Tetrakaidecahedron

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. 

Tetrahydropyridines, 21 111 Tetrahydroquinolines, 21 187,192,198-199 Tetrahydroxyalkylethylenediamine titanate complexes, 25 95 Tetrahydroxyborate anion, 4 256-258 Tetrahydroxycuprate, 7 770 Tetrahymena, ribozyme from, 17 618, 619 Tetrairidium dodecacarbonyl, 16 63 Tetraisopropyl titanate, 25 92 Tetrakaidecahedron structure, 12 8 Tetrakis(2,4-di-lerl-butylphenyl)-4,4 -biphenylenediphosphonite, 3 114 Tetrakis(2-chloroethyl)... 

Fig. 1. Rhomboidal dodecahedron (RDH) (a), tetrakaidecahedron (TKDH) (b), rhomboidal dodecahedral packing (c) and tet-rakaidecahedral packing (d)...

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... 

Fig. 4. Regular pentagonal dodecahedron (RPD) (a), Kelvin s minimal tetrakaidecahedron (Kelvin s cell) (b) and /J-tetrakaidecahedron (c)...

Figure 2.5 Three cavities in gas clathrate hydrates (a) pentagonal dodecahedron (512), (b) tetrakaidecahedron (51262), (c) hexakaidecahedron (51264), (d) irregular dodecahedron (435663), and (e) icosahedron (51268).

The Kelvin tetrakaidecahedron and its construction by truncation of an octahedron by a cube. The edges of the tetrakaidecahedron are one third as long as the edges of the octahedron. 

Of the 14 faces, 6 have four edges and 8 have six edges. The average number of edges per face is (6 x 4 + 8 x 6)/14 = 5 /7 This is very close to the results of experiments on ft brass, vegetable cells, and soap bubbles, as shown in Figure 1.9. For the Kelvin tetrakaidecahedron the ratio of surface area to that of a sphere of the same volume is 1.099. Most other shapes have much higher ratios. 

Waire and Phelan report that space filling is 0.3% more efficient with an array of of six polyhedra with 14 faces and two polyhedra with 12 faces than with the Kelvin tetrakaidecahedron. (This calculation allows faces in each to be curved.) The 14-faced polyhedra have 12 pentagonal and 2 hexagonal faces, while the 12-faced polyhedra have distorted pentagons for faces. The average number of faces per polyhedra = (6 x 14 + 2 x 12)/8 = 13.5. 

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. 

Princen and Levinson [6] and Reinelt and Kraynik [7] give the following values for the volume and surface area of Kelvin s tetrakaidecahedron A = 26.5094a2 and A = 26.7419a2, respectively. Compared to the orthic polyhedron the calculated surface here is decreased by is 0.18% [6] and 0.159% [7]. 

Eqs. (4.15) and (4.16) are obtained empirically from experimental data about (p H) dependence for emulsions. Eq. (4.17) is derived theoretically by using rhombic dodecahedron. For other polyhedra (minimal tetrakaidecahedron and pentagonal dodecahedron) these dependences are the same. 

Shear stress for three-dimensional foams using the Kelvin s tetrakaidecahedron model is given in [29], The value of Young s modulus (modulus of extension) was calculated to be... 

Similarly, the grain/pore geometry of the intermediate stage should be reasonably approximated by Kelvin s tetrakaidecahedron with three coordinate cylindrical pores along its edges. 

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