De Levie’s equations

The analysis of the results indicates that the resistance at (o = 0, Rp obtained using the correct analysis is twice that found from de Levie s equation. In addition, the plot of squared impedances produces a deformed ellipsoid instead of a perfect semicircle. It has been shown " that the CNLS fit of the simulated impedances to the de Levie equation (195) is not good, there being systematic differences between these two curves. However, when the CPE is used instead of the double-layer capacitance, the approximation is good. The values obtained for the parameter < ) are between 0.91 and 0.93. " In this case the use of the CPE only hides the inadequacy of the model. 

I is the groove depth (normal to the surface), and Z ei is the double-layer impedance per unit of the true surface area. Equation (9.14) reduces to the impedance of a perfectly flat surface for ji = 90° and to the impedance of cylindrical porous electrode for p = 0°. Gunning [414] obtained an exact solution of the de Levie grooved surface not restricted to a pseudo-one-dimensional problem in the form of an infinite series. Comparison with de Levie s equation (9.14) shows that the deviations arise at higher frequencies or, more precisely, at high values of the dimensionless parameter Q. = coC ialp, where a is half of the distance of the groove opening, a = Itan p (Fig. 9.8). 

Equation (9.19) differs from de Levie s equation, Eq. (9.7), by the presence of one additional term. It should be noted that when r = 0, Eq. (9.19) reduces to Eq. (9.6) or (9.7). When the frequency m oo, A -> oo, the second term in Eq. (9.19) goes to zero and the first term becomes Wa,p, which corresponds to the average harmonic resistance of the solution and the electrode. When w 0, the real part of the first term in parentheses goes to 2/3 and the first term becomes 2/ n,p/3, while the real part of the second term goes to R o,p/3. This means that the low-frequency impedance is real and equal to 2R o,p/3 + R o,p/3. At low frequencies the imaginary parts of both terms go to infinity and the electrode displays its capacitive behavior. The complex plane plots of the total impedance, as well as those of the first and second terms, are compared in Fig. 9.15. The second term in Eq. (9.19), Fig. 9.15c, shows a complex plane plot that is similar to that in the absence of the electrode resistance (de Levie s solution). The first term shows a... 

Because the correct numerical solution for a porous electrode in the presence of a potential gradient in pores demands knowledge of the pore parameters and more complex mathematics, in practice, a simplified de Levie equation (9.7) is used in approximations, which means that the experimentally measured impedances are fitted to Eq. (9.7). As was shown earlier, for the same pore and kinetic parameters, de Levie s equation underestimates low-frequency impedances by up to 100%, which might not be that important for very porous electrodes characterized by a very large surface area. 

Fig. 9.26 Complex plane plots at porous electrode in presence of redox proeess at eonstant overpotential rjO = 0.2 V and different exehange current densities continues lines - simulatirais, Eq. (9.46), dashed line - according to de Levie s equation (9.7) Cji = 20 uF cm, other pore parameters as in Fig. 9.22 (From Ref. [72] with kind permission from Springer Seience and...

As was mentioned earlier, the square of the semi-infinite-length pore impedance, Eq. (9.27), produces a perfect semicircle however, simulated impedances in the presence of a constant current lead to somewhat distorted semicircles [407]. This fact is clear from the CNLS approximations of the simulated impedances by de Levie s equation [435]. Such a fit is displayed in Fig. 9.29. However, when the CPE was used in A in Eq. (9.23),... 

Another method is to assume a certain distribution of pore parameters. Song et al. [449 52], in a series of papers, considered the distribution of pore parameters for electrodes in the absence of electroactive species, i.e., for purely capacitive electrodes. De Levie s equation (9.7) is applicable to individual pores, but for different pores different values of the parameter A are obtained. The dimensionless penetrability parameter, a, was defined as... 

Under some conditions, it is necessary to calculate the current distribution in the system from first principles, rather than by merely assuming a transmission-line-like behavior. This will be the casewhen the pores are not long compared with their radius. This type of electrode has been called a rough electrode by de Levie (27). The equipotential surfaces cannot be assumed to be perpendicular to the axis of the pores. Here, it is necessary to solve Laplace s equation... 

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