用细长的棒来缓解50年之痒
Scratching a 50-year itch with elongated rods
摘要：
液晶弹性的经典奥森-弗兰克理论是基于实验验证的事实，即存在三种独立的扭曲模式，每种模式都有其相关的弹性常数。另一方面，可以说更多的第一性原理序参量为基础的朗道-德热讷理论只涉及两个独立的弹性模式。由此产生的“弹性常数问题”在理论家中引起了相当大的麻烦。在世纪之交的一系列论文中，福田和横山提出，这个问题的解决可以在自由能理想部分的非局部效应的适当处理中找到。他们使用了一种巧妙但技术复杂的技术，该技术基于半柔性聚合物的场论方法。在这里，我们重温他们的想法，但现在在更容易理解的刚性粒子密度泛函理论的框架。我们的工作恢复了他们对棒状粒子的主要结果，即一般来说，一个有序的非相互作用的棒的集合有三个独立的弹性常数与之相关，所有的尺度都是粒子长度的平方，并遵守不等式$K_2 < K_1 < K_3$。我们还考虑了盘状粒子的情况，然后发现符合预期的是$K_3 < K_1 < K_2$。
英文摘要：
The classical Oseen-Frank theory of liquid crystal elasticity is based on the experimentally verified fact that there are three independent modes of distortion, each with its associated elastic constant. On the other hand, the arguably more first-principles order parameter-based Landau-de Gennes theory only involves two independent elastic modes. The resulting 'elastic constants problem' has led to a considerable amount of vexation among theorists. In a series of papers at the turn of the century, Fukuda and Yokoyama suggested that the resolution of this problem could be found in the proper treatment of non-local effects in the ideal part of the free energy. They used an ingenious, but technically complex, technique based on a field-theoretic approach to semi-flexible polymers. Here we revisit their idea but now in the more accessible framework of density functional theory of rigid particles. Our work recovers their main results for rod-like particles, in that generically an ordered assembly of non-interacting rods has three independent elastic constants associated to it that all scale as the square of the length of the particles and obey the inequalities $K_2 < K_1 < K_3$. We also consider the case of disc-like particles, and then find in line with expectations that $K_3 < K_1 < K_2$.
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