Wrinkles in Polytetrafluoroethylene on Polystyrene ... - ACS Publications

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Wrinkles in Polytetrafluoroethylene on Polystyrene: Persistence Lengths and the Effect of Nanoinclusions Jeffrey T. Paci,*,†,‡ Craig T. Chapman,† Won-Kyu Lee,§ Teri W. Odom,†,§ and George C. Schatz*,† †

Department of Chemistry and §Department of Materials Science and Engineering, Northwestern University, 2145 Sheridan Road, Evanston, Illinois 60208-3113, United States ‡ Department of Chemistry, University of Victoria, P.O. Box 3065, Victoria, British Columbia, Canada V8W 3V6 S Supporting Information *

ABSTRACT: We characterize wrinkling on the surfaces of prestrained polystyrene sheets coated with thin polytetrafluoroethylene skins using a combination of mechanical strain measurements and 3D finite element simulations. The simulations show that wrinkle wavelength increases with skin thickness, in agreement with a well-known continuum model and recent experiments. The wrinkle amplitudes also increase with strain. Nanoinclusions, such as holes and patterned lines, influence wrinkle patterns over limited distances, and these distances are shown to scale with the wrinkle wavelengths. Good agreement between experimental and simulated influence distances is observed. The inclusions provide strain relief, and they behave as if they are attracting adjacent material when the sheets are under strain. The wrinkles have stiffnesses in much the same way as do polymers (but at different length scales), a property that is quantified for polymers using persistence lengths. We show that the concept of persistence length can be useful in characterizing the wrinkle properties that we have observed. However, the calculated persistence lengths do not vary systematically with thickness and strain, as interactions between neighboring wrinkles produce confinement that is analogous to the kinetic confinement of polymers. KEYWORDS: wrinkles, plasma treatment, thin films, finite element simulations, polystyrene, persistence length, strain relief, Teflon

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INTRODUCTION

The processes associated with wrinkling are inherently nonlinear. A flat sheet composed of a substrate and skin initially behaves in a linear way when compressed, i.e., doubling the force doubles the strain. However, under sufficiently large compression, an elastic instability occurs,12 and the sheet behaves nonlinearly as soon as wrinkles form. Wrinkling occurs because the energy necessary to bend the skin and stretch the substrate to which it is attached is lower than the energy that would be necessary to compress the skin and substrate. Nonlinear theory for defect-free systems has been presented.13 It predicts that the wrinkle wavelength is

Wrinkles form in a wide range of materials, and they tend to form when hard skins on soft substrates are subject to compression.1,2 Wrinkling on nano- and micrometer length scales can occur when there is a significant mismatch in the thermal expansion coefficients of materials,3 or when prestrain is relieved in substrates.4 Stripes, herringbone patterns, and labyrinths have been observed. To control the orientation and ordering of wrinkles at the local level, researchers have introduced strain relief features. The relief structures tend to cause wrinkles neighboring the structures to straighten or flatten.3,5 Wrinkling provides a means of altering the properties of a surface to, for example, change its hydrophobicity6,7 or instill antifouling properties.8 It also enables the development of materials for flexible electronic devices,9,10 devices for mechanical measurements11 and so on, so it is important to understand and control factors such as wrinkle wavelength and orientation that influence wrinkle patterns. The introduction of nanoinclusions facilitates control over wrinkle patterns with resolution of 100 to a few hundred nanometers. We show that the implications of these inclusions can be accurately determined experimentally and modeled using finite element simulations. © XXXX American Chemical Society

⎛ E (1 − ν 2) ⎞1/3 b ⎟ λ = 2πh⎜ s 2 3 E (1 − ν ⎝ b s )⎠

(1)

where h is the thickness of the skin, Es and Eb are the Young’s moduli of the skin and substrate, and νs and νb are the Poisson’s ratios. In a recent set of experiments,5 Huntington et al. exposed prestrained polystyrene (PS) to plasma in a reactive ion etching Received: November 17, 2016 Accepted: February 17, 2017

A

DOI: 10.1021/acsami.6b14789 ACS Appl. Mater. Interfaces XXXX, XXX, XXX−XXX

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ACS Applied Materials & Interfaces

Figure 1. (a) Fabrication process used in forming nanowrinkles in the presence of various types of nanoinclusions. (b) Orientation of wrinkles perpendicular to patterned lines of strain relief and (c) orientation of wrinkles by strain relief due to holes in the skin layer.

the Kelvin model and a typical linear isotropic relaxation modulus, respectively, were used to approximately account for viscoelasticity. The authors acknowledge that additional work is necessary to further elucidate the behavior of such systems. We show that by making an appropriate choice for the Young’s modulus, the behavior of PS in the PTFE/PS wrinkle system can be accurately modeled as elastic, for the small-strain regime (≤10% strain) that is of interest for this work. In this paper, we use theoretical and computational methods and experiments to examine the influence of nanoinclusions on the formation of wrinkles. Our computational analysis includes the structures considered in ref 5 as well as new experiments that provide additional details concerning the interplay between wrinkle wavelength and nanoinclusion. We use finite-element simulations to model these systems at small strains. The PTFE on PS wrinkle system provides wavelength control and high aspect ratios. The inclusion of strain relief points is crucial, as they provide control over the order/disorder of wrinkles at the local level, across all length scales. The wrinkles have stiffnesses in much the same way as do polymers, and we demonstrate that the application of the notion of persistence length that is often used for polymers can be used to characterize wrinkles.17 In addition, we show that nanoinclusions influence wrinkle patterns over limited distances that are related to the wrinkle wavelengths. The results of our simulations and modeling are in good correspondence with the experimental studies.

