Axial ratio

The spherical geometry assumed in the Stokes and Einstein derivations gives the highly symmetrical boundary conditions favored by theoreticians. For ellipsoids of revolution having an axial ratio a/b, friction factors have been derived by F. Perrin, and the coefficient of the first-order term in Eq. (9.9) has been derived by Simha. In both cases the calculated quantities increase as the axial ratio increases above unity. For spheres, a/b = 1. 

We designate the length of the ellipsoid along the axis of rotation as 2a and the equatorial diameter as 2b to define the axial ratio a/b which characterizes the ellipticity of the particle. By this definition, a/b > 1 corresponds to prolate ellipsoids, and a/b < 1 to oblate ellipsoids. 

Based on these ideas, the intrinsic viscosity (in 0 concentration units) has been evaluated for ellipsoids of revolution. Figure 9.3 shows [77] versus a/b for oblate and prolate ellipsoids according to the Simha theory. Note that the intrinsic viscosity of serum albumin from Example 9.1-3.7(1.34) = 4.96 in volume fraction units-is also consistent with, say, a nonsolvated oblate ellipsoid of axial ratio about 5. 

Figure 9.3 Intrinsic viscosity according to the Simha theory in terms of the axial ratio for prolate and oblate ellipsoids of revolution.

The effect of ellipticity also increases [77] above the 2.5 value obtained for spheres. Analytical functions as well as graphical representations like Fig. 9.3 are available to describe this effect in terms of the axial ratios of the particles. In principle, therefore, a/b values for nonsolvated, rigid particles can be estimated from experimental [77] values. 

We shall see in Sec. 9.10 that sedimentation and diffusion data yield experimental friction factors which may also be described-by the ratio of the experimental f to fQ, the friction factor of a sphere of the same mass-as contours in solvation-ellipticity plots. The two different kinds of contours differ in detailed shape, as illustrated in Fig. 9.4b, so the location at which they cross provides the desired characterization. For the hypothetical system shown in Fig. 9.4b, the axial ratio is about 2.5 and the protein is hydrated to the extent of about 1.0 g water (g polymer)". ... 

The quantitative analysis of this problem results in a set of contours in terms of the axial ratio a/b and the solvation m /mj for constant values... 

All that can be concluded from the data given in the preceding example is that the particle is not an unsolvated sphere. However, when an appropriate display of contours is examined for f/fo (e.g.. Ref. 2), the latter is found to be consistent with an unsolvated particle of axial ratio about 4 1 or with a spherical particle hydrated to the extent of about 0.48 g water (g polymer). Of course, there are a number of combinations of these variables which are also possible, and some additional experimental data—such as the intrinsic viscosity—are needed to select that combination which is consistent with all experimental observations. 

Using 1.32 g cm as the density of the polymer, estimate the axial ratio for these molecules, using Simha s equation ... 

Protein molecules extracted from Escherichia coli ribosomes were examined by viscosity, sedimentation, and diffusion experiments for characterization with respect to molecular weight, hydration, and ellipticity. These dataf are examined in this and the following problem. Use Fig. 9.4a to estimate the axial ratio of the molecules, assuming a solvation of 0.26 g water (g protein)"V At 20°C, [r ] = 27.7 cm g" and P2 = 1.36 for aqueous solutions of this polymer. 

Verify —or revise, if necessary— he axial ratio estimated in the last problem for this protein. 

Miller indiees are used to numerieally define the shape of erystals in terms of their faees. All the faees of a erystal ean be deseribed and numbered in terms of their axial intereepts (usually three though sometimes four are required). If, for example, three erystallographie axes have been deeided upon, a plane that is inelined to all three axes is ehosen as the standard or parametral plane. The intereepts Z, Y, Z of this plane on the axes x, y, and z are ealled parameters a, b and c. The ratios of the parameters and Irx are ealled the axial ratios, and by eonvention the values of the parameters are redueed so that the value of b is unity. 

Beryllium is a light metal (s.g. 1 -85) with a hexagonal close-packed structure (axial ratio 1 568). The most notable of its mechanical properties is its low ductility at room temperature. Deformation at room temperature is restricted to slip on the basal plane, which takes place only to a very limited extent. Consequently, at room temperature beryllium is by normal standards a brittle metal, exhibiting only about 2 to 4% tensile elongation. Mechanical deformation increases this by the development of preferred orientation, but only in the direction of working and at the expense of ductility in other directions. Ductility also increases very markedly at temperatures above about 300°C with alternative slip on the 1010 prismatic planes. In consequence, all mechanical working of beryllium is carried out at elevated temperatures. It has not yet been resolved whether the brittleness of beryllium is fundamental or results from small amounts of impurities. Beryllium is a very poor solvent for other metals and, to date, it has not been possible to overcome the brittleness problem by alloying. 

