A. Unit cell structure selection and lattice design

Due to the different arrangements of cell struts, cellular structures can deform by either bending or stretching of the cell struts and can be classified into bending-dominated or stretch-dominated structures, respectively. Maxwell ^{Reference Maxwell27} proposed an algebraic rule setting out the condition for a pin-jointed frame of b struts and j frictionless joints to be both statically and kinematically determinate. In three dimensions, the condition is

(1) $$M = b - 3j + 6 = 0\quad .$$

If its joints are locked (rigid joint) and M > 0, its members carry tension or compression when loaded, and it becomes a stretch-dominated structure. A stretch-dominated unit cell structure is substantially more mechanically efficient than its bending-dominated (M < 0) counterpart because the slender structure is much stiffer when stretched than when bent. Thus, we choose stretch-dominated structures to fabricate architected PDC metamaterials.

The octet-truss unit cell, a stretch-dominated structure first proposed by Fuller, ^{Reference Fuller28} is used in this study. The octet-truss structure is a method for filling 3D space with a structurally efficient truss structure of arbitrary cell size [Fig. 2(a)]. The cell has a regular octahedron as its core, surrounded by eight regular tetrahedra distributed on its faces. All the strut elements have identical aspect ratios, with 12 solid rods connected at each node. The cubic symmetry of the cell’s f _cc structure generates a material with nearly isotropic behavior. When made from high specific modulus and strength materials, the octet-truss lattice is therefore a weight efficient, stress supporting cellular topology. The cuboctahedron unit cell, composed of a periodic arrangement of octahedra, is also used in this study [Fig. 2(b)]. While it is not fully rigid, the octahedral sub-units are rigid. It is therefore defined to be a periodically rigid topology. ^{Reference Meza, Phlipot, Portela, Maggi, Montemayor, Comella, Kochmann and Greer29}

FIG. 2. Computer-aided design models for (a) the octet-truss lattice, unit cell, (b) the cuboctahedron lattice, unit cell, and (c) their struts.

The struts of the unit cells are designed to have a square cross-section, thickness, t, and node-to-node length, l. The unit cell size equals $\sqrt 2 l$ and the cubic lattices, consisting of 3 × 3 × 3 unit cells, are constructed by periodic packing of these two kinds of unit cells along their three principal directions. Here, we fabricated samples with relative densities ranging from 1 to 22%. The relative densities were tuned by modifying the aspect ratio (t/l) within the periodic unit cells. More details about the relative density calculation are illustrated in Sec. II.B. Among all these lattices, the length, l, of each strut and hence the overall dimension of the samples have been kept constant. In addition, octet-truss lattices with the same relative density (same aspect ratio, t/l) but varying cell size, i.e., strut thickness and strut length, are also fabricated and tested.

B. Relative density design

The analytical expressions for the relative density of octet-truss and cuboctahedron lattices are derived. For low relative density, the higher order term of the strut aspect ratio, i.e., t/l, due to the nodal effect, is negligible; as the strut aspect ratio increases, the nodal effect plays a more significant role in relative density calculation and neglecting this effect renders noticeable overestimation of the relative density. For octet-truss and cuboctahedron lattice with square cross-sections, the node is considered as a cube and hence the volume of the node is taken as V _node = t ³. The end-to-end length [distance between the edges of two nearest node as shown in Fig. 2(c)] is defined by l′ = l − t, where l is typically called the node-to-node strut length. By considering a repeating unit cell, the relative density accounting for the nodal effect is then expressed as

(2) $$\bar{\rho } = {\rho \over {{\rho _{\rm{s}}}}} = {{{N_{{\rm{strut}}}}{t^2}l' + {N_{{\rm{node}}}}{t^3}} \over {{L^3}}}\quad ,$$

where N _strut and N _node are the number of struts and nodes within a repeating unit cell and L is the edge length of a repeating cubic unit cell.

