Liquid Crystal Model Research Articles

Regularity for a Fractional Liquid Crystal Model with Anomalous Dissipation and Thermal Effects

Regularity for a Fractional Liquid Crystal Model with Anomalous Dissipation and Thermal Effects

A novel discontinuous Galerkin projection scheme for the hydrodynamics of nematic liquid crystals

This paper is focused on the numerical approximations for the hydrodynamic model of nematic liquid crystals. Under the framework of a splitting projection method, we propose a novel interior penalty discontinuous Galerkin (DG) method for solving the coupled system, which is employed by combining the scalar auxiliary variables (SAV) approach, implicit-explicit (IMEX) treatments and a rotational pressure-correction method. One prominent feature of the developed scheme here is by introducing an additional stabilization term artificially in liquid crystal equation to balance the explicit treatment for the coupling term, so that the computations of vector field from velocity field are decoupled. Hence a linear, fully decoupled and unconditionally energy-stable DG scheme can be achieved in a fully discrete manner. We show that the resulting scheme is uniquely solvable and unconditional energy stable, and the optimal error estimates of the proposed scheme are further proved theoretically. Finally, numerical results are carried out to demonstrate the accuracy and energy stability of our scheme.

Dynamic Behavior and Optical Soliton for the M-Truncated Fractional Paraxial Wave Equation Arising in a Liquid Crystal Model

The main purpose of this article is to investigate the dynamic behavior and optical soliton for the M-truncated fractional paraxial wave equation arising in a liquid crystal model, which is usually used to design camera lenses for high-quality photography. The traveling wave transformation is applied to the M-truncated fractional paraxial wave equation. Moreover, a two-dimensional dynamical system and its disturbance system are obtained. The phase portraits of the two-dimensional dynamic system and Poincaré sections and a bifurcation portrait of its perturbation system are drawn. The obtained three-dimensional graphs of soliton solutions, two-dimensional graphs of soliton solutions, and contour graphs of the M-truncated fractional paraxial wave equation arising in a liquid crystal model are drawn.

Strong Well-Posedness of the Q-Tensor Model for Liquid Crystals: The Case of Arbitrary Ratio of Tumbling and Aligning Effects ξ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\xi $$\\end{document}

The Beris–Edwards model of nematic liquid crystals couples an equation for the molecular orientation described by the Q-tensor with a Navier–Stokes type equation with an additional non-Newtonian stress caused by the molecular orientation. Both equations contain a parameter ξ∈R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\xi \\in \\mathbb {R}$$\\end{document} measuring the ratio of tumbling and alignment effects. Previous well-posedness results largely vary on the space dimension n and the constraints of the parameter ξ∈R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\xi \\in \\mathbb {R}$$\\end{document}. This work addresses strong well-posedness of this model, first locally and then globally for small initial data, both in the Lp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^p$$\\end{document}-L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^2$$\\end{document}-setting for p>44-n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p > \\frac{4}{4-n}$$\\end{document}, in the general cases, i.e., for n=2,3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n = 2, 3$$\\end{document} and without any restriction on ξ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\xi $$\\end{document}. The approach is based on methods from quasilinear equations and the fact that the associated linearized operator admits maximal Lp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^p$$\\end{document}-L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^2$$\\end{document}-regularity. The proof of the latter property relies on techniques from sectorial operators, Schur complements and J\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {J}$$\\end{document}-symmetry.

A Second-Order SO(3)-Preserving and Energy-Stable Scheme for Orthonormal Frame Gradient Flow Model of Biaxial Nematic Liquid Crystals

A Second-Order SO(3)-Preserving and Energy-Stable Scheme for Orthonormal Frame Gradient Flow Model of Biaxial Nematic Liquid Crystals

Hysteresis Elimination for an Anisotropic Liquid-Crystal Model via Molecule Design and Replica-Exchange Optimization.

The phenomenon of hysteresis in simulations, in which a system's current state is correlated to previous states and inhibits the transition to a more stable phase, may often lead to misleading results in physical chemistry. In this study, in addition to the replica exchange method (REM), a novel approach was taken by combining an evolution strategy based on the evolutionary principles of nature to predict phase transitions for the Hess-Su liquid-crystal model. In this model, an anisotropy term is added to the simple 6-12 Lennard-Jones model to intuitively reproduce the behavior of liquid crystals. We first applied the pressure-temperature REM to the Hess-Su model and optimized the replica spacing for the energy distribution to gain the maximum advantage from the REM. We then used the same approach as for the Hamiltonian REM, seeking to optimize the replica spacing in the same way. Based on both results, we attempted to predict this coarse-grained liquid-crystal model's exact phase transition point. In the Hamiltonian REM, replicas were prepared with different molecular aspect ratios corresponding to the values of the anisotropy terms in the potential function. The Hess-Su liquid-crystal model, which undergoes a direct transition from the nematic to the solid phase without going through a smectic phase, is a challenging research target for understanding phase transitions. Despite the tremendous computational difficulty in overcoming the strong hysteresis present in this system, our method could predict the phase transition point clearly and significantly reduce the extent of hysteresis. Our approach is beneficial when simulating more complex systems and, above all, shows great potential for more accurate and efficient phase transition predictions in the field of molecular simulation in the future.

