Abstract
Though beambased lattices have dominated mechanical metamaterials for the past two decades, low structural efficiency limits their performance to fractions of the HashinShtrikman and Suquet upper bounds, i.e. the theoretical stiffness and strength limits of any isotropic cellular topology, respectively. While platebased designs are predicted to reach the upper bounds, experimental verification has remained elusive due to significant manufacturing challenges. Here, we present a new class of nanolattices, constructed from closedcell platearchitectures. Carbon platenanolattices are fabricated via twophoton lithography and pyrolysis and shown to reach the HashinShtrikman and Suquet upper bounds, via in situ mechanical compression, nanocomputed tomography and microRaman spectroscopy. Demonstrating specific strengths surpassing those of bulk diamond and average performance improvements up to 639% over the best beamnanolattices, this study provides detailed experimental evidence of plate architectures as a superior mechanical metamaterial topology.
Introduction
The HashinShtrikman (HS) and Suquet upper bounds represent the theoretical topological stiffness and strength limits of isotropic cellular solids. The effective Young’s modulus (E_{HSU}) and yield strength (σ_{y,SU}) of a material at those theoretical limits are expressible as^{1,2,3}:
where E_{s}, σ_{y,s}, and v represent the Young’s modulus, yield strength and Poisson’s ratio of the constituent material, respectively, and \(\bar \rho\) is the relative density of the cellular material, i.e., the volume fraction of the solid constituent.
Over the past decade, nanolattices^{1,4,5} successfully utilized ultrastrong constituent materials which approach the theoretical material strength limit; however, lattice topologies were largely limited to inefficient beamgeometries. Mechanical size effects enable nanostructures, including pyrolytic carbon octet lattices^{6}, nickel double gyroids^{7}, silica inverse opals^{8,9}, and gold foams^{10,11,12} to exceed their corresponding monolithic bulk solids in strength^{1}; however, their nonoptimal opencell topologies exhibit maximal performances well below the HashinShtrikman and Suquet upper bounds. The discussion on mechanical efficiency of cellular materials has classically focused on the difference between stretchingdominated or bendingdominated behavior (e.g., whether lattice elements predominantly undergo uniaxial tension/compression or bending under macroscopically applied stresses, respectively), with the former being stronger and stiffer^{13}. Although octet truss and isotropic truss lattices are stretchingdominated, ideal analytical models, even neglecting their nodal knockdowns, show that their stiffness and strength are limited to half the upper bounds, with the octet truss also being anisotropic^{1,14,15}. In practice, stress concentrations at nodes and nodal bending further reduce their actual performance^{16}, to around 25% and 20% of the HS and Suquet upper bounds, respectively. Moreover, geometric manufacturing imperfections, particularly for complex and thin features, may impose further knockdowns^{17}. In fact, no opencell topology is capable of reaching the stiffness or strength^{18,19} upper bounds, and while honeycombs are mechanically efficient, their response is vastly anisotropic, restricting their use to cores of sandwich structures^{20}.
Recently, closedcell architectures, consisting of plates arranged corresponding to the closest packed planes of crystal structures, have computationally been predicted to reach the HS and Suquet upper bounds^{18,21,22}; however, their manufacturing complexity has thus far prevented any experimental validation. The simplest such topology is the cubic+octet (CO) platelattice, also known as the simple cubic + face centered cubic configuration (SCFCC), characterized by a cubicplate unit cell embedded in an octetplate unit cell. CO platelattices in theory reach the HS upper bound approaching 0% relative density and remain within 90% of the bound^{18,21} at higher \(\bar \rho\). For any given loading direction, platelattices have higher structural efficiency, i.e. they store strain energy more uniformly among members, and have a higher volume fraction of members favorably oriented in the loading direction, compared to a corresponding beamlattice^{18}. Moreover, the threedimensional intersections of plates prevent the formation of kinematic mechanisms, thus ensuring the platelattice topology is always stretchingdominated, which may not be true in a corresponding beam topology. However, these advantages come at the cost of dramatically increased fabrication complexity. The closedcell topologies of threedimensional platelattices make most conventional fabrication routes, like assembly techniques^{16,20}, impractical, and leave additive manufacturing (AM) as the only suitable approach. Nevertheless, removing raw materials enclosed within cells remains challenging. Furthermore, synthesis of a highly voided platelattice generally requires printing features both near the resolution and maximum build volume limits. For example, a 10% relative density CO platelattice requires wall features over 100 times thinner than the unit cell size, and even in fairly high\(\bar \rho\) platelattices, thin wall features are susceptible to warping. While the above challenges are a general obstacle to the application of platelattices, they especially complicate synthesis approaches at the nanoscale where size effects are exploitable.
In this paper, we introduce a class of nanolattices, constructed from closedcell platearchitectures. These platenanolattices are the only materials to experimentally achieve the HashinShtrikman and Suquet upper bounds for isotropic elastic stiffness and strength, respectively. We manufacture structures with cubic+octet design from pyrolytic carbon via twophotonpolymerization direct laser writing (TPPDLW) and subsequent pyrolysis. Several critical fabrication challenges are overcome, including removal of excess raw material pockets, design of printing strategies to ensure homogeneous material properties for plates of different orientations and thicknesses, and management/optimization of shrinkage during pyrolysis. Structures are characterized via microRaman spectroscopy, nanocomputed tomography (nanoCT), and in situ mechanical compression. Approaching the theoretical limits in both the topology and constituent material properties, our platenanolattices demonstrate an average stiffness and strength improvement of up to 522 and 639% at a given relative density, respectively, compared to pyrolytic carbon octettruss^{6,23,24} and isotropic truss^{23} nanolattices, the most advanced mechanical metamaterials reported to date. Our platenanolattices have the highest specific stiffness of any reported architected material and outperform all known bulk materials in compressive strength at their given densities. Reaching specific strengths of 3.75 GPa g^{−1} cm^{3}, they are the only cellular materials to surpass certain diamond systems^{25}. Our study clearly shows a tremendous performance improvement to be gained from platebased topologies compared to stateoftheart beambased designs and provides crucial fabrication insights to enable additive manufacturing of highperformance platelattice materials.
