Translational invariance in nucleation theories: theoretical formulation.

The consequences of spontaneously broken translational invariance on the nucleation-rate statistical prefactor in theories of first-order phase transitions are analyzed. A hybrid, semiphenomenological approach based on field-theoretic analyses of condensation and modern density-functional theories of nucleation is adopted to provide a unified prescription for the incorporation of translational-invariance corrections to nucleation-rate predictions. A connection between these theories is obtained starting from a quantum-mechanical Hamiltonian and using methods developed in the context of studies on Bose-Einstein condensation. An extremum principle is used to derive an integro-differential equation for the spatially nonuniform mean-field order-parameter profile; the appropriate order parameter becomes the square root of the fluid density. The importance of the attractive intermolecular potential is emphasized, whereas the repulsive two-body potential is approximated by considering hard-sphere collisions. The functional form of the degenerate translational eigenmodes in three dimensions is related to the mean-field order parameter, and their contribution to the nucleation-rate prefactor is evaluated. The solution of the Euler-Lagrange variational equation is discussed in terms of either a proposed variational trial function or the complete numerical solution of the associated boundary-value integro-differential problem. Alternatively, if the attractive potential is not explicitly known, an approach that allows its formal determination from its moments is presented.