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Mechanical properties of graphyne and its family – A molecular dynamics investigation
Computational Materials Science 61 (2012) 83–88
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Mechanical properties of graphyne and its family – A molecular dynamics investigation Yulin Yang a, Xinmiao Xu b,⇑ a b
Mathematics and Physics Department, Xiamen University of Technology, Xiamen, Fujian 361024, China School of Mechanical Engineering and Automation, Fuzhou University, Fuzhou, Fujian 350001, China
a r t i c l e
i n f o
Article history: Received 28 January 2012 Received in revised form 16 March 2012 Accepted 27 March 2012 Available online 2 May 2012 Keywords: Mechanical properties Molecular dynamics simulation Graphyne Tensile deformation Fracture
a b s t r a c t In this work a series of carbon allotropes related to graphene, called graphyne, graphdiyne, gaphene-3, graphene-4 and graphene-5 are constructed by connecting two adjacent hexagonal rings with different number of acetylenic linkages. Mechanical properties of these monolayer networks are investigated through acting tensile loads on the architectures and molecular dynamics simulations are performed to calculate the fracture strains and associated ultimate stresses. In the armchair loading case, the fracture strain remains nearly unchanged whereas the ultimate strength degrades gradually with longer acetylenic chains. In the zigzag loading situation, the ultimate strength remains nearly the same whereas the fracture strain improves by a little amount with longer acetylenic chains. Furthermore, Young’s moduli of all the investigated architectures are computed to analyze the material stiffness at the near equilibrium regime. The obtained results show that these structures are mechanically stable with high strength and stiffness. The unique mechanical property variations of graphyne family against armchair and zigzag loads suggest flexible designations towards functional use of this novel material, especially in the direction-dependent nanomechanical applications. Ó 2012 Elsevier B.V. All rights reserved.
1. Introduction Due to their intriguing electronic, thermal, and mechanical properties, carbon-based nanostructures have been a subject of intensive research in recent years [1–10]. Carbon’s various hybridization states (sp, sp2, and sp3) can lead to numerous carbon allotropes; such as diamond (sp3), graphite (sp2), fullerene (sp2), carbon nanotubes (sp2), and graphene (sp2). Most of these materials are associated with superlative mechanical strength [11–13]. In the nanometer scale, the vast knowledge gained so far has established graphene as one of the strongest materials ever tested [14]. Ultrahigh elastic stiffness (1.0 TPa from both MD simulations [15] and DFT predictions [16]) and tensile strength (100 GPa for zigzag orientation and 90 GPa for armchair orientation from MD simulations [17], 130 ± 10 GPa from experimental measurement [18]) have been reported within this monolayer structure. The fracture strains for armchair and zigzag loads with the AIREBO potential were computed to be about 0.2 and 0.13, respectively [19]. Beyond the already known and well studied sp2-bond networks, the intercalation of sp, sp2 and sp3 carbon atoms have motivated significant research interests. The intercalation compounds can present diverse attractive properties as opposed to their pristine ones [20–22]. In ⇑ Corresponding author. Tel.: +86 0591 22866792; fax: +86 0591 83465373. E-mail address: [email protected] (X. Xu). 0927-0256/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.commatsci.2012.03.052
the all-carbon chemistry, it has been suggested that acetylenic linkages (AC„CA) can be inserted into suitable bonds in a molecular to expand the system. Exotic carbon allotropes, graphyne and its family, can then be obtained. Predicted by Baughman et al. [23], graphyne can be considered as a structure where one-third of the CAC bonds in the graphene are replaced with one acetylene unit and has been found to be structurally stable and synthetically approachable. The presence of acetylene groups is found to reduce the binding energy and modulate optical and electronic properties in a versatile style [24–26]. Thereafter, many follow-up theoretical works examined the properties of graphyne and its related structures, such as graphyne nanotubes [27], a-graphyne-like carbon nanotubes [28], graphyne sheet and its BN analog [29]. The second member in graphyne family, graphdiyne, has two acetylenic (diacetylenic) linkages between carbon hexagons. This material was estimated to have low formation energy [24] and recently, large area of graphdiyne film was generated via a cross-coupling reaction using hexaethynylbenzene [30]. The electronic, elastic and transport properties of graphdiyne nanoribbons are all found to be altered by the carbomerization process [31]. Although large regular sheets of graphyne have not been achieved, a significantly large molecular segment of graphyne, dehydrobenzoannulenes (DBAs), has been assembled [32]. With advances on synthetic tools, one can expect that graphyne can also be obtained experimentally in the near future.
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Y. Yang, X. Xu / Computational Materials Science 61 (2012) 83–88
Currently, there are limited theoretical works examining the structural and electronic properties of graphyne and graphdiyne and even few reports on their mechanical properties. Very recently, the mechanical stability and strength of graphyne were investigated using MD simulations with ReaxFF potential and high stiffness (224.0 N/m) was observed. The obtained tensile strength and fracture strain are 48.2 GPa and 0.082 respectively for armchair load and 107.5 GPa and 0.132 respectively for zigzag load [33]. But how the general graphyne family would perform under mechanical tests is not well established yet. In this work we apply molecular dynamics (MDs) simulations to address the mechanical property modulations of such two dimensional frameworks. We find that the addition of acetylene groups have disparate impacts on the mechanical performance in the armchair direction as opposed to zigzag direction. New avenues are therefore opened for advanced mechanical applications with graphyne-based functional materials. 2. Models and methodology 2.1. Models As illustrated in Fig. 1, monolayer graphyne and its family are constructed by replacing one-third of carbon–carbon bonds in two nearest-neighbor hexagonal rings in graphene sheet by acetylenic linkages AC„CA, where single acetylenic group case corresponds to graphyne situation, di-acetylenic groups case corresponds to graphdiyne circumstance, and n acetylenic group case corresponds to graphyne-n situation. In the present work we investigate five graphyne-like sheets with n varying from one to five, i.e., graphyne, graphdiyne, gaphene-3, graphene-4 and graphene-5. We found that the general trends can be well displayed with n up to five. The sizes of the investigated graphyne systems are all around 10 nm 10 nm. Edge effects can be well diminished through proper analysis for samples at this size scale. A prototype full atomistic model test specimen for mechanical characterization consisting of 10 nm by 10 nm graphyne sheet is illustrated in Fig. 1a. Similar to the edge definition
in graphene, the edges of graphyne and its family are designated armchair or zigzag based on the orientation of the crystalline lattice. For the future discussion on the bond length variations, we differentiate these bonds into eight types (from type A to type E0 ), as depicted in Fig. 1c. There exist three kinds of constituent bonds in the acetylene linked aromatic structures of graphyne family, namely, aromatic (type A and B), single (type C, C0 , E, and E0 ) and triple bonds (type D and D0 ). The single bonds at both extremes of the acetylene chains are defined as C and C0 , and the single bonds inside the chains are associated with type E and E0 . 2.2. Computational method The MD simulations are performed using the massively parallelized modeling code LAMMPS software package [34], and the atomic interactions are described by the AIREBO potential [35]. In the scheme of mechanical tests, the interactions between the nearest neighbors must be cut off before the second-nearest neighbor distance through appropriate switching functions. To describe the bond breakage and re-hybridization properly the cut-off of the REBO part of the potential was set as 2.0 [36]. Periodic conditions were applied in x- and y-axis. All the MD simulations are subject to an NPT ensemble, carried out at a background temperature of 300 K (Nóse-Hoover thermo bath coupling [37]). The VelocityVerlet time stepping method is adopted and the integration time step is set as 0.1 fs. The structural optimization was performed using the Polak–Ribière version of the conjugated gradient algorithm [38]. After the equilibrium states are achieved, tensile tests are loaded under deformation-control method [39] in the armchair (oriented along the y-axis) and zigzag (oriented along the x-axis) directions until complete failure of each sample. The engineered strain rate is 0.001/ps and strain increment is applied every 0.1 ps. Uniaxial mechanical tests are implemented to derive a simplified set of parameters to mechanically characterize graphyne sheets and its family. Three associated parameters, namely, Young’s modulus, tensile strength, and fracture strain, are calculated. The aforementioned samples are not of the same structure, thus the van der Waals
Fig. 1. (a) Snapshot of a 10 nm 10 nm graphdiyne sheet, along with the definitions of edges. (b) Schematic illustration of the units of graphyne family, constructed by repeating acetylene groups between aromatic benzene rings. Different number of acetylene linkages gives rise to various members in the family. a corresponds to the included angle of two acetylene chains. (c) Definitions of bond types. The bond types in acetylene linkages can be classified as triple and single carbon bonds. The single bonds adjacent to aromatic rings are defined as C (armchair orientation) and C’ (zigzag orientation), and the single bonds inside the acetylene chains are designated as E and E0 .
