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. © 2022 Wiley Periodicals LLC.

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各向同性向列相变和液晶滴

液晶滴在物理学和应用中引起了极大的兴趣。严格的数学分析具有挑战性，因为问题涉及调和图（或一般的 Oseen-Frank 能量最小化器）、自由界面和拓扑缺陷（可能位于液滴内部或其表面）以及一些有趣的边界锚定条件（用于定向配置） 。本文基于埃里克森模型对液晶各向同性态和向列态之间的相变进行了研究，我们可以发现，当液滴的尺寸远大于弗兰克常数与液晶的比值时，表面张力，极小值的 Γ 收敛定理。当小参数 ε 时，该 Γ 极限实际上是各向同性区域和向列区域之间相变的尖锐界面极限，对应于过渡层宽度，趋于0。这个限制过程不仅提供了人们所期望的液滴形状的几何描述，而且令人惊讶的是，它还给出了液晶表面上液晶取向的锚定条件。液滴取决于材料常数。特别是，只要表面张力、弗兰克-埃里克森常数在合适的范围内，早期唯象模型中假设的垂直、切向甚至自由边界条件就会自然出现。© 2022 Wiley 期刊有限责任公司。令人惊讶的是，它还根据材料常数给出了液滴表面液晶取向的锚定条件。特别是，只要表面张力、弗兰克-埃里克森常数在合适的范围内，早期唯象模型中假设的垂直、切向甚至自由边界条件就会自然出现。© 2022 Wiley 期刊有限责任公司。令人惊讶的是，它还根据材料常数给出了液滴表面液晶取向的锚定条件。特别是，只要表面张力、弗兰克-埃里克森常数在合适的范围内，早期唯象模型中假设的垂直、切向甚至自由边界条件就会自然出现。© 2022 Wiley 期刊有限责任公司。