Sunday, May 2, 2010

Why coffee stains are ring-shaped

There's a group of clever people at the James Franck Institute (UChicago) who study some of the overlooked and bizarre regularities of everyday life -- e.g. Tom Witten, Sid Nagel, Heinrich Jaeger, and Wendy Zhang. Their work has always struck me as very beautiful and "classical" in its spirit: it is the sort of thing the founders of the Royal Society, or for that matter Euler or the Bernoullis, might have studied and would have appreciated. They work on things like the flow of sand out of thin nozzles, the fact that liquids do not splash on top of Mt Everest, and the extent to which a droplet of water that pinches off a nozzle remembers the shape of the nozzle.

An esp. nice result that I heard about at a talk yesterday was Witten's group's theory of why coffee-stains have the sharp-edged, ring-like shapes they do. [The relevant references are Nature 389, 827 (1997) and Phys. Rev. E 62, 756 (2000).] The answer goes something like this: the edge of the coffee-bearing water droplet snags on surface roughnesses and gets pinned. (On sufficiently smooth surfaces, e.g. Teflon, coffee stains are not ring-shaped but uniform.) As water evaporates the bubble has to shrink; the water evaporates at roughly the same rate everywhere so ceteris paribus the bubble would want to shrink everywhere, but it can't because that would decrease its diameter and require the edge to move, and the edge can't move because it's pinned. The only way to keep the edge where it is while all of the surface evaporates at the same rate is for fluid to flow from the center of the droplet to the edge, to replenish the water lost from the edge. So over time most of the water in the bubble evaporates from the edge, and most of the solute gets deposited at the edge, so the stain is ringlike.

Under certain conditions, the edge is "almost" pinned but periodically unsnags ("depins"), moves some distance, and then snags again; this leads to terrace-like stain patterns.

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