When more d orbitals are available, six stronger equivalent eigenfunctions with their maxima directed towards the comers of a trigonal prism of unit axial ratio can be formed1). The bond-strength of these is 2.985, nearly equal to the maximum 3.000 possible for dsp orbitals. 

Trigonal-Prism Radii. In molybdenite and the corresponding tungsten sulfide the metal atom is surrounded by six sulfur atoms at the corners of a right trigonal prism of axial ratio unity, the bond orbitals involved being those discussed in an earlier section. Prom the observed interatomic distances the values 1.37 and 1.44 A are calculated for Mo and IPIV in such crystals (Table XV). 

Because of the anomalous axial ratio of TeC>2 (Table XVI), no attempt was made to correct the radius. 

Bollnow29 has evaluated the Coulomb energy of the rutile arrangement as a function of the axial ratio c/a, making the assumption that u is given by Equation 16. His results may be given by the equation... 

The configuration of the twelve ligands about the small MnlV atom is that of an approximately regular icosahedron. The thirteen-cornered and sixteen-cornered coordination polyhedra about Mnl, Mnll, and Mnlll are appropriate to axial ratios slightly greater than unity. 

Zinc crystallizes in a deformed A3 structure with large axial ratio, causing the six equatorial neighbors (at 2.660 A.) to be nearer than the six neighbors in adjacent planes (at 2.907 A.). The bond numbers are 0.54 and 0.21, respectively, leading to 72(1) = 1.249 A. 

AuSn has the nickel arsenide structure, B8, with abnormally small axial ratio (c/a = 1.278, instead of the normal value 1.633). Each tin atom is surrounded by six gold atoms, at the corners of a trigonal prism, with Au-Sn = 2.847 A. and each gold atom is surrounded by six tin atoms, at the corners of a flattened octahedron, and two gold atoms, at 2.756 A., in the opposed directions through the centers of the two large faces of the octahedron. 

For example, tin, with v = 2-5, crystallizes with a unique atomic arrangement, in which each atom has six ligates, four at 3-016 A and two at 3-175 A. These distances have been used (1947) in assigning the bond numbers 0-48 and 0-26 to these bonds. It is clear that these bond numbers can be taken as and and that the choice of the structure and the value of its axial ratio (which determines the relative lengths of the two kinds of bonds) are the result of the effort of the tin atom to use its valency 2-5 in the formation of stable bonds with simple fractional bond numbers. 

Zinc and cadmium have the A 3 structure, but with abnormally large axial ratio (1-856 for zinc) instead of the value 1-633 corresponding to close packing. From the distances 2-660 A (for six bonds) and 2-907 A (for the other six) the bond numbers 0-54 and 0-21 have been deduced. If the axial ratio were normal each of the twelve bonds formed by a zinc atom with v = 4-5 would have bond number f. The assumption of the distorted structure permits a split into two classes with the more stable bond numbers and (or, probably, with the average valency of zinc equal to 4). 

Three forms of titanium dioxide, Ti02, are known. Of these the crystal structures of the two tetragonal forms, rutile and anatase, have been thoroughly investigated2) in each case only one parameter is involved, and the atomic arrangement has been accurately determined. The third form, brookite, is orthorhombic, with axial ratios... 

The numbers in parentheses correspond to 1 estimated standard deviation (esd), except those for the Cpf Cn ratio, which correspond to 3 esd. The values of n and m shown are the smallest integers for which the ratio (n + m)/n lies within the experimental error limits of the measured axial ratio < Fe-<=R — t + e. The c parameters in the bottom line refer to the supercells corresponding to the underlined (n + ro)/n values ... 

Scheraga-Mandelkern equations (1953), for effective hydrodynamic ellipsoid factor p (Sun 2004), suggested that [rj] is the function of two independent variables p, the axial ratio, which is a measure of shape, and Ve, the effective volume. To relate [r ] to p and Ve, introduced f, the frictional coefficient, which is known to be a direct function of p and Ve. Thus, for a sphere we have... 

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