Charge transfer Randles circuit

The impedance data have been usually interpreted in terms of the Randles-type equivalent circuit, which consists of the parallel combination of the capacitance Zq of the ITIES and the faradaic impedances of the charge transfer reactions, with the solution resistance in series [15], cf. Fig. 6. While this is a convenient model in many cases, its limitations have to be always considered. First, it is necessary to justify the validity of the basic model assumption that the charging and faradaic currents are additive. Second, the conditions have to be analyzed, under which the measured impedance of the electrochemical cell can represent the impedance of the ITIES. 

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. 

Under this electrochemical configuration, it is commonly accepted that the system can be expressed by the Randles-type equivalent circuit (Fig. 6, inset) [23]. For reactions on the bare Au electrode, mathematical simsulations based on the equivalent circuit satisfactorily reproduced the experimental data. The parameters used for the simulation are as follows solution resistance, = 40 kS2 cm double-layer capacitance, C = 28 /xF cm equivalent resistance of Warburg element, W — R = 1.1 x 10 cm equivalent capacitance of Warburg element, IF—7 =l.lxl0 F cm (

charge-transfer resistance, R = 80 kf2 cm. Note that these equivalent parameters are normalized to the electrode geometrical area. On the other hand, results of the mathematical simulation were unsatisfactory due to the nonideal impedance behavior of the DNA adlayer. This should... 

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... 

Electrochemical reactions consist of electron transfer at the electrode surface. These reactions mainly involve electrolyte resistance, adsorption of electroactive species, charge transfer at the electrode surface, and mass transfer from the bulk solution to the electrode surface. Each process can be considered as an electric component or a simple electric circuit. The whole reaction process can be represented by an electric circuit composed of resistance, capacitors, or constant phase elements combined in parallel or in series. The most popular electric circuit for a simple electrochemical reaction is the Randles-Ershler electric equivalent... 

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. 

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... 

When the polymer flhn is oxidized, its electronic conductivity can exceed the ionic conductivity due to mobile counterions. Then, the film behaves as a porous metal with pores of limited diameter and depth. This can be represented by an equivalent circuit via modified Randles circuits such as those shown in Figure 8.4. One Warburg element, representative of linear finite restricted diffusion of dopants across the film, is also included. The model circuit includes a charge transfer resistance, associated with the electrode/fllm interface, and a constant phase element representing the charge accumulation that forms the interfacial double... 

Let us consider the double-layer model circuit as shown in Fig. 3.4. This circuit can be modified based on Randles circuit [2], a prevalent circuit in electrochemistry [7]. It consists of an active electrolyte resistance Rg in series with the parallel combination of the double-layer capacitance Cj and an impedance of a faradaic reaction. The faradaic reaction consists of an active charge transfer resistance R and Warburg resistance Rw- Hence, the electrical equivalent circuit can be modified as shown in Fig. 3.5. 

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.

Transmission line models can be used for inert electrodes and it is a modification of the Randles model (Fig. 6.3). Since the Randles-circuit can be used to describe a nondistributed system, the transmission line models invokes a finite diffusional Warburg impedance, Z, in place of concentration hindered impedance (Fig. 6.4). Randles model is concerned with Qi (the double layer capacitance), [the resistance to charge transfer) and Z by describing the processes occurring in the film. The expression of total impedance, Ztot, is given by following equation ... 

Flgure 6.3 A Randles circuit where is the solution resistance, Qi is the double-layer capacitance, R t is the charge transfer, and Z is the diffusion-hindered impedance. 

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... 

Randles analysis was based on the fact that, for transient responses to electrical perturbation, the electrical properties of electrodes at which the simple activation-controlled charge transfer is the rate-determining step (rds) can be represented by the equivalent circuit shown in Figure 6, where is a nonlinear and overpotential-dependent Faradaic resistance, which can be derived from Eq. (23). At very small values of 17 for which the exponential terms can be linearized (17 < 10 mV), this is obtained as... 

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.

Cathodic Electrochromic Materials—Tungsten Trioxide. Figure 4.3.20 shows electrochemical impedance spectra on both amorphous and crystalline Li containing WO3 films together with fits to the Randles circuit (Strpmme Mattsson [2000]). For the amorphous film the high frequency semicircle overlaps with the diffusion response. In the case of the crystalline film, only a part of the semicircle due to Cdi and Ra, can be observed. As is obvious from the displayed spectra, the charge transfer resistance is much larger for the crystalhne sample than for the disordered one at an equilibrium potential of 2.9 V vs. the Li reference electrode used in the experiment. Impedance spectra were taken at several equihbrium potentials, and in all cases the impedance response corresponded to that of the Randles circuit with a Zd of semi-infinite type. 

Figure 4.5.7. Randles equivalent circuit. represents the series resistance, R the charge transfer resistance of the electrochemical reaction, and Z , sohd state diffusion and other subsequent reactions.

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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