Proceedings of the 14th IEEE
International Conference on Nanotechnology
Toronto, Canada, August 18-21, 2014
Effects of Free Edges and Vacancy Defects on the
Mechanical Properties of Graphene
M. A. N. Dewapriya and R. K. N. D. Rajapakse
School of Engineering Science, Simon Fraser University, Burnaby, BC V5A1S6
Email: mandewapriya@sfu.ca
Abstract — Defects are unavoidable during synthesizing
and fabrication of graphene based nanoelecromechanical
systems. This paper presents a comprehensive molecular
dynamics simulation study on the mechanical properties of
finite graphene with vacancy defects. We characterize the
strength and stiffness of graphene using the concept of surface
stress in three-dimensional crystals. Temperature and strain
rate dependent atomistic model is also presented to evaluate the
strength of defective graphene. Free edges have a significant
impact on the stiffness; the strength, however, is less affected.
The vacancies exceedingly degrade the strength and the
stiffness of graphene. These findings provide a remarkable
insight into the strength and the stiffness of defective graphene,
which is critical in designing experimental and instrumental
applications.
Index Terms – Graphene fracture, vacancy defects,
molecular dynamics, nanomechanics, effects of free edges.
I. INTRODUCTION
The extraordinary electromechanical properties of
graphene have drawn remarkable attention from scientists
and engineers. Graphene based nanoelectromechanical
systems (NEMS), such as resonators [1], have demonstrated
intriguing applications in various engineering disciplines
from telecommunication [2] to biomedicine [3]. However, as
in many crystalline materials, defects are unavoidable during
synthesizing and fabrication of graphene based NEMS [3].
Defects, such as vacancies (missing atoms), drastically
reduce the strength and stiffness of graphene that critically
influence the performance of NEMS [4]. On the other hand,
edges and interfaces present in a finite, narrow sheet change
the thermo-mechanical properties and even influence the
stability of graphene [5].
Classical continuum mechanics break down at the
nanoscale [6]. The modified continuum models such as nonlocal elasticity [7] are also not applicable to systems made of
graphene, which is a single atomic layer. These nanoscale
systems can be analyzed by using first-principle methods.
Such simulations are computationally very expensive (often
impractical) when applied to systems with several thousands
of atoms. Graphene based systems can be effectively
modelled using atomistic methods such as molecular
dynamics (MD) to compromise between the accuracy and
the computational cost.
978-1-4799-4082-0/$31.00 ©2014 IEEE
908
This paper presents a comprehensive MD simulation
study that investigates the effects of vacancies on the
mechanical properties of finite graphene. We also show that
the strength and the stiffness of defective graphene can be
characterized by using the concept of surface stress in threedimensional crystals. Temperature and strain rate dependent
atomistic model is also presented to evaluate the strength of
defective graphene.
II. MOLECULAR DYNAMICS SIMULATIONS
We performed MD simulations using LAMMPS
package [8] with adaptive intermolecular reactive empirical
bond order (AIREBO) potential field [9].
A. AIREBO Potential Field
The AIREBO potential consists of three sub-potentials,
which are the reactive empirical bond order (REBO),
Lennard-Jones, and torsional potentials. The REBO potential
gives the energy stored in atomic bonds; the Lennard-Jones
potential considers the non-bonded interactions between the
atoms, and the torsional potential includes the energy from
torsional interactions between the atoms.
According to the REBO potential [10], the energy stored
in a bond between atom i and atom j can be expressed as
EijREBO = f ( rij )!"VijR + bijVijA #$ ,
(1)
where VijR and VijA are the repulsive and the attractive
potentials, respectively; bij is the bond order term, which
modifies the attractive potential depending on the local
bonding environment; rij is the distance between the atoms i
and j; f(rij) is the cut-off function. The cut-off function in
REBO potential [10], given in (2), limits the interatomic
interactions to the nearest neighbors.
