Surfactant elasticity number

The first term in both Equations 17 and 18 is the constant surface-tension contribution and the second term gives the first-order contribution resulting from the presence of a soluble surfactant with finite sorption kinetics. A linear dependence on the surfactant elasticity number arises because only the first-order term in the regular perturbation expansion has been evaluated. The thin film thickness deviates negatively by only one percent from the constant-tension solution when E = 1, whereas the pressure drop across the bubble is significantly greater than the constant-tension value when E - 1. 

R E, radius independent surfactant elasticity number, m f f., dimensionless shifted axial coordinate... 

Flow of trains of surfactant-laden gas bubbles through capillaries is an important ingredient of foam transport in porous media. To understand the role of surfactants in bubble flow, we present a regular perturbation expansion in large adsorption rates within the low capillary-number, singular perturbation hydrodynamic theory of Bretherton. Upon addition of soluble surfactant to the continuous liquid phase, the pressure drop across the bubble increases with the elasticity number while the deposited thin film thickness decreases slightly with the elasticity number. Both pressure drop and thin film thickness retain their 2/3 power dependence on the capillary number found by Bretherton for surfactant-free bubbles. Comparison of the proposed theory to available and new experimental... 

E is one of several elasticity numbers characterizing the stabilizing effect which adsorbed surfactant molecules have on an interface during mass-transfer processes (22). Note that E is inversely proportional to the capillary radius so that the effect of soluble surfactants on the bubble-flow resistance is larger for smaller capillary radii. 

Figure 8 reveals that the few data available for surfactant-laden bubbles do confirm the capillary-number dependence of the proposed theory in Equation 18. Careful examination of Figure 8, however, reveals that the regular perturbation analysis carried out to the linear dependence on the elasticity number is not adequate. More significant deviations are evident that cannot be predicted using only the linear term, especially for the SDBS surfactant. Clearly, more data are needed over wide ranges of capillary number and tube radius and for several more surfactant systems. Further, it will be necessary to obtain independent measurements of the surfactant properties that constitute the elasticity number before an adequate test of theory can be made. Finally, it is quite apparent that a more general solution of Equations 6 and 7 is needed, which is not restricted to small deviations of surfactant adsorption from equilibrium. 

The surface tension gradient in the thinning film, which is created by the efflux of liquid from the film and the sweeping of surfactant along the film surfaces to the Plateau borders (Figure 5), can be characterized by the dimensionless elasticity number, Es, which is defined for one surface-active component (21) by... 

A considerable number of experimental extensions have been developed in recent years. Luckliam et al [5] and Dan [ ] review examples of dynamic measurements in the SFA. Studying the visco-elastic response of surfactant films [ ] or adsorbed polymers [7, 9] promises to yield new insights into molecular mechanisms of frictional energy loss in boundary-lubricated systems [28, 70]. 

The subscript G specifies elasticity determined from isothermal equilibrium measurements, such as for the spreading pressure-area method, which is a thermodynamic property and is termed the Gibbs surface elasticity, EG. EG occurs in very thin films where the number of molecules is so low that the surfactant cannot restore the equilibrium surface concentration after deformation. 

Many surfactant solutions are normal Newtonian liquids even up to rather high concentrations. Their viscosities are very small as compared with the viscosity of the solvent water. This is particulary the case for micellar solutions with concentrations up to 20% W/W in which spherical micelles are present. Even in the presence of rodlike micelles the viscosities can be rather low. Systems with rodlike micelles have recently been studied extensively. Missel et al. (1-2) studied alkylsulfate solutions and showed how the lengths of the rods can be varied by the addition of salt or by the detergent concentration. Under all these conditions the solutions are of rather low viscosity. On the other hand, we have studied a number of cationic detergent solutions in which rodlike micelles were formed and which all became quite viscous at rather low concentrations. In addition some of these solutions had elastic properties. The phenomenon of viscoelasticity in detergent solutions is not new. Exten-... 

