Elastic moduli of single-walled carbon nanotubes and their ropes

Abstract

We report the results of a model calculation for studying the effects of pressure on a bunch of carbon nanotubes as well as individual nanotubes. Carbon nanotubes are extremely rigid in the axial direction. At pressures that we work with, the deformation in the axial direction comes out to be negligibly small in comparison to that in the transverse direction. We use the six-exponential and Brenner potentials to account for inter- and intratube interactions, respectively. Using second derivatives of potential, Young's modulus for single-walled armchair, zigzag and chiral tubes of different radii have been calculated. The values found by us in this simple model turn out to be in good agreement with other theoretical and experimental values. The strain dependence of Young's modulus has also been studied. We have also calculated the Poisson ratio and shear modulus of various single-walled nanotubes. We find that hydrostatic pressure is an ideal probe to study the radial deformations of the nanotubes. The nanotubes are considered to be flexible, identified by a flattening of cylinders under pressure through a parameter $f$. We calculate the total energy of the bunches having faceted tubes. The free energy thus calculated enables us to calculate phase changes at various pressures. From our calculations, we find the phase transformation to occur at about $5\phantom{\rule{0.3em}{0ex}}\mathrm{GPa}$. Young's modulus of nanoropes has also been calculated at various pressures and at the phase transition we obtain a discontinuity in the curve. A much simplified form of the Brenner potential has been suggested and results are compared with those obtained from original form of Brenner potential.