Polyhedral foams

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,... 

These local stmctural rules make it impossible to constmct a regular, periodic, polyhedral foam from a single polyhedron. No known polyhedral shape that can be packed to fiU space simultaneously satisfies the intersection rules required of both the films and the borders. There is thus no ideal stmcture that can serve as a convenient mathematical idealization of polyhedral foam stmcture. Lord Kelvin considered this problem, and his minimal tetrakaidecahedron is considered the periodic stmcture of polyhedra that most nearly satisfies the mechanical constraints. 

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... 

The most important parameters characterising a polyhedral foam are expansion ratio, dispersity and foam stability. The expansion ratio n is the ratio between the foam volume V and the volume of the liquid content Vl in it... 

At the onset of formation by barbotage methods the foam represents a gas emulsion. The rate of its transformation into a polyhedral foam depends on the velocity of bubble rise and the consequent drainage of the excess" liquid from the foam thus formed. Bubble size,... 

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. ... 

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. 

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)... 

Dispersity of gas emulsions and polyhedral foams is a very important parameter which determines many of their properties and processes occurring in them (diffusion transfer, drainage, etc.) and, therefore, their technological characteristics and areas of application. The kinetics of changes in dispersity indicates the rate of foam inner destruction resulting from coalescence and diffusion transfer. In real foams bubble size varies in a wide range (from micrometers to centimetres). Only by means of special methods it is possible to obtain foam in which bubble size varies in a narrow interval, i.e. foam that can be regarded as monodisperse. 

For a polyhedral foam this is the radius of a sphere with an equivalent volume. 

The dependence expressed in Eq. (1.35) is strictly fulfilled only when V> = Vc, i.e. when the gas volume fraction (p — 1. In order to calculate the excess (internal) pressure in bubbles of a fully polyhedral foam it is necessary to introduce two correction [84] ... 

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. 

For a polyhedral foam in which only the borders conduct electricity, i.e. the liquid in films is neglected, B = 3. Vice versa, if the conductivity is determined by the films and the contribution of borders is neglected, then B = 1.5. For the intermediate cases of liquid distribution in a foam the value of B depends on the ratio of liquid volume in films and borders = V/V ... 

Very useful for the kinetic studies of dispersity changes in a foam is the method based on simultaneous measurement of local expansion ratio and border pressure at one and the same level in the foam column [5]. This method can be applied to polyhedral foams in which the liquid is mainly in the borders. A formula for the length of a dodecahedron edge is obtained when Eqs. (1.43), (1.45) and (4.10) are solved together... 

A set up for evaluating optical density of foams is described in Section 8.4. (see Fig. 8.3). A technique for determination of the average bubble size in non-polyhedral foams employing the dependence of extinction on the foam layer thickness has been proposed by Nekrassov [64] (see Section 8.4.). 

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))... 

The radius of border curvature in a polyhedral foam which conforms the condition r/a 1 is the easiest to determine. If a micromanometer is used to measure the capillary pressure pa, then the radius of border curvature r can be calculated by combining Eqs. (1.40) and (1.45)... 

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... 

Liquid flow through polyhedral foams with different types of foam films... 

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]... 

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... 

An intrinsic property of a polyhedral foam is the reduced pressure in its Plateau borders. At the moment of foam formation the pressure in the borders depends mainly on the foam expansion ratio, dispersity and surface tension (see Eqs. (4.9) and (4.10)). At hydrostatic equilibrium the border pressure is expressed by Eqs. (1.37) and (1.38). 

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 order to investigate the influence of external pressure (disjoining pressure) experiments with single foam films have been carried out using the Thin Liquid Film -Pressure Balance Technique, described in Section 2.1.8 [e.g. 47,48], The radius of the microscopic foam films was close to the initial film radius in a real polyhedral foam (about 0.2 to 0.3 mm). Fig. 7.7 presents a histogram of the distribution by size (diameter) of films in the foam. The most probable film size (under these conditions) has permitted us to choose a suitable radius for the single foam films for these experiments. 

The foam behaviour at low surfactant concentrations is the same as the described above. However, the formation of polyhedral foam at the upper parts of the foam column occurs at much lower surfactant concentrations. It should be noted that these concentrations are considerably lower than those at which form CBF and NBF. This is related to the effect of surfactant concentration in the foam and depends mainly on the surface activity of the substances and on the foam film thickness [53,54,121]. The higher surface activity of... 

Since the densities of gas and oil are quite different, foamed emulsions can separate into concentrated emulsions and a polyhedral foam in which no oil (or very little) is present. The ability to control such instability proves to be the determining factor in the practical application of such systems. In that respect the only study known to the authors is [126]. 

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],... 

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 experimental data given in [45] reveal that at foam expansion ratio higher than 200 the value of B approaches 3. According to the data in [54] for foams with n from 80 to 900, B from 2.6 to 2.8, which corresponds to liquid content in borders of 85 to 95%. In the range of intermediate expansion ratios, corresponding to the transition from spherical to polyhedral foam, B depends on the degree of polyhedricity and on the liquid distribution between films, borders and vertexes. 

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