(
1,
*
*
" π ( r − R (1) ) %
*
ij
',
f (rij )= ) 1+ cos$ (2)
$# ( R − R (1) ) '&
*
*
0,
*+
rij < R (1)
R (1) < rij < R (2)
R (2) < rij
,
(2)
where R(1) and R(2) are the cut-off radii, which are 1.7 Å and
2 Å, respectively. The values of cut-off radii are defined
based on the first and the second nearest neighboring
distances of the relevant hydrocarbon. The cut-off function,
however, causes a non-physical strain hardening in carbon
nanostructures [11]. Therefore, modified cut-off radii,
ranging from 1.9 Å to 2.2 Å, have been used to eliminate
this non-physical strain hardening [12]. In this study, we
used a truncated cut-off function ft(rij), given in (3) [13], to
eliminate this strain hardening.
! 1, r < R
#
ij
ft (rij )= "
#$ 0, rij > R
Stress in MD simulations has been interpreted using either
the Cauchy stress [5,12] or the virial stress [15]. The Cauchy
stress is computationally efficient than the virial stress.
However, the Cauchy stress induces a non-physical initial
stress (at zero strain) at higher temperatures, whereas the
virial stress gives the initial stress as zero [15].
The Cauchy stress is the gradient of the potential energy
per unit volume vs strain curve; the virial stress [16], σij, is
defined as
σ ij =
(3)
where the value of R is 2 Å. Similar cut-off functions have
been used in [12] and [14] to simulate the fracture of
graphene.
B. Simulation Parameters
Length of graphene sheets was 10 nm; periodic
boundary conditions were used along the longitudinal
direction while the transverse edges were kept free. Width of
the sheets were changed from ~1 nm to 25 nm. Fig. 1 shows
armchair and zigzag graphene sheets. The sheets were
allowed to relax over 30 ps before applying strain; the time
step was 0.5 fs. During the relaxation period, the pressure
component along the transverse direction was kept at zero
using NPT ensemble implemented in LAMMPS.
The NPT ensemble controls the temperature by using
Nośe-Hoover thermostat, which induces a non-physical
thermal expansion in graphene [5]. This thermal expansion
was eliminated by introducing an initial random out-of-plane
displacement perturbation (~0.05 Å) to the carbon atoms.
The simulation temperature was 300 K. Strain was applied
by pulling the sheet along the longitudinal direction at a
strain rate of 109 s-1. Stress perpendicular to the pulling
direction was kept at zero to simulate uniaxial tensile test.
(a)
(b)
Fig. 1. (a) armchair and (b) zigzag graphene sheets. The size of the sheets is
50 nm × 10 nm. The arrows indicated the direction of the applied strain.
C. Calculation of Stress
909
#1 N
&
1
% ∑ ( Riβ − Riα ) Fjαβ − mα viα vαj (
∑
α
V
%$ 2 β =1
('
,
(4)
where i and j are the directional indices (x, y, and z); α is a
number assigned to an atom; β is a number assigned to
neighbouring atoms of atom α which varies from 1 to N; Riβ
is the position of atom β along the direction i; Fjαβ is the
force along the direction j on atom α due to atom β; mα and
vα are the mass and the velocity of atom α, respectively; V is
the total volume.
The definition of volume in the virial stress, however, is
ambiguous; the virial stress is quite similar to the Cauchy
stress when instantaneous volume is used in the virial
calculation [15]. In this work, we used the instantaneous
volume to calculate the virial stress. Thickness of graphene
was assumed 3.4 Å, which is the interlayer spacing of
graphene in graphite.
Five MD simulations, with different randomly
distributed vacancies, were performed for each vacancy
concentration and a given width. The strength and the
stiffness are less sensitive (<5%) to the distribution of
vacancies in the sheet. Therefore, the average strength and
stiffness of these five simulations were used for the analysis.
III. RESULTS AND DISCUSSION
A. Effects of Free Edges and Defects
The stress-strain curve of graphene is nonlinear as shown in
Fig. 2. Therefore, we obtained the stiffness by considering
the stress-strain curve up to 0.03 strain, where the curve is
linear. Fig. 2(a) shows that the free edges do not have a
significant effect on the tensile strength of graphene, which
is indicted by the insignificant change in the tensile strength
as the width increases. However, the width has a great
influence on the stiffness. This width effect is not significant
beyond 6 nm.
Figure 2(b) shows the influence of vacancy defects on
the stress-strain curve of a 12 nm wide graphene sheet,
where the effect of width does not prevail. The figure shows
that vacancies greatly reduce the strength of graphene. The
stiffness is also significantly affected.
