Geometric packing models

The distance of each reflection from the center of the pattern is a function of the fiber-to-film distance, as well as the unit-cell dimensions. Therefore, by measuring the positions of the reflections, it is possible to determine the unit-cell dimensions and, subsequently, index (or assign Miller indices to) all the reflections. Their intensities are measured with a microdensitometer or digitized with a scanner and then processed.8-10 After applying appropriate geometrical corrections for Lorentz and polarization effects, the observed structure amplitudes are computed. This experimental X-ray data set is crucial for the determination and refinement of molecular and packing models, and also for the adjudication of alternatives. 

In the packing model [50,62,68] the entanglement distance is interpreted by the gradual build-up of geometrical restrictions due to the existence of other chains in the environment or, more precisely, the entanglement distance is determined by a volume which must contain a defined number of different chains. This approach is based on the observation that, for many polymer chains, the product of the density of the chain sections between entanglements is... 

Particle size distributions often produce denser packing structures because smaller particles may fill void spaces between larger ones. Based on this assumption several researchers have described densest packings. Table 6 shows porosities that can be obtained with binary or ternary particle mixtures (i.e. consisting of two or three particle sizes). Recently, a geometrical mathematical model for calculating the porosity of randomly packed binary mixtures of spherical particles has been developed. It could be shown that the absolute deviation between theoretical and experimentally obtained data... 

Some geometric transport models are based on solid characteristics rather than on properties of the pore space itself. By assuming a particular packing arrangement it is possible to infer the pore space geometry from information on the size and shape of the solid particles (Coelho et al., 1997). While this approach may be applicable to sieved and repacked soil columns, it is often inappropriate for undisturbed samples, with pore characteristics that depend more on soil structure than on soil texture. Thus, models to predict solute dispersion from the properties of particles in packed beds (e.g., Aris Amundson, 1957 Koch Brady, 1985 Ras-muson, 1985) are not the main focus of this review. 

Figure 18. Oxygen packing model of unit cell of faujasite-type framework of zeolites X ana Y, showing packing of T.O. and D-6 ting groups (left) and solid geometric representation of structure (right). Dark balls represent sodium cations in structure.

The situation is different for the indirect packing effects. They can be calculated by the ab initio method if a structural model predicts the packing distortions with sufficient precision. Naturally, it will be hard to develop such a geometric-statistical model, but once the geometries are known they may be used as an input for the IGLO calculations and thus for the simulation of the -NMR spectrum. 

Although the classical picture of a micelle is that of a sphere, most evidence indicates that spherical micelles are not the rule and may in fact be the exception. As a result of geometric packing requirements and analyses (to be discussed below), ellipsoidal, disk-shaped, and rodUke structures may be the more commonly encountered shapes. However, from the standpoint of providing a concept of micelles and micelle formation for the nonspeciaUst, the Hartley spherical model remains a useful and meaningfiil tool. 

The close-packed-spheron theory of nuclear structure may be described as a refinement of the shell model and the liquid-drop model in which the geometric consequences of the effectively constant volumes of nucleons (aggregated into spherons) are taken into consideration. The spherons are assigned to concentric layers (mantle, outer core, inner core, innermost core) with use of a packing equation (Eq. I), and the assignment is related to the principal quantum number of the shell model. The theory has been applied in the discussion of the sequence of subsubshells, magic numbers, the proton-neutron ratio, prolate deformation of nuclei, and symmetric and asymmetric fission. 

---