The kinetic boundary layer around an absorbing sphere and the growth of small droplets

Deviations from the classical Smoluchowski expression for the growth rate of a droplet in a supersaturated vapor can be expected when the droplet radius is not large compared to the mean free path of a vapor molecule. The growth rate then depends significantly on the structure of the kinetic boundary layer around a sphere. We consider this kinetic boundary layer for a dilute system of Brownian particles. For this system a large class of boundary layer problems for a planar wall have been solved. We show how the spherical boundary layer can be treated by a perturbation expansion in the reciprocal droplet radius. In each order one has to solve a finite number ofplanar boundary layer problems. The first two corrections to the planar problem are calculated explicitly. For radii down to about two velocity persistence lengths (the analog of the mean free path for a Brownian particle) the successive approximations for the growth rate agree to within a few percent. A reasonable estimate of the growth rate for all radii can be obtained by extrapolating toward the exactly known value at zero radius. Kinetic boundary layer effects increase the time needed for growth from 0 to 10 (or 2 1/2) velocity persistence lengths by roughly 35% (or 175%).