Figure 3(a) shows that the stiffness gradually decreases
with the increase of vacancy concentration. At all the
considered vacancy concentrations, the stiffness reduces by
~50% as the width decreases up to ~1 nm. However, the
edge effects become insignificant at widths larger than ~5
nm as the number of atoms at the edges is negligible
compared to those in the bulk.
σ (ε, w) =
$
2τ ! 2Es
+#
+ E b &ε .
%
w " w
(6)
Therefore, the effective elastic moduli (Eeff) of a finite sheet
can be written as
90
width = 1 nm
w = 2 nm
w = 3 nm
w = 6 nm
80
Stress (GPa)
70
We obtained Es (GPa nm) and Eb (GPa) by regression
analysis, and the corresponding values are given in Table 1.
The best-fit curves, in the form of (7), are plotted in Fig.
3(a), and these curves capture the effects of free edges quite
well.
Figure 3(b) shows that free edges do not have a
significant effect on the strength as observed in Fig. 2(b);
however, the vacancies drastically reduce the strength. Even
a single vacancy reduces the strength by ~15%, whereas the
stiffness is not affected. In the case of single vacancy, the
vacancy percentage decreases with increasing width due to
the increase in the number of atoms, thereby the widthstrength relationship is quite different compared to the other
curves in Fig. 3(b).
Similar to (7), the strength σult can be expressed as
60
50
40
30
20
10
0
(a)
0
0.05
0.1
Strain
90
pristine
single vac.
0.5% vac.
1%.
2%
80
Stress (GPa)
70
60
σ ult =2σ s,ult w + σ b,ult ,
40
30
20
(b)
10
0
0.02 0.04 0.06 0.08
0.1
(8)
where σs,ult (GPa nm) and σb,ult (GPa) are the representative
ultimate tensile strengths of the surface and the bulk,
respectively; the values are given in Table 1.
Table 1 shows that σs,ult of zigzag sheets are positive,
except in the case of single vacancy, which indicates that the
strength increases as the width decreases. However, σs,ult of
zigzag sheets are not significant compared to σb,ult; therefore,
the increase in strength is not significant.
50
0
(7)
Eeff = 2Es w + Eb .
0.12
900
Strain
B. Continuum Modeling of Edge Effect
When a finite graphene sheet of width w is subjected to an
axial strain ε, the potential energy per unit length can be
expressed using the concept of surface stress in a threedimensional crystal as [5]
U(ε, w) = U 0 + 2τε + Esε + Ebε w 2 ,
2
2
(5)
where U0 is the potential energy at zero strain; τ is the edge
stress which arises from the difference of the energies in the
edge and interior atoms; Eb and Es are the bulk and the edge
elastic moduli, respectively. The stress in the sheet is given
by
910
800
Stiffness (GPa)
Fig. 2. Stress-strain curves of graphene with (a) various widths and (b)
vacancy concentrations.
700
600
pristine
single vac.
0.5% vac.
1%
2%
(a)
500
400
300
0
5
10
15
Width (nm)
20
25
however, is a good approximation to the durability function
[12].
The Arrhenius equation [20] expresses the temperature
dependent rate of a chemical reaction (k) as k =
A×exp[ΔE/(kBT)], where A is a constant that depends on the
chemical bonding; ΔE is the activation energy barrier; kB is
the Boltzmann constant. When a mechanical force F is
applied to a molecule, the activation energy barrier reduces
by an amount of FΔx, where Δx is the change in the atomic
coordinates due to F [21]. We defined a durability function
for graphene in the form of Arrhenius equation as
90
Strength (GPa)
80
70
60
50
(b)
40
0
5
10
Width (nm)
τ (T, t ) =
15
Fig. 3. Variations in (a) the stiffness and (b) the ultimate tensile strength of
armchair graphene with width and vacancy concentration. The both (a) and
(b) have the same legend. The curves in Fig. 3(a) and (b) represent (7) and
(8), respectively.
