Pattern Driven Stress Localization in Thin Diblock Copolymer Films

Article pubs.acs.org/Macromolecules

Pattern Driven Stress Localization in Thin Diblock Copolymer Films Andrew B. Croll*,† and Alfred J. Crosby*,‡ †

Department of Physics, North Dakota State University, Fargo, North Dakota 58102, United States Department of Polymer Science & Engineering, University of Massachusetts, Amherst, Massachusetts 01003, United States

‡

ABSTRACT: When an elastic plate is fixed to a soft substrate and compressed, it accommodates applied strain by buckling and forming a sinusoidally wrinkled topography. At large strains, the regular wrinkled pattern is replaced by sharp, localized folds. Such folds are ubiquitous in biology; however, strains are not necessarily large in all cases, suggesting that a different mechanism may contribute to the formation of folds. In this work, we use thin films of a symmetric diblock copolymer coupled to thick elastomer substrates to explore the progression from isotropic wrinkles to localized folds in films with secondary structure. The block copolymer molecules organize into lamellae parallel to the elastomer substrate, and the balance of film thickness, lamellar dimensions, and elasticity dictate the development of topographic structures in a systematic manner. This “self-assembled” topography in the bounding film leads to stress localization when the pattern has a lateral spacing of the same order as the wrinkling wavelength. This first systematic exploration of pattern driven localization reveals the importance of a new emergent length scale which also appears in more traditional localization experiments.

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many different cell types, hair follicles, collagen fibers each with different average spacing).16 Taking our inspiration from the complexity found in nature, we identify a new physical trigger for the localization of stress in a thin rigid filmthe interference between two co-occurring patterns. Notably, despite the new mechanism, the essential physics is still captured by the same geometric theory used elsewhere. We examine an elastomer capped with a polymer plate having a self-assembled, discrete, periodic variation in thickness. Such a patterned elastic plate localizes, even before wrinkling, and a new interfold length scale emerges. This “localization length” is proportional to the wavelength of an unpatterned plate subject to similar compression. Consider a simple model system with a rigid film fixed to a soft substrate, as illustrated in Figure 1. When stress is applied parallel to the surface of the composite, it deforms isotropically until a critical stress is exceeded, after which the system buckles and deforms out-of-plane.7,17 As is common in most simple pattern formation, two effects with different intrinsic length scales compete to select the pattern size, λ0. In this particular case, the bending of a rigid plate favors long length scales (Fb ∼ 1/l, where Fb is a free energy per unit length and l is the size of the plate), and the deformation of the soft substrate opposes large amplitude deformation (Fs ∼ A, where A is the deformation amplitude). The pattern is in equilibrium and not dynamically selected and the analysis is simple: the end state is the minimum of the total free energy. The composite’s free energy is given by the sum of the bending energy of the

INTRODUCTION Self-assembling patterns which develop from isotropic initial states are an undeniable cornerstone of modern soft-matter research. This avenue of study is driven by the desire to understand the complex physics of the varied structures found in nature and the technological interest in functional materials that may be derived through biomimicry.1−6 A well-known example of bio-inspired pattern formation arises when skin is squeezed together. As the skin is essentially an elastic plate fixed to a softer substrate, the applied stress is accommodated by the buckling, or wrinkling, of the plate. This phenomena has recently garnered significant scientific attention, particularly in the thin polymer film community.5,7−9 Sinusoidal wrinkles, although satisfying in their simplicity, are only one possible morphology that can relieve the applied stress, and it is now known that higher stress causes the sinusoidal pattern to evolveslowly localizing the stress in sharp crevasses into the foundation.10−13 Understanding the localization of stress is critical for the growing flexible electronic industry where focused bending leads to circuit failure,14 for insight into biological morphogenetic processes where stress often influences growth,15 and for understanding the mechanical stability of thin polymer films. More importantly, from a fundamental point of view the localization leads to the emergence of a fundamentally different pattern equally worthy of study. The fold pattern has its own characteristic length scale which may be more realistic in compressed biological systems than a wrinkle wavelength, λ, due to the unlikelihood of a mobile biological sample maintaining a constant stress state. An important feature of real systems that has, until now, been ignored is its inherent variation (for example, the skin morphology accommodates © XXXX American Chemical Society

