An atomistic methodology of energy release rate for graphene ......an atomistic simulation methodology to unveil the fracture mechanism of graphene at nanoscale. In our model, the

An atomistic methodology of energy release rate for graphene at nanoscaleZhen Zhang, Xianqiao Wang, and James D. Lee

Citation: Journal of Applied Physics 115, 114314 (2014); doi: 10.1063/1.4869207 View online: http://dx.doi.org/10.1063/1.4869207 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/115/11?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Communication: Different behavior of Young's modulus and fracture strength of CeO2: Density functional theorycalculations J. Chem. Phys. 140, 121102 (2014); 10.1063/1.4869515 Temperature and strain-rate dependent fracture strength of graphene J. Appl. Phys. 108, 064321 (2010); 10.1063/1.3488620 Effect of the electrical boundary condition at the crack face on the mode I energy release rate in piezoelectricceramics Appl. Phys. Lett. 94, 081902 (2009); 10.1063/1.3088855 Toughening and reinforcing alumina matrix composite with single-wall carbon nanotubes Appl. Phys. Lett. 89, 121910 (2006); 10.1063/1.2336623 Electrical fracture toughness for conductive cracks driven by electric fields in piezoelectric materials Appl. Phys. Lett. 76, 126 (2000); 10.1063/1.125678

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An atomistic methodology of energy release rate for graphene at nanoscale

Zhen Zhang,1 Xianqiao Wang,2 and James D. Lee1,a)

1Department of Mechanical and Aerospace Engineering, the George Washington University, Washington,DC 20052, USA2College of Engineering, University of Georgia, Athens, Georgia 30602, USA

(Received 3 October 2013; accepted 10 March 2014; published online 21 March 2014)

Graphene is a single layer of carbon atoms packed into a honeycomb architecture, serving as a

fundamental building block for electric devices. Understanding the fracture mechanism of

graphene under various conditions is crucial for tailoring the electrical and mechanical properties

of graphene-based devices at atomic scale. Although most of the fracture mechanics concepts, such

as stress intensity factors, are not applicable in molecular dynamics simulation, energy release rate

still remains to be a feasible and crucial physical quantity to characterize the fracture mechanical

property of materials at nanoscale. This work introduces an atomistic simulation methodology,

based on the energy release rate, as a tool to unveil the fracture mechanism of graphene at

nanoscale. This methodology can be easily extended to any atomistic material system. We have

investigated both opening mode and mixed mode at different temperatures. Simulation results

show that the critical energy release rate of graphene is independent of initial crack length at low

temperature. Graphene with inclined pre-crack possesses higher fracture strength and fracture

deformation but smaller critical energy release rate compared with the graphene with vertical pre-

crack. Owing to its anisotropy, graphene with armchair chirality always has greater critical energy

release rate than graphene with zigzag chirality. The increase of temperature leads to the reduction

of fracture strength, fracture deformation, and the critical energy release rate of graphene. Also,

higher temperature brings higher randomness of energy release rate of graphene under a variety of

predefined crack lengths. The energy release rate is independent of the strain rate as long as the

strain rate is small enough. VC 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4869207]

I. INTRODUCTION

Graphene, a two-dimensional material system, is the

building block of fullerenes, carbon nanotubes, and graphite.

Since Geim and his colleagues1 discovered a simple but

novel method to isolate single atomic layers of graphene

from graphite in 2004, graphene has attracted considerable

interest and became the focus of extensive research due to its

favorable mechanical, electronic, thermal, and optical

properties.1–9 Its extraordinary properties make graphene

suitable for a great number of promising applications such as

nanoelectromechanical systems (NEMSs),10 electronic cir-

cuitry,11 and biodevice.12 Even though atomically perfect

nanoscale materials can be mechanically tested to deforma-

tions well beyond the linear regime, monolayer graphene

membrane is intrinsically brittle at room temperature as cata-

strophic fracture is observed during the indentation.13 On

account of the brittleness, there is a need to thoroughly

understand fracture mechanics of graphene at atomic scale

especially when it is used as nanostructured materials.

