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We present an alternative representation of integer and fractional electrical elements in the Laplace domain for modeling electrochemical systems represented by equivalent electrical circuits. The fractional derivatives considered are of Caputo and Caputo-Fabrizio type. This representation includes distributed elements of the Cole model type. In addition to maintaining consistency in adjusted electrical parameters, a detailed methodology is proposed to build the equivalent circuits. Illustrative examples are given and the Nyquist and Bode graphs are obtained from the numerical simulation of the corresponding transfer functions using arbitrary electrical parameters in order to illustrate the methodology. The advantage of our representation appears according to the comparison between our model and models presented in the paper, which are not physically acceptable due to the dimensional incompatibility. The Markovian nature of the models is recovered when the order of the fractional derivatives is equal to 1.

Electrochemical Impedance Spectroscopy (EIS) is widely used to investigate the interfacial and bulk properties of materials, interfaces of electrode-electrolyte, and the interpretation of phenomena such as electrocatalysis, corrosion, or behavior of coatings on metallic substrates. This technique relates directly measurements of impedance and phase angle as functions of frequency, voltage, or current applied. The stimulus is an alternating current signal of low amplitude intended to measure the electric field or potential difference generated between different parts of the sample. The relationship between the data of the applied stimulus and the response obtained as a function of frequency provides the impedance spectrum of samples studied [

Fractional calculus (FC) is the investigation and treatment of mathematical models in terms of derivatives and integrals of arbitrary order [

Measurements of properties of materials, interfaces of electrode-electrolyte, corrosion, tissue properties of protein fibers, semiconductors and solid-state devices, fuel cells, sensors, batteries, electrochemical capacitors, coatings, and electrochromic materials have shown that their impedance behavior can only be modeled by using Warburg elements in conjunction with resistors, dispersive inductors, or CPE elements [

Unlike the work of the authors mentioned above, in which the pass from an ordinary derivative to a fractional one is direct, Gómez-Aguilar et al. in [

The paper is organized as follows: Section

The use of Caputo Fractional Derivative (CD) in Physics is gaining importance because of the specific properties: the derivative of a constant is zero and the initial conditions for the fractional order differential equations can be given in the same manner as for the ordinary differential equations with a known physical interpretation [

The CD is defined as follows [

The Laplace transform of the CD has the following form [

The Caputo-Fabrizio fractional derivative (CF) is defined as follows [

If

The Laplace transform of (

From this expression we have

Regarding the equivalent circuits, there is a diversity of models used. The most common is to adjust the system to a simple model or one that includes a bilayer structure, either with RC groups in parallel or in series model, although there are cases where it is appropriate to include the Warburg impedance element to consider possible diffusive processes on the surface. Generally, using a complex equivalent circuit is not necessary to obtain a good characterization of the real system; commonly a simple electrical circuit is the first choice and increases its complexity when knowledge of the electrochemical behavior of the system also increases.

The Cole impedance model is based on replacing the ideal capacitor in the Debye model [

On electric structures RC type, bias resistor

In electrochemical systems the diffusion can create an impedance called Warburg impedance [

Equation (

One of the problems of the fractional representation is the correct sizing of the physical parameters involved in the differential equation, to be consistent with dimensionality and following [

The model of polarizable electrode, also known as faradaic reaction, provides a simple description of the impedance of an electrochemical reaction on electrode surfaces. The equivalent circuit is represented by Figure

Equivalent electrical circuit for the polarizable electrode model.

Considering initial conditions equal to zero, the equivalent impedance is found by the following equation in the complex frequency domain:

Consider (

Consider that the values of the parameters of the circuit shown in Figure

Nyquist and Bode diagram for the model of polarizable electrode, Caputo derivative approach, in (a) and (b):

Now consider that the values of the parameters of the circuit shown in Figure

Nyquist and Bode diagram for the model of polarizable electrode, Caputo-Fabrizio derivative approach, in (a) and (b):

Several studies use equivalent circuits considering pure capacitances for adjusting the impedance spectra and thus describe phenomena as deterioration of materials due to its porosity or the effects of exposure and surface preparation of substrates on the impedances [

This model describes a heterogeneous reaction which occurs in two stages with absorption of intermediates products and absence of diffusion limitations. This circuit has been commonly used to model a porous electrode coating or a defective electrolyte interface and has recently been applied in the assessment of passive metal-electrolyte or metal-electrolyte interfaces with hard coating [

Equivalent electrical circuit for the heterogeneous reaction model.

