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Wrinkle Patterns of Anisotropic Crystal Films on Viscoelastic Substrates
In this paper we analyze evolution of wrinkle patterns of anisotropic crystal films on viscoelastic substrates. The effects of the residual stress state in the film and the anisotropic elastic property are emphasized. Analytical solutions for the initial growth kinetics and the equilibrium states are presented along with numerical simulations based on nonlinear evolution equations. Compared to wrinkling of isotropic elastic films, more ordered wrinkle patterns are predicted, including orthogonal, zigzag, parallel, and checkerboard patterns. Tranistion of the wrinkle patterns under various stress states is elucidated. Some related experimental works are referred to, but quantitative comparisons between the model the experiments await further studies.
Please cite this work as: Im, S.H., Huang, R., J. Mech. Phys. Solids 56, 3315-3330 (2008).
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excellent work
Dear Rui: This is an excellent work for the general area of wrinkling of thin films on compliant substrates. It is interesting to see that (1) the crystal anisotropy does not strongly affects the equilibrium wavelength of one-dimensional strips, but (2) determines the evolution of patterns for two-dimensional films. Rogers' experiments for buckled single crystal silicon ribbons on PDMS substrates verifies the first statement. I am just curious how to experimentally verify the second statement. It will be extremely challenging to well control the pre-strain with about 0.001% accuracy.
crystal anisotropy effect
Dear Hanqing,
Thanks for your comments. Indeed the effect of crystal anisotropy on the equilibrium wavelength shown in this paper is no more than a few per cent. This has to be a result of the relatively weak anisotropy of the SiGe crystal. The effect on the evolution and final wrinkle pattern for two-dimensional films is not surprising either. To elucidate the effect, we deliberately used a relatively small stress in Figs. 8 and 9. As you see from Fig. 10, the effect is much reduced by the stress isotropy at higher stress levels. I believe most of the experiments by Rogers' group were for very high stress or strain levels, in which case material nonlinearity of the substrate plays a role too as you and your colleagues have shown. A couple of other experiments (Peterson, 2006; and Yu et al., 2005) did show clearly the orientation of the wrinkles at both the early stage and the final pattern. It shall also be noted that the wrinkle pattern may depend on the loading history for both isotropic and anisotropic films, because there seem to exist many energetically equivalent (or similar) states. For example, the energies of the ordered zigzag pattern and the disordered labyrinth pattern in isotropic films were found to be very close (see Z. Huang, Hong, and Suo, Phys. Rev. E 70, 030601R, 2004).
RH
energetically similar states
Dear Rui: I agree that there exist many energetically similar states. Song et al. also showed that the energies of the checkerboard
and herringbone modes are very close at very small strains, which suggests that small imperfection may trigger different buckling patterns.
orthogonal pattern
Dear Prof. Huang,
It is a very good work! I think the orthogonal pattern you mentioned in the paper is very similar to the herringbone pattern. Am I right? What is the main difference between them? As you know, I have studied the anisotropic effect on the herringbone mode. Our results showed (1) the anisotropy has little effect on the equilibrium wavelength of the herringbone mode for Si/PDMS system (2) when [100] and [010] are the wavevectors, the system has the lower energy than the case when [110] and [-110] are the wavevectors, which agrees with your conclusion.
Jizhou Song
Re: orthogonal pattern
Dear Jizhou,
Thanks for your comment. Yes, the orthogonal pattern is similar to the herringbone pattern. There are two main differences: (1) a herringbone pattern does not have to be orthogonal, namely, the jog angle can be oblique or anything, such as the pattern we show in Fig. 9 (second and third in the last row); (2) an orthogonal pattern need not to be a perfect herringbone either. The overall pattern consists of stripes in two orthogonal directions, which does not necessarily have the long-range order as the herringbone pattern. Locally they may resemble an orthogonal herringbone, but with abundant defects here and there, as you can see from the last pattern in Fig. 8.
I beleive that our results of the equilirbium wavelength and energy agree closely with each other, although we made some different assumptions in the shear stress and displacements. One tricky issue is that the observed wrinkle patterns may not solely depend on the energetics. For example, the loading history and initial defects may affect the formation of the wrinkle patterns. Furthermore, a global energy minimum state (with all possible wrinkle patterns, ordered and disordered) is almost impossible to reach, both theoretically and practically. It may be an interesting topic to study the defect dynamics in the wrinkle patterns for both isotropic and anisotropic elastic films.
Best regards,
RH