Warburg impedance finite

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 ... 

So, the term [A0a0 + AR aR ] co-1/2 (1 — i) resembles the Warburg impedance corresponding to diffusional mass transport of A, O and R, with a mobile equilibrium between A and 0, i.e. kQ -> °°, whereupon the term in g = kQ /co would vanish. If, however, kQ has a finite value, the faradaic impedance is enlarged by the Gerischer impedance expressed by the term containing g. 

Termination in a large resistance, i.e. a blocked interface such as a metal totally covered with oxide or a highly resistive membrane used in ion exchange selective electrodes (Section 13.6) (Fig. 11.18c). This is sometimes referred to as the finite Warburg impedance. 

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. 

If there are deviations from the ideal semi-infinite linear diffusion process, the bounded Randles cell can also be modified by replacing the Warburg impedance with a CPE. The structure of the model is shown in Figure 4.20a. This modification is applied when the transport limitations appear in a layer of finite thickness. 

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. 

Remember 11.2 The Warburg impedance, equation (11.52), applies for diffusion in an infinite stagnant domain. This expression applies as a high-frequency limit for diffusion in a finite domain. 

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

Transmission line models can be used for inert electrodes and it is a modification of the Randles model (Fig. 6.3). Since the Randles-circuit can be used to describe a nondistributed system, the transmission line models invokes a finite diffusional Warburg impedance, Z, in place of concentration hindered impedance (Fig. 6.4). Randles model is concerned with Qi (the double layer capacitance), [the resistance to charge transfer) and Z by describing the processes occurring in the film. The expression of total impedance, Ztot, is given by following equation ... 

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... 

Historically, the Warburg impedance, which models semi-infinite diffusion of electroactive species, was the first distributed circuit element introduced to describe the behavior of an electrochemical cell. As described above (see Sect. 2.6.3.1), the Warburg impedance (Eq. 38) is also analogous to a uniform, semi-infinite transmission line. In order to take account of the finite character of a real electrochemical cell, which causes deviations from the Warburg impedance at low frequencies. 

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...

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... 

In cases 1-3 a conventional Warburg impedance cannot arise, as discussed earlier. However, there are certain situations where the equivalent circuit of figure 11 (except for the effects of finite cell thickness) would be appropriate. These include ... 

It should be noted that the aforementioned model deals with the semi-infinite diffusion, which is presented by Warburg impedance. As the diffusion layer of finite thickness 5 is formed at the RDE surface, it is necessary to meet certain relations between 5, specified by Eq. (3.5), and the depth of penetration of the concentration wave Xq = y/2Dfco. Investigations in this field show [25,26] that the two aforementioned diffusion models are in harmony when < 0.15. At Schmidt number Sc = v/D = 2000, this condition is satisfied when co > 6 2 (or/ > 0.1m, where m is rotation velocity in rpm). 

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