Nonlinear Mechanics of Monolayer Graphene
Rui Huang
Center for Mechanics of Solids, Structures and MaterialsDepartment of Aerospace Engineering and Engineering Mechanics
The University of Texas at Austin
Acknowledgments
Qiang Lu (postdoc)
Zachery Aitken (undergraduate student)
Prof. Marino Arroyo (Spain)y ( p )
Prof. Li Shi (UT/ME)
Funding: DoE and NSF (ARRA)
What is special about graphene?• So much about carbon nanotubes, let’s play with graphene now.☺• Can two-dimensional crystals exist in our 3D space? Rippling or not?• Uniqueness of 2D: electron transport, Quantum Hall effect, etc.q p , Q ,• Any interesting mechanics?
Novoselev et al. 2005
Graphene Based DevicesP tt d h id f bi l
Suspended nanoribbons for NEMS devicesPatterned graphene on oxide for bipolar p-n-p junctions
Garcia-Sanchez, et al., 2008.Ozyilmaz et al., 2007.
Mechanical properties of monolayer graphene
• Young’s modulus?• Ultimate tensile strength?• Bending modulus?
What is the thickness of a graphene monolayer?Interlayer spacing in graphite: 0.335 nmC-C covalent bond length: 0.142 nmgMechanically reduced thickness: 0.05 – 0.09 nm
We consider monolayer graphene as a 2D membrane with no thickness.
Nonlinear Continuum Mechanics of 2D Sheets
2D-to-3D deformation gradient:
d d=x F XX2 xixF ∂=
In-plane deformation: 2D Green-Lagrange strain tensor
X1 x1
x3 x2JiJ X
F∂
=
( )JKiKiJJK FFE δ−=21
Bending: 2D curvature tensor (strain gradient)2F x∂ ∂
No need to define any thickness!iI i
IJ i iJ I J
F xn nX X X∂ ∂
Κ = =∂ ∂ ∂
( )∫
any thickness!
Strain energy (hyperelasticity): ( )∫ Φ=A
dAU ΚE,
Lu and Huang, Int. J. Applied Mechanics 1, 443-467 (2009).
Stresses and Moments in 2D
2nd Piola Kirchoff stress and moment2nd Piola-Kirchoff stress and moment (work conjugates)
IJIJ E
S∂Φ∂
=IJ
IJMΚ∂Φ∂
=
Tangent moduli: KLIJKL
IJIJKL EEE
SC∂∂Φ∂
=∂∂
=2
KLIJKL
IJIJKL
MDΚ∂Κ∂Φ∂
=Κ∂∂
=2
KLIJIJ
KL
KL
IJIJKL EE
MSΚ∂∂Φ∂
=∂∂
=Κ∂∂
=Λ2 Intrinsic coupling between
tension and bending
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
ΚΚΚ
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
ΛΛΛΛΛΛΛΛΛ
+⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
12
22
11
333231
232221
131211
12
22
11
333231
232221
131211
12
22
11
22 ddd
dEdEdE
CCCCCCCCC
dSdSdSAn incremental form of
the generally nonlinear and anisotropic behavior: ⎠⎝⎦⎣⎠⎝⎦⎣⎠⎝ 123332311233323112
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
ΛΛΛΛΛΛΛΛΛ
+⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
ΚΚΚ
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
22
11
322212
312111
22
11
232221
131211
22
11
22 dEdEdE
ddd
DDDDDDDDD
dMdMdM
and anisotropic behavior:
⎟⎠
⎜⎝⎥⎦⎢⎣ ΛΛΛ⎟
⎠⎜⎝ Κ⎥⎦⎢⎣
⎟⎠
⎜⎝ 123323131233323112 22 dEdDDDdM
Lu and Huang, Int. J. Applied Mechanics 1, 443-467 (2009).
