Collaborative Research: Morphogenesis of First-Order Phase Transitions in Polar and Apolar Nematic Liquid Crystals

液晶是由棒状或圆盘状分子组成的，这些分子使这些材料在某些方面表现得像液体，而在另一些方面又表现得像固体。这个项目是对分析在液晶相变中观察到的结构的基本问题的探索。原则上，该项目将阐明不同阶段之间的界面形状是如何由弹性力和表面张力的平衡决定的。这些结果将在数学、物理和材料科学中具有重要意义，因为共存相中各种形态的发展是一种普遍存在的自然现象。所获得的知识将带来新的和强大的工具来实验控制这些形态，导致新的和令人兴奋的应用。这个跨学科的项目进一步提供了机会来教育学生在液晶的数学和物理。该项目将侧重于表征和描述软凝析油的整体形状和内部结构，这些软凝析油代表由具有不同弹性模量和各向异性表面张力的体弹性形成的定向有序相的核。提出的工作将考虑广泛的液晶，包括由不可定向的指向线场描述并经历过渡到部分位置有序相的向列和胆甾液晶，以及新发现的由指向和向量值电极化描述的铁电向列液晶。数学重点是分析二维和三维矢量和张量值场上定义的能量的平稳变分问题以及相关的梯度流，以便描述液晶的平衡构型和界面在大量相变中的动力学。校长新奇的工作是,它将考虑奇异液晶的极限能量施加严重的几何刚度容许类的函数。分析的主要挑战是由于目标空间的高维数和需要考虑导致高度非线性约束的奇异渐近极限。分析/实验相结合的方法将是必不可少的，因为它将为分析预测的成功提供明确的检查；这些预测反过来将指导实验。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Liquid crystals are composed of rod-like or disc-like molecules that enable these materials to behave in some ways like a liquid and in others like a solid. This project is an exploration of the fundamental problem of analyzing the configurations observed in phase transitions in liquid crystals. Principally, the project will elucidate how the shape of the interface between different phases is determined by a balance of elastic forces and surface tension. The results will be of importance in mathematics, physics, and materials science since development of various morphologies in coexisting phases is a ubiquitous natural phenomenon. The acquired knowledge will bring about new and powerful tools to experimentally control these morphologies, leading to new and exciting applications. This interdisciplinary program further provides opportunities to educate students in the mathematics and physics of liquid crystals. The project will focus on the characterization and description of the overall shape and internal structure of soft condensates representing nuclei of the orientationally ordered phases shaped by bulk elasticity with disparate elastic moduli and anisotropic surface tension. The proposed work will consider a broad spectrum of liquid crystals, embracing nematic and cholesteric liquid crystals that are described by non-orientable director line fields and experience transitions into partially positionally ordered phases, and newly discovered ferroelectric nematic liquid crystals that are described by the director and a vector-valued electric polarization. The mathematical emphasis is on the analysis of stationary variational problems for energies defined over vector- and tensor-valued fields in two and three dimensions and associated gradient flows, so as to describe the equilibrium configurations of liquid crystals and dynamics of interfaces in a plethora of their phase transitions. The principal novelty of the proposed work is that it will consider singular limits of liquid crystal energies that impose severe geometric rigidity on admissible classes of functions. The primary challenges to the analysis arise due to high dimensionality of a target space and the need to consider singular asymptotic limits that result in highly nonlinear constraints. The combined analytical/experimental approach will be essential in that it will provide a clear check of success for the analytical predictions; these predictions, in turn, will guide the experiments.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.