## Coffee Drops: dispersions and evaporative flows

Since the mid nineties the Chicago group has been interested in how a liquid flows in an evaporating droplet and how this flow gives a distinctive deposition of suspended solids. This page summarizes our ongoing work. We aim to include other related work, too, but we're less dilligent about that. If you know of recent work on evaporative flow that would be good to include on this page, by all means drop me a line

--- T. Witten, February 2003

## Coffee Caustics

 An informal web page about a puzzling singular flow, April 2004

## What will evaporation deposits do next??

 Redner patterns Coffee-Stain Rose From tinheart via Daniel Herbst, October 2015

## next step in pitchfork patterns

detail from "Lux Aeterna II" by Makoto Fujimura, courtesy Catherine Crouch, Swarthmore College.

# a Workshop

, Lorentz Center, Leiden, Netherlands, May, 2010

Laurent Limat, Vincenzo Vitelli, Thomas Witten, organizers

# Papers

## 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.

4 mbytes pdf

## Robust fadeout profile of an evaporation stain

Authors: T. A. Witten
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.

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

Author: Rui Zheng (University of Chicago)
Comments: 25 pages, 22 figures, submitted to Physical Review E
PhD thesis 1.2 megabytes
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

Authors: Yuri O. Popov (University of Chicago, University of Michigan)
Comments: 34 pages, 14 figures, LaTeX; submitted to Physical Review E
Subj-class: Soft Condensed Matter
Journal-ref: Physical Review E, 71, 036313 (2005)
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

Authors: Rui Zheng, Yuri O. Popov, Thomas A. Witten
Comments: 14 pages, 9 figures, submitted to Physical Review E
Subj-class: Soft Condensed Matter
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

Authors: Yuri O. Popov (University of Chicago, University of Michigan)
Comments: 37 pages, 21 figures, LaTeX; submitted to Physical Review E
Subj-class: Soft Condensed Matter
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

Authors: Yuri Popov (University of Chicago)
Comments: 94 pages, 40 figures, LaTeX; Ph.D. thesis
Subj-class: Soft Condensed Matter
Journal-ref: Ph.D. dissertation (University of Chicago, Dept. of Physics, 2003)
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.

### Characteristic Angles in the Wetting of an Angular Region: Deposit Growth

 As was shown in an earlier paper [1], solids dispersed in a drying drop migrate to the (pinned) contact line. This migration is caused by outward flows driven by the loss of the solvent due to evaporation and by geometrical constraint that the drop maintains an equilibrium surface shape with a fixed boundary. Here, in continuation of our earlier paper [2], we theoretically investigate the evaporation rate, the flow field and the rate of growth of the deposit patterns in a drop over an angular sector on a plane substrate. Asymptotic power laws near the vertex (as distance to the vertex goes to zero) are obtained. A hydrodynamic model of fluid flow near the singularity of the vertex is developed and the velocity field is obtained. The rate of the deposit growth near the contact line is found in two time regimes. The deposited mass falls off as a weak power Gamma of distance close to the vertex and as a stronger power Beta of distance further from the vertex. The power Gamma depends only slightly on the opening angle Alpha and stays between roughly -1/3 and 0. The power Beta varies from -1 to 0 as the opening angle increases from 0 to 180 degrees. At a given distance from the vertex, the deposited mass grows faster and faster with time, with the greatest increase in the growth rate occurring at the early stages of the drying process.

### Characteristic Angles in the Wetting of an Angular Region: Surface Shape

Authors: Yuri O. Popov, Thomas A. Witten (University of Chicago)

European Physical Journal E, 6, pp. 211-220 (2001)

Left: Experimental setup for the refraction demonstration.

Right: A set of parallel lines as seen through acute and obtuse wedges. The spacing between the undistorted lines is about $1.6$~mm. The opening angles of the acute and obtuse wedges are approximately $51^{\circ}$ and $124^{\circ}$ respectively. (Photo by Itai Cohen.)

### Physical Review E (Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics) -- July 2000 -- Volume 62, Issue 1, pp. 756-765

Robert D. Deegan, Olgica Bakajin, Todd F. Dupont, Greg Huber, Sidney R. Nagel, and Thomas A. Witten
James Franck Institute, 5640 South Ellis Avenue, Chicago, Illinois 60637