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The measuredwavelength increases for tension and the measured amplitudedecreases, reaching zero once the tensile strain reaches the pre-strain. The finite-deformation buckling theory agrees well withexperiments for both amplitude and wavelength. The existingmechanics models also capture the amplitude trend but deviatefrom the experimental results for large tensile strain (
10%).Comparison with Accordion Mechanics. The finite deformationmodel introduced here captures quantitatively all of the experi-mental observations. A point of interest, however, is that thevariations in amplitude and wavelength, with both prestrain andapplied strain, can be captured with reasonable accuracy, for thesystems studied here, with the very simple accordion model. In thismodel, the wavelength varies according to a simple rule of 0(1 pre applied), where 0 is the buckling wavelength at theonset of buckling given by Eq. 1. The value of 0 cannot, of course,be determined with this accordion model, but it can be treated asa fitting parameter to describe experimentally measured data. Forthe amplitude, the accordion model assumes a constant contourlength that provides an equation in integral form to compute theamplitude via0 1 4 2A2 2 sin2 2 x dx 0.Figs. 4 and 6 show the predicted variations in wavelength andamplitude with pre and applied, based on this model. Although theresults do not match exactly the experiment, the degree of agree-ment is remarkable, thereby indicating that the accordion picture ofthe mechanics of this system provides a good approximation of itsqualitative behavior.ConclusionsIn summary, this work presents experimental data that reveal manydetails of the mechanical behavior of buckled thin films on com-pliant supports, with a focus on ribbons of single-crystal silicon onPDMS. Theoreticalmodeling, performed in amanner that removescertain approximations implemented in previous models of thisclass of system, quantitatively reproduces the observations. Theresults show that the structures behave, approximately, with themechanics of an accordion bellows, in which strains are accommo-dated through changes in the amplitudes and the wavelengths of thebuckled geometries. These conclusions and the detailed analysesare important for themany envisioned applications for buckled thinfilm/substrate systems.Materials and MethodsFabrication and Measurements. The single-crystal Si (100) ribbonswere derived from SOI wafers (SOItec), with top Si thicknessesbetween 20 and 300 nm. The first step in the fabrication involvedpatterning a layer of photoresist (AZ5214) in the geometry of theribbons (2–100 m wide, separated by 2–100 m; 5–15 mm inlength) on top of an SOI wafer using conventional photolitho-graphic methods (Karl Suss MJB-3 contact mask aligner). Etchingthe exposed top Si layer by SF6 reactive ion etching (PlasmaTherm)defined the ribbons. Undercut etching of buried oxide layer withHF released the Si ribbons and left them resting on the underlying0 L () 0 1 pre L ε +0 L1 x3 x1 x3 xStretch Release strain and bucklingZero strain energy in film Zero strain energy in substrate1 x′3 x′Fig. 7. Three sequential configurations for the thin film/substrate bucklingprocess. (Left) Undeformed substrate with the original length L0, which rep-resents the zero strain energy state. (Middle) Substrate deformed by theprestrain and the integratedfilm,which represents its zero strain energy state.(Right) Deformed (buckled) configuration.01020300.00.51.01.52.0Membrane strain) % ( n i a r t SPrestrain εpre (%)Peak strainAB0102030101520( h t g n e l r u o t n o C µ ) mPrestrain εpre (%)Fig. 5. Strains and contour length of the buckled thin film. (A) Membraneand peak strain in the Si as a function of prestrain for a system of buckled Siribbons (100 nm thickness) on a PDMS substrate. The membrane strain is asmall and constant throughout this range. (B)Measured contour length of thebuckled Si structures as a function of prestrain. The nearly constant contourlength is consistent with a small membrane strain.0120 5 10 15051015 Experiment Finite-Deform. Previous Model Accordion ModelApplied strain εapplied (%)e d u t i l p m A A ( µ ) m h t g n e l e v a W λ ( µ ) mλ AFig. 6. Wavelengthandamplitudeofbuckledstructuresof Si (100nmthickness)onPDMS formedwithaprestrainof 16.2%, as a functionof the appliedstrain. Themeasured wavelength increases for tensile strain and the measured amplitudedecreases, reaching zero once the tensile strain reaches the prestrain. The finite-deformation buckling theory yieldswavelengths and amplitudes that both agreewell with experiments. Results from previous mechanics models (i.e., small de-formation limit) and the simple accordion model are also shown.APPLIED PHYSICALSCIENCES Si substrate. The PDMS (Sylgard 184;Dow) substrates were formedby casting and thermally curing (70°C for