Results
Fabrication procedure
While manufacturing constraints have limited nanoarchitected materials to opencell designs, like beamlattices, we demonstrate fabricability of virtually closedcell platenanolattices with dramatically improved mechanical properties over beamnanolattices (Fig. 1 and Supplementary Note 1). TPPDLW and subsequent pyrolysis at 900 °C were applied to create pyrolytic carbon platenanolattices with cubic+octet design. Relative densities of 25–60% were obtained by scaling the unit cell sizes while maintaining constant wall thicknesses of t_{c} = 260 nm and t_{o} = 150 nm of the cubic and octet plates, respectively. The ratio \(t_{\mathrm{c}}/t_{\mathrm{o}} = \sqrt 3\) corresponds to full isotropy at \(\overline \rho\) = 40%^{21}, which is the approximate center of our examined \(\overline \rho\) range. Given constraints on the overall build volume, all platenanolattices were printed at the resolution limit of the TPPDLW process to achieve the above wall thicknesses and relative density range. This necessitated individual printing strategies for plates of different orientations to ensure octet and horizontal and vertical cubic walls had the desired thicknesses, despite the ellipsoidal voxel shape of TPPDLW^{26} (Supplementary Fig. 2). Introducing small holes with diameters of 100–160 nm at the center of the plate faces allowed infiltration with a developer and removal of unpolymerized resin after the TPPDLW step, despite the closedcell topology. NanoCT scans confirmed that the pyrolytic carbon structures were completely voided and free from residual excess material (Supplementary Movies 1–3). Final pyrolytic carbon structures with \(\bar \rho\) > 37.5% had near ideal, undistorted topologies. With decreasing \(\bar \rho\), deformations occurred during pyrolysis and became more prominent at the lowest relative densities. However, nanoCT scans revealed that most deformations were limited to the specimen surfaces.
Finite element analysis showed that holes in the plate faces, introduced due to manufacturing constraints, do not significantly reduce the elastic properties of the platelattices nor induce any significant anisotropy (Fig. 2). As in the manufactured structures, the hole size was held constant for all modeled relative densities. Although a small performance knockdown emerged at higher \(\bar \rho\), where the ratio of hole sizetounit cell size was greatest, the stiffnesses with respect to the ideal models without holes did not fall below 96 and 93% for relative densities of 40 and 60%, respectively. All structures essentially remained isotropic despite the presence of holes, as the Zener anisotropy ratio was approximately 98% for relative densities 25–45% and around 95% at \(\bar \rho\) = 60%. Additionally, variation of the Poisson’s ratio of the constituent material between 0.17 and 0.3 had no significant effect (Supplementary Fig. 9). In previous work, face holes were also reported not to significantly affect the yield strength of platelattices^{18}. We numerically showed that the same applies to the buckling strength (see Supplementary Fig. 10).
Experimental mechanical behavior
CO platenanolattices with relative densities between 25 and 57.5% were in situ mechanically tested under uniaxial compression, using a nanoindentation system equipped with a flat punch tip. Figure 3 shows compressive stressstrain curves for relative densities of 57.5, 50, 37.5, and 25%, with curves recorded at all densities presented in the insets. Nonlinear behavior at small strains is attributed to slight misalignment between the structures and the indenter. A loadingunloading cycle between 4 and 10% strain was used to ensure accurate stiffness measurement, with the specimens fully seated on their supporting structures. At high relative densities, we found elasticplastic behavior with effective stiffnesses up to 21.6 GPa, followed by brittle fracture at stresses as high as 3 GPa. Yield strains remained constant at ~5%, while failure strains, predominantly elastic, gradually decreased with decreasing relative density (Supplementary Fig. 7). Near \(\bar \rho\) = 37.5%, the deformation behavior transitioned to a more progressive failure (Fig. 3c, d and Supplementary Movies 4–7) and stiffness and strength gradually decreased to approximately 5.9 GPa and 0.5 GPa respectively, at \(\bar \rho\) = 25%.
Figure 4 shows the stiffness and yield strength results of all tested CO platenanolattices. The lines represent the HashinShtrikman and Suquet upper bounds for different constituent material Young’s moduli and yield strengths, whereby the shaded regions provide confidence intervals on the bounds given uncertainties on the material properties of nano/microscale pyrolytic carbon. For stiffness, the lower and upper limits of the shaded region correspond to constituent Young’s moduli of 41 and 62 GPa, as measured from compression of pyrolytic carbon micropillars printed using the same process parameters as for the vertical cubic wall (Supplementary Fig. 6) and the highest reported value for nanoscale pyrolytic carbon^{27}, respectively. Likewise, for the yield strength the applicable bound range is given by the minimum and maximum yield strengths measured from our micropillars, 2.2 GPa and 2.7 GPa, respectively, which are in good agreement with literature data on the tensile and compressive strength of nanoscale pyrolytic carbon^{28,29}.
Our CO platenanolattices experimentally reach both the HashinShtrikman and Suquet upper bounds of an isotropic cellular material. Our most pristine structures, \(\bar \rho \ge 37.5\%\), lie at the upper limit of the shaded stiffness region of Fig. 4a. For yield strength, the highestdensity structures predominantly lie in the shaded region of Fig. 4b, achieving the Suquet bound; the lowest relative density structures (\(\bar \rho\, <\, 37.5\%\)) mostly lie below the bound region.
Consistent with the observed transition from linearelastic to progressive failure and reduced mechanical properties around \(\bar \rho = 37.5\%\), we identified two characteristic scaling relations of the measured strength and stiffness with the relative density (Supplementary Table 4). The effective stiffness of cellular materials (E) can be related to the relative density (\(\bar \rho\)), as \(E \propto \bar \rho ^b\) ^{30}, with the scaling exponent (b). The effective yield strength (σ_{y}) can be described as \(\sigma _{\mathrm{y}} \propto \bar \rho ^c\) with the scaling exponent (c). Least square fitting of our results with the above scaling relations gave b = 1.05 and c = 0.85 for \(\bar \rho \ge 37.5\%\), and b = 2.22 and c = 2.78 for \(\bar \rho\, <\, 37.5\%\). Nearlinear scaling of both stiffness and strength with \(\bar \rho\), for \(\bar \rho \ge 37.5\%\), generally indicates efficient stretchingdominated behavior^{21}. While scaling and deformation behavior correlation becomes less accurate above \(\bar \rho = 30\%\)^{30}, the observed shift to reduced scalings around \(\bar \rho = 37.5\%\) still reveals two distinct deformation regimes with different degrees of efficiency in terms of strength and stiffness.
