Extended Data Fig. 6: Theoretical calculations complementing the densification discussion in the main text.

a, Pilling-Bedworth ratios^33,58 (\({R}_{{\rm{PB}}}\)) for all the possible oxides involved, including Fe₂O₃, Fe₃O₄, FeO, and NiO. Such a quantity reflects the molar volume ratio between a pure metal and its oxide: \({R}_{{\rm{PB}}}={V}_{{\rm{oxide}}}/n{V}_{{\rm{metal}}}\), here \(n\) accounts for the metal atom stoichiometry per molar volume of the oxide. For the reduction process considered here (turning oxide to metal), the relative volumetric shrinkage can be theoretically predicted as: \((\Delta V\,/{V}_{{\rm{oxide}}})=1-1/{R}_{{\rm{PB}}}\). By opting a simple composite theory, we further estimated that the maximum possible volumetric shrinkage when per unit mass of Fe₂O₃ + NiO mixtures is completely reduced to metallic state is 46.7 %. b, Comparison between the measured volumetric shrinkage of the pellet at different reduction stages and the theoretical maximum possible value. Excessive Volumetric shrinkage is clearly seen, which accounts for more than 1/3 of the total volumetric shrinkage, confirming the pronounced effect of sintering-driven densification during the reduction process. c, comparison between the Fe-Ni interdiffusion coefficient and the Ni self-diffusion coefficient. Here the interdiffusion coefficient is determined through Darken’s second equation⁴³: \(\widetilde{{\mathscr{D}}}={c}_{{\rm{Fe}}}\,{D}_{{\rm{Ni\; in\; Fe}}}+{c}_{{\rm{Ni}}}\,{D}_{{\rm{Fe\; in\; Ni}}}\), where \({D}_{{\rm{Ni\; in\; Fe}}}\) and \({D}_{{\rm{Fe\; in\; Ni}}}\) are the tracer diffusion coefficient of Ni in Fe and Fe in Ni. The cation self-diffusion coefficient in NiO is also included as a reference. Raw data for the calculations shown in c are obtained from the literature^41,59,60.