Diffusion impedance convection

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)-... 

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... 

As shown in Figure 11.2, there exists a significant parameter space for film-covered electrodes in which the diffusion impedance must account for both convective diffusion associated with the external imposed flow and diffusion through a stagnant layer. Following Deslouis et al., the net diffusion impedance can be expressed as being composed of contributions from film and convective-diffusion terms... 

The mathematical models for the convective-diffusion impedance associated with convective diffusion to a disk electrode are developed here in the context of a generalized framework in which a normalized expression accoxmts for the influence of mass transfer. 

Remember 11.4 The formula pr impedance obtained under the Nemst hypothesis, as given by equation (11.70), provides a poor model for convective-diffusion impedance. 

Several authors have addressed the influence of a finite vedue of the Schmidt number on expressions for the convective-diffusion impedance. Levart and Schuh-mann showed that the concentration term could be expressed as a series expansion in Sc i.e.. 

The convective-diffusion impedance can be tabulated directly as a function of the Schmidt number as... 

A similar development was provided by Tribollet and Newman for electro-hydrodynamic impedance. The use of look-up tables facilitates regression of models to experimental data that take full accoimt of the influence of a finite Schmidt number on the convective-diffusion impedance. Use of only the first term in equation (11.97) yields a numerical solution for an infinite Schmidt number. Tribollet and Newman report use of the first two terms in equation (11.97) The low level of stocheistic noise in experimental data justifies use of the three-term expansion reported here. 

In principle, development of the convective-diffusion impedance for this system requires solution in the frequency domain to... 

Theoretical developments show that it is possible to deduce hydrodynamic information from the limiting current measiuement, either in quasi-steady state where /(f) cx py t) or, at higher frequency, in terms of spectral analysis. In the latter case, it is possible to obtain the velocity spectra from the mass-transfer spectra, where the transfer function between the mass-transfer rate and the velocity perturbation is known. However, in most cases, charge transfer is not infinitely fast, and the analysis also requires knowledge of the convective-diffusion impedance, i.e., the transfer function between a concentration modulation at the interface and the resulting flux of meiss under steady-state convection. 

The dimensionless impedance of a small electrode can be defined by summing the effects of die local convective-diffusion impedance, i.e.. 

Figure 13.4 Normalized global convective-diffusion impedance for a small rectangular electrode, The solid line represents the low-frequency solution (equation (13.37)), and the dashed line represents the high-frequency solution (equation (13.40)). Overlap is obtained for 6 < K < 13, with the dimensionless frequency K, given by equation (13.34). (Taken from Delouis et al. and reproduced with permission of The Electrochemical Society.)...

The term Zd represents the convective-diffusion impedance with its dimensionless form -1/0.(O) (see Chapter 11). Considering now equations (14.27), (14.28) and (15.47), one obtains the relation between observable quantities corresponding to the general form of equation (14.22), i.e.. 

Example 16.3 Evaluation of Double-Layer Capacitance Find the meaning of the effective capacitance obtained using equation (16.41) for the convective-diffusion impedance expressed as equation (11.20), i.e.,... 

Solution At high-frequencies, all models for convective diffusion to a rotating disk approach the Vfarburg impedance, given as equation (11.52). Thus, the convective diffusion impedance can be expressed as Zo co) = Zd(0)/y/Jcor. Following Example 1.7, which... 

A graphical method was reported by Tribollet et al. that can be used to extract Schmidt numbers from experimental data in which the convective-diffusion impedance dominates. 3 The technique accounts for the finite value of the Schmidt... 

A Nemst stagnant-diffusion-layer model was used to accovmt for the diffusion impedance. This model is often used to account for mass transfer in convective systems, even though it is well known that this model caimot ac-coimt accurately for the convective diffusion associated with a rotating disk electrode. 

Elbicki etal. 984) reviewed the optimum configurations for each of the above electrodes (thin-layer, tubular, and wall-jet) based on a mathematical treatment of the diffusive and convective phenomena in force. Boundary conditions on such physical restraints as electrode area, cell dimensions, and inlet configuration were established. Some confusion in the past has resulted from misinterpreting these equations (Weber, 1983) they are derived for cells in which the boundary layer may freely grow unencumbered. In certain cells (e.g., low-volume wall-jet or long-channel electrodes), walls, nozzles, etc. may impede the growth of the diffusion layer and bias the output current expected. Under these conditions, the wall-jet electrode behaves virtually as a thin-layer cell (if the nozzle spacing is small and the nozzle acts as a point source). Both detectors were concluded to yield output currents of... 

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... 

The kinetic impedance Zj represents the faradaie impedance in the absence of a concentration overpotential. In the simplest case, the kinetie impedance corresponds to the transfer resistance Rt, but in more complicated situations it may include several circuit elements. The diffusion impedance Z describes the contribution of the concentration overpotential to the faradaic impedance and therefore depends on the transport phenomena in solution. In the absence of convection, it is referred to as the Warburg impedance and, in the opposite case, as the Nernst impedance Z. ... 

In Chapter 1, we calculated the electrochemical impedance of an electrode. It comprises three terms for each electrode two terms of impedance of diffusion and convection - one for the change in concentration of the reduced form j... 

In this chapter, the circuit models which have been proposed to represent ac polarization impedance are quite simple. They do, however, give a good fit to experimentally determined data. Esthetically, the simple models are appealing and relatively easy to relate to the physical processes of charge migration, diffusion, and convection described in Chapter 3. More sophisticated models could be proposed, but they would not prove any more useful experimentally than the simple circuits. 

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