Three-sphere model

For the N-layer model in particulates, which is an improvement of the three-sphere model, it has been shown5), by taking into consideration the boundary conditions between phases, that ... 

Figures 1 a and 1 b represent the two-phase and the three-phase models respectively in the representative volume element of the composite. In the modified model three concentric spheres were considered with each phase maintaining a constant volume 4). The novel element in this model is the introduction of the third intermediate hybrid phase, lying between the two principal phases.

Thus, in the three-layer model, with the intermediate layer having variable physical properties (and perhaps also chemical), subscripts f, i, m and c denote quantities corresponding to the filler, mesophase, matrix and composite respectively. It is easy to establish for the representative volume element (RVE) of a particulate composite, consisting of a cluster of three concentric spheres, that the following relations hold ... 

Figure 4.4 The points-on-a-sphere model. The most probable arrangements of two, three, four, five, and six points on the surface of a sphere that maximizes their distance apart.

A section in the full-bed models was isolated that was comparable to the WS model. The layout of these different sections was identical, except that the WS model had a two-layer periodicity and the full-bed models had a six-layer periodicity. To be able to make direct comparisons of velocity profiles, several sample-points needed to be defined. In the three different models seven tangential planes were defined and on each plane three axial positions were defined. This reduced the data to single radial velocity profiles at corresponding positions in all three models, as shown in Fig. 10, for the WS model. Identical planes were defined in the full-bed models. Some spheres and sample planes 4 and 5 are not displayed to improve the visibility of the sample planes and lines. In the right-hand part of the figure, plane 4 is shown with the axial positions at which data were taken and compared. 

The data from the WS model in some cases deviated slightly from the full-bed models. This could be explained by the slightly different layout of the WS model. Some spheres had to be relocated in the WS model to create a two-layer periodicity from the six-layer periodicity in the full-bed models. The differences in velocity magnitudes were mainly found in the transition area between the wall layers and the center layers. The effect of slightly larger gaps between spheres from the nine-sphere wall layers and the three-sphere central layers, due to the sphere relocations, had a noticeable effect on the velocity profile. Differences were also found in the central layer area where the sphere positions were not identical. 

These three structures are the predominant structures of metals, the exceptions being found mainly in such heavy metals as plutonium. Table 6.1 shows the structure in a sequence of the Periodic Groups, and gives a value of the distance of closest approach of two atoms in the metal. This latter may be viewed as representing the atomic size if the atoms are treated as hard spheres. Alternatively it may be treated as an inter-nuclear distance which is determined by the electronic structure of the metal atoms. In the free-electron model of metals, the structure is described as an ordered array of metallic ions immersed in a continuum of free or unbound electrons. A comparison of the ionic radius with the inter-nuclear distance shows that some metals, such as the alkali metals are empty i.e. the ions are small compared with the hard sphere model, while some such as copper are full with the ionic radius being close to the inter-nuclear distance in the metal. A consideration of ionic radii will be made later in the ionic structures of oxides. 

Figure 3.8. Crystal structure of CsCl. The positions of the centres of the atoms in the unit cell are shown in (a). In (b) the same cell is described by means of its characteristic sections taken at the height 0, A, and 1 of the third axis. In (c) a projection of the cell on its square basis is presented the values of the third (fractional) coordinate are indicated. In (d) the shortest interatomic distances are shown dCs-ci = a)3/2 = 411.3 X 0.866025. = 356.2. In (e) the subsequent group of interatomic distances (d = a = 411.3) involving six atoms in the adjacent cells is presented. A group of eight cells is represented in (f) to suggest that the actual structure of CsCl corresponds to a three-dimensional infinite repetition of unit cells and to show that the coordination around the white atoms is similar to that around the black ones shown in (d). The unit cell of the CsCl structure is shown as a packed spheres model in (g).

Finally, we consider Model V by describing two examples of outer-sphere electron-transfer in solution. Figures 7 and 8 display results for the diabatic electronic population for Models Va and Vb, respectively. Similar to the mean-field trajectory calculations, for Model Va the SH results are in excellent agreement with the quantum calculations, while for Model Vb the SH method only is able to describe the short-time dynamics. As for the three-mode Model IVb discussed above, the SH calculations in particular predict an incorrect long-time limit for the diabatic population. The origin of this problem will be discussed in more detail in Section VI in the context of the mapping formulation. 

