graphene science handbook mechanical and chemical properties

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Graphene Science HandbookMechanical and Chemical PropertiesMahmood Aliofkhazraei, Nasar Ali, William I. Milne, Cengiz S. Ozkan,Stanislaw Mitura, Juana L. Gervasoni

Effects of Vacancies, Nitrogen Atoms, and sp Bonds onMechanical Properties of Graphene Using MolecularDynamics Simulations

Publication detailshttps://www.routledgehandbooks.com/doi/10.1201/b19674-5

Akihiko Ito, Shingo OkamotoPublished online on: 25 Apr 2016

How to cite :- Akihiko Ito, Shingo Okamoto. 25 Apr 2016, Effects of Vacancies, Nitrogen Atoms, andsp Bonds on Mechanical Properties of Graphene Using Molecular Dynamics Simulations from: GrapheneScience Handbook, Mechanical and Chemical Properties CRC PressAccessed on: 16 Feb 2022https://www.routledgehandbooks.com/doi/10.1201/b19674-5

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3 Effects of Vacancies, Nitrogen Atoms, and sp3 Bonds on Mechanical Properties of Graphene Using Molecular Dynamics Simulations

Akihiko Ito and Shingo Okamoto

ABSTRACT

Mechanical properties of graphene containing atomic size defects, such as vacancy, nitrogen atom, and sp3 bond are discussed in this chapter. Molecuar dynamics (MD) simu-lations on tensile and shear loadings of defective graphene were performed to estimate the mechanical properties, such as strength and modulus. We showed the usefulness of MD simulation for understanding the relationship between nano-structures and mechanical properties.

3.1 INTRODUCTION

Defects often affect the mechanical and electronic properties of carbon materials quite significantly. This, therefore, has led to more studies having been recently conducted on defects (i.e., vacancies [1,2], sp3-type defect [2], dislocations [3], and grain boundaries (GB) [4]) in graphene and graphite. It is crucial to clarify the mechanical properties of graphene and graphite, the

basic structures in carbon materials, in order to develop high-performance carbon materials. Molecular dynamics (MD) is the most suitable method for investigation of nanostructures.

Recently, studies aiming to clarify the relationship between atomic-scale defects and mechanical properties using molec-ular simulation have increased in number. Literature on the mechanical properties of carbon materials including carbon nanotubes (CNT) and diamonds are summarized in Table 3.1. There is a large number of research on the mechanical proper-ties of CNTs that were carried out before 2007. Consequently, the reports on graphene have increased in number. The tensile properties of graphene and CNTs containing multiple Stone –Wales (SW) defects have been investigated by Xiao et al. using MD simulations [5]. These studies have clarified the relation-ship between the number of defects and the mechanical prop-erties. The influence of GBs on the tensile strength of graphene has been investigated by Shenoy et al. [6]. Pao et al. studied the effect of GBs on the shear properties of graphene using MD simulations [7]. The MD simulations on the tensile loadings of

CONTENTS

Abstract ....................................................................................................................................................................................... 413.1 Introduction ....................................................................................................................................................................... 413.2 Computational Method ...................................................................................................................................................... 43

3.2.1 Potential ................................................................................................................................................................. 433.2.2 Analysis Model ...................................................................................................................................................... 44

3.2.2.1 Pristine Graphene ................................................................................................................................... 443.2.2.2 Vacancy-Containing Graphene ............................................................................................................... 443.2.2.3 Nitrogen-Containing Graphene .............................................................................................................. 443.2.2.4 Graphene Containing Both Vacancies and Nitrogen Atoms .................................................................. 453.2.2.5 Graphene Containing Both Interlayer sp3 Bonds and Nitrogen Atoms .................................................. 45

3.2.3 MD Simulation ...................................................................................................................................................... 463.3 Result and Discussion ........................................................................................................................................................ 47

3.3.1 Validation of Calculation Method ......................................................................................................................... 473.3.2 Vacancy .................................................................................................................................................................. 483.3.3 Nitrogen ................................................................................................................................................................. 493.3.4 Comparison of Effects between Vacancy and Nitrogen ........................................................................................ 513.3.5 Comparison of Effects between sp3 Bond and Nitrogen ....................................................................................... 543.3.6 Comparison of Effects between Vacancy and sp3 Bond ........................................................................................ 56

3.4 Conclusions ........................................................................................................................................................................ 57References ................................................................................................................................................................................... 58

© 2016 by Taylor & Francis Group, LLC

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42 Graphene Science Handbook

graphene with SW defects or vacancies have been performed by Ansari et al. [8]. The influence of the single and double vacancies on the tensile strength has been investigated through molecular mechanics (MM) calculations by Zhang et al. [9]. They compared their results obtained using MM calculations with Mielke’s results obtained using quantum mechanics

calculations [10]. The mechanical properties of graphene and graphite-containing vacancies have been investigated by the authors [11] who have clarified the effects of the size and distribution of vacancies on the tensile and shear proper-ties of graphene and graphite. In addition, the carbon materi-als generally contain impurities such as oxygen, nitrogen, or

TABLE 3.1Literature on the Mechanical Properties of Carbon Materials Using Molecular Simulations

Author Year Method Materials Defect Load

Xia et al. [12] 2002 MD CNT SW TensionOgata and Shibutani [13] 2003 DFT CNT − Tension, compression

Sammalkorpi et al. [14] 2004 MD CNT Vacancy TensionMielke et al. [10] 2004 DFT, QM, MM CNT Vacancy TensionZhang et al. [9] 2005 QM, MM CNT Vacancy TensionPapanikos et al. [15] 2006 FEM CNT Vacancy TensionKomanduri et al. [16] 2006 MD CNT Vacancy TensionSchatz et al. [17] 2007 QM/MM CNT Vacancy TensionHan et al. [18] 2007 MD CNT Vacancy CompressionTserpes and Papanikos [19] 2007 FEM CNT SW TensionShen and Chen [20] 2007 MD Diamond Nitrogen TensionLi et al. [21] 2007 DFT Graphene − Tension

HaiYang and XinWei [22] 2008 MD CNT Vacancy, sp3 Tension, compressionXiao et al. [5] 2009 FEM Graphene, CNT SW TensionJiang et al. [23] 2009 MD Graphene Isotopic carbon TensionHaiYang and XinWei [24] 2009 MD CNT sp3 ShearShen and Chen [25] 2009 MD Diamond Nitrogen TensionAluru et al. [26] 2009 MD Graphene − Tension

Zhao and Aluru [27] 2010 MD Graphene Vacancy TensionWong [28] 2010 MD CNT Vacancy TensionKim et al. [29] 2010 MD, MM Graphene sp3 ShearShenoy et al. [30] 2010 MD Graphene sp3 TensionShenoy et al. [6] 2010 MD, DFT Graphene GB TensionNeek-Amal and Peeters [31] 2010 MD Graphene Vacancy TensionTsai and Tu [32] 2010 MD Graphene − Tension, shear

Shokrieh and Rafiee [33] 2010 CM Graphene, CNT − Tension

Anifantis et al. [34] 2010 FEM Graphene − Tension, shear

Ansari et al. [35] 2011 MD Graphene Vacancy TensionZhang et al. [36] 2011 MD Graphene sp3 Tension, shearMin and Aluru [37] 2011 MD Graphene − Shear

Peón-Escalante et al. [38] 2012 FEM Graphene Vacancy Tension, shearAhzi et al. [39] 2012 MD Graphene Vacancy, nitrogen TensionYan et al. [40] 2012 MD Graphene Vacancy, SW TensionWong and Vijayaraghavan [41] 2012 MD Graphene Vacancy TensionAnsari et al. [8] 2012 MD Graphene Vacancy, SW TensionCao and Qu [42] 2012 MD Graphene GB TensionCao and Yuan [43] 2012 MD Graphene GB TensionOkamoto and Ito [44] 2012 MD Graphene Nitrogen TensionIto and Okamaoto [11] 2012 MD Graphene, Graphite Vacancy TensionCao and Qu [45] 2013 MD Graphene GB TensionChang et al. [46] 2013 MD Graphene GB TensionWu and Wei [47] 2013 MD Graphene GB TensionPao et al. [7] 2013 MD Graphene GB ShearOkamoto and Ito [48] 2013 MD Graphene Nitrogen Tension, shearIto and Okamoto [49] 2013 MD Graphene Vacancy Tension, shearLe and Batra [50] 2014 MD Graphene Crack TensionRajabpour, A. et al. [51] 2014 MD Graphene sp3 ShearIm et al. [52] 2014 MD Graphene GB Tension

Liu et al. [53] 2014 MD Graphene SW Tension


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43Effects of Vacancies, Nitrogen Atoms, and sp3 Bonds on Mechanical Properties of Graphene

hydrogen atoms, and these impurities may affect the mechani-cal and electronic properties of the materials. Recently, Shen et al. [20,25] investigated the effects of nitrogen (N) doping on the mechanical properties of ultrananocrystalline diamond (UNCD) films, using MD simulations. They demonstrated that the strength of the N-doped UNCD films decreases with increase in number and density of the dopant N atoms. The MD simulations of N-containing graphene were also performed by Ahzi et al. [39] and the authors [44,48], who have clarified the relationships between the tensile properties and N content in graphene. In addition, the authors investigated the effect of the distributional forms of N atoms in graphene.

