Evaporative Deposition in Receding Drops

Julian Freed-brown
http://arxiv.org/abs/1410.0639 Soft Matter 10 9506-9510 (2014)

We present a framework for calculating the surface density profile of a stain deposited by a drop with a receding contact line. Unlike a pinned drop, a receding drop pushes fluid towards its interior, continuously deposits mass across its substrate as it evaporates, and does not produce the usual "coffee ring." For a thin, circular drop with a constant evaporation rate, we find the surface density of the stain goes as $\eta(r) \propto \left(\left(r/a_0\right)^{-1/2}-r/a_0\right)$, where $r$ is the radius from the drop center and $a_0$ is the initial outer radius. Under these conditions, the deposited stain has a mountain-like morphology. Our framework can easily be extended to investigate new stain morphologies left by drying drops.

Ehrenfest lecture, Leiden, May 2010

Robust fadeout profile of an evaporation stain

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.

Fadeout effect

Slides shown at DeGennesDays meeting, Paris, May 2008 pdf, 3 megabytes

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

We consider contact line deposition and pattern formation of a pinned evaporating thin drop. We identify and focus on the transport dynamics truncated by the maximal concentration, proposed by Dupont [1], as the single deposition mechanism. The truncated process, formalized as "pipe models", admits a characteristic moving shock front solution that has a robust functional form and depends only on local conditions. By applying the models, we solve the deposition process and describe the deposit density profile in different asymptotic regimes. In particular, near the contact line the density profile follows a scaling law with respect to the square root of the concentration ratio, and the maximal deposit density/thickness 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, we identify the power-law decay of the deposition profile. In comparison, our work is consistent with and extends the previous results[2]. We are also able to comment on the depinning process and multiple-ring patterns within our models, and our predictions are consistent with the empirical evidence.

Evaporative Deposition Patterns Revisited: Spatial Dimensions of the Deposit

A model accounting for finite spatial dimensions of the deposit patterns in the evaporating sessile drops of colloidal solution on a plane substrate is proposed. The model is based on the assumption that the solute particles occupy finite volume and hence these dimensions are of the steric origin. Within this model, the geometrical characteristics of the deposition patterns are found as functions of the initial concentration of the solute, the initial geometry of the drop, and the time elapsed from the beginning of the drying process. The model is solved analytically for small initial concentrations of the solute and numerically for arbitrary initial concentrations of the solute. The agreement between our theoretical results and the experimental data is demonstrated, and it is shown that the observed dependence of the deposit dimensions on the experimental parameters can indeed be attributed to the finite dimensions of the solute particles. These results are universal and do not depend on any free or fitting parameters; they are important for understanding the evaporative deposition and may be useful for creating controlled deposition patterns.

Deposit Growth in the Wetting of an Angular Region with Uniform Evaporation

Solvent loss due to evaporation in a drying drop can drive outward flows and solute migration. The whole process is characterized by the evaporation rate, as well as the geometrical restriction. In this paper, as a continuation of earlier investigations [5, 6], we consider a simplified, yet physically feasible model, with uniform evaporation. We study the flow velocity field near the singularity of the angular region, and find the rate of the deposit growth along contact lines in early and intermediate time regimes. Compared to the diffusion-controlled evaporation profile, uniform evaporation yields more singular deposition at early time regime, and nearly uniform deposition profile is obtained for a wide range of opening angles in the intermediate time regime. Uniform evaporation also shows a more pronounced constrast between acute opening angles and obtuse opening angles.

Evaporative Deposition Patterns Revisited: Spatial Dimensions of the Deposit

A model accounting for finite spatial dimensions of the deposit patterns in the evaporating sessile drops of colloidal solution on a plane substrate is proposed. The model is based on the assumption that the solute particles occupy finite volume and hence these dimensions are of the steric origin. Within this model, the geometrical characteristics of the deposition patterns are found as functions of the initial concentration of the solute, the initial geometry of the drop, and the time elapsed from the beginning of the drying process. The model is solved analytically for small initial concentrations of the solute and numerically for arbitrary initial concentrations of the solute. The agreement between our theoretical results and the experimental data is demonstrated, and it is shown that the observed dependence of the deposit dimensions on the experimental parameters can indeed be attributed to the finite dimensions of the solute particles. These results are universal and do not depend on any free or fitting parameters; they are important for understanding the evaporative deposition and may be useful for creating controlled deposition patterns.

Singularities, Universality, and Scaling in Evaporative Deposition Patterns

The theory of solute transfer and deposit growth in evaporating sessile drops on a plane substrate is presented. The main ideas and the principal equations are formulated. The problem is solved analytically for two important geometries: round drops (drops with circular boundary) and pointed drops (drops with angular boundary). The surface shape, the evaporation rate, the flow field, and the mass of the solute deposited at the drop perimeter are obtained as functions of the drying time and the drop geometry. In addition, a model accounting for the spatial extent of the deposit arising from the non-zero volume of the solute particles is solved for round drops. The geometrical characteristics of the deposition patterns as functions of the drying time, the drop geometry, and the initial concentration of the solute are found analytically for small initial concentrations of solute and numerically for arbitrary initial concentrations of solute. The universality of the theoretical results is emphasized, and comparison to the experimental data is made.