Symmetry and the Order Parameter

The identification of the appropriate order parameter for nematic liquid crystals is aided by a consideration of the observed structure and symmetry of the phase. As in any liquid, the molecules in the nematic phase have no translational order i.e., the centers of mass of the molecules are distributed at random throughout the volume of the liquid. Experiments of many varieties, however, do demonstrate that the nematic phase differs from ordinary liquids in that it is anisotropic. The symmetry, in fact, is cylindrical that is, there exists a unique axis along which the properties of the phase display one set of values, while another set of values is exhibited in all directions perpendicular to this axis. The symmetry axis is traditionally referred to as the director . The optical properties of nematics provide an example of how the cylindrical symmetry is manifest. For light passing parallel to the director, optical isotropy is observed, while for all directions perpendicular to the director, optical birefringence is observed. Rays polarized parallel to the director have a different index of refraction from those polarized perpendicular to the director. 

Many experiments demonstrate that the anisotropy of nematics arises because of the tendency of the rod-like molecules in the fluid to align their long axis parallel to the director. This is shown schematically in Fig. 1(a). The director is denoted by the symbol n the rod-like molecules are represented by the short lines. Note that at finite temperatures, the thermal motion of the molecules prevents perfect alignment with n the orientations of the molecules are in fact distributed in angle, but with the director as the most probable, or the most populated, direction. 

If we look more closely at the orientation of a single molecule with respect to the director, we find that the cylindrical symmetry of the phase requires just a single order parameter to describe the structure. In Fig. 1(b), we let the director lie along the a -axis of a fixed rectangular coordinate system. The orientation of the rod-like molecule can then be described using the three Eulerian angles shown. Because 

By tradition, the order parameter in any order-disorder problem is always taken such that it is unity in the perfectly ordered phase and vanishes for the completely disordered phase. Examination of the average values described above shows that the proper order parameter for the nematic liquid crystal is 

The stability of the nematic liquid crystal results from interactions between the constituent molecules. Without going into their nature at this time, it is clear that there must exist interactions that cause the molecules to prefer to align parallel to each other (and to the mean direction of alignment, the director). In the spirit of the mean field aproximation, we can attempt to mimic these intermolecular interactions with an effective single-molecule potential function V. 

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