of a convenient but inexact direct correlation function for the liquid in three dimension; (c) the use of an order parameter expansion which neglects vibrational motion in the solid; and (d) the truncation of the order parameter expansion for the density difference between phases after a few terms. We successfully predict the existence of both the liquid-solid transition and a limiting density for the system without the use of auxiliary thermodynamic criteria. For the hard sphere fluid-solid transition the predicted transition parameters are in satisfactory agreement with (computer) experimental data. A universal bifurcation point is found which gives densities for the liquid and the solid phases which are very close to the dense random close packed and the crystal close packed values, respectively. We interpret this bifurcation point as signaling the end of possible compression in the system. A physical interpretation of all the bifurcation results is given.