Prediction of Young’s modulus of hexagonal monolayer sheets based on molecular mechanics

Abstract

The present work investigates Young’s modulus of hexagonal monolayer sheets based on molecular mechanics. A repeating unit cell of the sheet has been chosen. Harmonic force field is adopted to model atomic interactions. The total energy of the unit cell is established as a function of the force constants and atomic displacements. A closed-form expression is formulated for Young’s modulus of the sheet by minimizing the total energy of the unit cell under uniaxial tension in equilibrium state. Molecular dynamics simulations were also carried out to consider Young’s modulus of graphene, boron nitride, silicon carbide, aluminum nitride, and boron antimonide monolayer sheets. The accuracy of the proposed formula is verified and discussed with results obtained by molecular dynamics simulations and available data in the literature for these 5 sheets.

Appendix

Relationships between variations of bond length and angle with displacements of atoms

The bond length l _ij can be determined by coordinates (x _i, y _i) of atoms i and (x _j, y _j) of atoms j as below:

$$ \left( {l_{ij} } \right)^{2} = \left( {x_{j} - x_{i} } \right)^{2} + \left( {y_{j} - y_{i} } \right)^{2} $$

(18)

Differentiating both sides of Eq. (18) gives:

$$ l_{ij} \delta l_{ij} = \left( {x_{j} - x_{i} } \right)\left( {\delta x_{j} - \delta x_{i} } \right) + \left( {y_{j} - y_{i} } \right)\left( {\delta y_{j} - \delta y_{i} } \right) $$

(19)

It is noted that δx _i  = u _i and δy _i  = v _i are displacements of atom i in the x and y-directions, respectively. Equation (19) is rewritten as below:

$$ \delta l_{ij} = \frac{1}{{l_{ij} }}\left[ {\left( {x_{j} - x_{i} } \right)\left( {u_{j} - u_{i} } \right) + \left( {y_{j} - y_{i} } \right)\left( {v_{j} - v_{i} } \right)} \right] $$

(20)

At the initial state, we have l _ij  = l ₀. Using Eq. (20) and regarding the conditions u ₄ = u ₅ = v ₄ = v ₅ = 0,δl ₄₁ andδl ₅₂ can be expressed by Eq. (7a) and (7b), respectively.

Regarding the geometric relations in Fig. 1c, we have

$$ AH = AB'\sin \left( {{{\alpha_{1} } \mathord{\left/ {\vphantom {{\alpha_{1} } {2 + \delta \alpha }}} \right. \kern-0pt} {2 + \delta \alpha }}} \right) $$

(21)

$$ AH = l_{0} \sin \left( {{{\alpha_{1} } \mathord{\left/ {\vphantom {{\alpha_{1} } 2}} \right. \kern-0pt} 2}} \right) + u_{1} $$

(22)

$$ AB' = l_{0} + \delta l_{41} $$

(23)

It is noted that sin (α ₁/2 + δα) ≈ sin (α ₁/2) + δα cos (α ₁/2)for very small δα. Substituting Eq. (22) and (23) into Eq. (21) yields

$$ \delta \alpha \left[ {l_{0} \cos \left( {{{\alpha_{1} } \mathord{\left/ {\vphantom {{\alpha_{1} } 2}} \right. \kern-0pt} 2}} \right)} \right] + \delta l_{41} \sin \left( {{{\alpha_{1} } \mathord{\left/ {\vphantom {{\alpha_{1} } 2}} \right. \kern-0pt} 2}} \right) = u_{1} . $$

(24)

Noting that α ₁ = 120⁰, Eq. (7c) can be deduced by combining Eq. (7a) and (24). By conducting a similar analysis as above, Eq. (7d) can be also obtained.