  The freezing/melting transition is at the heart of many natural and
industrial processes. In the classical picture, the transition proceeds via the
nucleation of the new phase, which has to overcome a barrier associated to the
free energy cost of the growing nucleus. The total nucleation rate is also
influenced by a kinetic factor which somehow depends on the number of attempts
to create a nucleus, that translates into a significant density of proto-nuclei
in the system. These transient tiny nuclei are not accessible to experiments,
but they can be observed in molecular simulations, and their number and size
distributions can be acquired and analysed. The number distributions are
carefully characterized as a function of the system size, showing limited
finite size effects. It is also shown that the proto-nuclei do exist even in
the stable phase, in agreement with the fact that the volume contribution to
their free energy is negligible in the first stages of nucleation. Moreover,
the number and size distributions vary continuously between the stable and the
metastable phases, in particular when crossing the coexistence temperature. The
size distributions associated to \emph{any} nucleus and to the \emph{largest}
one have also been calculated, and their relation recently established for
bubbles in a liquid [J. Puibasset, J. Chem. Phys. 157, 191102 (2022)] has been
shown to apply here. This is an important relation for free energy barrier
calculations with biased molecular simulations.
