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Geometry-Dependent Stripe Rearrangement Processes Induced by Strain on Preordered Microwrinkle Patterns Takuya Ohzono*,† and Masatsugu Shimomura†,‡ Dissipative-Hierarchy Structures Laboratory, Spatio-Temporal Function Materials Research Group, Frontier Research System, RIKEN, 2-1 Hirosawa, Wako, Saitama 351-0198, Japan, and Research Institute for Electronic Science, Hokkaido University, N21W10, Sapporo 001-0021, Japan Received February 7, 2005. In Final Form: June 9, 2005 A preordered microwrinkle pattern on a metal-capped surface of a soft elastomer is employed to elucidate the elementary buckling phenomenon during strain-induced stripe rearrangement processes. The preordered one-dimensional stripe tends to align perpendicular to the direction of strain reversibly when lateral compressive strain is applied on the substrate at some angle φ with respect to the stripe orientation. For any value of strain, the film surface can be decomposed in domains containing stripes with two different orientations, namely the original and applied strain orientations. As strain is increased, the domains of the second type of stripes progressively grow and invade the whole surface. Interestingly, the domain shapes during growth are composed of parallelogram units that simply depend on φ and stripe wavelength. Moreover, domain growth proceeds in characteristic directions depending on the shape of the domain unit.
1. Introduction Many present and future technological challenges involve the micro- or nanopatterning of a surface and the dynamic control of the pattern with the aid of spontaneous processes, for example, to achieve desired optical effects, such as diffraction or reflection,1 to provide a wettabilitycontrolled surface for direction-changeable flow channels in a microfluidic system,2 or to prepare a geometrycontrolled substrate for biological cell patterning.3,4 One class of methods for achieving such patterning relies on a physical instability with an intrinsic length scale. This method enables us to produce ordered patterns whose characteristic length scale is controlled by simply making changes in the experimental conditions, removing the need for the use of direct surface patterning techniques.5-15 A length scale arises from a competition between factors favoring large and small length scales. An example of * To whom correspondence should be addressed. E-mail: ohzono@ riken.jp. † RIKEN. ‡ Hokkaido University. (1) Ibn-Elhaj, M.; Schadt, M. Nature 2001, 410, 796. (2) Campbell, C. J.; Klajn, R.; Fialkowski, M.; Grzybowski, B. A. Langmuir 2005, 21, 418. (3) Jiang, X.; Takayama, S.; Qian, X.; Ostuni, E.; Wu, H.; Bowden, N.; LeDuc, P.; Ingber, D. E.; Whitesides, G. M. Langmuir 2002, 18, 3273. (4) Nishikawa, T.; Ookura, R.; Nishida, J.; Arai, K.; Hayashi, J.; Kurono, N.; Sawadaishi, T.; Hara, M.; Shimomura, M. Langmuir 2002, 18, 5734. (5) Cross, M. C.; Hohenberg, P. C. Rev. Mod. Phys. 1993, 65, 851. (6) Seul, M.; Andelman, D. Science 1995, 267, 476. (7) Bowman, C.; Newell, A. C. Rev. Mod. Phys. 1998, 70, 289. (8) Ball, P. The Self-Made Tapestry; Oxford University Press: New York, 2001. (9) Rabinovich, M. I.; Ezersky, A. B.; Weidman, P. D. The Dynamics of Patterns; World Scientific: Singapore, 2000. (10) Mo¨hwald, H. Thin Solid Films 1988, 159, 1. (11) Seul, M.; Wolfe, R. Phys. Rev. Lett. 1992, 68, 2460. (12) Harrison, C.; Adamson, D. H.; Cheng, Z.; Sebastian, J. M.; Sethuraman, S.; Husse, D. A.; Register, R. A.; Chaikin, P. M. Science 2000, 290, 1558. (13) Higgins, A. M.; Jones, R. A. L. Nature 2000, 204, 476. (14) Shimomura, M.; Sawadaishi, T. Curr. Opin. Colloid Interface Sci. 2001, 6,11. (15) Sharp, J. S.; Jones, R. A. L. Adv. Mater. 2002, 14, 799.