Units for 2D quantities
( )ΚE,Φ strain energy density function: J/m2
Φ∂ 2D stress: N/mIJ
IJ ES
∂Φ∂
= 2D stress: N/m
IJSC Φ∂=
∂=
2
2D in-plane modulus: N/mKLIJKL
IJKL EEEC
∂∂=
∂= 2D in-plane modulus: N/m
IJM Φ∂= moment intensity: (N m)/m
IJIJM
Κ∂ moment intensity: (N-m)/m
KLIJKL
IJIJKL
MDΚ∂Κ∂Φ∂
=Κ∂∂
=2
bending modulus: N-m KLIJKL Κ∂Κ∂Κ∂ g
KLIJIJKL EE
MSΚ∂∂Φ∂
=∂∂
=Κ∂∂
=Λ2
coupling modulus: N KLIJIJKL EE Κ∂∂∂Κ∂ p g
Analogous to the 2D plate/shell theories.
(n,n): armchair
Atomistic Modeling of Graphene• Rectangular unit cell in an arbitrary ( , ): a c a
direction, subject to in-plane stretching and bending.
(n,0): zigzag
(2n,n)
α• 2nd-generation REBO potential (Brenner
et al., 2002)Bond angle effect (second nearest neighbors)– Bond angle effect (second-nearest neighbors)
– Dihedral angle effect (third-nearest neighbors)– Radical energetics (defects and edges)
• Energy minimization (molecular mechanics) for equilibrium states. ) q
• Stress and moment calculationsE d i ti– Energy derivation
– Virial stress calculations– Direct force evaluation
Pure Bending of Graphene Roll up a rectangular graphene sheet into a
⎟⎟⎞
⎜⎜⎛
⎟⎠⎞
⎜⎝⎛ 02cos2 1XR ππ
n1
=κ
p g g pcylindrical tube without in-plane stretch (E11 = 0)
⎟⎟⎟⎟
⎠⎜⎜⎜⎜
⎝⎟⎠⎞
⎜⎝⎛
⎟⎠
⎜⎝
=
02sin210
1
LX
LR
LL
ππF
Rn
⎥⎥⎦
⎤
⎢⎢⎣
⎡−⎟
⎠⎞
⎜⎝⎛= 12
21 2
11 LRE π
0.15
om)
B1, armchair
0.6
nN)
B1, armchair
⎠⎝ ⎠⎝ LL ⎥⎦⎢⎣ ⎠⎝2 L
0.1
er a
tom
(eV
/ato
m B1, armchair
B1, zigzag
B2, armchair
B2, zigzag
B2*, armchair
B2*, zigzag 0.3
0.4
0.5
nt p
er le
ngth
(nN
B1, armchair
B1, zigzag
B2, armchair
B2, zigzag
B2*, armchair
B2*, zigzag
0.05
trai
n en
ergy
per B2*, zigzag
0.1
0.2
0.3
endi
ng m
omen
t B2*, zigzag
0 0.5 1 1.5 2 2.5 30
Bending curvature (1/nm)
Str
0 0.5 1 1.5 2 2.50
0.1
Bending curvature (1/nm)
Ben
Bending Modulus of Graphene
0.2
0.25
−nm
)
ik
Θq3Θ 2
0.1
0.15
mod
ulus
(nN
−
B1, armchair
j
θijk
θjil
Θq4Θq1
Θq2
0.05
0.1
Ben
ding
m B1, armchair
B1, zigzag
B2, armchair
B2, zigzag
B2*, armchair
B2*, zigzag
lθq4
θq2θq1
θq3
0 0.5 1 1.5 2 2.50
Bending curvature (1/nm)
B2*, zigzag
• Bending moment-curvature is nearly linear, with slight anisotropy. • Including the dihedral effect leads to higher bending energy and bending
modulus.
Lu, Arroyo, and Huang, J. Phys. D: Appl. Phys. 42, 102002 (2009).
