Floating and Sinking of a Pair of Spheres at a Liquid-Fluid Interface

Spheres floating at liquid-fluid interfaces cause interfacial deformations such that their weight is balanced by the resultant forces of surface tension and hydrostatic pressure while also satisfying a contact angle condition. Determining the meniscus shape around several floating spheres is a complicated problem because the vertical locations of the spheres and the horizontal projections of the three-phase contact lines are not known a priori. Here, a new computational algorithm is developed to simultaneously satisfy the nonlinear Laplace-Young equation for the meniscus shape, the vertical force balance, and the geometric properties of the spheres. We implement this algorithm to find the shape of the interface around a pair of floating spheres and the horizontal force required to keep them at a fixed center-center separation. Our numerical simulations show that the ability of a pair of spheres to float (rather than sink) depends on their separation. Similar to previous work on horizontal cylinders, sinking may be induced at close range for small spheres that float when isolated. However, we also discover a new and unexpected behavior: at intermediate inter-particle distances, spheres that would sink in isolation can float as a pair. This effect is more pronounced for spheres of radius comparable to the capillary length, suggesting that this effect is a result of hydrostatic pressure, rather than surface tension. An approximate solution confirms these phenomena and shows that the mechanism is indeed the increased supporting force provided by the hydrostatic pressure. Finally, the horizontal force of capillary attraction between the spheres is calculated using the results of the numerical simulations. These results show that the classic linear superposition approximation (due to Nicolson) can lose its validity for relatively heavy particles at close range.