Figure 1a shows the fabrication of the inverse opal material. First, monodisperse polystyrene (PS) particles of diameter 260–930 nm were self-assembled onto a gold/chromium coated substrate in a face centered cubic (FCC) orientation. The PS was sintered at 96 °C to improve stability and increase the interconnect diameter between PS spheres. Nickel, 99.9%, was electrodeposited into the voids of the PS structure, followed by PS etching in tetrahydrofuran. The result was an open cell inverse opal material having interconnected spherical pores in a face-centered cubic orientation. For some samples, conformal electrodeposition of either additional nickel or rhenium-nickel alloy (80 wt% rhenium) increased the solids volume fraction, strut diameter, and mass. The additional deposited nickel had the same composition as the nickel used to fabricate the inverse opal structure. Mechanically tested samples with rhenium are labeled Re. The average grain sizes of the nickel film and nickel inverse opals were 12.4 nm and 15.1 nm, calculated from the XRD data shown in the supplementary information using the Scherrer equation. The grain size was similar for coated and un-coated nickel samples. A 15 nm grain size in pure nickel corresponded to a ≈6.4 GPa hardness33,34, near the peak hardness values of pure nickel, which agreed well with our electrodeposited nickel thin film nanoindentation hardness measurements of 5.8 GPa. The self-assembly based fabrication method can produce material samples larger than 100 mm2, while allowing ~10 nm control of structure and chemistry by altering the polystyrene diameter, processing temperature, and electroplating parameters. In comparison, most methods for fabricating nanostructured cellular solids with controlled and repeated unit cell morphologies are limited to sample sizes with areas smaller than about 5 mm2 1,11,12,15,27,28,35. Figure 1b–g show SEM images of the material with 500 nm pores fractured on the (111) plane. Figure 1b,c show the inverse opal material with no coating and 0.16 solids volume fraction. Figure 1d,e show the material uniformly coated with 21 nm of nickel, which increased the solids volume fraction from 0.16 to 0.35. Figure 1f,g show the nickel inverse opal material uniformly coated with 25 nm of rhenium-nickel alloy, which increased the solids volume fraction to 0.46. Figure 1h shows a material with 500 nm pores, 2 cm2 area, and 15 µm thickness fabricated on a gold/chromium coated glass slide. Figure 1i shows a material with 300 nm pores and 20 µm thickness fabricated on gold/chromium-coated polyimide. The material is flexible and when prepared on a polyimide sheet could be bent past a 0.5 cm radius (Fig. 1i). Table 1 shows the fabricated materials and their geometric attributes such as coating thickness and volume fraction.

Figure 1 (a) The fabrication process for a unit cell of the nickel inverse opal material. (b–g) Cross section SEM images of nickel inverse opal material. (b,c) A nickel inverse opal with no coating. (d,e) A nickel inverse opal material with a 21 nm coating of additional electrodeposited nickel. (f) A nickel inverse opal material with a 25 nm coating of additional electrodeposited rhenium-nickel. (g) A closer image of one of the struts in (f). (h) A 2 cm2 nickel inverse opal material with 500 nm pores and 15 µm thickness grown on a gold/chromium coated glass slide. (i) A nickel inverse opal material with 300 nm pores grown on gold/chromium coated 20 µm thick polyimide. Full size image

Table 1 Mechanical and physical measurements of inverse opal cellular materials. Full size table

The high specific strength of the inverse opal material results from two mechanical processes: porosity-based weakening, which is independent of size, and strengthening of the pore struts, which is size-dependent at the nanometer-scale. Porosity-based weakening is the reduction in load that the material can support due to reduced volume fraction of solid material, and mediated by bending dominated deformation of the material struts36. Size-dependent strengthening increases the local strength of the constituent material because of dislocation starvation in the struts, for strut widths smaller than about 1 µm2,9,19,37. The inverse opal material yield strength, σ*, is related to the solids volume fraction, (ρ*/ρ s ), and strut yield stress, σ y , by

$${\sigma }^{\ast }={C}_{1}\,{\sigma }_{y}(d){(\frac{{\rho }^{\ast }}{{\rho }_{s}})}^{{C}_{2}},$$ (1)

where σ y depends on the effective strut diameter, d, and can be greater than the bulk material strength due to size-dependent strengthening, ρ* is the density of the porous inverse opal, and ρ s is the density of the bulk solid8,36. The geometric coefficients, C 1 and C 2 , are unique to the inverse opal geometry and independent of both pore size and solid chemistry. Once C 1 and C 2 are known for a given lattice geometry, the size-dependent strengthening effect σ y (d) can be determined by measuring σ*, d, and ρ*/ρ s 3,8,9.

