Mechanical Properties of Vacancy-containing Graphene and Graphite using Molecular Dynamics Simulations

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1 Mechanca Propertes of Vacancy-contanng Graphene and Graphte usng Moecuar Dynamcs Smuatons Akhko Ito and Shngo kamoto Abstract We nvestgated the mechanca propertes of graphene and graphte contanng vacances under tense oadng usng moecuar dynamcs (MD) smuatons. In the MD smuatons, we used two types of potenta functons: the second-generaton reactve emprca bond-order (REB) potenta for covaent C C bond, and the Lennard-Jones potenta for the nterayer nteracton of graphte. The nfuence of the sze and the dstrbutona form of vacances on the mechanca propertes of graphene and graphte were studed. It was found that the tense strength of graphene havng randomy dstrbuted vacances wth a vacancy densty of 4%, decreased by 59%. Index Terms Graphene, graphte, moecuar dynamcs, vacancy C I. INTRDUCTIN ARBN-based materas can have exceent mechanca and eectrca propertes. Therefore, there s much nterest around ther use n appcatons n structura sub-assembes and nano-eectro mechanca systems such as eectrochemca eectrodes and fed emsson. Carbon materas such as damond, graphene, carbon nanotubes (CNT), and fuerenes, have a wde range of exceent propertes thanks to ther dfferent types of bonds and atomstc structures. In partcuar, graphene has rgdty and strength that are neary equa to those of damond, as we as nove eectronc propertes ncudng hgh eectron mobty. Thus, studes on graphene and graphte made of graphene ayers have recenty ntensfed [] [3]. Defects often affect the mechanca and eectronc propertes of materas. There have been reports of expermenta studes on defects (.e., vacances [4], dsocatons [5], and gran boundares [6]) n graphene ayers. It s mportant to carfy the nfuence of defects on the mechanca and eectrca propertes of graphene and graphte n order to produce hgh-performance carbon materas. Recenty, studes amng to carfy the reatonshp between Manuscrpt receved December 4, 2. Ths work was supported n part by the Rng-Rng project of JKA. Akhko Ito s wth the Mechanca Engneerng Course, Graduate Schoo of Scence and Engneerng, Ehme Unversty, 3 Bunkyo-cho, Matsuyama , Japan. He s aso wth the Composte Materas Research Laboratores, Toray Industres, Inc., Masak-cho , Japan (e-ma: Akhko_Ito@nts.toray.co.jp). Shngo kamoto s wth the Mechanca Engneerng Course, Graduate Schoo of Scence and Engneerng, Ehme Unversty, 3 Bunkyo-cho, Matsuyama , Japan (e-ma: okamoto.shngo.mh@ehme-u.ac.jp). atomc-scae defects and mechanca propertes have ncreased n number. The tense propertes of graphene and carbon nanotubes contanng mutpe Stone-Waes (SW) defects have been nvestgated usng moecuar dynamcs (MD) smuatons by ao et a. [7]. These studes have carfed the reatonshp between the number of defects and the mechanca propertes. The nfuence of gran boundares on the tense strength of graphene has been nvestgated by Grantab et a. [8]. The MD smuatons of tense oadngs of snge-waed carbon nanotubes wth vacances have been performed by Wong et a. [9]. The nfuence of the snge and doube vacances on the tense strength has been nvestgated through moecuar mechancs (MM) cacuatons by Zhang et a. []. Zhang et a. compared ther resuts obtaned usng MM cacuatons wth Meke s resuts obtaned usng quantum mechancs (QM) cacuatons []. However, the nfuence of vacances on the mechanca propertes of graphene and graphte were negected. In ths study, we nvestgated the nfuence of vacancy sze on the mechanca propertes of graphene and graphte through MD smuatons. In addton, we carfed the reatonshp between the dstrbutona form of vacances and the mechanca propertes. A. Potenta Functon II. METHD In the present study, we used two types of nteratomc potentas: the second-generaton reactve emprca bond order (2 nd REB) [2], and Lennard-Jones potentas. The 2 nd REB potenta for covaent C-C bonds s expressed as * [ VR ( rj ) BjVA ( rj )], E = () REB j> where r j represents the dstance between atoms and j. The B j * represents the bond-order term. The terms V R (r j ) and V A (r j ) represent the par-addtve nteractons that refect nteratomc repusons and attractons, respectvey as n V V R A Q + r ( r) f ( r) Aexp( r) = c 3 ( r) = fc ( r) B n exp( β nr) n= α, (2)

