**2.1 Graphical representations of impedance measurements**

As mentioned before, data obtained in electrochemical impedance spectroscopy tests are reported by the potentiostat equipment in the following two ways:

a. Real component of total impedance (Z real) and imaginary component of total impedance (Z imaginary), as shown in **Figure 9**.

Although the International Union of Pure and Applied Chemistry (IUPAC) conventions hold that the real part should be represented by Z<sup>0</sup> and the imaginary part denoted by Z″, the use of this notation can also be found as Z with a subscript notation of "r" for the real part and "j" for the imaginary part.

b. Impedance modulus (|Z|) and phase shift (angle) (°) versus the frequency (*f*), as shown in **Figure 10**.

These two methods of describing impedance measurements are the basis for two common ways of presenting data, called Nyquist and Bode plots, respectively.

**Figure 9.** *Impedance data presented in -Zimaginary (Z″)* versus *Zreal (Z*<sup>0</sup> *) representation.*

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

**Figure 10.** *Impedance data presented in phase shift and total impedance* versus f *representation.*

Nyquist plots show the correlation between the real and imaginary parts of the impedance when the frequency varies. The difficulty of Nyquist plots is that frequency information cannot be directly shown. However, Bode plots show how the magnitude and phase angle of impedance change as a function of the frequency [6]. Because of this, it is important to have a complete understanding of both Nyquist and Bode plots, when analyzing impedance data.

The impedance modulus |Z|, the phase shift, ∅ (°), and the real and imaginary components of the total impedance are related to each other according to the following expressions.

$$\left|\mathbf{Z}\right|^2 = \mathbf{Z}'^2 + \mathbf{Z}''^2. \tag{10}$$

$$\tan \mathfrak{Q} = \frac{Z''}{Z'}.\tag{11}$$

$$Z' = |Z|\cos\mathfrak{D}.\tag{12}$$

$$Z'' = |Z|\text{sen}\mathfrak{D}.\tag{13}$$

The previous analysis has not considered the fact that all electrodes show a capacitance, called "double layer capacitance" (Cdl), which is independent of Faradaic reactions. On the other hand, an electrical resistance, associated with the resistance of the electrolyte (Rsol), exists between the point at which the potential is measured (usually the tip of the Luggin capillary) and the working electrode. This resistance will also manifest itself in the total impedance of the system.

#### **2.2 EIS data manipulation and fitting**

All Kramers–Kronig-consistent impedance spectra may be fitted to an equivalent circuit, which is the point of using a measurement model. The effects of Cdl and Rsol can be considered in impedance analysis if their magnitudes are known. They can also be determined by measurements in the absence of the pair of electroactive species. However, determining the Cdl and Rsol values separately increases the complexity of experimentation and information analysis. An analysis method that avoids the need to separate measurements is derived from a process widely used in other areas, such as electrical engineering, and adapted for electrochemical applications. This method is called "complex plane impedance analysis."

Considering a simple series circuit of resistance and capacitance, the total impedance is equal to

$$Z = R + \frac{1}{j\alpha \mathbf{C}},\tag{14}$$

where the real part of Z is simply *R* and the corresponding imaginary part is <sup>1</sup> *<sup>j</sup>ωC*.

If the behavior described by Eq. (14) is represented in a diagram of Z = Z0 + Z″ (Argand diagram), where Z<sup>0</sup> = real component of total impedance and Z″ = imaginary component of total impedance, the graph of **Figure 11** will be obtained. In this case, the corresponding graph is a series of points at different values of *ω*, where the value of the imaginary impedance component (Z″) tends to zero as the frequency tends to infinity. Here, the capacitance can be considered as short-circuited. Additionally, it is necessary to mention that, in electrochemical studies, the imaginary component of the total impedance (Z<sup>0</sup> ) is usually multiplied by �1. This is because, in strict mathematical rigor, in most of these systems, Z″ has negative a value (as shown in **Figure 9**).

The ohmic resistance-adjusted phase angle has an asymptotic value of �90° at high frequency when an electrode is considered "ideally polarizable." If there is a constant-phase element (CPE), the asymptotic value at high frequency would be lower than 90°. Thus, plots of the ohmic resistance-adjusted phase angle present a direct demonstration of a capacitive behavior or frequency dispersion behavior.

The constant-phase element (CPE) is frequently used to improve the fit of models of impedance data. Then, a capacitance may be obtained from a distribution of time constants along the electrode surface. However, not all time-constant distributions lead to a CPE. The ohmic resistance-corrected phase angle gives a useful method to establish whether a time-constant distribution is represented by a CPE or a Cdl.

Every graphical result can be described in an equivalent circuit. For the results shown in **Figure 11**, two main elements appear: (i) Rsol, which corresponds to the resistivity of the electrolyte, and (ii) a capacitor in series to the Rsol, which assumes

**Figure 11.** *Representation of Eq. (14).*

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

**Figure 12.** *Electrical equivalent circuit consisting of Rsol in series to a capacitor (Cdl).*

the formation of the double layer or the deposition of charged species over the electrode surface. The equivalent circuit is shown in **Figure 12**.

In other cases, the real and imaginary components of the total impedance can behave as parallel combinations of resistors and capacitors. Thus, the response is characterized by the presence of a semicircle. At low frequencies, the impedance is purely resistive because the reactance of the capacitor is very large. The diagram in **Figure 13** corresponds to the simplest analogy of a Faradaic reaction on an electrode with an interfacial capacitance, Cdl. Graphically, in a Nyquist plot, the initial point of the semicircle corresponds to the Rsol, while the final point that reaches the *x*-axis minus the value of the Rsol, corresponds to an Rct. Here, it is important to consider that, for a redox reaction to occur, the species must be close enough to allow the electron transfer. Then, a capacitor (Cdl) or CPE in parallel will always be used in this circuit model.

