Leslie-Ericksen Equations Research Articles

Nematodynamics and random homogenization

We study the homogenization problem for the system of equations of dynamics of a mixture of liquid crystals with random structure. We consider a simplified form of the Ericksen–Leslie equations for an incompressible medium with inhomogeneous density with random structure. Under the assumption that randomness is statistically homogeneous and ergodic, we construct the limit problem and prove almost sure convergence of solutions of the original problem to the solution of the limit (homogenized) problem.

Compressible hydrodynamic flow of nematic liquid crystals with vacuum

In this paper, we first consider the free boundary problem for a simplified version of Ericksen–Leslie equations modeling the compressible hydrodynamic flow of nematic liquid crystals in dimension one which connects continuously to vacuum. We obtain the existence of global weak solutions. Furthermore, we establish the life-span of smooth solutions to the compressible nematic liquid crystal model with the support of density growing sublinearly in time direction.

Equivalent Theories of Liquid Crystal Dynamics

There are two competing descriptions of nematic liquid crystal dynamics: the Ericksen-Leslie director theory and the Eringen micropolar approach. Up to this day, these two descriptions have remained distinct in spite of several attempts to show that the micropolar theory comprises the director theory. In this paper we show that this is the case by using symmetry reduction techniques and introducing a new system that is equivalent to the Ericksen-Leslie equations and includes disclination dynamics. The resulting equations of motion are verified to be completely equivalent, although one of the two different reductions offers the possibility of accounting for orientational defects. After applying these two approaches to the ordered micropolar theory of Lhuiller and Rey, all the results are eventually extended to flowing complex fluids, such as nematic liquid crystals.

A free boundary problem for compressible hydrodynamic flow of liquid crystals in one dimension

In this paper, we consider the free boundary problem for a simplified version of Ericksen–Leslie equations modeling the compressible hydrodynamic flow of nematic liquid crystals in dimension one. We obtain both existence and uniqueness of global classical solutions provided that the initial density is away from vacuum.

Response time improvement of liquid–crystal wavefront corrector using optimal cell gap of numerical computation

The optimal cell gap of nematic liquid–crystal wavefront corrector (LCWFC) is proved theoretically by numerical computation of the Ericksen–Leslie equation based on finite differences and confirmed experimentally. The numerical results of optimizing cell gap overcome the constraints of many approximations e.g. the small angle approximation and agree much better with the experimental data than those of analytical solution. Therefore, this method supplies a useful tool to achieve the optimal cell gap of a nematic LCWFC with high precision. Together with high birefringence and low rotational viscosity liquid crystal (LC) material effect, a nematic LCWFC with response time of sub-millisecond can be achieved in the visible spectral region.

Decoupling of the Ericksen–Leslie equations

We examine the hydrodynamic behaviour of a nematic liquid crystal confined between two parallel electrodes and subject to a non-uniform electric field across the layer. By decoupling the Ericksen–Leslie equations for the nematic, we are able to derive a dynamic equation for the director orientation which is not explicitly dependent on the fluid velocity, rather a non-local term contains the effects of flow. The flow velocity and electric potential are determined subsequently from the calculated director profile. Our numerical scheme enables us to predict an effective rotational viscosity which takes account of fluid shear viscosities and allows us to establish the behaviour in special symmetry cases. Significantly, our calculations also demonstrate how kickback can be avoided, and the switch-off time optimised, by restricting the applied voltage to an appropriate range.

Stokes-Einstein diffusion of colloids in nematics

We report the experimental observation of anisotropic diffusion of polystyrene particles immersed in a lyotropic liquid crystal with two different anchoring conditions. Diffusion is shown to obey the Stokes-Einstein law for particle diameters ranging from 190 nm up to 2 μm. In the case of prolate micelles, the beads diffuse four times faster along the director than in perpendicular directions, D||/D[Symbol: see text] ≈ 4. In the theory part we present a perturbative approach to the Leslie-Ericksen equations and relate the diffusion coefficients to the Miesovicz viscosity parameters η(i). We provide explicit formulas for the cases of uniform director field and planar anchoring conditions which are then discussed in view of the data. As a general rule, we find that the inequalities η(b) <η(a) <η(c), satisfied by various liquid crystals of rodlike molecules, imply D||>D[Symbol: see text].

