What is Coarsening?

What do salad dressing, cast iron, and rocks have in common? Coarsening is crucial to all three.

When you shake up the salad dressing, the oil and vinegar get jumbled together in small droplets. When you stop shaking, the tiny droplets merge into bigger ones, gradually making for a coarser and coarser mixture until all the oil is on the top.

Molten iron, before it is cast, has a fair percentage of carbon dissolved in it. As it cools, this dissolved carbon precipitates out (just like water droplets condense out of damp air as it cools, forming clouds and rain and snow and other precipitation). Because it cools over a period of seconds to hours, the carbon doesn't have a chance to float to the top: it stays dispersed through the iron in small particles. The hardness and brittleness of cast iron depends on the size and form of these carbon particles, and thus depends on how the iron is cooled.

Rocks often have lots of tiny grains of different materials: quartz, alkali feldspar, and plagioclase in granite; plagioclase feldspar and calcium-rich pyroxene in basalt, ... Different rocks have different sizes of these grains. Rocks formed from lava of erupting volcanos have very fine grains: molten rock cooling over eons deep underground form large grains.

Much recent progress has been made by studying simple models. The most thoroughly studied system is shown in the following pictures. Think of the black regions as oil and the white regions as vinegar: the model changes pixels from black to white if they have more vinegar neighboring pixels than oily neighbors, and vice-versa.

Starting from a random configuration, the black and white regions separate from one another. The animation at left shows the steps one at a time.

This picture at right shows the system after two hundred more time steps. Notice that it looks roughly like the first picture, except magnified by a factor of about three. The animation shows the evolution from the first picture to the second one, but speeded up to ten steps at a time.

This picture looks like the first picture blown up by a factor of ten. (Or, it would if we could do a 3200x2400 size simulation!) It's coarser by ten because it's had more time for the black and white regions to separate. It's had 100x the amount of time to separate: the length scale grows as the square root of time. The animation shows the evolution speeded up to 100 steps per snapshot. (Actually, it's had 111 times as much time to separate: sorry, I should have had 18 frames in the last two movies... Also, the periodic boundary conditions may have caused the coarsening to accelerate at the end.)

In this model, to reach a given length we need to wait a time that goes like the length times itself. (After 10x10 time steps, the length grew by a factor of 10; to get to a length 30 would take 900 time steps.) This can be explained (to a physicist, anyhow) using the fact that the boundaries between the two phases become smoothly curved at late times. Regions which curve inward tend to grow; regions which poke out shrink; small droplets shrink because they're surrounded by boundaries with mostly positive curvature. The growth gets slower as the length increases because the curvatures of the remaining black and white (oil and vinegar) regions becomes smaller and smaller. In a more realistic model (which doesn't let oil transform into vinegar, but just lets it move around) to get to a length 10 takes a time of 10x10x10: to get to 30 would take 27,000 time steps. In this model (where the number of each particle is conserved) coarsening is often associated with spinodal decomposition.

You might think that this is only the beginning: after all, black and white squares are far removed from molecules of oil and vinegar, not to mention quartz and orthoclase. Actually, in many ways this absurdly simple model is almost good enough: it is believed that the patterns of black and white in our model (at least in a more realistic version) begin to look just the same as those found in oil and water, after the domains have enough time to grow big! (Rocks are now thought to be more complicated.) The fact that things are simpler when zillions of atoms all do their own things is one of the basic truths that make science possible.

If the simplest model is so successful, what more is there to do? Actually, this is still a rather lively field: just within our group at LASSP we've had several exciting developments:

Logarithmic Coarsening.
At low temperatures, the natural shapes of the crystalline regions won't be smooth. If the regions naturally have sharp corners, the curvature argument for growth breaks down, and the growth becomes much slower: you might have to wait a time that goes like 2^10 ~ 1000 to reach a size 10, and 2^30 ~ 1,000,000,000 to reach a size 30.

Coarsening With External Forces.
The question of how grains grow if you heat a material while pounding on it is almost untouched. The carbon particles form while the horseshoe is red hot and the blacksmith is hammering away on it: the hammering is important precisely not only to shape the horseshoe but also to change the carbon particles. Lisa Wickham and I have been studying a simpler, but related problem of how the domains grow when the oil is being pulled upward and the vinegar downward. (Gravity does this for salad dressing, which is a big problem for experimentalists trying to confirm the simple theories described above).
Hydrodynamics and Coarsening.
Actually, our model doesn't describe oil and vinegar very well, because it doesn't allow the black and white regions to flow around like liquids. One of us (Eric Siggia) did some foundation work on coarsening in around 1979. Including the flow of the oil and vinegar, he showed that at late times the growth speeds up, so one reaches a coarsening length 1000 after only 1000 time steps.
Faster algorithms for Coarsening.
It's slow work simulating these systems: the animation shown above took all evening on a modern, fast workstation. I spared you the agony by splitting the animation into three parts, with each part ten times the speed of the previous: the computer had to step one time step at a time (each time asking each of the 76800 pixels whether it wanted to change color)! That's the whole idea, of course: the coarsening slows down as the lengths get large. John Marko and Gerard Barkema here, by changing details of how the black and white pixels move around, speed up the system by enormous amounts without changing the important behaviors.
Coarsening in Polymer Systems.
At long times when the coarsening length is much larger than the molecules, the coarsening behavior is independent of whether the molecules form two metals or two rocks or two liquids (until the drops start flowing). What kind of molecule is diffusing can change the behavior, though: if the molecules are very long and wiggly (polymers, DNA, ...) the coarsening obeys different laws. John Marko has been studying some polymer coarsening problems in thin films and near surfaces.

Other Coarsening

References


Last modified: June 14, 2006

Jim Sethna, sethna@lassp.cornell.edu

Statistical Mechanics: Entropy, Order Parameters, and Complexity, now available at Oxford University Press (USA, Europe).