Before downing your next beer, pause to contemplate the bubbles. You’ll find that they grow and shrink in odd, hard-to-predict ways. A mathematician and an engineer have found a simple and surprising equation to describe this process, using a field of mathematics no one expected to be relevant.
Now, new simulations are building on that result to illuminate more than just foams. Metals and ceramics are made of crystals that grow and shrink the same way that beer bubbles do, affecting the properties of the materials. The new work may thus lead to more resilient airplane wings, more reliable computer chips and stronger steel beams.
Over time, the bubbles in foam tend to consolidate, becoming fewer and larger. The physics driving this process has long been understood: When two bubbles adjoin one another, gas tends to pass from the bubble with higher pressure to the one with lower pressure. The higher-pressure bubble bulges into the lower-pressure one, so the shape of a bubble reflects the pressure in all the bubbles around it. As a result, the shape of the bubble should be enough to determine how fast it’s going to grow or shrink.
In 1952, the mathematician John von Neumann used this principle to find an elegant equation that predicts the growth rate of a bubble in a two-dimensional spread of bubbles. But the three-dimensional case — which is the one scientists and beer drinkers most care about — proved far more difficult. Some researchers even believed the problem unsolvable.
What was needed, it turned out, was to introduce another field of mathematics: topology. Topology is essentially the study of connections. For a topologist, two objects are the same if one can be shrunk or stretched into the same shape as the other, but without punching holes or gluing anything together, since that would change the way the parts of the object connect to one another. So, for example, a doughnut and a coffee cup are the same shape to a topologist: Squish down the cup part of the coffee cup, and you’ll end up with a doughnut-shaped ring. A doughnut is topologically different, however, from a beach ball.
Topology would seem to offer little help in determining how quickly bubbles in foam grow or shrink, because squishing a bubble into a different shape will change its growth rate. But several years ago researchers found an equation involving the Euler characteristic, a property of a shape that stays the same no matter how the shape is stretched or smushed. “It’s really beautiful,” says Robert MacPherson of the Institute for Advanced Study in Princeton, N.J., the mathematician on the project, “because the Euler characteristic shouldn’t have anything to do with it.”
To calculate the Euler characteristic, first imagine slicing a bubble in different directions. Usually, the resulting shape would be roughly circular. But suppose that the bubble had a couple of little bumps on its surface. If your slice went through these bumps, you could end up with two disconnected circles. Or, if the bubble had a small divot in it and your slice went through the divot, you could end up with a little hole inside your slice. The Euler characteristic of the slice is the number of disconnected pieces it contains, minus the number of holes within it.
The equation that MacPherson and David Srolovitz of the Agency for Science, Technology and Research in Singapore developed shows that bubbles grow quickly when the beer is warm and when the bubbles have divots in them rather than bumps, or when they are connected to lots of other bubbles.
Now, the team is taking the equation one step further, using it to create computer simulations of foams, metals and ceramics. The work is a way to expose a material’s inner structures that are not normally visible, but that influence the material’s overall properties. “If you beat an egg white enough, it becomes almost a paste and you can make peaks out of it,” MacPherson says. “If you looked at an individual egg white bubble, you wouldn’t expect that. We want to understand these collective properties.”
“Topology is going to be a very powerful tool for understanding these structures,” says Jeremy Mason, a materials scientist at the Institute for Advanced Study who has joined the research team. He points out that the behavior of foam as a whole is unlikely to change dramatically if you stretch or squish the bubbles a bit without changing the way they’re connected to one another, even though changing the shape of an individual bubble will affect its growth rate. Focusing on the connections — that is, the foam’s topology — may then allow researchers to home in on its most crucial aspects.
“We’re awash in data, and the challenge is to identify what is meaningful and compare it between two different structures,” Mason says. “Measuring topological characteristics may give us a language for that.”