Dip-coating flow in the presence of two immiscible liquids

Abstract

Dip coating is a common technique used to cover a solid surface with a thin liquid film, the thickness of which was successfully predicted by the theory developed in the 1940s by Landau & Levich (Acta Physicochem. URSS, vol. 17, 1942, pp. 141–153) and Derjaguin (Acta Physicochem. URSS, vol. 20, 1943, pp. 349–352). In this work, we present an extension of their theory to the case where the dipping bath contains two immiscible liquids, one lighter than the other, resulting in the entrainment of two thin films on the substrate. We report how the thicknesses of the coated films depend on the capillary number, on the ratios of the properties of the two liquids and on the relative thickness of the upper fluid layer in the bath. We also show that the liquid/liquid and liquid/gas interfaces evolve independently from each other as if only one liquid were coated, except for a very small region where their separation falls quickly to its asymptotic value and the shear stresses at the two interfaces peak. Interestingly, we find that the final coated thicknesses are determined by the values of these maximum shear stresses.