Our artificial mechanical systems, chips, and computers need enough stiffness and strength1–3, while the bio-systems and naturally intelligent brain are soft with wrinkles and folds4–7. When stiff meets soft, stiff/soft bilayers are prone to structural instability8–10 and form a variety of morphological instability modes under compression11. Sinusoidal wrinkles are commonly the primary instability mode of the stiff/soft bilayer systems10,12. Secondary instabilities could occur in period-wrinkling membranes upon further compression9, breaking the sinusoidal symmetry and resulting in period-doubling9,11, folding13–15, and creasing16,17. The effect of the modulus ratio and the adhesion energy between the two layers on wrinkle evolution has been investigated experimentally and theoretically18–21. However, the effect of soft foundation thickness has long been considered negligible12,22,23, except for the thickness contrast lower than ten24. Although the existing experimental and theoretical reports are self-consistent, a comprehensive understanding of the whole picture of wrinkle evolution remains challenging. Here, we find that the thickness contrasts over several orders in magnitude still govern the bifurcation behavior of wrinkle evolution in stiff/soft bilayers. A universal scaling law is proposed for wrinkle evolution in bilayer systems, and validated by comprehensive experiments for different materials and a wide range of thickness contrasts.

To have a clear picture of the wrinkle evolution in stiff/soft bilayers, we first designed and conducted experiments with polydimethylsiloxane (PDMS) microfilms, whose top layer was hardened by oxygen plasma. The stiff/soft bilayer was fabricated by placing a polydimethylsiloxane (PDMS) microfilm on a rigid substrate and then exposing it to oxygen plasma to create a stiff SiO1.8 coating nanolayer (Fig. 1a, top), as has been reported previously25. The thickness of the PDMS film, *t*s, ranges from 20 to 300 µm, while that of the SiO1.8 membrane, *t*m, is ranged in 30–281 nm. The Young’s modulus of the stiff SiO1.8 membrane is *E*m = 30 GPa, about four orders of magnitude higher than that of the soft PDMS films (*E*s = 1.8 MPa). Wrinkles appear in the stiff/soft bilayer when laterally compressed to a strain *ε* over a threshold value *ε*0 under uniaxial stretching (see Methods). As the thickness contrast *t*s/*t*m is high enough, the intrinsic periodic length *λ*0 at the threshold of the primary wrinkling instability (1st order instability) is proportional to *t*m simply11

Where \(\overline {{{E_i}}}\)= *E**i*/(1-*µ**i*2) with *i* = s of m; *E*m and *E*s is Young’s modulus of stiff membrane and soft film, respectively, *µ*m and *µ*s are Poisson ratios of stiff membrane and soft film; and *t*m is the thickness of the stiff membrane. This independence of the periodic length *λ*0 of the primary wrinkle to *t*s for the bilayer systems tested in this work is shown in Fig. S8a.

When the wrinkle-to-fold, or wrinkle-to-fold-to-crease transition occurs under increasing compressive strain, the surface perimeter of one period of the finally formed wrinkles, or folds, or creases, is defined as

*λ* = *nλ*0, *n* = 1, 2, 4, 8,……, (2)

where, *n* = 1, 2, 4, 8… is for single-, double-, quadruple- and so on period buckling modes, corresponding to 1st, 2nd, 4th, and 8th order instability of buckling evolution, respectively. For example, as a result of 2nd order instability, one period-doubling developed from two primary wrinkles (Figs. 1a and 1b), and, hence, its surface perimeter will stay as *λ* = 2*λ*0, regardless of the magnitude of the compressive strain. Similarly, one quadruple-period fold is formed by merging four primary wrinkles under increasing compressive strain as a result of 4th order instability, holding *λ* = 4*λ*0. At high enough compressive strain, the folds will collapse at the bottom to form creases (Fig. 1b), and finally, the stiff membrane will be broken at the bottom of each crease (Fig. 1c).

