Why is that? It’s because the edges of the spill are where the fluid is at its thinnest. A spill sitting on a flat surface is something like a pancake. Big in the middle and thin on the rim. As the thin edges evaporate, they are replenished by liquid and particles of coffee from the denser interior. That starts an outward push going in the puddle. The more it dries, the more particles get drawn out to the edges. When all of the liquid has evaporated we’re left with a ring of coffee particles making a stain.