2 where Q, A, α, B n, and β n represent constant parameters. The functon f c (r) represents the cutoff functon that decreases monotonousy from to as n f c ( r), π = + cos R, ( r R ) mn R mn / 2, R mn r < R < r < R r > R mn, (3) 5A 3A 52A : Carbon atom (a) Armchar tenson mode (b) Zgzag tenson mode Fg. 2 Confguratons of graphene used under zgzag tenson. 3A Force (nn) rgna (R mn =.7A ) Ths work (R mn = 2.A ) Bond ength r j (A ) Fg. Interatomc forces for 2 nd REB potenta wth orgna R mn and R mn used n ths work. 3A 52A 24A Fg. 3 Confguraton of graphte used under zgzag tenson. A 3.35A B Z where R mn =.7A and R = 2.A n orgna 2 nd REB potenta. It s known that for the orgna 2 nd REB potenta, the nteratomc force ncreases dramatcay at r = R mn and reaches zero at r = R owng to the dscontnuty n the second dervatve of the cutoff functon, as shown n Fg.. Ths dramatc ncrease n the nteratomc force wth the orgna 2 nd REB potenta may greaty affect tense strength. Therefore, n ths work, the cutoff parameter s set to 2. Å n order to avod the dramatc ncrease n the nteratomc force [3]. The other parameters, except for R mn, are set to the vaues proposed by Brenner [2]. The Lennard-Jones potenta for the nterayer nteracton n the graphte mode s expressed as A Fg. 4 Schematc of graphte structure. (a) Snge - vacancy carbon atom 2 6 LJ 4 r r V = ε. (4) rj rj The 2 nd REB potenta and the Lennard-Jones potenta are swtched accordng to the nteratomc dstance and bond order [4]. The vaue of ε s set to.284 ev and r s set to Å so that the nterpanar spacng of graphte at 3 K s 3.35 Å, whch s a known expermenta vaue. B. Anayss mode Frsty, the anayss modes of graphene used under zgzag tensons consst of 588 carbon atoms wth dmensons equa to those of the rea crystate n a typca carbon matera, as shown n Fg. 2. No perodc boundary condtons are mposed n our case. The anayss modes consst of two parts. ne s referred to as (b) Doube - vacancy (c) Sextupe - vacancy Fg. 5 Anayss mode for the graphene contanng a custer-type vacancy. the actve zone n whch the atoms move accordng to the nteractons wth ther neghborng atoms. The other encosed wthn the boxes (as shown n Fg. 2) s referred to as the boundary zone n whch the atoms are restraned. The thckness of the boundary zone s 3.a for the armchar tenson mode and.5 3a for the zgzag tenson mode, where a s the ength of the C=C bond n graphene. The anayss mode of graphte used under zgzag tenson conssts of 4,6 carbon atoms wth dmensons equa to those of the rea crystate n typca carbon matera, as shown n Fg. 3. The graphte mode s made of seven ayers of graphene sheets, whch are stacked n an AB-type sequence wth an nterayer spacng of 3.35 Å, as shown n Fg. 4. The anayss modes of graphene wth custer-type vacances are shown n Fg. 5. These modes of graphte

3 (b) 2% - vacances (a) % - vacances (c) 4% - vacances Fg. 6 Anayss mode for the graphene contanng unformy dstrbuted vacances. revea that the graphene sheet wth a custer-type vacancy s aways the center ayer. The anayss modes of graphene contanng unformy dstrbuted vacances are shown n Fg. 6. Each vacancy s a snge vacancy and set so that the dstance between neghborng vacances s dentca. Cacuatons for three vaues of vacancy densty, namey, 2, and 4%, are performed. The anayss modes of graphene contanng randomy dstrbuted vacances are set by removng carbon atoms n the actve zone usng a pseudorandom number generator. C. Moecuar dynamcs smuatons We nvestgated the mechanca propertes of vacancy-contanng graphene