However, the mechanical properties of graphene that con-tain both vacancies and N atoms have yet to be fully clarified. In the present study, the authors investigated the influences of the vacancies and N atoms on the tensile strength and Young’s modulus of graphene when these two types of defects are present together in graphene. In addition, the authors dis-cussed which defect of vacancy or N atom greatly affects the mechanical properties of graphene.

Perfect graphene consists of sp2 bonds which, mutually, make up an angle of 120° and formulate a planar structure. On the other hand, real carbon materials also contain sp3 bonds derived from raw materials, or are generated during the man-ufacturing processes. The mechanical properties of bilayer graphene sheets coupled by sp3 bonding were investigated by Zhang et al. [36]. It was verified from the MD simulations by Zhang et al. that interlayer sp3 bonds decrease both the tensile strength and Young’s modulus. However, the mechani-cal properties of graphene containing both sp3 bonds and N atoms have not been investigated so far. In the present study, the authors clarified the influence on the tensile strength when both sp3 bonds and N atoms are present together in graphene.

3.2 COMPUTATIONAL METHOD

3.2.1 Potential

In the present study, three types of interatomic potentials are used: the second-generation reactive empirical bond order (2nd REBO) [54], Tersoff [55,56], and Lennard–Jones poten-tials. The 2nd REBO potential for covalent C─C bonds is given in Equation 3.1:

E V r B V rR ij ij A ij

j ii

REBO = −>

∑∑ [ ( ) ( )],* (3.1)

where the terms VR(rij) and VA(rij) denote the pair-additive interactions that reflect interatomic repulsions and attractions, respectively. The Bij

* denotes the bond-order term.The Tersoff potential for covalent C─N bonds is given in

Equation 3.2:

V f r A r b f r B rC ij ij ij ij ij C ij ij ij ij

i j

= − − −≠∑1

2[ ( ) exp( ) ( ) exp( )]λ µ ,,

(3.2)

where the parameter bij is the bond-order term that depends on the local environment.

bij ij

nijn n

i ii= +( )−

χ β ζ11 2/

,

(3.3)

ζ θij C ik ijk

k i j

f r g=≠∑ ( ) ( ),

, (3.4)

g

c

d

c

d hijk

i

i

i

i i ijk

( )( cos )

,θθ

= + −+ −

12

2

2

2 2

(3.5)

where θijk is the angle between bonds ij and ik.The parameters Aij, Bij, λij, and µij depend on the atom type,

namely, carbon or nitrogen. For atoms i and j (of different types), these parameters are

A A A B B Bij i j ij i j= × = ×( ) , ( ) ,/ /1 2 1 2

(3.6)

λ

λ λµ

µ µij

i jij

i j=+

=+( )

,( )

,2 2

(3.7)

where the parameters with a single index represent the inter-action between atoms of the same type.

The parameter χij in Equation 3.3 is determined similar to the previous work [44,48].

The 2nd REBO and Tersoff potentials contain the same cutoff function fC(r), as given in Equation 3.8. It is known that for these potential functions, the interatomic force increases dramatically when r exceeds Rmin owing to the discontinu-ity of the second derivatives of the cutoff function, and this dramatic increase in the interatomic force greatly affects the tensile strength. In the author’s works, the cutoff length Rmin of the 2nd REBO potential is set to 2.0 Å in the studies using graphene models with vacancies to avoid the dramatic increase in interatomic forces, similar to the previous work by Brenner et al. [57]. And then, the cutoff length Rmin of both the 2nd REBO and Tersoff potentials is set to 2.1 Å in the studies using graphene models with N atoms.

f r

r R

r R R RR r R

r

C ( )

,

{ cos[ ( ) ]},

,

min

min max minmin max=

<+ − − < <

1

12

0

π /

>>

Rmax

(3.8)

The Lennard–Jones potential for interlayer interaction in the bilayer graphene models is given in Equation 3.9:

Vr

r

r

rLJ

ij ij

=

−

4 0

12

0

6

ε .

(3.9)


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The ε is set to 0.00284 eV, and the equilibrium distance r0 is set to 3.2786 Å, so that the interplanar spacing of graph-ite at 300 K is 3.35 Å, which is a known experimental value [58]. The use of the 2nd REBO and Lennard–Jones potentials is switched according to the interatomic distance and bond order, similar to the method used in the adaptive intermolecu-lar REBO potential [59]. The use of the Tersoff and Lennard–Jones potentials is also switched in the same way.

3.2.2 analysis Model

The analysis models used in the present studies are shown below.

3.2.2.1 Pristine GrapheneAnalysis models, namely, zigzag graphene ribbon (ZGR) and armchair graphene ribbon (AGR) models of pristine gra-phene used under zigzag and armchair tensions are shown in the previous research [11], as shown in Figure 3.1. The analy-sis models of ZGR with a double length in the zigzag tensile direction, as shown in Figure 3.2 are also used under tensile loadings in order to investigate which defect of vacancy or N atom affects greatly the mechanical properties. The analysis

models of pristine graphene used under two types of shear loadings have the same number of atoms and dimensions as the analysis models used under tensile loadings, as shown in Figure 3.1. The details of the analysis models are explained in the previous papers [49] by the authors.

3.2.2.2 Vacancy-Containing GrapheneThe analysis models of the AGR with a cluster-type vacancy are shown in Figure 3.3. Periodic boundary conditions and boundary zones are defined in a way similar to the method for the pristine AGR model. The analysis models with randomly distributed vacancies are defined in the previous papers [11] by the authors.

3.2.2.3 Nitrogen-Containing GrapheneThe analysis models with randomly distributed N atoms are defined in the previous papers [44,48] by the authors. The analysis models of the ZGR containing two N atoms that are located at an interval d are shown in Figure 3.4. Three cases with the differing intervals, namely, 1.76 Å, 2.38 Å, and 7.31 Å, are investigated. Periodic boundary conditions and boundary zones are defined in a way similar to the method for the pristine ZGR model.

(a) (b)

Y

XO

Y

XO

: Carbon atom

l

52 Å

30 Å

l l l

51 Å

30 Å

: Tensile loading: Shear loading

FIGURE 3.1 Configurations of graphene used under tensile and shear loadings. (a) ZGR and (b) AGR models.

l

104 Å

30 Å

l

Y

XO

FIGURE 3.2 Configurations of ZGR with a double length in the tensile direction.


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3.2.2.4 Graphene Containing Both Vacancies and Nitrogen Atoms

The example of the analysis model of the ZGR containing both vacancies and N atoms which are randomly distributed in graphene, is shown in Figure 3.5. The positions of the vacan-cies and N atoms are set so that the clustered vacancies and adjoining N atoms are not generated. Five cases with different vacancy density and N contents, namely, 1% vacancies—4% N atoms, 2% vacancies—4% N atoms, 4% vacancies—1% N atoms, 4% vacancies—2% N atoms, and 4% vacancies—4% N atoms, are investigated.