Considering the cases of octet-truss and cuboctahedron unit cells, N _strut is 24 and 12 for octet-truss and cuboctahedron unit cells, respectively; N _node is 4 and 3 for octet-truss and cuboctahedron unit cells, respectively; L is equal to $\sqrt 2 l$ . Therefore, the relative density expressions are given by

(3) $${\bar{\rho }_{{\rm{octet}}}} = 6\sqrt 2 {{{t^2}} \over {{l^2}}}\left( {1 - {5 \over 6}{t \over l}} \right)\quad ,$$

(4) $${\bar{\rho }_{{\rm{cubo}}}} = 3\sqrt 2 {{{t^2}} \over {{l^2}}}\left( {1 - {3 \over 4}{t \over l}} \right)\quad ,$$

where ${\bar{\rho }_{{\rm{octet}}}}$ and ${\bar{\rho }_{{\rm{cubo}}}}$ are the relative densities of octet-truss and cuboctahedron lattices with nodal correction, respectively. The corresponding expressions without nodal correction simply neglects the term in the parenthesis in Eqs. (3) and (4). The discrepancy between relative densities with and without nodal correction starts to emerge when $\bar{\rho } > 20\%$ . ^{Reference Gibson and Ashby30} In our case, we neglect the nodal volume effect since the relative density range used for experiments is 1–22%.

C. Fabrication of 3D microarchitected PDCs with microsized ligaments

While the previous work has demonstrated the possibility of printing PDCs, the instability and volatile nature make it challenging for further reduction of feature sizes below 100 micrometers. Here, both vinyl- and thiol-siloxanes were chosen due to their high vinyl and thiol content, respectively. This increases the crosslink density of the system, which simultaneously increases the printed preceramic structure stiffness, critical for low-density fabrication, while also eliminating excessive burn-off and shrinkage during pyrolysis to ceramics. Previous PDC studies have utilized thiol–ene chemistries, but either did not report their monomers or utilizes unstable alkoxy silane precursors, which limits shelf life stability. ^{Reference Eckel, Zhou, Martin, Jacobsen, Carter and Schaedler25,Reference Zanchetta, Cattaldo, Franchin, Schwentenwein, Homa, Brusatin and Colombo31} The monomers were combined with a photoinitiator and photo-absorber for high-resolution printing. UV curing occurred through the free radical thiol–ene click reaction. A radical quencher was also added to the preceramic monomer to maintain resin stability before printing. ^{Reference Rakas and Jacobine32,Reference Lowe and Bowman33} Thiol–ene click chemistry is a highly efficient chemistry, and mixing of monomers without this stabilizer or other stabilizers can readily lead to premature “dark” crosslinking within light-blocked containers, even without the photoinitiator. This dark polymerization has been reported to be likely caused by peroxide impurities and also an ambient oxygen ground state charge transfer process; hence, stabilizers are nearly always required.

A high-resolution, large-area stereolithography system was used for the “green part” (preceramic parts before pyrolysis) fabrication, as shown in Fig. 1(a). The 3D models were first built using a custom code and then sliced into 2D patterns. These 2D patterns are sequentially transmitted to a spatial light modulator, 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 2D image, and the substrate on which it rests is lowered to reflow a thin film of liquid over the cured layer. The image projection is then repeated, with the next image slice forming the subsequent layer. To increase the resolution, the layer thickness during the 3D printing process must be well controlled. Experiments were conducted to study the effect of polymerization depth against light power projected onto the liquid surface, demonstrating the reliable printing layer thickness control below 10 micrometers.

To further expand the scalability of the architected metamaterial and examine the size effects under a wide range of structural feature size control, the spatial light modulator is coordinated with an optical scanning system to produce large-scale parts with microscale resolution. As the mirrors scan, 2D patterns are reflected onto a new area next to the previously exposed area. The pattern change on the projector is coordinated with the scanning rate of the scanning mirror system. A customized focusing lens is used below the scanning optics to project the image onto the liquid surface. The frame is updated as the image is moved via the scanning optics to effectively create a continuous image in the photosensitive material. As this scanned image is much larger than a single image of the projector, it enables small feature sizes over a large area. This technique allows the fabrication of preceramic parts, hundreds of millimeters in size, with multiscale 3D architected features down to the 10-μm scale [Figs. 1(b) and 1(c)], which is uniquely suitable for fabricating lattice materials with a broad range of feature sizes.

