Wrinkling of thin films is an easy-to-implement and low-cost technique to fabricate stretch-tunable periodic micro and nanoscale structures. However, the tunability of such structures is often limited by the emergence of an undesirable period-doubled mode at high strains. Predictively tuning the onset strain for period doubling via existing techniques requires one to have extensive knowledge about the nonlinear pattern formation behavior. Herein, a geometric prepatterning-based technique is introduced that can be implemented even with limited system knowledge to predictively delay period doubling. The technique comprises prepatterning the film/base bilayer with a sinusoidal pattern that has the same period as the natural period of the system. This technique has been verified via physical and computational experiments on the polydimethylsiloxane (PDMS)/glass bilayer system. It is observed that the onset strain can be increased from the typical value of 20% for flat films to greater than 30% with a modest prepattern aspect ratio (2·amplitude/period) of 0.15. In addition, finite element simulations reveal that (i) the onset strain increases with increasing prepattern amplitude and (ii) the delaying effect can be captured entirely by the prepattern geometry. Therefore, one can implement this technique even with limited system knowledge, such as material properties or film thickness, by simply replicating pre-existing wrinkled patterns to generate prepatterned bilayers. Thus, geometric prepatterning is a practical scheme to increase the operating range of stretch-tunable wrinkle-based devices by at least 50%.

Introduction

The generation of wrinkled patterns via compression of supported thin films is a scalable and affordable technique for fabricating periodic micro and nanoscale features over large areas [19]. The mechanism of wrinkle formation is similar to buckling of columns under compressive loads with one significant difference—the wrinkle period is independent of the in-plane length of the film. Instead, the wrinkle period depends only on the thickness of the film and the ratio of mechanical properties (Young’s modulus and Poisson’s ratio) of the film and base at the onset of buckling [10,11]. Consequently, wrinkled patterns exhibit a distinct scale-independent “natural” period that can be well-controlled by tuning the film thickness and the elastic moduli ratio. In addition, the natural period and the amplitude of the wrinkles can be further tuned by controlling the magnitude of strain [12,13]. Thus, well-controlled stretch-tunable periodic micro/nanoscale patterns can be generated via wrinkling.

In the past, single-period sinusoidal wrinkles that were generated via uniaxial compression have been used as stretch-tunable functional features for applications such as tunable nanofluidic channels [14,15] and tunable diffraction gratings [1618]. The degree of tunability of such devices is limited by the maximum strain that can be applied. Often, this maximum strain for tuning wrinkle patterns is limited by the onset of period doubling at high strains [1319]. Period doubling refers to the emergence of an additional deformation mode in the wrinkle pattern, wherein alternate valleys become progressively shallower with increasing strain. This behavior is illustrated in Fig. 1(a). Emergence of the period-doubled mode leads to a complex pattern that is structurally, and often functionally, distinct from the initial single-period sinusoidal pattern. Thus, when a large design space for stretch-tunability of single-period wrinkles is desired, one must suppress the onset of period doubling.

Although the phenomenon of period doubling at high compressive strains is well-known [13,1923], practical techniques to predictively and independently tune the onset strain are not available. Period doubling in wrinkling was first reported by Brau et al. [19] who empirically demonstrated the phenomenon and proposed an analytical model. Although their model predicts the amplitude of wrinkles postdoubling, their prediction that the onset strain is strongly dependent on only the Poisson’s ratio of the base is impractical to implement. Since then, a more practical approach that is based on prestretching the base has been demonstrated to control the onset strain [2224]. Although practical to implement, this prestretch-based technique requires one to have extensive knowledge about the nonlinear mechanics of pattern formation to be able to predict the change in the onset strain. In addition, the degree of tunability of onset strain is coupled to and limited by the specific stress–strain constitutive relationship of the material. Herein, an alternate technique based on geometric prepatterning is proposed that enables one to predict and determine the change in onset strain simply by knowing/selecting the size of the prepattern. Due to the universality of the first-order relationship between the nondimensional prepattern size and the change in onset strain, this technique can be implemented to predictively tune the onset strain even when the material properties of the bilayer or the film thickness are unknown. Thus, this geometric prepatterning technique enables one to predictively and independently tune the onset strain for a variety of bilayer wrinkling systems.

