**2. Electrochemical impedance spectroscopy: a theoretical review**

The electrochemical impedance spectroscopy (EIS) technique is an electrochemical method used in many electrochemical studies, which is based on the use of an alternating current (AC) signal that is applied to the working electrode, WE, and determining the corresponding response. In the most common experimental procedures, a potential signal (E) is applied to the WE and its current response (I) is determined at different frequencies. However, in other cases, it is possible to apply a certain current signal and determine the potential response of the system. Thus, the potentiostat used processes the measurements of potential *vs.* time and current *vs.* time, resulting in a series of impedance values corresponding to each frequency

*Electrochemical Impedance Spectroscopy and Its Applications DOI: http://dx.doi.org/10.5772/intechopen.101636*

analyzed. This relationship of impedance and frequency values is called the "impedance spectrum." In studies using the EIS technique, the impedance spectra obtained are usually analyzed using electrical circuits, made up of components such as resistors (R), capacitances (C), inductances (L), combined in such a way as to reproduce the measured impedance spectra. These electrical circuits are called "equivalent electrical circuits."

Impedance is a term that describes electrical resistance (R), used in alternating current (AC) circuits. In a direct current (DC) circuit, the relationship is between current (I) and potential (E) is given by Ohm's law Equation. (1), where E is in volts (V), I is amperes (A), and R is Ohms (Ω):

$$I(A) = \frac{E}{R} \frac{\langle V \rangle}{\langle \Omega \rangle}. \tag{1}$$

In the case of an alternate signal, the equivalent expression is as follows:

$$I(A) = \frac{E}{Z} \frac{\langle V \rangle}{\langle \Omega \rangle}. \tag{2}$$

In Eq. (2), *Z* represents the impedance of the circuit, in units of Ohms. It is necessary to note that, unlike resistance, the impedance of an AC circuit depends on the frequency of the signal that is applied. The frequency (f) of an AC system is expressed in units of hertz (Hz) or the number of cycles per second (s�<sup>1</sup> ). In this way, it is possible to define the admittance (Y) of an AC circuit. Admittance is the reciprocal of impedance and is an important parameter in mathematical calculations involving the technique and on the other hand, the equipment used in EIS studies measures admittance.

The impedance of a system at each frequency is defined by the ratio between the amplitude of the alternating current signal and the amplitude of the alternating potential signal and the phase angle. A list of these parameters at different frequencies constitutes the "impedance spectrum" (**Figure 4**). The mathematical development of the theory underlying the EIS technique allows describing the impedance of a system in terms of a real component and an imaginary component (associated with the square root of �1).

#### **Figure 4.**

*Periodic perturbation signal with amplitude (ΔE) is applied between the WE and RE from high to low frequencies.*

Since the EIS technique is based on the study of electrical networks, there is a great deal of information in the literature regarding electrical circuits. Thus, in understanding the theory that supports the EIS technique, it is convenient to describe current and voltage as rotating vectors or "phasor," which can be represented in a complex plane or "Argand Diagram." For instance, a sinusoidal voltage can be represented by the following expression.

$$E(V) = \Delta E(V) \text{ sen ot.}\tag{3}$$

where E is the instantaneous value of the potential, ΔE is the maximum amplitude, and *ω* is the angular frequency, which is related to the frequency *f* according to

$$
\mu = 2\mathfrak{g}'.\tag{4}
$$

In this case, ΔE can be understood as the projection, on axis 0 of phasor E in a polar diagram, as shown in **Figure 5**.

In most cases, the current (I) associated with a sinusoidal potential signal is also sinusoidal, with the same frequency (*ω*) but with a different amplitude and phase than the potential. This can be represented according to Eq. (5).

$$I(A) = \Delta I(A) \text{ sen } (\alpha t + \mathcal{Q}). \tag{5}$$

This means that, in terms of phasors, the rotating vectors are separated in the polar diagram by an angle ∅ in degrees (°). This can be illustrated as shown in **Figure 6**.

The response to a potential E, of a simple circuit with a pure resistance R, can be described by Ohm's law Equation (1). This, in terms of phasors, corresponds to a situation where the phase angle is equal to zero.

