Distribution of orientation

Fig. 10. Distribution of orientation densities (ODF) for Zn phase (left) and for A1 phase (right) of AIZn78 alloy...

Usually, dilute polymer solutions are isotropic systems, i.e. macromolecular chains can exist in these solutions independently of each other with a random distribution of orientations of the long axes of coils. The solutions of flexible-chain polymers remain isotropic when the solution concentration increases whereas in concentrated solutions of macromolecules of limited flexibility the chains can no longer be oriented arbitrarily and some direction of preferential orientations of macromolecular axes appears, i.e. the mutual orientations of the axes of neighboring molecules are correlated. This means that... 

The distribution of orientation of the structural units can be described by a function N(0, solid angle sin 0 d0 dtp d Jt. It is most appropriate to expand this distribution function in a series of generalised spherical harmonic functions. 

Molecules adsorbed at interfaces are in constant motion, both from the point of view of their orientation and of the dynamic exchange with the bulk solution. The distribution of orientation angle is characterized by the parameter D, see Eq. (13), or the angle of orientation Q. Complete randomness for the angle 0, the angle of rotation around the principal... 

The orientation is not strictly identical for all structural units and is rather spread over a certain statistical distribution. The distribution of orientation can be fully described by a mathematical function, N(6, q>, >//), the so-called ODF. Based on the theory of orthogonal polynomials, Roe and Krigbaum [1,2] have shown that N(6, generalized spherical harmonics that form a complete set of orthogonal functions, so that... 

The ° mn coefficients are the mean values of the generalized spherical harmonics calculated over the distribution of orientation and are called order parameters. These are the quantities that are measurable experimentally and their determination allows the evaluation of the degree of molecular orientation. Since the different characterization techniques are sensitive to specific energy transitions and/or involve different physical processes, each technique allows the determination of certain D mn parameters as described in the following sections. These techniques often provide information about the orientation of a certain physical quantity (a vector or a tensor) linked to the molecules and not directly to that of the structural unit itself. To convert the distribution of orientation of the measured physical quantity into that of the structural unit, the Legendre addition theorem should be used [1,2]. An example of its application is given for IR spectroscopy in Section 4. 

Problems related to the use of a guest dye can be reduced if the polymer contains a fluorescent chemical group. Gohil and Salem [70] took advantage of such intrinsic fluorescence to characterize the in-plane distribution of orientation in biaxially drawn PET films. In these experiments, the chain-intrinsic fluorescent label is due to the formation of dimers by two terephthalic moieties, exclusively within the noncrystalline regions. A comparison between sequential and simultaneous drawing along the MD and TD directions was undertaken for a fixed MD draw ratio of 3.5 and various TD draw ratios. The orientational order was characterized by two "orientation ratios" Rmd and RTD such that... 

Although the (P2) coefficient is a convenient index to characterize the orientation and is commonly measured by WAXD for semicrystalline polymers, it is only an average value and different distributions of orientation could give the same (P2) coefficient (Section 2). It has been shown that higher (P ) coefficients are required to fully characterize the distribution of orientation [83]. This is... 

Fig. 2 Schematic representation of the 13C NMR signal of a single crystal containing the functional group AB, oriented (A) perpendicular to the applied field, and (B) parallel to the applied field. The lineshape in (C) represents the NMR signal of a polycrystalline sample with a random distribution of orientations yielding the chemical shift anisotropy pattern displayed. (From Ref. 15.)...

The simple collision theory for bimolecular gas phase reactions is usually introduced to students in the early stages of their courses in chemical kinetics. They learn that the discrepancy between the rate constants calculated by use of this model and the experimentally determined values may be interpreted in terms of a steric factor, which is defined to be the ratio of the experimental to the calculated rate constants Despite its inherent limitations, the collision theory introduces the idea that molecular orientation (molecular shape) may play a role in chemical reactivity. We now have experimental evidence that molecular orientation plays a crucial role in many collision processes ranging from photoionization to thermal energy chemical reactions. Usually, processes involve a statistical distribution of orientations, and information about orientation requirements must be inferred from indirect experiments. Over the last 25 years, two methods have been developed for orienting molecules prior to collision (1) orientation by state selection in inhomogeneous electric fields, which will be discussed in this chapter, and (2) bmte force orientation of polar molecules in extremely strong electric fields. Several chemical reactions have been studied with one of the reagents oriented prior to collision. ... 

