Advances in Mathematical Physics

Volume 2016, Article ID 9720181, 15 pages

http://dx.doi.org/10.1155/2016/9720181

## Equivalent Circuits Applied in Electrochemical Impedance Spectroscopy and Fractional Derivatives with and without Singular Kernel

^{1}CONACYT-Centro Nacional de Investigación y Desarrollo Tecnológico, Tecnológico Nacional de México, Interior Internado Palmira S/N, Colonia Palmira, 62490 Cuernavaca, MOR, Mexico^{2}Facultad de Ingeniería Mecánica y Eléctrica, Universidad Veracruzana, Avenida Venustiano Carranza S/N, Colonia Revolución, 93390 Poza Rica, VER, Mexico^{3}Facultad de Ingeniería Electrónica y Comunicaciones, Universidad Veracruzana, Avenida Venustiano Carranza S/N, Colonia Revolución, 93390 Poza Rica, VER, Mexico

Received 2 February 2016; Accepted 26 April 2016

Academic Editor: Alexander Iomin

Copyright © 2016 J. F. Gómez-Aguilar et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

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.

#### 1. Introduction

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 [1]. Transfer function analysis is a mathematical approach to relate an input signal (or excitation) and the system’s response. The ratio formed by the pattern of the output and the input signal makes it possible to find the zeros and poles, respectively. To analyze the behavior of the transfer function in the frequency domain, several graphical methods were used, such as Bode plots that provide a graphical representation of the magnitude and phase versus frequency of the transfer function and the Nyquist diagrams that are polar plots of impedance modulus and phase lag. It is very common in the literature to analyze impedance results by a physical model; this model is expressed by a mathematical representation and usually is represented by equivalent electrical circuits composed of elements such as resistors, capacitors, inductors, the Constant Phase Element (CPE), and Warburg elements (which is a special case of the CPE). Although the resulting model is not necessarily unique, it describes the system with great precision in the range of frequencies studied [1].

Fractional calculus (FC) is the investigation and treatment of mathematical models in terms of derivatives and integrals of arbitrary order [2–5]. In the last years, the interest in the field has considerably increased due to many practical potential applications [6–13]. In the literature, a number of definitions of the fractional derivatives have been introduced, namely, the Hadamard, Erdelyi-Kober, Riemann-Liouville, Riesz, Weyl, Grünwald-Letnikov, Jumarie, and the Caputo representation [2–5]. For example, for the Caputo representation, the initial conditions are expressed in terms of integer-order derivatives having direct physical significance [14], this definition is mainly used to include memory effects. Recently, Caputo and Fabrizio in [15] present a new definition of fractional derivative without a singular kernel; this derivative possesses very interesting properties, for instance, the possibility to describe fluctuations and structures with different scales. Furthermore, this definition allows for the description of mechanical properties related to damage, fatigue, and material heterogeneities. Properties of this new fractional derivative are reviewed in detail in [16].

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 [17]. In this context, FC allows the investigation of the nonlocal response of electrochemical systems, this being the main advantage when compared with classical calculus. Some researches concerning EIS introduce FC; for example, the authors of [18] study the electrical impedance of vegetables and fruits from a FC perspective; the experiments are developed for measuring the impedance of botanical elements and the results are analyzed using Bode and Nyquist diagrams. In [19] a technique is presented to extract the parameters that characterize a dispersion Cole-Cole impedance model. Oldham in [20] describes how fractional differential equations have influenced the electrochemistry. Other applications of fractional calculus in electrochemical impedance are given in [21–26].

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 [12] analyze the ordinary derivative operator and try to bring it to the fractional form in a consistent manner. Following this idea 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 order of the fractional equation is . In this representation an auxiliary parameter is introduced; this parameter characterizes the existence of the fractional temporal components and relates the time constant of the system.

The paper is organized as follows: Section 2 explains the basic concepts of the FC, Section 3 presents the examples considered and the interpretation of typical diagrams, and the conclusions are given in Section 4.

#### 2. Introduction to Fractional Calculus

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 [4].

The CD is defined as follows [4]:where is a CD with respect to , is the order of the fractional derivative, and represents the gamma function.

The Laplace transform of the CD has the following form [4]:

The Caputo-Fabrizio fractional derivative (CF) is defined as follows [15, 16]:where is a CF with respect to and is a normalization function such that ; in this definition the derivative of a constant is equal to zero, but, unlike the usual Caputo definition (1), the kernel does not have a singularity at .

If and , the CF fractional derivative, , of order , is defined by

The Laplace transform of (3) is defined as follows [15, 16]:For this representation in the time domain it is suitable to use the Laplace transform [15, 16].

From this expression we have

#### 3. Electrochemical Impedance and the Interpretation of Typical Diagrams

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 [27, 28]. Cole model is represented by a series resistor , a capacitor , and a resistor in parallel . In general it reflects the electrical resistance of the interface sample-electrode and maintains a negligible value with respect to ; is the order of the power that best fits the model obtained, , giving an ideal capacitor when it is . Using the algebraic representation of the circuit can be said to represent the total impedance aswhere .

On electric structures RC type, bias resistor represents the charge transfer resistance and capacitance of double layer; the CPE is a component that models the behavior of a double layer capacitor in actual electrochemical cells, that is, an imperfect capacitor, and the impedance is represented asEquation (8) describes the deviation from ideal capacitors. Considering and the constant (the inverse of capacitance), this equation describes a capacitor. For a CPE, the exponent is less than one [29].

In electrochemical systems the diffusion can create an impedance called Warburg impedance [26], commonly used to describe phenomena such as diffusion, adsorption, or desorption of electroactive substances at interfaces metal/coating; this impedance depends on perturbation frequency: at high frequency a small Warburg impedance results and at low frequency a higher Warburg impedance is generated. The Warburg impedance parameter indicates the existence of diffusive processes that can be related to the release of the dissolved species. This parameter only reports the blocking ability of the passive layer, so it is not possible to know the nature of the species which are dissolved by electrochemical impedance spectroscopy technique. The equation for the infinite thickness of the Warburg impedance is given bywhere is a Warburg coefficient. On a Nyquist plot the infinite Warburg impedance appears as a diagonal line with a slope of ; on a Bode plot, the Warburg impedance exhibits a phase shift of 45° [29].

Equation (9) is valid if the diffusion layer has an infinite thickness. Quite often this is not the case. If the diffusion layer is bounded, the impedance at lower frequencies no longer obeys (9). For the Warburg impedance with finite thickness, we get the formwhere is the Nernst diffusion layer thickness; is an average value of the diffusion coefficients of the diffusing species. This equation is more general and is called finite Warburg [29].

##### 3.1. Equivalent Representation Based on Fractal Capacitors

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 [12] we introduce an auxiliary parameter in the following way:orwhere is an integer and when . Expressions (11) and (12) become a classical derivative; the auxiliary parameter has the dimension of time (seconds). This nonlocal time is called the cosmic time in the literature [30]. Another physical and geometrical interpretation of the fractional operators is given in Moshrefi-Torbati and Hammond [31]. Parameter characterizes the fractional temporal structures (components that show an intermediate behavior between a conservative system and dissipative; such components change the time constant of the system) of the fractional temporal operator [32]. In the following we will apply this idea to construct the fractional equivalent circuits and examples are analyzed.

##### 3.2. Polarizable Electrode

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 1.