Adsorption kinetics model mixed diffusion-kinetic-controlled

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.),... 

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). 

The presence of impurities in surfactant solutions can give very misleading results. In a recent paper, experimental dynamic surface tensions of sodium dodecyl sulphate (SDS) solutions were interpreted by Fainerman (1977) on the basis of a mixed diffusion-kinetic-controlled adsorption model. As the result a rate constant of adsorption k j as a function of time was obtained (cf. Fig. 5.5, ), although this parameter was assumed to be a constant. 

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. 

For the practical use a criterion is needed to decide whether it is justified that such a kinetic equation is applied. This criterion must ensure that the kinetic constants are independent of the parameters of the adsorption process, mainly the surfactant concentration and the monolayer coverage. Experimental data for various surfactants show that for surface lifetimes shorter than 20 ms the reduced desorption rate constant k , = kj /T is nearly constant and of the order of 100 s [16]. This important result allows to define a simple criterion for a non-diffusional adsorption mechanism by comparing the characteristic times of diffusion and adsorption kinetics according to the model of Eq. (4.15). The condition for mixed or kinetic controlled... 

For a modelling of adsorption processes the well-known integro-differential equation (4.1) derived by Ward and Tordai [3] is used. It is the most general relationship between the dynamic adsorption r(t) and the subsurface concentration e(0,t) for fresh non-deformed surfaces and is valid for kinetic-controlled, pure diffusion-controlled and mixed adsorption mechanisms. For a diffusion-controlled adsorption mechanism Eq. (4.1) predicts different F dependencies on t for different types of isotherms. For example, the Frumkin adsorption isotherm predicts a slower initial rate of surface tension decrease than the Langmuir isotherm does. In section 4.2.2. it was shown that reorientation processes in the adsorption layer can mimic adsorption processes faster than expected from diffusion. In this paragraph we will give experimental evidence, that changes in the molar area of adsorbed molecules can cause sueh effectively faster adsorption processes. 

The same steps as discussed above for the case of isotope exchange (diffusion in liquid film, surface reaction, intraparticle diffusion) were considered in a kinetic model [771] of metal ion adsorption from solution. This model was presented in a book with diskettes (FORTRAN program, rate controlled by reaction, by transport or mixed control). 

The equilibrium and dynamic aspects of surface tension and adsorption of surfactants at the air-water interface are important factors in foam film stability [82]. Dynamic adsorption models with the diffusion-controlled and mixed-kinetic mechanisms are discussed in some surfactant solution litera-... 

In diffusion-controlled adsorption models, one assumes that there is no activation energy barrier to the transfer of surfactant molecules between the subsurface and the surface [85]. Thus diffusion is the only mechanism needed in establishing adsorption equilibrium. The time required for the molecules to transfer from the bulk to the subsurface is much longer than the time required for equilibration between the surface and the subsurface. On the contrary, if the adsorption or desorption rate at the interface is slow or comparable to the diffusion rate, the adsorption process is significant. This model is called the mixed-kinetic adsorption model. This condition may depend not only on the properties of the system but also on the diffusion length and possibly on convection conditions. The diffusion-controlled model of Eqs. (3) and (4) have been given by Fainerman et al. [86,87]. 

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