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Mechanics of Thin Films:
Wrinkling, Delamination and Fracture
Rui Huang
Center for Mechanics of Solids, Structures and Materials
Department of Aerospace Engineering and Engineering Mechanics
The University of Texas at Austin
January 27, 2012
Research Activities
• Wrinkling of Thin Films on Soft Substrates (NSF)
• Fracture and Delamination in Thin-Film Structures (SRC)
• Surface Morphological Evolution and Self-Assembly of Quantum
Dots (DoE)
• Reliability in Advanced Interconnects and Packaging (SRC)
• Nonlinear Mechanics of Graphene (NSF)
• Nonlinear Mechanics of Hydrogels and Soft Materials
http://www.ae.utexas.edu/~ruihuang/research_topics.html
Mechanics of Thin Films:
Stretch and Compress
Soft substrate
Suo, 1995Li et al, 2005
SiGe film on BPSG: viscous wrinkling
BPSG
Si
SiGe
Heat to
750°C
• Hobart et al., 2000
• Yin et al., 2002 & 2003
• Peterson et al., 2006
• Yu et al., 2005 & 2006
Au film on PDMS: thermoelastic wrinkling
• Bowden et al., 1998 & 1999
• Huck et al., 1999
• Efimenko et al., 2005
Elastic wrinkling vs Viscous wrinkling
film/substrate materials elastic/elastic elastic/viscous
Onset of wrinkling Critical stress or strain
depends on the elastic
modulus and thickness of
substrate.
Unstable with any
compressive stress in a thin
film (L/h >> 1).
Wrinkle amplitude Equilibrium amplitude
increases with the
stress/strain.
Amplitude grows over time.
Wrinkle wavelength Equilibrium wavelength
depends on the elastic
modulus and thickness of
substrate.
The fastest growing
wavelength dominates the
initial growth, then
coarsening over time.
Energetics Spontaneous energy
minimization.
Energy dissipation by viscous
flow.
Elastic film on viscoelastic substrate
2/1
1
−=
c
feq hAε
ε
Cross-linked polymers
Compressive
Strain
Wrinkle
Amplitude
0
Evolution of wrinkles:
(I) Viscous to Rubbery
(II) Glassy to Rubbery
Rubbery State
Rε
Glassy State
Gε
Huang, JMPS 2005.
3/1
32
=
s
f
feqE
Ehπλ
Equilibrium states at the
elastic limits (thick substrate):3/2
3
4
1
−=
f
sc
E
Eε
(I)
(II)
Scaling of evolution equation
( ) RwwFwKt
w−∇⋅⋅∇+∇∇−=
∂
∂σ
22
s
RRη
µ=
Fastest growing mode:122 Lm πλ =
=
1
04
expτ
tAA
2
4
F
KRc −=σcritical stress:
Viscoelastic layer
Elastic film
Rigid substrate
sfs
sffs hhK
ηνν
µν
)1)(1(12
)21( 3
−−
−=
s
fs
s
shh
Fην
ν
)1(2
21
−
−=
01
σF
KL −=
( )20
1σ
τF
K=Length and time scales:
meqR
Kλπλ >
=
4/1
2
−= 1
3
20
c
feqhA
σ
σEquilibrium state:
)( 0 cσσ <
Viscoleastic Wrinkle Evolution
Time
wavelength
Stage I Stage II Stage III
t1 t2
amplitude
I: Exponential growth, dominated by the fastest growing mode
II: Coarsening (λm → λeq, A↑)
III: Nearly equilibrium (with local ordering)
Im and Huang, JAM 2005 and PRE 2006.
Viscoelastic layer
Rigid substrate
Elastic film
H
h
Viscoelastic layer
Rigid substrate
z
x
Power-law Coarsening
• On a compressible, thin viscoelastic substrate: λ ~ t1/4;
• On an incompressible, thin viscoelastic substrate, λ ~ t1/6;
• On a thick VE substrate, λ ~ t1/3.
• The substrate elasticity slows down coarsening as it approaches the equilibrium state.
