Director Field Research Articles

Electrically tunable cholesteric liquid crystal lines defects

In the work, the pattern formation in a planar-oriented Cholesteric Liquid Crystals (CLC) was studied for various thicknesses to pitch (d/p) ratios under the influence of an electric field directed perpendicular to the plane of the cell. In addition, the features of the director's fields and defects lines were studied. Indeed, we present the splitting of dislocation cores into disclinations with a nonsingular core, the geometry of director patterns associated with dislocation bends. Experimentally, three planar cells with the gap thicknesses d corresponding to three Grandjean zones were prepared with d/p ≈ 1/2, 1, and 3/2. The results show that the dynamic pattern formation of the grating stripes was performed in a different manner. For d/p ≈ 1/2 and d/p ≈ 1, the stripes were showed through the whole cell surface parallel and perpendicular to the rubbing direction, respectively. As well, their contrast was increased with time during formation. In this case, the defect core structure split into a λ−1/2 λ+1/2 pair disclinations for a burger vector (b) equal to the pitch (p) dislocation. Besides that, when d/p ≈ 3/2, the grating stripes were performed near the boundaries of the surfaces and then prolonged to the entire cell surface along the rubbing direction. Here, the defect core structure split into a τ−1/2 τ+1/2 pair disclinations for a b = p dislocation.

Kink- and double π-forms of distortions in a microsized nematic capillaries

We have described a novel mechanism of a kink-like and double π- forms of distortions of the director field n^ in a microsized nematic volume, confined between two infinitely long horizontal cylinders with a radial temperature gradient ∇T, under the effect of a voltage U applied between cylinders. It is found that, on the one hand, under certain conditions, in terms of curvature of cylinders κ and the voltage U, the torques and forces acting on the director n^ may excite the kink-like distortion wave spreading along the normal to both cylindrical boundaries, whose resemblance to a kink-like distortion wave depends on the value of applied voltage U and the curvature of the inner cylinder. It has been worked out, on the other hand, the conditions, in terms of κ and U, producing the distortion mechanism of the n^ in the double π-form, with the intermediate relaxation wall.

A Quasilinear System Related with the Asymptotic Equation of the Nematic Liquid Crystal’s Director Field

In this paper, the author studies the local existence of strong solutions and their possible blow-up in time for a quasilinear system describing the interaction of a short wave induced by an electron field with a long wave representing an extension of the motion of the director field in a nematic liquid crystal’s asymptotic model introduced in [Saxton, R. A., Dynamic instability of the liquid crystal director. In: Current Progress in Hyperbolic Systems (Lindquist, W. B., ed.), Contemp. Math., Vol.100, Amer. Math. Soc., Providence, RI, 1989, pp.325–330] and [Hunter, J. K. and Saxton, R. A., Dynamics of director fields, SIAM J. Appl. Math., 51, 1991, 1498–1521] and studied in [Hunter, J. K. and Zheng, Y., On a nonlinear hyperbolic variational equation I, Arch. Rat. Mech. Anal., 129, 1995, 305–353], [Hunter, J. K. and Zheng, Y., On a nonlinear hyperbolic variational equation II, Arch. Rat. Mech. Anal., 129, 1995, 355–383] and in [Zhang, P. and Zheng, Y., On oscillation of an asymptotic equation of a nonlinear variational wave equation, Asymptotic Anal., 18, 1998, 307–327] and, more recently, in [Bressan, A., Zhang, P. and Zheng, Y., Asymptotic variational wave equations, Arch. Rat. Mech. Anal., 183, 2007, 163–185].

Chaotic dynamics of active topological defects

ABSTRACT Topological defects are interesting phenomena which can be observed in ordered phases such as oriented active fluids or nematic liquid crystals. Topological defects are determined by the overall structure of the director field in an active fluid in nematic phase and by exerting force to the units of the active particles they can interact or cause motion in the environment. Studying them as particles with dynamical equations, instead of studying the director field of the nematic environment, would provide us the power to study the characteristics of their motion. The equations of motion for multi-defect systems have been previously studied and in this work we focus on the chaotic properties of the dynamics and analyze the effect of activity on the transition of such systems to chaos by numerically estimating their Lyapunov exponents. We show the importance of the defects for this transition (as they are active in the sense of having self-propulsion) and numerically derive critical activity values for multi-defect systems in which this transition occurs.

Nematic tactoid population.

