Phase diagrams of cholesteric liquid crystals obtained with a generalized Landau-de Gennes theory

Phase diagrams of chiral nematic liquid crystals are studied within the framework of a generalized Landau-Ginzburg-de Gennes theory. Using the parametrization of Grebel, Hornreich, and Shtrikman for the tensor order parameter Q, all relevant elastic terms are included for the helicoidal phase and the blue phases of chiral nematic liquid crystals up to fourth order in Q and its gradient ∂Q. The influence of the additional elastic terms on the phase diagrams of the chiral nematic phases is then investigated. The theory correctly describes the variation of the pitch with temperature and the induced biaxiality of the cholesteric phase. The results resolve the discrepancies encountered by Hornreich and Shtrikman in the comparison of experiment and theory. New features in the topology of the phase diagrams of blue phases, like re-entrant phase transitions, are predicted.     

Phase diagrams of cholesteric liquid crystals obtained with a generalized Landau-de Gennes theory

Abstract

Phase diagrams of chiral nematic liquid crystals are studied within the framework of a generalized Landau-Ginzburg-de Gennes theory. Using the parametrization of Grebel, Hornreich, and Shtrikman for the tensor order parameter Q, all relevant elastic terms are included for the helicoidal phase and the blue phases of chiral nematic liquid crystals up to fourth order in Q and its gradient ?Q. The influence of the additional elastic terms on the phase diagrams of the chiral nematic phases is then investigated. The theory correctly describes the variation of the pitch with temperature and the induced biaxiality of the cholesteric phase. The results resolve the discrepancies encountered by Hornreich and Shtrikman in the comparison of experiment and theory. New features in the topology of the phase diagrams of blue phases, like re-entrant phase transitions, are predicted.     

高分子液晶条带织构的计算机模拟

The relationship between the vector version and tensor version of Frank distortion energy for the liquid crystal was established, through which the relationship between the Frank elastic constants and the expansion coefficients of spatial derivatives for the orientational order parameter Sij of liquid crystalline polymer was obtained. Ginzberg-Landau equation for the orientational order parameter Sij was numerically solved by using the cell dynamical system, which involves a free energy functional containing Landau-de Gennes orientational free energy, Maier-Saupe anisotropic interaction free energy and Frank distortion free energy. Incase of the splay elastic constant being much lager than the bend elastic constant, the evolution process of band textures in liquid crystalline polymer during the shear relaxation and its small angle light scattering patterns were simulated. The influence of preshear rate on the band forming and band structure was investigated. It was found that the formation of longitudinal band textures is due to the much rapider longitudinal splay relaxation than the latitudinal bend relaxation, and the induced times of band forming and the characteristic length of band structure decrease with the increasing of preshear rate, which is consistent with the experimental results.     

Determination of the elastic constants of nematic liquid crystals by means of the affine transformation model

In this paper the affine connection approach will be used to calculate the elastic constants of nematic liquid crystals. Following this approach, which was originally conceived to compute the nematic viscosity coefficients, an expression for the elastic constants, without adjustable free parameters, will be formulated in terms of a temperature dependent metric, whose non‐isotropic part is proportional to the tensorial order parameter of the nematic phase. The dependence of the elastic constants on the scalar order parameter, in the geometry of the nematic molecules, and in the anisotropic part of the molecular interaction, will be determined.     

Determination of the elastic constants of nematic liquid crystals by means of the affine transformation model

In this paper the affine connection approach will be used to calculate the elastic constants of nematic liquid crystals. Following this approach, which was originally conceived to compute the nematic viscosity coefficients, an expression for the elastic constants, without adjustable free parameters, will be formulated in terms of a temperature dependent metric, whose non-isotropic part is proportional to the tensorial order parameter of the nematic phase. The dependence of the elastic constants on the scalar order parameter, in the geometry of the nematic molecules, and in the anisotropic part of the molecular interaction, will be determined.     

