Equivalent circuits Kinetic

In the case of reactions that are not completely irreversible (or not completely reversible), we must account for both the kinetic factors (e.g., the polarization resistance R and the concentration changes (the Warburg impedance). The simplest equivalent circuit for this case is shown in Fig. 12.15c, while Fig. 12.17c shows the impedance diagram for this circuit (AJS = 10 = 1 Q the other parameters... 

In particular, the coupling between the ion transfer and ion adsorption process has serious consequences for the evaluation of the differential capacity or the kinetic parameters from the impedance data [55]. This is the case, e.g., of the interface between two immiscible electrolyte solutions each containing a transferable ion, which adsorbs specifically on both sides of the interface. In general, the separation of the real and the imaginary terms in the complex impedance of such an ITIES is not straightforward, and the interpretation of the impedance in terms of the Randles-type equivalent circuit is not appropriate [54]. More transparent expressions are obtained when the effect of either the potential difference or the ion concentration on the specific ion adsorption is negli-... 

Electrochemical impedance measurements of the physical adsorption of ssDNA and dsDNA yields useful information about the kinetics and mobihty of the adsorption process. Physical adsorption of DNA is a simple and inexpensive method of immobilization. The ability to detect differences between ssDNA and dsDNA by impedance could be applicable to DNA biosensor technology. EIS measurements were made of the electrical double layer of a hanging drop mercury electrode for both ssDNA and dsDNA [34]. The impedance profiles were modeled by the Debye equivalent circuit for the adsorption and desorption of both ssDNA and dsDNA. Desorption of denatured ssDNA demonstrated greater dielectric loss than desorption of dsDNA. The greater flexibility of the ssDNA compared to dsDNA was proposed to account for this difference. 

P. Dolin and B. Erschler, Acta Phys. Chem. 13 747 (1940). First use, equivalent circuit, in electrode kinetics. 

In a situation where a charge transfer is also influenced by diffusion to and from the electrode, the Warburg impedance will be seen in the impedance plot. This circuit model presents a cell in which polarization is controlled by the combination of kinetic and diffusion processes. The equivalent circuit and the Nyquist and Bode plots for the system are all shown in Figure 2.40. It can be seen that the Warburg element is easily recognizable by a line at an angle of 45° in the lower frequency region. 

Figure 2.40. Graphic presentations of a mixed kinetic and diffusion control circuit a equivalent circuit, b Nyquist plot, c Bode magnitude plot, d Bode phase plot (Rei = 100 2, Rct = 100Q,Q= 0.001 F, a= 20 Qsm)...

We have discussed in the above sections Faradaic impedance and the correlation between Faradaic impedance and kinetic parameters. In general, one desires to separate the Faradaic impedance from Rel and Cd. Now we will focus on the extraction of Zf and the kinetic parameters from direct impedance measurements. This is based on the transformation between equivalent circuits in series and equivalent circuits in parallel. 

Ahn et al. have developed fibre-based composite electrode structures suitable for oxygen reduction in fuel cell cathodes (containing high electrochemically active surface areas and high void volumes) [22], The impedance data obtained at -450 mV (vs. SCE), in the linear region of the polarization curves, are shown in Figure 6.22. Ohmic, kinetic, and mass transfer resistances were determined by fitting the impedance spectra with an appropriate equivalent circuit model. 

Understanding the oxidation mechanism is important. Impedance spectroscopy was recently used to study methanol electrooxidation, and kinetic parameters can be deduced from impedance spectra. Figure 6.58 shows an equivalent circuit that was developed for methanol oxidation on a Pt electrode, but which is common for all electrochemical reactions. In this circuit, a constant phase element was used rather than a double-layer capacitance, since a CPE is more realistic than a simple capacitor in representing the capacitive behaviour. 

If C8S/Ch is relatively small, or caCssi ss 1, this will approximate to our original formula, since either the effect of the surface states on the potential distribution is insignificant or the kinetics are too slow to allow the occupancy of the surface state to be significantly altered during a potential cycle. If, however, coCaaRsa < 1, the equivalent circuit will again resemble that discussed above, but the factor [(CSS/CH) + 1] will premultiply the capacitive part. The effect will be to reduce the apparent admittance by this factor, which will, in turn, reduce the apparent capacitance by [1 + (Css/ CH)] and increase the apparent resistances by the same factor. 

H. J. ENGELL The paper of Efimov and Erusalimchik is very brief. They give no equivalent circuit nor do they discuss their method of measurement thus, it is difficult to say just what they have measured. Their measurements were made, however, at a potential of about -500 mv in acid solutions. Under these conditions the rate of hydrogen evolution is relatively fast therefore, it is possible that their minimum in the capacity could have been influenced by kinetic effects. I should also point out that we obtained higher values of the capacity when we did not etch the surface by applying an anodic current before each experiment. 

