Two-dimensional foams

With three bubbles, the septa must meet at 120° if the system is to be mechanically stable. A fourth bubble could now be added as shown in Fig. XIV-14, but this would not be stable. The slightest imbalance or disturbance would suffice to move the septa around until an arrangement such as in Fig. XIV-14h resulted. Thus a two-dimensional foam consists of a more or less uniform hexagonal type of network. 

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.)...

The theoretical analysis for two-dimensional foams and emulsions has recently been expanded to three dimensions [38], with Kelvin s minimal tet-rakaidecahedron as the unit cell. The system is subjected to a uniaxial extensional strain. As the elastic limit, or yield point, is approached, the cell shape tends towards a rhombic dodecahedron however, at the yield point, the shrinking quadrilateral faces of the polyhedron have finite (albeit small) area. 

Fig. 3.42 Region of two-dimensional foam formed by a PSi35-PNaA26 Langmuir-Blodgett film within the confines of a planar surface micelle domain (Meszaros et al 1994). 

Calculations of the elastic constants and yield stress of disordered two-dimensional foams as a function of the gas volume fraction have been reported by Weaire et al. [26-28]. 

A study of the flow of a polyhedral foam in a regime of slip at the tube walls has been conducted [39]. It has been established that the rise in the dynamic viscosity of the foaming solution leads to diminishing the flow rate but to a much lesser extent at t0 = 1.25 Pa. Thus, a two fold increase in viscosity causes a 1.3 times decrease in the flow rate, while a 6 times increase in the dynamic viscosity only a 2.23 times decrease. This is probably related to the expanding of the effective thickness of the liquid layer 8 (ca. 3 times). The transition from plug flow (slip regime) to shear flow occurs at To = 9-10 Pa. This value of the shear stress is much smaller than the one obtained from Princen s formula for a two-dimensional foam (Eq. (8.18)) at a given expansion ratio and correlates well with To calculated from Eq. (8.24) and the experimental data of Thondavald and Lemlich [23],... 

Figure 3. Two dimensional foam drainage for C AOS with 1 wt% NaCl.

Plateau borders. Euler s theorem then relates the numbers of polygons (P), sides (S), and vertices (V) by P — 5 -f V = 2. This equation, combined with the observation that three sides meet at every vertex, and every side connects two vertices, so that 3V = 25, leads to the conclusion that for an infinite two-dimensional foam the average number of sides per polygon is six. 

The earliest analysis of the deformation properties of a liquid foam is that of Princen (1983), who modeled two-dimensional foams and dense emulsions at rest by an array of regular hexagons (see Fig. 9-32a). While Princen s model was limited to hexagons with a particular orientation relative to the imposed shearing flow, this restriction was lifted in the work of Khan and Armstrong (1986) and Kraynik and Hansen (1986). Polydisperse cell sizes have also been considered (Weaire et al. 1986 Khan and Armstrong 1987 Weaire and Fu 1988 Kraynik et al. 1991 Okuzono et al. 1993), as well as wet foams with... 

At a molecular area that both Kim and Canned, and Pallas and Pethica agree should be in the G-LE two-phase region, the film is clearly heterogeneous there are circular dark domains, which can be associated with the gas phase, and a bright, continuous LE phase. If the monolayer is expanded, the bubbles of gas increase in size and the bright phase between them becomes increasingly thinner until the structure resembles a two-dimensional foam (see Section VII). When the expansion is stopped, the LE phase very slowly reorganizes into droplets. 

As already noted, when the LE phase is expanded, bubbles of gas grow and thin lamellae of fluid are left between them, forming a two-dimensional foam structure (Fig. 10a) in which the size of the cells and the distribution of the number of sides changes with time. Dendritic structures (Fig. 21) are observed in the LE-LC transition region when the monolayer is compressed, or in temperature quenches from the LE one-phase region into the two-phase region. In the case of fatty acids,only circular islands are observed at low temperature, even with rapid compression. The temperature threshold for the appearance of dendritic patterns is quite sharp. Such structures are also found in phospholipid monolayers. ... 

Later on Khan [225] also theoretically obtained the following estimate for the shear modulus of dry two-dimensional foam under the assumption that the foam cells are regular hexagonal cells and the deformation is not small ... 

Yield stress and plastic viscosity. The most important rheological characteristic determining the foam behavior ( solid-shaped or fluid-shaped ) is the yield stress To. This variable was calculated in [379] for a two-dimensional foam model ... 

The phenomena just mentioned lead to formation of a polyhedral foam the shape of the air cells approximates polyhedra. For cells of equal volume, the shape would be about that of a regular dodecahedron (a body bounded by 12 regular pentagons), and the edge q then equals about 0.8/-, where r is the radius of a sphere of equal volume. Actually, the structure is less regular, because of polydispersity. Moreover, close packing of true dodecahedrons is not possible. In a two-dimensional foam, say, one layer of bubbles... 

Debregeas G, Tabuteau H, di MegUo JM (2001) Deformation and flow of a two-dimensional foam under continuous shear. Phys Rev Lett 87(17) 178305... 

FIGURE 4.22 Image of two-dimensional foam prepared from saline solution of 4 wt.% of eommercial ethoxylated alcohol surfactant containing emulsified octane where oil drops have drained out of foam films into Plateau borders. (Reprinted from J. Colloid Interface Sci., 150, Koczo, K., Lobo, L., Wasan, D., 492. Copyright 1992, with permission from Elsevier.)... 

Recently a relatively successful approach is to study the two-dimensional foam steadily flowing through a construction (see Figure 4.26). 

Two-dimensional foam flowing downward through a construction. The wide field of view shows only the end of the 1-m-long horizontal channel. 

G. Katgert, M. E. Mohius, and M. van Hecke. Rate dependence and role of disorder in linearly sheared two-dimensional foams. Phys. Rev. Lett., 101 058301, 2008. 

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