Evaluation of Impedance Spectra

The concentrations follow the diffusion equation. The derivation of this effect leads to an additional impedance (Warburg impedance) as given by 

Sometimes more than one semicircle occurs in the impedance spectrum as well as the Warburg impedance. The origin of the second semicircle is usually due to 

It should be emphasized that impedance measurements are mainly used for measuring space charge capacities. They are usually performed in a frequency range of 10 kHz up to nearly 1 MHz depending on the Faraday current. 

Cjc and / -, can only be evaluated from the complex Z data if the equivalent circuit is known. One simple circuit which is valid for a one-electron redox process, has been shown in Fig. 4.8. Any equivalent circuit can be tested by measuring Z and Z over a large frequency range. It is common to plot the imaginary Z values versus the cor- 

As already mentioned, the equivalent circuit given in Fig. 4.8 is the simplest case. If the impedance is also investigated in a potential range where large currents are observed, then one has to take into account that the concentration of the ox or red species may be lower at the surface than in the body. In such a case, the concentration of the involved species will also be modulated when a small a,c. voltage is applied. Thus, we have instead of Eq. (4,15) 

For details and an exact derivation of the reader is referred to ref. [13]. The derivation also shows that Z is in series with as shown in Fig. 4.13a. Typically, the Warburg impedance leads to a linear increase of Z with rising Z and the slope is 45° as also shown in Fig, 4.13a. In this case, Z has been calculated assuming an infinite thickness of the diffusion layer. Any convection of the liquid limits the thickness of the diffusion layer. The latter is limited to a well defined value when a rotating disc electrode is used (see Section 4.2.3). In this case, the impedance spectrum is bent off at low frequencies as shown in Fig. 4.13b. The Z branch i.s only linear at its high frequency end where it shows a slope of 45°. 

Sometimes more than one semicircle occurs in the impedance spectrum as well as the Warburg impedance. The origin of the second semicircle is usually due to a two-step reaction process, i.e. an intermediate state is involved. This can occur, for instance, if an adsorbed molecule participates in the reaction, or if energy states within the energy gap at the semiconductor surface are involved, or if just more than one electron occurs in the reaction. In these cases, becomes a complex quantity and we have to replace by a complex Faraday impedance Zp, as illustrated in Fig. 4.14. Such a Faraday impedance depends on the reaction mechanism. One can derive Zp from a kinetic model proposed for a reaction process. First we derive AJ, which depends finally on rate constants and on various derivatives, such as Acjn,ermediates ot Ap where 

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