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