Polyhedra space-filling

The composition of clathrates with crystal structures built-up of space-filling polyhedrons (e.g. clathrates I, II, HI and IV) can be calculated using the formalism mentioned above for the sodalite type of structure. For these clathrates, the composition results from the fact that each framework atom is shared by four polyhedral cages. For the first observed intermetallic clathrate I, NagSUe [17, 31-33] with two NaSiao and six NaSi24 polyhedrons in the unit cell, the content of the unit cell is... 

It may often be convenient to describe the crystal structure in terms of the domains of the atoms [40], The domain is the polyhedron enclosed by planes drawn midway between the atom and each neighbor, these planes being perpendicular to the lines connecting the atoms. The number of faces of the polyhedral domain is the coordination number of the atom and the whole structure is a space-filling arrangement of such polyhedra. 

Silanation reaction of silane (SiELi) with OH groups Sorption uptake of liquid or gas by a microporous material Space-filling spatial arrangement of polyhedra such that each polyhedron shares all its faces with other polyhedra Stacking fault misalignment of layer(s) arising from a fault plane... 

Crystal structures may be described in terms of the coordination polyhedra MX of the atoms or in terms of their duals, that is, the polyhedra enclosed by planes drawn perpendicular to the lines M-X joining each atom to each of its neighbours at the mid-points of these lines. Each atom in the structure is then represented as a polyhedron (polyhedral domain), and the whole structure as a space-filling assembly of polyhedra of one or more kinds. We can visualize these domains as the shapes the atoms (ions) would assume if the structure were uniformly compressed. For example, h.c.p. and c.c.p. spheres would become the polyhedra shown in Fig. 4.29. These polyhedra are the duals of the coordination polyhedra illustrated in Fig. 4.5. These domains provide an alternative way of representing relatively simple c.p. structures (particularly of binary compounds) because the vertices of the domain are the positions of the interstices. The (8) vertices at which three edges meet are the tetrahedral interstices, and those (6) at which four edges meet are the octahedral interstices. Table 4.9 shows the octahedral positions occupied in some simple structures c.p. structures in which tetrahedral or tetrahedral and octahedral sites are occupied may be represented in a similar way. (For examples see JSSC 1970 1 279.)... 

Two space-filling models have been offered to explain the bicontinuous structure [110]. One of them is the so-called Talmon-Prager model [111-113] in which Voronoi polyhedra have been used to fill a given space. The filling is proposed in such a way that one can (i) divide the polyhedra in two types, i.e. oil and water and (ii) go on increasing the number of one type of polyhedron (oil or water), correspondingly decreasing the other, so that one droplet microemulsion can smoothly transform into the other via a state where two continuous phases coexist, but are separated. In real cases, of course, a film of surfactant molecules separates the two phases. In the model, the surfactant volume is not taken into consideration. 

As indicated above, the clathrate hydrates adopt a number of different structures. These are often visualized as consisting of one or more polyhedra, which share faces or edges to fill three-dimensional space. Each polyhedron has fonr-connected hydrogen-bonded water molecules at each vertex, and hence clathrate hydrates can be considered as ices. Some typical hydrate cages are shown in Figure 1. 

Fig. 29.9 The structure (determined by X-ray diffractirai) of a picket-fence porphyrin model compound with an azido ligand bound to the Fe(II) centre (a) a stick representation illustrating the protection of the [N3] ligand by the four fence-posts of the porphyrin ligand, and (b) a space-filling model looking down into the cavity. [Data I. Hachem e( al. (2009) Polyhedron, vol. 28, p. 954.] Coloin code Fe, gretai N, blue O, red C, grey H, white.

Inasmuch as the inscribed sphere corresponds to only 226 electrons per unit cube, it seems likely that the density of energy levels in momentum space has become small at 250.88, possibly small enough to provide a satisfactory explanation of the filled-zone properties. However, there exists the possibility that the Brillouin polyhedron is in fact completely filled by valence electrons. If there are 255.6 valence electrons per 52 atoms at the composition Cu6Zn8, and if the valence of copper is one greater than the valence of zinc, then it is possible to determine values of the metallic valences of these elements from the assumption that the Brillouin polyhedron is filled. These values are found to be 5.53 for copper and 4.53 for zinc. The accuracy of the determination of the metallic valences... 

Microstructures consist of three-dimensional networks of cells or grains that fill space. Each cell is a polyhedron with faces, edges, and corners. Their shapes are strongly influenced by surface tension. However, before examining the nature of three-dimensional microstructures, the characteristics of two-dimensional networks will be treated. 

The polyhedron corresponding to the Neovius surface has the same arrangement of points as that for the infinite semi-regular polyhedral surface 6.43 discussed above, but the spaces between the points are differently filled with polygons so that each of the 48 points per cubic cell has the configuration of 8.4.8.6 and this leads to a surface of genus of 9. This surface has two kinds of flat points and is thus not regular (Mackay Terrones 1991). 12 tubes in the [110] directions connect cavities. 

The most probable number of faces of polyhedra filling up the space was determined by Schwarz [71] who employed the stereometric formula of Euler and Plateau s laws. For a convex polyhedron the following ratio is valid... 

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