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Graphene: Continuum Mechanics and Atomistic Modeling
Rui Huang
Center for Mechanics of Solids, Structures and Materials
Department of Aerospace Engineering and Engineering Mechanics
The University of Texas at Austin
Acknowledgments
� Qiang Lu (postdoc)
� Wei Gao (graduate student)
� Zachery Aitken (undergraduate student, now at Caltech)
� Prof. Marino Arroyo (Spain)
� Prof. Li Shi (UT/ME)
� Funding: NSF (ARRA)
From Wikipedia:
Graphene is a one-atom-thick planar sheet of sp2-bonded carbon
atoms that are densely packed in a honeycomb crystal lattice. The term
Graphene was coined as a combination of graphite and the suffix -ene
by Hanns-Peter Boehm, who described single-layer carbon foils in
1962. Graphene is most easily visualized as an atomic-scale chicken
wire made of carbon atoms and their bonds.
Graphene Based Devices
Garcia-Sanchez, et al., 2008.
Suspended nanoribbons for NEMS devicesPatterned graphene on oxide for bipolar
p-n-p junctions
Ozyilmaz et al., 2007.
Mechanical properties of monolayer graphene
• High in-plane stiffness
• High tensile strength
• High bending stiffness (relative to its own weight)
� Isotropic, linearly elastic under infinitesimal deformation
� Anisotropic, nonlinear under finite deformation
� Graphene nanoribbons: edge effects? size dependent?
� Substrate-supported graphene: adhesive interactions?
Nonlinear Continuum Mechanics of 2D Sheets
2D-to-3D deformation gradient:
In-plane deformation: 2D Green-Lagrange strain tensor
Bending: 2D curvature tensor (strain gradient)
d d=x F X
X1
X2
x1
x3x2
J
iiJ
X
xF
∂
∂=
( )JKiKiJJK FFE δ−=2
1
2
iI iIJ i i
J I J
F xn n
X X X
∂ ∂Κ = =
∂ ∂ ∂
Strain energy (hyperelasticity): ( )∫ Φ=A
dAU ΚE,
Lu and Huang, Int. J. Applied Mechanics 1, 443-467 (2009).
2D Stresses and Moments
2nd Piola-Kirchoff stress and moment
(work conjugates)
Tangent moduli:
IJ
IJE
S∂
Φ∂=
IJ
IJMΚ∂
Φ∂=
KLIJKL
IJIJKL
EEE
SC
∂∂
Φ∂=
∂
∂=
2
KLIJKL
IJIJKL
MD
Κ∂Κ∂
Φ∂=
Κ∂
∂=
2
KLIJIJ
KL
KL
IJIJKL
EE
MS
Κ∂∂
Φ∂=
∂
∂=
Κ∂
∂=Λ
2Intrinsic coupling between
tension and bending
Κ
Κ
Κ
ΛΛΛ
ΛΛΛ
ΛΛΛ
+
=
12
22
11
333231
232221
131211
12
22
11
333231
232221
131211
12
22
11
22 d
d
d
dE
dE
dE
CCC
CCC
CCC
dS
dS
dS
ΛΛΛ
ΛΛΛ
ΛΛΛ
+
Κ
Κ
Κ
=
12
22
11
332313
322212
312111
12
22
11
333231
232221
131211
12
22
11
22 dE
dE
dE
d
d
d
DDD
DDD
DDD
dM
dM
dM
An incremental form of
the generally nonlinear
and anisotropic behavior:
Lu and Huang, Int. J. Applied Mechanics 1, 443-467 (2009).
Units for 2D quantities
( )ΚE,Φ strain energy density function: J/m2
IJ
IJE
S∂
Φ∂= 2D stress: N/m
KLIJKL
IJIJKL
EEE
SC
∂∂
Φ∂=
∂
∂=
2
2D in-plane modulus: N/m
IJ
IJMΚ∂
Φ∂= moment intensity: (N-m)/m
KLIJKL
IJIJKL
MD
Κ∂Κ∂
Φ∂=
Κ∂
∂=
2
bending modulus: N-m
KLIJIJ
KL
KL
IJIJKL
EE
MS
Κ∂∂
Φ∂=
∂
∂=
Κ∂
∂=Λ
2
coupling modulus: N
�Analogous to the 2D plate/shell theories (but no thickness).
