Atomistic simulation and continuum modeling of graphene nanoribbons under uniaxial tension
ABSTRACT Atomistic simulations are performed to study the nonlinear mechanical behavior of graphene nanoribbons under quasistatic uniaxial tension, emphasizing the effects of edge structures (armchair and zigzag, without and with hydrogen passivation) on elastic modulus and fracture strength. The numerical results are analyzed within a theoretical model of thermodynamics, which enables determination of the bulk strain energy density, the edge energy density and the hydrogen adsorption energy density as nonlinear functions of the applied strain based on static molecular mechanics simulations. These functions can be used to describe mechanical behavior of graphene nanoribbons from the initial linear elasticity to fracture. It is found that the initial Young's modulus of a graphene nanoribbon depends on the ribbon width and the edge chirality. Furthermore, it is found that the nominal strain to fracture is considerably lower for graphene nanoribbons with armchair edges than for ribbons with zigzag edges. Molecular dynamics simulations reveal two distinct fracture nucleation mechanisms: homogeneous nucleation for the zigzag-edged graphene nanoribbons and edge-controlled heterogeneous nucleation for the armchair-edged ribbons. The modeling and simulations in this study highlight the atomistic mechanisms for the nonlinear mechanical behavior of graphene nanoribbons with the edge effects, which is potentially important for developing integrated graphene-based devices.
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ABSTRACT: Doping as one of the popular methods to manipulate the properties of nanomaterials has received extensive application in deriving different types of graphene derivates, while the understanding of the resonance properties of dopant graphene is still lacking in literature. Based on the large-scale molecular dynamics simulation, reactive empirical bond order potential, as well as the tersoff potential, the resonance properties of N-doped graphene were studied. The studied samples were established according to previous experiments with the N atom’s percentage ranging from 0.38%-2.93%, including three types of N dopant locations, i.e., graphitic N, pyrrolic N and pyridinic N. It is found that different percentages of N-dopant exert different influence to the resonance properties of the graphene, while the amount of N-dopant is not the only factor that determines its impact. For all the considered cases, a relative large percentage of N-dopant (2.65% graphitic N-dopant) is observed to introduce significant influence to the profile of the external energy, and thus lead to an extremely low Q-factor comparing with that of the pristine graphene. The most striking finding is that the natural frequency of the defective graphene with N-dopant’s percentage higher than 0.89% appears larger than its pristine counterpart. For the perfect graphene, the N-dopant shows larger influence to its natural frequency. This study will enrich the current understanding of the influence of dopants on graphene, which will eventually shed lights on the design of different molecules-doped graphene sheet.Applied Mechanics and Materials 05/2014; 553:3-9.
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Atomistic simulation and continuum modeling of graphene nanoribbons under uniaxial tension
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IOP PUBLISHING
MODELLING AND SIMULATION IN MATERIALS SCIENCE AND ENGINEERING
Modelling Simul. Mater. Sci. Eng. 19 (2011) 054006 (16pp)doi:10.1088/0965-0393/19/5/054006
Atomistic simulation and continuum modeling of
graphene nanoribbons under uniaxial tension
Qiang Lu, Wei Gao and Rui Huang
Department of Aerospace Engineering and Engineering Mechanics, University of Texas at
Austin, Austin, TX 78712, USA
Received 5 January 2011, in final form 19 April 2011
Published 23 June 2011
Online at stacks.iop.org/MSMSE/19/054006
Abstract
Atomisticsimulationsareperformedtostudythenonlinearmechanicalbehavior
of graphene nanoribbons under quasistatic uniaxial tension, emphasizing the
effects of edge structures (armchair and zigzag, without and with hydrogen
passivation) on elastic modulus and fracture strength. The numerical results
are analyzed within a theoretical model of thermodynamics, which enables
determination of the bulk strain energy density, the edge energy density and the
hydrogen adsorption energy density as nonlinear functions of the applied strain
based on static molecular mechanics simulations. These functions can be used
todescribemechanicalbehaviorofgraphenenanoribbonsfromtheinitiallinear
elasticity to fracture. It is found that the initial Young’s modulus of a graphene
nanoribbon depends on the ribbon width and the edge chirality. Furthermore,
it is found that the nominal strain to fracture is considerably lower for graphene
nanoribbonswitharmchairedgesthanforribbonswithzigzagedges. Molecular
dynamics simulations reveal two distinct fracture nucleation mechanisms:
homogeneousnucleationforthezigzag-edgedgraphenenanoribbonsandedge-
controlled heterogeneous nucleation for the armchair-edged ribbons.
modeling and simulations in this study highlight the atomistic mechanisms
for the nonlinear mechanical behavior of graphene nanoribbons with the edge
effects,whichispotentiallyimportantfordevelopingintegratedgraphene-based
devices.