In the previous studies [44,48], it was found that the decrease in strength becomes large when two N atoms are present and adjoining each other in graphene. It is worth clarifying which defect of two adjoining N atoms or vacancy

significantly affects the mechanical properties of graphene. Therefore, two types of the analysis models are used in order to compare the effects of vacancy and N atoms on the fracture behavior of graphene. One is the ZGR model containing a pair of two adjoining N atoms and a cluster-type vacancy. In that model, three types of vacancies namely, single, double, and sextuple vacancies are investigated. In the case of the single vacancy, the configuration is shown in Figure 3.6. The other model contains a single vacancy and a pair of N atoms which are located parallel to the tensile axis at a different interval, namely, 1.76 Å, 2.38 Å, and 7.31 Å, as shown in Figure 3.7. No periodic boundary conditions are imposed. The bound-ary zones are defined in a way similar to the method for the pristine ZGR model.

3.2.2.5 Graphene Containing Both Interlayer sp3 Bonds and Nitrogen Atoms

The analysis models of the bilayer ZGR containing both interlayer sp3 bonds and N atoms are used in order to inves-tigate which defect of interlayer sp3 bond or N atom greatly affects the mechanical properties of graphene. Two types of bilayer ZGR which are constructed by two layers of ZGR are used. One is the bilayer ZGR containing two N atoms per layer, which is located parallel to the tensile axis at a different interval d, namely, 1.76 Å, 2.38 Å, and 7.31 Å, and an inter-layer sp3 bond as shown in Figure 3.8. The other model con-tains randomly distributed interlayer sp3 bonds and N atoms. The positions of the interlayer sp3 bonds and N atoms are set so that the clustered interlayer sp3 bonds and the adjoin-ing N atoms are not generated. Four cases with different

Y

XO

(a) (b) (c)

FIGURE 3.3 Analysis models of AGR containing a cluster-type vacancy. (a) Single, (b) double, and (c) sextuple vacancies.

Y

X 1.76 Å 2.38 Å 7.31 ÅO

: Nitrogen atom

FIGURE 3.4 Analysis models of ZGR containing two N atoms that are located at an interval d.

Y

XO : Nitrogen atom

FIGURE 3.5 Analysis model of ZGR containing both vacancies and N atoms, which are randomly distributed.


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N contents, namely, 0, 1, 4, and 10%, at the interlayer sp3 bond density of 5%, are investigated. The interlayer sp3 bonds are generated in a way similar to the method by Zhang et al. No periodic boundary conditions are imposed. The boundary zones are defined in a way similar to the method used for the pristine ZGR model.

3.2.3 Md siMulation

The velocity Verlet method is used for the time integral of the equations of motion of atoms. The velocities of all atoms are adjusted simultaneously using the velocity scaling method [60] so that the temperature of the object can be maintained at

Y

XO : Nitrogen atom

FIGURE 3.6 Analysis model of ZGR containing a pair of two adjoining N atoms and a cluster-type vacancy.

d

Y

XO : Nitrogen atom

FIGURE 3.7 Analysis model of ZGR containing a single vacancy and a pair of two N atoms which are located parallel to the tensile axis at a different interval d.

X

Y

Z

X

Z

Y

d

sp3 bond

: Nitrogen atom

(a)

(b)

FIGURE 3.8 Analysis model of bilayer ZGR containing two N atoms per layer, which are located parallel to the tensile axis at a different interval d, and an interlayer sp3 bond. Viewed in the (a) Z direction and (b) Y direction.


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the preset temperature TSET. The mass of a carbon atom, mC, and the mass of a nitrogen atom, mN, are 1.9927 × 10−26 kg and 2.3253 × 10−26 kg, respectively. The time step is 0.2 fs.

The atomistic and global stresses are obtained on the basis of the Virial theorem shown in the previous papers [11] by the authors.

The details of the computational conditions are summa-rized in Table 3.2. In the case of not containing N atoms, the methods of tensile and shear loadings are as follows. The initial positions of the atoms are given so that the analysis model represents the crystal structure of graphene at a preset temperature. First, the atoms in the active zone of the analy-sis model are relaxed in unloaded states for 7000 MD steps. The atoms in the boundary zone are fixed. After constant displacements are applied to the atoms in both the boundary zones to simulate uniaxial tensile loading in the X direction or shear loading in the Y direction, the atoms in the active zone are relaxed for 7000 MD steps. The tensile strain increment, ΔεXX, is 0.004. The shearing strain increment, ΔγXY, is 0.0023. The output stresses are sampled for the last 2000 MD steps for each strain and are averaged. Young’s and shearing moduli are obtained from the slopes of the straight lines in the range where the relationship between the stress and strain is lin-ear, and tensile and shearing strengths are given by the largest peak of the nominal stress–nominal strain curves.

In the cases containing N atoms, the methods of tensile and shear loadings are as follows, because larger MD simulation steps are required to stabilize the initial structures compared with the case of not containing N atoms. The initial positions of the atoms are given such that the analysis model becomes identical to the crystal structure of N-containing graphene

at 300 K. First, the atoms of the analysis model are relaxed until the stresses are stabilized for 10,000–35,000 MD simu-lation steps. The atoms in the active zone are relaxed in all the directions. The atom shown by a gray circle (see Figures 3.1a and 3.2) is relaxed in only the X direction. The atoms in the left-hand side boundary zone are relaxed in only the Y direction. The atoms in the right-hand side boundary zone except the atom shown by a gray circle are relaxed in only the X and Y directions. After the atoms are relaxed, constant displacements are applied to the atoms in the boundary zones to simulate uniaxial tensile loading in the X direction or shear loading in the Y direction. The atoms in the boundary zones are restrained in the X and Z directions, and in all the directions in the case of tensile and shear loadings, respec-tively. The atoms in the active zone of the analysis model are relaxed for all the directions in the case of both tensile and shear loadings for 7000 MD simulation steps. The atoms in the boundary zones of the analysis model are relaxed for only the Y direction in the case of tensile loading. The tensile strain increment ΔεXX is 0.004. The shearing strain increment ΔγXY is 0.0023. The Young’s and shearing moduli are obtained by the same way as the above mentioned method in the case of not containing N atoms.

3.3 RESULT AND DISCUSSION

3.3.1 Validation oF calculation Method

The authors performed the MD simulations on tensile and shear loadings of pristine graphene at 300 K to verify the pro-priety of their method of calculation. The results of tensile

TABLE 3.2Computational Conditions of Molecular Dynamics Simulations

Without N Atoms With N Atoms

Initial position of atoms Given so that the analysis model can become the crystal structure of graphene at a preset temperature

Relaxation method in unloaded states

Directions of relaxation Active zone Boundary zone Active zone Boundary zone

All directions Clamped All directions Left-hand side:Y directionRight-hand side:X and Y directions

Relaxation time 7000 MD steps 10,000–35,000 MD steps

Method of loadings Directions of enforced displacements

Tensile: X direction, shearing: Y direction

Incremental strain Tensile: ΔεXX = 0.004, shearing: ΔγXY = 0.0023

Relaxation method in loaded states

Directions of relaxation Active zone Boundary zone Active zone Boundary zone

All directions Clamped All directions Tensile:Y directionShearing:Clamped

Relaxation time 7000 MD steps 7000 MD steps

Method to determine mechanical properties

Stress Sampled and averaged for last 2000 MD steps for each strain

Modulus Obtained from the slopes of the straight lines in the range where the relationship between the stress and strain is linear

Strength Given by the largest peak of the nominal stress–nominal strain curves


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loadings are shown in Table 3.3. The results in the literature are also shown for comparison in Table 3.3.

The computed tensile strength is 83 GPa, which nearly agrees with the 121 GPa calculated by Shenoy et al. through MD simulations [30] and the experimentally obtained value of 130 GPa [61]. The computed Young’s modulus is 0.836 TPa, which is within the range of results obtained by the DFT method [21] (1.05 TPa), by the MD method [30] (0.86 TPa), and by experiment (0.5 TPa [62] and 1 TPa [61]). It is esti-mated that the lower value obtained in this work is due to the effect of size on the elastic properties of graphene [23]. On the other hand, the results of shear loadings are shown in Table 3.4. The computed shear strength is 51 GPa, which nearly agrees with the 60 GPa calculated by Min and Aluru [37] using MD simulations. The computed shear modu-lus is 0.412 TPa, which also nearly agrees with the value of 0.421 TPa calculated by Pao et al. through MD simula-tions and the experimentally obtained value of 0.434 TPa by Blakslee et al. [61].