σ0 = -0.02 (∆), -0.01 (o), -0.005 (□), -0.002 (◊), -0.001 (∇)
Blue: uniaxial; Red: equi-biaxial
Huang and Im, PRE 2006.
01
σF
KL −=
( )20
1σ
τF
K=
Length and time scaling:
Numerical simulation: uniaxial wrinkling
t = 104 t = 105
t = 106 t = 107
t = 0
0=f
R
µ
µ
0,01.0)0(
22)0(
11 =−= σσ
Huang and Im, PRE 2006.
Numerical simulation: equi-biaxial wrinkling
t = 104 t = 105
t = 106 t = 107
t = 0
0=f
R
µ
µ
01.0)0(
22)0(
11 −== σσ
Huang and Im, PRE 2006.
Effect of Substrate Elasticity510−=
f
R
µ
µ
σ0 = -0.02 (∆), -0.01 (o), -0.005 (□)
Near-Equilibrium Patterns
Same wavelength
Different patterns
0
005.0
)0(22
)0(11
=
−=
σ
σ
01.0)0(
22)0(
11 −== σσ
005.0)0(
22)0(
11 −== σσ
02.0)0(
22)0(
11 −== σσ
510−=f
R
µ
µ
Wrinkle patterns under biaxial stresses
0 σx
σy
σc
σc
Parallel wrinkles
Para
llel w
rinkle
s
10 µmσ1
σ2
f
cx
cyν
σσ
σσ=
−
−Transition
stress:
Wrinkle patterns under non-uniform stresses
( )( )
−=
aL
axx
2/cosh
/cosh10σσ0σσ =y
)(xfx =σ
L
1D shear-lag model:
L = 3750 hf L = 3250 hf L = 2500 hf L = 1250 hf
R
fsf hhEa
µ=
x/hf
Str
ess,
σx/E
f
005.0/0 −=fEσ0== yx σσoutside (green):
Wrinkling of rectangular films
Experiments (Choi et al., 2007): Simulations (Im and Huang, 2009):
� Stress relaxation in both x and y
directions from the edges.
� The length of edge relaxation
depends on the elastic modulus of
the substrate.
� No wrinkles at the corners, parallel
wrinkles along the edges, and
transition to zigzag and labyrinth in
large films.
500025001250
Wrinkling of periodically patterned films
Experiments (Bowden et al., 1998):
Simulations (Im, 2009)
Wrinkling of Single-Crystal Thin Films
30 nm SiGe film on BPSG at 750°C
(R.L Peterson, PhD thesis, Princeton, 2006)
100 nm Si on PDMS (Song et al., 2008)
(100)(010)
Evolution of orthogonal wrinkle patterns
t = 0:
RMS = 0.0057
λave= 38.8
t = 105 :
RMS = 0.0165
λave= 44.8
t = 5 × 105 :
RMS = 0.4185
λave= 47.1
t = 106 :
RMS = 0.4676
λave= 50.8
t = 107 :
RMS = 0.5876
λave= 56.4
t = 108 :
RMS = 0.5918
λave= 56.6
003.021 −== σσ
Im and Huang, JMPS 2008.
Effect of stress magnitude
Im and Huang, JMPS 2008.
Checkerboard wrinkle pattern
0
20
40
60
80
0
20
40
60
80
−1
−0.5
0
0.5
1
00178.021 −== σσ RMS = 0.05286 and λave= 55.6
Im and Huang, JMPS 2008.
� Transition of local buckling mode from spherical to cylindrical as the
compressive stress/strain increases or as the wrinkle amplitude increases.
Wrinkling vs Buckle-delamination
Wrinkling Buckle-delamination
Stafford et al., Macromolecules 39, 5095 (2006).
Moon et al., Acta Mat.
50, 1219 (2002).
Concomitant wrinkling and buckle-delamination
Optical micrographs of a 120 nm PS thin film on a PDMS substrate.
Mei et al., Appl. Phys. Lett. 90, 151902 (2007).
Which buckling mode occurs first?
2b
Buckle-delamination
Wrinkling
Mei et al., Appl. Phys. Lett. 90, 151902 (2007).
