Topological Metadefects: Tangles of Dislocations

Abstract

The concept of topological defects is universal. In condensed matter, it applies to disclinations, dislocations or vortices that are fingerprints of symmetry breaking during phase transitions. Using as a generic example the tangles of dislocations, we introduce the concept of topological metadefects i.e. defects made of defects. We show that in cholesterics, dextrogyre and levogyre primary tangles are generated through the D2 → C2 symmetry breaking from the coplanar dislocation pair called Lehmann cluster submitted to a high enough tensile strain. The primary tangles can be wound up individually into double-helices. They can also annihilate in pairs or associate into tangles of higher orders following simple algebraic rules.

Escape into the third dimension in cholesteric liquid crystals

“…To obtain a three-dimensional Q-tensor with the right topological structure we can simply add an extra row and column of zeros. Taking v to be the vector field derived from the complex function (18) with q = 0 then gives the desired splitting of a +1 singularity into a pair of +1/2 disclinations that are linked q ′ times. A simulation of this structure in a cylindrical capillary with normal anchoring is illustrated in figure 9 for q ′ = 1.…”

“…In cholesterics, escaped structures are λ-lines, defects in the material's optical axis. They are important for photonic properties and lie at the heart of structures observed in many recent experiments [10,11,[15][16][17][18]. In addition, many structures of biopolymers have been identified with cholesteric, or chiral, liquid crystal textures [19][20][21][22], and in this vein the properties of cholesterics may find applications in tissue mechanics and developmental biology [23,24].…”

“…Via a novel application of contact topology, we show that escape in a cholesteric may be frustrated by the requirement that the director maintain a uniform sense of handedness. Frustration leads to a wider array of metastable structures than can be observed in an achiral system, including stable strings of point defects in spherical droplets [32], shells [33] and cylindrical capillaries [28], as well as the knotted and twisted structures that appear in spherical droplets [32,34], at the core of heliknotons and Hopfions [10,11], and have been observed forming from a Lehmann cluster in recent experiments [16][17][18]. Our methodology offers a new perspective on these structures and provides analytic models for describing the processes that underlie their formation, as well as defect splitting processes more generally.…”