Entropic Effects of Thermal Rippling on van der Waals Interactions between Monolayer Graphene and a Rigid Substrate

Graphene monolayer, with extremely low flexural stiffness, displays spontaneous rippling due to thermal fluctuations at a finite temperature. When a graphene membrane is placed on a solid substrate, the adhesive interactions between graphene and the substrate could considerably suppress thermal rippling. On the other hand, the statistical nature of thermal rippling adds an entropic contribution to the graphene-substrate interactions. In this paper we present a statistical mechanics analysis on thermal rippling of monolayer graphene supported on a rigid substrate, assuming a generic form of van der Waals interactions between graphene and substrate at T = 0 K. The rippling amplitude, the equilibrium average separation, and the average interaction energy are predicted simultaneously and compared with molecular dynamics (MD) simulations. While the amplitude of thermal rippling is reduced by adhesive interactions, the entropic contribution leads to an effective repulsion. As a result, the equilibrium average separation increases and the effective adhesion energy decreases with increasing temperature. Moreover, the effect of a biaxial pre-strain in graphene is considered, and a buckling instability is predicted at a critical compressive strain that depends on both the temperature and the adhesive interactions. Limited by the harmonic approximations, the theoretical predictions agree with MD simulations only for relatively small rippling amplitudes but can be extended to account for the anharmonic effects.