Methane adsorbed on graphite. II. A model of the commensurate-incommensurate transitions

A two-structure model is presented for the pressure- and temperature-driven commensurate-incommensurate ($C-I$) transitions of an adlayer of methane adsorbed on the basal plane of graphite. The statistical-thermodynamic properties of the system are calculated from an empirical intermolecular potential with substrate-mediated effects. Free-energy constructions predict transition temperatures and pressures in agreement with experiment. The quantitative anharmonic finite-temperature model demonstrates directly that the zero-temperature configuration is a $\sqrt{3}\times \sqrt{3}$ registered phase. The $C-I$ transitions of the model are first order, and the incommensurate phases (compressed and expanded) are free-floating two-dimensional "solids" without domain walls. The model results are consistent with existing static theories of "misfit dislocations" in that a transition to a free-floating unwalled phase preempts the system from reaching the critical misfit necessary for the spontaneous formation of domain walls.