## Citation

Albertson, Theodore Glenn
(2020)
*Simulations of Conic Cusp Formation, Growth, and Instability in Electrified Viscous Liquid Metals on Flat and Curved Surfaces.*
Dissertation (Ph.D.), California Institute of Technology.
doi:10.7907/thhf-5478.
https://resolver.caltech.edu/CaltechTHESIS:05182020-133853408

## Abstract

It is well known that above a critical field strength sufficiently large to overcome damping by capillary forces, the free surface of a perfectly conducting liquid will spontaneously deform into one or more sharp protrusions known as conic cusps. Such cusps undergo tip sharpening while rapidly accelerating toward regions of highest electric field strength, eventually giving rise to beams of ions and/or charged droplets . These charged beams form the basis for liquid metal ion sources (LMIS) commonly used in focused ion beam systems, scanning ion microscopy, micromilling, ion mass spectrometry, implantation, and lithography. During the past few decades, there has been growing interest in optimizing the formation, growth, and stability of conic cusps in liquid metals for a new class of efficient and highly miniaturizable satellite micropropulsion devices consisting of microarrays of externally wetted solid needles coated with a film of liquid metal propellant. The thrust levels generated by such microarrays is suitable for propulsion of small satellites and precision pointing maneuvers for larger satellites.

This thesis addresses the formation, growth, and instability of conic cusp formations in perfectly conducting, electrified viscous liquids on flat and curved surfaces. We use finite element simulations based on the arbitrary Lagrangian-Eulerian (ALE) method for coupling the vacuum and liquid domains across the accelerating interface. The simulations in Chapters 2–4 describe the evolution of liquid flow subject to electric field distributions generated by opposing flat parallel and solid electrodes. In particular, we examine in Chapter 2 the growth of a small liquid protuberance on an otherwise flat viscous liquid layer of perfectly conducting fluid subject to an initial uniform electric field. Previous studies in the literature have established that tip sharpening proceeds via a self-similar process in two distinct limits: the Stokes regime at Re = 0 and the inviscid regime Re → ∞. These simulations, conducted at fixed capillary number Ca and for 0.1 ≤ Re ≤ 50,000, which span the viscous to inviscid regimes, demonstrate that the conic tip *always* undergoes self-similar growth irrespective of Reynolds number. Field self-enhancement due to conic cusp tip sharpening is shown to generate divergent power law growth in finite time (so-called blowup behavior) of the interfacial and volumetric forces acting at the advancing tip. The computed blow up exponents at the tip surface associated with the various terms in the Navier-Stokes equation and interface normal stress condition reveal the different forces at play as Re increases. Rescaling of the tip shape by the capillary stress exponent yields excellent collapse onto a universal conic tip shape with interior half-angle dependent on the magnitude of the Maxwell stress. The simulations clearly show that the interior cone angle adopts values both above and below the Taylor cone angle value of 49.3°. Additional details of the modeled flow dispel prevailing misconceptions that dynamic cones resemble conventional Taylor cones or that viscous stresses at finite Re can be neglected. In Chapter 3, we demonstrate how the rapid acceleration of the curved liquid interface also generates a thin surface boundary layer with very high local strain rate in the vicinity of the conic tip. The value of the surface vorticity along the moving interface is shown to be in excellent agreement with theoretical predictions. More importantly, the results in Chapters 2 and 3 demonstrate that the velocity streamlines are always at an oblique angle to the moving interface, contrary to commonly held belief that the streamlines always lie tangent to the moving boundary. In Chapter 4, we extend the simulations to include variation of the capillary number and find that for sufficiently high Re and Ca, the advancing interface develops significant oscillations. Fourier analysis of these interface oscillations indicates that the extracted instability wavelength characteristic of flows at smaller values of Re tends to exceed the simplified theoretical prediction based on inviscid flow. By contrast, the extracted instability wavelength for the largest values of Re examined tends to fall below the inviscid prediction.

In Chapter 5, we explore the effect of substrate curvature on the flow and stability of electrified films by examining the behavior of a thin viscous film of perfectly conducting liquid on two types of curved surfaces. These shapes, which include a solid conical needle with a spherical cap tip and a solid parabolic needle, are intended to mimic substrates used in some externally wetted microemitter arrays in LMIS systems. For the simulations in Chapter 5, the needle is situated below a counter electrode perforated with a circular aperture. The films are shown to develop both on-axis and off-axis cusp-like protrusions depending on the parameter range examined. In particular, the formation of off-axis protrusions are directly traced to substrate shapes which manifest an abrupt change in curvature, as present in a solid conical needle with a spherical cap tip. The simulations reported here are anticipated to help optimize fabrication of externally wetted needle shapes for use in a variety of LMIS systems.

Item Type: | Thesis (Dissertation (Ph.D.)) | ||||||
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Subject Keywords: | Fluid mechanics, Electrocapillary, Self-Similarity, Instability, LMIS, Electric Propulsion | ||||||

Degree Grantor: | California Institute of Technology | ||||||

Division: | Engineering and Applied Science | ||||||

Major Option: | Applied Physics | ||||||

Thesis Availability: | Restricted to Caltech community only | ||||||

Research Advisor(s): | - Troian, Sandra M.
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Thesis Committee: | - Schwab, Keith C. (chair)
- Bellan, Paul Murray
- Marrese-Reading, Colleen M.
- Polk, James E.
- Troian, Sandra M.
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Defense Date: | 10 September 2019 | ||||||

Funders: |
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Record Number: | CaltechTHESIS:05182020-133853408 | ||||||

Persistent URL: | https://resolver.caltech.edu/CaltechTHESIS:05182020-133853408 | ||||||

DOI: | 10.7907/thhf-5478 | ||||||

Related URLs: |
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Default Usage Policy: | No commercial reproduction, distribution, display or performance rights in this work are provided. | ||||||

ID Code: | 13708 | ||||||

Collection: | CaltechTHESIS | ||||||

Deposited By: | Theodore Albertson | ||||||

Deposited On: | 01 Jun 2020 21:59 | ||||||

Last Modified: | 07 May 2021 18:49 |

## Thesis Files

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- Final Version
Restricted to Caltech community only until 1 June 2021. See Usage Policy. 14MB |

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