Geometry-driven folding of a floating annular sheet
Predicting the large-amplitude deformations of thin elastic sheets is difficult due to the complications of self-contact, geometric nonlinearities, and a multitude of low-lying energy states. We study a simple two-dimensional setting where an annular polymer sheet floating on an air-water interface is subjected to different tensions on the inner and outer rims. The sheet folds and wrinkles into many distinct morphologies that break axisymmetry. These states can be understood within a recent geometric approach for determining the gross shape of extremely bendable yet inextensible sheets by extremizing an appropriate area functional. Our analysis explains the remarkable feature that the observed buckling transitions between wrinkled and folded shapes are insensitive to the bending rigidity of the sheet.
The mechanics of thin sheets at fluid interfaces is a current frontier of elasto-capillary phenomena Roman and Bico (2010). In contrast to thicker films that balance the liquid-vapor surface tension by generating moderate strain Lester (1961); Marchand et al. (2012); Style et al. (2013); Andreotti et al. (2016) or curvature Py et al. (2007); Neukirch et al. (2007); Pokroy et al. (2009); Duprat et al. (2012), very thin sheets strongly resist in-plane stretching but are readily curled, wrinkled, or folded under capillary forces King et al. (2012); Paulsen et al. (2016). In the asymptotic regime
where , are the bending and stretching moduli and is a characteristic length, the liquid surface energy becomes the only dominant energy, rendering the elasto-capillary problem into a purely geometric area minimization. This nontrivial class of “asymptotic isometries” was demonstrated in the partial wrapping of a liquid drop by an ultrathin circular sheet, where axial symmetry is spontaneously broken Paulsen et al. (2015).
Our understanding of this field is still in its infancy, and many basic questions remain. What classes of gross shapes are possible, and what is the nature of the transitions between them? In general, transitions in microstructure – such as the wrinkle-fold transition in 1D systems Hunt et al. (1993); Pocivavsek et al. (2008); Brau et al. (2013) – are driven by competing energies. Are there situations in which microstructure is dictated by geometrical, rather than mechanical constraints?
Here we study a simple, near-planar system which exhibits: (i) a variety of gross shapes with continuous and discontinuous transitions between them, (ii) coexistence of distinct microstructural elements, and (iii) a wrinkle-fold transition governed by geometric constraints. We find that purely geometric considerations determine the gross shape, which may dictate a specific microstructure. If more than one microstructure is possible, then mechanical energies may select one.
Experiment.— We work in a geometry first experimentally investigated by Piñeirua et al. Piñeirua et al. (2013), but with much thinner films ( nm) in order to probe the asymptotic regime of Eq. (1). We spin-coat polystyrene films ( GPa) on glass substrates, and cut into an annular shape with radii and (Fig. 1a), where mm mm, and mm mm. The film is floated onto water in a Langmuir trough, and surfactant (perfluorododecanoic acid) is added outside the film. Surfactant concentration is varied by translating barriers to reduce the available area. This controls the tension on the outer boundary of the annulus, (Fig. 1b). The barrier speed is sufficiently slow so that the surfactant is in equilibrium, and hydrochloric acid is added to the water so that the surfactant is insoluble and cannot migrate through the bulk to the inner interface. As we will show, the two dimensionless parameters of interest are and . The thickness and Young’s modulus of the sheet, the density of the liquid, and gravity only play a minor role.
Experimental studies of this problem Huang et al. (2007); Schroll et al. (2013) (known as the Lamé setup) have typically focused on , where only part of the sheet is wrinkled Davidovitch et al. (2011); Piñeirua (2011); Taylor et al. (2015). For the sheet is under nearly isotropic tension. As increases above a -dependent threshold Taylor et al. (2015), hoop compression is developed near the inner edge, giving rise to radial wrinkles. The wrinkled zone expands until it reaches the outer edge when Piñeirua (2011); Taylor et al. (2015), as shown in Fig. 1c.
For , the net radial force on an angular sector is negative, signaling an inward collapse. However, for a highly bendable film, multiple deformations are possible. Figure 1d shows the deformation for , where the sheet forms two folds Toga (2014). As increases further, these folds accumulate more material (Supplementary Video 1 SI ()).
