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.

Figure 1

Conventional colloidal elastic dipoles in CNLCs of equilibrium pitch $p=33$ μm prepared by doping a chiral additive S-811 into a nematic host AMLC-0010. (a–c) Optical micrographs of an unwound homeotropic cell with a 4-μm silica microsphere inducing an elastic dipole orthogonal to cell substrates obtained (a) between crossed polarizer (P) and analyzer (A) marked by blue double arrows, (b) between the crossed polarizer and analyzer with an additional 530-nm phase retardation plate having its slow axis marked by the green double arrow, and (c) without polarizers or other optical elements. (d, e) 3PEF-PM vertical $xz$ cross sections showing elastic dipoles with two opposite orientations and points defects (d) above and (e) beneath the beads. The cell thickness is visible from 3PEF-PM cross sections relative to the 4-μm particle size. (f) The n(r) configuration induced by a colloidal microsphere obtained using an ansatz from Ref. [26], showing a good qualitative agreement with the experimental images depicted in (d, e) in homeotropic confined cell; the hyperbolic bulk point defect is shown by a red filled circle.

Figure 2

CNLC colloidal dipoles with twisted solitons induced by 4-μm silica spheres. (a) Polarizing optical micrograph (obtained between crossed polarizer and analyzer marked by blue arrows) showing a topological colloidal dipole in a homeotropic unwound CNLC cell of thickness $d$ $\approx$ 10 μm filled with a CNLCs of equilibrium pitch 10.6 μm based on AMLC-0010 nematic host. (b–d) In-plane $xy3\text{PEF}$-PM images of the colloidal dipole obtained (b) for circularly polarized 3PEF-PM excitation light and (c, d) for two mutually orthogonal linear polarizations of 3PEF-PM excitation light (marked by red double arrows) in a cell of thickness $d$ $\approx$ 30 μm filled with CNLCs of equilibrium pitch 33 μm based on AMLC-0010 nematic host. (e–j) 3PEF-PM vertical $xz$ cross sections of the colloidal dipoles in cells of different thickness obtained for circularly polarized 3PEF-PM excitation light; the thickness is 10 μm in (e), 7 μm in (f, g), and 30 μm in (h–j). (k) Computer-simulated n(r), depicted using cylinders, around a particle with homeotropic anchoring; note that the small disclination loop above the particle is equivalent to a hyperbolic point defect. (l) The corresponding computer-simulated 3PEF-PM texture in the vertical plane for circular polarization of the 3PEF-PM excitation light.

Figure 3

Structures of skyrmion-decorated topological dipoles in CNLC cells. The spatial patterns of n(r) are depicted using cylinders and the small defect loops are presented by isosurfaces of the reduced nematic order parameter [39]. (a–d) Colocated director structures (shown using cylinders) and 3PEF-PM images (gray-scale intensity plots) shown in (a, c) in-plane $\left(xy\right)$ cross sections containing the singular defect loops and (b, d) corresponding vertical $\left(xz\right)$ cross sections of the CNLC cell for different ratios between the cell thickness $d$ and the particle diameter $2R$.

Figure 4

Imaging of n(r) induced by colloidal dimers formed by two antiparallel elastic dipoles induced by 4-μm silica spheres in a cell of thickness $d$ $\approx$ 7 μm filled with CNLCs of pitch $p$ $\approx$ 10.5 μm prepared by doping a chiral additive S-811 in AMLC-0010 nematic host. (a) POM obtained between crossed polarizer and analyzer marked by blue arrows and (b) bright-field micrographs of a self-assembled dimer formed by two antiparallel elastic dipoles. (c, d) In-plane and (e, f) vertical 3PEF-PM cross sections obtained for (c, e) linearly and (d, f) circularly polarized excitation light.

Figure 5

Examples of computer-simulated director configurations arising from self-assembly of colloidal elastic dipoles induced by spheres with perpendicular surface anchoring. The spatial patterns of n(r) are depicted using cylinders and the small defect loops are shown using the bend -splay visualization [39]. (a, c, d) Computer-simulated n(r) of self-assembled colloidal dimers as depicted (a) in the in-plane $x\text{−}y$ cross section in the cell midplane and (c, d) vertical cross sections in two perpendicular (c) $xz$ and (d) $yz$ planes according to the coordinate system shown in (b). (b) The corresponding in-plane computer-simulated 3PEF-PM image for circularly polarized excitation light and n(r) shown in (a); the inset shows the corresponding experimental image. Note that the induced bulk point (small ring) defects of a particle dimer shown in (c) shift away from vertical axes passing through the particle's centers.

