Aggregation in the adsorption layer

The model for aggregation in the adsorption layer with the arbitrary aggregation number n is described by the following equation of state and adsorption isotherm... 

For a surfactant forming aggregates in the adsorption layer the equations of state and adsorption isotherms were derived in Chapter 2. In the limiting case of large aggregates these equations expressed via the molar area of monomers cd and the critical adsorption of aggregation Fic, can be written in the form of Eqs. (2.107)-(2.108). 

For the ideal mixture (both in the bulk and at the surface) of two surfactants with similar molar areas, where only component 1 is able to form large two-dimensional aggregates in the adsorption layer, the following equations were derived in [95]... 

To derive an adsorption kinetics model the Ward and Tordai equation (4.1) is again the main relationship between the dynamic adsorption r(t) and the subsurface concentration c(0,t). As it was described in detail in paragraph 4.1.2, an adsorption isotherm as additional function r(c) is needed for a kinetic model. The isotherm equations (2.110) - (2.112) given in Chapter 2 represent such type of function, which accounts for a 2D-aggregation in the adsorption layer [48]. The set of equation is too complex to find an analytical solution. Only for the short time range and for low adsorption layer coverage, the following approximation is valid [65]... 

For arbitrary times, surface layer coverage and critical adsorption Fc, the set of equations (2.110) - (2.112) together with the transport equation has to be solved numerically. This model assumes that the aggregation process itself does not require additional time, i.e. there is always equilibrium between monomers and aggregates in the adsorption layer. To solve this set of equations numerically, first-order finite difference schemes can be applied as described in... 

Considerable effort has been spent to explain the effect of reinforcement of elastomers by active fillers. Apparently, several factors contribute to the property improvements for filled elastomers such as, e.g., elastomer-filler and filler-filler interactions, aggregation of filler particles, network structure composed of different types of junctions, an increase of the intrinsic chain deformation in the elastomer matrix compared with that of macroscopic strain and some others factors [39-44]. The author does not pretend to provide a comprehensive explanation of the effect of reinforcement. One way of looking at the reinforcement phenomenon is given below. An attempt is made to find qualitative relations between some mechanical properties of filled PDMS on the one hand and properties of the host matrix, i.e., chain dynamics in the adsorption layer and network structure in the elastomer phase outside the adsorption layer, on the other hand. The influence of filler-filler interactions is also of importance for the improvement of mechanical properties of silicon rubbers (especially at low deformation), but is not included in the present paper. 

The adsorption kinetics of a surfactant to a freshly formed surface as well as the viscoelastic behaviour of surface layers have strong impact on foam formation, emulsification, detergency, painting, and other practical applications. The key factor that controls the adsorption kinetics is the diffusion transport of surfactant molecules from the bulk to the surface [184] whereas relaxation or repulsive interactions contribute particularly in the case of adsorption of proteins, ionic surfactants and surfactant mixtures [185-188], At liquid/liquid interface the adsorption kinetics is affected by surfactant transfer across the interface if the surfactant, such as dodecyl dimethyl phosphine oxide [189], is comparably soluble in both liquids. In addition, two-dimensional aggregation in an adsorption layer can happen when the molecular interaction between the adsorbed molecules is sufficiently large. This particular behaviour is intrinsic for synergistic mixtures, such as SDS and dodecanol (cf the theoretical treatment of this system in Chapters 2 and 3). The huge variety of models developed to describe the adsorption kinetics of surfactants and their mixtures, of relaxation processes induced by various types of perturbations, and a number of representative experimental examples is the subject of Chapter 4. 

Figure 2.14 illustrates the effect of the aggregation number for the case where the surface layer is free of monomers (F < 10 mol/m ). In this case, when the aggregation number exceeds a certain value (n > 20), the curves also become independent of n. Here, if the critical adsorption F is not too small, the curves exhibit a characteristic Iracture which indicates the formation of clusters in the adsorption layer (cf Fig. 2.15). However, for n>50 (that is, for very large clusters) the shape of the curves becomes independent of n.

As one can see the Frumkin model does not reflect all the details of the adsorption process. In some cases, the reorientation and aggregation models lead to better results. The scope of the data available is still insufficient to formulate a criterion for the best choice of the adsorption model. However, it follows from the results summarised in this chapter, that the reorientation model can be successfully applied to oxyethylated surfactants, and also for surfactants with relatively large molar area (o > 2.5T0 m /mol). At the same time, the surfactant molecules with relatively high values of the Frumkin constant and low molar area (m < 2.5-10 m /mol) are more capable for aggregation in the surface layer. 

If processes happen in the adsorption layer, such as changes in the orientation or conformation or the formation of aggregates due to strong intermolecular interaction, additional fluxes in the adsorption layer have to be considered. These fluxes are also shown schematically in Fig. 4.1. 

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. 

Another possibility discussed in Chapter 2 is the formation of small clusters or aggregates within the adsorption layer. It was shown that all those surfactants, which had been discussed as strongly interacting in the adsorption layer adsorb according to the surface aggregation model of Eqs. (2.110) - (2.112). The process of aggregation formation/dissolution as... 

To demonstrate the effect of the model parameters on the shape of the surface pressure isotherms, the process of dimer formation in the adsorption layer at various Fc is shown in Fig. 8. The smaller Fc is, the more pronounced is the difference between the aggregation and Langmuir isotherm (n = 1). For Fc < 10 mol/m the shape of the surface pressure isotherm becomes independent of Fc because the adsorption layer contains only aggregates. 

Also the aggregation number has a direct influence on the isotherm shape, however, for n > 20 it becomes also independent of n. If the critical adsorption Fc is sufficiently large, the curves exhibit a characteristic kink which indicates the formation of clusters in the adsorption layer. 

The adsorption of surface active molecules at an interface is a dynamic process. In equilibrium the adsorption flux to and desorption flux from the interface are in balance. If the actual surface concentration is smaller than the equilibrium one, T < To, the adsorption flux predominates, if r > To, the desorption flux prevails. If processes happen in the adsorption layer, such as changes in the orientation or conformation of adsorbed molecules, or the formation of aggregates due to strong intermolecular interaction, additional fluxes within the adsorption layer have to be considered. 

Above adsorption isotherms where discussed for surfactants able to form small aggregates (dimers or trimers) in the adsorption layer. There are quite a number of surfactants, which can be described by this model perfectly, for example the homologous series of fatty acids or alcohols or the alkyl sulphates (Fainerman et al. (2000). 

The aggregation number n is an additional parameter in the denominator, leading to a slower adsorption caused by the formation of aggregates within the adsorption layer. For the whole time interval, the set of equations together with the diffusion equation has to be solved numerically. 

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