Close packing hard sphere model

The threshold volume fraction of percolation (( >,) is guided by the amphiphile shell length and the overall volume fraction of the dispersed phase. For zero shell length and no interparticle attractive interaction, according to the randomly close-packed hard sphere model, 4, = 0.65 systems with strong attractive interactions end up with (f), being lowered from 0.65 to 0.10. 

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. 

The electrical properties of a composite material composed of insulating media and conductive particles can always be modeled as a function of concentration or fraction of the conducting particles 4>. It was demonstrated in a modeling study of composite polycrystalline media composed of two typ>es of closed packed hard spheres [2], where the impedance structure of the system... 

An even more general and correspondingly less detailed atomic model of amorphous oxide surfaces has been called the Bernal surface (BS)[3, 21]. It is based upon the fact that many oxides and halides can be regarded as close-packed arrays of large anions with much smaller cations occupying interstitial (usually tetrahedral or octahedral) positions (see., e.g. Ref. [4]). In line with this point of view, the BS is a surface of a collection of dense randomly packed hard spheres, a sphere representing an oxide anion. The cations in interstitial positions between hard spheres are excluded from the simulation since they do not attract adsorbed molecules due to their small polarizability. Thus only the atomic structure of the oxide ions is considered. This is called the Bernal structure and has been used for modelling simple liquids and amorphous metals [15]. 

Repulsive Forces. As atoms approach one another or surfaces very closely, the electron clouds of the interacting atoms begin to overlap, with the result that repulsion (known as Bom repulsion) becomes the dominant force and closer approach becomes impossible. For this reason, the hard sphere model can be used to describe ionic solids. The individual atoms have well-defined radii that determine the distances of closest approach in close-packed arrangements. In equation 1.3, this distance is found to be 3.14 A for KCl. 

Mg " ). The metal-oxygen bonds are largely ionic in nature, so that the hard sphere model of atom packing can be used to predict likely atomic arrangements. In fact, for structures with pure ionic bonding, the oxyanions pack together in the most space-efficient manner possible (termed the close-packed arrangement), and the metal cations fit into the interstices of these anions. 

Its exponential behavior makes this term the dominant one when short atom-atom distances between the interacting molecules are produced. Consequently, this term prevents molecules from getting closer than some limiting distance this is the physical principle behind the hard-sphere model and Kitaigorodsky s close-packing principles [2]. Moreover, the shortest atom-atom distances that one can find in different intermolecular interactions for the same pair of atoms always fall in a restricted range, a fact that allows one to define atomic radii. They differ in ionic and neutral crystals due to the different electronic structure of ionic and neutral species, as easily shown when comparing the contours at 90% probability in electron density maps for isolated atoms and their ions. 

On the basis of the hard sphere model, close-packing represents the most efficient use of space with a common packing efficiency of 74%. The bcc structure is not much less efficient in packing terms, for although there are only eight nearest neighbours, each at a distance x (compared... 

A close-packed lattice contains tetrahedral and octahedral interstitial holes (see Figure 5.5). Assuming a hard-sphere model for the atomic lattice, one can calculate that an atom of radius 0.41 times that of the atoms in the close-packed array can occupy an octahedral hole, while significantly smaller atoms may be accommodated in tetrahedral sites. 

Throughout our discussion, we refer to ionic lattices, suggesting the presence of discrete ions. Although a spherical ion model is used to describe the structures, we shall see in Section 6.13 that this picture is unsatisfactory for some compounds in which covalent contributions to the bonding are significant. Useful as the hard sphere model is in describing common crystal structure types, it must be understood that it is at odds with modem quantum theory. As we saw in Chapter 1, the wavefunction of an electron does not suddenly drop to zero with increasing distance from the nucleus, and in a close-packed or any other crystal, there is a finite electron density everywhere. Thus all treatments of the solid state based upon the hard sphere model are approximations. 

The atomic bonding in this group of materials is metallic and thns nondirectional in natnre. Conseqnently, there are minimal restrictions as to the nnmber and position of nearest-neighbor atoms this leads to relatively large nnmbers of nearest neighbors and dense atomic packings for most metallic crystal structnres. Also, for metals, when we nse the hard-sphere model for the crystal structure, each sphere represents an ion core. Table 3.1 presents the atomic radii for a number of metals. Three relatively simple crystal structures are found for most of the common metals face-centered cubic, body-centered cnbic, and hexagonal close-packed. 

Ostwald first modelled catastrophic inversions as being caused by the complete coalescence of the dispersed phase at the close packed condition (corresponding to a dispersed phase fraction of 0.74 in Ostwald s uniform hard sphere model). Other studies, e.g. Marzall, have shown that catastrophic inversions (though these inversions were not called catastrophic inversions by that author) can occur over a wide range of WOR. It has been suggested that this may be due to the formation of double emulsion drops (0/W /0), boosting the actual volume of the dispersed phase. 

Figure Bl.21.1. Atomic hard-ball models of low-Miller-index bulk-temiinated surfaces of simple metals with face-centred close-packed (fee), hexagonal close-packed (licp) and body-centred cubic (bcc) lattices (a) fee (lll)-(l X 1) (b)fcc(lO -(l X l) (c)fcc(110)-(l X 1) (d)hcp(0001)-(l x 1) (e) hcp(l0-10)-(l X 1), usually written as hcp(l010)-(l x 1) (f) bcc(l 10)-(1 x ]) (g) bcc(100)-(l x 1) and (li) bcc(l 11)-(1 x 1). The atomic spheres are drawn with radii that are smaller than touching-sphere radii, in order to give better depth views. The arrows are unit cell vectors. These figures were produced by the software program BALSAC [35]-...

The results of Equation (3.56) are plotted in Figure 3.14. It can be seen that shear thinning will become apparent experimentally at (p > 0.3 and that at values of q> > 0.5 no zero shear viscosity will be accessible. This means that solid-like behaviour should be observed with shear melting of the structure once the yield stress has been exceeded with a stress controlled instrument, or a critical strain if the instrumentation is a controlled strain rheometer. The most recent data24,25 on model systems of nearly hard spheres gives values of maximum packing close to those used in Equation (3.56). 

Widom [9] realized the importance of this problem for statistical mechanics and showed that the centers of the particles of a hard disk gas, in an equilibrium position, are not uniformly random distributed. The available area for a nevt particle power series in particle density 6 = Nnr2/A, where N is the number of adsorbed panicles, r their radius and A the total area of the surface. The coefficients of the series terms are identical up to the second power of 9 for the equilibrium and the RSA models. The differences in the higher powers coefficients lead for RSA to jamming for Op = 0.76, 0.547 and 0.38 for the ID (segments on a line), 2D (disks on a surface) and 3D (spheres on a volume), respectively, while for the equilibrium configurations the close-packing occurs at 9 = 1, 0.91 and 0.74, respectively. 

In protein crystals, due to the large size of the molecule, the empty space can have cross sections of 10-15 A or greater. The empty space between the protein molecules is occupied by mother liquor. This property of protein crystals, shared by nucleic acids and viruses, is otherwise unique among the crystal structures. In fact, the values of the packing coefficient of protein crystals range from 0.7 to 0.2, but the solvent molecules occupy the empty space so that the total packing coefficient is close to 1 [37]. Nevertheless, a detailed theoretical study has been carried out to examine the models of DNA-DNA molecular interactions on the basis of hard-sphere contact criteria. The hard-sphere computations are insufficient for qualitative interpretation of the packing of DNA helices in the solid state, but... 

---