Physical Origin of Bending ModulusBending modulus of a thin elastic plate:Bending modulus of a thin elastic plate:
3~ YhddMDκ
= h
For monolayer graphene, bending moment and
dκ
bending stiffness result from multibody interatomic interactions (second and third nearest neighbors).
0 0( ) 142 3
ijA
ijk
bV r TDσ π
θ
−⎛ ⎞∂= −⎜ ⎟⎜ ⎟∂⎝ ⎠
• D = 0.83 eV (0.133 nN-nm) by REBO-1• D = 1.4 eV (0.225 nN-nm) by REBO-2
j⎝ ⎠
( ) y• D = 1.5 eV (0.238 nN-nm) by first principle
Lu, Arroyo, and Huang, J. Phys. D: Appl. Phys. 42, 102002 (2009).
Coupling between bending and stretching
0.117
0.118
(eV
/ato
m)
( ) ( ) ( ) 2200
20 2
121
21, εκκσεκκκεκ CDMDW +−++−+≈
0.115
0.116
rgy
per
atom
(eV
nm 397.00 =R
0.114
0.115
Str
ain
ener
gy
0 0.005 0.01 0.015 0.020.113
Strain ε = R/R0 − 1
RThe tube radius increases upon relaxation, leading to simultaneous bending and stretching.
Lu, Arroyo, and Huang, J. Phys. D: Appl. Phys. 42, 102002 (2009).
be d g d s e c g.
−6
Uniaxial Stretch of Monolayer GrapheneU i i l t t h
−6.5
)
C
0LL
=λ
Uniaxial stretch:
m (e
V)
−7
−6.5
Ene
rgy
(eV
)
BNominal strain:
0
1λ per a
tom
−7
A
D
Green-Lagrange strain:
1−= λε
Ener
gy
0 0.05 0.1 0.15 0.2 0.25 0.3−7.5
Nominal strain ε
0EE
( )121 2
11 −= λEC D
B01222 == EEA
B
Lu and Huang, Int. J. Applied Mechanics 1, 443-467 (2009).
Biaxial Stresses under Uniaxial Stretch (Poisson’s effect)
25
30
35
40
(N/m
)
P11
20
25
30
s (N
/m)
S11
10
15
20
25
Nom
inal
str
ess
(N
P22
5
10
15
Mem
bran
e st
ress
(N
S22
0 0.05 0.1 0.15 0.2 0.25 0.3−5
0
5
Nominal strain ε
N
P22
P12
and P21
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35−5
0
Green−Lagrange strain E11
M
S12
Nominal strain ε Green−Lagrange strain E11
1111 dE
dS Φ=Energy derivation:
εddP Φ
=1111
Virial stress calculation for nominal stress: ( )∑≠
−=nm
mni
mJ
nJiJ FXX
AP )()()(
21
εd
212112
12222211
11 ,1
,,1
PSPSPSPS =+
==+
=εε
Relationship between nominal stresses and 2nd P-K stress:
Anisotropic tangent moduli250
300
350
α = 0 (zigzag)
α = 10.89o
o
0
50
100
150
200
C11
(N
/m) α = 19.11o
α = 30o (armchair)
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛=
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
22
11
232221
131211
22
11
2dEdEdE
CCCCCCCCC
dSdSdS
(n,n): armchair
0 0.05 0.1 0.15 0.2 0.25 0.3−50
0
100
150
⎠⎝⎠⎝⎠⎝ 1233323112 2dECCCdS
0
50
100
C21
(N
/m)
20
30
(n,0): zigzag
(2n,n)
α 0 0.05 0.1 0.15 0.2 0.25 0.3−50
0
−10
0
10
20
C31
(N
/m)
Graphene is linear and isotropic under infinitesimal deformation but becomes nonlinear Coupling between
0 0.05 0.1 0.15 0.2 0.25 0.3−30
−20
Green−Lagrange strain E11
infinitesimal deformation, but becomes nonlinear and anisotropic under finite deformation.
