by Monte Basgall
Where do physicists go for help when they're stumped? When the French physicists Anne Marie Cazabat and Xavier Fanton couldn't explain why their experimental thin film was forming no "fingers," they went to Andrea Bertozzi, a Duke associate professor of mathematics and physics.
Bertozzi is an expert in the theory of the pesky drips that form when thin films, such as paints, are applied to walls. Inevitably, no matter how hard paint companies search for dripless formulations, the edges of such films form finger-like rivulets instead of flowing evenly down a wall's surface.
The Paris-based researchers were studying the opposite situation: what happens when a thin film moves up a wall. Specifically, they used a heat source positioned below a cold source to drive a thin film of a silicon oil called polydimethylsiloxane up the inclined surface of a silicon wafer.
As long as the silicon wafer was set at a completely vertical angle, Cazabat and Fanton found that the film's upper edge also formed finger patterns under gravity's influence precisely the expected behavior.
The unexpected came when they positioned the wafer at shallower angles. Then the usual fingering pattern inexplicitly vanished. "They seemed to have a found a stable coating process, which is great!," Bertozzi recalled in an interview. "So the question was, why is this happening?"
Scrambling for explanation themselves, the physicists looked for the absence of a "capillary ridge," a normal raised bump at the film's edge that "has been traditionally thought to play a significant role in the formation of fingers," Bertozzi said.
"You would think that if they didn't see fingers, then there must be no ridge. But they found a very pronounced ridge. So they threw up their hands in the air and started sending their data to Duke to get some help," she said.
Thus began a mathematical adventure, also involving Duke post-doctoral mathematics associate Andreas Muench and North Carolina State University mathematics professor Michael Shearer, that culminated in a December 7 report in the journal Physical Review Letters as well as a follow-up article in the news pages of the journal Science.
"We needed to combine a number of different branches of mathematics to solve this problem," Bertozzi said.
One key to their unorthodox solution was the fact that changing the wafer's angle also changed the film's thickness, Bertozzi noted. Thinnest in the vertical position, the film grew increasingly thick as angles grew shallower. And gravity's influence also grew as the film thicknesses increased, thus putting gravity in increasing competition with the heat-driven force known as the Marangoni Stress that pushed the film upwards.
Using computer simulations, the mathematicians found this competition leads to the formation of two distinct types of shock waves. One type of shock called "compressive" is the normal kind that forms when a jet breaks the sound barrier.
But computer simulation also revealed another type of shock wave separating out and moving ahead of the first at the advancing fluid's edge. Called an "undercompressive shock," it is one that, while mathematically describable, was not thought to exist in nature, she said.
"What you see is that there are these two shock fronts that are separating from each other," she said. "As this gravitational force becomes stronger, you get into a regime where you are able to pick up these undercompressive fronts."
The addition of the undercompressive shock front appears to prevent finger formation, she added. "As the film gets thicker, all of a sudden there is a transition from compressive fronts to undercompressive fronts, and from films that finger to those that don't."
Back in Paris, Bertozzi said, the experimental physicists are now busy designing experiments to test out this new mathematical theory. "It was an honor to get experimentalists of such high caliber to share their data with us and to ask for help," she added. "In this case, the mathematics really explained what was going on. There was no intuitive way of understanding what was going on through experiments alone.
"We were very proud to play a role in the understanding of some new and unusual physics."