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.
Highlights
III B, we outline the original electrohydrodynamic analysis based on the surface Bernoulli equation by Zubarev, which reveals a one-parameter family of asymptotic self-similar solutions for the potential functions and interface shape
We examine in detail the dynamic evolution of an axisymmetric protrusion in an electrically stressed liquid metal
Based on previous work by Zubarev and co-workers, as well as interface dynamics known to occur in capillaryinertial systems just prior or subsequent to pinchoff, the analysis predicates that protrusion growth is governed by a self-similar blowup process in which the kinetic, capillary, and Maxwell pressure contribute to leading order
Summary
The strong surface distortion accompanying electrically stressed liquids has fascinated researchers for centuries dating back to experiments in the early 1600s by Gilbert [1], who reported emission of a fine jet of liquid when water was attracted to a highly charged piece of amber,. Zubarev determined the asymptotic form of the electric potential, velocity potential, and interface shape applicable to distances far from the tip apex for an inviscid, perfectly conducting liquid and demonstrated power-law divergence in finite time of the surface pressure, a characteristic of blowup phenomena in self-focusing singularity flows. III B, we outline the original electrohydrodynamic analysis based on the surface Bernoulli equation by Zubarev, which reveals a one-parameter family of asymptotic self-similar solutions for the potential functions and interface shape We show why these solutions capture only flow configurations for which the surface kinetic energy per unit volume is negligible in comparison to the Maxwell and capillary pressure. V, we provide the complete self-similar solutions, valid throughout the near and far field from the liquid tip, obtained by a boundary integral patching technique, which directly incorporates the asymptotic solutions into the formulation
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