Molecular Dynamics Analysis on Effects of Vacancies upon Mechanical Properties of Graphene and Graphite

Transcription

1 Engneerng Letters, 2:3, EL_2_3_9 Moecuar Dynamcs Anayss on Effects of Vacances upon Mechanca Propertes of Graphene and Graphte Akhko Ito and Shngo kamoto Abstract The mechanca propertes of graphene and graphte contanng vacances under tense oadng were nvestgated usng moecuar dynamcs smuatons. Two types of potenta functons were used n the smuatons: the second-generaton reactve emprca bond-order (REB) potenta for covaent C C bonds, and the Lennard-Jones potenta for the nterayer nteracton n graphte. The nfuence of the sze and dstrbuton of the vacances on the mechanca propertes of graphene and graphte were studed. It was found that the tense strengths of graphene and graphte are sgnfcanty decreased when they contan randomy dstrbuted vacances. Index Terms Graphene, graphte, moecuar dynamcs, vacancy C I. INTRDUCTIN ARBN-based materas can have exceent mechanca and eectrca propertes. Consequenty, ther appcaton to structura subassembes and nanoeectromechanca systems such as eectrochemca eectrodes and fed emsson has attracted consderabe nterest. Carbon materas such as damond, graphene, carbon nanotubes (CNTs), and fuerenes have a wde range of exceent propertes thanks to the dfferent types of bonds and atomstc structures contaned wthn them. In partcuar, graphene has rgdty and strength amost on a par to that of damond, as we as nove eectronc propertes that ncude hgh eectron mobty. Thus, the study of graphene and graphte made of graphene ayers has recenty ntensfed [1] [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 Manuscrpt receved May 1, 212. Ths work was supported n part by the Rng-Rng project of JKA. A. 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). S. 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). n order to produce hgh-performance carbon materas. Studes amng to carfy the reatonshp between atomc-scae defects and mechanca propertes have recenty ncreased n number. For exampe, the tense propertes of graphene and CNTs contanng mutpe Stone-Waes (SW) defects have been nvestgated usng moecuar dynamcs (MD) smuatons by ao et a. [7]. Such studes have carfed the reatonshp between the number of defects present and the mechanca propertes of a system. The nfuence of gran boundares on the tense strength of graphene has been nvestgated by Grantab et a. [8], whe MD smuatons on the tense oadngs of snge-waed CNTs wth vacances have been performed by Wong et a. [9]. The nfuence of snge and doube vacances on the tense strength has been nvestgated through moecuar mechancs (MM) cacuatons by Zhang et a. [1]. Zhang et a. compared ther resuts obtaned usng MM cacuatons to those obtaned by Meke usng quantum mechancs (QM) cacuatons [11]. However, the nfuence of vacances on the mechanca propertes of graphene and graphte has yet to be fuy carfed. Recenty, we eucdated the effect of vacancy sze on the mechanca propertes of graphene through MD smuatons [12]. In the present study, we aso nvestgate the effects of the sze and dstrbutona form of vacances n graphte on these same propertes. A. Potenta Functons II. METHD In the present study, we used two types of nteratomc potenta: the second-generaton reactve emprca bond order (2 nd REB) [13] and Lennard-Jones potentas. The 2 nd REB potenta for covaent C-C bonds s expressed as * [ VR ( rj ) BjVA ( rj )], E = (1) 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 (Advance onne pubcaton: 27 August 212)

