The very irregular state of motion observed by Robert Brown for small pollen grains suspended in water initiated one of the the most fascinating fields of science. The importance of such discovery – the so-called diffusion process – is immeasurable; it has been found in many contexts and is widespread in nature. A characteristic feature of this random motion is the linear growth with time exhibited by the mean square displacement, which is typical of a Markovian process. In contrast with this situation, a large class of systems and processes present a diffusion behaviour characterised by a nonlinear time dependence of the same quantity, thus constituting what is called anomalous diffusion behaviour.

The last decades have witnessed an increased interest in the anomalous diffusion processes that seem to be indeed present in a variety of experimental scenarios in physics, chemistry, biology, and several other branches of engineering; it is a rapidly growing field of research, attracting the attention of the scientific community. This happens from the theoretical side – due to the new mathematical problems evoked – but also from the point of view of experimental or practical applications. It is noteworthy that the number of studies reporting experimental problems dealing with anomalous diffusion has strongly increased – this attests to the ubiquity of a phenomenon initially considered a rare event.

The power of the mathematical tools based on fractional calculus, on the other hand, has also attracted the attention of the community working with pure and applied mathematics. The association of these techniques with the diffusional problem represents in practice a new field of research. It was shown in several ways that fractional calculus, if it is not unique, is nevertheless a suitable or even the natural mathematical framework to use to face the high complexity represented by anomalous diffusion phenomena. One powerful way of using these mathematical tools to analyse diffusion processes leads naturally to the necessity to search for solutions of fractional linear and nonlinear diffusion equations.

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].

In this chapter, we consider the influence of the surfaces or membranes on diffusive processes. The main aim is to investigate how the surface may modify the diffusive process of a system governed by a fractional diffusion equation.

In the first part of the chapter, we analyse the one-dimensional problem characterised by time-dependent boundary conditions, showing how they influence the diffusive process in the system for an arbitrary initial condition, i.e., the quantities related to the diffusion process, such as the first passage time, which may have an anomalous behaviour. A similar analysis is carried out for the two-dimensional case with inhomogeneous and time-dependent boundary conditions. These results show the potential of this formalism to analyse other physical scenarios, such as describing the molecular orientation and the anchoring problem in liquid crystals confined to a cylindrical region, taking into account the adsorption phenomena at the interfaces.

The second part of the chapter is dedicated to investigating situations in which the processes occurring on the surface are coupled to the bulk dynamics by means of the boundary conditions. As a first application, we consider a surface in which, besides the adsorption–desorption process, a reaction process may occur and the system presents anomalous diffusion behaviour. Another application refers to the transport through a membrane of definite thickness, for which the processes occurring on the surface also couple with the diffusion equations governing the bulk dynamics. In all cases, the system may exhibit an anomalous diffusive behaviour for which surface effects play a remarkable role.

1D and 2D Cases: Different Diffusive Regimes

Surface effects are present in a variety of real scenarios of interest in engineering [112, 136], biological systems [137], and physics [138, 139] as a fundamental feature of several processes. For example, industrial and biochemical reactions can have the reaction rate or the sorption of reagents limited by the mass transfer between the fluid phase and the catalyst surface. In biological systems [140], the surfaces (or membranes) are responsible for the selectivity of particles by means of sorption and desorption processes, and, consequently, the particles transfer from one region to the other. Other contexts can also be found in physics such as the electrical response of water [141] or liquid crystals [142] in which the effects of the interface between electrode and fluid play an important role.

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.

The preceding chapters dealt with the fractional diffusion equation with spatial and temporal fractional derivatives, diffusion coefficients with space and time dependencies, external forces, and surface effects in finite length situations. Remarkable consequences appear also when we consider the diffusion process in the presence of anisotropy.

To analyse the anisotropic case, we first face a problem in which suspended or dispersed particles diffuse through an anisotropic semi-infinite medium. The process is described in the framework of the usual diffusion equation, but anomalous diffusion behaviour arises in the system because the phenomenon of adsorption– desorption of particles occurs at the interface, and the conservation of the number of particles in the system has to be imposed.

The second problem is to consider a fractional diffusion equation subjected to an anisotropy, with a nonsingular spatial and temporal diffusion coefficient. We will show that the distribution governed by the equation is not separable in terms of space and time variables as in the usual diffusion, which is an unexpected behaviour since the fractional operator is linear. As a specific application, the chapter closes with the search for the solutions to the comb model with integer and fractional derivatives, and also with a drift term. This model is a simplified picture of highly disordered systems and can be connected with a rich class of diffusive processes due to geometric constraints.

The Adsorption–Desorption Process in Anisotropic Media

We consider first the diffusion problem in a semi-infinite anisotropic medium in contact with a solid substrate at which an adsorption–desorption process takes place [114, 210]. Initially, a defined number of particles is suspended or dispersed in the medium and an anisotropic diffusive process starts. The particles reaching the solid substrate can be adsorbed and desorbed in such a way that the kinetics of this process is governed by a typical balance equation characterising a chemical reaction of first kind (Langmuir approximation) as the one considered in Section 5.4. The conservation of the number of particles is then invoked and the profiles of the surface as well as of the bulk density of particles are analytically obtained by means of Laplace–Fourier techniques. The results for the momentum distribution show that the system exhibits anomalous diffusion [211] behaviour, according to the values of the characteristic times entering the problem.

This chapter starts with a brief history of the approaches to diffusion phenomena, by emphasising the first investigations of Brownian motion, i.e., stochastic motion, the random walk problem, and its connection with the diffusion processes. Subsequently, the concepts of anomalous diffusion and continuous-time random walk are introduced. Some formal aspects of the dynamics in normal and anomalous diffusion are presented. The link between these formalisms is established by introducing memory effects in the diffusion processes. In this enlarged scenario, non-Markovian behaviour and temporal memory are incorporated into the description of the diffusive processes in the presence of external fields, thus opening the whole approach to consider the possibility of application of fractional calculus.

Historical Perspectives on Diffusion Problems

The term diffusion comes from the Latin diffusio, diffusionem, connected with the verb diffundere, meaning “to scatter”, “to pour out”, and is formed by dis- “apart, in every direction” plus fundere “pour”. In physics, this term is applied to molecular diffusion, i.e., the random molecular motion by which matter is transported from places of higher to places of lower concentrations.

Pioneering Studies

The pioneering investigations of the diffusion process are usually attributed to the Scottish chemist Thomas Graham (1805–1869), who is also one of the founders of the Chemical Society of London and its first president (1841–1843). An important paper on gaseous diffusion appeared in 1829, in the Quarterly Journal of Science, under the title “A short account of experimental researches on the diffusion of gas through each other, and their separation by mechanical means”. The first lines of the article state [68]: