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The approaches developed for studying the polarization of molecules and the dynamics of ions in dielectric materials are usually considered separately. The two effects are often believed to take place in different frequency ranges. The low frequency response is usually dominated by ionic migration, whereas the high frequency response is played by molecular polarization. The goal here is to clarify the interplay between free and bound charge densities and their influences on permittivity and impedance profiles by proposing a version of the Poisson–Nernst–Planck (PNP) model that allows to include the effect of a frequency-dependent (and thus not instantaneous) polarizability.

We present a theoretical analysis of the current–voltage characteristics of electrolytic cells using their electrical impedance response to a simple periodic external voltage of small amplitude. The examined cell is a thin-film electrolyte (solid, liquid, or gel) sandwiched between flat electrodes at which Ohmic conduction or adsorption–desorption phenomena occur. The basic equations of the Poisson–Nernst–Planck (PNP) model are analytically solved for the case of two diffusing mobile ions (ambipolar diffusion), yielding the charge densities in thermal equilibrium and the profile of the electric potential across the sample. Proper boundary conditions permit us to explore the influence of blocking, Ohmic and adsorbing-desorbing electrodes on the final I(V) curves. Remarkable, it is possible to establish an equivalence between the Ohmic and adsorption–desorption boundary conditions for a general set of physically relevant values of the parameters characterizing the electrodes. This equivalence is a natural way to further incorporate a frequency dispersion caused by the surfaces on the whole response of the cell. This makes the approach well-tailored to investigate the role of the different diffusion coefficients for cations and anions as well as the surface effects on the voltammetry measurements in insulators containing ionic impurities.

We investigate possible connections between two different implementations of the Poisson–Nernst–Planck (PNP) anomalous models used to analyze the electrical response of electrolytic cells. One of them is built in the framework of the fractional calculus and considers integro-differential boundary conditions also formulated by using fractional derivatives; the other one is an extension of the standard PNP model presented by Barsoukov and Macdonald, which can also be related to equivalent circuits containing constant phase elements (CPEs). Both extensions may be related to an anomalous diffusion with subdiffusive characteristics through the electrical conductivity and are able to describe the experimental data presented here. Furthermore, we apply the Bayesian inversion method to extract the parameter of interest in the analytical formulas of impedance. To resolve the corresponding inverse problem, we use the delayed-rejection adaptive-Metropolis algorithm (DRAM) in the context of Markov-chain Monte Carlo (MCMC) algorithms to find the posterior distributions of the parameter and the corresponding confidence intervals.

We analyze the electrical impedance response established in terms of the time-fractional approach formulation of the Poisson-Nernst-Planck model by considering a general boundary condition. The total current across the sample is solenoidal, as the Maxwell equations require, and the boundary conditions can be related to different scenarios. We also study the behavior of the electrical conductivity and its connection with the ionic spreading in the sample to establish the diffusion regime present in the system in the low-frequency limit. Furthermore, we investigate impedance spectroscopy measurements of a nematic liquid crystal cell and observe a good agreement between the experimental data and the model.

A connection between the impedance spectroscopy response of an anomalous Poisson–Nernst–Planck (PNPA) diffusional model and of equivalent circuits containing constant phase elements (CPEs) is established for a typical electrolytic cell. The analysis is carried out in the limit of low frequency in order to highlight the surface effects and to explore how they can be connected to the presence of CPEs in the circuit. It is shown that, depending on the choice of the equivalent circuit, the action of these elements can be the same as the one obtained using integro–differential boundary conditions to describe anomalous diffusive processes in the framework of PNPA models. The predictions are also compared with experimental data obtained from an electrolytic solution.

The electrical impedance data of different nematic liquid-crystal cells are analyzed in the framework of a model in which the diffusion of mobile ions in the bulk is governed by a fractional diffusion equation of distributed order. The boundary conditions at the electrodes limiting the sample are described by an integro-differential equation governing the kinetic at the interface that embodies, in particular, the usual kinetic equation for describing the adsorption–desorption process at the electrodes but is expressed in terms of a temporal kernel that can be chosen to cover scenarios that are not suitably described within the usual framework of blocking electrodes. The analysis is carried out by supposing that the positive and negative ions have the same mobility and that the electric potential profile across the sample satisfies the Poisson’s equation. The results cover a rich variety of scenarios, including the ones connected to anomalous diffusion.