Warburg impedance semi-infinite diffusion

The real and imaginary parts of Warburg impedance for the planar, semi-infinite -> diffusion are equal ... 

The total faradaic impedance, consists of three terms The first one comes from the derivative di/dE and is called the charge-transfer resistance, Rc, the two others, which are contributions from dUd are called impedances of mass transfer in the case of semi-infinite diffusion it is called a semi-infinite Warburg impedance. 

In Section 4.2.1 the method of impedance spectroscopy was introduced. The polarization of an electrode with an alternating potential of small amplitude is also influenced by restricted mass transport. The Warburg impedance for semi-infinite diffusion describes the diffusion process. The Warburg impedance is a complex quantity with real and imaginary parts of equal magnitude. The impedance is given by the equation... 

Figure 5.7 Warburg impedance for semi-infinite diffusion, (a) Nyquist plot and (b) Bode plot.

Historically, the Warburg impedance, which models semi-infinite diffusion of electroactive species, was the first distributed circuit element introduced to describe the behavior of an electrochemical cell. As described above (see Sect. 2.6.3.1), the Warburg impedance (Eq. 38) is also analogous to a uniform, semi-infinite transmission line. In order to take account of the finite character of a real electrochemical cell, which causes deviations from the Warburg impedance at low frequencies. 

It should be noted that the aforementioned model deals with the semi-infinite diffusion, which is presented by Warburg impedance. As the diffusion layer of finite thickness 5 is formed at the RDE surface, it is necessary to meet certain relations between 5, specified by Eq. (3.5), and the depth of penetration of the concentration wave Xq = y/2Dfco. Investigations in this field show [25,26] that the two aforementioned diffusion models are in harmony when < 0.15. At Schmidt number Sc = v/D = 2000, this condition is satisfied when co > 6 2 (or/ > 0.1m, where m is rotation velocity in rpm). 

Considering the surface roughness and inhomogeneities of Sn electrodes, capacitances and Cji were replaced by the respective CPEs and Qj[. To account for diffusive mass transport, we tested two elements the Warburg impedance W that represents a semi-infinite diffusion and the element O that follows from the description of the so-called transmissive boundary [106]. It is supposed in the latter case that diffusion occurs in the 6-thick layer. Their impedances are, respectively [106] ... 

For a semi-infinite diffusion process at cathode represented by Warburg impedance, the Nyquist plot appears as a straight line with a slope of 45°. The impedance increases linearly with decreasing frequency. The infinite diffusion model is only valid for infinitely thick diffusion layer. For finite diffusion layer thickness, the finite Warburg impedance converges to infinite Warburg impedance at high frequency. At low frequencies or for small... 

The boundary conditions for the Warburg impedance, Zw, previously discussed were such that semi-infinite diffusion prevails. However, as we have already seen in connection with voltammetry and other techniques for film-modified electrodes, diffusion in these cases is bounded and is restricted to a thin layer of thickness d. This problem has been independently addressed by three different groups [110-113] and leads to essentially the same end result, namely that the phase angle begins to increase at very low frequencies due to the onset of finite length effects. Figure 20.27a illustrates the complex plane impedance plot obtained in this instance. 

Cottrell equation) as well as it determines the shape of cyclic voltammograms. In these cases the diffusional flux varies with time. The diffusion currenf densify also depends on the size of the electrode (-> microelectrodes). In - electrochemical impedance spectroscopy the -> Warburg impedance corresponds to the semi infinite diffusion of the charged particles. 

Diffusion resistance Zp,yy to current flow carried by electroactive species can create impedance, frequently known as the Warburg element [23, p. 376]. If the diffusion layer Lp is assumed to have an unlimited thickness within the experimental AC frequency range, than a "semi-infinite" diffusion may become the rate-determining step in the Faradaic kinetic process. In the "semiinfinite" diffusion model the diffusion layer thickness Lp is assumed to be always much smaller than the total thickness of the sample d (Lp d. The equation for the "semi-infinite" Warburg impedance Z m) is a function of concentration-driven potential gradient dV/rfC. The "semi-infinite" diffusion limitation is modeled by characteristic resistance and a Warburg infinite diffusion component Z that can be derived [8] as ... 

EHie to the assumption of semi-infinite diffusion made by the Warburg impedance for the derivation of the diffusion impedance, it predicts that the impedance diverges from the real axis at low frequencies. The DC impedance of the diffusion-limited electrochemical cell would be infinitely large. The Warburg impedance can be represented by a semi-infinite transmission line (TLM) composed of capacitors and resistors (Figure 5-6) [1, p. 59]. 

If there are deviations from the ideal semi-infinite linear diffusion process, the bounded Randles cell can also be modified by replacing the Warburg impedance with a CPE. The structure of the model is shown in Figure 4.20a. This modification is applied when the transport limitations appear in a layer of finite thickness. 

Experiments carried out on monocrystalline Au(lll) and Au(lOO) electrodes in the absence of specific adsorption did not show any fre-quency dispersion. Dispersion was observed, however, in the presence of specific adsorption of halide ions. It was attributed to slow adsorption and diffusion of these ions and phase transitions (reconstructions). In their analysis these authors expressed the electrode impedance as = R, + (jco iJ- where is a complex electrode capacitance. In the case of a simple CPE circuit, this parameter is = T(Jcaif. However, an analysis of the ac impedance spectra in the presence of specific adsorption revealed that the complex plane capacitance plots (C t vs. Cjnt) show the formation of deformed semicircles. Consequently, Pajkossy et al. proposed the electrical equivalent model shown in Fig. 29, in which instead of the CPE there is a double-layer capacitance in parallel with a series connection of the adsorption resistance and capacitance, / ad and Cad, and the semi-infinite Warburg impedance coimected with the diffusion of the adsorbing species. A comparison of the measured and calculated capacitances (using the model in Fig. 29) for Au(lll) in 0.1 M HCIO4 in ths presence of 0.15 mM NaBr is shown in Fig. 30. 

Let us now consider a semi-infinite linear diffusion of charged particles from and to the electrode. The Faraday impedance is defined as the sum of the charge-transfer resistance R and the Warburg impedance W corresponding to the semiinfinite diffusion of the charged particles... 

The faradaic impedance consists of one real part arising from the derivative Eq. (4.17) and is called the charge transfer resistance, R, and the second part containing/ is called the mass transfer impedance, Zw, which, in the case of semi-infinite linear diffusion, is called the Warburg impedance and is composed of two parts Zw,o andZw.R, [8, 30, 145]... 

As was shown earlier, the presence of the CPE of fractal impedance produces a distribution of the time constants. In addition, other elements such as the Warburg (semi-infinite or finite-length) linear or nonlinear diffusion, porous electrodes, and others also produce a dispersion of time constants. Knowledge about the nature of such dispersion is important in the characterization of electrode processes and electrode materials. Such information can be obtained even without fitting the experimental impedances to the corresponding models, which might be still unknown. Several methods allow for the determination of the distribution of time constants [378, 379], and they will be briefly presented below. 

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... 

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