Motivated by the self-assembly of natural systems, researchers have investigated the stimulus-responsive curving of thin-shell structures, which is also known as self-folding. Self-folding strategies not only offer possibilities to realize complicated shapes but also promise actuation at small length scales. Biaxial mismatch strain driven self-folding bilayers demonstrate bifurcation of equilibrium shapes (from quasi-axisymmetric doubly curved to approximately singly curved) during their stimulus-responsive morphing behavior. Being a structurally instable, bifurcation could be used to tune the self-folding behavior, and hence, a detailed understanding of this phenomenon is appealing from both fundamental and practical perspectives. In this work, we investigated the bifurcation behavior of self-folding bilayer polygons. For the mechanistic understanding, we developed finite element models of planar bilayers (consisting of a stimulus-responsive and a passive layer of material) that transform into 3D curved configurations. Our experiments with cross-linked Polydimethylsiloxane samples that change shapes in organic solvents confirmed our model predictions. Finally, we explored a design scheme to generate gripper-like architectures by avoiding the bifurcation of stimulus-responsive bilayers. Our research contributes to the broad field of self-assembly as the findings could motivate functional devices across multiple disciplines such as robotics, artificial muscles, therapeutic cargos, and reconfigurable biomedical devices.

From the wrinkling of our skin1 to the propelling motion of wheat awns into the ground,2 biological systems organize themselves for form and function through self-folding across multiple length and time scales. Self-folding generally refers to the mechanisms through which a structure utilizes the differential behavior of its constituents to generate spontaneous curvature changes in response to an external stimulus;3 a well-known example is the bending of a bi-metallic strip thermostat subjected to temperature changes.4 The potential to overcome the limitations of traditional planar lithographic techniques and actuate minuscule architectures has made self-folding a topic of substantial technological importance. Researchers have utilized the self-folding behaviors of thin structures in response to stimuli such as temperature, pH, solvent, enzyme, light, and electric field to develop functional devices for sensing, electronics, medicine, energy storage, robotics, and microfluidic applications.3,5,6

Owing to their thin-shell configurations, the behaviors of self-folding structures are often dominated by geometric nonlinearities. Since Stoney's7 work to determine the force-curvature relationship of the most fundamental class of self-folding structures—a residually stressed bilayer system curving under isotropic mismatch strains—efforts have been made to incorporate geometric nonlinearities arising from the large displacements of thin bilayers.8–11 The nonlinear analyses revealed bifurcation, a phenomenon where the deformation of a bilayer system transitions from one mode to another one at a critical mismatch strain. At low mismatch strains, the bilayers undergo a combination of bending and stretching to achieve quasi-axisymmetric configurations with non-zero Gaussian curvatures.9 Since stretching is energetically more expensive than bending12 for thin sheets, the bilayers bifurcate to an approximately singly curved state beyond some critical value of the mismatch strains.9 While mismatch strain-driven bifurcation of circular13 or rectangular14 bilayers has been studied, a detailed investigation encompassing other geometries within a comprehensive framework has not been performed and will complement the current understanding of the self-folding behavior. In this paper, through a combination of finite element modelling and experiments, we demonstrate the effects of the initial shape, structural nonlinearity, and edge layer on the stimulus-responsive shape transformation behavior (pre- and post-bifurcation) of bilayers. We also explore a scheme to generate gripper-like architectures from tether-less stimulus-responsive actuation of freestanding bilayers.

We considered a family of regular convex polygonal bilayers (one layer expands isotropically in response to a stimulus, while the other remains passive) that share the same circumscribing circle [Fig. 1(a)]. Starting from a triangle and followed by a square, the polygons approached their circumscribing circle with increasing number of edges. We performed finite element modelling of the bilayer morphing behavior as this technique has been used to analyze stimulus-responsive shape transformation problems involving geometric nonlinearities.15,16 In our experiments, we relied on the fact that Polydimethylsiloxane (PDMS) elastomers demonstrate tunable swelling behavior (higher cross-link densities resulting in lower swelling and vice versa) in organic solvents.17 To remove the dependence of simulation results on specific physical units and the chosen numeric, we non-dimensionalized the calculated quantities. Details of our modelling, experimentation, and non-dimensionalization schemes are provided in the supplementary material.

FIG. 1.

(a) A family of regular convex polygons. In each case, the center of the circumscribing circle (red dot) coincides with the polygon centroid. The green and blue arrows for a given polygon bound all the unique major curvature axes for the post-bifurcated singly curved state. The acute angles between the two arrows depend on the polygon (30° for a triangle, 45° for a square, 18° for a pentagon, and so on). (b) shows the dependence of bifurcation mismatch strains on polygonal shapes.