3.3.2 Vacancy

The moduli and strengths of vacancy-containing graphene obtained at 300 K are listed in Tables 3.5 and 3.6, respec-tively. The nominal stress–nominal strain curves under the shear loading are given in Figure 3.9.

The results for pristine graphene are also shown for refer-ence. Both the tensile and shear strength of AGR and ZGR containing a vacancy decrease greatly. On the other hand, both the Young’s and shear modulus hardly change with the vacancy size.

The relation between the vacancy size and relative strength, that is, the strength of the crack-containing mate-rial relative to the strength of pristine graphene under tensile and shear loadings, is shown in Figure 3.10. The ratio of the strength of vacancy-containing graphene to that of pristine graphene under the tensile loading decreases as the vacancy size increases following the rule of Griffith [11]. Moreover, the relation between the relative strength under the shear load-ing and vacancy size is almost identical to that during the ten-sile loading. Snapshots of AGRs during the shear loadings are shown in Figure 3.11. In the vacancy-containing graphene, the concentration of stress occurs around the vacancy just before the fracture. The direction of fracture progression is at an angle of about 150° to the X-axis in all cases. In the case of the tensile loadings [11], a fracture in the zigzag direction was

TABLE 3.3Comparison of the Computed Tensile Properties of Pristine Graphene and the One Reported by the Literature

Author MethodYoung’s

Modulus (TPa)Tensile

Strength (GPa)

Present work MD 0.836 83

Lee et al. [61] Experimental 1 130

Frank et al. [62] Experimental 0.5 −Aluru et al. [26] MD 1.01 99

Shenoy et al. [30] MD 0.86 121

Jiang et al. [23] MD 1.1 −Shen et al. [63] MD 0.933 −Neek-Amal and Peeters [31]

MD 0.501 ± 0.032 −

Tsai and Tu [32] MD 0.912 −Li et al. [21] DFT 1.05 116

Shokrieh and Rafiee [33]

CM 1.04 −

Xiao et al. [5] FEM 1.13 110

Anifantis et al. [34] FEM 1.367 −Sakhaee-Pour [64] FEM 1.025 −

TABLE 3.4Comparison of the Computed Shear Properties of Pristine Graphene and the One Reported in the Literature

Author MethodShear

Modulus (TPa)Shear Strength

(GPa)

Present work MD 0.412 51

Blakslee [65] Experimental 0.434 −Min and Aluru [37] MD − 60

Pao et al. [7] MD 0.421 60

LiJun and TienChong [66]

MD 0.08–0.21 6–45

Shen et al. [63] MD 0.403 −Kim et al. [29] MD 0.284 16.6

Tsai and Tu [32] MD 0.358 −Faccio et al. [67] MD 0.408 −Anifantis et al. [34] FEM 0.280 −Sakhaee-Pour [64] FEM 0.225 −

TABLE 3.5Moduli of the Vacancy-Containing Graphene

Materials Young’s Modulus (TPa) Shear Modulus (TPa)

AGRPristine 0.879 0.408

Single vacancy 0.868 (−1.2%) 0.399 (+0.5%)

Double vacancy 0.870 (−1.0%) 0.392 (–1.3%)

Sextuple vacancy 0.848 (−3.6%) 0.385 (–3.9%)

ZGRPristine 0.794 0.442

Single vacancy 0.782 (−1.5%) 0.418 (–2.1%)

Double vacancy 0.765 (−3.6%) 0.420 (–1.6%)

Sextuple vacancy 0.767 (−3.4%) 0.402 (–5.8%)

Note: The values in parentheses represent the differences between the pristine and vacancy-containing materials.


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propagated more easily than the armchair one. The results of the present work agree with those results.

The tensile properties of graphene containing randomly distributed vacancies are summarized in Table 3.7 [11]. The Young’s modulus and tensile strength significantly decrease as the density of vacancies increases. The decline in tensile strength is 59% at the density of 4%.

3.3.3 nitrogen

The computed mechanical properties of graphene containing randomly distributed N atoms are listed in Tables 3.8 and 3.9 [44,48]. The same trends can be seen in the cases of tensile and shear loadings. The modulus hardly changes as N content increases. On the other hand, the strength decreases espe-cially when containing adjoining N atoms.

(a)

0

10

20

30

40

50

60

Nominal strain, γXY

Nom

inal

stre

ss, τ

XY (G

Pa)

PristineSingle vacancyDouble vacancySextuple vacancy

0

10

20

30

40

50

60

0 0.1 0.2 0.3 0 0.3Nominal strain, γXY

Nom

inal

stre

ss, τ

XY (G

Pa)

PristineSingle vacancyDouble vacancySextuple vacancy

0.20.1

(b)

FIGURE 3.9 Stress–strain curves of graphene containing a cluster-type vacancy under the shear loading. (a) AGR and (b) ZGR.

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0 2 4 6 8

Rela

tive s

tren

gth

Number of atom defects

Tension

Shear

FIGURE 3.10 Relative strengths and vacancy size, namely, the number of atom defects under tensile and shear loadings.

TABLE 3.6Strength of the Vacancy-Containing Graphene

Materials

Tensile Strength (GPa) Shear Strength (GPa)

MD(This Work)

MM [9](Zhang et al.)

QM [10](Mielke et al.)

MD (This Work)

AGRPristine 76 87.9 124 51

Single vacancy 61 (–19%) 64.8 (–26%) 101 (–18%) 36 (–30%)

Double vacancy 62 (–18%) 64.4 (–26%) 107 (–13%) 36 (–30%)

Sextuple vacancy 51 (–32%) 33 (–34%)

ZGRPristine 91 105.5 135 53

Single vacancy 75 (–17%) 70.4 (–33%) 100 (–26%) 42 (–17%)

Double vacancy 64 (–29%) 71.3 (–32%) 105 (–22%) 39 (–23%)

Sextuple vacancy 65 (–28%) 34 (–33%)

Note: The values in parentheses represent the differences between the pristine and vacancy-con-taining materials.


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The effects of the distance between N atoms on the mechanical properties were investigated. The stress–strain curves of the ZGR containing two N atoms that are located at an interval d under the tensile and shear loading are shown in Figure 3.12.

The calculated tensile and shear properties are listed in Table 3.10.

It is found that both the tensile and shear strengths decrease with decreasing distance between two N atoms. The decrease in the tensile strength relative to that of pris-tine ZGR is 25% when two N atoms adjoin each other, that is, when d is 1.76 Å. Then, the decrease in the shear strength relative to that of pristine ZGR is 29% at the same d. On the other hand, both the Young’s modulus and shear modulus hardly change.

The relations of the tensile and shear strengths with the distance between two N atoms are shown in Figure 3.13. The larger the distance between two N atoms, the higher are both the tensile and shear strengths. These relations are almost constant when d is over 2.38 Å.

Snapshots taken during the shear loadings are shown in Figure 3.14. In the case of d = 1.76 Å ((a-1)–(a-4)), fracture occurs at the point where two N atoms adjoin each other, sim-ilar to the case of tensile loading [39]. In the case of d = 2.38 and 7.31 Å ((b-1)–(b-4) and (c-1)–(c-4), respectively), fracture starts at the cleavage of the C─C bond adjoining the N atom and not at a C─N bond, similar to the case of tensile loading. In all the cases, fractures propagate in the direction of 120° to the X-axis.

010203040506070

(GPa) ixy

Y

XO

Initial structure ofthe pristine AGR

Fracture progressedJust before fractureoccurred

Initial structure of the AGR with a single vacancy


Initial structure ofthe AGR with a

sextuple vacancy

Fracture progressedJust before fracture occurred

Initial structure of the AGR with a double vacancy


Fracture started

Fracture started

Fracture started

Fracture started

(a-1) (a-2) (a-3) (a-4)

(b-1) (b-2) (b-3) (b-4)

(c-1) (c-2) (c-3) (c-4)

(d-1) (d-2) (d-3) (d-4) τ

FIGURE 3.11 Stages of fracture progression in the AGRs containing a cluster-type vacancy under shear loading. (a-1)–(a-4): pristine, (b-1)–(b-4): single vacancy, (c-1)–(c-4): double vacancy, (d-1)–(d-4): sextuple vacancy.

