Page 1

Abstract—We investigated the mechanical properties of

graphene and graphite containing vacancies under tensile

loading using molecular dynamics (MD) simulations. In the MD

simulations, we used two types of potential functions: the

second-generation reactive empirical bond-order (REBO)

potential for covalent C–C bond, and the Lennard-Jones

potential for the interlayer interaction of graphite. The influence

of the size and the distributional form of vacancies on the

mechanical properties of graphene and graphite were studied. It

was found that the tensile strength of graphene having randomly

distributed vacancies with a vacancy density of 4%, decreased

by 59%.

Index Terms—Graphene, graphite, molecular dynamics,

vacancy

I. INTRODUCTION

ARBON-based materials can have excellent mechanical

and electrical properties. Therefore, there is much

interest around their use in applications in structural

sub-assemblies and nano-electro mechanical systems such as

electrochemical electrodes and field emission. Carbon

materials such as diamond, graphene, carbon nanotubes

(CNT), and fullerenes, have a wide range of excellent

properties thanks to their different types of bonds and

atomistic structures. In particular, graphene has rigidity and

strength that are nearly equal to those of diamond, as well as

novel electronic properties including high electron mobility.

Thus, studies on graphene and graphite made of graphene

layers have recently intensified [1]–[3].

Defects often affect the mechanical and electronic

properties of materials. There have been reports of

experimental studies on defects (i.e., vacancies [4],

dislocations [5], and grain boundaries [6]) in graphene layers.

It is important to clarify the influence of defects on the

mechanical and electrical properties of graphene and graphite

in order to produce high-performance carbon materials.

Recently, studies aiming to clarify the relationship between

Manuscript received December 14, 2011. This work was supported in

part by the Ring-Ring project of JKA.

Akihiko Ito is with the Mechanical Engineering Course, Graduate School

of Science and Engineering, Ehime University, 3 Bunkyo-cho, Matsuyama

790-8577, Japan. He is also with the Composite Materials Research

Laboratories, Toray Industries, Inc., Masaki-cho 791-3193, Japan (e-mail:

Akihiko_Ito@nts.toray.co.jp).

Shingo Okamoto is with the Mechanical Engineering Course, Graduate

School of Science and Engineering, Ehime University, 3 Bunkyo-cho,

Matsuyama 790-8577, Japan (e-mail: okamoto.shingo.mh@ehime-u.ac.jp).

atomic-scale defects and mechanical properties have

increased in number. The tensile properties of graphene and

carbon nanotubes containing multiple Stone-Wales (SW)

defects have been investigated using molecular dynamics

(MD) simulations by Xiao et al. [7]. These studies have

clarified the relationship between the number of defects and

the mechanical properties. The influence of grain boundaries

on the tensile strength of graphene has been investigated by

Grantab et al. [8]. The MD simulations of tensile loadings of

single-walled carbon nanotubes with vacancies have been

performed by Wong et al. [9]. The influence of the single and

double vacancies on the tensile strength has been investigated

through molecular mechanics (MM) calculations by Zhang et

al. [10]. Zhang et al. compared their results obtained using

MM calculations with Mielke’s results obtained using

quantum mechanics (QM) calculations [11]. However, the

influence of vacancies on the mechanical properties of

graphene and graphite were neglected. In this study, we

investigated the influence of vacancy size on the mechanical

properties of graphene and graphite through MD simulations.

In addition, we clarified the relationship between the

distributional form of vacancies and the mechanical

properties.

II. METHOD

A. Potential Function

In the present study, we used two types of interatomic

potentials: the second-generation reactive empirical bond

order (2nd REBO) [12], and Lennard-Jones potentials. The 2nd

REBO potential for covalent C-C bonds is expressed as

( )

∑∑

>

iij

( )

ij

r

[

V

],

*

ij

−=

A ijR REBO

EVBr

(1)

where rij represents the distance between atoms i and j. The

Bij

VA(rij) represent the pair-additive interactions that reflect

interatomic repulsions and attractions, respectively as in

( ) ( )

=

cR

rfrV

* represents the bond-order term. The terms VR(rij) and

()

( )

r

( )

r

()

∑

=

n

−=

−

+

1

3

1

cA

exp

exp

nn

rBfV

rA

r

Q

β

α

, (2)

Mechanical Properties of Vacancy-containing

Graphene and Graphite

using Molecular Dynamics Simulations

Akihiko Ito and Shingo Okamoto

C

Proceedings of the International MultiConference of Engineers and Computer Scientists 2012 Vol I,