The progression for narrow annuli, shown in Fig. 2a-c, is very different: as we increase , the sheet again forms axisymmetric wrinkles, but then collapses to a closed state (Supplementary Video 2 SI ()), similar to previous observations on thicker films Piñeirua et al. (2013). This shape, which we call a “collapsed racetrack”, consists of multiple folds coexisting with wrinkled zones.
Yet another mode of symmetry breaking is shown in Fig. 2d-f. Here, nonuniform wrinkles change the radius of curvature of the boundary (Supplementary Video 3 SI ()), in a manner that cannot be accomplished with a finite number of localized folds. This provides a concrete example where the gross shape can be generated by only a particular microstructure.
Phase diagram.— The different morphologies are shown in a phase diagram, Fig. 3a, where the points indicate the instability of the axisymmetric wrinkled state upon increasing . Noting that the data cluster around the line , we investigate the dependence on thickness by plotting at the transition as a function of thickness in Fig. 3b. No systematic trend is observed. Wrinkle-fold transitions in 1D systems generally exhibit strong dependence on thickness, indicating an energetic competition between substrate deformation and bending resistance Milner et al. (1989); Hunt et al. (1993); Pocivavsek et al. (2008); Reis et al. (2009); Holmes and Crosby (2010); Leahy et al. (2010); Diamant and Witten (2011); Audoly (2011); Brau et al. (2011); Ebata et al. (2012); Brau et al. (2013). By contrast, Fig. 3b signals a totally different mechanism for a wrinkle-fold transition, governed by geometric constraints.
Geometric model.— The observation in Fig. 3b motivates a geometric model for the wrinkle-fold transition, similar to the one introduced in Ref. Paulsen et al. (2015) for the wrapping of a droplet with a very thin sheet. This model relies on a separation of energy scales, Eq. (1), whereby wrinkles, crumples, and folds enable compression of the sheet at no cost, allowing the sheet to become “submetric” to its initial flat state Ligarò and Barsotti (2008); Pak and Schlenker (2010), whereas surface energies are not sufficient to stretch the sheet significantly. Thus, the gross shape (which ignores small-scale features) is obtained by minimizing the surface energy of the exposed liquid interfaces, under a constraint of inextensibility of the sheet. Subtracting a constant term proportional to the area of the trough, the energy becomes:
where is the liquid surface area enclosed within the annulus, and is the change in liquid surface area outside. We now consider the energies in three distinct families of gross shapes: axisymmetric contraction, the 2-fold shape, and a racetrack shape.
Axisymmetric gross shape.— For an axisymmetric deformation, all points of the annulus move radially inwards a distance by forming wrinkles (Fig. 4a). Hereafter, we render all quantities dimensionless by rescaling with and , and the energy [Eq. (2)] thus becomes . Minimization yields for , and for , where . Our experiments do not find this axisymmetric mode at large .
Two folds.— Inspired by Fig. 1d, we introduce a 2-folds ansatz where two halves of the sheet translate rigidly toward each other over a distance , gathering excess material into two folds, as illustrated in Fig. 4b. This ansatz breaks axial symmetry minimally by stitching together flat sections of the initial shape. This is similar to the polygonal shapes obtained upon wrapping a liquid droplet with a circular sheet, where flat petals curl around the drop Paulsen et al. (2015). The energy is , where . The predicted displacement for is:
For any , we find that : the 2-folds ansatz is energetically favorable, consistent with our experimental observations.
Following a similar principle to the droplet-wrapping problem Paulsen et al. (2015), the 2-folds ansatz can be generalized to -folds, by dividing the initial annulus into angular sectors. Here we find that the optimal number of folds is always .
Observed gross shapes.— To quantify the shapes observed in the experiment, we denote by the widest distance in the inner boundary of the sheet; the diameter along the bisector of that line is (Fig. 4b). Figure 4c shows versus for sheets with five different values of ; it agrees with the prediction (solid line), and deviates significantly from its value for an axisymmetric shape. The progression of versus is in qualitative agreement with the prediction for two folds (Fig. 4e), although contraction is observed for , namely, before wrinkles are predicted to reach the outer edge. This last observation indicates a shortcoming of our purely geometric model, which we discuss in the concluding paragraphs.
