Thermodynamics of Fe₃O₄-Co₃O₄ and Fe₃O₄-Mn₃O₄ spinel solid solutions at the bulk and nanoscale

High temperature oxide melt solution calorimetry has been performed to investigate the enthalpies of mixing (Δ_mixH) of bulk and nanophase (1 - x)Fe₃O₄-xM₃O₄ (M = Co, Mn) spinel solid solutions. The entropies of mixing (Δ_mixS) were calculated from the configurational entropies based on cation distributions, and the Gibbs free energies of mixing (Δ_mixG) were obtained. The Δ_mixH and Δ_mixG for the (1 - x)Fe₃O₄-xCo₃O₄ system are negative over the complete solid solution range, for both macroscopic and nanoparticulate materials. In (1 - x)Fe₃O₄-xMn₃O₄, the formation enthalpies of cubic Fe₃O₄ (magnetite) and tetragonal Mn₃O₄ (hausmannite) are negative for Mn₃O₄ mole fractions less than 0.67 and slightly positive for higher manganese content. Relative to cubic Fe₃O₄ and cubic Mn₃O₄ (stable at high temperature), the enthalpies and Gibbs energies of mixing are negative over the entire composition range. A combination of measured mixing enthalpies and reported Gibbs energies in the literature provides experimental entropies of mixing. The experimental entropies of mixing are consistent with those calculated from cation distributions for x > 0.3 but are smaller than those predicted for x < 0.3. This discrepancy may be related to the calculations, having treated Fe²⁺ and Fe³⁺ as distinguishable species. The measured surface energies of the (1 - x)Fe₃O₄-xM₃O₄ solid solutions are in the range of 0.6-0.9 J m^-2, similar to those of many other spinels. Because the surface energies are relatively constant, the thermodynamics of mixing at a given particle size throughout the solid solution can be considered independent of the particular particle size, thus confirming and extending the conclusions of a recent study on iron spinels.