Physics Letters A 376 (2012) 1942–1947

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Physics Letters A

www.elsevier.com/locate/pla

Mechanical deformation and fracture mode of polycrystalline graphene:Atomistic simulations

Feng Hao a, Daining Fang a,b,∗a Department of Engineering Mechanics, Tsinghua University, Beijing 100084, Chinab College of Engineering, Peking University, Beijing 100087, China

a r t i c l e i n f o a b s t r a c t

Article history:Received 28 February 2012Received in revised form 17 April 2012Accepted 19 April 2012Available online 25 April 2012Communicated by R. Wu

Keywords:Polycrystalline grapheneMolecular dynamicsBucklingFracture mode

Mechanics of polycrystalline graphene are studied through molecular dynamics simulations. Localbuckling forms the ridge or funnel centering on pentagon, and fluctuating stress occurs under smalltensile strain due to out-of-plane distortion. In addition, brittle breaking is initialized from heptagonsand ends with fracture of pentagons.

© 2012 Elsevier B.V. All rights reserved.

1. Introduction

Graphene, as the extremely atomic membrane, attracts inten-sive research interests more recently owing to its outstandingmechanical [1–3], thermal [4,5] and electronic [6,7] properties.For example, the Young’s modulus and strength of graphene sheetreach the order of 1 TPa and 100 GPa respectively [8] and thethermal conductivity of monolayer graphene sheet is found tobe 5500 W/(m K) by experiments [9,10]. Apparently, the key ofthese novel properties relies on structural perfection of the hexag-onal graphene lattice and strong interatomic sp2 bond within thegraphene sheet. However, defects such as vacancies and Stone–Wales (SW) dislocations [11–13], unavoidably exist in the graphenesynthesized by mechanical peeling or chemical vapor deposition.Effects of vacancies and pentagon–heptagon pairs on the mechan-ical, thermal and electronic properties are investigated, and it isdemonstrated that properties of graphene are very sensitive to itsmicrostructures [14–16].

Recently, polycrystalline graphene receives rising attention con-tinuously. In fact, polycrystalline graphene is the only solutionto achieve large-scale applications of graphene currently. Indi-vidual pentagon–heptagon pair is identified in transmission elec-tron microscopy (TEM) [17], and polycrystalline graphene includ-ing pentagon–heptagon pairs is characterized using aberration-corrected annular dark-filed scanning transmission electron mi-

* Corresponding author. Tel.: +86 010 62772923; fax: +86 010 62772933.E-mail address: fangdn@pku.edu.cn (D. Fang).

0375-9601/$ – see front matter © 2012 Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.physleta.2012.04.040

croscopy (ADF-STEM) [18,19], which verifies the presence of a tiltgrain boundary (GB). Moreover, GBs in other forms are also ob-served and studied at an atomic resolution [20–23]. In both ex-periments and simulations, it is found that GBs have significantimpacts on graphene, by reducing in electronic [24–26] and ther-mal conductivities [27,28]. More recently, Yazyev and Louie inves-tigate energy and density of states of tilt GBs [29], and Grantabet al. reveal anomalous strength characteristics of tilt GBs ingraphene [30].

In order to study polycrystalline graphene with tilt GBs, herewe pursue molecular dynamics (MD) simulations to obtain in-sights into the underlying mechanisms of deformation and frac-ture. We confirm that local out-of-plane buckling occurs [31],whose center locates in the 5–6–6 rings rather than pentagon–heptagon pair. Structural failure under tension begins from break-ing of specific bonds in initial stretched heptagons, similar to [32],and ends with the fracture of pentagons.