Geometric Prepatterning Technique

The geometric prepatterning technique for suppression of period doubling is illustrated in Fig. 1(b). Suppression of period doubling is achieved by prepatterning the base layer with a sinusoidal pattern that has a period identical to the natural period of the equivalent flat bilayer system. The equivalent flat bilayer system is identical to the prepatterned system except for the absence of geometric prepatterning, i.e., the material properties, thin-film thickness, and applied stretch in the flat and prepatterned systems are identical. The natural period of the equivalent flat bilayer system is the period of the wrinkles that emerge immediately at the onset of buckling. Upon compression of the prepatterned bilayer, it is observed that the natural period persists at high strains and period doubling begins at a strain that is higher than that for the equivalent flat bilayer. As illustrated in Fig. 2, this behavior has been computationally and experimentally verified for the polydimethylsiloxane/glass bilayer system.

Physical Demonstration.

The experiments performed here separately identify the effect of prepatterning on onset strain by decoupling it from the effect of material properties and strain. Distinct regions of period-doubled and single-period wrinkles were observed at the edge of the prepatterned and flat sections of the same bilayer (Fig. 2). The prepatterned region was fabricated by replicating a pre-existing wrinkled surface on top of PDMS during the curing process [25]. Wrinkled patterns were fabricated by (i) uniaxially stretching the PDMS base on a custom tensile stage[13], (ii) generating a thin glassy layer on top of the prestretched base by exposing it to air plasma [25,26], and (iii) releasing the stretch in the base layer to generate uniaxial compressive strain in the glassy thin film. As the entire bilayer was subjected to the same stretch and was exposed to air plasma at once, the two regions differ only in the presence or absence of the prepatterns.

Fabrication Technique.

PDMS films were fabricated by casting and thermally curing a two-part polydimethylsiloxane (PDMS) silicone elastomer mix that is commercially available from Dow Corning, Corning, NY (Sylgard 184). The two parts were mixed by combining 15 parts of resin and one part of curing agent by weight. A nonstandard curing ratio of 15:1 was used because this combination was observed to generate PDMS films with the desirable mechanical properties of low Young’s modulus and high failure stretch [27]. After degassing the mixture, curing was performed via a two-step thermal curing process so as to minimize the volumetric shrinkage in the film. Alignment features were generated on the bottom surface of the films by casting and curing the mixture in custom-made aluminum molds. These alignment features were later used to align the direction of stretch with the wrinkle prepatterns. The Young’s modulus of the cured PDMS was measured on an Instron tensile tester and found to be 1.893 ± 0.033 MPa [27].

The cured PDMS films were manually cut into individual coupons that were approximately 20 mm wide, 1.9–2.2 mm thick, and had a clamped length of 37.5 mm. These coupons were then mounted and stretched on a custom-made precision tensile stage [13]. The accuracy of the clamped length was ensured by mating the alignment features on the coupons to the corresponding features on the stage. The entire stage with the stretched coupon was then inserted into a vacuum chamber and exposed to low-pressure radio frequency air plasma. The air plasma chemically modifies the surface and generates a glassy thin film on top of the PDMS layer that has a measured Young’s modulus of 3.2 ± 0.78 GPa [27] and is 10–100 nm thick. The thickness of the glassy film can be tuned by controlling the duration of the plasma exposure. The plasma oxidation process was calibrated to link the observed period to the duration of exposure [27]. After plasma oxidation, wrinkles were generated by gradually releasing the stretch in the base layer thereby causing the top glassy layer to compress and buckle.

Prepatterned regions on the film were fabricated by imprinting wrinkled surfaces onto the base during the thermal curing process. Imprinting was performed by gradually and “gently” placing the coupons with the wrinkled surfaces on top of the curing material after the onset of curing but before gelation. Alignment of the prepatterns to the subsequent direction of stretch was achieved by visually sensing and then aligning the alignment marks on the coupons with the alignment marks on the mold. Post curing, the imprinted wrinkled patterns were carefully separated from the cured PDMS base to expose the prepatterned region of the base. The protocol for curing and imprinting is described in detail elsewhere [28]. As the oxidized glassy layer generated via plasma oxidation generates a thin film that is chemically bonded to the base PDMS, no delamination of the film was observed during detachment of the prepattern mold from the cured base film.

Demonstration of Delayed Doubling.

Several experiments were performed to verify the delayed onset of period doubling by varying the prestretch in the base and the prepattern amplitude. The results of these experiments are summarized in Fig. 3. Wrinkles were fabricated on three different bilayer samples; each sample comprised a pair of flat and prepatterned regions (pairs (a) and (d), (b) and (e), and (c) and (f) in Fig. 3). The flat and prepatterned regions were fabricated on the same PDMS base to ensure that plasma oxidation and stretch were identical for the two regions. To ensure that both flat and prepatterned regions were obtained on the same base: (i) the prepattern coupons were cut into pieces that were each smaller than a single stretched PDMS coupon, and (ii) during curing, only part of the PDMS base was imprinted with the prepattern to generate the prepatterned region, while the rest of the base surface was left untouched to generate the flat region. As illustrated in Fig. 3, period doubling was observed only on the flat regions of the sample (panels in top row of Fig. 3) but not on the prepatterned regions of the same sample (panels in bottom row of Fig. 3). The natural period (λn) was evaluated from the observed period in the images (λo) and the applied strain (ε) using an empirically verified approximation that the number of wrinkles does not change during compression as [13]

 
λn=λo(1+ε)
(1)

Computational Demonstration

Finite Element Modeling Technique.

