effects of loading mode and orientation on deformation ...molecular dynamics simulation is performed...
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Effects of loading mode and orientation on deformation mechanismof graphene nano-ribbons
Y. J. Sun,1 F. Ma,1,2,a) Y. H. Huang,3 T. W. Hu,1,2 K. W. Xu,1,4,a) and Paul K. Chu2,a)1State Key Laboratory for Mechanical Behavior of Materials, Xi’an Jiaotong University, Xi’an 710049,Shaanxi, China2Department of Physics and Materials Science, City University of Hong Kong, Tat Chee Avenue, Kowloon,Hong Kong, China3College of Physics and Information Technology, Shaanxi Normal University, Xi’an 710062, Shaanxi, China4Department of Physics and Opt-electronic Engineering, Xi’an University of Arts and Science, Xi’an 710065,Shaanxi, China
(Received 8 August 2013; accepted 25 October 2013; published online 7 November 2013)
Molecular dynamics simulation is performed to analyze the deformation mechanism of graphene
nanoribbons. When the load is applied along the zigzag orientation, tensile stress yields brittle
fracture and compressive stress results in lattice shearing and hexagonal-to-orthorhombic phase
transformation. Along the armchair direction, tensile stress produces lattice shearing and phase
transformation, but compressive stress leads to a large bonding force. The phase transformation
induced by lattice shearing is reversible for 17% and 30% strain in compressive loading along the
zigzag direction and tensile loading along the armchair direction. The energy dissipation is less
than 10% and resulting pseudo-elasticity enhances the ductility. VC 2013 AIP Publishing LLC.[http://dx.doi.org/10.1063/1.4829480]
Since the successful exfoliation of single-layer gra-
phene in 2004,1 two-dimensional materials have become
one of the research focuses in physics, chemistry, and mate-
rials science.2–5 Thereafter, many approaches have been
developed to fabricate graphene and graphene nanoribbons,
including mechanical exfoliation,6,7 reduction of graphite
oxide,8,9 chemical vapor deposition, decomposition of
SiC,10 and un-zipping of carbon nanotube (CNT).11,12
Generally, residual stress exists in the as-prepared samples
as well as those after substrate transfer,13–15 and it is closely
related to the localized ripples, edges, grain boundaries,
adatom adsorption, and Stone-Wales defects produced in
the synthesis and post thermal processes.16–18 The residual
stress might influence the structural stability of this ultra-
thin system in practice. In addition, graphene possesses me-
chanical strength of about 130 GPa,19 which is almost 100
times larger than that of steels, and so graphene is believed
to be an ideal reinforcing component in composite materi-
als.20,21 Graphene sheets are typically distributed randomly
in the composite host and when the materials are subjected
to an external load along different orientations, the gra-
phene sheets may be activated differently. Hence, it is of
scientific and technological interest to determine the de-
pendence of their deformation behavior on the loading
mode and loading orientation.
It is generally accepted that the competition between
bond rotation and bond rupture determines the deformation
mechanism of graphene. By conducting molecular dynam-
ics (MD) simulation, Grantab et al.22 found that graphenesheets with large-angle tilt boundaries having a high density
of defects were as strong as the pristine one and much
stronger than those with low-angle boundaries having fewer
defects. They ascribed the abnormal behavior to the ability
of the large-angle tilt boundaries to better accommodate the
strained rings via bond rotation.22 Bond-rotation-related
mechanical deformation has been observed from graphene
with Stone-Wales (SW) defects.23–25 Similar to plane slips
in bulk metals, bond rotation depends on the loading direc-
tion and can be described by a physical parameter resem-
bling the Schmidt factor.26 In bulk metals, the deformation
mechanism is determined by the loading modes, namely,
compression and stretching and hence, both the loading
direction and loading mode affect the mechanism of me-
chanical deformation. In this work, MD simulation is per-
formed to determine the effects of compressive/tensile
loading along the zigzag and armchair directions on the de-
formation mechanism.
MD simulation is carried out on the large-scale atomic/
molecular massively parallel simulator (LAMMPS).27 The
interaction between carbon atoms is described by the adaptive
intermolecular reactive bond order (AIREBO) potential,
which can accurately capture the interactions between carbon
atoms as well as bond breaking and re-forming.28,29 The cut-
off parameter describing the short-range C-C interactions is
chosen to be 2.0 Å in order to avoid spuriously high bonding
force and nonphysical results at large deformation.30 A lattice
constant of 1.426 Å is adopted as the initial value and the layer
separation of graphite of 3.4 Å is taken as the effective thick-
ness of the mono-layer graphene.31 A Poisson’s ratio of 0.165
is used.32 Prior to the simulation, the graphene sheets with
periodic conditions in the two in-plane directions are relaxed
to an equilibrium state in the isothermal-isobaric (NPT)
ensembles for 1 000 000 MD steps with a time step of 1 fs.
