## Liesegang rings

Liesegang rings are a phenomenon which is widely known due to the work of Raphael Eduard Liesegang in 1896 although he was probably not the first one to make such a observation. An experimental set-up which can be used is the following. A Petri dish is covered with a layer of a gel containing potassium dichromate. Then a drop of silver nitrate solution is deposited at the centre of the dish. It is observed that over a period of hours coloured rings appear which are centred on the point where the drop of solution was. It turns out that similar observations can be made in many other chemical systems. Another typical set-up is to take a test tube filled with gel containing one chemical and put a solution of another chemical on top of the gel. In this case horizontal bands are produced. Part of the fascination of the Liesegang phenomenon is the striking visual patterns which accompany it. Of course it should not be forgotten that not everything which looks the same must have the same underlying mechanism. This kind of phenomenon appears to be widespread in chemistry and has also been invoked in connection with certain biological phenomena. See for instance the discussion and pictures of fungal growth in http://seedsaside.wordpress.com/2008/02/21/liesegang-rings/

It is tempting to try and find a mathematical explanation of the Liesegang phenomenon. It has been observed that the position of the rings follows a geometric progression, that the time of their appearance goes as the square root of the distance and empirical laws for their thickness have also been stated. Thus there are some definite things which a mathematical model could try to reproduce.
There have been many attempts to give a theoretical (and mathematical) account of the effect. Here, for reasons of personal preference, I will concentrate on models which can be formulated in terms of systems of partial differential equations.

On 6th May Arnd Scheel from the University of Minnesota gave a talk on Liesegang patterns at the Free University in Berlin. I had already been interested in the subject for several months and so I naturally wanted to attend. Due to an overlap with the time of my course on general relativity I was only able to hear about half of the lecture but fortunately the speaker took time to explain some of his insights into the subject to me afterwards. The models he uses are reaction diffusion equations, i.e. systems of the form $\frac{\partial u}{\partial t}=D\frac{\partial^2 u}{\partial x^2}+f(u)$where the unknown $u$ is vector-valued, $D$ is a diagonal matrix and $f$ is some smooth function. Here the spatial dimension is one so that the model is describing bands rather than rings. There are models in the literature where the function $f$ is very irregular (in particular discontinuous). Here I want to restrict to the case that $f$ is smooth. Rather than trying to describe the dynamics of the formation of the rings it is easier to concentrate on the final steady state. Then the time derivative of $u$ vanishes and the equation reduces to a system of ordinary differential equations which can be studied using methods from the theory of dynamical systems. One of the main things I learned from Scheel’s talk was that the existence of Liesgang patterns can be associated to the presence of homoclinic orbits in this dynamical system. In other words, there is a time independent solution of the dynamical system and another solution which converges to it both in the past and in the future. The Liesegang bands correspond not to the homoclinic orbit itself (which would just give a single band) but to other solutions of the dynamical system which approach it asymptotically. Whether appropriate solutions exist depends on the form of $f$ and the chemical literature contains a huge variety of choices. Since the phenomenon is so widely observed it must expected that it has a certain stability. If $f$ is perturbed a little then the bands should survive. This is somewhat surprising due to the fact that homoclinic orbits are usually not stable under general perturbations of a dynamical system. The explanation proposed by Scheel is that the systems of relevance for chemistry have a special structure which has a definite chemical significance. Only perturbations should be considered which preserve this additional structure and it can happen that perturbations of this special type do not destroy the homoclinic solution.

The special structure of the dynamical systems just mentioned reminds me of a comment made by Karl Sigmund in a plenary talk he gave at the International Congress of Mathematicians in Berlin in 1998 This talk was one of the factors which got me interested in mathematical biology. Sigmund’s point was that the general theory of dynamical systems is not well adapted to many of the problems arising in population dynamics. For good reasons the general theory concentrates on generic systems, i.e. on those whose qualitative properties are preserved under small perturbations. Since experimental measurements are never exact this makes good sense from the point of view of the applicability of the results. It may, however, happen that there is some aspect of the system which should be exempted from perturbations since it has a definite meaning for the applications of interest. In other words, it makes sense to consider systems whose properties are stable under perturbations which are generic modulo an invariant manifold which is kept fixed during the perturbation. In population models whis has to do with the fact that the population of some species being zero has an absolute significance which is independent of the details of the population dynamics. This reminded me of a research interest of my own, cosmology, where the aim is to produce simple models of the dynamics of the universe which also give rise to dynamical systems. In that case if the extension of the universe in one spatial direction tends to zero as the big bang singularity is approached than this has an absolute significance like the extinction of a species in a population model. Restricting to perturbations which preserve some submanifold tends to make homoclinic and heteroclinic solutions more common.

My first contact with Liesegang rings came through a talk by Benoit Perthame I heard at a conference on chemotaxis which took place at the Radon Institute in Linz last December. He mentioned some work on modelling a rare neurological disease called Balo’s concentric sclerosis. In the literature on this subject I found that this disease is characterized by alternating concentric rings of damaged and relatively intact tissue and that a connection to Liesegang rings had been suggested. I intend to return to this topic in a future post.