such systems is buckling produced by residual mechanical stress on surface-modified elastic or viscoelastic soft substrates.16-20 The surface skin layer becomes harder than the substrate by the modification. Thus, a multilayer structure is formed. The typical morphology consists of surface wrinkles or corrugations. In this case, supposing the hard surface area is almost constant, the surface bending potential energy prefers a small curvature, i.e., a large wavelength of wrinkles with a large amplitude. This is the case of simple Euler’s buckling of a plate without support. Meanwhile, the deformation potential energy in the soft substrate prefers a small displacement, i.e., a short wavelength with a small amplitude. The resultant wavelength arises from the compromise of these effects and is related to the mechanical properties and configurations, such as modulus, Poisson’s ratio, and the effective thickness of the hard surface. Although most studies have focused on the wrinkle formation process21-26 and resultant stationary states,16-19,27-29 little is known about the pattern dynamics under external mechanical stimulation. Regarding this point, we have shown that the complex labyrinthine pattern, which is spontaneously produced by the isotropic (16) Bowden, N.; Brittain, S.; Evans, A. G.; Hutchinson, J. W.; Whitesides, G. M. Nature 1998, 393, 146. (17) Bowden, N.; Huck, W. T. S.; Paul, K. E.; Whitesides, G. M. Appl. Phys. Lett. 1999, 75, 2557. (18) Chua, D. B. H.; Ng, H. T.; Li, S. F. Y. Appl. Phys. Lett. 2000, 76, 721. (19) Huck, W. T. S.; Bowden, N.; Onck, P.; Pardoen, T.; Hutchinson, J. W.; Whitesides, G. M. Langmuir 2000, 16, 3497. (20) Ohzono, T.; Shimomura, M. Phys. Rev. B. 2004, 69, 132202. (21) Moldovan, D.; Golubovic, L. Phys. Rev. Lett. 1999, 82, 2884. (22) Sridhar, N.; Srolovitz, D. J.; Suo, Z. Appl. Phys. Lett. 2001, 78, 2482. (23) Huang, R.; Suo, Z. J. Appl. Phys. 2002, 91, 1135. (24) Yoo, P. J.; Lee, H. H. Phys. Rev. Lett. 2003, 91, 154502. (25) Yoo, P. J.; Suh, K. Y.; Kang, H.; Lee, H. H. Phys. Rev. Lett. 2004, 93, 34301. (26) Huang, Z.; Hong, W.; Suo, Z. Phys. Rev. E 2004, 70, 30601(R). (27) Stafford, C. M.; Harrison, C.; Beers, K. L.; Karim, A.; Amis, E. J.; Vanlandingham, M. R.; Kim, H.; Volksen, W.; Miller, R. D.; Simonyi, E. E. Nat. Mater. 2004, 3, 545. (28) Harrison, C.; Stafford, C. M.; Zhang, W.; Karim, A. Appl. Phys. Lett. 2004, 85, 4016. (29) Uchida, N. Physica D 2005, in press.
10.1021/la0503449 CCC: $30.25 © 2005 American Chemical Society Published on Web 07/07/2005
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residual stress, is gradually ordered to form a simple stripe by applying compressive strain, and the original pattern gradually recovers upon unloading, which indicates the existence of the shape memory of the initial pattern.20 The stripe ordering (rearrangement) process consists of a series of local buckling, i.e., changing the wrinkle wave direction to that of strain locally. This novel property is a promising candidate for dynamic control of micro- or nanostructures geared toward various technological applications by a very simple uniaxial compression process. From the analysis in the previous work,20 however, it is only known that the domains with rearranged stripes by strain grow almost isotropically and subsequently merge with each other upon loading. That is, it remains unknown on what locations around the domain (or in what directions) elementary local buckling takes place during domain growth. The difficulty of such analysis is due to the complexity in the initial pattern, which is a randomly packed mosaic of stripe domains with various orientations. Therefore, the misfit angle, which is defined as the angle between the original stripe orientations of the domains and the direction of compression, varies depending on the domains. As a result, it is assumed that the misfit-angledependent characteristic features of elementary local buckling are intricately mixed and difficult to decompose. Thus, a simpler stripe pattern for the initial state is required to elucidate the elementary local buckling process during the domain growth. Moreover, the use of a simpler pattern might allow us to decompose the previously reported hysteretic behavior during a loading-unloading cycle20 into the misfit-angle-dependent factors. In this study, the preordered microwrinkle pattern, which shows a simple stripe configuration, is employed to elucidate the feature of elementary local buckling during the stripe rearrangement process induced by applying and unloading lateral compressive strain. The misfit angle φ between the direction of the preordered stripe and that of the strain is parametrized to extract the angledependent features. First, we show that the stripe is gradually rearranged in the strain direction on loading with a slight increase in stripe wavelength and recovers to its original direction upon unloading for any value of the misfit angle, as expected from the previous study.20 Then, the φ-dependent hysteretic behavior is described. Finally, we describe the dynamics of domain growth and shrinkage due to strain changes, namely the shapes of domains and their changes. We discuss them in terms of the characteristic geometry of the domain shapes and the configuration of the topological defects, both of which are related to the misfit angle and stripe wavelength. 