Polyhedral packing

Fig. 14.3 Polyhedral packing plots for the two-dimensional layers of [RE(P2S6),/2(PS4)P in the series of solids A2RE(P2S6)i/2(PS4), where A=K, Cs RE = Y, La. Rare-earth polyhedra are striped PS4 polyhedra are black phosphorous atoms in P2S6 are shown as black circles. Alkali atoms are not shown for clarity. Although these phases have distinctly different structures based on space group symmetry and atomic positions, the compounds are clearly related upon close inspection of the building blocks.

Herbert Hauptman with one of his polyhedral packing models (courtesy of Herbert Hauptman). 

Fic. 96. Polyhedral packing diagram of [Cu(Lios)2lBF1 (a) and schematic view of the feldspar structure viewed down the a-axis (b). (From Fig. 2 in Keller, S. W. Angew. Chem., Int. Ed. Engl. 1997,36, 247.)... 

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. 

Table II lists several physical properties of various phosphors as compared to those for an ideal phosphor. The refractive index should be fairly low so that in a polymeric system (polymer Rl 1.5) the number of scattering phenomena are decreased. In this respect, BaFCI Eu is best. To reduce structure mottle and improve phosphor packing, spherical or polyhedral shaped particles are most desireable. Further, to improve phosphor packing, narrow particle distributions are best.

The phosphors should possess low refractive indices to reduce scattering. Polyhedral shapes promote better packing and reduced structure mottle. 

Figure 2. (a) ORTEP diagram (at 50% probability) of II showing the atom-labeling scheme, (b) Polyhedral representation of Z.I1 S 0,N. nnit and the one-dimensional chain, (c) Packing diagram, viewed along —6-axis of II. 

In a monodisperse foam the deformation of spherical bubbles and formation of films at the places of their contact starts when the gas content in the system reaches - 50% (vol.) for simple cubic bubble packing or 74% for close (face-centred) cubic or hexagonal packing (foam expansion ratio - 4). In a polydisperse foam the transition to polyhedral structure starts at expansion ratio n - 10-20, according to [ 10] but, as reported in [51], this can occur at n < 4, the latter being more probable. The structure which corresponds to the transition of bubbles from spherical to polyhedral shape is called occasionally honeycomb structure. 

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). 

However, the VL F value for a spherical foam is substantial and amounts to 48% of the total foam volume for a simple cubic packing and to 26% for a hexagonal close packing. For the latter the foam expansion ratio varies from 2.01 to 3.85 which may introduce large errors into the calculation of the //IIlln value. In a polyhedral foam the liquid volume can be neglected with respect to the foam volume but for the determination of //min( ) more detailed information on the structure of the foam is needed. 

All of these types of structure possess exposed triangular faces and a very common mode of cluster enlargement is by capping one or more of these faces by another metal atom [e.g., by an M(CO)3 group], thus generating a tetrahedron fused to the main or central unit. An example is Os7(CO)21. An additional metal atom (plus some ligands) may also associate itself with one of the edges of a polyhedral or close-packed central array. 

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