The
(Some figures in this article are in colour only in the electronic version)
1. Introduction
Graphene ribbons with nanoscale widths (W < 20nm) have been produced recently, either
by lithographic patterning [1–3] or by chemically derived self-assembly processes [4], with
potential applications in nanoelectronics and electromechanical systems.
graphene nanoribbons (GNRs) can be zigzag, armchair or a mixture of both [5]. It has
been theoretically predicted that the special characteristics of the edge states lead to a size
effect in the electronic state of graphene and control whether the GNR is metallic, insulating
The edges of
0965-0393/11/054006+16$33.00© 2011 IOP Publishing LtdPrinted in the UK & the USA1
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Modelling Simul. Mater. Sci. Eng. 19 (2011) 054006Q Lu et al
or semiconducting [5–8]. The effects of edge structures on deformation and mechanical
properties of GNRs have also been studied to some extent [9–18]. On the one hand, elastic
deformation of GNRs has been suggested as a viable method to tune the electronic structure
and transport characteristics in graphene-based devices [15,16]. On the other hand, plastic
deformationandfractureofgraphenemayposeafundamentallimitforreliabilityofintegrated
graphene structures.
The mechanical properties of bulk graphene (i.e. infinite lattice without edges) have been
studied both theoretically [19–21] and experimentally [22]. For GNRs, however, various edge
structures are possible [23,24], with intricate effects on the mechanical properties. Ideally, the
mechanicalpropertiesofGNRsmaybecharacterizedexperimentallybyuniaxialtensiontests.
Todate,however,nosuchexperimenthasbeenreported,althoughsimilartestswereperformed
forcarbonnanotubes(CNTs)[25]. Theoretically,previousstudiesonthemechanicalproperties
of GNRs have largely focused on the linear elastic properties (e.g. Young’s modulus and
Poisson’s ratio) [11–15]. While a few studies have touched upon the nonlinear mechanical
behavior including fracture of GNRs [12,13,16], the effect of edge structures in the nonlinear
regime has not been well understood. In this study, by combining atomistic simulations with a
thermodynamics-based continuum model, we systematically investigate the nonlinear elastic
deformation of GNRs under quasistatic uniaxial tension, emphasizing the effects of edge
structures in both linear and nonlinear regimes.
Thepaperisorganizedasfollows. Section2describesthemethodofatomisticsimulations.
A thermodynamics model is presented in section 3 for analysis of the numerical results.
Section4discussestheedgeeffectoninitialYoung’smodulusofGNRs,andsection5discusses
fracture of graphene. In section 6, the effect of hydrogen adsorption is analyzed. Section 7
summarizes the results.
2. Atomistic simulation
The second-generation reactive empirical bond-order (REBO) potential [26] is used in this
study for atomistic simulations.Briefly, the potential energy of an atomistic system is
calculated as
?
where rijis the interatomic distance between atoms i and j, VRand VAare pairwise potential
functionsfortherepulsiveandattractiveinteractions, respectively, and¯bijisabond-orderterm
that depends on the number and types of neighbors to account for many-body interactions.
In particular, the bond-order function,¯bij, in the second-generation REBO potential takes
into account the local bonding environment up to the third nearest neighbors, through its
dependence on both bond angles and dihedral angles [27]. With this, the REBO potential
allows the influence of atomic re-hybridization on the binding energy to change as chemical
bonds break and reform over the course of atomistic simulation. The complete form of the
REBO potential for both carbon–carbon (C–C) and carbon–hydrogen (C–H) interactions is
given in [26].
To limit the range of covalent interactions, a cutoff function is typically used in atomistic
simulations. The originally suggested form of the cutoff function for the REBO potential is
2
? =
i
?
j>i
[VR(rij) −¯bijVA(rij)],
(1)
fc(r) =
1
1
2
0
r < D1
?
1 + cos
?(r − D1)π
D2− D1
??
D1< r < D2
r > D2
,
(2)
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Modelling Simul. Mater. Sci. Eng. 19 (2011) 054006Q Lu et al
Figure1. RectangularGNRswith(a)zigzagand(b)armchairedges, subjectedtouniaxialtension.
where D1and D2are the two cutoff distances for a smooth transition from 1 to 0 as the
interatomic distance (r) increases. For C–C interaction, D1= 1.7Å and D2= 2.0Å were
suggested [26]. However, as noted in several previous studies [12,28–30], such a cutoff
function typically generates spurious bond forces near the cutoff distances, an unphysical
result due to discontinuity in the second derivative of the cutoff function. This artifact shall
be avoided in the study of nonlinear mechanical properties of graphene under relatively large
strains. As suggested by the developers of the original REBO potential [28], using a larger
cutoff distance could remove the unphysical responses. However, to keep the pair interactions
within the nearest neighbors, the cutoff distance must not be too large. In this study, the cutoff
function is taken to be 1 within a cutoff distance (D1= 1.9Å) and zero otherwise. It is found
that the numerical results up to fracture of GNRs are unaffected by the choice of the cutoff
distance within the range between 1.9 and 2.2Å.
