nonlinear dynamics of wrinkle growth and pattern formation

Nonlinear Dynamics of Wrinkle Growth and Pattern Formation in Stressed Elastic Thin FilmsPattern Formation in Stressed Elastic Thin Films

Se Hyuk Im and Rui Huang

Center for Mechanics of Solids, Structures and MaterialsDepartment of Aerospace Engineering and Engineering Mechanics

Th U i it f T t A tiThe University of Texas at Austin

----- Work supported by NSF CAREER Award (CMS-0547409) -----

Au film on PDMS: thermoelastic wrinkling• Bowden et al., 1998 & 1999• Huck et al., 1999• Efimenko et al., 2005

SiGe film on BPSG: viscous wrinkling• Hobart et al 2000Hobart et al., 2000• Yin et al., 2002 & 2003• Peterson et al., 2006• Yu et al., 2005 & 2006

BPSGSi

SiGe

Si

Heat to 750°C

Elastic wrinkling vs Viscous wrinkling

film/substrate materials elastic/elastic elastic/viscous

Onset of wrinkling Critical stress or strain Unstable with any gdepends on the elasticmodulus and thickness of substrate.

ycompressive stress in a thin film (L/h >> 1).

Wrinkle amplitude Equilibrium amplitude increases with the stress/strain.

Amplitude grows over time.

W i kl l th E ilib i l th Th f t t iWrinkle wavelength Equilibrium wavelength depends on the elasticmodulus and thickness of substrate.

The fastest growing wavelength dominates the initial growth, then coarsening over time.g

Energetics Spontaneous energy minimization.

Energy dissipation by viscous flow.

(Yoo and Lee, 2003-2005)

Al film on PS: viscoelastic wrinkling

A model for viscoelastic wrinklingViscoelastic relaxation modulusViscoelastic relaxation modulus Wrinkle

Amplitude Rubbery State

(II)Glassy State

Equilibrium states at the (I)

(II)

Compressive0

Rε Gε

Equilibrium states at the elastic limits (thick substrate):

3/23

41

⎟⎟⎠

⎞⎜⎜⎝

⎛−= s

c EEε

(I)

Compressive StrainEvolution of wrinkles:

(I) Viscous to Rubbery

R G

2/1

1⎟⎟⎠

⎞⎜⎜⎝

⎛−=

cfeq hAεε

4 ⎠⎝ fE

(I) Viscous to Rubbery

(II) Glassy to RubberyHuang, JMPS 2005.

⎠⎝3/1

32 ⎟⎟

⎠

⎞⎜⎜⎝

⎛=

s

ffeq E

Ehπλ

Scaling of evolution equationElastic film

( ) RwwFwKtw

−∇⋅⋅∇+∇∇−=∂∂ σ22

Viscoelastic layer

Elastic film

Rigid substratehh)21( 3

s

RRημ

=

Fastest growing mode: ⎟⎞

⎜⎛ t

Rigid substrate

sfs

sffs hhK

ηννμν

)1)(1(12)21( 3

−−

−=

s

fs

s

s hhF

ηνν

)1(221−−

=

Fastest growing mode:122 Lm πλ = ⎟⎟

⎠

⎞⎜⎜⎝

⎛=

10 4exp

τtAA

K K

4KR

01 σF

KL −= ( )201

στ

FK

=Length and time scales:

24FKR

c −=σcritical stress:

K ⎞⎛4/1 ⎞⎛2Equilibrium state:

meq RK λπλ >⎟⎠⎞

⎜⎝⎛= 2 ⎟⎟

⎠

⎞⎜⎜⎝

⎛−= 1

32 0

cfeq hA

σσEquilibrium state:

)( 0 cσσ <

Three-Stage Viscoleastic Wrinkle Evolution

Stage I Stage II Stage III

wavelengthwavelength

amplitude

Timet1 t2

I E ti l th d i t d b th f t t i dI: 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.

Power-law Coarsening

1 σFKL −=

Length and time scaling:

0σF

( )201

στ

FK

=

σ0 = -0.02 (Δ), -0.01 (o), -0.005 (□), -0.002 (◊), -0.001 (∇)

Blue: uniaxial; Red: equi-biaxial

• On a compressible thin viscoelastic substrate: λ ~ t1/4;

q

On a compressible, thin viscoelastic substrate: λ t ;

• 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.

