Topic F — Film Stress, Wafer Bow and Wrinkling

thermal-mismatch stress and Stoney bow · wrinkling of a stiff film on a soft substrate (linear buckling with SLEPc) · tier: standard → advanced

Motivation

Thin films deposited at high temperature or cured with shrinkage carry residual stress: on a stiff wafer it bends the wafer (the bow measured by wafer-curvature tools), on a soft substrate it makes the film wrinkle — a defect in displays and devices, but also a tool for bio-inspired surfaces with controlled wetting, adhesion and optics. You will compute both, from the same film/substrate model.

Related fields: thin films for semiconductor and display devices: bio-inspired wrinkled and hierarchical structures, organic semiconductors, solution-processed films. Course weeks: 3 (thermal stress, generalized plane strain), 6 (linear buckling with SLEPc), 8 (plates), 9 (periodic constraints, shells).

Model

Part 1 — wafer bow (standard). Film (thickness h_f, E_f,\nu_f,\alpha_f) on a substrate (h_s\gg h_f), cooled by \Delta T. Plane strain / generalized plane strain strip (Week 3) or an axisymmetric disk; compute the curvature and the film stress. Compare with Stoney and with the exact bilayer (Timoshenko) formula as h_f/h_s grows.

Part 2 — wrinkling (advanced). A film on a compliant substrate (thickness H_s\gg h_f, E_s\ll E_f), in-plane compressive pre-strain \varepsilon_0 in the film (from cooling or swelling). Pre-buckling solve, geometric stiffness, (-K_G)\phi=\theta K\phi (Week 6), on a 2D plane-strain cell of width \ell with periodic lateral boundaries (dolfinx_mpc, Week 9) — the critical strain \varepsilon_c(\ell), its minimum over \ell gives the wrinkle wavelength. Stretch: post-buckling (large-strain, load steps from the mode as an imperfection).

Design outputs: bow and film stress against h_f/h_s and \Delta T; critical strain and wavelength against E_f/E_s.

Suggested parameters

Bow: Si wafer 500 µm, film 0.5–50 µm, \Delta T=-300 K, oxide/metal/polymer films. Wrinkling: E_f/E_s = 10²–10⁵ (PDMS substrate 1–3 MPa, polymer or oxide film), h_f = 100 nm–10 µm, substrate depth at least 5 wavelengths, \nu_s\approx0.48 (mind locking).

Required tasks

  1. Bilayer strip under \Delta T: curvature against h_f/h_s; Stoney and Timoshenko comparison.
  2. Wrinkling cell: \varepsilon_c(\ell) for one stiffness ratio, the minimum and the wavelength.
  3. Stiffness-ratio study of \lambda and \varepsilon_c; effect of a finite substrate depth.
  4. Mesh independence (elements across the film thickness!).

Verification specific to this topic

  • Stoney: \kappa=\frac{6\,\sigma_fh_f(1-\nu_s)}{E_sh_s^2} for h_f\ll h_s; the exact bilayer formula (Timoshenko 1925) for any thickness ratio.
  • Wrinkling (semi-infinite substrate): \lambda=2\pi h_f\Big(\frac{\bar E_f}{3\bar E_s}\Big)^{1/3}, \varepsilon_c=\frac14\Big(\frac{3\bar E_s}{\bar E_f}\Big)^{2/3}, \bar E=E/(1-\nu^2) (Chen & Hutchinson 2004; Huang, Hong & Suo 2005) — valid for E_f/E_s\gg1; report the deviation as the ratio decreases.
  • Stress-free test: equal expansion coefficients give no stress and no bow (Week 3).
  • MMS for the bilayer elasticity with a discontinuous coefficient (interface aligned with the mesh: optimal rates).

Stretch goals

  1. Post-buckling amplitude A=h_f\sqrt{\varepsilon_0/\varepsilon_c-1} (Huang et al.) from a large-strain computation.
  2. Shell model of the film (Week 9) on a solid substrate.
  3. Delamination of the film edge with a cohesive zone (Week 10).

Milestones

M1 bilayer bow with Stoney/Timoshenko · M2 periodic cell pre-buckling · M3 SLEPc buckling and \varepsilon_c(\ell) · M4 stiffness-ratio study · M5 depth effect.

Pitfalls

  • One element across the film with P1: the film locks in bending (Week 1) and \varepsilon_c is far too high — P2 and at least two cells across.
  • \nu_s\to0.5: volumetric locking in the substrate — P2, or a mixed (\mathbf u,p) formulation, and a check with \nu_s=0.45.
  • Without periodic boundary conditions, edge effects select the wavelength — compare with the periodic cell before believing a pattern.

References

  • X. Chen, J. W. Hutchinson, “Herringbone buckling patterns of compressed thin films on compliant substrates”, J. Appl. Mech. 71 (2004) 597–603.
  • Z. Y. Huang, W. Hong, Z. Suo, “Nonlinear analyses of wrinkles in a film bonded to a compliant substrate”, J. Mech. Phys. Solids 53 (2005) 2101–2118.
  • L. B. Freund, S. Suresh, Thin Film Materials, Cambridge 2003 — Stoney, bilayer curvature.
  • S. Timoshenko, “Analysis of bi-metal thermostats”, J. Opt. Soc. Am. 11 (1925) 233–255.