Geometry and Mechanics in the Opening of Chiral Seed Pods

Motion in plants is a highly nontrivial process. It is often based on anisotropic swelling and shrinkage, driven by the relatively slow variation of water content within the tissue (1). Swelling and shrinkage may lead to the accumulation of elastic energy and the buildup of stress. This stress can be released via mechanical instabilities, such as fracture, buckling, and snapping (2, 3). The buildup of stress may also occur in sclerenchymal tissue (tissue made of dead cells) and is thus a process that can be analyzed from a purely mechanical point of view. Sclerenchymal tissue typically consists of fiber cells whose walls are made of layered cellulose fibrils with a preferred orientation. When absorbing/expelling water, the tissue expands/shrinks anisotropically, perpendicularly to the fibrils’ orientation [(1), p. 200]. Changes in air humidity induce such uniaxial swelling/shrinkage that drive, for example, the opening and closure of a pine cone (4) and the penetration of wheat seeds into soil (5).

Chiral pod opening is a nontrivial example of hygroscopic (humidity-driven) motion in sclerenchymal tissue. In this process, two initially flat pod valves curl into helical strips of opposite handedness. A detailed study of the structure of pod valves in over 300 species of Leguminosae shows a wealth of architectures (6). We studied pod opening in Bauhinia variegate (Fig. 1, A and B), whose pod valves are known to consist of two fibrous layers, oriented roughly at ±45° with respect to the pod’s longitudinal axis [supporting online material (SOM) text and fig. S1, x-ray scattering measurements and fig. S2]. It turns out that the mere presence of two layers that shrink in perpendicular directions is sufficient to drive the flat-to-helical transition in pod opening.

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As a proof of concept, we constructed a mechanical analog of the Bauhinia pod that copies its geometry. We stretched two identical thin latex sheets uniaxially by the same elongation factor. We then glued one on top of the other in perpendicular directions (Fig. 1C). Once released from the external stretching, the composite object underwent the same shrinkage profile as the Bauhinia valve: The two layers shrunk uniaxially in perpendicular directions. Elongated strips cut out from the composite sheets were found to curl into helical configurations (Fig. 1F).

It is well known that uniaxial shrinkage of one layer in a bilayered sheet induces curvature, which accommodates the difference in length of the two layers. When the sheet is thin enough, it bends accordingly. Strips with a single intrinsic curvature have been studied extensively; large sheets adopt a cylindrical configuration, whereas narrow strips adopt helical configurations that are “cut from a cylinder” (7–10).

A qualitatively different situation occurs when two layers shrink perpendicularly. In such a case, the sheet “wants” to bend in two opposite and perpendicular directions—that is, they locally assume a saddle-like configuration (Fig. 1, D and E). This tendency to bend into a saddle creates a metric incompatibility: On the one hand, the in-plane rest lengths between surface elements require the surface to be metrically flat, to have zero Gaussian curvature. On the other hand, a saddle-like configuration has nonzero Gaussian curvature. Thus, the body cannot assume a stress-free configuration and must contain residual stress. The key feature of residually stressed bodies is that their configuration changes when they are dissected into smaller parts, reflecting the release of stored elastic energy.

Geometrically incompatible materials are well known in the mechanical literature and are described by the theory of “incompatible elasticity” (11, 12). A mathematical theory for incompatible thin sheets was recently derived in (13, 14). In this theory, thin sheets are modeled as two-dimensional surfaces. The configuration of a surface is fully characterized by two tensors: a metric tensor, a, which contains all information about lateral distances between points, and a curvature tensor, b, which contains information about the local curvature. An elastic sheet is characterized by two additional tensors that are intrinsic to its structure: a reference metric, a¯, and a reference curvature, b¯. These tensors represent the lateral distances and curvatures that would make the sheet locally stress-free. They are determined by the swelling/shrinkage distribution within the sheet. When a≠a¯, the sheet contains in-plane stretching deformations, and when b≠b¯, the sheet contains bending deformations. The total elastic energy is the sum of the stretching and bending energies, E=Es+EB∼t∫[(1−v)|a−a¯|2+vtr2(a−a¯)]dS+t3∫[(1−v)|b−b¯|2+vtr2(b−b¯)]dS(1)where t is the thickness of the sheet, dS is the infinitesimal surface element, and v is the Poisson ratio of the material. The ∼ relation stands for proportionality, where the coefficients are constant material parameters. Given the material parametersa¯ andb¯, the system is postulated to minimize energy by choosing a configuration with optimal a and b.

Naïvely, one would think that the sheet can eradicate its elastic energy by settling on a configuration that satisfies a=a¯ and b=b¯. This, however, is not always possible: Every surface in space must fulfill a set of constraints—the Gauss-Codazzi-Mainardi equations—that connect between the metric and curvature tensors (15). If the local shrinkage is such that the resultinga¯ andb¯ do not satisfy these constraints, there cannot be any configuration in which a=a¯ and b=b¯ (SOM text). This is the mathematical formulation of the metric incompatibility discussed above. In such a case, the configuration of minimum energy is determined by a competition between the stretching and bending energies.

