strength of graphenes containing randomly dispersed vacancies

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Acta Mech 223, 669678 (2012)DOI 10.1007/s00707-011-0594-8

Konstantinos I. Tserpes

Strength of graphenes containing randomly dispersedvacancies

Received: 1 July 2011 / Revised: 22 November 2011 / Published online: 18 December 2011 Springer-Verlag 2011

Abstract In the present work, the tensile strength of graphenes containing randomly dispersed vacancies ispredicted using an atomistic-based continuum progressive fracture model. The concept of the model is based onthe assumption that graphene, when loaded, behaves like a plane-frame structure. The finite element method isused to model the structure of graphene and the modified Morse interatomic potential to simulate the nonlinearbehavior of the CC bonds. Randomly dispersed vacancies (1 missing atom) are introduced into grapheneusing a random numbers algorithm. Graphenes are subjected to incremental uniaxial tension. The model iscapable of simulating fracture evolution considering defect interaction. The effects of size, chirality, defectdensity and defect topology on the Youngs modulus, strength and failure strain of graphenes are examined.Computed results reveal that vacancies may counterbalance the extraordinary mechanical properties of graph-ene, since 4.4% of missing atoms, corresponding to 13.2% of missing bonds, result in a 50% reduction inYoungs modulus and tensile strength of the material. Also found is a secondary effect of defect topology.

1 Introduction

Graphene is an atomic-scale honeycomb lattice composed of carbon atoms. Although graphene is consideredto be the parent material of carbon nanotube, which is probably the most famous material of the twenty-firstcentury, only recently, it has been obtained in a stable crystalline form by Geim and Novolesov [1]. Thisachievement is so important that was awarded by the Nobel Prize in Physics for 2010.

Graphene is not only the thinnest material ever but also the strongest. Recently, Leeet al.[2] have measured,by nanoindentation in an atomic force microscope, the Youngs modulus and intrinsic strength of defect-freegraphene monolayer to be 1.0 TPa and 130 GPa, respectively. This finding gives certainly a boost to the inves-tigation being conducted for establishing graphene and carbon nanotubes as reinforcements, since it is the firsttime that the suspected extraordinary mechanical properties of these materials are experimentally verified. Inthe past, similar values have been also reported for carbon nanotubes but only by simulations (Fig. 1). Contrary

to simulations, experiments give a Youngs modulus of 0.45TPa and a strength of 40GPa for carbon nanotubesas can be seen in Fig. 1 [3]. The large deviation between simulations and experiments has been found to beattributed to the presence of defects in the nanotube structure [3,68]. The mechanical properties of defect-freecarbon nanotube have not been measured yet.

Hashimoto et al. [9] have provided a direct experimental evidence for the existence of defects in graphenelayers. Using high-resolution transmission electron microscopy, they observed numerous topological defects,vacancies and adatoms in graphene layers. The presence of these defects is expected to degrade the mechanical

K. I. Tserpes (B)Laboratory of Technology and Strength of Materials,Department of Mechanical Engineering and Aeronautics, University of Patras, 26500 Patras, GreeceE-mail: [email protected].: +30-2610-969498Fax: +30-2610-997190


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0

20

40

60

80

100

120

140

0 5 10 15 20

stress(GPa)

strain (%)

10% weakening of 1 atom [4]

10% weakening of 1 atom, inflection point at 19% strain

1 missing atom [4]

1 missing atom, inflection point at 19% strain

Experiments [5]

Fig. 1 Comparison of predicted tensile stressstrain curves [5] for the (20,0) tube with theoretical and experimental curves fromthe literature

properties of graphene as happens in carbon nanotubes. Therefore, a detailed knowledge about the exact effectof defects is essential. The existing works on this field have treated defects in graphenes as cracks. Khare et al.[10] found, by performing coupled quantum mechanical/molecular mechanical calculations, that the fracturestress of graphene sheet containing defects in the form of slits decreases monotonically and sharply with theslit size. Zhao and Aluru [11] came to the same conclusion using molecular dynamic simulations. Jin andYuan [12] have calculated J-integral in atomic models in order to study crack growth in 2D graphene. Tsaiet al. [13] have characterized fracture behavior of a graphene sheet containing a center crack using atomisticsimulations and continuum mechanics. Their study was focused on the applicability of the strain energy releaseconcept in modeling fracture of graphene sheet rather on the degradation of graphenes mechanical propertiesdue to crack growth. Although these studies have adopted very interesting computational approaches and haveprovided very useful results regarding the crack growth behavior of graphenes, their findings cannot be directly

related to the effect of defects on the mechanical behavior of graphenes since the mechanisms of crack (orslits) formation from defects are not available. In other words, there is no evidence that mechanics of fractureevolution due to defect interaction are similar to mechanics of crack growth. The only reported work on thebehavior of graphene in the presence of actual defects is the work of Xiao et al. [14]. The authors reported thatthey studied the effects of multiple Stone-Wales (5-7-7-5) defects on the tensile behavior of carbon nanotubesand graphene; however, they presented only fracture patterns of graphene without giving any results on itsmechanical properties.

