Drainage of a Thin Liquid Film Confined between Hydrophobic Surfaces

We investigate theoretically the drainage of a thin liquid film between two undeformed hydrophobic spheres. The role ofhydrophobicity is revealed in the apparent slippage of liquid over the solid. The origin of the slippage effect is probably linked with a decrease in viscosity in the very thin near-to-wall layer. The solution is obtained for arbitrary values of slip lengths (from zero to infinity) as well as for arbitrary radii of curvature of approaching surfaces. The main result consists in that the pressure and the drag force yield the product of corresponding expressions for similar hydrophilic spheres and some corrections for slippage. These corrections depend only on the relationships between the gap and the slip lengths. As a result, at distances that are much greater than both slip lengths of approaching surfaces, the liquid flow is the same as that for hydrophilic surfaces. If the gap width exceeds considerably only one of the slip lengths then the pressure and the resistance will be equal to those experienced by hydrophilic sphere moving toward the free bubble surface. If the gap is much smaller than both slip lengths, the flow will be like that which arises when two bubbles approach each other. In the latter case, the hydrodynamic drag is not inversely dependent on the gap but is inversely proportional to the slip lengths and only logarithmically dependent on the gap. The correction for slippage plays a dramatic role in the coagulation processes. The main result for coagulation consists in the possibility for collision to occur at a finite time. Also, this correction needs to be taken into account when the various properties of confined liquids (first of all the hydrophobic attractive force) are investigated with the drainage technique.