Recent advances in three-dimensional (3D) graphene assembly have shown how we can make solid porous materials that are lighter than air. It is plausible that these solid materials can be mechanically strong enough for applications under extreme conditions, such as being a substitute for helium in filling up an unpowered flight balloon. However, knowledge of the elastic modulus and strength of the porous graphene assembly as functions of its structure has not been available, preventing evaluation of its feasibility. We combine bottom-up computational modeling with experiments based on 3D-printed models to investigate the mechanics of porous 3D graphene materials, resulting in new designs of carbon materials. Our study reveals that although the 3D graphene assembly has an exceptionally high strength at relatively high density (given the fact that it has a density of 4.6% that of mild steel and is 10 times as strong as mild steel), its mechanical properties decrease with density much faster than those of polymer foams. Our results provide critical densities below which the 3D graphene assembly starts to lose its mechanical advantage over most polymeric cellular materials.

Keywords

In the literature, the strength of a 3D graphene assembly has been measured using previous tensile tests ( 12 , 14 , 16 ). A material with a bulk density of 1.5 to 110 mg/cm 3 has been measured to have a tensile strength of 0.011 to 11 MPa ( 12 , 14 , 16 ). Considering that its building block is graphene, which has a density of 2300 mg/cm 3 and a strength of 130 GPa ( 1 ), its strength is estimated to be 2.2 to 1360 MPa by using the same scaling laws as the polymeric open-cell foam, because the strength-density plot has a slope of 3 / 2 on a double logarithmic plot ( 20 , 21 ). This estimation is several orders of magnitude larger than what is measured in previous experiments ( 12 , 14 , 16 ). Such a sharp difference suggests that the mechanics of the 3D graphene assembly are very different from those of conventional polymeric foams.

Graphene is one of the stiffest and strongest materials ( 1 , 2 ). Besides its outstanding mechanical properties, its atomic thickness and large surface area make it ideal for many engineering applications ( 3 – 5 ), albeit materials-by-design examples of using graphene to form bulk materials and achieve targeted material properties remain rare. A single piece of graphene is too delicate to generate mechanical functions; it is beneficial to have graphene preassembled as a three-dimensional (3D) scaffold, inheriting high stiffness and strength from the 2D graphene as the building block. This assembly by design is applicable to materials science, energy and environmental innovations, and many other fields of study ( 6 – 10 ). There is recent progress in making graphene-based porous materials, and several different experimental methods have been used to produce the material ( 7 , 10 , 11 ). These materials combine lightness with strength, and it is claimed that they can be lighter than air ( 12 ). It is intriguing to ask whether the material, in vacuum, can be a substitute for helium in unpowered flight; in this case, the material must be sufficiently strong to avoid being crushed by the surrounding air pressure. Although the mechanical behavior of graphene per se has been well characterized, the relationship between its porous structure and its mechanics is largely unknown, and the experimental measurements for its stiffness and strength vary widely ( 12 – 19 ). Moreover, the failure mechanisms are not well understood, and it is not clear how its mechanical properties relate to its structure from the nanoscale to larger scales. Mechanical models derived from computational modeling and experiments can guide design strategies to improve the mechanical performance of this material. Here, we focus especially on studying the material architecture that allows graphene to form stable 3D porous bulk materials, implemented at different scales in the material, to achieve a set of target bulk material properties including stiffness and tensile and compression strength, which are the mechanical properties most involved with engineering applications.

