Theory Of Nematic Liquid Crystals Research Articles

Director Field Model of the Primary Visual Cortex for Contour Detection

We aim to build the simplest possible model capable of detecting long, noisy contours in a cluttered visual scene. For this, we model the neural dynamics in the primate primary visual cortex in terms of a continuous director field that describes the average rate and the average orientational preference of active neurons at a particular point in the cortex. We then use a linear-nonlinear dynamical model with long range connectivity patterns to enforce long-range statistical context present in the analyzed images. The resulting model has substantially fewer degrees of freedom than traditional models, and yet it can distinguish large contiguous objects from the background clutter by suppressing the clutter and by filling-in occluded elements of object contours. This results in high-precision, high-recall detection of large objects in cluttered scenes. Parenthetically, our model has a direct correspondence with the Landau - de Gennes theory of nematic liquid crystal in two dimensions.

Diffusive transport in two-dimensional nematics

We discuss a dynamical theory for nematic liquid crystals describing the stage of evolution in which the hydrodynamic fluid motion has already equilibrated and the subsequent evolution proceeds via diffusive motion of the orientational degrees of freedom. This diffusion induces a slow motion of singularities of the order parameter field. Using asymptotic methods for gradient flows, we establish a relation between the Doi-Smoluchowski kinetic equation and vortex dynamics in two-dimensional systems. We also discuss moment closures for the kinetic equation and Landau-de Gennes-type free energy dissipation.

Liquid Crystal Elastomers and Phase Transitions in Actin Rod Networks

In this article, we construct and analyze models of anisotropic crosslinked polymers employing tools from the theories of nematic liquid crystals and liquid crystal elastomers. The anisotropy of these systems stems from the presence of rigid rod molecular units in the network. We construct energy functionals for compressible and incompressible elastomers as well as for rod-fluid networks. The theorems on the minimization of these energies combines methods of isotropic nonlinear elasticity with the theory of lyotropic liquid crystals. Two of the theorems refer to incompressible elastomers, in the cases that the bulk liquid crystal energy is given by the well-known polynomial form of the Landau--de Gennes theory and also in the case of a singular potential. Another theorem refers to compressible elastomers, and in the last row, the rod density is taken as a main field of the model. We apply our results to the study of phase transitions in networks of rigid rods, in order to model the behavior of actin filament systems found in the cytoskeleton. Our results show a good agreement with the molecular dynamics experiments reported in the literature as well as with some laboratory experiments. The model does not include polydispersity effects due to variable rod shape and size, and it does not account either for phase transitions to lamellar phases.

Stability of the vortex defect in the Landau–de Gennes theory for nematic liquid crystals

We analyze the radially symmetric solution corresponding to the vortex defect (the so-called melting hedgehog) in the Landau–de Gennes theory for nematic liquid crystals. We prove the existence, uniqueness and stability results of the melting hedgehog.

Conservative solutions to a system of asymptotic variational wave equations

This paper is concerned with an asymptotic variational wave system which models weakly nonlinear waves for a system of variational wave equations arising in the theory of nematic liquid crystals and a few other physical contexts. By constructing a global semigroup, we establish the well-posedness of the initial–boundary value problem within the class of energy-conservative solutions for initial data of finite energy.

Maier–Saupe nematogenic fluid interacting with an isotropic and an anisotropic Yukawa potentials: field theory description

We adopt a field-theoretical approach to study equilibrium properties of a homogeneous Maier–Saupe nematogenic fluid interacting with an isotropic and an anisotropic Yukawa potentials. In the mean field approximation, we retrieve the standard Maier–Saupe theory for nematic liquid crystals. Beyond the mean field approximation, we show that the repulsive isotropic potential plays an important role in the description of the nematic phase. In the Gaussian approximation analytical expressions for the correlation functions, the free energy, the pressure, and the elasticity constant are derived. We establish a simple condition on the parameters of the isotropic potential for the correlation functions and thermodynamic properties to be defined in the nematic phase. Subsequently, we find corrections to the single-particle distribution function and the order parameter due to fluctuations.

Global energy conservative solutions to a system of variational wave equations

This paper is concerned with a system of variational wave equations which is the Euler–Lagrange equations of a variational principle arising in the theory of nematic liquid crystals and a few other physical contexts. The global existence of an energy-conservative weak solution to its Cauchy problem for initial data of finite energy is established by using the method of energy-dependent coordinates and the Young measure theory.

Multistability in planar liquid crystal wells

A planar bistable liquid crystal device, reported in Tsakonas et al. [Appl. Phys. Lett. 90, 111913 (2007)], is modeled within the Landau-de Gennes theory for nematic liquid crystals. This planar device consists of an array of square micrometer-sized wells. We obtain six different classes of equilibrium profiles and these profiles are classified as diagonal or rotated solutions. In the strong anchoring case, we propose a Dirichlet boundary condition that mimics the experimentally imposed tangent boundary conditions. In the weak anchoring case, we present a suitable surface energy and study the multiplicity of solutions as a function of the anchoring strength. We find that diagonal solutions exist for all values of the anchoring strength W ≥ 0, while rotated solutions only exist for W ≥ W_{c}>0, where W_{c} is a critical anchoring strength that has been computed numerically. We propose a dynamic model for the switching mechanisms based on only dielectric effects. For sufficiently strong external electric fields, we numerically demonstrate diagonal-to-rotated and rotated-to-diagonal switching by allowing for variable anchoring strength across the domain boundary.

