Lehmann rotation of cholesteric droplets subjected to a temperature gradient: role of the concentration of chiral molecules

Chemical Leslie effect in a chiral smectic-C^{★} film: Nonsingular target patterns

We analyze experimentally and theoretically the winding and unwinding of the c[over ⃗] director in a chiral smectic-C^{★} film crossed by an ethanol flow. This leads to a target pattern under crossed polarizers when the +1 defect imposed by the boundary conditions is pinned on the edge of the film. We show that the target is deformed at the center of the film when it is subjected to a flow of ethanol because of the presence of two recirculation vortices of chemohydrodynamical origin. This deformation and the two vortices disappear during the unwinding of the target when the ethanol flow is stopped. This unambiguously shows that the target deformation is due to the vortices and not to the elastic anisotropy. These two points are confirmed theoretically. An estimate of the two so-called chemomechanical and chemohydrodynamical Leslie coefficients is also derived from this study.

Steady rigid-body rotation of cholesteric droplets and their dumbbell-shaped aggregates driven by heat flux along the helical axes

We investigated the steady unidirectional rotation of cholesteric (Ch) droplets driven by a heat flux. The droplets coexisted with the isotropic (Iso) phase and possessed a helical molecular arrangement. When a heat flux was transported along the helical axis, the droplets and their dumbbell-shaped aggregates exhibited steady rigid rotation. Our results are in contrast with those of previous reports in which Ch droplets in the same geometry exhibited pure director rotation. The fact that Ch droplets and their aggregates prefer rigid rotation can be ascribed to the orientational elasticity combined with the anchoring force at the Ch-Iso interface, which locks the director to the rotational flow in the droplets.

Anomalous Lehmann Rotation of Achiral Nematic Liquid Crystal Droplets Trapped under Linearly Polarized Optical Tweezers

Continuous rotation of a cholesteric droplet under the heat gradient was observed by Lehmann in 1900. This phenomenon, the so-called Lehmann effect, consists of unidirectional rotation around the heat flux axis. We investigate this gradient heat effect using infrared laser optical tweezers. By applying single trap linearly polarized optical tweezers onto a radial achiral nematic liquid crystal droplet, trapping of the droplet was performed. However, under a linearly polarized optical trap, instead of stable trapping of the droplet with slightly deformed molecular directors along with a radial hedgehog defect, anomalous continuous rotation of the droplet was observed. Under low power laser trapping, the droplet appeared to rotate clockwise. By continuously increasing the laser power, a stable trap was observed, followed by reverse directional rotation in a higher intensity laser trap. Optical levitation of the droplet in the laser beam caused the heat gradient, and a breaking of the symmetry of the achiral nematic droplet. These two effects together led to the rotation of the droplet under linearly polarized laser trapping, with the sense of rotation depending on laser power.

Optical Textures and Orientational Structures in Cholesteric Droplets with Conical Boundary Conditions

Cholesteric droplets dispersed in polymer with conical boundary conditions have been studied. The director configurations are identified by the polarising microscopy technique. The axisymmetric twisted axial-bipolar configuration with the surface circular defect at the droplet’s equator is formed at the relative chirality parameter N0≤2.9. The intermediate director configuration with the deformed circular defect is realised at 2.9<N0<3.95, and the layer-like structure with the twisted surface defect loop is observed at N0≥3.95. The cholesteric layers in the layer-like structure are slightly distorted although the cholesteric helix is untwisted.

A thermomechanical coupling in cholesteric liquid crystals: Unidirectional rotation of double-twist cylinders driven by heat flux

We made aggregates of cholesteric liquid crystalline Cylinders with Double-Twist orientational structure (DTC) and investigated their rigid-body rotation under a temperature gradient, focusing on how the rotational speed should depend on the cylinder size. The experimental results showed that the angular velocity of the DTC aggregates linearly increased with the height of the cylinders and was inversely proportional to the base area. With a phenomenological equation, we analyzed the torque caused by the heat flux and its balance with the viscous friction, and found that the simple analysis well explained the size-dependence of the rotation of the DTC aggregates. The coupling constant between the heat flux and the torque to drive the rigid-body rotation was in the same order of magnitude as that for the director rotation.

Formation and dynamics of the aggregates of cholesteric double-twist cylinders

We succeeded in driving the unidirectional rigid-body rotation of cholesteric (Ch) double-twist cylinder (DTC) droplets under a heat flux along the cylindrical symmetry axis. To directly observe the rigid-body rotation of DTC droplets, in each of which the center of the rotation and the symmetry axis of the structure correspond, we fabricated DTC aggregates that comprise several DTCs with intact structures. Given a steady heat flux, the DTC aggregates metastabilized by the shape and the surface anchoring show a unidirectional rigid-body rotation with a constant angular velocity. The rotational direction is determined by the molecular chirality and the direction of the heat flux, and the rotational velocity increases with the temperature gradient and decreases with the aggregation number N of the DTCs as 1 + 2/sin2(π/N). The behavior agrees with a simple model based on the linear phenomenological equation.

Role of anchoring energy on the texture of cholesteric droplets: Finite-element simulations and experiments

We present a numerical method to compute defect-free textures inside cholesteric domains of arbitrary shape. This method has two interesting properties, namely a robust and fast quadratic convergence to a local minimum of the Frank free energy, thanks to a trust region strategy. We apply this algorithm to study the texture of cholesteric droplets in coexistence with their isotropic liquid in two cases: when the anchoring is planar and when it is tilted. In the first case, we show how to determine the anchoring energy at the cholesteric-isotropic interface from a study of the optical properties of droplets of different sizes oriented with an electric field. This method is applied to the case of the liquid crystal CCN-37. In the second case, we come back to the issue of the textural transition as a function of the droplet radius between the double-twist droplets and the banded droplets, observed for instance in cyanobiphenyl liquid crystals. We show that, even if this transition is dominated by the saddle-splay Gauss constant K_{4}, as was recently recognized by Yoshioka et al. [Soft Matter 12, 2400 (2016)1744-683X10.1039/C5SM02838H], the anchoring energy does also play an important role that cannot be neglected.

