Perpendicular Boundary Conditions Research Articles

Giant director fluctuations in liquid crystal drops.

We report the discovery and elucidation of giant spatiotemporal orientational fluctuations in nematic liquid crystal drops with radial orientation of the nematic anisotropy axis producing a central "hedgehog" defect. We study the spatial and temporal properties of the fluctuations experimentally using polarized optical microscopy, and theoretically, by calculating the eigenspectrum of the Frank elastic free energy of a nematic drop of radius R_{2}, containing a spherical central core of radius R_{1} and constrained by perpendicular boundary conditions on all surfaces. We find that the hedgehog defect with radial orientation has a complex excitation spectrum with a single critical mode whose energy vanishes at a critical value μ_{c} of the ratio μ=R_{2}/R_{1}. When μ<μ_{c}, the mode has positive energy, indicating that the radial hedgehog state is stable; when μ>μ_{c}, it has negative energy indicating that the radial state is unstable to the formation of a lower-energy state. This mode gives rise to the large-amplitude director fluctuations we observe near the core, for μ near μ_{c}. A collapse of the experimental data corroborates model predictions for μ<μ_{c} and provides an estimate of the defect core size.

Director Fluctuations in Two-Dimensional Liquid Crystal Disclinations

We present analytical calculations of the energies and eigenfunctions of all normal modes of excitation of charge +1 two-dimensional splay (bend) disclinations confined to an annular region with inner radius R1 and outer radius R2 and with perpendicular (tangential) boundary conditions on the region’s inner and outer perimeters. Defects such as these appear in islands in smectic-C films and can in principle be created in bolaamphiphilic nematic films. Under perpendicular boundary conditions on the two surfaces and when the ratio β=Ks/Kb of the splay to bend 2D Frank constants is less than one, the splay configuration is stable for all values μ=R2/R1. When β&gt;1, the splay configuration is stable only for μ less than a critical value μc(β), becoming unstable to a “spiral” mixed splay-bend configuration for μ&gt;μc. The same behavior occurs in trapped bend defects with tangential boundary conditions but with Ks and Kb interchanged. By calculating free energies, we verify that the transition from a splay or bend configuration to a mixed one is continuous. We discuss the differences between our calculations that yield expressions for experimentally observable excitation energies and other calculations that produce the same critical points and spiral configurations as ours but not the same excitation energies. We also calculate measurable correlation functions and associated decay times of angular fluctuations.

Linked topological colloids in a nematic host

Geometric shape and topology of constituent particles can alter many colloidal properties such as Brownian motion, self-assembly, and phase behavior. Thus far, only single-component building blocks of colloids with connected surfaces have been studied, although topological colloids, with constituent particles shaped as freestanding knots and handlebodies of different genus, have been recently introduced. Here we develop a topological class of colloids shaped as multicomponent links. Using two-photon photopolymerization, we fabricate colloidal microparticle analogs of the classic examples of links studied in the field of topology, the Hopf and Solomon links, which we disperse in nematic fluids that possess orientational ordering of anisotropic rod-like molecules. The surfaces of these particles are treated to impose tangential or perpendicular boundary conditions for the alignment of liquid crystal molecules, so that they generate a host of topologically nontrivial field and defect structures in the dispersing nematic medium, resulting in an elastic coupling between the linked constituents. The interplay between the topologies of surfaces of linked colloids and the molecular alignment field of the nematic host reveals that linking of particle rings with perpendicular boundary conditions is commonly accompanied by linking of closed singular defect loops, laying the foundations for fabricating complex composite materials with interlinking-based structural organization.

Topology and self-assembly of defect-colloidal superstructure in confined chiral nematic liquid crystals.

We describe formation of defect-colloidal superstructures induced by microspheres with normal surface anchoring dispersed in chiral nematic liquid crystals in confinement-unwound homeotropic cells. Using three-dimensional nonlinear optical imaging of the director field, we demonstrate that some of the induced defects have nonsingular solitonic nature while others are singular point and line topological defects. The common director structures induced by individual microspheres have dipolar symmetry. These topological dipoles are formed by the particle and a hyperbolic point defect (or small disclination loop) of elementary hedgehog charge opposite to that of a sphere with perpendicular boundary conditions, which in cells with thickness over equilibrium cholesteric pitch ratio approaching unity are additionally interspaced by a looped double-twist cylinder of continuous director deformations. The long-range elastic interactions are probed by holographic optical tweezers and videomicroscopy, providing insights to the physical underpinnings behind self-assembled colloidal structures entangled by twisted solitons. Computer-simulated field and defect configurations induced by the colloidal particles and their assemblies, which are obtained by numerically minimizing the Landau-de Gennes free energy, are in agreement with the experimental findings.

Colloidal spirals in nematic liquid crystals.

