Theoretical treatment and numerical simulation of potential and concentration profiles in extremely thin non-electroneutral membranes used for ion-selective electrodes

The applicability of extremely thin non-electroneutral membranes for ion-selective electrodes (ISEs) is investigated. A theoretical treatment of potential and concentration profiles in space-charge membranes of ≪1 μm thickness is presented. The theory is based on the Nernst–Planck equation for ion fluxes, which reduces to Boltzmann’s formula at equilibrium, and on the Poisson relationship between space-charge density and electric field gradient. A general solution in integral form is obtained for the potential function and the corresponding ion profiles at equilibrium. A series of explicit sub-solutions is derived for particular cases. Membrane systems with up to three different ion species are discussed, including trapped ionic sites and co-extracted ions. Solid-contacted thin membranes (without formation of aqueous films at the inner interface) are shown to exhibit a sub-Nernstian response. The theoretical results are confirmed by numerical simulations using a simplified finite-difference procedure based on the Nernst–Planck–Poisson model, which are shown to be in excellent agreement.