Helmholtz Free Energy of a Phase Containing a Sparse Ensemble of Heterophase Clusters with Application to Nucleation Theory

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Abstract
A decomposition of the Helmholtz free energy of a phase containing a sparse ensemble of heterophase clusters is derived based on classical statistical mechanics and on the general physical characteristics of such systems. It is not assumed that the phase is an ideal gas. The building blocks of this decomposition are the Helmholtz free energies of the constituents (phase and stationary heterophase clusters) and, for every cluster species, a volume Vkcm, which is of the magnitude of the thermal fluctuation volume of the center of mass of the stationary cluster containing k monomers. A definition of Vkcm is given in terms of the configuration integrals of the clusters. Vkcm is evaluated for k ≫ 1, with the result that Vkcm is proportional to k−1/2 and is a function of temperature, the specific volume, and the isothermal compressibility of the phase in the cluster. A thermodynamically consistent expression for the work to form a stationary cluster, which reads as Δgk/(kBT) = −ak + (3/2)bk2/3 + 3ck1/3 + d, is derived. The coefficients a, b, c, and d depend on the thermodynamic properties of the homogeneous phases, on the surface tension, and on one additional phenomenological material function of temperature and pressure. The description is general and covers a wide class of materials. It is shown that the heterogeneous system represents the thermodynamic equilibrium and not the pure phase without clusters. The resulting expression for the equilibrium particle number, which is different from the one used in classical nucleation theory, is by a standard procedure input for the calculation of the stationary Becker-Döring nucleation rate and entails a correction factor for the classical nucleation rate. Comparison with experiments is provided for nucleation onset measurements of argon and for measurements of the homogeneous nucleation rate of water. Measurements and theory can be brought to match within the limits of experimental precision in both cases.
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