Time constant distribution

It must be emphasized that the mathematical simplicity of equations (13.1) and (13.2) is the consequence of a specific time-constant distribution. As shown in this chapter, time-constant distributions can result from nonuniform mass transfer, geometry-induced nonuniform current and potential distributions, electrode porosity, and distributed properties of oxides. At first glance, the associated impedance responses may appear to have a CPE behavior, but the frequency dependence of the phase angle shows that the time-constant distribution differs from that presented in equation (13.7). 

Remember 13.2 Not all time-constant distributions give rise to a CPE. 

Time-constant distributions were described in Section 13.1.1 as having either a 2-D or a 3-D character. Under what conditions could a system show CPE behavior resulting from a distribution in only the axial direction ... 

Figure 9.20 Time constant distribution for four different Cole (immittance) cases, and in addition the log-normal Gaussian) and rectangular distributions. They are all symmetrical around the characteristic time constant Tc, and correspond to circular arcs (Cole) or nearcircular arcs. a = 0 is a single time constant (Debye) case.

Equation (8.4), defining the time constant distribution function, is known as the Fredholm integral equation of the first kind ... 

Ivers-Tiffe and coworkers proposed a modification of the solution of Eq. (8.4) using a Fourier transform [384]. They applied their method to determine the time constant distribution in solid electrolytes and batteries [384-387]. An example of the application of this technique to solid oxide fuel cells (SOFCs) is shown in Fig. 8.20. The complex plane plot shows several poorly separated semicircles, while the time constant distribution function displays five peaks the authors assigned these time constants to different processes. 

A modification of this procedure was proposed in the literature [389] and applied to determine the time constant distribution function [379]. This method is based on the predistribution of time constants uniformly on the logarithmic scale, and to improve the quality of the analysis, a Mmite Carlo technique was used to increase the number of analyzed time constants. Approximation was carried out using a constrained least-squares method and led to a continuous distribution function. This procedure converted the nonlinear problem to a linear one from which versus r , were obtained and produced positive values of the distribution function. The procedure was also applied to the distribution of the dielectric constants [379,389]. 

Two types of time-constant distributions around the electrode, such as radial current distribution along the electrode s surface and distribution of kinetic rates of adsorption-controlled electrochemical reaction, may lead to the CPE behavior [8, 9,10,11]. It has been demonstrated that for the expressions relating the effective capacitance and the CPE parameter Q values Eqs. 3-3 through 3-6 may become more or less accurate depending on the nature of... 

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