# 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.

**0**Bookmarks

**·**

**138**Views

- [Show abstract] [Hide abstract]

**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.

Page 1

Atomistic simulation and continuum modeling of graphene nanoribbons under uniaxial tension

This article has been downloaded from IOPscience. Please scroll down to see the full text article.

2011 Modelling Simul. Mater. Sci. Eng. 19 054006

(http://iopscience.iop.org/0965-0393/19/5/054006)

Download details:

IP Address: 218.23.88.210

The article was downloaded on 23/06/2011 at 19:22

Please note that terms and conditions apply.

View the table of contents for this issue, or go to the journal homepage for more

HomeSearchCollectionsJournalsAbout Contact usMy IOPscience

Page 2

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

Page 3

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)

Page 4

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

Page 5

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

Page 6

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

Page 7

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

Page 8

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

Page 9

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

8

Page 10

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

9

Page 11

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

Page 12

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

Page 13

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)

Page 14

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)

13

Page 15

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

#### View other sources

#### Hide other sources

- Available from Rui Huang · Jun 5, 2014
- Available from Rui Huang · Jun 5, 2014
- Available from Rui Huang · Jun 5, 2014
- Available from utexas.edu