Generalized drift velocity of a cholesteric texture in a temperature gradient

We propose a general method to calculate the drift velocity of cholesteric textures subjected to a temperature gradient when the backflow effects are negligible. The textures may be Translationally Invariant Configurations (TICs) or localized structures such as cholesteric droplets or cholesteric fingers. For the TICs and for the droplets, the drift is rotational while for the fingers, the drift is translational. We show that for the TICs, the drift is only due to the thermomechanical coupling terms of Leslie (classical term) and of Akopyan and Zel'dovich (which are additional texture-dependent terms). For the localized structures, we show that another mechanism involving the temperature variations of the elastic constants and the existence of a transverse temperature gradient can lead to a drift which adds to the one due the classical thermomechanical effects.

Continuous Rotation of Achiral Nematic Liquid Crystal Droplets Driven by Heat Flux

Suspended droplets of cholesteric (chiral nematic) liquid crystals spontaneously rotate in the presence of a heat flux due to a temperature gradient, a phenomenon known as the Lehmann effect. So far, it is not clear whether this effect is due to the chirality of the phase and the molecules or only to the chirality of the director field. Here, we report the continuous rotation in a temperature gradient of nematic droplets of a lyotropic chromonic liquid crystal featuring a twisted bipolar configuration. The achiral nature of the molecular components leads to a random handedness of the spontaneous twist, resulting in the coexistence of droplets rotating in the two senses, with speeds proportional to the temperature gradient and inversely proportional to the droplet radius. This result shows that a macroscopic twist of the director field is sufficient to induce a rotation of the droplets, and that the phase and the molecules do not need to be chiral. This suggests that one can also explain the Lehmann rotation in cholesteric liquid crystals without introducing the Leslie thermomechanical coupling-only present in chiral mesophases. An explanation based on the Akopyan and Zeldovich theory of thermomechanical effects in nematics is proposed and discussed.

Do Lehmann cholesteric droplets subjected to a temperature gradient rotate as rigid bodies?

We performed a Fluorescence Recovery After Photobleaching (FRAP) experiment during the Lehmann rotation of cholesteric droplets in thermodynamic coexistence with the isotropic liquid and subjected to a temperature gradient. By creating and tracking bleached spots near the surface of banded droplets (in which the cholesteric helix is perpendicular to the gradient) and concentric circle droplets oriented by an electric field (in which the helix is parallel to the gradient), we found that neither type of droplet rotates as a solid. This result shows that the texture rotation is mainly due to the local director rotation.

Thermomechanical coupling in coarse grained cholesteric liquid crystal model systems with pitches of realistic length

Cholesteric liquid crystal where the director is rotated by a temperature gradient.

Lehmann rotation of cholesteric droplets: Role of the sample thickness and of the concentration of chiral molecules

We study the role of the sample thickness d and of the concentration C of chiral molecules during the Lehmann rotation of cholesteric droplets of radius R subjected to a temperature gradient G→. Two configurations are studied depending on how the helix is oriented with respect to G→. The first result is that, at fixed C and R, the rotation velocity ω increases with d when the helix is parallel to G→, whereas it is independent of d when the helix is perpendicular to G→. The second result is that, for a given C,ω0=limR→0ω(R) is the same for the two types of droplets independently of d. This suggests that the, as yet unknown, physical mechanism responsible for the droplet rotation is the same in the two types of droplets. The third result is that the Lehmann coefficient ν[over ¯] defined from the Leslie-like relation ω0= G¯G/γ1 (with γ_1 the rotational viscosity) is proportional to the equilibrium twist q. Last, but not least, the ratio R¯=ν ¯/q depends on the liquid crystal chosen but is independent of the chiral molecule used to dope the liquid crystal.

Lehmann rotation of the cholesteric helix in droplets oriented by an electric field

We study the Lehmann rotation of the cholesteric helix in droplets of the liquid crystal N-(p-methoxybenzilidene)-p-butylaniline doped with a small amount of the chiral molecule R811 when they are subjected to a temperature gradient. We show that the helix rotates much faster when it is parallel to the temperature gradient than when it is perpendicular to it. The first configuration is obtained by submitting the droplets to an ac electric field parallel to the temperature gradient, whereas the second one is observed at zero field. We show that the rotation velocity of the helix strongly depends on the droplet radius, even when the helix is parallel to the temperature gradient. This observation supports the idea that the Leslie thermomechanical coupling cannot explain alone the Lehmann effect.

Microscopic vs. macroscopic origin of the Lehmann effect in cholesteric liquid crystals

In a recent letter (EPL 97, 36006 (2012)), we have shown that the Leslie thermomechanical coupling cannot alone explain the Lehmann effect (namely the rotation of cholesteric droplets when they are subjected to a temperature gradient). This result was obtained by measuring in a compensated cholesteric mixture the "Lehmann coefficient" as a function of temperature both below and at the transition to the isotropic liquid. In this article, we detail these experiments and present new ones performed with other compensated mixtures and a diluted cholesteric mixture. The new results confirm the macroscopic origin of the Lehmann effect, in contrast to the Leslie thermomechanical effect that is clearly of microscopic origin.