Y. Yang, X. Xu / Computational Materials Science 61 (2012) 83–88
interlayer interaction distance cannot be the same. The ambiguity for the thickness of monolayer structure poses difficulty in the definition of stress and moduli in force per unit area (N/m2 or Pa). Therefore, we report the strength and stiffness in force per unit length (N/m). In-plane stiffness, Young’s modulus Y, can be defined as,
Yi ¼
1 @ 2 U A0 @ e2i
ð1Þ ei ¼0
where i is the direction (x- or y-) of the applied strain, U is the total strain energy, e is the linear strain, and A0 is the equilibrium rectangular area of the monolayer sheet. The stress tensors for each individual carbon atom are first calculated [40] and then averaged over all the atoms on the sheet to obtain the macroscopic stress. Noise is reduced by averaging the results over latter half of the relaxation period. Herein, the stress–strain evolutions are also presented in force per unit length (N/m). Fracture strain eF and maximum strength rc can be derived from the obtained force–strain curves, where the critical point can be determined from the spontaneous drop in the strain energy. Our simulation method is validated by calculating rc and eF of a 100 Å 100 Å pristine graphene sheet stressed along the armchair direction. We obtained a fracture strain of 0.136, which is consistent with the previous report of 0.137 [39]. The obtained Young’s modulus is 1.07 ± 0.02 TPa, which also agree well with the nanoindentation measurements of 1.0 TPa [41] The obtained results confirmed the validity of the simulation methods used here. 3. Results and discussions 3.1. Bond length analysis The bond lengths in the equilibrated and minimized structures are first investigated. Under a background temperature of 300 K, all the samples are fully relaxed to its energy minimum states and the direct bond length measurements are taken. All the bonds within the interior sections are averaged, resulting in near-constant values of 1.405 Å, 1.398 Å, 1.340 Å, and 1.240 Å for aromatic, single (C and C0 ), single (E and E0 ), and triple bonds respectively (Table 1). The variations of bond lengths among different structures in the family are seem to be small and random. It is noteworthy that C or C0 bonds are obviously shorter than the typical single bond of around 1.53 Å, whereas the D or D0 ’ bonds are some what a little longer than the typical triple bond of around 1.20 Å. The average of the bonds suggests that the carbomerized structures of graphyne family are still conjugated systems. The obtained rules of bond length alteration are consistent with the various reports in the open literature [25,31,33]. 3.2. Stress–strain evolutions We now turn to analyze the stress–strain relations of graphyne and its family to see how the general mechanical performance they Table 1 Equilibrium bond lengths (Å). Background temperature is 300 K in our works. Bond types
A, B
C, C0
D, D0
Graphyne Ref. [32] Ref. [24] Graphdiyne Ref. [30] Graphyne-3 Graphyne-4 Graphyne-5
1.405 1.49 1.419 1.405 1.440 1.406 1.405 1.406
1.398 1.48 1.401 1.396 1.400 1.398 1.398 1.396
1.240 1.19 1.221 1.240 1.239 1.240 1.239 1.239
E, E0
Remarks
1.340 1.341 1.340 1.339 1.338
MD, AIREBO potential MD, ReaxFF potential DFT, LSDA MD, AIREBO potential DFT, GGA-PBE MD, AIREBO potential MD, AIREBO potential MD, AIREBO potential
85
can be and how the elastic properties can be altered with increasing number of acetylenic linkages. Constructed from a hexagonal supercell, graphyne and its family structures have an in-plane hexagonal symmetry and exhibit characteristic zigzag edges. For the armchair edges, the triple-bond linkages give them a linear topology of repeating acetylenic and aromatic units, which is different from offset aromatic units of graphene. Nevertheless, they still show some vernacular of the graphene and are designated as armchair edges. Prismatic bar tensile deformations are axially loaded on the armchair- and zigzag-oriented monolayer sheets and the obtained stress–strain relations are shown in Fig. 2, along with data from pristine graphene case for comparison. In the armchair load situation, the stress–strain curves present somewhat parabolic increasing trend, this should be attributed to the an-harmonic terms in the carbon–carbon interaction potential. In the zigzag load case, the stress increases slower in the small strain regime and faster in the large strain regime. One observes an obvious structural deformation at the beginning of the loading test. The included angle of two acetylene chains, defined as angle a (Fig. 1b), reduces dramatically as the sheet is stretched. This helps to release the external force and lower the total stress within the sheet. In the latter half of the loading test, bonds begin to elongate and the total stress enlarges quickly. The calculated fracture strain eF and corresponding maximum stress rc are listed in Table 2. It is interesting to observe that in the armchair orientation the elastic region for all graphyne family members extends to about 0.11, whereas the associated rc degrades gradually from 14.437 N/m to 4.950 N/m for acetylenic groups increases from one to five (from graphyne to graphyne-5). On the contrary, in the zigzag orientation the ultimate strength for the five conformations remain nearly the same while the fracture strain is improved by a small amount with more acetylenic groups inserted in the hexagonal grid. The sheets are found to expand elastically up to 0.177–0.224 and then rupture, with corresponding maximum stress all around 21 N/m. To make more apt assessment with the widely studied graphene sheet we also calculated eF and rc for a graphene membrane with the same size as that of graphyne sheet. The maximum stress at the fracture point for the graphene sheet is 30.69 N/m for armchair loads and 36.58 N/m for zigzag loads. Therefore one can see that the tensile strengths of graphyne family are less than half of those of graphene sheet. The lower critical strain and strength for the armchair case is expected because in the armchair deformation circumstance part of the bonds are subjected directly to the load at the beginning of the elongation process, whereas in the zigzag situation the external loads can be dissipated through both bond elongation and bond angle variation. Finally, one more interesting thing should be noticed. Graphyne-3, graphyne-4 and graphyne-5 architectures exhibit a mechanically strengthening-like behavior in their stress–strain