Boundary layer effects on the rate of diffusion-controlled reactions

The description using the Smoluchowski equation, on which the traditional theory of diffusion-controlled reactions is based, presupposes local equilibrium. Hence it must break down in the vicinity of an absorbing wall, where a kinetic boundary layer is formed. We explore in this paper the effects of this boundary layer on the reaction rate. We start from a description using the Klein-Kramers equation and use two approximate treatments of the boundary layer that gave satisfactory results for the analogous one-dimensional problem. The simplest one leads to a simple analytic correction factor in the Smoluchowski-Debye reaction rate that becomes appreciable for particle radii not very large compared to a typical length, the velocity persistence length. The correction factor always decreases the reaction rate. For constant of piecewise constant potentials between the reacting particles we also work out a somewhat more elaborate approximation, based on Grad's thirteen-moment equations. It yields effects of the same order of magnitude as the simple theory; the differences between the two approximate treatments should give an indication of the errors still contained in each of them.