A triatomic molecule A—B—C is modeled as three spheres connected by two springs and in addition, the bond angle 9 between A—B and B—C has an equilibrium value of 00- For instance, water has an equilibrium bond angle of 104.5°, so that if it is bent to another value the molecular energy is increased according to the quadratic equation... 

The coordination number of a cation depends on the number of anions or ligand atoms that can be fit around it in three dimensions (Fig. 2). In the hard-sphere model the coordination number is determined by the ratio of the radius of the cation to that of the anion... 

Equation (8.3.14) is not an asymptotically exact result for the black sphere model due to the superposition approximation used. When deriving (8.3.14), we neglected in (8.3.11) small terms containing functionals I[Z], i.e., those terms which came due to Kirkwood s approximation. However, the study of the immobile particle accumulation under permanent source (Chapter 7) has demonstrated that direct use of the superposition approximation does not reproduce the exact expression for the volume fraction covered by the reaction spheres around B s. The error arises due to the incorrect estimate of the order of three-point density p2,i for a large parameter op at some relative distances ( f — f[ < tq, [r 2 - r[ > ro) the superposition approximation is correct, p2,i oc ct 1, however, it gives a wrong order of magnitude fn, oc Oq2 instead of the exact p2,i oc <7q 1 (if n — r[ < ro, fi — f[ < ro). It was... 

Certainly, the approach to the hard sphere density will be closer for the regularly branched three-functional model. A glance at Fig. 19 reveals a behavior of P (q2)-1 which is fully consistent with this picture. 

Fig. 24. a The three-functional regularly branched chain model with Gaussian behavior of the subchains, called the soft sphere model s). b The Berry-plot of the reciprocal particlescattering factor of the soft sphere model. Compare also Figs. 19, 25 and 27... 

The change of electronic conductivity G(r) over diameter of such two-sphere model composition as element in a system of contacting particles is shown in a Figure 10.6b. The transfer of electron across this composition consists of three stages electron tunneling over the interspace — Rq is replaced by the M/SC conductivity across a particle with subsequent electron tunneling over the further interspace R — Rq. The probability of electron tunneling falls down exponentially with increase in distance from the surface of particle. 

This conclusion is further strengthened considerably by the theoretical calculation of CBE originally performed by Pearson and Gray (102) and later on somewhat modified by Pearson and Mawby (8). Values of CBE are calculated according to three models, viz. the hard sphere model, the polarizable ion model and the localized molecular orbital model. Only the last one, treating the bonds as covalent, is able to account in a satisfactory way for the values found experimentally for such halides as HgCl2 and CdCl2. For LiCl and NaCl, on the other hand, an acceptable fit with the experimental values is obtained already by the hard sphere model, which certainly indicates a predominantly electrostatic interaction. 

Three models were used to simulate the observed extraction rates for the oily components. Each of employed the effective diffusivity De as an adjustable parameter. They were the Characteristic Time Model (3) two other published models here designated Single Sphere Models I (4,5) and II (6). In these models it is assumed that the solute is extracted from a particulate bed composed of porous... 

We shall consider in detail the predictions of the hard-sphere model for the viscosity, thermal conductivity, and diffusion of gases indeed, the kinetic theory treatment of these three transport properties is very similar. But first let us consider the simpler problem of molecular effusion. 

The Kerner equation, a three phase model, is applicable to more than one type of inclusion, Honig (14,15) has extended the Hashin composite spheres model to include more than one inclusion type. Starting with a dynamic theory and going to the quasi-static limit, Chaban ( 6) obtains for elastic inclusions in an elastic material... 

A metallic crystal can be pictured as containing spherical atoms packed together and bonded to each other equally in all directions. We can model such a structure by packing uniform, hard spheres in a manner that most efficiently uses the available space. Such an arrangement is called closest packing (see Fig. 16.13). The spheres are packed in layers in which each sphere is surrounded by six others. In the second layer the spheres do not lie direotlv over those in the first layer. Instead, each one occupies an indentation (or dimple) formed by three spheres in the first layer. In the third layer the spheres can occupy the dimples of the second layer in two possible ways. They can occupy positions so that each sphere in the third layer lies directly over a sphere in the first layer (the aba arrangement), or they can occupy positions... 

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