Light as a feather, strong as an ox

Nature has found a way to achieve mechanically efficient materials by evolving cellular structures. Natural cellular materials, including honeycomb (1) (wood, cork) and foamlike structures, such as trabecular bone (2), plant parenchyma (3), and sponge (4), combine low weight with superior mechanical properties. For example, lightweight balsa has a stiffness-to-weight ratio comparable to that of steel along the axial loading direction (5). Inspired by these naturally occurring cellular structures, human-made lightweight cellular materials fabricated from a wide array of solid constituents are desirable for a broad range of applications including structural components (6, 7), energy absorption (8, 9), heat exchange (10, 11), catalyst supports (12), filtration (13, 14), and biomaterials (15, 16).

However, the degradation in mechanical properties can be drastic as density decreases (17, 18). A number of examples among recently reported low-density materials include graphene elastomers (19), metallic microlattices (20), carbon nanotube foams (21), and silica aerogels (22, 23). For instance, the Young’s modulus of low-density silica aerogels (22, 23) decreases to 10 kPa (0.00001% of bulk) at a density of less than 10 mg/cm³ (<0.5% of bulk).

This loss of mechanical performance is because most natural and engineered cellular solids with random porosity, particularly at relative densities less than 0.1%, exhibit a quadratic or stronger scaling relationship between Young’s modulus and density as well as between strength and density. Namely, E/Es ∝ (ρ/ρs)n and σy/σys ∝ (ρ/ρs)n, where E is Young’s modulus, ρ is density, σyis yield strength, and s denotes the respective bulk value of the solid constituent material property. The power n of the scaling relationship between relative material density and the relative mechanical property depends on the material’s microarchitecture. Conventional cellular foam materials with stochastic porosity are known to deform predominantly through bending of their cell walls and struts (24). This type of deformation results in relative stiffness scaling with n = 2 or 3. A number of approaches in recent years have aimed to reduce this coupling between mechanical properties and mass density (5, 17, 18, 20, 25–31). Among these, few fabrication processes are capable of building arbitrary three-dimensional microarchitectures with controlled micro- and nanostructure across a wide range of mass density and material constituents. The desired material properties are thus limited to a narrow density range and specific loading directions.

Improved mechanical properties can arise from a material that contains micro- and nanoscale building blocks arranged in an ordered hierarchy. Among these new designs are metallic microlattices with high recoverability when compressed (20, 26), TiN nanotrusses (32, 33), and ceramic composite trusses (34) that show enhanced fracture toughness of coating materials when the thickness of coating materials is reduced to the nanoscale.

We report a group of ultralight mechanical metamaterials that maintain a nearly linear scaling between stiffness and density spanning three orders of magnitude in density, over a variety of constituent materials. We use the term “mechanical metamaterials” to refer to materials with certain mechanical properties defined by their geometry rather than their composition. The materials described here are highly ordered, nearly isotropic, and have high structural connectivity within stretch-dominated, face-centered cubic (fcc) architectures. The ultralow-density regime is accessed by fabricating microlattices with critical features ranging from ~20 μm down to ~40 nm. The densities of samples produced in this work ranged from 0.87 kg/m³ to 468 kg/m³, corresponding to 0.025% to 20% relative density.

A stretch-dominated unit cell structure, consisting of b struts and j frictionless joints and satisfying Maxwell’s criterion, M = b – 3j + 6 > 0, is substantially more mechanically efficient—with a higher stiffness-to-weight ratio (defined as E/ρ)—than its bend-dominated counterpart. This is attributed to its struts carrying load under compression or tension rather than bending (17). A fundamental lattice building block of this type is the octet-truss unit cell (Fig. 1A), whose geometric configuration was proposed by Deshpande et al. (35). The cell has a regular octahedron as its core, surrounded by eight regular tetrahedra distributed on its faces (fig. S1). All the strut elements have identical aspect ratios, with 12 solid rods or hollow tubes connected at each node. The cubic symmetry of the cell’s fcc structure generates a material with nearly isotropic behavior (36). The relative density of such octet-truss unit cells can be approximated by ρ = 26.64(d/L)2 (35), where Land d are the length and diameter of each beam element. On the macroscale, under uniaxial compressive loading, the relative compressive stiffness and yield strength of these structures theoretically show linear scaling relationships: E/Es ∝ (ρ/ρs) and σ/σs ∝ (ρ/ρs) (35). A cubic lattice is readily constructed by periodic packing of the unit cell along its three principal directions (Fig. 1, B and C) (37, 38). Alternate orientations of the bulk lattice relative to the unit cell’s principal axes can likewise be constructed (fig. S2), with the fundamental tessellation of space by the unit cell remaining the same.

