A cluster-integral series is developed for the calculation of the droplet and crystallite partition functions involved in the classical theory of homogeneous nucleation. These formal series reduce the calculation of droplet free energies to the consideration of few-body integrals similar to those which determine the virial coefficients of an imperfect gas. Theories are proposed which correspond to infinite-order series resummations directly analogous to the Percus-Yevick and hypernetted-chain integral equations for simple liquids. The cluster-integral expansion coefficients through sixth order are evaluated for argon droplets over a wide range of temperatures by a very efficient specialized Monte Carlo sampling technique. These coefficients are used in two distinctly different ways: direct truncation of a suitable rearrangement of the series (star-tree approximation), and formation of Padé approximants for a generating function identified as being well suited for this purpose. By comparing the results obtained from these different procedures, it is concluded that the predicted droplet free energies are quite accurate in the range of droplet size of special interest in homogeneous nucleation theory, i.e., the inaccuracy appears to be less than 3% in the free energy per particle for droplets of 80 argon atoms.

• Received 28 January 1983

DOI:https://doi.org/10.1103/PhysRevA.28.2482