Archive for September 8th, 2006

Allen-Cahn I: Fick’s laws, where we introduce D!

September 8, 2006

Sometime back, I wrote about Pattern formation, Turing and Allen-Cahn equations. Towards the end of that post, I wrote:

Probably, the reaction diffusion equation that is mentioned is the Allen-Cahn equation — I have to do a bit of homework on that, and then, I will post the details.

While doing the homework, I realised that the proper way of going about the issue is to follow the advice of Lewis Carroll’s King:

“Begin at the beginning,” the King said gravely, “and go on till you come to the end; then stop”.

So, in a series of posts, I will write about the history of Allen-Cahn equations and associated concepts, primarily from a materials scientist’s point of view. The last post in the series will be about the connections between Allen-Cahn equations and the pattern formation studies of Turing.

Let us, then, begin at the beginning, and talk about Fick’s laws of diffusion. There are two laws; the first law is about steady state diffusion. The second law is obtained by combining the law of mass conservation along with the first law. The wiki page gives these details along with the biological perspective.

Last year was the sesquicentennial of Fick’s laws of diffusion; and, go here to read Fick himself on liquid diffusion — the essay reads extremely well — don’t miss it. By the way, that 1995 issue of Journal of membrane science is the 100th volume of the journal; and to celebrate the same, the issue reprinted selected papers pertaining to the early history of membrane science. Among other things, the papers by Graham and Knudsen are also published in the same issue; go, take a look!

According to the first law, when there are no changes in concentration within the diffusion volume with respect to time, the diffusion flux is proportional to the gradient on concentration of the diffusing species. The constant of proportionality is diffusivity D. The negative sign indicates that the flux always tends to decrease the gradients in concentration. The second law, as we noted, is just a combination of the first law with the law of conservation of mass.

The diffusivity, as defined above, is then a positive constant. If it is not, the diffusive flux will be against the concentration gradient. In the next post, we will discuss such cases, and the resulting modification to Fick’s laws. In the meanwhile, here is some recommended reading:

  1. Prof. Bhadeshia‘s Homepage
  2. Prof. Carter’s course notes