Stress-Induced Wrinkling in Thin Films
Rui Huang
Center for Mechanics of Solids, Structures and Materials
Department of Aerospace Engineering and Engineering Mechanics
The University of Texas at Austin
Wrinkles
“Wrinkles occur on scales varying from a few nanometers (in thin films) to hundreds of kilometers (on the surface of the earth), in a variety of natural phenomena (see above).”
(From http://www.deas.harvard.edu/softmat/)
Wrinkling in Thin Films
Applications of Wrinkling
(Jones et al., MRS Symp. Proc. 769, H6.12, 2003 )
- Stretchable interconnects/electrodes for flexible electronics
- Optical scattering, grating, and waveguide structures
- Mechanical characterization of polymer thin films
- Reliability of integrated devices containing soft organic materials
Mechanics of Wrinkling
• Elastic film on elastic substrate– Equilibrium and Energetics
• Elastic film on viscous substrate– Non-equilibrium and Kinetics
• Elastic film on viscoelastic substrate– Evolution of wrinkle patterns
Freestanding film: Euler bucklingCritical load:
22
112
L
hc
Other equilibrium states: energetically unfavorable
• Buckling relaxes compressive stress
• Bending energy minimizes at long wavelength
On elastic substrates• Deformation of the substrate
disfavors wrinkling of long wavelengths and competes with bending to select an intermediate wavelength
Elastic substrate
Wrinkling: short wavelength, on soft substrates, no delamination
Buckling: long wavelength, on hard substrates, with delamination
Critical Condition for Wrinkling
3/2
21
3
4
1
f
scf E
E
0 0.002 0.004 0.006 0.008 0.010
0.005
0.01
0.015
0.02
0.025
Stiffness Ratio, Es/E
f
Co
mp
ress
ive
Str
ain
, -
wrinkling
flat film
Thick substrate (hs >> hf):
The critical strain decreases as the substrate stiffness decreases.
In general, the critical strain depends on the thickness ratio and Poisson’s ratios too.
In addition, the interface must be well bonded.
Equilibrium Wrinkle Wavelength
Thick substrate (hs >> hf):
3/1
32
s
f
f E
Eh
The wrinkle wavelength is independent of compressive strain.
The wavelength increases as the substrate stiffness decreases.
In general, the wavelength depends on thickness ratio and Poisson’s ratios too.
0 0.002 0.004 0.006 0.008 0.010
20
40
60
80
100
Stiffness Ratio, Es/E
f
Wrin
kle
Wav
elen
gth,
/
hf
Measure wavelength to determine film stiffness
Equilibrium Wrinkle Amplitude
Thick substrate (hs >> hf):
2/1
21 1
c
fhA
0 2 4 6 8 100
0.5
1
1.5
2
2.5
3
Compressive Strain, /c
Wrin
kle
Am
plitu
de, A
/hf
Measure amplitude to determine film stress/strain.
The wrinkle amplitude increases as the compressive strain increases.
For large deformation, however, nonlinear elastic behavior must be considered.
Equilibrium Wrinkle Patterns
In an elastic system, the equilibrium state minimizes the total strain energy.
However, it is extremely difficult to find such a state for large film areas.
More practically, one compares the energy of several possible patterns to determine the preferred pattern.
How does the pattern emerge?
How to control wrinkle patterns?
Kinetics: on a viscous substrate
• Viscous flow controls the growth rate: long-wave wrinkling grows slowly, and an intermediate wavelength is kinetically selected.
Viscous layer
Rigid substrate
Fastest mode
mc 0
GrowthRate s
Euler buckling
sm
1
f
m
h
stAA exp0
(For hs >> hf)
Kinetically Constrained Equilibrium Wrinkles
Infinitely many: each wavelength ( > c) has an equilibrium state
Energetically unstable: longer wavelength lower energy
Kinetically constrained: flow is very slow near the equilibrium state
•Elastic film is bent in equilibrium. •Viscous layer stops flowing.
Huang and Suo, J. Appl. Phys. 91, 1135 (2002).
