Model polyhedral foam

In [64] the possible stable configurations of films in polyhedral foams is discussed from the thermodynamic point of view that any disperse system tends to minimum surface energy. Almgren and Taylor [64] modelled the shape of the films and the angles between them with wire devices and studied several film configurations. They established that only film configurations which obey Plateau laws are stable with respect to minor deformations. 

The correctness of the above hydrodynamic model of a polyhedral foam with border hydroconductivity and constant radius of border curvature can be confirmed by comparing the velocity Q calculated from Eqs. (5.9) and (5.11) with the volumetric liquid rate obtained by Krotov s theory [7]. Thus, the liquid flow under gravity at r = const is... 

Considering a dodecahedral model of a polyhedral foam, the relation between its expansion ratio and dispersity is given by Eq. (4.9). The ratio between the initial foam expansion ratio and the expansion ratio at a given time is... 

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. 

Actual foam contains bubbles whose shape is intermediate between spheres and polyhedra. Such foam is said to be cellular [214, 280]. The distinction between the cellular and polyhedral kinds of foam is rather conventional and is determined by very low moisture contents (of the order of some tenth of per cent). Nevertheless, the polyhedral model of foam cells is used rather frequently [38,125,244,438,480],... 

The most important geometric parameters of polyhedral foam are the length of a Plateau border and its cross-section area. These quantities can be found in [214, 244]. For example, for the pentagonal dodecahedron model, the length of the Plateau border is... 

Here p and p are the liquid and gas densities, respectively, g is the vector of the gravitational acceleration, and AP is the capillary rarefaction given by (7.1.10) and (7.1.15). The kinetic coefficient H was called the coefficient of hydroconductivity and calculated for polyhedral foam models [245, 246]. Generally speaking, the variable H is a tensor, but usually the isotropic approximation is used, where this parameter is a scalar. Various expressions for the coefficient H were proposed and made more precise in [125, 214, 245]. Thus, different approaches used to calculate the coefficient of hydroconductivity were analyzed in [488]. For example, the structure of spherical and cellular foam was studied under the assumption that liquid flows through a porous layer according... 

Rupture of foams. In [242], the concept of critical thickness is suggested as a criterion of the rupture of foams. This means that polyhedral foam will rupture as soon as the thickness of some films making the faces of the foam polyhedron cells attains a critical value. Adopting the channel model of the foam, i.e., assuming that the liquid completely resides in Plateau borders, and using Eqs. (7.1.21) and (7.1.23), one can represent the volume fraction of the liquid phase, V (the reciprocal of the foam multiplicity K) in the form... 

Finally, returning briefly to foams, we note that the number of cells in a polyhedral foam can decrease by a mechanism other than him rupture by the mechanisms discussed above. As the pressure is largest for gas in the smallest cells, a driving force exists for transfer of gas from small to large cells by diffusion across the thin liquid Aims separating them. Ultimately this process leads to the disappearance of small cells and growth of large cells. Lemlich (1978) and Ranadive and Lemlich (1979) present a model of gas diffusion in foams. 

The expansion ratio profile of a continuously generated foam has been computed using various hydrodynamic models [87,88] but here again several significant simplifications are introduced. For example, a model of polyhedral bubbles was employed for all foam layers situated at different levels which, however, is not the real state in the lower foam layers. 

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). 

Preliminary remarks. Models of the foam cell. The polyhedral shape of foam cells is the limit shape as the foam multiplicity grows infinitely. At the same time, this is a rather convenient structural model for actual foam with finite multiplicity. A polyhedron constructed of liquid films must satisfy the following two rules, stated by Plateau [9, 379, 407] ... 

It was repeatedly proposed to use Kelvin s tetrakaidecahedron (that is, minimal truncated octahedron) [381, 407, 479] with eight hexagonal and six quadrangular faces as the polyhedral model of a foam cell and of a cell of any three-dimensional biological tissue. Note, however, that it was statistically shown [195] that Kelvin s tetrakaidecahedron is encountered in biological tissues among other tetrakaidecahedral cells only in 10% of the cases. 

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... 

Having described the structural elements of foams approaching the dry-foam limit (O —> 1), it is still a daunting task to describe the structure and properties of the system as a whole. The task is even more difficult for systems in which O Q is exceeded, but the polyhedral regime has not yet been reached. In this case, the drops have exceedingly complex shapes, and linear and tetrahedral Plateau borders, as defined above, are not present. Much can be learned about the qualitative behavior by considering 2-D model systems, in which the drops do not start out as spheres but as parallel circular cylinders, and tetrahedral Plateau borders do not arise. We shall first consider the particularly simple monodisperse case, with a subsequent gradual increase in complexity. 

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