Foams Plateau borders

The reduced pressure in foam Plateau borders can be measured with a capillary manometer, developed by Khristov et al. [31,32]. A number of modifications of such a micromanometer are available. 

The main element of any micromanometer is a capillary hermetically welded to a porous plate (usually a sintered glass filter) with a suitable size of pores. This filter ensures a contact between the liquid in the capillary and the foam Plateau borders. 

The foam expansion ratio can be characterised by the liquid volume fraction in the foam, which is the sum of the volume fractions of the films, plateau borders and vertexes. Alternatively, the foam density can be used as a measure of the foam expansion ratio. The reduced pressure in the foam plateau border can be measured using a capillary manometer [4], while the bubble size and shape distribution in a foam can be determined by microphotography of the foam. Information about the liquid distribution between films and plateau borders is obtained from the data on the border radius of curvature, the film thickness, and the film-to-plateau border number ratio obtained in an elementary foam cell. 

Plateau Border The region of transition at which thin fluid films are connected to other thin films or mechanical supports such as solid surfaces. For example, in foams Plateau borders form the regions of liquid situated at the junction of liquid lamellae. Sometimes referred to as a Gibbs ring or Gibbs—Plateau Border. 

Fig. XIV-16. A photomicrograph of a two-dimensional foam of a commercial ethox-ylated alcohol nonionic surfactant solution containing emulsified octane in which the oil drops have drained from the foam films into the Plateau borders. (From Ref. 234.)...

J. A.F. Plateau, who first studied their properties. It is the Plateau borders, rather than the thin Hquid films, which are apparent in the polyhedral foam shown toward the top of Figure 1. Lines formed by the Plateau borders of intersecting films themselves intersect at a vertex here mechanical constraints imply that the only stable vertex is the one made from four borders. The angle between intersecting borders is the tetrahedral angle,... 

A real foam has further degrees of freedom available for estabHshing local mechanical equiHbrium the films and Plateau borders may curve. In fact, curvature can be readily seen in the borders of Figure 1. In order to maintain such curvature, there must be a pressure difference between adjacent bubbles given by Laplace s law according to the surface free energy of the film and the principle radii of curvature of the film AP = ) Note that the... 

Fig. 3. Two-dimensional schematic illustrating the distribution of Hquid between the Plateau borders and the films separating three adjacent gas bubbles. The radius of curvature r of the interface at the Plateau border depends on the Hquid content and the competition between surface tension and interfacial forces, (a) Flat films and highly curved borders occur for dry foams with strong interfacial forces, (b) Nearly spherical bubbles occur for wet foams where...

Fig. 4. Schematic representation of a two-dimensional model to account for the shear modulus of a foam. The foam stmcture is modeled as a coUection of thin films the Plateau borders and any other fluid between the bubbles is ignored. Furthermore, aH the bubbles are taken to be uniform in size and shape.

Column Operation To assure intimate contact between the counterflowing interstitial streams, the volume fraction of liquid in the foam should be kept below about 10 percent—and the lower the better. Also, rather uniform bubble sizes are desirable. The foam bubbles will thus pack together as blunted polyhedra rather than as spheres, and the suction in the capillaries (Plateau borders) so formed vidll promote good liqiiid distribution and contact. To allow for this desirable deviation from sphericity, S = 6.3/d in the equations for enriching, stripping, and combined column operation [Lemhch, Chem. E/ig., 75(27), 95 (1968) 76(6), 5 (1969)]. Diameter d still refers to the sphere. 

Measuring the radius of a foam lamella plateau border where it initially contacts the oil or of an emulsified drop... 

To explain the role of the medium capillary pressure upon foam coalescence, consider a flat, cylindrical, stationary foam lamella of thickness, 2h, circa 1000 A, and radius, R (i.e., 50 to 100 /xm), subject to a capillary pressure, P, at the film meniscus or Plateau border, as shown in Figure 3. The liquid pressure at the film meniscus is (P - P ), where P is the gas pressure. g c g... 

It is now possible to explain the origin of a critical capillary pressure for the existence of foam in a porous medium. For strongly water-wet permeable media, the aqueous phase is everywhere contiguous via liquid films and channels (see Figure 1). Hence, the local capillary pressure exerted at the Plateau borders of the foam lamellae is approximately equal to the mean capillary pressure of the medium. Consider now a relatively dry medium for which the corresponding capillary pressure in a... 

At this point, the concept of the linear collapsed Plateau border is introduced. The Plateau border is the area of bulk continuous phase between three adjacent droplets or cells in an emulsion or foam respectively. The collapsed border is, therefore, an extremely thin version, which can be represented macro-scopically, as the line of intersection of three films of zero thickness, at angles of 120°. 

In a later study [56], the effect of gas volume fraction (foam rheology was investigated. Two models were considered one in which the liquid was confined to the Plateau borders, with thin films of negligible thickness and the second, which involves a finite (strain-dependent) film thickness. For small deformations, no differences were observed in the stress/strain results for the two cases. This was attributed to the film thickness being very much smaller than the cell size. Thus, it was possible to neglect the effect of finite film thickness on stress/strain behaviour, for small strains. 

This work shows that high shear rates are required before viscous effects make a significant contribution to the shear stress at low rates of shear the effects are minimal. However, Princen claims that, experimentally, this does not apply. Shear stress was observed to increase at moderate rates of shear [64]. This difference was attributed to the use of the dubious model of all continuous phase liquid being present in the thin films between the cells, with Plateau borders of no, or negligible, liquid content [65]. The opposite is more realistic i.e. most of the liquid continuous phase is confined to the Plateau borders. Princen used this model to determine the viscous contribution to the overall foam or emulsion viscosity, for extensional strain up to the elastic limit. The results indicate that significant contributions to the effective viscosity were observed at moderate strain, and that the foam viscosity could be several orders of magnitude higher than the continuous phase viscosity. 

To understand drainage we have to discuss the pressure inside the liquid films. At the contact line between liquid films, a channel is formed. This is called the Plateau border. Due to the small bending radius (rP in Fig. 12.18), there is a significant Laplace pressure difference between the inside of the compartment and the liquid phase. The pressure inside the liquid is significantly smaller than in the gas phase. As a result, liquid is sucked from the planar films into the Plateau s border. This is an important effect for the drainage of foams because the Plateau borders act as channels. Hydrodynamic flow in the planar films is a slow process [574], It is for this reason that viscosity has a drastic influence on the evolution of a foam. Once the liquid has reached a Plateau border the flow becomes much more efficient. The liquid then flows downwards driven by gravitation. 

Foams are always thermodynamically stable. The stability of liquid foams is largely determined by the repulsion between surfactants and the viscosity of the liquid. They decay by drainage driven by the negative Laplace pressure in the Plateau borders. 

Figure 3.26 Pressure differences across the curved surfaces in a foam lamella leading to Pa> Pb and liquid flow towards Plateau borders at the expense of film thinning.

Figure 1.5 shows an example of an aqueous foam with emulsified and imbibed crude oil droplets residing in its plateau borders. Using a simple model the degree of emulsification and imbibition has been found to correlate quite well with foam sensitivity to oil for a wide variety of foams, oils, and conditions [114]. A limitation of emulsification/imbibition models is that they will only be important for foam lamellae that are thick enough to accommodate realistic emulsion droplet sizes. Typical foam lamellae in porous media appear to have thicknesses on the order of tens of nm [70,71,114,328],... 

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