TABLE 3.7Tensile Properties of Vacancy-Containing Graphene

Density of Vacancies (%) Young’s Modulus (TPa) Tensile Strength (GPa)

0 0.794 91

1 0.752 (−5.4%) 56 (−38%)

2 0.657 (−17%) 48 (−48%)

4 0.617 (−22%) 37 (−59%)

Note: The values in parentheses represent the differences between the pristine and vacancy-containing materials.


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3.3.4 coMParison oF eFFects between Vacancy and nitrogen

The mechanical properties of graphene containing both vacancies and N atoms are summarized in Tables 3.11 and 3.12.

The tensile and shear strengths decrease greatly as the den-sity of vacancies increases at the constant N content of 4%. On the other hand, the tensile and shear strengths hardly change as the N content increases at the constant vacancy density of 4% as shown in Table 3.12.

0102030405060708090

100

Nominal strain, εX

Nom

inal

stre

ss, σ

X (G

Pa)

Pristined = 1.76 Åd = 2.38 Åd = 7.31 Å

0

10

20

30

40

50

60

0 0

Nom

inal

stre

ss, τ

XY (G

Pa)

Pristined = 1.76 Åd = 2.38 Åd = 7.31 Å

0.05 0.1 0.15 0.2

(a) (b)

0.05 0.1 0.15 0.2 0.25Nominal strain, γXY

FIGURE 3.12 Stress–strain curves of graphene containing two N atoms located at an interval d. (a) Tensile and (b) shear loadings.

TABLE 3.8Moduli of the N-Containing Graphene

Nitrogen Content (%)

Young’s Modulus (TPa) Shear Modulus (TPa)

Without Adjoining N Atoms

With Adjoining N Atoms



0 0.786 0.786 0.442 0.442

1 0.773 (−1.6%) 0.770 (−2.0%) 0.465 (+5.2%) 0.452 (+2.3%)

2 0.801 (+1.9%) 0.776 (−1.3%) 0.479 (+8.4%) 0.462 (+4.5%)

4 0.822 (+4.6%) 0.823 (+4.7%) 0.466 (+5.4%) 0.466 (+5.4%)

Note: The values in parentheses represent the differences between the pristine and N-containing materials.

TABLE 3.9Strength of the N-Containing Graphene


Tensile Strength (GPa) Shear Strength (GPa)





0 94 94 53 53

1 89 (−5.3%) 64 (−32%) 52 (−1.8%) 40 (−25%)

2 87 (−7.4%) 69 (−27%) 51 (−3.8%) 41 (−23%)

4 82 (−13%) 69 (−27%) 50 (−5.7%) 42 (−21%)

Note: The values in parentheses represent the differences between the pristine and N-containing materials.

TABLE 3.10Mechanical Properties of Graphene Containing Two N Atoms Located at a Distance d

d (Å)

Young’s Modulus

(GPa)

Tensile Strength

(TPa)

Shearing Modulus

(TPa)

Shearing Strength

(GPa)

1.76 0.795 71 0.452 37

2.38 0.796 84 0.450 50

7.31 0.809 90 0.570 51


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The relation between the tensile strength and vacancy density is shown in Figure 3.15a. The relation between the tensile strength and N content is also shown in Figure 3.15b. For the randomly distributed vacancies and N atoms, the aver-age values of the two results calculated using the models with different vacancy and N atom arrangements, is plotted. The error bar in the graph represents the range between these two values. The tensile strength decreases by 52% at the vacancy density of 4% and N content of 4%.

Snapshots taken during the tensile and shear loadings are shown in Figures 3.16 and 3.17, respectively. In all the cases, a fracture occurs due to cleavage of the C─C bond, which is on the periphery of the vacancy.

The mechanical properties of graphene containing both a cluster-type vacancy and a pair of adjoining N atoms were investigated. The relation between the relative strength, that is, the ratio of the tensile strength of graphene containing

0

20

40

60

80

100

0 2 4 6 8Distance d between nitrogen atoms ()

Stre

ngth

(GPa

)

Tensile loading

Shear loading

FIGURE 3.13 Relations of tensile and shear strengths with the distance between two N atoms.

Initial structure

Initial structure Progress of fracture

Progress of fracture

Beginning of fracture

Beginning of fracture

Before fracture

Before fracture

Initial structure Before fracture

Progress of fractureBeginning of fracture

: Nitrogen atom

Y

XO

(a-1) (a-2) (a-3) (a-4)

(b-1) (b-2) (b-3) (b-4)

(c-1) (c-2) (c-3) (c-4)

FIGURE 3.14 Stages of fracture progress under shear loading in graphene containing two N atoms at different intervals d: 1.76 Å ((a-1)–(a-4)), 2.38 Å ((b-1)–(b-4)), and 7.31 Å ((c-1)–(c-4)).

TABLE 3.11Effect of Vacancy on the Strength of N-Containing Graphene

Density of Vacancies (%)


Tensile Strength (GPa)

Shear Strength (GPa)

0 4 83 (−12%) 42 (−21%)

1 4 64 (−32%) 38 (−28%)

2 4 62 (−34%) 32 (−40%)

4 4 45 (−52%) 27 (−49%)

The values in parentheses represent the differences between the pristine and defect-containing materials.

TABLE 3.12Effect of N Atom on the Strength of Vacancy-Containing Graphene

Density of Vacancies (%)


Tensile Strength (GPa)

Shear Strength (GPa)

4 0 45 (−52%) 27 (−49%)

4 1 44 (−5.3%) 27 (−49%)

4 2 45 (−5.2%) 27 (−49%)

4 4 45 (−52%) 27 (−49%)

The values in parentheses represent the differences between the pristine and defect-containing materials.


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defects to that of pristine graphene, and vacancy size, is shown together with the results [11] for the ZGR containing only the vacancy in Figure 3.18.

There is no difference in the behavior of relative strength when the vacancy size increases for the two cases containing both a vacancy and N atoms, and only a vacancy.

Snapshots of ZGR containing both a vacancy and a pair of adjoining N atoms taken during the tensile loading are

shown in Figure 3.19. In the case of the single vacancy ((a-1)–(a-4)), a fracture occurs on the periphery of the adjoining N atoms. On the other hand, in the other cases, a fracture occurs around the vacancy, instead of the periphery of the adjoining N atoms.

Thus, it was found that the effect of adjoining N atoms is larger than that of the single vacancy, and smaller than that of the double vacancy.

0102030405060708090

Stre

ngth

(GPa

)

Vacancy density (%)

TensionShear

0

10

20

30

40

50

60

0 1 2 3 4 5 0 1 2 3 4 5

Stre

ngth

(GPa

)

Nitrogen content (%)

TensionShear

(a) (b)

FIGURE 3.15 Relation between strength and density of defects. Strength with (a) vacancy density and (b) N-content.

Initial structure

Y

XO: Carbon atom : Nitrogen atom

(a-1) (a-2) (a-3) (a-4)

(b-1) (b-2) (b-3) (b-4)

(c-1) (c-2) (c-3) (c-4)

(d-1) (d-2) (d-3) (d-4)

Cleavage of C–C Fracture developed Tearing of the sheet

Tearing of the sheetFracture developedCleavage of C–CInitial structure

Initial structure Cleavage of C–C Fracture developed Tearing of the sheet

Tearing of the sheetFracture developedCleavage of C–CInitial structure

FIGURE 3.16 Stages of fracture progressions of ZGRs containing vacancies and N atoms randomly under tensile loading. (a-1)–(a-4): 1% vacancies and 4% N atoms, (b-1)–(b-4): 4% vacancies and 1% N atoms, (c-1)–(c-4): 4% vacancies and 4% N atoms, (d-1)–(d-4): 4% vacan-cies and 0% N atoms.


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Next, the effects of distance between two N atoms when containing the single vacancy and two N atoms were investigated.

The stress–strain curves of the ZGRs containing a single vacancy and a pair of two N atoms which are located parallel to the tensile axis at a different interval d, are shown in Figure

3.20. The larger the distance of separation between two N atoms, the higher is the tensile strength as shown in Figure 3.21, similar to the case of containing only two N atoms [44].

