Quantized wrinkles and fracture of stiff membranes on soft lms

Yancheng Meng Nanjing University of Aeronautics and Astronautics Henggao Xiang Nanjing University of Aeronautics and Astronautics Jianqiang Zhang Nanjing University of Aeronautics and Astronautics Zhili Hu Nanjing University of Aeronautics and Astronautics Jun Yin Nanjing University of Aeronautics and Astronautics Liqiang Li Tianjin University https://orcid.org/0000-0001-8399-3957 Zhuhua Zhang Nanjing University of Aeronautics and Astronautics https://orcid.org/0000-0001-6406-0959 Wanlin Guo (  wlguo@nuaa.edu.cn ) Nanjing University of Aeronautics and Astronautics https://orcid.org/0000-0002-2302-8044

The stiff/soft bilayer was fabricated by placing a polydimethylsiloxane (PDMS) micro lm on a rigid substrate and then exposing to oxygen plasma to create a stiff SiO 1.8 coating layer (Fig. 1a), a process similar to a previous report 23 . The thickness of the soft PDMS lm, t s , ranges from 20 to 300 μm, while that of the SiO 1.8 membrane, t m , is ranged in 30-281 nm. The Young's modulus E m of the SiO 1. 8

layer is 30
GPa, about four orders of magnitude higher than that (E s = 1.8 MPa) of the PDMS micro lm. Stretching such a stiff/soft bilayer in one direction fractures the SiO 1.8 layer perpendicularly into strips, which simultaneously form wrinkles due to compression induced by the Poisson effect (Fig. S3). The intrinsic periodic length λ 0 at the threshold of the wrinkling instability is proportional to the thickness of the stiff membrane t m , expressed as λ 0 = At m , where A is a constant related to elastic properties of the bilayer (see SI). Enhancing the stretch makes the strips more compressed, driving wrinkle-to-fold transitions and foldcollapses. Yet, the double and quadruple folds as well as the corresponding fold-collapses appear randomly under the uniaxial stretch (Fig. S3), and no control on the instability modes has been achieved thus far.
We achieve the morphological control by perpendicularly peeling the bilayer from the rigid substrate (Fig.   1a). The bilayer under perpendicular peeling is compressed in its inner side, along with a gradient from the at, unpeeled region to the bent region. Accordingly, wrinkles, double folds, quadruple folds, and quadruple fold-collapses simultaneously appear in respective regions of the stiff membrane, as shown in Fig. 1b for a bilayer with t s = 112 μm and t m = 155 nm. The instability modes encoded by local curvatures of the bilayer form a stark contrast to a randomly occurred, single instability mode under the uniaxial stretch (Fig. S3). More interestingly, the quadruple fold-collapse holds a deep-set, steep bottom (i, Fig. 1c and Fig. S4) that has been partially cracked at the apex (ii, Fig. 1c) due to highly concentrated stress thereof 24 . Membrane fracture occurs at the collapsed bottom as the local curvature further increases (iii, Fig. 1c). As such, the fracture route is prede ned along the fold line, and, thus, strictly straight and controllable, in contrast to normal fracture that is dominated by structural defects. As the peeling proceeds, the wrinkles dynamically evolve and the quadruple fold-collapses occur progressively throughout the whole bilayer until the bilayer is completely peeled off, leaving the membrane evenly fractured with a pattern period of 4λ 0 (Fig. 1d ).
The fracture pattern is deterministically controlled by the thickness of the soft micro lm t s . For two bilayers with the same t m = 155 nm as above, we nd that wrinkles in the one with t s = 33 μm evolves only into single-period fold-collapses, resulting in a fracture pattern with a period of 1λ 0 (Fig. 1b ). Note that the single-period folds and collapses are a missing mode in previous theory and experiments 2,9 . Our peeling method can access this mode because the wrinkles experience a localized evolution driven by an extremely high compressive strain that is mediated by t s . The other bilayer with t s = 90 μm is fractured with a 2λ 0 period due to its double-period folds and ensuing fold-collapses (Fig. 1b I). Thus, three distinct buckling evolution routes starting from wrinkles and ending with fracture emerge: wrinkles to single-folds and to single-fold-collapses as route 1 (Fig. 1b ); wrinkles to double-folds, and to double-fold-collapses as route 2 (Fig. 1b I); wrinkles to double-folds, quadruple-folds, and to quadruple-fold-collapses as route 3 ( Fig. 1d ). At a given t m , which evolution route will be taken by a peeling bilayer is controlled by t s , so does the number of buckling modes (Fig. S5).