(n,n): armchair
(n,0): zigzag
(2n,n)
αααα
Atomistic Modeling of Graphene
• 2nd-generation REBO potential (Brenner et al., 2002)
– Bond angle effect (second-nearest neighbors)
– Dihedral angle effect (third-nearest neighbors)
– Radical energetics (defects and edges)
• Molecular Mechanics: energy minimization for static equilibrium states.
• Stress and moment calculations
– Energy derivation
– Virial stress calculations
– Direct force evaluation
0 0.05 0.1 0.15 0.2 0.25 0.3−7.5
−7
−6.5
−6
Nominal strain ε
En
erg
y (
eV
)
A
B
C
D
Uniaxial Stretch of Monolayer GrapheneE
ner
gy p
er a
tom
(eV
)
Lu and Huang, Int. J. Applied Mechanics 1, 443-467 (2009).
0 0.05 0.1 0.15 0.2 0.25 0.30
5
10
15
20
25
30
35
40
Nominal strain ε
No
min
al str
ess P
11 (
N/m
)
α = 0 (zigzag)
α = 10.89o
α = 19.11o
α = 30o (armchair)
A
C D
B
εε
d
dP
Φ=)(11
Nominal stress-strain
relation:
Anisotropic tangent moduli
0 0.05 0.1 0.15 0.2 0.25 0.3−30
−20
−10
0
10
20
30
Green−Lagrange strain E11
C3
1 (
N/m
)
(n,n): armchair
(n,0): zigzag
(2n,n)
αααα
Graphene is linear and isotropic under
infinitesimal deformation, but becomes nonlinear
and anisotropic under finite deformation.
Coupling between
tension and shear
0 0.05 0.1 0.15 0.2 0.25 0.3−50
0
50
100
150
200
250
300
350
C11 (
N/m
)
α = 0 (zigzag)
α = 10.89o
α = 19.11o
α = 30o (armchair)
0 0.05 0.1 0.15 0.2 0.25 0.3−50
0
50
100
150
C21 (
N/m
)
=
12
22
11
333231
232221
131211
12
22
11
2dE
dE
dE
CCC
CCC
CCC
dS
dS
dS
Lu and Huang, Int. J. Applied Mechanics 1, 443-467 (2009).
Fracture strength under uniaxial stretch
0 0.05 0.1 0.15 0.2 0.25 0.30
5
10
15
20
25
30
35
40
Nominal strain ε
Nom
inal str
ess P
11 (
N/m
)
α = 0 (zigzag)
α = 10.89o
α = 19.11o
α = 30o (armchair)
Nom
inal fr
actu
re s
train
0 5 10 15 20 25 300.1
0.2
0.3
0.4
0 5 10 15 20 25 3030
32
34
36
Chiral angle (degree)
Nom
inal fr
actu
re s
tress (
N/m
)
02
2
=∂
Φ∂=
∂
∂
εε
PFracture occurs as a result of
intrinsic instability of the
homogeneous deformation:
� The nominal stress and strain to fracture depend on the direction
of uniaxial stretch.Lu and Huang, Int. J. Applied Mechanics 1, 443-467 (2009).
Monolayer graphene under uniaxial tension
0 0.05 0.1 0.15 0.2 0.25 0.30
10
20
30
40
Nominal strain
Nom
inal 2D
str
ess (
N/m
)
zigzag (MM/REBO)
armchair (MM/REBO)
zigzag (Wei et al., 2009)
armchair (Wei et al., 2009)
0 0.05 0.1 0.15 0.2 0.25 0.30
100
200
300
400
Nominal strain
2D
Young's
modulu
s (
N/m
)
zigzag (MM/REBO)
armchair (MM/REBO)
zigzag (Wei et al., 2009)
armchair (Wei et al., 2009)
• Disagreement: the REBO potential underestimates the initial
Young’s modulus.
• Agreement: the fracture stress/strain is higher in the zigzag
direction than in the armchair direction.
