Abstract

Non-axisymmetric rigid molecules have proved to exhibit rich liquid-crystalline phases that are barely understood theoretically, especially for the connection between the molecular architecture and macroscopic phase behaviors. A promising route is to derive macroscopic theory from molecular theory, which has shown its success in the study of rod-like and bent-core molecules.

We discuss several essential ingredients in this route to construct free energy for general rigid molecules: molecular symmetry; order parameters to describe mesoscopic states; expansion of interaction kernels; entropy. The molecular symmetry is characterized by a point group. We follow three steps below to arrive at the macroscopic free energy. 

1) From the proper rotations in the point group, we identify, with explicit expressions, the invariant tensors, which are nonvanishing when averaged by the density function. It is from the averages of these invariant tensors that the order parameters are chosen.

2) By expanding the interaction kernels into coupling of tensors, the interaction free energy can be constructed. The improper rotations, although having no effect on order parameters, impose extra conditions on the expansion of interaction kernels. 

3) The entropy term can be converted into a function of tensor order parameters by finding the maximum entropy state. We further propose a simple function as an approximation that maintains the essential properties of the original entropy, while avoiding the integrals in the original entropy.