Diffusion coefficient impedance

The movement of the fast electrons leads to the fonnation of a space-charge field that impedes the motion of the electrons and increases the velocity of the ions (ambipolar diffusion). The ambipolar diffusion of positive ions and negative electrons is described by the ambipolar diffusion coefficient... 

The movement of the analyte is an essential feature of separation techniques and it is possible to define in general terms the forces that cause such movement (Figure 3.1). If a force is applied to a molecule, its movement will be impeded by a retarding force of some sort. This may be as simple as the frictional effect of moving past the solvent molecules or it may be the effect of adsorption to a solid phase. In many methods the strength of the force used is not important but the variations in the resulting net force for different molecules provide the basis for the separation. In some cases, however, the intensity of the force applied is important and in ultracentrifugal techniques not only can separation be achieved but various physical constants for the molecule can also be determined, e.g. relative molecular mass or diffusion coefficient. 

As expected, the impedance responses obtained in practice do not fully match that of the model of Fig. 9.13. However, as shown by the typical case of Fig. 9.14 which illustrates the response obtained for a 5 mol% ClO -doped polypyrrole electrode in contact with a LiC104-propylene carbonate solution (Panero et al, 1989), the trend is still reasonably close enough to the idealised one to allow (possibly with the help of fitting programmes) the determination of the relevant kinetics parameters, such as the charge transfer resistance, the double-layer capacitance and the diffusion coefficient. 

At high frequencies, a semicircle is expected as a result of a parallel combination of R and Cg. At low frequencies a Warburg impedance may be found as part of the interfacial impedance. In some cases it may dominate the interfacial impedance as in Fig. 10.13(a), in which case only the diffusion coefficient of the salt will be determinable. It should be noted that, in the absence of a supporting electrolyte, the electroactive species, in this case Li, cannot diffuse independently of the anions. 

When the characteristic time for charge diffusion is lower than the experiment timescale, not all the redox sites in the film can be oxidized/reduced. From experiments performed under these conditions, an apparent diffusion coefficient for charge propagation, 13app> can be obtained. In early work choroamperometry and chronocoulometry were used to measure D pp for both electrostatically [131,225] and covalently bound ]132,133] redox couples. Laviron showed that similar information can be obtained from cyclic voltammetry experiments by recording the peak potential and current as a function of the potential scan rate [134, 135]. Electrochemical impedance spectroscopy (EIS) has also been employed to probe charge transport in polymer and polyelectrolyte-modified electrodes [71, 73,131,136-138]. The methods... 

Olesen T, Moldrup P, Yamaguchi T, Rolston DE. 2001. Constant slope impedance factor model for predicting the solute diffusion coefficient in unsaturated soil. Soil Science 166 89-96. 

Staunton S. 1990. A comparison of the surface impedance factors of Ca, Na, Rb and Cs derived from their self-diffusion coefficients in various soils. Journal of Soil Science 41 643-653. 

Figure 29. Comparison of the oxygen vacancy diffusion coefficient (Dy) in LSC (x= 0.2) determined from permeation measurements vs that extracted from impedance measurements using the model in Figure 26. Data are from refs 190 and 28. (Adapted with permission from ref 28. Copyright 1998 Elsevier.)...

In systems where diffusion phenomena are of significance, the mechanistic study is facilitated by using the general expression for Impedance Z (26). This equation shows for instance how the Warburg coefficient can be evaluated by conducting impedance studies at very low frequencies. These coefficients in turn enable the evaluation of diffusion coefficients for the diffusing species. 

The diffusion coefficient for water was found to be rather insensitive to the proportion of the dimethacrylate crosslinker, especially in sorption (Fig. 7). This insensitivity might be rationalized in a number of ways. One of these is consistent with the view that these systems have a lightly crosslinked matrix which does little to Impede the diffusion of water molecules. 

The hydrodynamic repulsion of molecules in solution has already been discussed (see Chap. 8, Sect. 2.5) and is analysed in much more detail in Sect. 3 of this chapter. It appears to be a very significant complication, yet it can be analysed relatively straightforwardly. The basic effect arises when one molecule approaches another. Solvent molecules between these two molecules have to be squeezed out in a direction approximately perpendicular to the motion of the approaching molecules. Their approach (and similarly their separation) is impeded and the effective diffusion coefficient is less than that when the molecules are more widely separated. 

While eqn. (211) is a bit complex, the similarity to the more familiar diffusion equation [e.g. eqns. (43), (44), (158) or (197)] is apparent. The diffusion coefficient has to be replaced by a position-dependent tensor which couples (or connects) the motion of one of the particles, e.g. the particle k, with that of the other particles, e.g. a particle j. When these particles are a long way apart, the solvent between them can be squeezed out easily. As the particles approach, this is no longer true because the two particles block certain directions for escape of the solvent as the particles approach. Increasingly, the solvent has to be squeezed out of the way in a direction perpendicular to that of the approach of the particles and this causes the solvent to impede the particle approach more and more effectively. When j = k, the effect on the same particle of its own motion is negligible. Hence, the diffusion coefficient tensor elements, Tjj = kBT/%, are the same as the diffusion coefficient for the particle... 

As mentioned, all reaction models will include initially unknown reaction parameters such as reaction orders, rate constants, activation energies, phase change rate constants, diffusion coefficients and reaction enthalpies. Unfortunately, it is a fact that there is hardly any knowledge about these kinetic and thermodynamic parameters for a large majority of reactions in the production of fine chemicals and pharmaceuticals this impedes the use of model-based optimisation tools for individual reaction steps, so the identification of optimal and safe reaction conditions, for example, can be difficult. 

In the diffusion region the reorientational motion of the molecules is impeded by a frictional force exerted by a medium considered structureless (continuum). For a spherical molecule, the rotational diffusion coefficient, D, is given by the Stokes-Einstein-Debye equation42... 

On a RDE, in the absence of a surface layer, the EHD impedance is a function of a single dimensionless frequency, pSc1/3. This means that if the viscosity of the medium directly above the surface of the electrode and the diffusion coefficient of the species of interest are independent of position away from the electrode, then the EHD impedance measured at different rotation frequencies reduces to a common curve when plotted as a function of p. In other words, there is a characteristic dimensionless diffusional relaxation time for the system, pD, strictly (pSc1/3)D, which is independent of the disc rotation frequency. However, if v or D vary with position (for example, as a consequence of the formation of a viscous boundary layer or the presence of a surface film), then, except under particular circumstances described below, reduction of the measured parameters to a common curve is not possible. Under these conditions pD is dependent upon the disc rotation frequency. The variation of the EHD impedance with as a function of p is therefore the diagnostic for... 

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