The three dynamical regimes of a droplet driven by thermocapillarity

MAC INTYRE J. R.; CORREA, PABLO; GOMBA J. M.; SELLIER M,; PERAZZO, CARLOS ALBERTO

Christchurch

Simposio; IUTAM Symposium on Recent Advances in Moving Boundary Problems in Mechanics; 2018

The migration of droplets over a solid substrate is a long-standing topic of interest in academia[4, 7] and crucial in several situations of practical importance, such as coating flow technology,ink-jet printing, microfluidics and micro-electronics, and medical diagnostics [6, 2, 3, 1]. A dropletover a non-uniformly heated surface experiments a shear-stress that moves it from warmer tocolder regions in a quite complex way, and although there are various theoretical, experimentaland numerical works in the literature, a more detailed understanding of these thermally drivenflows is needed, in particular about the mechanisms underlying the dynamics of the three-phasecontact line. Here, the fate of droplets of partially wetting liquids subject to a thermo-capillarydriving force is investigated numerically with a particular emphasis on the effects of the form ofthe molecular interaction potential. A non-zero contact angle is imposed through a disjoining-conjoining pressure term, and the gradient of temperature along the substrate is constant. Thedroplet dynamics model is based on the lubrication approximation and the resulting partial differ-ential equation is solved in the Finite Element package COMSOL. The numerical results for twodifferent molecular interactions are compared: on the one hand, London-van der Waals molecularand ionic-electrostatics molecular interactions that account for polar liquids; on the other hand,long and short-range molecular forces that model molecular interactions of non-polar fluids. Inboth cases, the study confirms earlier observations [5] that three regimes exist depending on thevalue of the contact angle.For small contact angles, the film regime is observed for which the droplet stretches with theadvancing contact line moving substantially faster than the receding one leading to an increas-ing droplet footprint and a decreasing maximum thickness. The balance between viscous andMarangoni stresses produces a linear profile behind the capillary ridge at the front. This regimeis well described by the similarity solution given by [5] irrespective of the molecular interactionpotential which is a consequence of the fact that for small contact angles, the contribution of thedisjoining pressure term is negligible.For intermediate values of the contact angle, the transition regime occurs. In this regime, thedroplet either travels with a constant profile which is different from the initial one or breakupsinto a series of smaller droplets. In the former case, the droplet is found to travel with a constantvelocity which scales as the 1/3 power of the thermocapillary driving stress, and small differencesin the contact angle produces noticeable effects on the velocity. In this case, the outcome is foundto be strongly dependent of the molecular interaction potential: for polar liquids breakup occursmore readily than for non-polar liquids. In other words, it takes a smaller value of the drivingthermo-capillary stress to breakup the initial droplets into smaller ones for polar liquids than fornon-polar ones. This feature is explained by a linear stability analysis of the uniform film whichdevelops as the droplet evolves in the intermediate regime. This linear stability analysis revealsthat the maximum growth rate of the instability is larger for polar liquids than for non-polar onesand occurs for shorter wavelengths, so that the instability for non-polar liquids takes longer timesand larger distances to develop. The stability analysis also allows a prediction of the number ofdroplets the parent droplet breakups into and the distance between them.For larger values of the contact angle, the model confirms earlier observations that the droplettravels with a constant profile and at constant speed towards the colder part of the substrate. Thetravelling velocity is found to be greater for non-polar than for polar liquids and scales linearlywith the thermo-capillary driving force.This study provides new insights on the fate of partially-wetting droplets subject to a non-uniform,linearly varying temperature field. It also provides new guidelines on how to select fluid propertiesand operating conditions to achieve a desired wetting outcome which may be of importance in thecontext of droplet actuation.