[en] 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.