and graphte usng the MD smuatons under constant voume and temperature, that s, a canonca (NVT ) ensembe. The Veret method s used for the tme ntegra of the equatons of moton of atoms. The veoctes of a atoms are adjusted smutaneousy usng the veocty scang method [5] so that the temperature of the object can be mantaned at the preset temperature T SET. The mass of a snge carbon atom, m, s kg. The tme step s. fs. The atomc stress actng on each atom s cacuated to obtan the stress-stran curves and to vsuaze the stress dstrbuton durng tense oadngs. The atomc stress σ J for each of the,, and Z drectons of J s gven by cacuatng the knetc energes of, the nteratomc force actng on, and the voume occuped by atom as n σ ( J = m V JV J + J F J ), (5) Ω where Ω represents the voume occuped by atom, whch s referred to as the atomc voume. The atomc voume s cacuated by averagng the voume over a atoms n the nta structure of each system. The nteratomc force actng on atom due to ts neghborng atoms s represented by F. The goba stress of an anayss mode s cacuated by averagng over a carbon atoms n each system. Method of tenson oadng The nta postons of the atoms are gven so that the anayss mode represents the crysta structure of graphene or graphte at a preset temperature. Frst, the atoms n the actve zone of the anayss mode are reaxed n unoaded states for 7, MD steps. The atoms n the boundary zone are fxed. After constant dspacements are apped to the atoms n both of the boundary zones to smuate unaxa tense oadng n the drecton, the atoms n the actve zone are reaxed for 7, MD steps. The stran ncrement, ε, s.4. The output stresses are samped for the ast 2, MD steps for each stran and are averaged. oung s modu are obtaned from the sopes of the straght nes n the range where the reatonshp between the stress and stran s near, and tense strengths are gven by the ast peak of the nomna stress-nomna stran curves. III. RESULTS AND DISCUSSIN A. Vadaton of cacuaton method We performed the MD smuatons on tense oadngs of prstne graphene at 3 K to verfy the proprety of our cacuaton method. The resuts are presented n Tabe I and Fg. 7. The average tense strength s 83 GPa, whch s n agreement wth the 2 GPa cacuated by Pe et a. through MD smuatons [6] and the expermentay obtaned vaue of 23.5 GPa []. The average oung s moduus s 836 GPa, whch s wthn the range of resuts obtaned by the DFT method [7] (,5 GPa) and by experment [8] (5 GPa and TPa). It s estmated that the ower vaue obtaned n ths work s due to the effect of sze on the eastc propertes of graphene [8]. TABLE I MECHANICAL PRPERTIES F PRISTINE GRAPHENE Drecton Tense strength oung s moduus (GPa) (GPa) Armchar Zgzag Average Nomna stress σ x (GPa) Zgzag Armchar Nomna stran Fg. 7 Stress-stran curves of prstne graphene under Armchar or Zgzag tenson. B. Mechanca propertes of vacancy-contanng graphene The mechanca propertes of vacancy-contanng graphene obtaned at 3 K are sted n tabes II and III, together wth the resuts from prevous studes on carbon nanotubes

4 [9] []. The nomna stress-nomna stran curves of the graphene are gven n Fg. 8. The resuts for prstne graphene are aso gven for reference. The decrease n the tense strength s arger for graphene wth doube vacancy, foowed by that of the sextupe and then snge vacancy. In addton, the fracture stran for graphene wth doube vacancy s the east. The decrease n tense strength reatve to that of prstne graphene s 29, 28, and 7%, respectvey. Nevertheess, the oung's moduus hardy changes wth the vacancy sze. When compared wth the resuts of prevous studes on carbon nanotubes usng MD [9], moecuar mechancs (MM) [], and quantum mechancs (QM) [] cacuatons, the reductons n the tense strength of graphene wth a snge and doube vacancy n ths work are cose to the resuts obtaned wth the MM and QM cacuatons. Snapshots of the tense oadngs are shown n Fg. 9. For prstne graphene, the dstrbuton of stress