Equivalent circuit Randles

FIG. 6 Randles equivalent circuit for the ITIES Zq is the interfacial capacitance, Zy)v are the faradaic impedances of the charge transfer reactions, and is the solution resistance. 

Since the ion transfer is a rather fast process, the faradaic impedance Zj can be replaced by the Warburg impedance Zfy corresponding to the diffusion-controlled process. Provided that the Randles equivalent circuit represents the plausible model, the real Z and the imaginary Z" components of the complex impedance Z = Z —jZ " [/ = (—1) ] are given by [60]... 

The reduction of Cd(II) ions on DME was also investigated in 1 M perchlorate, fluoride and chloride solutions using dc, ac admittance, and demodulation methods [27]. It was found that in the perchlorate supporting electrolyte, the reduction mechanism is also CEE, and that the rate constant of the chemical step is quite close to the value characteristic for fluoride solutions. The theories available at present could not be applied to the Cd(II) reduction in chloride solution because of the inapplicability of the Randles equivalent circuit. 

A renewal of interest in the other rate-controlling processes started in those groups who were developing the impedance method [49, 53] and the a.c. polarographic method [12, 25], probably because it was found that, in many cases, Randles equivalent circuit did not hold and also because the appropriate mathematics are more tractable in the frequency domain. Still, it is recommended that the a.c. studies are combined with the diagnostic results which can be obtained from steady-state techniques and/or cyclic voltammetry. 

Zac represents the ac impedance of the usual Randles equivalent circuit. 

Occasionally, the impedance spectra of diamond electrodes are well described by the Randles equivalent circuit with a frequency-independent capacitance (in the 1 to 105 Hz range) [66], Shown in Fig. 11 is the potential dependence of the reciprocal of capacitance squared, a well-known Mott-Schottky plot. Physically, the plot reflects the potential dependence of the space charge region thickness in a semiconductor [6], The intercept on the potential axis is the flat-band potential E whereas the slope of the line gives the uncompensated acceptor concentration NA - Nd in what follows, we shall for brevity denote it as Na ... 

The experimental impedance is always obtained as if it were the result of a resistance and capacitance in series. We have already seen in (11.20) and (11.21) the relation between an RC series combination and the Rct + zw combination. It can be shown for the full Randles equivalent circuit for this simple charge transfer reaction, see Fig. 11.4, on separating the in-phase and out-of-phase components of the impedance, that... 

This approach, and the Randles equivalent circuit , have been very widely used in the years since 1947. The impact of Randles paper can be gauged by the fact that it continues to be cited, typically attracting 10 citations a year in the period since 1981, and by the breadth of research fields on which it has impacted. Thus in the last few years alone Randles paper has been cited in papers on biosensors, corrosion, physiology, displays, and materials research d as well, of course, as in electrochemistry. 

Determination of impedance components (calculated according to the Ershfcr-Randles equivalent circuit) at different concentrations of solvated electrons... 

Here, the substitution p = jco has been made to obtain the impedance. Equation (22) leads to the Randles equivalent circuit for an electrode reaction and is equivalent to eqn. (70) given below. In all electrode reactions, the impedance due to the reaction must have a differential capacity in parallel with it, to give the impedance of the interface... 

Figure 11. Randles equivalent circuit used in the calculation of the inte facial capacitance. C, inte facial capacitance, Rs, bulk solution resistance Zw,...

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. 10 Electrochemical impedance for a diffusion layer of infinite thickness, a Randles equivalent circuit b Scheme of the impedance in the complex plane...

Differential capacitance measurements by Niki et for cytochrome C3 from D. vulgaris, strain Miyazaki, were consistent with irreversible, diffusion-limited adsorption for 4-s drop times above a concentration of 10 fiM. The surface excess of cytochrome C3 was calculated to be 0.92 x 10 " mole/cm. Niki etal also investigated the a.c. polarographic behavior of cytochrome C3 at the reversible half-wave potential. The capacitive peak height was frequency independent while the resistive peak height decreased with increasing frequency to a value of zero above 2000 Hz. These results were fit to a Laitinen-Randles equivalent circuit yielding an n value of... 

The so-called Randles equivalent circuit describes diffusion-controlled charge transfer processes (Figure 5.8). 

Figure 5.8 Randles equivalent circuit for a diffusion-controlled charge transfer process, Qj double layer capacitance, R charge transfer resistance, electrolyte resistance, and W Warburg impedance.

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

The impedance of a FET with the gate immersed in solution and potential applied to a reference electrode in solution may be represented by the equivalent circuit shown in Fig. 6.5. The circuit consists of the silicon resistance Rsi, space-charge capacitance Csc, oxide capacitance Cox of the FET, and the Randles equivalent circuit for the double layer, where Zw has been omitted since there are no redox molecules in solution. In the absence of redox molecules. Ret is large and Zj ag can be considered to result from the series combination of the three capacitances. 

Figure 6.4 Impedance plane plot for a Randles equivalent circuit with charge transfer resistance and Warburg impedance. First region is a kinetics-governed semicircle tall. Last region Is a mass transfer-capacitive tall. Region between two Is a diffusion governed. Reproduced from Ref. 161 with permission of The Royal Society of Chemistry.

The Randles equivalent circuit is used to describe a simple electrode reaction, where the solution resistance Ra) is in series with the charge transfer resistance (Act) and the Warbiug impedance (Zw) expressing the diffusion of the electroactive species, and the double-layer capacitance (Cji) is in parallel with Act and Zw (Zp = Act + is called the Faraday impedance). 

Complexation of alkali metal ions by 15-crown-5 and 12-crown-4 terminated SAMs was studied by EIS [78,80]. The impedance of SAM-covered electrodes in the presence of a redox probe (usually Ru(NH3)g ) was described by the Randles equivalent circuit, and the charge-transfer resistance Rct changed systematically with the metalion concentration. Binding of Na" " ions... 

Figure 6. Randles equivalent circuit and complex plane diagram for activation control.

Figure 2.1.14. The Randles equivalent circuit, which describes the response of a single-step charge-transfer process with diffusion of reactants and/or products to the interface.

Determination of Parameters from Randles Circuit. Electrochemical three-electrode impedance spectra taken on electrochromic materials can very often be fitted to the Randles equivalent circuit (Randles [1947]) displayed in Figure 4.3.17. In this circuit R /denotes the high frequency resistance of the electrolyte, Ra is the charge-transfer resistance associated with the ion injection from the electrolyte into the electrochromic film and Zt, is a Warburg diffusion impedance of either semi-infinite, or finite-length type (Ho et al. [1980]). The CPEdi is a constant phase element describing the distributed capacitance of the electrochemical double layer between the electrolyte and the film having an impedance that can be expressed as... 

Figure 4.3.17. The Randles equivalent circuit, with resistance of a charge-transfer process and the diffusional impedance Z. R f and CPE i are the high frequency resistance and the double layer distributed capacitance, respectively.

---