Landau-de Gennes Model Research Articles

Determination of Temperature Dependence of Kerr Constant for Nematic Liquid Crystal

Nematic liquid crystals are the preferred materials for linear and nonlinear optics, where frequently required photosensitivity is provided by appropriate doping. This investigationis concerned with the determination of the Kerr constant of nematic liquid crystal (9CHBT). The temperature dependence of the Kerr constant B for azo dye doped nematic liquid crystalshas been studied. The sample has a positive Kerr constant which increases with decreasing of temperature. The linear dependence of the Kerr constant on (T – T*)−1is found to be ingood agreement with the predications of the Landau-de Gennes model.

Investigation of the Kerr effect and third-order susceptibility constants in a nematic liquid crystal

In this paper, the static Kerr effect was investigated for the W-1680 at temperatures above the nematic–isotropic transition temperature and compared with 9CHBT. We experimentally measured the electro-optical Kerr constant and the pre-transitional behavior. The Kerr law has been confirmed and the variation of the inverse Kerr constant, with a temperature above the smectic–isotropic transition temperatures, were determined. The linear dependence of (T − T*) − 1 on the Kerr constant is found to be in good agreement with the predications of the Landau–De Gennes model. Additionally, the temperature dependence of third order susceptibility has been studied. This value for these samples will reduce when the temperature increases.

Characterising texture formation in fibre lattices embedded in a nematic liquid crystal matrix

A two-dimensional computational study is performed on the texturing of fibre-filled nematic liquid crystals using the Landau-de Gennes model describing the spatio-temporal evolution of the second moment of the orientation distribution function or quadrupolar tensor order parameter. The investigation is performed on a consistent computational domain comprising a square array of four circular fibres embedded within a unit square containing a uniaxial low molar mass calamitic liquid crystal. Interest is focused on the role of temperature, boundary conditions and their effect on the nucleation and evolution of defect structures. Thermal effects are characterised below and above the temperature at which the nematic state is stable. Simulations in the stable nematic state serves as a scenario for investigating the effect of imposing different external boundary conditions, namely periodic and Dirichlet; the former describes a square lattice array of fibres embedded in a nematic liquid crystal, and the latterdescribes a four-fibre arrangement in an aligned nematic material. In each case, the influence of temperature is characterised, with defect structures forming and either remaining or splitting into lower strength defects. For fibre lattices, splitting transitions of defects at the centre of the domain occur at a critical temperature, but for the four-fibre arrangement, defect transitions occur continuously over a temperature range. The discontinuous defect splitting transition in fibre arrays occurs at lower temperatures than the continuous defect transformation in the four-fibre arrangement. At sufficiently low temperatures, the four-fibre arrangement and the fibre lattice give the same texture consisting of two disclination lines close to each fibre. The evolution of the texture with respect to temperature can be characterised as a change from single-fibre mode at low temperature to a collective mode with a centre-located heterogeneity at higher temperature. At higher temperatures, in the stable isotropic state, it is shown that surface-induced ordering arising from the fibre/liquid crystal interaction propagates into the bulk forming thin disclination lattices around the four-fibre configuration.

Texture formation mechanisms in faceted particles embedded in a nematic liquid crystal matrix

This paper presents a computational study of filled nematics with the aim of characterizing novel texturing processes that occur when the embedded particles have geometric singularities such as edges. As a generic texturing process due to interacting material and geometric singularities, two dimensional numerical simulations of a single square particle embedded in a calamitic thermotropic nematic liquid crystal were performed using the Landau–de Gennes model and material properties corresponding to 5CB. The results were condensed into texture phase diagrams in terms of temperature and particle size. In addition to the usual bulk defect modes found in filled nematics with smooth geometries, the square corners introduce surface defects that interact with each other or with bulk defects. The net result is that for faceted particles three modes are possible: (i) string defect modes, (ii) mixed surface/bulk defect modes, and (iii) surface defect modes. The modes' stability and transitions are explained in terms of defect energies. Accurate simulations that target transitions between the modes demonstrate critical slowing down such that long lived unstable complex textures are pinned. The present results offer new routes to engineered texturing of nematic liquid crystals by embedding faceted particles.

