Frequency dispersion in the fractional Langmuir approximation for the adsorption–desorption phenomena

1. Introduction

To investigate the electrochemical impedance spectroscopy response of a liquid material, usually an AC voltage of small amplitude, V₀, and variable frequency, f = ω/2π, is applied to the sample [1]. This permits one to measure the impedance, Z(ω), for a large interval of frequency, as well as to monitor the frequency dependence of other parameters used to electrically characterize the sample. In the low-frequency region, the role of the mobile ions on the measured impedance is normally conspicuous [2]. To account for the influence of these ions, a useful theoretical analysis is performed in the framework of a continuous model in which the continuity equations for the positive and negative ions are solved simultaneously with the equation of Poisson for the actual electric potential across the sample. This is the so-called Poisson–Nernst–Planck model [3]. Actually, the whole response of the electrolytic cell strongly depends on the role played by the electrodes limiting it [4]. In a few words, the theory consists mainly in the mathematical solution of these differential equations for various boundary and initial conditions.

These boundary conditions have to be judiciously chosen in order to encompass the relevant interfacial effects occurring in the system. What is more, among these surface effects, the role of blocking and non-blocking electrodes has been invoked to shed some light on the problem [5]. Both situations can be analysed by considering the adsorption process in which some memory effects could play a relevant role [6,7]. Another possible interfacial mechanism may be the accumulation of ions near the surface, both in the absence or in the presence of adsorption phenomena. In some cases, this accumulation should be connected with the necessity to modify Fick’s Law and to solve a modified diffusion equation. In this regard, fractional diffusion equations together with fractional derivative approaches have been considered as an appropriated theoretical framework to take into account a similar class of effects on the electrical impedance [8]. Several pioneering works, dedicated to addressing the electrochemical impedance by means of fractional calculus, provided convincing indication of the necessity to extend the use of fractional derivatives to the boundary conditions [9–14].

Anyway, there seems to be no doubt that the adsorption phenomenon has a central role in the electrochemical impedance response [2], but it cannot be considered alone to account for the features found in the experimental systems. Indeed, the influence of a random distribution of potential energy connected with the adsorption phenomenon on the effective adsorption coefficient and desorption time in the context of the Langmuir kinetic equation has been recently investigated [15]. The analysis has been carried out by invoking a homogeneous distribution of relaxation times occurring at the adsorbing electrodes when Debye relaxation processes are considered. This procedure allows for a generalization of the concept of Langmuir isotherm, yielding a dependence on the frequency for the phenomenological parameters entering the kinetic equation, in good agreement with the experimental data. A different generalization of the kinetic equation may also be implemented aiming at a description of non-usual or non-Debye relaxation phenomena, if the kinetic equation contains a fractional time derivative and a temporal kernel associated with the desorption rate, which in turn may be connected with memory effects [16]. The presence of the fractional time derivative naturally produces an anomalous relaxation. The two approaches are not mutually exclusive but, on the contrary, they can be shown to represent valid and alternative ways to tackle the complex phenomena occurring at the interfaces of these media.

