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Research of Materials Science December 2013, Volume 2, Issue 4, PP.50-57

Mechanical Properties of Graphene with

Vacancy Defects Yulin Yang

Mathematics and Physics Department, Xiamen University of Technology, Xiamen, Fujian 361024, China

Email: yulinyangyulin@126.com

Abstract

Defects are generally believed to degrade the mechanical robustness and reduce the strength of graphene sheet. In this work we

investigated the mechanical properties of monolayer graphene sheet with randomly distributed vacancy defects. Molecular

dynamics simulations are carried out to elucidate the atomic-level structures and tensile and shear deformations are applied.

Ultimate strengths and fracture strains are calculated and the effect of defect ratio is analyzed. Interestingly, super-ductility is

observed in the high defect ratio situation. The obtained results as demonstrated here provide new insights in understanding the

mechanical performance of graphene based nano-materials where defects are indispensible.

Keywords: Graphene; Mechanical Properties; Tensile Test; Molecular Dynamics Simulation

1 INTRODUCTION

The amazing mechanical behavior and properties of graphene-based nanomaterials has attracted significant research

interests in recent years, due to their promising prospects in versatile branches such as micromechanics,

microelectronics, and thermal applications [1-4]

. Graphene is known to have ultra-high stiffness and strength, yet a

wide scatter have been observed in the mechanical properties. Pristine graphene sheet was reported to have high

Young's modulus rivaling that of graphite (~1.0 TPa), and its superior strength (90~100 GPa for tensile load and

50~60 GPa for shear load) arises from a combination of high stiffness and unusual flexibility and resistance to

fracture [5-7]

. However, the second law of thermodynamics dictates the presence of a certain amount of defects and

disorders in crystalline materials [8]

. Also, the imperfections of material manufacturing process, device or composite

production [9,10]

, chemical treatment[11]

, particle irradiation [12, 13]

and mechanical loading[14]

can all induce defects.

In most of the application situations, these unavoidable defects can affect their material performance, especially the

mechanical properties, with the impact more or less significant for different defect types, defect concentrations and

charilities. For single-wall carbon nanotube (SWCNT), single vacancy defect was reported to lead to about 35%

reduction in the fracture strain [15]

. Furthermore, Sammalkorpi et al. demonstrated that the strength reduction caused

by single vacancy depends on whether bond reconstruction occurs prior to loading. They found that vacancies in

SWCNT can reduce the tensile strength and fracture strain by 40% and 50%, respectively, whereas influence the

Young's modulus by only a negligible percentage [16]

. The presence of multiple defects makes nanotubes even

weaker, which reasonably explains the difference between the comparatively low SWCNT fracture strength (13-52

GPa[17]

) observed in experiments and the high theoretical predictions (above 100 GPa)[3,18]

. Using molecular

simulations with modified Brenner's potential, Yang et al. predicted the normalized strength of SWCNT reduced

from 185 GPa to less than 50 GPa for defect concentration varying from zero to 7.5% [19]

. The Stone-Wales defect

on the armchair SWCNT serves to reduce the failure stress and strain by 20% to 40%, whereas the SW defect on the

zigzag SWCNT has negligible effect on the mechanical performance [20]

. The role of thermodynamically

unavoidable atomistic defects in the design of carbon nanotube based space elevator megacable was investigated and

the strength was expected to be reduced by a factor of at least ~70% [21]

. For the graphene sheet, Dettori et al. are the

first to examined the effect of point defect on the mechanical properties of graphene and found that the defect-

induced stress field is the basin of mutual interaction between two nearby defects. The obtained Young's modulus

and Poisson ratio showed a decreasing trend with respect to defect density for vacancy defects [22]

. Moreover, focus

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has been placed on line defects such as grain boundaries [23]

, which are found to reduce the strength, but the effect is

more pronounced for some boundary angles [24]

and less pronounced for others [25]

. Further continuum mechanics

theoretical improvements revealed that the detailed arrangement of defects plays the major role in increasing or

decreasing the strength with tilt angle [26]

.

While structural defects may deteriorate the performance of graphene-based devices, the deviation from perfection

can also be utilized and be careful engineered to achieve new functionalities. Therefore, a good understanding of the

material performance for graphene with defects is useful for further improvement of graphene-based nanotechnology.

