Isotropic-Nematic Phase Transition and Liquid Crystal Droplets

Liquid crystal droplets are of great interest from physics and applications. Rigorous mathematical analysis is challenging as the problem involves harmonic maps (or Oseen-Frank energy minimizers in general), free interfaces, and topological defects which could be either inside the droplet or on its surface along with some intriguing boundary anchoring conditions for the orientation configurations. In this paper, through a study of the phase transition between the isotropic and nematic states of liquid crystal based on the Ericksen model, we can show, when the size of a droplet is much larger in comparison with the ratio of the Frank constants to the surface tension, a Gamma-convergence theorem for minimizers. This Gamma-limit is in fact the sharp interface limit for the phase transition between the isotropic and nematic regions when the small parameter epsilon, corresponding to the transition layer width, goes to 0. This limiting process not only provides a geometric description of the shape of the droplet as one would expect, and surprisingly it also gives the anchoring conditions for the orientations of liquid crystals on the surface of the droplet depending on material constants. In particular, homeotropic, tangential, and even free boundary conditions as assumed in earlier phenomenological modelings arise naturally, provided the surface tension, Frank-Ericksen constants are in suitable ranges. (c) 2022 Wiley Periodicals LLC.

Automated summary

This paper focuses on the phase transition between isotropic and nematic states of liquid crystal droplets based on the Ericksen model. It demonstrates the existence of a Gamma-convergence theorem for the geometric description of droplet shape and anchoring conditions for liquid crystal orientations. The findings have significant implications for the physics and applications of liquid crystal droplets.