Hard sphere packing model

Mass Diffusivity in Liquid Metais and Ailoys. The hard-sphere model of gases works relatively well for self-diffusion in monatomic liquid metals. Several models based on hard-sphere theory exist for predicting the self-diffusivity in liquid metals. One such model utilizes the hard-sphere packing fraction, PF, to determine D (in cm /s) ... 

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... 

There are two classes of solids that are not crystalline, that is, p(r) is not periodic. The more familiar one is a glass, for which there are again two models, which may be called the random network and tlie random packing of hard spheres. An example of the first is silica glass or fiised quartz. It consists of tetrahedral SiO groups that are linked at their vertices by Si-O-Si bonds, but, unlike the various crystalline phases of Si02, there is no systematic relation between... 

Hard-sphere models lack a characteristic energy scale and, hence, only entropic packing effects can be investigated. A more realistic modelling has to take hard-core-like repulsion at small distances and an attractive interaction at intennediate distances into account. In non-polar liquids the attraction is of the van der Waals type and decays with the sixth power of the interparticle distance r. It can be modelled in the fonn of a Leimard-Jones potential Fj j(r) between segments... 

The C, values for Sb faces are noticeably lower than those for Bi. Just as for Bi, the closest-packed faces show the lowest values of C, [except Bi(lll) and Sb(lll)].28,152,153 This result is in good agreement with the theory428,429 based on the jellium model for the metal and the simple hard sphere model for the electrolyte solution. The adsorption of organic compounds at Sb and Bi single-crystal face electrodes28,152,726 shows that the surface activity of Bi(lll) and Sb(lll) is lower than for the other planes. Thus the anomalous position of Sb(lll) as well as Bi(lll) is probably caused by a more pronounced influence of the capacitance of the metal phase compared with other Sb and Bi faces28... 

The accuracy of the theory diminishes somewhat for models where the packing is dissimilar from a hard-sphere fluid. Yethiraj and co-workers have... 

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). 

Amongst the assumptions we have made in developing the model are the following that Pick s law is applicable to the diffusion processes, the gel particles are isotropic and behave as hard spheres, the flow rate is uniform throughout the bed, the dispersion in the column Ccui be approximated by the use of an axial dispersion coefficient cuid that polymer molecules have an independent existence (i.e. very dilute solution conditions exist within the column). Our approach borrows extensively many of the concepts which have been developed to interpret the behaviour of packed bed tubular reactors (5). 

The first term in this expression, Av0, represents the frequency shift resulting from short range repulsive packing forces. These many-body repulsive forces are in general expected to lead to a non-linear dependence of frequency on density. However, this complex non-linear behavior can be accurately modeled using a hard-sphere reference fluid, with appropriately chosen density, temperature and molecular diameters (see below). 

The degree of short-range order in an amorphous material can be characterized by a hard sphere model if the basic structure of an amorphous material is approximated by spheres. The density of packing of atoms around a reference atom is described by the number of atom centers per volume that lie in a spherical shell of thickness, dr, and radius about the reference atom. In a hard sphere model, the number, n, of neighboring spheres with centers between r and dr is measured as a function of r. 

In the hard sphere model, atoms may not overlap so g(r) = 0 for r < D, where D is the atom diameter. For completely random packing, g(r) is constant for r > D. In a crystalline material, g(r) = 0 except at discrete interatomic distances. Figure 15.4 shows a schematic plot of g(r) versus r for random packing, a crystal,... 

If elements A and B are completely miscible in all compositions in their liquid state, it is reasonable to assume nearly equal interactions among the three possible cases A-A, B-B and A-B in their liquid state. Let us assume the atomic radii of A and B are about equal to one another (i.e., RA Rb)- With this assumption, the system is essentially a hard sphere or a Bernal model [9]. In the absence of gravitational or other external effects, the liquid solution will reach a dynamic equilibrium where the system is random-dense-packed and homogeneous throughout. Under these conditions we ask the following question pertaining to the atomic configuration. 

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