Dynamics of the interaction of multiple evaporating droplets on compliant substrates

TL;DR

This paper presents a comprehensive theoretical model for the interaction of two evaporating droplets on soft substrates, considering thermal, elastic, and vapor diffusion effects, revealing complex behaviors like attraction, repulsion, and symmetry breaking.

Contribution

The study introduces a coupled lubrication and elasticity model that accounts for evaporative cooling, Marangoni stresses, and vapor diffusion, providing new insights into droplet interactions on compliant surfaces.

Findings

Droplets interact via substrate deformation and vapor phase.

Marangoni stresses delay droplet coalescence and extend lifetime.

Spontaneous symmetry breaking occurs on intermediate stiffness substrates.

Abstract

The dynamics of the interaction of a system of two thin volatile liquid droplets resting on a soft viscoelastic solid substrate are investigated theoretically. The developed model fully considers the effect of evaporative cooling and the generated Marangoni stresses due to the induced thermal gradients, while also accounting for the effect of the gas phase composition and the diffusion of vapour in the atmosphere of the droplets. Using the framework of lubrication theory, we derive evolution equations for both the droplet profile and the displacement of the elastic solid, which are solved in combination with Laplace's equation for the vapour concentration in the gas phase. A disjoining-pressure/precursor-film approach is used to describe contact-line motion. The evolution equations are solved numerically, using the finite-element method, and we present a thorough parametric analysis to investigate the physical properties and mechanisms that affect the dynamics of droplet interactions. The results show that the droplets interact through both the soft substrate and the gas phase. In the absence of thermocapillary phenomena, the combined effect of non-uniform evaporation due to the increased vapour concentration between the two droplets and elastocapillary phenonema determines whether the drop-drop interaction is attractive or repulsive. The Marangoni stresses suppress droplet attraction at the early stages of the drying process and lead to longer droplet lifetimes. For substrates with intermediate stiffness, the emergence of spontaneous symmetry breaking at late stages of evaporation is found. The rich dynamics of this complex system is explored by constructing a detailed map of the dynamic regimes.