Received: January 26, 2012 Revised: April 2, 2012

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Figure 1. Illustration of the relevant geometry and experimental observations. (a) Schematic of an unpatterned film under compressive strain (left), an atomic force micrograph (right), and a representative cross-sectional trace (center). The schematic shows uniaxial compression for clear definition of wrinkle amplitude (A) and wrinkle wavelength (λ0) whereas the experiment shows a PS-b-P2VP film under equibiaxial compression (applied strain is 5.9 × 107, plate thickness is 93 nm, and the image is 40 × 40 μm in dimension). (b) Amplitude and interfold distance, llocal, in a localized sample. AFM shows the same film as in (a) after ordering, and again a representative cross-sectional trace is shown (center).

plate and the deformation energy of the substrate, F = Fb + Fs.5,8 Following the notation of Cerda, which is particularly amenable to problems of localized bending, a plate of dimension l has a bending energy which scales as5,8 Fb ∼ BlA2/2λ04, where B = E̅ft3/12, E̅ f = Ef/(1 − ν2), Ef is the elastic modulus, ν is the Poisson ratio, and t is the capping film thickness. The stretching energy is given by Fs ∼ KlA2/2, where the stiffness, K, scales as E̅s/λ0, with Es the substrate elastic modulus.5 Assuming the plate is inextensible, the amplitude is

A ∼ (λ 0 / π ) Δ / l

rigid plate with a water or elastomer foundation), localization only occurs at extremely high applied strains (ΔL/l ∼ 0.2−0.3). For example, the dramatic period-doubling and eventual localization observed in soft elastic films on softer substrates is well described by considering a nonlinear foundation in the usual wrinkling theory. In short, localization is caused by a nonlinear elastic foundation. With an elastic plate coupled to an inhomogeneous foam foundation, localization occurs at lower applied strain (ΔL/l ∼ 0.04) due to the local collapse of cells in the foam.12 Despite the obvious heterogeneity of a foam, the resulting structure of the plate is still that of sharp folds, similar to the localization seen in experiments on homogeneous foundations. The appearance of similar folds in such varied systems suggests that unifying similarities exist; the physics that dominates the final state is due to the sharp bending of the thin plate, not the particular cause of that bending. In their work on floating plates, Pocivavsek et al. present a purely geometric nonlinearity in their description of the plate, which leads to a similar wrinkle to fold transition.10,13 In this case no material nonlinearity is required; the geometry of the bend simply cannot be described without nonlinear terms. Such a geometric cause for localization is quite general; however, because of the extreme strain applied to the system, it is hard to definitively rule out material nonlinearity (i.e., modulus, plastic deformation, cracking, etc.). Hence, evidence of a fundamental transition to folding in thin films is still lacking. Not only is the pattern driven localization of stress described in this paper of interest for its novelty, but it represents a better experimental system for describing the general transition from a wrinkled to a localized state. This is for two reasons: (1) The

(1)

and Δ = σl/E̅f is the displacement of the film boundaries and σ the applied stress. Minimization yields the wavelength λ 0 ∼ (B /K )1/4

(2)

which is notably free of dependence on the applied stress and has been well characterized experimentally.7,9,18 This scaling reduces to the more common form of λ 0 ∼ t(Ef̅ /Es̅ )2/3

(3)

upon substitution of B and K as defined above. It is noteworthy that the bending energy is strongly linked to the plate thickness; hence, the modulation in thickness explored in this work can loosely be thought of as a modulation of the plate’s bending energy. As compression increases beyond a certain point, ΔL/l, the smooth oscillations of surface wrinkles as described by λ and A, transition to sharp folds.10−13,19,20 Studies of the transition have ranged from elastic or surfactant sheets floating on a water foundation10,11,19 to elastic sheets on foams12 or elastomeric foundations.13 When the system is homogeneous (e.g., uniform B

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applied strain can remain vanishingly low; hence, any material nonlinearity can be ruled out. (2) Because of the application of thermal strain, the plate is sheltered from unwanted mechanical contributions (unaccounted for shear, clamps, boundaries, etc.) which would otherwise dominate the morphology.20 In the following article we begin with a description of our experiment and then discuss our main result: a thickness pattern can trigger a polymer film to localize stress into sharp folds, but only when the spacing of the thickness variation is commensurate with that of the linear wrinkling instability. Second, a new length scale dominates the resulting morphology, and this new length scale mimics the spacing of the wrinkling instability. Finally, we show how pattern driven localization can be adequately compared with a general, geometric scaling theory.

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Figure 2. A schematic showing an “as cast” block copolymer film (a) and the same film in the ordered state after a long anneal (b). Both films will buckle due to the thermal stress generated during cooling.

