Particle-based and continuum models for confined nematics in two dimensions.

We use the particle-based stochastic multi-particle collision dynamics (N-MPCD) algorithm to simulate confined nematic liquid crystals in regular two-dimensional polygons such as squares, pentagons and hexagons. We consider a range of values of the nematicities, U, and simulation domain sizes, R, that canvass nano-sized polygons to micron-sized polygons. We use closure arguments to define mappings between the N-MPCD parameters and the parameters in the continuum deterministic Landau-de Gennes framework. The averaged N-MPCD configurations agree with those predicted by Landau-de Gennes theory, at least for large polygons. We study relaxation dynamics or the non-equilibrium dynamics of confined nematics in polygons, in the N-MPCD framework, and the kinetic traps bear strong resemblance to the unstable saddle points in the Landau-de Gennes framework. Finally, we study nematic defect dynamics inside the polygons in the N-MPCD framework and the finite-size effects slow down the defects and attract them to polygon vertices. Our work is a comprehensive comparison between particle-based stochastic N-MPCD methods and deterministic/continuum Landau-de Gennes methods, and this comparison is essential for new-age multiscale theories.

Tunable templating of photonic microparticles via liquid crystal order-guided adsorption of amphiphilic polymers in emulsions.

Multiple emulsions are usually stabilized by amphiphilic molecules that combine the chemical characteristics of the different phases in contact. When one phase is a liquid crystal (LC), the choice of stabilizer also determines its configuration, but conventional wisdom assumes that the orientational order of the LC has no impact on the stabilizer. Here we show that, for the case of amphiphilic polymer stabilizers, this impact can be considerable. The mode of interaction between stabilizer and LC changes if the latter is heated close to its isotropic state, initiating a feedback loop that reverberates on the LC in form of a complete structural rearrangement. We utilize this phenomenon to dynamically tune the configuration of cholesteric LC shells from one with radial helix and spherically symmetric Bragg diffraction to a focal conic domain configuration with highly complex optics. Moreover, we template photonic microparticles from the LC shells by photopolymerizing them into solids, retaining any selected LC-derived structure. Our study places LC emulsions in a new light, calling for a reevaluation of the behavior of stabilizer molecules in contact with long-range ordered phases, while also enabling highly interesting photonic elements with application opportunities across vast fields.

Elastic anisotropy in the reduced Landau-de Gennes model.

We study the effects of elastic anisotropy on Landau-de Gennes critical points, for nematic liquid crystals, on a square domain. The elastic anisotropy is captured by a parameter, L 2 , and the critical points are described by 3 d.f. We analytically construct a symmetric critical point for all admissible values of L 2 , which is necessarily globally stable for small domains, i.e. when the square edge length, λ , is small enough. We perform asymptotic analyses and numerical studies to discover at least five classes of these symmetric critical points-the WORS , Ring ± , C o n s t a n t and p W O R S solutions, of which the WORS , Ring + and C o n s t a n t solutions can be stable. Furthermore, we demonstrate that the novel C o n s t a n t solution is energetically preferable for large λ and large L 2 , and prove associated stability results that corroborate the stabilizing effects of L 2 for reduced Landau-de Gennes critical points. We complement our analysis with numerically computed bifurcation diagrams for different values of L 2 , which illustrate the interplay of elastic anisotropy and geometry for nematic solution landscapes, at low temperatures.

Tailored nematic and magnetization profiles on two-dimensional polygons.

We study dilute suspensions of magnetic nanoparticles in a nematic host, on two-dimensional polygons. These systems are described by a nematic order parameter and a spontaneous magnetization, in the absence of any external fields. We study the stable states in terms of stable critical points of an appropriately defined free energy, with a nemato-magnetic coupling energy. We numerically study the interplay between the shape of the regular polygon, the size of the polygon, and the strength of the nemato-magnetic coupling for the multistability of this prototype system. Our notable results include (1) the coexistence of stable states with domain walls and stable interior and boundary defects, (2) the suppression of multistability for positive nemato-magnetic coupling, and (3) the enhancement of multistability for negative nemato-magnetic coupling.

Quantum-dots-dispersed bent-core nematic liquid crystal and cybotactic clusters: Experimental and theoretical insights.