Goal of this figure

This figure depicts the proposed structure, a lattice based on a stretch-dominated unit cell and compares it with another existing structure, a lattice based on a bend-dominated unit cell.

Panels A and D

A comparison of the mechanical responses of the two structures:

The ultralight and ultrastiff material proposed by the authors is based on an octet-truss unit cell. When loaded, the struts of this structure will mainly be subjected to compression or tension. On the other hand, the structure proposed previously by another team is based on a tetrakaidecahedron unit cell. The struts of this structure carry an applied load under bending.

B,C & E,F

A comparison of the lattices obtained from these two unit cells:

A large lattice is constructed by periodic packing of the cubic unit cell along its three principal directions (B & E). Larger lattices were also obtained with a different packing of unit cells (C & F).

To study how the loading direction and lattice orientation of an octet-truss lattice affects its E-ρ scaling relationship, we analyzed, fabricated, and tested them in a variety of orientations (39) (figs. S1 to S5). In addition to these stretch-dominated lattices, as a point of comparison, a bend-dominated tetrakaidecahedron unit cell (40, 41) of the same size scale was generated and the corresponding cubic-symmetric foams (known as Kelvin foams) were fabricated with a variety of densities (Fig. 1, D to F).

The fabrication of these microlattices is enabled by projection microstereolithography, a layer-by-layer additive micromanufacturing process capable of fabricating arbitrary three-dimensional microscale structures (42, 43). In contrast to other three-dimensional (3D) rapid prototyping methods such as 3D printing and ultraviolet (UV) projection waveguide systems (44), this type of fabrication technology is ideal for 3D lattices with high structural complexity and with feature sizes ranging from tens of micrometers to centimeters. By combining projection microstereolithography with nanoscale coating methods, 3D lattices with ultralow relative densities below 0.1% can be created. The process begins with a photosensitive polymer resin bath; we use either 1,6-hexanediol diacrylate (HDDA) or poly(ethylene glycol) diacrylate (PEGDA). Shown schematically in Fig. 2A, the apparatus uses a spatial light modulator—in this case a liquid-crystal-on-silicon chip—as a dynamically reconfigurable digital photomask. A three-dimensional CAD model is first sliced into a series of closely spaced horizontal planes. These two-dimensional image slices are sequentially transmitted to the reflective liquid-crystal-on-silicon chip, which is illuminated with UV light from a light-emitting diode array. Each image is projected through a reduction lens onto the surface of the photosensitive resin. The exposed liquid cures, forming a layer in the shape of the two-dimensional image, and the substrate on which it rests is lowered, reflowing a thin film of liquid over the cured layer. The image projection is then repeated, with the next image slice forming the subsequent layer. Our polymer microlattices were fabricated in tens of minutes and have features spanning size scales from 10 to 500 μm. For mechanical testing purposes, all materials described here were fabricated as blocks of various sizes consisting of multiple unit cells (table S1). Scanning electron microscopy (SEM) images of the as-built polymer lattice and unit cell are shown in Fig. 2, B and F.

Goal of this figure

This figure presents the process of manufacturing the structures and some of the structures actually created.

Panel A

How to manufacture the lattice:

The lattices were manufactured using the projection microstereolithography technique, a layer-by-layer printing technique. The desired structure is modeled in 3D using a CAD software, this model is divided in different layers.