In this work we focus on vacancy defects and perform molecular dynamics (MD) simulations to investigate the

mechanical behaviour and properties of graphene sheets with different incipient defect ratios.

2 MODELS AND METHODS

2.1 Models

The size of the monolayer graphene sheet we considered is 100 Å×100 Å. The incipient vacancy defects are

randomly dispersed on the graphene basal planes according to a prescribed defect ratio ϕ, which is defined as the

ratio of missing atoms versus total atoms on the entire pristine sheet. Graphene with vacancy defects at incipient

defect ratio x% will be abbreviated as VD-x%.

2.2 Methods

MD simulations are performed using the massively parallelized modelling code LAMMPS software package[27]

, and

the atomic interactions are described by the AIREBO potential[28]

, where the cutoff parameter of the REBO part of

the potential was modified as 2.0 to avoid nonphysical high force [29,30]

. Periodic boundary conditions are applied in

the in-plane directions. The Velocity-Verlet integration time step is set as 0.1 fs. Structural optimizations are

performed using the Polak-Ribière version of the conjugated gradient algorithm [31]

. The MD simulations are

performed with a background temperature of 300 K (Nose-Hoover thermo bath coupling [32]

). After the equilibrium

states are achieved, uniaxial tensile tests are performed under NPT ensemble to study the mechanical properties of

the defective graphene. The engineered strain rate is 0.001 ps-1

and the strain increment is applied every 1000 time

steps. Both armchair and zigzag orientations are investigated.

The mechanical tests are implemented to derive the stress-strain relations and the associated parameters, namely,

maximum strength σc and fracture strain εF. Young’s modulus is not analyzed because we focus on the mechanical

properties of the defective graphene sheets under heavy loads. The macroscopic stress is obtained by averaging the

atomic virial stress over all the atoms on the sheet [33]

. Noise is reduced by averaging the results over the latter half of

the relaxation period. The volume of graphene sheet is computed by multiplying the in-plane area of the simulation

model with a thickness of 3.35 Å, which is the generally accepted van der Waals interlayer interaction distance. Our

simulation methods are validated by calculating the maximum strengths and fracture strains of a 100 Å×100 Å

pristine graphene sheet. The obtained parameters agree well with the experimental measurements as well as other

theoretical reports as listed in Table 1.

TABLE 1 MECHANICAL PROPERTIES OF PRISTINE GRAPHENE SHEET. THE RESULTS FROM THE PREVIOUS EXPERIMENTAL

MEASUREMENTS AND THEORETICAL REPORTS ARE ALSO LISTED FOR COMPARISON

c (GPa) F Remarks

Arm Zig Arm Zig

Our work 91.4 107.5 0.136 0.203 MD

[5] 130 10 Nano-indenting

[34] 90 107 0.13 0.20 MD

[17] ~60 SWCNT(experiments)

3 RESULTS AND DISCUSSIONS

In this section, we will investigate the mechanical properties of defective graphene sheets under tensile and shear

deformations. Both armchair and zigzag orientations are discussed. Focus will be placed on the mechanical response

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under heavy loads. Furthermore, to depress the possible fluctuation created by randomness in the arrangements of

the defects, we create 20 independent samples for each defect ratio ϕ and perform the associated MD simulations

accordingly.

3.1 Structural deformations at the equilibrium state

0 2 4 6 8 100.00

0.01

0.02

0.03

MS

D (

An

gs

.)

Defect Ratio (%)

MSD

(a) @0K

0 2 4 6 8 10 120.3

0.6

0.9

1.2

1.5

1.8

h (

An

gs

.)

Defect Ratio (%)

(b) @300K

h

h

FIG. 1 (A) MEAN SQUARE DISPLACEMENT AS A FUNCTION OF DEFECT RATIO FOR GRAPHENE SHEETS WITH VACANCY

DEFECTS AT ZERO TEMPERATURE AFTER EQUILIBRATION. THE INSET SHOW THE SNAPSHOT OF GRAPHENE WITH DEFECTS,

WHERE THE DEFECTS ARE HIGHLIGHTED IN BLUE. (B) AVERAGED OUT-OF-PLANE FLUCTUATION (RIPPLE HEIGHT h , AS

SHOWN IN THE INSET) OF DEFECTIVE GRAPHENE AT 300 K AFTER EQUILIBRATION, AS A FUNCTION OF DEFECT RATIO.

