Nucleation theorems, the statistical mechanics of molecular clusters, and a revision of classical nucleation theory

The nucleation theorems relate the temperature and supersaturation dependence of the rate of nucleation of droplets from a metastable vapor phase to properties of the critical molecular cluster, the size that is approximately equally likely to grow or decay. They are derived here using a combination of statistical mechanics and cluster population dynamics, using an arbitrary model cluster definition. The theorems are employed to test the validity of the classical theory of homogeneous nucleation and its ``internally consistent'' form. It is found that the properties of the critical cluster for these models are incorrect, and it emerges that this occurs because the classical theory employs the free energy of a fixed droplet, rather than one free to take any position in space. Thus a term representing positional, or mixing, entropy is missing from the cluster free energy. A revised model is proposed, based on the capillarity approximation but with such a term included, and it is shown that it is fully consistent with the nucleation theorems. The model increases classical rates by factors of approximately 10^4--10^6. Other nucleation models should be tested for internal consistency using the same methods. Finally, the nucleation theorems are used to extract the excess internal energies of molecular clusters from experimental data for several substances.