Packing density of spheres

PROBLEM 7.4.4. For a monoatomic cubic crystal consisting of spherical atoms packed as close as possible, given the choices of a simple cubic crystal (SCC atom at cell edges only this structure is rarely used in nature, but is found in a-Po), a body-centered cubic crystal (BCC, atom at comers and at center of body), and a face-centered cubic crystal (FCC body at face comers and at face centers), show that the density is largest (or the void volume is smallest) for the FCC structure (see Fig. 7.12). In particular, show that the packing density of spheres is (a) 52% in a simple cubic cell (b) 68% for a body-centered cell (c) 71% for a face-centered cubic cell. 

Fedors (214) took into account the maximum packing density of spheres in suspension to develop an equation for very high concentrations. The volume fraction of random dense-packed spheres is 0.63. 

Apart from chemical composition, an important variable in the description of emulsions is the volume fraction, outer phase. For spherical droplets, of radius a, the volume fraction is given by the number density, n, times the spherical volume, 0 = Ava nl2>. It is easy to show that the maximum packing fraction of spheres is 0 = 0.74 (see Problem XIV-2). Many physical properties of emulsions can be characterized by their volume fraction. The viscosity of a dilute suspension of rigid spheres is an example where the Einstein limiting law is [2]... 

Metal atoms tend to behave like miniature ball-bearings and tend to pack together as tightly as possible. F.c.c. and c.p.h. give the highest possible packing density, with 74% of the volume of the metal taken up by the atomic spheres. However, in some metals, like iron or chromium, the metallic bond has some directionality and this makes the atoms pack into the more open b.c.c. structure with a packing density of 68%. 

The surface chemistry of coesite and stishovite was studied by Stiiber (296). The packing density of hydroxyl groups was estimated from the water vapor adsorption. More adsorption sites per unit surface area were found with silica of higher density. Stishovite is especially interesting since it is not attacked by hydrofluoric acid. Coesite is dissolved slowly. The resistance of stishovite is ascribed to the fact that silicon already has a coordination number of six. Dissolution of silica to HaSiFg by hydrogen fluoride is a nucleophilic attack. It is not possible when the coordination sphere of silicon is filled completely. In contrast, stishovite dissolves with an appreciable rate in water buffered to pH 8.2. The surface chemistry of. stishovite should be similar to that of its analog, rutile. 

A special form of homofimctional hnking utihzes so called dendrimers. Dendrimers are nanospherical structures for which the exact size depends on the number of branching points and which carry reactive functional units in their periphery (for example aldehyde-, thiol-, epoxy groups etc). The structure of dendrimers is similar to a tree, and their ramifications consist of repetitive units. It should be noted that their size is limited due to the fact that the packing density of their terminal groups increases. With increasing size, their macroscopic structure approximates the form of a sphere. 

Fig. 1.1 Relative atomic packing density of a binary mixture of spheres with radius ratios R ranging from 0.2 to 0.8, based on developments of Zheng et al. (1995) (from Miracle et al. (2003) courtesy of Taylor and Francis).

The atoms are densely packed, with most adjacent atoms in van der Waals contact (Section 13.3). Internal atomic packing densities average around 75% (v/v), which is similar to the maximum packing density of equally sized hard spheres. For comparison, water and cyclohexane have packing densities of 58% (v/v) and 44% (v/v), respectively, at 298 K and 1 atm. 

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