Huang and Im, PRE 2006.

Numerical simulation: uniaxial wrinklingg

t = 104 t = 105t = 0 t = 104 t = 105t 0

t = 106 t = 107

0,01.0 )0(22

)0(11 =−= σσ

0=f

R

μμ

, 2211

fμ


Numerical simulation: equi-biaxial wrinkling

t = 104 t = 105t = 0 t = 104 t = 105t 0

t = 106 t = 107

010)0()0( −==σσ

0=R

μμ

01.02211 ==σσ

fμ


Effect of Substrate Elasticity 510−=f

R

μμ

σ0 = -0.02 (Δ), -0.01 (o), -0.005 (□)

Near-Equilibrium Patterns 510−=f

R

μμ

0

005.0)0(

22

)0(11

=

−=

σ

σ 005.0)0(22

)0(11 −== σσ

Same wavelength

Different patterns01.0)0(22

)0(11 −==σσ

02.0)0(22

)0(11 −==σσ


Wrinkle patterns under biaxial stressesσy

nkle

sar

alle

l wrin

σc

Pa

0 σxσc

Parallel wrinkles

cy νσσ

=−Transition

σ2

fcx

νσσ

=−stress:

10 μm σ1

Wrinkle patterns under non-uniform stresses1D h l d l

0σσ =

)(xfx =σ

σ x/E

f( )( )⎥⎦

⎤⎢⎣

⎡−=

aLax

x 2/cosh/cosh10σσ

1D shear-lag model:

0σσ y

Stre

ss,

R

fsf hhEa

μ=

Lx/hf

005.0/0 −=fEσ0== yx σσoutside (green):

L = 3750 hf L = 3250 hf L = 2500 hf L = 1250 hf

Wrinkling of rectangular filmsE i t (Ch i t l 2007) Si l ti (I d H 2009)Experiments (Choi et al., 2007): Simulations (Im and Huang, 2009):

500025001250

Stress relaxation in both x and y directions from the edges.The length of edge relaxation g gdepends on the elastic modulus of the substrate.No wrinkles at the corners, parallel wrinkles along the edges andwrinkles along the edges, and transition to zigzag and labyrinth in large films.

Wrinkling of periodically patterned filmsE i t (B d t l 1998)Experiments (Bowden et al., 1998):

Simulations (Im and Huang, 2009)

Wrinkling of Single-Crystal Thin Films30 nm SiGe film on BPSG at 750°C30 nm SiGe film on BPSG at 750 C(Hobart et al., 2000; Yin et al., 2002 & 2003;R.L Peterson, PhD thesis, Princeton, 2006)

100 nm Si on PDMS (Song et al., 2008)100 nm Si on PDMS (Song et al., 2008)

(100)(010)

Spectra of initial growth rate

σ2[010] θ =0: θ = 45°:

σ1[100]

Im and Huang, JMPS 2008.

Energy Spectra of Equilibrium Wrinkles

σ2[010] θ =0: θ = 45°:

σ1[100]

Evolution of orthogonal wrinkle patterns003.021 −==σσ 003.021 σσ

t = 0:RMS = 0.0057λave= 38.8

t = 105 :RMS = 0.0165λave= 44.8

t = 5 × 105 :RMS = 0.4185λave= 47.1

t 106 t = 107 : t = 108 :t = 106 :RMS = 0.4676λave= 50.8

t = 107 :RMS = 0.5876λave= 56.4

t = 108 :RMS = 0.5918λave= 56.6


Effect of stress magnitude


Checkerboard wrinkle pattern

0

0.5

1

0

20

40

0

20

40

−1

0.5

4040

00178.021 −==σσ RMS = 0.05286 and λave= 55.6

Transition of local buckling mode from spherical to cylindrical as the compressive stress/strain increases and/or as the wrinkle amplitude increases.


Transition of wrinkle patterns


Summary• Viscoelastic wrinkling

– Three-stage evolutionThree stage evolution– Power-law coarsening

Near equilibrium patterns– Near-equilibrium patterns

Di i kl tt• Diverse wrinkle patterns– Anisotropic patterns under biaxial stresses– Nonuniform patterns– Wrinkling of anisotropic crystal thin films

nonlinear dynamics of wrinkle growth and pattern formation

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