Shape Instabilities Driven by Topological Defects in Nematic Polymer Networks

We explore the structure of nuclei and topological defects in the first-order phase transition between the nematic (N) and isotropic (I) phases in lyotropic chromonic liquid crystals (LCLCs). The LCLCs are formed by self-assembled molecular aggregates of various lengths and show a broad biphasic region. The defects emerge as a result of two mechanisms: (1) surface-anisotropy that endows each N nucleus (‘tactoid’) with topological defects thanks to preferential (tangential) orientation of the director at the closed I–N interface, and (2) Kibble mechanism with defects forming when differently oriented N tactoids merge with each other. Different scenarios of phase transition involve positive (N-in-I) and negative (I-in-N) tactoids with nontrivial topology of the director field and also multiply connected tactoid-in-tactoid configurations. The closed I–N interface limiting a tactoid shows a certain number of cusps; the lips of the interface on the opposite sides of the cusp make an angle different from π. The N side of each cusp contains a point defect-boojum. The number of cusps shows how many times the director becomes perpendicular to the I–N interface when one circumnavigates the closed boundary of the tactoid. We derive conservation laws that connect the number of cusps c to the topological strength m of defects in the N part of the simply connected and multiply connected tactoids. We demonstrate how the elastic anisotropy of the N phase results in non-circular shape of the disclination cores. A generalized Wulff construction is used to derive the shape of I and N tactoids as a function of I–N interfacial tension anisotropy in the approximation of frozen director field of various topological charges m. The complex shapes and structures of tactoids and topological defects demonstrate an important role of surface anisotropy in morphogenesis of phase transitions in liquid crystals.

Photoresponsive liquid crystalline polymer networks with macroscopically uniaxial molecular orientation were prepared by polymerizing mixtures of liquid crystalline mono and di- acrylates, and an azobenzene compound in a homogeneous glass cell at a nematic phase. The polymer networks showed an enantiotropic phase transition from an anisotropic phase to an isotropic phase as well as high transparency. The photoisomerization of the azobenzene compound resulted in a change in a birefringence. The photochemical change in the birefringence was investigated by using a Xe lamp and a single pulse light from a Nd:YAG laser at light sources. We observed a response time in a range of a few microseconds and a decay time in a range of a few microhundreds of seconds. The optical switching behavior depended on the cross-linking density of the liquid crystalline polymer networks.

Photoresponsive liquid crystalline polymer networks with macroscopically uniaxial molecular orientation were prepared by polymerizing mixtures of liquid crystalline mono- and diacrylates and an azobenzene compound in a homogeneous glass cell at a nematic phase. The polymer networks showed an enantiotropic phase transition from an anisotropic phase to an isotropic phase as well as high transparency. The photoisomerization of the azobenzene compound resulted in a change in a birefringence. The photochemical change in the birefriengence was investigated by using a Xe lamp and a single pulse light from a Nd:YAG laser as light sources. We observed a response time in the range of a few microseconds and a decay time in the range of a few hundred microseconds. The optical switching behavior depended on the cross-linking density of the liquid crystalline polymer networks.

Topological Defects Induced High-Spin Quartet State in Truxene-Based Molecular Graphenoids

In the so-called quasiplanar texture of a nematic layer confined between parallel plates with homeotropic anchoring conditions, the director field rotates by π between limit surfaces so that field lines have the shape of a dowsing Y-shaped wooden tool. The orientation of the director field at midheight of the layer is arbitrary for symmetry reasons and is thus very sensitive to perturbations. We point out that contrary to accepted ideas the quasiplanar texture can be preserved infinitely in spite of its metastability with respect to the homogeneous homeotropic texture. We propose to call such a long-lived version of the quasiplanar texture the dowser texture. We demonstrate both experimentally and theoretically that in samples of variable thickness, the director field is sensitive to the gradient of the sample thickness through a linear coupling term. As a result, it has a tendency to follow the direction of the thickness gradient. Because of its sensitivity to perturbations we propose to call the midplane director field the dowser field and its tendency to follow the thickness gradient cuneitropism. Under effect of the gradient field, the dowser field obeys the sine-Gordon equation and exhibits domain walls that correspond to the well-known solitonic solutions of the sine-Gordon model.

Active nematics are dense systems of rodlike particles that consume energy to drive motion at the level of the individual particles. They exist in natural systems like biological tissues and artificial materials such as suspensions of self-propelled colloidal particles or synthetic microswimmers. Active nematics have attracted significant attention in recent years due to their spectacular nonequilibrium collective spatiotemporal dynamics, which may enable applications in fields such as robotics, drug delivery, and materials science. The director field, which measures the direction and degree of alignment of the local nematic orientation, is a crucial characteristic of active nematics and is essential for studying topological defects. However, determining the director field is a significant challenge in many experimental systems. Although director fields can be derived from images of active nematics using traditional imaging processing methods, the accuracy of such methods is highly sensitive to the settings of the algorithms. These settings must be tuned from image to image due to experimental noise, intrinsic noise of the imaging technology, and perturbations caused by changes in experimental conditions. This sensitivity currently limits automatic analysis of active nematics. To address this, we developed a machine learning model for extracting reliable director fields from raw experimental images, which enables accurate analysis of topological defects. Application of the algorithm to experimental data demonstrates that the approach is robust and highly generalizable to experimental settings that are different from those in the training data. It could be a promising tool for investigating active nematics and may be generalized to other active matter systems.