FIG. 1.

(a) A family of regular convex polygons. In each case, the center of the circumscribing circle (red dot) coincides with the polygon centroid. The green and blue arrows for a given polygon bound all the unique major curvature axes for the post-bifurcated singly curved state. The acute angles between the two arrows depend on the polygon (30° for a triangle, 45° for a square, 18° for a pentagon, and so on). (b) shows the dependence of bifurcation mismatch strains on polygonal shapes.

Close modal

At low mismatch strains, the bilayers demonstrate quasi-axisymmetric configurations where the deformation behavior is axisymmetric near the bilayer centroids but due to edge effects, the corners and edges deform in a slightly different manner. Bifurcation to an approximately singly curved state with a unique major curvature axis (direction along which the bilayer is bent) occurs at critical strains. Representative pre- and post-bifurcation behaviors of triangular and circular bilayers are shown in Figs. S1–S3 of the supplementary material. The dependence of mismatch strain during bifurcation on polygonal shapes is shown in Fig. 1(b). Due to non-dimensionalization, results from different geometric families (varying in circumscribing circle radii “R” and bilayer thickness “t”) merge onto a single value for a given polygon. As shown, triangles require the highest mismatch strain for bifurcation followed by squares, and as the number of edges increases, bifurcation occurs at lower strains and the critical strain approaches the limiting value for a circle. This trend could be understood by considering the sizes of the mid-section near the centroid of the polygons. Prior to its bifurcation, a triangle sustains high mismatch strains and develops a strongly curved quasi-axisymmetric configuration as its easier to stretch the small mid-section near the triangle centroid (the radius of the inscribing circle being Rcos(π3)). With the increasing number of polygonal edges, stretching becomes difficult due to the increase in mid-section sizes (Rcos(π4) for a square, Rcos(π6) for a hexagon, Rcos(π10) for a decagon, and R for a circle in the limiting case) and the polygons eventually bifurcate at lower values of curvatures (Fig. S1 of the supplementary material) and mismatch strains. The effects of different geometric and material properties on the actuation of bilayer structures are shown in Fig. S4 of the supplementary material.

Our model also predicted the emergence of preferred polygonal bending directions beyond bifurcation. As shown in Fig. 2(a), regular polygons prefer to orient their major curvature axes parallel to their undeformed edge (for polygons with an odd number of edges such as triangle and pentagon) or edges (for polygons with an even number of edges such as square and hexagon). Polygons with a higher number of edges such as octagon and decagon also demonstrated similar preferences to orient their major curvature axes (not shown). For convenience, future references to this preferential bending will be mentioned as mode 1. We performed additional simulations to understand the energetics behind the preferential bending. Figure 2(b) shows some results of our analysis where the strain energy for polygonal bending mode 1 was subtracted from that of a different configuration which is here referred to as bending mode 2 (individual energetics for the 2 modes of a square bilayer are shown in Fig. S5 of the supplementary material). The major curvature axis of mode 2 [blue arrows in Fig. 1(a)] is farthest away from that of mode 1. It is evident that for all the polygons, there is no energy difference before bifurcation. Beyond bifurcation, the energy difference for a given polygon reaches a peak with increasing mismatch strains and then gradually drops to a lower value. While triangles, squares, and pentagons maintain positive values of energy difference (mode 1 is still favorable), hexagons demonstrate negative values of energy difference (meaning that mode 2 becomes favorable) at high mismatch strains.

FIG. 2.

(a) Preferred bending directions (mode 1) for triangles, squares, pentagons, and hexagons. The stimulus-responsive and passive layers are denoted by purple and green, respectively. (b) The energy differences between bending modes 2 and 1 for squares (purple), triangles (blue), pentagons (orange), and hexagons (red) are plotted against the mismatch strain. The red (hexagon), purple (square), and blue (triangle) double-headed arrows denote the major curvature axis directions for each mode.

FIG. 2.

(a) Preferred bending directions (mode 1) for triangles, squares, pentagons, and hexagons. The stimulus-responsive and passive layers are denoted by purple and green, respectively. (b) The energy differences between bending modes 2 and 1 for squares (purple), triangles (blue), pentagons (orange), and hexagons (red) are plotted against the mismatch strain. The red (hexagon), purple (square), and blue (triangle) double-headed arrows denote the major curvature axis directions for each mode.

