  The interior of a neutron star (NS) is usually assumed to be made of cold
catalyzed matter. However, the outer layers are unlikely to remain in full
equilibrium during the formation of the star and its cooling, especially after
crystallization. We study the cooling and equilibrium composition of the outer
layers of a NS down to crystallization. Here the impurity parameter, usually a
free parameter in cooling simulations, is calculated self-consistently using a
microscopic model for which a unified equation of state has recently been
determined. We follow the evolution of the nuclear distributions of the
multi-component Coulomb liquid plasma (MCP) fully self-consistently, adapting a
general formalism originally developed for the description of supernova cores.
We calculate the impurity parameter at the crystallization as determined in the
one-component plasma (OCP) approximation. Our analysis shows that the sharp
changes in composition obtained in the OCP approximation are smoothed out when
a full nuclear distribution is allowed. The Coulomb coupling parameter at
melting is found to be reasonably close to the canonical value of 175, except
for specific pressures for which supercooling occurs in the OCP approximation.
Our MCP treatment leads to non-monotonic variations of the impurity parameter
with pressure. Its values can change by several orders of magnitude reaching
about 50, suggesting that the crust may be composed of an alternation of pure
(highly conductive) and impure (highly resistive) layers. The results presented
here complement the recent unified equation of state obtained with the same
model. Our self-consistent approach to hot MCP shows that the presence of
impurities in the outer crust of a NS is non-negligible and may have a sizeable
impact on transport properties. In turn, this may have important implications
for the cooling of NS and their magneto-rotational evolution.