Figure 6

Multiparticle assemblies of elastic dipoles in a cell of thickness of $d$ $\approx$ 15 μm filled with CNLCs of pitch $p$ $\approx$ 20 μm prepared by doping a chiral additive S-811 in E-31 nematic host. (a–d) Bright-field optical micrographs showing typical examples of self-assembly of antiparallel elastic dipoles into (a) dimers, (b) tetramers, (c) pentamers, and (d) hexamers. The insets in (a) show relative locations of the 4-μm silica microspheres and the particle-induced point (small ring) defects, which determine the orientations of elastic dipoles. (e, f) 3PEF-PM vertical cross-sectional images, which were obtained along lines marked on (b, c), showing the details of the assembly of defects and particles obtained for linear polarizations of light having directions depicted by red arrows.

Figure 7

Colloidal pair interactions. (a, b) Center-to-center separation versus time between two (a) antiparallel and (b) parallel elastic dipoles probed using videomicroscopy when they are released by laser tweezers. Insets in (a) and (b) show bright-field optical micrographs taken at the corresponding initial and final center-to-center distances as well as schematically depict orientations of the elastic dipoles. The interactions were probed for 4-μm silica spheres in a cell of thickness $d$ $\approx$ 15 μm containing a CNLC with pitch $p$ $\approx$ 20 μm prepared by doping a nematic host E-31with a chiral additive S-811.

Figure 8

Diversity of field and defect configurations of colloidal self-assemblies due to plastic particles of diameter $\approx 6$ μm dispersed in CNLCs of pitch 10.5 μm prepared by doping S-811 chiral additive to AMLC-0010 nematic host filled in a cell of thickness $d$ $\approx$ 10 μm: (a–f) dimers, (g–j) trimers, (k–p) tetramers; the particles and structures are imaged using polarizing optical microscopy (top images) and bright-field optical microscopy (bottom images). (q) Polarizing and (r) bright-field optical micrographs of 6-μm spacers surrounded by looped double-twist-cylinder solitons in an unwound homeotropic CNLC cell. The orientations of crossed polarizer (P) and analyzer (A) are shown using blue double arrows.

Figure 9

(a–o) Polarizing optical micrographs of multiparticle assemblies of elastic dipoles with twisted solitons induced by 4-μm silica microspheres in a cell of thickness diameter $\approx 10$ μm filled with CNLCs of pitch 10.5 μm prepared by doping a chiral additive S-811 in AMLC-0010 nematic host. The orientations of crossed polarizer (P) and analyzer (A) are shown using blue double arrows.

Figure 10

Computer-simulated n(r) and 3PEF-PM textures around colloidal dimers formed by spheres with perpendicular surface anchoring and inducing both solitonic nonsingular and singular defects. (a, b) Simulated n(r) and defects of colloidal dimers depicted in vertical cross sections passing through the sphere centers. Note that the particles are interspaced by two different closed loops of half-integer disclinations and are also surrounded by nonsingular director twist distortions around them. The spatial patterns of n(r) are depicted using cylinders. (c, d) 3PEF-PM textures (shown as gray-scale intensity patterns) superimposed with n(r) depicted using cylinders for (c) in-plane and (d) vertical cross sections of the structure shown in (b). Disclination lines are presented by isosurfaces of the splay and bent deformations exposing their threefold symmetry [39].

Figure 11

Computer-simulated n(r) around colloidal trimers formed by spheres with perpendicular surface anchoring and inducing both solitonic nonsingular and singular defects. (a–d) Simulated n(r) and defects for two different structures depicted in the (a, c) in-plane cross section corresponding to the cell midplane and (b, d) vertical cross sections passing through the sphere centers. Note that the particles are interspaced by closed loops of singular defect lines and are also surrounded by nonsingular director twist distortions around them. The spatial patterns of n(r) are depicted using cylinders and the defect lines are shown using the bend-splay visualization [39].

Figure 12

Examples of computer-simulated n(r) around colloidal trimers formed by spheres with perpendicular surface anchoring and inducing single or multiple closed defect loops. (a, b) All three particles are entangled by a single defect. (c, d) Two particles are entangled by a single defect loop and one particle in (c) is inducing a separate defect loop next to. The spatial patterns of n(r) are depicted using cylinders and the defect lines are shown using the bend-splay visualization [39].