Leslie Coefficients Research Articles

Long‐time dynamics of Ericksen–Leslie system on 𝕊2

In this paper, we study the long‐time behavior of full Ericksen–Leslie system modeling the hydrodynamics of nematic liquid crystals between two dimensional unit spheres. Under a weaker assumption for Leslie's coefficients, we give the key energy inequality for the global weak solution. At last, inspired by the conditions on the simplified system, we establish several sufficient conditions which guarantee the uniform convergence of the system in and spaces as time tends to infinity under small initial data.

Chemical Leslie effect in a chiral smectic-C^{★} film: Nonsingular target patterns.

We analyze experimentally and theoretically the winding and unwinding of the c[over ⃗] director in a chiral smectic-C^{★} film crossed by an ethanol flow. This leads to a target pattern under crossed polarizers when the +1 defect imposed by the boundary conditions is pinned on the edge of the film. We show that the target is deformed at the center of the film when it is subjected to a flow of ethanol because of the presence of two recirculation vortices of chemohydrodynamical origin. This deformation and the two vortices disappear during the unwinding of the target when the ethanol flow is stopped. This unambiguously shows that the target deformation is due to the vortices and not to the elastic anisotropy. These two points are confirmed theoretically. An estimate of the two so-called chemomechanical and chemohydrodynamical Leslie coefficients is also derived from this study.

Anisotropic viscous effects of local flow by a rotating microparticle in nematic liquid crystal.

The presented study opens a perspective to investigate the effects of local flow on nematic liquid crystals. A particle rotated in nematic fluids typically generates a rotationally symmetric local flow, which causes a change in the director orientation. The director above the threshold velocity has a particular angle determined by the ratio of Leslie coefficients, α_{2}/α_{3}. In 5CB liquid crystals, this director angle with respect to the flow is approximately 13^{∘}. The angle is calculated through Ericksen-Leslie theory. The angle is not dependent on rotation frequency or particle size but temperature. The area of the influenced region increases with the rotation frequency and particle size. The changes in radius of the influenced region are calculated theoretically using Ericksen number. Further, an interference pattern appears at the edge of the influenced region by the refractive indexes mismatch between the influenced region and the rest. We experimentally obtain the thickness of the influenced region analyzing intervals of the pattern.

Dynamics of the Ericksen–Leslie Equations with General Leslie Stress II: The Compressible Isotropic Case

In this article, the non-isothermal compressible Ericksen–Leslie system for nematic liquid crystals subject to general Leslie stress is considered. It is shown that this system is locally well-posed within the $$L_q$$ -setting and that for initial data close to equilibria points (which are identical with the ones for the incompressible situation), the solution exists globally. Moreover, any global solution which does not develop singularities converges to an equilibrium in the topology of the natural state manifold. Note that no structural assumptions on the Leslie coefficients are imposed and, in particular, Parodi’s relation is not being assumed. The results can be viewed as an extension of the studies in Hieber and Pruss (Math Ann 369:977–996, 2017) for the incompressible case to the compressible situation.

On well-posedness of Ericksen–Leslie’s parabolic–hyperbolic liquid crystal model in compressible flow

We study the well-posedness of the Ericksen–Leslie’s parabolic–hyperbolic liquid crystal model in compressible flow. Inspired by our study for incompressible case [N. Jiang and Y.-L. Luo, On well-posedness of Ericsen–Leslie’s hyperbolic incompressible liquid crystal model, preprint (2017), arXiv:1709.06370v1 ] and some techniques from compressible Navier–Stokes equations, we first prove the local-in-time existence of the classical solution to the system with finite initial energy, under some natural constraints on the Leslie coefficients which ensure that the basic energy law is dissipative. Furthermore, with an additional assumption on the coefficients which provides a damping effect, and the smallness of the initial energy, the existence of global solution can be established.

On Well-Posedness of Ericksen--Leslie's Hyperbolic Incompressible Liquid Crystal Model

We study the Ericksen-Leslie's hyperbolic incompressible liquid crystal model. Under some constraints on the Leslie coefficients which ensure the basic energy law is dissipative, we prove the local-in-time existence and uniqueness of the classical solution to the system with finite initial energy. Furthermore, with an additional assumption on the coefficients which provides a damping effect, and the smallness of the initial energy, the unique global classical solution can be established.

Dynamics of the Ericksen–Leslie equations with general Leslie stress I: the incompressible isotropic case

The Ericksen–Leslie model for nematic liquid crystals in a bounded domain with general Leslie and isotropic Ericksen stress tensor is studied in the case of a non-isothermal and incompressible fluid. This system is shown to be locally strongly well-posed in the $$L_p$$ -setting, and a dynamical theory is developed. The equilibria are identified and shown to be normally stable. In particular, a local solution extends to a unique, global strong solution provided the initial data are close to an equilibrium or the solution is eventually bounded in the topology of the natural state manifold. In this case, the solution converges exponentially to an equilibrium, in the topology of the state manifold. The above results are proven without any structural assumptions on the Leslie coefficients and in particular without assuming Parodi’s relation.

