A study of the evaporative deposition process: Pipes and truncated transport dynamics

Abstract

We consider contact line deposition of an evaporating thin drop. Following Dupont's proposal (unpublished), we focus on transport dynamics truncated by a maximal concentration as the single deposition mechanism. The truncated transport process, formalized as the "pipe model", admits a characteristic shock front that has a robust functional form and depends only on local hydrodynamic properties. By applying the pipe model, we solve the density profile in different asymptotic regimes. In particular, we find that near the contact line the density profile follows a scaling law that is proportional to the square root of the concentration ratio defined as the initial solute volume concentration divided by the maximal concentration. The maximal deposit density occurs at about 2/3 of the total drying time for uniform evaporation and 1/2 for diffusion-controlled evaporation. Away from the contact line, areal density decays exponentially with the radial distance to the power of -3 for the uniform evaporation and -7 for the diffusion-controlled evaporation.

Robust fadeout profile of an evaporation stain

“…For uniform evaporation with J(0) =J, we find α = 1/2 and Σ ∼ x −3 , again in agreement with Ref. [13]. Similarly, for a gravity-flattened drop with h(0) =h with J(0) = 1 2J , we find α = 1/2 so that again Σ ∼ x −3 .…”

“…Recognizing that solute motion must be inhibited as its concentration increases, he postulated that the solute simply stops moving when its local volume fraction φ exceeds a threshold value φ c , while solute at lower volume fraction is carried passively. Following earlier work by Popov [12], Zheng has recently analyzed the surface density profile resulting from the Dupont mechanism in two important cases [13]. Zheng defines the surface density Σ as the volume of solvent per unit surface area measured after the solvent has evaporated.…”

“…In time ∆t, material is advected to r i from a point r i − v(r i , t)∆t just inside r i . Thus [3,13]…”

The contact line of an evaporating droplet over a solid wedge and the pinned–unpinned transition

“…If the contact line, which is circular, is pinned during the whole process of evaporation, then the shape of an evaporating droplet is merely a spherical cap that is turning flat. This is a simple and usual case that can describe the interface motion of evaporating sessile droplets, which have been studied by, for example, Deegan et al (1997), Popov (2005), Zheng (2009) and Gelderblom et al (2012). However, the smallness condition of Ca breaks down in the vicinity of the contact line, where the liquid velocity increases rapidly.…”

The Model of Drying Sessile Drop of Colloidal Solution

“…In the works 7,8 authors suggested that fixed solute stops outward flows and evaporation flux from the surface of the deposit region. In our opinion, models proposed in the works 5,6,9,10 are more adapted to inorganic deposition and models 7,8 are adapted to colloidal gels. Now the most complicated problem relates to determining the rate of evaporation from the free surface of the drop.…”