Equilibrium Liquid Crystal Configurations: Energetics, Singularities and Applications

这项研究涉及液晶的数学描述，液晶是介于固体和液体之间的物质的中间状态。液晶具有重要的技术价值，因为它们是许多电子显示器的工作材料。单轴向列相液晶由棒状分子组成，这些分子在每个点上倾向于沿特定方向排列，就像火柴盒中的火柴一样。这个方向会受到电场的影响。方向是通过求解描述液晶力学的控制偏微分方程组来确定的。这些方程特别取决于对液晶的能量密度假定的形式。这项提议有不同的目的。第一个是检查何时可以将箭头始终如一地分配到首选方向。这样的箭头不是物理的，因为分子可以首尾相接地倒置，而不会改变它们的机械性质。然而，在一些广泛使用的理论中假设了这样一个箭头的存在，我们将研究这些理论的预测与其他没有指定这样的箭头的理论相比如何。第二个目的是调查不同于通常使用的能量密度的形式是否有助于描述液晶在缺陷处的行为，即在排列方向没有唯一定义的点上。与此相关的是如何在实验上确定能量密度的问题。第三个目标是在没有给首选方向分配非物理箭头的理论背景下，研究方程何时有平衡解，以及这些解是否有缺陷。最后，我们将把我们的工作应用于新型显示器件的原型研究，例如惠普实验室正在开发的后对准双稳态向列(PABN)器件。

The research concerns the mathematical description of liquid crystals, an intermediate state of matter between solids and liquids. Liquid crystals are of technological importance because they are the working material of many electronic displays. A uniaxial nematic liquid crystal consists of rod-like molecules which at each point tend to align in a particular direction, like matches in a matchbox. This direction can be influenced by electric fields. The direction is determined by solving the governing partial differential equations that describe the mechanics of the liquid crystal. These equations depend in particular on the form that is assumed for the energy density of the liquid crystal. The proposal has various aims. The first is to examine when an arrow can be consistently assigned to the preferred direction. Such an arrow is not physical, since the molecules can be inverted end-to-end without changing their mechanical properties. However the existence of such an arrow is assumed in some widely-used theories, and we will study how the predictions of such theories compare to other theories in which no such arrow is assigned. The second aim is to investigate whether forms of the energy density different from those usually used could help to describe the behaviour of liquid crystals at defects, that is at points where the alignment direction is not uniquely defined. Related to this is the question of how the energy density can be determined experimentally. The third aim is to study when the equations have equilibrium solutions, in the context of theories which do not assign an unphysical arrow to the preferred direction, and whether these solutions have defects. Finally we will apply our work to the study of prototype new display devices, such as the Post Aligned Bistable Nematic (PABN) Device being developed at Hewlett-Packard laboratories.