This representation allows obtaining a simile of an electrochemical reaction and increases the complexity of the analysis. Then, if the Rsol or Re value is high, the semicircle will be shifted to higher values of the *x*-axis of the graph. The corresponding electrical circuit is shown in **Figure 14**.

Another representation that is common to see in practice is a diffusional component in series to the Rct. This diffusional component is called "Warburg element" (denoted as W) and it is shown as a 45° linear response right after the semicircle

#### **Figure 13.**

*Representation of the real and imaginary components of the total impedance of a parallel combination of a resistor and a capacitor.* α*, a parameter associated with the CPE.*

#### **Figure 14.**

*Equivalent circuit in which a mechanistic interpretation of the system under study is used to extract a meaning for the faradaic impedance, Rct; double-layer constant-phase element, CPEdl; and ohmic resistance of the solution, Re.*

**Figure 15.** *Equivalent circuit corresponding to the Rsol in series with a Warburg element, and a capacitor in parallel.*

closes at the *x*-axis (**Figure 15**). Warburg semi-infinite diffusion is an impedance element, which describes the diffusion behavior of the electrolyte in the absence of convection with a diffusion layer that can spread to infinity.

When this electrical component is shown, the equivalent circuit is called the "Randles equivalent circuit" (**Figure 16**).

The circuit representation of the process model shown in **Figure 16** has a corresponding mathematical expression given as

$$Z = R\_{\epsilon} + \frac{R\_{\epsilon t} + W\_o}{1 + (j\omega)^a Q (R\_{\epsilon t} + W\_o)},\tag{15}$$

where Re is the ohmic resistance, Rct is the charge-transfer resistance, Wo is the diffusion impedance, and α and Q are parameters for a CPE, that is, Z = ((jω) α Q) CPE�<sup>1</sup> . The diffusion impedance is expressed in terms of a diffusion resistance, Rd, and a dimensionless diffusion impedance, �1/θ<sup>0</sup> (0), as:

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

#### **Figure 16.**

*Electrical circuit corresponding to a high-frequency constant-phase element (CPE) behavior, which is associated with the double layer. Re (Rsol) is the ohmic resistance; Rct is the charge-transfer resistance associated with electrode kinetics; Wo, diffusion impedance is associated with the transport of reactive species to the electrode surface.*

**Figure 17.** *Nyquist plot with more than one Rct denoted by semi-circles.*

$$\mathcal{W}\_o = R\_d \left( \frac{-1}{(\theta)'(0)} \right),\tag{16}$$

where θ is the dimensionless concentration phasor scaled to its value at the electrode surface, and θ<sup>0</sup> (0) is the derivative with respect to the position evaluated at the electrode surface [7].

Once we obtain the measured data and the corresponding Nyquist and Bode plots, the data can be fitted with these corresponding electrical equivalent circuits. Most of the software that are in the potentiostats can fit the results by using common electrical circuits. Nevertheless, it is highly important to know what kind of materials we are using in practice to give the electrical circuit a physical meaning. The fitting software, such as ZView/ZPlot from Scribner Associates ®, can approach the fit of the results by graphing an electrical circuit and getting the "theoretical" graph out of the model, in addition to the precision of the electrical circuit. The quality of the fit can be defined using a graphical comparison of the impedance data or by the weighted χ<sup>2</sup> statistic. Both simplex regression and Levenberg–Marquardt regression strategies are used [7, 8].

Finally, when two or more semicircles are shown, it is possible to modify the electrical equivalent circuit shown in **Figures 14** and **16**, by adding more Rcts in the fitting. Usually, the Rcts that are away from the Rsol are the charge transfers occurring on the proximities of the surface of the electrode and are in the low-frequency zone (**Figure 17**). Here, many possibilities appear as options for fitting the data as shown in **Figure 18**.

**Figure 18.**

*Different examples of equivalent circuits when two charge transfers are involved plus the resistance of the electrolyte (in boxes).*

**Figure 19.** *Bode plot from circuit presented in Figure 14.*

Regarding the Bode plots for these examples, **Figure 19** shows the expected diagram for the Randles equivalent circuit in **Figure 14**.

As it is observed in **Figure 19**, a typical Bode plot is shown for a simplified Randles circuit (without the diffusional component). In this case, the frequency values go from the lowest to the highest, which means that the first points correspond to the impedance of the electrolyte if we analyze the graph from the right to the left. When a charge transfer occurs, the |Z| *vs. f* graph shows a slope (red dashed line), while the phase shift *vs. f* shows a valley at the same frequency value. The flattened areas correspond to the rearrangements of the double layer until we observe another slope and a semi-valley. At these frequencies, the electrical behavior may be attributed to diffusional effects. In an ideal system, where no resistance or impedance is contributing to the electrochemical system, the phase shift tends to zero, but, as mentioned earlier, all electrodes present a resistance toward the charge transfer, as well as the electrolyte, where the reaction is taking place. For instance, neither of these parameters can be ignored.

#### **2.3 Applications**

Electrochemical impedance spectroscopy can be used in many areas, and its application is highly important when describing the interface electrode/solution. As the results are represented in equivalent circuits, it must be considered that this

electrical behavior occurs between the WE and theCE, where a differential in current is taking place.
