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Energy dissipatio

aState Key Laboratory for Mechanical Behavi

Xi'an 710049, Shaanxi, China. E-mail: mafei

cnbCollege of Physics and Information Techn

710062, Shaanxi, ChinacDepartment of Physics and Materials Scienc

Avenue, Kowloon, Hong Kong, China. E-maidDepartment of Physics and Opt-electronic

Science, Xi'an 710065, Shaanxi, China

Cite this: RSC Adv., 2016, 6, 60856

Received 27th February 2016Accepted 15th June 2016

DOI: 10.1039/c6ra05167g

www.rsc.org/advances

60856 | RSC Adv., 2016, 6, 60856–608

n in mechanical loading of nano-grained graphene sheets

Zhi Yang,a Yuhong Huang,b Fei Ma,*ac Yaping Miao,ac Hongwei Bao,a Kewei Xu*ad

and Paul K. Chu*c

A molecular dynamics (MD) simulation illustrates that different from single-crystal graphene sheets, the

loading and unloading stress–strain curves of nanocrystalline ones do not coincide with each other,

indicating substantial energy dissipation due to irreversible structural changes in the grain boundaries. An

energy dissipation coefficient is proposed to quantitatively describe the effects of the grain size,

temperature and strain rate dependent irreversible breaking and reforming of bonds in GBs, realignment

of grain orientation, lattice-shearing-induced phase transformation, and formation of Stone–Wales

defects and vacancies near GBs. The energy dissipation coefficient increases as the grain size decreases,

especially at high temperature and low strain rate, and consequently, the reversibility of nanocrystalline

graphene sheets under mechanical loading deteriorates compared to single-crystal graphene.

1. Introduction

Graphene has emerged as one of the most active research eldssince its discovery in 2004 and it has many potential applica-tions because of its unique properties.1 Graphene has a Young'smodulus up to 1.0 TPa, an elastic strain limitation of 20% andan intrinsic strength of 130 GPa,2 high thermal conductivityover 5000 W mK�1, and a fast carrier mobility of 200 000 cm2

V�1 s�1.3 Furthermore, the physical properties are tunablethrough elastic strain engineering. Strain in lattice couldchange the energy band structure and the electronic states,resulting in excellent electrical and optical properties.4 A strainparallel to the C–C bonds of about 12.2% will open a band gapof 0.486 eV, which is indispensable for graphene to be appliedin eld effect transistors.5,6 Levy et al. found that graphenenano-bubbles on a platinum (111) surface exhibited an enor-mous pseudo-magnetic eld of about 300 tesla due to straineffect.7 Moreover, graphene is also an excellent candidate forthe applications in next generation touch screen, mechanicalresonators and wearable electronics.8 Sang-Hoon Bae fabricateda transparent strain sensor based on graphene by using reactiveion etching and stamping techniques, and demonstrated thepiezoresistive response up to a strain of 7.1% induced by the

or of Materials, Xi'an Jiaotong University,

@mail.xjtu.edu.cn; kwxu@mail.xjtu.edu.

ology, Shaanxi Normal University, Xi'an

e, City University of Hong Kong, Tat Chee

l: paul.chu@cityu.edu.hk

Engineering, Xi'an University of Arts and

61

motion of ngers.9 Keun-Young Shin developed a simpleapproach to fabricate exible and transparent graphene lms inthe large-scale via inkjet printing and vapor deposition process(VDP).10 In addition, the nanocomposites of polymer and gra-phene oxide exhibit a dramatic improvement in both electronicand mechanical properties.8,11 In order to fully make use ofthe excellent mechanical–electrical and mechanical–opticalcoupling effects, it is indispensable to explore and understandthe deformation and mechanical behaviors of graphene sheets.

