| Abstract: | Geometric shape and topology of constituent particles can alter many colloidal properties such as Brownian motion, self-assembly, and phase behavior. Thus far, only single-component building blocks of colloids with connected surfaces have been studied, although topological colloids, with constituent particles shaped as freestanding knots and handlebodies of different genus, have been recently introduced. Here we develop a topological class of colloids shaped as multicomponent links. Using two-photon photopolymerization, we fabricate colloidal microparticle analogs of the classic examples of links studied in the field of topology, the Hopf and Solomon links, which we disperse in nematic fluids that possess orientational ordering of anisotropic rod-like molecules. The surfaces of these particles are treated to impose tangential or perpendicular boundary conditions for the alignment of liquid crystal molecules, so that they generate a host of topologically nontrivial field and defect structures in the dispersing nematic medium, resulting in an elastic coupling between the linked constituents. The interplay between the topologies of surfaces of linked colloids and the molecular alignment field of the nematic host reveals that linking of particle rings with perpendicular boundary conditions is commonly accompanied by linking of closed singular defect loops, laying the foundations for fabricating complex composite materials with interlinking-based structural organization.Interlocking closed loops in physical field lines (1–3), small molecules (4), DNA and synthetic polymer chains (5), and various vortices (6–8) can lead to new physical behavior, biological functionality, and material properties that largely stem from the underlying topology (1). For example, linking looped lines of the liquid crystal (LC) molecular alignment field n(r) (3), which describes spatial changes in local average orientations of constituent rod-like molecules (9), causes formation of topologically protected particle-like structures resembling mathematical Hopf and Seifert fibrations (1, 3, 10, 11). Similar field configurations with linked closed loops or linked torus knots of field lines are also predicted to exist in electromagnetic fields (2, 12, 13), in Bose-Einstein condensates (14, 15), and in magnetization of various ferromagnets (16–18), often defining novel types of physical behavior that arise from topological stabilization of such field configurations. However, the implications of topological linking on behavior of colloidal particles have not been considered thus far, neither experimentally nor theoretically, although many types of complex-shaped colloidal particles have been recently fabricated (8, 19, 20).In this work, we fabricate micrometer-sized colloidal particles with differently linked components shaped as closed solid polymeric rings with disconnected surfaces that, when dispersed in a fluid host like water or LC, undergo Brownian motion both relative to each other and as a whole. In a nematic LC host (8, 19–23), these particles induce a large variety of field configurations and looped and linked vortex lines that entangle the linked components of the colloidal particles, resulting in elastic coupling between them. Using a combination of 3D nonlinear optical imaging, videomicroscopy, and noncontact laser manipulation (8, 24, 25), we characterize the interplay between topologies of colloidal surfaces, n(r) configurations, and defects, as well as probe the strength of elastic coupling between the colloidal particle’s components. We supplement these experiments with a theoretical analysis based on numerical minimization of the bulk Landau-de Gennes and surface anchoring free energies (26–29) that yields configurations topologically homeomorphic to experimental counterparts. Finally, we discuss the prospects for interlocking-based assembly of composite materials and for experiment-driven fundamental explorations of topological interaction of physical links with nonpolar fields. |