TABLE I
SURFACE AND BULK PROPERTIES OF GRAPHENE
Vacancy
concentration
σs,ult
σb,ult
0% (ac)
-3.8
87.4
Es
-202
Eb
963
0.5% (ac)
-5.3
63.3
-209
903
827
1% (ac)
-4.5
58.2
-207
2% (ac)
-5.3
49.6
-232
764
Single vac. (ac)
-10.5
76.4
-223
964
4.3
105.5
-268
867
70.5
-263
813
783
681
856
0% (zz)
0.5% (zz)
1.4
1% (zz)
1.1
63.1
-247
2% (zz)
Single vac. (zz)
1.4
-1.1
53.8
80.3
-217
-291
dt
,
(11)
where α is the vacancy percentage. Even the presence of a
single vacancy reduces the strength drastically; this strength
reduction is considered by the constant k. The values of k are
1.13 and 1.21 for armchair and zigzag sheets, respectively
[17]. γ = vq, where v is the activation volume, which is 8.25
Å3; the value of v is close to the representative volume of a
carbon atom in graphene, which is 8.6 Å3. q is a directional
constant that takes into account the different bond
orientation along the armchair and zigzag directions [17]; q
is 1 for armchair sheets and it is 91.7/108.9 (0.82) for zigzag
sheets, where 91.7 and 108.9 are the tensile strengths, in
GPa, of armchair and zigzag sheets at 300 K, respectively.
σ(t) is the stress at time t, which we expressed in terms of
the strain rate ε as
We recently used the Arrhenius equation and the Bailey’s
criterion to model the temperature and strain rate dependent
fracture strength of defective graphene [17]; an overview of
this model is presented below. This model, however, does
not take into account the effects of free edges.
The Bailey’s criterion of durability [18] provides a basis
to estimate the lifetime of materials at various temperatures
[19]. The criterion is expressed as
∫ τ (T, t ) = 1,
(10)
where τ0 is the vibration period of atoms that is 5 fs for
carbon in graphene [5]; n is the number of bonds in the
sheet; U0 is the interatomic bond dissociation energy that is
4.95 eV for a carbon-carbon bond [10]; β represents the
reduction of average bond dissociation energy due to
presence of vacancies; we defined β, using MD simulations
at 300 K, as
!#
1,
α=0
β ="
0.165
α
+
k,
α>0
$#
C. Kinetic Modeling of Strength
tf
" U β − γσ (t ) %
τ0
exp $ 0
',
n
k BT
#
&
(9)
0
where tf is the time (t) taken to the fracture; τ(T,t) is the
durability function at temperature T, which is generally
determined by experiments [18]. The Arrhenius equation,
911
2
σ (t ) = a (εt ) + b (εt ) ,
(12)
where a and b are the second and the third order elastic
moduli, respectively; the values of a and b were obtained
from regression analysis of the stress-strain curves given by
MD simulations at 300 K, where, a and b are 1.11 TPa and 3.20 TPa for armchair sheet, the corresponding values for
zigzag sheet are 0.91 TPa and -1.90 TPa.
We calculated the failure time tf by numerically solving
(9). We obtained the fracture stress σ(tf) by substituting the tf
into (12). Fig. 4 shows that the fracture strength given by the
proposed model agrees quite well with the MD simulations
results. The proposed model is computationally quite
efficient than molecular dynamics simulations.
REFERENCES
110
2% model
2% MD
100
Strength (GPa)
90
80
pristine model
pristine MD
single vac. model
single vac. MD
1% model
1% MD
70
60
50
40
30
200
(a)
400
600
800
1000
1200
1400
1600
Temperature (K)
130
2% model
2% MD
120
Strength (GPa)
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100
pristine model
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single vac. model
single vac. MD
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90
80
70
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(b)
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Fig. 4. Comparison of the model predicted strength of (a) armchair and (b)
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IV. CONCLUSIONS
In summary, we used molecular dynamics simulations to
study the influence of free edges and vacancy concentration
on the strength and stiffness of graphene. Results reveal that
vacancy defects have a profound impact on the strength and
stiffness. We also present an atomistic model to assess the
temperature and strain rate dependent fracture strength of
defective graphene. The model is computationally very
efficient and quite accurate compared to the molecular
dynamics simulations.
ACKNOWLEDGMENT
This work was financially supported by the Natural
Sciences and Engineering Research Council (NSERC) of
Canada. Computing resources were provided by WestGrid
and Compute/Calcul Canada.
912
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