2. Materials and methods

In this work we focus on polycrystalline graphene with tilt GBsas shown in Fig. 1(a), where neighboring grains orientated in cer-tain directions merge together by the edge dislocations includingpairs of pentagon and heptagon. Though the grain size can be con-trolled, we assume here that each grain has nine perfect hexagonsin the width. Each two neighboring edge dislocations are equallyspaced in parallel along the opposite directions, colored by red andblue, respectively. The (1,0) dislocations, (a1,0) denoted by lat-tice vectors a1 and a2, are immersed into a semi-infinite ribbon

F. Hao, D. Fang / Physics Letters A 376 (2012) 1942–1947 1943

Fig. 1. Geometry of polycrystalline graphene, and the misorientation angle θ = 13.2◦ of symmetric grain boundaries including pentagon–heptagon pairs to form edge dislo-cations, (a) where the parallel equally spaced 5–7 dislocation boundaries are colored by red and blue, respectively; (b) ridges and funnels are observed resulting from theout-of-plane buckling along the grain boundaries; the center of buckling occurs around three atoms shared by a pentagon and two hexagons seen from partial enlargeddrawing. (For interpretation of the references to color in this figure, the reader is referred to the web version of this Letter.)

along armchair directions in graphene [29], leading to a crystallo-graphic misorientation angle θ , which can be defined by Frank’sequation [33]

θ = 2 arcsin|b(1,0)|

2d, (1)

where b is Burger vector, a topological invariant representingdistortion of crystalline configuration. For the GB studied here,|b(1,0)| = 2.46 Å. d is the distance between the neighboringpentagon–heptagon pairs, yet separated by a hexagon in the fol-lowing studies. Thus the misorientation angle θ = 13.2◦ .

In our MD simulations, we use the large-scale atomic andmolecular massively parallel simulator (LAMMPS) package [34].Periodic boundary condition is applied to a rectangular poly-crystalline graphene sheet with length L = 9 nm and widthD = 10 nm. The adaptive intermolecular reactive empirical bond-order (AIREBO) potential functions and parameters are used forthe interatomic interactions between carbon atoms [35]. This ap-proach successfully predicts the mechanical and thermal transportproperties of graphene materials and their derivatives [14,27,30].

The structure is firstly equilibrated at ambient conditions (tem-perature T = 300 K and pressure P = 1 atm) under a Nosé–Hooverthermostat for 1.5 ns using a time step of 0.5 fs. After the struc-tures reach thermal equilibration, a strain rate of 1 ns−1 is applied.Tensile loads that are perpendicular and parallel to the GB, de-fined as x-axis and y-axis respectively, are both investigated. Dur-ing the uniaxial tension, stresses in other directions are relaxed.The deformation, potential energy change and atomic virial stress

are recorded and calculated with a strain up to the critical value,which leads to structural failure at 300 K.

3. Results and discussion

The equilibrium configuration of polycrystalline graphene isfirstly obtained by geometrical optimization. Fig. 1(b) shows ridgesand funnels along the GB, indicating out-of-plane buckling ofthe two-dimensional graphene sheet. For this non-polar geometry,probabilities of ridge and funnel are identical in theory. However,as shown in Fig. 1(b), the appearances of the ridges and the fun-nels are not random, straight wrinkles are also formed with thesame buckling directions.

Interestingly, Fig. 1(b) shows that centers of ridges or fun-nels locate in 5–6–6 rings, rather than pentagon–heptagon pairs.Local out-of-plane bucklings are also observed in other refer-ences [29,31,36,37]. Lusk et al. prove that localized defects out ofthe graphene sheet reduce its total energy. However, a mechanis-tic understanding in detail of this phenomenon remains unclearly.To clarify the intrinsic mechanism, we investigate interatomic po-tential energy of atoms, and atomic virial stress that is defined by

Siαβ = 1

Ω i

(1

2mi vi

α viβ +

∑j=1,N; j �=i

ri jβ f i j

α

), (2)

where, α and β denote the indices of stress tensor along Carte-sian coordinate axis, while i and j are the atomic indices, Ω i,mα

and vα are the atomic volume, mass and velocity of atom i, and

1944 F. Hao, D. Fang / Physics Letters A 376 (2012) 1942–1947

Fig. 2. (a) Potential energy of each atom, higher potential of pentagons compared to other atoms, especially the three atoms, where the center of the buckling occurs.(b) and (c) show the atomic stresses along x-axis and y-axis, respectively. Atoms of pentagons are subjected to compressive stresses, while some atoms of heptagon beartensile stresses. (For interpretation of the references to color in this figure, the reader is referred to the web version of this Letter.)

