Generalized Warburg Element

Macdonald 18 introduced a generalized finite-length Warburg element described as 

A similar equation hut containing the function coth was used hy Inzelt and L ng to descrihe the diffusional impedance of conducting polymers under reflective conditions [see Section III.6(ii) and Eq. (99)]. An electrical model containing this element accounted well for the impedance spectra, with a minimum number of free parameters. 

Although models including impedance represented by Eq. (226) may well describe some experimental data, the physical significance of the parameter Aq for ( ) 0.5 is not clear. 

The impedance plots are shown in Fig. 4.12. The mass transfer impedance displays a straight line at 45° at high frequencies because the penetration length of the ac signal is smaller than the layer thickness and at low frequencies the imaginary part goes to infinity as the constant current cannot flow in the system. At very thin layers or low frequencies a capacitive behavior is observed. In fact, because 

The finite-length transmissive impedance may be found in ZView as Warburg Element (short) Ws and the finite-length reflective impedance as Warburg Element (open) Wq. Examples of simulations in Mathematica are shown in files in the exercises. 

In practical applications of EIS it is often found that the experimental data for the finite-length diffusion cannot be approximated by Eq. (4.72) or Eq. (4.83). For example, in the case of hydrogen absorption in Pd the low-frequency reflective impedance is not strictly capacitive, or in the transmissive case the complex plane plot is slightly depressed [154-156]. In such cases one should use a so-called generalized finite-length Warburg element for transmissive 

4 Impedance of the Faradaic Reactions in the Presence of Mass Transfer 

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Analogously, the generalized Warburg equation, representative of the response of constant phase elements in electrochemical impedance spectroscopy experiments, becomes (Nyikos and Pajkossy, 1990 Dassas and Duby, 1995) ... 

Greszczuk et al. [252] employed the a.c. impedance measurements to study the ionic transport during PAn oxidation. Equivalent circuits of the conducting polymer-electrolyte interfaces are made of resistance R, capacitance C, and various distributed circuit elements. The latter consist of a constant phase element Q, a finite transmission line T, and a Warburg element W. The general expression for the admittance response of the CPE, Tcpr, is [253]... 

We will here employ a more general expression than Warburg s by involving an element with a constant phase, which takes into account the material s... 

Warburg impedance — Warburg impedance is a transport-related interfacial impedance element originated from the diffusion of the electroactive species taking part in the electrode process. In general,... 

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