TY - JOUR

T1 - Isotropic-Nematic Phase Transition and Liquid Crystal Droplets

AU - Lin, Fanghua

AU - Wang, Changyou

N1 - Publisher Copyright:
© 2022 Wiley Periodicals LLC.

PY - 2023/9

Y1 - 2023/9

N2 - 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 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 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.

AB - 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 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 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.

UR - http://www.scopus.com/inward/record.url?scp=85128231102&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85128231102&partnerID=8YFLogxK

U2 - 10.1002/cpa.22050

DO - 10.1002/cpa.22050

M3 - Article

AN - SCOPUS:85128231102

SN - 0010-3640

VL - 76

SP - 1728

EP - 1792

JO - Communications on Pure and Applied Mathematics

JF - Communications on Pure and Applied Mathematics

IS - 9

ER -