Dancing Water Droplets and Electro-Elasto-Capillarity

And Also, How to Read a Science Paper, Hands-On Edition

While doing some light research for a frivolous personal post concerning dance, I came across a paper entitled Tap dance of a water droplet. Naturally, although I have zero expertise in the subject (does a semester of undergrad electromagnetism count?), I had to read it, and then write about it here. And naturally, because again, I have zero expertise here, this will be briefer-than-usual with less detail than the usual journal club installment.

As an outsider, the first question I have to ask is, why study droplets of water in electric fields? A quick google search yields an interesting college news article that explains the history of studying droplets in electric fields: how research in the field led to a reduction of lightning strikes (yes, that does sound oddly like controlling the weather), and that almost impossible-to-solve differential equations have been used to describe them. It also reports that by diverging from the status quo (that is, describing droplet size by volume instead of by height) physicist Justin Beroz & colleagues were able to simplify these equations, making them calculable by a simpler pen-and-paper math problem. This matched an unexpectedly simple distribution of the data they found, as the physicists watched droplets burst under the strength of an electric field.

So anyway, studying water droplets and how they behave in electric fields has some fascinating math behind it and big implications for engineers working with systems of liquid and electricity. (To address the weather-controlling surprise above, think power lines. Studying water and electricity led to engineers insulating power lines; previously, when raindrops approached the uncovered cords, their electric field burst the approaching droplets, increasing instances of lightning strikes.)

Clearly this is a scientific problem with real-world applications. So, let’s continue reading. The abstract continues with the phrase “progress of electrowetting on a movable substrate,” and claims that studying this will have big implications for the medical field. Ok, so what is electrowetting? In this video, an MIT researcher tells us how electrowetting is the process by which an electric field can modify the wetting properties of a hydrophobic surface, making it a means of moving small drops of liquids. The researcher explains how this can be used in biology labs to manage liquid transfers without wasting pipette tips (thus, saving money). By extension, this means cheaper drug development, the primary impact concern of this researcher; it also means cheaper experiments with less plastic waste across many biological sciences.

Essentially, they are using electric fields to move & mix droplets of samples on an electric surface.

Suddenly, the rest of the abstract is making a lot more sense to me; they are running experiments to test how this process works, with the hope that it can improve drugs and advance nanotechnology (the ‘microelectromechanical systems’ mentioned at the end of the abstract.) Using electricity to control drops of water also gives us a way to control microscopic technologies, i.e. nanotech.

With that discovered, we are ready to move into the main body of the text.

The Problem

We can use capillary action and electrowetting to control small quantities of liquids, which has applications for bioengineering, medicine, physics, chemistry, and microelectromechanical systems. However, these methods leave liquid droplets exposed to the environment – is there a way to protect them from evaporation or contamination?

Elastic film (such as graphene, PDMS, and silicon film) can wrap up a water droplet via capillary force alone, but the surface tension built between liquid and film means that the film will now hold on to its droplet forever (or at least, until it evaporates). So there is interest in a method that can wrap – and unwrap – droplets as needed.

So we use capillary forces to wrap the drop in the first place, but want it to be elastic (that is, reversible), and guess what? We can use electricity to do this. Hence the term electro-elasto-capillarity, or EEC. This papers studies the effects of EEC under direct (DC) and alternating current (AC).

The Experiment

Fig. 1 depicts the experimental set up. We place a PDMS film (our elastic film of choice) over a superhydrophobic substrate. This substrate insulates the droplet from a current, and allows the PDMS film to wrap the droplet via adhesion. All this is placed on an electrode that has been hooked up to a voltage source, with the second electrode inserted into the water droplet. In other words, they built a capacitor – charge can build up on the electrode & water droplet, but a superhydrophobic insulator between them keeps the charges built up on each side.

So the film spontaneously wraps the droplet due to adhesion forces, but what happens when a voltage is applied to this system? When the voltage increased, the film unwrapped. Finally, a high-speed camera was set up to record the movement of the film (and measuring θ, the angle of the film raised up over the substrate) under the different voltages of AC and DC currents.

Fig. 2 shows the droplet under DC. The droplet starts (at 0 V) wrapped up by its film, but as you increase the voltage, θ increases until it reaches 180° (flat, on the substrate). At this point, the electric force has cancelled out the capillary force between the film and the droplet.

Next? Try it under AC. The alternating current means that it is alternating between the positive and negative voltage it is set to. (10V of DC is a steady stream 10V; 10V of AC swings between +10V and -10V.) You can see this cyclic motion in a sinusoidal wave – sine and cosine functions are often used to describe cycles like this. Fig. 3 shows what happens when the droplet is exposed to the AC current. We get the tap dancing water droplet promised to us in the title. The droplet is exposed to maximum voltage (here, -650V or +650V) at 0 and 10 milliseconds. The photographs at those times show a flat, unwrapped droplet; in between max voltage, and the capillary force takes over while the droplet is wrapped by its film.

Thus, the scientists have used EEC to wrap & unwrap water droplets, just as they set out to do.

Discoveries

The scientists also wanted to study the physics of water droplets dancing to electricity, and they began with a model of their experimental set up shown in Fig. 5A. The equations listed for this computational model describe how stuff like the energy and angle change with varying frequency. Some variables needed include the geometry of the set-up, physical properties of PDMS (including it’s Young’s modulus and Poisson’s ratio), and the radius of the droplet. Energy can be stored in the kinetic energy of the film (which depends on the moment of inertia of each wing), the bending energy of the film, the surface energy of the droplet, and the electric energy stored in the capacitor.

These equations can be plugged into a Lagrangian equation (Eq. 2.8), which is used to study dynamic systems. Equation 2.8 is graphically depicted in Fig. 5B. It predicts the movement on the film as the angle changes under a particular voltage & frequency, and it predicts that angle quite closely.

At that particular frequency, the scientists also discovered that the response frequency emitted by the droplet was twice that of the input frequency – an octave higher, as shown in Fig. 4. They seem to have gotten a little carried away with the whole tap dancing metaphor – not only do the droplets tap dance to an alternating current, they sing an octave higher when exposed to a musical frequency.

They also measured vibration frequency of the droplet when exposed not just to one frequency, but a whole range of them, as played from a piano scale. They don’t really tell us the application of this, and stick to reporting these physical phenomena they observed, as well as show us the pretty diagrams that result from the experiment (including a discovery of resonance peaks).

Conclusion

The ability to control and reverse droplet wrapping has direct applications for technologies used in biology, physics, chemistry, and technology. Thus, this discovery-driven paper contributes research to a field still growing as an applied science. In between the mathematical descriptions of dancing water droplets and harmonic responses to musical stimuli, I’d say the scientists had a little bit of fun putting this work together.

And I cannot possibly tell you how happy it makes me that I got to learn this – and refresh on some concepts from those physics classes – as a serendipitous side-effect of looking up the history of tap dance.

Photo credit

References

Wang, Z., Wang, F. C., & Zhao, Y. P. (2012). Tap dance of a water droplet. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 468(2145), 2485–2495. https://doi.org/10.1098/rspa.2011.0679