378361 Capillary Attraction and Hydrodynamic Resistance Between Two Floating Spheres at a Viscous Oil-Water Interface
The aggregation of colloids floating at a fluid/fluid interface due to attractive capillary forces arising from the deflection of the interface around the particles has drawn significant interest. This 2D phenomenon plays a critical role in self-assembly in pollination, the formation of dense particle laden interfaces in Pickering emulsions, froth flotation processes in mining and the nanoscale assembly of materials. The rate at which floating colloids approach is a balance between the capillary attractive force and the viscous hydrodynamic resistance experienced by the colloids as they move along the interface. Research has focused principally on the capillary force, but the hydrodynamics of the particle motion, which is typically a Stokes flow without inertia, and the effect of the flow resistance on the aggregation has received less attention.
As a particle straddling an interface moves along the surface, the phases bounding the interface exert a viscous resistance, which is proportional to the viscosities of the individual phases, and the relative immersion of the particle in the phases. When the viscosities of the bounding phases are close to one another, the viscous resistance can be approximated by the familiar Stokes resistance of a sphere in an infinite medium. However under most cases of interest, the viscosities of the bounding phases differ significantly, as for example particles floating on an air/water surface or a viscous oil/water interface. In this case the hydrodynamic resistance is a strong function of the immersion depth and viscosities.
In this presentation, we first present experimental measurements of the resistance as a function of immersion and viscosity stratification for the capillary attraction of two spherical colloids at an interface between an oil overlying an aqueous phase. We measure the time for aggregation of pairs of Teflon particles of identical sizes (approximately 1– 2 mm in diameter) on oil/water interfaces with oils of different viscosities (10 – 100 times that of water). The aggregation times are fit to a theoretical model in which the capillary attraction is set equal to the hydrodynamic resistance. The capillary attraction is computed in the small slope approximation, with the contact angle of the oil/water interface at the particle surface, necessary for the calculation of the attraction, measured directly. The viscous resistance is expressed as a resistance (frictional) coefficient f multiplied by the Stokes drag on the particle in an infinite medium with a viscosity equal to that of the upper (oil) phase. The coefficient f is a function of the viscosity ratio of the water to the oil phase, k, and the immersion depth d (calculated from the vertical force balance) of the particle into the lower aqueous phase to the particle radius a. We find that f(k,d/a) can decrease significantly for very small increases in the immersion depth of the particle into the less viscous aqueous phase for the particular particle sizes. These measurements of f are then compared with the theoretical calculations of the hydrodynamic resistance, which is obtained by solving the Stokes equations in toroidal coordinates. The hydrodynamic resistance is calculated by integrating an analytical solution for the pressure and velocity fields obtained by solving a Laplace equation in toroidal coordinates using a Mehler-Fock transform.