just before the fracture s unform and the eve of stress s hgh. In comparson, n the vacancy-contanng graphene, the concentraton of stress occurs around the vacancy just before the fracture, whch emerges from the crcumference of the vacancy. TABLE II TENSILE STRENGTHS F VACANC-CNTAINING GRAPHENE AND CNT Graphene CNT(Carbon Nanotube) σ B σ B [9] σ B [] σ B [] (MD, GPa) (MD, GPa) (MM, GPa) (QM, GPa) Prstne Snge vacancy 75 (-7%) 3 (-%) 7.4 (-33%) (-26%) Doube vacancy 64 (-29%) (-3%) 7.3 (-32%) 5 (-22%) Sextupe vacancy 65 (-28%) σ B s the tense strength. Vaues n parentheses represent the dfferences between the prstne and vacancy-contanng materas. MD, MM, and QM refer to Moecuar Dynamcs, Moecuar Mechancs, and Quantum Mechancs, respectvey. The [5,5] CNTs whose tense drecton agrees wth the zgzag tenson are used for a of the carbon nanotube resuts. TABLE III THE UNG S MDULI F VACANC-CNTAINING GRAPHENE (UNIT: GPA) Prstne 794 Snge vacancy 782 (-.5%) Doube vacancy 765 (-3.6%) Sextupe vacancy 767(-3.4%) Vaues n parentheses represent the dfferences between the prstne and vacancy-contanng materas. Nomna stress σ x (GPa) Prstne Snge vacancy Doube vacancy Sextupe vacancy Nomna stran Fg. 8 Stress-stran curves of the graphene contanng a custer-type vacancy under zgzag tenson. (a-) Inta structure of the prstne graphene (b-) Inta structure of the graphene wth a snge vacancy (c-) Inta structure of the graphene wth a doube vacancy (d-) Inta structure of the graphene wth a sextupe vacancy (a-2) Just before fracture (b-2) Just before fracture (c-2) Just before fracture (d-2) Just before fracture (a-3) Fracture (b-3) Fracture (c-3) Fracture (d-3) Fracture σx (GPa) Fg. 9 Stages of fracture progresson n graphene contanng custer-type. vacancy. (a-) (a-3): prstne, (b-) (b-3): snge vacancy, (c-) (c-3): doube vacancy, (d-) (d-3): sextupe vacancy. Reatve strength MD cacuaton Grffth s Grffth's crteron method Number of atom defects Fg. Reatve strengths and szes of vacancy, namey, the number of atom defects obtaned usng MD cacuaton and Grffth s crteron. We compared the cacuated resuts wth the Grffth s crteron n order to verfy the proprety. The theoretcay dea strength σ for brtte fracture s expressed as Eγ σ s =, (6) d where E s oung s moduus, γ s s the surface energy, and d s the nteratomc dstance. Then, the strength of materas contanng a fracture of ength 2C accordng to the Grffth s crteron s expressed as 2Eγ σ s f =, (7) πc The reatve strength σ re, that s, the strength of the materas wth a fracture reatve to the theoretcay dea strength s obtaned by dvdng σ f by σ as 2d σ re =, (8) πc A pot of the reatve strength aganst the number of atomc

5 Proceedngs of the Internatona MutConference of Engneers and Computer Scentsts 22 Vo I, IMECS 22, March 4-6, 22, Hong Kong defects s shown n Fg.. The resuts of MD cacuatons agree we wth the predcted vaues usng the Grffth s crteron. Tense strength (GPa) C. Infuence of dstrbutona form of defects For the graphene wth unformy or randomy dstrbuted vacances, the reatonshp between the tense strength and the densty of vacances s shown n Fg.. For the random vacancy dstrbuton, the average vaues of the two resuts cacuated usng the modes wth dfferent vacancy arrangements are potted. The error bar (Ｉ) represents the range between two vaues. The tense strength decreases wth the ncrease n vacancy densty. The reducton n the tense strength s 59% at a densty of 4% for the random vacancy dstrbuton. Ths s neary twce that of the reducton n the tense strength of hydrogen (H)-functonazed graphene [6]. In comparson, the oung s moduus sghty decreases wth the ncrease n the vacancy densty (see Fg. 2). The reducton n the oung s moduus s 2% at a densty of 4%. Ths s neary 4 tmes that of the reducton n the oung s moduus of H-functonazed graphene. It s reasonabe to assume that graphene s more senstve to vacances than to H coverage, because a vacancy