Landau-de Gennes Theory Research Articles

Effects of carbon nanotubes on the isotropic to nematic phase transition in nematic liquid crystal elastomer

The isotropic to nematic phase transition on nematic liquid crystal elastomer - carbon nanotubes composites has been studied theoretically. The model free energy density has been constructed using Flory Huggins theory of isotropic mixing and Landau-de Gennes theory. The effects of carbon nanotubes and external mechanical stress on the isotropic to nematic phase transition of the nematic liquid crystal elastomer-carbon nanotubes composites have been discussed. The volume fraction of carbon nanotubes dependence of the isotropic to nematic transition temperature, mechanical strain and strain birefringence has been calculated. The weak first-order character of the isotropic to nematic phase transition on nematic liquid crystal elastomer-carbon nanotubes composites has been predicted.

Effects of quantum dots on the nematic to smectic-A phase transition

ABSTRACT Experiments on the composites of liquid crystals and quantum dots showed the significant role of quantum dots on the properties of the nematic–smectic-A phase transition. The effects of quantum dots on the nematic-smectic-A phase transition have been discussed using Flory- Huggins theory of isotropic mixing and Landau-de Gennes theory. The impact of quantum dots on the smectic-A order parameters, nematic–smectic-A transition temperature, splay and bend elastic constants has been discussed. The smectic-A order parameters, transition temperature and elastic constants decrease with the increase of volume fraction of quantum dots.

Magnetic field induced isotropic ferrofluid to ferronematic phase transition in thermotropic liquid crystal

Experiments on liquid crystals doped with ferromagnetic nanoparticles predicted the isotropic ferrofluid to ferronematic phase transition. The magnetic field induced isotropic ferrofluid to ferronematic phase transition in thermotropic liquid crystal has been studied theoretically. Theoretical model is based on Flory–Huggins theory of isotropic mixing and Landau-de Gennes theory. The impact of ferromagnetic nanoparticles and magnetic field on the isotropic ferrofluid to ferronematic phase transition and transition temperature has been discussed. Theoretical results predict the magnetic field induced critical point of the isotropic ferrofluid to ferronematic phase transition. The temperature and volume fraction of ferromagnetic nanoparticles dependence of the magnetic birefringence and Cotton-Mouton constant are calculated.

Theoretical models for the nematic twist-bend phase and the Landau de Gennes theory of Longa and Tomczyk

ABSTRACT We present a critical analysis of the existing phenomenological theories of the nematic twist-bend phase including the models proposed by Dozov and Selinger and the more general Landau de Gennes theory of L. Longa and W. Tomczyk. The role of the flexoelectric polarisation is considered in detail as well as the stabilisation of the nanoscale helical structure of the twist-bend phase. In particular we focus into the mechanism of the reduction of the bend elastic constant determined by negative flexoelectric corrections and discuss why the splay elastic constant is not reduced at all. A molecular theory of elasticity of bent-core nematics, which also enables one to describe the reduction of the bend elastic constant, is considered in detail as well. Finally we consider two different tensor orientational order parameters of the twist-bend phase which describe all kinds of uniaxial and local biaxial ordering of bent-core molecules in the twist-bend phase.

Particle-based and continuum models for confined nematics in two dimensions.

We use the particle-based stochastic multi-particle collision dynamics (N-MPCD) algorithm to simulate confined nematic liquid crystals in regular two-dimensional polygons such as squares, pentagons and hexagons. We consider a range of values of the nematicities, U, and simulation domain sizes, R, that canvass nano-sized polygons to micron-sized polygons. We use closure arguments to define mappings between the N-MPCD parameters and the parameters in the continuum deterministic Landau-de Gennes framework. The averaged N-MPCD configurations agree with those predicted by Landau-de Gennes theory, at least for large polygons. We study relaxation dynamics or the non-equilibrium dynamics of confined nematics in polygons, in the N-MPCD framework, and the kinetic traps bear strong resemblance to the unstable saddle points in the Landau-de Gennes framework. Finally, we study nematic defect dynamics inside the polygons in the N-MPCD framework and the finite-size effects slow down the defects and attract them to polygon vertices. Our work is a comprehensive comparison between particle-based stochastic N-MPCD methods and deterministic/continuum Landau-de Gennes methods, and this comparison is essential for new-age multiscale theories.

