Spheres, close-packing, differing size

The absorption spectrum of isolated indium spheres differs from that of closely packed oblate spheroids in that the peak shifts from 2230 A to 4100 A about half of this shift is attributable to particle shape and half to particle interaction. Indium particles on immunological slides are not identical, however, but are distributed in size and shape about some mean this tends to broaden the spectrum. 

As the atoms of an element are all equal sized, the structures of many elements correspond to the CCP or HCP array. By contrast, many ionic compounds can be described as a close-packed array of anions G rge spheres), with cations (smaller spheres) located in the hoUows between the anions. The hoUows, which are called interstitial sites, come in two different sizes as described above. Tetrahedral sites are coordinated by four anions, and octahedral sites are coordinated by six anions, as shown in Figure 3.2. For... 

For metallic elements, only one type of sphere is needed. However, in compound oxides two or more spheres of different sizes must be accommodated. To do this, the interstices of the hard sphere lattice are utihzed and since is normally the larger ion, it is generally considered to comprise the close-packed array. Within any such array, there are two types of interstices those with tetrahedral symmetry and those... 

Pilot plant smdied have also been performed by Larsen et al. [37], who obtained stable operation and more than 95% SO2 removal from flue gas streams with a gas-side pressure drop of less than 1000 Pa. The importance of the membrane structure on the SO2 removal has been studied by Iversen et al. [6], who calculated the influence of the membrane resistance on the estimated membrane area required for 95% SO2 removal from a coal-fired power plant. Authors performed experiments on different hydrophobic membranes with sodium sulfite as absorbent to measure the SO2 flux and the overall mass-transfer coefficient. The gas mixture contained 1000 ppm of SO2 in N2. For the same thickness, porosity, and pore size, membranes with a structure similar to random spheres (typical of stretched membranes) showed a better performance than those with a closely packed spheres stmcture. 

At concentrations at which the swollen dendrimer volume fraction was below 0.64 (the critical volume fraction for close packing of hard spheres), which corresponds to a dendrimer weight fraction of 0.25, the dendrimers in solution behaved as a dispersion of uniform soft spheres with no significant interpenetration between the segments of different dendrimers. At higher concentrations, however, the dendrimers collapsed, their size decreasing with increase of number density so that the volume fraction of the solution was maintained at approximately 0.64. 

The manufacture of molded articles is usually carried out with mixtures of aluminum oxides with different particle size distributions. This is particularly important when pore-free end-products are required, because this enables a higher volume concentration of aluminum oxide to be obtained than the 74% of ideally cubic close packed spheres by filling the gaps with smaller particles. The particle size distributions used in practice are usually determined using empirically determined approximate formulae (Andreasen or Fuller distribution curves) which take into account the morphology of the individual particles. 

Clusters are characterized by their size-dependent large surface/volume ratio which may be estimated using different models. If we assume that a cluster containing N atoms is built up of spheres each with a radius r = lA and closely packed together to a bigger sphere with a radius R, one obtains for the volume of the cluster V = and for the volume of each atom v = I 7rr the relation, V = =iVu = iVj x(r) ... 

A factor of great importance in ionic crystals is the difference in size of the positive and negative ions. Pauling has shown how the geometric requirements for close packing of spheres of different sizes can be simply expressed in terms of the radius ratio p = rjvi, defined as the ratio of the radius of the smaller ion, r, to that of the larger ion,... 

When spheres of a given size are close-packed, the spaces between the layers of spheres (the voids or interstices) can be filled with smaller spheres. If the spheres represent cations and anions, the structures of ionic solids can be visualized. There are two types of interstices between layers of close-packed atoms - tetrahedral holes or interstices and octahedral holes or interstices. Tetrahedral holes are formed when one sphere in a layer fits over or under three spheres in a second layer. Octahedral holes are formed when three spheres in one layer fit over or under three spheres in a second layer. The two types of holes have different numbers per close-packed sphere, different sizes, and different coordination numbers and coordination geometries. The coordination number of the anion would be the number of cations in contact with the anion. The coordination geometry of the anion would be the geometrical arrangement of the cations which surround the anion. Related statements can be made regarding the coordination number and coordination geometry of the cation. 

Like metallic solids, ionic solids tend to adopt structures with symmetric, close-packed arrangements of atoms. However, important differences arise because we now have to pack together spheres that have different radii and opposite charges. Because cations are often considerably smaller than anions —= (Section 7.3) the coordination numbers in ionic compounds are smaller than those in close-packed metals. Even if the anions and cations were the same size, the close-packed arrangements seen in metals cannot be replicated without letting ions of like charge come in contact with each other. The repulsions between ions of the same type make such arrangements unfavorable. The most favorable structures are those where the cation—anion distances are as close as permitted by ionic radii but the anion—anion and cation—cation distances are maximized. 

The atomic packing in disordered solids was investigated first by Bernal (1964), who considered the problem in the context of a model of a simple liquid that consisted of randomly close-packed hard spheres of uniform size and described the structure as a distribution of five different canonical polyhedra with well-defined volume fractions. 

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