Kinetic-controlled models

Further models of adsorption kinetics were discussed in the literature by many authors. These models consider a specific mechanism of molecule transfer from the subsurface to the interface, and in the case of desorption in the opposite direction ((Doss 1939, Ross 1945, Blair 1948, Hansen Wallace 1959, Baret 1968a, b, 1969, Miller Kretzschmar 1980, Adamczyk 1987, Ravera et al. 1994). If only the transfer mechanism is assumed to be the rate limiting process these models are called kinetic-controlled. More advanced models consider the transport by diffusion in the bulk and the transfer of molecules from the solute to the adsorbed state and vice versa. Such mixed adsorption models are ceilled diffusion-kinetic-controlled The mostly advanced transfer models, combined with a diffusional transport in the bulk, were derived by Baret (1969). These dififiision-kinetic controlled adsorption models combine Eq. (4.1) with a transfer mechanism of any kind. Probably the most frequently used transfer mechanism is the rate equation of the Langmuir mechanism, which reads in its general form (cf. Section 2.5.), 

Under the condition p l, i.e. at small interfacial coverage, the so-called Henry mechanism 

The definition of k,j and k , the rate constants of adsorption and desorption, respectively, 

Through the subsurface concentration c(0,t) the Eqs (4.33) and (4.1) are coupled and an integro-differential equation system results. Such systems can be solved only numerically by different means (Miller Kretzschmar 1980, Chang et al. 1992, Chang Franses 1992). 

Another attempt to combine the diffusion theory with transfer mechanisms was made by Fainerman et al. (1987). He derived approximate solutions by averaging the rate of the two different processes in the following way 

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Adsorption Kinetics Diffusion and Kinetic Controlled Models... 

Active Oxygen Method (AOM), 535, 544 Adsorption, and interfacial properties diffusion and kinetic controlled models, 617-618 (figs.), 620-622 Gibbs adsorption isotherm, 617-619 kinetics of surface-active substances, 639... 

This kinetically controlled model can be modified if we assume different deposition rates for (a) the nudeation of a bare site, (b) the horiTontal growth around a metal deposition, and (c) the vertical growth on top of a metal deposition. For the... 

The above diffusion-controlled model assumes transport by difiusion of the surface-active molecules to be the rate-controlled step. The so-called kinetic controlled model is based on the transfer mechanism of molecules from solution to the adsorbed state, and vice-versa [17]. 

Models which consider diffusion in the bulk as the only rate-controlling process are called diffusion controlled. If the diffusion is assumed to be fast in comparison to the transfer of molecules between the subsurface and the interface the model is called kinetic-controlled or barrier-controlled. Both steps are taken into account in mixed diffusion kinetic controlled models. 

Although this is a very complex equation, it allows to take into consideration any function of R(t), and consequently A(t), resulting from experiments with growing drops or bubbles. In combination with an adsorption isotherm (diffusion-controlled case) or a transfer mechanism (mixed diffusion-kinetic-controlled model) it describes the adsorption process at a growing or even receding drop. Eq. (4.48) can be applied in its present form only via numerical calculations and an algorithm is given by MacLeod Radke (1994). 

Models considering diffusion in the bulk as the only rate controlling process are called pure diffusion controlled. When the diffusion is assumed to be fast in comparison to the transfer of molecules between the subsurface and the interface the model is called kinetic-controlled or barrier-controlled. Both steps are taken into account in so-called mixed diffusion kinetic controlled models. Van den Tempel proposed processes within the adsorption layer to be considered instead of hypothetical adsorption barriers [18, 19, 20]. We believe that such models, which account for actual physical processes within adsorption layers, such as reorientation of molecules, their dimerisation and formation of clusters, although explanations for all known cases of anomalous adsorption kinetics do not exist yet, have to be preferred over any formal model. However, reliable experimental evidence for a slower surface tension decrease caused by aggregation within the adsorption layer does not allow the conclusion that this is an exclusive mechanism. 

Eq. 93 can be derived from kinetically controlled model systems (Figure 17) as well as from equilibrium models (Figure 18), indicating that the bilinear model is valid under diffusion control as well as under equilibrium or pseudoequilibrium conditions [175, 345, 440, 448, 449]. 

The term a gives the slope of the left-hand ascending side of the curve and (a - b) that of the right-hand descending side. The non-linear parameter jS, which must be estimated by a stepwise iteration procedure, relates to the volume ratio of the aqueous and lipid phases in the system. Setting jS = 1 and b 2a produces the original McFarland model. Kubiny s bilinear model can be derived from kinetically controlled model systems as well as from equilibrium models, indicating that it is valid under diffusion control as well as under equilibrium or pseudo-equilibrium conditions. For many data sets, the bilinear function aptly fits the experimental observations. Difficulties in calculations may arise from unbalanced data sets, which often occur in environ-... 