Fig. 3. (a) Stress–strain curves along x-axis and y-axis, and in the inserted figure, configuration under tension along x-axis. (b) The profile of axial stress flux under smallexternal tensile force along x-axis, the maximum stresses occur in the heptagons.

ri j and f i j are the distance and force between atom i and j. Virialstress S describes the magnitude and direction of the atom inter-actional forces.

As shown in Fig. 2(a), atoms in pentagons have higher po-tential energies compared to other atoms, especially three atoms

shared by a pentagon and two hexagons, where the center of alocalized buckling locates. Moreover, Fig. 2(b) and (c) show thatatomic virial stresses Sxx along x-axis and S yy along y-axis, re-spectively. Some atoms in pentagons are largely subjected to com-pressive stresses (positive values as colored by red), while some

F. Hao, D. Fang / Physics Letters A 376 (2012) 1942–1947 1945

Fig. 4. (a) Mechanical analysis of bonds Z1, Z2 and Z3 in a qualitative way. Fracture modes under uniaxial stretching: (b) and (c) show the initial breaks, beginning fromheptagon, for mode I relating to x-axis stretching and mode II relating to y-axis stretching; (d) and (e) are the final configurations before total failure for mode I and IIrespectively.

atoms of heptagons bear tensile stresses (negative values as col-ored by blue). These results are similar to the work of Yakobsonand Ding [32]. The aforementioned three atoms lead to the for-mation of localized buckling domain due to its highly compressivestress state. As a result, two-dimensional graphene sheet becomesunstable under excessive local compressive stress. To reduce totalelastic energy, out-of-plane distortion must occur to reach lowerenergy state.

Next, to investigate the effective mechanical properties of poly-crystalline graphene with tilt GBs, we calculate stress–strain rela-

tions from tensile simulations along x-axis and y-axis. As shownin Fig. 3(a), the results exhibit different mechanical behaviors de-pending on the stretching directions. Increases both in fracturestrength and strain limit are observed for tension along x-axis incomparison with y-axis. Furthermore, for small strain values alongx-axis, as shown in the insert of Fig. 3(a), stresses are fluctuat-ing at a low value, which corresponds to geometrical distortion ofGBs. In the initial stage of tension, graphene sheet with wrinklesis firstly stretched to a flattened planar geometry at low tensilestrain. Subsequently stress increases as the strain is up to 0.02.

1946 F. Hao, D. Fang / Physics Letters A 376 (2012) 1942–1947

By fitting the linear stress–strain relation we obtain Young’s mod-ulus Y = 620 GPa and 880 GPa for tension in x-axis and y-axis,respectively. These values are slightly reduced compared to thatof pristine graphene [1,8]. We plot axial atomic stress, as a vectorSN = (S11, S22, S33), under small tensile strain in Fig. 3(b). The di-rection of the arrow represents that of vector SN, and the size ofarrow represents the magnitude of stress. It is clearly indicatedthat uniform stress is distributed inside graphene grains, whilelarger disorder stress arises in pentagon–heptagon pairs on theGBs. Stress SN of the pentagon is in the opposite direction to thestretching force because of the unreleased compressive strain, con-trarily, the heptagon bears maximal stress caused by initial tensilestrain and the following analysis in Fig. 4(a) and (b).

Finally, we focus on the fracture of graphene lattice under loadexceeding its elastic limit. Two fracture modes are observed, de-pending on the direction of applied loads. Here we define mode Ias for tension along x-axis and mode II along y-axis, of which de-tailed mechanisms are discussed in the following paragraphs. Asshown in Fig. 4(c) and (d), the breaking of graphene lattice startsfrom some specific bonds of heptagons in the GB regardless ofwhich mode, which is also observed for pristine graphene [38] andpolycrystalline graphene [30]. It is also noted that initial brokenbonds are different in mode I and II.

