Diffusion impedance Complicated

Impedance Spectroscopy. Impedance spectroscopy has been carried out on devices with WO3 as the cathodic electrochromic layer, counter electrodes of iridium oxide, polyaniline or Prussian blue, and polymers as electrolytes (Katsube et al [1986], Friestad et al [1997]). The equivalent circuit for a whole device becomes very complicated. In the works quoted above simplified, Randles-type circuits were used for the two electrochromic layers, while the ion conductor was modeled by a pure resistance, or neglected. Extraction of device parameters from the data fitting was reported. However, it is clear that in many cases it will be difficult to distinguish the contributions from the different layers in a device, in particular if the migration impedances, ion diffusion impedances, etc. are of the same order of magnitude. When it comes to characterizing electrochromic devices, impedance spectroscopy is a very time-consuming process, since a spectrum down to low frequencies should be taken at a number of equilibrium potentials. Thus we believe that transient current measurements in many cases offer a faster alternative that sometimes allows a simple determination of diffusion coefficients. 

The kinetic impedance Zj represents the faradaie impedance in the absence of a concentration overpotential. In the simplest case, the kinetie impedance corresponds to the transfer resistance Rt, but in more complicated situations it may include several circuit elements. The diffusion impedance Z describes the contribution of the concentration overpotential to the faradaic impedance and therefore depends on the transport phenomena in solution. In the absence of convection, it is referred to as the Warburg impedance and, in the opposite case, as the Nernst impedance Z. ... 

In a packed column, however, the situation is quite different and more complicated. Only point contact is made between particles and, consequently, the film of stationary phase is largely discontinuous. It follows that, as solute transfer between particles can only take place at the points of contact, diffusion will be severely impeded. In practice the throttling effect of the limited contact area between particles renders the dispersion due to diffusion in the stationary phase insignificant. This is true even in packed LC columns where the solute diffusivity in both phases are of the same order of magnitude. The negligible effect of dispersion due to diffusion in the stationary phase is also supported by experimental evidence which will be included later in the chapter. 

A complication that occurs on a low at.% Ru electrode is that, owing to the low Faradaic currents (low Ru content) and hence large Rt value, currents due to other trace redox reactions, e.g. oxygen reduction, become more detectable. This reveals itself in a phase-angle of 45° as co 0 as trace oxygen reduction would be diffusion-controlled. The impedance corresponding to this situation can be shown to be the same as that in Equation 5.3, with U(p) expressed by the relationship ... 

The hydrodynamic repulsion of molecules in solution has already been discussed (see Chap. 8, Sect. 2.5) and is analysed in much more detail in Sect. 3 of this chapter. It appears to be a very significant complication, yet it can be analysed relatively straightforwardly. The basic effect arises when one molecule approaches another. Solvent molecules between these two molecules have to be squeezed out in a direction approximately perpendicular to the motion of the approaching molecules. Their approach (and similarly their separation) is impeded and the effective diffusion coefficient is less than that when the molecules are more widely separated. 

However, although powerful numerical analysis software, e.g., Zview, is available to fit the spectra and give the best values for equivalent circuit parameters, analysis of the impedance data can still be troublesome, because specialized electrochemical processes such as Warburg diffusion or adsorption also contribute to the impedance, further complicating the situation. To set up a suitable model, one requires a basic knowledge of the cell being studied and a fundamental understanding of the behaviour of cell elements. 

Two impedance arcs, which correspond to two relaxation times (i.e., charge transfer plus mass transfer) often occur when the cell is operated at high current densities or overpotentials. The medium-frequency feature (kinetic arc) reflects the combination of an effective charge-transfer resistance associated with the ORR and a double-layer capacitance within the catalyst layer, and the low-fiequency arc (mass transfer arc), which mainly reflects the mass-transport limitations in the gas phase within the backing and the catalyst layer. Due to its appearance at low frequencies, it is often attributed to a hindrance by finite diffusion. However, other effects, such as constant dispersion due to inhomogeneities in the electrode surface and the adsorption, can also contribute to this second arc, complicating the analysis. Normally, the lower-frequency loop can be eliminated if the fuel cell cathode is operated on pure oxygen, as stated above [18],... 

