Discrete structure of pancake graphene nanobubbles.

The recent experiments and computer simulations observe various geometrical formations of the nanobubbles in the van der Waals heterostructures. Among well studied spherical and tent geometries, there is yet least understood pancake (or flat islands) graphene nanobubbles (GNB). This more exotic form exhibits discrete values of the vertical sizes around just a few diameters of the molecules which are trapped inside the GNBs. We develop a model based on the membrane theory and the confined fluids thermodynamics. Our approach describes the equilibrium properties of such flat GNBs. We show that discrete pancake geometry is a result of the disjoining (solvation) pressure induced by the trapped fluid inside GNB. The calculated total energy defines a discrete series of the metastables states with the pancake heights, which are multiple to molecular diameter. We observe that the value and the distribution of the total energy minimums crucially depend on the temperature. More precisely, the energy barriers between metastable states and their depths decrease as the temperature becomes larger. Also, we demonstrate that the pancake forms are favorable in the cases of sufficiently low membrane-substrate adhesion energy and the number of trapped molecules. These properties are in agreement with the published simulations and experiments. The numerical comparison of our result with molecular dynamics results additionally shows the adequacy of the proposed model.