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.

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.

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.

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.

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.

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.

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.

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.

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.

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".

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.

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.

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.

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.

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.

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.

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.

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.

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.

Modeling smectic layers in confined geometries: order parameter and defects.

We identify problems with the standard complex order parameter formalism for smectic-A (SmA) liquid crystals and discuss possible alternative descriptions of smectic order. In particular, we suggest an approach based on the real smectic density variation rather than a complex order parameter. This approach gives reasonable numerical results for the smectic layer configuration and director field in sample geometries and can be used to model smectic liquid crystals under nanoscale confinement for technological applications.

Polymer-disordered liquid crystals: susceptibility to an electric field.

When nematic liquid crystals are embedded in random polymer networks, the disordered environment disrupts the long-range order, producing a glassy state. If an electric field is applied, it induces large and fairly temperature-independent orientational order. To understand the experiments, we simulate a liquid crystal in a disordered polymer network, visualize the domain structure, and calculate the response to a field.

Brownian motion of boomerang colloidal particles.

We investigate the Brownian motion of boomerang colloidal particles confined between two glass plates. Our experimental observations show that the mean displacements are biased towards the center of hydrodynamic stress (CoH), and that the mean-square displacements exhibit a crossover from short-time faster to long-time slower diffusion with the short-time diffusion coefficients dependent on the points used for tracking. A model based on Langevin theory elucidates that these behaviors are ascribed to the superposition of two diffusive modes: the ellipsoidal motion of the CoH and the rotational motion of the tracking point with respect to the CoH.

Shape and chirality transitions in off-axis twist nematic elastomer ribbons.

Using both experiments and finite element simulations, we explore the shape evolution of off-axis twist nematic elastomer ribbons as a function of temperature. The elastomers are prepared by cross-linking the mesogens with planar anchoring of the director at top and bottom surfaces with a 90° left-handed twist. Shape evolution depends sensitively on the off-axis director orientation at the sample midplane.

Statistical mechanics of bend flexoelectricity and the twist-bend phase in bent-core liquid crystals.

We develop a Landau theory for bend flexoelectricity in liquid crystals of bent-core molecules. In the nematic phase of the model, the bend flexoelectric coefficient increases as we reduce the temperature toward the nematic to polar phase transition. At this critical point, there is a second-order transition from high-temperature uniform nematic phase to low-temperature nonuniform polar phase composed of twist-bend or splay-bend deformations.