From Microscopic Theory to Macroscopic Theory: a Systematic Study on Modeling for Liquid Crystals

In this paper, we propose a systematic way of liquid crystal modeling to build connections between microscopic theory and macroscopic theory. In the first part, we propose a new Q-tensor model based on Onsager's molecular theory for liquid crystals. The Oseen-Frank theory can be recovered from the derived Q-tensor theory by making a uniaxial assumption, and the coefficients in the Oseen-Frank model can be examined. In addition, the smectic-A phase can be characterized by the derived macroscopic model. In the second part, we derive a new dynamic Q-tensor model from Doi's kinetic theory by the Bingham closure, which obeys the energy dissipation law. Moreover, the Ericksen-Leslie system can also be derived from new Q-tensor system by making an expansion near the local equilibrium.