Extensive measurements on a series of stiff/soft bilayers reveal that all the acquired fracture patterns fall into three periods following the three routes, i.e., 1λ 0 , 2λ 0 , and 4λ 0 (Figs. 1e and S6). Speci cally, the bilayers with t m = 155 nm exhibits a 1λ 0 fracture period when t s /λ 0 < 2.5, a 2λ 0 period when 2.5 < t s /λ 0 < 9.5, and a 4λ 0 period when t s /λ 0 > 9.5 (Fig. 1e). Further increasing t s /λ 0 over 40 can result in an 8λ 0 fracture period (Fig. S7), although the corresponding bilayers are too thick to realize a steady peeling process. This stepped thickness dependence of fracture periods can be viewed as a discretization of wave-like wrinkles upon large compression, much like quantized energy of electronic wavefunctions induced by spatial con nement. The critical t s /λ 0 for the transitions between distinct periods, denoted as t sN /λ 0 with N = 1, 2, and 3 in Fig. 1e, are constants. As λ 0 is proportional to t m , t sN linearly increases with t m as well, consistent with our extensive measurements on bilayers with different t m (Fig. S6).
Accordingly, the ribbon width W and t m are linearly related by W = nAt m (n = 1, 2, 4), which again agrees with our experiments (Fig. S8). Of technical importance is that W is as small as 8 μm (2λ 0 ) when peeling a bilayer with t m = 42 nm and t s = 76 μm (Fig. S8). Note that the peeling angle of no less than 90° is crucial to the quantized fractures in the membranes. Otherwise, the bending-induced compression does not su ce to trigger the wrinkle-to-fold transitions; instead, nonperiodic fracture appears in the peeled part of the bilayer due to the in-plane stretch applied for peeling (Fig. S9).
The stepped thickness dependence of fracture patterns can be understood by a continuum model, in which the soft micro lm under peeling is approximated as a cantilever beam with large de ection 25 (see SI). According to this model, the strain increases with moving away from the free end and reaches a maximum at the clamped point. Yet, the beam model cannot correctly describe the strain distribution in the region near the clamped point since the strain must be continuous across this point. As such, the practical maximum strain point deviates from the clamped end by a certain distance. This deviation indeed exists in our experimental samples under peeling, with magnitudes in a range 0.12t s -0.18t s (see SI) and an average of 0.16t s . Therefore, we correct our model by taking this deviation into account for determining the maximal strain (see SI).
The bending-induced strain in the soft micro lm is transferred into the stiff membrane. Here, we focus on the membrane region where the fracture occurs due to the maximal strain. For a fold-collapsed membrane with a period of nλ 0 (n = 1, 2, 4), the total strain within a period is localized to the collapsed region and can be assessed through integrating the strain over the same period length of the micro lm prior to the buckling (Fig. 2a). This integrated strain Δ n is just a shortened length of the micro lm normalized by λ 0 . Since the overall compressive strain in the micro lm is proportional to t s -1/2 (see SI), Δ n scales as where = E s /(1-μ s 2 ), f 0 is the adhesive force per unit width between the bilayer and the substrate, and E s and μ s are the Young's modulus and Poisson ratio of the soft micro lm, respectively. The details on deriving the Eq. 2 are provided in SI. Based on experimentally measured critical t s (Fig. 1e), the tted critical Δ n reads = (c n t m 1/2 )/λ 0 for the nλ 0 -period fracture, where c n =10 -3/2 m 1/2 (see SI) . Surprisingly, is a constant independent of n, thus denoted as Δ c , for all the bilayers with a given t m . Δ c ≌ 0.9 when t m = 155 nm (Fig. S10), close to 0.85 from experimental measurements on three bilayers with t m = 155 nm and t s = t sN (N = 1, 2 and 3, see Fig. 2b, olive spheres). These results enable us to constitute a criterion, Δ n ≥ 0.9, for the fracture of SiO 1.8 /PDMS bilayers with a period of nλ 0 . For example, the bilayers with 2.5 <t s /λ 0 < 9.5 can only be fractured with a period of 2λ 0 , because its Δ 1 is below 0.9 while Δ 2 meets this criterion (Fig. 2b).
