The low-frequency limiting behavior of ambipolar diffusive models of impedance spectroscopy

Abstract

The Poisson–Nernst–Planck (PNP) diffusional model is a successful theoretical framework to investigate the electrochemical impedance response of insulators containing ionic impurities to an external ac stimulus. Apparent deviations of the experimental spectra from the predictions of the PNP model in the low frequency region are usually interpreted as an interfacial property. Here, we provide a rigorous mathematical analysis of the low-frequency limiting behavior of the model, analyzing the possible origin of these deviation related to bulk properties. The analysis points toward the necessity to consider a bulk effect connected with the difference in the diffusion coefficients of cations and anions (ambipolar diffusion). The ambipolar model does not continuously reach the behavior of the one mobile ion diffusion model when the difference in the mobility of the species vanishes, for a fixed frequency, in the cases of ohmic and adsorption–desorption boundary conditions. The analysis is devoted to the low frequency region, where the electrodes play a fundamental role in the response of the cell; thus, different boundary conditions, charged to mimic the non-blocking character of the electrodes, are considered. The new version of the boundary conditions in the limit in which one of the mobility is tending to zero is deduced. According to the analysis in the dc limit, the phenomenological parameters related to the electrodes are frequency dependent, indicating that the exchange of electric charge from the bulk to the external circuit, in the ohmic model, is related to a surface impedance, and not simply to an electric resistance.