This chapter describes some analytical results obtained by means of a pioneering application of fractional diffusion equations to the electrochemical impedance technique employed to investigate properties of condensed matter samples. The first part of the chapter focuses on some basic aspects of the impedance spectroscopy and the continuum Poisson–Nernst–Planck (PNP) model governing the behaviour of mobile charges. In this model, the fundamental equations to be solved are the continuity equations for the positive and negative charge carriers coupled with Poisson's equation for the electric potential across the sample. The diffusion equation is then rewritten in terms of fractional time derivatives and the predictions of this new model are analysed, emphasising the low frequency behaviour of the impedance by means of analytical solutions. The model is reformulated with the introduction of the fractional equations of distributed order for the bulk system. As a step further, the proposition of a new model – the so-called PNPA model, where “A” stands for anomalous – is built by extending the use of fractional derivatives to the boundary conditions, stated in terms of an integro-differential expression governing the interfacial behaviour. Some experimental data are invoked just to test the robustness of the model in treating interfacial effects in the low frequency domain.

Impedance Spectroscopy: Preliminaries

The electrochemical impedance technique is used to investigate electrical properties of liquid materials [312]. The sample is submitted to an ac voltage of small amplitude to assure that its response to the external signal is linear. Thus, the impedance, Z(ω), is measured as a function of the frequency f = ω/2π of the applied voltage, V(t), with a typical amplitude V0. In the low frequency region, of particular importance is the role of the mobile ions regarding the value of the measured impedance because they contribute to the electrical current [152].

In this frequency region, the theoretical analysis of the influence of the ions on the electrical impedance is usually performed by solving the continuity equations for the positive and negative ions and the equation of Poisson for the actual electric potential across the sample. This is the so-called PNP model.