The as-prepared graphene sheets commonly contain variousdefects,12–16 and the graphene sheets fabricated by chemicalvapor deposition (CVD) are usually polycrystalline. The grainboundaries (GBs) play dominant roles in the mechanical, elec-trical and chemical properties.17 Nano-indentation illustrateda slight reduction in the mechanical strength due to GBs,18,19

but some found that nanocrystalline graphene sheets havealmost constant fracture strain and stress, independent on thegrain size.20 The complex deformation behavior of nano-crystalline graphene sheets may be related to the structuralevolution in GBs. Similar to metals, the GBs in polycrystallinegraphene sheets can be considered as an array of edge dislo-cations but are usually disrupted by defects, and the dislocationdensity increases with increasing misorientation angle betweenthe GBs.21,22 A large misorientation angle induces overlapping ofneighboring dislocations and the strain eld is compensated.23

As a result, large-angle tilt boundaries that have a large densityof defects are as strong as the pristine one and even strongerthan those with low-angle boundaries having fewer defects.24,25

Using nonequilibrium molecular dynamics (NEMD), Cao andBagri illustrated that the thermal conductivity and Kapitzaconductance is misorientation angle dependent.26,27 The rst-principles calculations and classical molecular dynamics

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simulations have revealed an enhanced defect reactivityinduced by an inhomogeneous strain eld at grain bound-aries.28 For nanocrystalline graphene sheets with GBs disruptedby various kinds of defects, the situation is more complicated.In this paper, the energy dissipation during mechanical defor-mation of nanocrystalline graphene sheets is described and anenergy dissipation coefficient (h) is proposed to quantitativelydescribe the stability of the GBs in nanocrystalline graphene. Itshows that energy dissipation increases with the grainboundary density, temperature and tensile strain, but isdependent on the strain rate little.

2. Simulation and modeling

The models of nanocrystalline graphene sheets with randomlydistributed grain size and orientation are constructed by Vor-onoi tessellation.29 A Voronoi tessellation represents a collec-tion of convex polygons isolated by planar cell wallsperpendicular to lines connecting neighboring nucleation sites.Each cell is lled with randomly oriented graphene domainsand the atoms adjacent to the planar cell walls are taken as ingrain boundaries (GBs). The initial C–C bond length is set as1.42 A that is the same as the experimental value. If the sepa-ration between two atoms in GBs is smaller (<1.41 A), one ofthem will be removed. But an atom will be added if there isa large void in GBs. In such a scheme, the GBs as well as theatomic congurations in GBs are distinct for each model evenwith the same average grain size, and accidental errors might be

Fig. 1 (a) Typical nanocrystalline model under tensile loading; (b) enlargatoms in the grain boundaries are red; (c) grain boundary densities in nanmisorientation angle between adjacent grains.

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produced in evaluating the energy dissipation in GBs and themechanical properties. So ve models are built for each averagegrain size, they are all used in the simulation of deformation,and the simulated physical parameters are averaged to avoid theaccidental errors. All the nanocrystalline graphene sheets are 20nm� 20 nm in size. A set of Voronoi tessellations with N� N (Ntakes 2, 3,., 10) grains are adopted to represent the nano-crystalline graphene sheets with an average grain size of 10.0nm, 6.66 nm, 5.0 nm, 4.0 nm, 3.33 nm, 2.86 nm, 2.50 nm, 2.22nm, and 2.0 nm, respectively. A typical atomic congurationwith 3.33 nm grains is exhibited in Fig. 1(a). Fig. 1(b) shows theenlarged image of the regions marked by the red rectangle inFig. 1(a) and the grain boundaries are shown in red. Larger andsmaller carbon rings rather than hexagonal ones that affect thedeformation behavior can be observed from the GBs. Fornanocrystalline graphene sheets, the GBs are indeed linedefects and are usually composed of topological 5–7 ring pairs.So the width of GBs is almost a constant, which is approximatelyequal to the width of 5–7 ring pairs of 2.84–3.26 A and atoms inthese regions are taken as the ones in GBs. Accordingly, thegrain boundary density (r) is dened as the length of GBs perunit area to describe the amount of GBs involved in nano-crystalline graphene sheets in this work [Fig. 1(c)]. Although thedenition is not very exact, this quantity could be used to studythe size dependent behaviors in nanocrystalline graphene tosome degree. Fig. 1(d) displays the distribution of misorienta-tion angles in the simulation models, they are almost randomlydistributed in the range of 0–30� for any model and the average

ed image of the region shown by the red rectangle in panel (a) and theocrystalline graphene sheets as a function of grain size; (d) statistic of