In the authors knowledge, there is no reported work on the effect of vacancies on the mechanical propertiesof graphene. Such a study is accomplished in this work using an atomistic-based continuum progressive frac-ture model. For the study to be as realistic as possible, a random distribution of vacancies (1 missing atom) isconsidered. Besides the effect of defect density, the effects of size, chirality and defect topology on the Youngsmodulus, tensile strength and failure strain of graphenes are also examined. The paper is formulated as follows.Following the introduction, Sect. 2 describes the progressive fracture model used by giving emphasis to the

FE analysis of graphene, the implementation of the Morse potential and the algorithm of the model. In Sect. 3,modeling of defects is presented. The results of the complementary study on the size and chirality effect arepresented in Sect. 4, while the main results of the paper are presented and discussed in Sect. 5. The paper isconcluded in Sect. 6.

2 The progressive fracture model

The progressive fracture model has been developed in [3] and has been successfully implemented to simulatethe fracture behavior and predict tensile strength of pristine and defected single-walled carbon nanotubes in[3,8]. The model uses the FE method to simulate the atomic structure of graphene and the Morse potential tosimulate the nonlinear behavior of the CC bonds.


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Fig. 2 Beam FE modeling of graphene

2.1 FE model

Carbon atoms in graphene are bonded together with covalent bonds forming an hexagonal 2D lattice. Thesebonds have a characteristic bond length and bond angle. The displacement of the individual atoms under anexternal force is constrained by the bonds. Therefore, the total deformation of graphene is the result of theinteractions between the bonds. By considering the bonds as connecting load-carrying elements, and atoms as

joints of the connecting elements, graphene may be simulated as a plane-frame structure. By treating graph-ene as a plane-frame structure, its mechanical behavior can be analyzed using classical structural mechanicsmethods such as the FE method. The progressive fracture model is implemented by means of a 3D FE modelof graphene, which has been developed using the ANSYS commercial FE code. For the modeling of the CCbonds, the 3D elastic BEAM4 ANSYS element was used. Figure 2 depicts how the hexagon, which is theconstitutional element of graphene nanostructure, is simulated as structural element of a space-frame. In thesame way, the entire graphenes lattice is simulated. The simulation leads to the correspondence of the bondlength to the element length as well as the wall thickness with the element thickness. By assuming a circularcross-sectional area for the element, as in Fig. 2, wall thickness corresponds to element diameters.

2.2 Simulation of the nonlinear behavior of CC bonds

The nonlinear behavior of the CC bonds is simulated using the modified Morse interatomic potential. Thispotential seems not appropriate for describing the behavior of carbon nanotubes and graphene when bonds arebroken, since it does not allow for reconfiguration of bonds. However, as shown by Belytschko et al. [ 4], thefracture strength of carbon nanotubes depends primarily on the inflection point of the interatomic energy andis almost independent of dissociation energy. Therefore, since the inflection strain occurs substantially beforethe strain associated with bond breaking, where the formation of other bonds is expected, the independence offracture strength to the dissociation energy provides some confidence that the modified Morse potential cangive a correct picture of graphene fracture in cases of moderate temperatures (0500K). Such is the currentcase where the specific potential was used to study the effect of vacancies on the tensile behavior of graphene.

The validity of the modified Morse potential in predicting the mechanical performance of carbon nano-tubes has been proven in a series of works. Recently, Wernik and Meguid [15] have sucessfully employed

the potential to simulate the nonlinear response of armchair and zigzag nanotubes under tensile and torsionalloading conditions. In a study similar to the present one, Parvaneh and Shariati [ 16] have used the potentialto study the effects of defects on the Youngs modulus of single-walled nanotubes but not on the strength andfracture behavior of the nanotubes. Regarding the accuracy of the potential against the famous multi-bodyREBO potential [17], Duan et al. [18] have shown, through a comparative study, that the modified Morsepotential gives more accurate predictions of tensile strength and fracture strain for carbon nanotubes.

