Self-consistent theory of spatial distribution of an assembly of charged drops in an electrodynamic balance

Abstract

Levitation of several charged drops in the air by applying oscillating quadrupole electric fields in an electrodynamic (ED) balance offers opportunities for improving our understanding of the role played by inter-particle forces on pattern formation. In these systems, stable structures arise as a balance between the time-averaged quadrupole force, the drag force and the collective force of mutual repulsion acting on individual drops. In order to characterize these structures, we formulate a general theory for the spatial arrangement of the levitated charged drops via a many-body Mathieu-Poisson equation. Although this system is not amenable to an exact analytical treatment, we develop an approximate analytical theory by invoking a jellium picture to predict the charge density profiles of the 2D planar configurations attained by the droplets. This approach reduces the system of equations into a set of dual integral equations, which is solved self-consistently via Hankel transform techniques. The theory predicts functional forms for the droplet density profiles and the size of the planar cluster as a function of the number of charged drops and the system parameters. As a part of validating the theory, numerical integration of the exact equations of motion of the interacting many-drop system has been carried out. Additionally, experiments were also conducted by levitating a large number of drops in an ED balance previously developed in our laboratory. A comparison of the predictions of the theory with those of the numerical simulations shows excellent agreement between the two. The experimental data on the spatial density distribution also seems to be consistent with the predictions of the theory. The results are further discussed.