To print the i-th layer of the structure, the digital mask will be programmed so light is delivered only where there are solid parts in the layer. The light cures the resin in the parts not blinded by the mask, thus creating the i-th layer of the solid. Then, the elevator moves the solid being created in order to cure the i+1-th layer.

This constitutes the first step of the process: A polymer solid is obtained (image of the polymer octet-truss unit cell). Other steps of manufacturing can be added to create hollow-tube metallic objects.

Panels B-E

Structures obtained:

Different octet-truss microlattices were fabricated with different constituent materials and different processes. After the first step, the microstereolithography, a solid polymer HDDA microlattice is obtained (B).

After a step of electroless nickel plating and removal of the polymer core, this polymer lattice is converted into a hollow-tube Ni-P (C).

The solid polymer (B) can also be coated with ceramic using a different technique (atomic layer deposition). Hollow-tube ceramics are thus obtained (D) following removal of the polymer.

Finally, nanoparticles can be incorporated in the initial resin to create solid ceramic microlattices (E).

Panels F-I

Magnified views:

The size and aspects of the struts of the different microlattices can be compared using these magnified views, which correspond to the 4 objects (B-E).

Although projection microstereolithography requires a photopolymer, other constituent materials such as metals and ceramics can be incorporated with additional processing. Using the base polymer lattice as a template, we are able to convert the structures to metallic and ceramic microlattices. Metallic lattices were generated via electroless nickel plating on the as-formed HDDA. The thickness of the metal coating is controlled by the plating time, yielding metal films from 100 nm to 2 μm. The polymer template is subsequently removed by thermal decomposition, leaving behind the hollow-tube nickel-phosphorus (Ni-P) stretch-dominated microlattice shown in Fig. 2, C and G.

A similar templating approach is used to generate hollow-tube aluminum oxide (amorphous Al₂O₃, alumina) microlattices; however, the coating is produced by atomic layer deposition (ALD), a gas-phase process, rather than liquid-phase processing. The resulting hollow-tube microlattices have alumina thicknesses from ~40 to 210 nm, with an example shown in Fig. 2, D and H, with corresponding material weight density ranging from less than 1 kg/m³ to 10.2 kg/m³.

Loading the resin bath with nanoparticles can further expand the base material set. Solid Al₂O₃ ceramic lattices were prepared in the microlithography system by using photosensitive PEGDA liquid prepolymer loaded with ~150-nm alumina nanoparticles (Baikowski Inc., ~12.5% alumina by volume). The same sequential lithographic exposure process produced a microlattice made of a hybrid of solid PEGDA and alumina nanoparticles. These hybrid lattices are converted to pure Al₂O₃ octet-truss microlattices through a sintering procedure (39). An example of this structure is shown in Fig. 2, E and I. The parameters and properties for a selection of our stretch-dominated mechanical metamaterials and bend-dominated foams are summarized in table S1. The densities of all samples were calculated by measuring the weight and fabricated dimensions of the completed microlattices.

The microstructured mechanical metamaterials were tested to determine their Young’s modulus E and uniaxial compressive strength σy, defined as the crushing stress of the material. Uniaxial compression studies of all microlattices with the same cubic dimensions were conducted on an MTS Nano Indenter XP, equipped with a flat punch stainless steel tip with a diameter of 1.52 mm. During 20 consecutive compression cycles up to 10% strain, we observed typical viscoelastic behavior for the polymer microlattices with pronounced hysteresis and loading rate–dependent Young’s modulus. The Young’s moduli for all polymer microlattices and foams were extracted at a loading rate at 87.2 nN/s, corresponding to a strain rate of 10^-3 s^-1. Uniaxial compression of these structures is shown in movies S1 to S3. Representative stress-strain curves from uniaxial compression crushing tests for determining the compressive strength of octet-truss microlattices made of solid HDDA polymer, hollow-tube Ni-P metal, and solid alumina are shown in fig. S7, A and B, and fig. S8A, respectively. Bulk HDDA polymer, cured by UV cross-linking a solid sample of similar dimensions to the octet-truss lattices, was determined to have Young’s modulus Es = 530 MPa and yield stress σys = 86 MPa. [See (39) for detailed measurement methods and bulk property values for other constituent materials.]

