Models of Liesegang pattern formation

In this article different mathematical models of the Liesegang phenomenon are exhibited. The main principles of modeling are discussed such as supersaturation theory, sol coagulation and phase separation, which describe the phenomenon using different steps and mechanism beyond the simple reaction scheme. We discuss whether the underlying numerical simulations are able to reproduce several empirical regularities and laws of the corresponding pattern structure. In all cases we highlight the meaning of the initial and boundary conditions in the Ferenc Izsák & István Lagzi 2 Correspondence/Reprint request: Ferenc Izsák, Department of Applied Analysis and Computational Mathematics, Eötvös Loránd University, Budapest, Pázmány sétány 1/C, Hungary E-mail: [email protected] corresponding mathematical formalism. Above the deterministic ones discrete stochastic approaches are also described. As a main tool for the control of pattern structure the effect of an external electric field is also discussed. Introduction In the first decades after the pioneering observation of Liesegang similar phenomena have been reported in number of chemical systems. The research was restricted mainly to the analysis of the experiments. Beyond the simple or even sophisticated laboratory experiments the most beautiful patterns can be found in the nature as agate rocks, see shells and diverse patterns of animal coats [1,2]. Accordingly, study of pattern formation became a well established research topic in the computational biology, computational chemistry and computational geosciences. Figure 1. The Liesegang pattern formation. To understand the common origin of these phenomena one needs a general model which is applicable in the broad range of the observations above. A deep understanding of the phenomena could contribute not only to have better insight into the mechanism of the pattern formation, but it paves the way to reproduce them. Besides of the good progress in the understanding of the Liesegang phenomena, there are still some open problems providing motivation for the future research. In this study we provide an overview of the recent models. The summary reflects the today status of the topic keeping in mind that their extension in many directions is desirable. In most of the studies one dimensional models are developed corresponding to real experiments in a tube, where an inner electrolyte is present and an outer electrolyte is invading at the junction point. Under Models of Liesegang pattern formation 3 some circumstances precipitation bands are formed at time t1, t2,.... at the locations x1, x2,..., measured from the junction point of the reagents. The width of the bands are denoted with w1, w2,.... Frequently, dimensionless spatial and time coordinates are used such that qualitative properties of the phenomenon can be reproduced. These qualitative properties are based on regularities of the Liesegang phenomenon (see [3]), which we summarize as follows: (P1) Spacing law