EIS simulations for ion-selective site-based membranes by a numerical solution of the coupled Nernst–Planck–Poisson equations

The numerical simulations of steady-state potentials and impedance spectra of ion-selective membranes electrodes (ISEs) with ionic sites are presented. The set of Nernst–Planck–Poisson and continuity equations is solved numerically by means of the finite difference method and the Rosenbrock solver. Transient and steady-state solutions for ion-selective electrodes under open- or closed-circuit conditions are calculated. The Fourier transformation of the potential-time response to a current perturbation is used for determination of complex impedances. The paper presents simulations of ISEs as a function of varying the ionic concentrations in the “bathing” solution or the rates of ion transfer across the membrane/solution interfaces. It is shown that the non-Nernstian behaviour of passive membrane electrodes is a result of kinetic constraints at the interfaces, which are manifested in the appearance of an additional arc between the high-frequency bulk and the low-frequency (Warburg) arcs. The presented approach directly relates the transport properties of the bulk and interfaces (ionic diffusivities and heterogeneous rate constants of transport across interfaces) to the characteristic features of complex impedances (dimensions and characteristic radial frequencies), and allows non-linear effects to be simulated.