The first step in modeling the shaping of Bauhinia pods (or the mechanical model) is to determine the reference tensors a¯ and b¯. Each of the fibrous layers that construct the valve is separately planar. This implies that the reference metric is Euclidean and can be written in the forma¯=(1001)(2)Measurements on segments of Bauhinia pods cut along the two directions of principal curvature exhibit oppositely curved arcs of approximately the same radius. This validates our assertion that b¯ exhibits at each point two principal curvatures of the same magnitude but of opposite signs. Adopting coordinates (x, y) that are parallel to the directions of principal curvature, we obtain a saddle-like (hyperbolic) reference curvature tensor of the formb¯=(κ000−κ0)(3)where κ₀ is the magnitude of the reference curvature. In Bauhinia pods, the directions of principal curvature form an angle of 45° with respect to the longitudinal axis. In strips cut out from latex sheets, this angle can be determined at will. A strip is characterized by the following parameters: a thickness t, a width w, a length L, an induced curvature κ₀, and an angle θ between the strip’s longitudinal axis and a direction of principal curvature.

A quantitative study of strips cut out from Bauhinia pods and from latex sheets at an angle of θ = 45° exhibits two different regimes: Wide strips adopt a configuration close to a cut from a cylindrical envelope (“cylindrical helices”), whereas narrow strips exhibit a “pure twist,” where the strip’s centerline is straight (“twisted helices”). At the coarsest level, we may characterize the shape of the strips by two parameters: the radius r and the pitch p of the strip’s mid-curve. Figure 2 shows results for a collection of strips cut from both Bauhinia pods and latex sheets. For each sample, we measured the parameters t, w, r, p, and κ₀ (κ₀ was estimated by measuring the curvature of very thin strips cut along principal directions). We plotted a dimensionless pitch p˜=pκ0 and a dimensionless radius r˜=rκ0 as functions of a dimensionless width w˜=wκ0/t. With this scaling, both pod tissue and latex data were found to collapse to a single functional behavior. We found a critical transition between the two regimes. For w˜<wcrit≈2.5, the radius r˜ is nearly 0, exhibiting a forward bifurcation at the critical value. The pitch p˜ attains a maximum at a value of w˜ of about w˜crit.

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The theoretical study consists of minimizing the energy (Eq. 1) of a strip with reference tensors a¯ andb¯ as given by (Eq. 2) and (Eq. 3). Dimensional analysis indicates that the stretching energy scales like tw(κ₀w)⁴, whereas the bending energy scales like t3wκ02. Thus, one expects the occurrence of a stretching-dominated regime for large w and a bending-dominated regime for small w. The transition between the two regimes is expected to occur when the two energies are of comparable magnitude, when w=t/κ0, or w˜≈1.

When w˜≫1, the stretching term dominates, and a≈a¯=I, except within a bending-dominated boundary layer of characteristic width t/κ0. Thus, b aims to optimally approximateb¯, under the constraints that the metric be Euclidean. At equilibrium, one of the principal curvatures is found to coincide with the reference curvature (up to a multiplicative factor depending on the Poisson ratio v), whereas the second principal curvature is 0, b=(1−v)(κ0000) or b=(1−v)(000−κ0).

The two solutions are equally stable. The configuration looks like strips cut out from the envelope of a cylinder, with a centerline that is a helix of radius and pitch (Fig. 2, right inset):

Such cylindrical helices are similar to configurations observed in strips with a single intrinsic curvature (7–10). Yet, there is a fundamental difference between the two cases because in the case of a single curvature,a¯ andb¯ are compatible, and there is no mechanical frustration. In the presently studied case,a¯ andb¯ are incompatible, resulting in stored mechanical energy. Two manifestations of this geometric frustration are the bistability of the equilibrium and the configurational change occurring when the strip is dissected.

When w˜≪1 (narrow strips), the equilibrium configuration is bending-dominated, b≈b¯. Now, it is the metric a that best approximates (in a mean square sense) the reference metrica¯, among all metrics compatible withb¯. It can be shown that at equilibrium, a≈a¯ in the vicinity of the mid-curve, which is all that is needed for predicting the shape of the mid-curve.

The configurations in the narrow regime are twisted helices and are similar to configurations found in (stretch-free) liquid membranes (16, 17). Once again, there is a fundamental difference between the two cases because twisted helices in liquid membranes are stress-free. In the present study, they result from a compromise between stretching and bending energies.

The radius and pitch in the narrow regime are r=0 and p=2πκ0(5)The limiting values (Eq. 4) and (Eq. 5) are represented in Fig. 2 as dotted lines and are found to agree with measured data. Lastly, in the intermediate regime both stretching and bending are comparably important, and one has to minimize the total energy with respect to both a and b. The calculation of a global minimizer can be done numerically, as described in the SOM text. The predicted dimensionless radius and pitch are displayed as functions of w˜ in Fig. 2B (solid line) and show good agreement with experimental data.

