Legendre polynomials, polymer orientation

Photoisomerization was studied from a purely photochemical point of view in which photo-orientation effects can be disregarded. While this feature can be true in low viscosity solutions where photo-induced molecular orientation can be overcome by molecular rotational diffusion, in polymeric environments, especially in thin solid film configurations, spontaneous molecular mobility can be strongly hindered and photo-orientation effects arc appreciable. The theory that coupled photoisomerization and photo-orientation processes was also recently developed, based on the formalism of Legendre Polynomials, and more recent further theoretical developments have helped quantify coupled photoisomerization and photo-orientation processes in films of polymer. 

From the knowledge of the order parameter from the measurement of the variation of the optical absorption spectrum due to poling, one can estimate the x parameter (Eq. (43)) intervening in the above developments. For poled polymers one can use also another alternative description of linear and nonlinear optical susceptibilities by expanding the orientation distribution function in the series of Legendre polynomials, where the expansion coefficients are order parameters ... 

In addition to the polar order produced by the electric field, Ep, axial order may be present due to either intrinsic ordering (e.g., in liquid-crystalline polymers) or mechanically induced ordering (e.g., uniaxial elongation). For rod-like molecules with dominant dipole and hyperpolarizability directed along the molecular axis, i.e. /u and only, in uniaxial systems, the orientational distribution function G (0) can be expanded in terms of Legendre polynomials... 

Legendre polynomial of (cos (p where

disorientation angle between the chain axis and the director. It can range from 0 for random orientation to 1 for perfect parallel orientation. Order parameters for nematic solutions of PpBA in N,N-dimethylacetamide + 3% (w/w) LiCl were determined in the composition range extending from the lower limit for stability of the pure mesophase, which is just below the critical concentration, to the solubility limit of the polymer. The experimental values of the order parameter, ranging from 0-76 to 0 83, were found to lie between the theoretical predictions of Doi and those of Flory and Ronca. ... 

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