system. The PS was exposed to CHF3 gas, which resulted in the growth of a layer of polytetrafluoroethylene (PTFE) similar to Teflon. Layers 10 nm to a few 10 s of nm thick were deposited. Strain-relief patterns were created by applying masks patterned with arrays of recessed posts or lines, the details of which can be found in ref 5. The chemically treated PS was then heated to above the glass transition temperature, Tg, for PS, shrinking the film. The detailed fabrication processes to form nanowrinkles on the skin layers patterned with different nanoinclusions are shown in Figure 1a. Figure 1b, c shows examples of the types of the resulting wrinkle patterns created in those experiments. These systems have been shown to facilitate wrinkling with a wide range of wavelength tunability.4,5 Modeling is important to provide a deeper understanding of the processes involved. The PTFE forms thin stiff skins, and when the PS is heated above Tg the sheets shrink, resulting in the formation of wrinkles.5 Polystyrene above Tg is a viscoelastic material. As such, its shear and bulk moduli, which are related to its Young’s modulus, change with time once a stress has been applied. This behavior can be modeled using a Prony series, but the timedependence of the series makes its application challenging. Wrinkling on viscous substrates has been modeled using lubrication theory.14 Computational effort is a challenge in such simulations, because, given a sufficient amount of time, flow in a viscous system will result in the substrate storing no energy. The bending energy of the film would cause the wrinkles to coarsen leaving only the largest wrinkles (e.g., a half period of a sinusoidal curve).14,15 However, smaller wrinkles can persist for long times if they are kinetically constrained. In refs 15 and 16, B


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EXPERIMENTAL METHODS

how to define time for a relaxation time for Abaqus/Standard as it is for a Prony series in Abaqus/Explicit. Motivated by the dimensions of the samples shown in Figure 1, the models are typically 4 μm × 2 μm × 500 nm (l × w × h) for 20 and 30 nm skin thicknesses, and 2 μm × 1 μm × 500 nm for 10 nm skins. System size is one variable that impacts computer run time, and doubling the system volume doubles that time. Mesh dimensions also affect the run time, and finer meshes are necessary for the 10 nm skin calculations as discussed below, which is the reason for the use of the smaller system size for this case. Geometric nonlinearity is allowed. The base of the PS is always fixed in the direction perpendicular to the surface. The mesh dimensions are 25 and 12.5 nm for the large and small sheets, respectively, except for the thicknesses of the skins. 50 nm meshes produce results with features that appear unphysical for the 20 and 30 nm skins. Analogous problems occur for the 25 nm mesh for the 10 nm skins. Halving the mesh size increases computer run times by approximately a factor of 16. In the experiments, there is a transition region between the skin and substrate ∼2−5 nm thick. Unlike the case of gold on PS,23 for which the transition region is on the order of hundreds of nanometers, we expect the transition region for the plasma-treated cases to have only a minor impact on the wrinkling because of its small thickness. Therefore, this region was not included in our models. C3D8R elements were used. These elements are potentially susceptible to the hourglassing form of locking. Testing indicated that the elements used were sufficiently small that hourglassing had no appreciable effect on the results. We pursue solutions for which a decrease in strain rate would have no effect on the wrinkle pattern, as we can not computationally afford to apply strain as slowly as in the experiments. Pressures were applied by gradually moving the walls at the edges of the sheets, and were applied to the substrates and skins that contact those edges. A schematic of a typical sheet geometry, the boundary conditions and loading directions is shown in Figure 2a. A rough calculation for our

Fabrication of 2D Wrinkles. Prestrained PS films were treated with a CHF3 plasma in a reactive ion-etching (RIE) system (Samco) for different exposure times at a flow rate of 25 sccm, at a power of 70 W and a pressure of 25 Pa. The resulting skin thickness on the PS was controlled by changing the treatment time.4 The skin-deposited PS was then heated in a convection oven at 130 °C until the substrate shrunk to the desired size. The strain was determined by measuring the area of a box drawn on the substrate before and after shrinking and using the equation ϵ = (A0 − Af)/A0, in which A0 is the initial area and Af is the final area. PS surfaces with nanowrinkles were then coated with a layer of AuPd with a thickness of approximately 8 nm for SEM imaging. Fabrication of Strain-Relief Structures. Composite poly(dimethylsiloxane) (PDMS) masks patterned with arrays of recessed posts or lines were prepared for inverse solvent-assisted nanoscale embossing (inSANE).18 Directed nanowrinkles by strain relief structure on the PS substrate were formed by (1) patterning of the top of the PS films with photoresist in ethanol by inSANE; (2) treatment of the PS films with a plasma in a RIE system (Samco) for different exposure times of CHF3 as the gas at a flow rate of 20 sccm, and at a power of 70 W and a pressure of 25 Pa; (3) deposition of a thin (4 nm) layer of Cr by using an electron beam deposition system and lifting off of the photoresist by the use of isopropanol; (4) treatment of the PS films with a plasma in a RIE system (Samco) for 30 s with the gases Ar (30 sccm) and O2 (5 sccm) at 70 W and 20 Pa; and (5) removal of Cr with chromium etchant (Transene Company Inc.) and heating of the PS in a convection oven at 130 °C until the films had shrunk to the desired size.