Anchoring effect on switching dynamics of a nematic liquid crystal intensity modulator

The director reorientation relaxation time of a novel nematic liquid crystal is measured through fitting the decay curve of intensity modulation by Ericksen–Leslie equation. Application of the so-called hard anchoring condition results in simplifying the mentioned equation as well as ignoring the reorientation of boundary molecules in switching procedure. This paper investigates the hardness of 1294-1b nematic liquid crystal's anchoring on rubbed PVA and the effect of boundary molecules on response time and viscoelastic coefficient calculation by experimentally obtaining the decay times of different sample thicknesses.

An Overview on Numerical Analyses of Nematic Liquid Crystal Flows

The purpose of this work is to provide an overview of the most recent numerical developments in the field of nematic liquid crystals. The Ericksen-Leslie equations govern the motion of a nematic liquid crystal. This system, in its simplest form, consists of the Navier-Stokes equations coupled with an extra anisotropic stress tensor, which represents the effect of the nematic liquid crystal on the fluid, and a convective harmonic map equation. The sphere constraint must be enforced almost everywhere in order to obtain an energy estimate. Since an almost everywhere satisfaction of this restriction is not appropriate at a numerical level, two alternative approaches have been introduced: a penalty method and a saddle-point method. These approaches are suitable for their numerical approximation by finite elements, since a discrete version of the restriction is enough to prove the desired energy estimate.The Ginzburg-Landau penalty function is usually used to enforce the sphere constraint. Finite element methods of mixed type will play an important role when designing numerical approximations for the penalty method in order to preserve the intrinsic energy estimate.The inf-sup condition that makes the saddle-point method well-posed is not clear yet. The only inf-sup condition for the Lagrange multiplier is obtained in the dual space of H 1(Ω). But such an inf-sup condition requires more regularity for the director vector than the one provided by the energy estimate. Herein, we will present an alternative inf-sup condition whose proof for its discrete counterpart with finite elements is still open.

Vortex flow in free-standing smectic C films driven by elastic distortions

The effects of coupling between director and flow dynamics are well-known in nematic liquid crystals, but such interactions are often disregarded in smectic materials. Experimental studies of the relaxation of distorted director fields in free-standing smectic C films suggest that a neglect of flow effects may lead to quantitative errors in the evaluation of material parameters and even to qualitative misinterpretations. A quantitative analysis of flow coupling in particularly simple experimental geometries is presented here. In thin planar films, the smectodynamic theory can be approximated by the two-dimensional Ericksen-Leslie equations of nematodynamics, comprising two elasticity and five viscosity coefficients. This represents one of the simplest hydrodynamic problems in anisotropic fluids in 2D. We solve the full coupled system of incompressible quasi Navier–Stokes and director torque balance equations numerically for circular films with and without topological defects in the film center. The coupled evolution of director and flow fields during the relaxation of initial distortions is analyzed, and consequences of the nonlinearity of the dynamic equations are discussed.

Numerical approximation of nematic liquid crystal flows governed by the Ericksen-Leslie equations

Numerical approximation of the flow of liquid crystals governed by the Ericksen-Leslie equations is considered. Care is taken to develop numerical schemes which inherit the Hamiltonian structure of these equations and associated stability properties. For a large class of material parameters compactness of the discrete solutions is established which guarantees convergence.

A numerical method for solving the dynamic three-dimensional Ericksen–Leslie equations for nematic liquid crystals subject to a strong magnetic field

A finite difference technique, based on a projection method, is developed for solving the dynamic three-dimensional Ericksen–Leslie equations for nematic liquid crystals subject to a strong magnetic field. The governing equations in this situation are derived using primitive variables and are solved using the ideas behind the GENSMAC methodology (Tomé and McKee [32]; Tomé et al. [34]). The resulting numerical technique is then validated by comparing the numerical solution against an analytic solution for steady three-dimensional flow between two-parallel plates subject to a strong magnetic field. The validated code is then employed to solve channel flow for which there is no analytic solution.

