Liquid-gas phase transition of Ar inside graphene nanobubbles on the graphite substrate.

Graphene nanobubbles (GNBs) are formed when a substance is trapped between a graphene sheet (a 2D crystal) and an atomically flat substrate. The physical state of the substance inside GNBs can vary from the gas phase to crystal clusters. In this paper, we present a theoretical description of the gas-liquid phase transition of argon inside GNBs. The energy minimization concept is used to calculate the equilibrium properties of the bubble at constant temperature for a given mass of captured substance. We consider the total energy as a sum of the elastic energy of the graphene sheet, the bulk energy of the inner substance and the energy of adhesion between this substance, the substrate and graphene. The developed model allows us to reveal a correlation between the size of the bubble and the physical state of the substance inside it. A special case of a GNB that consists of argon trapped between a graphene sheet and a graphite substrate is considered. We predict the 'forbidden range' of radii, within which no stable GNBs exist, that separates bubble sizes with liquid argon inside from bubble sizes with gaseous argon. The height-to-radius ratio of the bubble is found to be constant for radii greater than 200 nm, which is consistent with experimental observations. The proposed model can be extended to various types of trapped substances and 2D crystals.