Computed mechanical behavior
Figure 5a compares the stiffnesses of pyrolytic carbon platenanolattices and cubic+octet finite element models with ideal and predeformed unit cells. For \(\bar \rho \ge 37.5\%\), experiments and ideal finite element results excellently agree, with the computed stiffnesses coinciding with the measured stiffness\(\bar \rho\)scaling. The experimentally observed steepening in stiffness scaling for \(\bar \rho \,<\, 37.5{\mathrm{\% }}\) is well captured by simulations with increasing predeformations, with magnitudes in agreement with experimental observations. Figure 5b compares the measured yield and compressive strength to the computed elastic buckling strength of ideal models. The shaded region is bounded by the first eigenmode, occurring in the octet walls, and the vertical wall buckling mode most visible in the experiments. The computed buckling strength notably exceeds the measured yield strength in the high\(\bar \rho\) regime and well correlates with the measured yield strength reduction below \(\bar \rho = 37.5{\mathrm{\% }}\), indicating a transition from yielding to elastic buckling near that relative density. Across the entire \(\bar \rho\) range the measured compressive strength trends with the computed buckling failure. Given that pyrolytic carbon only shows a modest change in stiffness between elastic and plastic deformation, the elastic buckling simulations provide a reasonable approximation for the plastic buckling strength, suggesting that high\(\bar \rho\) platenanolattices fail by plastic buckling following initial yielding. As yield and buckling strength deviate with increasing \(\bar \rho\), prolonged postyield deformation results in increasing failure strain with increasing \(\bar \rho\) (Supplementary Fig. 7).
Comparison with literature data
With average compressive strength and stiffness improvements up to 639% and 522%, respectively, compared to the best beamnanolattices^{6,23}, our platenanolattices are the only reported materials to lie at the theoretical compressive strength limits and the only architected material to exceed synthetic macroscale cellular materials, like ceramic foams^{25}, in stiffness, thus representing the strongest and stiffest existing architected materials to date. Figure 6 compares the compressive strength and stiffness data from this work with those of other architected and bulk materials^{6,7,8,9,10,11,12,16,23,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45}. The theoretical limits^{1,6,23} are taken as regions bounded by a linear scaling of graphene (the strongest and stiffest known material, albeit in two dimensions and at the nanoscale) and bulk diamond (the strongest and stiffest bulk material at the macroscale). With densities below 0.792 g/cm^{3}, and compressive strengths up to 3 GPa, our platenanolattices achieve specific strengths up to approximately 3.75 GPa g^{−1} cm^{3}, which surpass all known bulk metallic, polymeric and composite materials, and all technical ceramics, including certain diamond systems^{25}. With stiffnesses up to 21.6 GPa, our platenanolattices are the stiffest architected material reported to date. Compared to pyrolytic carbon octet truss and isotropic truss nanolattices^{23} of the same relative density range, the average improvement is between 137 and 522% for stiffness and between 89 and 639% for strength, at densities of 0.792 and 0.349 g/cm^{3}, respectively. Considering only the best performing pyrolytic carbon beamnanolattices, we retain a 100% improvement in strength and a 33–89% improvement in stiffness. For additional comparison, normalized Ashby charts of stiffness and strength show that the cubic+octet topology of our platenanolattices clearly outperforms both beam and shell architectures (Supplementary Fig. 8).
Discussion
Synthesis of nanolattices from mechanically strong and stiff ceramics or metals requires sophisticated multistep processes that are complicated to apply to closedcell topologies and have, so far, mostly been limited to nonoptimal beamlattice designs. Highresolution 3D additive manufacturing processes are generally limited to viscoelastic polymers, but demonstration of nanolattice performance at the HashinShtrikman upper bound requires linear elastic material properties^{19,21} like those of ceramics and metals (Supplementary Note 5). Ceramic and metallic nanolattices are manufacturable by conversion^{1} from polymer templates such as those printed by TPPDLW. However, closedcell designs impose process restrictions, complicating the adoption of atomiclayerdeposited ceramics^{32,34} as well as electroless^{36} and electroplated^{40} metals, even for the synthesis of composites^{34,35,36} where templates are not removed. Pyrolysis is the only alternative, and requires structures to be fabricated such that they survive extreme linear shrinkage of up to 90%^{6}.
Here, we overcame several critical manufacturing challenges, leading to fabrication of highly efficient virtually closedcell ceramic platetopologies (Supplementary Note 1). As with most additive manufacturing techniques, TPPDLWprinting of fully enclosed cellular geometries results in trapped excess liquid monomer and/or rupture of thin membranes during postprint development. We show that nanometersize pores are sufficient to eliminate residual monomer from assemblies of even tens of micrometersize cells, while retaining mechanical performance on par with fully closed cell topologies. In contrast to TPPDLWderived beamnanolattices, platenanolattices cannot simply be printed from individual line features in one threedimensional trajectory pattern. To address this challenge, we developed an orientationspecific layerbylayer hatching strategy (Supplementary Fig. 2) to combine the highest surface quality with smallest possible wall thicknesses, and fully exploited sizedependent strengthening of the constituent material. Material properties of hatched TPPDLWderived structures are highly sensitive to printing parameters^{46}; thus, carefully selected combinations of laser average power, scan speed and hatching distances were adopted herein to ensure identical constituent properties throughout our nanoarchitected material (Supplementary Fig. 3). To demonstrate property uniformity, for each plate orientation, microRaman spectroscopy measured nearly identical degree of conversion (DC) of prepyrolysis polymeric structures and nearly identical degree of graphitization (DG) for all pyrolytic carbon structures (Supplementary Figs. 3, 5; Supplementary Tables 1, 2). While pyrolysis of TPPDLW printed beamlattices has been demonstrated, the much larger ratio of unit cell dimensions to feature size in platelattices drastically complicates this conversion by pyrolysis, e.g., the overall size of our CO platenanolattices is up to 5 times larger than that of same\(\bar \rho\) octet beamlattices^{6,23} with comparable feature sizes. Given structures with larger surface areas tend to shrink more during pyrolysis, support pillars were coarsely hatched to introduce controlled porosity to lessen the shrinkage mismatch between the large surface area cubic+octet platenanolattices and the monolithic support pillars.
