Periodic Precipitation in Hele-Shaw Cells: Mechanism Insights from Experiments and Numerical Simulation Considering Layer Thickness Effects

Liesegang patterns are self-organized macroscopic structures formed by precipitates. Traditional methods for producing these patterns rely on gel phases, which inhibit flow but introduce complications, such as undesired chemical interactions. This study presents a strategic approach utilizing a "Hele-Shaw cell", consisting of two glass plates that create a narrow gap (30-225 μm). This setup prevents flow due to friction between glass surfaces without the use of a gel. We identified two critical thresholds for the gap thickness: one that inhibits flow and another that facilitates discrete precipitate bands. For aqueous solutions of CuCl₂ and K₂CrO₄, these thresholds were determined to be 110 and 150 μm, respectively. Below the first threshold, the period of precipitate bands increased with a greater gap thickness. This thickness dependency was successfully reproduced using our proposed mathematical model based on the microscopic dynamics of precipitation processes instead of the phenomenological step function model commonly used. Our numerical simulations indicate that the nucleation rate of precipitates significantly influences the formation of bands, with the nucleation rate having an inverse effect on band spacing. It is assumed that the nucleation rate is proportional to the specific surface area of the glass plate, and thus, the practical nucleation rate was proportional to the inverse of the thickness. This research provides a simplified experimental method and a robust mathematical model for Liesegang pattern formation, potentially paving the way for the further exploration of self-organization applications.