Atomic-scale finite element modelling of mechanical behaviour of graphene nanoribbons

Abstract

Experimental characterization of Graphene NanoRibbons (GNRs) is still an expensive task and computational simulations are therefore seen as a practical option to study the properties and mechanical response of GNRs. Design of GNR elements in various nanotechnology devices can be approached through molecular dynamics simulations. This study demonstrates that the atomic-scale finite element method (AFEM) based on the second generation REBO potential is an efficient and accurate alternative to the molecular dynamics simulation of GNRs. Special atomic finite elements are proposed to model graphene edges. Extensive comparisons are presented with MD solutions to establish the accuracy of AFEM. It is also shown that the Tersoff potential is not accurate for GNR modeling. The study demonstrates the influence of chirality and size on design parameters such as tensile strength and stiffness. Graphene is stronger and stiffer in the zigzag direction compared to the armchair direction. Armchair GNRs shows a minor dependence of tensile strength and elastic modulus on size whereas in the case of zigzag GNRs both modulus and strength show a significant size dependency. The size-dependency trend noted in the present study is different from the previously reported MD solutions for GNRs but qualitatively agrees with experimental results. Based on the present study, AFEM can be considered a highly efficient computational tool for analysis and design of GNRs.

Appendix

1.1 Tersoff potential parameters

$$f_{c}^{T} \left( {r_{{ij}} } \right) = \left\{ {\begin{array}{*{20}l} {1,} \hfill & {r_{{ij}} < R - D} \hfill \\ {{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2} - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}\sin \left[ {{{{\pi \mathord{\left/ {\vphantom {\pi 2}} \right. \kern-\nulldelimiterspace} 2}(r_{{ij}} - R)} \mathord{\left/ {\vphantom {{{\pi \mathord{\left/ {\vphantom {\pi 2}} \right. \kern-\nulldelimiterspace} 2}(r_{{ij}} - R)} D}} \right. \kern-\nulldelimiterspace} D}} \right]} \hfill & {R - D < r_{{ij}} < R + D} \hfill \\ {0,} \hfill & {r_{{ij}} > R + D} \hfill \\ \end{array} } \right.$$

(15)

The parameters R and D are not systematically optimized but are chosen so as to include the first-neighbor shell only. For C–C bonds, Tersoff (1988) presented a set of suitable values of R and D that are given below. As the parameters R and D are chosen to include only the first-neighbor interaction, thus, the cut-off function, f ^T _c , goes from 1 to 0 within a cut-off distance R − D < r_ij < R + D.

$$\zeta_{ij}^{{n_{T} }} = \sum\limits_{k \ne i,j} {f_{c} \left( {r_{ik} } \right)g\left( {\theta_{ijk} } \right)} ;g\left( {\theta_{ijk} } \right) = 1 + \frac{{c^{2} }}{{d^{2} }} - \frac{{c^{2} }}{{\left[ {d^{2} + \left( {h - cos\theta_{ijk} } \right)^{2} } \right]}}$$

(16)

The bond angle θ_ijk is defined as shown in Fig. 11.

Fig. 11

Definition of angles in Tersoff potential

For carbon–carbon interactions these parameters are A = 1393.6 eV, B = 346.74 eV, λ¹ = 3.4879, λ² = 2.2119, R = 1.95 Å, D = 0.15 Å, $\beta^{{n_{T} }}$ = 1.5724 × 10⁻⁷, n_t = 0.72751, c = 3.8049 × 10⁴, d = 4.3484 and h = − 0.57058 (Tersoff 1988).

1.2 Second generation REBO potential parameters

$$f_{c}^{R} \left( {r_{{ij}} } \right) = \left\{ {\begin{array}{*{20}l} {1,} \hfill & {r_{{ij}} < R^{{\left( 1 \right)}} } \hfill \\ {\left[ {1 + \cos \left[ {\frac{{\pi \left( {r_{{ij}} - R^{{\left( 1 \right)}} } \right)}}{{\left( {R^{{\left( 2 \right)}} - R^{{\left( 1 \right)}} } \right)}}} \right]} \right]} \hfill & {R^{{\left( 1 \right)}} < r_{{ij}} < R^{{\left( 2 \right)}} } \hfill \\ {0,} \hfill & {R^{{\left( 2 \right)}} < r_{{ij}} } \hfill \\ \end{array} } \right.$$

(17)

The term B^R corresponds to the bond order term. It’s related with the number of neighbors and the angle, which it’s related with the forming and breaking of the bonds between of the atoms. The expression for B^Ris:

$$B^{R} = \frac{1}{2}\left[ {b_{ij}^{\sigma \pi } + b_{ji}^{\sigma \pi } } \right] + b_{ij}^{\pi }$$

(18)

$$b_{ij}^{\pi } = \varPi_{ij}^{rc} + b_{ij}^{dh}$$

(19)

The term b ^σπ _ij is composed by covalent bond interactions, and by the angular function g(cosθ_jik), which include the contribution from the second nearest neighbour according to the cosine of the angle of the bonds between atoms ik and ij.

$$b_{ij}^{\sigma \pi } = \left[ {1 + \sum\limits_{k \ne i,j} {f_{ik} \left( {r_{ik} } \right)g\left( {\cos \theta_{jik} } \right)e^{{\lambda_{ijk} }} + P_{ij} \left( {N_{i}^{C} ,N_{i}^{H} } \right)} } \right]^{{ - \frac{1}{2}}}$$

(20)

