Statistical mechanics of defects in polymer liquid crystals

We develop a continuum theory for the statistical mechanics of thermally activated point defects in the nematic and hexagonal phases of polymer liquid crystals. In the nematic phase, there are elementary splay defects (chain ends and hairpins), and in the hexagonal phase, there are both splay defects and twist defects. In the nematic phase, splay defects are free in the limit of large separation; i.e., their binding energy is finite. By contrast, in the hexagonal phase, both types of defects are bound in $+-$ pairs. We derive expressions for two correlation functions, the structure factor and the director fluctuation spectrum, in the presence of defects, and we use these correlation functions to define macroscopic Frank constants and elastic moduli. In the nematic phase, the presence of ionized splay defects causes the macroscopic splay constant $\hat{K}_1$ to be finite. It is large and strongly temperature-dependent in the low-temperature regime, but smaller and temperature-independent in the higher-temperature Debye-Hückel regime. By contrast, in the hexagonal phase, the macroscopic splay and twist constants $\hat{K}_1$ and $\hat{K}_2$ are infinite, just as in harmonic theory. These effects should be observable in x-ray and light-scattering experiments on polymer liquid crystals