Liquid Crystal Director Field Research Articles

On the Existence and Partial Stability of Standing Waves for a Nematic Liquid Crystal Director Field Equations

AbstractIn this paper, following the studies in Amorim et al. (Partial Differ Equ Appl 4, 36, 2023), we consider some new aspects of the motion of the director field of a nematic liquid crystal submitted to a magnetic field and to a laser beam. In particular, we study the existence and partial orbital stability of special standing waves, in the spirit of Cazenave and Lions (Commun Math Phys 85:549–561, 1982) and Hadj Selem et al. (Milan J Math 82:273–295, 2014) and we present some numerical simulations.

Dynamically reconfigurable complex liquid crystal emulsions

ABSTRACT Complex emulsions, containing liquid crystals (LCs) and immiscible oils such as fluorocarbon or silicone oils, present innovative functionalities in soft materials. These emulsions exhibit dynamic reconfiguration in droplet morphology and internal LC organisation, expanding applications in optics, sensing, and active matter. This review highlights the evolution and potential of LC-containing complex emulsions. Emphasis is placed on emulsions featuring dynamic and reversible morphological transformations and LC reorganisation, underpinning their suitability for tunable micro-lenses, chemical sensing, and biosensing. Nematic LCs within complex emulsions allow for topological defect-driven functionalization, attaching antibodies to induce substantial changes in the LC director field upon interfacial recognition events. This feature may facilitate the development of ultrasensitive chemical and biological sensors. Moreover, cholesteric LCs, through their selective reflection, enable the rapid and sensitive detection of foodborne pathogens. The controlled assembly of magnetic nanoparticles at interfaces illustrates advancements towards magnetic field-responsive and active colloids. Future perspectives include exploiting these properties for intelligent colloidal systems capable of autonomous behaviour, and furthering applications in dynamic photonic devices, high-security identification tags, and responsive lens systems.

Characteristic boundary layers in the vanishing viscosity limit for the Hunter-Saxton equation

The Hunter-Saxton equation models the propagation of weakly nonlinear orientation waves in a massive director field of the nematic liquid crystal. In this paper, we study the vanishing viscosity limit for an initial boundary value problem of the Hunter-Saxton equation with the characteristic boundary condition. By the formal multiscale analysis, we first derive the characteristic boundary layer profile, which satisfies a nonlinear parabolic equation. On the base of the Galerkin method along with a compactness argument, we then establish the global well-posedness of the boundary layer equation. Finally, we prove the global stability of the boundary layer profiles together with the optimal convergence rate of the vanishing viscosity limit by the energy method.

Luminescent DNA-origami nano-rods dispersed in a lyotropic chromonic liquid crystal

ABSTRACTAnisometric DNA nanoparticles with precisely tailored size and shape are prepared by three-dimensional folding of DNA – DNA origami – marked with a fluorescent intercalating dye (YOYO-1), stabilised with silica, and dispersed in a lyotropic chromonic liquid crystal (an aqueous solution of disodium cromoglycate). Observations by total internal reflection fluorescence microscopy indicate that the luminescent nanoparticles are stable in the liquid crystal dispersion and accumulate at the positions of point defects in the liquid crystal director field. The observations indicate that silica-stabilisation of the nanoparticles prevents the dye from leaking into the liquid crystal host and that the luminescent particles assemble at disclinations.

Topological Defect Guided Order Evolution across the Nematic-Smectic Phase Transition.

Topological defects usually emerge and vary during the phase transition of ordered systems. Their roles in thermodynamic order evolution keep being the frontier of modern condensed matter physics. Here, we study the generations of topological defects and their guidance on the order evolution during the phase transition of liquid crystals (LCs). With a given preset photopatterned alignment, two different types of topological defects are achieved depending on the thermodynamic process. Because of the memory effect of LC director field across the Nematic-Smectic (N-S) phase transition, a stable array of toric focal conic domains (TFCDs) and a frustrated one are generated in S phase, respectively. The frustrated one transfers to a metastable TFCD array with a smaller lattice constant, and further changes to a crossed-walls type N state due to the inheritance of orientational order. A free energy on temperature diagram and corresponding textures vividly describe the phase transition process and the roles of topological defects in the order evolution across the N-S phase transition. This Letter reveals the behaviors and mechanisms of topological defects on order evolution during phase transitions. It paves a way for investigating topological defect guided order evolution which is ubiquitous in soft matter and other ordered systems.