According to the expressions of Δ n and Δ c , t sN can be expressed as t The buckling evolution in the bilayers induced by mechanical peeling is further supported by nite element simulations (see details in SI). Figure 2c presents the simulation results of a stiff/soft bilayer with t m = 155 nm and t s = 60 μm. When the bilayer is slightly bent, wrinkles appear in its compressed side. The wrinkles gradually evolve into double folds as the bilayer is more bent, similar to the experimental results and amenable to the above criterion. The calculated von Mises stress in the stiff membrane exhibits a relatively regular distribution for the wrinkles (Fig. 2d), but becomes scattered with a pronounced peak when the double folds appear ( Fig. 2d and inset). The peak lies at the fold bottom that is deviated from the very peeling front, just as what is observed in our experiments. This deviation further justi es the correction of our above continuum model. Thus, the ultrahigh stress at the fold bottom leads to eventual fracture of stiff membranes. The simulated results are robust against changing t s in a wide range 20-300 μm and agree with experiments in terms of the buckling period (Figs. 1e and Fig. S11) and its evolution (Fig. S12).
The quantized folds and fracture in the stiff/soft bilayers are not unique to the peeling method but a general behavior to any form of loading that can effectively compress the bilayers. Regarding the uniaxial stretch method that compresses the bilayers by Poisson effect (Fig. 3, inset), the local fold-collapse and facture period in the most compressed region still follows a stepped dependence on the local compressive strain ε, although buckling modes occur randomly in the whole bilayer (Fig. S3). If we de ne an equivalent strain as (see SI), the stepped dependence becomes uniform for all the bilayer with different t m and λ 0 (Fig. 3). Moreover, the critical for the transitions between distinct fracture periods nearly reproduce those determined from above peeling experiments, as compared by the black and gray lines in Fig. 3. Notably, the fold-collapse periods collected from refs. 8 and 22, where the bilayers are directly squeezed, can also be quanti ed by our scaling law. These results further verify the intrinsic behavior of quantized wrinkles in stiff/soft bilayers.
In 2008, Pocivavsek et al. reported that wrinkles can evolve into a local fold in a soft polyester lm resting on water under lateral compression, but into double or quadruple folds when the lm is on a gel substrate 2 . Later, a model analysis based on nonlinear oscillator dynamics unraveled that the wrinkle-tofold transitions were triggered by a period-doubling bifurcation 9 . Subsequent nite element simulations and analyses underscored the role of a relatively small modulus ratio of stiff/soft bilayers in inducing the buckling folds rather than outward ridges 3,21,26 . Similar results were analyzed for bilayers with small thickness contrast 22 . However, these studies suggested unexpected complexity in describing high-order fold transitions 3,9 , and experimental access to these transitions has been restricted by limited strain that can be effectively applied with traditional loading methods. Actually, these studies reported only a part of wrinkle-fold transitions along a speci c evolution route (either route 2 or 3). Our results feature three advancements over existing studies: i) a complete map of three buckling evolution routes, that is controllable transitions from wrinkles to 2 n-1 -folds and eventually to fractures, with n=1, 2, or 3 depending on the soft micro lm thickness; ii) fractures of the stiff membranes with a preset pattern due to the quantized thickness dependence of the fracture period as well as the extreme stress concentration beyond the reach of traditional methods; ii) a concise scaling law to quantify the quantized wrinkles and factures.