Finite element modeling was performed by developing 2D models of wrinkling using the structural mechanics module of the comsol 5.1 software package and the matlab R2015a software package for pre and postprocessing. These models were developed by implementing buckling of wide plates under the plane strain condition with the top film modeled as a linear-elastic material and the bottom layer as a nonlinear Neo-Hookean material. A nonlinear strain–displacement relationship was used for both the layers to account for large angles during wrinkling. The bilayer was uniaxially compressed by simultaneously compressing the top and bottom layers. Modeling of wrinkle formation in flat bilayers was performed in two steps: (i) linear prebuckling analysis to predict the mode shapes required for generating a perturbed mesh and (ii) a nonlinear post-buckling analysis on the perturbed mesh to predict the shape and amplitude of the wrinkles after buckling bifurcation. Modeling of wrinkle formation in prepatterned bilayers was performed on the perturbed mesh via a single-step nonlinear analysis. The perturbed mesh was generated from the prepattern geometry using a custom mesh-perturbation toolbox described in detail elsewhere [25,29].

The boundary conditions for the finite element modeling of wrinkles are illustrated in Fig. 4(a). The height of the base layer was chosen to be sufficiently high (5 μm) to simulate semi-infinite boundary condition at the lower edge. The length of the base layer was selected to be exactly ten times the natural period so that no errors due to boundary discretization are introduced during the simulation. This proportional base length condition was ensured by first evaluating the natural period via linear buckling analysis of a base of length 22.5 μm and then regenerating a new model with base length ten times the evaluated natural period.

Demonstration of Delayed Doubling.

The effect of prepatterning on the period doubling onset strain is demonstrated in Fig. 4(b). The predictions of finite element simulations shown in Figs. 2 and 4 are consistent with the physically observed period doubling suppression behavior summarized in Fig. 2. During simulations, the onset strain was evaluated by tracking the growth of amplitude of the single-period and period-doubled modes as the film and base layers were compressed simultaneously. After the onset of period doubling, alternate valleys become progressively shallower with increasing strain. Therefore, the onset strains (ε2,0 for flat bilayers and ε2,p for prepatterned bilayers) were numerically evaluated as the strain at which the smaller amplitude (A2 in Fig. 4(b)) reaches its maximum value. It was observed that the applied strain (ε = 0.249) for the fabricated pattern (Fig. 2) was higher than the predicted onset strain for the flat bilayer (ε2,0 = 0.19) but less than the onset strain for the prepatterned bilayer (ε2,p = 0.31). Thus, one expects to observe two distinct patterns at the prepattern/flat boundary with the single-period pattern lying on the prepatterned side. This expected behavior was observed during the experiments, thereby verifying that prepatterning suppresses the onset of period doubling.

Characterization of Onset Strain Tunability

To explain the effect of geometric prepatterning on onset strain, one must first identify the cause of the period doubling phenomenon. Although small-strain linear-elastic models can accurately predict the onset of sinusoidal wrinkles, they fail to predict onset of period doubling. Instead, one must account for the nonlinearity in the base at high strains to explain emergence of the period-doubled mode.

Origin of Tunability Via Geometric Prepatterning.

Wrinkles are formed as a result of the competition between the deformation energy due to compression of base versus bending of top film. For linear materials, the energy of base (Ub) is directly related to the period of wrinkles and strain (Ub ∼ ελ), whereas energy of the top film (Ut) is inversely related to the square of the period (Ut ∼ ε/λ2). These scaling relationships are obtained from the linear-elastic deformation model as [11] 
Uw=εαλ+εβλ2
(2)
Here, Uw is the total strain energy of a flat bilayer system that undergoes wrinkling bifurcation (Uw = Ub + Ut), the term with α is the contribution due to the compression of the base, and the term with β is the contribution due to the bending of the film. The parameters α and β depend on the film thickness and material properties of the base and the film and are given by [11] 
α=Es3π
(3)
 
β=π24Efh3(1ν2f)
(4)

Here, h is the thickness of the top film, νf is the Poisson’s ratio of the top film, Ef and Es are the Young’s moduli, and the subscripts f and s refer to the top film and the base, respectively.