Equilibrium locations of defects in two-dimensional configurations of the NLC director field

ABSTRACT We revisit the problem of finding the equilibrium location of the defect in two-dimensional configurations of the nematic liquid crystal (NLC) director field within the framework of the Frank-Ozeen formalism. Our analytical approach can be used for a wide class of nematic-filled regions, on the boundary surfaces of which there is a strong anchoring for the director with a piecewise constant pretilt. First, we use complex analysis to find a nematic director profile as a superposition of contributions from a set of virtual defects with appropriate topological charges acting on the defect in question. Then, we use the net zero force principle to find the equilibrium location of the defect. Our method generates the set of virtual defects automatically without using ad hoc symmetry considerations as it is in the standard method of images. We reproduce some known results using very simple calculations. We also consider several simple yet original problems. Namely, we find the equilibrium location of a disclination line near a long cylindrical particle with strong combined homeotropic-circular anchoring on its lateral surface and the equilibrium location of the disclination line in the NLC bounded by a parabolic cylinder with strong circular anchoring.

TADA: The Topology-Accommodating Direction Assignment Algorithm for Liquid Crystals.

Despite the fact that topological defects are a hallmark of liquid crystalline materials, current computational techniques for identifying topological defects in particle-based simulations of these materials─which rest upon Q-tensor theory─do not leverage topological features of the system. In this work, we describe the topology-accommodating direction assignment (TADA) algorithm, a novel approach for identifying disclination cores in liquid crystalline materials, which is sensitive to topology: this method assigns to each mesogen a unique vector, thereby extending the concept of the liquid crystal director field down to the scale of mesogens. In systems containing disclination cores, TADA identifies line segments along which this assigned vector field is discontinuous, with cores located at the interior termination points of these line segments. The mere presence of defects can be identified by searching far away from them. We validate this approach by comparing its results to those obtained using the scalar order parameter for a variety of liquid crystalline assemblies sourced from molecular-dynamics simulations. We also discuss several benefits of the TADA algorithm over existing approaches for identifying topological defects in liquid crystalline materials.

Double-twisted nematic director configurations in cylindrical capillaries with a photocontrollable angle of twist.

The orientational configurations of thermotropic nematic liquid crystal in cylindrical capillaries with nondegenerated planar surface anchoring are investigated. The boundary conditions were determined by a photoaligning coating on the inner wall of the capillary treated with a linearly polarized UV light while rotating the capillary around its long axis, thus providing the easy alignment axis perpendicular to the polarization direction of illuminating light. By changing the angle between the incident light polarization and the capillary axis, this procedure allows us to realize axially symmetric twisted structure with any angle of twist, ranging from a trivial axial alignment with zero twist (when the UV polarization is perpendicular to the capillary axis) to the 180^{∘}-twisted configuration (when the UV polarization is along the capillary). These two extreme configurations together with an intermediate configuration induced by UV light polarized at an angle of 45^{∘} to the capillary axis were produced and their director profiles were studied using polarizing microscopy techniques. UV light intensity and polarization distribution over inner capillary surface covered with photoaligning layer were analyzed using optical ray tracing calculations. An analytical model of the observed configurations of the liquid crystal director field is proposed, which takes into account the finite anchoring on the capillary surface. By comparing the theoretically calculated and experimentally estimated director profiles, the liquid crystal anchoring energy on the photoaligned capillary surface was estimated.