According to the modified Morse potential, the potential energy of graphene system is expressed as

E= Estretch + Eangle, (1)

Estretch = De

1 e(rr0)

2 1

, (2)

Eangle =1

2k( 0)

2

1+ ksextic( 0)4

, (3)


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Fig. 3 Forcestrain curve of the modified Morse interatomic potential

where Estretch is the bond energy due to bond stretching and Eangle the bond energy due to bond angle-bending,r is the current bond length and is the current angle of the adjacent bond. The parameters of the potential are

[4]:

r0 = 1.421 1010 m, De = 6.03105 10

19 Nm,

= 2.625 1010 m1, 0 = 2.094rad, k = 0.9 1018 Nm/rad2, ksextic = 0.754rad

4.

This is the usual Morse potential except that the bond angle-bending energy has been added and the parame-ters have been slightly modified by Belytschko et al. [4] so that it corresponds with the REBO potential [17] forstrains below 10%. As bond stretching dominates graphenes fracture and the effect of angle-bending potentialis very small, only the bond stretching potential is considered. By differentiating Eq. (2), the stretching forceof atomic bonds is obtained in the molecular force-field as

F= 2De(1 e(rr0))e(rr0). (4)

Figure 3 plots the relationship between force F and bond strain for the CC bonds. The strain of the bondis defined by = (r r0)/r0. As may be seen, the forcestrain relation is highly nonlinear at the attractionregion especially at large strains. The inflection point (peak force) occurs at 19% strain. The repulsive force( < 0) increases rapidly as the bond length shortens from the equilibrium length with less nonlinearity thanthe attractive force.

2.3 Algorithm of the model

For modeling the CC bonds, the 3D elastic ANSYS BEAM4 element was used. The nonlinear behavior of theCC bonds, as described by the interatomic potential, was assigned to the beam elements using the stepwiseprocedure of progressive fracture modeling, which is briefly described in the following lines. Initially, the stiff-ness of the beam elements is evaluated from the initial slope of the forcestrain curve of the modified Morse

potential (Fig. 3) using the elements cross-sectional area A. The initial stiffness is 1.16 TPa. The graphene isloaded by an incremental displacement at one end with the other end fixed. At each load step, the stiffness ofeach element is set equal to F/A, where is the axial strain of the element as evaluated from the FE modeland F is the interatomic force calculated using Eq. (4) given the . The next displacement increment is thenapplied to the graphene, and this iterative procedure goes on until catastrophic failure of the nanotube takesplace.

3 Modeling of defects

Defects of several forms, such as vacancies (missing atoms), topological defects (mainly Stone-Wales defect)and adatoms, appear in a graphene sheet in a random way [ 9]. In the present work, we consider the presenceof randomly dispersed vacancies (1 missing atom corresponding to three missing bonds). These defects have


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Fig. 4 Graphene with 10 missing atoms indicated by arrows. Numbers of the deleted nodes were given by the random number

generator. Zigzag and armchair directions are also indicated

been proven to significantly affect the mechanical properties of graphene [10,11]) as they act as cracks andholes, which they progressively enlarge due to defect interaction.

Creation of random vacancies in graphene structure is done using a random numbers algorithm. A set ofrandom numbers equal to the number of defects are created using a Random Integer Generator (http://www.random.org) in which randomness comes from atmospheric noise, which for many purposes is better than thepseudo-random number algorithms typically used in computer programs. Random numbers correspond to thecarbon atoms or nodes in the FE model, which are removed from graphene together with the three adjacentbonds. The procedure is illustrated in Fig. 4 for the creation of 10 defects at a graphene with 217 atoms.

The effect of defects is measured in terms of defect density defined as

Dd =

Dn

An , (5)

where Dn is the number of defects and An is the total number of atoms in graphene.

4 Size and chirality effect

Prior to the main analyses of the present work, a study on the possible effects of size and chirality on themechanical properties of graphene has been conducted. The need for accomplishing this study has emanatedfrom the findings on the existence of size effects on the mechanical properties of carbon nanotubes. Severalworks have been published in this area. In the frame of a systematic work published in three papers, Shenand Li [1921] have shown using an energy approach based on a molecular mechanics framework that thereare considerable effects of tube diameter and length on the Youngs modulus, shear modulus, bend-angle and

strain energy of single- and multi-walled nanotubes.Figure 5 plots the predicted stressstrain curves of pristine zigzag and armchair graphenes of different size.

