Think small with phase field models

Note: This was originally posted on 6 January 2005.

Phase field (PF) models are truly great for studying a wide variety of problems and phenomena. Given all the buzz (not to mention the beautiful microstructures), more and more people are getting interested in using them in their work. More importantly, the buzz is not just confined to academia; it has spread to others (particularly, those in industrial research centers) as well. Clearly, the PR has been fantastic!

We now have a larger community of people who will be applying PF models to lots of new, practical and industrially relevant problems. These new problems, together with the new ideas that are generated for solving them will broaden as well as deepen the field. There is no doubt that this is a positive development.

Much of the buzz arises from the impression that any materials problem involving migrating interfaces (or other structural discontinuities, such as dislocations) is ripe for PF models. There is certainly a strong basis for this impression. You just have to look at the range of materials problems for which PF models have been applied: solidification, diffusional and diffusionless phase transformations, recrystallization and grain growth, sintering, dislocations. Still, I want to pose this question: is this impression correct? In other words, are there problems that PF models are not good for?

The way I see it, PF models are good for two broad kinds of problems: those that examine some key (minute or microscopic) details of evolution of a microstructural feature, and those that require large scale simulations. Studies on the former type of problems can be called “small-scale” or “local", while those of the latter type, “large-scale” or “global".

Local studies would focus, for example, on:

• details (morphology, growth rate) of a single microstructural entity such as a dendrite or a precipitate
• topological transitions during grain growth
• nucleation morphologies of multiple variants in a martensitic-like transformation

In each case, the aim is to examine the microscopic details of the process to elucidate the role of different factors during nucleation and further development of the microstructure.

Global studies, on the other hand, focus on the microstructure in a large region. Clearly, the larger the simulation box, the better. With larger simulations, statistics become better; more importantly, effects of long range interactions (electrical, elastic, diffusional) can be better studied, and finite-size effects are smaller.

However, even in the large-scale studies, the simulation box is not truly large or macroscopic - we will consider an example below - they are “large-scale” only in relation to the current computational capabilities. With this understanding, it is now clear that we can use such studies only for those problems where it is reasonable to assume that the simulation box is representative of the entire sample. This “representative-box” assumption is not unique to PF models, however; it is central to many experimental studies. For example, in detailed microscopic characterization of materials (and TEM in particular), we implicitly assume that what we see in one area is truly representative of the entire sample. For this assumption to be fulfilled, the obvious requirement is that the experimental conditions experienced by every sub-region in a macroscopic sample must be identical; so, for example, we ensure that the sample is kept under constant temperature and pressure.

To get back to the question: are there problems that PF models are not good for? The answer, unfortunately, is that there indeed are. Many problems of practical interest fail this “representative-box” test. Examples include solidification in a casting or a weld, phase transformations or grain growth under temperature gradients, etc. For problems of this kind, PF models are just inappropriate. This assertion follows from a fundamental feature of all PF models: their characteristic length scale, which is the width of the interface, ‘w’. For practical problems where microstructures change (i.e., exhibit gradients) over length scales that are too large compared to ‘w’, we have no hope of using PF models. Let me take an example problem: grain growth.

In using PF models, we are basically interested in probing microstructures with a characteristic length scale. For example, in a typical polycrystalline metal, the grain size is about 1 micron, and we are interested in studying how, and how fast, this microstructure evolves. For this we rig up a PF model with an interface width, ‘w’. In simulations, in the interest of stable numerical computation, we want at least four grid points to span this width, so we have w = 4 * Delta x. In our example, we would further like two parallel boundaries (which we may take as d, the grain size) to be separated at least by several (say, 10 times) widths: thus, d = 10 * w = 40 * Delta x. If we have a 2D simulation box with 1000 x 1000 grid points with a grid spacing of Delta x, we will have approximately n=25 grains to a side, to yield a total of about n-squared = 625 grains in the simulation box.

This brings us to the key point: the simulation box is only about 25*d = 25 microns wide! What if the problem involves microstructural gradients which extend over hundreds or thousands of microns? We have to come to the sad conclusion that PF models are just not good enough for such problems. Remember, we are not even talking about the third dimension that is so important to all of us! In 3D, the impact of this conclusion is even stronger.

It is going to take a giant leap in computational capabilities (well beyond the already fantastic, exponential progress that follows from Moore’s law) before we can contemplate solving practical problems involving graded microstructures. Until then, we just have to live with this fundamental limitation.

So, you now have the answer: with PF models, you can only think small! I bet you will (probably truthfully) say that you knew it all along …