# Mathematics Of Beer Bubbles 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.

We often take for granted many of the scientific wonders that exist in our world. But did you ever think there was something amazing going on in your beer. Well maybe if youýve had too many. But really, there is a lot brewing in that brew.

ýIf you look closely, there is beautiful physics and beautiful mathematics involved. Some bubbles grow, some bubbles shrink,ý physicist David Srolovitz, Ph.D., said. And the average bubble size goes up, while the number of bubbles shrink, until there are no more. By adapting mathematical formulas that already exist in two dimensions to three, Dr. Srolovist can predict how the bubbles will react.

So who cares? Engineers do, because metal is made up of a pattern of grains that act like bubbles!

ýIf we envision that each of these grains just like a bubble, the way each grain grows or shrinks is exactly the same theory. By changing grain size, you can make a material stronger or weaker,ý Dr. Srolovitz said.

Whether itýs for parts of an airplane or a bridge, heat-treating metal controls grain size and gives engineers the ability to manipulate it. But Dr. Srolovitz says the beverage industry is also interested in this.

ýOne thing you want to do if you are in the beer business is that what you produce is consistent from glass to glass,ý Dr. Srolovitz said.

So next time you chug down that bubbly brew, you can appreciate the science in the suds.

BACKGROUND: A pair of mathematicians at the Institute of Advanced Study in Princeton are applying math to the froth in a beer glass ý 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 ý separated by liquid walls moving under surface tension ý 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 ý 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 ýwallsý 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 ýcoarsening.ý 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.

Note: This story and accompanying video were originally produced for the American Institute of Physics series Discoveries and Breakthroughs in Science by Ivanhoe Broadcast News and are protected by copyright law. All rights reserved.

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