It has been found that a stiff membrane on the deep soft substrate will lose its instability to form wrinkle at strain over its primary critical value *ε*0, period doubling at strain over critical value *ε*1, and period quadrupling at strain over *ε*2, and so on (data from ref. 11 in Fig. 1d), and may finally fracture as shown by our experiment (Fig. 1c). However, even with a thickness contrast *t*s/*t*m = 193, our experiment shows that only single-period wrinkle, fold and crease occur and finally the stiff membrane fractures with a single period length of *λ* = *λ*0. This means that even with deep soft substrates and a large thickness contrast of 193, the wrinkle evolution of the SiO1.8/PDMS bilayer holds a single wavelength (data in the blue square in Fig. 1d), i.e., remains in the 1st order instability with *λ* ≡ *λ*0 (Fig. 1e). Only for thickness contrasts larger than 230, the period-doubling bifurcation can be activated, i.e., the 2nd order instability occurs and achieves the final mode with *λ* = 2*λ*0 (Fig. 1e). Period-quadrupling bifurcation will not occur until the thickness contrast is greater than 740, which corresponds to the final mode of 4th order instability with *λ* = 4*λ*0 (Fig. 1e).

It is difficult to apply high enough compression onto the micron-thick bilayers using traditional loading methods, such as pre-stretching, direct compression, and stretching because the thin bilayer will bend as a whole under a large amount of stretching or compression. This restriction challenges the control of the wrinkle evolution of the stiff membrane in bilayers with finite thickness contrast and, further, prevents us from systemically investigating the dependence of wrinkle evolution on thickness contrasts, materials, and loading modes.

We find that a simple peeling strategy can efficiently overcome this difficulty. Instead of fighting with the overall bending deformation, we make use of it by perpendicularly peeling the bilayer from a rigid substrate (Fig. 2a). Under an appropriate adhesion strength *f*0 between the bilayer and the rigid substrate (*f*0 = 10 N/m for all peeling experiments in this work), the bilayer under perpendicular peeling can be controllably compressed on its inner side to enough high level so that all the possible orders of wrinkle evolution can be experienced by its stiff membrane. The local compressive strain is local-curvature dependent, increased from zero in the flat, unpeeled region to a maximum, *ε*, in the highly bending region, and then turn from compression to tension in the perpendicular region (Note S1. 1). Moreover, the maximum compressive strain *ε* in the stiff layer can be easily controlled by changing *t*s, since this thickness is inversely correlated with the overall curvature of the bending region (Note S1. 1). Evidently, the 90° peeling strategy is a more reliable method to create large compression into micro-bilayer films. Such a peeling strategy enables unprecedented precision and direct control of folding instabilities up to the highest folding order in the SiO1.8/PDMS bilayers with *t*s/*t*m ranging from 71 to 3388, which is difficult to achieve with conventional loading methods.

To well investigate the dependence of the instability order on thickness contrast in a wide range, we performed in situ peeling experiments in scanning electron microscopy (SEM) (Fig. 2b). Figure 2c presents the in situ observed buckling morphologies of three peeling bilayers with a given *t*m = 155 nm yet different *t*s. For the thin bilayer with *t*s = 29 µm (*t*s/*t*m = 187), buckling in the stiff membrane evolves from wrinkles, via single folds, ending with single-period creases with increasing local curvature of the bilayer (equivalent to increasing compressive strain), i.e., confined to the 1st order instability. In contrast, the bilayer with *t*s = 90 µm (*t*s/*t*m = 580) exhibits a period-doubling bifurcation, i.e., the 2nd order instability occurs, in which wrinkle, period-doubling, and double crease are experienced. The bilayer with *t*s = 114 µm (*t*s/*t*m = 736) shows buckling evolution from wrinkle, via period-doubling and then quadruple fold, finally ending with quadruple crease and fracture, which includes period-quadrupling bifurcation, i.e., there are 1st, 2nd, and 4th order instabilities. Obviously, the final instability order (*n*) of the wrinkle evolutions in the three bilayers are 1, 2, and 4, respectively. Extensive in situ experiments on a series of stiff/soft bilayers with thickness contrasts ranging from 187 to 1000 reveal that the surface perimeters of all the acquired final crease patterns fall into three values, i.e., 1*λ*0, 2*λ*0, and 4*λ*0 (Fig. S9). Although different buckling modes coexist in the bending regions of some peeled bilayers, the specific final instability order can be easily recognized by the surface perimeter of arrested creases, *λ* = *nλ*0, with *n* = 1, 2, and 4 (Fig. 2d). Higher order instabilities can be observed in bilayers with larger thickness contrasts as will be discussed later and shown in Fig. S10a.

