Evaporation of a thin film: diffusion of the vapour and Marangoni instabilities

TL;DR

This paper investigates the stability of evaporating thin liquid films, considering thermal Marangoni effects and vapour diffusion, revealing how these factors influence film stability and pattern formation.

Contribution

It introduces a non-local lubrication model incorporating diffusion and Marangoni effects, providing new insights into film stability and pattern bifurcation.

Findings

Evaporation is diffusion limited for moderately thick films.

Marangoni stresses destabilize the film, while capillarity and evaporation stabilize it.

The bifurcation from flat film state is supercritical.

Abstract

The stability of an evaporating thin liquid film on a solid substrate is investigated within lubrication theory. The heat flux due to evaporation induces thermal gradients; the generated Marangoni stresses are accounted for. Assuming the gas phase at rest, the dynamics of the vapour reduces to diffusion. The boundary condition at the interface couples transfer from the liquid to its vapour and diffusion flux. A non-local lubrication equation is obtained; this non-local nature comes from the Laplace equation associated with quasi-static diffusion. The linear stability of a flat film is studied in this general framework. The subsequent analysis is restricted to moderately thick films for which it is shown that evaporation is diffusion limited and that the gas phase is saturated in vapour in the vicinity of the interface. The stability depends only on two control parameters, the capillary and Marangoni numbers. The Marangoni effect is destabilising whereas capillarity and evaporation are stabilising processes. The results of the linear stability analysis are compared with the experiments of Poulard et al (2003) performed in a different geometry. In order to study the resulting patterns, the amplitude equation is obtained through a systematic multiple-scale expansion. The evaporation rate is needed and is computed perturbatively by solving the Laplace problem for the diffusion of vapour. The bifurcation from the flat state is found to be a supercritical transition. Moreover, it appears that the non-local nature of the diffusion problem unusually affects the amplitude equation.