Scaling in steady-state cluster-cluster aggregation

Abstract

The diffusion-limited cluster-cluster aggregation model is investigated under conditions which for long times lead to steady-state coagulation. Single particles are added to the system at a constant rate and the larger clusters appearing as a result of the aggregation process are removed according to various rules. Our results show that the dependence of the number of clusters, N(t), on the feed rate κ in a unit volume and the time t can be well represented by a scaling form N(t)∼${\kappa }^{\alpha }$f(${\kappa }^{\beta }$t) with a scaling function f(x)∼x for x≪1 and f(x)=1 for x≫1. The exponents α and β are found to depend on the spatial dimension d, of the system, but within the statistical errors they always satisfy the relation α+β=1 in accordance with the prediction of a generalized rate equation discussed by Rácz (see the companion paper). The values we have obtained for α and β are consistent in two and three dimensions with the corresponding results of the Smoluchowski equation approach but inconsistent in one dimension. This can be considered as an indication of the fact that the upper critical dimension for the kinetics of the diffusion-limited cluster-cluster aggregation model is 2.