# 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 flows from regions with low surface tension to regions with high surface tension. The authors ask, within the drop, what direction is this Marangoni flow? Using both theoretical and experimental approaches the authors determine conditions which result in two different flow patterns.

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 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 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.

Ristenpart *et. al.* determined 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 four different evaporating fluid drops.