Smoluchowski fluctuation theory of ion transport in solutions

Abstract

We developed a Smoluchowski-level theory of time-dependent ion concentration fluctuations at equilibrium, including higher-order correlations, but with hydrodynamic interactions suppressed. A partial sum over the higher-order correlations, a chain sum, eliminates divergences. From the resulting renormalized self-van Hove function we recover Onsager’s limiting law for the self-diffusion coefficients of the ions. The limiting law comes from the chain sums that are neglected in a ‘‘linearization’’ approximation used in some earlier work. The corresponding development for the distinct van Hove functions is merely indicated. We apply these results to a solution of a single spherical polyion with many small ions, to obtain the self-diffusion coefficient for the polyion in near-limiting law conditions, and to simple models for aqueous NaCl and CuSO4 solutions, to calculate the self-diffusion coefficients of the ions and the Maxwell (Debye–Falkenhagen) relaxation, all in the concentration range up to 1 M.