### Coffee Rings

December 10, 2010

Paul Erdos accomplished so much in mathematics that he's credited for at least one thing he didn't do. He's often cited as the source of the quotation, "A mathematician is a device for turning coffee into theorems," since he consumed large quantities of the liquid. This quotation is believed to have been spoken by another prominent mathematician, Alfréd Rényi. In any case, coffee is important enough to be taken seriously by mathematicians, and my own experience in research indicates that it's a respected tool of the trade for physical scientists. It's no wonder that physicists have taken to looking closely at coffee rings on their desktops.

The first observations of coffee rings were done in 1997 by Robert D. Deegan and his colleagues at the James Franck Institute and the University of Chicago.[1,2] They discovered that the mechanism for formation of a coffee ring is as follows:
1. The liquid surface is pinned at its contact line with the substrate.
2. The liquid evaporating at the exterior edge, which is the larger liquid reservoir, is replenished by liquid from the interior.
3. There is a constant flow of liquid from the interior to the exterior.
4. This flow carries nearly all dispersed solids towards the edge.

Of course, physicists never stop at just a qualitative explanation, so they determined that there is a particular power-law that describes the growth of the mass of the ring as a function of time. This power-law is independent of the substrate type, carrier fluid, or the dispersed solids, which means it happens not just for coffee, but for other colloidal fluids on other surfaces as well.[1] Rings form even when droplets are dried upside-down. Along with coffee, the Chicago team verified the effect with red wine, milk, tea and soup.

The ring effect is lessened for fluids with a high Marangoni number, since Marangoni flow will transport some material back to the center of the ring.[3] It's been shown that for some organic fluids, preferential deposition occurs at the center of the ring because of the Marangoni flow caused by an evaporation-induced surface tension gradient.[2] Even in the absence of Marangoni flow, there's competition between the various physical processes taking place in the colloidal droplet, so you would expect the coffee ring behavior to vary quite a bit between materials. If the liquid evaporates much faster than the particle movement, there will be no ring.[3] Size matters, since it was found experimentally that for 100 nm particles, the minimum coffee ring size is 10 μm.[4]

Washington's Coffee ad. Rotogravure image from the New York Tribune, June 22, 1919

Can coffee ring dynamics produce useful structures? A paper presented at last month's 63rd annual meeting of the American Physical Society Division of Fluid Dynamics showed that periodic band structures can be formed with proper choice of initial conditions. The paper, "Coffee ring deposition in bands," by Shreyas Mandre (Brown University), Ning Wu (Colorado School of Mines), Joanna Aizenberg (Harvard University) and Lakshminarayanan Mahadevan (Harvard University) reports that if the particle concentration is below a threshold, the deposit appears as periodic bands oriented parallel to the contact line.[5,6] This effect is a result of a competition between the speed of evaporation speed and the speed of deposition. The authors were able to develop a mathematical model that predicts the pattern that's formed. Similar patterns have been seen in non-colloidal droplet drying (see figure).[7]

Concentric rings observed in salt deposits from a drying two mm droplet.
Fig. 1 of Ref. 7, via arXiv.

A video of ring formation that includes data on the avalanche of particles at the edge can be found at Ref. 8. The experiment shown in the video used a droplet containing one micrometer particles of fluorescent polystyrene. The particles were in a concentration 7x109 particles per cc in 3μl of water on a glass slide at 23o at 30% relative humidity.[8]

The stimulative effect of coffee is primarily from caffeine, (C8H10N4O, 1,3,7-trimethyl-1H-purine-2,6(3H,7H)-dione). The concentration of this varies according to coffee type and preparation, as follows (approximate values):
•Espresso: 30-40 mg per ounce
•Drip coffee: 16-25 mg per ounce
•Instant: 10-15 mg per ounce
•Decaf, brewed: 0-0.1 mg per ounce

Mathematicians and scientists should stick to coffee and eschew the new caffeinated alcoholic energy drinks. The US Food and Drug Administration issued a warning on November 17, 2010, that it considers caffeine to be an "unsafe food additive" when added to alcoholic malt beverages.[9]

### References:

Linked Keywords: Paul Erdos; coffee; theorem; Coneheads; consume large quantities; Alfréd Rényi; physical scientists; physicists; coffee rings; James Franck Institute; University of Chicago; evaporation; power-law; colloid; Marangoni number; Marangoni flow; organic compound; Washington's Coffee; New York Tribune; American Physical Society; Division of Fluid Dynamics; Brown University; Colorado School of Mines; Harvard University; fluorescent; polystyrene; relative humidity; caffeine; caffeinated alcoholic energy drinks; Food and Drug Administration.