evolutions. With a close examination of the spatial bond distributions we found that this phenomenon originates from the spatial bond re-hybridization after the first bond breakage, which suggests that more ductile material can be attained from these kinds of conformations. The variation of fracture strain and ultimate stress of graphyne family presents disparate rules for armchair and zigzag load tests. To give more insight into this phenomenon, the bonds in the samples are differentiated into eight groups (Fig. 1c) and the average bond elongation for each groups are analyzed. The bond elongation evolutions of graphdiyne under tensile tests are presented in Fig. 3a and b. One can see that in the armchair loading case, the main bond elongation is experienced by bond C, next come bond D and E. This indicates that the major stretching takes place on the acetylene chain. From the atomic stress distribution as illustrated in Fig. 3c one can further see that the major stress accumulates on the ligature atoms between aromatic rings and acetylene chains. Bond length variation and atomic stress distribution for
Y. Yang, X. Xu / Computational Materials Science 61 (2012) 83–88
Graphene Graphyne Graphdiyne Graphyne-3 Graphyne-4 Graphyne-5
(a) Armchair Stress (N/m)
30
20
10
0 0.00
0.05
0.10
0.15
0.20
(b) Zigzag 30
Stress (N/m)
86
20
10
0 0.00
0.25
0.05
Strain
0.10
0.15
0.20
0.25
Strain
Fig. 2. Stress–strain relations of the graphyne family under tensile tests, results of graphene membrane with the same size are also presented for comparison. (a) Armchair direction. (b) Zigzag direction. Legends for both panels are the same and are presented only once in (a).
be more and more in-chain atoms, whereas the number of ligature atoms is the same. rc represents the ultimate stress averaged among all the atoms. Since the major stress accumulates on the ligature atoms, the averaged total stress will be reduced as the chain length increases. Therefore the ultimate strength degrades from graphyne to graphyne-5. The situation in the zigzag case is not the same. From Fig. 3d one can see that the atomic stress distribution is rather uniform on both aromatic rings and acetylene chains subjected to the external strain. Therefore the averaged total stress cannot vary with the chain length or the architectures. Except for bond B, all the other types of bonds take part in the elongation procedure and the maximum stretching ratios are similar for all the architectures. From this point of view the fracture strain might be the same for all the structures. However, a detailed analyses of the bond angle variation reveals that architectures with longer acetylene chains encompass higher degree of angle deformation, i.e., the reduction of angle a is larger in graphyne-3 to graphyne5 and smaller in graphyne and graphdiyne. Smaller angle a gives rise to not only released stress which results in slower stress augmentation, but also wider strain range the system can under go. Thus the larger fracture strains as observed in graphyne-3 to graphyne-5 can be understood. Furthermore, it is expected that
Table 2 Fracture strain eF and corresponding ultimate strength rc for the various systems investigated. System ID
Graphene Graphyne Graphdiyne Graphyne-3 Graphyne-4 Graphyne-5
Armchair direction
Zigzag direction
eF
rc (N/m)
eF
rc (N/m)
0.136 0.112 0.109 0.109 0.108 0.108
30.643 14.437 9.538 7.313 5.910 4.950
0.207 0.177 0.208 0.223 0.224 0.224
36.580 20.471 20.835 20.922 20.928 20.927
the other structures are also analyzed and the maximum bond stretching ratio for all the conformations are listed in Table 3, suggesting that the general trends are the same for the whole graphyne family. These results can well explain the unique fracture strain and ultimate strength variations as observed in Fig. 2. The length of acetylene chain enlarges from graphyne to graphyne-5. Since the major stretching takes place on the acetylene chain, the fracture strain, defined as a relative elongation ratio, cannot be influenced by the chain length, thus should be the same for all the architectures. Contrary, as the chain length increases there will
1.56
(a) Armchair
1.52
Bondlength (Angs.)
Bondlength (Angs.)
1.56
1.48 1.44 1.40
Bond-A Bond-B Bond-C Bond-D Bond-E
1.36 1.32 1.28 0
2
4
6
8
10
12
1.48 1.44 1.40
Bond-A Bond-B Bond-C' Bond-D' Bond-E'
1.36 1.32 1.28 0
14
5
10
15
20
25
Strain (%)
Strain (%)
(c)
(b) Zigzag
1.52
40GPa 32 24 16 8 0
(d)
α 90GPa 72 54 36 18 0
Fig. 3. Bond elongation evolutions of graphdiyne under tensile tests. (a) Armchair direction. (b) Zigzag direction. And atomic stress distributions before sheet rupturing. (c) Armchair direction. (d) Zigzag direction.
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Y. Yang, X. Xu / Computational Materials Science 61 (2012) 83–88 Table 3 Bond stretching ratio, defined as (ll0)/l0, where l is the average bond length at the fracture point and l0 the average bond length at the equilibrium state. Orientation
Armchair
Zigzag
Bond types
A (%)
B (%)
C (%)
D (%)
E (%)
A (%)
B (%)
C0 (%)
D0 (%)
E0 (%)
Graphyne Graphdiyne Graphyne-3 Graphyne-4 Graphyne-5
2.86 2.84 2.59 2.65 3.04
4.62 3.63 3.91 4.16 3.49
11.34 11.31 11.38 9.43 10.63
8.41 8.66 8.71 8.02 8.93
8.44 8.89 7.93 8.71
8.42 10.13 9.15 9.25 8.48
0.56 0.72 0.73 0.43 0.92
11.30 10.52 9.61 9.51 8.94
8.52 7.53 7.25 6.82 6.51
7.83 6.97 6.60 6.24
calculated results showed that these architectures are mechanically stable with high strength and stiffness. With the increase of acetylene groups, the obtained ultimate stress is found to depress gradually upon armchair loads whereas remains nearly unchanged upon zigzag loads. Fracture strain is observed to enlarge by a little amount with more inserted acetylene groups under zigzag deformations whereas remains the same under armchair deformations. The disparate influence of acetylene groups on the mechanical performance can be explained by the unique bond elongation and atomic stress distribution among the various conformations. Moreover, upon armchair deformations an interesting mechanically strengthening-like behavior is observed in architectures with more than two acetylene groups between two nearest-neighbor benzene rings. The versatile mechanical performance of graphyne and its family upon armchair and zigzag deformations can facilitate the design of advanced composites with intriguing direction-dependent properties and propel novel nanomechanical applications in carbon-based nanostructures.