Bending Modulus of Monolayer Graphene
0 0.5 1 1.5 2 2.5 30
0.05
0.1
0.15
Bending curvature (1/nm)
Str
ain
en
erg
y p
er
ato
m (
eV
/ato
m)
B1, armchair
B1, zigzag
B2, armchair
B2, zigzag
B2*, armchair
B2*, zigzag
0 0.5 1 1.5 2 2.50
0.05
0.1
0.15
0.2
0.25
Bending curvature (1/nm)
Bendin
g m
odulu
s (
nN
−nm
)
B1, armchair
B1, zigzag
B2, armchair
B2, zigzag
B2*, armchair
B2*, zigzag
• Bending moment-curvature is nearly linear, with slight
anisotropy.
• Including the dihedral angle effect leads to higher
bending energy and bending modulus.
Lu, Arroyo, and Huang, J. Phys. D: Appl. Phys. 42, 102002 (2009).
0 0.5 1 1.5 2 2.50
0.05
0.1
0.15
0.2
0.25
Bending curvature (1/nm)
Bendin
g m
odulu
s (
nN
−nm
)
B1, armchair
B1, zigzag
B2, armchair
B2, zigzag
B2*, armchair
B2*, zigzag
Effect of dihedral angle
i
j
k
l
θθθθijk
θθθθjil
θθθθq4
θθθθq2θθθθq1
θθθθq3
ΘΘΘΘq3
ΘΘΘΘq4ΘΘΘΘq1
ΘΘΘΘq2
Lu, Arroyo, and Huang, J. Phys. D: Appl. Phys. 42, 102002 (2009).
)()( ijAijijRij rVbrVV −=
RC
ij
DH
ijjiijij bbbb Π+++= −− )(21 πσπσ
( )[ ]∑≠
Θ−=),(,
20 )()(cos12 jilk
jlcikcijkl
DH
ij rfrfT
b ijljikijkl nn ⋅=Θcos
Physical Origin of Bending Modulus
Bending modulus of a thin elastic plate:
For monolayer graphene, bending moment and
bending stiffness result from multibody interatomic
interactions (second and third nearest neighbors).
• D = 0.83 eV (0.133 nN-nm) by REBO-1
• D = 1.4 eV (0.225 nN-nm) by REBO-2
• D = 1.5 eV (0.238 nN-nm) by first principle
0 0( ) 14
2 3
ijA
ijk
bV r TD
σ π
θ
− ∂= − ∂
3~ Ehd
dMD
κ= h
Lu, Arroyo, and Huang, J. Phys. D: Appl. Phys. 42, 102002 (2009).
Coupling between bending and stretching
0 0.005 0.01 0.015 0.020.113
0.114
0.115
0.116
0.117
0.118
Strain ε = R/R0 − 1
Str
ain
energ
y p
er
ato
m (
eV
/ato
m)
R
Lu, Arroyo, and Huang, J. Phys. D: Appl. Phys. 42, 102002 (2009).
( ) ( ) ( ) 2200
20
2
1
2
1
2
1, εκκσεκκκεκ CDMDW +−++−+≈
� The tube radius increases upon
relaxation, leading to simultaneous
bending and stretching.
nm 397.00 =R
AIREBO vs REBO
AIREBO: Adaptive intermolecular REBO potential (Stuart et al.
2000); implemented in LAMMPS.
• Add a LJ term for long-range weak interactions
• Add a torsional term to improve the dihedral angle effect of
single bond (C-C and C-H)
• Modified parameters in the spline function of REBO
Effects on mechanical properties of graphene:
• The equilibrium bond length is slightly shorter (1.397 Ǻ vs
1.420 Ǻ)
• Increase the in-plane stiffness (good)
• Screw up the bending modulus (bad)
Excess Edge Energy and Edge Force
1.391
1.425
1.420
1.420
1.367
1.412
1.425
1.398
1.425
1.4201.421
1.420
Edge energy (eV/nm ) Edge force (eV/nm) r0
Armchair Zigzag Armchair Zigzag (nm)
DFT [17]
(GPAW)9.8 13.2 - - 0.142
DFT [18]
(VASP)10 12 -14.5 -5 0.142
DFT [22]
(SIESTA)12.43 15.33 -26.40 -22.48 0.142
MM [20]
(AIREBO)- - -10.5 -20.5 0.140
MD [21] - - -20.4 -16.4 0.146
MM
(REBO)10.91 10.41 -8.53 -16.22 0.142
Zigzag edge:
Armchair edge:
fZfZ
fAfA
0 0.2 0.4 0.6 0.8 1−7.4
−7.3
−7.2
−7.1
−7
−6.9
−6.8
1/W (nm−1
)
Energ
y p
er
ato
m (
eV
)
armchair, un−relaxed
armchair, 1−D relaxation
zigzag, un−relaxed
zigzag, 1−D relaxation
Eq. (6), γ = 11.1 eV/nm
Eq. (6), γ = 10.6 eV/nm
Lu and Huang, Phys. Rev. B 81, 155410 (2010).
