Incorporating ionic size in the transport equations for charged nanopores

Nanopores with fixed charges show ionic selectivity because of the high surface potential and the small pore radius. In this limit, the size of the ions could no longer be ignored because they occupy a significant fraction of the pore and, in addition, they would reach unrealistic concentrations at the surface if treated as point charges. However, most models of selectivity assume point ions and ignore this fact. Although this approach shows the essential qualitative trends of the problem, it is not strictly valid for high surface potentials and low nanopore radii, which is just the case where a high ionic selectivity should be expected. We consider the effect of ion size on the electrical double layer within a charged cylindrical nanopore using an extended Poisson-Boltzmann equation, paying special attention to (non-equilibrium) transport properties such as the streaming potential, the counter-ion transport number, and the electrical conductance. The first two quantities are related to the nanopore selectivity while the third one characterizes the conductive properties. We discuss the nanopore characteristics in terms of the ratio between the electrolyte and fixed charge concentrations and the ratio between the ionic and nanopore radii showing the experimental range where the point ion model can still be useful. Even for relatively small inorganic ions at intermediate concentrations, ion size effects could be significant for a quantitative estimation of the nanopore selectivity in the case of high surface charge densities.