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EXPERIMENTAL SECTION

RESULTS AND DISCUSSION With the system described above, we can systematically explore the influence of uniform thickness variations on compression imposed deformations by simply increasing the annealing time. For films that are initially smooth, i.e., where t ≪ τ or h is an integer of the lamellar dimension, smooth wrinkles are observed upon thermally induced compression (Figures 1a and 3a). This deformation mechanism has been well described in the literature for films that are composed of a large variety of materials.7,9−13,19 For materials with thickness variation (t > τ), the response is strikingly different. As the variable thickness sample is cooled, strain develops but deformations appear only in discrete sitesthe stress has localized. As seen in Figure 1, a film that has localized will display a series of sharp folds in contrast to smooth sinusoidal wrinkles. The morphological details of the block copolymer surface play a role in the strain localization process as can be seen in Figure 3. Samples of varying capping layer thickness were prepared and annealed so as to ensure complete phase separation. When a film is commensurate with an integer number of layers, it remains topologically smooth during long anneals and wrinkles isotropically. When the thickness is increased and islands form, weak localization is observed. The localization is seen to grow in strength (more of the sample is flat) as material is added to the film. Ultimately, the thickness can be increased a full lamella, resulting again in a smooth film and isotropic wrinkles. Not only does this observation mean that the strength of the localization is controlled by the block copolymer morphology, but it verifies that localization is not simply related to average plate thickness. For precision, we denote any feature with a curvature, κ ≫ Ah/λ02 as a fold; i.e., if the curvature is greater than is expected for a sinusoid, a feature is considered a fold. Beyond the morphological differences of folds and wrinkles displayed in Figures 1 and 3, the amplitude of these features is observed to scale differently as a function of applied stress. Figure 4 shows the amplitude as a function of compression for a typical wrinkle and a typical fold. For wrinkles, we observe that A scales with the square root of applied stress beyond the critical stress, similar to multiple previous reports on wrinkle amplitude in the literature and consistent with the predicted relationship in eq 1. For folds, we observe that A scales linearly beyond the critical stress, also similar to other localizing systems described in the literature.10 In contrast to these previous reports, we do not observe a single fold, but rather a series of folds, whose spacing defines a new length scale, llocal. Assuming that the thin film can still be considered inextensible,

We examine the folding transition through the effect of periodic plate thickness variations. In essence, we add a controlled variation in the strength of the bending energy, leading to a new type of nonlinearity in the system. We use a common elastomer (Dow-Corning, Sylgard 184) and a capping film of diblock copolymer (104 kg/mol polystyrene-b-poly(2-vinylpyridine), Polymer Source). This material system offers several key advantages for the controlled study of a thickness variation on the morphology of the compressed film/ elastomer composite. First, this polymer is comprised of blocks which have equal glass transition temperatures (∼373 K) and nearly equal elastic moduli. The two blocks also have a high interaction parameter, making the two blocks strongly immiscible. We create thin films by spin-coating diblock copolymer solutions on mica substrates. The spin-coating process occurs at a high speed, which traps the diblock copolymers in an isotropic, glassy state (i.e., the film is homogeneous and of uniform thickness). When subsequently annealed above its glass transition temperature, the molecules arrange to minimize contact between different block segments. For the symmetric molecules studied here, lamellae are formed.2 The key feature of this microphase separation for our study is how the material arranges under the constraint of the film thickness. The ratio of initial film thickness to lamellar thickness (here tLamella ∼ 53 ± 2 nm) defines the surface structure in a well-understood manner (Figure 1).21 When f = t/tLamella is equal to an integer, n, the film remains flat; however, for a slightly thinner film n − 0.5 < f < n there is not enough volume to maintain both the lamellar structure and a uniform film thickness. The result is a film surface decorated by lamella deep “holes”. At a thickness of f = n − 0.5 a complex interconnected pattern of islands and holes appears, and when n − 1 < f < n − 0.5, the surface is decorated by “islands” (Figure 1). The surface pattern can be characterized by an average “island”, “bicontinuous”, or “hole” width, λp, which can be coarsened as annealing continues.22 It is the patterned film thickness, where each hole or island is an identical discrete step in thickness, that allows us to locally manipulate the bending energy in a highly controlled manner. To apply equibiaxial stress, we exploit the mismatch in thermal expansion (δα) between the block copolymer film and elastomer substrate; here δα = 2.2 × 10−4 K−1.23,24 This creates a local stress in the capping sheet σT = δαδTlE̅ , where δT is the change in temperature. Typically, a film is placed on a substrate, and the composite is annealed above the glass transition temperature of the block copolymer; hence, stress due to heating or processing is relieved. Next the sample is slowly cooled to room temperature (see Figure 2). If the film is only held above its glass transition temperature for a short time (t ≪ τ, where τ is the time to microphase separate and is a complex function of chain length, temperature, viscosity, and interaction strength25), no phase separation occurs, and the film remains isotropic. When cooled, these samples form sinusoidal wrinkles that have no preferred direction and can easily be examined with optical profilometry (Zygo NewView 7300) or atomic force microscopy (Veeco Dimension 3100). Typical results can be seen in Figure 1a. C

