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# Shape-shifting structured lattices via multimaterial 4D printing

Edited by John A. Rogers, Northwestern University, Evanston, IL, and approved September 3, 2019 (received for review May 27, 2019)

## Significance

Thin shape-shifting structures are often limited in their ability to morph into complex and doubly curved shapes. Such transformations require both large in-plane expansion or contraction gradients and control over extrinsic curvature, which are hard to achieve with single materials arranged in simple architectures. We solve this problem by 4-dimensional printing of multiple materials in heterogeneous lattice designs. Our material system provides a platform that achieves in-plane growth and out-of-plane curvature control for 4-material bilayer ribs. The lattice design converts this into large growth gradients, which lead to complex, predictable 3-dimensional (3D) shape changes. We demonstrate this approach with a hemispherical antenna that shifts resonant frequency as it changes shape and a flat lattice that transforms into a 3D human face.

## Abstract

Shape-morphing structured materials have the ability to transform a range of applications. However, their design and fabrication remain challenging due to the difficulty of controlling the underlying metric tensor in space and time. Here, we exploit a combination of multiple materials, geometry, and 4-dimensional (4D) printing to create structured heterogeneous lattices that overcome this problem. Our printable inks are composed of elastomeric matrices with tunable cross-link density and anisotropic filler that enable precise control of their elastic modulus (*E*) and coefficient of thermal expansion

Shape-morphing structured systems are increasingly seen in a range of applications from deployable systems (1, 2) and dynamic optics (3, 4) to soft robotics (5, 6) and frequency-shifting antennae (7), and they have led to numerous advances in their design and fabrication using various 3-dimensional (3D) and 4-dimensional (4D) printing techniques (8, 9). However, to truly unleash the potential of these methods, we need to be able to program arbitrary shapes in 3 dimensions (i.e., control the metric tensor at every point in space and time), thus defining how lengths and angles change everywhere. For thin sheets, with in-plane dimensions that are much larger than the thickness, this is mathematically equivalent to specifying the first and second fundamental forms of the middle surface. These quadratic forms describe the relation between material points in the tangent plane and the embedding of the middle surface in 3 dimensions and thus, control both the intrinsic and extrinsic curvature of the resulting surface (10, 11). From a physical perspective, arbitrary control of the shape of a sheet requires the design of material systems that can expand or contract in response to stimuli, such as temperature, humidity, pH, etc., with the capacity to generate and control large in-plane growth gradients combined with differential growth through the sheet thickness (12, 13). Such systems are difficult to achieve experimentally; hence, most current shape-shifting structures solutions rarely offer independent control of mean and Gaussian curvatures (14, 15). We address this challenge by 4D printing a lattice design composed of multiple materials.

Beginning at the material level, we created printable inks based on a poly(dimethylsiloxane) (PDMS) matrix, an elastomeric thermoset that exhibits a large operating temperature window and a high thermal expansion coefficient (16). Although the inks are printed at room temperature, the broad range of polymerization temperatures for PDMS enables us to cure the resulting structures at much higher temperatures. On cooling to room temperature, these cured matrices achieve maximal contraction, hence transforming into their deployed states. The same base elastomer is used in all inks to facilitate molecular bonding between adjacent ribs and layers. To create inks with reduced thermal expansivity, we fill the elastomer matrix with short glass fibers (*A*–*C*) (8, 17). To impart rheological properties suitable for direct ink writing (i.e., a shear yield stress, shear thinning response, and plateau storage modulus) (*SI Appendix*, Fig. S1), we add fumed silica (20 to 22% wt/wt) (*SI Appendix*) to these ink formulations. As an added means of tuning their coefficient of thermal expansion α and their elastic modulus E, we vary the cross-link to base weight ratio (x-link:base by weight) (Fig. 1*D* and *SI Appendix*) within the elastomeric matrices. Unlike prior work, we generate a palette of 4D printable inks that span a broad range of properties (i.e., α ranges from *E* and *F* and *SI Appendix*, Figs. S2 and S3).

To go beyond the geometric limitation associated with uniform isotropic or anisotropic sheets, we used multimaterial 4D printing to first create simple bilayers (*SI Appendix*, Fig. S4*A*), the basic functional unit of our multiplexed bilayer lattices. The curvature response of these bilayers to a temperature change *SI Appendix*. We use Eq. **1** to delineate a space of dimensionless curvature increments *SI Appendix*, Fig. S4*B*). The curvature change of our experimental bilayers is reversible and repeatable as demonstrated by thermal cycling experiments (*SI Appendix*, Fig. S4*C*). However, these simple bilayer elements alone do not provide a path to significantly altering the midsurface metric, which is necessary for complex 3D shape changes. For example, the maximum linear in-plane growth that can be achieved with this set of materials, for

To overcome this limitation, we arranged the bilayers into an open cell lattice (19) via multimaterial 4D printing (Fig. 2 *A*–*C*). We consider this lattice a mesoscale approximation of the underlying continuous surface, in which an average metric can now be rescaled with significantly larger growth factors than the largest thermally realizable linear growth of the constituent materials (19, 20). Specifically, if the initial distance between the lattice nodes is denoted **1** leads to a new distance between lattice nodes L. The linear growth factor, *SI Appendix*). In Fig. 2 *B* and *C*, we show **1**). The measured linear growth for the different **2** (Fig. 2*C*) and demonstrates a tremendous range from *SI Appendix*, Fig. S5) reveals that, for *SI Appendix*).

