Modelling the capillary locking of point defects in nematic liquid crystals

DEDICATED TO JERRY ERIKSEN: It has long been known that the equilibrium configuration withminimum energy of a nematic liquid crystal within a cylindersubject to homeotropic conditions on the lateral boundary isescaped along the axis of the cylinder, and there is no singularityof the orientation field. So problems of explaining the presenceof point defects in capillary tubes and of exploring their stabilityarise. There is enough evidence to believe that the menisciplay a central role in preventing the orientation field arounda point defect from unwinding towards the escaped configuration.We propose a variational model which describes how a point defectinteracts with a meniscus. This interaction fades away at afinite distance. When active, it is two-sided, being repulsiveat first and then attractive when the defect comes closer tothe centre of curvature of the meniscus. Thus, when a defectis enclosed between two menisci they can become antagonists,so that there is a metastable equilibrium position where thedefect could be locked in.     

On weak solution to the steady compressible flow of nematic liquid crystals

In this paper, we study the three‐dimensional‐simplified Ericksen‐Leslie system for the steady compressible flow of nematic liquid crystals in a bounded domain. It is proved that the existence of a weak solution for the adiabatic exponent γ > 1 provided the initial direction field in the upper hemisphere.     

On the number of singular points to the flow of nematic liquid crystals

In this paper, we investigate the singular points of suitable weak solutions of the system modeling the flow of liquid crystals. We obtain an upper bound for the number of singular points at any fixed time.     

On the well‐posedness for the heat flow of harmonic maps and the hydrodynamic flow of nematic liquid crystals in critical spaces

In recent paper, we prove the well‐posedness for the heat flow of harmonic maps with initial data and for the hydrodynamic flow of nematic liquid crystals with initial data . Copyright © 2011 John Wiley & Sons, Ltd.     

On nematic liquid crystals with variable degree of orientation

Energy minimizing static configurations of nematic liquid crystals with variable degree of orientation can be described by a map which minimizes certain degenerate variational integrals. This mathematical model was recently posed by J. L. Erickson. In this paper, we shall prove the existence, local uniqueness, and global regularity of minimizers. We shall also estimate the size of possible defects.     

Moment singularities at angular corners in liquid crystals of nematic type

Summary A linear theory of axial plane reorientation in liquid crystals of nematic type is utilized to examine principal moment singularities which may occur at the vertex of corner boundaries. Reorientation free, moment free, and mixed boundary conditions are considered, and the order of the singularity as a function of the corner angle for these three cases is obtained.

---

Zusammenfassung Eine lineare Theorie der Achsenebenen-Neuorientierung an Flüssigkristallen vom nematischen Typ wird verwendet, um die Hauptmoment-Singularität zu untersuchen, die im Scheitel von Begrenzungen durch Ecken auftreten kann. Neuorientierungsfreie, momentenfreie, und gemischte Grenzbedingungen werden behandelt, und die Ordnung der Singularität als Funktion des Eckenwinkels wird für diese drei Fälle bestimmt.

---

---

This work was supported by the United States Energy Research and Development Administration.     

The stability of nematic liquid crystals under crossed electric and magnetic fields

The alignment of the director in nematic liquid crystals is known to be influenced by electric and/or magnetic fields. The stability of known traveling wave solutions is examined for a best fit cubic equation which approximates the nonlinear sinusoidal terms in the dynamic equation for the director when it is subjected to crossed electric and magnetic fields.     

A relaxed model and its homogenization for nematic liquid crystals in composite materials

We analyse a model for equilibrium configurations of composite systems of nematic liquid crystal with polymer inclusions, in the presence of an external magnetic field. We assume that the system has a periodic structure, and consider the relaxed problem on the unit length constraint of the nematic director field. The relaxation of the Oseen–Frank energy functional is carried out by including bulk as well as surface energy penalty terms, rendering the problem fully non‐linear. We employ two‐scale convergence methods to obtain effective configurations of the system, as the size of the polymeric inclusions tends to zero. We discuss the minimizers of the effective energies for, both, the constrained as well as the unconstrained models. Copyright © 2004 John Wiley & Sons, Ltd.     

On the long-time behavior of some mathematical models for nematic liquid crystals

A model describing the evolution of a liquid crystal substance in the nematic phase is investigated in terms of two basic state variables: the velocity field u and the director field d, representing the preferred orientation of molecules in a neighborhood of any point in a reference domain. After recalling a known existence result, we investigate the long-time behavior of weak solutions. In particular, we show that any solution trajectory admits a non-empty ω-limit set containing only stationary solutions. Moreover, we give a number of sufficient conditions in order that the ω-limit set contains a single point. Our approach improves and generalizes existing results on the same problem.     

On temporal decay estimates for the compressible nematic liquid crystal flow in

We use a general energy method recently developed by [Guo Y, Wang Y. Decay of dissipative equations and negative sobolev spaces. Commun. Partial Differ. Equ. 2012;37:2165–2208.] to prove the global existence and temporal decay rates of solutions to the three-dimensional compressible nematic liquid crystal flow in the whole space. In particular, the negative Sobolev norms of solutions are shown to be preserved along time evolution, and then the optimal decay rates of the higher order spatial derivatives of solutions are obtained by energy estimates and the interpolation inequalities.     

Existence of minimal energy configurations of nematic liquid crystals with variable degree of orientation

Summary We prove existence of minimizers of the functional recently suggested by Ericksen [8] for the statics of nematic liquid crystals. A set of necessary conditions for the minimizers and a monotonicity formula are also found.     

奇数维可压缩液晶流方程解的逐点估计

首先, 本文利用标准的能量估计方法得到高维(3 维及以上) 的液晶流方程组小初值经典解的整体存在性. 然后, 本文运用Green 函数方法, 得到奇数维情形(3 维及以上) 该解的逐点估计. 该结果表明, 密度ρ和动量m同Navier-Stokes 方程组一样满足一般Huygens 原理, 而单位向量场d则没有这种现象, 其有着与热方程的解类似的时空估计.