Numerical analysis for a nematic droplet in polymer dispersed liquid crystals

Starting from the Landau-de Gennes free energy expression, the author has numerically analysed the director pattern in a nematic droplet of polymer dispersed liquid crystals. The nematic director has been understood as the eigenvector, which corresponds to the largest eigenvalue of the tensor order parameter. To investigate the droplet structure influence, all equations have been treated on the curvilinear coordinate system which is generated along the droplet boundary. In the case of spherical and spheroidal droplets with normal strong anchoring, the director exhibits an axial configuration and a disclination ring. The ring radius and the capactiance of the system change without hysteresis with the applied voltage.     

Generalized nematostatics

The generalized equations of bulk and interfacial nematostatics in terms of the tensor order parameter are derived using calculus of variations, taking into account long and short range nematic bulk free energies as well as anchoring and saddle-splay surface free energies. A general expression for the surface stress tensor order parameter for a nematic liquid crystal/isotropic fluid (NLC/I) interface has been derived, and found to represent normal, shear, and bending stresses. It is shown that the surface stress tensor is asymmetric. It is also found that anchoring energy contributes to bending and normal stresses, while saddle-splay energy contributes to normal and shear stresses. The rotational identifies governing the bulk and surface stress tensors are derived and used to show that the equations of nematostatics are fully consistent with the general balance equations of polar fluids. The equations presented provide a theoretical framework for solving interfacial problems involving NLCs that is applicable to cases where variations in liquid crystalline order and saddle-splay energy play significant roles.     

Generalized nematostatics

The generalized equations of bulk and interfacial nematostatics in terms of the tensor order parameter are derived using calculus of variations, taking into account long and short range nematic bulk free energies as well as anchoring and saddle-splay surface free energies. A general expression for the surface stress tensor order parameter for a nematic liquid crystal/isotropic fluid (NLC/I) interface has been derived, and found to represent normal, shear, and bending stresses. It is shown that the surface stress tensor is asymmetric. It is also found that anchoring energy contributes to bending and normal stresses, while saddle-splay energy contributes to normal and shear stresses. The rotational identifies governing the bulk and surface stress tensors are derived and used to show that the equations of nematostatics are fully consistent with the general balance equations of polar fluids. The equations presented provide a theoretical framework for solving interfacial problems involving NLCs that is applicable to cases where variations in liquid crystalline order and saddle-splay energy play significant roles.     

A surface-induced transition in polymeric nematics

New polymeric liquid crystals can be treated as standard nematic liquid crystals when only their bulk properties are at issue, but they exhibit peculiar surface properties. The most striking one is that biaxial distributions may be induced on a confining surface. On continuously varying the surface anchoring conditions, we find a first-order phase transition from planar to homeotropic alignment in the bulk. Moreover, the decay towards these uniaxial states is radically different in the two cases: it is asymptotically exponential in the former, whereas it happens abruptly at a finite depth in the latter. There is precisely one surface biaxial distribution that induces bistability between these decay modes: it depends on the elastic constants in the Landau-de Gennes free energy functional. The analysis of the model we propose can prove useful in detecting the sign of the difference between splay and bend constants.     

Landau-de Gennes theory of the core structure of a screw dislocation in smectic A liquid crystals

We present details of calculations of the core structure of a screw dislocation in a smectic A liquid crystal, using the phenomenological Landau-de Gennes free energy functional. The order parameter frustration created by topological constraints far from the dislocation core is resolved in one of three qualitatively different ways. The three types of dislocation core solution are the DT (double twist), CL (classical), and BP (broken polar symmetry) solutions, respectively. The stability requirements for these structures are discussed, as a function of temperature, smectic elastic properties, and coupling between smectic and nematic order. The effect of possible inhomogeneity between left- and right-handed conformers is also examined.     

A molecular theoretical derivation of the Landau free energy of chiral smectic C liquid crystals

A molecular theoretical derivation of the Landau free energy of chiral smectic C liquid crystals is presented on the basis of a molecular theoretical approach. The excess free energy concerned with the ferroelectric helicoidal structure is derived within a mean field approximation. Several material constants such as the Curie temperature, the elastic constant, the helical pitch, the piezo- and flexoelectric constants and the dielectric susceptibility are expressed in terms of some effective intermolecular interaction parameters between molecules. A molecular theoretical consideration of the chiral smectic C-smectic A transitions is given in comparison with a few previous works. By comparing with some available experimental data for DOBAMBC, effective interaction parameters are numerically estimated.     

