With a key mathematical insight, a pair of theorists has solved a 5-decades-old puzzle as easily as you might burst a soap bubble with a pin. The new result lets researchers predict whether a bubble in foam will grow or shrink. More than a mere curiosity, the mathematical relation could aid engineers designing foamy materials, biologists studying the architecture of tissues, and physicists probing how crystalline grains are arranged within a solid.
Foam looks simple, but researchers can't explain how it evolves as bubbles grow, shrink, and merge--a process known as coarsening. In 1952, famed mathematician John von Neumann deciphered one aspect of 2-dimensional foams, such as soap bubbles squeezed between glass plates. Whether a bubble grows or shrinks depends on the sum total of the curvature of its faces. But Von Neumann reduced the messy problem of adding up curvature to the much simpler task of counting a bubble's sides. He proved that, regardless of their sizes or shapes, 2-D bubbles with five or fewer sides shrink, those with seven or more grow, and those with six remain the same. For half a century, researchers have struggled to extend von Neumann's result to 3 dimensions.
Now, mathematician Robert MacPherson of the Institute of Advanced Study in Princeton, New Jersey, and theoretical materials scientist David Srolovitz of Yeshiva University in New York City have cracked the problem. What made it so difficult is that bubbles' surfaces can curve in complicated ways like saddles or potato chips. However, MacPherson realized that he could succinctly describe the curvature using a mathematical concept called the Euler characteristic. When an object is sliced in two, the Euler characteristic is the tally of surfaces revealed minus the number of holes in them--one for a croquet ball, zero for a hollow tennis ball. "After that insight, we were able to knock out the rest of it relatively quickly," Srolovitz says.
Using the Euler characteristic, MacPherson and Srolovitz also invented an abstract "mean width" that they could calculate for any object regardless of its shape. In 3 dimensions, a bubble's faces meet at distinct edges, and the researchers found that a bubble will grow if the sum of the lengths of its edges is greater than 6 times its mean width. If the sum of all the edge lengths is smaller, the bubble will shrink, as the team reports tomorrow in Nature. The researchers have shown that in 2 dimensions their result reduces to von Neumann's rule and have extended the relation to hypothetical bubbles in 4 or more dimensions.
Other researchers had already developed empirical relations that, on average, tied the growth of a bubble to the number of faces, and the new, exact result might be useful for to putting those rules of thumb on a firmer theoretical foundation, says Sascha Higlenfeldt, an applied mathematician at Northwestern University in Evanston, Illinois. "It's very satisfying to have this formulation to work with," he says. "You know you're on safe ground now." James Glazier, a physicist at Indiana University in Bloomington, says the new work is "a beautiful piece of mathematics." He notes, however, that a tougher problem is describing how the overall structure of the foam develops as bubbles disappear and merge. "We still have many more years of difficult work ahead before we can truly say we understand coarsening foams."