Most natural materials exhibit positive thermal expansion, but large thermal deformation causes serious damage in various engineering applications, especially in precision mechanical systems such as space optical systems, adaptive interconnect components, MEMS, and so on. Mechanical metamaterials can achieve extraordinary thermal expansion characteristics, such as negative thermal expansion and near-zero thermal expansion response, through their special periodic microstructures. However, limited by very few available cell topologies and complicated non-linear structural deformation, most existing thermal expansion metamaterials can only achieve orthogonally isotropic negative/zero/positive thermal expansion (NTE/ZTE/PTE) within a mild range, especially the three-dimensional (3D) ones, which greatly restricts their potential applications in space optical systems and micro-electromechanical systems.

        Recently, the team of Professor Yan Chen from the school of mechanical engineering of Tianjin University, in cooperation with Professor Guowu Wei from University of Salford and Professor Zhong You from University of Oxford, proposed a series of isotropic and orthotropic three-dimensional thermal expansion metamaterials based on a single-degree-of-freedom kirigami octahedral structure. This innovation achieved an exceptionally large range of tunable thermal expansion coefficients (-2354.3 to 3006.7 ppm/°C) at the three-dimensional scale and enabled independently programmable negative, zero, and positive thermal expansion responses in three orthogonal directions within a single metamaterial. This research was published online on October 22, 2024, in Advanced Materials. The corresponding authors of the paper are Professor Yan Chen and Professor Jiayao Ma from Tianjin University, with postdoctoral fellow Yuanqing Gu and doctoral student Zhibo Wei as co-first authors. The study was supported by the National Natural Science Foundation of China and New Cornerstone Science Foundation through the XPLORER PRIZE.
        Researchers started with the truncated octahedral kirigami unit cell and analyzed its single-degree-of-freedom kinematic characteristics. They found that the two sets of dihedral angles maintain a linear relationship throughout the folding process. The isotropic unit cell has a constant Poisson's ratio of -1, enabling uniform contraction and expansion in three directions. Building on this, by breaking the symmetry of the kirigami polyhedra, they obtained the design of an orthotropic single-degree-of-freedom origami unit cell. By sharing square panels, both isotropic and orthotropic unit cells were extended in three directions to form a metamaterial array (Figure 1).

Fig. 1 Isotropic and orthotropic kirigami cells design and their tessellated array

        Having obtained the polyhedral cells with isotropy and orthotropy, we subsequently develop thermal expansion metamaterials by replacing the folding creases of the cells with curved bi-material strips comprising of two layers with high and low coefficient of thermal expansion (CTE) respectively. This design leverages the differential thermal expansion properties of the bi-material strips. When heated, these strips bend either positively or negatively, allowing for the creation of unit cells with negative thermal expansion and positive thermal expansion responses. It was proved that the bending angle of the bi-material strips under heat is independent of their initial curvature. By relying on the linear variation of the movement angles and ensuring that other design parameters are the same (such as the thickness t and arc length L of the bi-material strips), the bi-material strips can achieve the same degree of bending deformation upon heating, resulting in a coordinated thermal deformation consistent with the mechanism design. Furthermore, by adjusting the structural parameters of the NTE and PTE cells, a wide range of programmable thermal expansion coefficients can be produced. Experiments were conducted using bimetallic strips (Ni36/Mn75Ni15Cu10) and aluminum plates (substrate material) to create cells with NTE (video 1) and PTE (video 2) responses, respectively, validating the reliability of the theoretical model.

Fig. 2 NTE and PTE cells design and parameter analysis

Video 1 NTE cell

Video 2 PTE cell

        An Ashby plot of CTE versus specific modulus is illustrated in Figure 3, incorporating this work and previously reported 3D metamaterials, where the natural and engineering materials are also included for reference. The designs reported in this paper can achieve the ultra-wide CTE variation by regulating parameters such as the thickness of the bi-metal strips, significantly expanding the range to a maximum of -2354.3 to 3006.7 ppm/°C, can also offer widely tunable specific modulus.

Fig. 3 Ashby plot of 3D thermal expansion metamaterials

        As shown in Figure 4, based on the programmability of the NTE cells, zero thermal expansion (ZTE) can be realized by cancelling out the normal thermal expansion of the substrate panels through folding of the bi-metal creases. Alternatively, the ZTE metamaterial can also be constructed by a 3D hybrid tessellation of PTE and NTE cells, by producing zero boundary displacement in all three orthogonal directions. A 2×2×2 ZTE metamaterial was produced for experimental verification, as shown in video 3,with a thermal expansion coefficient of only 4.01 ppm/℃.

Fig. 4 The ZTE metamaterial design

Video 3 ZTE cell and ZTE array/p>

        As shown in Figure 5, taking the orthotropic kirigami polyhedron as a unit cell, we can also construct metamaterials with orthotropic thermal expansion following the similar strategy as the isotropic ones. Similar to the linear angle variation relationship in isotropic unit cells, orthotropic unit cells also exhibit coordinated thermal deformation. Video 4 demonstrates an orthotropic NTE cell design. By combining orthotropic cells with a hybrid method, a programmable orthotropic metamaterial was designed to exhibit PTE, ZTE, and NTE in three orthogonal directions. Video 5 shows the thermal expansion response of a 2×2×2 array, with CTE of 77.3 ppm/℃, -718.8 ppm/℃, and 3.93 ppm/℃ in three orthogonal directions, respectively. For large metamaterial array, the thermal expansion response of a 4×4×4 array was simulated using finite element analysis, with CTE of 144.8 ppm/°C, -304.3 ppm/°C, and 0.27 ppm/°C in the orthogonal directions, respectively.

Fig. 5 The orthotropic metamaterial design

Video 4 Orthotropic cell

Video 5 Orthotropic 2×2×2 array

        To conclude, this study proposed a kinematics-based design framework for thermal expansion metamaterials, and developed a family of 3D programmable metamaterials with isotropic and orthotropic thermal expansion properties based on one-DOF kirigami foldable polyhedrons. Such enriched isotropic and orthotropic thermal expansion properties are tailorable over an ultra-wide range and thus offer potentially powerful options for applications in space-based mirrors, precision instruments, thermal actuators, MEMS, and others that demand operations in a broad range of temperatures.

Gu Y#, Wei Z#, Wei G, You Z, Ma J*, Chen Y*. Kirigami-inspired three-dimensional metamaterials with programmable isotropic and orthotropic thermal expansion. Advanced Materials, 2024, 2411232.
(http://dx.doi.org/10.1002/adma.202411232)