Despite the variability in the properties of pyrolytic carbon at small scales, our cubic+octet platenanolattices clearly reside within the bound bands of Fig. 4. Size effects strongly increase both the strength and stiffness of pyrolytic carbon, with decreasing characteristic length^{27,28,47}. In the range of 1 µm and 5 nm, Young’s moduli around 40 GPa^{28} and 62 GPa^{27} are reported, respectively; the former agrees well with our 6 µmdiameter micropillar compression results (Supplementary Fig. 6). Consequently, the Young’s moduli of the constituent material in our platenanolattices, with wall thicknesses of 150–260 nm, are expected to lie within the above upper and lower limits. Nonlinear least square fitting of the stiffnesses of our most pristine specimens (\(\bar \rho \ge 37.5\%\)) with Eq. (1), assuming a Poisson’s ratio v = 0.17^{48} and considering a correction function to account for the structure’s face holes (Supplementary Note 4), estimated a constituent material stiffness (E_{s}) of 62 GPa. This agrees remarkably well with the literature utilized upper limit in Fig. 4a, confirming performance of our platenanolattices at the HS upper bound. Regarding strength, literaturereported compressive properties^{28} of pyrolytic carbon with characteristic dimensions down to 600 nm agree well with our pillar experiments. Literaturereported ultimate tensile strengths of 2 µmsize specimens of up to 2.75 GPa^{29} are also in good agreement, indicating that our pillarmeasured yield strengths represent the constituent yield strength of our platenanolattices well.
Platenanolattices fully exploit the topological advantages of their plate architecture in the higher relative density range. Although all details of the computer models are not fully reflected in our high\(\bar \rho\) platenanolattices given the limitations of nanomanufacturing, numerical results confirmed that the structural quality of these lattices was sufficient to perform on par with theoretical and numerical predictions, albeit with some experimental scatter. Correspondingly, experiments with polymeric cubic+octet platemicrolattices with nearideal geometries (Supplementary Note 5) correlated well with the pyrolytic carbon results, but with less scatter. Despite manufacturing imperfections, our pyrolytic carbon data has one of the lowest variabilities among the nanolattices presented in Fig. 6. Outliers in stiffness remain within ±6.9% of the average upper bound predictions.
While platelattices clearly outperform beamlattices in the higher relative density range, our results reveal a tradeoff between performance and manufacturability at lower relative densities. Finite element analyses (Fig. 5) showed that the underperformance of the low\(\bar \rho\) platenanolattices with respect to the theoretical upper bounds in Fig. 4 is related to small plate thicknesstosize ratios which promote warping during pyrolysis and hence introduce plate curvature via pyrolysisinduced buckling (Fig. 1). Simulations showed that precurvature is the key mechanism diminishing stiffness, with a yieldingtobuckling transition at \(\bar \rho \approx 37.5\%\) also having a strong effect on the strength. As buckling ideally should not affect stiffness measurements, this explains the notably stronger reduction in yield strength compared to stiffness for the low\(\bar \rho\) platenanolattices. In agreement with the simulations, nearideal polymer platemicrolattices did not show a transition to worse stiffness scaling at lower relative densities, although the lowest relative density sample displayed reduced strength (Supplementary Note 5). The combination of imperfections and premature buckling of low\(\bar \rho\) platenanolattices led to convergence of our data with theoretically less efficient pyrolytic carbon beamnanolattices^{6}, particularly in terms of strength (Fig. 6). While the manufacturing complexity of nanoarchitected materials may amplify these effects, low\(\bar \rho\) platelattices are, regardless of scale, far more difficult to manufacture without significant geometric defects than beamlattices due to the more extreme aspect ratios of individual features. Moreover, the theoretical performance of platelattices improves as the number of plate elements per unit cell is increased (as in the simple cubic + body centered cubic (SCBCC) design^{18}); however, in practice each additional plate element further reduces the lowest manufacturable relative density, and the simplest topologies like the cubic+octet design may be most practical.
Plate curvature may mitigate brittle failure in platelattices. While our high\(\bar \rho\), predeformationfree specimens failed in a brittle manner, increasing plate curvature at low \(\bar \rho\) coincided with a more progressive failure behavior, albeit at the cost of reduced mechanical properties (Fig. 3). In previous work, highly curved, thin shell topologies made of brittle constituent materials, such as pyrolytic carbon, have been shown to exhibit progressive failure with high stress plateaus up to 80% strain^{45}. This behavior arises from an advantageous crack propagation mechanism displayed by shells with high radius of curvaturetothickness ratios. In platenanolattices, plate curvature induced during pyrolysis also produces such behavior. Interestingly, nonpredeformed polymer cubic+octet platemicrolattices exhibited the same transition from catastrophic to progressive failure (Supplementary Fig. 12) as a result of the introduction of plate curvature by postyield buckling. Therefore, preexistent plate curvature is not necessary to induce progressive failure, and elastic buckling may well have contributed to progressive failure in our pyrolytic carbon structures. The implication is that nonpredeformed thinwalled pyrolytic carbon platenanolattices, if manufacturable, may potentially still exhibit beneficial progressive failure without compromising stiffness and yield strength. Platedesigns with more plate elements per unit cell and thinner features, such as the SCBCC design, may exploit postbuckling curvatureinduced progressive failure most efficiently. At the same time, though, thin wall design is complicated by manufacturing limitations and potentially exacerbates defectinduced buckling strength reductions. Ultimately, both preexisting and bucklinginduced curvature are expected to produce a tradeoff between compressive strength and deformability.
