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Schematic illustration of the process for fabricating buckled, orwavy,single-crystal Si ribbons (silver) on a PDMS (blue) substrate (upper two frames).The lower two frames illustrate the response of these structures to strainsapplied to the PDMS.0Fig. 2. Optical (A), scanning electron (B), and atomic force (C) micrographsof wavy, single-crystal Si ribbons. For A and C, 20- m-wide and 100-nm-thickSi ribbons were used with PDMS prestrained to 28%for A and 23%for C.For B, 30- m-wide, 150- m-long, and 100-nm-thick Si ribbonswere usedwitha PDMS prestrain of 15%. c 14 3E sE f 2/3is defined as the critical buckling strain, or the minimum strainneeded to induce buckling.Because of the small deformation approximations and linearstress–strain behavior used to deriveEqs. 1 and 2, these results applynot only to the buckling process itself, but also to the response ofthe buckled system to applied strains, applied,by pre applied.Inother words, Eq. 1 suggests not only that the buckling wavelengthis independent of prestrain, but also that this wavelength will notchange with strain (tensile or compressive) applied ( applied)tothesystem after formation of the buckled structure, provided that thetotal strain ( pre applied) is larger than c. Eq. 2 can be used todescribe this postbuckling behavior by simply replacing pre by pre applied.Eqs. 1 and 2 imply displacements that are tangential to the localsurface relief, yielding a displacement trajectory that has the shapeof a wave whose wavelength is fixed. The prestrain-independentwavelength in Eq. 1 and, by implication, the wavy motion trajectoryhave been widely applied to many experimental systems, with somelevel of qualitative or, in some cases, claimed quantitative agree-ment. These previous studies do not, however, provide sufficientprecision to test rigorously the predictions of Eqs. 1 and 2 becausethey involve experimental uncertainties due to some combinationof factors including poorly defined film/substrate interfaces, un-knownmechanical properties in the films or substrates, poor spatialuniformity in the critical dimensions and mechanical properties,and/or the formation ofmicro- or nanocracks during filmdepositionor strain relaxation. The system of Figs. 1 and 2 avoids theselimitations because of the highly controlled nature of the single-crystal films and the strong bonding to the elastomeric supports.The most direct experimental test of the existing models involvesthe measurement of wavelengths in the well controlled systems ofFigs. 1 and 2 at various pre. All strains, for this case and the othersthat follow, were determined from the measured contour contourand wavelengths of the buckled ribbons and given by ( contour )/ . Fig. 3 shows the results for the case of ribbons of single-crystalsilicon with thicknesses of 100 nmon PDMS. Themean wavelength, 15 m, is comparable to o ( 18 m) evaluated by using Eq. 1with literature values for the mechanical properties (Ef 130 GPa,Es 1.8 MPa, vf 0.27, vs 0.48) (25, 48). The measurementsshow, however, a qualitative behavior characterized by a clear andsystematic decrease in wavelength with increasing prestrain, con-trary to the prediction of Eq. 1. Hints of similar variations inwavelength have also been reported for layers of polystyrene onPDMS substrates when the prestrain varies from 0% to 10% (1),and in platinum films on rubber substrates for prestrains of 400%(32). This strain-dependent wavelength behavior has also beenobserved in postbuckling studies of single-crystal Si and GaAsribbons in the layouts of Figs. 1 and 2, where the wavelength variessystematically and in linear proportion to the applied strains (10,49). These discrepancies between existing theory and experimentshave been attributed to various effects, including nonlinearities inthe stress–strain responses of the film or substrate materials (1),partial delamination of the films from the substrates, and finite sizeeffects in the films (10). Detailed experimental studies indicate,however, that none of these explanations is valid for the case of thesingle-crystal systems of the type presented here. First, nonlineari-ties in the stress–strain behavior of the materials might be expectedto lead to nonsinusoidal displacement profiles in the wavy struc-tures, in contrast to the highly sinusoidal behavior observed inexperiment, such as that shown in Fig. 3. In addition, independentmeasurements show that the elastic modulus of PDMS is constant,to a good approximation, for strains of up to several tens of apercent (50). Silicon and gallium arsenide single crystals exhibitlinear responses up to strains that approach the fracture point (51).Second, detailed imaging studies such as those in Fig. 2 show thatbonding in well designed systems can be extremely good. Third,finite size effects are likely unimportant because qualitativelysimilar variations in wavelength are observed in systems with ribbonwidths between 2 and 100 m, with thicknesses between 20 and 320nm, with lengths between 5 and 15 mm, and on substrates withthicknesses between 0.5 and 5 mm.In the following, we present a buckling theory that accounts forfinite deformations and geometrical nonlinearities to yield a quan-titatively accurate description of the system. This buckling theory isdifferent from previous models in the following three importantaspects:1. The initial strain-free (or stress-free) configurations for thesubstrate and film are different (i.e., the film is free of strainin the top frame of Fig. 1, whereas the substrate is free ofstrain in the second frame of Fig. 1, except near the film–substrate interface).B0 20406080-3.0-1.50.01.53.0 t h g i e H ( µm)Distance ( µm)0.63.36.811.616.328.430 µmFig. 3. The wavelength decreases as the prestrain increases, and the behav-ior of the buckled thin film is highly sinusoidal. (A) Atomic force micrographsof buckled Si ribbons (100 nm thickness) on PDMS, formed with variousprestrains (indicated on the left in percent). The red and green triangles andthe vertical dashed lines define particular relative locations on the samples, toillustrate more clearly the changes. The wavelength systematically decreasesas the prestrain pre increases. (B) Line cut profiles of a representative ribbonfor the cases of pre 28.4% (blue) and pre 0.6% (red). The symbols aremeasured data; the lines are sinusoidal fits. 2. The strain-displacement relation in the substrate (as well as thefilm) is nonlinear.3. The stress–strain relation in the substrate is characterized by thenonlinear neo-Hookean constitutive law.Some details of this analysis, with derivations of expressions for thewavelengths and amplitudes of the wavy structures, appear inMaterials and Methods.Wavelength and Amplitude in Initial Buckling. As with the previousanalyses, the new theory predicts purely sinusoidal buckling dis-placements when the prestrain, pre, exceeds the critical strain, c.The wavelength, however, is different from that given by existingtheories and can be written 0 1 pre 1 1/3 , [3]where 0 is the wavelength in Eq. 1 and 5 pre(1 pre)/32. Asshown in Fig. 4, depends on prestrain and quantitatively agreeswith the experimental data without any parameter fitting, when thefollowing values are used for the film thickness and modulus andsubstrate modulus: h 100 nm, Ef 130 GPa, Es 1.8 MPa, vf 0.27, vs 0.48. An intuitive understanding of Eq. 3 is as follows: 0/(1 pre) represents the change of wavelength expected basedon simple accordion bellowsmechanics; 1/(1 )1/3, which dependsonly on prestrain, results from the geometrical nonlinearity (finitedeformation) and nonlinear constitutivemodel. For small prestrain,the value of approaches 0, although it retains the same functionalvariation with pre down to prestrains arbitrarily close to
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