To explain the results discussed above, taking mode I as an ex-ample. We now firstly consider perfect graphene lattice as an elas-tic structure and analyze internal forces in the bonds, as shown inFig. 4(a) and (b) for mode I. According to the symmetry, bonds Z1and Z2 are equivalent. Axial force N3 less than N1 applied to Z1owning to the absence of load along Y axis, and the stress responsefrom bending moment M1 caused by shear force Q 1 is much largercompared to axial force N equivalent to Q 1. As an antisymmet-ric load, Q 1 does not exist in bond Z3 for a symmetric structure,thus M1 is also zero in bond Z3. It is noted that bond Z3 herein isregarded as the center of the symmetric structure because of theperiodic boundary condition applied to graphene lattice. The rela-tions between these internal forces are given as√

3

2N1 = σ wt,

N1 = N2,√3Q 1 = N1 + N3,

M1 = Q 1l/2, (3)

where σ is external applied stress, l is the bond length; w and tare width and thickness of graphene lattice unit respectively.As studied here in Fig. 1(a), grains are orientated in the directionsθ/2 = 6.8◦ . Thus, the difference of forced stages between perfectgraphene and inside grain is not significant for this so small mis-orientation angle. By aforementioned statics analysis, it is shownthat Z1 and Z2 bear the maximum load. Furthermore, as shown inFig. 2(c), atoms in heptagon that are far from pentagons are sub-jected to initial tensile stress, resulting in that maximum stressoccurs in specific bonds of heptagons, as indicated in Fig. 3(b).These arguments explain the mechanism of the origin in structuralfailure in Fig. 4(c) and direction of crack propagation as observedin Fig. 4(e).

Now, back to Fig. 4(e) and (f), there exist two fracture modes.In mode I, bond breaking starts from specific bonds in heptagonsand extends along the direction formed by similar broken bondsin hexagons as analyzed in Fig. 4(b). In mode II, likewise, bondbreaking develops to a crack along the crystallographic direction,and as tensile strain increases, this crack propagates and opens tillthe failure of whole material. For both fracture modes, it is shownthat the center of bucking, three atoms shared by shared by onepentagon and two hexagons, is preserved well. This result can be

explained as follows. The local buckling bears initially compres-sive stress as shown in Fig. 2(b) and (c), similar to [32], after thispart of deformation energy is released under certain tensile strain,these local domains begin to be stretched. As a result, the centerof buckling is superior in endurance strength and fractures at thelast moment.

4. Conclusion

In summary, we construct polycrystalline graphene with tiltgrain boundaries and investigate its mechanical properties usingmolecular dynamics simulations. Local bucklings, in the form ofridges or funnels, resulting from highly compressive stress statein pentagons, are observed. To reduce deformation energy andreach a low energy state, out-of-plane distortion must be permit-ted. Fluctuating stress occurs remarkably at small tensile strain inperpendicular to GBs due to the non-planar geometrical distortion.Two fracture modes are identified depending on loading directions.Bond breaking starts from heptagons subjected to initial tensilestress in the both fracture modes. Simulation results are discussedwith internal force and stress distribution analysis. Disorder stresson the GBs is observed in contrast to other regions where stressis uniformly distributed. Methods employed here can be appliedto reveal the mechanisms of similar results, for example, the localbuckling of graphene with other defects [35–37].

Acknowledgements

This work is supported by the Natural Science Foundationof China under Grant Nos. 11090330, 11090331 and 11072003.We also acknowledge support from Special Funds for the MajorState Basic Research Program of China (#2010CB832701).

References

[1] A.K. Geim, Science 324 (2009) 1530.[2] Q. Wang, Phys. Lett. A 374 (2010) 1180.[3] S.W. Cranford, M.J. Buehler, Carbon 49 (2011) 4111.[4] A.A. Balandin, S. Ghosh, W. Bao, I. Calizo, D. Teweldebrhan, F. Miao, C.N. Lau,

Nano Lett. 8 (2008) 902.[5] B.D. Kong, S. Paul, M. Buongiorno Nardelli, K.W. Kim, Phys. Rev. B 80 (2009)

033406.[6] A.H. Castro Neto, F. Guinea, N.M.R. Peres, K.S. Novoselov, A.K. Geim, Rev. Mod.