Tactoids are pointed, spindlelike droplets of nematic liquid crystal in an isotropic fluid. They have long been observed in inorganic and organic nematics, in thermotropic phases as well as lyotropic colloidal aggregates. The variational problem of determining the optimal shape of a nematic droplet is formidable and has only been attacked in selected classes of shapes and director fields. Here, by considering a special class of admissible solutions for a bipolar droplet, we study the prevalence in the population of all equilibrium shapes of each of the three that may be optimal (tactoids primarily among them). We show how the prevalence of a shape is affected by a dimensionless measure α of the drop's volume and the ratios k_{24} and k_{3} of the saddle-splay constant K_{24} and the bending constant K_{33} of the material to the splay constant K_{11}. Tactoids, in particular, prevail for α⪅16.2+0.3k_{3}-(14.9-0.1k_{3})k_{24}. Our class of shapes (and director fields) is sufficiently different from those employed so far to unveil a rather different role of K_{24}.

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.

Thermal and optical manipulation of morphology in cholesteric liquid crystal microdroplets constrained on microfibers

Liquid crystals (LCs) are distinctive materials that are applicable to a wide range of disciplines, such as continuum mechanics, optics of anisotropic media, statistical physics, and crystallography. The diverse structures and eye-catching topological defects of LCs in confined geometries are affected by nematic elasticity, chirality, and surface anchoring. Herein, we report the formation and detailed configuration of cholesteric LC (CLC) microdroplets with different pitches pierced by electrospun poly(methyl methacrylate) microfibers. Two kinds of surface anchoring, namely, homeotropic anchoring at the air–CLC droplet interface and planar anchoring at the fiber–CLC droplet interface, coexist in this system. By controlling temperature and light irradiation, the system exhibits thermal- and photo-dependent LC morphological and topological evolutions. The observed structures are complemented by numerical simulations of possible director fields decorated by defects. The externally controllable CLC necklaces constitute extraordinary systems for exploring the morphology and topological defects and open a route for applications in topological remote control, nanoscience, biomedical research, and the development of devices based on topologically structured soft media.

Textures of Nematic Liquid Crystal Cylindric-Section Droplets Confined by Chemically Patterned Surfaces

The director fields adopted by nematic liquid crystals (LCs) that are confined by the surface to form long, thin droplets are investigated using polarising optical microscopy. Samples are produced by de-wetting of the LC on a surface patterned with alternating high-surface energy and low-surface energy stripes of 10–30 μm width. The droplets obtained are expected to adopt a profile which is that of a longitudinal section of a cylinder and, as this suggests, the director fields observed are variants in the case where the LC is constrained in a cylindrical capillary or fibre. Hence, when there is normal anchoring at the air interface, the textures observed are related to the well-known escaped radial texture (for the nematic LC mixture E7) or plane polar texture (for the LC mixture MLC6609). More surprising is the observation that the nematic LC mixture MLC7023, which is anchored in a planar or tilted manner at the air interface, also gives what appears to be an escaped radial director field. As an exploration of the possibility of using these systems in creating sensors, the effects of adding a chiral dopant and of adding water to the substrates are also investigated.

Electrically driven nematic flow in microfluidic capillary with radial temperature gradient.

An electrically driven fluid pumping principle and a mechanism of kinklike distortion of the director field n[over ̂] in the microsized nematic volume has been described. It is shown that the interactions, on the one hand, between the electric field E and the gradient of the director's field ∇n[over ̂], and, on the other hand, between the ∇n[over ̂] and the temperature gradient ∇T arising in a homogeneously aligned liquid crystal microfluidic channel, confined between two infinitely long horizontal coaxial cylinders, may excite the kinklike distortion wave spreading along normal to both cylindrical boundaries. Calculations show that the resemblance to the kinklike distortion wave depends on the value of radially applied electric field E and the curvature of these boundaries. Calculations also show that there exists a range of parameter values (voltage and curvature of the inner cylinder) producing a nonstandard pumping regime with maximum flow near the hot cylinder in the horizontal direction.

Lyotropic nematic liquid crystals: interplay between a small twist elastic constant and chirality effects under confined geometries