Equation (2) is valid for all wrinkle periods, i.e., it holds for both the natural and the period-doubled modes. For strain-controlled boundary condition, the natural period of the system can be obtained by minimizing the energy of the wrinkles with respect to the period; thus, the natural period (λn) is given by 
λn=(2βα)13
(5)

For linear materials at low strains, this natural period is independent of the strain; thus, no period doubling would occur at high strains. In contrast, the deformation energy of the base for a nonlinear material, such as PDMS, is determined by a nonlinear coupling between the strain and the period. Due to this, the period corresponding to the minimum deformation energy state deviates away from the natural period with increasing strain. The onset strain for period doubling is the strain at which the 2λn mode becomes energetically favorable over the λn mode.

Geometric prepatterning delays the onset of period doubling by altering the nonlinear dependence of the base deformation energy on the period and amplitude of wrinkles. For a linear material, the ratio of deformation energy in the base for the 2λn mode to that for the λn mode (Ub,2λ/Ub,λ) is equal to 2 at all strains as predicted by Eq. (2). This ratio for a linear base remains unchanged for both the flat and the prepatterned systems. However, the ratio Ub,2λ/Ub,λ drops with an increase in strain for a nonlinear base material such that the 2λn mode becomes energetically favorable at high strains. The rate of decrease in the energy ratio with strain is lower for the case of a prepatterned base. This manifests as an increase in the onset strain for the prepatterned system.

To verify the effect of prepatterning on onset strain, the deformation energy in a nonlinear base material has been computationally evaluated. As illustrated in Fig. 5, the ratio of deformation energy (Ub,2λ/Ub,λ) decreases at a slower rate for the prepatterned bilayer as compared to the flat bilayer. Thus, the period doubling onset strain is higher for a prepatterned bilayer. For these simulations, both linear-elastic and nonlinear Neo-Hookean material models were used. To separately evaluate the energy in the base, a sinusoidal displacement (v = vo·cos(2πX/λ)) was applied to the top boundary of the base. This displacement is identical to the amplitude of the wrinkles; thus, the strain energy of this system is same as one would observe in the base of a bilayer undergoing wrinkling. This indirect evaluation technique for base energy has been extensively used in the literature to develop analytical models for bilayer wrinkling modes [11,12]. It has also been used during computational modeling of wrinkling to compare the effect of base nonlinearity on the wrinkling process [30]. Energy in the base was not evaluated via direct measurements of bilayer wrinkling systems because those measurements combine the contribution due to wrinkling and axial compression of the base. Thus, it is not possible to separately evaluate the base energy from post-buckling simulations of bilayer wrinkling.

To compare the results of this finite element simulation to that of bilayer wrinkling, one must link the applied displacement boundary condition to the equivalent strain for bilayer wrinkling. This is achieved by linking the displacement to the equivalent strain via kinematic relationships for bilayer wrinkling. For a flat bilayer, the amplitude of wrinkles (A) is kinematically related to the applied compressive strain (ε) and the period of wrinkles (λ) as 
ε=(πAλ)2
(6)
For a prepatterned bilayer, the amplitude of sinusoidal wrinkles is kinematically related to the strain as 
ε=(πAλp)2(πApλp)2
(7)
Here, λp is the period of the prepattern, and Ap is the amplitude of the prepattern, i.e., the amplitude at zero strain. A derivation of Eq. (7) based on the strain–displacement kinematic relationship for the thin film is summarized in the Appendix. Equations (6) and (7) are based on the approximation that there is negligible axial strain in the top film during wrinkle formation. This approximation is accurate for large Young’s moduli ratio (Ef/Es) [29]. The second term on the right-hand side of Eq. (7) is a nondimensional parameter (εp) that quantifies the size of the prepattern. The amplitude of the displacement that is applied at the top boundary of the base was evaluated as vo = A − Ap. Thus, for the simulations of Fig. 5, the equivalent strain was varied over a range, and the corresponding amplitude of the applied displacement (vo) was evaluated as 
vo=λpπ(ε+(πApλp)2)0.5Ap
(8)

These simulations (summarized in Fig. 5) demonstrate that the energy penalty for the period-doubled mode increases when the base is prepatterned, i.e., the ratio of energy of the 2λ mode to that of the λ mode increases upon prepatterning. Thus, prepatterning the bilayer delays the onset of period doubling by affecting the deformation energy.

Prediction of Onset Strain.

