Anomalous diffusion and electrical impedance response: Fractional operators with singular and non-singular kernels

Abstract

We revisit the electrochemical impedance spectroscopy problem in the framework of the Poisson-Nernst-Planck model for a semi-infinite electrolytic cell, using the Gouy-Chapman interface approximation. The drift-diffusion problem is formulated in terms of fractional differential operators having singular and non-singular kernels. These operators are considered as extending the diffusion equations related to the ions’ motion in the sample to the arbitrary order. The solutions to these equations are searched in the small AC linear approximation for the applied voltage. In this way as well, we show that the resulting electrical impedance, taking into account anomalous diffusion effects, naturally predicts a constant-phase elements behavior emerging from the physicochemical parameters, without considering equivalent circuits. The whole approach yields different, non-usual scenarios for the electrical impedance response of the cell, which may be connected to anomalous diffusion behavior of the mobile charges.