Close-packed spheres

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. 

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

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. 

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) ... 

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 ... 

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. 

FIGURE 1.5 (a) Two layers of close-packed spheres with the enclosed octahedral holes shaded (b) a computer representation of an octahedral hole. 

Figure 4.6 Placement of a layer B of close-packed spheres on top of a layer A, generating octahedral (O) and tetrahedral (T) interstices between the layers.

VTi. Keenan E. Dungey, George MJII Lisensky, and S. Michael Condren, "Kixium Monolayers A Simple Alternative to the Bubble Raft Model for Close-Packed Spheres," /. Chem. Educ., Vol. 76, 1999, 618-619. 

An important quantity, which characterizes a macroemulsion, is the volume fraction of the disperse phase 4>a (inner phase volume fraction). Intuitively one would assume that the volume fraction should be significantly below 50%. In reality much higher volume fractions are reached. If the inner phase consists of spherical drops all of the same size, then the maximal volume fraction is that of closed packed spheres (fa = 0.74). It is possible to prepare macroemulsions with even higher volume fractions volume fractions of more than 99% have been achieved. Such emulsions are also called high internal phase emulsions (HIPE). Two effects can occur. First, the droplet size distribution is usually inhomogeneous, so that small drops fill the free volume between large drops (see Fig. 12.9). Second, the drops can deform, so that in the end only a thin film of the continuous phase remains between neighboring droplets. 

Hydraulic number for close-packed spheres. We consider a volume V which contains n spheres. Here, n is assumed to be a large number so that it is practically continuous. The volume filled by particles is V) = n 4/3 nR3. The total surface area of all particles is A = n 4ttR2. The free ( void ) volume is Vv = 0.26 V = 0.26 n 4/3 ttR3/0.74. Thus the hydraulic number is... 

A possible approach that we have presented in detail elsewhere C6, 7) is to consider the centroids of these globular segments to be distributed as the centres of randomly close-packed spheres. 

Trumble, K.P., Spontaneous infiltration of non-cylindrical porosity close packed spheres , Acta. Mater., 1998 46(7) 2363-2367. 

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