Our strategy of controlling the buckling folds and fracture by peeling is powerful to tailor a series of largearea stiff membranes into microstructures with devisable shapes. Figure 4a shows an array of square SiO 1.8 micro akes that are fabricated by sequentially peeling the stiff/soft bilayer along two perpendicular directions. The side length of the square micro akes can be ne-controlled to be nλ 0 (n = 1, 2 and 4) by varying t s (Figs. 4a and 4d). Similar sequential fractures along two selected directions can yield rhombic micro akes (Fig. 4b). For the bilayer with t s = t s1 , both rectangular micro akes with a size of 1λ 0 ×2λ 0 and square micro akes with a size of either 1λ 0 ×1λ 0 or 2λ 0 ×2λ 0 can be produced by sequential fractures along two perpendicular directions (Fig. 4c). Nevertheless, the speci c size of formed micro akes is random at this critical point. Aside from the SiO 1.8 membranes, our strategy is applicable to fracture many other brittle materials, as demonstrated by tailoring a polymethyl methacrylate (PMMA) membrane (Fig. S13a) into micro-ribbons. It can even be used to make an Au nanomembrane with topographical patterning at controlled scale (Fig. S13b), although brittle fracture cannot occur due to the high ductility of gold. The mechanically induced nano/micro-scale troughs in the Au lm without using lithographic techniques may nd potential applications in, for example, nanoparticle sieving 6 and molecular manipulation 27 . Even monolayer graphene can be fractured into micro-/nano-ribbons (Fig. S13c), although more complex physical process may be involved here to induce the fracture, such as facture anisotropy 28,29 and edge warping 30,31 .
We have developed a simple peeling strategy that can control the wrinkle-to-fold transitions with a preset period and then drive ordered fracture in stiff/soft bilayers. In principle, our strategy is applicable to fracture various two-dimensional materials of keen current interest. If technically matured, it will represent a new way for mass production of ribbons and akes with customized edge orientations and nanometer sizes. For example, peeling a MoS 2 monolayer (t m = 0.6 nm) resting on a PDMS micro lm with t s = 0.5 μm from a rigid substrate would result in an array of nanoribbons with a uniform width of only 90 nm according to our scaling law. This mechanical tailoring method will be complementary to previous fabrication methods based on ion etching 32,33 and chemical processing [34][35][36] . Moreover, the thicknessdependent wrinkle evolutions in stiff/soft bilayers may help us understand the ageing mechanism of skins as well as cerebral cortexes with strongly convoluted sulci and gyri that have proved crucial for achieving advanced intelligent functions 14,15 .  Mechanism of quantized wrinkles and fracture in SiO 1.8 /PDMS bilayers. a, Schematic illustration of a bent cantilever beam and compression-induced buckling in a bent bilayer. The shortened length of the micro lm (Δ n ) in one period length of the fold collapse is depicted on the right. The stress distribution in the beam is color-encoded. "com" and "ten" represent compressive and tensile stresses, respectively. b, Normalized shortened lengths of the micro lm induced by single-fold collapse (Δ 1 /λ 0 , squares), doublefold collapse (Δ 2 /λ 0 , circles), and quadruple-fold collapse (Δ 4 /λ 0 , triangles) as functions of sqrt (λ 0 /t s ); the critical shortened length for fracture occurrence Δ c /λ 0 is about 0.9 by theory and 0.85 on average by experiments. c, Three-step illustrations of morphological evolution from wrinkles to double folds by nite-   Application demonstrations of peeling-induced fracture of stiff/soft bilayers. a, An array of square SiO 1.8 micro akes with a size of 1λ 0 ×1λ 0 . b, An array of rhombic SiO 1.8 micro akes. c, An array of rectangular SiO 1.8 micro akes with a size of 2λ 0 ×1λ 0 , fractured from a bilayer with t s = t s1 . d, An array of square SiO 1.8 micro akes with a size of 2λ 0 ×2λ 0 .

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