We first analyze the structural deformations of graphene sheets with defects. The defective graphene sheets are

created from pristine graphene membranes by removing atoms. After structural relaxation at zero temperature, the

atoms on the sheet tend to re-arrange their local positions to balance the spatial stress. From the mean square

displacement (MSD) of the defective graphene before and after relaxation as presented in Fig. 1(a), one can see that

MSD increases gradually with the increasing ϕ, which can be understood from the more disturbed atomic positions

in the larger ϕ case. However, the MSD decreases when ϕ is greater than 8%, which indicates that in the high ϕ case

the intensive local deformation can lead to unusual atomic interactions and therefore interesting mechanical

properties. To further analyse the intrinsic ripple structures, the graphene sheets are equilibrated at 300 K, and the

averaged out-of-plane fluctuations h are calculated. The amplitude of h obeys the relation h L with 0.6 0.8

for graphene. Assuming 0.6 , our estimation of the ratio of /h L is 0.032 for pristine graphene sheet, in good

agreement with 0.035 reported in ref. [35]. Topological defects in graphene are found to be energetically favorable to

deform out-of-plane and increase the ripple height [36]

. From Fig. 1(b) one can see that h increase dramatically with

increasing defect ratio. Because dense vacancy defects can lead to reduced inter-atomic confinement among adjacent

carbon atoms, thus the higher ripple amplitude in the high ϕ range can be understood.

3.2 General Mechanical Responses

0.00 0.03 0.06 0.09 0.120

2

4

6

8

10

E

TO

T (

X10

9 J

/m3) Graphene

VD-0.05%

VD-8.5%

Tensile Strain

0.00 0.05 0.10 0.150

30

60

90

Str

es

s (

GP

a)

Tensile Strain

(a) Armchair

0.00 0.04 0.08 0.12 0.16 0.200

4

8

12

16

Graphene

VD-0.05%

VD-8.5%

E

TO

T (

X10

9 J

/m3)

Tensile Strain

0.00 0.05 0.10 0.15 0.200

30

60

90(b) Zigzag

Str

ess

(G

Pa

)

Tensile Strain

FIG.2 TOTAL ENERGY INCREMENTS PER UNIT VOLUME OF GRAPHENE SHEETS UNDER TENSILE DEFORMATIONS ALONG THE

ARMCHAIR (A) AND ZIGZAG (B) DIRECTIONS. RESULTS OF PRISTINE GRAPHENE, GRAPHENE SHEET WITH LOW AND HIGH

DEFECT RATIO (ϕ =0.05% AND 8.5%) ARE PRESENTED. THE UPPER-LEFT INSET FIGURES ARE THE CORRESPONDING STRESS-

STRAIN RELATIONS. THE MIDDLE-LEFT INSET SNAPSHOTS SHOW THE LOADING DIRECTIONS.

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We next characterize the general mechanical response for the defective graphene under tensile deformations. To

have a close inspect of the load-deformation rules, the total energy method is implemented, since the increment of

the total energy should be equal to the external work. Fig.2 illustrates the total energy increment per unit volume for

pristine graphene, graphene with vacancy defects in low ϕ (single defect) and high ϕ deformed along armchair/

zigzag directions, and the insets show the stress-strain relations. With single defect presented, both total energy

increment rules and stress-strain relations well reproduce those of pristine graphene but characterize much earlier

fracture points, indicating the defect-activated weakening of the system, which will decrease the stiffness and

strength of the nanomaterial. This kind of brittle fracture has been observed in graphene sheet with single defect [37]

or nanocrystalline grains [38]

. While similar, the responses are much complicated for high defect ratio situations

(ϕ=8.5%). Starting with a much slower increasing rate in stress versus strain, small reductions are occasionally

observed, leading to a serrated curve as shown in the inset figures of Fig. 3. Failure along the weakest path is not

immediately catastrophic. The stress-strain relations exhibit multiple stress peaks and an overall multiple fracturing

behaviour. The material becomes weaker but more ductile, with reduced ultimate strengths and enlarged fracture

strains. The small drops in stress-strain relations are believed to originate from geometric rearrangement on the sheet

(to dissipate the accumulated loads). This kind of fracturing had also been observed in extended graphynes, where

secondary fracture occurs due to the mobility of the acetylene linkages [39]

. The effect of multiple defects on strength

depends on the residual dangling bonds induced by vacancy defects, which weakens the bond structure significantly

and enhances the mobility of carbon atoms strongly. The multiple stress peaks pose difficulty in determining the

fracture point. However, the total energy of the system increases with increasing strain before the final sharp drop.

