Foam structure polyhedral

Because of the non-regularity of the polyhedral foam structure (lack of long-range order) the foam becomes macroscopically isotropic, the specific surface area (per unit volume) accepting the luminous flux, is uniformly distributed in direction normal to the films... 

Thus, in approximate modeling it is possible to use the following polyhedral model of the foam structure [214] (see Figure 7.1) ... 

Foam is a disperse system in which the dispersed phase is a gas (most commonly air) and the dispersion medium is a liquid (for aqueous foams, it is water). Foam structure and foam properties have been a subject of a number of comprehensive reviews [6, 17, 18]. From the viewpoint of practical applications, aqueous foams can be, provisionally, divided into two big classes dynamic (bubble) foams which are stable only when gas is constantly being dispersed in the liquid 2) medium and high-expansion foams capable of maintaining the volume during several hours or even days. In general, the basic surface science rules are established in foam models foam films, monodisperse foams in which the dispersed phase is in the form of spheres (bubble foams) or polyhedral (high-expansion foams). Meanwhile, real foams are considerably different from these models. First of all, the main foam structure parameters (dispersity, expansion, foam film thickness, pressure in the Plateau-Gibbs boarders) depend... 

Foam structure and dynamics. Surface layers surrounding the bubbles in a foam act as a membrane or skin that can stretch and relax in response to the lateral forces acting on it. At first, drainage of liquid taking place at the surface layer is entirely hydrod)niamic, but once spherical bubbles are in contact, flat walls develop between them, and polyhedral cells appear in the foam (Fig. 14.9c). Capillary forces be-... 

These local structural rules make it impossible to construct a regular, periodic, polyhedral foam from a single polyhedron. No known polyhedral shape that can be packed to fill space simultaneously satisfies the intersection rules required of both the films and the borders. There is thus no ideal structure that can serve as a convenient mathematical idealization of polyhedral foam structure. Lord Kelvin considered this problem, and his tet-rakaidecahedron is still considered the best periodic structure of identical polyhedra that can nearly satisfy the mechanical constraints, while providing the smallest surface energy (or area). A more efficient structure for minimizing the surface energy, has more recently been proposed (known as the Weaire-Phelan structure (10)), but it consists of bubbles of two different types, and whether it is the optimal structure remains an open question. 

Figure 3.12 Foam structure. Left spherical bubbles. Right polyhedral cells...

Near the surface, a high (>70 vol.%) gas content structure, polyhedral foam is usually formed. Below the surface is a low (<70 vol.%) gas content structure called a spherical foam (or kugelschaum or gas emulsion). 

Figure 13.5 Foam structure during formation and drainage in a vertical column. Gas is injected from below. Polyhedral foam ("Polyderschaum") has a high gas content (>70 vol.%). The liquid in polyhedral foam structure is distributed between films and Plateau borders (i.e. channels that form where films meet). Reprinted from Pugh (1996), with permission from Elsevier...

There appear to be two stages in the collapse of emulsions flocculation, in which some clustering of emulsion droplets takes place, and coalescence, in which the number of distinct droplets decreases (see Refs. 31-33). Coalescence rates very likely depend primarily on the film-film surface chemical repulsion and on the degree of irreversibility of film desorption, as discussed. However, if emulsions are centrifuged, a compressed polyhedral structure similar to that of foams results [32-34]—see Section XIV-8—and coalescence may now take on mechanisms more related to those operative in the thinning of foams. 

While most vesicles are formed from double-tail amphiphiles such as lipids, they can also be made from some single chain fatty acids [73], surfactant-cosurfactant mixtures [71], and bola (two-headed) amphiphiles [74]. In addition to the more common spherical shells, tubular vesicles have been observed in DMPC-alcohol mixtures [70]. Polymerizable lipids allow photo- or chemical polymerization that can sometimes stabilize the vesicle [65] however, the structural change in the bilayer on polymerization can cause giant vesicles to bud into smaller shells [76]. Multivesicular liposomes are collections of hundreds of bilayer enclosed water-filled compartments that are suitable for localized drug delivery [77]. The structures of these water-in-water vesicles resemble those of foams (see Section XIV-7) with the polyhedral structure persisting down to molecular dimensions as shown in Fig. XV-11. 

Let us discuss the structure of a metastable polyhedral foam in a bit more detail. In pioneering experimental studies Joseph Plateau5 established some simple rules in the second half of the 19th century. Three of these... 

When the number and volume of the polyhedral compartments are given, the optimal structure of the foam is the one that creates the smallest total film area. This condition constitutes a formidable but straightforward mathematical optimization problem. Solution as an average, the polyhedra consist of 13.4 sides. Experimentally it was indeed found that the polyhedra most commonly found in foams have 14 sides, followed by 12 sides as a second choice. 

A foam is a coarse dispersion of gas in liquid, and two extreme structural situations can be recognised. The first type (dilute foams) consist of nearly spherical bubbles separated by rather thick films of somewhat viscous liquid. The other type (concentrated foams) are mostly gas phase, and consist of polyhedral gas cells separated by thin liquid films (which may develop from more dilute foams as a result of... 

In a monodisperse foam the deformation of spherical bubbles and formation of films at the places of their contact starts when the gas content in the system reaches - 50% (vol.) for simple cubic bubble packing or 74% for close (face-centred) cubic or hexagonal packing (foam expansion ratio - 4). In a polydisperse foam the transition to polyhedral structure starts at expansion ratio n - 10-20, according to [ 10] but, as reported in [51], this can occur at n < 4, the latter being more probable. The structure which corresponds to the transition of bubbles from spherical to polyhedral shape is called occasionally honeycomb structure. 