The green parts are then pyrolyzed to form the PDC lattices (Fig. 3). Pyrolysis was carried out in a tube furnace based on previous reports, ^{Reference Eckel, Zhou, Martin, Jacobsen, Carter and Schaedler25} ramping at 1 °C/min, holding at 1000 °C for 1 h, and then ramping to room temperature at 3 °C/min under ultra-high purity argon. Siloxane conversion is accompanied by water, methane, and low-level alkene degasification leaving behind a blend of silica (SiO₂), carbon, and SiOC depending on the resin carbon content and processing. ^{Reference Eckel, Zhou, Martin, Jacobsen, Carter and Schaedler25}

FIG. 3. (a and b) Scanning electron micrographs of the octet-truss lattice and the cuboctahedron lattice after pyrolysis. (c and d) Scanning electron micrographs for the unit cell of the octet-truss lattice and the cuboctahedron lattice.

D. Experimental design

The mass and volume of all the ceramic lattices are measured after pyrolysis. The relative density is calculated as the ratio between the mass and volume and compared with the designed volume fraction. The as-fabricated SiOC lattices were tested at ambient temperature in free compression along [001] direction, as shown in Fig. 2, at a nominal strain rate of 8 × 10⁻⁴ s⁻¹ on Instron 5944 using standard flat compression plates (T1223-1022 Instron, Norwood, Massachusetts). The peak load divided by the cross-sectional area of the lattices was defined as the effective strength of the lattices. The Young’s modulus was extracted from the steady slope of the stress strain curves. The tested strength, Young’s modulus of lattices, and relative density of the lattices are used to plot the scaling relationship and size effect analysis in Sec. III.

A. Polymerization depth characterization

Traditional liquid resin compositions used for stereolithography usually consist of liquid monomer, photoinitiator, and passively absorbing dye. The photoinitiator will release radicals when exposed to UV light. Solidification of the liquid monomer occurs as a result of cross-linking when it reacts with radicals within the preceramic monomers. Passively absorbing dye such as Sudan is used for absorbing the UV light to well control the curing depth of the liquid monomer. We characterized the working curves of the preceramic monomer through optimization of the photo-absorber and vinyl-thiol siloxane as well as the addition of the stabilization quencher. The working curve equation ^{Reference Tumbleston, Shirvanyants, Ermoshkin, Janusziewicz, Johnson, Kelly, Chen, Pinschmidt, Rolland, Ermoshkin, Samulski and DeSimone34} is used to describe the curing depth of the preceramic resin, C _d, in a form of

(5) $${C_{\rm{d}}} = {1 \over \alpha }\ln \left( {{{{\alpha _{\rm{I}}}E} \over {{E_{\rm{c}}}}}} \right)\quad ,$$

where α = α_I + α_D is the resin absorption coefficient, α_I is the absorption coefficient of the photoinitiator, α_D is the coefficient of the passively absorbing dye, E is the actual incident energies, and E _c is the critical incident energy needed for cure, in units of mJ/cm².

To characterize the polymerization depth of a free layer, an array of overhanging bridge structures was fabricated with an array of photon energy [Fig. 4(a), inset]. Each bridge layer represented one polymerization layer, and the mean thickness of a given layer was taken to be the polymerization depth. Figure 4(a) shows the variation of the polymerization depth as a function of total exposure energy (mJ/cm²) received by the liquid resin. The results indicate that the polymerization depth is linearly proportional to the natural logarithm of UV exposure energy, which is in good agreement with the numerical model. The experimental results confirmed the needed exposure energy for reducing the polymerization depth, which are needed for reliably creating final feature size below 10 μm.

FIG. 4. (a) Photo flux versus polymerization depth. The curing depth increases linearly with natural logarithm of exposure energy. (b) EDS of a single strut inside the lattice.

B. Polymer-derived ceramic characterization

The pyrolysis procedure was accompanied by 37% weight loss and 34% linear shrinkage. Scanning electron microscopy with energy dispersive X-ray analysis (EDS) was performed. The result [Fig. 4(b)] shows that the sample has a composition of 20.44 atomic percent (at.%) carbon, 38.6 at.% oxygen, and 40.96 at.% silicon, or SiO_0.9C_0.5 The content of oxygen is similar to the previous studies; however, the content of carbon is lower and the content of silicon is higher compared to the previous studies. ^{Reference Eckel, Zhou, Martin, Jacobsen, Carter and Schaedler25} Previous reports have shown the significant dependence of SiOC on composition. ^{Reference Du, Wang, Lin and Zhang35} These effects can be generalized as the SiOC becomes closer in composition to silicon carbide (SiC), i.e the elimination of oxygen, the greater the mechanical properties, Young's modulus and hardness. The mechanical properties of SiC are approximately double that of SiOC. Variation in composition can lead to an approximate doubling in properties.

