Ellipsoids of revolution

Frenkel D and Mulder B 1985 The hard ellipsoid-of-revolution fluid. 1. Monte-Carlo simulations Mol. Phys. 55 1171-92... 

The intrinsic viscosity of a solution of particles shaped like ellipsoids of revolution is given by the expression... 

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. 

In the last section we noted that Simha and others have derived theoretical expressions for q pl(p for rigid ellipsoids of revolution. Solving the equation of motion for this case is even more involved than for spherical particles, so we simply present the final result. Several comments are necessary to appreciate these results ... 

The totally symmetrical sphere is characterized by a single size parameter its radius. Ellipsoids of revolution are used to approximate the shape of unsymmetrical bodies. Ellipsoids of revolution are characterized by two size parameters. 

The ellipsoid of revolution is swept out by rotating an ellipse along its major or minor axis. When the major axis is the axis of rotation, the resulting rodlike figure is said to be prolate when the minor axis is the axis of rotation, the disklike figure is said to be oblate. 

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.

At 37°C the viscosity of water is about 0.69 X 10"3 kg m" sec" the difference between this figure and the viscosity of blood is due to the dissolved solutes in the serum and the suspended cells in the blood. The latter are roughly oblate ellipsoids of revolution in shape. 

The intrinsic viscosity of poly(7-benzyl-L-glutamate) (Mq = 219) shows such a strong molecular weight dependence in dimethyl formamide that the polymer was suspected to exist as a helix which approximates a prolate ellipsoid of revolution in its hydrodynamic behaviorf ... 

R/Ro)soiv(f/fo)ellip = n + (mib/m2)(P2/Pi)] (f/fo)eiiip-Briefly justify this expansion of the (f/fo oiv factor. Assuming these particles were solvated to the extent of 0.26 g water (g protein)", calculate (f/fo)eiHp-For prolate ellipsoids of revolution (b/a < 1), Perrin has derived the following expression ... 

Observed properties of many nuclei have been interpreted as showing that the nuclei are not spherical but are permanently deformed (4). The principal ranges of deformation are neutron numbers 90 to 116 and 140 to 156. Most of the deformed nuclei are described as prolate ellipsoids of revolution, with major radii 20 to 40 percent larger than the minor radii. 

Once a general conformation type or preliminary classification has been established it is possible to use sedimentation data to obtain more detailed information about polysaccharide conformation. For example, the low value of ks/[v 0 25 found for the bacterial polysaccharide xylinan has been considered to be due to asymmetry [115]. If we then assume a rigid structure the approximate theory of Rowe [36,37] can be applied in terms of a prolate ellipsoid of revolution to estimate the aspect ratio p L/d for a rod, where L is the rod length and d is its diameter) 80. 

The first of these was by Vieillard-Baron [5] who investigated a system of spherocylinders but failed to detect a liquid crystal phase primarily because the anisometry, L/D, of 2 was too small [37]. He also attempted to study a system of 2392 particles with the larger L/D of 5 but these simulations had to be abandoned because of their large computational cost. However, in view of the ellipsoidal shape of the Gay-Berne particles it is the behaviour of hard ellipsoids of revolution which is of primary relevance to us. 

The quantity riV/RT is equal to six times the rotational period. The rotational relaxation time, p, should he shorter than the fluorescence lifetime, t, for these equations to apply. It is possible to perform calculations for nonspherical molecules such as prolate and oblate ellipsoids of revolution, but in such cases, there are different rotational rates about the different principal axes. 

The first attempt to the problem of the hydration for ellipsoids of revolution, suggests a combination graphic Ua-b with the contribution of the form of the Perrin fimction "p" (ratio of friction). This was followed in for Flory and Scheraga-Mandelkern describing and analytical combination of Ua-b with p to yield a function P, which, with [t]] in cm / g is given by ... 

Figure 11.6 Fits of the G10 PAMAM dendrimer data to a distribution of spheres and a single ellipsoid of revolution...

In the case of an ellipsoid of revolution for which the absorption or emission... 

For a single-domain ferromagnet, any nonspherical particle shape gives rise to shape anisotropy due to the internal magnetostatic energy. The magnetostatic energy, for an ellipsoid of revolution, is equal to... 

This group comprises bodies generated by rotating a closed curve around an axis. Spheroidal particles (also called ellipsoids of revolution) are of particular interest, since they correspond closely to the shapes adopted by many drops and... 

Another complication is the demagnetization correction due to the geometry of the specimen. Demagnetization (or the equivalent depolarization problem for dielectric bodies in an electric field) can only be solved analytically for an ellipsoid of revolution (27X28). When He is applied parallel to one of the three axes of revolution, the magnetization is parallel to He, but the internal field H is given by (29) ... 

Figure 4.17 (a) Prolate and (b) oblate ellipsoids of revolution, showing the relationship... 

Exact expressions of [>/], s0, and D0 are available for compact sphere and compact ellipsoid of revolution (62). Those for rigid rod are still beyond our... 

The length L and diameter d of a rigid rod are related to the major axis a and minor axis b of the equivalent ellipsoid of revolution by a=L and b=(3/2)ll2d. It should be noted that, for actual molecules in solution, the quantity d can be defined only vaguely. With these relations, Simha s equation (64) for the intrinsic viscosity of an elongated ellipsoid of revolution can be rewritten for a thin rod (L/d> 1)... 

Fig. 22 Theoretical relations between Af2/[ j] and In L/d for rigid rods ellipsoid Simha equation (D-l) for equivalent ellipsoid of revolution Y-F Yamakawa-Fujii theory for straight cylinders. Dashed lines indicate asymptotes to respective solid curves...

Perrin s theory (108) for prolate ellipsoids of revolution gives for thin rigid rods of L/d> 10... 

Applequist and Mahr (114) proposed the use of Buckingham s equation (see the next subsection) for ellipsoids of revolution to calculate vacuum of rodlike molecules. They found for poly-L-tyrosine in quinoline that the values of 1/2 so computed from experiment varied linearly with molecular weight and yielded (4.94 0.014) D for fa. In this case, the molecular weights of the samples were indirectly estimated from the observed rotational relaxation times with the assumption of the relation for rigid rods. 

Yet there is no theoretical justification for the use of Buckingham s equation for macromolecules other than rigid ellipsoids of revolution. Probably, it is insuperably difficult to calculate theoretically the factor qjfg in Eq. (E-8) for polymer molecules of arbitrary conformation. Facing this difficult situation, Omura et al. (117) took a tentative step in which pin Eq. (E-l 1) is related to fN by... 

Asymmetry as well as solvation can cause a friction factor to have a value other than /0. Next let us consider the ratio///, which, according to Equation (14), accounts for the effect of particle asymmetry on the friction factor. We saw in Section 1.5 that ellipsoids of revolution are reasonable models for many asymmetric particles. 

Jean Perrin derived expressions for the ratio /// for ellipsoids of revolution in terms of the ratio of the equatorial semiaxis to the semiaxis of revolution b/a. The following expressions were obtained ... 

We noted above that either solvation or ellipticity could cause the intrinsic viscosity to exceed the Einstein value. Simha and others have derived extensions of the Einstein equation for the case of ellipsoids of revolution. As we saw in Section 1.5a, such particles are characterized by their axial ratio. If the particles are too large, they will adopt a preferred orientation in the flowing liquid. However, if they are small enough to be swept through all orientations by Brownian motion, then they will increase [17] more than a spherical particle of the same mass would. Again, this is very reminiscent of the situation shown in Figure 2.4. 

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