Fig. 4. Young’s modulus of graphyne family, along with the data of pristine graphene for comparison.
covalently bonded graphyne could mechanically behaves as an atomistic spring-system [33]. There exist three types of bonds in all architectures, single, double and triple carbon–carbon bonds. Single bonds are not as stiff as the other two. Among all the single bonds, the bonds connecting the benzene rings and the acetylene groups are inferior than the others. In both armchair and zigzag tests, the first bond breakage is always observed to occur among bond C, i.e., in the linkage of benzene ring and the acetylene group. 3.3. In-plane stiffness Having examined the fracture process of graphyne family, we now proceed further to investigate their elastic properties at the near equilibrium regime. From the uniaxial tensile loads, the Young’s moduli for all the aforementioned samples are computed through Eq. (1) and the results are summarized in Fig. 4. The variation of Young’s modulus Y with respect to sheet orientation is tiny. Yzig is lower than Yarm by 7–10 N/m for all the conformations. In both armchair and zigzag circumstances, Y represents a general degrading trend with the increase of acetylene groups. This is reasonable because the in-plane honeycomb structure formed by hexagonal rings is the stiffest structure ever tested. Inserting acetylene chains into the nearest-neighbor benzene rings could only deteriorate the stability of the original conformation, and the longer the chain is, the higher the stiffness is. Compared to the 320 N/m in-plane stiffness for graphene sheet, the Young’s modulus for graphyne family is about 10–50% of that of graphene. 4. Conclusions In this paper the mechanical properties of graphyne and its family are investigated through molecular dynamics simulations. The
Acknowledgments This work was financially supported by NSF of China (No. 11074039) and the 2009 Project for Scientific and Technical Development of Xiamen (No. 3502Z20099007) and Fujian Education Bureau (No. JB11022). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28]
K.M. Liew, X.Q. He, C.H. Wong, Acta Mater. 52 (2004) 2521. K.M. Liew, Y. Sun, Phys. Rev. B 77 (2008) 205437. Q.X. Pei, Y.W. Zhang, V.B. Shenoy, Carbon 48 (2010) 898–904. P.S. Raux, P.M. Reis, J.W.M. Bush, C. Clanet, Phys. Rev. Lett. 105 (2010) 044301. Q. Zheng, Y. Geng, S. Wang, Z. Li, J.-K. Kim, Carbon 48 (2010) 4315–4322. C.G. Liu, Z.N. Yu, D. Neff, A. Zhamu, B.Z. Jiang, Nano Lett. 10 (2010) 4863. F. Diederich, M. Kivala, Adv. Mater. 22 (2010) 803. A. Hirsch, Nature Mater. 9 (2010) 868. L. Xu, Y. Zheng, J. Zheng, Phys. Rev. B 82 (2010) 195102. N. Wei, L. Xu, H.Q. Wang, J.C. Zheng, Nanotechnology 22 (2011) 105705. H.W. Kroto, J.R. Heath, S.C. O’Brien, R.F. Curl, R.E. Smalley, Nature 318 (1985) 162. S. Iijima, Nature 354 (1991) 56. K.S. Novoselov, A.K. Geim, S.V. Morozov, D. Jiang, Y. Zhang, S.V. Dubonos, et al., Science 306 (2004) 666. I.W. Frank, D.M. Tanenbaum, A.M. van der Zande, P.L. McEuen, J. Vacuum Sci. Tech. B 25 (2007) 2558. N. Wei, Z. Fan, Y. Zheng, L. Xu, H. Wang, J. Zheng, Nanoscale 4 (2012) 785. F. Liu, P. Ming, J. Li, Phys. Rev. B 76 (2007) 064120. Y.P. Zheng, N. Wei, Z.Y. Fan, L.Q. Xu, Z.G. Huang, Nanotechnology 22 (2011) 405701. C. Lee, X. Wei, J.W. Kysar, J. Hone, Science 321 (2008) 385. L. Xu, N. Wei, Y. Zheng, Z. Fan, H. Wang, J. Zheng, J. Mater. Chem. 22 (2012) 1435. R.M. Minyaev, V.E. Avakyan, Dokl. Chem. 434 (2) (2010) 253. X.L. Sheng, Q.B. Yan, F. Ye, Q.R. Zheng, G.Su. Phys, Rev. Lett. 106 (2011) 155703. A.K. Nair, S.W. Cranford, M.J. Buehler, Europhys. Lett. 95 (2011) 16002. R.H. Baughman, H. Eckhardt, M. Kertesz, J. Chem. Phys. 87 (1987) 6687. N. Narita, S. Nagai, S. Suzuki, K. Nakao, Phys. Rev. B 58 (11) (1998) 009. N. Narita, S. Nagai, S. Suzuki, K. Nakao, Phys. Rev. B 62 (11) (2000) 146. N. Narita, S. Nagai, S. Suzuki, Phys. Rev. B 64 (2001) 245408. V.R. Coluci, S.F. Braga, S.B. Legoas, D.S. Galvão, R.H. Baughman, Nanotechnology 15 (2004) S142. A.N. Enyashin, Y.N. Makurin, A.L. Ivanovskii, Carbon 42 (2004) 2081.
88
Y. Yang, X. Xu / Computational Materials Science 61 (2012) 83–88
[29] J. Zhou, K. Lv, Q. Wang, X.S. Chen, Q. Sun, P. Jena, J. Chem. Phys. 134 (2011) 174701. [30] G.X. Li, Y.L. Li, H.B. Liu, Y.B. Guo, Y.J. Li, D.B. Zhu, Chem. Commun. 46 (2010) 3256. [31] H.C. Bai, Y. Zhu, W.Y. Qiao, Y.H. Huang, RSC Adv. 1 (2011) 768–775. [32] M.M. Haley, Pure Appl. Chem. 80 (3) (2008) 519. [33] S.W. Cranford, M.J. Buehler, Carbon 49 (2011) 4111–4121. [34] S. Plimpton, J. Comput. Phys. 117 (1995) 1–19. [35] W.B. Donald, O. A Shenderova, J.A. Harrison, S.J. Stuart, B. Ni, S.B. Sinnott, J. Phys.: Condens. Matter. 14 (2002) 783.
[36] O.A. Shenderova, D.W. Brenner, A. Omeltchenko, X. Su, L.H. Yang, Phys. Rev. B 61 (2000) 3877. [37] W.G. Hoover, Phys. Rev. A 31 (1985) 1695. [38] E. Polak (Ed.), Optimization: Algorithms and Consistent Approximations, Springer, New York, 1997. [39] Y.P. Zheng, L.Q. Xu, Z.F. Fan, N. Wei, Y. Lu, Z.G. Huang, Curr. Nanosci. 8 (2011) 89. [40] Q.X. Pei, Y.W. Zhang, V.B. Shenoy, Nanotechnology 21 (2010) 115709. [41] Y. Geng, J. Li, S.J. Wang, J.K. Kim, J. Nanosci. Nanotechnol. 8 (2008) 6238.
