THE EXISTENCE AND UNIQUENESS OF TIME-PERIODIC SOLUTIONS TO THE NON-ISOTHERMAL MODEL FOR COMPRESSIBLE NEMATIC LIQUID CRYSTALS IN A PERIODIC DOMAIN

In this paper,we are concerned with a three-dimensional non-isothermal model for the compressible nematic liquid crystal flows in a periodic domain.Under some smallness and structural assumptions imposed on the time-periodic force,we establish the existence of the time-periodic solutions to the system by using a regularized approximation scheme and the topological degree theory.We also prove a uniqueness result via energy estimates.

ENERGY CONSERVATION FOR THE WEAK SOLUTIONS TO THE 3D COMPRESSIBLE NEMATIC LIQUID CRYSTAL FLOW

In this paper,we establish some regularity conditions on the density and velocity fields to guarantee the energy conservation of the weak solutions for the three-dimensional compressible nematic liquid crystal flow in the periodic domain.

GLOBAL EXISTENCE OF WEAK SOLUTIONS TO THE NON-ISOTHERMAL NEMATIC LIQUID CRYSTALS IN 2D

In this article, we prove the global existence of weak solutions to the non- isothermal nematic liquid crystal system on T2, on the basis of a new approximate system which is different from the classical Ginzburg-Landau approximation. Local in space energy inequalities are employed to recover the estimates on the second order spatial derivatives of the director fields locally in time, which cannot be derived from the basic energy balance. It is shown that these weak solutions satisfy the temperature equation, and also the total energy equation but away from at most finite many ＂singular＂ times, at which the energy concentration occurs and the director field losses its second order derivatives.

Determining the sum of flexoelectric coefficients in nematic liquid crystals by the capacitance method

A detailed theoretical analysis of determining the sum of flexoelectric coefficients in nematic liquid crystals using the capacitance method is given. In the strong anchoring parallel aligned nematic （PAN） and hybrid aligned nematic （HAN） cells, the dependences of the capacitance on the sum of flexoelectric coefficients and the applied voltage are obtained by numerical simulations, and the distributions of the director and the electric potential for different applied voltages and flexoelectric coefficients are also given. Based on this theoretical analysis, we propose an experimental design for measuring the capacitance of a liquid crystal cell using the improved precision LCR meter E4980A （Agilent）. Through comparing the experimental data with the simulated results, the sum of flexoeletric coefficients can be determined.

ASYMPTOTIC BEHAVIOR OF SOLUTIONS TO THE COMPRESSIBLE NEMATIC LIQUID CRYSTAL SYSTEM IN R3

In this paper,we study a nematic liquid crystals system in three-dimensional whole space R3 and obtain the time decay rates of the higher-order spatial derivatives of the solution by the method of spectral analysis and energy estimates if the initial data belongs to L1（R3） additionally.

LPS'S CRITERION FOR INCOMPRESSIBLE NEMATIC LIQUID CRYSTAL FLOWS

In this paper we derive LPS＇s criterion for the breakdown of classical solutions to the incompressible nematic liquid crystal flow, a simplified version of Ericksen-Leslie system modeling the hydrodynamic evolution of nematic liquid crystals in R^3. We show that if 0 〈 T 〈 ＋∞ is the maximal time interval for the unique smooth solution u ∈ C^∞（[0, T）,R^3）,then ｜u｜＋｜△d｜∈L^q（[0,T],L^p（R^3））,where p and q satisfy the Ladyzhenskaya-Prodi-Serrin＇s condition：3/p＋2/q=1 and p∈（3,＋∞].

BLOW-UP CRITERION OF CLASSICAL SOLUTIONS FOR THE INCOMPRESSIBLE NEMATIC LIQUID CRYSTAL FLOWS

In this paper, we consider the short time classical solution to a simplified hydro-dynamic flow modeling incompressible, nematic liquid crystal materials in R3. We establisha criterion for possible breakdown of such solutions at a finite time. More precisely, if （u, d）is smooth up to time T provided that ∫T 0‖△×u（t, ·）‖BMO（R3） ＋‖△d（t, ·）‖8L4（R3）dt 〈∞.

COMPENSATION EFFECT OF POLYIMIDE THIN FILMS ON NORMALLY WHITE TWISTED NEMATIC LIQUID CRYSTAL DISPLAYS

The disadvantages of Normally White Twisted Nematic Liquid Crystal Display (NW-TN-LCD) were discussed. The reason that the negative birefringent polyimide thin films were used to compensate NW-TN-LCD to decrease off-axis leakage, improve contrast ratios and enlarge viewing angles was explained in this paper. A certain polyimide thin film was taken as an example to show compensation effect on NW-TN-LCD.

