  The possible presence of amorphous and heterogeneous phases in the inner
crust of a neutron star is expected to reduce the electrical conductivity of
the crust, with potentially important consequences on the magneto-thermal
evolution of the star. In cooling simulations, the disorder is quantified by an
impurity parameter which is often taken as a free parameter. We aim to give a
quantitative prediction of the impurity parameter as a function of the density
in the crust,performing microscopic calculations including up-to-date
microphysics of the crust. A multi-component approach is developed at finite
temperature using a compressible liquid drop description of the ions with an
improved energy functional based on recent microscopic nuclear models and
optimized on extended Thomas-Fermi calculations. Thermodynamic consistency is
ensured by adding a rearrangement term and deviations from the linear mixing
rule are included in the liquid phase. The impurity parameter is consistently
calculated at the crystallization temperature as determined in the
one-component plasma approximation for the different functionals. Our
calculations show that at the crystallization temperature the composition of
the inner crust is dominated by nuclei with charge number around $Z \approx
40$, while the range of the $Z$ distribution varies from about 20 near the
neutron drip to about 40 closer to the crust-core transition. This reflects on
the behavior of the impurity parameter that monotonically increases with
density up to around 40 in the deeper regions of the inner crust. Our study
shows that the contribution of impurities is non-negligible, thus potentially
having an impact on the transport properties in the neutron-star crust. The
obtained values of the impurity parameter represent a lower limit; larger
values are expected in the presence of non-spherical geometries and/or fast
cooling dynamics.
