Formation free energies of clusters at high supersaturations

The Helmholtz free energy of a constrained supersaturated vapor with a cluster size distribution consisting of clusters of various sizes is modeled as a mixture of hard spheres of various sizes attracting each other. This model naturally takes into account monomer-monomer and monomer-cluster interactions, so it implicitly pertains to nonideal gases, unlike prior work. Based on this model, the expressions for the equilibrium concentration and the formation free energies of clusters in a metastable supersaturated vapor have been derived. These results indicate that the widely used formula, n_i = n₁exp(-βΔG_i), that computes the formation free energy of a cluster does not work at high supersaturations. As an example, the formation free energies of clusters with Stillinger's physical cluster definition in metastable, highly supersaturated vapors interacting via Lennard-Jones potential are studied using these expressions. Noticeable differences have been found for both the formation free energies of clusters and sizes of the critical clusters computed from our proposed expressions vs those from the formula n_i = n₁exp(-βΔG_i).