Randles equivalent circuit Nyquist plot

By taking into account the double-layer capacity, Q, and the electrolyte resistance, Re, one obtains the Randles equivalent circuit [150] (Fig. 10), where the faradaic impedance Zp is represented by the transfer resistance Rt in series with the Warburg impedance W. It can be shown that the high-frequency part of the impedance diagram plotted in the complex plane (Nyquist plane) is a semicircle representing Rt in parallel with Cd and the low-frequency part is a Warburg impedance. 

Fig. II.5.3 Scheme of the impedance of the Randles equivalent circuit in the complex impedance plane (Nyquist plot)...

Figure 19-6. Randles equivalent circuit and the corresponding parameters obtained from Nyquist plot.

A typical electrochemical cell where polarization is due to a combination of kinetic and diffusion processes is modeled by a "full" Randles equivalent circuit composed of a parallel addition of a double-layer capacitance Cp and Faradaic impedance [23, p. 385] (Figure 5-8). This circuit also includes solution resistance element R. An example of the Nyquist and Bode plots for this circuit... 

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)...

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 7.10. [A) An unlabeled stem-loop structure immobilized on gold electrode opens up in the presence of target DNA, forming a film of matched and mismatched ds-DNA, respectively. (B) Nyquist plots shows in the increase in the chaise transfer resistance of the DNA film after hybridization Ra of hairpin (a], mismatched duplex (b) and matched duplex (c]. Inset shows the modified Randle s equivalent circuit used to fit the electrochemical data. (C] Relationship between ARa and the concentration ofthe target strand showing sensitivity up to lOpM. Y. Wang, C. Li, X. Li, Y. Li, H.-B. Kraatz, Anal. Chem., 2008, 80, 2255-2260. Copyright 2008 American Chemical Society.

In general, the impedance of solid electrodes exhibits a more complicated behavior than predicted by the Randles model. Several factors are responsible for this. Firstly, the simple Randles model does not take into account the time constants of adsorption phenomena and the individual reaction steps of the overall charge transfer reaction (Section 5.1). In fact the kinetic impedance may include several time constants, and sometimes one even observes inductive behavior. Secondly, surface roughness or non-uniformly distributed reaction sites lead to a dispersion of the capacitive time constants. As a consequence, in a Nyquist plot the semicircle corresponding to a charge-transfer resistance in parallel to the double-layer capacitance becomes flattened. To account for this effect it has become current practice in corrosion science and engineering to replace the double layer capacitance in the equivalent circuit by a... 

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