Partial orientation function

In this chapter we describe the unified generation of stereoisomers including conform-ers of a molecular structure [102,103,105]. This method has the potential to generate stereoisomers that cannot be described in terms of stereocenters, stereogenic double bonds or single bond rotations. Fundamentals such as the concept of a (partial) orientation function are discussed, and mathematical tools such as Radon partitions and binary Grassmann-Plucker relations are used to construct tests for abstract orientation functions. Some simple examples are treated in detail. 

The partial orientation function (POP) with respect to S, a subset of the atom quadruples of the conformation considered, is the orientation function x of the conformation restricted to the subset S,... 

Example (Partial orientation functions for figure 4.2) We obtain POFs for the conformations shown in Figure 4.2, which have the following values on quadruples of type (i) and (ii) above ... 

Example (Figure 4.2c, cont.) We already mentioned that a central carbon atom and its four neighbors form a minimal Radon partition, such as in exeimple 4.2c. Thus, A, B = 0, (1,2,3,4 is a minimal Radon partition of conformation Ic. According to Remark 4.16, A, B is also an abstract minimal Radon partition of its orientation function. Knowing only the partial orientation function shown in the subsection on partial orientation functions, we have the foUowing situation, corresponding to the quintuple (0,1,2,3,4) ... 

The matrix IR spectra of la and several isotopomers (cU-la, l80-la) reveal details of the electronic structure of the carbene.23 In particular the red-shift of the C=0 stretching vibration (compared to p-benzoquinone) below 1500 cm-1 indicates a substantial contribution of the phenoxyl/phenyl resonance structure to the wave function of la. The C2V symmetry of the carbene was experimentally revealed by measuring the IR dichroism of partially oriented samples of matrix-isolated la. The orientation of la in an argon matrix was achieved by irradiation with linearly polarized light. 

Brown and Chuah [84] and Oppermann el ol. [36] studied the spinning of partially oriented yam (POY) as a function of take-up speed from 500 to 5000m/min. Figures 11.18 and 11.19 show, respectively, development of the tenacity and elongation as a function of spinning speed. Tenacity increases with increasing... 

Figure 13.6 (a) Elongation as a function of wind-up speed for partially oriented yarn, (b-d) Stress-strain curves of fibers of PET blends with 3% copolyester of 1,4-phenyleneterephthalate and p-oxybenzoate (CLOTH) and 3% copolymer of 6-oxy-2-naphthalene and p-oxybenzoate (CO), spun at 3500, 4000 and 4500 m/min (1) PET control (2) 3 % CLOTH (3) 3 % CO the loci of the theoretical extensions of the PET control are shown as dashed curves [17]. From Orientation suppression in fibers spun from melt blends, Brody, H., J. Appl. Polym. Sci., 31, 2753 (1986), copyright (1986 John Wiley Sons, Inc.). Reprinted by permission of John Wiley Sons, Inc. 

A partitioning function for a system of rigid rod-like particles with partial orientation around an axis is derived from the use of a modified lattice model. The free energy of mixing is shown as a function of the mole numbers, the axis ratio of the solute particles and a disorientation parameter this function passes through a minimum with increase in the disorientation parameter. The chemical potentials display discontinuities at the concentration at which the minimum appears and then separation into an isotropic phase and a somewhat more concentrated anisotropic phase arises. The critical concentration, v, is given in the form 13) ... 

Table 1. The number of transitions in a 1-quantum spectrum of a partially oriented spin system as a function of the number of spin 1/2 nuclei...

Fig. 18. Simulated spectral width as a function of molecular orientation for the 6Q (A), and 7Q (B) spectra of indene (1) partially oriented in liquid crystalline solution. (Reproduced with permission from ref. 62.)...

The equations which predict the elastic constants of a partially oriented polymer involve orientation functions to define the orientation of the aggregate units. For example, the average extensional compliance S33 for a transversely isotropic aggregate of transversely isotropic structural units is given by S 33 = Su sin 0+S33Cos" 0-l-(2Si3-l-544) sin 0cos 9. 

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