RESULTS

Full atomic models of the 3D graphene assembly To understand this difference, we built full atomic models of the 3D graphene assembly in molecular dynamics (MD) simulations by mimicking the synthesis of the porous material. Large-scale simulations based on a reactive force field (22–24) are performed to simulate the process of fusing graphene flakes together into the 3D assembly (see fig. S1 and Supplementary Materials). The initial model, as illustrated in Fig. 1A, is composed of graphene flakes with dimensions following the lognormal distribution to avoid negative values for the dimensions and spherical inclusions of a constant diameter that mimic the effect of water clusters in freeze-casting porous graphene materials (13). The graphene flakes have no functional groups, and the edges are modeled without hydrogen for efficient formation of covalent bonds between flakes, which result in a 3D graphene assembly close to the polycrystalline graphene from chemical vapor deposition (CVD) (25, 26). It has been shown that CVD graphene in low hydrogen pressure is not terminated by hydrogen or functional groups (27). These graphene flakes are randomly distributed and oriented inside the simulation box (Fig. 1B). This initial state is close to a gas with a density of 3.9 mg/cm3. We design a cyclic protocol to condense the material. Each of the cycles is composed of four stages, and we control both the temperature and pressure for each stage, as shown in Fig. 1C. Repeating the cycle enables us to obtain the condensed graphene flakes, as shown in Fig. 1E. We find that, by repeating the cycle more than eight times, we are able to obtain an equilibrated structure and the total number of covalent bonds N C–C converges from 1.22 to 1.4 for each carbon atom (Fig. 1G), which is close to that of the ideal graphene with an infinitely large size (1.5 per atom). After removing the inclusions and equilibrating the carbon material under ambient conditions, we are able to obtain a stable structure of the 3D graphene assembly that closely represents the irregular experimental sample, as shown in Fig. 1F. It has a density of 366.2 mg/cm3, which is 4.6% that of mild steel. The diameter of the spherical inclusions can be tuned during synthesis to adjust the density and make the material lighter. We note that most of the walls are curved adjacent to the junction where several of them meet (Fig. 1F). This curvature is induced by defects in the form of pentagons and heptagons at grain boundaries (25), which cause out-of-plane deformations (28, 29) and achieve defined 3D architectures composed of nondevelopable surfaces (8). Fig. 1 Computational synthesis of the 3D graphene assembly. (A) Initial model composed of 500 randomly distributed rectangular graphene flakes and spherical inclusions. (B) Schematics of the graphene with L dimensions that follows a lognormal distribution as given below and spherical inclusion with uniform d in diameter. (C) The targeting temperature T as a function of simulation time in the alternative NPT-NVT ensemble during each equilibration cycle. (D) The targeting pressure p as a function of simulation time in the alternative NPT-NVT ensemble during each equilibration cycle, which is only applicable to the first stage from 0 to 25 ps. (E) The closely packed graphene-inclusion structure obtained after cyclic equilibrations. (F) The equilibrated structure of the 3D graphene assembly after removing the spherical inclusions with dimensions of 11 nm × 11 nm × 11 nm, and the SEM image of a graphene assembly [reproduced from Wu et al. (8)]. Scale bar, 20 μm (inset). (G) The total number of covalent bonds counted at the end of each anneal cycle, averaged by the total number of carbon atoms in the system.

