Warburg impedance infinite-length

Equation (IL5.36) shows that the Warburg impedance cannot be represented as a series combination of frequency-independent elements in an equivalent circuit. This is possible, however, by a semi-infinite resistive-capacitive transmission line with a series resistance R per unit length and a shunt capacity C per unit length (Fig. IL5.4). 

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. 

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. 

Thus, the diffusion impedance expressions depend on the electrode separations d at low frequencies. One way to identify the finite Warburg impedance is to use measurements at various values of the electrodes separation d. When LpCO / D 3 (at oo °o), the tank term approaches imity, the diffusion length is negligible compared to the whole region avaUable for diffusion d, and Zpjj-j-approaches infinite length Warburg Zy ... 

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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