Dynamic Scaling in Aggregation Phenomena

Abstract

The dynamics of the diffusion-limited cluster-cluster aggregation model (CCA) is reviewed through the temporal evolution of the cluster size distribution n_s(t), which is the number of clusters of size s at time t. In recent studies it was shown that the results of the Monte-Carlo simulations can be well represented by a dynamic scaling function of the form n_s(t)~s^-2f(s/t^z) where f(x) is a scaling function which depends on the spatial dimension and the microscopic details of the aggregation process. Two basic factors which affect the aggregation process are the mobility and the reactivity of the aggregating clusters. The diffusion constant of a cluster of size s is assumed to be proportional to s^γ. Depending on such factors as the chemical reactivity, kinetic energy, mass, etc. of the aggregates, the coagulation of two clusters may or may not take place. These effects are simulated by assuming that the probability that two clusters of sizes i and j irreversibly stick together is proportional to (ij)^σ. The overall behavior of n_s(t) and its moments have been determined for a set of values of γ and σ. We find that the results are consistent with the scaling theory, and the exponents in n_s(t) depend continuosly on γ and σ. Moreover, there is a critical value of γ for a given σ, and vica versa, at which the shape of the cluster size distribution crosses over from a monotonically decreasing function to a bell-shaped curve. This phenomenon, which is consistent with experiments, can be described by the above scaling form for n_s(t) with appropriate forms for f(x) in the two regimes.