The results of these mechanical tests, together with the bend-dominated tetrakaidecahedron-based Kelvin foams fabricated from the base HDDA polymer, are summarized in Fig. 3, A and B, which respectively plot relative Young’s modulus and strength against relative mass density. Figure 4 shows the location of these material properties on the stiffness versus density material selection chart, together with other recently reported ultralight materials for comparison. The stretch-dominated microlattices populate the highly desirable ultralight, ultrastiff space toward the upper left of the chart (17) and have stiffness-to-weight ratios that do not substantially degrade as density decreases by several orders of magnitude. In contrast to the common bend-dominated E/Es ∝ (ρ/ρs)2 scaling of open-cell stochastic foams such as silica aerogels and carbon foams, our stretch-dominated microlattice materials demonstrate the desired linear relationship of E/Es ∝ ρ/ρs, approaching the theoretical limit, and exhibit this remarkable scaling relationship over three orders of magnitude in density and across all constituent materials studied. These octet-truss lattice materials are highly isotropic, so the scaling of stiffness with density does not vary with the orientation of the lattice (fig. S4), as confirmed by our studies of different loading directions. These lattices have the highest specific stiffness when the lattice is loaded normal to the (111) plane, which is closest-packed within the fcc architecture.

Question asked

How the material properties of the proposed structures evolve with density? Is this relation identical to the one of previously reported objects (the Kelvin foams)?

Experiment

Two types of structures were manufactured: the proposed structure (based on an octet-truss unit cell) and the previously reported Kelvin foam.

Five different objects with the same proposed structure but different materials or processing techniques were created (see Fig. 2 B-E). The densities of these objects were calculated by measuring the dimensions and weighting the samples.

The material properties were assessed by mechanical testing with an uniaxial compressive test.

Results

The proposed materials all exhibit a linear relationship between the relative Young modulus and the relative density. The previously reported structure presents a quadratic relation between stiffness and density.

(A) The study of the compressive strength versus density graph (B) reveals that the hollow-tube and solid ceramic octet-truss lattices present different compressive behaviors.

Conclusion

The proposed objects with the octet-truss unit cell satisfy the desired criterion of exhibiting a linear relationship between stiffness and density whatever the constituent material and for a broad range of densities.

Question asked

How the material properties of the proposed objects compare with those of previously reported ultralight materials?

Method

The Young modulus versus density chart is used to select a material according to the desired characteristics. The dotted lines in this plot indicate contours with linear relationship between stiffness and density.

Data obtained in the present study (red, blue, and green squares or circles) are plotted in this graph as well as data from recent studies reporting ultralight materials (black crosses and stars).

Results

The proposed structures are the only ones to exhibit the desired linear relationship between stiffness and density. For a given density range, the proposed objects are stiffer than the previously reported ultralight materials.

In the ultralow-density regime (relative density <0.1%), we observed markedly different compression behavior in hollow-tube ALD ceramic octet-truss microlattices relative to solid ceramic lattices at higher relative densities (8 to 20%). The hollow-tube ceramic microlattices with nanoscale wall thicknesses showed smoother behavior with progressively fewer discontinuities in their stress-strain curves (fig. S8, A and B), in contrast to solid microstrut ceramic lattices with catastrophic, fracture-dominated behavior. The loading-unloading curves of hollow-tube Al2O3 lattices revealed elastic behavior followed by a nonlinear response on each loading cycle. Although relative compressive stiffness and relative density initially follow a nearly linear scaling law, the transition from conventional brittle behavior (in low-density ceramic materials) to more “ductile” mechanical behavior (in ultralight materials with nanoscale wall thicknesses) suggests a transition from a fracture-dominated failure mode to a buckling-dominated failure mode, with suppression of the catastrophic failure seen in solid Al₂O₃ octet-truss lattices.

