This article re-examines a classic question in liquid-crystal physics: What are the elastic modes of a nematic liquid crystal? The analysis uses a recent mathematical construction, which breaks the director gradient tensor into four distinct types of mathematical objects, representing splay, twist, bend, and a fourth deformation mode. With this construction, the Oseen-Frank free energy can be written as the sum of squares of the four modes, and saddle-splay can be regarded as bulk rather than surface elasticity. This interpretation leads to an alternative way to think about several previous results in liquid-crystal physics, including: (1) free energy balance between cholesteric and blue phases, (2) director deformations in hybrid-aligned nematic cells, (3) spontaneous twist of achiral liquid crystals confined in a torus or cylinder, and (4) curvature of smectic layers.

本文重新审视了液晶物理学中的一个经典问题：向列液晶的弹性模式是什么？该分析使用了最新的数学构造，该构造将指向矢梯度张量分为四种不同类型的数学对象，分别表示张开，扭曲，弯曲和第四变形模式。通过这种构造，可以将Oseen-Frank自由能表示为这四个模的平方和，并且可以将鞍形展开看作是体积而不是表面弹性。这种解释导致了另一种思考液晶物理学先前结果的方法，其中包括：（1）胆甾相和蓝相之间的自由能平衡，（2）杂合排列向列型细胞中的指向矢变形，（3）自发扭曲限制在圆环或圆柱体中的非手性液晶，