Diffusion-controlled aggregation on two-dimensional square lattices: Results from a new cluster-cluster aggregation model

Abstract

A series of related models for diffusion-limited aggregation in which particle clusters as well as individual particles are allowed to move has been investigated. In the limit of low particle concentrations ($\rho$), these models lead to aggregates with a fractal or Hausdorff dimensionality ($D$) which is distinctly smaller than the Euclidean dimensionality ($d$). At finite particle concentrations, there is a crossover from a fractal structure on short length scales to a uniform structure ($D=d$) on longer length scales. If the mobility of small clusters (or single particles) is equal to or greater than the mobility of larger clusters, a limiting ($〈\rho 〉\to 0$) fractal dimensionality of $D=1.46\mathrm{}\pm 0.04$ (for $d=2\mathrm{}$) is estimated from both the density-density correlation function [$C\left(r\right)\mathrm{}$] and the dependence of the radius of gyration ($R_g$) on cluster size ($N$) for intermediate clusters formed during the simulations. This result indicates that the new aggregation models are fundamentally distinct from the Witten-Sander (particle-cluster) aggregation model. The numerical value of the fractal dimensionality seems to be insensitive to the relationship between cluster diffusion coefficient and cluster size, provided that large clusters do not have a higher mobility than small ones. The structures produced by these models closely resemble the structures observed in actual flocculated colloid systems. In the limit where only the largest clusters can move, our model is equivalent to the Witten-Sander model for diffusion-limited aggregation, and we find $D\approx \frac{5}{3}$ for $d=2\mathrm{}$. At finite particle concentrations, the clusters generated by this version of the model have a fractal dimensionality of $D\approx \frac{5}{3}$ on short-length scales, and a uniform structure ($D=d=2\mathrm{}$) on long length scales. Very similar results have been obtained from a model in which the cluster diffusion coefficient is proportional to the square of the cluster size ($N^2$).