# Influence of Substrate Conductivity on Circulation Reversal in Evaporating Drops

## Overview

• [1] Ristenpart, W., Kim, P., Domingues, C., Wan, J., & Stone, H. Physical Review Letters. 99, 234502-1 - 234502-4 (2007).
• Keywords: Evaporation, Marangoni Effect, Contact Angle, Thermal Conduction, Surface Tension

## Summary

Figure 1 from [1].

In this paper, Ristenpart et. al. study the flow patterns inside liquid drops on flat surfaces. Non-unifrom evaporation from a drop's surface leads to a thermal gradient and, thus, a gradient in surface tension. Displaying the Marangoni effect, fluid in a drop flows from regions with low surface tension to regions with high surface tension. The authors set out to answer: What is the flow pattern and direction? Using both theoretical and experimental approaches, the authors determine conditions which lead to two different flow directions.

Figure 1 is a diagram of a drop with important parameters labeled. $k_L$ and $k_S$ are the thermal conductivities of the liquid and the substrate respectively. $\theta_c$ labels the contact angle.

Figure 4 from [1].

The authors determine that the flow moves in either of two directions. The form of the flow's streamlines are independent of direction and are approximated by those shown in figure 4 (a). The direction of the flow depends on the contact angle and the ratio of thermal conductivities $k_R=\frac{k_S}{k_L}$.

The researchers determined theoretically that for $k_R$ above two, the fluid flows radially inward across the free surface of the drop and radially outward over the surface of the substrate. Below $k_R$ of 1.45, the fluid flows in the opposite direction. In between $k_R$ of 1.45 and 2, the flow direction depends on the contact angle. Ristenpart et. al. determine that $k_R^{crit}$ depends on the contact angle as follows:

$k_{R}^{crit}=tan(\theta_{c})cot\left(\frac{\theta_c}{2}+\frac{\theta_c^2}{\pi}\right)$. Under all flow conditions, particles gather in the middle of the drop (at stagnation points of the flow) leaving interesting patterns after the drop evaporates completely (figure 4).

The researchers check their predictions of flow direction by observing the flow in evaporating drops of four different fluids: methanol, ethanol, isopropanol, and chloroform. By observing suspended particles in the fluids, the scientists make qualitative observations of flow direction which coincide with the predictions of the $k_{R}^{crit}$-dependent flow.

## Soft Matter Details

Experimental Techniques:

The scientists place liquid drops on a clear substrate (PDMS) through which they can observe suspended particles with a microscope from below. This direct method of observing the direction of flow is limited to clear substrates. Looking at the flow from above might disturb the natural evaporation rate. The authors suspend polystyrene particles in three of their test fluids and silica particles in the forth.

Theoretical Techniques:

The researchers are interested in flow and particle deposition. Particularly, the flow near the contact line is of interest. This is similar to the case of a receding or advancing contact angle. Determining the flow right near the triple point where liquid, substrate, and air meet is very difficult. Maybe some of the methods in this paper could be used to study receding and advancing contact lines if they have not been used already.

Critical Value:

Soft matter is full of critical values which delineate the border between two qualitatively different regimes. In this case, the critical value indicates the complete reversal of flow direction! The authors look at four fluids with different contact angles and thermal conductivities. An interesting experiment would be to dynamically vary either the contact angle or the thermal conductivity to see if one can observe the transition from flow in one direction to flow in the opposite direction. Also interesting would be to observe drops very close to $k_{R}^{crit}$.

Open Questions:

At the end of their paper, the authors suggest that their methods could be used to analyze the influence of surfactant gradients on Marangoni flow in drops- particularly in water and biological systems.

This article is listed on the homepage under charged substrates, but the conductivity mentioned in the title is actually thermal conductivity, not electrical conductivity. Using a charged substrate and suspended charged particles or an ionic solution would add complexity to the system, but may also allow greater control over the deposition of particulates.