Coupling between tension and shear
Lu and Huang, Int. J. Applied Mechanics 1, 443-467 (2009).
Fracture strength under uniaxial stretch
25
30
35
40
1 (N
/m)
str
ain
0.3
0.4
34
36
ress
(N
/m)
10
15
20
25
Nom
inal
str
ess
P11
(
α = 0 (zigzag)
α = 10.89o
Nom
inal
frac
ture
st
0.2
0.3
32
34
min
al fr
actu
re s
tres
0 0.05 0.1 0.15 0.2 0.25 0.30
5
10
Nominal strain ε
No
α = 10.89
α = 19.11o
α = 30o (armchair)
No
0 5 10 15 20 25 300.1
0 5 10 15 20 25 3030
Chiral angle (degree)
Nom
Nominal strain ε Chiral angle (degree)
2Φ∂∂PFracture occurs as a result of
02 ≤∂Φ∂
=∂∂
εεPintrinsic instability of the
homogeneous deformation:
The nominal stress and strain to fracture depend on the direction of uniaxial stretch.
Graphene Nanoribbons: Edge Effects
Garcia-Sanchez, et al. (2008).
• Various edge states: zigzag/armchair/mixed H-terminated reconstructed
Koskinen, et al. (2008)
Various edge states: zigzag/armchair/mixed, H terminated, reconstructed…• Edge effect leads to size-dependent electronic properties of GNRs.• Are mechanical properties of GNRs size dependent?
Excess Edge Energy and Edge ForceZigzag edge: −6.8
armchair, un−relaxed
1.391
1.425
g ag edge
fZfZ
−7.1
−7
−6.9
er a
tom
(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
1.420
1.420−7.3
−7.2
−7.1
Ene
rgy
per
Edge energy (eV/nm ) Edge force (eV/nm) r0
Armchair edge:
0 0.2 0.4 0.6 0.8 1−7.4
1/W (nm−1)
1.367
1.412
1.425
1.398
1.425
Edge energy (eV/nm ) Edge force (eV/nm) 0
Armchair Zigzag Armchair Zigzag (nm)DFT [17] (GPAW) 9.8 13.2 - - 0.142
DFT [18] 10 12 14 5 5 0 142
fAfA
1.4201.421
1.420
[ ](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(AIREBO) 10.5 20.5 0.140
MD [21] - - -20.4 -16.4 0.146MM
(REBO) 10.91 10.41 -8.53 -16.22 0.142Lu and Huang, arXiv:0910.0912.
Edge buckling of GNRs10.29
10.28
10.285
10.29
ener
gy (
eV/n
m)
y = 2e−05*x4 − 0.00071*x3 + 0.0093*x2 − 0.054*x + 10Zigzag GNR
2 4 6 8 10 1210.27
10.275
Buckle wavelength (nm)
Exc
ess
en
Intrinsic wavelength ~ 6.2 nm
Buckle wavelength λ (nm)
10.885
10.89
/nm
)
Armchair GNR
10.875
10.88
10.885
Exc
ess
ener
gy (
eV/n
4 3 2
2 4 6 8 10 12 14 1610.87
Buckle wavelength λ (nm)
Ex
y = 2.2e−06*x4 − 0.00011*x3 + 0.0019*x2 − 0.014*x + 11
Intrinsic wavelength ~ 8.0 nm
Lu and Huang, arXiv:0910.0912.
The wavelengths for edge buckling do not scale with D/f.
