Unifying monopolar and quadrupolar capillarity in turbulent interfacial suspensions

Particles at fluid interfaces aggregate via capillary interactions and, at sufficient concentration, can form a single cluster percolating throughout the entire domain. The aggregation process differs between capillary forces dominated by monopolar mode (long-ranged and gravity-driven, as in the "Cheerios effect") and quadrupolar mode (short-ranged and contact-line-driven). Here we systematically vary the diameter of spherical particles in turbulent interfacial suspensions, spanning the parameter space from quadrupole-dominant to monopole-dominant interactions. We first establish mode dominance criteria through dedicated pair-approach experiments. We then demonstrate how monopole-dominant particles at the interface of turbulent liquid layers cluster more readily and percolate at lower concentrations, due to the slower spatial decay of their mutual attraction compared to the quadrupole-dominant cases. We define an effective interaction length scale balancing capillarity and strain-driven drag, and an associated Bond number. This allows us to unify aggregation behaviors from both modes onto a universal phase map. Furthermore, we demonstrate that percolation occurs when the mean inter-particle distance becomes shorter than the interaction length. This simple, parameter-free criterion accurately predicts observed thresholds, facilitating practical prediction and control in environmental and industrial applications.