Towards a molecular theory of freezing

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Abstract
The subject of this article is the fluid–solid transition and, in particular, an analysis of crystallization in terms of quantities which describe the average local arrangements of molecules in a fluid. We determine whether it is possible to predict the existence of crystalline solutions for the local molecular density from a Hamiltonian which is invariant under all translations and rotations. Crystallization is studied using the singlet probability density, the pair correlation function, and the intermolecular potential energy. An integral equation is obtained for these quantities, and we pursue the existence of crystalline (i.e., periodic but nonconstant) solutions for the singlet probability which branch from the fluid (i.e., constant) solution which is the number density. The phenomenon of crystallization, that is, the existence and determination of these solutions, can then be represented as a nonlinear eigenvalue problem. The analysis is applied to hard sphere systems in one, two, and three dimensions. Crystallization to close‐packed lattices is found in two and three dimensions when the isotropic media are overcompressed by amounts which depend on the structures to which the fluids crystallize. That is, the fluid persists into a portion of the metastable region. The nature of the crystalline solutions is analyzed in the neighborhood of the branching eigenvalues, and the relation between these special eigenvalues and equilibrium freezing points is discussed. The stability of these crystalline solutions is determined by comparing the values of a free energylike functional on these solutions with its value for the fluid.