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

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 $\mathrm{}\left(N\right)\mathrm{} R_g=N^{\beta }$. The corresponding Hausdorff dimensionality ($D=\frac{1}{\beta }$) is related to the Euclidean dimensionality $d$ by the relationship $D\sim \frac{5}{6}d$ for $d=3\mathrm{}$ and 4. For the two-dimensional case we find that $\frac{D}{d}$ has a value about 2% smaller (0.847 ± 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 $\beta$ is insensitive to lattice details and in both two- and three-dimensional simulations $\beta$ is insensitive to the sticking coefficient ($S$) over the range $1.0\ge S\ge 0.1$.