Orientation distribution function

Typical shapes of the orientation distribution function are shown in figure C2.2.10. In a liquid crystal phase, the more highly oriented the phase, the moreyp tends to be sharjDly peaked near p=0. However, in the isotropic phase, a molecule has an equal probability of taking on any orientation and then/P is constant. 

Here the bar indicates an average over the orientational distribution function.Here cos — 4)is the... 

Figure C2.2.10. Orientational distribution functions for (a) a highly oriented liquid crystal phase, (b) a less well...

This can be inserted in equation (02.2.3) to give tlie orientational distribution function, and tlius into equation (02.2.6) to deteniiine the orientational order parameters. These are deteniiined self-consistently by variation of tlie interaction strength iin equation (c2.2.7). As pointed out by de Gemies and Frost [20] it is possible to obtain tlie Maier-Saupe potential from a simple variational, maximum entropy metliod based on tlie lowest-order anisotropic distribution function consistent witli a nematic phase. 

Fig. 2. Schematic representation of the orientational distribution function f 6) for three classes of condensed media that are composed of elongated molecules A, soHd phase, where /(0) is highly peaked about an angle (here, 0 = 0°) which is restricted by the lattice B, isotropic fluid, where aU. orientations are equally probable and C, Hquid crystal, where orientational order of the soHd has not melted completely.

The quantitative texture investigations (based on the orientation distribution function) al-... 

Pulsed deuteron NMR is described, which has recently been developed to become a powerftd tool for studying molectdar order and dynamics in solid polymers. In drawn fibres the complete orientational distribution function for the polymer chains can be determined from the analysis of deuteron NMR line shapes. By analyzing the line shapes of 2H absorption spectra and spectra obtained via solid echo and spin alignment, respectively, both type and timescale of rotational motions can be determined over an extraordinary wide range of characteristic frequencies, approximately 10 MHz to 1 Hz. In addition, motional heterogeneities can be detected and the resulting distribution of correlation times can directly be determined. 

As a prelude to the discussion it is necessary to consider the definition of orientation in terms of the Euler angles, and the definition ofan orientation distribution function in terms ofan expansion ofLegendre functions. These definitions set the scene for examining the information which can be obtained from different spectroscopic techniques. In this review, infra-red and Raman spectroscopy and nuclear magnetic resonance, will be considered. 

Deformation Schemes and Theoretical Consideration of Orientation Distribution Functions... 

The anisotropy of the liquid crystal phases also means that the orientational distribution function for the intermolecular vector is of value in characterising the structure of the phase [22]. The distribution is clearly a function of both the angle, made by the intermolecular vector with the director and the separation, r, between the two molecules [23]. However, a simpler way in which to investigate the distribution of the intermolecular vector is via the distance dependent order parameters Pl+(J") defined as the averages of the even Legendre polynomials, PL(cosj r)- As with the molecular orientational order parameters those of low rank namely Pj(r) and P (r), prove to be the most useful for investigating the phase structure [22]. 

We begin, however, with the singlet orientational distribution function which is shown for the three liquid crystal phases in Fig. 6. In each phase the distribution is peaked at cos of 1 showing that the preferred molecular orientation is parallel to the director. The form of the distribution function is well represented by the relatively simple function... 

History. Wilke [129] considers the case that different orders of a reflection are observed and that the orientation distribution can be analytically described by a Gaussian on the orientation sphere. He shows how the apparent increase of the integral breadth with the order of the reflection can be used to separate misorientation effects from size effects. Ruland [30-34] generalizes this concept. He considers various analytical orientation distribution functions [9,84,124] and deduces that the method can be used if only a single reflection is sufficiently extended in radial direction, as is frequently the case with the streak-shaped reflections of the anisotropic... 

Figure I. Schematic representation of statistical orientational distribution functions in isotropic (I) and nematic (N) potentials. Dashed curves depict adjusted distributions under electric field poling.

Liquid crystal display technology, 15 113 Liquid crystalline cellulose, 5 384-386 cellulose esters, 5 418 Liquid crystalline conducting polymers (LCCPs), 7 523-524 Liquid crystalline compounds, 15 118 central linkages found in, 15 103 Liquid crystalline materials, 15 81-120 applications of, 15 113-117 availability and safety of, 15 118 in biological systems, 15 111-113 blue phases of, 15 96 bond orientational order of, 15 85 columnar phase of, 15 96 lyotropic liquid crystals, 15 98-101 orientational distribution function and order parameter of, 15 82-85 polymer liquid crystals, 15 107-111 polymorphism in, 15 101-102 positional distribution function and order parameter of, 15 85 structure-property relations in,... 

Orientational distribution function, of liquid crystalline materials, 15 82-85 Orientation distribution function (ODF), 26 430, 431... 

Note 5 Even for molecules with cylindrical symmetry, does not provide a complete description of the orientational order. Such a description requires the singlet orientational distribution function which can be represented as an expansion in a basis of Legendre polynomials with L an even integer. The expansion coefficients are proportional to... 

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