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FINITE ELEMENT SIMULATIONS We use Abaqus/Explicit19 for the simulations, and dynamic/ explicit steps. The compression of the sheets in the experiments is expected to result in the selection of low energy buckling modes. In an attempt to model this process as closely as possible, we compressed our sheets in an analogous way, allowing for the selection of modes. Teflon has a Young’s modulus of 0.4 GPa, a Poisson’s ratio of 0.5, a density of 2.20 × 103 kg/m3, and is treated as being purely elastic. The density of PS above Tg is 1.05 × 103 kg/m3. One might expect the PS to start to quickly shrink as soon as it is released (unclamped) while at a temperature well above Tg as in the experiments. Instead the shrinking occurs very slowly though (over minutes), so the PS does not seem to be behaving as a liquid (Poisson’s ratio ∼0.5). The PS is flowing, but it is just barely flowing. On the basis of these considerations, and the works of refs 20 and 21, we used a Poisson’s ratio of 0.35 for the PS unless otherwise stated. The Poisson’s ratio of a material can have a profound impact on its behavior in wrinkle systems.13 The Young’s modulus of PS is known to drop by a factor of 1000 or more when heated from below to above Tg.16,20,22 On this basis, we approximate the viscoelasticity by modeling PS as an elastic material with a Young’s modulus of 4 MPa except as noted. Abaqus/Explicit uses a Prony series to model viscoelasticity. To perform simulations using a series, the simulations need to be done in or in nearly the time it takes for the actual process. In the experiments, this is minutes. The simulations have strains applied in on the order of 1 × 10−7 to 1 × 10−5 s, and increasing this time to minutes is not computationally affordable. Also, our experimental systems are not under stress at time zero; they are under prestrain. Therefore, it is unclear how to treat time in the series in our case. Stress causes the wrinkling, and the time at which the stress arises is not well-defined. So, for example, it is as unclear

Figure 2. (a) Schematic illustration of a typical simulation sheet showing the geometry, boundary conditions and loading directions. (b) Typical initial geometries and initial locations of a pair of holes. All holes in the simulations initially had diameters of 100 nm. (c) Persistence length variable definitions: Tangent vectors at positions A and B separated by the distance s along the contour of a polymer chain. θ is the angle between tangent vectors. C


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Figure 3. Results of the application biaxial strain of (a) 5% to a 2 μm × 1 μm × 500 nm sheet with a 10 nm skin, (b) 10% to a 4 μm × 2 μm × 500 nm sheet with a 20 nm skin, and (c) 10% to a 4 μm × 2 μm × 500 nm sheet with a 30 nm skin. Each sheet initially contained a 100 nm hole. (d) The application of 5% biaxial strain to a 2 μm × 1 μm × 500 nm sheet with a 10 nm skin, with an additional 2 μm × 150 nm of substrate attached. (e, f) Application of 10% biaxial strain to a 4 μm × 2 μm × 500 nm sheet with a (e) 20 and (f) 30 nm skin, with an additional 4 μm × 300 nm of substrate attached. The scale bars are approximate, because the sheets have been tilted for clarity. Figure 1b, c show typical experimental results for patterned lines and holes, respectively, that can be compared with the simulations.

contours were drawn for each figure panel. Tangents were calculated along the lengths of the contours, and three estimates for the cosine as a function of s were made for each contour, starting from positions well separated from each other. This resulted in 30 estimates for the cosine as a function of s for each panel. The results were averaged using s bins that were 5 nm wide, and the ln of the cosines were fitted to straight lines out to 100 nm.

systems suggests such solutions may result from pressures applied over as little as ∼1 × 10−7 s. Testing indicates that the peak stresses resulting from applying 10% strain over 1 × 10−7 s are different than 2 × 10−7s results by approximately 50% for a 2 μm × 1 μm by 500 nm model (maximum speed = 2 μm × 10%/1 × 10−7 s = 2 m/s). The wrinkle patterns were essentially the same. Peak stress should be independent of the time over which pressure is applied, provided the pressure is applied slowly enough. Similarly, the stresses in 4 × 10−7s calculations are different than 2 × 10−7s calculations by approximately 20%. Calculation run times scale linearly with pressure application times. Strain speeds were restricted to 2 m/s or less for the results reported in this work. The larger the distance over which strain relief occurs, the smaller the acceptable speed. The stability of an Abaqus/Explicit procedure is ensured provided that a sufficiently small time increment is used. Abaqus/Explicit uses a conservative estimate in setting the default value for this increment for a given problem. We used this default value. The stiffness of a polymer chain can be quantified using its persistence length, Lp.17 For lengths of less than ∼Lp, the chain behaves in a stiff, elastic, rodlike fashion. For larger lengths, the chain behaves flexibly, with motions describable using concepts associated with a random walk. The length is defined as24 cos(θ(s)) = exp( −s /2Lp)

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SIMULATION RESULTS We begin by considering the simulation sheet with a 20 nm skin and a central hole 100 nm in diameter placed under 10% biaxial strain (see Figure 3b). This shows that the surface becomes wrinkled in a labyrinth pattern,3,13 and the hole orients the wrinkles in a way analogous to what has been observed experimentally (see Figure 1c).3,5 The labyrinth is an efficient means of relieving compression in all directions.13 The wrinkle wavelength is λ ≈ 150 nm, and the hole orients wrinkles to a radius of ∼500 nm. The wavelength was estimated by counting wrinkle periods, and the radius was estimated by eye. Analogous results for 10 and 30 nm skins are shown in Figure 3a, c, respectively. The 10 nm skin has a wavelength of ∼80 nm and the hole is oriented to a radius of ∼300 nm, while the 30 nm skin has a wavelength of ∼200−225 nm and the hole is oriented to a radius of ∼700 nm. The wrinkle wavelengths are approximately consistent with, although somewhat smaller than, those observed experimentally (see Figure 1c and ref 4), where wavelengths of ∼100, 200, and 300 nm for 10, 20, and 30 nm skins were observed. Also, the wavelengths are on the same order as those predicted in our modeling and experiments for gold on PS, and they increase with skin thickness in approximately the same way, consistent with a widely known continuum model.23 For the 20 nm skin, eq 1 suggests λ = 400 nm. Thus, the model on which the equation is based appears sufficient to provide semiquantitative wavelength estimates for our systems. The finite thicknesses of the substrates in our sheets are possible sources of the discrepancies.13 This level of agreement is consistent with what has been observed experimentally for

