论文标题
各向同性纽马的相变和液晶液滴
Isotropic-Nematic Phase Transition and Liquid Crystal Droplets
论文作者
论文摘要
液晶液滴引起了物理和应用的极大兴趣。严格的数学分析是具有挑战性的，因为问题涉及谐波图（通常是Oseen-Frank模型），自由接口和拓扑缺陷，这些缺陷可能位于液滴内部或其表面上，以及一些有趣的边界锚定条件，以实现方向配置。在本文中，通过研究基于埃里克森模型的液晶的各向同性和列态之间的相变，我们可以证明，与弗兰克常数与表面张力的比率相比，液滴的大小要大得多，而$γ$ - convergence convergence convergence convergence for Minimimizizers。实际上，当与过渡层宽度相对应的小参数$ \ varepsilon $时，这种$γ$ limit实际上是各向同性和列中区域之间相变的急剧界面限制。这个限制过程不仅可以像人们期望的那样对液滴的形状进行几何描述，而且令人惊讶的是，它还为液晶表面上液晶方向的锚定条件取决于材料常数。特别是，前提是早期现象学模型中假定的同源，切向甚至自由边界条件自然出现，前提是表面张力，弗兰克和埃里克森常数处于合适的范围。
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 $\varepsilon$, 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.
