Deriving Flow Velocity and Initial Concentration From Liesegang-Like Patterns

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 ease of recognition, the underlying mechanisms of pattern-forming processes remain elusive. We introduce a novel advection-dominated phase-field model that effectively replicates the Liesegang-like patterns observed in Zebra rocks. This numerical model leverages the concept of phase separation, a well-established principle governing Liesegang phenomena in a two-dimensional setting. Our findings reveal that initial solute concentration and fluid flow velocity are critical determinants in pattern morphologies. 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.