Bifurcation between pre- and post-nucleation models in fluorapatite Liesegang patterns
Liesegang patterns, characterized by periodic precipitation, typically follow either a pre-nucleation or post-nucleation model. However, the factors governing the selection of either pathway in a single chemical system are poorly understood. In this study, we demonstrated a concentration-driven bifurcation between the pre-nucleation and post-nucleation models during the formation of fluorapatite (FAp) Liesegang patterns in an agarose matrix using two systematic sets of experimental conditions. First, a three-variable matrix systematically varying the concentrations of the three involved electrolytes (i.e., calcium chloride, phosphate buffer, and sodium fluoride) revealed a distinct difference between the pre- and post-nucleation pathways. Using time-lapse imaging, line profile analysis, and X-ray diffraction, we monitored the evolution of amorphous calcium phosphate (ACP) intermediates into other species such as crystalline FAp. At certain concentration conditions, the system followed a pre-nucleation model, in which discontinuous ACP growth and subsequent crystallization-induced shrinkage produced discrete bands. By contrast, other conditions triggered a post-nucleation pathway, in which a spatially continuous ACP phase first precipitated and then separated into discrete bands upon structural conversion. Second, by fixing the outer electrolyte (calcium chloride) concentration and systematically varying the two inner electrolyte concentrations (phosphate buffer and sodium fluoride), we successfully mapped the continuous crossover between these two regimes and identified a transition boundary where the features of both pathways co-existed. The study proposes that the bifurcation between the two models would be governed by the kinetic balance between the generation of metastable ACP and its structural conversion. These findings provide a broadly applicable framework for understanding pattern formation in mineral systems, which progress through metastable intermediates.