Solution of the time-dependent Onsager problem

Abstract

We obtain the Green’s function solution of the Smoluchowski equation with a Coulomb potential and an electric field, corresponding to a general boundary condition at the origin. In the low-field limit the solution exhibits diffusive behavior for long times, but above a critical field the long-time decay becomes purely exponential. We derive expressions for the time evolution of quantities such as the rate of geminate recombination and the survival probability, and for t→∞, a→0 (where a is the radius of a perfectly absorbing sphere at the origin) we recover Onsager’s results. We consider a variety of initial conditions, such as a δ function, isotropic and constant distributions, as well as a more complicated distribution applicable to fluorescence quenching in the presence of an electric field. For small values of the electric field and the Laplace transform variable we obtain an expression for the probability of a scavenging reaction taking place in competition with geminate recombination. Using the prescribed diffusion approximation, we also obtain a simple expression for the time dependence of the survival probability, valid for an isotropic initial distribution, and we discuss the results of the numerical evaluation of various quantities of physical interest. Throughout the paper we have obtained parallel results for both Coulomb attraction and repulsion between the pair of charges.