Why does an impurity remain in the liquid during the freezing of many materials?

In the context of zone melting, impurities remain in the liquid phase during freezing because the atoms or molecules of the impurity prefer the liquid state over the solid state at the liquid-solid interface.1

This preference is quantified by the distribution coefficient, $k$, which represents the ratio of the impurity concentration in the newly forming solid to that in the liquid. When $k$ is less than unity, the solid that crystallizes is purer than the liquid from which it forms. As the material freezes, the rejected impurity accumulates in the liquid just ahead of the advancing solid interface. Consequently, the liquid becomes increasingly enriched with impurities as the molten zone travels through the material, eventually concentrating these impurities at the end of the ingot. Furthermore, because the presence of impurities typically depresses the freezing point of a substance, the liquid phase remains molten longer than the pure material, allowing the impurities to be effectively swept toward the end of the charge.12