The analysis of electrode impedances complicated by the presence of a constant phase element

TL;DR: In this article, it was shown that the electrical double-layer at a solid electrode does not in general behave as a pure capacitance but rather as an impedance displaying a frequency-independent phase angle different from 90°.

About: This article is published in Journal of Electroanalytical Chemistry.The article was published on 1984-09-25 and is currently open access. It has received 2602 citations till now. The article focuses on the topics: Constant phase element & Electrode.

Summary

(I) INTRODUCTION

• This ideal behaviour is experimentally best observed at mercury/ solution interfaces, for example at a dropping mercury electrode (DME), a hanging mercury drop electrode (HMDE), or a mercury pool electrode.
• This means that, leaving the fundamental background unexplained, the authors will try to analyse mathematically some possible cases of distributed behaviour, and discuss their consequences for procedures for analysing experimental data.

(II. 1) The Meally polarized electrode

• The basic assumption in tins section will be that the impedance Z of the ideally polarized electrode behaves in conformity with the complex plane plot represented by the dashed straight line in Fig. la throughout the whole frequency range o < w < o¢, where w is the angular frequency (s-l).
• Comparison of eqns. (la) and (4a) leads to if it is postulated that in their case the observed behaviour is due to inhomogenmty of the electrode surface in such a way that only the value of the double-layer capacity is distributed along the interface.
• This means that the following relations hold: rG(r) = F(s) with s = In(r/%) ( 7) EQUATION where H(x) is the empirical or phenomenologmal function that describes the dependence of the imaginary component Y" on x = ln(1/~0ro), i.e. in their case: the "ideal" behaviour).
• The idea that it is just an intrinsic property of the electrical double-layer, the atoms, molecules and ions in it being subject to extremely high electric field strengths and fluctuations therein.
• As Cole and Cole have pointed out [13, 15] , the special property of this process should be the frequency independence of the ratio of maximum energy stored to the energy dissipated per cycle.

(I12) Charge density and capacttance of the ideally polartzed electrode

• Apart from the question which model is more probable, the CPE behavlour, if observed, causes a problem with respect to the study of the double-layer.
• Normally, capacity measurements are frequently employed to determine double-layer properties such as charge density, adsorption, etc. as a function of the dc potential.
• From eqn. (3) it appears that the CPE admittance contains a resistive component parallel to the capacitive component, so that on application of a potential perturbanon from a certain initial state, charge will pass the interface perpetually.
• Also in this respect it has to be concluded that the present state of knowledge about the physical properties of the solid metal solution interface is as yet unsatisfactory.

II.3 The non-ideally polarized electrode (11.3.1) Irreversible urn form charge transfer

• R is proceeding, its rate being controlled by slow charge transfer, so that diffusion or any other kind of mass transfer need not be accounted for.
• Then, in the classical case, a transfer resistance Rot appears m the Randles's equivalent circuit parallel to the double-layer capacitance C d.
• Thus far in the literature, one intuitively replaced C d by the CPE m cases where the CPE behaviour was observed in the absence of the faradalc process, or more generally in cases where a fit assuming normal behaviour of the double-layer proved to be impossible.
• In other words, impedance or admittance analyses are carried out adopting the equivalent circuit shown in Fig. 4a .
• Then the total admtttance is most convemently expressed by EQUATION ).

3.2.) Irreverstble distributed charge transfer

• Next, the question arises whether it is philosophically correct to assume the rate of charge transfer to be uniform along the interface, whereas it is assumed that the most characteristic property of the interface, the potential dependence of the charge density, is distributed.
• Again, the lack of basic understanding prevents a fundamental discussion and even a decision whether the answer to this question is yes or no.
• Yet it seems useful to consider from the phenomenological point of view what can be expected in the case where the value of Rct is distributed.
• Two possiblhties will be envisaged, namely the case where Rot and C a are distributed independently and the case where the distributions are coupled.

(II. 3. 3) Non-trreverstble charge transfer

• Algebraic manipulations as in the foregoing sections will be more tedious, but it can be inferred that the conclusions and considerations will be similar and analogous.
• For, replacing R~ I by Yv in eqns. (19) and (20) introduces an extra frequency dependence incorporated in Q-1, whereas this will not occur in the more simple eqn. (15) .
• In other words, if the distribution model makes sense, this causes a more complex frequency dependence of the term due to the distributed capacity.

