Explicit demonstration of geometric frustration in chiral liquid crystals.

Many solid materials and liquid crystals exhibit geometric frustration, meaning that they have an ideal local structure that cannot fill up space. For that reason, the global phase must be a compromise between the ideal local structure and geometric constraints. As an explicit example of geometric frustration, we consider a chiral liquid crystal confined in a long cylinder with free boundaries. When the radius of the tube is sufficiently small, the director field forms a double-twist configuration, which is the ideal local structure. However, when the radius becomes larger (compared with the natural twist of the liquid crystal), the double-twist structure cannot fill space, and hence the director field must transform into some other chiral structure that can fill space. This space-filling structure may be either (1) a cholesteric phase with single twist, or (2) a set of double-twist regions separated by a disclination, which can be regarded as the beginning of a blue phase. We investigate these structures using theory and simulations, and show how the relative free energies depend on the system size, the natural twist, and the disclination energy. As another example, we also study a cholesteric liquid crystal confined between two infinite parallel plates with free boundaries.

Modulated phases of nematic liquid crystals induced by tetrahedral order.

Recent theoretical research has developed a general framework to understand director deformations and modulated phases in nematic liquid crystals. In this framework, there are four fundamental director deformation modes: twist, bend, splay, and a fourth mode Δ related to saddle-splay. The first three of these modes are known to induce modulated phases. Here, we consider modulated phases induced by the fourth mode. We develop a theory for tetrahedral order in liquid crystals, and show that it couples to the Δ mode of director deformation. Because of geometric frustration, the Δ mode cannot fill space by itself, but rather must be accompanied by twist or splay. Hence, it may induce a spontaneous cholesteric phase, with either handedness, or a splay nematic phase.

Coarse-grained theory for motion of solitons and skyrmions in liquid crystals.

Recent experiments have found that applied electric fields can induce motion of skyrmions in chiral nematic liquid crystals. To understand the magnitude and direction of the induced motion, we develop a coarse-grained approach to describe dynamics of skyrmions, similar to our group's previous work on the dynamics of disclinations. In this approach, we represent a localized excitation in terms of a few macroscopic degrees of freedom, including the position of the excitation and the orientation of the background director. We then derive the Rayleigh dissipation function, and hence the equations of motion, in terms of these macroscopic variables. We demonstrate this theoretical approach for 1D motion of a sine-Gordon soliton, and then extend it to 2D motion of a skyrmion. Our results show that skyrmions move in a direction perpendicular to the induced tilt of the background director. When the applied field is removed, skyrmions move in the opposite direction but not with equal magnitude, and hence the overall motion may be rectified.

Alignment of a topological defect by an activity gradient.

As a method for controlling active materials, researchers have suggested designing patterns of activity on a substrate, which should guide the motion of topological defects. To investigate this concept, we model the behavior of a single defect of topological charge +1/2, moving in an activity gradient. This modeling uses three methods: (1) approximate analytic solution of hydrodynamic equations, (2) macroscopic, symmetry-based theory of the defect as an effective oriented particle, and (3) numerical simulation. All three methods show that an activity gradient aligns the defect orientation, and hence should be useful to control defect motion.

Geometry and mechanics of disclination lines in 3D nematic liquid crystals.

In 3D nematic liquid crystals, disclination lines have a range of geometric structures. Locally, they may resemble +1/2 or -1/2 defects in 2D nematic phases, or they may have 3D twist. Here, we analyze the structure in terms of the director deformation modes around the disclination, as well as the nematic order tensor inside the disclination core. Based on this analysis, we construct a vector to represent the orientation of the disclination, as well as tensors to represent higher-order structure. We apply this method to simulations of a 3D disclination arch, and determine how the structure changes along the contour length. We then use this geometric analysis to investigate three types of forces acting on a disclination: Peach-Koehler forces due to external stress, interaction forces between disclination lines, and active forces. These results apply to the motion of disclination lines in both conventional and active liquid crystals.

Annihilation trajectory of defects in smectic-C films.

In a two-dimensional liquid crystal, each topological defect has a topological charge and a characteristic orientation and hence can be regarded as an oriented particle. Theories predict that the trajectories of annihilating defects depend on their relative orientation. Recently, these predictions have been tested in experiments on smectic-C films. Those experiments find curved trajectories that are similar to the predictions, but the detailed relationship between the defect orientations and the far-field director is different. To understand this difference, we extend the previous theories by adding the effects of elastic anisotropy and find that it significantly changes the curved trajectories.

Theory of the splay nematic phase: Single versus double splay.

Recent experiments have reported a novel splay nematic phase, which has alternating domains of positive and negative splay. To model this phase, previous studies have considered a one-dimensional (1D) splay modulation of the director field, accompanied by a 1D modulation of polar order. When the flexoelectric coupling between splay and polar order becomes sufficiently strong, the uniform nematic state becomes unstable to the formation of a modulated phase. Here we reexamine this theory in terms of a recent approach to liquid crystal elasticity, which shows that pure splay deformation is double splay rather than planar single splay. Following that reasoning, we propose a structure with a two-dimensional (2D) splay modulation of the director field, accompanied by a 2D modulation of polar order, and show that the 2D structure generally has a lower free energy than the 1D structure.