Classical molecular mechanics (MM) simulations are performed for GNRs subjected to
quasistatic uniaxial tension. For each MM simulation, a rectangular GNR of width W and
length L is first cut out from the ground state of an infinite graphene lattice, as shown by
two examples in figure 1. Next, by holding the length of the GNR with periodic boundary
conditions at both ends, edge relaxation is simulated to obtain the equilibrium state of the
GNR at zero strain (ε = 0). As shown in a previous study [18], the ribbon width reduces
slightly upon edge relaxation. Subsequently, by gradually increasing the ribbon length, a
longitudinal tensile strain (ε > 0) is applied. At each strain level, the statically equilibrium
lattice structure of the GNR is calculated to minimize the total potential energy by a quasi-
Newton algorithm [31]. For each GNR, the average potential energy per carbon atom at the
equilibrium state is calculated as a function of the nominal strain until it fractures, as shown
in figure 2. In all simulations, periodic boundary conditions are applied at both ends of the
GNR, whereas the two parallel edges (zigzag or armchair) of the GNR are free of external
3
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Modelling Simul. Mater. Sci. Eng. 19 (2011) 054006Q Lu et al
0 0.10.20.3
-7.5
-7
-6.5
-6
-5.5
Nominal strain
Energy per carbon atom (eV)
W = 1.3 nm
W = 2.6 nm
W = 4.3 nm
W = 8.5 nm
bulk graphene
00.050.1 0.150.2
-7.5
-7
-6.5
-6
Nominal strain
Energy per carbon atom (eV)
W = 1.2 nm
W = 2.5 nm
W = 4.4 nm
W = 8.9 nm
bulk graphene
(a)
(b)
Figure 2. Potential energy per carbon atom as a function of the nominal strain for GNRs under
uniaxialtension, with(a)zigzagand(b)armchairedges, bothunpassivated. Thedashedlinesshow
the results for bulk graphene under uniaxial tension in the zigzag and armchair directions.
constraint. Forcomparison,themechanicalbehaviorofinfinitegraphenelatticeunderuniaxial
tension is also simulated by applying periodic boundary conditions at all four edges, in which
lateral relaxation perpendicular to the loading direction is allowed in order to achieve the
uniaxialstresscondition. Tostudytheeffectofhydrogenadsorptionalongthefreeedges, MM
simulations of GNRs with both bare and hydrogen-passivated edges are performed.
The critical strain (or stress) to fracture as predicted by the static MM simulations may be
consideredtheidealstrengthofthedefect-freeGNRsatzerotemperature(T = 0K).However,
the process of fracture nucleation and crack growth are typically not observable in the MM
simulations. On the other hand, molecular dynamics (MD) simulations at finite temperatures
can be used to study the fracture process. In this study, to qualitatively understand the fracture
mechanisms, classical MD simulations of GNRs under uniaxial tension are performed at
relatively low temperatures (from 0.1 to 300K). The temperature control is achieved using
an Anderson thermostat [32]. Each GNR is loaded by increasing the nominal strain, with a
dwelling period of about 2ps (or 2000 time steps) at each strain level. The strain increment
is adjusted so that increasingly smaller increments are used as the total strain increases, with
a minimum increment at 0.0005. The velocity-Verlet scheme is used for time integration
with a time step of around 1fs. We note that MD simulations are often sensitive to the
temperaturecontrolandtheloadingrate. Inthisstudy,theMDsimulationsprovideaqualitative
understanding of the fracture mechanisms, consistent with the static MM calculations. The
quantitative nature of the MD simulation is not essential for this purpose.
3. Thermodynamics
To understand the numerical results from atomistic simulations, we adopt a simple
thermodynamics model for GNRs under uniaxial tension. For a GNR of width W and length
L, the total potential energy as a function of the nominal strain consists of contributions from
deformation of the interior lattice (i.e. the bulk strain energy) and from the edges (i.e. the edge
energy), namely
?(ε) = NU0+ U(ε)WL + 2γ(ε)L,
where ε is the nominal strain in the longitudinal direction of the ribbon (relative to the bulk
graphene lattice at the ground state), U0is the potential energy per carbon atom at the ground
(3)
4
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Modelling Simul. Mater. Sci. Eng. 19 (2011) 054006Q Lu et al
0 0.1
Nominal strain
0.20.3
0
1
2
3
4
5
6
7
8
Strain energy per area (J/m2)
Zigzag direction
Armchair direction
0 0.1
Nominal strain
0.20.3
7
8
9
10
11
Edge energy per length (eV/nm)
Zigzag edge
Armchair edge
(a)
(b)
Figure 3. (a) Bulk strain energy density of monolayer graphene under uniaxial tension in the
zigzag and armchair directions; (b) edge energy density of GNRs under uniaxial tension. The open
symbolsareobtaineddirectlyfromtheatomisticsimulations, andthesolidlinesarethepolynomial
functions in equations (6) and (8).
state of graphene, N is the number of carbon atoms, U(ε) is the bulk strain energy density of
monolayer graphene (per unit area) and γ(ε) is the edge energy density (per unit length of the
free edges). The average potential energy per carbon atom is thus
¯?(ε) =?(ε)
√3r2
N
= U0+ U(ε)A0+2A0
W
γ(ε),
(4)
whereA0=3
is the equilibrium bond length of graphene. As shown in figure 2, the average potential energy
increases as the ribbon width (W) decreases, an effect due to the contribution of the edge
energy (i.e. the third term on the right-hand side of equation (4)).