Mechanical response of the 3D graphene assembly under loading We estimate the mechanical properties of this material by simulating uniaxial tension and compressive tests. A series of snapshots of the structure during the simulation is given in Fig. 2A, and the full force extension–compression curve is given in Fig. 2B. The material has a Young’s modulus of 2.8 GPa (given by the slope of the stress-strain curve at the zero strain point) and a tensile strength of 2.7 GPa (the peak stress in the stress-strain curve), which is an order of magnitude higher than that of mild steel. We observe that the strain stiffening behavior of the material in tension is governed by the bending of the graphene walls, aligning them toward the loading direction at large deformations before rupture (Fig. 2A, iii). The compressive strength of the material is measured to be 0.6 GPa, which corresponds to the point before the more significant increment of the stress (measured by averaging the stress between 0.1 and 0.3 strain), and it is found to be governed by the buckling of the walls in compression (Fig. 2A, i). We find that, for small deformations (less than 0.02 strain), Poisson’s ratio is measured to be 0.3 in both tensile and compressive loadings (Fig. 2C). For large compressive deformations, there is almost zero strain in directions perpendicular to the loading direction once buckling occurs (Fig. 2C and fig. S2 for volume change). This agrees with what has been observed in recent experiments (8, 30). Note that we did not consider functional groups in the current 3D graphene structures because our model is constructed on the basis of the CVD graphene. Additional chemical groups could be expected to affect both the nonbonded interaction between two facing layers and the material strength of the graphene itself; these effects may play roles that could affect the mechanics of the graphene but are not included in our current model. Fig. 2 Mechanical tests on the 3D graphene assembly. (A) Simulation snapshots of the full atomic graphene structure in tension and compressive tests that are taken at ε x = −0.5, 0.0, 0.6, and 1.0 for (i) to (iv), respectively. The atomic stress and its distribution at different strain states are computed and included in fig. S3. The symmetric distribution of positive and negative stress suggests that the graphene is largely bent under deformation. Insets show schematics for the different mechanisms of the material behavior under compression and tension. (B) Full stress-strain curve of the material under compression and tension force. (C) The average strains in the two directions other than the loading direction as a function of ε x ; for |ε x | < 0.02, the slope of the curve is measured to be −0.3. For larger deformations, the three linear fits on the plot have slopes of 0.03, −0.6, and 0.04 from left to right of the curve.

Scaling laws of the mechanics of the 3D graphene assembly We repeat the mechanical tests in computational simulations but use different material samples with their mass density varying between 80 and 962 mg/cm3 by altering the inclusion diameter during the material preparation. We measure the Young’s modulus (E), tensile strength (σ T ), and compressive strength (σ C ) obtained in our simulations (data in table S1), and we normalize these results by the mechanical features of graphene (ρ s , E s , and σ Ts ), as summarized in Fig. 3. In analogy to the mechanics of polymeric open-cell foam (20, 21), we plot the data points as functions of the material density on double logarithmic graphs, which allow us to determine the scaling laws of the mechanical properties of the material as (1) (2) (3) Fig. 3 The normalized Young’s modulus (A), tensile strength (B), and compressive strength (C) of the 3D graphene assembly as a function of its mass density.The data points include mechanical test results of the full atomic 3D graphene assembly (PG), the full atomic gyroid graphene (GG), and the 3D-printed polymer samples (3D-printed). The solid curves are plotted according to scaling laws obtained in the study with slopes of 2.73, 2.01, and 3.01 for (A), (B), and (C), respectively. ρ s = 2300 mg/cm3, E S = 1.02 TPa, and σ Ts = 130 GPa correspond to the density, Young’s modulus, and tensile strength of graphene for its in-plane mechanics, which are used to normalize the properties of graphene materials (PG, GG, and references mentioned). ρ s = 1175 mg/cm3, E S = 2.45 GPa, and σ Ts = 50 MPa correspond to the density, Young’s modulus, and tensile strength of the bulk material properties of polymer material for 3D printing, which are used to normalize the results of 3D-printed samples. Compared to previous experiments on 3D graphene assembly (the so-called graphene aerogel) in the literature, our results show an overall agreement with the results obtained from different measurements (12–19). They follow the same scaling law, suggesting that their mechanics as a function of the density are mainly dominated by their structures, and our simulation model captures the essential mechanism of their deformation under mechanical loading up to failure. However, we note that the experiments described in the literature use several different techniques to prepare the material samples, such as CVD and freeze-drying (7, 11, 19), leading to different defect forms, and thereby, the results are more dispersed than our modeling results. Considering the power index of the functions of mechanics versus density, Young’s modulus (2.73 ± 0.09), tensile strength (2.01 ± 0.05), and compressive strength (3.01 ± 0.01) are overall larger than those of conventional polymeric open-cell foams (2, 1.5, and 2 for Young’s modulus, tensile strength, and compressive elastic strength, respectively) (20, 31), suggesting that the microstructures of the graphene assembly are very different from those of conventional polymeric open-cell foams. From conventional polymeric foams, it has been learned that a bending-dominated foam structure generally gives a larger power index than a stretch-dominated foam structure (21). This point is supported by the distribution of atomic stress in our simulations, which shows that the stress distribution is quite symmetric for atoms under compression (negative stress) and tension (positive stress), as shown in fig. S3. Therefore, the results suggest that the high power indexes of the 3D graphene assembly are caused by the bending-dominated behavior that may be even stronger than the ideal bending-dominated polymeric open-cell foams. Our findings would be applicable to predicting the overall mechanical properties of graphene aerogels. It can be further used not only to decide the aerogel density according to the specific mechanical requests but also to provide a well-defined trade-off between density and mechanics. This knowledge will be useful for designing complex structures and composite materials by assigning materials with an optimized material distribution that leads to the highest mechanical strength with the least total material usage.

