Origami enables the creation of three-dimensional structures from flat two-dimensional shapes. While the traditional Japanese art uses paper as its medium, materials scientists are employing this concept to construct three-dimensional structures from other materials but so far only with flat surfaces. Now, scientists from Delft University of Technology (TU Delft) have devised a novel strategy that enables the creation of curved surfaces.

Natural and engineered materials like sponges, foams, and bone tissue boast internal and external curved surfaces. In mathematical terms, a surface that is curved along its two principal or orthogonal directions, forming a sphere or saddle shape, is known as an intrinsically curved surface. One of the most important types of intrinsically curved surfaces are minimal surfaces, which are found widely in natural systems from soap films to butterfly wings. Structures containing minimal surfaces show remarkable physical properties including very large surface-to-volume ratios, high yield stresses, low elastic moduli, high fatigue resistance, and bone-mimicking transport properties.

“Since classic origami does not have a solution to the problem [of folding minimal surfaces from a flat state], we have invented an alternative approach that enables us to start from a flat state and self-fold intrinsically curved surfaces including minimal surfaces,” explains Amir A. Zadpoor, senior author of the study.

The team’s approach creates curved minimal surfaces from flat structures comprising rigid foldable frames with pre-strained sheets. Rather like a pop-up tent, the flat design folds itself into a three-dimensional structure with curved surfaces. The structures boast a particular type of curved surface known as a triply periodic minimal surfaces (TPMS), which extend infinitely in three directions and divide the space into two, intertwined labyrinths. By designing foldable patches that can be connected together, the team are able to build up larger structures with very complex architectures. Not only did the scientists model how these structures could be formed, they also created actual three-dimensional objects from 3D-printed foldable polymer frames with pre-strained latex membranes.

“The novelty of our approach lies in the idea of constructing these complex morphologies ‘patch-by-patch’,” says Sebastien J.P. Callens, first author of the study. “By creating a network of identical, foldable patches that simultaneously fold into their final shape [we can create] a minimal surface structure. Our approach enables topologically more complex structures to be formed in a surprisingly simple fashion.”

Image shows a folding sequence for a TPMS structure based on a type of minimal surface called ‘Schwarz D’. Image credit: Sebastien J.P. Callens.
Image shows a folding sequence for a TPMS structure based on a type of minimal surface called ‘Schwarz D’. Image credit: Sebastien J.P. Callens.

As well as the scalability of the approach, which allows the patching together of multiple types of TPMSs into larger complex structures, the other advantage of 2D-to-3D fabrication over direct 3D fabrication is that the flat starting materials are very amenable to surface functionalization.

“You could use surface functionalizing processes, such as nanolithography,” points out Callens. “After functionalizing the surface, you could fold it into the desired 3D structure using an origami-like approach enabling the fabrication of metamaterials with vastly augmented functionalities.”

The scientists believe that their simple technique could be applied to a wide range of materials, as long as the recipe of a rigid, foldable frame combined with a pre-strained flexible material is followed. Many applications could benefit from the complex porous geometries offered by TPMS structures.

“One example are bone-substituting biomaterials used in the treatment of large bone defects,” points out Callens. “For these porous biomaterials, it is desirable to have both a complex porous geometry mimicking natural bone tissue (such as TPMS-based geometries) and also nanoscale surface texture to promote tissue regeneration and prevent infections. Our work is the first step towards such applications using TPMS-based geometries.”

The team are now working on ways to realize small-scale, self-folding TPMS structures practically. While they have achieved macroscale prototypes, the researchers want to move towards more extensive small-scale structures, which might require new developments in 3D printing and small-scale fabrication techniques. Callens adds that the team also wants to explore how cells respond to these foldable biomaterial structures.

Reference: S.J.P. Callens et al., Applied Materials Today 15 (2019) 453-461, https://doi.org/10.1016/j.apmt.2019.03.007