We used finite element simulations of an inverse opal unit cell to understand how the geometry relates to the material modulus and yield strength and to solve for C 1 and C 2 (Fig. 2). Finite element simulations improved upon previous work, which used idealized relationships to calculate the size-dependent strengthening in nanoporous gold3,8,9, by more accurately relating the applied load to the deformation and stress distribution in the inverse opal geometry. The simulations considered an inverse opal unit cell with isotropic elasticity and J2 plasticity loaded in the [111] direction. In order to mimic the experiments and control the solids volume fraction, the simulations included 0–33 nm nickel coatings, where the coating has the same material properties as the base nickel. Figure 2a shows repeat unit cells used in the finite element simulations; the colors correspond to Von Mises stresses at 1.2% engineering strain. The maximum stress occurs in the narrow region of the struts parallel to the displacement direction. Figure 2b shows corresponding stress-strain curves for nickel inverse opals with various coating thicknesses. The yield strengths, σ*, were calculated at a 0.2% strain offset and compared to the solids volume fraction to solve for the coefficients in Eq. 1 38. The cellular solid yield strength is

$${\sigma }^{\ast }=0.78\,{\sigma }_{y}{(\frac{{\rho }^{\ast }}{{\rho }_{s}})}^{1.52}.$$ (2)

Figure 2 (a) Unit cells of the nickel inverse opal material used to determine the yield strength and elastic modulus through finite element simulations. The unit cell on the right has an additional 19 nm nickel coating. The colors correspond to Von Mises stress at a 15 nm displacement, near the yield point. (b) Calculated stress-strain curves of nickel inverse opal material with 0–33 nm thick nickel coatings. Full size image

For each material, the elastic modulus, E*, was calculated from the initial slope of the linear stress-strain region in Fig. 2b. The elastic modulus is related to the solids volume fraction and bulk nickel elastic modulus, E = 171 GPa, by

$${E}^{\ast }=1.0\,E{(\frac{{\rho }^{\ast }}{{\rho }_{s}})}^{1.73}.$$ (3)

The polynomial relationship between the mechanical properties and solids volume fraction indicates that the cellular solid deforms primarily through bending39. This bending-dominated deformation is expected, since the inverse opal cell, with 32 struts and 21 joints, does not satisfy the Maxwell criterion40. The stress and modulus relations compare closely with measurements described below, and are valid for coated and un-coated inverse opals made from any one material.

We measured mechanical properties of the fabricated materials using nanoindentation and micropillar compression measurements, which are well established techniques for measuring nanostructured cellular material compressive strengths and moduli1,3,4,6,7,8,9,10,11,12,13,14,15,16,17,18,19. The micropillar compression tests measured the yield strength, σ*17, while nanoindentation measurements allowed us to determine elastic modulus, E*, and hardness for all materials. Figure 3 shows micropillar compression test results for materials made from 500 nm diameter PS particles and 0, 19, and 33 nm nickel coatings. Figure 3a shows SEM images of 4 µm diameter micropillars before and after compression tests. Failure predominately occurred at the narrowest region of the struts and in the [111] direction parallel to the micropillar axis, consistent with the finite element simulations, which are orientated in the same geometry. Figure 3b shows the measured engineering stress-strain data for several inverse opal and solid nickel micropillars loaded to an engineering strain of about 20%. Table 1 shows the measured yield strength, which was measured using the 0.2% offset method.

Figure 3 (a) SEM images of inverse opal micropillars with 500 nm pores, before and after compression testing. The uncoated sample is 84% porous and the 19 nm nickel coated sample is 58% porous. Failure occurs in the [111] direction, parallel to the compression axis. Scale bars are 3 µm. (b) Engineering stress, σ*, versus engineering strain for micropillars under compression. The nickel sample is bulk electroplated nickel. (c) Strut yield strength, σ y , as a function of strut diameter, d, for nickel inverse opal materials with 260, 500, and 930 nm pores. Blue data are nickel inverse opals measured with micropillar compression tests. Green data are nickel inverse opals measured with nanoindentation. Orange data are 500 nm pore nickel inverse opals coated with 19 and 33 nm of additional nickel. Shaded regions represent standard deviations. The fit line was only applied to nickel samples with no coatings. Full size image