ЭКВИВАЛЕНТНОСТЬ УРАВНЕНИЯ ХАНТЕРА-САКСОНА И ОБОБЩЕННОГО УРАВНЕНИЯ ФЕРРОМАГНЕТИКА ГЕЙЗЕНБЕРГА

Integrable systems play an important role in modern mathematics, theoretical and mathematical physics. The display of integrable equations with exact solutions and some special solutions can provide important guarantees for the analysis of its various properties. The Hunter-Saxton equation belongs to the family of integrable systems. The extensive and interesting mathematical theory, underlying the Hunter-Saxton equation, creates active mathematical and physical research. The Hunter-Saxton equation (HSE) is a high-frequency limit of the famous Camassa-Holm equation. The physical interpretation of HSE is the propagation of weakly nonlinear orientation waves in a massive nematic liquid crystal director field. In this paper, we propose a matrix form of the Lax representation for HSE in 𝑠𝑢ሺ𝑛 ൅ 1ሻ/𝑠ሺ𝑢ሺ1ሻ ⊕ 𝑢ሺ𝑛ሻሻ - symmetric space for the case 𝑛 ൌ 2. Lax pairs, introduced in 1968 by Peter Lax, are a tool for finding conserved quantities of integrable evolutionary differential equations. The Lax representation expands the possibilities of the equation we are considering. For example, in this paper, we will use the matrix Lax representation for the HSE to establish the gauge equivalence of this equation with the generalized Heisenberg ferromagnet equation (GHFE). The famous Heisenberg Ferromagnet Equation (HFE) is one of the classical equations integrable through the inverse scattering transform. In this paper, we will consider its generalization. Andalso the connection between the decisions of the HSE and the GHFE will be presented.

Lump, lump-one stripe, multiwave and breather solutions for the Hunter–Saxton equation

Abstract The aim of this article was to address the lump, lump-one stripe, multiwave and breather solutions for the Hunter–Saxton equation with the aid of Hirota bilinear technique. This model concerns in a massive nematic liquid crystal director field. By choosing the function f in Hirota bilinear form, as the general quadratic function, trigonometric function and exponential function along with appropriate set of parameters, we find the lump, lump-one stripe, multiwave and breather solutions successfully. We also interpreted some three-dimensional and contour profiles to anticipate the wave dynamics. These newly obtained solutions have some arbitrary constants and so can be applicable to explain diversity in qualitative features of wave phenomena.

Electric-Field-Induced Instabilities in Nematic Liquid Crystals

Systems involving nematic liquid crystals subjected to magnetic fields or electric fields are modeled using the Oseen-Frank macroscopic continuum theory, and general criteria are developed to assess the local stability of equilibrium solutions. The criteria take into account the inhomogeneity of the electric field and the mutual influence of the electric field and the liquid-crystal director field on each other. The criteria show that formulas for the instability thresholds of electric-field Freedericksz transitions cannot in all cases be obtained from those for the analogous magnetic-field transitions by simply replacing the magnetic parameters by the corresponding electric parameters, contrary to claims in standard references. This finding is consistent with [Arakelyan, Karayan, Chilingaryan, Sov. Phys. Dokl., 29 (1984) 202-204]. A simple analytical test is provided to determine when an electric-field-induced instability can differ qualitatively from the analogous magnetic field-induced instability. For the systems we study, it is found that taking into account the full coupling between the electric field and the director field can either elevate or leave unchanged an instability threshold (never lower it), compared to the threshold provided by the magnetic-field analogy. The physical mechanism that underlies the effect of elevating an instability threshold is the added free energy associated with a first-order change in the ground-state electric field caused by a perturbation of the ground-state director field. Examples are given that involve classical Freedericksz transitions and also periodic instabilities. The inclusion of flexoelectric terms in the theory is studied, and it is found that these terms are not capable of altering the instability thresholds of any of the classical Freedericksz transitions, consistent with known results for splay transitions.

Liquid crystal director fields in three-dimensional non-Euclidean geometries

This paper investigates nematic liquid crystals in three-dimensional curved space, and determines which director deformation modes are compatible with each possible type of non-Euclidean geometry. Previous work by Sethna et al showed that double twist is frustrated in flat space , but can fit perfectly in the hypersphere . Here, we extend that work to all four deformation modes (splay, twist, bend, and biaxial splay) and all eight Thurston geometries. Each pure mode of director deformation can fill space perfectly, for at least one type of geometry. This analysis shows the ideal structure of each deformation mode in curved space, which is frustrated by the requirements of flat space.

