A numerical experiment approach to modeling impedance: Application to study a Warburg-type spectrum in a membrane system with diffusion coefficients depending on concentration

Abstract

A new approach to theoretical description of low-frequency electrochemical impedance spectra is proposed. The method is based on a numerical experiment. First, a mathematical model simulating a time-dependent electrochemical process is developed. The calculations simulating real impedance measurements are carried out as follows. A current density is specified as a function of time: a direct current (DC) during the time needed to system to achieve a steady state, then as an alternative signal of a given frequency imposed on the DC bias applied from the beginning. The potential difference (PD) across the system as the response to the applied current is calculated, allowing the PD amplitude and the phase shift to be computed and obtaining the real and imaginary parts of the impedance. The method is applied to study the impedance of an ion-exchange membrane with two adjacent diffusion layers. A full agreement is found with the spectra calculated by applying the phasor representation of functions in the case where the electrolyte diffusion coefficient (D) is assumed constant. However, if the concentration dependence of D is taken into account, the phasor method is not so easy applicable, while there is no difficulty to introduce a D=D(c) dependence into the equations used in the “numerical experiment”. Applying this method for treatment of impedance experimental data leads to the conclusion that the difference between the diffusion layer thicknesses (δ) found from the experiment in the conditions of an imposed steady state DC and theoretically by applying the Lévêque equation is higher than it was believed when assuming D=const. As a consequence, the contribution of current-induced convection partially reducing δ and enhancing mass transfer in depleted solution near a membrane was underestimated earlier.