Van der Waals’ square gradient theory

OK, I have encountered this theory again, after many years. The idea is to describe separation between two phases as the minimum of a free energy with respect to an order parameter \phi:

g(\phi) = \frac a2 \phi^2 + \frac b4 \phi^2

When becomes negative, the minimum of g changes from 0 to other values, \pm \phi_0. The function then has the celebrated double minimum feature, which features prominently in many symmetry-breaking theories, including of course the appearance of particle mass mass and, you know, the Big Bang and the Universe.

But here we are just considering phase separation in materials. The interface between two coexisting phases must have some associated cost, and the simplest (lowest order) way to include it is by introducing a total free energy functional

F = \int dr \,  f(\phi, \nabla\phi)

f(\phi) = g(\phi) + \frac c2 (\nabla\phi)^2 .

This is also called a London-Linzburg-Landau free energy, also appearing in their theory of superconductors.

Now, parameters ab, and c are not easy to measure (or, at least, estimate) experimentally, but they are related to: the surface tension, the width of the interface, and the magnitude of the bulk equilibrium order parameter (i.e. \phi_0 ). Here I show how to obtain it in a slightly more general setting, since I was not able to find it on the internet (it can be found e.g, in the book by Rowlinson & Widom).

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TED, and Steven Strogatz

This is a very interesting website: TED. A collection of interesting talks, all of them downloadable and in the public domain (CC license).  There isn’t much on physics yet, and certainly not in my area. There is a very nice talk, though, by Steven Strogatz on synchronization. Strogatz is a remarkable speaker, I had the pleasure of attending a set of lectures on “living polymers” that were very interesting, and I remember he used some of this examples on his introduction. Some of his demonstrations are quite ingenuous, and not so difficult to reproduce.

Er, I forgot to embed the video in the first version of this entry, which is funny since I came across this site from a wordpress announce. There it goes:

La movida superficie de un líquido

(Yet another post in Spanish, since the news article is in this language! ). Se insiste mucho en esto, y por fin nos hemos decidido a tenerlo en cuenta: la divulgación de los resultados de la investigación. Aqui está La movida superficie de un líquido, que divulga nuestro último artículo en el JCP: en madrid+d y en plataformasinc. Continue reading


SklogWiki is a wiki for people interested in simple liquids, complex fluids, and soft condensed matter. SklogWiki is particularly oriented towards theoretical and computational studies.

That seems to wrap it up. Speciallized wikis are a great idea (see wikia), and this one was created for topics that interest me a lot. The name comes from the famous formula for the entropy S = k log W, by Boltzmann (even if it seems he never wrote it this way).

Statistical Mechanics, by Ben Widom

Statistical Mechanics, coverI love Dr. Benjamin Widom’s writing style. This recent book by him is an excellent introduction to the topic of statistical mechanics. The subtitle, A Concise Introduction for Chemists, makes clear some previous knowledge of quantum mechanics, thermodynamics is needed. And some spectroscopy, not much. A new classic, to shelve along Molecular Theory of Capillarity (by B. Widom and J. S. Rowlinson).