In order to investigate the dependence of the final instability order (*n*) on the thickness contrast domination, we further carry out peeling experiments with five groups of SiO1.8/PDMS bilayers. The thicknesses of the stiff SiO1.8 membranes for the five groups of samples are *t*m = 138, 155, 194, 232, and 281 nm. The thickness of soft microfilm varies from *t*s = 20 to 300 µm in each group. Figure 2e shows the statistical dependences of *λ*/*λ*0 on *t*s/*t*m. *λ* is the surface perimeter of cease pattern, which is obtained by measuring the fracture period length after the peeling. *λ*/*λ*0 should be equal to *n* in principle. Nevertheless, due to possible residual strain and measuring errors, the measured fracture period length is not strictly equal to integer multiples of *λ*0, or *nλ*0. Meanwhile, *λ*0 is calculated by Eq. 1 according to the measured *t*m, and, thus, there is also measuring error here. Therefore, there are errors between *λ*/*λ*0 and *n* in Fig. 2e. The bilayers exhibit *λ* \(\cong\) 1*λ*0 creases pattern when *t*s/*t*m ≤ 220, *λ* \(\cong\) 2*λ*0 pattern when 220 ≤ *t*s/*t*m ≤ 733, and *λ* \(\cong\) 4*λ*0 pattern when *t*s/*t*m > 733 (Fig. 2e). Here, 220 and 733 are two critical thickness contrast *t*s*n*/*t*m for the tuning of final instability order, which is denoted as *t*s1/*t*m and *t*s2/*t*m, respectively. *t*s1 and *t*s2 are the corresponding two critical thicknesses of soft film, as shown in Fig. S4a and b. These results are similar to those obtained in the stretching experiments shown in Fig. 1e. The critical values of *t*sn/*t*m obtained from experimental measurements are concentrated within a small range (Fig. 2e). This is because *t*s is several orders of magnitude larger than *t*m. Even small measuring errors can have a significant impact on the results of *t*s/*t*m. Evidently, stepwise dependences of *λ*/*λ*0 on *t*s/*t*m reveal a scaling law between the final instability order (*n*) and the thickness contrast (Fig. 2e). The specific buckling evolutions at different final instability orders are schematically illustrated in Fig. 2f up to *n* = 4. The buckling evolution in the bilayers revealed by the peeling experiments is also supported by finite element simulations. The simulations can reproduce the wrinkle-to-fold-to-crease transitions in a series of stiff/soft bilayers, which agree with experiments in terms of final instability order and the evolution process (Note S2, Figs. S6 and S7).

It should be noted that, although the peeling treatment involves bending deformation of the bilayer film accompanied by curvature radius in the bending region, the final instability order is still determined by the thickness contrast rather than the curvature radius. There are two key pieces of evidence. As mentioned above, *t*s is three orders of magnitude larger than *t*m in this work. Therefore, curvature radius is mainly determined by *t*s (Note S1. 2). The dependence of the final instability order on *t*s is still stepwise (Fig. S4a and b). However, the critical values of *t*s, i.e, *t*s*n*, for the tuning of final instability order are different according to different *t*m values. We can then find that the buckling instability can terminate in different orders in two peeled bilayer films with the same *t*s, i.e, the same curvature radius, yet different *t*m, as shown in Fig. S4. In addition, direct stretching does not involve curvature radius, but the scaling law was also produced in this way in bilayer films. Moreover, the critical thickness contrast for the transitions between distinct fracture periods nearly reproduce those determined from peeling experiments. Evidently, the curvature radius in the peeled bilayer film has no effect on the scaling law.

As above mentioned, if *t*m changes, *t*s*n* shift accordingly. This result suggests that there may be an internal relationship between *t*s*n* and *t*m to determine the final instability order. Meanwhile, the buckling evolution is related to compression. In order to find the relationship between *t*s*n* and *t*m, we convert *t*s into compressive strain in the stiff membrane by approximating a peeling bilayer as a bent beam (Note S1.1- S1.2, Figs. S1-S3). We then calculate the shrinkage of the inside of the peeling soft microfilm in one period of three final creases with the period wavelength of 1*λ*0, 2*λ*0, and 4*λ*0, and find that the critical shrinkages for the onset of the three creases are approximately equal to a constant (Note S1.3, Fig. S5). The critical shrinkage corresponds to *t*s*n*, as shown in Figs. S3 and S4. Based on the above approximate equation, we further derived and finally obtained the relationship between *t*s*n* and *t*m (Note S1.3)

*t*s*n* = 0.028*n*2*A*2*t*m, (3)

