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 …
