Correlations between particles can lead to subtle and sometimes counterintuitive phenomena. We analyze one such case, occurring during the sudden expansion of fermions in a lattice when the initial state has a strong admixture of double occupancies. We promote the notion of quantum distillation: during the expansion and in the case of strongly repulsive interactions, doublons group together, forming a nearly ideal band insulator, which is metastable with low entropy. We propose that this effect could be used for cooling purposes in experiments with two-component Fermi gases.