C. Strength and Young’s modulus scaling in high-temperature ceramic lattices

The testing results of Young’s modulus and strength of both octet-truss and cuboctahedron are summarized in Fig. 5. The compressive strength versus relative density is plotted for each architecture in Fig. 5(a) and compared to other octet-truss lattices with different parent solids. ^{Reference Zheng, Lee, Weisgraber, Shusteff, DeOtte, Duoss, Kuntz, Biener, Ge, Jackson, Kucheyev, Fang and Spadaccini17,Reference Dong, Deshpande and Wadley36}

FIG. 5. Scaling law of effective strength and Young’s modulus of the octet-truss and cuboctahedron lattices. The experimental results of previous studies are shown as a comparison.

Well-developed theories ^{Reference Dong, Deshpande and Wadley36,Reference Deshpande, Fleck and Ashby37} show that on the macroscale, under uniaxial compressive loading, the compressive stiffness and yield strength of these structures theoretically show linear scaling relationships: $E{\rm{/}}{E_{\rm{s}}} \propto \bar{\rho }$ and $\sigma {\rm{/}}{\sigma _{\rm{s}}} \propto \bar{\rho }$ , where E _s is the Young’s modulus of the base material and σ_s is the strength of the base material. However, as indicated in Fig. 5(a), the previously reported experimental results ^{Reference Zheng, Lee, Weisgraber, Shusteff, DeOtte, Duoss, Kuntz, Biener, Ge, Jackson, Kucheyev, Fang and Spadaccini17,Reference Dong, Deshpande and Wadley36} show that under uniaxial compression, the scaling factors for stretch-dominated lattices are normally in the range of 1.06–1.21, or even as high as 2.4, larger than the theoretical value, since there are fabrication imperfections weakening the performance of these lattices.

Interestingly, in testing results of the PDC lattices, the apparent strength (σ)–relative density $\left( {\bar{\rho }} \right)$ scaling in the as-fabricated ceramic lattices shows a scaling power that is smaller than 1 for both octet-truss lattices and cuboctahedron lattices, outperforming theoretically predicted scaling powers. By contrast, the obtained scaling relationship of Young’s modulus (E) and relative density $\left( {\bar{\rho }} \right)$ in Fig. 5(b) is consistent with previously reported scaling values as well as predicted by theory on stretch-dominated lattices. It should be commented that the cuboctahedron lattice here is considered to be a stretch-dominated lattice for its periodic rigidity. We hypothesize that the strength of these architected lattices are influenced by possible strong size-dependent effects as the smallest ligament becomes slender in the microscale range (i.e., the reduction of strut size), which will be further investigated in the following sections.

D. Strut thickness-dependent PDC strength

To investigate the size dependency of PDC strength, the strength of the parent solid that comprises the individual solid strut members for each sample is estimated by normalization of the effective compressive strength of the lattices with its topology-dependent factor. The effective strength of cellular materials can be approximated by the first-order scaling law ^{Reference Mocingkharnklang, Elzey and Wadley38} :

(6) $${\sigma _{{\rm{eff}}}} = \Sigma {\sigma _{\rm{s}}}\quad ,$$

where Σ is a lattice topology and geometry dependent scaling factor, and σ_s is the fracture strength of the base material. The scaling factor Σ, accounting for both the lattice geometry and relative density, can be calculated by

(7) $$\Sigma = C{\bar{\rho }^\alpha }\quad ,$$

where C is a geometric parameter depending on the lattice topology as well as the loading direction, $\bar{\rho }$ is a function of (t/l), and the exponent α is 1 for stretch-dominated behavior, theoretically. ^{Reference Deshpande, Ashby and Fleck39} Based on previous studies, the experimental results of the geometric parameter, C, for octet-truss lattices and cuboctahedron lattices are 0.3 and 0.16 ^{Reference Meza, Phlipot, Portela, Maggi, Montemayor, Comella, Kochmann and Greer29,Reference Deshpande, Fleck and Ashby37} for [001] direction loading, respectively.

