  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.
