Miha Ravnik

Reconfigurable knots and links in chiral nematic colloids.

Colloidal particles inside a liquid crystalline solvent can be manipulated to create knots of arbitrary shape and complexity. Tying knots and linking microscopic loops of polymers, macromolecules, or defect lines in complex materials is a challenging task for material scientists. We demonstrate the knotting of microscopic topological defect lines in chiral nematic liquid-crystal colloids into knots and links of arbitrary complexity by using laser tweezers as a micromanipulation tool. All knots and links with up to six crossings, including the Hopf link, the Star of David, and the Borromean rings, are demonstrated, stabilizing colloidal particles into an unusual soft matter. The knots in chiral nematic colloids are classified by the quantized self-linking number, a direct measure of the geometric, or Berry’s, phase. Forming arbitrary microscopic knots and links in chiral nematic colloids is a demonstration of how relevant the topology can be for the material engineering of soft matter.

Three-dimensional colloidal crystals in liquid crystalline blue phases

Applications for photonic crystals and metamaterials put stringent requirements on the characteristics of advanced optical materials, demanding tunability, high Q factors, applicability in visible range, and large-scale self-assembly. Exploiting the interplay between structural and optical properties, colloidal lattices embedded in liquid crystals (LCs) are promising candidates for such materials. Recently, stable two-dimensional colloidal configurations were demonstrated in nematic LCs. However, the question as to whether stable 3D colloidal structures can exist in an LC had remained unanswered. We show, by means of computer modeling, that colloidal particles can self-assemble into stable, 3D, periodic structures in blue phase LCs. The assembly is based on blue phases providing a 3D template of trapping sites for colloidal particles. The particle configuration is determined by the orientational order of the LC molecules: Specifically, face-centered cubic colloidal crystals form in type-I blue phases, whereas body-centered crystals form in type-II blue phases. For typical particle diameters (approximately 100 nm) the effective binding energy can reach up to a few 100 kBT, implying robustness against mechanical stress and temperature fluctuations. Moreover, the colloidal particles substantially increase the thermal stability range of the blue phases, for a factor of two and more. The LC-supported colloidal structure is one or two orders of magnitude stronger bound than, e.g., water-based colloidal crystals.

Assembly and control of 3D nematic dipolar colloidal crystals.

Topology has long been considered as an abstract mathematical discipline with little connection to material science. Here we demonstrate that control over spatial and temporal positioning of topological defects allows for the design and assembly of three-dimensional nematic colloidal crystals, giving some unexpected material properties, such as giant electrostriction and collective electro-rotation. Using laser tweezers, we have assembled three-dimensional colloidal crystals made up of 4 μm microspheres in a bulk nematic liquid crystal, implementing a step-by-step protocol, dictated by the orientation of point defects. The three-dimensional colloidal crystals have tetragonal symmetry with antiparallel topological dipoles and exhibit giant electrostriction, shrinking by 25-30% at 0.37 V μm(-1). An external electric field induces a reversible and controllable electro-rotation of the crystal as a whole, with the angle of rotation being ~30° at 0.14 V μm(-1), when using liquid crystal with negative dielectric anisotropy. This demonstrates a new class of electrically highly responsive soft materials.

Landau–de Gennes modelling of nematic liquid crystal colloids

Phenomenological Landau–de Gennes modelling based on the free energy of nematic liquid crystal colloids is reviewed. Nematic phase, gradient of order, and surface anchoring contributions to the total free energy are used. The numerical finite difference relaxation technique is explained as an efficient tool for the minimisation of the free energy. Effects of the mesh and mesh allocation are discussed. Various conceptually different colloidal structures are calculated to show the universality of the model. Single particles, dipolar–quadrupolar dimers, entangled dimers, dimers bound by escaped hyperbolic rings, and hierarchically patterned Saturn-ring colloidal superstructures are presented.

