## Abstract

The close-form expressions of the Young's moduli and the fracture stresses of cyclicgraphene, graphyne and supergraphene along their armchair and zigzag directions are derived based on a molecular mechanics model. Checking against present finite-element calculations of their Young's moduli shows that the explicit solutions are reasonable. The obtained analytical solutions should be of great help for understanding the mechanical properties of the graphene-like materials.

## 1. Introduction

When graphene was exfoliated out by Novoselov *et al.* at first in 2004 [1], it has been widely considered as a very promising material for applications in nanoengineering due to their excellent mechanical and electronic properties [2–5]. Recently, the future application prospects of graphene-like materials have attracted much interest and stimulated numerous studies [6–9]. Although the large homogeneous sheets of other carbon allotropes, such as supergraphene, cyclicgraphene and graphyne have not been reported so far, great efforts have been made to synthesize and assemble a precursor and subunit of graphyne from a new chemical method [10,11]. In particular, their physical and chemical properties such as stability, thermal conductivity and electronic properties have been investigated systematically by density functional theory [12–15].

Despite their importance and the studies of available molecular dynamics (MD) simulations and continuum modelling [16], the link between molecular and continuum descriptions of their mechanical properties is not established yet.

In order to overcome the limitations of the atomistic simulations and continuum modelling, both of the ‘stick-spiral’ model (SSM) [17] and the beam model (using finite-element (FE) method) [18,19] based on the interatomic potentials are effectively developed to characterize the mechanical properties of the three types of carbon allotropes (cyclicgraphene, graphyne and supergraphene) in this paper. Furthermore, the close-form expressions of their Young's moduli and fracture stresses in armchair and zigzag sheets are derived based on the molecular mechanics model. The explicit solutions of the SSM are further validated from current FE calculations based on the beam model.

## 2. Young's moduli of three types of carbon allotropes in zigzag and armchair sheets based on the molecular mechanics model

The molecular structures for three carbon allotropes are shown in figure 1. The network of cyclicgraphene is composed of C3 and C12 circles. Carbon atoms in C3 circle are interconnected by carbon–carbon bonds and there is an olefinic bond (−C=C−) between C3 circle and C3 circle [12]. The network of graphyne can be formed by connecting each hexagon C6 by linear carbon chain which is formed by inserting acetylenic bond (−C≡C−) into carbon–carbon bond [12]. Supergraphene contains two kinds of chemical bonds, in which the bonds at the apex of the hexagon are more stable than the other bonds [12]. All of the three allotropes consist of two groups via the classification of Heimann *et al.* [20] based on valence orbital hybridization (*N*). The cyclicgraphene belong to the system with pure sp^{2} hybridization type (*N*=2), while the others belong to the mixed *sp*^{2}+*sp*^{1} hybridization type (1<*N*<2). The detailed types of the bonds are shown in figure 2.

In the SSM of the molecular mechanics, the total energy, *U*, at small strains can be expressed as the sum of energies [17]
*U*_{ρ} and *U*_{θ} are energies associated with bond stretching and angle variation, *K*_{i} and *C*_{j} are the force constants associated with bond stretching and angle variation, d*b*_{i} and d*θ*_{j} are the elongation of bond *i* and the variance of bond angle *j*, respectively.

In the SSM [17], the twisting moment *M*_{j} resulting from bond angle variation and the stretching force *F*_{i} resulting from bond elongation can be expressed as
*a* (−C−C−), *b* (−C=C−), *c* (aromatic bond), *d*(−C−C−), *e* (−C≡C−), *g* (analogical aromatic bond), *h* (−C=C−) and *i* (−C=C−) [16] and ten bond angles *α*, *β*, *θ*, *φ*, *ω*, *η*, *ϕ*, *γ*, *δ* and *ξ*. The longitudinal external tensile stress will result in bond elongation d*a*, d*b*, d*c*, d*d*, d*e*, d*g*, d*h* and d*i* and bond angle variances d*α*, d*β*, d*θ*, d*φ*, d*ω*, d*η*, d*ϕ*, d*γ*, d*δ* and d*ξ*. In the model, an elastic stick with an axial stiffness of *K* is used to address the relationship of external force versus bond length variation of carbon–carbon bond. The spiral spring with a stiffness *C* is employed to describe the twisting moment resulting from an angular distortion of bond angle. Three types of carbon allotropes subjected to a tensile force along the zigzag direction are shown in figure 3.

