Imaging and quantifying mixing in a model droplet micromixer

Rapid mixing is essential in a variety of microfluidic applications but is often difficult to achieve at low Reynolds numbers. Inspired by a recently developed microdevice that mixes reagents in droplets, which simply flow along a periodic serpentine channel [H. Song, J. D. Tice, and R. F. Ismagilov, "A microfluidic system for controlling reaction networks in time," Angew. Chem. Int. Ed. 42, 767 (2003)], we investigate a model "droplet mixer." The model consists of a spherical droplet immersed in a periodic sequence of distinct external flows, which are superpositions of uniform and shear flows. We label the fluid inside the droplet with two colors and visualize mixing with a method we call "backtrace imaging," which allows us to render cross sections of the droplet at arbitrary times during the mixing cycle. To analyze our results, we present a novel scalar measure of mixing that permits us to locate sets of parameters that optimize mixing over a small number of flow cycles.