Most spraying processes work under dynamic conditions and improvement of their efficiency requires the use of surfactants that lower the liquid surface tension yLv under these dynamic conditions. The interfaces involved (e.g. droplets formed in a spray or impacting on a surface) are freshly formed and have only a small effective age of some seconds or even less than a millisecond. The most frequently used parameter to characterize the dynamic properties of liquid adsorption layers is the dynamic surface tension (that is a time dependent quantity). Techniques should be available to measure yLv as a function of time (ranging firom a fraction of a millisecond to minutes and hours or days). To optimize the use of surfactants, polymers and mixtures of them specific knowledge of their dynamic adsorption behavior rather than equilibrium properties is of great interest [28]. It is, therefore, necessary to describe the dynamics of surfeictant adsorption at a fundamental level. The first physically sound model for adsorption kinetics was derived by Ward and Tordai [29]. It is based on the assumption that the time dependence of surface or interfacial tension, which is directly proportional to the surface excess F (moles m ), is caused by diffusion and transport of surfeictant molecules to the interface. This is referred to as the diffusion controlled adsorption kinetics model . This diffusion controlled model assumes transport by diffusion of the surface active molecules to be the rate controlled step. The so called kinetic controlled model is based on the transfer mechanism of molecules from solution to the adsorbed state and vice versa [28]. 

An open-chain transition state model, based on the improved Cram model (Section A.2.), was proposed for the prediction of the stereochemical outcome1 2. Under kinetic control, if the substituent R1 in the benzylic position is of medium size, the syn-isomer is formed as the major product2. 

In evaluating the kinetics of copolymerization according to the chemical control model, it is assumed that the termination rate constants k,AA and A,Br are known from studies on homopolymerization. The only unknown in the above expression is the rate constant for cross termination (AtAB)- The rate constant for this reaction in relation to klAA and kmB is given by the parameter . 

Two general models can describe the kinetics of adsorption. The first involves fast adsorption with mass transport control, while the other involves kinetic control of die system. Under the latter (and Langmuirian) conditions, the surface coverage of tlie adsorbate at time t, Tt, is given by. 

These models indicate that propylene gas phase polymerization with a highly active TiCil3 catalyst shifts from kinetic control at short reaction times to diffusion control at longer times as the catalyst yield exceeds about 4000 g.PP/g.TiCil3. Measures to reduce this limitation would significantly benefit the process. 

A more detailed picture of the temperature dependence of the growth is given in Figure 2.4, where the island density is plotted as a function of temperature. It can be seen that only in the temperature range from 207 to 288 K the growth is perfectly template controlled and the number of islands matches the number of available nucleation sites. This illustrates the importance of kinetic control for the creation of ordered model catalysts by a template-controlled process. Obviously, there has to be a subtle balance between the adatom mobility on the surface and the density of template sites (traps) to allow a template-controlled growth. We will show more examples of this phenomenon below. 

The results for Pd and V lead to the conclusion that the high-symmetry sites of the alumina film on Ni3Al(l 1 1) can act as template for the growth of nanostructured model catalysts. They also prove that kinetic control of the growth is of utmost importance. 

Presently, the quantitative theory of irreversible polymeranalogous reactions proceeding in a kinetically-controlled regime is well along in development [ 16,17]. Particularly simple results are achieved in the framework of the ideal model, the only kinetic parameter of which is constant k of the rate of elementary reaction A + Z -> B. In this model the sequence distribution in macromolecules will be just the same as that in a random copolymer with parameters P(Mi ) = X =p and P(M2) = X2 = 1 - p where p is the conversion of functional group A that exponentially depends on time t and initial concen-... 

The orbital coefficients obtained from Hiickel calculations predict the terminal position to be the most reactive one, while the AMI model predicts the Cl and C3 positions to be competitive. In polyenes, this is true for the addition of nucleophilic as well as electrophilic radicals, as HOMO and LUMO coefficients are basically identical. Both theoretical methods agree, however, in predicting the Cl position to be considerably more reactive as compared to the C2 position. It must be remembered in this context that FMO-based reactivity predictions are only relevant in kinetically controlled reactions. Under thermodynamic control, the most stable adduct will be formed which, for the case of polyenyl radicals, will most likely be the radical obtained by addition to the C1 position. 

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