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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 - 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 (Jones et al., MRS Symp. Proc. 769, H6.12, 2003 ) 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 buckling Critical load: h c 121 L 2 • Buckling relaxes compressive stress • Bending energy minimizes at long wavelength Other equilibrium states: energetically unfavorable 2 On elastic substrates Elastic substrate • Deformation of the substrate disfavors wrinkling of long wavelengths and competes with bending to select an intermediate wavelength Wrinkling: short wavelength, on soft substrates, no delamination Buckling: long wavelength, on hard substrates, with delamination Critical Condition for Wrinkling Compressive Strain, - 0.025 Thick substrate (hs >> hf): 0.02 0.015 wrinkling 1 3Es 1 f 2 c 4 E f 0.01 flat film 0.005 0 0 0.002 0.004 0.006 0.008 Stiffness Ratio, E /E s 0.01 f 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. 2/3 Equilibrium Wrinkle Wavelength 100 Wrinkle Wavelength, /h f Thick substrate (hs >> hf): 80 1/ 3 Ef 2h f 3 E s 60 40 Measure wavelength to determine film stiffness 20 0 0 0.002 0.004 0.006 0.008 Stiffness Ratio, E /E s 0.01 f 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. Equilibrium Wrinkle Amplitude Wrinkle Amplitude, A/h f 3 Thick substrate (hs >> hf): 2.5 1/ 2 1 2 A h f 1 c 2 1.5 1 Measure amplitude to determine film stress/strain. 0.5 0 0 2 4 6 Compressive Strain, / 8 10 c 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 Fastest mode Growth Rate s sm Viscous layer Rigid substrate 0 c m Euler buckling A A0 expst m h f 1 (For hs >> hf) • Viscous flow controls the growth rate: long-wave wrinkling grows slowly, and an intermediate wavelength is kinetically selected. 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 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). Wrinkling on Viscoelastic Substrates Cross-linked polymers Wrinkle Amplitude Rubbery State Glassy State 0 R G Evolution of wrinkles: (I) Viscous to Rubbery (II) Glassy to Rubbery Compressive Strain (Lee at al., 2004) Wrinkling Kinetics I: R G Growth Rate Fastest mode Wrinkles of intermediate wavelengths grow exponentially; The fastest growing mode dominates the initial growth. 0 m A(t ) A0 expst For hs >> hf : m h f 1 The kinetically selected wavelength is independent of substrate! Wrinkling Kinetics II: G Instantaneous wrinkle at the glassy state: 1/ 3 Ef 0 2h f 3 E G 1/ 2 A0 h f 1 G Kinetic growth at the initial stage: A(t ) A0 Bexp(t ) 1 Long-term evolution: 1/ 3 0 Ef 2h f 3 E R 1/ 2 A0 A h f 1 R Numerical Simulation t=0 w/h f 0.1 0 -0.1 0 200 400 600 0 x/hf 0.1 Growing wavelengths f w/h t= 1104 0 -0.1 0 200 0 w/h f 200 400 600 x/hf 0 50 100 Wavelength, L/hf Equilibrium wavelength f w/h 50 100 Wavelength, L/hf Coarsening 2 t= 600 0 -2 0 1107 400 x/hf 2 t = 1105 50 100 Wavelength, L/hf 0 -2 0 200 400 600 0 50 100 Wavelength, L/h Evolution of Wrinkle Wavelength 70 50 /Ef=0.0001 /Ef=0.00001 Wavelength, L/h f Wavelength, L/h f /Ef=0.0001 40 30 Lm = 26.9hf 20 0 2 4 6 Normalized time, t/ 8 10 x 10 4 60 Leq = 60.0hf /Ef=0.00001 50 40 Leq = 33.7hf 30 Lm = 26.9hf 20 4 10 10 5 10 6 Normalized time, t/ Initial stage: kinetically selected wavelengths Intermediate stage: coarsening of wavelength Final stage: equilibrium wavelength at the rubbery state 10 7 Evolution of Wrinkle Amplitude 1.5 1 /Ef=0.0001 Aeq = 1.63hf 1 0.1 RMS RMS /Ef=0.00001 Aeq = 0.619hf 0.5 /Ef=0.0001 0.01 /Ef=0.00001 0 2 4 6 Normalized time, t/ 8 10 x 10 4 0 4 10 10 5 6 Normalized time, t/ Initial stage: exponential growth Intermediate stage: slow growth Final stage: saturating 10 10 7 2D Wrinkle Patterns I t=0 t = 106 t = 104 t = 107 t = 105 2D Wrinkle Patterns II t=0 t = 5X106 t = 105 t = 2X107 t = 106 2D Wrinkle Patterns III t=0 t = 106 t = 104 t = 107 t = 5X105 On a Patterned Substrate t=0 t = 104 t = 106 t = 107 t = 105 Circular Perturbation t=0 t = 5105 t = 104 t = 106 t = 105 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, crosslinking – Other effect at nanoscale? • Nonlinear elastic/viscoelastic behavior – Nested wrinkles?