This model has also been used in the study of coated metals. In this case,

Following the same methodology from previous example, the fractional impedance in the Caputo sense of this circuit is

In the Caputo-Fabrizio sense the fractional impedance is given by

Consider that the values of the parameters of the circuit shown in Figure

Nyquist and Bode diagram for the heterogeneous reaction model, Caputo derivative approach, in (a) and (b):

Now consider that the values of the parameters of the circuit shown in Figure

Nyquist and Bode diagram for the heterogeneous reaction model, Caputo-Fabrizio derivative approach, in (a) and (b):

Some authors make changes to the ideal capacitors including elements of Warburg [

Nyquist and Bode diagram for the heterogeneous reaction model, Caputo approach in (a) and (b) and Caputo-Fabrizio approach in (c) and (d). For (a) and (b),

The circuit shown in Figure

This model describes the polarization of an electrode considering limiting the diffusion. This circuit models a cell where polarization is due to a combination of kinetic and diffusion processes,

Equivalent electrical circuit for the limited diffusion model.

Following the same methodology from previous example, the fractional impedance in the Caputo sense of this circuit is

In the Caputo-Fabrizio sense the fractional impedance is given by

The capacitor

Consider that the values of the parameters of the circuit shown in Figure

Nyquist and Bode diagram for the limited diffusion model, Caputo derivative approach, in (a) and (b):

Now consider that the values of the parameters of the circuit shown in Figure

Nyquist and Bode diagram for the limited diffusion model, Caputo-Fabrizio derivative approach, in (a) and (b):

Replacing capacitor

If

An impedance spectrum which is obtained in response to the small amplitude signal excitation is often interpreted in terms of an equivalent electrical circuit. This one is based on a physical model that may represent and characterize elements whose electrochemical properties and structural features or the physicochemical processes are taking place in the studied being. The resulting spectrum described in a wide frequency range might present one, two, or more time constants, depending on the monolayer or multilayer structure, porosity, or diffusive limitations caused by the charge transfer process. The existence of a time constant indicates a homogeneous layer structure; if two different constants are presented, this may indicate the existence of two sublayers.

In the field of biomaterials, the electrochemical impedance spectroscopy technique proves to be useful in characterizing roughness or heterogeneous surfaces, in coatings analysis, as cell suspensions, in studying the adsorption of protein, and in characterizing the performance of materials, mainly with regard to electrocatalysis and corrosion. In this context, FC allows the investigation of the nonlocal response of electrochemical systems. FC has been used successfully to modify many existing models of physical processes; the representation of equivalent models in integer-order derivatives provided a good approximation of the electrochemical response of the model.

On the basis of Cole’s proposal to add an extra degree of freedom in order to solve the RC circuits for characterization purposes and improve the correlation in the adjustment to experimental data, we have developed analytical arguments to derive this result based on integration in weighted individual relaxation processes. However, the distributions of relaxation times involve complex functions and are difficult to measure. This study has shown a pattern in which the Cole type behavior appears as a result of competition between a capacitive and resistive behavior within the sample, characterized by the fractional order derivative of the applied voltage. This combination of stored and dissipated energy is conveniently based on the representation of linear viscoelastic behavior; this dissipation is known as internal friction. In the literature it is common to characterize based on least-squares fit of equivalent electrical circuit models on experimental data, including Cole models. From the description of the fractional differential equation models it can be noted that the representation of Cole models is derived as a particular solution to the RC circuit under FC.

Since some authors replace the derivative of a fractional entire order in a purely mathematical context, the physical parameters involved in the differential equation do not have the dimensionality obtained in the laboratory and from the physical point of view of engineering this is not entirely correct. In this representation an auxiliary parameter

The Caputo representation has the disadvantage that their kernel had singularity; this kernel includes memory effects and therefore this definition cannot accurately describe the full effect of the memory. Due to this inconvenience, Caputo and Fabrizio in [

The authors declare that they have no competing interests.

The authors would like to thank Mayra Martínez for the interesting discussions. J. F. Gómez-Aguilar acknowledges the support provided by CONACYT: catedras CONACYT para jovenes investigadores 2014.