Lattice Boltzmann scheme for modeling liquid-crystal dynamics: Zenithal bistable device in the presence of defect motion

Multicomponent flow on curved surfaces: A vielbein lattice Boltzmann approach

We develop and implement a finite difference lattice Boltzmann scheme to study multicomponent flows on curved surfaces, coupling the continuity and Navier-Stokes equations with the Cahn-Hilliard equation to track the evolution of the binary fluid interfaces. The standard lattice Boltzmann method relies on regular Cartesian grids, which makes it generally unsuitable to study flow problems on curved surfaces. To alleviate this limitation, we use a vielbein formalism to write the Boltzmann equation on an arbitrary geometry, and solve the evolution of the fluid distribution functions using a finite difference method. Focusing on the torus geometry as an example of a curved surface, we demonstrate drift motions of fluid droplets and stripes embedded on the surface of the torus. Interestingly, they migrate in opposite directions: fluid droplets to the outer side while fluid stripes to the inner side of the torus. For the latter we demonstrate that the global minimum configuration is unique for small stripe widths, but it becomes bistable for large stripe widths. Our simulations are also in agreement with analytical predictions for the Laplace pressure of the fluid stripes, and their damped oscillatory motion as they approach equilibrium configurations, capturing the corresponding decay timescale and oscillation frequency. Finally, we simulate the coarsening dynamics of phase separating binary fluids in the hydrodynamics and diffusive regimes for tori of various shapes, and compare the results against those for a flat two-dimensional surface. Our finite difference lattice Boltzmann scheme can be extended to other surfaces and coupled to other dynamical equations, opening up a vast range of applications involving complex flows on curved geometries.

Multiparticle collision dynamics for tensorial nematodynamics

Liquid crystals establish a nearly unique combination of thermodynamic, hydrodynamic, and topological behavior. This poses a challenge to their theoretical understanding and modeling. The arena where these effects come together is the mesoscopic (micron) scale. It is then important to develop models aimed at capturing this variety of dynamics. We have generalized the particle-based multiparticle collision dynamics (MPCD) method to model the dynamics of nematic liquid crystals. Following the Qian-Sheng theory [Phys. Rev. E 58, 7475 (1998)1063-651X10.1103/PhysRevE.58.7475] of nematics, the spatial and temporal variations of the nematic director field and order parameter are described by a tensor order parameter. The key idea is to assign tensorial degrees of freedom to each MPCD particle, whose mesoscopic average is the tensor order parameter. This nematic MPCD method includes backflow effect, velocity-orientation coupling, and thermal fluctuations. We validate the applicability of this method by testing (i) the nematic-isotropic phase transition, (ii) the flow alignment of the director in shear and Poiseuille flows, and (iii) the annihilation dynamics of a pair of line defects. We find excellent agreement with existing literature. We also investigate the flow field around a force dipole in a nematic liquid crystal, which represents the leading-order flow field around a force-free microswimmer. The anisotropy of the medium not only affects the magnitude of velocity field around the force dipole, but can also induce hydrodynamic torques depending on the orientation of dipole axis relative to director field. A force dipole experiences a hydrodynamic torque when the dipole axis is tilted with respect to the far-field director. The direction of hydrodynamic torque is such that the pusher- (or puller-) type force dipole tends to orient along (or perpendicular to) the director field. Our nematic MPCD method can have far-reaching implications not only in modeling of nematic flows, but also to study the motion of colloids and microswimmers immersed in an anisotropic medium.

Bulk chirality effect for symmetric bistable switching of liquid crystals on topologically self-patterned degenerate anchoring surface

We demonstrate a bistable switching liquid crystal (LC) mode utilizing a topologically self-structured dual-groove surface for degenerated easy axes of LC anchoring. In our study, the effect of the bulk elastic distortion of the LC directors on the bistable anchoring surface is theoretically analyzed for balanced bistable states based on a free energy diagram. By adjusting bulk LC chirality, we developed ideally symmetric and stable bistable anchoring and switching properties, which can be driven by a low in-plane pulsed field of about 0.7 V/µm. The fabricated device has a contrast ratio of 196:1.