On the uniqueness of weak solution for the 2-D Ericksen--Leslie system

In this paper, we prove the uniqueness of weak solutions to the two dimensional full Ericksen-Leslie system with the Leslie stress and general Ericksen stress under the physical constrains on the Leslie coefficients. This question remains unknown even in the case when the Leslie stress is vanishing. The main technique used in the proof is Littlewood-Paley analysis performed in a very delicate way. Different from the earlier result in [28], we introduce a new metric and explore the algebraic structure of the molecular field.

Global existence of weak solution for the 2-D Ericksen–Leslie system

We prove the global existence of weak solution for two dimensional Ericksen–Leslie system with the Leslie stress and general Ericksen stress under the physical constrains on the Leslie coefficients. We also prove the local well-posedness of the Ericksen–Leslie system in two and three spatial dimensions.

Well-Posedness of the Ericksen–Leslie System

We prove the local well-posedness of the Ericksen–Leslie system, and the global well-posedness for small initial data under a physical constraint condition on the Leslie coefficients, which ensures that the energy of the system is dissipated. Instead of the Ginzburg–Landau approximation, we construct an approximate system with the dissipated energy based on a new formulation of the system.

Global weak solution and blow-up criterion of the general Ericksen–Leslie system for nematic liquid crystal flows

In this paper we investigate the three dimensional general Ericksen–Leslie (E–L) system with Ginzburg–Landau type approximation modeling nematic liquid crystal flows. First, by overcoming the difficulties from lack of maximum principle for the director equation and high order nonlinearities for the stress tensor, we prove existence of global-in-time weak solutions under physically meaningful boundary conditions and suitable assumptions on the Leslie coefficients, which ensures that the total energy of the E–L system is dissipated. Moreover, for the E–L system with periodic boundary conditions, we prove the local well-posedness of classical solutions under the so-called Parodiʼs relation and establish a blow-up criterion in terms of the temporal integral of both the maximum norm of the curl of the velocity field and the maximum norm of the gradient of the liquid crystal director field.

Geometrical content of Leslie coefficients

In this work, we will study how the effective geometry acquired by nematic molecules under thermal vibration contribute to the determination of the Leslie coefficients. To do this, we will divide this work in two sections. In the first section, we present the geometrical fundamentals of the so-called Hess-Baalss (HB) approach [D. Baalss and S. Hess, Phys. Rev. Lett. 57, 86 (1986)] where we show that its basic assumptions can be understood as a geometrical interpretation of de Gennes' passage from the microscopic to the macroscopic order parameter. In the second section, we use an extended version of the HB approach [M. Simões, K. Yamaguti, and A. J. Palangana, Phys. Rev. E 80, 061701 (2009)] to obtain the geometrical contribution to each Leslie coefficient. Our results will be compared with experimental data, and we will show that the Miesowicz's coefficients are connected as long as the ratio α(3)/α(4) between these Leslie coefficients can be considered small.

General Orientational Dependences of Nematic Liquid Crystal Viscosities under Uniform Shear Flow

General orientational dependences of the viscosity coefficients of nematic liquid crystals are derived in the zero-shear-rate limit, where the relative orientation of perfectly aligned ellipsoids to the shear flow geometry can be taken arbitrary, by using the Kirkwood-Smoluchowski equation for the pair-correlation function. For some special orientations the results are consistent with previous reports on Miesowicz viscosity coefficients, Leslie coefficients and others.

Non‐equilibrium molecular‐dynamics measurement of the Leslie coefficients of a Gay–Berne nematic liquid crystal

Using non‐equilibrium molecular‐dynamics (MD) simulations, we have measured the six Leslie coefficients of a nematic liquid crystal composed of molecules interacting via the Gay–Berne potential. In the presence of a simple shear flow, an alignment field is applied to control the molecular orientation and a uniform director is stabilized in the central region of the channel in which the liquid crystal is confined and sheared. With the director tuned by varying the applied field, a number of orientational states are stabilized in the presence of the shear flow and various viscous stress components are measured in these states of differently oriented directors. The six Leslie coefficients α i are determined by interpreting the MD measurement data for viscous stress according to the constitutive relations in the Ericksen–Leslie–Parodi theory. Our measurement of the Leslie coefficients shows the Parodi relation α2+α3 = α6−α5 is well satisfied. Given the values of the Leslie coefficients, liquid crystal orientations are evaluated for different alignment fields and shear rates, and then compared with those directly measured in MD simulations, demonstrating a quantitative agreement. Our simulation results show that in the Gay–Berne nematic liquid crystal, the viscous stress and the coupling between orientation and flow are well described by the Ericksen–Leslie–Parodi theory.

