Introduction

Graphene, a two-dimensional monolayer in the form of a honeycomb lattice, has sparked an avalanche of research activities due to a series of outstanding properties and potential applications. It is considered among the strongest materials ever reported, with an in-plane stiffness C = 345 N/m equivalent to 1 TPa1,2 and tensile strength of more than 100 GPa.3,4,5,6 On the other hand, graphene is flexible against out-of-plane deformation because of its low bending stiffness of ~1.5 eV.1,7,8 The unique combination of the extremely high in-plane stiffness with small out-of-plane bending modulus suggests graphene as a unique building material component for fabricating origami mechanologic in flexible electronics9 as well as in nanoelectromechanical devices.10

Recently, an experimental work shows that graphene can act as a paper, suitable for folding nontrivial origami structures, which allows the use of graphene to build 3D microscopic structures with tunable mechanical properties and designed functionalities.11 In the produced origami structures, graphene unavoidably undergoes various deformations, such as stretching and bending. In particular, the bending deformations have a rich variety of forms and are crucial to the stability of graphene origami, a pre-designed or arbitrarily crumpled 3D object made of planar 2D sheet-material, confined within a volume ~ L3. First, folding graphene inevitably needs to overcome the bending energy concentrated in the folds-ridges (bilayer edges12) of length scale ~ L. The stabilization of such folded graphene is by energy gain from van der Waals (vdW) attraction in the contact-stacking regions, scaling as ~ –L2. Since the vdW energy is proportional to the folded area, there should exist a critical size of graphene above which the energy change (const × L – const′ × L2) turns negative, and folded structures can be thermodynamically favorable or at least metastable. To the best of our knowledge, the question of what this critical size is remains unanswered. Second, the construction of more complicated origami structures often requires more than single folding of graphene. A double fold involves much more mechanical distortion when formed, and there will be an additional energy cost for the four-layer folding corner, or vertex (circle in Figure 1b), which is positive and tends to destabilize the double-folded structure. Although the contribution of such corners scales as only ~ L0, their energy is quite high and cannot be overlooked. Similarly, it contributes in a generic origami structure such as origami waterbomb, whose bistability is central for mechanologic.9 Evaluating the smallest length scale from graphene’s physical properties should in turn define the memory or other origami-associated functionality spatial density, or “spatial resolution” ~ L of the smallest features in any complex origami sustained mechanically.

Figure 1.
figure 1

(a) The schematic figure of the flat square-shaped graphene used to construct the origami structure. The x- and y-directions are along armchair (AC) and zigzag (ZZ) edges. The number of units in the x-direction is 3n, and a is the lattice constant of graphene; I and II indicate the folding sequence. The crease pattern is shown as dashed lines for valley folds, and dashed-dotted lines for mountain fold, with a cartoon of the final folded origami shown in the inset. (b) The optimized atomic structure of double-folded origami: top view (upper left), side view along x (right), and side view along y (bottom), respectively, with free edges highlighted:13 AC, red; ZZ, blue. r1 (r2) represents the curvature radius of the bent single-layer graphene edge (the middle plane of the bent bilayer graphene edge). The circle marks the folding vertex.

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A question arises as to how to quantify the contributions of different scaling with the size L, and, in particular, what is the energy cost of the double-fold corners/vertices? To shed light on this problem, it is necessary to perform extensive computations, due to dimensions involved much larger than atomic, for energy analysis on the folded graphene structure, and construct a comprehensive map of the size-dependent folding. Here we combine atomistic computations with a continuum model to evaluate the feasibility of folding a graphene sheet. Our result shows that a single fold is not favorable when the size is below 67 Å, and only when the size is more than 210 Å is the double fold more favorable. Overall, it puts the length scale of the smallest origamis into L ≳ 25–30 nm.

Results and discussion

Our origami-like elementary structures are made from a square-shaped graphene sheet, as shown in Figure 1a. We define the edge in the AC-direction to be along the x-axis, and that in the ZZ-direction to be along the y-axis. Note that x, y can be along any orthogonal directions, because from a linear elasticity standpoint, graphene is an isotropic material.14 The edge along y has a length of L = 5na, and that along the x-direction is set to be 3√3na, which is equal to 3√3/5 L ≃1.04 L. Here n is the number of repeated units along the x- or y-direction, and a = 2.42 Å is the lattice constant of graphene.

By folding this graphene sheet in half along the middle line in the y-direction (the so-called valley fold), and then further folding it at the middle along the x-direction (Figure 1a), we obtain the double-folded origami structure. It is a key element (Figure 1b) in more complex origami. It represents the simplest possible order-4 vertex,15 where three valley folds meet with a single mountain fold (in accordance with Maekawa’s theorem for flat-foldable origami16,17).