Toward Programmed Complex Stress-Induced Mechanical Deformations of Liquid Crystal Elastomers

We prepare a liquid crystal elastomer (LCE) with a spatially patterned liquid crystal director field from an all-acrylate LCE. Mechanical deformations of this material lead to a complex and spatially varying deformation with localised body rotations, shears and extensions. Together, these dictate the evolved shape of the deformed film. Using polarising microscopy, we map the local rotation of the liquid crystal director in Eulerian and Lagrangian frames and use these to determine rules for programming complex, stress-induced mechanical shape deformations of LCEs. Moreover, by applying a recently developed empirical model for the mechanical behaviour of our LCE, we predict the non-uniform stress distributions in our material. These results show the promise of empirical approaches to modelling the anisotropic and nonlinear mechanical responses of LCEs which will be important as the community moves toward realising real-world, LCE-based devices.

Difference in the interaction of nano-diameter rod and tubular particles with a disclination line in a nematic liquid crystal.

In the presence of a disclination line, inclusions within an aligned nematic liquid crystal (LC) are first attracted and ultimately trapped in it. The kind of orientational distortion created by the inclusions is fundamental in determining the trapping. In the present work, we observe differences in the trapping behaviour, onto a ½ defect line in a nematic LC, of two types of particles both elongated but different in their actual geometry. Even if both types have cylindrical shape, aggregates of Mo6S2I8 nanowires (rod-like shape) and multiwall carbon nanotubes (tubular shape, i.e. hollow) trap differently although still due to deformations induced in the LC director field. Attractive forces are stronger on elongated bundles of nanowires than on similarly sized bundles of multi-wall carbon nanotubes. The reason is the difference in the attraction forces originating from different types of distortions of the LCs. The hollow and the full cylinders are not homotopically equivalent and this inequivalence holds also for the liquid crystal around them. The nanowires induce defects in the LC close-by their surfaces as shown for microrods, topologically equivalent to spheres. In contrast, multi-wall carbon nanotubes, being hollow, do not form defects close to their ends. However, the tubes are strongly bent and the strong planar anchoring of LC at the surface induces deformation in the LC enabling attraction forces with the defect line. HiPco single wall carbon nanotubes could not be trapped because their bundles looked much straighter and smaller than the ones of MWCNTs and thus neither defects nor standard strong deformations are expected. In conclusion, even if the shape of both types of particles is cylindrical, the topological difference between rods and tubes has profound consequences on the physical behaviour and on the presence and type of defect-mediated nematic attraction forces.

Electric field driven transformations of orientational structure in chiral nematic systems with large flexoelectricity

Transformation of the orientational structure in a plane-parallel cell of a cholesteric liquid crystal, was studied depending on the magnitude of the voltage applied to the boundary planes of the cell. The equilibrium configuration of the liquid crystal director field was found minimizing the free energy both analitically and numerically. We took into account the orientational elastic energy in the volume and at the cell boundaries, the energy of an inhomogeneous electric field, and the flexoelectric effect. Systems with large flexoelectric coefficient were studied in detail. It was found, that when small voltages are applied to the cell boundaries, a continuous Fredericks transition occurs first, and then, when certain characteristic voltage is achieved, a significant jump-like change in the orientational structure takes place, and this structure gradually changes with the further voltage increase. It is shown that various scenarios of transformation of the orientational structure can be implemented depending on the actual values of the set of material parameters of the system studied. The discovered ability to control the structure of a liquid crystal in a cell can find use in various technical devices.

Dynamic Complex Liquid Crystal Emulsions.