Dynamics of the Kerr effect in the isotropic phase of comb-shaped nematic polymers

The equilibrium Kerr effect and its dynamics are investigated in isotropic melts of the comb-shaped LC polymer poly(4-[(4′-cyano-1,1′-biphenyl-4-yl)oxy]butyl-2-acrylate) and its copolymers with acrylic acid or methyl methacrylate and isotropic melts of the LC ionomer containing rubidium ions in a wide temperature range. A sharp change in the temperature dependence of relaxation times for the induced macroscopic orientational order is found for the LC copolymers and the LC ionomer in the temperature range 10–15 K higher than the temperature of the nematic-isotropic phase transition. In contrast, the phenomena caused by the small-scale dynamics, such as electric conductivity and dipole relaxation, are approximated by the conventional Arrhenius type dependence, and the equilibrium electrooptical properties of the copolymers are well characterized via the Landau-De Gennes model. The nonclassical behavior of macroscopic relaxation times may be associated with temperature variations in the contribution of main polymer chains to the orientation macroscopic dynamics of the melt, in which the decisive role is played by side mesogenic groups.

Scaling of the elastic contribution to the surface free energy of a nematic liquid crystal on a sawtoothed substrate

We characterize the elastic contribution to the surface free energy of a nematic liquid crystal in the presence of a sawtooth substrate. Our findings are based on numerical minimization of the Landau-de Gennes model and analytical calculations on the Frank-Oseen theory. The nucleation of disclination lines (characterized by non-half-integer winding numbers) in the wedges and apexes of the substrate induces a leading order proportional to q ln q to the elastic contribution to the surface free-energy density, with q being the wave number associated with the substrate periodicity.

Layer Undulations in Smectic A Liquid Crystals

We investigate instabilities of smectic A liquid crystals when a magnetic field is applied in the direction parallel to smectic layers. We use the Landau-de Gennes model of smectic A liquid crystals to characterize the critical magnetic field. When smectic A liquid crystals are confined between parallel plates, we derive analytic estimates for the magnetic field strength, at which the undeformed state loses its stability. We also present numerical simulations to confirm the Helfrich-Hurault effect due to the applied magnetic field.

Edge dislocation core structure in lamellar smectic-A liquid crystals

The core of the smectic-A edge dislocation is resolved via simulation of a previously validated high-order Landau–de Gennes model. Simulations reveal that the smectic-A edge dislocation core is composed of a biaxial orientation field, in addition to a singularity in translational order, resulting from frustration between orientational and translational elasticity.

Liquid crystal models of biological materials and processes

This paper presents an overview of liquid crystal (LC) models of phase diagrams, phase transitions, self-assembly, interfaces, defects, and rheology and their integrated applications to biological mesophase materials and processes. Biological liquid crystals, classified into analogues (helicoidal plywoods), biopolymer solutions (in vitro DNA, polypeptides, collagen solutions) and in vivo LCs (membranes, silk, DNA), are discussed in terms of molecular characteristics and the symmetry of the thermodynamic phases. The thermodynamics and self-assembly of biological liquid crystals (BLCs) are discussed in terms of the Doi–Maier–Saupe lyotropic model and its extensions to chiral phases, showing the role of excluded volume and chirality. The defect physics of BLCs is described using the Landau–de Gennes model of chiral and achiral nematostatics to identify (i) thermodynamic phases, and (ii) observed textures and defect lattices under confinement and flow. The rheology and flow properties of BLCs are described using the Leslie–Ericksen and the Landau–de Gennes models of nematodynamics. The applications of integrated thermodynamic/defect/rheology modeling to the experimental characterization of several BLCs, including collagen and DNA solutions, are shown to provide organizing principles and quantitative tools to establish the properties of these natural materials. The phase transitions, tactoidal spherulites, flow-birefringence in dilute solutions, and banded textures in sheared concentrated solutions of collagen show how the principles of LC physics operate in BLC materials. Drying and spreading drops of DNA solutions leading to the formation of nematic monodomain aligned along the contact line are shown to follow self-assembly structuring under the action of interfacial, bulk, and contact line torques. Finally modeling of spider and silkworm LC spinning is presented as an example of biological polymer processing, where a high performance fiber material is produced through a lyotropic LC protein solution. The concerted structuring action of capillary confinement, strong anchoring, and nematic flow leads to predictions in agreement with the reported textural transitions in the duct of spiders and silkworms. The quantitative description of BLC materials and processes using mesoscopic models provides another tool to develop the science and future biomimetic applications of these ubiquitous natural anisotropic soft materials.

