Kinetics of Deposition in the Diffusion-Controlled Limit

The adsorption of particles diffusing in a half-space bounded by the substrate and irreversibly sticking to the substrate upon contacts is investigated. We show that when absorbing particles are planar disks diffusing in the three-dimensional half-space, the coverage approaches its saturated jamming value as $t^{-1}$ in the large time limit [generally as $t^{-1/(d-1)}$ when the substrate is $d$ dimensional and $d>1$, and as $e^{-t/\ln(t)}$ when $d=1$]. We also analyze the asymptotic behavior when particles are spherical and when particles are planar aligned squares.