It has to be mentioned that such equivalent circuits as circuits (Cl) or (C2) above, which can represent the kinetic behavior of electrode reactions in terms of the electrical response to a modulation or discontinuity of potential or current, do not necessarily uniquely represent this behavior that is other equivalent circuits with different arrangements and different values of the components can also represent the frequency-response behavior, especially for the cases of more complex multistep reactions, for example, as represented above in circuit (C2). In such cases, it is preferable to make a mathematical or numerical analysis of the frequency response, based on a supposed mechanism of the reaction and its kinetic equations. This was the basis of the important paper of Armstrong and Henderson (108) and later developments by Bai and Conway (113), and by McDonald (114) and MacDonald (115). In these cases, the real (Z ) and imaginary (Z") components of the overall impedance vector (Z) can be evaluated as a function of frequency and are often plotted against one another in a so-called complex-plane or Argand diagram (110). The procedures follow closely those developed earlier for the representation of dielectric relaxation and dielectric loss in dielectric materials and solutions [e.g., the Cole and Cole plots (116) ]. 

In the treatment which follows, we assume that discharge of the doublelayer capacitance drives the reaction, and therefore use C = in Eq. (41). The effects of changes in coverage of the adsorbed intermediate are then taken into account by combining Eq. (41) with the kinetic equations for steps in the mechanism. In this method, no assumptions need then be made about the equivalent circuit or the nature of the pseudocapacitance, and the transient current during potential decay is not assumed to be equal to the steady-state current. The results then enable all three definitions of [Eqs. (46)-(48)] to be evaluated and compared, as illustrated in Fig. 10. 

In 1940, Frumkin explored the relationships among the double-layer structure on mercury electrodes, the capacitance measured by use of a Wheatstone bridge, and the surface tension, following the theoretical underpinnings of the Lippmann equation. Grahame ° expanded this treatment of the mercury electrode, providing a fundamental understanding of the structure of the electrical double layer. Dolin and Ershler applied the concept of an equivalent circuit to electrochemical kinetics for which the circuit elements were independent of frequency. Randles developed an equivalent circuit for an ideally polarized mercury electrode that accounted for the kinetics of adsorption reactions. ... 

The above analysis shows that in the simple case of one adsorbed intermediate (according to Langmuirian adsorption), various complex plane plots may be obtained, depending on the relative values of the system parameters. These plots are described by various equivalent circuits, which are only the electrical representations of the interfacial phenomena. In fact, there are no real capacitances, inductances, or resistances in the circuit (faradaic process). These parameters originate from the behavior of the kinetic equations and are functions of the rate constants, transfer coefficients, potential, diffusion coefficients, concentrations, etc. In addition, all these parameters are highly nonlinear, that is, they depend on the electrode potential. It seems that the electrical representation of the faradaic impedance, however useful it may sound, is not necessary in the description of the system. The systen may be described in a simpler way directly by the equations describing impedances or admittances (see also Section IV). In... 

Direct use of equivalent circuits may lead to analysis of more complex data. For example, for a system containing one adsorbed species, Eq. (139) may be described by the ladder circuit shown in Fig. 21. The parameters Ra and Ca describing the faradaic impedance [Eq. (141)] are complex functions of the parameters A, B, and C while direct use of Eq. (135) leads to simpler data analysis (i.e., parameters A, B, and C are simpler functions of the kinetic parameters than the electric parameters Ra and Co). 

IMPS uses modulation of the light intensity to produce an ac photocurrent that is analysed to obtain kinetic information. An alternative approach is to modulate the electrode potential while keeping the illumination intensity constant. This method has been referred to as photoelectrochemical impedance spectroscopy (PEIS), and it has been widely used to study photoelectrochemical reactions at semiconductors [30-35]. In most cases, the impedance response has been fitted using equivalent circuits since this is the usual approach used in electrochemical impedance spectroscopy. The relationship between PEIS and IMPS has been discussed by a number of authors [35, 60, 64]. Vanmaekelbergh et al. [64] have calculated both the IMPS transfer function and the photoelectrochemical impedance from first principles and shown that these methods give the same information about the mechanism and kinetics of recombination. Recombination at CdS and ZnO electrodes has been studied by both methods [62, 77]. Ponomarev and Peter [35] have shown how the equivalent circuit components used to fit impedance data are related to the physical properties of the electrode (e.g. the space charge capacitance) and to the rate constants for photoelectrochemical processes. 

Plasma processing reactors normally operate with the wafer biased at radio frequencies, typically in the range 0.1 to 13.56 MHz. Even if the ions injected at the sheath edge were monoenergetic, an lED would result in an RF (time-dependent) sheath, even in the absence of collisions. The literature on RF sheaths is voluminous. Both fluid [170-175] and kinetic (e.g., Monte Carlo) [176-180] simulations have been reported. One of the most important results of such simulations is the lED. The ion angular distribution (IAD) [74, 75] and sheath impedance (for use in equivalent circuit models) [32] are also of importance. 

Figure 8.7.2 Equivalent circuit of cell with (a) Rfi, the solution resistance, Cd, the doublelayer capacitance, and Zf, the faradaic impedance. The faradaic impedance represents the effect of the heterogeneous electron-transfer process. Often Zf is broken down into the components shown in (b where the charge-transfer resistance R manifests the kinetics of heterogeneous charge transfer, and the components of the Warburg impedance,...

Impedance methods have been more useful in studying electron-transfer kinetics in electroactive monolayers in the absence of an electroactive solution species (71-73), such as alkylthiol layers with tethered electroactive groups (Section 14.5.2). The equivalent circuit adopted is shown in Figure 14.3.18, where the adsorbed layer is represented by Cads = (F AT)/4RT and the electron-transfer kinetics by = (2RT)/F ATkf, so that... 

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