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misorientation angle is about 15.95�. The statistic results of ourmodels are similar to those obtained by others.29,30

MD simulation is carried out using the LAMMPS (Large-scaleAtomic/Molecular Massively Parallel Simulator) package. Theinteractions between carbon atoms are described by the adap-tive intermolecular reactive empirical bond order (AIREBO)potential which can accurately describe the interactionsbetween carbon atoms as well as bond breaking and reform-ing.31 The cutoff parameter describing the short-range C–Cinteraction is selected to be 2.0 A in order to avoid spuriouslylarge bonding forces and nonphysical results at large defor-mation.32 The Nose–Hoover thermostat is utilized to account forthe thermal effect.33 Uniaxial tensile loading is applied alongthe x axis using the deformation-control method.34 The atomsare allowed to move freely along the y axis and periodicboundary conditions are applied along the two in-plane direc-tions. The interlayer separation of graphite, 3.4 A, is taken as theeffective thickness of the monolayer graphene. On the atomiclevel, the stress is computed according to the virial theorem.35

Prior to uniaxial tensile loading, the nanocrystalline graphenesheets are fully relaxed to an equilibrium state in theisothermal–isobaric ensembles at 3000 K for 20 ps initially andthen relaxed at a given temperature for 600 ps. A time step of 1 fsis employed in the MD simulation and a Poisson's ratio of 0.165is used.36 The reversibility is evaluated by releasing the nano-crystalline graphene sheets from a tensile strain of 5–25% ata given temperature and strain rate.

3. Results and discussion

Fig. 2(a) presents the stress–strain curves of the nanocrystallinegraphene sheet with a grain size of 3.33 nm under uniaxialloading. As shown by the sharp reduction in stress at a strain of21.04% (the black solid line), destructive brittle fracture takesplace in the nanocrystalline graphene sheet. The fracture stresssF is about 75 GPa, which is smaller than that of single-crystalgraphene. The red dotted line shows the stress–strain curve inthe unloading process from a strain of 15%. The stress–straincurve in the unloading stage does not coincide with that in theloading stage, and a residual strain of 2.45% is observed aer

Fig. 2 (a) Typical loading–unloading stress–strain curves; (b) energy dissiloading–unloading stress–strain curves of graphene sheets with differen

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one cycle of loading and unloading. It means that the strainedgraphene sheet cannot recover to the initial state completely.This is different from the observation from an ideal single-crystalline graphene in which the stress–strain curves in theloading and unloading processes are completely coincidentwith each other and there is no residual strain if the strainedgraphene is unloaded from a strain smaller than the fracturestrain. The irreversibility should be related to the irreversiblestructural evolution in the GBs such as, bond breaking andreforming, bond angle adjustment, and out-of-plane stretchingand it will be discussed later.