As geometric prepatterning affects the dependence of deformation energy on strain, it is expected that the onset strain can be tuned by varying the amplitude of the prepattern. This hypothesis was verified via finite element simulations of the bilayer wrinkling model. The results for 85 separate prepatterned bilayers are summarized in Fig. 6. It is observed that up to a limit, the onset strain increases with an increase in the size of the prepattern. Interestingly, for small to moderately large prepatterns, this increase in onset strain is independent of the Young’s moduli ratio and the film thickness. Instead, the onset strain depends only on the nondimensional prepattern size (εp). Thus, geometric prepatterning is an elegant technique for suppression of period doubling; it can be successfully implemented even when one has limited knowledge about the material properties and/or thin-film thickness—as is the case during wrinkle fabrication via plasma oxidation. It is important to note that despite its apparent universality, the relationship between onset strain and prepattern size is accurate only for those bilayer systems for which the Young’s moduli ratio is sufficiently high (>100) and the period doubling mode is the preferred next higher-order mode. This condition is satisfied by common engineered bilayer systems such as PDMS/glass and PDMS/metal bilayers but is not satisfied by biological bilayer systems with Young’s moduli ratio of less than 10, wherein creases are the preferred next higher-order mode [21,31]. As engineered bilayer systems seldom have such low Young’s moduli ratio, the prepatterning technique presented here is applicable for design and fabrication of a variety of functional wrinkled structures.

The nondimensional prepattern size (εp) is obtained from the prepattern geometry as 
εp=(πApλp)2
(9)

Here, Ap is the prepattern amplitude, and λp is the prepattern period that is equal to the natural period (λn). Equation (9) is identical to the kinematic relationship that relates period, amplitude, and compressive strain during formation of wrinkles in flat bilayers. Thus, the nondimensional prepattern size may be interpreted as the compressive strain that is required to generate the prepattern via wrinkling of an equivalent flat bilayer. The onset strain can be increased by a factor of 1.5 from ∼20% to ∼30% with a moderate prepattern size of 0.057. This corresponds to an aspect ratio (2Ap/λp) of 0.15, i.e., an amplitude of 150 nm for a period of 2 μm. As this nondimensional prepattern size (εp) is significantly lower than the period doubling strain for flat bilayers, the prepatterns can themselves be generated via wrinkling of flat bilayers.

The universality of the relationship linking the onset strain to the nondimensional prepattern size (εp) at small to moderately large εp values can be explained in terms of another physical interpretation of εp. The nondimensional prepattern size quantifies the “prestrain” in the bilayer system because it represents the fractional increase in the curved length of the bilayer surface due to the prepattern (Eq. (7)). This initial prestrained state corresponds to a zero stress state, i.e., a state of zero deformation energy. As the subsequent single-period deformation mode is identical to the prepatterned sinusoidal mode, this prestrain quantitatively reduces the deformation energy but maintains the same qualitative deformation energy versus strain relationship. In addition, the period-doubled mode emerges when the ratio of the base deformation energies in the natural and period-doubled modes (Ub,2λ/Ub,λ) drops below a fixed threshold value with increasing strain. For small to moderately large prestrains, the combination of (i) same qualitative energy–strain relationship and (ii) ratio-based threshold suggests that the period doubling onset strain would be defined entirely by the prestrain value (εp). This expectation is supported by the results of computational modeling summarized in Fig. 6. For large prestrains (i.e., εp > 0.1), this expectation of universality is not valid anymore; instead, the period doubling onset strain varies with the material properties and film thickness.

For large prepatterns, the period doubling onset strain approaches an optimum value and then starts decreasing with further increase in the prepattern size. This observation is consistent with the deformation energy versus equivalent strain behavior of the base that is shown in Fig. 5. At very large strains, the ratio of Ub,2λ/Ub,λ for flat bilayers exceeds the ratio for prepatterned bilayers. Thus, for large prepatterns, the 2λn mode becomes energetically more favorable in the prepatterned bilayers than in the flat bilayers. This reversal of energy versus prepattern size trend then manifests as a reversal in the period doubling suppression behavior of prepatterns above prepattern sizes (εp) of ∼0.1. Nevertheless, the onset strain can be deterministically tuned via geometric prepatterning in the range of ∼20% to ∼30% strain before this reversal limit is encountered.

Enhanced Tunability of Wrinkle Geometry.

During fabrication of stretch-tunable sinusoidal wrinkled structures, delayed onset of period doubling leads to a wider range over which the geometry of the wrinkles can be tuned. As both the period and the amplitude of the wrinkles depend on the compressive strain in the thin film, an increase in the period doubling onset strain increases the range over which period and amplitude can be tuned. This increase in the geometric tunability of the wrinkles due to delayed onset of period doubling is quantified here.

Tunability of Period.