These differences in compressive behavior between solid and hollow-tube ceramic octet-truss lattices are primarily attributed to local buckling induced by the high aspect ratio of the strut length to nanoscale wall thickness, in contrast to nanoscale TiN trusses (32) and ceramic composite (34), whose aspect ratios are low enough to allow the nanoscale strengthening effect of the wall thickness to dominate. For example, the ratio of strut length to nanoscale wall thickness in fig. S8B is about 1400:1 and contributes to its large compression strain, governed by ε ∝ (1/ρ)0.5 (17). Thus, the relative compressive strength makes a transition from the nearly linear scaling law governing the stretch-dominated failure mode at an approximate density near 0.08% to a scaling power of 2.7, as indicated in Fig. 3B. In the same figure, a similar transition from yielding-dominated to buckling-dominated failure at an approximate relative density of 0.2% is evident in Ni-P lattices, consistent with the trend observed for bend-dominated metallic microlattices (26).

When an ultralow-density metallic microlattice is bend-dominated, its stiffness degrades substantially with reduced density. An example of this is the Ni-P lattice reported by Schaedler et al. (20), whose specific stiffness (stiffness-to-weight ratio) degrades from 0.23 × 106 m²/s² to 0.05 × 106 m²/s² as density is reduced from 40 mg/cm³ to 14 mg/cm³ (45). By contrast, our Ni-P stretch-dominated metallic lattice is not only much stiffer in the same density range, its specific stiffness stays nearly constant, measured as 1.8 × 106 m²/s² and 2.1 × 106 m²/s² at densities of 14 mg/cm³ and 40 mg/cm³, respectively. Similarly, in a recent report of high-strength microarchitected ceramic composites (34), their strength performance approaches the linear scaling relationship over a narrow density range, and only when loaded in a direction optimized for their anisotropic architecture. Our metamaterials, in contrast, maintain their mechanical efficiency over a broad density regime without substantial degradation in specific stiffness, owing to the nearly linear E-ρ scaling relationship.

We have shown that these high mechanical efficiencies are possible across a range of constituent materials. Fabricating ordered lattice structures at these length scales brings them into the regime in which it becomes possible to design microstructured functional materials with superior bulk-scale properties.

Supplementary Materials

www.sciencemag.org/content/344/6190/1373/suppl/DC1

Materials and Methods
Figs. S1 to S9
Table S1
Movies S1 to S3
References (46–55)

