Spheres cubic packing

Uniform spheres, cubic packing = 0.524 Uniform spheres, random packing = 0.621 Uniform spheres, hexagonal packing ... 

This bismuth-III structure is also observed for antimony from 10 to 28 GPa and for bismuth from 2.8 to 8 GPa. At even higher pressures antimony and bismuth adopt the body-centered cubic packing of spheres which is typical for metals. Bi-III has a peculiar incommensurate composite crystal structure. It can be described by two intergrown partial structures that are not compatible metrically with one another (Fig. 11.11). The partial structure 1 consists of square antiprisms which share faces along c and which are connected by tetrahedral building blocks. The partial structure 2 forms linear chains of atoms that run along c in the midst of the square antiprisms. In addition, to compensate for the... 

Germanium forms the same kinds of modifications as silicon at similar conditions (Fig. 12.4). Tin, however, does not exhibit this diversity )3-tin transforms to a body-centered cubic packing of spheres at 45 GPa. Lead already adopts a cubic closest-packing of spheres at ambient pressure. 

The space filling in the body-centered cubic packing of spheres is less than in the closest packings, but the difference is moderate. The fraction of space filled amounts to ns/3 = 0.6802 or 68.02 %. The reduction of the coordination number from 12 to 8 seems to be more serious however, the difference is actually not so serious because in addition to the 8 directly adjacent spheres every sphere has 6 further neighbors that are only 15.5 % more distant (Fig. 14.3). The coordination number can be designated by 8 + 6. 

Corresponding to its inferior space filling, the body-centered cubic packing of spheres is less frequent among the element structures. None the less, 15 elements crystallize with this structure. As tungsten is one of them, the term tungsten type is sometimes used for this kind of packing. 

Unit cell of the body-centered cubic packing of spheres and the coordination around one sphere... 

The CsCl type offers the simplest way to combine atoms of two different elements in the same arrangement as in body-centered cubic packing the atom in the center of the unit cell is surrounded by eight atoms of the other element in the vertices of the unit cell. In this way each atom only has adjacent atoms of the other element. This is a condition that cannot be fulfilled in a closest-packing of spheres (cf. preceding section). 

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...

FIGURE 3.9 Cubic packing of spheres with a porosity of 47.65% (A) vs. rhombohedral packing of spheres with a porosity of 25.95% (B). 

Figure 3.35. Position of the holes in closest packing. Unit cell projections are shown for the cubic and hexagonal sphere closest packing. Coordinates of the spheres and of the tetrahedral and octahedral holes are given. The values indicated inside the drawing correspond to the third coordinate (along the vertical axis) when two values are given, these correspond to two positions along the same vertical line.

Although equation 4.7 is reasonably true for random packings, it does not apply to all regular packings. Thus with a bed of spheres arranged in cubic packing, e = 0.476, but the fractional free area varies continuously, from 0.215 in a plane across the diameters to... 

Equation (48) has been derived under the assumption that the volume fraction can reach unity as more and more particles are added to the dispersion. This is clearly physically impossible, and in practice one has an upper limit for , which we denote by max. This limit is approximately 0.64 for random close packing and roughly 0.71 for the closest possible arrangement of spheres (face-centered cubic packing or hexagonal close packing). In this case, d in Equation (46) is replaced by d(j>/[ 1 — (/m[Pg.169]

PE-PEP diblock were similar to each other at high PE content (50-90%). This was because the mechanical properties were determined predominantly by the behaviour of the more continuous PE phase. For lower PE contents (7-29%) there were major differences in the mechanical properties of polymers with different architectures, all of which formed a cubic-packed sphere phase. PE-PEP-PE triblocks were found to be thermoplastic elastomers, whereas PEP-PE-PEP triblocks behaved like particulate filled rubber.The difference was proposed to result from bridging of PE domains across spheres in PE-PEP-PE triblocks, which acted as physical cross-links due to anchorage of the PE blocks in the semicrystalline domains. No such arrangement is possible for the PEP-PE-PEP or PE-PEP copolymers (Mohajer et al. 1982). 

Fig. 6.11 TEM images showing a sequence of morphologies on increasing PS homopolymer (M = 5.9kgmor ) concentration (wt%) in blends with a PS-PI diblock (Mr = 48.7kgmol-1,/PS = 0.51) (Winey et al. 1991c). The blends were annealed at 125 °C. (a) 10% PS, lamellae, (b) 30% PS, bicontinuous cubic, (c) 50% PS, hexagonal-packed cylinders, (d) 70% PS, cubic-packed spheres.

UsingTEM to identify blend morphology, two diblocks with/ps 0.8 that form cubic-packed spherical phases and cylindrical phases respectively in the pure copolymer were found not to macrophase separate in a blend with d = 2.2, but to form single domain structures (cylinders or spheres) in the blend (Koizumi et al. 1994c). Similarly, blending a diblock with fK = 0.26 with one with fK = 0.64 (d = 1.2) led to uniform microphase-separated structures, with a lamellar phase induced in the 50 50 blend. Vilesov et al. (1994) also observed that blending two PS-PB diblocks with approximately inverse compositions (i.e. 22wt% PS and 72 wt% PS) induces a lamellar phase in the 50 50 blend. These examples all correspond to case (i). 

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