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\begin{document}

\title{Coefficients of reaction-diffusion processes derived from patterns in
rocks}



\author[1]{Chong Liu}%
\author[2]{Victor Calo}%
\author[2]{Klaus Regenauer-Lieb}%
\author[3]{Manman Hu}%
\affil[1]{The University of Hong Kong}%
\affil[2]{Curtin University}%
\affil[3]{University of Hong Kong}%


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  \date{June 18, 2024}


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\begin{abstract}
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.%
\end{abstract}%



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