Such that the effective strength of the lattice can be normalized by the geometric factor to estimate the strength of the individual ligament

(8) $${\sigma _{\rm{n}}} = {{{\sigma _{{\rm{eff}}}}} \over \Sigma }\quad ,$$

where σ_n can be seen as an estimation of the strength of the parent solid.

Here, as relative density reduces, the only changing parameter for both octet-truss lattices and cuboctahedron lattices is strut thickness while the strut length has been kept constant. The normalized strength of the parent solid excluding all geometric factors is plotted against the only variable strut thickness in Fig. 6. The size-dependent mechanical property of lattices with differing relative density from Fig. 5(a) can now be visualized in Fig. 6. The means of estimated PDC strength are calculated using each 10-μm interval for strut thickness for strut volume to filter the scattering of the data. ^{Reference Flores, Bordia, Bernard, Uhlemann, Krenkel and Motz40} The strength of the PDC solid carries a power-law relationship with the strut thickness as there is no characteristic length for the brittle PDC strut. ^{Reference Flores, Bordia, Bernard, Uhlemann, Krenkel and Motz40–Reference Tsu, Mugele and Mcclintock43} A power-law fitting is applied to the calculated mean strength of the ceramic solids as a function of strut thickness and yields the following relationship:

(9) $${\sigma _{\rm{s}}} = D{t^{ - \beta }}\quad ,$$

where β is the scaling factor of the strut thickness and is found to be 0.36, from our measured experimental results, while D is a constant. The normalized PDC strength increases as the strut thickness decreases, as shown in Fig. 6, indicating a size dependency of the strength of the PDC solids possibly due to the reduced size and number of cracks.

FIG. 6. Effect of strut diameter on the normalized strength of relative-density-controlled ceramic lattices. The means of estimated PDC strength are calculated using each 10-μm interval for strut thickness for strut volume to filter the scattering of the data.

Previous studies also show that the strength of brittle ^{Reference Riley44–Reference Chantikul, Bennison and Lawn45} materials typically increases with decreasing dimensions. Based on the relationship proposed by Griffith between the fracture strength, σ_f, and the critical size of a flaw, c, for brittle materials such as ceramics, ^{Reference Griffith46} we know that

(10) $${\sigma _{\rm{f}}} \propto {1 \over {\sqrt c }}\quad .$$

A flaw cannot be larger than the component in which it is located. Assuming c correlates with the strut thickness, t, ^{Reference Gao, Ji, Jager, Arzt and Fratzl47} the relationship can be written as

(11) $${\sigma _{\rm{f}}} \propto {t^{ - 0.5}}\quad .$$

Even though the scaling factor from our measured experimental results is lower than the value proposed by Griffith, which may be caused by the insufficient number of measurements and experimental errors, our experimental results indicate a size dependency of the strength of the PDC solids.

This size effect as a function of reduced strut thickness in density-controlled sample explains the apparent scaling law of ${\sigma _{{\rm{eff}}}} - \bar{\rho }$ . By combining Eqs. (6), (7), and (9), the effective strength of the lattice can then be modified as

(12) $${\sigma _{{\rm{eff}}}} = CD{\bar{\rho }^\alpha }{t^{ - \beta }}\quad .$$

Substitute relative density expression without nodal correction into Eq. (12), we get

(13) $${\sigma _{{\rm{eff}}}} = CD{\bar{\rho }^{\alpha - {\beta \over {2 }}}}{l^{ - \beta }}\quad .$$

Equation (13) can be further modified as

(14) $${\sigma _{{\rm{eff}}}} \sim {\bar{\rho }^{\alpha - {\beta \over {2}}}}\quad .$$

From Eq. (14), we know that the scaling between the effective strength and relative density of the lattice is reduced by a factor of $- {\beta \over {2}}$ , which is caused by the size dependency of the PDC strength. As density further reduces, this ${\beta \over {2}}$ will further increases, which will result in higher specific strength in lower density solid architected ceramics as compared to higher density samples, i.e., an apparent scaling power smaller than 1 as density decreases further.

