Cluster-growth processes on a two-dimensional lattice

Abstract

Computer simulations of cluster formation on a two-dimensional lattice have been carried out with the use of a model in which each occupied lattice site in the cluster exerts an independent (multiplicative) screening effect on further growth. The model used in this work is very similar to that investigated by Rikvold and is also related to the models of Eden and of Sawada, Ohta, Yamazaki, and Honjo. If the screening length ($\lambda$) is finite (as is the case for the Eden, Rikvold, and Sawada-Ohta-Yamazaki-Honjo models), we find $D=d=2\mathrm{}$, where $D$ is the Hausdorff dimensionality of the cluster and $d$ is the Euclidean dimensionality. However, for some model parameters the structure of finite clusters on a length scale less than approximately $\lambda$ may be characterized to a good approximation by an effective Hausdorff dimensionality $D^{\prime }$ ($D^{\prime }<2\mathrm{}$). For the case of an infinite screening length, the screening effect at distance ${\overrightarrow{\mathrm{r}}}_n$ from the $n\mathrm{th}$ occupied lattice site is given by $S\left({\overrightarrow{\mathrm{r}}}_n\right)=\exp (-\frac{A}{{\left|{\overrightarrow{\mathrm{r}}}_n\right|}^{\epsilon }})$. In this case, the Hausdorff dimensionality is no longer 2, and its value depends on the exponent $\epsilon$. By analyzing density-density correlation functions and the dependence of the radius of gyration on the cluster size (number of occupied lattice sites), we find that in the $\lambda \to \infty$ limit $D\simeq \epsilon$. $\epsilon \sim \frac{5}{3}$ produces clusters which resemble those formed in the Witten-Sander model of diffusion-limited aggregation.