Minimization principle for shear alignment of liquid crystals.

If a static perturbation is applied to a liquid crystal, then the director configuration changes to minimize the free energy. If a shear flow is applied to a liquid crystal, then one might ask: Does the director configuration change to minimize any effective potential? To address that question, we derive the Leslie-Ericksen equations for dissipative dynamics and determine whether they can be expressed as relaxation toward a minimum. The answer may be yes or no, depending on the number of degrees of freedom. Using theory and simulations, we consider two specific examples, reverse tilt domains under simple shear flow and dowser configurations under plane Poiseuille flow, and we demonstrate that each example shows relaxation toward the minimum of an effective potential.

Electric field-induced crossover from 3D to 2D topological defects in a nematic liquid crystal: experimental verification.

A substrate was patterned with two pairs of half-integer strength topological defects, (+1/2, +1/2) and (+1/2, -1/2). In a sufficiently thick cell, a disclination line runs in an arch above the substrate connecting the two half integer defects within each pair. The director around the disclination line for the like-sign pair must rotate in 3D, whereas for the opposite-sign defect pair the director lies in the xy-plane parallel to the substrate. For a negative dielectric anisotropy nematic, an electric field applied normal to the substrate drives the director into the xy-plane, forcing the arch of the disclination line of the like-sign pair to become extended along the z-axis. For sufficiently large field the arch splits, resulting in two nearly parallel disclination lines traversing the cell from one substrate to the other. The opposite-sign defect pair is largely unaffected by the electric field as the director already lies in the xy-plane. Experimental results are presented, which are consistent with numerical simulations.

Theory of defect motion in 2D passive and active nematic liquid crystals.

The motion of topological defects is an important feature of the dynamics of all liquid crystals, and is especially conspicuous in active liquid crystals. Understanding defect motion is a challenging theoretical problem, because the dynamics of orientational order is coupled with backflow of the fluid, and because a liquid crystal has several distinct viscosity coefficients. Here, we suggest a coarse-grained, variational approach, which describes the motion of defects as effective "particles". For passive liquid crystals, the theory shows how the drag depends on defect orientation, and shows the coupling between translational and rotational motion. For active liquid crystals, the theory provides an alternative way to describe motion induced by the activity coefficient.

Comparing skyrmions and merons in chiral liquid crystals and magnets.

When chiral liquid crystals or magnets are subjected to applied fields or other anisotropic environments, the competition between favored twist and anisotropy leads to the formation of complex defect structures. In some cases, the defects are skyrmions, which have 180^{∘} double twist going outward from the center, and hence can pack together without singularities in the orientational order. In other cases, the defects are merons, which have 90^{∘} double twist going outward from the center; packing such merons requires singularities in the orientational order. In the liquid crystal context, a lattice of merons is equivalent to a blue phase. Here we perform theoretical and computational studies of skyrmions and merons in chiral liquid crystals and magnets. Through these studies, we calculate the phase diagrams for liquid crystals and magnets in terms of dimensionless ratios of energetic parameters. We also predict the range of metastability for liquid crystal skyrmions and show that these skyrmions can move and interact as effective particles. The results show how the properties of skyrmions and merons depend on the vector or tensor nature of the order parameter.

Active Brownian particles near straight or curved walls: Pressure and boundary layers.

Unlike equilibrium systems, active matter is not governed by the conventional laws of thermodynamics. Through a series of analytic calculations and Langevin dynamics simulations, we explore how systems cross over from equilibrium to active behavior as the activity is increased. In particular, we calculate the profiles of density and orientational order near straight or circular walls and show the characteristic width of the boundary layers. We find a simple relationship between the enhancements of density and pressure near a wall. Based on these results, we determine how the pressure depends on wall curvature and hence make approximate analytic predictions for the motion of curved tracers, as well as the rectification of active particles around small openings in confined geometries.

Theory of liquid crystal elastomers and polymer networks : Connection between neoclassical theory and differential geometry.

In liquid crystal elastomers and polymer networks, the orientational order of liquid crystals is coupled with elastic distortions of crosslinked polymers. Previous theoretical research has described these materials through two different approaches: a neoclassical theory based on the liquid crystal director and the deformation gradient tensor, and a geometric elasticity theory based on the difference between the actual metric tensor and a reference metric. Here, we connect those two approaches using a formalism based on differential geometry. Through this connection, we determine how both the director and the geometry respond to a change of temperature.

Orientation of topological defects in 2D nematic liquid crystals.

