Model for Predicting Color that May be Produced When Light Interacts with Structure of Droplets

A study by engineers from MIT and Penn State University has revealed that under ideal conditions, normal droplets of clear water can produce vibrant colors on a transparent surface, without the need to add dyes or inks.

In a paper published in the Nature journal on February 27^th, 2019, the engineers have described that a surface covered by a fine mist of transparent droplets and illuminated with a single lamp should be able to produce a vibrant color if the size of each tiny droplet is exactly the same.

This iridescent effect is caused by “structural color,” which induces an object to generate color merely by the way light interacts with its geometric structure. The effect could help explain some of the iridescent phenomena, such as the colorful condensation inside a water bottle or on a plastic dish.

The engineers have created a model for predicting the color that will be produced by a droplet, under particular optical and structural conditions. It would be possible to use the model as a design guide to create, for instance, droplet-based litmus tests, or color-changing inks and powders in makeup products.

Synthetic dyes used in consumer products to create bright colors might not be as healthy as they should be. As some of these dyes are more strongly regulated, companies are asking, can we use structural colors to replace potentially unhealthy dyes? Thanks to the careful observations by Amy Goodling and Lauren Zarzar at Penn State and to Sara’s modeling, which brought this effect and its physical explanation to light, there might be an answer.

Mathias Kolle, Assistant Professor, Department of Mechanical Engineering, MIT

Kolle’s co-authors on the paper are Sara Nagelberg of MIT, along with lead author Goodling, Zarzar, and others from Penn State.

Follow the Rainbow

In 2018, Zarzar and Goodling were analyzing transparent droplet emulsions produced from a mixture of oils with distinct densities. They observed the interactions of the droplets in a clear Petri dish and noticed that the drops seemed astonishingly blue. They captured an image of this and sent it to Kolle with a question: Why is there color here?

At first, Kolle was of the view that the color could be caused by the same effect by which rainbows are produced, where sunlight redirected by raindrops cause individual colors to be separated into different directions. In physics, the manner in which a plane of electromagnetic waves (for example, incoming sunlight) is scattered by spheres like raindrops is described by the Mie scattering theory. However, the droplets observed by Zarzar and Goodling were not spheres. In contrast, they were domes or hemispheres on a flat surface.

“Initially we followed this rainbow-causing effect,” stated Nagelberg, who led the modeling effort to try and describe the effect. “But it turned out to be something quite different.”

According to her, hemispherical droplets of the team broke symmetry, that is, they were not perfect spheres—an apparently evident fact but yet a significant one, as this indicates that light should behave in a different way in spheres versus hemispheres. Particularly, the hemisphere’s concave surface enables an optical effect that is not feasible in perfect spheres—total internal reflection (TIR).

TIR is a phenomenon where an interface between a medium with lower refractive index (like air) and a medium with high refractive index (such as water) is struck by light at a high angle such that 100% of the light is reflected. This is the effect that enables light to be carried by optical fibers for several kilometers with minimal loss. Light entering into a single droplet is reflected along its concave interface by TIR.

Indeed, as soon as light enters a droplet, Nagelberg discovered that it had the ability to take different paths, bouncing two, three, or even more times before exiting at a different angle. The manner in which light rays add up while exiting governs whether or not a droplet will produce color.

For instance, when two rays of white light that contain all visible light wavelengths enter at the same angle and exit at the same angle, they could take completely different paths inside a droplet. In case one of the rays bounces three times, its path is longer than a ray that bounces twice, so it lags behind a bit before exiting the droplet. If the waves of the two rays stay in phase (that is, crests and troughs of the waves are aligned) due to the phase lag, the color corresponding to that wavelength will be visible. This interference effect, which eventually produces color in otherwise clear droplets, is considerably stronger in small droplets when compared to large droplets.

When there is interference, it’s like kids making waves in a pool. If they do whatever they want, there’s no constructive adding up of effort, and just a lot of mess in the pool, or random wave patterns. But if they all push and pull together, you get a big wave. It’s the same here: If you get waves in phase coming out, you get more intensity of color.

Mathias Kolle, Assistant Professor, Department of Mechanical Engineering, MIT

A Carpet of Color

The color produced by the droplets is also based on structural conditions, for example, the curvature and size of the droplets, along with the refractive indices of the droplet.

By incorporating all these parameters into a mathematical model, Nagelberg attempted to predict the colors that would be produced by the droplets under specific optical and structural conditions. Then, Zarzar and Goodling tested the predictions of the model against actual droplets produced by them in the lab.

Initially, the engineers improved their initial experiment by developing droplet emulsions with sizes that could be precisely controlled with the help of a microfluidic device. According to Kolle, they produced a “carpet” of droplets of precisely the same size, in a clear Petri dish, and then illuminated it with a single, fixed white light. Subsequently, the droplets were recorded with a camera that revolved around the dish, and it was observed that the droplets displayed vibrant colors that shifted when the camera circled around. This showed how the angle at which light is observed to enter the droplet has an impact on the color of the droplet.

The engineers also produced droplets of different sizes on a single film and noticed that when viewed from a single viewing direction, the color would turn redder with increase in the droplet size, and would then loop back to blue and cycle through again. According to the model, this holds good as larger droplets would give more room for the light to bounce, thereby forming larger phase lags and longer paths.

The engineers demonstrated the significance of curvature in the color of a droplet by produced water condensation on a transparent film treated with a hydrophobic (water-repelling) solution, where the droplets formed the shape of an elephant. While more concave droplets were produced by the hydrophobic parts, the rest of the film produced shallower droplets. When compared to the shallow droplets, light was able to more easily bounce around in the concave droplets. The outcome was a very brilliant elephant pattern against a black background.

Apart from liquid droplets, the engineers 3D-printed solid, tiny caps and domes from different polymer-based, transparent materials and observed an analogous colorful effect in these solid particles, that could be predicted using the model created by the team.

Kolle anticipates that the model might be used to develop particles and droplets for a wide range of color-changing applications.

There’s a complex parameter space you can play with. You can tailor a droplet’s size, morphology, and observation conditions to create the color you want.

Mathias Kolle, Assistant Professor, Department of Mechanical Engineering, MIT

The study was partially supported by the National Science Foundation and the U.S. Army Research Office through the Institute for Soldier Nanotechnologies at MIT.