We report a new class of dynamically reconfigurable complex colloids comprising immiscible liquid crystals (LCs) and fluorocarbon oils. Producing stable spherical droplets requires the utilization of appropriately designed surfactants to reduce the high intrinsic surface tension between the LCs and the fluorocarbon oils that initially lead to nonspherical, "snowman-shaped" Janus droplets. After stabilizing the interfaces via surfactants, the LC droplet morphology can be dynamically switched between LC-in-fluorocarbon-in-water double emulsions (LC/F/W), spherical Janus emulsions, and inverted double emulsions (fluorocarbon-in-LC-in-water, F/LC/W) in response to changes in the surrounding surfactants. These stabilization methods can be extended to smectic LCs to create droplets with more complex internal arrangements and expand the range of LC emulsions that can be prepared. In addition, by using new mesogenic surfactants that control the LC director field at each LC interface, we prepare LC complex colloids exhibiting different internal configurations. The ability to control the LC anchoring conditions made it possible to create topological singularities as powerful templates for the precision assembly of antibodies at the droplets' interface. These dynamic LC complex colloids of controllable morphology and LC orientation are rich soft material platforms that will find utility in a variety of sensing applications.

Topologically Protected Generation of Stable Wall Loops in Nematic Liquid Crystals.

The nematic liquid crystal (LC) director field can contain defects that are both singular and nonsingular, but nonsingular defects with an integer winding number of the director are typically metastable because of their high energy. We demonstrate topology-mediated generation and stabilization of nonsingular wall loops in a sandwich-type LC cell by combining a patterned substrate with a planar substrate. We implement a design which imposes a topological constraint on a singular disclination loop such that it irreversibly annihilates upon application of a field, and it results in the generation of a stable nonsingular wall loop when the field is removed. Theoretical modeling agrees with experimental observations, providing insight into the wall generation mechanism and its stability. The concept to stabilize high-energy structures through orientation-patterning-defined topological constraints extends our ability to control orientationally ordered matter.

Realignment of Liquid Crystal Shells Driven by Temperature-Dependent Surfactant Solubility.

We investigate dynamic director field variations in shells of the nematic liquid crystal (LC) compound, 4-cyano-4′-pentylbiphenyl, suspended in and containing immiscible aqueous phases. The outer and inner shell interfaces are stabilized by the cationic surfactant, cetyl trimethyl ammonium bromide (CTAB), and by the water soluble polymer, poly(vinyl alcohol) (PVA), respectively. PVA and surfactant solutions normally promote tangential and orthogonal alignments, respectively, of the LC director. The rather high Krafft temperature of CTAB, TK ≈ 25 °C, means that its solubility in water is below the critical micelle concentration at room temperature in most labs. Here, we study the effect of cooling/heating past TK on the LC shell director configuration. Within a certain concentration range, CTAB in the outer aqueous phase (and PVA in the inner) switches the LC director field from hybrid to uniformly orthogonal upon cooling below TK. We argue that the effect is related to the migration of the surfactant through the fluid LC membrane into the initially surfactant-free aqueous PVA solution, triggered by the drastically reduced water solubility of CTAB at T < TK. The results suggest that LC shells can detect solutes in the continuous phase, provided there is sufficient probability that the solute migrates through the LC into the inner aqueous phase.

Interference of nonpolarized light in liquid crystal domains on a polymer surface

The interference of nonpolarized light transmitted through an ensemble of nematic liquid crystal domains formed on a polycarbonate surface has been investigated. A model based on the superposition of the ordinary and extraordinary beams passing through domains with a radial structure has been developed. Expressions for the phase difference and intensity of the interfering beams have been derived that take into account the distribution of the liquid crystal director field. The dependences of the optical transmittance of a domain layer on the applied voltage have been calculated with regard to the material and optical constants of a liquid crystal and structural features and averaged morphological parameters of individual domains in the experimental sample. The results of the calculation are consistent with the experimental data, which confirms the validity of the proposed model.

Liquid crystal metasurfaces on micropatterned polymer substrates.

Formation of photonic liquid crystal metasurfaces on rubbed polyimide substrates patterned by focused ion beam is demonstrated. Modulation of the surface anchoring conditions with periods from 1 to 6 micrometers gives rise to periodic deformation of the nematic liquid crystal director field. The exact periodicity is confirmed by the light diffraction measurements. Distinct colors originating from the specific zero-order diffraction spectra are observed and qualitatively explained in terms of an analytical model within the one-constant approximation. Quantitatively accurate optical spectra are obtained by the full scale numerical simulations taking into account all relevant material parameters. The results pave the way for hybrid liquid-crystal-based metasurfaces with tunable optical transmission, diffraction, and lasing.