Finite-length Warburg diffusion

Figure 10.6 shows that the overall impedance of the system decreases after addition of plasticizer. The data are in agreement with the increase observed in ionic conductivity. From the parameters obtained by fitting the experimental data shown in Fig. 10.6, the apparent diffusion coefficient can be estimated using equation 10.7,where 4 is the thickness of the electrolyte film and 5 is a parameter related to the element O in the equivalent circuit proposed, which accounts for a finite-length Warburg diffusion (Zd), which represents a kind of resistance to mass transfer. 

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... 

The diffusion impedance of a bulk electrolyte can be described by a finite length Warburg impedance with transmissive boundary (Eq. (7)). A transmissive boundary is appropriate because an ion produced at the cathode is consumed at the anode and vice versa during battery electrochemical processes. A more precise treatment, using... 

Particular attention should be paid to process Pia- As will be shown later, in the equivalent circuit this process is modeled by a generahzed finite length Warburg (G-FLW) element accounting for the diffusion loss within the anode substrate. 

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 ... 

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...

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. 

Although there is no complete derivation of a generahzed CPE yet available which arises from nonuniform diffusion (NUD) in a finite-length region, one may heuristically modify the CPE and Warburg diffusion expressions in such a way as to generalize them both. The result is... 

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... 

This semicircle is an instance of Case discussed above. On the other hand when C >> tt, the GR arc may be of virtually the same shape as the finite-Warburg diffusion arc. The possibility of confusion between these two arcs thus arises but can be resolved from their different dependencies on length, il. The size of the unnormalized GR arc is independent of Jl, as it should be. Data which include even a part of the GR arc should allow the GR parameters to be estimated. 

The boundary conditions for the Warburg impedance, Zw, previously discussed were such that semi-infinite diffusion prevails. However, as we have already seen in connection with voltammetry and other techniques for film-modified electrodes, diffusion in these cases is bounded and is restricted to a thin layer of thickness d. This problem has been independently addressed by three different groups [110-113] and leads to essentially the same end result, namely that the phase angle begins to increase at very low frequencies due to the onset of finite length effects. Figure 20.27a illustrates the complex plane impedance plot obtained in this instance. 

Diffusion of Particles in Finite-Length Regions - the Finite Warburg Impedance... 

Thus, the diffusion impedance expressions depend on the electrode separations d at low frequencies. One way to identify the finite Warburg impedance is to use measurements at various values of the electrodes separation d. When LpCO / D 3 (at oo °o), the tank term approaches imity, the diffusion length is negligible compared to the whole region avaUable for diffusion d, and Zpjj-j-approaches infinite length Warburg Zy ... 

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