CHAMPAIGN, Ill. -- By deriving an equation of state for compressible foam, and then simulating it numerically, University of Illinois researchers predict a dramatic morphological change that will occur as the surface tension is increased or, equivalently, the volume of the foam is greatly expanded. Foams are ubiquitous in nature and widely used in industry, from foamy foods such as bread and ice cream to foamy materials such as plant stems, bones, magma and foam rubber. All foams have one characteristic in common: the bubble-delimiting films minimize surface energy by encapsulating the largest volume using the least amount of material.
"In a common liquid foam, like a soap froth, the elastic energy in the films is negligible compared to the work required to compress the air in the bubbles," said Hassan Aref, professor and head of the theoretical and applied mechanics department. "The individual bubbles, which are often of roughly comparable size, retain constant volumes, except for the slow redistribution of gas by diffusion or the rupturing of films between bubbles. If you imagine greatly enhancing the surface tension, however, the elastic energy in the films will compress most of the bubbles, leading to a very different structure." To investigate this phenomenon, Aref and graduate student Dmitri Vainchtein first derived the equation of state for compressible foam. "This equation shows that foam with a free boundary will expand to a maximum volume if the external pressure is lowered at constant temperature," Aref said. "The equation also suggests that the same foam -- when enclosed in a container -- can be expanded further, but will become unstable at a certain volume that we can predict."
Though difficult to explore experimentally, the nature of the instability was revealed in a series of numerical simulations. "We found that as the surface tension increased, the overall structure of the foam changed dramatically," Aref said. "We observed what may be described as two 'phases' of foam. In one phase we have a large number of small bubbles clustered together. In the other phase we must then have a small number of much larger bubbles that occupy most of the space in the container."
The increased surface tension appears to compress most of the bubbles, forcing the remaining bubbles to expand to fill the space, Aref said. This phenomenon might provide a model for the undesirable formation of large voids in solidifying foams, including those that form when baking bread. "As bread is baked, the bubble membranes begin to harden, which may roughly correspond to an increase in surface tension," Aref said. "The resulting segregation instability results in a loaf that contains clusters of tiny bubbles embedded in a background of a few much larger bubbles."
While of little consequence in bread, the formation of large voids can be a nuisance in products like foam rubber, where a homogeneous texture is desired. "A better understanding of this phenomenon could lead to more precise process control in the manufacturing environment," Aref said.
The researchers reported their findings in the December issue of Physics of Fluids.
Journal
Physics of Fluids