2 Engneerng Letters, 2:3, EL_2_3_9 Force (nn) rgna (R mn = 1.7 Å) Ths work (R mn = 2. Å) Bond ength r j (Å) Fg. 1. Interatomc forces for the 2 nd REB potenta wth orgna R mn and modfed R mn (ths work).. ( ) ( ) Q V + R r fc r 1 Aexp( α r) r, (2) V A = 3 ( r) = fc ( r) B n exp( β nr) n= 1 where Q, A, α, B n, and β n represent constants. The functon f c (r) represents the cutoff functon that decreases monotonousy from 1 to as n f c ( r) 1, π = 1+ cos R, ( r R ) max mn R mn / 2, R mn r < R < r < R r > R where R mn = 1.7 Å and R max = 2. Å n the orgna 2 nd REB potenta. It s known that for the orgna 2 nd REB potenta, the nteratomc forces ncrease dramatcay at r = R mn and reach zero at r = R max because of the dscontnuty n the second dervatve of the cutoff functon, as shown n Fg. 1. Ths dramatc ncrease n the nteratomc force wth the orgna 2 nd REB potenta may greaty affect the tense strength. Therefore, n ths work, the cutoff parameter was set to 2. Å to avod any dramatc ncrease n the nteratomc force [14]. The other parameters, except for R mn, were set to the vaues proposed by Brenner [13]. The Lennard-Jones potenta for the nterayer nteracton n the graphte mode s expressed as mn max max, (3) 12 6 LJ 4 r r V = ε. (4) rj rj used accordng to the tense drectons. The anayss modes of perfect ZGR and AGR consst of 588 and 576 carbon atoms, respectvey, wth dmensons equa to those of a rea crystate n a typca carbon matera, as shown n Fg. 2. No perodc boundary condtons are mposed n our case and the anayss modes consst of two parts. The frst s referred to as 52Å 3Å : Carbon atom (a) ZGR mode (b) AGR mode Fg. 2. Confguratons of graphene used under zgzag and armchar tenson. ZGR: Zgzag graphene, AGR: Armchar graphene 3Å 52Å Fg. 3. Confguraton of graphte used under zgzag tenson n the drecton. A B A 3.35Å 51Å 24Å carbon atom Fg. 4. Schematc of the structure of graphte, wth the nterayer spacng shown. (a) ZGR-Snge vacancy Z (d) AGR-Snge vacancy 3Å The use of the 2 nd REB potenta and Lennard-Jones potenta are swtched accordng to the nteratomc dstance and bond order [15]. The vaue of ε was set to.284 ev and r was set to Å so that the nterpanar spacng n graphte at 3 K s 3.35 Å, whch s a known expermenta vaue [16]. (b) ZGR-Doube vacancy (e) AGR-Doube vacancy B. Anayss Mode Two types of graphene modes, referred to as zgzag graphene (ZGR) and armchar graphene (AGR) modes, are (c) ZGR-Sextupe vacancy (f) AGR-Sextupe vacancy Fg. 5. Anayss modes for graphene contanng custer-type vacances. ZGR: Zgzag graphene, AGR: Armchar graphene (Advance onne pubcaton: 27 August 212)

3 Engneerng Letters, 2:3, EL_2_3_9 (a) 1% - vacances 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 1 σ ( J = m V JV J + J F J ), (5) Ω (b) 2% - vacances (c) 4% - vacances Fg. 6. Anayss modes for the ZGR (zgzag graphene) contanng unformy dstrbuted vacances. the actve zone n whch the atoms move accordng to the nteractons wth ther neghborng atoms. The other encosed wthn the boxes shown n Fg. 2 s referred to as the boundary zone n whch the atoms are constraned. The thckness,, of the boundary zone s 3.a for the AGR mode and 1.5 3a for the ZGR mode, where a s the ength of the C=C bond n graphene. The anayss mode of perfect graphte used under zgzag tenson n the drecton, conssts of 4,116 carbon atoms, wth dmensons equa to those of a rea crystate of a typca carbon matera, as shown n Fg. 3. The graphte mode s made of seven ayers of graphene sheets that are stacked n an AB-type sequence wth an nterayer spacng of 3.35 Å, as shown n Fg. 4. We conducted two nvestgatons on the effects of vacances. The frst s on the sze of the vacancy. The anayss modes used of graphene wth custer-type vacances are shown n Fg. 5. These modes of graphte revea that the ZGR sheet wth a custer-type vacancy s aways the centra ayer. The dstrbutona form of the vacancy s aso nvestgated