Young’s modulus prediction of hexagonal nanosheets and nanotubes based on dimensional analysis and atomistic simulations

Abstract

The present work investigates Young’s modulus of hexagonal nanosheets and nanotubes based on dimensional analysis and molecular mechanics. Using second derivatives of the strain energy density revealed from molecular dynamics simulations at 0 K (i.e., molecular mechanics) with harmonic potentials for various combinations of force constants, Young’s modulus have been computed for single-walled armchair and zigzag nanotubes of different radii. This parametric study with the aid of dimensional analysis allows explicitly establishing Young’s modulus of (n, n) armchair and (n, 0) zigzag nanotubes as functions of the force constants, bond length and chiral index n. Proposed formulae are applied to estimate Young’s modulus of graphene, boron nitride, silicon carbide sheets and their nanotubes. The accuracy of the proposed formulae are verified and discussed with available data in the literature for these three sheets and their nanotubes.

Appendices

Appendix 1

See Table 6

Table 6 The material’s parameters used in the computations

Appendix 2: Estimation of the force constant from Tersoff and Tersoff-like potentials

The energy E of a bond with the bond angle kept constant is developed in a Taylor series for small bond extension Δr = r − r ₀:

$$ E\left( r \right) = E\left( {r_{0} } \right) + \frac{{\partial E\left( {r_{0} } \right)}}{\partial r}\Delta r + \frac{1}{2}\frac{{\partial^{2} E\left( {r_{0} } \right)}}{{\partial r^{2} }}\left( {\Delta r} \right)^{2} $$

(19)

E(r ₀) = 0, since the system is in equilibrium in unstrained state at r = r ₀.

$\frac{{\partial E\left( {r_{0} } \right)}}{\partial r} = 0$, since E(r) have a minimum at r = r ₀.

Using the notation Δr = δl _ij , and regarding Eq. (2) and (19) yield

$$ K = \frac{{\partial^{2} E\left( {r_{0} } \right)}}{{\partial r^{2} }} . $$

(20)

Hence K can be evaluated when E(r) is known from Tersoff or Tersoff-like potentials.