mpes the ack of an atom, whereas H-coverage refers to the converson of oca carbon bondng from sp2 to sp3 hybrdzaton. Snapshots of the graphene wth unformy dstrbuted vacances durng tense oadng are shown n Fg. 3. In every case, the concentraton of the stress occurs around each vacancy just before the fracture n the same manner as for the graphene wth a snge vacancy. Then, fractures occur startng from a vacancy and progress toward neghborng vacances. The progresson of the fracture drecton s perpendcuar to the tense axs n a cases. Conversey, snapshots of the graphene wth randomy dstrbuted vacances durng the tense oadng are shown n Fg. 4. The fracture starts from the area where the vacances gather. The progresson drecton of the fracture s then random. (a-3) Fracture (b-) Inta structure of (b-2) Just before graphene wth 2% unformy dstrbuted vacances (b-3) Fracture σ x (GPa) 4 (c-) Inta structure of (c-2) Just before graphene wth 4% unformy dstrbuted vacances (c-3) Fracture Fg. 3 Stages of fracture progresson n graphene contanng unformy dstrbuted vacances. The densty of vacances s % ((a-) (a-3)), 2% ((b-) (b-3)), and 4% ((c-) (c-3)). (a-) Inta structure of (a-2) Just before graphene wth % randomy dstrbuted vacances (a-3) Fracture (b-) Inta structure of (b-2) Just before graphene wth 2% randomy dstrbuted vacances (b-3) Fracture σ x (GPa) 4 (c-) Inta structure of (c-2) Just before graphene wth 4% randomy dstrbuted vacances (c-3) Fracture Fg. 4 Stages of fracture progresson n graphene contanng randomy dstrbuted vacances. The densty of vacances s % ((a-) (a-3)), 2% ((b-) (b-3)), and 4% ((c-) (c-3)). Unform Random Densty of vacances (%) Fg. Tense strength of graphene aganst vacancy densty. oung's moduus (GPa) (a-2) Just before (a-) Inta structure of graphene wth % unformy dstrbuted vacances Unform Random Densty of vacances (%) Fg.2 oung s moduus of graphene aganst vacancy densty. ISBN: ISSN: (Prnt); ISSN: (nne) 5 D. Mechanca propertes of vacancy-contanng graphte The stress and stran curves of graphte wth a custer-type vacancy at 3 K are shown n Fg. 5. The resuts for prstne graphte are aso gven for reference. In every case, reductons n stress occur before the fracture. For the snge vacancy, the reducton occurs twce before the fracture, whch occurs durng the ast stress peak; for the other vacances, the reducton occurs ony once. Snapshots of the graphte wth a custer-type vacancy are shown n Fg. 6. It was found that the reducton n stress before the fracture was due to a tear n the graphene sheet. For the snge vacancy, the frst reducton n stress s due to the tearng of the vacancy-contanng center ayer ((a-) and (b-)). Then, the second reducton s due to the tearng of a neghborng ayer ((a-2) and (b-2)). In ths case, the atom n the broken pece of the center ayer reacts wth the atom at the edge of the neghborng ayer and eads to the tearng of the neghborng ayer by dsturbng the zgzag edge surface (see IMECS 22

6 Fg. 7). The reatonshp between the tense strength of the center ayer wth a custer-type vacancy n the graphte and the tense strength of the graphene wth a custer-type vacancy s shown n Fg. 8. For a types of vacancy, the tense strength of the center ayer s amost equa to that of the graphene wth a smary szed vacancy. Ths means that the nterayer nteracton hardy affects the tense strength of the vacancy- Nomna stress σ x (GPa) Prstne Snge vacancy Doube vacancy Sextupe vacancy Nomna stran Fg. 5 Stress-stran curves of graphte contanng custer-type vacancy under zgzag tenson. Z (a-) Tearng of the center ayer wth a snge vacancy (a-2) Tearng of the neghborng ayer (a-3) Fracutre of graphte Z (b-) Tearng of the center ayer wth a snge vacancy (b-2) Tearng of the neghborng ayer (b-3) Fracutre of graphte σ x (GPa) Fg. 6 Stages of fracture progresson of graphte wth snge vacancy ((a-) (a-3): vewed n the drecton, (b-) (b-3): vewed n the drecton of sant.) Reacton