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Mechanical properties of graphyne and its family – A molecular dynamics investigation Yulin Yang a, Xinmiao Xu b,⇑ a b
Mathematics and Physics Department, Xiamen University of Technology, Xiamen, Fujian 361024, China School of Mechanical Engineering and Automation, Fuzhou University, Fuzhou, Fujian 350001, China
a r t i c l e
i n f o
Article history: Received 28 January 2012 Received in revised form 16 March 2012 Accepted 27 March 2012 Available online 2 May 2012 Keywords: Mechanical properties Molecular dynamics simulation Graphyne Tensile deformation Fracture
a b s t r a c t In this work a series of carbon allotropes related to graphene, called graphyne, graphdiyne, gaphene-3, graphene-4 and graphene-5 are constructed by connecting two adjacent hexagonal rings with different number of acetylenic linkages. Mechanical properties of these monolayer networks are investigated through acting tensile loads on the architectures and molecular dynamics simulations are performed to calculate the fracture strains and associated ultimate stresses. In the armchair loading case, the fracture strain remains nearly unchanged whereas the ultimate strength degrades gradually with longer acetylenic chains. In the zigzag loading situation, the ultimate strength remains nearly the same whereas the fracture strain improves by a little amount with longer acetylenic chains. Furthermore, Young’s moduli of all the investigated architectures are computed to analyze the material stiffness at the near equilibrium regime. The obtained results show that these structures are mechanically stable with high strength and stiffness. The unique mechanical property variations of graphyne family against armchair and zigzag loads suggest flexible designations towards functional use of this novel material, especially in the direction-dependent nanomechanical applications. Ó 2012 Elsevier B.V. All rights reserved.
1. Introduction Due to their intriguing electronic, thermal, and mechanical properties, carbon-based nanostructures have been a subject of intensive research in recent years [1–10]. Carbon’s various hybridization states (sp, sp2, and sp3) can lead to numerous carbon allotropes; such as diamond (sp3), graphite (sp2), fullerene (sp2), carbon nanotubes (sp2), and graphene (sp2). Most of these materials are associated with superlative mechanical strength [11–13]. In the nanometer scale, the vast knowledge gained so far has established graphene as one of the strongest materials ever tested [14]. Ultrahigh elastic stiffness (1.0 TPa from both MD simulations [15] and DFT predictions [16]) and tensile strength (100 GPa for zigzag orientation and 90 GPa for armchair orientation from MD simulations [17], 130 ± 10 GPa from experimental measurement [18]) have been reported within this monolayer structure. The fracture strains for armchair and zigzag loads with the AIREBO potential were computed to be about 0.2 and 0.13, respectively [19]. Beyond the already known and well studied sp2-bond networks, the intercalation of sp, sp2 and sp3 carbon atoms have motivated significant research interests. The intercalation compounds can present diverse attractive properties as opposed to their pristine ones [20–22]. In ⇑ Corresponding author. Tel.: +86 0591 22866792; fax: +86 0591 83465373. E-mail address: [email protected] (X. Xu). 0927-0256/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.commatsci.2012.03.052
the all-carbon chemistry, it has been suggested that acetylenic linkages (AC„CA) can be inserted into suitable bonds in a molecular to expand the system. Exotic carbon allotropes, graphyne and its family, can then be obtained. Predicted by Baughman et al. [23], graphyne can be considered as a structure where one-third of the CAC bonds in the graphene are replaced with one acetylene unit and has been found to be structurally stable and synthetically approachable. The presence of acetylene groups is found to reduce the binding energy and modulate optical and electronic properties in a versatile style [24–26]. Thereafter, many follow-up theoretical works examined the properties of graphyne and its related structures, such as graphyne nanotubes [27], a-graphyne-like carbon nanotubes [28], graphyne sheet and its BN analog [29]. The second member in graphyne family, graphdiyne, has two acetylenic (diacetylenic) linkages between carbon hexagons. This material was estimated to have low formation energy [24] and recently, large area of graphdiyne film was generated via a cross-coupling reaction using hexaethynylbenzene [30]. The electronic, elastic and transport properties of graphdiyne nanoribbons are all found to be altered by the carbomerization process [31]. Although large regular sheets of graphyne have not been achieved, a significantly large molecular segment of graphyne, dehydrobenzoannulenes (DBAs), has been assembled [32]. With advances on synthetic tools, one can expect that graphyne can also be obtained experimentally in the near future.
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Currently, there are limited theoretical works examining the structural and electronic properties of graphyne and graphdiyne and even few reports on their mechanical properties. Very recently, the mechanical stability and strength of graphyne were investigated using MD simulations with ReaxFF potential and high stiffness (224.0 N/m) was observed. The obtained tensile strength and fracture strain are 48.2 GPa and 0.082 respectively for armchair load and 107.5 GPa and 0.132 respectively for zigzag load [33]. But how the general graphyne family would perform under mechanical tests is not well established yet. In this work we apply molecular dynamics (MDs) simulations to address the mechanical property modulations of such two dimensional frameworks. We find that the addition of acetylene groups have disparate impacts on the mechanical performance in the armchair direction as opposed to zigzag direction. New avenues are therefore opened for advanced mechanical applications with graphyne-based functional materials. 2. Models and methodology 2.1. Models As illustrated in Fig. 1, monolayer graphyne and its family are constructed by replacing one-third of carbon–carbon bonds in two nearest-neighbor hexagonal rings in graphene sheet by acetylenic linkages AC„CA, where single acetylenic group case corresponds to graphyne situation, di-acetylenic groups case corresponds to graphdiyne circumstance, and n acetylenic group case corresponds to graphyne-n situation. In the present work we investigate five graphyne-like sheets with n varying from one to five, i.e., graphyne, graphdiyne, gaphene-3, graphene-4 and graphene-5. We found that the general trends can be well displayed with n up to five. The sizes of the investigated graphyne systems are all around 10 nm 10 nm. Edge effects can be well diminished through proper analysis for samples at this size scale. A prototype full atomistic model test specimen for mechanical characterization consisting of 10 nm by 10 nm graphyne sheet is illustrated in Fig. 1a. Similar to the edge definition
in graphene, the edges of graphyne and its family are designated armchair or zigzag based on the orientation of the crystalline lattice. For the future discussion on the bond length variations, we differentiate these bonds into eight types (from type A to type E0 ), as depicted in Fig. 1c. There exist three kinds of constituent bonds in the acetylene linked aromatic structures of graphyne family, namely, aromatic (type A and B), single (type C, C0 , E, and E0 ) and triple bonds (type D and D0 ). The single bonds at both extremes of the acetylene chains are defined as C and C0 , and the single bonds inside the chains are associated with type E and E0 . 2.2. Computational method The MD simulations are performed using the massively parallelized modeling code LAMMPS software package [34], and the atomic interactions are described by the AIREBO potential [35]. In the scheme of mechanical tests, the interactions between the nearest neighbors must be cut off before the second-nearest neighbor distance through appropriate switching functions. To describe the bond breakage and re-hybridization properly the cut-off of the REBO part of the potential was set as 2.0 [36]. Periodic conditions were applied in x- and y-axis. All the MD simulations are subject to an NPT ensemble, carried out at a background temperature of 300 K (Nóse-Hoover thermo bath coupling [37]). The VelocityVerlet time stepping method is adopted and the integration time step is set as 0.1 fs. The structural optimization was performed using the Polak–Ribière version of the conjugated gradient algorithm [38]. After the equilibrium states are achieved, tensile tests are loaded under deformation-control method [39] in the armchair (oriented along the y-axis) and zigzag (oriented along the x-axis) directions until complete failure of each sample. The engineered strain rate is 0.001/ps and strain increment is applied every 0.1 ps. Uniaxial mechanical tests are implemented to derive a simplified set of parameters to mechanically characterize graphyne sheets and its family. Three associated parameters, namely, Young’s modulus, tensile strength, and fracture strain, are calculated. The aforementioned samples are not of the same structure, thus the van der Waals
Fig. 1. (a) Snapshot of a 10 nm 10 nm graphdiyne sheet, along with the definitions of edges. (b) Schematic illustration of the units of graphyne family, constructed by repeating acetylene groups between aromatic benzene rings. Different number of acetylene linkages gives rise to various members in the family. a corresponds to the included angle of two acetylene chains. (c) Definitions of bond types. The bond types in acetylene linkages can be classified as triple and single carbon bonds. The single bonds adjacent to aromatic rings are defined as C (armchair orientation) and C’ (zigzag orientation), and the single bonds inside the acetylene chains are designated as E and E0 .
