Forest M Gregory, Zhou Ruhai, Wang Qi
Department of Mathematics, University of North Carolina, Chapel Hill, North Carolina 27599, USA.
Phys Rev E Stat Nonlin Soft Matter Phys. 2002 Sep;66(3 Pt 1):031712. doi: 10.1103/PhysRevE.66.031712. Epub 2002 Sep 30.
The Doi theory has successfully modeled the monodomain shear flow problem for rigid, rodlike nematic polymers. Numerical simulations of the Smoluchowski equation for the orientational probability distribution function (PDF) predict monodomain attractors in regions of nematic concentration N and shear rate gamma. Theoretical work has focused on approximate constructions of PDF solutions in linear flow regimes. Here we develop a collection of simple observations, expressed by symmetries of the Smoluchowski equation, which imply global properties that all PDF solutions must obey. The well-known orientational degeneracy of quiescent nematics is a continuous O(3) symmetry. In simple shear, a discrete reflection symmetry survives that is evident in recent numerical simulations and implies bistability of out-of-plane attractors; and rodlike and discotic nematic liquids of reciprocal aspect ratio respond identically up to a fixed rotation of the PDF. Finally, we show the orientational effects due to varying molecular aspect ratio in any linear flow are equivalent to varying the straining component of the flow field.
多伊理论已成功地对刚性棒状向列型聚合物的单畴剪切流问题进行了建模。针对取向概率分布函数（PDF）的斯莫卢霍夫斯基方程的数值模拟预测，在向列相浓度N和剪切速率γ的区域中存在单畴吸引子。理论工作主要集中在线性流动状态下PDF解的近似构造上。在此，我们通过斯莫卢霍夫斯基方程的对称性得出了一系列简单的结论，这些结论暗示了所有PDF解都必须遵循的全局性质。静态向列相中众所周知的取向简并是一种连续的O(3)对称性。在简单剪切中，一种离散的反射对称性仍然存在，这在最近的数值模拟中很明显，并且意味着面外吸引子的双稳性；相互纵横比的棒状和盘状向列型液体在PDF的固定旋转之前表现相同。最后，我们表明，在任何线性流动中，由于分子纵横比变化而产生的取向效应等同于改变流场的应变分量。
