Packings contacts between spheres

Referring now to the shape of the void as shown in Figure 28, the rhombohedral andletrahedral cells are denoted by the letters R and T. The T-spaces correspond to the voids between four spheres in closest packing with one sphere nested in the hollow formed by the other three. In this space there are six points of contact between spheres and four lines or branches. These branches connect one cell with another. 

Packing of uniform spheres with at least four contacts between spheres and not all the contacts on the same hemisphere. 

Column Operation To assure intimate contact between the counterflowing interstitial streams, the volume fraction of liquid in the foam should be kept below about 10 percent—and the lower the better. Also, rather uniform bubble sizes are desirable. The foam bubbles will thus pack together as blunted polyhedra rather than as spheres, and the suction in the capillaries (Plateau borders) so formed vidll promote good liqiiid distribution and contact. To allow for this desirable deviation from sphericity, S = 6.3/d in the equations for enriching, stripping, and combined column operation [Lemhch, Chem. E/ig., 75(27), 95 (1968) 76(6), 5 (1969)]. Diameter d still refers to the sphere. 

Attention will be restricted here to fluorine compounds of the three d-transition series elements, in which the metal ion is octahedrally coordinated. Octahedral coordination, however, is found in nearly all these cases, which is quite reasonable considering the sizes of the ions in question. A close-packed octahedron of fluoride ions of radius 1.33 A adapts a size of its octahedral interstice appropriate to a sphere of radius 0.55 A. Cations having this size and larger ones meet the conditions of a contact between cations and anions. Thus stability is predicted for octahedral coordination until such contacts of ions become possible for coordination numbers higher than 6. For a coordination of 8 fluoride ions this is only the case if the radii of the cations are as large as 0.86 A (square antiprism) or 0.97 A (cube) resp. 

Among the structural models of cellular or porous materials those characterized by differently ordered packing of balls or spheres of the same diameters have most widely been used. In this approach either the spheres have been considered as real cells or the cell (pore) models have been derived from an analysis of assumed spacings between the contacting solid spheres. However, no system of packed spheres would adequately describe the properties of any real cellular system which never exhibit a regular packing. It is also impossible to describe the structure of most cellular systems via models assuming spheres of equal size. 

From the fact that the smallest kind of interstice between spheres in contact is a tetrahedral hole it follows that we should expect to find coordination polyhedra with only triangular faces, in contrast to those in, for example, cubic closest packing which have square in addition to triangular faces. Moreover, it seems likely that the preferred coordination polyhedra will be those in which five or six triangular faces (and hence five or six edges) meet at each vertex, since the faces are then most nearly equilateral. It follows from Euler s relation (p. 61) that for such a polyhedron, 1)5 -h Oug = 12, where 1)5 and are the numbers of vertices at which five or six edges meet, so that starting from the icosahedron (vg = 12) we may add 6-fold vertices to form polyhedra with more than twelve vertices. 

Figure 11.20 (a) In a close-packed layer, each sphere is in contact with six others, p) Spheres In the second layer fit into the depressions between the first-layer spheres, p) In the hexagonal close-packed structure, each third-layer sphere is directly over a first-layer sphere, (d) In the cubic close-packed structure, each third-layer sphere fits into a depression that is directly over a depression in the first layer. 

---