ABSTRACT
The deformation, drainage, and rupture of an axisymmetrical film between colliding drops in the presence of insoluble surfactants under the influence of van der Waals forces is studied numerically at small capillary and Reynolds numbers and small surfactant concentrations. Constant-force collisions of Newtonian drops in another Newtonian fluid are considered. The mathematical model is based on the lubrication equations in the gap between drops and the creeping flow approximation of Navier-Stokes equations in the drops, coupled with velocity and stress boundary conditions at the interfaces. A nonuniform surfactant concentration on the interfaces, governed by a convection-diffusion equation, leads to a gradient of the interfacial tension which in turn leads to additional tangential stress on the interfaces (Marangoni effects). The mathematical problem is solved by a finite-difference method on a nonuniform mesh at the interfaces and a boundary-integral method in the drops. The whole range of the dispersed to continuous-phase viscosity ratios is investigated for a range of values of the dimensionless surfactant concentration, Peclét number, and dimensionless Hamaker constant (covering both "nose" and "rim" rupture). In the limit of the large Peclét number and the small dimensionless Hamaker constant (characteristic of drops in the millimeter size range) a fair approximation to the results is provided by a simple expression for the critical surfactant concentration, drainage being virtually uninfluenced by the surfactant for concentrations below the critical surfactant concentration and corresponding to that for immobile interfaces for concentrations above it. Copyright 2000 Academic Press.
ABSTRACT
The hydrodynamic model of steady wetting developed by Boender et al. is extended to include the effect of a (nonionic) surfactant in the limiting case of negligible diffusion and low concentrations, confining attention to steady wetting between parallel plates. The approximation that the meniscus inclination becomes equal to the static contact angle at a distance from the solid of the order of a molecular dimension is extended to take account of the local surfactant concentration, making use of Young's law. A second inner boundary condition, provided by a surfactant balance at the contact line, places a restriction on the speed at which the interface is shed, leading to surfactant accumulation and partial or almost total immobilization of the interface which reduces the wetting speed. Under certain conditions, this immobilization is self-stabilizing, leading to hysteresis effects. Both these effects and the reduced wetting speed correspond with results reported in the literature. Copyright 1998 Academic Press.
ABSTRACT
The same approach used by Boender, Chesters, and van der Zanden in the context of an advancing liquid-gas meniscus in a capillary tube is extended to the case of spontaneous spreading of a droplet on an ideal solid surface. The result is an ordinary differential equation for the droplet profile which can be solved if the meniscus inclination phi0 is specified at some distance lambda from the solid. As in the capillary-tube case, good agreement is obtained with experimental data obtained by the authors and by others if phi0 is set equal to the static contact angle (zero in cases investigated experimentally), taking lambda of the order of a molecular dimension (1 nm). A comparison of predicted dynamic contact angles in the spreading-drop and capillary-tube cases for given values of the capillary number indicates only a weak dependence of the behavior on the system geometry. De Gennes and co-workers have predicted that during the final stages of spreading the inner length scale lambda should be determined by the effects of disjoining pressure in the thin film adjacent to the contact line rather than by molecular dimensions. The lambda value implied by their model is derived, thereby establishing the regime of spreading in which such effects should be dominant. The observed behavior in this regime is found to correspond somewhat better with a lambda value of the order of a molecular dimension, although the differences are small. Although the explanation probably lies in the nonideality of even the smoothest surfaces, this result suggests that the simplest model, based on a single lambda value of the order of 1 nm, should provide an excellent predictive tool. Copyright 1998 Academic Press.
