Deformation and breakup of a stretching liquid bridge covered with an insoluble surfactant monolayer

Abstract

The breakup of surfactant-laden drops and jets is of technological interest and fundamental scientific importance. Surfactants are routinely used to control the breakup of drops and jets in applications ranging from inkjet printing to crop spraying. Accurate computation of breakup of surfactant-laden drops and jets is often the key to the development of new applications and to providing a rational fundamental understanding of both existing and emerging applications. While highly accurate algorithms for studying the breakup of surfactant-free drops and jets are well documented and much is now known about the dynamics in such situations, little is known by contrast about the closely related problem of interface rupture when surfactant effects cannot be neglected. The deformation and breakup of a stretching liquid bridge of an incompressible Newtonian fluid whose surface is covered with an insoluble surfactant monolayer are analyzed here experimentally and computationally. In the experiments, high-speed visualization is used to capture the transient deformation of a bridge. The dynamic shapes of bridges (captive between two rods of 3.15 mm diameter) are captured and analyzed with a time resolution of 1 ms. The bridge lengths are 3.15 mm initially and about 4–7 mm at breakup, which occurs after stretching for about 0.1–0.2 s, depending on the volume and viscosity of the liquid and the surface density of spread monolayers. The dynamics of a surfactant-covered bridge is governed by the Navier-Stokes and convection-diffusion equations. First, these equations are solved with a three-dimensional, but axisymmetric, or two-dimensional (2D), finite element algorithm using elliptic mesh generation. Second, the governing set of 2D equations is reduced to a set of one-dimensional (1D) equations by means of the slender-jet approximation and the resulting set of 1D equations is solved with a 1D finite element algorithm. The presence of surfactant results not only in the lowering of surface tension and the capillary pressure, but also in surface tension gradients and Marangoni stresses, both of which affect the transient dynamics leading to breakup. In particular, the role of Marangoni stresses in delaying bridge breakup and on formation of satellite droplets is investigated as a function of the initial surface density and surface activity of the surfactant, and surface Peclet number that measures the importance of convection relative to diffusion. The predictions of the 2D algorithm are confirmed to be faithful to the physics by demonstrating that the computed results accord well with the experiments and existing scaling theories. In the pinch-off region, the surfactant is swept out of a thinning neck by strong convection. The calculations thus reveal that the scaling behavior in the presence of surfactant parallels that observed in the absence of surfactant, in accordance with recent reports by others. The 2D computations and the experiments are used in tandem to identify regions in the space of governing parameters where the 1D equations can be used with confidence.