(2)

where ⟨cos(θ(s))⟩ is the average cosine of the angle, θ, between the tangent vectors of the polymer, and s is the contour distance between them (see Figure 2 c). Wrinkles also have associated stiffnesses, so we explore the application of the concept of Lp to our systems. The factor of 2 in the exponential in eq 2 accounts for the fact that we use two-dimensional images in the analysis. Persistence lengths were calculated following the method described in ref 25 for polymers, with contours drawn along the wrinkles using ImageJ V1.50b26 and the freehand lines tool, in regions remote from the holes. The contours were then smoothed as in ref 27 using the weighted average of five contiguous coordinates centered around a given point. Ten D


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ACS Applied Materials & Interfaces wrinkles on ultraviolet/ozone radiation- and plasma-treated PDMS.28,29 The influence patterned lines have on wrinkle patterns was also investigated. Figure 3e shows the result of applying 10% biaxial strain to a sheet with a 20 nm skin. This was a 4 μm × 2 μm by 500 nm sheet to which 4 μm × 300 nm of substrate was added. Strain relief adjacent to the 300 nm patterned line results in the formation of stripes, and a herringbone-like wrinkle pattern.29,30 Analogous results for a 10 and 30 nm skins are shown in Figure 3d−f. The wrinkle patterns are qualitatively the same as in the 20 nm case. A herringbone pattern has been shown to constitute the minimum energy configuration among a set of competing periodic modes (one-dimensional, checkerboard, herringbone), for hard films on thick elastomeric substrates undergoing biaxial strain relief.30 This pattern produced the lowest average elastic energy per unit area for the set for this type of system, for stresses well above the critical stress for wrinkling. The pattern relaxes the biaxial stress in the film in all directions, while generating very little stretch energy. Its presence in the cases in which there are patterned lines, which provide significant anisotropic strain relief, and the presence of labyrinths in the cases of holes, which provide approximately isotropic strain relief, is consistent with the works of refs 13 and 15. The herringbone pattern instead of longer stripes may result from the use of a Young’s modulus of 4 MPa for the PS in the simulations rather than the fully viscoelastic materials used in the experiments of ref 5. In the experiments, the substrate may participate in the wrinkling process at a lower modulus. This may result in more strain relief in the direction perpendicular to the patterned lines in the experiments, which is something we examine in detail below. The patterned lines provide strain relief that is analogous to the holes. These lines result in adjacent wrinkles that are highly ordered for lengths of 300, 500, and 700 nm, for the 10, 20, and 30 nm skins, respectively (see Figure 3). These lengths are approximately the same as the corresponding radii over which holes align wrinkles. We refer to these lengths as Lr. The implications for wrinkling of pairs of holes in sheets were also examined, as pairs of holes provide insight into details of inclusion-inclusion interactions. Figure 2b illustrates typical hole geometries and locations. These sheets were 4 μm × 2 μm × 500 nm with 20 nm skins, and the holes were 100 nm in diameter. The result for an initial hole separation of 300 nm is shown in Figure 4a. The wrinkle wavelength was ∼150 nm as in the analogous case of a single hole, and the holes orient wrinkles along the full distance between the holes, as might be expected. The analogous result for a pair of holes initially separated by 500 nm is shown in Figure 4b. The holes orient wrinkles along the full distance between the holes in this case as well. For a pair of holes initially 2000 nm apart (see Figure 4c), each hole orients wrinkles to a radius of ∼500 nm, as in the analogous single-hole case.

Figure 4. Application of 10% biaxial strain to a 4 μm × 2 μm × 500 nm sheet with a 20 nm skin, and containing a pair of 100 nm holes initially separated by (a) 300, (b) 500, and (c) 2000 nm. The scale bars are approximate, because the sheets have been tilted for clarity. Figure 1c shows typical experimental results for holes that can be compared with the simulations.

Table 1. Persistence Lengths of the Wrinkles in Figure 3a−c, along with Standard Errors of the Means (SE), and R2 Values for the Fits to the ln of the Average Cosines versus s of eq 2a skin thickness (nm)

Lp (nm)

SE (nm)

R2

Lr (nm)

λ (nm)

10 20 30

48 158 154

9 29 28

0.95 0.87 0.96

300 500 700

80 150 200−225

a Lr is the radius to which a hole aligns wrinkles, which is approximately the same as the distance to which patterned lines align wrinkles. λ is the average wavelength of the wrinkles.

straight-length segments than wrinkles near sources of strain relief, where some of this confinement has been relieved. The average highly ordered lengths of wrinkles around holes and adjacent to patterned lines, Lr, are much larger than the persistence lengths (see Table 1). The Lr suggest the persistence length of the wrinkles in the 30 nm skin would be significantly larger than the persistence length for the 20 nm skin, in the absence of confinement due to adjacent wrinkles. However, wrinkles remote from sources of strain relief are confined in ways that are analogous to how polymers are sometimes confined due to kinetic effects by strong adhesion to substrates.31 As a result, wrinkles span between holes that are separated by much more than twice the persistence lengths. The maximum distance that can be spanned is approximately the diameter to which an individual hole orients wrinkles for a given skin thickness (2Lr), which is the same as twice the highly ordered length of wrinkles adjacent to patterned lines. As for polymers, the persistence length of a wrinkle is a convolution of its stiffness and level of confinement. Wrinkles are always associated with adjacent wrinkles. Adjacent wrinkles confine in such a way that they impose bends that make Lp

