Conditions of diffusion control

As discussed earlier, it is generally observed that reductant oxidation occurs under kinetic control at least over the potential range of interest to electroless deposition. This indicates that the kinetics, or more specifically, the equivalent partial current densities for this reaction, should be the same for any catalytically active feature. On the other hand, it is well established that the O2 electroreduction reaction may proceed under conditions of diffusion control at a few hundred millivolts potential cathodic of the EIX value for this reaction even for relatively smooth electrocatalysts. This is particularly true for the classic Pd initiation catalyst used for electroless deposition, and is probably also likely for freshly-electrolessly-deposited catalysts such as Ni-P, Co-P and Cu. Thus, when O2 reduction becomes diffusion controlled at a large feature, i.e., one whose dimensions exceed the O2 diffusion layer thickness, the transport of O2 occurs under planar diffusion conditions (except for feature edges). 

Historically, the potential sweep technique and cyclic voltammetry were developed for analysis (as successors to polarography) and much of the theoretical development is concerned with the situation under conditions of diffusion control, for that is where the analytical applications are most readily made. In many of these approaches, the underlying assumption is that the electron transfer that must necessarily occur at the interface is a fast process and plays little part in determining the dependence of the observed current upon potential or upon the concentration of the reactant. However, these assumptions may not always apply. 

When intraparticle diffusion occurs, the kinetic behaviour of the system is different from that which prevails when chemical reaction is rate determining. For conditions of diffusion control 0 will be large, and then the effectiveness factor tj( 1/ tanh 0, from equation 3.15) becomes. From equation 3.19, it is seen therefore that rj is proportional to k Ul. The chemical reaction rate on the other hand is directly proportional to k so that, from equation 3.8 at the beginning of this section, the overall reaction rate is proportional to k,n. Since the specific rate constant is directly proportional to e"E/RT, where E is the activation energy for the chemical reaction in the absence of diffusion effects, we are led to the important result that for a diffusion limited reaction the rate is proportional to e E/2RT. Hence the apparent activation energy ED, measured when reaction occurs in the diffusion controlled region, is only half the true value ... 

Fig. 2.12. Schematic diagram to illustrate the growth process of two chemical compound layers under conditions of diffusion control (x > x -1...

The same applies to Pt-Pt2Si-PtSi-Si specimens. Under conditions of diffusion control the Pt2Si layer grows only at the expense of diffusing platinum atoms, while the PtSi layer grows only at the expense of diffusing silicon atoms, though in those compounds taken alone both components may well happen to diffuse at close rates. 

In the case of two compound layers, even growing under conditions of diffusion control, application of Matano s analysis and calculation of... 

An unambiguous criterion to distinguish between the growth regimes of any compound layer is the availability or lack of diffusing atoms of a given kind for other layers of a multiphase binary system. Under conditions of reaction (chemical) control these atoms are still available, while under conditions of diffusion control already not, and this is all that is necessary to explain the absence of some part of compound layers from the A-B reaction couple. 

Under conditions of diffusion control, the TiAl3 layer, known to be the first to occur between titanium and aluminium, consumes all the aluminium atoms diffusing across its bulk exclusively for its own growth, not sharing them with the other intermetallic compound layers. Therefore, those or at least one of them can grow only at the expense of diffusion of the titanium atoms. 

During the whole course of annealing the A-B couple under pressure, contacts between initial and occurring phases may well be lost and renewed several times, giving rise to a hardly tractable microstructure of the A-B transition zone. Thus, in many cases the compound-layer formation actually takes place in a few independent couples. Though in each of those couples no more than two compound layers can grow under conditions of diffusion control, multiple compound layers will ultimately be seen between A and B. Evidently, the newly occurred layers can only grow at the expense of the former ones whose thickness must therefore decrease. 

Under conditions of diffusion control, all other compound layers of a multiphase binary system, located between the two growing ones, are kinetically unstable. If these other layers were initially missing from the A-B couple, they will not occur in it until at least one of initial substances (either A or B) is completely exhausted. If present, they must disappear... 

One inert marker only indicates the diffusing species in that compound layer in which it is embedded or with which it borders. If this layer grows under conditions of diffusion control, then the very presence of other compound layers provides in itself evidence that another component is diffusing across their bulks. 

Clearly, under conditions of diffusion control the rate of dissolution expressed in terms of the concentration of dissolving elements in the melt does not depend upon the atomic packing density of the crystallographic faces of any substance under investigation. Therefore, dissolution of single crystals of different orientation (line 2 in Fig. 5.7) is characterised by the... 

From equations (5.6) and (5.7), it follows that under conditions of diffusion control the dissolution-rate constant should depend linearly on the square root of the angular speed of the disc rotation. As seen in Fig. 5.9, this is indeed the case (see also Refs 197, 300, 301, 303, 304, 307, 308). 

Hie most commonly found shape of catalyst particle today is the hollow cylinder. One reason is the convenience of manufacture. In addition there are often a number of distinct process advantages in the use of ring-shaped particles, the most important being enhancement of the chemical reaction under conditions of diffusion control, the larger transverse mixing in packed bed reactors, and the possible significant reduction in pressure drop. It is remarkable (as discussed later) that the last advantage may even take the form of reduced pressure losses and an increased chemical reaction rate per unit reactor volume [11]. 

We note that under conditions of diffusion control, the current depends on t . Thus, a small change in drop time, resulting from a change in surface tension with potential, does not produce a large difference in the diffusion current. If the error caused by this effect is considered troublesome, it is possible to knock the drops off at fixed intervals, yielding drops of exactly equal size, irrespective of the surface tension. This mode of operation becomes of particular importance for kinetic studies conducted at the foot of the polaro-graphic wave, since the activation-controlled current is proportional to the surface area, which is itself proportional to (the volume increases linearly with time). 

In enzyme electrodes, which are deliberately operated under conditions of diffusion control, the diffusion limits the sensitivity. Here, the coupling of cyclic enzyme reactions gives rise to a sensitivity enhancement by overcoming the limit set by diffusion. The excess of enzyme present in the membrane is included in the substrate conversion. On the other hand, the upper limit of linearity and the operational stability are decreased. 

Diffusion rates are significantly greater in wool fiber than in human hair [33], This effect is due to the lower disulfide content of wool fiber relative to human hair. Therefore, one might anticipate a more rapid rate of reduction for wool fiber than for human hair under conditions of diffusion-controlled reduction. 

Under the conditions of diffusion control, a plot of E versus log [(—/l/0 1] should be made instead of the regular Tafel plot and the relationship should be linear with the slope of 23RTIaJ ). 

Fig. 10,3 - Absorbance-time curve for the formation of a coloured species under conditions of diffusion control.

Examples of the application of Eq. (3.3.17) to spherical particles include the work of Carter [9] for the oxidation of nickel, Kawasaki et al. [10] for the reduction of iron oxides, and Weisz and Goodwin [11] for the combustion of coke deposits on catalysts. In the latter two cases, the solid was initially a porous pellet when diffusion controls the overall rate, however, the relationships above may be used. We will discuss this in more detail when the system of a porous reactant solid is presented. Many studies on the oxidation of metals have been made in one-dimensional geometries [12,13]—hence the term parabolic law for the rate of progress of such oxidation reactions under conditions of diffusion control [see Eq. (3.3.14)]. Hutchins [14] has verified Eq. (3.3.16) experimentally using a cylindrical system. 

The effective diffusivity in the product layer may be determined by plotting, according to Eq. (4.3.7), the conversion data obtained under conditions of diffusion control. This method was applied to spherical systems by Kawasaki et al. [38] for the reduction of iron oxide pellets, by Weisz and Goodwin... 

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