We shall show here that the main results for the electric response of an electrolytic cell, in which a distribution of relaxation times characterizes the interfacial behaviour of the system, can be mathematically faced using the tools of fractional calculus. To accomplish this task, we use a generalization of the kinetic balance equation of the Langmuir approximation expressed in terms of a time derivative of fractional order. One of the motivations to treat the problem in this way is to consider that, in the adsorption of ions at a solid electrode, an elastic scattering occurs when there is no loss of translational energy during the collision; however, if the ion is still in a weakly bound state, even if it is on the surface, the thermal motion of the surface atoms can cause the ion to desorb. In the sequence, when the ion collides with a surface, it loses energy and is converted into a state where it remains on the surface for a reasonable time, i.e. it is physically adsorbed. As a consequence, the actual position of the ion on the electrode may have a memory of its incoming state, eventually modifying the adsorption–desorption rates [16]. These effects may also be connected to the roughness of the surface, which in the low-frequency limit leads us to obtain Y = 1/Z (ω) ∝ (iω)^γ, where Y denotes the admittance and γ is a parameter connected to the roughness of the surface [17–20] and may also be connected with the order of the fractional time derivative. In a similar manner, but using a different perspective, we may consider that the adsorption–desorption mechanism at the interface could be handled in such a way as if the spatial inhomogeneities of the surfaces were responsible for the presence of different relaxation processes across the whole interface. To account for these processes, a random distribution of relaxation times in the usual Debye sense could be incorporated into the Langmuir approach for the adsorption–desorption phenomenon. This analysis is similar to the one proposed for the conduction in disordered solids, employing the concept of diffusion time distribution [21]. In §2, we recall the main findings of this kind of approach, pointing towards possible extensions of the kinetic equation by modifying the kernel of the integral equation governing the surface density of particles. In §3, we discuss a simple extension of the kinetic equation using fractional derivatives in order to underline the natural incorporation of a frequency dispersion in the phenomenological parameters. A discussion of the results accompanied by a comparison between the predictions of the approaches presented in the preceding sections are the main issues of §4. Some concluding remarks together with some perspectives for immediate physical applications of this powerful formalism are briefly sketched in §5.

2. Kinetic equation: distribution of relaxation times

In the usual Langmuir approximation for the adsorption–desorption phenomena occurring at the interface of two different media (e.g. a solid phase representing the surfaces in contact with a liquid phase representing the bulk), the kinetic equation usually states that the time variation of the surface density of adsorbed particles at a given surface, σ = σ(t), depends on the bulk density of particles just in front of this surface, n(t), and on the surface density of particles already sorbed. This can be represented by the simple balance equation in the time domain

in which κ and τ are the adsorption coefficient and the desorption time, respectively. This equation is valid in the limit in which the density of adsorbed particles is very small with respect to the number of adsorbing sites at the surface. In the Laplace domain, the solution for this equation can be found; it is given by If we assume that the initial surface density of adsorbed particles is zero, i.e. σ(t ≤ 0) = 0, for s = iω, equation (2.2) yields the solution in the frequency domain for a harmonic regime. By performing the inverse Laplace transform or a direct integration of equation (2.1), the previous equation in the time domain can be formally written as follows: We note in addition that, after some calculations, equation (2.3) can be written as indicating that the effective density of particles close to the surface is formed by two terms. The first term is connected with the adsorption of the particles in the first layer close to the surface; the second term accounts for the desorption whose relaxation times follow a simple exponential behaviour. This is nothing but an integral version of the kinetic equation, equation (2.1). By rewriting it in this way, we may also identify the last term of the right side as an integrodifferential operator, which permits us to put equation (2.4) in the form Equation (2.5) may be seen as an integrodifferential operator, whose kernel in the differential operator is of the exponential type. A detailed discussion about the integrodifferential operators’ memory effects and its connection with fractional calculus may be found in [22–26]. The presence of a kernel of exponential type is expected here because it is related to the type of relaxation manifested in equation (2.1), which is essentially a Debye relaxation. However, other types of relaxations [27–29] may be associated with different operators, if one observes that the solution given by equation (2.4) may be formally written as [15] where the quantity ϕ(u) = (1/τ) exp ( − u/τ) plays the role of a response function related to the differential operator present in the relaxation equation. A straightforward generalization of equation (2.6) may be stated as in which K = κτ represents a typical adsorption length connected with the range of the effective forces responsible for the adsorption process [30], and ϕ(u) may have a very general profile, where different types of relaxation processes could be considered. The fractional relaxation equation will be treated in the next section. Now, we analyse how the superposition of Debye relaxation processes may be used to obtain non-usual response functions with experimental data.

