Model For Nematic Liquid Crystals Research Articles

Stochastic rotation dynamics for nematic liquid crystals.

We introduce a new mesoscopic model for nematic liquid crystals (LCs). We extend the particle-based stochastic rotation dynamics method, which reproduces the Navier-Stokes equation, to anisotropic fluids by including a simplified Ericksen-Leslie formulation of nematodynamics. We verify the applicability of this hybrid model by studying the equilibrium isotropic-nematic phase transition and nonequilibrium problems, such as the dynamics of topological defects and the rheology of sheared LCs. Our simulation results show that this hybrid model captures many essential aspects of LC physics at the mesoscopic scale, while preserving microscopic thermal fluctuations.

Instability of point defects in a two-dimensional nematic liquid crystal model

We study a class of symmetric critical points in a variational 2D Landau–de Gennes model where the state of nematic liquid crystals is described by symmetric traceless 3×3 matrices. These critical points play the role of topological point defects carrying a degree k2 for a nonzero integer k. We prove existence and study the qualitative behavior of these symmetric solutions. Our main result is the instability of critical points when |k|≥2.

Uniform well-posedness and low Mach number limit to the compressible nematic liquid crystal flows in a bounded domain

We prove global-in-time and uniform-in-ϵ of the strong solutions to the 3D compressible nematic liquid crystal flows in a bounded domain, where ϵ is the Mach number. Consequently, we obtain the strong solution of compressible nematic liquid crystal model that converges to that of incompressible nematic liquid crystal model. This is the first result on the low Mach number limit for compressible nematic liquid crystal flows in 3D bounded domain. Our proof relies on the dedicated estimates on the solutions and the subtle use of the boundary conditions.

Propagating director bend fluctuations in nematic liquid crystals.

We show, by molecular simulation, that for a range of standard, coarse-grained, nematic liquid crystal models, the director bend fluctuation is a propagating mode. This is in contrast to the generally accepted picture of nematic hydrodynamics, in which all the director modes (splay, twist, bend, and combinations thereof) are overdamped. By considering the various physical parameters that enter the equations of nematodynamics, we propose an explanation of this effect and conclude that propagating bend fluctuations may be observable in some experimental systems.

An Energy-Minimization Finite-Element Approach for the Frank--Oseen Model of Nematic Liquid Crystals

This paper outlines an energy-minimization finite-element approach to the computational modeling of equilibrium configurations for nematic liquid crystals under free elastic effects. The method targets minimization of the system free energy based on the Frank--Oseen free-energy model. Solutions to the intermediate discretized free elastic linearizations are shown to exist generally and are unique under certain assumptions. This requires proving continuity, coercivity, and weak coercivity for the accompanying appropriate bilinear forms within a mixed finite-element framework. Error analysis demonstrates that the method constitutes a convergent scheme. Numerical experiments are performed for problems with a range of physical parameters as well as simple and patterned boundary conditions. The resulting algorithm accurately handles heterogeneous constant coefficients and effectively resolves configurations resulting from complicated boundary conditions relevant in ongoing research.

Non-Newtonian rheological properties of shearing nematic liquid crystal model systems based on the Gay-Berne potential.

The viscosities and normal stress differences of various liquid crystal model systems based on the Gay-Berne potential have been obtained as functions of the shear rate in the non-Newtonian regime. Various molecular shapes such as regular convex calamitic and discotic ellipsoids and non-convex shapes such as bent core molecules and soft ellipsoid strings have been examined. The isotropic phases were found to be shear thinning with the shear rate dependence of the viscosity following a power law in the same way as alkanes and other non-spherical molecules. The nematic phases turned out to be shear thinning but the logarithm of the viscosity proved to be an approximately linear function of the square root of the shear rate. The normal stress differences were found to display a more or less parabolic dependence on the shear rate in the isotropic phase whereas this dependence was linear at low to intermediate shear rates in the nematic phase.

Compressible hydrodynamic flow of nematic liquid crystals with vacuum

In this paper, we first consider the free boundary problem for a simplified version of Ericksen–Leslie equations modeling the compressible hydrodynamic flow of nematic liquid crystals in dimension one which connects continuously to vacuum. We obtain the existence of global weak solutions. Furthermore, we establish the life-span of smooth solutions to the compressible nematic liquid crystal model with the support of density growing sublinearly in time direction.

