Finite-length diffusion impedance

In the second case (limit of fast kinetics at the gas-solid interface), the film becomes entirely bulk transport limited, corresponding to the limit of Hebb— Wagner polarization. Since electronic conduction is fast, this situation yields a Warburg impedance for finite length diffusion ... 

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

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.. 

Total electrode impedance consists of the contributions of the electrolyte, the electrode solution interface, and the electrochemical reactions taking place on the electrode. First, we consider the case of an ideally polarizable electrode, followed by semi-infinite diffusion in linear, spherical, and cylindrical geometry and, finally a finite-length diffusion. 

Fig. II.5.5 Nyquist impedance plot due to finite length diffusion with a transmissive boundary condition...

A finite length diffusion layer thickness cannot only be caused by constant concentrations of species in the bulk of the solution but also by a reflective boundary, that is, a boundary that cannot be penetrated by electroactive species (dc/dr = 0). This can happen when blocking occurs at the far end of the diffusion region and no dc current can flow through the system, for example, a thin film of a conducting polymer sandwiched between a metal and an electrolyte solution [6]. The impedance in this case can be described with the expression... 

In practical applications of EIS it is often found that the experimental data for the finite-length diffusion cannot be approximated by Eq. (4.72) or Eq. (4.83). For example, in the case of hydrogen absorption in Pd the low-frequency reflective impedance is not strictly capacitive, or in the transmissive case the complex plane plot is slightly depressed [154-156]. In such cases one should use a so-called generalized finite-length Warburg element for transmissive... 

Fig. 4.13 Cranplex plane plots fOT reflective finite-length diffusion left -Warburg impedance, right -total impedance dashed line - ideal case, continuous line - generalized Warburg with = 0.94...

Macdonald and coworkers [158-164] obtained an exact solution of the finite-length diffusion impedance in unsupported conditions where the Nemst-Planck and continuity equations for both negative and positive mobile charges were solved and involved full satisfaction of Poisson s equation. One can expect such conditions in diluted electrolytic solutions and in poorly conductive solids. 

Let us consider now diffusion inside a sphere neglecting the diffusion gradient outside the sphere. Such a case might be observed for hydrogen absorption or Li intercalation into spherical particles. Diffusion inside the sphere can go only to the sphere center and is called finite-length internal spherical diffusion. In the steady state in which the impedance measurements are carried out, dc concentration inside the sphere is uniform, and no dc current is flowing. Ac perturbation causes oscillations of concentration at the sphere surface, which diffuse inside the sphere. In such a case, two boundary conditions in Eq. (4.94) are changed ... 

In the case of cylindrical diffusion at low frequencies, impedance becomes parallel to the real axis and never approaches the real axis. This is related to the fact that in chronoamperometry the current never goes to zero. Jacobsen and West [167] also considered a case of finite-length cylindrical diffusion. 

In such a case the diffusion layer thickness / is replaced by in the finite-length transmissive diffusion impedance, Eq. (4.72) ... 

Fig. 7.8 Electrical equivalent circuits of faradaic impedance corresponding to (a) indirect, Eq. (7.81), and (b) direct, Eq. (7.83), hydrogen absorption reaction with finite-length linear diffusion of hydrogen...

The total impedance complex plane plot for indirect hydrogen absorption without hydrogen evolution, including solution resistance and double-layer capacitance, is displayed in Eig. 7.9. It shows two semicircles due to the — Cdi and / ab Cp coupling followed by the finite-length reflective linear diffusion displaying a line at 45° followed by a capacitive line at 90°. 

As was shown earlier, the presence of the CPE of fractal impedance produces a distribution of the time constants. In addition, other elements such as the Warburg (semi-infinite or finite-length) linear or nonlinear diffusion, porous electrodes, and others also produce a dispersion of time constants. Knowledge about the nature of such dispersion is important in the characterization of electrode processes and electrode materials. Such information can be obtained even without fitting the experimental impedances to the corresponding models, which might be still unknown. Several methods allow for the determination of the distribution of time constants [378, 379], and they will be briefly presented below. 

Figure 2.1.13. Complex plane representations of the impedance due to a finite-length diffusion process with (a) reflective, (b) transmissive boundary conditions at x = 1.

But an electrolytic cell or dielectric test sample is always finite in extent, and its electrical response often exhibits two generic types of distributed response, requiring the appearance of distributed elements in the equivalent circuit used to fit IS data. The first type, that discussed above, appears just because of the finite extent of the system, even when all system properties are homogeneous and space-invariant. Diffusion can lead to a distributed circuit element (the analog of a finite-length transmission line) of this type. When a circuit element is distributed, it is found that its impedance cannot be exactly expressed as the combination of a finite number of ideal circuit elements, except possibly in certain limiting cases. 

Determination of Parameters from Randles Circuit. Electrochemical three-electrode impedance spectra taken on electrochromic materials can very often be fitted to the Randles equivalent circuit (Randles [1947]) displayed in Figure 4.3.17. In this circuit R /denotes the high frequency resistance of the electrolyte, Ra is the charge-transfer resistance associated with the ion injection from the electrolyte into the electrochromic film and Zt, is a Warburg diffusion impedance of either semi-infinite, or finite-length type (Ho et al. [1980]). The CPEdi is a constant phase element describing the distributed capacitance of the electrochemical double layer between the electrolyte and the film having an impedance that can be expressed as... 

We will exemplify this approach for the finite-length-diffusion case. The impedance function for this case is described in Section 2.1.3, Eq. (135) for a reflective boundary and Eq. (136) for a transmissive one. The first case corresponds to diffii-... 

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