Radial-velocity profile along the surface of evaporating liquid droplets

Abstract

The deposit pattern of drying liquid droplets has a close relationship with the radial-velocity profile along the droplet surface. In this paper, the surface temperature of the droplet is first solved numerically and approximated by a simple analytical form, and an analytical expression for the radial-velocity of the surface flow is then obtained by using the lubrication analysis developed by Hu and Larson. The theoretical analysis indicates that the outward surface flow will be reduced by the thermal Marangoni stress along the surface. When the Marangoni number is below a critical value, Ma_Crit, all the surface liquid will move outward and a dense, ring-like deposit will be formed. When above Ma_Crit, a stagnation point, within which the surface flow is inward and beyond which the surface flow is outward, will appear on the droplet surface. In such case, the particles transported to the surface beyond the stagnation point will move to the droplet edge to form the ring deposit, and the others will deposit on the central region of the droplet. Numerical results indicate that the critical Marangoni number decreases in a power law with the contact angle. The theory for the radial-velocity of the surface flow will be helpful to predict and control the deposit patterns from the drying droplets.