usng the ZGR mode. The anayss modes of the ZGR contanng unformy dstrbuted vacances are shown n Fg. 6. Each vacancy s a snge vacancy, set so that the dstance between neghborng vacances s dentca. Cacuatons for three vaues of vacancy densty, namey 1, 2, and 4%, were performed. The anayss modes of graphte wth 1, 2, and 4% vacances were constructed usng seven ayers of the ZGR mode wth the correspondng densty of vacances. The anayss modes of graphene and graphte contanng randomy dstrbuted vacances were set by removng carbon atoms n the actve zone by usng a pseudorandom number generator. C. Moecuar Dynamcs Smuatons We nvestgated the mechanca propertes of vacancy-contanng graphene and graphte usng MD smuatons under constant voume and temperature,.e., a canonca (NVT) ensembe. The equatons of moton of the atoms were tme-ntegrated usng the veocty Veret method. The veoctes of a atoms were adjusted smutaneousy usng the veocty scang method [17] so that the temperature of the object was mantaned at the preset temperature, T SET. The mass of a snge carbon atom, m, s kg. The tme step used was 1. fs. The atomc stress actng on each atom was cacuated to obtan the stress-stran curves and to vsuaze the stress 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 can then be cacuated by averagng over a carbon atoms n each system. Method of Tenson Loadng The nta postons of the atoms were such 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 were reaxed n unoaded states for 7, MD steps. The atoms n the boundary zone were then fxed n a drectons for the graphene modes and n ony the and drectons for the graphte modes. After constant dspacements were apped to the atoms n both of the boundary zones to smuate unaxa tense oadng n the drecton, the atoms n the actve zone were reaxed for 7, MD steps. The stran ncrement used, ε, was.4. The output stresses were samped for the ast 2, MD steps for each stran and then averaged. oung s modu were 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 MD smuatons on the tense oadngs of prstne zgzag graphene (p-zgr) and prstne armchar graphene (p-agr) at 3 K to verfy the vadty of our cacuaton method. The resuts are presented n Tabe I and Fg. 7. An average tense strength of 83 GPa was obtaned, whch s n cose agreement wth the 121 GPa cacuated by Pe et a. through MD smuatons [18] and wth the expermentay obtaned vaue of GPa [1]. The average oung s moduus of 836 GPa s wthn the range of resuts encompassed by those obtaned by densty functona theory [19] (1,5 GPa) and by experment [2] (5 GPa and 1 TPa). It s estmated that the ower vaue obtaned n ths work TABLE I MECHANICAL PRPERTIES F PRISTINE GRAPHENE Matera Tense strength oung s moduus (GPa) (GPa) p-agr p-zgr Average p-agr: prstne armchar graphene, p-zgr: prstne zgzag graphene. (Advance onne pubcaton: 27 August 212)

4 Engneerng Letters, 2:3, EL_2_3_9 Nomna stress, σ x (GPa) ZGR AGR Nomna stran Fg. 7. Stress-stran curves of prstne graphene under Armchar or Zgzag tenson. s due to the effect of mode sze on the eastc propertes of graphene [2]. B. Effect of Vacancy Sze on Mechanca Propertes of Graphene The mechanca propertes of vacancy-contanng zgzag graphene (v-zgr) and vacancy-contanng armchar graphene (v-agr) obtaned at 3 K are sted n Tabes II to IV, together wth the resuts from prevous studes on CNTs [9] [11]. The nomna stress-nomna stran curves for the v-zgr and v-agr are gven n Fgs. 8 and 9, respectvey. The resuts for prstne graphene are aso provded for reference. For the v-zgr, the decrease n tense strength s argest for graphene wth a doube vacancy, foowed by that for the sextupe, and then snge vacancy speces. In addton, the fracture stran for graphene