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PERSISTENCE LENGTHS The results of persistence length estimates for the wrinkles shown in Figure 3a−c are shown in Table 1, with the associated R2 values for the fits. The standard errors of the means (SE) are also reported. The Lp for the 10 nm skin is significantly smaller than for the 20 and 30 nm skins. This is consistent with the 10 nm skin having smaller bending stiffness. The fits for all three Lp values are good, as indicated by the R2 values. Wrinkles remote from holes and patterned lines are confined by their neighbors, so it is expected that these wrinkles will have shorter E


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increase from ∼10 nm at 1% strain (aspect ratio = 0.1) to ∼30 nm at 10% strain (aspect ratio = 0.4). The Lp fits at 1 and 2.5% strains are poor and those for 5− 10% strains are good, as can be seen from the R2 values in the table and in Figure 5. This suggests that the Lp values of the wrinkles at the two lowest strains are reflective of the fact that those wrinkles do not exhibit the statistics associated with welldefined persistence lengths. The wrinkles are significantly more confined by neighboring wrinkles in some regions than they are in others. At larger strains, the wrinkles butt up against their neighbors, with the R2 values associated with the Lp suggesting all wrinkles are approximately equally confined. As strain increases, the segment lengths (i.e., Lp) have been predicted to decrease for analogous defect-free systems,13 a result expected to apply to our defect-containing systems should larger strains be applied. As shown in Table 1, the Lr values are significantly larger than λ for a given skin thickness. The continuum model of ref 3, which assumes only tangential forces along lines of strain relief and a single wrinkle wavelength, suggests

shorter than what it would be for an isolated wrinkle. Similarly, neighboring wrinkles confine in a way that makes Lr as large as it is, and perhaps larger than what would be the persistence length of a wrinkle in the absence of confinement. The bending stiffness of a wrinkle is related to the stiffness of a plate, which is proportional to its thickness cubed.12,30

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PERSISTENCE LENGTHS AS A FUNCTION OF STRAIN Let us now consider the dependence of persistence lengths as a function of applied strain. This is illustrated using Figure 3b and related models. As discussed above, Lp = 158 ± 29 nm when this sheet is under 10% strain. As shown in Table 2, Lp is Table 2. Persistence Length versus Strain for the 20 nm Skins, along with Standard Errors of the Means (SE), R2 Values for the Fits to the ln of the Average Cosines versus s of eq 2, Average Wrinkle Amplitudes, A; Average Wavelength of the Wrinkles, λ; and Aspect Ratioa

⎛ E (1 − ν 2) ⎞ b ⎟ Lr ≈ 0.3h⎜ s 2 E (1 − ν ⎝ b s )⎠

strain (%)

Lp (nm)

SE (nm)

R2

A (nm)

λ (nm)

aspect ratio

0.5 1 2.5 5 7.5 10

N/A 903 139 186 147 158

N/A 165 25 34 27 29

N/A −0.66 −0.21 0.82 0.96 0.87

N/A 10 20 20 30 30

N/A 150 150 150 150 150

N/A 0.1 0.3 0.3 0.4 0.4

(3)

This gives Lr ≈ 17 λ. Our modeling suggests Lr ≈ 3 or 4λ for our system. The Fourier transforms of the wrinkle patterns and their radial dependence are shown in Figures S1 and S2, and they provide insight into a possible source of the disagreement. There are distributions of wavelengths associated with our wrinkles as shown in Figure S2. The wavelengths associated with the lowest characteristic wavelengths are expected to dictate Lr, suggesting better agreement with the continuum model than is indicated by the use of wavelengths associated with the distribution of amplitude maxima. When the size of the nanoinclusion is comparable to the wavelength, then this leads to the strongest coupling. The wrinkle wavelengths are independent of the sizes of the nanoinclusions. The persistence lengths are also independent of nanoinclusion size, provided the lengths are calculated outside the distances over which the inclusions influence wrinkle patterns, as in the case of this work. Lr is a function of the sizes of the nanoinclusions. Lr takes on its asymptotic value for an inclusion with dimensions that exceed ΔL, the change in length of the sheet. Lr also has this value for smaller sizes, as is limited by the stiffnesses of the skin and substrate. For sufficiently small inclusions, Lr goes to zero monotonically, as the size of the nanoinclusion goes to zero.

a

There were no wrinkles at 0.5% strain. Images of wrinkle patterns are also shown above. The scale bar is approximate.

approximately the same at smaller strains including 2.5%. The differences in the wrinkle patterns are subtle between 2.5 and 10% strain, a result that is consistent with the invariance of relative energy ordering of different wrinkle patterns with applied stress above a certain threshold,30 and with nonlinear theory.13 For the most part, strain is accommodated by way of an increase in the average wrinkle amplitude (A) as strain is increased. This is consistent with experimental observations for wrinkles on plasma-treated PDMS,29 and with theoretical predictions.13,32 This average increase is expected to continue until eventual saturation caused by nonlinear effects in stretching and shearing the substrate, at which point a new generation of larger-wavelength wrinkles may arise.28 In all cases, the holes orient wrinkles to radii ∼500 nm. At 1% strain, Lp = 903 ± 165 nm, and the wrinkles have small amplitudes and are approximately parallel to the sheet walls. The pattern represents an approximate checkerboard mode, a 2D wrinkle mode that is of low energy for small stresses.13,30 The Lp for the 1% strain results are not well determined, so the results for 1% strain in Table 2 are not significant. There are no wrinkles at 0.5% strain. The increase in A with strain can be seen in the table, where A is defined as half the difference in the height of the wrinkle crests versus troughs. The amplitudes in the table