2. Experimental Section 2.1. Preparation of a Preordered Wrinkle Pattern. A 5-mm-thick elastomeric poly(dimethylsiloxane) (PDMS) of a right circular cylinder with a diameter of 8 mm are prepared (Dow Corning Sylgard Elastomer 184) (Figure 1a). To obtain the preordered microwrinkle pattern, the surface is modified through Pt deposition by applying a uniaxial compressive strain (7%) (Figure 1b). The deposition of Pt with a thickness of approximately 8 nm is conducted over a flat surface with a circular shape. An ion sputter (E-1030, Hitachi) is used for the deposition with a current of 5 mA, a pressure of 10 Pa, a distance of 30 mm between samples and the Pt target, and a deposition time of 180 s. During the deposition under uniaxial compression, the PDMS surface is assumed to be modified to have a silica-like rigid nature16-19 due to the collision energy of active species. Uniaxial compression anisotropically deforms the PDMS surface, where expansion in the perpendicular direction also spontaneously takes place. The compressive strain is unloaded soon after deposition. As surface temperature decreases to room temperature, the ordered micro-
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Figure 1. (a) Sample features, (b) experimental setup for preparation of preordered stripe pattern of wrinkles, and (c) experimental setup for mechanical perturbation (lateral compressive loading and unloading). The small vise is used for compression. wrinkle pattern, whose stripe direction is perpendicular to the direction of compression, appears over the surface except in the area close to the edge in 1-2 min. The reason is that the residual stress becomes anisotropic, where the main stress axis is identical to the spontaneously expanded direction during compression. The stripe wavelength of the current sample is approximately 0.9 µm. A similar preordered pattern, which is formed on a curved surface without applying strain during surface modification, has also been reported3 and bears the same principle, where anisotropy in residual stress plays a role. Note that the preordered wrinkle pattern is remembered to some extent at room-temperature, according to the previous study.20,30 Thus, an irreversible process, such as material flow or plastic deformation, possibly occurs during pattern formation. This suggests that the pattern is stable or fixed, i.e., the waves of wrinkles do not travel, under ambient conditions without any mechanical perturbation. Although the stripe pattern might be modulated by applying compressive strain, the original ordered stripe is expected to recover after unloading, which is described later. 2.2. Loading and Unloading. Our experimental setup for the mechanical perturbation is similar to that used previously (Figure 1b).20 Uniaxial compressive strain is exerted and unloaded stepwise to the PDMS cylinder with the preordered pattern in a direction with the misfit angle φ, using a small vise under the optical microscope at 293 K. The rate of changing strain is (1%/ min. The step size is (1%, thus it takes one minute for changing strain for a step. The system is equilibrated for 5 min after every strain change. The images are acquired after equilibrations. The observations are conducted in the same area near the center of the sample surface during a loading-unloading cycle to avoid edge effects which caused anomalous stripes16 and a nonuniform stress distribution. The responses during each loading-unloading cycle with five different misfit angles (φ1 ) 22.1, φ2 ) 34.2, φ3 ) 52.3, φ4 ) 65.3, and φ5 ) 84.4°) are evaluated for an identical sample. Note that the values are in the range of 0 to 90°, which is sufficient to elucidate the misfit angle dependency corresponding to the symmetry property. The maximum value for compressive strain is 13% except for the case with φ5 () 84.4°), where the value is 18%. The loading-unloading cycle is repeated for 2-3 times with each given misfit angle and, as a result, the φ-dependent features are consistent within negligible error in the patterns and statistical analysis that is explained later in this section. Thus, the typical data of a cycle for each misfit angle are shown and discussed in this report. It is confirmed, by atomic force microscopy, that the white and black parts of the optical (30) Ohzono, T.; Shimomura, M. Jpn. J. Appl. Phys. 2005, 44, 1055.