To give insights into the material’s behavior at the fun-

damental level, researchers have investigated fracture

mechanics of graphene at atomic scale through molecular

dynamics (MD) simulations. Jack et al.14 found specific pat-

terns of vacancies could control fracture surface geometries

of graphene under uniaxial tensile load. Zhao and Aluru15

tested the variation in fracture strength of monolayer pristine

graphene with temperature by performing the uniaxial tensile

test. Simulation results showed that the fracture strength of

graphene decreased with the increase of temperature. Stress

intensity factor and J-integral are widely used to describe the

fracture mechanical behavior at continuum level. However,

at atomic scale, there is no crack tip singularity. Therefore,

the stress intensity factor which is closely associated with

singularity is not applicable in MD simulation. Jin and

Yuan16 developed an atomistic approach to evaluate the

path-independent J-integral of discrete atomic system. The

J-integral developed by Rice and Rosengren17 is considered

as an essential parameter to evaluate fracture mechanical

behavior of materials. It assumes the crack propagates self-

similarly. However, crack propagation does not always fol-

low this rule at atomic scale. Sen et al.18 found that when

tearing graphene sheets from adhesive substrates, the two

initially parallel crack notches propagate towards each other

and finally form a non-symmetric horned edge. Brommer

and Buehler19 showed that the crack propagation of graph-

diyne deviated from the initial direction even at simplest

opening mode loading. Not only that, but all other graphynes

have been found similar phenomena.20 These findings are in

disagreement with the macroscopic continuum theory.

In fracture mechanics, Griffith introduced a fundamental

physical quantity, energy release rate, to evaluate the mate-

rial property in 1921.21 Energy release rate is defined as the

energy dissipated during fracture per unit of newly created

fracture surface area. This macroscopic fracture parameter is

well formulated from the continuum mechanics approach.a)Electronic mail: [email protected]

0021-8979/2014/115(11)/114314/8/$30.00 VC 2014 AIP Publishing LLC115, 114314-1

JOURNAL OF APPLIED PHYSICS 115, 114314 (2014)

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Different from J-integral, energy release rate does not

assume crack grows self-similarly, and it is a more general

and more fundamental physical quantity to characterize the

fracture mechanical property of materials. Jin and Yuan22

developed two different methods, the global energy method

and the local force method, to calculate the energy release

rates in atomic systems. The global energy method was

based on the change of total potential energy of two gra-

phene sheets, of which the only difference is central crack

length, while the local force method is based on the virtual

work that is required to prohibit the crack extension. Both

methods are based on a static model which is not suitable for

investigating the temperature effects. In the study of flaw

insensitive fracture in nanocrystalline graphene, Zhang

et al.23 first estimated the fracture surface energy of nano-

crystalline graphene through the facture strength of a center-

cracked strip according to the classical Griffith model. But

there remains lack of the formulation of energy release rate

through atomistic approach at nanoscale, which should be

different from the one we are familiar with in continuum

mechanics. Therefore, it is necessary and worthwhile to

establish the new formulation of energy release rate when

we investigate the fracture behavior at the atomic scale.

In this paper, energy release rate of a single layer gra-

phene system with slit crack will be studied by performing

MD simulations. Following Griffith’s theory, which focuses

on the global energy balance during crack growth, we develop

an atomistic simulation methodology to unveil the fracture

mechanism of graphene at nanoscale. In our model, the

energy release rate is related to the total work done externally.

Since the Griffith criterion of fracture is based on the energy

balance of two metastable states along the fracture path,

which differs by lattice spacing in crack length at atomistic

model,24 the strain rate has to be small enough to simulate a

pseudo static process. Energy release rates of graphene at dif-

ferent small strain rates are investigated. We study both open-

ing mode (external loading perpendicular to the slit crack)

and mixed mode at nearly absolute zero temperature, room

temperature (300 K), and high temperature (1000 K).

II. SIMULATION MODEL AND COMPUTATIONAL

In our MD simulations, a square-shaped single-layer gra-

phene sheet is placed in the XY plane and the Z axis is defined

normal to the graphene plane and a slit crack along the Y axis

is predefined on graphene sheet (cf. Fig. 1(a)). This displace-

ment boundary condition applies to the carbon atoms within

the bounds of 0.21 nm from the two edges parallel to the X

axis. The graphene sheet is set at different temperatures ini-

tially using the Nose-Hover thermostat.25 Then we perform

uniaxial tension by applying a constant strain rate. The time

step is set as 0:483 fs. The initial crack length is defined as dis-

tance between the edge and the last void atom (cf. Fig. 1(b)).

In mixed mode cases, we examine a single-layer graphene

sheet with an inclined slit crack (cf. Fig. 1(c)). The crack angle

h is defined as the angle between the crack and the X-axis.

Because of the discontinuity at atomic scale and the hexagonal

structure of graphene, we only consider two cases: h ¼ 60�

and h ¼ 30�. The boundary conditions and the time step in

mix mode case are the same as in opening mode case.