ABSTRACT There are two classes of liquid crystals (LCs): Thermotropic LCs where the organic molecule represents the anisotropically shaped mesogen and lyotropic LCs, where the mesogens are non-spherical supramolecular assemblies dispersed in a solvent. Various kinds of lyotropic nematic liquid crystals (LLCs) exist, differing in shape and composition of the aggregates: Chromonic, polymeric or micellar LLCs.However, micellar LLCs are the most ubiquitous kind of LLCs, occurring in soaps and biological systems. Although these nematic lyotropics are so different, they exhibit – contrary to thermotropic nematics – spontaneous reflection symmetry breaking under confinement. We discuss spontaneous mirror symmetry breaking under capillary confinement for a standard micellar ND phase and give a detailed director field structure of the twisted polar configuration. Furthermore, we measure for the first time a complete set of viscoelastic properties of a micellar nematic LLC, finding a small twist elastic constant K 22, which is at least one order of magnitude smaller than K 11 and K 33. As demonstrated by the following work, a small K 22 and therefore spontaneous mirror symmetry breaking under confinement is rather the rule than the exception in lyotropic nematics and constitutes a unique difference between lyotropic and typical thermotropic nematics, for which the well-known one-constant approximation holds. Abbreviations: LC: Liquid Crystal; LLC: Lyotropic liquid crystal; TER: twisted escaped radial director configuration; TP: twisted polar director configuration; : director; K 11: splay elastic constant of the nematic phase; K 22: twist elastic constant of the nematic phase; K 33: bend elastic constant of the nematic phase; s: strength of a defect/disclination; ER: escaped radial director configuration; CDEAB: N,N-ethylhexadecylammonium bromide; DOH: decan-1-ol; ND: nematic phase of disk-shaped building blocks; NC: nematic phase of rod-shaped building blocks; H-field: magnetic field; PR: planar radial director configuration; PP: planar polar director configuration; RR: right handed – right handed; LL: left handed – left handed; RL: right handed – left handed; LR: left handed – right handed.

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.

Memory effects, arches and polar defect ordering at the cross-over from wet to dry active nematics.

We use analytic arguments and numerical solutions of the continuum, active nematohydrodynamic equations to study how friction alters the behaviour of active nematics. Concentrating on the case where there is nematic ordering in the passive limit, we show that, as the friction is increased, memory effects become more prominent and +1/2 topological defects leave increasingly persistent trails in the director field as they pass. The trails are preferential sites for defect formation and they tend to impose polar order on any new +1/2 defects. In the absence of noise and for high friction, it becomes very difficult to create defects, but trails formed by any defects present at the beginning of the simulations persist and organise into parallel arch-like patterns in the director field. We show aligned arches of equal width are approximate steady state solutions of the equations of motion which co-exist with the nematic state. We compare our results to other models in the literature, in particular dry systems with no hydrodynamics, where trails, arches and polar defect ordering have also been observed.

Nematic shells: new insights in topology- and curvature-induced effects.

Within the framework of continuum theory, we draw a parallel between ferromagnetic materials and nematic liquid crystals confined on curved surfaces, which are both characterized by local interaction and anchoring potentials. We show that the extrinsic curvature of the shell combined with the out-of-plane component of the director field gives rise to chirality effects. This interplay produces an effective energy term reminiscent of the chiral term in cholesteric liquid crystals, with the curvature tensor acting as a sort of anisotropic helicity. We discuss also how the different nature of the order parameter, a vector in ferromagnets and a tensor in nematics, yields different textures on surfaces with the same topology as the sphere. In particular, we show that the extrinsic curvature governs the ground state configuration on a nematic spherical shell, favouring two antipodal disclinations of charge +1 on small particles and four +1/2 disclinations of charge located at the vertices of a square inscribed in a great circle on larger particles.

Hydrodynamics of Anisotropic Liquid Crystals in an Applied Magnetic Field

We consider a magnetic effect on hydrodynamic flows of nematic liquid crystals with unequal elastic constants in dimension two. Under a planar ansatz for the director field and velocity, we derive a reduced Ericksen--Leslie system which consists of a quasilinear parabolic equation and a Navier--Stokes equation involving critical nonlinearity due to elastic anisotropy. When unequal elastic constants are sufficiently close, we establish the global existence of smooth solutions to the reduced system and show convergence to an equilibrium as $t$ tends to $\infty$. A magnetic field-induced instability of the aligned state under the hydrodynamic flow is discussed.

Computing with non-orientable defects: Nematics, smectics and natural patterns

Defects are a ubiquitous feature of ordered media. They have certain universal features, independent of the underlying physical system, reflecting their topological origins. While the topological properties of defects are robust, they appear as ‘unphysical’ singularities, with non-integrable energy densities in coarse-grained macroscopic models. We develop a principled approach for enriching coarse-grained theories with enough of the ‘micro-physics’ to obtain thermodynamically consistent, well-set models that allow for the investigations of dynamics and interactions of defects in extended systems. We also develop associated numerical methods that are applicable to computing energy driven behaviors of defects across the amorphous-soft-crystalline materials spectrum. Our methods can handle order parameters that have a head-tail symmetry, i.e. director fields, in systems with a continuous translation symmetry, as in nematic liquid crystals, and in systems where the translation symmetry is broken, as in smectics and convection patterns. We illustrate our methods with explicit computations.