The existence and uniqueness of time-periodic solutions to the non-isothermal model for compressible nematic liquid crystals in a periodic domain

The existence and uniqueness of time-periodic solutions to the non-isothermal model for compressible nematic liquid crystals in a periodic domain

Deviation from the prediction of the Maier-Saupe theory of mixtures of charged discotic Gay-Berne and Lennard-Jones particles

ABSTRACT We report an analysis of the orientational order and orientational distribution functions of mixtures of discotic charged Gay-Berne (GB) and Lennard-Jones (LJ) particles as models of ionic liquid crystals and colloidal suspensions. We have studied three different stoichiometries (GB:LJ of 1:2, 1:1 and 2:1), with the opposite charges on the two particles selected to guarantee electroneutrality. We have found that for relatively low values of the scaled charge the dependence of the fourth-rank orientational order parameter, <P 4>, on the second-rank orientational order parameter, <P 2>, is in very good agreement with the prediction of the Maier-Saupe theory. In contrast, for relatively high values of the charge a deviation is observed which is in opposite directions for mixtures with an excess for GB particles and mixtures with an excess of LJ particles.

Asymptotic Log-Harnack inequality for the 2D Ericksen–Leslie model system with degenerate noise and damping

In this paper, we consider a 2D liquid crystal model with damping and examine some asymptotic behaviors of the strong solution. More precisely, we establish the asymptotic log-Harnack inequality for the transition semigroup associated with a simplified liquid crystal model driven by an additive degenerate noise via the asymptotic coupling method. As consequences of the asymptotic log-Harnack inequality, we derive the gradient estimate, the asymptotic irreducibility, the asymptotic strong Feller property, the asymptotic heat kernel estimate and the ergodicity.

Complete integrability and equilibrium thermodynamics of biaxial nematic systems with discrete orientational degrees of freedom

We study a discrete version of a biaxial nematic liquid crystal model with external fields via an approach based on the solution of differential identities for the partition function. In the thermodynamic limit, we derive the free energy of the model and the associated closed set of equations of state involving four order parameters, proving the integrability and exact solvability of the model. The equations of state are specified via a suitable representation of the orientational order parameters, which imply two-order parameter reductions in the absence of external fields. A detailed exact analysis of the equations of state reveal a rich phase diagram where isotropic versus uniaxial versus biaxial phase transitions are explicitly described, including the existence of triple and tricritical points. Results on the discrete models are qualitatively consistent with their continuum analogue. This observation suggests that, in more general settings, discrete models may be used to capture and describe phenomena that also occur in the continuum for which exact equations of state in closed form are not available.

The Dynamic Model of the Active-Inactive Cell Interface

To explore the morphodynamics of the active-inactive cell monolayer interfaces by using the active liquid crystal model. A continuum mechanical model was established based on the active liquid crystal theory and the active-inactive cell monolayer interfaces were established by setting the activity difference of cell monolayers. The theoretical equations were solved numerically by the finite difference and the lattice Boltzmann method. The active-inactive cell interfaces displayed three typical morphologies, namely, flat interface, wavy interface, and finger-like interface. On the flat interfaces, the cells were oriented perpendicular to the interface, the -1/2 topological defects were clustered in the interfaces, and the interfaces were negatively charged. On the wavy interfaces, cells showed no obvious preference for orientation at the interfaces and the interfaces were neutrally charged. On the finger-like interfaces, cells were tangentially oriented at the interfaces, the +1/2 topological defects were collected at the interfaces, driving the growth of the finger-like structures, and the interfaces were positively charged. The orientation of the cell alignment at the interface can significantly affect the morphologies of the active-inactive cell monolayer interfaces, which is closely associated with the dynamics of topological defects at the interfaces.

Stochastic effects on solution landscapes for nematic liquid crystals

ABSTRACT We study the effects of additive and multiplicative noise on the solution landscape of nematic liquid crystals confined to a square domain within the Landau-de Gennes framework, as well as the impact of additive noise on the symmetric radial hedgehog solution for nematic droplets. The introduction of random noise can be used to capture material uncertainties and imperfections, which are always present in physical systems. We implement random noise in our framework by introducing a Q -Wiener stochastic process to the governing differential equations. On the square, the solution landscape for the deterministic problem is well understood, enabling us to compare and contrast the deterministic predictions and the stochastic predictions, while we demonstrate that the symmetry of the radial hedgehog solution can be violated by noise. This approach of introducing noise to deterministic equations can be used to test the robustness and validity of predictions from deterministic liquid crystal models, which essentially capture idealised situations.

