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# A simple formula that could be useful for air purification, space propulsion and molecular analyzes

When a drop of rain falls through a storm cloud, it is subjected to powerful electric fields that shoot and shoot the droplet, like a soap bubble in the wind. If the electric field is strong enough, the droplet may burst, creating a thin, electrified fog.

Scientists began to notice droplet behavior in electric fields in the early 1900s, when lightning had damaged newly constructed power lines. They soon realized that the electric fields of the lines caused the raindrops to burst, providing a path for lightning. This revelation led engineers to design thicker coatings around power lines to limit lightning impacts.

Today, scientists understand that the stronger the electric field, the more likely it is that a droplet breaks out. However, calculating the exact field strength that will burst a particular droplet has always been a complex mathematical task.

Researchers at MIT have discovered that the conditions under which a droplet bursts in an electric field boil down to a simple formula, which the team first established.

With this new simple equation, researchers can predict the exact force that an electric field should have in order to burst a droplet or keep it stable. The formula applies to three cases previously analyzed separately: a droplet pinned to a surface, sliding on a surface or floating freely in the air.

Their results, published today in the journal Letters of physical examination, can help engineers adjust the electric field or droplet size for a range of applications depending on the electrification of droplets. These include technologies for air or water purification, space propulsion and molecular analysis.

"Before our result, engineers and scientists had to perform intensive computer simulations to assess the stability of an electrified droplet," says lead author Justin Beroz, a graduate student in MIT's mechanical engineering and physics departments. . "With our equation, we can predict this behavior immediately, with a simple paper-and-pencil calculation.This presents a great practical advantage for engineers working with, or trying to design, any system involving liquids and electricity. "

The co-authors of Beroz are A. John Hart, Associate Professor of Mechanical Engineering, and John Bush, Professor of Mathematics.

"Something simple unexpectedly"

Droplets tend to form small, perfect spheres because of the surface tension, the cohesive force that binds water molecules to the surface of a droplet and pulls the molecules inward. . The droplet may deform from its spherical shape in the presence of other forces, such as the force of an electric field. While the surface tension acts to hold a droplet together, the electric field acts as an opposite force, pulling the droplet outward as the charge is formed on the surface.

"At some point, if the electric field is strong enough, the droplet can not find a shape that balances the electric force, and at that moment, it becomes unstable and bursts," Beroz explains.

He and his team were interested in the moment just before the burst, when the droplet was deformed to reach its critical shape. The team set up an experiment in which they slowly distributed droplets of water on an electrified metal plate to produce an electric field and used a high-speed camera to record the deformed shapes of each droplet.

"The experience is really boring at first." You watch the droplet slowly change shape, then all of a sudden, it bursts, "says Beroz.

After experimenting with droplets of different sizes and under different electric field intensities, Beroz isolated the video image just before each explosion, then presented its extremely stable shape and calculated several parameters such as volume, height and radius of the droplet. He plotted the data from each drop and discovered, to his surprise, that they were all falling down a line that was invariably straight.

"From a theoretical point of view, it was a surprisingly simple result given the mathematical complexity of the problem," Beroz says. "He suggested that there might be a neglected but simple way to calculate the bursting criteria for the droplets."

Volume above the height

Physicists have long known that a liquid droplet in an electric field can be represented by a set of coupled nonlinear differential equations. These equations, however, are incredibly difficult to solve. To find a solution, it is necessary to determine simultaneously the configuration of the electric field, the shape of the droplet and the pressure inside it.

"This is commonly the case in physics: it is easy to write the governing equations but very difficult to solve them," says Beroz. "But for droplets, it turns out that if you choose a particular combination of physical parameters to set the problem upfront, a solution can be inferred in a few lines, otherwise it's impossible."

Physicists who have tried to solve these equations in the past have made taking into account, among other parameters, the height of a droplet – an easy and natural choice to characterize the shape of a droplet. But Beroz made a different choice, reframing the equations in terms of droplet volume rather than height. This is the essential interest in reformulating the problem into an easy-to-solve formula.

"Over the last 100 years, the convention was to choose the height," says Beroz. "But when a droplet is deformed, its height changes and, therefore, the mathematical complexity of the problem is inherent in height.On the other hand, the volume of a droplet remains fixed regardless of the way it is deformed in the electric field. "

By formulating the equations using only "fixed" parameters in the same direction as the volume of a droplet, "the complicated and insoluble parts of the equation cancel, leaving a simple equation that matches to experimental results, "says Beroz.

More specifically, the new formula developed by the team combines five parameters: the surface tension, the radius, the volume, the intensity of the electric field and the electric permittivity of the air surrounding the droplet. By plugging four of these parameters into the formula, you will calculate the fifth.

According to Beroz, engineers can use the formula to develop techniques such as electrospray, which involves popping a droplet held at the orifice of an electrified nozzle to produce a fine spray. Electrospraying is commonly used to aerosolize biomolecules from a solution, so that they can pass through a spectrometer for detailed analysis. The technique is also used to produce thrust satellites and propel in space.

"If you are designing a system involving liquids and electricity, it is very convenient to have an equation of this type that you can use every day," says Beroz.

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