wth a doube vacancy s the east of those studed. The decrease n tense strength reatve to that of prstne graphene s 29, 28, and 17% for the doube, sextupe, and snge v-zgr, respectvey. For the v-agr, the decrease n tense strength s argest for graphene wth a sextupe vacancy, foowed by the snge then doube vacancy v-agr. The decrease n tense strength reatve to that of prstne graphene s 32, 19, and 18% for the sextupe, snge, and doube vacancy v-agr, respectvey. Compared wth the resuts of prevous studes on CNTs usng MD [9], MM [1], and QM [11] cacuatons, the reductons n the tense strength of the v-zgr and v-agr wth a snge and doube vacancy n ths work are smar to those obtaned wth the MM and QM cacuatons and not cose to those obtaned prevousy by MD cacuatons. For both the v-zgr and the v-agr, the oung s moduus shows no sgnfcant change wth the change n vacancy sze. Snapshots of the tense oadngs of the ZGR and AGR are shown n Fgs. 1 and 11, respectvey. For the p-zgr (Fg. 1(a-2)), the dstrbuton of stress mmedatey before fracture s unform and the eve of stress s hgh. In comparson, n the v-zgr, the majorty of the stress occurs around the vacancy just before fracture, whch emerges from the crcumference of the vacancy. For a cases of AGR (Fg. 11), the fractures progress perpendcuar to the tense axs. We compared the cacuated resuts wth the Grffth s TABLE II TENSILE STRENGTHS F VACANC-CNTAINING ZGR AND [5,5] CNT ZGR [5,5] CNT(Carbon Nanotube) Matera σ B σ B [9] σ B [1] σ B [11] (MD, GPa) (MD, GPa) (MM, GPa) (QM, GPa) Prstne Snge vac. 75 ( 17%) 13 ( 1%) 7.4 ( 33%) 1 ( 26%) Doube vac. 64 ( 29%) 11 ( 3%) 71.3 ( 32%) 15 ( 22%) Sextupe vac. 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. TABLE III TENSILE STRENGTHS F VACANC-CNTAINING AGR AND [1,] CNT AGR [1,] CNT(Carbon Nanotube) Matera σ B σ B [9] σ B [1] σ B [11] (MD, GPa) (MD, GPa) (MM, GPa) (QM, GPa) Prstne Snge vac. 61 ( 19%) ( 26%) 11 ( 18%) Doube vac. 62 ( 18%) 85 ( 5.5%) 64.4 ( 26%) 17 ( 13%) Sextupe vac. 51 ( 32%) 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. TABLE IV THE UNG S MDULI F VACANC-CNTAINING GRAPHENE (UNIT: GPA) Matera ZGR AGR Prstne Snge vacancy 782 ( 1.5%) 868 ( 1.2%) Doube vacancy 765 ( 3.6%) 87 ( 1.%) Sextupe vacancy 767 ( 3.4%) 848 ( 3.6%) 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 ZGR (zgzag graphene) contanng a custer-type vacancy under zgzag tenson. Nomna stress, σ x (GPa) Prstne Snge vacancy Doube vacancy Sextupe vacancy Nomna stran Fg. 9. Stress-stran curves of the AGR (armchar graphene) contanng a custer-type vacancy under zgzag tenson. (Advance onne pubcaton: 27 August 212)

5 Engneerng Letters, 2:3, EL_2_3_9 (a-1) Inta structure of the prstne ZGR (a-2) Just before fracture (a-3) Fracture (c-1) Inta structure of the ZGR wth a doube vacancy (b-1) Inta structure of the ZGR wth a snge vacancy (b-2) Just before fracture (b-3) Fracture (d-1) Inta structure of the ZGR wth a sextupe vacancy (b-1) Inta structure of the AGR wth a snge vacancy (a-2) Just before fracture (b-2) Just before fracture (a-3) Fracture (b-3) Fracture (c-1) Inta structure of the AGR wth a doube vacancy (a-1) Inta structure of the prstne AGR (d-1) Inta structure of the AGR wth a sextupe vacancy σ x (GPa) (c-2) Just before fracture (d-2) Just before fracture (c-3) Fracture (d-3) Fracture Fg. 1. Stages of fracture progresson n the ZGR (zgzag graphene) contanng custer-type vacances. (a-1) (a-3): prstne, (b-1) (b-3): snge vacancy, (c-1) (c-3): doube vacancy, and (d-1) (d-3): sextupe vacancy. σ x (GPa) (c-2) Just before fracture (d-2) Just before fracture (c-3) Fracture (d-3) Fracture (6) 1.2 Reatve strength Eγ s, d Fg. 11. Stages of fracture progresson n the AGR (armchar