The effect of seed geometry and size on the mechanism of the pattern formations in two-dimensional space by Monte Carlo simulation

Abstract

The effect of the seed geometry and size on the pattern formation mechanism in two-dimensional space was investigated by Monte Carlo (MC) simulation using the fractal concept and particles density. The patterns obtained from the square seed has symmetrical main brunches whereas the circle seed patterns pose the denser and higher number of brunches with the increase of the dimension of the seed, and hence they have both long-range correlations. The critical exponents, α and the fractal dimensions, Df were computed using the power-law. The results were compared with a model based on the diffusionlimited aggregation (DLA) obtained by MC. It is seen that the morphology of the patterns is greatly affected by the geometry and size of the seeds. Therefore, in order to quantify the change of Df with geometry of the seeds, the fractal dimension function, Df (l) and Df (r) for length of the square and circular seeds were introduced, respectively. The results are expected to provide an expansion of the DLA model in terms of geometry and size of the seeds.