One of the central experimental efforts in nematic colloids research aims to explore how the interplay between the geometry of particles along with the accompanying nematic director deformations and defects around them can provide a means of guiding particle self-assembly and controlling the structure of particle-induced defects. In this work, we design, fabricate, and disperse low-symmetry colloidal particles with shapes of spirals, double spirals, and triple spirals in a nematic fluid. These spiral-shaped particles, which are controlled by varying their surface functionalization to provide tangential or perpendicular boundary conditions of the nematic molecular alignment, are found inducing director distortions and defect configurations with non-chiral or chiral symmetry. Colloidal particles also exhibit both stable and metastable multiple orientational states in the nematic host, with a large number of director configurations featuring both singular and solitonic nonsingular topological defects accompanying them, which can result in unusual forms of colloidal self-assembly. Our findings directly demonstrate how the symmetry of particle-generated director configurations can be further lowered, or not, as compared to the low point group symmetry of solid micro-inclusions, depending on the nature of induced defects while satisfying topological constraints. We show that achiral colloidal particles can cause chiral symmetry breaking of elastic distortions, which is driven by complex three-dimensional winding of induced topological line defects and solitons.

Two-dimensional skyrmions and other solitonic structures in confinement-frustrated chiral nematics.

We explore spatially localized solitonic configurations of a director field, generated using optical realignment and laser-induced heating, in frustrated chiral nematic liquid crystals confined between substrates with perpendicular surface anchoring. We demonstrate that, in addition to recently studied torons and Hopf-fibration solitonic structures (hopfions), one can generate a host of other axially symmetric stable and metastable director field configurations where local twist is matched to the surface boundary conditions through introduction of point defects and loops of singular and nonsingular disclinations. The experimentally demonstrated structures include the so-called "baby-skyrmions" in the form of double twist cylinders oriented perpendicular to the confining substrates where their double twist field configuration is matched to the perpendicular boundary conditions by loops of twist disclinations. We also generate complex textures with arbitrarily large skyrmion numbers. A simple back-of-the-envelope theoretical analysis based on free energy considerations and the nonpolar nature of chiral nematics provides insights into the long-term stability and diversity of these inter-related solitonic field configurations, including different types of torons, cholestric-finger loops, two-dimensional skyrmions, and more complex structures comprised of torons, hopfions, and various disclination loops that are experimentally observed in a confinement-frustrated chiral nematic system.

COMPUTER SIMULATION OF THE HOMEOTROPIC TO FOCAL CONIC TRANSITION IN CHOLESTERICS

Previous reports have proposed mechanisms for the homeotropic to focal conic transition in cholesterics. In this report, we present quantitative numerical modeling of this transition. We will show that the simulation shows the existence of the transient planar state, agreeing with previous experimental findings. We will also demonstrate that the simulations never show energy densities high enough to form isotropic regions, demonstrating that this transition is a continuous process. Instead of introducing isotropic regions, the system transforms through an undulation distortion which strongly resembles a Helfrich-Hurault distortion. This simulation is compared with two experimental samples: one with parallel boundary conditions and one with perpendicular boundary conditions. The simulations are compared with dynamical capacitive measurements and microscope photographs, and found to agree very well. We will present detailed drawings of the director configuration during this transition. The agreement of this simulation with the experimental data shows that the mechanism of this transition is now well understood

Computer simulation evidence of the transient planar state during the homeotropic to focal conic transition in cholesteric liquid crystals

Previous experimental evidence has suggested the existence of a transient planar state during the transition from a homeotropic state to a focal conic state. This does not appear at first to be physically reasonable. Here we show a quantitative numerical model of this transition that shows the existence of the transient planar state and agrees well with measured data. We compared this simulation with two experimental samples: one with parallel boundary conditions, and one with perpendicular boundary conditions. Both samples had thickness to pitch ratios of 13.9. The agreement of the simulation with the experimental data gives further evidence that the system passes through the transient planar during the transition. Also shown here are preliminary results for the second part of the transition, the transition from the transient planar state to the focal conic state, that suggest a Helfrich-like undulation deformation.

Electric-Field-Induced Director-Field Transition of Nematic Liquid Crystal in a Closed Conical Cavity

Continuum theory and numerical method are utilized in this study to report the dependence of director-field transition induced by electric field on the elastic constants of nematic liquid crystal (NLC) with negative dielectric anisotropy in a closed conical cavity with perpendicular boundary conditions of NLC. Normal umbilic structure (NU), twisted umbilic structure (TU), normal hyperbolic point structure with a disclination ring (NH), twisted hyperbolic point structure with a disclination ring (TH), and normal Poincare structure (NP) are found to be stable structures in electric-field-induced transitions. Moreover, five typical sequences of transition, TNU→TU→TH, TNU→TU→NH→TH, TNU→TU→NH, TNU→NH, and TNU→NP→NH are found in the range of applied voltage less than 10 V. The transition sequences and the transition voltages are strongly dependent on the ratios of elastic constants K11/K33 and K22/K33.

Electric-field effects on nematic droplets with negative dielectric anisotropy.

Using polarization microscopy, we have investigated the director configurations of droplets of a nematic liquid crystal with negative dielectric anisotropy and perpendicular boundary conditions subjected to electric fields. The droplets were suspended in liquid poly(dimethyl siloxane) and electric fields were applied both parallel and perpendicular to the viewing direction. The resulting director pattern is deduced by comparing the experimental transmission patterns with patterns calculated from various director models. For low fields, we observe a director field which is essentially radial, but with some azimuthal twist superimposed and a radial point (hedgehog) defect at the center. At intermediate fields, the hedgehog defect moves away from the center along the electric-field direction, while at the highest fields a line defect appears which lies along the diameter parallel to the field. Comparison of the transmission patterns with model calculations yields good agreement for the low- and intermediate-field cases, and reasonable agreement for the high-field case.