Two-Dimensional Spatial Solitons in Nematic Liquid Crystals

We study the propagation of spatial solitons in nematic liquid crystals, using the self-similar method. Analytical solutions in the form of self-similar solitons are obtained exactly. We confirm the stability of these solutions by direct numerical simulation, and find that the stable spatial solitons can exist in various forms, such as Oaussian solitons, radially symmetric solitons, multipole solitons, and soliton vortices.

Temperature dependence of ratio between dielectric anisotropy and order parameter in fluorinated nematic liquid crystals

Temperature dependence of ratio between dielectric anisotropy and order parameter of fluorinated nematic liquid crystal is investigated by using a semi-empirical molecular orbital package that can accurately calculate an angle between molecular dipole moment and long axis. We optimize the molecular conformations with three semi-empirical Hamiltonians AM1, PM3 and PM5, and then make a comparison between computational results and experimental measurements. It is shown that the results obtained from AM1 method are in good agreement with the measurements. The present study offers an applicable method to predict the dielectric properties of liquid crystal material.

Calculating the dielectric anisotropy of nematic liquid crystals:a reinvestigation of the Maier-Meier theory

This paper investigates the average dielectric permittivity （^-ε） in the Maier-Meier theory for calculating the dielectric anisotropy （△ε） of nematic liquid crystals.For the reason that ^-ε of nematics has the same expression as the dielectric permittivity of the isotropic state,the Onsager equation for isotropic dielectric was used to calculate it.The computed ^-ε shows reasonable agreement with the results of the numerical methods used in the literature.Molecular parameters,such as the polarizability and its anisotropy,the dipole moment and its angle with the molecular long axis,were taken from semi-empirical quantum chemistry （MOCPAC/AM1） modeling.The calculated values of △ε according to the Maier-Meier equation are in good agreement with the experimental results for the investigated compounds having different core structures and polar substituents.

Molecular theory of nematic liquid crystals viewed as effect of collective excitation in ferromagnetic systems

We develop a microscopic theory of the nematic phase with consideration of the effect of the collective excitation on properties of nematic liquid crystals. The model is based on the Heisenberg＇s exchange model of the ferromagnetic materials. Since the orientation of the molecular long axis and the angular momentum of the molecule rotating around its long axis have the same direction, operators can be introduced to research the nematic liquid crystals. Using the lattice model and the Holstein-Primakoff transformation, the Hamiltonian of the system can be obtained, which has the same form as that of the ferromagnetic substance. The relation between the order parameter and reduced temperature can be gotten. It is in good agreement with the experimental results in the low temperature region, the accordance is better than that of the molecular field theory and the computer simulation. In high temperature region close to the transition point, by considering the effect of the higher-order terms in the Hamiltonian, theoretical prediction is in better agreement with the experiment. That indicates the many-body effect is important to nematic liquid crystals.

Lattice Boltzmann Simulation of 3D Nematic Liquid Crystal near Phase Transition

Phase transition between nematic and isotropic liquid crystal is a very weak first order phase transition.We avoid to use the normal Landau-de Gennes's free energy that reduces a strong first order transition, and set up adata base of free energy calculated by means of Tao-Sheng Lin's extended molecular field theory that can explain theexperiments of the equilibrium properties of nematic liquid crystal very well. Then we use the free energy method oflattice Boltzmann developed by Oxford group to study the phase decomposition, pattern formation in the flow of theliquid crystal near transition temperature.

Asymmetric dynamic phase holographic grating in nematic liquid crystal

A new scheme for recording a dynamic phase grating with an asymmetric profile in C60-doped homeotropically aligned nematic liquid crystal（NLC） was presented.An oblique incidence beam was used to record the thin asymmetric dynamic phase holographic grating.The diffraction efficiency we achieved is more than 40%,exceeding the theoretical limit for symmetric profile gratings.Both facts can be explained by assuming that a grating with an asymmetric saw-tooth profile is formed in the NLC.Finally,physical mechanism and mathematical model for characterizing the asymmetric phase holographic grating were presented,based on the photo-refractive-like（PR-like） effect.

On the Uniqueness of Heat Flow of Harmonic Maps and Hydrodynamic Flow of Nematic Liquid Crystals

For any n-dimensional compact Riemannian manifold (M,g) without boundary and another compact Riemannian manifold (N,h), the authors establish the uniqueness of the heat flow of harmonic maps from M to N in the class C([0,T),W1,n). For the hydrodynamic flow (u,d) of nematic liquid crystals in dimensions n = 2 or 3, it is shown that the uniqueness holds for the class of weak solutions provided either (i) for n = 2, u ∈ Lt∞ L2x∩L2tHx1, ▽P∈ Lt4/3 Lx4/3 , and ▽d∈ L∞t Lx2∩Lt2Hx2; or (ii) for n = 3, u ∈ Lt∞ Lx2∩L2tHx1∩ C([0,T),Ln), P ∈ Ltn/2 Lxn/2 , and ▽d∈ L2tLx2 ∩ C([0,T),Ln). This answers affirmatively the uniqueness question posed by Lin-Lin-Wang. The proofs are very elementary.