graphene) contanng custer-type vacancy. (a-1) (a-3): prstne, (b-1) (b-3): snge vacancy, (c-1) (c-3): doube vacancy, and (d-1) (d-3): sextupe vacancy. crteron n order to verfy ther vadty. The theoretcay dea strength σmax for brtte fracture s expressed as σ max = MD cacuaton 1. Grffth's crteron where E s oung s moduus, γs s the surface energy, and d s the nteratomc dstance. Consequenty, accordng to the Grffth s crteron, the strength of a matera contanng a crack of ength 2C s expressed as Number of atom defects Fg. 12. Reatve strengths and szes of vacancy,.e., the number of atom defects n the zgzag graphene (ZGR) obtaned usng MD and Grffth s crteron. 2 Eγs. πc (7) 1.2 The reatve strength, σre,.e., the strength of the crack-contanng matera reatve to the theoretcay dea strength s obtaned by dvdng σf by σmax as Reatve strength σf = MD cacuaton 1. Grffth's crteron σ re = 2d. πc. (8) A pot of the reatve strength of the ZGR and AGR aganst Number of atom defects Fg. 13. Reatve strengths and szes of vacancy,.e., the number of atom defects n the armchar graphene (AGR) obtaned usng MD and Grffth s crteron. (Advance onne pubcaton: 27 August 212)

6 Engneerng Letters, 2:3, EL_2_3_9 the number of atomc defects s shown n Fgs. 12 and 13, respectvey. The resuts of the MD cacuatons agree we wth the predcted vaues usng Grffth s crteron for both the ZGR and AGR. C. Effect of Vacancy Sze on Mechanca Propertes of a Graphene Sheet n Graphte The mechanca propertes of a vacancy-contanng graphene sheet n graphte are sted n Tabe V. The stress-stran curves of the graphte are gven n Fg. 14. The resuts for prstne graphte are aso provded for reference. In every case, reductons n stress occur before the fracture. For the custer-type vacancy, the tense strengths of the centra ayer n the graphte and the graphene are shown n Fg. 15. For a types of vacancy studed, the tense strength of the centra ayer s amost equa to that of graphene wth a smary szed vacancy. Ths means that the nterayer nteracton has neggbe effect on the tense strength of the vacancy-contanng centra ayer. Snapshots of the graphte wth a sextupe vacancy are shown n Fg. 16. It was found that the reducton n stress before fracture was due to a tear n the graphene sheet. For the TABLE V MECHANICAL PRPERTIES F VACANC-CNTAINING GRAPHITE Matera σ B E (GPa) (GPa) Prstne Snge vacancy 78 ( 14%) 815 (.1%) Doube vacancy 65 ( 28%) 84 ( 1.4%) Sextupe vacancy 7 ( 23%) 788 ( 3.4%) σ B s the tense strength. E s the oung s moduus. 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. 14. Stress-stran curves of graphte contanng custer-type vacances under zgzag tenson. Tense strength of the graphene sheet wth a vacancy (GPa) Graphene Graphte Number of atom defects Fg. 15. Tense strength of graphene wth a vacancy and of the vacancy-contanng centra ayer n graphte, whch depend on the sze of the vacancy,.e., the number of atomc defects. sextupe vacancy, the reducton n stress s due to the tearng of the vacancy-contanng centra ayer ((a-1) and (b-1) n Fg.16). The stress then reached a maxmum before the other ayers caused fractures. In ths case, the atom n the broken pece of the centra ayer reacts wth the atom at the edge of the neghborng ayer, eadng to the tearng of the neghborng ayer by dsturbng the zgzag-edge surface (see Fg. 17). D. Infuence of Dstrbutona Form of Vacances The reatonshp between the tense strength and densty of vacances for both graphene and graphte wth unformy or randomy dstrbuted vacances s shown n Fg. 18. For the randomy dstrbuted vacances, the average vaues of the two resuts cacuated usng the modes wth dfferent vacancy arrangements s potted. The error bar n the graph represents the range between