# Frothy Physics: The Math of Foams

A new mathematical model describes the complex evolution of bubbly foams

Few of us have not paused at one time or another to marvel at the beauty of a soap bubble. The iridescent, evanescent orbs, which can persist for minutes before vanishing in an instant, have captivated bubble-blowing children and pensive bathtub recliners alike.

They have also caught the eye of physicists and mathematicians, who have strived for hundreds of years to understand and predict the properties of bubbles at a fundamental level. Clusters of multiple bubbles carry a particular mathematical allure: they obey a series of geometric rules (such as bubble surfaces always meeting at certain angles) and behave as a kind of rudimentary computer, constantly shifting and rearranging to solve an optimization problem—how to limit the surface area of the bubbles.

Now a computer model that describes the behavior of ensembles of bubbles—better known as foams—may give researchers a better handle on bubble physics, which could in turn lead to better fire retardants, bicycle helmets and other foam-based products.

The new model, devised by two mathematicians at the University of California, Berkeley, breaks down the evolution of a foam into three discrete stages: First the foam rearranges its macroscopic structure as surface tension and the flow of air push the bubbles around until the foam settles into a stable configuration. Then liquid drains from the thin membranes, known as lamellae, that encase the individual bubbles until one of those lamellae is too weak to support the bubble. Finally, in the third stage, the thinned lamella breaks and the bubble pops, knocking the entire foam out of equilibrium and starting the process over. The research appeared in the May 10 issue of Science.

Each of the three stages of foam evolution plays out on its own scale of space and time. The microscopic thinning of a bubble's lamellae, for example, takes place very slowly, sometimes over hundreds of seconds, explains study co-author James Sethian, professor of mathematics at U.C. Berkeley. Then the rupture of those lamellae “happens at hundreds of meters per second,” he adds. One of the hurdles in simulating the dynamics of a foam is capturing the requisite detail of the small-scale processes without bogging down the simulation on less salient details.

The solution put forth by Sethian and his co-author Robert Saye treats each scale differently—in effect, zooming in on the small-scale processes when they occur and zooming back out during the slower, macroscopic processes. “You can deal with them separately and then couple them together,” says physicist Denis Weaire of Trinity College Dublin. The end results of each stage of the simulation feeds into the next stage in a feedback loop—the macroscopic motions of bubbles in a foam influence the microscopic draining of fluid from lamellae, which in turn triggers the rapid rupture of a thin lamella, thereby setting the bubbles in motion once more. But the simulation handles each of those processes in isolation. “It's way beyond anything we've attempted before,” Weaire says.

Relatively static foams, such as the bubbles in a head of beer—“foams that are just sitting there”—have been studied extensively, Weaire remarks. But there has been little progress on foams in flux, he adds, since he and a co-author published more than a decade ago a book called The Physics of Foams, in which they urged colleagues to advance the understanding of dynamic processes. The new work “is a step in that direction. It is a first step.” He notes that the model has some limitations, such as dealing only with so-called dry foams—those with relatively low liquid contents.

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