Puzzle Pair: Two brothers tackle the physics of shampoo

For chemical engineers, puzzles can be found everywhere—even in everyday materials. Joseph Peterson, an assistant professor of chemical and biomolecular engineering at the UCLA Samueli School of Engineering, says he often discovers interesting puzzles in hair care products – that is, the underlying physics that describe how these liquids flow.

“Shampoo and other liquid soaps are fascinating materials,” Peterson said. “At the molecular level, soap molecules organize themselves into very long but somewhat fragile, rope-like structures, similar to polymers. However, because these polymers are constantly breaking apart and stitching themselves back together, shampoo can behave very differently from other liquids.”

Soap molecules arrange themselves into long rope-like structures.  Sometimes the ends of these structures will stick together (left-to-right), and sometimes a rope will spontaneously break into (right-to-left).  Together, these processes change how a liquid soap flows.

Two time zones and nearly 1,800 miles away, Purdue University professor of mathematics Jonathon Peterson, Joseph’s older brother, is also intrigued by puzzles—albeit of a slightly different nature.

“I study random walks, which are simple models for random motion,” Jonathon Peterson said. “The classical model for random walks is well understood by now, but I study models with additional self-interacting complications that often lead to interesting and unexpected behaviors.”

While both work in academia — the younger specializing in physics of liquids, the older, a pure mathematician – the two never really thought they’d work together professionally… but back to shampoo. 

Scientists have been building mathematical models of shampoo-like materials for more than 30 years. In principle, a complete understanding of the complicated physics could help in the design of better materials or improved manufacturing processes for a wider range of polymers and plastics, not just shampoo.  In practice, however, liquid soaps have attracted (and will likely continue to attract) the attention of researchers simply because they are fascinating, many-layered puzzles.

Despite decades of research, some pieces of Joseph Peterson's shampoo puzzle still remained unresolved—or at least difficult to deal with. When thinking about the physics of chains that break apart and re-attach, one useful analogy is to think of cutting and splicing in terms of shuffling a deck of cards. Cutting and splicing is a mathematically complex process compared to simply tossing the cards in the air and restacking the deck randomly.

“After a while, either method will give you a well-mixed deck—but mixing the deck all at once makes the math a lot easier,” Joseph Peterson noted. Despite the simplification, Joseph Peterson's "shuffling model" would still require significant mathematical expertise to understand and implement. Peterson hoped to someday find a simpler solution—something that requires no specialized knowledge to understand or compute. However, after three decades of scientific exploration, such a find seemed unlikely.

Sometimes, for especially tricky puzzles, one must go backward to move forward—taking things apart in new ways to discover better ways of putting them back together. Last spring, Joseph Peterson began experimenting with novel approaches to breaking down his equations. By pulling on one particular thread, the shuffling model unraveled in a manageable way. Steps that had previously seemed difficult became straightforward, leaving only the task of reassembling the pieces.

Delegating to three UCLA graduate students in his Complex Fluids Processing research group, Joseph Peterson started to see progress. On his team, Vickie Chen helped reassemble the model, Charlie Drucker validated the results, and Claire Love wrote a program to automate comparisons with experimental data.

“I enjoyed the fact that this felt like a quintessential inter- and intra-group collaboration,” Drucker said. “I hope to see more of this kind of teamwork in the future.”

However, there was still one significant problem. In reconstructing the puzzle, Joseph Peterson and Chen changed the order in which certain steps were performed. While such mathematical interchanges are often valid, they can sometimes produce incorrect results.

“The interchange of these mathematical operations is exactly the sort of technical detail mathematicians focus on,” Jonathan Peterson said. “It’s a major topic in the graduate-level real analysis course I had just finished teaching.”

When Joseph first showed his brother the calculations, Jonathon thought checking the interchange would be a straightforward exercise, but it turned out to be more complex than expected. After several attempts, he was able to mathematically prove that his brother’s approach was indeed valid.

The collaboration culminated in a manuscript titled “Analytic Solution for the Linear Rheology of Living Polymers,” published open access in the Journal of Non-Newtonian Fluid Mechanics. The authors hope the work will help experimentalists replace outdated rules of thumb with detailed comparisons against a reliable model.

“Sometimes we forget there’s a human side to scientific progress,” Joseph Peterson said. “As engineers, we don’t always use the best tools available–sometimes we defer to those that are easier to use. In such cases, good science means closing the gap between doing things the right way and the easy way. That’s what we hope this paper achieves.”

As for what is next for the Peterson brothers, only time will tell if another collaboration might be in the books.

“It was fun to collaborate, but collaborations between pure mathematicians and engineers are generally rare,” Jonathon Peterson said. “This could be the first of many papers, or it could be the last.” His younger brother agreed, joking that the next shuffling project the duo will do will most likely be a game of pinochle.