Phase diagram and electro-optical Kerr effect of nematic mixtures containingtolane and biphenyl cores

An experimental phase diagram for two types of nematic liquid crystals and their binary mixtures was established by polarised optical microscopy and differential scanning calorimetry methods. The mixtures comprised a tolane-base liquid crystal, 4-heptyl-3-fluoro-4-isothiocyanatotolane (7TOLF), and a biphenyl-base nematogen, 4-heptyl-3-fluoro-4-isothiocyanatobiphenyl (7BF). The static Kerr effect and third-order non-linearity were investigated for 7TOLF and 7BF and their nematic mixtures within the isotropic phase. Both the compounds have a positive and large Kerr constant. The second-order phase-transition temperatures, T*, were determined for these mixtures. The linear dependence of (T - T*)− 1 on the Kerr constant is found to be in good agreement with the predications of the Landau–de Gennes model. The third-order non-linear susceptibility,χ(3), values were determined for these mixtures.

Detection of alignment changes at the open surface of a confined nematic liquid crystal sensor

An open surface optical sensor based on nematic liquid crystal contained in square shaped wells was fabricated. Structures and structural transitions associated with changes in surface anchoring conditions were optically detected using polarizing microscopy. A phenomenological Landau–de Gennes model was applied to model the response of the sensor. Simulated optical micrographs were compared with experiments to understand the nematic profiles and to control and optimize the sensor in a specific regime. Distinct optical patterns were detected when the sensor was exposed to air and to water. A structural transition in the nematic structure was observed, driven by changes in the anchoring strength at the open surface of the sensor. Anchoring strength was varied by adding a homeotropic surfactant (sodium dodecyl sulfate) to strongly planar de-ionized water. The transition can be controlled using sensor parameters such as well depth and aspect ratio.

Kerr Effect Studies on Mixtures of the Two Tolane Base Liquid Crystals

The static Kerr effect and third order non-linearity were investigated for mixtures of the tolane-base liquid crystals within the isotropic phase in the neighborhood of the nematic phase. The nematic mixtures are characterized by components having similar molecular structure, i.e., the mixtures comprised by 4′-pentyl-3-fluoro-4-isothiocyanatotolane (5TOLF) and 4′-heptyl-3 fluoro-4-isothiocyanatotolane (7TOLF) which have a positive and large Kerr constant. The nematic to isotropic transition temperatures and nematic to crystalline temperatures for these mixtures were measured using DSC and polarizing microscopy methods. The nematic mixture of M3 (%50 5TOLF + %50 7TOLF) shows the largest nematic range. The linear dependence of (T − T*)−1 on the Kerr constant is found to be in good agreement with the predications of the Landau-De Gennes model. The hypothetical second order phase transition temperatures and susceptibility values were determined for these compounds.

