Zebra rocks, characterized by their striking reddish-brown stripes, rods, and spots of hematite (Fe-oxide), showcase complex self-organized patterns formed under far-from-equilibrium conditions. Despite their recognition, the underlying mechanisms remain elusive. We introduce a novel advection-dominated phase-field model that effectively replicates the Liesegang-like patterns observed in Zebra rocks. This model leverages the concept of phase separation, a well-established principle governing Liesegang phenomena. Our findings reveal that initial solute concentration and fluid flow velocity are critical determinants in pattern selection and transition. We quantitatively explain the spacing and width of a specific Liesegang-like pattern category. Furthermore, the model demonstrates that vanishingly low initial concentrations promote the formation of oblique patterns, with inclination angles influenced by rock heterogeneity. Additionally, we establish a quantitative relationship between band thickness and geological parameters for orthogonal bands. This enables the characterization of critical geological parameters based solely on static patterns observed in Zebra rocks, providing valuable insights into their formation environments. The diverse patterns in Zebra rocks share similarities with morphologies observed on early Earth and Mars, such as banded iron formations and hematite spherules. Our model, therefore, offers a plausible explanation for the formation mechanisms of these patterns and presents a powerful tool for deciphering the geochemical environments of their origin.

Mineral precipitation can form complex patterns under non-equilibrium conditions, in which two representative patterns are rhythmic Liesegang stripes and fractal dendrites. Interestingly, both patterns occur in the same rock formations, including various dendritic morphologies found in different rocks, such as limestone and sandstone. However, the underlying mechanism for selecting the vastly different mineral precipitation patterns remains unclear. We use a phase-field model to reveal the mechanisms driving pattern selection in mineral precipitation. Simulations allow us to explore the effects of diffusion parameters on determining the dendritic morphologies. We also propose a general criterion to distinguish the resulting dendrites in simulations and field observations based on a qualitative visual distinction into three categories and a quantitative fractal dimension phase diagram. Using this model, we reproduce the classified dendrites in the field and invert for the key parameters that reflect the intrinsic material properties and geological environments. This study provides a quantitative tool for identifying the morphology selection mechanism with potential applications to geological field studies, exploration for resource evaluation, and other potential industrial applications.

The formation of mineral deposits in mesothermal quartz veins is a complex process that has been the subject of much research. The classical fault-valve hypothesis suggests that mineralization occurs when metamorphic fluids are injected during a brittle event and then locked in to mineralize, but this hypothesis does not fully explain the regular spacing of repeated mineralized patterns that are often observed. This paper proposes a new mechanism for mineralizing systems based on the theory of cnoidal waves in solids. Cnoidal waves are standing waves that can persist for long times in materials under compressive and extensional regimes. We investigate mineral deposits by analytical and numerical methods and show that the cnoidal wave instability theory provides a plausible alternative mechanism for mineralizing systems. This study opens a new avenue for field studies to demonstrate that the mechanism-based cnoidal waves play an essential role in the formation of mineral deposits.

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.