Stability of the Melting Hedgehog in the Landau–de Gennes Theory of Nematic Liquid Crystals

We investigate stability properties of the radially symmetric solution corresponding to the vortex defect (so called "melting hedgehog") in the framework of the Landau - de Gennes model of nematic liquid crystals. We prove local stability of the melting hedgehog under arbitrary $Q$-tensor valued perturbations in the temperature regime near the critical supercooling temperature. As a consequence of our method, we also rediscover the loss of stability of the vortex defect in the deep nematic regime.

Space Dependent Mean Field Approximation Modelling

It is shown that the self-consistency condition which is the basic equation for calculating the mean-field order parameter of any mean-field model Hamiltonian can be replaced by the standard Metropolis Monte Carlo scheme. The advantage of this method is its ease of implementation for both the homogeneous mean-field order parameter and the heterogeneous one. To be specific, the mean-field version of the Ising model spin system is discussed in detail and the resulting magnetization is the same as in the case of solving the respective mean-field self-consistency equation. In addition, it is shown that if a high temperature phase of such system is quenched below critical temperature then the mean field experienced by spins develops into a network of domains in analogous way as it happens with the spins in the case of the exact many-body Hamiltonian system and the coarsening processes start to take place. To show that the introduced Metropolis Monte Carlo method works also in case of the continuous variables the order parameter for the Maier-Saupe model for nematic liquid crystals has been calculated.

Well-Posedness of a Fully Coupled Navier--Stokes/Q-tensor System with Inhomogeneous Boundary Data

We prove short-time well-posedness and existence of global weak solutions of the Beris--Edwards model for nematic liquid crystals in the case of a bounded domain with inhomogeneous mixed Dirichlet and Neumann boundary conditions. The system consists of the Navier--Stokes equations coupled with an evolution equation for the $Q$-tensor. The solutions possess higher regularity in time of order one compared to the class of weak solutions with finite energy. This regularity is enough to obtain Lipschitz continuity of the nonlinear terms in the corresponding function spaces. Therefore the well-posedness is shown with the aid of the contraction mapping principle using that the linearized system is an isomorphism between the associated function spaces.

Director alignment relative to the temperature gradient in nematic liquid crystals studied by molecular dynamics simulation.

The director alignment relative to the temperature gradient in nematic liquid crystal model systems consisting of soft oblate or prolate ellipsoids of revolution has been studied by molecular dynamics simulation. The temperature gradient is maintained by thermostating different parts of the system at different temperatures by using a Gaussian thermostat. It is found that the director of the prolate ellipsoids aligns perpendicularly to the temperature gradient whereas the director of the oblate ellipsoids aligns parallel to this gradient. When the director is oriented in between the parallel and perpendicular orientations a torque is exerted forcing the director to the parallel or perpendicular orientation. Because of symmetry restrictions there is no linear dependence of the torque being a pseudovector on the temperature gradient being a polar vector in an axially symmetric system such as a nematic liquid crystal. The lowest possible order of this dependence is quadratic. Thus the torque is very weak when the temperature gradient is small, which may explain why this orientation phenomenon is hard to observe experimentally. In both cases the director attains the orientation that minimises the irreversible entropy production.

A review of mathematical analysis of nematic and smectic-A liquid crystal models

We review the mathematical analysis of some uniaxial, liquid crystal phases. Firstly, we state the models for the two different studied phases: nematic and smectic-A liquid crystals. The spatial and temporal profiles of the liquid crystal configurations will be described by means of strongly nonlinear parabolic partial differential systems, which are presented at the same time. Then we will state some results about existence, regularity, time-periodicity and stability of solutions at infinite time for both models. It is our aim to show that, although nematic and smectic-A phases have different physical properties and are modelled by different nonlinear parabolic problems, there exists a common mathematical machinery to rewrite the models and obtain analytical results.

Estimation of viscosity coefficients and rheological functions of nanocrystalline cellulose aqueous suspensions

This paper presents a methodology to calculate different rheological functions and viscosity coefficients for lyotropic liquid crystals (LCs) using analytical calculations and experimental rheological data. By implementing Doi and Larson models for lyotropic nematic liquid crystals, the Leslie viscosity coefficients for nanocrystalline cellulose (NCC) aqueous suspensions have been calculated as a function of concentration of molecules per unit volume. The Landau viscosity coefficients from the symmetric viscous stress tensor have been calculated based on the mapping between Landau-de Gennes theory and the Leslie–Ericksen theory. Various parameters and dimensionless numbers from the Landau-de Gennes theory were calculated and related to experimental data. The validation was done using numerical simulations of the Landau-de Gennes equations for transient simple shear flow between parallel plates of 7 wt NCC aqueous suspensions and experimental rheological data. Apparent viscosity and shear stress from numerical simulations have been compared with the experimental results for the same concentration and found a good agreement. A cholesteric pattern was observed for low shear rates and flow-aligning regime for higher shear rates.

