# Mathematics of Beer Bubbles

## Physicists and Mathematicians Analyze Beer Bubbles to Peer Deeper into the Structure of Materials

December 1, 2007

Mathematicians built a formula to explain the behavior of beer bubbles in three dimensions and found that it can be applied to other materials like metals. Explaining the behavior of beer bubbles is complex and related to the granular structure of metals. In beer, the bubbles merge and coarsen until the head disappears. Research shows that controlling the surface tension of bubbles offers the opportunity to make more bubbles, which interests both brewers and engineers.

## Science Insider

BACKGROUND: A pair of mathematicians at the Institute of Advanced Study in Princeton are applying math to the froth in a beer glass specifically, how a beer head changes over time by studying its frothy networks of gas-filled bubbles.

WHAT THEY FOUND: The two mathematicians found that the mathematics of how beer bubbles behave is similar to how grains in metals grow. The grain-like structures in metals get more coarse as their boundaries move. Similarly, the bubbles in a beer head separated by liquid walls moving under surface tension merge and coarsen the foamy structure until the head disappears. Specifically, they found that how fast the beer head collapses depends on the widths of the bubbles rather than the number of adjacent bubbles. By carefully controlling the surface tension properties of the bubbles, they discovered they could figure out how to make more bubbles, giving brewers more control over the levels of beer heads in their products.

TINY BUBBLES: Foams are examples of so-called soft matter: they donýt flow freely like a true liquid, but neither do they assume the definite shape of a solid, like diamond. Foams arise when some form of mechanical agitation ý say, a chef beating egg whites with a wire whisk to produce a fluffy meringue thrusts air into a liquid, forming bubbles of many different sizes. Initially, each bubble is a sphere: a volume of air encased in a very thin liquid skin that isolates each bubble from its neighbors. They owe their geometry to the phenomenon of surface tension, a force that arises from molecular attraction. The greater the surface area, the more energy that is required to maintain a given shape, which is why the bubbles seek to assume the shape with the least surface area: a sphere.

COARSER AND COARSER: The pull of gravity gradually drains the liquid downward, causing the bubbles to press more tightly against each other. As the amount of liquid in the foam decreases, the walls of the bubbles become very thin, so that smaller bubbles gradually are absorbed by larger ones. So over time, the tiny bubbles that make up foam become larger. The combination of these two effects is called coarsening. As the coarsening continues over time, the bubbles begin to resemble soccer balls.

The American Mathematical Society and the Mathematical Association of America contributed to the information contained in the TV portion of this report.

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TO GO INSIDE THIS SCIENCE:
David J. Srolovitz
Yeshiva University
New York, NY
srol@yu.edu
212-960-5214