According to the principle of work and energy, themechanical work W in the loading process is transferred intothe elastic strain energy in the lattice and chemical energy in theGBs. The elastic strain energy in the lattice is released in theunloading process, but the chemical energy due to bondbreaking and reforming in the GBs cannot be released.37 Thearea enclosed by the stress–strain curves in the loading andunloading processes describes the dissipated energy dW and anenergy dissipation coefficient (h ¼ dW/W) is proposed toquantitatively describe the process. The loading and unloadingstress–strain curves of nanocrystalline graphene sheets withdifferent grain sizes are plotted in the inset of Fig. 2(b).Accordingly the energy dissipation coefficient is calculated andthe results are shown in Fig. 2(b) as a function of grainboundary densities. As the grain size decreases, the density ofGBs increases, and the irreversible structural evolution duringthe loading process becomes more prominent. Consequently,the energy dissipation coefficient increases. For instance, in thegraphene sheet with 10 nm grains corresponding to a grainboundary density of 0.016 nm�1, the average energy dissipationcoefficient is 0.152. It increases to 0.19 by 25% in the graphenesheet with 2 nm grains and grain boundary density of 0.235nm�1. Similar energy dissipation in mechanical loading hasalso been observed from traditional metals but it is produced bymotion of dislocations,38 domain wall movement,39 grainboundary sliding,40 deformation twinning,41 and phasetransformation.42

Fig. 3(a) displays the evolution of the atomic conguration inthe graphene sheets with a grain size of 4.0 nm during the

pation coefficients (h) as a function of grain boundary densities with thet grain boundary densities presented in the inset.

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Fig. 3 (a) Snapshots of the atomic configurations in a loading–unloading cycle; (b) structural evolution in local regions during the loading andunloading processes.

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loading and unloading processes. The map is colored accordingto the potential energy of each atom. The potential energyincreases with tensile strain up to 15% gradually, particularly atGBs, and then decreases upon unloading. When the stress isreduced to zero, a residual strain about 2.1% remains. Fig. 3(b)presents the enlarged atomic congurations of a representativeregion at three stages: (1) at a strain of 0% before loading, (2)loaded up to a strain of 15%, and (3) unloaded down to 2.1%.The red, black, and blue arrows track the evolution of theatomic congurations in the vicinity of the three defects whichare metastable and can be maintained at low temperature,29 butthey may be changed to low-energy congurations at a hightemperature or under mechanical loading.30 When the strain isincreased up to 15%, a new bond is formed at the le of thelarge ring marked by the red arrow to promote the formation ofstable pentagon rings, while bond breaking and reforming takeplace subsequently in the local region indicated by the bluearrow. This results in the transformation from a heptagon–octagon topological defect into a pentagon–heptagon one. Inthe region indicated by the black arrow, a distorted octagon ringchanges to a regular one during tensile loading. Moreover,lattice-shearing-induced phase transformation from hexagonalstructure to orthorhombic one occurs, accompanied withStone–Wales defects and vacancies. Since the atoms in GBs havehigher energy, and they tend to reconstruction, which mightpromotes the grain boundary sliding and realignment, similarto that in metals. Hence, the nal atomic congurations in thelocal regions cannot recover to the original states uponunloading owing to: (1) breaking and reforming of bonds inGBs, (2) realignment of grain orientation, (3) lattice-shearing-induced phase transformation, (4) formation of Stone–Walesdefects and vacancies near GBs. These result in a residual strainaer unloading, but the residual strain is reduced in the second

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loading and unloading to some degree. Besides grain bound-aries, impurities and voids have also been observed in poly-crystalline graphene sheets fabricated by chemical vapordeposition, and so on. The movements of impurities and voidsshould also result in energy dissipation. However, in the work ofthis manuscript, we mainly focus on the inuences of GBs onthe plastic deformation as well as on the energy dissipation innanocrystalline graphene sheets, and thus only GBs areproduced in the initial simulation models. The effects of otherdefects induced in the deformation process cannot becompletely avoided, but it is difficult to be evaluated.

