  The heavy element ashes of rp-process hydrogen and helium burning in
accreting neutron stars are compressed to high density where they freeze,
forming the outer crust of the star. We calculate the chemical separation on
freezing for a number of different nuclear mixtures resulting from a range of
burning conditions for the rp-process. We confirm the generic result that light
nuclei are preferentially retained in the liquid and heavy nuclei in the solid.
This is in agreement with the previous study of a 17-component mixture of
rp-process ashes by Horowitz et al. (2007), but extends that result to a much
larger range of compositions. We also find an alternate phase separation regime
for the lightest ash mixtures which does not demonstrate this generic
behaviour. With a few exceptions, we find that chemical separation reduces the
expected $Q_{\rm imp}$ in the outer crust compared to the initial rp-process
ash, where $Q_{\rm imp}$ measures the mean-square dispersion in atomic number
$Z$ of the nuclei in the mixture. We find that the fractional spread of $Z$
plays a role in setting the amount of chemical separation and is strongly
correlated to the divergence between the two/three-component approximations and
the full component model. The contrast in $Y_e$ between the initial rp-process
ashes and the equilibrium liquid composition is similar to that assumed in
earlier two-component models of compositionally driven convection, except for
very light compositions which produce nearly negligible convective driving. We
discuss the implications of these results for observations of accreting neutron
stars.