Y. Yang, X. Xu / Computational Materials Science 61 (2012) 83–88
interlayer interaction distance cannot be the same. The ambiguity for the thickness of monolayer structure poses difficulty in the definition of stress and moduli in force per unit area (N/m2 or Pa). Therefore, we report the strength and stiffness in force per unit length (N/m). In-plane stiffness, Young’s modulus Y, can be defined as,
Yi ¼
1 @ 2 U A0 @ e2i
ð1Þ ei ¼0
where i is the direction (x- or y-) of the applied strain, U is the total strain energy, e is the linear strain, and A0 is the equilibrium rectangular area of the monolayer sheet. The stress tensors for each individual carbon atom are first calculated [40] and then averaged over all the atoms on the sheet to obtain the macroscopic stress. Noise is reduced by averaging the results over latter half of the relaxation period. Herein, the stress–strain evolutions are also presented in force per unit length (N/m). Fracture strain eF and maximum strength rc can be derived from the obtained force–strain curves, where the critical point can be determined from the spontaneous drop in the strain energy. Our simulation method is validated by calculating rc and eF of a 100 Å 100 Å pristine graphene sheet stressed along the armchair direction. We obtained a fracture strain of 0.136, which is consistent with the previous report of 0.137 [39]. The obtained Young’s modulus is 1.07 ± 0.02 TPa, which also agree well with the nanoindentation measurements of 1.0 TPa [41] The obtained results confirmed the validity of the simulation methods used here. 3. Results and discussions 3.1. Bond length analysis The bond lengths in the equilibrated and minimized structures are first investigated. Under a background temperature of 300 K, all the samples are fully relaxed to its energy minimum states and the direct bond length measurements are taken. All the bonds within the interior sections are averaged, resulting in near-constant values of 1.405 Å, 1.398 Å, 1.340 Å, and 1.240 Å for aromatic, single (C and C0 ), single (E and E0 ), and triple bonds respectively (Table 1). The variations of bond lengths among different structures in the family are seem to be small and random. It is noteworthy that C or C0 bonds are obviously shorter than the typical single bond of around 1.53 Å, whereas the D or D0 ’ bonds are some what a little longer than the typical triple bond of around 1.20 Å. The average of the bonds suggests that the carbomerized structures of graphyne family are still conjugated systems. The obtained rules of bond length alteration are consistent with the various reports in the open literature [25,31,33]. 3.2. Stress–strain evolutions We now turn to analyze the stress–strain relations of graphyne and its family to see how the general mechanical performance they Table 1 Equilibrium bond lengths (Å). Background temperature is 300 K in our works. Bond types
A, B
C, C0
D, D0
Graphyne Ref. [32] Ref. [24] Graphdiyne Ref. [30] Graphyne-3 Graphyne-4 Graphyne-5
1.405 1.49 1.419 1.405 1.440 1.406 1.405 1.406
1.398 1.48 1.401 1.396 1.400 1.398 1.398 1.396
1.240 1.19 1.221 1.240 1.239 1.240 1.239 1.239
E, E0
Remarks
1.340 1.341 1.340 1.339 1.338
MD, AIREBO potential MD, ReaxFF potential DFT, LSDA MD, AIREBO potential DFT, GGA-PBE MD, AIREBO potential MD, AIREBO potential MD, AIREBO potential
85
can be and how the elastic properties can be altered with increasing number of acetylenic linkages. Constructed from a hexagonal supercell, graphyne and its family structures have an in-plane hexagonal symmetry and exhibit characteristic zigzag edges. For the armchair edges, the triple-bond linkages give them a linear topology of repeating acetylenic and aromatic units, which is different from offset aromatic units of graphene. Nevertheless, they still show some vernacular of the graphene and are designated as armchair edges. Prismatic bar tensile deformations are axially loaded on the armchair- and zigzag-oriented monolayer sheets and the obtained stress–strain relations are shown in Fig. 2, along with data from pristine graphene case for comparison. In the armchair load situation, the stress–strain curves present somewhat parabolic increasing trend, this should be attributed to the an-harmonic terms in the carbon–carbon interaction potential. In the zigzag load case, the stress increases slower in the small strain regime and faster in the large strain regime. One observes an obvious structural deformation at the beginning of the loading test. The included angle of two acetylene chains, defined as angle a (Fig. 1b), reduces dramatically as the sheet is stretched. This helps to release the external force and lower the total stress within the sheet. In the latter half of the loading test, bonds begin to elongate and the total stress enlarges quickly. The calculated fracture strain eF and corresponding maximum stress rc are listed in Table 2. It is interesting to observe that in the armchair orientation the elastic region for all graphyne family members extends to about 0.11, whereas the associated rc degrades gradually from 14.437 N/m to 4.950 N/m for acetylenic groups increases from one to five (from graphyne to graphyne-5). On the contrary, in the zigzag orientation the ultimate strength for the five conformations remain nearly the same while the fracture strain is improved by a small amount with more acetylenic groups inserted in the hexagonal grid. The sheets are found to expand elastically up to 0.177–0.224 and then rupture, with corresponding maximum stress all around 21 N/m. To make more apt assessment with the widely studied graphene sheet we also calculated eF and rc for a graphene membrane with the same size as that of graphyne sheet. The maximum stress at the fracture point for the graphene sheet is 30.69 N/m for armchair loads and 36.58 N/m for zigzag loads. Therefore one can see that the tensile strengths of graphyne family are less than half of those of graphene sheet. The lower critical strain and strength for the armchair case is expected because in the armchair deformation circumstance part of the bonds are subjected directly to the load at the beginning of the elongation process, whereas in the zigzag situation the external loads can be dissipated through both bond elongation and bond angle variation. Finally, one more interesting thing should be noticed. Graphyne-3, graphyne-4 and graphyne-5 architectures exhibit a mechanically strengthening-like behavior in their stress–strain