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

## TL;DR

This study reveals how iron oxides form in a sequence of distinct layers when hydroxide diffuses into a gel, showing how chemical reactions and diffusion control mineral transformation.

## Contribution

The paper introduces a quantitative framework to resolve sequential precipitation and transformation of iron oxides under diffusion-limited conditions.

## Key findings

- Three distinct iron oxide phases form in sharp, spatially separated fronts: goethite, green rust, and magnetite.
- Front positions follow power-law kinetics with high accuracy (R² ≥ 0.97) across all tested conditions.
- The alkalinity-demand hierarchy explains the transformation sequence and widening of the goethite region.

## Abstract

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, d

i
(t) = α

i

t

βi

, with βi
spanning 0.39–0.56
and high goodness-of-fit (R
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 ΛG/ΛGR/ΛM ≈ 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.