Topological defects are an essential part of the structure and dynamics of all liquid crystals, and they are particularly important in experiments and simulations on active liquid crystals. In a recent paper, Vromans and Giomi [Soft Matter, 2016, 12, 6490] pointed out that topological defects are not point-like objects but actually have orientational properties, which strongly affect the energetics and motion of the defects. That paper developed a mathematical formalism which describes the orientational properties as vectors. Here, we agree with the basic concept of defect orientation, but we suggest an alternative mathematical formalism. We represent the defect orientation by a tensor, with a rank that depends on the topological charge: rank 1 for a charge of +1/2, rank 3 for a charge of -1/2. Using this tensor formalism, we calculate the orientation-dependent interaction between defects, and we present numerical simulations of defect motion.

Modeling out-of-plane actuation in thin-film nematic polymer networks: From chiral ribbons to auto-origami boxes via twist and topology.

Various experimental and theoretical studies demonstrate that complex stimulus-responsive out-of-plane distortions such as twist of different chirality, emergence of cones, simple and anticlastic bending can be engineered and pre-programmed in a liquid crystalline rubbery material given a well-controlled director microstructure. Via 3-d finite element simulation studies, we demonstrate director-encoded chiral shape actuation in thin-film nematic polymer networks under external stimulus. Furthermore, we design two complex director fields with twisted nematic domains and nematic disclinations that encode a pattern of folds for an auto-origami box. This actuator will be flat at a reference nematic state and form four well-controlled bend distortions as orientational order changes. Device fabrication is applicable via current experimental techniques. These results are in qualitative agreement with theoretical predictions, provide insight into experimental observations, and demonstrate the value of finite element methods at the continuum level for designing and engineering liquid crystal polymeric devices.

Theory of helicoids and skyrmions in confined cholesteric liquid crystals.

Cholesteric liquid crystals experience geometric frustration when they are confined between surfaces with anchoring conditions that are incompatible with the cholesteric twist. Because of this frustration, they develop complex topological defect structures, which may be helicoids or skyrmions. We develop a theory for these structures, which extends previous theoretical research by deriving exact solutions for helicoids with the assumption of constant azimuth, calculating numerical solutions for helicoids and skyrmions with varying azimuth, and interpreting the results in terms of competition between terms in the free energy.

Electrically Induced Twist in Smectic Liquid-Crystalline Elastomers.

As an approach for electrically controllable actuators, we prepare elastomers of chiral smectic-A liquid crystals, which have an electroclinic effect, i.e., molecular tilt induced by an applied electric field. Surprisingly, our experiments find that an electric field causes a rapid and reversible twisting of the film out of the plane, with a helical sense that depends on the sign of the field. To explain this twist, we develop a continuum elastic theory based on an asymmetry between the front and back of the film. We further present finite-element simulations, which show the dynamic shape change.

Dynamic Theory of Polydomain Liquid Crystal Elastomers.

When liquid crystal elastomers are prepared without any alignment, disordered polydomain structures emerge as the materials are cooled into the nematic phase. These polydomain structures are often attributed to quenched disorder in the cross-linked polymer network. As an alternative explanation, we develop a theory for the dynamics of the isotropic-nematic transition in liquid crystal elastomers, and show that the dynamics can induce a polydomain structure with a characteristic length scale, through a mechanism analogous to the Cahn-Hilliard equation for phase separation.

Predicting a polar analog of chiral blue phases in liquid crystals.

In liquid crystals, if flexoelectric couplings between polar order and director gradients are strong enough, the uniform nematic phase can become unstable to the formation of a modulated polar phase. Previous theories have predicted two types of modulation: twist bend and splay bend; the twist-bend phase has been found in recent experiments. Here, we investigate other types of modulation, using lattice simulations and Landau theory. In addition to twist bend and splay bend, we also find polar blue phases, with 2D or 3D modulations of both the director and the polar order. We compare polar blue phases with chiral blue phases, and discuss opportunities for observing them experimentally.

Brownian motion of arbitrarily shaped particles in two dimensions.

We implement microfabricated boomerang particles with unequal arm lengths as a model for nonsymmetric particles and study their Brownian motion in a quasi-two-dimensional geometry by using high-precision single-particle motion tracking. We show that because of the coupling between translation and rotation, the mean squared displacements of a single asymmetric boomerang particle exhibit a nonlinear crossover from short-time faster to long-time slower diffusion, and the mean displacements for fixed initial orientation are nonzero and saturate out at long times. The measured anisotropic diffusion coefficients versus the tracking point position indicate that there exists one unique point, i.e., the center of hydrodynamic stress (CoH), at which all coupled diffusion coefficients vanish. This implies that in contrast to motion in three dimensions where the CoH exists only for high-symmetry particles, the CoH always exists for Brownian motion in two dimensions. We develop an analytical model based on Langevin theory to explain the experimental results and show that among the six anisotropic diffusion coefficients only five are independent because the translation-translation coupling originates from the translation-rotation coupling. Finally, we classify the behavior of two-dimensional Brownian motion of arbitrarily shaped particles into four groups based on the particle shape symmetry group and discussed potential applications of the CoH in simplifying understanding of the circular motions of microswimmers.