epl draft Robust fadeout profile of an evaporation stain

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)/h̄)(J̄/J(0)), where h̄ and J̄ are averages over the drop surface. Introduction. – Recent years have witnessed startling new mechanisms for creating predictable, selforganized structures from long-known classical principles [1]. Among these newly recognized mechanisms is the deposition of solute in an evaporating drop of liquid [2]. Evaporation entails a specific flow pattern of the liquid which in turn concentrates the solute onto the perimeter of the drop. Many universal features of the resulting deposition have been tested [3, 4] and exploited for applications [5–9]. However, several prevalent features of the density profile of the deposition still await explanation [10]. Here we address one such feature: the decrease or fadeout of density with distance from the perimeter, as shown in Fig. 1. Our mechanism predicts a power law fadeout whose exponent depends on two controllable features of the evaporation profile and droplet shape. The drop of Fig. 1 contained a volatile solvent that partially wet the solid substrate; this solvent leaves the drop over time via evaporation. The drop also contained nonvolatile molecular or colloidal solutes. These may be carried along by any lateral flow of the solvent as it dries, but they are not carried away when the solvent evaporates. Such lateral flow occurs because a) the perimeter or contact line is typically pinned by irregularities in the surface and previous solute deposition, b) the free surface of the liquid takes the equilibrium shape of constant mean curvature dictated by its surface tension, and c) the local change of volume dictated by the thinning of this equilibrium shape is not matched by the local loss due to evaporation. To supply the volume needed for evaporation, lateral flow is required. Near the perimeter the evaporative loss Fig. 1: Optical micrograph of the edge of a 5 mm drop of black ink diluted with water, deposited on a glass microscope slide and dried in air. Continuous decrease of image density occurs as one moves downward from the edge of the drop towards the center. The discrete particles and other patterns are not addressed here. greatly outweighs the supply due to local thinning; thus, the lateral flow is strong. It is sufficiently strong to carry any point of the drop’s interior to the perimeter during the drying time [2]. For a thin, circular drop with evaporation controlled by air diffusion, the flow field may be readily determined and the consequent accumulation of solute with time deduced [3]: in this base case the mass deposited at the perimeter in time t varies initially as the p-1 ar X iv :0 90 3. 49 19 v2 [ co nd -m at .s of t] 2 7 M ay 2 00 9