Bending of Multilayer van der Waals Materials

Out-of-plane deformation patterns, such as buckling, wrinkling, scrolling, and folding, formed by multilayer van der Waals materials have recently seen a surge of interest. One crucial parameter governing these deformations is bending rigidity, on which significant controversy still exists despite extensive research for more than a decade. Here, we report direct measurements of bending rigidity of multilayer graphene, molybdenum disulfide (${\mathrm{MoS}}_2$), and hexagonal boron nitride (hBN) based on pressurized bubbles. By controlling the sample thickness and bubbling deflection, we observe platelike responses of the multilayers and extract both their Young’s modulus and bending rigidity following a nonlinear plate theory. The measured Young’s moduli show good agreement with those reported in the literature ($E_{\text{graphene}}>E_{\text{hBN}}>E_{{\text{MoS}}_2}$), but the bending rigidity follows an opposite trend, $D_{\text{graphene}}<D_{\text{hBN}}<D_{{\text{MoS}}_2}$ for multilayers with comparable thickness, in contrast to the classical plate theory, which is attributed to the interlayer shear effect in the van der Waals materials.

Figure 1

The schematics (a) and corresponding electron microscope images (b) illustrate various deformation modes. (Reproduced with permission from Refs. [4, 8, 14, 15, 17]). Panels (c) and (d) illustrate the microstructural deformation upon bending of multilayer 2D materials with perfectly glued (c) and ultralubricated (d) interfaces.

Figure 2

(a) The schematic of pressurized bubbles with two distinct shape characteristics at small and large deflections. (b) Normalized deflection profiles measured from two sets of $0.5\text{−}\mu \mathrm{m}$-radius bubbles. We observe a membranelike shape with a kinked edge for $h/t\gtrsim 2$ (orange markers) and a platelike shape with a smooth edge for $h/t\lesssim 1.5$ (blue markers).

Figure 3

We show $\mathrm{\Delta }p{a}^4/h^{3}t$ as a function of $h/t$ for graphene (a), hBN (b) and ${\mathrm{MoS}}_2$ (c), respectively. The solid lines represent the idealized nonlinear plate solution, and horizontal dot-dashed lines are the membrane solution. In the color system, the darker symbol denotes the thicker sample. Extracted Young’s modulus (d) and bending rigidity (e) as a function of thickness. At a given thickness, AFM measurements of up to 17 different bubbles produce the results plotted as symbols. The dash-dotted lines in (d) indicate the average values, while the dashed and solid lines in (e) are the theoretical predictions for the perfectly glued ($D\sim N^3$) and ultralubricated ($D\sim N$) cases, respectively.

Figure 4

(a) The mechanical response of monolayer and bilayer graphene bubbles to pressure in MD simulations. The solid line represents the idealized nonlinear plate solution, while the dashed line is the membrane solution. For bilayer samples, a stronger resistance to pressure is observed by enhancing the interlayer interaction strength. The error bars are from thermal fluctuations in MD simulations. (b) Radial distribution of the atomic slip length ($s$) in the bulged bilayer graphene with the height-thickness ratio $h/t$ of $\sim 1.2$ and 1.8, respectively. The right panel of (b) shows the area distribution of $s$ for bilayer graphene with $h/t$ of $\sim 1.2$. Enhancing the strength of interlayer interactions can restrain interlayer sliding, leading to reduced slip lengths and higher bending rigidity.

Figure 5

The layer-dependent function to measure the effect of interlayer coupling on the bending rigidity of multilayered graphene, hBN, and ${\mathrm{MoS}}_2$. The black lines are theoretical predictions of the limiting cases for graphene. Our experimental results are denoted by colored markers, sitting between the two limits.