## 2 Impedance Spectroscopy
### Basics of Impedance Spectroscopy and Modelling
Thus, the impedance can be described in a phase diagram by a phasor with module \(|Z|\) and phase angle \(\theta\), with corresponding horizontal and vertical components representing the resistive and capacitive terms, respectively (the impedance \(Z\) can be written as \(Z=Z^{\prime}+iZ\)\", where \(i=\sqrt{-1}\), being the real part of \(Z\), \(Z^{\prime}\) or \(ReZ\), given by \(|Z|\times\cos(\theta)\), and the imaginary part of \(Z\), \(Z^{\prime}\) or \(ImZ\), described by \(|Z|\times sin(\theta)\)). In the literature, it is usual to plot instead \(Z=Z^{\prime}-iZ\)\", ensuring that the graphic representation stays in the first quadrant of the polar/complex representation. The impedance graph that relates its real and imaginary parts (\(ImZ\) versus \(ReZ\)) is called the Nyquist diagram (Figure 1a). From the graph, it is possible to obtain relevant information about the system, such as the bulk resistance (\(R_{\Omega}\)) and charge transfer resistance (\(R_{ct}\)) of the material, as well as to detect the occurrence of diffusion processes. Classical electrical components can be used to represent the known contributions to the total electrical conduction of a system in the equivalent electrical circuit model. For instance, it is common to use a resistor to represent the conduction paths along the grains of a material. However, the most common circuit component is a parallel association of a resistor and a capacitor. This fact is due to the diversity of conduction mechanisms that can be represented by it; among them are the conduction mechanisms along the grain boundaries caused by, for example, ion hopping or conduction along the surface of the grains/material (mechanisms of an electronic nature) [100; 101] in the frequency domain \(Z_{R/C}=\frac{R}{(1+iWRC)}\). Figure 1b shows the trace of this element in the frequency domain using an Nyquist plot. Also, the electrical double layer formed in the interface between the material surface and the measuring electrodes, applied to measure the material impedance, can be represented by an R//C circuit. In theory, at high frequencies, the interception of the Nyquist graph with the \(x\)-axis represents the ohmic resistance (\(R_{\Omega}\)), or simply the internal resistance of the system. In addition, the interception of the graph at low frequencies returns information about the value of charge transfer resistance (\(R_{ct}\)) at the bulk/electrode interface. However, often, a depressed semicircle can be observed as a signature of the non-ideal behavior of the system under study (Figure 1c). This behavior may be related to environmental factors (temperature and pressure) and/or to different internal factors, such as the presence of a polycrystalline or multiphase structure (with different grain and sub-grain boundaries and grain sizes), the existence of functional groups at the grain/electrolyte interface, electrode effects (presence of porosity and/or different grain sizes), or to additional resistance of the electrolyte. This behavior may represent reactions with heterogeneous kinetics and different relaxation times in the sample (distribution of relaxation constants), determined from modelling using equivalent electrical circuits [102; 1; 2]. Constant-phase elements, CPEs, are also commonly used elements in model building. These elements are used to depict phenomena due to surface or the normal-to-surface distribution of properties (which combination can yield a surface-position-dependent local impedance, resulting from capacitance distribution in polycrystals [103], electrode geometry [104], and normal-to-surface distributions, such as resistivity or permitivity in oxide layers, passive films, and coatings [105]), electric double layers in the interfaces of systems under evaluation (such as electrode-electrolyte ones, which almost never behave  
Figure 1: (**a**) Illustrative Nyquist diagram (\(-\)Z” versus \(Z\)); (**b**) R//C circuit frequency domain Nyquist plot; (**c**) depressed semicircle model (x \(\
eq\) y) in a Nyquist plot.