A linear mixed finite element scheme for a nematic Ericksen–Leslie liquid crystal model

In this work we study a fully discrete mixed scheme, based on continuous finite elements in space and a linear semi-implicit first-order integration in time, approximating an Ericksen–Leslie nematic liquid crystal model by means of a Ginzburg–Landau penalized problem. Conditional stability of this scheme is proved via a discrete version of the energy law satisfied by the continuous problem, and conditional convergence towards generalized Young measure-valued solutions to the Ericksen–Leslie problem is showed when the discrete parameters (in time and space) and the penalty parameter go to zero at the same time. Finally, we will show some numerical experiences for a phenomenon of annihilation of singularities.

Blow-up criteria for 3D nematic liquid crystal models in a bounded domain

Abstract In this paper we prove some blow-up criteria for two 3D density-dependent nematic liquid crystal models in a bounded domain. MSC:35Q30, 76D03, 76D09.

Trajectory attractors for the Sun–Liu model for nematic liquid crystals in 3D

In this paper we prove the existence of a trajectory attractor (in the sense of Chepyzhov and Vishik) for a nonlinear PDE system obtained from a 3D liquid crystal model accounting for stretching effects. The system couples a nonlinear evolution equation for the director d (introduced in order to describe the preferred orientation of the molecules) with an incompressible Navier–Stokes equation for the evolution of the velocity field u. The technique is based on the introduction of a suitable trajectory space and of a metric accounting for the double-well type nonlinearity contained in the director equation. Finally, a dissipative estimate is obtained by using a proper integrated energy inequality. Both the cases of (homogeneous) Neumann and (non-homogeneous) Dirichlet boundary conditions for d are considered.

Modeling of molecular reorientation and beam propagation in chiral and non-chiral nematic liquid crystals

The exact molecular reorientation model for nematic liquid crystals taking into account all diagonal Frank elastic constants and using two angles to describe director orientation is presented. Solutions and simplified equations are shown for the most common planar and chiral configurations. Gaussian beam propagation simulated using fully vectorial Beam Propagation Method in nonlinear case is also provided. Detailed comparison between exact solutions and single Frank constant approximation is made. However, no significant differences between these two models were found neither in beam propagation nor in polarization distribution, some difficulties may occur in choosing single Frank constant especially when it comes to quantitative results. Presented results correspond to a propagation of a beam of the Gaussian or topologically similar shapes.

A New Approach to Non-Isothermal Models for Nematic Liquid Crystals

We introduce a new class of non-isothermal models describing the evolution of nematic liquid crystals and prove their consistency with the fundamental laws of classical Thermodynamics. The resulting system of equations captures all essential features of physically relevant models, in particular, the effect of stretching of the director field is taken into account. In addition, the associated initial-boundary value problem admits global-in-time weak solutions without any essential restrictions on the size of the initial data.

Effect of external field on coherent backscattering in nematics

For a one-elastic-constant model of nematic liquid crystal the optical theorem is shown to produce an explicit relationship between the scattering length of extraordinary wave mode and magnetic coherence length. The Monte Carlo simulation of coherent backscattering is performed accounting for the long-range orientational fluctuations and scattering length anisotropy; the coherent backscattering peak is shown to change quite weakly while the magnetic field varies several orders.

On the long-time behavior of some mathematical models for nematic liquid crystals

A model describing the evolution of a liquid crystal substance in the nematic phase is investigated in terms of two basic state variables: the velocity field u and the director field d, representing the preferred orientation of molecules in a neighborhood of any point in a reference domain. After recalling a known existence result, we investigate the long-time behavior of weak solutions. In particular, we show that any solution trajectory admits a non-empty ω-limit set containing only stationary solutions. Moreover, we give a number of sufficient conditions in order that the ω-limit set contains a single point. Our approach improves and generalizes existing results on the same problem.