Racetrack.— Figure 2a-c, 4d, and Ref. Piñeirua et al. (2013) motivate another ansatz, depicted in Fig. 5a,b, wherein the sheet is divided into four angular sectors. Two opposite sectors are straightened by wrinkles whose amplitude is tailored to collect an increasing amount of azimuthal length towards the outer edge, while the inner edge remains flat. The other two sectors are bent with an inner radius ; there, the inner edge is wrinkled and the outer edge is flat. The relative amount of material compressed into wrinkles is schematically represented by the shading in Fig. 5a,b. There is a conceptual difference between the racetrack and the 2-folds states: the latter is constructed by stitching together pieces of the annulus, whereas the former requires deformation at all points.
A simple calculation shows that the length of the straightened parts is given by: . The racetrack energy is: .
For (wide annuli), the optimal departs continuously from . For any , the energy of the racetrack ansatz is higher than in the 2-folds ansatz, consistent with our experiments. The racetrack can be favorable for even larger (dashed line in Fig. 3a), but the pathway from two folds to a racetrack is not obvious (see SI ()).
For (narrow annuli), the flat state () is a local minimum for (Fig. 5c). For , the only energy minimum is at , corresponding to a collapsed racetrack (Fig. 5b). For these narrow annuli, . Thus, the model predicts a sudden collapse from the initial flat state to a closed shape; jumps from 1 to 0 as exceeds (Fig. 5d and SI ()). Our measurements are in qualitative agreement with this prediction. (The surfactant dynamics is much quicker than the motion of the barrier, but may not equilibrate over the short time-scale of the collapse.)
The predicted values of at which the racetrack appears is shown as a solid line on the phase diagram (Fig. 3a), corresponding to the local minimum of the energy. If the annulus could activate over the energy barrier (in Fig. 5c) and reach the global minimum, the racetrack shape would occur for slightly smaller (dotted line in Fig. 3a, see also SI ()). We point out that the collapsed racetrack we observe is often curved, as in Fig. 2a-c. A simple calculation shows that the curving of the midline is a soft mode of the racetrack state.
Conclusion.— We have shown that an annular sheet subjected to differential tensile loads displays a rich, shape-dependent behavior. This behavior is captured by a geometric model, in which the liquid surface area is minimized while the sheet is assumed infinitely bendable and inextensible. The 2-folds and racetrack shapes emerge as qualitatively distinct solutions of the geometric model – the first one is reminiscent of the flat annulus, and is constructed by “stitching” together portions of it, whereas the second type substantially deforms the sheet far from the axisymmetric state.
We have shown that purely geometric considerations select gross shapes whenever interfacial area is the dominant energy. This occurs in 2D and 3D, under boundary constraints (such as the partial wrapping of a liquid drop Paulsen et al. (2015)), or topological constraints (such as accommodating a hole in the present work). This geometric approach links these interfacial phenomena to problems where a nearly-inextensible membrane such as a mylar balloon or a parachute is inflated Taylor (1963); Paulsen (1994); Ligarò and Barsotti (2008); Pak and Schlenker (2010).
The selected gross shape constrains the type of microstructure in the sheet, and may even determine it uniquely (e.g., two large folds for wide annuli).
However, a given gross shape may be compatible with multiple microscopic configurations (e.g., wrinkles with different wavelengths), whereby sub-dominant energies due to bending and substrate deformation can then govern the fine microscopic features.
More dramatically, if several gross shapes are nearly degenerate in area, these sub-dominant energies can select the gross shape.
For example, the racetrack ansatz is made of wrinkles and completely collapses, whereas the observed shape, made of many folds, does not (Fig. 2a).
Another example arises for the thickest film we studied ( nm): Finite contraction is observed in the axisymmetric wrinkled state (top right corner of Fig. 4f).
Then, for larger , a nonuniform distribution of wrinkles mimics the 2-folds ansatz, before folds have actually formed (Fig. 4f).
This calls for a more complete understanding of the relationship between gross shapes and their underlying microstructure.
Acknowledgements.We thank H. Bermudez for use of a Langmuir trough. J.D.P. gratefully acknowledges support from the ESPCI Paris Joliot Chair. This work was supported by the W. M. Keck Foundation.
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