Thermomechanical and Interfacial Properties of Graphene Membranes

Rui Huang

Center for Mechanics of Solids, Structures and MaterialsDepartment of Aerospace Engineering and Engineering Mechanics

The University of Texas at Austin

February 20, 2014

Acknowledgments Wei Gao (graduate student)

Peng Wang (graduate student)

Qiang Lu (postdoc)

Zachery Aitken (undergraduate student)

Prof. Ken Liechti (UT/ASE-EM)

Prof. Rod Ruoff (UT/ME)

Prof. Graeme Henkelman (UT/Chemistry)

Prof. Yong Zhu (NCSU)

Funding: National Science Foundation

Membrane Materials or Structures Mechanical (solid) membranes: no bending rigidity

Bio-membranes (lipid bilayers, fluid membranes, etc.)

2D crystal monolayers: graphene, h-BN, MoS2, etc.

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 dx F X

X1

X2

x1

x3 x2J

iiJ X

xF

JKiKiJJK FFE 21

2iI i

IJ i iJ I J

F xn nX X X

Strain energy: A

dAU ΚE,

Lu and Huang, Int. J. Applied Mechanics 1, 443-467 (2009).

Temperature effect?

xw

xw

xu

xu

21

xxw

2

Small deformation at T = 0 K (Statics)

κεκε ,

1)1(2E

212

2121 GDD

The basic elastic properties (E, ν, D, and DG) can be determined by DFT or molecular mechanics (statics) calculations. They are NOT directly related to each other!

Linear elastic strain energy:

xw

xw

xu

xu

21

In-plane stretching:

Bending:

Moderately nonlinear kinematics:

Elastic properties of graphene at T = 0 K

0 0.05 0.1 0.15 0.2 0.25 0.30

10

20

30

40

Nominal strain

Nom

inal

2D

stre

ss (N

/m)

zigzag (MM/REBO)armchair (MM/REBO)zigzag (Wei et al., 2009)armchair (Wei et al., 2009)

0 0.5 1 1.5 2 2.5 30

0.05

0.1

0.15

Bending curvature (1/nm)

Str

ain

energ

y per

ato

m (

eV

/ato

m)

B1, armchair

B1, zigzag

B2, armchair

B2, zigzag

B2*, armchair

B2*, zigzag

Method E (N/m) ν E/(1-ν) D (eV)

DFT 345 0.149 406 1.5

MM (REBO-2) 243 0.397 403 1.4

MM (AIREBO) 277 0.366 437 1.0

T

AT

,

T > 0 K: a hybrid approach

Continuum mechanics

Statistical mechanics

Molecular dynamics

Thermo-dynamics

ZTkA B ln

TkU

ZP

B

exp1

t

Harmonic analysis of thermal fluctuation

k

ik

keww rqqr )(ˆ)( k

kkkb wwqDLU )(ˆ)(ˆ2

2Im

2Re

420 qq

)0(42

01Im1Re )(ˆ)(ˆexp

ykk k

B

B

bb qDL

TkwdwdTk

UZeq

qq

420

1Im1Re22 )(ˆ)(ˆexp)(ˆ1)(ˆ

k

B

B

bk

bk qDL

TkwdwdTk

UwZ

w

qqqq

k k

kB

k qDL

Tkwh 420

22 )(ˆ q

DTkLhh Bn

402

16

Basically a linear plate model, independent of in-plane stiffness or Gaussian curvature.

Effect of pre-tension (still harmonic)

)0(

2Im

2Re

20

*420

20

20

* )(ˆ)(ˆykk

kkkksb wwqEDqLLEUUUeq

qq

Gao and Huang, JMPS, in press. 00

*

024.0

nm

EDl

L0 (T = 0 K)

00LL

0 20 40 60 80 1000

0.5

1

1.5

2

L0 (nm)

h(A

)

ε0 = 0

ε0 = 0.05%

ε0 = 0.1%

ε0 = 1%

A small pre-tension can considerably suppress thermal fluctuation, resulting in a different scaling behavior.

DLE

ETk

DqEq

DLTkh

B

k kk

B

2

200

*

0*

1

20

*4

20

2

41ln

4

1

Nonlinear thermoelasticity

TZTkTA B ,ln, 00

)0(42

0

1

20

*20

20

*

0 1exp),(ykk k

B

kB qDLTk

DqE

TkLETZ

eq

TEAL

TT

,~21, 00

*

020

0

1

020

*

2**

0

* 416

~

LED

DTkEEE B

Partition function:

Helmholtz free energy:

Stress:

Tangent modulus:

Fluctuation induced tension

Thermal softening and strain stiffening; size dependent?

Gao and Huang, JMPS, in press.

Thermal expansion

0 10 20 30 40 50-10

-5

0

5

10

T (K)

CTE

, (x

10-6

K-1

)

2D

10 nm

20 nm

3 nm

5 nm

LD T

22

L0 (T = 0 K)011 LL

L (T > 0 K)

021 TD

Anharmonic in-plane oscillations result in a positive thermal expansion, nearly independent of temperature (up to 1000 K).

