Ellipsoids and Spheres

Once Dt for a macromolecule is determined, it can be compared to theoretical values for prolate ellipsoids and spheres using Equations (4.12) and (4.13) for standard conditions of 20°C and the viscosity of water and a value for the axial ratio can be estimated. 

Figure 13. Schematic presentation of the structure of the best-characterized chemotaxis-specific receptor, Tar. The regions of interaction with the ligands and the proteins in the receptor complex are shown. For simplicity, helix superooiling is omitted, and the a-helices are represented by cylinders. Components that dock to the receptor in the assembled complex are shown schematically as ellipsoids and spheres. Cytoplasmic sites of methylation and demethylation (adaptation sites, residues 295, 3Q2, 309, and 491] are shown as small ovals. (Taken with permission from Falke and Hazelbauer (223], with slight modifications made by J.J. Falke.]...

The shape and size of macromolecules together with the segment distribution within these forms determine the excluded volume of the polymer. Compact molecular shapes such as helices, ellipsoids, and spheres have only an external (intermolecular) excluded volume the space occupied by a given volume in space cannot be occupied by others, and so is an excluded volume for other molecules. Coils, on the other hand, with their loose internal structure, also have, additionally, an internal (intramolecular) excluded volume, since the space occupied by one segment is not available to another segment of the same molecule. 

As the SCC of LCE depended on the size of the LC domains, which was usually in the micrometer range, it was researched whether SCC could be realized for nanoscale shaped bodies [144]. In MCLP nanoparticles based on the nematic main chain polyether l-(4-hydroxy-4 -biphenyl)-2-(4-hydroxyphenyl)butaneandin other LC main-chain moieties a shape change between ellipsoids and spheres could be observed. This effect was only observed if the particle size in these polymer systems was below a critical size. This size related effect resulted from an quasi-equilibrium between the intrinsic shape of the entangled MCLP and the thermodynamically most stable form in the isotropic phase, the sphere [144]. 

Volume of circumscribed parallelepiped, ellipsoid, and sphere Area of circumscribed ellipsoid Symmetry index... 

In addition to an array of experimental methods, we also consider a more diverse assortment of polymeric systems than has been true in other chapters. Besides synthetic polymer solutions, we also consider aqueous protein solutions. The former polymers are well represented by the random coil model the latter are approximated by rigid ellipsoids or spheres. For random coils changes in the goodness of the solvent affects coil dimensions. For aqueous proteins the solvent-solute interaction results in various degrees of hydration, which also changes the size of the molecules. Hence the methods we discuss are all potential sources of information about these interactions between polymers and their solvent environments. 

External-pressure failure of shells can result from overstress at one extreme or n om elastic instability at the other or at some intermediate loading. The code provides the solution for most shells by using a number of charts. One chart is used for cylinders where the shell diameter-to-thickness ratio and the length-to-diameter ratio are the variables. The rest of the charts depic t curves relating the geometry of cyhnders and spheres to allowable stress by cui ves which are determined from the modulus of elasticity, tangent modulus, and yield strength at temperatures for various materials or classes of materials. The text of this subsection explains how the allowable stress is determined from the charts for cylinders, spheres, and hemispherical, ellipsoidal, torispherical, and conical heads. 

Fig. 46 Schematic diagram of elemental process during transition from Hex cylinder to bcc sphere (i) undulation of interface (a, b), (ii) break-up of cylinders into ellipsoids (b, c), (iii) relaxation of domains from ellipsoids into spheres (c, d), and (iv) relaxation in junction distribution to attain uniform distribution (d, e). Pole where concentration of junction points is low may work as memory of grain conservation upon reverse transition from bcc sphere to Hex cylinder. Small arrows in part (b) indicate diffusion of chemical junctions along interface in process (ii). From [136], Copyright 2000 American Chemical Society...

The three-dimensional probability density that represents the shape of the orbital consists of a pair of distorted ellipsoids, and not two tangent spheres. The shape of the complex orbitals p2p l derives from... 