We study a quantum-dots (QDs) dispersed bent-core nematic liquid crystalline system in planar geometry and present experimental measurements of the birefringence (Δn), order parameter (S), dielectric dispersion, absorption spectra, and optical textures with attention to variations with temperature. A bent-core liquid crystal (LC) 14-2M-CH_{3} is used as the host material and CdSe/ZnS core-shell type QDs are used as the dopant. The nematic (N) phase of the pristine (undoped) LC 14-2M-CH_{3} contains cybotactic clusters, which are retained by its QDs incorporated LC nanocomposite. Our experimental findings support: (i) reduced orientational order parameter of the QDs dispersed LC system compared to its pristine counterpart at fixed temperatures, (ii) reduced cybotactic cluster sizes due to the incorporation of QDs, and (iii) increased activation energies related to reduced cluster sizes. We complement the experiments with a novel Landau-de Gennes-type free energy for a doped bent-core LC system that qualitatively captures the doping-induced reduced order parameter and its dependence on the properties of the QDs and its variation with temperature.

Tailored morphologies in two-dimensional ferronematic wells.

We focus on a dilute uniform suspension of magnetic nanoparticles in a nematic-filled micron-sized shallow well with tangent boundary conditions as a paradigm system with two coupled order parameters. This system exhibits spontaneous magnetization without magnetic fields. We numerically obtain the stable nematic and associated magnetization morphologies, induced purely by the geometry, the boundary conditions, and the coupling between the magnetic nanoparticles and the host nematic medium. Our most striking observations pertain to domain walls in the magnetization profile, whose location can be manipulated by the coupling and material properties, and stable interior and boundary nematic defects, whose location and multiplicity can be tailored by the coupling too. These tailored morphologies are not accessible in uncoupled systems and can be used for multistable systems with singularities and stable interfaces.

Magnetic nanoparticles in a nematic channel: A one-dimensional study.

We study a dilute suspension of magnetic nanoparticles in a nematic-filled channel and how the spatial magnetization M can be tailored by the nematic anisotropy. We study the spatial configurations as stable critical points of a generalized phenomenological energy for a dilute ferronematic in the absence of external magnetic fields. We show how spatial inhomogeneities in the equilibrium nematic profile, induced by confinement and boundary effects, generate nonzero spatially inhomogeneous magnetization profiles in the system. Depending on the magnetonematic coupling energy, M can either follow the nematic profile for large coupling or exhibit distinct polydomain structures separated by defect lines for weak coupling and low temperatures. Some exact solutions for prototypical situations are also obtained.

One-dimensional theoretical analysis of coupling and confinement effects on the cybotactic clusters of bent-core nematic liquid crystals.

The bent-core liquid crystals (LCs) are highly regarded as the next-generation materials for electro-optic devices. The nematic (N) phase of these LCs possesses highly ordered smecticlike cybotactic clusters which are promising for electro-optic applications. We have studied a one-dimensional Landau-de Gennes model of spatially inhomogeneous order parameters for the N phase of bent-core LCs. We investigate the effects of spatial confinement and coupling (between these clusters and the surrounding LC molecules modeled by a coupling parameter γ) on the order parameters. The coupling is found to increase the cluster order parameter significantly, suggesting enhancement in cluster formation, and also predicts a transition to a phase with weak nematiclike ordering above the nematic supercooling temperature.

Nematohydrodynamics for colloidal self-assembly and transport phenomena.

HYPOTHESIS: Colloidal particles in a nematic liquid crystal (NLC) exhibit very different behaviour to that observed in an isotropic medium. Such differences arise principally due to the nematic-induced elastic stresses exerted due to the interaction of NLC molecules with interfaces, which compete with traditional fluid viscous stresses on the particle. THEORY: A systematic mathematical analysis of particles in an NLC microfluidic channel is performed using the continuum Beris-Edwards framework coupled to the Navier-Stokes equations. We impose strong homeotropic anchoring on the channel walls and weak homeotropic anchoring on the particle surfaces. FINDINGS: The viscous and NLC forces act on an individual particle in opposing directions, resulting in a critical location in the channel where the particle experiences zero net force in the direction perpendicular to the flow. For multi-particle aggregation we show that the final arrangement is independent of the initial configuration, but the path towards achieving equilibrium is very different. These results uncover new mechanisms for particle separation and routes towards self-assembly.

Uniaxial versus biaxial character of nematic equilibria in three dimensions.

We study global minimizers of the Landau-de Gennes (LdG) energy functional for nematic liquid crystals, on arbitrary three-dimensional simply connected geometries with topologically non-trivial and physically relevant Dirichlet boundary conditions. Our results are specific to an asymptotic limit coined in terms of a dimensionless temperature and material-dependent parameter, t and some constraints on the material parameters, and we work in the t → ∞ limit that captures features of the widely used Lyuksyutov constraint (Kralj and Virga in J Phys A 34:829-838, 2001). We prove (i) that (re-scaled) global LdG minimizers converge uniformly to a (minimizing) limiting harmonic map, away from the singular set of the limiting map; (ii) we have points of maximal biaxiality and uniaxiality near each singular point of the limiting map; (iii) estimates for the size of "strongly biaxial" regions in terms of the parameter t. We further show that global LdG minimizers in the restricted class of uniaxial Q -tensors cannot be stable critical points of the LdG energy in this limit.