3/2
3
4
1
=
f
sW
E
Eε
= βαξ
πε ,,
12
22
h
b
b
hB
Yu and Hutchinson, Int. J. Fracture 113, 39-55 (2002);
FE eigenvalue analysis
1000=s
f
E
E
b=h
b=3h
b=10h
First eigen mode
3
1== sf νν
( )βαξπ
ε ,12
22
=
b
hB
3/2
3
4
1
=
f
sW
E
Eε
Mei et al., Mechanics of Materials 43, 627-642 (2011).
Comparison of critical strains
For intermediate values of b/h and Ef/Es, the previous solutions for
wrinkling and buckle-delamination both over-estimate the critical
strain for onset of buckling.Mei et al., Mechanics of Materials 43, 627-642 (2011).
=
WW
B bf
λε
ε
Postbuckling analysis – without delamination
0 0.005 0.01 0.015 0.020
0.5
1
1.5
Nominal strain, ε
Wrinkle
am
plit
ude,
AW
/h
Analytical solution
FEA, A0/h = 10-4
FEA, A0/h = 0.01
FEA, A0/h = 0.05
1−=c
W
h
A
ε
ε c
s
ssm
Eq εε
ν
ν−
−
−=
43
)1(4 2
m
s
sm q
)1(2
21
ν
ντ
−
−−≈
1000=s
f
E
E
3
1== fs νν
Mei et al., Mechanics of Materials 43, 627-642 (2011).
Postbuckling analysis – with delamination
0 20 40 60 80 100 120-1
0
1
2
3
4
5
x/h
Dis
pla
cem
ent,
w/h
ε = 0.002
ε = 0.004
ε = 0.006
ε = 0.008
ε = 0.01
0 0.002 0.004 0.006 0.008 0.010
1
2
3
4
5
6
7
Nominal strain, εB
uckle
am
plit
ude,
AB/h
b/h = 1
b/h = 5
b/h = 10
b/h = 20
The buckling amplitude is sensitive to the defect size.
b = 10h
AB/h
1000=s
f
E
E
Mei et al., Mechanics of Materials 43, 627-642 (2011).
Effect of model size L/b
ε
h
H
L
The critical strain for onset of buckling is insensitive to the model
size, but the post-buckling amplitude increases as the model size
increases.
b 1000=s
f
E
E
Mei et al., Mechanics of Materials 43, 627-642 (2011).
Coexistent buckling modes
0 200 400 600 800 10000
5
10
15
20
25
x/h
w/h
b/h = 50L/b = 20
ε = 0.0118
Image courtesy of Chris Stafford at NISTMei et al., Mechanics of Materials 43, 627-642 (2011).
Progressive wrinkling and delamination
h Film
Substrate
Film
Substrate
Start with a perfectly bonded film. Wrinkling
occurs first. Significant wrinkling nucleates
delamination cracks, which grows subsequently
to form large buckles.
0
A
B
C
cδ
0σ
K0
δ
σ
(1-D)K0
Damage Initiation
Final failure
Cohesive zone model for interfacial delamination
K0: initial elastic stiffness
σ0: cohesive strength
δc : critical separation for failure
( )( )0max
0max
δδδ
δδδ
−
−=
c
cDDamage parameter:
Damage initiation criterion (strength):
Final failure criterion (toughness):
0σσ =
cG δσ 02
1=Γ=
Bilinear Traction-Separation (ABAQUS)
( ) ( )[ ] δδδσ 01 KD−=
Evolution of buckle amplitude
A
B
C
A
B
C
Initiation of wrinkle-induced delamination
0
2
43
)1(4σεε
ν
ν=−
−
−= c
s
ssm
Eq
A strength criterion:2
0
2)1(4
43
−
−+=
ss
scWID
E
σ
ν
νεε
Subsequent growth of interfacial delamination
Unstable
growth
stable growth
Co-evolution of
wrinkling and buckle-delamination
Wrinkling of thin sheets
One-tenth scale model NGST sunshield
from NASA Goddard Space Flight Center
2-quadrant 10m
solar sail model
2m solar
sail models
Gossamer spacecraft test articles at NASA
Langley Research Center
Stretch-induced wrinkling
L0/W0 = 2.5 and W0/t0 = 1000
Nayyar et al., IJSS 48, 3471-3483 (2011).
