A cell model of a liquid droplet

An expression for the free energy of a droplet composed of attracting hard spheres is found using a simple cell model. The hard-sphere repulsion is assumed to act only between molecules in the same cell, whereas attraction extends over many cells. A maximum term analysis gives rise to a mean-field free energy which includes terms proportional to the first and second power of the droplet radius R with coefficients which can be related to the planar surface tension and Tolman length. Certain Gaussian fluctuations about the maximum term are also considered, corresponding to droplet translation and capillary wave fluctuations. Inclusion of these fluctuations is necessary to ensure that the nucleation rate is proportional to the system volume. They also reduce the planar surface tension and introduce a logarithmic term, -4/3 ln R, into the free energy. The inclusion of other fluctuations and the relationship between these equations and those arising in density-functional theories of nucleation is discussed.