Pattern formation is a ubiquitous phenomenon in animate and inanimate systems generated by mass transport and reaction of chemical species. The Liesegang phenomenon is a self-organized periodic precipitation pattern always studied in porous media such as hydrogels and aerogels for over a century. The primary consideration of applying the porous media is to prevent the disintegration of the precipitation structures due to the sedimentation of the precipitate and induced fluid flow. Here, we show that the periodic precipitation patterns can be engineered using a Hele–Shaw cell in a confined liquid phase, restricting hydrodynamic instability. The patterns generated in several precipitation reaction systems exhibit spatiotemporal properties consistent with patterns obtained in solid hydrogels. Furthermore, analysis considering the Rayleigh–Darcy number emphasizes the crucial role of fluidity in generating periodic precipitation structures in a thin liquid film. This exploration promises breakthroughs at the intersection of fundamental understanding and practical applications.

中文翻译：

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受限液体层中的周期性沉淀

图案形成是由化学物质的质量传递和反应产生的生命和无生命系统中普遍存在的现象。 Liesegang 现象是一种自组织周期性沉淀模式，一个多世纪以来一直在水凝胶和气凝胶等多孔介质中进行研究。应用多孔介质的首要考虑是防止由于沉淀物的沉降和诱导流体流动而导致沉淀结构的崩解。在这里，我们表明可以在有限的液相中使用 Hele-Shaw 池来设计周期性降水模式，从而限制流体动力学的不稳定性。几种沉淀反应系统中产生的模式表现出与固体水凝胶中获得的模式一致的时空特性。此外，考虑瑞利-达西数的分析强调了流动性在薄液膜中产生周期性沉淀结构的关键作用。这一探索有望在基础理解和实际应用的交叉点上取得突破。