Pattern formation is a frequent phenomenon in physics, chemistry, biology, and materials science. Bottom-up pattern formation usually occurs in the interaction of the transport phenomena of chemical species with their chemical reaction. The oldest pattern formation is the Liesegang phenomenon (or periodic precipitation), which was discovered and described in 1896 by Raphael Edward Liesegang, who was a German chemist and photographer who was born 150 years ago. The purpose of this feature article is to provide a comprehensive overview of this type of pattern formation. Liesegang banding occurs because of the coupling of the diffusion process of the reagents with their chemical reactions in solid hydrogels. We will discuss several phenomena observed and discovered in the past century, including reverse patterns, precipitation patterns with dissolution (due to complex formation), helicoidal patterns, and precipitation waves. Additionally, we will review all existing models of the Liesegang phenomenon including pre- and postnucleation scenarios. Finally, we will highlight several applications of periodic precipitation.

Nature uses self-organized spatiotemporal patterns to construct systems with robustness and flexibility. Furthermore, understanding the principles underlying self-organization in nature enables programmable design of artificial patterns driven by chemical energy. The related mechanisms are however not clearly understood because most of these patterns are formed in reaction–diffusion (RD) systems consisting of intricate interaction between diffusion and reaction. Therefore, comprehensive understanding of the pattern formation may provide critical knowledge for developing novel strategies in both natural science and chemical engineering. Liesegang patterns (LPs) are one of the typical programmable patterns. This study demonstrates that appropriate tuning of gel concentration distribution is a key programming factor for controlling LP periodicities. The gel distribution was realized in bi- or multilayered gels constructed by stacking agarose gels of different concentrations. Thus, exceptional LP periodicities were achieved locally in bilayered gels. Furthermore, RD simulations revealed that the nucleation process modulated by the gel distribution determines the LP periodicity in bilayered gels. Finally, based on this concept, desired LP periodicities were successfully realized by programming gel distributions in multilayered gels. Thus, deep insights into the fundamental role of nucleation in designing LPs can lead to the practical applications of LPs and the understanding of self-organization in nature.

Pattern formation based on the Liesegang phenomenon is considered one of the useful models for gaining a mechanistic understanding of spontaneous spatiotemporal pattern formations in nature. However, for more than a century, the Liesegang phenomenon in chemical systems has been investigated by using electrolytes as both the reaction substrate and aggregation promoter, which has obfuscated the role of the electrolyte. Here, we distinguish the electrolyte (NaSO) from the reaction substrates (Ag ion and citrate), where NaSO does not participate in the reaction step and acts as an aggregation promoter. The addition of NaSO in Ag-citrate-type Liesegang rings gave well-resolvable clear bands with a larger spacing coefficient. The observed changes were discussed by using the classical DLVO (Derjaguin-Landau-Verwey-Overbeek) theory, where the role of the electrolyte is to shield the electrostatic repulsive interaction among the reaction products. Furthermore, the numerical simulation of the reaction-diffusion equation with different aggregation thresholds reproduced the salt-dependent change in the spacing coefficient. We expect that an understanding of the exact role of the electrolyte as the aggregation promoter reported here will offer novel insight into how nature spontaneously forms beautiful spatiotemporal patterns.

Many types of periodic patterns can spontaneously form in nature across wide spatiotemporal scales. Construction of a chemical model that mimics these periodic patterns are of considerable interest from both scientific and technological viewpoints. The Liesegang phenomenon is one of the chemical models to form periodic patterns with well-defined periodicity. However, the parameters that influence the mechanism and resultant pattern geometry are not completely known. In this study, we use surface chemistry methods to evaluate the influence of nucleation threshold on the geometry of Liesegang patterns. Cysteine was used as an additional ligand for the precursors (Agn nuclei and/or Ag nanoparticles in the present system) to reduce their surface free energy and thus the nucleation free energy. As a result, the formed Liesegang patterns had smaller spacing coefficient (i.e., finer periodic patterns), a phenomenon that was also reproduced using reaction-diffusion simulation with lowered nucleation threshold. The small spacing coefficient at lowered nucleation threshold was discussed in terms of a slower rate of Ostwald ripening. Similar control of the nucleation threshold through surface chemistry can be applied to various precipitation systems, as well as gaining insight into the comprehensive mechanism underlying various animate and inanimate patterns formed in nature.

Spontaneous pattern formation is common in both inanimate and living systems. Although the Liesegang pattern (LP) is a well-studied chemical model for precipitation patterns, various recent LP systems based on artificial control could not be easily evaluated using classical tools. The Matalon–Packter (MP) law describes the effect of the initial electrolyte concentration, which governs the diffusion flux (F diff), on the spatial distribution of LP. Note that the classical MP law only considers F diff through the initial concentration of electrolytes, even though it should also depend on the volume of the reservoir used for the outer electrolyte because of the temporal change in the concentration therein due to diffusion. However, there has been no report on the relationship between the MP law, the reservoir volume, and F diff. Here, we experimentally demonstrated and evaluated the effect of the reservoir volume on LP periodicity according to the classical MP law. Numerical simulations revealed that the reservoir volume affects the temporal modulation of F diff. By expressing the MP law as a function of estimated F diff after a certain period of time, we provide a uniform description of the changes in periodicity for both small and large reservoir volumes. Such modification should make the MP law a more robust tool for studying LP systems.