Analysis of electrolyte transport through charged nanopores

We revisit the classical problem of the flow of an electrolyte solution through charged capillaries (nanopores). In the limit where the length of the capillary is much larger than its radius, the problem can be simplified to a one-dimensional averaged flux-force formalism that relates the relevant fluxes (electrical current, salt flux, fluid velocity) to their respective driving forces (difference in electric potential, salt concentration, pressure). Calculations in literature mainly consider the limit of non-overlapping electrical double layers (EDLs) in the pores and the absence of salt concentration gradients in the axial direction. In the present work these simplifications are relaxed and we discuss the general case with overlapping EDLs and nonzero axial salt concentration gradients. The 3x3 matrix that relates these quantities exhibits Onsager symmetry and for one of the cross coefficients we report a new significant simplification. We describe how Onsager symmetry is preserved under change of variables which we illustrate by one example of a different flux-force matrix given by Gross and Osterle (1968). The model is well-suited to physically represent membranes consisting of charged nanopores for electrokinetic energy conversion and water desalination. We analyse the energy conversion of a salt concentration difference into electrical power using an efficiency vs. power diagram. Because of the non-zero axial gradient in salt concentration that we allow in our calculations, under wide ranges of conditions, partial loops in current, salt flux or fluid flow are predicted in the pore