Virtual isotropic‐nematic transitions in alkyl cyanobiphenyls

Pretransitional fluctuations in the isotropic phase of liquid crystalline and non‐liquid crystalline alkyl cyanobiphenyls have been investigated using light scattering and magnetic birefringence measurements. We find evidence for a virtual isotropic‐nematic phase transition in short‐chain alkyl cyanobiphenyls with no observable nematic phase. The measured temperature dependence of fluctuations is well‐described by mean‐field theory. Virtual phase transition temperatures extrapolated from separate light scattering and magnetic birefringence experiments are in good agreement. Landau–de Gennes model parameters for the compounds investigated are calculated from the experimental results.

Reduced Landau–de Gennes functional and surface smectic state of liquid crystals

In [X.B. Pan, Landau–de Gennes model of liquid crystals and critical wave number, Comm. Math. Phys. 239 (1–2) (2003) 343–382], based on the de Gennes analogy between liquid crystals and superconductivity [P.G. de Gennes, An analogy between superconductors and smectics A, Solid State Commun. 10 (1972) 753–756], the second author introduced the critical wave number Q c 3 (which is an analog of the upper critical field H c 3 for superconductors) and predicted the existence of a surface smectic state, which was supposed to be an analogy of the surface superconducting state. In a surface smectic state, the bulk liquid crystal is in the nematic state, and a thin layer of smectic appears in a helical strip on the surface of the sample. In this paper we study an approximate form of the Landau–de Gennes model of liquid crystals, and examine the behavior of minimizers, in particular the boundary layer behavior. Our work shows the importance of the joint chirality constant qτ, which is the product of wave number q and chirality τ and also appears in the work of [P. Bauman, M. Calderer, C. Liu, D. Phillips, The phase transition between chiral nematic and smectic A ∗ liquid crystals, Arch. Rational Mech. Anal. 165 (2002) 161–186] and [X.B. Pan, Landau–de Gennes model of liquid crystals and critical wave number, Comm. Math. Phys. 239 (1–2) (2003) 343–382]. The joint chirality constant of a liquid crystal is useful to predict whether the liquid crystal is of type I or type II, and it is also useful to examine whether the liquid crystal is in a surface smectic state. The results in this paper suggest that a liquid crystal with large Ginzburg–Landau parameter κ and large joint chirality constant qτ exhibits type II behavior, and it will be in the surface smectic state if q τ ∼ b κ 2 for some β 0 < b < 1 , where β 0 is the lowest eigenvalue of the Schrödinger operator with a unit magnetic field in the half space, and 0 < β 0 < 1 . We also show that a liquid crystal with small qτ exhibits type I behavior.

Electro-optical Kerr effect of nematic mixtures comprised of 6CHBT and 6BOBT

AbstractStatic Kerr effect and the pre-transition temperature were investigated for the 6CHBT, 6BOBT and the nematic mixtures, M1 (40% 6CHBT: 60% 6BOBT), M2 (30% 6CHBT: 70% 6BOBT) and M3 (20% 6CHBT: 80% 6BOBT) at temperatures above the nematic-isotropic transition temperature. The linear dependence of (T-T*)−1 on the Kerr constant is found to be in good agreement with the predications of the Landau-De Gennes model. The hypothetical second order phase transition temperatures and susceptibility values were determined for these compounds.

Computational modelling of nematic phase ordering by film and droplet growth over heterogeneous substrates

This paper presents a computational study of defect nucleation associated with the kinetics of the isotropic‐to‐nematic phase ordering transition over heterogeneous substrates, as it occurs in new liquid crystal biosensor devices, based on the Landau–de Gennes model for rod‐like thermotropic nematic liquid crystals. Two regimes are identified due to interfacial tension inequalities: (i) nematic surface film nucleation and growth normal to the heterogeneous substrate, and (ii) nematic surface droplet nucleation and growth. The former, known as wetting regime, leads to interfacial defect shedding at the moving nematic‐isotropic interface. The latter droplet regime, involves a moving contact line, and exhibits two texturing mechanisms that also lead to interfacial defect shedding: (a) small and large contact angles of drops spreading over a heterogeneous substrate, and (b) small drops with large curvature growing over homogeneous patches of the substrate. The numerical results are consistent with qualitative defect nucleation models based on the kinematics of the isotropic–nematic interface and the substrate–nematic–isotropic contact line. The results extend current understanding of phase ordering over heterogeneous substrates by elucidating generic defect nucleation processes at moving interfaces and moving contact lines.

