Theory Of Nematic Liquid Crystals Research Articles

Orientational Order Parameter Undecyloxy-Cyanobiphenyl (11OCB) and DodecyloxyCyanobiphenyl (12OCB) by using Optical Birefringence Method

High resolution optical birefringence studies throughout the nematic and smectic A phases of undecyloxy-cyanobiphenyl (11OCB) and dodecyloxy-cyanobiphenyl (12OCB) have been performed to evaluate orientational order parameter (<P2>) as a function of temperature at wavelength λ=632.8 nm. The temperature dependence of the birefringence (n) data have been exploited to assess <P2> using a well-known Haller’s extrapolation method. Effectively, in the nematic and smectic A phases the values of <P2> have been fitted with the estimated values from Maier-Saupe mean field theory for nematic liquid crystals and McMillan mean field theory for smectic liquid crystals. Experimentally obtained <P2> values have been compared with those achieved from Maier-Saupe mean field theory and McMillan mean field theory. The agreement between experimental and calculated <P2> values at nematic-isotropic (N-I) is much higher than experimental <P2> values.

The solvability for a nonlinear degenerate hyperbolic–parabolic coupled system arising from nematic liquid crystals

Abstract This paper focuses on the Cauchy problem for a one-dimensional quasilinear hyperbolic–parabolic coupled system with initial data given on a line of parabolicity. The coupled system is derived from the Poiseuille flow of full Ericksen–Leslie model in the theory of nematic liquid crystals, which incorporates the crystal and liquid properties of the materials. The main difficulty comes from the degeneracy of the hyperbolic equation, which makes that the system is not continuously differentiable and then the classical methods for the strictly hyperbolic–parabolic coupled systems are invalid. With a choice of a suitable space for the unknown variable of the parabolic equation, we first solve the degenerate hyperbolic problem in a partial hodograph plane and express the smooth solution in terms of the original variables. Based on the smooth solution of the hyperbolic equation, we then construct an iterative sequence for the unknown variable of the parabolic equation by the fundamental solution of the heat equation. Finally, we verify the uniform convergence of the iterative sequence in the selected function space and establish the local existence and uniqueness of classical solutions to the degenerate coupled problem.

On symmetry of energy minimizing harmonic-type maps on cylindrical surfaces

<abstract><p>The paper concerns the analysis of global minimizers of a Dirichlet-type energy functional in the class of $ \mathbb{S}^2 $-valued maps defined in cylindrical surfaces. The model naturally arises as a curved thin-film limit in the theories of nematic liquid crystals and micromagnetics. We show that minimal configurations are $ z $-invariant and that energy minimizers in the class of weakly axially symmetric competitors are, in fact, axially symmetric. Our main result is a family of <italic>sharp</italic> Poincaré-type inequality on the circular cylinder, which allows for establishing a nearly complete picture of the energy landscape. The presence of symmetry-breaking phenomena is highlighted and discussed. Finally, we provide a complete characterization of in-plane minimizers, which typically appear in numerical simulations for reasons we explain.</p></abstract>

Violation of Ericksen Inequalities in Lyotropic Chromonic Liquid Crystals

By analyzing elastic theory for nematic liquid crystals, we distinguish three regimes of elastic constants. In one regime, the Ericksen inequalities are satisfied, and the ground state of the director field is uniform. In a second regime, certain necessary inequalities are violated, and the free energy is thermodynamically unstable. Between those possibilities, there is an intermediate regime, where the Ericksen inequalities are violated but the system is still stable. Remarkably, lyotropic chromonic liquid crystals are in the intermediate regime. We investigate the nonuniform structure of the director field in that regime, show that it depends sensitively on system geometry, and discuss the implications for lyotropic chromonic liquid crystals.

Recent analytic development of the dynamic $ Q $-tensor theory for nematic liquid crystals

<abstract><p>Liquid crystals are a typical type of soft matter that are intermediate between conventional crystalline solids and isotropic fluids. The nematic phase is the simplest liquid crystal phase, and has been studied the most in the mathematical community. There are various continuum models to describe liquid crystals of nematic type, and $ Q $-tensor theory is one among them. The aim of this paper is to give a brief review of recent PDE results regarding the $ Q $-tensor theory in dynamic configurations.</p></abstract>

Existence and Stability of Kayaking Orbits for Nematic Liquid Crystals in Simple Shear Flow

We use geometric methods of equivariant dynamical systems to address a long-standing open problem in the theory of nematic liquid crystals, namely a proof of the existence and asymptotic stability of kayaking periodic orbits in response to steady shear flow. These are orbits for which the principal axis of orientation of the molecular field (the director) rotates out of the plane of shear and around the vorticity axis. With a small parameter attached to the symmetric part of the velocity gradient, the problem can be viewed as a symmetry-breaking bifurcation from an orbit of the rotation group mathrm{SO}(3) that contains both logrolling (equilibrium) and tumbling (periodic rotation of the director within the plane of shear) regimes as well as a continuum of neutrally stable kayaking orbits. The results turn out to require expansion to second order in the perturbation parameter.