Fig. 10.20. Theoretical spectral patterns for NMR of solid powders. The top trace shows the example of high symmetry, or cubic site symmetry. In this case, all three chemical shift tensor components are equal in value, a, and the tensor is best represented by a sphere. This gives rise to a single, narrow peak. In the middle trace, two of the three components are equal, so the tensor is said to have axial site symmetry. This tensor is best represented by an ellipsoid and gives rise to the assymetric lineshape shown. If all three chemical shift components are of different values, then the tensor is said to have low-site symmetry. This gives rise to the broad pattern shown in the bottom trace.

The convergent method also allows the synthesis of dendrimers of regular shapes other than spheres. Ellipsoid- and rod-shaped dendrimers can be synthesized hy using appropriate core molecules [Tomalia, 2001]. 

The mechanism of mass transfer to the external flow is essentially the same as for spheres in Chapter 5. Figure 6.8 shows numerically computed streamlines and concentration contours with Sc = 0.7 for axisymmetric flow past an oblate spheroid (E = 0.2) and a prolate spheroid (E = 5) at Re = 100. Local Sherwood numbers are shown for these conditions in Figs. 6.9 and 6.10. Figure 6.9 shows that the minimum transfer rate occurs aft of separation as for a sphere. Transfer rates are highest at the edge of the oblate ellipsoid and at the front stagnation point of the prolate ellipsoid. 

A turbulent eddy can be visualized as a large number of different-sized rotating spheres or ellipsoids. Each sphere has subspheres and so on until the smallest eddy size is reached. The smallest eddies are dissipated by viscosity, which explains why turbulence does not occur in narrow passages there is simply no room for eddies that will not be dissipated by viscosity. 

It is interesting to compare and contrast an isotropic ellipsoid and an anisotropic sphere the polarizability of both particles is a tensor, the principal values of which are... 

At relatively low concentrations of surfactant, the micelles are essentially the spherical structures we discussed above in this chapter. As the amount of surfactant and the extent of solubilization increase, these spheres become distorted into prolate or oblate ellipsoids and, eventually, into cylindrical rods or lamellar disks. Figure 8.8 schematically shows (a) spherical, (b) cylindrical, and (c) lamellar micelle structures. The structures shown in the three parts of the figure are called (a) the viscous isotropic phase, (b) the middle phase, and (c) the neat phase. Again, we emphasize that the orientation of the amphipathic molecules in these structures depends on the nature of the continuous and the solubilized components. 

The pure-rotational Raman spectrum of a polyatomic molecule provides information on the moments of inertia, hence allowing a structural determination. For a molecule to exhibit a pure-rotational Raman spectrum, the polarizability must be anisotropic that is, the polarizability ellipsoid must not be a sphere. As noted in Section 5.2, a spherical top has a spherical polarizability ellipsoid, and so gives no pure-rotational Raman spectrum. Symmetric and asymmetric tops have asymmetric polarizabilities. The structures of several nonpolar molecules (which cannot be studied by microwave spectroscopy) have been determined from their pure-rotational Raman spectra these include F2, C2H4, and C6H6. 

The geometrical features of shear deformation are shown in Fig. 24.5. Here, the shear is on the K plane in the direction of d. The initial unit sphere is deformed into an ellipsoid and the Ki plane is an invariant plane. The K2 plane is rotated by the shear into the K 2 position and remains undistorted. A reasonable slip system to assume for the lattice-invariant shear deformation is slip in a (111) direction on a 112 plane in the b.c.t. lattice, which corresponds to slip in a (110) direction on a 110 plane in the f.c.c. lattice. 

Figure 11. Our new explanation is that to the traveling observer, her sphere of observation is by the velocity transformed into an ellipsoid of observation. However, to the traveler this ellipsoid appears spherical because A and B are focal points of the ellipsoid and thus ACB — ADB — AEB. Thus, to her the car appears cubic because she observes that all sides of the car are touched by this sphere simultaneously, just as when the car was stationary. To the stationary observer this simultaneity is, however, not true.

Figure 5. Molecular structure of methyl 2,3,4,6,2, 4, 6 -hepata-0-acetyl-fS-l>-laminarabioside (19). The upper ring is glycosidic residue. Shaded and nonshaded ellipsoids, and small spheres represent C, O, and H atoms, respectively.

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