## 2 Impedance Spectroscopy
### Basics of Impedance Spectroscopy and Modelling
as pure capacitors [106]), dispersion effects caused by ion adsorption/diffusion phenomena (for instance, charge transport phenomena occurring along open pores filled with water molecules and along the interface between the material and the measuring electrodes are examples of this type of mechanism [100; 101; 107]), or system-surface roughness [108]. In the frequency domain, \(Z_{EFC}=Af^{-n}\big{(}cos\big{(}\frac{n\pi}{2}\big{)}-jsin\big{(}\frac{n\pi}{2} \big{)}\big{)}\), where A is the impedance value for the angular velocity of 1 rad/s, and f is the frequency. Figure 2 shows a trace of this type of element in the frequency domain using the Nyquist representation. The Warburg impedance is a constant phase element with an angle of 45\({}^{\circ}\). It typically represents diffusion mechanisms due to diffusion phenomena of reactive species or ions, such as the case where water is present (so, diffusion can be the consequence of conduction along open pores filled with water [109; 110]). It should be noted that the Warburg impedance increases with decreasing frequency or with decreasing diffusion coefficient (the diffusion coefficient in the case of pores filled with water molecules represents the charge-transport-capacity provided by the interaction between a solute, in this case water, and a solvent, the porous medium filled with water molecules). In the frequency domain, \(Z_{War}=\frac{A}{\big{(}w^{\frac{1}{2}}\big{)}}-j\frac{A}{\big{(}w^{\frac{1}{2 }}\big{)}}\) (where A is the Warburg coefficient, inversely proportional to the \((diffusion\,coefficient)^{\frac{1}{2}}\)) [99], and w is the angular velocity). Figure 2 shows the trace of this element in the frequency domain using the Nyquist representation. The last most commonly used element in the development of equivalent circuits is the finite-length Warburg one, also referred to as O-element, which is mainly used to characterize diffusion processes in electrodes [111]. Mathematically, it is defined in the frequency domain by \(Z_{O}=\frac{tanh\big{(}B\times\sqrt[2]{iw}\big{)}}{Y_{0}\times\sqrt[2]{iw}}\), where \(B=\frac{\delta}{D}\) (\(\delta\) is the thickness of the diffusion layer and D is the diffusion coefficient), \(Y_{0}=\frac{1}{\sqrt[2]{2}\times A}\) (\(A\) is Warburg coefficient), and \(w\) is the angular velocity; the Nyquist plot can be seen in Figure 3.  
Figure 3: Illustrative Nyquist diagram of an O-element. Figure 2: Nyquist plot illustrating a CPE or a Warburg element in the frequency domain. To complement the Nyquist plots, Z' and Z'' as frequency functions can also be plotted, known as Bode diagrams (Figure 4). The combined analysis of the Nyquist diagram and the Bode curves provides complete information on the electrical behavior of the system. In general, researchers combine elementary circuit elements to create their equivalent circuit models for representing the occurring phenomena in the under-evaluation systems by IS. To exemplify this procedure, we can mention the simplified Randles cell (which accounts for a solution resistance, a double layer capacitor and a charge transfer, or polarization resistance), that is typically the starting point for the development of other more complex models, the equivalent circuit and Nyquist plot of which are schematized in Figure 5.