Complex faradaic impedance

The complex faradaic impedance is derived from eqn. (224) by substituting s = ico followed by splitting into real and imaginary parts. The result is [53, 175]... 

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]... 

Fig. 39. Complex plane diagram of the faradaic impedance in the case of a preceding chemical reaction (CE) with an equilibrium constant of KA = 1. Solid line feA — °° broken lines feA has a finite value, decreasing from top to bottom.

Figure 12.5 Complex-plane impedance plots for p-Si in 1.0 mol dm NH4F (pH 4.5). Note that the faradaic resistance (derived from the low-frequency intercept on the real axis) can be positive or negative. The negative value at 1.75 V indicates that the current decreases with increasing potential in this region of the current-potential curve (see Fig. 12.4). Adapted from Bailes et al (1998).

The Warburg impedance has a minimum at 1/2. The mass-transfer impedance is a vector containing real and imaginary components that are identical, that is, the phase angle (p = atan(Z" v/Z w) = atan(-l) = 5°. The faradaic impedance is shown in Fig. 11(b) (dashed line). On the complex plane plot, it is a straight line with a slope of 1 and intercept The total electrode impedance consists of the solution resistance, R, in series with the parallel connection of the double-layer capacitance, Qi,... 

In this case all the elements are positive and the faradaic impedance represents one semicircle on the complex plane plots (see Fig. 22). When Cj, the total impedance represents two semicircles (Fig. 22). When A B /C the faradaic impedance is equal to R. . The complex plane plots are analogous to those shown in Fig. 4 and represent one capacitive semicircle. 

Figure 23. Complex plane plot for the case of one adsorbed species, B < 0 and C-R IBI = 0 continuous Une, total impedance dashed line, faradaic impedance. Parameters used R. = 100 2, C, = 2 x 10 F,q = 2xl0 F,/f,= 10a...

When 5 = 0, the faradaic impedance is real and equals One semicircle is observed in the complex plane plots (Fig. 4). 

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... 

Depending on the value of the parameter the poles of the second term of Eq. (167) are real or imaginary. Taking into account Eq. (167), there are 54 theoretically different cases of poles and zeros. They were considered systematically in Ref. 95. The faradaic impedance may be represented by many different equivalent circuits, depending on the sign of parameters B and C and relative values of all the parameters. Its complex plane plots display different forms from two capacitive semicircles through various capacitive and inductive loops to two inductive loops. In order to obtain the total impedance, the double-layer capacitance and solution resistance should be added to the faradaic impedance. Some examples of complex plane plots of faradaic impedances are presented in Eig. 26. 

In the presence of the faradaic reaction, assuming that the faradaic impedance can be expressed as a simple equivalent resistance, the complex plane plots represent a rotated semicircle p ig. 28(b)], instead of a semicircle centered on the Z axis. Similarly, the double-layer capacitance in the presence of the faradaic reaction may be obtained as... 

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). 

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. 

The nse of faradaic impedance spectroscopy as a means to identify affinity complexes between the aptamers and small molecules is, however, more difficult... 

Double-layer charging of the pores only (non-faradaic process) and inclusion of a pore-size distribution leads to complex plane impedance plots, as in Fig. n.5.7, i.e. at high frequencies, a straight line results in an angle of 45° to the real axis and, at lower frequencies, the slope suddenly increases but does not change to a vertical line [16]. 

The complex plane plots obtained in this case are shown in Fig. 4.12. The faradaic impedance displays a straight line at 45° at high frequencies where the ac diffusion, that is, the oscillations of the concentration, are limited to the zone around the electrode and diffusion behaves in a semi-infinite linear manner. At higher frequencies oscillations of the concentrations arrive at the back wall and a... 

To obtain the total impedance, the faradaic impedance, Eq. (5.19), must be inserted into the total impedance (Fig. 4.1b). The complex plane and Bode plots of the total impedance are as in Fig. 2.35. The circuit parameters R i and Cp depend on the potential, as illustrated in Fig. 5.1. The charge transfer resistance displays a minimum at Ep and its logarithm is linear with the potential further from the minimum, while the pseudocapacitance displays a maximum. These values at the potential Ep are... 

When B < 0, the faradaic impedance is described by Eq. (5.54). It describes resistance in series with a parallel connection between Cp and Rp (Fig. 5.2, left). The complex plane plots in this case display two semicircles (Fig. 5.3a). 

The circuit in Eq. (5.65) can be represented by that in Eq. (5.54) assuming that Cp < 0, Bp < 0 and Bet > Rp- The complex plane plots are displayed in Eig. 5.3e. Note that the faradaic impedance is always positive and the total impedance displays a plot with capacitive and inductive loops. 

In this case, the faradaic impedance is represented as a semicircle starting tUly/Bju and ending at A /B/u. When combined with the double-layer capacitance, the electrode impedance displays two semicircles of the same radius on the complex plane plots. Examples of complex plane plots at different concentrations of the active species and the same potential are displayed in Fig. 9.33. In this case, the ratio of the highest to the lowest faradaic impedance is constant however, at low concentrations, the doublelayer capacitance causes poor visibility of the first semicircle. Two distinct semicircles are well formed at higher concentrations of the electroactive species. 

The influence of the electrode potential, E — E°, on the complex plane plots is shown in Fig. 9.34. With an increase in the negative potential the faradaic impedance changes and tends to the condition in Eq. (9.78). The total impedance always displays two semicircles however, their separation is most visible at around E - E° = 0. 

The influence of the electrode porosity is displayed in Fig. 9.35. At very low porosities, i.e., for very shallow pores (I = 0.005 cm), the electrode behaves practically as flat, and one semicircle is observed on the complex plane plots. With an increase in the pore length, two semicircles are observed (I = 0.05 cm), and with further increases, one semicircle is observed identical to the faradaic impedance and the influence of the double-layer capacitance disappears. 

Fig. 9.33 Complex plane plots on porous electrode at various concentrations of oxidized species in absence of potential gradient in pores parameters E — ° = 0V, / = 0.05 cm, r = 10 cm, D = 10" cm s", k° = 10 cm s", o = 1.58 concentrations indicated in graph continuous line total impedance, dashed lines faradaic impedances...

Fig. 9.34 Influence of electrode potential, E - E°, on complex plane plots for porous electrode in absence of potential gradient, Cq = 0.1 M other parameters as in Fig. 9.33 continuous line total impedances, dashed lines faradaic impedances...

Fig. 8.13 - Complex plane impedance diagram for a parallel / C circuit. This is the simplest possible analogue of a Faradaic reaction at an electrode with an interfacial capacitance Qi.

Mathias and Haas recently extended the analysis of the impedance response of a redox polymer film to consider electromigration processes and redox site interactions. The analysis is rather complex, so we present only a summary of the results here. The Faradaic impedance is given by... 

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