mechanics of thin films: wrinkling, delamination and fractureruihuang/talks/corning2012.pdf ·...

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