Snapshots taken during the tensile loading are shown in Figure 3.22. In the case of the single vacancy and the pair of adjoining N atoms ((a-1)–(a-4)), a fracture occurs on the periphery of the adjoining N atoms. On the other hand, in the other cases, that is, when the distance of separation between two N atoms becomes large, a fracture occurs around the vacancy.

Thus, graphene is more sensitive to vacancies than N atoms, except when two N atoms adjoin each other in graphene.

3.3.5 coMParison oF eFFects between sP3 bond and nitrogen

The stress–strain curves of the bilayer ZGRs containing an sp3 bond in the interlayer and two N atoms per layer are shown in Figure 3.23. The relations of the tensile strengths with the distance between two N atoms are shown in Figure 3.24. The greater the distance between two N atoms, the higher is the tensile strength, similar to the results of graphene with vacan-cies and N atoms.


Initial structure Cleavage of C–C Tearing of the sheet


Y

XO : Carbon atom : Nitrogen atom

(a-1) (a-2) (a-3) (a-4)

(b-1) (b-2) (b-3) (b-4)

(c-1) (c-2) (c-3) (c-4)

(d-1) (d-2) (d-3) (d-4)


Fracture developed

FIGURE 3.17 Stages of fracture progressions of ZGRs containing vacancies and N atoms randomly under shear loading. (a-1)–(a-4): 1% vacancies and 4% N atoms, (b-1)–(b-4): 4% vacancies and 1% N atoms, (c-1)–(c-4): 4% vacancies and 4% N atoms, (d-1)–(d-4): 4% vacan-cies and 0% N atoms.

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0 2 4 6 8

Rela

tive s

tren

gth

Number of defects

Vacancy and N-atomsOnly vacancy

FIGURE 3.18 Relative strength and number of atom defects, that is, vacancy size in graphene containing a cluster-type vacancy and a pair of adjoining N atoms, or only a cluster-type vacancy.


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Snapshots taken during the tensile loading are shown in Figure 3.25. In the case where the d = 1.76 Å ((a-1)–(a-3), (a′- 1)–(a′-3)), that is, the pair of adjoining N atoms, a frac-ture occurs on the periphery of the adjoining N atoms. On the other hand, when the d becomes larger (d = 7.31 Å ((c-1)–(c-3), (c′-1)–(c′-3))), a fracture occurs around the sp3 bond.

Finally, the mechanical properties of bilayer ZGRs con-taining both randomly distributed N atoms and sp3 bonds were investigated. The relation between the relative strength, that is, the ratio of the tensile strength of bilayer ZGR with

defects to the tensile strength of perfect graphene and N con-tents, is shown by filled diamonds in Figure 3.26. The result [44] in the case of containing only N atoms is also shown for reference by open diamonds. The tensile strength decreases by about 20% when containing sp3 bonds at the density of 5%, but it hardly changes as the N content increases. It is found that the decrease in strength when both sp3 bonds and N atoms are present together becomes greater than that in the case of including only N atoms.

Snapshots of the bilayer ZGR with 10% N atoms and 5% sp3 bonds taken during the tensile loading are shown in Figure 3.27. A fracture occurs due to cleavage of the C─C bond, which is in proximity to both a C─N bond and interlayer sp3

Initial structure Initial structure

Beginning of a fracture

Progression of the fracture

Tearing of the sheet

Beginning of a fracture Beginning of a fracture

Progression of the fracture Progression of the fracture

Tearing of the sheet Tearing of the sheet

: Nitrogen atom: Carbon atom

Y

XO

(a-1) (b-1) (c-1)

(a-2) (b-2) (c-2)

(a-3) (b-3) (c-3)

(a-4) (b-4) (c-4)

Initial structure

FIGURE 3.19 Stages of fracture progressions of ZGR containing a cluster-type vacancy and a pair of adjoining N atoms. (a-1)–(a-4): single vacancy and N atoms, (b-1)–(b-4): double vacancy and N atoms, (c-1)–(c-4): sextuple vacancy and N atoms.

0

20

40

60

80

100

120

0 0.2Nominal strain, εx

Nom

inal

stre

ss, σ

x (G

Pa)

Pristined = 1.76 Åd = 2.38 Åd = 7.31 Å

0.05 0.1 0.15

FIGURE 3.20 Stress–strain curves of ZGRs containing a single vacancy and a pair of two N atoms which are located parallel to the tensile axis at a different interval d.

0

20

40

60

80

100

0 2 4 6 8

Tens

ile st

reng

th (G

Pa)

Distance between nitrogen atoms ()

FIGURE 3.21 Tensile strength and distance between N atoms in ZGR containing a single vacancy and a pair of adjoining N atoms which are located parallel to the tensile axis at the different interval d.


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bond as shown in Figure 3.28. In the case of containing only N atoms [44], it was found that a fracture occurs due to cleav-age of the C─C bond which is in proximity to a C─N bond. On the other hand, in the case of containing both N atoms and sp3 bonds, it was found that the C─C bond which is in proximity to both a C─N bond and interlayer sp3 bond is more easily cleaved. The greater decrease in strength when both sp3 bonds and N atoms are present together seems to be owing to those behaviors. Thus, graphene is slightly more sensitive to sp3 bonds than N atoms, except when two nitrogen atoms adjoin each other in graphene.

3.3.6 coMParison oF eFFects between Vacancy and sP3 bond

In the case of containing randomly distributed vacancies, the reduction in the tensile strength stands at about 59% at the vacancy density of 4% as shown in Table 3.7. On the other hand, the reduction stands at 17% at the sp3 bond den-sity of 5.81% by Zhang et al. [36]. Recently, Zandiatashbar et al. investigated the effect of vacancy and sp3 bond on the mechanical properties of graphene using AFM nanoindenta-tion [2]. They reported that vacancy affects the mechanical properties of graphene more significantly than the sp3 bond. Thus, graphene must be more sensitive to vacancy than the sp3 bond.

0

20

40

60

80

100

0 2 4 6 8

Tens

ile st

reng

th (G

Pa)

Distance between nitrogen atoms ()

FIGURE 3.24 Tensile strength and distance between N atoms in bilayer ZGR containing an interlayer sp3 bond and two N atoms per layer, which are located parallel to the tensile axis at a different interval d.

: Nitrogen atom: Carbon atom

Beginning of a fracture

Progression of the fracture

Tearing of the sheet

Progression of the fracture Progression of the fracture

Beginning of a fracture Beginning of a fracture

Tearing of the sheet Tearing of the sheet

Y

XO

(a-1) (b-1) (c-1)

(a-2) (b-2) (c-2)

(a-3) (b-3) (c-3)

(a-4) (b-4) (c-4)

Initial structure Initial structure Initial structure

FIGURE 3.22 Stages of fracture progressions of ZGR containing a single vacancy and a pair of adjoining N atoms which are located parallel to the tensile axis at a different interval d : 1.76 Å ((a-1)–(a-4)), 2.38 Å ((b-1)–(b-4)), and 7.31 Å ((c-1)–(c-4)).

0

20

40

60

80

100

120

0 0.05 0.1 0.15 0.2Nominal strain, εx

Nom

inal

stre

ss, σ

x (G

Pa)

Pristined = 1.76 Åd = 2.38 Åd = 7.31 Å

FIGURE 3.23 Stress–strain curves of bilayer ZGRs containing an interlayer sp3 bond and two N atoms per layer which are located parallel to the tensile axis at a different interval d.


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3.4 CONCLUSIONS

The authors performed the MD simulations of tensile or shear loadings on graphene containing vacancies, nitrogen atoms,

or sp3 bonds, to investigate the influence of these defects on the mechanical properties.

It was found that for the cluster-type vacancy, the relation-ship between the vacancy size and the shear strength agrees with that during the tensile loading at any size of vacancy. Fractures develop in the zigzag direction, similar to the results of tensile loading.

It was also found that the shear strength decreases greatly when two N atoms adjoin each other, similar to the results of tensile loading. In addition, fracture starts at the cleavage of the C─C bond adjoining the N atom and not at a C─N bond, similar to the case of tensile loading.

The authors compared the effects of vacancies and N atoms using the analysis models containing both vacancies and N atoms. It was clarified that graphene is more sensitive to vacancies than N atoms, except when two nitrogen atoms adjoin each other in graphene.

