Diffusion impedance Finite

On the basis of this model and the equivalent circuit shown in Figure 4.5.67, the changes and differences, depending on the used anode in the fuel cell (Pt/C or PtRu/C) in the impedance spectra during the experiment, are dominated by the changes of the charge transfer resistance of the anode (Raj), the surface relaxation impedance (Rg, tg) and the finite diffusion impedance (Z ). 

A finite-length diffusion impedance of charged particles is represented by Zq parameter. The resulting "finite length" diffusion-impedance response does not have the -45° line, instead displaying a depressed semicircle or a vertical -90° line. The circuit is representeid by a parallel combination of a CPE and an ideal resistor which also strongly depends on the electrochemical potential. The universal expression for finite diffusion impedance (/to) was... 

Fig. 11.18. Variation of impedance for diffusive systems (a) Semi-infinite diffusion (b) Reflective finite diffusion (c) Transmissive finite diffusion.

It should be noted that the presence of diffusion controlled corrosion processes does not invalidate the EIS method but does require extra precaution. In the case of a finite diffusional impedance added in series with the usual charge transfer parallel resistance shown in Fig. 3b, the frequency-dependent diffusional impedance can be described as (21)... 

Finite diffusion — Finite (sometimes also called -> limited) diffusion situation arises when the -> diffusion layer, which otherwise might be expanded infinitely at long-term electrolysis, is restricted to a given distance, e.g., in the case of extensive stirring (- rotating disc electrode). It is the case at a thin film, in a thin layer cell, and a thin cell sandwiched with an anode and a cathode. Finite diffusion causes a decrease of the current to zero at long times in the - Cottrell plot (-> Cottrell equation, and - chronoamperometry) or for voltammetric waves (see also - electrochemical impedance spectroscopy). Finite diffusion generally occurs at -> hydrodynamic electrodes. 

An important example of the system with an ideally permeable external interface is the diffusion of an electroactive species across the boundary layer in solution near the solid electrode surface, described within the framework of the Nernst diffusion layer model. Mathematically, an equivalent problem appears for the diffusion of a solute electroactive species to the electrode surface across a passive membrane layer. The non-stationary distribution of this species inside the layer corresponds to a finite - diffusion problem. Its solution for the film with an ideally permeable external boundary and with the concentration modulation at the electrode film contact in the course of the passage of an alternating current results in one of two expressions for finite-Warburg impedance for the contribution of the layer Ziayer = H(0) tanh(icard)1/2/(iwrd)1/2 containing the characteristic - diffusion time, Td = L2/D (L, layer thickness, D, - diffusion coefficient), and the low-frequency resistance of the layer, R(0) = dE/dl, this derivative corresponding to -> direct current conditions. 

Figure 3.9. Variation of impedance for diffusive systems a semi-infinite diffusion b reflective finite diffusion c transmissive finite diffusion...

The following model is a bounded Randles cell also accounting for a linear but finite diffusion, with a homogeneous layer of finite thickness. The structure of the model is shown in Figure 4.18a. The corresponding impedance is... 

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],... 

As an exercise, the reader can verify that equation (2.73) satisfies both real and imaginary parts of equation (2.70). This development represents the starting point for both the Warburg impedance associated with diffusion in a stationary medium of infinite depth and the diffusion impedance associated with a stationary medium of finite depth. 

Since the specie Mg " diffuses toward the electrode surface, the resulting concentration perturbation (0) is obtained from the finite-length diffusion impedance, represented... 

Diffusion through a stagnant layer of finite thickness can also yield a uniformly accessible electrode. The diffusion impedance response of a coated (or film-covered) electrode, imder the condition that the resistance of the coating to diffusion is much larger than that of the bulk electrol5M e, is approximated by the diffusion impedance of file coating. This problem is also analyzed in Section 15.4.2. 

Equation (11.70) can be considered to be a finite-length diffusion impedance. As tanh(oo) = 1, the impedance response asymptotically approaches the response for an infinite domain at high frequencies, i.e.. 

Several authors have addressed the influence of a finite vedue of the Schmidt number on expressions for the convective-diffusion impedance. Levart and Schuh-mann showed that the concentration term could be expressed as a series expansion in Sc i.e.. 

A similar development was provided by Tribollet and Newman for electro-hydrodynamic impedance. The use of look-up tables facilitates regression of models to experimental data that take full accoimt of the influence of a finite Schmidt number on the convective-diffusion impedance. Use of only the first term in equation (11.97) yields a numerical solution for an infinite Schmidt number. Tribollet and Newman report use of the first two terms in equation (11.97) The low level of stocheistic noise in experimental data justifies use of the three-term expansion reported here. 

A graphical method was reported by Tribollet et al. that can be used to extract Schmidt numbers from experimental data in which the convective-diffusion impedance dominates. 3 The technique accounts for the finite value of the Schmidt... 

The residual errors are presented in Figures 20.11(a) and (b) for the real and imaginary parts of the impedance, respectively. The dashed lines represent the experimentally determined noise level of the measurement. The scales used to present the results in Figure 20.11 are in stark contrast to the scales used in Figure 20.5. The residual error plots show that the measurement model provides a substantially better fit to the data than does the finite-diffusion-length model. 

The quantitative and qualitative analysis presented in Section 20.2.1 demonstrates that the finite-diffusion-layer model provides an inadequate representation for the impedance response associated with a rotating disk electrode. The presentation in Section 20.2.2 demonstrates that a generic measurement model, while not providing a physical interpretation of the disk system, can provide an adequate representation of the data. Thus, an improved mathematical model can be developed. 

As described in Sections 20.2.1 and 20.2.2, the quality of the regressions can be assessed to varying degrees of success by inspection of plots. The Nyquist or complex-impedance-plane representation given in Figure 20.13 reveals the difference between the finite-diffusion-length model and the models based on numerical solution of the convective-diffusion equation, but cannot be used to distinguish the models based on one-term, two-term, and three-term expansions. 

At first glance, it may not be obvious that such an approach should work. It is well known, for example, that the impedance spectrum associated with an electrochemical reaction limited by the rate of diffusion through a stagnant layer (either the Warburg or the finite-layer diffusion impedance) can be approximated by an infinite number of RC circuits in series (the Voigt model). In theory, then, a measurement model based on the Voigt circuit should require an infinite number of parameters to adequately describe the impedance response of any electrochemical system influenced by mass transfer. 

So far unlimited diffusion has been assumed. When the diffusion layer has finite dimension, with a boundary of constant concentration of the diffusing species and permeable for these species, however, a different expression for the diffusion impedance is derived which is the so-called Nemst impedance. This impedance is given by the equation... 

The complete expression for the Warburg impedance corresponding to finite diffusion with reflective boundary condition is [5]... 

Because of the assumption of semiinfinite diffusion made by Warburg for the derivation of the diffusion impedance, it predicts that the impedance diverges from the real axis at low frequencies, that is, according to the above analysis, the dc-impedance of the electrochemical cell would be infinitely large. It can be shown that the Warburg impedance is analogous to a semi-infinite transmission line composed of capacitors and resistors (Fig. 8) [3]. However, in many practical cases, a finite diffusion layer thickness has to be taken into consideration. The first case to be considered is that of enforced or natural convection in an... 

Fig. 9 Impedance spectra for diffusion-limited behavior (a) semi-infinite diffusion (b) finite diffusion with unhindered ion transfer at the far end of the diffusion region and (c) finite diffusion with blocked ion transfer at the far end of the diffusion layer.

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