Robust fadeout profile of an evaporation stain

TL;DR

This paper explains the fading density profile of stains left by drying droplets, linking the power-law decay to flow stagnation points and local evaporation rates within the droplet.

Contribution

It introduces a theoretical model connecting the stain density decay to flow stagnation points and local evaporation characteristics in drying drops.

Findings

Density decreases as a power of distance from the edge.

Power depends on local evaporation rate and liquid depth at stagnation point.

Provides a formula relating decay power to local and average properties.

Abstract

We propose an explanation for the commonly-seen fading in the density of a stain remaining after a droplet has dried on a surface. The density decreases as a power $p$ of the distance from the edge. For thin, dilute drops of general shape this power is determined by a flow stagnation point in the distant interior of the drop. The power $p$ depends on the local evaporation rate J(0) at the stagnation point and the liquid depth $h(0)$ there: $p = 1 - 2 (h(0)/\bar h)(\bar J/J(0))$, where $\bar h$ and $\bar J$ are averages over the drop surface.