Princeton math professor Robert MacPherson and Yeshiva University dean and materials science professor David Srolovitz have constructed a formula for predicting the rate of change of the volume of bubbles over time, a formula with practical applications varying from metal processing to beer manufacturing.
The two professors first met when Srolovitz was a mechanical and aerospace engineering professor at Princeton and MacPherson asked to sit in on his graduate course on materials science.
"He came to pretty much every class," Srolovitz said. "So about halfway through the course, I asked him about how the [course] material related to his work in geometry, and he told me that since most of his work in mathematics relates to problems that go back centuries, he was looking for new types of geometry problems."
At the end of the course, Srolovitz covered the von Neumann-Mullins theory of grain growth. The theory, developed about 50 years ago, states that the rate of change of the surface area of a bubble or grain is proportional to its number of sides minus six.
Since the discovery of the von Neumann-Mullins theory, researchers in different fields have been working to extend the theory to a law for three dimensions, which would allow them to analyze the rate of change of volume instead of area.
"The von Neumann relation is really beautiful because it is purely topological," meaning that the relation is based only on the geometry of the bubble, Srolovitz said. "And although the materials we all care about — or the beer we all care about — depends on 3D analysis, for 50 years we weren't able to find an analogous 3D formula."
Working together since January 2006 on what Srolovitz called ongoing research, the two have managed to find the 3D formula. Though currently the only real-world applications of the formula are in three dimensions, the law will also work for all higher dimensions.
In order to solve the problem, MacPherson and Srolovitz implemented an advanced technique of geometric probability — integration over space of planes — that allowed them to integrate the curvature of a bubble over the entire surface area of the bubble.
"In developing this formula, we introduced certain concepts that have only ever been used in pure math," Srolovitz said.
They discovered that the rate of change of a bubble's (or grain's) volume is proportional to both a two-dimensional measurement called the mean width, which is the average of all the possible measurements of the bubble that can be made with a caliper, and the total length of all the edges where a bubble touches another bubble.
The formula can be applied to metal processing by calculating the strength and brittleness of metals. The strength of a metal is proportional to the inverse of the square root of the size of the grains in the metal, and the stronger a metal is, the more brittle it becomes. Different applications call for metals of differing strength-to-brittleness ratios, which is determined by the size of the grains in the metal. MacPherson and Srolovitz's formula will allow for better metallurgical analysis to calculate how long and at what temperature metal should be heated to allow for optimally sized grains.
Another application of the formula relates to beer manufacturing.
"All the talk about beer was used to create hype for the discovery," Srolovitz said. "But the word beer appears in our article only once."
The formula can help beer manufacturers analyze the consistency and longevity of the head of foam on their beers. Srolovitz said that a beer like Guinness, known for having a very smooth, long-lasting head, is the result of lots of tiny bubbles, whereas beers with coarser heads that don't last as long usually result from fewer, larger bubbles.
The formula will ultimately help beer manufacturers understand the most important factors for creating their beer heads, allowing them to engineer the desired foam head more accurately.
Metal processing and beer manufacturing are only two of the many applications for the formula, but despite the practical applications, the formula is essentially based in theoretical math and therefore required the alliance of a mathematician and an applied scientist.
"MacPherson and I have very complementary backgrounds for this project," Srolovitz said. "His is in pure math and mine is more in materials. Our [fields of] expertise are almost orthogonal."
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