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ISBN: 978-988-19251-1-4

ISSN: 2078-0958 (Print); ISSN: 2078-0966 (Online)

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where Q, A, α, Bn, and βn represent constant parameters. The

function fc(r) represents the cutoff function that decreases

monotonously from 1 to 0 as in

, 1

RR

( )

r

()

,

, 0

, 2/ cos1

max

maxmin

min max

min

min

>

<<

−

−

+

<

=

Rr

RrR

Rr

Rr

fc

π

(3)

-5

0

5

10

15

20

25

30

35

1.4 1.6

Bond length rij(Å)

1.82 2.2

Force (nN)

This work (Rmin= 2.0Å)

Original

(Rmin= 1.7Å)

Fig. 1 Interatomic forces for 2nd REBO potential with original Rmin and Rmin

used in this work.

where Rmin = 1.7Å and Rmax = 2.0Å in original 2nd REBO

potential.

It is known that for the original 2nd REBO potential, the

interatomic force increases dramatically at r = Rmin and

reaches zero at r = Rmax owing to the discontinuity in the

second derivative of the cutoff function, as shown in Fig. 1.

This dramatic increase in the interatomic force with the

original 2nd REBO potential may greatly affect tensile

strength. Therefore, in this work, the cutoff parameter is set to

2.0 Å in order to avoid the dramatic increase in the

interatomic force [13]. The other parameters, except for Rmin,

are set to the values proposed by Brenner [12]. The

Lennard-Jones potential for the interlayer interaction in the

graphite model is expressed as

ij

r

.4

6

0

12

0

−

=

ij

LJ

r

rr

V

ε

(4)

The 2nd REBO potential and the Lennard-Jones potential

are switched according to the interatomic distance and bond

order [14]. The value of ε is set to 0.00284 eV and r0 is set to

3.2786 Å so that the interplanar spacing of graphite at 300 K

is 3.35 Å, which is a known experimental value.

B. Analysis model

Firstly, the analysis models of graphene used under zigzag

tensions consist of 588 carbon atoms with dimensions equal to

those of the real crystallite in a typical carbon material, as

shown in Fig. 2.

No periodic boundary conditions are imposed in our case.

The analysis models consist of two parts. One is referred to as

ll

Y

X

51Å

30Å

O

: Carbon atom

l

52Å

30Å

l

(a) Armchair tension model (b) Zigzag tension model

Fig. 2 Configurations of graphene used under zigzag tension.

52Å

30Å

52Å

ll

ll

24Å

XXX

ZZZ

YYY

OOO

24Å

30Å

Fig. 3 Configuration of graphite used under zigzag tension.

AAA

BBB

AAA

3.35Å

3.35Å

carbon atom

Fig. 4 Schematic of graphite structure.

(a) Single - vacancy

(b) Double - vacancy

Fig. 5 Analysis model for the graphene containing a cluster-type vacancy.

the active zone in which the atoms move according to the

interactions with their

other—enclosed within the boxes (as shown in Fig. 2)—is

referred to as the boundary zone in which the atoms are

restrained. The thickness l of the boundary zone is 3.0a for the

armchair tension model and 1.5×

model, where a is the length of the C=C bond in graphene.

The analysis model of graphite used under zigzag tension

consists of 4,116 carbon atoms with dimensions equal to those

of the real crystallite in typical carbon material, as shown in

Fig. 3. The graphite model is made of seven layers of

graphene sheets, which are stacked in an AB-type sequence

with an interlayer spacing of 3.35 Å, as shown in Fig. 4.

The analysis models of graphene with cluster-type

vacancies are shown in Fig. 5. These models of graphite

(c) Sextuple - vacancy

YY

XX

OO

neighboring atoms. The

a3 for the zigzag tension

Proceedings of the International MultiConference of Engineers and Computer Scientists 2012 Vol I,

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ISBN: 978-988-19251-1-4

ISSN: 2078-0958 (Print); ISSN: 2078-0966 (Online)

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(a) 1% - vacancies

(b) 2% - vacancies(c) 4% - vacancies

YY

XX

OO

Fig. 6 Analysis model for the graphene containing uniformly distributed

vacancies.

reveal that the graphene sheet with a cluster-type vacancy is

always the center layer.

The analysis models of graphene containing uniformly

distributed vacancies are shown in Fig. 6. Each vacancy is a

single vacancy and set so that the distance between

neighboring vacancies is identical. Calculations for three

values of vacancy density, namely 1, 2, and 4%, are

performed.

