Interaction Potential and Partition Function

Consider the simplest case, namely, the nematic phase consisting of uniaxial rodlike molecules. Generally, the intermolecular interaction again consists of the repulsive and attractive parts but both of them become anisotropic. The potential of pair molecular interaction can be written in the following general form  

Note that Euler angle F is not considered due to the rod-like form of a molecule the other angles are shown in Fig. 6.19a. Vector ri2 connects the gravity centers of rods. If the particular form of is known, it can be used for calculation of the 

Again this function includes the repulsive multiplier Zhr and the attractive part. The repulsive part may be considered as a reference for calculation of the thermal average necessary for the attractive part. To find Zhr, we may operate with an excluded volume, although even for hard rods (suffix HR) it is very difficult to calculate it. The total procedure is enormously complicated because, even for Zhr known, it requires multiple averaging over (a) all orientations of molecule 1, (b) all orientations of molecule 2, and (c) all distances ri2- 

Below we shall consider two extreme cases, long hard rods without attraction and rod-like molecules without repulsion. The first approach (by Onsager) may be applied to very long molecules like tobacco mosaic viruses and calls for hard mathematics [20]. We shall discuss it very schematically in the next section. The second one (by Maier and Saupe [21 ]) appeared to be simpler but very powerful and can be applied to many typical nematic liquid crystals. We shall consider it in the subsequent Section 6.7.3. 

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