A comparison of different methods of Young’s modulus determination for single-wall carbon nanotubes (SWCNT) using molecular dynamics (MD) simulations

The computed values of Young’s modulus (Y) of single-wall carbon nanotubes given by four common methods based on (i) the determination of stress for a fixed value of strain, (ii) the determination of strain energy for a fixed value of strain, (iii) the longitudinal vibrations, and (iv) the transverse vibrations, and a new method (v) based on the determination of strain for a fixed value of stress have been compared to check the consistency of different methods. The computed values of Y are found to be in agreement with each other with the exception that results of the transverse vibration method differ from those given by other methods when the aspect ratio, namely, the ratio of length to the radius of the tube, is small; the results of the transverse vibration method for the tubes of small diameter are also found to differ from those given by other methods when the commonly used value of thickness of the tube, 3.4 Å, is assumed. The solutions of these problems are discussed in terms of an appropriate consideration for the value of thickness and diameter dependent end-correction in the length of the tube. Effect of defects in the form of vacancies, van der Waals (VDW) interactions, chirality, and diameter of the carbon nanotubes on Y has also been investigated. Y is found to be sensitive to the number of vacancies. Y is found to decrease by ∼1% when VDW interactions between carbon atoms are ignored. Y is also found to be lower for an armchair tube compared to a zigzag tube of the same diameter. As regards the dependence of Y on diameter, we found that as the diameter increases from ∼7 Å to ∼25 Å, Young’s modulus drops by 4% and 8%, respectively, for armchair and zigzag tubes. These results are discussed and compared with other experimental and computed results reported in the literature.