Transcript Document

Wrinkling of thin films on compliant substrates
Rui Huang
Center for Mechanics of Solids, Structures and Materials
The University of Texas at Austin
Wrinkling of thin films
Au films on PDMS
(Bowden et al., Nature 393, 146, 1998)
SiO2 on Si
(Courtesy of David Cahill)
More wrinkling thin films
Stretchable interconnects for large-area flexible electronics
(Jones et al., MRS Symp. Proc. 769, H6.12, 2003 )
Wrinkling of skins (Cerda and Mahadevan, PRL 90, 074302, 2003)
Outline
• Elastic film on elastic substrate
– linear and nonlinear analyses
• Elastic film on viscous substrate
– Kinetic process of wrinkling
• Elastic film on plastic substrate
– Ratcheting-induced wrinkling
Freestanding film: Euler buckling
Critical load:

h
c 
 
121    L 
2
2
Other equilibrium states: energetically unfavorable
Effect of an elastic substrate
Elastic substrate
• Wrinkling relaxes compressive strain
• Bending energy prohibits wrinkling of short wavelengths
• Deformation of the elastic substrate penalizes wrinkling
of long wavelength
Linear analysis
0
Elastic substrate
Small perturbation:
Elastic substrate
w  A sin kx
Strain energy change per unit area:
2

E f A2 k 2 h 
8
(
1


f ) Es 1
2 2
U 
 12(1   f ) 0  k h 

2 
48(1  f ) 
Ef
kh 
0.1
0.01
0.08
Compressive strain, - 
0
Normalized energy change
Wrinkling Stability
0.005
0.005
0.06
0.0033
0
0.04
0.002
0.001
-0.005
 = -0.01
s
0.02
E /E = 0
s
E /E = 0.01
f
-0.01
0
0.005
0
0.002
f
0
 = 0.3
0
0.1
0.2
0.3
0.4
Normalized wave number, kh
 Es

E
 f
2 1  f 3/ 2

 
0

1  f
c
0.5
f
 = 0.3
f
0.2
0.4
0.6
0.8
Normalized wave number, kh
 1  f E 
s 
c  
 2 1  f E f 


2/3
1
Stability Map
Compressive strain, -  0
0.025
0.02
wrinkle
0.015
 1  f E 
s 
c  
 2 1  f E f 


0.01
no wrinkle
0.005
 = 0.3
f
0
0
0.002
0.004
0.006
Stiffness ratio, E /E
s
f
0.008
0.01
2/3
Nonlinear Analysis
Nonlinear effect: large deflection of the film
2

E f A2 k 2 h 
8
(
1


3E f h
f ) Es 1
2 2
4 4
U 

12
(
1


)


k
h


k
A


f
0
2
2
48(1  f ) 
Ef
kh  64(1  f )
Energy minimization leads to the energetically favored wave number
and the corresponding equilibrium amplitude:

1
2 Es
ke  4(1  f ) 
h 
E f 
1/ 3
 8(1   )  E 
f
f


Ae  h  0
2 2 / 3  4E 
 3 1  f
 s


2/3
2
 
3

1/ 2
The energetically favored mode is independent of the compressive strain.
90
9
80
8
70
7
Normalized amplitude, A/h
Normalized wavelength,
 /h
Constrained Equilibrium State
60
50
40
30
20
 = 0.3
10
0
0
4000
6000
Stiffness ratio, E /E
f
8000
s
E /E = 10000
f
s
5000
5
4
2000
1000
3
2
 = 0.3
1
f
2000
6
10000
0
0
f
0.02
0.04
0.06
Compressive strain, -
0.08
0
0.1
0.5
m
h
0.01
Most unstable wave number, k
Normalized energy change
Most unstable mode
0.005
0.005
0.0033
0
0.002
0.001
-0.005
 = -0.01
E /E = 0
s
f
0
 = 0.3
f
-0.01
0
0.1
0.2
0.3
0.4
Normalized wave number, kh
0.5
0.4
0.01
0.3
0.005
0.002
0.2
0.001
0.1
E /E = 0
s
0
0
 = 0.3
f
f
0.01
0.02
0.03
Compressive strain, -
0
Kinetics effect: growth rate depends on the driving force
Other nonlinear effects: plasticity, large deformation of substrate
0.04
Effect of a viscous underlayer
Viscous layer
Rigid substrate
• Wrinkling relaxes compressive strain;
• Bending energy prohibits wrinkling of short wavelengths;
• Viscous flow controls the growth rate: wrinkling of long
wavelength is kinetically constrained.
Wrinkling Kinetics
Linear perturbation analysis:
 Et 
w  A0 exp s  sin kx
  
Fastest growing mode
H
k m h  f ( 0 , , f )
h
-3
1.5
x 10
H/h = 
1
Euler buckling
kc h  12 0 (1   f )
10
0.5
5
2
0
-0.5
-1.5
Slow growing
long-wave mode
 0 = -0.012
-1
0
0.1
0.2
0.3
0.4
0.5
Wave number, kh
Huang and Suo, Int. J. Solids Struct. 39, 1791 (2002).
Kinetically Constrained Equilibrium Wrinkles
A
ln
A0
w  A(t ) sin kx
 Et 
A  A0 exp s 
  
Viscous layer
Rigid substrate
2

1  kc 
Aeq  h    1
3  k 

E
t

•Elastic film is bent in equilibrium.
•Viscous layer stops flowing.
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
Huang and Suo, J. Appl. Phys. 91, 1135 (2002).
Simultaneous Expansion and Wrinkling
Viscous layer
Rigid substrate
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).
Wrinkle-Induced Fracture
s
tension
compression
Tensile stress at the equilibrium state:
2
 




2 E  h   1 
1




s

1



1  2     3  c 
6


 

2
2


Wrinkle-induced cracks
A 200m by 200m SiGe island on BPSG annealed for
90 minutes at 790°C.
Huang et al., Acta Mechanica Sinica 18, 441, (2002)
Thin Film Ratcheting
Uni-directional shear 
s
cyclic stress
metal film
cyclic temperature
substrate

Y
3
Ratching-creep analogy:
d
  R
dN
p
Strain per cycle 
1


Em
Em T
R 
 2 

12(1 vm ) (1 vm )Y

p
Huang, Suo, Ma, Acta Materialia 49, 3039-3049 (2001).
Ratcheting-Induced Wrinkling
Elastic film
Plastic ratcheting
Metal layer
Cyclic temperature
Substrate
Amplitude growth per cycle:
Equilibrium amplitude:
Ef
hm
dA
 s(khf ,  f , , f )
A
dN
hf
R
2


1  kc 
Aeq  h    1
3  k 

Huang et al., in preparation.
Summary
• Wrinkling of a compressed thin film on an elastic substrate
is constrained: a critical strain exists; the wavelength of the
equilibrium state is independent of the strain.
• Flow of a viscous underlayer control the wrinkling
kinetics: a fastest growing mode exists, and the equilibrium
state is kinetically constrained.
• Plastic ratcheting of a metal layer subject to cyclic
temperatures can induce wrinkling of a compressed cap
layer.
Acknowledgement
Zhigang Suo (Harvard University)
Zhenyu Huang (Harvard University)
Haizhou Yin (Princeton University)
James C. Sturm (Princeton University)
Jim Liang (Intel Corp.)
Se Hyuk Im (University of Texas-Austin)