A variety of force fields have been developed to describe

the interaction of graphene system. The reactive empirical

bond order (REBO) potential developed by Tersoff26

accounts for the many-body interatomic forces and well

describes the bonding in a wide range of carbon nanostruc-

tures. The adaptive intermolecular reactive empirical bond

order (AIREBO) potential,27 which furthermore including the

long-range interaction and torsional potential, has been

FIG. 1. (a) Computational model of

the armchair graphene with dimension

of 11:11 nm � 10:58 nm. Slit crack

locates on the edge, and the initial

crack length is a ¼ 0:85 nm. The exter-

nal loading is applied along the X axis

on atoms in green colored region; (b)

illustration for definition of initial

crack length; (c) computational model

of the armchair graphene with an

inclined crack of angle of 60�.

114314-2 Zhang, Wang, and Lee J. Appl. Phys. 115, 114314 (2014)

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performed to describe graphene fracture behavior well.28 The

first-principles-based reactive force field ReaxFF29 is based

on highly accurate and benchmarking density functional stud-

ies and is widely used to characterize fracture properties of

graphene.18 The Tersoff potential is employed here since it is

fairly enough to show the general principles of energy release

rate in our methodology with high computation efficiency.

We calculated the interatomic force Fi acting on the ith atom

based on the total potential energy E in our previous work.30

When stretching out the sample slowly with a constant

rate, we consider that the graphene sheet reaches equilibrium

state at each time step. From the Newton’s third law, the

external force acting on the ith atom, f i, is equal to the oppo-

site of the interatomic force Fi, i.e., f i ¼ �Fi. To simplify

the plot, we define the total external force as

F ¼

XNB

i¼1

f i � vi

v; (1)

where vi is the stretching velocity of the ith atom in the

boundary region and NB is the number of atoms in that

region. In our simulation, the strain rate is fixed, namely,

jvij ¼ v. Besides, we define fracture strength Fmax as the

maximal external loading force before the crack grows; frac-

ture deformation Umax is the maximal displacement of atoms

at boundary before the crack grows. Both of them can be

found at the critical moment when the crack is about to prop-

agate from its initial state. In order to obtain the critical

energy release rate, one needs to accumulate the total exter-

nal work before the crack starts to grow. Therefore, the total

external work W is given by

W ¼ð

t

XNB

i¼1

f iðtÞ � vidt ¼ð

t

FðtÞvdt: (2)

In this study, an interlayer separation distance of graphite,

which is 0.34 nm, is defined as the effective thickness. Then

the critical energy release rate is given by

FIG. 2. Critical energy release rate vs. initial crack length for zigzag graphene

sheets at different strain rates with temperature 0 K under opening mode.

FIG. 3. Snapshots of crack propagation of armchair graphene under opening mode at nearly 0 K. The initial crack length is a ¼ 2:13 nm. (a) Initial moment;

(b) before the crack begins to grow; (c) after the first chemical bond breaks near crack tip; (d) and (e) crack keeps on growing; (f) at the end, the specimen

breaks into two pieces.

114314-3 Zhang, Wang, and Lee J. Appl. Phys. 115, 114314 (2014)

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GC ¼ �dW

2t � da¼Wa �WaþDa

2t � Da; (3)

where Wa and WaþDa are the critical values of total external

work with initial crack length a and aþDa, respectively.

III. RESULTS AND DISCUSSIONS

In our methodology, the pseudo static process requires

small strain rate. We test four different strain rates on the

same graphene sample; the results are depicted in Fig. 2.

When the strain rate is 8:26� 10�3 ps�1, the critical energy

release rate GC is slightly overestimated. When it is smaller

than a critical value, the strain rate barely has no effect on

GC. Therefore, for the purpose of illustration, we use the

same strain rate 4:13� 10�3 ps�1 for all the following

simulations.

First, we consider opening mode case at nearly 0 K. We

investigate three armchair graphene sheets with different

sizes: sample A with dimension 5:82 nm � 5:29 nm, sample

B with dimension 11:11 nm � 10:58 nm, and sample C with

dimension 16:40 nm � 15:88 nm. The initial crack lengths

range from a ¼ 0:58 nm to a ¼ 2:27 nm, from a ¼ 0:85 nm

to a ¼ 4:69 nm, and from a ¼ 0:85 nm to a ¼ 6:10 nm for

samples A, B, and C, respectively. The simulation results

show that the crack propagates straightly along the Y axis

through the graphene sheet. Even though the crack surface

can be a little bit rough, the propagation direction trends to

be perpendicular to the applying direction of the external

force (cf. Fig. 3). The crack propagation patterns of both

armchair and zigzag graphene are similar, regardless of gra-

phene dimension or initial crack length. The total external

force keeps on increasing as graphene sheet being stretched,

and the curve collapses at the critical moment when the

crack starts to grow, as shown in Figure 4. To hold the same

deformation, a larger total external force is applied on gra-

phene sheet with a smaller initial crack length. As depicted

in Figure 4, graphene with a larger initial crack length has a

tendency to break earlier than the one with a smaller initial

crack length.