If four similar bubbles are brought into contact the four films formed (Fig. 1,6,b) can be balanced when the angle between them is 90° but this structure is unstable. The slightest change in pressure in any bubble disturbs force equilibrium and the contact area of these four films is transformed into a system with two Plateau borders, where three films meet (Fig. 1.6,c). Thus the monolayer of polyhedral foam which can be formed from identical bubbles between two plates will have symmetrical and regular structure with a hexagonal packing (Fig. 1.6,d). 

The geometry of three-dimensional polyhedral foam is more complex. Plateau experimented with soap bubbles and found that in equilibrium polyhedral structure (first law of Plateau) at each vertex of the polyhedron (cell) six faces (films) and four Plateau borders meet. The angle between borders equals lOO (second law of Plateau). This has been proved by Matzke who studied real foams [61-63], Plateau borders meet at the so-called vertex. Other configurations of the borders are unstable. ... 

The polyhedral foam consists of gas bubbles with a polyhedral shape the faces of which are flat or slightly bent liquid films, the edges are the Plateau borders and the edge cross-points are the vertexes (see Chapter 1). In the study of the physicochemical characteristics of foams there are several techniques that involve the analytical dependences of these characteristics and the structural parameters of foams. 

A semi-quantitative estimation of the influence of the structural parameters and physicochemical properties of the foaming solution on the initial drainage rate can be obtained from the equation describing the drainage in a homogenous polyhedral foam, the liquid of which flows out only through the borders [7]... 

Individual structural elements of the foam, such as films and borders, can be under hydrostatic equilibrium and can correspond to a true metastable state. Therefore, when there is no diffusion expansion of bubbles in a monodisperse foam, its state can be regarded as metastable in the whole disperse system. Krotov [5-7] has performed a detailed analysis of the real hydrodynamic stability of polyhedral foam by solving two problems determination of... 

In a polyhedral foam the liquid is distributed between films and borders and for that reason the structure coefficient B depends not only on foam expansion ratio but also on the liquid distribution between the elements of the liquid phase (borders and films). Manegold [5] has obtained B = 1.5 for a cubic model of foam cells, assuming that from the six films (cube faces) only four contribute to the conductivity. He has also obtained an experimental value for B close to the calculated one, studying a foam from a 2% solution of Nekal BX. Bikerman [7] has discussed another flat cell model in which a raw of cubes (bubbles) is shifted to 1/2 of the edge length and the value obtained was B = 2.25. A more detailed analysis of this model [45,46] gives value for B = 1.5, just as in Manegold s model. 

The extinction of the luminous flux passing through a foam layer occurs as a result of light scattering (in the processes of reflection, refraction, interference and diffraction from the foam elements) and light absorption by the solution. In a polyhedral foam there are three structural elements, clearly distinct by optical properties films, Plateau borders and vertexes. The optical properties of single foam films have been widely studied (see Section 2.1.3) but these of the foam as a disperse systems are poorly considered. 

The initial expansion ratio and dispersity of polyhedral foams are related through the quantitative dependence, given by Eq. (4.9). There at Ap > 103 Pa the content of the liquid phase in the films can be neglected. Thus, the connection of the structure parameters n, a and r can be expressed by the simple relation in Eq. (4.10). It follows from it that under given foaming conditions a definite expansion ratio can be reached by changing the border pressure, foam dispersity and surface tension of the foaming solution. 

However, the VL F value for a spherical foam is substantial and amounts to 48% of the total foam volume for a simple cubic packing and to 26% for a hexagonal close packing. For the latter the foam expansion ratio varies from 2.01 to 3.85 which may introduce large errors into the calculation of the //IIlln value. In a polyhedral foam the liquid volume can be neglected with respect to the foam volume but for the determination of //min( ) more detailed information on the structure of the foam is needed. 

As mentioned in Chapter 5, the analytical dependence of the initial volumetric rate of drainage w0 on the structural parameters of a foam is unknown. Therefore, a semi-quantitative estimation of the effect of both the structural parameters of a foam and the properties of a foaming solution can be done using the Eq. (5.60) of the flow rate in a homogeneous polyhedral border foam [59]. 

Provided that G > Gp (for liquid foams x of solutions > x of air) we obtain Eq. (39) from Eq. (41) by substitution of k instead of G. In contrast to Wagner s formula, Odelevsky s formula holds for all concentrations of the disperse phase (gas) and for all types of gas-filled systems gaseous emulsions (d < 0.74), spherical (0.74 < d< 0.9) and polyhedral ( > 0.9) foams. It requires isotropy of the matrix structures and equal diameters of the disperse phase inclusions. Therefore, the dependence of the ratio of the foam to the solution electroconductivity on the degree of foaming in the general form is given by equation... 

The actual structure of highly foamed systems is polyhedral therefore, the models proposed by Bikerman (Fig. 19b) and Chistyakov and Chemina (Fig. 19c) are more frequently used in case of high voltages, for example for lining high-voltage transformers. 

At high shear rates in some systems, the onions become large and very monodisperse in size, and they then order into a macrocrystalline packing. At rest, it is clear that the onions are not spherical, but polyhedral, because they must fill space. In the perfectly ordered macrocrystalline state, the typical shape of the space-filling onions appears to be that of the Kelvin tetrakaidecahedron, which is a model structure for liquid foams (see Section 9.5.1). These well-defined MLVs might be important as encapsulants in the pharmaceutical or cosmetics industries (Roux and Diat 1992). 

A foam is a disperse system that consists of gas bubbles separated by liquid layers. Because of the significant density difference between the gas bubbles and the medium, the system quickly separates into two layers, with the gas bubbles rising to the top, which may undergo deformation to form polyhedral structures this process will be discussed in detail below. 

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