Topological defects in liquid crystals not only affect the optical and rheological properties of the host, but can also act as scaffolds in which to trap nano or micro-sized colloidal objects. The creation of complex defect shapes, however, often involves confining the liquid crystals in curved geometries or adds complex-shaped colloidal objects, which are unsuitable for device applications. Using topologically patterned substrates, here we demonstrate the controlled generation of three-dimensional defect lines with non-trivial shapes and even chirality, in a flat slab of nematic liquid crystal. By using the defect lines as templates and the electric response of the liquid crystals, colloidal superstructures are constructed, which can be reversibly reconfigured at a voltage as low as 1.3 V. Three-dimensional engineering of the defect shapes in liquid crystals is potentially useful in the fabrication of self-healing composites and in stabilizing artificial frustrated phases.

Spatially confined liquid crystals exhibit non-uniform alignment, often accompanied by self-organised topological defects of non-trivial shape in response to imposed boundary conditions and geometry. Here we show that a nematic liquid crystal, when confined in a sinusoidal microwrinkle groove, exhibits a new periodic arrangement of twist deformations and a zigzag line defect. This periodic ordering results from the inherent liquid crystal elastic anisotropy and the antagonistic boundary conditions at the flat liquid crystal–air and the curved liquid crystal–groove interfaces. The periodic structure can be tuned by controlling the groove geometry and the molecular chirality, which demonstrates the importance of boundary conditions and introduced asymmetry for the engineering of topological defects. Moreover, the kinks in the zigzag defects can trap small particles, which may afford a new method for manipulation of colloids. Our system, which uses easily fabricated microwrinkle grooves, provides a new microfabrication method based on the arrangement of controllable defects.

To account for azimuthal surface anchoring of a nematic liquid crystal, Berreman [ Phys. Rev. Lett. 28, 1683 (1972)] proposed a simple model attributing the surface anchoring to the elastic distortion of the liquid crystal induced by the grooves of a surface. He showed that the surface anchoring energy is proportional to ${sin}^2\varphi$, with $\varphi$ being the angle between the director at infinity and the direction of the surface grooves. We argue that his assumption of negligibly small azimuthal distortion of the nematic is not valid. Proper treatment of the azimuthal distortion reveals that the Berreman’s model should yield a surface anchoring energy proportional to ${sin}^4\varphi$. This implies that surface grooves alone cannot contribute to the surface anchoring coefficient in the usual Rapini-Papoular sense.

We numerically investigate the interaction between two spherical particles in a nematic liquid crystal mediated by elastic distortions in the orientational order. We pay attention to the cases where two particles with equal radii $R_0$ impose rigid normal anchoring on their surfaces and carry a pointlike topological defect referred to as a hyperbolic hedgehog. To describe the geometry of our system, we use bispherical coordinates, which prove useful in the implementation of boundary conditions at the particle surfaces and at infinity. We adopt the Landau–de Gennes continuum theory in terms of a second-rank tensor order parameter $Q_{ij}$ for the description of the orientational order of a nematic liquid crystal. We also utilize an adaptive mesh refinement scheme that has proven to be an efficient way of dealing with topological defects whose core size is much smaller than the particle size. When the two “dipoles,” composed of a particle and a hyperbolic hedgehog, are in parallel directions, the two-particle interaction potential is attractive for large interparticle distances $D$ and proportional to $D^{-3}$ as expected from the form of the dipole-dipole interaction, until the well-defined potential minimum at $D\simeq 2.46R_0$ is reached. For the antiparallel configuration with no hedgehogs between the two particles, the interaction potential is repulsive and behaves as $D^{-2}$ for $D\lesssim 10R_0$, which is stronger than the dipole-dipole repulsion $\left(\sim D^{-3}\right)$ expected theoretically as an asymptotic behavior for large $D$.