The faradaic impedance

In more than one respect, the small-amplitude sinuosoidal a.c. method can be superior to the large-amplitude step methods for the study of coupled homogeneous reactions. First, the wide range of frequencies at which meaningful data can be obtained will correspond to an equally wide range of rate constants on which, in principle, information can be obtained. Second, the a.c. perturbation can be superimposed on a large-amplitude d.c. or step perturbation so that information in the time scale of the latter is incorporated as well. Moreover, this affords an internal check on the reliability of data interpretations. Finally, it is important 

In view of the dualism explained above, it may be useful to recall the mutual relationship between the general parameters used in the two approaches. In Smith s treatments, the a.c. polarographic current, I (cot) = AAjF, is generally formulated by [159] 

In view of the earlier treatment of the a.c. method in terms of the fara-daic impedance or admittance [see Sect. 2.3.2(b), eqns. (61)—(65)), it can be stated that 

Note that in the normal case of control by charge transfer and diffusion, treated in Sect. 2.3.2(b), we have V — p + 1 and U = 1. However, if control by homogeneous chemical reactions is incorporated, more complex expressions for V and U will result. Furthermore, the function F(tm ), and thus a, will be influenced by the parameters of the chemical reactions. In the succeeding sections, this will be discussed briefly for the mechanisms included in Table 9. Similar methodologies can be applied to the more complex mechanisms ECE, etc. Because of the lengthy algebra, we will not treat these here, but refer readers to the original literature [154, 156—158]. 

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Figure 1.7 ORR impedance plots for a Pt electrode at several cathode DC currents, showing a low ftequency branch corresponding to slow increase of the rate (slow lowering of the faradaic impedance) following cathodic perturbation [Makharia et al., 2005].

The first estimations of for photoinduced processes were reported by Dvorak et al. for the photoreaction in Eq. (40) [157,158]. In this work, the authors proposed that the impedance under illumination could be estimated from the ratio between the AC photopotential under chopped illumination and the AC photocurrent responses. Subsequently, the faradaic impedance was calculated following a treatment similar to that described in Eqs. (22) to (26), i.e., subtracting the impedance under illumination and in the dark. From this analysis, a pseudo-first-order photoinduced ET rate constant of the order of 10 to 10 ms was estimated, corresponding to a rather unrealistic ket > 10 M cms . Considering the nonactivated limit for adiabatic outer sphere heterogeneous ET at liquid-liquid interfaces given by Eq. (17) [5], the maximum bimolecular rate constant is approximately 1000 smaller than the values reported by these authors. 

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. 

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]... 

Returning to the fundamental ac harmonic in Fig. 3.42, we wish to establish the relationship between I and the faradaic impedance Z( instead of considering a combination of a series resistance Rs and a pseudo-capacity C8, the alternative is to separate a pure resistance of charge transfer Rct and a kind of resistance to mass transfer Zw, the Warburg impedance the derivation of the polarogram39 then (for AEtc < 8/remV) leads to the equation... 

There are several ways to present the Faradaic impedance data obtained at an electrode immobilized with an immunocomplex in the presence of a redox probe. For example, ZIm is plotted vs ZRe as a function of decreasing frequency to obtain a... 

Obviously, the faradaic impedance equals the sum of the two contributions f ct, the charge transfer resistance, and Zw = aco-1/2 (1 — i), the Warburg impedance. Again, the meaning of the parameters Rct and a is still implicit at this stage of the treatment and explicit expressions have to be deduced from an explicit rate equation, e.g. the expressions given in eqns. (51). 

Fig. 17. (a) The components of the faradaic impedance plotted against to. (b) The components of the faradaic admittance plotted against to1/2. System parameters fict = lficm1, p = 0.03. The solid parts of the plots indicate the frequency range that is normally accessible for meaningful analysis of data with the simple theory described in this section... 