γγ
W
SU
NUWU 0
00
2)( +=+=
Edge buckling of GNRs
2 4 6 8 10 1210.27
10.275
10.28
10.285
10.29
Buckle wavelength λ (nm)
Excess e
nerg
y (
eV
/nm
)
y = 2e−05*x4 − 0.00071*x
3 + 0.0093*x
2 − 0.054*x + 10
2 4 6 8 10 12 14 1610.87
10.875
10.88
10.885
10.89
Buckle wavelength λ (nm)
Excess e
nerg
y (
eV
/nm
)
y = 2.2e−06*x4 − 0.00011*x
3 + 0.0019*x
2 − 0.014*x + 11
Zigzag GNR
Armchair GNR
Intrinsic wavelength ~ 6.2 nm
Intrinsic wavelength ~ 8.0 nm
�The wavelengths for edge buckling do not scale with D/f.
Lu and Huang, Phys. Rev. B 81, 155410 (2010).
Zigzag GNRs Armchair GNRs
GNRs under Uniaxial Tension
FF
( ) ( ) ( )LWLU εγεε 2+Φ=
δεδ FLU =
ε
γ
εεσ
d
d
Wd
d
d
dU
WLW
F 21+
Φ===
GNRs under Uniaxial Tension
( )ε
γ
εεσ
d
d
Wd
d
W
F 2+
Φ==
FF
Zigzag GNRs Armchair GNRs
Lu and Huang, arXiv:1007.3298 (2010).
2D Young’s Moduli of GNRs
( )ε
γ
εεσ
d
d
Wd
d 2+
Φ=
( )2
2
2
2 2
ε
γ
εε
d
d
Wd
dE +
Φ=
Young’s modulus under
infinitesimal strain:
edgebulkE
WEE 000
2+=
The nonlinear dependence of the excess edge energy on strain leads
to anisotropic, width-dependent Young’s modulus for GNRs.
Lu and Huang, arXiv:1007.3298 (2010).
Fracture of graphene nanoribbons
Zigzag edge
Armchair edge
Lu and Huang, arXiv:1007.3298 (2010).
Graphene on Oxide Substrates
Ozyilmaz et al., 2007.
HR-STM image (Stolyarova et al., 2007)
The 3D morphology is important for the transport properties of graphene-based devices.
Ishigami et al., 2007.
RMS2
Correlation
length
Van der Waals Interaction
Lennard-Jones potential for
particle-particle interactions:
Monolayer-substrate interaction
(energy per unit area):
−
Γ−=
9
0
3
00
2
1
2
3)(
h
h
h
hhU
hr h
U
0h0
-Γ0
Aitken and Huang, J. Appl. Phys. 107, 123531 (2010).
1 2
6 12( )
LJ
C CW r
r r= − +
Van der Waals Thickness and Energy
Gupta et al., 2006.
� Interlayer spacing in graphite ~ 0.34 nm;
� AFM measurements of h0 for graphene on oxide range from 0.4 to 0.9 nm;
� The adhesion energy (Γ0) has not been measured directly;
� Theoretically estimated values for Γ0 range from 0.6 to 0.8 eV/nm2.
Sonde et al., 2009.
Flat Graphene on Flat Surface
( )3 9
0 00
3 1
2 2vdW
h hU z
z z
= −Γ −
max 230 MPaσ =
0 0.6h nm=
Representative values:
( )
−
Γ=
10
0
4
0
0
0
2
9
z
h
z
h
hzvdWσ
maxσ
z
Interfacial
strength:0
0max 466.1
h
Γ=σ
Initial
stiffness: 2
0
00
27
hk
Γ=
2eV/nm 6.00 =Γ
MPa/nm 72000 =k
Aitken and Huang, J. Appl. Phys. 107, 123531 (2010).