Close modal

The preferential post-bifurcation bending of rectangles has been attributed to the edge effects which make bending along the longer axis energetically favorable.14 Our analysis builds upon that argument and establishes the preferred bending directions for regular polygons with different shapes. The elastic strain energy density maps for the 2 triangular bending modes are shown in Fig. S6 (supplementary material). Of the 2 bending modes, mode 1 is preferable as it demonstrates an edge layer with low energy density (as shown at a mismatch strain away from the bifurcation point). However, at very high mismatch strains, the energy densities become relatively uniform for both the bending modes. The emergence of low energy regions near the corners and edges could be explained through their reduced lateral constraints. Before bifurcation, the polygons form quasi-axisymmetric configurations with equal curvatures (near the centroid) along the two orthogonal directions. Beyond bifurcation, although the cylindrical bending type of deformation along one direction becomes favorable, doubly curved regions (Fig. S7 of the supplementary material) with low energy densities emerge near the edges and corners (the reduced constraints enable them to curve along both the directions). With increasing mismatch strains, the double curvatures gradually vanish, and consequently, the energy variations become minute [Fig. 2(b)]. Our calculations showed that square-shaped bilayers have the largest energy differences within the polygonal family at high mismatch strains. We ascribe this characteristic to the large angular distance between the two major curvature axes of the respective modes [45° as shown in Fig. 1(a), largest within the family]. As shown in Fig. S8 (supplementary material), the difference in orientation between the bending modes of square bilayers led to differences in curvature distributions and eventually resulted in large energy differences. In Fig. S9 (supplementary material), we present the energy density contours for the two modes of hexagonal bending. As shown, changes in the number and location of edge layers are responsible for the change in the preferred bending mode at high mismatch strains. The preferred bending mode for a given polygon strongly depends on its geometry. We found that for isosceles triangles, mode 2 is preferred when the inclined edges are just 1% larger than their horizontal counterpart (Fig. S10 of the supplementary material). It is worth mentioning that the elastic strain energy density contours reported in this work have been obtained through the finite element method, whose calculation schemes vary from those reported in the literature to calculate shell energetics.9,10 A brief discussion on this topic has been included in the supplementary material.

We performed experiments with bilayer polymer samples to validate our finite element model predictions (Fig. 3). As shown in Fig. 3(c), triangles required the highest critical mismatch strain for their bifurcation and polygons with a higher number of edges bifurcated at lower strains. Our experiments also verified the preferential bending modes of triangles (mode 1), squares (mode 1), and hexagons (mode 2) at high mismatch strains beyond their bifurcation [Figs. 3(d), 3(e), and 3(g)]. The energetic difference between different bending modes of a pentagon becomes very small (close to zero) at high mismatch strains [Fig. 2(b)]. This fact, coupled with the presence of imperfections in the experimental samples (local variations in thickness and material properties), resulted in the observed variations in the bending modes of the pentagons [Fig. 3(f)]. While some variations were also observed for triangles and hexagons, squares demonstrated a perfect agreement with our model predictions owing to their large energy differences between the two modes at high strains [Fig. 2(b)].

FIG. 3.

Top (a) and cross-sectional (b) views of the bilayer PDMS samples. (c) shows the bifurcation mismatch strains from experiments and computational analysis. Histograms demonstrate the percentage of triangles (d), squares (e), pentagons (f), and hexagons (g) assuming different post-bifurcation bending directions at a mismatch strain of 0.052. The results are presented in terms of angular deviations from the preferred bending mode of each polygon. The scale bars for (a) and (d)–(g) are 10 mm.

FIG. 3.

Top (a) and cross-sectional (b) views of the bilayer PDMS samples. (c) shows the bifurcation mismatch strains from experiments and computational analysis. Histograms demonstrate the percentage of triangles (d), squares (e), pentagons (f), and hexagons (g) assuming different post-bifurcation bending directions at a mismatch strain of 0.052. The results are presented in terms of angular deviations from the preferred bending mode of each polygon. The scale bars for (a) and (d)–(g) are 10 mm.

Close modal

It is worth mentioning that the swelling of elastomers in organic solvents is a complex phenomenon. Within the framework of nonlinear finite elements, this process has been implemented by treating the polymeric gel as a compressible hyperelastic solid,18 which requires a number of experimentally determined parameters such as the free swelling stretch of the network in the solvent, small strain shear modulus of the dry network, density of polymeric chains, mixing enthalpy between solvent and the elastomer, and volume of the solvent molecules and their chemical potential. Since the goal of our computational analysis was to understand the mechanics of the stimulus-responsive bilayer behavior, we focused on the nonlinear geometric effects relevant to these structures and replaced the complex process of organic solvent induced elastomer swelling by a simple thermal expansion driven shape transformation of linear elastic materials (Figs. 1, 2, and 4). This approach allowed us to constitute the general predictions without incurring excessive computational costs (due to material nonlinearities) and experimental efforts (required for material characterization). To demonstrate the applicability of our model, we computationally investigated the behavior of the polygons from our bifurcation experiments (Rt=41.88) with a second-order Ogden material model for dry PDMS (see the supplementary material for details). Our calculations accurately captured the experimentally observed bifurcation strains [Fig. 3(c)].

