Reaction-limited cluster-cluster aggregation in dimensionalities2–10

Abstract

Simple models for reaction-limited aggregation (the random addition of pairs of clusters) have been investigated in dimensionalities 2–10 and 20. Simulations have been carried out using both hypercubic lattice models and off-lattice models. For the hierarchical models in which the added clusters are all of the same size, the results are in good agreement with the theoretical ideas of R. C. Ball and T. A. Witten [J. Stat. Phys. 36, 873 (1984)] and of S. P. Obukhov (Zh. Eksp. Teor. Fiz. 87, 2024 (1984) [Sov. Phys.—JETP 60, 1167 (1984)]), which indicate that above a critical dimensionality of $d_c$=2$D^{\mathrm{°}}$ [$D^{\mathrm{°}}$=ln(4)/ln((3/2)] the clusters are mutually transparent and have a fractal dimensionality of $D^{\mathrm{°}}$. Polydisperse models have also been used to investigate both the cluster structure and the kinetics of aggregation. For the case d=2 the mean cluster size (S) and cluster number (N) exhibit an algebraic time dependence and the cluster-size distribution has a peaked form with a cutoff at both large and small cluster sizes. For d>2 the cluster-size distribution can be described quite well by a power-law form with an exponential cutoff at large sizes. For the polydisperse models the simulation results also indicate an upper critical dimensionality for the embedding space or lattice. A polydisperse version of the ‘‘Sutherland’s ghost’’ (transparency) model of Ball and Witten leads to the result $D^{\mathrm{°}}$≃4.0, $d_c$≃8.0, which is consistent with the simulation results. For d<$d_c$ the simulation results are not in good agreement with recent theoretical predictions.