Theoretical Aspects of Liquid Crystals and Liquid Crystalline Polymers

Liquid crystallinity refers to an intermediate state of matter where the molecules exhibit a degree of order that is between that of ordinary liquids and solids. Typically, these molecules have an elongated shape and an intrinsic tendency toward alignment. Yet, the inter-molecular forces are not so strong as to bind them into a regular lattice as in crystalline solids. Thus, the orientational order among the molecules makes the material anisotropic and ‘‘crystalline,’’ while the lack of strong positional order allows the material to flow like ordinary fluids. This liquid–solid duality gives rise to much of the ‘‘anomalous’’ behavior of liquid crystalline materials. When understood, however, their unique dynamics can be harnessed to produce high-performance materials with unique properties. In this entry, we review the state of the art in theoretical modeling and computation of the flow and rheology of low-molecular-weight liquid crystals (LCs) and liquid crystalline polymers (LCPs). The latter can be viewed as macromolecular liquid crystals, and the significance of the molecular weight will be made clear shortly. We restrict our scope to nematics, one of the several types of LCs with no positional order altogether; they seem to have the most important applications and have been the focus of theoretical efforts. Owing to the long-standing academic and industrial interest in these materials, their dynamics have been summarized in several reviews, e.g., Refs. While overlapping those in certain aspects, we emphasize constitutive modeling and numerical simulation of defects and texture. The latter nicely demonstrates the interplay between macroscopic flow and mesoscopic orientation, a hallmark of liquid crystalline dynamics. Due to space limitation, we will attempt to construct a coherent framework instead of an exhaustive literature review. Based on their conceptual origins, we discuss continuum theories and molecular theories separately. In addition, a theory for liquid crystalline materials has been proposed based on nonequilibrium thermodynamics. As the theoretical framework and mathematical representation are quite independent of the other theories, we will not discuss this theory but refer the reader to Beris and Edwards’ monograph. Because of similar molecular attributes, LCs and LCPs share enough common features to be discussed in this single entry. A significant difference is that the large LCP molecules have a much longer relaxation time. Thus, the molecular conformation of LCPs is easily disturbed by flow and deformation. Rheologically, therefore, the material exhibits ‘‘molecular viscoelasticity’’ as macromolecular fluids normally do. Small-molecule LCs relax so fast that their molecular configuration remains almost always at equilibrium; there is no viscoelasticity. Both LCs and LCPs resist spatial distortion to their orientational pattern. This tendency is known as ‘‘distortional elasticity. ’’ The continuum theories were originally developed for small-molecule LCs, capturing distortional elasticity but not molecular viscoelasticity. Molecular theories, on the other hand, have evolved to contain both ingredients. Not surprisingly, a properly constructed molecular theory should reduce to a continuum theory in the limit of vanishing molecular relaxation time.