In Fourier’s space, equation (2.7) may be put in the form

in which If the relaxation function is assumed as being formed by a superposition of simple relaxation phenomena, a well-defined profile for the relaxation function could be as follows: in which G(τ_r) represents a possible distribution function of relaxation times. The normalization condition requires that The Fourier transform of the relaxation function (2.10) is The expression (2.12) represents a possible generalization of the kinetic equation (2.7) in the Fourier space incorporating a different distribution of relaxation times. A simple generalization of this kind opens the possibility to analyse the role of the interfacial behaviour on the bulk response of the sample in the framework of the impedance spectroscopy technique. In particular, it permits to incorporate a frequency dependence on the effective parameters characterizing the adsorption–desorption phenomena in a simple and direct way. This can be easily checked because, if we put Φ(ω) = R(ω) − i I(ω), having in mind equation (2.2) (with s = iω), the effective adsorption coefficients, κ_e and τ_e, may be defined in general by the condition which yields These general forms permit one to expect that κ_e = κ_e(ω) and τ_e = τ_e(ω), as we demonstrate in the three illustrative and physically relevant cases we shall consider below.

Let us consider first the simple case in which G(τ_r) is given by the following distribution [15]:

This distribution predicts the existence of a cut-off in the relaxation time, τ_M, evidencing that the value of the relaxation time has an upper bound. This is similar to what happens in the Debye theory of specific heat in the frequency domain [31]. In this case, from equation (2.12), we obtain

Another relevant case is the normalized Gaussian function of width $\mathrm{\Delta }\tau ={\tau }_{\mathrm{M}}/\sqrt{2}$, accounting for the limiting behaviour of a random distribution of relaxation times across the surface undergoing the adsorption–desorption phenomenon, in the form

In this case, the formal solution from equation (2.12) is found to be in which are, respectively, the error and the incomplete gamma functions [8]. We may also consider other possibilities for the distribution of the relaxation times, in particular, the one-side Lévy distributions, which are characterized asymptotically by power laws [29].

For the simplest case represented by equation (2.16), closed expressions are easily obtained from equation (2.13), in which the frequency dependence can be analytically handled by means of simple calculations. These are

From these expressions, using equation (2.14), we eventually obtain and As it is evident from equations (2.21) and (2.22), if the relaxation is due to different mechanisms, the effective adsorption parameters depend on the frequency: there is dispersion. The same feature is expected when the surface is inhomogeneous, in the sense that a geometrical non-homogeneity may be responsible for a dispersion in the adsorption parameters. As pointed out above, such inhomogeneities may be connected to the roughness of the surface or may result from another kind of failure influencing its effective dimension. In this perspective, these spatial inhomogeneities of the surfaces may be responsible for the presence of different relaxation processes across the whole interface. For the processes governed by equation (2.18), which are related to a Gaussian relaxation, it is possible to show that the real and imaginary parts are defined as follows: and By using the previous equations, we get and Similarly to equations (2.21) and (2.22), even in this case there is dispersion in τ_e and κ_e.

3. Fractional kinetic equation

A possible different way to tackle this complex behaviour, in which the effective adsorption parameters incorporate a frequency dependence, is to generalize the kinetic equation by using fractional calculus. A very useful procedure is to assume that the adsorption–desorption phenomenon is governed by a general kinetic equation, written in terms of a fractional derivative of order γ < 1, in the form

where τ₀ is an intrinsic time, connected with an internal dynamics. If γ = 1, equation (3.1) obviously reduces to equation (2.1). To solve the problem in the time domain, we may choose to represent the fractional operator in equation (3.1) as Caputo’s one [32], defined as which reduces to the usual derivative if γ = n, with n an integer and non-negative. This fractional operator has been applied throughly in physics because it makes the process of working with differential equations of fractional order similar to the method employed for usual differential equations. Its role in the investigation of the relaxation processes is discussed in detail in [33]. This fact is connected with the existence of integration constants in the process of solution [8]. A formal solution of a problem represented by a generalized kinetic equation as in equation (3.1) may be found in terms of the H-function of Fox or the two-parameter Mittag-Leffler function [34,35]. Let us consider this problem for a moment, using the common notation for the fractional operator [36] For simplicity, we assume τ = τ₀, and rewrite the kinetic equation (3.1) as Saxena et al. [34] consider, instead, the following fractional integral kinetic equation of order γ where τ > 0, γ > 0 and n(t) is an integrable function in the interval [0, b], where b > 0. In equation (3.5), the Riemann–Liouville fractional integral of order γ is defined as For the integral equation (3.5), the solution found is where ${\mathrm{H}}_{1,2}^{1,1}(\cdots )$ is the H-function of Fox [37]. In the case we are considering here, γ < 1. This means that the integral appearing in equation (3.6), does not diverge if σ(0) = 0 as required by the initial conditions. In this case, the solution of our problem for the fractional differential kinetic equation may be obtained from equation (3.7) just by changing γ to −γ, namely In addition, this particular H-function of Fox may be connected with the two- or three-parameter Mittag–Leffler function and the problem is formally solved. These general mathematical aspects of the problem concerning a fractional kinetic equation and its formal solutions deserve a specific treatment and will be considered elsewhere in order to achieve a more general formulation of the problem.