The authors also compared the effects of interlayer sp3 bonds and N atoms using the analysis models containing both sp3 bonds and N atoms. It was clarified that graphene

X

Y

Z

X

Y

Z

X

Y

Z

sp3 bond

sp3 bond

sp3 bond

Initial structure(viewed in the Z direction)

Initial structure(viewed in the Y direction)

Beginning of a fracture(viewed in the Z direction)

Beginning of a fracture(viewed in the Y direction)

Tearing of the layer(viewed in the Z direction)

Tearing of the layer(viewed in the Y direction)






X

Z

Y ×

X

Z

Y

Y

×

X

Z

×






(a-1) (a-2) (a-3)

(a′-1) (a′-2) (a′-3)

(b-1) (b-2) (b-3)

(b′-1)(b′-2) (b′-3)

(c-1) (c-2) (c-3)

(c′-1) (c′-2) (c′-3)



FIGURE 3.25 Stages of fracture progressions of bilayer ZGRs containing an interlayer sp3 bond and two N atoms per layer which are located parallel to the tensile axis at a differing interval d: 1.76 Å ((a-1)–(a-3)), 2.38 Å ((b-1)–(b-3)), and 7.31 Å ((c-1)–(c-3)).

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0 2 4 6 8 10 12Nitrogen content (%)

Rela

tive s

tren

gth

: N-atoms and 5 % sp3 bonds: Only N-atoms

FIGURE 3.26 Relative strength as a function of nitrogen content in bilayer ZGR with N atoms and 5% sp3 bonds, or in ZGR with only N atoms.


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is slightly more sensitive to sp3 bonds than N atoms. It was also found that a fracture occurs due to cleavage of the C─C bond which is in proximity to both a C─N bond and inter-layer sp3 bond.

In summary, it was found that the effect on the strength of graphene increases in the order of N atom, sp3 bond, and vacancy.

REFERENCES

1. Thrower, P. A. 1969. The study of defects in graphite by trans-mission electron microscopy. Chem. Phys. Carbon 5:217.

2. Zandiatashbar, A., Lee, G.-H., An, S. J., Lee, S., Mathew, N., Terrones, M., Hayashi, T., Picu, C. R., Hone, J., and Koratkar, N. 2014. Effect of defects on the intrinsic strength and stiff-ness of graphene. Nat. Commun. 5, article 3186.

3. Hashimoto, A., Suenaga, K., Gloter, A., Urita, K., and Iijima, S. 2004. Direct evidence for atomic defects in graphene layers. Nature 430:870–873.

4. Osing, J. and Shvets, I. V. 1998. Bulk defects in graphite observed with a scanning tunnelling microscope. Surf. Sci. 417:145–150.

5. Xiao, J. R., Staniszewski, J., and Gillespie Jr., J. W. 2010. Tensile behaviors of graphene sheets and carbon nano-tubes with multiple Stone-Wales defects. Mater. Sci. Eng. A 527:715–723.

6. Grantab, R., Shenoy, V. B., and Ruoff, R. S. 2010. Anomalous strength characteristics of tilt grain boundaries in graphene. Science 330:946–948.

7. Liu, T.-H., Pao, C.-W., and Chang, C.-C. 2013. Thermal response of grain boundaries in graphene sheets under shear strain from atomistic simulations. Comp. Mater. Sci. 70:163–170.

8. Ansari, R., Ajori, S., and Motevalli, B. 2012. Mechanical prop-erties of defective single-layered graphene sheets via molecu-lar dynamics simulation. Superlatt. Microstruct. 51:274–289.

9. Zhang, S., Mielke, S. L., Khare, R., Troya, D., Ruoff, R. S., Schatz, G. C., and Belytschko, T. 2005. Mechanics of defects in carbon nanotubes: Atomistic and multiscale simulations. Phys. Rev. B 71:115403.

10. Mielke, S. L., Troya, D., Zhang, S., Li, J.-L., Xiao, S., Car, R., Ruoff, R., Shatz, G. C., and Belytschko, T. 2004. The role of vacancy defects and holes in the fracture of carbon nanotubes. Chem. Phys. Lett. 390:413–420.

11. Ito, A. and Okamoto, S. 2012. Molecular dynamics analysis on effects of vacancies upon mechanical properties of gra-phene and graphite. Eng. Lett. 20:271–278.

12. Xia, Y., Zhao, M., Ma, Y., Ying, M., Liu, X., Liu, P., and Mei, L. 2002. Tensile strength of single-walled carbon nanotubes with defects under hydrostatic pressure. Phys. Rev. B 65:155415.

13. Ogata, S. and Shibutani, Y. 2003. Ideal tensile strength and band gap of single-walled carbon nanotubes. Phys. Rev. B 68:165409.

14. Sammalkorpi, M., Krasheninnikov, A., Kuronen, A., Nordlund, K., and Kaski, K. 2004. Mechanical properties of carbon nanotubes with vacancies and related defects. Phys. Rev. B 70:245416.

15. Tserpes, K. I., Papanikos, P., and Tsirkas, S. A. 2006. A pro-gressive fracture model for carbon nanotubes. Composites: Part B 37:662–669.

16. Agrawal, P. M., Sudalayandi, B. S., Raff, L. M., and Komanduri, R. 2006. A comparison of different methods of Young’s modulus determination for single-wall carbon nano-tubes (SWCNT) using molecular dynamics (MD) simulations. Comp. Mater. Sci. 38:271–281.

17. Khare, R., Mielke, S. L., Paci, J. T., Zhang, S., Ballarini, R., Schatz, G. C., and Belytschko, T. 2007. Coupled quantum mechanical/molecular mechanical modeling of the fracture of defective carbon nanotubes and graphene sheets. Phys. Rev. B 75:75412.

18. Xin, H., Han, Q., and Yao, X.-H. 2007. Buckling and axially compressive properties of perfect and defective single-walled carbon nanotubes. Carbon 45:2486–2495.

19. Tserpes, K. I. and Papanikos, P. 2007. The effect of Stone–Wales defect on the tensile behavior and fracture of single-walled carbon nanotubes. Compos. Struct. 79:581–589.

20. Shen, L. and Chen, Z. 2007. An investigation of grain size and nitrogen-doping effects on the mechanical properties of ultrananocrystalline diamond films. Int. J. Sol. Struct. 44:3379–3392.

sp3 bond

Nitrogen

×

Cleavage

×

Carbon

Carbon

(a) (b)

FIGURE 3.28 Enlargements of the boxed section shown in Figure 3.27b. (a) Just before a cleavage occurs and (b) the cleavage occurred.

X

Z

Y×

(a) (b) (c)

FIGURE 3.27 Snapshots of the bilayer ZGR with 10% N atoms and 5% sp3 bonds during tensile loading. (a) Initial structure, (b) just before a cleavage occurs, and (c) fractures occurred.


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ded

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10.

3.98

.104

At:

22:3

6 16

Feb

202

2; F

or: 9

7814

6659

1240

, cha

pter

3, 1

0.12

01/b

1967

4-5


21. Liu, F., Ming, P., and Li, J. 2007. Ab initio calculation of ideal strength and phonon instability of graphene under tension. Phys. Rev. B 76:064120.

22. Hai-Yang, S. and Xin-Wei, Z. 2008. Molecular dynamics study of effects of sp3 interwall bridging and initial vacancy-related defects on mechanical properties of double-walled carbon nanotube. Phys. B 403:3798–3802.

23. Jiang, J.-W., Wang, J.-S., and Li, B. 2009. Young’s modu-lus of graphene: A molecular dynamics study. Phys. Rev. B 80:113405.

24. Hai-Yang, S. and Xin-Wei, Z. 2009. Molecular dynamics study of effects of sp3 interwall bridging upon torsional behavior of double-walled carbon nanotube. Phys. Lett. A 373:682–685.

25. Shen, L. and Chen, Z. 2009. A study of mechanical proper-ties of pure and nitrogen-doped ultrananocrystalline diamond films under various loading conditions. Int. J. Sol. Struct. 46:811–823.

26. Zhao, H., Min, K., and Aluru, N. R. 2009. Size and chirality dependent elastic properties of graphene nanoribbons under uniaxial tension. Nano Lett. 9:3012–3015.

