Ellipsoids, prolate

Although both the and 6-functions are too insensitive and uncertain for the determination of axial ratios, the length, L, of the equivalent ellipsoid (prolate) can be calculated with confidence from either the intrinsic viscosity at zero gradient or the more commonly known flow birefringence. This can be shown as follows Eq. (22), (19a and c), and (18a and c) can be rewritten as... 

Uniaxial tensile deformations give prolate (needle-shaped) ellipsoids, and compressive or biaxial deformations give oblate (disc-shaped) ellipsoids. Prolate particles can be thought of as a conceptual bridge between the roughly spherical particles used to reinforce elastomers and the long fibers frequently used for this purpose in thermoplastics and thermosets. Similarly, oblate particles can be considered as analogs of clay platelets used to reinforce a variety of materials with dimensions that are controllable. " The orientation of nonspherical particles is also of considerable importance because of the anisotropic reinforcements such particles provide. 

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. 

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 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 ... 

Based on a lot of experimental observations, criteria for the drop stability can be defined as below the U curve, namely We < We.cn, the interfacial stress can equilibrate the shear stress, and the drop will only deform into a stable prolate ellipsoid. Above this curve, the viscous shear stress becomes larger than the interfacial stress. The drop is at first extended and finally breaks up into smaller droplets. 

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. 

An ellipsoidal nucleus with two spherons in the inner core has major radius greater than the minor radii by the radius of a spheron. about 1.5 f, which is about 25 percent of the mean radius. The amount of deformation given by this model is accordingly in rough agreement with that observed (18). In a detailed treatment it would be necessary to take into account the effect of electrostatic repulsion in causing the helions to tend to occupy the poles of the prolate mantle, with tritons tending to the equator. 

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 way forward was proposed by Berne and Pechukas [11] many years later. Their important idea was to consider the overlap between two prolate ellipsoidal gaussian distributions. From the expression for this overlap they evaluated a range parameter which was taken to be the contact distance g and a strength parameter which was set equal to the well depth, e. If the orientations of the two rod-like molecules in the laboratory frame are represented by the unit vectors Ui and Uj and the orientation of the intermolecular vector by the unit vector f then the expression for the angular dependence of the contact distance is... 

Fig. 3a,b. The phase behaviour of a system of hard ellipsoids, both prolate and oblate, as a function of the ellipticity, alb, plotted against a the packing fraction, p b the scaled number density, p ... 

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. 

Figure 4.16 Double bond (a) Lewis model of two tetrahedra sharing an edge, (b) Domain model the two single electron pair domains of the double bond are pulled in toward each other by the attraction of the two carbon cores forming one four-electron double-bond domain with a prolate ellipsoidal shape, thereby allowing the two hydrogen ligands to move apart.

Figure 10 Deformation of spherical filler particles into prolate (needle-shaped) ellipsoids see text for details. 

It is simple to understand the connection between the shear modulus and a. A sphere can be deformed into a prolate ellipsoid either by mechanical stress, or by an electric field. The input work required is measured by G = shear modulus in the first case and by a in the second case. Equating the input work needed in each case and solving for G, yields ... 

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