Strain-Induced Instability
� The competition between the elastic strain energy of graphene and the van der Waals
interaction energy sets a critical strain for instability as well as the equilibrium
corrugation wavelength beyond the critical strain.
44344214342143421PlaneIn
gg
BendingPlaneIn
g
CDCU
−−
+
+
≈ 4
4
2
422
64
32
4
2
4
~δ
λ
πδ
λ
π
λ
πε
+
+−Γ≈
4
0
2
0
08
675
4
271
~
hhU
gg
vdW
δδ
Elastic strain energy
of graphene:
Van der Waals
Interaction Energy:
h0
substrate
z
x
ελ
+=
λ
πδ
xhz gg
2sin0
vdWg UUU~~~
+=Total free energy:
Aitken and Huang, J. Appl. Phys. 107, 123531 (2010).
Strain-Induced Corrugation of Graphene
41
0
2
0
272
Γ=
Dhc πλ
0
036
Ch
Dc
Γ−=ε
0074.0−=cε
nmc 68.2=λ
Representative values:
N/m353=C eV5.1=D
0 0.6h nm=2
eV/nm 6.00 =Γ
� On a flat surface, a graphene monolayer is flat and stable below a critical
compressive strain;
� Beyond the critical strain, the monolayer corrugates with increasing amplitude
and decreasing wavelength.
Aitken and Huang, J. Appl. Phys. 107, 123531 (2010).
Graphene on a Corrugated Surface
( ) ( ) ( ) ( )2
0
2
2
0
2
0
1,~~
hhU
hhhUhUhU
sgsg
vdWgvdW
δδδδδ +
+
+≈
( ) ( )hkh
h
h
h
hhU vdW
42
5
2
9 2
0
11
0
5
001 =
+
−
Γ=
( )
−
Γ=
λ
π
λ
π
λ
π
λπ
hK
h
hhK
h
hhU
2
24
29 656
11
0
3
323
5
00
3
2
substrate
z
x
λ
h
h
s
and find to
energy total the minimize
and Given
gδ
λδ ,
=
λ
πδ
xz ss
2sin
+=
λ
πδ
xhz gg
2sin
van der Waals
interaction energy:
vdWg UUU~~~~~
+=
Total free energy:
Aitken and Huang, J. Appl. Phys. 107, 123531 (2010).
Substrate Induced Corrugation of Graphene
ΓΓ=
0
2
0000
,;,εδλ
δ
δ C
h
D
hhf s
s
g
� Conformal at long wavelengths.
� Non-conformal at short wavelengths.
� Transition between the two states depends on the amplitude of substrate
surface corrugation, becoming more abrupt with increasing amplitude.
ΓΓ=
0
2
00000
,;,εδλ C
h
D
hhg
h
h s
Aitken and Huang, J. Appl. Phys. 107, 123531 (2010).
Effect on Adhesion Energy
� The adhesion energy decreases as the corrugation wavelength decreases,
approaching a plateau at the short‐wavelength limit.
� The adhesion energy decreases with increasing amplitude of the substrate
surface corrugation.
� Better adhesion at the conformal state than the non-conformal state.
Aitken and Huang, J. Appl. Phys. 107, 123531 (2010).
Effects of Mismatch Strain
� A tensile strain tends to flatten the supported graphene, while a
compressive strain tends to increase the corrugation amplitude.
� A “snap-through” instability occurs at a critical tensile strain.
� A “buckling” instability occurs at a critical compressive strain.
Aitken and Huang, J. Appl. Phys. 107, 123531 (2010).
Critical Strain for Buckling Instability
• With increasing amplitude of the substrate surface corrugation:
– The critical compressive strain for buckling decreases.
– The corresponding buckling wavelength increases.
Aitken and Huang, J. Appl. Phys. 107, 123531 (2010).
Summary
� Nonlinear continuum mechanics for 2D graphene monolayer
� Atomistic modeling of graphene under bending and stretching
� Excess edge energy, edge forces, and induced edge buckling
� Graphene nanoribbons under uniaxial tension: edge effects on elastic modulus and fracture
� Graphene on substrates: van der Waals interaction and corrugation; strain-induced corrugation
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