Dynamics of Surface Pattern Evolution 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

Self-Assembled Surface Patterns Yang, Liu and Lagally, Granados and Garcia, Tabe et al., 2002 & 2003.

Wrinkle Patterns Bowden et al., 1998 & Muller-Wiegand et al., Stafford et al., Cahill et al., 2002.

Part I: Surface Diffusion-Controlled Patterns

Surface Instability of Stressed Solid Asaro and Tiller (1972); Grinfeld (1986); Srolovitz (1989)…… Competition between surface energy and strain energy leads to a critical wavelength: (~300nm) Chemical potential on surface: Surface evolution: Linear analysis: Nonlinear analysis: develop crack-like grooves or cusps.

Instability of Epitaxial Films Spencer, Voorhees and Davis (1991); Freund and Jonsdottir (1993); Gao (1993)…… The film is stressed due to lattice mismatch between the film and the substrate (e.g., Ge on Si). Stress relaxation leads to formation of dislocations and/or surface roughening. Linear analysis: similar to that of stressed solids Nonlinear analysis: self-assembly of quantum dots How to control the size and order of quantum dots?

The Base Model Surface chemical potential: Surface flux: Equation of surface evolution: Nonlinear terms arise from wavy surface as the boundary condition for the stress field and from the wetting effect.

Stress Analysis Boundary condition on the surface: (In the order of ) Zeroth-order: First-order:(B.C.)

Linear Evolution Equation Fourier transform Length scale: Time scale: Critical wavelength: Fastest growing wavelength:

Nonlinear Stresses Second-order:(B.C.) Nonlinear evolution equation:

Spectral Method Fourier transform of the nonlinear equation: Numerical simulations: Calculate spatial differentiation in the Fourier space Calculate nonlinear terms in the physical space Communicate between physical and Fourier spaces via FFT and its inverse Semi-implicit integration:

1D Simulations Consideration of nonlinear stress leads to unstable evolution and formation of deep grooves. Linear equationNonlinear equation

2D Simulations Downward blow-up instability: nanopits? t = 50 t = 85 t = 0 t = 20

Effect of Wetting Linear evolution equation with wetting: Transition of surface energy (Spencer, 1999):

Linear Analysis Critical film thickness: Thick films: no effect; Very thin films: stabilized. Typical values:

1D Simulations Stable growth Coarsening Blow-up instability

2D Simulations Upward blow-up instability: nano whiskers? t = 0t = 50t = 200 t = 259t = 260

Nonlinear Stress + Wetting: 1D Simulation Stable growth Coarsening No blow-up instability!

t = 0 t = 1000t = 500 t = 200 t = 250 t = 50 Nonlinear Stress + Wetting: 2D

Part I: Summary Nonlinear stress field leads to downward blowup instability. Wetting effect leads to upward blowup instability. Combination of nonlinear stress and wetting stabilizes the evolution. Nonlinear analysis of surface diffusion-controlled pattern evolution in strained epitaxial films:

Part II: Compression-Induced Wrinkle Patterns

(Lee at al., 2004)

Freestanding film: Euler buckling Critical load: Other equilibrium states: energetically unfavorable Buckling relaxes compressive stress Bending energy favors 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 Thick substrate (h s >> h 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.

Equilibrium Wrinkle Wavelength Thick substrate (h s >> h 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. Measure wavelength to determine film stiffness

Equilibrium Wrinkle Amplitude Thick substrate (h s >> h f ): 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?

Wrinkling on Viscoelastic Substrates Cross-linked polymers Compressive Strain Wrinkle Amplitude 0 Evolution of wrinkles: (I)Viscous to Rubbery (II)Glassy to Rubbery Rubbery StateGlassy State

Wrinkling Kinetics I: Fastest mode m 0 Growth Rate Wrinkles of intermediate wavelengths grow exponentially; The fastest growing mode dominates the initial growth. For h s >> h f : The kinetically selected wavelength is independent of substrate!

Wrinkling Kinetics II: Instantaneous wrinkle at the glassy state: Kinetic growth at the initial stage: Long-term evolution:

Evolution Equations Polymer Substrate Metal film

t = 0 t = 1  10 4 Numerical Simulation t = 1  10 5 t = 1  10 7 Growing wavelengths Coarsening Equilibrium wavelength

Evolution of Wrinkle Wavelength Initial stage: kinetically selected wavelengths Intermediate stage: coarsening of wavelength Final stage: equilibrium wavelength at the rubbery state

Evolution of Wrinkle Amplitude Initial stage: exponential growth Intermediate stage: slow growth Final stage: saturating

t = 0t = 10 4 t = 10 5 t = 10 7 t = D Wrinkle Patterns I

t = 0t = 10 5 t = 2X10 7 t = 10 6 t = 5X10 6 2D Wrinkle Patterns II

t = 10 7 t = 5X10 5 t = 10 6 t = D Wrinkle Patterns III t = 0

t = 10 4 t = 10 5 t = 10 6 t = 10 7 On a Patterned Substrate

Circular Perturbation t = 0t = 10 4 t = 10 5 t = 5  10 5 t = 10 6 t = 10 7

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

Large-Cell Simulation t=3X10 7 t=1X10 7 t=1X10 6 t=1X10 5 t=8X10 4 t=1X10 4 t=3X10 4 t=5X10 4

Acknowledgments Co-workers: Se Hyuk Im, Yaoyu Pang, Hai Liu, S.K. Banerjee, H.H. Lee, C.M. Stafford Funding: NSF, ATP, Texas AMRC Thank you !