--- T. Witten, February 2003
|
| An informal web page about a puzzling singular flow, April 2004 |
 | Redner patterns |
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.
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.
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.
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.
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.
Authors:
Yuri O. Popov,
Thomas A. Witten (University of Chicago)
 | 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. |
European Physical Journal E, 6, pp. 211-220 (2001)
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.)
(Received 14 December 1998)
Solids dispersed in a drying drop will migrate to the edge of the drop and form a solid ring. This phenomenon produces ringlike stains and occurs for a wide range of surfaces, solvents, and solutes. Here we show that the migration is caused by an outward flow within the drop that is driven by the loss of solvent by evaporation and geometrical constraint that the drop maintain an equilibrium droplet shape with a fixed boundary. We describe a theory that predicts the flow velocity, the rate of growth of the ring, and the distribution of solute within the drop. These predictions are compared with our experimental results. ©2000 The American Physical Society
URL: http://publish.aps.org/abstract/PRE/v62/p756
PACS: 81.15.-z, 68.10.Jy, 47.55.Dz, 83.70.Hq Additional Information