While this homogeneous lattice design can achieve an isotropic rescaling of the Euclidean metric, inducing intrinsic (Gaussian) curvature in an initially planar sheet requires spatial gradients of the metric tensor along the surface. To achieve this, we used a heterogeneous lattice in which the initial sweep angle of every rib is considered an independent degree of freedom and is, therefore, indexed within the lattice (Fig. 2*D*). From a conformal map of the desired target shape to the plane, we can compute the required growth factor for each rib and invert Eq. **2** to find the corresponding value of *SI Appendix*). With this approach, we theoretically show the maximum possible opening angle of a grown spherical cap as a function of *SI Appendix*, Fig. S6, demonstrating the theoretical capability of this approach to create a flat lattice that can morph into a complete sphere when *SI Appendix*). We test the efficacy of this approach by transforming flat, square lattices into freestanding spherical caps (Fig. 2 *D* and *E* and *SI Appendix*, Fig. S7) and saddles (*SI Appendix*, Fig. S8). As a preview of multishape possibilities (polymorphism), we leveraged the fact that these PDMS matrices swell when exposed to a variety of solvents (21). We show that a planar lattice programmed to transform into a spherical cap (positive Gaussian curvature) through a negative temperature change can also be transformed beyond its printed configuration to adopt a saddle-shaped geometry (negative Gaussian curvature) when immersed in a solvent (*SI Appendix*, Fig. S9 and Movie S2). Using multimaterial 4D printing, we produced different spherical caps by parametrically varying the number of printed filaments along the width and height of the lattice ribs (*SI Appendix*, Fig. S7*A*). At a scaling level, the nondimensional sagging deflection *SI Appendix*). Our experiments confirm that sagging lattices (i.e., when *SI Appendix*, Fig. S7*A* and Table S1). For the nonsagging structures *SI Appendix*)*SI Appendix*, Fig. S7*B*). However, increasing the slenderness of the ribs by either increasing *SI Appendix*, Fig. S7*B* and Table S1) as expected.

To fully control 3D shape requires the ability to program both the intrinsic curvature and extrinsic curvature. We achieved this by introducing multiplexed pairs of bilayers as ribs within heterogeneous lattices that exploit the large range of α and E values exhibited by our ink palette. Specifically, 4 different materials are used in the cross-sections of each rib, which allows us to control expansion across their thickness and width according to Eq. **1**. We can direct normal curvature up or down by interchanging the top and bottom layers and discretely control its magnitude by transposing the materials in the cross-section as shown in Fig. 3.

Altogether, our multiplexed bilayer rib lattice yields a shape-changing structural framework with 2 significant novelties compared with other motifs. First, these lattices exhibit a substantial amount of local linear in-plane growth (40% growth to 79% contraction as currently demonstrated; 57% times growth to 100% contraction in theory), which can be independently varied across the lattice as well as in each of the 2 orthogonal directions of the lattice. This capability can be generalized to lattices of different scales, materials, and/or stimuli. Second, the out-of-plane bending control reduces elastic frustration, which simplifies their inverse design and expands the range of shapes that can be achieved compared with prior work (8, 22).

To test the capability of our approach to create dynamic functional structures, we innervate these freestanding lattices with liquid metal features composed of a eutectic gallium indium ink that are printed within selected ribs throughout the lattice (Fig. 4*A* and *SI Appendix*) to yield a shape-shifting patch electromagnetic antenna (Fig. 4 *A* and *B*). We printed a 2-material lattice *B* and Movie S3). This trend continues until the functional lattice reaches a critical temperature of ∼150 °C, where internal forces can no longer support its weight or provide the appropriate out-of-plane buckling force (i.e., it rapidly returns to its flat-printed geometry). As expected, we observe a sharp increase in the fundamental resonant frequency when this transition occurs. While shape-shifting patch antennae are of interest for wireless sensing and dynamic communication, the ability to integrate conductive ribs could be harnessed for other shape-shifting soft electronic and robotic applications.