The transition between the two limiting regimes—wide versus narrow strips—is also manifested in the geometry of the surface. One way to characterize a surface is via the local values of its mean curvature, H, and Gaussian curvature, K, which are the respective mean and product of the two principal curvatures. In Fig. 3, we plot H² and K as function of the distance from the edge for a collection of latex strips differing only in width. These data were extracted by measuring the surface topography of the latex strips with optical profilometry. Each scan provides us with a map of the surface height, from which we computed the local values of H and K by discrete differentiation. We then averaged H and K over points that are at equal distance from the edge of the strip. Narrow strips, w˜≤w˜crit, are approximately minimal surfaces with mean curvatures of uniformly 0 as dictated byb¯. Above the critical width, H is no longer uniformly 0. For wide strips, H reaches the predicted value of (1 – v)κ₀ far enough from the edge (dotted line) while remaining 0 at the edge, with a boundary layer connecting the two regions (14). For wide strips, K ≈ 0 as determined by (the Euclidean)a¯, whereas for narrow ones, K tends toward the asymptotic value of −κ02 (–1 in the units used in the plot) as w → 0. Thus, our model is capable of predicting not only the global shape of the helical strips but their surface topography as well.

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A natural question is whether the transition between the narrow and wide regimes is continuous or not; more precisely, is the change in equilibrium configuration continuous as w˜ crosses the critical value of w˜crit? Both our calculations and measurements indicate that the transition is continuous, although sharp. For example, Fig. 3A shows that as w˜ approaches from above the value of w˜crit, the mean curvature tends continuously to zero.

Our theoretical and mechanical models capture well the behavior of Bauhinia valves and link quantitatively between the microscopic structure and the macroscopic configurations. Yet, they have an even wider scope of applicability. We next studied the dependence of the equilibrium configuration on the angle θ. Configurations of latex, Bauhinia, and theoretically predicted strips, respectively, are shown in Fig. 4, A to C, for fixed width and varying θ. The dependence of the radius and the pitch on the angle for latex strips are shown in Fig. 4D. The parameters correspond to the “narrow” regime, w˜<w˜crit. For θ = 0° and θ = 90°, the strips’ longitudinal axis coincides with a direction of principal curvature. As a result, the shapes are circular, with curvature approximately equal to the corresponding principal curvature. As θ is increased from 0° to 45°, the pitch increases, and the radius decreases, until a pure twist is obtained at 45°. The same behavior is obtained when the angle decreases from 90° to 45°. The handedness of the configurations is reversed for strips cut at angles between 90° and 180° (not shown). This demonstrates that the handedness of the helices is not a bulk material property but rather depends on the global geometry of the sample.

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We have also conducted an experimental study of the dependence of the equilibrium configuration on the angle θ in the “wide” regime, w˜<w˜crit (data not shown). In this regime, two bistable configurations emerge, each being flat along another direction of principal curvature. Both states have the same handedness and the same bulk energy density, so that the globally stable state is determined by boundary effects. The two metastable states are indistinguishable at θ = 45°. For θ = 0°,90°, the two bistable states correspond to the two states of tape springs, which were studied in (18), for an example.

The only inputs in our model are a¯ and b¯ and not any specific swelling/shrinkage profiles. Many different combinations of anisotropic layered growth would lead to the same intrinsic geometry—the samea¯ andb¯. For example, the same form of b¯ is achieved if the angle between the swelling/shrinkage directions is not 90° or even under isotropic growth of one layer together with an anisotropic growth of the other (SOM text and figs. S3 and S4). Thus, the mechanism we have presented is quite general and is expected to manifest in different systems with different internal structures, consistently with the diverse valve architectures reported in (6).

The present study is potentially applicable to systems of different scales and contexts. For example, many organic and inorganic macromolecules are found in different helical configurations (9, 16, 17, 19). In our model, the only nontrivial property of the sheet is its saddle-like reference curvature tensor. It can therefore be relevant to any sheet with such spontaneous curvatures, including monolayers and sheets with different microstructures. Macromolecules made of chiral molecules are often modeled by introducing a term in the bending energy that accounts for a spontaneous twist. It can be shown that geometrically, a sheet with a spontaneous twist is equivalent to one with hyperbolic reference curvature tensor (SOM text). Indeed, a transition qualitatively similar to the one presented in Fig. 2 was predicted in (20, 21). After translating the parameters of (21), we found their predicted value for w˜crit to be similar to ours. It would be interesting to see whether our model, which accounts for a wider range of intrinsic geometries, could describe configurations of macromolecules that are currently not well understood (19). Lastly, mechanisms such as the one studied here can be implemented by using inducible responsive materials. Such bio-mimicking designs have an applicative potential as soft actuators (for a proof of concept, see movies S1 and S2).