Levich relation

Apparently, the current in the ascending part of the wave and the pseudo-limiting current are mostly determined by transport-independent, kinetic factors. The limiting current of the second wave, obtained after correction for the IR voltage drop, satisfies the Levich relation (Chapter 1, Equation 1.15) and is hence determined by the transport rate of hydrogen peroxide to the electrode surface. This wave will not be further discussed since it is of no use for the aim of this investigation. 

This result shows that the transport of the particles may be treated by the usual convective diffusion equation, using ce as the variable and supports the analysis of the system in terms of the cascade model simple Tafel and Levich relations will hold for the system of particles. Thus, in the dark... 

The anodic behavior of p-type Si electrodes is quite different for lower HF concentrations. The current increases, but not really exponentially, with rising anodic polarization, it passes a maximum and increases again slowly at higher anodic potentials [8] (Fig. 8.6). The current increases with the rotation speed to of the electrode. Since the current does not follow a tu /z dependence (Levich relation [11]) the relationship cannot be determined entirely by diffusion. At electrode potentials below the peak, silicon is dissolved again in the divalent state, as already reported above in the case of high HF concentrations. Here also H2 formation was observed. At electrode potentials beyond the current peak, as shown in Fig. 8.6, the dissolution was found to occur via the tetravalent state of Si and the H2 evolution disappeared at p-type electrodes [8]. These results were confirmed 25 years later [12]. Experiments performed using the thin slice arrangement (see Chapter 4) have shown that the anodic reactions occur only via the valence band at all electrode potentials [8]. 

For the surface of a low-viscosity liquid, Levich [12] has derived the following dispersion relation ... 

In work by Okada et al. (03) on a rotating-disk flow, Eqs. (10a) and (10b) in Table VII, the electrolyte was completely enclosed between the rotating disk and the counterelectrode. Mass transfer was measured at the rotating as well as at the stationary disk, and the distance between disks was varied. At low rotation rates, the flux at the rotating disk was higher than predicted by the Levich equation, Eq. (la) in Table VII. The flux at the stationary disk followed a relation of the Levich type, but with a constant roughly two-thirds that in the rotating-disk equation. 

We also need to develop the theories for hquid film coefficient to use in the aforementioned equations. For drops that are close to spherical, without separation, Levich (1962) assumed that the concentration boundary layer developed as the bubble interface moved from the top to the bottom of a spherical bubble. Then, it is possible to use the concepts applied in Section 8.C and some relations for the streamlines around a bubble to determine Kl. ... 

From the preliminary research described in the previous section, it appears that the small oxidation wave with half-value potential (EV2) of ca. 0.21V vs. SCE offers favourable perspectives for the amperometric determination of relatively high hydrogen peroxide concentrations. In contrast to the second oxidation wave with I2=0.76 V vs. SCE, the pseudo-limiting currents obtained in the prewave do not satisfy the relation of Levich (Chapter 1, Equationl.15). However, they are almost completely independent of the rotation rate of the electrode, revealing that these limiting currents are not controlled by transport of electroactive species but by (an) other process(es). This is illustrated in Fig.4.5. 

In flow systems that necessitate consideration of two-dimensional geometry, Flanagan and Marcoux did some early work [247]. They examined a variety of conditions, among them the importance of axial diffusion in a tube. They found that neglecting axial diffusion is justified for most flows except the slowest. This is because transport due to the flow dominates in the axial direction, and this holds for electrode lengths that are small compared with the tube radius. This is often called the Levich approximation. Levich [362] related the diffusion layer thickness to the tube radius. It is a function of distance x along the electrode and flow velocity,. The condition can then be reduced to the condition... 

The relation between Ki and the particle diameter under some typical conditions obtained in this manner is shown in Fig. 9-26. For Levich s asymptote at high Peclet number, Eq. (9-51) becomes... 

Estimation of diffusion layer thickness ( S ) S for a moving particle is related to the velocity of motion (u) of the particle through the water. For a sphere of radius a moving through the water at a constant velocity it can be shown using the equations given by Levich (34, p. 84-85) that the average dif-... 

The relevant frequencies u>k (the summation may also be continuous) are related to the inverse of the relaxation times (Levich, 1966). In many cases, it may be acceptable to use a simpler expression of this expansion (the so-called Pekar separation Pekar, 1951)... 

The film thickness that results from the H-dipping process may be explained by the description of the associated drag-out problem suggested by Landau and Levich (Landau Levich, 1942). Based on their description, for a small capillary number (C 1), a useful relationship may be obtained that relates the thickness of the film emerging from a coating bead to the radius of the associated meniscus and carrying speed, U (Landau Levich, 1942, Park Han, 2009) ... 

A. N. Frumkin and V. G. Levich, Zh. Fiz. Khim. 21, 1183 (1947) (in Russian). This work, as well as related research on the motion of drops and bubbles in fluids, is summarized in the textbook (translated from the Russian) V G. Levich, Physicochemical Hydrodynamics (Prentice-Hall, Englewood Cliffs, NJ, 1962). 

Theory for chronopotentiometry of reversible channel electrode reactions under the Levich approximation has been presented by Aoki and Matsuda [73]. The transition time in flowing solution, tc, was found to be related to that in stationary solution, t0, via the approximations... 

If the denominator in Eq. (8.47) is set equal to 1, F(0) is given by a sinB, with a = F(0) / sin(0), and an equation for Xb is obtained which only differs from Eq. (8.35) by a numerical factor. Hence, if we introduce an addition condition for a slight variation of the adsorption along the surface, both the sinusoidal velocity distribution and the relation for the retardation coefficient proposed by Levich (1962) can be verified. To compare this additional condition with the condition of strong retardation of the surface T) Xb, the following conditions result (cf Eqs (8.40), (8.47) and (8.48)) ... 

The simplest case involves a monotonic variation of as a function of time (e.g., oc and an automatic plotting of / d v. These automated Levich plots can be of value compared to manual (and possibly more precise) point-by-point measurements, especially when the electrode surface is changing with time (e.g., during an electrodeposition, or with impurity or product adsorption) and a rapid scan is needed. This technique and related methods have been reviewed (28). 

Following Landau C Levich, the transition to the lubrication regime must begin when the liquid film is almost parallel to the plate that is, when 6 —>0 or when, according to Eq. (10.3.13), x— 2alpg) In this limit the curvature is given from Eq. (10.3.12) by the asymptotic relation... 

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