The droplet deformation increases with increases in the Weber number which means that, in order to produce small droplets, high stresses (i.e., high shear rates) are require. In other words, the production of nanoemulsions costs more energy than does the production of macroemulsions [4]. The role of surfactants in emulsion formation has been described in detail in Chapter 10, and the same principles apply to the formation of nanoemulsions. Thus, it is important to consider the effects of surfactants on the interfacial tension, interfacial elasticity, and interfacial tension gradients. 

The partition of surfactant molecules between the oil and aqueous phases. With higher surfactant concentrations, the molecules with shorter EO chains (i.e., lower HLB number) may accumulate preferentially at the O/W interface. This may result in a reduction of the Gibbs elasticity, which in turn would cause an increase in the Ostwald ripening rate. 

Qualitatively, the Middle Atlantic Bight samples were very similar to those collected in the California Bight. The predominant components were the same in both locales, suggesting that a limited number of compound classes are dominant in microlayer films, but present in varying proportions. The specific mixtures of surfactants in the microlayers sampled in this study strongly influenced air-sea interfacial quasi-static elasticity. Pre-... 

Thus, the important features of the structural-mechanical barrier are the rheological properties (See Chapter IX,1,3) of interfacial layers responsible for thermodynamic (elastic) and hydrodynamic (increased viscosity) effects during stabilization. The elasticity of interfacial layers is determined by forces of different nature. For dense adsorption layers this may indeed be the true elasticity typical for the solid phase and stipulated by high resistance of surfactant molecules towards deformation due to changes in interatomic distances and angles in hydrocarbon chains. In unsaturated (diffuse) layers such forces may be of an entropic nature, i.e., they may originate from the decrease in the number of possible conformations of macromolecules in the zone of contact or may be caused by an increase in osmotic pressure in this zone due to the overlap between adsorption layers (i.e., caused by a decrease in the concentration of dispersion medium in the zone of contact). 

The first surface elasticity measurements appear to have been the Gibbs elasticity measurements made by Mysels et al. (25) for thin-films that are supported by glass frames. Their results indicated that for a number of surfactant systems EG (film) 10 mN/m, but the rigid films produced by the addition of a suitable alcohol could increase the elasticity of dodecyl sulfate to EQ (film) 100 mN/m. Lucassen-Reynders (23) reported a similar range of values of EM, from near-zero to about 70 mN/m, and suggested that adding simple electrolyte into an anionic surfactant solution could increase the elasticity by factors of 2 to 3. 

The observed stabilizing effect of surfactants toward convection induced by surface tension has been confirmed theoretically in a recent paper by Berg and Acrivos (B13), in which the stability analysis technique and the physical model were the same as Pearson s except that the free-surface boundary condition [(iii) of Table III] took into account the presence of surface active agents. Critical values for the Thompson number were computed as functions of two dimensionless parameters, one embodying the surface viscosity and the other the surface elasticity. ... 

Some consequences which result from the proposed models of equilibrium surface layers are of special practical importance for rheological and dynamic surface phenomena. For example, the rate of surface tension decrease for the diffusion-controlled adsorption mechanism depends on whether the molecules imdergo reorientation or aggregation processes in the surface layer. This will be explained in detail in Chapter 4. It is shown that the elasticity modulus of surfactant layers is very sensitive to the reorientation of adsorbed molecules. For protein surface layers there are restructuring processes at the surface that determine adsorption/desorption rates and a number of other dynamic and mechanical properties of interfacial layers. 

Note that relation (5.281) does not contain any kinetic characteristics of micellisation. If the micellar concentration is low and the formation (disintegration) of micelles is sufficiently fast (tj o), the adsorption rate and, consequently, the dynamic surface elasticity depend only on the efficiency of the surfactant transfer by micelles from the bulk to the surface, and, therefore, on the diffusion coefficient of micelles and the mean aggregation number. This means that the micellar size can be determined from dynamic surface properties. Really, if approximation (5.231) for the diffusion coefficient of micelles is used, it follows from Eq. (5.281)... 

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