Theoretical study of electrolyte transport in nanofiltration membranes with constant surface potential/charge density

The pressure-driven electrolyte transport through nanofiltration membrane pores with constant surface potential or charge density is investigated theoretically. Two approaches are employed in the study. The first one is based on one-dimensional Nernst-Planck equation coupled with electroneutrality, zero current, and Donnan equilibrium conditions. This model is extended to account for interfacial effects by using a smooth approximation of step function for the volume charge density. The second approach is based on two-dimensional Nernst-Planck, Poisson, and Navier-Stokes equations, which are solved in a high aspect ratio nanopore connecting two reservoirs with much larger diameter. The modification of equations on the basis of Slotboom transformation is employed to speed up the convergence rate. The distributions of potential, pressure, ion concentrations and fluxes due to convection, diffusion, and migration in the nanopore and reservoirs are discussed and analyzed. It is found that for constant surface charge density, the convective flux of counter-ions in the nanopore is almost completely balanced by the opposite migration flux, while for constant surface potential, the convective flux is balanced by the opposite diffusion and migration fluxes. The co-ions in the nanopore are mainly transported by diffusion. A particular attention is focused on describing the interfacial effects at the nanopore entrance/exit. Detailed comparison between one- and two-dimensional models is performed in terms of rejection, pressure drop, and membrane potential dependence on the surface potential/charge density, volume flux, ion concentration, and pore radius. A good agreement between these models is found when the Debye length is smaller than the pore radius and the surface potential or charge density are sufficiently low.