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Figure 2. Photographs of microscopy images (upper panels, 17.5 × 17.5 µm2) and of corresponding spatial domain distributions (lower panels, 35 × 35 µm2) during loading-unloading cycle with five different misfit angles. Each square in the lower panels indicates the area shown in each upper panel. The five cases with different misfit angles are shown in (a) φ ) φ1, (b) φ2, (c) φ3, (d) φ4, and (e) φ5. (φ1 ) 22.1, φ2 ) 34.2, φ3 ) 52.3, φ4 ) 65.3, and φ5 ) 84.4°.) The same area is shown for a given misfit angle through the cycle. microscopy images correspond to the topography of the wrinkled surface (Figure 2). 2.3. Image Analysis. The obtained images (35 × 35 µm2) are analyzed in terms of the stripe orientation using a method proposed by Egolf et al.31 and others,32 where derivatives of the pattern intensity are used to extract the local wave vector.20 We can approximate the stripe pattern z(x, y) locally by Bcos[ω (x, y)], for which the local wave vector k(x, y) is defined via k(x, y)
≡ ∇ω (x, y). Although the details of the formalism are not shown here, the local wave vector k(x, y) is calculated using normal and mixed second-order derivatives of the pattern intensity field z(x, (31) Egolf, D. A.; Melnikov, I. V.; Bodenschatz, E. Phys. Rev. Lett. 1998, 80, 3228. (32) Bazen, A. M.; Gerez, S. H. IEEE Trans. Pattern Anal. Mach. Intell. 2002, 24, 905.
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Figure 3. Relative average spatial wavelength of fully rearranged stripes at maximum strain to that of original stripes with respect to misfit angle. y). The local orientation of the wave θ(x, y) is defined through the relation exp[iθ(x, y)] ) k(x, y)/|k(x, y)|, where k ) kx + iky. In a real field, the wave vector (kx, ky) is equivalent to (-kx, -ky), so their signs are chosen to hold a condition, kx g 0, resulting in wave orientation values ranging from -90 to +90°. Note that the strain direction, i.e., the x direction (see Figure 1c), corresponds to the orientation expressed by θ ) 0. The histograms of the area corresponding to the various stripe orientations at each degree of strain during a loading-unloading cycle are analyzed. To investigate how the original stripe orientation changes, an image showing spatial domain distribution is created for each microscopy image. This is conducted by extracting the points that show θ within the range of -φ ( 10°, whitening them in the image, and blackening the rest. That is, the surface with pattern is decomposed in domains containing stripes with two different orientations, namely the orientation of the original stripes and an orientation that is more parallel to the applied strain. The average spatial wavelength of the initial stripe, λ0, and that of the stripe under strain, λs, are obtained by detecting the dominant peak of the 2D power spectra of the pattern intensity field z(x, y).
3. Results and Discussions 3.1. Overviews of Stripe Rearrangement Processes. In this section, we provide overviews of the stripe rearrangement process and the hysteretic behavior observed in the present system, which are qualitatively similar to our previous results.20 The upper panels in Figure 2 show photographs of microscopy images during a loading-unloading cycle with five different misfit angles. The microscopy images indicate that the original stripe orientation gradually rearranges upon loading, approximately completes the rearrangement at specific compressive strain, and gradually recovers its original orientation upon unloading for all cases with different misfit angles. It is clear that the change in the local strain state, which is due to applied strain to whole sample, destabilizes the original surface profile locally and induces local buckling in the direction of applied strain. The stripe rearrangement process starts from random locations, which are apart from each other with the distances ranging from several to 10 times of λ0. In some cases, it starts from topological defects in the original stripe pattern, such as dislocations.7 As loading proceeds, the stripe-rearranged domains grow by inducing the next stripe rearrangement (local buckling) in the neighborhood. The stripe rearrangement process can be clearly seen in the domain distribution images (see lower images of each panels of Figure 2). The black domains, which are the parts with rearranged wrinkles, show characteristic shapes running diagonally down the images depending on the misfit angle. Here, we briefly note the similarity of this process to general nucleation and growth process. Figure 3 shows that the average spatial wavelengths of the fully rearranged stripes at maximum strain slightly increase compared with those of the initial preordered
Figure 4. Changes in stripe orientation histograms during loading-unloading cycle. The five cases with different misfit angles are shown in (a) φ ) φ1, (b) φ2, (c) φ3, (d) φ4, and (e) φ5. (φ1 ) 22.1, φ2 ) 34.2, φ3 ) 52.3, φ4 ) 65.3, and φ5 ) 84.4°).