The period of wrinkles in the postbifurcation deformed coordinate system (λd) deviates away from the undeformed prepattern period (λp) when the compressive strain in the thin film is increased. This postbifurcation period corresponds to the physical period of the wrinkles in the deformed state and can be measured from the physical geometry of the wrinkles in the deformed state. This period can be evaluated using the approximation that the number of wrinkles per unit undeformed length does not change during compression of the film. This approximation has been empirically verified elsewhere [13] and has been verified here during finite element simulations. For finite element simulations, this verification is trivial as it requires one to simply verify that the number of wrinkles in the simulated domain does not change during compression of the bilayer. Before the onset of period doubling, the number of wrinkles in the simulated domain does not change from the value of 10 set here (Fig. 4(a)). Thus, for the case of bilayer compression, the period of the wrinkles in the deformed state (λd) can be quantified as 
λd=λp(1ε)
(10)

This ability to tune the period of the wrinkles with strain can be harnessed to fabricate stretch-tunable functional structures that rely on the single-period structure of wrinkles for functionality. For example, such period tunability is desirable during fabrication of tunable diffraction gratings, wherein grating period tunability leads to stretch-tunable functions such as the ability to steer the diffracted beam for sensing or imaging applications. In these applications, the increase in period doubling onset strain has a linear effect on the change in wrinkle period as predicted by Eq. (10). Thus, an increase in the period doubling onset strain by 50% leads to an increase in the range of period tunability by 50%.

Tunability of Amplitude.

As the amplitude of wrinkles is kinematically related to the compressive strain in the film, the amplitude of the wrinkles can be tuned via the compressive strain. For a prepatterned bilayer, the amplitude can be obtained from Eqs. (7) and (9) as 
A=λpπ(ε+εp)0.5
(11)

Verification of the kinematic relationships represented by Eqs. (7) and (11) has been performed against finite element simulations, as shown in Fig. 7(a). The data summarized in Fig. 7 correspond to a subset of the bilayer simulations that are summarized in Fig. 6. From finite element simulations, it is evident that the amplitude of the wrinkles in the prepatterned bilayer is higher due to a combination of the suppression of period doubling (higher ε term in Eq. (11)) and direct geometric effect of the prepattern (εp term in Eq. (11)). At high strains, the mismatch between the theoretical estimates and the finite element analysis results is due to the small slope approximations and due to dropping off the second-order strain terms during derivation of the simplified form of Eqs. (7) and (11) (details in the Appendix). Nevertheless, prepatterning the bilayer increases the amplitude of the wrinkles by at least 35% for a modest prepattern size of εp = 0.1.

Increase in Aspect Ratio.

For several applications, such as in microfluidics, the ability to fabricate deep channels with high aspect ratio may be more desirable than the ability to tune the geometry over a wide range. For wrinkled structures, the aspect ratio increases with an increase in the compressive strain in the thin film. This is because, the amplitude of wrinkles increases with increasing strain (Eq. (11)), whereas the period of wrinkles decreases with increasing strain (Eq. (10)). Thus, delayed onset of period doubling enables one to fabricate wrinkles with a higher aspect ratio than those available through wrinkling of flat bilayers. Using Eqs. (10) and (11), the aspect ratio of the wrinkles (ra = 2 A/λd) can be evaluated in terms of the compressive strain as 
ra=2π(11ε)(ε+εp)0.5
(12)

Equation (12) has been verified against finite element simulations, as shown in Fig. 7(b). The mismatch between the theoretical estimate and the finite element simulations arises due to the mismatch in the amplitude for the two cases. Nevertheless, geometric prepatterning with a modest prepattern size of εp = 0.1 increases the aspect ratio of the wrinkles by at least 57%. Thus, by delaying the onset of period doubling through geometric prepatterning, one can increase the aspect ratio of wrinkles to values not achievable in flat bilayers.

The maximum aspect ratio for single-period wrinkles is obtained when the strain in the thin film is equal to the period doubling onset strain, i.e., when ε = ε2,p. When geometric prepatterning is used to delay the onset of period doubling, high aspect ratio wrinkles are obtained due to the combined effect of three factors: (i) geometry of the prepattern (εp term in Eq. (12)), (ii) the effect of prepattern on the increased period doubling onset strain (ε term in the numerator of right-hand side of Eq. (12)), and (iii) the effect of strain on the period of the wrinkles (ε term in the denominator of right-hand side of Eq. (12)). The maximum aspect ratio for several prepatterned bilayers is shown in Fig. 8. The data summarized in this figure were obtained by analyzing the geometry of the wrinkles in the same set of 85 bilayers that were simulated to generate the data summarized in Fig. 6.

