Geometry of Bend: Singular Lines and Defects in Twist-Bend Nematics

Abstract

We describe the geometry of bend distortions in liquid crystals and their fundamental degeneracies, which we call β lines; these represent a new class of linelike topological defect in twist-bend nematics. We present constructions for smecticlike textures containing screw and edge dislocations and also for vortexlike structures of double twist and Skyrmions. We analyze their local geometry and global structure, showing that their intersection with any surface is twice the Skyrmion number. Finally, we demonstrate how arbitrary knots and links can be created and describe them in terms of merons, giving a geometric perspective on the fractionalization of Skyrmions.

Anatomy of a Discovery: The Twist–Bend Nematic Phase

“…However, this constraint on experiments does not inhibit theoretical studies. For example, a model for bend distortions in the twist-bend phase based on the Frank free energy has been developed [59], which might add to our understanding of complex structures in thin films.…”

Helical microstructures in molluscan biomineralization are a biological example of close packed helices that may form from a colloidal liquid crystal precursor in a twist–bend nematic phase

“…While we agree that this question needs to be addressed, to avoid confusion, here we are using the more commonly used former term. Singularities and defects in twist-bend nematic liquid crystals have been investigated in a recent study [41]. There is now considerable excitement over potential technological uses of the twist-bend nematic phase in electro-optical devices.…”

Intrinsic Geometry and Director Reconstruction for Three-Dimensional Liquid Crystals

“…In fact there are several possible choices; we present our subsequent analysis only for the Frenet-Serret frame of the director field. This is the frame associated to the integral curves of the director and defined by the bend: the bend b = ∇ n n = −n × (∇ × n) is a vector that is everywhere perpendicular to the director, b • n = 0; we write b = κ e 1 , where κ is the (magnitude of the) curvature of the director integral curves and e 1 is their Frenet-Serret normal [8]. We then define e 2 = n × e 1 to complete the Frenet-Serret frame associated to the director field.…”

“…As quadratic functions of the director gradients, they do not appear in the Oseen-Frank energy, but would appear in a higher-order energy functional. Energy functionals that have been proposed for twist-bend nematic materials make use of an additional direction, the polarisation, whose gradients appear in the energy [8,20]; in such materials the gradients of polarisation would also serve to provide the required structure functions.…”

“…Indeed, Frank's original derivation [7] of the free energy of liquid crystals is founded upon the geometry of their elastic distortions; splay, twist, bend and saddle-splay. Geometric methods continue to furnish insights into liquid crystals phases: the geometry of bend distortions gives a description of defects and textures in the twist-bend nematic phase of bent-core molecules [8], while a geometric anaylsis of the packing of diabolos offers a similar insight into the formation of structures in the splay-twist phase [9,10], and a further recent application has been to the analysis of defect lines in active liquid crystals [11,12].…”