Geometry-guided colloidal interactions and self-tiling of elastic dipoles formed by truncated pyramid particles in liquid crystals

The progress of realizing colloidal structures mimicking natural forms of organization in condensed matter is inherently limited by the availability of suitable colloidal building blocks. To enable new forms of crystalline and quasicrystalline self-organization of colloids, we develop truncated pyramidal particles that form nematic elastic dipoles with long-range electrostaticlike and geometry-guided low-symmetry short-range interactions. Using a combination of nonlinear optical imaging, laser tweezers, and video microscopy, we characterize colloidal pair interactions and demonstrate unusual forms of self-tiling of these particles into crystalline, quasicrystalline, and other arrays. Our findings are explained using an electrostatics analogy along with liquid crystal elasticity and symmetry breaking considerations, potentially expanding photonic and electro-optic applications of colloids.

Figure 1

Polygonal truncated pyramids in a nematic LC. (a)–(d) Bright field, (e) and (g) polarizing, and (i)–(l) 3PEF-PM images of (a) and (j) convex and (c) concave PTPs with induced n(r) distortions. Here $P,A$, and $Z$ mark, respectively, the polarizer, analyzer and a slow axis of a 530-nm phase retardation plate. A magenta double arrow shows linear polarization of 3PEF-PM excitation light. (f), (h), and (m) Schematic of n(r) (green lines) around PTPs, which corresponds to the plane of a disclination loop (a red solid line): Nails represent molecules tilted out of the plane of an image and their length is proportional to a tilt angle; the head of the nails is below the plane of the image and their points are directed towards the observer. (n) Histograms of PTP displacements along and perpendicular to ${\mathbf{n}}_0$. (o) Bright field and (p) 3PEF-PM images and (q) n(r) schematic for a self-assembled dimer of dipoles formed by a microsphere and PTP.

Figure 2

Side-to-side elastic pair interactions of PTPs in a homeotropic nematic cell. (a) Sequence of optical bright field micrographs showing the attractive pair interaction between antiparallel dipoles. (b) The n(r) around interacting PTPs. (c) The 3PEF-PM cross-section obtained along a dashed yellow line shown in (a). (d) The n(r) and defects between self-assembled PTPs within the encircled area marked in (c). (e) Separation between antiparallel dipoles shown in (a) versus time. (f) The $U_{s\text{−}s}-U_0$ versus $S$; the inset shows a log-log plot of $F_{s\text{−}s}$ versus $S$ fitted (solid red line) to a power law with an exponent −4.0 ± 0.3. (g) Bright field micrographs showing repulsive pair interactions between parallel dipoles. (h) The $S\left(t\right)$ for dipoles shown in (g).

Figure 3

(a) Bright field micrographs showing base-to-base attraction between elastic dipoles in a planar nematic cell. (b) Schematic of self-assembled PTPs and defect loops. (c) Vertical $zx$ and (d) in-plane $xy3\text{PEF}$-PM images of self-assemblies. (e) The $S\left(t\right)$ for dipoles shown in (a). (f) The $U_{b\text{−}b}$ versus $S$ for parallel elastic dipoles in (a); the inset shows the corresponding force $F_{b\text{−}b}$ versus $S$. (g) Bright field micrographs showing interactions between two elastic dipoles in a homeotropic cell when pushed towards each other; the elapsed time is marked on the images.

Figure 4

Examples of assemblies of (a)–(f) convex and (h) and (i) concave pentagonal PTPs in a homeotropic nematic cell. (c) Optical images and (d) a schematic of a ringlike assembly. (e) Diamondlike assembly. (f) Micrographs of a Penrose tiling fragment and (g) the corresponding schematic; magenta pentagons represent particles missing in the experimental assembly. (h) and (i) Dimers of concave PTPs with different binding.

Figure 5

Bright field and polarizing micrographs of n(r)-distortions and colloidal assemblies in a homeotropic nematic cell due to rhombic PTPs with internal angles of bases equal to (a) 36°/144° and (b) 72°/108°. (c)–(e) Examples of rhombic PTP tiling.