Diffusion layer of finite thickness

The appropriate boundary conditions for a diffusion layer of finite thickness are that... 

Diffusion layer of finite thickness (diffusion + convection). We now use the Nemst hypothesis, which assumes that the concentration of the reacting species that diffuse changes linearly in a layer of thickness 5n and is constant thereafter. 

Because it is difficult to satisfy the above simplifying assumptions, we were not able to apply the method of Worrell et al. to polypyrrole films (30). A more rigorous theoretical analysis is needed if a current-pulse method is to be applied to electronically conductive pol)rmers. Equations applicable to a finite diffusion situation, starting from a diffusion layer of finite thickness, can be obtained from the heat transfer literature (50). These equations form the basis of this new theoretical analysis. 

It should be noted that the aforementioned model deals with the semi-infinite diffusion, which is presented by Warburg impedance. As the diffusion layer of finite thickness 5 is formed at the RDE surface, it is necessary to meet certain relations between 5, specified by Eq. (3.5), and the depth of penetration of the concentration wave Xq = y/2Dfco. Investigations in this field show [25,26] that the two aforementioned diffusion models are in harmony when < 0.15. At Schmidt number Sc = v/D = 2000, this condition is satisfied when co > 6 2 (or/ > 0.1m, where m is rotation velocity in rpm). 

FIGURE 5-10 Electrochemical impedance for Randles equivalent circuit in the complex plane for A. diffusion layer of infinite thickness B. diffusion layer of finite thickness... 

Fig. 4. Schematic diagram of the layered model for a pore (47). The two nuclear spins diffuse in an infinite layer of finite thickness d between two flat surfaces. The M axes are fixed in the layer system. The L axes are fixed in the laboratory frame, with Bq oriented at the angle P from the normal axis n. The cylindrical polar relative coordinates p, (p, and z are based on the M axis. The smallest value of p corresponding to the distance of minimal approach between the two spin bearing molecules is 5.

The following model is a bounded Randles cell also accounting for a linear but finite diffusion, with a homogeneous layer of finite thickness. The structure of the model is shown in Figure 4.18a. The corresponding impedance is... 

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. 

Laviron55 has recently noted that linear potential sweep or cyclic voltammetry does not appear to be the best method to determine the diffusion coefficient D of species migrating through a layer of finite thickness since measurements are based on the shape of the curves, which in turn depend on the rate of electron exchange with the electrode and on the uncompensated ohmic drop in the film. It has been established that chronopotentiometric transition times or current-time curves obtained when the potential is stepped well beyond the reduction or oxidation potential are not influenced by these factors.55 An expression for the chronopotentiometric transition has been derived for thin layer cells.66 Laviron55 has shown that for a space distributed redox electrode of thickness L, the transition time (r) is given implicitly by an expression of the form... 

DIFFUSION KINETICS OF PLANE LAYER SWELLING Consider two stages of swelling process in a plane layer - the initial and final. In the initial stage, the influence of the opposite layer boundary on the swelling process is inessential and therefore diffusion in a layer of finite thickness at sufficiently small values of time can be considered as the diffusion in half-space. 

The hydrated layer has finite thickness, therefore the exchanging ions can diffuse inside this layer, although their mobility is quite low compared to that in water (n 10-11cm2s-1 V-1). As we have seen in the liquid junction, diffusion of ions with different velocities results in charge separation and formation of the potential. In this case, the potential is called the diffusion potential and it is synonymous with the junction potential discussed earlier. It can be described by the equation developed for the linear diffusion gradient, that is, by the Henderson equation (6.24). Because we are dealing with uni-univalent electrolytes, the multiplier cancels out and this diffusion potential can be written as... 

Figure 7-8 Single sided diffusion from a finite thick layer into a finite layer of the same material.

Three scenarios are generally to be considered, depending on the thickness of the diffusion layer semi-infinite thickness, finite thickness in the presence of convection (or Nemst s hypothesis) and finite thickness through a thin film in an open circuit. 

For a semi-infinite diffusion process at cathode represented by Warburg impedance, the Nyquist plot appears as a straight line with a slope of 45°. The impedance increases linearly with decreasing frequency. The infinite diffusion model is only valid for infinitely thick diffusion layer. For finite diffusion layer thickness, the finite Warburg impedance converges to infinite Warburg impedance at high frequency. At low frequencies or for small... 

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