Follow the methodology as described in Sec. II, the criti-

cal energy release rate GC of armchair graphene can be

obtained as shown in Figure 5(a). For each sample, GC

barely varies with initial crack length. Besides, the sample

dimension nearly has no influence on the critical energy

release rate. The average values and standard deviations of

sample A, sample B, and sample C are listed in Table I.

Overall, GC of armchair graphene at nanoscale is 33:02

6 0:79 J=m2, and it is independent of the sample dimension

and initial crack length. From the results in Figure 5(b) and

Table II, for sample B, the critical energy release rate of arm-

chair graphene is 33:18 J=m2, while that of zigzag graphene

is 29:45 J=m2. Therefore, when the orientation changes, the

critical energy release rate varies.

FIG. 4. Force-displacement relation for sample B under opening mode with

various initial crack lengths a at temperature 0 K.

FIG. 5. (a) Critical energy release rate vs. initial crack length for armchair grapheme with different dimensions; (b) critical energy release rate vs. initial crack

length for sample B at different orientations.

TABLE I. Comparison of the critical energy release rate for armchair gra-

phene sheet with different dimensions.

Sample A Sample B Sample C

Average value (J=m2) 32.56 33.18 33.19

Standard deviation 1.11 0.47 0.82

114314-4 Zhang, Wang, and Lee J. Appl. Phys. 115, 114314 (2014)

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For mixed mode case at 0 K, all simulations are

performed on graphene with dimension of 11:11 nm

� 10:58 nm. The processes of crack propagation are depicted

in Figure 6. No matter how large the initial angle is, the

crack eventually propagates through graphene sheet perpen-

dicular to the direction of external loading, same as in open-

ing mode case. The opening mode case can be considered as

the case that the crack angle is 90�. The simulation results

tell us that when the initial crack length is the same, as the

crack angle decreases, the fracture deformation and the frac-

ture strength increase. Therefore, decreasing the angle of the

initial crack can improve the ability of resisting large defor-

mation and large external loading. The critical energy release

rate still remains constant with changing initial crack length

TABLE II. Comparison of the critical energy release rate for graphene sheet

(sample B) with different directions.

Armchair direction Zigzag direction

Average value (J=m2) 33.18 29.45

Standard deviation 0.47 0.07

FIG. 6. Snapshots of crack propagation of armchair graphene under mixed mode at nearly 0 K. The initial crack length is a ¼ 2:13 nm and the crack angle is

60�. (a) Initial moment; (b) before the crack begins to grow; (c) after the first chemical bond breaks near crack tip; (d) and (e) crack keeps growing; (f) at the

end, the specimen breaks into two pieces.

FIG. 7. Critical energy release rate vs. initial crack length for grapheme sheets with different orientations under mixed mode. (a) 60� crack angle; (b) 30� crack

angle.

114314-5 Zhang, Wang, and Lee J. Appl. Phys. 115, 114314 (2014)

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FIG. 8. Close-up views of crack tip of

9 different cases at 0 K. Pink color

highlights the atoms and bonds which

deform easily due to the voids and the

horizontal external loading. (a)

Opening mode case for armchair gra-

phene type I; (b) opening mode case

for armchair graphene type II; (c)

opening mode case for zigzag gra-

phene; (d) mixed mode case (30�

crack) for armchair graphene type I;

(e) mixed mode case (30� crack) for

armchair graphene type II; (f) mixed

mode case (60� crack) for armchair

graphene; (g) mixed mode case (30�

crack) for zigzag graphene; (h) mixed

mode case (60� crack) for zigzag gra-

phene type I; (i) mixed mode case (60�

crack) for zigzag graphene type II.

FIG. 9. Snapshots of crack propagation of armchair graphene under opening mode at room temperature. The initial crack length is a ¼ 2:13 nm. (a) Initial

moment; (b) before the crack begins to grow; (c) after the first chemical bond breaks near crack tip; (d) and (e) crack keeps growing; (f) at the end, the speci-

men breaks into two pieces.