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POLYMER−HOLE INTERACTIONS Holes do not have boundary conditions associated with them that would lead to attraction or repulsion of material. However, the collective system response around a hole can “act” as if the holes attract or repel. For example, when the stress of the surface element at the right edge of the hole in Figure 3a is plotted versus simulation time, the stress along the radial direction of the hole (σ11) is negative most of the time, i.e., the hole is apparently “attracting” that piece of the material during most of the simulation. The materials are placed under strain or are undergoing the release of prestrain, so, for the most part, the holes allow for strain relief, i.e., material is forced into the holes, and the skin bends easily. The skin is stiff in tension, so it does not stretch easily. This means that holes do not “repel” material very efficiently. The F


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Figure 5. Straight-line fit to ln < cos(θ(s)) > versus s for (a) 2.5%, (b) 5%, (c) 7.5%, and (d) 10% strains.

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DIRECT COMPARISON WITH EXPERIMENTS Images from new experiments that we have performed of wrinkles in 10, 20, and 30 nm PTFE skins under 10% biaxial strain are shown in Figure 7. The wrinkles in the simulations and experiments are similar to each other in that they consist of labyrinth patterns. However, the experimental wrinkles are less uniform, perhaps the result of irregularities in the surfaces that were present prior to wrinkling. The radial dependence of the Fourier transforms of the wrinkle patterns in Figure 7 is shown in Figure S3. Estimates of Lp were made based on the experimental images, using the method described above. The estimates are 30 ± 6, 220 ± 40, and 1000 ± 180 nm for the 10, 20, and 30 nm skins, respectively. The associated fits are shown in Figure 7. Note that data out to 40 nm was fit for the 10 nm result, as the associated Lp is small. As can be seen in the figure, the fits for the 10 and 20 nm skins are good, while the fit for the 30 nm skin is poor. Therefore, the Lp for the 30 nm skin is not well determined, perhaps because the average length of the wrinkles is close to the persistence length estimate, so this Lp result is of marginal significance. New experimental results for strain relief due to patterned lines with 10 nm skins are shown in Figure 8(a). Biaxial strain was 10%, and strain relief over Lr ≈ 600 nm was observed. This is twice the value estimated from our simulations. As discussed above, this difference could be due to the use of Eb = 4 MPa for PS in the simulations, which is something we explore in detail in the next section. Strain relief in analogous experiments with 20 and 30 nm skins produced stripes that are analogous to the 10 nm experiments, but that are ∼2000 and 4000 nm long, respectively. As in the case of the 10 nm skin, these values are significantly larger than those in the analogous simulations. In the simulations, Lr scales with h to a power less than one, while in the experiments, Lr scales with h to a power greater than one. The bending stiffness of a plate is proportional to its thickness cubed.12,30 An Lr = 600 nm for the 10 nm skin and h3 scaling suggests Lr = 4800 and 16200 nm for 20 and 30 nm skins,

result shown in Figure 6 can be used as an illustrative example. In this calculation for a 10 nm skin, 5% strain was applied in the

Figure 6. Application of 5% strain to a 2 μm × 1 μm × 500 nm sheet with a 10 nm skin. Strain was applied along the long axis.

long dimension, resulting in wrinkles of wavelength ∼70 nm (hole diameter to wavelength ratio of 100:70). The hole does not stretch to any appreciable extent in the direction parallel to the wavefront (≪ 1%), as one would expect would occur for a stretchy material. Without stretch, the holes can not “repel” adjacent material. Analogous calculations for 20 and 30 nm skins produce wrinkle wavelengths of 130 and 180 nm, respectively, and they correspond to hole diameter to wavelength ratios of 100:130 and 100:180. The results are otherwise analogous to those of the 10 nm skin calculation. The strongest coupling of holes to polymer wrinkling occurs when the ratio is close to one. If the hole is much smaller than the wrinkle wavelength, then it will not influence the wrinkling much, whereas if it is much larger, then the hole acts like a macroscopic boundary condition, and the wrinkles will conform to the hole. In between these limited cases, when the ratio is one, there is significant perturbation of the wrinkle pattern, but only close to the hole. G


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Figure 7. Experimental wrinkles in (a) 10, (b) 20, and (c) 30 nm skins. The surfaces are under 10% biaxial strain. Associated straight-line fits to ln < cos(θ(s)) > versus s are also shown. For the experiments, Lp = 30 ± 6, 220 ± 40, and 1000 ± 180 nm for 10, 20, and 30 nm skins, respectively. The corresponding numerical estimates are 48 ± 9, 158 ± 29 and 154 ± 28 nm for Eb = 4 MPa and 160 ± 30, 150 ± 30, and 300 ± 50 for Eb = 1 MPa.

