Impedance models convective diffusion

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. 

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. 

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 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. 

Figure 20.11 Normalized residual errors for the fit of the convective-diffusion models presented in Figure 20.12 to impedance data obtained for reduction of ferricyanide on a Pt rotating disk electrode a) real and b) imaginary.

Refined models for mass transfer to a disk electrode are presented in Section 11.6. The equivalent circuit presented in Figure 20.12 was regressed to the impedance data. The mathematical formulation for the model is given as equation (17.1). Four models were considered for the convective-diffusion term Zd (/) ... 

The three-term convective-diffusion model provides the most accurate solution to the one-dimensional convective-diffusion equation for a rotating disk electrode. The one-dimensional convective-diffusion equation applies strictly, however, to the mass-transfer-limited plateau where the concentration of the mass-transfer-limiting species at the surface can be assumed to be both uniform and equal to zero. As described elsewhere, the concentration of reacting species is not uniform along the disk surface for currents below the mass-transfer-limited current, and the resulting nonuniform convective transport to the disk influences the impedance response. ... 

As described in Sections 20.2.1 and 20.2.2, the quality of the regressions can be assessed to varying degrees of success by inspection of plots. The Nyquist or complex-impedance-plane representation given in Figure 20.13 reveals the difference between the finite-diffusion-length model and the models based on numerical solution of the convective-diffusion equation, but cannot be used to distinguish the models based on one-term, two-term, and three-term expansions. 

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. 

---