To test the capability of our approach to create shapes with complex geometrical features on multiple scales, we printed a planar lattice that transforms into a human face (Fig. 5 and Movies S4 and S5). We chose to create a simulacrum of the face of C. F. Gauss, father of modern geometry, starting with a 3D target surface mesh (Fig. 5*A*) generated from a painting of Gauss through a machine learning algorithm (23) (*SI Appendix*). We conformally project the face to the plane and discretize the planar projection using a lattice with **2** (*SI Appendix*, Fig. S10). Furthermore, since large parts of the face have almost zero Gaussian curvature, we can rely on our multiplexed bilayer technique to influence the normal curvature independent of the metric transformation (Fig. 5 *B* and *C*). The normal curvature changes sign most prominently near the eye sockets, where the mean curvature undergoes a sign change compared with the rest of the face. With these choices, the lattice dimensionless sagging parameter is *D*) in a salt water tank, in which density is only slightly less than the average lattice density (*SI Appendix*), prevents sagging yet allows the lattice to sink to the bottom of the tank (Movie S5). After its shape transformation, the printed lattice shows a clear correspondence to the target geometry (Fig. 5*E*), where the more prominent features of the face, such as the nose, chin, and eye sockets; the finer features associated with the lips and cheeks; and the subtle curvature transitions are all visible. This multiplexed bilayer approach is successful in directing spherical regions up (nose and chin) or down (eye sockets) as required and achieves significant mean curvature near the forehead and around the perimeter of the face, where the Gaussian curvature is small. To test the accuracy of our printed lattice, we obtained a 3D reconstruction of the transformed face (Fig. 5*F*) using a laser-scanning technique (*SI Appendix*). By fitting the scanned data to the target surface (*SI Appendix*), we can compute the smallest distance from each point on the scanned face to the target shape. We use this distance as an error metric and normalize it as *G*). The distribution of the normalized error (Fig. 5*H*) exhibits a 95% confidence interval within

The lattice designs described here are applicable to a wide range of materials, length scales, and stimuli. For example, a lattice with ribs of equal widths

The lattice architecture can be extended to larger scales of freestanding structures by designing materials with similar α and larger E (and/or smaller ρ) so that, to first approximation, the lattice-based structure would still provide the desired metric changes. Moreover, this approach can be applied to smaller microscale structures by utilizing smaller fillers and nozzle diameters. As these lattice architectures are agnostic to the mode of stimulus, one can imagine designing general lattices with local, variable actuation for myriad materials systems and stimuli [e.g., pneumatics (24, 25), light (26), temperature (27), pH (28), solvent (8, 29), electric field (6), or magnetic field (30)] to transform the same lattice reversibly and dynamically into one or multiple complex shapes.

Our inverse design procedure can be broadened not only to include arbitrary geometries of the underlying lattice but also, to multiplexed bilayers with different material compositions, each of which can be independently varied in space. While we have restricted ourselves to conformal maps and square lattice cells here, our method can be generalized for other projections, spatially varying cell sizes, different tessellations of the plane, and other 2-dimensional (2D) and 3D open cell lattice designs (31⇓–33). By using temperature as stimulus, these lattices can be repeatedly and rapidly (as quickly as *SI Appendix*, Table S2) actuated in a continuous, well-controlled, and predictable manner. Altogether, our multimaterial 4D printed lattice provides a versatile platform for the integrated design and fabrication of complex shape-morphing architectures for tunable antennae, dynamic optics, soft robotics, and deployable systems that were previously unattainable.

## Acknowledgments

We thank L. K. Sanders and R. Weeks for assistance with manuscript preparation and useful discussions. We acknowledge support from the NSF through Harvard Materials Research Science and Engineering Center Grant DMR-1420570, NSF Designing Materials to Revolutionize and Engineer our Future Grant 15-33985, and Draper Laboratory. W.M.v.R. thanks the Swiss National Science Foundation for support through a postdoctoral grant and the American Bureau of Shipping for support through a Career Development Chair at Massachusetts Institute of Technology. J.A.L. thanks GETTYLAB for their generous support of our work. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the NSF.

## Footnotes

↵

^{1}J.W.B. and W.M.v.R. contributed equally to this work.- ↵
^{2}To whom correspondence may be addressed. Email: jalewis{at}seas.harvard.edu or lmahadev{at}g.harvard.edu.

Author contributions: J.W.B., W.M.v.R., J.A.L., and L.M. designed research; J.W.B., W.M.v.R., C.L., M.N.H., R.L.T., and A.K. performed research; J.W.B., W.M.v.R., C.L., M.N.H., R.L.T., A.K., and L.M. contributed new reagents/analytic tools; J.W.B., W.M.v.R., J.A.L., and L.M. analyzed data; and J.W.B., W.M.v.R., J.A.L., and L.M. wrote the paper.

Conflict of interest statement: J.A.L. is a cofounder of Voxel8, Inc., which focuses on 3-dimensional printing of materials.

Data deposition: STL files for the 3-dimensional surface mesh used as target shape for Gauss’ face and the conformal projection of this face to the plane have been deposited at https://github.com/wimvanrees/face_PNAS2019. The numbering of the faces is consistent between the 2 files, which provides the necessary information to reconstruct the mapping between the 2 shapes.

This article is a PNAS Direct Submission.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1908806116/-/DCSupplemental.

Published under the PNAS license.

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