Nonlinear Dynamics of Wrinkle Growth and Pattern Formation in Stressed Elastic Thin Films Se Hyuk Im and Rui Huang Center for Mechanics of Solids, Structures and Materials Department of Aerospace Engineering and Engineering Mechanics The University it 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., 2000 Yin et al., 2002 & 2003 Peterson et al., 2006 Yu et al., 2005 & 2006 SiGe BPSG 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 depends on the elastic modulus and thickness of substrate. 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 i wavelength The fastest t growing depends on the elastic modulus and thickness of substrate. wavelength dominates the initial growth, then coarsening over time. Energetics Spontaneous energy minimization. Energy dissipation by viscous flow.
Al film on PS: viscoelastic wrinkling (Yoo and Lee, 2003-2005)
A model for viscoelastic wrinkling Viscoelastic relaxation modulus Wrinkle Amplitude Rubbery State (II) Glassy State Equilibrium states at the elastic limits (thick substrate): ε c 1 3 E 4 E f = s A eq = h f λ = 2πh eq ε ε f c 2 / 3 1 E f 3E s 1/ 2 1/ 3 0 ε R (I) ε G Evolution of wrinkles: (I) Viscous to Rubbery (II) Glassy to Rubbery Compressive Strain Huang, JMPS 2005.
K w t 2 = K s Scaling of evolution equation 3 f (1 2ν s ) μ f h hs = 12(1 ν )(1 ν ) η f 2 w + s F F ( σ w) Rw 1 2ν h s sh = 2(1 ν ) η s s f μ R = R η s Elastic film Viscoelastic layer Rigid substrate Fastest growing mode: λ = m 2 2πL 1 A = A 0 t exp 4 τ 1 Length and time scales: L 1 = K Fσ 0 τ 1 = K ( Fσ ) 2 0 critical stress: σ c = 4KR F 2 Equilibrium state: ( σ 0 < σ c ) 1/ 4 K λ eq = 2 2 σ π > λm R 0 A eq = h f 1 3 σ c
Three-Stage Viscoleastic Wrinkle Evolution Stage I Stage II Stage III wavelength amplitude t 1 t 2 Time I: Exponential growth, dominated d by the fastest t growing mode II: Coarsening (λ m λ eq, A ) III: Nearly equilibrium (with local ordering) Im and Huang, JAM 2005 and PRE 2006.
Power-law Coarsening Length and time scaling: L 1 = K Fσ 0 τ 1 = K ( Fσ ) 2 0 σ 0 = -0.02 (Δ), -0.01 (o), -0.005 ( ), -0.002 ( ), -0.001 ( ) Blue: uniaxial; Red: equi-biaxial On a compressible, thin viscoelastic substrate: λ ~ t 1/4 ; On an incompressible, thin viscoelastic substrate, λ ~ t 1/6 ; On a thick VE substrate, λ ~ t 1/3. The substrate elasticity slows down coarsening as it approaches the equilibrium state. Huang and Im, PRE 2006.
Numerical simulation: uniaxial wrinkling t = 0 t =10 4 t =10 5 t = 10 6 t = 10 7 (0) (0) σ = 0.01, σ 0 11, 22 = μr μ μ f = 0 Huang and Im, PRE 2006.
Numerical simulation: equi-biaxial wrinkling t = 0 t =10 4 t =10 5 t = 10 6 t = 10 7 σ (0) (0) 11 = σ 22 = = 0.0101 μr μ μ f = 0 Huang and Im, PRE 2006.
Effect of Substrate Elasticity μ R μ f = 10 5 σ 0 = -0.02 (Δ), -0.01 (o), -0.005 ( )
(0) = 0 R 5 Near-Equilibrium Patterns σ11 = 0.005 (0) (0) σ = σ = 0. 005 σ (0) 22 11 22 μ μ f = 10 Same wavelength σ (0) (0) 11 = σ 22 = 0.01 Different patterns Huang and Im, PRE 2006. σ (0) (0) 11 = σ 22 = 0.02
Wrinkle patterns under biaxial stresses σ y Pa arallel wrin nkles σ c Parallel wrinkles 0 σ c σ x σ 2 Transition stress: σ σ y x σ σ c c = ν f 10 μm σ 1
Wrinkle patterns under non-uniform stresses σ x = f (x) σ y = σ 0 1D shear-lag model: σ x ( x / a) ( ) L / 2 cosh = σ 0 1 cosh a E a = f h μ s R h f σ x /E f Stress, L outside (green): σ x = σ y = 0 σ 0 / E f = 0.005 x/h f L = 3750 h f L = 3250 h f L = 2500 h f L = 1250 h f
Wrinkling of rectangular films Experiments (Choi et al., 2007): Simulations (Im and Huang, 2009): 1250 2500 5000 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.
Wrinkling of periodically patterned films Experiments (Bowden et al., 1998): Simulations (Im and Huang, 2009)
Wrinkling of Single-Crystal Thin Films 30 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) (010) (100)
Spectra of initial growth rate σ 2 [010] θ =0: θ = 45 : [100] σ 1 Im and Huang, JMPS 2008.
Energy Spectra of Equilibrium Wrinkles σ 2 [010] θ =0: θ = 45 : [100] σ 1
Evolution of orthogonal wrinkle patterns σ 1 = σ 2 = 0.003 003 t = 0: RMS = 0.0057 λ ave = 38.8 t = 10 5 : RMS = 0.0165 λ ave = 44.8 t = 5 10 5 : RMS = 0.4185 λ ave = 47.1 t = 10 6 : t = 10 7 : t = 10 8 : RMS = 0.4676 λ ave = 50.8 RMS = 0.5876 λ ave = 56.4 RMS = 0.5918 λ ave = 56.6 Im and Huang, JMPS 2008.
Effect of stress magnitude Im and Huang, JMPS 2008.
Checkerboard wrinkle pattern 1.5 0 0.5 1 0 20 40 40 20 0 σ = σ = 0.00178 RMS = 0.05286 and λ ave = 55.6 1 2 Transition of local buckling mode from spherical to cylindrical as the compressive stress/strain increases and/or as the wrinkle amplitude increases. Im and Huang, JMPS 2008.
Transition of wrinkle patterns Im and Huang, JMPS 2008.
Summary Viscoelastic wrinkling Three-stage evolution Power-law coarsening Near-equilibrium equilibrium patterns Diverse wrinkle patterns Anisotropic patterns under biaxial stresses Nonuniform patterns Wrinkling of anisotropic crystal thin films