Many two-dimensional (2D) materials consist of layers of atoms held together in-plane by strong covalent bonds and between planes by weak van der Waals forces. Under stress, these van der Waals materials are susceptible to out-of-plane deformations like buckling and folding. As reported in Physical Review Letters, researchers from China and the United States have now experimentally determined a key deformation parameter of three prototypical van der Waals materials. Their results contradict classical predictions and suggest that interlayer shear plays an important role in determining the mechanical properties of multilayered van der Waals materials.

The bending rigidity of a material layer describes its resistance to deformations that involve a change in curvature. In thin film materials, the parameter is often determined using classical plate theory—an approach that assumes the atomic layers are perfectly glued together. Classical plate theory yields a bending rigidity that scales cubically with layer number (or thickness). If all of the layers are assumed to be ideally lubricated instead of perfectly glued together, the other limiting case, the bending rigidity scales linearly with layer number (or thickness).

“With relatively weak van der Waals forces between [the layers], the actual bending rigidity of various two-dimensional materials could be anywhere between the two widely different theoretical limits,” says Rui Huang, a professor at the University of Texas at Austin (UT Austin) and one of the lead scientists of this new research.

To better understand the bending rigidity of such materials, researchers from the National Center for Nanoscience and Technology in China collaborated with researchers from UT Austin, Tsinghua University in China, and the University of Science and Technology of China on a series of pressurized microbubble experiments. They studied atomically thin sheets of multilayer graphene, hexagonal boron nitride (hBN), and molybdenum disulfide (MoS₂), representative of metallic, insulating, and semiconducting van der Waals materials, respectively. 

The samples were prepared on a silicon substrate covered with a layer of silicon dioxide (SiO₂). The researchers patterned holes in the SiO₂ with radii ranging from 0.5 μm to 1.5 μm. Then the researchers covered and sealed the holes with a multilayer sheet of graphene, hBN, or MoS₂. The nanosheets ranged in thickness from 7 to 70 layers, as confirmed by atomic force microscopy (AFM) and Raman spectroscopy.

Leveraging high pressure in a tunable pressure device, the researchers diffused nitrogen molecules into the 2D material-sealed hole slowly through the substrate. This created a pressure difference across the multilayer sheet, causing it to bulge upward. The bubble-shaped deformation was characterized in terms of height—as measured by AFM—and thickness. The process was repeated over a series of pressure differences for each hole. From the data, the researchers calculated the in-plane Young’s modulus and bending rigidity for each material as a function of thickness. 

As expected, the in-plane Young’s modulus was independent of thickness. The values were in good agreement with measurements made by AFM indentation of monolayers and followed the trend MoS₂ < hBN < graphene.

The bending rigidities increased with thickness, as expected, but did not scale with either of the two theoretical limits. Furthermore, in materials with comparable thickness, the bending rigidity followed a trend opposite that of Young’s moduli, graphene < hBN < MoS₂. This was puzzling at first, Huang says, because it directly contradicted classical plate theory. However, the team soon realized that the reversal could be due to interlayer shear, or slippage.

To explore the effect of interlayer shear, the researchers used molecular dynamics simulations to analyze monolayer and bilayer graphene bubbles. Their results show that the mechanical response—and therefore the bending rigidity—of a bilayer depends on the parameters of the shear interactions between atomic layers. “We conclude that the interlayer shear or slippage between 2D materials [layers] competes with intralayer deformation (stretching and bending) and can modulate the overall mechanical responses,” the authors write. Informed by their results, the team has suggested an approach to calculating effective bending rigidity that incorporates an interlayer shear factor.

Andres Castellanos-Gomez leads a research team investigating the properties of atomically thin materials at the Materials Science Institute of Madrid, part of the Spanish Research Council. “[T]he authors tackle a very fundamental problem in mechanics at the nanoscale,” he says. “Are 2D materials membranes or elastic plates?” This question is complicated because of the strong anisotropic mechanical properties of van der Waals materials, Castellanos-Gomez says. Furthermore, he notes that it has not been clear whether atomic layers slide in response to out-of-plane deformation. He cites this research as clarifying these questions.

“Interestingly, [the authors] found that the mechanical response of all these materials is in between the one expected for a membrane and that expected for a plate,” Castellanos-Gomez says. “Moreover, they found that slippage of atomic layers indeed plays an important role upon out-of-plane deformation.”

As multilayer 2D materials are designed and considered for applications, such as stretchable electronics, understanding the parameters governing their out-of-plane deformation is essential. “Researchers dealing with bending deformations of multilayer 2D materials should be aware of the different scaling of the bending rigidity and the underlying interlayer shear effects,” Huang says. “The simple theoretical limits should be used with caution.”  

Read the abstract in Physical Review Letters.