For an infinite graphene monolayer (W → ∞), the bulk strain energy density function,
U(ε), can be obtained directly from the MM calculations, namely
4
0istheareapercarbonatomatthegroundstateofgrapheneandr0= 1.42Å
U(ε) =
¯?(ε;W → ∞) − U0
A0
.
(5)
Figure 3(a) shows the calculated bulk strain energy density versus the nominal strain in the
zigzagandarmchairdirections. Foreachcase,thenumericalresultsfromatomisticsimulations
are fitted with a polynomial function up to eighth order of the nominal strain, namely
U(ε) = a2ε2+ a3ε3+ a4ε4+ a5ε5+ a6ε6+ a7ε7+ a8ε8,
where the coefficients are listed in table 1. The eighth-order polynomial function in (6)
is necessary to achieve a satisfactory fitting with the second derivative of the strain energy
density function. The leading term of the polynomial function is necessarily quadratic so
that the strain energy is zero and a minimum at the ground state (ε = 0). Furthermore, the
hexagonalsymmetryofthegraphenelatticeatthegroundstatedictatesthatitisisotropicunder
an infinitesimal strain (ε ? 1). Thus, the quadratic term in equation (6) is independent of
the loading direction. However, the symmetry is broken under a finite deformation, leading
to nonlinear, anisotropic elastic properties [19–21], as represented by the high-order terms on
the right-hand side of equation (6). Consequently, the coefficients listed in table 1 are different
for the two loading directions except for the quadratic term (a2).
(6)
5
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Modelling Simul. Mater. Sci. Eng. 19 (2011) 054006Q Lu et al
Table 1. Coefficients of the polynomial fitting in equation (6) for the bulk strain energy density
function of graphene subject to uniaxial tension in zigzag and armchair directions (unit: Jm−2).
ZigzagArmchair
a2
a3
a4
a5
a6
a7
a8
121.65
144.06
−2947.21
14517.28
−41544.88
66883.97
−46193.34
121.65
1175.81
−23584.89
219264.35
−1189116.03
3459762.95
−4159339.72
Table 2. Coefficients of the polynomial fitting in equation (8) for the edge energy density of GNRs
with zigzag and armchair edges (unit: eVnm−1).
ZigzagArmchair
b0
b1
b2
b3
b4
b5
b6
b7
b8
10.41
−16.22
25.99
−123.40
1387.77
−6306.31
16090.44
−29257.32
26649.06
10.91
−8.53
11.39
−2034.17
37377.27
−374309.95
2144425.42
−6538094.57
8061231.96
For GNRs, the edge energy density function is determined by subtracting the bulk energy
from the total potential energy of the GNR based on equation (4), i.e.
γ(ε) =
W
2A0[¯?(ε) − U(ε)A0− U0].
(7)
Figure 3(b) shows the calculated edge energy density versus the nominal strain for the zigzag
and armchair edges. The results are essentially independent of the ribbon width in the range
considered for this study (1nm < W < 10nm). Similar to the bulk strain energy density,
a polynomial function up to eighth order of the nominal strain is used to fit the edge energy
density, namely
γ(ε) = b0+ b1ε + b2ε2+ b3ε3+ b4ε4+ b5ε5+ b6ε6+ b7ε7+ b8ε8,
where the coefficients for the zigzag and armchair edges are listed in table 2. The first term on
the right-hand side of equation (8) is independent of the nominal strain, which represents the
excess edge energy at zero strain (ε = 0) as discussed in the previous study [18]. The second
term varies linearly with the strain, which gives the residual edge force or edge stress at zero
strain [18]. In general, however, the edge energy is a nonlinear function of the nominal strain.
Next we consider variation of the potential energy. Under uniaxial tension, the GNR
is subjected to a net force (F) in the longitudinal direction. At each strain increment, the
mechanical work done by the longitudinal force equals the increase in the total potential
energy, which can be written in a variational form, i.e.
(8)
δ? = FLδε.
