The theory of liquid-crystal alignment

The most important theoretical models for attempting to accoimt for liquid-crystal alignment are the lattice model, introduced by Flory in 1956, and the mean-field model introduced about 1960 by Maier and Saupe. Of these, the former has proved to be more readily applicable to polymers. It is based on the rather obvious idea that, as the number of rigid impenetrable rods in a given volume is increased, they must eventually become aligned approximately parallel to each other, at least in local regions, and it attempts to predict the concentration at which this will tend to happen. In this model there is no specific interaction between the rods except for the short-range repulsive force that corresponds to their mutual impenetrability. 

The lattice model concerns itself with finding the number of ways that a given number of rigid rods can be arranged within a given volume. In order to make the problem mathematically tractable the rods are idealised to have a square cross-section and the length of each rod is assumed to be an integral multiple X of its cross-section, so that x is the aspect ratio of the rod. The rod can thus be considered to consist of x units, each of which is a cube. 

The volume in which the rods are placed is considered to be made up of cubic cells, or sites, of the same size as these units, arranged on a lattice with one edge parallel to the director of the liquid-crystal arrangement that arises when the rods are packed sufficiently densely. A rod at an angle f to the director is then represented in the simplest case by y sections each of xjy (assumed integral) units and each section is assumed to occupy xjy cells of the lattice that are adjacent to each other in the direction parallel to the director, as shown in fig. 12.21. The problem is to calculate how the number of ways that the rods can be placed on the lattice varies with the degree 

21 (a) A rod oriented at angle f to the director nn of a domain and (b) the representation of the rod as y sequences of cubes each parallel to nn. (Re-drawn with permission from Taylor and Francis Ltd.) 

The simplest assumption that can be made about the distribution of orientations of the rods is that all orientations within a cone of semiangle with its axis parallel to the director are equally likely. For lAmax small it is easy to show (see problem 12.7) that Aloriem is equal to where C is a constant. The entropy S of the system is equal to k In N, so that 

---

Starting from equation (12.16a), show that according to the version of the theory of liquid-crystal alignment presented in section 12.4.4 the minimum aspect ratio for the rods of a material that exhibits thermotropic liquid crystallinity is 5.44. 

---