Graphene nanoribbons (GNRs) along the zigzag/armchair ori-
entations and with a size of 50 � 120 Šwith 2240/2279atoms are created by deleting atoms from the outside part of
the nanoribbons and a vacuum region 15 Å in width is added
a)Authors to whom correspondence should be addressed. Electronic addresses:
mafei@mail.xjtu.edu.cn; kwxu@mail.xjtu.edu.cn; and paul.chu@cityu.edu.hk
0003-6951/2013/103(19)/191906/5/$30.00 VC 2013 AIP Publishing LLC103, 191906-1
APPLIED PHYSICS LETTERS 103, 191906 (2013)
http://dx.doi.org/10.1063/1.4829480http://dx.doi.org/10.1063/1.4829480http://dx.doi.org/10.1063/1.4829480mailto:mafei@mail.xjtu.edu.cnmailto:kwxu@mail.xjtu.edu.cnmailto:paul.chu@cityu.edu.hkhttp://crossmark.crossref.org/dialog/?doi=10.1063/1.4829480&domain=pdf&date_stamp=2013-11-07
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perpendicular to the length direction and normal to graphene
sheets, so that the atoms near one edge of the GNRs do not
interact with those near the opposite edge because of the peri-
odic boundary conditions. Simulation is performed under
compressive/tensile loading along the width direction of the
GNRs in the NVT ensemble using the deformation-control
method. A strain increment of about 10�4 corresponding to a
strain rate of 109 s�1 is applied in every time step to the load-
ing and controlled unloading processes. Four combinations
including two orientations (armchair and zigzag) and two
loading modes (tensile and compressive) are considered, that
is, compressive loading along the zigzag direction (CZZ), ten-
sile loading along the zigzag direction (TZZ), compressive
loading along the armchair direction (CAC), and tensile load-
ing along the armchair direction (TAC). The temperature is
kept at 5 K using the Nose-Hoover thermostat. Only in-plane
deformation is allowed so that wrinkles can be avoided thus
enabling the study on deformation behavior of an ideal
two-dimensional system. The strain and stress values are cal-
culated according to Ref. 30. The evolution of atomic configu-
rations and statistical bond lengths are analyzed in different
stages by ATOMEYE.33
Fig. 1 shows the TZZ, TAC, CZZ, and CAC stress-
strain curves of the graphene nanoribbons. The positive and
negative strain values indicate tensile and compressive load-
ing, respectively. During TZZ and CAC, the absolute stress
increases with strain smoothly but abruptly becomes zero at
strain of 30% and 10%, corresponding to the destructive
structural changes in the GNRs. On the other hand, during
CZZ and TAC, three deformation stages are involved. In the
first stage denoted by I, the stress also increases smoothly
with strain, but stress oscillation corresponding to periodic
change in the structure is observed from the second stage
denoted by II (strain of about 12.5% and 20%). In the third
stage denoted by III, the stress increases sharply initially and
then suddenly becomes zero at strain of about 19% and 47%.
In comparison with TZZ and CAC, the ductility in CZZ and
TAC is enhanced by about 111% and 56.7%. The fracture
strain in TAC is the largest (about 47%) whereas that in
CAC is the smallest (10%). In the two compressive loading
processes of CZZ and CAC, the atomic separation is short-
ened and atomic interaction increases thereby producing the
positive second-order elastic constants (second-order deriva-
tive of stress with respect to strain) in stage I. However, in
the two tensile loading processes of TZZ and TAC, the grad-
ually reduced atomic interaction yields negative second-
order elastic constants. The stress terrace in the strain range
between 20% and 45% in TAC resembles elongation of the
yield point induced by Cottrell atmospheres around disloca-
tions and grain boundaries in low-carbon steels.34 It is thus
obvious that the deformation mechanism of graphene sheets
is determined by the combined effect of the loading mode
and orientation.
In order to understand the stress oscillation as well as
enhanced ductility during CZZ and TAC, we systemically
analyze the evolution of the atomic configuration and the
typical snapshots are depicted in Fig. 2. Fig. 3 displays the
FIG. 1. Stress-strain curves of graphene nanoribbons subjected to a strain
rate of 10�4 ps�1 along different orientations (armchair or zigzag) and underdifferent loading modes (tensile or compressive) at 5 K. CZZ, TZZ, CAC
and TAC represent compressive loading along the zigzag direction, tensile
loading along the zigzag direction, compressive loading along the armchair
direction, and tensile loading along the armchair direction, respectively, and
I, II and III denote the elastic, shearing, and hardening deformation stages,
respectively.