Here, for simplicity, we come back to equation (3.1), with τ₀ ≠ τ, to immediately explore the physical implications of the fractional equation, for what concerns the incorporation of a frequency dependence in the phenomenological parameters characterizing the adsorption–desorption phenomenon. We move ourselves to the frequency domain where the problem is easier to tackle. This procedure allows for a direct comparison of its predictions with the results obtained in the preceding section using the same simplified treatment. Indeed, by means of calculations entirely similar to the ones carried out before, equation (3.1) may be cast in the form

To obtain (3.10), we have used the familiar property of the Fourier transform of a derivative of integer order n, which also holds for a fractional derivative of order γ, namely It follows that the Fourier transform of the relaxation function now becomes in which α = cos (γπ/2) and β = sin (γπ/2). This is the fractional extension of the equation (2.12). To explore the analogy between the two approaches, we consider again the case corresponding to equation (2.16). If we put, as before, Φ = R − i I, we easily obtain and In this case, the effective phenomenological parameters are given by From equations (3.15), it follows that a Langmuir isotherm of fractional order may be considered as equivalent to a dispersion of the effective adsorption parameters. Indeed, the frequency dependence of these parameters is naturally accounted for in this generalized formalism. In addition, it is possible to show that, by means of a proper choice of γ and τ₀, the fitting process with κ_e and τ_e may be put in correspondence with the fitting with κ_eff and τ_eff. This result indicates that the problem formulated in terms of a kinetic equation with a derivative of fractional order may be an alternative way to describe the adsorption–desorption phenomena using a distribution of relaxation times in the framework of a Debye relaxation process. In addition, the generalized formalism permits one to consider different relaxation regimes, e.g. non-Debye ones, expressed in terms of non-exponential relaxation functions or accounting for a power-law behaviour. This conclusion holds in general, because the H-function of Fox (fractional case) is asymptotically governed by a power-law behaviour, in contrast to the behaviour found in the non-fractional case, which is typically characterized by an exponential relaxation.

4. Results

Let us explore in simple illustrative terms the potentialities of both kinetic approaches discussed before to account for the frequency dispersion of the phenomenological parameters entering the kinetic equation. In figure 1, the behaviour of the effective relaxation time is shown as a function of the frequency for equations (2.21), (2.25) and (3.15). We note that the curves exhibit similar behaviour in the high-frequency region, where the effective relaxation time tends to vanish, whereas for low frequency, there is net discrepancy between the expressions. Indeed, while equation (3.15) exhibits a general tendency to increase, equations (2.21) and (2.25) tend to a plateau when ω → 0. An opposite behaviour is found for the trends of κ_eff, shown in figure 2. In this case, the agreement between the trends is good for intermediate values of frequency, but the differences become more conspicuous in the high as well as in the very low-frequency region. Again, we note that for very high frequency, all the approaches predict a monotonic increasing of the effective parameters, but while the predictions of equations (2.22) and (2.26) are similar, equation (3.15) predicts a more rapid increasing rate.

