Self-organizing diffusion-reaction systems naturally form complex patterns under far from equilibrium conditions. A representative example is the rhythmic concentration pattern of Fe-oxides in Zebra rocks; these patterns include reddish-brown stripes, rounded rods, and elliptical spots. Similar patterns are observed in the banded iron formations which are presumed to have formed in the early earth under global glaciation. We propose that such patterns can be used directly (e.g., by computer-vision-analysis) to infer basic quantities relevant to their formation giving information on generalized chemical gradients. Here we present a phase-field model that quantitatively captures the distinct Zebra rock patterns based on the concept of phase separation that describes the process forming Liesegang stripes. We find that diffusive coefficients (i.e., the bulk self-diffusivities and the diffusive mobility of Cahn-Hilliard dynamics) play an essential role in controlling the appearance of regular stripe patterns as well as the transition from stripes to spots. The numerical results are carefully benchmarked with the well-established empirical spacing law, width law, timing law and the Matalon-Packter law. Using this model, we invert for the important process parameters that originate from the intrinsic material properties, the self-diffusivity ratio and the diffusive mobility of Fe-oxides, with a series of Zebra rock samples. This study allows a quantitative prediction of the generalized chemical gradients in mineralized source rocks without intrusive measurements, providing a better intuition for the mineral exploration space.