114314-6 Zhang, Wang, and Lee J. Appl. Phys. 115, 114314 (2014)

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under mixed mode deformation; however, it decreases as the

initial crack angle decreases, opposite to the fracture defor-

mation and fracture strength. As stated before, GC of arm-

chair graphene under opening mode is 33:18 J=m2 (Figure 5).

From Figs. 7(a) and 7(b), it can be obtained that GC of arm-

chair graphene are 27:40 J=m2 and 18:13 J=m2 when crack

angles are 60� and 30�, respectively. These results indicate

that under the same loading, crack grows earlier under open-

ing mode case than mixed mode case; once the crack starts

propagation, graphene under opening mode has stronger abil-

ity of resisting against crack propagation than mixed mode.

Besides, with other factors being the same, armchair gra-

phene always has larger energy release rate than zigzag gra-

phene, which can be seen from Fig. 7.

Fig. 8 shows close-up views of the bond breaking con-

figurations near the crack tip in opening mode and mixed

mode. First of all, bond breaking always occurs at the atom

close to the last void. In opening mode cases, the atom can

be found along initial crack alignment, as depicted in Figs.

8(a)–8(c). In mixed mode cases, crack growth happens at the

weakest bonded atom around the crack tip, on the side close

to the midline of the graphene sheet, which is the left side in

our simulations. On the effort of horizontal external loading,

all atoms around the crack tip surface are forced to align

with the horizontal direction, and bond rotation mainly

serves as a cushion. Later, bond stretching bears most of the

loading and finally triggers the bond breaking. Here, we can

conclude that the occurrence of bond breaking depends on

the interplay of bond energy, external loading, and local

geometry.

Based on the opening mode case, we study on sample B

before, when we increase the initial temperature to 300 K,

the propagation direction is still perpendicular to the direc-

tion of the external force, as shown in Fig. 9. A similar phe-

nomenon is observed at 1000 K. From Table III, keeping all

other factors unchanged, the fracture deformation and frac-

ture strength decrease as the temperature increases. The

results demonstrate that the temperature plays an important

role in determine the critical energy release rate. The higher

the temperature is, the lower the energy release rate is.

Moreover, high temperature brings significant fluctuation of

the value of GC, as seen in Table IV and Fig. 10. It is com-

mon sense that in MD simulations the real-time temperature

of the system with controlled temperature ensemble fluctu-

ates itself around the desired temperature. Meanwhile, as the

temperature increases, the mechanical rippling instability

increases too. Therefore, it is reasonable that the energy

release rate oscillates fiercely with high temperature.

IV. CONCLUSION

This paper presents a new methodology to study the

fracture mechanics of graphene at atomic scale. It is based

on Griffith’s energy release rate, without any other assump-

tion. Therefore, this methodology can be used for any atomic

system. The findings from the simulations can be summar-

ized as:

(1) The crack path in graphene sheet does not depend on

crack angle, temperature, or graphene orientation, which

is mainly determined by the direction of external

loading.

(2) The critical energy release rate of a finite size graphene

sheet is a constant for small strain rate.

(3) The critical energy release rate of a finite size graphene

sheet is a constant for both opening mode and mixed

mode.

(4) Even though decreasing crack angle can improve the

ability of resisting large applied loading in comparison

with the case in opening mode, the critical energy release

rate of graphene in opening mode is greater than that in

mixed mode.

(5) Armchair graphene always has larger critical energy

release rate than zigzag graphene.

(6) Under the same circumstances, graphene sheet is easier

to break at higher temperature; the fracture deformation,

the fracture strength, and the critical energy release rate

decrease as the temperature increases.

These findings lend compelling insights into the atomis-

tic mechanism of graphene fracture and provide useful

guideline for the design of graphene-based nanodevices.

TABLE III. Comparison of fracture deformation and fracture strength of

armchair graphene under opening mode at different temperatures.

Initial crack

length (nm)

Umax (nm) Fmax (nN)

0 K 300 K 1000 K 0 K 300 K 1000 K

1.56 0.649 0.586 0.450 1010 912 820

2.56 0.624 0.555 0.250 972 864 703

3.69 0.610 0.579 0.251 950 902 633

TABLE IV. Comparison of the critical energy release rate for armchair gra-

phene sheet at different temperature.

0 K 300 K 1000 K

Average value (J=m2) 33.18 28.04 19.17

Standard deviation 0.47 1.71 4.02

FIG. 10. Critical energy release rate vs. initial crack length for grapheme

sheets at different temperatures under opening mode.

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