# Nanometric pitch in modulated structures of twist-bend nematic liquid crystals

## TL;DR

This paper investigates the nano-metric pitch of twist-bend nematic liquid crystals, revealing how elastic and boundary conditions influence their stable periodic structures, aligning theoretical predictions with experimental observations.

## Contribution

It introduces a theoretical model explaining the nano-metric pitch in twist-bend nematic phases considering elastic energy and boundary effects.

## Key findings

- The twist-bend nematic phase has a nano-metric heliconical structure.
- The wave vector depends on elastic parameters, anchoring energy, and sample thickness.
- The model aligns with experimental results on twist-bend nematic structures.

## Abstract

The extended Frank elastic energy density is used to investigate the existence of a stable periodically modulate structure that appears as a ground state exhibiting a twist-bend molecular arrangement. For an unbounded sample, we show that the twist-bend nematic phase $N_{TB}$ is characterized by a heliconical structure with a pitch in the nano-metric range, in agreement with experimental results. For a sample of finite thickness, we show that the wave vector of the stable periodic structure depends not only on the elastic parameters but also on the anchoring energy, easy axis direction, and the thickness of the sample.