Origin of resonant electrical impedances in membranes induced by osmosis: Analytical solutions of the AC Nernst–Planck, Poisson and continuity equations as functions of water velocity

Analytical solutions of the Nernst–Planck, Poisson and continuity equations for a membrane undergoing osmosis in a cross-flow system reveal that the flow of alternating ionic charge induced in the membrane during impedance measurements is actively assisted by the flow of water. The solutions yield expressions for the impedance of the membrane that characterize the dielectric, conduction and geometrical properties (Maxwell–Wagner model) as well as the actively assisted contributions that are functions of the velocity of water and the electrical field values at the membrane boundaries. The consistency of this theory with experimental results reported in literature provides a basis for interpreting in situ impedance characterizations of membrane systems, and in particular the viability of the membranes and the extent of fouling during reverse osmosis