Long-time behavior for nonlinear hydrodynamic system modeling the nematic liquid crystal flows

We study a simplified system of the original Ericksen-Leslie equations for the flow of nematic liquid crystals. This is a coupled non-parabolic dissipative dynamic system. We show the convergence of global classical solutions to single steady states as time goes to infinity by using the Łojasiewicz-Simon approach. Moreover, we provide an estimate on the convergence rate.

Tensorial Form of Leslie-Ericksen Equations and Applications

We propose a system of dynamical equations coupling a tensor Qij related to the molecular order with the flow velocity, whose structure is very similar to the Leslie-Ericksen equations. In particular, the Leslie viscosities and the Frank elastic constants appear explicitly, which contrasts with constitutive equations derived from the microscopic Doi theory and using the full order parameter tensor Sij. The description in terms of the tensor Qij may be seen as intermediate between the director of Leslie-Ericksen theory and the order parameter tensor Sij of microscopic theories. With our approach we take advantage of the fact that the Leslie viscosities and the Frank elastic constants are well-established for several real systems (as for example 5CB, 8CB, or MBBA). Our equations, that become manifestly equivalent to Leslie-Ericksen equations when a director may be defined, are used in order to regularise the director field at the position of the field singularities in a simple way. Our objective is to simulate the macroscopic behaviour of nematics containing a large assembly of defects under several experimental conditions (under shear and/or magnetic field). Since we are essentially concerned by the macroscopic response, the exact structure of the defects is secondary, it is why we believe this approach to be pertinent in studying the dynamics of the texture induced by defects. The model is applied to a spinning nematic (5CB) subjected to a magnetic field, and a relaxation mechanism involving topological defects is evidenced.

Non-classical scaling for forced wetting of a nematic fluid on a polymeric fiber

We report on the forced wetting of a liquid crystal on a polymer fiber in the nematic and isotropic phase under partial wetting conditions. As the cylindrical fiber exits from the fluid reservoir, the fluid forced-coated on the fiber is immediately broken into droplets due to capillary-driven instability, even in the wetting conditions. For the nematic fluid, the initial film thickness, h, before breakup, scales almost linearly with the capillary number, Ca, as h ∼ Ca0.94, while h ∼ Ca2/3 for the isotropic fluid, as predicted for a Newtonian fluid. The amount of the fluid coated on a fiber is larger in the nematic phase than in the isotropic phase at a given velocity within the velocity range studied. Analysis using Ericksen-Leslie equations shows that Frank elasticity plays no role in increasing the coating thickness for the nematic fluid, while viscous anisotropy is the source of observed rescaling, h ∼ Ca. This non-classical scaling is attributed to the deformation-stress cross-coupling and the existence of extensional kinematics in the meniscus formation region.

A Novel Approach to the Normal Force in Shear of Nematic Liquid Crystals in the Cone-and-Plate Rheometer

We reexamine the interpretation of the normal force measured with a cone-and-plate rheometer for nematic liquid crystals. We point out the fact that the quite widespread belief that the normal force is directly related to the well-known rheological function 𝒩1 (i.e., the first normal stress difference) fails for these complex fluids. After a brief presentation of the theoretical bases leading to the general expression of the normal force, this new approach is applied to nematic liquid crystals within the framework of Leslie-Ericksen theory. In order to avoid heavy numerical computations the full Leslie-Ericksen equations describing the three-dimensional flow within the cone-and-plate cell are reduced, thanks to reasonable approximations, to an effective one-dimensional problem (i.e., a system of partial differential equations with spatial derivatives with respect to one single coordinate). Results for low molecular weight liquid crystals are presented and discussed. The main feature evidenced in this work is the notable difference between the normal force calculated in this way and the 𝒩1 function, in particular the normal force function becomes negative for high shear rate while 𝒩1 stays positive.

