Nature establishes structures and functions via self-organization of constituents, including ions, molecules, and particles. Understanding the selection rule that determines the self-organized structure formed from many possible alternatives is fundamentally and technologically important. In this study, the selection rule for the self-organization associated with a reaction−diffusion system was explored using the Liesegang phenomenon, by which a periodic precipitation pattern is formed as a model system. Experiments were conducted by systematically changing the mass flux. At low mass fluxes, a vertically periodic pattern was formed, whereas at high mass fluxes, a horizontally periodic pattern was formed. The results inferred that a structural vertical-to-horizontal periodicity transition occurred in the selforganized periodic structure at the crossover flux at which the entropy production rate reversed. Numerical analyses attributed the as-observed flux-dependent structural transition to the selection of the self-organized pattern with a higher entropy production rate. These findings contribute to our understanding of how nature controls self-organized structures and geometry, potentially facilitating the development of novel designs, syntheses, and fabrication processes for well-controlled organized functional structures.

The Liesegang phenomenon can be used for micro- and nanofabrication processes to yield materials with periodic precipitation of diverse types of materials. Although there have been several attempts to control the periodicity of the Liesegang patterns, it remains unclear whether the periodic precipitation of AgCl in gel medium causes regular- or revert-type patterns. To confirm the periodicity of the AgCl pattern, we conduct one-dimensional experiments under various ion concentration conditions. From microscopic observations, three different precipitation modes were observed, i.e., continuous precipitation with a sharp front, periodic precipitation and continuous precipitation with a gradual front. For these three modes, numerical analyses of the pattern geometry are performed for the periodic precipitation. It was confirmed that the regular-type pattern appeared for all concentration conditions conducted in the present experiments. Furthermore, the pattern was found to obey the spacing law and the Matalon–Packter law. From our experiments, we concluded that AgCl forms regular-type Liesegang patterns, regardless of the dimension of diffusion.