Classics in Materials Science: Potts model and its relevance to simulation of microstructures

A soap bubble is an extraordinarily beautiful thing and yet it requires virtually no skill to produce. This is because surface tension does all the work for you, making sure that a perfect spherical membrane is produced every time. In fact it is impossible to blow imperfect bubbles. Even if you try to blow the bubble through a noncircular orifice you may at first achieve a temporary nonequilibrium shape [...] but the end result is always a perfect sphere.

In this chapter we will show how such beautiful but tyrannical surface tension effects can be investigated using the computational equivalent of soap bubble solution: the Potts model. Like soap bubble solution is is easy to use and provides fundamental insights into surface tension phenomena; but also like soap bubble solution, it can lead to a sticky mess.

– Mark Miodownik, Chapter 3. Monte Carlo Potts Model, in Computational Materials Engineering: An introduction to microstructure evolution, Edited by K G F Janssens, D Raabe, E Kozeschnik, M A Miodownik, and B Nestler, Academic Press, 2007.

Potts model is widely used in the materials science literature to simulate a wide variety of microstructural features and phenomena: be it grain growth — normal and abnormal, recrystallization or Zener pinning (and, the article by Miodownik, from which I quote above, discusses all these and much more). In fact, to quote Miodownik again,

At the moment for large 3D systems, with complicated textures and pinning phases, the model has no equal.

In this Classics in Materials Science post, I would like to talk about Potts model and the paper in which Potts outlined it [1].

Potts model, as wikipedia notes, is a model describing spins that are interacting on a crystalline lattice:

The model is named after Renfrey B. Potts who described the model near the end of his 1952 Ph.D. thesis. The model was related to the “planar Potts” or “clock model”, which was suggested to him by his advisor Cyril Domb. The Potts model is sometimes known as the Ashkin-Teller model (after Julius Ashkin and Edward Teller), as they considered a four component version in 1943.

Not surprisingly, the paper of Potts is communicated to Cambridge Philosophical society by Cyril Domb who is also thanked in the acknowledgement for suggesting the problem and helpful suggestions.

The Antecedents

The paper of Potts begins with a reference to a result obtained by Kramers and Wannier [2,3] for 2D Ising models:

In considering the statistics of the ‘no-field’ square Ising lattice in which each unit is capable of two configurations and only nearest neighbours interact, Kramers and Wannier [...] were able to deduce an inversion transformation under which the partition function of the lattice is invariant when the temperature is transformed from a low to a high (‘inverted’) value. The important property of this inversion transformation is that its fixed point gives the transition point of the lattice.

The result that is being alluded to, in the words of Kramers and Wannier is this:

… which permits location of the Curie temperature if it exists and is unique. It lies at
\frac{J}{kT_c} = 0.8814
if we denote by J the coupling energy between neighbouring spins.

That is, Kramers and Wannier, calculated the Curie temperature for a 2D Ising lattice in terms of the coupling constant for spins, and showed that it is unique. (The reference to ‘no-field’ by Potts means that there was no externally applied magnetic fields).

One restriction that Kramers and Wannier put on the spins (which is what makes their model a Ising model) is the following:

Let each of the spins be capable of two orientations which we characterize by \mu_i = +1 and \mu_i = -1.

Potts relaxed that condition and solved for the Curie temperatures:

It is the purpose of the present paper to consider the inversion transformations for a square lattice in which each unit is capable of a general number r of configurations.

Contribution of Potts

Potts was, of course, not the first one to introduce more than two possible spins per lattice site; that was done, apparently, by Domb. Further, much earlier, Ashkin and Teller [4], introduced a system in which each lattice can have any one of the four different types of spins, and calculated the Curie temperature for such a lattice (as part of Julius Ashkin’s Ph D thesis, I understand — which is a reason why Potts model is sometimes also known as Ashkin-Teller model); more interestingly, Ashkin and Teller even talked of four different types of atoms when they considered the four possible spins:

We have considered a two-dimensional square net consisting of four kinds of atoms supposing that only nearest neighbours interact and that there are only two distinct potential energies of interaction, one between like and one between unlike atoms.

Thus, the main contribution of Potts seems to be that (a) he considered a general model in which the spins can take ‘r’ possible values (r = 2, 3, 4, …) and (b) obtained the Curie temperature for the lattice for the cases r=2, 3, 4 (See the figure below):

Potts-solution

The current relevance

A tutorial review written in early eighties notes [5]:

The problem attracted little attention in its early years. But in the last ten years or so there has been a strong surge of interest, largely because the model has proven to be very rich in its contents. It is now known that the Potts model is related to a number of outstanding problems in lattice statistics; the critical behaviour has also been shown to be richer and more general than that of the Ising model. In the ensuing efforts to explore its properties, the Potts model has become an important testing ground for the different methods and approaches in the study of the critical point theory.

In addition to such inherent interest, as we noted earlier, Potts model has also been found to be of great use to those who try to simulate microstructures. In the case of microstructural evolution, our interest in the model is not so much towards obtaining analytical solutions of the type noted above, but to get an idea of the microstructure as a function of time — go to this page, for example, for an Open Source Potts model code for grain growth simulations in 3D.

I can do no better than quote Miodownik [6] on the beauty of Potts models (with liberal sprinkling of modelling philosophy) to end this post:

The beauty of the Potts model is that it is a simple way to model complex systems by modeling local physics (in particular, the effect of surface tension phenomena on the development and evolution of microstructure). … At all times we have been concerned with comparing the model with theory and experimental data; this instinct is essential to any modeler. It is easy to make a model yield pretty pictures that appear to have a correspondence with a “real” phenomenon, but quantification of the simulation is the only way to use the model as a method to gain physical insgihts and understanding. Finally it is important to note that a model is just that, a model, and the benefit of it is as much to guide experiments and to hone the intuition about physical phenomena as it is to make predictions. The major role of computer models is to reduce the number of experiments that need to be carried out and to highlight what variables are key to understanding the results. The Potts model should be seen in this light; it is a guide to the intuition, and above all it is a medium of communication between experimentalists and theoreticians.

References

[1] R B Potts, Some generalized order-disorder transformation, Proceedings of the Cambridge Philosophical Society, Vol. 48, pp. 106−109, 1952.

[2] H A Kramers and G H Wannier, Statistics of the two-dimensional ferromagnet. Part I, Physical Review, Vol. 60, pp. 252-262, 1941.

[3] H A Kramers and G H Wannier, Statistics of the two-dimensional ferromagnet. Part II, Physical Review, Vol. 60, pp. 253-276, 1941.

[4] J Ashkin and E Teller, Statistics of two-dimensional lattices with four components, Physical Review, Vol. 64, pp. 178-184, 1943.

[5] F Y Wu, The Potts model, Reviews of Modern Physics, Vol. 54, pp. 235-268, 1982.

[6] M Miodownik, Chapter 3. Monte Carlo Potts Model, in Computational Materials Engineering: An introduction to microstructure evolution, Edited by K G F Janssens, D Raabe, E Kozeschnik, M A Miodownik, and B Nestler, Academic Press, 2007.

About Guru

I am an Assistant Professor in a Metallurgical Engineering and Materials Science Department; I also pursue research in the broad area of computational materials science.
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