The mathematical description of molecular orientation

In general it is not meaningful to talk about the orientation of a polymer molecule as a whole it is therefore useful at the outset to refer to the orientation of structural units. Such a unit may be a crystallite, a regular 

The polymer sample is assumed to have at least orthotropic symmetry, i.e. it contains three mutually perpendicular directions such that if it is rotated through 180° about any one of these directions its macroscopic properties are unchanged. Axes OZ1Z2A3 are chosen parallel to these three symmetry directions of the sample. The orientation of a particular structural unit can then be specified in terms of three Euler angles, 6, / and jr, as shown in fig. 10.7. 

The discussion of the most general form of biaxial orientation involves fairly complicated mathematics. Further discussion in this section and in the sections on the characterisation of orientation is therefore largely restricted to a special simple type of distribution of orientations of the structural units. This distribution is the simplest type of uniaxial orientation, for which the following conditions apply. 

For the simplest type of uniaxial orientation the distribution function reduces to N(6), where N(6)do) is the fraction of units for which OX3 lies within any small solid angle dco at angle 6 to 0X3. By characterising the distribution is meant finding out as much as possible about N 6) for the various types of structural unit that may be present in the polymer. 

Most experimental methods give only limited information about N 6). Many give only (cos 0), the average value of co 6, or give only (cos 0) 

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