between edge atoms Neghborng ayer Center ayer Neghborng ayer Fg. 7 Enargement of crced secton shown n Fg. 6(b-). Tense strength of the graphene sheet wth a vacancy (GPa) Graphene Graphte Number of atom defects Fg. 8 Tense strength of graphene wth vacancy and of the vacancy-contanng center n graphte, dependng on the sze of the vacancy, namey, the number of atomc defects contanng center ayer. IV. CNCLUSIN We performed MD smuatons of tense oadngs on vacancy-contanng graphene and graphte to nvestgate the nfuence of vacances on the mechanca propertes. It was found that for the custer-type vacancy, the reatonshp between the sze of the vacancy and the tense strength agree wth the reatonshp predcted usng Grffth s crteron. We demonstrated that the dfference n the dstrbutona form of vacances affects the tense strength. In addton, t was found that there s not a arge dfference between the tense strengths of vacancy-contanng graphene and that of a graphene sheet n graphte contanng a vacancy of a smar sze. REFERENCES [] C. Lee,. We, J. W. Kysar, J. Hone, Measurement of the Eastc Propertes and Intrnsc Strength of Monoayer Graphene, Scence, vo. 32, pp , Ju., 28. [2] H. L. Zhang, S. F. Wang, R. Wang, J. Jao, The Dsocatons n Graphene wth the Correcton form Lattce Effect, European Physca Journa B,vo.73, pp , Feb., 2. [3] S. K. Georgantznos, G. I. Gannopouos, N. K. Anfants, Numerca Investgaton of Eastc Mechanca Propertes of Graphene Structures, Materas and Desgn, vo. 3, pp , Dec., 2. [4] P. A. Thrower, The Study of Defects n Graphte by Transmsson Eectron Mcroscopy, Chemstry and Physcs of Carbon, vo.5, pp. 27, 969. [5] A. Hashmoto, K. Suenaga, A. Goter, K. Urta, and S. Ijma, Drect Evdence for Atomc Defects n Graphene Layers, Nature,vo.43, pp , Aug., 24. [6] J. sng and I. V. Shvets, Buk Defects n Graphte bserved wth a Scannng Tunneng Mcroscope, Surface Scence, vo.47, pp.45 5, Nov., 998. [7] J. R. ao, J. Stanszewsk, J. W. Gespe Jr., Tense Behavors of Graphene Sheets and Carbon Nanotubes wth Mutpe Stone-Waes Defects, Materas Scence and Engneerng A, vo. 527, pp , Jan., 2. [8] R. Grantab, V. B. Shenoy, R. S. Ruoff, Anomaous Strength Characterstcs of Tt Gran Boundares n Graphene, Scence, vo. 33, pp , Nov., 2. [9] C. H. Wong, Eastc Propertes of Imperfect Snge-waed Carbon Nanotubes under Axa Tenson, Computatona Materas Scence, vo. 49, pp , Jun., 2. [] S. Zhang, S. L. Meke, R. Khare, D. Troya, R. S. Ruoff, G. C. Schatz, and T. Beytschko, Mechancs of Defects n Carbon Nanotubes: Atomstc and Mutscae Smuatons, Physca Revew B, vo. 7, pp. 543, Mar., 25. [] S. L. Meke, D. Troya, S. Zhang, J-L L, S. ao, R. Car, R. Ruoff, G. C. Shatz, and T. Beytschko, The Roe of Vacancy Defects and Hoes n the Fracure of Carbon Nanotubes, Chemca Physcs Letters, vo. 39, pp. 43, Apr., 24. [2] D. W. Brenner,. A. Shenderova, J. A. Harrson, S. J. Stuart, B. N, S. H. Snnott, A Second-generaton Reactve Emprca Bond rder (REB) Potenta Energy Expresson for Hydrocarbons, Journa of Physcs Condensed Matter, vo. 4, pp , Jan., 22. [3]. A. Shenderova, D. W. Brenner, A. metchenko,. Su, L. H. ang and M. oung, Atomstc modeng of the fracture of poycrystane damond, Physca Revew B, vo. 6, no. 6, pp , Feb., 2. [4] S. J. Stuart, A. B. Tuten and J. A. Harrson, A reactve potenta for hydrocarbons wth ntermoecuar nteractons, Journa of Chemca Physcs, vo. 2, no. 4, pp , Jan., 2. [5] L.V. Woodcock, Isotherma moecuar dynamcs cacuatons for qud sats, Chemca Physcs Letters, vo., pp , Aug., 97. [6] Q.. Pe,.W. Zhang, V.B. Shenoy, A moecuar dynamcs study of the mechanca propertes of hydrogen functonazed graphene, Carbon, vo. 48, pp , Mar., 2. [7] F. Lu, P. Mng, and J. L, Ab nto cacuaton of dea strength and phonon nstabty of graphene under tenson, Phys. Rev. B, vo. 76, pp. 642, Aug., 27. [8] J.-W. Jang, J.-S. Wang and B. L, oung s moduus of graphene: A moecuar dynamcs study, Phys. Rev. B, vo. 8, pp. 345, Sep., 29.