Global well‐posedness to the compressible inertial Qian–Sheng model of liquid crystals

In this paper, we study the well‐posedness of the Cauchy problem to the compressible inertial Qian–Sheng model. Inspired by Jiang–Luo–Ma–Tang's work in [J. Hyperbolic. Differ. Equ 18:221‐256, 2021], we introduce a novel Condition (H) which ensures the energy dissipation, and then, we prove that when both the viscosity coefficients and Condition (H) are satisfied, the local‐in‐time solutions exist for large initial data. Otherwise, we can only obtain small data local solutions. Finally, the local solution can be extended globally under at least one of the two conditions: or .

Dynamic behavior of optical self-control soliton in a liquid crystal model

The Paraxial Model, a foundational concept in optics, provides a simplified yet effective approach to understanding the behavior of light in optical systems. This research investigates optical soliton solutions within the truncated time M-fractional Paraxial wave equation using the extended tanh method and the modified extended tanh method. Addressing the challenge of fractional order, we employ the shortened M-fractional derivative to eliminate it from the governing model, facilitating a detailed analysis. Through strategic manipulation of free parameters, soliton solutions are expressed in terms of hyperbolic and trigonometric functions, revealing a spectrum of soliton phenomena such as Lump soliton, rough wave, periodic soliton, and king wave for specific parameter values that are connected to the dispersal effect, the Kerr non-linearity effect and the diffraction effect of the governing equation. To enhance the understanding of these solutions, we utilize Matlab for graphical representations, providing a visual interpretation of the intricate dynamics. This study not only uncovers diverse soliton behaviors in the context of fractional wave equations but also offers valuable insights into their formation and complexity, bridging theoretical analysis with computational visualization.

Dimensional reduction and emergence of defects in the Oseen-Frank model for nematic liquid crystals

Dimensional reduction and emergence of defects in the Oseen-Frank model for nematic liquid crystals

Boundary regularity of a heat flow problem from Ericksen's model for nematic liquid crystals

Boundary regularity of a heat flow problem from Ericksen's model for nematic liquid crystals

Molecules structure and viscosity relationship of nematic liquid crystal and BPNN-QSAR model

Nematic liquid crystal materials designed for optics, microwave communication tuning, etc. need high response speed, which is related to the rotational viscosity and the birefringent index of the liquid crystal. In order to achieve a wide tuning range of phase modulation, the nematic liquid crystals often employ large π-electron conjugated systems and large polar groups to enhance the birefringence and dielectric anisotropy of the liquid crystal molecule, which, however, increases the viscosity of the liquid crystal material, deteriorating the response speed of the microwave device. Herein, we explore the viscosity of the nematic liquid crystal from the perspective of liquid crystal compound structure by testing the viscosity of our designed and synthesized forty-two different nematic liquid crystals by using a rotating rheometer at 25 ℃. To the best of our knowledge, the BPNN-QSAR quantitative structure-activity model between nematic liquid crystal molecular structure and viscosity is established for the first time. The correlation coefficient between the predicted value and the experimental value is &lt;i&gt;q&lt;/i&gt;&lt;sup&gt;2&lt;/sup&gt; = 0.607 &gt; 0.5, indicating that the model can be used to predict the viscosity performances of liquid crystal compounds. Besides, the molecular structure descriptors affecting the viscosity properties are explored. Based on the practical application and this model, seven liquid crystal molecules of two series with large birefringent index are designed and tested. The viscosity predicted by the BPNN model is smaller than that of the molecules of the same type and matches with the measured viscosity.

Well-posedness and the zero activity limit for the active nematic liquid crystal model

Well-posedness and the zero activity limit for the active nematic liquid crystal model

A priori and a posteriori error analysis for semilinear problems in liquid crystals

In this paper, we develop a unified framework for the a priori and a posteriori error control of different lowest-order finite element methods for approximating the regular solutions of systems of partial differential equations under a set of hypotheses. The systems involve cubic nonlinearities in lower order terms, non-homogeneous Dirichlet boundary conditions, and the results are established under minimal regularity assumptions on the exact solution. The key contributions include (i) results for existence and local uniqueness of the discrete solutions using Newton–Kantorovich theorem, (ii) a priori error estimates in the energy norm, and (iii) a posteriori error estimates that steer the adaptive refinement process. The results are applied to conforming, Nitsche, discontinuous Galerkin, and weakly over penalized symmetric interior penalty schemes for variational models of ferronematics and nematic liquid crystals. The theoretical estimates are corroborated by substantive numerical results.

Tensorial time derivatives on moving surfaces: General concepts and a specific application for surface Landau-de Gennes models

Observer-invariance is regarded as a minimum requirement for an appropriate definition of time derivatives. We systematically discuss such time derivatives for surface tensor field and provide explicit formulations for material, upper-convected, lower-convected and Jaumann/corotational time derivatives which all lead to different physical implications. We compare these results with the corresponding time derivatives for tangential tensor fields. As specific surface 2-tensor fields we consider surface Q-tensor fields and conforming surface Q-tensor fields and apply the results in surface Landau-de Gennes models for surface liquid crystals.