“Compared with the giants of quantum physics, we soft-matter theorists look like the dwarfs of German folk tales,” says Pierre Gilles de Gennes. “We are strongly motivated by industrial purposes. We see fundamental problems emerging from practical questions.” Such humility is taken to new extremes by Sidney Nagel and co-workers (Deegan, R. D. et al. Nature 389, 827–829; 1997), who find insight into the physics of soft matter by means of an eminently familiar, even mundane, phenomenon: the coffee stain.

Credit: R. D. DEEGAN ET AL.

Readers could be forgiven for assuming that the physics of an evaporating coffee droplet would be buried in the annals of a previous century. But how liquid droplets flow, spread and dissipate is imperfectly understood. The statics of the situation are explained by Thomas Young's famous condition for wetting (1805), based on the balance of surface tensions at the droplet edge. But what of the dynamics?

The dynamical situation is often complicated by the fact that the edge of the droplet (the contact line) gets ‘pinned’ at points on the surface, generally as a result of surface heterogeneities. This is what underlies the work of Nagel et al., who propose an explanation for why coffee stains have pronounced boundaries, where most of the material is deposited. This material is uniformly distributed in the initial drop, so why does it get concentrated around the edges?

If the edge of the evaporating droplet is pinned in place, evaporation cannot simply shrink the droplet's lateral dimensions. Instead, there must be a net flow towards the edge to replenish liquid lost by evaporation while keeping the contact line in place. Suspended material is carried along with this flow, as shown by video microscopy of polymer microspheres in a drying droplet.

The idea can be made quantitative by calculating the evaporative flux from the surface of a circular droplet as a function of radial distance. It turns out that, in a dilute solution, the flux and flow velocity become infinite at the contact line. That implies complete transfer of the solute to the perimeter by the time the droplet dries out, leaving a precisely sharp ring stain. In practice, this is modified in a concentrated solute, which leaves a finite ring width and thus the sort of graduated stains seen here.

Industrial processes that might benefit from these ideas are the drying of ink droplets in printing and of surface coatings applied as liquids; and perhaps the efforts of amateur water-colourists, to which de Gennes has also likened soft-matter theorists: “spending their Sunday afternoons in the park, and capturing a few simple scenes.”