# Electrostatic interactions and electrokinetic transport in ion-exchange membranes

Electrostatic interactions and electrokinetic transport in ion-exchange membranes have been investigated using analytical and computational methods. First, electrostatic interactions of electric double layers bounded by curved concave surfaces were studied using the Poisson-Boltzmann equation. Unlike the case of two charged parallel plates, the electric double layer force inside spherical cavities or cylindrical pores was predicted to diverge as surface potential goes to infinity. For typical values of surface potential, the double layer forces for curved surfaces were found to be significantly larger than the corresponding force between two parallel plates for most values of salt concentration and separation distance. It was found that at constant surface charge density, the double layer forces for curved surfaces decrease with increasing dielectric constant, opposite to the trend between two parallel plates. The dielectric constant also plays a special role in a force decomposition which led to a reformulation of the double-layer force for different geometries, as weighted averages of the hydrostatic pressures The electrokinetic transport of ions and water in ion-exchange membranes has been investigated using a new mathematical model. The model takes into account ion/fixed-charge site electrostatic interactions, water dipole orientation, ion hydration forces, and concentration dependent transport properties inside a membrane pore. Electric potential, solvent dielectric constant variations, water velocity, and cation and anion concentration profiles in the radial and axial pore directions were computed by solving Poisson's Equation, Booth's Equation, the Navier-Stokes Equations, and the Nernst-Planck Equations, respectively. A numerical technique, which can be applied to general cases of variable pore radius and wall charge density, has been devised by approximating the pore with small segments of straight cylinders that are then coupled through the Donnan relation and mass conservation equation. The computer model was tested by comparison with experimental data for Donnan dialysis separations with DuPont Nafion 117 cation-exchange membrane, where the membrane separates an aqueous acid solution from a mixture of monovalent/divalent cation salts. Both computer predictions and experimental measurements showed that monovalent ions with a larger hard sphere radius was selectively absorbed in and transported across the membrane in a multicomponent separation system. In addition to predicting ion transport for monovalent/monovalent and monovalent/divalent salt mixtures, the transport model also predicted accurately water transport through membranes for multicomponent component as well as for single salt systems. The multicomponent transport model was also applied to ion and water transport in ion-exchange membranes with non-uniform pore properties, where it was found theoretically that a variation in membrane pore radius changed significantly the water flux across the membrane, while the ion fluxes were relatively independent of pore size variation. Interesting ion transport, osmosis and electroosmosis behaviors in a multicomponent system have been observed and interpreted based on the transport model