stripes. Moreover, the higher the misfit angle, the higher the increase in the wavelength. Although detailed discussion is difficult from the present result, one possible reason for the φ-dependent change in the wavelength is anisotropy of the effective modulus of the near surface for buckling,27 which originates from the initial preordered simple wavy profile. Here, we only note that the rate of increase in the average wavelength after rearrangement is about 1.2 at most. Figure 4 shows changes in the stripe orientation histograms during a loading-unloading cycle. The histograms on the middle range of strain show two peaks, which correspond to the initial (θ ) -φ) and rearranged orientations (θ ) 0). That is, each location on the surface is in either original or rearranged state, thus, a certain local buckling (rearrangement) process is an abrupt structural change in the current experimental time scale and each local buckling occurs at different values of applied strain. Although it is not shown in this report, the realtime observation also indicates that the rearrangement process stops within 1 s after cessation of the stepwise strain increase in most cases. Thus, the abrupt change, which is local buckling, is completed within such time scale. Meanwhile, the main reason each local buckling occurs at different value of applied strain is speculated to be a heterogeneous distribution of strain (or stress). If there is such a distribution, local buckling takes place depending on the location where effective strain becomes a critical value. Such strain distribution arises from (1) the inherent roughness, density distribution, and initial topological defects, and (2) the transitory surface profiles, namely domain structures during domain growth. The former factors are related to the manner for the initial local buckling (nucleation of rearranged domains), which occurs
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Figure 5. (a) Changes in area accounted for original stripes with θ of -φ ( 10° during loading (filled circles)-unloading (open circles) cycle for φ ) φ2 () 34.2°) (solid line) and φ5 () 84.4°) (gray line). (b) Degree of hysteresis with respect to misfit angle, which is area enclosed by each loop in (a).
at random locations and/or randomly positioned topological defects in the original stripe pattern, as described previously. The latter factors are important for the domain growth process. It is clear from the images in Figure 2 that there is a misfit-angle-dependent regular manner for the domain shapes and growth. The details are analyzed and discussed in the next section, which is the main aim of this study. In the following paragraphs in this section, we describe the misfit-angle-dependent hysteretic behavior. Upon unloading, the stripe also gradually rearranges back to the original. However, the process is not the reverse of that on loading. It is recognized in Figure 2 that the patterns at a certain degree of strain on loading and unloading are different from each other. Even if we compare the patterns that show almost identical histograms of the stripe orientation on loading and unloading, the spatial domain distribution is different. The reason for the difference in the domain structures upon loading and unloading is attributed to the irreversibility in the evolution of the distribution of strain upon a cycle of loading and unloading. The site for a local buckling event at a certain time is assumed to be determined through a delicate balance of local residual strain. The overall stripe orientations as well as the inherent imperfections, such as roughness and density distribution, affect the strain distribution. Thus, a small difference in the strain distribution during loading and unloading may trigger a local buckling at a different site, resulting in the different domain structure. One possible origin of the small difference is uncontrollable mechanical noise/fluctuation upon changing external strain, which may cause a slightly different distribution of residual strain. In addition, the effect of the inherent imperfections on the delicate strain distribution upon loading may differ from that upon unloading. In either case, the evolution of the distribution of strain becomes irreversible upon a cycle of loading and unloading. The hysteretic behaviors are also characterized quantitatively by analyzing the changes in the histograms in Figure 4 and the results are shown in Figure 5. As typical examples of the low and high degrees of hysteresis, the
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changes in the area corresponding to the original stripes with θ within the range of -φ ( 10° during a loadingunloading cycle for φ ) φ2 () 34.2°) and φ5 () 84.4°) are shown in Figure 5a. The results indicate that the statistical hysteretic behavior also exists in addition to that of the domain structures that is described previously. Figure 5b indicates that the degree of hysteresis at high misfit angles is remarkable. It is known from the structural mechanics that the bending rigidity of the film surface with one-dimensional wavy corrugation becomes anisotropic, where the bending rigidity increases as the misfit angle increases. Thus, the results suggest that, for the cases with higher misfit angles, the strain (or stress) required for local buckling upon