27. Zhao, H. and Aluru, N. R. 2010. Temperature and strain-rate dependent fracture strength of graphene. J. Appl. Phys. 108: 64321.

28. Wong, C. H. 2010. Elastic properties of imperfect single-walled carbon nanotubes under axial tension. Comp. Mater. Sci 49:143–147.

29. Zheng, Q., Geng, Y., Wang, S., Li, Z., and Kim, J.-K. 2010. Effects of functional groups on the mechanical and wrinkling properties of graphene sheets. Carbon 48:4315–4322.

30. Pei, Q. X., Zhang, Y. W., and Shenoy, V. B. 2010. A molecu-lar dynamics study of the mechanical properties of hydrogen functionalized graphene. Carbon 48:898–904.

31. Neek-Amal, M. and Peeters, F. M. 2010. Linear reduction of stiffness and vibration frequencies in defected circular mono-layer graphene. Phys. Rev. B 81:235437.

32. Tsai, J.-L. and Tu, J.-F. 2010. Characterizing mechanical properties of graphite using molecular dynamics simulation. Mater. Design 31:194–199.

33. Shokrieh, M. M. and Rafiee, R. 2010. Prediction of Young’s modulus of graphene sheets and carbon nanotubes using nanoscale continuum mechanics approach. Mater. Design 31: 790–795.

34. Georgantzinos, S.K., Giannopoulos, G.I., and Anifantis, N.K. 2010. Numerical investigation of elastic mechanical proper-ties of graphene structures. Mater. Design 31:4646–4654.

35. Ansari, R., Motevalli, B., Montazeri, A., and Ajori, S. 2011. Fracture analysis of monolayer graphene sheets with double vacancy defects via MD simulation. Solid State Commun. 151:1141–1146.

36. Zhang, Y.Y., Wang, C.M., Cheng, Y., and Xiang, Y. 2011. Mechanical properties of bilayer graphene sheets coupled by sp3 bonding. Carbon 49:4511–4517.

37. Min, K. and Aluru, N.R. 2011. Mechanical properties of gra-phene under shear deformation. Appl. Phys. Lett. 98:013113.

38. Tapia, A., Peón-Escalante, R., Villanueva, C., and Avilés, F. 2012. Influence of vacancies on the elastic properties of a graphene sheet. Comp. Mater. Sci. 55:255–262.

39. Mortazavi, B., Ahzi, S., Toniazzo, V., and Rémond, Y. 2012. Nitrogen doping and vacancy effects on the mechanical prop-erties of graphene: A molecular dynamics study. Phys. Lett. A 376:1146–1153.

40. Wang, M. C., Yan, C., Ma, L., Hu, N., and Chen, M. W. 2012. Effect of defects on fracture strength of graphene sheets. Comp. Mater. Sci. 54:236–239.

41. Wong, C. H. and Vijayaraghavan, V. 2012. Nanomechanics of free form and water submerged singlelayer graphene sheet under axial tension by using molecular dynamics simulation. Mater. Sci. Eng. A 556:420–428.

42. Cao, A. and Qu, J. 2012. Study on the mechanical behavior of tilt bicrystal graphene by molecular dynamics simulations: Bulk verse nanoribbons. J. Appl. Phys. 112:43519.

43. Cao, A. and Yuan, Y. 2012. Atomistic study on the strength of symmetric tilt grain boundaries in graphene. Appl. Phys. Lett. 100:211912.

44. Okamoto, S. and Ito, A. 2012. Effects of nitrogen atoms on mechanical properties of graphene by molecular dynamics simulations. Eng. Lett. 20:169–175.

45. Cao, A. and Qu, J. 2013. Atomistic simulation study of brittle failure in nanocrystalline graphene under uniaxial tension. Appl. Phys. Lett. 102:71902.

46. Yi, L., Yin, Z., Zhang, Y., and Chang, T. 2013. A theoreti-cal evaluation of the temperature and strain-rate dependent fracture strength of tilt grain boundaries in graphene. Carbon 51:373–380.

47. Wu, J. and Wei, Y. 2013. Grain misorientation and grain-boundary rotation dependent mechanical properties in poly-crystalline graphene. J. Mech. Phys. Solids 61:1421–1432.

48. Okamoto, S. and Ito, A. 2013. Molecular dynamics analysis on tensile and shearing properties of nitrogen-containing gra-phene. J. Comm. Comp. 10:19–27.

49. Ito, A. and Okamoto, S. 2013. Tensile and shearing properties of vacancy-containing graphene using molecular dynamics simulations. J. Comm. Comp. 10:9–18.

50. Le, M.-Q. and Batra, R. C. 2014. Crack propagation in pre-strained single layer graphene sheets. Comp. Mater. Sci. 84:238–243.

51. Kheirkhah, A. H., Iranizad, E. S., Raeisi, M., and Rajabpour, A. 2014. Mechanical properties of hydrogen functionalized graphene under shear deformation: A molecular dynamics study. Solid State Commun. 177:98–102.

52. Han, J., Ryu, S., Sohn, D., and Im, S. 2014. Mechanical strength characteristics of asymmetric tilt grain boundaries in graphene. Carbon 68:250–257.

53. He, L., Guo, S., Lei, J., Sha, Z., and Liu, Z. 2014. The effect of Stone–Thrower–Wales defects on mechanical properties of graphene sheets—A molecular dynamics study. Carbon 75:124–132.

54. Brenner, D. W., Shenderova, O. A., Harrison, J. A., Stuart, S. J., Ni, B., and Sinnott, S. B. 2002. A second-generation reactive empirical bond order (REBO) potential energy expression for hydrocarbons. J. Phys.: Condens. Matter 14:783–802.

55. Tersoff, J. 1989. Modeling solid-state chemistry: Interatomic potentials for multicomponent systems. Phys. Rev. B 39:5566–5568.

56. Tersoff, J. 1998. Structural properties of amorphous silicon nitride. Phys. Rev. B 58: 8323–8328.

57. Shenderova, O. A., Brenner, D. W., Omeltchenko, A., Su, X., Yang, L. H., and Young, M. 2000. Atomistic modeling of the fracture of polycrystalline diamond. Phys. Rev. B 61: 3877–3888.

58. Ruland, W. 1965. X-ray studies on the structure of graphitic carbons. Acta Cryst. 18:992–996.

59. Stuart, S. J., Tutein, A. B., and Harrison, J. A. 2000. A reactive potential for hydrocarbons with intermolecular interactions. J. Chem. Phys. 112:6472–6486.

60. Woodcock, L. V. 1971. Isothermal molecular dynamics calcu-lations for liquid salts. Chem. Phys. Lett. 10:257–261.


Dow

nloa

ded

By:

10.

3.98

.104

At:

22:3

6 16

Feb

202

2; F

or: 9

7814

6659

1240

, cha

pter

3, 1

0.12

01/b

1967

4-5


61. Lee, C., Wei, X., Kysar, J. W., and Hone, J. 2008. Measurement of the elastic properties and intrinsic strength of monolayer graphene. Science 321:385–388.

62. Frank, I. W., Tanenbaum, D. M., van der Zande, A. M., and McEuen, P. L. 2007. Mechanical properties of suspended gra-phene sheets. J. Vac. Sci. Technol. B 25:2558–2561.

63. Shen, L, Shen, H.-S., and Zhang, C.-L. 2010. Temperature-dependent elastic properties of single layer graphene sheets. Mater. Design 31:4445–4449.

64. Sakhaee-Pour, A. 2009. Elastic properties of single-layered graphene sheet. Solid State Commun. 149:91–95.

65. Blakslee, O. L., Proctor, D. G., Seldin, E. J., Spence, G. B., and Weng, T. 1970. Elastic constants of compression-annealed pyrolytic graphite. J. Appl. Phys. 41:3373–3382.

66. LiJun, Y. and TienChong, C. 2012. Loading direction depen-dent mechanical behavior of graphene under shear strain. Sci. China—Phys. Mech. Astron. 55:1083–1087.

67. Faccio, R., Denis, P. A., Pardo, H., Goyenola, C., and Mombrú, Á. W. 2009. Mechanical properties of graphene nanoribbons. J. Phys.: Condens. Matter 21:285304.


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