Splay-bend elastic inequalities shape tactoids, toroids, umbilics, and conic section walls in paraelectric, twist-bend, and ferroelectric nematics

Abstract

Elastic constants of splay $K_{11}$, twist $K_{22}$, and bend $K_{33}$ of nematic liquid crystals are often assumed to be equal to each other in order to simplify the theoretical description of complex director fields. Here we present examples of how the disparity of $K_{11}$ and $K_{33}$ produces effects that cannot be described in a one-constant approximation. In a lyotropic chromonic liquid crystal, nematic droplets coexisting with the isotropic phase change their shape from a simply-connected tactoid to a topologically distinct toroid as a result of temperature or concentration variation. The transformation is caused by the increase of the splay-to-bend ratio $K_{11}/K_{33}$. A phase transition from a conventional nematic to a twist-bend nematic implies that the ratio $K_{11}/K_{33}$ changes from very large to very small. As a result, the defects caused by an externally applied electric field change the deformation mode of optic axis from bend to splay. In the paraelectric-ferroelectric nematic transition, one finds an inverse situation: $K_{11}/K_{33}$ changes from small to large, which shapes the domain walls in the spontaneous electric polarization field as conic sections. The polarization field tends to be solenoidal, or divergence-free, a behavior complementary to irrotational curl-free director textures of a smectic A.

KEYWORDS:

• Frank elastic constants
• splay and bend deformations
• nematic droplets
• domain walls in ferroelectric nematic
• twist-bend nematic, lyotropic chromonic liquid crystal

Disclosure statement

No potential conflict of interest was reported by the author(s).

Author contributions

O.D.L. wrote the paper.

Data availability statement

The data that support the findings of this study are available within the manuscript.

Additional information

Funding

This work was supported by National Science Foundation: [grant number DMS-2106675, ECCS-2122399, DMR-2215191].