Edge pinning and transformation of defect lines induced by faceted colloidal rings in nematic liquid crystals

Nematic colloids exhibit a large diversity of topological defects and structures induced by colloidal particles in the orientationally ordered liquid crystal host fluids. These defects and field configurations define elastic interactions and medium-mediated self-assembly, as well as serve as model systems in exploiting the richness of interactions between topologies and geometries of colloidal surfaces, nematic fields, and topological singularities induced by particles in the nematic bulk and at nematic-colloidal interfaces. Here we demonstrate formation of quarter-strength surface-pinned disclinations, as well as a large variety of director field configurations with splitting and reconnections of singular defect lines, prompted by colloidal particles with sharp edges and size large enough to define strong boundary conditions. Using examples of faceted ring-shaped particles of genus $g=1$, we explore transformation of defect lines as they migrate between locations in the bulk of the nematic host to edge-pinned locations at the surfaces of particles and vice versa, showing that this behavior is compliant with topological constraints defined by mathematical theorems. We discuss how transformation of bulk and surface defect lines induced by faceted colloids can enrich the diversity of elasticity-mediated colloidal interactions and how these findings may impinge on prospects of their controlled reconfigurable self-assembly in nematic hosts.

Figure 1

Defects transformation around torus-shaped colloidal particles in a nematic LC: (a,b,e) Bright-field optical and (c,d) 3PEF-PM micrographs of an orthogonally aligned (a,c) and tilted (b,d,e) particle. Cross-sectional $xz$ images in (c,d) were obtained along a yellow line in (a,b). Insets in (a,b) show corresponding polarizing microscopy textures obtained between crossed polarizers $P$ and $A$. Double arrows and circles with a dot in the center show the in-plane and out-of-plane directions of the far-field director ${\mathbf{n}}_0$, respectively. Red double arrows show the direction of polarization of the excitation beam parallel to the plane of the image. Black dashed lines in (b,e) show the particle's tilt axis. (f–i) Corresponding schematic diagrams of $\mathbf{n}\left(\mathbf{r}\right)$ and defects inside (f,g) and outside (h,i) the ring-shaped particle; only −1/4 defect propagating along edges is shown for clarity. A black thick arrow shows a normal s to the plane of the ring-shaped particle. Curved red arrows in (b,d,m,n) show the direction of tilt. White dashed line in (d) shows a plane of the ring-shaped particle and $\alpha$ marks an angle between s and ${\mathbf{n}}_0$. Black (a,b,e) and white [inset of (b)] thin arrows point to a bulk −1/2 defect loop (a) and to the −1/2 defect line branches (b,e), respectively. (j–n) Schematic diagrams showing $\mathbf{n}\left(\mathbf{r}\right)$ with only edge-pinned surface defects (j,k), with both bulk and edge-pinned defect lines (l) and absorption of a bulk defect at the edge upon tilting the particle (m,n). Solid red lines and filled circles in (f–n) mark the bulk −1/2 disclinations. Solid orange and dark-blue lines and squares show the surface-bound ±1/4 defect lines.

Figure 2

Edge-pinned defect lines around a particle in a twisted nematic cell: (a,b,d,e) Optical micrographs obtained between crossed (a,d,e) and parallel (b) polarizers $P$ and $A$. Orientation of a pair of bulk −1/2 defects (shown by black arrows) inside a particle's ring in (a) and (d) is different with respect to $s_{\perp }$. Inset in (b) shows right-handed twist of $\mathbf{n}\left(\mathbf{r}\right)$ between bottom and top substrates with orthogonal easy axes ${\mathbf{e}}_b$ and ${\mathbf{e}}_t$, respectively. (e) Optical texture showing a zoomed-in view of a bulk −1/2 defect line seen in (d) while traversing from one edge to another. (c,f) Schematic diagrams of defects and their transformation corresponding to particles shown in (a,d), respectively. The handle-shaped bulk defect lines are shown using thick red lines. The quarter-strength edge-bound surface defect lines of opposite strengths are shown using thin blue and orange lines, with the strength of opposite signs marked next to them.

Figure 3

Detailed schematic diagrams of $\mathbf{n}\left(\mathbf{r}\right)$ and defects in the region of transformation of disclinations inside (a) and outside (c) of a toroidal particle with square-shaped cross section (b). The handle-shaped bulk defect lines are depicted as thick red lines. The quarter-strength edge-bound surface defect lines of opposite signs are shown using thin blue and orange lines, with the strength of opposite signs marked next to them. The green lines depict the director field.

Figure 4

Edge-pinned defect lines around a ring-shaped particle in a planar nematic cell: (a) a 3PEF-PM texture for a sample $xz$ cross section across the thickness of the cell showing that a torus is tilted with respect to the middle of the cell and is levitating in the bulk of the sample. The inset shows a bright-field micrograph of a tilted particle. (b) 3D schematic diagram of bulk and edge-pinned surface defects that undergo transformation. (c,d) Schematic diagram of $\mathbf{n}\left(\mathbf{r}\right)$ and the induced defects inside (c) and outside (d) the particle. As in Fig. 2, the handle-shaped bulk defect lines are shown using thick red lines. The quarter-strength edge-bound surface defect lines of opposite strengths are shown using thin blue and orange lines, with the strength of opposite signs marked next to them. The green lines depict the director field.

Figure 5

(a–f) Nucleation, transformation, and absorption of bulk −1/2 defect lines at the edges of a tilted toroidal particle when locally quenching a nematic from an isotropic $I$ (a) to nematic $N$ (b–f) phase in a twist nematic cell upon switching laser tweezers off. The dark spot at the outer edge of the particle seen in (b–f) at around 9 o’clock corresponds to the debris attached to the particle below the top visible edge, which is not visible in the $I$ phase (a) because of a different particle tilt. (g–i) Optical trapping and stretching of the bulk −1/2 defects with laser tweezers in the exterior (g) and interior (h,i) of the particle's ring. Black arrows point to the bulk −1/2 defect lines. Yellow (bright) crosses in the middle of isotropic droplets of 5CB track the spatial position of a trapping laser beam when it is turned on.

Figure 6

Deformation of bulk disclinations around torus-shaped colloidal particles in a nematic LC. (a) Bright-field optical micrograph of a tilted particle with an outer bulk half-integer disclination that is not absorbed at the particle edges. Inset shows a corresponding polarizing microscopy texture obtained between crossed polarizers $P$ and $A$. Black and white (in the inset) thin arrows mark points where the deformed outer half-integer defect loop transitions between the top and bottom edges. (b) A schematic diagram showing the corresponding $\mathbf{n}\left(\mathbf{r}\right)$.