Quantifying dysprosium extraction kinetics via magnetic levitation

We present a new method to quantify the reaction kinetics of oil-in-water rare-earth solvent extraction systems. The method aims to analyze the trajectory of an oil drop rising in a paramagnetic aqueous phase when the drop is exposed to a magnetic stray field. The latter generates a repulsive Kelvin force on the oil droplet, counteracting its buoyancy, thereby enabling a magnetic levitation in a paramagnetic aqueous phase containing trivalent dysprosium cations Dy(III). With the extractant PC88A dissolved in the oil phase, Dy(III) cation exchange proceeds continuously at the oil-water interface. The depletion of Dy(III) in the aqueous phase and the concurrent formation of Dy-complexes in the organic phase alter both the droplet’s magnetic susceptibility and its density. Consequently, the force balance acting on the droplet shifts, driving an uni-directional ascend toward the magnetic pole. By tracking the droplet motion with far-field optical microscope, we develop a quantitative model that correlates the droplet displacement with the time-resolved Dy(III) extraction rate. Scaling analysis reveals a first-order dependence of the reaction rate on both the Dy(III) and extractant concentrations, and a negative first-order dependence on hydrogen ion concentration. The kinetic rate law is further extended to account for the reverse reaction using reaction equilibrium data. Numerical simulations of the droplet trajectory based on the complete rate law show excellent agreement with experiment. This work establishes a rapid and accurate method for quantifying rare-earth extraction kinetics using a non-contact, single-droplet approach. The method is resource-efficient and readily adaptable to high throughput kinetic analysis of magnetically susceptible systems. • Interfacial rare-earth extraction kinetics can be quantitatively inferred from droplet trajectories. • Magnetic levitation enables accurate determination of kinetic rate laws without physical contact. • Mathematical modelling resolves the coupling of mass transport and force balance. • Extraction kinetics are independent of droplet size, confirming interface-controlled reactions. • The framework enables resource-efficient, high-throughput kinetic screening relevant to rare-earth separation and recycling.