## 2 Impedance Spectroscopy
### Materials Modelling by IS  
Several examples of electrical circuit development based on IS application to interpret materials' electrical response for differentiating chemical elements/electrolytes are found in the literature. Continuing the discussion started in the previous section, it is vital to highlight details of models for the analysis of emerging nanostructured materials. For example, and as previously mentioned, the model of one resistor-capacitor in parallel (R//C) is known to fit the impedance of an ideal dielectric, a pristine material with unique and well-defined characteristics (from the micro to the macro scale in its entire volume), and/or only in a specific region of a sample with the same electronic characteristic. An increasing number of scientific and technological applications of functional materials consider using heterogeneous nanostructures due to the possibility of improving characteristics and electrical response (compared to using just one of the pure components). Even using a pristine nanomaterial, it is possible to observe different dielectric behaviors (dielectric discrimination) in different regions and interfaces of the system (involving bulk, grain boundary, and measurement electrodes--see Figure 5); this is mainly due to the enormous surface area/volume ratio in interaction with the environment of these materials. In addition, heterogeneous nanostructures can exhibit polycrystalline behavior with different relaxation times, constituting a non-ideal impedance response. In such cases, the electrical response obtained from material evaluation generally cannot be modelled by the simple R//C model, making it necessary to expand the interpretation of the different contributions (Figure 7). Consequently, the authors use the series approach to represent these aspects [112; 113; 114; 115; 116]. The electrical response of several functional nanomaterials, including metal oxides, metal nanoparticles, conducting polymers, carbon-based materials, and their hybrids has been explained from this perspective [117; 118; 119; 120; 121; 122; 123; 124; 125; 126]. For example, an essential contribution to the Materials Science domain was the development of a series of equivalent circuits as a universal model for carbon-based supercapacitors [127]. Activated carbon generally does not have a uniform grain size due to its synthesis process; consequently, a carbon electrode with multiple distributions of micro- and macro-pores in the nanostructure can be generated. So, to model the electrical response of the device, the authors of the work suggested an  
Figure 6: Possibilities of associating elements according to the classical rules of analysis of electrical circuits, considering the nonexistence of a contribution under certain conditions. (**a**) serial association (**b**) parallel. equivalent circuit with a vertical network of capacitance and resistance contributions from the n-grains in series with a single R//C combination, representing the bulk resistance and capacitance of the electrolyte solution. Among the diverse examples found in the literature using the serial combination of elements is the one from [PERSON] et al. [128], where they modelled lithium-ion capacitors based on the experimentally obtained Nyquist data. They followed a serial approach, for which they identified the following contributions (Figure 8): an internal ohmic resistance (\(R_{\text{o}}\)); a charge transfer between electrolyte and electrode corresponding to the embedment/ejection of internal lithium-ion capacitor ions and ion adsorption/desorption, represented by a parallel combination of \(R_{\text{ct}}\) and \(C_{\text{dl}}\), where \(R_{\text{ct}}\) is the charge transfer resistance or electrochemical reaction resistance and \(C_{\text{dl}}\) is the electrical double-layer capacitance; a constant phase element, \(Q_{\text{W}}\), in series with intercalation capacitance, \(C_{\text{W}}\), due to ion diffusion in the electrodes in a spherical structure, but not along a single direction.