In this section, a description of the state of the art is attempted by (i) a review of the most fundamental types of reaction schemes, illustrated by some examples (ii) formulation of corresponding sets of differential equations and boundary conditions and derivation of their solutions in Laplace form (iii) description of rigorous and approximate expressions for the response in the current and/or potential step methods and (iv) discussion of the faradaic impedance or admittance. Not all the underlying conditions and fundamentals will be treated in depth. The... 

In order to obtain a surveyable expression for the faradaic impedance, it is convenient to define the Warburg parameters a0 and oR by... 

So, the term [A0a0 + AR aR ] co-1/2 (1 — i) resembles the Warburg impedance corresponding to diffusional mass transport of A, O and R, with a mobile equilibrium between A and 0, i.e. kQ -> °°, whereupon the term in g = kQ /co would vanish. If, however, kQ has a finite value, the faradaic impedance is enlarged by the Gerischer impedance expressed by the term containing g. 

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.

There have been very few methods that establish the surface concentrations by directly manipulating the bulk solution concentrations. Essentially, the only technique that still employs this method is the faradaic impedance method, which will be discussed below. 

The faradaic impedance method is the grandparent of all other small-amplitude methods. The first known experiments were reported by Warburg in 1899 [2]. 

An equivalent circuit of the three-electrode cell discussed in Chapters 6 and 7 is illustrated in Figure 9.1. In this simple model, Rr is the resistance of the reference electrode (including the resistance of a reference electrode probe, i.e., salt bridge), Rc is the resistance between the reference probe tip and the auxiliary electrode (which is compensated for by the potentiostat), Ru is the uncompensated resistance between the reference probe and the working-electrode interphase (Rt is the total cell resistance between the auxiliary and working electrodes and is equal to the sum of Rc and Ru), Cdl is the double-layer capacitance of the working-electrode interface, and Zf is the faradaic impedance of the electrode reaction. 

The resistance of the electrolyte, Ra, represents the ohmic resistance of the complete column of electrolyte between both electrodes. Note that in this way of presentation, the Faradaic impedance cannot necessarily be correlated with an occurring process, therefore it is represented by a general symbol ZF. It can be seen in this case that the total impedance consists of contributions from the working and the counter/reference electrodes, and knowledge of the behaviour of only the working electrode needs manipulation. A simple possibility is to use a CE/RE electrode with much larger surface than the WE. However, in the work presented in this book, this manipulation is not done, because the system used here will be used in applications as a set of two identical electrodes. In addition, one of the parameters that we are interested in is the electrolyte resistance, a parameter that is suppressed when using electrodes of different surface areas. 

Zs is the faradaic impedance due to the substrate for a redox system it consists of a series combination of a charge-transfer resistance Rt and a convective diffusion impedance Zds (2s - ts+2os)-... 

Thus, the fundamental difference between N-NDR and HN-NDR systems is that the former s stationary polarization curve exhibits a range of negative real impedance, whereas for the latter the zero-frequency impedance is strictly positive in the potential region of interest. From this observation one might get the impression that the mechanisms of electrode reactions are fundamentally different for systems in the two groups. But in fact it is only a small step, or more precisely, one additional potential-dependent process, that transforms an N-NDR system into an HN-NDR system. Formally, any HN-NDR system is composed of a subsystem with an N-shaped stationary polarization curve whose NDR is hidden by at least one further slow and potential-dependent step of the interfacial kinetics of the total system. This step dominates the faradaic impedance at low perturbation frequencies, whereas at higher... 

The faradaic impedance, Zf, and the total impedance how to calculate Zf from experimental measurements... 

For the sake of simplicity usually the charging and the Faraday processes are treated independently, however, it is justified only in certain cases. This approximation is valid if a high excess of supporting electrolyte is present, i.e., practically only nonreacting ions build up the double layer at the solution side. In modelling the electrode - impedance almost always an - equivalent circuit is used in that the - double-layer impedance and the -> faradaic impedance are in parallel which is true only when these processes proceed independently. 

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