The analysis models of graphene containing randomly

distributed vacancies are set by removing carbon atoms in the

active zone using a pseudorandom number generator.

C. Molecular dynamics simulations

We investigated

vacancy-containing graphene and graphite using the MD

simulations under constant volume and temperature, that is, a

canonical (NVT ) ensemble. The 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 [15] so that the temperature of the

object can be maintained at the preset temperature TSET. The

mass of a single carbon atom, m, is 1.9927×10-26 kg. The time

step is 1.0 fs.

The atomic stress acting on each atom is calculated to

obtain the stress-strain curves and to visualize the stress

distribution during tensile loadings. The atomic stress σi

each of the X, Y, and Z directions of J is given by calculating

the kinetic energies of, the interatomic force acting on, and the

volume occupied by atom i as in

(

i

Vm

=Ω

the mechanical properties of

J for

),

1

J

ii

J

i

J

i

J

i

FJV

+

σ

(5)

where Ωi represents the volume occupied by atom i, which is

referred to as the atomic volume. The atomic volume is

calculated by averaging the volume over all atoms in the

initial structure of each system. The interatomic force acting

on atom i due to its neighboring atoms is represented by Fi.

The global stress of an analysis model is calculated by

averaging over all carbon atoms in each system.

Method of tension loading

The initial positions of the atoms are given so that the

analysis model represents the crystal structure of graphene or

graphite at a preset temperature. First, the atoms in the active

zone of the analysis model are relaxed in unloaded states for

7,000 MD steps. The atoms in the boundary zone are fixed.

After constant displacements are applied to the atoms in both

of the boundary zones to simulate uniaxial tensile loading in

the X direction, the atoms in the active zone are relaxed for

7,000 MD steps. The strain increment, ∆ε, is 0.004. The

output stresses are sampled for the last 2,000 MD steps for

each strain and are averaged. Young’s moduli are obtained

from the slopes of the straight lines in the range where the

relationship between the stress and strain is linear, and tensile

strengths are given by the last peak of the nominal

stress-nominal strain curves.

III. RESULTS AND DISCUSSION

A. Validation of calculation method

We performed the MD simulations on tensile loadings of

pristine graphene at 300 K to verify the propriety of our

calculation method. The results are presented in Table I and

Fig. 7. The average tensile strength is 83 GPa, which is in

agreement with the 121 GPa calculated by Pei et al. through

MD simulations [16] and the experimentally obtained value

of 123.5 GPa [1]. The average Young’s modulus is 836 GPa,

which is within the range of results obtained by the DFT

method [17] (1,050 GPa) and by experiment [18] (500 GPa

and 1 TPa). It is estimated that the lower value obtained in this

work is due to the effect of size on the elastic properties of

graphene [18].

TABLE I

MECHANICAL PROPERTIES OF PRISTINE GRAPHENE

Tensile strength

(GPa)

Armchair 76

Zigzag 91

Average 83

Direction

Young’s modulus

(GPa)

879

794

836

0

10

20

30

40

50

60

70

80

90

100

00.050.10.150.2

Nominal strain

Nominal stress σx (GPa)

Zigzag

Armchair

Fig. 7 Stress-strain curves of pristine graphene under Armchair or Zigzag

tension.

B. Mechanical properties of vacancy-containing

graphene

The mechanical properties of vacancy-containing graphene

obtained at 300 K are listed in tables II and III, together with

the results from previous studies on carbon nanotubes

Proceedings of the International MultiConference of Engineers and Computer Scientists 2012 Vol I,

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[9]–[11]. The nominal stress-nominal strain curves of the

graphene are given in Fig. 8. The results for pristine graphene

are also given for reference. The decrease in the tensile

strength is larger for graphene with double vacancy, followed

by that of the sextuple and then single vacancy. In addition,

the fracture strain for graphene with double vacancy is the

least. The decrease in tensile strength relative to that of

pristine graphene is 29, 28, and 17%, respectively.

Nevertheless, the Young's modulus hardly changes with the

vacancy size. When compared with the results of previous

studies on carbon nanotubes using MD [9], molecular

mechanics (MM) [10], and quantum mechanics (QM) [11]

calculations, the reductions in the tensile strength of graphene

with a single and double vacancy in this work are close to the

results obtained with the MM and QM calculations.