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The trends of κ_eff and τ_eff can also be analysed in another perspective. For instance, in figure 3, τ_eff versus κ_eff is shown for all the approaches and the profiles of the curves practically coincide in a wide range of physically meaningful values of the phenomenological parameters. In addition, in figure 4 the frequency behaviour of the quantity K_eff = κ_effτ_eff is shown. We recall that this quantity has the dimensions of a length and may be interpreted as the effective thickness near to the surfaces where the adsorption–desorption process takes place, i.e. it is probably connected with the ranges of the effective forces governing the process. In this perspective, the results exhibited in figure 4 reveal first of all that the behaviour of this quantity is non-monotonic; further, they reveal that, even if the curves are distinct for both kinetic approaches, they account in a similar way for the interfacial behaviour of the system, i.e. they exhibit a similar qualitative profile. In the high-frequency domain, this thickness tends to zero, indicating that the adsorption–desorption process involves only the particles that are present in a very small region close to the surface, and the process is highly localized. As the frequency decreases, this thickness changes drastically and, in the very low-frequency region, it becomes high and tends to a limiting value, different in the three cases. This similar qualitative behaviour permits us to conclude that the number of particles involved in the adsorption process in the low-frequency region is higher than the one in the high-frequency region. This is an expected result if these particles are mobile charges, whose response is always more prominent in the low-frequency than in the high-frequency region. What is more, the results of figure 4 indicate that this is true in general, and that this kind of response is strongly driven by the surface.

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5. Concluding remarks

We consider the adsorption–desorption process governed by a kinetic equation in the Langmuir perspective, i.e. the actual time dependence of the surface density of particles is formed by the difference between the density of adsorbed particles in the bulk close to the surface and the density of desorbed particles at a certain rate, characterized by a relaxation time, τ. The analysis focuses on two approaches; the first one uses a distribution of relaxation times in the context of a Debye relaxation process occurring at the interface. In this case, the kinetic equation is the one usually employed in the Langmuir approach, in the presence of two different distributions of the relaxation times. A noteworthy result arises in this way, showing a frequency dispersion in the phenomenological parameters that effectively intervene in the process, namely, κ_eff and τ_eff; the second approach formulates the problem in terms of a kinetic equation with a derivative of fractional order as a way to generalize the mathematical description of the adsorption process at the interface. Using simplified calculations in order to make the physical meaning of the results as clear as possible, it is shown that even this approach is responsible for the presence of a frequency dispersion in the values of the parameters effectively governing the process. A careful comparison of the trends exhibited by κ_eff and τ_eff are a good indication that using a fractional kinetic equation may be an efficient way to describe the adsorption–desorption phenomena incorporating the predictions coming from a distribution of relaxation times in the framework of a Debye relaxation process.

The comparison outlined above indicates that both methods are potentially useful to explain the frequency dependence of the parameters connected with the adsorption–desorption phenomena in the framework of the Langmuir approach. Their predictions coincide for a significant range of frequency values, pointing towards the necessity to consider a more complex underlying physics at the interfaces (random distribution of relaxation times, effective fractal dimensions of the surfaces, roughness, etc.). This can be done by considering fractional extensions of the kinetic equation as well as more general forms of the function G(τ), in equation (2.10), in order to cover the high complexity of the interfacial phenomena in these complex systems. An analysis along these lines is mandatory and the experimental results surely will be decisive to indicate the more appropriate way to tackle this important class of problems.

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Authors' contributions

G.B., L.R.E. and E.K.L. conceived of the study, designed the study and coordinated the study. G.B. and L.R.E. helped draft the manuscript. E.K.L. checked the calculations and made all the figures. All authors gave final approval for publication and agree to be held accountable for the work performed therein.

Competing interests

We declare we have no competing interests.

Funding

This work was supported by the MEPhI Academic Excellence Project (agreement with the Ministry of Education and Science of the Russian Federation of 27 August 2013, project no. 02.a03.21.0005) (G.B.) and by the Program of Visiting Professor of Politecnico di Torino (L.R.E.). E.K.L. and L.R.E. also thank the CNPq (Brazilian agency) for partial financial support.