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Figure 3. Experiments conducted with varying thickness, microphase-separated diblock copolymer films. (a) Optical micrographs of films after a short anneal and cooling cycle. Here all films show isotropic wrinkles. During a long anneal (b), the films microphase separate into islands (left image), bicontinuous (middle image), or holes (right image). When these films cool, they all localize but to varying degrees (c, AFM images). Optical images are 50 × 50 μm; AFM are 50 × 50 μm. (d) Topography of well-annealed samples (offset for clarity). The top trace is of a single lamella film. Moving down, we show traces for consecutively thicker films (island, bicontinuous, and hole morphologies), ending with a commensurate two lamellae thick film.

interfold length scale (llocal) is a physically different length scale than occurs in wrinkling; it is larger than, but is of the same order of magnitude as, the wrinkle wavelength. llocal can be physically interpreted as the distance over which the local stress in the elastic foundation becomes negligible. Light scattering or other Fourier space experiments (commonly used with surfactant or nanoparticle films) cannot distinguish this difference. If metrology is the goal, the scaling relations developed for sinusoidal wrinkles (eqs 1 and 2) cannot be applied; it is llocal which contributes most strongly to the scattering. In addition to understanding the amplitude growth of folds, it is critical to understand when and why folds are observed rather than smooth wrinkles. On the basis of the observations from above, it is clear that discrete thickness variations, such as islands or holes, will give rise to fold formation rather than smooth wrinkles, but it is not clear how the lateral dimensions of the thickness variations contributes to determining which deformation mechanism will occur. To explore the transition between the wrinkled and folded states, we create wellcoarsened samples in which the pattern spacing, λp, is very large compared to λ0 (Figure 5, inset c). In this case two isolated sets of sinusoidal buckles occurthe thick and thin regions buckle independently. The wavelengths of each region correspond to predictions based on film thickness, i.e., eq 2. As λp is reduced, localization sets in (Figure 5, inset b). If λp is reduced still further, there is a transition back to sinusoidal wrinkles (Figure 5, inset a). Qualitatively, the films localize if the average diblock pattern spacing, λp, becomes commensurate with the wavelength of a wrinkle in a smooth film of identical thickness (i.e., the “as cast” sample), λ0. The morphological state can be systematically explored by changing the substrate modulus (i.e., λ0) while holding thickness and annealing time constant (constant λp) or by annealing films on mica substrates and coarsening the diblock surface pattern before transferring to the elastomer substrate. To describe this tendency for films to fold or localize rather than wrinkle when λp ∼ λ0, we must consider the effect of the boundaries defining the thickness variations. Any buckling plate will be influenced by its boundaries, and such effects have been well documented for elastic plates on soft substrates.7,8,26 In

Figure 4. Amplitude of a typical wrinkle or fold during quasi-static cooling experiments (open circles, isotropic sample; black circles, phase-separated sample). Below a stress of ∼1.8 × 107 Pa the amplitude does not measurably change. The patterned films shows a nonzero amplitude at zero stress reflecting the variable plate thickness. Both films buckle above ∼1.8 × 107 Pa. Solid curves are the fits described in the text (blue, A ∼ σT; red, A ∼ σT1/2). (a) Optical profilometry of an isotropic sample of thickness t ∼ 80 nm at temperatures of 100, 70, and 30 °C. (b) A similar sample annealed above 100 °C for 30 min at similar temperatures during cooling. The stress in each is 0, 2.5 × 107, and 5.3 × 107 Pa, respectively. All images are 28 × 28 μm.

which is consistent with the proven assumptions of eq 1 for the wrinkled deformations, then we can describe A as a function of applied stress similar to previous reports on systems involving a single fold. In these previous studies, it was shown that the amplitude of a fold will scale linearly with the applied displacement for an inextensible film. In our system, the macroscopically applied displacement is divided evenly into the number of folds, n, distributed across the surface length. In other words, the local displacement applied to each fold is Δ/n. Therefore A ∼ Δ/n ∼ εl /n ∼ σTllocal /Ef

(4)

where llocal = l/n. The data in Figure 4 confirm the marked difference between a localized fold, where A scales linearly with stress A ∝ σT, and a wrinkle where A scales as a square root of stress A ∝ σT0.5. Further, the fit to eq 4 to the data yields llocal = 6.8 μm, which again agrees with direct measurement of the distance between folds. Radially averaged FFT of the image gives lloc = 7.8 μm. It is important to note that the emergent D

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Figure 6. (a) Numerical comparison of the two relevant free energies (solid curve, fold; dashed curve, wrinkle). Although the nature of the two intersecting trends is not very sensitive to the parameters that are chosen, the specific parameters used in this representation are: Δ/λ0 = 1.4. (b) llocal as a function of λ0 measured in several localized films (from FFT analysis). The solid line is a linear fit, with a slope of 2.0.