The radial hedgehog solution in the Landau–de Gennes theory: Effects of the bulk potentials

We study equilibrium configurations in spherical droplets of nematic liquid crystal with strong radial anchoring, within the Landau–de Gennes theory with a sixth-order bulk potential. The sixth-order potential predicts a bulk biaxial phase for sufficiently low temperatures, which the conventional fourth-order potential cannot predict. We prove the existence of a radial hedgehog solution, which is a uniaxial solution with a single isotropic point defect at the droplet centre, for all temperatures and droplet sizes, and prove that there is a unique radial hedgehog solution for moderately low temperatures, but not deep in the nematic phase. We numerically compute critical points of the Landau–de Gennes free energy with the sixth order bulk potential, with rotational and mirror symmetry, and find at least two competing stable critical points: the biaxial torus and split core solutions, which have biaxial regions around the centre, for low temperatures. The size of the biaxial regions increases with decreasing temperature. We also compare the properties of the radial hedgehog solution with the fourth-order and sixth-order potentials respectively, in terms of the Morse indices as a function of the temperature and droplet radius; the role of the radial hedgehog solution as a transition state in switching processes; and compare the bifurcation plots with temperature, with the fourth- and sixth-order potentials. Overall, the sixth-order potential has a stabilising effect on biaxial critical points and a de-stabilising effect on uniaxial critical points and we discover an altogether novel bulk biaxial critical point of the Landau–de Gennes energy with the sixth-order potential, for which the bulk biaxiality is driven by the sixth-order potential.

Theoretical study of the nematic to isotropic phase transition in the mixtures of nematic liquid crystal and quantum dots nanoparticles

ABSTRACTThe nematic to isotropic phase transition in the mixtures of nematic liquid crystal and quantum dots nanoparticles has been studied. Flory Huggins theory of isotropic mixing and Landau-de Gennes theory have been used to study the nematic to isotropic phase transition. The effects of quantum dots nanoparticles on the nematic order parameter, transition temperature, and splay elastic constant have been discussed. The decrease of nematic order parameter, transition temperature, and splay elastic constant with the increase in volume fraction of quantum dots nanoparticles has been observed.

Landau-de Gennes theory of the nematic to smectic—A phase transition under confinement conditions

We study the nature of the nematic to smectic-A phase transition under confinement conditions in the context of Landau-de Gennes theory. The effect of quenched disorder on the nematic to smectic-A transition temperature, order parameter, enthalpy, and thermal hysteresis are discussed followed by phase diagrams. The tricritical behavior of the nematic to smectic-A phase under confinement conditions is predicted.

Local chiral inversion of chiral nematic liquid crystals in cylinders.

On the basis of Landau-de Gennes theory and the finite-difference iterative method, the autonomic modulation of chiral inversion in a cylindrical cavity with degenerate planar anchoring is investigated. Under the applied helical twisting power (inversely related to the pitch P), a chiral inversion can be achieved due to the nonplanar geometry effect, and the inversion capacity rises with the increase of the helical twisting power. The combined effect of the saddle-splay K_{24} contribution (corresponding to the L_{24} term in Landau-de Gennes theory) and the helical twisting power are analyzed. It is found that the chiral inversion is more strongly modulated on the condition that the chirality of spontaneous twist is opposite to that of applied helical twisting power. Further, larger values of K_{24} will induce larger modulation of the twist degree and smaller modulation of the inverted region. The autonomic modulation of chiral inversion shows great potential for chiral nematic liquid crystal materials to be used in smart devices, such as light-controlled switches and nanoparticle transporters.

