Modeling of surface vs. bulk ionic conductivity in fixed charge membranes

A two-region model for describing the conductivity of porous fixed charge membranes is proposed. In the surface region, the conductivity is due to the mobile positive ions (counterions) around the negative fixed charges. In the pore center region, the conductive properties resemble those of the external electrolyte solution because the fixed charges are assumed to be effectively neutralized by the counterions in the surface region. Activation energies and surface diffusion coefficients are estimated by assuming that the counterion jump from a fixed charge group is the rate limiting process for surface transport. The barrier energy for this jump is calculated using a simple electrostatic model with two microscopic parameters, the sum of the counterion and fixed charge hydration radii and the local dielectric constant. The bulk conductivity is obtained from experimental data. The total membrane conductivity and the counterion transport number are then calculated as functions of the external solution concentration for several pore radii and membrane fixed charge concentrations. The results are compared with those given by the Donnan model for homogeneous membranes and by the numerical solution of a continuous model based on the Poisson Boltzmann equation extended to finite size ions. The study of the membrane conductivity for a series of electrolytes allows to distinguish clearly between the mechanism characteristic of the bulk ionic conductivity and that of surface conductivity. The surface conductivity is found to be significant for narrow pores at low external solution concentrations.