Wrinkle-induced fracture
SiGe film on glass
Yin, et al., J. Appl. Phys. 94, 6875 (2003).
( )( )
−
−
−=
6
11
3
1
12
2
2
2
k
kkhE ceq
νσ
Liang, et al., Acta Mater 50, 2933 (2002).
Channel cracks in thin films
Cross section: Top view:
Courtesy of Jun He of Intel Corp.
He et al., Proc. of the 7th Int. Workshop on Stress-Induced
Phenomena in Metallization (2004), pp. 3-14.
Film cracking with delamination
10 µm10 µm
1µm SiN on Si (Courtesy of Q. Ma,
Intel Corp.)
Tsui et al., JMR, 20, 2266 (2005).
• Under what conditions would interfacial delamination occur?
• If it occurs, how would interfacial delamination affect the fracture
driving force?
A “phase diagram” for delamination
0 1-1-0.89
I
IIAIIB
IIIA
IIIB0.5
0.99
h
E if
i 2
0σ
Γ=Γ
αcompliant film
stiff substrate
stiff film
compliant substrate
I: No delamination
II: Spontaneous delamination
III: Stable delamination
A for barrierless delamination,
and B with an initiation barrier
Mei, Pang, and Huang, Int. J. Fracture 148, 331 (2007).
Creep-modulated crack growth
V
Viscous layer
σ
h
H
K = KcCrack growth criterion:
Λ =Kc
σ
2Length scale:
Time scale:t0 =
Λ2
D=
Kc
σ
4 η
HhE
Liang et al., Experimental Mechanics 43, 269-279 (2003).
( )2
2
0
5.0c
ssK
HhE
tV
η
σνχ =
Λ=
Steady-state velocity:
Numerical simulation of crack growth
0 0.5 1 1.5 2 2.5 30
0.5
1
K/K
c
0 0.5 1 1.5 2 2.5 30
0.5
1
1.5
a / ΛΛ ΛΛ
0 0.5 1 1.5 2 2.5 30
0.2
0.4
0.6
Vt 0
/ ΛΛ ΛΛ
t / t0
Transient
state
Stationary
crack
( ) 4/1)( DttK σ∝
Steady state
growth
0tV
Λ∝∞
Swell induced surface instability in gels
• Southern and Thomas, 1965
• Tanaka et al, 1987
• Trujillo et al., 2008
� Critical condition for the onset of
surface instability?
� Any characteristic size?
� Effect of kinetics?
A linear perturbation analysis
∂
∂+
∂
∂
∂
∂
∂
∂+
=
100
01
01
~
2
2
1
2
2
1
1
1
x
u
x
u
x
u
x
u
h
h
λ
λ
F
=
100
00
001
hλF
Linear perturbation: ( ) ( )21222111 ,,, xxuuxxuu ==
Homogeneous swelling
Kang and Huang, J. Mech. Phys. Solids 58, 1582-1598 (2010).
Critical swelling ratio
βλλ
λ h
h
h 41
2
=
+
• The critical swelling ratio
depends on Nv and χ, ranging
between 2.5 and 3.4.
• For each χ, there exists a
critical value for Nv.
• For small Nv (< 10-4), the
critical swelling ratio is nearly
a constant (~3.4).
Kang and Huang, J. Mech. Phys. Solids 58, 1582-1598 (2010).
A stability diagram
χc = 0.63
Soft network in good solvent
Poor solvent
Stiff network
Kang and Huang, J. Mech. Phys. Solids 58, 1582-1598 (2010).
Finite element simulation
Nv = 0.001, χ = 0.4
Kang and Huang, J. Mech. Phys. Solids 58, 1582-1598 (2010).
Surface Evolution
Kang and Huang, J. Mech. Phys. Solids 58, 1582-1598 (2010).
Mechanics of Thin Films:
Stretch and Compress
Soft substrate
Suo, 1995Li et al, 2005