Theory of binary mixtures of a rodlike polymer and a liquid crystal

We present a mean field theory to describe phase separations in mixtures of a low molecular-weight-liquid crystalline (LC) molecule and a rigid-rodlike polymer (rod) such as carbon nanotubes (CNTs) and LC polymers. By taking into account two orientational order parameters of the rod and the LC, we find three nematic phases (N(0),N(1),N(2)) on the temperature-concentration plane, depending on the attractive or repulsive interactions between the rod and the LC. We discuss the phase behavior of the systems where the rod and the LC are oriented to be parallel or perpendicular with each other and find that the phase diagrams of the parallel alignment are different from that of the perpendicular one. We predict the appearances of a lower critical solution temperature (LCST) and an upper one, a tricritical point, a critical point, and the first- and the second-order nematic-isotropic phase transitions. The theory can qualitatively describe the phase diagram with a LCST observed in a rod/LC mixture. We also predict a variety of phase separations in CNT/LC mixtures.