The classical nonlinear mechanics of wrinkles in 2D materials

Two-dimensional (2D) materials are extraordinary for two main reasons. One is the set of mechanical properties that makes those materials a versatile playground where multiple modelling objects are possible, from the thinnest membranes with minuscule bending rigidity to very stiff plates, all with very reproducible parameters.

The other reason is that both the electronic and mechanical properties of 2D materials can be fine-tuned by assembling them into van der Waals heterostructures, which expands even more the range of materials parameters and boundary conditions.

The so-called van der Waals materials consist of two-dimensional layers bound by weak van der Waals forces. After the isolation of graphene, the field of two-dimensional van der Waals materials has experienced an explosive growth and new families of two-dimensional systems and block-layered bulk materials have been created.

The atomic thinness of the membranes makes it possible to switch between different regimes—from where the bending rigidity is much smaller than the Young modulus (monolayer membrane), to where they are comparable (in the case of few-layer membranes). This interplay may have an undesired consequence: the neutrally planar state is not stable and, with minute levels of compression, it wrinkles, sometimes creating complex patterns.

In materials with such a sensibility to structure, understanding the formation and spread of wrinkles is most important. For example, the formation of wrinkling patterns is usually associated with a non-trivial strain distribution, which, for piezoelectric 2D crystals such as molybdenum disulfide (MoS₂) and hexagonal boron nitride (h-BN), leads to a complex distribution of the electric field and can be used in respective devices. Another possibility is extracting valuable information from the emerging patterns, like the material parameters that characterize the elastic moduli of 2D membranes and their vdW interaction with the substrate, that can be used for metrological purposes.

Now, a team of scientists has studied ¹ patterns of radially oriented wrinkles induced by bubbles in vdW heterostructures that consist of h-BN or MoS2 layers on top of another 2D material. They find that he shape and wavelength of the wrinkles depend not only on the thickness of the two-dimensional crystal forming the bubble, but also on the atomistic structure of the interface between the bubble and the substrate, which can be controlled by their relative orientation.

The scientists argue that the periodic nature of these patterns emanates from an energetic balance between the resistance of the top membrane to bending, which favours large wavelength of wrinkles, and the membrane-substrate vdW attraction, which favours small wrinkle amplitude. Employing the classical “Winkler foundation” model of elasticity theory, they show that the number of radial wrinkles conveys a valuable relationship between the bending rigidity of the top membrane and the strength of the vdW interaction.

These findings demonstrate that classical nonlinear mechanics of solids, which describes strain-induced elastic instabilities, can be applied (with some modifications) to the description of pattern formation in complex vdW heterostructures and provide a metrological tool for characterizing the interplay between the vdW attraction and the bending modulus of monolayers and multilayer composites.

Author: César Tomé López is a science writer and the editor of Mapping Ignorance

Disclaimer: Parts of this article may have been copied verbatim or almost verbatim from the referenced research papers.