Lattice orientation and crack size effect on the mechanical properties of Graphene

Abstract

The effect of lattice orientation and crack length on the mechanical properties of Graphene are studied based on molecular dynamics simulations. Bond breaking and crack initiation in an initial edge crack model with 13 different crack lengths, in 10 different lattice orientations of Graphene are examined. In all the lattice orientations, three recurrent fracture patterns are reported. The influence of the lattice orientation and crack length on yield stress and yield strain of Graphene is also investigated. The arm-chair fracture pattern is observed to possess the lowest yield properties. A sudden decrease in yield stress and yield strain can be noticed for crack sizes <10 nm. However, for larger crack sizes, a linear decrease in yield stress is observed, whereas a constant yield strain of $\approx$0.05 is noticed. Therefore, the yield strain of $\approx$0.05 can be considered as a critical strain value below which Graphene does not show failure. This information can be utilized as a lower bound for the design of nano-devices for various strain sensor applications. Furthermore, the yield data will be useful while developing the Graphene coating on Silicon surface in order to enhance the mechanical and electrical characteristics of solar cells and to arrest the growth of micro-cracks in Silicon cells.

Appendix: Tersoff potential function

The mathematical expression of the bond energy based on the Tersoff potential is given in Eq. (1). The bond energy in the Tersoff framework is a combination of repulsive ($f_R$) energy function which is exponentially decaying and attractive ($f_A$) energy function that exponentially increases; with the increase of distance between the atoms. $f_c$ is a smooth spherical cutoff function around atom $\upalpha$ based upon the distance to the first nearest-neighbor shell. The function $f_c$ in Eq. (1) is defined as (Tersoff 1989):

$$\begin{aligned}&f_c(r_{\upalpha \upbeta })\nonumber \\&\quad = \left\{ \begin{array}{ll} 1 &{} \text {when}~r_{\upalpha \upbeta }< \mathcal {R}_{\upalpha \upbeta } \\ \frac{1}{2} + \frac{1}{2} \cos \left( \frac{\pi ( r_{\upalpha \upbeta }-\mathcal {R}_{\upalpha \upbeta })}{(\mathcal {S}_{\upalpha \upbeta }-\mathcal {R}_{\upalpha \upbeta })} \right) &{} \text {when}~\mathcal {R}_{\upalpha \upbeta }< r_{\upalpha \upbeta } < \mathcal {S}_{\upalpha \upbeta } \\ 0 &{} \text {when}~r_{\upalpha \upbeta } > \mathcal {S}_{\upalpha \upbeta } \end{array} \right. \nonumber \\ \end{aligned}$$

(3)

from Eq. (3), $f_c$ returns a value of 1 if $r_{\upalpha \upbeta }$ is less than $\mathcal {R}_{\upalpha \upbeta }$ and 0 when $r_{\upalpha \upbeta }$ greater than $\mathcal {S}_{\upalpha \upbeta }$. The values of the constants $\mathcal {R}_{\upalpha \upbeta }$ = $\sqrt{\mathcal {R}_\upalpha \mathcal {R}_\upbeta }$ and $\mathcal {S}_{\upalpha \upbeta }$ = $\sqrt{\mathcal {S}_\upalpha \mathcal {S}_\upbeta }$, (where $\upalpha$ and $\upbeta$ can be two different atom types, like Silicon and Carbon) are listed for Silicon and Carbon atoms in Tersoff (1989). The repulsive and attractive potential energies are tuned with the parameters $\mathcal {A}_{\upalpha \upbeta }$ and $\mathcal {B}_{\upalpha \upbeta }$, respectively. The repulsive potential energy is defined as (Tersoff 1989)

$$\begin{aligned} f_R(r_{\upalpha \upbeta }) = \mathcal {A}_{\upalpha \upbeta } e^{-\mathcal {D}_{\upalpha \upbeta } r_{\upalpha \upbeta } } \end{aligned}$$

(4)

and the attractive potential energy is estimated from Tersoff (1989)

$$\begin{aligned} f_A(r_{\upalpha \upbeta }) = -\mathcal {B}_{\upalpha \upbeta } e^{-\mathcal {E}_{\upalpha \upbeta } r_{\upalpha \upbeta } } \end{aligned}$$

(5)

where $\mathcal {A} = \sqrt{\mathcal {A}_\upalpha \mathcal {A}_\upbeta }$, $\mathcal {B} = \sqrt{\mathcal {B}_\upalpha \mathcal {B}_\upbeta }$, $\mathcal {D}_{\upalpha \upbeta } = (\mathcal {D}_\upalpha + \mathcal {D}_\upbeta )/2$ and $\mathcal {E}_{\upalpha \upbeta }=(\mathcal {E}_\upalpha + \mathcal {E}_\upbeta )/2$, in Eqs. (4) and (5) are constants. The variable $b_{\upalpha \upbeta }$ in Eq. (1) is designed to represent the bond strength of the potential. $b_{\upalpha \upbeta }$ is inversely proportional to the coordination number and is defined as (Tersoff 1989)

$$\begin{aligned} b_{\upalpha \upbeta } = \xi _{\upalpha \upbeta } \left( 1 + \mathcal {P}_\upalpha ^{q_\upalpha } {\upzeta }_{\upalpha \upbeta }^{q_\upalpha } \right) ^{-1/2q_\upalpha } \end{aligned}$$

(6)

where $\mathcal {P}$ and q are the constants. $\upzeta _{\upalpha \upbeta }$ provides a weighted measure of the number of other bonds ($\upgamma$) competing with the bond $\upalpha$-$\upbeta$, which is defined as (Tersoff 1989)

$$\begin{aligned} \upzeta _{\upalpha \upbeta } = \sum _{\upgamma \ne \upalpha ,\upbeta } f_c\left( r_{\upalpha \upgamma } \right) g\left( {\uptheta }_{\upalpha \upbeta \upgamma }\right) \end{aligned}$$

(7)

where $\upxi _{\upalpha \upbeta }$ is the strengthening or weakening factor of the hetero-polar bonds and $g\left( \uptheta _{\upalpha \upbeta \upgamma }\right)$ provides a measure of dependence on the bonding angle $\uptheta _{\upalpha \upbeta \upgamma }$, subtended at atom $\upalpha$ by atoms $\upbeta$ and $\upgamma$. The variable $g\left( \uptheta _{\upalpha \upbeta \upgamma }\right)$ is included to stabilize the atomic geometry under shear operations and to provide an effective coordination contribution based on the elastic energy of the current configuration, which is defined as (Tersoff 1989)

$$\begin{aligned} g\left( {\uptheta }_{\upalpha \upbeta \upgamma }\right) = 1 + \frac{c_\upalpha ^2}{d_\upalpha ^2} - \frac{c_\upalpha ^2}{d_\upalpha ^2 + \left( h_\upalpha -\cos \left( {\uptheta }_{\upalpha \upbeta \upgamma }\right) \right) ^2} \end{aligned}$$

(8)

where $c_\upalpha$, $d_\upalpha$ and $h_\upalpha$ are the constants.