The dependence of maximum aspect ratio on the nondimensional prepattern size demonstrates the same universality at small to moderate prepattern sizes as that observed in the period doubling onset strain versus prepattern size behavior (Fig. 6). This behavior is expected because the maximum aspect ratio depends on the onset strain. At large prepattern sizes, the onset strains for bilayers with different material properties are not identical; however, the change in onset strain is substantially smaller than the corresponding change in the prepattern size. Thus, the aspect ratio is dominated by the direct geometric effect of the prepattern size (i.e., by the εp term in Eq. (12)) for large prepatterns. Consequently, the universality of maximum aspect ratio versus prepattern size is valid over a wide range of prepattern sizes, i.e., one can accurately estimate the maximum aspect ratio simply by knowing the prepattern size even without having any knowledge about the material properties or the thin-film thickness. It is important to note here that when wrinkling of thin films is used to fabricate the prepatterned bilayers, only a finite section of the curve in Fig. 8 is physically accessible. This is because prepatterns with a size (εp) greater than ∼0.3 cannot be fabricated due to the onset of period doubling at a strain of ∼30%. Thus, the maximum aspect ratio is limited to about 0.7 for a prepattern size of εp = 0.3. This is more than twice the maximum aspect ratio that is achievable via compression of flat bilayers.

Limitations of Prepatterning Technique

Limitations of Fabrication Technique.

Despite its elegance, the need for the prepattern period to be “identical” to the natural period may be perceived as a limitation for practical implementation of this technique. When the prepattern period is not identical to the natural period, a multiperiod hierarchical pattern is expected [10]. However, this concern can be overcome by the mode lock-in phenomenon that occurs during compression of prepatterned bilayers [29]. We have recently identified that the emergence of the hierarchical mode is preceded by a mode locked state wherein the prepattern mode persists [25,29]. When the prepattern period is “substantially similar” to the natural period, it is possible for the single-period mode to persist beyond the period doubling onset strain, as illustrated in Fig. 9. The next step to developing this technique would be to quantify this substantial similarity regime. Nevertheless, it is evident from the experiments performed here that this limit is higher than the variations in λp/λn that arise due to fabrication errors; therefore, this scheme can be successfully implemented even with an imperfect match between the prepattern and natural periods.

Limitations of Computational Modeling.

Although a base prestretching-based technique has been used herein to generate film compressive strains during physical experiments, finite element modeling has been performed by simultaneously compressing the film and the base without any base prestretch. This is because of the inability to accurately implement a base prestretch-based computational technique in the comsol software package that was used here. For wrinkling computations using other packages, the standard technique for implementing prestretch involves first stretching the bilayer and then setting the stress in the top layer to zero. This condition then mimics the physical experiments, wherein an unstressed thin film is deposited on top of a prestretched base layer. When implemented in comsol, this technique leads to an inaccurate nonlinear strain–displacement relationship for the thin film as the strain is evaluated using the shorter initial unstretched configuration of the top film instead of the unstressed length of the film on a stretched base. In the absence of an accurate prestretching technique, a direct film compression technique has been used here. As the prestretch in the base affects the period doubling onset strain [2224], this approximation somewhat limits the quantitative applicability of the technique presented here. Specifically, the onset strain due to prepatterning may not be identical to those in Fig. 6 when this prepatterning-based technique is physically implemented in prestretched bilayer systems. Nevertheless, the following observations are still applicable: (i) prepatterns delay the onset of period doubling as verified by physical experiments performed here and (ii) universality of the onset strain versus prepattern size at small to moderately large prepattern sizes. Thus, when the prestretch is nonzero but held constant, one may still apply the results from Fig. 6 with the modification that the onset strain for flat prestretched bilayer is higher than the flat unstretched bilayer. To do so, one may experimentally observe the period-doubling onset strain in a flat bilayer at the applied prestretch and then use it as an offset to increase the values predicted by the curve of Fig. 6 to predict the effect of prepatterning. For typical prestretch values of ∼20%, such an approach is expected to require an increase of ∼6% strain [23] over the values for the onset strain obtained from Fig. 6 to generate realistic predictions of the onset strain in prestretched and prepatterned bilayers.

Conclusions

In summary, a practical technique for delaying the onset of period doubling via geometric prepatterning has been demonstrated here. The onset strain for period doubling can be increased from a typical value of 20% to >30% strain with a moderate prepattern aspect ratio of 0.15. This increase in the onset strain is accompanied with an increase in the range of period tunability by at least 50%, increase in amplitude of the wrinkles by at least 35%, and increase in the aspect ratio of the structures by at least 57%. For small to moderately large prepatterns, the suppression behavior is fully captured by the prepattern geometry. Thus, this technique can be implemented even with limited knowledge about the properties of the wrinkling bilayer system. Prepatterning with the natural period of the system maintains the single-period morphology of the wrinkles at high strains. Consequently, the operating range of a variety of stretch-tunable functional devices can be increased by at least 50% simply by replicating the wrinkled patterns and repeating the wrinkle fabrication steps on the prepatterned system.

