Packing problem

The structures of alloys are more complicated than those of pure metals because they are built from atoms of two or more elements with different atomic radii. The packing problem is like that of a storekeeper trying to stack oranges and grapefruit in the same pile. 

Mingos, D.M.P. and Rohl, A.L. (1991) Size and shape characteristics of inorganic molecules and ions and their relevance to molecular packing problems. Dalton Transactions, (12), 3419-3425. 

The simplest formulation of the packing problem is to give some collection of distance constraints and to calculate these coordinates in ordinary three-dimensional Euclidean space for the atoms of a molecule. This embedding problem - the Fundamental Problem of Distance Geometry - has been proven to be NP-hard [116]. However, this does not mean that practical algorithms for its solution do not exist [117-119]. 

In some cases, planar chain structures become impossible when replacing a hydrogen atom on the backbone of a macromolecule by a bulky group. Packing problems arise due to steric hindrance if the trans orientation of each bond is... 

The first point from this development and example is that, although the quasichemical approach is directed towards treating strong attractive - chemical - interactions at short range, it can describe traditional packing problems accurately. The second point is that this molecular-field idea permits us to go beyond the primitive quality noted above of the primitive quasichemical approximation, and specifically to account approximately for the influence of the outer-shell material on the equilibrium ratios Km required by the general theory. This might help with cases of delicate structures noted above with anion hydrates. 

Because of steric factors there is not a simple relationship between AG or AH or TAS and the ionic radius . There is then no simple explanation of such observations as tetrad effects, see below, since some such breaks could come about in an apparent series of ion size changes of equal increment. Tetrad breaks may occur at the expected place following ionic size differentials but may also occur elsewhere through the sensitivity of packing problems. 

Theoretically, no equation can completely describe the physical properties of a packing without taking into account particle-diameter, size-distribution, and other particulate parameters already mentioned. The reason for their omission in some equations is threefold (a) The equation may contain arbitrary constants related to particle-characteristics constituting the packing (b) the derivation of the equation may be in terms of an ideal or isotropic medium, and (c) the equations may be empirical. So far, statistical analysis seems to have played only a small role in studies of packing problems, and probably will continue to do so until the particulate properties of various materials are better understood and made subject to mathematical treatment. 

In order to clarify terms which will occur frequently in later portions of this text, their, definitions are given at this point. These definitions are taken from a study of systematic packings made by Graton and Fraser (1935), to which the reader is referred for an excellent analysis of packing problems. 

Figure 5. Encapsulation of hydride to prevent expansion and packing problems 1 = Al capsule 2 = flute 3 = porous filter 4 = hydride.

In order to characterize the shape, it is necessary to find some dominant relation among angles, i.e., the internal symmetry of the figure. Shapes can be compared with several geometrical known standards. In ref. [80], shape descriptors were defined and their applications to the packing problems were found to be interesting. 

The quantities K R) describe occupancy transformations fully involving the solution neighborhood of the observation volume. These coefficients are known only approximately. Building on the preceding discussion, however, we can go further to develop a self-consistent molecular field theory for packing problems in classical liquids. We discuss here specifically the one component hard-sphere fluid this discussion follows Pratt and Ashbaugh (2003). 

This relation says that the probability of n solvent centers occupying the excluded volume is the product of two probabilities the probability that the larger ( = 1) region has solvent centers multiplied by the conditional probability that the observation volume has n centers when the larger region has centers. For the packing problems the approximation obtained above amounts to p(n = 01 )... 

To address the hmitations of ancestral polymer solution theories, recent work has studied specific molecular models - the tangent hard-sphere chain model of a polymer molecule - in high detail, and has developed a generalized Rory theory (Dickman and Hall (1986) Yethiraj and Hall, 1991). The justification for this simplification is the van der Waals model of solution thermodynamics, see Section 4.1, p. 61 attractive interactions that stabilize the liquid at low pressure are considered to have weak structural effects, and are included finally at the level of first-order perturbation theory. The packing problems remaining are attacked on the basis of a hard-core model reference system. 

Thus, we first consider Eq. (8.10) for hard-core chain models, specifically tangent hard-sphere chain models (Dickman and Hall (1986) Yethiraj and Hall, 1991). Models and theories of the packing problems associated with hard-core molecules have been treated in Sections 4.3, 6.1, 7.5, and 7.6. We recall... 

For packing problems, the excess chemical potential as a function of available volume x, e.g. m x) = — ln(l —x) is the primordial available volume theory. 

In these cases the packing problem must be solved subject to the constraint of local electroneutrality, under which conditions the change of a single ion size parameter by a few percent can completely alter the preferred coordination scheme. 

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