Bond breaking and reforming in GBs are sensitive to thetemperature and strain rate in mechanical loading.43 Therefore,the energy dissipation coefficient should also be temperatureand strain rate dependent. Taking the graphene sheets withgrains in the sizes of 2.2 nm, 4.0 nm and 6.6 nm as examples,two sets of MD simulations are performed: (1) in the tempera-ture range of 10–1500 K but at a given strain rate of 108 s�1 and(2) in the strain rate range of 108 to 1010 s�1 but at a giventemperature of 300 K. Fig. 4(a) and (b) show the energy dissi-pation coefficients as a function of temperature and strain rate,respectively. As shown in Fig. 4(a), the energy dissipation coef-cients increase with temperature almost linearly, for example,from 0.075 at 10 K to 0.38 at 1500 K by 407% for the graphenesheet with 4.0 nm grains. Kinetically, the atoms in the GBsoscillate more at higher temperature. The energy barrier can beovercome more easily to promote bond breaking, rotation, andreforming in the GBs in addition to the formation of polygonalholes44 and buckling along the GBs.45 As shown in Fig. 4(b), theenergy dissipation coefficients decrease with increasing strainrate slightly. At lower strain rate, the atoms in the GBs haveenough time to rearrange during tensile loading and the irre-versible structural evolution is more substantial thereby leading

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Fig. 4 (a) Energy dissipation coefficients of nanocrystalline graphene sheets at different temperature; (b) logarithmic plots of the energydissipation coefficients a function of strain rate.

Fig. 5 (a) The loading stress–strain curves and those unloaded from different maximum strains; (b) energy dissipation coefficient at differenttemperatures.

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to a larger energy dissipation coefficient.44 However, thedependence on the strain rate is not appreciable. Furthermore,the graphene sheets with smaller grains have larger grainboundary densities [Fig. 1(a)] and irreversible bond rearrange-ment is more likely in the GBs. Hence, the energy dissipationcoefficient becomes more sensitive to the temperature andstrain rate in the graphene sheets with smaller grains.

Large strain usually leads to more structural rearrangementin the GBs and hence, energy dissipation is more substantial.Unloading from different strain values is simulated in thetemperature range of 10–1500 K. Fig. 5(a) shows the simulatedstress–strain curves of graphene sheets with a grain size of 4.0nm at 300 K. Elastic lattice distortion dominates at small strain,but bond rearrangement in the GBs becomes more signicantwith increasing strain. Consequently, the residual strainincreases from 0.23% to 2.1% and the enclosed area by theloading and unloading stress–strain curves is substantiallyenlarged. The strain from where the graphene sheets areunloaded increases from 5% to 15% and the energy dissipationcoefficients increase with the maximum unloading strain,particularly at a high temperature [Fig. 5(b)]. For example, theenergy dissipation coefficient at 300 K increases by 62.8% whenthe maximum unloading strain increases from 5% to 15%,whereas it increases by 115% at 1500 K. Tight-binding MDsimulation indicates that evaporation of carbon dimers and

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sequential Stone–Wales transformation of carbon bonds lead torapid motion and annihilation of the GBs particularly thosewith large carbon rings.46 The larger energy dissipation coeffi-cient can be attributed to enhanced reconstruction of the localatomic conguration in the GBs.

4. Conclusion

In summary, MD simulation is conducted to study the defor-mation behavior of nanocrystalline graphene sheets. Differentfrom single-crystal graphene sheets, the loading and unload-ing stress–strain curves do not coincide with each other andnonzero residual strain is usually observed despite unloadingfrom a strain much smaller than fracture strain. It indicatesenergy dissipation and irreversible structural changes. Themechanical work W in the loading process is converted intoelastic strain energy in the lattice and chemical energy in theGBs and the elastic strain energy in the lattice is released inthe unloading process, but the chemical energy due to bondbreaking and reforming in the GBs cannot be released. Theenergy dissipation increases with reducing grain size, espe-cially at a high temperature, and the deformation reversibilityof nanocrystalline graphene deteriorates consequently. Ourresults offer a potential guidance for NEMS design anddevelopment.

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Acknowledgements

This work was jointly supported by National Natural ScienceFoundation of China (Grant No. 51271139, 51471130,51302162), Fundamental Research Funds for the CentralUniversities, and City University of Hong Kong AppliedResearch Grant (ARG) No. 9667104.

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