The combination of an optimal topology at the HS and Suquet upper bounds, and ultrahigh strength nanoscale constituent pyrolytic carbon, makes our platenanolattices the only cellular material to lie above the theoretical specific strength limit for all bulk materials, as well as outperform all other architected materials in stiffness (Fig. 6). Beamlattices, such as the octet truss, in practice perform on the order of 25 and 20% of the HS and Suquet upper bounds, respectively; which is in good agreement with the found fivefold and sixfold improvement, respectively, of our platenanolattices over TPPDLWderived pyrolytic carbon octet truss and isotropic truss nanolattices^{23} in the density range of 0.35–0.79 g/cm³.
In summary, we demonstrated closedcell platenanolattices manufacturable via TPPDLW and pyrolysis by applying nanometersize face holes, which we showed to impose only small reductions in mechanical performance, orientation specific hatching strategies and optimization of support structures to minimize shrinkage mismatch with specimens during pyrolysis. This study experimentally demonstrates a cellular material reaching the theoretical strength and stiffness limits of an isotropic voided topology. At the same time, the constituent pyrolytic carbon approaches the theoretical material strength limit, thanks to mechanical sizeeffects. This combination of both optimal topology and ultrastrong constituent material culminates in the strongest and stiffest existing architected material, with up to 639% average performance improvement over the most efficient beamnanolattices. While beambased lattices have dominated structural metamaterials^{1} for the past two decades, this study’s comprehensive experimental characterization of platelattices provides important, scaleindependent and processindependent groundwork to establish platearchitecture as a superior design principle. Our results provide understanding of densitydependent performance and deformation behaviors in light of previous theoretical studies and highlight several critical areas of interest for future work, including imperfection sensitivity, and designdependent manufacturing limitations.
Although the TPPDLW/pyrolysis fabrication route developed in this work is limited to small specimens, platelattices with exceptional sizestrengthened constituent materials may be manufactured by more scalable techniques (e.g., projection microstereolithography^{49}) using preceramic resins which exploit size effects into the micrometer range coupled with dramatically reduced shrinkage during pyrolysis^{43}. Hence, platenanolattices deserve strong attention in the development of scalable highresolution additive manufacturing processes^{50,51}, which might in the future make carbonbased nanoarchitected metamaterials as ubiquitous as carbon composites have become in modern engineering applications.
Methods
Fabrication
Polymeric platenanolattices with 5 × 5 × 5 unit cells were fabricated by twophotonpolymerization direct laser writing (TPPDLW), with the photoresist IPDIP (Nanoscribe GmbH) on silicon substrates, using a Photonic Professional GT (Nanoscribe GmbH) DLW system. Vertical cubic walls were composed of six 100 nm hatched lines, printed with a laser average power of 12 mW. Horizontal cubic and octet walls were composed of single inplane voxel lines, printed with laser average powers of 15.25 and 16 mW, respectively. Both octet and vertical cubic walls had 50 nm slicing distances. All walls were printed at a scan speed of 5000 µm s^{−1}. To accommodate otherwise detrimental shrinkage during pyrolysis, structures were printed on multiline coiled springs attached to the top of 340 nmhatched and 680 nmsliced support pillars of aspect ratio 0.85. Pillars were printed at a scan speed of 20,000 µm s^{−1} and an average laser power of 30 mW. Springs were printed at 100 µm s^{−1} and an average laser power of 7.5 mW. Development was performed in a PGMEA bath for 40 min, whereby introduced plate face holes allowed infiltration of the virtually closedcell structures with the solvent and removal of enclosed excess raw material. A subsequent 20 min IPA bath was applied to remove residual PGMEA. The polymeric structures were dried in an Autosamdri 931 critical point dryer (Tousimis Research Corp. Inc.) to avoid collapse of the structures by surface tension during evaporation of the solvent. All structures were subsequently pyrolyzed at 900 °C for 1 h in vacuum^{6}, with a maximum ramp rate of 3 °C min^{−1}. During pyrolysis, lattices linearly shrank 78–80%. Separately, individual lattices were printed atop 0.76 mmdiameter tungsten pins and pyrolyzed for imaging by nanocomputed tomography. Pillars with 6 µmdiameters and aspect ratios of 3^{52}, printed with the same writing parameters of the vertical walls, were fabricated to be representative of the constituent material properties.
Mechanical characterization
Relative densities were determined using Solidworks 2018 (Dassault Systèmes SE) and cubic+octet platelattice models, including holes, with the SEM measured dimensions of the diagonallycrosssectioned polymer calibration structures (Supplementary Fig. 3c). The relative density is assumed to not change significantly, given consistent and isotropic shrinkage during pyrolysis^{6}. Lattice dimensions were optically measured from highresolution SEM using a FEI Magellan 400XHR (Thermo Fisher Scientific Inc.). All lattices were mechanically tested under uniaxial compression on an Alemnis Nanoindenter (Alemnis AG) equipped with a 100 µm diamond flat punch tip inside a FEI Quanta 3D FEG (Thermo Fisher Scientific Inc.) dual beam (SEM/FIB), using a constant strain rate of 0.001 s^{−1}. Frontview in situ SEM videos were used to correct raw load displacement curves for the compliance of the substrate and support pillars via an inhouse digital image correlation (DIC) algorithm. An unloadreload cycle was introduced at strains between 4 and 10%, depending on the relative density of the sample, to ensure suspended structures were fully seated on the support pillar surface while avoiding excessive straininduced sample damage; subsequently, all structures were loaded to failure. Young’s moduli were extracted from the maximal slope of linear elastic regimes of the first unloading segment, and yield strengths were determined from the 0.2% yield offset of the engineering stress–strain plots. The compressive strength was measured from the maximal stress before densification. Constituent material properties were measured from in situ pillar compression tests under the same conditions as the platenanolattices (Supplementary Fig. 6).
