近晶A相液晶的锯齿形
Sawtooth profile in smectic A liquid crystals
摘要：
我们研究了近晶A型液晶在$\mathbb{S}^2$ -值向量场上的de Gennes自由能，以理解在外加磁场存在下形成的锯齿形图案。我们确定了一个小的无量纲参数$ε$，并研究了当场强为$\mathcal{O}(ε^{-1})$阶时极小元的行为。在这个制度下，我们表明通过$Γ$收敛，一个人字形结构的导演连接两个最小状态的领域是有利的。我们还分析了Chen-Lubensky自由能，其中包括近晶序参量的二阶梯度，并获得了与de Gennes情况相同的一般行为。数值模拟示出了两种能量的人字形结构。
英文摘要：
We study the de Gennes free energy for smectic A liquid crystals over $\mathbb{S}^2$-valued vector fields to understand the chevron (zigzag) pattern formed in the presence of an applied magnetic field. We identify a small dimensionless parameter $ε$, and investigate the behaviors of the minimizers when the field strength is of order $\mathcal{O}(ε^{-1})$. In this regime, we show via $Γ$-convergence that a chevron structure where the director connects two minimum states of the sphere is favored. We also analyze the Chen-Lubensky free energy, which includes the second order gradient of the smectic order parameter, and obtain the same general behavior as for the de Gennes case. Numerical simulations illustrating the chevron structures for both energies are included.
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