Interaction of Multiple Drops and Formation of Toroidal-Spiral Particles

In the development of drug delivery technologies for treating complex diseases, encapsulating multiple compounds and manipulating their sustained-release kinetics independently (for optimal therapeutic effect) can be challenging. Toward this goal, we previously developed a fluid-dynamic technology based on multi-drop interactions to produce solid toroidal-spiral (TS) particles. During sedimentation in a miscible, viscous liquid, polymeric drops self-assemble into a reproducible and controllable TS structure, which can be solidified into particles by photo-initiated cross-linking of the polymer. The goal of encapsulating multiple drops of different physical properties (such as size and density) generally requires complicated and time-consuming laboratory iteration on the starting conditions, because all satellite drops (containing drugs) must catch up and coalesce simultaneously with the main drop that forms the surrounding matrix upon solidification. In this paper we consider a model system for multi-drop entrainment that features a main drop followed by three smaller satellite drops arranged in a horizontal, triangular array. Experiments visualized with a high-speed camera are used to validate computer simulations based upon a swarm-of-Stokeslets method. The simulations accurately track complex drop configurations involving intertwined interfaces. Replacing the actual starting drop shapes with suitably positioned, volume-equivalent spheres yields very similar configurations: the crucial deformations and interactions occur during sedimentation, as opposed to during the initial injection of the drops. The simulations are then used to formulate two robust rules of thumb by which further trial-and-error (whether in the laboratory or by computation) can be avoided toward encapsulating multiple satellite drops with different properties. The first rule applies to satellite drops of different properties but symmetric starting positions, and establishes the single-drop Hadamard-Rybczynski (HR) sedimentation velocity as the crucial parameter. The second rule makes use of a universal entrainment map by which three satellite drops of the same radius but different densities and asymmetric starting positions can all be encapsulated at an arbitrarily prescribed distance of sedimentation. Two final simulations demonstrate how both rules can be combined to successfully design an (asymmetric) injection geometry to encapsulate three satellite drops of different radii and densities, at an arbitrarily prescribed distance of sedimentation. Understanding fundamental hydrodynamics of interaction between multiple drops could lead to potential scale-up of production of TS particles and also impact applications of mixing and printing in general.