# Mathematical modeling of liquid metal dynamics

**Faculty Mentors: **

- Hangjie Ji (Mathematics, NCSU)
- Karen Daniels (Physics, NCSU)

**Prerequisites:** Calculus, differential equations, basic programming skills.

**Outline:** Eutectic Gallium Indium (EGaIn) is a room-temperature liquid metal alloy widely used in soft electronics/robots, self-healing wires, tunable antennas, and thermal paste [1]. When a positive voltage is applied to EGaIn in an electrolyte solution, the surface tension of the liquid metal decreases due to rapid oxidation, giving rise to various interfacial instabilities and pattern formation. Recently, together with Dr. Daniels’s, we experimentally identified a sequence of morphological changes in EGaIn droplets sliding down an inclined plane, where the liquid metal transitions from a Newtonian fluid to a viscoelastic fluid [2]. In this project, we will investigate a lubrication-type PDE model [3] for the dynamics of sliding droplets that involve partially unknown physics (the oxide concentration).

**Objectives:** Exploration of the PDE model to better understand the spatial instability of liquid metal dynamics. The results will be analyzed to estimate the influence of key factors such as the imposed voltage and inclination angle in the experiments.

**Outcomes:** A comprehensive numerical study of the PDE model for the dynamics of EGaIn droplets down an incline. This study will provide tools to estimate key parameters in the PDE model and compare the numerical results against experimental observations.

**References: **

*Overcoming Rayleigh–Plateau instabilities: Stabilizing and destabilizing liquid-metal streams via electrochemical oxidation.*PNAS, 2020. 11: p. 19026-19032. [2] Song, M., et al.,

*Interfacial tension modulation of liquid metal via electrochemical oxidation.*Adv Intell Syst, 2021. 3: p. 2100024. [3] Oron, A., S. Davis, and S. Bankoff,

*Long-scale evolution of thin liquid films.*Rev Mod Phys, 1997. 69: p. 931.