Stoichiometry polarization

4 Stoichiometry polarization a) Measurement with selectively blocking electrodes 

Before we go into the thermodynamics let us estimate the effects using the heuristic equivalent circuit in Fig. 7.23, an approximation that we shall later find is suprisingly 

At this point it is useful to lump together all the pure interfacial effects that take place at much smaller times (r- -) under a circuit in series, as done in 

If we denote the time constant of the stoichiometry polarization by r (cf. C ), the relation 

Hence at comparatively short times t C the capacitor (in Fig. 7.23) is permeable and the problem reduces to what has already been discussed. For times of the order of r, the capacitors characterized by 0 and are no longer permeable and our equivalent circuit is reduced to the key element (C in series with Rion both parallel to R on) in series with R-. We shall now consider the corresponding transient behaviour  

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Figure 37. The impedance spectrum of the cell Pt Ag2Te Pt displays the stoichiometry polarization (200°C, fine composition set up by Coulometric titration). The transition from a straight 45° line to a semicircle, before the maximum frequency, corresponds to the theoretical treatment given in Section III.3.227 Reprinted from R. Andreaus and W. Sitte, J. Electrochem. Soc., 144 (1997) 1040-1044. Copyright 1997 with permission from The Electrochemical Society, Inc.

The more accurate relationships obtained from the underlying microscopic current equations are discussed below (see Section III.3.iii). There we will see that the impedance that describes the stoichiometry polarization is composed of a so-called Warburg behavior (linear increase with a slope of 45° followed by a semicircular behavior, see Figure 39). Figure 37 shows an experimental example that comes close to the ideal situation. The detailed behavior around the maximum is also in accord with the precise treatment (see Section III.3.iii). 

What has been ignored so far and will only be briefly mentioned is that a stoichiometric polarization is also caused by grain boundaries if the ratio of ionic and electronic conductivities differs from the bulk value, as it is usually the case.230 Figure 41 gives a clear example of this. In the general case of blocking electrodes and grain boundaries we expect even two stoichiometry polarization processes. 

The decisive result for a steady-state stoichiometry polarization, instead of Eq. (67) and its ionic auologue,3 9 is now... 

There is a striking similarity between the experimentally observed and the theoretically calculated profiles, and all four characteristic features occur in both. The calculated location of the minimum, which mainly depends on the vacancy mobility, is close to the location observed in the experiment. The computed temperature dependence of the depth of the minimum corresponds with the results of the measurement. Obviously, the stoichiometry polarization model of resistance degradation correctly predicts the conductivity variations. In particular the almost quantitative agreement of the very characteristic shape of the conductivity distribution proves the validity of the existing model described above. It should be noted that in the calculations only the hole mobility is chosen such that the theoretically and the experimentally observed depth of the minimum is similar, but all other parameters used in the simulation are taken from literature [77, 336, 338],... 

However, this is only unproblematic for the relevant short time period, since a stoichiometry polarization occurs because of the finite nature of the electronic resistance in AgCl (see Section 7.3.4). 

As shown in Ref. [601] the situation is complicated by the fact that also grain boundaries (owing to the different transference numbers there compared to the bulk) can induce stoichiometry polarization. This is ignored here. 

In the case of cell E2 the stoichiometry polarization is faster by a factor of 4 because two blocking electrodes are used. The boimdary condition Eq. (7.91) now applies for both electrode contacts. The concentration function for the polarization is... 

F (A0I + A0ii). The term in the braces is Uqs- and refers exclusively to the sample, while A0i + A0II refers to the potential drops over the two ion conductor samples. If contact resistances also occur at other phase boundaries the drop in electrochemical potential does not equal zero in the stationary state it is given by the product of current and contact resistance. Since, on the time scale of the stoichiometry polarization, electrical bulk processes and charge transfer processes actually behave in a quasi stationary manner, we may write that U =. .. + SilRi, whereby i refers to all parts outside the sample. 

Adsorption or space charge eflfects lead to deviations from pure stoichiometry polarization, which expresses itself in the 45° angle (see Fig. 7.46). 

An exponential behaviom was obtained (see Section 7.3.4a) in the time domain for the switching-on measurements for long times in the case of selective ion or electron blocking electrodes (corresponding to pure stoichiometry polarization ) accordingly the diffusion impedance Z (Warburg impedance) for very small frequencies... 

Note that stoichiometry polarizations can also be induced by grain boundaries [601] (cf. also footnote 86 on page 445). 

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