Mechanics of a pristine gyroid graphene structure To understand the effect of defects, we built an idealized atomic 3D graphene structure by taking its geometry as a periodic gyroid porous structure, which accounts for the common geometric feature of the graphene structure because each unit cell is composed of a highly curved 2D graphene surface (20, 21, 31). The gyroid shape guarantees minimum density under a given periodicity because the structure is known to have a minimum surface area in a given volume (that is, minimal surface), which corresponds to the 2D nature of graphene in 3D geometry. We obtain equilibrated structures by three different processes (see Supplementary Materials and fig. S4), adopting external potential as (4)where ∑E C–C accounts for all interactions among carbon atoms in graphene, described by the adaptive intermolecular reactive empirical bond order (AIREBO) potential (23, 24); λ is a Lagrange multiplier for the constraint energy of the desired geometry; (x i ,y i ,z i ) is the Cartesian coordinate of each carbon atom; and L is the length constant of the periodic structure in all three directions, which directly affects the density of the graphene material as ρ ~ L− 1 (see table S2 for detailed data). As the result of iterative modifications of the geometry based on our algorithm, all carbon atoms have a bond number of three, which is the same number as the pristine graphene, with mainly heptagon and pentagon defects to foam curved surfaces. Their coordinates satisfy the minimum E graphene , and thus, the overall geometry approximates the mathematical form of the gyroid structure as (5)which has the geometry shown in Fig. 4A. On the basis of our new algorithm, we can design and build the atomic structures of the 3D graphene structure with different length constants (L), as shown in Fig. 4B. Notably, we ensure the convergence of the coordinate number and potential energy before deciding the final structures, and thus, the lattice structure is overall continuous without large holes, and all carbon atoms mainly have a pentagon, hexagon, or heptagon ring shape, which has been proven to not affect the fracture toughness of graphene (25). We investigate the mechanical properties of gyroid graphene structures by applying tensile and compression deformation to the atomic models and by recording the stress in the material (see the Supplementary Materials for the loading conditions and measurement of mechanical properties; see fig. S5 for the stress-strain curves of the loading tests). Their deformation and failure mechanisms largely represent those of the 3D graphene assembly (fig. S6), and their mechanical properties, including Young’s modulus and tensile and compression strength as a function of their material density, are summarized in Fig. 3 (and data in table S2). Fig. 4 Different atomistic and 3D-printed models of gyroid geometry for mechanical tests. (A) Simulation snapshots taken during the modeling of the atomic 3D graphene structure with gyroid geometry, representing key procedures including (i) generating the coordinate of uniformly distributed carbon atoms based on the fcc structure, (ii) generating a gyroid structure with a triangular lattice feature, and (iii) refinement of the modified geometry from a gyroid with a triangular lattice to one with a hexagonal lattice. (B) Five models of gyroid graphene with different length constants of L = 3, 5, 10, 15, and 20 nm from left to right. Scale bar, 2.5 nm. (C) 3D-printed samples of the gyroid structure of various L values and wall thicknesses. Scale bar, 2.5 cm. The tensile and compressive tests on the 3D-printed sample are shown in (D) and (E), respectively.