## 2 Impedance Spectroscopy
### Materials Modelling by IS
In addition to typical cases, this serial approach has recently been used to study the dynamics of biological materials in food and plant science. For example, [PERSON] et al. [116] mentioned researchers using this type of equivalent circuit-building methodology to represent the contributions of electrode-soil and electrode-stem contacts, as well as root medium, to study the state of plant roots. In contrast, few works can be found in the literature using a parallel association methodology for constructing electrical equivalent circuit models. [PERSON] et al. [129] reported a detailed impedance spectroscopy study of Ni-Mo- and Ni-Fe-based electrocatalytic materials deposited onto porous and compact substrates. The results were interpreted using equivalent circuit models. EIS measurements were carried out in a three-electrode cell at room temperature. The samples were used as working electrodes, while Ag/AgCl and Pt wire or wire mesh were used as reference and counter electrodes, respectively. The electrolytes used were 1 M NaOH or 1 M KOH (pH = 14). The measurements were performed in the frequency range between 0.7 Hz and 100 kHz, using a 10 mV peak amplitude AC voltage. One of the equivalent circuits used to model the electrical response is depicted in Figure 9a. The authors identified the contributions of an electrochemical  
Figure 7: Illustration of the different regions in the nanostructured sample between electrodes and the modelling of the electrical response of each of them (R\({}_{\text{1}}\)//C\({}_{\text{1}}\), R\({}_{\text{2}}\)//C\({}_{\text{2}}\), and R\({}_{\text{3}}\)//C\({}_{\text{3}}\), under the influence of the bulk, grain boundary and electrodes, respectively). double layer capacitance, \(C_{dl}\), a series resistor, \(R_{s}\) (accounting for the substrate and the electrolyte resistance), a resistor \(R_{ct}\) (representing the reaction charge transfer resistance), and a resistor in series with an inductor, \(L_{a}\)-\(R_{a1}\) (representing the additional part of the impedance response that is due to a changing surface coverage of adsorbed species). A representation of some of the fitted parameters for the proposed model can be seen in Figure 9b. [PERSON] and his collaborators [130] presented three-dimensional impedance-based biochemical sensors based on their work. The 3D sensors use suspended electrodes in contact with the analyte solution through their top and bottom surfaces, doubling the interface area. For characterizing the sensors they used EleIS, and they observed that a larger solution-electrode interface area allowed the electrolyte solution to interact with a larger electrode surface, which resulted in larger values of solution-sensitive parameters, such as double-layer capacitance and Warburg coefficient; consequently, using the results they improved the sensitivity of the sensor. When they compared the performance of one conventional 2D planar interdigitated sensor and two 3D sensors using different  
Figure 8: Experimental and fitted data using the proposed model circuit [128]. Figure 9: The parallel association approach carried out by [PERSON] et al. [129]. (**a**) Model schematic; (**b**) some of the fitted circuit parameters as a function of applied potential vs. Ag/AgCl (3 M KCl); charge transfer resistance R\({}_{\text{ct}}\), adsorption resistance R\({}_{\text{a1}}\), and inductance I\({}_{\text{a}}\). concentrations of di-ethylhexyl phthalate in deionized water, they found that the 3D sensors provided higher sensitivity. Using the obtained Nyquist plots (Figure 10a,b), they developed an electrical equivalent model (Figure 10c) using a parallel-based strategy; the fit adjustment was of reasonably good quality (Figure 10d,e).