Impact dynamics of a ferrofluid marble on surfaces under a vertical magnetic field: Pre-impact motion and maximum spreading

Liquid marbles are droplets coated with hydrophobic particles, offering unique advantages over standard droplets in digital microfluidics. Liquid marbles can be effectively manipulated using a ferrofluid core and magnetic actuation. This study provides the first comprehensive examination of the impact dynamics of a ferrofluid marble in the presence of magnetic fields, combining systematic experimentation alongside a rigorous theoretical framework. In experiments conducted on hydrophilic and hydrophobic substrates, three distinct post-impact regimes were identified: bouncing, non-bouncing, and rupture, which were mapped onto a governing phase diagram. We develop a robust theoretical framework grounded in energy conservation that encompasses both pre- and post-impact magnetic effects. To analyze the pre-impact phase, we derive modified expressions for impact velocity and the Weber number, incorporating gravitational and magnetic Bond numbers. For the post-impact phase, the theory culminates in an inclusive viscous dissipation model that accurately predicts the maximum spreading ratio (βmax) for ferrofluid droplets and liquid marbles, significantly outperforming existing theories. Through detailed energy budget analysis, we quantify the competing roles of kinetic, surface, magnetic, and dissipation energies, revealing the dual nature of magnetic field influence, which simultaneously promotes and inhibits spreading behavior. These findings elucidate foundational design principles for cutting-edge magnetofluidic applications.