(9)
Therefore, the force (F) can be obtained from the derivative of the potential energy function,
with which a two-dimensional (2D) nominal stress can be defined without ambiguity as the
6
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Modelling Simul. Mater. Sci. Eng. 19 (2011) 054006Q Lu et al
00.05 0.1 0.15
Nominal strain
0.20.25 0.30.35
0
10
20
30
40
Nominal 2D stress (N/m)
W = 1.3 nm
W = 2.6 nm
W = 4.3 nm
W = 8.5 nm
bulk graphene
00.05 0.10.15 0.2
-5
0
5
10
15
20
25
30
35
Nominal strain
Nominal 2D stress (N/m)
W = 1.2 nm
W = 2.5 nm
W = 4.4 nm
W = 8.9 nm
bulk graphene
(a)
(b)
Figure 4. Nominal stress–strain curves for GNRs under uniaxial tension, with (a) zigzag and (b)
armchair edges, both unpassivated. The dashed lines show the results for bulk graphene under
uniaxial tension in the zigzag and armchair directions.
force per unit width of the GNR, namely
σ(ε) =F
W=dU
dε+
2
W
dγ
dε.
(10)
Notethatwedonotassumeanyspecificthicknessforthemonolayergrapheneinthedefinition
of the 2D stress. When placed on a substrate, the thickness of a graphene monolayer depends
on the interaction between graphene and the substrate [33], which is not an intrinsic property
ofgrapheneitself. Asaresult,the2Dstressinequation(10)hasaunitofNm−1,differentfrom
the conventional 3D stress (Nm−2). Figure 4 shows the nominal stress–strain curves of the
GNRs,obtainedbynumericallytakingthederivativeofthepotentialenergyinfigure2. Nearly
identicalstress–straincurvescanbeobtainedanalyticallybyequation(10)withthepolynomial
functions in equations (6) and (8). Apparently, the stress–strain relation for graphene is
nonlinear in all cases, for which the tangent elastic modulus can be defined as
E(ε) =dσ
dε=d2U
dε2+
2
W
d2γ
dε2.
(11)
For an infinite monolayer graphene (W → ∞), the stress–strain relation is fully
determined by the bulk strain energy density function. With the polynomial function in
equation (6), an analytical expression for the stress–strain relation may be obtained. In
figure5(a)weplotthestress–straincurvesforinfinitegraphenesubjectedtouniaxialtensionin
the zigzag and armchair directions, comparing the results from the atomistic simulations with
first-principles calculations by Wei et al [20]. Figure 5(b) shows the corresponding tangent
modulus for bulk graphene. Apparently, the atomistic simulations with the REBO potential
considerably underestimate the stiffness of the graphene monolayer, even under infinitesimal
strain (ε ∼ 0). The initial Young’s modulus, E0= (dσ/dε)ε=0, is 243Nm−1by the REBO
potential and 345Nm−1by the first-principles calculation. This discrepancy is the major
shortcoming of the REBO potential in modeling mechanical behavior of graphene and CNTs,
as noticed previously [34–36]. Nevertheless, the REBO potential has been used extensively,
including this study, to qualitatively understand the mechanical behavior of low-dimensional
carbonmaterialsontheatomisticscale. SeveralmodificationstotheREBOpotentialhavebeen
suggested recently [37–39], which are yet to show consistent improvement in the prediction
of Young’s modulus of graphene.
7
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Modelling Simul. Mater. Sci. Eng. 19 (2011) 054006 Q Lu et al
00.05 0.10.15 0.20.250.3
0
10
20
30
40
Nominal strain
Nominal 2D stress (N/m)
zigzag (MM/REBO)
armchair (MM/REBO)
zigzag (Wei et al., 2009)
armchair (Wei et al., 2009)
00.050.1 0.15 0.20.250.3
0
100
200
300
400
Nominal strain
2D Young's modulus (N/m)
zigzag (MM/REBO)
armchair (MM/REBO)
zigzag (Wei et al., 2009)
armchair (Wei et al., 2009)
(a)
(b)
Figure 5. (a) Nominal stress–strain curves for monolayer graphene under uniaxial tension in the
zigzag and armchair directions; (b) tangent Young’s modulus as a function of the nominal strain.
For GNRs, due to the edge effect, the nominal stress–strain relation depends on the ribbon
width, as shown in figure 4. The difference between GNRs with zigzag edges and those with
armchairedgesisalsoappreciable, evenatrelativelysmallstrains. Wediscusstheedgeeffects
in the following sections.
4. Edge effect on the initial Young’s modulus
The nominal stress–strain curves in figure 4 show approximately linear elastic behavior of all
GNRs at relatively small strains (e.g. ε < 5%). Following equation (11), the initial Young’s
modulus of the GNRs in the linear regime can be written as
2
WEe
where Eb
Using the polynomial functions in equations (6) and (8), we have
?d2U
?d2γ
While bulk graphene is isotropic in the regime of linear elasticity, the initial edge modulus
depends on the edge chirality with different values for the zigzag and armchair edges. As a
result,theinitialYoung’smodulusoftheGNRdependsonbothedgechiralityandribbonwidth
(W), as shown in figure 6. The initial edge modulus obtained from the REBO potential in this
study is Ee
(∼ 23eVnm−1) for the unpassivated armchair edge. With positive moduli for both edges, the
Young’s modulus of unpassivated GNRs increases as the ribbon width decreases. Figure 6
showsthatthenumericalresultsfromtheatomisticsimulationsagreecloselywithequation(12)
using the polynomial fitting parameters for the bulk and edge modulus. As such, it is predicted
that the edge effect on the initial Young’s modulus of GNRs diminishes as the ribbon width
increases. A similar effect has been reported for nanowires and nanofilms, for which the
surface effect leads to size-dependent Young’s modulus [40–42].