evolutions. With a close examination of the spatial bond distributions we found that this phenomenon originates from the spatial bond re-hybridization after the first bond breakage, which suggests that more ductile material can be attained from these kinds of conformations. The variation of fracture strain and ultimate stress of graphyne family presents disparate rules for armchair and zigzag load tests. To give more insight into this phenomenon, the bonds in the samples are differentiated into eight groups (Fig. 1c) and the average bond elongation for each groups are analyzed. The bond elongation evolutions of graphdiyne under tensile tests are presented in Fig. 3a and b. One can see that in the armchair loading case, the main bond elongation is experienced by bond C, next come bond D and E. This indicates that the major stretching takes place on the acetylene chain. From the atomic stress distribution as illustrated in Fig. 3c one can further see that the major stress accumulates on the ligature atoms between aromatic rings and acetylene chains. Bond length variation and atomic stress distribution for
Y. Yang, X. Xu / Computational Materials Science 61 (2012) 83–88
Graphene Graphyne Graphdiyne Graphyne-3 Graphyne-4 Graphyne-5
(a) Armchair Stress (N/m)
30
20
10
0 0.00
0.05
0.10
0.15
0.20
(b) Zigzag 30
Stress (N/m)
86
20
10
0 0.00
0.25
0.05
Strain
0.10
0.15
0.20
0.25
Strain
Fig. 2. Stress–strain relations of the graphyne family under tensile tests, results of graphene membrane with the same size are also presented for comparison. (a) Armchair direction. (b) Zigzag direction. Legends for both panels are the same and are presented only once in (a).
be more and more in-chain atoms, whereas the number of ligature atoms is the same. rc represents the ultimate stress averaged among all the atoms. Since the major stress accumulates on the ligature atoms, the averaged total stress will be reduced as the chain length increases. Therefore the ultimate strength degrades from graphyne to graphyne-5. The situation in the zigzag case is not the same. From Fig. 3d one can see that the atomic stress distribution is rather uniform on both aromatic rings and acetylene chains subjected to the external strain. Therefore the averaged total stress cannot vary with the chain length or the architectures. Except for bond B, all the other types of bonds take part in the elongation procedure and the maximum stretching ratios are similar for all the architectures. From this point of view the fracture strain might be the same for all the structures. However, a detailed analyses of the bond angle variation reveals that architectures with longer acetylene chains encompass higher degree of angle deformation, i.e., the reduction of angle a is larger in graphyne-3 to graphyne5 and smaller in graphyne and graphdiyne. Smaller angle a gives rise to not only released stress which results in slower stress augmentation, but also wider strain range the system can under go. Thus the larger fracture strains as observed in graphyne-3 to graphyne-5 can be understood. Furthermore, it is expected that
Table 2 Fracture strain eF and corresponding ultimate strength rc for the various systems investigated. System ID
Graphene Graphyne Graphdiyne Graphyne-3 Graphyne-4 Graphyne-5
Armchair direction
Zigzag direction
eF
rc (N/m)
eF
rc (N/m)
0.136 0.112 0.109 0.109 0.108 0.108
30.643 14.437 9.538 7.313 5.910 4.950
0.207 0.177 0.208 0.223 0.224 0.224
36.580 20.471 20.835 20.922 20.928 20.927
the other structures are also analyzed and the maximum bond stretching ratio for all the conformations are listed in Table 3, suggesting that the general trends are the same for the whole graphyne family. These results can well explain the unique fracture strain and ultimate strength variations as observed in Fig. 2. The length of acetylene chain enlarges from graphyne to graphyne-5. Since the major stretching takes place on the acetylene chain, the fracture strain, defined as a relative elongation ratio, cannot be influenced by the chain length, thus should be the same for all the architectures. Contrary, as the chain length increases there will
1.56
(a) Armchair
1.52
Bondlength (Angs.)
Bondlength (Angs.)
1.56
1.48 1.44 1.40
Bond-A Bond-B Bond-C Bond-D Bond-E
1.36 1.32 1.28 0
2
4
6
8
10
12
1.48 1.44 1.40
Bond-A Bond-B Bond-C' Bond-D' Bond-E'
1.36 1.32 1.28 0
14
5
10
15
20
25
Strain (%)
Strain (%)
(c)
(b) Zigzag
1.52
40GPa 32 24 16 8 0
(d)
α 90GPa 72 54 36 18 0
Fig. 3. Bond elongation evolutions of graphdiyne under tensile tests. (a) Armchair direction. (b) Zigzag direction. And atomic stress distributions before sheet rupturing. (c) Armchair direction. (d) Zigzag direction.
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Y. Yang, X. Xu / Computational Materials Science 61 (2012) 83–88 Table 3 Bond stretching ratio, defined as (ll0)/l0, where l is the average bond length at the fracture point and l0 the average bond length at the equilibrium state. Orientation
Armchair
Zigzag
Bond types
A (%)
B (%)
C (%)
D (%)
E (%)
A (%)
B (%)
C0 (%)
D0 (%)
E0 (%)
Graphyne Graphdiyne Graphyne-3 Graphyne-4 Graphyne-5
2.86 2.84 2.59 2.65 3.04
4.62 3.63 3.91 4.16 3.49
11.34 11.31 11.38 9.43 10.63
8.41 8.66 8.71 8.02 8.93
8.44 8.89 7.93 8.71
8.42 10.13 9.15 9.25 8.48
0.56 0.72 0.73 0.43 0.92
11.30 10.52 9.61 9.51 8.94
8.52 7.53 7.25 6.82 6.51
7.83 6.97 6.60 6.24
calculated results showed that these architectures are mechanically stable with high strength and stiffness. With the increase of acetylene groups, the obtained ultimate stress is found to depress gradually upon armchair loads whereas remains nearly unchanged upon zigzag loads. Fracture strain is observed to enlarge by a little amount with more inserted acetylene groups under zigzag deformations whereas remains the same under armchair deformations. The disparate influence of acetylene groups on the mechanical performance can be explained by the unique bond elongation and atomic stress distribution among the various conformations. Moreover, upon armchair deformations an interesting mechanically strengthening-like behavior is observed in architectures with more than two acetylene groups between two nearest-neighbor benzene rings. The versatile mechanical performance of graphyne and its family upon armchair and zigzag deformations can facilitate the design of advanced composites with intriguing direction-dependent properties and propel novel nanomechanical applications in carbon-based nanostructures.
