  We discuss the growth process of a crystalline phase out of a metastable
over-compressed liquid that is brought into contact with a crystalline
substrate. The process is modeled by means of molecular dynamics. The particles
interact via the Lennard-Jones potential and their motion is locally
thermalized by Langevin dynamics. We characterize the relaxation process of the
solid-liquid interface, showing that the growth speed is maximal for liquid
densities above the solid coexistence density, and that the structural
properties of the interface rapidly converge to equilibrium-like properties. In
particular, we show that the off-equilibrium dynamic stiffness can be extracted
using capillary wave theory arguments, even if the growth front moves fast
compared to the typical diffusion time of the compressed liquid, and that the
dynamic stiffness converges to the equilibrium stiffness in times much shorter
than the diffusion time.
