Gnomonious Projections for Bend-Free Textures: Thoughts on the Splay-Twist Phase

Abstract

The Hopf fibration has inspired any number of geometric structures in physical systems, in particular in chiral liquid crystalline materials. Because the Hopf fibration lives on the three sphere, ${\mathbb{S}}^3$, some method of projection or distortion must be employed to realize textures in flat space. Here, we explore the geodesic-preserving gnomonic projection of the Hopf fibration, and show that this could be the basis for a new liquid crystalline texture with only splay and twist. We outline the structure and show that it is defined by the tangent vectors along the straight line rulings on a series of hyperboloids. The phase is defined by a lack of bend deformations in the texture, and is reminiscent of the splay-bend and twist-bend nematic phases. We show that domains of this phase may be stabilized through anchoring and saddle-splay.

arXiv comment6 pages, 6 figures

References in corpus (5)

• Nematic twist-bend phase with nanoscale modulation of molecular orientation
• Twist-bend heliconical chiral nematic liquid crystal phase of an achiral rigid bent-core mesogen
• Statistical mechanics of bend flexoelectricity and the twist-bend phase in bent-core liquid crystals
• Comparing skyrmions and merons in chiral liquid crystals and magnets
• Iterated Moire Maps and Braiding of Chiral Polymer Crystals

Cited by in corpus (4)

• On the molecular origins of the ferroelectric splay nematic phase
• Intrinsic Geometry and Director Reconstruction for Three-Dimensional Liquid Crystals
• The Thermodynamic Origins of Chiral Twist in Monolayer Assemblies of Hard Rod-like Colloids
• Good Fibrations: Packing Rules for Diabolic Domains