A Multiple Scales Approach to Evaporation Induced Marangoni Convection

This paper considers the stability of thin liquid layers of binary mixtures of a volatile (solvent) species and a non-volatile (polymer) species. Evaporation leads to a depletion of the solvent near the liquid surface. If surface tension increases for lower solvent concentrations, sufficiently strong compositional gradients can lead to Bénard-Marangoni-type convection that is similar to the kind which is observed in films that are heated from below. The onset of the instability is investigated by a linear stability analysis. Due to evaporation, the base state is time dependent, thus leading to a non-autonomous linearised system, which impedes the use of normal modes. However, the time scale for the solvent loss due to evaporation is typically long compared to the diffusive time scale, so a systematic multiple scales expansion can be sought for a finite dimensional approximation of the linearised problem. This is determined to leading and to next order. The corrections indicate that sufficient separation of the top eigenvalue from the remaining spectrum is required for the validity of the expansions, but not the magnitude of the eigenvalues themselves. The approximations are applied to analyse experiments by Bassou and Rharbi with polystyrene/toluene mixtures [Langmuir 2009 (25) 624–632].