**Figure 5.** *Phasor diagram corresponding to the alternating potential of Eq. (3).*

*Electrochemical Impedance Spectroscopy and Its Applications DOI: http://dx.doi.org/10.5772/intechopen.101636*

**Figure 6.** *Phasors of current (I) and potential (E) separated by a phase angle* ∅*.*

When a capacitor is considered in the electrical circuit, different aspects must be known. The concept of "capacitance" (C) can be defined as the relationship between the potential applied (between the capacitor plates) and the total charge (*q*), according to Eq. (6):

$$q = \text{CE}.\tag{6}$$

Considering the current I(A) flowing through the capacitor, the current can be expressed as

$$I(A) = \frac{dq}{dt} = C\frac{dE}{dt}.\tag{7}$$

Thus, considering that current is obtained as a sinusoidal response, we can describe it in terms of the potential, capacitance, and angular frequency, as shown in Eq. (8):

$$I(A) = \text{or } \text{C} \Delta E(V) \text{ } \cos \text{ at.} \tag{8}$$

As Ohm's equation described in Eq. (1), it is possible to rewrite the equation in terms of Eqs. (3) and (5), giving the following expression:

$$Z(\Omega) = \frac{\Delta E(V) \text{ sen at}}{\Delta I(A) \text{ sen } (at + \mathcal{Q})}. \tag{9}$$

In mathematical terms, the real and imaginary components of phasor E and phasor I can be represented in an Argand diagram, with the abscissa corresponding to the real component and the ordinate axis referring to the imaginary component.

**Figure 7.**

*Phasor representation of current (I) and potential (E) with time (t), for a relationship between current and potential in a circuit with a capacitive reactance of (a) phase angle (*∅*) = 0, and (b) phase angle (*∅Þ *= 90°.*

**Figure 7a** shows the representation of phasor E and I for a purely resistive circuit and **Figure 7b** shows a circuit with a reactive capacitance.

In this regard, when an electrode is highly conductive, it is expected that the phase angle tends to zero, due to limited resistivity for the electron transfer to occur. However, in practice, the electrodes present a delay in the current response, which in this technique is represented by the phase angle (phase shift). This means that the system presents a certain resistivity toward the electron transfer, or in actual terms, impedance.

In **Figure 8**, we can observe the final representation of the terms explained in this section.

Therefore, to understand the practical applications, it is important to consider the following. Assume that we apply a sinusoidal potential excitation. The response to this potential is an AC signal. This current signal can be analyzed as a sum of sinusoidal functions (a Fourier series). Concisely, a conventional electrochemical impedance experimental setup involves an electrochemical cell, a potentiostat, and a frequency response analyzer (FRA). The FRA applies the sine wave and examines the response of the system to determine its impedance. The quality of the

*Electrochemical Impedance Spectroscopy and Its Applications DOI: http://dx.doi.org/10.5772/intechopen.101636*

**Figure 8.** *Electrochemical response to the potential perturbation measured in the linear domain.*

impedance measurement is controlled by a series of parameters, including the selection of potentiostatic or galvanostatic modulation, the perturbation amplitude, the frequency range, and the number of cycles used to measure each frequency. The parameters used are highly dependent on the experimental setup and are influenced by factors such as material type, electrolyte, and electrolyte concentration. According to Gamry Instruments [5], a simple explanation of these parameters is described below:


Therefore, the initial frequency, final frequency, and points per decade parameters can help us to calculate the total number of data points acquired. It is recommended that the frequency range used is as wide as possible. Preferably, this implies a range of 6–7 decades (e.g., 10<sup>2</sup> –10<sup>5</sup> Hz), if mathematical tool such as Kramers–Kroning (KK) analysis is used. However, many systems do not allow analysis over a wide frequency range, without obtaining a significant amount of noise.


the study at hand. There are some studies where the DC voltage is set up at the open circuit potential (OCP), which is the potential where the total current in the system is zero. Other potentials can be chosen from the voltammetric profiles obtained from the initial studies, where the capacitive and faradaic zones can be recognized. The AC voltage is summed with the DC voltage.

vi. Estimated Z: It corresponds to a user-entered estimate of the cell's impedance at high frequencies. It is used to limit the number of trials required while the system optimizes the potentiostat settings for improvement, current range, offset, etc. Before taking the first data point, the system chooses the potentiostat settings that are ideal for the estimated Z value. If the estimate is precise, the first (or the following one) attempt to determine the impedance will succeed. Nevertheless, if the estimate lacks accuracy, the system will take a few trials while it optimizes the potentiostat settings.