values are easiest to estimate adjacent to patterned lines, where they are more uniform than in the images without strain relief. Dependence of Results on Choice of Polymer Modulus. We repeated the 10 nm skin strain relief calculations for the patterned lines, using the Young’s modulus for PS corresponding to the lowest value it is expected to eventually attain when heated above Tg (0.2 MPa).20 A Poisson’s ratio of 0.495 was used,20 reflecting the liquid-like behavior of PS in this limit. The 5% biaxial strain results are shown in Figure 8(b). Stripes the full width of the sheet, i.e., 1D wrinkles, were observed and λ = 130 nm. This suggests strain relief over ≥950 nm for this model. Results of simulations in which 7.5% biaxial strain was applied are the same, aside from small differences in wrinkle amplitudes (not shown). Thus, the experiments indicate the 0.2 MPa result is an overestimate of the strain relief. The simulations suggest the possibility of adjusting the Young’s modulus of the PS to reproduce the experimental Lr and thus predict λ and perhaps Lp. Equation 3 indicates that decreasing Eb from 4 MPa by ∼2 should double Lr. A simulation using this modulus and νb = 0.35 results in an increase in strain relief by ∼20 nm. The wavelength of the wrinkles is ∼80 nm as compared to ∼70 nm for Eb = 4 MPa. Thus, it is not straightforward to use eq 3 to fine-tune an estimate for the optimal Eb. A number of Eb values between 0.2 and 4 MPa were also tested, with Poisson’s ratio set to 0.495. This ratio was used because PS becomes liquid-like as its modulus value decreases over this range.20,21 The use of Eb = 1 MPa was found to result in an Lr for the 10 nm skin that is close to the analogous Lr observed experimentally (∼600 nm). The result is shown in Figure 9a, where strain relief over ∼500 nm can be seen. This modulus and ratio were also used in calculations with 20 and 30 nm skins, and they resulted in the strain reliefs shown in Figure 9b, c, suggesting Lr ≈ 2500 and 3300 nm, respectively. These Lr values also compare well to those observed in the respective experiments. As for the experimental results, it is impossible to assign strain relief distances without ambiguity.

Figure 8. (a) Experimental strain relief adjacent to a patterned line for a 10 nm skin. The 1D wrinkles are ∼600 nm in length. (b) Simulation results corresponding to the asymptotic values in time above PS Tg of the Young’s modulus and Poisson’s ratio. The 1D wrinkles are ∼950 nm in length, the full skin width for the simulation sheet.

values greater than those observed. For the experimental wrinkles, counting gives wrinkle wavelengths of ∼100, ∼ 200, and ∼300 nm, for 10, 20, and 30 nm skins, respectively. These H


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sizes of the sheets in the simulations. The wrinkle wavelengths are ∼100, 200, and 300 nm, respectively, which are approximately the same as those observed in the respective experiments. The radial dependence of the Fourier transforms is shown in Figure S4. The Lp values are 160 ± 30 nm, 150 ± 30 nm, and 300 ± 50 nm, and they are well-defined (R2 = 0.86, 0.92, and 0.85, respectively). These Lp are different than the respective experimental values as well as those determined base on the Eb = 4 MPa results. This is consistent with the energy of herringbone patterns being relatively insensitive to the distances between jogs.30 Labyrinths are simply disordered herringbones.15 The persistence lengths do not show a systematic trend with skin thickness in the simulations. Interestingly, for the experiments, using the cubic scaling of the stiffness of a plate with thickness, Lp = 30 nm for a 10 nm skin suggests Lp is ∼240 and ∼810 nm for 20 and 30 nm skins, which are close to the values calculated from the experimental images. This suggests the potential utility of Lp as a measure of the stiffness of a wrinkle pattern. Lp ≈ 30, 220, and 1000 nm for 10, 20, and 30 nm skins in the experiments. The wavelengths are 100, 200, and 300 nm, which means Lp scales with approximately λ3, so any change in material properties that impacts λ (see eq 1) should also affect Lp. The conclusions are also expected to be meaningful for different films with either smaller or larger Young’s modulus. Gold films on PS have also been studied, and gold has a larger modulus than PTFE (78 GPa versus 0.4 GPa). Equation 1 is useful in interpreting those results as well, provided proper considerations of the polymer melting that applies to the Au/ PS system are made.23 The full range of phenomenon examined in this work has not been explored for that system, but the behavior of Au/PS is expected to be consistent with what we have found. Consistency for other systems is also expected, provided the film is stiff relative to the substrate.

Figure 9. (a) Application of biaxial strain to a 2 μm × 1 μm × 500 nm sheet with a 10 nm skin, with an additional 2 μm × 150 nm of substrate attached. (b, c) Application of biaxial strain to a 2 μm × 6 μm × 500 nm sheet with a (b) 20 nm and (c) 30 nm skin, with an additional 2 μm by 1 μm of substrate attached. Eb was 1 MPa. The scale bars are approximate, because the sheets have been tilted for clarity.

The large Lr values make it computationally impractical to examine strain relief due to holes, so pristine sheets with 10, 20, and 30 nm skins were examined. The wrinkle patterns resulting from applying 5% biaxial strain to a sheet with a 10 nm skin and 10% biaxial strains to sheets with 20 and 30 nm skins are shown in Figure 10. The patterns are more ordered than those of the analogous experiments, perhaps a result of the relatively small

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CONCLUSION We have demonstrated that wrinkling on sheets of prestrained polystyrene covered with thin skins of polytetrafluoroethylene can be accurately modeled using the finite element method. Polystyrene above its glass transition temperature (Tg) is viscoelastic, which is a challenge for modeling because of the time dependence of its elastic properties. We demonstrated that by choosing a Young’s modulus that is approximately that of PS above Tg, accurate wrinkle patterns can be obtained by treating the material as elastic, without explicitly including the time dependence in the models. The wavelengths of the wrinkles increase with skin thickness, and this increase is in agreement with a well-known continuum model and with recent experiments. Wrinkle amplitudes increase with strain. Nanoinclusions, such as holes and patterned lines, provide strain relief over limited distances. They do not have associated boundary conditions that attract or repel material, but inclusions do act as if they attract material. The distances scale with the wrinkle wavelengths. The concept of persistence length can be applied to wrinkles. Skin thicknesses and strain conditions produce confinement, which is analogous to the kinetic confinement of polymers, and that effects the interactions between neighboring wrinkles.