Lattice Boltzmann simulation of asymmetric flow in nematic liquid crystals with finite anchoring

Liquid crystals (LCs) display many of the flow characteristics of liquids but exhibit long range orientational order. In the nematic phase, the coupling of structure and flow leads to complex hydrodynamic effects that remain to be fully elucidated. Here, we consider the hydrodynamics of a nematic LC in a hybrid cell, where opposite walls have conflicting anchoring boundary conditions, and we employ a 3D lattice Boltzmann method to simulate the time-dependent flow patterns that can arise. Due to the symmetry breaking of the director field within the hybrid cell, we observe that at low to moderate shear rates, the volumetric flow rate under Couette and Poiseuille flows is different for opposite flow directions. At high shear rates, the director field may undergo a topological transition which leads to symmetric flows. By applying an oscillatory pressure gradient to the channel, a net volumetric flow rate is found to depend on the magnitude and frequency of the oscillation, as well as the anchoring strength. Taken together, our findings suggest several intriguing new applications for LCs in microfluidic devices.

Competition of lattice and basis for alignment of nematic liquid crystals

Due to elastic anisotropy, two-dimensional patterning of substrates can promote weak azimuthal alignment of adjacent nematic liquid crystals. Here we consider how such alignment can be achieved using a periodic square lattice of circular or elliptical motifs. In particular, we examine ways in which the lattice and motif can combine to favor differing orientations. Using Monte Carlo simulation and continuum elasticity we find, for circular motifs, that the coverage fraction controls both the polar anchoring angle and a transition in the azimuthal orientation. If the circles are generalized to ellipses, arbitrary control of the effective easy axis and effective anchoring potential becomes achievable by appropriate tuning of the ellipse motif relative to the periodic lattice patterning. This has possible applications in both monostable and bistable liquid crystal device contexts.

Beam-splitter switches based on zenithal bistable liquid-crystal gratings

The tunable optical diffractive properties of zenithal bistable nematic liquid-crystal gratings are theoretically investigated. The liquid-crystal orientation is rigorously solved via a tensorial formulation of the Landau-de Gennes theory and the optical transmission properties of the gratings are investigated via full-wave finite-element frequency-domain simulations. It is demonstrated that by proper design the two stable states of the grating can provide nondiffracting and diffracting operation, the latter with equal power splitting among different diffraction orders. An electro-optic switching mechanism, based on dual-frequency nematic materials, and its temporal dynamics are further discussed. Such gratings provide a solution towards tunable beam-steering and beam-splitting components with extremely low power consumption.

Liquid crystal alignment induced by micron-scale patterned surfaces

Induced bulk orientation of nematic liquid crystal in contact with micron-scale patterned surfaces is investigated using the Landau-de Gennes theory by means of three-dimensional simulations. The effect of the size and spacing of square cross-sectional well and post patterns is investigated and shown to influence the orientation of the liquid crystal bulk, far removed from the surface. Additionally, the effective anchoring strength of the induced alignment is estimated using a modified version of the torque balance method. Both azimuthal and zenithal multistability are shown to exist within unique ranges of feature sizes.

Backflow-mediated domain switching in nematic liquid crystals

We study the dynamics of the nematic liquid crystal kickback effect upon removal of a primary electric field and its amplification by a perpendicular secondary electric field resulting in the formation of domains with a reverse director orientation. Using computational fluid dynamics, we show that the domain formation is a robust phenomenon that takes place also in the complex case of multiple irregular random Freedericksz domains in three dimension as they appear in a realistic experimental situation. We propose domain switching by kickback amplification as a tool for self-insertion of shell-like inhomogeneities into an otherwise perfectly uniform director field configuration.

