Transformation Cascades in Iron Oxides: Quantitative Resolution of Sequential Precipitation Using the Reaction-Diffusion Framework

Sequential phase transformations under transport limitation govern mineral formation, corrosion, and diffusion-driven synthesis, yet equilibrium phase diagrams and well-mixed experiments largely obscure transient intermediates, spatial segregation, and kinetic hierarchies. Here, we use precipitation-diffusion in 1.0 wt % agar hydrogels to resolve the transformation cascade of iron oxides as hydroxide (1.0-3.0 M NaOH) diffuses into Fe²⁺/Fe³⁺-loaded gels, producing three sharp, spatially separated fronts: yellow goethite (α-FeOOH), green rust (Fe²⁺-Fe³⁺ LDH), and black magnetite (Fe₃O₄). Quantitative tracking shows that front positions follow power-law kinetics, <i>d</i> _<i>i</i> (<i>t</i>) = <i>α</i> _<i>i</i> <i>t</i> ^<i>β_i</i> , with <i>β_i</i> spanning 0.39-0.56 and high goodness-of-fit (<i>R</i> ² ≥ 0.97; typically >0.99) across all tested conditions. A coupled Stefan moving-boundary analysis links the parabolic front kinetics to phase-specific hydroxide consumption, yielding an alkalinity-demand hierarchy Λ_G/Λ_GR/Λ_M ≈ 1:1.3:1.9, which rationalizes both the transformation sequence and the progressive widening of the goethite region. Increasing outer hydroxide accelerates all fronts, whereas increasing total iron loading slows propagation; after 96 h, fronts typically penetrate 3-12 mm into the gel. Microscopy and spectroscopy support a solution-mediated dissolution-reprecipitation pathway, and Fe²⁺-rich conditions drive a transition from steady fronts to oscillatory Liesegang banding, demonstrating how diffusion-reaction balance controls both cascade formation and periodic precipitation.