Pattern formation in Landau–de Gennes theory

We study the spherical droplet problem in 3D Landau–de Gennes theory with finite temperature. By rigorously constructing the biaxial–ring solutions and split–core–segment solutions, we theoretically confirm the numerical results of Gartland–Mkaddem in [15]. The structures of disclinations are also addressed.

Numerical simulation of the 2D lid-driven cavity flow of chiral liquid crystals

ABSTRACT In this study, the two-dimensional (2D), lid-driven cavity flow of chiral liquid crystals (LCs) was modelled using the Landau de-Gennes (LdG) theory. Parametric studies investigating the effect of Er (Ericksen number) and Θ (chiral strength) on the microstructure of chiral LCs were performed. In this study, we observed that an increase in Θ caused the chiral texture to have more striations and shorter pitch lengths as causes for an increased number of defects. When Θ was held constant, an increase in Er disrupted the chiral structure and even broke it at very high Er. Interestingly, a transition from low Er (10) to moderate Er (1,000) increased the number of defects; however, further increases in Er reduced the number of defects since much of the chiral structure was destroyed by the high viscous flow effects. In particular, even at very high Er, the chiral structure and defects were still present as a vortex was always present with lower velocities, where the viscous flow effect was smaller. We also found that a hexagonal structure with penta-hepta defects formed at high chiral strengths.

Dynamics of chiral liquid crystals between cylinders using Landau-de Gennes theory

ABSTRACT This study employs the Landau-de Gennes theory to investigate the behaviour of chiral liquid crystals (CLCs) between concentric and eccentric cylinders under different flow conditions. This theory was implemented in dynamic finite element simulations to solve the evolution of the microstructure of CLCs and couple it with linear momentum balance equation (modified Navier–Stokes equation) to capture the structure of CLCs. The study focused on the microstructure formation of CLCs and their performance as lubricants under various chirality strengths (θ), Deborah numbers (De), and eccentricity of eccentric cylinders. The hexagonal structure of the CLCs was found between cylinders at low De (0.001), where the chiral term predominates. Moreover, the number of hexagonal structures between cylinders was shown to be significantly correlated to the chiral strength. We showed that increasin De resulted in vanishing hexagonal pattern of CLCs and the De value at which the hexagonal pattern vanishes depends on the chiral strength. We also observed that a higher eccentricity ratio significantly increased the pressure force per length on the inner cylinders and caused more non-uniformity in the structure. Based on the findings, we concluded that the performance of CLCs as lubricants is greater at higher θ and eccentricity and lower De.

Stability of a split-core configuration induced by saddle-splay elasticity in a submicron nematic liquid crystal spherical droplet

ABSTRACT We use Landau–de Gennes theory to investigate the configurations of liquid crystals confined to a submicron spherical droplet subject to weak homeotropic anchoring. The results show that in a one-constant approximation, there are three equilibrium configurations: radial hedgehog, ring disclination and uniform, where the radial hedgehog configuration is a stable state. With increasing elastic anisotropy, which is essentially an increase in the splay elastic constant, the radial hedgehog configuration transforms into a split-core configuration, which is metastable. The split-core configuration becomes the lowest-energy state induced by saddle-splay elasticity within appropriate ranges of the radius and elastic anisotropy.

Coupling between splay deformations and density modulations in splay-bend phases of bent colloidal rods.

Using a grand-canonical Landau-de Gennes theory for colloidal suspensions of bent (banana-shaped) rods, we investigate how spatial deformations in the nematic director field affect the local density of twist-bend and splay-bend nematic phases. The grand-canonical character of the theory naturally relates the local density to the local nematic order parameter S. In the splay-bend phase, we find S and hence the local density to modulate periodically along one spatial direction. As a consequence the splay-bend phase has the key symmetries of a smectic rather than a nematic phase. By contrast we find that S and hence the local density do not vary in space in the twist-bend phase, which is therefore a proper nematic phase. The theoretically predicted one-dimensional density modulations in splay-bend phases are in agreement with recent simulations.

