A one-dimensional continuous model for carbon nanotubes

TL;DR

This paper derives a one-dimensional elastic model for carbon nanotubes from a 2D continuum theory, enabling analysis of large deformations and predicting stable helical and super helical shapes consistent with experiments.

Contribution

It introduces a strictly reduced 1D curvature elastic model for CNTs, aligning with Kirchhoff rod theory and extending previous models to include large deformation features.

Findings

Helical shapes have lower energy than straight shapes.

The model predicts a chiral angle difference of about π/6 between equilibrium states.

Predicted super helical shapes match experimental observations.

Abstract

The continuous two-dimensional (2D) elastic model for single-walled carbon nanotubes (SWNTs) provided by Tu and Ou-Yang in [Phys. Rev. B \textbf{65}, 235411 (2003)] is reduced to a one-dimensional (1D) curvature elastic model strictly. This model is in accordance with the isotropic Kirchhoff elastic rod theory. Neglecting the in-plane strain energy in this model, it is suitable to investigate the nature features of carbon nanotubes (CNTs) with large deformations and can reduce to the string model in [Phys. Rev. Lett. \textbf{76}, 4055 (1997)] when the deformation is small enough. For straight chiral shapes, this general model indicates that the difference of the chiral angle between two equilibrium states is about $\pi/6$, which is consistent with the lattice model. It also reveals that the helical shape has lower energy for per atom than the straight shape has in the same condition. By solving the corresponding equilibrium shape equations, the helical tube solution is in good agreement with the experimental result, and super helical shapes are obtained and we hope they can be found in future experiments.