Microstructural characterization
Raman spectra were acquired from baked polymeric and pyrolyzed carbon structures (Supplementary Figs. 3a, d, 5) using an inVia (Renishaw plc) confocal Raman microscope with a 50× objective, operated at an excitation wavelength of 532 nm, with a laser intensity of 10% and an exposure time of 1 s over 20 acquisitions. The degree of conversion (DC) was extrapolated from Raman spectra taken of individual polymeric cubic+octet unit cells, pillars printed under the same conditions as the vertical walls, singlevoxel horizontal sheets, and octet platenanolattice unit cells. DC values were calculated using
with I_{C=C} and I_{C=O}, the integrated intensities of carbon–carbon and carbon–oxygen double bond peaks (1635 cm^{−1} and 1730 cm^{−1}, respectively) in the polymerized resin, and \(I^\prime _{{\mathrm{C}} = {\mathrm{C}}}\) and \(I^\prime _{{\mathrm{C}} = {\mathrm{O}}}\), the integrated intensities of the same peaks in the unpolymerized resin^{53} (Supplementary Table 1). The relative degree of graphitization (DG) of pyrolyzed CO plate lattices and pillars was measured via R2 parameter, defined as
where I_{G} (1580 cm^{−1}), I_{D1} (1360 cm^{−1}), and I_{D2} (1620 cm^{−1}) are the integrated intensities of the graphite, firstorder defect 1, and firstorder defect 2 infrared (IR) bands, respectively, as well as via the halfwidthhalfmaximum (HWHM) of the D1 spectral band^{54} (Supplementary Fig. 5; Supplementary Table 2).
Nanocomputed tomography (NanoCT)
Samples were scanned in a Xradia Ultra 810 (Carl Zeiss AG) operating with a rotating chromium anode Xray source with 5.4 keV beam energy, using the phasecontrast imaging mode where the undiffracted Xray beam is phaseshifted by 3π/2 using a gold phasering positioned near the back focal plane of the zone plate. A total of 901 projections were obtained over 180°, with exposure time and detector binning depending on the sample and field of view (Supplementary Table 3). Using the software XMReconstructor (Carl Zeiss AG), the image reconstruction was performed by a filtered backprojection algorithm, the projections were realigned for movement compensation using the adaptive mode compensation (AMC) measurements, and the rotation center was adjusted before obtaining the image stacks datasets. The datasets were processed using the software Avizo v. 9.4.0 (Thermo Fisher Scientific Inc.), being first denoised using its windowed version of the nonlocal means filter applied in 3D with a search window of 21 voxels, voxel neighborhood of 5 voxels and 1.0 similarity factor, run once for the large field of view (LFOV) 30 and 40% sample datasets and twice for the high resolution (HRES) 60% sample dataset. The segmentation of the image was performed using the watershed algorithm, using marker seeds selected through contrast thresholding, and outside/inside of the carbon cell markers selected via the contrastbased tridimensional selection propagation using the magic wand tool, given that phasecontrast gives similar gray levels for the inside and the outside of the carbon cell. Spurious selections due to streaking or thin walls missing from the selection were manually corrected along the sample using the same magic wand tool. The models were subsequently generated from the resulting binary stack using the generate surface module using constrained smoothing 3, which uses an adapted Gaussian filter to smooth voxel edges without losing thin regions in the process, and this surface was remeshed using the Avizo remesh option with high regularity^{55}, 100% size and fixed contours. The remeshed surface was then exported as an STL file.
Finite element analysis
Models of ideal cubic+octet platelattices with and without face holes were constructed in Solidworks 2018 (Dassault Systèmes SE) with SEMmeasured wall and hole dimensions of the fabricated polymeric structures. Finite element analysis was performed in ABAQUS (Dassault Systèmes SE) for relative densities of approximately 25, 30, 40, 50, and 60%. Results were determined to be within 1% from a mesh sensitivity study. At mid and low relative densities, computed mechanical properties of shell and solid models do not differ noticeably (Supplementary Fig. 10). Simulations comparing ideal and imperfect, predeformed models as well as eigenmode analyses to calculate buckling strengths were carried out in ANSYS 16.2 (Ansys Inc.) with shell models (shell181) with an element size of 2.5% of the unit cell size. Predeformations of imperfect models were applied as a percentage of the vertical wall buckling eigenmode of perfect models with zeroed macroscopic displacement. The degree of deformation was adjusted by fitting the computed and experimentally measured average stiffnesses. In all simulations, constituent material properties were representative of nanoscale pyrolytic carbon, with the average yield strength of 2.5 GPa measured from the pyrolytic carbon micropillars, the literature reported 62 GPa^{27} Young’s modulus and a Poisson’s ratio of ν = 0.17^{56}. An additional set of calculations was performed with ν = 0.3, to facilitate comparison with literature values^{21}, and to assess the effects of Poisson’s ratio on mechanical performance. All models were subject to periodic boundary conditions in the form of uniform macroscopic strains^{57,58}.
Data availability
The authors declare that all data supporting the findings of this study are available within the paper and its supplementary information files. The source data underlying Figs. 2a, b, 3a–d, 4a, b, 5a, b, 6a, b and Supplementary Figs. 3a, b, 3d, 5, 6, 7a, b, 8a, b, 9a–d, 10a, b, 12a–f, 13a, b are provided in the Supplementary Data file.
Change history
08 May 2020
An amendment to this paper has been published and can be accessed via a link at the top of the paper.
References
 1.
Bauer, J. et al. Nanolattices: an emerging class of mechanical metamaterials. Adv. Mater. 29, 1701850 (2017).
 2.
Hashin, Z. & Shtrikman, S. A variational approach to the theory of the elastic behaviour of multiphase materials. J. Mech. Phys. Solids 11, 127–140 (1963).
 3.
Suquet, P. M. Overall potentials and extremal surfaces of power law or ideally plastic composites. J. Mech. Phys. Solids 41, 981–1002 (1993).
 4.
Zhang, X., Wang, Y., Ding, B. & Li, X. Design, fabrication, and mechanics of 3D micro‐/nanolattices. Small https://doi.org/10.1002/smll.201902842 (2019).
 5.