these two vaues. There s no dfference n the tense strength of graphene and graphte. The tense strength decreases wth an ncrease n vacancy densty. The reducton n the tense strength stands at about 6% for a vacancy densty of 4% wth the random vacancy dstrbuton. Ths s neary doube the reducton observed for the tense strength of hydrogen (H)-functonazed graphene [18]. In comparson, the oung s moduus sghty decreases wth the ncrease n vacancy densty for both graphene and graphte (see Fg. 19). The reducton n the oung s moduus s about 2% for a vacancy densty of 4%. Ths s neary 4 tmes greater than the reducton n oung s moduus prevousy determned for H-functonazed graphene. It s therefore reasonabe to assume that the propertes of graphene and graphte are more senstve to vacances than to H-coverage, snce whereas a vacancy mpes the ack of an atom, H-coverage refers to the converson of oca carbon bondng σ x Z (a-1) Tearng of the centra ayer wth a snge vacancy (a-2) Tearng of the neghborng (GPa) ayer (a-3) Fracutre of graphte Z (b-1) Tearng of the centra ayer wth a sextupe vacancy (b-2) Tearng of the neghborng ayer (b-3) Fracutre of graphte Fg. 16. Stages of fracture progresson of graphte wth a sextupe vacancy ((a-1) (a-3): vewed n the drecton, (b-1) (b-3): vewed n the drecton of sant.) Neghborng ayer Centra ayer Neghborng ayer Reacton between edge atoms Fg. 17. Enargement of the crced secton shown n Fg. 16(b-1). (Advance onne pubcaton: 27 August 212)

7 Engneerng Letters, 2:3, EL_2_3_9 from sp2 to sp3 hybrdzaton. Snapshots of the graphene wth unformy dstrbuted vacances durng tense oadng are shown n Fg. 2. For every case studed heren, the majorty of the stress occurs around each vacancy just before the fracture, n the same manner wtnessed for the graphene wth a snge vacancy. Fractures then occur, startng from a vacancy and progressng towards neghborng vacances. The progresson drecton of the fracture s perpendcuar to the tense axs n a cases. Conversey, the snapshots of graphene wth randomy dstrbuted vacances durng the tense oadng (Fg. 21) show that the fracture starts from the area where the vacances gather. Thereafter, the progresson drecton of the fracture s then random. IV. CNCLUSIN Tense strength (GPa) We performed MD smuatons of tense oadngs on vacancy-contanng graphene and graphte to nvestgate the nfuence of vacances on ther mechanca propertes. It was found that for custer-type vacances, 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 the vacances affects the tense strength. In addton, no sgnfcant dfference was found between the tense strengths of vacancy-contanng graphene and of a graphene sheet n graphte contanng a smary szed vacancy. For the unformy or randomy dstrbuted vacances, there s tte dfference n the tense strengths between graphene and graphte. 1 (a-3) Fracture (b-1) Inta structure of (b-2) Just before graphene wth fracture 2% unformy dstrbuted vacances (b-3) Fracture (c-1) Inta structure of (c-2) Just before graphene wth fracture 4% unformy dstrbuted vacances (c-3) Fracture σ x (GPa) Fg. 2. Stages of fracture progresson n graphene contanng unformy dstrbuted vacances. The densty of vacances s 1% ((a-1) (a-3)), 2% ((b-1) (b-3)), and 4% ((c-1) (c-3)). (a-1) Inta structure of (a-2) Just before graphene wth fracture 1% randomy dstrbuted vacances (a-3) Fracture (b-1) Inta structure of (b-2) Just before graphene wth fracture 2% randomy dstrbuted vacances (b-3) Fracture (c-1) Inta structure of (c-2) Just before graphene wth fracture 4% randomy dstrbuted vacances σ x (GPa) (c-3) Fracture Graphene - Unform Graphene - Random Graphte - Unform Graphte - Random Densty of vacances (%) Fg.18. Tense strengths of graphene and graphte aganst vacancy densty. Error bars ustrate the range used to cacuate the average strengths shown. oung's moduus (GPa) (a-2) Just before (a-1) Inta structure of graphene wth fracture 1% unformy dstrbuted vacances Graphene - Unform Graphene - Random Graphte - Unform Graphte - Random 8 [1] [2] 7 6 [3] REFERENCES Fg. 21. Stages of fracture progresson n graphene contanng randomy dstrbuted vacances. The densty of vacances s 1% ((a-1) (a-3)), 2% ((b-1) (b-3)), and 4% ((c-1) (c-3)). 