Therefore, we define the fracture point as the highest energy point. This phenomenon also indicates that for defective

graphene under large strain, although the strength remains almost unchanged, the potential energy of the system can

still be increased.

3.3 Mechanical properties under tensile deformations

0 2 4 6 8 10 120.06

0.09

0.12

0.15

0.18

0.21

Fra

ctu

re S

tra

in

Defect Ratio (%)

Armchair(c)

0 2 4 6 8 10 12

30

45

60

75

90

105

(b)

Ma

xim

um

Str

en

gth

(G

Pa

)

Defect Ratio (%)

Zigzag

0 2 4 6 8 10 12

30

45

60

75

90

105

(a)

Ma

xim

um

Str

en

gth

(G

Pa

)

Armchair

Defect Ratio (%)

0 2 4 6 8 10 120.06

0.09

0.12

0.15

0.18

0.21

Fra

ctu

re S

tra

in

Defect Ratio (%)

Zigzag(d)

FIG.3 VARIATIONS OF THE MAXIMUM STRENGTH σ (A,B) AND FRACTURE STRAIN εF (C,D) WITH RESPECT TO ϕ FOR DEFECTIVE

GRAPHENE SHEETS UNDER TENSILE TESTS ALONG THE ARMCHAIR (A,C) AND ZIGZAG (B,D) DIRECTIONS.

We now turn to analyze the mechanical properties at fracture point for defective graphene. It should be noted that the

ultimate strength is the maximum stress in the stress-strain curves, while the fracture strain is determined from the

spontaneous large drop of the total energy increment curves. Fig. 3 displays the ultimate strength and fracture strain

for defective graphene with respect to defect ratio ϕ under tensile tests, with both armchair and zigzag charilities

been considered. Without defect the ultimate tensile strength is 91.4 GPa and 107.5 GPa for armchair and zigzag

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graphene, respectively. With single vacancy defect presented, the maximum strength is significantly degraded (σarm

is 77.8 0.7GPa and σzig is 86.0 1.3 GPa). The difference between armchair and zigzag charility is narrowed from

16.1 GPa to 8.2 GPa. Similar narrowing of ultimate strength difference between armchair and zigzag charilities has

also been observed in CNTs with single SW defect [40]

. With multiple defects appear the maximum strength

decreases gradually and saturates at higher defect ratiorange (ϕ >7%). The strength difference between armchair and

zigzag sheet is further narrowed in high ϕ circumstances, which is less than 2.0 GPa when ϕ is greater than 6%

(Fig.4). The strength for the armchair graphene decreases slower than that of zigzag graphene, indicating the knock-

down effect of defects on maximum strength is more pronounced for zigzag tests and less pronounced for armchair

tests. Similar descending-saturating trend had been observed in exploring the failure strength of SWCNTs with

respect to defect ratio under tensile tests [41]

, wherein the normalized strength was reduced from 180 GPa to 50 GPa

with a vacancy concentration of ~ 7.6%. Also, this phenomenon reasonably explains the difference between the

comparatively low SWCNT fracture strength (13-52 GPa [17]

) observed in experiments and the high theoretical

predictions (above 100 GPa).

Besides ultimate strength, the fracture strain is another important parameter characterizing the mechanical properties

under heavy load. Fracture strain of defective graphene are found to exhibit an unusual degrading-saturating-

improving trend with increasing ϕ, with the same rule holds for both armchair and zigzag charilities (Fig.3 (c,d)).

Similar to the ultimate strength, with single VC defect presents the fracture of the sheet is initiated much earlier. εFarm

is reduced from 0.136 to 0.109 0.001, and εFzig is reduced from 0.203 to 0.124 0.003. The difference between

εFarm and εFzig is significantly narrowed. The fracture strain has been reduced by about 21%~39%, the same level as

that reported for single-wall carbon nanotube with single vacancy defect(~35% reduction in the fracture strain[15]

).

