Faradaic frequency dependence

Under potentiostatic conditions, the photocurrent dynamics is not only determined by faradaic elements, but also by double layer relaxation. A simplified equivalent circuit for the liquid-liquid junction under illumination at a constant DC potential is shown in Fig. 18. The difference between this case and the one shown in Fig. 7 arises from the type of perturbation introduced to the interface. For impedance measurements, a modulated potential is superimposed on the DC polarization, which induces periodic responses in connection with the ET reaction as well as transfer of the supporting electrolyte. In principle, periodic light intensity perturbations at constant potential do not affect the transfer behavior of the supporting electrolyte, therefore this element does not contribute to the frequency-dependent photocurrent. As further clarified later, the photoinduced ET... 

A constant phase element (CPE) rather than the ideal capacitance is normally observed in the impedance of electrodes. In the absence of Faradaic reactions, the impedance spectrum deviates from the purely capacitive behavior of the blocking electrode, whereas in the presence of Faradaic reactions, the shape of the impedance spectrum is a depressed arc. The CPE shows power law frequency dependence as follows129 130... 

Formulations of the general expressions for YF (s) and of some limiting cases can be found in ref. 53. It is to be noted that this faradaic admittance is connected in parallel to the double-layer admittance Yc (s). The latter may well be influenced by the presence of the adsorbed intermediate. Nevertheless, the quite specific frequency dependence of Yf can be applied to detect the presence of a mechanism of the type discussed here [53, 132]. 

The above-described situation is but an exception rather than the rule. Generally, the diamond electrode capacitance is frequency-dependent. In Fig. 12 we show a typical complex-plane plot of impedance for a single-crystal diamond electrode [69], At lower frequencies, the plot turns curved (Fig. 12a), due to a finite faradaic resistance Rp in the electrode s equivalent circuit (Fig. 10). And at an anodic or cathodic polarization, where Rf falls down, the curvature is still enhanced. At higher frequencies (1 to 100 kHz), the plot is a non-vertical line not crossing the origin (Fig. 12b). Complex-plane plots of this shape were often obtained with diamond electrodes [70-73],... 

Faradic impedance (//) is directly related to the rates of charge transfer reactions at and near the electrode/electrode interface. As shown in Figure 3.1, the Faradaic impedance acts in parallel with the double-layer capacitance Cd, and this combination is in series with the electrolyte resistance Rei The parameters Rei and Cd in the equivalent circuit are similar to the idea of electrical elements. However, X/ is different from those normal electrical elements because Faradaic impedance is not purely resistive. It contains a capacitive contribution, and changes with frequency. Faradaic impedance includes both the finite rate of electron transfer and the transport rate of the electroactive reagent to the electrode surface. It is helpful to subdivide Zj into Rs and Cs, and then seek their frequency dependencies in order to obtain useful information on the electrochemical reaction. 

The general expression (10.3) guides development of impedance models from proposed reaction sequences. The reaction mechanisms considered here include reactions dependent only on potential, reactions dependent on both potential and mass transfer, coupled reactions dependent on both potential and surface coverage, and coupled reactions dependent on potential, surface coverage, and mass transfer. The proposed reaction sequence has a major influence on the frequency dependence of the interfacial Faradaic impedance described in Qiapter 9. 

The transfer function V can be determined from the Faradaic impedance Zdisk and the transfer function M (v). It is an important kinetic parameter that allows evaluation of the frequency dependence of the amoimt of charge stored at the electrode surface. 

Here, the impedance response is independent of the working point, and the frequency dependence is determined solely by the material parameters of the composite. For / <linear branch appears only at frequencies co > a/Cfr). Doublelayer charging and proton transport dominate the overall electrode response in this regime, whereas Faradaic processes are insignificant due to the high frequencies. An equivalent representation of this system is an RC-transmission line [130], Since no fractality or branching of the network is assumed, the response resembles that of a Warburg impedance with a characteristic proportionality Z a where... 

The measurement of the cell characteristics in a bridge yields values of R and Cb that in series are equivalent to the whole cell impedance, including the contributions from Rq and Q, which are often not of interest in studies focused on the faradaic process. In general, one desires to separate the faradaic impedance from Rq and Q. It is possible to do so by considering the frequency dependencies of R and Cb, or by evaluating Rq and from separate experiments in the absence of the electroactive couple. Techniques for making such determinations are considered in Section 10.4. For the moment, let us assume that the faradaic impedance, expressed as the series combination R and Cg, is evaluable from the total impedance (see Figure 10.1.14). 

