nonlinear mechanics of monolayer graphene - …ruihuang/talks/graphene_tamu2009.pdf · nonlinear...
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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
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Acknowledgments
Qiang Lu (postdoc)
Zachery Aitken (undergraduate student)
Prof. Marino Arroyo (Spain)y ( p )
Prof. Li Shi (UT/ME)
Funding: DoE and NSF (ARRA)
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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
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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.
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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.
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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).
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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).
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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.
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(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
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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
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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).
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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).
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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.
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−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).
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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:
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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).
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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.
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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?
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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.
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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.
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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
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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
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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
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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 +
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Fracture of graphene ribbonsZi dZigzag edge
Armchair edge
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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).
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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.
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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
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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.
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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
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Graphene on Rough SubstratesConformal or non conformal?Conformal or non-conformal?
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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