Cylinders, scattering

In Chapter 8 we shall derive the field scattered by an infinite cylinder of arbitrary radius and refractive index we shall also consider scattering by a finite cylinder in the diffraction theory approximation. Although the finite cylinder scattering problem is not exactly soluble, we can obtain analytical expressions for the amplitude scattering matrix elements in the Rayleigh-Gan s approximation. 

Fig. 2.10. Collision cylinder. Scattering in the ci molecular frame, where the C2 molecules have velocity g2i.

Micellar structure has been a subject of much discussion [104]. Early proposals for spherical [159] and lamellar [160] micelles may both have merit. A schematic of a spherical micelle and a unilamellar vesicle is shown in Fig. Xni-11. In addition to the most common spherical micelles, scattering and microscopy experiments have shown the existence of rodlike [161, 162], disklike [163], threadlike [132] and even quadmple-helix [164] structures. Lattice models (see Fig. XIII-12) by Leermakers and Scheutjens have confirmed and characterized the properties of spherical and membrane like micelles [165]. Similar analyses exist for micelles formed by diblock copolymers in a selective solvent [166]. Other shapes proposed include ellipsoidal [167] and a sphere-to-cylinder transition [168]. Fluorescence depolarization and NMR studies both point to a rather fluid micellar core consistent with the disorder implied by Fig. Xm-12. 

Figure A3.1.7. Direct and restituting collisions in the relative coordinate frame. The collision cylinders as well as the appropriate scattering and azimuthal angles are illustrated.

Separate regions in Figure 6.33 account for scatter of velocities of cylinders and spheres separating into 2, 10, or 100 fragments. The assumptions used in deriving the figure are from Baker et al. (1983), namely,... 

Separate regions in the figure account for the scatter of velocities for spheres and cylinders separating into 2, 10 or 100 fragments. The number of fragments must first be chosen, usually on the basis of scaled energy. 

Experimentally, the stretching of block copolymer chains has been addressed in two ways by measuring L as a function of N, and by measuring the components of Rg of the block chains both parallel and perpendicular to the interface. The domain dimensions have been studied most extensively for styrene-isoprene and styrene-butadiene block copolymers X-ray and neutron scattering are the methods of choice. The predicted SSL scaling of L N2/3 has been reported for spheres, cylinders and lamellae [99,102-106], but not in all cases. For example, Bates et al. found N0 37 for styrene-butadiene spheres [100], and Hadziioannou and Skoulios observed N0 79 for styrene-isoprene lamellae [107], In the sphere case, kinetic limitations to equilibration were felt to be an important factor [100],... 

Fig. 2. Schematic representation of the supramolecular cylinders of the dendrimer derived from macromonomer 9 (R=OC12H25,n=3) in the Qh mesophase atop view of a cylinder containing six repeat units in a stratum with the alkyl tails melted to match the average column radius determined by X-ray scattering experiments b side view of a cylinder containing 30 repeat units of the polymer assembled with melted alkyl tails. Reproduced with permission from references 5 a...

Synopsis of Experiment and Results. The material is irradiated during straining and relaxation. The example shows that a nanostructure which is hard to interpret from a series of scattering patterns may clearly reveal its complex domain structure after transformation to the CDF. Different structural entities are identified which respond each in a different way on mechanical load. The shape of the basic particles is identified (cylinders). The arrangement of the cylinders is determined. Thus the semi-quantitative analysis of the CDF provides the information necessary for the selection and definition of a suitable complex model which is required for a... 

For an ensemble of uncorrelated ID particles (cylinders, layers) with a Gaussian96 particle thickness distribution the ID scattering intensity is [197]... 

If the structural entities are lamellae, Eq. (8.80) describes an ensemble of perfectly oriented but uncorrelated layers. Inversion of the Lorentz correction yields the scattering curve of the isotropic material I (5) = I (s) / (2ns2). On the other hand, a scattering pattern of highly oriented lamellae or cylinders is readily converted into the ID scattering intensity /, (53) by ID projection onto the fiber direction (p. 136, Eq. (8.56)). The model for the ID intensity, Eq. (8.80), has three parameters Ap, dc, and <7C. For the nonlinear regression it is important to transform to a parameter set with little parameter-parameter correlation Ap, dc, and oc/dc. When applied to raw scattering data, additionally the deviation of the real from the ideal two-phase system must be considered in an extended model function (cf. p. 124). 

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