Diffusion impedance

J. G. Powles, M. J. D. Mallett, G. Rickayzen, W. A. B. Evans. Exact analytical solutions for diffusion impeded by an infinite array of partially permeable barriers. Proc Royal Soc London 457 391, 1992. 

The impedance with its components R and C is known as the Warburg diffusion impedance, and constant as the Warburg constant. In the equivalent circuits for electrochemical reactions, a Warburg impedance is represented by the symbol -W- as shown in the lower part of Fig. 12.15b. 

Figure 2.3 Relation between diffusion impedance factor, /l, and moisture content, 6>l, in a range of soils. Numbers shown are % clay contents. Mean of six soils volcanic ash soil. (After Tinker and Nye, 2000 Olesen et al., 2001). Reproduced by permission of Oxford University Press...

Figure 2.4 Relation between diffusion impedance factor, /l, and bulk density, p, in four water-saturated rice soils. Dotted line is the theoretical relation between /l and p for a mixture of different-sized spherical particles (Kirk et al., 2003). Iloilo Epiaquult clay 21% org C 1.04% pH 3.93. Maahas Haplaquoll clay 54% org C 1.83% pH 5.89. Nueva Ecija Epiaquert clay 35 % org C 1.57 % pH 5.25. Tarlac Tropaquept clay 33 % org C 1.06% pH 6.02. Reproduced by permission of Blackwell Publishing...

The corollary is that, when intraparticle diffusion impedes reaction, maximum selectivity is achieved by operating with small pellets and large diameter pores. 

ZD is the convective diffusion impedance with its dimensionless form -1/0J(O) [41]. 

Zs is the faradaic impedance due to the substrate for a redox system it consists of a series combination of a charge-transfer resistance Rt and a convective diffusion impedance Zds (2s - ts+2os)-... 

The Warburg impedance has been substituted by ZD, the diffusion impedance ... 

Nonlinear regression analysis, described in Chapters 19 and 20, was developed for impedance spectroscopy in the early 1970s. The models were cast in the form of electrical circuits with mathematical formulas added to accoimt for the diffusion impedance associated with simplified geometries. 

Process models Nemst dielectrics (1894) Warburg diffusion (1901) Finkelstein Solid film (1902) Randles double layer and diffusion impedance (1947) Gerischer two heterogeneous steps with adsorbed intermediate (1955) De Levie porous electrodes (1967) Schuhmann homogeneous reactions and diffusion (1964) Gabrielli generalized impedance (1977) Isaacs LEIS (1992)... 

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. 

The electrolyte resistance Re is added in series with the previous impedance. If the electrochemical reaction is mass-trai3sport limited, the previous equivalent circuit is still valid, but the Faradaic impedance includes a diffusion impedance as described in Chapter 11. 

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

Equation (11.15) is the convection diffusion impedance. The electrode potential measured with respect to the potential of a reference electrode, following equation (10.18), is given by... 

Following equation (11.15), the expression of the convective-diffusion impedance can be found to be... 

Diffusion through a stagnant layer of finite thickness can also yield a uniformly accessible electrode. The diffusion impedance response of a coated (or film-covered) electrode, imder the condition that the resistance of the coating to diffusion is much larger than that of the bulk electrol5M e, is approximated by the diffusion impedance of file coating. This problem is also analyzed in Section 15.4.2. 

The concentration at the interface between the film and the bulk electrolyte indicates whether the diffusion impedance of the system can be assumed to be that of the stationary film. The interface concentration, given by... 

Example 11.4 Continuation of Example 11.3 Can the diffusion impedance of a film with e = 1.0 he ignored ... 

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