Numerical simulation of liquid crystalline polymer flow into two‐dimensional thin cavity moulds

AbstractIn this paper, flows of liquid crystalline polymers into two‐dimensional thin cavity moulds are simulated. The flows are modelled by Ericksen–Leslie equations of motion in the high viscosity limit. An elliptic pressure equation is derived under Hele–Shaw approximations, and the non‐isothermal natures of the flow are modelled. The equations are solved using the finite‐difference technique. A new boundary‐mapping technique is developed in this study to solve the difficulty in the finite‐difference treatment of arbitrarily shaped boundaries, which possess no natural coordinate system. This new method avoids the difficult mesh control in the body‐fitted mapping process and makes the mapping process easy to implement. It can also solve the problems caused by the uneven distribution of grid nodes in the traditional body‐fitted mapping technique. Copyright © 2008 John Wiley &amp; Sons, Ltd.

Evaluation of translational friction coefficients of macroscopic probes in nematic liquid crystals

We present a study of the translational friction coefficients of spherical and ellipsoidal probes in nematic liquid crystalline fluids, based on the numerical treatment of Leslie-Ericksen equations [Q. J. Mech. Appl. Math. 19, 357 (1966); Adv. Liq. Cryst. 4, (1979); Trans. Soc. Rheol. 5, 23 (1961); Adv. Liq. Cryst. 2, 233 (1976)] for incompressible nematic fluids. Simulations of director dynamics in a local environment surrounding the moving probe are presented, and the dependence of translational diffusion on liquid crystal viscoelastic parameters is discussed. The time evolution of the director field is studied in the presence of an orienting magnetic field in two characteristic situations: Directors of motion parallel and perpendicular with respect to the field. In the particular case under investigation, a detailed analysis is given for the case of spherical, prolate, and oblate ellipsoidal probes in rectilinear motion in nematic (4-methoxibenzylidene-4'-n-butylaniline), together with a comparison with other nematogens, namely, 4,4'-dimethoxuazoxy benzene and (4'-n-pentyl-4-cyanobiphenil). A discussion of the general methodology presented in this work is given for the case of colloidal dispersions in nematic liquid crystals, which are considered as model systems of dispersions of particles in host media with anisotropic physical properties.

Evaluation of translational friction coefficients of micro-sized spherical probes in nematic liquid crystals

We study the translational friction coefficients of a spherical micrometric probe moving in nematic liquid crystalline fluids, by solving numerically the constitutive hydrodynamic equations of uncompressible isothermal nematic fluids (Leslie–Ericksen equations). The nematic medium is described by a vector field, which specifies the director orientation at each point and by the velocity vector field. Simulations of director dynamics surrounding the moving probe are presented, and the dependence of translational diffusion upon liquid crystal viscoelastic parameters is discussed. The time evolution of director field is studied in the presence of an orienting magnetic field in two characteristic situations, i.e. direction of motion parallel and perpendicular to field. In particular, a detailed analysis is given for the case of a spherical probe in rectilinear motion in nematic MBBA (4-methoxibenzylidene-4′-n-butylaniline), together with a comparison with other nematogens.

Automatic Extraction of an Equivalent Circuit from a TFT-LCD Unit Cell for LCD TV Application

In this article, we present a novel scheme for automatic generation of an equivalent circuit for unit cell of LCD-TV. In order to automatically extract a circuit model, a computer program generates an electrical connectivity of resistors and capacitors from the layout information through a so-called pattern analysis with electrode and port information. Thereafter, the Ericksen-Leslie's equation and two types of Laplace's equation in the dielectric and conductor domain are solved by finite element method (FEM). Finally, we can obtain a resistive equivalent circuit and a capacitive equivalent circuit in an independent manner. For combining two types of independent equivalent circuits, we propose a node insertion algorithm which enables us to generate an equivalent RC circuit without increasing the capacitive elements.