When espresso evaporates: the physics of coffee rings

Original paper: Capillary flow as the cause of ring stains from dried liquid drops

Figure 1. A 2-cm dried drop of coffee with a stain around the perimeter, forming a coffee ring. Adapted from Deegan et. al.

I’ve spilled a lot of coffee over the years. Usually not a whole cup, just a drop or two while pouring. And when it’s just a drop, it’s easy to justify waiting to clean it up. When the drop dries on the table, it forms a stain with a ring around the edges (Figure 1), giving it the look of a deliberately outlined splotch of brown in a contemporary art piece (when I say “coffee ring” I mean the small-scale, spontaneously formed stain around the edge of the original drop, rather than the imprint left on a table from the bottom of a wet coffee cup). But the appearance of these stains is simply a result of the physics happening inside the drop. Coffee is made of tiny granules of ground up coffee beans suspended in water, so the ring must mean that these granules migrate to the edge of the droplet when it dries. Why do the granules travel as they dry? Today’s paper by Robert D. Deegan, Olgica Bakajin, Todd F. Dupont, Greb Huber, Sidney R. Nagel, and Thomas A. Witten provides evidence that coffee rings arise due to capillary flow–the flow of liquid due to intermolecular forces within the liquid and between the liquid and its surrounding surfaces.

contact angle
Figure 2. Diagrams of contact angles for different droplets. From left to right, the first is exhibits poor wetting, with a large contact angle. The next has good wetting, with a smaller contact angle. The last has perfect wetting, with a contact angle of zero, and coffee grains suspended in this solvent would not be able to form a ring upon drying.

The researchers found that these rings don’t just form in coffee. Their observations showed that the rings form in a wide variety of solutes (the suspended coffee granules), solvents (the water), and substrates (the table you spill on) as long as a few conditions are met. First of all, the droplet has to have a non-zero contact angle[1] (See Figure 2). In other words, the droplet doesn’t spread out into a completely flat puddle on the table. Second, the contact line has to be pinned. This means that the surface has irregularities or roughness that cause the edge of the droplet to get stuck in place. Last, the solvent has to evaporate; the ring won’t form if the droplet never dries.

So now we know the conditions required for rings to form, but we want to know how they form. Deegan and his colleagues found that the rings are caused by a geometrical constraint. Here’s how it works: The pinning of the contact line means that the perimeter of the droplet cannot move, so the diameter of the droplet has to remain constant. But if the water in the droplet is evaporating, the droplet’s height will be reduced at every point (Figure 3a). Along the edges, where the droplet is thinnest, the height would be reduced to zero, and the droplet would shrink.

But the contact line pinning means that droplet can’t shrink. To prevent this shrinkage, liquid must flow out to replenish the liquid at the droplet edge as it evaporates. This flow brings with it the suspended coffee granules (or whichever solute is suspended in the solvent), pushing them outward until they pack at the edge of the droplet to form a ring (Figure 3b).

droplet cross sections
Figure 3. (a) Diagram showing the cross-section of a droplet on a surface. The shaded region shows how the droplet will shrink due to evaporation after a small amount of time if the contact line is not pinned. (b) Now, a black line is added to show how the droplet will shrink if the contact line is pinned. The arrows indicate that more liquid must flow to the outside of the droplet to replace what is lost to evaporation. Adapted from Deegan et. al.

By calculating how quickly water evaporates from the surface of a droplet, the researchers derived an expression for the mass of the ring as a function of time. It takes the form of a power law, which can be shown as a straight line on a log-log plot. Equipped with a quantitative prediction, the researchers set about performing experiments to test their model. Instead of using coffee, they opted for plastic microspheres suspended in drops of water. They placed the drops on glass slides and used a video microscope to image the droplets as they dried, recording the particles moving to the edges of the droplet (Figure 4).

Figure 4. Particles flowing to the edge of a droplet during evaporation to form a ring. Video from [2] and produced by Deegan et. al.
The researchers knew the mass of the individual particles, so they were able to calculate the mass of the ring as a function of time by counting the particles as they traveled to the edges. The results were shifted by an offset time t0 to account for early times where the power law prediction doesn’t hold and were shifted by mass M0 to account for the particles deposited during this initial stage. From the plot comparing the data and theory (Figure 5), we can see that the prediction shows good agreement with the data.

M vs T
Figure 5. Plot of mass in the ring as a function of time. The mass is plotted in units of particle number, so the plot shows how the number of particles grows over time. The three lines correspond to three different droplets. The upper curve overlapped with the middle so was shifted up for clarity. The circles show data and the solid lines show the theoretical prediction. The slope of 1.37 is the exponent of the power law predicted by the theory; On a log-log plot, a power law is a line with the exponent as the slope. Adapted from Deegan et. al.

In the twenty years since this paper was published, the study of drying droplets has continued in full force [3]. Scientists have discovered various particle patterns that can form under different drying conditions. Why do we care so much about these drying droplets? If the beauty of the physics isn’t motivation enough, then maybe the applications will convince you. The physics of drying is essential to inkjet printing, and a better understanding of the drying process could help make more precise printers [4]. Drying patterns can be used to identify the presence of certain proteins, making this a potential tool for disease detection [5]. Maybe next time you spill some coffee, you’ll take a moment to think of the physics of the drying droplet before you wipe it away.

[1] The contact angle is the angle where a liquid-gas interface meets a solid surface. The smaller the contact angle, the better the wetting of the surface.

[2] https://mrsec.uchicago.edu/research/highlights/coffee-ring-effect

[3] https://www.nature.com/uidfinder/10.1038/550466a

[4] Soltman, D. & Subramanian, V. Langmuir 24, 2224–2231 (2008).

[5] Trantum, J. R., Wright, D. W. & Haselton, F. R. Langmuir 28, 2187–2193 (2012)

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