We will see below that R [ is primarily determined by the heterogeneous charge-transfer kinetics, and we have already observed above that the terms crlo) and Hcro) come from mass-transfer effects. Recognition of this situation has led to a division of the faradaic impedance into the charge-transfer resistance, R i, and the Warburg impedance, Z, as shown in Figure 10.1.14. Equations 10.2.25 and 10.2.26 demonstrate that this latter impedance can be regarded as a frequency-dependent resistance, = alo), in series with the pseudocapacitance, = Cg = Thus the total faradaic impedance,... 

Devise and justify an equivalent circuit for a system in which O and R are bound to the surface of the electrode as the result of a chemical modification. Follow the steps in Sections 10.2 and 10.3 to evaluate the expected frequency dependence of the faradaic impedance for the case in which the electrode reaction is nemstian. What phase angle is expected ... 

Electric Double Layer and Fractal Structure of Surface Electrochemical impedance spectroscopy (EIS) in a sufficiently broad frequency range is a method well suited for the determination of equilibrium and kinetic parameters (faradaic or non-faradaic) at a given applied potential. The main difficulty in the analysis of impedance spectra of solid electrodes is the frequency dispersion of the impedance values, referred to the constant phase or fractal behavior and modeled in the equivalent circuit by the so-called constant phase element (CPE) [5,15,16, 22, 35, 36]. The frequency dependence is usually attributed to the geometrical nonuniformity and the roughness of PC surfaces having fractal nature with so-called selfsimilarity or self-affinity of the structure resulting in an unusual fractal dimension... 

Electroactive species adsorption is sensitively indicated by this method, which thus determines whether redox system properties are obscured by a specific type of interaction with the electrode. Strongly absorbable electroinactive compounds yield peaks on the AC ciuwes located at potentials of the adsorption-desorption process. However, they are much narrower and their frequency dependence differs markedly from the Faradaic peaks. Such peaks were also used for determination of sruface-active... 

Example 14.4 The faradaic impedance of a hydrogen evolution reaction can be represented by four different circuits displaying the same values of impedances and frequency dependence [211]. These circuits are displayed in Fig. 14.6. The faradaic... 

The frequency dependence of Yp components follows from Eqs. (11)-(13) and is expressed by the proportionality g l/>/. The limiting behavior should clarify the information content of each vector component. At sufficiently high frequencies the faradaic admittance becomes purely resistive and the imaginary part Yp vanishes... 

These equations can be used for a more adequate method of determination of the heterogeneous rate k (equal to kf + kt) than on the basis of a single frequency measurement. By the combination of Eqs. (13) and (23, 24) we can write the frequency dependence of the faradaic phase... 

Fig. 9. A typical frequency dependence of the faradaic phase angle. The slope determines the heterogeneous rate constant. The example given here is for the reduction of methyl viologen in methanol with k° = 0.038 cm .

The frequency dependence of a simple response of an electrochemical cell consists of three contributions (i) double-layer charging with a linear dependence on co (via the term coC) (ii) the frequency independent faradaic charge transfer (Ret) (hi) the diffusional contribution with dependence and, finally, (iv) the solution resistance acting in series with all contributions listed (i-iii). 

Similar to EIS, SWV (square-wave voltammetry) is another frequency-dependent electrochemical technique that could also be used in label-free Faradaic immunosensing [167]. In this case, a train of potential pulses is superimposed on a staircase potential signal with the latter centered between a cathodic pulse and an anodic pulse of the same amplitude. During each cathodic pulse, the analyte diffuses to the electrode surface and it is immediately reduced. During the anodic pulse, analyte that was just reduced is reoxidized. The current is sampled just before and at the end of each pulse and the current difference between these two points is then plotted against the staircase potential in a SW voltammogram. A linear potential scan in SWV is faster than EIS record and a familiar peak-shaped signal is more easily interpreted. 

Though the magnitude and the frequency dependence of Fp are essentially controlled by four parameters (see Eq. (8.23)), the faradaic subcircuit contains more (five) elements. For this reason, the specific link between faradaic EC elements occurs ... 

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