E. Strut volume-dependent PDC strength analogous to Weibull size effect

We then proceed to investigate the size strengthening effect by measuring the strength of lattices of identical relative densities but with uniform reduction of minimal strut volumes. While the benefit of reducing the coating thickness of ceramics through ALD has been experimentally demonstrated in previous studies in Refs. Reference Liontas and Greer48 and Reference Meza, Zelhofer, Clarke, Mateos, Kochmann and Greer49 and explained by the weakest link theory, ^{Reference Zok50} the role of the overall size, i.e., strut volume of each solid ligament that comprise the overall lattices, is not clear. The size from the length-direction, which contributes to the surface volume of individual solid strut members, in addition to the diameter and thickness of the strut could contribute the overall size effect of the lattice material. Here, to fully capture the size dependency of the PDC strength, we expand the study by simultaneously modifying the average strut thickness and length within the unit cells. The samples were fabricated with identical relative densities but with a serial reduction of single strut volumes. The Weibull statistical analysis is applied to interpret the relationship between strut volume and material strength.

Derived from the weakest link theory based on a chain model, ^{Reference Bazant41,Reference Bazant42} the risk of failure of the material is given by a probabilistic expression:

(15) $${P_{\rm{f}}}\left( \sigma \right) = 1 - \exp \left\{ { - {{P\left( \sigma \right)V} \over {{V_0}}}} \right\}\quad ,$$

where P _f(σ) represents the risk of failure of a material under stress σ; V is the volume of the material and V ₀ is a standardized reference volume; P(σ) is defined as the probability of material failure under stress σ for a given material volume V. Weibull ^{Reference Weibull51,Reference Weibull52} proposed a power-law solution for P(σ), vanishing at a critical value of σ_c as $P\left( \sigma \right) = {\left( {{{\sigma - {\sigma _{\rm{c}}}} \over {{\sigma _0}}}} \right)^m}$ , where σ_c is the critical strength below which the probability of failure is zero, for brittle material σ_c = 0 since for brittle materials, any tensile stress can cause brittle fracture; σ₀ is the characteristic strength at P _f(σ) = 63.2%; m is the well-known Weibull modulus. ^{Reference Hertzberg53}

Therefore,

(16) $${P_{\rm{f}}}\left( \sigma \right) = 1 - \exp \left\{ { - {{\left( {{\sigma \over {{\sigma _0}}}} \right)}^m}{V \over {{V_0}}}} \right\}\quad .$$

For the same risk of failure under two different stresses, we obtain ${{{\sigma _1}} \over {{\sigma _2}}} = {\left( {{{{V_2}} \over {{V_1}}}} \right)^{{1 \over m}}}$ , which leads to the expression of nominal strength of material,

(17) $${\bar{\sigma }_{\rm{f}}} \propto {\left( {{1 \over V}} \right)^{{1 \over m}}}\quad .$$

In the present case, V = t ² l represents the strut volume. Volume is a function of strut thickness (t) as defined in Fig. 2.

According to the theorem as in Eq. (17), the strength of PDC relates to the strut volume by a simple power law from a statistical viewpoint as shown in Fig. 7, which plots the normalized PDC strength versus the strut volume (V). The mean of the estimated strength was calculated using each order of magnitude for the strut volume. The scaling power in Fig. 7 shows the trend of the size dependency of strength as a function of strut volume reduction over three decades: decrease of the strut volume leads to the increase of the PDC strength. For 3D architected brittle ceramic metamaterials, the failure of the metamaterials is determined by successive failure of multiple struts similar to the case in Ref. Reference Genet, Couegnat, Tomsia and Ritchie58. The failure of individual strut is catastrophic due to the brittle nature of PDC, which suggests the applicability of the Weibull analysis and weakest link theory only on the strut volume instead of the total volume of the architected material. ^{Reference Weibull52} However, to obtain the Weibull modulus and size effect for 3D microarchitected brittle ceramic metamaterials, a nonclassical probabilistic model is needed to account for structural factors, which is discussed in Sec. IV.

FIG. 7. Effect of strut volume on the normalized strength of volume-controlled ceramic lattices. The mean of the estimated strength was calculated using each order of magnitude for the strut volume.