Out-of-plane fluctuation leads to in-plane contraction (negative thermal expansion) –anharmonic effects TBD

022 LL 0),( 02 LT

MD Simulations (T > 0 K)• Free-standing graphene in NPT ensemble (zero stress)• Constrained graphene in NVT ensemble (zero strain)• Biaxially strained graphene in NVT ensemble

Thermal rippling Thermal expansion/contraction Thermal stress Temperature dependent

mechanical properties

5 10 20 30 40 50

0.5

1

2

5

L0(nm)

h(A

)

NVT (zero−strain), ζ = 0.45NPT (zero−stress), ζ = 0.67Harmonic analysis, ζ = 1

Thermal Rippling

Harmonic approximation:

Significant anharmonic effects due to coupling between bending and stretching (ζ < 1), similar to biomembranes!

Lh ~

0 200 400 600 800 10000

0.5

1

1.5

2

2.5

3

T (K)

h(A

)

NVT (zero−strain)NPT (zero−stress)Harmonic analysis

DTkLhh Bn

402

16

Gao and Huang, JMPS, in press.

NPT: Thermal Expansion/Contraction

• Negative thermal expansion at low T, and positive at high T.• Thermal expansion/contraction is size dependent!• By suppressing out-of-plane fluctuations, 2D simulations

predict a constant positive CTE (size-independent).

Gao and Huang, JMPS, in press.

NVT: Thermal Stress at Zero Strain

• As expected, negative thermal expansion leads to tensile stress at low T, and the opposite is true at high T.

• However, thermal rippling differs under NPT and NVT.

NVT: biaxially strained graphene

Nonlinear elasticity due to two effects: (1) intrinsic strain softening (large strain

behavior); (2) strain stiffening due to thermal rippling

(small strain behavior)

L0 = 20 nm and T = 300 K

Gao and Huang, JMPS, in press.

Biaxial modulus at zero strain

Due to the effect of thermal rippling, the elastic modulus becomes both temperature and size-dependent at zero strain; this effect however is largely beyond harmonic.

Gao and Huang, JMPS, in press.

Biaxial modulus at 1% strain

The effect of thermal rippling reduces as the strain increases; The tangent elastic modulus decreases linearly with temperature

(almost harmonic), and becomes size independent for relatively large membranes.

%1

1

020

*

2*** 4

16~

LED

DTkEEE B

Gao and Huang, JMPS, in press.

Graphene on substrate

300 400 500 600 700 800 900 10000.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

T (K)

Rip

ple

ampl

itud

e(A

)

αs = 0

αs = 1 x 10−6 K−1

αs = 3 x 10−6 K−1

αs = 5 x 10−6 K−1

300 400 500 600 700 800 900 10000

0.2

0.4

0.6

0.8

1

T (K)

Stre

ss (N

/m)

s = 0

s = 1 x 10-6 K-1

s = 3 x 10-6 K1

s = 5 x 10-6 K-1

0TTs

Assume that the in-plane dimension of graphene follows thermal expansion of the substrate (full strain transfer).

Gao and Huang, JMPS, in press.

Interfacial strain transfer

0 0.5 1 1.5 2 2.5 3 3.5-1.5

-1

-0.5

0

0.5

1

1.5

2

Applied strain m (%)

Gra

phen

e st

rain

(%)

Sample #3Sample #4

D

cp E

L

22

MPa 5.0~c

Jiang, Huang, and Zhu, Adv. Funct. Mater., 2014.

Interfacial properties: Adhesion and Friction

z

10

04

0

0

0

29

zh

zh

hzvdW

3 9

0 00

3 12 2vdW

h hU zz z

Aitken and Huang, J. Appl. Phys. 107, 123531 (2010).

20

0

0

0

000max h

phhh

f s

Zero friction on a perfectly flat surface (assuming vdW).

Friction strength depends on surface roughness and adhesion.

Temperature effect?

A blister test

• CVD grown graphene was transferred to a copper substrate.

• The graphene/photoresist composite film was pressurized with deionized water.

• Deflection profiles were measured by a full field interference method.

• Energy release rate was calculated as a function of delamination growth to obtain fracture resistance curves.

• The delamination path was confirmed by Raman spectroscopy.

Cao et al., Carbon 2014.

p CuCu

SU-8 (~30μm)

~800μm

Delamination Resistance Curves

Without graphene

With graphene

Compare to other measurements: Graphene/SiO2: 0.2-0.45 J/m2 (Scott Bunch) Graphene/Cu film: 0.72 J/m2 (Yoon et al.)

Cao et al., Carbon 2014.

Graphene/SiO2: effect of surface structures

Graphene/SiO2: vdW-DFT

The vdW adhesion energy is reduced by surface hydroxylation and further reduced by adsorption of water molecules.

Gao et al., submitted.

Graphene/SiO2: capillary forces?

cavitation

Breaking of water bridge

Summary A statistical mechanics approach is employed to study

thermal rippling and thermoelasticity of graphenemembranes; the current analysis is limited by harmonic analysis.

MD simulations show significant anharmonic effects at zero stress or zero strain, but nearly harmonic behavior when the membrane is subjected to pre-tension.

Graphene on substrate: strain transfer (shear) and adhesion have been measured, but the underlying mechanisms (vdW or capillary) require further studies.