The phase transition for the aggregation model in the effective-medium approach

Abstract

The one-particle model on a lattice is proposed to account for the formation of a spanning cluster in the aggregation process. In the framework of that model, effective-medium theory is described to calculate the effective diffusion rate, sigma , and effective adsorption rate, v, within the aggregation front. A sharp increase in v is found at a very small concentration of adsorbed particles if the elementary rate of attachment, Gamma , measured in units of the elementary diffusion rate, is above a critical value Gamma * approximately=2. This peculiarity in behaviour of v signals the occurrence of rare and ramified structure. The model can also be applied to explore the concentration dependence of absorption rate. A unit rate of attachment, Gamma , corresponds to instantaneous local absorption. Then, the Smoluchowski result is reproduced for a small concentration of traps. Values Gamma <1 account for the reaction-limited mode. The concentration dependences of the rates v and sigma are found for any value Gamma in the steady-state regime.