Drops immobile interface

For the case of immobile interfaces and long bubbles (L- 0(2R) or greater), the principal contribution to the pressure drop is the viscous drag in the uniform film region. The pressure drop relation is then ... 

The function accounts for the effect of the surface mobility. For large interfacial elasticity one has d —>0, see Eq. (53) then T —>1 and Eq. (61) acquires a simpler form, corresponding to drops of tangentially immobile interfaces. In the other limit, small interfacial elasticity, one has d 1 and in such a case T oc 1/ln J, i.e., T decreases with the increase in d, that is, with the decrease in Eq. A numerical solution to this problem is reported in Ref 24. The effect of the interfacial viscosity on the transitional distance, which is neglected in Eq. (61), is examined in Ref 141. It is established therein that the critical distance, h, can be with in about 10% smaller than hp... 

The critical film thickness for rupture is of the order of 50 A. If the interaction time of the drops is too short to reach the critical film thickness, the drops will not coalesce. The drainage of the film is the rate-determining step in coalescence of deformable drops in polymer blends. Various models have been proposed to describe the film drainage. One model assumes fully mobile interfaces, another model assumes immobile interfaces, and a third model assumes partially mobile interfaces. The mobility of the interfaces is strongly dependent on the presence of impurities, such as surfactants. Surfactants reduce the mobility of the interfaces due to interfacial tension gradients [315]. 

In the limiting case of solid particles, we have x oo, and Equation 4.294 reduces to the Taylor equation (Equation 4.271). Note that in the case of close approach of two drops ( 1), the velocity ]7 is proportional to 4h. This implies that the two drops can come into contact Qt = 0) in a finite period of time (t < oo) under the action of a given force, F, because the integral in Equation 4.284 is convergent for hf = 0. This is in contrast to the case of immobile interfaces 1), when V oih and T 00 ior hf 0. 

A higher continuous phase viscosity increases the resistance to film drainage by partially immobilizing the drop-fluid interface. This reduces coalescence probability. If two similar volumes of immiscible liquids are dispersed, the fluid having the higher viscosity will normally become the continuous phase. The first attempts to produce suspension polymers used sugar to thicken the suspending phase and to retard coalescence. 

For drops having immobile interfaces, Coulaloglou and Tavlarides (1977) show the drainage time for unequal size drops to be... 

Mass transfer rates for skirted bubbles in polyvinyl alcohol solutions have been measured by Guthrie and Bradshaw (G9) and Davenport et al (D4). When a skirt is present the transfer rate increases, but not in proportion to the increase in surface area. Davenport attributes this to the accumulation on the surface of the skirt of surface-active impurities which immobilize the interface and reduce the transfer rate. Presumably transfer rates from skirted bubbles or drops in very pure liquids would be appreciably higher than from fluid particles without skirts. 

Colloids The Thickness of the Double Layer and the Bulk Dimensions Are of the Same Order. The sizes of the phases forming the electrified interface have not quantitatively entered the picture so far. There has been a certain extravagance with dimensions. If, for instance, the metal in contact with the electrolyte was a sphere (e.g., a mercury drop), its radius was assumed to be infinitely large compared with any dimensions characteristic of the double layer, e.g., the thickness K-1 of the Gouy region. Such large metal spheres, dropped into a solution, sink to the bottom of the vessel and lie there stable and immobile. 

Steric Hindrance. Another form of stabilization is relatively independent of ionic strength the oil droplets are prevented from making contact by simple steric hindrance. This may take two forms, either an immobilized water layer at the interface or a solid interfacial film. Emulsion stabilization by proteins, gums, and polyoxyethylene derivatives occurs by the first mechanism. Hydrophobic parts of the stabilizers adsorb at the oil surface, but adjacent large hydrophilic segments are hydrated and form an immobilized layer on the order of 10-100 nm thick (Figure 9). As mentioned, these hydrated segments frequently interact to cause flocculation, while coalescence of the oil drops themselves is prevented. Such emulsions are frequently used as carriers for oil-soluble flavors, essences, and colorants. 

The Gibbs elasticity characterizes the lateral fluidity of the surfactant adsorption monolayer. For high values of the Gibbs elasticity the adsorption monolayer at a fluid interface behaves as tangentially immobile. Then, if two oil drops approach each other, the hydro-dynamic flow pattern, and the hydrodynamic interaction as well, is the same as if the drops were solid particles, with the only differenee that under some conditions they could deform in the zone of contact. For lower values of the Gibbs elastieity the... 

In a first approximation, one can assume that the viscous dissipation of kinetic energy happens mostly in the thin liquid film intervening between two drops. (In reality, some energy dissipation happens also in the transition zone between the film and the bulk continuous phase.) If the drop interfaces are tangentially immobile (owing to adsorbed surfactant), then the velocity of approach of the two drops can be estimated by meanss of the Reynolds formula for the velocity of approach of two parallel solid disks of radius R, equal to the film radius (142) ... 

By contrast with the flow-induced dispersion, the flow-induced coalescence is not as well described. For example, there is a critical capillarity number for breakup, but not for coalescence. Three approaches have been used, based on (i) the minimization of energy [129], (ii) the frequency of collisions assumed proportional to the shear energy, ycri2 [292], and (iii) the instability of the liquid layer trapped between two drops with an immobile or mobile interface [125, 293-297] ... 

The potential at the boundary between the Stern layer and the diffuse part of the double layer is called the zeta potential ( ) and has values ranging from 0-100 mV. Because the charge density drops off with distance from the surface, so does the zeta potential the distance from the immobile Stern layer to a point in the bulk liquid at which the potential is 0.37 times the potential at the interface between the Stern layer and the diffuse layer, is defined as the double layer thickness and is denoted 8 (Figure 3.26). The equation describing 8 (Knox 1987) is ... 

The influence of the contact force F is not intuitively obvious one would expect a higher contact force to lead to faster coalescence. According to the models with immobile and partially mobile interfaces, the coalescence time increases with the contact force. This is due to the fact that the flattened area between the two deformable colliding drops increases with F, requiring more film material to be drained over a longer distance. 

A distinction can be made among the available methods between static and dynamic contact angle determination methods. In the case of a static determination the contact angle of a drop with an immobile solid/liquid/gas interface is determined microscopically (sessile drop). In the captive bubble method the contact angle of an air bubble, which is located under the solid surface in contact with the liquid, is determined. In contrast to the sessile drop method, in the captive bubble method the contact angle is measured at a completely wet surface. 

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