Kinetically Limited Film Thickness

The constrained equilibrium description discussed up to now conveys the impression that a possible oxidation of the catalyst surface in the O-rich environments of oxidation catalysis would rather Meld bulklike thick oxide films on Ru but thin surface oxide structures on the more noble 4d metals. This reflects the decreasing heat of formation of the bulk oxides over the late TM series, and seems to suggest that it is primarily at Pd and Ag where oxide formation in the reactive environment could be self-limited to nanometer or subnanometer thin overlayers. Particularly for the case of Ru, Fig. 5.12 shows that the gas phase conditions t3q)ical for technological CO oxidation catalysis fall deep inside the stability regime of the bulk oxide, indicating that thermodynamically nothing should prevent a continued growth of the once formed oxide film. 

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The changes in a and f> can be determined very rapidly, allowing changes in film thickness and optical properties to be determined in real time, hence the technique can be used to gain kinetic information. The speed of data collection is limited only by the response of the instrument and this may be as low as a few tens of microseconds. 

Equations 2.26 and 2.27 carmot be solved analytically except for a series of limiting cases considered by Bartlett and Pratt [147,192]. Since fine control of film thickness and organization can be achieved with LbL self-assembled enzyme polyelectrolyte multilayers, these different cases of the kinetic case-diagram for amperometric enzyme electrodes could be tested [147]. For the enzyme multilayer with entrapped mediator in the mediator-limited kinetics (enzyme-mediator reaction rate-determining step), two kinetic cases deserve consideration in this system in both cases I and II, there is no substrate dependence since the kinetics are mediator limited and the current is potential dependent, since the mediator concentration is potential dependent. Since diffusion is fast as compared to enzyme kinetics, mediator and substrate are both approximately at their bulk concentrations throughout the film in case I. The current is first order in both mediator and enzyme concentration and k, the enzyme reoxidation rate. It increases linearly with film thickness since there is no... 

Radial thickness uniformity The very good within-wafer uniformity of film thickness is, as already mentioned, one of the notable features of LPCVD polysilicon layers. This good uniformity arises, also as already mentioned, because of the lack of any mass transfer limitations on growth. This can be shown to be the case very simply if we consider the rate of heterogeneous kinetics (/,), for... 

Additional deviations from the Nernst law [Eq. (4)] can come from kinetic effects in other words, if the potential scan is too fast to allow the system to reach thermal equilibrium. Two cases should be mentioned (1) ion transport limitation, and (2) electron transfer limitation. In case 1 the redox reaction is limited because the ions do not diffuse across the film fast enough to compensate for the charge at the rate of the electron transfers. This case is characterized by a square-root dependence of the current peak intensity versus scan rate Ik um instead of lk u. Since the time needed to cross the film, tCT, decreases as the square of the film thickness tCT d2, the transport limitation is avoided in thin films (typically, d < 1 xm for u < 100 mV/s). The limitation by the electron transfer kinetics (case 2) is more intrinsic to the polymer properties. It originates from the fact that the redox reaction is not instantaneous in particular, due to the fact that the electron transfer implies a jump over a potential barrier. If the scan... 

The limits at which one or the othertype of diffusion is controlling have been determined by Tsai.When > 50, the rate is controlled by film diffusion. When Klc b < 0.005, the rate is controlled by particle diffusion. In these relationships, K is the distribution coefficient, Ic is the diffusivity ratio (Dp/Df), 8 is the relative film thickness on a resin particle with a radius of a. Between these two limits, the kinetic description of ion exchange processes must include both phenomenon. 

First, the overall mass transfer coefRcient k a of the microreactor was estimated to be 3-8 s [43]. For intensified gas liquid contactors, kj a can reach 3 s while bubble columns and agitated tanks do not exceed 0.2 s Reducing the flow rate and, accordingly, the liquid film thickness is a means of further increasing kj a, which is limited, however, by liquid dry-out at very thin films. Despite such large mass transfer coefficients, gas-liquid microreactors such as the falling film device may still operate between mass transfer and kinetic control regimes, as fundamental simulation studies on the carbon dioxide absorption have demonstrated [44]. Distinct concentration profiles in the liquid, and even gas, phase are predicted. 

No doubt in all real cases there will be a distribution of diameters d. This difficulty can be avoided by using Equ. 3.28 to calculate an average diameter dp for a particular distribution function (Atkinson and Ur-Rahman, 1979). The further application of dp for evaluating the extent of transport-limiting factors is described in Sect. 4.5.2. Biofilm reactors with mechanical or hydro-dynamic control of film thickness exhibit a uniform dp and thus are more suitable than floe bioreactors for process kinetic analyses (see Sects. 3.6 and 4.3). 

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