5 Densty of vacances (%) Fg. 19. oung s modu of graphene and graphte aganst vacancy densty. Error bars ustrate the range used to cacuate the average modu shown. [4] [5] [6] C. Lee,. We, J. W. Kysar, and J. Hone, Measurement of the eastc propertes and ntrnsc strength of monoayer graphene, Scence, vo. 321, pp , Ju. 28. H.-L. Zhang, S.-F. Wang, R. Wang, and J. Jao, The dsocatons n graphene wth the correcton from attce effect, Eur. Phys. J. B, vo.73, pp , Feb. 21. S. K. Georgantznos, G. I. Gannopouos, and N. K. Anfants, Numerca nvestgaton of eastc mechanca propertes of graphene structures, Mater. Desgn, vo. 31, pp , Dec. 21. P. A. Thrower, The study of defects n graphte by transmsson eectron mcroscopy, Chem. Phys. Carbon, vo. 5, pp. 217, A. Hashmoto, K. Suenaga, A. Goter, K. Urta, and S. Ijma, Drect evdence for atomc defects n graphene ayers, Nature, vo. 43, pp , Aug. 24. J. sng and I. V. Shvets, Buk defects n graphte observed wth a scannng tunneng mcroscope, Surf. Sc., vo. 417, pp , Nov (Advance onne pubcaton: 27 August 212)

8 Engneerng Letters, 2:3, EL_2_3_9 [7] J. R. ao, J. Stanszewsk, and J. W. Gespe, Jr., Tense behavors of graphene sheets and carbon nanotubes wth mutpe Stone-Waes defects, Mater. Sc. Eng. A, vo. 527, pp , Jan. 21. [8] R. Grantab, V. B. Shenoy, R. S. Ruoff, Anomaous strength characterstcs of tt gran boundares n graphene, Scence, vo. 33, pp , Nov. 21. [9] C. H. Wong, Eastc propertes of mperfect snge-waed carbon nanotubes under axa tenson, Comp. Mater. Sc., vo. 49, pp , Jun. 21. [1] 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, Phys. Rev. B, vo. 71, pp , Mar. 25. [11] 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 fracture of carbon nanotubes, Chem. Phys. Lett., vo. 39, pp , Apr. 24. [12] A. Ito and S. kamoto, Mechanca propertes of vacancy-contanng graphene and graphte usng moecuar dynamcs smuatons, Lecture Notes n Engneerng and Computer Scence: Proceedngs of The Internatona MutConference of Engneers and Computer Scentsts 212, IMECS 212, Hong Kong, March, 14 16, 212, pp [13] D. W. Brenner,. A. Shenderova, J. A. Harrson, S. J. Stuart, B. N, and S. H. Snnott, A second-generaton reactve emprca bond order (REB) potenta energy expresson for hydrocarbons, J. Phys. Condens. Mat., vo. 14, pp , Jan. 22. [14]. A. Shenderova, D. W. Brenner, A. metchenko,. Su, L. H. ang, and M. oung, Atomstc modeng of the fracture of poycrystane damond, Phys. Rev. B, vo. 61, no. 6, pp , Feb. 2. [15] S. J. Stuart, A. B. Tuten, and J. A. Harrson, A reactve potenta for hydrocarbons wth ntermoecuar nteractons, J. Chem. Phys., vo. 112, no. 14, pp , Jan. 2. [16] W. Ruand, -ray studes on the structure of graphtc carbons, Acta Cryst., vo.18, pp , Jun [17] L. V. Woodcock, Isotherma moecuar dynamcs cacuatons for qud sats, Chem. Phys. Lett., vo. 1, pp , Aug [18] Q.. Pe,. W. Zhang, and V. B. Shenoy, A moecuar dynamcs study of the mechanca propertes of hydrogen functonazed graphene, Carbon, vo. 48, pp , Mar. 21. [19] 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. 6412, Aug. 27. [2] J.-W. Jang, J.-S. Wang, and B. L, oung s moduus of graphene: a moecuar dynamcs study, Phys. Rev. B, vo. 8, pp , Sep. 29. (Advance onne pubcaton: 27 August 212)