Triangular nozzles provide the tiniest droplets, say researchers in Harvard University’s Division of Engineering and Applied Sciences who used a mathematical algorithm to determine that a miniature three-sided tap could produce drips some 21 percent smaller than a conventional round nozzle.

The miserly taps – which could, in theory, create drops just 8 billionths of a millimeter in size – might prove a boon for technologies that employ sprays of costly materials. For instance, triangular taps could boost the resolution of ink-jet printers, which work by squirting fine droplets of ink onto surfaces. They could also cut the size of traditional silicon chips and biochips, both of which feature patterns that are sometimes produced by a tightly controlled spray of droplets.

“Round nozzles are perfectly good for most applications,” says Henry Chen, a graduate student in physics who presented the work at a recent meeting of the American Physical Society’s Division of Fluid Dynamics. “Most nozzles don’t need to perform with exacting volume or pressure, so it may not even have occurred to anyone to try anything other than a circular opening.”

The minuscule triangular nozzle envisioned by Chen and his adviser, Michael P. Brenner, allows just one tiny drop to squeeze through. The tap’s three corners reduce a drop’s curvature and in turn the pressure needed to eject it from the nozzle. Compared to droplets from a tiny round tap, the scientists found, drops from an equally small triangular nozzle require less pressure to spray them out, easing the toll on pipes.

Chen and Brenner came across the new and improved tap shape – a triangle with slightly concave sides – using an algorithm developed to optimize the shape of mechanical devices, including ones familiar from everyday experience.

“We hope that the theoretical methods we used to answer this problem will prove broadly applicable,” says Brenner, Gordon McKay Professor of Applied Mathematics and Applied Physics. “We are trying to develop the general mathematical methods that are needed for carrying out mathematical optimizations of structures used in engineering.”

In addition to taps, Chen and Brenner see their mathematical methods applied to a number of other examples, including a new switch with a shape deemed optimal and a coffee cup whose form facilitates boiling and convection.