Within all the investigated situations, εF is found to decreases gradually for low defect ratio and enlarges after further

increasing of defect ratio (Fig. 3 (c,d)). The unusual enlargement of εF in high ϕ range shows that although the

strength of the material is reduced, the ductility is greatly improved within this ϕ range. This kind of improvement

shows a super-ductile behaviour in the defective graphene with dense vacancy defects. Overall, for a given number

of defect ratios, the armchair configuration has less strength and lower fracture strain, but more certainty compared

with the zigzag sheets. Similar trend has also been reported previously for carbon nanotubes with randomly

occurring Stone-Wales defects [42]

.

0.00 2 4 6 8 10

30

45

60

75

90

105

highly defective graphene,

tiny difference

graphene with single vacancy defect,

small difference between arm

c and

zig

c

Ma

xim

um

Str

en

gth

(G

Pa

)

Defect Ratio (%)

Armchair

Zigzag

pristine graphene,

large difference between arm

c0 and

zig

c0

FIG.4 MAXIMUM STRENGTH DIFFERENCE BETWEEN ARMCHAIR AND ZIGZAG CHIRALITIES FOR MONOLAYER GRAPHENE SHEET

WITH VACANCY DEFECTS, AS A FUNCTION OF DEFECT RATIO.

To determine the statistical distribution of the uncertain maximum strength and fracture strain, we fit the simulation

data by Weibull distributions. The Weibull statistics are known to well characterize the material behaviour when

failure is governed by the weakest link (as in our situation) and had been well demonstrated to estimate the strength

of CNTs both theoretically [42]

and experimentally [43]

. The Weibull strength distribution can be expressed as [44]

0

( ) 1 exp

m

F

(1)

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Where m is the Weibull modulus and σ0 is the scaling parameter. For sufficiently large m, the relative width of the

strength distribution decreases and σ0 approximates the ensemble average strength. From a series of pre-measured

strength σi, the parameter m can be determined by maximum likelihood method from the following equation [44]

11

1

ln 1 1ln

mNi i Ni

iimNii

m N

(2)

By iterative searching from Eq. (2) we obtained m for the defective graphene under tensile deformations, as shown in

Fig. 5. We can see that the Weibull modulus presents a decreasing trend with the increase of defect ratio and the

lowest m lies in the VD-8.5% defective graphene deformed along zigzag orientation, indicating the more uncertainty

for the obtained fracture strength within this circumstance. Generally speaking, the obtained Weibull modulus

confirms that the obtained maximum strength and fracture strain lie within the confidence interval. Thus the

conclusions as drew above are reliable.

0 2 4 6 8 10 124

6

8

10

12

14

Weib

ull

mo

du

lus

Defect Ratio (%)

Armchair

Zigzag

FIG. 5 WEIBULL MODULUS (SHAPE PARAMETER) M OF DEFECTIVE GRAPHENE SHEETS, AS A FUNCTION OF DEFECT RATIO.

4 CONCLUSIONS

In summary, through molecular dynamics simulations we have demonstrated the effect of multiple vacancy defects

on the mechanical properties of monolayer graphene sheets. The maximum strength is observed to degrade with

increasing defect ratio and converges to a finite value when the ratio is high. However, the fracture strain is observed

to decrease in the low defect ratio range and increases in the high ratio range. For randomly and uniformly

distributed multiple defects, the fracture initiated at quite random locations and the crack grew irregularly, super

ductility is observed in the high defect ratio range. For a given number of defect coverage, the armchair

configuration has lower strength and smaller fracture strain, but more certainty compared with the zigzag ones. The

difference between armchair and zigzag chiralities is significantly narrowed with the presence of single defect and

nearly disappeared when the defect ratio is high. Our study as demonstrated here provides valuable insights in

understanding the mechanical properties of graphene based nanomaterials where defects are unavoidable.

ACKNOWLEDGMENT

This work was financially supported by Fujian Education Bureau (No. GA11020).

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AUTHOR

Yulin Yang, male, was born in 1980. He obtained the Master degree from Xiamen University in the field of theoretical physics.

Currently his research interests including computational physics and material science. Email: yulinyangyulin@126.com