Size-dependent bending modulus of nanotubes induced by the imperfect boundary conditions.

Jin Zhang

doi:10.1038/srep38974

Abstract

The size-dependent bending modulus of nanotubes, which was widely observed in most existing three-point bending experiments [e.g., J. Phys. Chem. B 117, 4618–4625 (2013)], has been tacitly assumed to originate from the shear effect. In this paper, taking boron nitride nanotubes as an example, we directly measured the shear effect by molecular dynamics (MD) simulations and found that the shear effect is not the major factor responsible for the observed size-dependent bending modulus of nanotubes. To further explain the size-dependence phenomenon, we abandoned the assumption of perfect boundary conditions (BCs) utilized in the aforementioned experiments and studied the influence of the BCs on the bending modulus of nanotubes based on MD simulations. The results show that the imperfect BCs also make the bending modulus of nanotubes size-dependent. Moreover, the size-dependence phenomenon induced by the imperfect BCs is much more significant than that induced by the shear effect, which suggests that the imperfect BC is a possible physical origin that leads to the strong size-dependence of the bending modulus found in the aforementioned experiments. To capture the physics behind the MD simulation results, a beam model with the general BCs is proposed and found to fit the experimental data very well.

Highlights

The bending modulus of nanotubes obtained from the three-point bending (TPB) test is expected to be equivalent to the Young’s modulus measured by the direct tensile test
This discrepancy makes us believe that a direct measurement of the transverse shear deformation is desired and some new theories may need to be formulated to reveal the physics behind the size-dependence of the bending modulus observed in the TPB tests
Based on molecular dynamics (MD) simulations and the continuum mechanics theories we will reveal the correlation between the imperfect BCs and the size-dependent bending modulus of nanotubes reported in existing experiments[10,15]

Summary

Introduction

The bending modulus of nanotubes obtained from the TPB test is expected to be equivalent to the Young’s modulus measured by the direct tensile test. The shear modulus obtained by fitting the shear theory to the experimental results was usually found to be two orders of magnitude lower than the results measured via the torsion tests[18] and the results calculated by the theoretical simulations[19] This discrepancy makes us believe that a direct measurement of the transverse shear deformation is desired and some new theories may need to be formulated to reveal the physics behind the size-dependence of the bending modulus observed in the TPB tests. The Euler-Bernoulli beam model together with the non-ideal BCs was proposed to account for the physics of the observed phenomena This modified beam model is found to fit the size-dependent bending modulus of nanotubes measured in the TPB experiments[10,15] very well

Methods

Results

Conclusion