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?