Pattern formation in porous media via the Liesegang ring mechanism

A model for the spontaneous formation of patterned deposits in porous media, reminiscent of Liesegang rings (LR) formation in gels, is presented. Simulations show that under certain conditions the frontal deposition of a continuous solid film changes to a quasi-periodic pattern formation of bands. Bifurcation analysis of a simpler, skeleton model explains LR formation as instability of a uniformly propagating plane reaction front to a time periodic solution. A theoretical stability criterion is developed that suggests that key parameters in pattern formation are the critical concentration for nucleation and the speed of the autocatalytic growth reaction. While our theory is not aimed at explaining all LR formation experiments in colloidal systems, it can be applied in the case of strong concentration gradients. Using our theory, the uneven spacing law of LR bands is explained as a consequence of the time-varying velocity of the moving reaction front. Some apparent contradictions of previous LR models are also explained, and suggestions for experimentally controlling pattern formation are made.