References and Notes

1. K. Ando, H. Onda, J. Wood Sci. 45, 120–126 (1999).

2. T. M. Ryan, C. N. Shaw, Proc. R. Soc. B 280, 20130172 (2013).

3. P. Van Liedekerke et al., Phys. Biol. 7, 026006 (2010).

4. J. H. Shen, Y. M. Xie, X. D. Huang, S. W. Zhou, D. Ruan, Key Eng. Mater. 535–536, 465–468 (2013).

5. J. A. Kepler, Compos. Sci. Technol. 71, 46–51 (2011).

6. L. Valdevit, A. J. Jacobsen, J. R. Greer, W. B. Carter, J. Am. Ceram. Soc. 94, s15–s34 (2011).

7. N. Lyu, B. Lee, K. Saitou, J. Mech. Des. 128, 527 (2006).

8. T. A. Schaedler et al., Adv. Eng. Mater. 16, 276–283 (2014).

9. K. J. Maloney et al., APL Mater. 1, 022106 (2013).

10. L. Valdevit, A. Pantano, H. A. Stone, A. G. Evans, Int. J. Heat Mass Transfer 49, 3819–3830 (2006).

11. M. Q. Li, X. Shi, Rare Metal Mater. Eng. 36, 575 (2007).

12. O. Y. Kwon, H. J. Ryu, S. Y. Jeong, J. Ind. Eng. Chem. 12, 306 (2006).

13. R. A. Olson, L. C. B. Martins, Adv. Eng. Mater. 7, 187–192 (2005).

14. P. Day, H. Kind, Mod. Cast. 74, 16 (1984).

15. S. Baudis et al., Biomed. Mater. 6, 055003 (2011).

16. K. Arcaute, B. K. Mann, R. B. Wicker, Tissue Eng. C 17, 27–38 (2010).

17. J. L. Gibson, F. M. Ashby, Cellular Solids: Structure and Properties (Cambridge Univ. Press, Cambridge, 2001).

18. H. Fan et al., Nat. Mater. 6, 418–423 (2007).

19. L. Qiu, J. Z. Liu, S. L. Y. Chang, Y. Wu, D. Li, Nat. Commun. 3, 1241 (2012).

20. T. A. Schaedler et al., Science 334, 962–965 (2011).

21. M. A. Worsley, S. O. Kucheyev, J. H. Satcher, A. V. Hamza, T. F. Baumann, Appl. Phys. Lett. 94, 073115 (2009).

22. T. M. Tillotson, L. W. Hrubesh, J. Non-Cryst. Solids 145, 44–50 (1992).

23. S. O. Kucheyev et al., Adv. Mater. 24, 776–780 (2012).

24. L. J. Gibson, MRS Bull. 28, 270–274 (2003).

25. K. Tantikom, Y. Suwa, T. Aizawa, Mater. Trans. 45, 509–515 (2004).

26. A. Torrents, T. A. Schaedler, A. J. Jacobsen, W. B. Carter, L. Valdevit, Acta Mater. 60, 3511–3523 (2012).

27. C. Q. Dam, R. Brezny, D. J. Green, J. Mater. Res. 5, 163–171 (1990).

28. J. K. Cochran, K. J. Lee, D. McDowell, T. Sanders, Multifunctional metallic honeycombs by thermal chemical processing. In Processsing and Properties of Lightweight Cellular Metals and Structures, A. K. Ghosh, T. H. Sanders, T. D. Claar, Eds. (Minerals, Metals and Materials Society, Seattle, WA, 2002), p. 127–136.

29. J. Zhang, M. F. Ashby, Int. J. Mech. Sci. 34, 475–489 (1992).

30. D. Rayneau-Kirkhope, Y. Mao, R. Farr, Phys. Rev. Lett. 109, 204301 (2012).

31. K. C. Cheung, N. Gershenfeld, Science 341, 1219–1221 (2013).

32. D. Jang, L. R. Meza, F. Greer, J. R. Greer, Nat. Mater. 12, 893–898 (2013).

33. L. Meza, J. Greer, J. Mater. Sci. 49, 2496–2508 (2014).

34. J. Bauer, S. Hengsbach, I. Tesari, R. Schwaiger, O. Kraft, Proc. Natl. Acad. Sci. U.S.A. 111, 2453–2458 (2014).

35. V. S. Deshpande, N. A. Fleck, M. F. Ashby, J. Mech. Phys. Solids 49, 1747–1769 (2001).

36. J. D. Renton, Elastic Beams and Frames (Horwood, Chichester, UK, ed. 2, 2002).

37. S. Hyun, J. E. Choi, K. J. Kang, Comput. Mater. Sci. 46, 73–82 (2009).

38. H. L. Fan, F. N. Jin, D. N. Fang, Mater. Des. 30, 511–517 (2009).

39. See supplementary materials on Science Online.

40. W. Y. Jang, S. Kyriakides, A. M. Kraynik, Int. J. Solids Struct. 47, 2872–2883 (2010).

41. Y. Takahashi, D. Okumura, N. Ohno, Int. J. Mech. Sci. 52, 377–385 (2010).

42. X. Zheng et al., Rev. Sci. Instrum. 83, 125001 (2012).

43. C. Sun, N. Fang, D. M. Wu, X. Zhang, Sens. Actuators A 121, 113–120 (2005).

44. A. J. Jacobsen, W. Barvosa-Carter, S. Nutt, Adv. Mater. 19, 3892–3896 (2007).

45. L. M. Moreau et al., Nano Lett. 12, 4530–4539 (2012).