  Solid-liquid phase transitions are fundamental physical processes, but
atomically-resolved microscopy has yet to capture both the solid and liquid
dynamics for such a transition. We have developed a new technique for
controlling the melting and freezing of 2D molecular layers on a graphene
field-effect transistor (FET) that allows us to image phase transition dynamics
via atomically-resolved scanning tunneling microscopy. Back-gate voltages
applied to a F4TCNQ-decorated graphene FET induce reversible transitions
between a charge-neutral solid phase and a negatively charged liquid phase.
Nonequilibrium molecular melting dynamics are visualized by rapidly heating the
graphene surface with electrical current and imaging the resulting evolution
toward new equilibrium states. An analytical model has been developed that
explains the observed equilibrium mixed-state phases based on spectroscopic
measurement of both solid and liquid molecular energy levels. Observed
non-equilibrium melting dynamics are consistent with Monte Carlo simulations.
