Session X37: Liquid Crystals: Defects, Theory and Simulations

Topological defects in liquid crystals as templates for molecular self-assembly.

Topological defects in liquid crystals (LCs) have been widely used to organize colloidal dispersions and template polymerizations, leading to a range of elastomers and gels with complex mechanical and optical properties. However, little is understood about molecular-level assembly processes within defects. This presentation will describe an experimental study that reveals that nanoscopic environments defined by LC defects can selectively trigger processes of molecular self-assembly. By using fluorescence microscopy, cryogenic transmission electron microscopy and super-resolution optical microscopy, key signatures of molecular self-assembly of amphiphilic molecules in topological defects are observed - including cooperativity, reversibility, and controlled growth of the molecular assemblies. By using polymerizable amphiphiles, we also demonstrate preservation of molecular assemblies templated by defects, including nanoscopic o-rings synthesized from Saturn-ring disclinations. Our results reveal that topological defects in LCs are a versatile class of three-dimensional, dynamic and reconfigurable templates can direct processes of molecular self-assembly in a manner that is strongly analogous to other classes of macromolecular templates.   

Molecular nature of liquid crystal defects

Distortion of liquid crystal (LC) director creates defects which can be easily observed and controlled experimentally. Liquid crystal defects, which are a consequence of symmetry breaking, have significant influence on the macroscopic properties of liquid crystals, and many of the features that make LCs particularly exciting for new applications can be traced back to the existence of liquid crystal defects. However, the molecular nature of liquid crystal defects remains largely unknown. In this work, we perform large-scale atomistic simulations of 5CB/8CB mixture with a cylinder with strong homeotropic anchoring. The presence of the cylinder distorts the nematic field around the cylinder surface and creates two line defects. The local order parameter and biaxiality are calculated to characterize these defects. The results of this study indicate that, the density and the order parameter are correlated, and at the defect both parameters are low.   

\textbf{Topological defects and self-assembly of cuboidal colloidal particles with sharp edges in a nematic liquid crystal}.

The geometry of colloidal particles defines the topology and self-assembly of colloidal superstructures in nematic liquid crystals. Past research has largely focused on the defects that arise around spherical colloids, and the defect-induced aggregation between them. In this work, we examine experimentally and theoretically, the effect of edge curvature of colloidal particles on their defect configurations and self-assembly in a nematic liquid crystal (5CB). The polarized images of the particles with homeotropic surface anchoring in 5CB show that the presence of sharp edges can reshape completely the defect ring. The defect makes sharp turns and follows the edge of the cube particles, which significantly affects the interaction between particles and their eventual self-assembly. In agreement with our experimental results, our computational studies indicate that the gradual increase of the edges sharpness that occurs as we transition from spheres to cubes, changes the defect structure from a Saturn ring to a twisted ring, which is pinned to the edges of the cube particle. The wide variety of topological defects achievable by changing the curvature could provide new tools to tune colloidal self-assembly.   

Nematic liquid crystal bridges

We study the effects of confining a nematic liquid crystal between two parallel glass plates with homeotropic boundary conditions for the director at all bounding surfaces. We find that the free surface of the nematic bridge is a surface of constant mean curvature. In addition, by changing the distance between the plates and the contact angle with the glass plates, we transition between loops and hedgehogs that can be either radial or hyperbolic.   

Fine structure of the topological defect core: Disclination in lyotropic chromonic liquid crystal

Topological defects represent an important concept in many branches of modern physics ranging from cosmology and optics to hard and soft matter. One of the most difficult problems is the fine structure of the so-called core region of defects, where the deformations of the order parameter are so strong that the phenomenological description valid in the far field becomes invalid. Experimental exploration of the fine core structure is usually hindered by the small size (atomic/molecular level) of the core, where optical or even electron microscopy techniques are invalid. In this work, we take advantage of the peculiar nature of the so-called lyotropic chromonic liquid crystals (LCLC) of a nematic type that carry disclinations with a core extending over macroscopic distances (tens of micrometers), large enough to explore their spatial variation by optical and electron microscopy. We demonstrate that the director and the scalar order parameter (associated with the degree of orientational order) show a profound change in the core region. In particular, as one approaches the center of the defect, the azimuthal dependency of the director field changes dramatically and the scalar order parameter shows a strong dependence on the strength of splay and bend deformations.   

Lassoing the Saddle-Splay: Harnessing $K_{24}$ Distortions to Line Up Disclinations

Systems with holes, such as colloidal handlebodies and toroidal droplets, have been studied in the nematic liquid crystal (NLC) 4-cyano-4’-pentylbiphenyl (5CB). It was found that point or ring topological defects occur within each hole and around the system, such that the overall topological charge of the system is conserved. However, what has not been fully appreciated is the ability of the hole geometry with homeotropic (perpendicular) anchoring conditions to induce a saddle-like deformation in the NLC bulk. We exploit this by creating an array of many holes suspended in an NLC cell with oriented planar (parallel) anchoring at the cell boundaries. Through simulations and experiments, we study how the bulk saddle deformations of each hole interact to create novel defect structures, including an array of $\frac{1}{2}$ disclination lines, reminiscent of those that occur in LC blue phases. The locations of these disclination lines are tunable via the geometry of the cell and hole array, which has potential for controlled, three-dimensional self-assembly in NLC’s.   

