Criterion for reversal of thermal Marangoni flow in drying drops

The thermal Marangoni flow induced by nonuniform surface temperature has been widely invoked to interpret the deposition pattern from drying drops. The surface temperature distribution of a drying droplet, although being crucial to the Marangoni flow, is still controversial. In this paper, the surface temperature in the drop central region is analyzed theoretically based on an asymptotic analysis on the heat transfer in such region, and a quantitative criterion is established for the direction of the surface temperature gradient and the direction of the induced Marangoni flow of drying drops. The asymptotic analysis indicates that these two directions will reverse at a critical contact angle, which depends not only on the relative thermal conductivities of the substrate and liquid, but also on the ratio of the substrate thickness to the contact-line radius of the droplet. The theory is corroborated experimentally and numerically, and may provide a potential means to control deposition patterns from drying droplets.