Lattice Boltzmann method coupled with the Oldroyd-B constitutive model for a viscoelastic fluid

We developed a lattice Boltzmann method coupled with the Oldroyd-B constitutive equation to simulate a viscoelastic fluid. In this work, the flow field of the solvent is solved using an incompressible lattice Boltzmann Bhatnagar-Gross-Krook (BGK) model, while the advection operator of the polymer stress tensor is directly calculated with the help of the particle distribution functions. Specifically, we present a numerical scheme for the advection of the polymer stress tensor through the truncation of second-order Taylor expansion, which does not need to introduce the extra distribution functions and has better numerical accuracy. We consider two types of numerical tests to examine the performance of the presented method, including a two-dimensional (2D) channel flow and the 4:1 contraction problem. Our numerical results for the 2D channel flow agree well with the analytical results and some experimental results reported in the previous studies. Moreover, the numerical results also indicate that the current method can capture some complex rheological behaviors of the 4:1 contraction flow.

Independent control of polar and azimuthal anchoring

Monte Carlo simulation, experiment, and continuum theory are used to examine the anchoring exhibited by a nematic liquid crystal at a patterned substrate comprising a periodic array of rectangles that, respectively, promote vertical and planar alignment. It is shown that the easy axis and effective anchoring energy promoted by such surfaces can be readily controlled by adjusting the design of the pattern. The calculations reveal rich behavior: for strong anchoring, as exhibited by the simulated system, for rectangle ratios ≥2 the nematic aligns in the direction of the long edge of the rectangles, the azimuthal anchoring coefficient changing with pattern shape. In weak anchoring scenarios, however, including our experimental systems, preferential anchoring is degenerate between the two rectangle diagonals. Bistability between diagonally aligned and edge-aligned arrangement is predicted for intermediate combinations of anchoring coefficient and system length scale.

Interfacial motion in flexo- and order-electric switching between nematic filled states

We consider a nematic liquid crystal, in coexistence with its isotropic phase, in contact with a substrate patterned with rectangular grooves. In such a system the nematic phase may fill the grooves without the occurrence of complete wetting. There may exist multiple (meta)stable filled states, each characterized by the type of distortion (bend or splay) in each corner of the groove and by the shape of the nematic-isotropic interface, and additionally the plateaux that separate the grooves may be either dry or wet with a thin layer of nematic. Using numerical simulations, we analyse the dynamical response of the system to an externally-applied electric field, with the aim of identifying switching transitions between these filled states. We find that order-electric coupling between the fluid and the field provides a means of switching between states where the plateaux between grooves are dry and states where they are wetted by a nematic layer, without affecting the configuration of the nematic within the groove. We find that flexoelectric coupling may change the nematic texture in the groove, provided that the flexoelectric coupling differentiates between the types of distortion at the corners of the substrate. We identify intermediate stages of the transitions, and the role played by the motion of the nematic-isotropic interface. We determine quantitatively the field magnitudes and orientations required to effect each type of transition.

Dynamic coupling between a multistable defect pattern and flow in nematic liquid crystals confined in a porous medium

When a nematic liquid crystal is confined in a porous medium with strong anchoring conditions, topological defects, called disclinations, are stably formed with numerous possible configurations. Since the energy barriers between them are large enough, the system shows multistability. Our lattice Boltzmann simulations demonstrate dynamic couplings between the multistable defect pattern and the flow in a regular porous matrix. At sufficiently low flow speed, the topological defects are pinned at the quiescent positions. As the flow speed is increased, the defects show cyclic motions and nonlinear rheological properties, which depend on whether or not they are topologically constrained in the porous networks. In addition, we discover that the defect pattern can be controlled by controlling the flow. Thus, the flow path is recorded in the porous channels owing to the multistability of the defect patterns.

Mesh-free modeling of liquid crystals using modified smoothed particle hydrodynamics

We present a generalization of the modified smooth particle hydrodynamics simulation technique capable of simulating static and dynamic liquid crystalline behavior. This generalization is then implemented in the context of the Qian-Sheng description of nematodynamics. To test the method, we first use it to simulate switching in both a Fréedericksz setup and a chiral hybrid aligned nematic cell. In both cases, the results obtained give excellent agreement with previously published results. We then apply the technique in a three-dimensional simulation of the switching dynamics of the post aligned bistable nematic device.

