Q-Tensor models of defect dynamics in pure and doped liquid crystals

液晶以其在现代平板屏幕中的应用而闻名，但其他技术上重要的应用包括色敏温度计，温度敏感涂料，光电设备如快门和传感器，材料如超轻防弹衣，以及最近的微流体芯片上的设备。它们也是非常重要的材料，因为它们表现出一些类似液体的性质（它们流动）和一些类似固体的性质（它们的性质取决于方向）。尽管在过去的40年里进行了大量的理论工作，但新的应用仍然需要对液晶模型有更基本的数学理解，以便快速预测器件。液晶理论中一个令人头痛的问题就是所谓的“缺陷”问题。液晶通常表现出一个优选的方向，但这个优选的方向可以从一个地方到另一个地方改变。缺陷是优选方向在其附近溶解的线或点。当一个人梳理头发时可以看到一个类比：在一个球体上均匀地梳理头发是不可能的;在图案中总是会有一些洞。宇宙学家也使用液晶缺陷作为早期宇宙的模型。缺陷的内部图案还为各种类型的液晶提供了引人注目的和美丽的特征光学特征。 描述它们给液晶的数学理论带来了很大的困难。标准理论（由苏格兰数学家弗兰克·莱斯利（Frank Leslie）提出）只是忽略了它们，而替代方法（“Q张量”理论）通常在优先方向变化缓慢的区域计算效率低下。在后一种情况下，即使存在液晶性质的计算模型，也可能难以提供物理图像。最终的结果是，新器件的性能无法可靠和快速地预测。 我们的计画将提供一个更复杂的有缺陷液晶的运动模型。我们将建立在我们最近发展起来的一种新的数学近似法的基础上。这种方法，我们标记为无缺陷的Q张量近似（DFQTA），使用莱斯利和Q张量方法的最佳功能。在非缺陷液晶中，这种方法导致了解决一些基准问题所需的计算时间的显着改善（因子100的顺序），而没有任何精度损失。我们的新模型将处理缺陷运动修补在一起的解决方案，在无缺陷和有缺陷的地区。将使用DFQTA处理无缺陷区域。缺陷本身将被从问题中剔除，并使用一系列近似来单独处理，这些近似是由于包含它们的区域很小而成为可能的。将使用渐近分析对两个区域进行匹配。我们的方法将结合联合收割机的工程师（谁往往淡化数学分析的必要性）的算法要求，与复杂的数学近似工具，以获得一个准确的和计算效率高的模型。一个特定的工程应用涉及悬浮在液晶溶液中的纳米颗粒。纳米颗粒的尺寸为微米或更小，即使在非常低的浓度（远小于1%体积）下，它们也可以显着改变溶剂的性质。这些粒子正被用于新一代的设备，包括液晶和其他地方。但在液晶中，纳米粒子被缺陷吸引。纳米粒子的运动和光学特征似乎都受到捕获它们的缺陷的控制。我们的新理论将使这样的系统被成功地处理在计算上可访问的时间尺度。

Liquid crystals are best known for their applications in modern flat screens, but other technologically important applications include colour-sensitive thermometers, temperature-sensitive paints, optoelectronic equipment such as shutters and sensors, materials such as ultra-light body armour, as well as very recently in microfluidic devices-on-a-chip. They are also materials of great fundamental interest because they exhibit some liquid-like properties (they flow) and some solid-like properties (their properties depend on direction). Despite intense theoretical work over the last forty years, it is still the case that new applications require a more fundamental mathematical understanding of liquid crystalline models in order that rapid device prediction can be made. A major headache in liquid crystal theory is the so-called "defect" problem. Liquid crystals usually exhibit a preferred direction, but this preferred direction can change from place to place. The defects are lines or points near which the preferred direction dissolves. An analogy can be seen when one combs one's hair: it is impossible to comb hair uniformly over a sphere; there will always be some holes in the pattern. Cosmologists have also use the liquid crystal defects as models of the early universe. The internal patterns of the defects also provide the dramatic and beautiful characteristic optical signatures for individual types of liquid crystal. Describing them has caused much difficulty in mathematical theories of liquid crystals. The standard theory (due to the Scottish mathematician Frank Leslie) just omits them, while alternative approaches ("the Q-tensor" theory) often are computationally inefficient in regions in which the preferred direction changes only slowly. In the latter case, even when there is a computational model for the liquid crystal properties, it may be difficult to provide a physical picture. The end result is that properties of new devices cannot be predicted reliably and quickly. Our project will provide a more sophisticated model of the motion of defected nematic liquid crystals. We shall build on a new mathematical approximation which we have recently developed. This approach, which we have labelled the Defect-free Q-tensor approximation (DFQTA), uses the best features of the Leslie and Q-tensor approaches. In non-defected liquid crystals, this approach has led to dramatic improvements (of the order of a factor of 100) in the computational time required to solve some benchmark problems, without any loss of accuracy. Our new model will treat defect motion by patching together solutions in the defect-free and defected regions. The defect-free region will be treated using DFQTA. The defect itself will be cut out of the problem, and treated separately using a range of approximations made possible by the fact, amongst others, that the regions that contains them are small. The two regions will be matched using asymptotic analysis. Our approach will combine the algorithmic requirements of engineers (who often downplay the necessity for mathematical analysis), with sophisticated mathematical approximation tools to obtain an accurate and computationally efficient model. One particular engineering application concerns nanoparticles suspended in liquid crystal solutions. Nanoparticles are of micron size or smaller, and even at very low concentrations (much less than 1% by volume), they can change the properties of the solvent significantly. These particles are being used in a new generation of devices, both in liquid crystals and elsewhere. But in liquid crystals the nanoparticles are attracted to defects. Both the motion and the optical signature of the nanoparticles seem to be governed by defects which trap them. Our new theory will enable such systems to be treated successfully on computationally accessible time-scales.