The homotopy theory of topological defects is a powerful tool for organizing and unifying many ideas across a broad range of physical systems. Recently, experimental progress has been made in controlling and measuring colloidal inclusions in liquid crystalline phases. The topological structure of these systems is quite rich but, at the same time, subtle. Motivated by experiment and the power of topological reasoning, we review and expound upon the classification of defects in uniaxial nematic liquid crystals. Particular attention is paid to the ambiguities that arise in these systems, which have no counterpart in the much-storied XY model or the Heisenberg ferromagnet.

We investigate the hydrodynamic interactions between micro-organisms swimming at low Reynolds number. By considering simple model swimmers, and combining analytic and numerical approaches, we investigate the time-averaged flow field around a swimmer. At short distances the swimmer behaves like a pump. At large distances the velocity field depends on whether the swimming stroke is invariant under a combined time-reversal and parity transformation. We then consider two swimmers and find that the interaction between them consists of two parts: a passive term, independent of the motion of the second swimmer, and an active term resulting from the simultaneous swimming action of both swimmers. The swimmer-swimmer interaction is a complicated function of their relative displacement, orientation, and phase, leading to motion that can be attractive, repulsive, or oscillatory.

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 k B T , 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.3D self-assembly | colloids | chirality | mesoscopic modeling T he major limitation of self-assembly approaches to building 3D photonic materials (1-6) is the difficulty of scaling the materials up from the nanoscale to the device dimensions. The approaches typically rely on the manipulation of particles by van der Waals or screened electrostatic forces (7) and the manipulation proceeds either under ultra high vacuum conditions or in water dispersions. The technological complexity lies in controlling these forces over macroscopic length scales. Technologies for 3D self-assembly are developed, based on electrically driven particle manipulation (8), sedimentation (9), growth on patterned surfaces (10), tuning of electrostatic interactions between oppositely charged particles (11, 12), and DNA-guided crystallization (13). In liquid crystal colloids, the self-assembly is demonstrated for two-dimensional structures of clusters, chains, and two-dimensional crystals (14-17). There have been recent advances in the synthesis and characterization of building blocks, but these have not been matched by similar progress in organizing the building blocks into assemblies and materials (18). Therefore, using cholesteric blue phases represents a unique and clear route to the 3D self-assembly of colloids. Recently, blue phases are receiving greater attention because materials with a broad stability range have been discovered (19,20). In addition application of a polymer-stabilized blue phase enabled development of a display w...