Greer, J. R. & Deshpande, V. S. Threedimensional architected materials and structures: design, fabrication, and mechanical behavior. MRS Bull. 44, 750–757 (2019).
 6.
Bauer, J., Schroer, A., Schwaiger, R. & Kraft, O. Approaching theoretical strength in glassy carbon nanolattices. Nat. Mater. 15, 438–443 (2016).
 7.
Khaderi, S. N. et al. The indentation response of Nickel nano double gyroid lattices. Extrem. Mech. Lett. 10, 15–23 (2017).
 8.
do Rosário, J. J. et al. Selfassembled ultra high strength, ultra stiff mechanical metamaterials based on inverse opals. Adv. Eng. Mater. 17, 1420–1424 (2015).
 9.
do Rosário, J. J., Berger, J. B., Lilleodden, E. T., McMeeking, R. M. & Schneider, G. A. The stiffness and strength of metamaterials based on the inverse opal architecture. Extrem. Mech. Lett. 12, 86–96 (2017).
 10.
Biener, J., Hodge, A. M., Hamza, A. V., Hsiung, L. M. & Satcher, J. H. Nanoporous Au: a high yield strength material. J. Appl. Phys. 97, 1–5 (2005).
 11.
Volkert, C. A., Lilleodden, E. T., Kramer, D. & Weissmüller, J. Approaching the theoretical strength in nanoporous Au. Appl. Phys. Lett. 89, 061920 (2006).
 12.
Hodge, A. M. et al. Scaling equation for yield strength of nanoporous opencell foams. Acta Mater. 55, 1343–1349 (2007).
 13.
Deshpande, V. S., Ashby, M. F. & Fleck, N. A. Foam topology bending versus stretching dominated architectures. Acta Mater. 49, 1035–1040 (2001).
 14.
Deshpande, V. S., Fleck, N. A. & Ashby, M. F. Effective properties of the octettruss lattice material. J. Mech. Phys. Solids 49, 1747–1769 (2001).
 15.
Messner, M. C. Optimal latticestructured materials. J. Mech. Phys. Solids 96, 162–183 (2016).
 16.
Dong, L., Deshpande, V. & Wadley, H. Mechanical response of Ti6Al4V octettruss lattice structures. Int. J. Solids Struct. 60, 107–124 (2015).
 17.
Pasini, D. & Guest, J. K. Imperfect architected materials: mechanics and topology optimization. MRS Bull. 44, 766–772 (2019).
 18.
TancogneDejean, T., Diamantopoulou, M., Gorji, M. B., Bonatti, C. & Mohr, D. 3D platelattices: an emerging class of lowdensity metamaterial exhibiting optimal isotropic stiffness. Adv. Mater. 30, 1803334 (2018).
 19.
TancogneDejean, T. & Mohr, D. Elasticallyisotropic truss lattice materials of reduced plastic anisotropy. Int. J. Solids Struct. 138, 24–39 (2018).
 20.
Wadley, H. N. Multifunctional periodic cellular metals. Philos. Trans. R. Soc. A 364, 31–68 (2006).
 21.
Berger, J. B., Wadley, H. N. G. & McMeeking, R. M. Mechanical metamaterials at the theoretical limit of isotropic elastic stiffness. Nature 543, 533–537 (2017).
 22.
Schwaiger, R., Meza, L. R. & Li, X. The extreme mechanics of micro and nanoarchitected materials. MRS Bull. 44, 758–765 (2019).
 23.
Zhang, X., Vyatskikh, A., Gao, H., Greer, J. R. & Li, X. Lightweight, flawtolerant, and ultrastrong nanoarchitected carbon. Proc. Natl. Acad. Sci. USA 116, 6665–6672 (2019).
 24.
Li, X. & Gao, H. Mechanical metamaterials: smaller and stronger. Nat. Mater. 15, 373–374 (2016).
 25.
CES Material Property Database. (Granta Design Limited, Cambridge, UK, 2019). https://grantadesign.com/.
 26.
LaFratta, C. N., Fourkas, J. T., Baldacchini, T. & Farrer, R. A. Multiphoton fabrication. Angew. Chem. Int. Ed. 46, 6238–6258 (2007).
 27.
Manoharan, M. P., Lee, H., Rajagopalan, R., Foley, H. C. & Haque, M. A. Elastic properties of 46 nmthick glassy carbon thin films. Nanoscale Res. Lett. 5, 14 (2010).
 28.
Albiez, A. & Schwaiger, R. Size effect on the strength and deformation behavior of glassy carbon nanopillars. MRS Adv. 4, 133–138 (2019).
 29.
Zhang, X. et al. Theoretical strength and rubberlike behaviour in microsized pyrolytic carbon. Nat. Nanotechnol. 14, 762–769 (2019).
 30.
Gibson, L. J. & Ashby, M. F. Cellular Solids: Structure and Properties. (Cambridge University Press, 1999).
 31.
Jacobsen, A. J., Mahoney, S., Carter, W. B. & Nutt, S. Vitreous carbon microlattice structures. Carbon N. Y. 49, 1025–1032 (2011).
 32.
Meza, L. R., Das, S. & Greer, J. R. Strong, lightweight and recoverable three—dimensional ceramic nanolattices. Science 345, 1322–1326 (2014).
 33.
Zheng, X. et al. Ultralight, ultrastiff mechanical metamaterials. Science 344, 1373–1377 (2014).
 34.
Bauer, J., Hengsbach, S., Tesari, I., Schwaiger, R. & Kraft, O. Highstrength cellular ceramic composites with 3D microarchitecture. Proc. Natl. Acad. Sci. USA 111, 2453–2458 (2014).
 35.
Bauer, J., Schroer, A., Schwaiger, R. & Kraft, O. The impact of size and loading direction on the strength of architected lattice materials. Adv. Eng. Mater. 18, 1537–1543 (2016).
 36.
Mieszala, M. et al. Micromechanics of amorphous metal/polymer hybrid structures with 3D cellular architectures: size effects, buckling behavior, and energy absorption capability. Small 13, 1602514 (2017).
 37.