A unit cell of graphyne consists of five atomic structure models, which is shown on the left side of figure 3*b* and the other three atomic structure models is shown on the right side of figure 3*b*. As shown in figure 2, we denote the force *f* contributed by bond *b*, *c*, *e* and *g* along the external tensile direction so that the total force of cyclicgraphene, graphyne and supergraphene on the sheets is 2*f*, 6*f* and 2*f*, respectively. For cyclicgraphene, we define the force along the vertical bond is *f*_{1}, while the force along the bond *a* is *f*_{2}. According to the force balance, it can be derived that
*f* into two directions for three carbon allotropes, of which one is along the bond direction and the other perpendicular to the bond, force equilibrium of bond extension leads to
*b*, *c*, *e* and *i* into two halves, the twisting moment on the right half by the left half is *ε*_{1}, *ε*_{2} and *ε*_{3}. Using equations (2.3)–(2.8), together with the relation *θ*=*ϕ*=2*π*/3, *α* = *π*/3, *β*=5*π*/6, *φ*=*γ*=*η*=*ω*=2*π*/3 and *δ*=*ξ*=2*π*/3, the axial strains along the zigzag direction can be defined as
*L*_{1}, *L*_{2} and *L*_{3}. From geometrical relationships along the zigzag, it can be shown that
*L* of unit cell perpendicular to the tension direction is subjected to a tensile force *F*, Young's modulus *Y* can be defined as
*t* is the thickness of the sheet, and *F*=2*f*, 6*f* and 2*f* for cyclicgraphene, graphyne and supergraphene, respectively. The thicknesses *t* of cyclicgraphene, graphyne and supergraphene are 3.47, 3.46 and 3.64 Å, respectively [16].

Substituting equations (2.9) and (2.10) into equation (2.11), together with the thickness *t*_{1}, *t*_{2} and *t*_{3}, the Young's moduli *Y* _{1}, *Y* _{2} and *Y* _{3} of cyclicgraphene, graphyne and supergraphene can be derived as
*K*=742 nN nm^{−1}, *C*=1.42 nN nm^{−1} [17], *a*=1.59 Å, *b*=1.30 Å, *c*=1.40 Å, *d*=1.39 Å, *e*=1.33 Å, *g*=1.40 Å, *h*=1.34 Å, *i*=1.34 Å, *t*_{1}=3.47 Å, *t*_{2}=3.46 Åand *t*_{3}=3.64 Åare given [16], the Young's moduli of cyclicgraphene, graphyne and supergraphene are derived as 461.19, 448.53 and 84.81 GPa, respectively.

For three types of carbon allotropes subjected to a tensile force along the armchair direction, their molecular structures are shown in figure 4.

For armchair sheets, the similar calculations can be obtained

## 3. Young's moduli of three types of carbon allotropes along the zigzag and armchair directions based on finite-element method

The molecular mechanics-based space-frame structure model was proposed and was successfully used to calculate the equivalent macro mechanical properties of carbon nano-materials such as carbon nanotube [18,19,21,22], graphene [23] and Boron Nitride [24]. The FE beam structures of the three typical cyclicgraphene, graphyne and supergraphene sheets can be easily built from the coordinates of the MD structures [16]. In order to reduce the boundary effect on their Young's moduli, the three typical rectangular sheets with the sizes of 111.35×94.73 Å , 125.67×105.71 Å and 124.03×103.64 Å are modelled [7] (figure 5). The Young's moduli of the beams and their circular cross-section radii are determined as [23],
*K*_{r}=742 nN nm^{−1}, *K*_{θ}=1.42 nN nm [17] and *r* is the bond length (in §2, there are seven chemical bond lengths *a*, *b*, *c*, *d*, *e*, *g*, *h* and *i* in the three typical sheets). As the equivalent Young's modulus of graphene nanoribbon is almost independent on Poisson's ratio *v* of the beam [25], here we choose *v*=0.1 of the beam in all the following FE calculations.

All the present FE calculations are performed using the commercial ANSYS 14.0 package with 2-node BEAM 188 element. One end part of each FE model is fixed and the other end part is given by a displacement of 0.1 Åalong the armchair and zigzag directions (figure 6), respectively. The relationship of reaction force versus displacement at the end part along the armchair and zigzag directions of the three structures are captured in the simulations, as shown in figure 7.

Based on the obtained results of reaction force versus displacement, the equivalent Young's modulus could be calculated by equation (3.3) as follows:
*L* is the displacement at the end part of the sheet, *F* is the reaction force, *W* and *L* are the width and the length of the sheet, respectively.