Snapshots of the tensile loadings are shown in Fig. 9. For

pristine graphene, the distribution of stress just before the

fracture is uniform and the level of stress is high. In

comparison, in the vacancy-containing graphene, the

concentration of stress occurs around the vacancy just before

the fracture, which emerges from the circumference of the

vacancy.

TABLE II

TENSILE STRENGTHS OF VACANCY-CONTAINING GRAPHENE AND CNT

Graphene

σB

(MD, GPa) (MD, GPa)

Pristine 91

104

Single vacancy 75 (-17%)

103 (-1%)

Double vacancy 64 (-29%)

101 (-3%)

Sextuple vacancy 65 (-28%)

σB is the tensile strength. Values in parentheses represent the differences

between the pristine and vacancy-containing materials. MD, MM, and QM

refer to Molecular Dynamics, Molecular Mechanics, and Quantum

Mechanics, respectively. The [5,5] CNTs whose tensile direction agrees with

the zigzag tension are used for all of the carbon nanotube results.

TABLE III

THE YOUNG’S MODULI OF VACANCY-CONTAINING GRAPHENE (UNIT: GPA)

Pristine

Single vacancy

Double vacancy 765 (-3.6%)

Sextuple vacancy 767(-3.4%)

Values in parentheses represent the differences between the pristine and

vacancy-containing materials.

CNT(Carbon Nanotube)

σB [9]

σB [10]

(MM, GPa)

105.5

70.4 (-33%) 100 (-26%)

71.3 (-32%) 105 (-22%)

－

－

σB [11]

(QM, GPa)

135

－

794

782 (-1.5%)

0

10

20

30

40

50

60

70

80

90

100

00.050.10.150.2

Nominal strain

Nominal stress σx (GPa)

Pristine

Single vacancy

Double vacancy

Sextuple vacancy

Fig. 8 Stress-strain curves of the graphene containing a cluster-type vacancy

under zigzag tension.

0

20

40

60

80

100

120

140

(GPa)

i

x

σ

σσ

σ

(a-1) Initial structure of

the pristine graphene

(a-3) Fracture occurred

(a-2) Just before fracture

occurred

(b-1) Initial structure of

the graphene with a

single vacancy

(b-3) Fracture occurred

(b-2) Just before fracture

occurred

(c-1) Initial structure of

the graphene with a

double vacancy

(c-3) Fracture occurred

(c-2) Just before fracture

occurred

(d-1) Initial structure of

the graphene with a

sextuple vacancy

(d-3) Fracture occurred

(d-2) Just before fracture

occurred

YY

XX

OO

Fig. 9 Stages of fracture progression in graphene containing cluster-type.

vacancy. (a-1)–(a-3): pristine, (b-1)–(b-3): single vacancy,

(c-1)–(c-3): double vacancy, (d-1)–(d-3): sextuple vacancy.

0.0

0.2

0.4

0.6

0.8

1.0

1.2

02468

Number of atom defects

Relative strength

MD calculation

Griffith's method

Griffith’s criterion

Fig. 10 Relative strengths and sizes of vacancy, namely, the number of atom

defects obtained using MD calculation and Griffith’s criterion.

We compared the calculated results with the Griffith’s

criterion in order to verify the propriety. The theoretically

ideal strength σmax for brittle fracture is expressed as

,

max

d

E

s γ

σ

=

(6)

where E is Young’s modulus, γs is the surface energy, and d is

the interatomic distance. Then, the strength of materials

containing a fracture of length 2C according to the Griffith’s

criterion is expressed as

,

2

C

E

π

s

f

γ

σ =

(7)

The relative strength σrel, that is, the strength of the

materials with a fracture relative to the theoretically ideal

strength is obtained by dividing σf by σmax as

,

2

π

C

d

rel

σ

=

(8)

A plot of the relative strength against the number of atomic

Proceedings of the International MultiConference of Engineers and Computer Scientists 2012 Vol I,

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ISBN: 978-988-19251-1-4

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defects is shown in Fig. 10. The results of MD calculations

agree well with the predicted values using the Griffith’s

criterion.

C. Influence of distributional form of defects

For the graphene with uniformly or randomly distributed

vacancies, the relationship between the tensile strength and

the density of vacancies is shown in Fig. 11. For the random

vacancy distribution, the average values of the two results

calculated using the models with different vacancy

arrangements are plotted. The error bar (Ｉ) represents the

range between two values. The tensile strength decreases with

the increase in vacancy density. The reduction in the tensile

strength is 59% at a density of 4% for the random vacancy

distribution. This is nearly twice that of the reduction in the

tensile strength of hydrogen (H)-functionalized graphene [16].