E0= Eb
0+
0,
(12)
0is the initial Young’s modulus of bulk graphene and Ee
0is the initial edge modulus.
Eb
0=
dε2
?
?
ε=0
= 2a2,
(13)
Ee
0=
dε2
ε=0
= 2b2.
(14)
0= 8.33nN (∼52eVnm−1) for the unpassivated zigzag edge and Ee
0= 3.65nN
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Modelling Simul. Mater. Sci. Eng. 19 (2011) 054006Q Lu et al
Figure 6. Initial Young’s modulus versus ribbon width for GNRs with unpassivated and hydrogen-
passivated edges. The horizontal dotted–dashed line indicates the initial Young’s modulus of bulk
graphene predicted by the REBO potential.
It is noted in figure 4 that the nominal stress is not zero for GNRs at zero nominal strain.
This is due to the presence of a residual edge force (or edge stress) at zero strain. As discussed
in the previous study [18], relaxation of the edge bonds results in a compressive edge force
due to a mismatch in the equilibrium bond lengths. The edge force can be obtained as the first
derivative of the edge energy function, namely
f(ε) =dγ
dε.
(15)
With equation (8) for the edge energy density, the edge force at ε = 0 equals the coefficient b1,
which is negative (compressive) for both zigzag and armchair edges as listed in table 2. As a
result, the nominal stress of the GNRs as defined in equation (10) is negative at zero strain and
is inversely proportional to the ribbon width. The compressive edge force may lead to edge
buckling [18], which would partly relax the nominal stress and potentially affect the initial
stress–strain behavior for the GNRs. This effect is found to be negligible as the edge buckling
is typically flattened under uniaxial tension with the nominal strain beyond a fraction of 1%.
5. Fracture of GNRs
Without any defect, the theoretical strength of monolayer graphene (infinite lattice) is dictated
by intrinsic lattice instability. As shown in several previous studies [19–21,30,43], the critical
strain to fracture for graphene varies with the loading direction. Under uniaxial tension, as
shown in figure 5, the graphene monolayer fractures at the maximum nominal stress, when the
tangent modulus becomes zero (i.e. d2U/dε2= 0). At a finite temperature, however, fracture
may occur much earlier due to thermally activated processes [12]. It is noted that both the
MM simulations and first-principles calculations predict higher tensile strength in the zigzag
direction than in the armchair direction. However, the REBO potential underestimates the
theoretical strength (fracture stress) of graphene in both directions. This discrepancy may be
a result of the discrepancy in the predictions of the initial Young’s modulus of graphene by the
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Modelling Simul. Mater. Sci. Eng. 19 (2011) 054006Q Lu et al
Figure 7. Fracture strain versus ribbon width for GNRs under uniaxial tension, with (a) zigzag
and (b) armchair edges. The horizontal dashed line in each figure indicates the fracture strain of
bulk graphene under uniaxial tension in the same direction.
two methods. On the other hand, the REBO potential overestimates the fracture strain in the
zigzag direction, whereas the predicted fracture strain in the armchair direction agrees closely
with the first-principles calculation.
For GNRs, the lattice structure becomes inhomogeneous due to edge relaxation, which
leadstotwodistinctfracturemechanismsforGNRswithzigzagandarmchairedges. Asshown
in figure 4(a), the GNRs with zigzag edges fracture at a critical strain very close to that of
bulk graphene loaded in the same direction. In contrast, figure 4(b) shows that the GNRs with
armchairedgesfractureatacriticalstrainconsiderablylowerthanbulkgraphene. Inbothcases,
the fracture strain slightly depends on the ribbon width, as shown in figure 7. The apparently
different edge effects on the fracture strain imply different fracture nucleation mechanisms for
the zigzag- and armchair-edged GNRs, which are revealed by MD simulations.
To qualitatively understand the fracture processes of GNRs under uniaxial tension, MD
simulations are performed at different temperatures (0 < T < 300K). Figure 8 shows two
examples of fractured GNRs at 50K. For the GNR with zigzag edges (figure 8(a)), fracture
nucleationoccursstochasticallyattheinteriorlatticeoftheGNR.Asaresult,thefracturestrain
is very close to that of bulk graphene strained in the same direction, consistent with the MM
calculations (figure 7(a)). However, for the GNR with armchair edges (figure 8(b)), fracture
nucleation occurs exclusively near the edges. Thus, the armchair edge serves as the preferred
location for fracture nucleation, leading to a considerably lower fracture strain compared
with bulk graphene, as seen also from the MM calculations (figure 7(b)). Therefore, two
distinct fracture nucleation mechanisms are identified as interior homogeneous nucleation for
the zigzag-edged GNRs and edge-controlled heterogeneous nucleation for the armchair-edged
GNRs. In both cases, the fracture process is essentially brittle. The formation of suspended
atomicchainsisobservable,mostlyneartheedges,intheMDsimulationsasshowninfigure8.