Figure 5. A state diagram denoting the range of composites affected by pattern driven localization. The upper left region (high λ0, low λp) wrinkles as though no pattern were present (inset a). Lower right (low λ0, high λp) films form regions of large and small wrinkles (inset c). Only in the region around λp = λp (the solid line) is there strong localization (inset b). As λp → ∞, one finds isotropic wrinkles (inset d). Inset images are 50 × 50 μm. Localized samples are shown as squares, and wrinkled samples are shown as circles. Each region of the state diagram is labeled accordingly.

which we find to be well fit by llocal ∼ cλ0, with c a fit constant equal to 2.0 (Figure 6b). The interfold length scale is proportional to the ordinary wrinkle wavelength but is twice as large.

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these previous studies, it is commonly observed that lambda is stretched or compressed to accommodate λp or the distance between edges. This change in wavelength is not observed in our measurements, rather a complete change in morphology. In our system, the collection of boundaries acts in concert and creates a new larger scale effect. Hence, we propose the following ansatz: the bending plate forms nodes at step edges (boundaries). With this hypothesis we can describe the morphology using the same elastic theory derived elsewhere.10,20 Localization can be understood by reconsidering the same simple energy balance as was used to describe the sinusoidal samples; however, because of the steepness of the localized structure, greater than linear geometric contributions are now required.10,20 In this case, the bending energy is negligible everywhere except over the width of the fold (∼W) and is therefore of order Fb ∼ B /W

CONCLUSIONS In summary, we observe that a lamellar forming diblock copolymer film bound to a soft PDMS substrate under compression can be driven from a simple wrinkled topography to a complex state of stress localization when the spacing between surface features is commensurate with that of a natural buckling wavelength. Remarkably, we find this localization at vanishingly small applied stress, the limit in which current elastic theory is valid. The stress localization depends on pattern morphology, localizing most strongly when the pattern consists of small isolated thin regions. Furthermore, localization depends strongly on the imposed pattern spacing which we characterize with a state map, identifying two morphological regions: wrinkles and sharp localizations or folds. Despite the complexity, these observations can be understood with energybased scaling arguments. The experiments identify the importance of an interfold length scale, related to but greater than the natural length scale of the wrinkling. The new length scale has important implications for the use of thin film buckling metrology in systems with multiple internal length scales (particularly if measured with scattering methods). This new type of localization is expected to play a large role in the complex biological theater, where it is common for several different patterns to coexist.

(5)

The deformation of the substrate is also confined to this small region and can be written as Fs ∼ KWσT 2llocal 2/Ef̅ 2 − KσT 3llocal 3/Ef̅ 3

(6) 10

and again the total energy is Ffold = Fb + Fs. Ordinarily, L will minimize the total fold energy, but given our ansatz above L is not free to vary but is pinned at the edge of a depression in the capping film, i.e., W = λp.26 Naturally, localization will occur whenever Ffold is less than Fwrinkle. Assuming that σT is small, the folding energy is found to be lower for a small range of pattern spacing (λp) centered on the minima of Fwrinkle 2

λp ∼ Ef̅ λ 0 /σTllocal

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected] (A.B.C.); [email protected]. umass.edu (A.J.C.).

(7)

Notes

The authors declare no competing financial interest.

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We show the numerical comparison of Ffold and Fwrinkle in Figure 6a, which shows the small range over which Ffold < Fwrinkle. Fitting a general power law (i.e., λ0 = kλpα) to the data in the state diagram reveals that α ∼ 1 or λ0 more likely scales as λp and not as λp0.5. Given eq 7, this implies that llocal must also scale as λ0. This conclusion can be verified by direct comparison with measurements when llocal is plotted as a function of λ0,

ACKNOWLEDGMENTS Funding for this work was provided by EPSCoR (EPS0814442), NSF MRSEC (NSF DMR-0820506), and NSF DMR-0907219. Some of the work for this manuscript was completed at the Aspen Center for Physics. E

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