Review: knots and other new topological effects in liquid crystals and colloids

Abstract

Humankind has been obsessed with knots in religion, culture and daily life for millennia, while physicists like Gauss, Kelvin and Maxwell already involved them in models centuries ago. Nowadays, colloidal particles can be fabricated to have shapes of knots and links with arbitrary complexity. In liquid crystals, closed loops of singular vortex lines can be knotted by using colloidal particles and laser tweezers, as well as by confining nematic fluids into micrometer-sized droplets with complex topology. Knotted and linked colloidal particles induce knots and links of singular defects, which can be interlinked (or not) with colloidal particle knots, revealing the diversity of interactions between topologies of knotted fields and topologically nontrivial surfaces of colloidal objects. Even more diverse knotted structures emerge in nonsingular molecular alignment and magnetization fields in liquid crystals and colloidal ferromagnets. The topological solitons include hopfions, skyrmions, heliknotons, torons and other spatially localized continuous structures, which are classified based on homotopy theory, characterized by integer-valued topological invariants and often contain knotted or linked preimages, nonsingular regions of space corresponding to single points of the order parameter space. A zoo of topological solitons in liquid crystals, colloids and ferromagnets promises new breeds of information displays and a plethora of data storage, electro-optic and photonic applications. Their particle-like collective dynamics echoes coherent motions in active matter, ranging from crowds of people to schools of fish. This review discusses the state of the art in the field, as well as highlights recent developments and open questions in physics of knotted soft matter. We systematically overview knotted field configurations, the allowed transformations between them, their physical stability and how one can use one form of knotted fields to model, create and imprint other forms. The large variety of symmetries accessible to liquid crystals and colloids offer insights into stability, transformation and emergent dynamics of fully nonsingular and singular knotted fields of fundamental and applied importance. The common thread of this review is the ability to experimentally visualize these knots in real space. The review concludes with a discussion of how the studies of knots in liquid crystals and colloids can offer insights into topologically related structures in other branches of physics, with answers to many open questions, as well as how these experimentally observable knots hold a strong potential for providing new inspirations to the mathematical knot theory.

Transformation between elastic dipoles, quadrupoles, octupoles, and hexadecapoles driven by surfactant self-assembly in nematic emulsion

“…A spherical glycerol droplet has a genus equal to zero and Euler characteristic χ Euler = 2. Consistent with topological theorems ( 2 , 3 , 13 , 22 , 23 ), the hedgehog charge of the three-dimensional topological defects π 2 ( 2 /ℤ 2 ) = ℤ induced in the nematic LC bulk [the hyperbolic hedgehog point defect and the equivalent to it Saturn ring of π 1 ( 2 /ℤ 2 ) = ℤ 2 disclination] that compensates the hedgehog charge of surface boundary conditions at the drop surface is equal ±χ Euler /2 = ±1; the sign of the induced defect can be determined upon decorating the nonpolar director field with a unit vector field along one of the two antiparallel directions ( 23 , 40 ). The net winding number of π 1 ( 1 ) LC defects in the two-dimensional interfacial director field at the LC-glycerol interface is equal to χ Euler = 2, with the two boojums each having a winding of unity [ Figs.…”

Curvature Potential Unveiled Topological Defect Attractors

“…Furthermore, TDs could efficiently trap appropriate nanoparticles [31,32]. Note that TDs in liquid crystals could be relatively easily manipulated [33][34][35][36][37][38][39][40][41], which offers an indirect path to manipulate assemblies of trapped NPs. Finally, understanding the curvature-enabled stabilization mechanisms of TDs and their assemblies might shed light on still unresolved problems in fundamental physics [42].…”

Topological Defects in Nematic Liquid Crystals: Laboratory of Fundamental Physics

“…[ 52 ] Several consequent studies revealed that similar structures appear also in LCs. [ 53–58 ] Such structures are referred to as “textures” and are also characterized by conserved topological numbers. However, they are not singular in the respective physical field.…”