The double-folded origami consists of four layers, with four edges that have highly different morphologies and terminations. Two adjacent edges are free, and the other two are folded. The optimized (relaxed to minimal energy) geometry of double-folded graphene origami is shown in Figure 1b. Notably, the two free edges are not straight, as expected. One of the edges has the two inner layers protrude outward with respect to the two outer (top and bottom) layers. An analogous but weaker effect occurs for the other free edge, where the two middle layers appear to be pulled slightly inward in vicinity of the free AC|ZZ free corner. This is due to the asymmetry of the edges, especially the presence of a double-folded edge and two single-folded edges. As for the two folded edges, one can be created from a folded bilayer origami (Figure 1b, bottom), yet the bending along this edge is coupled with a minor structural buckling attributed to the compressive strain in the inner graphene layer; the other is two folded single-layer graphene sheets stacked together (Figure 1b, right).

The single-folded origami structures are created using the same set of graphene sheets. By valley-folding the graphene sheet in half along the middle line in the y-direction, we obtain the single-folded origami. This hairpin-like bilayer structure has three free edges, and the other edge is folded; a representative optimized geometry is shown in Figure 2a. Details of the computational setup are given in the Methods section.

Figure 2.
figure 2

(a) The optimized single-folded origami geometry. L stands for the length in the y-direction, and r′ represents the curvature radius of the bent single-folded armchair (AC) edge. (b) Computed deformation energy Edeform (symbols) of single-folded origami structures as a function of the size L. The dotted line is a polynomial fit to the calculated data points according to Equation 7.

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The deformation energy of double-folded origami has three contributions. One is the bending energy from the two folded edges, which manifests as two folded single-layer graphene sheets stacked together, and one folded bilayer graphene sheet, respectively. The bending energy should be proportional to L. Another term is the energy gain due to vdW attraction between layers, which is proportional to the stacked surface area, thus scaling as –L2. The third one is the energy cost of forming the origami vertex (the circled corner in Figure 1b), which should be a constant. Adding up these contributions, the deformation energy can be tentatively written as a quadratic function of L:

$$ E_{{{\text{deform}}}} = aL^{{2}} + bL + c. $$
(1)

The deformation energies calculated as the total energies of an origami geometry relative to that of the respective flat square graphene patch are shown as data points in Figure 3 and indeed display an inverted parabolic dependence on L. The second-order polynomial in Equation 1 provides an excellent fit, resulting in a = –0.0157 eV/Å2, b = 1.88 eV/Å, and c = 7.9 eV. In the formal limit L → 0, Equation 1 should give the energy cost of the origami vertex (or corner) cc together with energy (negative) at the triple tip-contacts, evaluated below c′ (i.e., together is c = cc + 3c′ = 7.9 eV). Note that c ≫ kBT but is roughly equivalent in magnitude to the energy of just a single covalent carbon–carbon triple bond and seems to be optimistic, not too prohibitive, for fabricating double-folded origami structures.

Figure 3.
figure 3

The deformation energy Edeform of double-folded origami (inset image) as a function of the size L. The bending energy per unit area of graphene Ebend as a function of the curvature radius r is given in the inset plot, calculated from carbon nanotubes of different diameters 2r. Dotted lines are polynomial fits to the calculated data points (according to Equation 1 for Edeform and linear for Ebend).

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We further analyze Edeform by breaking it into more specific contributions, not only by different scaling in Equation 1. Analytically, the deformation energy of a double-folded graphene origami can be written as

$$ E_{{{\text{deform}}}} = E_{{{\text{bend1}}}} + E_{{{\text{bend2}}}} + E_{{{\text{vdW}}}} + E_{0} , $$
(2)

where Ebend1 is the bending energy of the two stacked folded single-layer graphene, Ebend2 is the bending energy corresponding to the folded bilayer graphene edge, EvdW is the overall interfacial energy due to vdW interaction between layers (total 4, that is 3 vdW interfaces), and E0 is the energy of the folding corner. Ebend1 can be expressed as

$$ E_{{{\text{bend}}1}} \; = \;1/2k_{b} r_{1}^{ - 2} S, $$
(3)

where kb is the bending stiffness of graphene that can be determined by calculating the bending energy in straight carbon nanotubes (CNTs), see the Methods section. The bending energy of a CNT as a function of r−2 is plotted in the inset of Figure 3. By fitting the LAMMPS results, we obtain kb = 0.95 eV for both zigzag and armchair CNTs, very close to previous empirical-potential values ranging from 0.83 eV to 1.17 eV.18,19 S is the equivalent area of the bending section at the edge. Specifically, we take this area as two 3/4 cylinder surfaces with a curvature radius of r1 ≃ 5 Å and a length of L/2 (Figure 1b). Thus, S can be expressed as S ≃ ¾\(\cdot \) 2πr1L. Similarly, Ebend2 can be expressed as