Surface free energies for nematic shells

We propose a continuum model to describe the molecular alignment in thin nematic shells. By contrast with previous accounts, the two-dimensional free energy, aimed at describing the physics of thin films of nematics deposited on curved substrates, is not postulated, but it is deduced from the conventional three-dimensional theories of nematic liquid crystals. Both the director and the order-tensor theories are taken into account. The so-obtained surface energies exhibit extra terms compared to earlier models. These terms reflect the coupling of the shell extrinsic curvature with the nematic order parameters. As expected, the shape of the shell plays a key role in the equilibrium configurations of nematics coating it.

Conservative solutions to a system of variational wave equations

We investigate a system of variational wave equations 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 an energy-conservative weak solution to its Cauchy problem for initial data of finite energy. The main difficulty arises from the possible concentration of energy. We construct the solution by introducing a new set of variables depending on the energy, whereby all singularities are resolved.

The radial-hedgehog solution in Landau–de Gennes' theory for nematic liquid crystals

We study the radial-hedgehog solution in a three-dimensional spherical droplet, with homeotropic boundary conditions, within the Landau–de Gennes theory for nematic liquid crystals. The radial-hedgehog solution is a candidate for a global Landau–de Gennes minimiser in this model framework and is also a prototype configuration for studying isolated point defects in condensed matter physics. The static properties of the radial-hedgehog solution are governed by a non-linear singular ordinary differential equation. We study the analogies betweenGinzburg–Landauvortices and the radial-hedgehog solution and demonstrate a Ginzburg–Landau limit for the Landau–de Gennes theory. We prove that the radial-hedgehog solution is not the global Landau–de Gennes minimiser for droplets of finite radius and sufficiently low temperatures and prove the stability of the radial-hedgehog solution in other parameter regimes. These results contain quantitative information about the effect of geometry and temperature on the properties of the radial-hedgehog solution and the associated biaxial instabilities.

Zenithal bistable device: Comparison of modeling and experiment

A comparative modeling and experimental study of the zenithal bistable liquid crystal device is presented. A dynamic Landau de Gennes theory of nematic liquid crystals is solved numerically to model the electric field induced latching of the device and the results are compared with experimental measurements and theoretical approximations. The study gives a clear insight into the latching mechanism dynamics and enables the dependence of the device latching on both material parameters and surface shape to be determined. Analytical approximation highlights a route to optimize material selection in terms of latching voltages and the numerical model, which includes an accurate surface representation, recovers the complex surface shape effects. Predictions of device performance are presented as a function of both surface anchoring strength and surface shape and grating pitch. A measurement of the homeotropic anchoring energy has been undertaken by comparing the voltage response as a function of cell gap; we find the homeotropic anchoring energies can be varied in the range 0.5 to 4 ( 10^{-4} J m^{-2} ).

Excluded-Volume Short-Range Repulsive Potential for Tetrahedral Molecules

Tetrahedral symmetry in nematic liquid crystals has been studied for several years, since the seminal paper by Fel [1]. Amongst statistical theories of nematic liquid crystals, a molecular mean-field model has proven to be quite effective in predicting phase sequences for thermotropic biaxial nematic molecules endowed with D2h symmetry, as a function of the parameters entering their interaction potential [2,3]. This model has highlighted the role of a partially repulsive quadrupolar potential of mean torque in promoting condensed phases. It has been shown that the quadrupolar approximation to the excluded-volume interaction between hard spherocuboids can be written precisely as the superposition of two London interactions: one repulsive and one attractive [4]; furthermore, polar steric interactions have been shown to be capable of inducing orientationally ordered states possibly unexpected [5]. By adapting a numerical code available to the scientific community, we evaluate the analytical excluded volume function for non-convex tetrahedral molecules, modelled as chains of tangent hard spheres. Since this function is overly complicated, we expand it over a suitable set of Symmetry Adapted Wigner Functions (SAWFs).

Effective Rotational Viscosity of Vertical Alignment Nematic Liquid Crystal Cells

An effective rotational viscosity of vertical alignment (homeotropic) nematic liquid crystal (NLC) cells with negative dielectric anisotropy is derived from a theory of NLCs in which flow effects under the free-slip boundary condition are taken into account. The effective rotational viscosity is a function of the Leslie viscosity coefficients and is much smaller than the rotational viscosity at the initial stage of the director reorientation induced by external electric field to the NLC cells. This is the origin of fast response of vertical alignment of NLC cells.

