Foam cells pentagonal dodecahedron

Support for this postulation came from work done on the shape of the ideal foam cell [32-40]. Ross and co-worker [34,35] proposed three minimal geometric structures, i.e. those which will subdivide space with minimum parti-tional area. These were the pentagonal dodecahedron, the minimal tet-rakaidecahedron, originally suggested by Thomson (Lord Kelvin), and the P-tetrakaidecahedron (Fig. 4). 

The pentagonal dodecahedron, however, is not entirely space-filling, i.e. a close-packed array of such figures has a number of interstitial voids. On the other hand, Kelvin s tetrakaidecahedron and the P-tetrakaidecahedron are. The latter requires 4% more surface area, so a system of such figures would spontaneously rearrange to the more stable array of Kelvin cells. Thus, it would seem that Kelvin s tetrakaidecahedron is the ideal candidate nevertheless, this is not observed in real systems Pentagonal faces are shown on foam cells. These... 

Pentagonal dodecahedron [76-80], compact tetradecahedron [73,80,82] and minimal tetrakaidecahedron [67,68] are most often used as models of foam cells in the calculation of foam electrical conductivity and hydroconductivity, foam dispersity and in the process of adsorption accumulation of foam. 

The shape of foam films and border profiles in large interval of foam expansion ratio from 10 to 1500 has been experimentally studied in [83], A regular pentagonal dodecahedron made up of transparent organic glass with an elastic rubber balloon inside it which took the shape of a sphere at inflation (Fig. 1.10) was used as a model of foam cell. 

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))... 

Considering a pentagonal dodecahedron model for foam cells Budansky and Kimmel [25] have derived a value for G which is between those obtained from the above discussed two models. 

The ratio of the interfacial area to the liquid volume in a foam can be expressed via the unit cell parameters of the foam. For example, if a pentagonal dodecahedron is taken as the unit cell, this ratio will be [24,47]... 

Figure 7.1. A foam cell model (a) pentagonal dodecahedron, (b) section of the Plateau border...

This compares to values of 1.0990 for the planar tetrakaidecahedron 1.1053 for the rhombic dodecahedron and 1.0984 for the regular pentagonal dodecahedron. The latter - though often considered as a unit cell in foam modeling - is not really a viable candidate either, as it not only violates Plateau s laws but is also not space filling.)... 

As pointed out by Reinelt and Kraynik (54), however, the idealized vertex does not adequately represent an equilibrium structure. Similar reservations apply to the work of Budiansky and Kimmel (95), who considered the behavior of an isolated foam cell in the form of a rectangular pentagonal dodecahedron and obtained a shear modulus between the two above values. 

The compressive strength is defined by the elastic instability of the thin cell wall membranes. The closed-cell foam is presumed to have a cell geometry represented by a pentagonal dodecahedron. Based on the buckling load for a simple supported thin circular plate which has the same area as the pentagonal faces, the final result for the beginning of the collapse plateau is ... 

The most important structural variables are again polymer composition, density, and cell size and shape. Structural foams have relatively high densities (typically >300 kg/m ), and cell structures similar to those in Figure 2d that are primarily composed of holes in contrast to a pentagonal dodecahedron t5q>e of cell structure in low density plastic foams. Since structural foams are generally not imiform in cell structure, they exhibit considerable variation in properties with particle geometry (100). 

The constant C, which can be obtained by fitting the experimental data to the equation, accounts for the cell shape. For example, C takes a value of 3.46 for pentagonal dodecahedrons and a value of 3.35 for tetracaidecahedra. The other way to estimate the cell shape is by doing a permanganate etching to the foam. Thin cell walls are removed, and the shape of the cells can be observed directly with a microscope (Fig. 6). Pentagonal dodecahedrons and tetracaidecahedra are usually observed. 

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... 

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