Border Plateau

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.)...

Make an estimate of the hydrostatic pressure that might be present in the Plateau border formed by the meeting of three thin black films. Make the assumptions of your calculation clear. 

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... 

Figure 3. Schematic of flowing lamellae in a periodic constricted pore. The porous medium imposes a capillary pressure, P, at each Plateau border. c...

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... 

Upon moving from the pore constriction (f=0) to the pore body (f = A/2), the lamella is stretched as it conforms to the wall. To achieve the requisite volume rearrangement a radial pressure differential is induced which thins the film but results in no net fluid efflux into the Plateau borders. The converse occurs when the film is squeezed upon moving from a pore body to a pore constriction. If R /R, or equivalently a, is large enough... 

M.P. Aronson and H.M. Princen Geometry of Clusters of Strongly Coagulated Fluid Drops and the Occurence of Collapsed Plateau Borders. Colloids Surfaces 4, 173 (1982). 

Thus we would expect water to climb up the walls of a clean (i.e. water-wetting) glass vessel for a few millimetres but not more, and we would expect a sessile water droplet to reach a height of several mm on a hydrophobic surface, before the droplet surface is flattened by gravitational forces. The curved liquid border at the perimeter of a liquid surface or film is called the Plateau border after the French scientist who studied liquid shapes after the onset of blindness, following his personal experiments on the effects of sunlight on the human eye. 

Regular dodecahedral structure of a bubble and cross-sectional view of a plateau border. 

The pressure within the plateau border can be related to the pressure in the dispersed phase pd and the plateau border suction Pc through... 

The radius of curvature of plateau border can be related to the area of the plateau border Op and the film thickness xp for a regular dodecahedral arrangement from geometric considerations by (9),... 

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