Threshold of Bénard-Marangoni instability in drying liquid films

Abstract

-We here show how evaporation/condensation processes lead to efficient heat spreading along a liquid/gas interface, thereby damping thermal fluctuations and hindering thermocapillary flows. This mechanism acts as an effective thermal conductivity of the gas phase, which is shown to diverge when the latter is made of pure vapor. Our simple (fitting-parameter-free) theory nicely agrees with measurements of critical conditions for Bénard-Marangoni instability in drying liquid films. Heat spreading is also shown to strongly affect wavelength selection in the nonlinear regime. In addition to providing a quantitative framework for analyzing transitions between complex evaporation-driven patterns, this also opens new perspectives for better controlling deposition techniques based on drying.

Genesis of Bénard–Marangoni Patterns in Thin Liquid Films Drying into Air

“…As described in [2], the mass of the liquid meniscus against the lateral wall has to be taken into account, together with the mass of the vapor contained within the cylinder, in order to obtain an accurate estimation of the instantaneous liquid thickness e(t). The evaporation rate itself, E, is simply computed from the time derivative of the total mass, using a linear fit.…”

“…This estimation assumes a linear (purely conductive) temperature distribution in the liquid layer, which actually corresponds to the usual definition of the Marangoni number Ma = −γ T ∆ Te/ηκ (where γ T is the surface tension variation with temperature, η is the liquid dynamic viscosity and κ is the liquid thermal diffusivity) characterizing the destabilizing effect of thermocapillarity. The supercriticality can then be computed as ε = (Ma − Ma c )/Ma c , where Ma c is the critical Marangoni number corresponding to the transition between convective and conductive states (observed at the critical thickness e c , see [2] for details).…”

“…When the gas phase is much thicker than the liquid depth, as considered hereafter, the Biot number is simply proportional to the wavenumber, with a coefficient depending only on thermo-physical properties of the fluids involved (including the diffusion coefficient of the vapor in the air) and on ambient temperature and pressure. In view of these recent results, the preliminary answer to question iii) above seems to be positive, in that the two-layer problem of a liquid evaporating into a non-soluble inert gas can indeed be reduced, under quite reasonable assumptions [6,2], to a much simpler problem involving liquid phase quantities only. The only noteworthy difference compared to the Pearson's problem of a liquid layer heated from below is that the free surface heat transfer is described by a (non-local) generalization of Newton's cooling law accounting for the phase change process.…”

“…Note that the last of these questions has recently been investigated by the authors [2], who showed that the instability threshold is quite accurately described by a generalized Pearson's theory [13], provided the heat transfer coefficient at the free surface (i.e. in dimensionless form, the Biot number) is suitably defined to incorporate heat spreading along the interface by vapor diffusion in the gas and associated latent heat exchange.…”

“…in dimensionless form, the Biot number) is suitably defined to incorporate heat spreading along the interface by vapor diffusion in the gas and associated latent heat exchange. Interestingly, this effect formally turns out to act as an effective thermal conductivity of the gas phase, which can be quite large for very volatile liquids [2]. Such reduction to a one-sided approach is actually possible thanks to the smallness of relaxation time scales in the gas (compared to those in the liquid layer), allowing to slave the dynamics of temperature and concentration fluctuations in the gas to the interfacial temperature fluctuations (see also [3,6]).…”