Cooling a spherical nematic shell.

Within the framework of Landau-de Gennes theory for nematic liquid crystals, we study the temperature-induced isotropic-nematic phase transition on a spherical shell under the assumption of degenerate tangential anchoring. Below a critical temperature, a thin layer of nematic coating a microscopic spherical particle exhibits nonuniform textures due to the geometrical frustration. We find the exact value of the critical threshold for the temperature and determine exactly the nematic textures at the transition by means of a weakly nonlinear analysis. The critical temperature is affected by the extrinsic curvature of the sphere, and the nematic alignment is consistent with the Poincaré-Hopf index theorem and experimental observations. The stability analysis of the bifurcate textures at the isotropic-nematic transition highlights that only the tetrahedral configuration is stable.

Nematic reorientation effects on resonant modes, wavelength mismatch, and slow-light phenomena in one-dimensional magnetophotonic crystals with a dual anisotropic defect.

The present study is devoted to the investigation of spectral properties of an alternated sequence of magnetic and dielectric layers containing a dual defect based on magnetic and nematic layers. Combining the Hydrodynamic Continuum Theory for nematic liquid crystals and Berreman's formalism, we determine how the nematic ordering affects the light localization, polarization rotation, and slow-light phenomena observed in the magnetophotonic system. In particular, we analyze the effects associated with a field-induced reorientation of the director in a nematic defect with strong planar boundary conditions. Our results reveal that field-induced reorientation of the nematic ordering can be used as an efficient mechanism to tune and control the spectral properties of magnetophotonic structure, anomalies in group velocity, and the wavelength mismatch between resonant mode and maximum polarization. The effects of nematic layer thickness are also analyzed.

Symmetry and Multiplicity of Solutions in a Two-Dimensional Landau\u2013de Gennes Model for Liquid Crystals

We consider a variational two-dimensional Landau–de Gennes model in the theory of nematic liquid crystals in a disk of radius R. We prove that under a symmetric boundary condition carrying a topological defect of degree frac{k}{2} for some given even non-zero integer k, there are exactly two minimizers for all large enough R. We show that the minimizers do not inherit the full symmetry structure of the energy functional and the boundary data. We further show that there are at least five symmetric critical points.

Singularity and existence for a multidimensional variational wave equation arising from nematic liquid crystals

This article is focused on a multidimensional nonlinear variational wave equation that is the Euler-Lagrange equation of a variational principle arising from the theory of nematic liquid crystals. By using the method of characteristics, we show that the smooth solutions for the spherically symmetric variational wave equation break down in finite time, even for an arbitrarily small initial energy. The key point is the energy equation derived from the variational wave equation, which is also used to establish the existence of energy-conservative weak solutions in a finite period.

Computational molecular field theory for nematic liquid crystals.

Nematic liquid crystals exhibit configurations in which the underlying ordering changes markedly on macroscopic length scales. Such structures include topological defects in the nematic phase and tactoids within nematic-isotropic coexistence. We discuss a computational study of inhomogeneous configurations that is based on a field theory extension of the Maier-Saupe molecular model of a uniaxial, nematic liquid crystal. A tensor order parameter is defined as the second moment of an orientational probability distribution, leading to a free energy that is not convex within the isotropic-nematic coexistence region, and that goes to infinity if the eigenvalues of the order parameter become nonphysical. Computations of the spatial profile of the order parameter are presented for an isotropic-nematic interface in one dimension, a tactoid in two dimensions, and a nematic disclination in two dimensions. We compare our results to those given by the Landau-de Gennes free energy for the same configurations and discuss the advantages of such a model over the latter.

Sphere-valued harmonic maps with surface energy and the K13 problem

AbstractWe consider an energy functional motivated by the celebrated {K_{13}} problem in the Oseen–Frank theory of nematic liquid crystals. It is defined for sphere-valued functions and appears as the usual Dirichlet energy with an additional surface term. It is known that this energy is unbounded from below and our aim has been to study the local minimisers. We show that even having a critical point in a suitable energy space imposes severe restrictions on the boundary conditions. Having suitable boundary conditions makes the energy functional bounded and in this case we study the partial regularity of the global minimisers.

Nonlinear nematic elasticity

The nonlinear elastic properties of nematic liquid crystals have acquired new interest with the recent experimental observation of bulk modulated nematic phases which are composed by achiral molecules. We extend the Oseen-Zocher-Frank's elastic theory for nematic liquid crystals by including gradients of the nematic strain tensor in the elastic deformation energy. The invariants of the elastic tensor fields, up to the fourth order in the nematic director spatial derivatives, are calculated. An alternative approach that consists in the extension of the linear elastic energy to higher powers of the nematic strain tensor, as in classical elasticity of solids, is also developed. The twist-bend nematic modulated phase is investigated by both approaches and the results are critically compared. The conical angle of the twist-bend phase is calculated as function of the elastic constants. Surface-like effects are considered. Finally, we demonstrate that a splay-bend nematic phase with small oscillations of the nematic director around an axis is prohibited.