Torrents, A., Schaedler, T. A., Jacobsen, A. J., Carter, W. B. & Valdevit, L. Characterization of nickelbased microlattice materials with structural hierarchy from the nanometer to the millimeter scale. Acta Mater. 60, 3511–3523 (2012).
 38.
Valdevit, L., Godfrey, S. W., Schaedler, T. A., Jacobsen, A. J. & Carter, W. B. Compressive strength of hollow microlattices: experimental characterization, modeling, and optimal design. J. Mater. Res. 28, 2461–2473 (2013).
 39.
Schaedler, T. A. et al. Ultralight metallic microlattices. Science 334, 962–965 (2011).
 40.
Wendy, Gu,X. & Greer, J. R. Ultrastrong architected Cu mesolattices. Extrem. Mech. Lett. 2, 7–14 (2015).
 41.
Cheng, I.C. & Hodge, A. M. Strength scale behavior of nanoporous Ag, Pd and Cu foams. Scr. Mater. 69, 295–298 (2013).
 42.
Hayes, J. R., Hodge, A. M., Biener, J., Hamza, A. V. & Sieradzki, K. Monolithic nanoporous copper by dealloying MnCu. J. Mater. Res. 21, 2611–2616 (2006).
 43.
Bauer, J. et al. Additive manufacturing of ductile, ultrastrong polymerderived nanoceramics. Matter 1, 1–10 (2019).
 44.
Eckel, Z. C. et al. Additive manufacturing of polymerderived ceramics. Science 351, 58–62 (2016).
 45.
Guell Izard, A., Bauer, J., Crook, C., Turlo, V. & Valdevit, L. Ultrahigh energy absorption multifunctional spinodal nanoarchitectures. Small 15, 1903834 (2019).
 46.
Bauer, J., Guell Izard, A., Zhang, Y., Baldacchini, T. & Valdevit, L. Programmable mechanical properties of twophoton polymerized materials: from nanowires to bulk. Adv. Mater. Technol. 4, 1900146 (2019).
 47.
Ji, Y. et al. Confinementinduced super strong PS/MWNT composite nanofibers. EPL 84, 56002 (2008).
 48.
Zhao, J. X., Bradt, R. C. & Walker, P. L. J. The fracture toughness of glassy carbons at elevated temperatures. Carbon 23, 15–18 (1985).
 49.
Sun, C., Fang, N., Wu, D. M. & Zhang, X. Projection microstereolithography using digital micromirror dynamic mask. Sens. Actuators A Phys. 121, 113–120 (2005).
 50.
Jonušauskas, L. et al. Mesoscale laser 3D printing. Opt. Express 27, 15205–15221 (2019).
 51.
Zheng, X. et al. Multiscale metallic metamaterials. Nat. Mater. 15, 1100–1106 (2016).
 52.
ASTM International. Standard Test Method for Compressive Properties of Rigid Plastics. Annual Book of ASTM Standards 1–8 (2008).
 53.
Jiang, L. J. et al. Twophoton polymerization: investigation of chemical and mechanical properties of resins using Raman microspectroscopy. Opt. Lett. 39, 3034–3037 (2014).
 54.
Sadezky, A., Muckenhuber, H., Grothe, H., Niessner, R. & Pöschl, U. Raman microspectroscopy of soot and related carbonaceous materials: spectral analysis and structural information. Carbon 43, 1731–1742 (2005).
 55.
Zilske, M., Lamecker, H. & Zachow, S. Adaptive remeshing of nonmanifold surfaces. In Eurographics 2008 Annex to the Conference Proceedings. 207–211 (Eurographics Association and Wiley, 2008).
 56.
Garion, C. Mechanical properties for reliability analysis of structures in glassy carbon. World J. Mech. 4, 79–89 (2014).
 57.
Berger, J. B. The Design and Modeling of Periodic Materials with Novel Properties. (University of California, Santa Barbara, 2014).
 58.
Danielsson, M., Parks, D. M. & Boyce, M. C. Threedimensional micromechanical modeling of voided polymeric materials. J. Mech. Phys. Solids 50, 351–379 (2002).
Acknowledgements
This work was financially supported by the Office of Naval Research (program Manager: D. Shifler, Grant No. N000141712874). J.B. gratefully acknowledges partial financial support from the Deutsche Forschungsgemeinschaft (DFG), grant BA 5778/11. SEM imaging and insitu mechanical testing was performed at the UC Irvine Materials Research Institute (IMRI), using instrumentation funded in part by the National Science Foundation Center for Chemistry at the SpaceTime Limit (CHE082913). Raman measurements were conducted at the UCI Laser Spectroscopy Lab. The authors are thankful to Dimitry Fishman for useful discussions on Raman spectroscopy. J.M.S.S. acknowledges the Deutsche Forschungsgemeinschaft (DFG) (WE4051/211).
Author information
Affiliations
Contributions
J.B. and L.V. designed the research. C.C. and J.B. developed the manufacturing procedure. C.C. performed microRaman spectroscopy. C.C., J.B., and A.G. performed mechanical experiments. C.C. and J.B. analyzed experimental data. C.S.O. and J.M.S.S. performed computed tomography and processed scans. J.B. and J.B.B. carried out finite element analyses. C.C., J.B., and L.V. interpreted results and wrote the manuscript.
Corresponding author
Ethics declarations
Competing interests
The authors declare no competing interests.
Additional information
Peer review information Nature Communications thanks Xiaoyan Li and the other anonymous reviewer(s) for their contribution to the peer review of this work. Peer reviewer reports are available.
Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Supplementary information
Source Data
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Crook, C., Bauer, J., Guell Izard, A. et al. Platenanolattices at the theoretical limit of stiffness and strength. Nat Commun 11, 1579 (2020). https://doi.org/10.1038/s41467020154342
Received:
Accepted:
Published:
Further reading

Thermal postcuring as an efficient strategy to eliminate process parameter sensitivity in the mechanical properties of twophoton polymerized materials
Optics Express (2020)

Bending dynamics of viscoelastic photopolymer nanowires
Applied Physics Letters (2020)
Comments
By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.