Percolation in Cluster-Cluster Aggregation Processes

TL;DR

This paper investigates the morphology of clusters in diffusion-limited and reaction-limited aggregation processes, revealing a crossover from aggregation to percolation behavior at a critical concentration in two-dimensional systems.

Contribution

It introduces the concept of a crossover length scale and identifies a critical concentration where the cluster structure transitions to standard percolation.

Findings

A crossover length scale $L_c$ exists between aggregation and percolation regimes.

At a critical concentration $c_p$, the spanning cluster exhibits standard percolation properties.

Long-range correlations resemble percolation, but structural links diminish, indicating a homogeneous small-scale regime.

Abstract

Numerical simulations of Diffusion-Limited and Reaction-Limited Cluster-Cluster Aggregation processes of identical particles are performed in a two-dimensional box. It is shown that, for concentrations larger than a characteristic gel concentration, the morphology of the resulting spanning cluster at the gel time $t_g$ exhibits a crossover length $L_c$ between percolation ($l>L_c$) and aggregation ($l<L_c$). $L_c$ vanishes when increasing $c$, and, at a critical concentration value $c_p$ (where $L_c \rightarrow 0$) the entire spanning cluster scales as the percolating cluster obtained by standard percolation. Even if for $c>c_p$ the long-range correlations are similar to that of percolation, the vanishing links in the structure suggest that an homogeneous regime appears at small scales.