According to Brenner et al. (2002) the parameters P_ij and λ_ijk are taken to be zero for solid-state carbon. The following equations show the angular function in three regions of bond angle θ,

For 0° < θ < 109.476°

$$g\left( {\cos \theta_{jik} } \right) = G\left( {\cos \theta_{jik} } \right) + Q\left( {N_{i}^{t} } \right)\left[ {\gamma \left( {\cos \theta_{jik} } \right) - G\left( {\cos \theta_{jik} } \right)} \right]$$

(21)

$$\begin{aligned} G\left( {\cos \theta_{jik} } \right) & = 0.5024\cos^{5} \left( \theta \right) + 1.4297\cos^{4} \left( \theta \right) + 2.0313\cos^{3} \left( \theta \right) \\ & \quad + 2.2544\cos^{2} \left( \theta \right) + 1.4068\cos \left( \theta \right) + 0.3755 \\ \end{aligned}$$

(22)

$$\begin{aligned} \gamma \left( {\cos \theta_{jik} } \right) & = - 0.0401\cos^{5} \left( \theta \right) + 1.272\cos^{4} \left( \theta \right) - 0.5597\cos^{3} \left( \theta \right) \\ & \quad - 0.4331\cos^{2} \left( \theta \right) + 0.4889\cos \left( \theta \right) + 0.2719 \\ \end{aligned}$$

(23)

For 109.476° < θ < 120°

$$g\left( {\cos \theta_{jik} } \right) = G\left( {\cos \theta_{jik} } \right)$$

$$\begin{aligned} G\left( {\cos \theta_{jik} } \right) & = 36.2789\cos^{5} \left( \theta \right) + 71.8829\cos^{4} \left( \theta \right) + 57.5918\cos^{3} \left( \theta \right) \\ & \quad + 24.0970\cos^{2} \left( \theta \right) + 5.6774\cos \left( \theta \right) + 0.7073 \\ \end{aligned}$$

(24)

For 120° < θ < 180°

$$g\left( {\cos \theta_{jik} } \right) = G\left( {\cos \theta_{jik} } \right)$$

(25)

$$\begin{aligned} G\left( {\cos \theta_{jik} } \right) & = - 1.3424\cos^{5} \left( \theta \right) - 4.928\cos^{4} \left( \theta \right) - 6.83\cos^{3} \left( \theta \right) \\ & \quad - 4.346\cos^{2} \left( \theta \right) - 1.098\cos \left( \theta \right) + 0.0026 \\ \end{aligned}$$

(26)

The function Q(N ^t _i ) is given by

$$Q\left( {N_{i}^{t} } \right) = {\text{ }}\left\{ {\begin{array}{*{20}l} 1 \hfill & {N_{i}^{t} < 3.2} \hfill \\ {{{\left[ {1 + \cos \left( {2\pi \left( {N_{i}^{t} - 3.2} \right)} \right)} \right]} \mathord{\left/ {\vphantom {{\left[ {1 + \cos \left( {2\pi \left( {N_{i}^{t} - 3.2} \right)} \right)} \right]} 2}} \right. \kern-\nulldelimiterspace} 2}} \hfill & {3.2 < N_{i}^{t} < 3.7} \hfill \\ 0 \hfill & {N_{i}^{t} > 3.7} \hfill \\ \end{array} } \right.$$

(27)

The term N ^t _i is the sum of the carbon atoms number and the hydrogen atoms number, in this case N ^H _i is zero,

$$N_{i}^{t} = N_{i}^{C} + N_{i}^{H}$$

(28)

$$N_{i}^{C} = \sum\limits_{{k\left( { \ne i,j} \right)}}^{carbon atoms} {f_{ik} \left( {r_{ik} } \right)}$$

(29)

The term Π ^rc _ij is a three-dimensional cubic spline, which depends on the number of carbon atoms that are neighbors of atoms i and j and the nonconjugated bonds.

$$\varPi_{ij}^{rc} = F_{ij} \left( {N_{i}^{t} ,N_{j}^{t} ,N_{ij}^{conj} } \right)$$

(30)

$$N_{ij}^{conj} = 1 + \left[ {\sum\limits_{{k\left( { \ne i,j} \right)}}^{carbon atoms} {f_{ik} \left( {r_{ik} } \right)F\left( {x_{ik} } \right)} } \right]^{2} + \left[ {\sum\limits_{{l\left( { \ne i,j} \right)}}^{carbon atoms} {f_{jl} \left( {r_{jl} } \right)F\left( {x_{jl} } \right)} } \right]^{2}$$

(31)

$$F\left( {x_{{ik}} } \right) = \left\{ {\begin{array}{*{20}l} 1 \hfill & {x_{{ik}} < 2} \hfill \\ {{{\left[ {1 + \cos \left( {2\pi \left( {x_{{ik}} - 2} \right)} \right)} \right]} \mathord{\left/ {\vphantom {{\left[ {1 + \cos \left( {2\pi \left( {x_{{ik}} - 2} \right)} \right)} \right]} 2}} \right. \kern-\nulldelimiterspace} 2}} \hfill & {2 < x_{{ik}} < 3} \hfill \\ 0 \hfill & {x_{{ik}} > 3} \hfill \\ \end{array} } \right.$$

(32)

$$x_{ik} = N_{k}^{t} - f_{ik} \left( {r_{ik} } \right)$$

(33)

where k, l, and j are the neighbors of atoms.

The term b ^dh _ij is zero for graphene due to its planar configuration. All the parameters considered can be found in Stuart et al. (2000).