Euler Characteristic is Key to Explaining Coarsening of Three-Dimensional Foams

May 3, 2007

A new model, which takes advantage of an object's Euler characteristic, lets researchers explain when bubbles in foams will grow or shrink.

In 1952, John von Neumann deciphered the mystery of why two-dimensional bubbles grow, shrink, or merge — a process called coarsening. Von Neumann proved that the answer depends on how many sides a 2D bubble has: When it has five or fewer sides, it shrinks; with seven or more, it grows; with six, it remains unchanged. Such foams form when bubbles are squeezed between glass plates.

The answer for the coarsening of three-dimensional bubbles has been much more elusive. Now, mathematician Robert MacPherson, of the Institute for Advanced Study, and theoretical materials scientist David Srolovitz, of New York City's Yeshiva University, have solved the riddle. Their findings appear in the April 26 issue of Nature.

The key ingredients are an object's Euler characteristic— the number of surfaces minus the number of holes in a sliced object — and what the researchers term the object's "mean width." A three-dimensional bubble will grow if the sum of the lengths of its edges is greater than six times its mean width. Otherwise it shrinks.

The new equation works in two dimensions, three dimensions, or any number of dimensions, vastly expanding the areas to which it can be applied. "This is one of those rare occasions where some really beautiful, pure mathematics could be applied to an important problem in the sciences wholesale," Srolovitz says.

The discovery will likely help scientists investigating, for example, the stability of foams or the arrangement of crystalline grains in a solid.

Sources: Nature, April 26, 2007; Science Now, April 25, 2007; Yeshiva University