High r factors are, however, not without some other complications since they imply porosity of materials. Porosity can lead to the following difficulties (a) impediment to disengagement of evolved gases or of diffusion of elec-trochemically consumable gases (as in fuel-cell electrodes 7i2) (b) expulsion of electrolyte from pores on gas evolution and (c) internal current distribution effects associated with pore resistance or interparticle resistance effects that can lead to anomalously high Tafel slopes (132, 477) and (d) difficulties in the use of impedance measurements for characterizing adsorption and the double-layer capacitance behavior of such materials. On the other hand, it is possible that finely porous materials, such as Raney nickels, can develop special catalytic properties associated with small atomic metal cluster structures, as known from the unusual catalytic activities of such synthetically produced polyatomic metal clusters (133). 

Probably one of the most serious objections to the above theoretical model for the cell-impedance controlled lithium transport is the use of the conventional Pick s diffusion equation even during the phase transition, because lithium diffusion inside the electrode should be influenced by the phase boundary between two different phases. However, the contribution of the phase boundary to lithium transport is complicated and not well known. For instance, one can neither know precisely the distribution nor the shape of the growing/shrinking phase during the phase transition. 

The theory of electrode impedance in the case of a faradaic process proceeding at the surface has been thoroughly formulated and many special cases have been solved [1,2, 14, 15]. The formulation is based on Pick s laws with appropriate boundary conditions which take into account the periodic time-dependence of the perturbation voltage. Since the mathematics involved are rather complicated, they will be omitted here. Instead, a general solution will be given [16] for a simple, slow electrode reaction of the first order and the plane diffusion... 

Though in the general case, mathematical expressions of the Nernst model are more complicated than of those semi-infinite diffusion, stationary mass transport is described by a rather simple Eq. (3.12). In this connection, there occurs an interesting possibility to use superposition of both models, which is convenient to apply when i is the periodic time function. Perturbation signals of this type are considered in the theory of electrochemical impedance spectroscopy. In this case, i(t)... 

When the impedance plot is sufficiently detailed and the various contributions resolved, one can determine the various resistances, capacitance, and diffusion process parameters. Care must be taken in analyzing the data as different equivalent circuits can yield the same or very similar impedance plots. ° It may therefore be meaningless to use sophisticated programs for fitting complicated equivalent circuits comprising many elements and to look for best fit if the data are not sufficiently accurate, which is usually the case. It is then advisable to guess an as simple equivalent circuit as possible that can be justified electro-chemically for describing the supposed processes in the cell and fit its parameters. 

Other more complicated forms of impedance-frequency dependencies can be revealed when a combination of diffusion process and homogeneous bulk solution reaction or recombination of electroactive species is considered [34]. In the case of homogeneous bulk solution and electrochemical reaction, the corresponding charge-transfer impedance Z can be simplified by a kinetic resistance R. Another parameter, —critical frequency of reaction or recom-... 

A more complicated expression for pore impedance develops when Faradaic charge transfer is present [68], such as that for the impedance of an electrode consisting of cylindrical pores in the absence of DC current (that is, in the absence of diffusion) and represented by a parallel combination of charge-... 

Then, the two models give equivalent results. This calculation was also given by Buck without the electroneutrality hypothesis (i.e. Cp 0). The transmission line approach is often called the porous model of a conducting polymer as electrons are supposed to cross the polymer (phase 1) and ions are supposed to move into pores, filled by electrolyte, represented by the second branch of the transmission line. It is noticeable that the transmission line approach allows more complicated kinetics to be tested for a two-species problem, e.g. charge transfer in parallel to the capacity Ce(x) and C,- (x), or diffusion of the ion in the ionic pores , i.e. to introduce complex impedances instead of the real resistance p and/or p2, of the pure capacitances Ci and/or C2. It also allows position-dependent parameters to be introduced to mimic concentration gradients in the polymer [Cj(x) constant]. 

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