Zenithal bistable device: Comparison of modeling and experiment

A comparative modeling and experimental study of the zenithal bistable liquid crystal device is presented. A dynamic Landau de Gennes theory of nematic liquid crystals is solved numerically to model the electric field induced latching of the device and the results are compared with experimental measurements and theoretical approximations. The study gives a clear insight into the latching mechanism dynamics and enables the dependence of the device latching on both material parameters and surface shape to be determined. Analytical approximation highlights a route to optimize material selection in terms of latching voltages and the numerical model, which includes an accurate surface representation, recovers the complex surface shape effects. Predictions of device performance are presented as a function of both surface anchoring strength and surface shape and grating pitch. A measurement of the homeotropic anchoring energy has been undertaken by comparing the voltage response as a function of cell gap; we find the homeotropic anchoring energies can be varied in the range 0.5 to 4 ( 10^{-4} J m^{-2} ).

Defect trajectories and domain-wall loop dynamics during two-frequency switching in a bistable azimuthal nematic device

Bistable azimuthal nematic alignment textures have been created in micrometer-scale channels for which one sidewall is smooth and straight and the other possesses a symmetric sawtooth morphology. The optical textures have been observed during dynamic switching between the two stable states in response to dual frequency ac waveform driving of a highly dispersive nematic liquid crystal. The switching processes involves collapsing of filamentlike director reorientation (tilt-wall) loops and the associated motion and annihilation of surface defects along and close to the edge at the sawtooth sidewall. The predictions from both the n-director-based Ericksen-Leslie theory and the Q-tensor theory are in good agreement with the experimental observations.

Consistent numerical evaluation of the anchoring energy of a grooved surface

We evaluate the azimuthal anchoring energy of a grooved surface by calculating numerically the Frank elastic energy of a nematic cell composed of the grooved surface and a flat one with rigid azimuthal anchoring, where the director is fixed along the phi direction. We pay attention to the surface anchoring induced by elastic distortions of the director due to its contact with a nonflat surface, which impose local planar degenerate anchoring. Surface anchoring of this kind was analyzed analytically for shallow grooves by Berreman [Phys. Rev. Lett. 28, 1683 (1972)] and critically reexamined by the present authors [Phys. Rev. Lett. 98, 187803; 99, 139902(E) (2007)]. We consider two types of surface. one is a surface with one-dimensional sinusoidal parallel grooves, and the other is a surface with two-dimensional square patterns whose surface height is given by a sum of two sinusoidal functions with orthogonal wave vectors. The total energy is the sum of the anchoring energy and the twist energy in the bulk. For the calculation of the twist energy to be eliminated and the evaluation of the azimuthal-angle dependence of the anchoring energy, the "average" azimuthal angle at the bottom, phi(0), must be determined. We adopt two methods to determine phi(0). One is a simple extrapolation of the twist deformation in the bulk. The other relates phi(0) to the variation of the total Frank elastic energy with respect to phi. Our calculations indicate that both methods give essentially the same results, which indicates the consistency of those two methods. We also show that, for a surface with square patterns, the agreement between theory and numerical calculations is quite good even when the maximum of the surface slope is around 0.4, which theory assumes is much smaller than unity. When the surface slope is of order unity, the deviation of numerical results from theory crucially depends on the the surface elastic constant K24.

Anchoring of a nematic liquid crystal induced by surface grooves: a numerical study

To examine the anchoring energy of a surface with one-dimensional grooves of sinusoidal shape, we carry out numerical calculation of the Frank elastic energy of a nematic cell composed of such a grooved surface and a flat surface. We evaluate the anchoring energy of the grooved surface by carefully eliminating the contribution from a uniform twist deformation in the bulk. When qA < or = 0.2 , with q and A being the wave number and the amplitude of the surface groove, we find that the azimuthal-angle dependence of the calculated anchoring energy agrees perfectly with our previous analytical result under the assumption of qA<< 1[Fukuda, Phys. Rev. Lett. 98, 187803 (2007); 99, 139902(E) (2007)]. Even when qA approximately 0.6 or 1, we observe an unexpectedly good agreement between the calculated and the analytical anchoring energies, indicating the wide applicability of the analytical anchoring energy in spite of the assumption of qA<<1 in its derivation.