# 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

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

Figure 1 is a diagram of a drop with important parameters labeled. The magnitude of the arrows directed away from the drop shows the relative rates of evaporation at different places on the drop's surface. <math>k_L</math> and <math>k_S</math> are the thermal conductivities of the liquid and the substrate respectively. <math>\theta_c</math> labels the contact angle.

The authors determine that there are two qualitatively different flow patterns. Both patterns have the same streamlines which are shown in figure 4. However, the direction of the flow depends on the contact angle and the relative thermal conductivities <math>k_L</math> and <math>k_S</math>.

The researchers determined theoretically that the critical ratio of thermal conductivities <math>k_R=\frac{k_S}{k_L}</math> above which fluid flows radially inward across the free surface of the drop and radially outward over the surface of the substrate. Below <math>k_R</math> the fluid flows in the opposite direction. Under both flow conditions, particles gather in the center of the drop leaving interesting patterns after the drop has evaporated completely. (coffee rings?)

Ristenpart *et. al.* determine that <math>k_R</math> depends on the critical angle as follows: <math>k_{R}^{crit}=tan(\theta_{c})cot\left(\frac{\theta_c}{2}+\frac{\theta_c^2}{\pi}\right)</math>.

The researchers check this equation by observing the flow in evaporating drops of four different fluids: methanol, ethanol, isopropanol, and chloroform. The qualitative observations of flow direction by observing suspended particle flow follow the predictions of the critical thermal conductivity ratio.

## Soft Matter Details

**Experimental Techniques:**

The scientists place the liquid drops on a clear substrate (PDMS) through which they can observe suspended particles with a microscope. The 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 (because polystyrene is not soluble in the forth fluid?).

**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 already.

**Critical Value:**

Soft matter is full of critical values which signify the border between two qualitatively different regimes. In this case, the critical value separates flow directions completely reverse! 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 you 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 <math>k_{R}^{crit}</math>.

**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 under charged substrates, but the conductivity here is actually thermal conductivity, not electrical conductivity. Using a charged substrate and charged particulates would add complexity to the system, but may also allow greater control over the deposition of particulates.