Diffusion-Controlled Deposition: Cluster Statistics and Scaling

Abstract

Diffusion-controlled deposition in dimension $d=2\mathrm{}$ is studied by Monte Carlo simulation, and the number of clusters of size $s$ is found to scale as $n_s\sim s^{-\tau }$ with $\tau \approx 1.35$. The inequality $\tau <2\mathrm{}$ is shown to imply for a deposit of $N$ particles per nucleation site that the exponents in the scaling Ansatz $n_s\mathrm{}\left(N\right)\mathrm{}\sim s^{-\tau }f\left(\frac{s^{\sigma }}{N}\right)$ satisfy the scaling law $\sigma =2-\tau$. If the scaling properties of deposits on a surface are related to those of an aggregate grown on a seed particle, $\tau =1+\frac{\mathrm{}(d-1\mathrm{})\mathrm{}}{D}$ is obtained, where $D$ is the fractal dimension of the aggregate.