Cluster-size distributions of ionic and colloidal systems

We develop a method for calculating the physical and geometric properties of atomic clusters in a very general aggregation model. Our method accommodates an arbitrary interparticle potential. It also allows the use of a very wide range of pairing criteria for specifying which configuations are considered to form clusters. We develop an exact differential equation satisfied by the generating function for the quantities {${\mathit{n}}_{\mathit{c}}$(k)}, where ${\mathit{n}}_{\mathit{c}}$(k) is the mean number per unit volume of clusters containing exactly k atoms. We discuss economic methods of solving this equation; a second paper is planned to discuss numerical solutions in detail. We show that the analog, for continuum percolation, of the virial theorem is an exact rate equation of Smoluchowski type for the cluster densities, in which the time is replaced by the total particle density. In this equation, the rate constants are given by contact values of connectedness functions. Applications of this rate equation are discussed. We also give an exact equation for the generating function for cluster volumes. Extensions of the method to calculate cluster free energies economically are described.