## 2 Materials and Methods
### Measuring Impedance in Non-Newtonian Materials
#### 2.2.2 An Example: Randle's Circuit  
Any electrochemical cell can be represented in terms of an equivalent electrical circuit that comprises a combination of resistances and capacitances (inductances only for very high frequencies) [4; 5]. This circuit should contain at the very least components to represent:  
1. The double material layer: a pure capacitor of capacity \(C_{d}\),
2. The impedance of the faradaic process \(Z_{f}\),
3. The uncompensated resistance, \(R_{\Omega}\), which is usually the solution resistance between working and reference electrodes. Figure 1: Generic principle of a phase-sensitive detector functionality. The textbook combination of these elements is shown in Figure 2 and known as _Randle's circuit_, with \(Z_{f}\) and \(C_{d}\) in parallel arrangement. The impedance \(Z_{f}\) can be further subdivided in two equivalent ways:  
1. Subdivision into a resistance, \(R_{s}\), in series with a pseudo-capacitance \(C_{s}\),
2. Subdivision into a resistance, accounting for the resistance to charge transfer, \(R_{ct}\), and an impedance that measures the difficulty of mass transport of the electroactive species, called the _Warburg impedance_, \(Z_{w}\). The last equivalence is often used because of its physical meaning: for kinetically-favoured reactions \(R_{ct}\to 0\) and \(Z_{w}\) predominates, while for difficult reactions \(R_{ct}\to\infty\) and predominates. More detail about mathematical formulations, based on electrochemical reaction, of the _Warburg impedance_ and thus of the faradaic impedance can be found in [4; 5]. The impedance for [PERSON]'s circuit will be a complex number which can be represented in its polar plot formalism for all tested frequencies. The explicit real and imaginary parts are given by:  
\[ \begin{array}{ll}Z^{\prime}=&R_{\omega}+\frac{R_{ct}+\sigma\omega^{-0.5}}{( \sigma\omega^{0.5}C_{d}+1)^{2}+\omega C_{d}^{2}(R_{ct}+\sigma\omega^{-0.5})^{ 2}}\\ Z^{\prime\prime}=&-\frac{\omega C_{d}(R_{yt}+\sigma\omega^{-0.5})^{2}+\sigma^ {2}C_{d}+\sigma\omega^{-0.5}}{(\sigma\omega^{0.5}C_{d}+1)^{2}+\omega^{2}C_{d} ^{2}(R_{ct}+\sigma\omega^{-0.5})^{2}}\end{array}  \tag{14}\]  
where \(\sigma\) is a coefficient depending on a combination of different material constants, and \(\omega\) is the frequency in rad/s. Notice the presence of a fractional-order exponent in the model which allows a frequency-dependent real part of impedance (as opposed to classical integer-order impedance whose linear components have a real part constant with frequency). It is further useful to look at the polar plot of the functions in (14), also known in chemistry as Cole-Cole plot, as shown in Figure 3. From this figure is possible to appreciate the division in two regions: one represents the behaviour of a kinetically favoured reaction, while the other one shows the predominance of the mass transfer effect. It is also interesting to discuss the polar plot in Figure 3, bearing in mind the (14) and studying their limiting forms:  
1. \(\omega\to 0\) \[ \begin{array}{ll}Z^{\prime}&=R_{\omega}+R_{ct}+\sigma\omega^{-0.5}\\ Z^{\prime\prime}&=-\sigma\omega^{-0.5}-\sigma^{2}C_{d}\end{array} \] (15) The low-frequency behaviour is represented by the straight line of the unit slope and corresponds to a reaction controlled by sole diffusion, the impedance though assumes the form of the Warburg impedance, which dominates. 2. \(\omega\to\infty\) \[ \begin{array}{ll}Z^{\prime}&=R_{\omega}+\frac{R_{ct}}{1+\omega^{2}C_{d}^{2 }R_{ct}^{2}}\\ Z^{\prime\prime}&=-\frac{\omega C_{d}R_{ct}^{2}}{1+\omega^{2}C_{d}^{2}R_{ct}^{ 2}}\end{array} \] (16) For higher frequencies, the control of the reaction is purely kinetic and it traduces in the fact that \(R_{ct}\gg Z_{w}\). Hence the electrical circuit in Figure 2 can be simplified in a  
Figure 2: Randle’s circuit: equivalent electrical circuit of an electrochemical cell for a simple electrode process. \(Z_{f}\) can be substituted with the series of \(R_{ct}\) and \(Z_{w}\). simple RC parallel.