The weirdest martensite: Smectic liquid crystal microstructure and Weyl-Poincar\'e invariance

We propose a generalization of the mathematical theory of martensites to describe the complex assembly of focal conics in smectic liquid crystals. Smectics are remarkable, beautiful examples of materials microstructure, with ordered patterns of geometrically perfect ellipses and hyperbolas. The solution of the complex problem of filling three-dimensional space with domains of focal conics under constraining boundary conditions yields a set of strict rules, which are similar to the compatibility conditions in a martensitic crystal. Here we present the rules giving compatible conditions for the concentric circle domains found at two-dimensional smectic interfaces with planar boundary conditions. Using configurations generated by numerical simulations, we develop a clustering algorithm to decompose the planar boundaries into domains. The interfaces between different domains agree well with the smectic compatibility conditions. We also discuss generalizations of our approach to describe the full three-dimensional smectic domains, where the variant symmetry group is the restricted Weyl-Poincar\'e group of Lorentz boosts, translations, rotations, and dilatations.   

4 x 4 Matrix Method Simulations of Swinging Nematic Liquid Crystals

We present the results of numerical simulations of swinging nematic liquid crystal (SNLC) systems using the 4 x 4 Berreman matrix method. SNLCs are a special class of cholesteric liquid crystals that periodically change handedness when propagating along the helical axis. Unlike standard cholesterics which can only reflect one circular polarization state allowing the other to pass, SNLCs are able to reflect both simultaneously. Our simulations explore the advantages and disadvantages of various periodic functions (sinusoidal, square wave, triangular, etc\textellipsis ), the influence of pitch and optical birefringence on the reflection central wavelength and bandwidth, as well as the overall impact incidence angle has on the reflection spectra.   

Phase transitions and order parameters of complex liquid crystalline ordered systems

Liquid crystalline states of matter possess rich phase diagrams, exotic topological defects and unique responses to external fields. Traditionally the focus has been on liquid crystal phases with uniaxial $D_{\infty h}$ symmetry and biaxial $D_{2h}$ symmetry. However, in full generality liquid crystalline orders are associated with breaking the $O(3)$ rotational symmetry to any point group symmetry $G \subset O(3)$. We present a general theory for arbitrary three dimensional point group symmetries that allows to derive order parameters and investigate phase transitions of liquid crystalline states. The theory is constructed on symmetry grounds, and adapts to the description of both thermal and quantum liquid crystal systems. The realization of the model in experimental systems are also discussed.   

Coarsening Dynamics of Inclusions and Thermocapillary Phenomena in Smectic Liquid Crystal Bubbles

The Observation and Analysis of Smectic Islands in Space (OASIS) project comprises a series of experiments that probe interfacial and hydrodynamic behavior of thin spherical-bubbles of smectic liquid crystal in microgravity. Smectic films are the thinnest known stable condensed phase structures, making them ideal for studies of two-dimensional (2D) coarsening dynamics and thermocapillary phenomena in microgravity. The OASIS flight hardware was launched on SpaceX-6 in April 2015 and experiments were carried out on the International Space Station using four different smectic A and C liquid crystal materials in separate sample chambers. We will describe the behavior of collective island dynamics on the bubbles, including temperature gradient-induced themomigration, and the diffusion and coalescence-driven coarsening dynamics of island emulsions in microgravity. This work was supported by NASA Grant No. NNX-13AQ81G, and NSF MRSEC Grants No. DMR-0820579 and DMR-1420736.   

Competition of lattice and basis for alignment of nematic liquid crystals

Due to elastic anisotropy, two-dimensional patterning of substrates can promote weak azimuthal alignment of adjacent nematic liquid crystals. We consider a periodic square lattice of elliptical motifs to examine ways in which the lattice and motif can combine to favor differing orientations. Using semi-analytic elastic continuum theory and Monte Carlo simulations, we find, for circular motifs, that the coverage fraction controls both the polar anchoring angle and a transition in the azimuthal orientation. If the circles are generalized to ellipses, arbitrary control of the effective easy axis and effective anchoring potential becomes achievable by appropriate tuning of the orientation of the ellipse motif relative to the lattice vectors. To determine the behavior of liquid crystals near the domain boundaries, we additionally formulate and solve the full 3D Euler-Lagrange equations directly. We additionally comment on the role of weak anchoring and saddle-splay elasticity.   

Particles and curvatures in nematic liquid crystals.

Elastic interactions in anisotropic fluids can be harnessed to direct particle interactions.~A strategy to smoothly manipulate the director field in nematic liquid crystals is to vary the topography of the bounding surfaces. A rugged landscape with peaks and valleys create local deformations of the director field which can interact with particles in solution. We study this complex interaction in two different settings. The first consists of an array of shallow pores in a poly-dimethyl-siloxane (PDMS) membrane, whose curvature can be tuned either by swelling the PDMS membrane or by mechanical stretching. The second is a set of grooves with wavy walls, fabricated by photolithography, with various parameters of curvature and shapes. In this contexts we study how the motion of colloidal particles in nematic liquid crystals can be influenced by their interaction with the peaks and valleys of the bottom substrate or of the side walls. Particles with different associated topological defects (hedgehogs or Saturn rings) behave differently as they interact with the topographical features, favoring the docking on peaks or valleys. These experimental systems are also ideal to study the ``lock and key'' mechanism of particles in holes and to investigate a possible route for particle sorting.   

Effect of an applied electric field on a weakly anchored non-planar Nematic Liquid Crystal (NLC) layer

We consider a mathematical model that consists of a NLC layer sandwiched between two parallel bounding plates, across which an external field is applied. We investigate its effect on the director orientation by considering the dielectric and flexoelectric contributions and varying parameters that represent the anchoring conditions and the electric field strength. In particular, we investigate possible director configurations that occur in weakly anchored and non-planar systems. We observe that non-planar anchoring angles destroy any hysteresis seen in a planar system by eliminating the fully vertical director configuration and the ''saturation threshold'' seen in weakly anchored planar Freedericksz cells.