thermomechanical and interfacial properties of graphene membranesruihuang/talks/graphene... ·...
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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
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
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.