To exclude the effect of graphenes aspect ratio, almost square graphenes of a constant aspect ratio have beenconsidered. Tensile loading is modeled by fully constraining the nodes at one end and applying an incrementalaxial displacement at the other end of graphene. Stress is taken as the ratio of the total reaction force at theconstrained nodes to the constrained area equals to the constrained length times the thickness of graphene,which for the present application was taken equal to 0.34nm. The comparison between the curves reveals asignificant effect of size on the Youngs modulus, tensile strength and failure strain of graphene. The predictedvalues of Youngs modulus, taken from the initial slope of the curve, and tensile strength are listed in Fig. 6.With increasing the graphenes size, the Youngs modulus and tensile strength of the material decreases; therate of reduction is decreasing as the size of graphene increases. The range of variation is between 1.56 and1.30 TPa for the Youngs modulus of zigzag graphenes, 1.56 and 1.32 TPa for the Youngs modulus of armchairgraphene, 128 and 99 GPa for the tensile strength of zigzag graphenes and 171 and 122GPa for the tensile
http://www.random.org/http://www.random.org/http://www.random.org/http://www.random.org/http://www.random.org/http://www.random.org/


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Fig. 5 Predicted tensile stressstrain curves of pristine zigzag and armchair graphenes of different size

Fig. 6 Size and chirality effect on a the Youngs modulus and b tensile strength of graphenes

strength of armchair graphenes. Regarding the chirality effect, this stands only for the tensile strength and notfor the Youngs modulus: armchair graphenes possess a 30% higher tensile strength than zigzag ones. Fromthe size of 9.35nm 7.96 nm, the mechanical properties of graphene become almost size-independent; theYoungs modulus is stabilized at the value of 1.3 TPa, while the tensile strength at 100 GPa for the zigzaggraphenes and at 125 GPa for the armchair graphenes. Comparison of these values with the experimental mea-surements of Lee et al. [2], being 1.0 0.1 TPa for the Youngs modulus and 130 10 GPa at a strain of 0.25for the tensile strength, reveals an overestimation of the present model for the Youngs modulus and a goodagreement for the tensile strength and failure strain. The overestimation in the prediction of Youngs modulus

is attributed to the large Youngs modulus of the CC bonds imposed by the modified Morse potential.Based on the above findings, to exclude the size effect from the study on the vacancies effect, the larger

graphene (14.52nm 12.20 nm) is adopted. Due to the difference found in the tensile strength between thetwo different chiralities, both loading directions will be considered hereafter.

5 Effect of vacancies

Analysesof the 14.52nm12.20 nm graphene containing different number of vacancies have been performed.Figure 7 plots the predicted tensile stressstrain curves of the defected zigzag and armchair graphenes. It isshown that the presence of defects dramatically degrade the mechanical performance of graphenes by reducingtheir Youngs modulus, tensile strength and failure strain. The contradiction between the significant reductionof Youngs modulus, predicted herein and the insignificant reduction, predicted by Parvaneh and Shariati [16]


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Fig. 7 Predicted tensile stressstrain curves of defected a zigzag and b armchair graphenes

Fig. 8 Variation ofa the Youngs modulus and b the tensile strength of defected graphenes as functions of defect density

for nanotubes, reveals that the magnitude of influence of defects is governed by defect density. In [16], theworst scenario considered are triple vacancies, which represent a much smaller defect density than the densitiesconsidered in the present study. The variation of Youngs modulus and tensile strength of defected graphenes

are plotted against defect density in Fig. 8. A similar linear reduction is observed for the Youngs modulus forboth loading directions. The tensile strength reduces in a bi-linear way: the reduction rate is larger in smalldefect densities. For a defect density of 4.5%, a 50% reduction in Youngs modulus and tensile strength in bothtypes of graphenes occurs. This is a very important finding because it reveals that vacancy defects, appearingin considerable densities in graphenes, may counterbalance the extraordinary mechanical properties of thematerials.

The obtained effect of vacancies on the strength of graphenes was more or less expected since this typeof defects act as holes, thus seriously degrading the load-carrying capability of the material. However, theimportance of the finding is the extent of the effect as a function of defect density. As defect density increases,a multi-site damage state is developed in graphene. Moreover, the possibility for the creation of holes due todefect aggregation prior to loading and early defect interaction during loading increases dramatically. Figure 9ashows the 14.52nm 12.20 nm graphene containing defects of a 3.3% density. As can be seen, the presenceof this amount of defects has created several large holes (gray areas) in the structure due to defect aggregation.


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Fig. 9 a Formation of holes (dark areas) due to defect aggregation prior to loading at the 14 .52nm 12.20 nm with a 3.3%defect density, b early failure of graphene due to the formation of a large crack (dark area)

These holes propagate very quickly during loading to interact and form cracks perpendicular to the loadingdirection, which extend through the width of the structure leading to an early failure, as can be seen in Fig. 9b.

