Dynamic cluster-size distribution in cluster-cluster aggregation: Effects of cluster diffusivity

Abstract

The dynamics of the diffusion-limited model of cluster-cluster aggregation is investigated in two and three dimensions by studying the temporal evolution of the cluster-size distribution $n_s$(t), which is the number of clusters of size s at time t. In a recent study it was shown that the results of the two-dimensional simulations for mass-independent diffusivity can be well represented by a dynamic-scaling function of the form $n_s$(t)∼$s^{\mathrm{-}2}$f(s/$t^z$), where f(x) is a scaling function with a power-law behavior for small x, namely f(x)∼$x^{\delta }$ for x≪1 and f(x)≪1 for x≫1. In this paper we extend the calculations of the cluster-size distribution to three dimensions and to the case of the cluster diffusivity depending on the size of the clusters. The diffusion constant of a cluster of size s is assumed to be proportional to $s^{\gamma }$. The overall behavior of $n_s$(t) and the exponents δ and z have been determined for a set of values of γ. We find that the results are consistent with the scaling theory, and the exponents in $n_s$(t) depend continuously on γ. Moreover, there is a critical value of γ [${\gamma }_c$(d=2)≃-(1/4), ${\gamma }_c$(d=3)≃-1/2] at which the shape of the cluster-size distribution crosses over from a monotonically decreasing function to a bell-shaped curve which can be described by the above scaling form for $n_s$(t), but with a scaling function f̃(x) different from f(x).