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Figure 10. (a) Results of the application of 5% biaxial strain to a 2 μm by 1 μm by 500 nm sheet with a 10 nm skin. (b, c) Application of 10% biaxial strain to a 4 μm × 2 μm × 500 nm sheet with a (b) 20 nm and (c) 30 nm skin. Eb was 1 MPa.

ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acsami.6b14789. I


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(14) Huang, R.; Suo, Z. Wrinkling of a Compressed Elastic Film on a Viscous Layer. J. Appl. Phys. 2002, 91, 1135−1142. (15) Huang, Z. Y.; Hong, W.; Suo, Z. Evolution of Wrinkles in Hard Films on Soft Subtrates. Phys. Rev. E 2004, 70, 030601. (16) Huang, R. Kinetic Wrinkling of an Elastic Film on a Viscoelastic Substrate. J. Mech. Phys. Solids 2005, 53, 63−89. (17) Landau, L. D.; Lifshitz, E. M. Statistical Physics, Course of Theoretical Physics, 3rd ed.; Elsevier: Burlington, MA, 1980; Vol. 5. (18) Lee, M. H.; Huntington, M. D.; Zhou, W.; Yang, J. C.; Odom, T. W. Programmable Soft Lithography: Solvent-Assisted Nanoscale Embossing. Nano Lett. 2011, 11, 311−315. (19) Abaqus−Simulia; Dassault Systemes: Vélizy-Villacoublay, France; http://www.3ds.com/products-services/simulia/products/. (20) Chan, E. P.; Page, K. A.; Im, S. H.; Patton, D. L.; Huang, R.; Stafford, C. M. Viscoelastic Properties of Confined Polymer Films Measured via Thermal Wrinkling. Soft Matter 2009, 5, 4638−4641. (21) Greaves, G. N.; Greer, A. L.; Lakes, R. S.; Rouxel, T. Poisson’s Ratio and Modern Materials. Nat. Mater. 2011, 10, 823−837. (22) Kumar, A.; Gupta, R. K. Fundamentals of Polymer Engineering, Revised and Expanded; Marcel Dekker: New York, 2003. (23) Chapman, C. T.; Paci, J. T.; Lee, W. K.; Engel, C. J.; Odom, T. W.; Schatz, G. C. Interfacial Effects on Nanoscale Wrinkling in GoldCovered Polystyrene. ACS Appl. Mater. Interfaces 2016, 8, 24339− 24344. (24) Frontali, C.; Dore, E.; Ferrauto, A.; Gratton, E.; Bettini, A.; Pozzan, M. R.; Valdevit, E. An Absolute Method for the Determination of the Persistence Length of Native DNA from Electron Micrographs. Biopolymers 1979, 18, 1353−1373. (25) Mücke, N.; Klenin, K.; Kirmse, R.; Bussiek, M.; Herrmann, H.; Hafner, M.; Langowski, J. Filamentous Biopolymers on Surfaces: Atomic Force Microscopy Images Compared with Brownian Dynamics Simulation of Filament Deposition. PLoS One 2009, 4, e7756. (26) ImageJ; National Institutes of Health, Bethesda, MD; freely downloadable from http:://imagej.nih.gov/ij/. (27) Mücke, N.; Kreplak, L.; Kirmse, R.; Wedig, T.; Herrmann, H.; Aebi, U.; Langowski, J. Assessing the Flexibility of Intermediate Filaments by Atomic Force Microscopy. J. Mol. Biol. 2004, 335, 1241− 1250. (28) Efimenko, K.; Rackaitis, M.; Manias, E.; Vaziri, A.; Mahadevan, L.; Genzer, J. Nested Self-Similar Wrinkling Patterns in Skins. Nat. Mater. 2005, 4, 293−297. (29) Yang, S.; Khare, K.; Lin, P. C. Harnessing Surface Wrinkle Patterns in Soft Matter. Adv. Funct. Mater. 2010, 20, 2550−2564. (30) Chen, X.; Hutchinson, J. W. Herringbone Buckling Patterns of Compressed Thin Films on Compliant Substrates. J. Appl. Mech. 2004, 71, 597−603. (31) Rivetti, C.; Guthold, M.; Bustamante, C. Scanning Force Microscopy of DNA Deposited onto Mica: Equilibration Versus Kinetic Trapping Studied by Statistical Polymer Chain Analysis. J. Mol. Biol. 1996, 264, 919−932. (32) Ohzono, T.; Shimomura, M. Ordering of Microwrinkle Patterns by Compressive Strain. Phys. Rev. B: Condens. Matter Mater. Phys. 2004, 69, 132202.

Fourier transforms of wrinkles (PDF)

AUTHOR INFORMATION

Corresponding Authors

*E-mail: jeff[email protected]. *E-mail: [email protected]. ORCID

Jeffrey T. Paci: 0000-0003-4341-752X Craig T. Chapman: 0000-0001-5349-5359 Teri W. Odom: 0000-0002-8490-292X Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This research was supported by the Office of Naval Research (N00014-13-1-0172). We thank Mark D. Huntington for the use of an SEM image. JTP thanks Matthew G. Moffat for insightful discussions near the beginning of this work. W.-K. Lee gratefully acknowledges support from the Ryan Fellowship and the Northwestern University International Institute for Nanotechnology.

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REFERENCES

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