FIG. 2. Typical snapshots showing the
local atomic structures of graphene
nanoribbons during loading of differ-
ent modes: (a) TZZ at strain of 0% (I),
10% (II), 20% (III), and 30% (IV); (b)
CAC at strain of 0% (I), 5% (II), 8%
(III), and 9% (IV); (c) TAC at strain of
0% (I), 10% (II), 20% (III), and 47%
(IV); (d) CZZ at strain of 0% (I), 10%
(II), 12.5% (III), and 19% (IV).
191906-2 Sun et al. Appl. Phys. Lett. 103, 191906 (2013)
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statistical bond length in local regions with time. During
TZZ, the length of bond 1 and bond 2 increases with strain
but at a strain of 30%, the bond length decreases sharply to
that of the ideal crystal lattice [Fig. 3(a)] as a result of brittle
fracture as illustrated in Fig. 2(a). During CAC, the length of
the two types of bonds decreases with strain and the rate is
larger for bond 1 than bond 2 [Fig. 3(b)]. At about 10%
strain, the length of bond 1 is 1.2 Å and the system is sub-
jected to spuriously high bonding force yielding nonphysical
results as shown in Fig. 2(b). If out-of-plane movement is
allowed, wrinkles should occur in this case. A large lattice
change is also observed during TAC and CZZ. Lattice shear-
ing is activated from the edges of the GNRs along certain
directions at 20% and 12.5% strain and then extends inwards
gradually [Figs. 2(c) and 2(d)]. During lattice shearing, the
bond length in the sheared regions changes suddenly. For
instance, in TAC, the length of bond 1 diminishes from 1.7 Å
to 1.42 Å and it is just as long as the length of bond 2 in the
unsheared region. That of bond 2 increases dramatically
from 1.42 Å to 1.63 Å, which is close to the length of bond 1
in the unsheared region [Fig. 3(b)]. Similar abrupt changes
in the bond length are also observed from the sheared regions
during CZZ. The lattice change corresponds to phase trans-
formation from hexagonal to orthorhombic.26,35 Lattice
shearing proceeds gradually and an energy barrier is over-
come resembling plane slipping in metals. This leads to
stress oscillation in Stage II as shown in Fig. 1. Bond rupture
at local regions occurs after completion of the phase transfor-
mation in the whole system and fracture of GNRs occurs
finally. Our results indicate that the phase transformation
induced by lattice shearing is the origin of stress oscillation
and enhanced ductility during CZZ and TAC.
The orientation-dependent deformation behavior can be
understood by comparing to bulk metals. The Schmidt factor
is suggested to bridge the applied normal stress r and theresolved shear stress s along a given shearing direction in this
two-dimensional system. It can be expressed as sin u � cos u,in which u denotes the angle between the loading and shear-ing directions.26 Since there is the largest atomic line density
and the largest separation between atomic arrays along arm-
chair directions, they are believed to be the possible shearing
directions. In the TAC process, the Schmidt factor has a large
value of about 0.48 and the resolved shear stress exceeds the
critical stress (42.2 GPa (Ref. 26)) required by lattice shear-
ing. Lattice shearing occurs after elastic deformation and
before destructive fracture, as schematically shown in Fig.
4(a). However, in the TZZ process, the Schmidt factor has a
value of 0.44 and decreases with strain gradually. Hence, the
resolved shear stress along the closely packed direction can-
not activate the lattice shearing. Instead, brittle fracture takes
place immediately after elastic deformation, as schematically
shown in Fig. 4(b). Because of the orthorhombic relationship
between the zigzag and armchair directions, CZZ is equiva-
lent to TAC as a result of Poisson’s effect, while CAC is
equivalent to TZZ. Therefore, lattice-shearing-enhanced duc-
tility is found from both the TAC and CZZ processes, but not
in the CAC and TZZ processes as discussed above. Table I
summarizes the main results.
The reversibility of the phase transformation is illus-
trated during both CZZ and TAC when the lattice-sheared
GNRs are unloaded at a controlled strain rate of 109 s�1 from
a strain smaller than the fracture strain. Figs. 5(a) and 5(b)
depict the stress-strain curves of a loading-unloading cycle
during CZZ and TAC for maximum loading strain of 17%
and 30%, respectively. In the low strain range between 0%
and 9% in CZZ and 0% to 12% in TAC, the stress-strain
curves in the unloading process almost coincide with those
in loading process suggesting elastic deformation. However,
at large loading strain, lattice-shearing-induced phase trans-
formation from hexagonal to orthorhombic occurs and
Stone-Wales defects and vacancies are produced sometimes.
