Application of the charge regulation model to transport of ions through hydrophilic membranes: one-dimensional transport model for narrow pores (nanofiltration).

The charge regulation concept is combined with the Navier-Stokes and Nernst-Planck equations to describe the ion retention of nanofiltration membranes consisting of narrow cylindrical pores. The charge regulation approach replaces the assumption of a constant charge or a constant potential at the membrane pore surface, and accounts for the influence of pH, salt concentration, and type of electrolyte on ion retention. In the current model, radial concentration and potential gradients are considered to be negligibly small (valid for narrow enough pores), resulting in a one-dimensional transport description. The model describes typical experimental data for nanofiltration membranes, such as the change of ion retention with pore radius, ion concentration, pH, and pressure both for monovalent and multivalent ions. For a constant solvent velocity, the model in some cases predicts an optimum pore size for retention. Nonequal retentions for anions and cations are predicted at low and high pH values, as well as a minimum solvent velocity for very low salt concentrations. For higher salt concentrations, and at a fixed pressure difference, an increase in solvent velocity with increasing ion concentrations is predicted, in agreement with other one-dimensional transport descriptions found in the literature, but in contrast to some experimental data.