$$ E_{{{\text{bend}}2}} \; = \;1/2k^{\prime}_{b} r_{2}^{ - 2} S^{\prime}, $$
(4)

where k′b is the bending stiffness of the folded bilayer graphene, whose value cannot be written immediately but can be obtained later by comparing it with the fitting, and S′ is the area of 3/4 cylinder surface. Here we take the curvature radius r2 ≃ 8.5 Å as the value in the middle plane of the bent bilayer graphene, and the length of S′ is equal to 1.04L/2 (Figure 1b). Thus \(S^{\prime} \simeq 3/4 \cdot 2\pi r_{2} \cdot \frac{1.04}{2}L\). On the other hand, there are stacked four layers of square graphene of size \(\frac{L}{2}\) ×1.04 L/2. Edge corrections need to be considered due to the aforementioned protrusion of the two inner layers with respect to the two outer layers. The area of the protruded bilayer is in the shape of an acute isosceles triangle with an apex angle of ≃5°. Therefore, the overall interfacial energy can be expressed as

$$ E_{{{\text{vdW}}}} = \left[ {3\frac{L}{2}\frac{1.04L}{2} + 2\left( {\frac{1.04L}{2}} \right)^{2} \sin 5^\circ } \right]\gamma , $$
(5)

where γ is interfacial energy per unit area, which is –18.7 meV/Å2 by our calculation. By inserting Equations 3–5 into Equation 2, we analytically obtain the deformation energy of the double-folded origami structure as

$$ E_{{{\text{deform}}}}^{a} \; = \;{-}0.0154L^{2} \; + \;\left( {0.448\; + \;0.144k^{\prime}_{b} } \right)L\; + \;E_{0} . $$
(6)

The coefficient of the quadratic term (–0.0154) agrees very well with the direct fitting result above (a = –0.0157), supporting the validity of the aforementioned fitting. If we assume the same coefficient of the linear term and the constant term (i.e., 0.448 + 0.144 k′b = 1.88 eV/Å), we would even obtain k′b = 9.94 eV specifically for the bilayer, and again E0 = 7.9 eV. The calculated k′b is about 10 times higher than kb. It is reasonable because k′b is supposed to be larger than kb due to the increased thickness, but smaller than the no-slip bending stiffness of bilayer graphene (exceeding kb by ~ 100 times20), since in our case of origami folding, the two layers can slip relative to each other.

Besides the double-folded origami element, we have also evaluated the energetics of single-fold structures, for comparison purposes. Similarly, the energy cost comes from the bending energy of the folded edge, the energy gain originating from the vdW interaction between the two layers, and possibly nonscalable (with L) contribution from the still interacting corner-tips of graphene, c′. The deformation energy of single-folded origami can be written as

$$ E_{{{\text{deform}}}} \; = \;a^{\prime}L^{2} \; + \;b^{\prime}L\; + \;2c^{\prime}. $$
(7)

Calculated results for the left-hand side of Equation 7 are shown in Figure 2b and display a parabolic dependence on L. Fitting using the quadratic function as the right-hand side of Equation 7 gives a′ = –0.0097 eV/Å2, b′ = 0.69 eV/Å, and 2c′ = –3.5 eV. The latter negative value of intercept implies that the edge corner-tips of graphene attract each other, causing some reduction in energy relative to its planar form.

To verify the numerical results, we perform further analysis of E′deform. The deformation energy of a single-folded graphene origami can be analytically expressed as

$$ E_{{{\text{deform}}}}^{{a^{\prime}}} \; = \;E^{\prime}_{{{\text{bend}}}} \; + \;E^{\prime}_{\text{vdW}} \; + \;E^{\prime}_{0} , $$
(8)

where E′bend is the bending energy of the folded single-layer graphene, E′vdW is the overall interfacial energy due to vdW interaction between layers, and E′0 is the energy result from the coupling of bending to the edge stress. E′bend can be expressed as

$$ E^{\prime}_{{{\text{bend}}}} \; = \;1/2k_{b} r^{\prime-2} S^{\prime}, $$
(9)

where kb = 0.95 eV, r′ ≃ 3.5 Å, S′ ≃ (3/4 2π 3.5 Å)×L, and therefore E′bend = 0.64 eV/Å × L. The interfacial energy due to vdW interaction can be written as

$$ E^{\prime}_{\text{vdW}} \; = \;\gamma L1.04L/2,$$
(10)

hence, the deformation energy can be expressed as

$$ E_{{{\text{deform}}}}^{{a^{\prime}}} \; = \; - 0.0097L^{2} + \, 0.64L + E^{\prime}_{0}. $$
(11)

The coefficient of the quadratic term (–0.0097) and the linear term (0.64) agrees perfectly with the fitting result (a′ = –0.0097 eV/Å2, b′ = 0.69 eV/Å). Apparently, E′0 = –3.5 eV. With negative energy associated with the two corner-tip contacts (2c′ = –0 3.5 eV), we can now recover the energy of the double-fold corner as cc = c – 3c′ ≃ 13 eV.