Dynamically morphing microchannels in liquid crystal elastomer coatings containing disclinations

Liquid crystal elastomers (LCEs) hold a major promise as a versatile material platform for smart soft coatings since their orientational order can be predesigned to program a desired dynamic profile. In this work, we introduce temperature-responsive dynamic coatings based on LCEs with arrays of singular defects-disclinations that run parallel to the surface. The disclinations form in response to antagonistic patterns of the molecular orientation at the top and bottom surfaces, imposed by the plasmonic mask photoalignment. Upon heating, an initially flat LCE coating develops linear microchannels located above each disclination. The stimulus that causes a non-flat profile of LCE coatings upon heating is the activation force induced by the gradients of molecular orientation around disclinations. To describe the formation of microchannels and their thermal response, we adopt a Frank–Oseen model of disclinations in a patterned director field and propose a linear elasticity theory to connect the complex spatially varying molecular orientation to the displacements of the LCE. The thermo-responsive surface profiles predicted by the theory and by the finite element modeling are in good agreement with the experimental data; in particular, higher gradients of molecular orientation produce a stronger modulation of the coating profile. The elastic theory and the finite element simulations allow us to estimate the material parameter that characterizes the elastomer coating's response to the thermal activation. The disclination-containing LCEs show potential as soft dynamic coatings with a predesigned responsive surface profile.

A convergent finite element scheme for a fourth-order liquid crystal model

Abstract In this manuscript we propose and analyse a fully discrete, unconditionally stable finite element scheme for a recently developed director model for liquid crystalline flows (Metzger, S. (2020) On a novel approach for modeling liquid crystalline flows. Commun. Math. Sci., 18, 359–378). The model consists of nonlinear fourth-order partial differential equations describing the evolution of the director field and Navier–Stokes equations governing the velocity field. We employ a stable splitting approach to reduce the computational complexity by decoupling the update of the director field from the update of the velocity field. We also perform a rigorous passage to the limit as the spatial and temporal discretization parameters simultaneously tend to zero, and show that a subsequence of finite element approximations converges towards a weak solution of the original model. Passing to the limit in the nonlinear terms requires us to derive the strong convergence of the gradient of the director field from uniform bounds for its discrete Laplacian. Furthermore, we present simulations underlining the practicability of the proposed scheme, investigate its convergence properties and discuss the differences between the underlying model and already established Ericksen–Leslie-type models.

Shape Programming by Modulating Actuation over Hierarchical Length Scales.

Many active materials used in shape-morphing respond to an external stimulus by stretching or contracting along a director field. The programming of such actuators remains complex because of the single degree of freedom (the orientation) in local actuation. Here, texturing this field in zigzag patterns is shown to provide an extended family of biaxial active stretches out of an otherwise single uniaxial active deformation, opening a larger parameter space. By further modulating the zigzag patterns at the larger scale of the structure, its deployed shape can be controlled. This notion of texturing over hierarchical length scales follows geometrical principles, and is robust against changes in size and materials. The robustness of the approach is demonstrated by considering three different responsive materials: inextensible flat fabrics, channel-bearing elastomer (respectively, contracting and expanding perpendicularly to the director field when actuated pneumatically), and 3D-printed thermoplastic (composed of extruded filaments that contract when heated). It is shown that large-scale shape-morphing structures can be generated and that their geometry can be controlled with highaccuracy.

Differential rotation in cholesteric pillars under a temperature gradient

Steady rotation is induced in cholesteric droplets dispersed in a specific liquid solvent under a temperature gradient. In this phenomenon, two rotational modes have been considered: (1) collective rotation of the local director field and (2) rigid-body rotation of the whole droplet structure. However, here we present another rotational mode induced in a pillar-shaped cholesteric droplet confined between substrates under a temperature gradient, that is, a differential rotation where the angular velocity varies as a function of the radial coordinate in the pillar. A detailed flow field analysis revealed that every pillar under a temperature gradient involves a double convection roll. These results suggested that the differential rotation in the cholesteric pillars was driven by the inhomogeneous material flow induced by a temperature gradient. The present experimental study indicates that the coupling between the flow and the director motion plays a key role in the rotation of the cholesteric droplets under the temperature gradient.

Helicoidal dynamics of biaxial curved rods in twist-bend nematic phases unveiled by unsupervised machine learning techniques.

Uniaxial rods in a nematic phase diffuse preferentially in the direction parallel to the nematic director n[over ̂]. The nematic director field n[over ̂](r) of a chiral twist-bend nematic (N_{TB}) phase of achiral banana-shaped particles, recently discovered experimentally, displays a heliconical twist of given handedness and periodicity. Using simulations, we investigate the long-time macroscopic diffusion in N_{TB} phases, and find that the predilection of curved rods to diffuse in the direction of the twisting n[over ̂](r) yields a fascinating chiral dynamics along helices, even though achiral curved rods display Brownian motion with a nontrivial rototranslational coupling. We devise a machine learning protocol to characterize the helicoidal particle trajectories, finding that their pitch and radius are determined by the pitch and conical angle of the N_{TB} phase thereby connecting its structural and dynamical properties.