The Young's moduli of cyclicgraphene, graphyne and supergraphene along their armchair and zigzag directions are derived based on the SSM and FE calculations in table 1. The FE results agree well with those of SSM for cyclicgraphene and graphyne, in which the maximum relative error is around 17%. However, the difference between the FE results (beam model) and SSM is more than 40% for supergraphene. In our previous work [25], the difference of the two methods (SSM and beam model) is clarified in detail, in which the SSM overestimates and the beam model underestimates the mechanical properties in narrow graphene nanoribbons under in-plane bending condition by comparison with MD simulations (the maximum difference can be up to 300%) [25]. In figures 3*c* and 4*c*, 20 bonds bear the in-plane bending even if the supergraphene is subjected to a uniaxial tension, which is the main reason to lead to the large difference. The Young's moduli of graphyne (170 N m^{−1} (armchair) and 224 N m^{−1} (zigzag)) were obtained from MD simulations in previous work [7], while the Young's moduli of graphyne from our FE calculations are 159.5 N m^{−1} (armchair) and 158 N m^{−1} (zigzag) (The Young's modulus is the product of the conventional Young's modulus with the thickness), respectively. We stressed that the previous MD results were obtained using REAXFF potential, in which the functions of the REAXFF potential are completely different with present harmonic potential (equation (2.1)). The comparison between previous MD results with present SSM or FE results is not suitable. Therefore, we only compare the FE results with our molecular mechanics model as the parameters of the FE model are obtained from the present harmonic potential.

## 4. Fracture stresses of three carbon allotropes

To obtain the relationship between the fracture stresses of different carbon allotropes and the tensile strength of different bonds, the bond strengths of carbon allotropes must be well studied. If a chemical bond is more unsaturated than other chemical bonds, it will break more easily. For cyclicgraphene, the olefinic bond is more fragile than the single bond (−C−C−) when they are subjected to a same force. Similarly, the acetylenic bond breaks more easily than others of graphyne, and the olefinic bond is the most fragile bond for supergraphene.

As shown in figure 1, each sheet can be formed by a number of unit cells. For example, supergraphene is constructed from some regular hexagonal supercell. Along the zigzag direction, the vertical carbon–carbon bond (bond *a*, figure 2) and slanting olefinic bond (bond *b*, figure 2) are most probably to break for cyclicgraphene, while slanting acetylenic bond and olefinic bond are most fragile for graphyne and supergraphene, respectively. For cyclicgraphene, we assume that the vertical carbon–carbon bond break earlier than slanting olefinic bond.

We define fracture force of the most fragile bond as *f*_{0}, and then the tensile strength of a bond can be defined as

When the number of the unit cell is equal to *n*, the total external force *F*_{1}, *F*_{2} and *F*_{3} of cyclicgraphene, graphyne and supergraphene can be derived as
*L*_{1}, *L*_{2} and *L*_{3} of cyclicgraphene, graphyne and supergraphene can be defined as
*σ*_{s1}, *σ*_{s2} and *σ*_{s3} along the zigzag direction as follows:

However, if the slanting olefinic bond is the first to break for cyclicgraphene, then the total external force *F*_{1} of cyclicgraphene should be expressed as
*σ*_{s1} of cyclicgraphene should be expressed as

## 5. Conclusion

In summary, the close-form expressions of the Young's moduli and the fracture stresses of cyclicgraphene, graphyne and supergraphene along their armchair and zigzag directions are derived based on a molecular mechanics model. Checking against present FE calculations of their Young's moduli shows that the explicit solutions are reasonable. The obtained analytical solutions should be of great help for understanding the mechanical properties of the graphene-like materials.

## Authors' contributions

B.Z. and J.S. performed all the calculations, interpreted the initial results and wrote the initial manuscript. P.Y. and J.Z. interpreted the final results and edited the final manuscript. All authors gave final approval for publication.

## Competing interests

We declare we have no competing interests.

## Funding

The support provided by the National Natural Science Foundation of China (grant nos. 11572140 and 11302084), the Programs of Innovation and Entrepreneurship of Jiangsu Province, the Fundamental Research Funds for the Central Universities (grant no. JUSRP11529, JG2015059), the Research Fund of State Key Laboratory of Mechanics and Control of Mechanical Structures (NUAA) (grant no. MCMS-0416G01), the Open Fund of Key Laboratory for Intelligent Nano Materials and Devices of the Ministry of Education (NUAA) (grant no. INMD-2015M01) is kindly acknowledged.

## Acknowledgements

We gratefully acknowledge support by ‘Thousand Youth Talents Plan’.

- Received September 5, 2015.
- Accepted January 14, 2016.

- © 2016 The Author(s)

Published by the Royal Society. All rights reserved.