Asimilarchainformationwasobservedinexperiments[44]andinafirst-principlesstudy[16].
It is evident from figure 8 that the cracks preferably grow along the zigzag directions
of the graphene lattice in both cases. By the Griffith criterion for brittle fracture [45], this
suggests lower edge energy in the zigzag direction of graphene as opposed to the armchair
direction,whichisconsistentwithourcalculationsoftheedgeenergyinthepreviousstudy[18].
However,severalfirst-principlescalculations[17,23,46,47]havepredictedloweredgeenergy
for the armchair edge, opposite to the calculations using empirical potentials [9,18]. On the
other hand, other first-principles calculations [19,20] have predicted lower fracture strain and
10
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Modelling Simul. Mater. Sci. Eng. 19 (2011) 054006Q Lu et al
Figure 8. Fracture of GNRs under uniaxial tension. (a) Homogeneous nucleation for a zigzag
GNR; (b) edge-controlled heterogeneous nucleation for an armchair GNR. The circles indicate
the nucleation sites, and the arrows indicate the directions of crack growth. Color indicates the
potential energy of the carbon atoms.
stress for bulk graphene under uniaxial tension in the armchair direction (see figure 5(a)), in
qualitative agreement with the MM calculations in this study. A more quantitative study on
the fracture process of graphene is left for future work.
Inadditiontothefracturestrain,thenominalfracturestress(i.e.uniaxialtensilestrength)of
theGNRscanbedeterminedfromthestress–straincurvesinfigure4. Asshowninfigure9,the
fracture stress increases as the ribbon width increases for GNRs with unpassivated edges. The
edge effect is relatively small for the zigzag-edged GNRs, with all the fracture stresses around
36Nm−1,veryclosetothatofbulkgraphene. Forthearmchair-edgedGNRs,thefracturestress
is considerably lower, e.g. 27.5Nm−1for an unpassivated GNR with W = 2.5nm, compared
with 30.6Nm−1for bulk graphene under uniaxial tension in the armchair direction. Again,
the lower fracture stress for the armchair-edged GNRs can be attributed to the edge-controlled
heterogeneous nucleation mechanism shown in figure 8(b).
In this study we have focused on the fracture of defect-free GNRs. It is expected that
interior defects of graphene lattice, such as vacancies, dislocations and grain boundaries,
could have significant effects on the fracture of graphene. A similar effect has been studied
for CNTs [48,49]. Recently, Terdalkar et al [50] have presented atomistic simulations of the
kinetic processes of bond breaking and bond rotation near a crack tip in graphene. Grantab
et al [51] have demonstrated by atomistic calculations an anomalous effect of tilt grain
11
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Modelling Simul. Mater. Sci. Eng. 19 (2011) 054006Q Lu et al
Figure 9. Nominal fracture stress versus ribbon width for GNRs under uniaxial tension, with (a)
zigzag and (b) armchair edges. The horizontal dashed line in each figure indicates the fracture
stress of bulk graphene under uniaxial tension in the same direction.
boundaries on the strength of graphene. Further studies on fracture of GNRs may consider
interactions between the interior defects and the edge structures.
6. Effects of hydrogen adsorption
The edges of GNRs are often passivated with hydrogen (H) atoms. Hydrogen adsorption
changes the bonding environment and the energetics of the edges. Subject to uniaxial tension,
the potential energy of a GNR now includes the contribution from hydrogen adsorption at the
edges, namely
?(ε) = NU0+ U(ε)WL + 2γ(ε)L − 2γH(ε)L,
where γH(ε) is the adsorption energy per length for hydrogen passivated edges. The negative
sign for the last term in equation (16) indicates typically reduced edge energy due to hydrogen
adsorption [17,23]. By comparing the calculated potential energies for the GNRs with and
without H-passivation, the adsorption energy can be determined as a function of the nominal
strain for both armchair and zigzag edges. At zero strain (ε = 0), our MM calculations predict
the hydrogen adsorption energies to be 20.5eVnm−1and 22.6eVnm−1for the zigzag and
armchair edges, respectively, which agree closely with the first-principles calculations [23].
Under uniaxial tension, the adsorption energy varies with the nominal strain, as shown in
figure 10. The calculated H-adsorption energy is fitted with an eighth-order polynomial
function, namely
(16)
γH(ε) = c0+ c1ε + c2ε2+ c3ε3+ c4ε4+ c5ε5+ c6ε6+ c7ε7+ c8ε8,
where the coefficients are listed in table 3. The first three terms on the right-hand side of
equation (17) directly affect the edge energy, edge force and edge modulus at infinitesimal
strain, respectively, whereasthehigherordertermsaccountforthenonlineareffectswithfinite
strain. The effect of elastic deformation (strain) on the adsorption energy demonstrates an
intrinsic coupling between mechanics and chemistry on the atomistic scale.
