Sphere pack models

C. G. Wilson and F. J. Spooner, A Sphere-Packing Model for the Prediction of Lattice Parameters and Order in o Phases, Acta Cryst., 39A, 342 (1973). 

Since the dominant feature of packings of spheroidal particles is the constrictions between the tetrahedral cavities formed by the Alumina microspheres, a more realistic model is required, based on the random sphere packing models. Such models are obviously more complex. Conversely, they permit a more realistic representation of the pore space among the spheroidal particles. A preliminary model has been reported for sorption [20] and relative permeability Pr [21]. 

These facts have already been discussed in the examples of diamond/graphite and red/white phosphorus (see Sect. 4.3). The varying properties can be shown by differences in chemical structures. However, these structures are not easy to understand. Because it is possible to correctly demonstrate the arrangement of metal atoms using closest-sphere packing models, it is useful to look at metal structures with regard to the property structure relationships and try to address the above-mentioned misconceptions. 

In order to arrive at the important sodium chloride structure, one could look at and discuss the interstices in the cubic closest sphere packing model [2] interstices that are octahedral and tetrahedral can be found (see E5.8). If one fills the octahedral interstices with smaller spheres, one ends up with the sodium chloride structure (see Fig. 5.13). The larger spheres symbolize the chloride ions, the smaller spheres the sodium ions. The structure can be described as the cubic closest packing of chloride ions, whose octahedral interstices are completely filled by sodium ions. For other salt structures, only part of the octahedral sites are filled, as in aluminum oxide where the ratio of ions 2 3 applies [2],... 

Problem The structure of many metal crystals can best be described by using closest sphere packing models. Because there are two possibilities of systematically sphere packings, these should be introduced first (see also Figs. 5.3 and 5.4). Both packings can be differentiated by the layering sequence ABAB and ABCA - with respect to the densely packed triangular layers. 

Fig. 13. Experimental dependence of maximum mobilization pressure drops on contact angle from the sphere pack model.

Experimentally measured mobilization behavior in the sphere-pack model shows an increase in the necessary force levels as the receding surface contact angle is increased. This is in qualitative agreement with porous media displacement studies. 

A combination of characterization techniques for the pore structure of meso-and microporous membranes is presented. Equilibrium (sorption and Small Angle Neutron Scattering) and d)mamic (gas relative permeability through membranes partially blocked by a sorbed vapor) methods have been employed. Capillary network and EMA models combined with aspects from percolation theory can be employed to obtain structural information on the porous network topology as well as on the pore shape. Model membranes with well defined structure formed by compaction of non-porous spherical particles, have been employed for testing the different characterization techniques. Attention is drawn to the need for further development of more advanced sphere-pack models for the elucidation of dynamic relative permeability data and of Monte-Carlo Simulation for the analysis of equilibrium sorption data from microporous membranes. 

The geometiy of the sphere pack model. After the early paper of Hara (1935), Gassmann (1951a) published his classic paper about the elasticity of a hexagonal packing of spheres. 

Implementing Eqs. (6.93) and (6.94) into Eqs. (6.5) and (6.6) results in compressional and shear wave velocities. For sphere pack models, both wave velocities show identical dependence on porosity and presstue. Thus, the ratio Fp/Fs is independent of porosity and pressure and only controlled by Poisson s ratio of the solid material v. ... 

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