In comparison, the Young’s modulus slightly decreases with

the increase in the vacancy density (see Fig. 12). The

reduction in the Young’s modulus is 20% at a density of 4%.

This is nearly 4 times that of the reduction in the Young’s

modulus of H-functionalized graphene. It is reasonable to

assume that graphene is more sensitive to vacancies than to

H–coverage, because a vacancy implies the lack of an atom,

whereas H-coverage refers to the conversion of local carbon

bonding from sp2 to sp3 hybridization.

Snapshots of the graphene with uniformly distributed

vacancies during tensile loading are shown in Fig. 13. In

every case, the concentration of the stress occurs around each

vacancy just before the fracture in the same manner as for the

graphene with a single vacancy. Then, fractures occur starting

from a vacancy and progress toward neighboring vacancies.

The progression of the fracture direction is perpendicular to

the tensile axis in all cases. Conversely, snapshots of the

graphene with randomly distributed vacancies during the

tensile loading are shown in Fig. 14. The fracture starts from

the area where the vacancies gather. The progression

direction of the fracture is then random.

0

20

40

60

80

100

012345

Density of vacancies (%)

Tensile strength (GPa)

Uniform

Random

Fig.11 Tensile strength of graphene against vacancy density.

0

200

400

600

800

1000

012345

Density of vacancies (%)

Young's modulus (GPa)

Uniform

Random

Fig.12 Young’s modulus of graphene against vacancy density.

(b-1) Initial structure of

graphene with

2% uniformly

distributed vacancies

(b-3) Fracture occurred(b-2) Just before

fracture occurred

(a-1) Initial structure of

graphene with

1% uniformly

distributed vacancies

0

20

40

60

80

100

120

140

YY

XX

OO

(a-3) Fracture occurred(a-2) Just before

fracture occurred

(c-3) Fracture occurred(c-2) Just before

fracture occurred

(c-1) Initial structure of

graphene with

4% uniformly

distributed vacancies

Fig. 13 Stages of fracture progression in graphene containing uniformly

distributed vacancies. The density of vacancies is 1% ((a-1)–(a-3)),

2% ((b-1)–(b-3)), and 4% ((c-1)–(c-3)).

(GPa)

i

x

σ

σσ

σ

(b-1) Initial structure of

graphene with

2% randomly

distributed vacancies

(b-3) Fracture occurred(b-2) Just before

fracture occurred

(a-1) Initial structure of

graphene with

1% randomly

distributed vacancies

(a-3) Fracture occurred(a-2) Just before

fracture occurred

(c-3) Fracture occurred(c-2) Just before

fracture occurred

(c-1) Initial structure of

graphene with

4% randomly

distributed vacancies

Fig. 14 Stages of fracture progression in graphene containing randomly

distributed vacancies. The density of vacancies is 1% ((a-1)–(a-3)),

2% ((b-1)–(b-3)), and 4% ((c-1)–(c-3)).

0

20

40

60

80

100

120

140

YY

XX

OO

(GPa)

i

x

σ

σσ

σ

D. Mechanical properties of vacancy-containing graphite

The stress and strain curves of graphite with a cluster-type

vacancy at 300 K are shown in Fig. 15. The results for pristine

graphite are also given for reference. In every case, reductions

in stress occur before the fracture. For the single vacancy, the

reduction occurs twice before the fracture, which occurs

during the last stress peak; for the other vacancies, the

reduction occurs only once.

Snapshots of the graphite with a cluster-type vacancy are

shown in Fig. 16. It was found that the reduction in stress

before the fracture was due to a tear in the graphene sheet. For

the single vacancy, the first reduction in stress is due to the

tearing of the vacancy-containing center layer ((a-1) and

(b-1)). Then, the second reduction is due to the tearing of a

neighboring layer ((a-2) and (b-2)). In this case, the atom in

the broken piece of the center layer reacts with the atom at the

edge of the neighboring layer and leads to the tearing of the

neighboring layer by disturbing the zigzag edge surface (see

Proceedings of the International MultiConference of Engineers and Computer Scientists 2012 Vol I,

IMECS 2012, March 14 - 16, 2012, Hong Kong

ISBN: 978-988-19251-1-4

ISSN: 2078-0958 (Print); ISSN: 2078-0966 (Online)

IMECS 2012