Packed spheres

A model which is attracting increasing attention, because of its relevance to actual solids composed of globular particles (Section 1.6), is the packed sphere model. By applying the same general principles as those outlined... 

The pores in question can represent only a small fraction of the pore system since the amount of enhanced adsorption is invariably small. Plausible models are solids composed of packed spheres, or of plate-like particles. In the former model, pendulate rings of liquid remain around points of contact of the spheres after evaporation of the majority of the condensate if the spheres are small enough this liquid will lie wholly within the range of the surface forces of the solid. In wedge-shaped pores, which are associated with plate-like particles, the residual liquid held in the apex of the wedge will also be under the influence of surface forces. 

While not overcrowded, the polyethylene structure uses space with admirable efficiency, the atoms filling the available space with 73% efficiency. For contrast, recall that close-packed spheres fill space with 74% efficiency, so polyethylene does about as well as is possible in its utilization of space. 

Fig. 1. Methods for representing SiO and AlO tetrahedra by means of (a) baH-and-stick model, (b) soHd tetrahedron, (c) skeletal tetrahedron, and (d) spare-filling of packed spheres (1). (e) Linking of four tetrahedra in a four-membered ring, (f) Secondary building unit called tmncated octahedron as...

FIGURE 22.6 Two types of interstitial holes between layers of closed-packed spheres. 

The degree of ordering of the microspheres was estimated by using the radial distribution function g(D) of the P4VP cores of the microspheres (Fig. 11). As previously described, for hexagonal packed spheres, the ratio of the peaks of the distances between the centers of the cores would be For the film at r = 0.5, the... 

Now consider adding a third layer of close-packed spheres. This new layer can be placed in two different ways because there are two sets of dimples in the second layer. Notice in Figure ll-29b that the view through one set of dimples reveals the maroon spheres of the first layer. If spheres in the third layer lie in these dimples, the third... 

If the lattices are viewed as close-packed spheres, the fee and the hep lattices have the highest density, possessing about 26% empty space. Each atom in the interior has 12 nearest neighbors, or in other words, an atom in the interior has a coordination number of 12. The bcc lattice is slightly more open and contains about 32% empty space. The coordination number of a bulk atom inside the bcc lattice is 8. 

Fig. 1 Morphologies of diblock copolymers cubic packed spheres (S), hexagonal packed cylinders (C or Hex), double gyroid (G or Gyr), and lamellae (L or Lam). Inverse phases not shown. From [8], Copyright 2000 Wiley...

The corresponding unit cells are shown in Figure 1.1 and an examination of simple ball-and-stick models (which the reader is strongly urged to carry out) shows that the face-centred cubic (fee) and hexagonal close-packed (hep) structures correspond to the only two possible ways of close-packing spheres, in which each sphere has twelve nearest neighbours. 

A rough measure for the overlap concentration 4>ov is that volume fraction of polymer at which close-packed spheres with radius rg just touch. Then 4> =0.74 rl /(4Trr /3). Taking r — A((J> - 0) ... 

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).

Several cubic structures, therefore, in which (besides 0, 0, 0 0, K, M M, 0, M M, M, 0) one or more of the reported coordinate groups are occupied could be considered as filled-up derivatives of the cubic close-packed structures. The NaCl, CaF2, ZnS (sphalerite), AgMgAs and Li3Bi-type structures could, therefore, be included in this family of derivative structures. For this purpose, however, it may be useful to note that the radii of small spheres which fit exactly into tetrahedral and octahedral holes are, respectively, 0.225. and 0.414... if the radius of the close-packed spheres is 1.0. For a given phase pertaining to one of the aforementioned types (NaCl, ZnS, etc.) if the stated dimensional conditions are not fulfilled, alternative descriptions of the structure may be more convenient than the reported derivation schemes. 

As no value of the voidage is available, e will be estimated by considering eight closely packed spheres of diameter d in a cube of side 2d. Thus ... 

For an idealised bed of uniform rhombohedrally packed spheres of radius r, for example, the waists are of radius 0.155r, from Table 16.2, and the maximum theoretical suction potential of which such a waist is capable is ... 

The pore shape is determined by the particle shape. Plate-shaped particles lead to plate-shaped pores in the case of regular packing. Sphere-shaped particles favor cylindrical or sometimes ink-bottle-type pores. 

Proteins often form gels at concentrations between 20 and 30 percent. Likewise the "gel concentration" for colloidal particles should be equivalent to a value for close-packed spheres between 60 and 75 percent. 

Close-packed spheres occupy 74.04% of a total volume, hence the hard-sphere radius of I" in these 2 1 salts in 2.03 A. Correction for the electrostatic attraction alone would give a monovalent iodide radius of about 2.24, an opposite repulsion-correction for the different co-ordination number would reduce this to about 2.10 A for the monovalent sodium-chloride type (see Appendix). Such values are consistent with our earlier estimates, but incompatible with the electron-density minimum value (4) of 1.94 A. 

When used in equal volume, which will have the higher viscosity (a) a suspension of loosely packed spheres, or (b) a suspension of tightly packed spheres ... 

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