Eq. 3 demonstrates that *t*s*n* is related to *t*m and *A* (material constant, see Eq. 1), that is, the mechanical parameters of the bilayer itself. After *t*s*n* is divided by all the mechanical parameters of the bilayer, we can obtain a new variable that depends only on the final instability order *n*

*t*n = *A*−2*t*s*n*/*t*m = 0.028*n*2, *n* = 1, 2, 4, 8, 16……. (4)

Following the above treatment, we converted *t*s into a more general variable *t* = *A*− 2*t*s/*t*m, which is can be defined as equivalent thickness contrast. Obviously, *t* is applicable to all stiff/soft bilayer films, because it contains all mechanical parameters of stiff membrane and soft film. *t*n is critical value of *t*, below which the buckling instability terminates at order *n* (Note S1.3). The value of *t*n is 4 times change, that is *t**n*=4*t**n*−1. Therefore, tuning final instability order *n* requires *t* to accumulate to a specific critical value, i.e, the dependence of *n* on *t* is stepwise, which is the scaling law. The model can also be applied to the data in Figs. 2e by calculating the critical values of *t*s*n*/*t*m = *A*2*t**n*, as shown by dashed lines.

To examine the universality of the scaling law in different material systems, we then conducted peeling experiments on a series of polymethyl methacrylate (PMMA)/PDMS (modulus ratio = 1111) bilayers and gold film/PDMS (modulus ratio = 33332) bilayers. We find that *t = A**−* 2 *t*s/*t*m can well unify the buckling evolution in different bilayer systems (Fig. 3). It is also found that the crease surface parameters available in literature collapse completely to our *n ~ t* curves9,11,19,24, fully supporting the scaling law. The thickness contrasts in this work and available in the literature range from 2.5 to 3388, and the modulus ratios range from 54 to 33332, (see details in Table S1). Obviously, the scaling law for *n ~ t* is independent of the loading method and holds effective in bilayer systems over a large span of thickness contrasts and modulus ratios, and, hence, is universal in all the discussed cases. Therefore, wrinkle evolution in stiff/soft bilayers can be characterized by a single parameter *t*.

An additional product of our peeling experiments is the controllable fracture of stiff membranes. Detailed characterizations show that the steep bottom of the crease has actually been partially cracked (left panel of Fig. 4a, Fig. S11a) due to highly concentrated stress26. Membrane fracture occurs along the bottom lines of creases in the region with a large curvature (Fig. 4a, right panel). The fracture route is along the crease line (Fig. S11b), and, thus, strictly straight, in contrast to normal fracture that is governed by randomly distributed defects. The period of as-produced membrane ribbons can be well controlled among 1*λ*0, 2*λ*0, and 4*λ*0 by presetting *t*, according to the scaling law (Fig. 4b).

Furthermore, the ordered fracture induced by peeling can be used to tailor the stiff membranes into microstructures with designed shapes over a large area. In Fig. 4c, image i shows an array (1*λ*0 × 1*λ*0) of square SiO1.8 microflakes that are fabricated by sequentially peeling the stiff/soft bilayer along two perpendicular directions. As *t* (= 0.027) is just less than the first critical value *t*1 = 0.028, only the first order creases *n* = 1 form and fracture. Similar sequential fractures along two selected directions can yield rhombic microflakes (ii in Fig. 4c). For the bilayer with *t* around the critical value *t*1, both first order (*n* = 1) and second order (*n* = 2) crease instabilities can occur. As a result, 1*λ*0 × 2*λ*0 rectangular microflakes and 1*λ*0 × 1*λ*0 and 2*λ*0 × 2*λ*0 square microflakes can be produced by sequential fractures along two perpendicular directions (iii in Fig. 4c). Nevertheless, the specific size of formed microflakes is random at this critical point. Further increasing *t* beyond *t*1, the quadrangular SiO1.8 microflakes with a size of 2*λ*0 × 2*λ*0 can be stably prepared (iv in Fig. 4c).

In summary, our extensive experiments and analyses have revealed a universal scaling law for wrinkle evolution with increasing compressive strain. The final instability order (*n*) that can be achieved in wrinkle evolution of a bilayer is determined by a single parameter *t* (*= A*− 2*t*s/*t*m) consisting of the thickness contrast (*t*s/*t*m) and a material constant (*A*). Combining the scaling law with the peeling strategy, stiff membranes with thicknesses across multiple orders of magnitude can be cut into regular ribbons and flakes with designed sizes and shapes, shedding light on a new method for nano-manufacture.