Phys. 81 (2009) 109.[7] C. Berger, Z. Song, X. Li, et al., Science 312 (2006) 5777.[8] Z. Xu, J. Comput. Theor. Nanosci. 6 (2009) 625.[9] J.H. Seol, I. Jo, A.L. Moore, et al., Science 328 (2010) 5975.

[10] S. Ghosh, I. Calizo, D. Teweldebrhan, et al., Appl. Phys. Lett. 92 (2008) 151911.[11] J. Kotakoski, A.V. Krasheninnikov, U. Kaiser, J.C. Meyer, Phys. Rev. Lett. 106

(2011) 105505.[12] B.I. Yakobson, Appl. Phys. Lett. 72 (1998) 918.[13] B.I. Yakobson, Ph. Avouris, Topics Appl. Phys. 80 (2001) 287.[14] F. Hao, D. Fang, Z. Xu, Appl. Phys. Lett. 99 (2011) 041901.[15] F. Scarpa, R. Chowdhury, S. Adhikari, Phys. Lett. A 375 (2011) 2071.[16] J. Haskins, A. Kinaci, C. Sevik, H. Sevincli, G. Cuniberti, Tahir Cagin, ACS Nano 5

(2011) 3779.[17] A. Hashimoto, K. Suenaga, A. Gloter, K. Urita, S. Lijima, Nature 430 (2004) 870.[18] P.Y. Huang, C.S. Ruiz-Vargas, A.M. van der Zande, et al., Nature 469 (2011) 389.[19] K. Kim, Z. Lee, W. Regan, C. Kisielowski, M.F. Crommie, A. Zettl, ACS Nano 5

(2011) 2142.[20] L.B. Biedermann, M.L. Bolen, M.A. Capano, D. Zemlyanov, R.G. Reifenberger,

Phys. Rev. B 79 (2009) 125411.[21] S. Malola, H. Hakkinen, P. Koskinen, Phys. Rev. B 81 (2010) 165447.[22] A. Reina, X. Jia, J. Ho, D. Nezich, H. Son, V. Bulovic, M.S. Dresslhaus, J. Kong,

Nano Lett. 9 (2009) 30.[23] B. Wang, Y. Puzyrev, S.T. Pantelides, Carbon 49 (2011) 3983.[24] O.V. Yazyev, S.G. Louie, Nat. Mater. 9 (2010) 806.[25] S. Bae, H. Kim, Y. Lee, et al., Nat. Nanotechnol. 5 (2010) 574.[26] X. Li, W. Cai, J. An, et al., Science 324 (2009) 1312.

F. Hao, D. Fang / Physics Letters A 376 (2012) 1942–1947 1947

[27] A. Bagri, S. Kim, R.S. Ruoff, V.B. Shenoy, Nano Lett. 11 (2011) 3917.[28] W. Cai, A.L. Moore, Y. Zhu, X. Li, S. Chen, L. Shi, R.S. Ruoff, Nano Lett. 10 (2010)

1645.[29] O.V. Yazyev, S.G. Louie, Phys. Rev. B 81 (2010) 195420.[30] R. Grantab, V.B. Shenoy, R.S. Ruoff, Science 330 (2010) 946.[31] Y. Liu, B.I. Yakobson, Nano Lett. 10 (2010) 2178.[32] B.I. Yakobson, F. Ding, ACS Nano 5 (2011) 1569.

[33] J.P. Hirth, J. Lothe, Theory of Dislocation, Wiley, New York, 1982.[34] S. Plimpton, J. Comput. Phys. 117 (1995) 1.[35] D.W. Brenner, O.A. Shenderova, J.A. Harrison, S.J. Stuart, B. Ni, S.B. Sinnott,

J. Phys.: Condens. Matter 14 (2002) 783.[36] M.T. Lusk, D.T. Wu, L.D. Carr, Phys. Rev. B 81 (2010) 155444.[37] M.T. Lusk, L.D. Carr, Phys. Rev. Lett. 100 (2008) 175503.[38] H. Bu, Y. Chen, M. Zou, H. Yi, K. Bi, Z. Ni, Phys. Lett. A 373 (2009) 3359.