This chapter presents the pathway towards the construction of Poisson–Nernst– Planck anomalous (PNPA) models proposed to connect the anomalous diffusion phenomenon with the impedance spectroscopy. The first part is dedicated to analysing the conceptual links between a PNPA model and equivalent electrical circuits containing constant-phase elements (CPEs) in the low frequency domain. It is demonstrated, on analytical grounds, that the effect of a CPE in an equivalent electrical circuit may be represented by an appropriate term added to the boundary conditions of PNP or PNPA models. The second part recalls the fundamental equations of the PNPA models, including also reaction terms and Ohmic boundary conditions to account for more complex bulk and interfacial behaviour. It is shown that the formulation based on the fractional diffusion equations establishes on general theoretical grounds a connection between the PNPA models with an entire framework of continuum models and equivalent circuits with CPEs to analyse impedance data.

PNPA Models and Equivalent Circuits

As we have seen in Chapter 9, the continuum models frequently used to analyse the data are essentially based on diffusion-like equations for the ions, satisfying the Poisson's equation requirement for the electric potential (Poisson–Nernst–Planck or PNP model), or on equivalent electrical circuits [312]. There are various distributed circuit elements that can be incorporated into equivalent circuits [342]. However, a careful analysis is necessary before reaching to general conclusions about the data, since the incorrect choice of the equivalent circuit can lead to deceptive conclusions about the process that occurs in the sample [343].

Even more powerful and useful general models, such as ordinary (PNP) or anomalous diffusion (PNPA) ones, are not free from ambiguities [344]. On one hand, the PNPA models aim at incorporating behaviours that may not be well described in terms of usual diffusive PNP models. On the other hand, an important extension used in the framework of equivalent circuits is the CPE, whose presence can be connected with the necessity of describing unusual effects in many solid electrode–electrolyte interfaces. For instance, it has been pointed out that simple elements cannot describe frequency dispersion often found in the solid electrode– electrolyte interfacial region [345]. This behaviour can be related to surface disorder and roughness [346–349], electrode porosity [350], and electrode geometry [351].