FIG. 4.

(a) Triangle and square from the same polygonal family with constrained circular areas (filled with black patterns) near their centroids. (b) Bifurcation mismatch strain and constraint requirements to achieve gripper-like configurations. Deformed shapes achievable at different levels of the constraint are also attached. (c) Patterned triangular bilayers with only the purple layer near the centroid. Depending on the pattern sizes, the samples demonstrated bifurcated (i) and dog-ear (ii) configurations. The scale bars are 5 mm.

FIG. 4.

(a) Triangle and square from the same polygonal family with constrained circular areas (filled with black patterns) near their centroids. (b) Bifurcation mismatch strain and constraint requirements to achieve gripper-like configurations. Deformed shapes achievable at different levels of the constraint are also attached. (c) Patterned triangular bilayers with only the purple layer near the centroid. Depending on the pattern sizes, the samples demonstrated bifurcated (i) and dog-ear (ii) configurations. The scale bars are 5 mm.

Close modal

Motivated by convex and star-shaped polygonal grippers for robotic manipulations,19–22 we explored the possibilities to design self-actuating tether-less grippers from stimulus-responsive bilayers. In contrast to pneumatic and electronic means of actuation that often requires heavy accessories such as pumps, compressors, fluid supply systems, and/or power sources, our envisioned design relies on the programmable morphing of freestanding bilayers in response to changes in their surroundings. In principle, the gripping action arises from a particular mode of polygonal deformation (often referred to as the dog-ear shape14) where the corners undergo significant out-of-plane displacements and the central areas remain unchanged. This type of deformation is difficult to achieve from isotropic mismatch strain actuated bilayers as they are prone to bifurcation. Since bifurcation occurs due to the mismatch strain induced stretching of polygonal mid-sections, we propose to constrain a portion of the mid-section [Fig. 4(a)] to achieve gripper-like configurations [Fig. 4(b)]. Central areas of self-actuating structures are often constrained through attachment14 and/or connection20 mechanisms for practical purposes. Hence, our proposed scheme would be applicable toward a variety of functional griping devices across multiple length-scales. Our analysis suggests the existence of a critical constraint to avoid bifurcation and generate selective polygonal deformations near the corners. The critical constraint of square-shaped bilayers is higher (owing to their larger mid-section sizes, as discussed earlier) than that of their triangular counterparts (as shown in Fig. S11 of the supplementary material). Below the critical values, the polygons bifurcate into an approximately singly curved state. But due to added constraints, the bifurcation strains are higher than their unconstrained counterparts. Square bilayers demonstrate an additional mixed-mode bifurcated configuration (half dog-ear and half singly curved) near their critical constraint. This type of shape has been experimentally observed with micro-fabricated bilayers under certain conditions.14,23 To realize the gripper-like shapes within our experimental framework, we fabricated patterned bilayer samples of triangular configurations [Fig. 4(c)] where a circular portion of the passive layer in the center of the triangle was removed. This design ensured that there was no mismatch strain within the central region (because of the lack of differential swelling) and thus enabled us to perform our experiments without the need of external constraints. As shown, we were able to induce the bilayers to achieve either the bifurcated or the dog-ear shapes depending on the center region size at the same value of mismatch strain.

In conclusion, we investigated the morphing behavior of self-folding bilayers in response to an external stimulus. Through computational modelling, we developed mechanistic understandings of the bilayer behavior and predicted their characteristics before and after bifurcation. Our experiments with polymeric bilayers confirmed the model predictions. This work contributes to the broad field of self-assembly as the proposed design principles could be incorporated into stimulus-responsive actuators, therapeutic cargos, artificial muscles, and reconfigurable biomedical and chemo-mechanical devices.

See supplementary material for a detailed description of methods and referred supplementary figures.

This work was supported by the U.S. Department of Energy, Office of Science, Basic Energy Sciences, under Award No. # DE-FG02-07ER46471. Arif M. Abdullah acknowledges support from the FMC Educational Fund Fellowship through the University of Illinois at Urbana-Champaign.

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Supplementary Material