5.1 Fracture progression

As stated previously, vacancies in graphene act as holes. The bonds that are adjacent to vacancies are highlyloaded and their fracture leads to hole enlargement. During loading, many holes enlarge simultaneously dueto fracture of the bonds forming larger holes or cracks, which eventually lead to a separation of graphene.Figure 10 illustrates fracture progression at the 14.52nm 12.20 nm graphene containing 10 defects as afunction of applied strain. The location of defects is shown in Fig. 10a. Fracture initiated at 11% strain fromall defects. Fracture severity is larger around the defects 4, 5 and 6 as they are close to the load applicationarea and also their vicinity results in a locally high defect density area. When the bond between defects 5 and

6 fails, a large hole is created that soon interacts with defects 4 and 3 to cause a large crack perpendicularto the loading direction, which separates graphene. A more or less similar fracture process is observed at alldefected graphenes examined herein depending on the defect density and defect topology.

5.2 Effect of defect topology

In Fig. 8b, which shows the effect of defect density on the tensile strength of armchair graphenes, there are twopoints that represent a harsher reduction in strength from the one expected at the specific defect densities. Thisexception is due to the effect of defect topology. Positioning of defects in graphene is very important sinceholes can be created either prior to loading due to defect aggregation or at an early stage of loading due to defectinteraction. As creation of defects in graphenes is done randomly, both phenomena are highly possible due tothe proximity of defects. Therefore, besides the primary effect of defect density, there is also the secondary

effect of defect topology that must be examined. It can be easily understood that the effect of defect topology ismore significant at high defect densities. To assess and demonstrate the effect of defect topology, the presenceof 20 defects (0.44% defect density) under two different topologies has been considered. Figure 11 comparesthe tensile stressstrain curves predicted for these two cases. Although no effect has been predicted for theYoungs modulus of graphene, there is a 3.24% difference in the predicted failure stress and a 4% in the failurestrain. Considering that these differences, attributed to defect topology, occur at a relatively low defect density,the overall effect of defect topology becomes evident.

6 Conclusions

The scope of this work was to obtain information about the effect of vacancies on the mechanical properties ofgraphene and ascertain if defects may counterbalance its extraordinary strength as stands for carbon nanotubes


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Fig. 10 Fracture progression as a function of applied strain at the 14.52nm 12.20 nm graphene containing 10 defects

Fig. 11 Predicted stressstrain curves of the 14.52nm12.20, nm graphene containing 20 defects (0.44% defect density) of twodifferent topologies A and B

[22]. To this end, an atomistic-based continuum progressive fracture model, established through its applica-tion to carbon nanotubes, has been implemented to predict tensile behavior of graphenes in the presence ofrandomly distributed vacancies (1 missing atom). Vacancies are created using a random numbers algorithm inwhich randomness comes from atmospheric noise. The model is able to simulate fracture progression, takinginto account defect interaction, and predict Youngs modulus, tensile strength and failure strain of graphenes.

A preliminary study on the size and chirality (loading direction) effect on the mechanical properties ofgraphene has been conducted. Computations show a significant effect of size: with increasing the graphenessize, the Youngs modulus and tensile strength of the material decreases; the rate of reduction is decreasingas the size of graphene increases. From a certain size, the Youngs modulus is stabilized at the value of 1.3TPa for both chiralities, while the tensile strength at 100 GPa for the zigzag graphenes and at 125 GPa for thearmchair graphenes.


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The effect of vacancies has been studied by performing analyses of defected graphenes under differentdefect densities varying from 0.22 to 4.4%. It was found that the presence of vacancies dramatically degradesthe mechanical properties of graphene: 4.4% of missing atoms, corresponding to 13.2% of missing bonds,result in a 50% reduction in Youngs modulus and tensile strength of the material. A secondary effect of defect

topology has also been ascertained. This comes from the fact that vicinity of defects can result in creation ofholes either prior to loading due to defect aggregation or at an early stage of loading due to defect interaction.As the obtained effect of vacancies on the strength of graphenes was more or less expected, the impor-

tance of the findings of the present work lies to the extent of the effect as a function of defect density. Thedegree of obtained strength degradation is a clear indication that vacancies may counterbalance the extraordi-nary mechanical properties of graphene. Based on these findings, experimental evidence about the amount ofvacancies that may be present in graphene as well as experimental measurements of the mechanical propertiesof defected graphene would be very useful to the research community.

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