Free energy pathways of a multistable liquid crystal device.

The planar bistable device [Tsakonas et al., Appl. Phys. Lett., 2007, 90, 111913] is known to have two distinct classes of stable equilibria: the diagonal and rotated solutions. We model this device within the two-dimensional Landau-de Gennes theory, with a surface potential and without any external fields. We systematically compute a special class of transition pathways, referred to as minimum energy pathways, between the stable equilibria that provide new information about how the equilibria are connected in the Landau-de Gennes free energy landscape. These transition pathways exhibit an intermediate transition state, which is a saddle point of the Landau-de Gennes free energy. We numerically compute the structural details of the transition states, the optimal transition pathways and the free energy barriers between the equilibria, as a function of the surface anchoring strength. For strong anchoring, the transition pathways are mediated by defects whereas we get defect-free transition pathways for moderate and weak anchoring. In the weak anchoring limit, we recover a cusp catastrophe situation for which the rotated state acts as a transition state connecting two different diagonal states.

Perspectives in active liquid crystals.

Active soft matter is a young, growing field, with potential applications to a wide variety of systems. This Theme Issue explores this emerging new field by highlighting active liquid crystals. The collected contributions bridge theory to experiment, mathematical theories of passive and active nematics, spontaneous flows to defect dynamics, microscopic to continuum levels of description, spontaneous activity to biological activation. While the perspectives offered here only span a small part of this rapidly evolving field, we trust that they might provide the interested reader with a taste for this new class of non-equilibrium systems and their rich behaviour.

Colloidal liquid crystals in rectangular confinement: theory and experiment.

We theoretically and experimentally study nematic liquid crystal equilibria within shallow rectangular wells. We model the wells within a two-dimensional Oseen-Frank framework, with strong tangent anchoring, and obtain explicit analytical expressions for the director fields and energies of the 'diagonal' and 'rotated' solutions reported in the literature. These expressions separate the leading-order defect energies from the bulk distortion energy for both families of solutions. The continuum Oseen-Frank study is complemented by a microscopic mean-field approach. We numerically minimize the mean-field functional, including the effects of weak anchoring, variable order and random initial conditions. In particular, these simulations suggest the existence of higher-energy metastable states with internal defects. We compare our theoretical results to experimental director profiles, obtained using two types of filamentous virus particles, wild-type fd-virus and a modified stiffer variant (Y21M), which display nematic ordering in rectangular chambers, as found by confocal scanning laser microscopy. We combine our analytical energy expressions with experimentally recorded frequencies of the different equilibrium states to obtain explicit estimates for the extrapolation length, defined to be the ratio of the nematic elastic constant to the anchoring coefficient, of the fd-virus.

Transitions through critical temperatures in nematic liquid crystals.

We obtain estimates for critical nematic liquid crystal (LC) temperatures under the action of a slowly varying temperature-dependent control variable. We show that biaxiality has a negligible effect within our model and that these delay estimates are well described by a purely uniaxial model. The static theory predicts two critical temperatures: the supercooling temperature below which the isotropic phase loses stability and the superheating temperature above which the ordered nematic states do not exist. In contrast to the static problem, the isotropic phase exhibits a memory effect below the supercooling temperature in the dynamic framework. This delayed loss of stability is independent of the rate of change of temperature and depends purely on the initial value of the temperature. We also show how our results can be used to improve estimates for LC material constants.

Multistability in planar liquid crystal wells.

A planar bistable liquid crystal device, reported in Tsakonas et al. [Appl. Phys. Lett. 90, 111913 (2007)], is modeled within the Landau-de Gennes theory for nematic liquid crystals. This planar device consists of an array of square micrometer-sized wells. We obtain six different classes of equilibrium profiles and these profiles are classified as diagonal or rotated solutions. In the strong anchoring case, we propose a Dirichlet boundary condition that mimics the experimentally imposed tangent boundary conditions. In the weak anchoring case, we present a suitable surface energy and study the multiplicity of solutions as a function of the anchoring strength. We find that diagonal solutions exist for all values of the anchoring strength W ≥ 0, while rotated solutions only exist for W ≥ W_{c}>0, where W_{c} is a critical anchoring strength that has been computed numerically. We propose a dynamic model for the switching mechanisms based on only dielectric effects. For sufficiently strong external electric fields, we numerically demonstrate diagonal-to-rotated and rotated-to-diagonal switching by allowing for variable anchoring strength across the domain boundary.