Accurate prediction of the gel point for real polymer networks is a long-standing challenge in polymer chemistry and physics that is extremely important for applications of gels and elastomers. Here, kinetic Monte Carlo simulation is applied to simultaneously describe network topology and growth kinetics. By accounting for topological defects in the polymer networks, the simulation can quantitatively predict experimental gel point measurements without any fitting parameters. Gel point suppression becomes more severe as the primary loop fraction in the networks increases. A topological homomorphism theory mapping defects onto effective junctions is developed to qualitatively explain the origins of this effect, which accurately captures the gel point suppression in the low loop limit where cooperative effects between topological defects are small.

We solve the forward and inverse problems associated with the transformation of flat sheets with circularly symmetric director fields to surfaces of revolution with non-trivial topography, including Gaussian curvature, without a stretch elastic cost. We deal with systems slender enough to have a small bend energy cost. Shape change is induced by light or heat causing contraction along a non-uniform director field in the plane of an initially flat nematic sheet. The forward problem is, given a director distribution, what shape is induced? Along the way, we determine the Gaussian curvature and the evolution with induced mechanical deformation of the director field and of material curves in the surface (proto-radii) that will become radii in the final surface. The inverse problem is, given a target shape, what director field does one need to specify? Analytic examples of director fields are fully calculated that will, for specific deformations, yield catenoids and paraboloids of revolution. The general prescription is given in terms of an integral equation and yields a method that is generally applicable to surfaces of revolution.

Invariant integrals applied to nematic liquid crystals with small Ericksen number and topological defects

Owing to the sustained energy input, active nematics can spontaneously generate topological defects, which generally undergo chaotic motions ruled by active hydrodynamics. Here, we combine theory and simulations to investigate the structure and dynamics of topological defects in three-dimensional (3D) active nematics under boundary twisting, similar to twisted bilayer graphene. We show that increasing twist angle will lead to the transition of defect patterns from wedge-twist defect loops to pure-twist defect lines. These defect lines can organize into ordered networks, which oscillate persistently and propagate as a wave. Intriguingly, the wave speed of defect networks is significantly faster than that of local active flows, in contrast to the traditional active defects. We identify that the fast propagation of 3D topological defects results from the activity-driven defect pattern transition and the vertical oscillations mediated by the nematic elasticity. These findings reveal the marked influence of boundary perturbations on active nematics and could also provide a simple strategy to program 3D defect dynamics.

We study the topology of the c-director field near topological defects with point core and with a droplet in the core of the defect in nonpolar smectic-C and ferroelectric smectic-C* freestanding films using polarized optical microscopy. Free and confined geometry of topological defects and droplets with strong outer boundary condition are compared. The c-director field can be remarkably different around a point defect and a droplet with the same topological charge S=+1. In ferroelectric films, splay deformation of the c-director transforms into bend deformation after droplet nucleation. Heating a ferroelectric film with an S=+1 droplet leads to a dramatic change of the c-director topology from bend to splay. In confined geometry we found spiral structures in which the c-director has opposite direction of rotation along the inner and outer boundaries of the island. Our observations are discussed on the basis of theories taking into account both the influence of polarity and of confined geometry on elasticity and topology of the c-director field.

Rotation of topological defects by trapped micro-rods in the nematic phase of a liquid crystal

The equilibrium swelling and phase behavior of monodomain nematic networks in nonmesomorphic solvents have been investigated as a function of temperature. The gels thermoreversibly undergo a phase transition between the swollen isotropic and shrunken monodomain nematic states with accompanying an anisotropic shape change. The nematic ordering occurs continuously within a narrow temperature range, which induces a significant reduction in gel volume as well as an elongation in the director direction. The length of the gels along the director axis in the nematic phase becomes larger than that in the isotropic phase, despite a considerable volume decrease upon nematic ordering. As temperature decreases, the shape anisotropy of the nematic gels increases without appreciable volume change, and becomes almost constant at the sufficiently low temperatures. A theory is proposed to describe the anisotropic swelling and phase equilibria of the monodomain nematic networks in isotropic solvents. The theory successfully describes the features of the swelling behavior observed.

We use a Monte Carlo algorithm to simulate the director field around a spherical inclusion in a uniform nematic liquid crystal matrix. The resulting structure crucially depends on the relative strength of the nematic bulk elasticity and the director anchoring on the particle surface. When this anchoring is weak, the director field perturbations are small and have quadrupolar symmetry. With increasing strength of anchoring two topologically nontrivial situations are possible: a dipolar configuration with a satellite point defect (hedgehog) near the particle pole, or a quadrupolar configuration with a ``Saturn ring'' of disclination around the particle equator.

We analyze topological solitons in the noncommutative plane by taking a concrete instance of the quantum Hall system with the SU(N) symmetry, where a soliton is identified with a skyrmion. It is shown that a topological soliton induces an excitation of the electron number density from the ground-state value around it. When a judicious choice of the topological charge density $J_{0}(\mathbf{x})$ is made, it acquires a physical reality as the electron density excitation $\Delta \rho ^{\text{cl}}(\mathbf{x})$ around a topological soliton, $\Delta \rho ^{\text{cl}}(\mathbf{x})=-J_{0}(% \mathbf{x})$. Hence a noncommutative soliton carries necessarily the electric charge proportional to its topological charge. A field-theoretical state is constructed for a soliton state irrespectively of the Hamiltonian. In general it involves an infinitely many parameters. They are fixed by minimizing its energy once the Hamiltonian is chosen. We study explicitly the cases where the system is governed by the hard-core interaction and by the noncommutative CP$^{N-1}$ model, where all these parameters are determined analytically and the soliton excitation energy is obtained.