Randles cell

For an electrochemical reaction (Equation 4.40) at open circuit potential, Ra can be expressed as 

The total impedance is calculated by adding the individual impedances of the elements  

If an extra element is added to a Randles cell, a modified Randles model can be formed. 

the impedance can be expressed with real and imaginative parts separated as 

Based on the calculated result of Model D8 (Equation 4.36), the total impedance can be expressed as 

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The simplest and most common model of an electrochemical interface is a Randles circuit. The equivalent circuit and Nyquist and Bode plots for a Randles cell are all shown in Figure 2.39. The circuit includes an electrolyte resistance (sometimes solution resistance), a double-layer capacitance, and a charge-transfer resistance. As seen in Figure 2.39a, Rct is the charge-transfer resistance of the electrode process, Cdl is the capacitance of the double layer, and Rd is the resistance of the electrolyte. The double-layer capacitance is in parallel with the charge-transfer resistance. 

Figure 2.39. Graphic presentations of the Randles cell a equivalent circuit, b Nyquist plot, c Bode magnitude plot, d Bode phase plot (Re/ = 20 2, Rct = 80 Q, CdI = 0.001 F)...

The Nyquist plot of a Randles cell is always a semicircle. At high frequencies the impedance of Cdl is very low, so the measured impedance tends to Reh At very low frequencies the impedance of Cm becomes extremely high, and thus, the measured impedance tends to Rct + Rd. Accordingly, at intermediate frequencies, the impedance falls between Rd and Rct + Rd. Therefore, the high-frequency intercept is associated with the electrolyte resistance, while the low-frequency intercept corresponds to the sum of the charge-transfer resistance and the electrolyte resistance. The diameter of the semicircle is equal to the charge-transfer resistance. 

The Bode plot contains a magnitude plot and a phase angle plot. For a Randles cell, the values of the electrolyte resistance and the sum of the electrolyte resistance and the polarization resistance can easily be identified from the horizontal line in the magnitude plot. At high or low frequencies, the phase angles are close to 0°. Otherwise, at intermediate frequencies, the phase angles fall between 0° and 90°. 

The Randles cell model is not only useful but also serves as a starting point for more complex models, created by adding more components. 

The Nyquist plot is presented in Figure 4.9b. At high frequencies (real axis at a value of R. At low frequencies ( 0), it intercepts the real axis at a value of R+R0. Note that the bounded Warburg impedance is easily recognized from its Nyquist plot. At high frequencies, this circuit element looks like a traditional Warburg impedance and shows a 45° line on the Nyquist plot. At low frequencies, it looks like the semicircle of a Randles cell,... 

If a resistor is added in series with the parallel RC circuit, the overall circuit becomes the well-known Randles cell, as shown in Figure 4.11a. This is a model representing a polarizable electrode (or an irreversible electrode process), based on the assumptions that a diffusion limitation does not exist, and that a simple single-step electrochemical reaction takes place on the electrode surface. Thus, the Faradaic impedance can be simplified to a resistance, called the charge-transfer resistance. The single-step electrochemical reaction is described as... 

The Nyquist plot in a complex-plane diagram of the Randles cell with capacitor replaced by a CPE is a depressed semicircle, as shown in Figure 4.136. If co —> the intercept of the depressed semicircle at the real axis equals Reh and rf m — 0, the intercept equals the value of Rei + Rct. The deviation of the n value describes the... 

Figure 4.14. a Randles cell with an extra capacitor in series with Rcl (Model D13) b Nyquist plot of Randles cell with an extra capacitor in series with Rct over the frequency range 1 MHz to 1 mHz (Model D13 Rd = 100 Q, R, l = 300 Q, C,u = 0.001 F, Cad = 1 F)... 

Figure 4.15a shows the structure of a modified Randles cell. This circuit models a cell in which polarization is due to a combination of kinetic and diffusion processes with infinite thickness. 

Figure 4.156 shows the simulated Nyquist plot of the modified Randles cell with a combination of kinetic and diffusion processes plus infinite thickness. As in the example, the Warburg coefficient is assumed to be er = 5 Qs 12. Other... 

The following model is a bounded Randles cell also accounting for a linear but finite diffusion, with a homogeneous layer of finite thickness. The structure of the model is shown in Figure 4.18a. The corresponding impedance is... 

The complex-plane impedance diagram of the bounded Randles cell is given in Figure 4.186. In this example, the parameters of the forward and backward reactions and diffusion coefficients are assumed to be equal. Impedance diagrams with variations in the parameters are presented in Appendix D (Model D17). 

Figure 4.20. a Equivalent circuit of modified Randles cell with bounded CPE in series with Rc, (Model D19) b Nyquist plot of modified Randles cell having a bounded CPE in series with Rc, over the frequency range 6 kHz to 1 mHz (Model D19 Rei = 10 2, Rcl = 20 Q, R0 =... 

Figure D.30. Nyquist plot of Randles cell over the frequency range 1 MHz to 1 mHz (Rct =100 Q., C = 0.001 F)...

Figure D.32. Randles cell with Rct replaced by CPE (Figure 4.12a)...

Figure D.40. Modified Randles cell with an extra capacitor in series with Rct (Figure...

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