Active zone of growing clusters: Diffusion-limited aggregation and the Eden model in two and three dimensions

Abstract

Our previous studies of the growing interface of two-dimensional clusters produced by the Eden process and by diffusion-limited aggregation are expanded and extended to d=3 dimension. Growth is described in terms of the motion of an active zone defined as the region of the clusters where the probability of collecting new particles is nonzero. If the description is limited to spherically averaged properties then this active region can be characterized by the probability P(r,N)dr that the Nth particle is deposited a distance r from the center of mass of the existing cluster. As in our previous Monte Carlo simulations of the d=X2 case, we find that, for large N, P(r,N)=(√2π ${\xi }_N$ ${)}^{\mathrm{-}1}$exp[-(r-r${¯}_N$ ${)}^2$/ 2${\xi }_N^2$] with r${¯}_N$∼$N^{\nu }$∼$N^{1/D}$, where D is the fractal dimension of the infinite cluster and ${\xi }_N$∼$N^{\nu ¯}$ where ν¯<ν, indicating the presence of a second diverging length in these growth processes. We also study the center-of-mass motion of growing clusters. We find that one has to be especially careful with simulations of diffusion-limited aggregation because discarding particles which wander too far away from the cluster generates an effective force on the center of mass, resulting in a motion which can obscure the intrinsic structure of the clusters.