In both cases, an energy barrier should be overcome
FIG. 3. Evolution of bond length as a
function of strain during loading along
(a) zigzag and (b) armchair directions
with the positive and negative strain val-
ues indicating tensile and compressive
loading, respectively. When lattice
shearing appears locally, “Sheared”
denotes the bond length evolution in
these regions and “Un-Sheared” in other
regions.
FIG. 4. The schematic models depict-
ing the lattice evolution and the defor-
mation mechanisms in two typical
loading modes: (a) TAC and (b) TZZ.
191906-3 Sun et al. Appl. Phys. Lett. 103, 191906 (2013)
-
correspondingly and the mechanical energy from the outside
is stored as chemical energy in graphene. Upon unloading,
the stored chemical energy will be dissipated suddenly as
thermal energy but not mechanical energy. This leads to the
difference in the stress-strain curves in the loading and
unloading processes, and stress-strain hysteretic loops are
formed. The magnified loops in grey are shown in the insets
of panels (a) and (b) and the integrated areas reflect the
energy dissipation. The calculated energy dissipation is less
than 10% and quite small. It can be ascribed to the extraordi-
narily low energy barrier of about 50 meV for the inverse
phase transformation.35
Fig. 6 displays the typical snapshots of the atomic config-
uration during loading and unloading and pseudo-color is
employed to indicate the magnitude of the shear strain on
each atom. Lattice shearing occurs mainly along the armchair
directions which are believed to have the highest resolved
shear stress.26 During CZZ, the phase transformation induced
by lattice shearing is nearly completed at 17% strain and the
perfect hexagonal lattice is recovered upon controlled unload-
ing [Fig. 6(a)]. No defects are produced in the circle. If com-
pressive loading is applied again, the phase transformation
takes place. The repeatable behavior endows graphene with
pseudo-elasticity similar to that possessed by metal
nanowires.36–38 During TAC as shown in Fig. 6(b), the bond
length is significantly increased and bond fracture and rotation
in TAC become easier compared to CZZ. Therefore, some
SW defects are produced locally as shown in the inset of Fig.
6(b). The defects are maintained during controlled unloading
and prevent subsequent phase transformation under tensile
loading. Therefore, the pseudo-elasticity at large tensile strain
is degenerated, although the ductility along the armchair
direction is the best. At an elevated temperature, bond fracture
and rotation become easier and so is formation of defects.
Accordingly, the ductility of graphene at high temperature is
not good and it is contrary to the behavior of conventional
bulk metals. This is also the reason why lattice shearing is sel-
dom observed from graphene sheets. Since graphene sheets
are usually distributed randomly in the composite host,
different loading modes along various orientations may be
activated at the same time. The resulting ductility of graphene
sheets can be effectively utilized to improve the mechanical
strength of composites.
In summary, MD simulation is conducted to study the
mechanical behavior of two-dimensional graphene and the
deformation mechanism is determined by both the loading
orientation and loading mode. When the load is applied
along the zigzag orientation, tensile stress gives rise to brittle
fracture, whereas compressive stress results in lattice shear-
ing and phase transformation from hexagonal to orthorhom-
bic. However, when the load is applied along the armchair
direction, tensile stress leads to lattice shearing and phase
transformation, but compressive stress yields large bonding
force. The phase transformation induced by lattice shearing
is reversible even at large strain of 17% and 30% during
CZZ and TAC and the energy dissipation is less than 10%.
As a result, pseudo-elasticity exists along some orientations
under certain loading modes and the ductility is enhanced.
Graphene sheets can be utilized as strengthening materials in
composites to accomplish high elasticity, large strength, and
low energy dissipation.
FIG. 6. Evolution of atomic structures during loading and unloading: (a)
CZZ and (b) TAC. The inset in (b) illustrates nucleation and development of
SW defects.
TABLE I. Deformation mechanisms of GNRs in four loading modes.
Zigzag Armchair
Tensile Brittle fracture Shearing enhanced ductile
Compressive Shearing enhanced ductile Non-physical result
FIG. 5. Loading-unloading stress-strain
curves of graphene nanoribbons at 5 K:
(a) CZZ and (b) TAC. The dissipation
energy can be evaluated from the mag-
nified grey area in the inset.
191906-4 Sun et al. Appl. Phys. Lett. 103, 191906 (2013)
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This work was jointly supported by Key Project of
Chinese National Programs for Fundamental Research and
Development (Grant No. 2010CB631002), National Natural
Science Foundation of China (Grant Nos. 51271139,
51171145, and 51302162), New Century Excellent Talents
in University (NCET-10-0679), Fundamental Research
Funds for the Central Universities, and Hong Kong Research
Grants Council (RGC) General Research Funds (GRF) No.
CityU 112212, and City University of Hong Kong Applied
Research Grants (ARG) 9667038 and 9667066.
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