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

Abstract

Liquid-metal ion sources (LMIS) are widely used for ion implantation in semiconductors, focused ion-beam systems for lithographic patterning and now even small in-space thruster units for precision pointing. Above a critical field strength, a liquid metal just prior to ion emission forms a conical protrusion, which undergoes accelerated sharpening of the tip due to field self-enhancement. Despite decades of interest in this phenomenon, the influence of inertial effects on the shape and flow field prior to emission remain poorly understood. Zubarev [Formation of conic cusps at the surface of liquid metal in electric field, JETP Lett. 73, 544 (2001)] showed that when local Maxwell and capillary forces prevail at an electrified tip of a perfectly conducting liquid in inviscid flow, the asymptotic behavior of the potential fields and interface shape far from the tip can be described by a one-parameter family of self-similar series solutions describing a dynamic Taylor cone with radially convergent flow toward the tip. The Maxwell and capillary pressure undergo divergent growth in finite time, characteristic of blowup phenomena in self-focusing singularity flows. Suvorov and Zubarev [Formation of the Taylor cone on the surface of liquid metal in the presence of an electric field, J. Phys. D: Appl. Phys. 37, 289 (2004)] later found solutions that retained inertial effects and showed that the self-focusing, divergent dynamics leading to power-law growth are preserved. In this work, we focus especially on the influence of inertial effects and extend the analysis to include time-reversal symmetry. We provide an interweaved procedure for calculating the coefficients of the series expansions for the potential fields and interface shape and use these to deduce a compact relation for bounds on the minimum and maximum liquid apex height achievable. We then develop a boundary integral patching technique, which yields the complete self-similar solution, valid throughout the near- and far-field domain. The resulting two-parameter family of numerical solutions reveals a multiplicity of fluid configurations, which we coin subconical, superconical, and mixed conical, that exhibit not only tip sharpening but tip bulging, tip separation flow, interface stagnation points, and receding shapes reminiscent of recoil after capillary pinchoff. The existence of such multiple configurations may ultimately help explain experimental observations during LMIS operation involving tip pulsation, droplet emission, liquid recoil and collapse.

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