Nematic excluded volume effects

It is obvious that in this approximation we do take into account some short-range steric correlations between rigid molecules. We note that these correlations contribute not only to the internal energy but also to the entropy of the nematic. Excluded volume effects restrict the molecular mo-... 

On going from the isotropic to the anisotropic LC state, the orientation-dependent attractive interactions come into play [125,126] while the steric interactions (the excluded-volume effect) between the mesogenic rods are relaxed [127]. In the LC state, all molecules are required to take an asymmetric shape. Accordingly, chain segments adopt a unique conformer distribution called a nematic conformation [26,93-95,102,103,105]. The Y-Vsp relation determined in the aforementioned studies may be used to examine the mean-field potentials effective in nematic as well as isotropic hquids. After Frank [128] and HUdebrand and Scott [2,129], the intermolecular interaction potentials such as... 

As discussed in the previous subsection, it is important to take into account the excluded volume effects even in a simple mean-field theory based on an anisotropic attraction interaction. The excluded volume effects are determined by hard-core repulsion that does not allow molecules to penetrate each other. It is interesting to note that by doing so we already go beyond the formal mean-field approximation. Indeed, with excluded volume effects the internal energy of the nematic phase can be written as... 

The computer simulation studies of Frenkel et al. [45] indicate that the excluded volume effects for molecular hard cores must play an important role in the stabilization of the smectic phase. In particular, it has been shown that hard spherocylinders, interacting only via hard-core repulsion, can form nematic, smectic A, and columnar phases. 

Figure 10.16 shows the phase diagram of the mean field theory calculations [52]. The vertical axis is temperature and the horizontal axis represents the volume fraction of rod-shaped molecules. The attractive interaction parameter C12 between the liquid crystal and the rod-shaped molecules is 0.3 (Fig. 10.16a) and 0.4 (Fig. 10.16b). The solid line is the coexistence curve and the dotted line shows the NI phase transition. Because of the excluded volume effect, there is a phase separation between an isotropic and a nematic phase (I-i-No) on the high-temperature side in Fig. 10.16. With the decrease of temperature, the concentration difference in the coexistence region of I -1- No is reduced, and triple point of No -1-1 -I-No appears in Fig. 10.16a. On the low-temperature side of the triple point, the nematic phase splits into two nematic phases (No and No ) with different concentrations of liquid crystal molecules.

Figure 10.17 shows schematically the biaxial nematic phase Njb and the uniaxial nematic phase Ni [55]. The orientatiOTi of the rodlike molecules in a plane perpendicular to the direction of the major director (n) is random in the uniaxial N1 phase. However, with increasing cmicentration of the rodlike molecules, the rodlike molecides may orient by mutual attraction and the excluded volume effect in a second direction with the minor director (m) as shown in the figure on the right. Thus, the novel biaxial nematic phase Nib may be possible. In the same fashion, in the N2 phase, it may be possible to have a biaxial nematic N2b phase, where the additional ordering of liquid crystals appears in the minor director (in) perpendicular to the major director (h) of rodlike molecules. 

PHIC extrapolate to roughly 6.7, which is close to the value predicted by the Flory theory in the melt. This suggests that even for bulk HPC, the nematic-isotropic transition is driven primarily by excluded-volume, or packing, effects and only secondarily by anisotropic van der Waals interactions. The temperature dependence of the axial ratio could be incorporated into the Maier-Saupe potential by suitably adjusting the temperature dependence of the coefficient 17ms-... 

Orientation in crosslinked elastomers primarily reflects the configurational entropy and intramolecular conformational energy of the chains. However, as first shown by deuterium NMR experiments on silicone rubber (Deloche and Samulski, 1981 Sotta et al., 1987), unattached probe molecules and chains become oriented by virtue of their presence in a deformed network. This nematic coupling effect is brought about intermolecular interactions (excluded volume interactions and anisotropic forces) which can cause nematic coupling (Zemel and Roland, 1992a Tassin et al., 1990). The orientation is only locally effective, so it makes a negligible conttibution to the stress (Doi and Watanabe, 1991), and the chains retain their isotropic dimensions (Sotta et al., 1987). 

The mesomorphic behavior of a solution of rigid rods has been studied by several authors.i-iZ-Ll In particular, for long rods of length b and radius a, OnsagerL has shown that the effective excluded volume per rod in the isotropic phase is of order b a. The actual volume occupied per rod is ira b. A lyotropic transition to a nematic phase occurs at a rod concentration O given by... 

The nematic phase being the liquid crystal of highest symmetry, its condensation from the isotropic liquid should be the simplest to describe. Indeed, molecular theories convincingly explain the natural onset of nematic ordering in a population of anisotropic molecules with excluded volume interaction (Onsager) or in mean field theory (Maier-Saupe). Regarding the effect of symmetry on the isotropic to nematic (I-N) phase transition, the phenomenological approach is useful too. 

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