Assembly of polyhedra

Several intermetallic phase structures can be described in terms of an assembly of building blocks consisting of a collection of coordination polyhedra. A classification of types of intermetallic structures based on the coordination number, configurations of coordination polyhedra and their method of combination has been given by Kripyakevich (1963). 

According to Kripyakevich, a coordination polyhedron of an atom is the polyhedron, the vertices of which are defined by the atoms surrounding this atom a coordination polyhedron should have a form as close as possible to a sphere, that is, it should be convex everywhere and have the maximum number of triangular faces. 

At the vertices of a coordination polyhedron of a given atom (in addition to atoms of different elements) there can also be atoms of the same kind. A considerable variety of coordination polyhedra exists. In some cases, plane coordination polygons have to be considered. The number of vertices may vary from 3 to 24. Generally, the structure consists of atoms with different coordination numbers according to Kripyakevich, structures are most conveniently classified according to the type of coordination polyhedron of the atoms with the lowest coordination number. 

An important contribution to the structure analysis of intermetallic phases in terms of the coordination polyhedra has been carried out by Frank and Kasper (1958). They described several structure types (Frank-Kasper structures) as the result of the interpenetration of a group of polyhedra, which give rise to a distorted tetrahedral close-packing of the atoms. Samson (1967, 1969) developed the analysis of the structural principles of intermetallic phases having giant unit cells (Samson phases). These structures have been described as arrangements of fused polyhedra rather than the full interpenetrating polyhedra. 

Examples of structures described by packing of one polyhedron type are  

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In certain 3D nets there are well-defined polyhedral cavities, and the links of the net may alternatively be described as the edges of a space-filling assembly of polyhedra. At least four links must meet at every point of such a net, and the most important nets of this kind are, in fact, 4-connected nets. Space-filling arrangements of polyhedra leading to such nets are therefore described after we have dealt with the simpler 4-connected nets. 

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.)... 

Damascene, PF, Engel, M., Glotzer, S.C. (2012). Predictive self-assembly of polyhedra into complex structures. Science, 337, 453-457. 

Another way of description of the structures [55] uses, instead of single polyhedra, assemblies of polyhedra (Fig. 2c) that, by translation/rotation in the three directions of space, regenerate the whole framework. These are called secondary building units (SBU) and the concept has been very useful for defining the large pore strategy. 

The way of the best choice to model PS s structure on both molecular and supramolecular levels begins with allocation of primary building units (PBUs), which without gaps and overlaps would fill a 3D space occupied by a PS. An universal method for allocation of such PBUs in both ordered and randomly arranged PSs, formed of packings of convex particles (or pores), is based on the construction of the assembles of Voronoi polyhedra (V-polyhedra) and Delaunay simplexes (or D-poly-hedra), which form Voronoi-Delaunay tessellation [100],... 

If the emulsion consisted of an assembly of closely packed uniform spherical droplets, the dispersed phase would occupy 0.74 of the total volume. Stable emulsions can, however, be prepared in which the volume fraction of the dispersed phase exceeds 0.74, because (a) the droplets are not of uniform size and can, therefore, be packed more densely, and (b) the droplets may be deformed into polyhedra, the interfacia] film preventing coalescence. 

Figure 10.23 Examples of self-assembled coordination polyhedra and their descriptors (Copyright Wiley-VCH Verlag GmbH Co. KGaA. Reproduced by permission).

Supramolecular assembly of silver(l) polyhedra with embedded... 

He Y et al (2008) Hierarchical self-assembly of DNA into symmetric supramolecular polyhedra. Nature 452 198-201... 

The third question in 3D MESA - the dimension of the smallest pieces that can self-assemble - motivated experiments with metallic, polyhedral, 10- xm-sized plates [ref. 63]. The fabrication of 10- xm-sized polyhedra is difficult, so these experiments examined the self-assembly of flat plates, which were easily obtained at this scale. Photolithography, electrodeposition, and electron beam evaporation provided a convenient way to form plates having surfaces whose properties could be modified selectively. The surfaces of these plates consisted of either gold or chromium. Treatment with a long-chained alkanethiol formed a hydrophobic SAM on the gold with a phosphonic acid-terminated thiol, a hydrophilic one. Similarly, a phosphonic acid-terminated thiol formed a hydro-phobic SAM on chromium (through coordination of the acid with surface Cr... 

Planar exo-multidentate ligands (molecular panels, e.g. 34) coordinate to palladium to give a variety of polyhedra. Molecular cages have also been assembled using ds - PdL2 comers see Self-assembled Inorganic Architectures). 

Application of the lessons learnt in the self-assembly of molecular squares has led to successful strategies for preparing further molecular polygons (and, in particular, new triangles, rectangles and hexagons) as well as examples of particular other polyhedra. ... 

Figure 7.35 Generation of the chiral channel in the space group P6 j22. The polyhedra represent T04 tetrahedra, the cylinder represents the predefined forbidden zone. Assembly of T04 tetrahedra outside the forbidden zone by the 6, operation will generate a chiral channel. Reproduced with permission from [38]. Copyright (2005) American Chemical Society...

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