(III. 1) Potenttal dependence

• To this end, the authors formulate the following premises: (i) The rate expression for the electrode reaction O + n e-~.
• For dc irreversible reactions, the second term in eqn. ( 30) is negliDble because of the large value of exp(-ep) in the faradaic region.
• With the aid of eqn. (27), the integral of the right-hand side of eqn. (31) can be computed as a function of the theoretical kf(E) values.
• The left-hand side can be determined from R~t 1 vs. E data and thus enables us to find kf(E).
• For dc reversible reactions, both terms containing Jv on the right-hand side of eqn. (30) vanish and so kf(E) can be obtained directly from R~t 1.

(111.2) Frequency dependence," ac Irreversible case

• If eqn. (15) or (19) ~s supposed to hold, it is easily derived that on varying the frequency (at fixed potential) the well-known circular arc, as first described by Sluyters [19] , will be obtained, but its centre is shifted vertically underneath the horizontal axis (see Fig. lc ).
• As usual, the evaluation of the system parameters in the whole frequency region will start by calculating the "interfacial admittance" YE by means of eqn. (14) , where for the ohmic resistance R a a proper value is substituted, e.g. determined by extrapolation of measured data to infinite frequency, preferably at a potential outside the faradaic region.
• From eqns. ( 14) and (15) , it follows that both the real and the imaginary components of YE are functions of co following.

(111.3) Frequency dependence; general case

• As has been argued in Section (II. 3.3) , to what extent this influences the eventual frequency dependence of IrE now depends on the origin of the CPE.
• The analysis of YE data, calculated as before from Y and R~, requires numerical procedures rather than graphical ones.
• If the fit is successful, this may be considered as an argument supporting the validity of the model underlying eqn. (37).

(IV) EXPERIMENTAL (IV.l) Reductton of protons

• Also the electrodes--monocrystalline and polycrystalline gold--were the same as those used before.
• The electrodes were contacted to the solution by the pendant meniscus method.
• Reproducibility and stability of the surface state were frequently checked by cyclic voltammetry.
• For the dc and ac measurements, the potenttal step procedure with intermittent sweeps into the oxygen region was applied [20] .
• All potentials are referred to the saturated sodium chloride calomel electrode (SSCE).

(1V.2) Reductton of trls-oxalato ferric tons

• The solutions were prepared by dissolving oxahc acid and potassium carbonate (both Merck suprapur) in freshly twice-distilled water.
• The solutions were deaerated by means of argon.
• In oxalate solutions it is not possible to clean the electrode surface electrochemically by sweeping into the oxygen region, since then the gold dissolves to a considerable extent.
• Therefore the potential was swept only to + 0.25 V vs. SCE and back to the initial potential E, = 0 V vs. SCE (saturated calomel electrode), which was found to suffice in this medium.
• The frequency in the impedance measurements was varied between 80 Hz and 10 kHz (twelve frequencies).

(V. 1) The proton reductton from perchlorw acid

• The curves are circular arcs with their centres somewhat below the horizontal axis.
• It appears that at the polycrystalline electrode a is slightly tugher than at the single crystal electrodes.
• In order to examine the dispersion along the loci in Fig. 6b , the values of Qwere determined by plotting log Y~' vs. log o~ (see eqn. 36).
• The R~t a data were analysed following the procedures described in Section (III) and combined with results from dc current responses to potential step perturbations as described in the previous paper [20] .
• Gerlscher and Mehl tried to verify their theory at copper and silver electrodes and in fact observed a frequency dispersion in Y~, which, however, was said to be "spurious" rather than a confirmation of the theory.

(V.2) The reduction of trts-oxalato Fe(III)

• This system is of interest for the present treatment because it can be expected to be quasi-irreversible and therefore the frequency dependence of Y~ and Y~' may originate from the faradalc contribution as well as from the CPE.
• The results of such a procedure, however, appeared unsatisfactory, especially at potentials close to the polarographic half-wave potential.
• The conclusion is that also at the higher frequencies, the factor containing p in eqn. (37) is different from umty (the ~rreversible hmit).
• The data obtained are satisfactordy consistent, as can be deduced from Table 2 , where the results from the two procedures are compared.
• The re-calculated Y~' vs. Y~ plots are shown m Fig. 9a as the drawn curves.

(vI) CONCLUSION

• The most important result of the present work is, in their opinion, that it is shown how severely the CPE influences the frequency dispersion of the interfacial admittance even in the case of quite small rotation angles (2-4 o ) .
• In addition, reported values of the double-layer capacity, obtained by "Co"= Y~'/w at some frequency, are in fact values of an irrational quantity in the case where CPE behaviour occurs.
• Unfortunately, the circumstances of the experiments discussed here did not meet this requirement, mostly because of the irreversible nature of the two systems studied.