  We study, by means of computer simulations, the crystal-melt interface of
three different systems: hard-spheres, Lennard Jones and the TIP4P/2005 water
model. In particular, we focus on the dynamics of surface waves. We observe
that the processes involved in the relaxation of surface waves are
characterized by distinct time scales: a slow one related to the continuous
recrystallization and melting, that is governed by capillary forces; and a fast
one which we suggest to be due to a combination of processes that quickly cause
small perturbations to the shape of the interface (like e. g. Rayleigh waves,
subdiffusion, or attachment/detachment of particles to/from the crystal). The
relaxation of surface waves becomes dominated by the slow process as the
wavelength increases. Moreover, we see that the slow relaxation is not
influenced by the details of the microscopic dynamics. In a time scale
characteristic for the diffusion of the liquid phase, the relaxation dynamics
of the crystal-melt interface of water is around one order of magnitude slower
than that of Lennard Jones or hard spheres, which we ascribe to the presence of
orientational degrees of freedom in the water molecule. Finally, we estimate
the rate of crystal growth from our analysis of the capillary wave dynamics and
compare it with previous simulation studies and with experiments for the case
of water.