Mutually tangled colloidal knots and induced defect loops in nematic fields

Colloidal dispersions in liquid crystals can serve as a soft-matter toolkit for the self-assembly of composite materials with pre-engineered properties and structures that are highly dependent on particle-induced topological defects. Here, we demonstrate that bulk and surface defects in nematic fluids can be patterned by tuning the topology of colloidal particles dispersed in them. In particular, by taking advantage of two-photon photopolymerization techniques to make knot-shaped microparticles, we show that the interplay of the topologies of the knotted particles, the nematic field and the induced defects leads to knotted, linked and other topologically non-trivial field configurations. These structures match theoretical predictions made on the basis of the minimization of the elastic free energy and satisfy topological constraints. Our approach may find uses in self-assembled topological superstructures of knotted particles linked by nematic fields, in topological scaffolds supporting the decoration of defect networks with nanoparticles, and in modelling other physical systems exhibiting topologically analogous phenomena.

Mesoscopic modelling of colloids in chiral nematics

We present numerical modelling of colloidal particles in chiral nematics with cubic symmetry (blue phases) within the framework of the Landau-de Gennes free energy. The interaction potential of a single, nano-sized colloidal particle with a -1/2 disclination line is calculated as a generic trapping mechanism for particles within the cholesteric blue phases. The interaction potential is shown to be highly anisotropic and have threefold rotational symmetry. We discuss the equilibration of the colloidal texture with respect to particle positions and the unit cell size of the blue phase. We also describe how preservation of the liquid crystal volume and the number of particles allows blue phase colloidal structures with different unit cell sizes and configurations to be compared numerically.

Geometrical frustration of chiral ordering in cholesteric droplets

Frustration of chiral ordering is explored in cholesteric liquid crystal droplets with planar degenerate anchoring using numerical modeling. Droplets of variable pitches are studied, demonstrating the role of a gradually increasing cholesteric pitch and the corresponding equilibrium structures. All previously known structures are identified but with notable differences. The structures presented with director fields are complemented with a detailed description of the defect regions. The characteristic half-diameter +2 disclination from previous studies is found to be in fact a double-helix of two λ+1 disclination lines, whereas the full-diameter +1 disclination is composed of an alternating series of τ−1/2 and λ+1/2 disclination rings. Finally, two new meta-stable cholesteric structures -Lyre and Yeti- are found, which are characterised by complex compositions of cholesteric disclinations.

Nematic colloids entangled by topological defects

Assembly of colloidal particles in nematic liquid crystals is governed by the symmetry of building blocks and type of defects in the liquid crystalline orientation. Particles in a nematic act as nucleation sites for topological defect structures that are homotopic to point defects. The tendency for a minimal deformation free energy and topological constraints limit possible defect configurations to extended and localized defect loops. Here we report on a recently discovered colloidal binding, where particles are entangled by disclination loops. Nematic braids formed by such disclinations stabilize multi-particle objects and entrap particles in a complex manner. Observed binding potentials are highly anisotropic showing string-like behavior and can be of an order of magnitude stronger compared to non-entangled colloids. Controlling the assembly based on entangled disclination lines one can build multi-particle structures with potentially useful features (shapes, periodic structure, chirality, etc.) for photonic and plasmonic applications.

Shape-tuning the colloidal assemblies in nematic liquid crystals

Colloidal platelets are natural building blocks for the shape-controlled assembly of two-dimensional periodic lattices and can form versatile optical elements. Using three-dimensional (3D) numerical modelling, we demonstrate the self-assembly of triangular, square and pentagonal sub-micrometer sized platelets in a thin layer of nematic liquid crystal. Torques acting on individual platelets are calculated, showing that platelets with quadrupolar symmetry (squares, hexagons, etc) are, orientationally, more strongly bound than platelets with dipolar symmetry (triangles, pentagons) which is important for switching applications. Inter-platelet potentials are shown to depend in a complex way on the orientations of the platelets, exhibiting easy and hard reorientation axes and multiple minima. Linear chains of elastic dipoles form into two-dimensional periodic lattices via interesting rotational and translational shifts, to minimize the distortions in the surrounding nematic medium.

Confined active nematic flow in cylindrical capillaries.

We use numerical modeling to study the flow patterns of an active nematic confined in a cylindrical capillary, considering both planar and homeotropic boundary conditions. We find that active flow emerges not only along the capillary axis but also within the plane of the capillary, where radial vortices are formed. If topological defects are imposed by the boundary conditions, they act as local pumps driving the flow. At higher activity, we demonstrate escape of the active defects and flow into the third dimension, indicating the importance of dimensionality in active materials. We argue that measuring the magnitude of the active flow as a function of the capillary radius allows determination of a value for the activity coefficient.