Fast phase modulation using dual‐frequency nematic liquid crystals

Abstract— Experimental study and numerical simulations of optical phase modulators based on nematic liquid crystals characterized by frequency sign inversion of the dielectric anisotropy has been performed. The received results point out an extreme role of back‐flow effects and electro‐hydrodynamic instability, which make restrictions for creating high‐speed phase modulators with large switchable values of phase retardation. It is shown by numerical simulations that the hydrodynamic destabilization can be suppressed by decreasing the ratio of Leslie coefficients |α2/α4,5|.

A method to estimate the Leslie coefficients of liquid crystals based on MBBA data

A new effective approach for estimating the Leslie coefficients of liquid crystals based on MBBA data is presented. Literature values of the temperature‐dependent Leslie viscosity coefficients of MBBA are used to fit with the analytical functions of Leslie coefficients based on the Imura and Okano theory, and a set of constants in these functions are determined. The same functions are then applied to two high birefringence liquid crystal mixtures, E7 and UCF‐2, to extract their Leslie coefficients based on their order parameters. Using the extracted Leslie coefficients, the simulation results agree very well with the experimental data.

Rheological properties in discotic liquid crystalline phases of 2,3,7,8,12,13-hexa(n-tetradecanoyloxy)truxene.

Measurements of rheological properties, including the electrorheological effect, are made in liquid crystalline phases, discotic nematic (ND) and discotic disordered rectangular columnar (Drd) phases, of 2,3,7,8,12,13-hexa(n-tetradecanoyloxy)truxene. It is found that the fluidity of the ND phase is Newtonian and an application of high electric field perpendicular to the flow direction (velocity gradient direction) causes a decrease of the viscosity. These results indicate that a flow alignment of the director occurs with its orientation near the velocity gradient direction and the signs of the Leslie coefficients alpha2 and alpha3 are positive. From the temperature dependence of alpha2, instability of the flow alignment structure in ND and Drd phases is also discussed.

ANALYTICAL SOLUTION FOR THERMAL CONVECTION IN A LAYER OF LIQUID CRYSTAL WITH FREE SURFACE

An Oberbeck-Bussinesq type model for convection in isothermically incompressible liquid crystals (nematics and cholesterics) is proposed. For its derivation, the nonequilibrium thermodynamics and asymptotic methods are employed. The model is based on the Leslie-Ericksen-Parodi (LEP) continuum theory [1–4] taken with some minor modifications. The problem of the thermal gravity convection in a plane horizontal layer with ‘free undeformable’ boundary and under the action of external magnetic or electric field is considered. Solution of this problem is obtained and presented in an explicit analytical form. Existence of the explicit analytical solution is very useful, since it allows us to investigate the influence of various parameters (Leslie coefficients, Frank moduli, anisotropic thermo-diffusivity, etc.) at the onset of convection.

TRANSIENT DIRECTOR ORIENTATION PATTERN OF CREEPING PLANE POISEUILLE FLOW OF A NEMATIC POLYMER

The time-dependent Leslie-Ericksen equation in numerically solved for creeping plane poiseuille flow of a nematic polymer (Poly Benzyl Glutamate). The Leslie coefficients and Frank orientation curvature elasticity constants of the nematic polymer reported in the literature were used for the computations of three dimensional director time (t) evolution over the two-dimensional (x-y) plane. The flow is along the x-axis, and the thickness axis the y-axis. On the bounding plates assumed strong homeotropic anchoring condition. On the inlet and outlet flow boundaries, zero surface couple condition is used. Although poiseuille flow has a zero shearing line in the middle with changing sing of shear rate when going through the mid gap region, unlike simple shear flow, directors escape the shear (x-y) plane as flow rate increases, which this computation study shows in accordance with various experimental studies. As the center velocity (U) increases, complex 3-D orientation structure emerges. When U is further increased, tube orientation walls reminiscent of the tube walls of shear flow emerges, not through pinching and reconnection of orientation wall pair, but through emanation from the central backbone orientation wall structure.

Frank Matthews Leslie. 8 March 1935 – 15 June 2000

Frank Leslie was a distinguished applied mathematician, who was above all foremost in creating the modern continuum theory of nematic liquid crystals in the late 1960s. This theory is now known as the Ericksen-Leslie theory, and the crucial elements in it as Leslie coefficients. After developing the hydrodynamic theory of nematic liquid crystals, he went on to perform a similar task in the 1990s for smectic liquid crystals. He also actively collaborated with experimentalists and engineers involved in liquid crystal applications, and his work has been extremely influential in the development of liquid crystal display (LCD) device technology.