Periodic precipitation or the “Liesegang phenomenon” is a special type of chemical pattern formations. It was discovered by a German chemist and photographer, Raphael Eduard Liesegang in 1896 but did not have any general explanation more than a century ago.
Patterns in Nature
In the last decades of the 20th century different kinds of chemical, physical and biological pattern formations have excited an ever increasing interest in the scientific community. In chemical patterning one of the most intensively investigated area was the so-called Belousov-Zhabotinsky reaction, but there were many publications about viscous fingering, diffusion limited aggregation, morphogenesis of fungal colonies and some other simple living bodies, and last but not least patterning during electrochemical deposition too.
Although at first sight the above mentioned systems are quite different there are many similarities in the way they form the corresponding patterns, and the methods by that they can be handled. All of them contain at least one or several diffusion-limited steps, while the formation of the spatial or spatiotemporal order is always a result of a complicated interplay of these and the underlying chemical, physical or biological processes.
Mathematics and the modeling of reaction-diffusion processes
Mathematical description of such systems consists of so-called reaction-diffusion differential equations. Unfortunately these are usually systems of coupled nonlinear partial differential equations, that cannot be treated by standard analytical methods. The only viable way is the application of different numerical methods. Numerical solution of such systems of equations is computationally very demanding, moreover it is sometimes computationally prohibitive even nowadays.
The story of the so-called Liesegang phenomenon is good example for this problem.
Liesegang patterning is a special type of chemical pattern formation in which the spatial order is formed by density fluctuations of a weakly soluble salt. From analytical chemistry we know many different reactants that form a precipitate (sparingly soluble salt) when they react with each other. A good example for this behavior is the reaction of silver-nitrate (AgNO3) and potassium-dichromate (K2Cr2O7).
If one of these components is evenly distributed in a swollen gel (e.g. in gelatine), and the solution of the other diffuses into it, the spatial distribution of the slowly forming precipitate will not be continuous. A series of precipitate zones (bands or rings depending on the geometry of the experimental setup) will form according to some simple scaling laws.