Liquid-crystalline order in polymer systems: Basic models

The liquid crystalline (LC) order appears in a variety of polymer systems, such as solutions of rod-like molecules (DNA, TMV), solutions of semiflexible molecules (long fragments of DNA), block-copolymer melts, main-chain and side-chain liquid crystalline polymer melts etc. Many LC phases have been observed in these systems: the most common being: the nematic, cholesteric, smectic or lamellar, hexagonal, and double gyroid (in block copolymers) phases. We will discuss in detail some of them and give their quantitative description in terms of order parameters. We will also present various theoretical models used to study LC ordering in the systems. The models discussed in this paper are as follows: Onsager model and its extension within the Density Functional Theories (DFT), Khohlov-Semenov model for semiflexible polymers, Kratky-Porod model, combination of the Kratky-Porod model and Maier-Saupe model, self-consistent field theoretical model and finally the Landau-Ginzburg models and their connection with the Edwards model for polymer systems. We will also briefly discuss the elasticity of polymer systems in the case of nematic ordering. We present in a pedagogical manner the general ideas which are behind various models and give references to the papers which contain the technical details.