Micellar shape dependence of cholesteric pitch in lyotropic liquid crystals

A generalized Landau-Ginzburg-de Gennes theory of cholesteric pitch in lyotropic chiral liquid crystals is presented. Using the non-critical order parameter, introduced by Toledano and Figueiredo Neto, the question of pitch dependence on micellar shape is addressed. This extra order parameter accounts for the shape change of the micelles. The predictions agree qualitatively with the unique experimental observations of Lopes and Figueiredo Neto.     

Micellar shape dependence of cholesteric pitch in lyotropic liquid crystals

A generalized Landau-Ginzburg-de Gennes theory of cholesteric pitch in lyotropic chiral liquid crystals is presented. Using the non-critical order parameter, introduced by Toledano and Figueiredo Neto, the question of pitch dependence on micellar shape is addressed. This extra order parameter accounts for the shape change of the micelles. The predictions agree qualitatively with the unique experimental observations of Lopes and Figueiredo Neto.     

Nematic to smectic-A phase transition in mixture of liquid crystal and nonmesogenic impurities: existence of tricritical point

Experiments on the mixture of liquid crystals and nonmesogenic impurities showed the significant role of nonmesogenic impurities on the nematic–smectic-A phase transition. Using both Flory–Huggins theory of isotropic mixing and Landau–de Gennes theory, we present a phenomenological theory that discusses the role of such impurities on the nematic–smectic-A phase transition in a mixture of smectic liquid crystal and nonmesogenic impurities. We discuss the impact of nonmesogenic impurities on the order parameters, Frank elastic constants (splay and bend) and transition temperature of the nematic–smectic-A phase transition. Our theoretical results show that there exists a tricritical point for which the second-order nematic–smectic-A phase transition becomes first order at a tricritical point by increasing the concentration of nonmesogenic solute. We find a remarkable agreement between theoretical and experimental results.     

Pattern formation from consistent dynamical closures of uniaxial nematic liquid crystals

Pattern formation in uniaxial polymeric liquid crystals is studied for different dynamic closure approximations. Using the principles of mesoscopic non-equilibrium thermodynamics in a mean-field approach, we derive a Fokker-Planck equation for the single-particle non-homogeneous distribution function of particle orientations and the evolution equations for the second and fourth order orientational tensor parameters. Afterwards, two dynamic closure approximations are discussed, one of them considering the relaxation of the fourth order orientational parameter and leading to a novel expression for the free-energy like function in terms of the scalar order parameter. Considering the evolution equation of the density of the system and values of the interaction parameter for which isotropic and nematic phases coexist, our analysis predicts that patterns and traveling waves can be produced in lyotropic uniaxial nematics even in the absence of external driving.     

Fourth-order nematic elasticity and modulated nematic phases: a poor man’s approach

We propose an extension of Frank-Oseen’s elastic energy for bulk nematic liquid crystals which is based on the hypothesis that the fundamental deformations allowed in nematic liquid crystals are splay, twist and bend. The extended elastic energy is a fourth-order form in the fundamental deformations. The existence of bulk spontaneous modulated or deformed nematic liquid crystal ground states is investigated. The analysis is limited to bulk nematic liquid crystals in the absence of limiting surfaces and/or external fields. The non deformed ground state is stable only when Frank-Oseen’s elastic constants are positive. In case where at least one of them is negative, the ground state becomes deformed. The analysis of the stability of the deformed states in the space of the elastic parameters allows to characterise different types of deformed nematic phases. Some of them are new nematic phases, for instance a twist – splay nematic phase is predicted. Inequalities between second-order elastic constants which govern the stability of the twist–bend, splay–bend, and splay–twist states are obtained. Their stability in respect to triple splay–bend–twist deformations is investigated.