Numerical Simulations of Hydrodynamics of Nematic Liquid Crystals:Effects of Kinematic Transports

In this paper,we investigate the effects of kinematic transports on the nematic liquid crystal system numerically and theoretically.The model we used is a"1+2"elastic continuum model simplified from the Ericksen-Leslie system.The numerical experiments are carried out by using a Legendre-Galerkin spectral method which can preserve the energy law in the discrete form.Based on this highly accurate numerical approach we find some interesting and important relationships between the kinematic transports and the characteristics of the flow.We make some analysis to explain these results.Several significant scaling properties are also verified by our simulations.

GLOBAL EXISTENCE AND LONG-TIME BEHAVIOR FOR THE STRONG SOLUTIONS IN H2 TO THE 3D COMPRESSIBLE NEMATIC LIQUID CRYSTAL FLOWS

In this paper, we investigate the global existence and long time behavior of strong solutions for compressible nematic liquid crystal flows in threedimensional whole space. The global existence of strong solutions is obtained by the standard energy method under the condition that the initial data are close to the constant equilibrium state in H2-framework. If the initial datas in Ll-norm are finite additionally, the optimal time decay rates of strong solutions are established. With the help of Fourier splitting method, one also establishes optimal time decay rates for the higher order spatial derivatives of director.

Simulation and optimization of spatial light modulation of twisted-nematic liquid crystal display

The approaches to obtaining desired intensity or phase modulation by twisted-nematic liquid crystal display （TN-LCD） have been extensively studied based on the knowledge of the LCD＇s internal structure parameters. Generally, the TN-LCD placed between two linear polarizers （P） produces coupled intensity and phase modulation. To obtain the commonly used phase-only modulation, quarter wave plates （QWPs） are often used in front of and/or behind the LCD. Here we present a method to optimize the optical modulation properties of the TN-LCD to obtain phase-only modulation in the configuration of P-QWP-LCD-QWP-P each with proper orientation. Our method is based on the macroscopical Jones matrix descriptions for the LCD, the QWPs, and the linear polarizers. Through Jones matrix calculations, the orientations of the polarizers and QWPs can be optimized to satisfy differently desired modulation demands. In contrast to the traditional method, which requires knowledge of the LCD＇s internal structure parameters, our method simplified the complicated theory analysis and can work in the absence of information on the LCD＇s internal structure parameters, which are usually not available for the commercial products.

A Quasilinear System Related with the Asymptotic Equation of the Nematic Liquid Crystal’s Director Field

In this paper, the author studies the local existence of strong solutions and their possible blow-up in time for a quasilinear system describing the interaction of a short wave induced by an electron field with a long wave representing an extension of the motion of the director field in a nematic liquid crystal’s asymptotic model introduced in[Saxton, R. A., Dynamic instability of the liquid crystal director. In: Current Progress in Hyperbolic Systems(Lindquist, W. B., ed.), Contemp. Math., Vol.100, Amer. Math.Soc., Providence, RI, 1989, pp.325–330] and [Hunter, J. K. and Saxton, R. A., Dynamics of director fields, SIAM J. Appl. Math., 51, 1991, 1498–1521] and studied in [Hunter, J. K.and Zheng, Y., On a nonlinear hyperbolic variational equation I, Arch. Rat. Mech. Anal.,129, 1995, 305–353], [Hunter, J. K. and Zheng, Y., On a nonlinear hyperbolic variational equation II, Arch. Rat. Mech. Anal., 129, 1995, 355–383] and in [Zhang, P. and Zheng,Y., On oscillation of an asymptotic equation of a nonlinear variational wave equation,Asymptotic Anal., 18, 1998, 307–327] and, more recently, in [Bressan, A., Zhang, P. and Zheng, Y., Asymptotic variational wave equations, Arch. Rat. Mech. Anal., 183, 2007,163–185].

A FINITE ELEMENT ALGORITHM FOR NEMATIC LIQUID CRYSTAL FLOW BASED ON THE GAUGE-UZAWA METHOD

In this paper,we present a finite element algorithm for the time-dependent nematic liquid crystal flow based on the Gauge-Uzawa method.This algorithm combines the Gauge and Uzawa methods within a finite element variational formulation,which is a fully discrete projection type algorithm,whereas many projection methods have been studied without space discretization.Besides,error estimates for velocity and molecular orientation of the nematic liquid crystal flow are shown.Finally,numerical results are given to show that the presented algorithm is reliable and confirm the theoretical analysis.