loading is dictated by the high bending rigidity. Upon unloading, the strain (or stress) required for local unbuckling is not influenced by the film’s high rigidity; this is supported by Figure 5a where the unloading curve for high misfit is quite similar to lower misfit (i.e., lower rigidity) samples. Thus, the strain (or stress) for local buckling is smaller than for buckling at high misfit and it is reasonable that large hysteresis emerges for high misfit. One observation supporting the role of rigidity is that the maximum strain required for full rearrangement is higher for the high misfit case than for the lower misfit cases. Other results supporting this argument concerning the characteristic domain shapes will be described in the next section. However, the effect of the high bending rigidity for high misfit cases upon unloading might be small, considering the observation described bellow. In Figure 5a, the unloading curve for the high misfit case essentially overlaps the unloading curves for low misfit cases for strain less than 10%. This indicates that the statistical unloading processes are similar for different misfit values once the applied strain is lower than 10%. One might explain this situation by considering the effect of the memory of the initial preordered stripe pattern. Upon unloading, the memory of the initial preordered pattern reduces the bending rigidity of the strain-induced ordered pattern. Thus, the memory assists the rearrangement process. The memory of the pattern (surface topography) works as the initial imperfections that trigger buckling back to the original stripe pattern.30 Therefore, if the system had no memory effects, a higher degree of hysteresis might be observed. The qualitative results independent of the misfit angle described in this section are same as the previous result, where the labyrinthine pattern is used as the initial sample for the loading-unloading experiment.20 Thus, the phenomenon of the reversible stripe ordering by loading is probably independent of the initial wrinkle pattern. However, the present experimental system enables us to decompose the misfit-angle-dependent factors, which are mixed in the previous study. The misfit-angle-dependent hysteretic behavior is also described in this section, which is a statistical aspect of the dynamics. Meanwhile, in the next section, we will describe detailed dynamics of domain growth and shrinkage due to strain changes, namely the shapes of domains and their changes. 3.2. Evolution of Geometry-Dependent Domain Shapes. Figure 6 shows typical patterns found on the stripe rearrangement processes. Type A (Figure 6a) is mainly observed for cases with low misfit angles. The type A domain of the rearranged stripes induced by loading is an array of rearranged wrinkle crests (or valleys) that bridge the original neighboring stripes. Type B (Figure 6b) is mainly observed for cases with high misfit angles upon unloading. The type B domain is an array of rearranged wrinkle crests (or valleys) that ride over
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Figure 6. Images (5.9 × 5.9 µm2) with typical patterns found on stripe rearrangement processes. Figure 8. Schematics of strain-induced rearranged domain growth for (a) the type A and (b) B domains, and (C) for process of type C pattern occurrence. The left state on each panel is changed to the right state as compressive strain is increased. The arrows indicate the direction of defect migration.
Figure 7. (a) Simplified geometry of stripe pattern around type A domain. The gray parallelogram corresponds to the unit of the rearranged domain. (b) Theoretical value of ld according to eq 1 (dashed line) and the experimental averages (filled circles), which are taken including the data of both loading and unloading processes. (c) Experimental averages of ψ + φ calculated from the images obtained upon loading (filled circles) and unloading (open circles) are separately shown with the theoretical values including [eq 2, filled triangles] and excluding [eq 2a, dashed line] the changes in the wavelength shown in Figure 3. The experimental values are calculated by extracting the type A domains from all of the data.
multiple original stripes. Type C (Figure 6c) is mainly observed for cases with high misfit angles upon loading. The type C pattern is a single valley that ranges over multiple original stripes. Depending on the strain change, the domains grow or shrink in a specific way. To understand the factors that characterize the rearranged domain shape, particularly that of type A, the stripe pattern is simplified and its geometry is shown in Figure 7a. The gray parallelogram in Figure 7a corresponds to the unit of the rearranged domain with θ ) 0. Although the model is shown only for the type A, note that all types of domains shown in Figure 6 are composed of an ordered assembly of the unit domains. By a simple consideration of the geometry, if λ0, λs, and φ are given, the length ld of the wrinkle crest (or valley) that bridges the neighboring original stripes and the angle ψ between the direction of the domain boundary and that along the original wrinkle crest (or valley) can be calculated as
ld ) λ0/sin φ
(1)
ψ ) 90 - arctan[(λs - λ0 cos φ)/(λ0 sin φ)]
(2)
ψ ) 90 - φ/2 (for λs ) λ0)
(2a)
and
where the angles are in the units of degrees.
