Nonconvective Ionic Flow in Fixed‐Charge Systems

We have considered the movement of an ion under the influence of Brownian motion, an externally applied constant electric field, and the spatially inhomogeneous potential set up by an assemblage of fixed charges. The resulting equation of motion is solved when the lattice of fixed charges is periodic, and when the fixed‐charge potential is small. Expressions are obtained for the macroscopic friction constant of the mobile ions as a function of the fixed‐charge potential, and, equivalently, for the relaxation field exerted by the fixed charges. The Einstein relation is used to obtain the macroscopic diffusion constant. The fixed‐charge potential is approximated by a periodic solution of the Poisson—Boltzmann equation in linearized form. The results are markedly dependent on the geometry of the fixed‐charge lattice. By varying the lattice parameters we present models for both polyelectrolyte solutions and simple ion‐exchange membranes, which give, in spite of the approximation of small fixed‐charge potential, semiquantitative agreement with data from self‐diffusion experiments on these systems.