Diffusion-controlled cluster formation in 2—6-dimensional space

Abstract

Diffusion-controlled cluster formation has been simulated on lattices of dimensionality 2-6. For the case of a sticking probability of 1.0 at nearest-neighbor sites, we find that the radius of gyration (${R}_{g}$) of the cluster is related to the number of particles ($N$) by ${R}_{g}\ensuremath{\sim}{N}^{\ensuremath{\beta}}$ (for large $N$). The exponent $\ensuremath{\beta}$ is given by $\frac{\ensuremath{\beta}\ensuremath{\sim}6}{5d}$, where $d$ is the classical (Euclidean) dimensionality of the lattice. These results indicate that the Hausdorff (fractal) dimensionality ($D$) is related to the Euclidean dimensionality ($d$) by $D\ensuremath{\approx}\frac{5d}{6}$ ($d=2\ensuremath{-}6$). Similar results can be obtained from the density-density correlation function in two-dimensional simulations. Nonlattice simulations have also been carried out in two- and three-dimensional space. The radius-of-gyration exponents ($\ensuremath{\beta}$) obtained from these simulations are essentially equal to those obtained in the lattice model simulations. We have also investigated the effects of sticking probabilities ($S$) less than 1.0 on diffusion-limited cluster formation on two- and three-dimensional lattices. While smaller sticking probabilities do lead to the formation of denser clusters, the radius-of-gyration exponents are insensitive to sticking coefficients over the range $0.1\ensuremath{\le}S\ensuremath{\le}1.0$.