Mode Selection of Diffusion-limited Aggregation

Abstract

The diffusion-limited aggregation (DLA) is a pattern formation governed only by the nature of diffusion field. Irreversible sticking process of particles gives rise to isotropic random fractal structure of DLA. Since Witten and Sander proposed such a simulation model, many researchers have been devoted to various structures of offlattice DLA, such as self-similarity, active zone, multifractality, branching distribution, fractal dimension, and so on. However, many authors have indicated complex structures of DLA cluster. Mysterious one among them is near pentagonal symmetry with the outline shape of DLA cluster. A wedge model was proposed from the viewpoint of harmonic measure, which gives the fractal dimension D 1:714 for a symmetric cone of pentagon. The details of mysterious symmetry in DLA cluster are studied by numerical simulations. In order to study the exterior shape of DLA, we measure the coordinates of 10 sticking particles without cluster growth that is the same manner as the study of active zone. The sticking simulations are performed on the 2500 offlattice clusters composed of 3 10 particles. Here, simulation methods with high accuracy are used as reported in ref. 14. Averaged power spectrum of sticking distribution in angle direction is shown in Fig. 1 as a function of spectrum mode. The peak position of mean spectrum can be estimated as the mode of 5:4 by the fitting of parabolic curve near the peak. Thus, the mysterious symmetry of DLA is characterized by the near pentagon as indicated before reports. However, this peak mode is slightly increasing with increasing of cluster size in the present simulation up to 3 10 particles. Above ordinary DLA simulation is started from the initial condition of a seed particle with the radius RS 1⁄4 0. In order to clarify the stability of near pentagonal symmetry, following DLA simulations are performed on the initial conditions of circular seed up to RS 1⁄4 500. As shown in Fig. 2 for RS 1⁄4 500, it is obvious that the near pentagonal mode in angle direction is extremely stable for large size of cluster independent of initial condition. Cluster density as a function of angle is analyzed by power spectrum, and then spectrum peak is obtained from the mean data for 1000 clusters (30000 clusters for RS 1⁄4 0). An example of power spectrum along the cluster growth from the seed of RS 1⁄4 500 is drawn in Fig. 3(a), which shows decreasing of peak mode with increasing of cluster size. As shown in Fig. 3(b), the Fig. 1. Power spectrum of angle distribution for sticking particles onto the surface of 2500 ordinary DLA clusters composed of N 1⁄4 3000000 particles. The peak position is estimated as the mode of 5:4.