Et
sAA exp0
0
lnA
A
t
Viscous layer
Rigid substrate
kxtAw sin)(
1
3
12
k
khA c
eq
Simultaneous Expansion and Wrinkling
Expansion starts at the edges and propagates toward center
Wrinkle grows before expansion relaxes the strain
Long annealing removes wrinkles by expansion
Liang et al., Acta Materialia 50, 2933 (2002).
Viscous layer
Rigid substrate
Wrinkling on Viscoelastic SubstratesCross-linked polymers
Compressive Strain
Wrinkle Amplitude
0
Evolution of wrinkles:
(I) Viscous to Rubbery
(II) Glassy to Rubbery
Rubbery State
R
Glassy State
G
(Lee at al., 2004)
Wrinkling Kinetics I: GR
Fastest mode
m 0
GrowthRate
Wrinkles of intermediate wavelengths grow exponentially;
The fastest growing mode dominates the initial growth.
1
f
m
h
For hs >> hf :
The kinetically selected wavelength is independent of substrate!
stAtA exp)( 0
Wrinkling Kinetics II: G
Instantaneous wrinkle at the glassy state:
2/1
0 1
G
fhA
3/1
0 32
G
f
f E
Eh
Kinetic growth at the initial stage:
1)exp()( 0 tBAtA
Long-term evolution: 3/1
32
R
f
f E
Eh
0
2/1
1
R
fhA
0A
t = 0
t = 1104
Numerical Simulation
0 200 400 600-0.1
0
0.1
x/hf
w/h
f
0 200 400 600-0.1
0
0.1
x/hf
w/h
f
0 200 400 600-2
0
2
x/hf
w/h
f
0 200 400 600-2
0
2
x/hf
w/h
f0 50 100
W avelength, L/hf
0 50 100W avelength, L/h
f
0 50 100W avelength, L/h
f
0 50 100W avelength, L/h
f
t = 1105
t = 1107
Growing wavelengths
Coarsening
Equilibrium wavelength
Evolution of Wrinkle Wavelength
0 2 4 6 8 10
x 104
20
30
40
50
Normalized time, t/
Wa
vele
ng
th,
L/h f
/E
f=0.0001
/E
f=0.00001
Lm
= 26.9hf
104
105
106
107
20
30
40
50
60
70
Normalized time, t/
Wa
vele
ng
th,
L/h f
/E
f=0.0001
/E
f=0.00001
Lm
= 26.9hf
Leq
= 33.7hf
Leq
= 60.0hf
Initial stage: kinetically selected wavelengths
Intermediate stage: coarsening of wavelength
Final stage: equilibrium wavelength at the rubbery state
0 2 4 6 8 10
x 104
0.01
0.1
1
Normalized time, t/
RM
S
/E
f=0.0001
/E
f=0.00001
104
105
106
107
0
0.5
1
1.5
Normalized time, t/
RM
S
/E
f=0.0001
/E
f=0.00001
Aeq
= 0.619hf
Aeq
= 1.63hf
Evolution of Wrinkle Amplitude
Initial stage: exponential growth
Intermediate stage: slow growth
Final stage: saturating
t = 0 t = 104 t = 105
t = 107t = 106
2D Wrinkle Patterns I
t = 0 t = 105
t = 2X107
t = 106
t = 5X106
2D Wrinkle Patterns II
t = 107
t = 5X105
t = 106
t = 104
2D Wrinkle Patterns IIIt = 0
t = 0 t = 104 t = 105
t = 106 t = 107
On a Patterned Substrate
Circular Perturbationt = 0 t = 104 t = 105
t = 5105 t = 106 t = 107
Evolution of Wrinkle Patterns• Symmetry breaking in isotropic system:
– from spherical caps to elongated ridges
– from labyrinth to herringbone.
• Symmetry breaking due to anisotropic strain– from labyrinth to parallel stripes
• Controlling the wrinkle patterns– On patterned substrates
– By introducing initial defects
What else?• Ultra-thin films
– Effect of surface energy and surface stress– Effect of thickness-dependent modulus– Effect of temperature, molecular weight, cross-
linking– Other effect at nanoscale?
• Nonlinear elastic/viscoelastic behavior– Nested wrinkles?