The measured micropillar yield strength, σ*, provides insight into the size-dependent strength of the inverse opal struts, σ y (d). The supplementary methods section describes measurements of effective strut diameter, d, and solids volume fraction, (ρ*/ρ s ), which are combined with Eqn. 2 for this analysis. Figure 3c shows strut yield strength, σ y , as a function of effective strut diameter at the narrowest region of the strut, approximately where the failure occurred, for nickel inverse opals with 260, 500, and 930 nm pores. Blue data points show the strength of nickel inverse opals measured with micropillar compression. Green data points are nickel inverse opals measured with nanoindentation, where the yield strength was approximated by the hardness. Orange data points are 500 nm pore nickel inverse opals coated with 19 and 33 nm of additional nickel and measured with micropillar compression. In general, the strut strength increased as the effective strut diameter decreased. The largest value of strut yield strength measured by micropillar compression was 8.07 GPa for the nickel inverse opal material with 260 nm pore size, 0.10 solids volume fraction, and 17 nm effective strut diameter. This strength was a 4X increase over the yield strength of bulk electrodeposited nickel, which is 1.98 GPa33. Importantly, the 1.98 GPa yield strength of the bulk nanocrystalline nickel was near the maximum reported for pure nickel33,34, so the large strength enhancement must be due to the pore structure introduced by the inverse opal geometry. Similar strengthening has previously been reported in solid single crystal micro and nanopillar compression experiments, where the strength of sub-μm diameter pillars approached the theoretical strength of the material2,9,19,20,23,37,41,42, as well as in nanoporous gold and mesoporous copper lattices with nanocrystalline and polycrystalline microstructure3,8,28. The −0.6 slope of the best fit line for the uncoated nickel samples agreed remarkably well with the approximately −0.6 slope experimentally measured in multiple single crystal FCC metals including nickel, aluminum, gold, and copper17, as well as with the −0.66 slope measured in nanocrystalline nickel nanopillars24. This good agreement indicates that the strengthening mechanism and initial dislocation density in the nickel inverse opal samples were similar to previous single crystal and nanocrystalline nanopillar samples. The strengthening is derived from the small grain or feature size, which leaves no mechanism for dislocation build up and the dominant form of dislocation nucleation becomes surface nucleation24. Similar to single crystal materials, the smaller strut diameters are stronger because the decreased probability of a critical dislocation compared to larger struts and bulk materials24. The results in Fig. 3c show a clear size-dependent strength enhancement in inverse opal materials.

Figure 3b and Table 1 show that the inverse opal strength increased in nickel inverse opals coated with additional nickel. The inverse opal strength increase is due to the strong dependence of the material strength on relative density (Eqn. 2), despite the strut yield strength decreasing as the strut diameter increased (Fig. 3c). Interestingly, the strut yield strength exceeds the strength predicted by the fit line in Fig. 3c, which indicates that coatings could provide a method for enhancing the total strength of inverse opal composites. Currently, the exact mechanism for the strength enhancement and the interaction between coated layers is not clear.

The high strength of the inverse opal struts combined with the regular porous architecture imparts unique macroscopic material properties including high hardness, high strength at a given weight, and a controllable modulus while maintaining high specific strength. Figure 4 shows inverse opal material properties and compares them to other high strength engineering materials. Figure 4a shows hardness as a function of solids volume fraction for nickel and nickel-coated nickel inverse opal materials. Hardness measures the resistance to local plastic deformation and is important for several applications including tribology. The hardness was measured with nanoindentation and increased from 0.082 to 2.4 GPa as the solids volume fraction increased from 0.1 to 0.48. The most dramatic hardness increase, from 0.3 to 2.4 GPa, occurred between 0.28 and 0.48 solids volume fractions, where the inverse opal hardness approached the linear scaling of the bulk hardness and solids volume fraction. The hardness predicted from finite element simulations of the nanoindentation tests are shown in black and have good agreement with the measured hardness at 0.2 to 0.3 solids volume fractions41,43; however, the simulations under predict the hardness at low (less than 0.2) and high (greater than 0.3) solids volume fractions. Nanoindentation simulation details can be found in the supplementary methods section. The larger than predicted material hardness at high solids volume fractions results from a larger than predicted plastic Poisson’s ratio. This can be seen by comparing the measured and calculated nominal hardness, hardness normalized by yield strength, which depends strongly on the plastic Poisson’s ratio (Fig. S2d). Figure 4b shows that the measured and predicted nominal hardness both increase with solids volume fraction, but the measured nominal hardness increases dramatically at greater than 0.35 solids volume fractions while the predicted nominal hardness remained ~1.3. This jump in the measured nominal hardness is likely due to rapid densification of the inverse opal geometry during large plastic strains, which would increase the plastic Poisson’s ratio and increase the nominal hardness. Interestingly, the dramatic hardness increase in inverse opal materials with greater than 0.3 solids volume fractions corresponds to the same volume fractions where previous simulations of inverse opal materials predicted a rapid increase of modulus towards the Hashin and Shtrikman upper bound for isotropic materials29,44,45. At greater than 0.3 solids volume fraction, simulations show that the inverse opal structure can outperform the state-of-the-art octet- and isotropic-truss geometries in isotropic Young’s modulus, shear and bulk modulus, as well as in structural efficiency (total stiffness)16,29. At smaller than 0.3 solids volume fraction, the inverse opal hardness is a good predictor of the yield strength.