Effects of viscosities on the transient current in homeotropic nematic liquid crystal cells

Effects of the Leslie viscosity coefficients α i ( i = 1, 2, …, 5) of nematic liquid crystals (NLCs) with negative dielectric anisotropy (Δ ε < 0) on the electric-field-induced director reorientation in homeotropic NLC cells have been studied from the analysis of the transient current induced by step voltage application. The transient current in a homeotropic NLC cell with Δ ε < 0 was well reproduced by computer simulation, based on the theory of NLCs in which the flow effects and the free-slip boundary condition are taken into account. It is found that the response time of vertical alignment NLC displays is dominantly governed by α 2 and α 4 + α 5 of NLCs with Δ ε < 0.

Measurement of viscosities from the transient current of nematic liquid crystals with negative dielectric anisotropy

A method for measuring the complete set of the Leslie viscosity coefficients of nematic liquid crystals (NLCs) with negative dielectric anisotropy (Δ ε < 0) is proposed. The method is based on the analysis of the transient current, induced by step voltage excitation, in homeotropic NLC cells. It has been shown that the transient current in homeotropic NLC cells with Δ ε < 0 can be explained by the theory of NLCs in which the flow effects and the free-slip boundary condition are taken into account [Y. Iwata et al., Jpn. J. Appl. Phys. 43 (2004) L 1588]. It is demonstrated that the complete set of the Leslie viscosity coefficients of NLCs with Δ ε < 0 can be simultaneously and accurately measured by fitting the transient current calculated by the theory to the experimental data with a differential correction method.

Algebraic Theory of Linear Viscoelastic Nematodynamics

This paper consists of two parts. The first one develops algebraic theory of linear anisotropic nematic “N-operators” build up on the additive group of traceless second rank 3D tensors. These operators have been implicitly used in continual theories of nematic liquid crystals and weakly elastic nematic elastomers. It is shown that there exists a non-commutative, multiplicative group N6 of N-operators build up on a manifold in 6D space of parameters. Positive N-operators, which in physical applications hold thermodynamic stability constraints, do not generally form a subgroup of group N6. A three-parametric, commutative transversal-isotropic subgroup \({\text{S}}_{\text{3}}\subset {\text{N}}_{\text{6}} \) of positive symmetric nematic operators is also briefly discussed. The special case of singular, non-negative symmetric N-operators reveals the algebraic structure of nematic soft deformation modes. The second part of the paper develops a theory of linear viscoelastic nematodynamics applicable to liquid crystalline polymer. The viscous and elastic nematic components in theory are described by using the Leslie–Ericksen–Parodi (LEP) approach for viscous nematics and de Gennes free energy for weakly elastic nematic elastomers. The case of applied external magnetic field exemplifies the occurrence of non-symmetric stresses. In spite of multi-(10) parametric character of the theory, the use of nematic operators presents it in a transparent form. When the magnetic field is absent, the theory is simplified for symmetric case with six parameters, and takes an extremely simple, two-parametric form for viscoelastic nematodynamics with possible soft deformation modes. It is shown that the linear nematodynamics is always reducible to the LEP-like equations where the coefficients are changed for linear memory functionals whose parameters are calculated from original viscosities and moduli.

Necessary conditions for a minimum at a radial cavitating singularity in nonlinear elasticity

It is still not known if the radial cavitating minimizers obtained by Ball [J.M. Ball, Discontinuous equilibrium solutions and cavitation in nonlinear elasticity, Phil. Trans. R. Soc. Lond. A 306 (1982) 557–611] (and subsequently by many others) are global minimizers of any physically reasonable nonlinearly elastic energy. We therefore consider in this paper the related problem of obtaining necessary conditions for these radial solutions to be minimizers with respect to nonradial perturbations. A standard blowup argument applied to either an inner or an outer variation yields an apparently new inequality that, for most constitutive relations, has yet to be verified. However, in the special case of a compressible neo-Hookean material, W(F)=μ2|F|2+h(detF), we show that the inequality produced by an outer variation clearly holds whilst that produced by an inner variation is a well-known inequality (first proven by Brezis, Coron, and Lieb [H. Brezis, J.-M. Coron, E.H. Lieb, Harmonic maps with defects, Comm. Math. Phys. 107 (1986) 649–705] ) which arises in the theory of nematic liquid crystals:∫B|∇n(x)|2dx⩾∫B|∇(x|x|)|2dx=8π for all n∈W1,2(B,∂B) (so that |n|=1 a.e.) that satisfy n(x)=x/|x| on ∂B, where B is the unit ball in R3.

Nonlocal model for nematic liquid-crystal elastomers

We have developed a fully nonlocal model to describe the dynamic behavior of nematic liquid-crystal elastomers. The free energy, incorporating both elastic and nematic contributions, is a function of the material displacement vector and the orientational order parameter tensor. The free energy cost of spatial variations of these order parameters is taken into account through nonlocal interactions rather than through the use of gradient expansions. We also give an expression for the Rayleigh dissipation function. The equations of motion for displacement and orientational order are obtained from the free energy and the dissipation function by the use of a Lagrangian approach. We examine the free energy and the equations of motion in the limit of long-wavelength and small-amplitude variations of the displacement and the orientational order parameter. We compare our results with those in the literature. If the scalar order parameter is held fixed, we recover the usual viscoelastic theory for nematic liquid crystals.

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.