Fig. 4. Young’s modulus of graphyne family, along with the data of pristine graphene for comparison.
covalently bonded graphyne could mechanically behaves as an atomistic spring-system [33]. There exist three types of bonds in all architectures, single, double and triple carbon–carbon bonds. Single bonds are not as stiff as the other two. Among all the single bonds, the bonds connecting the benzene rings and the acetylene groups are inferior than the others. In both armchair and zigzag tests, the first bond breakage is always observed to occur among bond C, i.e., in the linkage of benzene ring and the acetylene group. 3.3. In-plane stiffness Having examined the fracture process of graphyne family, we now proceed further to investigate their elastic properties at the near equilibrium regime. From the uniaxial tensile loads, the Young’s moduli for all the aforementioned samples are computed through Eq. (1) and the results are summarized in Fig. 4. The variation of Young’s modulus Y with respect to sheet orientation is tiny. Yzig is lower than Yarm by 7–10 N/m for all the conformations. In both armchair and zigzag circumstances, Y represents a general degrading trend with the increase of acetylene groups. This is reasonable because the in-plane honeycomb structure formed by hexagonal rings is the stiffest structure ever tested. Inserting acetylene chains into the nearest-neighbor benzene rings could only deteriorate the stability of the original conformation, and the longer the chain is, the higher the stiffness is. Compared to the 320 N/m in-plane stiffness for graphene sheet, the Young’s modulus for graphyne family is about 10–50% of that of graphene. 4. Conclusions In this paper the mechanical properties of graphyne and its family are investigated through molecular dynamics simulations. The
Acknowledgments This work was financially supported by NSF of China (No. 11074039) and the 2009 Project for Scientific and Technical Development of Xiamen (No. 3502Z20099007) and Fujian Education Bureau (No. JB11022). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28]
K.M. Liew, X.Q. He, C.H. Wong, Acta Mater. 52 (2004) 2521. K.M. Liew, Y. Sun, Phys. Rev. B 77 (2008) 205437. Q.X. Pei, Y.W. Zhang, V.B. Shenoy, Carbon 48 (2010) 898–904. P.S. Raux, P.M. Reis, J.W.M. Bush, C. Clanet, Phys. Rev. Lett. 105 (2010) 044301. Q. Zheng, Y. Geng, S. Wang, Z. Li, J.-K. Kim, Carbon 48 (2010) 4315–4322. C.G. Liu, Z.N. Yu, D. Neff, A. Zhamu, B.Z. Jiang, Nano Lett. 10 (2010) 4863. F. Diederich, M. Kivala, Adv. Mater. 22 (2010) 803. A. Hirsch, Nature Mater. 9 (2010) 868. L. Xu, Y. Zheng, J. Zheng, Phys. Rev. B 82 (2010) 195102. N. Wei, L. Xu, H.Q. Wang, J.C. Zheng, Nanotechnology 22 (2011) 105705. H.W. Kroto, J.R. Heath, S.C. O’Brien, R.F. Curl, R.E. Smalley, Nature 318 (1985) 162. S. Iijima, Nature 354 (1991) 56. K.S. Novoselov, A.K. Geim, S.V. Morozov, D. Jiang, Y. Zhang, S.V. Dubonos, et al., Science 306 (2004) 666. I.W. Frank, D.M. Tanenbaum, A.M. van der Zande, P.L. McEuen, J. Vacuum Sci. Tech. B 25 (2007) 2558. N. Wei, Z. Fan, Y. Zheng, L. Xu, H. Wang, J. Zheng, Nanoscale 4 (2012) 785. F. Liu, P. Ming, J. Li, Phys. Rev. B 76 (2007) 064120. Y.P. Zheng, N. Wei, Z.Y. Fan, L.Q. Xu, Z.G. Huang, Nanotechnology 22 (2011) 405701. C. Lee, X. Wei, J.W. Kysar, J. Hone, Science 321 (2008) 385. L. Xu, N. Wei, Y. Zheng, Z. Fan, H. Wang, J. Zheng, J. Mater. Chem. 22 (2012) 1435. R.M. Minyaev, V.E. Avakyan, Dokl. Chem. 434 (2) (2010) 253. X.L. Sheng, Q.B. Yan, F. Ye, Q.R. Zheng, G.Su. Phys, Rev. Lett. 106 (2011) 155703. A.K. Nair, S.W. Cranford, M.J. Buehler, Europhys. Lett. 95 (2011) 16002. R.H. Baughman, H. Eckhardt, M. Kertesz, J. Chem. Phys. 87 (1987) 6687. N. Narita, S. Nagai, S. Suzuki, K. Nakao, Phys. Rev. B 58 (11) (1998) 009. N. Narita, S. Nagai, S. Suzuki, K. Nakao, Phys. Rev. B 62 (11) (2000) 146. N. Narita, S. Nagai, S. Suzuki, Phys. Rev. B 64 (2001) 245408. V.R. Coluci, S.F. Braga, S.B. Legoas, D.S. Galvão, R.H. Baughman, Nanotechnology 15 (2004) S142. A.N. Enyashin, Y.N. Makurin, A.L. Ivanovskii, Carbon 42 (2004) 2081.
88
Y. Yang, X. Xu / Computational Materials Science 61 (2012) 83–88
[29] J. Zhou, K. Lv, Q. Wang, X.S. Chen, Q. Sun, P. Jena, J. Chem. Phys. 134 (2011) 174701. [30] G.X. Li, Y.L. Li, H.B. Liu, Y.B. Guo, Y.J. Li, D.B. Zhu, Chem. Commun. 46 (2010) 3256. [31] H.C. Bai, Y. Zhu, W.Y. Qiao, Y.H. Huang, RSC Adv. 1 (2011) 768–775. [32] M.M. Haley, Pure Appl. Chem. 80 (3) (2008) 519. [33] S.W. Cranford, M.J. Buehler, Carbon 49 (2011) 4111–4121. [34] S. Plimpton, J. Comput. Phys. 117 (1995) 1–19. [35] W.B. Donald, O. A Shenderova, J.A. Harrison, S.J. Stuart, B. Ni, S.B. Sinnott, J. Phys.: Condens. Matter. 14 (2002) 783.
[36] O.A. Shenderova, D.W. Brenner, A. Omeltchenko, X. Su, L.H. Yang, Phys. Rev. B 61 (2000) 3877. [37] W.G. Hoover, Phys. Rev. A 31 (1985) 1695. [38] E. Polak (Ed.), Optimization: Algorithms and Consistent Approximations, Springer, New York, 1997. [39] Y.P. Zheng, L.Q. Xu, Z.F. Fan, N. Wei, Y. Lu, Z.G. Huang, Curr. Nanosci. 8 (2011) 89. [40] Q.X. Pei, Y.W. Zhang, V.B. Shenoy, Nanotechnology 21 (2010) 115709. [41] Y. Geng, J. Li, S.J. Wang, J.K. Kim, J. Nanosci. Nanotechnol. 8 (2008) 6238.