Figure 4 (a) Nanoindentation hardness measurements versus total solids volume fraction of the inverse opal materials. The black dots are numerical predictions of the hardness measurements from finite element simulations. (b) Nominal hardness (hardness normalized by the yield strength) versus total solids volume fraction of several uncoated and nickel-coated nickel inverse opal materials. The flow stress at 10% strain is used as the yield strength for the nominal hardness measurements. The black dots are numerical predictions of the nominal hardness. (c) Ashby plot of material strength, calculated at a 0.2% strain offset, versus density for the fabricated materials and several reference materials. (d) Ashby plot of specific modulus and specific strength of the fabricated materials and several reference materials. For comparison in (c) and (d), common Ti, Al, Ni, and Fe high strength alloys are labeled as follows 1 - CP Ti, 2 - 2024-T4, 3 - Inconel 718, 4 - 7075-T6, 5 - HSSS steel, 6 - Ti-6Al-4V, 7 - Ti-10V-2Fe-3Al31,32,67,68. Error bars show standard deviations. Full size image

Figure 4c shows the strength versus density of our materials along with data from other materials for comparison purposes, including high strength Ti, Ni, Al, and steel alloys. The nickel inverse opal materials with no coatings had the strength of aluminum alloys and titanium (σ* = 90–240 MPa), and density close to that of water (ρ* = 880 to 1400 kg m−3), filling a new material space in the strength-density relationship. The two lightest inverse opal materials had the same average solids volume fractions ((ρ*/ρ s ) = 0.1); however, the sample with smaller effective strut diameters (d = 17 vs. 59 nm) had an increased strut yield strength (σ y (d) = 10 vs. 4.6 GPa) that more than doubled the material yield strength (σ* = 194 vs. 90 MPa). The presence of a coating further increased the material strength. The strength and density of the coated inverse opal materials was comparable to high-strength, lightweight alloys such as 7075-T6 aluminum, Ti-6Al-4V titanium, and Ti-10V-2Fe-3Al titanium. These material properties were achieved despite the inverse opal material being fabricated from dense engineering materials (Ni = 8,900 kg m−3 and Re = 21,020 kg m−3).

The unique relationship between the strut strength and inverse opal geometry enables independent control of the specific modulus and specific strength. Figure 4d shows the specific strengths versus specific moduli of the nickel inverse opal materials. The average specific strengths varied between 102–230 MPa/(Mg/m3) and the average specific elastic moduli varied between 9–25 GPa/(Mg/m3). Uncoated nickel inverse opal materials had solid volume fractions between 0.10 and 0.28 and average specific moduli between 15 and 20 GPa/(Mg/m3). Coating additional layers of nickel onto the inverse opal materials increased the solids volume fractions up to 0.48 and increased the average specific moduli to 25 GPa/(Mg/m3). In some cases, the specific moduli exceeded the bulk nickel specific modulus (19.2 GPa/(Mg/m3)), likely due to the non-negligible contribution of the stiff native oxide. As additional layers of nickel were coated, the specific strengths remained near 200 MPa/(Mg/m3) because an increased solids volume fraction increased the geometric strength but also increased the strut diameter, which reduced the benefit of size-based strengthening. As described by the arrows in Fig. 4d, the specific modulus can be controlled independent of the specific strength by increasing the total solids volume fraction; whereas, the specific strength can be independently controlled by changing the strut effective diameter while maintaining a constant solids volume fraction.