GNRs under Uniaxial Tension( ) ( ) ( )LWLU 2Φ
FF
( ) ( ) ( )LWLU εγεε 2+Φ=
δεδ FLU =
εγ
εεσ
dd
Wdd
ddU
WLWF 21
+Φ
===
Zigzag GNRs Armchair GNRs
88
77
66
55
44
33
2210)( εεεεεεεεε aaaaaaaaa ++++++++=Φ
Interior Energy Function876543210)(
)(Φ
Zigzag Armchaira0 -45.234 -45.234a1 0 0
)(εΦ a2 125.0 125.0a3 75.1973 1027.83a4 -2168.75 -20029.5a5 9640.78 175524.2a6 -24439.74 -900733.6a7 35398.35 2489046.8a8 -22491.77 -2847924.5
εddΦ
2d Φ2εd
d Φ
Edge energy function8
87
76
65
54
43
32
210)( εεεεεεεεεγ bbbbbbbbb ++++++++=
Zigzag Armchairb0 10.41 10.91b1 -16.22 -8.53
)(εγ b2 22.14 4.43b3 20.72 -1627.70b4 -562.63 26874.17b5 7611.89 -232656.13b6 -38231.50 1104605.34b7 80531.93 -2596214.47b8 -63317.61 2012549.08
γd 2γdεγ
d 2εd
2D Young’s Moduli of GNRs
( )εγ
εεσ
dd
Wdd 2
+Φ
=
( ) 2
2
2
2 2εγ
εε
dd
WddE +Φ
=
Young’s modulus under i fi it i l t iinfinitesimal strain:
baE 220
42 +=W
aE 20 2 +
Fracture of graphene ribbonsZi dZigzag edge
Armchair edge
MD simulations of fracture (300 K)Zigzag edge Armchair edge
Zigzag GNRs: fracture starts away from the edges; fracture strain same as that for an infinite graphene (PBC);g p ( )Armchair GNRs: fracture starts near the edges; fracture strain lower than that for an infinite graphene (PBC).
Graphene on Oxide SubstratesHR STM image (Stol aro a et al 2007)HR-STM image (Stolyarova et al., 2007)
RMS2
Ozyilmaz et al., 2007.
The 3D morphology is important for
Correlation length
p gy pthe transport properties of graphene-based devices.
Ishigami et al., 2007.
Van der Waals Interaction
Lennard-Jones potential for particle-particle interactions:
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎠⎞
⎜⎝⎛−⎟
⎠⎞
⎜⎝⎛−=
120
60
0 2)(rr
rrVrV
⎥⎦⎢⎣ ⎠⎝⎠⎝
⎤⎡ ⎞⎛⎞⎛93Monolayer-substrate interaction
(energy per unit area): ⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎠⎞
⎜⎝⎛−⎟
⎠⎞
⎜⎝⎛−=
90
30
0 21
23)(
hh
hhUhU
h
U
hr h0
h0
-U0
Van der Waals Thickness and Energy
Gupta et al., 2006.
Sonde et al., 2009.
Interlayer spacing in graphite ~ 0.34 nm;
AFM measurements of h0 for graphene on oxide range from 0.4 to 0.9 nm;
Th dh i (U ) h t b d di tlThe adhesion energy (U0) has not been measured directly;
Theoretically estimated values for U0 range from 0.6 to 0.8 eV/nm2.
Strain-Induced Corrugation
kxAhh sin0 +=
graphenevdWtotal UUU +=
Linear perturbation analysis:
⎥⎦
⎤⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛+−≈ 2
0
2
010 1
hA
hfUUvdW
λ0050~ε −⎦⎣ ⎠⎝ 00 hh
224
4 ACDU graphene⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎠⎞
⎜⎝⎛+⎟
⎠⎞
⎜⎝⎛≈
λπε
λπ nm3~
005.0λε c
g p⎥⎦⎢⎣ ⎠⎝⎠⎝ λλ
Graphene on Rough SubstratesConformal or non conformal?Conformal or non-conformal?
Summary
Nonlinear continuum mechanics for 2D graphene monolayermonolayerAtomistic modeling of graphene under bending and stretchingExcess edge energy, edge forces, and induced edge bucklingGraphene nanoribbons under uniaxial tension:Graphene nanoribbons under uniaxial tension: edge effects on elastic modulus and fractureGraphene on oxide: van der Waals interaction pand corrugation