To compare the simple theoretical results with those of the experiments, the type A domains are extracted from the images at the middle range of strain and their average values for ld andψ are calculated. (The type A domain is hardly found in the images obtained upon loading with the high misfit angle.) Figure 7b shows that the theoretical values of ld according to eq 1 are in agreement with the experimental averages, where the averages are taken including the data of both loading and unloading processes. In Figure 7c, the experimental averages of ψ + φ calculated from the images obtained upon loading and unloading are separately shown with the theoretical values including [eq 2] and excluding [eq 2a] the changes in average wavelength shown in Figure 3. The qualitative increasing tendency of ψ + φ with respect to the misfit angle obtained in experiments is in agreement with that obtained by theory. Although the details are not clear, the difference in those values of ψ + φ between loading and unloading is possibly associated with the hysteresis described in the previous section. The corresponding slight differences are also recognized from the black domain shapes appearing in the domain distribution images in Figure 2 (for example, compare the domain shapes in two images under 4% strain in Figure 2b). The above results indicate that the observed shape of the rearranged unit domain is approximately determined by eqs 1 and 2. It is considered that the relationship is valid for the unit domains that compose the type B and C domains. Next, we describe how the domains evolve due to the change in applied strain. Figure 8, panels a and b, shows the growth directions of type A and B domains. The type A domain has a pair of regions, which are topological defects of dislocation7 of the striped system, with point symmetry with respect to the domain center. The strain increase induces local buckling in such regions by consuming a dislocation. At the same time, a new dislocation is created right next to the new unit of the rearranged domain. In other words, this process is the migration of a dislocation in a specific direction. The type B domain shows similar domain growth with a pair of larger defective regions (Figure 8b). In this case, the defective regions consist of a pair of amplitude grain boundaries7 of the striped system. The new wrinkle is induced by strain at the defective regions. At the same time, the larger defective region migrates to the neighbor. As the result, the type A and B domains show characteristic shapes running diagonally down the images depending on the misfit angle (see Figure 2). The defective regions with dislocations or amplitude grain boundaries are considered to be in an energetically less favorable state than those with a simple wavy
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corrugation because of the additional distortion (shear strain) at the near surface of such defective regions. Thus, the reason for this growth process can be understood by considering the number of defects created through a local buckling process as follows. Local buckling takes place when local strain (or stress) becomes a critical value, changing the system toward the energetically favorable state. The number of the defects created through a local buckling process should be minimized. Most of the observed domain growth process at the middle range of applied strain produce no new defects through a local buckling process (Figure 8, panels a and b). Meanwhile, if we only consider the parallelogram shape of the domain unit, there are other possible directions of domain growth, for example, a direction perpendicular to that of applied strain. However, the process of the addition of one domain unit to the original domain in such directions creates at least one defect, resulting in less energetically favorable states which are hardly realized. Thus, once a rearranged domain takes place, domain growth proceeds in two opposite directions on a straight line, which is geometrically determined through eq 2. The origin of the difference between type A and B domains depends on the first nucleation process of the rearranged domain. The lower frequency of appearance of the type B domains in the case with low misfit angles probably suggests that low strain required for local buckling due to the small bending rigidity leads to the type A domain. Meanwhile, high strain required for local buckling due to the high bending rigidity probably leads to the type C pattern. This qualitative difference is related to the high statistical hysteresis for the high misfit cases (Figure 5b). We observe a characteristic pattern that leads to the type C (and the nucleus of the type B) pattern. It is recognizable in Figure 2e (or 2d) that there are rippled patterns (Figure 8c) before the formation of the type C (or B) pattern. Similar zigzag or herringbone patterns of wrinkles under anisotropic strain conditions have been reported theoretically26 and experimentally for relatively large irreversible wrinkles with a wavelength of 5-15 mm.33 The rippled patterns range over multiple original stripes with an identical phase. The strain increase first induces the rippled patterns to relax strainasmuch as possible. This is probably due to the smaller amount of energy required for the formation of the rippled patterns than that for the resultant buckling. Then, the valley abruptly appears along the crests of the ripples in phase and rides over multiple stripes, when local strain reaches the critical value for buckling of the rippled pattern. The type C pattern also has defective regions as indicated in Figure 8c. Although it is somewhat irregular, a similar growth manner to that of the type A and B domains, which is the geometry-dependent domain growth, is observed during loading. On the final steps to the maximum strain states, the original wrinkles between rearranged domains are also rearranged for all cases shown in Figure 2. However, the rearrangement proceeds in a commensurate manner. The origin is the spatial randomness in the location of the nucleation of the rearranged wrinkle domains. Each domain grows in a specific geometry-dependent direction in the middle range of applied strain. Thus, the spacings between neighboring matured domains also remain random. Meanwhile, the geometry-dependent specific size of the unit domain rarely fit the resultant spacings. As the result, the patterns compromise by creating defects (33) Ghosh, S. K.; Ramberg, H. Tectonophysics 1968, 5, 89.