Acknowledgment

This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract No. DE-AC52-07NA27344. The author utilized the postdoctoral funding for independent research available at LLNL to write this manuscript (#LLNL-JRNL-683274). The author also thanks Professor Martin Culpepper at MIT for access to plasma etcher and Dr. Prakash Govindan at Gradiant Corporation for access to laboratory workspace.

Appendix: Derivation of Kinematic Relationships

The kinematic relationships represented by Eqs. (6), (7), and (11) have been obtained from the sinusoidal geometry of the wrinkles and the strain–displacement relationship of the base and the thin film. Before the onset of wrinkles, the film deforms through the axial compressive mode, i.e., through lengthwise shortening of the film. At the first bifurcation strain, the axial compressive mode becomes energetically unfavorable, and the film starts deforming via bending. Thus, shortening of the film is limited to the initial axial compression. For flat bilayers with Young’s moduli ratio greater than 1000, this first bifurcation strain is ∼0.5% and may be neglected for applied strains on the order of 15%. For prepatterned bilayers, this shortening is less than that observed in the corresponding flat bilayer [29]. Thus, the curved length of the thin film can be approximated as an invariant parameter during strain-driven wrinkle growth. This invariance is used here to relate the geometry of the wrinkles to the compressive strain in the thin film.

For Flat Bilayers.
Without loss of generality, the domain can be restricted to a single wrinkle period to derive the kinematic relationships. The sinusoidal profile of the wrinkles is given by 
y=Acos(2πxλd)
(A1)

Here, x and y are the spatial coordinates in the deformed coordinate system, and λd is the wrinkle period in the deformed coordinate system. The natural period and the deformed period are related through the kinematic relationship as λd = (1 − ε)λn.

In the undeformed state, the length of the thin film over one period is equal to the natural period λn. In the deformed state, the curved length of the film (Lc) is given by 
Lc=0λd(1+(dydx)2)0.5dx
(A2)
This integral can be simplified by expanding the integrand using the generalized Binomial expansion and by dropping the higher-order terms as 
Lc=0λd(1+0.5(dydx)2)dx
(A3)
This simplification is based on the approximation that the slope of the wrinkles is small enough so that the third-order and higher terms (i.e., (dy/dx)3 and higher terms) may be neglected. For a wrinkle of amplitude 150 nm and a period of 2 μm, this slope is 0.05 so that such an approximation is reasonable. By substituting for the known geometry of the wrinkles (Eq. (A1)), one obtains 
Lc=λd+π2A2λd
(A4)
As the length of the thin film is an invariant during wrinkle growth, the curved length is identical to the natural period. Thus, one obtains 
A=(λnλd)λdπ
(A5)
By substituting for λd, the kinematic relationship for wrinkle formation during compression of a flat bilayer is obtained as 
A=λnπε(1ε)
(A6)
For small to moderate strains, the second-order ε2 term is often dropped from Eq. (A6) to obtain the well-known amplitude versus strain scaling as 
A=λnπε
(A7)
For Prepatterned Bilayers.
For a prepatterned bilayer, the geometry of the wrinkle is still represented by Eq. (A1). In addition, the constancy of the length of the thin film during wrinkle growth also holds for prepatterned bilayers but with the modification that the undeformed length of the thin film is equal to the curved length of the prepatterned geometry. Thus, following the mathematical steps for Eq. (A4), the length of the film in the undeformed state (Lo) is obtained in terms of the prepattern geometry (Ap and λp) as 
Lo=λp+π2Ap2λp
(A8)
Similarly, the length of the film in the deformed state is obtained as 
Lc=λd+π2A2λd
(A9)
In the prepatterned bilayer, the deformed period and the prepattern period are kinematically related by Eq. (10). Thus, the invariance of the length of thin-film thickness (i.e., Lo = Lc) results in the kinematic relationship 
π2A2λp2=(1ε)(ε+π2Ap2λp2)
(A10)
As the order of magnitude of the term π2Ap2/λp2 is same as that of the strain ε, one may simplify Eq. (A10) by dropping the second-order strain terms as 
π2A2λp2=ε+π2Ap2λp2
(A11)

This simplified form results in an approximate kinematic relationship represented by Eq. (11) that is analogous to the well-known kinematic relationship for flat bilayers represented by Eq. (A7).

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