Blowup Rate Estimates of a Singular Potential and Its Gradient in the Landau-de Gennes Theory

Blowup Rate Estimates of a Singular Potential and Its Gradient in the Landau-de Gennes Theory

Hydrodynamics of fringing-field induced defects in nematic liquid crystals

Consumer demand for high resolution and high refresh-rate displays has naturally led to the fabrication of liquid crystal displays with ever smaller pixels. As a consequence, fringing fields between adjacent pixels grow in magnitude, leading to abrupt changes in orientation. Electric field strengths above some threshold can lead to order melting and, in turn, disclinations. This paper presents accurate modeling of disclinations induced by fringing fields due to interdigitated electrodes in a nematic liquid crystal calculated by means of the Landau–de Gennes theory. Disclination paths are determined while taking into account the flow of the liquid crystal. Making use of interdigitated electrodes, precise electrical control over the creation and positioning of defects is demonstrated for homeotropic, planar, hybrid, and in-plane surface alignments.

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.

Tricritical Phenomena and Cascades of Temperature Phase Transitions in a Ferromagnetic Liquid Crystal Suspension

We consider temperature-driven phase transitions occurring in a liquid crystal suspension of ferromagnetic particles within the Landau–de Gennes theory. The temperature dependences of the order parameters in the uniaxial model with a vector order parameter for the magnetic subsystem are obtained. The dimensionless expression for the free energy density of the suspension has been used for the study of the phase behavior general regularities of the system. Phase state diagrams of the suspension and temperature dependences of the order parameters of the liquid crystal and the ensemble of magnetic particles for different values of the phenomenological expansion coefficients are constructed. It is shown that the considered model admits the existence of a cascade of temperature phase transitions: isotropic phase–superparamagnetic nematic phase–ferromagnetic nematic phase. We have shown that in the mesomorphic state of the liquid crystal, the spontaneous magnetization can appear in a continuous way or by a jump with decreasing temperature, which corresponds to the tricritical behavior. The values of temperature and expansion coefficients corresponding to the tricritical and triple points are numerically found.

Liquid crystal droplets under extreme confinement probed by a multiscale simulation approach

ABSTRACT In this work, we computationally investigate liquid crystal (LC) droplets in the size range 0.03–1 μm, confined within shells of combined anchoring conditions. Two different types of surface were defined to promote homeotropic and planar degenerate anchoring, respectively. We identified the LC behaviour within the nanoscale droplets using a bespoke multiscale simulation approach. To study 30 nm droplets, we used coarse grained simulations within the dissipative particle dynamics formalism; to study 0.1 μm and larger droplets, we used a finite element method based on the Landau–de Gennes theory. Good agreement between the two methods was observed in our prior analysis and was confirmed in the present work. We explicitly study droplets of size 0.1 and 1 μm by using continuum mechanics calculations. Our results for the largest droplet are consistent with those available in the literature, suggesting that the extension to smaller droplets presented here is realistic, and therefore can be helpful for innovations in which device intensification could be achieved using LC nanodroplets.

Decomposition of topological defects in liquid crystal wells with homeotropic anchoring

ABSTRACT The decomposition of ±1 topological defects inside shallow wells of nematic liquid crystals, subject to homeotropic strong anchoring on the sides, was investigated using the Landau-de Gennes theory. For the decomposition of a − 1 defect, two different two-half-defect structures were obtained, depending on the defect position; while for the decomposition of a + 1 defect, only one of two two-half-defect structures were obtained. Furthermore, the relative stabilities of the two two-half-defect structures were analysed theoretically and compared with a diagonal structure. With strong anchoring, the two two-half-defect structures were always metastable for any aspect ratio of the well. With weak anchoring, the second two-half-defect structure was still metastable, whereas the first two-half-defect structure could be in the ground state at a certain aspect ratio with adequate anchoring. The results could be used to stabilise, localise, or even to destabilise structures bearing strongly charged topological defects.