Polyhedral compressibility

Hazen and Finger (1979) extended equation 1.110 to mean polyhedral compressibility (mean compressibility of a given coordination polyhedron within a crystal structure), suggesting that it is related to the charge of ions in the polyhedron through an ionicity factor, analogous to what we have already seen for thermal expansion—i.e.. 

Table 1.16 Polyhedral compressibility moduli in selected natural and synthetic compounds (adapted from Hazen and Finger, 1982). Data are expressed in megabars (IMbar = 10 bar). Str = type of structure. K = 1/Pr-. ...

Experimental values of mean polyhedral compressibility modulus in various compounds are listed in table 1.16. 

Figure 1,14 Mean polyhedral compressibility for different coordination states. Interpo-lant is equation 1.111. From R. M. Hazen and L. W. Finger, Comparative Crystal Chemistry, copyright 1982 by John Wiley and Sons. Reprinted by permission of John Wiley Sons.

The polyhedral compressibility of the various olivine compounds conforms satisfactorily to the generalizations of Anderson (1972) and Hazen and Finger... 

The present case also involves changes is polyhedral compressibilities across the structural anomaly (for a detailed account, see Wentzcovitch et fl/[23]). Since the cation polyhedra link the tetrahedral chains, this behavior... 

There appear to be two stages in the collapse of emulsions flocculation, in which some clustering of emulsion droplets takes place, and coalescence, in which the number of distinct droplets decreases (see Refs. 31-33). Coalescence rates very likely depend primarily on the film-film surface chemical repulsion and on the degree of irreversibility of film desorption, as discussed. However, if emulsions are centrifuged, a compressed polyhedral structure similar to that of foams results [32-34]—see Section XIV-8—and coalescence may now take on mechanisms more related to those operative in the thinning of foams. 

Garekani, H. A., Ford, J. L., Rubinstein, M. H., and Rajabi-Sahboomi, A. R. (1999), Formation and compression characteristics of prismatic, polyhedral and thin plate-like crystals of paracetamol, Int. J. Pharm., 187,77-89. 

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.)... 

Also Tablet. A compressed agglomerate made of particulate solids, specifically, in pharmacy, a small compact of a medicated particulate formulation usually in the shape of a disc or a flat polyhedral body. (See also briquette.) The process of forming tablettes. 

In Table 3, the compressibility of the lattice parameters and of the polyhedral building units are reported, along with other structural parameters useful to understand the high-pressure behavior of Cs-annite, Rb-annite and phlogopite. The points outlined below show the effect of how different structural units of trioctahedral micas relate with one another, and then respond as a whole to pressure ... 

Table 3. Polyhedral bulk modulus for Tl, T2, Ml, M2 and interlayer sites, in kbar. Axial and volume compressibilities (fia,b,c,v) ir kbari, and first derivative versus P of p/Po, in kbar . Bulk modulus at P = 0 (ATo, in kbar) calculated by the Birch-Mumaghan EoS, constraining its first derivative versus P (K o) to be equal to 4. Tetrahedral rotation angle (a) and tetrahedral tilting (Az) first derivative versus P, in °kbar and Akbar % respectively.

The concept, emphasizing the dominating role of the cations for the formation of particular structures, can also be of use for the interpretation of structure transformations under high pressure. The compressibility of the cation-oxygen polyhedra (P) is inversely proportional to the polyhedral charge density z/cP, where d is the distance between the central cation and the O atoms (Hazen Prewitt, 1977). The behaviour of the Si,0 tetrahedra under pressure depends thus also on the compressibility of the M,0 polyhedra. 

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