Isotropic-Nematic Phase Transition and Liquid Crystal Droplets

Abstract

Liquid crystal droplets are of great interest from physics and applications. Rigorous mathematical analysis is challenging as the problem involves harmonic maps (and in general the Oseen-Frank model), 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 droplet is much larger in comparison with the ratio of the Frank constants to the surface tension, a Γ-convergence theorem for minimizers. This Γ-limit is in fact the sharp interface limit for the phase transition between the isotropic and nematic regions when the small parameter ε, corresponding to the transition layer width, goes to zero. 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 that the surface tension, Frank and Ericksen constants are in suitable ranges.

Nematic-Isotropic phase transition in Liquid crystals: a variational derivation of effective geometric motions

“…It is worthy to mention that recently Golovaty et al [15,14] studied a model problem based on highly disparate elastic constants. Most Recently, Lin-Wang [22] studied isotropic-nematic transitions based on an anisotropic Ericksen's model. (2.16)…”

Phase transition of anisotropic Ginzburg--Landau equation

“…The proof here is inspired by the density argument in [23]. It is worth mentioning that the use of density argument to prove weakly lower-semincontinuity like (5.20) was first employed in [12].…”

“…On the other hand, Theorem 1.1 can be considered as a dynamical version of a recent work by Golovaty et al [13]. Concerning the method we use, at least in obtaining the anchoring boundary condition (1.17c) which is one of the main novelty in this work, we are inspired by a recent work of Lin-Wang [23]. There the authors studied isotropic-nematic transitions based on an anisotropic Ericksen's model.…”

Two dimensional liquid crystal droplet problem with tangential boundary condition

“…It is worth to point out that the Problems A, B and C thus presented were all derived from a phenomenological theory, see [24] and [6]. In a recent work [21], it is shown that one can rigorously establish these model problems from a general theories of Ericksen (for liquid crystals with variable degree of orientations [7]) or from the de Gennes-Landau model of liquid crystals [8] in suitable physical regimes.…”