On the Cauchy problem for a nonlinear variational wave equation with degenerate initial data

This paper is focused on a one-dimensional nonlinear variational wave equation which is the Euler–Lagrange equations of a variational principle arising in the theory of nematic liquid crystals and a few other physical contexts. We establish the local existence and uniqueness of classical solutions to its Cauchy problem with initial data given on the parabolic degenerating line.

Oseen–Frank-type theories of ordered media as the Γ-limit of a non-local mean-field free energy

In this work we recover the Oseen–Frank theory of nematic liquid crystals as a [Formula: see text]-limit of a particular mean-field free energy as the sample size becomes infinitely large. The Frank constants are not necessarily all equal. Our analysis takes place in a broader framework however, also providing results for more general systems such as biaxial or polar molecules. We start from a mesoscopic model describing a competition between entropy and a non-local pairwise interaction between molecules. We assume the interaction potential is separable so that the energy can be reduced to a model involving a macroscopic order parameter. We assume only integrability and decay properties of the macroscopic interaction, but no regularity assumptions are required. In particular, singular interactions are permitted. The analysis becomes significantly simpler on a translationally invariant domain, so we first consider periodic domains with increasing domain of periodicity. Then we tackle the problem on a Lipschitz domain with non-local boundary conditions by considering an asymptotically equivalent problem on periodic domains. We conclude by applying the results to a case which reduces to the Oseen–Frank model of elasticity, and give expressions for the Frank constants in terms of integrals of the interaction kernel.

Symmetry Breaking and Restoration in the Ginzburg–Landau Model of Nematic Liquid Crystals

In this paper we study qualitative properties of global minimizers of the Ginzburg-Landau energy which describes light-matter interaction in the theory of nematic liquid crystals near the Friedrichs transition. This model is depends on two parameters: $\epsilon>0$ which is small and represents the coherence scale of the system and $a\geq 0$ which represents the intensity of the applied laser light. In particular we are interested in the phenomenon of symmetry breaking as $a$ and $\epsilon$ vary. We show that when $a=0$ the global minimizer is radially symmetric and unique and that its symmetry is instantly broken as $a>0$ and then restored for sufficiently large values of $a$. Symmetry breaking is associated with the presence of a new type of topological defect which we named the shadow vortex. The symmetry breaking scenario is a rigorous confirmation of experimental and numerical results obtained in our earlier work.

The Oseen–Frank Limit of Onsager’s Molecular Theory for Liquid Crystals

We study the relationship between Onsager's molecular theory and the Oseen-Frank theory for nematic liquid crystals. Under the molecular setting, we consider the free energy that includes the effects of nonlocal molecular interactions. By imposing the strong anchoring boundary condition on the second moment of the number density function, we prove the existence of global minimizers for the free energy. Moreover, when the re-scaled interaction distance tends to zero, the corresponding global minimizers will converge to a uniaxial distribution whose orientation is described by a minimizer of Oseen-Frank energy.

Existence of smooth solutions to a one-dimensional nonlinear degenerate variational wave equation

This paper is focused on a one-dimensional nonlinear variational wave equation which is the Euler–Lagrange equations of a variational principle arising in the theory of nematic liquid crystals and a few other physical contexts. We establish the global existence of smooth solutions to its degenerate initial–boundary value problem under relaxed conditions on the initial–boundary data. Moreover, we show that the solution is uniformly C1,α continuous up to the degenerate boundary and the degenerate curve is C1,α continuous for α∈(0,12).

Liquid relaxation: A new Parodi-like relation for nematic liquid crystals.

We put forward a hydrodynamic theory of nematic liquid crystals that includes both anisotropic elasticity and dynamic relaxation. Liquid remodeling is encompassed through a continuous update of the shear-stress free configuration. The low-frequency limit of the dynamical theory reproduces the classical Ericksen-Leslie theory, but it predicts two independent identities between the six Leslie viscosity coefficients. One replicates Parodi's relation, while the other-which involves five Leslie viscosities in a nonlinear way-is new. We discuss its significance, and we test its validity against evidence from physical experiments, independent theoretical predictions, and molecular-dynamics simulations.

Radial symmetry on three-dimensional shells in the Landau–de Gennes theory

We study the radial-hedgehog solution on a three-dimensional (3D) spherical shell with radial boundary conditions, within the Landau–de Gennes theory for nematic liquid crystals. We prove that the radial-hedgehog solution is the unique minimizer of the Landau–de Gennes energy in two separate regimes: (i) for thin shells when the temperature is below the critical nematic supercooling temperature and (ii) for a fixed shell width at sufficiently low temperatures. In case (i), we provide explicit geometry-dependent criteria for the global minimality of the radial-hedgehog solution.