Determining how much of an electric field a water droplet can handle before it bursts has long tormented scientists.

The concept appears easy but finding a simple mathematical formula that explains this phenomenon has proved difficult - until now.

A team of MIT scientists has determined a simple formula to assess the stability of an electrified droplet. It's so basic that working it out can be done with pencil and paper.

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“Before our result, engineers and scientists had to perform computationally intensive simulations to assess the stability of an electrified droplet,” says lead author Justin Beroz, a graduate student in MIT’s Department of Mechanical Engineering and Physics.

“With our equation, one can predict this behavior immediately, with a simple paper-and-pencil calculation. This is of great practical benefit to engineers working with, or trying to design, any system that involves liquids and electricity.”

## Discovery could help new space travel methods

The team behind the discovery say the new formula could lead to new advancements in a variety of fields from space propulsion to mass spectrometry, high-resolution printing, air purification, and molecular analysis.

Water droplets form as little spheres thanks to surface tension. This force binds water molecules at a droplet's surface and tugs the molecules inward, forming the shape.

This perfect shape might get distorted if it is exposed to other forces such as an electrical field. The surface tension tries to hold the droplets form but the opposing force of the field pulls the droplet out of shape.

“At some point, if the electric field is strong enough, the droplet can’t find a shape that balances the electrical force, and at that point, it becomes unstable and bursts,” Beroz explains.

Beroz says his team was interested in the moment - just before the droplet bursts and when it is at its most distorted shape.

## Boring experiment yields results

To examine this, the researchers set up an experiment where they slowly dispersed water droplets onto an electrified plate and used a super high-speed camera to record the droplets.

“The experiment is really boring at first — you’re watching the droplet slowly change shape, and then all of a sudden it just bursts,” Beroz says.

Initially, the team just documented a range of drops changing the size of the drop and the strength of the electric field. Later, each frame of the droplet was isolated to examine the shift in the droplet's shape as it was distorted by the field. Beroz outlined each droplet's critically stable shape just before it burst and calculated several parameters such as the droplet’s volume, height, and radius.

He then plotted this data and found that it fell along a straight line.

“From a theoretical point of view, it was an unexpectedly simple result given the mathematical complexity of the problem,” Beroz says.

“It suggested that there might be overlooked, yet simple, way to calculate the burst criterion for the droplets.”

## Keep it simple

The key to the discovery of the simple equation was to disregard the droplet's height and instead focus on its volume.

“For the last 100 years, the convention was to choose height,” Beroz says.

“But as a droplet deforms, its height changes, and therefore the mathematical complexity of the problem is inherent in the height. On the other hand, a droplet’s volume remains fixed regardless of how it deforms in the electric field.”