The 2D nominal stress–strain relation for a GNR with H-passivated edges can then be
obtained as
?dγ
12
(17)
σ(ε) =dU
dε+
2
W
dε−dγH
dε
?
,
(18)
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Modelling Simul. Mater. Sci. Eng. 19 (2011) 054006Q Lu et al
00.05 0.1
Nominal strain
0.150.2 0.250.3
16
17
18
19
20
21
22
23
24
Adsorption energy (eV/nm)
Zigzag edge
Armchair edge
Figure 10. Hydrogen adsorption energy of GNRs under uniaxial tension. The open symbols
are obtained directly from atomistic simulations, and the solid lines are the polynomial fitting in
equation (17).
Table 3. Coefficients of the polynomial fitting in equation (17) for the hydrogen adsorption energy
of GNRs with zigzag and armchair edges (unit: eVnm−1).
ZigzagArmchair
c0
c1
c2
c3
c4
c5
c6
c7
c8
20.53
−16.14
0.3798
144.49
−577.04
5109.55
−31512.41
84961.78
−83158.52
22.61
−8.25
21.66
−297.69
1755.68
19851.33
−342205.25
1835393.08
−3589655.15
and the tangent modulus is
E(ε) =d2U
dε2+
2
W
?d2γ
dε2−d2γH
dε2
?
.
(19)
Figure 11 compares the stress–strain curves for H-passivated GNRs, unpassivated GNRs and
bulk graphene. At infinitesimal strain, the initial Young’s modulus follows equation (12), but
with a modified edge modulus due to H-adsorption, namely
?d2γ
As shown in figure 6, H-adsorption has a negligible effect on the initial Young’s modulus for
GNRs with zigzag edges. In contrast, the effect is significant for GNRs with armchair edges.
The edge modulus as defined in equation (20) becomes negative for the H-passivated armchair
edge. Consequently, by equation (12), the initial Young’s modulus of the GNR decreases as
the ribbon width decreases, opposite to the unpassivated GNRs.
Ee
0=
dε2
?
ε=0
−
?d2γH
dε2
?
ε=0
= 2b2− 2c2.
(20)
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Modelling Simul. Mater. Sci. Eng. 19 (2011) 054006Q Lu et al
0 0.050.1 0.15 0.20.25 0.30.35
0
10
20
30
40
Nominal strain
Nominal 2D stress stress (N/m)
unpassivated z-GNR
H-passivated z-GNR
bulk graphene
00.05 0.10.15 0.2
0
5
10
15
20
25
30
35
Nominal strain
Nominal 2D stress (N/m)
unpassivated a-GNR
H-passivated a-GNR
bulk graphene
Figure 11. Comparison of nominal stress–strain curves under uniaxial tension for bulk graphene,
GNRswithunpassivatededgesandGNRswithhydrogen-passivatededges: (a)zigzag-edgedGNR
(W = 4.3nm) and (b) armchair-edged GNR (W = 4.4nm).
The effect of hydrogen adsorption on fracture strain is shown in figure 7. Hydrogen
passivation of the edges leads to slightly lower fracture strains for zigzag GNRs, but slightly
higherfracturestrainsforarmchairGNRs. Theeffectisrelativelysmallinbothcases. Figure9
shows that H-adsorption slightly increases the fracture stress for both zigzag- and armchair-
edgedGNRs. Thesamefacturemechanismsshowninfigure8areobservedinMDsimulations
for GNRs with H-passivated edges.
7. Summary
This paper presents a theoretical study on the effects of edge structures on the mechanical
properties of graphene nanoribbons (GNRs) under uniaxial tension. Both the bulk strain
energydensityandedgeenergydensity(withoutandwithhydrogenpassivation)arecalculated
from atomistic simulations as functions of the nominal strain. Due to the edge effect, the
initial Young’s modulus of GNRs under infinitesimal strain depends on both the chirality
and ribbon width. Furthermore, it is found that the strain to fracture is considerably lower
for armchair-edged GNRs than for zigzag-edged GNRs. Two distinct fracture nucleation
mechanisms are identified, homogeneous nucleation for GNRs with zigzag edges and edge-
controlled heterogeneous nucleation for those with armchair edges. Hydrogen adsorption
along the edges is found to have relatively small effects on the mechanical behavior of zigzag-
edged GNRs, but its effect is more significant for armchair-edged GNRs. Finally, we note that
several reconstructions have been predicted for the edge structures of graphene [23,46,47]
and reconstructed edges may have different effects on the mechanical properties of GNRs [11]
compared with the pristine edges considered in this study.
Acknowledgments
The authors gratefully acknowledge funding of this work by the National Science Foundation
throughGrantNo0926851. TheythankDrJeffreyWKysarofColumbiaUniversityforhelpful
discussions and for providing the first-principles results for comparison in figure 5.
14
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