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Recent experiments and computer simulations observe various geometricalformations of nanobubbles in van der Waals heterostructures. Among the wellstudied dome and tent geometries, there is yet least understood pancakegraphene nanobubbles (GNB). This more exotic form exhibits discrete values ofvertical sizes around just a few diameters of the molecules trapped inside theGNBs. We develop a model based on the membrane theory and confined fluidsthermodynamics. Our approach describes the equilibrium properties of such flatGNBs. We show that discrete pancake geometry is the result of disjoiningpressure induced by the trapped fluid inside GNB. The calculated total energydefines a discrete series of the metastable states with the pancake heights,which are multiple to molecular diameter. We observe that the value and thedistribution of the total energy minima crucially depend on the temperature.The energy barriers between metastable states decrease as the temperaturebecomes larger. Also, we demonstrate that the pancake forms are favorable inthe cases of sufficiently low membrane-substrate adhesion energy and the smallnumber of trapped molecules. These properties are in agreement with thepublished simulations and experiments. The numerical comparison of our resultwith molecular dynamics results additionally shows the adequacy of the proposedmodel.

Disjoining pressure oscillations causing height discretization in graphene nanobubbles