@article{53807539fa8249ad8f2c8c207054cef7,
title = "An analytical study of two-dimensional buckling of thin films on compliant substrates",
abstract = "A stiff thin film on a heated compliant substrate may buckle when the system is cooled due to the thermal expansion mismatch between the film and substrate. Highly ordered and disordered herringbone patterns (wavy structures) then emerge as the system continues to cool. We have established an analytic approach to study one-dimensional, checkerboard, and ordered herringbone buckling patterns. The analytical approach gives the buckle wave length and amplitude in terms of the thin film and substrate elastic properties, thin film thickness, and the thermal strain. It is shown that the herringbone mode has the lowest energy, which explains why this mode is frequently observed in experiments. These classes of materials might be interesting as a route to high performance electronics with full, two-dimensional stretchability.",
author = "J. Song and Hanqing Jiang and Choi, {W. M.} and Khang, {D. Y.} and Y. Huang and Rogers, {J. A.}",
note = "Funding Information: We acknowledge the support from the National Science Foundation under Grant No. DMI-0328162, and the U.S. Department of Energy, Division of Materials Sciences under Award No. DEFG02-91ER45439, through the Frederick Seitz MRL and Center for Microanalysis of Materials at the University of Illinois at Urbana-Champaign. H.J. acknowledges the support from NSF No. CMMI-0700440, and Y.H. acknowledges the support from NSFC. FIG. 1. Schematic illustration of the process for fabricating two-dimensional wavy Si nanomembranes on a PDMS substrate: (a) Si membrane is bonded on the stretched PDMS; (b) formation of 2D wavy patterns when PDMS is relaxed; (c) Herringbone mode; and (d) top-down view of the herringbone mode. The parameters are illustrated in (c) and (d), including the short wavelength λ , long wavelength λ 2 , and jogs wavelength λ 1 , the amplitude B of the jogs in the plane of the film, and the jog angle θ . FIG. 2. Schematic illustrations of different buckling modes: (a) 1D mode, (b) checkerboard mode from Eq. (11) , and (c) herringbone mode from Eq. (25) with A = 1.0 μ m , k 1 = 0.309 μ m − 1 , B = 10 μ m , and k 2 = 0.139 μ m − 1 . FIG. 3. (a) Optical, (b) atomic force, and (c) scanning electron micrographs of a 2D wavy Si nanomembrane on PDMS. The thickness of the silicon is 100 nm . These images highlight the highly periodic nature of the wavy patterns. The parameters involved are the short wavelength λ , long wavelength λ 2 , and jogs wavelength λ 1 , the amplitude B of the jogs in the plane of the film, and the jog angle θ . FIG. 4. (a) Optical micrographs of a 2D wavy Si nanomembrane on PDMS and its different locations [(b)–(d)]. The thickness of the silicon is 100 nm . These images highlight the large variation of long wavelength in the same sample. FIG. 5. Ratio of total energy in the buckled state to its counterpart in the unbuckled state U total ∕ U 0 vs the long wavelength λ 2 for herringbone mode under 2.4% prestrain. The film thickness is 100 nm . FIG. 6. (a) Amplitude of jogs B , (b) jog angle θ , (c) short wavelength λ , and (d) amplitude A vs long wavelength λ 2 for herringbone mode under 0.5%, 1.5%, and 2.4% prestrains. The film thickness is 100 nm . FIG. 7. Ratios of energy in the buckled state to that in the unbuckled state U 0 vs the prestrain ε pre for 1D, checkerboard, and herringbone modes; (a) Total energy U total in the Si film∕PDMS substrate system; (b) strain energy U s , in the PDMS substrate; (c) bending energy U b in the Si film; and (d) membrane energy U m in the Si film. ",
year = "2008",
doi = "10.1063/1.2828050",
language = "English (US)",
volume = "103",
journal = "Journal of Applied Physics",
issn = "0021-8979",
publisher = "American Institute of Physics Publising LLC",
number = "1",
}