Marangoni instability of liquid–liquid systems with a surface-active solute

Abstract

The onset of Marangoni instability in partially miscible liquid–liquid systems, such as organic–aqueous systems, in the presence of surface-active solutes is studied. The organic phase is initially pre-saturated with water and diffuses in the aqueous phase after both phases are contacted. The aqueous phase is contaminated with a surface-active solute which is also soluble in the organic phase and therefore, is transferred across the interface between the liquids. Marangoni instability in these systems may be induced by interfacial tension gradients, due to the heat of solution and the solute interfacial activity. In the present work, interfacial-tension-lowering solutes are considered. A linear stability analysis is carried out for both cases of stationary and oscillatory perturbations and the corresponding characteristic equations are derived in analytical form. They are analysed numerically for typical values of the physical parameters involved. It is shown that the stability conditions depend mainly on three physical parameters of the systems: the solute diffusivity ratio, the kinematic viscosity ratio and the product of the heat of solution and the thermal interfacial tension coefficient. In the stationary case, systems are stable when the solute diffusivity ratio is larger than one, independently of the magnitude of the viscosity ratio, and instability may appear if solute diffusivity is lower in the aqueous phase than in the organic phase. In the oscillatory case, systems are stable when the solute diffusivity ratio is smaller than one. If solute diffusivity is higher in water, the oscillatory stability depends on the kinematic viscosity ratio and the sign of the product of the heat of solution and the thermal interfacial tension coefficient. Systems with negative product are stable for the viscosity ratio smaller than one and could be unstable for the viscosity ratio larger than one. Instability may set in as overstability in systems with positive product for arbitrary viscosity ratio. In the case of equal solute diffusivities in both phases, the stationary stability of the systems depends on the sign of this product. Systems with negative product are stable whereas in systems with positive product stationary convection may appear. The oscillatory stability of systems with equal solute diffusivities depends only on the viscosity ratio. They are stable when kinematic viscosity is lower in the aqueous phase than in the organic phase and oscillatory instability may occur if kinematic viscosity is higher in the aqueous phase. The stability criteria are applied to three liquid–liquid systems contaminated with two surfactants and the theoretical predictions are compared against experimental observations.