Magnetic Field‐Induced Instabilities in Liquid Crystals

We use the Landau–de Gennes model to investigate the magnetic field‐induced instabilities in liquid crystals. In particular, we examine the change of weak and strong stabilities in the pure smectic states and in the pure nematic states. Motivated by de Gennes’ discovery on the analogies between liquid crystals and superconductors, we introduce critical magnetic fields $H_{\text{\rm s}}$ and $H_{\rm sh}$. The pure smectic states lose their global minimality (strong stability) at $H_{\text{\rm s}}$ and lose their local minimality (weak stability) at $H_{\rm sh}$. We also examine the change of stability in the pure nematic states. We show in the case of equal elastic coefficients that a liquid crystal in a sufficiently strong magnetic field will not be in a pure nematic state, which exhibits a significant difference between the Landau–de Gennes model for liquid crystals and the Ginzburg–Landau model for superconductivity.

Growth regimes in phase ordering transformations

Growth, shape and texturing dynamics of single 2D spherulites are analyzed using the Landau-de Gennes (LdG) liquid crystal model of isotropic-nematic phase ordering. Direct numerical simulation shows that non-circular nucleation due to anisotropy in the interfacial tension results in non-circular shapes in the early stages of growth. However, interfacial heterogeneities in growth and structure then lead to interfacial defect nucleation and shedding and a reshaping of the interface into a circular shape. The formulated dynamics show that a growing spherulite in the early stage of phase ordering also acquires topological higher charges than expected from the well-known Kibble mechanism. In agreement with experiments under strong quenching, the predicted growth dynamics of a spherulite of characteristic radius R is linear: $R $~$ t$. To better understand these computational results, a dynamic shape equation is obtained from the LdG model and is shown to have the same form as the growth and shape equations of crystal growth. The shape equation is used to reveal the mechanisms involved in shape transformations, interfacial defect shedding, and growth dynamics computed by direct numerical simulation of the bulk LdG equations.

Perturbative approach to the structure of a planar interface in the Landau–de Gennes model

The structure of nearly static planar interfaces is studied within the framework of the Landau-de Gennes model with the dynamics governed by the time-dependent Ginzburg-Landau equation. To account for the full elastic anisotropy the free energy expansion is extended to include a third order gradient term. The solutions corresponding to the in-plane or homeotropic director alignment at the interface are sought. For this purpose a consistent perturbative scheme is constructed which enables one to calculate successive corrections to the velocity and the order parameter of the interface. The implications of the solutions are discussed. The elastic anisotropy introduces asymmetry into the order parameter and free energy profiles, even for the high symmetry homeotropic configuration. The velocity of the interface with the homeotropic or in-plane alignment is enhanced or reduced, respectively. There is no reorientation of the optical axis in the boundary layer. For the class of nematogens with approximate splay-bend degeneracy the temperature dependence of the interface velocity is weakly affected by the remaining twist anisotropy.

Electro-optic characterization of novel tolane-based nematic liquid crystals

Four new tolane-based liquid crystals, 4′-pentyl-, 4′-hexyl-, and 4′-heptyl-3-fluoro-4-isothiocyanatotolane and 4′-hexyloxy-3′-fluoro-4-isothiocyanatotolane have been investigated for the first time to determine their electro-optical behavior and third order non-linearity by the static Kerr effect method. The temperature dependence of electric Kerr constant in the isotropic phase and the pretransitional behavior, related to order parameter fluctuations, have been investigated for four new high birefringence nematic liquid crystals, laterally fluorine substituted isothiocyanatotolanes in the isotropic phase. All the compounds had a positive and large Kerr constant which increased with decreasing temperature. The Landau–De Gennes model was obeyed for these compounds.