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 (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 -convergence theorem for minimizers. This -limit is in fact the sharp interface limit for the phase transition between the isotropic and nematic regions when the small parameter , corresponding to the transition layer width, goes to . 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.

Topological transformations of a nematic drop

“…Note that the energy in Eq. 14 can be thought as a Landau–de Gennes analog of Ericksen’s model for nematic liquid crystals with variable degree of orientation ( 38 ), also considered recently in ( 39 ). Equation 14 reduces to the energy functional in ( 39 ), with coefficients that may depend on the degree of orientation s as long as Eq.…”

Phase Transition of Parabolic Ginzburg--Landau Equation with Potentials of High-Dimensional Wells

“…A generalization to matrix-valued case has been done by Laux-Liu [21] to study the isotropic-nematic transition in Landau-De Gennes model of liquid crystals, which essentially corresponds to the case when m + = RP 2 and m − = 0. More recently, in [26] the author used these methods, together with those developed recently by Lin-Wang [23], to attack the convergence problem of an anisotropic 2D Ginzburg-Landau model. In particular, he derived some delicate convergence results of the level sets of the solutions, which are crucial to obtain anchoring boundary conditions of the limiting solutions on the moving interface.…”

Study on concentric configuration of nematic liquid crystal droplet by Landau-de Gennes theory

“…Among the most popular are the surface mesh method [3] that discretizes the free boundary with a finite-element mesh known as the interface segregating different phases, and the diffuse-interface method [61] that assumes a continuous variation of the phase within the interfacial regions described by a smooth field (spacedependent function) known as the phase field. In recent decades, there has been a multitude of research in free-boundary NLCs with fairly diverse modelling approaches, both in free boundary and in NLC [28,66,36,16,63,30,41,11,20,2,37]. The common practice is to design energy functionals with respect to the shape and the NLC order parameter, and find the stable configuration with energy minimization methods.…”