Multiplicity of Inertial Self-Similar Conical Shapes in an Electrified Liquid Metal

Abstract

Liquid metal ion sources (LMIS) are widely used in applications ranging from local ion implantation in semiconductors, to focused ion beam systems for milling and nanolithography, to space micropropulsion devices being developed by NASA. Above a critically large field strength, an electrically stressed liquid metal develops one or more cuspidal protrusions which undergo accelerated conic tip sharpening with runaway field self-enhancement. Zubarev (2001) first predicted from an inviscid model that the electric stresses at the liquid apex undergo self-similar divergent growth in finite time. The inviscid assumption is appropriate to liquid metals since the viscous boundary layer extends only a few tens of nanometers from the moving interface. In this work, we examine in more depth a two-parameter family of far-field self-similar solutions incorporating inertial, electrical and capillary effects, which to leading order describe electric and velocity potential fields corresponding to a rapidly accelerating \textit{dynamic} Taylor cone. These far field solutions are incorporated self-consistently into boundary integral simulations which reveal the entire liquid shape in the near field. By invoking time reversal symmetry inherent to inviscid flow, we unmask an entire family of novel self-similar conic modes exhibiting features such as inertial recoil, tip bulging from accelerated advance and tip counter-current flow as well as multiple interface stagnation points. These dynamic configurations help explain for the first time the origin of decades old experimental observations that have reported phenomena such as tip oscillation, pulsation and breakup during operation. The various liquid tip shapes accessible to such systems should help correct persistent misconceptions of pre- and post-emission behavior in LMIS systems and related technologies.