Ohzono and Shimomura
or readjusting the local wavelength, resulting in the peak broadening of the rearranged stripes under maximum strain in the orientation histogram shown in Figure 4. The simple stripe pattern after a loading-unloading cycle recovers the original sharpness of the peak in the histograms. This is simply because the memorized original preordered stripe pattern is recovered. In terms of the domain shapes and the manner of their changes, we mention the hysteretic behavior again. The main reason for the hysteresis in the domain structure during a loading-unloading cycle is possibly the difference in the evolution of spatial distribution of strain upon loading and unloading, as described previously. Upon loading, strain required for buckling increases as the misfit angle increases due to the increase in the effective bending rigidity. Upon unloading, while this dependency is the same, the effect of the memory of the initial preordered pattern reduces the effective bending rigidity of the rearranged pattern, and assists in local unbuckling in all locations. As a result, the nucleation of domains on the strain-induced aligned stripe patterns upon unloading becomes easier than that upon loading. For the higher misfit cases, the reduction of the bending rigidity is probably larger. Thus, the characteristic domain shapes observed upon unloading for higher misfit cases frequently show the type A and B domain shapes, which are quite different from those observed upon loading. Consequently, the statistical hysteresis shown in Figure 5 is observed. Meanwhile, if the system had no memory effect, the type C pattern might be observed also during unloading, thus, further degree of hysteresis might be observed. 4. Conclusions The preordered microwrinkle pattern is employed to study the reversible stripe rearrangement processes induced by applying and unloading lateral compressive strain. The dependencies of statistical properties and the characteristic domain shapes on the misfit angle φ enable us to decompose the elementary processes of the stripe rearrangement, which were hardly derived from the previous study.20 The original stripe orientation gradually rearranges upon loading. The shapes of the rearranged stripe domain at the initial stage differ depending on the misfit angle. The low and high misfit angles result in wrinkles that ride over neighboring and multiple original stripes, respectively. Each domain is composed of parallelogram units and has defective regions with a point symmetry relative to the domain center. The shapes of the domain unit and configuration of defective regions are simply determined through the stripe geometry. The increase in strain induces new buckling at the defective regions and makes the regions migrate in a geometry-dependent specific direction, resulting in a characteristic domain shape. The strain-induced stripe rearrangement completes the rearrangement at a specific compressive strain. Upon unloading, the original stripe pattern gradually recovers its original orientation. However, the process is not a reverse of that upon loading. The main reason for the hysteresis is assumed to be the difference in evolution of the spatial distribution of residual strain upon loading and unloading. Further theoretical studies might be required for the justification of our discussions, particularly on the relationship between the value of strain required for buckling and the misfit angle φ. However, this study provides important observations on the cooperative and ordered response of the wrinkled surface to the global mechanical
Stripe Rearrangement Processes
strain. The ordered anisotropic domain growth process is one of the most important results. We believe that such a novel property will pave the way to dynamic control of the surface topographic pattern on the microscopic scale
Langmuir, Vol. 21, No. 16, 2005 7237
with the aid of spontaneous processes (or self-organization) toward many technological applications. LA0503449