Figure 5. Distribution of orientations of molecules state selected by an inhomogeneous hexapole electric field.

In addition to imposing spatial restrictions on the distributions of adjacent water molecules, the various groups in the solute sugar molecule also impose orientational structuring upon these solvent molecules. Figure 10 displays the distributions of orientations for water molecules aroimd the methylene carbon C6. 

Figure 10. Distribution of orientations for water molecules adjacent to the exocyclic methylene carbon atom C6 as calculated from a molecular dynamics simulation of a-D-glucopyranose in aqueous solution. The function plots the frequency of occurrence of an angle between the water OH bond vectors and the vector from the carbon atom to the water oxygen atom. A value of cos( ) of 1.0 corresponds to an OH bond vector pointing directly away from the carbon atom. (Reproduced from Ref. 32. Copyright 1989 American Chemical Society.)...

Figure 11. Distribution of orientations for water molecules adjacent to the hydrogen-bonding 03 hydroxyl oxygen atom, as calculated from an MD simulation of a-D-glucopyranose, as in Figure 10. (Reproduced from Ref. 32. Copyright 1989 American Chemical Society.)...

Studies on crystalline CggO [39] using calorimetry and high-resolution X-ray powder diffraction show a face centered cubic lattice (a = 14.185 A) with an orientational disorder at room temperature. An orientational ordering transition occurs at 278 K, upon which a simple cubic phase develops. At 19 K this phase, which is similar to the orientational ordered phase of Cgg itself, shows additional randomness due to a distribution of orientation of the oxygens in CggO. 

In general, structure solution from powder XRD data has a good chance of success only if the experimental powder XRD pattern contains reliable information on the intrinsic relative intensities of the diffraction maxima, which requires that there is no preferred orientation in the powder sample. Preferred orientation arises when the crystallites in the powder sample have a nonrandom distribution of orientations, and this effect can be particularly severe when the crystal morphology is strongly anisotropic (e.g. long needles or flat plates). When a powder sample exhibits preferred orientation, the measured relative peak intensities differ from the intrinsic relative diffraction intensities, limiting the prospects for determining reliable structural information from the powder XRD pattern. In order to circumvent this... 

Peak intensities. If the material exhibits preferred orientation (i.e. a nonrandom distribution of orientations of the crystallites within a powder), the relative intensities of peaks in the powder XRD pattern will deviate from the intrinsic relative intensities that are characteristic of the crystal structure, and hence the powder XRD patterns recorded for two samples of the same material but exhibiting different degrees of preferred orientation may appear substantially different. This issue is particularly pertinent in comparing an experimental powder XRD pattern with a simulated powder XRD pattern for a known crystal structure, as there are implicitly no effects due to preferred orientation in the latter case. 

Orientation Once the particles are dispersed in the polymer, they must be oriented so that the flat surface of the clay is parallel to the surface of the packaging material to maximize the barrier effect. Several models have been developed in order to describe the mass transfer within the nanocomposites. Most models assume that the platelets have a regular and uniform shape (rectangular, sanidic, or circular) and form a regular array in space. They are either parallel to each other or have a distribution of orientations, with the... 

Consideration of the tetrahydroiso-a-acids depends on their precursors. They are manufactured commercially from, ultimately, either a- or /3-acids. The chromatogram of those from the a-acid route demonstrate a chromatographic profile similar in detail to conventional iso-a-acids, with similar ratios of stereoisomers. Those derived from /3-acids look quite different. Because the /3-acids are racemic, there is a random distribution of orientations about the two adjacent chiral centers, which in practice results in the two peaks of about equal area for each of the... 

Because of the great chemical importance of those methods of trapping radicals which result in a completely randomized distribution of orientations, the task of interpreting their electron spin resonance spectra will now be considered. 

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