Universal kinetics in reaction-limited aggregation.

Abstract

Reaction-limited cluster aggregation is modeled with the kinetic rate (Smoluchowski) equations, using a kernel determined intrinsically by the clusters fractal geometry. The kernel scales with cluster mass as ${\mathrm{M}}_{1}$${\mathrm{M}}_{2}^{\ensuremath{\lambda}\mathrm{\ensuremath{-}}1}$ (${\mathrm{M}}_{1}$\ensuremath{\gg}${\mathrm{M}}_{2}$), and ${\mathrm{M}}_{1}^{\ensuremath{\lambda}}$ (${\mathrm{M}}_{1}$\ensuremath{\approxeq}${\mathrm{M}}_{2}$), with \ensuremath{\lambda}=1 in three dimensions, resulting in exponential kinetics and a cluster mass distribution ${\mathrm{C}}_{\mathrm{M}}$\ensuremath{\sim}${\mathrm{M}}^{\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\tau}}}$, with \ensuremath{\tau}=(3/2), in excellent accord with experiments. The singular nature of this solution forces the adjustment of the cluster fractal dimension, ${\mathrm{d}}_{\mathrm{f}}$, thereby determining its value.