Combining the energy analysis of single- and double-folded graphene structures together, one is able to construct a “phase” diagram, a morphology preference map, as shown in Figure 4, which describes at which size the L form is energetically preferred: flat-planar, single-folded, or double-folded. The purple and green curves are the fitted deformation energies for single-folded and double-folded origami elements, respectively. This diagram reveals a strong size dependence of graphene folding feasibility. We find that all the graphene origamis with different folding multiplicity are energetically stable only when the size of the graphene sheet is larger than the critical value. When L > 210 Å, the double-folded origami structure is the most energetically favorable. While within 67 Å < L < 210 Å, single-folded, but not higher, origami structures are most stable. For L < 67 Å, not even the simplest folded origami structure can form; the graphene energy is dominated by elasticity, and it remains flat. We note that all the numerical results obtained are based on AIREBO potential, which is known to underestimate the stiffness,21 kb. From more realistic density-functional theory (DFT) calculations, it is approximately twice as large (also somewhat varying, depending on the exchange–correlation functionals used). Consequently, the length scale in the map of Figure 4 can be larger, toward L > 10 nm for a single fold and L > 35 nm for the double fold, an element definitive for any nontrivial origami design.

Figure 4.
figure 4

Diagram of graphene folds based on the deformation energy as a function of size L. No origami can form in the shaded region. Single-folded origami (purple squares, see cartoons) is energetically favorable in the middle region, L ~ 7–21 nm, whereas double-folded origami (green circles) can exist stabilized in the larger-L region, L ≳ 21 nm.

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In the previous analysis, we have only discussed pristine, ideal (single-crystal) graphene at zero temperature (finite temperature and geometry effects on graphene conformation have been studied, for example, in22). These results can be extended to a more general case considering that the stability of origami increases with the strength of the vdW interaction and decreases with the bending stiffness. Since increasing the temperature reduces the effective interlayer contact area but tends to increase the effective bending stiffness of graphene,23 the origami becomes overall less stable at a higher temperature. Similarly, polycrystallinity in graphene would disfavor the effective interlayer contact and increase the bending stiffness, thereby lowering the overall stability. It is worth noting that compared to the typical paper origami that are created through irreversible plastic deformation, the graphene origami is entirely different in that they are folded by the ubiquitous vdW adhesion. Therefore, a graphene origami can revert to a perfect graphene sheet upon unfolding.

Conclusion

We found that for nano-origami from graphene, its very name could only be justified to some degree since there is the lower bound, as previously revealed. At too small a length scale, determined by inherent material properties (mostly bending stiffness and surface energy), even the simplest origami element cannot be achieved. We have determined the threshold size for forming single- and double-folded graphene origami elements by large-scale simulations using the AIREBO potential. In particular, the additional energy cost for the folding corner in double-folded origami is estimated to be ≃13 eV (for comparison, this value is nearly twice the energy of a covalent C≡C triple bond). Since the repeated controlled folding of graphene along arbitrary directions appears to be feasible in practice (e.g., by the tip of a scanning tunneling microscope24), the obtained size-dependent stability domains of graphene origami opens a possibility to guide the formation of origami structures in experiment and invites similar analysis for graphene-derivative-based origami as well.25,26

Methods

All geometry optimization and energy calculations are performed in LAMMPS27 using the AIREBO potential.28 The latter has been shown to accurately capture the bonded interactions between carbon atoms. For the long-range Lennard–Jones term, a cutoff scale factor of 3.0 is used. All structures are fully relaxed using the conjugate gradient method until the force on each atom is less than 10–15 eV/Å. We set the vacuum region to adjacent images in periodic boundary conditions to be over 30 Å to avoid any spurious interactions. For constructing the origami, we used large graphene square patches, as shown in Figure 1a. To test the influence of the edge termination on the energetics, we have compared graphene sheets with pristine bare edges and hydrogen-terminated edges, finding no quantitative changes in the results. Therefore, we focus on results obtained by using graphene with bare, unpassivated edges. For calculating the bending stiffness, single-walled CNTs with different diameters are routinely used,1,19 with periodic images laterally separated by a vacuum region of over 30 Å. Although the bending stiffness may exhibit some anisotropy in empirical-potential calculations (differing for armchair and zigzag directions),29 the relevant curvatures in this work are deemed too small to cause any nontrivial effect.