4 h) a 10:1 (weight)mixture of base resin to crosslinking agent against a surfacefunctionalized silicon wafer. Flat slabs of PDMS (0.5–5 mm thick)formed in this manner served as the substrates. Precision transla-tion stages were used for stretching and compressing these sub-strates. The process for integrating the Si ribbons with the PDMSsubstrates began with exposure of the PDMS to UV (low-pressuremercury lamp, BHK, 173  W/cm2 from240 to 260 nm, at a distanceof  10 mm) induced ozone for 1–3 min to create surface –OHgroups, with the substrates in the stretching stage. Placing aprocessed SOI wafer against the PDMS and then removing thewafer transferred the Si ribbons to the PDMS through the action ofstrong, covalent –O–Si–O– bonds that form at the interface. Toavoid bonding between the PDMS and the silicon wafer of the SOIsubstrate (i.e., the handle wafer), the contact between the PDMSand the processed SOI was limited to  1 min. For this contactduration, the PDMS/Si adhesion was sufficiently strong that re-moving the PDMS lifted the Si ribbons fromthe Si handle wafer butsufficiently weak that the PDMS did not stick to the exposed regionsof the wafer. After peeling back the PDMS with the Si ribbons onits surface, the bonding was allowed to run to completion (

10 minat room temperature) before releasing or applying strains. Thetranslation stage for controlling the strain was designed to fit intooptical, scanning electron, and atomic force microscopes, for thepurpose of in situ observation. Optical microscopy was used todetermine the wavelength by measuring the distance between twopoints in the image and piding by the number of waves in between.Atomic force microscopy (DI 3100; Veeco) was used to determinethe amplitudes and to verify the wavelength measurements.Analysis. Fig. 7 defines the parameters and variables for three stagesof controlled buckling in Fig. 1. Fig. 7 Left (stage I) illustrates theinitial state of the PDMS before prestretching with original lengthL0, which represents its zero strain energy configuration. Fig. 7Middle (stage II) shows the stretched PDMS attached to anundeformed Si film. After prestretching, the length of the PDMSis L0(1    pre), which is also the original length of the undeformedSi film. Fig. 7 Middle also gives the zero strain energy state of theSi. Relaxing the prestrain will buckle the Si film, as shown in Fig.7 Right (stage III). The positions in the substrate in stage I (and III)are related to stage II by x 

 x(1  pre). The normal displacementon the substrate is w   A cos(2 x/ ). The thin-film membraneenergy and bending energy per unit length can be obtained fromthevon Karman plate theory (52) and are given byUm   hE  f2  2A2 2 1  pre2   pre 2andUb    4h3E  f3A2 4 1  pre4 ,respectively. [Details are given in the supporting information (SI).]The total energy in the film is (Um   Ub)L0(1    pre), where L0(1   pre) is the film length at its strain-free stage.The substrate is subjected to the surface displacement w   Acos(2 x/ ) and prestrain  pre. For large deformation, the Greenstrains EIJ in the substrate are related to the displacements u by (53)EIJ   (uI,J   uJ,I   uK,IuK,J). The stress–strain relation of polymeris nonlinear and is usually characterized by the neo-Hookeanconstitutive law for the second Piola–Kirchhoff stress TIJ andGreenstrain (53). The force equilibrium equations (53) are (TJKFiK),J   0,where FiK is the deformation gradient.The governing equations for the substrate become highly non-linear. Because the amplitude A is much less than the wavelength ,weidentify A/  as a small parameter and expand the displace-ment field to the order of A/ ,(A/ )2and (A/ )3. For an incom-pressible substrate, the energy per unit length is Us   ( /3)Es(A2/ )(1 (5/32)( 2A2/ 2)). (Details are given in the SI.)Minimizationof total energy per unit length, Utot   (Um   Ub)(1    pre)   Us,i.e.,