Foams polyhedral cells

Fig. 25.8. Poly meric foams, showing the polyhedral cells. Some foams hove closed cells, others hove cells which ore open.

The so-called bi-liquid foams described by Sebba and Vincent [17-20] will not be discussed since they are not true colloidal emulsions, but resemble conventional foams more closely, possessing polyhedral cells of centimetre dimensions. 

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... 

Figure 12.13 Photograph of a foam showing the polyhedral cells bounded by plane faces.

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-... 

Figure 1 shows the cellular structure of a typical PP foam, which was manufactured by molding PP foam beads with superheated steam. Each bead consists of closed polyhedral cells, with no visible accumulation of material in the comers. The PP foam beads themselves are melted together at their surfaces and form a secondary cellular structure. 

Foam is an agglomeration of a large number of different bubbles (Plate 4.13). Each bubble in the foam is a polyhedral cell with a number of different faces. Each face is curved as a result of the excess pressure across it. If the pressure in two adjacent cells is the same, the excess pressure across the surface is zero and the separating surface is planar. The foam will contain, only, three film surfaces intersecting along lines at 120° and four lines of soap film meeting at a point with adjacent lines intersecting at 109° 28. ... 

Now applying this result to the average polyhedral cell of foam, with = 30 = 3(109.47 °), ... 

A foam is defined as a coarse dispersion of a gas in a liquid, where the volume fraction of gas is greater than that of the liquid. Solid foams (for example foam rubber or polystyrene foam) are also possible, but here we focus on more common liquid foams. These are always formed by mixtures of liquids (usually containing a soap or surfactant) and never by a pure liquid. If the volume fraction of gas is not too high, the bubbles in the foam are spherical, but at higher gas volume fractions the domains are deformed into polyhedral cells, separated by thin films of liquid (Fig. 3.12). Typically the gas bubbles are between 0.1 and 3 mm in diameter. 

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

The cells in foams are polyhedral, like grains in a metal (Fig. 25.8). The cell walls, where the solid is concentrated, can be open (like a sponge) or closed (like a flotation foam), and they can be equiaxed (like the polymer foam in the figure) or elongated... 

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... 

Similarly, some solid foams (e.g. foam rubber) consist of spherical gas bubbles trapped within a solid network, whereas others (e.g. expanded polystyrene) consist of as little as 1 per cent solid volume and are composed of polyhedral gas cells separated by very thin solid walls. 

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. ... 

In case of a fully polyhedral foam R/r 1 when the contribution of the vertex (by volume and length) can be neglected compared to the total volume of the liquid in one cell, the border along the whole length between the vertexes has identical area of the cross-section, and a single radius of curvature. The area of the cross-section of border A is determined by the radius of curvature of the border (r) and the film thickness (h) (Fig. 1.9)... 

Complete information about the liquid distribution between films and Plateau borders is supplied from the data about the border radius curvatures, the film thickness and the films to Plateau borders number ratio in an elementary foam cell. For a polyhedral foam consisting of pentagonal dodecahedron cells the ratio of film liquid volume and border volume can be expressed by the formulae (see Eqs. (4.7) - (4.9))... 

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],... 

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. 

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