Diffusion-controlled cluster formation in two, three, and four dimensions

Abstract

Diffusion-controlled cluster formation has been simulated in two-, three-, and four- dimensional space. The radii of gyration (${R}_{g}$) of the resulting clusters have a power-law dependence on the number of particles in the cluster $(N) {R}_{g}={N}^{\ensuremath{\beta}}$. The corresponding Hausdorff dimensionality ($D=\frac{1}{\ensuremath{\beta}}$) is related to the Euclidean dimensionality $d$ by the relationship $D\ensuremath{\sim}\frac{5}{6}d$ for $d=3$ and 4. For the two-dimensional case we find that $\frac{D}{d}$ has a value about 2% smaller (0.847 \ifmmode\pm\else\textpm\fi{} 0.01). However, a value of $\frac{5}{6}$ (0.833) is only just outside the 95% confidence limits and cannot be completely ruled out. In the two-dimensional simulations $\ensuremath{\beta}$ is insensitive to lattice details and in both two- and three-dimensional simulations $\ensuremath{\beta}$ is insensitive to the sticking coefficient ($S$) over the range $1.0\ensuremath{\ge}S\ensuremath{\ge}0.1$.