In a dark room with only a black light and a computer screen offering visibility, we watch the spreading of a small fluorescent blob. As the experiment progresses, minutes turn into hours. I ask myself, what am I, a math grad student, doing in this physics laboratory?
The fluorescent blob that we are so interested in consists of surfactant molecules spreading on a surface of liquid. Surfactants lower surface tension. You use them every day. Shampoo, soap, and laundry detergent lower the surface tension between dirt and your skin or clothes so that water is able to wash the dirt away. A more recent example is surfactant replacement therapy, a medical procedure credited with saving thousands of lives of newborn babies over the last few decades. In the therapy, surfactant is introduced into the lungs of premature babies in order to facilitate their first breaths. Better understanding of the mechanics of spreading surfactant might lead to a more reliable medical procedure.
Spreading occurs because liquid moves from where surface tension is low to where it is high. So when surfactant is placed on a liquid, a liquid wave moves away from that location. But where is the surfactant in relation to the wave? Can we quantify the mechanism that connects the moving wave and the spreading surfactant? Together with physics professor Karen Daniels and physics graduate student Dave Fallest, my advisor Michael Shearer and I investigate the answers to these questions. My involvement in this research is funded by a National Science Foundation grant for training students in the mathematics of materials.
How is mathematics connected to this experimental project? As any mathematician loves to argue, everything can be boiled down to a mathematical equation. In this case, a system of partial differential equations (PDE) models the evolution of the liquid surface and the surfactant. The system of equations has terms that incorporate the physical influences such as gravity, capillarity, and diffusion. The equations are complicated, but as mathematicians we are free to ignore some of the terms in order to examine the impact of each influence. Based on results of the numerical and theoretical study of the PDE system, we suggest possible experiments to our physics counterparts. Conversely (as we mathematicians like to say), the experiments suggest interesting mathematical questions.
Our group of collaborators extends beyond the campus. In January 2009, researchers from NC State, Duke, UNC, and Harvey Mudd College gathered at NC State for a Thin Films Day, at which we shared recent results and discussed directions for future work. The departments represented included mathematics, physics, and mechanical and aerospace engineering.
The NC State thin films group is part of a larger research group that investigates nonlinear systems both mathematically and physically. In weekly lab meetings, we discuss research projects in fluid flow and granular materials. These meetings help mathematicians and physicists to understand each other’s language and ways of thinking, sometimes leading to unexpected revelations.
Popping the mathematical bubble to collaborate with our physics colleagues has led to a better understanding of liquid systems with surfactant. As the saying goes, “Two heads are better than one.” By combining the approaches of two disciplines, we are better able to explore the significant scientific issues. Using analysis, numerical simulation, and experiment simultaneously in cooperation with our physics collaborators has facilitated a deeper understanding of the physical behavior.
Participants in the Thin Films Day held at NC State in January 2009.
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